International Association of Geodesy Symposia Ivan l. Mueller, Series Editor

International Association of Geodesy Symposia Ivan 1. Mueller, Series Editor

Symposium 101: Global and Regional Geodynamics Symposium 102: Global Positioning System: An Overview

Symposium 103: Gravity, Gradiometry, and Gravimetry Symposium 104: Sea Surface Topography and the Geoid

Symposium 105: Earth Rotation and Coordinate Reference Frames Symposium 106: Determination of the Geoid: Present and Future

Symposium 107: Kinematic Systems in Geodesy, Surveying, and Remote Sensing

Kinematic Systems in Geodesy, Surveying, and Remote Sensing

Symposium No. 107 Banff, Alberta, Canada, September 10-13, 1990

Convened and Edited by

Klaus-Peter Schwarz Gerard Lachapelle

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona

Klaus-Peter Schwarz Department of Surveying Engineering University of Calgary Calgary, Alberta T2N IN4 Canada

Series Editor Ivan I. Mueller Department of Geodetic Science & Surveying The Ohio State University Columbus, OH 43210-1247 USA

For information regarding previous symposia volumes contact: Secretaire General Bureau Central de I'Association Internationale de Geodesie 138, rue de Grenelle 75700 Paris France

Library of Congress Cataloging-in-Publication Data

Gerard Lachapelle Department of Surveying Engineering University of Calgary Calgary, Alberta T2N 1 N4 Canada

Kinematic systems in geodesy, surveying, and remote sensing / Klaus­Peter Schwarz, Gerard Lachapelle, editors.

107) p. cm. - (International Association of Geodesy Symposia; v.

Papers from a symposium held in Banff, Alta., Sept. 1990. Includes bibliographical references. ISBN-13:978-0-387-97465-1 e- ISBN· 13: 978-1- 4612 -3102-8 DOI: 10.1007/978-1-4612-3102-8

1. Geodesy-Mathematical models-Congresses. 2. Surveying­Mathematical models-Congresses. 3. Earth-Surface-Remote sensing-Congresses. 4. Global Positioning System-Congresses. 5. Inertial navigation systems-Congresses. I. Schwarz, Kiaus-Peter. II. Lachapelle, Gerard. III. Series: International Association of Geodesy Symposia; symposium 107. QB275.K56 1991 526'.1 '015118-dc20 91-9325

Printed on acid-free paper.

© 1991 Springer-Verlag New York Inc. Copyright not claimed for works of U.S. Government employees. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the li"ade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Permission to photocopy for internal or personal use, or the internal or personal use of specific clients, is granted by Springer­Verlag New York, Inc. for libraries registered with the Copyright Clearance Center (Ccq, provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress St., Salem MA 01970, USA. Special requests should be addressed directly to Springer-Verlag New York, 175 Fifth Avenue, New York, NY 10010, USA. ISBN·13:978-0-387-97465-1I1991 $0.00 + 0.20

Camera-ready copy provided by the editors.

9 8 7 6 5 4 3 2 1

ISBN-I3: 978-0-387-97465-1

Preface

It is with great pleasure that we present the proceedings of the First Symposium on Kinematic Systems in Geodesy, Surveying, and Remote Sensing to the scientific community. They follow a series of three symposia on inertial techniques the proceedings of which were published in 1977, 1981, and 1985, and which in many ways can be seen as forerunners of this symposium. The change in title signals a change in direction, however. While for a long time inertial techniques were the only ones giving precise kinematic information, this has changed dramatically over the last decade. Precise kinematic positioning by GPS is fast becoming a standard technique and attitude determination by multi-antenna GPS is in the testing stage. Other systems for the precise tracking of moving objects are revolutionizing precise engineering surveys and new technology is bound to further change the picture over the next few years. It seemed therefore appropriate to extend the framework beyond inertial techniques and to include all methods which allow the accurate determination of objects moving with respect to the Earth's surface. The term kinematic techniques seemed to capture these applications well.

The symposium was structured in such a way that features common to all kinematic systems were treated first before specific applications were considered. Common features included the mathematical framework connecting the observables obtained from specific sensors and the estimation techniques common to all applications. Only after these common features have been treated, specific technologies are presented. The technological development is still very rapid and major contributions in this area can be expected in years to come. The last five sessions of the symposium were devoted to major application areas. They show a wide variety of applications involving both inertial and GPS satellite techniques. Once the potential of these techniques are better understood by different user groups, a further diversification of application areas can be expected.

A symposium of this size requires the cooperation of many people. It is therefore a pleasant task to thank all those who made this meeting a success. Thanks are due to the chairmen, speakers, and attendees for creating an atmosphere that stimulated new ideas and made discussion an essential part of the symposium. This is especially true of the session chairmen who took an active part in planning and organizing the sessions. Our appreciation is expressed to the sponsors of this meeting for their support, specifically to the International Association of Geodesy; the Federation International des Geometres; Energy, Mines and Resources, Canada, and Transport Canada. The organization of this symposium would not have been possible without the active support of faculty, staff, and graduate students of the Department of Surveying Engineering. Among them, M. Anderson, who did the day-to-day organizational work, and H.E. Martell and D. Lapucha,

v

who organized the field demonstration, deserve special thanks. The cooperation with Springer-Verlag New York in preparing the proceedings for print was highly professional and equally enjoyable.

The editors are pleased to present the results of this joint effort to the scientific community in the hope that it will advance research and development in an exciting field.

Klaus-Peter Schwarz and Gerard Lachapelle

vi

Contents

Preface

Session 1 Theory and Modelling Chairman, P. Teunissen

Kinematic Modelling--Progress and Problems K.P. Schwarz

GPS Observables and Error Sources for Kinematic Positioning Gerard Lachapelle

Modelling Inertial Positioning Problems in Covariant Formalism G. Bartha, M. Doufexopoulou, and R. Korakitis

An Observability Analysis of a GPS-Aided Geodetic Inertial Strapdown Measurement Unit Margit Bolcsvolgyi-Ban and Dietrich Schroder

Mathematical Analysis of the Geodetic Space-Stabilized INS Bernd Eissfeller

Session 2a Equipment Trends and Measurement Procedures Chairman, M. Hadfield

v

3

17

27

37

47

Gyroscopes: Current and Emerging Technologies 59 Robert B. Smith and John A. Weyrauch

Accelerometers: Current and Emerging Technology 70 Brian Norling

ULISS 30: A New Generation Inertial Survey System Enhancing Flexibility and Security of Operation, Performance and Reliability 85 Frederic Mazzanti and Clive de la Fuente

Design for an Airborne GPS-Inertial Surveyor 95 J. Arnold Soltz, James l. Donna, and Glenn Mamon

vii

Basic Geometric Considerations for a Self-Calibration of Strapdown Inertial Sensor Blocks by Tumbles 105 Ernst H. Knickmeyer and Elfriede T. Knickmeyer

Session 2b Equipment Trends and Measurement Procedures Chairman, K. McDonald

Trends in INS Development--A Mini-Panel Presentation and Discussion Michael J. Hadfield, James Huddle, and Lorc Camberlein

Strapdown Inertial Surveying for Internal Pipeline Surveys Paul L. Hanna, Michael E. Napier, and Vidal Ashkenazi

Geodetic Application of a Laser-Inertial Strapdown System Dieter Keller, Stefan Rohrich, and Matthias Becker

Attitude Determination via GPS Jerry Knight and Ron Hatch

A Real-Time GPS-Based Differential Positioning System Ken Doucet, Yola Georgiadou, Alfred Kleusberg, and Richard Langley

Session 3a Estimation Methods and Quality Control Chairman, J.R. Huddle

Some Aspects of Real-Time Model Validation Techniques for Use in Integrated Systems P.J.G. Teunissen

Fault Detection and Estimation in Dynamic Systems M. Wei, D. Lapucha, and H. Martell

MDB: A Design Tool for Integrated Navigation Systems Martin Salzmann

Use of Spectral Methods in Strapdown ISS Data Processing Joseph Czompo

Wave Method in Processing Navigation Information in Survey Systems D.S. Salytchev and A.B. Bykovsky

Session 3b Estimation Methods and Quality Control Chairman, R.V.C. Wong

119

140

154

168

178

191

201

218

228

238

Detection and Modelling of Coloured Noise for Kalman Filter Applications 251 Martin Salzmann, Peter Teunissen, and Michael Sideris

Research on Closed-Loop Kalman Filter Technology of SINS/GPS for Surveying 261 Wan Dejun, Ren Tianmin, and Dong Bin

viii

Reliability Analysis Applied to Kinematic GPS Position and Velocity Estimation 273 Gang Lu and Gerard Lachapelle

Oil a Measure for the Discernibility Between Different Ambiguity Solutions in the Static-Kinematic GPS-Mode 285 H. -J. Euler and B. Schaffrin

Session 4 Algorithms and Software Trends Chairman, c.c. Goad

Instantaneous Ambiguity Resolution Ron Hatch

Inertially Aided Lane Recapture Mter GPS Carrier Lock Loss P. V. W. Loomis and G.J. Geier

Centimeter Level Surveying in Real-Time Kendall E. Ferguson and Ellis R. Veatch II

Combining Pseudo-Range and Phase for Dynamic GPS V. Ashkenazi, T. Moore, and J. Westrop

Comparison of Heterogeneous Unsynchronised Data by Transforming Coordinate Independent Functionals Alain Geiger and Marc Cocard

Session 5a Positioning and Navigation Applications Chairman, L. Camberlein

Experience with the ULISS-30 Inertial Survey System for Local Geodetic and

299

309

319

329

341

Cadastral Network Control· 351 Rene Forsberg

ULISS 30: Results from Interpolation and Extrapolation Measurements 363 J. -M. Becker and M. Lidberg

GPS/INS Trajectory Determination for Highway Surveying 372 Dariusz Lapucha

Kinematic GPS Surveying in Cyprus 382 Richard M. Haines

Geodine 30: A Real-Time Inertial Surveying System for Land Geophysical Surveys 393 Jean-Marie Doizi

Session 5b Positioning and Navigation Applications Chairman, M. Avni

Integrated System for Automatic Landing Using Differential GPS and Inertial Measurement Unit Thomas Jacob

ix

405

EUROFIX: A Synergism of Navstar/GPS and Loran-C 423 Durk van Willigen

Open Concept: Integrated Navigation for Airborne Remote Sensing Image Georeferencing 433 D.1. Ross, L. Hil~ and E.M. Senese

A Discussion of GPS/INS Integration for Airborne Photogrammetric Applications 443 M.E. Cannon and K.P. Schwarz

Experimental Research on a Strapdown Inertial Navigation Device Working Underwater 453 Zhang Shuxia and Sun Jing

Session 6a Gravity and Attitude Applications Chairman, R. Forsberg

The Role of GPS/INS in Mapping the Earth's Gravity Field in the 1990's 463 Oscar L. Colombo

An Integrated Precise Airborne Navigation and Gravity Recovery System 477 Klaus Hehl, Gunter W. Hein, Herbert Landau, Michael Ertel, Jurgen Fritsch, and Peter Kewitsch

GPS and Airborne Gravimetry: Recent Progress and Future Plans 488 John M. Brozena

Accuracy of Gravity Vector Recovery with the LTN 90-100 RLG Strapdown System 498 Elfriede Knickmeyer

A Method to Determine Increments of Vertical Deflections 510 O.S. Salytchev and A.B. Bykovsky

Session 6b Gravity and Attitude Applications Chairman, G. Hein

Gravity Field Modelling for INS 523 Nguyen Chi Thong

Smoothing and Desmoothing in the Fourier Approach to Spherical Coefficient Determination 533 Paul A. Zucker

Airborne Gravity Surveying: An Effective Exploration Tool 543 William Gumert, Victor Gratero~ Gerald Washcalus, and John Kratochwill

An Evaluation of Alignment Procedures for Strapdown Inertial Systems 553 K.P. Schwarz and Ziwen Liu

x

Application of Strapdown Inertial Surveyor in Determination of Hoist Skip and Mine Shaft Trajectory 565 H.E. Martel~ M. We~ K.P. Schwarz, W. Griffin, andA. Peterson

Author Index 575

xi

SESSION 1

THEORY AND MODELLING

CHAIRMAN P. TEUNISSEN

DELFr TECHNICAL UNIVERSITY DELFr, THE NETHERLANDS

KINEMATIC MODELLING - PROGRESS AND PROBLEMS

ABSTRACT

K.P. Schwarz Department of Surveying Engineering

The University of Calgary 2500 University Drive N.W.

Calgary, Alberta, Canada, T2N IN4

The paper looks at kinematic modelling from a conceptual point of view and formulates a general mathematical framework for the four tasks of interest to surveying: positioning, attitude determination, curvature computation, and vector gravimetry. It then examines the inertial and GPS observables and their suitability for the above tasks. The generality of the model is shown by outlining a number of typical applications. Finally, some of the remaining problems are briefly discussed.

I. FUNDAMENTAL IDEAS

In this paper, kinematic modelling is understood as the determination of a vehicle trajectory in space from measurements. It thus combines elements of modelling, estimation and interpolation. Modelling relates the observables to the trajectory. Estimation uses actual observations, i.e. it adds an error process to the model and solves the resulting estimation problem in some optimal sense. Interpolation connects the discrete points resulting from the estimation process and obtains a trajectory by formulating some appropriate smoothness condition.

It has been argued that interpolation is not really necessary in this process and that the measurements are more appropriately transformed into a string of discrete points in space. This is true for some applications. However, the underlying physical problem, the movement of a vehicle in space, is obviously a trajectory problem, and due to this fact, trajectory information is often required from the model. In many cases, a primary sensor, such as a multi-band scanner, has to be positioned and oriented in space at a given instant of time. Since this instant usually does not coincide with one of the instants at which a discrete trajectory point has been determined, interpolation becomes necessary. The determination of a string of discrete positions is therefore considered as a subset of the more general case of trajectory interpolation.

Rigid body motion in three-dimensional space can be described by six parameters which are typically chosen as three translational and three rotational parameters. Besides position, orientation and curvature can be obtained from such a trajectory model. The three­dimensional rotation information can be expressed with respect to any given set of coordinates. Two very useful sets of coordinates, often used in geodetic applications, are those refering to the local-level frame and to the body frame. The ftrst deftnes orientation in an earth-ftxed coordinate system which is a good approximation to the 'natural' coordinate system of north, east and up. The second deftnes orientation changes with respect to a vehicle-ftxed coordinate frame. Both will be used in the following. Curvatures are trajectory derivatives. They are either given with respect to time or to trajectory length. They are usually expressed as orientation changes and are thus closely related to angular velocities of the vehicle.

3

The vehicle trajectory is detennined from measurements. If these measurements contain components not due to motion, they have to be modelled out Typical examples are gravity in case of accelerometer measurements and atmospheric effects in case of satellite range measurements. On the other hand, differencing the trajectory models derived from two different sets of measurements, eliminates the motion and results in a map of the non-kinematic components in the two measurements. This is the principal idea behind vector gravimetry. By differencing trajectory information from inertial measurements which contain the effect of gravity and trajectory information from GPS where this information has been eliminated, the gravity vector can in principle be determined. As always when measurements are used, errors complicate the simple principle and the estimation and interpolation problems are in this case much more severe than the modelling problems.

In the following, four major application areas of kinematic modelling will be distinguished. They are positioning, attitude determination, curvature estimation, and vector gravimetry. This distinction is made for reasons of conceptual clarity. In practice, more than one of these tasks has often to be addressed in a given application. Thus, airborne remote sensing usually requires positioning and attitude determination, while pipeline pigging and the related railroad survey problem require positioning and curvature determination. From a mathematical point of view, the flrst three tasks are simply a re-parametrization of the trajectory problem. In practice, they are quite different, however, as far as the estimation problem is concerned. Vector gravimetry is obviously different, even from a conceptual point of view because it requires trajectory information from two distinctly different data sets. The presentation in the following is at an elementary level, discussing some major aspects rather than treating them mathematically.

2. TRAJECTORY MODELS

In the following, rigid body motion in three-dimensional Euclidean space will be used as the basic kinematic model. In this context, motion can be described by three translations and three rotations, i.e. by an equation of the form

r(ro, t) = r(t) + R(t)ro (1)

Figure 1 illustrates this well-known relationship. The coordinate system {el,e2,e3}, in which the translation vector r(t) is expressed, is in principle arbitrary and can thus be chosen to simplify the problem formulation. The vector ro which is used to formulate the rotational motion refers to any flxed vector in the body. Its rotation is equivalent to the rotation about the centre of mass of the rigid body, i.e. in our n,ase to the centre of mass of the vehicle. It is typically given by an angular velocity vector ot which gives rotation rates relative to an inertial system of reference. R(t) can be parametrizea in different ways, as for instance by direction cosines, Euler angles, or quaternions. For a discussion see for instance Hughes (1986) or Scciehien (1986). It can be shown that in case of a rigid body, R(t) can be derived by using ~ (t) and the known rotation rate of the earth. The coordinate frame of R(t) is usually chosen in such a way that it solves the problem at hand in a convenient manner.

GPS positioning with either pseudo-range or phase data provides essentially the vector r(t), i.e. the position of the antenna centre in space. It does not give attitude changes of the vehicle on which the antenna is mounted. For this, a multi-antenna system or an inertial reference unit is needed. In that case, r(r o,t) can be determined, i.e. all six trajectory parameters can be derived from measurements. However, even this may not be sufficient in some cases. The sensing device and the primary sensor have to experience the same translational and rotational motion, i.e. no relative motion should occur between them. This is not as trivial a requirement as it may sound because it is often not possible to rigidly connect

4

the rotation sensor and the primary sensor. If the primary sensor is for instance an aerial camera mounted to the floor of the aircraft and the rotation sensor is a multi-antenna system mounted on top for best satellite visibility, it is difficult to secure the condition of no relative motion.

Figure 1: Rigid body motion

3 . MODELLING OBSERV ABLES

Tile two types of observables considered in the following will be called INS observables when obtained from an inertial measuring unit (IMU) and GPS observables when obtained from a pair of GPS receivers or an array of GPS antennas. Other sensors can be used and can easily be expressed in a similar scheme.

A strapdown IMU outputs three components of the specific force vector and three componenw of the angular velocity vector in the body frame system. They will be denoted by fb and n ~ in the following. The subscript indicates the frame in which the vectors are expressed, the superscripts the direction of the rotation, for instance the superscripts ib would denote the rotation of the b-frame with respect to the i-frame. The state of the system can be described by the well-known trajectory model

re ve

ve = Rebfb - 2n~ve + ge (2)

Reb Re~~b

which expresses changes in position r, velocity v , and rotation rate ro by systems of three differential equations each. In equation (2) the earth-fixed Conventional Terrestrial System has been chosen to coordinatize the equations, for details see Wei and Schwarz (1990). The observables appear on the right-hand side of the equations. To solve these equations for the state vector components on the left-hand side, the vectors ge and role are needed. The gravity vector is usually approximated by the normal gravity vector Ye ana further improved in the estimation process. The earth rotation vector is known with sufficient accuracy. Parameters required for the solution of the positioning, the attitude and the curvature problem can all be derived from the state vector. The solution of the vector gravimetry problem is possible by

5

rearranging the second set of equations in (2) with respect to ~ and obtaining the acceleration vector ve from an independent source, for details see Schwarz and Wei (1990).

GPS observables are either of the pseudorange type p or of the carrier phase type <1>. A discussion of code and carrier phase observables and the errors affecting them is given in Lachapelle (1990 this volume), which is referenced for all details. Models to transform the resulting range equations into positions and velocities are well-known, see for instance Wells et al (1986). In the process, orbital models as well as atmospheric models are needed and the earth rotation rate is again assumed to be known. To facilitate comparison with the INS­trajectory model, the GPS-trajectory equations will not be expressed in terms of the original observables but in terms of position and velocity which can be considered as GPS pseudo­observables. The trajectory model is then of the form

(3)

for pseudorange measurements with a single receiver or for differential pseudorange or carrier phase measurements for a pair of receivers. In case a multi-antenna system with a minimum of three antennas is available, the state equation would be of the form

re Ve

ve 0 = (4)

Reb Re neb b b

where the rotation matrix R can be obtained by differencing between antennas. In this case, distances between antennas must be considered as constant and accurately known. In both trajectory models (3) and (4), constant velocity between measurement epochs is assumed. Other possible models are discussed in Schwarz et al (1989). The parameters needed for the positioning problem can be derived from either of the two sets of equations, the parameters for the attitude and curvature problem can only be obtained from equation (4).

4. INTEGRATION ISSUES

GPS and INS are in many ways complementary systems for accurate positioning and attitude determination. GPS positioning by differential carrier phase is superior in accuracy as long as no cycle slips occur. GPS relative positions are therefore ideally suited as INS updates and resolve the problem of systematic error growth in the INS trajectory. On the other hand, INS measurements are very accurate in the short term and can thus be used to detect and eliminate cycle slips and multipath effects of short duration. For these tasks and for the resolution of the term R(t) ro, a medium accuracy INS is fully sufficient. Besides the practical integration problems, such as synchronization and stability of the time systems, the method of integration, the detection and correction of cycle slips and the 'in-flight' calibration of INS parameters are the main modelling problems.

6

Table 1 shows the four major possibilities for INS/GPS integration and the Kalman filter approaches associated with them. In GPS-aiding, the two data streams are processed through separate Kalman filters and the GPS fl).ter results are used to regularly update the INS. This method corresponds to the decentralized Kalman filter. In the integrated approach data from both systems are fed into a common filter. This corresponds to the centralized Kalman filter. In the open loop concept, no feedback is applied, i.e. no information from the previous filtering step is used to correct the measurements. This corresponds to the standard Kalman filter. In the closed loop concept, feedback is applied and the results of the previous step are used to minimize the trajectory approximation errors. This corresponds to the extended Kalman filter concept. Since the two data streams are completely independent and can thus be used for mutual checking, the closed loop system appears to be the better solution as long as a reliable cycle slip detector can be designed. As to the use of a centralized versus a decentralized filter, there seems to be no clear preference. Because of the structure of equations (2) to (4), the number of states for the largest state vector in each approach is about the same. Thus, no major reduction in the computational work can be expected. Results in Wei and Schwarz (1990) indicate that there is no difference in accuracy, while results in Dayton and Nielson (1989) give the centralized filter a small edge.

GPS - aiding (decentralized K.F.)

Open loop (standard KF.) I Closed loop (extended KF.)

Integrated (centralized K.F.)

Open loop (standard KF.) I Closed loop (extended KF.)

Table 1: INS/GPS integration option

An optimal INS/GPS integration for survey applications has to give a good symbiosis between cycle slip detection and correction and 'in-flight' calibration of inertial system parameters. The first provides INS-aiding to the GPS system while the second improves INS modelling by optimally using GPS data. The key to avoiding a bootstrap procedure lies in the fact that cycle slips are multiple integers of the two basic wavelengths. Cycle slip detection and correction of GPS carrier phase data by INS is based on the fact that the accuracy of INS data over short time spans is considerably better than 0.5 cycles and thus allows the detection and correction of cycle slips as long as the GPS data rate is fast enough. Typically, the measured integer cycle number is compared to the number predicted from INS. If a cycle slip occurred, the respective carrier phase can be corrected by resetting it to the integer value closest to the predicted value. The validity of the predicted value can be tested by statistical means, see for instance Teunissen (1990) and Wei et al (1990) in this volume or, ifthe computations are done post mission, by computing the ambiguity function.

Obviously, the method will work best for short time intervals. In environments were shading effects are frequent, as for instance urban and forested areas, the time interval may be determined by the length of the shading rather than by the GPS output rate. In these cases, it is important to know the accuracy of the INS prediction as a function of the length of the time interval. Lapucha (1990) has published RMS differences for the range error between INS and GPS as a function of time. They are plotted in Figure 2. It has been assumed in this case that the GPS errors are negligible. The figure indicates that for reliable cycle slip correction, i.e. for range differences smaller than 10 cm, predictions by INS should be restricted to intervals of about five seconds. This value seems to be extremely small. However, the Kalman filter used in this case had not been optimized for this bridging problem and no attempt had been made to

7

calibrate certain INS parameters during periods of good GPS data. This can be done by including scale factor and bias parameters of the INS in the state vector and by designing an adaptive filter which estimates these parameters during phases of good GPS data. It is expected that in this way the bridging interval could be considerably increased and the reliability of the method considerably improved.

70~------------------------~r-~

60

E 50 o -~ 40 c: ~ 30 ~ o 20

10

O+-~--.---....---r-----r--......----r---"""

o 10 20 30

Time Interval (sec)

Figure 2: RHS range differences for GPS-INS as a function of time

5. CLASSIFICATION OF MAJOR APPLICATION AREAS

Conceptually, four major tasks can be distinguished in kinematic modelling: Positioning, attitude determination, curvature estimation, and vector gravimetry. Although they may overlap or occur together in a specific application, they are a useful conceptual framework to distinguish the major application areas.

Kinematic positioning

Kinematic positioning can either be done by stand-alone GPS or INS, or by combining the two measurement systems. The point to be positioned is the centre of the measuring system, i.e. the centre of the accelerometer proof masses in one case and the GPS antenna centre in the other. Their position is described by the term r(t) in equation (1) or the corresponding vector in equations (2) to (4). If a point other than the measuring system centre is required, i.e. if the vector r(ro,t) in equation (1) has to be determined, then the vector ro between the centre and the point to be determined as well the attitude changes of this vector have to be known. This is always necessary when some primary sensor has to be positioned or when a INS/GPS integration is required. It will be assumed in this section that the rotation matrix R(t) and the vector ro are available.

The kinematic positioning model of an INS is obtained by augmenting equation (2) by a number of error states and estimating the total state vector by a Kalman filter approach. The integration of velocity, position and attitude from the actual measurements is done implicitly in the coordinate frame chosen. Typically, either the Conventional Terrestrial frame which is a three-dimensional Cartesian frame or the local-level frame which is a frame in curvilinear ellipsoidal coordinates will be used for strapdown systems. Operational and numerical

8

advantages of chosing either one of these frames are discussed in Wei and Schwarz (1990). For platform systems the local-level frame is the norm. The error behaviour of inertial positioning systems is well understood. Since initial errors are twice integrated with respect to time for positioning, the error growth is rapid and regular updating is needed to obtain accurate positioning results. In the semi-kinematic mode this is achieved by regular zero velocity updates at stops. In the kinematic mode, it can be done by external velocity or position updates, coming for instance from GPS. It appears that the use of INS as a stand-alone system for precise positioning will be rather restricted in the future. It will either be used in conjunction with GPS to increase reliability or in applications where GPS is not viable, as for instance underground or underwater.

The kinematic positioning model for GPS is in principle given by equation (3), although sometimes bias terms with a Gauss-Markov structure are added to model cycle slips in carrier phase measurements, see Wong et al (1988), or orbital biases in pseudorange measurements. Many errors in the measurements are effectively eliminated by differencing techniques and the accuracy of kinematic GPS is essentially limited only by receiver noise, multipath effects and the correct resolution of cycle slips in case of carrier phase observations. Currently achievable accuracies are discussed in Lachapelle (1990). With velocity and position as input to equation (3), a Kalman filter is used for the estimation and interpolation process. As has been pointed out, this model implies constant velocity between position fixes. This is obviously justified if the trajectory dynamics does not contain frequencies higher than those that can be resolved from the given data rate. This is a problem in some applications. However, with output rates increasing, it can be expected that this will only be a temporary difficulty. Semi-kinematic applications as well as kinematic applications with inverse photogrammetric control indicate that accuracies in the decimetre range and better can be achieved when using differential carrier phase methods, see for instance Mader (1986), Cannon (1989), Landau (1989), Baustert et al. (1989), Cannon and Schwarz (1990, this volume). However, the reliable resolution of cycle slips, the influence of the distance from the monitor station on the accuracy, and the influence of multi path effects require further study.

As has already been pointed out, the integration of INS and GPS resolves some of the above problems and also gives the R(t) matrix with sufficient accuracy. Other approaches to resolve the cycle slip problem are the availability of high accuracy P-code receivers and measurements to a large number of satellites, for instance with an integrated GPS/GWNASS receiver. These possibilities will gain increased attention over the next few years. Applications of kinematic positioning are numerous and include point densification, photogrammetry without ground control, airborne line imaging, laser profiling, hydrographic surveying, highway and railway monitoring systems as well as pipeline monitoring system. A number of these applications are discussed in later sessions.

Attitude determination

Attitude determination is needed with medium accuracy to link the primary sensor to the attitude sensor and with very high accuracy when attitude is required as an independent output. This is for instance the case in some of the emerging multi-spectral scanning applications. Medium accuracy is essentially required to determine the term R(t) ro in equation (1). Usually an accuracy of 20 arcminutes is quite sufficient. It can be obtained from a medium accuracy inertial reference unit or a GPS multi-antenna system. High accuracy applications require attitude changes of the primary sensor usually in the body frame to correct the sensed data. In this case, an accuracy in the range of 10 arc seconds is required which currently can only be obtained from a high performance inertial system. In this case the data smoothing has to be consistent with the trajectory dynamics.

9

Attitude determination with strapdown inertial units is by now a standard procedure and the algorithms have been thoroughly studied and compared. For a discussion of the important aspects, see for instance Van Bronkhorst (1978) and Savage (1984). To achieve the high accuracy mentioned above, smoothing of the high rate data is usually necessary. Typically, the data acquisition rate is 50 Hz or higher. This rate is usually not required to model the aircraft trajectory. Figure 3 shows the spectrum for the roll of a typical small survey aircraft as computed from the raw angular velocities. The angular velocities above 2 Hz show white noise behaviour. Data for azimuth and pitch are even smoother than those for roll. On the other hand, to obtain a standard deviation of 10 arc sec. from a navigation grade RLG, averaging times of at least two seconds are needed. Thus, data smoothing has to be optimized with view to the essential frequencies of the trajectory spectrum.

Data smoothing can be done by spectral methods before attitude computations are done in formula (2). This obviously requires a post-mission approach. It can also be done by designing optimal band-path filters for specific trajectory dynamics which are then incorporated in the Kalman filter based on an extended equation (2). First results on the spectral approach are given in Czompo (1990, this volume), while the second approach is currently investigated. Application of this method is very sensitive to changes in the mean value and care has to be taken not to introduce biases in this way.

2.0

1.5

0 C/) a.. 1.0 (5 a:

0.5

0.0 0 1 234

Frequency (Hz) 5

Figure 3: Roll motion of a small survey aircraft

6

For medium accuracy applications, the use of an inertial reference unit is straightforward. It has the major advantage that it can be directly attached to the primary sensor and thus minimizes the error of the term R(t) roo Once the current development of light-weight reference units based on small ring laser or fibre optic gyros reaches the production line, this approach will become very attractive.

The development of GPS multi-antenna systems is entering the production stage and first results are encouraging, see for instance Brown and Ward (1990) and Kruczynski et al. (1989). The basic idea is quite simple. Carrier phase measurements between a minimum of three antennas on a fixed base are differenced and relative displacements are used to derive attitude information. By carefully minimizing measurement noise, accuracies of arc minutes have been achieved in laboratory tests, even for short base lines of about 1 m. In general, results in an actual working environment have been poorer. This may be due to vibrations of

10

the carrier which would be especially damaging for such a device or non-optimal data resolution due to orientation of the multi-antenna system or satellite configuration. Since this development is still in its infancy, major contributions can be expected over the next few years. Results from the attitude computation can directly be used in equation (4). At present, the data rate of these devices is still too low for a complete resolution of the trajectory dynamics.

Curvature determination

Representation of space curves can either be done in a convenient three-dimensional Cartesian system by a position vector r(t) or in a local orthogonal triad moving along the curve, such as the Frenet frame. The latter consists of the tangent vector t(s), the principal normal vector n(s), and the binormal vector b(s); see e.g. Stoker (1969) for a definition of terms. The plane spanned by the first two vectors is called the normal plane, that spanned by the last two the osculating plane. The rate at which the normal plane turns about the binormal vector is called curvature and is usually denoted by K. The rate at which the osculating plane turns about the tangent vector is called torsion and is usually denoted by 'to The parameters K and 't as functions of the arclength s of the curve can be used to completely describe a spatial curve. This representation is particularly convenient when one is interested in deriving curvature and torsion directly from measurements.

Assume that the triad {t,n,b} rotates about its origin while it moves along the curve. The corresponding angular velocity vector ro can be expressed in terms of the three unit vectors as

The time change of the unit vectors is given by the crossproduct of ro and the base vectors and is expressed in the usual way

t

n

b

o

=

0>3

o

t

n (5)

o b

where the dot indicates the time derivative. The corresponding Frenet equations for space curves are of the form

t' o K 0 t

n' = -K o n (6)

b' o -'t 0 b

where the prime indicates differentiation with respect to the arc length. Thus, for constant velocity, the two matrices should be the same and the angular velocity vector takes the following form

ro = 'tt + Kb (7)

11

The vector (7) is often called the Darboux vector and expresses the fact that by choosing an appropriate coordinate frame, the angular velocity vector is completely described by rotation rates in two specific planes, the normal plane and the osculating plane. These rotation rates are related to curvature K and torsion t which are intrinsic parameters of the space curve. In order for equation (7) to hold, the arc length s must be interpreted as time. Thus, constant velocity has to be assumed.

To relate measured angular velocities to curvature and torsion as given by equation (6), the transformation between the measurement frame and the Frenet triad must be known. If at least one axis of the body axes system is parallel to one of the three Frenet vectors, then K and t can directly be expressed by two of the angular velocity measurements. This is usually the case for strapdown inertial systems because the forward direction of the vehicle is one axis of the body frame and corresponds to the tangent direction of the Frenet frame. Thus t, expressed as rotation rate in the osculating plane, can be derived from angular velocity measurements about the two axes orthogonal to t. It should be noted that the axes of the body frame in the osculating plane are not identical to the axes of the Frenet frame. Similarly, two measured angular velocities are required to determine K as the rotation rate in the nonnal plane.

Angular velocity measurements are typically given as functions of time, for instance oo(t). Transformation from time t to arclength s is unique as long as the curve is regular. Since trajectories are at least piecewise continuous, this transformation can always be done. It becomes especially simple if constant velocity can be assumed. It should be noted, however that the vector i-(t) which is tangent to the space curve is different in length from the unit vector t(s) which is also tangent. It is therefore necessary to scale the output of an inertial system by the linear system velocity to obtain K and t directly, for details see Knickmeyer et al. (1988).

The parameters K and t are invariant measures characterizing the space trajectory locally and they have a strongly intuitive geometrical interpretation. Invariance applies to both, parameter and coordinate transformations. They are especially useful when the measurement system is guided along a given trajectory as in case of a pipeline monitoring system, see Schwarz et al (1990) and Porter (1990), or in case of rail-bound systems or mining elevators, S6e Martell et al (1990, this volume). In some of these cases, the trajectory change with respect to an ellipsoidal coordinate system is also required. This can be obtained by using the attitude information from the state equations or by using the components of the gravity vector in the body frame to get a quick estimate of pitch and roll, see e.g. Porter (1990).

The potential of multi-antenna systems for curvature determination appears to be more limited than for attitude determination. Although a similar approach seems to be feasible in principle, current data rates and accuracies will make the extraction of reliable local parameters much more difficult.

Kinematic vector gravimetry

Kinematic vector gravimetry is defined as the determination of the full gravity disturbance vector from a moving vehicle. Partial solutions to the problem are given by techniques applied in marine and airborne gravimetry where the magnitude of this vector is determined and in inertial gravimetry where the surface deflections of the vertical are determined at ZUPT points; for a comprehensive discussion of recent work, see Torge (1989). The full solution seems to be in reach now by combining measurements from a highly stable and highly accurate INS with carrier phase measurements from GPS. Kinematic motion can in this case be eliminated by differencing the two measurement sets in some suitable way. This application is different from the previous cases of GPS/INS integration because it requires both types of measurements for a solution. A different approach to kinematic vector gravimetry is

12

gravity gradiometry where acceleration differences are measured on a stabilized platform and second-order gravity gradients are derived from these measurements. This application will not be discussed in the following. Reference is made to Forward (1985), lekeli (1984), and Schwarz and Wei (1990) for theoretical and practical details.

Modelling problems connected to vector gravimetry by GPS/INS can be discussed in an elementary way by re-ordering the second set of equations in (2) in the following manner

(8)

where ge has been approximated by the normal gravity vector Ye and the gravity disturbance vector Bge. This equation gives the gravity disturbance vector directly in t~rms of observed quantities v~eve,fb, the rotation matrix Reb derived from measurements ro;; and the known quantities Qe and Yeo Table 2 outlines in detail how the different quantities are obtained.

Variable Obtained from by

Ve GPS carrier phase rate differentiation of measurements

Ve GPS carrier phase 'measurement'

Reb inertial gyros and knowledge of integrated measurements earth rate

fb inertial accelerometers measurement

Qie e known earth rate and position computation

Ye known gravity model and position computation

Table 1: Computation of formula (8)

The main contribution of GPS is acceleration ve and velocity Ve obtained from carrier phase measurements. In inertial gravimetry these quantities are assumed zero at ZUPT stations which provides a simple solution of equation (8).

The implementation of the basic idea for airborne applications is just at its beginning. A recent investigation by Knickmeyer (1990) shows that the measurement accuracies required to reliably obtain 1 mgal for the magnitude of Bg and 1 arc second for its direction over a two minutes averaging period are extremely difficult to achieve with current technology. Accelerometer bias stability at the 0.5 mgallevel, gyro bias drifts in the range of 0.001 to 0.0001 deg/h and very low noise characteristics would be required for the inertial sensors. Velocity errors of the GPS measurements should be below 0.3 mm/s. Averaging times of two minutes will cause no problem in geodetic applications. For geophysical applications, however, averaging times of 10 seconds would be desirable.

13

Questions that need further investigation are the level of differencing, the choice of filtering algorithms most suitable for this problem, the optimal combination of time and spectral methods, and the elimination of the more stringent requirements for deflection determination. Differencing of INS and GPS measurements can in principle take place at either one of three levels: acceleration, velocity, position. Since INS measurements are strongly affected by errors at the acceleration level while GPS measurements are typically influenced by range errors, combination at the velocity level seems intuitively the best solution. It would minimize the distortion of errors by either integration or differentiation. It would also be the solution which uses none of the original measurements and this may be a drawback, at least, as long as the error spectra are not known to a good approximation.

Knowledge of the error spectra and the high frequency gravity spectrum are also problems in the design of appropriate filter algorithms and in deciding whether to use time or frequency domain methods. Work in this area is just starting. Problems that will have to be addressed are filter design for a band-limited gravity spectrum and specific aircraft dynamics, their effect on error spectra and the estimation accuracy possible with current data rates. It appears that a mix of spectral and time domain methods offers the best chance for the optimal extraction of gravity field information.

Since requirements for the precise determination of deflections of the vertical seem to be much more stringent than those for the magnitude of gravity the use of scalar gravity for deflection determination is a viable alternative. It appears that with appropriate operational methods and a reasonable profile density, deflection determination from airborne gravity seems to be feasible at the 1 arc second level.

OUTLOOK

Kinematic modelling in survey-related applications has experienced rapid development during the last five years. In many ways, technology was the driving force and a systematic assessment of different approaches is just at its beginning. Much work is needed to extend current methodology, to express new observables within a consistent framework, to classify the numerous applications and to further develop statistical techniques. The continuing rapid progress in hardware development makes this an exciting field for theoretical as well as applied studies.

ACKNOWLEDGEMENTS

Many of the ideas and implementations discussed in this paper are the result of research conducted by a number of individuals at the University of Calgary. Visiting scientists, research fellows and graduate students contributed to the understanding of the problems discussed. Individual contributions are acknowledged in the references but they do not always reflect the interactions within a research group. Financial support for this work has been obtained through an NSERC Operating grant.

References

Baustert, G., et al. (1989). "German-Canadian experiment in airborne INS-GPS integration for photogrammetric applications." Proc., General Meeting of the Int. Assoc. of Geodesy (lAG), Symp. 102, Edinburgh, Scotland, Springer-Verlag, New York, N.Y.

14

Brown, R. and P. Ward (1990). A GPS receiver with built-in precision pointing capability. Proc. PLANS 1990, IEEE AES, Piscataway, N.J.

Cannon, M.E. (1989). "High accuracy GPS semi-kinematic positioning: modelling and results." Navigation, 37(1), 53-64. .

Cannon, M.E. and K.P. Schwarz (1990). A discussion of GPS/INS integration for airborne photogrammetric applications. This volume.

Czompo, S. (1990). Use of spectral methods in strapdown ISS data processing. This volume.

Dayton, R.B. and J.T. Nielson (1989). A flight test comparison of two GPS/INS integration approaches. Proc. ION GPS-89, Colorado Springs, Sept. 27-29, 1989.

Hughes, P.C. (1986). Spacecraft Attitude Dynamics. John Wiley & Sons, New York.

Jekeli, C. (1984). Analysis of airborne gravity gradiometer survey accuracy. Manuscripta Geodaetica, 9, 4, 323-379.

Knickmeyer, E.H., Schwarz, K.P., and Teunissen, P.J.G. (1988). "Strapdown - ein Tragheitsnavigationskonzept fUr Ingenieuranwendungen." Proc. X. Int. Kurs fUr Ingenieurvermessung, DUmmler, Bonn.

Knickmeyer, E.T. (1990). Vector gravimetry by a combination of inertial and GPS satellite measurements. Ph.D. thesis, University of Calgary.

Kruczynski, L.R., P.C. Li, A.G. Evans, B.R. Hermann (1989). Using GPS to determine vehicle attitude: USS Yorktown test results. Proc. ION GPS-89, Colorado Springs, Sept. 27-29, 1989.

Lachapelle, G. (1990). GPS observables and error sources for kinematic positioning. This volume.

Landau, H. (1989). "Precise kinematic GPS positioning." Bulletin Geodesique,63(1), 85-96.

Lapucha, D. (1990). Precise GPS/INS positioning for a highway inventory system. M.Sc. thesis, University of Calgary.

Mader, G. (1986). "Dynamic positioning using GPS carrier phase measurements." Manuscripta Geodaetica, 11(4), 272-277.

Martell, H.E., M. Wei, K.P. Schwarz, W. Griffin, A. Peterson (1990). Application of a strapdown inertial surveyor for the determination of hoist skip and mime shaft trajectory. This volume.

Porter, T.R. (1990). Application of strapdown inertial systems for precise pipeline monitoring. M.Sc. thesis, University of Calgary.

Savage, P.G. (1984). Strapdown system algorithms. AGARD Lecture Series No. 133, Neuilly sur Seine.

Schiehlen, W. (1986). Technische Dynamik. B.G. Teubner Stuttgart.

15

Schwarz, K.P. and M. Wei (1990). A framework for modelling the gravity vector by kinematic measurements. Bulletin Geodesique,64(4).

Schwarz, K.P., Cannon, M.E., and Wong, R.V.C. (1989). "A comparison of GPS kinematic models for the determination of position and velocity along a trajectory." Manuscri pta Geodaetica, 14(5), 345-353.

Schwarz, K.P., E.H. Knickmeyer, H. Martell (1990). The use of strapdown technology in surveying. ACSGC,44, 1, 29-37.

Schwarz, K.P., M.E. Cannon, R.V.C. Wong (1989). A comparison of GPS kinematic models for the determination of position and velocity along a trajectory. Manuscripta Geodaetica, 14(2), 345-353.

Stoker, 1.1. (1969). Differential Geometry. Wiley Interscience, New York.

Torge, W. (1989). Gravimetry. Walter de Gruyter, Berlin.

Van Bronkhorst, A. (1978). Strapdown systems algorithms. AGARD Lecture Series No. 95, Neuilly sur Seine.

Wei, M. and K.P. Schwarz (1990). A strapdown inertial algorithm using an earth-fixed Cartesian frame. Navigation, 37, 2, 153-167.

Wells, D.E., N. Beck, D. Delikaraoglou, A. Kleusberg, E.l. Krakiwsky, G. Lachapelle, R.B. Langley, M. Nakiboglou, K.P. Schwarz, I.M. Tranquilla, P. Vanicek (1986). Guide to GPS Positioning, Canadian GPS Associates, Fredericton, New Brunswick.

Wong, R.V.C., K.P. Schwarz, M.E. Cannon (1988). High accuracy kinematic positioning by GPS-INS. Navigation, 35(2), 275-287.

16

GPS OBSERV ABLES AND ERROR SOURCES FOR KINEMATIC POSITIONING

Gerard Lachapelle Department of Surveying Engineering

The University of Calgary, Calgary, Alberta, Canada, T2N IN4

ABSTRACT

The characteristics of GPS code and carrier phase observables are reviewed. Associated error sources, which include orbital, multipath, Selective Availability, atmospheric, and measuring errors are quantified. In particular, the effect of multipath on code observations is discussed and its impact on the related use of code measurements to estimate the carrier phase ambiguities is analysed. The estimation of C/ A code noise under field conditions is discussed together with results obtained with Ashtech receivers. Error reduction through differencing and/or combination of code and carrier phase obsetvables is reviewed.

INTRODUCTION

The technical characteristics of GPS are summarized in Table 1. The carrier frequencies selected for the system, in the UHF part of the electromagnetic spectrum, have properties well suited for all-weather precise satellite navigation, namely minimal attenuation due to atmospheric gases, rain and fog, virtually straight line-of-sight propagation, and a relatively short wavelength of 19 and 24 cm for Ll and L2 respectively, well suited for high measurement resolution. The two frequencies can be used to correct for the dispersive effect of the ionosphere in that part of the spectrum while tropospheric refractivity can be accounted for using spatial models based on surface measurements and assumed vertical behavior of the temperature, pressure and water vapour. The properties of the Pseudo Random Noise (PRN) code, which is bi-phase modulated on the carrier frequencies, are interesting both from a civilian and military aspect. Such properties include high measurement accuracy, resistance to multipath, and spreading of the moduled frequencies over bandwidths of 2 and 20 MHz respectively for the CIA and P codes. The high code measurement accuracy is due to the characteristics of the time autocovariance function of a PRN sequence which has sharp peaks and permits a high measurement resolution (e.g., Brown 1983). The higher frequency of the P code, at 10.23 MHz, results in a measurement resolution 10 times higher than that on the C/ A code.

FUNDAMENTAL OBSERVABLES

The two fundamental obsetvables most frequently used for kinematic positioning are the code pseudo-range p and the carrier phase 4>:

p = p + dp + c (dt - dT) + dion + dtrop + £(P)

4> = P + dp + c (dt - dT) + AN - dion + dtrop + £(4))

17

(1)

(2)

Table 1: GPS Technical Characteristics

Satellites 21 satellites + 3 active spares

Satellite Satellites broadcast siWla1s autonomously Constellation Orbital 6 planes, 4 satellites per plane

Characteristics 55 deg inclination, 12-hourperiod, 20,231 km altitude Frequencies Dual L-band (1575.42 MHz, 1227.6 MHz)

Digital Spread spectrum PRN, CIA code @ 1.023 MHz, P code

Signal @ 1O.23MHz Structure Signal Continuous navigation messaJ!e @ 50 Hz

Other Code Division Multiule Access signal seuaration CoveraJ!e Waidwide

Position Standard POsitioning Service: 100 m 2DRMS Accurw;y Precise Positioning Service: 16 m SEP

Velocity 0.1 m/s Time 100 ns

where <Il = - Acl>measured (<Il in m and <l>measured in cycles)

P = II r-RII dp = dPn + dpSA

dPn nominal (broadcast) orbital error component

dpSA orbital error component due to Selective Availability (SA)

dt, dT satellite & receiver clock errors respectively

dion, dtrop ionospheric and tropospheric delays, respectively

e(p), e(<Il) measurement noise

and e(p) = f{e<Prx), e(pmult)} (3)

e(<Il) = f{e(<Ilrx) , e(<Ilmult)}

e(prx) = receiver code measuring noise

e(prx) = f (receiver components, tracking bandwidth, code) e(<Il) = f{e(<Ilrx), e(<Ilmult)}

e(<Ilrx) = receiver measuring noise

e(<Ilrx) = f (receiver components, tracking bandwidth)

Another fundamental observation which is sometimes measured with GPS receivers and used for precise instantaneous velocity estimation is the instantaneous Doppler frequency fd or d<ll/dt (e.g., Cannon et al1990). The Doppler frequency, when available, is measured on the code phase.

18

ERROR SOURCES

Orbital Error and Selective Availability (S.A.): The nominal broadcast orbital error term dPn is usually of the order of 5 to 25 m with error peaks which have reached 80 m in the past. The error is expected to remain within 5 to 10 m once GPS is declared operational. The dPSA term, which is due to S.A., was initially estimated at about 100 m with a correlation time of 3 minutes (Kremer et al. 1989); values well in excess of 100 m were observed during Summer 1990, prior to S.A. being turned off in late August. Post­mission precise orbital services provide orbits that are not affected by S.A. and which are accurate to 5 m or better. Orbital errors can be reduced by differential GPS (DGPS). The level of orbital error invoked by S.A., together with its short correlation time, results in a higher differential correction update rate requirement in OGPS mode. Figure 1 shows accuracy degradation versus update rate under S.A. (Kalafus et al. 1986).

10

Station Error Fixed

o 10 20 30 40

Time Since Update (Seconds)

Figure 1: Accuracy Versus Update Rate in DGPS Mode with S.A. On.

Timing Errors: The terms dt and dT are the satellite and receiver clock errors. When satellites run on cesium time and frequency standards, the magnitude of dt is of the order of five to 10 ns, i.e, 1 to 3 m. The magnitude of dT depends on the type of clock used inside the receiver. Geodetic GPS receivers are normally equipped with good quality ovenized quartz clocks and the stability of dT is similar to that of dt over periods of up to several seconds, the Allan variance of ovenized quartz clocks over such periods being as good or better than the satellite's cesium clocks. Both terms can be solved for or eliminated by differencing. S.A. results in satellite clock dithering which adversly affects the accuracy with which the velocity of the platform can be estimated. The dithering is implemented through the injection of errors in the at satelite clock term:

(4)

with

dt = Atsv + Et (residual errors). (5)

19

Atmospheric Errors: The terms ~on and du'O are due to the effect of the ionosphere and troposphere. The ionospheric error can De ~ractically eliminated if dual frequency measurements are available. This is one of several advantages of the P code. The opposite signs of the ionospheric correction on code and carrier, i.e., group delay versus carrier phase advance, can be used to average out the effect of the ionosphere with single frequency measurements, although the accuracy is limited by CIA code noise (e.g., Goad 1990). The group delay is a function of the level of ionospheric activity and can reach 50 m at the zenith in extreme cases. The corresponding effect near the horizon is 150 m.

The ionosphere is generally correlated over distances of up to 1,000 km and time periods of three hours. OOPS can generally be used to reduce its effect. Ionospheric scintillation results in a rapid fluctuation of the Total Electron Content and a corresponding variation of the 'Doppler shift (e.g., Klobuchar 1983). Under severe ionospheric activities, this can result in frequent losses of phase lock. Geomagnetic activity forecasts can be used to avoid observations during intense ionospheric scintillation periods.

The effect of tropospheric refractivity can be estimated with an accuracy of a few dm in most cases when the mask angle is set at 50 or greater. The vertical distribution of water vapour is less predictable than that of dry temperature and pressure and is the major cause of the prediction error. The use of water vapour radiometers and stochaster modeling has been used successfully in static mode (e.g., Tralli & Lichten 1990) but is of limited use for kinematic applications. OOPS over distances of 20 km or less appears to be the most cost effective method to limit the effect of the troposphere below the few cm-Ievel.

Receiver Measurement Errors: These errors consist mostly of thermal noise intercepted by the antenna or produced by the internal components of the receiver. Their magnitude depends on parameters such as tracking bandwidth, carrier-to-noise density ratio, code and code tracking mechanization parameters (Martin 1980). The accuracy of carrier phase measurements, although affected by the tracking bandwidth, is generally better than 1 cm. Major improvements have been made in receiver design through the 80's and manufacturers now claim an accuracy of the order of 1 m for CIA code pseudo-ranges in static mode when the tracking bandwidth can be maintained at a level of one Hz. The ratio of 10 between CIA and P code chipping rates will increase the measuring accuracy of P code pseudo-ranges substantially. The accuracy improvement ratio will not be as high as 10 however due to the lower signal strength of the P code (Ward 1989) and an accuracy of 20 cm seems more realistic. The use of averaging techniques applied on P code pseudo­ranges could nevertheless yield the accuracy required for phase ambiguity resolution.

Multipath: Multipath affects both code and carrier measurements. Carrier phase multipath does not exceed 0.25A.., but can result in a large position error if the GDOP is poor (e.g., Georgiadou & Kleusberg 1989). Pseudo-range multipath is limited to a maximum of one chip length of the PRN code, Le., 293 m with the CIA and 29.3 m with the P code. This is another advantage of the P code. In static mode, multipath is systematic in nature and difficult to detect or reduce using filtering techniques. An example of static CIA code multipath is shown in Figure 2. This multipath signature was obtained with a TI4100 receiver in a high multipath environment (Lachapelle et al. 1989). In this case, an amplitude of 20 m was observed with systematic effects having periods of up to several minutes. The subsequent use of an absorption ground plane reduced multi path substantially. In kinematic mode, multipath is more random in nature. The degree of randomness will obviously depend on the dynamics of the platform. Figure 3 shows the effect of CIA multipath on a Norstar 1000 antenna mounted on an oscillating mast in a high multipath environment (ibidem). The amplitude was 20 m. In the best of cases, kinematic multipath will degrade the accuracy of code measurements to several metres. Antenna characteristics, location and RF absorbent ground planes will reduce multipath.

20

TI4166 PHASE SMOOTHED RANGE MINUS CIA CODE PSEUDO RHG DAY 313 1988 U of C Monitor Station Without Absorbant Ground Plane

- Prn

; ::! .t::~:~:::'~;:;;!#:,t:-t~1*::1t:'~J:;~;_:~J,(i\;~, 13

12

z: w 15 ... ... -c:a w (.) z: ([ I':

8 1-

_ Vertical Scale 1 Tick" B ..

I I tl'l.B8 t14.25 tH.se

CPS TillE (Hours)

9

6

Ll Data Rat .. " 3 .... c rlax W1 : 99.99 :.r.

I +14.75 +15.88

Figure 2: Static CIA Code Multipath: A ~ 20 m, T ~ 3 minutes

HS1886 PHASE SMOOTHED RAHGE MI HUS C/A CODE PSEUDO RHG DAY 223 1988 R Anten ..... Pos ition " IIASt PRH

'16 8~~~~~~~~~~~~~~~~~~~~~

- 16

8r---------------------------------~~~~~~~

Vertica I Sea Ie 1 Tick" 8 ..

+21.89

nata Rat .. = 5 Hz

+21. 25

Figure 3: Kinematic CIA Code Multipatb: A ~ 20 m, Quasi-Random

DERIVED OBSERV ABLES

Differenced Observables:

This well known technique is used to reduce several of the error sources discussed in the previous Section. The most widely used differencing modes are (i) the pseudo-range single difference between receivers, (ii) the pseudo-range and carrier phase double difference between receivers and satellites and (iii) the carrier phase triple differences where double differences are differenced over time. The pseudo-range single difference between receivers is

Ap = Ap + Adp - cAdT + Adion + Adtrop + Ae(p) (6)

21

where the common satellite clock error dt has cancelled out. Adp, Adion and Adtro are relatively small quantities which can usually be neglected when the distance betweeC the monitor and the platform is kept relatively short. The differential error Ae(p), due to the receveir noise and multipath, i.e.,

(7)

is not reduced by differentiation, but is increased due to the amplification of Ae(p ) by a factor of "2, since e(PrJ's are uncorrelated between receivers. Equation (6) is us~m one form or another for real-time differential applications. Time averaged dp and dp/dt are transmitted in real time from one or several monitors to the platform at the interval required to maintain a specified level of accuracy under S.A. (See Figure 1). The differential clock parameter dT is still present in (6) and has to be carefully modelled, especially under S.A .. Real-time CIA code single differences can yield, in the absence of large multipath effects, an accuracy of 5 m when a good geometry is available. The use of phase smoothed pseudo-ranges will yield a corresponding accuracy of the order of 2 to 3 m.

Double difference observations have the form

VAp = VAp + VA~on + VAdtrop + VAe(p)

V A<1l = V Ap + A. VAN - V Adion + V Adu-op + V Ae(p)

(8)

(9)

where the differential receiver clock term dT has cancelled out. This method is extensively used for the processing of carrier phase observations in kinematic mode because the integer nature of the ambiguity can be exploited. The double difference method would also be advantageous for real-time applications to eliminate the differential clock term dT still present in single difference observations (Equation 3). This would require that raw observations be transmitted at the monitor and the VA's be formed at the mobile.

Combined Code and Carrier Phase Observables:

In this case, code measurements are used implicitly to estimate the carrier phase ambiguities. If the platform can be held in a stationary position at a point accurately known with respect to the monitor for a few minutes, the ambiguities can be determined after one minute of observations and code measurements are not required provided no cycle slips occur once the platform goes in the kinematic mode. For the general case, this is not possible and code measurements are used to obtain an estimate of the ambiguities. The recursive filter formulation initially proposed by Hatch (1982) was used successfully in a variety of shipborne and airborne applications where positioning at the metre accuracy level were reported (e.g., Keel et al. 1989). The use of CIA code measurements will generally result in the estimation of carrier phase ambiguities to within a few cycles. The main disadvantage of this method is the discontinuity which occurs when the recursive filter is reset. Kalman filters which do not have this limitation have been proposed (e.g., Schwarz et al. 1989; Hwang & Brown 1989; Goad 1990). The use of the dual frequency P code would have major advantages, such as a noise at the dm-Ievel in the absence of multipath and the option of using wide laning techniques and/or all-in-view satellite redundancies (e.g., Hatch 1989, 1990)

22

CODE FIELD NOISE ESTIMATION

The receiver C/ A code measuring error claimed by manufacturers is of the order of 1 m as previously discussed. This claim can be verified by comparing the code against the much more accurate carrier phase measurements. Since the latter are accurate to the mm level, they can be assumed errorless for this purpose. Referring to Equation (3), one can write the error variance of a C/ A code measurement in the presence of multipath as

(10)

where cr2(Prx[C/A]) is the receiver variance and cr2(Pmult[C/AJ) , the variance caused by multipath. We will assume that, in the present case, multipath IS quasi-random in nature to

avoid biases. The quantity to be estimated is cr2(Prx[C/A])' However, unless we can

reasonably estimate ~(Pmult[C/A])' we cannot separate cr2(Prx[C/A]) from cr2(pmult[C/A])

and we can only estimate ~(Pc/A)'

In order to eliminate orbital and atmospheric errors, observations are collected in differential mode over a short distance, preferably less than 10 lan. At each measurement epoch, either in static or kinematic mode, the following quantity is computed:

B = [AVPC/A - AV<1l].

The B is mostly due to e(AVPc/a) since e(AV<1l) < 1 cm. We can further write

and

= [L(AVPC/A - AV<1l)]/n

a2(AVPC/A) ::; [L(AVPC/A - AV<1l)2]/(n - 1)

if the quasi-random multipath is known.

(11)

(12)

(13)

(14)

(15)

The above method was tested using DGPS data collected in semi-kinematic mode with two Ashtech LD-Xll receivers (Cannon et al. 1990) during the Summer of 1990. The maximum distance between monitor and mobile did not exceed 10 lan. Some 20 minutes of data were collected in static mode and some 45 minutes in kinematic mode along a road at a velocity of 80 lan h-l. Seven satelllites were tracked simultaneously. Time series of the Bls are shown in Figures 4 and 5 for both the static and kinematic case respectively.

23

SV 13 was used as the reference satellite to fonn the double differences in both cases. The ~mean of each satellite varied between -0.5 m and 0.9 m in the static case and between 0.5 m and 0.8 m in the kinematic case. The overall average was 0.5 m in the static case and 0.0 m in the kinematic case. This is not surprising since, in the kinematic case, the code multipath affecting the antenna mounting on the vehicle's roof was expected to be more random than in the static case. The a(A V~c/a>'s varied between 4.3 m and 6.4 m in the static case and between 3.5 m and 5.2 m in me kinematic case, with overall values of 5.7 m and 4.9 m respectively. The use of SV 6 as reference satellite produced slightly lower values, the cause likely being the effect of multipath at the monitor. The estimated standard deviation of a single pseudo-range measurement, a(Pc/a)' is obtained from a(AVPc/a) by applying eq. (12). This gives

a(Pe/A) = 2.8 m (static case)

a(pC/A) = 2.5 m (DOPS kinematic case; monitor static)

The internal measuring accuracy of the receiver could be found by applying eq. (15), provided multipath could be estimated. An examination of the individual satellite results, e.g., the ~mean values, clearly indicates the presence of multipath at both the monitor and

the vehicle. In order to obtain a a(Prx[c/a]) of 1 m, a(Pmult[c/a]) would have to be 2.6 m in the static case. This is the correct order of magnitude, in view of the multipath effects detected on individual satellites. This level of multipath has also been observed by other investigators (e.g., Evans & Hennann 1989). Ashtech's claim that its receiver internal noise is of the order of 1 m appears therefore to be justified.

20

.-E 10 -~ ~ .... c :z 0 ~ ~

;> C'-l -10

-20~---r--~~--~--~---r--~----~--~

493000 494000 495000 496000 497000 GMT (sec)

Figure 4: Observed AV CIA Code Noise and Multipath - Static Case

24

20

-e 10 -~ fIl .-c:> Z 0 ~ ~

;> 00 -10

-20+---~----r---~--~----~--~----r---~

493000 494000 495000 496000 497000 GMT (sec)

Figure 5: Observed II V CIA Code Noise and Multipath - Kinematic Case

CONCLUSIONS

A review of the GPS observables used for kinematic positioning was presented, together with an error analysis of there error sources. The use of differenced observations eliminates many of these errors. The code noise experiment described herein showed that CIA code noise appears to be of the order of 1 m in the absence of multipath. One can speculate that forthcoming P code receivers will deliver a pseudo-range accuracy of the order of 20 cm, a level likely sufficient for carrier phase ambiguity resolution in kinematic mode. The challenge will be to deal with the effect of multipath.

Acknowledgment. The authorwould like to thank Ms. M.E. Cannon for assisting with the analysis of the code field noise data presented in this paper.

REFERENCES

Brown, R.G. (1983) Introduction to Random Signal Analysis and Kalman Filtering. John Wiley & Sons, New York.

Cannon, M.E., Lachapelle, G., Ayers, H., and Schwarz, K.P. (1990) A Comparison of SEMIKIN and KINSRVY for Kinematic Applications. Proc. of the Satellite Division's Intern. Techn. Meeting GPS '90, ION, Washington, D.C. (in press).

Evans, A.G., and Hermann, B.R. (1989) A Comparison of Several Techniques to Reduce Signal Multipath from the Global Positioning System. lAG Symposium 102 on Global Positioning System: An Overview. Springer Verlag New York, pp. 74-81.

Georgiadou, Y., and Kleusberg, A. (1989) Multipath Effects in Static and Kinematic GPS Surveying. lAG Symposium 102 on Global Positioning System: An Overview, pp. 82-89, Springer Verlag New York.

25

Goad, C.C. (1990) Optimal Filtering of Pseudo-ranges and Phases from Single Frequency GPS Receivers. Navigation, The Institute of Navigation, Washington, D.C.(in press).

Hatch, R. (1982) The Synergism of GPS Code and Carrier Measurements. Proceedings of Third International Geodetic Symposium on Satellite Doppler Positioning, DMA/NGS, pp. 1213-1232, Washington, D.C.

Hatch, R. (1989) Ambiguity Resolution in the Fast Lane. Proc. of the Satellite Division's Intern. Techn. Meeting GPS '89, ION, pp. 45-50,Washington, D.C.

Hatch, R. (1990) Instantaneous Ambiguity Resolution. Proceedings of International Symposium on Kinematic Systems for Geodesy, Surveying and Remote Sensing, Spring Verlag New York (in press).

Hwang, P.Y.C., Brown, R.G. (1989) GPS Navigation: Combining Pseudo-range with Continuous Carrier Phase Using a Kalman Filter, Proc. of the Satellite Division's Intern. Techn. Meeting GPS '89, ION, pp. 185-190, Washington, D.C.,

Kalafus, R.M., Van Dierendonck, A.J., and Pealer, N.A. (1986) Special Committee 104-Recommendations for Differential GPS Service. Navigation, Vol. 33, No 1, pp.307-312, Washington, D.C.,

Keel, G., Jones, H., Lachapelle, G., Moreau, R., and M. Perron (1989) A Test of Airborne Kinematic GPS Positioning for Aerial Photography. Photogr. Eng. and Remote Sensing, Vol. 55, No. 12, pp. 1727-1730, ASPRS, Washington, D.C.

Klobuchar, J. (1983) Ionospheric Effects on Earth-Space Propagation. Report No. ERP-866, Air Force Geophysics Laboratory, Hanscom AFB, Mass.

Kremer, G.T., Kalafus, R.M., Loomis, P.V.W., and Reynolds, J.O. (1989) The Effect of Selective Availability on Differential GPS Corrections. Proc. of the Satellite Division's Intern. Techn. Meeting GPS '89, ION, pp. 343-347, Washington, D.C.

Lachapelle, G., Falkenberg, W., Neufeldt, D., and Kielland, P. (1989) Marine DGPS . Using Code and Carrier in a Multipath Environment. Proc. of the Satellite Division's Intern. Techn. Meeting GPS '89, ION, Washington, D.C., pp. 343-347.

Martin, E.H. (1980) GPS User Equipment Error Models. Global Positioning System, Vol 1, The Institute of Navigation, Washington, D.C., pp. 109-118.

Schwarz, K.P., Cannon,M.E., Wang,R.V.C. (1989) A comparison of GPS kinematic models for the determination of position and velocity along a trajectory. Manuscripta Geodaetica, Vol. 14, No.5, pp. 345-353.

Tralli, D.M., and Lichten, S.M. (1990) Stochastic Estimation of Tropospheric Path Delays in Global Positioning System Geodetic Measurements. Bull. Geod., Vol. 64, No.2, pp. 127-159.

Ward, P. (1989) Dispelling Some Popular Myths about GPS Receivers for Military Applications. Proc. of the Satellite Division's Intern. Techn. Meeting GPS '89, ION, pp. 343-347, pp. 281-290, Washington, D.C.

26

MODELLING INERTIAL POSITIONING PROBLEMS IN COVARIANT FORMALISM

G. Bartha Geodetic and Geophysical Research Institute

Hungarian .Academy of Sciences P.O.Box 5, H-9400 Sopron, Hungary

M. Doufexopoulou, R. Korakitis Higher Geodesy Laboratory NTU of Athens

Heroon Polytechniou 9, Zografos GR-157 73 Athens, Greece

INTRODUCTION

The physical backbone of inertial positioning is the motion equation (specific force equation) of the inertial vehicle which is expressed in different coordinate frames depending on the technical implementation (space stabilized, earth-slaved, strapdown system). The result of this adaptation is a set of mechanization equations for data processing or error analysis.

Several models were introduced for different mechanizations and several error sources were investigated in the history of inertial positioning (far from a complete list: Britting, 1971; Schwarz, 1983; Gonthier, 1984, Wong, 1982, Schroeder et al, 1988; etc). All of them had a common feature: the motion equation is given in geometrical (mainly spherical or ellipsoidal) representation using coordinates of a rotating frame.

In the first part of the paper a covariant (coordinate-independent) form of the motion equation in a rotating reference frame is derived by means of the Hamiltonian formalism. The covariant equation contains generalized coordinates Q and their metric matrix M. For a specific problem the generalized coordinates are replaced by specific holonomic coordi­nates, such as spherical, ellipsoidal or natural coordinates, and M is replaced by the corre­sponding metric matrix.

In the second part of the paper the effectivity of this type of modelling is demonstrated by an error study of the effect of the local gravity field. In gravity field modelling the higher order terms (n>36) in the spherical harmonic expansion describe the variations in the local gravity field. These terms are approximated by different sets of coefficients in different models. Therefore this part of the spherical expansion was considered here as the "not eliminable" gravity effect and approximated by a generally accepted local gravity field model with quasi stochastic description.

From the covariant motion equation an earth-slaved mechanization equation is deduced using natural coordinates (A, <1>, 6. W, i.e. astronomical longitude and latitude, geopo­tential). The acceleration is considered to be measured in a platform system slaved to the 10-cal level Cartesian system (coordinates X) and the computed position is given by geodetic coordinates (A., <p, h; i.e. geodetic longitude, latitude, height). The platform coordinate system does not necessarily coincide with the .tangential coordinate system of any geometric reference surfaces - in contrast to models generally used - because its vertical axis lies in

27

the direction of the local vertical, which differs from the "vertical" direction of the geomet­ric surfaces by the deflection of vertical (~, 11).

The effect of the "not eliminable" local gravity field was investigated by using the derived mechanization equation. Accelerometer measurements distorted by local gravity were simu­lated for a moving vehicle on the surface of the reference ellipsoid along a predetermined path. The simulated measurements were processed by using the mechanization equation on the equipotential surface of normal potential, i.e. only local gravity was considered as an error source.

1. COVARIANT MOTION EQUATION IN ROTATING FRAME

The inertial kinetic energy T expressed by coordinates Q in a frame rotating by constant an­gular velocity 0 is given by

T=~(QT +OT)M(Q +0) =~(QTMQ) +OTMQ +~(OTMO) (1.1)

where Q is a contravariant column vector of the generalized holonomic coordinates of the moving vehicle in a reference frame and M is the corresponding metric matrix (Lanczos, 1966; Budo, 1965; Hotine, 1969; Simmonds, 1982; Arnold, 1985).

The Lagrangian function of the motion problem is

1 L=T- U=2(QTMQ) +OTMQ- Urn +W

Urn=U+V; (1.2)

where U is the potential of the inertial dynamic force which makes the vehicle move, Urn is the potential of the inertial acceleration measured in the vehicle, V and W are the gravita­tional and gravity potentials, respectively.

The generalized impulse conjugated to Q is a covariant row vector pT:

aL pT= -=M(Q+O) (1.3)

aQ

The total potential energy (Urn - W) is not conservative therefore the Hamiltonian function is derived as

under the condition 0 = 0 (Bekessy et al, 1962).

Then the canonical motion equation set is:

aH Q=-=M-lpT - 0

apT

28

(1.4)

(1.5)

Since the vector Q contains the generalized, arbitrary holonomic cc>ordinates of a rotating frame, Eq. (1.5) can be considered as the covariant form of the mechanization equation of an inertial measuring system on the rotating earth.

Replacing Q and M in Eq. (1.5) by

Q=[~l M=[ r2ct~ ~ ?] (1.6)

or

Q=[~l M=[ (N+h)2COS2<p o o

o (M+h)2

o (1.7)

one gets the mechanization equation in spherical or ellipsoidal representation of the coordi­nates (Britting, 1971; Schroeder et al., 1988). In equations (1.6) and (1.7) A and <p are the spherical (or geodetic) longitude and latitude, r is the radius of the reference sphere, M is the radius of the reference ellipsoid in the direction of the meridian, N is the radius of the reference ellipsoid taken in the direction of the prime vertical.

2. MODELLING OF THE LOCAL GRAVITY FIELD USING NATURAL COORDINATES

2.1. Mechanization and Error Equation of Earth-Slaved System in Natural Coordinates

The actual form of,Q and Q is given as

,Q - [&>] . - 0 ' Q=[~t]=[~}q+£ (2.1)

where ~ W = g <lX3 = g db; q = [H E = [~~] and M is the corresponding metric matrix

(Heiskanen and Moritz, 1967). The metric matrix can be computed from the following relation between the holonomic

coordinates (Dermanis, 1985; Hotine, 1969):

aQ dQ = -dX = M-l/2 dX (2.2)

ax In our case dQ can be written as

aq ax ae dQ=dq+de =---dX + -dX (2.3 a)

ax ax ax or, in component form

29

+

and from this

M-l/2 = m-l/2 + ~

~ ~ aXl aX2

~= ~ ~ aX l aX2

0 0

and

l/(N+h)coscp

o o

~ ~

o l/(M+h)

o

~ aXl aX2 aX3

[~] ~ ~ ~

dX2

aXl aX2 aX3 dX3

0 0 g

m-l{l :[

l/(N+h)coscp 0

0 l/(M+h)

0 0

~ " aX3 (N+h)coscp

a~ ~ -+ aX3 (M+h)

'Y + .1g

o 1

-~

0 ] 0

0

(2.3b)

(2.4)

M-l = M-1/2 (M-l/2)T = m-1/2 (m-1/2)T + m-1/2 ~T + ~ (m-1/2)T + ~~T = m-1 + l! (2.5)

where 'Y and g are nonnal gravity and gravity (g = 'Y + .1g = dW/dX3); x is the vector of Cartesian coordinates in the tangential triad of the geometric reference ellipsoid which coincides with the physically defined equipotential surface of the nonnal potential; X is the vector of Cartesian coordinates of the Local Level System; q is the vector of curvilinear "natural" coordinates (A., cp,.1U) of the equipotential surface of the nonnal potential ('Y = dU/dh = dU/dX3); Q is the vector of curvilinear natural coordinates (A, <ll, .1 W) of the geoid.

The error effect of the local gravity field is derived as follows. Geometrically the vehicle is moving on the reference ellipsoid either along the meridian (northward) or parallel to the equator (eastward) but its platfonn is slaved to the nonnal of the physically defmed geoid. Therefore the mechanization equation set (used for simulation of the accelerometer measurements foJ is

Q=M-1pT - n ; (2.6a)

30

Ffm=oUm=OUm oX =M-I12(fm OQT oX OQT

Without the local gravity field these equations are:

. 1 am-1 10m oUm oU _pT=2:{pT(_)p} +2:{QT(-)Q}+---o~ o~ o~ o~

ci=m-IpT - Q ;

Substituting the real measurements simulated by Eq. (2.6) into (2.7) one gets

(2.6b)

(2.7)

1\ 1\

4 = ~-I pT - Q _pT = !{pT(am-\p} + !{QT(om)Q}+ Fm _ oU (2.8) 2 oqT 2 oqT oqT

Then the equations for positioning error caused by the local gravity field is: . . ~ I T 1\ I\T T 1\ nT 1\ T Aq = q - q = m- p - m-Ip =m-I p - m-I C" +m-1 Ap

1\ 1\ . . ~ 1 o.M,-1 am-1 10M am _ApT=- (PT - pT>=2:{PT( ___ )P) +_{QT( ___ )Q}

oQT oqT 2 OQT oqT

1\ 1\ 1 om-1 om-1

+2:{ApT(_)Ap} - PT(-)Ap oqT oqT

(2.9)

The predetennined path of the vehicle is described by the function q( t) with respect to the reference ellipsoid, or by Q(t) with respect to the geoid:

Q(t) = q(t) + e(t) = [8] {i] (2.10)

From this we get:

pT=m(ci +O)=m ![~] +m [~]

pT=m(Q +O)=m (ci +£+O)=m :t[~]+m ![i}m{~] . dEdX. de T' e=dX dx X=dX e x (2.11)

where x is the ellipsoidal tangential velocity of the vehicle. The matrix m-I is the metric matrix of the natural coordinates on the equipotential surface

with normal gravity:

31

[

1/(N+h)2cos2q>

m- l = 0

o

o l/(M+h)2

o where normal gravity is given by

'Y = 9.780 327 (1 + 0.005 279 041 sin2q> + ... )

(2.12)

while m- l is computed by replacing q in Eq. (2.12) by q(t) + Aq(t-1). The matrix M-I is given by Eq. (2.5). .

The matrices m and M can be analytically derived from m-l and M-I, respectively, in or­der to avoid numerical matrix inversions and to save computer time. Their differentiation can also be analytically performed, using the following approximations:

a a (N+h)cosq> -

a a

aA aX I aA. aA.

a a a a a a (2.13) - "'" (M + h)- - "'"

aQ a<1l aX2 aq aq> aq>

a 1 a 1 a a 1 a -----aw g aX3 g ah au 'Y ah

The components of the deflection of vertical (11, ;) are expressed with respect to the co­ordinates of the Local Level System, see Eq. (2.2).

Error simulation was performed by numerical integration using equations (2.10. - 2.13.) and the kinematic and gravity field parameters given in Table 1.

2.2 Modelling of Local Gravity Field Variations

Variations of the local gravity field in the space domain can be described by modelling the ;, 11 and Ag variations similarly to the local gravity field modelling in the previous chapter. The anomalous gravity vector is given by "predicted" values along the vehicle path.

The prediction of;, 11 and Ag is done under the assumption of quasi-stochastic variations of the gravity field, i.e. supposing a local gravity field of stochastic character. By comput­ing the amplitude spectrum of the the remaining part (error term) in the spherical harmonic expansion of;, 11 and Ag above degree of n=36 one can validate the stochastic behaviour.

The reference field of low degree (n<37) is preferred against the higher degree expansion for two reasons:

The numerical values of the coefficients provided by different spherical harmonic models are similar up to degree and order of 36,

low degree reference field introduces only minor distortions into the high degree variations in the local gravity field.

The stochastic character of the;, 11 and Ag variations for wavelengths above the reference field characterized by the lower degree (n<37) terms in the spherical expansion is a standard assumption in geodetic gravity field modelling. Linear prediction of these parame~ers is done using their low order statistical moments. The prediction model is the

32

geodetic form of the least norm approximation with uniform data, the least squares prediction. The form of this model using information provided by known quantities is

[ ~p] [C;px] l1p = CTIPx C-IxxX Agp CI'MPx

(2.14)

where ~P' ~p and Agp are the predicted values; C.x and Cxx are the cross- and autoco­variance matrices, respectively; x is the known gravity field information (in our example it is given at the endpoints of the vehicle path).

Due to the elimination of the low order harmonic components, the spherical "shape" is mostly removed from the gravity field. Additionally, the extension of the vehicle route is small relative to the earth radius, so the elements of the C matrices can be computed using a planar (not spherical) exponential covariance model (Moritz, 1976):

C(s) = Co exp(s/So) (2.15)

where s is the distance between a known point and a predicted point; Co is the variance; So is the correlation distance. For spherical covariance models see Moritz (1980); Jordan (1978).

The correlation distance can be interpreted as a dominating wavelength in the spectrum of the quantity to be predicted. The results of various numerical investigations (Schwarz et al., 1985; Doufexopoulou et aI., 1989) were used to select a realistic value for the correlation distance Smooth local fields tend to give longer correlation distances than rapidly changing fields. An "optimistic" value of 45 Ian was selected for a smooth field while a "pessimistic" 15 Ian for a rough field.

3. RESULTS AND DISCUSSION

3.1. Kinematic and Local Gravity Field Parameters and Results

The model parameters are summarized in Table 1. Model #1 and Model #2 did not give significant errors. The effect of the constant deviation between the ellipsoidal and geoidal gravity (Model #3) is shown on Fig. 1. The effect of the gradient of 11 and ~ is shown on Fig. 2 through Fig.4 (error in Xl, X2 and X3, respectively). The effect of the different correlation distances is depicted on Fig. 5 (for Xl) and Fig. 6 (for X2).

Model code number 1 2 3 4 5 6 7 8 Direction of vehicle motion N E N N N N E E Acceleration (m/s2) At = 10 sec-,-s = 50 m 1 1 1 1 1 1 1 1 Uniform velocity (m/s) At = 5000 sec, s = 50 Ian 10 10 10 10 10 10 10 10 Deceleration (m/s2) At = 10 sec, s = 50 m -1 -1 -1 -1 -1 -1 -1 -1 ~ at start (arcsec) 0 0 1 0 0 0 0 0 E. at 45 Ian distance (arc sec) 0 0 1 1 5 10 1 1 11 at start (arc sec) 0 0 1 0 0 0 0 0 11 at 45 Ian distance (arcsec) 0 0 1 1 5 10 1 1 Ag at start (mgal) 0 0 3 0 0 0 0 0 Ag at 45 Ian distance (mjtal) 0 0 3 3 15 30 3 3 Correlation distance (Ian) 0 0 0 15 15 15 15 45

Table 1. Model parameters

33

(5

6.X2 --.., (0 E '--

!o-5 0

!o-!0-

k!

0-

-5 -'\ i , iii i I I I Iii i I I ' I , I T--r--r-r---.--,

o (0000 20000 30000 40000 50000 60000 Distance (m)

Fig. 1. Error behaviour in case of constant gravity deflection

500

--.., 0r-........,~::::::=:::::::::=-____ ~'" / 3mgal

E '-- -500

~ -fOOO !o-!o­k!-(500

-2000

5" / 15mgal

10" / 30mgal

-2500 nr-r......--.---,-".---,-,-.--,-,---,-r-r..--.-,-,.--.-r-r-.---r---'--"""--'-l o (0000 20000 30000 40000 50000 60000

Di st ance (m)

Fig. 2. Error in Xl for different gradients

6.X2 10" / 30mgul

(0000 --.., E '--

!o-0 5" / 15mgal !o- 5000 !0-k!

1" / 3mgal

10000 20000 30000 40000 50000 60000 Distance (m)

Fig. 3. Error in X2 for different gradients

34

200 ~X3 fO" / 30mgal

tOO ........ 5" / ISmgal E "- 0

!o- -tOO 0 !o-!o- -200 '" / 3mgal k!

-300

-400

-500 0 tOOOO 20000 30000 40000 50000 60000

Distance (m)

Fig. 4. Error in X3 for different gradients

2000 ~ Xl So= ts;/'

E't500

"-

500

tOOOo 20000 30000 40000 50000 60000 Distance (m)

Fig. 5. Error in Xl for different correlation distances

2000 ~X2

~ tOOO !o-!0-k!

500

So = t5km

O~rr~-'""rr"~'T"-rro'-""~,,,, o tOOOO 20000 30000 40000 50000 60000 Distance (m)

Fig. 6. Error in X2 for different correlation distances

35

3.2. Discussion

Since there are no computation errors in the modelling, all the errors in the position come from the simulated gravity field model. Without gravity field variations (11 and ~ are constant)· the AX I and AX2 errors remain in the order of few meters for the whole mission. However, errors in X2 are slightly larger than in Xl. The presence of a simulated gravity field strongly affects the position errors (Fig. 2, 3,4). Even for the smallest differences (1 ",3 mgal) the errors are extremely large. For the optimistic model, errors in the direction of motion are one order of magnitude larger than errors in the other coordinate.

A comparison between the "optimistic" and "pessimistic" models (Fig. 5 and 6) shows that the correlation distance affects the magnitude of the position errors. The "pessimistic" model gives faster error build-up, especially in X2.

These first numerical results clearly show that the gravity field strongly affects position­ing accuracy (as was expected) and the azimuth of the survey affects the error behaviour in different coordinates. Generally the errors of X3 are larger than those of Xl and X2.

The high dependence on the gradient of 11, ~ and .1g shows that inertial positioning will get worse when done in a rough gravity field (mainly of tectonic origin). This is verified by the dependence of the errors on the correlation distance. When the terrain is the reason for short correlation distance, one can expect a smooth gravity field after the terrain removal. This is not the case if the field has tectonic origin.

REFERENCES

Arnold (1985), A mechanika matematikai modszerei, Muszaki Kiado, Budapest, Hungary Bartha (1986), Bulletine Geodesique, 60, pp 121-128 Bekessy et al (1962), Elmeleti Fizikai Feladatok, Tankonyvkiado, Budapest, Hungary (in

Hungarian) Britting (1971), Inertial Navigation Systems Analysis, Wiley-Interscience, New York Budo (1963), Mechanika, Tankonyvkiado, Budapest, Hungary (in Hungarian) Dermanis, A. (1985) Personal communications Doufexopoulou et al. (1989), Acta Geodaetica Geophysica Mont. of Hung. Acad. 24/3/4

pp.230-236 Flanders (1963), Differential Forms, Academic Press, New York, USA Gonthier (1984), Smoothing Procedures for Inertial Survey Systems of Local Level Type,

UCSE Report No. 20008, The University of Calgary, Canada Grafarend (1973), OSU Report No. 202, Columbus, Ohio, USA Heiskanen-Moritz (1967), Physical Geodesy, Freeman, San Francisco, USA Hotine (1969), Mathematical Geodesy, ESSA, Washington, USA Jordan (1978), JGR BIO/No. 20. Kearsley et al. (1985), White Sands Revisited, A Comparison of Techniques to Predict

Deflections of the Vertical, UCSE Report No. 30007, The University of Calgary, Canada Lanczos (1966), The Variational Principles of Mechanics, University of Toronto, Canada Misner et al. (1973), Gravitation, Freeman, San Francisco, USA Moritz (1976), OSU Report No. 242, Columbus, Ohio, USA Moritz (1980), Advanced Physical Geodesy, Wichmann Rozsa (1974), Linearis Algebra, Muszaki, Budapest, Hungary (in Hungarian) Schroeder et al. (1988), Manuscripta Geodaetica, 13, pp 224-248 Schwarz (1983), Rev. of Geophys. and Space Physics, 21/4, pp 878-890 Simmonds (1982), A Brief on Tensor Analysis, Springer, Berlin, Germany Vassiliou (1984), Manuscripta Geodaetica, 9/4, pp 281-306 Wong (1982), A Kalman Filter-Smoother for an Inertial Survey System of Local Level

Type, UCSE Report No. 20001, The University of Calgary, Canada

36

AN OBSERVABILITY ANALYSIS OF A GPS-AIDED GEODETIC INERTIAL STRAPDOWN MEASUREMENT UNIT

Margit Bolcsvolgyi-Ban College for Surveying and Country Planning

1-3 Pirosalma Str., H-8002 Szekesfehervar, Hungary &

Dietrich Schroder Department of Geodetic Science

University of Stuttgart, Keplerstr. 11, D-7000 Stuttgart 1, F. R. G.

INTRODUCTION

A widely used tool to combine data of different measurement devices in real time is the Kalman-Filter-Technique. To guarantee a uniformly asymptotically globally stable filter, the system model upon which the Kalman filter is based has to be both observable and controllable. Thus a first step to design a filter is to check the observability and controllability of the system model. In this paper we restrict ourselves to the observability problem, but because of the Kalman duality theorem the results can easily be extended to the controllability problem. The structure of the system model is not only important in real time filtering, but also for the post­mission adjustment, if, e.g., the state vector is introduced as an unknown quantity in the adjustment.

Here the observability problem is outlined for a hybrid INS/GPS measuring device. First the mathematical model of the combined system is introduced, which can be looked upon as the dynamic equations of a differential system. After some notes on the definition of observability for deterministic and stochastic systems, numerical meth­ods are discussed to determine observability. If the system model is unobservable, the splitting up of the state space is discussed by means of constructing orthonormal bases for the observable and unobservable subspaces, respectively.

For an INS local level mechaniziation the elements of the dynamic matrix extended for bias and drift parameters of the sensors only vary slowly in time and a good approximation of such a system is a stepwise time invariant system model. This assumption no longer holds for a strapdown mechanization. Here the decomposition into the subspaces only applies if their dimension is not changing with time. Since the theoretical background of the observability problem and the decomposition into subspaces of a general time varying system are so far not known in detail, the general problem is only dealt with approximately.

37

THE MATHEMATICAL MODEL

It is well-known that the functional relationship between the observables of a strap­down inertial measurement unit (IMU) and the unknown parameters of inertial po­sitioning can be reduced to a non-linear system of first order differential equations:

d3jdt = f(3(t), n(t)) (1) If the IMU is combined with other measuring devices, for example with the roving receiver of a kinematic GPS survey unit, the equation above has to be supplemented by another model equation of the additional observables r , r E iR!"':

r(t) = h(3(t)) (2)

Eq. (1) may be considered as the state equation and (2) as the output equation of a dynamical differential system. The unknown parameter vector 3 E F is composed of the state variables, which are primarily the position, velocity and orientation pa­rameters, whereas the input vector n E 1R? contains all the variables whose values we have to know, in addition to the initial state 3(to), to solve the state equation.

The state vector may be augmented in view of nuisance parameters neglected so far, for example inertial sensor model coefficients or GPS bias parameters. In doing so we want to pass over to a linear model, assuming that the effect of those parameters on the state vector is small. Usually the "true" values of all the input variables as well as the "true" values of the initial states are not available, but instead only an estimate W(t) and X(to) of them, with

E{X(to)} = 3(to) , E{(X(to) - 3(to))(X(to) - 3{to))T} = EX(to)

E{W(t)} = n(t) , E{(W(t) - n(t))(W(t) - n(t))T} = Ew(t)8(t, t') ,

where 8(t, t') denotes the Dirac Delta-Function. In the following it is assumed that the known input-variables are included in the functional model as time-varying or constant parameters and are thus discarded from the input vector. Now an approx­imate stochastic trajectory X(t) can be defined by the numerical solution of the state equation with respect to the given realizations X(to) and W(t). Given the observations Y of the output r, i.e.

Y - e = h(3(t)) ,e I"V (0, Ee) (3)

we can linearize the state equation (1) and the observation equation (3) with respect to the approximation X ( t) and W ( t) :

dxjdt = F{t)x + G(t)w(t) ,y - e = H{t)x (4)

with x(t) := 3(t) - X(t) , w{t) := n(t) - W(t) , y{t) := Y(t) - h(30(t)) and the Jacobians F(t) := af ja30 , G(t) := af jano and H(t) := ahja30 whereby 3 0 := X +0 and no := W +0 denote the numerical equivalent but non-random forms of the random approximations X(t) and W(t), respectively. As the observations y are usually available only at discrete times ti , i E IN, we want to convert the state equation also into a discrete form. In doing so for the time interval [ti, ti+1] we obtain formally

X(ti+1) = X{ti) + 1,ti+1 F{T)X{T)dT + liti+1 G{T)W(T)dT (5)

The first integral of the right hand side (rhs) of equation (5) exists as a Riemann inte­gral in the mean square sense, if x(t) is a continuous function in the mean square sense. Having assumed W(t) to be an unbiased estimate of n(t) and the estimates to be

38

uncorrelated for different times, i.e. E{w(t)} = 0 and C{w(t), w(t')} = Ew(t)a(t, t') it results that the stochastic process given formally by w(t)dt defines a Brownian motion. Thus the second integral on the rhs exists as a Wiener integral and equation (5) can be rewritten to

Xi+1 := X(tHl) = CP(tHI, ti)Xi + Ui+1 with

i ti+1

Ui+1:= ti CP(ti+b T)G(T)W(T)dT ,E{Ui+d = 0

and T iti+! T T E{ Ui+1UHtl = EUi+1 = ti CP(ti+1' T )G( T )Ew( T)G( T) CP(ti+1' T) dT

The linear dynamic equations are finally transformed to

(6)

(7)

(8)

Xi+1 = Aixi + Di+1Wi+1 ,Yi = Hixi + Fiwi (9) with Ai := CP(tHb ti) and Wi '" (0, In), i.e. the observation equation has been trans­formed to unity covariance of the measurement error by multiplying the original observation equation with RT , where RT results from a Cholesky factorization of E~l, i.e. RT ei = Fiwi with Fi = (Irn 0) accounting for the different dimen­sions of ei and Wi. Di results from a correspondent factorization of EUi such that E{(DiWi)(DiWif} = DiDT = EUi holds.

ON STOCHASTIC OBSERVABILITY

Having now defined the state equation as well as the observation equation of our linear model, the question arises which information about the state variables is contained in the observed data and whether all the states can be determined from it. Let us first look at the degenerate dynamic equations (9), i.e. Fi = Di+1 = 0:

ei+1 = Aiei "i = Hiei (10) The deterministic discrete-time linear dynamic system (10) is called completely ob­servable in the interval [tj, tj+1' ... , tj+N] , if, given the sequence of output values Ii , i E I : {j, j + 1, ... , j + N}, it is possible to uniquely hdetermine the state ei for all i E I. To be completely observable the structure of a model defined by the matrices Ai and Hi must be such that the output is affected in some manner by the change of any single state variable and this.- effect must be distinguishable from the effect of any other state variable. Obviously observability is a property of the structure of a specific state representation rather than of the system itself. Thus looking from the point-of-view of the input-output behaviour of the system model, certain state space models will be more suitable for estimation than others, even though both might accurately portray the input-output characteristics of the dynamical system. However, we have to remark that the structure of our dynamical system is defined by the mathematical model of the measuring devices and is not at our disposal for a realization or identification problem.

In the following we restrict ourselves without loss of generality to the observability in the interval [to, it, ... , tN} and to checking the deducibility of the initial state eo from the output sequence Ii , i E 10 : {O, 1, ... , N}. A necessary and sufficient condition for

39

a system to be completely observable in the interval 10 is the existence of an integer value k ~ N such that the observability matrix defined by

k

Gk := (HJ' A~ Ai A~ Hi ... II AT HI)T ,Ao:= In (11) i=l

has the rank n. In the non-degenerate stochastic model only an estimate of the unknown state

vector can be given, applying some criteria of optimality. The possibility of deducing the initial state ~o in the degenerate deterministic model is equivalent to the existence of a linear unbiased estimator for Xo based only on the measurements Yi , i E 10•

A necessary and sufficient condition for such an estimator applying the optimality criteria of minimum variance, i.e. the existence of a linear unbiased minimum variance estimator (LUMVE), is the positive-definiteness of the matrix

k i i

Mk := L:(II Af)HTE-;/ Hi(II AT? (12) i=O j=O j=O

for some i E 10 • Obviously this condition fails if E;/ = 0 or if E;/ is infinite over the finite time interval considered. Thus defining stochastic observability by the non­singularity of (12) is not compatible with complete observability for the degenerate deterministic system. To overcome this problem Chen (1980) has given another condition of stochastic observability, i.e. for the existence of a LUMVE of Xo.

In the sense of Chen the stochastic system is called stochastic observable in the inter­val {to, ... , tN} if there exists an integer value k such that GtGk = In or rank(GK ) = n, where G+ denotes the pseudo-inverse of Gk • In particular he proved that the observ­ability oi the degenerate deterministic model implies stochastic observability of the non-degenerate stochastic system. We have to remark, however, that the regular­ity of Mk is for our "homogenized" model (9) purely numerically equivalent to the condition of complete observability of the degenerate deterministic system.

Consider now the deterministic (or degenerate stochastic) system (10). To check observability for the sequence 10 , we rewrite the system equation to

7"ik = Gk~O (13)

with "ik = (/'6, ,[, ... , ,If and the k . m x n observability matrix defined by (11). Note that even the deterministic system is only solvable if the consistency condition 7"ik E R(Gk) holds, where R(·) denotes the range of the observability matrix. If the consistency condition is not fulfilled, we have to introduce the least-squares problem of finding ~o such that

C'Yk - Gk~O?C'Yk - Gk~O) -+ min. (14) As the dimension of the nullspace of Gk is given by dim(N(Gk)) = n - r, where rank(Gk ) = r, we obtain a unique solution of (13) or (14) only if r = n. If r < n we can add a vector ~8 E N(Gk ) to a given solution ~b without changing the output

7k = Gk(~h + C) = Gk~h (15) or alternatively the minimum condition, as ~8 fulfills the homogeneous equation. Thus a linear combination of the original states which is an element of N(Gk ) is an unobservable linear combination. If r < n we want not only to determine the dimension of the nullspace, i.e. the number of independent unobservable linear com­binations of the original states, but also which linear combinations are unobservable

40

state space output space

Fig.1. Subspaces defined by the observability matrix Gk •

(N(.).L and R(·).L denote the orthogonal complement of the subspaces.)

or observable. This can be achieved by constructing an orthonormal basis for the unobservable and observable subspaces. Consider an orthogonal decomposition of the observability matrix

Gk = (PH Pk2) (~k ~) ( ~I: ) (16)

with the regular r x r matrix Rk and the orthogonal transformation matrices Pk and Qk, which are partitioned corresponding to Rk. Obviously the block matrices of Pk and Qk provide orthonormal bases for the subspaces given in Fig. 1 according to Klema and Laub (1980). The map between the four fundamental subs paces is realized by the observability matrix and its pseudo-inverse. In the following we consider some methods of constructing orthonormal bases of the subspaces.

NUMERICAL METHODS

In this study only numerical methods of determining the observability are considered. Analytic methods, even the evaluation of the formulae on a computer by means of a formula manipulating program system are from our experience not tractable for a dynamical system of the dimension as considered here.

A first method for checking observability is to check the regularity of the sym­metric matrix Gr Gk , i.e. of the Gauss-transformed observability matrix. For this purpose one can determine the determinant of the matrix, but this method is not recommended due to the problems of the numerical determinant determination by means of a digital computer. Furthermore, only a statement can 'be made about the observability, but no splitting of the state space is provided.

The computation of the eigenvalues and eigenvectors of the symmetric matrix presents as a rule no problems, but in spite of the squaring of the matrix it is less sensitive than the investigation of the observability matrix itself.

Here three methods are considered which avoid these problems and which provide on the other hand an orthonormal basis for the subs paces of the state space. In the following only the least-square problem (14) is considered.

QR- Factorization. A first method for constructing an orthonormal basis is the QR-Decomposition. If rank(Gk ) = n the observability matrix can be transformed by means of orthogonal transformations to

41

(17)

where R~ denotes an n x n upper triangular matrix. Remember that an orthogonal transformation will not change the norm of a vector, thus the least-square functional is rewritten to

J = (Qt'Yk - QfGkeo)T(Qt'Yk - QfGkeo) = ('Yk1 - R~eo)T('Yk1 - R~eo) + 'Y'b'Yk2

(18)

by a corresponding partitioning of Qf'Yk = 'Yk. The least square approximation of the initial state can easily be computed by e~s = R'k-1'Yk1. There are several algorithms for constructing the orthogonal transformation matrices (see e.g. Golub and van Loan (1983)), here we only want to mention Householder and Givens transformations. If on the other hand rank(Gk) = r < n, the Householder and Givens transforma­

tions do not necessarily produce an orthogonal basis for N ( G k).1 , and the transforma­tion has to be supplemented by another transformation Ih consisting of elementary permutations such that

GkIh = QkRk with Rk = (~~ ~Z) (19)

where R~ is an upper r x r triangular matrix. Thus the observability can be checked by computing R~, but if r < n we still have to split up the non-observable subspace. For this purpose we have to apply another transformation such that RZ vanishes. This can be achieved by another orthogonal transformation matrix Zk composed of another set of Householder and Givens transformations, such that

(R~ R~)Zl = (Tk 0) (20)

where Tk is an upper triangular r x r matrix. Thus we arrive for the least-square functional at

J = (Qf'Yk - QfGkIIkZkzlIIfeo)T(Qf'Yk - QfGkIIkZkZlIIfeo) = ('Yk1 - TkeO)T('Ykl - Tkeo) + 'Y12'Yk2

(21)

where ZlIIfeo = (e~ e~)T. Obviously IIkZk provides the desired orthonormal basis of N(Gk).1. To compute the complete orthogonal decomposition QrGkIIkZk by means of Householder and Givens transformations, the rank of the observability matrix has to be determined. It was pointed out by Golub and van Loan (1983) that a QR-factorization algorithm can fail to detect the rank. Thus another method is recommended.

Singular Value Decomposition (SVD). The SVD algorithm is another pos­sibility of constructing a complete orthogonal decomposition of the observability ma­trix. It is well-known, that for every m x n matrix A with rank(A) = r < max(m, n) there exists an m x m orthogonal matrix U and an n x n orthogonal matrix V, such that

A = UEVT = (U1 U2 ) E (~~) (22)

with the diagonal matrix E = diag(O'I, ... , O'r, 0, ... , 0). The ordered numbers 0'1 ~ 0'2 ~ ••• ~ O'r ~ 0 are called the singular values of A. Note that Vt and V2 are

42

orthonormal bases of N(A) and N(A).L, respectively. Looking at our least-square functional, we arrive with the SVD of the observability matrix at

J = (Ul'fk - Ek Vreo)T(Ul"ik - Ek Vreo) = (7k1 - diag(O"I, ... , O"r){o)T(7kl - diag(O"t, ... , O"r){O) + 7~7k2 (23)

with

( ~1 ) eo = ( f:) ~~ -r (24)

and again {o denotes the least-square approximation of the observable linear com­binations of the original state vector. For a discussion of the different methods of constructing the transformation matrices we once again refer to Golub and van Loan (1983). Note that near rank deficiency of the observability matrix in contrast to the QR-factorization cannot escape detection when applying the SVD.

It is well-known that due to approximations, rounding errors etc., it is very un­likely that true zero singular values will occur in a numerical evaluation of the SVD algorithm. It is, therefore, common to neglect singular values smaller than a certain threshold T, e.g. T = f II A 1100 where f denotes the machine precision, calling if, for which 0"1 ~ ••• ~ 0"1' > T ~ 0"1'+1 ~ ••• ~ O"n holds, the numerical rank of the matrix. But this choice of the threshold T is useful only if the numerical rank is well determined with respect tOT, i.e. if there is a well defined gap between 0"1' and 0"1'+1.

Otherwise we have to apply other more sophisticated methods when determining the threshold T based on a perturbation analysis of the SVD and the evaluation of the matrix G".

Point - by - Point System Transformation. Another method of splitting up the state space starts from the assumption that the dimensions of the subs paces as well as their orthonormal bases only change slowly with time. The system can then be looked upon as time-invariant for small time intervals, and we can compute the subspaces for each single epoch. The algorithm introduced by Patel (1981) is closely related to the minimal realization theory of Kalman (1963). Here the splitting of the state space into controllable and uncontrollable subspaces is not considered, but only the splitting into an observable and unobservable one. The starting point of the algorithm is a singular value decomposition of the output design matrix Hi:

Ii = Hiei = UioEiOv:~ei (25) Here we only consider the case rank(Hi) = ro < n, such that ei cannot be determined by the observations at epoch ti alone. To split up the observable part of the system, we choose the correspondingly partitioned singular vectors of Vio as a basis for the state space and apply a coordinate transformation Zi = ViOei such that

(26)

holds. We note that Vi2 is a basis of the nullspace of Hi, i.e. Hi Vi2 = O. If now N(Hi) is Ai-invariant, it follows from V:[Vi2 = 0 that F12 = 0 and thus the observable linear combinations are given by

43

t ( ~i) }rO Vio<"i = t }n - ro (28)

If now N(Hi) is not Ai-invariant, we have F12 =J 0 and thus ~i is not decoupled from the other transformed states. We proceed by a SVD of F12 where now F22

takes over the part of Ai in the previous step. The algorithm is performed unless rank(Fj,j+d = rj = 0 ,E{=o ri < n or we have found n non zero singular values. In the former case the system is unobservable, whereas in the latter it is observable. The overall transformaton matrix is constructed by multiplying the matrices

S (Iv 0 ) Vi = Vio n 0 vr

3=1 13

(29)

and s is the number of SVD steps performed for the matrices Fj,j+I with its rank not being equal to zero.

SIMULATION RESULTS

The Simulated System. The observability analysis was performed for a sim­ulated INS/CPS measuring device. Concerning the INS a strapdown mechanization was simulated. Starting from an assumed time progress of the velocity magnitude, as well as of the pitch and yaw angles of a frame of reference attached to the moving vehicle, a reference trajectory as well as the output increments of the inertial sen­sors are computed. For the sensors a specific model was introduced consisting of up to 15 different coefficients for the accelerometers and gyros (SchrOder et al (1988), Schroder(1989)). By means of the velocity and angle increments the navigation com­putation is performed, giving an approximate trajectory, which will differ from the reference trajectory due to the additional sensor coefficients and algorithm errors. This approximate trajectory was used to linearize the state equation and the discrete state transition matrix was computed by numerically solving the initial value problem for the state transition matrix.

Regarding CPS only carrier phase observations in the differential mode are con­sidered. The carrier phases were simulated by a modification of the Lindlohr (1988) CPS software, such that one antenna, the fixed site, remains at the initial point of the IMU-trajectory, whereas the position of the roving antenna coincides with the moving IMU reference point. Based on the IMU reference trajectory and an assumed satellite constellation the carrier phases are computed.

There were two different approaches simulated. One assumes a separated evaluation of the CPS data, providing position updates for the INS. The second approach uses double differenced phase observations, where the observation equation was linearized with respect to the INS-computed approximated trajectory. For both approaches an observation time interval of 3 s was assumed.

Numerical Results. For all the numerical methods applied to check observabil­ity, the simulated system proved to be completely observable for both CPS observa­tion approaches. The problem of the numerical determination of the observability is outlined by means of the evaluation of the singular value spectra of the observability matrices in the interval [to, til where Ti was increased stepwise, see Fig. 2. In this example the primary state vector was augmented by three accelerometer biases and three gyro drift parameters. CPS position updates were introduced as observations.

44

0 X ..-... 0 x b xxx - XXXX b -- --.... b -4

. ... b -4 .......- '-"

0 ~ - Xx XXX b() b() 0 -8 .9 -8 X

-12 -12 T

T

-16 -16 0 3 6 9 12 15 0 3 6 9 12 15

a) ti = 6 s b) ti = 9 s

-- 0 X ..-... 0 X b xx b Xx -- xxx .... -- xXxx b -4

. ... - b -4 0

.......-x 0

b() -0 xx b() XXX -8 .9 -8 Xx Xx x Xx

-12 T -12 T

-16 -16 0 3 6 9 12 15 0 3 6 9 12 15

c) ti = 12 s d)ti=15s

0 x 0 X b Xx b X -- xxxxx -- xx .... .... xxxxxx t:::: -4 b -4 -- .........-

0 0 .- Xx ~ x b()

.::: -8 Xx 0 -8 X xxx Xx

x -12 -12 T

T

-16 -16 0 3 6 9 12 15 0 3 6 9 12 15

e) ti = 18 s f) ti = 45 s

Fig.2. The singular value spectra of the observability matrices in the interval [to, til T = f \I Gk \lex» f denotes the machine precision.

45

Obviously, looking at the magnitudes of the singular values, some linear combinations are "better" observable than others. Looking at the transformation matrix VT, it is interesting to remark that especially the linear combinations involving the azimuth and both horizontal velocity states do not belong to this dominant subspace.

CONCLUSIONS

The SVD of the observability matrix is recommended as a numerical stable algorithm to check observability for a linear time-variant system and to construct orthonormal bases for the subspaces of the state space. One of the major problems in the practical application of the algorithm is to choose an appropriate threshold for the numerical rank, which can only be reliably determined by a careful perturbation analysis of the underlying model.

In the case of the investigated combined INS / GPS model, the SVD as well as the other methods applied showed that all the states introduced so far are observable.

One possibility to continue the research is to introduce the "true" sensor model with all the individual coefficients and to check them for observability in order to obtain a reasonable bias design for the combined INS/GPS model.

Acknowledgement. The research was carried out within the framework of the Sonderforschungsbereich "High Precision Navigation" supported by the Deutsche Forschungsgemeinschaft·

REFERENCES Chen, H.-F. (1980). On stochastic observability and controllability, A utomatica,16,

179-190. Golub, G.H. and C.F. van Loan (1983). Matrix computations, North Oxford Aca­

demic, Oxford. Kalman, R.E. (1963). New methods in Wiener Filtering theory, in J.L. Bogdanoff

and F.Kozin (ed.), Proc. 1st Symp. on Eng. Appl. of Random Function Th. and Probability J. Wiley and Sons, 270-288.

Klema, V.C. and A.J.Laub (1980). The singular value decomposit·ion: Its computa­tion and some applications, IEEE Trans. Aut. Control, AC-25, 164-176.

Lindlohr, W. (1988). PUMA. Processing of Undifferenced GPS carrier beat phase Measurements and Adjustment computations, Technical Report 5, Institute of Geodesy, University of Stuttgart, Stuttgart.

Patel, R.V. (1981). Computation of minimal order state-space realizations and ob­servability indices using orthogonal transformations, Int. J. Control, 33, 227-246.

Schrooer, D., Nguyen Chi T., S. Wiegner, E.W. Grafarend and B. Schaffrin (1988). A comparative study of local level and strapdown inertial systems, manuscripta geodaetica, 13, 224-248.

Schrooer, D. (1989). Comparative study of inertial measurement systems for geodetic applications, in K. Linkwitz and U. Hangleiter (ed.), High Precision Navigation, Springer, 611-627.

46

MATHEMATICAL ANALYSIS OF THE

GEODETIC SPACE-STABILIZED INS

Bernd Eissfeller Kayser-Threde GmbH

Wolfratshauser Str. 48, D-8000 Munchen 70 FRG

ABSTRACT

The topic of this paper is the analysis of the space-stabi­lized geodetic INS Honeywell GEO-SPIN II and its integration with GPS in a Kalman filter. The basic concept of analysis is a rigorous physical description of the inertial sensors and the platform. The analysis results in error models of the ESG gyros, of the platform servo-loops and the fluid-filled pendulous accelerometers. The anomalous gravity field is mo­deled by a five state shaping-filter based on a 2nd order Gauss-Marcov process for the disturbing potential. In compa­rison to the existing error ~odel of the GEO-SPIN II a large cross-coupling term and a g -drift of the gyros are intro­duced. The platform servo-loop errors consider a mass unbal­ance of the inner element, friction of the bearings, and the damping of the motors. The error model of the accelerometers is extended by a scalefactor asymmetry, a cubic scale factor error and a cross-coupling term. After introducing the main system errors, the navigational errors and the GPS-biases a 31 states Kalman filter is derived.

INTRODUCTION

In the last years a lot of work was done in the post-mission analysis of inertial data (Boedecker, 1987; Koenig, 1988). In many papers very sophisticated stochastic adjustment mod­els have been applied to the data. But most of the results have shown that the main problem are unmodelled systematic effects in the data (Caspary, 1987; Koenig, 1988). There are several reasons for this situation :

1. The error theory of geodetic INS was developed analogous to inertial navigation although we talk about cm-accuracy in inertial geodesy

2. In part only suboptimal filtering and smoothing was used

47

based on more or less deterministic error models

3. Platform servo-loop (stabilization) errors and vibratio­nal effects are completely neglected in the error models

4. Other higher order errors and physical effects are negle­cted although they contribute to the error budget on cm­level

In order to improve this situation it is tried to set-up a detailed physical error model for a specific INS.

Because of its accuracy potential (Adams and Hadfield,1981) the Honeywell GEO-SPIN II is investigated. This specializa­tion is necessary too because meaningful physical models can only be derived by mapping the hardware errors on sensor and system level into detailed mathematical models. As a result a variety of model refinements and new mathe­

matical basefunctions for postmission adjustment are derived. Although the GEO-SPIN II was investigated the approach is

of general character and can be applied to other inertial platform systems.

THE SPACE-STABILIZED INS

A block diagram of the space-stabilized INS is presented in Fig.l.

PLATFOIUI

9 J" oj = 1 •..• 4 C b -P

vn rn p q Az - ,- t , I

VEHICLE

Fig. 1. Block diagram of the space-stabilized INS.

On the stable element of the system the following sensors are mounted :

48

1. Three pulse-rebalanced pendulous accelerometers Honeywell GG-177

2. Two electrostatically suspended gyroscopes (ESG)

The stable element is isolated from the vehicle motion by aid of a four-cardan system including the platform control and servo motors. The basic system requirement is to keep the inner element non-rotating with respect to inertial axis. This means that the accelerations are measured in inertial coordinates.

In the following a summary of the error models derived in Eissfeller (1989) for all the system components is presented.

ERROR MODEL OF THE ESG

The ESG is a high accuracy two-axis free rotor gyro.The main component is a hollow beryllium rotor (r - 19 mm, f - 720 Hz) which is suspended in an electrical field by aid r of six servo-controlled electrodes against the specific force . A schematic viewgraph of the ESG is shown in Fig.2. For more detail see Savage (1978).

Optic&l Pick-Off, pol&[

Electrodes (6)

Driving Coil

Optical Pick-Off,

equatorial

Vacuum Pump

Fig.2. Built-up of the Electrostatically Suspended Gyroscope.

The starting point of an error model is Newton's law of ro­tational motion for the rotor in body fixed coordinates (b)

49

.b H ~ib + H

b w' b --1

b m ( 1 )

where ~ is the angular rate;~ its skew-symmetric form; H the inertia tensor and m the vector of disturbing torques.start­ing with the physical principle o~ the ESG (Phelps, 1968) the following disturbing torques m have to be considered in eq. (1) : -

1. Electrostatic torques, i.e. geometry errors of the rotorl electrode system and the coupling of the geometry errors with a rotor displacement.

2. Magnetic torques, i.e. a typical eddy current effect

3. Massunbalance torques of the rotor

4. Misalignment of the redundant axis torquer (RAT) of the second ESG

5. Vibrational rectification effects caused by terms of the form f. "f. (cross-coupling) and linear vibrations of the sensor 1 blotk

6. Temperature dependent torques

7. Random walk torque

A detailed derivation of all the above error torques can be found in Eissfeller (1989, p. 54-134). In order to arrive at the final drift model the high frequency nutation of the gy­ro was neglected. The error drift of the ESG No. i is of the following form

where ~ is the drift angle; h is the angular momentum; ~kl are the disturbing torque coefficients; ~ is white noise ana fk are the components of the specific force vector.

ERROR MODEL OF PLATFORM STABILIZATION LOOPS

Because of the high stiffness of the platform stabilization loops this type of error is usually neglected in navigation. In navigation a motion of the inner element is attributed to to the dift of the gyros only. However, a detailed analysis shows that even the control loops are not free of errors.

Following Broxmeyer(1964) and Fernandez and Macomber (1962) and starting again with Newton's law of rotational motion of

50

the 4-cardan system we find by Taylor expansion for the at­titude error ~ of the inner element the dynamic model

( 3 )

where ~p is the inertia tensor of the cardans; ~ are the cardan d1sturbing torques; ~M are the motor control torques and ~ are the four cardan angles. Assuming the servo elec­tronics as simple lead networks and assuming linear position and rate feed-back control circuits the attitude error of the inner element is found in the L-domain as

~(p) =

+ Q"(p)

+ Q"(p)

Q"(p) Q2'(P) K ~O / P

Q2'(P) ! { C ~(p) + ~ ~(p) + ~ / p }

. * 1 ( P ~p ~O + ~p ~O ) + G"(p) Q1'(P) R ~p-

( 4 )

~p(p)

where Q" is the closed loop transfer matrix and Q1', Q2', ! are gain matrices. A steady state solution can be obtained by applying the final value theorem of the L-transform ~(~)­lim p ~(p)lp" 0

+ C ~ + D A + d + !p ~p ( 5 )

where ~O are the initial alignment errors; ~ the gyro error angles; ~ is the rotor miscentering ; m are the gimbal dis­turbing torques and f, ~ are system matrices. The matrix !p is defined as

0 - sin9 1/(J2 w 2) 2 COS 9 1/(J3 w 2)

2 0

0 COS9 1/(J2 w 2) sin9 1/(J3 w 2) 0 !p

2 2 =

w 2) 1/(J1 0 0 0 1

0 0 0 4 COS9 3/(3J3 w 2) 4

( 6 )

where 9 1 is the heading angle; 9 3 is the pitch angle; w. are the undamped natural frequencies of the closed loop cohtrol circuits; J. are the gimbal moments of inertia.

The following disturbing torques ~p have been modelled :

1. Mass unbalance of the inner element

2. Friction of the gimbal ball bearings and slip rings

3. Viscous damping of the servo motors by a back e.m.f.

51

ERROR MODEL OF PENDULOUS ACCELEROMETER

The accelerometers are the primary data source in a platform INS for the computation of velocity and position information. Therefore a realistic error model of the accelerometers is extremely important.

The basic relation between measured specific force f and in­put specific force f is given as

2 fi = fi + Sf i = k fi + ~l fj - ~j fl + Q/(J wb ) fi fl + b

( 7 ) error; b is the bias error; ~ are about the axis l,j. Typical error

sensor like the GG-177 are derived scale factor error we find

where k is the scale factor the misalignment angles models for a fluid-filled in Savage (1978). For the

2 k - kO + k1 fill Ifil I + k2 ti + k3 fi + k4 (T - TO) (8)

where kO is the linear; k1 the asymmetry; k2 the quadratic; k3 the cubic and k4 the temperature dependent scale factor error. The accelerometer bias error model is decomposed of five different terms

+ b 2 f. f. ~ J

( 9 )

where b O is the g-independent bias; b 1 is the vibrational bias; b 2 is the anisoelastic error term; b is the tempera­ture sensitive bias; and €b is the acceler6meter white noise. Further on Q is the pendulositYi J is the moment of inertia of the proof mass; wb is the closed-loop characteristic fre­quency of the undamped pedulum.

SHAPING FILTER OF THE ANOMALOUS GRAVITY FIELD

The anomalous gavity field is still a serious problem in in­ertial surveying and geodesy, if we talk about cm-accuracies. Therefore the introduction of a realistic state-space model is mandatory. Because in the usual application field of in­ertial geodesy the local gravity field is unknown in the re­gion of a traverse, a stochastic modelling in the sense of statistical gravimetry (Moritz,1980) based on Autocovariance Functions (ACF) is one possible solution. The aim is to de­rive a simple, if possible time-invariant, shaping filter based on a local and planar covariance model. A further re­quirement is that the covariance model has a realistic Power Spectral Density Function, which approximates the local spec­tral properties of the gravity field. Taking this into ac­count a 2nd order Gauss-Marcov anomalous potential ACF (Vas­siliou and Schwarz, 1987) is used

2 b -~ P CTT,(P) = aT (1 + p IPI) e (10)

where aT is the variance of the disturbing potential; P is

52

the horizontal distance and ~ is the correlation parameter. Applying the basic approach of shaping filter design as out­lined in Jordan et ale (1981) and introducing higher-order functionals of the disturbing potential leeds to the follow­ing state-space model

T

d Tx

dt Ty

r-~: 0

sin A

= V -~ cos A

Tz o

Tzr o L

sin A

-~(1+sin2 A)

-~/2 sin 2A

o

o

+ V

cos A

-~/2 sin 2A

-~(1+cos2

o

o

A)

o o

o

o

o

o

o

T

(11)

where T, T , T are the east, west and down components of the gravIty ~ist~rbance vector; A is the azimuth of the in­ertial traverse; V is the velocity of the measuring vehicle; ~g is the input white noise.

OVERALL DYNAMIC ERROR MODEL

In the chapters above the error models of the sensors, of the platform stabilization loops and of the anomalous gravi­ty field are presented. The basic question is now, how these errors propagate into velocity and position errors. In order to answer this question the position errors sr and the veloc­ity errors Sv are introduced.

,....------4 w~ 13 ~re~ -11 14-----.....

Fig.3. Error propagation in the space-stabilized INS.

53

Of course their coupling based on the system dynamics of the space-stabilized INS (Britting, 1971) is used. For the posi­tion and velocity errors earth-centered and earth-fixed co­ordinates are used. P

In Fig.3 s~ is the attitude error of the inner element; Sf is the accelerometer error; s~n is the gravity disturbance vector; g. is the skew-symmetric form of the earth rotation vector; w~eis the Schuler frequency.

The derived dynamic model (Eissfeller, 1989) is of the fol­lowing standard form

x = F x + u + G ~ (12)

where F is the dynamics matrix; x is the error state vector; u is the vector of deterministic-disturbances; ~ white noise and G is the is the noise input matrix.

!!!o (13)

In eq.(13) the state vector x consisting of 31 elements is defined, where ~ is the gyro-drift angle error; Sr and Sv is the position and velocity error; Yg are the five shaping-fil­ter states of the anomalous gravity field; s£o are the ini­tial alignment erros of the inner element; ~o is the accel­erometer bias error vector; !!!o are the ESG bias torques; €o and €1 are the GPS receiver clock offset and clock drift; ~A are tfie carrier phase ambiguities of the GPS signal.

Dynamics Matrix

The dynamics matrix F is written down in a block matrix no­tation where the indices show the location of each block ma­trix F .. different from zero

-1)

!- mat(!11 !17 !23 !31 !32 !33 !34 !35 !36 !44 !66 !77 F99 )

( 14) where

!11 = A

!17 hr -1 =

!23 = I

e !31 = C.

-1

2 !32 = w

S

!33 = - 2

!34 = [ 0

I

i r fp L

( 3 e -r ne -ie

I fn e

F -p I * e -r -

0 ]

1 0 J -

I

C

54

(15 )

(16)

(17)

( 18)

( 19)

(20 )

( 21 )

F35 e C i [ !p I Q ] (22) - C. -1 -p

!36 e C i (23) - C. -1 -p

!44 - v F -g (24)

F66 = -diag( ~b1 ~b2 ~b3 ) (25)

!77 - -diag( ~m1 ~m2 13 m3 ) (26)

F99 = - ~E1 (27)

In the eq.(15-27) c i e is the transformation matrix from in­ertial to earth-fixed coordinates; £D1 is the transformation matrix from platform to inertial cooTdinates; F - skew(f ); F is the dynamic matrix in eq.(ll) of the anomilous gravfty f~eld shaping filter; ~ are the correlation parameters of 1st order Gauss-Marcov processes.

Deterministic Disturbances

The deterministic disturbance vector consists of those ele­ments not included into the state-vector (13). It is of the following form

u = r u - L-IC. o o o o ] (28)

where ~IC. is the vector of deterministic ESG errors; u are the comprised system and sensor errors, which are inpu~ di­rectly to the velocity errors.

-1 ~IC. = hr (!IC. - !o ) (29)

U v = £i e Cp i [!p I Q ] D ~ + d + Kp mp)

+ fie £pi ( S!p - ~o (30)

White Noise Input Matrix

The white noise input matrix G is derived as

Q - mat( G11 Q22 Q43 Q64 Q75 G96 )

where

Q11 -1 I = hr

G22 C. e £p

i = -1

Q43 = V Fg

55

(31)

(32)

(33)

(34)

Q64 - -diag( Gb1 Gb2 Gb3

G75 - -diag( Gm1 Gm2 Gm3

G96 - 1

For more detail see Eissfeller (1989).

CONCLUSION

(35)

(36)

(37)

The derived error model of the space-stabilized INS is not only of theoretical nature. It can be also applied to post­mission analysis in order to compensate for systematic ef­fects. This requires in comparison to the approach usually done that the specific force vector and the resolver outputs (gimbal angles) are stored on disk or tape as auxiliary data.

REFERENCES

Adams, G. and Hadfield, M. (1981). GEO-SPIN / IPS-2 Improve­for Precision Gravity Measurement, Proc. of the 2nd Int. Symposium on Inertial Technology for Surveying and Geodesy, Banff, Canada.

Boedecker, G. (1987). Zur Ausgleichung von Inertialdaten, Zeitschrift fUr Vermessungswesen, Vol. 112, No.1.

Britting, K. (1971). Inertial Navigation Systems Analysis, Wiley-Interscience, New-York.

Broxmeyer, C. (1964). Inertial Navigation Systems, Mc Graw­Hill, New-York.

Eissfeller, B. (1989). Analyse einer raumstabilisierten In­ertialplattform und Integration mit GPS, Schriftenreihe Studiengang Vermessungswesen, Universitat der Bundeswehr, Heft 37, Neubiberg.

Fernandez, M. and Macomber, G. (1962). Inertial Guidance En­gineering, Prentice Hall, Englewood Cliffs, NJ.

Jordan, S., Moonan, P., Weiss, J. (1981). State Space Models of Gravity Disturbance Gradients, IEEE, Vol. AES-17, No.5.

Koenig, R. (1988). Zur Fehlertheorie und Ausgleichung iner­tialer Positionsbestimmungen, Schriftenreihe Studiengang Vermessungswesen, Universitat der Bundeswehr, Heft 32, Neu­biberg.

Moritz, H. (1980). Advanced Physical Geodesy, Wichmann Ver­lag, Karlsruhe.

Phelps, P. (1968). Research in Electrically Supported Vacu­um Gyroscope, Vol. I : Summary, Honeywell Inc., Minneapolis.

Savage, P. (1978). Strapdown Sensors, AGARD Lecture Series No. 95. .

Vassiliou, A.A. and Schwarz, K.P. (1987). Study of the High­Frequency Spectrum of the Anomalous Gravity Potential, Vol. 92, No. B1.

56

SESSION2a

EQUIPMENT TRENDS AND MEASUREMENT PROCEDURES

CHAIRMAN M.HADFIELD

HONEYWELL, INC. ST. PETERSBURGH, FLORIDA

GYROSCOPES: CURRENT AND EMERGING TECHNOLOGIES

ABSTRACT

Robert B. Smith and John R. Weyrauch Honeywell Systems and Research Center

Minneapolis, Minnesota 55418

Gyroscopes and accelerometers are unique devices, the critical input sensors for inertial navigation and surveying systems. Among the many types of gyroscopes only a few have proved adequate for precise inertial applications. These are the ring laser gyroscope (RLG), electrostatic suspended gyroscopes (ESG), air-bearing floated-gimbal spinning mass gyroscopes, and dynamically tuned rotor gyroscopes (DTG). All have been developed to a high art among precision instruments, with supremely low noise and bias errors. Extensive acceptance of inertial navigation systems in commercial flight service, particularly using RLGs, has also required and stimulated development of practical and competitve reliability, cost, and lifetime.

A few currently emerging gyroscope technologies might play a role in future inertial systems, but it is not yet clear whether they can advance to be competitive in performance and cost with current gyroscopes. Most notable is the fiber-optic gyroscope (FOG), another form of optical gyroscope, which in the last several years has advanced to a state suitable for engineering prototypes in aircraft attitude heading and reference systems. Another is the hemispherical resonator gyroscope (HRG), which uses the precession of vibrations on the rim of a "wine glass" shaped quartz resonator.

INTRODUCTION

Gyroscopes are critical input sensors that together with accelerometers make inertial navigation and surveying possible. The performance accuracy of the system depends fundamentally on the accuracy and stability with which gyroscopes can measure rotation rates in inertial space. In addition, very practical issues can often dominate over the performance issue in determining acceptance for different applications. Currently, initial cost, cost of ownership, reliability, lifetime, suitability for sttapdown systems, environmental sensitivity, and survivability all can be the primary competitive issues between different gyroscope technologies and manufacturers.

Gyroscopes are a very old (100 years) and very mature technology, with a wide variety of different forms, both successful and unsuccessful, being developed over the years. Only a few types of gyroscopes have survived to be used in inertial systems, which place primary importance on extremely small and stable errors. These gyroscopes have in common extremely good isolation of the internal physics of inertial rotation sensing, from all disturbances that rotate with the case and vehicle.

The audience is directed to the references at the end of this paper for more extensive background and details. For the important issue of strapdown systems and many gyroscope descriptions, see Savage, 1978. A mid-1980s review of inertial technology is given by Ragan, et al., 1984. A recent and very extensive bibliography on gyroscopes is

59

Cousins, 1988; and Rueger, 1982, is a monograph on gyroscopes written for a surveying engineering audience.

Here we discuss the different gyroscopes that have served well for inertial applications, the evolution from the earlier used gyroscopes to the domination by RLGs in new applications, and two emerging technologies that might playa role in the future.

GYROSCOPES FOR INERTIAL NAVIGATION

Inertial navigation systems utilize clusters of gyroscopes and accelerometers plus computers to autonomously detennine the host vehicle position, velocity, and orientation in three-dimensional space. The function of the set of gyroscopes is to establish a frame of reference for the accelerometers and the measurement processing that bears a known relationship to inertial space. Gyroscopes provide a self-contained frame of reference onboard the vehicle, which is equivalent to the exterior frame of reference provided by the stars in celestial navigation. Early gyroscope designs used spinning masses and the well understood principles of rotational mechanics to measure angular rotation rates. Current angular momentum wheel devices have refmed these techniques to a very high level New gyroscope approaches use quite different physical principles, ranging from optical interactions to mechanical vibration standing waves.

The performance accuracy of inertial systems is loosely labeled "navigation grade" if the growth rate of the position error for the system is less than 1 nmi/hr. Typical gyroscope characteristics for this type of system are the following:

• Bias stability of 0.01 deg/hr • Wide-band random rate noise of 0.002 deglroot-hr • Scale factor accuracy of 50 ppm

Gyros with characteristics better than this now exist and are used in systems sometimes referred to as precision navigation systems. For surveying applications, performance at the top end of navigation-grade systems is required.

To understand the forces that motivate gyroscope technology development, we must frrst look at the dominant markets and users. Most of the new technology development funding in the United States has been and is coming from the Department of Defense (DoD) for missiles, aircraft, smart munitions, and land vehicles. These applications have a wide range of performance requirements, but all share the following goals:

• Increased reliability • Reduced cost-both acquisition and life cycle • Improved operational readiness (e.g., reduced alignment time) • Reduced size, weight, and power • Operation over environmental extremes (e.g., temperature, vibration)

There are two fundamental mechanizations for inertial navigation systems: gimbaled platform and strapdown. Gimbaled platform approaches use electromechanically driven gimbals to isolate the gyroscope and accelerometer clusters from the host vehicle. The gyroscope signals are used to drive gimbal torquers to maintain the platform in an inertially stationary orientation in space regardless of vehicle angular rotations. The complexity, weight, reliability, and cost of the electromechanical gimbaled platform put such systems at a significant competitive disadvantage compared to strapdown systems.

In strapdown systems, the sensors are not isolated from the vehicle. In these systems, the vehicle-mounted gyroscopes measure attitude changes in the vehicle frame of reference, and then the navigation computer transforms the vehicle-mounted accelerometer

60

measurements into the local-level coordinate frame for velocity and position computations. The strapdown mechanization places more severe performance requirements on the gyroscopes in terms of dynamic range, bandwidth, and scale factor. To meet the DoD technology goals, navigation system developments have concentrated on strapdown systems because of the lighter weight and higher reliability. Gyro technology efforts have, therefore, been concentrated on developing sensors that are best suited for the strapdown systems to meet the system goals described above (Savage, 1978). In addition there is always intense competitive pressure from both military and commercial markets to reduce cost and increase reliability and lifetime.

CURRENT TECHNOLOGIES

Almost by definition current technologies are the mature gyroscopes, which have been developed over many years, provide excellent performance, and are successes in the technical and economic marketplace competition. We briefly describe four, one that is still advancing and three that have reached a plateau in their development and applications.

Ring Laser Gyroscope (RLG)

The ring laser gyroscope represented a truly new technology as it went through evolutionary development through the late 1960s and mid 1970s. A milestone was the acceptance in 1978 of RLG-based inertial navigation systems in commercial passenger aircraft.

Fig. 1 shows the basic structure of an RLG. A triangular optical path is defined by three (or more) mirrors mounted on a quartz block for stability. The optical path is closed, forming an optical ring resonator. Within the block there is a bored passage for the light beam, and it is also filled with the low pressure He-Ne gas. High voltage electrodes and discharge over a symmetric portion of the path provides the laser gain, which sustains two counter-propagating beams. The RLG is a laser. The input rotation axis is perpendicular to the plane of the ring-path (the plane of the figure). Under rotation, the Sagnac differential phase shift results in different optical frequencies for the two beams. One of the mirrors is partially transmitting, by which samples of the two beams are extracted and superimposed to form an interference pattern on a pair of detectors. Signals from the two detectors permit up and down frequency counting of the difference frequency, which is proportional to rotation rate. Typically, for rates of degrees per second, the frequency is in the audio range, and a single pulse represents a few arc seconds of resolvable rotation. Up and down counting and accumulation of pulses provides an accurate measure of angular position; th~t is, very accurate rate integration is achieved without depending on physical properties of material (as in a floated gyroscope) or electronic analog signals and integrators.

A ubiquitous feature of the basic RLG is that there is a rotation rate below which the frequencies of the two beams lock together and there is no indicated output rate. This is due to backscatter between the two beams, usually at the mirrors. This is a quantitative issue; regardless of how low the backscatter is, there will always be a corresponding "lock­in" threshold rate. The most common technique for eliminating this problem is to maintain a rotational vibration or "dither" about the input axis. This "dither motor" is an integral part of the gyroscope mount and is driven piezoelectrically. The frequency output oscillates over a large range, but with exact frequency counting techniques the average frequency is accurately proportional to rotation rate, even through zero rate, and with a scale factor accuracy on the order of a few parts per million.

61

Laser Discharge

Detector

Transducer Sensor

L pa=ngth I --I Control ~

Clra,lIt

Inpu;\ Rate.)

Fig 1. Schematic of an RLG. The triangular optical path for two counter­propagating beams is defmed by three mirrors.

There is a vast literature on RLGs, and there is room only for a few suggestions. For an early description of RLGs and their applications in strapdown systems, see Savage, 1976. A brief description of the operating principles is given by Coccoli, et al., 1984. Chow, et al., 1985, is an excellent review of RLG theory. They describe several different types ofRLGs, particularly different methods for solving the "lock-in" problem. Upton, 1990, is suggested as a very recent report on how RLG development has continually advanced to achieve precision inertial navigation.

The virtues of the RLG are many. It is basically simple in concept and construction, having much reduced parts count, lower precision requirements, and reduced assembly costs over electromechanical gyroscopes. There is a dramatic difference in construction complexity of an RLG compared to DTGs or floated gyroscopes. (ESGs have a similar basic simplicity.) There are no bearings and fewer wear-out mechanisms, and the lifetime has increased to far exceed that of alternative gyroscopes. The RLG is ideally suited for strapdown systems, which fact played a major role in its acceptance.

Nevertheless, there is considerable art in the design and manufacture of RLGs. One key attribute is supremely good mirrors, which must be virtually perfect, particularly to assure low backscatter between the two beams. Lifetime of the mirrors is a significant issue, as are cleanliness and good vacuum techniques to assure long life and stability of the gain characteristics. Also, a means to accurately control the optical path length is needed;

62

Fig 2. Photographs of two RLGs, having leg length of 4.2 in. (left) and 5.7 in. (right).

considerable experience is needed in the choice of design parameters that affect optical beam and the laser gain characteristics; and costs must be kept low, always.

Some criticism is made of RLGs for the presence of dither vibration, high voltages, and the need for high-vacuum techniques. For some applications, these can be significant issues. But high-performance gyroscopes face a very large and complex set of requirements; there is no perfect gyroscope possessing all desired attributes. In fact, the dither motor has never been a problem with reliability and lifetime, and high voltage and vacuum have presented no major problems. These features of an RLG are a small price to pay for its exceptional performance and advantages.

In the last decade, RLGs have dominated other gyroscopes for new inertial systems applications.

Electrostatically Suspended Gyroscope (ESG)

The electrostatically suspended gyroscope is a spinning-mass gyroscope in which a spherical rotor is suspended in a vacuum (within the gyroscope case) by an electrostatic field generated by case-fixed electrodes (Savage, 1978; Pondrom, 1984; and Hadfield, 1984). Hence, there is no physical contact of the rotor with the case. Pickoffs on the case

63

sense the orientation of the case relative to the rotor axis, which remains stationary in inertial space. The ESG has an advantage in having a two-axis output. The rotor is initially spun up by spin coils in the case that generate a rotating field across the rotor. Once the rotor is spinning, sustaining spin-motor drive is generally not needed.

This gyroscope has seen good production success in high-performance applications­specifically strategic bomber navigation and high-accuracy marine navigation. It has also been used in Honeywell's GEO-SPIN land navigation/survey system. Though it is a very high performance device, continued development of this type of gyroscope has been greatly reduced because of the cost/performance, manufacturing, and reliability advantages of optical gyroscopes, plus the trend toward multifunction strapdown systems. ESGs measure attitude change directly and are not well suited for strapdown applications where rotation rate information is also needed.

Air-Bearing Floated Gyroscope

The floated rate integrating gyroscope has had the longest production history and is the original high-accuracy inertial navigation system gyroscope. The device consists of a cylindrical hermetically sealed momentum wheeVspinmotor assembly (the float), mounted on delicate pivots in a cylindrical case (Savage, 1978; Ahn, 1984; Pondrom, 1984b). The cavity between the case and float is ftlled with a high viscosity fluid that serves the dual purpose of suspending the float at neutral buoyancy to eliminate loads on the gimbal pivots and providing damping to resist relative float/case angular motion about the pivots. The device senses angular rate through the gyroscopic reaction torque generated about the output axis when the gyroscope is rotated about its input axis. The reaction torque is provided by the viscosity of the fluid which generates a torque proportional to the relative rotation rate developed between the float and case. Thereby, displacement of the float becomes proportional to the integrated rate. The design and components for this type of gyroscope have been refined many times. Today, it is still the most accurate mechanical gyroscope and is in production for high-accuracy ballistic missile guidance systems. Air­bearing floated gyroscopes are of questionable interest for surveying applications because of their complexity (both gyroscope and system mechanization) and high cost.

Dynamically Tuned Rotor Gyroscope (DTG)

A dynamically tuned rotor gyroscope consists of a high-moment-of-inertia rotor supported by a flexible, two degree-of-freedom gimbal, connected with a case-fixed drive shaft and driven by a synchronous-hysteresis spin motor. The gimbal is composed of torsionally elastic but laterally rigid members, which allow the instantaneous spin axis of the rotor to be tilted with respect to the drive axis and gyroscope case. The elasticity of two gimbal axes is adjusted so that the resonant frequency for the rotor tilt is "tuned" to be the same as the spin rate, in which case there is perfect isolation of the rotor from the case.

A two-axis pickoff is included that measures the angular deviation of the rotor axis relative to the case. Also included is a two-axis torque generator assembly that allows the rotor axis to be torqued relative to the case by a command current as needed for alignment of the axes (Savage, 1978; Carrol, 1984). Thus, the DTG provides a two-axis rate output.

These gyroscopes also have a long and continuing production history and have very well understood operating characteristics. They are widely used in a variety of inertial navigation systems and are particularly well suited for medium accuracy (1 nmi/hr) gimballed-platform applications and for lower performance strapdown applications. Because of their design maturity and because RLGs have significant operating and cost advantages in strapdown applications, little is being done to further enhance DTGs.

64

EMERGING TECHNOLOGIES

There are always competitive motivations and pressures to develop new and better gyroscopes. or simply alternative gyroscopes. New gyroscope concepts always face very tough competition with existing gyroscopes. To compete, new gyroscopes must offer not just equivalent but significantly better petfonnance and cost benefits. Any completely new technology has always taken 15 to 20 years to develop and be accepted. Development efforts in new gyroscope concepts have continued and some may eventually emerge as competitive technologies.

Fiber-Optic Gyroscope (FOG)

In terms of time and resources invested. the most obvious emerging technology today is that of fiber-optic gyroscopes. As with RLGs, FOGs use the rotation-induced Sagnac optical phase shift between counter-propagating beams. But the optical path is defined instead by a coil of single-mode optical fiber. An important distinction is that, rather than itself being a laser, the fiber coil is excited from a separate external light source, usually a solid-state diode laser. Bergh, et al., 1984. and Culshaw and Giles, 1983, are two review articles that describe the principles and variety ofFOOs possible. Smith. 1989, provides a volume of reprint papers from the published literature on FOOs and contains a complete bibliography. Because of the freedom to choose the length of fiber used, as well as choosing the light source, detector, and signal processing scheme, there is a much larger variety of FOGs possible than for RLGs.

Interferometer Fiber-Optic Gyroscope (IFOG). Fig. 3 shows the layout of a basic fiber-optic gyroscope. The bold line is the optical fiber light path, with the rotation sensing coil on the right, joined by a fiber coupler/beamsplitter, which divides the incoming light equally into the two counter-propagating beams. The same coupler serves as a beam combiner to form the phase-detecting optical intetference signal proceeding to the left. It may be seen that this is a single-pass ring interferometer, rather than a recirculating ring resonator. A similar coupler on the left serves to separate the input light from the source on the left and the returning output signal, which is directed to the detector. The detector output is a sine-like function of the optical phase shift and rotation rate.

A wide variety of signal processing schemes is used to recover t.'le rotation signal from the optical phase shift. Included in all schemes is some form of optical phase modulation or frequency shift deliberately introduced in the fiber coil path, together with phase synchronous detection at the output

Since the basic output is a sine function of rotation rate, it is linear (constant scale factor) only over a limited range of input rotation rates. This basic form of fiber-optic gyroscope is proposed primarily for low rotation rates, as a null-rate sensor, or in low­accuracy applications. A different form of fiber-optic gyroscope uses a "closed-loop" signal processing configuration wherein the basic output is used to control a nonreciprocal optical frequency shifter of some form placed in the ring path. With changing rotation rate, the frequency is shifted as needed to keep the basic output at null; and the consequent drive voltage to the frequency shifter, or the measured frequency itself, then becomes the output signal. This will be an accurately linear function of rotation rate over a very wide range of rotation rates. Only this form of FOO will be suitable for inertial navigation systems.

Many of the optical functions required, such as couplers, frequency shifters, polarizer, etc. are implemented better and at lower net cost on multi-function integrated optics "chips." Integrated optics, being the key to compact and low cost gyroscopes in volume production, will playa very important role in future fiber-optic gyroscopes.

65

Light Source Polarizer OffICial Fiber Coil

+ ~ Multi-Turn, -100m

~

~ Coupler Coupler

::::=--:::::: :::::--=:::::

Detector ~ ~ ~

+ X X Optical Phase

Modulator Bias

Signal - Modulation Processing Analog Voltage Output fm

(Rotation Rate)

COM 132-0981

Fig. 3. Basic fiber-optic gyroscope configuration. The bold line represents the optical fiber light path. The arrows show direction of light propagation.

~

The pace of development has been somewhat determined by developments in optical fiber communications technology. Specialty fiber, such as polarization-maintaining fiber and polarizing fiber, different forms of diode laser light sources, etc., were all milestones that made advances in FOGs possible. The random rate noise has been easy to achieve; numbers as low and lower than 0.001 deglroot-hr have been reported. Bias stability of 0.01 to 0.001 deg/hr has also been reported over limited environmental changes. The scale factor accuracy depends on the particular configuration, but in some closed-loop cases has approached 50 ppm. Probably the most difficult requirements to meet are performance over the extremes of the required temperature range, particularly bias error stability.

The fiber-optic gyroscope is being championed for inertial navigation, but it is too early to tell whether it will offer competitive performance in the marketplace (where "niche applications" is often mentioned). For lower accuracy applications product prototypes are being built. Flight tests by several companies have been reported in attitude heading and references systems (e.g., for autopilots).

Ring Resonator Fiber-Optic Gyroscope (RFOG). In addition to the IFOG just described, there has been demonstrated a passive ring resonator FOG. The fiber path is closed to form a optical ring resonator similar to that of an RLG. But this resonator is excited by an external light source, with frequency shifts introduced externally before launching the counter-propagating beams. When these frequency shifts are accurately controlled to track the resonance frequencies of the ring, then the frequency difference is the same as in an RLG. The RFOG offers some advantages over the IFOG, but it is much more complex, earlier in its development, and will probably not be a contender for inertial applications.

66

Hemispherical Resonator Gyroscope (HRG)

The hemispherical resonator gyroscope is an example of a vibratory gyroscope, wherein the Coriolis acceleration acts on the displacement velocity of a vibrating structure rather than rotational velocity. In the HRG the vibrating structure is the rim of a spherical half­shell supported by a stem, as in a wine glass. The HRG has been under development by the Delco Corp., proceeding through several generations (Lynch, 1984; Loper and Lynch, 1983).

Fig. 4 shows a view of the separated gyroscope structure, with the active hemispherical shell facing down and supported by an integral double-ended stem. The axis of the stem and assembly is the input rotation axis of the gyroscope. Shown at the bottom are the set of eight inner capacitive pickoff electrodes, which detect the position of the vibrational pattern. Above is the conical part of the housing, containing a similar set of sixteen electrostatic "forcer" electrodes that sustain the vibrations of the shell. Metalization of the surface of the shell completes the capacitive circuit and isolates output signals from the drive voltage. The whole is fabricated from fused quartz, bonded together, sealed, and evacuated. In vacuum the shell is a resonator with an exceptionally high Q-approximately a million. Resonant frequency is 2 to 9 kHz.

Getter \ __ Cap

\, Ring Forcer Electrode

Pickoff Electrodes

Fig. 4. Drawing of separated hemispherical resonator gyroscope assembly.

The rim of the shell vibrates in an elliptically shaped mode with displacement velocities perpendicular to the rotation axis. Under rotation in inertial space, the Coriolis acceleration causes the node and antinode axes to precess; interestingly, the precession is only 0.3 of the input rotation. The outer forcer electrodes are used to start and sustain the vibration at a very stable amplitude. It is critically important that the modal vibration pattern be sustained equally well regardless of the position changes due to rotation. The shell must be isotropic. The drive circuit should cause no preferred position for the vibration mode. The eight inner pickoff electrodes, respective amplifiers, and phase-referenced detection yield measurements the direction cosines of the nodes and antinodes of the vibration. Thus,

67

given a model for scale factor and other errors, the rotation is measured with respect to some initial angular position.

Several advantages are claimed for the HRG. It is quite suitable for strapdown applications: all "solid state" with no moving parts or wear out mechanism; quite stable; and relatively insensitive to environmental effects (for example, no plausible magnetic sensitivity). It can operate over a wide temperature range, but it is thermally sensitive. A good model for the temperature effects is necessary and possible. To compensate for imperfection errors and temperature effects ("the model"), a microprocessor is provided as an integral part of each gyroscope.

Dickinson and Strandt, 1990, report on results from flight tests of a hexad system described in Mason, 1988. Six gyroscopes and accelerometers are used for purposes of redundancy, reliability, and ability to detect gyroscope failures. (Similar hexad systems using RLGs exist.) The results approached good inertial system performance. RMS errors of about 2.0 nmi/hr were reported, with championship data from the best subset of three gyroscopes predicted to give 0.8 mni/hr.

Although this gyroscope appears to be closer in its development to navigation grade performance than the FOG, it is too early to predict whether this gyroscope can be cost competitive with existing RLG systems or offer sufficient advantages in perfonnance.

SUMMARY

Three types of current "spinning mass" gyroscopes have been developed to a high state of refmement for inertial navigation and are suitable for surveying systems. Currently though advances in development of these have reached a plateau where further improvements become increasingly difficult, are not needed because of strong trends toward strapdown systems and competition from the ring laser gyroscope, and for which funding is difficult to obtain. These gyroscopes continue to be produced and existing navigation systems continue to operate well in service. The ring laser gyroscope has come to dominate in new strapdown systems, and there is continuing investment in its development for different perfonnance ranges and applications. There is little question that the ring laser gyroscope has become and will continue to be the dominant type of gyroscope for inertial systems in the foreseeable future. Two distinctly different gyroscope technologies, the fiber-optic gyroscope and the hemispherical resonator gyroscope, have advanced to the point where they can be viewed as emerging technologies. They work well as gyroscopes, but it is too early to predict whether their performance, cost, and particular advantages are dramatic enough to pose a threat to current technologies in the marketplace.

REFERENCES/BIBLIOGRAPHY

Gyroscopes Cousins, F.W. (1988). The Anatomy of the Gyroscope (edited by J.L. Hollington),

NATO/AGARDograph No. 313, Report No. AGARD-AG-313, NATO, Neuilly-sur­Seine, France, 285 pp. (A topical listing of published literature references and patent numbers on gyroscopes and applications, not annotated.)

Regan, R.R., Editor, et al., (1984). Inertial technology for the future, IEEE Trans. on Aerospace and Electronics Systems, AES-20 (20), 414-443. (Review article with multiple authored sections, some individually referenced below.)

Reuger, J.M. (1982). Inertial Sensors Part 1: Gyroscopes, UCSE Reports No. 30002, Div. Surveying Eng., Univ. Calgary, Calgary, Alberta, Canada. (Note similar monograph on accelerometers: UCSE Report No. 30008, 1986.)

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Savage, P.G. (1978). Strapdown Sensors, in NATO/AGARD Lecture Series 95, Strapdown Inertial Systems-Theory and Application, NATO/AGARD, Neuilly-sur­Seine, France, pp. 22-1 to 2-46.

Ring Laser Gyroscope Savage, P.G. (1976). Laser Gyros in strapdown inertial navigation systems, Proc. IEEE

Position, Location, & Navigation Symposium, (San Diego, No. 1-3, 1976), IEEE, NY.

Coccoli, J.D., Feldman, J.D., and Helfant, S. (1984). Part Vill: Ring laser gyros, in Ragan, 1984, above, pp. 426-428.

Chow, W.W., Gea-Banachloche, J., Pedrotti, L.M., Sanders, V.E., Schleich, W. and Scully, M.O. (1985). The ring laser gyro, Rev. Modern Phys. 57 (1),61-104.

Upton, R.W., Jr. (1990). The next frontier for strapdown RLG inertial systems, Proc. IEEE Position, Location, & Navigation Symposium, (Las Vegas, March 20-23, 1990), IEEE, New York, pp. 537-542.

Electrostatically Suspended Gyroscope Hadfield, MJ. (1984). Part VI: Hollow rotor ESG technology, in Ragan, 1984, above,

pp. 424-425. Pondrom, W.L. (1984a). Part V: Electrostatically suspended gyroscope, in Ragan, 1984,

above, pp. 422-424.

Air-Bearing Floated Gyroscope Ahn, B.-H. (1984). Part I: Floated inertial instruments, in Ragan, 1984, above, pp. 414-

417. Pondrom, W.L. (1984b). Part IV: Gas film supported free-rotor gyroscope, in Ragan,

1984, above, pp. 421-422.

Dynamically Tuned Gyroscope Carroll, R. (1984). Part VII: Dynamically tuned gyroscope, in Ragan, 1984, above, pp.

425-426.

Fiber-Optic Gyroscope Bergh, R.A., Lefevre, H.C., and Shaw, H.J. (1984). An overview of fiber-optic

gyroscopes, J. Lightwave Technology, LT-2 (2), 91-107. (Review Article) Cui shaw , B. and Giles, lP. (1983). Fiber optic gyroscopes, J. Phys. E.: Sci. Instrum.

16,5-15. (Review Article) Smith, R.B. (1989). Selected Papers on Fiber Optic Gyroscopes, SPIE Milestone

Series Vol. MS-8, SPIE, Bellingham, Wash. (A volume of 107 selected reprints; includes an essentially complete bibliography of the literature up to October 1989.)

Hemispherical Resonator Gyroscope Dickinson, J.D. and Strandt, C.R. (1990). HRG Strapdown Navigator, Proc. IEEE

Position, Location, & Navigation Symposium, (Las Vegas, March 20-23, 1990), IEEE, New York, pp. 110-117.

Loper, E.J. and Lynch, D.O. (1984). Hemispherical resonator gyro: Status report and test results, Proc. [nst. Nav., Natl. Tech. Mtg, (San Diego, Jan. 17-19, 1984), Inst. Nav., Wash. DC, 1984, pp. 105-107.

Lynch, D.O. (1984). Part X: Hemispherical resonator gyro, in Ragan, 1984, above, pp. 432-433.

Mason, L.E. (1988). Fault tolerant Solid-State Attitude Reference, Proc. Guidance and Control Conference, (Keystone, Colo., Jan. 30 - Feb. 3, 1988), AAS Pub. Office, San Diego, paper AAS 88-017, pp. 189-202.

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ACCELEROMETERS: CURRENT AND EMERGING TECHNOLOGY

ABSTRACT

Brian Norling Sundstrand Data Control, Inc. Redmond, Washington, USA

This paper provides a perspective on and survey o~ accelerometer technologies for use in high-accuracy inertial navigation and survey systems. The role o~ accelerometers in these systems is discussed with emphasis on performance requirements and the e~fects of system implementation. A survey of current technologies includes physical principles of each technology, information on accelerometers from key suppliers, and ongoing evolution in each technology. Also included is a review of emerging accelerometer technologies and the technolo­gical forces that are shaping these trends. The summary compiles other influences on the accelerometer industry and the implications of manufacturing system technology advances.

INTRODUCTION

Many new accelerometer technologies are emerging which have the potential to displace some of the current technologies. In the past, the thrust of accelerometer design was for increasingly higher accuracies. Extremely high accuracies are now achievable. The emphasis has shifted to the achievement of higher performance-to-cost ratios while minimizing weight and power consumption. This is a time of rapid technological growth and change.

The behavior of all accelerometers is defined by Newton's second law of motion: force equals mass times acceleration.

F = ma A seismic mass is the key element for the measurement of

acceleration. When a mass is accelerated, a force proportional to the magnitude of acceleration must exist. By measuring this force, the acceleration can be determined.

Accelerometers do not measure exclusively acceleration. They measure specific force, which is a combination of the force necessary to produce acceleration and the force due to the gravitational field. A stationary accelerometer, with its input axis oriented vertically, will indicate a specific force of 1 g, but it is not accelerating. The force required to produce real linear acceleration is indistin­guishable from the force of gravitational attraction per Einstein's

70

equivalence postulate. This sustained 1 g acceleration has ramifications in accelerometer design and testing, which must be accounted for in inertial navigation algorithms.

Accelerometers have evolved substantially over past decades to become more accurate, less expensive, smaller, lighter, and to consume less power. Much of this is a direct result of the substantial strides in semiconductor electronics. The advances in computer technology have also allowed the accelerometer to be simpler through the replacement of mechanical and electronic error cancellation with digital error compensation.

ACCELEROMETERS FOR SURVEYING AND INERTIAL NAVIGATION

An inertial navigation system utilizes the inertial properties of accelerometers and gyroscopes to provide, continuously, the information necessary to compute the navigation algorithms. The vehicle's instantaneous acceleration is measured in vehicle coordi­nates by the accelerometers. The gyros provide a continuous, stable angular reference whereby the acceleration may be referred to a suitable navigation frame.

In an inertial navigation or survey system, the sensitive axes of three accelerometers are typically oriented to be mutually orthogonal. The outputs of these accelerometers are combined with angular orientation information from the gyroscopes to determine the acceleration vector in 3 dimensional space. In exoatmospheric applications, this space is inertial. The first integration of the acceleration vector over time provides a velocity vector and the second integration provides position information. The dominant error sources in accelerometers then produce position errors which grow as the square of time. In practice, the systems are substantially more complex, but the essence of acceleration measurement remains the same.

In aircraft and ship navigation systems, as well as survey systems, the reference frame is a noninertial frame attached to the surface of the rotating earth. Such systems are aided by having a continuous measure of the direction of local vertical. Errors grow more slowly so that position error is roughly proportional to elapsed journey time. By aiding this system with zero velocity updates or GPS position updates, both position and velocity errors can be sUbstant­ially reduced. For the purpose of providing a representative accelerometer error budget, a fully autonomous, mutually orthogonal, strapdown INS with a 1 nautical-mile-per-hour (nmph), 1 sigma error is assumed. A typical allocation between a ring laser gyro and a pendulous force-balance accelerometer is also assumed. The precise accuracy is highly mission sensitive; therefore, a range of performance is shown in Table 1.

For system error propagation less than 1 NMPH, the values in Table 1 must be proportionally improved. The state of the art at this writing is better than 0.1 NMPH. Many accelerometer errors drive INS position errors which have a sinusoidal nature with a period corresponding to the ei~htY-four minute cycle of a Schuler pendulum. A bias error 3 , which is an accelerometer null offset in either of the horizontal accelerometers, manifests itself in the same way as an INS platform tilt error. One micro-g of accelerometer bias is equal to one microradian of platform tilt. A sudden bias shift

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Table 1. Typical accelerometer error budget to achieve -1.0 NMPH, 1a CEP

Bias 40 - 75 ",g

Scale Factor 200 - 300 ppm

Input Axis Alignment 30 - 60 ",Rad

Vibration Rectification 10 - 50 ",g/g2

Turn-on Transient -0.1 ",g/sec

Noise -0.5 ",g/{hr

during autonomous navigation initiates a one-sided sinusoidal Schuler tuned position error. A 100 micro-g bias error results in a 0.7 mile peak-to-peak oscillation in indicated ~osition.

An accelerometer scale factor error! produces a direct error in velocity immediately after a vehicle acceleration. But with time, this velocity error will tend to alternate sinusoidally about zero. A scale factor error of 1000 ppm during acceleration from 0 to 500 nmph will initiate a -0.5 Knot velocity error.

Many system configurations other than strapdown are used and each has a unique impact on allocation of accelerometer errors. Strapdown implementation imposes stringent bias modelability and frequency response requirements on accelerometers. Ring laser gyros are sometimes (but infrequently) rate biased by rotating the system platform (carouselling or "May tagging' ) to prevent lock-in. This has the added effect of easing the accelerometer bias and alignment performance requirements. Fully gimballed, inertially stabilized platforms, which are becoming uncommon, often had the ability to do a gimbal flip to recalibrate accelerometer bias and scale factor. Therefore, the specifications had to be held only for the duration of the navigation/survey run rather than for 5 or 10 years as in a strapdown navigator. Fault tolerant IMUs5.6•7 often include redundant skewed axis accelerometers which see a continuous partial component of earth's gravity. This requires better scale factor performance.

CURRENT ACCELERO:METER TECHNOLOGY

Over the years, many different mechanizations have been used to achieve high accuracy accelerometer performance. Only the commercially active technologies will be discussed in this section. These include the pendulous integrating gyroscope accelerometer (PIGA) at the highest performance level and a variety of force­balance accelerometers which provide the next class of performance at a substantially lower cost.

Pendulous Integrating Gyroscope Accelerometer

A conceptual illustration of a pendulous integrating gyro accelerometer is shown in Figure 1. The fundamental principal by which the PIGA operatesD is to utilize torque developed by gyroscopic precession to balance the pendulous torque. The gyro wheel is spun at a high angular velocity to create a large angular momentum. This gyro functions conventionally except for the effects of the mass

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imbalance purposely designed into the gimbal. The overhung mass creates pendulosity which generates a torque proportional to acceleration. This torque causes gyro wheel to precess about the input axis. Error torques from bearing friction cause an orthogonal precession resulting in a displacement of the gimbal from null position which is detected by the position pickoff. This signal is ampl,.ified and used to drive the servo actuator. This actuator applies torque to the trunion which, due to the gyroscopic precession effect, applies a torque on the innermost gimbal to restore it to the null position. At null, this torque just cancels the friction torque. The amount of rotation required is a direct measure of intecjrated acceleration over the period of measurement. The net resul t is that angle of the outermost gimbal is proportional to change in velocity.

OUTPUT

TRUNION ----4----~'"

BEARING ---+--~:l':)I.1

GIMBAL

GYRO WHEEL

POSITION PICKOFF

SERVO ACTUATOR

AMPLIFIER

FEEDBACK

PHASE SHIFTING NETWORK

POSITION PICKOFF

IMBALANCE (PENDULOUS) MASS

Figure 1. Conceptual illustration of the PIGA

A primitive version of the first PIGA type instrument was developed by the Germans in World War II for the V2 missile. Charles Stark Draper Labs later developed this concept into the highest precision accelerometer in the world. Two manufacturers are now producing PIGAs--Honeywell and Litton--and other companies have produced it in the past. Different versions of this instrument are produced in France and the Soviet Union. These instruments are highly accurate and also very costly. The high complexity, high precision tolerances, and high piece part count limit the lowest achievable cost level.

An accelerometer with a similar mode of operation using a gyro wheel is called the Pulsed Integrating Pendulous Accelerometer (PIPA)D. These are also used in high accuracy, high cost guidance systems.

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Force-Balance Accelerometers

There are two basic classes of force balance accelerometers. The pendulous mass type, havinq an unbalanced pivotal mass with anqu1ar displacement, is the most common for hiqh accuracy applications. The linear translational mass type, havinq a mass which is displaced linearly, tends to be more difficult to fabricate and requires substantial desiqn tradeoffs to suppress undesirable vibrational modes.

The force balance type accelerometer utilizes a seismic mass which is levitated in a fixed position relative to the structure by a closed loop servo system, which is an electrical equivalent mechanical sprinq. The force balance sensor consists of a position detector, an amplifier and an electromechanical system (see Fiqure 2). These elements, in combination, convert the mechanical force resu1tinq from acceleration into a proportional current or vo1taqe which is, in turn, converted back into an equal but opposinq mechanical force necessary to balance the input inertia caused by acceleration. The seismic mass is qenera11y constrained to a sinq1e axis of motion by flexures or pivots. The position of the mass is measured by a hiqh1y sensitive position detector. An acceleration induced chanqe in mass position is detected and a siqna1 is fed to an amplifier which drives the force qenerator to reposition the mass to null position. The output of the sensor is the current throuqh the force qenerator which is linearly proportional to input accelera­tion. This output must then be converted to a diqita1 word representinq acceleration or chanqe in velocity.

PIVOT

FEEDBACK CURRENT

PHASE SHIFTING NETWORK

Fiqure 2. Pendulous force-balance accelerometer

ANALOG VOLTAGE OUTPUT

The position detection is accomplished by many different means such as capacitive, inductive, optical, and resistive. capacitive tends to be the most popular for precision accelerometers because it has no mechanical contact, it is hiqh1y stable, and it provides useful mechanical "squeeze film" dampinq in dry sensors by "squeeze film" pumpinq of the backfill qas.

The force qenerators qenera11y rely on electromagnetic forces usinq torquer coils and permanent maqnets. The excellent linearity and high force capability of this type of system account for their popular usaqe. Electrostatic force generators are typically used only in extremely low g-ranqe applications or in very small size devices where the relative forces become substantial.

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The servo feedback systems typically include phase shifting networks to provide electronic damping in conjunction with the squeeze film damping and eddy current damping. The dynamic response characteristics can be shaped by the feedback characteristics (loop gain) and are therefore independent of mechanical restraint (springs) . This allows very low mechanical constraint in the sensitive axis which reduces errors due to instabilities in these cons"traints.

The accelerometer mechanisms are filled with either gas or liquid. Liquid fill, such as silicon oil, is used to provide higher damping for ~ given plate size and to provide buoyancy of the proof mass.

Force balance accelerometers have a number of strengths which can be summarized as follows:

o Broad useful frequency range due to closed loop dynamic response, especially in dry versions.

o Electronic damping achievable. o Low stiffness constraints for minimal bias errors. o Minimal displacement of seismic mass for low cross coupling. o Low sensi ti vi ty to amplifier stabili ty due to closed loop

response. o Ability to "store" velocity change during momentary overloads

or power outages, especially in fluid filled versions. Force balance accelerometers have certain fundamental disadvantages

as well: o Higher complexity of mechanism and servo electronics. o High power consumption dictated by the energy products of

permanent magnetics. o Power consumption and self heating change as a function of

acceleration. Many manufacturers from around the world produce force balance

accelerometers of an inertial navigation level of accuracy. Some are supplied only in systems and others are available as individual components. Information from many of the key suppliers is listed below.

The Bell Model XI accelerometer is the heart of all precision accelerometer systems produced by Bell Aerospace Textron. The single-axis, pendulous proof mass, force rebalance device uses a capaci ti ve bridge pick-off to detect acceleration forces on the flexure supported proof mass. Rebalancing of the proof mass is accomplished by electromagnetic forces. The self-contained, shielded, hardened electronics provide low noise output and low EMI and radiation susceptibility. The double hinge flexure, plus high servo loop gain, provides extremely small cross-axis acceleration coupling errors. Both fluid filled and dry gas damped versions are available.

Sundstrand Data Control produces the Q-Fle~ accelerometer product line including the QA700, QA1200, QA2000, and the recently developed QA3000. The Q-Fle~ is a linear, force-balance, closed loop, pendulous, servo accelerometer. It features a fused quartz (silica), dual flexure, pendulum suspension system for a completely elastic restraint with high stability. Moving capacitor plates on the proof mass provide squeeze film gas damping and the differential capacitive position detection. Rebalance torque is provided by a dual permanent

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magnet "voice coil" type system. The hybrid servo electronics utilize a current feedback network to allow the use of a wide range of load resistors and filtering without effecting servo loop response. Thermal modeling of bias, scale factor, and axis alignment is used when peak performance is required. The Q-Flex* product line continues to evolve in order to provide higher performance and/or lower cost. The Q-Fle~ is used when high performance, rugged construction, long-term stability, and accurate repeatability are required.

The A-4 accelerometer is Litton's current production instrument and it is employed on a variety of systems (e.g., from Attitude Heading Reference Systems (AHRS) to subnautical mile per hour inertial navigators). A sensing element is suspended in a pendulous manner from a frame by a pair of hinges which define the axis of rotation. Acceleration causes the sensitive element (proof mass) to rotate relative to the case. The amplitude and polarity of the motion is sensed by an optical pickoff that produces a dc signal--the difference of two currents flowing in two photo diodes. This signal is input to an Accelerometer Restoring Amplifier (ARA). The ARA provides a precision restoring current to the A-4' s torquer to maintain the proof mass at null. The dry pendulous A-4 is O.S" x 0.3" X 0.2", weighs 5 grams, and is accurately temperature modeled over a -65° to 170°F range. The A-4 is normally packaged in a triad configuration (i.e., three instruments mounted in a common enclosure for efficient handling, mounting, and calibration). The A-4 triad is 1.7" x 1.4" x O.S" and weighs 68 grams.

Kearfott is a pioneer in inertial navigation quality accelerometers having developed force rebalance accelerometers as early as 1950.

The current production MOD VII single-axis accelerometer is a flexure suspended, pendulous, force rebalance device used in both gimballed inertial navigation systems having performance in the 0.1 NMPH class to the O.S NMPH class and in unheated strapdown systems having performance in the 0.4 NMPH inertial class. Tactical missile applications accommodate 60 g sustained inputs. The MOD VII is also utilized in a three-axis assembly for satellite navigation where extremely low noise and time resolution are required for orbital controls. Kearfott has produced over 20,000 MOD VII accelerometers.

Traditionally, Systron Donner has used pivot and jewel or a "torsion bar" type of suspension in a servo force balance mechaniza­tion. Fluid-filled, partially-floated accelerometers are provided for applications demanding high performance and/or exposures to extreme levels of shock and vibration. Dry accelerometers utilizing electrical or servo damping are provided for requirements driven by low cost, light weight, and mid-to-Iow performance objectives.

Systron Donner is developing a series of accelerometers using fused quartz flexure suspensions to augment their traditional line of pivot and jewel technologies. One series of devices utilizes servo force balance techniques combined with hybrid electronics in a miniature (less than 1 cubic inch) envelope. Intended for general application in instrumentation, guidance, navigation, and control systems, the accelerometer also finds use in a line of six-degree-of-freedom (6-DOF) measurement systems in development at Systron Donner.

Rockwell produces a high accuracy, quartz flexure, pendulous force balance accelerometer for use in their own inertial navigators. similarly, Northrop produces a force-balance accelerometer for internally produced inertial navigators.

Other manufacturers of navigation grade, force-balance accelerometers in the world also deserve mention. These include the

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Ferranti Model FA24 and FA4 from the United Kingdom, the JAE Model JA-S from Japan, the Sagem Model from France, and in the spirit of Glasnost, accelerometers from the soviet union and the Peoples Republic of China also deserve mention.

Although force-balance accelerometer technology is fairly mature, evolution is constantly in progress to improve magnetic circuits, flexures, servo electronics, analog-to- digital conversion, and other key aspects. Permanent magnet technology now includes new materials such as doped rare earth magnets with beneficial temperature coefficients and neodinium-boron-iron magnets with extremely high energy products.

The output of the accelerometer must be converted to a digital word before the INS ~-processor can use the information. This is driving a trend toward direct digitally compatible output. Vibrating beam accelerometers are simpler to "digitize" because fully digital electronic counters can be used to convert from the analog frequency domain to digital domain. Traditional analog current or voltage output accelerometers require analog electronics in the digitizing process. Some techniques are within the servo loop such as pulse rebalancing. 1 Others are outside the loop using various techniques to convert analog-to-digital. Commercially available A-to-D converter chips are approaching the required accuracy. But there are serious technical difficulties with providing the large measurement range and extremely small quantization steps.

EMERGING ACCELERO:METER TECHNOLOGY

The pace of new accelerometer technology development appears to be on the increase. Many new designs are in development with the promise of providing better than 1 NMPH navigation accuracy. The following list discusses those technologies which are publicly disclosed as approaching inertial navigation accuracies.

Quartz Vibrating Beam Accelerometers

The vibrating beam accelerometer (VBA) technology has been in development for over 20 years and vibrating string accelerometers even precede these efforts. Significant breakthroughs have been made recently in the fabrication techniques and materials technology which have allowed the quartz VBAs to mature rapidly. VBAs are highly desirable because they provide a frequency output which is directly compatible with digital signal processing electronics. They also typically consume less power than force-balance accelerometers.

VBAs typically utilize a pair of crystalline quartz resonating beams loaded in tension and compression by a seismic mass as shown in Figure 3. The high stiffness of the axially loaded resonating beams allows sensors to be operated in the open-loop mode and yet achieve broad frequency response with high scale factor sensitivity. The seismic mass is constrained in all other degrees-of-freedom by a flexure system. The resonating beams convert force-to-frequency when mated with crystal controlled oscillator electronics. The output of a VBA is typically two frequency signals whose difference is linearly proportional to acceleration.

The Sundstrand Data Control VBAs utilize double-ended tuning fork geometry resonators as depicted in Figure 3. The first Accelerex4D VBA, model RBAS0021, is now in production. This 0.60 inch diameter

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by 0.42 inch long VBA targets performance in the tactical missile market. Sundstrand also continues development of the SuperflexTM VBA20

which is demonstrating performance consistent with <1 NMPH navigation accuracy.22 The SuperflexTM is a quartz flexure constrained, translat­ing mass with gas damping. It is housed in a 1.0 inch diameter x 1.0 inch long hermetic package including hybrid oscillator electronics.

...... '-"

--r-'

L.l..t.L '.Ll

~~ l CRYSTAL CONTROLLED •• OSCILLATOR L..

rREOUENCY PUlSE UICROPROCESSOR

r. COUNTERS I'-' ~ MASS

r. r-' t

~ CRYSTAL '2 U CONTROLLED CLOCX

OSCILLATOR

rt " (I

Figure 3. conceptual illustration of a VBA with signal processing

Kearfott Guidance and Control is developing a VBA which targets <1 NMPH accuracy. The Kearfott VBA is based on single tine resonators with tuned, isolating end conditions. The Kearfott VBA utilizes a separate pendulous mass to load each of two resonators.

Kearfott has developed two classes of solid-state accelerometers based on a quartz vibrating beam. A strategic class VBA has such low noise that it can sense gravity changes due to lunar/solar tidal variations. It has a demonstrated dynamic range of 2 x 109 with a maximum range of ±60 gs and a noise level of 30 nano-gs. The high-g VBA has been tested by the u.s. Army up to 800 gs.

Systron Donner is presently developing a VBA in the 500 ~g class. It is an open loop, gas damped, direct digital output accelerometer, using a set of fused quartz flexures and crystalline quartz force sensors. This subminiature accelerometer is intended for use in the Microminiature Inertial Measurement Unit (Micro-IMU) also being developed by Systron Donner.

Other companies in Europe, Japan, and the United states are also working on VBAs but public information is not yet available.

Electrostatically Suspended Accelerometers

Electrostatic forces tend to be small relative to forces achievable with electromagnetic systems. Therefore, electrostatically suspended accelerometers tend to have a low maximum g-range capability. The importance of this type of accelerometer is increasing as the instrumentation of satellites becomes more common and this relates peripherally to geodesy and surveying.

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Bell Aerospace produces the Miniature Electrostatic Accelerometer (MESA) .14 This accelerometer is designed specifically for low 9 applications. Full scale is typically ±1 milli-g up to ±25 milli-gs. What is significantly different about this accelerometer is that it has no mechanical spring. In other accelerometers used for low-g measurements, the mechanical spring is a large contributor to the bias instability and the time-dependent drift and temperature coefficient of the bias. The MESA does not have a mechanical spring; the MESA's proof mass is electrostatically suspended in all three axes. When it is in operation, there is absolutely no physical contact between the proof mass and any other part of the accelero­meter. A single proof mass can sense accelerations in three axes simultaneously. (Single axis versions have also been produced.) It can survive high-g pyrotechnic-generated shocks and launch environments while unpowered.

Litton also produces a micromachined electrostatically suspended accelerometer which is discussed later.

Silicon Micromachined Accelerometers

The field of silicon microsensor development is one of the fastest growing and provides the greatest potential for cost reduction in the accelerometer industry. Low performance silicon accelerometers are already available from many manufacturer's such as Nova Sensors, IC Sensors, and Endevco. Many of these utilize piezoresistive bridge technology to sense acceleration and are generally limited to 0.01 percent accuracy. Open loop capacitive sensing can provide better accuracy. Closed loop capacitive sensing with an electrostatic restoring force can achieve inertial navigation accuracy. Vibrating beam silicon accelerometers are also demonstrating inertial navigation accuracy.

Silicon is a nearly ideal material for constructionl8 of an accelerometer. Single-crystal silicon has a high yield and fracture strength (>300,000 psi), a high modulus of elasticity, is perfectly elastic at operating temperatures, has low thermal expansion properties, is nonmagnetic, and can be made selectively conductive by doping. The absence of grain boundaries or defects in high purity single-crystal silicon virtually eliminates mechanical hysteresis.

The Litton silicon accelerometer, shown in Figure 4, is a closed­loop pendulous accelerometer, micromachined from silicon and glass wafers using processes and techniques developed for the integrated circuit industry. It uses a capacitive pickoff to sense pendulum motion and electrostatic force rebalance. The accelerometer pendulum assembly is micromachined (anisotropically etched) from a silicon wafer as a monolithic unit comprising proof mass, hinges, and electrical contacts. The accelerometer is assembled in three layers: top and bottom glass layers, and the intermediate pendulum assembly layer incorporating the proof mass, hinges, and electrical contacts. These three layers are anodically bonded together to form an accelerometer assembly without using epoxies or other organic adhesives. The assembled sensor assembly measures 0.215 x 0.328 x 0.052 inches (0.0037 cubic inch volume) and weighs 0.11 grams. The silicon accelerometer has been developed for AHRS and tactical missile applications.

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Figure 4. Litton micromachined silicon accelerometer

Sundstrand is developing a silicon micromachined vibrating beam accelerometer for use in inertial navigation, midcourse guidance, and instrumentation. The extensive technological base developed on quartz micromachined VBAs is being mated with the producibility gains available with a monolithic silicon micromachined accelerometer.

Kearfott is currently developing a micro-machined VBA (MVBA) utilizing thin film silicon technology combined with vibrating beam technology. The MVBA has already demonstrated bias stability of 15 J.Lgs.

Charles Stark Draper Lab is developing a silicon micromachined accelerometer using a capacitive/electrostatic servo system.

Superconducting Accelerometers

The recent advances in high temperature superconductors have made superconducting accelerometers practical in many environments. Present versions still require cryogenic temperatures and the support equipment which this implies is prohibitive for routine inertial navigation applications, but there are side benefits in material stability and thermal noise which make them attractive for applications such as gravimetric surveys.

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Accelerometer design approaches are numerous, a design developed by the University of Marylandll is used here as a representative example. This superconducting accelerometer utilizes a force-balance servo system with superconducting position sensing circuits and restoring force elements. Restoring force is generated with electro­magnetic coils of superconducting wire placed in close proximity to the superconducting proof mass. The magnetic flux from the coils is excluded from the proof mass due to the Meissner effect. This redistribution of magnetic flux generates a force on the proof mass which is analogous to a "magnetic spring". Position detection utilizes the Miessner effect to modulate the inductance of sensing inductors as a function of superconducting proof mass proximi ty . Superconducting Quantum Interference Device (SQUID) amplifiers are used for gain elements in the feedback loop. with a theoretical acceleration sensitivity of 10-12 g Hz -112, this design is expected to serve gravi ty survey, gravi ty gradiometry, and inertial guidance applications in the future.

Multiple Sensor Systems For Acceleration And Angular Rate Measurement

Sensor systems which measure both acceleration and angular rate with the same transducers are emerging in many forms due to a market demand for small, low cost inertial measurement units. Many benefits can be obtained by combining these functions in order to reduce the number or complexity of transducers required. Typically, the signal processing complexity is increased. with the continued rapid advances in J.'-processors and evolving digital signal processing chips, the multiple sensor processing has become more practical and cost effective.

Several companies are producing multiple sensor systems for angular rate and acceleration measurement. Kearfott is manufacturing the Mul tisensorTM for low-cost tactical applications. Accelerometer channel calibration accuracies of 241 J.'gs have been published. s The Kearfott design approach utilizes piezoelectric benders attached to a rotating trunion with demodulation electronics to obtain frequency response down to true dc. Rockwell Collins produces a multiple sensor system based on similar principles, which has been marketed as part of a proprietary AHARS package.

Many of the multiple sensors utilize coriolis forces to measure angular rate. Coriolis force is the perpendicular force developed due to straight line motion in a rotating reference frame. Gener­ally, accelerometers are subjected to a periodic motion perpendicular to their input axes. Coriolis forces, sensed by the accelerometers, are proportional to the product of lateral velocity and angular rate. The coriolis forces are separated from the linear acceleration by synchronous demodulation. Sundstrand Data Control has several designs operating based on this principal. ll Vibrating beam accelero­meters permit all processing to be done directly in the digital domain. Acceleration channel calibration accuracy better than 100 J.'g and rate stability better than 10o/hr have been demonstrated. Litton also has a coriolis multiple sensor system in development. n

Charles Stark Draper Laboratory has demonstrated an inertial navigation accuracy multisensor. It is a single-degree-of-freedom floated integrating gyroscope with the output axis purposely unbalanced to obtain pendulosity. The accelerometer bias and scale factor results have been reported to be better than the 100 J.'g and 50 ppm goals by a factor of 3 to 5.

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Mitsubishi is marketing a "Multi-sensor" with acceleration bias stability of 2000 ~g for 30 minutes and 5000 ~g day-to-day stability. JAE manufactures a similar multiple sensor.

The field of multiple sensor systems development is broadening and per-axis-cost advantages appear to be achievable. Development is complicated by the need to separate angular rate from linear acceleration signals, as the coriolis accelerations are typically small. Rapid advances in digital processing chips are making this separation feasible and economical.

SUMl\fARY

The emphasis in accelerometer design has shifted from the previous drive for higher performance to an emphasis on higher performance-to­cost ratios. Performance suitable for 0.1 nmph navigation or better is available. The real need is to provide more cost effective accelerometers in all performance ranges.

silicon micromachined accelerometers are a revolution in manufacturing technology and hold the greatest promise for unpar­alleled performance-to-cost ratios. Advances in the manufacturing system technologies are driving down production costs of discretely assembled devices. The Sundstrand Q-Fle~ accelerometer, for example, is produced in a "Just-In-Time" (JIT) inventory system which uses demand pull to result in overall producibility gains. Honeywell is utilizing "computer Integrated Manufacturing" (CIM3) to produce SFIRs more efficiently.

The level of automation in precision accelerometer manufacturing is certainly increasing. However, the relatively low manufacturing volumes limit the level of automation which is cost effective. Piece part fabrication often has a high level of automation. Batch processing, N/C machining, and robotics are commonplace. Precision assembly of piece parts tends to require extreme accuracy and delicacy which is difficult to achieve with robotics. Although, in some cases, cleanliness of the assembly process demands automation to eliminate the prime source of contamination--people. Accelero­meter design tends to be a lengthy process demanding precise feedback of results achieved which, in turn, demands high accuracy and minimal seismic disturbance in the test lab. These factors drive the common usage of fully automated test equipment to perform functional testing and calibration of most performance parameters.

The computer revolution has also had a profound effect on accelerometer technology. The analysis and simulation tools facili­tated better initial design and subsequent refinement. Finite element analysis of stresses, strains, resonant modes, heat transfer, magnetic fields, and electric fields all result in more precise designs. Electronics analysis and simulation results in better optimization. Systems dynamics simulation improves stability margins, frequency response, and transient dynamic behavior. More powerful navigation computers have allowed the accelerometers to become simpler and more stable in many cases. Instead of correcting error sources mechanically or electronically, this compensation can be done in software, which is totally stable. It is now commonplace to correct for bias, scale factor, and alignment sensitivities digitally in high-accuracy navigation and survey systems.

Accelerometer technology and performance requirements continue to evol ve in response to navigation and survey system changes. Many new

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accelerometer technologies are emerging which have the potential to substantially reshape the industry.

System level trends are also reshaping accelerometer error budgets. Fault tolerant inertial navigators6 require tighter accelerometer performance to "self detect" failing accelerometers. GPS/INS navigators9 and survey systems utilize GPS position updates to comp~nsate for some accelerometer errors, resulting in looser accelerometer error budgets. Survey systems with periodic zero­velocity updates14 or doppler velocity measurement" have a similar loosening effect.

The field of accelerometer development is rapidly evolving, and many significant changes are expected in the coming years.

REFERENCES

lSkalon Izmeritel'naya Tekhika, A. 1., "CUrrent Trends in The Design of Self-Balancing Accelerometers With Digital output", No. 6, pp. 41-43, June, 1982

2"Litton Makes First Flight Test of Fiber optic Gyro", Avionics Report, p. 7, June 2, 1989

3"Assembly CIM Launches Big Benefits", Assembly Engineering, pp. 17-19, July, 1989

4Smithson, T. G., "A Review of The Mechanical Design and Development of a High Performance Accelerometer", IMech E, C49/1987

.5Jeerage, Dr. Mahesh K., "Reliability Analysis of Fault-Tolerant lMU Architectures with Redundant Inertial Sensors", IEEE AES Magazine, July 1990

~anderwerf, K. and Wefald, K., "Fault Tolerant Inertial Navigation system", DASC 1988 Conference Proceedings

7Barnard, G., et al., "Integrated Inertial Reference Assembly (lIRA) Analysis, Trade Studies, System Definition, Test Plan", Final Report, AFWAL-TR-82-1151

8pick, T., "The Multisensor™ Inertial Measurement unit (MlMU): A Low Cost Technology For Tactical Applications", AlAA Guidance, Navigation, and Control Conference, August 1986

~itland, J. and Spalding, K., "Impact of Inertial System Quality on GPS - Inertial Performance in a Jamming Environment", AlAA 1987 #87-2594

1()Matthews, A. and Welter, H., "Cost-Effective, High Accuracy Inertial Navigation", Navigation: Journal of the Institute of Navigation, Vol. 36, No.2, Summer 1989

"palk, H. et al., "Development of a Superconducting Six-Axis Accelerometer", University of Maryland, USAF report #AD-A215 948, July 1989

12peters, R. and Foote, S., "Computer-Automated Characterization of a High Production Volume, Inertial Grade Accelerometer", 1982

13IEEE Standard Inertial Sensor Terminology, IEEE Std 528-1984, New 'lork, USA

14Kr iegsman, B. A. and Mahar, K. B., "Gravity-Model Errors in Mobile Inertial-Navigation Systems", J. Guidance, Vol. 9, No.3, May­June 1986, pp. 312-318

1.5Huddle, J. "Theory and Performance For Position and Gravity Survey With an Inertial System", Journal of Guidance, Control, and Dynamics, Vol. 1, May-June 1978, pp. 183-188

16schwarz, K., "Inertial Surveying and Geodesy", Reviews of Geophysics and Space Physics, Vol. 21, May 1983, pp. 878-890

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17Benson, D. "A Comparison of Two Approaches to Pure-Inertial and Doppler-Inertial Error Analysis", IEEE Transactions on Aerospace and Electronics Systems, Vol. AES-11, July 1975, pp. 447-455

18Leonard, Milt, "IC Fabrication Techniques Sculpt Silicon Sensors", Electronic Design, October 26, 1989

l~ueger, J. M., "Inertial Sensors Part 2: Accelerometers, UCSE Report No. 30002, Div. Surveying Eng., Univ. Calgary, Canada 1986

~orling, B., "Superflex: A Synergistic Combination of Vibrating Beam And Quartz Flexure Accelerometer Technology", Navigation: Journal of The Institute of Navigation, Vol. 34, No.4, Winter 1987-88

21Holdren, F. and Norling, B., "Introduction of Quartz Vibrating Beam Accelerometer Technology Providing Capability For Low Cost, Fully Digital Navigation", Symposium Gyro Technology Proceedings, Stuttgart, Sept. 1989

22Norling, B., "Precision Gravity Measurement utilizing Accelerexa Vibrating Beam Accelerometer Technology", Proceedings of The IEEE PLANS, Las Vegas, Nevada, March, 1990

llwrigley, Holester, Denhart, "Gyroscope Theory, Design, and Instru­mentation, MIT Press, Cambridge, Mass., p. 113.

24Lange, W. and Dietrich, R., "The MESA Accelerometer For Space Application", Workshop Proceedings; Measurement and characteriza­tion of the acceleration environment on board the space station, Guntersville, AL, Aug. 1986 (NASA)

ACKNOWLEDGEMENfS

The written and informational contributions of the following people were key to preparation of this paper and are greatly appreciated. Richard Cimera of Kearfott, Steve Saks of Litton, Brad Sage of Systron Donner, James Olchawski of Bell Aerospace, Mike Ash of Charles Stark Draper Labs, Paul Savage of Strapdown Associates, and Rex Peters of Sundstrand Data Control, Inc.

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ULISS 30 A NEW GENERATION INERTIAL SURVEY SYSTEM ENHANCING

FLEXIBILITY AND SECURITY OF OPERATION, PERFORMANCE AND RELIABILITY

Frederic Mazzanti, Clive de la Fuente

SAGEM 6, Avenue d'Iena, 75783 PARIS CEDEX 16

Tel (1) 40706363 Fax (1) 40706488

ABSTRACT I INTRODUCTION

ULISS 30 is a new generation of inertial survey system (ISS). It has been developed to meet the most extended requirements of survey applications. To achieve high flexibility and security of operation for survey applications, ULISS 30 is integrated in the GEODlNE 30 configuration, which allows powerful data processing, data storage and new specific survey functions. This configuration is also capable of combining three complementary surveying tools : a Total Station (mechanically aligned with the inertial system), a GPS receiver and a ULISS 30 ISS. This association provides the most integrated, versatile and productive surveying system. Specific software have already been developed for survey, trajectography and seismic applications. This paper 1) describes the ULISS 30 inertial hardware, electronics and packaging, 2) discusses the survey software and its main features. Throughout it emphasizes the features designed to increase ease and security of operation, performance and reliability, productivity, user friendliness, integration capability with an opened architecture and a range of options (integrated total station, lap top PC, GPS , ... ). Finally and 3) it illustrates the range of possible applications by describing two specific configurations already developed: trajectographyand seismic survey.

ULISS 30, A NEW GENERATION INERTIAL SURVEY SYSTEM

ULISS family

ULISS 30 is a new version of the ULISS inertial system family. Almost 1500 ULISS inertial navigation system have been manufactured for aircraft programs only. These systems are in service with 14 different types of aircraft for 12 air forces and navies.

A state-of-the-art inertial platform

ULISS 30 benefits from this unique combination of experience to offer the most modern inertial platform technology with the highest standard of performance. It implements a

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three-axis-four-gimbal inertial platfonn equipped with two two-axis dry tuned gyroscopes (S040) and three gaz damped accelerometers (A310).

Figure 1 : ULISS 30, a rugged and accurate inenial survey system

Integrated electronics

The electronic modules benefit from the high level of integration of the ULISS family. These modules use extensively hybrid and LSI components to achieve low power consumption, a high functional integration and a high reliability. The electronic modularity (a complete function implemented in a single module) pennits an easy maintenance and a high rate of built in failure detection.

The computer (UT382-50) has a high computation throughput (400 000 opts) and allows the use of elaborate software, such as a real time Kalman fIlter.

The serial data interface module (RS422 type) has a capability of 4 input/outputs which makes it possible to integrate ULISS 30 into a multi-function system such as GEODINE 30. For instance, ULISS 30 may be coupled with a Total Station, a GPS receiver, a microcomputer (PC), a control and display unit (CDU), a plotter, etc.

A mechanically and thermally rugged system

The chassis consists of a waterproof sealed casing provided with shock absorbers and a mounting tray to secure onto the vehicle.

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The electronics and inertial platform use forced air cooling in a closed loop fashion. The heat is transferred from the inside radiator to the outside radiator by a set of Peltier devices. In conjunction with the low power consumption, the excellent efficiency of this heat pump mechanism always keeps the internal temperature low and therefore allows a high MTBF.

Simple, flexible and powerful operation

The operation of ULISS 30 is based on three main modes:

Standby is entered when the system is switched on and allows for mission preparation. Specific parameters such as ellipsoid constants, ZUPT time, etc can be initialized.

Alignment is selected by the operator and lasts less than 17 ·minutes. It implements the inertial platform thermal stabilization, gyros start up and initializes the attitude of the platform by a two position gyrocompassing (automatic vertical and north finding). During the alignment, both X and Y day to day gyro drifts are estimated and compensated.

Na~igation is selected at the end of alignment. In this mode, the system provides attitude (heading, pitch and roll), velocity components in a geographical frame (North, West, Up) and position coordinates in the X, Y and Z axes.

In order to achieve the very high accuracy which is required, the vehicle must be stop at regular intervals in order to carry out ZUPT's (zero velocity updates). The ZUPT submode is automatically selected when a vehicle's stop has been detected by the system. During the ZUPT, the system also checks in real time that the vehicle is truly stopped, and rejects instantaneously the ZUPT when a significant movement occurs. This automatic ZUPT selection and rejection secures the use of the system and avoids a 'possible user error. It also increases the system's performance because each stop is automatically used to carry outaZUPT.

At the end of each ZUPT sequence, a position update may be implemented on a known geodetic point. After each position update, a smoothing of all the previous surveyed points since the last position update (or alignment) is carried out. This smoothing is very useful when the ULISS 30 is not integrated in the GEODINE 30 configuration.

Finally, heading and position updates may also be implemented at the same time. This procedure has to be used to achieve high heading accuracy (better than 20 arc seconds) or to align accurately the system above 70 degrees of latitude (the initial heading error is inversely proportional to the latitude's cosine).

Powerful software

An optimal real time Kalman filter has been developed to take into account the ZUPT. An interesting feature of this Kalman filter is that there is a permanent autocalibration of the azimuth gyro drift which keeps its error below 5.10-3 0/hr. This allows high performance over long time missions (10 hours for instance) without re-alignment. To avoid estimation errors due to a position update on a wrong geodetic point, the real time Kalman filter only takes into account the ZUPT.

Another Kalman fllter is used to smooth the surveyed points by exclusively using the position updates.

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Performance

The main perfonnance characteristics of ULISS 30 are summarized in the following table.

Attitude 0,7 mrdRMS Heading, pitch, roll

Horizontal position 4.10-4 x lID RMS Without position update (*)

Altitude 2.10-4 x lID RMS Without position update

Horizontal position 4.10-5 x lID RMS After pos. update and smoothing

Altitude 2.10-5 x lID RMS After pos. update and smoothing

Alignment 17 minutes

Navigation TIme 10 hours Without re-alignment

ZUPT < 30 seconds

Output rate 6.25 Hz Up to 50 Hz if necessary

Power consumption <180W Cold start: 700 W during 2 min

Size 520 x 450 x 255 mm < 60 liters (including mount)

Weight 45 kg Including mount

MTBF 2000 hours

(*) lID is the straight horizontal distance between the last position update and the present position.

GEODINE 30, A VERSATILE AND POWERFUL CONFIGURATION

The following is a summary of existing functions and indicates what is operationally used by two customers.

GEODINE 30 configuration

The GEODINE 30 configuration is illustrated in figure 2 and consists of a ULISS 30 inertial survey system, a Total Station (Geodimeter 420), a PC compatible (COMPAQ 386/20 MHz), a graphic navigation indicator, a control and display unit, and optionally, a GPS receiver which can be linked to ULISS 30.

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I CDU I ~ PRINTER

• 1 RS422

r·· ... ••·•••••••••••••••••••••••

~ GPS

l.~~.~.~!.~~.~

·!RS422 RS42~ ~

i .. .. FLOPPY !- ULISS 30 ... PC ..

J r- DISK

•

" RS422 RS232

r······TOTAL··········l ~ NAVIGATION i STATION i INDICATOR , ...........•..•..•.................... ,

Figure 2 : GEODINE 30, a versatile and powerful survey configuration

INS and Total Station integration

Both ULISS 30 and the Total Station must be fastened to the same mechanical frame and aligned. The Inertial System provides in real time a vertical and heading reference to the Total Station, and the Total Station provides to the Inertial System access to virtually any points around the vehicle.

Vehicle installation

A vehicle installation example is given in figure 3.

The ULISS 30 is fixed in the back of the vehicle on a casing including back up batteries (2 x 12 V DC). The Total Station is mounted on a tripod with a view through the skylight. The PC compatible may be put anywhere and is isolated from vehicle vibrations by shock absorbers. The navigation indicator has to be installed on the dash-board.

Several reference points are defmed around the vehicle and may be used to survey a point or to implement a position update. It must be noted that a vehicle reference point is used to defme the location of a GPS receiver antenna.

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Figure 3 : A typical installation of GEODINE 30 in a survey vehicle showing the simple and practical integration of the Total Station and Workstation

(by courtesy of Compagnie Generale de Geophysique)

Standard software functions

Software overview

To increase the data storage capacity and the flexibility of use, all the functions are transferred into the PC. Accordingly, ULISS 30 is only used to implement the ZUPT's and to provide the position coordinates of the inertial cluster. Position updates and smoothing are carried out in the PC.

To facilitate the operation while in motion, the function keys are used extensively. A standard screen has been designed to optimize the user interface. Six easy to use windows have been defmed and are respectively dedicated to current function title, user dialog, real time data, date and time, warning message and function key titles (see figure 4).

Mission preparation

To prepare the mission the user can define, if necessary, the ellipsoid semi-major axis and flattening (or one of those already memorized), the geographical coordinates to rectangular coordinates conversion (UTM, Lambert, etc), the lever arm coordinates in the vehicle frame of the vehicle reference points (from the center of gravity of the inertial cluster to the reference point), the name of the mission, the date, etc.

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SAGEM - 09/13/90 - 15 h 12 mn 48 s .... ;.; ...... ,'

SURVEY OF A GEODETIC POINT

.,.:: ....... '

:. Warning: next ZUPT before 43 seconds

RP 1

- TRAJECTOGRAPHY -

is ON Ref. Pt : 1

Pts since Pos. Upd. : 4523 Pts since ZUPT : 890 Marks 14

Memory Place

Nav. Time ULISS Mode

: 56%

5 h 12 mn OK

: NAV

ISS accuracy RMS (m): aE=2.23 aN=O.45 aZ=O.64

Figure 4 : Example of GEODINE 30 standard screen

Total Station mechanical alignment and vehicle reference point initialization

Two interesting procedures have been developed in order to mechanically align or to "harmonize" the total station with the ULISS 30 and to estimate accurately the lever arm coordinates of the vehicle's reference points.

For the ftrst operation, the user has to roughly estimate the lever arm coordinates of one reference point (an accuracy of less than 5 cm is enough and easily achieved). Then, the harmonization procedure is started and consists of a) surveying any point by using the vehicle's reference point, b) travelling a few hundred meters from this surveyed point and c) surveying several times and at different vehicle headings the previously surveyed point by using the Total Station.

Each measurement is used in real time by a least squares ftlter to estimate the new three harmonization angles and their associated accuracies. The procedure may be used to harmonize the Total Station or to check the previous harmonization angles (once a week for instance). This procedure which does not depend on ULISS 30 attitude error, is very accurate (less then 1 milliradian) and fast (about 5 minutes).

Finally, the Total Station is used to precisely estimate the lever arm coordinates of all the vehicle reference points. This procedure consists of surveying several times each reference point with the Total Station. Then, the reference point coordinates in the vehicle frame are computed. A coordinate accuracy better than one centimeter is easily achieved.

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Geodetic point data base

A large library of known geodetic points can be registered. These geodetic points are used for instance to implement a position update. However, two types of geodetic points are stored in the data base : the primary points which have been introduced by the user and the secondary points which have been surveyed by using ULISS 30. A point recording contains the position coordinates and the associated accuracies. If a point has been surveyed several times, the system computes its optimal coordinates by using the weighted mean method.

Real time functions (6,25 Hz)

A real time window on the PC displays the present position, the velocity components (in geographical frame), the vehicle's velocity (km/h), the attitudes (heading, pitch and roll), the ULISS 30 mode, and a position error estimation. The position error estimation is used to check a position update and therefore increases the security of operation.

A minute before the next ZUPT is required, a buzzer is operated and a warning message is sent to the screen.

The user may select a way-point in the data base and the navigational guidance parameters (steering, bearing and distance) are computed and displayed on a navigation indicator.

Finally, the software prevents any hard disk access during the vehicle motion.

Static point surveying

a) Using a vehicle reference point

Once a reference point on the vehicle is selected, the user can introduce an eccentric lever arm from this point in cartesian or cylindrical coordinates. The eccentric lever arm is useful when the surveyed point is within a few meters range of the vehicle.

Whenever, the surveyed point is already in the data base, its assumed coordinates in the vehicle's frame are computed and can be used to recover a lost point if necessary.

b) Using the Total Station (direct mode and triangulation mode)

When a point cannot be reached by the vehicle, the user has to use the Total Station.

Whenever, the surveyed point is already in the data base, its assumed polar coordinates (distance, azimuth and elevation) are computed and can be used to recover a lost point if necessary.

If the reflecting device (a prism for instance) can be put above the point, the total station provides directly the polar coordinates which are sent to the PC via the serial link. Then, the software asks for the height of the reflecting device.

In some cases, the point cannot be reached even with a small prism (at the top of a church for instance). In this case, the total station provides only the direction (azimuth and elevation) and a single aiming is not enough to survey the point To solve this problem, the user has to aim at the point from different positions (up to 5). Then an automatic least squares filter computes the point's coordinates and the associated accuracy.

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Position upate

A position update consists of surveying a known point which is stored in the data base. Then the position error is computed and the present position is updated. Horizontal position and altitude can be updated separately if necessary.

Post-processing

A powerful post-processing software which implements a static Kalman Filter has been developed. It takes into account the position updates and the double surveyed points. The horizontal position and the altitude can be smoothed separately if necessary. A position update can be corrected or cancelled whenever an operational error has been detected.

A static surveyed point can be converted to a position update if its true coordinates are known. This function is useful in order to take into account a GPS surveyed point whose coordinates are only known at the end of the mission.

Specific trajectography software

This application was developed in order to reply directly to the particular needs of the National Land Survey of Sweden. It allows the storage in real time, at a frequency as high as 6.25 Hz, of the trajectory of one of the reference points on the side of the vehicle or a theoretical point displaced to one side of the vehicle. This feature gives the ability to trace the kerb-line, or the median line of the road without the necessity to drive over them exactly. It is equally possible to pick-up detail information while the vehicle is in motion. This function is of particular use for the registration of road profiles and kerb-side utilities especially for input into Land Information Systems database. It is important to note that the precision of the trajectory and the points picked up while moving are identical to those taken while static. In effect, by using the information in the internal Kalman filter, a fIrst smoothing of the data is carried out at the end of each ZUPf; this takes place as soon as the last segment of trajectory has been registered on the hard disk. Additionally the same smoothing as described above (using position updates) is used to provide the final coordinates of the points.

Other interesting software developments have been made. For example the calculation of the area within a closed loop (with an accuracy of better than 5 m2/Ha), or graphical functions, or the "vectorization" of a trajectory, thus compacting the amount of data stored.

Specific seismic software

This application was developed according to the specifications of the Compagnie Generale de Geophysique in order to reply to the survey requirements of 2D and 3D land seismic operations. The specific software was written in order to optimize the number of seismic points picked up in a day (several hundred) as well as giving precision estimates in real time.

According to the type of mission chosen the user can define the theoretical seismic trace (broken line, slalom line, or grid iron pattern). Afterwards numerous guidance functions are available to aid the user in the setting out of the theoretical points, by using either the reference points on the side of the vehicle, or the total station.

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At the end of each mission the coordinates of the seismic points are immediately available in the standard fonnat UKOOA Pl/84.

CONCLUSION

The complementary nature of the three main components of GEODINE 30 is easy to visualize. The Total Station provides access to virtually any point around the vehicle while the ISS avoids the levelling of the Total Station on the ground. The accurate attitude provided by ULISS 30 also allows numerous guidance functions to a chosen way-point. Whatever the mode of using the GPS receiver (static differential or kinematic), it provides a very high position accuracy and can therefore remove the necessity of having a geodetic network of known points. However the GPS receiver will be interrupted by satellite obstructions and therefore needs ULISS 30 in order to provide centimeter.1evel trajectory, to overcome its transient interruptions and to filter the GPS noise. But to optimize the production rate and the ease of use, it is indispensable to have a software tailored to the specific requirements of each user as described above. Because of these features and the use of ULISS 30, rugged, accurate and state-of-the-art ISS, GEODINE 30 is a multipurpose highly productive and versatile surveying system which can respond to all types of surveying applications.

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DESIGN FOR AN AIRBORNE GPS-INERTIAL SURVEYOR

INTRODUCTION

J. Arnold Soltz James I. Donna

Glenn Mamon

The Charles Stark Draper Laboratory, Inc. 555 Technology Square

Cambridge, MA. 02139 USA

The Aerial Profiling of Terrain System (APTS), designed and built at C.S. Draper Laboratory under the sponsorship of the US Geological Survey, achieved accuracies of 60 cm horizontal and 15 cm vertical in terrain profiling. APTS used a high accuracy inertial navigation system periodically updated by measurements from a laser tracker, which used surveyed retroreflectors located on the ground. A laser profiler fixed to the IMU measured distance to the terrain along a known direction. With a full constellation of Global Positioning System (GPS) satellites soon to be available, comparable accuracies could be achieved with a medium-accuracy inertial navigator updated by differential carrier-phase GPS measurements. Additional flexibility could be provided by gimballing the profiler. For appplications in which precise survey of ground points is desired, improvements in the laser tracker would make it possible to acquire retroreflectors that are genuinely unsurveyed. The design provides for the inclusion of additional optional sensors such as imaging devices, and for the use of the surveyor as a test bed for other navigation systems. The design incorporates technology capable of recording data at rates of several megabits per second.

SECTION 1 - REVIEW OF APTS

APTS consisted of an inertial measurement unit (IMU), a laser tracker, a laser profiler, digital computer and peripherals, (CRT display, magnetic tape recorders and line printer), and a real-time control program. The system was permanently installed in a Twin-Otter aircraft modified for use with the system. In operation, the IMU gimbal angles plus the laser tracker formed a three-axis navigation datum; the laser tracker measured the vector distance to three or more surveyed retroreflectors in the project area. This non-drifting independent measurement of aircraft position was used to update the real-time navigation and to refine the IMU navigation during

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postprocessing. The laser profiler was hard mounted to the aircraft and measured distances to the terrain referenced to the IMU stable member.

The APTS real-time software program acquired and recorded system data and performed navigation computations sufficient to permit the continuous acquisition of retroreflectors during a mission of two to three hours in length. To achieve the required profile data accuracy, the time between successive locks had to be approximately 3 to 10 minutes. After acquiring three surveyed retroreflectors, the coordinates of unsurveyed retroreflectors were updated in real time and stored in the computer to be used for subsequent passes over these retroreflectors. The real-time software also provided system moding such as ground calibration and alignment of the IMU, transfer to navigation upon motion detection, initiation of search and acquisition of the retroreflectors and waypoint guidance to the autopilot. IMU and tracker gimbal servo control was mechanized in the real-time software as well as the IMU stable member temperature control. A multiplexed AID convertor continuously monitored important instrument parameters such as temperatures, heater power and power supply voltage. An important feature of the real-time software was determining the time of validity of the data received from the IMU, tracker, and profiler.

In APTS a sophisticated post-flight data reduction facility entailed optimally combining the measurement data from the IMU with that from the laser tracker to produce a best estimate of position. Flight data was passed through a data editor program and reformatted for further processing. A renavigation program solved the appropriate navigation equations to reconstruct a time history of the inertial­navigation-indicated values of velocity and position. Concurrently, values of velocity and position during laser tracking were computed. These values were subtracted from the corresponding inertially derived values to obtain difference data. A data compression program applied a polynomial fit to the difference data. Outputs of the data compression program served as input to a Kalman filter, an optimal estimation algorithm, which processed the data both forward and backward in time. The separate resulting error estimates from the forward and backward filters were then combined in an optimal smoother program to produce the best possible error estimates. A recombiner program used the smoothed and interpolated estimated position errors to correct the inertial-navigator-indicated values of position from the renavigation program, thereby producing a best estimate of aircraft position throughout the survey flight. Terrain profiles were produced from the vertical position estimates combined with the profiler range measurements, and corrected for deviation from the vertical direction.

The performance of the APTS was tested over a calibration range set up by the US Geological Survey. The relative locations of 15 retroreflector sites were established by Doppler satellite surveying techniques. Conventional surveying techniques were used to tie the retroreflector sites to both the vertical and horizontal North American datums. Orthophoto maps at 1:800 scale with I-ft interval contours were prepared for several of these sites. To prove system performance the profile data collected by APTS was compared to the contours on the map. The standard error in lattitude was ±25 cm and in longitude was ±38 cm. The standard error in the vertical coordinate was ±13 cm. Part of the horizontal error is caused by noise in the Doppler surveys, estimated to be ±1O to 20 cm.

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SECTION 2 - APMS SYSTEM DESCRIPTION

The Airborne Precision Mapping System (APMS) is intended to be an operational version of APTS. The heart of the system is a navigation system consisting of an IMU and laser tracker and GPS. The IMU and laser tracker function in the same manner as in APTS. The GPS receiver is used to bound the growth of errors in real time sufficient to permit the acquisition of retroreflectors and in post-processing in the navigation filter. Using GPS makes the possibility of using a commercially available IMU feasible. Operational procedures are to be simplified as much as possible. Reduced ground alignment times, tolerance of larger ambient temperature ranges, minimization of size, power and weight, minimization of required aircraft modificaitons, and simplification of installation and removal of the equipment are desirable design goals. A scanning laser profiler are used instead of a hard mounted profiler so that a larger area can be profiled, reducing mission flying time to cover a given area. Reliability and maintainability shall be major considerations in the design.

The APMS takes advantage of the technological improvements derived from the testing and applicaiton of the prototype APTS. It shall be designed to operate in a relatively small fixed wing aircraft, flying at altitudes of 600 to 1500 meters, and at air speeds of approximately 100 knots. It is capable of determining the position of the aircraft to an accuracy of ±15 cm in the vertical coordinate and ±60 cm in the horizontal coordinates (90 percent confidence numbers) in its most precise operational mode. It is designed to serve as a reference platform for a variety of sensors and operational scenarios relating to earth science data collection. These can include but are not limited to such tasks as terrain profiling, point positioning, providing a platform for aerial cameras, airborne gravity measurements, map accuracy testing and as a reference platform for measuring the performance of other navigation systems. The APMS consists of the following major subsystems:

1. A navigation system consisting of an IMU augmented by a radio receiver for the GPS, and a laser tracker with passive ground based retroreflectors. The GPS receiver serves to bound the real-time error growth of the IMU, facilitating the real-time acquisition of retroreflectors by the laser tracker, as a minimum. The use of GPS will also permit missions to be accomplished at higher altitudes when the laser devices are not needed. The laser tracker will be used during those missions requiring the highest accuracy and where the survey data is requred on a local datum. During these missions, the laser tracker will be capable of being precisely pointed with respect to the IMU. By successively acquiring, tracking, and ranging to retroreflectors placed over surveyed control points, the laser tracker provides essentially independent measurements of the aircraft position, complementing the inertial measurement of position.

2. A scanning laser profiler, capable of being precisely pointed with respect to the IMU and measuring ranges to the terrain below. The laser profiler will be capable of operating in three modes: (a) point always to the nadir, (b) scan perpendicular to the flight path with a scan width of ±14 degrees, and (c) profile a predetermined path.

3. An ima&in~ subsystem consisting of a video camera, boresighted with the profiler, a video recorder, and a time code generator. The time code generator

97

superimposes system time on the video image while simultaneously providing a compatibly formatted time signal to be recorded on the audio channel of the video recorder. The video image will be used to edit the collected mapping data.

4. Computer(s) and Peripherals - A main mission computer, data recorder(s), and display(s) will be used for system moding and control, real-time data processing and recording, and monitoring of the important parameters of each phase of a mission flight. Subsystems, such as the IMU, laser tracker and laser profiler may incorporate local processors and recorders for control and preprocessing of the data from the respective subsystems. The design of the system takes advantage of the recent advances in technology, such as the use of distributed microprocessors, and data bus transfer techniques. Further study and a preliminary design are required before an exact approach can be selected for the separation of computer functions. The selection of the subsystems, such as the laser tracker and laser profiler, and the definition of the data rates will help determine the exact distribution of computing functions. In all probability, the IMU and the GPS receiver, if selected from commercially available equipment, will have their own dedicated processors. The approach taken in the specification, as shown in Figure 1, APMS data flow, therefore assumes a particular partitioning of computers and processors, which is only one possible implementation.

5. Optional sensors - Provisions are made for incorporating optional sensors. Typical alternate sensors are radio navigation systems for navigation updates, thermal imagers, terrain reflectivity sensors, multispectral sensors or stereo vision systems.

6. ControllDisplay Unites) - The operator will control, operate, and monitor the system performance during a mission through a suitable controVdisplay unit(s).

7. Aircraft Interfaces - A system power subsystem will be mechanized. There will be both a flight power unit and a ground power unit (GPU) for operating, testing and calibrating the system on the ground. It may be necessary to use an auxiliary power unit (APU) for flight to isolate the system from the aircraft engines. Air pressure and temperature sensors will be incorporated and interfaces to the aircraft autopilot will be provided.

8. Postprocessin~ system - A dedicated postprocessing system will be developed, consisting of the necessary computers, recorders and software to process the recorded real-time flight data and provide the deliverable flight data in a standard format compatible with ground processing equipment being used by government and commercial agencies.

98

-SCANNING PROFILER .-.

LASER TRACKER

,- OniiR-'

RANGE .-. COUNTER

--1

PROFILER PROCESSOR

TRACKER PROCESSOR

, °7JiAL ' __ -I_~ HIGH SPEED , RADAR' DATA

SENSORS PROCESSOR L ___ ,

HIGH SPEED DATA

STORAGE

--

CLOCK. & TIMING

VIDEO CAMERA

VIDEO TOO

VCR

-IMU

Fig. 1. APMS data flow.

_ ...

1 IMU CDU

lEU ~

MISSION COMPU1ER

99

- -

-.

-, 1--1 TRISPONDER .... , ... ~---...,I

GPS AIDING _

~ ________ ~ GPSDATA

AUTO­PILOT

CDU

AIR DATA

COMPUTER

---jDITERNAL

INTERFACE

-- -_EXTERNAL

INTERFACE

IMU IMUDATA -COMPUTER HI SPEED _

~L..-. ________ --1DAT A, W's, e~

DATA STORAGE

DATA BUS

.-

SECTION 3 - PERFORMANCE ANALYSIS

An important question in the design of APMS is the choice of an inertial navigator. Reference 1 lists the specifications for three commercially available INSs: the gimballed Honeywell GEOSPIN, the gimballed Singer High Accuracy Inertial Navigation System (HAINS), and the strapdown Honeywell Ring Laser Gyro Navigator (RLGN). Reference 2 lists general specifications for four categories of INS, divided into the classes gimballed vs. strapdown and medium accuracy vs. high accuracy. Reference 3 gives additional details on the RLGN.

Included in the gimballed medium accuracy category is the Litton LN-39; the strap down medium accuracy category includes the Litton LN-93 and Honeywell H-4030; the gimballed high-accuracy category includes the Singer SKN-2440.

To determine the suitability of commercial systems, covariance studies of three such systems were performed using the racetrack trajectory flown by APTS on May 30, 1985 (Reference 4). The flight path is shown in Figure 2. Retroreflectors are spaced approximately 8 km (160 seconds apart).

Preparing input parameters for the covariance analysis requires some interpretation of the parameters listed in References 1, 2, and 3. For example, the 50 Jlg accelerometer bias stability parameter very likely refers to the long-term stability, or stability from tum-on to tum-on, rather than to the stability during a two- to three-hour flight that would be relevant for APMS. A more relevant parameter for APMS is the accelerometer random noise listed in Reference 2 as having an rms value of 3 Jlg and a time constant of 60 seconds; that value was used in this study. Table 1 lists the plan and measurement noise parameters used to describe the three INSs in this study. The usual APTS gravity disturbance collocation model was used.

• Nagog 7000

-9000

Lagend:. Retroreflector site X lOG-second Ume mark

EAST METERS 103

Figure 2. Flight path on May 30, 1985.

100

Table 1. Plant and measurement noise parameters used in this study.

Parameter Fl..GN HAINS GEOSPIN Units • 10 4 10-4 10 4 m2 /s 2 HZ e

oV 3 x 10-16 3 x 10-16 3 x 10 -16 m2/s4HZ

• -12 7 x 10-13 1 x 10 -16 rad2 /s2 HZ 'II 2x 10 •

3 x 10-19 1.5 x 10-19 6 x 10-21 rad2 /s4 HZ £

· 1.5 x 10-11 1.5 x 10-11 1.5 x 10-11 m2 /s 6 HZ Ba

Range 0.01 0.01 0.01 meter

Angle 10 -5 1.5 x 10 4 6 x 10-5 radian

Figure 3 is an example using the RLGN, showing the standard error in the vertical error predicted by the filter covariance matrix with tracker locks about every 160 seconds. All three systems easily meet the horizontal spec, and they also stay within the 15-cm vertical spec except for a few excursions near the beginning and end of the flight. Figures 4 and 5 show the standard error when measurements from the Sudbury retroreflector are omitted, leaving a 320-second gap. In this case, all three INSs fail to meet the vertical spec about equally badly, with peak vertical errors around 0.5 m. All three meet the horizontal spec, but the GEOSPIN performs noticeably better in the horizontal than the other two INSs.

References 5 and 6 indicate the potential for accuracy at the level of a few centimeters when, GPS is used in differential mode with receivers that track and record GPS carrier phase. Even with differential GPS, an INS would still be required to provide an attitude reference for the pro filer, and a tracker and at least three retroreflectors would be required to tie WGS84 to the local geoid. Nevertheless, with GPS it should be possible to meet the APMS accuracy specification over a region several tens of kilometers on a side using only three retroreflectors.

The use of a commercial gimballed or strapdown system may require a larger number of inertial instrument error states than we now have in the APTS postprocessor. Because we frequently perform full calibrations on the current IMU, we need only six inertial error states in the present postprocessor: three accelerometer bias and three gyro bias. The inertial instruments in strapdown systems, for example, cannot be put into all the attitudes experienced by the instruments in the current IMU; thus some states, like accelerometer scale factor and misalignment, will not have been calibrated since the system left the factory. The same may be true of gimballed systems for which only a short preflight alignment is performed.

101

I! o I!! a w IIUE

SMOOTHER COVARIANCE 18:50:32 05128187

0.20

0.15

0.10

0.05

0.0 -+1-t---t--t-_I~+--+--+_II--+--t---+-5~2~·OO~-+-+--+1 -1,-+--+--+1 ......,1--+ 2800 3800 4400 '000 '800

TIME SECONDS

Figure 3. Predicted standard error for the RLGN.

SMOOTHER COVARIANCE

0.'

Zf/I !U: 0.4 ow a~ w iii:'

0.2

'0.0

TIME SECONDS

01:02:05 OS/28187

Figure 4. Predicted standard error for the RLGN omitting Sudbury measurements.

102

SECTION 4 - CONCLUSION

We find that the strapdown Honeywell RLGN, the gimballed Singer HAINS, and the gimballed Honeywell GEOSPIN all meet the APMS vertical and horizontal accuracy requirements (15 cm and 60 cm respectively) when retroreflectors are spaced 160 seconds (8 kilometers) apart, even without the use of GPS measurements. When retroreflector spacing is increased to 320 seconds (16 kilometers), all three systems fail to meet the vertical accuracy requirement if GPS measurements are not used. If differential GPS measurements are available with receivers (such as the TI 4100) that implement continuous carrier phase tracking and permit frequent recording of carrier phase, it should be possible to meet APMS accuracy requirements over a region several tens of kilometers on a side with as few as three surveyed retroreflectors to provide a tie-in to the local geoid.

REFERENCES

1. Mamon, G., Memo No. APTS II-Oll "Further Considerations on IMU," Sept. 30, 1986.

2. Graham, W. R., et aI., "Standard Integration Filter Functional Design and Architecture," AFWAL-TR-86-1607, June 1986, TASC, Reading, MA.

3. Bachman, K. L., "Ring Laser Gyro Navigator (RLGN) Flight Test Results," in Proceedings of the National Aerospace Meeting, p. 171, 1981.

4. Greenspan, R. L., and J. I. Donna, "APTS/GPS Measurement Task Final Technical Report," CSDL-R-1845, February 1986.

5. Remondi, B. W., "Performing Centimeter-Level Surveys in Seconds with GPS Carrier Phase: Initial Results," Navigation, Vol. 32, No.4, p. 386, 1986.

6. Remondi, B. W., "GPS Carrier Phase: Description and Use," NOAA Technical Memorandum NOS NGS-42, 1985.

7. CSDL staff, Aerial Profilin~ of Terrain System Phase V Fabrication. Assembly and Testing. Vols. VA-l and VA-2, The Charles Stark Draper Laboratory Report R-1451, July 1982.

8. Brown, R. H., (with contributions by W. H. Chapman, W. F. Hanna, C. E. Mongan, US Geological Survey, and J. W. Hursh, The Charles Stark Draper Laboratory, Inc.), "Inertial Instrument System for Aerial Surveying," ll..£ Geological Survey Open-File Report 85-184.

9. Hursh, J. W., "Aerial Profiling of Terrain System (APTS), a Laser-Inertial Airborne Surveyor," Proceedings of the National Technical Meeting, Institute of Navigation, San Diego, CA, pp. 156-161, January 1985.

103

10. Hursh, J. W., G. Mamon, and J. A. Soltz, "Aerial Profiling Terrain", Proceedin~s of the 1st International Symposium on Inertial Technology for Surveyin~ and Geodesy, Ottawa, Canada, pp. 121-130, October 1977.

11. Soltz, J. A., G. Mamon, and W. H. Chapman, "Aerial Profiling of Terrain System Implementation," Proceedin~s of the 2nd International Symposium on Inertial Technolo~y for Surveyin~ and Geodesy, Banff, Canada, June 1981, (Addendum).

12. Mamon, G., and R. C. Rogers, "APT System Design and Real-Time Software," Proceedin~s of the 3rd International Symposium on Inertial Technology for Surveyin~ and Geodesy, Banff, Canada, pp. 423-440, September 1985.

104

Basic Geometric Considerations for a Self-Calibration of Strapdown Inertial Sensor Blocks by Tumbles

Ernst H. Knickmeyer1 and Elfriede T. Knickmeyer2

1 Pulsearch Consolidated Technology Ltd. Suite 700, 10201 Southport Road S.W. Calgary, Alberta, Canada T2W 4X9

Abstract

2 Dept. Surveying Eng., Univ. Calgary 2500 University Drive N.W. Calgary, Alberta, Canada T2N 1 N4

Strapdown Inertial Systems are usually calibrated by means of very accurate and expensive turn tables. This high accuracy is not required if self-calibration techniques are employed which make proper use of less accurate reference information and the uncalibrated system output itself. Starting from fundamental calibration prinCiples, such a technique is stepwise evaluated for geometric parameters. The steps cover the general three-dimensional calibration model, solvability considerations, the self-calibration of pairs and triplets of sensors -separate for accelerometers and single-degree-of-freedom gyroscopes, the essential deter­mination of the rotation between them, and finally the combined self-calibration of both triplets. Pros and cons are discussed in view of the calibration principles.

1 Introduction

The large-scale calibration of Strapdown Inertial Sensor Blocks (SISB) is performed effectively by their manufacturers, /21. Users often face the task to check a manufacturer's calibration, to determine single time varying or suspicious error compensation parameters, or to calibrate a subset of SISB sensors. Some concepts and principles, which underly a calibration and enhance its understanding, are considered in the sequel. Geometric calibration parameters are emphasized and the considerations are restricted to a calibration by tumbles and to SISBs consisting of near parallel triplets of almost orthogonal accelerometers and single-degree-of­freedom gyroscopes.

SISBs are usually designed to sense two vectors, the angular velocity with respect to inertial space and the specific force which is the difference: acceleration with respect to inertial space minus gravitation. The calibration of an SISB is possible only when these "input" vectors change their orientation with respect to the SISB, which can be achieved e.g. by accurate turn tables, /4/, 151. For the determination of geometric target parameters the only requirements are that a rotation axis does not change its direction in the local-level frame t during a turn (coordinate frames see Appendix 1), and that the SISB's orientation is known approximately in t. The latter leads to the class of "self-"calibrations, where the uncalibrated system determines its own attitude. The rotation axes need not be orthogonal nor perfectly aligned with l nor need the rotation angles of the turn table be accurate or accurately known. This allows the user to perform a calibration by means of much simpler turn tables, /6/.

Before the self-calibration model is treated, assumptions and target parameters are specified.

105

What is Calibration?: Calibration is the comparison of the measured uncalibrated sensor output, or functions of it, with the known properties of the ideal SISB output to determine a sensor error compensation model.

Known are the properties of the ideal output (in brackets the number of quantities): 1. Mutual orthogonality of the axes of a triad (2x3) 2. Magnitude and equality of scale within each triad (2x3) 3. Zero biases (2x3) 4. Parallelism between orthogonalized gyro and accelerometer triads (3) 5. Rotation of a common orthogonalized sensor coordinate system s to b, defined by

the housing (3) and further: 6. The magnitude of gravity, necessary for accelerometer scale factor determination.

The magnitude of the angular velocity need not be known because the gyro scale factor is determined differently.

7. The approximate attitude B! s or b a That the unknown direction of a rotation axis is fixed in I during a turn.

Measured are the angular velocity and the specific force by the uncalibrated sensors.

Target parameters of the calibration are an affine transformation matrix and a bias vector for gyro and accelerometer triad, each. These 2x12 parameters are equivalent to the 24 properties of the ideal 515B output and are invariant with respect to a changing attitude or translational motion of the 515B.

Errors of a calibration can be minimized following the

Principles of Calibration 1. Determine as few errors of a single sensor as possible at a time. 2. Exclude other errors as far as possible. 3. Calibrate the instrument in that range in which it is used. 4. Exclude transient effects and trends by a proper sequence and timing of the mea­

surements. 5. Assure that both triads are related to the same coordinate frame s or b.

The first four points are rather general. 1. and 2. minimize smearing effects of remaining model errors and make each target parameter a function of as few measurements as possible. 5. is a point which is easily achieved in case the attitude al s or b is known. It requires special attention when gyro and accel triads are calibrated separately.

2 General Three-Dimensional Geometric calibration Model

The calibration model is evaluated in three steps. Firstly the compensation model is stated which transforms the uncalibrated into the calibrated measurements; secondly the affine transformation matrix is examined; thirdly the calibration model (observation equation) is derived from the compensation model.

The compensation model is (notation see Appendix 2):

(1a,b)

106

where the coordinate frame b might be replaced by s for the self-calibration. The specific force is in the stationary case

(1 c)

and g! = (O,O,g) is the known gravity vector. Other sensor errors are not considered here but can

be included in (1). ta and m9 are the uncompensated measurements.

An affine transformation matrix a can be decomposed into a rotation R. an orthogonaliza­tion Q (direction cosines of a in s) and a diagonal scaling matrix S:

(2)

Combinations are

M = B . Q ... Misalignment , U = Q . S ... Deformation. (3a,b)

The 3 free parameters of each, R, Q, and S, depend on the order of R, Q, S and on the order of the rotations within Rand Q.. The choice of a particular order defines a certain sequence of intermediate coordinate systems and thus the corresponding transformation parameters. The model (1) is linear in all target parameters; the replacement of A by the nine independent parameters of R . Q . S leads to a non-linear model for which the decomposition is useful.

The separate self-calibration of the triads of an SISB is based on the determination of the 9 free parameters of fi, Q, and S and deals with a non-linear model. In the non-linear case A can be decomposed e.g. by a polar decomposition:

(4a)

where .u and Y... are matrices containing different eigenvectors with det II = det Y... = 1, and K,2 is a diagonal matrix of common eigenvalues.

The diagonal elements of S are the lengths of the columns of D and

Q=~-1D.

If a is close to identity the linearized decomposition becomes:

A= 1+11a , a-1 ~1-aA

s. ~ diag (A11,A22,A33) , M = A .. ~-1 R = I + (M - MT) 12 , Q. = (M + MT) 1 2

(4b,c)

(4d)

(5a)

(5b,c)

(Sd,e)

(5d,e) belong to a particular definition of intermediate coordinate frames, others are possible, /7, p.20/.

The calibration model for the accelerometers is

107

fa = - (Agba)-1 . Rb! . ~! + tma (non-linear)

fa -fb,,;, + ,dAaba . Rb! .~! + baa (linearized with 5a),

and reordering leads to

[2M. T] [vec(2M. T)]

,,;, + (13 I UbTI1]) . vec baT = + [(13 IfbT) 113] . b;

(6a)

(6b)

(6c,d)

with the Kronecker product I and the vectorization operator vec, 11/, and can be applied also directly to the inverse matrix of (6a).

In principle the gyroscopes could also be calibrated stationary with an analogous model based on the earth rate as input. Practically, this is impossible for Ag.. The input vector !!lie is too "short" and thus the signal to noise ratio too small. In order to excite Ag, the sensed angular rate has to be larger which is a requirement also of calibration principle no. 3.

It follows naturally that this larger co is that which turns the SISB from one fixed attitude, which is required for the accelerometer calibration, to the next. The integrated sensed angular velocities are treated as observations, since the exact magnitude of the input,m is assumed to be unknown, see also Chapter 4.

The calibration model for the gyroscopes is

t2 t2 t2 ftg(t1.t2) = J,mg dt = (Ag,bg)-1. J !!lj~ dt + J bgg dt

t1 t1 t1

= (Ag.bg)-1 . J RbI (t) dt . !!li~ + (Ag.bgt 1 .!! + bg.g . (t2 - t1)

with ~~ = Rbl (t) . (!!li~ + ~b) and !! = J !!ltbb dt .

(7a)

(7b)

(7c)

In the stationary case !!ltbb is zero and the determination of the bias is "disturbed" only by

!!2j~. Longer periods increase the coefficient of bg, thus improving its determination. This advan­

tage has to be balanced against an increasing influence of a random walk and other influences not considered here.

The magnitude of !! is required from an independent source to determine the gyro scale factor. It can be determined, apart from second-order errors, by two measurements of the same accelerometer before and after a turn of about 21t about a horizontal axis, where the accelerometer is orthogonal to the rotation axis. The turn has to start and end with this accelerometer close to the horizon. Then the measured specific forces are small so that a scale factor plays no role, the bias cancels, and small misalignments enter only with their cosine. The

108

independent rotation angle is

I~I == 21t + (fend - fstart}/g . (8)

The described model is complete if al s or b is known with an accuracy which is sufficient not to spoil the determination of the target parameters. Otherwise the attitude at each stop or some parameters of it become additional auxiliary unknowns. The reordering of (6c,d) applies also to (7b).

3 A First Glance at Solvability and Optimization

A necessary condition for solvability is that the number of observations is larger or equal than the number of target parameters. 15 of the 24 target parameters (12 accelerometer parameters and 3 gyro biases) have to be determined stationary, 9 by tumbles. With 3 observations per attitude and assuming that the attitudes are known, the minimum number n of attitudes is

n . 3 specific forces ~ 12 aecel target parameters or n ~ 4. (9a)

For the remaining 9 gyro parameters with 3 angles observed per turn, the minimum number m of turns is

m . 3 angles ~ 9 or m ~ 3 . (9b)

Thus, n = 4 is the minimum but does not necessarily lead to solvability. Such a solution - if it exists - will not consist of attitudes apart by multiple 1t/2 rotations. It also will not comply with calibration principles 1 and 2. However n=4 and m=3 match perfectly and it is a fancy task to search for this minimum set of attitudes and turns and also to optimize the solution. Additional attitude parameters enlarge the set of target parameters when the exact attitude is unknown. Only 2 auxiliary parameters per attitude for levelling have to be added, assuming that the {!lie­term of (7b) is still accurate enough with an approximate azimuth in ab"

The minimum number n of attitudes is then

n . 3 specific forces ~ 12 + n ·2 or n ~ 12, (10)

and the minimum number of turns is still m ~ 3 as in (9b).

Whether or not a sequence of attitudes determines all target parameters is decided by the observation equation matrix

[ b T ] 13 I f1.att I 13

Ii- ~.~ 'rJt 113 . (11 )

109

It contains the coefficients of equation (6c,d) for each attitude. Sufficient for solvability is that Ii has full column rank. An optimization of the calibration, 18/, is based on a modification of the attitudes and thus of H.

Optimization in a more general sense can strive for the objectives:

1. Minimum number of attitudes, which leads to a set of target parameters in the shortest time.

2. Maximum separability of the target parameters in the sense of calibration principles 1 and 2. This facilitates the detection of model errors and allows the separate determination of specific subgroups of sensor errors.

3. Certain properties of the normal equation matrix liTIi or its inverse; the following objectives might concur.

a) Maximum separability of the target parameters in the statistical sense. A diagonal covariance matrix can be achieved in case of known attitudes spread evenly over a circle; for approximately known attitudes, it becomes a diagonally dominant matrix. The disadvantage of such a procedure is that all sensor errors are excited at each attitude. An advantage is that the calibration attitudes cover the set of all possible input attitudes more densely and thus might excite hidden model errors which otherwise remain undetected.

b) Minimum trace of the covariance matrix of the target parameters. Only few modifi­cations are possible to approach this optimum for a given turn table and 818B. They are to vary the attitude-turn sequence (first-order design), to vary the stationary and turning periods in order to reduce the variances of the measurements (second­order design), and simply to repeat the calibration.

c) Maximum condition of HT1::I..

4 Separate Self·calibration of Accelerometer and Gyroscope Triads and Their Relative Orientation

The separate self-calibration of a triad makes use only of its own output and of the assumptions that the input vector does not change its direction in the sensor frame during sensor readout, and that its magnitude is known. The magnitude of m is not known, see remarks before (7) and (8) on the gyro scale factor. This lack is overcome by treating integrated angular velocities and comparing these angles with those derived by accelerometers before and after a turn.

A change of the direction of m during a turn is critical for the gyro non-orthogonality and other parameters. This is why the turn itself is required to be free of coning and other mechanical errors. During an imposed angular acceleration from zero to a certain maximum (e.g. 200/h) the direction of 1!l moves from the celestial pole close (ca. 0.5 mrad = 0.30) to the imposed rotation axis. For an east-pointing direction this deviation takes its maximum. With a constant imposed input angular velocity the sensed angular velocity (7a) can be used directly as an observation for the gyro non-orthogonality, similar to the treatment of an accelerometer non-orthogonality.

The calibration of a sensor pair is performed preferably in some plane parallel to the input vector in order to use the full range from zero to its magnitude. A nearly orthogonal pair has 5 calibration target parameters: 2 biases, 2 scale factors, 1 non-orthogonality E.

110

The observation equations of an accelerometer pair are:

(12)

Scale factor and bias of one sensor are determined by two observations at orientations parallel (a = 0) and antiparallel (a = 7t) to the input vector. The bias is simply half the sum of the two observations, the scale factor is the double input magnitude, divided by their difference. For these computations a deviation of the sensor from the input direction enters the determination only with the cosine of the deviation. The same effect prohibits the determination of the non-orthogonality in this orientation. The sensors have to be turned to a position where the measurement of both of them is sensitive to the orientation, obviously the optimal a is near 7t/4. The two simultaneous observations, corrected for bias and scale factor, determine a and E. A deviation of the input vector from the plane spanned by the sensor axes again enters the computation only as a second-order effect. The resultant 5 target parameters and the orientations can be improved iteratively in a non-linear model, especially when the full triad is read out each time. Another rotation sequence for a sensor pair is to evenly distribute the angles a on a full circle. Then the 5x5 target parameter block of the normal equations becomes diagonal and the unknowns become uncorrelated if the attitudes are known.

The repetition of one of these procedures for each combination of 2 sensors results in a compensation model consisting of scale factors, biases, and non-orthogonalities for gyroscopes and accelerometers, each. They provide a bias-free, scaled, and orthogonalized sensor output in two separate coordinate frames og and oa, respectively, which are related by a rotation.

The only geometric pieces to which both triads have access are the rotation axes of the table. The crucial question is now about which direction the SISB has to be turned to allow a simultaneous determination of this direction in both frames. The answer is: close to vertical. Then the two nearly horizontal specific forces, say f~a and flya, will describe biased and phase shifted

sine curves during the turn, and the angles J3x and J3y between rotation axis and directions oax and oay are given as a function of maxima and minima of these curves:

asin (~a(max) I g) + asin (toxa(min) I g) = 7t - 2 J3x

J3x = {7t - asin (toxa(max) I g) - asin (foxa(min) I g) } I 2 . (13)

The equation also holds for y. The specific forces have to be corrected for a centrifugal force, if necessary. The rotation axis in oa is

lliUoa = [cos J3x, cos J3y, (1 - cos2J3x - cos2J3y)1/2 ]T. (14a)

Neglecting the earth rate, the gyro rotation axis is simply

III

An integration of angular velocities, which considers mie, delivers attitudes ~a(t1) and a:a(t2)

before and after the turn. Then

(14b)

and the rotation axis lml0g follows from a transformation of (:~~? into Euler axis (and angle).

This transformation is straightforward in case the attitudes are delivered in form of quaternion elements instead of direction cosines.

The procedure has to be repeated for a second rotation axis !llIQ1 which should be nearly perpendicular to lml in the orthogonalized frames. The two directions given in oa and og finally yield R090a

lr210a T

mLQ10a T

(ir210a x !illQ1oa) T

= iro+og T . Roa T .!.UU - og

= .!!l!Q1og T . Boa og T

= (ir210g x .!!l!Q1og )T . Roaog T

BOaog =[ :;;gTT ]-1.[ :;o~ ~ ] (ir210g x illIQ10g ) T (ir210a x illIQ1oa) T .

(15)

The remaining 3 of the overall 24 target parameters are those for a rotation of s, which is usually identified with oa, to b. The b-frame can be defined and realized optically by mirrors or mechanically by some pins or edges of the housing. Parts of the input vector or the rotation axes must be known in b. This can be achieved by edges, planes, or tiltmeters, aligned in b. The procedure is simple but somewhat awkward. It can be implemented on a turn table only with difficulties and is required only for special applications, e.g. attitude transfers.

5 Combined Self·Calibration of the Sensor Triads of an SISB

A separate self-calibration makes sense only for the accelerometer triad; for the gyroscopes external information is required, and an integration which deals with the earth rate, for instance an extract of the local-level mechanization equations. A very efficient way to use these in combination with a rotation sequence of only 12 attitudes is described in 121, 13/, and forms the basis of the following considerations.

Neither the gyro measurements nor functions of them are treated as observations explicitly, see Fig.1; rather, the gyro errors propagate through the integration of the measured angular velocity mg into the attitudes B~. This attitude is then used to rotate the measured specific force fa, which contains the accelerometer errors, from b to I. The specific force in I thus is a function of accel and gyro errors. It is reduced by the gravity to yield the linear acceleration in I, which should be zero for the stationary turn table. Deviations from zero are caused by the sensor errors. The 12 attitudes are connected by 9 x-turns about near horizontal and vertical axes and 2 auxiliary turns. The differences of accelerations before and after a turn are treated as observations. No x/4 attitude is used and the compensation model is not decomposed. In the

112

linearized case the normal equations fall apart to a large degree so that the target parameters are obtained as simple linear combinations of the observations. The rotation sequence even allows the determination of some asymmetry errors. Frame s is defined with respect to the accelerometers, a rotation from s to b is not determined by the sequence.

Accel errors in f a

Rg. 1: Propagation of Sensor Errors into Acceleration:;. The acceleration in I should be zero in the stationary case. Deviations from zero are functions of the sensor errors.

The underlying non-linear model, which propagates the sensor errors analytically, is based on the differential equation

(16)

where ~~ is replaced by measurements filg with (1 b). Note that fil is integrated into direction

cosines rather than angles (7b). Two subsequent attitudes are related by

t2

= Bit>(t1) + f Btb(t) dt t1

(17)

(18a)

For the following separation of time-dependent and -independent quantities the index notation is used. The frames are omitted. It is assumed that M and Aa are close to identity (Sa). Quadratic terms of target parameters are neglected (linearization) and the tensor property of Ejl k is used.

113

= J Bij mm dt . £jmk - J fink dt . £imn m(~) (1Sc)

' __________ ~~~--------~J QBik

+ J Bijmmdt'£j1k'~m - J Bijdt'£jmk'bQm

(17) formally reads with (1Sc):

v Eiktm

(19)

with four- and three-dimensional quantities E and F, respectively. The first two terms of (19) represent B(t2, mech) as computed by the mechanization equations. The accelerations at t1 and t2 are

y./ (t1 ) = R!b (t1)

= R!b (t1) . 1a(t1) - gl

+ R!b (t1) . QAaba . 1a(t1) - R!b (t1) . baa (20a)

yl (t2) = ( Bft, (t2, mech) + Eikt m . dAgr m + Fikm . ~m ) . Aaba (fa(12) -baa) - 'II = R!b (t2, mech) . 1a(t2) - gl

+ (Eikt m . rJ.Mt m + Fikm . bgm ) . 1a(12) + Bft, (t2, mech) . ,dAaba . fa(t2) - Bft, (t2, mech) . baa (20b)

In these equations la are uncompensated specific forces. The mechanization equations generate flft, (t2,mech) as a function of mg, and Y! as a function of both fa and mg. E and F result from separate integrations. Significant simplifications occur for the forementioned x-turns.

Reordering the difference of (20a) and (20b) similar to (6c,d) brings the target parameters into vector form. The collection of these equations for proper pairs of attitudes leads to a system of linear equations which is solvable for the target parameters, if s :: b is defined with respect to the sensors by 3 independent condition equations for the accelerometer misalignments. Equation (20) shows that s is the common frame for accels and gyros and calibration principle nO.5 does not require special attention. However, both types of sensors should determine rotation angles with the same accuracy because the gyro target parameters depend also on the accel accuracy

114

and vice versa. The non-rigorous treatment of stochastic errors of the measured angular velocity in an estimation based on ~ is partly made up by repetitions of the calibration procedure which provide an outer accuracy of the target parameters.

The elegant procedure described in this chapter is not restricted to geometric sensor errors. It applies also to other functional target parameters, which have to be propagated analytically into the acceleration in I, and for which a specific sequence of tumbles has to be designed, see Chapter 3.

6 ConclIsions

Some concepts and techniques have been viewed which underly the calibration of an SISB or subsets of its sensors. Geometric considerations were emphasized and formulated algebraically. A self-calibration of SISBs is possible with relatively simple turn tables. While a separate self­calibration of accelerometers is possible with the gravity as input vector, the gyroscopes cannot be calibrated easily in an analogous way. The required gyro-external information can be delivered by accurate turn tables or by accelerometers, the latter leading to an efficient self­calibration of an SISB as a whole. Many considerations of this investigation hold also for the calibration of other sensors in vector fields.

Acknowledgements

Mr. D.J. Hogan, INS Engineering Product Specialist, Litton Systems Canada Ltd., stimulated this investigation in 1988 during the first author's stay at Litton facilities in Etobicoke, Ont.. The encouragement of Prof. Dr. K.P. Schwarz, The University of Calgary, and valuable discussions with Prof.dr.ir. P.J.G. Teunissen, Delft University of Technology, are also acknowledged.

LHerature

111 Brewer, J.W.: Kronecker Products and Matrix Calculus in System Theory. IEEE Trans. on Circuits and Systems, Vol. CAS-25, No.9, Sept. 1978, pp.772-781.

121 Diesel, J.W.: Calibration of a Ring Laser Gyro Inertial Navigation System. 13th Biennial Guidance Test Symposium, Holloman Air Force Base, New Mexico, Oct. 1987, Vol. I, W·1-37.

131 Diesel, J. W.: Calibration of Strapdown Inertial Systems. In: Chinese Society of Inertial Technology (ed.), P.O. Box 3913, Beijing 100854, People's Rep. of China: First Int. Symp. on Inertial Technology in Beijing, China, May 1989, pp. 109-112.

141 Grewal, M.S., V.D. Henderson, and R.S. Miyasako: Application of Kalman Filtering to the Calibration and Alignment of Inertial Navigation Systems. In: The Institute of Electrical and Electronics Engineers (IEEE), Position, Location, and Navigation Symp., Record 86 CH 2365-5, Las Vegas, NV, Nov 4-7,1986, pp.65-72.

151 Joos, D.K.: Identification and Determination of Strapdown Error-Parameters by Laboratory Testing. In: H. Sorg (Ed.): Advances in Inertial Navigation Systems and Components, AGARD-AG-254, Neuilly sur Seine, April 1981.

161 Knickmeyer, E.H.: Calibration, Handling, and Use of a Cardan Frame with the Litton LTN90-100 Inertial Reference System. Publ. no. 30011, Dept. of Surveying Eng., The University of Calgary, 1989.

l15

/7/ Mark, J., D. Tazartes, and T. Hilby: Fast Orthogonal Calibration of a Ring Laser Strapdown System. In: Symp. Gyro Technology, Stuttgart 23,/24. Sept. 1986.

/8/ Mondsheim, L.F.: Observability of Accelerometer Test-Input Errors, AIAA, 80-1766, 1980, W.290-296.

Appendix 1: Definition of Coordinate Frames

Abbr. Description Origin Direction of Axes

l local-level, centre of proof- 1: East North oriented masses of 2: North

accelerometers 3: Zenith

b body asl 1: right of a device or craft onto which 2: forward the sensor block is mounted 3: upward

s sensor block asl definition with respect to sensor input axes, orthogonalized common frame of all sensors

OJ orthogonalized not defined definition with respect to gyro input axes, scaled gyroscopes and orthogonal

oa orthogonalized asl definition with respect to accel input axes, scaled accelerometers and orthogonal

g gyroscopes not defined sensitive input axes of gyros, non-orthogonal

a accelerometers asl sensitive input axes of accels, non-orthogonal

Appendix 2: Notation

't IDb £ijk

Specific force Affine transformation matrices from a and g to s Biases of accelerometers and gyros, respectively Rotation matrix from l to s

Acceleration

Angular velocity of b with respect to i, coordinated in l

Epsilon tensor; [m xl ~£ikj IDk, £ijk Bil Bjm Bkn = £ann

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SESSION2b

EQUIPMENT TRENDS AND MEASUREMENT PROCEDURES

CHAIRMAN K.McDONALD

NA VTECH SEMINARS INC. ARLINGTON, vmGINIA, USA

TRENDS IN INS DEVELOPMENT - A MINI-PANEL PRESENTATION AND DISCUSSION

Chairman - Michael J. Hadfield. Honeywell. Inc.. St. Petersburg. Florida. USA

Panel Member - Dr. James Huddle. Litton Guidance and Control Systems. Woodland Hills. California, USA

Panel Member - Dr. Loic Camberlein. SAGEM. Cergy Pontoise. Cedex. France

1. ABSTRACT

Many advances have been made in the past 15-20 years on the application of inertial navigation systems (INSs) to the fields of geodesy. surveying. and remote sensing. This paper addresses those applications and experiences gained where the INS was used in a direct measurement role. However. INSs have also been used to stabilize or provide motion compensation data for other primary sensors. such as radars. lasers. and other electro-optical devices. This aspect is also touched upon.

The gains made during this period were particularly impressive in the performance area. Also included was the flexibility with which the same basic equipment could be adapted. with mostly software and procedural changes. to a wide variety of uses. After summarizing the initial experiences. which used mostly gimballed inertial systems. the paper moves into the era of strapdown INSs. during the mid-to-late 1980's. An example of one of these systems. the U.S. Army's Modular Azimuth Position System (MAPS) is described. along with summaries of very extensive. open loop survey and positioning tests.

Honeywell started development of its H-726 Modular Azimuth Position System (MAPS) in September 1984. This system represented a major advancement in the field of inertial pointing and positioning systems for combat vehicles. with the introduction of the rugged. highly accurate. and extremely reliable laser gyro.

The Dynamic Reference Unit (DRU). the heart of the MAPS. is an all-axis strap-down. inertial navigation system. which has been adapted for land navigation. positioning and pointing. Over the last five years. the H-726 has been thoroughly evaluated and field tested over a wide variety of conditions in the US. UK. FRG. Sweden. Norway. Switzerland. France and Japan. Applications ranged from positioning and pointing of self-propelled howitzers to steering coal mining machines. and artillery battery surveying. At this writing. Production contracts are underway to supply the H-726 MAPS to the US. UK and Swedish Armies.

But what does the future hold? In the very near term. there is the potential for applying RLG strapdown INS technology to closed loop surveying and positioning for improved accuracies. Several levels of performance capability are available with different grades of existing gyros and accelerometers. as well as various system mechanization and software improvements. The combination of INS and GPS systems. in either stand alone or fully integrated and embedded (packaged together) ways. is also achievable.

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Later in the 1990's, it will be possible to reduce the size, weight, and power of strapdown RLG INSs without sacrificing performance, compared to MAPS systems capabilities. Other inertial sensor and electronics technologies may also become available. All of these possibilities are discussed in the paper.

2. INTRODUCfION AND SUMMARY

In preparing for this topic, much thought was given to the method of presentation. To give the topic a broad prospective with regard to both experience and new technologies, it was felt that a representative from only one manufacturer would not do it justice. Therefore, we decided to convene a panel of several leading people from as many companies to each present his approach to the subject. We were very fortunate in obtaining the talents of the three people listed above. They have a combined experience in the inertial survey and mapping fields of over 45 years with most of the existing and new technologies.

The following pages present a summary of the presentation each panel member made, with several of the graphics used in that presentation. Following the presentations, a general discussion with the KIS-90 audience ensued, with many interesting questions and opinions being volunteered.

120

SUMMARY PRESENTATION BY M. HADFIELD:

Honeywell's inertial survey involvement began in the mid 1970's, with a feasibility demonstration program for ESG technology, funded by the U.S. Army Engineer Topographic Laboratories, Ft. Belvoir, VA. The program began in 1976 and culminated in a field demonstration series at White Sands Missile Range, New Mexico, in November, 1977. Since that time, Honeywell has fielded ESG inertial systems, Azimuth Position System (MAPS), using its 46 lb. H-726 Dynamic Reference Unit (DRU), based on ring laser gyro technology, and most recently shown the feasibility of adapting its newest and smallest inertial system, the small common INS, a 20 lb. class model H-764, to the inertial survey and positioning application. Figure MH-l shows how these systems fit within the broad spectrum of a wide ranging product line.

Principal emphasis during the presentation, with regard to current technology, was on ring laser gyro systems, typified at present by the H-726 and in the future (2-3 years hence) by the H-764 systems. The principal advantage of these systems, in comparison to earlier generation conventional gyros, DTGs, ESGs etc., is in lower acquisition costs, much higher reliability, and lower costs of ownership - operation, maintenance and repair. GPS compatibility is another strong plus for systems like these, with incorporation of GPS receiver functions into the inertial unit to allow very tight integration of the two technologies for maximum benefits. Photographs of the H-726 DRU and the H-764C INU are shown in Figures MH-2 and MH-3, respectively.

Performance accuracy is not being overlooked, however, with the transition to RLG technology. High performance RLG systems, with accuracies approaching those of the earlier ESG systems, are typified by the Honeywell model H-774, shown in Figure MH-4. Just slightly larger than the H-726 and about 24% heavier, at 65 Ibs., it can be fitted with gyros and accelerometers that give performance rivaled only by the ESTG systems, but in a pure strapdown mechanization instead of the gimballed ones used by ESG and other gyro technologies.

Moving from the RLG technology update to the broader question of the overall technology outlook for inertial systems for the next five years, this presentation offered the following ten predictions:

1. Strapdown mechanizations will replace the gimballed inertial systems.

2. Optical gyros will predominate -- with ring laser gyros as the strong leaders, and the fiber optic gyros beginning to emerge but at lower accuracies.

3. Inertial units will be available in wider ranges of size, weight, power and performance.

4. Acquisition costs will reduce and then stabilize, compared to earlier technologies.

5. Ownership costs (operation, maintenance and repairs) will reduce.

6. Unique survey modifications (mostly software) must be "pay as you go" because of the generally small order quantities involved.

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7. RLG inertial unit reliability will continue increasing (thousands of hours MTBF will become commonplace).

8. INSs will be easier to use operationally.

9. Integrated INS/GPS will become commonplace. This will include both integral to one unit and use of separate units. Complimentary functions of INS and GPS in survey, mapping etc., will receive increased attention.

10. Integral computers will continue to make dramatic increases in computing power - both speed (throughput) and memory capacity.

These ten predictions are the challenges and food for thought offered to the KIS-90 audience for its' future consideration and planning in the field of inertial survey and mapping.

122

HI-PREC SID

STRATEGIC NAV

Honeywell HONEYWELL MILITARY AVIONICS RLG PRODUCT LINE

SNU-84-1 MAPS

FNU-85-1

USAF Fl

, @M C-130

USAF lAPS SPECIAL APPLICATIONS

~, Ml09 HOWITZER

TACTICAL AND TRANSPORT NAV MOTION

LAND NAV COMPENSATION

Figure 1.

Honeywell H-726 MAPS DRU

SIZE: 15" W x 8_75" H x 11" L

WEIGHT: 46.7 LBS

POWER: 24 VDC, 103 WATTS

Figure 2.

123

H-700-3

f\ CONVENllONAl

• TACTICAL MISSILES

RLG INSIAHRS

/::.

4!0-

$Mti~ 'I'}j

ARMY TACMS

HELICOPTER NAVI

GUIDANCE AND CONTROL

FIXED WING

Honeywell

GG1320 GYRO

Honeywell

SIZE: 13" W x 9" H x 10" L

WEIGHT: 65 LBS

POWER: 115V, 400 HZ 140WATIS

H-764 EVOLUTION

Figure 3.

H·774 HIGH PERFORMANCE RLG INS

Figure 4.

124

F3 (USAF/AR MV STANDARD INS)

BOARDS

H-764C INU

Trends in Inertial. system Developaent and

Application to SUrvey and Geodesy

J.R. Huddle Litton Guidance & Control Systems Division

5500 Canoga Avenue

Introduction

Woodland Hills, Cali£ornia USA 9~367 Tel 8~8-7~5-3264

Litton Guidance and Control Systems Division was a pioneer in the application of inertial techno~~ to surveying and geodesy beginning this development in 1966 '. Since that time several hundred· systems have been produced under a number of acronyms: PADS - position and Azimuth Determining System, ISS - Inertial Survey System, IPS - Inertial Positioning System, RGSS - Rapid Geodet~c survey System and LASS I & II - Litton Auto Surveyor System. The basic hardware employed in these systems uses floated, spinning-wheel gyros and fluid-filled and dry accelerometers with the instrument cluster being isolated from the three axes of vehicle angular motion by a high performance gimbal stabilization system. It has been well-established in the literature that this technology obtains positioning accuracy on the order of 5 to 10 ppm over traverses gre~~f than 10km and sub­decimeter accuracy over shorter traverses.' Further, for 1 to 2 hour traverses this equipment. has demonstrated ~avi ty vector measurement capability to the sub-arc-second level.

Inertial Systea Development Trends

The development of inertial systems naturally follows the technology that can be exploited for development of the gyros and accelerometers on which these systems are based. For the past three decades there has been persistent military and commercial market demand for inertial systems which have the following attributes:

• Less: Weight, cost, size, power and reaction (alignment) time

• More: Ruggedness, maintainability and reliability

while preserving a medium level of navigation performance in the one mile per hour error category. Besides this major market demand there also exists a limited number of applications which require a higher level of performance in the .2 mile per hour class.

125

As a consequence of these market reali ties, the inertial system industry has transitioned from fully-stabilized gimballed platform systems which have exploited the fluid-filled, pendulous­accelerometer technology and the floated spinning-wheel gyro technology of the late 1950's and early 1960's, to newer evolving technologies which are more compatible with these market demands.

Acceleroaeter Trends

Accelerometers for the most part are still mechanized as force rebalanced pendulums but have been miniaturized to the point where currently a full triad for medium performance applications is on the order of 70 grams. An example of this dry technology is depicted in Figure 1 which shows the Litton A-4 accelerometer triad.

Further accelerometer miniaturization will occur over the next decade as etched silicon chip technology is exploi ted to obtain cost reduction afforded by batch processing techniques. This technology which still employs the force rebalanced pendulum principle, potentially offers performance suitable for medium accuracy applications of inertial equipment.

Gyro Trends

Since the time of the spinning-wheel floated gyro, the inertial industry has seen the introduction of the dry, spinning wheel tuned-rotor gyro technology and the dithered ring laser gyro technology. The dithered ring laser gyro technology is now highly mature and serves along with the miniaturized dry accelerometer as the basis for the so-called "standard" strapdown inertial systems which are in a high rate of production in the current time period.

The spinning-wheel, floated and tuned-rotor gyros employed in gimballed inertial systems still remain in production but the volume of such production will rapidly diminish over the next few years. Small spinning-wheel, tuned-rotor gyros employed in lower quali ty strapdown systems for specialized applications such as antenna stabilization, aircraft attitude and heading reference systems (AHRS) and short-range missile guidance systems will remain in production for a longer period of time. However in general, the major thrust of the industry is to replace the spinning wheel gyro technology with optical gyro technology. This thrust is not only directed toward introducing smaller versions of the dithered ri~ laser gyro, but the newer di ther-Iess or Zero Lock Gyro ( ZLG) technology and also the fiber optic gyro (FOG) technology. Except for specialized applications requiring extremely high accuracy, these optical gyro technologies will be employed in conventional strapdown inertial system configurations.

126

An example of the dither less gyro technology unique to Litton is shown in Figure 2 which depicts the Litton 18.4 cm (5-18) ZLG which will be an element of future standard inertial systems. Since this gyro has no dead-band in its operating region as does the conventional ring laser gyro, it requires no dither mechanism. Consequently the acoustic noise generated by the dither mechanism which disturbs the accelerometer outputs and excites resonant modes in the instrument assembly is absent. An additional feature of the ZLG is that its resolution can be enhanced by special processing below the typical .3 arc-sec level by two orders of magnitude with no significant senescence.

A number of approaches to realizing a gyro with fiber optic technology have been examined over the past few years. At the present time ] it appears that the interferometric (IFOG) versus the resonant (RFOG) is preferred to obtain high performance levels. Further, closed-loop (versus open-loop) mechanizations to reduce sensitivity to variation in various parameters inherent in the gyro design and improve its dynamic range are required to realize the highest potential performance. An example of the IFOG technology that will challenge the ring laser gyro technology over the next decade is shown in Figure 3 which depicts a Litton gyro which achieves 1 degree per hour performance suitable for AHRS applications.

FOG technology is being aggressively pursued world-wide for several reasons. First it is extremely rugged as exemplified by successfully demonstrated systems to be used for guiding gun­launched smart projectiles. Further the technology is inherently simpler and consequently more reliable than ring laser gyros with significantly reduced manufacturing cost. When FOG is employed with the etched silicon chip accelerometer, complete inertial measurement units should be available for less than $30,000 in current year dollars as high volume production is realized.

The Next Generation standard Inertial System

In the parlance of the inertial ~ystem industry, the "standard" inertial navigation system 1.S one which has a performance capability of .8 nautical miles per hour. The next generation of this fully-militarized equipment which will soon enter high volume production is shown in Figure 4. This system which is called the Litton LN-100, will have a weight of less than 20 lbs, a size of 7x7x10 inches, consume 40 watts and require no external cooling. This system is currently in full scale development and scheduled for installation in existing military vehicles.

127

Future Generation standard Inertial Systems

As a function of the progress in the development of closed loop IFOG, future generations of standard inertial systems will become available with IFOG and miniaturized accelerometers. This evolution will take place by first introducing conventional strapdown versions of these systems into the market place for application to AHRS, stabilization of antenna and other sensors and short range missile guidance. An example of this class of system is shown in Figure 5 which depicts a prototype version of the Litton LN-200 which is currently being developed as a c'Hdidate for a number of near-term, high volume production programs. Once such production is realized, a number of specialized applications discussed at this symposium such as machinery alignment, remote vehicle guidance, pipeline curvature measurement and control of drilling, tunneling and other special tools will be supported.

Litton is currently involved in a funded development program for the Global positioning System Guidance Package (GGP) whose specific long-term objective over the next few years is to realize a strapdown system using closed loop IFOGs and silicon accelerometers capable of achieving .8 mile per hour performance. The system when packaged with a multi-channel GPS receiver is to obtain a size of less than 100 cubic inches, a weight of 6 pounds, consume 30 watts and require no external cooling.

Application of Inertial Technology to surveying and Geodesy

The geodesy and surveying community has in the past presented the inertial navigation system industry with challenging requirements due to the extraordinarily high accuracy that this type of work requires relative to most commercial and military applications. However this market is relatively small and consequently if it is to be served, it must rely on the industry developments which are driven by the large military and commercial markets which support high volume production. Over the past two decades the high accuracy needs for position and gravity vector survey have been met by the gimballed inertial system using floated, spinning-wheel gyro technology. This was possible because the u.s. Army Positioning and Azimuth Determining System (PADS) entered volume production. Since this technology does not permit low cost manufacturing, the use of inertial technology by the surveyor to date has been limited.

Reviewing the current developments in inertial technology discussed in this paper in conjunction with the advent of full operational capability of the Global Positioning System, one can

128

operational capability of the Global Positioning System, one can speculate on what the future may hold for the surveyor-geodesist. From the papers and discussions that have taken place at this symposium, it appears two classes of inertial equipment are of interest to the surveying community.

Low to Xedium Accuracy Inertial systems

The first class of inertial systems that is of interest to the surveyor-geodesist is of the low to medium level of accuracy with attendant implied lower acquisition cost than formerly available. Such a system would be employed in a tightly-coupled fashion with a GPS receiver operating in the (differential) carrier phase tracking mode. Conventional strapdown systems employing optical gyro and miniaturized accelerometer technologies should adequately serve applications which permit this mode of operation. There is however a caveat in this conclusion. Strapdown systems inherently have greater transient error behavior than gimballed systems using equivalent performance inertial instruments, due to the exposure of the instruments to vehicle angular motion. This characteristic coupled with low to medium accuracy inertial system performance may not be sufficiently effective in cycle-slip repair to serve the practical needs of the position surveyor. with this caveat in mind, the integrated GPS-Inertial equipment being developed for the GGP should be an excellent candidate for a total system solution over the long-term. For the nearer-term, the medium accuracy LN-100 and lower accuracy LN-200 systems will be candidates for such applications.

High Accuracy Inertial Systems

The second class of inertial systems that is of interest to the surveyor-geodesist is of the high accuracy performance class. Such systems are useful for two purposes:

• For enabling repair of cycle slips during GPS carrier phase tracking for greater time periods than obtainable with low or medium accuracy inertial equipment used for position survey

• For realizing airborne survey systems which are capable of measuring the gravity vector to better than 1 arc-second

"High accuracy" in the parlance of the inertial system industry generally refers to systems capable of achieving .2 mile per hour or better performance for aircraft navigation. Such a

129

characteristic is somewhat simplistic in that it pertains more to gyro performance than to accelerometer performance. This follows as accelerometer errors tend to create zero-mean Schuler oscillations in velocity error which cause slower divergence in position over the long-term (hours) than gyro drift rate which creates non-zero mean velocity errors. In fact in inertial surveying, the requirement for high performance accelerometers is quite critical as acceleration measurement error creates velocity and position errors more rapidly over the short-term (minutes) than do gyro drift rates. Further measurement of the gravity vector to the sub-arc-second level has direct implications as to the effective performance of the accelerometers in the system. Consequently while conventional strapdown inertial systems may improve over time to the point where .2 mile per hour performance in position error divergence is obtainable over the long term, such systems may not be suitable for position survey and geodesy applications demanding the highest accuracy. This leads us to the consideration of more sophisticated system mechanizations which incorporate the accelerometer and optical gyro technology that is now or will be in high volume production in the future.

A prototype system that has been developed and demonstrated for high accuracy applications called the LN-94R. The inertial instruments employed in this system are standard 28cm conventional ring laser gyros and A-4 accelerometers. This system is "rate­biased" which means the instrument cluster is rotated in a continuous manner about the vertical or vehicle yaw axis as shown in Figure 6. The rotation about this single axis achieves two objectives:

• Translates low frequency inertial instrument noise errors which reside in the plane orthogonal to the rotation axis to high frequency noise such that their integrated effect on position and attitude error is reduced

• Permits operation of the ring laser gyros without dither as the rate bias rotation rate is high enough so that the zero-rate dead zone is not encroached. This feature eliminates gyro random walk due to dither and permits the optimum quantum noise source limit of random walk for the gyro to be achieved. It further eliminates error in the accelerometer outputs induced by the high frequency mechanical dither associated with dithered gyros in conventional ring laser gyro strapdown system mechanizatons

While the LN94R has implemented and demonstrated the feasibility of a novel approach to high accuracy system performance

130

. t' I . I [8.9] f th' t' uSl.ng conven l.ona rl.ng aser gyros, ur er l.mprovemen s l.n performance still are obtainable. For example a more sophisticated approach could employ ZLG's on a fully-stabilized platform. Such an approach has a number of attributes:

• No di ther induced noise in theYM gyro or accelerometer outputs since the Zero Lock Gyros require no dither

• Complete isolation of the instrument cluster from transient vehicle angular motion obtaining elimination of consequent transient errors induced by such motion. Note since the instruments employed are capable of operating in strapdown environments, the requirements on the stabilization system are not particularly stringent which enables the use of a stabilization system with lower performance and cost

• virtual elimination of quantization noise in the gyro outputs by use of enrancement techniques which are obtainable with the ZLG

• Translation of low frequency accelerometer and gyro noise errors to high frequency by controlled rotation about one or more axes of the gimbal system as is achieved by rotation about the vertical axis in the LN94R. Note again since the gyros employed are capable of operating in the full dynamic range of a strapdown environment with high accuracy, they are much more tolerant of such controlled motion than spinning wheel gyros whose performance degrades under such condition for a variety of reasons such as bias variation due to heating under high torquing rates, drift sensitivity to change in orientation with respect to the force of gravity, relatively poor scale factor, etc.

The technology for this latter type of system is available today and has been demonstrated to a significant degree. Whether such a system will become available to the surveying and geodesy communi ty remains an open question for the future. The answer depends on the extent of the total market for a system whose performance potential is so high.

131

Reference.

1. Clark, W.J., "Final Technical Report on the Application of Inertial Techniques to surveyinq." USAETL Report AQ22107E66, Litton G&CS, Woodland Hills, CA, Nov. 1966.

2. Huddle, J.R. and Mauqhmer, R.W., "The Application of Error Control Techniques in the Desiqn of an Advanced Auqmented Inertial Surveyinq System," 28th Annual Meeting of the Institute of Navigation, West Point, N.Y., 1972.

3. Huddle, J .R., "Historical Perspective on Estimation Techniques for Position and Gravity Survey with Inertial Systems", Proceedings of the 3rd International symposium on Inertial Technology for Surveying and Geodesy, Banff, Canada, September, 1985, Pp. 215-240.

4. Rueqer, J .M., "Hiqh Accuracy in Short ISS Missions", Proceedings of the 3rd International Symposium on Inertial Technology for Surveying and Geodesy, Banff, Canada, September, 1985, pp. 579-592.

5. Huddle, J.R., "The Rapid Geodetic Survey System (RGSS)", Chapman Conference on Progress in the Determination of the Earth's Gravity Field, Fort Lauderdale, Florida September, 1988, Pp. 60-63.

6. Pavlath, G.A. and Suman, M.C., "Fiber optic Gyroscopes: Advances and Future Developments", Journal of the Institute of Navigation, Vol. 31 No.2, Summer, 1984, Pp. 70-83.

7 • Pavlath, G. A. and Klemes, M. S., "Production Development of Small Fiber optic Gyros", 46th Annual Meeting of the Institute of Navigation, Atlantic City, New Jersey, June, 1990.

8. Matthews, A. and Welter, H., "Cost-Effective, Hiqh Accuracy Inertial Naviqation", Institute of Navigation Technical Meeting, San Mateo, California, January 1989

9. Huddle, J .R., "Advances in Strapdown Systems for Geodetic Applications", High Precision Navigation, K. Linkwitz and U. Hanqleiter editors, Sprinqer Verlaq Berlin, 1989, Pp. 496-530.

132

FIGURE 1. LITTON GCS A·4 ACCELEROMETER TRIAD

FIGURE 3. LITTON GCS INTERFEROMETRIC FIBER OPTIC GYRO (I FOG) FOR

AHRS APPLICATIONS

FIGURE 5. PROTOTYPE OF LITTON GCS LN·200 IFOG INERTIAL SYSTEM

133

FIGURE 2. LITTON GCS 18.4·CM S -18 ZERO­LOCK LASER GYRO (ZLG)

FIGURE 4. LITTON GCS LN-1 00 ZLG PROTOTYPE INERTIAL SYSTEM

FIGURE 6. LITTON GCS LN·94R RATE·BIAS RLG INERTIAL SYSTEM

ROTATING SENSOR ASSEMBLY

TRENDS IN INERTIAL TECHNOLOGY AND SYSTEMS FOR KINEMATIC GEODESY

INTRODUCTION

Loic Camberlein

SAGEM 27, Rue Leblanc

75512 PARIS Tel (1) - 40 70 63 63 Fax (1) - 40 70 63 S2

The analysis of trends in inertial technology and systems must, of course, be made keeping in mind the applications of concern, i.e. kinematic geodesy at large including photogrametry, gravimetry ... , and the specific characteristics of inertial systems with respect to these fairly special applications.

Inertial systems have the unique caracteristics among other positioning equipment of being based on very accurate and relatively high passband measurement of rotations (with gyroscopes) and specific forces (with accelerometers). They are also basically self-contained and particularly well suited to kinematic operation. They can be used either stand-alone or combined with other equipment when complementary characteristics or measurements are required. This is the case, for example, in kinematic geodesy when inertia and GPS combining may be necessary to ftlter out severe satellite masking and cycle slip occurences. This may also be the case in gravimetry where specific force and absolute acceleration must be very accurately measured independandy.

Therefore trends in inertial technology and systems must be analyzed through:

1) trends in inertial sensors, i.e. gyroscopes and accelerometers, and 2) trends in inertial systems,

keeping in mind the specific practical requirements of kinematic geodesy in terms of operation, performance and price.

This paper is a contribution to this analysis.

TRENDS IN INERTIAL SENSORS

Gyroscopes

Gyroscope technology is a lively research and development area because of the variety of applicable principles and technics. As an example, figure l.a. shows a family of inertial grade gyroscope technologies which include floated gyros, electrically suspended gyros, dry-tuned gyros, ring laser gyros and fiber optic gyros. According to their specific characteristics these gyroscopes are used in different types of systems for space, aircraft, marine or land applications.

134

Floated Fluid damped

Electrally suspended

Gas damped

Dry-tuned

P.I.G .A.

Ring laser

Vibrating beam

Fibre-optic

Figure 1. SAGEM's families of inertial grade a) gyroscopes b) accelerometers

135

If the scope of interest is reduced to kinematic geodesy and to its main requirements in terms of operation, performance and price, then in brief, floated gyros begin to be outdated, electrically suspended gyros, which are the most accurate gyros, are restricted to strategic applications, and fiber optic gyros are emerging and not yet of an adequate level of performance.

However dry-tuned gyros (DTG) are in full production and have steadily improved and reached an excellent performance stability. This is the case of the S040 gyro used in the ULISS systems. Recent tests over a batch of 35 of the latest evolution of this gyro have demonstrated a random drift of about 0.0004°/h rms over 5 minutes and of about 0.001°/h rms over a few hours. Moreover the random drift of such DTG is low and about 0.OOOO5°/vh.

The ring laser gyro (RLO) technology has matured over the last fifteen years and its performance has considerably improved, especially in random-walk as the mirror technology has been refined. Sagem's current family of RLGs includes the OSL 32 with a triangular cavity of 32 cm, the OLC 16 and the OLC 8 with square cavities of respectively 16 and 8 cm. Only but the OSL 32 is adequate for high performance geodesy with a drift rate of 0.001 ° /h and a random-walk below O.OOI°/vh with the very high quality mirrors manufactured by Sagem.

Accelerometers

Accelerometer technology is also diversified and evolving as shown in figure 1.b. which presents a family of inertial grade accelerometers.

Here again if the scope is restricted to kinematic geodesy, then in short, fluid damped accelerometers are aging and PIOAs, which are the most accurate accelerometers, are restricted to strategic applications. However gas damped accelerometers are in full production and have been regularly perfected. For example, the bias short term stability of gas damped accelerometers for gimballed and thermally controlled systems has significantly improved, as well as the bias long term and thermal sensitivity of gas damped accelerometers for strapdown and not thermally controlled RLO systems. Thus the A310 accelerometer of the ULISS gimballed system demonstrates in its latest evolution a bias stability of about 1 micro g over a few hours in a thermally controlled environment, while the A600 accelerometer for strap-down systems, such as SlOMA, shows a residual bias of about 5 micro gs after thermal corrections.

The vibrating beam accelerometer (VBA) technology is emerging and in limited production. Its demonstrated excellent performance, relatively simple electronics and low cost are very promising for new systems.

TRENDS IN INERTIAL SYSTEMS

The variety of inertial systems is as broad as the variety of applications and gyro/accelerometer technologies. As an example, figure 2 shows Sagem's family of inertial systems limited to the two types of applications - land vehicules and aircraft - which are of particular interest for geodesy. In this figure two technologies are also represented : dry-tuned gyro gimballed systems and ring laser gyro strapdown systems.

136

NSM 20 (a)

G~BALLEDSYSTEMS

SIGMA RL (b)

ULISS (b)

STRAPDOWN SYSTEMS

Figure 2. SAGEM's family of inertial systems for land vehicles (a) and aircraft (b)

137

ULISS 30 (a)

SIGMA 30 (a)

GIMBALLED SYSTEM OR STRAPDOWN SYSTEM?

For high accuracy geodesy the type of system - gimballed or strapdown - is not without importance for several well known reasons summarized in Table 1. In every case mentionned in this table the residual error is smaller in a state-of-the-art DTG gimballed system than in a state­of-the-art RLG strapdown system.

GIMBALLED SYSTEM STRAPOOWN SYSlEM

- Temperature stabilized gyrosIaccelemeters - No thermal stabilization of gyros/accelerometers

- Modelizationlcorrection of gyros/accelerometers' temperature sensitivity

- Attitude stabilized gyros/accelemeters - Strapdown gyros/accelemeters - Modelizationlcorrection of

gyros/accelemeters' dynamic errors - Attitude numerical integration (coning,

sculling errors)

- DTG random walk: < 0.0001 °/vh - RLG random walk: ... O,(X)1 0 Iv h

Table 1 : For high accuracy geodesy, significant differences between gimballed systems and RLG straptdown systems lead to smaller residual errors in favor of gimballed systems.

This is why a gimballed system like ULISS 30 (see figure 3) is basically preferable for high accuracy inertial geodesy, see (1) and (3). Furthermore its recent design and technology provide a) small size, weight and power needs, b) excellent reliability and ease of operation, see (2), and c) optional integrated Total Station and GPS, see (4) and (5).

REFERENCES

(1) 1.M BECKER & M. LIDBERG, Experience with the ULISS 30 System, KIS 90.

(2) 1.M DOIZY, Geodine 30, a Real-Time Surveying System for Land Geophysical Survey, KIS 90.

(3) R. FORSBERG, Experience with the ULISS 30 Survey System for Local Geodetic and Cadastral Network Control, KIS 90.

(4) F. MAZZANTI & C. de K. FUENTE, ULISS30, a New Generation Inertial Survey System Enhancing Flexibility and Security of Operation, Perfonnance and Reliability, KIS 90.

(5) P. LIORET, Inertial + Total Station + GPS for High Productivity Surveying, PLANS 89.

138

Figure 3. ULISS 30 inertial survey system with optional integrated Total Station (picture) and GPS

139

STRAPDOWN INERTIAL SURVEYING

FOR

INTERNAL PIPELINE SURVEYS

Paul L. Hanna, Michael E. Napier and Vidal Ashkenazi

Institute of Engineering Surveying and Space Geodesy University of Nottingham

UK

1 Introduction

The oil, gas, water, sewerage and chemical industries all operate vast networks of under­ground services to transport their products. The efficient and economic effectiveness of these services is supported by a continual program of pipeline inspection, monitoring and subsequent maintenance. A variety of inspection methods such as echo sounding, ultrasonic techniques, radiography and the use of television cameras can determine considerable infor­mation describing pipeline condition by mounting the necessary sensors on an instrumented vehicle propelled by the product flow through the pipeline. Fault location, an integral part of the pipeline investigation, is determined by the positioning capability of the inspection vehicle.

The various industries are obliged legally to carry out periodic inspections of their pipeline networks. The economic, political and environmental consequences of not detect­ing potential hazards, such as pipeline corrosion, are all too great and have been heightened by the recent ecologically-aware climate. This trend is likely to continue with the result that pipeline maintenance and monitoring will take on increased importance. The limitations of the current vehicle location techniques are such that any improvement would benefit greatly the pipeline industry. The possible improvement in pipeline inspection vehicle location by the application of engineering surveying techniques formed the basis of a PhD research pro­gram carried out by the author whilst a member of the Department of Civil Engineering and the Institute of Engineering Surveying and Space Geodesy at the University of Nottingham, UK.

2 Pipeline Pigging

Pipeline pigging! refers to the passage of a vehicle, the so-called pig, through a pipeline where the driving force is provided by the differential pressure of the product flow across the length of the pig. Pigs are constructed in a variety of sizes to inspect pipelines of diameters 8-42 inches (~ 200-1050 mm). Intelligent pigs incorporate various sensors to allow an analytic approach to be made to defect inspection and repair assessment. British

lsee, for example, Scott [9].

140

Product Flow ----

! Odometer wheels

! I

Sensors Plastic seals

Figure 1: The British Gas Intelligent Pig

! Support wheels

Gas pIc have been at the forefront of intelligent pigging and have perfected their metal-loss pig, which relies on a magnetic flux leakage device to record a three dimensional fingerprint of the entire pipewall as the pig moves. By comparing off-line the depth, axial length and circumferential width of the detected feature with a library of known fingerprints, the structural significance may be assessed thereby allowing any required remedial action to be determined. The British Gas intelligent pig, complete with inertial navigation system as described later, is illustrated schematically in Figure 1

Pig positioning, necessary to locate detected features, is reviewed by Kershaw [6]. Most intelligent pigs incorporate one or more odometers (velocity wheels) which run along the internal surface of the pipeline giving an indication of the pig's longitudinal velocity. By numerically integrating this velocity, the pig's displacement along the pipeline relative to some reference point may be estimated. A further pig locating technique is ti detect the vibrations caused by the passage of the pig past the pipewelds and other known features using geophones clamped to the exterior of the pipeline. Provided that detailed as-laid survey information exists, then the pig may be continually tracked throughout its inspection and British Gas research has determined that ranges of up to 80 kilometres are attainable by such techniques.

3 Limitations of Pig Location Methods To examine a fault detected in a large diameter pipeline buried with 1 metre of cover, it is usually necessary to excavate a three-metre length of pipe to give full access. Consequently the ability to detect features to within ±1.5 metres is desirable to avoid unnecessary ex-

141

•

cavation. This is a stringent positioning requirement considering that many pipelines are over 300 kilometres long. Gas pipelines are usually laid from nominal twelve metre pipe lengths welded together and so the weld counting technique described above could locate the pig to, at best, the correct twelve metre pipelength. The odometer-derived displacement is accurate at most to about 1 % of the total distance travelled from a known feature and to achieve the afore-stated accuracy, these features would be required every 150 metres, which would be difficult in many pipelines since even the best available as-laid surveys are not usually sufficiently detailed.

A serious limitation of these positioning techniques is that they offer only a uni-axial indication of position and particularly in long sections of non-straight buried pipeline, it is difficult to ascertain the accurate three dimensional location of the pig. It should also be emphasised that the as-laid surveys for many older pipelines are often inaccurate and so in these cases the weld counting technique is an inadequate position determining tool.

There is also a requirement to determine pipeline flexure and, in particular the migration and spanning of pipelines laid in offshore environments over periods of time. This informa­tion is necessary so that the operator can predict potential pipeline failure and assess any preventative action necessary. It is clear, therefore, that any improvement in pig positioning capability would be of great benefit to the pipeline industry.

4 The Inertially-Aided Pig

The problems of accessibility severely restrict the range of conventional survey techniques which can be used to improve the pig's positioning. The autonomous property of inertial navigation, however, would seem to provide a potential solution. An inertial navigation system (INS) mounted onboard a pipeline pig could be used not only to compute the three dimensional position of the pig throughout the survey, but also to estimate the pig-to­inertial space attitude. With suitable pig-to-pipe attitude information, the attitude of the pipe relative to inertial space or, in practice, relative to some geographic reference frame could be estimated throughout the survey. It should be noted that, unlike in most inertial surveying applications, the carrying vehicle, the pig, cannot be stopped once the survey has commenced and so zero velocity updates are not available to damp the pig's velocity errors. The information necessary to damp the propagated errors inherent in inertial navigation could, however, be provided by the continuously available odometer velocity measurements.

The space available onboard a pig to package an INS and associated data handling equipment and power supplies is very small and even for a 36 inch diameter pipeline pig corresponds to a cylindrical cavity of about 10 inches diameter by 36 inches long. These technical specifications led to the choice of strapdown inertial sensors and since real time positioning is not required, then a strap down INS which could be used to record the inertial measurements for subsequent off-line processing was an ideal choice. A further requirement of this research was to investigate the possibility of using a cheap low-grade INS in an effort to reduce the cost of a complete strap down inertial surveying system (SDISS).

5 Computer Simulation Description

The large costs associated with pipeline inspection necessitated a feasibility study approach to the investigation and so the means of improving the positioning capability of a pig by the integration of an inertial navigation solution to odometer velocity measurements was investigated by computer simulation. The formulation of the computer simulation is described in detail in Hanna [4] and in component form, the simulation package may be

142

considered to comprise

1. Data Generation Stage. By defining typical pipeline topography and pig dynamics, the measurements typically recorded by the inertial sensors (gyroscopes and accelerome­ters) and the odometer wheels were generated. The facility to corrupt these data and thus include the effects of the major sensor error sources2 was also made available. The advantage of carrying out the entire investigation by simulation was that the data generation stage allowed the user access to the, by definition, absolute naviga­tional truth of the pig throughout the survey. This information is never available in practice. In addition, the flexibility of the computer simulation package allowed a variety of different surveys to be investigated which could not have been be achieved in practice.

2. Data Processing Stage. This stage processes the inertial measurements to compute the navigation quantities3 of the pig. By combining this navigation solution with the odometer-indicated velocity data an optimum navigation solution is envisaged.

3. Analysis Stage. This stage comprises a set of data handling and evaluation algo­rithms which can assess any improvement in pig location and attitude determination introduced by integrating the inertial navigation system to the odometer.

It should be noted that gyroscopes have been used for several years to enhance the attitude determination of pipeline pigs and to estimate pipeline flexure between successive surveys. The recently developed inertial geometry pig by Pulsearch of Calgary, (Adams, St John Price & Wade [1]), has shown that three dimensional position and attitude information can be successfully computed from an inertial navigation system mounted onboard an oil pipeline pig. The necessary error bounding measurements are provided by almost continuous high-accuracy velocity measurements as well as the weld counting technique which gives a frequent measure of the pig's chainage. The Pulsearch pig can be located to better than ten metres and radii of curvature of up to 100 metres or more may be determined.

The research undertaken by the author at Nottingham was to ascertain whether an INS could enhance the positioning capability of a pig in situations where the only external information available was in the form of odometer velocity measurements and occasional position fixes. Consequently the weld counting technique was not investigated. In addition, the positioning specification was set to ± 1.5 metres so as to minimise pipeline excavation, as stated earlier. A peculiar feature of the gas pipeline pigs examined in this research is the constant rotation of the pig about its longitudinal forward-facing axis throughout the inspection. Gas pipeline pigs are also subject to high levels of shock and acceleration caused by the vehicle striking pipewelds at high speed whereas oil-damping creates a more benign dynamic environment for oil pipeline pigs.

6 Data Processing Algorithms

6.1 Strapdown Processing

The strap down inertial data processing comprises two main calculations: the attitude de­termination and the navigation determination. The attitude calculation is peculiar to strap­down inertial systems and involves the frequent updating of the attitude rotation matrix

2these errors included fixed scale factor errors, biases and sensitive axis misalignments as well as random time-varying biases for the accelerometers, gyroscopes and odometers.

3 velocity, position and attitude.

143

which relates the body and geographic frames. This calculation was based on the split-cycle integration algorithm presented by Savage [8]. The navigation calculation which is generic to inertial navigation was based on equations presented in Britting [2].

At the start of the data processing stage, the attitude rotation matrix was initialised in a least squares alignment procedure. This involved processing data collected over a short period whilst the pig was stationary prior to the survey proper. In this research it was assumed that some form of sophisticated sensor calibration process had already been carried out and so the sensor errors modelled in the data generation stage referred to the residual post-calibration uncertainties associated with the calibration. Consequently only a simplified least squares alignment procedure was deemed necessary to initialise the rotation matrix.

6.2 Error Modelling

Kalman filtering techniques were included in the data processing stage to mix the inertial and the odometer navigation solutions in a stochastically sensible way to optimally model the pig's navigation errors. The Kalman filter mechanisation equations are presented in Gelb [3] and the inertial navigation error equations are derived in, for example, Wong [11]. The linear first order differential equation to be solved has the form

X(t) - F(t)x(t) + G(t)W(t) (1) where

x(t) = the system 'state vector

w(t) - the forcing function

and F(t) & G(t) are the necessary formulation matrices. The state vector x comprises a set of parameters which fully describe the unforced motion of the system being modelled and the vector W comprises all the perturbing forcing effects. In this research, the Kalman Filter was formulated to model the pig's geographic-referenced attitude, velocity and position error vectors as well as the uncalibrated gyro drift states. The state vector of parameters to be estimated comprises:

where

x = {<PN' <PE, <PD, 8VN, 8VE, 8PN, 8PE, 15k, 8h, €x, €Y' €z} T

<P N, <P E, <P D - the geographic referenced attitude error

8V N, 8VE - the geographic referenced north and east velocity errors

8PN,8PE - the geographic referenced north and east position errors

8k,8h - the vertical velocity and corresponding altitude errors

€x, €Y' €z - the body referenced residual gyro drift

(2)

and the F matrix is presented in Figure 2. The forcing function and associated formulation matrix are derived in Hanna [4].

6.2.1 The Measurements The external measurements necessary to update the state vector estimate and its covariance matrix have the form

(3)

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and comprise vectors of velocity differences between the odometer-indicated velocity and the inertially-computed velocity. Although the odometer indicates the pig's forward velocity, with the assumption of zero lateral motion, a three dimensional velocity measurement is available. Occasional position measurements, which comprised position differences between the three dimensional coordinates of known control points and the inertially-computed position, were also used to update the state vector. The formulation matrices H and the measurement covariance matrices4 are derived in Hanna [4] for both types of update. In addition, the Rauch- Tung-Striebel smoothing algorithm described in [7] were implemented to make full use of the available measurements.

6.2.2 Suboptimal Filter Design

Although the coding of the filter mechanisation equations was straightforward, the filter's fine-tuning and performance analysis required a major effort. The pig's constant rotation rate, although small when compared to that of a missile, significantly increases the computer processing burden since the attitude-dependent matrices F and G have to be frequently updated to take account of the vehicle's attitude change. To obtain maximum accuracy from the filter error estimate, these matrices were recomputed at each filter prediction point. Consequently, simplifications were made in other parts of the filter5 where possible to enable the processing stage to be executed over a practicable time scale. In particular, most of the error sources were lumped together and modelled as white noise processes in the forcing function. Additional noise was included to account for inadequacies in the model and thereby desensitise the filter. Tests to determine the various update rates and the allocation of values to the various parameters are discussed in detail in Hanna [4].

6.2.3 Adaptive Filtering

Adaptive filtering can give a useful insight as to whether a filter is functioning optimally and is based upon the properties of the innovations sequence. If after a measurement update at epoch k, X( +) is the true optimal estimate, then there should be no information available in the residual vector v, where

(4)

The collection of these vectors from the sequential measurement epochs is referred to as the innovations sequence which, for an optimal filter, is a zero-mean Gaussian white noise sequence. The innovations sequence was inspected at each measurement update and the local and global slippage tests suggested by Teunissen & Salzmann [10] were implemented to try and determine whether the filter was behaving optimally.

7 The Inertially-Derived Navigation Solution Results from two main tests are presented which compare and contrast the pig navigation accuracy attainable by the integration to the velocity measurements of a low-grade and a high-grade strapdown inertial navigation system. The simulations centred on the inspection

4necessary to describe the errors associated with the measurements. 5 and so a suboptimal filter in which simplifications were made was designed.

146

Error Parameter I Units I High-Grade Low-Grade

Gyroscope data Scale factor ppm 5 500 Misalignment angle arc seconds 3 100 Constant drift °/h 0.01 1 Drift spectral density (O/h?/h 4 x 10-6 1 X 10-5

Drift correlation time s 0.05 3600 Measurement noise °/h - 0.025 Pulse size arc seconds 1 0.1 Accelerometer data Scale factor ppm 50 500 Misalignment angle arc seconds 3 100 Constant bias m/s2 0.0005 0.01 Bias spectral density (m/s2)2 Is 1 X 10-13 1 X 10-11

Bias correlation time s 3600 3600 Measurement noise m/s 2 2.4 X 10-6 2.4 X 10-5

Pulse size m/s 0.0005 0.0005 Odometer data Scale factor ppm 5 100 Misalignment angle arc seconds 20 120 Bias spectral density (m/s)2/s 1 X 10-11 4 X 10-6

Bias correlation time s 3600 3600 Measurement noise m/s 0.001 0.015

Table 1: Sensor Error Parameters

of a fifteen kilometre south-north oriented pipeline through which the pig travelled at a. speed of about 3 metres per second over a period of one hundred minutes. The pig's roll rate was set to 0.011 radians per second (about one revolution every ten minutes) which is commensurate with typical pig dynamics.

Table 1 presents the sensor error parameters modelled in the data generation stage and it should be noted that these quantities refer to the residual post-calibration error sources and do not represent any particular sensors commercially available.

The low-grade gyroscope's error behaviour were characterised by the random walk drifts which would be expected of conventional spinning-mass gyroscopes. The high-grade gyro­scopes were modelled as ring laser gyros which are characterised by the short correlation times of their random drifts. Typical accelerometer error information is generally not well documented and stochastic data are particularly scarce. In both grades of accelerometer, the time-varying bias was modelled by a random walk profile of the same form as that mod­elled for the low-grade gyro drift. The parameters describing the systematic error sources were abstracted from the relevant literature and are supposedly representative of typical component performances. The odometer error model was chosen to yield a position error which varied with time but which was accurate to about 1% of the total distance travelled from the start of the survey. In addition, a much higher grade of odometer was investi­gated to assess the improvement in pig positioning when the available velocity data were essentially error-free.

Raw navigation errors were derived by differencing the INS computed navigation quan-

147

Residual Error High-Grade INS Low-Grade INS

(metres) mean q mean q

Latitude hI{) 3.26 2.18 30.42 42.30

Longitude h>" 3.29 3.65 97.01 81.28

Altitude hh 2.20 1.60 9.99 7.39

The standard deviation u, relates to the standard deviation of the sample of residual errors.

Table 2: Residual Position Errors (Low-Grade Odometer)

tities with the known truth values from the data generation stage. The raw position errors were of the order of tens of kilometres (hundreds of kilometres for the low-grade INS) over a period of 100 minutes, which illustrates the very limited use of inertial systems over ex­tended periods of time. The usefulness of the simulation approach is that these error profiles were available throughout the survey and not just at control points.

8 Results

By differencing the smoothed Kalman Filter error estimate with the raw errors, a residual navigation error which described the accuracy of the integrated navigation system was available. In addition, the state vector's smoothed covariance matrix gave a statistical measure of the quality of the fix. Since small variations in many of the numerous parameters of the simulation will lead to a variety of different solutions, attention is focussed upon the trends introduced by varying the key parameters.

Figure 3 illustrates the residual latitude errors6 for different combinations of INS and odometer data. These results were obtained by processing velocity updates at a rate of one measurement per minute and high-accuracy position fixes every twenty minutes. The first graph shows that the combination of low-grade INS/low-grade odometer brought about a large reduction in latitude error from an initial value of several hundred kilometres to generally below the one hundred metre level. The second graph for the combination of high-grade INS/low-grade odometer illustrates that the smoothed latitude error is seen to be generally below the ten metre level. This better error modelling is expected since the unmodelled error sources in the Kalman filtering stage have a much smaller effect than for the low-grade INS.

Table 2 illustrates the difference between the two grades of INS for the low-grade odome­ter data and although it may be inferred that the high-grade INS can be located to the single pipelength level (< ± 6 metres), the standard deviation of the Kalman filter latitude error estimate rose to fifteen metres between the position fixes and so further error mod­elling improvements are required. For the low-grade INS, the residual position errors and associated standard deviations are very large and of little use for the intended application. It was noted, however, that the initial instability of the smoothed residual errors formed a major component of these large average errors which is discussed later.

Investigations were carried out to weight the smoothed residual error estimates in ac­cordance with their computed precisions. This analysis proved to be too sensitive to small

6in this case, these are the along-track errors.

148

250

e 200

I 150 A o -g 100 A ~

~ 50 j

-50

e -1 I o -20

"t1 3 -30 ~ j -4-0

-50

-60

300

E 250

1 200

~ 150

3 100 ~

30,

Filtered Curve Smoothed Curve

mins

lcw-gracE lNS/Lo.'l-gracE od::rreter

Hi~~-r.rade !NS/La~-gra(e odometer

~.(7{!-gracE INS/High-grare od:m:!ter

:3 50

.~~~~~~~~ -50

8

E 6 1 04-

"t1

Time - mins

Hich-gracE INS/High-grade oc'cm:~ter

32 ~ ~~--~--~--~~~--~~~~--~~~~

30

-2

Figure 3: Residual Latitude Errors

149

fluctuations in the initial values of the filter parameters and it was concluded that mislead­ing information was being computed. It is intended, however, to determine the covariance matrices between different points throughout the survey in an attempt to derive further information indicating the quality of the navigation solution.

The lower two graphs illustrate the residual latitude errors for the two grades of INS combined with the high-grade odometer. In addition to significantly improving the velocity error modelling between position fixes, the use of effectively error-free velocity data also improves the position error modelling. It is clear, however, that the unmodelled error sources of the low-grade INS are still preventing the integrated navigation system from attaining the desired accuracy. For the high-grade INS, the improvement in position error modelling was significant and the residual latitude error is observed to consist of a Schuler oscillation with an amplitude of less than three metres. More significantly, the root mean square value of the maximum uncertainty associated with the filter error estimate is 4.62 metres indicating that the single pipelength precision has been obtained.

Three techniques are now examined to improve the positioning accuracy of the inertially­aided pig:

1. In the previous tests, the velocity difference measurements have been integrated into the Kalman Filter at the fairly slow rate of one measurement per minute. It was observed that trebling this update rate, for example, brought about modest improve­ments in the residual errors' mean values and associated standard deviations. This improvement was, however, at the expense of a large increase in the necessary com­putations needed to process the additional measurements and consequently it was decided not to employ a rate faster than the currently adopted rate of one measure­ment per minute.

2. Although the position measurement interval of twenty minutes corresponding to ap­proximately three kilometres moved by the pig is a relatively large interval for inertial surveying, more frequent accurate position information is often not available. The position fix interval was reduced to ten minutes, however, and it was observed that the residual position errors of the low-grade INS were reduced by a factor of three, irrespective of the quality of the odometer data. There were also significant position­ing improvements for the high-grade INS. Increasing the frequency of position fixes will soon reach the point, however, where the mixture of position and odometer data becomes sufficient to locate the vehicle to the required accuracy and due to the general unavailability of known points along the pipeline, this technique was not examined further.

3. The tests were repeated with a non-rotating vehicle to examine the effects of vehicle roll on the navigation solution. Although the removal of the vehicle roll had little effect on the raw navigation errors of the high-grade INS, the raw errors of the low­grade INS were reduced by more than a factor of three. The main reason for this large reduction was that the scale factor error of the forward-facing gyroscope takes on a far reduced role in the vehicle error budget of a non-rotating pig. The roll's removal also enabled far better error modelling for the low-grade INS and it can be seen in Table 3 that a reduction of almost an order of magnitude for the residual position errors has been achieved. In the case where position fixes were available every ten minutes, the low-grade INS/low-grade odometer combination almost reached the single pipelength positioning level.

150

Residual Error High-Grade INS Low-Grade INS (metres) mean q mean q

Latitude 6c.p 3.20 2.02 6.73 7.74

Longitude 6>- 2.41 2.67 12.95 12.74

Altitude 6h 0.60 0.51 0.59 0.44

The standard deviation, u, relates to the standard deviation of the sample of the residual errors.

Table 3: Residual Position Errors for Non-Rotating Pig (Low-Grade Odometer)

It should be noted, however, that pig rotation can often be advantageous since it more evenly distributes the wear of the pig's support structure which is in contact with the pipe wall. Pig rotation can also be useful for the calibration or removal of some of the sensors' (inertial and odometer) errors. The main drawback of the pig's rotation, however, is the large additional processing requirement in the Kalman filtering stage necessary to update the formulation matrices and a reduction in this processing would enable increases in the velocity update rates to be made.

The residual attitude errors were considerably smaller than the raw attitude errors for all of the tests conducted. For both grades of INS, the residual roll error was about five times greater than the corresponding pitch error which reflects the effect of the unmodelled scale factor error of the forward-facing gyroscope. Problems associated with filter instability were frequently experienced during the fine-tuning of the filter and caused divergence of the smoothed error estimate close to the start of the survey. In an attempt to try and improve the attitude error modelling, the entire data processing stage was repeated for each test using the true pig initial alignment. This step, however, only shifted the origin of the residual attitude errors from which it was concluded that the unmodelled error sources, and not the initial alignment, are limiting the error modelling.

For both grades of INS, the systematic error sources modelled in the data generation stage were replaced by quantities much smaller than those stated in Table 1 to see if the error modelling in the Kalman filter stage could be improved. Results of these investigations were presented recently by the author (see Hanna & Napier [5]) from which it was verified that significant error modelling improvements were obtained and, in particular, the residual gyro drift estimated by the filter closely approximated the known error profile from the data generation stage. This led to better attitude error modelling. It is hoped to carry out further research to improve the pig (and therefore the pipe) attitude determination. More detailed results describing the attitude error modelling are presented in Hanna [4].

9 Conclusions A software package to simulate the strap down inertially-aided survey of a pipeline by an instrumented vehicle, the pig, has been successfully written, coded and tested. As expected, it is concluded that the combination of the inertial navigation system and the odometer gives a better positioning and attitude determination capability than that attainable by the individual navigation aids. The combination of the INS and the odometer navigation data reduced the position errors from several hundreds of kilometres to, at worst, the hundred metre level and in many cases to below the ten metre level. This improvement is significant

151

when it is appreciated that these figures relate to the three dimensional position errors whereas the odometer only gives uni-axial information.

The nature of simulation results, however, makes it impossible to allocate a single param­eter to describe the navigation solution improvement since there is the inherent danger of modelling all the known error sources in the data processing stage and thereby re-computing the original truth. The suboptimal Kalman Filter designed in this research enabled large reductions in the simulation's execution time and by increasing the process noise of the filter model, the neglected effects could be stochastically allowed for. The main use of the simulation results, therefore, is for the examination of trends introduced by varying the nu­merous parameters and with the limitations of the simulation's assumptions, the following general conclusions are drawn:

1. Provided that occasional accurate three-dimensional position fixes are available, the combination of the high-grade INS and the already-available pig odometer velocity will allow the pig to be positioned to a precision of within 15 metres (10'). By intro­ducing very accurate velocity measurements, such as can be provided by a doppler velocity sensor, the positioning capability of the high-grade inertially-aided pig can be enhanced to within 4 metres, which is within the semi-pipelength level (± 6 metres). By increasing the frequency of the position updates, sub-metre positioning capability is confidently expected.

2. The uncompensated systematic error sources severely limit the positioning capability of the low-grade INS and the simulation results indicated that even with relatively frequent position updates corresponding to every 1.5 kilometres travelled by the pig, single pipelength positioning was not achieved. A more elaborate alignment scheme, in which the unmodelled systematic errors of the inertial sensors could potentially be estimated, would aid the error modelling stage by allowing either the inertial measurements to be compensated before processing or alternatively, by including the determined error sources in the forcing function of the Kalman Filter.

3. If it is possible to prevent the pig from rotating, a much better positioning capability and attitude determination of the vehicle are possible. In addition, the removal of the vehicle rotation will significantly reduce the necessary computations in the error modelling stage, which will allow more effort to be placed on modelling the neglected error sources or on increasing the velocity update rate. The simulation indicated that with the higher rate of position information and the low-grade odometer, the low-grade INS based pig could be located to almost semi-pipelength level.

10 Acknowledgements

The author wishes to acknowledge British Gas pIc and the Science and Engineering Re­search Council who, between them, funded the project. The author also wishes to express his thanks to Dr M.E. Napier and Professor V. Ashkenazi of the Institute of Engineering Sur­veying and Space Geodesy (IESSG) at the University of Nottingham and to Mr E. Holden of British Gas pIc for their roles in supervising the research project. The author is also indebted to the IESSG for funding to attend the International Symposium on Kinematic Systems at Banff, Canada, 1990.

152

References

[1] Adams, J. R., Price, P. St. J. and Wade, R. In-Situ Geometry Pigging and Structural Data Analysis Systems. Conference Proceedings of Pipeline Pigging Technology-1989. Houston, USA, February 1989.

[2] Britting, Kenneth. R. Inertial Navigation Systems Analysis. USA, Wiley-Interscience, 1971.

[3] Gelb, Arthur. (editor). Applied Optimal Estimation. Sixth printing, USA, The M.LT. Press, September 1980.

[4] Hanna, P. L. Inertial Positioning for Internal Pipeline Surveys. PhD Thesis, University of Nottingham, UK, May 1990.

[5] Hanna, P. L. and Napier, M. E. Applications of Strapdown Inertial Systems to En­gineering Surveying. Conference Proceedings of Gyro Symposium '89. Stuttgart, Ger­many, September 1989, pp. 1.0-1.12.

[6] Kershaw, C.F. Pig Tracking and Location-Onshore and Offshore. Conference Pro­ceedings of Pipeline Pigging Technology-1986. Newcastle, UK, February 1986.

[7] Rauch, H. E., Tung, F. and Striebel, C. T. Maximum Likelihood Estimates of Linear Dynamic Systems. AIAA Journal, August 1965, Vol. 3, No.8, pp. 1445-1450.

[8] Savage, Paul. G. Strap down System Algorithms, Advances in Strap-Down Inertial Sys­tems. (Lecture Series Director, Schmidt, George. T.), AGARD-LS-133, London, Tech­nical Editing and Reproduction Ltd, April 1984, pp. 3.1-3.30.

[9] Scott, J. M. The reasons for pigging. Conference Proceedings of Pipeline Pigging Technology-1986. Newcastle, UK, February 1986.

[10] Teunissen, P. J. G. and Salzmann, M. A. Performance Analysis of Kalman Filters. Report No. 88.2 of the Faulty of Geodesy, Department of Mathematical and Physical Geodesy, Delft University of Technology, Delft, The Netherlands, September 1988.

[11] Wong, Richard. V. C. Development of a RLG Strapdown Inertial Survey System. UCSE Report No. 20027, Department of Surveying Engineering, University of Calgary, Cal­gary, Canada, December 1988.

153

GEODETIC APPLICATION OF A LASER-INERTIAL STRAPDOWN SYSTEM

Dieter Keller, Stefan Rohrich and Matthias Becker

Institute of Physical Geodesy Technical University Darmstadt Petersenstr. 13 6100 Darmstadt, F.R.G.

This paper describes the development of a Laser-Gyro strapdown inertial measurement system for the geodetic use and gives first results of test measurements.

A Honeywell LASERNAV II inertial navigation system was modified for an ex­tended output of sensor data with increased precision. Based on this data a geodetic software for inertial positioning with zero velocity and coordi­nate updates was developed. This includes the strapdown-computations with integration, transformation to the navigation frame and a Kalman-Filter for estimation of filtered coordinates and error handling. The post mission ad­justment is made in terms of a special combination model estimating the initial state vector and coordinates. One of the main problems, the high internal noise-level of the strapdown sensor data, is discussed.

Laboratory test measurements in stationary mode and on a turn-table as well as surveys of known profiles are analysed in view of the precision, the ac­curacy and the potential for stand-alone geodetic use. Furthermore the pos­sibility of a combination with GPS, from GPS-aided INS to full intgration, is discussed.

154

1. INTRODUCTION

In the recent years inertial survey systems could benefit from new develop­ments in technology resulting in the potential and the demand for new ap­plications. On one hand the use of conventional gimbaled systems for posi­tioning and gravity field determination is continuously investigated and still newly developed systems like SAGEM's ULISS30, become available, see e.g. (Mazzanti and de la Fuente, 1990). On the other hand, strapdown sy­stems, equiped with ring laser gyros were found to have a potential for geodetic use. In spite of their principal deficiencies, like limited accu­racy of gyros and high noise level of the data, the possibilities of using the body fixed sensor data for straightforward analytical computations of all quantities needed for geodesy, without the dependance on mechanical feedback loops, made them ideally suited for a use in geodesy. In 1984 Mueller and Adams published test results of a Honeywell HG1050 Inertial Re­ference System (IRS) in land vehicle application (Mueller, Adams, 1984). They showed, that standard aircraft navigation systems, without hardware modification, reached accuracies of better than 10 m for a 1 hour mission with the IRS mounted on a tank. Based on this promising experience and on the fact that Ring-Laser-Gyro (RLG) Strapdown Systems are available at about 25% or less of the costs of gimbaled systems, the Special Research Group "High Precision Navigation" at the Universities of Stuttgart and Darmstadt purchased a Honeywell Lasernav II Inertial Navigation System. The research topics within the group range from the pure stand alone geodetic surveying with INS, to gravity field determination, GPS/INS combination to position and attitude referencing for photogrammetry and laser profiling, see (Linkwitz and Hangleiter, 1989) for further references. The Lasernav II was delivered in early 1989 and the group at the Institute of Physical Geo­desy (IPG) is in charge of the conversion of the navigation system to a geodetic survey system applicable to the fields mentioned above. At this first stage a similar approach as described by Wong (1988) for the Litton LTN-90-l00 system is used. In the sequel the hardware and its modificati­ons, the software developed both for Kalman Filtering and Post-Mission ad­justment are described. First results of laboratory calibrations and test runs on a traverse are discussed with respect to the applications desired.

2. HARDWARE CONFIGURATION

The 1aser Inertial ~urvey ~stem (LISS) is based on a standard Honeywell 1asernav II Inertial Navigation ~stem (LINS). The LINS is equipped with Honeywell GG1342AE Laser gyros and Sundstrand QA2000 accelerometers. The output of the system is modified to provide the compensated sensor data on the standard ARINC-429 serial output. Table 1 gives rate and resolution of the geodetically interesting quantities.

For the applications to be described below the most important output are the body rates and body accelerations. Obviously the resolution is much better than the accuracy to be expected based on the resolution and perfor­mance of the sensors themselves, like 1.997 arcsec/pulse for the laser gy­ros or 10 ~g for the accelerometers. However due to internal data proces­sing as e.g. dynamical corrections (coning and sculling) or temperature corrections prior to the output, the 32 bit word length of the internal data bus was put out.

155

Parameter Rate Range Resolution

Latitude 5 Hz ± 900 16 m Longitude 5 Hz ±1800 16 m Roll-Angle 50 Hz ±1800 39.6 arcsec Pitch Angle 50 Hz ± 900 39.6 arcsec Azimuth 20 Hz ±180° 19.8 arcsec East-Velocity 10 Hz ±2107 mls 0.064 mls North-Velocity 10 Hz ±2107 mls 0.064 mls Angular increments 50 Hz ± 80 2·l0- 5 arcsec Velocity incre-ments 50 Hz ± 3.8 mls 4.6.l0- 9 m/s 9 Sensor Temperatures 10 Hz ± 112°C 0.07°C

Table 1: Modified Output of the LISS

All data of the ARINC-BUS is converted to IEEE-488 standard parallel bus, recorded on a HP9320 computer and stored on a Bering Bernoulli exchangeable Hard-Disk of 2 x 20 MByte capacity. Additional information, like ZUPT-ti­mes, event-marks, station names etc. can be entered via the keyboard. About 10 MB of data are recorded during a one hour mission, for details see (Kel­ler, 1989). The whole system is mounted in a small van. Offset measurements are presently made by protractor and ruler tape with sufficient accuracy.

3. PRINCIPLES OF DATA EVALUATION

The basic idea of using the LINS as geodetic system is the use of the com­pensated sensor data in a specially developed software which is designed for surveying applications. The standard output will later be used only for control purposes. The processing is done in two steps, both off-line, with the possibility of a future on-line implementation of the first step.

The first step comprises the alignment and Kalman Filtering. The second step uses the filtered data for a Post-Mission adjustment.

3.1 NAVIGATION AND KALMAN FILTER SOFTWARE

The algorithms used follow Savage (1984) and are similar to those used by Wong (1988), for details see (Keller, 1990). At first the horizontal velo­city and gyro data are used for levelling and azimuth determination of the sensor block, i.e. the initialisation of the transformation matrix from body frame to navigation frame.

The rate data display a high noise level and a spectral analysis shows di­stict peaks at frequencies of 15 and 24 Hz. There are almost the same alia­sing effects due to dithering of the laser-gyros as detected by Wong (1988) in Litton LTN-90-l00 data. We therefore use a 2nd order Butterworth low

156

pass filter (1 Hz cutoff frequency) and a following median estimation over 200 samples prior to coarse attitude computation. The median turned out to be much less sensitive against noisy data than the mean value. The coarse align takes less than 1 minute to converge to an accuracy of better than 1 degree in azimuth and 20 arcsec in roll and pitch angle, being sufficient for the initialisation of the body to navigation transformation matrix and for begining with quaternion and velocity integration. The coarse align time is adjusted, depending on the environmental vibration level.

A 15 state Kalman Filter is used for fine alignment and later for the esti­mation of errors of raw navigation quantities with time.

The basic equations are (Ge1b, 1974):

~(t) = I(t)~(t) + Q(t) y(t) + Q(t) ~(t) (1)

with:

I(t) time dependent dynamics-matrix of the system

y(t) [(Sw)B' (Sv)B' (h)]T = disturbing vector

SWB sensor errors of the gyros in the body frame SVB sensor errors of the accelerometers in the body frame h gravity disturbance vector

Q(t) coefficient matrix of the disturbing vector y(t)

! 1 ~(t) = stochastic noise vector

(2)

The dynamics matrix used is that deduced by Schroder et al.(1988). The coefficient matrix Q(t) is build up by the body to navigation transforma­tion matrix .Q~ .

This complete model, which is described by Hausch et a1. (1990) for the use with Ferranti Fils platform data, was further modified for strapdown system application. The deterministic parts of the disturbance vector are included in the state vector ~(t) and the number of states is reduced from 33 to 15 by omitting sensor-nonorthogonalities and scale factors. The new state vec­tor is

~*(t) I*(t)~*(t) + Q*(t) ~(t)

~*(t)

with bg and b a being the bias stabilities of gyros and accelerometers respectively.

157

(3)

The modified noise vector li*(t) contains quasi stochastic sensor errors and sensor noise,

(4)

and new dynamics and disturbance matrices are

[

f(t) (9,9)

f*(t) = Q

(6,9) 1 i 1 P = lit correlation time of gyros and accelerometers.

The solution is in the general case:

s..* (t) t

~*(t,to) s..*(to ) + ~*(t,to) J ~*(to,T) Q*(T)~*(T)dT to

For small time differences (ti -tj ) (Hausch et a1, 1990, Vasi1iou,1984)

with the approximation:

N

I n=O

s..*(to ) is the initial state vector at time to .

(5)

(6)

In the first approach of Kalman filter for the LISS implementation also the gravity disturbance terms of ~*(t) are omitted.

By using discrete Kalman Filter equations, see e.g. (Ge1b, 1974), predicted state, covariance extrapolation and updates are computed. The update measu­rements may be zero velocity updates (ZUPT) and coordinate updates (CUPT) with the observation equations:

ZUPT: y = Q (7)

Yin! velocity computed by the INS in the navigation frame

CUPT: y = fref - fins' (8)

fref' fins = reference and computed INS positions respectively

During fine align mode ZUPTs are performed, but contrary to the normal im­plementation ZUPT a closed loop Kalman Filter is used. This means, that the

158

estimated states for velocity errors are added in the velocity integration and orientation errors are transformed to the body frame and added to the quaternion integration. At the end of the fine align, after typically 12 min, the position is also reset and the open loop configuration is enabeled for the survey. In case a CUPT is available, the filter is set to closed loop configuration again and all 9 states are fed back to the navigation module for the duration of the stop. The Kalman filter loop runs with 2 Hz and ZUPT and CUPT take about 20 seconds of static position with no distur­bing environmental vibrations, respectively.

3.2 POST-MISSION ADJUSTMENT

The post-mission adjustment is done with the Gauss-Markov model, given by (Koch, 1980; Wolf, 1975)

Ax=l+n with Q(l) = a2 £-1 . (9)

In the INS we are using the Kalman-filter update-equations (7), (8) which can be reformulated for the least squares adjustment as observation equations

==> =>

y(t) y(t) y(t)

Yi n 5 (t) - v 0 v 0

~v ~*(ti) ~v m."'(ti'to) ~"'(to)

for ZUPTs, and for CUPTs in addition:

=> =>

ll:e.(t i )= ll:e.(t i )= ll:e.(t i )=

o at ZUPT's

Here ti is the observation time of a ZUPT or CUPT measurement.

(10)

(11)

~v' ~p are matrices, which map the part of the state-vector ~(t) of velo­city v or the positions p to the observation vector 1. To use these equati­ons in a post-mission least squares adjustment the following definitions are made:

~ unknown deterministic parameter, the initial state vector ~"'(to) of the system equ.(3).

1 velocity measurements in the ZUPTs and additional coordinate-differences in the CUPTs

Q(l) - variance-covariance matrix of the measurements

The design-matrix is build up like:

159

The stochastic integrated white noise of equation (5) is represented by the residual vector n. With n - (nl •......• ~)T. ~ - n(~) we have:

~ - £*(~.to) ?£*(to.T) Q*(Tht(T)dT to

(12)

So we solve for the initial state-vector and its covariance-matrix by using all measurements at the end of a survey.

Contrary to a Kalman filter. the complete set of update measurements influences the determination of the initial state vector ~*(to) and its co­variance matrix Q(~*(to» in a single step. This could be compared to ad­ding an optimal smoothing process after Kalman Filtering. Starting values for ~*(to) and Q(~*(to» are set to zero. The estimated state vector. however. is not comparable to the initial state vector in a Kalman-filter.

In a second step. we obtain the coordinates of new points (all new points are ZUPTs) using equ. (11)

(13)

The approach used here is a simplification of the combined model which al­lows estimation of state-vector. unknown coordinates and gravity field pa­rameters by use of collocation in one step. see (Hausch et al .• 1990).

4. NUMERICAL RESULTS

4.1 SYSTEM CALIBRATION

The LASERNAV II system was factory calibrated and the biggest parts of sen sor biases are removed internally. However. small changes may occur in time

Accelerometer Dim. Mean St. Dev. Stability Specified

Biaserror X [~g] -6.4 ±3.2 ±60 Biaserror Y [~g] -14.2 ±12.2 ±60 Biaserror Z [~g] -3.2 ±7.8 ±60

Scalefactorerror X [ppm] -3.6 ±O.S ±lS0 Scalefactorerror Y [ppm] 10.9 ±2.S ±lS0 Scalefactorerrcir Z [ppm] 14.9 ±4.7 ±lSO

Misalignment YX [~rad] -5.1 +23.1 ±SO Misalignment zx [~rad] 8.1 ±13.7 ±SO Misalignment ZY [~rad] 5.2 ±lS.6 ±SO

Table 2a: System Calibration - Accelerometers

160

IASER -Gyro Dim Mean St. Dev. Stability Specified

Biaserror X [Deg/h] 0.001 ±0.OO3 +O.OOB Biaserror Y [Deg/h] 0.001 ±0.002 ±O.OOB Biaserror Z [Deg/h] 0.002 ±0.001 ±O.OOB

Sca1efactorerror X [ppm] -7.B ±l1.B ±5 Sca1efactorerror Y [ppm] 6.2 ±13.0 ±5 Sca1efactorerror X [ppm] -0.1 ±B.B ±5

Misalignment XY [J£rad] 19.B ±2B.9 ±25 Misalignment XZ [J£rad] 20.6 ±22.4 ±25 Misalignment YX [J£rad] -O.B ±lB.1 ±25 Misalignment YZ [J£rad] 3.5 ±36.6 ±25 Misalignment ZX [J£rad] -20.2 ±30.B ±25 Misalignment ZY [J£rad] -15.1 ±16.B ±25

Table 2b: System Calibration - Gyros

and little information released by the manufacturer caused us to perform a system calibration. Additionally repeated calibration-runs are needed for determining turn on to turn on stabilities. With strapdown systems this can be done easily using the INS-data themselves, see e.g. (Savage, 1977). We implemented a calibration procedure published by Diesel (19BB) which is ba­sed on a 12 position setup. Measurements were made on a precision 3-axis turn table at the DLR, Braunschweig, F.R.G. Table 2 shows the average re­sults of 4 calibration runs and compares them to specifications given by Honeywell for Lasernav type systems. It can be seen that indeed there pre­sently are no significant residual biases and standard deviation indicates reasonable stabilities well within the specifications. Bigger instabilities in gyro scale factors may be due to the special setup of the tests.

4.2 KALMAN FILTER RESULTS

At present we only have test runs of the LISS on a small set of reference sites at the university campus in flat terrain. The duration of a forward and backward run is about 35 min. Table 3 shows the standard deviations used in the Kalman filter for initializing the state vector covariance ma­trix and for the measurement noise matrix.

The spectral densities in the system noise matrix are determined by an analysis of the sensor data to be 5.56 10- 7 m2 /s5 for the accelerometer biases and 4.2 arcsec2 /s3 for gyro biases under the assumption of a Gauss­Markov process and a correlation time of 40 h. They were determined within the setup of the LISS in the survey van.

Fig. 1 to 4 show results of Kalman filtered data on the test profile for two different runs. The first two pictures are using only ZUPT updates in

161

Quantity

fE' ION

fup

vE ' vN ' vup tP, oX, h Gyro biases accel. biases coordiante updates velocity at ZUPTs

S.D.

20 arcsec 1 deg 0,001 mls 20 m 0,003 degjh 10 p.g 0,10 m 0,0015 mls

Source

Coarse Align Coarse Align (Wong, 1988) (Wong, 1988) Calibration Calibration Empirical value Empirical value

Tab. 3: Standard Deviations for State Vector and Measurement Noise Matrix

addition to known start- and end-point coordinates. The residual errors found in comparison to the true reference coordinates show a systematic in­crease, but especially height errors are relatively small. Errors are below 15 m. When introducing an additional coordinate update in the middle, Fig. 3 and 4 with the same runs, error growth is controlled much better. The error of the filtered INS-coordinates are in the range of less than 10 m with the additional CUPT.

4.3 POST MISSION RESULTS

First tests of this model use a LISS survey made at the testfie1d on the campus. A1ltogether there are 9 points which can be used as CUPTs and ZUPTs. The first and the last point are identical. The total time for this measurement was about 35 minutes. During the driving mode the data output of velocities and Euler angles of the LASERNAV II was in 10 Hz rates. So the transition-matrix is computed in 0.1 sec intervals. So we get a conti­nous update of the actual transition matrix. For the A-matrix in the Gauss­Markov model we use the transition matrix at times of the ZUPT observati­ons. The 1-vector is computed in the CUPTs and ZUPTs from mean values of data outputs during the ZUPT measurement.

In this example we used an unit matrix for f, but weighting according to . the Kalman Filter system noise matrix may be introduced.

Three kinds of update measurements were tested. 1. all points as CUPT, 2. 3 CUPTSs, the first, the last and the 6th point, 3. only the first and the last stop used as CUPT.

After the estimation of ~*(to) coordinates and velocities of all 9 points (also the CUPT points) were computed using equ. (13). The accuracy of these coordinates fluctuates between 1 and 5 m. Then these coordinates were com­pared with given reference coordinates and the velocities with the obser­vations in the ZUPTs. The results of case 1 show a good estimation of the state vector. The differences in the coordinates are smaller than 1 m (Fig. 5). In this case they are identical to the residuals of the Gauss~Markov­model. In the other cases we can see a smaller deviation at points tem­porally close to CUPTs either prior or after the CUPT. This is the distinc­tion to a Kalman-filter-algorithm because in this model the complete set of

162

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In this tests it is difficult to Judge about systematic parts in the resi­duals. For the coordinates (case 2 and expectially case 3) we can see re­sidual systematic errors (Fig. 6 and 7). When there are more CUPTs, the in­fluence of these observations inproves the estimation of the state vector and raises the accuracy of the coordinates due to more degrees of freedom in the determination of ~·(to) and a better system control.

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164

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5. CONCLUSION AND DISCUSSION

In this paper the first numerical results for the Honeywell LISS are given. At the present state of software development and optimization the 15 state Kalman filtering with ZUPT and CUPT updates resulted in coordinate errors below 10 to 15 m for a 1 hour mission. The combined adjustment approach which can be considered as a way of combining filtering and smoothing in a least squares adjustment was applied with integrated rate data as well as with Lasernav II velocity data. Both data sets delivered residuals for coordinates in the order of 1 to 2 m in test runs. Due to the sub-optimal tuning of the software the results have to be considered as intermediate values only. After the full implementation the limits of the LISS data ac­curacy will be approached.

Based on the results available now, it is estimated, that coordinates will be determined with accuracies of about half a meter or slightly better for a 2 hour run.

The completion of the software will include the possibility to use GPS data. First tests were made using a Trimble 4000 receiver synchronized to the LISS data recording system. Coordinates or differential coordinates can be introduced as CUPTs to Kalman Filter or adjustment. The final goal in our application of LISS is the full integration of GPS and inertial data at the level of velocities as described e.g. in (Eissfeller, Spietz, 1989), with an extended Kalman Filter state vector including GPS related quanti­ties.

165

ACKNOWLEDGMENT

We are thankful to the German Research Foundation, DFG, for sponsoring the research within the framework of the Sonderforschungsbereich 228 "Hoch­genaue Navigation".

REFERENCES

Diesel, J.W., 1988: "Calibration of a Ring Laser Gyro Inertial Navigation System", 13th Biennial Guidance Test Symposium

Eissfeller, B., P. Spietz, 1989: "Basic Filter for the Integration of GPS and an Inertial Ring Laser Gyro Strapdown System", Hanuscripta Geo­daetica, Vol. 14, No 3, p. 166-182

Gelb, A., 1974: "Applied Optimal Estimation", The H.I.T. Press, Mass. Inst. of Technology, Cambridge (Mass.), USA

Hausch, W., St. Rohrich, E. Groten, 1990: "Kombinierte Ausgleichung bei der post-mission Analyse von Inertialdaten", Allgemeine Vermessungsnach­richten 5, p. 165-175

Keller, D., 1989: "First Results of the Honeywell Lasernav II System in Geodetic Application", Proc. of the Symposium on Gyro Technology, Stuttgart

Keller, D., 1990: "Aufbau eines geodiitischen Strapdown-Inertia1systems zur Punktbestimmung" , Darmstadt, Dissertation (in press)

Koch, K.R., 1980: "Parameterschiitzung und Hypothesentests in 1inearen Ho­de11en", Ferd. Dummler Verlag, Bonn,

Linkwitz, K., U. Hangleiter, (Eds.), 1988: "High Precision Navigation", Proceedings of an International Workshop, May 17-20, 1988 in Stutt­gart und Altensteig-Wart, FRG, organized by the SFB228 of the Stutt­gart University, Springer-Verlag, Berlin, Heidelberg, New York, Lon­don, Paris, Tokyo, HongKong

Mazzanti, F., C. de la Fuente, 1990:"ULISS 30, A New Generation Inertial Survey System Enhancing Ease and Security of Operation, Performance and Reliability", paper presented at the International Symposium KIS 1990, Banff, Canada

Mueller, C.E., G. Adams, 1984: "Laser Gyro Land Navigation System Perfor­mance Predictions and Field Results", IEEE Plans '84, p. 81-90,New York,

Savage, P.G., 1977: "Calibration Procedures for Laser Gyro Strapdown Iner­tial Navigation Systems", Proceedings of the 9th Annual Electro-Op­tics/Laser Conference and Exhibition, Anaheim California

166

Savage, P.G., 1984: "Strapdown Systems Algorithms, in Advances in Strapdown Inertial Systems", AGARD Lecture Series No. 133, NATO, Loughton (GB)

Schroder, D., NC. Thong, S. Wiegner, E. Grafarend, B. Schaffrin, 1988: "A comparative Study of Local Level and Strapdown Inertial Systems", Manuscripta Geodaetica 13, p. 224-248

Vassiliou, A., 1984: "Processing of Unfiltered Inertial Data", Publ. No. 20006 of the Division of Surveying Engineering, The University of Calgary

Wolf, H., 1975: "Ausgleichungsrechnung", Ferd. Diimmler Verlag, Bonn,

Wong, R.V.C., 1988, 1988, "Development of a RLG Strapdown Inertial Survey System", USeE Reports No. 20027, The University of Calgary

167

ABSTRACT

ATTITUDE DETERMINATION VIA GPS

Jerry Knight and Ron Hatch Magnavox Advanced Products and Systems Company

Torrance, CA 90503 USA

The principles of attitude determination using the Global Positioning System (GPS) are described. The specific techniques which can be used to resolve the initial carrier phase ambiguities which arise when the antennas are more than one-half wavelength apart are also described. The specific method used by Magnavox to solve the attitude problem is discussed. Finally, sample results obtained from the first prototype Magnavox ADS receiver are presented.

INTRODUCTION

A well known principle of geometry is two distinct points define a unique line and three points which do not all lie on the same line define a unique plane. This is the idea behind Global Positioning System (GPS) attitude determination systems (ADS).

In the ADS antennas are treated as points. Two or more collinear antennas define a line, three or more antennas which are not all collinear define a plane. If the relative positions of the antennas are known, the orientation of the lines and planes can be determined.

GPS broadcasts L-band signals at two frequencies commonly called Ll and L2. The wavelength is 0.190 m for Ll and 0.244 m for L2. The wavelength of the beat between Ll and L2 commonly called L2-Ll is 0.86 m. The signals are modulated with two orthogonal codes. One, called CIA-code, has a wavelength of 300 m and is freely available to the public. The other, called P-code, has a wavelength of 30 m and is subject to secret encryption. Generally, Ll only receivers use CIA code while P-code receivers track both Ll and L2. Furthermore, some receivers track only the code while others track both the code and the L-band carrier.

There are three types of ADS receivers: those that track only code, those that track CIA code and Ll carrier phase, and those that track both Ll and L2 carrier phase. The relative accuracy of code tracking receivers is one to five meters. The relative accuracy of carrier phase measurements are from 0.01 to 0.05 cycles of carrier phase (0.002 to .01 m for Ll and

Accuracy of the ADS is proportional to the relative accuracy of the receivers and to the spacing of the receivers. Since code-tracking-only receivers are more than 100 times more imprecise than carrier tracking receivers, an ADS using receivers which track only code must have more than 100 times the antenna spacing to achieve the same accuracy. As an example, to match the 0.5 degree accuracy of low cost commercial gyroscopes, an ADS using a receiver which tracks code to one meter accuracy must space its antennas 115 m apart, while an ADS using a receiver which tracks carrier phase to 0.01 cycles accuracy must only space its antennas by 0.23 m. Since few ADS applications can tolerate hundreds of meters of antenna spacing, the remainder of this paper will consider only carrier tracking measurements.

168

FIRST DIFFERENCE

The accuracy of the ADS depends only on the relative accuracy of the positions of the system antennas. The solution is only very weakly affected by errors in the absolute position (i.e. the elevation angle and azimuth of the satellites is only a weak function of receiver position). In fact, a kilometer of error in absolute system position produces a negligible attitude error.

Since only the relative positions of the antennas is important, we use "first difference" measurements to solve the ADS problem. For example, consider an ADS with two receivers which are tracking the same suite of satellites. Each receiver will provide a phase range to each of the satellites. Each phase range is distorted by the ionosphere, the troposphere and mUltipath. However, since the antennas are closely spaced the errors are substantially identical for both receivers. If the measured range from the first receiver to a given satellite is subtracted from the measured range from a second antenna to the same satellite, the difference will be only a function of antenna spacing, receiver clock differences, and receiver noise. In particular, the ionospheric, tropospheric and most multipath distortions are eliminated.

SATELLITE REQUIREMENTS

Three dimensional positioning with GPS requires measurements from four satellites, any three of which must not lie in a plane containing the receiver. The four measurements are required to solve for three position coordinates (e.g. north, east, up) and the time difference between the receiver's clock and the GPS system clock. In effect we require four first­difference measurements to solve for four unknowns.

If the antennas in the ADS are mounted on a rigid assembly such that the distance between the antennas remains constant as the assembly is rotated, then the known antenna spacing can be used as a solution constraint. Only three measurements are then required for the solution.

CLOCK SOLUTION

One of the unknowns in the GPS solution is the time difference between the receiver's clock and the GPS clock, or, in the case of first-difference measurements, the difference between the clocks in the two receivers. This difference is commonly called the clock error. The clock error is identical for each of the measurements. For example, if the clock error is 1/2 carrier cycle, then each of measurements the is 1/2 cycle too large.

The clock error is removed by tracking a synthetic clock calibration signal, using double difference measurements, or adding a clock error state to the solution filter. Addition of a synthetic clock signal reduces the number of required measurements to two at the expense of additional hardware, while a total of three measurements are required for the other two methods.

A synthetic clock calibration signal is relatively simple in concept (Figure 1). The receiver's reference clock oscillator (Osc. in the figure) is used to synthesize Ll and L2 frequency carrier signals. A CIA code (perhaps a pseudolite code or an unused satellite code) and a fixed data stream is modulated onto the Ll carrier so that the signals from the various receivers can be time normalized. These signals are then mixed together and injected into each antenna. The receivers then track the synthetic signal and report the corresponding phase range. Except for differences in cable lengths, all receivers should measure exactly the same range for the signal. The portion of the measurement due to cable lengths is unique to each cable configuration, but does not vary once the system is manufactured. This "factory" delay is easily determined by orienting the

169

CLOCK CALIBRATION SIGNAL

S nth

Figure 1 : Generation of a clock calibration signal from the reference oscillator. L 1 and L2 carrier signals are generated. CIA code and data is modulated onto L 1. The carriers are then mixed and injected at the ADS antennas.

antenna assembly at a known attitude and solving the inverse of the ADS solution. Once the constant "factory" delay is removed, deviation in the first synthetic phase range from zero is the clock error.

The clock bias may also be removed by using "double difference" measurements. As noted previously, the clock bias distorts all the first difference measurements equally. Therefore, if one of the first differences is subtracted from the others, the reSUlting "double difference" measurements are free of clock errors. This is the method of choice if a clock calibration signal is not available and has classically been used in differential GPS surveys.

Finally, the clock can be included as a solved for state in the solution filter. This method increases filter complexity and computational requirements, and, experimentation has shown that the additional degree of freedom to the solution greatly complicates and slows ambiguity resolution when compared to the double difference method.

PHASE AMBIGUITY

The structure of the GPS signal does not allow the receiver to distinguish one cycle of carrier phase from another. Therefore, only the fractional portion of the phase range is useful and the integer portion of the range first difference must be determined as part of the solution process. The possible range of the integer portion is a function of the antennas spacing. If the antennas are spaced by less than one hal f wavelength, then the phase difference can never be greater than one half wavelength, and therefore, there is no ambiguity. If, however, the antennas are separated by more than one half wave length, any particular fractional phase measurement could correspond to one of several true phase differences. The number of possible phase differences is given by the following formula:

possible solutions - 2 * floor(r) + 1

where r is the antenna separation in carrier wavelengths and the function floor(r) is the largest integer less than or equal to r. For an L1 only ADS using antennas with aIm spacing there are 11 possible solutions for the integer portion of the phase difference.

To illustrate the concept of phase ambiguity let us consider a two dimensional, two antenna example with antenna separation of five carrier wavelengths (Figure 2). The center of the circle corresponds to the the first antenna and the circumference corresponds to all the possible relative orientations of the second antenna. The graduated "satellite-axis"

170

PHASE AMBIGUITY ·1.0 00

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Figure 2: Phase ambiguity in 2 dimensions. The integer portion of each measurement is unknown. The measurement X.O is consistent with any of 20 attitudes.

passing through the center of the circle is the trace of the vector from the first antenna to the satellite and the line with the arrow shows the actual orientation of the antenna. As drawn, the phase difference measurement is X.O cycles, where X is one of 11 unknown integers from -5 to +5 inclusive. There are 20 possible orientations of the ADS consistent with this measurement. Each of these is represented by a small circle around the circumference of the big circle. Note that the fractional part of the "satellite-axis" coordinate of each possible solution is equal to the fractional part of the phase difference measurement .

The three dimensional pointing problem is analogous to the two dimensional problem just illustrated. Now, the set of all possible solutions define a sphere with radius equal to the antenna spacing. The ambiguous phase difference measurement from one satellite defines a family of planes which are perpendicular to the line connecting the first antenna and the satellite. Again these planes intersect the line connecting the antenna and the satellite at distances from the antenna whose fractiollal part equal the fractional phase difference measurement. The intersection of the planes with the sphere is the set of possible solutions. The intersections are a family of parallel circles whose center is on (and perpendicular to) the line to the satellite and whose circumference is on the sphere.

The phase measurements from a second satellite define a second family of planes which intersect those from the first satellite . The intersect i ons of the two sets of planes are a series of parallel lines perpendicular to the plane containing the receiver and the two satellites . Figure 3 is a stereographic representation of these intersections. In this example the antennas are spaced one meter apart (5.1 Ll wavelengths) and the two satellites are oriented 90 degrees from one another in the plane of the figure. Each of the lines in the grid is the projection of the intersection of the sphere with the measurement planes . Each grid lntersection represents two possible solutions, one above I and one below the plane of the figure. There are 178 possible solutions.

171

AMBIQUITY WITH 2 SATELLITES Satell ito 1

Figure 3: Stereographic projection 01 the measurement planes and solution sphere. Each grid point within the circle represents two possible solutions, aile above and one below the plane 01 the ligure. There are 178 possibte solutions.

AMBIGUITY RESOLUTION METHODS

PHASE AMBIGUIlY WI I H 3 SA TELU I ES

/" ( " /" ,

Figure 4: The grid intersectiOlls olligurC' 3 me shown as lectnnglcs to allow 101 Iloi~e. The line~ ,II ' · the !ncasuICmpnt planes lor a thirrl S;)tellite. Solutions not "rtersectcd tJy the fi.ms me discarded. 12001 170 possitJilities 3. c eliminated.

Measurements from additional satellites define additional families of planes. Any of the possible solutions which do not lie on the new families of planes may be eliminated from consideration. In theory, providing there is no redundancy in the satellite geometry, measurements from five satellites or four plus a clock calibration measurement should always define a unique solution. However, the real measurements are noisy so the planes must be treated as thin slabs and their intersections are not points but fuzzy spotS. In Figure 4, the possible solutions from Figure 3 are represented by small rectangles which allow for the noise in the first two measurements. The intersection of the planes representing a measurement from a third satellite are also plotted . This third satellite also lies in the plane of the figure. Possible solutions which are not "touched" by the third set of planes may be discarded. In this example, 120 of the 178 original solutions are eliminated leaving only 58. If the measurement noise was exactly zero and the rectangles were points, then only 6 possibilities would remain.

In summary, reducing antenna spacing reduces the number of ambiguities; adding more measurements resolves ambiguities; and, reducing the measurement noise resolves ambiguities. A variety of techniques have been used to achieve these goals.

First, consider methods of reducing antenna spacing . Since the accuracy on the ADS is proportional to the antenna spacing it 1s generally undesirable to place the systems antennas close together. However, a third collinear antenna can be added between two more widely spaced antennas. The measurements from this third antenna are used to resolve ambiguity while the measurements from the other antennas are used for precision. If the spacing between the first and third antennas is less than one half wavelength then the ambiguity is totally eliminated .

Since the number of ambiguous solutions is a function of antenna spacing divided by carrier wavelength, increasing the carrier wavelength is equivalent to reducing the antenna spacing. To achieve this purpose the Ll­L2 beat phase is used. The wavelength of Ll-L2 is 0.86 m versus 0 . 19 m for the Ll carrier. Thus, for a 1 meter antenna spacing, there are only three possible integers for each measurement, versus 11 for the Ll only. And, for the three-dimensional full solution there are 18 possibilities versus 178. Since both the Ll and Ll-L2 measurements may be used for each satellite, receivers which track both Ll and L2 not only have fewer ambiguities to consider but effectively double the number of measurements available for ambiguity resolution.

172

Several methods have been used to get more m('n.suremt~nt.s for nmbiguity resolution. This is done by tracking more satellites, adding additional receivers or by tracking the same satellites with a new geometry.

In practice simply tracking more satellites is not always possible due to the limited number of visible satellites at any given time and the limited number of channels available in a given receiver. As a general rule, though, ADS receivers should have as many channels as possible. A second technique for adding satellites is to use measurements from the GLONAS satellite system of the USSR. A combined GPS-GLONAS receiver is ideal for ADS. Use of extra receivers (antennas) is also equivalent to tracking more satellites. .

The final group of techniques produce more measurement planes by varying the geometry of the solution in one of two ways. First, the antenna assembly can be physically rotated. The rotation will affect the geometry between the antennas and each of the satellites differently, and in effect produces an entirely new suite of measurements. This method is cumbersome in cases where the antenna is permanently mounted on a vehicle, but is viable for mobile pointing applications. Second, the antenna may be held stationary for a period of time and the movement of the satellites will change the antenna-satellite geometry in effect producing a new measurement suite. This method is not useful for dynamic applications where a quick solution is required but is quite valuable for cases where the extra precision achieved by longer term averaging of measurements is desired.

Reduction of measurement noise is difficult to achieve and generally requires performance trade-offs. The measurement noise of ADS phase first differences is due primarily to remaining multipath and to receiver tracking noise.

Multipath is caused primarily by reflections of the GPS signal from below the antennas and is eliminated by adding ground planes and choke rings. This has the side effect of making the receiver unable to track low elevation satellites and in effect reduces the number of available measurements. In applications where the antenna assembly must be allowed to tilt, e.g. aboard ships and applications with large inclinations the low angle signals must be tracked and large ground planes cannot be used.

Phase tracking loop noise is a function of the bandwidth designed into the receiver. A trade-off is made in receiver dynamics and tracking noise. An extremely low noise receiver cannot be subjected to·any dynmnics, while a receiver designed to handle high dynamics will be noisy. Loop noise is also a function of the signal-to-noise-ratio of the GPS signal at the antenna. Strong signals are tracked with lower noise, weak signals with higher noise. Figure 5 is a plot of the standard deviation of the tracking loop noise for a receiver with a 20 Hz bandwidth versus input signal- to­noise ratio (C/NO). Receivers with this tracking bandwidth are typical of general purpose receivers designed to handle up to 100 mls velocity and one times gravity accelerations. If the tracking loops are made less noisy the receiver will loose phase lock when its moved at this speed and acceleration.

Finally measurement noise can be reduced by averaging phase residuals over time. The noise of the phase tracking loops is Gaussian and proportional to the inverse of the square root of the averaging time. Multipath errors are strongly correlated over short periods of time but are mean zero over several minutes. Thus, the phase error can be greatly reduced by averaging the phase residuals while the attitude is not changed.

MEASUREMENT CHARACTERISTICS

ADS phase measurements are a function of the angle between the antenna structure and the satellites. Since the satellites move the measurement vary with time. Figure 6 is a plot of the measurements one would expect to be observed by a noiseless ADS as a function of time (note that the top edge, 0.5 phase, and bottom edge, -0.5 phase, are eq~ivalent). The phase measurement and the time variation of the measurement is different for each

173

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COMPUTED PHASE DIFFERENCE

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satellite. Figure 7 is a similar plot for the same satellites over the same time period, but with the azimuth of the ADS rotated by 45 degrees. The phases and phase rates in Figure 7 are totally different from those in Figure 6. This is what allows us to determine the antenna attitude and resolve ambiguities using only the fractional phase measurements.

Measured phase differences, Figure 8, deviate from the theoretical phase differences for this configuration, Figure 6, due to noise. The standard deviations of the measured minus theoretical residuals from these curves is 0.02 cycles, however, the short term deviations due to mUltipath are as large as 0.1 cycle and persist for several minutes.

174

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The data presented in Figure 8 were gathered in a bad environment. The antenna was held fixed on a building roof top and the was surrounded by air conditioners and metal ducts . In most environments the size of the multipath distortions is much reduced. In dynamic environments the changing geometry of the satellites, antennas and mUltipath sources tend to minimize the effects of multipath.

175

SOLUTION METHODS

The equations describing the ADS problem are easily expressed in vector form. Choose one of the antennas to be the reference. The attitude of the vectors from this antenna to the other antennas will be determined. Let V(i) be the vector from the reference antenna to antenna i; let S(j) be the unit direction cosine vector from the reference antenna to satellite j; let r(i) be the distance of antenna i from the reference antenna; let p(i,j) be the phase difference measurement for satellite j at antenna i (with ambiguity resolved); and let d(i,k) be the known vector dot product describing the geometry between antennas i andk. Then the governing equations are:

V(i).S(j)

V(i) . V(i)

V(i).V(k)

p(i,j)

r(i)

d(i,k)

Measurement equations

Radial constraints

Geometry constraints

where V.V represents the vector dot product. The units in all the equations are cycles of carrier phase.

We wish to solve for V(i) for all i. The measurement equations are linear, while the radial and geometry equations are quadratic. The solution is straight forward but quite tedious. Generally speaking one chooses measurements from two satellites to construct an initial trial solution for each possible ambiguity using the full equations. Then, the equations are linearized and additional measurements are absorbed using a least squares, square root information, or Kalman filter. Solutions with large residuals and those which violate the radial or geometric constraints are eliminated.

A patented method by Hatch greatly simplifies the complexity of the initialization. First, the two satellites with angular separation closest to 90 degrees are chosen to initialize the solution. This is equivalent to choosing the best DOP (dilution of precision) for the solution. Next the coordinate system is rotated to an non-orthogonal system such that the y­and z-axis point at the two satellites. The coordinates ~f the possible solutions in this coordinate system are (X, M.y-phase, N.z phase) where M and N are ambiguity integers and y_phase and z_phase -are the phase measurements for the satellite on the y- and z-axis. X is defined from the spherical constraint x*x + y*y + z*z = r*r. There are two possible values of x for each y, z pair. One of these represents the solution which is above the y-z plane, the other is below the plane. Further measurements are employed immediately to discard as many of the ambiguous solutions as possible. The coordinate system is then returned to the normal orientation for measurement processing.

After the list of poss ible solutions has been cons truc ted the measurements are filtered and the measurement residuals computed for each of the possible solutions. Those solutions with a statistically large residual or sequence of residuals are discarded until all but one solution is eliminated. The final remaining solution is declared the correct solution.

Once the phase ambiguities are determined the ADS can compute the absolute relative phase error for each measurement. All subsequent measurements can then be used directly providing the receiver does not cycle slip or break phase lock. For example, suppose the difference in the measured phase ranges to the first satellite is 107.1 cycles and for the second satellite is -12.8. If the ADS solution determines the true phase differences to be - 3.1 and 1.8 respec ti ve ly , then the ADS records that measurement one requires a phase correction of -110 cycles, and measurement two requires a correction of +13 cycles. Thereafter, these corrections are applied directly to the raw phase differences and no further ambiguity resolution is required. If large filter residuals indicate one of the receivers may have lost phase lock and changed one or more of the corrections the phase correction for the suspect satellite must be

176

redetermined. If there are enough good satellites [or l\ simultlllleous ADS solution, then the entire ambiguity search does not need to be repeated.

When a new satellite is first tracked, or when a satellite breaks cycle lock after the ADS has already resolved the ambiguities, the satellite's phase correction can be determined directly from a simultaneous ADS solution. No ambiguity search is required. In this case the measurement equation for the satellite is solved for the integer portion of the satellite's phase difference. The difference between the computed and the measured phase ranges is the phase correction for the satellite.

EXPECTED ACCURACY AND RESULTS

The expected error for the ADS is a complex function of receiver attitude and satellite geometry and measurement noise. The affect of receiver attitude and satellite geometry is closely related to the classic GPS performance variable DOP (Dilution of Precision). This is the trace of the geometric covariance matrix and is an estimate of the ratio of position error divided by measurement error. When the full GPS constellation is implemented, the azimuth and elevation DOP will generally be less than three. Currently, however, DOP's are highly variable.

The performance of a two antenna ADS built by Magnavox for the U.S. Army Corps of Engineers is shown in Figure 9. This plot shows the variation in computed azimuth as a function of time (samples taken once every two seconds). During this experiment the ADS tracked four satellites and a clock calibration signal on both Ll and L2. DOP was less than three during the entire experiment which was terminated when one of the satellites set and the DOP increased to over 100. The heavy line at 111.62 degrees indicates the true azimuth. The thin lines at 111.68 and 111.56 degrees indicate one milliradian error. The elevation error during this experiment was similar to the azimuth error.

Cl (J)

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ADS AZIMUTH 111.75 ...-------------------------------.

111.7

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111.6

111.55 -----

o 300

SAMPLE NUMBER (2 sec)

Figure 9: Computed azimuth from the Magnavox ADS as function of time. Truth is marked by the heavy line, 1 mrad error by the light lines.

177

GOO

A REAL·TIME GPS·BASED DIFFERENTIAL POSITIONING SYSTEM

Ken Doucet, Yola Georgiadou, Alfred Kleusberg and Richard Langley Geodetic Research Laboratory, Dept. of Surveying Engineering

University of New Brunswick, P.O. Box 4400 Fredericton, New Brunswick, Canada E3B 5A3

(506) 453-5151, Fax (506) 453-4943

ABSTRACf

Hydrographers, surveyors, and geophysicists, are among the many groups who would benefit from precise position information either in real-time or with very little delay. The most promising technology for meeting this need is the Global Positioning System (GPS). Used in the differential mode with a communication link between a base station and a user, GPS should be capable of providing real-time relative positions over distances of hundreds of kilometres with accuracies approaching tens of centimetres. We have begun the development of a prototype of such a system at the University of New Brunswick. The system consists of two Ashtech XII Model L receivers with two Macintosh microcomputers handling receiver communications and decoding, data processing, and data transmission tasks. The system is capable of different levels of positioning accuracy depending on the application. In the differential pseudorange correction mode, the transmitting station will broadcast pseudorange corrections which can be applied at remote sites to obtain real-time position accuracies at the 5 metre level. For high precision kinematic positioning and site stability monitoring, the transmitting station will broadcast pseudorange and carrier phase observations. The format of these broadcast messages conforms to those proposed by Radio Technical Commission for Marine Services Special Committee 104 (RTCM SC-I04) although a new message type has been added for high precision applications. System design and processing methodology will be discussed and test results presented.

INTRODUCTION

The Global Positioning System (GPS), introduced in 1973, is a system designed by the United States military to provide continuous position and velocity information for a variety of land, sea, and air based military applications. Two levels of accuracy are available to users of GPS. The precise positioning service (PPS), restricted to authorized users, is intended to provide instantaneous position and velocity information at accuracies of 15 metres and 0.1 meters/second, respectively. The standard positioning service (SPS), available to the civilian community, is currently expected to provide instantaneous position and velocity information at accuracies of 100 metres and 0.3 meters/second, respectively. For most navigation applications, SPS levels of accuracy will be sufficient but there are many applications for which SPS will not suffice. For these applications, relative GPS techniques must be employed.

Relative positioning involves differencing observations collected at two sites to remove common biases. During post-mission analysis of data collected during marine trials, Lachapelle et al. (1984) obtained relative positions at the 5 to 10 metre level with differenced TI-4100 P-code pseudoranges. With improvements in hardware, this technique has been successfully employed to obtain relative positions at the 5 metre level with CIA code receivers. Due to communication link limitations, several alternative approaches to direct differencing of pseudoranges have been proposed for real-time applications. These approaches all rely on the concept of a tracking station at a known site broadcasting navigation corrections that may be used to obtain real-time relative

178

positions at the 5-10 metre level. The current standard for such a system, designed by Special Committee # 104 of the Radio Technical Committee for Maritime Services (RTCM) broadcasts estimated pseudorange errors for all satellites visible at a monitor site (RTCM 1990). Many commercial systems based on the RTCM standard are now available and are capable of providing relative positions of a moving platform at the 5 metre level. For low to medium accuracy requirements, these systems will undoubtedly replace many existing survey systems when the GPS constellation is fully deployed.

To fully exploit the potential of GPS for kinematic positioning requires the use of the more precise carrier phase observation. Differential processing of GPS carrier phase data collected at stationary sites over several hours has routinely resulted in relative positioning accuracies at the 2 part per million (ppm) level or better. Realizing that station occupation times of this length were necessary only to resolve the carrier phase integer ambiguities, Remondi (1985) proposed a kinematic survey mode which required occupying two stationary sites only long enough to resolve the integer ambiguities and then the subsequent deployment of one of the receivers as a rover. Based on this technique, Mader (1986) obtained relative vertical accuracies at the submetre level during airborne tests. During land based tests, Hein et. al. (1988) reported 2 cm relative position accuracies over 10 km baselines using dual frequency GPS receivers and station occupation times of 10-40 seconds. This approach, however, cannot be used for a variety of marine and airborne applications where it is impossible, or at least impractical, to remain stationary long enough to resolve the cycle ambiguities.

By combining P-code pseudorange and Doppler observations during post-mission processing, Kleusburg (1986) obtained relative positioning results at the submetre level for a moving platform over distances of 95 kilometres. This approach effectively removed the requirement for accurate initial positions by incorporating the geometric information available from pseudorange observations. Hatch (1986) employed a similar technique utilizing wide-lane carrier phase observations to smooth P-code pseudoranges in such a way that cycle ambiguities could be resolved very quickly even on moving platforms. This technique was used by Seeber (1989) to obtain 10-20 centimetre relative positions for marine and airborne applications. This level of accuracy, previously unattainable during kinematic surveys, surpasses the requirements of most of the currently envisioned applications of GPS. The main drawback of this method is the reliance on dual-frequency equipment capable of tracking P-code pseudoranges. For single and dual­frequency CIA code equipment this level of accuracy may only be possible over short baselines.

Potential kinematic applications of GPS may be categorized by the expected level of accuracy and whether or not this accuracy is required post-mission or in real-time. One of the major difficulties for real-time applications is the implementation of a reliable communication link between monitor and rover. As mentioned previously, for low to medium accuracy applications, this has been overcome by reducing the amount of data that must be transmitted and the frequency of these transmissions. For high accuracy real-time applications, however, implementation of a reliable communication link may well be one of the most difficult problems since all data collected at the monitor site must, in general, be transmitted. In addition, in contrast to postmission processing, obtaining high accuracy results in real-time is much more difficult since blunder detection and residual analysis techniques are much more difficult to implement. Table 1, based on material in Wells et al. (1986), summarizes some of the advantages and disadvantages of real-time and postmission processing.

The Geodetic Research Laboratory of the Department of Surveying Engineering at the University of New Brunswick has begun work on a real-time kinematic positioning system intended for medium to high accuracy applications. The components of the system, the status of the development, some system tests, and future plans are discussed in the following sections.

179

Real-ume Postmission Advantages Results available immediately More accurate

Reduced data storage problems Blunder detection easier

Detailed analysis of residuals

Model identification

No real-time data link needed

Disadvantages Less accurate Results cannot be used in the field

Measurements often not stored Data storage problems

Blunder detection difficult

Real-time data link

Table 1 Real-time versus Postmission Processing

SYSTEM OVERVIEW

Figure 1 illustrates the components of a real-time differential GPS system and the processing tasks required at the monitor and roving stations. Each of these components, as implemented in the UNB prototype, are explained in the following sections.

Tasks

Recel ver interface Data processing

Data logging RTCM encoder

Figure 1 Real-time Differential GPS System

180

Tasks Receiver Interface Data processing

Data logging RTCM decoder

GPS Receiver

The GPS receivers currently used for the UNB prototype are Ashtech L-Xll single frequency CIA code receivers with twelve independent channels for satellite tracking. Additional capabilities of these receivers are real-time communications and control, one second data rates and, internal storage of up to two megabytes of data. Each is also equipped with one internal and one external battery allowing for approximately eight hours of operation. Ground planes, provided with the antennas, help to reduce the effects of multipath on the pseudorange and carrier phase observations.

Computer Hardware and Software

The computer system on which the software for the real-time system is being developed is a Macintosh SEI30 equipped with a Motorola 68030 CPU, a 68881 math coprocessor and two buffered RS422 serial communications ports allowing data transfer rates of up to 57600 bits per seconds. Due to future portability considerations and the necessity for low level operations, all software modules are being written in C.

At the monitor and roving stations, receiver interface software enables receiver control and real­time data access. The Ashtech L-XII real-time communications option allows the controlling program to enable or disable satellites, modify the data rates and to set the satellite elevation mask angle. Each receiver can be set up to automatically transmit observation, ephemeris and navigation solution records as collected or when requested by the program. Once the data has been collected from the receiver, the receiver interface software then transforms the raw Ashtech observations records to obtain receive time tags, unambiguous CIA code pseudoranges, continuous carrier phase and Doppler observations for each satellite.

In order to satisfy low to high accuracy kinematic positioning requirements, the current prototype system has been designed based on the premise that all GPS observations must be transmitted from the monitor station. This means that the monitor station program is only required to compact the data collected from the receiver, log the data locally if required, and pass the data on to the transmitting device. The format of the compacted data is based on the RTCM standard (ibid) but does not use message type 4 which was designed for transmission of carrier phase data along with the usual pseudorange corrections. Instead, a new message, tentatively designated type 18, has been developed which allows for more accurate time tagging of the observations and a more complete observation set. As shown in Table 2, this message retains the 30 bit word structure and comprehensive parity checking of RTCM messages but allows for the reconstruction of a complete observation record from a transmitting station.

At the roving platform, data must be collected from both the receiver and the monitor data link, decoded, and then combined and processed to obtain the relative state of the platform. Depending upon parameters set at program start up, undifferenced, single differenced or, double differenced pseudorange, carrier phase and Doppler observations may all be incorporated in the state estimation process. Again, depending upon parameters set at program start up and the observations being processed, the state vector can consist of position, velocity, acceleration, clock offset, clock drift, clock drift rate, and cycle ambiguity terms for each tracked satellite. State prediction is based on appropriate transition and process noise covariance matrices while a Kalman filtre formulation is used for state estimation.

181

1 2 3 123456789012345678901234567890 ISVID IFISN IRTO I Parity RTO I Pseudorang~ I Parity Pseudorange (cont.) I CIP Complete I Parity instantaneous carrier phase (cont.) I Doppler I Parity Doppler (cont.) I Parity

P arameter #b' Its S al F c e actor R ange LSB* al fl sc e ISVID 6 1 1-64 --Frequency 1 -- 0- L1, 1 - L2 --ISNratio 5 4 0-124 --

or 5 0-155 --Time of reception 16 l/1.0e+5 0-0.65535 seconds t 5.0 Ils offset from mod-ified Z count Pseudorange 32 l/5.0e+1O 0-0.0859 seconds t 0.05 ns

or l/5.0e+9 0-0.859 seconds t 0.50 ns Complete Instan- 32 l/1.0e+3 t 2.2e+6 cycles t 0.5 mcycle taneous phase or l/1.0e+2 t 2.2e+ 7 cycles t 5.0mcycle Doppler 28 l/1.0e+4 t 1.3e+4 cycles/s t 0.05 mcycle/s

Table 2 Proposed RTCM Message Type 18

Communications

No specific data link hardware has been selected for the prototype system but a variety of radio frequencies, modulation techniques, and data protocols will be tested. These tests will involve stations at fixed sites as well as stations in the land and aeronautic mobile services. Initial tests will utilize the 800 MHz cellular phone networks using standard high speed modems employing multicarrier Packetized Ensemble Protocol error correction. Bit rates in excess of 16 kilobits per second should be possible. Subsequent tests will employ VHF and/or UHF frequencies using AM, narrow band FM, and PSK modulation techniques using bit rates appropriate to the transmission bandwidths. It is anticipated that the AX.25 packet protocol will be used in these later tests.

SYSTEM TESTS

For these tests, single frequency pseudorange, carrier phase, and Doppler observations were generated for a monitor site and a roving receiver traversing the perimeter of a circle 20 kilometres in radius and centred at the monitor site. The nominal velocity of the roving receiver was 50 rn/s (180 krn/hour) but variations in the receiver range and height with amplitudes of 2 and 20 metres and periods of 6 and 600 seconds, respectively, were superimposed on the nominal trajectory resulting in additional vehicle accelerations of t 2 rn/s2. Observations were generated at a one second sample rate based on broadcast ephemeris data for Julian day 109 of this year. As shown in Figure 2, during the 42 minutes required to traverse the route, a minimum of five and a maximum of six satellites were visible above a mask angle of 15 degrees. Observation biases were modelled in the simulations by introducing satellite orbital errors of 20 metres in X, Y, and Z, satellite clock errors computed from the broadcast model, and ionospheric delays estimated by assuming

182

the ionosphere to be a thin spherical layer, 350 kilometres in height, and with a vertical electron content (VEC) ofO.5x1018 m-2• Signal multipath effects were also modelled as simple sine waves with periods of five minutes. Table 3 summarizes the multipath amplitudes and observation noise levels for the four simulations. Note that the values for the first and second tests are similar to what might be expected for a CIA-code GPS receiver while those for the final two tests are similar to what might be expected for a P code receiver.

0

E 6 I 30

e v

(. a t i 60 0 n

270 90

\ 9P

'1 12

/ 19

Azimuth

180

Figure 2 Day 109 Satellite Distributions

Error source Test #1 #2 #3 #4 Ephemeris 20m 20m 20m 20m Ionospheric delay (VEC) 0.5x1018 m-2 0.5x1018 m-2 0.5xl018 m-2 0.5x1018 m2

Pseudorange multipath ------- 3 m, 300 s ------- 0.5 m, 300 s Carrier phase multipath ------- 10 mm, 300 s ------- 10 mm, 300 s Pseudorange noise 3m 3m 0.3 m 0.3 m Carrier phase noise IOmm IOmm IOmm IOmm Doppler noise 5mm 5mm 5mm 5mm

Table 3 Parameters for Kinematic Simulations

For all test, the state space model for the roving receiver was based on a constant acceleration model with additional states for cycle ambiguities. Initial values for the position, velocity, and acceleration components were in error by approximately 20 km, 50 m/s and 2 m/s2, respectively, and the associated variance information for the states reflected this situation. Observation variances were identical to those used by the observation generator. At each one second epoch, the state of the rover was first predicted using simple state transition and process noise covariance matrices and then updated based on double difference observations. After each update, each estimated

183

double difference carrier phase cycle ambiguity and it's standard deviation were examined to detennine if the ambiguity could be fIxed.

Figure 3 shows model errors for the north, east, and height components of the position state and for the double difference ambiguities based on the observations generated for the fIrst test. The standard deviation of the estimated parameters are also shown in these plots. As may be seen, these results indicate that submetre positioning should be achievable under conditions similar to those used for the simulation but, since the ambiguities could not be fIxed within the forty minutes flight time, subdecimetre positioning accuracies would not be easily realizable. Figure 4 shows model errors for the second simulation which was identical to the fIrst with the addition of multipath effects. As would be expected, multipath effects cause even longer convergence times for the fIltre with the result that submetre, let alone subdecimetre, positioning could be somewhat more diffIcult

! ~~~: ~J ~. ~;;;; * ~.sf;

fO~~ iii --=;;";;F1;;;~1 ~J~~~ li -0.5

.,

'~ ~ o.s "---

B 0 ---.---------------

Ii = I, U ' f.ItItI' 'J .....-::: il!-o,s_

.~ 730 7«1 ?SO 7110

~-

Figure 3 Test #1 Model Errors

Figure 4 Test #2 Model Errors

184

Figure 5 shows model errors for the north, east, and height components of the position state and for the double difference ambiguities based on the observations generated for the third test. As may be seen, the double difference ambiguities are all fixed within fIfteen minutes with the result that subdecimetre positioning should be possible under conditions similar to those used for the simulation. Figure 6 shows model errors for the fourth simulation which was identical to the third with the addition of multipath effects. As would be expected, multipath effects cause even longer convergence times for the ffitre with the result that double difference ambiguities can only be fIxed after thirty minutes.

i ~l 44

~ ~··r i::~l ___ ~ I ~.sr

-t

r x D.S

I .• f -: ~"" t , - 01 ,_, II

?20 T.JO 7. '7SD -r6O 'l1III0_ Figure 5 Test #3 Model Errors

:II o·o:~ ~~.~r.~I~I~_~~ ____ ~ ~ 44

.. ..a.1

i o:~~ I °V ;;:&0""""'" I ..,."

-. :I o·:L 1~r:·a.2~"~~M--4.1_.~~6 •. -'

-~ 7.10 ,., 'ISO Dot_ Figure 6 Test #4 Model Errors

185

Table 4 summarizes the root mean square (RMS) model errors for the four simulations.

State Test #1 #2 #3 #4 North (m) 0.13 0.27 0.12 0.14 East(m) 0.19 0.56 0.08 0.13 Height(m) 0.33 0.61 0.20 0.22 North velocity (m/s) 0.01 0.01 0.01 0.01 East Velocity (m/s) 0.01 0.01 0.01 0.01 Hebtht Velocity (m/s) 0.02 0.02 0.02 0.02 SV 12-9 ambiguity (cycles) 0.53 1.26 0.52 0.63 SV 12-16 ambiJruitv (cycles) 0.20 1.02 0.20 0.30 SV 12-17 ambiguity (cycles) 0.80 2.40 0.38 0.65 SV 12-19 ambiguity (cycles) 1.38 2.52 0.44 0.55

Table 4 RMS Model Errors for Kinematic Simulations

CONCLUSIONS

Except for the communication hardware, all components of the prototype real-time system have been implemented and are currently undergoing testing and refmement. Postmission processing of simulated kinematic single frequency datasets indicates that submetre accuracies should be achievable using CIA code or P-code receivers but subdecimetre accuracies will only be realizable if a low noise pseudorange, such as that from a P-code receiver, is available and multipath effects are controlled to some degree. With the addition of a communication link, field testing of the prototype system will take place in the near future.

ACKNOWLEDGEMENTS

The development of the prototype real-time system is funded by a Natural Sciences and Engineering Research Council Strategic Research Grant entitled Applications of Differential GPS Positioning.

REFERENCES

Hatch, R. (1986). "Dynamic differential GPS at the centimetre level", Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, New Mexico State University, Las Cruces, New Mexico, May, pp.1287-1298.

Kleusberg, A. (1986). "GPS positioning techniques for moving sensors." Proceedings ISPRS Symposium, Stuttgart, September, Vol. 13, pp. 201-205.

Lachapelle, G., J. Lathaby and M. Casey (1984). "Airborne single point and differential GPS navigation for hydrographic bathymetry." The Hydrographic Journal (U.K.), 37(3):200-208.

186

Mader, G.L. (1986). "Dynamic posItIoning using GPS carrier phase measurements.", Manuscripta Geodaetica, Vol. 11, pp. 272-277.

Seeber, G., and G. Wubbena (1989). "Kinematic positioning with carrier phases and 'on-the-way' ambiguity solution.", Proceedings of the Fifth International Geodetic Symposium on Satellite Positioning, New Mexico State University, Las Cruces, New Mexico, March, pp. 600-609.

RTCM Special Committee No. 104 (1990). "RTCM recommended standards for differential NAVSTAR GPS service.", Radio Technical Commision for Maritime Services, RTCM Paper 134-89/SC 104-68.

Remondi, B.W. (1985). "Performing centimetre accuracy relative surveys in seconds using GPS carrier phase.", Proceedings of the First International Symposium on Precise Positioning with the Global Positioning System, Rockville, Md, U.S.A., April, Vol II, pp. 789-798.

Wells, D.E., N. Beck, D. Delikaraoglou, A. Kleusberg, E.l. Krakiwsky, G. Lachapelle, R.B. Langley, M. Nakiboglou, K.P. Schwarz, I.M. Tranquilla, and P. VaniCek (1986). Guide to GPS Positioning. Canadian GPS Associates, Fredericton, N.B., Canada.

187

SESSION3a

ESTIMATION METHODS AND QUALITY CONTROL

CHAIRMAN J.R.HUDDLE

LIITON GUIDANCE & CONTROL SYSTEMS WOODLAND IDLLS, CALIFORNIA, USA

Abstract

SOME ASPECTS 01' REAL-TIME MODEL VALIDATION

TECHNIqUES I'OR USE IN INTEGRATED SYSTEMS

P.J.G. Teunissen Delft Geodetic Computing Centre (LGR)

Department of Geodeay. Delft University of Technology Thijll8weg 11. 2629 JA Delft. The Netherlands

The Kalman filter produces optimal estimaton. but the quality of the estimators is only guaranteed of course as long as the assumptions underlying the mathematical model hold. Misspecifications in the model will invalidate the results of filtering and thus also any conclusion based on them. It is therefore of importance to have ways to verify the validity of the assumed mathematical model. In this paper a brief review is given of a general real-time detection. identification and adaptation (DIA) procedure for use in integrated navigation systems.

I. INTRODUCTION

It is well-known that the standard real-time minimum mean squared error navigation filter produces optimal estimators with well defined statistical properties [lJ. The estimators are unbiased and they have minimum variance within the class of linear unbiased estimaton. The quality of the estimaton is however only guaranteed as long as the assumptions underlying the mathematical model hold. Misspecifications in the model will invalidate the results of filtering and thus also any conclusion based on them. It is therefore of importance to have ways to verify the validity of the assumed mathematical model.

The objective of the present paper is to give a brief review of the detection, identification and adaptation (DIA) procedure. that has been developed during the last few yean at the DeHt Geodetic Computing Centre [2-4J. It has been developed for the real-time model validation of integrated navigation systems and can be seen as the natural extension of the static quality con.trol theory as described in [5-7J. The present review is partly based on the course -Special studies in numerical methods in geodesy and related surveying sciences- which the author was invited to give at the Department of Surveying of the University of Calgary in March 1990. Derivations of results are avoided in the paper. Instead we keep to the basic ideas involved and try to motivate the main results by appealing to intuition and geometric interpretations. Some variations and extensions of the theory, which are important for certain applications are mentioned but not further dwelt upon. For these more advanced procedures the reader is refered to the referenced literature.

The DIA-procedure discussed in the present paper consists of the following three steps

1. Detection.: An overall model test is performed to diagnose whether unspecified model errors have occurred.

2. Iden.tification.: Mter detection of a model error. identification of the potential source of the model error is needed. This implies a search among the candidate hypotheses for the mOlt likely alternative hypothesis and their mOlt likely time of occurrence.

3. Adaptation.: Mter identification of an alternative hypothesis. adaptation of the recursive nav­igation filter is needed to eliminate the presence of state vector biases.

The DIA-procedure is completely recursive and avoids the explicit use of a parallel bank of aug­mented navigation filters. It is based on the concept of a uniformly-most-powerful-invariant test­statistic and is applicable in principle to any dynamic system that fits into the frame work of the state-space formalism.

191

ll. FILTERING AND BIAS ACCUMULATION

It will be assumed that the reader is familiar with the standard navigation filter in state-space formalism [11. Based on the usual model assumptions, the optimal recursive prediction and filtering equations for the state estimator read

~,"'-l = ~,.,,"'-l~-ll"'-l +!be , ~I'" = ~I"'-l + K,.,[1[,., - A"'~I"'_ll , with corresponding variance matrices

(1)

(2)

This filter can be shown to produce optimal estimators of the state vector with well defined statistical properties. The state estimators are unbiased, are Gaussian distributed and have minimum variance within the class of linear unbiased estimators. It is important to realize however, that optimality is only quaranteed as long as the assumptions underlying the mathematical model hold. Misspecifications in the model will invalidate the results of estimation and this also any conclusion based on them. It is therefore of importance to have ways to verify the validity of the working hypothesis Ho for which (1) and (2) are optimal

In order to verify the optimality of (1) and (2), the working hypothesis Ho will be opposed to a class of alternative hypotheses Ha. The specification of appropriate alternative hypotheses for a particular application is non-trivial and probably the most difficult task in the process of quality control. It depends to a great extent on experience and ones knowledge of the dynamic system. For the present discussion we restrict attention to misspecifications in the mean and assume that all second moments of the various random vectors are specified correctly and known. This restriction still leaves room for a sufficiently large class of alternative hypotheses that contains the most frequently occurring model errors. An extension of the theory for the case the second moments are only partially known is discussed in [8,91.

For a misspecification in the mean of the state vector we consider the following class of alternative hypotheses:

(3)

where E{.} is the mathematical expectation operator. The matrix Oz,. is assumed to be known and the vector V z is assumed to be unknown. Furthermore, it is assumed that Oz,. i= 0 for l ~ i ~ m and zero otherwise. The times l and m are unknown. Thus we assume to know the type of model error that may occur, but not the time period in which it occurs.

It will be clear that the slip Oz,. V z in the dynamic model can typically accommodate under­parametrizations in the state vector. Assume for instance that the dynamics of a moving vehicle is based on a constant velocity model under Ho. Then, if at time l the vehicle starts accelerating linearly the constant velocity model becomes inadequate and an additional parametrization in the form of Oz,.Vz is needed. In a way that is completely analogous to (3), one can also model under the alternative hypothesis Ha, slips in the measurement model. A slip 0 11,. V 11 in the measumement model can typically accommodate outliers in the data, sensor failures and instrumental biases. For instance, an outlier at time l in the j-th observable can be modelled as 0 11,1 = (0 ... 10 ... 0)* with the 1 on the j-th place and 0 11,. = 0 otherwise.

If Ha is true, filtering under Ho will generally result in biased estimators. It is therefore of importance to know how particular misspecifications in Ho manifest themselves as biases in the state vector or functions thereof. Knowledge of the impact of model errors can then be used to set acceptance criteria for the sizes of these model errors. This is of importance for the design of an appropriate navigation filter and for the design of a powerful enough DIA-procedure. Before considering the impact of model errors, one should first make clear on what functions of the state vector the impact is studied. This depends very much on the particular application for which the navigation filter is designed and it may range from just one single function of the state vector to all n elements of the state vector. For instance, the impact on instrumental parameters mayor may

192

not be of interest, or, one may be particularly interested in position but not in velocity, or, as is the case in some real-time CPS-applications, it is the horuontal solution which is of interest and not the pseudo-range bias. In all these cases one generally has to specify a set of linear(iJsed) functions of the state vector that is of particular interest. H we auume that these functions are collected in a matrix F*, then it is the bias in F* ~'k which is of interest. The bias in can be computed once a measure for V III is available. An appropriate measure for V III is provided by the minimal detectable bitu (MDB). The MDB is defined as the model error V III that can just be detected with the DIA-procedure at a fixed probability level. For more details on the concept of the MDB's and their relevance as a diagnostic tool, the reader is referred to [3,10].

With a measure for V III available, the corresponding bias in F*~'k can be shown to follow as

k k

rVik'k = F*~k'k+1[~)II ~Hl,i[l- KiAi])CIIII,i]Vs (4) i=1 i=i

The computation of (4) is straightforward and can be done recunively in a way that closely f01lows the recursion of the actual navigation filter. Once the bias of (4) is known, its significance can be tested with the following bitu-to-noise ratio:

(5)

Note that this scalar measure is invariant against reparametriJsations of the state vector. Also note that (5) can be interpreted geometrically as the square of the length of the vector that f01lows from an orthogonal projection of V~'k onto the orthogonal complement of the nu11space of F* (see figure 1).

N(F") ___________ _ \7xklk

I

I

'L N(FO).l

P};(FO) \7 Xk,k

Fig. 1. Pj}(F*) Vik'k is the orthogonal projection of Vik,k along the nullspace of F*, N(F*).

It can also be shown that (5) provides an upperbound for the bias-to-noise ratios of the individual elements of F*~'k' see [S] but also figure 1. In the DIA-theory, the bias-to-noise ratio (5) is used together with the MDB's to set acceptance criteria for the impact of model errors. This can be done at the designing stage of the navigation filter.

H the bias-to-noise ratio (5) turns out to be unacceptable large and increasing as time proceeds, the usefulness of the actual navigation filter is of course completely nullified if the alternative hypothesis Ho. is indeed the true hypothesis. This phenomenon is related to what is known in the Automatic Control literature as the divergence of the filter [11,12]: after a certain period of operation of the filter, the biases in the state estimators eventually diverge to values entirely out of proportions to the precision values predicted by the filter. This is illustrated for an elementary example in figure 2a ..

193

30.-----------------------------------,

20

••• •

•• • •• ••

•• .­.' / , .

• •• ./ . .

10 •• •• • ... ' y ........ ~. /.-• • ..... _ ..... --......•••••..

• • ... -•• r. • o+-------~--------~--------~------~

o 10 20 30 40 0 10 20 (~) fix number (b)

30.-----------------------------------,

20

10

'. ·0 4· •• •• /

• /e. , .

•• .. -y • . 0..... .... /e

-0 -.-0 - ---.-•• J. • • • o+-------~--------~--------~------~

o 10 20 30 400

.......... ll •••••.••••• .!. •

10 20 (C) fix number (d)

30 40

fix number

• •••

•• • •• .' • ./ ./ . .. / Y

30 40

fix number

Fig. 2. (a) Divergence after fix 20, (b) Filtering under Ha for k ~ 0, (c) Filtering with increased process noise, (d) DIA at fix 23.

In this example, ~",,"'-1 = 1 and A", = 1 with C""i =i - 20,i > 20 and zero otherwise. There are a number of empirical techniques available that lesson such degradations due to model errors. One, albeit drastic approach is to include all potential sources of model errors in the mathematical model. This at least assures that the filtered state vectors are unbiased. But the disadvantage of this approach is of course that it leads to overparametrizations for those periods of time where Ho holds. This is shown in figure 2b, where filtering is done under Ha for the complete time period. A more subtle approach is based on increasing the process noise of the dynamic model. By increasing the process noise, one in effect increases the variance matrix of the predicted state as it is used in the filter. This has as consequence that less weight is given to the predicted state and that the bias accumulation gets damped. This is shown in figure 2c. The advantage of this approach is its conceptual simplicity and ease of implementation. And it has proven its value for a number of important applications [131. The method has however also a number of drawbacks. First of all, one may question whether it is appropriate to use aecond moments as a tool to control the fird moments. Of course, the increase in process noise may bound the impact of model errors and thus eliminate divergence. But it will never eliminate the presence of bias completely. A second drawback of increasing the process noise is that one obtains a filtered state estimator that is noiser than it needs to be for those time periods where Ho holds. This would not be the case, if one would know at what time instant to increase the process noise. But knowing this, implies knowing the starting time I of the model error. This leads therefore to the necessity of having available a detector of model errors. Finally, a third drawback of increasing the process noise is that it in effect reduces the redundancy

194

information of the filter. Assume for instance that each measurement-update of the filter is based on only one measurement. A possible outlier in this measurement can then be detected by comparing the measured value with its predicted value based on the predicted state vector. The power of the outlier detection deteriorates however with an increase in the variancematrix of the predicted state. As a consequence, outliers in the data may pass unnoticed due to an increase in process noise and leave possible unacceptable biases in the filtered state vector.

A third approach of lessening degradations due to model errors is based on the concept of limited memory filtering [14,15]. The idea is here to base the filtering only on recent data in a moving window or fading window. In this approach the weight given to past information decreases as time proceeds. As a result the dependence of the present filtered estimator on past information decreases, with the effect that model errors are also damped out. The concept of limited memory filtering is closely related to the concept of increasing of the process noise. In fact, it can be shown that exponentially weighting of data is fully equivalent to an assumption of increased process noise [IS].

Although the above approaches for eliminating divergence have proven their value for many im­portant applications, it is the author's opinion that they are not really suited for handling alternative hypotheses as specified by (3). The methods are too empirical with no clearcut optimality prop­erties. Nevertheless, the basic ideas of these approaches are sound and will (perhaps ironically) be seen to reappear in the DIA-theory.

m. DIA: DETECTION, IDENTIFICATION, ADAPTATION

Detection

The objective of the detectionstep is to test the overall validity of the mathematical model Ho. Detection is only possible if the navigation system provides for redundant information. It is this surplus of information which enables one to test the statistical consistency of the data with the model. Fortunately, the mathematical model on which the navigation filter is based has a built in redundancy, because of the presence of the dynamic model. This implies that the redundancy of each measurement-update k is equal to the number of measurements, say m/u at this update. The statistical consistency checking of the data with the model is based on the predicted re6iduai6. The predicted residual at time k, llIc, is defined as the difference between the actual ml;-vector of observables at time k and the predicted vector of observables based on the predicted state vector: llIc = Ill; - AI;~II;_l' It can be shown that under the working hypothesis Ho, the sequence of predicted residuals constitute a Gaussian distributed white noise process [4]. It is exactly this knowledge of the distribution of the predicted residuals under Ho that enables one to test the validity of the assumed mathematical model. Our global overall model (GOM) teststatistic for testing at time k, the overall validity of the mathematical model Ho is given as

(6)

with the input ,£1; = !4cQ;;"llllc, where Qv" is the variancematrix of llIc, and gain gl; = l/[E~=1 mal where ma is the number of observables at time i (see figure 3).

The filter is initialized with r,l = O. Note that the GOM-teststatistic is computed recursively and that it is based on the predicted residuals which are readily available during each measurement update of the navigation filter. The GOM-teststatistic has been normalized, primarily for graphical purposes, such that its expectation under Ho is given as E{T.',I;IHo} = 1. Since its distribution under Ho follows a central F-distribution, the overall validity of Ho is rejected at time k and an unspecified model error is considered present in the time interval [l, k] if and only if 1",1; ~ Fa (91;, 00,0), where Q is the upper probability point of the central F-distribution with gl;,oo degrees of freedom.

195

Tk +/T\ gk +J''\ X/,k.

'-Y'- \. ,.1+

mk Delay

Fig. 3. The recursive GOM-teststatistic.

An important practical problem with the above GOM-test is the choice for I, the time that the model error is assumed to be starting to occur. Since the starting time of the model error is unknown a priori one has to start in principle with I = 1. But a &xed value for I, turns the filter (6) into a growing-memory filter, with the potential practical problem of a possibly long delay in time of detection. Rejection of Ho at time Ie, may imply namely that a global model error started to occur as early as time I = 1. In order to reduce the time of delay, a moving window of length N is introduced by constraining I to Ie - N + 1 5 ISle. When choosing N, one of course has to make sure that the detection power of the GOM-test is still sufficient. This is typical a problem one should take into account when designing the navigation filter. With the finite window of length N, the filter (6) is essentially reduced to a finite-memory filter.

Instead of using a finite window, one may alternatively use a fading window. By setting I equal to 1, and replacing the gain g1c = 1/[E:=, Frail by the gain g1c = w1c I[E:=, FraiW'I, with w ~ 1 the filter reduces to a lading-memory filter. Note that E{x',1c IHo} = 1 still holds with the fading window. The GOM-teststatistic however, instead offollowing an F-distribution now follows a linear combination of independent X2-distributions. With the fading window, the same recursive filter structure (6) is retained. This becomes advantageous when compared to the finite window, if a particular application requires the use of long windows. The weight factor w, which determines the nominal length of the fading window, is chosen on the basis of the detection power of the GOM-test. A useful approximation to the nominal length of the fading window is given by (w - 1) -1. A value of w = 1.2 would then correspond to a window length of N = 5. Although the type of window to use depends very much on the particular application at hand, one should keep in mind that the choice of the window length must always be based on the required detection power of the GOM-test.

The teststatistic (6) is termed the global overall model teststatistic, since it is designed to test the overall validity of the model and to detect unspecified global unmodelled trends. It can be shown that this teststatistic has the optimality property of being a uniformly-m08t-powerful-invariant teststatistic. Loosly speaking this means that one has with the GOM-test, the highest probability of correctly detecting unspecified model errors. In our applications of the DIA-theory, a distinction is made for practical reasons between local validity and global validity of the model. This distinction is introduced in order to have better detection and separation capabilities for model errors that have either a local or more global character. This implies that in the actual implementation of the theory detection consists of both a global overall model test and a local overall model test [2,4,91.

196

Identification

The next step after detection is the identification of the most likely model error. For identification, candidate alternative hypotheses need to be specified explicitly. For the present disc1lllion we restrict attention to the clall of alternative hypotheses as specified by (3). As with detection, identification gratefully makes use of knowledge of the distributional properties of the predicted residuals. In order to obtain a most sensitive teststatistic for identification purposes, we first need to know how a slip C.,i V. propagates into the predicted residuals. Let us denote the propagation of C.,i, for lSi S Ie, into the space of predicted residuals by the matrix C".. The matrix C". can then be found from simply following the recursion of the navigation filter. This is shown in figure 4.

C •. k + +

Fig. 4. The recursion of C" •.

Cu.

I I

:L 0---------9~~----~_

C"

*Q-1 F· 5 Th hal·· , Ie C,," lL 19.. e ort ogon proJect10n: I.' = (*Q-1 )1/2.

C" " Cv

With the matrix 0". available, we are now in the position to formulate the appropriate teststatis­tic. Let us aIIume for simplicity that the model error is one-dimensional and therefore that C". is an mle-vector, which will be denoted by the lower case kernel letter C" •• The appropriate slippage testst atistic for the identification of the slip C.,i V., lSi S Ie, fonows then from an orthogonal projection of the vector of predicted residuals lL onto the vector cu. This is shown in figure 5. The corresponding gloW .lippage (GS) teststatistic reads therefore

Ie ~ * Q-1 {-J C"i Vi ~

t,,1e = Ie .=, (7)

IE c;,Q;.1c",]1/2 i=1

As will be intuitively clear, it is the orthogonal projection that quarantees that the GS-teststatistic (7) is most sensitive for the model error Cs,iV s. Note that, with Q", and ~ readily available from the navigation filter and the e8icient recursion of C"i' the computation of the GS-teststatistic parallels that of the actual navigation filter.

Strictly speaking, the GS-teststatistic (7), has to be computed for each alternative hypothesis considered and for each Ie ~ I. However, since I is unknown a priori, one has to start in principle with I = 1. This implies that one has to compute Ie number of teststatistics per alternative hy­pothesis at the time of testing Ie. As a result one obtains a te"m(Jtns of increasing order with the GS-teststatistics as entries. This is shown in figure 6a. Clearly, this is unpractical, both from a computational point of view as well as because of the possible increase of the delay in time of iden­tification. Fortunately, not all entries of the testmatrix may be necessary if one studies the power of the test statistics. Although the power will increase theoretically for an increasing sile of the interval I', Ie], the gain could be negligible for all practical purposes. This motivates, in accordance with our disc1lllion of detection, the use of a moving window. This is shown in figure 6b for the case Ie - N + 1 SIS Ie and in figure 6c for the case Ie - N + 1 SIS Ie - M. The rationale behind this last constraint is that in some applications the GS-teststatistic may be too insensitive for global model errors if I > Ie - M. Instead of l18ing a finite window, the concept of a fading window, as discussed for detection can be applied to the teststatistic (7) as ·well.

197

~

T" t1,2 t1,S t 1,4.

[ t'·' t1,2 •

t~·l [" t 1,2 • •

t2,2 t2,3 t2,4 t2,2 t2,3 • t2,3 • t3,3 t3,4 t3,3 • t3,4

t4,4 t4,4 • . H. H. H.

Fig. 6. Testmatrix of GS-testatatistic with (a) no window, (b) a moving window with Ie - 2 < l ~ Ie and (c) a moving window with Ie - 2 < l ~ Ie - 1.

With the window8 introduced, we are now in the position to describe our identification procedure. At the time of testing Ie, one first determines per alternative hypothesis the value of l in the window for which the sample value of r,le is at a maximum. In other words, the kth column of the testmatrix is searched for the entry which is at a maximum. The corresponding row number of the teatmatrix then identifies l as the mOlt likely time of occurrence of the model error if the corresponding alternative hypothesis would be true. In order to find both the mOlt likely alternative hypothesis and mOlt likely value of l, the sample values of maSIe-N+1S'SIe-Mf,1e for the different alternative hypotheses are compared. The maximum of this set finally identifies the mOlt likely time of occurrence l and mOlt likely alternative hypothesis.

The procedure described above contains the minimum requirements for identification. For some applications it may be neces8ary to develop a more advanced identification procedure. For in8tance, it may be nece88ary to discriminate between local and global8lippages in the mean, or, to take care of p088ible masking effect8 due to different model errors, or to identify potential source8 of model errol's for which V s is known. For the changes and/or extention8 of the identification procedure which are required to accomodate these 8ituation8, the reader is referred to [4,8,9].

Adaptation

After identification of the mOlt likely alternative hypothesis, adaptation of the recur8ive navigation filter is needed to eliminate the presence of biases in the filtered 8tate of the navigation system. In order to be able to adapt the filter, one first needs an estimate of the identified model error V s' It will be intuitiviJy clear that the estimate of V s has to depend in lOme way on the predicted residuals. In fact one can 8how that the best estimator (in the sense of minimal variance) of the model error in the 8pace of predicted residuals, is given by the orthogonal projection of the predicted residuals onto the range-8pace 8panned by the column8 of the matrix formed by the C,,; [3]. With this re8ult and the whitene88 property of the predicted residuals follows then that the be8t estimator of V s can

be computed in a recursive form as 8hown in figure 7. This recursive filter is initialised with V'" = [C:,Q;,lC",]-l[CvIQ;,ll!l] and its gainmatrix is given as Gle = QV1 .• -1C:.[Q". +C".Qvl .• _1C:.]-1,

h Q . h' . fV .. "Ie-l w ere V' .• -l 18 t e V&rlancematnx 0 _ •

With the estimator V' ,Ie available, we are now in the position to correct the filtered 8tatevector at time Ie for the presence of bias. The adaptation cOnsist8 of 8ubtracting the 6iu accumulation due

to V,,1e from the 8tate filtered under Ho. The adapted filtered 8tatevector read8

.. /I .. 0 ~ -b,,1e ~11e = ~11e - Air,le..!:. • (8)

198

!LA: +J~ G"

+/"'I'\ Vi,,,

\.. ,/ \..

"'+

CIJIr. Delay

Fig. 7. The Recursion of *""'.

The estimator ~I'" is unbiased and corresponds with H 4 • This shows that explicit filtering under H4 is not necessary and that the results can be based on the nominal navigation filter. The variance

matrix of ~I'" follows from applying the error propagation law to (8). Since ~I'" and V"'" can be shown to be uncorrelated, the variance matrix of ~I'" follows simply as

(9)

The matrix xt", of (8) and (9) corresponds with the bias accumulation matrix of (4). Both

xL,,,, and V"'" of (8) can be computed efficiently in recursive form. This follows essentially from a combination of figure 4 and figure 7. The corresponding recursion is shown in figure 8. The recursion of xt", is initialised with X:" = II - K,A,]Cs ".

Vir. +f +./ {ll,k

'-V_ GIr. '+

- Cu. Delay ~

i

C.,1r. + -Air. KJr.

+,... Xi,k + "+

'--- ~Ir.,k-l Delay ~ -

Fig. 8. The recursion of V','" and xt",.

With (8), adaptation takes place at time k. This implies that strictly speaking the filtered states remain biased between I and k. Thus it would be theoretically more correct to adapt all filtered

199

states in the time interval [I, k]. This is posaible and may even be required for some applications. The corresponding adaptation involves smoothing, but is again based on an expression like (8). The reason why we have focussed in the present discussion on the above simple approach of only adapting ~Ik' is twofold. Firstly, in real-time applications it is the present estimate of the state vector which is of interest and not so much the past estimators. And secondly, the bias in the state vector for times between I and k may considered to be negligible if the built up of the model error is still too small to be detected with the GOM-teststatistic. Note that this again points out, that one should choose the window length and detection power in relation to the biases one is willing to accept.

References

[1] Gelb, A. (Ed.) (1974): Applied Optimal Estimation. The M.I.T. Press.

[2] Teunissen, P.J.G. and M.A. Sahmann (1988): Performance Analysis of Kalman Filters. Pr0-ceedings HYDR088, Special Publication, No. 23, Hydrographic Society, pp. 185-193.

[3] Teunissen, P.J.G. (1989): Quality Control in Integrated Navigation Systems. IEEE Aerospace and Electronic Systems Magazine, Vol. 5, No.7, pp. 35-41.

[4] Teunissen, P.J.G. and M.A. Sahmann (1989): A Recursive Slippage Test for Use in State-Space Filtering. Manuscripta Geodaetica, Vol. 14, No.6, pp. 383-390.

[5] Baarda, W. (1968): A Testing Procedure for Use in Geodetic Networks. Netherlands Geodetic Commision, Publications on Geodesy, New series, Vol. 2, No.5, Delft.

[6] Teunissen, P.J.G. (1984): Quality Control in Geodetic Networks. In: Optimization and Design of Geodetic Networks, Grafarend, E. and F. Sansa (Eds.), Springer Verlag, Berlin, pp. 526-547.

[7] Staff of DGCC (1982): The Delft Approach for the Design and Computation of Geodetic Networks. In: Forty Years of Thought, Delft, pp. 202-274.

[8] Teunissen, P.J.G. (1986): Adjusting and Testing with the Models of the Affine and Similarity Transformation. Manuscripta Geodaetica, Vol. 11, pp. 214-225.

[9] Teunissen, P.J.G. (1990): An Integrity and Quality Control Procedure for Use in Multi-Sensor Integration. Paper presented at ION Satellite Divison, 3rd International Technical Meeting, Colorado Springs, USA.

110] Sahmann, M.A. (1990): MDB: A Design Tool for Integrated Navigation Systems. Paper pre­sented at Int. Symp. on Kinematic Systems in Geodesy, Surveying and Remote Sensing, Banff, Canada.

Ill] Fitzgerald, R.J. (1971): Divergence of the Kalman Filter. IEEE Transactions on Automatic Control, Vol. AC-16, No.6, pp. 736-747.

[12] Schlee, F.H., C.J. Standisch and N.F. Toda (1966): Divergence of the Kalman Filter. AlA A Journal, Vol. 5, No.6, pp. 1114-1120.

[13] Jazwinski, A.H. (1970): Stochastic Processes and Filtering Theory. Academic Press, New York.

[14] Sorenson, H.W. and J.E. Sacks (1971): Recursive Fading Memory Filtering. Information Sci­ence,3.

[15] Anderson, B.D.O. (1973): Exponential Data Weighting in the Kalman-Bucy Filter. Information Science, 5, pp. 217-230.

200

FAULT DETECflON AND ESTIMATION IN DYNAMIC SYSTEMS

M. Wei, D. Lapucha and H. Martell

Department of Surveying Engineering The University of Calgary Calgary, Alberta, Canada

Abstract For state estimation in dynamic systems the standard Kalman filter requires a complete knowledge of the system model and the statistical information of the system. In this paper, a statistical technique for the detection and estimation of model errors caused by failures in the system model or the measurement model is discussed. The procedure is based on the residual characteristics of the Kalman filter. Statistical decision theory is used to derive hypothesis tests of the innovation sequence for fault detection and diagnosis. Failure parameters for the model errors is estimated via a two-step Kalman filtering technique, which is an adaptive algorithm. A numerical example illustrates the applications of the proposed method in kinematic positioning by using GPS and INS.

1. Introduction

The Kalman filter is often used to estimate states of a dynamic system. The estimate is optimal as long as. the dynamic system and the observations can be modeled quite accurately. The prior statistics for the stochastic noises of the system and the observations must also be well known. In many cases, the above assumptions are not fulfilled. The problem of estimating the unknown covariance of the system noises and measurement noises has been discussed by several authors (Jazwiski, 1970; Mehra, 1970).

The problem, which will be discussed in this paper, deals with a failure in the dynamic system or in the measurements, i.e. an incorrect model for the dynamic system and the sensor output. In many papers (Willsky, 1976; Willsky and Jones, 1976; Teunissen, 1990), the hypothesis test method is used to detect a failure of the dynamic system and the observations. In this paper, the failure in the dynamic system or in the sensor output is modeled by unknown bias terms. Using the hypothesis test method, a real-time procedure to detect the dynamic failure is discussed. The concern here is especially with failure due to abrupt changes in the system behaviour and observations and their effect on state estimation. The generalized testing algorithm can be specialized to isolate different failure sources based on the failure model.

201

Particular attention will be paid to an integrated kinematic positioning system using GPS and INS.

A two-step Kalman filtering technique, which was developed to estimate the states of a dynamic system with a constant, but unknown, bias vector (see Friedland, 1969; Ignagni, 1981) is used to estimate the failure parameters after fault detection and identification. This adaptive algorithm has advantages in terms of flexibility and efficiency for estimating the effect of the failure.

2. Failure Model and Error Analysis

It is well known that a dynamic system can be described by using the standard state spaCe model

Xk+l = cJ)k+l,k Xk + Wk (1)

where Xk is state vector of the dynamic system and Wk is the dynamic noise with covariance matrix Qk.

lf the states of the system are observable, the measurements can be written in the form

(2)

where Zk are the measurements and Vk is the measurement noise with covariance matrix Rk.

Based on the above model and assuming that the dynamic noise and measurement noise are white in nature, an optimal estimate of the states Xk is obtained by using the Kalman filter

(3)

(4)

with corresponding covariance matrices

(5)

(6)

where

(7)

202

The estimates X k are only optimal under the assumption that normal operating conditions exit, and that the model conforms to this given system behaviour. For the problem of fault detection via hypothesis testing, the normal operating condition is regarded as the null hypothesis Ho.

In the case that a failure in the dynamic system or in the measurements occurs, the above assumption is not valid. The estimates x k are therefore not an unbiased estimate of the true states Xk. The specification of failure is the most difficult component of the estimation procedure. Usually a failure in the dynamic system or in the measurements can be described by additional unknown biases in the dynamic model or in the measurement equations. These represent the uncertainty parameters of the failure model. This approach is particularly reasonable for an abrupt change of the system or the measurements.

A dynamic system which has undergone failure can be formulated by adding the unknown biases as follows

(8)

Zk = Hk Xk + Dkd + Vk (9)

where band d are unknown bias vectors describing the failure in the dynamic system or in the measurements. In the following, it is assumed that the statistical behaviour of the dynamic noises and measurement noises of the model (8) to (9), is well known.

H the bias vector b and d are assumed known, the optimal estimate of the states Xk are given by

(10)

(11)

It is apparently that if the dynamic system is under the normal condition, the estimates (lO) and (11) are reduced to (3) and (4). Equations (3) and (4) do not contain the bias terms therefore the estimates from (3) and (4) are termed bias-free.

Any errors in the nominally bias-free estimates given by (3) and (4) are apparent in the additional terms modelled in equations (8) and (9). As derived in Appendix I, errors in the bias-free estimates Xk(-) and Xk(+) are given by

Xk(-) - ik(-) = st(-)b + ~(-)d (12)

203

(13)

As mentioned before, the normal operating condition described by (1) and (2) are considered as the null hypothesis 1-10. The alternative hypothesis Ha can be represented by equations (8) to (9), that is the failure model. The problem of fault detection can thus be formulated as a problem in testing the null hypothesis 1-10 against the alternative hypothesis Ha.

The null hypothesis, 1-10 and the alternative hypothesis, Ha for detecting failure in the model given by (8) to (9) are formulated by

3. Innovation Sequence

Ho: b=O; d=O;

Ha: b~O; d~O.

(14)

(15)

To test the validity of the model under the normal condition, the predicted residual plays an important role. The predicted residual is defined as the difference of the actual system output and the predicted output of the bias-free estimates in the form

(16)

The predicted residual is not only an error signal as estimated by the Kalman filter, but also represents the new information brought by the latest observation Zk, updating the estimates x k(-). It is therefore called the innovation sequence.

Under the normal condition, i. e. the hypothesis Ho, it is well known that the innovation sequence rk is a zero mean Gaussian white noise sequence with covariance

(17)

where

(18)

The predicted residual fk is thus the Gaussian variable with distribution

I'k - N ( 0, PI'k ) . (19)

H, however, the model is incorrect, the predicted residual (16) becomes large and contains systematic trends because the model does not represent the

204

actual system. For the failure model (8) to (9), the predicted residual (16) of the bias-free estimates from equations (3) and (4) is given by

(20)

where Gf and Ge are the failure signature matrices which provide a relationship between the predicted residual rk and the unknown biases b, d (see Appendix I for the derivation of equation (20». Given the failure model with unknown bias terms (8) to (9), one can compute each effect of the different failure source on the predicted residual fk.

Equation (20) can be written in a simple form

(21)

where Gk = (Gt G~), u = (b, d )T. In equation (20), Uk is the actual predicted residual of the estimates (10)

and (11) based on perfect knowledge of the biases b and d in the form

(22)

Since the estimates Xk(-) and Xk(+) from equations (10) and (11) are optimal, the predicted residuals Uk, are a zero mean Gaussian white noise sequence with the same covariance as (17), that is

(23)

The reason that the covariance of both the predicted residuals rk from equations (3) and (4), and Uk from (10) and (11) are identical, is that the statistical behaviour of the noises for both models is assumed the same. The predicted residual rk, (20), for the failure model is thus a Gaussian sequence with the following distribution

fk - N ( Gt b+ G~ d, Pfk ) . (24)

Based on the statistical property of the predicted residual rk under both the null hypothesis Ho and the alternative hypothesis !fa, the hypotheses (14) and (15) can be reformulated as follows

(25)

(26)

205

4. Hypothesis Test

Based on the statistical property of the innovation sequence, a number of possible statistical tests can be performed (see Mehra and Peschon, 1971; Willsky, 1976; Teunissen, 1990).

To detect an unspecified system failure or gross errors in the measurements, a simple chi-squared test on the innovation sequence is performed first. The corresponding test statistic is given by

(27)

Under the null hypothesis (25), the test statistic (27) is a chi-squared variable

k Tl - X2 (:Lmi, 0 )

i=l

where mi is the dimension of the innovation vector rio

(28)

This test statistic can be used to test whether a failure occurs or not. At a certain level of significance a, a decision rule is given by

Tl < X2 no failure· a' , (29)

Tl ~ X~, failure.

This method simply provides a warning alarm, i.e. one can only detect whether a failure occurs and makes no attempt to identify the source of failure. This is usually the first step of the testing procedure.

After the fault alarm test, a failure identification which isolates the potential source of failure in the system model, or in the measurements, is needed. For this purpose, the generalized likelihood ratio approach is used. In order to test the null hypothesis Ho (25) against the alternative hypothesis Ha (26), the test statistic for the generalized likelihood ratio test is computed in the form

(30)

or

206

k k k T3 = (1: Gf Pi-} ri)T [ 1: Gf Pi-} Gi rl (I GT Pr} ri) . (31)

i=l i=l i=l

Following Teunissen and Salzmann (1989), the test statistic (30) is used for local model testing, while the test statistic (31) is computed for global model testing. The difference is that the first one, performed at time k, only depends on the predicted residual at the same time k while the latter uses the predicted residuals from multiple epoches. For real-time testing, the first test is more interest.

Due to the statistical properties of the predicted residuals rk, under the considered hypotheses, the test statistic T2 has the following distributions

(32)

where nk is the dimension of the biases u and the non centrality parameter Ak is given by

(33)

Given the significance level a, the decision rule for testing the null hypothesis Ho (25) against the alternative hypothesis Ha (26) is formulated as

(34)

This test procedure can be used to identify the different failure sources in the dynamic model or, in the measurements, if one specializes the test statistic (30) to test the null hypothesis Ho with a bias for each special failure source. For instance, to detect an outlier in one measu.rement z~ one formulates the hypotheses for one bias in the measurement zit as follows

(35)

Ha: di ~ o.

The failure model for the measurement equation (9) can be written as

D~ = (0, ... , 1, ... , O)T . (36)

207

This is similar to the data snooping approach in least squares adjustments techniques (Baarda, 1968), if one tests all hypotheses for each individual measurement zt (i = 1, ... , n).

It should be mentioned that using the generalized likelihood test to isolate the different failure sources requires a knowledge of each possible failure in the dynamic model or in the measurements.

5. Two-step Kalman Filter

Equation (21) provides a relationship between the predicted residual rk from the bias-free estimation (3) to (7) and the bias vector u, representing a possible failure in the dynamic system or in the measurements. It is apparent that the noises Uk in equation (21) are uncorrelated, if the dynamic noises and measurement noises are white and modeled quite accurately. Based on the above assumptions the maximum likelihood (ML) estimation method can be used to estimate the biases band d under the accepted alternative hypothesis Ha·

Using the the predicted residuals ri (i=l, ... , k), the ML estimate G1,k is computed by

k k G1,k = [L Gt Pi-} Gi r1 L Gt Pr} ri

i=l i=l

with the covariance matrix

k PU1,k = [ L Gr pr} Girl.

i=l

(37)

(38)

This procedure has a disadvantage in terms of delay of the detection procedure due to requirement of the predicted residuals from several epoches for estimating biases u. A real-time estimation procedure is suggested below.

One can consider the bias vector u as a state vector in the form

Uk+l = Uk (39)

with equation (21) as the corresponding measurement equation given by

(40)

associated with the noise matrix (23). The Kalman filter is therefore used to estimate the biases u. The estimate Gk is computed by

208

(41)

(42)

with the corresponding covariance matrices

(43)

(44)

where the Kalman gain is of the form

(45)

It is clear that the recursive forms (38) to (45) can be computed in real-time, if the hypothesis test is implemented in real-time. The equivalence of the ML solution (37) to (38) and the estimates (41) to (44) by using the Kalman filtering technique has been discussed by (Chang and Dunn, 1979).

The bias-free estimates Xk can be corrected, if the biases u are estimated. As discussed in Appendix I, the states Xk under the alternative hypothesis Ha can be computed by

(46)

with the covariance matrix

(47)

The above algorithm (41) to (47) together with the bias-free estimation (3) to (7) is termed two-step Kalman filter. At the first step, the states Xk are estimated by using the bias-free model (1) to (2) under the null hypothesis Ro. lf the null hypothesis Ho is rejected through the hypothesis test, the alternative model (8) to (9) will be assumed. It indicates that a failure in the dynamic system or in the measurements has occured. After failure identification, one can estimate the biases b, d in equations (8) and (9) and correct the estimates x k from the first step to get the estimates Q k under the accepted alternative hypothesis Ha. At the second step, the simple Kalman filtering procedure (41) to (45) is used to make an optimal estimate for the biases ~ and dk based on the predicted residual rk from the first step filter. It

209

is obvious that the Kalman filtering at the first step and at the second step can be implemented in parallel fashion.

One can perform a one step Kalman filter to estimate the states Xk and biases b, d of the model (8) and (9) simultaneously, if a failure is detected. This algorithm was used for GPS/INS integration (Wong et al., 1988) and for semi­kinematic GPS (Cannon, 1989). Compared to one step Kalman filter, the two­step Kalman filter has advantages in terms of flexibility and efficiency.

6. Application in Kinematic Positioning using GPS and INS

The fault detection and estimation procedure, discussed above, can be applied to kinematic positioning by using differential GPS and INS. For precise positioning, the double differenced phase measurements are used for GPS. Because of the frequent occurrence of cycle slips in the phase measurements in a dynamic environment, determining the unknown ambiguity in the phase measurements after cycle slips will be the major problem for kinematic GPS. A strapdown inertial system provides differential positioning, velocity, and attitude along the trajectory at high rate and with high accuracy over a short time period. Thus, using INS results for the cycle slip detection and correction will be feasible, if both GPS measurements and INS data are compatible.

In the following, applying the hypothesis test to kinematic GPS and to GPS/INS integration will be discussed.

1) Kinematic GPS

The error states for kinematic GPS are of the form

XGPS = (Sr, Sv)T

where Sr is the vector of positioning errors of GPS; Sv is the vector of velocity errors of GPS.

(48)

Equation (48) means that a constant velocity model for the GPS Kalman filter is assumed, as discussed by (Schwarz et al., 1990). In a real environment, a vehicle does not always move with constant velocity. In this case the model given by (48) will be incorrect and should be corrected by adding new unknown states, which represent the nonconstant velocity parameter. These are treated as biases b, in the dynamic equation.

The measurements for the model (48) are usually double differenced carrier phase and phase rate in the form

V.1~ = V.1p + AVLW (49)

210

(50)

Equation (49) shows that the ambiguity parameters remain in the phase measurements. This means that the occurrence of cycle slips will effect the phase measurements. One possible failure in the GPS measurements is then cycle slips in the phase data.

In order to apply the hypothesis test to the GPS filter for the dynamic model (48), with the measurements (49) and (50), the hypotheses for possible failure can be assumed as follows

Ho : b=a=Oi d=VadN=O (51)

Ha: b=a~Oi d=VadN~O (52)

where the nonconstant velocity bias b, will be an unknown acceleration a, and V AdN indicated double differenced cycle slips in the phase measurements.

The failure model in equations (8) and (9) for the hypotheses (51) and (52) can be written as

Bk = ( iat2, 0, 0, at, 0, 0 ) T (53)

for the acceleration bias, and

D~ = ( 0, ... , 1, ... , 0 ) T (54)

for cycle slips in the i-th measurement. Using such failure model given by (53) and (54), one can distinguish cycle

slips and the model error, and eliminate the effect of the model error on the cycle slip correction.

2) GPS/INS Integration

Usually, the strapdown INS is considered as the reference system, which provides position, velocity and attitude information. The GPS measurements are used to update the INS solution and estimate the error states of the INS.

To estimate the errors of a strapdown inertial system, a state vector with 15 error states can be used. It is of the form

XINS = (E, ~r, ~v, d, b) T (55)

where

211

£ is the vector of attitude errors in the local-level frame; Sr is the vector of position errors in the local-level frame; Sv is the vector of velocity errors in the local-level frame; d is the vector of gyro drifts about gyro axes in the body frame; b is the vector of accelerometer biases in the body frame. For a strapdown inertial system, the sensor failure is difficult to model and

isolate because of the high rate output of the inertial system. In order to detect a failure of an inertial system in general, biases for the coordinates or velocities from inertial system are assumed, instead of a more complicated model.

The GPS measurements, used for the GPS/INS integration, are double differenced phase and phase rate. As discussed above, the failure in the GPS measurements can be cycle slips. The hypotheses for applying hypothesis testing to GPS I INS can be

Ho : b = dr = 0; d = V L\dN = 0 (56)

Ha : b = dr ~ 0; d = V L\dN ~ 0 (57)

where dr is the INS coordinate bias and V L\dN is the ambiguity bias through cycle slips.

Through the hypothesis test, cycle slips in the phase measurements will be detected and isolated from other failure sources like model errors and sensor failures. Also, each gross model error and sensor failure will be detected. The two-step Kalman filtering algorithm is then used to estimate the failure biases. These algorithms can efficiently distinguish the different failure sources from the GPS and INS and estimate their effects, if the failure can be modeled correctly.

7. Test Results

To implement the hypothesis test method, software for using the fault detection and estimation procedure in kinematic GPS or in GPS/INS integration has been developed at the University of Calgary.

The data was collected via a vehicle moving with stable speed. The collected GPS data did not contain cycle slips. Artificial cycle slips were generated for the phase measurements from certain satellites and at different times. Using the method, discussed above, the cycle slips as well as the model errors for GPS and INS are tested.

Figures 1 to 3 show the results of the kinematic GPS survey. The positioning results after cycle slip detection and correction are compared with positioning results without cycle slips. Results show that an accuracy at two to three cycles level is achievable and that nonconstant velocity biases are detected and compensated.

212

-s -

'U ------------------ - - - --

O.1l '"'" I I

~.l "", I

-0.2 "" !.J.--... ....

--_ ...... _--'-.... ---~ ---.... ~ .... --

• -=---- -~-~ .......... -- ............... --

t.titude

10 ngJ tude

height

-0.4 '---------..&.-,--_____ '--, ______ ...J

o 1000 2000 3000

time (5)

Fig. 1 CycJe slips in two satellite phase data

01

~\ 0.0 \

\

i \

-e --E -0.2

. -.Q.4 . .... ---. -----.-.- -

-0.6 0 1000 3000

time (5)

Fig. 2 Cycle sUps in five satellite phase data

213

-I -I! ! "

-E -" ~ j -..,

1.0

0.5

0.0

.().5

-1.0

.. . -0

.. - - - -- - -1(0) 2(XX)

time (sec)

fi&. 3 Occurrence of cyde slips at differml epocha

0.4 ------------------------~-----•

G.2

0.0

-0.2

-0.4 • • --0.1

5(l) 1(0) 1500

time Wee)

r .. 4 Difference between INS predicted coordinatH and GPS coordinates

214

Figure 4 show the hypothesis test results for GPS/INS integration. Comparing INS predicted coordinates with the coordinates from GPS indicates that the hypothesis test gives a reasonably consistent result.

8. Conclusions

A fault detection and estimation procedure for dynamic system has been discussed. The principles are based on hypothesis testing and a two-step Kalman filter. In this paper, discussion focuses on abrupt changes in dynamic systems and observations.

Parallel processing for failure bias detection and estimation means that the above procedure is flexible enough to be implemented in real-time Kalman filtering.

A method for fault detection and estimation is applied to kinematic positioning by using differential GPS or GPS/INS integration. Test results show that the above method is a powerful tool for detecting and isolating gross sensor errors and model failure of GPS and INS. The two-step Kalman filter is an efficient algorithm for biases estimation, as well as estimating the states of a dynamic system.

As mentioned, fault detection using hypothesis testing is based on a prior knowledge of the statistical information of the system and measurement noises. In general, this requirement is not a problem, because the statistical behaviour of the noises is usually quite well known. If the stochastic component of a dynamic model is not well known, a so-called adaptive filtering technique for the dynamic system is needed. Study of this particular problem is being undertaken at the current time.

Acknowledgements

Dr. K.P. Schwarz is gratefully acknowledged for stimulating discussions and suggestions on this paper.

References

Cannon, M.E., High accuracy GPS semi-kinematic positioning: modelling and results. Proceedings of ION GPS-89, Colorado, USA, September, 1989.

Chang, C.B. and K.P. Dunn, On GLR detection and estimation of unexpected inputs in linear discrete systems. IEEE Transaction on Automatic Control, Vol. AC-24, 499-501, 1979.

Friedland, B. Treatment of bias in recursive filtering. IEEE Transaction on Automatic Control, Vol. AC-14, 359-367, 1969.

Gelb, A., Applied Optimal Estimation. The M.I.T. Press, Cambridge, Mass., 1974.

Ignagni, M.B., An alternate derivation and extension of Friedland's two-stage Kalman estimator. IEEE Transaction on Automatic Control, Vol. AC-26,746-750,1981.

215

Jazwinski, A.H., Stochastic Processes and Filtering Theory. New York: Academic, 1970.

Mehra, R.K., On the identification of variances and adaptive Kalman filtering. IEEE Transaction on Automatic Control, Vol. AC-15, 175-184, 1970.

Mehra, R.K. and I. Peschon, An innovations approach to fault detection and diagnosis in dynamic systems. Automatica, Vol. 7,637-640, 1971.

Schwarz, K.P., M.E. Cannon, and R.V.C. Wong, A comparison of GPS kinematic models for the determination of position and velocity along a trajectory. Manuscripta Geodaetica, 14: 345-353, 1989.

Teunissen, P.I.G., Quality control in integrated navigation systems. Proceedings of PLANS'90, Las Vegas, USA, March, 1990.

Teunissen, P.I.G. and M.A. Salzmann, A recursive slippage test for use in state-space filtering. To be published in Milnuscripta Geodaetica, 1990.

Willsky, A.S., A survey of design methods for failure detection in dynamic systems. Automatica, Vol. 12,601-611, 1976.

Willsky, A.5. and H.L. Jones, A generalized likelihood ratio approach to the detection and estimation of jumps in linear systems. IEEE Transaction on Automatic Control. Vol. AC-21, 108-112, 1976.

Wong, R. V.C., K.P. Schwarz, and M.E. Cannon, High-accuracy kinematic PosItioning by GPS-INS. Navigation: Journal of The Institute of Navigation, Vol. 35, 275-287, 1988.

Appendix I

Since the model (8) to (9) contains the bias terms b and d, the estimates Xk(-) and Xk(+) for states of dynamic equation contain systematic errors caused by biases band d. Comparing the estimates ik(-) and Xk(+) from (10) and (11) the correction for the bias-free estimates x k(-) and x k( +) can be made by

ik(-) = Xk(-) - ~(-)b - ~(-)d (AI)

(A2)

where 5t(-), st(+), se(-) and ~(+) are matrices which propagate the effect of unknown biases band d to the bias-free estimates Xk(-) and Xk(+) for the model (8) to (9) .

From (3), (10), (AI) and (A2), the correction for the estimate Xk(-) is computed by

Comparing both side of equation (A3) leads to

216

st(-) = cl>k,k-tS:_t(+) - Bk, (A4)

S~(-) = cI>k,k-t~_t(+) . (AS)

Similarly, comparing (4), (11), (AI) and (Al) results in

(A6)

S~(+) = ~(-) + Kk [Ok - Hk s1(-)] . (A7)

Equations (A4) to (A7) give a recursive algorithm for computing the matrices st(-), S~(+), S~(-) and ~(+) to correct the bias-free estimates (Al) and (A2) similar to the Kalman filtering algorithm.

Considering equation (AI), the predicted residual rk (16) for bias-free estimate of the states Xk of the model (8) and (9) is of the form

where

(A9)

With the following denotes

(AID)

G~ = (Ok - Hk~(-» (All)

equation (A8) is simply written as

fk = Gt b+ ~ d + 'Uk . (AI2)

217

MDB: A Design Tool for Integrated Navigation Systems

Martin Salzmann Geodetic Computing Centre

Delft University of Technology Thijsseweg 11

2629 JA Delft The Netherlands

Abstract

The design of an integrated navigation system requires that various aspects are taken into consideration. In this paper the aspect of reliability (in the statistical sense) is more closely investigated. A particular measure of reliability, the Minimal Detectable Bias (MDB), is considered. Its use as a design tool for integrated navigation systems is described and is illustrated by an example.

1 Introduction

The increasing demand for high precision dynamic positioning necessitates the use of in­tegrated navigation systems, which combine different positioning sensors. Not only the integration of different types of sensors but also the advanced parameter estimation tech­niques which underlie the data processing in an integrated navigation system render a careful design of the integrated positioning system inevitable. Actual surveys are costly and as a consequence it has to be ascertained in advance if the required performance bounds can be met. Throughout this paper we assume that data proce8lling jn the inte­grated navigation system is performed by means of the well-known Kalman filter.

Traditionally performance (of integrated navigation systems) is specified in terms of precision; how accurate can certain parameters (e.g. position, velocity) be estimated. The current trend towards quality control requires that also reliability is taken into account, i.e. the effect of possible model misspecifications on the estimation results. The precision and reliability (quality) requirements have to be reconciled with limiting conditions such as cost, available hardware, computer power, personnel, and time schedules. In this paper we will only consider some aspects of quality control, in particular reliability. At this point it is important to realize that precision and reliability can be analyzed in the design phase of an integrated navigation system.

Precision is usually represented by the covariance matrix of the unknowns or derived measures as point standard ellipses and will not be discussed here any further. Reliability is the sensitivity of the (position) estimation result to undetected model misspecifications. In this paper we will focus on a particular measure of reliability, namely the Minimal Detectable Bias (MOB).

The contents of this papers is as follows. First we briefly consider testing and relia­bility. Then we discuss the concept of Minimal Detectable Bias with special reference to

218

integrated navigation systems. The use of MDBs in integrated navigation system design is illustrated by an example. Finally some conclusions are given.

2 Testing and Reliability

Integrated Navigation Systems are primarily used in real-time environments and thus also performance analysis of the system has to be performed in real-time. Possible model misspecifications have to be detected in real-time and for this purpose a testing procedure, which parallels the Kalman filter algorithm, has been developed [Teunissen 1990J. The associated test statistics are all based on the so-called predicted residuals:

(1)

where llA: is the mk-dimensional vector of predicted residuals at time k; llk is the mk­

dimensional vector of observables; and Ak~klk-l is the vector of predicted observables. IT the model underlying the integrated navigation system is correct and based on the assumption that all data processing is based on the Kalman filter, the vector of predicted residuals at time k is defined as:

and is distributed under the null hypothesis Ho as

(2)

with Qv = diag(QvllQv2, ••• ,QVA:). Any model misspecification will affect the predicted residuals. For our purposes we assume that the alternative hypothesis (HIl ) can be specified as follows:

(3)

where Cv = (C:1 , C:2 , • •• ,C;A:)* is a known U::~=l mi X b )-ma"trix and V is a unknown b-dimensional vector.

The testing procedure mentioned above consists of three steps, namely 1. detection, 2. identification, and 3. adaptation (which is not considered here), and can be performed recursively in a manner that parallels the Kalman filter.

First unspecified model errors in the null hypothesis are detected using overall model tests. To obtain a sufficient detection power it might be necessary to perform the detection with a certain delay; one has to specify a window length for the detection test statistics. Once a model error has been detected the source of the model error has to be identified. In integrated navigation systems many alternative hypotheses can be specified. An im­portant aspect of the specification of HilS is that one not only has to specify the type of model misspecification, but also the time of its occurence. In [Teunissen and Salzmann, 1989] one-dimensional alternative hypotheses are considered and three different types of model misspecifications, namely an outlier in the observables, a permanent slip in a single observation channel, and a slip in the state vector are investigated. Also a window length for the identification test statistics has to specified.

219

For one-dimensional alternative hypotheses the matrix Cu reduces to the vector Cu and Til reduces to a scalar. The corresponding test statistic can then be written as:

(4)

where I is the time one assumes the misspecification starts, and k is the current moment. In [ibid.] efficient recursive schemes are derived to compute the vectors cu,'

Baarda [1968] introduced the concept of fixing the size of the model error that is detectable at a certain probability level by a certain test. Once a reference probability 'Yo for the power of the test 'Y (the probability of rejecting Ho when a Ha is true) and the level of significance a (the probability of rejecting Ho when in fact it is true) for a one-dimensional test ao have been chosen, the noncentrality parameter of the test can be computed from the inverse power function

AO = A(a = ao,b = 1,'Y = 'Yo) . (5)

For A = AO one can then compute the corresponding model error Cu Til.

2.1 Reliability

Besides testing for possible model misspecifications one is also interested in what size of biases are detectable with the tests described above (internal reliability). Furthermore one is interested in how well various alternative hypotheses can be separated (separability). And finally the influence of model errors on the estimation results in terms of the state vector is of interest (external reliability). External reliability can be diagnosed with the Bias-to-Noise ratio [Teunissen, 1990]. Internal reliability will be represented by Minimal Detectable Biases.

Before we proceed we want to stress that a model with redundant measurements is still unreliable if one does not test for possible model misspecifications.

3 Minimal Detectable Biases

In this section Minimal Detectable Biases (MDBs) are analyzed more closely. We choose to analyze MDBs as a reliability measure because they represent the bias that can be detected by the one-dimensional tests. MDBs can thus readily be interpreted. MDBs are defined for one-dimensional HaS and can be computed as:

"I: • Q-l L..Ji=1 cu, ui CUi

I ~ k , (6)

where AO is the fixed noncentrality parameter of the test, and I, k, and CUi are defined as in (4). A MDB is defined as the size of the model error Til that can be detected with a probability 'Yo with the one-dimensional test statistic (4). The definition of the MDB can be somewhat expanded if one also takes into account the time 10 at which the model error really occurs [Teunissen, 1990]. Then the MDB can also be used as a measure of

220

separability. In this paper we will limit ourselves to MDBs that are directly related to the tests that are actually performed.

Now the definition of the MDB has been given its use as a design tool is discussed. The MDB can be used to:

1. Verify the chosen test strategy and if necessary improve the design of the integrated navigation system.

2. Determine the window length and delay of the test statistics.

3. Compute measures of external reliability.

Ad(l): MDB computations may reveal that certain biases are poorly detectable. If one knows from experience the particular bias has little impact on the estimation result the corresponding test statistic may be deleted. More often, however, the effect of an ac­tual bias may be detrimental to (part of) the estimation result. Then the design of the integrated navigation system has to be reconsidered (by, for example, adding additional sensors). Ad(2): The MDB provides a useful tool for the setting of the window length and delay of the test statistics for various types of alternative hypotheses. If for instance the MDB does not decrease significantly with an increasing delay of the test statistic (k - I), the point at which no significant improvement occurs can be chosen as the window length. The general idea is illustrated in Figure 1. Ad(3): Measures of external reliability that correspond with a bias in the state vector estimate XA:IA: due to a model error of the size of the MDB in Ho can be computed in a straightforward manner from the equations that are needed to compute the MDBs. As such the MDBs are pivotal in reliability theory. (cf. [Teunissen, 1990]).

k -+ k -+ k -+

1 0 1 2 3 4 5 1 8.3 4.1 3.7 3.6 3.6 3.6 ·1 x x x ~ 0 1 2 3 4 ~ 8.3 4.1 3.7 3.6 3.6 ~ x x x

0 1 2 3 8.3 4.1 3.7 3.6 x x x 0 1 2 8.3 4.1 3.7 x x x

0 1 8.3 4.1 x x 0 8.3 x

(a) (b) (c)

Figure 1: Use of MDB to determine window length test statisticsj (a) Delay (k - I) in computing test statistics for 1 S k, with k time of testing and 1 time of occurence model misspecificationj (b) MDB-matrix IVI"A: for a hypothetical testj (c) Test statistics t"A: computed (x) with window length k -I = 2

For each computed test statistic t"A: the corresponding MDB can be computed. Com­parison of equations (4) and (6) shows that the computation of the MDBs closely parallels that of the one-dimensional test statistics and can be performed by the same software.

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3.1 Design Parameters

The design of integrated navigation systems involves a large number of parameters, most of which also have an impact on the MDBs. It is obvious that the design parameters listed below influence other quality parameters (precision, external reliability) as well. We limit ourselves, however, solely to the impact of the design parameters on the MDBs. MOBs are a function of:

• Stochastic model of the observables

- measurement precision

- correlation between observables at a single epoch

- correlation between observables in time

The stochastic model of the observables enters the definition of the MOBs directly via the variance matrix of the predicted residuals (Qv;) and indirectly via the vectors CVi (cf. (6)). Improvement of the measurement precision results in lower MDBs. The impact of correlation between observables at a single epoch and between epochs is harder to predict and depends on the type of alternative hypothesis considered.

• Measurement model. If more sensors are integrated into the navigation system the MOBs will generally decrease as the redundancy of the model is increased. How much the MOBs decrease depends largely on the type, number and precision of the additional sensors. With each additional observation the number of MOBs to be evaluated increases.

• Measurement geometry. Important in case land or space-based radiopositioning sys­tems are part of the sensor set. In case for example only one (radio-)positioning system is used the magnitude of the MOBs depends to a large extent on the trans­mitter geometry relative to the sensor geometry.

• Choice of the state-space model. As more sensors are integrated into the navigation system, the state space model can (or has to) be expanded by additional states (e.g. instrument biases). Furthermore additional states such as accelerations might be included in the state vector. Additional states lead to larger MOBs (due to lower model redundancy).

• System noise. System noise (which models the uncertainty in the time update of the Kalman filter) enters the variance matrix of the predicted residuals (Qv;) via the variance of the predicted observations and indirectly via the vectors CVi • Lower system noise leads to smaller MOBs.

• Filter concept; data processing may be performed using one central filter or be based on a decentralized filter approach [Carlson, 1988]. The data processing scheme has a large impact on the testing strategy and thus on the MOBs.

• Sample rate of sensors and filter cycle time. Increasing the measurement sample rate and filter cycle time leads to lower MOBs.

• Testing parameters.

222

- the noncentrality parameter AO - window length of the test

AO is a monotonic increasing function of the level of significance (Qo) and the power ho) of the test. The choice of Qo and 'Yo, however, influences all MDBs in a similar way. A larger window length (k -I) results in smaller MDBs.

It is difficult to quantify the actual effect of the design parameters discussed above on the MDBs. Therefore we will consider an example in the next section.

4 Example

As an example we consider the integration concept for the NAVGRAV survey [Haagmans et al., 1988] which took place in the North Sea in 1986. NAVGRAV was a mixed navigation and sea gravimetry experiment and thus both precise positioning and velocity determina­tion were required. The North Sea area is a good example of a region where integration can be considered as it is covered by various radiopositioning systems. For the NAVGRAV survey we considered the Syledis system and Hifix/6 (currently being replaced by the similar Hyperfix system). The Syledis system supports operation in range-range, pseudo­range, and so-called combined mode and as consequence is by itself already a subject of detailed design computations. Hifix/6 is a hyperbolic positioning system. Furthermore the integration of gyrocompass and log is considered. The design area is depicted in Figure 2.

o North Sea

Figure 2: Design Area; 0 Hifix transmitter, 0 Syledis transmitter, • test area.

In this example we try to give an insight in the properties of MDBs. Naturally not all design parameters discussed in the previous section can be illustrated in this brief example. We have chosen to oppose a non-integrated solution (that is positioning using Syledis only) and an integrated solution. First the design parameters on which the example is based are briefly given.

223

• Measurement model

- Syledis is used in range-range mode (up to 4 ranges)j measurements are assumed to be uncorrelated with a variance of 2.25m2.

- Two Hifix/6 hyperbolic lines of position (expressed in [metres)) are used with a variance of 6m2 and a correlation of 1.6m2 between lops.

- Gyro with a variance of 0.0625deg2 .

- Log in bottom track mode with a variance of 0.04m2/s2.

- Measurement interval 1 second.

• The state model is based on a constant velocity model with fixed instrument biasesj this results in the following state vector elements:

- Position (Easting, Northing)

- Velocity (East, North)

- Drift if gyro is included (this state includes the gyro error)

- Log bias if log is included

• System noise

- For the position and velocity states system noise is modelled as (decoupled) accelerations in East and North direction with a variance of 0.0625m2/s4

- Gyro drift system noise is modelled as a drift rate of 0.0025deg2/s2.

- Log bias system noise is modelled as a drift rate of 0.OOOlm2/s4.

• Testing parameters

- ao and 'Yo are chosen as 0.001 and 0.80 respectivily, resulting in a noncentrality parameter of >'0 = 17.07.

Design computations are performed at two locations in the design area. Location 1 (cf. Fig. 2) has excellent Syledis coverage (4 stations), whereas at location 2 only two Syledis ranges can be used. In the design computations the vessel is sailing a Northbound course at 5m/s.

In this example we consider MOBs related to alternative hypotheses for outliers in the observations, slips in observation channels, and slips in the state vector. MDBs for outliers and slips in the observations are expressed in [metres] for Syledis and Hifixj [deg] for gyrOj and [m/s] for the log. Slips in the state vector considered are: (a) slip in position and velocity due to an unmodelled acceleration along and across trackj (b) slip in gyro drift [deg]j (c) slip in log bias [m/s]. For six cases MOBs are given. Note that at a zero delay the MOBs related to HilS for outliers and slips in the observations coincide.

Case 1 No integrationj good Syledis coverage

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I Has related to observations I Has related to slip in state vector Ha type outlier slip Ha type slip delay (0) (4) (4) delay (1) (4) Syledis 1 6.95 6.45 3.43 acceleration along track 3.46 0.828 Syledis 2 7.44 6.56 4.17 acceleration across track 3.68 0.861

Syledis 1 and 2 refer to the best and worst Syledis range (out of 4) in terms of MDBs. The differences are fairly small and are due to the Syledis geometry. Note that with increasing delay the MOB improves considerably for a Ha related to a slip in the observations and an unmodelled acceleration, but hardly for an Ha related to an outlier. This indicates that the tests related to the first two HaS require a considerably larger window length than the tests related to outliers in the observations. 0

Case 2 No integration; good Syledis coverage; Syledis observations are assumed time correlated with a time constant of 5 sec.

I HaS related to observations

Ha type outlier slip delay (0) (4) (4) Syledis 1 3.15 2.22 2.92 Syledis 2 3.32 2.23 3.09

The correlation is modelled in the filter by means of state augmentation. It can be seen that the assumption of correlation in time has a major impact on the size of the MOBs. Note that now (cf. case 1) outliers can be detected better with increasing window length, whereas the MOBs related to slips in the observations do not improve significantly with increasing window length. 0

Case 3 Full integration; good Syledis coverage, Hifix, gyro, and log

I HaS related to observations I I HaS related to slip·in state vector

Ha type outlier slip Ha type slip delay (0) (4) (4) delay .(1) (4) Syledis (worst) 6.57 6.39 3.47 acceleration along track 0.904 0.502 Hifix (worst) 9.90 9.84 4.61 acceleration across track 0.741 0.465 gyro 12.1 8.5 7.8 gyro drift 8.1 3.1 log 1.46 1.16 0.86 log bias 0.84 0.31

First of all integration allows the computation of MDBs related to all subsystems. Full integration somewhat improves the MOBs of the Syledis ranges. The rather large MOBs of the gyro observations are due to the fact that the gyro observations are assumed un­correlated in time. The MOBs of the other systems seem quite acceptable. A major improvement is seen in the MOBs related to the Ha concerning a slip caused by an un­modelled acceleration (cf. case 1). Note that in contrast with Syledis and Hifix the MDBs for an outlier of gyro and log do considerably improve with increasing window length. 0

Case 4 Full integration; similar setup as for case 3 but Syledis operated in pseudo­range mode.

225

I HIJs related to observations

HIJ type outlier slip delay (0) (4) ('!l Syledis (worst) 7.42 6.95 4.52 Hifix (worst) 9.95 9.87 4.73 gyro 12.1 8.5 8.3 log 1.47 1.16 0.87

I HIJs related to slip in state vector

HIJ type slip delay (1) (4) acceleration along track 0.904 0.502 acceleration across track 0.741 0.465 gyro drift 8.1 3.4 log bias 0.84 0.31 clock offset [m] 3.56 2.00

The use of pseudo-ranges necissitates the addition of a state for the clock offset. For this example the clock offset system noise is modelled as a drift with a variance of 1.11E - 17 (which corresponds to 1m2/s2 ). It can be seen that the use ofSyledis in pseudo range mode instead of range-range mode does not degrade the internal reliability much (cf. case 3). Operating Syledis in pseudo range mode is very advantageous, as this mode supports an unlimited number of users. Pseudo range mode seems acceptable based on an analysis of the MOBs. 0

Case 5 Full integration; good Syledis coverage, measurement interval Syledis and Hifix 2 and 3 seconds respectively.

I HIJs related to observations I HIJs related to slip in state vector

HIJ type outlier slip HIJ type slip delay (0) (4) (4) delay (1) (4) Syledis (worst) 6.85 6.56 4.57 acceleration along track 0.905 0.502 Hifix (worst) 10.0 9.94 7.22 acceleration across track 0.741 0.465 gyro 12.1 8.6 8.9 gyro drift 8.1 3.5 log 1.47 1.17 0.95 log bias 0.85 0.34

At lower measurement rates it is more difficult to detect slips in the observations and thus the MOBs related to HIJs concerning slips in the observations get larger (cf. case 3). If small MOBs are required this requires larger window lengths for certain tests. 0

Case 6 Full integration; poor Syledis coverage (2 ranges; location 2 in Fig.2)

I HIJs related to observations I I HIJs related to slip in state vector

HIJ type outlier slip HIJ type slip delay (0) (4) (4) delay (1) 14) Syledis (worst) 6.56 6.38 3040 acceleration along track 0.903 0.501 Hifix (worst) lOA 10.1 5047 acceleration across track 0.741 0.465 gyro 12.1 8.7 10.0 gyro drift 8.3 4.1 log 1.46 1.15 0.81 log bias 0.83 0.29

It can be seen that the MOBs are hardly different from the case where good Syledis coverage is available (cf. case 3). This means that also outside the Syledis coverage area a high internal reliability can be maintained. However, in this particular case the precision of the East coordinate is significantly degraded due to the poorer geometry. 0

226

5 Concluding Remarks

In this paper we discussed the use of MOBs as a design tool in integrated navigation systems. MOBs are a measure of internal reliability and should be used in connection with external reliability and precision measures.

The MOBs are a measure of the size of the biases that can be detected by a particular test; as such MOBs are a measure of detect ability for certain model errors. The example clearly indicates that integration leads to lower MOBs.

The MOBs can be used to determine the window lengths of the tests. Our analysis shows that:

1. Tests related to HaS for slips in the state vector require large window lengths.

2. Window lengths of tests associated to various classes of observables vary significantly. It is shown that the identification of outliers for dead-reckoning type of observables requires a larger window length than for position related observables. The iden­tification of slips in the observations, however, always benefits by larger window lengths.

The link of the MOB as a measure of internal reliability with external reliability is discussed in Teunissen [1990]. MDBs constitute a powerful, albeit not the only, tool in integrated navigation system design.

References

Baarda, W. (1968). A Testing Procedure for Use in Geodetic Networks. Neth. Geod. Comm. Publ. on Geodesy, New Series, VoI.5(2).

Carlson, N.A. (1988). Federated filter for fault-tolerant integrated navigation systems. Proceedings IEEE PLANS 1988, pp.llO-119.

Haagmans, R. et a1. (1988). NAVGRAV Navigation and Gravimetric Experiment at the North Sea. Neth. Geod. Comm. Publ. on Geodesy, New Series, Vo1.32.

Teunissen, P.J.G. and M.A. Salzmann (1989). A recursive slippage test for use in State­Space filtering. Manuscripta Geodaetica VoI.14(6), pp.383-390.

Teunissen, P.J.G. (1990). Quality Control in Integrated Navigation Systems. Proceedings IEEE PLANS 1990, pp.158-165.

227

USE OF SPECTRAL METHODS IN STRAPDOWN ISS DATA PROCESSING

INTRODUCTION

Joseph Czompo Department of Surveying Engineering

The University of Calgary 2500 University Drive N.W.

Calgary, Alberta, Canada, T2N IN4

A modified Litton LTN-90-1oo Ring Laser Gyro Inertial Reference Unit, originally de­signed for civilian aircraft use and modified to satisfy the accuracy requirements for survey­ing (Wong, 1988; Schwarz et al., 1990) has been used for this study. It is a strapdown system with a large external data storage, so there is no need to process the inertial sensor measurements in real time. One can collect and record the sensor raw data and process them post-mission, using different, even complicated and time consuming methods.

This paper discusses processing the data in the frequency domain. This means frequency filtering and obtaining information from the spectra. It is preprocessing in the sense that it precedes the navigation/surveying software.

INERTIAL SENSOR DATA IN THE FREQUENCY DOMAIN

To observe the data in the frequency domain the Fast Fourier Transformation (FFf) is used to compute the amplitude spectra (Kanasewich, 1975). The number ofFFT points is 1024 which corresponds to a 16 second long data set

Dithers of the ring laser gyros cause a vibration of the sensor block. Also, the cooling fan and other environmental disturbances (e.g. from the survey vehicle engine) may cause vi­bration of the whole inertial system. All these effects may introduce noise to the sensor output causing different effects on the spectra.

Collecting Data

The data analyzed in the sequel were collected on a test traverse located in the Kananaskis region, 100 km from Calgary, on October 12, 1989 and May 4, 1990. By using data recorded during different parts of the test we can, to some extent, distinguish between dif­ferent noise sources and the sensor measurements generated by the motion only. For in­stance, the dither disturbance is expected to be dominant in the data which are collected during the initial alignment of the inertial system, when the vehicle is not moving and the engine is not running (static mode). In order to collect data on the engine disturbance with­out the vehicle motion effect we kept the engine running for several minutes after the vehi­cle stopped at the end of the May 4 test To see the effect of the cooling fan it was switched off during the May 4 mission for comparison.

228

Disturbances Caused by Dither

Dither frequencies are usually in the range of 380 - 420 Hz. Our data acquisition system collects the data at a 64 Hz sampling rate which causes an aliasing effect (Gold and Rader, 1969) on the dither frequencies; they show up in the range limited by the Nyquist frequen­cy of 32 Hz. Figures 1 through 3 show the FFf amplitude spectra of the x-gyro, z-gyro and z-accelerometer outputs in static mode of the inertial system (standing vehicle, engine off) for October 12. The peaks at about 9, 20 and 30.5 Hz correspond to the 3 aliased dither frequencies. Additional peaks in the z-accelerometer spectrum on Fig. 3 may be caused the sensitivity of the accelerometer to the product of vibrations in different directions.

250 X-Gyro FFT, Static

-(I) --u 200 CI) (I) u ... as 150 -CI)

100 'l:J :::J =: a.. 50 E ct

0 0 4 8 12 16 20 24 28 32

Frequency (Hz)

Fig. 1. Amplitude spectrum of the x -gyro output in static mode of the ISS

250 Z-Gyro FFT, Static

-(I) --u 200 -CI)

(I) (,) ... as 150 -CI)

100 'l:J :::J -a.. 50 -E ct

0 1

• • • • • • 0 4 8 12 16 20 24 28 32

Frequency (Hz)

Fig. 2. Amplitude spectrum of the z-gyro output in static mode of the ISS

229

Z-Accelerometer FFT, Static 0.03 ~---------------..;....--------....,

-N • ! 0.02

• ~ ~

.. 0.01 Q.

E C

0.00 -I-........... ,.... ... ~~~~~~~e:::t~~~~~!!!!!!~~.!!q o 4 8 12 16 20 24 28 32

Frequency (Hz)

Fig. 3. Amplitude spectrum of the z-accelerometer output in static mode of the ISS

Effect Caused by the Cooling Fan

Fig. 4 depicts the May 4 x-gyro spectrum at the start of the initial alignment (static mode) with the cooling fan switched off. It contains the same significant peaks as the spectrum on Fig.l when the fan was on. The vibration caused by the cooling fan can therefore be con­sidered negligible (Keller,1989). A closer look shows that the ~ikes in Fig. 4 are wider than those of Fig. 1, indicating that the dither frequencies were changing during the FFf time interval. This change is in connection with the instrument wann-up at the start of the mission. Since the fan was not turned on, the internal ISS temperature was changing faster than during the October 12 run. After a few minutes the frequencies are more stable, thus producing sharper spikes. By monitoring the dither frequencies it was found that they were changing over a wider band on May 4 (0.32 Hz) than on October 12 (0.18 Hz). This indi­cates the temperature stabilizing role of the fan.

250 X-Gyro FFT, Static, Fan off -• -u 200 • • u ...

• 150 -• 100 ~ ~ -Q. 50 E C

0 0 4 8 12 16 20 24 28 32

Frequency (Hz)

Fig. 4. Amplitude spectrum of the x-gyro output in static mode of the ISS, without cooling fan

230

Disturbance Caused by the Engine

Fig. 5 shows the x-gyro spectrum while the vehicle was standing but the engine was run­ning. The engine effect can be clearly identified in the 10 - 24 Hz frequency range.

250 X-Gyro FFT, Standing, Engine running

-., -Co) 200 • .,

Co) ... II 150 -• 100 'lJ :::J !:: a. 50 E c(

0 0 4 8 12 16 20 24 28 32

Frequency (Hz)

Fig. 5. Amplitude spectrum of the x-gyro output when the survey vehicle is standing but the engine is running

The wheel suspension system of the vehicle allows vibration about its "roll" and "pitch" axes but not around the vertical axis. This can be seen in the z-gyro spectrum (Fig. 6). There is no such difference between the three accelerometers; they are affected by the en­gine in the same way.

250 Z-Gyro FFT, Standing, Engine running -• -Co)

200 • • Co) ... II 150 -• 100 'lJ :::J !:: a. 50 E c(

0 0 4 8 12 16 20 24 28 32

Frequency (Hz)

Fig. 6. Amplitude spectrum of the z-gyro output when the survey vehicle is standing but the engine is running

231

Frequency Picture of the Vehicle Motion

The spectrum of the x-gyro output during vehicle motion is depicted on Fig. 7. The ampli­tude of the engine disturbance is increased due to the higher RPM and the road irregulari­ties. The motion frequencies appear in the the 0 - 6 Hz frequency band.

1000 FFT, Movin -., 900 -Co)

800 • ., 700 Co) .. • 600 - 500 • 400 ~

:::J 300 :!:: A. 200 E 100 c(

0 0 4 8 12 16 20 24 28 32

Frequency (Hz)

Fig. 7. Amplitude spectrum of the x-gyro output when the survey vehicle is moving

The y-gyro spectrum is similar to the x-gyro spectrum but the z-gyro spectrum is quite different (see Fig. 8).There is no high engine disturbance and the very low frequency com­ponents of the motion are dominant The latter indicates a much smoother rotation about the vertical body axis. This is quite natural for a land vehicle.

1000 Z-Gyro FFT, Moving System -., 900 -Co)

800 • ., 700 Co) ..

• 600 - SOO • 400 ~ :::J 300 -Q. 200 E 100 c(

0 0 4 8 12 16 20 24 28 32

Frequency (Hz)

Fig. 8. Amplitude spectrum of the z-gyro output when the survey vehicle is moving

232

FREQUENCY FILTERING

We have seen that frequencies related to the vehicle trajectory are concentrated in the 0 - 6 Hz region. It would be obvious to use a low pass filter to remove all the disturbing signals over 6 Hz. Unfortunately, the Butterworth filter and other similar digital filters (Terrell, 1988) do not work well for our purposes. They change the mean of the data set and intro­duce amplitude distortion below the cutoff frequency. Furthermore, they do not remove the frequencies over the cutoff threshold correctly. Inertial sensor data are very sensitive to these effects.

Another problem with the use of low pass fIlters is that the accelerometer outputs have higher frequency components over 6 Hz reflecting sudden acceleration changes (i.e. start, stop, beginning and end of curves, road surface irregularities). By filtering these frequen­cies out one may get however an incorrect trajectory. Strictly speaking this is also true for the gyros, but the rotations of a survey vehicle are much smoother than its acceleration.

Dither Spike Removal

A filtering method which does not have the disadvantages mentioned above is the Dither Spike Removal using FFT. This method removes only the dither frequencies from the spectrum. Dither spikes are generally not in the same frequency range as vehicle motion. By removing them one deletes only single distinct frequencies and not very much of the high frequency components of motion (if any). Also, the mean is not changed, so the dither removal is a gentle intervention into the spectrum.

The method consists of 4 main steps.

Cutting Time Slices. The data set is divided into time slices of a certain length. The length was chosen to be 128 seconds (8192 data points at 64 Hz sampling rate). It gives a frequency resolution high enough to reflect spectrum details but its size is still convenient.

Computing FFT. First the complex spectrum is computed then it is converted to ampli­tude and phase spectra for each sensor. The phase spectrum remains unchanged, only the amplitude of the spikes will be reduced.

Finding and Deleting Spikes. The processing program identifies the highest ampli­tude frequency line in predetermined frequency windows as dither spikes. The use of win­dows is convenient because we do not need to find the exact dither frequencies by hand. Furthermore, since they are changing in time all the possible frequencies for each time slice would have to be given if windows were not used. A window has to be chosen such that only one dither spike can appear in it for the whole data set. This can be easily done be­cause the spikes have a sufficiently different frequency.

Once the frequency is identified, the amplitude is reduced to the surrounding average amplitude. The reason for not using a zero amplitude is to leave the spectrum in its original fonn as much as possible.

Data Reconstruction by Inverse FFT. After the spike cut the amplitude and phase spectra are converted back to the complex spectrum, then an inverse FFT is applied to get data in the time domain. The new data have the dither frequencies removed.

The last 3 steps are repeated from slice to slice until the data set ends. Fig. 9 shows the amplitude spectrum of the x-gyro in static mode after spike removal. Its

FFf length is different from the length used in the spike removing program and it lies over

233

the boundary of two time slices discussed above. In this way, the efficient removal of the spikes can really be seen. Fig. 10 depicts a similar case in the presence of engine noise.

250 X-Gyro FFT, Static, Dither Filtered -• -C)

200 • • C) .. • 150 -• 100 'V :::J !:: a.. 50 E C

0 I I I I I I I

J. .... L _ A A...IL-Aa

0 4 8 12 16 20 24 28 32 Frequency (Hz)

Fig. 9. Amplitude spectrum of the x-gyro output in static mode, after dither spike re­moval

X-G 250

on, Dith. Filt. -• -C) 200 • • C) .. • 150 -

• 100 'V :::J !:: a.. 50 E C

0 0 4 8 12 16 20 24 28 32

Frequency (Hz)

Fig. 10. Amplitude spectrum of the x-gyro output when the survey vehicle is standing but the engine is running, after dither spike removal

Table 1. Statistical parameters of the inertial sensor data before and after dither filtering.

Before DitherFilt After Dither Filt. Mean RMS Mean RMS

x-gyro rate (arcsec/s) 8.72240 190.7 8.70193 87.4 y-gyro rate (arcsec/s) -2.61186 184.9 -2.61196 86.1 z-gyro rate (arcsec/s) -12.0899 201.5 -12.0588 77.3 x-acceleration (m/s2) -0.092702 0.0265 -0.092684 0.0197 y-acceleration (m/s2) 0.225211 0.0298 0.225192 0.0186 z-acceleration (m/s2) -0.007465 0.0360 -0.007458 0.0178

234

The Effect of the Dither Spike Removal

By removing relatively high amplitude sinusoids like the dither disturbance, one reduces the RMS of the sensor signal. Table 1 shows the main statistical parameters of the sensor signals in static mode before and after dither fIltering. The RMS is reduced by half after fIltering. H the mean is computed for a time period which does not exactly cover the FFf time slices discussed above, a small change in the mean is possible. In the present case the change is less than 8 x 10-6 deg/s for the gyros, and 2 x 10-5 m/s2 for the accelerometers. Both values are smaller than the corresponding calibrated biases by at least one order of magnitude.

Fig. 11 shows the Kalman-filtered position error in the October 12 test for the pure ISS (not aided by GPS) while Fig. 12 depicts the integrated GPS/ISS position difference be­havior in the May 4 test (GPS update was performed in every 32 seconds). Both graphs show that the dither filtered data results are in general somewhat better than raw data re­sults. Dither filtering reduces the noise (see Table 1). This can be taken into account in the Kalman Filter and improved results can be expected.

Position Error 3.----------------r================,---~--__,

-:- ::'-:~f -~ 2 ... o ... ... w

o 2 4

f'.-.~

6 8 10 12 14 16 ZUPT Numbers

Fig. 11. Position error of ISS before and after dither filtering

GPS-ISS Position Difference 1.50

_·_-tit-·_· - 1.25 E

1.00 .. ~

0.75 c .. ... .. 0.50 --Q 0.25

0.00 0 2 4 6 8 10 12 14 16 18 20 22 24 26

GPS Update Numbers

Fig. 12. GPS-ISS position difference before and after dither filtering

235

The probability distribution of the inertial sensor raw data in static mode is bimodal, which can cause problems with the noise model in the Kalman Filter. It looks like a sum of two Gaussian distributions located symmetrically around the mean (Fig. 13). This can be explained by the dithers. The gyros spend most of their time rotating at the dither angular rate in one direction then rotating back. This shifts the additional Gaussian noise up and down on the angular rate scale. After the dither spike removal the distribution of the cor­rected data becomes unimodal and similar to the nonnal distribution. The same effect can be observed on the accelerometer data. They are shaken by the dither mechanism in such a way that the probability distribution becomes bimodal.

>--.Q

CIS .Q 0 ~

Q.

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

X-Gyro Output Probability Distribution

Raw data Dither fill data

-500 -400 -300 -200 -100 0 100 200 300 400 500 Gyro Angular Rate (arcsec/s)

Fig. 13. X-gyro output discrete probability distribution before and after dither filtering

INFORMATION FROM SPECTRA

The spectral characteristics of the sensor data can be used for detecting different phases of the system motion. By checking the amplitude at certain frequencies or frequency bands the movement and/or the presence of engine noise can be automatically indicated. A growing amplitude can be used as a switch signal to change parameters in the Kalman Filter, accord­ing to the actual situation.

The noise from different sources can be estimated by the power spectral density continu­ously along the data set. This can help using better error models with continuously updated parameters.

CONCLUSIONS

Signals generated by the dither mechanism, survey vehicle engine and system movement can be clearly identified in the amplitude spectra computed from the inertial sensor output data. Except one dither frequency they are in different parts of the spectrum and are easily separable. The movement of the system and the engine disturbance can be characterized by several frequency components covering relatively wide frequency bands. The dither fre­quencies appear as frequency lines in the spectrum and they are temperature dependent. The effect of the cooling fan is negligible.

The spectrum of the z-gyro differs from the other two because the rotation about the ver­tical axis is relatively smooth. The z-gyro is not disturbed by the engine as much as the oth­ers, so the statistical properties of its output are different.

236

The dither frequencies can be removed from the sensor output by the Dither Spike Removal Technique. The resulting sensor signal is less noisy than the unprocessed raw data and its probability distribution is closer to the normal distribution. For some tests it gave slightly better results, and further improvement can be expected if the changed statisti­cal parameters are taken into account

Acknowledgement. This research was in part supported by the Industrial Alignment Project, jointly sponsored by NSERC and a number of companies under the leadership of Kadon Electro Mechanical Services Ltd. Prof. K. P. Schwarz is thanked for his guidance and support. The ISS and integrated GPS/ISS software was mainly developed by Dr. M. Wei, E. Cannon, H. Martell and D. Lapucha. Dr. M. Wei, D. Lapucha, H. Martell, and Z. Liu were taking part in conducting the tests. Their support is gratefully acknowledged.

REFERENCES

Gold, B. and Rader, C.M. (1969). Digital Processing of Signals, McGrow Hill, USA. Kanasewich, E.R. (1975). Time Sequence Analysis in Geophysics, University of Alberta

Press, Edmonton, Canada. Keller, D. (1989). First Results of the Honeywell Lasernav II System in Geodetic

Applications, Symposium Gyro Technology, Stuttgart, Germany. Schwarz, K. P., Knickmeyer, E. H. and Martell, H. (1990). The Use of Strapdown

Technology in Surveying, CISM Journal ACSGC Vol. 44, No.1, Spring 1990,29-37.

Terrell, T. J. (1988). Introduction to Digital Filters, MacMillan Education Ltd, London, England.

Wong, R. V. C. (1988). Development of a RLG Strapdown Inertial Survey System, Department of Surveying Engineering, Report No. 20027, The University of Calgary.

237

WAVE METHOD IN PROCESSING NAVIGATION INFORMATION IN SURVEY SYSTEMS

ABSTRACT

O.S. Salytchev and A.B. Bykovsky Bauman Moscow State Technical University

USSR

This article suggests to employ an original wave approach for the state vector estimation of a dynamic system. Such an approach will considerably increase the estimation accuracy because of the non statistical description of disturbances acting on the object (accelerometer errors, gyro platform drift rate, deflections of the vertical, etc.).

WAVE DESCRIPTION OF THE DYNAMIC SYSTEM DISTURBANCES

The concept of the wave process has fIrst been introduced by Johnson for the solution of problems connected with the control of multivariate dynamic systems (Leondes, 1980). This paper highlights the characteristics of the wave process description which make it possible to develop new algorithms estimating the "ideal accuracy".

The whole class of disturbances w(t) which are common to real systems can be divided into two categories: noise structure and wave structure type disturbances. The noise disturbances show a very pronounced random nature with breaks and jumps. Typical examples are background noise in radio engineering systems, thermal noise and atmospheric turbulences. The wave disturbances exhibit a very distinct wave form. Typical examples are shown in Figure 1. The effect of the noise can best be described in statistical terms, such as "white noise", "coloured noise", etc. In this case, statistical operators such as expectation, variance and correlation function can be applied.

A wave process can be described as follows:

w(t) = cl f1(t) + c2 f2(t} + ... + cn fn(t) (1)

where fi(t) ... known base functions, Ci . .. unknown coefficients which vary from one instant to the next.

The coeffIcients appear random with respect to time and value, and they change in a piecewise constant manner. It is important to note that, in general, the description of the wave process by equation (1) is not an expansion of w(t) into an orthogonal series.

Equation (1) is indeterminate since the values and time dependence of the coeffIcients are unknown. This indeterminacy indicates that, in general, w(t) does not relate to a deterministic process but to a random one. The base functions are known functions of time. They are determined from a visual or numerical analysis of the experimental representations of w(t}. Thus, the disturbances shown in Figure 1 can be given as

238

w(t)

1--P I I I I

t

w(t)

t

w(t)

t

Fig. 1. Examples of wave type disturbances.

w(t) = Cl, w(t) = Cl + c2 t ,and w(t) = cl + c2 exp(-13t).

The representation of the unknown disturbances w(t) in terms of a wave function gives information that is qualitatively different from that obtained by statistical methods. The statistical description of the disturbances gives the mean value and variance for the disturbances. The variance is usually not used for the description of short time statistical processes.

The wave description makes it possible to determine the nature of the variations of w(t) by predetermining the form of the base functions fi(t). The value of the disturbance remains indefinite since the values of the unknown coefficients Ci are indefinite, varying in a piecewise constant manner. Besides, the usual statistical properties of disturbances, such as expectation, variance, correlation function, etc., are based on long-period averaging (expectated behaviour within a relatively long time span). However, in many applications the analysis of disturbances does not take place over a long but rather a short time span. Then the statistical description is not very useful. In this case, the wave description of the unknown disturbances is effective.

During the analysis of a disturbance in a system, the designer can easily choose a particular form of base function. This choice is based on the form and nature of the variations in this particular realization of w(t) given in the specific time span. It is not based on the statistical properties of w(t) which may be known or unknown. For the design of systems it is often more convenient to describe the disturbances by a state model instead of in the form of equation (1). The state model is given by a differential equation which is satisfied everywhere by the function w(t). The simplest approach to determine such a differential equation in terms of the state variables is to find the Laplace transform of a linear combination of the weighted base functions (equation (1»

239

P(s) w(s) = CI fl (s) + c2 f2(S) + ... + cn fn(s) = Q(s)

and the representation of w(t) as an output variable of the fictitious linear dynamic system with transfer function

1 C(s) = Q(s)

and initial conditions w(O), W, W, .. which applied together with the Laplace transform gives the numerator polynomial P(s). Considering that the function Q(s) can be represented as

Q i i-I i-2 (s) = s + ri s + ri-I s + ... + rl '

the following differential equation describing the variations of w(t) can be obtained:

diw di-Iw di-2w -. + ri ---::--1 + ri-I ---::--2 + .. + rlw = 0 dtl dtl dtl

(2)

The coefficients ri are known from the Laplace transform of the base functions. The coefficients ci in equation (1) are piecewise constant. They vary randomly, however, at discontinuities of the function (breaks, jumps). The mathematical structure of the jump variations of the coefficients is taken into account by adding an input function consisting of a series of unknown impulse functions of random intensity (such as Dirac single, double, or triple functions). Then the state model for w(t) becomes:

diw di-Iw di-2w -. + ri --::} + ri-I ---==2 + .. + rlw = a(t) (3) dtl dtl dtl

Equation (3) may be written in state variables (xI=w)

Xl = x2 + al(t)

X2 = x3 + ~(t)

Xi = - rl Xl - r2 X2 - ... - ri Xi + ai(t)

or in matrix notation

x = Ax+O where ai(t). .. Dirac impulse function of unknown amplitude.

Equation (4) can be written in discrete form

analogue of a series of Dirac functions of unknown amplitude, 0*

• 5: lim 0 I.e. u = T->OT.

Physically, 0°;_1 is a pulse sequence of unknown value.

(4)

(5)

When equation (5), which describes the disturbances acting on an object, is added to the object equations, it becomes possible to represent any linear system as

xk = <I>tc,k-I xk-l + OOk_I (6)

240

Equation (6) differs from the traditional representation of a linear system by the right hand side. Here, the input white noise is replaced by a pulse sequence of unknown intensity.

Let us impose the following restriction on the impulse appearance time (~,o. Assume that

an impulse vector '00 different from zero appears every N time steps. This type of pulse sequence of unknown value is shown in Figure 2. In this case, the indetenninacy is

removed for the time when the pulses '00 appear. The assumption of such a restriction can easily be explained.

T

Fig. 2. Pulse sequence.

The possibility to alter the parameters ci (see equation (1» to unknown (random) time moments is used exclusively to ensure a free selection of the time intervals within which the approximation of the disturbances, such as w(t) = cl fl (t) + c2 f2(t) + '" + cn fn(t), is most convenient. This increases the approximation accuracy within the bounds of the selected base functions.

The assumption that the ci appear in equal time intervals leads to a decrease in approximation accuracy. However, it makes it possible to eliminate the indetenninacy resulting from the appearance of new values ci. Determine the error increase in the disturbance function caused by the restriction that the time intervals between neighbouring ci be constant. In this case, it is necessary to remember that not only the approximation accuracy is of interest in the estimation problem for the system state variables. Another important consideration is the accuracy of the output state variables of the physical system which depends mainly on the bandwidth. If the actual values of the approximation errors are significant, but their spectrum is such that they only increase the system bandwidth, then their influence on the accuracy of the output state variables will be negligible. Thus, high accuracy for the estimated state vector output components can be expected even in the case of a rough description of the input disturbances within the bounds of the wave structure.

The estimation of the state vector (6) using the present measurements

Zk = H xk + vk (7)

measurement vector, vector of measurement noise which is expected to be Gaussian white

noise with covariance matrix R = M[vvT],

241

must minimize the sum of the diagonal elements of the estimated error covariance matrix. Actually, the problem statement is similar to that of the Kalman filter, but here the initial

system is described by equation (6) with a sequence of unknown impulse functions fP as input.

Suppose that the initial values of the state vector are zero, i.e. XO=O. This is not a stringent restriction since the unknown initial values of the state vector may be taken into

account by an appropriate selection of the vector B8. In order to estimate the state vector in the first time interval of N time steps (k=1,2, .. N)

the initial value of the vector oS must be determined. Substituting equation (6) N times into expression (7) leads to

zI = H xl + vI = H cIlI.o Xo + H 08 + VI

z2 = H x2 + v2 = H cIl2•2 cIlI.o XO + H cIl2•1 08 + v2 (8)

zN ~"H'~~'~'~~"~ H cIlN•N-I ... cIl I•O Xo + H cIlN•N-I ... cIl I•O 08 + vN

Let us introduce a vector Sk which follows the recursion equation

Sk+I = <I>k+I 1 Sk . (9)

where SI = 08, k=1,2 •... N-l.

Using equation (9) and XO=O, equation (8) can be rewritten as:

zI = H SI + VI

z2 = H S2 + v2 (10)

ZN = HSN+vN

Let us estimate the vector Sk. Equation (9) describes the variations of the vector Sand equations (10) are in essence the observation equations of the vector S.

Estimating S using the formulas of the commonly known Kalman filter leads to

~k+I = <I>k+I.k ~k + Kk+I ( Zk+I - H <I>k+I.k ~0 (11)

where ~ 1 = 0; k=1,2, ... N-1

and Kk+I = Pk+I/k HT [ H Pk+I/k HT + Rk+I]-I,

Pk+I/k = <I>k+I.k Pk <I>Tk+I•k ,

Pk+I = (1- Kk+I H) Pk+I/k ,

where PI = M[SISI T]; Rk+I = M[Vk+IVTk+I].

After the estimation ~N is obtained at the end of N time steps (k=N), it becomes possible to estimate the state vector at the instant corresponding to N:

1\ 1\

XN = SN (12)

Let us now analyse the state vector estimation in the next time interval (with N time steps) from t = (N + l)T to t = 2NT . According to the problem statement a new unknown value

of the function O? appears at the instant k=N. This directly affects the value of xN+ 1.

242

The observation equations for the time interval from t = (N+ l)T to t = 2NT are

zN+l = H XN+l + VN+l = H <J)N+l,N xN + H O~ + VN+l

zN+2 = H xN+2 + vN+2 = H ~+2,N+l <J)N+l,N xN + H <J)N+2,N+l O~ + vN+2 (13)

Z2N = H x2N + v2N = H <J)2N,2N-l ···~+1,N xN + H <J)2N,2N-l ... <J)N+2,N+l O~ + V2N

Similar to the first time interval, the auxiliary vector Sk can be determined as

Sk+l = tf>k+l,k Sk (14)

where SN+l = O~, k=N+1,N+2, ... 2N-1. A

Using equation (14) and the estimated XN obtained in the previous time interval, equation (13) takes the form:

(15)

* A * z2N = z2N - H <J)2N,2N-l ···<J)N+l,N XN = H S2N + '2N

Estimate the vector S bearing in mind that equation (14) describes its change in time and equation (15) the vector measurements.

The respective equations in terms of the Kalman fIlter are

~k+l = tf>k+l,k ~k + Kk+l ( ~:l - H tf>k+l,k ~k) (16)

where ~N+l = 0; k=N+1,N+2, ... 2N-1

and Kk+l = Pk+l/k HT [H Pk+l/k HT + Rk+Il-l,

Pk+l/k = tf>k+l,k Pk tf>Tk+l,k ,

Pk+l = (1- Kk+l H) Pk+l/k ,

where PN+l = M[SN+lSN+l T]; Rk+l = M[l: 1 'k:l T].

After the estimate §2N is obtained it becomes possible to estimate the state vector at the instant k=2N:

A A A X2N = tf>2N,2N-l ... tf>N+l,N XN + ~2N (17)

For the third time interval of N time steps the estimation of §3N and ~3N is carried out in a similar manner.

According to the suggested procedure the estimation is carried out only at points which A

are multiples of N. In other words, the time sequence of the state vector estimation is XN, A A A X2N, X3N, X4N, ....

The number of time steps, N, is determined by the minimum number of integrations required to obtain a convergent estimate for the vector S.

The estimation procedure using the suggested algorithm can be described as follows: The whole estimation interval is broken up into time periods (which hereafter will be referred to as cycles). These cycles are multiples of the N time steps for the measurements. In each

243

cycle the auxiliary vector estimate of equation (14) is detennined. Mter the estimate of S has been obtained, the state vector estimation is perfonned according to equation (17). The state vector estimate is determined only in the basic time points of the cycle, i.e. only at those instants which correspond to multiples of N.

When it becomes necessary to restore the state vector estimates between the basic estimation points (i.e. between points N and 2N, 2N and 3N, 3N and 4N), the following procedure is applied: Assume that it is necessary to estimate the state vector in all points separated from one another by the sampling interval T within the interval determined by neighbouring basic points Nand 2N. The state vector estimate is obtained from

1\ 1\ A XN+l = <l>N+l.N XN + :SN+l 1\ 1\ A XN+2 = <l>N+2,N+l ~+1.N XN + :SN+2 1\ 1\ A XN+3 = <l>N+3.N+2 <l>N+2.N+l <l>N+l.N XN + :SN+3

with

~2N-l = <l>iA;2N-l ~2N ~2N-2 = <l>iJ-l;2N-2 ~2N-l

~N+l = <l>N;2;N+l ~N+2

(18)

(19)

The main characteristic of the suggested approach which distinguishes it from the traditional method is the following. Despite the fact that the estimation within each cycle is obtained by a set of Kalman filter equations, the auxiliary vector S has to be estimated as described in equation (14) with zero input noise. It is well known (Salytchev, 1987; Leondes, 1980) that the estimated error covariance matrix is determined by the level of the input noise. When there is no input noise, the estimated error covariance matrix tends to zero.

Let us analyse the maximum estimation accuracy using the suggested algorithm. Suppose that the object is described by the time-independent equation <l>k,k-l = <1>. This assumption does not affect the generality of the solution. It is made only to obtain more compact expressions. Omitting the derivations, the expressions for the current estimation error in an arbitrary cycle are:

N N Xk+N = { II (1- Kk+N+l-i H) <I> } Xk + { II (I - Kk+N+l-i H) <I> } Sk

~l ~l

where i > j

and

N-l j - l; { II (1- Kk+N+l-i H) <I> } Kk+N-j Vk+N-j

j=O i=l

j II (1- Kk+N+l-i H) <I> = 1 , i=l

. .. state vector error at the end of estimation cycle,

(20)

. .. Error caused by unknown amplitude vector 00 at beginning of cycle,

Xk ••• state vector error at beginning of estimation cycle.

It can easily be shown that the estimation error nonn tends to zero as the number N increases. This leads in the limit to an "absolute accuracy" of the wave algorithm. The above is true when the models fit the real disturbances adequately. Consequently, the suggested wave algorithm is in principle more accurate than the commonly known

244

statistical algorithms. However, this comparison is not quite correct since the suggested algorithm estimates the state vector as described by equation (6), while the Kalman algorithm estimate is affected by the white noise input. Thus, the comparison of estimation accuracies has to be made between the wave model (6) and the real physical processes taking place in the system. From this it follows that the effect of increasing the accuracy of the suggested algorithm in comparison to the Kalman fIlter is caused by the characteristics of the object description with the wave disturbances acting on the system. In fact, the wave description assumes that the disturbances are not random processes. It also assumes that their response to the system within a certain time interval can be represented by a deterministic function of unknown intensity which can be estimated.

It should be noted that the input noise on the right hand side of the object equations is applied not only to exactly describe the physical processes which take place in the system, but also to conveniently represent the mathematical object description in a traditional form with white input noise. This allows the known methods for state vector estimation to be widely employed without the necessity to describe disturbances by shaping fIlters for white noise input.

The question that remains to be answered is: For which systems is it expedient to employ the wave method? The answer depends on the selection of the values N (number of measurements in the estimation cycle) and T (sampling) for a particular system. On the one hand, they should provide a satisfactory estimation accuracy of S in each cycle, and on the other hand, the value NT should not exceed the time between the neighbouring appearances of the Dirac functions in the object description.

Given the values of the estimated error covariance matrix Pk+N and the measurement noise covariance matrix R, it becomes possible to calculate values N and T according to equation (20) for which the matrix elements of Pk+N = M[ik+Nil+Nl will not exceed the selected values.

It is also necessary that the value for NT be less or equal to the time between the neighbouring appearances of the Dirac functions in the object description. It is clear by intuition that the system to which it is expedient to apply the wave method must have a rather narrow bandwidth in the low frequency range. Then the response to the input disturbances has the fonn of slowly varying oscillations. A classical example of such a system is the inertial survey navigation system.

APPLICATION OF THE WAVE METHOD TO A SURVEY NAVIGATION SYSTEM

The error equations of an independent inertial navigation system are the object equations in the problem stated above (Bromberg, 1979). The sensor errors (accelerometer errors, gyro platform drift rate, deflections of the vertical) on the right hand side of the equations act as disturbances on the error equations of the inertial system. In the traditional statistical problem statement the sensor errors are modelled by shaping fIlters with white noise input. In this case, the output variable of the shaping filter has a prescribed correlation function (typically R = A2 exp(-~Itl). In order to employ the wave model it is necessary to construct a model of the disturbances whose input accepts a series of Dirac functions of

unknown intensity. The simplest wave model to analyse the disturbances is w(t)=o. In other words, the disturbances can be described by stepwise functions of unknown value whose time of variation depends on their frequency spectrum. A more complicated model

is a linear combination of steps and inclined lines, i.e. w(t)=o, or steps and exponential

curves, w(t)+~w(t)=o.

245

Using the wave model of actual disturbances, the error equation of the inertial system will

be reduced to x=A(t)x+O, or in discrete form Xk=<I>k,k-lXk-l+OkQt. where OkQl is a vector of pulses of unknown intensity.

When synthesizing the estimation wave algorithm it is necessary to choose the time NT of the impulse vector 00 between neighbouring appearances. This time is determined by the form of the selected disturbance wave model. Thus, the more complicated the disturbance model the less frequent the nonzero impulse 00 appears. Since the vector 00 consists of components 02,02, ... oq which may have different frequencies, the time t=NT for the

whole vector 00 is selected according to the component 09 with highest frequency. For

instance, if the simplest wave model w(t)=o is used, it will be possible to describe the gyro platform drift rate to a sufficient degree of accuracy within a time interval of 15 to 20 minutes, the accelerometer error within 25 to 30 minutes, and the deflection of the vertical within 15 to 17 minutes (at the object speed v=40 to 60 km/h). Hence, the time interval between subsequent occurrences of the nonzero vector 00 is NT=15 to 20 minutes when using this simple wave description.

In this case, the wave algorithm determines the system state vector (21) only in the basic points of the estimation cycles. In other words, the sequence of state vector estimates which must be stored in the onboard computer has the form ~(t=20), ~(t=40), ~(t=60), ... To estimate the state vector at each sampling point T, it is sufficient to employ equations (18) and (19). In this case, it is not necessary to smooth the state vector since the estimation accuracy is sufficiently high.

Figure 3 illustrates the results of processing the same data set by the Kalman filter and the wave method for the navigation information of a survey system. It is clear from a comparison of these graphs that the estimation accuracy of the wave algorithm is substantially higher than that of the Kalman fIlter.

Xk,X w [m]

6

3

20 40 60

Fig. 3. Position error with Kalman filter (x0 and wave algorithm (xw).

246

t [min]

REFERENCES

Bromberg, P.V. (1979). Theory of the inertial navigation systems, M. Nauka (in Russian).

Leondes, K.T. (ed.) (1980). Filtering and stochastic control in dynamical systems, M. Mir (in Russian).

Salytchev, O.S. (1987). Scalar estimation of multidimensional dynamic systems, M. Maschinostroienie (in Russian).

247

SESSION3b

ESTIMATION METHODS AND QUALITY CONTROL

CHAIRMAN R.V.C. WONG

WESTERN GEOPHYSICAL HOUSTON, TEXAS, USA

Detection and Modelling of Coloured Noise for Kalman Filter Applications

Martin Salzmann Peter Teunissen

Geodetic Computing Centre - Delft University of Technology Thijsseweg 11, 2629 JA Delft, The Netherlands

Michael Sideris Department of Surveying Engineering - The University of Calgary

2500 University Drive N.W., Calgary, Canada T2N IN4

Abstract

This paper gives an overview of detecting and modelling of coloured measurement noise for Kalman filter applications. Detection and modelling are important as the Kalman filter is based on the assumption of white (measurement) noise. Modelling is done by using autoregressive techniques, resulting in a shaping filter which can then be incorporated into the Kalman filter model. Modelling is performed in the time and frequency domain. The analysis is performed for single-channel systems.

1 Introduction

The 'classical' Kalman filter is based on the assumption that the data are uncorrelated in time. In practice, however, one often encounters coloured measurement noise. Coloured measurement noise is, for example, due to mechanical damping of measurement systems (e.g., conventional gyros) or data processing within the measurement system itself (e.g., GPS receivers). Because the Kalman filter is only optimal for the case of measurement noise uncorrelated in time, it is important to detect possible time correlations in the data. Once coloured noise has been detected it has to be modelled, so that it can be incorporated in the Kalman filter, or can be dealt with otherwise.

In this paper we consider the detection and modelling of coloured measurement noise for discrete-time data and single data channels. The modelling of coloured noise is based on autoregressive techniques. The Kalman filter itself is a real-time algorithm. The detection and modelling is addressed for batches of data samples. Thus, the procedures outlined here should be performed parallel to the Kalman filter operation.

The contents of the paper is as follows. First we give a brief outline and justification of the procedure to be followed in this paper (cf., Fig. 1). Then detection and modelling procedures are discussed. The modelling of coloured noise based on autoregressive tech­niques is discussed for the time and frequency domain. The result of the modelling is a so-called shaping filter, which can be incorporated into the Kalman filter model.

251

(sampled) Data

Kalman Filter

Detection

y

Time-Domain Analysis Frequency-Domain Analysis

Filter State Augmentation

Figure 1: General detection and modelling strategy

2 General Procedure

In the outset we assume the data are processed by a Kalman filter. The Kalman filter does not necessarily operate at an optimum because it is based on the assumption of white measurement noise. The processing of the data by a Kalman filter is essential, as for all further analysis it is tacitly assumed that the sequence of measurement residuals is wide sense stationary (WSS). Wide sense stationarity can only be guaranteed if all trends are removed from the data and this can be achieved by processing the data with a Kalman filter. In practice, the analysis of sequences is based on time averages so that we also assume that all sequences are ergodic. All subsequent techniques are based on the predicted residuals. The predicted residuals

Yk = ~ - Ak&klk-l for k = 1,2, ... (1)

(an underscore indicates the random character of a variable) and their covariance matrix

(2)

are generated automatically by the Kalman filter. In eq. (1) Yk represents the mk­

dimensional vector of predicted residuals at time k (Rk); ~k the mk-dimensional vector of

252

observables; and Ak&:lk-l (AkPklk-lAI> the vector of predicted observables (in paren­theses we give the corresponding covariance matrices). Under the null hypothesis the Kalman filter operates at an optimum and the sequence of predicted residuals is a zero mean, white noise sequence with known covariance and Gaussian if the data are assumed Gaussian.

The Kalman filter is a linear estimator and thus any correlation present in the mea­surements will also be visible in the predicted residuals.

In this paper we will further limit ourselves to single data channels. We will thus analyze scalar sequences of predicted residuals {!l[k]} and it is assumed that the detection and modelling is performed for batches of N data samples.

2.1 Detection

Coloured noise can be detected by using the autocorrelation sequence (ACS). The ACS of a WSS scalar process {Q[k]} is defined as (E{.} is the expectation operator):

ru [I] = E{Q[k + I]Q[k]} . (3)

For a limited batch of N data samples, an estimator of the ACS is given as

tv [I] tv [I]

1 N-l-l

N L Q[k+ I]Q[k] 0 ~ I ~ N 1:=0

fu[-/] -N ~ I ~ -1

(4)

The justification for choosing this estimator will be given later. The estimator given by eq. (4) is approximately Gaussian and under the null hypothesis the variance of the ACS estimator is approximately [Jenkins and Watts, 1968]:

D{iu[/]} ~ !(r![oj) for I i= 0 . (5)

As a consequence, the null hypothesis of uncorrelated residuals is rejected at the a level of significance if

I" []I ru[O] ( ) ru I > na/2 Vii for I i= 0 , 6

where na /2 is the a/2 probability point of the normal probability density function. White­ness of the sequence of predicted residuals can also be tested by non-parametric tests such as the run-test and the reverse arrangements test [Bendat and Piersol, 1986]; gaussianess can be tested with a Chi-square goodness-of-fit test [ibid.]; possible nonlinearities in the model (the Kalman filter is valid for linear models) can be detected by advanced proce­dures as given in [Brockett et al., 1988].

Equation (5) can be used to determine the sample size N. In the literature one often finds values of N = 500 or 1000 [Jenkins and Watts, 1968].

In practice, the monitoring of the predicted residuals will be executed parallel to a real-time Kalman filter. As it is unlikely that in dynamic environments the property of wide sense stationarity can be strictly maintained, it is advised to perform the above test at frequent intervals.

253

2.2 Modelling

H the residuals do not pass the whiteness test, and apparently the measurement noise is non-white, the following Kalman filter design strategies are available:

1. The sample interval can be enlarged so that the remaining data are uncorrelated; this may result in a very low frequency of measurement updates and a loss of information, but the original Kalman filter design can be maintained.

2. In some cases one can apply measurement differencing to obtain uncorrelated (de­rived) measurements at successive epochs.

3. One can model the noise in state-space and augment the Kalman filter model with the noise model.

The second and third option require explicit modelling of the noise characteristics. The second approach is not followed because it requires an alternative Kalman filter design, which is not the topic of this paper [Bryson and Henrikson, 1968]. In this paper we follow the third approach via state augmentation. In practice not all data channels will be subject to coloured noise.

We assume that coloured measurement noise is generated by passing white noise through a linear shaping filter. AB the Kalman filter is a linear filter, the predicted residuals will also be time correlated. H the shaping filter is invertable, one could devise a so-called whitening filter that decorrelates the data (see Fig. 2).

(linear) sampled

(linear) white noise data white noise

process ---+- shaping filter hitening filte process L(z) 1/ L(z)

Figure 2: Shaping filter with transfer function L(z) and whitening filter

Given the process of time correlated predicted residuals, one can identify the param­eters of the shaping filter. Once these parameters have been obtained, the shaping filter model for the time correlated observations can be incorporated in the Kalman filter. To arrive at tractable solutions we limit ourselves to filters which can be modelled by constant coefficient difference equations, in particular first and second order autoregressive (AR) models. We only consider low-order AR-models for the following reasons:

1. Various types of time correlated noise (e.g. Markov sequences) can be conveniently modelled by AR-models.

2. Efficient algorithms are available for the estimation of AR-model parameters (see, e.g. [Marple, 1987; Kay, 1988]).

3. State augmentation for time correlated noise described by AR-models is a straight­forward procedure.

254

3 Time Domain Approach

In this section we will discuss the modelling of shaping filters based on AR-models. An AR-model of order p for a discrete-time process b[k]} can be described by a constant coefficient difference equation as follows:

, Q[k] = - E a,Q[k - I] + n[k] (7)

1=1

for k > 0, where {n[k]} is a zero mean white noise process with variance u! and a" I = 1, ... ,p are the AR-parameters (ao is 1 by definition).

AR parameter estimation methods

In the following we briefly discuss two strategies to obtain estimates for the AR-parameters. Lea8t-Square8 Approach. If one considers a fixed batch of data, e.g. k = 1, ... , N, eq.

(7) can be used to obtain the following set of equations:

y=Ax+n, (8)

where

Note that the definition given above avoids the zeroing of data (Le. no assumptions have to made regarding Q[-(P+ 1)], ... ,Q[O]). Based on eq. (8) the estimation of the AR­parameters may be interpreted a Least-Squares problem based on a model with observation equations and the AR parameters may be estimated directly. Exact solutions are, however, not available due to the products of the parameters tli, i = 1, ... ,p and the random variables Q[i], i = 1, ... , N. In practice the variables Q[i] are replaced by their sample values. No definite guidelines for the choice of the vector yin eq. (8) can be given and, as a consequence, numerous (approximate) AR-parameter estimation schemes are available.

Autocorrelation (Yule- W4lker) Method. If both sides of eq. (7) are multiplied by Q[k - m] and subsequently the expectation operator is applied to both sides one obtains:

, E{Q[k]Q[k - m]} = E{ - E a,Q[k - I]Q[k - m]} + E{n[k]Q[k - m]} . (9)

l=1

Using the definition of the autocorrelation function E{Q[k - I]Q[k - m]} = r" [m - I] and observing that E{n[k]Q[k - m]} is zero for m > 0, eq. (9) can be written as

, r"[m]=-Er,,[m-I]a,, m>O.

l=1

(10)

Writing eq. (10) for m = 1, ... ,p one obtains the following linear set of equations, which are called the Yule-Walker equations

Ra= -r, (11)

255

where

and use is made of the fact that for real-valued processes r.,[-l] = r.,[l]. The matrix R is called the autocorrelation matrix and is semi-positive definite, Toeplitz, and (for real­valued processes) symmetric. A solution of the Yule-Walker equations follows immediately as

(12)

Due the special structure of the autocorrelation matrix, the inversion of the matrix R can be avoided by using a recursive scheme attributed to Levinson and Durbin [Box and Jenkins, 1976].

Writing eq. (9) for m = 0, and noting that E{n[k]!![k]} = u!, the variance of the white noise process {n[k]} is obtained as

p

o! = Ea,r.,[l] (ao = 1) . (13) '=0

In practice, the Autocorrelation method is a two-step procedure. The autocorrelation matrix R is generally not known and has thus to be estimated first. In the second step the ~parameters and the residual variance are estimated as follows (cf. eqs.(12) and (13»

p

a = -a-1r i U! = Ea,r.,[q (ao = 1) . (14) '=0

The ACS estimator given by eq. (4) and used in eq. (14) is a biased estimator. However this estimate guarantees a semi-positive definite autocorrelation matrix [Jenkins and Watts, 1968].

AR parameter estimator statistics

H the white noise process {n[k]} is assumed to be Gaussian it can be shown [Kay, 1988] that all AR parameter estimators are approximate Maximum Likelihood estimators. For these estimators (large sample) Cramer-Rao bounds are derived in [ibid.], namely

( A) (U!R-1 0) var{ ;. } ~ Nor ';! . (15)

Equation (15) should be used to determine the sample size N.

256

Practical Considerations

In order to obtain a stable causal filter, the estimated AR parameters are subject to certain conditions. For an AR(I) process this means that lall ::; 1. For an AR(2) model the conditions are [Hay kin, 1988]: -1 ::; al + a2 ; -1 ::; a2 - al ; and -1 ::; a2 ::; 1. Using the autocorrelation method for the estimation of the AR parameters these conditions are automatically fulfilled.

Usually the order of the AR process is not known in advance. To obtain tractable solutions for Kalman filter applications, it is advised to limit the order of the AR-process to 1 or 2. In the literature many order selection criteria are given. As we are only concerned with AR processes of order 1 or 2, it is advised to choose that model order for which the estimated residual variance 0-; is minimized.

State Augmentation

As the Kalman filter is a linear filter, the shaping filter which generates the coloured noise, can be modelled with the same (AR)-parameters. The shaping filter can then be used to augment the Kalman filter model by which the order of the Kalman filter is enlarged by the order ofthe shaping filter.

Consider the following Kalman (n-state) filter model with scalar measurement model:

~+1 - .A:+I.I:XI: + wI:

1l.1: - aT Xk + ~I: .

(16) (17)

If we assume that the measurement noise sequence {~[k]} is generated by a first order AR-model, the Kalman filter model can be augmented as

( ::: ) = (.~" _~,) ( : ) + (o~ ~) ( ::, ) (18)

1l.1: = (aT 1) ( : ) (19)

where 0 is a n X 1 vector of zeros and aT is a row of the design matrix AI: in eq. (1). For a second order AR-model the augmentation proceeds as

o o ~ ) ( ~:l ) + (~T ~) ( WI: )

.. OT 1 rll:+1 -al ~

(20)

l'.. = (aT 0 1) ( ~, ) (21)

Note that for eqs. (19) and (21) Db~l:} = 0; that is, state augmentation leads to a noise-free observation model.

257

4 Frequency Domain Approach

Before the frequency domain approach to AR modelling is discussed, the formal power spectral density (PSD) function of an AR process, which will be needed later on, will be gIven.

(12 PSDAR(fr) = --p--!.:.."----:-2 for - 1/2 :$ Ir :$ 1/2 (22)

1 + Lal exp( -j271Jrl) 1=1

where j = yCI, and Ir is a fraction of the sampling frequency IT, that is Ir = 1/ IT (IT = l/T [Hz] and T is the sampling interval in [sec». For real-valued discrete-time processes, the PSD is an even, periodic function. In Figure 3 a number ·of AR PSDs are given for first and second order AR-processes. Fractions of the sampling frequency close to zero correspond with low-pass filter characteristics, whereas fractions close to ±1/2 represent high-pass filter properties.

20 20 20 20

15 (a)

15 (b) 15 15 (d)

10 10 10 10

5 5 5 S

0 0 0 0

-5 -S -5 -S

-10 -10 -10 -10 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 O.S

20 20 20 20

15 15 (f)

IS (g) 15

(h) (e) 10 10 10 10

5 5 5 5

0 0 0 0

-5 -5 -5 -S

-10 -10 -10 -10 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

Figure 3: AR spectra; depicted is 10Iog10(PSD(fr» vs. the fraction of the sampling frequency Ir for (a) al = 0.6; (b) al = 0.6, a2 = 0.3; (c) al = 0.6, a2 = -0.3; (d) al = 0, a2 = 0.6; (e) al = -0.6; (f) al = -0.6, a2 = -0.3; (g) al = -0.6, a2 = 0.3; (h) al = 0, a2 = -0.6.

AR parameter estimation

The frequency domain approach to AR parameter estimation starts with the computation of the PSD of the sequence of predicted residuals. Two classes of methods are available

258

to compute the PSD, namely the periodogram methods, and the correlogram methods, which are extensively discussed in the literature [Marple, 1987; Kay, 1988]. The latter methods are based on the (discrete-time) Fourier transform relationship between the ACS and the PSD

00

PSDu(fr) = E ru[k] exp( -j27r/rk) for - 1/2 ~ Ir ~ 1/2 . (23) A:=-oo

IT one uses the FFT algorithm to compute the PSD from the sampled predicted residuals one obtains a PSD estimate at equally spaced frequencies. For real-valued processes, the resolution is aIr = aI/IT = 1/(2N).

The modelling in the frequency domain approach is based on the fact, well known from discrete-time linear system theory, that, if one assumes that the shaping filter is causally invertible, the power spectrum in the z-domain can be written as

(24)

for a white noise input with variance 0':. In eq. (24) z = exp(j21f'/r) and it is assumed that L(z) is a minimum phase system function (i.e. all poles and zeros lie within the unit circle in the z-plane) that represents the shaping filter.

Once a PSD estimate has been obtained, the modelling proceeds as follows. From the characteristics of the PSD it is determined by inspection whether a first or second order model has to be fitted to the data (cf. Fig. 3). Then the (AR)-parameters are obtained by computing a least-squares fit of the magnitude (squared) of the shaping filter transfer function IL(z)12 to the estimated PSD. N data samples lead to N nonlinear observation equations of the form

E{14} = a(x,i) i = 1, ... , N

where, with Ii = (i - 1)/(2N),

(25)

The parameters 0':, al (and a2 for second order models) represent the unknown param­eters. Mter linearization of the observation equations, the parameters can be estimated by Least-Squares adjustment. In practice, no covariance matrix for the observables y. is

'4

specified and thus no variance matrix of the unknowns can be given. As it is difficult to specify approximate values for the unknowns, the Least-Squares procedure may fail to converge. Therefore Held [1982] suggests to combine the least-squares method with a direct search method.

5 Summary and Concluding Remarks

In this paper we discussed the detection and modelling of coloured noise for Kalman filter applications (cf. Fig. 1). The modelling is based on autoregressive (AR) techniques and the sequence of predicted residuals. The time domain approach to AR parameter estimation can be summarized as:

259

1. Estimate the AR parameters using the least-squares or autocorrelation method. 2. Augment the filter model with the AR shaping filter.

Likewise the frequency domain approach is given as: 1. Compute the PSD of the sequence of predicted residuals. 2. Fit the computed PSD to the formal AR PSD. 3. Augment the filter model.

A number of remarks can be made regarding the procedure sketched in Figure 1. 1. Modelling is based on the assumption that the predicted residuals are zero

mean. Therefore the Kalman filter should be supported by a detection, identification, and adaptation procedure.

2. The sample size N should be based on variance measures as given by eqs.(5) and (15).

3. Both the frequency and time domain approaches are approximate methods. 4. The time domain approach is more direct. 5. State augmentation does not take the covariance matrix of the AR-parameters

into account. The estimates are considered to be constants in the augmented model. 6. State augmentation leads to an increase of the overall model order by the order

of the AR-model and to noise-free measurements. 7. The test of whiteness of the residuals should be performed at regular intervals.

References

Bendat J.S and A.G. Piersol (1986). Random Data, Analysis and Measurement Proce­dures, 2nd ed. Wiley-Interscience, New York.

Box, G.E.P. and G.M. Jenkins (1976). Time Series Analysis Forecasting and Control. Revised Edition, Holden-Day, San Francisco, CA.

Brockett, P.L., M.J. Hinch, and D. Patterson (1988), Bispectral-based tests for the detec­tion of Gaussianity and linearity in time series. JASA, Vol.83, No.403, pp.657-664.

Bryson, A.E. and L.J. Henrikson (1968). Estimation using sampled data containing sequentially correlated noise. J. Spacecraft and Rockets, Vol.5(6), pp.662-665.

Haykin, S. (1988). Adaptive Filter Theory. Prentice-Hall, Englewood Cliffs.

Held, V. (1982), Techniques and methodologies for the estimation of covariances, power spectra, and filter state augmentation. in: C.T. Leondes (ed.) Agardograph No.258, NATO, Neuilly sur Seine.

Jenkins, G.M. and D.G Watts (1968) Spectral Analysis and its Applications. Holden-Day, Oakland.

Kay, S.M. (1985), Broadband detection of signals with unknown spectra. Record 1985 IEEE ASSP Conference, pp.1263-1265.

Kay, S.M. (1988). Modern Spectral Estimation. Prentice-Hall, Englewood Cliffs.

Marple, S.L. (1987). Digital Spectral Analysis. Prentice-Hall, Englewood Cliffs.

260

RESEARCH ON CLOSED·LOOP KALMAN FILTER TECHNOLOGY OF SINS/GPS FOR SURVEYING

ABSTRACT

Wan Dejun Ren Tianmin

Dong Bin Control Engineering Department

Southeast University Nanjing, MD 210018

P.R. China

Most INS errors are oscillatory with three harmonic components (Schuler, Earth, and Foucault) and grow with time due to the random noise, such as gyro random walk, coning and sculling motion of the vehicle, and vibration of the IMU base. This is especially true for strapdown INS. Conventional open-loop Kalman filters used in ISS have the same characteristics as closed-loop Kalman filters if no random noise exists in the ISS. In practice, the ZUPT method is used as a standard procedure but it can only be employed if regular vehicle stops are possible.

In this paper the development of a closed-loop Kalman filter is discussed for on-line filtering of strapdown ISS/GPS. First, the state transfer feedback mechanization is designed with the aid of data from GPS; the system errors, such as sensor errors, attitude errors and navigation error parameters, which affect each other are fed back by "digital compensation" which is only feasible for strapdown INS. Also, the filtering stability of closed-loop Kalman filter is analyzed. Finally, simulation results are used to compare open-loop and closed-loop Kalman fIlters and show their characteristic features.

INTRODUCTION

Inertial surveying is becoming one of the major geodetic engineering methods. As the errors of inertial systems oscillate and grow with operating time, the ZUPT method is generally applied to correct the errors of the inertial system during the surveying process. However, this method requires that the vehicle carrying the inertial system stops at intervals of 3-5 minutes to perform the updating. Hence, integrated inertial surveying systems (e.g., INS/GPS hybrid systems) which can perform continuous navigation without stopping have received considerable attention among inertial researchers.

For positioning it is necessary to estimate the system errors of the inertial system by means of the Kalman filter technology using ZUPT or INS/GPS methods. In general, the Kalman filter mechanization uses open-loop estimation methods which are available for both platform and strapdown inertial systems. They do not depend on the navigation algorithm and can realize discrete processing and smoothing. But the open-loop Kalman filter cannot correct for system errors. Only under certain conditions, such as negligible random drift errors or short operating time, will the performance of the open-loop Kalman

261

filter approach that of the closed-loop filter. Under actual conditions, however, especially for missions of long duration, the estimation errors of the open-loop Kalman filter will probably cause divergence if not bounded by external measurements.

The implementation of the closed-loop Kalman filter has been made possible by the development of laser gyro strapdown inertial systems. The closed-loop Kalman filter cannot only estimate the system errors but also realize feedback correction. Therefore, the system error compensation is more reasonable, the mathematical model of the filter corresponds better to the actual system, and the results of estimation and compensation are better. In addition, the feedback correction is simpler and easily implemented due to the "mathematical platform" of the strapdown system.

This paper discusses the system mechanization, the physical mechanism and feedback mode of the open-loop and closed-loop Kalman filters, and compares typical features of these filters. Finally, a simulation study illustrates the effect of open-loop and closed-loop Kalman fIlters during continuous operating mode.

MECHANIZATION OF SYSTEM

Open-Loop Kalman Filter

The dynamic error equations of inertial surveying systems can be described by

X(t) = F(t) X(t) + G(t) W(t)

X(O) = Xo

The state vector of the system includes 15 states:

X = (<I>N, <l>E, <1>0, BVN, BVE, BVo, BA., &p, Bh, Ex, Ey, Ez, Ttx, Tty, Ttz)

(1)

(2)

where <I>N angular misalignment of the North axis of the mathematical platform of the strapdown system, angular misalignment of the East axis of the mathematical platform of the strapdown system,

<1>0 angular misalignment of the down axis of the mathematical platform of the strapdown system,

BVN ... North velocity error of the system,

BVE ... East velocity error of the system,

BVo ... down velocity error of the system,

BA. longitude error,

&!> latitude error,

Bh height error,

EX gyro drift along X axis of the vehicle body coordinate frame,

Ey gyro drift along Y axis of the vehicle body coordinate frame,

EZ gyro drift along Z axis of the vehicle body coordinate frame,

Ttx accelerometer bias along X axis of the vehicle body coordinate frame,

Tty accelerometer bias along Y axis of the vehicle body coordinate frame,

262

llz accelerometer bias along Z axis of the vehicle body coordinate frame, F(t) dynamics matrix of the system, G(t) noise shaping matrix, Wet) ... vector of system noise, Xo initial values of system errors.

The exact formulation of the matrix F(t) can be found in Yuan and Zheng (1985). If the gyro drift is modelled as random walk, G(t) will be a 15x3 matrix and Wet) a 3xl vector.

If the external measurement is a velocity or position vector, the observation equations may be expressed as

Z(t) = H(t) X(t) + Vet) (3)

where H(t) ... system design matrix of dimension 3x15,

Vet) ... vector of measurement noise.

Discretizing equations (1) and (3) leads to the following formulation of the Kalman filter equations:

XK = <I>K,K-I + rK-I WK-I

ZK = HKXK+VK

where

1 1 <I>K,K-I = 1+ F T + 2! p2 'f2 + 3! F3 T3 + '"

1 1 1 rK-I = T (I + 2! F T + 3! p2 'f2 + 4! F3 T3 + ... ) G

and T . .. filter update interval for <I> and r matrices.

The accuracy of the open-loop Kalman filter estimation depends mainly on the accuracy of the filter formulation and the accuracy of the external measurements. The open-loop Kalman filter cannot correct for system errors, which will freely follow their original characteristic function:

x = FX+GW

This is especially true when the system errors are random noise. Under this condition the response curves of the system are of the form shown in Figure 1 (see also Huang, 1986).

Yuan and Zheng (1985) have shown that the dynamics matrix F of the filter is a first-order approximation of the system errors, under the condition that the error 8 is small, i.e.

82=0. However, this condition is violated if the system errors are larger than that. Hence, the mechanization of the fllter is not ideal. It will result in larger deviations of the estimated values from theoretical values and the system error oscillations will have larger amplitudes or grow with time.

263

5

4

3

1

y

(j

o

Fig. 1. Error response energy curves of inertial system under excitation of random errors.

Closed-Loop Kalman Filter

The closed-loop Kalman fllter for an inertial surveying system may be written in discrete fonnulation as

XK = ~-l XK-l + rK-l WK-l + BK UK

ZK = HKXK+VK

(4) (5)

where BK is the system's feedback matrix and UK is the control input. The prediction and update equations can be expressed as

i K+1IK = <l»K+l,K iK + BK UK (6)

i K+1 = i K+1IK + KK+l (ZK+l - HK+l iK+!IK) (7)

Combining the above two equations, we obtain:

i K+1 = (<I»K+l,K - KK+l HK+l <I»K+l,K> iK + (I - KK+l) BK+l UK + KK+l ZK+l (8)

Now let

){K == XK -iK

and combine equations (4) and (8). Then we have

XK+l = XK+l - i K+1

= (<I»K+l,K - KK+IHK+l<l»K+l,K»){K + (rK - KK+IHK+lrK) WK - KK+l VK+l (9)

Here, we apply state variable feedback, so that the control input is a linear function of the state vector:

UK = CKiK

Substituting this expression into equation (4) gives

264

XK+l = (<I>K+l.K + BK CK) XK - BK CK}(K + rK WK

Combining equations (9) and (10), we can now write

(XK+l) _ (<I>K+l/K + BKCK -BKCK ) (XK) XK+l - 0 <l>K+l/K - KK+IHK+l<l>K+l/K XK

+ ( rK )WK _ (0 ) + VK+l rK - KK+IHK+lrK KK+l

(10)

(11)

In order to ensure that the state vectors XK-l and XK-l in equation (11) are close to zero, we take the simplified expression

(12)

The XK will approximate the zero state with the convergence speed of the optimal estimation. Then we have

1\

XK+l = <l>K+lKXK+rKWK-<I>K+lKXK . . ZK = HKXK+VK

(13)

(14)

Since equation (13) can be applied to the mathematical platform of the inertial system, the state vector may be numerically corrected with the previously estimated state vector.

The application of the closed-loop Kalman filter enables the system to carry out optimal estimation and a one-step correction of the error states, and it restricts the errors to small values. Hence, the first-order approximation of the F-matrix for the Kalman filter formulation can be kept, and the accuracy of the mathematical model for the filter is ensured. This eliminates errors due to model imperfections; an important feature for inertial surveying systems.

The closed-loop feedback correction is generally used in strapdown inertial systems. Its algorithm should be applied together with the attitude and navigation algorithms of the system. Figure 2 shows the block diagram for the closed-loop Kalman fIlter algorithm.

Attitude calculation

Altitude correction

Navigation computations

Correction of navigation parameters

Fig. 2. Block diagram of the closed-loop Kalman filter algorithm.

265

SIMULATION

Generation of Simulation Data

We simulate a vehicle travelling at a unifonn speed of 10 ms-1 from South to North in a horizontal plane. The disturbing part of the Earth's gravity field is neglected. We assume that the vehicle travels continuously for 30 minutes, covering a distance of 3600 metres. These are the conditions under which the hybrid inertial surveying system navigates using the open-loop or the closed-loop Kalman filter, respectively.

System Errors

We model the gyro drift as a combination of a random constant and a random walk with variances

crbias = crw = 0.00 1 0/h

Accelerometer errors are modelled as a bias:

~ = 10-5 g

System Measurements

GPS Velocities. The velocity measurements supplied by GPS are used as reference values. For the measurement errors the simplified white noise model

E(VV1) = (0.1 ms-1)2

is used.

GPS Positions. The positions supplied by GPS are used as reference values. For the measurement errors we use the simplified white noise model

E(VyT) = (0.1 arcsec)2

Initial Values for System Errors

The initial values of the system errors are assumed to be:

8VNo = 8VEo = 8VDo = 0.003 ms-1

8", = &!> = 8h = 5 m

M>No = B<l>Eo = 10"

M>Do = 30"

Initial Values for Filter

The initial values for the filter are set to:

266

PI,1 = P2,2 = (30")2

P3,3 = (100")2

P4,4 = P5,5 = (0.002 m)2

P6,6 = (0.005 m)2

P7,7 = Ps,S = (50")2

P9,9 = (20 m)2

PlO,lO = Pll,ll = P12,12 = (0.003 0/h)2

P13,13 = P14,14 = P15,15 = (10-5 g)2

Comparison of Simulation Results

The position, attitude and velocity errors of the inertial system integrated with velocity and position measurements are shown in Figures 3 to 8 and Figures 9 to 12, respectively. Error curves are plotted for open-loop and closed-loop Kalman filter results. The simulation results for the four methods are listed in Table 1.

o -10

-20

-30

-40

-SO

-60

-70

-80

Y {E-2, (arc sec)

6000 t iTlle t..c..c.}

Fig. 3. Longitude errors for open-loop and closed-loop Kalman filter integrated with velocity.

267

30

20

\0

o -10

-20

-30

-40

-so

YIE-41 {arc sec)

OPEN-LOOP

a..os£o-UXlP

Fig. 4. Latitude errors for open-loop and closed-loop Kalman filter integrated with velocity.

50

40

30

20

10

o

y tE l' (nrc sec)

2000 4000 6000 time ,a.ct

Fig. 5. Heading errors for open-loop and closed-loop Kalman filter integrated with velocity.

8

Y (i 0) (arc sec)

CLOSm-L(X)P

Fig. 6. Pitch angle errors for open-loop and closed-loop Kalman filter integrated with velocity.

268

40

20

o

-20

Y(B-4) (a/aec)

Fig. 7. East velocity errors for open-loop and closed-loop Kalman filter integrated with velocity.

40

20

o

-20

-40

-60

Y(E-4) 1m/sec)

Fig. 8. North velocity errors for open-loop and closed-loop Kalman filter integrated with velocity.

40

20

o

-zo

-40

-60

y (E-4) (arc sec)

Fig. 9. Longitude errors for open-loop and closed-loop Kalman filter integrated with position.

269

so

40

30

20

10

Y (E I)

(arc sec)

2000 4000

Fig. 10. Heading errors for open-loop and closed-loop Kalman filter integrated with position.

8

6

4

o

-2

v (E 0) lare sec)

40uO 600Q tilll4 <56')

Fig. 11. Pitch angle errors for open-loop and closed-loop Kalman filter integrated with position.

o

-It"}

-20

Geoa time ''''.c...

Fig. 12. East velocity errors for open-loop and closed-loop Kalman filter integrated with position.

270

Tab. 1. Comparison of errors for the four methods.

Error values ~n-loop filter ,-,losed-loop filter Open-loop filter ~losed-loop filter with velocity with velocity with position with position

Longitude 915.77x1O-3 1.4998x10-3 6.01lxl0-3 6.013x1O-3 [arcsec] Latitude 5.1079x10-3 1. 1 099x 10-3 6.525x10-3 6.522x1O-3 [arcsec] Course 620.396 618.354 614.53 586.74 [arc sec] Attitude 10.3309 6.9246 10.224 8.926 [arc sec] Velocity 5.6125x1O-3 5.4520xl0-3 -24.5lx1O-3 8.07x1O-3 [ms-I]

From the results of the simulation study we conclude that the open-loop Kalman filter integrated with external velocity measurements successfully reduces the time-dependent growth of the system's positioning errors. However, the position errors still increases gradually and depend principally on the internal error sources of the system and the accuracy of the external velocity measurements. This method requires high precision for the velocity updates. It allows continuous operation for longer times than the ZUPT method. GPS position aiding of the open-loop Kalman filter makes the estimated positioning errors dependent on GPS positioning accuracy, while the estimated velocity errors are larger than those obtained from GPS velocity aiding.

Analyzing the observability of the states, it is obvious that velocity or position updates in the Kalman filter will have little effect on the heading errors of the system. Therefore, the estimation of heading errors in the dynamic constant heading situation is not improved by external measurements. When external velocity measurements are used for updates, the estimated velocity errors depend principally on the accuracy of the external measurements. Therefore, the closed-loop Kalman filter results are similar to the open-loop data .. However, the accuracy of attitude and position improves remarkably. This is an important advantage of the closed-loop Kalman filter.

CONCLUSIONS

1. The integrated Kalman filter with closed-loop feedback correction is superior to the open-loop filter because it restricts the growth of the system errors more effectively. This makes it more useful for hybrid inertial surveying systems which operate for long time. This is especially true for laser gyro strapdown inertial systems.

2. Closed-loop mechanization can restrict the errors within a certain range and reduces the probability of time-dependent error growth. The positioning accuracy for a closed loop filter depends mainly on the accuracy of the external measurements.

3. To meet certain system accuracy requirements, the accuracy and characteristics of the inertial sensors may be decreased if a closed-loop Kalman filter is applied.

4. If heading measurements can be used as updates, then the observability of the heading errors and the error compensation can be improved.

271

5. Research on a closed-loop Kalman filter integrated with GPS differential position measurements as well as on the corresponding hybrid design of a system operating in continuous mode should be continued.

REFERENCES

Huang Deming (1986). Inertial navigation systems, National Defense Industry Publishing House, Beijing. .

Wong, R.V.C. (1981). A Kalman filter/smoother for the Ferranti Inertial Land Surveyor system, Proc. of the Second Inti. Symp. on Inertial Surveying, Canada.

Yuan Xin and Zheng E. (1985). Principle of strapdown inertial navigation, Nanjing.

272

RELIABILITY ANALYSIS APPLIED TO KINEMATIC GPS POSITION AND VELOCITY ESTIMATION

Gang Lu Gerard Lachapelle

Department of Surveying Engineering The University of Calgary, Calgary, Alberta, Canada, T2N IN4

ABSTRACT

Reliability analysis or quality control is an important aspect for real-time kinematic GPS positioning and velocity estimation. Due to the dynamic environment and few or no redundant observations, the reliability control is much more difficult in kinematic than in static GPS surveys. By using the testing theory of dynamic models, this paper investigates the reliability of kinematic GPS positioning and velocity estimation. Recursive formulas for reliability analysis and for the determination of bias influences on the fllter state vector are derived. The characteristics of different kinds of bias influences on position and velocity estimation are illustrated with numerical simulations. Some results of reliability analysis corresponding to single and multipIe'biases are also given as a function of the number of satellites observed simultaneously.

1. INTRODUCTION

It is known that by using GPS pseudo-range, phase and phase rate (Doppler) observations in a Kalman fllter, vehicle's position and velocity, which are the essential elements required in navigation applications, can be determined in real time [e.g., Schwarz et ale 1989; Hwang et al. 1989]. If the assumed dynamic and observation models are correct, the Kalman filter provides the optimal position and velocity estimates in a statistical sense. In a dynamic environment, the deviations or biases from the assumed models are often significant. For instances, the loss of phase lock on a satellite will cause the corresponding phase observation to jump abruptly or a pre-determined constant velocity model for vehicle motion may be invalidated due to vehicle acceleration in some parts of the trajectory. Such deviations or biases will lead to errors in the estimates provided by the filter. How do different biases affect the estimated positions and velocities? And what are the minimum detectable values of different kinds of biases by statistical testing in a prescribed kinematic GPS surveying model? These questions can be dealt with using reliability analysis for kinematic GPS applications.

Recently, several researchers have made significant progress towards establishing a general quality control theory based on the standard Kalman flltering model [Teunissen 1989, 1990; Salzmann et ale 1989]. The focus of this paper is on the modification and application of this general theory to the reliability analysis of estimated kinematic GPS positions and velocities. A more general recursive bias influence formulation on the filtered quantities is given by using Friedland's two stage Kalman fllter method. The influence characteristics of different biases on estimated kinematic GPS positions and velocities are investigated. Finally, the Minimum Detectable Biases (MDB) for kinematic GPS surveys are computed for selected satellite configurations.

273

2. BIAS INFLUENCE ON KALMAN FILTERING QUANTITIES

There are generally two ways to treat biases arising from the system model or the observation model in recursive filtering. Firstly, one can augment the state vector of the original model by adding components to represent the bias terms. The filter then estimates these terms as well as the original states. Secondly, the bias terms can be separately estimated by using the bias-free Kalman filtering results. This is the so-called two-stage Kalman filter [Friedland 1969; Ignagni 1981]. It is proven that these two methods are mathematically identical. But the advantages of the second method are that it is computation efficient and convenient for bias influence analysis on filtering results. Hence, it is more suitable for Kalman filter design. A Kalman filter model containing a constant bias vector can be expressed as

xk = ~xk-1 + Bkb + wk zk = Hkxk + Ckb + vk

(1)

(2)

where <D is the transition matrix, H the design matrix, Band C, bias design matrices and b, the constant bias vector. In the case when only observations have biases, Bk = O. Similarly, if only the dynamics have biases, Ck = O. If no biases are present in the model, Bk = Ck = O.

If the bias vector were perfectly known, as this is the case when we analyse the influence of a specified bias vector on the filtering results, the optimal estimates of the state vector x in equations (1) and (2) would be

while the predicted residuals with known biases would be

(3) (4)

(5)

where b is the true value of the constant bias vector. Kk is the Kalman gain matrix of the bias­free Kalman filter model. Since the bias influence on Kalman filter estimates is linear in nature

[Friedland 1969], we can formally write the differences between Xk( -), Xk( +) and their corresponding bias-free estimates Xk( -) ,Xk( +) as:

Xk(-) = ~(-) + Ukb

Xk(+) = Xk(+) + Vkb.

(6)

(7)

For the same reason, the difference between the biased predicted residuals Yk and bias-free predicted residuals Vk can be written as

(8)

where Uk ' Vk and Sk are the so-called sensitivity matrices [Ignagni 1981]. By simple

substitution of eqn. (3) and bias-free estimate Xk( -) into eqn. (6), we obtain:

274

Similarly, the recursive forms for Vk and Sk can be written as

Vk = Uk - KkSk Sk = HkUk + Ck-

(9)

(10) (11)

In a two-stage Kalman fllter, once the Sk is determined, the bias vector b can be estimated with a Kalman fllter algorithm or a sequential least squares method by using the system model btc-=hJ,.. -1 and the quasi-measurement model of eqn.(8) with the quasi-m~surement vector ~k (61as:rree predicted residuals) and its covariance matrix (Rk + HkPk( -)!lk ). The initial value for the recursive computation of Uk' V k and Skstarting at epocli n IS V n-l = O. Equations (9), (10) and (11) are of great importance in bi~ iIilluencx analysis. If a bias occurs, its influence on the bias-free Kalman filtering estimates Xk( +) and Vk , as shown in (7) and (8), are -V kb and Skb respectively. In Section 5, different kinds of bias influences on kinematic GPS position and velocity determination are investigated. Among them are carrier phase cycle slips, gross errors in pseudo-ranges and in phase rates, and vehicle motion biases in the assumed dynamic model.

3. RELIABILITY ANALYSIS IN KALMAN FILTERING

Reliability analysis is based on the use of statistical tests for bias detection in the assumed model. In Kalman filtering, the widely used quantities in statistical testing are predicted residuals or innovation sequence which are defined as the difference between actual and predicted observations. Under normal conditions or null hypothesis that no bias are present in the model, the predicted residuals,

...... ~T ...... T ...... T)T V = \Vl , Vl+l , •••• , Vk ,

are normally distributed as ~ - N(O, Qv)' where Q is a block-diagonal matrix since the predicted residuals are uncorrelated between epochs. However, if a bias vector b of dimension d is present, the mean of the predicted residuals at each epoch, known from eqn. (8), is no longer zero but biased by Skb. That is, the vector ~ is distributed as ~ - N(Vv, Qv) with

Vv = Sv· b,

where - ST ST ST T Sv - ( I' 1+1' ...... , k)· (12)

The test statistic for bias vector b is given by Teunissen [1989, 1990] as:

k k k

T = (L STQ,~Vi)T(L STQ,}Si)-l(L STQ,}Vi) - X2(d, A) (13) i=l i=l i=l

k A = bT(L sTQ,}svb (14)

i=l If k = 1, that is, only the predicted residuals of the current epoch are used to form the statistic, T is called the Local Test. When several epochs are used in the statistic, T is called the Global Test. The number of successive epochs included in the test is the window length of the test.

275

To test whether the considered bias vector is significant or not, one has to choose a significance level a and compare the actual computed value of the statistic T with the critical value of TO: In reliability analysis or Kalman filter design, however, what we are interested in is the mmimum detectable value of the bias vector with a given probability power l-~ in an a level test. By fixing ~ and a, the corresponding non-centrality parameter ?--o in eqn. (13) can be determined. Thus the minimum detectable bias (MDB) vector bo can be obtained by eqn. (14):

k

AO = b~(L STQ.}Si)bo (15) i=l

For a single bias, eqn. (15) reduces to

(16)

The minimum detectable bias vectors corresponding to the Local Test and the Global Test are referred to as Local MDB and Global MDB vectors respectively. For multiple biases, eqn. (15) describes a hyper-ellipsoid if the Ao is held to a constant value. The axes of this hyper-

k

ellipsoid are the inverses of the square root of the eigenvalues of the matrix L S T Q.} S i' i=l

For the convenience of Kalman filter design, we usually take the maximum axis of the ellipsoid (the inverse of the square root of minimum eigenvalue) as an overall measure of the system ability to detect the biases. The reliability analysis of estimated kinematic GPS positions and velocities will be given in Section 6.

4. KINEMATIC GPS POSITION AND VELOCITY ESTIMATION MODEL

In this paper, the Kalman fIlter model proposed by Schwarz et al. [1989] is adopted for the reliability analysis with one difference in treating the carrier phase cycle slips. In our approach, a cycle slip on a given satellite is treated as a constant bias to be recursively estimated and corrected by a two-stage Kalman filter. By using the GPS pseudo-range, phase and phase rate observations in a differential mode (double difference in our case), the instantaneous position and velocity of the vehicle can be simultaneously determined in this fIlter. The system state models assumed here are the constant velocity model and the constant acceleration model.

System model

In differential kinematic positioning, the initial baseline is occupied for a few of minutes in static mode to allow for a good determination of the ambiguities. Therefore, for the constant velocity kinematic model with double difference observation updates, the state vector is:

x = (&I>, BA, Bh, BVn, BVe, BVh) (17)

276

where cj) is the latitude, A. the longitude, h the height, V n the north velocity, Ve the east velocity and Vh, the up velocity. The phase ambiguity parameters which have been resolved in static mode arebeld fixed and thus do not appear in the state vector. The states oV n' OVe' oV h in (17) can be assumed to behave as a random walk or a fIrst-order Gauss-MarKov process, according to the vehicle motion.

Similarly, for the constant acceleration kinematic model with double difference observation updates, the states of the fIlter are:

(18)

where three additional acceleration states oan' oae, oah' corresponding to north, east and up accelerations respectively are added. These three states are also assumed to behave as a random walk or a Gauss-Markov process depending on the vehicle dynamics. For both models, the transition matrices <I» are given in (Schwarz et al. 1989).

Observation model

The simplified double difference (receiver-satellite) pseudo-range and phase observation equations are [Lachapelle 1990]:

VAp = VAp + VAdion + VAdtrop + £ (19)

V A<I»= V Ap + A. V AN - VAdion + V Adtrop + £ (20)

If the initial ambiguities are known and held fIxed, A. V AN disappears in (20). Furthermore, these two equations show that pseudo-range and phase observations are connected through the position states ocj), OA., oh. Therefore, these two kinds of observation are mainly contributing to the precise position determination in the fIlter. By differentiating the range equation p = II r - R II + Ot with respect to time and then linearizing with respect to cj), A., h, the range rate observation equation can be obtained as:

. dX dy dZ dX dy dZ dX dy dZ op = (a- + ~ + c-)Ocj) + (a-+ ~ + c-)OA. + (a- + ~ + c-)Oh

dcj) dcj) dcj) dA. dA. dA. dh dh dh

+ dPOVn + dP OVe + dPOVh + Ot dcj) R dA.Rcoscj) dh

(21)

where

Equation (21) is mainly contributing to the velocity updating in the fIlter and, to some limited extent, to the position determination. Double differencing (21) with respect to receiver-satellite forms the phase rate updating equation in this fIlter.

277

5. BIAS INFLUENCES ON POSITION AND VELOCITY ESTIMATION

This section investigates the influences of different kinds of biases on the position and velocity estimation. In kinematic GPS surveys, one of the most common biases in the observation (double difference) model are cycle slips, outliers in pseudo-ranges and outliers in phase rates. The likely biases when using the system model are the acceleration errors in constant velocity kinematic model and the acceleration disturbances in constant acceleration model. In order to inter-compare the magnitude of the different bias influences on position and velocity estimation, we defme some scalar terms as follows:

Local state Bias-to-Noise Ratio (LBNR) is the ratio between the bias VXi in a given state Xi, caused by bias vector b, and the its standard deviation (Jxi' i.e., LBNR=Vxi / (Jxi .

Global state Bias-to-Noise Ratio (GBNR) is the ratio between the t~al bias vector V~ in state vector and the state covariance matrix Pk(+), GBNR= V~T Pi: (+)V~.

In all the following computations, the ephemeris data of the 13 Block I and II satellites available on July 31,1990, were used. The simulated traverse was assumed to be in the Kananaskis Country, some 80 km west of Calgary. The vehicle velocity was assumed to be approximately 60 km h-1 and the kinematic model adopted was the constant velocity (Eq.17). The time window is between 18:00 to 19:00 (UTC), du:rjng which up to 7 satellites at an elevation ~150 were in view. The standard deviations assumed for the pseudo-range, carrier phase and phase rate observations were 2 m, 2 cm and 1.6 cm s-l, respectively. The interval between each epoch is 4 seconds.

1104 0 IiJ Vcp

• VVn 1102 ~ ~ 1100 ~

-10 c:Q C) 1098 ~

1096 -20 • . 1094 0 10 20 30 40 50 60

0 10 20 30 40 50 60

Number of epochs Number of epochs

Fig.l(a) GBNR Fig.1(b) LBNR for latitude and Vn

1 &-- VA. 30 .:..: ... ... ... ... ... -&- Vh 0 ... • • • • • VVe • VVh 20

~ -1 ~ 10 -2 ~ ...l 0 ... • • • • -3

-10 I I

-4 ..

0 10 20 30 40 50 60 0 10 20 30 40 50 60

Number of epochs Number of epochs

Fig.l(c) LBNR for longitude and Ve Fig.l(d) LBNR for height and Vh

Figure 1: Bias influences on estimated positions and velocities. Five satellites (PRN 9, 11, 12, 13 and 18) are available simultaneously. Satellites PRN 11 and 12 are assumed to have two cycle slips at epoch 3.

278

(a) Cycle slips in carrier phase observations

With double difference (receiver-satellite) carrier phase observations, cycle slips can be modelled as constant biases following the epoch at which they occur. Figure 1 shows one example of the influence of carrier phase cycle slips on the estimated positions and velocities. In this case, the pseudo-range and phase rate observations are assumed to be bias free.

From the above example and other test runs, we note that carrier phase cycle slips mostly affect the estimated positions as opposed to the estimated velocities. Also, the cycle slips have a permanent effect on the estimated positions due to the constant bias nature of the cycle slips. The magnitude of the effect on the estimated positions is a function, among others, of the number and the geometty of satellites affected by the cycle slips.

0.008 0.03 0 Veil

0.006 - I· 0.02 - L • Vv ~ 0::: n

0.004 Z ~

~ 0.01 -t.:)

0.002 -0.00

0.000 ;.

0 10 20 30 -0.01 I

Number of epochs 0 10 20 30 Number of epochs

Fig.2(a) GBNR Fig.2(b) LBNR for latitude and Vn

0.06 0.01 ~

0.05 ~ VA. t 0 Vh 0.00 -• We 4;7"

~ 0.04 • VV h ~ -0.01 -0.03 -0.02 - 5 -0.02 -

0.01 - -0.03 -

0.00 " -0.04 T I

0 10 20 30 0 10 20 30

Number of epochs Number of epochs

Fig.2(c) LBNR for longitude and Ve Fig.2(d) LBNR for height and Vh

Figure 2: Bias influences on estimated positions and velocities. Four satellites (PRN 9,12,13 and 18) are available simultaneously. A single outlier of 20 m is assumed on the pseudo-range observation of PRN 18 at epoch 3.

(b) Outliers in pseudo-range observations

We define an outlier herein as an instantaneous bias which only affects the epoch at which it occurs. Single and multiple pseudo-range outliers are assumed and analysed with respect to different satellite coverages. Figure 2 shows the influence of a pseudo-range outlier of 20 m on PRN 18 during a four-satellite coverage period.

From the single outlier case shown in Figure 2 and other test runs for multiple outliers, it can be seen that the influence of pseudo-range outliers on position and velocity estimation is limited to the current and subsequent few epochs when no bias affects the corresponding phase

279

measurements. That is, the Kalman filter can reduce the outlier influence on the states after a few epochs. The pseudo-range outliers mainly affect the positions; also, the magnitude of the influences (LBNRs) is much smaller than that due to cycle slips. This is caused by the lower precision of the pseudo-range observations.

( c) Outliers in the phase rate observations

Figure 3 shows the outlier influences on the estimated positions and velocities for an outlier of 0.25 cm s-1 in the phase rate observation of PRN 13 at epoch 3. A five-satellite coverage is available. We note two points from Figure 3. Firstly, the phase rate outlier mainly affects the estimated velocities. This is different from cycle slips influences which mainly affect the estimated positions. Secondly, the outlier influences on velocities are limited to the epoch at which the outlier occurs and to the following few epochs.The Kalman filter can effectively reduce the influences of the phase rate outlier after a few observation updates. The same conclusion also holds for multiple phase rate outliers which occur on different satellites simultaneously.

10 300 iii V,

~. 0- • VVn ; 200- ~ r 100 - ~ -10 -

0 + + -20 0 10 20 30 0 10 20 30

Number of epochs Number of epochs

Fig.3(a) GBNR Fig.3{b) LBNR for latitude and Vn

6 iii VA. 5

L 1 iii Vb

4 • Vv 0 • VVh ~

e

~ -1 3

~ 2 -2 1- -3 -0- -4-

-1 • -5 • • 0 10 20 30 0 10 20 30 Number of epochs Number of epochs

Fig.3(c) LBNR for longitude and Ve Fig.3(d) LBNR for height and Vh

Figure 3: Bias influences on estimated positions and velocities. A single outlier of 0.25 cm s-1 in the phase rate observation of PRN 13 has been introduced at epoch 3. A five-satellite (PRN 9,11,12,13 and 18) coverage is available.

280

(d) Bias in system model

Besides the biases or outliers in the observation model, the assumed system model may also deviate from the actual vehicle motion. This causes system model biases. Figure 4 shows one example of the system bias influences on position and velocity states. Here we assume that a single acceleration bias of 5 m s-2 occurs on the system state Ve at epoch 3 assuming the constant velocity model.

1.0e-2 0 V4I 0.03

~

~ O.Oe-tO r • VVn

~ 0.02 -

-1.0e-2 -0.01 •

110:--2.0e-2 • • 0.00 i- 0 10 20 30

0 10 20 30 Number of epochs

Number of epochs

Fig.4(a) GBNR Fig.4(b) LBNR for latitude and Vn

0.1 0.02 0 Vh 0 VA. 0.00

r -0.02 • VVh ~ 0.0 r • VVe ~ -0.04

~ -0.1 ~ -0.06 -0.08 -0.10 -

-0.2 • • -0.12 • • 0 10 20 30 0 10 20 30 Number of epochs Number of epochs

Fig.4(c) LBNR for longitude and Ve Fig.4(d) LBNR for height and Vh

Figure 4. Bias influences on estimated positions and velocities. An instantaneous acceleration bias of 5 m s-2 occurs on the system state Ve at epoch 3. A 7-satellite (PRN 02, 06,09, 11, 12, 13 and 18) coverage is available.

From Figure 4, one sees that the acceleration bias in a constant velocity model mainly affects the estimated velocities. This result is obvious because, over a short time interval (1 second data rate in this case), the acceleration causes very little position changes. In addition, we also notice that the influences of the instantaneous bias in system model are limited to the current and following few (2-3) epochs, which is similar to the influences of outliers in the observation model.

6. MDB FOR KINEMATIC GPS POSITION AND VELOCITY ESTIMATION

Reliability or Minimum Detectable Bias (MDB) analysis is an important tool for Kalman fllter design. It tells us the fllter's ability to detect some potential biases. In this section, we investigate the MDBs of some typical biases or outliers, e.g., carrier phase cycle slips and outliers in pseudo-range or in phase rate measurements, for the kinematic GPS position and velocity estimation model.

281

MDB for carrier phase cycle slips: Cycle slips are common biases occurring during carrier phase measurements. They are treated as constant biases in the receiver-satellite double difference obselVables. Figure 5(a) shows the local MDB improvements for different satellite coverages. Here we assume a single bias case, that is, only one satellite, PRN 18, has a cycle slip at epoch 3. Figure 5(b) gives the global MDB improvement with the widening of the testing window for the above single bias case in the 7 -satellite coverage situation.

60 50

e 40 ~

; 30 20 10

0 3 4 5 6 7 8

Number of satellites

Figure 5(a): Local MDB improvements with different satellite coverages for cycle slips in double difference obselVables on PRN 18 starting at epoch 3.

12 10

..-.. 8 e ~ 6

; 4 2 0

0 10 20 30 40 50

Number of epochs

Figure 5(b): Global MDB improvements with the widening of the testing window for cycle slips on PRN 18 starting at epoch 3( 7 sat. coverage).

Figure 5(a) shows that the MDB is much smaller when 7 satellites are tracked as opposed to the case when 4 satellites are being tracked. In the former case,the MDB is about 0.5 cycle (10 cm), while in the latter case, it goes up to 50 cm. For test runs with different 4-satellite combinations, the MDBs went up to 15 cycles (3 m), which is approximately the precision level assumed for the pseudo-ranges in our analyses. Therefore, five or more satellites are recommended for reliable cycle slip detection on any individual satellite. This assumes that the geometry is satisfactory.

Figure 5(b) shows that the global MDB for cycle slips improves rapidly during the first 10 to 20 epochs. Therefore, as far as the test power-to-computation burden is concerned, a window width of some 10 to 20 epochs would be appropriate for global test for kinematic GPS sUlVeys.

MDB for outliers in phase rate observations

Figure 6 shows the local and global MDBs for a single phase rate outlier occurring on one satellite. An outlier can be modelled as an instantaneous bias appearing at a specific epoch. From Figure 6(a), we also see that the obselVation of five or more satellites simultaneously leads to a significant improvement in the MDBs for the phase rate outlier. Another interesting result is shown in Figure 6(b); it appears that two epochs are sufficient for the global test. Including more epochs (Le. wider windows) does not improve the power of the global test for outlier detection.

282

20 18

i 16

.e 14

~ 12

~ 10 8 6

3 4 5 6 7 8

Number of satellites

Figure 6(a): Local MOB change with different satellite coverages for single outlier in phase rate on PRN 13 at epoch 3.

MDB for pseudo-range outliers

19 ~.!

18 -

i .e 17 -

~ 16

~ 15 ............... 14 I

0 10 20 30

Number of epochs

Figure 6(b): Global MOB improvements with the increase of epochs for single outlier in phase rate on PRN 13 at epoch 3 ( 4 sat­ellite coverage: 9,12,13, 18).

Pseudo-range outlier detection is one of the main concerns of GPS quality assurance. Using the reliability analysis method given in this paper, the minimum detectable pseudo-range outlier can easily be obtained. The following examples show the MOBs for a single outlier in the

double difference(V ~) pseudo-range observables.

9.0

8.8 .-.. g 8.6 ~

~ 8.4 m m m m

8.2

8.0 3 4 5 6 7 8

Number of satellites

Fig.7(a): Local MOB changes with different satellite coverages for single outlier in pseudo-range on PRN 18 occurring at epoch 3

9.0

8.8 g 8.6

; 8.4 ,,'ii-,,'Ii-fi'ii""

8.2

8.0 0 10 20 30

Number of epochs

Fig.7(b) Global MOB improvements with the increase of epochs for single outlier in pseudo-ranges occurring at epoch 3 on PRN 18 (5 satellite coverage: 9,11,12,13,18)

Figure 7 indicates that both the local and the global MOBs are all about 8.3 m, and that an increase in the number of satellites observed and/or window width does not improve the MDBs. This is due to the much lower precision of pseudo-ranges as compared to those of carrier phase and phase rate observations in the filter.

The MDB of the system model can also be investigated as above. The numerical results are not given here. Generally, the minimum detectable acceleration bias in the constant velocity

model, based on the parameters given in Section 5, is about 2 m s-2.

283

7 • CONCLUSIONS AND SUGGESTIONS

In this paper, the general recursive bias influence and MDB fonnulae are derived for Kalman fllter design and error analyses. The investigations show that carrier phase cycle slips and pseudo-range outliers mainly affect the estimated positions, while the phase rate outliers mainly affect the estimated velocities. Cycle slips or systematic biases have a severe long-tenn influences on position and/or velocity estimation. On the other hand, single outliers in the observation and system model only affect the current and following few epochs. This means that Kalman filter can reduce automatically the instantaneous outlier influences after a few epoch observation updates.

The MDB analysis indicates that, for kinematic position and velocity filtering, the Local (current epoch) test statistic seems powerful enough for single outlier detection and adaptation. For systematic biases, e.g.,cycle slips in double difference (V L\) observables, the Global test with a window length of some 10 to 20 epochs will have a better power-to-computation ratio.

The next steps will be to correlate the above findings with the geometry of the satellites being tracked and estimate critical thresholds for the number and configuration of satellites required. The above method will also be tested for code multipath detection, a major problem in GPS kinematic positioning [e.g., Lachapelle 1990].

Acknowledgment. The authors would like to thank Dr. M. Wei for a number of valuable discussions on this topic.

REFERENCES

Friedland,B. (1969). Treatment 0/ bias in recursive filtering, IEEE Trans. Automat. Contr., Vol. AC-14, Aug. 1969.

Hwang,P.Y.C., Brown,R.G. (1989) GPS Navigation: Combining Pseudo-range with Continuous Carrier Phase Using a Kalman Filter, Proceedings of the satellite division of ION GPS'89, Colorado Springs, Colorado.

Ignagni,M.B. (1981) An Alternate Derivation and Extension 0/ Friedland's Two-stage Kalman Estimator, IEEE Trans. Automat. Contr. Vol. AC-26, No.3.

Lachapelle, G. (1990) GPS Observables and Error Sources/or Kinematic Positioning. Proc. Intern. Symp. on Kinematic Systems for Geodesy, Surveying and Remote Sensing, Springer Verlag New York (in press).

Salzmann, M.A., Teunissen, P.I.G. (1989) Quality Control in Kinematic Data Processing, Paper presented at the second international symposium on land vehicle navigation, I uly 4-7, Munster, FRG.

Schwarz, K.P., Cannon,M.E., Wong, R.V.C. (1989) A comparison 0/ GPS kinematic models/or the determination o/position and velocity along a trajectory, Manuscripta Geodaetica, Vol. 14, 1989.

Teunissen, P.I.G.,Salzmann, M.A. (1989) A recursive slippage test/or use in state-space filtering, manuscripta geodaetica, Vol. 14, pp. 383-390, 1989.

Teunissen, P.I.G. (1990) Quality control in integrated navigation systems, PLANS '90, Las Vegas, U.S.A.

284

ON A MEASURE FOR THE DISCERNIBILITY BETWEEN DIFFERENT AMBIGUITY SOLUTIONS

IN THE STATIC-KINEMATIC GPS-MODE

H.-J. Euler and B. Schaffrin Department of Geodetic Science and Surveying

The Ohio State University, Columbus, OH 43210-1247 USA

ABSTRACT

The modeling of double-difference GPS phase observations for short baselines leads to the condition that the ambiguity parameters must be integers as long as residual biases can be neglected However, not always is the combination of those integers the best choice which lie closest to the unconstrained least-squares solution. Thus we may investigate quite a number of different ambiguity solutions and try to discern between them by means of statistical testing.

Since the discemibility between different hypotheses turns out to be basically a function of the respective non-centrality parameters we are able to define a corresponding measure, called AMBIGLOP, which reflects the geometric behavior of the satellites over time with respect to the instantaneous baseline under investigation, and which can somehow be related to the more popular measures like POOP or RDOP, for example.

INTRODUCTION

A crucial part, besides cycle slip fixing, in baseline determination using carrier beat phases of the Global Positioning System (GPS) is the recovery of the values of the phase ambiguities. After fixing the phase ambiguities, the solution of the baseline components gains, in general, a boost in accuracy.

In the literature one finds different strategies reaching from simple rounding of the estimated real value ambiguities to the nearest integer up to rounding only if the real values are within certain confidence limits [Dong and Bock, 1989] or looking for the combination with the smallest impact on the estimated variance. A description of the latter strategy can be found in Langley et al., (1984), Euler (1990), or Euler et al., (1990). It has the advantage that all real estimates are allowed as candidates. This might be important especially in small-scale networks with short observation times. However, the approach described in this paper is not restricted for such applications and might be used for large­scales, as well as kinematic measurements.

285

THE MODEL

The model used for adjustment of GPS observations might be expressed in a Gauss­Markoff model. The consistent equation system is as follows:

y + e = A~ with rk A = u e - N (0, cra p-1) (1)

where

y observation vector holding double difference phase observations e error vector of observations (n x 1) A coefficient matrix (n x u) ~ vector with unknown parameters (u xl)

The parameter vector can be estimated using:

~ = N-1 C = (ATpA)-l ATPy (2)

The sum of squares of residuals 0 will be determined and fulfills the following relationships:

2 ...... T ...... 2 ...... 2 o cro = (y - A~) P(y - A~) cro = (yTpy - C T~) cro -X2 (n-u) (3)

The values in the unknown parameter vector are sorted according to their meaning. The first part consists of all non-ambiguity parameters (e.g., coordinate unknowns, tropospheric scaling, etc.). The second part holds the phase ambiguity parameters.

In case the proper integer ambiguities are known, one can introduce them in the model and perform an adjustment using a Gauss-Markoff model with constraints. The influence of the constraints can be computed after an adjustment without constraints. For the given model, the constraints are formed by:

K ~ = leI with K = [0, IrJ , rk K = r where

o zero matrix (r x (u-r» Ir identity matrix (r x r) leI vector holding integer values (r x 1) r number of integers to be fixed

The changes for the parameter vector are given with:

~1 = ~ + N-1 KT (K N-1 KTrl (leI - K~) = ~ + N-1 KT S2 (leI - K~

with S2 = (K N-1 KTrl

The new sum of squares of the residuals can be determined with: ...... T ......

01 = n + R1 = n + (leI - K~) S2 (leI - K~) .

286

(4)

(5)

(6)

THE TESTS

The former section shows the adjustment model and how to fix estimated real value phase ambiguities to their proper integer value. However, one will not know the proper integers but want to know the most probable choice.

Following Koch (1987; pp. 307), a hypothesis test can be formulated.

With two different sets of integer combinations lq and le2 one gets for the fIrst set of integers

(01 - 0) G02 = Rl G02 = (lei - K~ T S2 (lei - K~) G02 .... "1.,2 (r; AI)

with the non-centrality parameter

Al = (lei - K~)T S2 (lei - K~) G02 = 0

under the null hypothesis

Hoi: K~ = lei .

Since 0 and Rl are independent, a test statistic

n-u Rl TI := -r-n"" F (r, n-u)

can be formed where F is the central F-distribution.

(7)

(8)

(9)

(10)

In case of the second set of integers K~ = le2, one gets the estimated unknowns with ..... ..... ..... ~ = ~ + N-l KT S2 (le2 - K~) . (11)

The respective quadratic form is distributed as follows:

2 2 ..... T ..... 2 '2 (02 - 0) Go = R2 Go = (le2 - K~) S2 (lC2 - K~) Go .... X (r; AU (12)

with A2 = (le2 - lel)T S2 (le2 -lei) G02 :I: 0

under the null hypothesis Hoi: K~ = lei .

The test statistic is

rr ._ n-u R2 .... F' ( . " _ \ .l.2 .--r- n r, n-u, /\;L/ ,

with p' as the non-central F-distribution.

Assuming R2> Rl the hypothesis HOI : K~ = lelwill be accepted for Tl ~ Fl-a; r, n- u with an error probability a. Hoi can be separated from the specific alternative hypothesis Ha : K~ = le2 with a power of 1 - ~(a,Au.

The procedure will test for significant changes in the sum of squares due to the introduced constraints. Under most circumstances one will find significant changes for all integer combinations and a decision to choose a particular set is not possible.

287

In the following a test which relies only on the separability of two different sets of integer numbers is described.

Having two sets of possible integer combinations Kl and K2 one can defme two different null hypotheses

HoI : K~ = Kl and Ho2 : K~ = K2. (14)

Both hypotheses might be tested against a third set of constraints building the specific alternative hypothesis. One can choose

"" K~ =Ka:= K~

and will get

~ = ~ + N-1J(fS 2 (Ka - K~) = ~ and

with the non-centrality parameter

A,3:= (Ka - K~)T S2 (Ka - K~) 002

For the hypotheses HoI and Ho2, the non-centrality parameters are:

A,31 = (Ka -K1)T S2 (Ka -K1) 0'02 * 0 under HoI : K~ = Kl

A,32 = (lea -K2)T S2 (Ka -KV Oc? * 0 under Ho2: K~ = K2

(15)

(16)

(17)

(18)

(19a)

(19b)

Without loss of generality, the hypotheses might be set up that R2(K2) > Rl(Kl) which results in A,32 > A,31. The goal is now to make a decision in favor of Kl.

The numbers 1-~1 or 1-~ are called the power of the test statistics Tl or T2, respectively.

J Fl-a; t, n-u

~i:= 0 F(r, n-u; A,3i) dT for i = 1,2. (20)

One can defme a limit It> (a, ~l, ~2) > 1 which has to be surpassed by A,3VA.3l, i.e.

R2" A,32 (R R) -R = ~ ~ 'Yo a, pt. p2 • 1 1\.31 (21)

If this inequality is true, the two different sets of integers are sufficiently discernible.

The inequality can be transformed to a ratio of the sum of squares as used in some investigations (Euler, 1990 and Euler et al., 1990). The ratio is defmed as:

TI2:= n +R2 n +Rl (22)

where

288

Rl minimal impact R2 next larger impact.

Combining equations (21) and (22) leads to:

T12=Q+R2 = 1 +R2- Rl ~ 1 + Rl'YO -Rl Q + Rl Q + Rl Q + Rl (23a)

T12~1+ Rl ('Yo-I) Q+Rl (23b)

In this test one has the target to majorize a certain value 'YO by the ratio of the non-centrality parameters.

Figure 1 shows the discernibility ratio 'YO(a, ~1, fu) for different choices of m and n. Test computations showed that a ratio between 5 and 10 should be chosen which leads to relative large ~1. ~1 describes the probability of making an error of the second kind. In hypothesis testing theory the error of the second kind is defined as the acceptance of a wrong null hypothesis. In case of the test criteria above, 131 is the probability connected to hypothesis HOI which has per definition less impact on the results in comparison to hypothesis Ho2. However, one has to test all combinations of possible integers until the chosen combination with discernible less impact than all the others is found. This will be the combination with the minimal impact (minimal R) of all combinations. Since it is known that one combination of integer exists, one can choose this combination as the one with proper integer value without difficulties. If none of the combinations is discernible with enough strength, the fixing of integers has to be rejected.

THE MEASURE OF GEOMETRIC IMPACT

For a given set of observations and the null hypothesis HoI the weakest separability ratio is obtained for a normalized non-centrality parameter

. TS rom z 2 z = Amm (Sv

z zTz which is represented by the minimum eigenvalue of S2.

A.min (S2) can be interpreted as a measure of the geometrical impact on the proposed discernibility test criteria. With respect to some other well-known measures of geometrical impact, A.min (S2) is called in the following AMBIGLOP. A relation to RDOP (Goad, 1988) can be obtained by partitioning the full normal equation matrix N

N=[ Nl1 N12] NT2 N22 (24)

where

Nll is connected to the non ambiguity parameters (e.g., coordinates) N22 is connected to the ambiguities.

Equation (5b) can be written as

S2 = N22 - N12 Nil N12

289

(25)

The determinant of the nonnal equation matrix is

det N = det N 11 • Al . A2 ... Amin

with det S2 = Al A2 ... Amin .

Solving for AMBIGLOP one gets

where

AMBIGLOP = A.mm (S:z) = det N ( 1 ) Al . A2 ... det Nll

_ det N tr (Adj (Nll» - Al . A2 ... tr <Nil )

det detenninant tr trace Adj adjoint

AMBIGLOP = det N Al . A2 ...

tr (Adj (Nll» ROOp2

RDOP is in this case the RDOP of the fixed solution.

THE RESULTS OF NUMERICAL COMPUTATIONS

(26)

(27)

Figure 2 shows the proposed AMBIGLOP out of a simulated computation for the full 21 +3 satellite constellation. Numerical values for the satellite orbits are taken from Lohmar (1989). An observation time of 18 minutes with an observation interval of 1 minute is assumed. The computed AMBIGLOP values are nonnalized by dividing with the number of epochs in order to compensate a dependency on the observation interval. However, the values are still dependent on the total length of observation time. With a linear change of total observation time, AMBIGLOP will change quadratically. In the simulation the values are reaching from 0.0002 to almost 0.001. One can find periods where the observation time has to be longer in order to reach a better performance in integer fixing. AMBIGLOP describes, similar to PDOP or RDOP, the geometrical impact on the solution. However, the possibility of integer fixing might not be guaranteed since the actual statistics will have an additional impact

Figures 3 and 4 show results of computations using the proposed discernibility test. The solution given in figure 3 is weaker due to fewer satellites during the beginning. At start, four satellites are available for about eight minutes until the number drops to three for about 18 minutes. The critical limit in the discernibility test is surpassed after receiving an additional satellite. However, the test ratio is weakened for a short period of time while two additional satellites are rising. Afterwards the ratio raises again above the limit and stays there until the end of the computation after accumulating two hours of observation time. The second part of the results are clipped in figure 3. This example shows that having too few measurements of one satellite might hurt the perfonnance of the test when their number of observations is considerably less than the number of observations on other satellites.

290

In figure 4, six satellites were always available during the period of the investigation. After using 16 minutes of data, the ratio exceeds the critical limit and keeps increasing until the end of accumulation of data. The progress of the latter example gives a more universal picture of the behaviour of the ratio test

The numbers describing the test criteria of discernibility are in these computations (X = 0.1, ~1 = 0.6 and ~2 = 0.01. The connected discernibility ratios are between 6 and 7. In computations with more than the minimum number of four satellites, the resulting critical limit (right side of Eq. 23b) is strong enough to avoid false judgements. However, when five or more satellites are available continuously the discernibility ratio might be lowered in order to allow an earlier verification of integer combinations without a risk.

After full installation of the satellite constellation, most of the time one will have five or more satellites above 15 degree elevation. In most cases the only concern are obstacles which might shade some satellites and cut the number of received signals to four. With only four satellites, the results rely more on small changes in the phase data which may result in real value ambiguities farther away from the proper integers. This occasion will lead to_a lower test ratio and rejection of integer fixing. However, the test criteria of discernibility, defined with the values given above, seems a little bit too weak to ensure that all false combinations will be rejected while measuring with the minimum number of four satellites. An acceptance of wrong integers occured occasionally when the total observation time was short. In order to avoid such problems, a higher discernibility limit is recommended for periods with only four satellites.

The figures 5 and 6 show AMBIGLOP related to the observation times required to confirm the proper integers for four and six satellites, respectively. Note, there is a major change of the size of AMBIGLOP between both pictures. In order to make a comparison possible, AMBIGLOP was computed for an observation time of 18 minutes for all baseline determinations even when the real observation time was much longer. In both cases a correlation between required observation time and AMBIGLOP can be recognized easily. Computations with a higher AMBIGLOP, reflecting a more favorable geometrical impact on integer recovery, resulted in shorter observation time.

SUMMARY AND CONCLUSIONS

The proper way for a statistical judgement on the separability of different, possible integer combinations is decribed. In an application one has to find the only integer combination which is superior, while discernible, to all other combinations. In contrast to the method proposed by Dong and Bock (1989), the integer candidates will be tested at once, and, in addition, values farther away from the real value ambiguities are allowed which might give some advantages especially in short-time applications. The evidence of applicability is provided in some numerical tests.

The introduced measure of geometrical impact AMBIGLOP is related to the more popular measure of geometrical impact on the coordinate precision RDOP. A strong correlation between size of AMBIGLOP and the required observation time for recovering the proper integer values was found. However, since the statistical behavior has an additional important impact on required observation times, more experiences are needed before giving final recommondations on the minimum value of AMBIGLOP required for a specific observation time.

291

ACKNOWLEDGEMENTS

We want to thank the authors of Hahn et al. (1989) for providing their subroutines for determining the F-distribution and related values. We appreciate the efforts of Steve Hilla and Larry Hothem of National Geodetic Survey for providing FGCC test data. Furthennore, H.-J. Euler acknowledges the funding support for this study from the NASA Commercial Center for Development of Space at The Ohio State University Center for Mapping and the Department of Transportation of 38 states through the Federal Highway Administration.

REFERENCES

Dong, D.-N., Bock, Y. (1989). Global Positioning System Network Analysis with Phase Ambiguity Resolution Applied to Crustal Deformation Studies in California, Journal of Geophysical Research, Vol. 94, No. B4 pp 3949-3966

Euler, H.-J. (1990). Untersuchungen zum rationellen Einsatz des GPS in kleinraeumigen Netzen, Deutsche Geodaetische Kommission, Reihe C (in print)

Euler, H.-J., Becker, M., Sauennann, K. (1990). Rapid Ambiguity Fixing in Small-Scale Networks, GPS'90, September 3-7, 1990, Ottawa, Ontario, Canada

Goad, C.C. (1988). Investigation of an Alternate Method of Processing Global Positioning Survey Data Collected in Kinematic Mode, GPS-Techniques Applied to Geodesy and Surveying, Groten, E. and Strauss, R. (editors), Springer-Verlag

Hahn, M., Heck, B., Jager, R., Scheuring, R. (1989). Ein Verfahren zur Abstimmung der SignitJ1canzniveaus filr allgemeine Fm,n - verteilte Teststatistiken -Teil I: Theorie, ZfV Vol. 114, pp. 234-248.

Koch, K.-R. (1987). Parameter Estimation and Hypothesis Testing in Linear Models, Springer-Verlag

Langley, R.B., Beutler, G., Delikaraglou, D., Nickerson, B., Santerre, R., Vanicek, V., Wells, D.E. (1984). Studies in the Application of the Global Positioning System to Differential Positioning, Technical Report No. 108, University of New Brunswick

Lohmar, F.I. (1989). Aktuelle Infonnationen aus dem GPS- Planungsbuero, 22. DVW­Seminar "Moderne Verfahren der Landesvermessung 12. bis 14. April 1989", Universitaet der Bundeswehr, Muenchen

292

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>-...... :0 c: ~

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V I * * o -L.....J

8.0

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10.0

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10 20

; • ---. 6 6-6-6-6 6 6 .. 6-6-6-6 6 ..

30 40 50 60 70 degree of freedom n

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80

Figure 1 Discernibility ratio 'Yo( a, ~1' ~2)

•

• 6

6

• 6 6

6

• 6 •• 6

6 66

6 6

III

0

o

6

6 • 6 • • 6

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.0 ·0

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6 6

6

6 ~

6 6~ 6

6 8

6 6 6 0

o ·6 ,,6

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0

12 14 16

time of day [hours]

18 20 22 24

Figure 2 AMBIGLOB simulation for future 21 +3 satellite constellation

293

25~----------------------------------~ - actual ratio - critical limit

20

15

o ..... E 10

o :;:;

5

o+-------~~~----------_+~--------~ o 1200 2400 observation time [sec]

Figure 3 Test results of TRIMBLE SST FGCC-test data station NBS3 to ORM!, three to six satellites available

3600

25~-----------------------------------r - actual ratio - critical limit

20

15

E 10

5

0+-----------+-----------4---------~3600 o 1200 2400

observation time [sec]

Figure 4 Test results of TRIMBLE SST FGCC-test data station NBS3 to OMRl, six satellites available

294

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0

0 0

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0 1000 2000 3000 4000

observation time [sec]

Figure 6 AMBIGLOB versus observation time needed for integer ambiguity confirmation. Solutions with six satellites available.

295

SESSION 4

ALGORITHMS AND SOFTWARE TRENDS

CHAIRMAN C.C.OOAD

THE OHIO STATE UNIVERSITY COLUMBUS, OHIO, USA

ABSTRACT

INSTANTANEOUS AMBIGUITY RESOLUTION

Ron Hatch Magnavox Advanced Products and Systems Company

Torrance, CA 90503 USA

With high quality dual-frequency receivers, it is possible to instantaneously resolve the whole-cycle carrier phase ambiguities in a kinematic scenario.

The various proposals for resolving ambiguities in a moving environment are reviewed and compared with the least squares search technique. The least squares technique is derived, and test results are described. As more capable and less expensive receivers become available and as the satellite constellation nears completion, the conditions for instantaneous ambiguity resolution can commonly be met; and the technique should find frequent use in applications requiring high accuracy in a moving environment.

INTRODUCTION

As people become aware of the potential for centimeter navigation, its application to diverse problems will become commonplace. The key to its widespread use, however, rests with the ability to resolve the lane ambiguity in near real time even while moving. The true power of differential kinematic positioning can only be achieved if a very fast and highly reliable method of resolving the carrier phase lane ambiguity can be found which works in a moving environment.

BACKGROUND

Differential GPS involves the use of a control or reference receiver at a known location to measure the systematic GPS errors; and, by taking advantage of the spatial correlation of the errors, the errors can then be removed from the measurements taken by roving or remote receivers located in the same general vicinity. There have been a wide variety of implementations described for effecting such a differential GPS system. It is the intent in this section to characterize various differential GPS systems and compare their strengths and weaknesses. Two general categories of differential GPS systems can be identified: (1) those which rely primarily upon the code measurements; and (2) those which rely primarily upon the carrier phase measurements.

Differential Code GPS

In its simplest form, differential code GPS simply involves the use of a reference receiver to obtain code measurements at a known location. These measurements are then subtracted from the expected value of the code measurements calculated using the known location and the transmitted orbital positions of the satellites. The difference of the expected value and the measured value then provides a correction which

299

can be transmitted in real time to any other receivers operating in the same general area. Alternatively, the correction can be generated and applied in a post-processing procedure, if the remote or roving receiver position is not needed in real time.

Differential GPS in this form can largely remove errors which are correlated in space. Orbital errors in the predicted ephemerides are correlated over very large distances and thus can be largely eliminated in "local" areas by the use of a reference receiver. Ionospheric and tropospheric errors are generally smaller than orbital errors but are not as strongly correlated. Thus, the errors from these sources can contribute to the residual error after even a few tens of kilometers. The ionospheric errors can be further minimized if two-frequency receivers are available.

A very significant improvement in the differential code GPS systems can be obtained at very little processing cost by using the carrier phase measurements in a supporting role. In a technique usually identified as "carrier-smoothed code" measurements (Hatch, 1982), the integrated carrier phase measurements can be used to obtain the system dynamics while the code measurements are used to determine the constant of integration of the carrier phase. This smoothing comes at very little cost and can provide a dramatic reduction in the normal code measurement noise.

The technique works best when measurements are available from both the Ll and the L2 frequencies. When measurements are available from only the Ll frequency, the smoothing of the code with the carrier becomes subject to corruption by systematic ionospheric effects if the time constant of the smoothing is too long. At Magnavox the single-frequency receivers generally use a carrier-smoothed code measurement which is obtained using a five-minute time constant exponential filter on the difference between the code and carrier. This time constant is chosen because it provides: (1) a significant amount of smoothing in the code measurement noise; while, (2) the typical amount of bias resulting from the ionospheric separation of code and carrier is usually substantially less than the residual measurement noise.

Others have described more sophisticated methods for smoothing single­frequency code measurements. Whether they are more effective remains questionable. Loomis et. al. (1989) have added a direct bias state with random walk characteristics to model the ionospheric separation of the code and carrier measurements. The principle effect of such a model is to shorten the effective time constant of the smoothing. Sennot and Spaulding (1990) have in similar fashion added states for multipath error. Goad (1990) has recently proposed a scheme which would adjust the gain (time constant) as a function of the ionospheric divergence actually encountered. This last proposal, by changing the effective gain or time constant as a function of the ionosphere actually encountered, has merit.

However, I believe, the price of GPS sets will eventually fall to a level where dual-frequency sets will become commonplace, and the ionospheric divergence problem with time will become moot. The residual ionospheric error with separation distance from the reference receiver will still be a problem because of the reduction in spatial correlation with separation distance.

Code differential GPS with carrier smoothing has repeatedly demonstrated results of a few meters of accuracy with separation distances as great as 1000 kilometers.

Differential Carrier GPS

Differential carrier GPS in a moving environment is now generally referred to as kinematic GPS. The code measurements are often called

300

upon to play a supporting role by: (1) solving for a more accurate receiver time and adjusting the carrier phase measurements to the same time epoch as the reference receiver; or (2) by obtaining an initial approximate position.

Navigation accuracies in the vicinity of one centimeter have been demonstrated for kinematic GPS. But there is a problem. The principal problem in using the carrier phase measurements in a ranging mode is the whole-cycle ambiguity. While carrier phase can be measured to a few degrees (a little over a millimeter of range), the number of whole cycles (19 centimeter lanes at the Ll frequency) between the satellite and the receiver is unknown. In order to use the carrier phase measurements as range measurements, this unknown number of whole cycles between the satellite and receiver must be determined. (The number of whole cycles in the range difference between the reference receiver and moving receiver is equivalent information.)

Remondi (1985) described the first use of carrier phase measurements in a moving environment. In order to determine the whole- cycle ambiguity, he used an antenna swap maneuver. This procedure is fine for a static environment; but in a moving environment, if the number of satellites tracked drops below four, the procedure must be repeated. This can represent an unacceptable constraint in many practical situations. Other methods, such as starting the moving receiver at a pre surveyed mark or using static survey techniques to determine the initial ambiguity values, are possible but suffer similar disadvantages in the moving environment.

A procedure using the P code measurements on Ll and L2 to determine the whole-cycle ambiguities has been described (Hatch, 1986) which overcomes the above limitations. It works well in the moving environment and only takes a few minutes to resolve the ambiguities. Nevertheless, the technique has not found widespread use because it is defeated by the military imposed selective availability. The denial of accurate P code measurements effectively defeats the technique. This is because the technique makes use of the carrier smoothing of the code to bring the code measurement noise below one half cycle of the difference frequency carrier. But denial of P code measurements means the code measurements are at least a factor of three poorer. In terms of noise averaging, this is a factor of about 10 times longer to achieve the same measurement noise.

Rather than use the P code measurements to resolve ambiguities in a moving environment, another method (with several implementation techniques) can be employed. Specifically, measurements from extra or redundant satellites can be used to determine the whole-cycle ambiguities even while moving. The various techniques for using redundant satellites found in the literature are considered briefly below.

The technique which has been around the longest is the ambiguity function method described by Counselman and Gourevitch (1981). It was not specifically developed for use with redundant satellites or for the moving environment. Although the use of extra or redundant satellites was spelled out, its use in a moving environment was not. However, no generalization or modification is required to apply it to the moving receiver.

The technique works reasonably well. Recent descriptions of results have been given by Mader (1990) in the moving environment and by Remondi (1990) in a pseudo-kinematic (i.e. intermittent static) environment.

The ambiguity function technique has what I consider to be two fundamental disadvantages compared to the least squares search technique to be described later. First, it has significant computational disadvantages. The technique is computationally intensive, and it is this disadvantage which has probably been responsible for its slow acceptance. The second disadvantage is not so easy to describe. The

301

discrimination between the proper solution and the false solutions is not as robust as the least squares search. The ambiguity function is a measure of the power or agreement between the measurements from the several satellites. By contrast, the least squares residuals in the least squares search technique is a measure of the disagreement between the measurements. The residuals are a much stronger and more robust method of discriminating between the solutions.

Apparently, the first paper which specifically called for the use of redundant satellites in a moving environment to resolve the ambiguities was that of Loomis (1989). The Loomis technique simply adds a state into the Kalman filter for each of the unknown ambiguity values. Unfortunately, the technique has one serious problem. It does not work very well. The reason that it does not work well is that there is no constraint imposed between the ambiguity parameters. The ambiguity values are treated as if they are independent, but only three of the (double difference) ambiguity parameters are independent. Once three of the ambiguities are determined, the position is determined; and, therefore, the other ambiguities are determined.

Hwang (1989), in a variation of the Loomis approach, recognized that three ambiguities were sufficient to solve the problem; but he still failed to take advantage of the fact by imposing any constraints. He simply picked the three ambiguity values which independently converged quickest.

The problems with the methods described above can be overcome by the least squares ambiguity search technique first described for a moving environment by the author (Hatch, 1989). It was suggested, for the static environment earlier (Melbourne, 1985; Hatch, 1986). Counselman (1989) in a recent article indicates that Gourevitch in an unpublished account suggested the equivalent in a static environment in 1982. No implementation details are given. Also, Frei and Beutler (1989) suggest, in passing, a bank of Kalman filters which would be equivalent to the least squares approach in a theoretical sense, though probably not nearly as efficient computationally.

It is time to consider the least squares ambiguity search technique in some detail.

THE LEAST SQUARES AMBIGUITY SEARCH TECHNIQUE

The least squares ambiguity search technique has already been described briefly (Hatch, 1989), and a patent application has been submitted. However, an even more efficient algorithm has been developed since that report. It is described below.

The algorithm is used as an off-line computation to establish the proper lane ambiguity values on startup or whenever the number of satellites has temporarily dropped below the minimum threshold of four. As long as there are five or more satellites, the algorithm can be run in the background to ensure that there has been no undetected cycle slip. This off-line process has two advantages: (1) In its normal operation the Kalman navigation filter is not burdened with the overhead associated with ambiguity uncertainty; and (2) cycle-slip monitoring becomes a simple process.

The kinematic GPS problem is often presented in the form of double­difference equations similar to those usually used in the static survey techniques. However, the processing is simpler and the essential features of the solution are more obvious if the control site data is processed separately. The function of the reference site is to remove common errors from the measurement. Thus, it can be processed independently to obtain corrections to be applied to the remote or roving receiver. By generating correction terms at the reference site and applying them to the measurements of the remote site, the necessity

302

of forming differences across sites is removed; and the remote site does not have to process the reference site data. The problem is thus split into two easier tasks. The reference site processes its data; and the remote site, after using the data supplied by the control site to correct its measurements, processes its data independently.

The remote site with its corrected measurements can form a difference across satellites to remove the receiver's clock phase from the solution or can simply solve for the clock as one of the solution parameters. Both of the approaches have their advantages. Some aspects of the solution process are more obvious in one form and some in the other. The specific approach employed has been to use a hybrid form which captures the advantages of the differencing without its normally associated disadvantage.

For the least squares ambiguity search solution, one needs to obtain an approximate solution together with a description of the volume (sphere, ellipsoid) over which the search is to be conducted. These parameters are not very critical, as will be seen below. The start-up procedure will be to use the differential code solution for the initial position, and the search can be done over the associated three-sigma or more uncertainty region surrounding that position.

A fundamental advantage in processing efficiency can also be gained by separating the satellites into two groups. The first or primary group consists of a set of four satellites, which are used to generate a set of potential solutions which lie within the uncertainty region. The remaining redundant satellites form a secondary group, which are then used to eliminate those potential solutions which are in disagreement with this secondary group of measurements.

There are two conflicting requirements for choosing which of the satellites should be included in the primary group of four satellites used to generate potential solutions. One would like to choose satellites with poor GOOP because the search will involve fewer potential solutions. This is because a high GOOP means that the solutions corresponding to specific ambiguity values are separated by greater distances. Thus, there are fewer solutions in any given volume of space. However, the GOOP cannot be so poor that the position uncertainty surrounding anyone specific solution includes more than one ambiguity value for the measurements from the secondary group of redundant satellites. The only safe procedure is to pick satellites in the primary group of four which have a reasonably good GOOP.

Potential Solutions

One might have assumed from the discussion of the Loomis and Hwang technique above that seven satellites provide a unique solution to the problem. Such is not the case. There are four variable parameters which change from epoch to epoch. These are the three components of position and the clock offset of the receiver. And there are three parameters which remain constant from epoch to epoch as long as no satellites lose lock. These are the ambiguity values of three of the first differences across satellites. But these seven parameters are not all independent. In fact, with only four satellites, specifying the three constant ambiguity values is sufficient to uniquely specify the position solution and to specify the clock phase within one whole cycle - - and nobody cares about the whole-cycle ambiguity of the clock solution.

The constraint between the ambiguity values and the position is abstract and cannot be imposed as a constraint equation. However, it is easy enough to specify the constraint explicitly by generating an explicit position solution for every ambiguity combination which leads to a position inside the uncertainty region to be searched. These explicit solutions for the primary group of four satellites are referred

303

to as potential solutions. The specific method for generating the potential solutions can now be

described. The direct undifferenced equations for the four satellite primary group solution can be written as the matrix equation:

~l+Nl C1 i C1

j C1 k

1- x

~2+N2 C2 i C2

j C2 k 1 Y

~3+N3 C3 i C~

J C3 k 1 z

~4+N4 C~ 1. C4

j ~ 1 ~c

where: ~ designates the measured carrier phase N is the error in the estimated number of whole cycles C represents the direction cosines to the satellites x, y, and z equal the correction to the estimated position ~~ equals the correction to the receiver clock tne superscripts designate the satellites

(1)

the subscripts i, j, and k designate the x, y, and z directions

The problem with equatipn (1) as it stands is that the number of whole-cycle ambiguities, N1, for each satellite, i, can be changed by a common amount without changing the position component of the solution. Only the whole-cycle value of the receiver clock would change. But the clock value itself is not of interest. By choosing the first satellite as a reference and subtracting it from the others to form a difference, the receiver clock is removed from the solution; and the extra clock ambiguity is removed.

The problem with the difference equations, however, is that, by forming differences with respect to a common satellite, the measurements acquire a positive correlation which may cause the solution to become biased. But the best of both forms can be obtained by a slight modification. Instead of solving for the receiver clock, we solve for the difference of the receiver clock phase and the measured phase of the reference satellite. The value of Nl is arbitrarily set to zero since it only affects the clock value. This gives:

0 C1 i C1

j C1 k 1 x

(~2_~1 ) + N2 C2 C2 C2 1 y i j k (2)

(~3_~1) + N3 C3 C3 C3 1 z i j k

(~4_~1) + N4 C~ 1.

C~ J

C4 k 1 (~c-~l)

In the form of equation (2), only three ambiguities arise; and the solution corresponding to any specific choice of the three values of the whole-cycle ambiguities results in a unique unbiased solution.

Rewriting equation (2) in terms of vectors and a matrix gives:

where: M is the four element measurement vector B is the matrix of direction cosines X is the four element solution vector the subscript p designates the primary group of satellites

The solution for X can now easily be obtained as:

304

(3)

(4)

where: the superscript -1 indicates the inverse of the matrix

But, for all potential solutions, X, corresponding to all the different choices of M which arise as a result of choosing different combinations of whole-cycle ambiguities, the value of the matrix inverse does not change. This allows the potential solutions to be generated as a simple vector summation of three basis vectors. To show this, choose three measurement vectors, M, as follows:

(0 1

(0 0

(0 0

00)

10)

01)

where: the superscript T stands for the transpose

(5)

(6)

(7)

the subscript designates the corresponding measurement difference

Using these measurement vectors in equation (4) gives the solution vectors:

Xl B-1 P M1 (8)

X2 B-1 P M2 (9)

X3 B-1 P M3 (10)

But the general measurement vector is given by:

(0 a f3 'Y) (11)

However, if equations (8), (9) , and (10) have been solved, the solution for the measurement given by (11) becomes:

(12)

It is easy to see that a nested loop with the value of a varying by inte~er whole cycles in an outer loop, followed by the value of f3 vary1ng in the middle loop, followed by a third loop where the value of 'Y varies across whole cycles, can be used to easily and economically generate an entire set of potential solutions covering an extended volume of space.

Eliminating Incorrect Potential Solutions

In order to eliminate storing unnecessary information, it would be nice if the secondary group of one or more satellites could be used to test the potential solutions as they are formed within the loop. Those which do not agree with the additional secondary measurements could be immediately eliminated. An efficient algorithm is indeed available which can be obtained from the algorithms for sequential least squares. There are many sources with derivations of sequential least squares algorithms. One such source is Melsa and Cohn (1978).

First, the innovations vector for the secondary group of satellites must be computed:

(13)

where: Y is the innovations vector the subscript s is used to designate the secondary group

A few comments are needed to clarify equation (13). If there are six

305

satellites total, the secondary group will include only two satellites. This means that Ys will be a two element vector, and so also the measurement vector Ms. Bs is a two by four matrix of the direction cosines of the two secondary satellites and x,., is the four element solution vector being updated and tested. From the derivation of~, it is clear that the innovations corresponding to the primary group of satellites would be zero.

The measurement vector of the secondary satellites includes an ambiguity value for each measurement. However, it is clear that the least squares residuals, after the solution is updated, will be smaller for smaller innovations values. The ambiguity value can therefore be chosen such that the associated innovations value is smallest. If whole­cycle wavelengths are used as measurement units, this is equivalent to rounding the innovations to the nearest whole value and assigning that value to the associated ambiguity. The residual fraction is then used as the adjusted innovations value.

The innovations vector can now be used to solve for a position update to the position obtained from the primary satellites:

(14)

where: the subscript c is used to indicate the complete set of primary and secondary satellites

The matrix C in equation (14) can be precomputed outside the potential solution loop so that it becomes a very efficient equation within the loop.

Now the residuals are needed in order to quantify the quality of the potential solution. The residual vector R is given by:

R (15)

The only element in this equation not completely defined is the complete vector of innovations values, Yc • This vector is simply the innovations of the secondary group, Ys ' obtained above appended to a set of four zeros which corresponds to the innovations of the primary group.

Finally, the estimated variance is used as a measure of the quality of the potential solution:

q (16)

where: n is the total number of satellites in the complete set

Only those potential solutions with q greater than a selected threshold are retained as potential solutions.

The greater the number of satellites the higher the probability that only the one true solution will remain as a solution which agrees with all of the measurement data. In addition, even when multiple solutions remain as viable solutions, only the true solution will continue to repeat as the satellite geometry changes. This true solution can be identified as the only solution which has the same ambiguity values from one epoch to another.

One of the best features of the algorithm developed above is that those solutions which do not satisfy the measurements are formed and then immediately rejected. Thus, they do not consume computer memory. Only viable solutions are saved, and the more satellites observed the fewer viable solutions. Thus, while many search algorithms have an exponential increase in memory usage with increased numbers of satellites, the above algorithm actually uses less memory with increased numbers of satellites.

306

TEST RESULTS

The above algorithm has been tested with both single-frequency and dua1-frequency static data processed as if it were kinematic. In addition, a number of kinematic tests with simulated data have been run. In all cases the algorithm has performed well, and it is possible to describe some of the general features of the algorithm.

First, the number of acceptable solutions behaves statistically. If the phase measurement noise is substantially less than plus or minus one-fortieth of a cycle (the new Magnavox digital receivers measure phase almost three times better than this), each redundant satellite can be expected to reduce the number of potential solutions included in the search region by one-twentieth. (The noise level of the measurements is very important.) Using a conservative estimate, each satellite reduces the number of valid potential solutions by a factor of ten. Thus, if six or seven satellites are visible, a sizable region of space can be searched with a high probability that only the one true solution will remain as a valid solution.

Second, when a false solution is retained as a potential solution, the amount of time it remains as an apparently valid solution is a linear function of its distance from the true solution. This has two significant implications. First, it means that a very large search region can be used, if one wishes, and then within a few seconds be progressively narrowed to the region immediately surrounding the true solution. Second, it quantifies the gain of using a two-frequency (wide lane) receiver, or, vice versa, the penalty of using a single-frequency (narrow lane) receiver. The single-frequency set will generally take 4.5 times longer to resolve the ambiguities than a dual-frequency receiver would. From the gain of additional satellites described in the preceding paragraph, it is apparent that statistically one additional satellite can make up for the use of a single-frequency set (assuming that ionospheric refraction is negligible).

Tests run to date indicate that a single epoch usually is sufficient to resolve the whole-cycle ambiguities (instantaneous resolution) under the following conditions: (1) dual-frequency receivers are available so that wide 1aning can be used; (2) distances are limited to a few tens of kilometers so that the ionospheric refraction effect is not too severe; and (3) there are seven or more total satellites available for tracking.

Even where these conditions are relaxed, the resolution can be very fast. If limitations (1) and (2) above apply but only six satellites are available, the ambiguities often are not instantaneously resolvable. However, the time required to properly resolve them will rarely exceed two minutes. If only five satellites are available, the time to resolve the ambiguities can occasionally reach five minutes or more, especially if one of the four satellite sub-groups has a very poor GDOP.

CONCLUSIONS

The test data indicate that the new least squares search ambiguity resolution technique can, under conditions which often apply, resolve the ambiguities instantaneously. Even when the ambiguities cannot be resolved at a single epoch, the time required is substantially shorter than that of rival techniques. Furthermore, the computational efficiency of the new algorithm is substantially better than that of many of the techniques employed by others. The new technique promises to provide the benefits of centimeter positioning accuracy in a moving environment.

307

REFERENCES

Counselman, C.C., III, and S.A. Gourevitch (1981), "Miniature Interferometer Terminals for Earth Surveying: Ambiguity and Mu1tipath With Global Positioning System," IEEE Transactions !ill Geoscience and Remote Sensing, Vol. GE-19, No.4, October 1981, pp. 244-252.

Counselman, C.C., III (1989), "Method of Resolving Radio Phase Ambiguity in Satellite Orbit Determination," Journal of Geophysical Research, Vol. 94, No. B6, June 10, pp. 7058-7064.

Frei, E. and G. Beutler (1989), "Some Considerations Concerning an Adaptive, Optimized Technique to Resolve the Initial Phase Ambiguities for Static and Kinematic GPS Surveying - Techniques," Proceedings of the Fifth International Geodetic Symposium !ill Satellite Positioning, DMA, DoD, NGS, NOAA, Las Cruces, N.M., March 13-17, New Mexico State University, Vol. 2, pp. 671-686.

Goad, C. (1990) , "Optimal Filtering of GPS Phase and Pseudo-range Measurements," Paper presented at Spring AGU meeting in Baltimore Md., May 29-June 1.

Hatch, R.R. (1982), "The Synergism of GPS Code and Carrier Measurements," Proceedings of the Third International Geodetic Symposium on Satellite Doppler Positioning, DMA, NOS, Las Cruces, N.M., Feb. 8-12, New Mexico State University, Vol. II, pp. 1213-1232.

Hatch, R.R. (1986), "Dynamic Differential GPS at the Centimeter Level," Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, DMA, NGS, Las Cruces, N.M., April 28-May 2, New Mexico State University, Vol. II, pp. 1287-1298.

Hatch, R.R. (1989), "Ambiguity Resolution in .the Fast Lane," Proceedings of ION GPS-89, Institute of Navigation, Colorado Springs, September 27-29, pp. 45-50.

Hwang, P.Y.C., (1990), "Kinematic GPS: Resolving Integer Ambiguities on the Fly," IEEE Position Location and Navigation Symposium, No. 90CH2811-8, Las Vegas, March 20-23, pp 579-586.

Loomis, P., G. Kremer, and J. Reynolds (1989), "Correction Algorithms for Differential GPS Reference Stations," Navigation, Vol. 36, No.2, Summer 1989, pp. 179-193.

Loomis, P. (1989) , "A Kinematic GPS Double-Differencing Algorithm," Proceedings of the Fifth International Geodetic Symposium on Satellite Positioning, DMA, DoD, NGS, NOAA, Las Cruces, N.M., March 13-17, New Mexico State University, Vol. 2, pp. 611-620.

Mader, G. L. (1990), "Ambiguity Function Techniques for GPS Phase Initialization and Kinematic Solutions," Paper presented at Spring AGU meeting in Baltimore, Md., May 29-June 1.

Melbourne, W.G. (1985), "The Case for Ranging in GPS-Based Geodetic Systems," Proceedings of the First International Symposium on Precise Positioning with the Global Positioning System, Rockville, Md., April 15-19, National Geodetic Information Center, NOAA, Vol. 1, pp. 373-386.

Me1sa J .L., and D.L. Cohn (1978), MCGraw-Hill, New York, pp. 216-219.

Decision and Estimation Theory,

Remondi, B.W. (1985), "Performing Centimeter-level Surveys In Seconds Wi th GPS Carrier Phase: Initial Results," NOAA Technical Memorandum NOS NGS-43, Rockville, Md., October 1985.

Remondi B. W. (1990), "Pseudo-Kinematic GPS Results using the Ambiguity Function Method," NOAA Technical Memorandum NOS NGS-52, Rockville, Md., May 1990.

Sennot, J., and J. Spalding (1990), "Mu1tipath Sensitivity Slip Tolerance of an Integrated Doppler DGPS Navigation IEEE Position Location and Navigation Symposium, No. Las Vegas, March 20-23, pp 638-644.

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and Carrier A1gori thm, " 90CH28ll-8,

INERTIALLY AIDED LANE RECAPTURE AFTER GPS CARRIER LOCK LOSS

INTRODUCflON

P.V.W. Loomis and GJ. Geier Trimble Navigation Limited

Kinematic surveys and positioning applications based on the Global Positioning System generally require maintenance of carrier lock on four or more GPS satellites for realization of centimeter level accuracies. Frequently, short periods of carrier lock loss for one or more GPS satellites is caused by masking from trees, buildings, and bridges, and radio frequency (rf) interference. If continuous track is not maintained on at least four satellites during the outage, the lane, or integer ambiguity, must be re-established after carrier lock is regained. The use of external sensors, most notably an Inertial Navigation System (INS), can aid in recapture of the integer ambiguity after relatively short periods of loss of lock. Research by Hein et. al. [1] indicates these periods may be on the order of twenty to thirty seconds for high quality inertial systems. This study extends this previous work somewhat to larger lanes and lower quality inertial systems.

The first section of this paper describes the recapture algorithm, which is an extension of the zero velocity update technique of terrestrial inertial surveying. In the next section of the paper, a covariance simulation is described which is used to determine the ability of different quality INSs to maintain carrier lane during simulated outages in GPS coverage. The error models utilized for both the INS and GPS are discussed in detail. In the third section, simulation results are presented based upon the covariance analysis predictions; measurement gaps of up to several minutes in length are considered, with no or three GPS satellites assumed available during the gap. The major results of this paper are summarized in the final section.

LANE RECAPTURE ALGORI1HM

Resolving the carrier lane is a major problem in performing accurate GPS surveys. Once it is resolved, the differential pseudorange between the base station (reference) receiver and the mobile survey (rover) receiver is known to within phase accuracy, better than a centimeter. Initializing the integer can be a time-consuming procedure [2], and re­initialization each time carrier lock is lost may be impractical. Consequently, alternative methods are sought to "recapture" the lane when lock is lost momentarily.

The carrier phase measurement is expressible as the sum of the lane integer N, the propagation distance (comprising range and atmospheric delays), receiver and transmitter clock offsets, and other errors as follows (units are wavelengths or cycles):

N + phase = range + tropo + iono + multipath + Bclockrx - Bclocktx:

Differencing the phase measurements from reference and rover reduces considerably the effect of atmospheric errors and the computed range errors due to incorrect orbit information. Although the absolute lane integer N is not observable in this difference, the differential lane integer AN is; fortunately, the differential integer contains all necessary range information and is therefore sufficient for positioning purposes. In the following equation, Il denotes differential (rover minus reference):

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AN + Aphase = Arange + Atropo + Aiono + Amultipath + Aclockrx

In survey processing, the phase is commonly "double-differenced" to further eliminate the effect of differential receiver clock error, &lockJX' At the beginning of this study, a comparison was made between the double-differencmg method, which is independent of clock error, and a differential or single-differencing method using receivers with an atomic frequency standard. Over the gap lengths being considered the atomic frequency is quite stable, much more so than an inertial system, and acts very much like an extra satellite. The two methods prove to be roughly comparable, except in the one common case. When the number of satellites tracked drops from four to three, the frequency standard acts as a "fourth satellite" and provides nearly complete positioning. The advantage is so substantial that the cost of the rubidium, some ten times less than the candidate high quality inertial system, is considered to be negligible relative to the benefit. As a consequence, this study concentrated on a differential system with atomic frequency standards at both reference and rover.

Once the integer AN is resolved for a satellite signal, it remains known until carrier track is lost. When the carrier is regained after a tracking gap, the integer must be resolved anew. By using an estimator that includes GPS bias states (e.g., AN, Atropo, Aiono) and inertial calibration states (gyro and accelerometer biases) as described in the next section, the lane integer is re-estimated after track is re-established. Recapture (estimation to the exact integer) is practical if the errors have not built up excessively during the carrier tracking gap. For short gaps, the quantities Atropo and Aiono, which are estimated prior to the gap, will not have changed substantially from their estimates at the start of the gap. Provided that the multipath effect is not large, and that position change has been tracked accurately during the gap, .the lane integer can be recaptured immediately. Of course, if at least four other satellites have been tracked during the gap, this can be done purely by GPS with the four carriers to establish the change in position.

This study deals with the case when fewer than four satellites were visible during the gap and an inertial system tracks the position change across the gap. Even though the inertial instruments are assumed to be calibrated by GPS prior to the gap, bias instability and noise in the inertial system degrade the position tracking. Eventually position accuracy degrades to worse than a lane width, at which point recapture becomes impossible. A properly designed estimator will extend the allowable gap duration by performing a velocity update of the inertial system after the gap using backward­differenced GPS carrier. This is homologous to the zero velocity update (ZUPT) of terrestrial inertial surveying, and decreases the rate of inertial position error growth considerably. (In contrast to [1], this study does not use GPS code information during recapture because of the possible contamination by code multipath.) The inertial position accuracy eventually degrades to the point that the integer must be resolved using the lane initialization techniques studied in [2].

ANALYSIS ME11IODOLOGY

Simulation Requirements

The basic concept in using INS aiding is that, once the INS has been calibrated during full GPS coverage, it should enable "coasting" during outages, i.e., maintenance of the correct integer counts for a period of time so that an integer re-initialization may not be necessary when the signals return. Use of either the barometric altimeter or the rubidium frequency standard bounds the error growth during GPS outages in a single direction. For the baro-altimeter, INS altitude error is limited to the residual bias error in the instrument (which has been calibrated by GPS prior to the outage), while the rubidium

310

clock limits the clock error growth. The rubidium frequency is inherently very stable compared to the inertial outputs, and its frequency offset will have been estimated by the estimator prior to the GPS outage.

The measurement gap filling simulation has been developed to determine the maximum gap lengths before re-initialization of the lane integer is required. High, medium, and low quality INSs are integrated with receivers capable of wide laning, half-wide laning, and normallaning. Ambiguity search methods [3] were not examined explicitly, but in general they should provide even better performance than wide-lane.

Simulation Description

Overview. An optimal covariance simulation was used to first determine the extent to which the raw inertial drift could be calibrated using GPS, and then to determine how the calibrated INS would perform with partial or complete loss of GPS. Figure 1 illustrates the major components of the simulation and their interaction. The vehicle motion model and the INS, GPS, and baro-altimeter error models provide the necessary parameters for running the optimal covariance equations, which generate the predicted 10' error values for each of the modelled states as a function of time. The estimator is first run for a 60 second period with complete GPS coverage, during which time the calibration of INS position and velocity errors reach a near steady-state condition. Following this, a measurement gap of specified length is simulated, during which time a specified number of GPS satellites are assumed obscured. At the completion of the simulated obscuration period, a 60 second period with full coverage is simulated, to enable the filter to re­establish the correct integer count.

Inlt. Pos., Vel., Hdg.

Maneuver Spec.

Inll. One Sigma Errors.

Error MOdel Pars.

Gap Durallon I

BARO ALTIMETER f--+ ERROR MODEL 1 Siale

41nll. One Sigma Errors

VEHICLE MOTION MODEL

Heave Mollon Model Pars.

•

Veh. Pos., Vel., Allllude

INS ERROR MODEL 33 Siaies

OPTIMAL COVARIANCE EQUATIONS

Error Propagallon Measuremenl Updale

One Sigma Error Hlslorles

GPS ERROR MODEL

5 SlaleslSalelllle 2 SlaleslClock

Receiver

Inll. One

Type Sigma Errors

'-- Error M Odel Pars.

Salelllie Geomelry

Fig. 1. Measurement Gap Filling Simulation Overview

The predicted 10' values output from the simulation represent the performance of a fully optimal Kalman filter, with complete statistical and dynamic knowledge of all sources of error. These predictions should be only slightly optimistic, since reduced state

311

filters can in general be designed to perfonn nearly as well as optimal filters through appropriate choice of process and measurement noise statistics models [4].

Each of the models used in the covariance simulation is described below: inertial, baro­altimetry, and GPS respectively. References for the error models and their parameters are provided wherever possible. The ship motion model is not described in detail; it included a sweeping tum during the signal outage and a heave motion of 1 meter amplitude and 20 second period.

INS Errors. This model is abstracted from [5]. It represents the nine position, velocity, and attitude errors in an east-north-up frame. A minor change in the model was implemented to reflect the fact that variation in altitude is severely restricted in the marine environment; feedback of position error into acceleration error was removed by modeling gravity magnitude as constant.

In addition to these basic nine error states, a total of 15 gyro error sources, six accelerometer error sources, and three gravity computation errors are modelled, resulting in a total of 33 INS error states.

The gyro errors include three g-insensitive drifts, three g-sensitive drifts, three gyro scale factor errors, and six input axis misalignments. Each g-insensitive drift is modelled as a random walk, with a power spectral density selected to produce an error growth which is 10% of the initial value over a 100 second period. The initial 1 (J value is a function of the INS quality considered (see Table 1, which summarizes the gyro error model parameters). Note that, in Table 1 (as well as in Table 2), the high quality INS numbers are derived from an error budget for the MAPS DRU RLG-based INS, while the medium and low quality numbers are representative of generic INSs of lower quality. Scale factor errors are modelled as Markov processes with 1800 second correlation times. The relatively long time constant reflects the near bias-like nature of the scale factor error, and is consistent with models presented elsewhere [6]. The process noise variance associated with each Markov process is computed from the initial state error variance and the correlation time based upon steady state conditions, as described in [4]. The g­sensitive drifts are expected to be significant only for mechanical gyros; i.e., RLGs are modelled with zero g-sensitive drifts. Thus, the three qualities of INSs represented in Table 1 are assumed to utilize RLGs. Gyro input axis misalignments are represented as random biases.

Table 1. Gyro error model parameters.

INS Drift SF G-SenslUvlty Axts Misalign Quality (deg/hr) (ppm) (deg/hrl/(m/sec2) (arc-min) Angle Walk

High 0.004 10 0.00 0.067 0.002 deg/hrl/2

Medium 0.100 250 0.00 l.000 0.02 deg/hr1/ 2

Low l.ooo 1000 0.00 l.ooo 0.20 deg/hr1/ 2

Modelled accelerometer error sources include accelerometer biases and scale factor errors (modelled as slowly varying Markov processes), and input axis misalignments, represented as random biases. Table 2 summarizes the accelerometer error source

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magnitudes as a function of INS quality. The horizontal components of gravity computation error (often referred to as deflections of the vertical) are modelled as Markov processes with 10 values of 10 arc-seconds, and correlation distances of 25 nm, while the vertical component (gravity anomaly) has a 10 value of 50 Jlg's and a correlation distance of 60 nm. [5, 7,8].

Table 2. Accelerometer error model parameters

INS Bias SF Axis Misallgn Quality (ug) (ppm) (arc-min)

High 40 120 0.167

Medium 200 250 1.000

Low 500 1000 1.000

Baro-altimetry Error. Barometric altimeter updates are processed each second by the filter to assist in stabilizing the INS vertical channel during GPS outages. When full GPS coverage exists, the bias error of the barometric altimeter can be calibrated. The differential baro-altimetry bias error model consists of a single Markov process, with a 10 value of 1 m, and a correlation time of 100 seconds, with a measurement noise standard deviation of 0.09 meters. GPS Error Model. Due to the relatively short time periods of interest, the GPS satellite geometry is assumed fixed. A single sample constellation was used in all the runs. The lines of sight were derived from an actual constellation in April 1989 with PDOP of 2.5, HDOP of 1.5, and VDOP of 2.0. Thus, a good nominal GPS geometry is assumed.

The GPS error model includes two states representing receiver clock phase and frequency error, and a maximum of five error states for each satellite; these errors include carrier multipath error, residual (i.e., post-differentially corrected) tropospheric, ionospheric and selective availability errors, and integer ambiguity error. Thus, a maximum number of 22 states can be modelled for four satellites. GPS carrier pseudo range measurements are processed each second with a 10 noise value of 1 cm.

The receiver clocks are rubidium frequency standards. The error model is derived fr~T [9], and consists of a random walk frequency error with an initial 10 value of lx1O- , drive~gy white noise with power spectral density computed from an Allan variance of 5x 10- over 2000 seconds. In addition, a phase noise with a corresponding power spectral density is included. This error model is considered adequate over time spans of ten to 200 seconds, roughly the length of signal loss being considered.

Carrier multipath error is modeled by a Markov process with a 5 minute correlation time and a 10 value of 1 cm. These values are consistent with survey data using Trimble 4000S receivers. Residual tropospheric, ionospheric, and Selective Availability (SA) errors are computed as a function of the separation distance between the ship and the reference station, increasing in size with distance. The tropospheric model is Markov over both distance and time; the ionospheric and orbit error models linear over distance and Markov over time. Table 3 summarizes the parameters of these error models. The values utilized for tropospheric and ionospheric error are consistent with parameters in [10, 11, 12, 13]. The parameters used for the orbit error model are representative of expected levels of Selective A vailabili ty.

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Table 3. GPS error model parameters Nominal Induced Corr. Dlst. COlT. TIme

Error (meters) (Ian) (sees)

Tropo 0.02 I sin (el) 50.0 600

Iono 3.0 I sin (el) 3000.0 1000

SA 20.0 20000.0 10000

Lane Recapture Test. Integer ambiguity error is modelled as a random bias state for each satellite, with a 10" value dependent upon the simulation conditions. As the simulation begins, these errors are zero, under the assumption that the integers are identified. During a period of continuous satellite coverage, they remain zero, assuming no cycle slips. Following a loss of carrier, all knowledge of the integers is effectively lost.

After the track is reinstated, carrier measurements resume. At this point, the estimator has only the inertially propagated estimates of the states. The lane integers, now unknown, are nominally equal to the phase corrected for differential propagation distance. The error in reconstructing the integers, then, involves the measurement errors in the phases and the state estimate errors. If the estimates of the error states (differential ionospheric and tropospheric model

error, orbit error) are accurate enough, the integer can be estimated precisely and declared known. The recapture test is to examine the estimate of the lane integer ~N for each pair of satellites, using the error variance of the integer estimate to determine whether it has converged on an integer. The test uses a confidence level of 99%, corresponding to an error standard deviation of roughly 0.2, requiring convergence to within a half-wavelength at the 2.56 0" level.

SIMULATION RESULTS

Cases Considered

Of primary interest in running the gap filling simulation is the determination of the maximum gap length which can be tolerated without permanent loss of the integer ambiguities over the full range of operating conditions. This includes separation distances ranging from 0 to 100 km, high, medium, and low quality INSs, wide laning, half wide laning, and normal laning receivers, and tracking none, one, two, or three GPS satellites during the simulated gap. The zero separation distance was included as a limiting case for which the residual spatially decorrelated errors could be completely removed.

To better understand the summary curves presented in this section, consider Figures 2 a. and b., which plot the 10" differential propagation errors for each satellite as a function of time, starting 60 seconds before the measurement gap begins. The lane recapture test operates directly on these errors. As illustrated in Figure 2a., the initial errors are very small following the 60 second calibration period with full GPS coverage. A gap of 50

314

Line 01 Sight

Error(m)

0.8

0.6

0.4

0.2

Carrier Lock Lost

• 1% Confidence Level lor Reacquisillon .••••• _ •• _-_ •• _ •••

Carrier Lock

Regained

Integers Reacquired O~==--P---~~--~~---+ ____ -+ ____ ~~ __ ~ ____ ~ __ ~

o 20 40 60 60 100 120 140 160 180 lime (sees)

Fig. 2a. Re-initialization of wide-lane integers with INS after 50 second gap

0.6

First Carrier Measurement 0.5

Secorld Carrier Measurement· Velocity EstImated

Un. 01 Sight 0.3

Error(m)

0.2 ••• '1 % Confidence Levellor Reacqulsillon ••••••••••••••••••••••• ~ • ...,.. • ......:lI~ ••••••••••••••••••••••

0.1

111 111 III 11 111 111 lIZ 113

lime (secs)

III 115 111

Integers Reacquired

117 III 111

Fig. 2b. Re-initiaIization of wide-lane integers with INS after 50 second gap

0.8 Carrier Lock

Carrier Regained

Lock

=A~ 0.6 Lost on

Position vertical AlISVs Error (m)

0.4

east Reacqwred

0.2 north

0

0 20 40 60 80 100 120 140 160

Tim. (seca)

Fig. 3. Position error history 50 second gap in GPS coverage

315

IlV

160

seconds with complete GPS loss is considered, using a high quality INS and wide laning GPS receiver. The separation distance for this case was 100 km. Note that each of the 10' errors along the four lines of sight converges to less than one-fifth the wide lane width (86 cm), indicating re-initialization of the correct set of integers for all 4 satellites. Figure 2b. is a close-up of the integer re-initialization sequence from Figure 2a., illustrating how the process begins with the second carrier measurement after lock is regained. This second measurement establishes velocity, which is used to improve the position estimate through the correlation between inertial position and velocity errors.

It should be emphasized that resolving the integers does not necessarily give position accuracy equal to carrier phase accuracy. This is because geometric range to the satellites is only one component of the double difference - atmospheric errors will cause position errors even if the integers are correctly identified. This is illustrated in Fig. 3, which plots the position error components at a separation distance of 100 km.

Complete Loss of GPS. Results for complete loss of GPS are summarized by Figure 4. The maximum tolerable gap length (in seconds) is plotted on the vertical axis, while the separation distance (in kilometers) is the horizontal scale. A total of 9 individual curves appear on the plot, corresponding to all possible combinations of INSs (high, medium, and low quality) and three receiver types: wide-lane dual-frequency P-code, wide-lane dual frequency with squared L2 carrier, and single frequency. Note the very significant variations in performance with INS quality. A high quality INS is able to tolerate gaps between 60 and 70 seconds when operating with a wide laning receiver, while a low quality INS can at best tolerate a gap of less than· 20 seconds. Also note that the separation distance has only a minor effect on maximum survivable gap length; a maximum variation over the full range of separation distances of roughly 12 seconds occurs for the high quality INS operating with the normal laning GPS receiver. The results at large separation distances are not completely rigorous, as they depend highly on the decorrelation times of the errors, which are not yet well-known.

Similar tests with partial loss of GPS (one or two satellites tracked through the gap) demonstrated virtually identical performance with marginally longer gaps.

70

60I--o--o---o---o----n

50

Maximum 40 ~~~-=-:~~==_=-~-----<:>-----_o__ Tolerable ,J, Gis;:~1h 30 -------0-------0-------0- _____ -0 ____ _

o 10 20 30 40 50 60 70 80 90 100

SeparaUon Dislance (km)

Fig. 4. Gap length v. separation distance loss of four GPS satellites

316

High quality INS Medium qualily INS

Low qualily INS

Wide lane Squared lane

Lllane

o o t:,.

Three GPS Satellites Tracked in Gap. Results when three satellites are tracked throughout the gap are summarized in Fig. 5. Note the significantly different behavior from the previous case: much longer measurement gaps can generally be tolerated, especially for the wide laning receiver; significant variation of gap length with separation distance occurs; and, for a given receiver type, results are relatively insensitive to INS quality. The longer gap lengths can be attributed to the loss of only one measurement, which is effectively replaced by the rubidium frequency standard. This is much more accurate in tracking time than the inertial system is in tracking motion, resulting in significantly longer allowable gap times. The more significant variation with separation distance can be attributed to the temporal decorrelation of the residual correlated errors, which are approaching the decorrelation times in Table 3.

300

250

200

Maximum Tolerable 150

Gap Lenglh (sees)

100

50

10 20 30 40 50 60 70 80 90 100

Separ.don Distance (11m)

Fig. 5. Gap length v. separation distance loss of one of four GPS satellites

SUMMARY OF RESULTS AND CONCLUSIONS

High qualily INS Medium qualily INS

Low qualily INS

Wide lane Squared lane

Ll lane

o o b.

When more than a single GPS satellite is lost, the use of a high quality INS can significantly increase the maximum tolerable gap length, permitting rapid reacquisition of the correct set of integer ambiguities. Maximum gap lengths range from 10 to 80 seconds for the high quality INS, as a function of separation distance and receiver type. Of these, receiver type plays a more important role than separation distance. Variation with separation distance for a given INS quality and receiver type is relatively minor; at most, 15 seconds over the range from 0 to 100 km is observed. On the other hand, variations with receiver type are significant for all INS qualities, e.g., a high quality INS can withstand a gap of 70 seconds when operating with a wide-Ianing receiver, but the length is reduced to roughly 40 seconds when using a half wide laning receiver.

Significantly different trends are observed when only a single satellite is lost. Here, only minor differences can be attributed to INS quality and greater variations with separation distance occur (especially for the longer gap lengths). Maximum gap lengths of more than 250 seconds can be accommodated, due to the assumed use of a rubidium frequency standard.

ACKNOWLEDGMENTS

This work was sponsored by the Engineer Topographic Laboratory of the U.S. Army Corps of Engineers, under the direction of Fred Gloeckler, Kevin Logan, and Steve DeLoach. The resources of Clyde Goad and Ben Remondi were crucial in creating a

317

simulation based on practical experience, as were the talents of AIgyte Cabak in implementing the software.

REFERENCES

1. Hein, G.W., Baustert, G., Eisfeler, B., and Landau, H., "High-Precision Kinematic GPS Differential Positioning and Integration of GPS with a Ring Laser Strapdown Inertial System, Navigation, Journal of the Institute of Navigation, Vol. 36, No.1, Spring 1989

2. Kleusberg, A., Georgiadou, Y. and Wells, D., University of New Brunswick, Fredericton, New Brunswick, Logan, K., U.S. Army Corps of Engineers, Fort Belvoir, Virginia, Geier, J. and Loomis, P., Trimble Navigation Ltd., Sunnyvale, California, "GPS Carrier Phase Ambiguity Resolution for Moving Receivers, GPSNI KIS 1990, Banff, Sep 10-13, 1990

3.. Hatch, R., Magnavox Advanced Products and Systems Co., Torrance, California, "Instantaneous Ambiguity Resolution", GPSNI KIS 1990, Banff, Sep 10-13, 1990

4. Gelb, A., &1plied Optimal Estimation, MIT Press, 1978.

5. Widnall, W.S., and Grundy, P.A., Inertial Navigation System Error Models, Intermetrics TR-03-73, 11 May 1973.

6. Nash, R.A., Jr., Kasper, J.A., Jr., Crawford, B.S., and Levine, S.A., "Application of Optimal Smoothing to the Testing and Evaluation of Inertial Navigation Systems and Components", IEEE Trans. on Automatic Control, Vol. AC-16, No.6, Dec. 1971, pp. 806-816.

7. Rice, D.A., "A Geoidal Section of the United States", XIIth General Assembly of the International Union of Geodesy and Geophysics, Helsinki, July - August 1960.

8. Levine, S.A., and Gelb, A., "Effect of Deflections of the Vertical on the Performance of a Terrestrial Inertial Navigation System", Journal of Spacecraft and Rockets, Vol. 6, No.9, Sept. 1969, pp. 978-984.

9. Precision Time and Freqyency Handbook, Ball Efratom Division, 1985.

10. Altshuler, Edward E., and Kalagham, Paul M., "Tropospheric Range Error Corrections for the Navstar System," Air Force Cambridge Research Laboratories, AFCRL-TR-74-0198, April 1974.

11. Greenspan, R.L., and J.I. Donna, "Measurement Errors in GPS Observables," Proceedings of the Forty-Second Annual Meeting of the Institute of Navigation, Seattle, Washington, June 1986.

12. Klobuchar, John A., "A First-Order, Worldwide, Ionospheric, Time-Delay Algorithm," Air Force Cambridge Research Laboratories, AFCRL-TR-75-0502, September 1975.

13. Jorgensen, P.S., "An Assessment of Ionospheric Effects on the GPS User," NAVIGATION, Journal of the Institute of Navigation, Vol. 36, No.2, Summer 1989.

318

CENTIMETER LEVEL SURVEYING IN REAL-TIME

ABSTRACT

Kendall E. Ferguson, Senior Software Engineer Ellis R. Veatch II, GPS Survey Advisor

Ashtech, Inc. 390 Potrero Avenue

Sunnyvale, California 94086

Since the advent of the Global Positioning System, several new positioning and surveying techniques have evolved. Within each method there are varying levels of achievable accuracies. Some of these methods are capable of real-time results but typically produce accuracies which are not acceptable for surveying applications. The most precise techniques have required post-processing of the field observation data.

The kinematic surveying technique is one of the most highly productive of those which required post -processing. This method has been shown to produce decimeter level, and often better, results. In March 1990, a test was performed employing the principles of the kinematic technique and centimeter level results were obtained in real-time.

This paper will provide a discussion of the tests performed as well as results obtained. Real-time kinematic results will be compared with those obtained from post­processing techniques. Additionally, a discussion of method constraints and foreseeable applications will be presented.

INTRODUCfION

To understand real-time kinematic procedures, one must understand the kinematic technique in general. The kinematic technique and field procedures are documented in numerous references [e.g., Remondi 1985a, Remondi 1985b, Remondi 1988, Veatch and Oswald 1989, Ferguson 1989]. Essentially, the kinematic technique requires that the cycle integer biases already be determined. Once these integers are known, a receiver, called the rover, can be moved. Because the ambiguities have already been resolved, the position of the antenna can be computed at each epoch [Remondi 1988]. Production can be greatly increased over static methods where

319

baselines are occupied for an hour or so, primarily to resolve the phase ambiguities. Until now, kinematic GPS has employed data post-processing. The data has been collected, downloaded in the office and the positions determined. Results can now be obtained in real-time.

Real-time allows for navigating to existing points as well as locating new points. One very nice feature of real-time is that it is immediately obvious when kinematic loss of lock has occurred. In a post-processed survey, bi-receiver loss of lock cannot be ascertained until after the data are collected. With a loss of lock during real-time, the operator simply returns immediately to the previous determined point and re­initializes. No lost points!!

Centimeter level kinematic accuracies have been obtained using post-processing techniques. It is expected and will be shown that real-time results should be identical.

EQUIPMENT

The current processing program design is personal computer based. As can be seen in Figure 1, the minimal equipment configuration consists of two GPS receivers, two communications links, and one processing computer. When the system is fully developed, the real-time processing software can be placed inside the receivers, eliminating the requirement for the external computer.

FIXED ROVER

L-R_8_C_8_IV_8_r -J\\ c··A LI.: .. // ...... ____ R_8_C_8_1V8_r-'

...... ~ ............ ,... ..i/

I c:m:umr I Figure 1

320

Choice of communication links is dependent upon the application. Speed, reliability, and cost effectiveness are major considerations. The computer can be positioned to best suit the particular application. When located with the fixed receiver, processing merely tracks the roving receiver. When located with the rover receiver, it can be used to navigate or position the rover and multi-rovers can be used with one transmitter.

METHOD

Both the real-time and the post-processing kinematic algorithms utilize the same underlying principles as originally developed by Dr. Benjamin RemondL Both methods require that the integer biases be known, typically determined at the beginning via a known baseline, such that they may subsequently be used to determine epoch-by-epoch positions. Additionally, they both require that at least 4 (more practically 5) satellites be continuously tracked while the rover antenna is in motion to maintain the integers [Remondi 1988]. The cycle-slip fIxing technique present in Ashtech's post-processing software is also present in the real-time software. Thus, provided the appropriate conditions are met, e.g., good geometry (PDOP) and at least 4 cycle-slip free double-difference observations at the epoch of the slip, cycle­slips are repaired in real-time.

Obviously, real-time kinematics are subject to some delays. There are delays from the time that data are sampled to the time that they are processed (Le., the time required to communicate the data and the time required to actually process the data). The communication delay provides the largest contribution. If an accurate position at an instant in time is required, it must be extrapolated. Ignoring the effect of the communication link, the current algorithm can provide a computed kinematic result for an epoch in approximately l/10th of a second. With some effIciency improvements, this can easily be improved by at least factor of 2.

The differences between the two approaches bring about potential changes in fIeld procedures. Typically, to initiate a real-time or a post-processed kinematic survey, an initial vector between the reference receiver and the rover receiver must be known. The difference lies in the time at which that vector must be known. In the post­processing technique, this vector must be known prior to actually processing the kinematic data, not at the time of collection as in the real-time application. For example, one could collect kinematic data on a day in which no sites occupied by the rover are known. All that is required is to determine a vector, typically through some other techniques such as antenna swap, static survey, or pseudo-kinematic, to anyone of the rover occupied sites prior to processing that kinematic data.

Additionally, kinematic data can be post-processed in chronological or reverse­chronological order. As an example, if kinematic loss of lock occurs somewhere

321

between the occupation of two known data points, the data can be processed from both directions up to the point of loss. In real-time, data are processed only in the order received. Therefore, if loss of kinematic lock occurs, one must immediately re­initialize.

Real-time kinematic processing can be more informative when loss of kinematic lock occurs. Again, for kinematic purposes, the common data between the flXed and roving receiver must be free of cycle slips on at least 4 satellites from epoch to epoch. Typically, GPS receivers provide indications when cycle slips occur. Many receivers can be configured such that they immediately inform the operator when loss of kinematic lock has occurred due to that receiver. They cannot provide an indication of the combined quality unless there is some type of real-time link. Real-time kinematic processes each measurement as they occur, and loss of kinematic lock can be immediately determined.

RESULTS

Presented here are results from two experiments. The first was our initial March 13th (day 073) test where the computer and receivers were connected by RS-232 cables. The second test was performed on August 29th (day 241) using radio modems. For this test, the computer was placed in the vehicle which transported the rover receiver. The flXed receiver communicated data to the processing computer via radio modem while the rover communicated via direct link (RS-232). It is important to note that each of the raw data sets contained cycle-slips which were repaired in both procedures.

Figures 2, 3, and 4 provide a comparison of the epoch per epoch solutions generated for day 073. These graphs depict the difference between the solution generated via post-processing and that generated during real-time. As can be seen from these figures, the variance between post-processed and real-time kinematic are small.

322

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Tables 1 through 6 show three sets of results for each day: sample stationary rover epoch-by-epoch solutions, sample moving rover epoch-by-epoch solutions, and vector summaries. For each of the three sets, both the real-time and post-processed results are provided.

DAY 073

Real-Time Post-Processed

EPOCH dX dY dZ dX dY dZ

04:47: 10 7.691 -4.387 o.no 7.691 -4.389 o.no 04:47:20 7.692 -4.387 0.720 7.692 -4.389 0.719

04:47:30 7.696 -4.382 0.713 7.696 -4.384 0.713

04:47:40 7.693 -4.385 0.715 7.693 -4.387 0.715

04:47:50 7.692 -4.385 0.716 7.692 -4.387 0.716

04:48:00 7.696 -4.383 0.716 7.696 -4.385 0.716

04:48: 10 7.698 -4.384 0.717 7.698 -4.386 0.717

04:48:20 7.694 -4.387 0.719 7.694 -4.389 0.718

04:48:30 7.696 -4.383 0.718 7.696 -4.385 0.717

Table 1. Stationary rover (site CORN) epoch-by-epoch solutions

324

Real-Time Post-Processed

EPOCH dX d'f dZ dX d'f dZ

05:06:10 25.172 -5.486 11.178 25.173 -5.487 11.177

05:06:20 25.187 -5.512 11.154 25.188 -5.513 11.154

05:06:30 21.499 -4.974 9.536 21.500 -4.975 9.536

05:06:40 9.477 -3.138 3.969 9.477 -3.139 3.968

05:06:50 -0.113 -1.702 -1.510 -0.113 -1.703 -1.511

05:07:00 -0.363 ·1.714 -1.770 -0.363 -1.716 -1.771

05:07:10 -0.719 -1.174 -1.800 -0.719 -1.175 -1.801

05:07:20 -0.719 -1.176 -1.799 -0.719 -1.177 -1.799

05:07:30 -0.717 -1.173 -1.802 -0.716 -1.174 -1.802

Table 2. Moving rover epoch-by-epoch solutions

Real-Time Post-Processed

SITE dX d'f dZ dX d'f dZ

0002 5.335 10.626 15.557 5.336 10.624 15.557

ASHT 25.168 -5.495 11.183 25.168 -5.498 11.183

CARD -4.705 -7.801 -11.828 -4.705 -7.803 -11.828

CORN 7.694 -4.385 0.718 7.694 -4.387 0.717

Table 3. Vector Solutions

DAY 241

Real-Time Post-Processed

EPOCH dX d'f dZ dX d'f dZ

16:44:40 48.840 -58.798 -28.709 48.840 -58.799 -28.709

16:44:50 48.841 -58.796 -28.713 48.841 -58.796 -28.713

16:45: 0 48.843 -58.798 -28.712 48.842 -58.798 -28.713

16:45: 10 48.841 -58.794 -28.718 48.841 -58.794 -28.719

16:45:20 48.840 -58.798 -28.712 48.840 -58.798 -28.712

16:45:30 48.841 -58.797 -28.713 48.840 -58.797 -28.714

16:45:40 48.844 -58.790 -28.720 48.843 -58.790 -28.720

Table 4. StationaIy rover (site POOS) epoch-by-epoch solutions

325

Real-Time Post-Processed

EPOCH dX dY dZ dX dY dZ

16:37:10 106.971 -239.284 -189.618 106.971 -239.284 -189.618

16:37:20 138.894 -297.272 -231.732 138.894 -297.272 -231.733

16:37:30 161.671 -348.352 -273.341 161.671 -348.352 -273.342

16:37:40 See Discussion Below 164.480 -354.412 -278.112

16:37:50 171.527 -359.142 -278.334 171.526 -359.142 -278.335

16:38: 0 171.469 -359.138 -278.362 171.469 -359.138 -278.363

16:38: 10 171.469 -359.134 -278.364 171.469 -359.135 -278.364

Table 5_ Moving rover epoch-by-epoch solutions

Real-Time Post-Processed

SITE dX dY dZ dX dY dZ

0001 173.024 -361.808 -281.561 173.022 -361.812 -281.560

P007 102.178 -181.062 -129.024 102.180 -181.066 -129.021

P008 48.842 -58.796 -28.715 48.841 -58.795 -28.716

P009 161.120 -314.960 -237.751 161.122 -314.960 -237.750

Table 6_ Vector Solutions

The epoch at time 16:37:40 on day 241 was not processed by the real-time kinematic algorithm. At that time, the communication channel corrupted the data for the fIxed receiver. The real-time processing algorithm detected the error and ignored that data. This type of intermittent communication error does not cause cycle-slips nor kinematic loss of lock. Subsequent samples were unaffected.

The real-time kinematic algorithm was also used to successfully navigate to known points. Figure 5 provides a sample of the screen display while in navigation mode. This screen has been designed to guide a user to a desired waypoint. The lower left portion of the display provides the course and heading information. The center of the large circle is the point being navigated to. The Large X depicts the current location of the rover receiver. The smaller x's depict a history of the rover positions. As the rover moves closer to the desired waypoint, the scale on the circle will automatically be magnifIed.

Also shown in the large circle are the satellites which where being processed by the software. The azimuth and elevation are graphically represented as would be found on a typical polar plot. The inner circle is the elevation cut-off angle which was used during processing (i.e., 0 degrees).

326

The left top portion of the screen provides a graphical representation of the elevation difference between the rover position and the waypoint position.

1.000

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For many applications it will be very useful to have the real-time kinematic software operational within the receiver. For surveying applications, this is especially true. The obvious scenario involves one or two men traveling from point to point setting marks as they go, where the additional computer equipment would be cumbersome.

Real-time techniques will allow for stake out and relocation surveys in addition to current collection type surveys. Computed coordinate positions will be downloaded into the receiver as waypoints, and will be navigated to and set in the field. Extending this capability, we will surely see the automated control of earthmoving equipment and dredges using real-time kinematic techniques. Grade and plan information will be fed into the on-board system, and blades and buckets will be adjusted to provide the proper cutting height and slope for instantaneous positions along the route. The operator will simply monitor the system and verify its operation.

327

High-precision real-time guidance of ground, sea and air vehicles along preplanned routes and true "just in time" production techniques will also be possible.

CONCLUSIONS

We have shown that the real-time kinematic technique provides accuracies equivalent to those provided by post-processed techniques. In order to successfully perform real-time kinematics a robust communication link, such as modem cellular phones or reliable radios, must be employed. We predict that real-time kinematic techniques will revolutionize many high-precision surveying, navigation, and monitoring applications.

ACKNOWLEDGEMENTS

The authors wish to acknowledge and commend the following Ashtech personnel for there support in the development and research of real-time kinematics. Reza Abtahi and Jean-Marie Eichner for promptly making the necessary receiver software enhancements. Peter Heinemann for assembling the radio modem equipment. Mike Evers, Jonathan Ladd, Richard Sauve and Jon Siegrist for participating in the test surveys. Doug Evans and Mehregan Ghazvini for such prompt response to hardware needs of the experiments.

BmuOGRAPHY

Ferguson K E., et. al., 1989: Kinematic and Pseudo-Kinematic Surveying with the Ashtech XII. Proceedings of ION GPS-89, Colorado Springs, Colorado, The Institute of Navigation, September 27-29, pp. 35-37.

Remondi B. W., 1985a: Performing centimeter accuracy relative surveys in seconds using carrier phase. Proceedings of the First International Symposium on Precise Positioning with the Global Positioning System, (NOAA), Rockville, Maryland. National Geodetic Information Center, NOAA, Rockville, Maryland 20852, April 15-19, 1985, pp 789-797.

Remondi B. W., 1985b: Performing Centimeter-Level Surveys in Seconds with GPS Carrier Phase: Initial Results. NOAA Technical Memorandum NOS NGS-43. National Geodetic Information Center, NOAA, Rockville Maryland 20852.

Remondi B. W., 1988: Kinematic and Pseudo-Kinematic GPS. Proceedings of the Satellite Divisions International Technical Meeting, Colorado Springs, Colorado, The Institute of Navigation, September 19-23, 1988, pp. 115-121.

Veatch E. R., Oswald J., 1989: The Kinematic GPS Revolution: "Surveying on the Move". ASPRS/ACSM Annual Convention Technical Papers, Baltimore, Maryland. ASPRS/ACSM, Vol. 5, pp. 288-297.

328

Combining Pseudo-Range and Phase for Dynamic GPS

V. Ashkenazi, T. Moore and J. We strop

Institute of Engineering Surveying and Space Geodesy University of Nottingham, England

Positioning with carrier phase undoubtedly gives higher precision GPS solutions than positioning with pseudo-ranges. However, the presence of cycle slips in carrier phase data, which are very difficult to detect and correct in a dynamic environment means it is impractical to use only carrier phase measurements. Gaps in the data, also associated with cycle slips, mean that there would be periods when no position information is available. The compromise to this problem is to use carrier phase combined with pseudo-range such that high precision positioning is obtained when the carrier phases are not affected by cycle slips, and there is only a small but graceful degradation in precsion when cycle slips do occur and cannot be corrected.

This paper firstly describes the results obtained when using pseudo-range solutions in undifferenced and differenced modes. This is followed by descriptions of combinations of pseudo-range and phase observables, which may be used to improve the positioning capability of GPS. Results are also given for these methods and they are compared in the fmal section.

Nottingham Field Trials

Field trials to primarily test the 'stop go' kinematic technique [Remondi, 1986] were conducted at Nottingham in February 1989. The data collected have subsequently been used to examine observables and methods that use both pseudo-range and carrier phase measurements. A small network consisting of 5 stations was used (see Figure 1).

A A

A /wm

A A Figure 1: Test Network

The maximum distance between stations did not exceed 50 m, allowing the roving receiver to be easily transported by one person. Horizontal angles, inter-station distances and orthometric heights for the network were established by a terrestrial survey. In addition, two sessions of GPS observations were undertaken to introduce absolute position, scale and orientation.

During the kinematic survey, two TRIMBLE 4000 SLD receivers were deployed, one at the centre, the other roving between the four outer points. A constellation of 5 satellites was observed and data were collected approximately every 5 seconds. A centred and levelled tribrach on a tripod was set up over each point prior to the survey. The roving antenna, mounted on a 0.5 m extension pole, was manually moved from point to point such that all of the points were visited at least three times. The antenna remained at each point for about 2 minutes.

329

There were only 2 cycle slips detected in the Ll observations. The decoding of the L2 observations indicated that cycle slips contaminated the data from every satellite whenever the roving receiver was in motion. This was caused by the inability of the TRIMBLE receiver to widen the tracking bandwidth for the L2 observations.

The positions computed for each point using double difference carrier phase observables [Ashkenazi et al, 1990] were expressed as vectors from the centre station. The mean and standard deviation for all common baseline components have been calculated. All of the average values agree with the ground truth to better than 7 mm. The largest standard deviation was 7.5 mm.

The positional errors of the three coordinate components, determined for each 5 second data epoch at a static point, are less than 15 mm and these do not reduce significantly with time, see Figure 2. This confirms that 1 or 2 epochs (5 or 10 seconds) of data would be sufficient to accurately recover an unknown baseline. However, the inclusion of additional epochs of data allows a statistical analysis to be more meaningful.

10 9 • • • • • ••• ..-. 8 • • • • • • E 7 • • • •

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-5 0 2 4 6 8 10 12 14 16 18 20 22

Epochs (- 5 secs)

Figure 2: Typical Component Positional Errors at a Static Point

Although the positional errors have been seen to be sub-centimetric for just the static points, they can be reasonably extended over the dynamic periods because of the good repeatability when re­visiting the same point. The positions of the mobile receiver were transformed from WGS84 cartesian coordinates to UK National Grid coordinates, and considered to be the 'true' positions.

Positioning With Pseudo-Ranges

Four simultaneously observed pseudo-ranges to different satellites allow the 3-D position of a user to be determined by range resection. Least squares observation equations are formed in terms of the unknown parameters; receiver coordinates and clock offset, and then solved to give the least squares estimates of these parameters. This procedure is performed within all GPS receivers used for positioning, and has been extensively documented.

330

The least squares solution for the unknown parameters requires the expected and observed pseudo-ranges to be compared. The expected pseudo-range is computed by assigning approximate values to all of the terms. Terms for the satellite clock offset, ionospheric and tropospheric delays are usually computed from models, ego a polynomial model for the satellite clock offset and the Klobluchar model for the ionospheric delay, which both use information broadcast in the satellite Navigation Message.

The alternative to modelling the terms is to eliminate them by differencing. There are four categories for differencing; with satellites, with time, with receivers and with frequency. The main disadvantage for positioning with differenced observables is that their measurement noise is higher than for undifferenced measurements. The potential elimination of the effects of an error by differencing must therefore be considered in light of the increase in the noise of the estimated positions. This factor prohibits the use of dual frequency measurements to remove the ionospheric delay because the decrease in positioning precision is a factor of 3-4 [Westrop, 1990]. Positioning results are given below for each type of differenced pseudo-range observable.

Pseudo-range Mean Error (Standard Deviation) (m) Observable E U -6.67 (5.96) S -6.67 (5.96) T -7.53 (5.94) S+T -7.53 (5.94) R 0.12 (7.63) R+S 0.12 (7.63) R+T -5.95 (7.64) R+S+T -5.95 (7.64)

Abbreviations U = Wldifferenced S = differenced with satellites T = differenced with time R = differenced with receivers

N 1.30 (4.35) 1.30 (4.35) 2.67 (3.70) 2.67 (3.70) 0.33 (4.27) 0.33 (4.27)

-6.81 (4.44) -6.81 (4.44)

R+S = differenced with receivers and satellites (conventional double difference)

Table 1: Differenced Pseudo-Range Positioning Results

Ht -53.29 (8.69) -53.29 (8.69) -9.10 (9.78) -9.10 (9.78) 0.41 (12.35) 0.41 (12.35)

-6.27 (12.34) -6.27 (12.34)

Figures 3 - 6 show the Easting, Northing and Height positional errors for the fIrst 20 minutes of the collected data, with respect to the truth, using the three single difference observables; S, T and R. The positional errors using the R + T observables are also given. It may be seen from these graphs that trends in the positioning are only removed by differencing with receivers.

Several observations may be noted from these results:

The results for U and S, T and S+ T, R and R+S and R+ T and R+S+ T are identical. This means that there is no benefIt in estimating the receiver clock offset, rather than eliminating it by differencing with satellites. These results are not surprising because the least squares estimate of the receiver clock offset is an amount which is common to all the pseudo-range observations, and differencing also accounts for any common effects. It must be noted that if the receiver clock offset were not modelled as a common effect in all simultaneously measured pseudo-ranges, then identical results would not occur.

331

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The U and S observables mainly exhibit higher mean errors than the other observables. This is because they contain the effects of errors at the satellites, ie. satellite orbital error and clock offset. Consequently, it may be clearly seen why these two observables should not be used for high accuracy dynamic positioning. This is especially applicable since Selective Availability was switched on.

The smallest mean errors occur when using the R and R +S observables, and this is because the largest error source (due to the satellites) is eliminated by using these observables. However, the standard deviations of the positions are large, and vary from 4 to 12 m.

It will be noted that the positions for the R and R+T observables exhibit similar trends. However, the positions for the R + T observables are biased from the mean. These offsets are caused by incorrect knowledge of the initial starting coordinates and clock offset for the mobile receiver [Remondi, 1986] and affect positions estimated with time differenced observables. The offsets are not identical for the T and R+ T positions in Figures 4 and 6 because the fIrst epoch is not the same

The correlation matrix for each differenced observable [Westrop, 1990] was used in the least squares estimation. Tests were conducted with the correlation matrix as the identity matrix, and this resulted in positional errors that were not identical pairs as shown in Table 2. Instead, the results varied according to which satellite was used to difference with the others. It was concluded from this that it is important to use the correct correlation matrix for differenced observables. This is in agreement with the conclusions made by Yau [1986] for various carrier phase observables.

The results for differential GPS, involving the computation of corrections to the observed pseudo-ranges at a monitor station, were found to be identical to those using receiver differenced observables. This is theoretically expected because solving for corrections at a known point is an equivalent process to fIxing these coordinates when using receiver differenced observables. This means that either the pseudo-ranges observed at the monitor, or the corrections computed at the monitor using these measurements, could be transmitted to the mobile receiver for use in the computation of its position. In either case, the estimated positions would be the same. Although the transmission of the observed data would enable a very simple relay monitor station to be employed, a slightly greater degree of computing power may be required at the mobile receiver to solve for the system of receiver differenced observation equations.

Positioning with Pseudo-Ranges and Carrier Phase

To a certain extent the characteristics of pseudo-ranges and carrier phase are complementary. For example, carrier phase, although very precise, is susceptible to cycle slips, whilst pseudo-ranges, with a measurement noise level at about a metre, do not contain them. Pseudo-ranges are absolute measurements whilst the carrier phases are not usually, unless the initial integer ambiguities have been determined.

There are essentially two approaches to using carrier phase as a 'fIlter' to improve the precision of the positioning. These are either to fIlter the pseudo-ranges or fIlter the positions estimated using the pseudo-ranges. Both of these approaches are similar and are illustrated below. Each of the dots may be considered as a pseudo-range or an absolute pseudo-range position solution and the arrow are the relative phase measurements or position solutions between measurement epochs.

334

-.-~-- . Figure 7: Combining Pseudo-Range and Carrier Phase: The Problem

Two filters have been examined in this study; the Kalman filter and an averaging filter, and both are examined below.

Kalman Filtering

A Kalman filter is a processing technique that is typified by particular measurement and dynamic models. Pseudo-range measurements may be input to the Kalman filter using the same least squares observation equations as previously described. Carrier phase observation equations may also be formed with respect to the unknown receiver coordinates and clock offset, or their second derivatives with time. Velocity, or range-rate, measurements may be derived from the carrier phase measurements at two epochs, i and i-I, by,

velocity = AI'Ot (<I>i - <I>i - t)

where 'Ot is the time interval between the two epochs, and A. is the wavelength.

Polynomial Dynamic Model

The dynamic model used during these studies was a polynomial model, which models position and velocity (pV model). It is commonly referred to as the 'constant velocity' model. Results are given below for positioning using a Kalman filter, with and without carrier phase range-rate measurements. Differential pseudo-range corrections, computed at the monitor station were applied to the position computation of the mobile receiver. In both cases, these corrections were the same. The variances of the measurements and dynamic model were also identical for both tests, such that no differences between the tests could be attributed to any factor, other than the inclusion of carrier phase. Further details about the parameters are given in Westrop [1990]. The results are given in tabular form below, and graphically in Figures 8 and 9.

Measurements Dynamic Mean Error (Standard Deviation) (m) Model E N Ht

PR PY 0.13 (4.07) 0.34 (2.72) 0.42 (5.15)

PR,<I> PY 0.67 (1.18) 0.17 (1.09) 0.17 (1.65)

PR <I> 0.10 (0.91) 0.29 (0.58) 0.71 (l.48)

Table 2: Kalman Filter Positioning Results

It may be seen that the precision of the positioning is considerably reduced by including phase measurements. This is expected because the phase measurements are contributing additional information that is much more precise than the pseudo-range data. It is also seen by comparing Tables 1 and 2 that there is an increase in precision, by a factor of two when using a Kalman filter with the PV model and only pseudo-range data. This indicates that the polynomial model is

335

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336

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andPVModel

helpful in reducing the noise in the positioning. It is also interesting to note that the mean errors for the positioning with R observables are very similar to those for Kalman filtering with pseudo­ranges only. The presence of the phase measurements has removed this similarity, and this may be partly explained by the requirement to assign a variance to the phase range-rate measurements, which has influenced the weighting within the Kalnan filter.

Phase Dynamic Model

An alternative to using the polynomial model, is to use the carrier phase relative positioning information to predict the position of the receiver with respect to the previous position, and to use the pseudo-ranges as noisy measurements to the filter. This approach has been adopted by Seeber et al [1986], Kleusberg et al [1986] and Hwang and Brown [1989], each having slight modifications.

Positioning results for this technique are also given in Table 2, and are shown graphically in Figure 10. It will be seen again, that the positioning precision has improved, although not to a large extent

Phase Smoothed Pseudo-Ranges

The technique of phase smoothing, initially described by Hatch [1982], and used sucessfully by many researchers, employs an averaging filter. A precise pseudo-range is formed at a reference epoch by averaging the observed pseudo-ranges over a period of time. The range-rate, as measured by the carrier phases, is subtracted from each of these observed pseudo-ranges, such that they appear to have been measured at the reference epoch. The carrier phase range rate is then added to the reference pseudo-range to give phase smoothed pseudo-ranges. For the purposes of this study, the phase smoothed pseudo-ranges were differenced with receivers to form R observables. Positioning results with these observables are given in Table 3, and are graphically presented in Figure 11.

Mean Error (Standard Deviation) (m) E N Ht

-0.22 (0.37) 0.37 (0.27) 0.18 (0.69)

Table 3: Phase Smoothed Pseudo-Range Positioning Results

It will be noticed from these results that the positioning precision has increased yet again, and is better by more than a factor of two, than Kalman filtering with a phase dynamic model. From Figures 10 and 11, it is particularly noticeable that the positioning errors for both methods are initially identical. These trends are due to the use of the pseudo-ranges for positioning. For phase smoothing, the influence of each pseudo-range diminshes as more are averaged to form the initial reference pseudo-range. For the previous method, each pseudo-range contributes with the same weight.

Comparison

Since the best accuracies, ie. small mean errors, occur by using receiver differenced observables or employing differential corrections, only these results are compared with respect to an improvement in positioning precision. The average standard deviations, computed from each

337

! 1.0

~ 0 O.S ~ ~ !>II

.... < z 0

E -O.S (I) 0 ~

" -1.0 z: i= ~ -l.S !>II

-2.0

! 1.0

~

~ 0.5 a: ra.l

.... < z: o E -0.5 <n ~ " -1.0 § t -l.S o z:

-2.0

- 2 ..=. ~ 0 1 ~ a: ra.l .... < z: 0 i= iii -1 0 ~

!: -2

" ~ -3

Figure 10: Kalman Filtering with Pseudo-Range, and Phase

Dynamic Model

338

! 1.0

~ 0 O.S ~ ~ !>II

.... < z: 0

E -O.S (I) 0 ~

" -1.0 z:

~ -l.S

-2.0

! 1.0

a: ~ 0.5 gj .... < z: o E -0.5 ill

~ " -1.0 !:i ~ -l.S o z:

-2.0

! 2

~ 0 a: a: !>II

.... < z: 0 i= iii -1 0 ~

!: -2

" ~ -3

TIME (sec., 41zeoo

\~

.~ INE (seco)

41280

Figure 11: Phase Smoothing

coordinate component, have been compared. These have been plotted in Figure 12, together with similar values using some static data .

...... E 7 ...... c 6 .2 - 5 • .2 -; 4 -., as Q - 3 en 'V 2 ... • 'V C • 0 -(/)

10

8 EI

.2 • -6 as E CD

4 c ~

2

0 a b c d e

Method a = Pseudo-Range Differenced with Receivers

b = Kalman Fiher, Pseudo-Range and PV Dynamic Model

c = Kalman Fiher, Pseudo-Range, Carrier Phase and PV Dynamic Model

d = Kalman Fiber with Phase Dynamic Model

e = Phase Smoothed Pseudo-Range

Figure 12: Comparison of Average Standard Deviations

Static

Kinematic

It is clearly evident that the average standard deviations have similar trends for both the static and kinematic data. In both cases, the average standard deviations are least, when using phase smoothed pseudo-ranges. Similar values are obtained when using carrier phase in a Kalman filter, either as velocity measurements or to predict the state. Theoretically the latter is expected to be better because actual observed motion is used to predict the state, rather than a hypothetical model.

In conclusion, it is recommended that the phase smoothing technique is used in preference to any other. The next best alternative method appears to be the use of a Kalman filter where the state is predicted by carrier phase infonnation.

References

Ashkenazi, Y., Hill, C., Summerfield, P. and Westrop, J. 1990. High Speed, High Precision Surveying by GPS, Proceedings of the Second International Symposium on Precise Positioning with the Global Positioning System, Ottawa, Ontario.

Hatch, R. 1982. The Synergism of GPS Code and Carrier Measurements, Proceedings of the Third International Geodetic Symposium on Satellite Doppler Positioning, DMA/NOS, Washington D.C. pp 1212-1231.

Hwang, P.e. and Brown, R.G. 1989. GPS Navigation: Combining Pseudo Range with Continuous Carrier Phase Using a Kalman Filter, Proceedings of the Second International Technical Meeting of the Satellite Division, Institute of Navigation, Colorado Springs, Colorado, pp 185-190.

339

Kleusberg, A., Quek, S., Wells, D., Lachapelle, G. and Hagglund, J. 1986. GPS Relative Positioning Techniques for Moving Platforms, Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, Austin, Texas, Vol 2, ppI299-131O.

Remondi, B. 1986. Performing Centimeter-Level Surveys in Seconds with GPS Carrier Phase: Initial Results, Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, Austin, Texas, Vol 2, pp 1229-1249.

Seeber, G., Schuchardt, A. and Wubbena, G. 1986. Precise Positioning Results with TI4100 GPS Receivers on Moving Platforms, Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, Austin, Texas, Vol 2, pp 1269-1285.

Westrop, J. 1990. Dynamic Positioning by GPS, Ph.D. Thesis in preparation, University of Nottingham.

Yau, J. 1986. Relative Geodetic Positioning using GPS Interferometry, Ph.D. Thesis, University of Nottingham.

340

Comparison of Heterogeneous Unsynchronised Data by Transforming Coordinate Independent Functionals

Abstract

Alain Geiger, Marc Cocard

Institute of Geodesy and Photogrammetry

Federal Institute of Technology ETH

CH-8093 Zurich, Switzerland

Tel: +411 377 3244

Fax: +411 371 2593

A major problem in testing kinematic results consists in the comparison of different data sets measured by different methods. In most cases these different measurements will be collected with different sampling rates, different data outages may also occur and different clocks may be used. Therefore the measurement epochs will not be synchronised. The coordinate systems to which the measurements are refered are normally not identical for all the data sets. For these reasons it is not possible to directly use a classical linear transformation approach to estimate some transformation parameters in a least square sense. This paper presents a possibility to determine transformational parameters by coordinate independent functionals. For this purpose correlation techniques are used for synchronisation. It is also shown that the spatial transformation parameters can be determined by the eigenvalue decomposition of coordinate independent geometrical functionals. This method has been applied to control kinematical data of GPS with Laser-Tracker measurements.

341

1. Introduction

It is obvious that for unsynchronised we can't speak of corresponding points in different data sets. Therefore any attempt to transform one set in another will fail when point to point correspondance has to be assumed. This is true e.g. for Helmert'-transform or similar approaches. But by looking to tracks of different data sets it seems quite easy to bring the plots of these sets to a reasonable alignement just 'by eye'. This means that a certain criterion of correspondance must exist. The proposition of this paper is to introduce coordinate independent functionals of the figures (data sets) and to derive transformation parameters between these sets of functionals.

2. Choice of the functionals

We may consider the data set as a I-dimensional figure parametrized by e.g. the time. This corresponds to 3-dimensional figure consiting of one wire.

For such a 'wire' the length L, the total mass M of the wire as well as center of mass s represent intrinsic, well dermed values (S: wire density):

and the center of mass

s = fSXdl M

For homogeneous wire densities S it follows

fXdl s =--

L =

[x.M] L

The latter value gives the discretizised form, with

[ ... ] = sum over all data points.

(1)

(2)

(3)

If two figures with homogeneous wire densities S should correspond, then the correspondence of the two centers of mass is compulsory.

For a certain figure we then define a function parametrized by e.g. the time t as:

r(t) = I r(t) I = I x(t) - s I (4)

This distance function r (t) is completely coordinate independent and completely defined by the parameter t. This function will be used for synchronisation. Another functional which is intrinsicely related to the figures is the moment of inertia. It is defined by

I = Trace fxTxSdl.E - fXxTSdl

In discrete form and disregarding the homogeneous density S we write:

342

(5)

a = Trace [XT X AI] E - [xxT AI] (6)

We now introduce the (3 x n) coordinate matrix X which is composed by the whole ensemble of n coordinate vectors of a data set in the following way:

(7)

For equal weight of every data point we then write:

(8)

This corresponds to the moment of inertia of a distribution of mass points with unit mass. For our purposes we use a simpler form which has similar properties with respect to the eigenvalues and eigenvectors.

This Matrix is symmetric and reads in diagonalized form:

;0 = u; UT

with U uT = E.

3. Time Synchronization

We come back to the distance-function (4).

r(t) = I x(t) - s I

(9)

(10)

(11)

If two coordinate sets with a certain synchronisation error At exist we can define the two distance functions

(12)

Since these two functions describe in principal (differences are due e.g. to measurement errors) the same figure the unknown At can be determined by correlation technique. We form the correlation function

(13)

It is well known that the maximum of <P will directly define the synchronization error At

max (<P12 ('t») = <P12 (At) (12)

The descrete calculation is carried out by the well known FFT-calculus where the convolution corresponds to a simple multiplication in the frequency domain w:

<I> (w) = Rl (w) Rz (w) (15)

343

4. Coordinate Transformation

Having solved the problem of synchronization we now try to find the coordiante transformation of different data sets. The problem is given by the fact, that the points are not (nominally) identical. H the data sets consist of corresponding points a 'HeImert' or a 'Similarity' transform could be applied.

We conturn the problem of non-existence of corresponding points by our basic asumption:

The eigenvectors of intrinsic tensor functions of different data sets must be parallel. (16)

This holds in the order of the measurement errors and the discretisation problems. For two data sets X and Y (see definition (7» we therefore have the diagonal forms (10) for both sets respectively

(17)

and

(18)

Because of the statement (16) the following relation must hold

T 110 = M • ~o M (19)

Where M is a scaling matrix

[:1 0

IJ M= ~ (20)

0

If we assume an isotropic scale factor the matrix degenerates to a scalar m. It follows:

(21)

Taking the diagonalization (10) into account we can rewrite (19)

(22)

or since V VT = E:

(23)

and setting

(24)

344

we arrive at:

(25)

We directly recognize a coordinate transformation T. For any coordinate set X we can calculate the corresponding set Y in the transformed system following

Y=TX

For an isotropic scale factor m a well known form of (26) appears

Y = VTMUX =mVTUX=mRX

Where, of course R is a rotation:

R RT = VT U UT V = VT E V = E

The scale follows from (21):

Trace ~o = m2 Trace ~

and

Trace (y yT)

Trace (X XT)

(26)

(27)

(28)

(29)

(30)

The following diagram summarizies the procedure to determine the coordinate transformation:

® X ~ ~ = XXT

U

~ ~o

tT tT tM (31)

Y ~ 11 =yyT V ~ 110

and T = VTMU.

U and V are rotations determined by the solution of the eigenvalue problem:

(~ - A E) e = 0

and (32)

(11 - Jl E) e = 0

resp. e are the corresponding eigenvectors and A, Jl the eigenvalues.

345

5. Example: Comparison between GPS and Laser-Tracker

In 1988 our institute carried out an experiment to check the GPS perfOImence in kinematic application. The GPS-measurements were controlled by a automatically tracking Laser system. This test has been documented more in detail by Cocard et al. (1989). The GPS equipment (Trimble 4OOOST) sampled at a rate of 15 sec and in a second experiment at 3 sec. The Laser tracker (automatic theodolite Geodimeter 14OT) carried out a complet vector measurement (vert. angle, horiz. angle, distance) every sec approximatively. The time was given by GPS in case of the GPS receiver and by the used Grid Laptop in case of the Laser Tracker. Since both clocks were unsynchronized the procedure described in chap. 3. has been applied to get a common time frame. The theodolite has been installed without any orientation to a fixed reference. The polar coordinates (vectors) were transformed in a local cartesian system. Therefore, it is clear that the transformation between both coordinate systems was completely arbitrary.

It was possible to apply the transforming procedures described in the previous paragraphs because the Laser-reflectors were mounted right untemeath the Trimble receiver. Therefore only a small height offset (- 15 cm) and a correction (for centering) in the lenght of the geodimeter vector ( - 8 em) had to be considered. After correcting this small offsets both sets of coordinate describe exactly the same geometrical track of the vehicle. In figure 1 and 2 the resulting positions of GPS and Laser-tracker respectively are depicted.

Figure 1: GPS-Track (diameter of the testfield approx. 2 km)

346

Figure 2: Laser Tracker result (diameter of the testfield approx. 2 km)

Figure 3 shows the sets transformed onto a map. For a final adjustement and for the calculation of residuals the denser (Laser-tracker) data have been interpolated at GPS epochs and compared to one another by an extended 'Helmert'-transform. Exended means that besides the classical parameters like rotation, scale, translation a very small synchronisation unknown (by considering the velocity) is taken into account. After all transformations a precision of about 2 - 3 m for a simple CIA-Code (differential) solution has been found.

6. Conclusion

The described methods to determine synchronization errors as well as coordinate transformations (orientation, scales, translations) are very well suited to compare different data sets which have no relation concerning time, orientation, sampling rate etc .. The methods are based on correlation technique and tensor transformation (eigenvalue solution) using geometric characteristics of the tracks under consideration.

7. References

Cocard, M., A. Geiger, U. Marti, M. MUller, B. Wirth (1989): Kinematische Positions­bestimmung mit GPS und Geodimeter. Internal Report, Institutfur Geodiisie und Photogrammetrie, ETH-ZUrich.

347

E 0 0 If) ,

:!! ~ ~

"Z .2", - .. . ", [;-

:~ E

...... ... -i.e = ... -'-~'" ..... ~i E-t:! !! .. ... .. "", " ..

E i! .. E a: ~

0

~

Figure 3: Combination 01 . Laser·Tracker results (Dots) and . GPS·Solution (Triangle). Overlay with cadastrial map.

348

SESSION 5a

POSITIONING AND NAVIGATION APPLICATIONS

CHAIRMAN L. CAMBERLEIN

SAGEM PARIS, FRANCE

EXPERIENCE WITH THE ULISS-30 INERTIAL SURVEY SYSTEM FOR LOCAL GEODETIC AND CADASTRAL NETWORK CONTROL

Rene Forsberg Geodetic-Seismic Division Kort- og Matrikelstyrelsen

Rentemestervej 8 DK-2400 Copenhagen NV

Denmark

The capability of the recently developed SAGEM ULISS-30 inertial survey system for performing local surveys at high accuracies have been tested in a field campaign carried out November 1989 on the island of Fyn. Denmark. in cooperation with the Swedish National Land Survey. In the test a number of lines between existing national geodetic control points were surveyed. along with points in the less reliably determined cadastral network. forming an irregular network pattern of 10-15 km extent. The survey involved frequent offset measurements (up to 50-100 m) with an ISS-integrated total station. The profile geometries were not particularly suited for inertial surveys. with narrow and rather winding roads. necessitating frequent vehicle turns. In addition to the pure inertial surveys a kinematic GPS/inertial test was also carried out. using a pair of Ashtech L-XII receivers. The inertial survey results. analyzed with a smoothing algoritm utilizing common points on forward/backward runs. indicate that 5-cm accuracies are possible on reasonably straight profiles of 5 km length. corresponding to a 10 ppm "best-case" accuracy for double-run traverses. On longer. more winding traverses error levels of 10-20 cm are typical. To handle the inertial data optimally. proper network adjustments are required. A discussion of suitable adjustment models of both conventional and collocation type is included in the paper.

INTRODUCTION

In this paper the preliminary results of an inertial survey project in a 10 x 15 km area on the danish island of Fyn is described, along with some more general comments on the adjustment of Kalman-filtered inertial survey data. The results are an update to earlier reported results (Forsberg, 1990), but still do not represent the final results, as the required inertial network adjustment software has not yet been completed.

The inertial project was a cooperation between Kort- og Matrikelstyrelsen (National Survey and Cadastre), Denmark, and the National Land Survey of Sweden (LMV). The inertial measurements were done in four working days in November 1989, utilizing LMV's newly acquired ULISS-30 Inertial Survey System (ISS). The purpose of the survey was primarily to assess the current obtainable accuracy of inertial surveying under typical danish conditions, primarily in order to evaluate the potential use of inertial survey systems in the ongoing revision and partial resurvey of the danish cadastral network ("moderniseringsplanen"). Another purpose of the survey was to acquire inertial data in a well-controlled network, in order to evaluate systematic system errors and provide improved error models, and to provide joint inertial and kinematic GPS test data as a base for further research into hybrid GPS/inertial systems.

The danish national geodetic network, a homogenous high-accuracy network consisting of some 35000 points, is generally well-suited as a base for inertial surveys. Many of

351

the points are located close to roads, necessitating only short (typically 10-50 m) offset measurements from the ISS vehicle. The ULISS-30 ISS with its build-in total station is well suited for these offset measurements, and allows the transfer of .coordinates from these geodetic points to the ISS without appreciable loss of accuracy.

Opposed to the geodetic network, the cadastral network of some 320000 points is much less accurate, inhomogenous, and frequently not properly tied to the geodetic system. Under the ongoing cadastral revision scheme - necessitated primarily by requirements from modern cartography and geographical information systems - the network is to be adjusted into a consistent, unified system with the national geodetic network. To aid in this process new measurements are required, and these measurements have so far nearly exclusively been obtained from photogrammetry through aerotriangulation. The accuracy of the photogrammetrically determined points is typically quoted to be from 5 cm (in urban areas) to 8 cm (rural areas). Compared to the tedious field preparation, processing, and administration involved in the photogrammetric methods, inertial methods could be very attractive, especially in rural areas with relatively few points: coordinates of the wanted points could be determined in near real time, the navigation capability of the ISS could be used to locate hard-to-find points, and in addition also heights are determined.

Despite the high costs involved, inertial surveying is thus an obvious candidate for solving the cadastral control problem, providing a sufficient accuracy is required. Considered earlier published accuracy claims for ISS, the use of high-accuracy platform systems can apparently meet the requirements of coordinate interpolation accuracies of 5-10 cm, for a review of inertial surveying principles and results see e.g. Forsberg (1988).

The used newly developed ULISS-30 ISS is a local-level platform system, of the same basic type as e.g. the Litton ISS systems. The system employs state-of-the-art technology, using two 2-DF dry-tuned gyros for platform stabilization, and three gas damped accelerometers for acceleration measurements. The initial alignment time for establishing the local-level (East, North, Up)-frame is some 20-30 min, with a typically 3 hr survey period following the alignment. A 12-state Kalman filter is used for real-time error control and platform stabilization. A more detailed desciption of the ULISS-30 system and the swedish experience with the system is given by Becker and Lidberg (1990).

THE INERTIAL PROJECT ON FYN

The inertial measurements on Fyn were planned primarily to take place along a 21 km L-shaped traverse, involving existing geodetic points spaced some 2 km apart. In addition a number of more or less irregular profiles were to be run in a smaller central 4 x 5 km area, in order to try to survey all cadastral points which had previously been measured by aerotringulation and/or readjusted in the cadastral revision scheme. The measurement program was expanded on the last day by additional "geodetic" traverses, resulting in a better interconnected inertial network. Figure 1 shows an outline of the surveyed inertial network, with the main L-shaped profile starting in south at point "A" (KMS geodetic point 41-18-805), turning west at "e" (41-16-802), ending in point "E" (34-02-808/34-02-7075). Table I outlines the performed ISS runs, with runs of "cadastral" type including the less accurate cadastral control as comparison "truth" coordinates, with other runs including only the more accurate geodetic points. Precise levelled heights were only available for a part of the geodetic points, with the heights of all points of the L-shaped traverse determined by first-order levelling.

352

Fli. 1. Inertial survey network on Fyn (SS~O'N, l0040'E). The L-shaped main traverse (ollows junction points ABCDE. Square UTM net on the map are with 1 km spaeing.

353

Table 1. Survey runs in the Fyn ISS project.

Survey Profile No of Date/ Survey time length measu- No of Profile run geometry (hr) (km) rements stations outline

10 A L-shaped 2.7 21 44 25 ACECA 10 B Cadastral line 1.4 5 31 18 BGHGB 10 C L-shaped GPS 2.0 18 24 13 ACDCA 10 D Straight GPS 1.0 6 13 6 AFA 11A Cadastr. loops 2.1 6 41 21 JKBJBKJ 13A Cad. irregular 2.8 13 48 27 KCLCKHJ 13B do 2.9 15 48 29 CLMBGMGL 14 A Geodetic lines 3.1 31 49 30 AFNFANE 14 B L-shaped 3.4 29 46 28 DNACECA

As seen from Table 1 and Fig. I, some runs were quite irregular in shape, a consequence in part of our wish to simulate a survey of all possible cadastral points in a given area in a minimum amount of time. The traverse times given in Table 1 are the actual survey times excluding system alignments at the initial point. This alignment was on Nov 13 and 14 augmented by a heading update, yielding significantly better real-time system performance. The profiles were run both forward and reverse, even for the somewhat irregular cadastral lines. Average productivity was 16 points/hour, with a maximum of 96 measurements in one day. This illustrates the extremely high productivity of ISS surveys, translating into overall costs of 350-400 DKK per point for single-run ISS profiles.

To aid in the evaluation of the ULISS-30 offset capability, a number of temporary offset points were established on the road surface in the L-shaped profile. These offset points were surveyed conventionally in the days prior to the inertial survey itself. The offset points were also required in order to perform a GPS/inertial integration test, which had to take place at night due to a limited GPS window. The accuracy of the offset points is comparable to the geodetic points, that is 1-2 cm in a relative sense. The use of the offset points points permits an evaluation of the errors in the coordinate transfer by the integrated total station in the ULISS-30. It appears that loss of accuracy due to use of the total station only occurs at the longest offsets, on shorter ranges the use of the total station might even be advantagous, as the ISS reference point on the vehicle (a protracter mounted behind the driver's door) may only be centered/offset measured to point with a limited accuracy.

On the narrow and somewhat winding rural roads of much of the area, the use of only one vehicle reference point (VRP) quickly proved to be a problem. Our use of a big vehicle (a Dodge van) neccessitated frequent 1800 vehicle turns to reach points at the "wrong" side of the road. Such vehicle turns are obviously not good for the ISS performance, best results are expected when the system is travelling on nearly straight lines. To circumvent the problem of the narrow roads, a second VRP point was improvised on the bumper of the ISS vehicle. This new VRP was located at the opposite side of the regular reference point near the driver's door. Unfortunately the use of two reference points caused caused some errors in the beginning (wrong VRP point number entered by the operator), so some observations on the "11 A" run have not yet been included in the processing.

354

The GPS/inertial integration experiment was carried out on the rainy night of Nov 10-II (runs 10C and 100, cf. Table I). A set of one stationary and one mobile Ashtech L-XII GPS receiver was used, with the mobile GPS antenna mounted on a survey pole at the rear of the ISS van. The GPS antenna was designated as VRP #2 in the ISS system for the test, and the offsets to the IMU centre measured by classical means. A hardware synchronization of the GPS unit and the ULISS-30 unit was not attempted. Instead satellite GPS time was used to "mark" the inertial data, which were collected at a rate of 6 Hz, providing the capability for synchronization of the ISS and GPS time at some fraction of a second. In the GPS/Inertial experiment ZUPTs were used as usual, and a longer break was introduced at the shorter profile end point to allow for changes in the GPS satellite constellation. Unfortunately the outcome of the integration test is not yet known. The kinematic GPS data alone are not processable due to a large number of loss of locks (and thus cycle slips) to the GPS satellites. This loss of lock was apparent allready in the field, and caused by numerous obstructions such as trees and buildings.

ADJUSTMENT OF THE INERTIAL DATA ON A PROFILE BASIS

Due to the somewhat unknown nature of the ULISS-30 real-time Kalman filter and the onboard post-mission smoothing smoothing program, we decided after the first two runs to perform the remaining inertial surveys without real-time coordinate updates on the profiles. The data have therefore up to now only been analyzed with a "customized" quite simple profile adjustment program ("ineradj"), originally developed to adjust Ferranti inertial profile data (Forsberg, 1986).

The adjustment program reads lists of Kalman filtered "raw" coordinates output by the system, and constrains these coordinates on selected known points, utilizing repeat observations (such as common points on the forward and backward runs of a traverse) as further constraints. The adjustment model is on purpose selected with rather few parameters, in order to yield robust estimates. The ULISS-30 real-time Kalman filter provides a good error control of platform level errors (platform relevelling is done at every ZUPT), so the dominant unresolved error sources in the horizontal coordinates will be related to azimuth errors and scale factors. The following 9-parameter adjustment model have been used for all runs, independent of the number of fixed points

SrN · ,1

SrE · ,1

1

= aNj s + aEj Q + . E (aEraEj_1)tj P + a + b tj J=I

i = aE. s - aN. Q - E (aN.-aN. l)t. P + c + d t.

1 1. J J-J 1 J=I

Sru ' = e + ft. ,1 1

(1)

where Sr is the coordinate error, aN and aE northing and easting coordinate differences relative to the initial point, t time, s a common scale factor, Q initial azimuth misalignment, P azimuth gyro drift, and a-f linear parameters absorbing offsets and parts of the random walk errors. Subscript Hi" is the sequential number of the ZUPT station, the sum term for P being an approximation of the integral yielding the position errors due to a constant azimuth gyro drift, see e.g. Britting (1971).

355

The adjustment model is obviously too simplistic, especially on longer, more winding traverses, where utilization of many fixed points in the adjustment would yield good determination of additional parameters, such as e.g. separate north- and east accelerometer scale factors (the common scale factor s serves more as a "nuisance" parameter, absorping errors "orthogonal" to the azimuth error). However, with more complex models the robustness of the estimation is diminished, yielding larger susceptibility to gross errors. Instead of expanding the adjustment model, a two-stage procedure is adopted for better results: a linear fit in time is applied to interpolate adjustment residuals between fixed stations, and the remaining "new" stations are corrected by the interpolated values, and the final coordinate values finally obtained after averaging.

Table 2 shows the results of the above adjustment for each of the survey runs of Table 1. In addition the table also show "subrun" results of the last two days, where the irregular traverses have been cut down into more regular straight or L-shaped traveres. The numbers shown to the right in the table are the r.m.s. difference between adjusted inertial coordinates and the known independent control, first after the simple 9-parameter adjustment, then after the linear residual fit in time, and finally for the averaged, double-run coordinates. The main L-shaped traverse forward/backward runs are lOA, 10C, and 14BA, with no total station excenter measurements done on the night run 10C. The selected subrun profiles are double run on the traverses with junction points KCL (#13AA), LGB (#13BA), AFN (#14AA), AF (#14AB), AND (#14AC), and ACE (#14BA), see Figure 1.

Table 2. Results of adjustment with profile software "ineradj"

One-way No of R.m.s. position accuracy ( m) Run/ length fixed After adjustm. After linear fit Averaged Subset (km) points N E U N E U N E U

10 A 21 3 15 21 27 14 19 24 15 19 19 10 B 5 4 14 7 - 16 9 - 15 7 -10 C 18 3 9 7 38 8 9 15 4 4 13 10 D 6 2 6 5 - 9 7 - 4 4 -llA 6 3 9 9 - 10 9 - 10 9 -13 A 13 6 17 15 - 17 12 - 19 12 -13 B 15 5 25 21 - 30 18 - 27 15 -14 A 31 5 31 17 21 12 11 12 12 11 11 14 B 29 6 11 12 20 11 15 18 12 12 21

13 AA 7 3 10 9 - 11 12 - 11 6 -13 BA 9 3 14 16 - 12 18 - 11 18 -

14 AA 16 3 11 17 - 7 10 - 6 8 -14 AB 6 2 10 10 - 11 14 - 11 12 -14 AC 23 3 15 17 - 18 13 - 13 12 -14 BA 21 3 10 11 16 9 15 9 7 12 7

356

Table 3. Results 01 azimuth/scalelactor parameters.

Run Scale (ppm) Az misalignment Az drift rate (deg/hr)

10 A 185 -97" -0.034 10 B 167 95" 0.023 10 C 124 68" 0.069 10 D 86 (582") 0.079 llA 189 9" -0.030 13 A 154 -12" -0.043 13 B 138 -10" 0.005 14 A 148 -47" -0.005 14 B 101 38" 0.013

Fig. 2, 3, and 4 shows the "raw" kalman-filtered coordinates, the adjusted residuals as a function of time, and the final averaged coordinates as a function of distance for an earlier adjustment of the L-shaped traverse (run 14 BA). The raw coordinates show very clear systematic trends due primarily to azimuth misalignments, yielding real-time coordinate errors up to 4 m (no real-time coordinate updates were done). After adjustment and linear fit (Fig. 3) errors are at 10 cm level, with some E outliers indicating unresolved systematic errors (or offset measurement errors). The overall error of around 10 em with 10 km between fixed points corresponds to an ISS coordinate interpolation capability of 10 ppm, a surprisingly good result, several times better than the manufacturer's actual specifications for the system.

In judging the overall performance of the system from the results of Table 2, the preliminary nature of the adjustment software should be taken into account, as well as the noise in the cadastral comparison coordinates, and possible excentricity errors. Judging from the subruns and the more regular runs on geodetic points it appears that the ISS is capable of giving 10 em accuracies provided the survey trajectory is not too irregular. It also appears that results are not consistent. Very good results have e.g. been obtained on the straight line A-F (run 10 D, 4 em r.m.s.), on the L-shaped traverse during the GPS test (run 10 C, also only 4 cm r.m.s. in the horizontal coordinates), and on the curved run A-F-N (run 14 AA, 8 em r.m.s. over 16 km). The cadastral runs appear to give relative lower accuracies than some of the runs only involving geodetic points. Apart from suspecting the accuracy of parts of the cadastral network control coordinates, the reason for the relatively poorer results may also be related to the slower average speed of the ISS survey, necessitated by the numerous points to be measured.

Another source of the difference in system performance is probably correlated to the quality of the initial alignment, and the use of a real-time heading update after the alignment. Table 3 shows the adjustment results for initial azimuth misalignment, azimuth drift, and scale factor. It is striking that the best performance of the system was obtained on days with good initial heading, which gives the built-in real-time Kalman filter the best conditions. The general alignment accuracy appears to be better than l', with the large a-value found in run 10D due to omission of alignment prior to the short run (continuing on the unaltered alignment of the previous run IOC). The scale factor s shows a surprisingly constant one-sided value around some 1 SO ppm. Preliminary network adjustments have yielded similar results for both the east and north scale factors separately. It is therefore probable that a systematic scale factor is inherent to all runs on Fyn. The source of this scale factor could of course be the

357

~ '=! z w a:: w t.. .... ~ o N

W .... a: z ~ o 0 0:: o o u ~

N I

"Raw" Kalman-filtered inertial coordinate differences

, ,

l.EGEND: -- NORTH

EAST UP

, \ ,

, , , .-------------\ -----

\ .... /

I I

I

;­/

:;:...-~- -- ... ---- _ ... ---

I

I

~ ~--------~F~OrR~W~A~R~D~--_r~?f------,_~B~A~C~K~W~A~R~D=r--------_. 1

0.0

:c "! 0

w u z W 0:: W .... .... ~ 0 0

w ..... ex: z 0 0:: "! 0 0 0 U I

~ 0

1

0.0

CJ

r "! 0

w u z w a:: w .... , .... 0

0 0

~ a: z 0 a:: "! 0 0 0 U I

~ 0

1

0

0.5 1.0 1.5 2.0

TIME !HRI

Coordinate differences after adjustment and linear fit

LEGEND: -- NORTH

EAST .--- UP

0.5

l.tGtNC, -- NORTH

EAST - --- UP

-

5

1 .0 1·5

TIME {HR I

Final averaged coordinate differences

, , , , I 'J

I

I

10

DISTANCE (KMI

358

2.0

15

2.5

2.5

20

Fig. 2.

Fig. 3.

Fig. 4.

accelerometers and their software-stored calibration factors, but the magnitude and sign of the error could also point to a trivial scale error in the definition of the real-time system UTM transformations used during the project (all coordinates were entered and output in UTM zone 32).

SOME NOTES ON NETWORK AND COLLOCATION-TYPE ADJUSTMENTS

The relatively simple adjustment procedure of the last section must of course be expanded into a true network adjustment for optimum results. Quantities to be solved for in such an adjustment would be physical quantities which may not be estimated (or only estimated very weakly) by the real-time Kalman filter. Such quantities include

- accelerometer scale factors sN' sE and sU' - accelerometer misalignments SNE' SEN' SNU' SEU' SUN' SUE' - initial azimuth error Q,

- azimuth gyro drift {J and possibly gyro torquer errors, - eccentricity elements for the VRP's relative to the IMU centre,

and could also be expanded to include terms relating to the gravity field changes. "Nuisance" parameters, like the time drifts of the simple profile model (I), and a common scale factor s, are usually also required to get residuals free of systematic errors. The azimuth and nuisance parameters should be estimated for every alignment (traverse), whereas accelerometer scale factors and misalignments, and offset vectors, may be assumed to be constant for the entire network, provided it was surveyed in a limited time period, like on Fyn. Network adjustments of this type has been implemented by numerous investigators, e.g. Schwarz et al (1984), Cross and Harrison (1986), and Hausch (1988).

The simultanous solution in a least squares adjustment of all of the above mentioned parameters is generally not possible. If e.g. a "nuisance" scale term s is solved for on every independent traverse, the simultanous solution for sN and sE will yield a singularity, requiring an additional constraint

(2)

for obtaining the common "physical" scale factors, and a similar constraint in azimuth would be required for solving for both SNE and SEN. Since height variations in the network are typically much smaller than the variations in the horizontal coordinates, a typical network parameter vector I! to be solved for could thus look like

(3)

where the first group of unknowns is for the entire network, and the subsequent groups for individual traverses, with a, b, c .. parameters like in (I). The complete set of unknowns is thus

(4)

with r the coordinates (N,E,U). The observation equations for the k'th station or ZUPT stop may be written in form

(5a)

359

(5b)

for an observation in a new (unknown) or fixed station, respectively, where II is the Kalman filtered output coordinates of the system (assumed to be only updated by zero velocity updates). Writing the complete set of observation equations as

(6)

the normal equations in the least squares solution

(7)

will have a simple block matrix structure

(8)

with the 3m x 3m submatrix 0 (for the unknown coordinates) being diagonal. This makes the least-squares solution of (7) by Cholesky reduction very efficient and fast. The Fyn data is currently being analyzed by such software. Preliminary results using 5 fixed points at the edges of the area yields results around

SN R1 137 ppm, sE R1 145 ppm, 9NE R1 30 J'rad

Coordinate results are 10 to 20 cm r.m.s., but some errors in the observation files have not yet been corrected, so the real accuracy of the system is better.

The collocation approach

On long inertial profiles, irregular drifts in the azimuth gyro and accelerometers appears to be a significant error source. Such drifts may be solved for using higher order terms, e.g. second second derivatives of azimuth error, linear drifts in accelerometer biases etc. To avoid solving for too many parameters unknowns, an alternative approach based on least-squares collocation may be used. The idea behind this approach is the following: Consider e.g. a double run profile between two known points PI and PN, with observations

fl -> P 2 -> P 3 -> .. -> fN -> .. -> P 3 -> P 2 -> fl

It is intuitively obvious that the utilization of coordinate differences I forward _ Ibackward

will provide a sampling of coordinate errors with increasing time intervals, and thus in principle allow recovery of a "coordinate drift" function, provided it is of a sufficiently long-wavelength nature. This should then also allow recovery of the underlying "system" errors. Such errors could for the horizontal components be represented by two functions of time, or states,

- an irregular azimuth error Q'(t), and - an along-track scale error s'(t)

This would yield observation equations corresponding to (Sa) of the form

360

fev""'(I) + v ... ·(t)) dl

to

fe-v ..... (I) + vB"(I» dl

to

(9)

where the parameter vector n only contains a few parameters. The velocity integrals may be approximated by sums of form

tk

J (v",,'(I) + v ... ·(I»dl .. k E «E.-E· 1)a.' + (N.-N. l)s.')

• JJ-J JJ-J J=1

(10)

to

This yield a general set of observation equations of form

Y=n3.+R~ (11)

which are of the general collocation type. Assuming a covariance matrix ~ for the signal z, and error variance matrix Eyy for the observations, the least-squares solution is

x = (BTl"' -lB\-l BTl"' -ly - - ~R ~ - ~R ' c = E + R"" RT -R -yy -~

As covariance functions for a' and s' e.g. first order Markov models of form

cov(a'(t1),a'(t2» = CO,lexP(-ltll t2- t1 1) cov(s'(t1),s'(t2» = Co,2exp( -1t2 1 t2-t1 I )

(12)

(13)

may be used, assuming a sufficiently long correlation time It is used, e.g. of magnitude an hour or longer.

The proposed collocation adjustment is equivalent to methods used in inertial gravimetry with good results (Forsberg, 1986). The method has the advantage of being able to handle both long and short traverses separately or in networks on a consistent basis. The drawback is the need for the solution of large set of linear equations, with as many equations as there are observations. The computational is, however, not as large as it seems, as the covariance matrix QR will be block diagonal. The covariances between a' and s', and between subsequent traverses, will be zero, thus allowing the use of sparse matrix techniques.

The collocation scheme has not yet been tested on inertial data. Tests on the Fyn data with high-order drift terms (second derivative of the azimuth error, and/or polynomial time drift terms) in the usual adjustment procedure have indicated that the "irregular" gyro drift and accelerometer biases might not be big enough to warrant the application of the collocation approach. Although e.g. a second-order azimuth azimuth a'(t) = a + fJt + 'Yl2 improves results in some cases (e.g. to 12 and 7 cm r.m.s. in Nand E

361

respectively for run lOA), in other cases results actually deterioate, as the redundancy in the observations become too low.

CONCLUSIONS

Apart from the more general adjustment remarks of the last section, this paper has treated a case story of actual inertial results with a new inertial survey system - the ULISS-30. From the results it appears that the system is capable of providing coordinates under typical danish conditions to the 10 cm-level, provided existing known control spaced at, say, 5 km is utilized, and provided too winding and looping traverses are avoided. There do not appear to be a very significant accuracy gain going from single-run to double-run traverses, but the use of double-run forward/backward traverses are of course preferable to increase the redundancy and guard against spurios errors. From an operational point of view the system has proven itself very efficient and useful, with the possibility of using more than one vehicle reference point, and the availability of an integrated total station, examples of factors improving the speed and ease of inertial survey campaigns.

Acknowledgements. The Fyn ISS project was a teamwork carried out by several colleagues of KMS. Denmark. especially S. Stampe-Villadsen and S. West-Nielsen. who participated both in the project preparation and field survey work. M. Aarrestrup helped in the GPS measurements. and the LMV system operators K. Nielson and M. Lidberg provided a friendly and flexible ISS service. Didier Burtin has implemented most of the new network adjustment software.

REFERENCES

Becker, J.-M. and M. Lidberg (1990): The Swedish Experience with Inertial Survey Systems. Proceedings of the 11 th Nordic Geodetic Commission Meeting, Copenhagen, May 1990, published by Kort- og Matrikelstyrelsen (in print).

Britting, K.R. (1971): Inertial Navigation System Analysis. Wiley Intersciences, New York.

Forsberg, R. (1986): Inertial Geodesy in a Rough Gravity Field. Department of Surveying Engineering, University of Calgary, report 30009.

Cross, P.A. and P. Harrison (1986): First Results from FILS3 Trials on the Edinburgh Inertial Test Network. Proceedings of the 3rd International Symposium on Inertial Technology for Surveying and Geodesy, Banff, September 1985. Publ. no. 60005, Dept. of Surveying Engineering, University of Calgary.

Forsberg, R. (1988): Inertial Surveying Methods. In: O. B. Andersen (Ed.): Modern Techniques in Geodesy and Surveying, Lecture Notes for the Nordic Research Summer School, Ebeltoft, Denmark, september 1988, Geodretisk-Seismiske Skrifter, 4. rk, vol. 1, pp 169-214, published by Kort- og Matrikelstyrelsen.

Forsberg, R. (1990): Inertial Surveys on Fyn - Some Preliminary Results. Proceedings of the 11 th Nordic Geodetic Commission Meeting, Copenhagen, May 1990, published by Kort- og Matrikelstyrelsen (in print).

Hausch, W. (1988): A refined adjustment model for inertial surveying. Manuscripta Geodaetica, ll, pp. 13-28.

Schwarz, K.P., D.A.G. Arden and J.J.H. English (1984): Comparison of Adjustment and Smoothing Methods for Inertial Networks. Dept. of Surveying Engineering, University of Calgary, report 30006.

362

KIS 1990 Banff, ALBERTA, Canada, September 10-13

ULISS 30: RESULTS FROM INTERPOLATION AND EXTRAPOLATION MEASUREMENTS.

J-M. Becker & M. Lidberg National Land Survey of Sweden

INTRODUCTION

Since June 1989 the National Land Survey of Sweden owns an Inertial Surveying System (ISS), ULISS 30 from SAGEM (France). This system is mounted in a custom-made Dodge van terrain vehicle with an openable roof. The complete system also includes a total station (Geotronics 440 LR), a PC (Compaq 386), a printer, and a navigation unit.

From the date of delivery this equipment has under­gone various tests, first to check that it fulfilled the technical specifications of the manufacturer, and second to familiarize our operators with the equipment. After this test period the ULISS 30 has been used in several pilot projects specifically designed to prove the ability of the ISS to carry out topographic sur­veys, either on its own or in combination with other geodetic techniques (classical or GPS).

Finally we made som tests using ISS in an "extra­polation mode", i.e. for the determination of points outside a known baseline. This can be useful for engi­neering purposes -such as underground constructions like the channeltunnel between UK and France.

The results obtained during our different tests and projects are very promising and are presented in this paper.

BASELINE TESTS

Near Gavle the NLS disposes of several calibration baselines. These baselines are straight and in a rela­tively flat terrain. They are situated along roads so that every survey point can be directly surveyed with the ISS car.

363

The "Long Baseline" is L-shaped with 2 branches, 15 km in an E-W direction and 58 km in N-S. It has 17 monumented points at intervals between 2 and 10 km.

The "Short Baseline" is 2.3 km, and is perfectly straight with high point density (every 10, 50 and 100m) •

A new baseline near Hofors-Fa1un has been established specially for altimetry tests. It is 20 km long and has large differences in elevation (about 175 m). The base­line is part of a line of the first order levelling network. It has vertical control points every km, and 10 of these points have horizontal (GPS) coordinates and gravity values.

Several series of measurements along all baselines have been accomplished in different configurations during the last year. Some results are shown below.

Table 1. Root Mean Square (RMS) and maximum deviation between measured and "known" positions for surveys with ULISS 30 along the baselines.

Length Fixed points No of at double

(km) (km) runs

Check points

RMS dev. (cm) max. dev. ( cm )

N E h -------------------------------------------------------

2.3 0-----2.3 2 2 x 16 2 4 4 3 8 9

5 0-------5 6 6 x 4 3 4 5 7 13 11

10 0------10 3 3 x 9 3 10 11 8 18 20

12.5 0----12.5 3 3 x 10 8 5 7 15 13 13

22.5 0----22.5 3 3 x 19 21 16 14 49 33 28

73 0--15--73 1 1 x 15 35 38 13 112 103 32

10"HF" 0------10 3 3 x 5 4 6 6 resp 11 11 15 14

-------------------------------------------------------

364

PILOT PROJECTS

ISS technology has never been used for civil purposes in Sweden before. Therefore we lacked experience of how to use it for surveying needs. To develop ISS working methods and to investigate applications, main pilot projects were designed and carried out in three speci­fic fields:

* Geodetic control networks. * Road surveys. * Cable surveys.

Geodetic control networks

In several types of cadastral and mapping surveys, low accuracy (an RMS error of 10 cm) is fully acceptable for the users needs. The NLS used the ISS technique for establishment of local networks, and densification and connection of existing networks.

ISS was most efficient in forest areas with broken terrain and with small and very curved roads through dense bush vegetation. In this kind of area the classi­cal surveying methods demand construction of many towers, intensive clearing of sight1ines, and a very dense number of setups. The use of GPS is nearly impos­sible, or very complicated because of the dense and high tree cover. Therefore, ISS is more favourable for this kind of survey. For the control of the accuracy of the ISS technique, several surveys were measured twice, and some distances between ISS-surveyed points were remeasured with electronic total stations. These remea­surements show that the errors are some cm for distan­ces up to 300 m, which indicates that the internal accuracy between ISS-surveyed pOints is good.

365

Table 2. Typical results from pilot projects. RMS calculated from deviations between inertial and classical measurements.

Length Fixed Comparison RMS dev. (cm) (km) points points N E h

--------------------------------------------------3.6 2 7 5 2 8 4.7 2 10 7 5 9 6.6 2 11 10 7 10

10.5 2 13 8 11 6 10.9 2 19 8 6 10

Road surveys

The measurements of road networks in the northern part of Sweden include about 3000 km in a mountaineous area where classical photogrammetric methods are too expen­sive in comparison with ISS-surveys. The accuracy required for the absolute position was 2 m (RMS error), and was easily obtained using known connection points at every 20-30 km. A daily production average of 60 km double run was achieved.

Cable surveys

The cable surveys were carried out for the Swedish Telecommunication Company. The purpose was positioning and mapping of fibre-optic cables, which cannot be detected with any electric or magnetic equipment.

The types of surveys executed during these pilot projects were of very different nature: from a simple survey of some isolated pOints along a traverse to a complete survey of all details about the road and other existing objects. In other terms, from one surveyed point every 200 meters to one every 10 meters. The required accuracy was about 1 dm (RMS error). All measurements had to be related to the national datum.

The results obtained were satisfying and show that ISS is competitive in this field.

3~

"EXTRAPOLATION" TESTS

At different occasions during our field work, the need to make "extrapolation measurements" has appeared, i.e to determine coordinates for points or lines outside the recommended area (less than 1/3 of the distance between two known pOints). This has been especially common in connection with the road surveys.

The second time this problem appeared, was in connec­tion with the underground measurements for the con­struction of the channel "Transmanche-Link" between France and UK. Prof. Groten (FRG) and Prof. Schelling (AU) were involved in this project and interested to use ISS for control of the results obtained with clas­sical surveying methods, which were affected by refrac­tion errors. The problem was to get the position within +/- 40 cm for the junction point in the middle of the tunnel, where both construction companies had to meet each other in the end of 1990.

The problem to solve consisted in "extrapolating" the coordinates from a short known baseline of 8 km (UK side) to about 18 km, and from a 3,5 km baseline (French side) to 16 km.

When the NLS was asked about this, we had no experi­ence and it was impossible to give a positive answer, especially for such a high request concerning accuracy. The only available results from similar "extrapola­tions" were from tests made by van den Herrewegen (B) in CERN with his FILS II. His results were not satis­fying and not accurate enough to solve the problem. Therefore we decided to make some tests on our own baselines: first extrapolation on "straight" lines, and second on "curved" lines.

The measurements were made in the same way as measu­rements for interpolation. The run starts with the fixed part, it continues with the extrapolation part, and completes with remeasurement of the extrapolation and the fixed part. In the post-processing, we used an error model wich solves for: one common scale factor, initial misalignment, and azimut drift.

It is surprising for us that we got such good res­ults from these tests, especially that the results from curved lines are as good as the resau1ts from straight lines (see fig. 1, and fig. 2).

367

023

Coordinate fixed part "extrapolation" errors in: 0.2.---------rr-----------------,

north o.o"'~=-_~~~~;;t:::;~ (m)

east (m)

height (m)

-t.2~----........ ----.-------.---------j

0.2..------rr-----------------,

-to I -.-.---•.. - ... -- - .- .. --.... --.... _ .... -.. .-.......... - ............ ---........ ----

·M'-----~----.,.-----.,..-.;'"T------!

0.2..------..-----------------,

-t.4 '-----..-Il------r------r--4------! 0.0 1.0 10.0 n.o 20.0 ( km )

*ztIl + IVI + Ull

Fig. 1. Extrapolation on straight lines (four different days).

368

o 1 2 3 4 5 (k .. )

Coordinate errors in:

north (m)

east (m)

fixed part "extrapolation"

0.2~--~---------"'"

0.0

'" ..... ;: -0.2 ........................... ...................... ~ ................... .. .................... ... ....... .

,-~

""--0.4 ............. ...................................... .................... .... ", .............................. ..

\ -0,6+---~---T-----r---~

0.2~--""""!!""---------....,

-0,2+----+-----..---~--~ 0.0 5.0 10.0 15.0 20.0. (km)

-& 20/3 * 2213 + 11/4

Fig. 2. Extrapolation on curved lines (three different days).

369

CONCLUSIONS

The experience gained so far during our different works with the Inertial Surveying technique, and especially with the ULISS 30 from SAGEM, allows us to conclude as follows:

1. The ISS technique has proved to be flexible and efficient if appropriate methodology and software are used for each different kind of survey.

2. The use of an electronic total station "harmonized" with the ISS unit facilitates highly the field opera­tions.

3. Prereconnaissance and good planning of the ISS mis­sions are time saving and necessary for high accuracy.

4. The ISS field operators have to be trained and familiarized with both ISS and classical survey tech­niques.

5. If double measurements are made and the measured lines are relatively straight it is possible to achieve an RMS error of:

* 5 cm for distances less than 2km. * 10 cm for distances less than 10km.

6. The internal accuracy betweeen ISS surveyed points is normally good: of the order of some cm for distances up to some hundred meters.

7. The accuracy of "straight extrapolation" measure­ments is high: better than 50 cm (RMS error), if the known baseline is at least 5 km, and if the extrapola­ted distance is less than 10 km.

Finally we found that there is much to be done to improve the software. With better error models for postprocessing of the Kalman-filtered inertial data, better results can be achieved.

It is our hope that the efforts made to hybridize ISS and GPS will continue and be successful, so that an accuracy of 1 cm (RMS error) will be achieved.

370

REFERENCES

Asenjo, E. and Sj6berg, L. (1989). Tr6ghets­positionering, TRITA GEOD 1017. Geodesi institutionen KTH.

Becker, J-M. (1988). Tr6ghetspositioneringstekniken, LMV-Rapport 1988:10, ISSN 0280-5731.

Becker, J-M. and Lidberg, M. (1990). The Swedish Experience with Inertial Survey Systems, NKG, 11th General Meeting ,Copenhagen.

Becker, J-M. (1990). Les experiences suedoises avec la technique inertielle pour les besoins topographiques du National Land Survey of Sweden, FIG:XIX Congress, Helsinki, Finland, 508. 3.

Becker, J-M. (1990). The swedish experience with the ISS Uliss 30 ,results from tests and pilot projects, LMV-Rapport 1990:8, ISSN 0280-5731

Caspary, W. F. (1990). Current state of Inertial Technology for Geodetic Applications, FIG:XIX Congress, Helsinki, Finland, 511.3.

Forsberg, R. (1988). Inertial Surveying Methods, Nordiska Forskarkurser,Ebeltoft,Danmark.

Forsberg, R. (1990). Inertial surveys on Fyn -some preliminary results, NKG, 11th General Meeting, Copenhagen.

Herrenwegen, M. Van den, and Vancraenenbroeck, J. (1989). Utilisation d'un systeme inertiel pour la topographie souterraine, Resultats de quelques essais faits dans la gal erie du LEP au CERN. Revue de l'AFT: "XYZ" no:40 pages 31-32.

Porte, T. R. and Martell, H. and Schwarz, K. P and Adams, J. R. (l990). The inertial pipeline pic', FIG:XIX Congress, Helsinki, Finland, 512.3

Rueger, J-M. (1984). Evaluation of an Inertial Surveing system, The Australian Surveyor ,vol 32.2

Sagem, (1987). ULISS 30 Inertial Land Navigation and Survey System. DA 321187.

Schwarz, K. P. (1985). Inertial Adjustment Models, an unified approach to post-mission processing of inertial data, Bulletin Geodesique Vol. 59.1.

Schwarz, K. P. and Knickmeyer, E. H. and Martell, H. (1990). The use of strapdown technology in surveying, CISM Journal ACSGC Vol 44:1.

Schwarz, K. P. and Lapucha, D. and Cannon, M. E and Martell, H. (1990). The use of GPSjINS in a highway inventory System ,FIG:XIX Congress, Helsinki, Finland, 508.4.

371

GPSIINS TRAJECTORY DETERMINATION FOR HIGHWAY SURVEYING

INTRODUCTION.

Dariusz Lapucha Dept. of Surveying Engineering

University of Calgary Calgary, Alberta, Canada

The conventional way of surveying is the establishment of the control network and then surveying the densifying points. This has a major drawback in a number of applications - its discrete feature. The determined positions relate uniquely to the measured points while positions outside these points have to be interpolated using certain terrain representation. In some applications, the exact determination of geometrical characteristics is crucial.This includes road and railroad profiling, pipeline profiling, determining the verticality of mining shafts. Applying the conventional methods for such tasks is very tedious and in some cases almost impossible. It was the reason while geodesists were dreaming long time about some kind of tool which would enable them near­continous positioning with sufficient level of accuracy.

With the advent of new high-technology systems like Global Positioning System (GPS) and Inertial Navigation Systems (INS) these dreams become true. The measuring unit placed on the vehicle driving particular profile could directly determine its trajectory and from that induce profile of interest.Both methods are faster than conventional ones, do not require visibility between points and are weather independent. Since each of these methods has nevertheless its own shortcomings, the integration of both systems gives best of all - eliminating the individual drawbacks while maintaining its advantages, resulting in a more accurate and more reliable system.

The purpose of this paper is to study the application of GPS/INS integrated system to highway surveying. The potential benefits are enormous: increased cost-effectiveness, near-continous road profile and surveyors safety. The GPS/INS survey could be done in any time ( once all GPS satellites will be deployed) without physical presence of surveyors on the road. Surveyor's safety is already a problem during work on the roads with a heavy traffic.

Project objectives

The work described here is part of on-going research on precise positioning of a Mobile Highway Inventory System. This video-logging system being in use at Alberta Transportation serves for the monitoring of highway. The description of the system can be found (Lapucha et aI, 1990). One of the requirements for its work is a knowledge of position, velocity and attitude. This has been done so far by combination of measurements from conventional instruments like: odometer, vertical gyroscope and compass. The low accuracy of the positioning part, listed in first column of Table 1, limited the use of data from MHIS to the qualitative analysis. In response to the need for more accurate positioning requirements, The University of Calgary proposed the use of GPS/INS system. The integrated GPS/INS offers potential improvement in determination of position, velocity, slope and curvature. The frrst part of the project is development of a post mission positioning package that combines GPS receivers in differential mode with an INS. It also includes testing of the existing GPS/INS prototype owned by the University of Calgary.

372

CURRENT FUTURE SURVEY MIUS MIUS SYSTEM

ACCURACY ACCURACY ACCURACY

Position ±lm ovetlkln ± 0.3 m over 1 kin ±5cmovetlkm

Speed <±5km1b ±0.4km1b N/A

Slope ± 1% of grade ± 0.5% of grade ± 0.1 % of grade

Azimuth ±2deg ±0.2deg ± 0.2 deg

Curvature ±2degper40m ± 0.2 deg per 40 m ± 0.2 deg per 40 m

Table 1. Accuracy requirements of highway survey systems.

In the recent paper (Lapucha et al, 1989) it was shown that integrated system could meet the medium accuracy requirements of the MIllS,given in second column of Table1. It was stated that even more stringent accuracy for all-purpose highway surveying system, given in third column., should be possible. This will be further validated in this paper with focussing on positioning component, which is of main interest. The analysis is based on results of new tests with modified GPS/INS prototype.

COMPLIMENTARITY OF GPS AND INS FOR TRAJECTORY DETERMINATION

The differential kinematic GPS is one of the methods that can be used for position and velocity detennination. The position and velocity of the vehicle carrying GPS receiver can be determined accurately, relatively to the receiver placed at stationary point using carrier phase and phase rate data, collected simultaneously at both receivers. Although it is an excellent positioning technique, its major drawback is the occurrence of cycle slips, that lead to degradation of accuracy especially if cycle slips occur on all observed satellites. Another disadvantage of the GPS for this application is relatively low rate of GPS measurements. In addition, the GPS system can work only in "open sky" environment, which is not always the case for typical survey jobs.

The INS is another survey method that can be used for trajectory determination. It is independent on electromagnetic wave propagation and therefore avoids all problems with signal reception which are so crucial for GPS. Of different systems available on the market, the strapdown systems are best suited for survey applications (Schwarz et al,1989). The strapdown system consists of three gyroscopes and three accelerometers which are mounted directly on the rotating platfonn. The gyro outputs - angular velocities are integrated in time to provide orientation changes of the platform relative to its initial orientation. The accelerometer outputs- acceleration rates are double integrated in time to provide position changes with respect to the initial position. The strapdown inertial system offers unique advantage over any surveying system - high-rate (64 Hz) position and attitude output. In addition, in comparison to other inertial systems, it has the advantages in terms of lower cost, higher reliability and high accuracy attitude parameters. Unfortunately the main drawback of INS technique is the systematic error growth in position when operating in anaided inertial mode. It therefore needs frequent updating to achieve the required accuracies. During the land survey, the vehicle stops provide an important update infonnation, namely the so-called zero

373

velocity updates ZVPT. Of course this method is not feasible if the vehicle has to be continuously in motion.

The integration of both systems offers therefore the maintaining the advantages of both systems while at the same time eliminating some of the limitations. In the absence of cycle slips , the excellent positioning accuracy of the differential GPS can be used to provide frequent updates for the inertial systems. On the other hand, inertial sensors provide orientation information for accurate slope and curvature determination and redundant precise position information for cycle slip detection and correction. It thus results in highly reliable system.

MATHEMATICAL MODEL

The problem requires the estimation of quasi-continuous trajectory from GPS and INS measurements. Each of the measurement systems has its error behaviour which is time dependent and has to be modelled in estimation process. The errors of such a system can be described in state space formulation

x (t) = F(x,t) x(t) + G(x,t) u(t) (1)

where x is the state vector, F is the dynamics matrix, G(x,t)u(t) is the forcing function, and the dot above a letter denotes differentiation with respect to time. The state vector x appearing inside the brackets denotes the general nonlinear case. The external measurements of the form

y = Hx + e (2)

where y is the vector of external measurements, H is the design matrix, and e is measurement noise, can be used to update the state vector.This is typical problem for Kalman filtering. Since in the GPS/INS case the process is non-linear, the extended Kalman filter is used. The INS measurements are used to compute the reference trajectory while the information stemming from GPS is used to update this trajectory. The prediction and update equations of the Kalman filter are well known (Gelb et al, 1974) anq will not be repeated here.

The estimation model has been expressed in general terms by equations (1), (2). This model is not unique. Choosing the particular set of H, F, G corresponds to the choice of a coordinate system. For the application discussed here it is advantageous to perform the integration in local­level frame. The advantages of such representation for the strapdown system are (Schwarz, Wei, 1989):

The geographic coordinates (q"A.,h) are obtained directly from the mechanization. The attitude of the body with respect to the local-level frame (roll, pitch, azimuth) is given directly.

Both sets of parameters are of direct interest for the trajectory determination. The strapdown system measures the specific force in the body frame, denoted by b and senses

angular velocities between the body frame and inertial frame, denoted by i. These measurements have to be transformed to the computational frame - local-level, denoted by 1, and then expressed in terms of local-level position, velocity and attitude. This can be expressed in the form (Schwarz, Wei,1988):

= (3a)

374

where

( (N+h) coscp 0 0 J

D = 0 (M+h) 0 o 0 1

(3b)

M and N are the meridian and prime vertical radii of curvature; r is a three-dimensional position vector, v is a three-dimensional velocity vector, g is the gravity vector, R is a rotation matrix giving the rotation between the two frames denoted by the subscripts, and .Q is a skew-symmetric matrix containing the three elements of the angular velocity vector between the frames denoted by the superscripts. The solution of these equations will result in position, velocity, and attitude of the vehicle, where attitude is expressed as vehicle attitude with respect to the geodetic (cp,A,h) system.

The set of differential equations is non-linear. This difficulty is overcome by linearization of the system about reference trajectory, determined by inertial measurements. This results in replacing the original set of parameters r, v, ro with their perturbations Br, Bv, and Bro, where ro is the vector of angular velocities relative to the inertial reference. They are combined together with additional states describing the model system errors to fonn the INS state vector

X INS = { Br, Bv, E, d, b }T (4)

where or and ov are position and velocity errors in cp, A, and h; e are misalignments in north, east, and up; d are residual drifts about the gyro axes; b are residual accelerometer biases.

The GPS kinematic model in state-space representation can be fonnulated in the same way (Schwarz et aI, 1989). The basic observable used here is phase double difference. The model equation is of the fonn (Wells, 1986):

V~<I> = V~p + V~N + V~da + eV.M) (5)

where V ~ means the double differencing with respect to satellites, V, and receivers, ~, and p is the spatial distance between satellite and receiver, N is the carrier phase ambiguity in metric units, e is measurement noise. The advantage of this observable is a dependence on geometric tenn only unless cycle slip occurs. If the phase rate is added, then both kinds of observables enable the estimation of GPS trajectory using simple state-vector representation:

x GPS = { Br, ov}T (6)

IMPLEMENTATION CONSIDERATIONS

In the filter design the concept of a decentralized filter has been applied where separate filters for INS and GPS are employed. The GPS/INS decentralized filter has the advantage of computational efficiency, simplicity and integrity (Wei,Schwarz, 1990). The two filters run parallel (Fig. 1), where the cycle slip corrected GPS position along with covariance matrix is used to update the INS filter every few seconds. The principle of cycle slip detection and correction is given in details in (Lapucha et aI, 1990). It uses the position output of INS at time of GPS observation for computation of predicted double difference and comparison with observed value. If cycle slip occurs then the ambiguity is reset to the new value detennined by last comparison. Between

375

moments of GPS observations the inertial system provides position, velocity and attitude at fast speed shown as upper part of Fig. 1.

Integrate: Position(t) Velocity(t) Attitude (t)

Filtered: INS Position (t)

.... UP_D....,.A_T_E-II--~ Velocity (t) Attitude (t)

GPS UPDATE

Filtered: Position (ftc) Velocity (tiJ

Fig. 1. GPS/INS Integration Flowchart.

These concepts were implemented in the program GPIN, developed by the author at the University of Calgary. The program consists of two principal parts: static part and dynamic part. The static part has two functions: determination of initial baseline and integer phase ambiguities from static GPS data as described in

(Cannon, 1989) determination of initial attitude parameters from INS data

During the dynamic part, two separate fllters are applied to the GPS and INS measurements as discussed before. The INS measurements are integrated in mechanization module at 64 Hz ( L TN-90-1(0) to provide position, velocity and attitude of the system. The algorithmic details of this mechanization are given in (Wong, 1989). When the GPS measurement matches the INS interval (Fig. 2) the integration is performed with interval &t', to compute the predicted position at the time of GPS measurement. Mter GPS position update, the position, velocity and attitude parameters are reset to the new values and integration is continued with interval &t", and then starts again with the usual INS interval cycle. The proper time-tagging of GPS and INS measurements is crucial for the accuracy of the integrated system.

376

GPS

1 time

~ .1.t" ... 1

... M

INS INS

Fig. 2 Time tagging of GPS and INS measurements.

GPS/INS PROTOTYPE

The GPS/INS hardware consists of CIA single frequency Trimble 4000 SX receiver, LTN-90-100 strapdown system and PC data collector computer with installed ARINC interface to the INS unit (Fig. 3). All individual elements run with direct current The system can be easily mounted and dismounted in about 5 minutes. The ARINC 429 interface and data logging software was developed by Pulsearch Ltd. and later the software part was modified by the Department of Surveying Engineering, University of Calgary.

SERIAL - .;::.

I/O :;:

PC ';~~ -PARAllEL ::~: ' I/O; .;::

~~~

D~ ~A D} ~A

GPS INS

Fig. 2 GPS/INS Prototype Scheme.

The GPS measurement data is transmitted to the computer through a serial port while the PPS (Pulse Per Second) information through a parallel port. The coming strings of GPS and INS data are time tagged with a reading from the computer clock. Since recorded computer time does not represent the time of individual measurements, it is refined in a post processing stage using the PPS output of Trimble receiver and taking into account the transmission delays.

TEST DESCRIPTION

Field tests of the GPS/INS system were carried out on May 3 and May 4, 1990 in the Kananaskis Region, west of Calgary. A 6 km test traverse going approximately South-North was selected, with accurate control points every 1 km. This particular stretch of highway has significant changes in grade, slope and curvature, making it a suitable traverse for testing the system. The traverse has

377

been recently densified by survey crew of Alberta Transportation with intermediate points established every 25 m. The points were surveyed using conventional techniques with 10 cm accuracy in latitude, longitude and height.

During the tests one GPS receiver was situated at a monitor site, 2 km from the first traverse point, and used for differential processing. The GPS/INS equipment was mounted in a van. The INS data were collected at 64 Hz rate while the GPS at 0.25 Hz rate. Each day the measurements were performed in two runs, on forward run with ZUPT -s and GPS static positioning performed at control points and on backward run without stops.The first run represented 'over-controlled' situation, where the GPS/INS was aided with ZUPT information. The second run represented the real production conditions.

There were three objectives of such schedule of tests:

assessment of general positioning accuracy of the system at static control points comparison of GPS/INS performance in ZUPT aided mode and in pure dynamic mode assessment if there is an accuracy degradation of the system with time due to unknown errors

At the start of forward run, a 10 minute period of static data GPS and INS were collected at the first control point of the traverse for initialization pmposes, as was explained before. The van was then driven at about 60 km h-1 to the next control point of the traverse. At the control points the GPS antenna was centered over control point and 2-3 minutes of static data were collected. In the backward direction the traverse was driven in one run, without any stops, with the first point occupied at the end of the session. 5 satellites were observed throughout the tests.

ANALYSIS OF TEST RESULTS

The analysis is limited to the positioning accuracy as was stated before.

-0-- Lat l( -0-- Lon ~ ______ .....

5

1

~ Hgt

2 3 4

Station 5 6 7

Fig. 4 Accuracy of integrated system at control points for May 3.

~ Lat 10 Lon

Hgt

1 2 3 4 5 6 7

Station

Fig. 5. Accuracy of integrated system at control points for May 4.

First, the overall positioning accuracy of the integrated system was assessed by comparing the filtered position results to the control coordinates at the static points along the traverse. Figures 4

378

and 5 show the results for May 3 and 4 respectively. Point 7 at these figures represents first point of the traverse occupied at the end of backward run. Results are below 10 cm and are well within those defmed for the MHIS in Table 1 and close to surveying system requirements. Part of the error, about 1-2cm, is stemming from the inaccurate centering of GPS antenna.

In order to assess the interpolating accuracy of the system, between GPS measurement epochs, the GPS/INS trajectory output was compared to the 'ground truth' heights, determined by conventional survey. Figures 6 and 7 show a comparison of May 3 and 4 height profiles for forward run, with Alberta Transportation surveyed heights. The surveyed heights for west side of the road ( backward direction) were not available during preparation of this paper. The rms height difference is ± 6.7 cm and ± 9.0 cm for May 3 and 4 respectively, and thus well within the accuracy requirements for the MIllS shown in Table 1. This value includes control heights errors itself, as well as identification and trajectory errors. It confirms also earlier results (Lapucha et aI, 1990), based on repeatibility analysis of heights from backward and forward runs.

0.2 - 0.2 5 -'-' 5 Q.I 0.1 C.I c Q.I

'"' ~ 0.0 ....

~ .....

'-' Q.I 0.1 C.I c Q.I

'"' ~ 0.0 .... ~

..c: ..... ~ -0.1 .c

~ -0.1 .Qj ::t:

-0.2

0 1 2 3 4 5

Distance (km)

Fig. 6 Comparison of height profile for May 3.

"Qj ::t:

-0.2

6 0 1 2 3 4 5

Distance (km)

Fig. 7. Comparison of height profile for May 4.

Additional numerical tests were performed to asses performance of INS part of the system during potential GPS lock loss. The raw INS coordinates, integrated over different GPS update intervals, were compared to GPS filtered coordinates. During these intervals the INS filter was not updated, as if the system was working in free inertial mode. It was assumed that GPS provides true reference trajectory. Although this does not give the true error, the complimentarity of INS part in the system can be investigated. Figure 8 shows coordinate difference between raw integrated INS position and GPS filtered position computed for 4 sec interval, which is standard interval of GPS measurements for Trimble 4000SX receivers. The position error is below 10 cm , which is approximately half cycle of L1 wavelength It demonstrates cycle slip detection and correction capability of the system as well as confirms interpolating accuracy of the system shown at Fig 6 and 7. The part of the plots on Fig. 8 and 9, related to 6-12 km distance values, represents the backward run

Next Fig. 9 shows the results of the same computations at 32 sec interval. The errors in latitude and longitude are up to 70 cm, while in height up to 30 cm . Such an accuracy is not sufficient for precise surveying but could fulfil the requirements of lower accuracy positioning, especially if height component is of main interest. On the other hand those results are still preliminary and should be improved with filter tuning.

379

6

0.2 ---------E

'-"

~ 0.1 u c ~ ... ~ eo-. .-CI ~ ... Q Q -0.1 U

-0.2 0 2 4 6 8 10 12

Distance (krn)

Fig. 8 Difference between GPS fIltered and INS predicted coordinates at 4 sec interval.

1.0 -E '-"

~ 0.5 u c ~ ... ~ eo-. 0.0 .-CI ~ ... Q Q -0.5 U

-1.0 0 2

r .... f:..·, - .,. ''\ I :...... v .... : . ,'\ ........ I I • ..., ,.

:\ ; I I \ I I I \1 I

I I 1 I J\ I I \ I ! ;' ! \ I

. I II : \ I I ....... I ,I I \ .. _ ••... 1 ..... I 70....: "

-...; ...;

4 6

Distance (krn) 8 10 12

Lat Lon Hgt

Lat Lon Hgt

Fig. 9 Difference between GPS fIltered and INS predicted coordinates at 32 sec interval.

The accuracy of the system in forward and backward run is at the same level (Fig 8 and 9).This proves that INS system is fully controlled by GPS in dynamic situation. The plots demonstrate important role of INS for integrity of the system, in which the INS provides position and velocity information during the GPS data gaps.

380

CONCLUSIONS

High accuracy of the integrated differential GPS and strapdown inertial system was demonstrated. Results from road tests show 5-10 em accuracy of the trajectory detennination. The INS part of the system is preserving the GPS positioning accuracy between GPS updates performed every 4 sec. Accuracy of the system is degrading with longer update interval but to the level which can be sufficient in MIllS application. The future work will be directed towards preserving the accuracy of the system during longer GPS data gaps.

The comparison of GPS/INS runs, unaided and aided with ZUPTs, shows that ZUPT update does not bring any improvement in positioning results. The GPS provides information of sufficient accuracy to bound the error growth of the inertial system. The performance of GPS/INS prototype in different dynamic conditions and longer missions, will be further tested.

The GPS/INS filtering program has been developed which uses the decentralized Kalman filter. This program can be applied in real time, for on-line cycle correction and filtering of data, provided data-link: between master and rover GPS receivers is available.

ACKNOWLEDGEMENTS

The project described in this paper has been funded by Alberta Transportation. Prof. K.P. Schwarz is thanked for his guidance and support. Valuable assistance in developing the software was obtained from Dr. M. Wei, E. Cannon and H. Martel. P. Beaulieu and Z. Liu were instrumental in unravelling the time tagging problem. Dr M. Wei, J. Czompo and Z. Liu were taking part in conducting the tests.Their support is gratefully acknowledged.

REFERENCES

Cannon, M.E. (1989), 'High Accuracy Semi-Kinematic Postioning: Modelling and Results', ION Meeting, Colorado Springs, Colorado, September 26-29.

Huddle, J.R. (1988), 'Advances in Strapdown Systems for Geodetic Applications', International Workshop'High Precision Navigation', SFB228 of Stuttgart University, FRG, May 17-20.

Gelb A. (ed) (1974); 'Applied Optimal Estimation', M.I.T. Press, Cambridge. Lapucha, D., K.P. Schwarz, M.E. Cannon and H. Martel (1989), 'The use of INS/GPS in a

Highway Survey System', Proceedings of the IEEE PLANS 1990. Schwarz, K.P., M.E. Cannon and R.V.C. Wong (1989), 'A Comparison of GPS Kinematic

Models for the Determination of Position and Velocity Along a Trajectory', Manuscripta Geodeatica, Vol. 14, No.5, pp. 345-353.

Schwarz, K.P., E.H. Knickmayer and H. Martel (1989),The Use of Strapdown Technology in Surveying', First International Symposium on Inertial Technology, Bejing, May 15-18.

Schwarz, K.P., Wei M. (1988), 'A framework for modelling kinematic measurements in gravity field applications, Proceedings of AGU Chapman Conference on Progress in the Determination of the Earth's Gravity Field, Ft. Landerdale, September 12-16.

VanBronkhorst, A. (1978), Strapdown System Algorithms. AGARD Lecture Series No. 95, Neuilly Sur Seine, France.

Wei, M. and K.P. Schwarz (1990), 'Testing a Decentralized Filter for GPS/INS Integration', Proceedings of the IEEE PLANS 1990.

Wells, D.E., N. Beck, D. Delikaraoglou, A. Kleusberg, E.J. Krakiwsky, G. Lachapelle, R.B. Langley, M. Nakiboglou, K.P. Schwarz, J.M. Tranquilla, P. Vanfcek (1987), Guide to GPS Positioning, University of New Brunswick Graphic Services, Fredericton.

Wong, R.V.C. (1988), Development of a RLG Strapdown Inertial Survey System, Department of Surveying Engineering, Report No. 20027, The University of Calgary.

381

KINEMATIC GPS SURVEYING

IN CYPRUS

Richard M. Haines Department of Lands

144 King William street, Adelaide, South Australia, 5000, Australia

ABSTRACT

The positioning community around the world has recognised the powerful capabilities of the Global Positioning System (GPS) in the kinematic mode. In particular surveyors and geodesists have capitalised on GPS as a viable alternative to conventional surveying methods. Th~y have predominantly used GPS in the static mode which has required observation times in the order of hours.

The kinematic GPS surveying technique has only become commercially available in recent years. This method has reduced observation times to a matter of minutes. Due to the continual developments of the technology and the limitations of the environment in which it can be used few organisations have utilised it in production yet.

This paper outlines the streamlined kinematic operation developed by the South Australian Department of Lands for high production surveying. The procedure is discussed with particular reference to a project undertaken in Cyprus.

INTRODUCTION

The South Australian Department of Lands is currently establishing a third order geodetic network throughout South Australia (S.A.). This is to support the introduction of a newly developed coordinated cadastre in that state. The "Tertiary Network", as it is commonly known, has been completed by conventional traversing in most urban areas of S.A. Rural areas have not yet been covered.

The kinematic GPS surveying technique was recognised as a procedure that would enable the rapid and cost efficient coverage of rural areas. During 1989 the method was thoroughly tested and a procedure developed for the coordination of survey marks to third order geodetic standards.

Following the research, SAGRIC International recognised

382

the potential of implementing the newly developed technique into its operations on overseas projects. This company predominantly carries out land management, surveying and mapping projects in developing countries. As with most projects of this nature there is usually a significant geodetic component required.

One particular project being managed by SAGRIC was a land information project in Cyprus. The geodetic component involved the strengthening of the primary geodetic network and the establishment of some third order control in various pilot areas. The kinematic operations as developed by the Dept. of Lands were successfully implemented on the project in Cyprus to coordinate approximately 60 points requiring only five days observations.

KINEMATIC GPS SURVEYING

The term "kinematic GPS surveying" is really a broad definition to describe the actual technique. It implies high precision in the relative positioning mode using carrier phase observations (Remondi,1988). The word "kinematic" suggests the continual determination of trajectory thus as applied to GPS surveying it would mean the solution to a baseline vector on an epoch by epoch basis.

The technique actually described here is a stop and go process and is often defined as "stop and go" or "semi-kinematic" GPS surveying. Although baseline vectors could be solved for at every epoch it is the period while a survey mark is occupied which is of primary interest.

SOUTH AUSTRALIAN DEPARTMENT OF LANDS KINEMATIC OPERATIONS

The Dept. of Lands has been using GPS since 1985. The state's primary geodetic network and extensive mapping control has been established using GPS in the static mode. In 1988 the Dept. purchased three Trimble 4000SLD receivers. These are equipped to perform kinematic GPS surveys using 10 channels on the single Ll frequency.

with the need to develop a tertiary network throughout S.A. the kinematic GPS technique was recognised as a viable alternative to static GPS and conventional methods. After an initial investigation, certain limitations were discovered. The requirement of maintaining lock on at least four satellites at all times and the problems associated with using post processing software which was continually developing were the largest concern. Despite these drawbacks it was believed S.A. was in the fortunate position of having an environment well suited to the kinematic technique due to the sparse coverage of trees in many areas. The Dept. subsequently initiated research into developing a kinematic procedure for its own applications.

383

Research

The kinematic technique has been well described by several authors (eg Mooyman and Quirion 1989). It has the nature of a survey by radiation. The result of the observations is a series of baseline measurements between the reference receiver (known point) and the roving receiver (unknown points).

The basic configuration is one without redundancy. To attain true redundancy a second round of observations must take place with the reference receiver on another known point. These second observations must be taken under totally different conditions to the first. This is to ensure that both baselines which are measured to the unknown points are totally independent to each other. It is for this reason that the configuration involving one rover measuring in conjunction with two reference receivers should never be used. The two baselines resulting from such a procedure are highly correlated since they are attained at the same instant of time, under an identical satellite geometry and with the same atmospheric conditions.

For the S.A. Dept. of Lands to utilise kinematic GPS for the coordination of geodetic networks an operation procedure had to be developed with redundant observations included. Figure 1 depicts the observation configuration necessary to do this.

Experimentation

A test range previously used for checking and calibrating aerial cameras was utilised to ascertain what accuracy could be achieved with this procedure. This range has over 100 marks at an average interspacing of 500m. They were coordinated by conventional techniques both horizontally and vertically to third order standards.

The kinematic procedure adopted allowed for a check on the internal consistency of the solution via a three dimensional adjustment of the observations. The use of the test range also allowed for a comparison with the existing coordinates in eastings, northings and height. The experiment involved two days observations with the use of one reference and two roving receivers. A total of 49 marks were coordinated each day during the 4.5 hour GPS window. All baselines measured were under 20km in length.

At the completion of the baseline computation stage certain accuracy limitations were revealed. If a baseline is computed at any stage of the survey using only four satellites then their geometry must be good. From the South Australian experience suitable baselines can only be attained if the measurements take place with a PDOP below 10 or equivalently with a RDOP below 3.

The results from the adjustment indicated that all 49 points had been horizontally and vertically positioned to an

3M

b. • (!]

LEGEND

Rm Order (ontrol Otupied Suney Mar\ - l"er , Otcvpied Sun,y Mar\ - 10m 2 OMenaliOllS, Day 1 OMerntiOllS, Day 2

"REDUNDAN(Y"

...... -

Fig. 1. Network coordination configuration

385

accuracy in the order of ±3cm (1 standard deviation S.D.). The statistical testing showed that all lines created from the procedure had met third order geodetic standards. The discrepancies when compared with the existing coordinates were below the 2cm (1 S.D.) level horizontally but were as high as 5cm vertically. No geoid analysis was undertaken in this exercise since the primary task was horizontal accuracy. Further analysis of height will be examined in the future.

The operation procedure was successful in attaining the required accuracy for tertiary network coordination. A system of planning, reconnaissance, pre-observation planning, observation and computation procedures were developed. The kinematic operation was subsequently implemented into the production activities of the S.A. Dept. of Lands.

SAGRIC INTERNATIONAL PTY. LTD.

SAGRIC International is a company incorporated in South Australia. It utilises the expertise developed by the South Australian Government on overseas projects. SAGRIC International's project experience is considerable and involves the management of a diverse range of activities throughout the world. The fields of land administration, land management, land information systems, surveying and mapping as developed by the S.A. Dept. of Lands have recently accounted for a large percentage of its overseas work.

The advent of the newly developed kinematic operations allowed SAGRIC International to capitalise on this latest technique. The company's most recent venture was a land information project being carried out in Cyprus. This project was recognised as a possibility for implementing the kinematic operations since third order control was required for cadastral and topographic surveys.

CYPRUS PROJECT

Cyprus is located in the north east corner of the Mediterranean Sea (See Figure 2). It has an area of 9251 square km and is the third largest Mediterranean Island. Its major topographic features include the Kyrenian Mountains in the north and the Troodos Massif in the south. A flat lowland known as the Mesaoria Plain lies between the two ranges and is open to the sea at both the east and the west.

The land information project being conducted in Cyprus required a detailed study of four pilot areas. This involved carrying out cadastral and topographic surveys in each area. To support these surveys third order control was required in all four areas. In addition to the pilot area studies the project also required the strengthening of the country's

386

25' 00

c

ORhOIl0(Y

Heraclion c1

K11om"l1Is' 1 00 00 310 410 1[llom"I1I'

Mn ••• 100 200 Miles

25'

Fig. 2. Cyprus location

35'

primary geodetic network to first order standards. The static GPS technique was recognised as being the most

suitable method for completing the first order survey. A preliminary study of each pilot area revealed that the environment was well suited to the kinematic GPS technique. Approximately 20 points at an average interspacing of 500m were to be coordinated in each pilot area. The third order surveys were subsequently carried out by the kinematic procedure developed by the S.A. Dept. of Lands.

Planning

The planning of any survey task requires the full understanding of the job requirement and a preliminary examination of the site to be surveyed. Aerial photography and the best available mapping will provide adequate details of the terrain and environment of the job site. At this stage the proximity of the existing horizontal and vertical control must also be assessed. For kinematic surveys a limit of 20km has been placed on the distance between the control and the job site.

surveying with GPS in these early stages of its development has also required a close inspection of the satellite availability. A 6 hour GPS window was available during the surveys on Cyprus. To carry out kinematic surveys further analysis of the geometry of the remaining 4 satellites at the tail ends of the window is also required. At the beginning of the window on Cyprus the geometry of the 4 satellites was not suitable for kinematic surveying. The PDOP was rapidly changing rather than being low thus the first hour was better suited to a static GPS survey. At the end of the window the geometry was suitable since the PDOP remained below 10. Kinematic surveys could thus take place

387

at this time. To fully utilise the GPS window it was planned to spend

the first hour carrying out a static survey. The baselines created would then be used to initialise the kinematic surveys. It is for this reason that antenna swaps were not used to initialise the kinematic surveys on Cyprus.

Maps were compiled at the conclusion of the planning showing the points to be surveyed. Location diagrams of available horizontal and vertical control were also attained ready for reconnaissance.

Reconnaissance

Reconnaissance involved visiting each point to be measured along with the control points to be used. Generally, GPS surveying requires a minimum 15 degree elevation mask for visibility to the sky at every point to be occupied. Kinematic surveying also requires that a similar visibility is possible while travelling between points to be surveyed. Due to this requirement reconnaissance involved the plotting of trees and other obstructions on the map of the area. suitable routes were then established based on the avoidance of these features. The reconnaissance on Cyprus was greatly assisted by the survey team who were very familiar with each pilot area.

Pre-observation Planning

Pre-observation planning is the logistics phase of the kinematic procedure. Once all of the reconnaissance and planning details were collated an observation schedule was devised based on the most rapid and efficient coverage of the survey area. This largely depended on optimising the use of three GPS receivers and the available personnel.

An observation schedule was made with the intention to complete each pilot area in one day. (Only one area required two days observations due to the increased number of marks there.) In order to achieve this, the three receivers were used such that one receiver remained fixed (reference) at the first control point and another receiver (rover) moved in kinematic mode to the points to be coordinated. The remaining receiver occupied various points in static mode which could not be measured by the kinematic technique. The roving receiver initialised and closed the kinematic survey on a known baseline. This allowed for the processing to be computed in a forward or reverse direction. This procedure was repeated from a second control point once the roving receiver had completed its first round of observations.

The above procedure was adopted because only one survey team was familiar with the kinematic routes to be taken. The remaining receiver was also useful for providing known baselines from which the roving receiver could initialise

388

the kinematic survey. The procedure was virtually identical to that of Fig. 1 since each point was occupied twice under a different satellite geometry resulting in independent baselines from two control points.

Observations

The observations simply involved carrying out the planned survey schedule. This schedule was strictly adhered to unless problems such as the total loss of lock occurred. In this case the roving receiver returned to a previously occupied point to reinitialise the biases.

Data was collected at a 15 second epoch interval and each point was occupied for only 2 minutes. The access to all marks was fast and reliable which allowed a measurement rate of approximately 7 minutes per mark. Radio communication was maintained throughout the survey between the reference and roving receivers. This was necessary to ensure all receivers started the various survey sessions together. If problems were encountered by either receiver this communication was essential. In particular if a power failure had occurred on the reference receiver the rover must be immediately notified to ensure it returns to a previously occupied point and does not continue on collecting useless data.

A total of five days observations were required to complete all of the kinematic surveys on Cyprus.

Co.putations and Results

All of the computed kinematic baselines satisfied the performance criteria set by the S.A. Dept. of Lands operation procedure. That is, all baselines were free from cycle slips, were observed under a geometry with a POOP below 10 and had RMS values below 0.05. Three dimensional adjustments for each pilot area were then undertaken. The results from the adjustments showed that all marks had been coordinated to an accuracy below the 2cm·level. All lines created from the procedure were tested to see if the adjustments had met certain geodetic standards. This was particularly useful to check what accuracy had been attained between adjacent marks since this line is never actually measured in the procedure. Each adjustment yielded results which satisfied third order geodetic standards.

The marks in one particular pilot area had also been measured by conventional traversing. This allowed for a comparison between coordinates derived by kinematic GPS and conventional techniques. Table 1 shows the discrepancies. These results were consistent with the solution from the adjustment and provided considerable confidence for the kinematic operations on Cyprus.

Table 1. Discrepancy between conventional and kinematic coordinates

Point No. Easting(m) Northing(m)

D504 -0.012 -0.009 D505 -0.003 0.002 D506 0.006 0.017 D507 -0.006 -0.015 D508 -0.011 -0.016 D509 -0.003 0.015 D510 0.021 -0.002 C266 0.011 0.009 C497 -0.003 -0.002

Operations Summary

Table 2 shows the total person hours required to complete the kinematic surveys in the four pilot areas. The entire job took approximately 2 weeks to complete at a cost in terms of personnel time of 5.3 hours per mark. This costing assumes all survey marks were pre-established and that the GPS equipment and vehicles were already available. This impressive rate of production clearly indicates the cost savings the kinematic GPS technique can provide.

To carry out the same surveys by conventional techniques would have been considerably difficult and time consuming. This is due to problems such as the lack of inter-visibility between survey marks and the necessity to provide secondary control.

Table 2. Operations summary

Activity Personnel Hours Person No. of Person-hrs. -hours Points per point

Planning 1 6 6 Reconnaissance 2 26 52 Pre-obs. plan. 1 16 16 Observations 6 30 180 Computations 1 60 60

Total 314 59 5.3

390

CONCLUSION

This paper described the research and experimentation conducted by the South Australian Department of Lands in utilising the kinematic GPS technique for high production surveying. In particular a procedure has been developed for the coordination of South Australia's third order geodetic network.

The kinematic operations were implemented on a project in Cyprus by the company SAGRIC International. A total of 59 survey marks were coordinated in two weeks which provided third order control for several cadastral and topographic projects. This project demonstrated that even when operating with the limited satellite constellation available at present, kinematic GPS surveying can be successfully used in production. The technique's significant rate of production when compared to conventional methods and its ability to be used in many suitable environments throughout the world make it a valuable tool for the surveyor and geodesist.

Acknowledgement. The support given by the SAGRIC project team on Cyprus and the Cyprus Department of Lands and Surveys is gratefully acknowledged. This project was only possible through the mutual cooperation of both these organisations. Global Surveys Limited, England are also acknowledged for providing considerable assistance during the preparation of this manuscript.

REFERENCES

Goad, C.G. (1989). Kinematic survey of clinton lake dam, Journal of surveying engineering, Vol. 115, No.1.

Hein, G.W. (1990). Kinematic differential GPS positioning: Applications in airborne photogrammetry and gravimetry, Fourth symposium on Geodesy in Africa, Tunis, Tunisia, May, 1990

Lapine, L.A. (1990). Practical photogrammetry control by kinematic GPS, GPS World, May/ June 1990.

Lunnay, c.w. and Haines, R.M. (1990). GPS in kinematic mode - tertiary network coordination in rural areas of South Australia, 32nd Australian Survey Congress, Canberra, Australia, April 1990.

Minkel, D.H. (1989). Kinematic GPS land survey -discriptions of operational test and results, Journal of surveying engineering, Vol. 115, No.1.

Mooyman, K. and Quirion, C.A. (1989). High production kinematic surveying, Trimble Navigation Ltd., Sunnyvale, CA94086., USA

Remondi, B.W. (1985). relative surveys in Proceedings of the Precise Positioning Rockville, MD., USA

Performing centimeter accuracy seconds using GPS carrier phase, First International symposium on with the Global Positioning System,

391

Remondi, B.W. (1986). Performing centimeter - level surveys in seconds with GPS carrier phase: initial results, Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, Austin, Texas, USA May 1986.

Remondi, B.W. (1988). Kinematic and pseudo-kinematic GPS, ION GPS88, Colarado Springs, Colarado, USA.

Wells, D.E. (1986). Guide to GPS positioning, Canadian GPS Associates, Fredricton, Canada.

392

Session 5

GEODINE 30 A REAL-TIME INERTIAL SURVEYING SYSTEM FOR LAND

GEOPHYSICAL SURVEYS

Jean-Marie DOIZI Compagnie Generale de Geophysique

Massy, France

INTRODUCTION

When defining a surveying methodology to respond to the demands of 2D and 3D land seismic surveys, a number of factors must be taken into account. These include increased positioning accuracy, the large number of points to be staked, the field geometry and the need for rapid determination of surveying coordinates.

To meet these conditions currently requires a large number of field crews, sophisticated methods and equipment, and large computing capacity.

The increasing cost of surveying operations for 3D land seismic surveys calls for re­assessment of the problem in the context of new operating methods which capitalize on the most recent equipment to be introduced on the market

The G60dine 30 system is based upon the combination of the innovative SAGEM Inertial Navigation System and the worldwide experience of eGG in land seismic.

G60dine 30 is designed to perform real-time surveying, lay-out and management of survey areas for 2D and 3D geophysical surveys and to handle the corresponding data points (geodetic points, receiver positions, shotpoints).

THE GEODINE 30 SYSTEM

Product structure r----------------, Printer

Inertial navigation unit Portable computer

Navigation indicator

Geodine30 comprises the following: - an Inertial Navigation Unit complete with a control and display unit - a total station directly interfaced with the platform and aligned with it - a portable personal computer complete with a navigation indicator. The system is installed in an all-terrain vehicle specially fitted out for this purpose.(Fig.2.)

393

Inertial platform and control display unit

The ULISS 30 system is composed of a UNI 30 inertial navigation unit and a CDU30 control display unit. This control display unit constitutes the man-machine interface which makes it possible to monitor the inertial system. (Fig.3.)

Total station

With the GDM 420 Geodimeter total station, offset points can be staked out and/or measured. When mounted on a land vehicle, the inertial system delivers in real time the coordinates of its location relative to the inertial platform or to any reference that is integral with the vehicle. In order to extend the capacity of the inertial system and to overcome the problem of having to be positioned at the points to be measured or staked out, the total station is interfaced with and tuned to the inertial platform. Points at a distance from the vehicle can then be staked out or measured within a 300-meter radius.

The total station is connected to the platform as follows: Low position: the total station is mounted on a tribach attached to the system casing.

From the low position,sightings can only be made through the open rear door of the vehicle.

High position: the total station is mounted on a tripod attached to the system casing. From the high position, sightings can be made in a 360° radius. The roof of the vehicle can be opened for this purpose. (Fig.I.)

The total station is tuned to the inertial platform, using a special "tuning procedure" developed by SAGEM.

Portable microcomputer and navigation indicator

The Compaq 386 portable microcomputer interfaces between the inertial navigation data and the requirements of surveying for geophysical operations. A real-time monitor has been specially developed to accommodate the real-time features of

the system. The Compaq hard disk can be activated only when the vehicle is stationary. The microcomputer is mounted on dampers. (Fig.5.)

The SAGEM MTPL+ navigation indicator is a computer peripheral installed on the vehicle dashboard. It informs the driver of distance and direction relative to a predetermined point and indicates the time available before the next zero-velocity update. (Fig.4.)

APPLICATION TO GEOPHYSICAL OPERATIONS. GEODINE 30 SOFTWARE FUNCTIONS:

Surveying for land seismic operations

Surveying for land seismic operations involves staking out in the field the source and receiver positions required for seismic data acquisition. The receiver positions (Geophone) are located at regular intervals along the profiles. The source positions (shotpoints) are situated near the receiver positions. Surveys may be single-line 2D (straight line dynamite, straight line vibroseis and slalom line) or multiline 3D (vibroseis or dynamite source), depending on the type of seismic operation envisaged. For present purposes, only the case of 3D surveys will be considered. The majority of 3D surveys are characterized by the notion of shotpoint and receiver grids. The receiver grid (complete set of geophone positions) consists of a network of straight, parallel profiles.

394

The shotpoint grid (complete set of shotpoint positions) may consist of a network of straight, parallel profiles at right angles to the receiver grid lines.

These grids cover from tens to hundreds of square kilometers. Required accuracy is usually to within a meter in planimetry. In altimetry it may be to the nearest ten centimeters. The complete set of actual coordinates is calculated in the geodetic system of the survey area.

Geodine 30 software functions

Geodine 30 was developed on the Compaq 386 microcomputer and makes it possible to apply inertial navigation data to the requirements of surveying for geophysical operations. The program was written by SAGEM in C-Ianguage and based on a real-time monitor, according to functional specifications defmed by COG. The GOOdine 30 system performs real-time lay-out and surveying and manages shotpoint

and receiver positions for 2D and 3D land seismic surveys.

The main software functions are as follows:

Increasing the density of primary geodetic network By increasing the density of the network, it is possible to define for the survey area a set of homogeneous points of comparable accuracy to the primary network. These coordinates may then serve as mandatory update points.

Conversion of coordinates Inertial data must be expressed in terms of the local geodetic and projection systems. The program therefore performs two-way conversion of geographic coordinates for a given datum in most known projection systems in the world.

Definition of theoretical source and receiver grid Detailed grid description enables Geodine 30 to define the theoretical coordinates of each shotpoint or receiver position.

Integration of lay-out specifications GOOdine 30 takes into account the geophysical specifications relative to staking out offset points. According to these specifications, various options are available to the operator, who selects the most suitable surveying position.

Lay-out and measuring of shot point and receiver positions Goodine 30 indicates each source and receiver position from pre-computed theoretical coordinates, so as to stake out the point and take Easting, Northing and Elevation measurements in real time. If a theoretical position is inaccessible in the field, an alternative position is established nearby, taking into account the offset constraints tolerated by the seismic specifications.

Lay-out and surveying are performed either by driving the vehicle to the position in question, or using the total station to make sightings from the inertial platform. In this case, the vehicle is parked at any given position, the point to be staked out or measured is selected from the data base and the microcomputer determines the parameters to be set on the total station.

395

Datil storage: G60dine 30 stores all the measured source and receiver positional data on MS-DOS diskette in UKOOA Pl/84 compatible format . These data can be entered directly into the G~micro* field data base and management system to produce location maps, elevation profiles, coverage diagrams, etc.

INERTIAL POSITIONING FOR THE ELFIMEILLON SURVEY

The Meillon 3D suney

The Meillon 3D survey was located in South West France and covered about 250 square kilometers. The area is a rural zone of varied relief, ranging from steep hills to fields of maize. A total of 12 000 source positions and 25 000 receiver positions were staked out and measured. The vibroseis method was used for seismic operations.

Inertial positioning method applied to the Meillon 3D survey (Fig.6.)

In view of the use of vibroseismic, all source positions had to be located on roads, tracks or lanes accessible to a 15-tonne four-wheel drive vehicle (i.e. a vibrator).

Consequently, following detailed reconnaissance of the survey area, all theoretical source positions were relocated at more feasible sites at right angles to the theoretical proflle.

Given the varied relief of the area and difficulty of access, very few of the receiver positions distributed along the proflles could be staked out and measured by the inertial positioning vehicle.

Instead, the inertial system was used to define only the receiver positions located at an intersection with a line of actual source positions. The remaining receiver positions were staked out according to conventional methods (theodolite and EDM).

Inertial positioning

Increasing the density 0/ the geodetic network

This operation involved increasing the density of the existing geodetic network in order to obtain a set of points spread evenly over the survey area which could serve as update points.

This set of points (270 in the survey) was defined according to two-way inertial navigations (between two and ten kilometers) observed and interpolated between known coordinates of the basic network.

Open-loop accuracy was as follows: In planimetry: 2.10-4 of the distance covered. In altimetry: 1.10-4 of the distance covered.

Closed-loop accuracy was 5.10-5 of the distance covered

(* COG trademark)

396

Staking out and measuring source and receiver positions (Fig.6)

Initialization of source and receiver grid parameters: This operation means that all grid coordinates can be pre-computed in the local projection system and each point can be identified in terms of LLLNNN, LLL indicating the line, NNN the position number on the line. When the operator has selected a position LLLNNN with the portable computer, the driver of the vehicle uses the navigation indicator to approach this position (See position 1 on Fig.6 Navigation towards known point). The vehicle then follows a route defined during reconnaissance, carrying out zero-velocity updates every three minutes. The system continuously displays the actual position of the vehicle in tenns of L, N and the distances to the nearest source and receiver positions. In addition, the system bleeps whenever the vehicle crosses a theoretical receiver line or a nonnal from a theoretical source location to the line. The driver then positions the vehicle on the intercepted point, the operator records the positional data and the assistant marks or flags the point. When visibility around the inertial platfonn is good, the computer can determine the data (angle and distance) for any theoretical grid point which the operator can then stake out and measure with the total station. Once a secondary network position has been reached, the operator perfonns a position update and the system updates the position data acquired since the previous position update.

Exploitation of data

As GOOdine 30 acts as the acquisition terminal for the G60micro field management system, inertial navigation data are transferred in order to update the Geomicro data base. G60micro can then begin quality control procedures and produce location maps, elevation profiles and 3D coverage diagrams.(Fig.7.)

CONTRmUTION OF GEODINE 30 TO THE MEILLON 3D SURVEY

Daily production levels achieved by the G60dine system for the Meillon survey were equivalent to the daily production of five conventional surveying crews. The G60dine 30 system provided definitive source and receiver position coordinates prior to seismic acquisition operations. The significant increase in production achieved by GOOdine 30 compared with conventional crews meant that two weeks were saved. In tenns of accuracy, discrepancies between theoretical and actual positions were less than 5Ocm. The average accuracy achieved in altimetry was +/- 3Ocm. The Geodine 30 system is limited only by the vehicle on which it is mounted, the speed of progress compared with that of a seismic acquisition crew, and cost.

CONCLUSION

GOOdine 30 represents a new concept in surveying for land geophysical operations. The system should reduce the cost of surveying and reduce the time required to defme the coordinates of the positions required for seismic data acquisition and processing.

397

REFERENCES

Doizi, 1M. (1989): "Systeme Geodine 30". Revue XYZ de l'Association Fran~aise de Topographie. SAGEM-CGG (1990): "Geodine 30".CGG Technical Series No. 585.90.03. CGG (1989): "Geomicro". Technical Series No. 581.89.09.

Fig. I. GDM 420 Geodimeter Total Station. High position.

398

Fig.2. GOOdine 30. All Terrain Vehicle.

FREOUENTS

Fig. 3. Inertial Navigation Unit

399

Fig.4. Sagem MTP + Navigation Indicator

Fig. 5. The Compaq 386 portable computer mounted on dampers

400

... PM

....... . ACTUAL ..... .

SOURCE IA POSITIONS POSITION

/",8 UPDATE

... PM

:II ················· · ··· ··········· ·m

(") m <: m :II C)

. . . .. . .... . ............................... :II 6 ..... ···· ·· ············ ··· ·······t············· ........ .

THEORETICAL { co::: --- ------

SOURCE .::: --------POSITIONS ::::. __ ___ _

..... .. . . .. .............. .. ... ... .. ..... . ............. .

.:~. NNN + 1

.:,:: NNN

1····souRce··iiR'ii ......... l\~~S~~I~~ 1 ··1 LLL - 1 LLL '&. LLL + 1

ALIGNMENT

... PM PERMANENT MARKER

8 SECONDARY NETWORK

... PM

+ POSITION UPDATE

Fig. 6. Inertial positioning for the MEILLON Survey

401

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402

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SESSION5b

POSITIONING AND NAVIGATION APPLICATIONS

CHAIRMAN M.AVNI

TRANSPORTATION DEVELOPMENT CENTRE MONTREAL, QuEBEC, CANADA

INTEGRATED SYSTEM FOR AUTOMATIC LANDING USING DIFFERENTIAL GPS

AND INERTIAL MEASUREMENT UNIT

Oipl.-Ing. Thomas Jacob

Institute for Guidance and Control of o. Prof. Or.-Ing. G. Schanzer

Technical University of Braunschweig Hans-Sommer Str. 66

0-3300 Braunschweig. West Germany

Phone 0531-391-3716, Telefax 0531-391-4587 Telex 952526tubsw

405

ABSTRACT

GPS-receivers have accuracy problems in high precIsion flight gui­dance applications. In dynamic flight maneuvers they show not only operational problems due to satellite masking but also a reduction in accuracy in accelerated flight and turn flight.

In the presented "Integrated System" those problems are solved by integrating GPS in differential mode with inertial measurment sensors into a hybrid system. This integrated system computes a high precise position, flight path and attitude information of a moving platform e.g. an aircraft.

The error behaviour in stationary and in dynamic applications is ex­plained. From the error behaviour a system concept of a hybrid Inte­grated Flight Guidance System is derived.

The position information, estimated in real-time, is used for a flight guidance value generator. These informations are fed to a flight di­rector instrument in the cockpit, which is used by the pilot for manual flight or is fed to an autopilot for automatic flight including automatic approach and touch down.

The system fulfills extreme accuracy requirements and can be used in approach and landing up to ICAO (International Civil Aviation Organi­zation) CAT III. It allows to perform landings even in bad weather conditions. As the integrated system is space based, it computes a landing aid which allows landing at any airfield, not equipped with con­ventional Instrument landing System or Microwave landing System.

In July 1989 the worldwide first automatic landing, using the presen­ted system, based on GPS has been performed by the Institute for Guidance and Control of the Technical University of Braunschweig.

The suitability of the concept (Kalman Filter coupling GPS and inertial measurement units (lMU», for flight path guidance and the accuracy of position finding (better than 1.3 m) will be presented by means of flight tests in a commuter aircraft (OORNIER DO 128) and also by si­mulator results. An inflight comparison of a reference Instrument landing System with the Integrated System shows the accuracy.

INTRODUCTION

When the American Global Positioning System is fully operational, there will be for the first time a navigation system available, which has a higher accuracy than any actual used long, medium or short ran­ge navigation system (except landing systems>. If it is possible, as shown in this paper, to improve position accuracy of GPS CIA code receivers in real-time applications to the order of 3 m or less, even an approach of an aircraft could be performed by using a space based "Integrated Navigation System". When GPS is available world­wide, approach and landing could be performed at any place not equipped

406

with conventional Instrument Landing Systems (ILS) or Microwave Lan­ding Systems (MLS), even in bad weather conditions. This can be advantageous for General Aviation and for feeder services, which normally operate from small airfields to international airports. Especially at small airfields no landings are allowed in bad weather conditions, because these fields in general are not equipped with ILS or MLS. Also most airports in the third world are not equipped with ILS or are equipped with systems which may not be reliable due to insufficient maintenance. Using the highly accurate Integrated Navigation System flight safety could be increased significantly at these fields.

The "Integrated Navigation System" as it is presented in this paper allows from a technical point of view to gUide an aircraft, by using ane system, from any terminal A to any terminal B including:

- runway - takeoff - approach - landing

guidance.

In developping an Integrated Guidance System for landing approach guidance, the following aspects:

- accuracy - dyn am ice r r or char ac t er i s tics - integrity

must be considered for the complete flight guidance system.

ACCURACY REQUIREMENTS FOR INSTRUMENT LANDING SYSTEMS

The accuracy and performance requirements for the ground equipment of Instrument Landing Systems (lLS) have been defined by the Inter­n at ion a I C i v iI A v i a t ion 0 r g ani sat ion (I C A 0) inA nne x 1 0 I 1 I. The r e qui -rements differ for horizontal and vertical guidance instruments. For each, a maximum bias value has been defined. That is an angular shift of the mean nominal path and a maximum beam bend, which is mainly due to multipath. The values are defined depending on the visibility conditions. For a horizontal visibility of 200 m and a vertical visibility of 0 m the ILS ground equipment has to fulfil the requirements for ca­tegory CAT III (The following values are calculated for a standard runway with a 3 degree glide path and 3000m runway length for CAT III ):

407

offset bend

horizontal [m] 3.0 2.4

vertical [m] 0.6 0.5

Tab!.@. 1~ ac curacy req uirement s for IlS ground equ ipment

The requirements for the guide beam characteristic are set in such a way. that the deviation of the aircraft due to errors of ground instru­ments at the runway threshold are less than:

CAT

horizontal vertical

III

5.0 m 1.2 m

Additionally. the roll and pitch attitude should not deviate more than 2 degrees from the reference values at the threshold due to course bends. For CAT III the signal quality has to be good enough to provide automatic flight.

If the Integrated Flight Guidance System should be applied in a com­mercial airplane for landings. it must satisfy the requirements of the ICAO depending on the weather conditions. Although GPS in the presented system is just one of many different sensors it is this sensor. that is responsible for stationary accuracy. Therefore. the error characteristics of a high precision 5 channel GPS C/ A code receiver in stationary as well as in dynamic flight tests have been analysed.

ERROR CHARACTERISTICS OF (iPS RECEIVERS IN REALTIME DIFFERENTIAL MODE

The typical error behaviour of GPS-measurements in stationary appli­cations using only block I satellites without selective availability (SV: 3.9.11,12.13) is shown in figure 1.

408

Nord­position

1m)

3

2

57'00 57600 57800 58000 58200 58'00 58600 58800 59000 tis)

Fig'yr.~ 1l Measurement of nort position using 5 channel GP5-CI A-code r e c e i v era n d b I 0 c k I sat e II i t e s wit ho u t 5 I A

These measurements have been made using the raw data of a 5 chan­nel CIA-code receiver, calculating a complementary filter of carrier phase and code measurements. This error behaviour in stationary flight can be modelled in the following way:

offset 15m drift 0.1 mlmin

two oscillations with

amplitude A1 0.2 m, period T1 16 s amplitude A2 0.6 m, period T2 250 s.

88 68 1&8 28 8

-28 -118 -68 -88

-188

North Position [m]

27888 28888 29889

lime [sec.l -

Fig'yr.~ 2.l Measurement of north pos it ion using 5-channal GP5 CI A­code receiver, block I and block II satellites

These natural errors are superposed by imitated errors - the selecti­ve availability (51 A) implemented on block II satellites. These errors are implemented into the system with the aim, to allow non-authorized users a maximum horizontal accuracy of 100 m (2drms) and a vertical

409

accuracy of 170 m. The influence of S/A in a measurement session versus 5 hours is shown on figure 2. In these measurements the same receiver with exactly the same software has been used operating with three satellite configurations: SV:9, 11, 12, 13, 18 and 9, 12, 13, 18, 20, and 3, 9, 12, 13, 20), A plot of north position versus east position of the same measurement is given in figure 3. It can be recognized from these measurements that a configuration with just one block II satellite produces a typical position error of 20 - 30 m, while confi­gurations with two block II satellites show a positon error of up to 140 m.

158· ft==t===jf====t====t===t====t==91

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- 118· H- -I---I

-21. --' .8 21.1 U . I 61 . ' .... I ....

F i gyr.~ ~.,;. Po sit ion plot u sin g 5 c han n e I G P S - C I A - cod ere c e i v e r, b I 0 c k I and block II satellites (same configuration as in fig. 2)

The typical error behaviour of stand alone GPS can be modelled as follows:

- the measured GPS-position has an offset versus the real position - the measured GPS-position drifts, due to the movement of the

satellites (change of constellation, GDOP and S/A) even when there is no aircraft movement

- typical measurements have a noise, of which the amplitude is de­pendent on receiver quality;

- an oscillation is superposed (receiver dependent, multipath, S/A) - when the selected satellite constellation is changed due to the

rising of a new satellite, the position measurements react nearly like a step function with an amplitude of several meters (errors in the order of 8 m have been measured)

410

As generally known, the offset and the time dependency of the offset can be extensively eliminated by using Differential-GPS-Technique (fig. 4>.

Regional-Airfield

Fig~I.~ !~ 0 ifferent ial GPS for precision approach guidance

As the position of the ground antenna is known, the range from the ground to each satellite can be calculated. By comparing the computed range with the measured range the actual system error can be deter­mined. Transmitting these errors to the aircraft via a telemetry, they can be corrected in the onboard position finding computation if the di­s tan c e tot h e g r 0 u n ds tat ion iss m a II en 0 ugh. T 0 use 0 - G P 5 for an a p­proach and landing guidance system the most interesting questions are:

- is it possible to correct the S/A - effect in real-time to an error level below 4m horizontal?

- which is the maximum distance between ground receiver and air­borne receiver, up to which the accuracy requirements are met.

Some measurements indicate that distances less than 200 km can be adequate. In optimal conditions (low range between user and reference ground station, no multipath. same propagation path satellite- reference sta­tion as satellite-user and no system degradation> this technique can reduce the position error up to the receiver residuals which are in the order of a meter or a few centimeter depending on the receiver type (fig.5>.

411

xgd il t 1.1 I m I -1.2

.... ~ 57111

Y9diftl.71~ Iml 1.11 ~

1.51 ~WIIMU1l UI ~==r.

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:!~;'Ij:~~~ 57611 577., 578.. 57911 5S... 581'1 582el 583.. 58U' gSiI 58811 g1ll1 5.8 •• tls1 _

Fig'yr.~ 5: Real-time O-GPS using bloc k I satellites (tau =0.6 s, v=O. mIs, 5 channel CIA code, L1 receiver)

In figure 6 results of a real time O-GPS technique, as developed in the Institute for Flight Guidance and Control, are calculated from the same measurements as in figure 3.

North [mJ

to' 4\

-2.' W 1'1

-6 . •

-: ---1.9

-9.9

- II.

- 13. - 13. ,-II. - 9.5 -7.! -6. 1 -... -2.1 ' .1

East [mJ

Fig'yr.~ .§..;, Real-time O-GPS using block I and block II satellites h=0.6 s, V=O. mIs, 5 channel CIA code, L1 receiver)

This has been a mainly stationary measurement versus 5 hours. In this session one antenna has been moved 8 m to the south and 1.5 m to the east. Although this measurements are stationary with an extremly short

412

baseline they show the potential of this real time technique. Although there has been the aim to degrade a CIA-code user by the SI A to a horizontal accuracy of 100m and vertical to 170 m, figure 6 shows, that it is possible

- by using the online real-time D-GPS technique, to correct the sy­stem degradation to

98 X horizontal 96 X vertical

- to fulfil the horizontal requirements of the ICAO for CAT III.

However using this correction technique, the dynamic error characteris­tics are not improved. In stationary tests, as well as in flight trials, tested GPS receivers have proved to be excellently accurate in the long term, however, a lower accuracy in dynamic maneuvers has been detected. Depending on the receiver type receivers show

- dynamic errors in flight phases with longitudinal accelerations as well as in phases with lateral accelerations. The reasons for the dynamic errors are:

- an influence of acceleration on the receiver clock (crystal oscil­lator) .

- an influence of changes in the measurement signals to the output due to transfer function of the code tracking loop (delay lock loop) and phase tracking loop of the carrier signal (costas loop).

- dynamic problems due to receiver integrated software and filter technique using low pass filters to reduce noise 12/.

- time lack due to signal processing up to 0.4s 12/. electromagnetic interferences in airborne applications with air­craft radio. This is a special problem of some two frequency recei­vers, when they are operating with an HF-Fllter for both frequen­cies with a too large bandwidth.

- operational problems in airborne applications. Flying a turn with a bank angle greater than the elevation of a satellite, a masking of the antenna is produced as:

- the hemispherical antenna is focused in such a direction that the locked satellites are undetectable. parts of the aircraft (e.g. wing, body, rudder or props) move in­to the line of sight between satellite and user antenna.

This effect produces always cycle slips in the phase measurement and of course a lock-off of the masked satellite. After reducing the bank angle again. receivers need up to 10 s - 30 s to relock again. In the most successful cases this effect produces an error of only several meters (of about the same order when changing the constellation) or at the worst case a total loss of GPS position information (fig. 7 point A.)

413

4000

i 3000 .-~

.---... -------......... '\ e n \\ #

oJ: 2000 A .... ~ lv 0 z

1000

(\( 0 --- ? -1000 -4000 -2000 0 . 2000. 4000

East m ~

Fi.9!!.!.~ I~ Ground Track measured with real-time O-GPS using block I satellites in a flight test <'r=0.6 s, V=60. mis, 5 channel CIA code, L1 receiver, 360 0 circ with bank angle at point A <I> =60 0 , at B standard curve <I> =20 0 )

The O-GPS measurements in a flight test are shown in figure 7. The pilot had the task to fly three flight patterns. In one flight pattern he had t 0 fly a 3 6 0 0 c i r c lew i t h a ban k an g leo f u p t 0 6 0 0 nor tho f the airport. In this extreme maneuver three satellites were masked. There­fore the GPS receiver was not able to calculate any position, not only during the maneuver but also for the next 45 sec. as he had to relock again. Therefore for several kilometers the pilot had no position infor­mation from GPS.

For a high precision flight guidance during a landing phase even in CAT I conditions none of these errors can be accepted. Altogether the errors must be for CAT III at least less than 3.0 m in the horizontal and 0.6 m in the vertical. In addition to the accuracy the ICAO re­quires integrity for the landing guidance system. In case of a malfunc­tion of a CAT IlLS glidepath transmitter the pilot has to have an "off­signal"· within 6 sec. All these requirements are met in the Integrated Flight Guidance System by system integration of complementary sen­sors (GPS and INS), including adequate filtering.

CONCEPT OF THE INTEGRATED FLIGHT GUIDANCE SYSTEM

To improve the dynamic behaviour of the system and also for safety reasons during a landing approach it is a need to generate additional position information without using GPS.

In dynamic maneuvers the aircraft position, (cp, A, h), the inertial ve­locity V, the attitude angles <1>, e, tV (bank, pitch, azimuth) and the bo-

414

dy fixed accelerations, can be measured and calculated by using an inertial measurement unit (IMU). IMU's have a good short term accu­racy, however. in the long term they have recognizable drifts. The long te-rm accuracy is dependent on the gyro drift which determines the coordinate axes misalignement En. Ea. Ed' With today high quality IMUs, a typical drift of 0.5 m/s can be obtained. Obviously a system concept which utilizes the good long term accuracy of the GPS and the good short term accuracy of an IMU would produce a good overall accu­racy. This concept is realized in the "Integrated Flight Guidance System" (fig. 8),

Pilot

CaImands

F i .9.Y.!:.~ ~.!. I n t e g rat e d F I i g h t G u ida n c e S y s t e m

The "Integrated Flight Guidance System" is basically composed of two parts:

- a position finding system and - a guidance generator.

To improve the dynamic characteristics of the entire system and to get sufficient information concerning the flight path during a breakdown of satellite information. the integrated flight gUidance system is coupled by Kalman filter technique with inertial sensors - gyros and accelerome­ters. Radio or barometric altitude sensors are used additionally to 0-GPS if a satellite masking is produced. In the Kalman filter there are im­plemented error models, that estimate the errors of system states as well as the errors of the sensor systems, as there are:

navigation coordinate misalignement gyrodrifts offsets of the accelerometers receiver clock error receiver clock error drift

415

By using these models the filter learns the errors of the inertial sen­sors in those phases of flight, where the GPS-receiver is in a good con­dition. If some satellites are masked in maneuver flight, the hybrid po­sitioning system calculates a position information using online calibra­ted inertial sensors. Therefore it is possible to use low cost inertial sensors instead of expensive inertial platform systems, and to calcu­late a position with high accuracy. To the output the system gives on­line calibrated

- position - velocity - acceleration - attitude angles - attitude angular rates

with an update rate of up to 50 Hz.

Although the best estimation of position is calculated using this tech­nique, no pilot is able to follow a nominal flight path using only a position information in latitude, longitude and height. It was advisible to display the position information to the pilot in a conventional manner. Therefore, a nominal flight path consisting of standard rate turns and linear parts using known coordinates of the target place (waypoint, airport threshold) is calculated. By calculating the nominal flight path s"tarting from the precisely known actual position a substantial advantage is achieved be­cause no intercept maneuver has to be flown.

Flight guidance data are calculated from the information of the actual position, the deviation and the attitude relative to the nominal flight path. With this procedure the pilot receives information on how to fol­low the nominal flight path, which may be curved in the horizontal as well as in the vertical plane and how to recover if he deviates from it. For the indication during the flight test an ILS cross deviation indica­tor or a flight director can be used.

SIMULATION AND FLIGHT TEST RESULTS

Fig. 9 presents simulation results. In the simulator the same flight procedure has been flown as in the flight test in fig. 5. The position errors of the subsystem GPS and INS (1 NM/h) as well as the errors of the integrated system are plotted versus time.

416

t 70

! 60 Xi ~ 50

40

30

30

10

0

0 60 120 180 240 300 360 t (sl-

Position error of INS (1Nm/h). GPS and Integrated Flight Guidance System (t=150 s ~=600. t=241s ~=200)

At t=140 sec a circle with a bank angle of 60 0 has been flown. While flying 60 0 bank angle a masking of 3 satellites has been produced for 100 sec. In this time GPS gives no position information. While the INS has produced a position error of 48 m. beginning at the alignment to the end of the masking. the error of the integrated system is 71 X (14 m) lower than the error of the INS. At t=260 sec the aircraft flew a turn returning to the extended center line (ECl)' In this phase of the flight. the satellites were in a bad constellation (Geometric oelution of Precision GoOP > 10). One satellite was masked. Here the error of the GPS grows up to 50 meters. The integrated system. however. has produced an error of up to 7m which is 90 X lower than the error of the Inertial Navigation System. although there had been no GPS posi­tion measurement for 65 s!

If there would be a loss of satellites during an approach the integra­ted system is able to recognize the satellite error. The system checks the errors of the satellites on ground. realizes failures. reduces their effect and gives a signal to the pilot. Considering the time until leaving CAT II accuracy requirements the pilot would have up to 40 5 to conti­nue the approach. to wait for relocking the satellites or to make his de­cision for a go around maneuver. Especially in this point the Integrated System has significant advantages compared to conventional IlS or MlS. It has a high integrity!

These simulations show. that by combining IMU and GPS in the discussed integrated system an overall accuracy can be achieved that is up to 90 X better than the accuracy of an INS standing alone. even in phases where no GPS is available. In phases where GPS is available the accuracy of alteration of position is in the dcm - order. using differential GPS phase measurements. however with an performance during high dynamic flight

417

phases which is much better than GPS standing alone.

4000

! 3000

~ 2000 '­o z:

1000

o

-1000~--~_4~0~00~--~~----~------~----~---- 2000 0 2000 4000

East m _

Figy'r.~ lQ..;. Ground Track computed by the "Integrated Flight Guidance System" (same flight test as in fig. 7)

Fig. 10 shows the ground track of the same flight as that displayed in fig. 7. The GPS-position and the position output of the position-fin­ding part of the "Integrated Flight Guidance System" are presented here. Based on the sensor errors, which are determined by the system, the position-finding part is able to determine the flight path and the attitude angles with a high precision. Even with a breakdown of the GPS signals, the filter algorithms still give the position, speed and attitude angle with a high precision for a limited time. A decisive prerequisite for that is a realistic mathematical model of the dynamic error charac­teristics of the inertial systems.

Which overall position accuracy can be achieved using the "Integrated Flight Guidance System"? The flight test results in automatic approach are shown for the lateral and vertical position in fig. 11. It can be recognized from these plots that an autopilot is able to fly the aircraft in a stable condition to the runway threshold using only the signals computed by the integrated navigation system. It should be emphasized that by using this integra­ted system the procedure of flying an intercept maneuver is not ne­cessary, because the aircraft is always under cover of the navigation system. Additionally this system is able to fly not only an automatic a p pro a c h but a Iso a n aut 0 mat i c to u c h dow n (i n c Iud i og aut 0 t h r 0 tt Ie) .

Figure 12 shows a comparison of the computed deviation from the glide path calculated by the "Integrated Flight Guidance System" with the measured Instrument Landing System of runway 27 R in Hannover (which is regarded by the ICAO as a reference system in DOC 8071 and meets the ICAO requirements of Annex 10 for CAT III ILS)' In se­veral approaches it could be verified that the difference of both glide path deviations (horizontal fig.12a and vertical fig. 12b) is within the requirements.

418

If the ILS is regarded as a reference system without any error the difference signal would be the total error of the integrated system. In reality, however, there is an error contribution by the ILS, so that the accuracy of the Integrated system is better than the difference sig­nal.

Artificial Glide Path

200 F1i;rtPath

0 0 1000 5000.

Distanct to touch down point [mJ -

4000 North Position Iml

3000

2000

1000

oL---------'----r Extended Centerline

o 1000 2000 3000 4000 5000 6000 East Position 1m)

Fi.Q.YI.~ 11.;. Ground Track and vertical flight path computed by the "In­tegrated Flight Guidance System" in automatic approach

419

I .... '.35

~'.3' C '.25

~ '.2' :i: '.15

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b: ~GPILS-~GPGPS and tJ. G P H r a d i 0 - ~ G P G P S)

420

SUMMARY

The flight tests which have been made with the "Integrated Flight Guidance System", developed at the Institute for Guidance and Control of the Technical University of Braunschweig have demonstrated good results by combining two sensor systems with different, complementary, time dependent, signal qualities: the inertial sensors, with their excellent short-term characteristics; and the GPS with excellent long-term c h a r act e r i s tic s. Wit h the K a I man fil t e r t e c h n i que it is po s sib Ie, eve n in high dynamic flight phases, to determine a position of high precision and reliability. The position determined in flight tests is better, than the precision of each system standing alone. While in real-time application the vertical accuracy of the "Integrated Flight Guidance System" is sufficient for CAT II landings the horizontal accuracy fulfills the de­mands of the ICAO for CAT III requirements.

It is possible to use the presented system in all high precision naviga­tion applications.

In offline applications of the position finding part of the "Integrated Flight Guidance System" accuracies in the order of 10cm can be achieved, by using special differential techniques (double differenti­al),

The first application of differential GPSIINS guidance will probably be the use as a high preCision position reference system and may be in regional air traffic the application as a landing gUidance system for such airports which have no IlS or MlS systems installed. It may be expected that a discussion about whether

- the Microwave landing System (MlS) might be out of date before it is generally used

- the Global Positioning System (GPS) can be used instead of MlS especially for business- or commuter aircraft operating into small airfields

will arise in the near future.

421

BIBLIOGRAPHY

/1/ Aeronautical Telecommunications, International Civil Aviation Organisation (ICAO) ANNEX 10

/2/ Avionics Navigation Systems Kay ton, M.; Fried, W.; John Wiley & Sons 1969

/3/ Richtlinien fur den Allwetterflugbetrieb nach Betriebsstufe " N a c h ric h ten fu r L u f tf a h r erN f L 1- 3 5 0 / 7 2 Bun des a n s t a I t fur Flugsicherung

/4/ Principle of Operation of NAVSTAR and System-characteristics Milliken, R.J; Zoller, C.J. AGARDograph AG 245

/5/ The NAVSTAR GPS System AGARD Lecture Series No. 161

/6/ Differential Operation of NAVSTAR GPS Kalafus, R.M. Journal of the Institute of Navigation Vol. 30, 1983

/7/ Flight Guidance" G. Schanzer, Technical University Braunschweig

/8/ Techniques of the development of Error Models for Aided Strapdown Navigation Systems Lechner, W. AGARDograph AG 256

/9/ Genauigkeitsanalyse von Tragheitsnavigationssystemen N. Lohl, Technical University Braunschweig 1981

/10/ Integrated flight guidance system using Differential-GPS for landing approach guidance Th. Jacob, AGARD Guidance and Control Panel Lissabon 1989

/11/ Approach flight guidance of a regional air traffic aircraft using GPS in differential mode Th. Jacob, ICAS Jerusalem 1988

422

ABSTRACT

EUROFIX: A SYNERGISM OF NAVSTAR/GPS AND LORAN-C

Dr. Durk van Willigen

Delft University of Technology

&

Reelektronika bv

The Selective Availability in the SPS-mode of Navstar/GPS, which largely degrades the potential accuracy, may be effectively countered by differential techniques. Wide-band (geo-stationary satellites) and narrow-band (terres­trial) DGPS services offer good accuracies and may additionally improve integrity. Unfortunately, it will not increase the availability of GPS and the cost aspects are not yet clear.

The EUROFIX concept integrates Navstar/GPS with Loran-C. Differential-GPS data are phase coded on the Loran-C signals. This narrow-band Loran-C communication link (5 .. 12 bps) transfers a full DGPS data set in about 40 .. 100 seconds. As reported by Kremer et al (Navigation, Spring 1990), the long message time delay causes correlation loss between the DGSP data and the actually measured SAed pseudoranges which in turn introduces range-error differences of 5 .. 12 meter, equivalent to about 15 •. 36 meter horizontal position errors (95%). Better results are feasible by temporary storing the measured pseudoranges (track history) in the receiver which makes re-synchronization of the pseudoranges with the delayed DGPS data possible. The derived DGPS positions are consequently 40 .. 100 seconds old which is generally acceptable for stationary users. Dynamic-user positioning is attainable by applying Loran-C as a dead-reckoning device which is updated by the computed DGPS positions every 40 .. 100 seconds. Sampling the Loran-C signals more towards (+40 Ils) the peak of the burst increases the SNR by ~12 dB yielding better tracking and data-link performances. Possible errors due to skywaves and ASF are made ineffective by the frequent DGPS updating. Analysis indicate that for the newly proposed North Sea Loran-C chain navigation accuracies of 25 meter (95%) are feasible.

The hybridization of GPS and Loran-C gives a better navigation availability than either of the systems may provide separately. Hybridization of the pseudoranges of both systems also eases RAIM procedures. The infra-structural costs are very modest as many Loran-C stations are operational in large areas of interest.

The EUROFIX user needs a standard-type GPS receiver, a modified Loran-C receiver for the data extraction and a software package to perform the position, velocity and integrity calculations.

423

INTRODUCTION

Navstar/GPS is undoubtedly the most promlsmg general purpose radio positio­ning system for the coming decades. The strong points of the system are widely published, while somewhat less attention is paid to items like single-nation military control, selective availability and possibly future user fees. However, all potential GPS users who intend to integrate the satellite system into air, sea and land transport systems should anticipate these aspects. However, selective availability (SA) in the Standard Positioning Service (SPS) mode and the single-nation military control are matters beyond the scope of any non-American country. We just have to accept the Slystem as it is, and because of the 3D-positioning, world-wide coverage and 24 hours service, most countries will even embrace it!

The SA-induced accuracy degradation in the SPS mode may to a large amount be made ineffective by application of differential techniques. Two main directions can be distinguished: the wide-band and the narrow-band solution. Wide-band means e.g. that the differential data is transmitted in the GPS frequency band by another satellite. By using equivalent pseudo-random-noise modulation (wide band), the GPS receiver is able to receive and decode the differential-data messages. Data formats are proposed by the RTCM-104 Special Committee [1]. With sufficient geo-stationary satellites all parts of the globe, with the exception of the polar regions, can be serviced. The narrow-band techniques use similar differential data formats, however this data is transmitted as narrow-band FSK signals by earth-bound stations e.g. in the LF frequency band. It is suggested by various sources to use e.g. coastal LF stations for this purpose [2,3]. Due to the moderate power levels of such stations, differential services are unfortunately limited to the coastal areas.

The differential information can be derived from a single GPS reference receiver at a precisely known site or from a number of spatially separated GPS receivers working together in a network. The latter solution may give improved differential-data quality over a very large area [4]. The question arises whether this services will be given free of user charges or that some form of user fees will be introduced. Where integrity and world-wide services are mandatory, e.g. for civil air transport and shipping, the wide-band service is highly attractive. Possibly user fees are there of less importance. However, for local shipping and land navigation user fees and world-wide coverage are not needed, so, the choice may be different.

High-risk cargo and passenger transports need a high level of integrity. By supplying sufficient pseudo ranges to the receiver it may determine the integrity of the system autonomously. However, if GPS is analyzed by the receiver as being inadequate, a second source of navigation is needed. This may be e.g. Glonass or Loran-C. The Glonass perspective is attractive as it has, just like GPS, global coverage. However, it should be mentioned that the error sources of the two almost identical satellite systems are not per definition uncorrelated which makes the gain in availability questionable.

For regional use, Loran-C offers a very useful alternative for backing up GPS. Loran-C has a high degree of . integrity and it is available over large areas of the northern hemisphere. Enge et al [5] demonstrate with a theoretical study

424

the large improvement in availability and in receiver-autonomous fault detection by hybridizing GPS and Loran-C pseudoranges. However, in its basic operational mode Loran-C is generally less accurate than GPS in the SA mode which is mostly caused by the insufficiently detailed ASF-tables available to the receiver. Highly accurate ASF tables and differential Loran-C may improve the absolute accuracy significantly. Unfortunately, due to grid warpage the differential Loran-C service area is rather limited.

Loran-C, is besides of being an adequate back-up system to GPS, is for many purposes also an interesting candidate for providing differential data-link services to an integrated GPSlLoran-C system. Phase modulating the Loran-C bursts makes data transmissions with a bit rate of 5 .. 10 bps U bit per 2 GRI) possible without disturbing the basic Loran-C positioning accuracy. However, this bit rate is just one-tenth of the proposed RTCM SC-104 50 bps which will inevitably introduce an intolerable differential-message delay of 50 .. 100 seconds. Nonetheless this large delay, GPS, Loran-C and a Loran-C data link can be efficiently integrated into a single hybridized and differential system which is called Eurofix. This cost-effective system shows increased accuracy and integrity over either of the systems separately without being dependent on a separate data channel.

EUROFIX CONCEPT

Because of the speed-limited differential Loran-C data link we must in the Eurofix concept consider two distinctly different user types: the very-low­dynamic or stationary user and the medium-dynamic user.

In one of the following paragraphs it is shown that the effective Baud rate of the data link is limited to about 5 bps. This means for the RTCM SC-104 data formats a ten fold increase in message delay which in turn implies a message-type-one (RTCM SC-104) transmission rate in the order of 40 .. 100 seconds for a set of 7 SV's. Kremer et al [6] indicate that for this message delay the real-time RMS range-error difference may increase to about 7 .. 12 meter. Recall that for the full GPS constellation the 95th percentile horizontal position error is roughly three times the one-sigma range error. So, with the above given range error, a two-sigma position error of 21..36 meter is obtained. Even with this very slow data-transfer rate, we still obtain every 40 .. 100 seconds a three-to-five fold improvement in accuracy over the basic GPS in the SPS-mode.

Much higher accuracies are possible by temporary storing the GPS pseudoranges until the full differential data set is received and decoded. The correction of the stored pseudoranges with the differential data is now not longer degraded by correlation loss due to time-synchronization errors. Although, the position update is now delayed up to 100 seconds, the full capability of the differential technique is saved. This back-calibration procedure is useful for stationary or slowly-moving objects but is, of course, less adequate for dynamic users. However, this situation may change completely if other positioning devices are available such as dead-reckoning or Loran-C. In that case the actual position is derived from an older, but accurately calibrated, position updated with the dead-reckoning device. The dead-reckoning time then

425

varies between 100 and 200 seconds. As Loran-C is used anyway for the differential data link, the same system is also available as a dead-reckoning position-update device. Normally, Loran-C suffers from a more noisy track than non-SAed GPS for equal tracking loop bandwidths. However, if Loran-C is only used as a dead-reckoning device it is then perfectly allowed to track the Loran-C bursts much closer to the peak of the burst. The resulting gain of about 12 dB in SNR gives an equivalent improvement in the SNR of the measured track. The penalty is that the groundwave signal may significantly be polluted by skywaves. So, actually, the receiver tracks now a composite signal with the groundwave and the skywaves as ingredients. The increase in SNR is illustra­ted in fig. 1 which depicts the TD-tracks of the transmitters at Lessay (master) and Soustons (secondary) of the Biscay chain. The receiver was at Delft which is 550 km away from Lessay and 1000 km from Soustons. The upper trace of fig. 1 shows the TD tracked at the standard sampling point position which is actually at 55 IlS from the start of the burst at the input of the bandpass filter. The SNR of the Master equals +6 dB and amounts to -6 dB for the secondary accordingly to the SNR definitions of the Minimum Performance Standards [7]. The lower trace depicts the improvement by tracking the signal 40 IlS later or at 95 Ils from the start of the burst. As can be seen in this 30-minute plot, during a 200-seconds time interval phase distortions due to changes in sky-wave interference are neglectable. The signals from Soustons entirely travel over land, while the Lessay path is a mixture of seawater and land. In a 200-second time interval, an one-sigma TD deviation of about 50 ns may be expected, which results in 7.5 m deviation (leT) at the base line. This is also confirmed by computer simulations of the receiver with its bandpass filter, the hard limiter and the sequentially-detecting second-order tracking loop. See fig. 2. The timing resolution is 25 ns, while the acceleration slew rate equals 2.5 m/s/s for SNR = 0 dB.

...... ..., .... :.; .......

Fig. 1 450 TD-measurements during 30 minutes in The Netherlands of the 8940 250 kW transmitters at Lessay (550 km) and at Soustons (1000 km/land path) in France. In the left-hand plot signals are tracked at the standard sampling point while in the right-hand plot signals are sampled 40 Ils later giving a 12 dB SNR-improvement. The standard deviation over the 30-minute interval is reduced from 117 ns to 51 ns. The vertical scale is 1 f.l.S per division. The leT SD-value in the lower plot in a 200-second interval is about 50 ns or 7.5 meter.

The newly proposed North Sea chain with the Master station in North-East England and with Lessay and Sylt as secondaries gives excellent coverage in one of the busiest waters of the world. Analyses of SNR's and chain configu-

426

ration yield for composite signal tracking 2a' standard position deviations of 21 m at Hoek van Holland, 22 m at Sheerness and 37 m at Le Havre. It must be noticed that in this dead-reckoning mode of Loran-C navigation, any ASF or skywave error is harmless as long as the ASF does not noticeably change during a 200-second position move.

us U.'I

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o .00 500.0 1000. 1500. 2000. 2500. 3000. 3500. '1000. '1500. GR I

Fig. 2 Computer simulation of the test receiver tracking signals from Soustons (SNR = -6 dB, upper track) and from Lessay (SNR = +6 dB, lower track). The simulated and the real receiver apply hard limiters and sequentially-detecting second-order tracking loops with the threshold set at 8. The exact zero crossing of the signal is at 95.016 IJ.S from the start of the burst.

The principle of EUROFIX is explained in fig. 3. The time intervals between tn, tn+l and tn+2 represent the message delays introduced by the narrow-band Loran-C data link. At time tn, the probability density functions of GPS in the SPS-mode, the groundwave of Loran-C without ASF corrections and finally the PDF of the Loran-C composite signal are shown. The differential GPS data are computed at the time tn and the Loran-C transmitter starts at the same time with the transmission of the differential data set. It takes (tn+l - tn) seconds to transfer the complete set to the Loran-C receiver. So, at the time tn+l, the receiver can calculate the precise position of the GPS receiver where it has been at the time tn.

As the Loran-C receiver also stores the found tracks of the groundwave and the composite Loran-C signals, it is possible to calibrate the 'groundwave­position' and the 'composite-position' at time tn. The next differential GPS data is fully received at time tn+2. In the time interval between tn+l and tn+2, the EUROFIX position is found by tracking the Loran-C composite signals. As the possible skywave content of the composite signal may slightly change during this time interval, the mean value of the PDF may also shift slightly.

427

The thick line close to the true-track line gives the mean value of the computed EUROFIX position. For clarity, the drift of the Loran-C composite signal is exaggerated. The PDF of the EUROFIX radial position error is equal to the PDF of the composite Loran-C signal.

t

PDF of composite Loran-C

PDF of Loran-C groundwave without /!SF correction

tn-position correction at time tn+1

Message delay ~

tn+1

PDF of EUROFIX

/ "-. ...

tn+1 -position correction at time tn+2

Composite Loran-C track

Loran-C groundwave track

EUROFIX track

--.... - Time

Fig. 3 The EUROFIX differential GPS/Loran-C concept. The Loran-C composite track represents the track of the Loran-C signal at 95 f.LS where the signal contains groundwave and skywave components. The groundwave signal is given without any ASF correction applied. The PDF of the GPS position is shown with Selective Availability switched on. The thick lime gives the obtained EUROFIX position track.

Summarizing, EUROFIX offers the following capabilities:

a - Stationary or slowly moving user has full DGPS capability, b - Dynamic user may get DGPS with 20 .. 30 meter 2«T accuracy, c - Loran-C data link provides integrity information about GPS, d - If GPS fails, Loran-C provides position with standard Loran-C accuracy, e - If Loran-C fails, GPS operates autonomously with SPS specifications, d - GPS and Loran-C can be hybridized in case of insufficient pseudoranges, e - Integration of GPS and Loran-C increases RAIM capabilities.

The EUROFIX concept is based on six functional parts, three at the Loran-C transmitter site, and three at the user site.

Loran-C transmitter site (see fig. 4): a - GPS reference receiver, b - Differential GPS data generator, c - Loran-C transmitter phase modulator for differential data link.

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User site (see fig. 5): d - Modified Loran-C receiver for differential data extraction, e - Standard GPS receiver, f - Software package for calculating the hybridized position.

\ II I Navstar/GPS DGPS Loran-C core receiver data generator transmitter

Atomic time reference

Fig. 4 Block diagram of an EUROFIXed Loran-C transmitting station. The reference GPS receiver is at the Loran-C transmitter site or is part of a GPS reference network.

Loran-C data

EUROFIX -DGPS data Position

Loran-G receiver EUROFIX

navigation - Velocity

software -GPS data Course

Navstar/GPS package

Aiding data -receiver

Integrity

Fig. 5 Block diagram of an EUROFIX receiver setup. The GPS receiver is a standard type, the modified Loran-C receiver decodes the EUROFIX data. The aiding data is optional for rate aiding applications.

EUROFIX DATA LINK

To transmit the differential GPS data to the user, the timing of the Loran-C bursts can be modulated with time advances and delays of about 1 f.Ls relative to the nominal time value. Conventional Loran-C receivers will hardly notice this modulation as long as the number of phase advances equals the number of phase delays. So, averaging the sampled phase data will suppress the applied modulation under the assumption that the lowest modulation frequency is still above the bandwidth of the position phase tracking loop. This modulation process is e.g. used in the Clarinet Pilgrim Loran-C Communication System. See fig. 6.

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no coding .... , ...• \.( positive code bit ( phase delay )

positive polarity

95 us zero crossing

negative polarity

negative code bit ( phase advance)

\, 0~ /\ .. ~ .. /\ ....

........

Fig. 6 EUROFIX phase coding. A 1 us phase delay gives a positive signal at the 95 us sampling point position and indicates a positive code bit. The opposite holds for a negative code bit.

To keep the hardware and the software complexity of the EUROFIX receiver low, a very simple to decode, yet efficient coding scheme is chosen. Additionally, the Bit Error Rate (BER) must be sufficiently low so that for normally encountered SNR's at ranges up to 1000 km, only very few differential messages are damaged. Finally, the fastest possible transmission rate is strived for.

As the Loran-C receiver is also used as a positioning device by tracking the groundwave and the composite signal, an adequately fast second-order phase-tracking loop with low tracking noise must be incorporated. The composite signal gives the low tracking noise while the perfectly balanced phase modulation prevents any groundwave tracking bias. As the tracking-loop bandwidth must be relatively wide for dynamic user applications, the modulation should also be balanced for short time intervals. To avoid dead zones in hard-limiting phase-tracking loops with high SNR values, the first two bursts of every GRI are excluded from modulation.

A simple, yet useful code with a code rate of 1/12 is depicted in Table 1 which shows that a single data bit is coded in 12 Loran-C bursts requiring 2 GRI's. The basic data transfer rate then equals 112 GRI or 5 .. 12 bits per second. The code balance is maximized for the shortest possible integration time. Note that the coding is different for the a and the b parts of the GRI. The receiver has only to test for two alternatives in every set of 12 bursts in 2 successive GRI's. The decoding is simple, as the code is fully synchro­nous with the GRI-a en GRI-b frames. It must be reminded that the code pattern shown in Table 1 does not include the normal phase code of the Loran-C bursts which indicates whether the signal is from the master or from a secondary station.

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Binary Modulation pattern data bit

value GRI-a GRI-b

1 0 o + - + - + - 0 o - + - + - + 0 0 0 - + - + - + 0 o + - + - + -

Table 1 Modulation pattern for code rate = 1/12, 0 = no time shift, + = positive code bit = 1 JlS phase delay, - = negative code bit = 1 JlS phase advance.

Is is interesting to investigate the Bit Error Rate (B£R) of such a simple coding method. In the service areas of the proposed Norwegian Sea Chain, the North Sea Chain, the Biscay Chain and the Iceland Chain, the user is generally within 600 km of a Loran-C transmitting station. For these relatively short ranges the SNR at the input of the antenna is +3 dB (MPS definition) or more while at the composite sampling point at the output of the bandpass filter of the receiver a SNR of better than +6 dB is expected. At a 1 IlS distance from the zero crossing the SNR amounts then consequently +4.4 dB. Assuming a hard-lim iter-type receiver and Gaussian noise, the polarity observation reliability - pobs - equals 0.9515. It is further. assumed that the 95 Ils zero crossing is perfectly tracked. The 12 detected polarities during GRI-a and GRI-b are, after being decoded with the 'binary one' data bit value (+ - + - + - - + - + - +), integrated in a counter. A positiive polarity increments the counter, while a negative polarity does not change the counter position. After having collected the 12 code-polarity samples of a data bit, the counter position N (0 ~ N ~ 12) tells us whether a logical zero or a logical one is received. N > 6 indicates a logical one, while N < 6 means a logical zero data bit has been transmitted. N = 6 is considered as an invalid data bit. From the binomial distribution function an erroneous decision probability - qdec - of 4.03· £-7 for a single code bit is found. The probability of finding an invalid code bit (N = 6) equals 8.92'£-6. As it takes 2 GRI periods, equivalent to 80 .. 200 ms, to transmit a single data bit, a complete RTCM SC-104 500-data-bit set needs 40 .. 100 seconds to be trans­ferred. The probability that one of the 500 data bits is not correct equals 500'9.3'£-6 = 0.0047 or 0.47 7.. The probability that an erroneous data bit is captured amounts only 500· 4.03' £-7 = 2.0· £-4 or 0.02 7.. The build in parity checks gives the receiver added capability to detect erroneous data bits. Due to the GRI frame structure, a good interleaving with respect to atmospherics is automatically established. It should be remarked that no forward-error­correction coding is used. Parity bits are only added to prevent false integrity alerts or erroneous differential corrections. More sophisticated coding may improve the data transmission reliability. Unfortunately, it will also reduce the receiver's decoding-algorithm simplicity.

CONCLUSIONS

The proposed £UROFIX concept may give differential GPS and integrity services without the need to erect new transmitting stations for the data transmis-

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sions. Loran-C is used for the differential data transfer by additionally phase modulating the signals without disturbing the normal Loran-C positioning accuracies. Only minor changes in the Loran-C stations are required. Loran-C is also used as a dead-reckoning device to counter the low data transmission rate. Improved tracking performance is obtained by tracking the signals more towards the peak of the bursts. Hybridizing GPS and Loran-C eases RAIM techniques, while the navigation availability is better than either system may provide separately. The EUROFIX Loran-C receiver is a standard type with added data-decoding facilities. Generally, no hardware changes are necessary. EUROFIX can give standard DGPS accuracies of 2 .. 10 m (20-) to the stationary or low-dynamic user, medium-dynamic users may expect 2o--values of 25 .. 35 meter.

Acknowledgements

The author would like to express his gratitude to Mr. Bart Hoogenraad, a former student of the Delft University of Technology, and to Prof. Borje Forssell of the Norwegian Institute of Technology at Trondheim. The author also acknowledges the encouraging discussions with Mr. Ares Lubbes and Mr. Owen Goodman from Intersite Surveys bv. The Loran-C project at the Delft University of Technology is supported by the Science Foundation under contract number DEL59.0912.

REFERENCES

[1] Recommendations of Special Committee 104 Differential Navstar/GPS Service, Radio Technical Commission for Maritime Services, February 20, 1985.

[2] P.K. Enge, R.M. Kalafus & M.F. Ruane, "Differential Operation of the Global Positioning System", IEEE Communications Magazine, Vol. 26, No.7, July 1988.

[3] D. Pietraszewski, J. Spalding, C. Viehweg & L. Luft, "U.S. Coast Guard Differential GPS Navigation Filed Test Findings", Navigation, Journal of The Institute of Navigation, Vol. 35, No. I, Spring 1988.

[4] A. Brown, "Extended Differential GPS", Journal of The Institute of Navigation, Vol. 36, No.3, Fall 1989.

[5] P.K. Enge, F.B. Vicksell, R.B. Goddard & F. van Graas, "Combining Pseudoranges from GPS and Loran-C for Air Navigation", Navigation, Journal of The Institute of Navigation, Vol. 37, No. I, Spring 1990.

[6] G.T. Kremer, R.M. Kalafus, P.V.W. Loomis & J.C. Reynolds, "The Effect of Selective Availability on Differential GPS Corrections", Navigation, Journal of The Institute of Navigation, Vol. 37, No.1, Spring 1990.

[7] Minimum Performance Standards Marine Loran-C Receiving Equipment, Report of Special Committee no. 70. Radio Technical Commission for Marine Services, December 20, 1977, Washington, D.C., 20554, USA.

Author's address: Prof.dr. Durk van Willigen Delft University of Technology Faculty of Electrical Engineering Mekelweg 4, 2628 CD Delft, The Netherlands tel +31-15-786186 / fax +31-15-781714

432

INTRODUCfION

Session 5b

OPEN CONCEPT INTEGRATED NAVIGATION FOR AIRBORNE

REMOTE SENSING IMAGE GEOREFERENCING

D.L Ross, L. Hill and E.M. Senese Ontario Centre for Remote Sensing

Surveys, Mapping and Remote Sensing Branch Ministry of Natural Resources

North York, Ontario MlN 3Al, Canada

There has long been interest in the provision of C()st effective means for the geographic referencing of airborne remotely sensing data. Corten (1976, 1984 and 1988) has described the concept of independent aerocontrol. This involves the use of one or more navigation sensors which under computer control form an integrated navigation system providing georeference data for remote sensor data independent of ground control. The benefit of this approach should be the reduction in cost and improved efficiency in data acquisition.

This concept is not new. It has been applied throughout the world in many civilian and military mapping programs. These required the specialized installation, ownership and maintenance of aerial survey radio trilateration systems. These systems were expensive to install and generally used in large mapping projects were economies of scale permitted.

The objective of this project is to investigate and exploit the technologies, economies, and applications of using the infrastructure of civilian and military air navigation aids to meet the aerocontrol requirements of remote sensing. The radionavigation aids under consideration in this project include: civilian and military distance measurement equipment (DME, TACAN and precision DME), NA VST AR-global positioning system (GPS), LORAN C, and OMEGAlVLF complemented by independent sensors such as Inertial Reference System (IRS) and precision barometric (pressure) altimetry. This project is at the early stage of development.

OPEN CONCEPT

Projects such as kinematic survey systems for the geographic referencing of airborne remote sensing data require significant capitalization and long term effort and are difficult to justify due to uncertainties of delivery times, costs and benefits.

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In Canada, experience in the georeferencing of remote sensing data by independent aerocontrol has largely occurred at the Federal level. This has involved large area photogrammetric mapping programs in the north, military, fisheries surveillance, airborne geophysics, laser bathymetry, electro-optical line imaging and synthetic aperture radar. In these applications, navigation sensors were highly integrated with the respective sensor packages. These applications and uses have largely been single purpose in design involving extensive engineering modifications in both hardware and software. The complexity and uniqueness of these systems has limited the direct transfer of aerocontrol technology to other applications with the exception of the skills and knowledge acquired.

To obtain project support for a development project of this scope within an operational resources management agency requires a business case which enables an open concept philosophy. The open concept refers to a number of current and anticipated applications over which investment costs can be amortized to as broad a base as possible. The open concept also relates to systems development activity and the selection process for procurement of hardware and software. The concept permits adjustment to plans to incorporate new opportunities such as improved technology. This assumes that changes can be integrated if the right technology path has been chosen (eg. software transportability).

PLANNING CRITERIA

To facilitate the open concept, several planning criteria were used.

Environmental Scan

To enable a broad perspective, activity in related industrial, technological and end user applications sectors must be continuously reviewed in a process which management consultants call the environmental or external scan.

Review of scientific and technical literature, trade journals, and manufacturers literature inside and outside of the geomatics sector is required. External review is important because it provides the relevant trends which keep the planning process open. Technology push trends occur in other industrial and scientific fields which can impact the plan. The fields of electronics, computer systems, and optics are a few examples. In the optics field, work on laser ring and fibre optics gyros were reported in optics journals long before they appeared in avionics publications as inertial navigation systems. It is important to recognize that technology is not relevant without the accompanying skills and knowledge acquired from basic and applied research in analytical tools (i.e. Kalman filters, modelling, etc.)

The most important component of the environmental scan is the pull from the end user who will directly benefit from the application of the technology. The client base served by the Ontario Centre for Remote Sensing involves a broad base of users in the business of managing and developing natural resources. These users are starting to rely on the use of Geographic Information Systems to manage their data for use in integrated resources management for sustainable development.

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The development of kinematic aerial survey tools provides a front end to the processing of geographic data for end use in integrated resources management. The tools must meet the requirements of the technical specifications/standards for these data bases and be cost effective alternatives to data collection by other means.

Standards

To minimize duplication of effort, development work cannot be undertaken in the dark. Existing specifications and standards must be considered and adapted. Several sectors are involved: surveys and mapping, civil aviation, computer industry, telecommunications, national and international standards associations, and the military.

Incorporating standards necessitates knowledge acquired by the ongoing acquisition of documentation. An extensive set of technical documents is required to understand modern digital avionics systems for which ARINC (Aeronautical Radio Inc., Annapolis, Maryland) and RTCA (Radio Technical Commission for Aeronautics, Washington, D.C.) are sources. The standards development process is lengthy, involving years to derive consensus. The benefit is a common base from which manufacturers can develop competitive products. The field of airborne kinematic survey systems is too small to justify the commercial development and manufacture of major integrated systems. Systems developers should recognize this reality. High costs necessitate adaptation of standards and technology from other sectors where manufacturers can afford to maintain infrastructure for reliable products and service. This advice is particularly true for airborne navigation systems, espeeially those approved as front line navigation by civil aviation authorities. One rule that we have adopted is to buy not build hardware. A hardware build policy effectively cuts a development program off from the rest of the industry, and can be problematic when interfacing systems. This rule was important in our procurement process for an inertial navigation system (INS). Older used gimbal systems (ARINC 561 INS, 1975) required computer interfacing which would require custom interfacing of synchro resolver outputs to a computer. Cost benefit analysis indicated that this option would involve higher long term cost and obsolescence than the use of a modern strapdown inertial navigation system (ARINC 704 IRS, 1983 with ARINC 429 digital outputs).

Civilian aircraft navigation systems, are designed to determine flight position in real time. In survey aircraft, navigation systems provide a dual role: flight navigation in real time and a means to annotate the geographic position of observations and measurements which are recorded for post time data reduction. It is desirable to integrate both of these requirements to minimize cost, space and power requirements. If navigation sensors are used in the flight deck as front line navigation sensors, they require approval from the appropriate civil aviation authority. Approaches to this configuration range from full integration of the flight deck with the remote sensor suite such as on modern fighter jets to that of an independent navigation system designed to support only the sensor suite. Various degrees of integration can be considered to complement both tasks.

In the commercial aviation sector, avionics manufacturers have responded to the needs of major airlines and general aviation by producing digital flight management systems (FMS ARINC Characteristic 702) and navigation management systems (NMS).

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These systems integrate navigation inputs from a variety of sensors; blend navigation data; compute course from a predetermined flight plan; and provide direct output to the autopilot to fly the aircraft from point to point. These systems are essential components of modern wide body passenger and cargo transport aircraft and are also used extensively in general aviation aircraft such as corporate executive jets. These systems have been designed for the specific purpose of efficient and safe point to point navigation and can involve capital costs of several hundreds of thousands to millions of dollars per aircraft. Certification for north Atlantic routes for commercial jets may require redundant systems to the extent of triple inertial and flight management systems. Specifications for current digital navigation systems of this class are covered in the 700 series specifications of ARINC (Aeronautical Radio Inc.). For aerial survey applications, the navigation sensors from the current generation of digital systems are of use. Unfortunately, many of the commercial flight and navigation management systems (FMS and NMS) are not directly suited for the needs of aerial survey aircraft where there is a requirement for georeferencing remote sensing data at high data rates. Some vendors provide approved software modifications which include routines for survey flight plan navigation; however, outputs for georeferencing are limited to non-precision slow data rate applications, e.g. annotation of oblique surveillance photography.

The most useful digital avionics interface data bus-standard at the present time is ARINC 429 - Digital Information Transfer System Mark 33, 429 (ARINC Specification 429, 1983; Spitzer, 1987). This unidirectional serial digital data bus, which operates at either low speed (12 to 14.5kHz) or high speed (100kHz) has the advantage that bipolar line drivers (transmit lines) permit up to 20 receivers to be attached to a single output line. This feature permits multiple line taps on a single output line from a navigation sensor to feed multiple receivers such as a NMS and a data logger. Serial output is provided in 32 bit word format, each word containing a unique label. Standard labels cover a wide menu of navigation parameters which are uniquely identified in the ARINC 429 standards document.

Wide adaption of these standards throughout the aviation industry has permitted the development of niche markets for which a number of manufacturers have developed specialized support products. One product development that has permitted this project is the availability of off the shelf ARINC 429 Interface cards on the IBM PC ISA bus. At least six manufacturers in North America are currently manufacturing the product to fill a demand from the avionics maintenance sector. Availability of a link between modern digital navigation sensors and IBM PC compatible computers offers great potential for the development of navigation applications in the PC environment where a rich set of software development tools exists. This configuration provides an open path for systems upgrading and growth since software is developed in a transportable environment at nominal cost. A longer life for the investment in sensors and software is suggested. Technology transfer and adaptability to other applications is enhanced since PC based applications can be readily transported including the use of laptop formats for field portability.

The current configuration used by the project team consists of an IBM PC compatible laptop computer. The navigation system can be quickly disconnected from the computer by a DB25 connector providing portability for development work offsite, and for bench and for inflight testing.

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A strong word of caution is provided on the use of PC's with high speed clock rateR no to 40 mHz) in aircraft; proper grounding, bonding and shielding of the system is required. Aircraft and computers are very noisy in the electromagnetic spectrum requiring the use of computer systems and peripherals which are packaged on shielded enclosures and the use of standard avionics installation procedures for wiring, shielding and bonding (ARINC 413, 1976; ARINC 404, 1974; Morrison, 1986). If done improperly, radio frequency interference can induce systems crashes and glitches inflight.

TIME

When acquiring digital remote sensing data, the common element which ties all the data together is the real clock base. In an open systems architecture the sensors cannot be internally synchronized to an external master clock, since each individual navigation sensor is autonomous. Handling asynchronous data streams requires a means whereby a common clock base can be used to tag all data streams before they are logged on mass storage. Synchronizing the data streams is a problem which must be addressed by software in the post mission processing of data. Time resolution is the governing factor which separates various applications. Applications requiring attitude adjustments in 3 axes such as electro­optical line imaging, synthetic aperture radar and laser ranging are the most time critical remote sensing applications since these instruments operate sampling frequencies on the order of 10 to 100Hz. Application of Nyquist sampling theory for the geometrical restitution of data from these instruments without aliasing would indicate sampling rates on the order of 100 to 200 Hz for attitude. These rates are considerably higher than those required for civil aviation navigation and survey flight line navigation. The time resolution of real time navigation systems are geared toward the feedback response of aircraft control systems on the order of a few hertz.

Airborne remote sensing technology can be classed in three broad types of sensors: frame (or area) imaging, line imagers and spot non-imagers. Examples of each type include photogrammetric mapping cameras, electro-optical line imagers (MEIS) and laser altimeters, respectively. Both the line and spot sensors are dependent on the motion of the aircraft for sampling, whereas the frame imager is independent of aircraft motion. To reconstitute data from line and spot sensors requires geometric correction of the observations prior calculating their geographic co-ordinates. This necessitates precise accessory measurements of the sensors pointing in three axes, as well as the respective angular rates. The time resolution for these observations is dictated by the dwell time of the sensor for which a single resolution element on the ground is viewed (instantaneous field of view, IFOV). For current civilian line imagers, the IFOV is on the order 0.5 to 1.0 milliradians. With frame imagers the geometry is fixed and governed by the geometric quality of the optics and the image plane. These sensors sample at rates that minimize image degradation due to motion. The image can be readily georeferenced through ground or aerocontrol. For a system which integrates navigation data for georeferencing of relJiote sensing data, the resolution of the common time base will be dictated by the most demanding sensor in the remote sensing suite with clock resolution in the range of milliseconds to microseconds.

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A common time base is required to tag all data for post-processing of asynchronous data to sychchronous data. Insertion of the clock data in digital data streams prior to recording poses some difficulties in design, particularly when black box subsystems are involved which have no internal real clock base. Insertion of clock by software during formatting of data to a storage device will result in variable timing errors due to varying execution of computer code. Timing uncertainty on the order of milliseconds to microseconds would be expected with software time insertion. Insertion of real clock into sensor black boxes is not suitable unless these systems provide suitable electronic interfaces. The third approach is to add time to the sensor output data streams when received at the input to a data logging computer provided appropriate interfaces are available (on receipt tagging).

The use of microprocessors in digital avionics systems results in time delays betwRn original measurements and processed output at the data bus. With the time resolution requirements for georeferencing, these time delay characteristics must be accounted for by bench testing of equipment. Known time delays can then be used during post flight data processing to correct for the time of actual measurement.

Options for the master clock include a local crystal oscillator set to a known time base which drives a digital time code generator with output such as IRIG (Inter-Range Instrumentation Group) format. Time base accuracy can be improved through the use of other local oscillators such as rubidium and a time code generator which enables transfer of time acquired from GPS or LORAN C. The preferred option is a portable time and frequency instrument which permits tagging of data to the co-ordinated universal time (UTC) base. Further advantage of an integrated local UTC time base from GPS or LORAN C is the more rigorous control of the local oscillator errors to improve accuracy of the local frequency by reference to external atomic clocks. Availability of the accurate frequency and time base can improve range accuracy of LORAN C, DME and Omega/VLF. One concept that may be desirable to obtain and log range data from all sensors with time tags (i.e. LORAN C, GPS, DME) for use in a combined pseudorange position calculations

At the present time the authors have not yet resolved the issue of how to integrate time/frequency standards into the systems design. This is a difficult area to technically resolve, particularly in light of the open concept design philosophy. Current effort is focusing on technology enabling tagging of data at time of receipt, using a master clock in extended UTC (to milliseconds) and with the availability of local error corrected frequency sources. With currently available technology, it appears that an integrated (GPS, LORAN, and local oscillator) master clock and frequency source can maintain daily accuracies on the order of 1110 microsecond UTC and 1/2 picosecond, respectively. A frequency source of this accuracy is useful for other instrumentation in remote sensing and navigation sensor suites, (e.g. source for range frequency counters used in DME measurements with a clock equivalent error of 0.2 mm).

CURRENT SYSTEMS

Presently, the project team is configuring a modular navigation system which is independent of an existing suite of IFR approved avionics in the flight deck of a remote sensing aircraft.

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Development work to this point has involved bench testing of interfaces and input/output software with avionics black boxes (ARINC 704 IRS and a 3 channel scanning DME with ARINC 429 110). Currently the system is being packaged in a DOT approved instrumentation rack for seat rail installation in Piper PA31-350, Beechcraft King Air and Bell 206L aircraft.

System Hardware and Software

The software development platform is an IBM PC compatible laptop computer in a shielded enclosure. The system uses an Intel 30386 CPU with 30387 math coprocessor, 1Mb main memory and a 40Mb hard disk. The computer provides two expansion slots on the IBM PC ISA bus, for which 1 slot is used for an ARINC 429 Interface card with 4 transmit and 4 receive channels. The system is configured under MS DOS with compiled BASIC used for testing, and evaluation of software concepts. Proven software are then translated to the 'C' language for efficiency~ Specific code has been developed for input/output of data from the two navigation sensors described below, as well as initial software for survey navigation. Depending on the airborne test results, work is anticipated to progress to the next stage of time/frequency tagging, adding peripheral support for logging navigation sensor data, and additional navigation sensors such as precision barometric altimetry.

Navigation Sensors

A strapdown inertial reference system with all digital inputs and outputs is used. (Litton LTN 90 with ARINC 429 DIGITAL 110). The IRS is not approved for front line navigation having internal software modified from the ARINC 704 specification to provide extended resolution outputs at higher output rates. The extended output specifications are based on software developed by Litton Systems Canada for the Canada Centre for Remote Sensing, EMR, Ottawa.

The second sensor is a digital scanning DME transceiver (Bendix/King DM441) which has all digital input/output in the ARINC 429 specification. Tuning of the station pairs in the DME for three channels is done through software on the host PC. Selection of this particular transceiver was based on a requirement for the availability of un rounded range data on the digital bus with a resolution of O.Olnmi.

The scanning process is sequential and similar in some respects to the operation of the geodetic satellite program SECOR (Sequential Correlation of Ranges). Scanning DME transceivers have been developed to compliment digital FMS and NMS systems; however, there are differences in how manufacturers of these systems treat DME data in software.

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POTENTIAL FOR SCANNING DME

Distance measurement equipment (DME) is a pulsed ranging system for aircraft navigation which operates in the frequency range of 960 to 1215 MHz. DME is part of a ground based network of radio navigation aids and are generally co-located with VOR or TACAN ground stations. DME provides a range to the beacon whereas VOR provides bearing, the combination of which yields a rho-theta scheme for determining aircraft position to the beacon. T ACAN is the military equivalent of VORIDME. The VOR azimuth from these beacons are of insufficient accuracy for survey applications. VORIDME are essential operational components of civilian radio aids to navigation throughout North America, for which the current and future status are described in the US Federal Radionavigation plan. (DOD-DOT, 1988). It is a line of sight range system which provides service to aviators in major airways where each beacon can be time shared by up to 100 users.

Scanning airborne DME technology is not extensively used by the aerial surveying community for a number of reasons. These include limited availability of rho-rho-rho geometry in remote areas, and limited low level service volume due to line of site coverage. This is restrictive in low altitude applications such as airborne geophysics. Many users also discard the potential for DME based on the stated range accuracy specification for civil aviation of ± 0.5nm or 3% of range (2 sigma), whichever is greater (RTCA, 1978).

There are areas where geographic distribution of ground DME beacons permit use in mid to high altitude aerial survey applications such as aerial photography. This includes areas such as the Great Lakes basin where a dense network of Canadian and U.S. beacons (90+ stations) exists to facilitate heavily used commercial airways. The conservative range accuracy specification of DME can be significantly improved upon through knowledge of DME systems characteristics and network calibration. There are three sources of systematic error in a DME range: clock and timing errors in the airborne transceiver (Rtf) and the ground transponder and atmosphere refraction. When the airborne unit transmits a pulse to the transponder a local clock counter begins counting until the retransmitted pulse on a second frequency is received from the ground station by the airborne receiver. The ground beacon (transponder) inserts a fixed 50 microsecond delay into the signal between the received and retransmitted signals. This delay is subtracted from the total transit time in the airborne transceiver. The maximum error of the beacon delay is 1 microsecond (or 0.2nm). As indicated earlier, it is possible to improve the accuracy of the airborne DME transceiver clock by substitution of the local crystal oscillator counter frequency with that from a more accurate source., thus improving range accuracy. In normal aircraft navigation, past observations are of no use and discarded. In survey navigation all navigation data are recorded and used as a basis for mathematical adjustments during post processing. Atmospheric corrections can be applied by modelling at that time. Through the processes of sampling, data logging and post flight adjustment, it is possible to account for the uncertainty in the ground transponder time delay error. This will improve range accuracy using rho-rho-rho radio trilateration. Assumptions can be based on a daily interval for flight calibration of the network for each remote sensing mission. During a mission of 3 to 5 hours, including transit to and from the data acquisition site, extensive DME range data would be acquired to perform post flight adjustments.

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Individual ground transponder time delay errors sbould be stable over tbat time given tbat drift of their local oscillators should be on tbe order of pico seconds/day.

Technology has permitted the development of frequency agile scanning airborne DME transceivers whicb are digitally programmable. Features of modern transceivers such the one used by the authors permit a higb sampling rate for range data wbere sequential range from beacons can be obtained at a maximum lock in rate of 30 ranges per second.

The advantage of these systems is tbe sampling base they can provide for adjustments. Assuming a survey aircraft ground speed of 120 to 180 knots, the sampling rate for a rho­rho-rho trilateration would correspond to a borizontal ground displacement of 20 to 30 ft. (6 to 9 m) between sequential range triplicate samples, respectively. For a remote sensing mission profile in an area of good DME coverage and geometry sucb as Southern Ontario, range data would be acquired through the entire flight. With a single 3 channel scanning DME transceiver and computer data logging this would represent sampling range triplets at a rate of 200 to 300 per nautical mile. In areas of extensive DME coverage, range data can be acquired by sampling other stations to expand observations in a network for adjustments or additional transceivers can be added to increase the number of stations sampled and to increase the sampling rate on the same stations. To date only one test flight has been conducted which has served to confirm the suitability of the DME concept. Further test flights using an operational version of the software in 'c' language are expected to commence this fall. It would be ideal to report achieving the DME network performance suggested by (Corten, 1988) of 5m to 15m.

SUMMARY

This project is at the early stage of development. To date a significant amount of effort has been involved in acquiring the basic skills and knowledge of the subject area. Although the project is in its infancy, the benefits of the open concept are apparent. This has involved the project teams skills and knowledge to undertake a wildlife inventory application using LORAN C.

Considerable work on adapting technology is required. The next phase of the project will involve the inclusion of a time/frequency base, a precision barometric altimeter and mass storage. The design of airborne calibration/verification methods will require some imagination prior to flight testing. Interest in the post processing of mission data has been expressed by academia as well as a number of other end users in government and industry.

441

REFERENCES

ARINC (1974). Air Transport Equipment Cases and Racking, ARINC Specification 404A, Annapolis, Maryland.

(1975). Air Transport Inertial Navigation System - INS, ARINC Characteristic No.561-11, Aeronautical Radio Inc., Annapolis, Maryland.

(1976). Guidance for Aircraft Electrical Power Utilization and Transient Protection, ARINC Report 413A, Aeronautical Radio, Inc., Annapolis, Maryland.

(1980). Flight Management Computer System, ARINC Characteristic 702-1, Aeronautical Radio Inc., Annapolis, Maryland.

(1981). Digital Information Transfer System Mark 33-DITS, ARINC Specification 429-5, Aeronautical Radio Inc., Annapolis, Maryland

(1983). Inertial Reference System, ARINC Characteristic 704-5, Aeronautical Radio Inc., Annapolis, Maryland.

CORTEN, F.LJ.H. (1976). Integrated Flight and Navigation Systems, Proc. XIIIth Congress of the International Society for Photogrammetry, Commission 1, Helsinki.

(1984) Navigation Systems and Sensor Orientation Systems in Aerial Survey, International Institute for Aerial Survey and Earth Sciences, Enschede, the Netherlands.

(1988). High Accuracy X, Y, Z - Positioning in Flight, Proc. XVlth Congress of the International Society for Photogrammetry and Remote Sensing, Commision 1, Kyoto, Japan.

MORRISON, R. (1986). Grounding and Shielding Techniques in Instrumentation, John Wiley and Sons, Inc., New York.

DOD-DOT (1988). Federal Radionavigation Plan, DOD Report No.4650 DOT Report No DOT-TSC-RSPA-88-4, National Technical Information Service, Springfield, Va.

RTCA (1978). Minimum Performance Standards Airborne Distance Measuring Equipment (DME) Operating within the Radio-Frequency Range of 960 to 1215 MHz, RTCA Doc. No.DO-151A, Washington, D.C.

SPITZER, C.R. (1987). Digital Avionics Systems, Prentice Hall, Englewood Cliffs, N.J.

DISCLAIMER - Mention of specific equipment and manufacturers is not to be construed as an endorsement by the Ministry of Natural Resources

442

A DISCUSSION OF GPS/INS INTEGRATION FOR AIRBORNE PHOTOGRAMMETRIC APPLICATIONS

M.E. Cannon K.P. Schwarz

Department of Surveying Engineering The University of Calgary

2500 University Drive N.W. Calgary, Alberta, Canada, T2N IN4

ABSTRACT

The concept of using integrated GPS/INS for precise aircraft positioning is discussed and the integration algorithm is briefly reviewed. Airborne tests jointly conducted by the University of the Federal Anned Forces, Gennany, The University of Calgary, and the Rheinbraun Company, Gennany are used to assess the accuracy of the integrated system. Two Trimble 4000SX receivers, an LTN 90-100 strapdown inertial system and a Zeiss RMK 15a/23 photogrammetric camera were used in the experiment allowing for a comparison of perspective centers of the camera detennined from traditional photogrammetric techniques and from GPS/INS integration. Results show that decimetre accuracy is achievable. The accuracy of the integrated system for attitude determination is also assessed but a conclusive result is not possible at this time. Relative angular velocities between the camera and the INS may have caused discrepancies in the respective attitudes.

INTRODUCTION

The integration of a Global Positioning System (GPS) receiver and an inertial navigation system (INS) for precise positioning has been demonstrated in land mode (e.g. Lapucha et al.,1990). Testing in an airborne environment is more difficult due to the unavailability of control in this dynamic setting. However, if the feasibility of using the integrated system for decimetre-accuracy positioning can be shown, many new applications using this technology will become a reality. One such application is aerotriangulation without ground control. Studies have shown that replacing conventional ground control with precise coordinates of the camera's perspective centre (PC) at exposure times, the need for ground control can be reduced, if not eliminated (see Schwarz et al. (1984) and Goldfarb (1987) for details). A test using GPS-only for aerotriangulation without ground control is reported in Keel et al. (1989), with GPS accuracies being 85 cm horizontally and 40 cm vertically. Only four satellites were tracked simultaneously during this test. With the advent of all-in­view GPS receiver technology, the increase in data rate and the addition of an INS, not only will these accuracies will be increased, but the reliability of the results will be strengthened.

The objective of this paper is to analyze GPS/INS data collected in an aircraft during a photogrammetric mission in Germany in 1988. With precise PC coordinates available from a bundle block adjustment, the accuracy of the integrated system can be assessed in pure kinematic mode. Cycle slip detection and correction capabilities of the GPS phase data by the processing scheme can also be verified. Finally, the feasibility of using this system for precise attitude detennination is discussed with regards to its contribution in a bundle adjustment.

443

GPS/INS MATHEMATICAL MODEL

The GPS and INS data are processed using a Kalman filter algorithm with the following states,

where

and

n, e, u

Ov b d

(1)

· .. refer to the north, east and up directions, respectively, · .. is the misalignment, ... are the corrections to latitude, longitude and height,

respectively, ... is the correction to velocity, · .. is the residual acceleration bias, · .. is the residual gyro drift.

GPS measurements are used as updates to the Kalman filter. Both carrier phase and Doppler frequency (phase rates) are used in a double difference mode. Kalman filter equations are not given here but can be found in Gelb (1974), for example. For details on the INS dynamics matrix and forcing function see Wong (1988).

Prediction Equations

Yes

Compute double differenced phase and

phase rates

Chec for cycle slips

in phase data

Kalman ~-----i Filter Update l"1li1-_'"

Equations

Fig. 1. GPS/INS integration scheme.

This approach of a centralized Kalman filter using measurements as updates as opposed to a de-centralized (cascaded) filter using GPS positions and velocities has several advantages. Lapucha et al. (1989) discuss the de-centralized approach where a six-state Kalman filter is used to process the GPS data estimating position and velocity. These are fed into the Kalman filter containing the 15 INS states (see Eq. 1). However, since the six GPS states are a subset of the INS states assuming double difference processing and known ambiguities, consolidating the two systems into one filter reduces processing time

444

significantly. The centralized approach can give superior results due to the 'tighter' integration and is especially beneficial for real-time applications where processing speed is a major concern (see Dayton and Nielson,1989). Fig. 1 gives a schematic of the integration approach where the INS gives the reference trajectory while the much lower rate GPS measurements are used as updates to that trajectory. The near-continuous data rate of the INS is well-suited for position-referencing external events such as camera exposures in this present case.

The success of the centralized filtering scheme described above is dependent on the ability of the integrated system to detect and correct cycle slips in the GPS carrier phase data. If this can be done properly, the GPS phase data (or alternatively ambiguity) can be corrected and no additional terms need be added to the state vector given in Eq. 1. The high relative accuracy of the INS can be used for this purpose. By using the predicted GPS antenna position at the measurement epoch in the computation of the 'approximate' double difference, it can be compared to the measured double difference and any significant misclosure could be attributed to cycle slips. If this is the case, the ambiguity on that particular double difference pair can be corrected by the number of cycles slipped. It is not even necessary to know on which satellite the slip occurred, i.e. the base or non-base satellite. The advantage of correcting the ambiguity instead of the raw data is that the correction is instantaneous, rather than correcting all subsequent measurements by the cycles slipped. Fig. 2 shows the cycle slip detection and correction scheme. The benefit of GPS/INS integration for cycle slip detection and correction is that the number of satellites that have cycle slips at anyone instant is irrelevant. In contrast, GPS-only positioning requires at least four cycle-slip free measurements to detect cycle slips on the redundant measurements, see Allison (1989) for example.

In order to correct cycle slips at the one cycle level (20 cm), the relative accuracy of the INS must be good to a few cm between GPS measurement epochs. Therefore, a high GPS data rate is beneficial to ensuring that this accuracy criterion is met. Periods of satellite shading that cause GPS data gaps may reduce the ability of the INS to correct cycle slips below the one cycle threshold.

Compute double difference using computed satellite coordinates and

INS-predicted position, l1 Vp

Form double difference using measured carrier

phase, AVC\>

Compute: 5= l1Vp-l1VC\>

Fig. 2. GPS/INS cycle slip detection/correction scheme.

AIRBORNE GPSIINS TEST DESCRIPTION

K a I m a n

u P d a t e

Airborne tests were carried out in August-September, 1988 near Cologne, Germany. Although three days of tests were performed, only the August 31 data are reduced due to

445

severe satellite shading problems on the other two days. The main hardware components used for the test were two Trimble 4000SX GPS receivers, a Litton L TN 90-100 strapdown inertial system and an RMK 15a/23 photogrammetric camera. Portable computers were also used for data collection and time-tagging purposes. The monitor GPS receiver was located in the photogrammetric area while the other was located along with the INS in the the fIxed wing Cessna aircraft. The GPS antenna was mounted on the fuselage of the aircraft.

At the start of the run, a sufficient amount of static data could not be collected at the initial baseline. However, at the end of the run a 20 minute static survey was performed. Due to this limitation, it was decided to process the data in a reverse sequence. Fig. 3 shows the reversed aircraft trajectory, i.e. from the end of the run to the start of the photography. Flying height was about 700 m and the photo scale was 1 :5000. Aircraft speeds reached 250 km/h (70 m/s). The airport is located at the north end of the fIgure. As the diagram shows, the initial baseline is approximately 30 km in length. Ideally, the initial baseline should be smaller, no more then 10 km to ensure that the correct integer ambiguities can be recovered. This may require the addition of multiple monitor receivers in the flight area. About 72 minutes of data were collected including the static data at the initial baseline.

51.28 _

51.18 _

50.98

Fig. 3. Aircraft trajectory.

Initial Rover '. • ••••• ' •••••.. '-.~£ Point '. ---. . '" ....... .

6.45 Longitude (deg)

'. '. '. ". '. ".

6.55

GPS data were collected at a 0.25 Hz (once every 4 seconds) and fIve satellites were tracked throughout the run. Due to the time period in which this campaign was held, Selective Availability (SA) was not a concern during data collection. INS data were logged at a 64 Hz rate. The data were time-tagged through the 1 PPS output of the GPS receiver and also through time marks that were interrogated from the GPS receiver. Time-tagging with the camera was accomplished through an HP portable computer. There were 13 strips

446

of photography taken of which 8 were processed. The time between exposures during a strip was about four seconds.

A photogrammetric block adjustment was performed by Rheinbraun using all the available ground control. No information from the GPS/lNS system was used in the bundle adjustment. Perspective centre (PC) coordinates of the camera at the exposure times were computed and then offset to the GPS antenna using the measured translations between the two. Camera orientation parameters were also output from the adjustment. Accuracy of the PC coordinates is a few cm and a few arc seconds in orientation.

POSITIONING RESULTS

The GPS and inertial data were processed using the GPSIN program developed at The University of Calgary which utilizes the centralized filtering approach discussed in the previous Sections. Since the run was processed in reverse sequence, all raw measurement files had to inverted and minor software changes had to be made to cope with descending time. Time-tagging problems between the GPS and INS data were evident mainly due to the poor timing information interrogated from the GPS receiver. The GPS time-tag was only accurate to ±20 ms which at the aircraft speed, is 1.4 m. Trial and error was used to find the best fit of the GPS and INS time scales. Time-tagging problems have been corrected since this test.

50

__ 25 e ~ ~

~ 0 f ~ Q -25

o

75

o 200 400 600 800

Kinematic Epoch

Fig. 4. Differences between measured and computed double differences for satellite 3 (using satellite 12 as the base satellite).

In order to assess the compatibility of the INS and GPS data, the misclosures of the computed and measured double differences were statistically analyzed. Fig. 4 shows the differences between the measured and computed double differences for satellite 13 (using satellite 12 as the base satellite) at each kinematic epoch. Also plotted on this figure is the aircraft speed. The misclosures are very small for the first 250 kinematic epochs (about 1000 seconds). However, once the aircraft begins some banked turns (illustrated by the sudden changes in aircraft velocity) the misclosures become larger and some exceed one

447

100 _

75 _ ..-e 50 _ ~ -~ 25 _ • ~ = • •• ,., ~ o. •• .. '" .. - • .. . .-.: ~ •

-25 • • • .• Q • -rI.I -50 = • ~

-75

·100 I I I • •

5 15 25 35 45 55 65 75 100 _ Photo

75 . ..-e 50 ~ ~ 25 ~ • ~ • .... = -: • ~ o. • •• y'll •• ~ • • • • .•

·25 _ Q • .c 1:: -50 _ 0 Z -75 _

-100 . . . · 5 15 25 35 45 55 65 75

100 • Photo .

75 • ..-e 50. ~ -~ 25 _ A ~ = ~ .- .. .; '" o. ~ .• 1~~ Q -25 _ -.c V ~ -50 _ .•

~

= -75 •

-100 I . •

5 15 25 35 45 55 65 75 Photo

Fig. 5. Comparison of GPS/INS and photogrammetric camera position.

448

Table 1. Summary of GPS/INS positioning results (for 31 points).

Coordinate Maximum RMS Total Difference (em) Difference (em) RMS (em)

North -55.6 14.6

East 31.5 14.6 18.5

Up_ -47.9 24.5

cycle. This is most likely due to time-tagging problems between GPS and INS but may also be attributable to filter overshooting. The rms and maximum difference of the misclosures is 5.7 cm and 35.7 cm, respectively. This shows that the cycle slip detection and correction of this particular data set is not at the one cycle level, as desired. Numerous cycle slips were, however, detected during the mission.

The coordinates of the GPS antenna and the attitude of the INS were output at the camera exposure times. Since these times did not coincide with either a GPS or INS measurement, INS rates were interpolated between two successive epochs to ensure that the integrated system would give position and attitude at the exact camera exposure time. Since the INS data rate is 64 Hz, this interpolation will not introduce significant error into the results.

Before GPS/INS positions could be compared with the photogrammetric results, a three parameter translation had to be applied to the integrated results to transform them to the local datum in which the photogrammetric results wer~ computed. Since no translation parameters were available between WGS-84 and this local datum, the first strip of the photogrammetric results was used to estimate the three translations. These parameters would then be independent of the remaining strips. Fig. 5 shows the differences between the GPS/INS and photogrammetric positions for 31 points after the translation has been applied. For each of the three dimensions, namely, north, east and height, the discrepancies are generally a few decimetres. A few outliers exist in the east component. A slight drift is evident in height and its cause is currently being investigated. Table 1 gives the maximum and rms differences between GPS/INS and the photogrammetry. The maximum difference is -55.6 cm in easting, 31.5 cm in northing and -47.9 cm in height. Rms differences are 14.6 cm, 14.6 cm and 24.5 cm for each the three components. When considering all three dimensions together, the rms is 18.5 cm. These results are slightly worse then those reported by Baustert et al. (1988) using TI 4100 P-code receivers in the same test area. This may be due to time-tagging problems with the Trimble receivers as outlined in the previous Section.

ATTITUDE DETERMINATION

The ability of the GPS/INS system to provide accurate and reliable attitude information will create many new and exciting applications of the system. For photogrammetric applications, accuracies of 10 arc seconds or better are required in order to benefit the block adjustment (Schwarz et al.,1984). In order to assess the feasibility of using GPS/INS for precise attitude determination in an aircraft environment, one strip of the flight was analyzed. Figs. 6 and 7 show the roll and azimuth of the aircraft for the strip (about 25 seconds of time) generated from GPS/INS results. Note that the roll component has a higher frequency than azimuth, which is to be expected; typical aircraft dynamics at the few Hz level is evident in roll.

Attempts to compare GPS/INS-derived attitudes with the photogrammetrically-derived orientation gave very large disagreements, especially in roll and pitch. This is currently

449

being investigated. One possible explanation is that the discrepancies are due to the hardware installation during data collection. If the camera is mounted in vibration shocks during flight, relative movements between the INS and the camera may be significant enough for the orientation of the two systems to be incompatible. There is also the possibility that the noise on the gyros is too high and precise attitude determination may not be feasible.

2

1 --~ Qj 0 '0 ----0 -1 ~

= ~ -2 s.. ~ .-< -3

-4 0 5 10 15 20 25

Time (sec)

Fig. 6. Aircraft roll for one photogrammetric strip generated from GPS/INS.

332

--~ ~ 330 --.c -§ .- 328 N

< = ~ s.. 326 y s.. .-<

324

0 5 10 15 20 25 Time (sec)

Fig. 7. Aircraft azimuth for one photogrammetric strip generated from GPS/INS.

In order to test this hypothesis, the noise of INS gyro measurements was computed using a Fast Fourier Transform (FFT) on the raw gyro data to compute the power spectral density

450

(PSD). Fig. 8 shows the PSD for the y gyro. Several distinct spectral ranges can be identified. For frequencies smaller than 3 Hz., aircraft dynamics dominate. Between 7 and 11 Hz., there is some engine vibration, apparently centered around one of the aliased dither frequencies at about 9 Hz. The other aliased dither frequencies are at approximately 22 and 24 Hz. Otherwise, there seems to be mainly system noise above 10 Hz.

0.08

0.06 Q 00 ~ Q 0.04 "" ~ ~ • ~ 0.02

0.00 0 8 16 24 32

Frequency (Hz)

Fig. 8. FFf of raw y-gyro INS output.

By eliminating the dither frequencies (Czompo,1990) and integrating only the noise above the aircraft dynamics, a fairly reliable estimate of the measurement noise can be obtained. By applying appropriate smoothing techniques for region below the dynamics threshold, a dependable estimate of the resolution of the aircraft attitude dynamics through INS can be obtained. Assuming the threshold to be 3 Hz., the 1 a-attitude errors are 59, 69 and 40 arcsec/sec for the x, y, and z gyros, respectively.

CONCLUSIONS

The feasibility of using an integrated GPS/lNS for precise aircraft positioning has been demonstrated. Test results show that decimetre accuracies are achievable using the GPS hardware and constellation from mid-1988. Improvements in system time-tagging and the operational GPS constellation will increase both the accuracy of the integrated system, and also the reliability of the results.

The accuracy of attitude angles determined from GPS/INS is dependent on the noise of the gyro measurements. Also, the relationship between the INS and the photogrammetric camera during the mission is of utmost importance if GPS/INS-derived attitudes are to be considered in a bundle block adjustment. Any relative movement between the two systems will be detrimental to the usefulness of the GPS/lNS attitude information. These two aspects are being investigated to determine the cause of the discrepancies between the independently derived orientation parameters.

In order to assess the practicality of aerotriangulation without ground control, the next phase is to use the GPS/INS-determined PC coordinates as control in the bundle block adjustment. This aspect is currently being investigated.

451

ACKNOWLEDGEMENTS

The authors would like to thank Joseph Czompo for computing the FFf's needed for the preparation of this paper. The University of the Federal Anned Forces, Munich and the Rheinbraun Company are also acknowledged for their assistance in the data collection and reduction.

REFERENCES

Allison, T. and R. Eshenbach (1989). Real-Time Cycle-slip Fixing during Kinematic Surveys, Proceedings of the 5th International Geodetic Symposium on Satellite Positioning, Las Cruces, March 13-17.

Baustert, G., M.E. Cannon, E. Dorrer, G. Hein, H. Krauss, H. Landau, K.P. Schwarz and Ch. Schwiertz (1989). German-Canadian Experiment in Airborne INS-GPS Integration for Photogrammetric Applications, Proceedings of the lAG Geodesy Symposia 102, Springer Verlag, New York.

Czompo, J. (1990), Use of Spectral Methods in Strapdown ISS Data Processing, Proceedings of the International Symposium on Kinematic Systems in Geodesy, Surveying and Remote Sensing, Banff,September 10-13, Sprioger-Verlag, New York.

Dayton, R.B. and J.T. Nielson (1989). A Flight Test Comparison of Two GPS/INS Integration Approaches, Proceedings of the ION GPS-89, The Institute of Navigation, Washington, D.C.

Gelb, A., ed. (1974). Applied Optimal Estimation, M.I.T. Press, Cambridge, Massachusetts.

Goldfarb, J.M. (1987). Aerotriangulation Using an INS-Differential-GPS, Report No. 20022, Department of Surveying Engineering, The University of Calgary.

Keel, G., H. Jones, G. Lachapelle, R. Moreau and M. Perron (1989). A Test of Airborne Kinematic GPS Positioning for Aerial Photography, Photogrammetric Engineering, Vol. 55, No. 12, pp. 1727-1730.

Lapucha, D., K.P. Schwarz, M.E. Cannon and H. Martell (1989). The Use of INS/GPS in a Highway Survey System, Proceedings of the IEEE PLAN'S 90, Las Vegas, March 20-23.

Lindenberger, J. (1989). Quality Analysis of Platform Orientation Parameters for Airborne Laser Proftling Systems, High Precision Navigation, Springer-Verlag, Berlin.

Schwarz, K.P., C.S. Fraser and P.C. Gustafson (1984). Aerotriangulation without Ground Control, International Archives of Photo gramme try and Remote Sensing. Vol. 25, Part AI, Rio de Janeiro, June 16-29.

Wong, R.V.C., K.P. Schwarz and M.E. Cannon (1988). High Accuracy Kinematic Positioning by GPS-INS, Navigation, Journal of the Institute of Navigation, Vol. 35, No.2, pp. 275-287.

Wong, R.V.C. (1988), Development of a RLG Strapdown Inertial Survey System, Report No. 20027, Department of Surveying Engineering, The University of Calgary.

452

EXPERIMENTAL RESEARCH ON A STRAPDOWN INERTIAL NAVIGATION DEVICE

WORKING UNDERWATER

ABSTRACT

Zhang Shuxia Sun Jing

Harbin Shipbuilding Engineering Institute People's Republic of China

Exploration as well as exploitation of ocean resources requires accurate underwater navigation for manned as well as remotely controlled vehicles. Thus, more and more attention is given to systems that can be guided to specific locations and can be kept there for any length of time. Such systems have to sense linear and angular velocity to provide position output and guidance. Strapdown inertial navigation systems, because of their small size and high precision, are ideally suited for this purpose. This paper discusses the following implementation problems of such a system:

• The design characteristics of the IMU (inertial measurement unit) with respect to heat transfer and static and dynamic performance.

• The design of a horizontal heading reference to attain Euler angles and velocity.

• A method for decoupling control of the two-degree-of-freedom dynamically tuned gyroscopes in the IMU in order to upgrade its controlability.

• Algorithm design for data collection, error compensation, real-time computation and communication with other control systems.

• An analysis of the error detection and separation capability, including a self-test of the redundant Z-axis monitor.

INTRODUCTION

Inertial navigation units provide heading and azimuth for systems working under water. The cost, reliability and maintenance of gimballed platform gyro compasses are not satisfactory while the strapdown scheme offers considerable savings in weight and cost. As inertial systems are installed directly on the vehicle, the velocity sensors need to have a wide dynamic range. Furthermore, a high speed computer with large capacity is needed.

453

OVERVIEW

Configuration

The proposed inertial navigation system uses strapdown technology with two dynamically tuned rotor gyros and three quartz hinged pivot accelerometers. It consists of the following subsystem components:

Inertial Measurement Unit (IMU), AID, D/A modules, Processors, Power supply with well-controlled output.

-b \\ bi

.. ...

.. ...

G

A

IMU ~ ....

... "

Fig. 1. System diagram.

...

A/D,D/A

T Power supply

v

! ! ... ... attitude !C, '1', cj)

Computer t.... Euler angles P, Q, R

.oil

The inertial navigation system is a heading and azimuth system aided by Earth-referenced velocity from a Doppler Sonar. It provides the following outputs:

Body angular rates, Body attitude, True heading, Euler angular rates, Euler angles.

Principle of Mechanization

The operating modes of the system are controlled by the computer. The operating modes are:

Optimal Self-Alignment (stationary, moving base), Transfer Alignment, Gyrocompassing, Basic Navigation.

454

Orientation of the Coordinate Systems

Inertial frame (i) OXiYiZi

°XeYeZe OENZ

Earth frame (e) Navigation frame (n)

Body frame (b) Gyro frame (g) Accelerometer frame (a)

OXbYbZb OXgYgZg OXaYaZa

(starboard, forward, upward) (input axis, output axis, spin axis) (input axis, output axis, z-axis)

-OZ oXbl OENZ ~ 0JbI'hI1t>1 ~ O~2'h2Lb2

Yaw (,q Pitch ('I')

OYb2 --~~ OXbybz"

Roll (cI»

The input axes of the three accelerometers are aligned with the ship's body axes. The gyro spin axes are oriented along the level axes. The gyro input axes are skewed from the vertical by 45° for self-test purposes and to provide a uniform thermal operating environment and a redundant Z-axis.

SOFTWARE

Software Block Diagram

The software uses the following algorithms:

Quaternions are propagated in time, i.e. the attitude matrix is formed by integration and the Euler angles are extracted.

The measured accelerations are transformed to the navigation frame.

Corrections for Earth rate are implemented.

Corrections for Coriolis accelerations are implemented and the velocities are damped by Doppler Sonar velocities.

The body Euler angles P, Q, R and their rates are calculated.

Algorithms

The quaternionQ is defmed by

. 1 b Q = 2 QCObi

i.e. it is a rotation operator Q = [QO,Q}'Q2,Q3]. The incremental gyro output is

T+~tl2

~eCi = J 0> Ci(t) dt T-~tl2

455

(1)

(2)

correction for "'0 Cariolis

~eration

error compensation

- b accelerometer Fbi --... output

gyro output

error compensation

damping loop

Fig. 2. Block diagram of software.

incremental angle in b

b AS bi

Euler angle p

computation Q R

Qextracted C n attitude b

1C 'II

'"

The angular rate of the navigation frame with respect to the inertial frame defined in the body frame is ro~ and the corresponding incremental angle is obtained from

AS~ = ~ ro~i At (3)

Equation (1) is implemented by using a discrete time updating algorithm

QK+l = QK M (4)

where M is the transition quatemion determined by integration. A third-order power series was used for the integration because it gave the optimal results in computer simulation studies.

The algorithm is of the fonn

di {M} - (1_IASI2 IAE>14. AS IAE>12 AS IASI4 AS) (5) ag - 8 + 384' 2 - 48 Ll + 3840 Ll

with AS = AS:n and IASI = " ASx2 + ASy2 + ASz2 .

Once the quatemions are integrated via equation (5), the quatemion attitude matrix can be constructed

( q02+q12_q22_q32 2(q}q2+qoq3) 2(qlq3-qoq2»)

C~ = 2(qlq2-q()C).3) Q02q12+q22_q32 2(Q2Q3+Qoqt) 2(QlQ3+qoq2) 2(Q2Q3-Qoql) Q02_Q12·Q22+q32

(6)

456

It is equivalent to the direction cosine matrix which can be derived using the body frame orientation defmed above

( cosKcos<j>+sinKsin'l'sin<j> cosKSin'l'sin<j>-sinKcos<j>

C~ = cos'l'sinK cOS'l'COSK-

cosKsin<j>-sinKsin'l'cos<j> -sinKSin<j>-cosKsin'l'cos<j>

The Euler angles can thus be extracted from equation (6) as

K = arctg C21 C22

'I' = arcsin C23

C31 <j> = - arctg C33

Approximation Errors

The approximation errors of the algorithms are as follows:

Drift error

Scale error

.1804 480 .18i

1 --.1804

192

where .18i is the sampling value and ~80 = 1~81.

Design Parameters

-sin~os'l' ) sm'l'

cos<j>cos'l'

(7)

(8)

The sampling period for the gyros and accelerometers is 8 ms. The quaternions are updated every 24 ms, and the attitude is output every 48 ms.

PERFORMANCE ANALYSIS

Analysis of Stationary Self-Alignment

In the stationary mode, the body Euler angles and the angular rates are derived from the outputs of the two accelerometers. The relationship between the level frame and the body frame is denoted by a and ~:

Let BAI and BA2 denote the outputs of the two accelerometers, then

~ = BA2

457

sinoA1 sina = ---

cosoA2

The accelerometer bias is 100 Jlg and the verticality error is 1<Pi1 = 0.'54.

(9)

When the body frame at alignment is level and at vehicle true heading, then the transformation matrix is defined by pitch and roll only, and the sensed Earth rate vector is resolved into forward and starboard components:

!'lx = Ole coscp sinx:

!'ly = Ole coscp cosx:

Heading can be obtained by

~ = _ arctan !'lx !'ly

Expressed as true value plus error term 0 gives

- (Ox + oOx) tan ( x: +0 x:) = -"'----'''------=:...

Oy+oOy

and simplified

£ ax: = ---

Olecoscp

where £ is 0.05/h when cp = 45°43'. This results in ax: = 4.77 mrad.

Analysis of Moving-Base Self-Alignment

(10)

(11)

(12)

Assume a moving system at a velocity of 20 knots and a misalignment angle of the Doppler Sonar of 0.°5, then the additive angular rate error is 0.003/h. This is much smaller than the gyro drift error £=0.05 ° /h and, therefore, the error caused by velocity is omitted.

Assuming a dynamic accelerometer bias variation of 1000 Jlg, the tilt performance is I <Pi I = 1 mrad. So the alignment error increases.

Under higher linear and angular acceleration the self-alignment performance degrades. Therefore, the moving base self-alignment should be performed under constant velocity. The IMU should be installed in the centre of motion.

Performance Overview

Heading error ~x: < 0.°5 (1a)

Roll and pitch error ~"', ~cp < 0.°5 (la)

458

HARDWARE

Casing

The casing is a high quality non-ferrous alloy cast with good mechanical properties and excellent corrosion resistance. The use of a cast allows the case to be one piece containing the mountings as well as the mounting surface for the sensor block, thus minimizing the cumulative effects of tolerance buildings and mechanical distortion.

This type of casing provides the following advantages:

Convenient for testing and sensor exchanges, Elimination of errors caused by improper mounting of individual sensors, Reduction of installation time and elimination of extra wiring, Reliability .

Two-degree-of-freedom dynamically tuned rate gyros

The gyros used are rate gyros with two measuring axes. Each gyro has two rebalance loops which are measuring control systems. Microprocessors are used in the rebalance loops. The microprocessor controlled module optimizes gyro performance and also influences the design scheme.

The main problem is the decoupling of the rebalance loop. A simple digital control method has been found to change the parameters of the decoupling control matrix in order to increase the dynamic performance and tracing ability of the gyro. The use of a mini­digital controller, which includes AID and D/A modules, is being researched.

Quartz-hinged pivot servo accelerometer

The accelerometer is a force rebalance servo system. It is characterized by high accuracy, stability, low drift, and heat stability. It consists of transducers, servo amplifiers and torquers.

Digital processor

Computer and IMU constitute a strapdown unit. Laboratory test can be easily performed on a rotation table. Hardware requirements for the strapdown computer are:

Fast and accurate data acquisition from the IMU, Real time error compensation and strapdown calculations, Communication with other control systems, Control of the rebalance loops.

In order to meet the above requirements, an AST/286 personal computer has been selected. It has a 1024 kb RAM with an 80287-8 NDP which performs all the floating point computations. Its main frequency is 10 MHz. The computer, AID and D/A modules, and the other systems are linked through buses.

459

RELIABILITY OF THE DEVICE

The device has been designed as a redundant system with a pair of two-degrees-of-freedom gyros and one redundant Z-axis monitor. The self-test capability has been realized. The configuration of the measuring axes is based on the following considerations:

simplicity of computations, reliability, minimization of measurement error.

Since there are four measuring vectors, one redundant signal can be used to test for and eliminate sensor errors.

CONCLUSIONS

The data analysis and test results show that the investigated inertial navigation device will have lower cost, smaller size and lower weight, and that it will be easier to service than conventional gimballed systems. With some hardware and software modifications the device becomes a strapdown navigator without compromising cost-effectiveness.

REFERENCES

Giardian, C.R., Heckathorn, J., Krasnjansk, D. (1981) A comparative study of strapdown algorithms, Journal of Navigation 28.

Pen Yuxiang and Xie Tianhuai (1989) Experimental investigation on strapdown inertial system, BISIT, May 1989.

Technical proposal. Strap-down gyrocompass and vertical reference system.

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SESSION6a

GRAVITY AND ATTITUDE APPLICATIONS

CHAIRMAN R.FORSBERG

NATIONAL SURVEY AND CADASTRE COPENHAGEN,DENMARK

THE ROLE OF GPS/INS IN MAPPING THE EARTH's GRAVITY FIELD IN THE 1990's

INTRODUCTION

Oscar L. Colombo University of Maryland Astronomy Program

NASA Goddard Space Flight Center, Code 926, Greenbelt, Maryland 20771, U.S.A.

In 1975, in Australia, the author was making his ftrst acquaintance with geodesy, in what he then believed to be a temporary departure from electrical engineering, by working all-night computer shifts to create a world data base of gravity values for the late Ron Mather. Much of the world is covered by oceans and, at that time, judging from the scant information going into Ron's data base, little was known of the gravity fteld over much of those oceans. Most data was from sparse and highly diverse ship and submarine surveys. Over the continents, the situation varied dramatically from region to region: one could see the outline of Australia in a density plot of gravity stations, thanks to the intensive geophysical prospecting that had been made there in search of minerals. But little could be seen of Africa, South America, and of far away Antarctica hardly anything at all.

That same year, however, something was to happen that would begin to change the situation dramatically, as far as the oceans were concerned: the launching of GEOS 3, the ftrst of NASA's altimeter satellites. Over the years, other altimeter missions followed, covering the oceans with detailed charts of the marine geoid: SEASAT, GEOSAT, with more still to come. As it is well known, the ocean surface does not coincide exactly with the geoid, because of the dynamic topography associated with geostrophic currents. From the oceanographer's point of view, a precise geoid derived independently from any altimetry whatsoever data remains today a fond wish. For a geophysicist, mostly interested in crustal signatures on the geoid of several meters in amplitude, the oceanographic signal is a rather small component of the noise, and geophysics has made great progress in understanding the structure of the crust and upper mantle in ocean regions thanks to satellite altimetry.

Over the continents, however, things remain patchy. Figure 1(a) shows a situation typical of many "well surveyed" regions (in this case, the area of Denver, Colorado). Most gravity measurements are taken in places easily accessible by road, and this region is no exception: the dots are gravity stations, and they show clearly the outline of an Interstate highway reaching a city built at the meeting of plains (right halt) and hilly country (left). The checkerboard layout of the streets, the wandering roads into the hills, and other characteristics of the city are clearly visible. This is what could be called the "highway effect".

Figure 1(b) shows a recent gravity survey from the air, with a fairly regular distribution of measurements in a systematic grid pattern over a large, swampy region, hard and slow to survey by conventional methods. The contrast shown here dramatizes the promise of airborne gravity measurements to provide fine resolution with dense coverage of uniform quality over all sorts of regions, including hard-to-get-to ones (ice caps, tops of mountain ranges, the oceans), at a reasonable cost.

463

Figure la Figure Ib

Figure 1. (a) The "highway effect" (gravity stations are normally set up along streets, roads, and similarly accessible places; from Milbert [1988]). (b) Airborne gravimetry promises uniform coverage of regions hard to survey by land, such as this low-lying, swampy region in eastern North Carolina (from (Brozena and Peters, 1988]).

BACKGROUND

In recent years, the results of several experiments have shown that with current differential, kine­matic GPS techniques, it is possible to determine the instantaneous position of a moving vehicle with decimeter accuracy at intervals of one second or less, relative to a fixed station [Krebill and Martin, 1987; Mader and Lucas, 1989]. Such results have been obtained by post-mission analysis of precise carrier phase data. Given the improvements in receiver capabilities, including that of tracking simultaneously eight or more GPS satellites, the availability of very powerful micro computers for data logging and analysis, and the appearance of continuously operating, world­wide nets of fiducial stations like the Cooperative International GPS Network (CIGNET) [Chin, 1989], one can expect, in the near future, to see such high accuracy navigation being realized regardless of distance travelled and velocity, at least in post-mission data analysis.

Knowing a vehicle's trajectory, it is possible to estimate its total inertial acceleration. The gravi­tational component of this acceleration can be obtained in three ways: using models of the non­gravitational forces to subtract them from the total motion; physically eliminating those forces; sensing them separately, with accelerometers and gyroscopes or star trackers. The first approach is that used in satellite geodesy to obtain gravitational information from tracking data of spacecraft high enough that air drag effects are small compared to gravitational ones, the data in this case be­ing the measurements ofGPS receivers in the spacecraft themselves [Melbourne and Tapley, 1983; Colombo, 1990a and 1990b]. The second method can be implemented with drag-free satellites (e.g., Gravity Probe-B) [Smith et al., 1988; Everitt et al., 1989. The likely gain in knowledge of the gravity through GPS tracking, at long wavelengths (here resolution is approximately half the height of the satellite), may be very great: more than two orders of magnitude improvement in accuracy over the best field models available today. In fact, because of the very high sensitivity of GPS tracking data to the longest wavelengths, it may be possible to map both in latitude and in longitude, for the first time, the broad geographical features of the faint variations in the gravity

464

field caused by seasonal and secular redistributions of mass in the atmosphere, the oceans and the solid Earth, some of them of particular interest in the study of global climatic change.

In space, the third technique, combining GPS and INS technology, has been proposed for some gravity experiments in the Space Shuttle [Jekeli, 1988; Jekeli and Upadhyay, 1990] and also for the planned European mission ARISTOTELES, while, on Earth, it is being tested in the form of GPS-assisted mobile gravimetry [Brozena et al., 1989].

In general, the nearer to the Earth's surface that measurements are taken, and the slower the ve­hicle, the greater the resolution of detail that can be achieved. IT the vehicle is an orbiting space­craft, then closely spaced, world-wide geographical coverage can be obtained in a few weeks, with a gravity field resolution of a few hundred kilometers. With aircraft, areas as large as Africa or Antarctica could be surveyed with a resolution of ten kilometers or better in a few months of total flying time and at a reasonable cost. Experimental evidence indicates that, with airplanes, the measured scalar value of the gravity vector g (i.e., the acceleration of gravity), is close to three milligals in accuracy and a resolution of 10-20 km [Brozena and Peters, 1988; Bell et al., 1988; Bell et al., 1990]. Such results have been achieved by using radar altimeters while flying over water, to estimate the high frequency part of the vertical non-gravitational acceleration. GPS navi­gation based on pseudorange has been used to obtain velocity, and from this the low frequency part of that acceleration, or Eotvos term (the vertical Coriolis effect caused by vehicle motion rela­tive to the curved Earth). The accuracy of the results is limited by that of the altimeter. Depending on the instrument, this accuracy lies between 0.1m and 1 m, and may be altitude­dependent. The estimated accelerations are obtained by differencing twice successive measurements and smoothing over a period of some 100 seconds. So, at aircraft speed of 100 m/sec., the resolution would be 10 km. A description of the airborne instrumentation used at the Naval Research Laboratory is given in [Brozena et al., 1986]

A further step would be to combine existing GPS and inertial navigation equipment to obtain the vector g in magnitude and direction (vector gravimetry) [Knickmeyer and Schwarz, 1990]. Integrating the horizontal component of this vector parallel to the instantaneous flight-path would give geoid proftles [Forsberg, 1989] that could be extended for some tens of kilometers at either side using the component normal to that path, into geoidal swaths. All this has important conse­quences for the geosciences, as well as for geodesy and geophysical prospecting. Extensive arguments concerning the role of gravity data in general, and satellite and airborne gravity in particular, can be found in [Lambeck, 1988], and in [Mueller and Zerbini, 1989].

AIRBORNE GRAVIMETRY, GPS, AND INS

The integration of GPS with inertial navigation systems (INS) is an area of active development, recently reviewed by G. Lachapelle and K.-P. Schwarz [1990]. The various advantages of combining both types of navigation have been summarized by R. G. Hartman [1988]. The use of INS updated with velocity or position values (when possible) at points along the way not just to refine navigation but to measure gravity as well (deflections of the vertical in particular), has had now a long career, somewhat limited in achievement. The reason, as in conventional airborne gravimetry, has been the frequent lack of high-quality update data. In cases where it is possible to stop the vehicle altogether, the velocity is known exactly (in earth-fixed coordinates, it is precisely zero), and results for traverses over land have been very successful with some specially built INS's [Huddle, 1988]. Having previously measured deflections at some of the stops also helps, when they are included in the post-mission analysis. Some or all of these elements may be present in a traverse along the roads of a well-surveyed region, but would be partially or completely lacking in most deserted regions, at sea, or in the air. This situation is now about to change drastically, for the same reason that conventional gravimetry from airplanes and other vehicles is suddenly becoming feasible. GPS data can be used to provide very precise position updates off­line. Conceptually, one could imagine a series of (x, y, z) vehicle coordinates that have been measured at equal time intervals with GPS. After numerically differencing twice each coordinate, one gets the (quasi) inertial components of acceleration in the frame of those coordinates. (This

465

assumes that the sampling rate for the coordinates is high enough that no significant acceleration aliasing takes place).

Smoothing the noisy accelerations over a sufficiently long interval, one should end up with a sequence of much more precise averaged values. Over the same period of time, one may also smooth the INS acceleration vector, determined from the accelerometers and gyroscopes. The two sets of smoothed accelerations can then be combined to get estimates of the gravitational vector, and from this both gravity anomalies and deflections of the vertical. Since the number of samples needed to achieve a certain accuracy after smoothing is constant, and so is the sampling rate of the GPS and INS devices, the time it takes to smooth out the errors sufficiently is also fixed. The velocity of the vehicle, however, may not be constant, nor are all vehicles capable of maintaining the same cruising speeds. The slower the vehicle moves, the less distance it covers while data is gathered to produce a single smoothed value of gravity; that distance is the resolution with which the gravity proflle is measured.

;I' = Inertial Frame: Acocleralion Veclor

iI v = Vehie le Frame:

il· .. = Rotational

g = Gravitational

a' FROM CPS (FUU. VECTOR ) OR GPS .AL T1MET[R (VERno.L ONLy)

il v , il·.. FROM lnenial N3YIgallon Syslem W,S) OR GRA VIMElU (VERTICAL ONLy)

SCALAR GRAVIMETER : MEASURES Igl

VECfOR GRAVIMETER MEASURES g (Ig~ t.g. E,. '1)

IF a IS AZIMUTH,

o = ~ cos a + T) sin a t = E, sin a + T) cos a

ARE DOWNTRACK, CROSSTRfCK DEFLECTIONS.

t.N = v 0 dt

IS DOWNTRACK GEOID UNDULATfoN (GEOID PROflLD'olG; 't'.cAN

EXTEND PRQFU..E INI'O SW A 1H)

DATA PROCESSING :

(Al FlL TERINGILOGGING RAW SENSOR + GPS DATA

(B) CALIBRATIONS: PRE-. POST-, IN-FLIGHT

(C) POST-PROCESSLNG OF ALL DATA FOR

og = g - y => ~, E" '1 • O. ~

(SINGLE TRAVERSES, FOLLOWED BY :z...D NETWORK ADJUSTMENT)

Figure 2. Basic principle behind vector gravimetry (scalar, or conventional mobile gravimetry may be thought of as the determination of just the vertical component of vector g)

In practice, there are reasons why this conceptually straightforward approach may not be implemented, and the original, undifferenced measurements may be analyzed instead_ However, as an explanation, it is useful for understanding the main characteristics, possibilities, and problems associated with scalar and vector gravimetry. Since in the latter not only the modulus but also the orientation of g are needed, it is clear that the quality of the gyroscopes used either to

466

maintain a known orientation (in a stable platfonn with mechanical gyros) or to measure it (in a strap-down system, usually with ring laser gyros), is quite critical to vector gravimetry. Orientation is much less important in scalar gravimetry, for simple geometrical reasons. The interplay of the various key factors determining the accuracy and resolution of the final gravity estimates (sensors' accuracies, velocity, sampling rates, roughness of the field along the traverse) can be understood on the basis of the few ideas just outlined, and illustrated in Figure 2 (conventional, scalar gravimetry could be thought of as being the use of similar means to determine only the vertical component of g). A very complete quantitative study of the influence of those various factors on the accuracy of the estimated deflections has been done recently by Knickmeyer and Schwarz [1990]. The potential of combining precise kinematic GPS with other airborne instrumentation is suggested in Figure 3, in which an airplane is used as instrumented platform for very precise remote sensing operations (one could think also of some instruments not shown here, such as an interferometric synthetic aperture radar (ISAR) for very precise and efficient topographic mapping, an ice­sounding radar to study the structure of glaciers in depth, or a photogramrnetric camera operated without the need for expensive ground control points). Since both north and south deflections of the vertical are measured, the horizontal component of gravity along the aircraft ground track can be found. Integrating that component gives the geoid profile along the same ground track. This profile can be extrapolated into a swath tens of kilometers wide by means of the gravity component normal to the path. A laser or radar altimeter measures the sea surface height colinear with the geoid profile; their difference is the dynamic sea surface topography (exceedingly laborious, time consuming and expensive to measure by oceanographic techniques). At the same time, an obliquely pointing Doppler radar could measure the aircraft-water relative velocity. As the aircraft velocity can be determined with great accuracy by differencing and smoothing the GPS position fixes, the total current speed can be estimated directly. Subtracting from this speed that of the geostrophic flow inferred from the dynamic topography yields the wind-driven component of the current.

INERTIAL NAV. UNIT Of AIRCRAfT

ALTII1ETER -MEASUREMEIIIT

GEOSTROPHIC CURRENT DYNAMIC TOpOGRAPHY

r GPS ANTENNA

SLA NT BEA..,

DOPPLER MEASUREMENT ,,-(TOTAL SURfACE CURRENT)

'" {-----~~~~~ INSTANTANEOUS SEA SURFACE FROM THE ALTIMETER

----GEOID SWATH

. --I . .

---------GEOID PROFILE (I NTEGRATED ALONG - TRACK

DEFLECTION) (PROFILE EXTRApOLATED ACCORDING TO ACROSS- TRACK DEFLECTION)

........ ELLIPSOID

Figure 3. Combining precise GPS kinematic GPS with other instruments in an airplane for geophysical exploration.

467

MEASUREMENT ACCURACY AND RESOLUTION

The accuracy of airborne methods depends on that of both GPS- and INS-derived accelerations. Current evidence indicates that there is little correlation among GPS measurements errors, so their power density spectrum is mostly flat, like that of white noise. These errors propagate into the coordinates of the instantaneous position ftxes, where spurious low frequency effects due to incorrect GPS ephemerides, refraction, etc., also appear. The effect of those low-frequency position errors on the estimated accelerations should be quite negligible, at the milligallevel, because they get differenced twice. So the error power spectrum of GPS-derlved accelerations may be expected to behave almost like that of a twice-differenced white noise, which is proportional to the fourth power of the frequency. The other element in the combination, INS, contributes to the total error mainly because of imperfect accelerometers and gyroscopes. The spectrum of the main component in the INS error is concentrated at low-frequencies (accelerometers and gyros' biases and drifts). Since the estimated gravitational accelerations are the differences between INS and GPS-derived accelerations, which are mutually uncorrelated, the gravity noise power spectral density should be the arithmetic sum of the INS and GPS contributions. While the inertial sensors have noise spectra that rise quickly with frequency, in practice the GPS-induced errors should be larger larger and prevail at high frequencies. For their part, since the errors of the INS sensors are not differenced twice like those of GPS, they should dominate the low end of the spectrum.

Both gyroscopes and accelerometers have, after the initial calibration, a random walk component that makes the total noise power spectrum rise as the inverse of £2 (frequency squared). In addition, some accelerometers and gyroscopes have a type of low frequency noise that can be described as a first order Markov process with correlation time T , where T is of the order of one or two hours [Knickmeyer and Schwarz, 1989]. For the accelerometers, this means that the low­frequency noise power rises as 1/£2 up to fT = Iff, and then flattens out towards f = O. For the gyroscopes, the bias is the slope of the orientation error, while it is the error in that orientation, or gyro drift, that affects the measured acceleration, particularly for the horizontal components of gravity, or deflections. So gyros contribute a noise component whose spectrum rises as the fourth power of inverse frequency up to fT, and then as the inverse of frequency squared towards f = o. Accelerometers influence mostly the vertical channels' output (gravity disturbances) of scalar and vector gravimeters. Gyro errors affect mostly the horizontal channels' measurements, the deflections of the vertical.

The total error spectrum for the gravity acceleration may be as shown in Figure 4, rising both towards zero frequency at one end, and towards inftnity at the other, as 1/£2 and as £2, respectively. In the middle range of frequencies, noise power density must reach a low point, rising again beyond it, as frequency decreases towards f = lIT, as 1/(2 (gravity disturbance), or as 1/£4 (deflections). In the Figure, the vertical axis actually shows the error r.m.s spectrum, rather than power spectrum, for deflection measurements. A term g/R ~ has been subtracted

from the error to eliminate the effect of Schuler tuning and simplify the plot (dx is horizontal position error, g is the acceleration of gravity, and R is the Earths radius). This could be done in practice, because dx can be obtained very precisely, at the decimeter level, by comparing the INS to the GPS navigation (it would change the noise spectrum at frequencies much higher or much lower than the Schuler frequency; this might matter only in very long flights).

Finally, beside the sensor errors, one should consider the effect of quantization noise in analog/digital signal conversion (this affects various kinds of sensors quite differently), of discrepancies between the INS clock and GPS time, and of various imperfect real-time corrections, pre-smoothing, etc. made by the INS computer previous to the output of the inertial data through an interface to an external data-logging computer for later post-processing. Vibrations that affect differently the GPS antenna and the INS sensor block, errors in the measured roll, pitch and yaw rotations, etc., can add to the overall "noise". Of particular concern is aliasing of high frequency

468

vibrations into the gravity signal band, because of the relatively low rate of measurement of the GPS receiver. It is hard to characterize the contributions of these other sources to the total spectrum of the elTOr; they should be kept as small as possible by proper engineering design.

(r.m.s. )

I V F

J J

v = 400 Km/h

= 10 bib

Figure 4. Notional plot of the power spectrum of the airborne gravity error due to the combined INS and GPS measurement noise, compared to that of the gravity signal at various flying speeds. The case shown here corresponds to a deflection of the vertical.

Factors that Influence the Spatial Resolution of Gravity Features

For the purpose of this discussion, the shon wavelength part of each of the three anomalous gravity components can be approximated by a simple first order stochastic process in space, with an r.m.s. of 20-30 mgal, and a correlation length ~ of some 20 km. For an airplane travelling in a straight line at constant velocity, the gravity signal detected by the GPS/INS combination appears as a temporal process, with a time constant 't given by: 't = &v ,where v is the velocity. At

greater speeds the value of 't will be smaller, and larger at lower speeds. After a change in velocity, the power density spectrum of the gravity signal, also shown in Figure 4, will be Doppler-shifted up or down in frequency, since the "knee" frequency fe = lit increases or decreases with v. At the same time, the spectral amplitude must increase or decrease in inverse proportion to v, to keep constant the total power of the signal (the area between the spectral plot

469

and the frequency axis). It is clear from the picture that the highest frequency fH below which the gravity signal still can be distinguished from the noise (i.e., the signal to noise ratio is more than one) is determined by the fast rising GPS acceleration noise. At any speed high enough for the flat portion of the signal spectrum to intercept the noise curve, the highest spatial resolution fH at which gravity can be reliably mapped is, approximately, rH = 0.5 v/fH (or half the equivalent spatial wavelength). Therefore, the lower the speed, the higher the resolution. At the low frequency end, resolution is limited by the fast rise in inertial sensor noise towards f = 0 (red noise).

High frequency resolution. One way to increase resolution is to travel slower while taking measurements. This could also be understood in terms of smoothing the instantaneous gravity accelerations, measured at a constant rate, over a given time interval: for slower motion, the same number of measurements will go into a smoothed value corresponding to a shorter distance, so the resolution length is also smaller.

Another way to increase resolution is to increase the rate of the measurements (the minimum number of measurements needed to obtain smoothed gravity values of a given accuracy will remain constant, but the distance covered while making them, at the same moving speed, will now be shorter). Since inertial sensors measure at very high rates, compared to the bandwidth of the signal, and since GPS errors dominate at high frequencies, it is the rate of measurement with GPS that is critical to resolution. Hence, a technological fIx is possible by using receivers that operate at higher rates. Currently most receivers in kinematic mode operate at rates as high as fIve times per second, which is probably as high as may be needed in most situations to get milligallevel accuracy with a resolution close to 1 km. Work done up to now indicates that with averaging times of about 60 seconds, top GPS rates of one measurement per second, accuracies of a few millimeters in phase-range, and flight speeds of 100 m/s, gravity anomalies can be measured with resolutions of 10-20 km and accuracies of 2-3 mgal.

Yet another way to increase the resolution is to increase the accuracy of the measurements, so fewer of these have to be averaged together to reach the same accuracy in acceleration. If both the rate of measurement and the flying speed stayed constant, resolution length should shorten in inverse proportion to the square root of measurement accuracy (as long as errors remain largely uncorrelated). Since GPS noise dominates at high frequency, it is the accuracy of the GPS measurements that matters the most. Here another technological fIx is possible, and this one requires an intimate integration of GPS with INS: the INS real-time navigation can be used (together with the broadcast ephemerides) to calculate the approximate Doppler shift in the GPS signals, even when the airplane moves very fast and unpredictably (the latter being unlikely for a gravity survey). This estimated Doppler shift can be used to drive the receiver's oscillator so that its frequency closely matches that of the incoming signal. The end result is that only a narrow band is needed in each channel phase-locked loop detector to track such a signal. Without the INS information, the bandwidth of the loop should be much wider, and the noise passing through it and into the GPS measurements, proportionally higher. This intimate integration of INS and GPS hardware is likely to contribute greatly to the overall navigation accuracy, and to the increased resolution of gravimetry. It makes particularly good sense in the context of vector gravimetry, where already a post-mission integration of both systems' measurements is needed.

Low frequency resolution. At the low-frequency end of the spectrum, the INS sensors' noise dominates. To be able to measure gravity over reasonably long traverses, this error has to be kept small enough for the flat portion of the gravity signal to intercept the noise spectrum at a sufficiently low frequency fL. The largest (or longest) feature that can be reliably resolved has a size rL = 0.5 v/fL . The longest meaningful duration for a straight line survey traverse flown at a more or less constant velocity is t L = 0.5/fL. Of course, other considerations may affect these limits, for example the accuracy, not of the disturbances or deflections, but of the integrated deflections, or geoid undulations. Although this integration is in space (i.e., along the line of flight), with constant flight velocity the result is similar to that of an integration in time, accentuating the low frequency errors while fIltering out the high frequency ones. Moreover,

470

geoid errors are proportional to velocity. Therefore, for relatively short flying times, r.m.s. geoid errors may be small (dominated by small high-frequency noise), but they may increase as traverses take longer (because of larger low-frequency noise).

THE STUDY OF GEOPHYSICAL FEATURES AND PROCESSES

10

..!! 8 Oceanic cu Uthosphefe 0)

E c 6 ~ 0 cu L..

::J 4 0 0 <

2

1000 100 10 1 Horizontal Resolution in km 'A

Figure 5. The value of airborne gravimetry to solid earth geophysics is illustrated here. The shaded areas correspond to 10% of the typical size of the gravity anomalies associated with the features named within, so anything on or above the error curve can be mapped to 10% or better. (Based on Mueller and Zerbini, 1989) .

Error law for airborne estimates of gravity anomalies of a certain size. From the characteristics of the GPS contribution to the total gravity error power spectrum, one can derive the error law represented by the curve in Figure 5. As the geophysical features whose anomalies are compared to the error due to GPS are flown over by an airplane at speed v, the power spectrum of the measurement error must be integrated over a frequency band ranging from f = 0 to

fmax = v().., where A. is the typical length of the feature, to obtain the power of the noise in that band. This can then be compared to the power of the signal (assumed to be more or less constant over that band). Since the density power spectrum of the GPS error rises as ~,this integral is approximately proportional to fmax5 (it would be exactly that if the feature were to vary along the traverse as: A Sinc [absolute(2x x()..)] , where x is the distance to the center of the feature and A is a proportionality constant). Since the curve represents the r.m.s. of the error, its law is proportional to fmax2.5, or to 1()..2.5

R.m.s. e"or law for smoothed gravity values. If the anomalous feature discussed in the previous paragraph had a constant amplitude, its optimal estimate would be the average of the independent gravity acceleration measurements made over the time interval /..Iv the airplane would take to fly over it. The 12.5 power law would apply, approximately, to the error in this estimate.

471

Therefore, if the GPS position errors were uncorrelated and had an r.m.s. of 10 cm, with a sampling rate of once per second, the instantaneous accelerations derived from these positions (by differentiating numerically twice) would have an r.m.s. of about 20000 mgal. Averaging over 60 seconds would reduce that error by a factor of about 6()2.S, to close to 0.6 mgal. From present evidence, the error is more like 2 mgal, in part because of the contribution of inertial sensor errors that follow a similar power law at high frequencies. The previous reasoning shows that smoothing can decrease the gravity acceleration error much faster, with increasing averaging time, than would be the case for a white noise error. On the other hand, the effect of the low frequency inertial sensor errors becomes larger with increasing averaging time, and with time since calibration. A plot of the Allan variance of the error in the time-averaged disturbances and deflections should be a typical V-shaped curve.

INERTIAL HARDWARE REQUIREMENTS

Accelerometers. For the vertical component of the anomalous gravity vector (i.e., the gravity disturbance), the error of the vertical accelerometer is the limiting factor at low frequency. Therefore, one should require this accelerometer to perfonn at a level of accuracy that is no worse than that specified for the whole INS/GPS combination. Typically, that would mean that the accelerometer should measure at the mgallevel, or better, after 60 seconds of averaging time. Accelerometers of this quality are commercially available, although they are quite expensive. One obvious example are those used in precise moving gravimeters.

For the horizontal components of the anomalous vector, or deflections of the vertical, measurements should be as accurate as those for the vertical channel (1 mgal) in order to achieve accuracies as low as 0.2 arcsec (comparable to good astrogeodetic results), so the accelerometers should be able to operate at least as well in the horizontal position as vertically. This is possible with some of the very precise accelerometers now available (e.g., with magnetically or electrostatically levitated proof masses, vibrating beams, etc.). Both for vertical and horizontal accelerometers, and for the gyroscopes, thennal compensation is necessary. In the case of aerial surveys, which may last only a few hours after calibrating the instrument, this thermal stabilization does not have to be as good as for marine instruments, that must operate and remain stable for many days. This could make a considerable difference in the cost of construction.

Gyroscopes. The two main types of gyroscopes available at present are ring-laser gyros (used mostly in strap-down units), and mechanical gyros (used largely in stabilized platfonns). Ring­laser gyros errors are dominated by random walk drift, which has a very unpredictable slope; mechanical gyros have some element of random walk, but their error is usually dominated by a more predictable type of drift. Moreover, some mechanical gyros (largely developed in the early Seventies) are at present the most accurate devices available, commercially or otherwise. Ring­laser gyros are now being developed that may, eventually, compare in accuracy to good mechanical ones.

The main component of gyroscope drift is rated at E degrees per hour (for mechanical gyros), or

E degrees per square root of hour (for ring-laser gyros). In either case, the error will be E ° after

one hour. In arc seconds, this error is E o/hour x 3600, and should not to exceed at any time 0.5 arc seconds (the minimum that would be of geodetic interest). By calibrating the inertial sensors both at the beginning and at the end of the traverse, and perhaps by flying forwards and then back along the same traverse, a gain of a factor of 4 or better in accuracy may be achieved. This means that the minimum acceptable drift for the gyroscope should be close to E = 0.0005 °/hour (or E = 0.0005 O/[hour]l/2 for ring-laser gyros). This type of accuracy may be reached by selecting carefully the gyros out of a large batch (as with the Litton Rapid Geodetic Surveying System, or RGSS), by using the best available gyros (electrostatically suspended gyros, or E.S.Go's, as in the Honeywell GEOSPIN inertial surveying system); or by external enhancements of lower-quality

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devices (e.g., "Maytag"-type velocity biasing for ring-laser gyros, as in the experimental Litton L TN 94-R). While better accuracies in the gyros are desirable, they will probably cease to matter below 0.0001 °/hour, because of all the other error sources. In fact, a system with gyros accurate to 0.0002 0/hour and a reasonably predictable drift (Le., using E.S.G.'s), may no longer be limited in its long-tenn petfonnance by the drift of the gyros.

The conclusions of the preceding discussion on INS hardware are summarized in Table 1. Other types of accelerometers and gyroscopes exist that may petfonn as well or better than those listed in the Table.

Table 1 Some inertial hardware suitable for airborne gravimetry

Device

Accelerometer

Gyroscope

Star-tracker

Kind

Vibrating beam,

Magnetic/electrostatically levitated proof mass

E.S.G.

Ring-laser ("Maytag" mount)

Accuracy

± 1 mgal (random scatter)

II II

Comments

Commercially available

" "

~ ± 0.0005 °/hour Comm. avail. (predictable drift) ~± 0.0005 o/[hr]l/2 (less predictable drift) Experimental,

may become available soon

A few arc minutes Day and night operation (limited by clouds) t

t Expensive. May supplement gyroscopes over very long traverses, providing frequent calibration on the go (e.g., on a ship).

DISCUSION AND CONCLUSIONS

Scalar Gravimetry or Vector Gravimetry?

Accuracy may be increased if a region has been surveyed with a series of traverses flown in a grid pattern, by applying techniques appropriate to the analysis of gravity data. Both disturbances and deflections are components of the gravity vector, the gradient of the geopotential, which is an harmonic function of position. Gravity data is usually reduced to a single sutface, such as the reference ellipsoid, by means of a downward continuation, preferably preceded by a terrain correction to smooth out the field.

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From disturbances alone, one can find geoidal undulations and deflections by computing well­known two-dimensional quadrature formulae on the reference surface. For this reason, it is natural to wonder if the direct measurement of deflections with a vector gravimeter is truly necessary. This is a good question, because scalar gravimeters might need relatively simple improvements to be used routinely in very accurate airborne surveys, while vector gravimeters would require further development, and may also cost twice as much as the scalar ones. While this should be investigated in more detail, a reasonable answer could be that, in the first place, vector gravimeters would yield more accurate results simply by gathering three times more statistically independent data than scalar ones, in the same period of time. But there is a deeper reason in favour of the vector gravimeter: since data is necessarily restricted to a small part of the Earth's surface, while the quadrature formulae used to transform anomalies into deflections and undulations require a global data coverage, an error due to the incomplete coverage would be added to the effect of data noise on the estimated deflections and undulations, particularly at the borders of the region surveyed. This undesirable effect would be absent from direct measurements of the deflections, and would be very marginal in the calculation of geoid profiles by integrating the deflections along the line of flight (here it would be a simple quadrature error, easily kept small by ensuring a sufficiently high spatial data density along the profile).

The determination of the geoid would be of special value to solid Earth geophysicists interested in the study of deep isostatic compensation of topographic features. Over land, direct determinations of geoid profiles or swaths with a vector gravimeter would give them the same sort of information as satellite altimetry does over the oceans, and this information could be augmented by simultaneous altimetric or ISAR terrain mapping from the same airplane. Glaciologists could use this type of information to understand the gravitational imbalances that drive glaciers and the flow of ice in the polar caps. Finally, in cases where obtaining a profile of the geoid is the main objective, this could be achieved in a single flight with a vector gravimeter, while a whole area surrounding that profile would have to be surveyed with a scalar one. This could make operations cheaper and faster (as well as results more accurate), and may be of particular interest to surveyors, who might replace tedious spirit levelling with levelling "on the wing" (at least for second order surveys), as well as to oceanographers, who would have an alternative to very slow and costly steric levelling by ship in the much cheaper and quicker combined geoid and altimeter traversing by aircraft.

Gravity in the 1990's

Making long-term predictions in an evolving field like geodesy, in todays' changing world, is risky business. So the author wishes to make it very clear that he will not be answering any complaints on the predictions that follow at least until the meeting of the KIS 2000.

Over the coming years, two developments are most likely to increase our knowledge of the gravity field of the Earth. At scales larger than 300 km, global surveys carried out from spacecraft equipped with GPS receivers may increase the accuracy with which we know that field by more than two orders of magnitude, perhaps even allowing us to detect the very small changes in gravity due to geophysical, oceanographic and atmospheric phenomena like glacial rebound, seasonal and secular changes in sea level, tides, etc., and to map them in longitude as well as in latitude. If so, such space techniques could become an important tool for monitoring the effects of global climate change. From low altitude spacecraft, the most sensitive surveys will take place. At the lowest altitudes, GPS/INS integration will become necessary to separate gravity information from the effect of drag, spacecraft thermal forces, solar radiation pressure, etc. Moreover, there remains the possibility of at least one gravity gradiometer mission before the end of the century, which could map the field on a global basis, to a resolution of some 80 km.

Airborne gravimetry (and moving-base gravimetry carried out in a variety of land and sea vehicles) will map the finest detail (1 - 10 km resolution) over large areas of the world otherwise difficult or inaccessible to surveyors. Scalar gravimetry may eventually give measurements of the vertical gravity disturbance at the sub-milligallevel. Today, the worthy goal of 1 mgal accuracy is

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just a factor of three in improvement away. Vector gravimetry, in addition to vertical disturbances, would measure deflections to an accuracy of 0.2 milliarc seconds, which would compare with very good astrogeodetic results, and provide decimeter-accuracy geoid profiles and geoid swaths "on the wing". This new form of gravity measurements complemented with other scientific and geodetic observations, should become available without any need to advance the existing technology.

Aircraft, land and sea vehicles carrying a variety of sensors, including gravimeters, shall be used very effectively as instrumented platforms for studying our planet.

One of the problems to be addressed, before the methods discussed in this paper can become effective tools for the long-range exploration of unsurveyed regions, is the refinement of truly long-distance kinematic GPS navigation to the same level of accuracy that is now achieved over distances of less than 200 km from a fixed reference receiver.

Ultimately, it will be a combination of government objectives and manufacturer's corporate policies, of perceived scientific and engineering needs, and of availability of funds for developing and purchasing new equipment, that will determine the course that space and airborne gravimetry will take in the coming decade. The technology itself is already available, as well as the knowledge needed to put it together and make it work.

Acknowledgement. This work has been supported by NASA through grant NAG-5245.

REFERENCES

Bell, R.E., I.M. Brozena, W.F. Haxby, and 1L. LaBrecque, 1988, Gravity field of the Western Weddell Sea, comparison of airborne gravity and GEOSAT derived gravity, Proceedings A.G.U. Chapman Conference on Progress in the Determination of the Earth's Gravity Field, September 1988, R.H. Rapp, editor, Ohio State University, Columbus, Ohio.

Bell, R.E., B.I. Coakley, and R.W. Stemp, 1990, Airborne gravity from a small twin engine aircraft over the Long Island Sound, Presented at the Spring 1990 Meeting of the A.G.U., Baltimore, Maryland.

Brozena, 1M., 1G. Eskinzes, and 1D. Clamons, 1986, Hardware design for a fixed-wing airborne gravity measurement system, Naval Research Laboratory Report 9000, Washington, D.C.

Brozena, I.M., and M.F. Peters, 1988, An airborne gravity study of eastern North Carolina, Geophysics, 53, 2, 245-253.

Brozena, 1M., G.L. Mader, and M.F. Peters, 1989, Interferometric Global Positioning System: Three-dimensional positioning source for airborne gravimetry, J. Geophys. Res., 94, B9, 12153-12162.

Chin, M. (editor), 1989, CIGNET Report, GPS Bulletin, 2, 3, 21-42, GPS Sub-commission, CSTG, Rockville, Maryland.

Colombo, O.L., 1989, The dynamics of GPS orbits and the determination of precise ephemerides, J. Geophys. Res., 94, B7, 9167-9182.

Colombo,O.L., 1990a, Mapping the Earth's gravity field with orbiting GPS receivers, Global Positioning System, an overview, Y. Bock and N. Leppard Eds., Proc. lAG Symp. 102 (Edinburgh, Scotland), lAG Symp. Series, 102, Springer-Verlag, N.Y.

Colombo,O.L., 1990b, Charting the gravity field with GPS receivers in moving vehicles, Proc. Second International Symposium on Precise Positioning with the Global Positioning System: GPS-90, Ottawa, September 1990.

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Everitt, C.W.F., Breakwell, J.V., Tapley, M., DeBra, D.B., Parkinson, B.W., Smith, D.E., Colombo, O.L., Pavlis, E.C., Tapley, B.D., Nerem, R.S., Yuan, N.D., 1989, Gravity Probe-B as a geodesy mission and its application for TOPEX. CSTG Bull., 11, pp. 55-67.

Forsberg, R, 1989, Inertial surveying methods, in Modern techniques in geodesy and surveying, O.B. Andersen (editor), Norsdiska Forskarkurser 18/1988, N.D.G., Ebeltoft, Denmark, in September 1988.

Hartman, RG., 1988, An integrated GPS/IRS design approach, Navigation, J.lnst. Navig., 35, 1, 121-134, Spring.

Huddle, J.R., The rapid Geodetic Survey System (RGSS), Proceedings A.G.U. Chapman Conference on Progress in the Determination of the Eanh's Gravity Field, September 1988, R.H. Rapp, editor, Ohio State University, Columbus, Ohio.

Jekeli, C., and T.N. Upadhyay, 1990, Gravity estimation from STAGE, a satellite-ta-satellite tracking mission, J. Geophys. Res., 95, B7, 10973-10985.

Knickmeyer, E., and K.-P. Schwarz, 1990, Geoid determination using GPS-aided inertial systems, Proceedings lAG General Meeting, Edinburgh, in 1989.

Krabill, W.B., and C.F. Martin, 1987, Aircraft positioning using Global Positioning System carrier phase data, Navigation, J. Inst. Navig., Spring.

Lachapelle, G., and K.-P. Schwarz, Kinematic applications of GPS and GPS/INS algorithms, procedures, and equipment trends, Global Positioning System, an overview, Y. Bock and N. Leppard Eds., Proc. lAG Symp. 102 (Edinburgh, Scotland), lAG Symp. Series, 102, Springer-Verlag, N.Y. .

Lambeck, K., 1988, Geophysical Geodesy--The Slow Deformations of the Earth. Clarendon Press, Oxford, England.

Mader, G.L., and lR Lucas, 1989, Verification of airborne position using Global Positioning System carrier phase measurements, J. Gephys. Res., 94, B8, 10175-10181.

Melbourne, W.G., and Tapley, B.D., 1983, The geopotential infonnation in GPS carrier­phase tracking from a low-orbiting gravimetric mission. EOS Trans. AGU, Vol. 64(45), page 680.

Milbert, D.G., 1988, Treatment of geodetic leveling in the integrated geodesy approach, Ph.D Disertation, Ohio State University Dept. Geod. Sc. and Surv., Rep. No. 396.

Mueller, 1.1., and Zerbini, S. (editors), 1989, The interdisciplinary role of space geodesy. Proceedings of an International Workshop held at the Ettore Majorana Center for Scientific Culture in Erice, Italy. Springer-Verlag, New York.

Smith, D.E., Lerch, F.J., Colombo, O.L., and Everitt, C.W.F., 1988, Gravity field infonnation from Gravity Probe-B. Proc. Chapman Con! Progress in Det. of Eanh's Gravity Field, pp. 159-163, Ft. Lauderdale, Florida.

476

AN INTEGRATED PRECISE AIRBORNE NAVIGATION AND GRAVITY RECOVERY SYSTEM

Verification of GPS-determined vertical disturbing acceleration accuracy -

Klaus Hehl, Giinter W. Hein, Herbert Landau and Michael Ertel

Institute of Astronomical and Physical Geodesy (IAPG) University FAF Munich

Werner-Heisenberg-Weg 39 D-8014 Neubiberg, F. R. Germany

Jiirgen Fritsch and Peter Kewitsch

Bundesanstalt fur Geowissenschaften und Rohstoffe Stilleweg 2

D-3000 Hannover, F. R. Germany

Abstract

In July I August 1989 a series of test-flights was carried out to investigate the possibility of integrating different sensors for precise position determination and gravity recovery. The tests were flown with a Dornier DO 128 fixed-wing research aircraft over the North Sea and also over a test area in the northern part of Germany.

Precise differential positioning of the aircraft was done using two different GPS receiver types simultaneously. A Sercel TR5SB CIA-code receiver and a TI4100 P-code receiver were installed in the aircraft while a third and fourth receiver of the same types were tracking satellites on the ground. The system on board consisted further of a ring-laser gyro strapdown system Honeywell LaserNav allowing precise position and attitude determination. For the purpose of airborne gravimetry the system was equipped with a Bodenseewerk KSS 31 Sea-Air Gravitymeter. The determination of high frequency height variations was possible using a precise laser altimeter Optech Rangefinder 501 SX.

The paper describes the experiment in detail and presents results of high precision position I velocity I acceleration determination using GPS. The computation of disturbing accelerations is the crucial point in airborne gravimetry, thus the two independent GPS solutions as well as the laser height measurements are compared and analysed. Inertial accelerations are removed by means of digital filtering and computation of first I second order derivatives of the measured velocities I heights.

It is shown, that under ideal, smooth flight conditions the disturbing vertical accelerations can be determined with an accuracy of < 0.5 mGal.

Paper presented at the International Symposium on Kinematic Systems in Geodesy, Surveying and Remote Sensing (KIS '90), September 10-13, 1990, Banff, Canada.

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1 Introduction

Mapping of the earth's gravity field is still a primary goal in geodesy, geophysics and geodynamics. Although various satellite missions could already achieve significant improvements, in particular using satellite altimetry over oceans (GEOS3, SEASAT, GEOSAT), and further missions are in planning stage (ERS-1, TOPEX, GP-B, ARISTOTELES), it concerns only the earth's gravity field over wavelengths larger than 50 ... 100 km. Terrestrial static gravity observations can cover the wavelengths up to approx. 3 km. However, due to economical and topographical reasons, the use of static gravity meters is limited to certain areas on land. Shipborne gravity measurements can achieve 1...2 mGal over short wavelengths « 1 km). Again, it is restricted to (small portions of the) oceans due to observation times and cost efficiency. Airborne gravity gradiometry (JEKELI 1988) could not yet show that it is applicable for routine measurements.

Airborne gravimetry stopped practically in the 1970's due to the inability of navigation tools to determine the disturbing vertical aircraft accelerations better than 10 mGal. Inertial gravimetry using inertial navigation systems did not yet show airborne results. Experiments on land are limited to the resolution of the accelerometers (see e.g. EISSFELLER et al. 1985).

Thus, we are faced with the situation that the earth's gravity field with wavelengths of 0.5 ... 50 km is nowadays only determined in very little regions. Special emphasis has to be put therefore on observation techniques to fill these gaps.

With the recently developed capability of kinematic observations to the Global Positioning System (GPS) to determine the position of moving sensors in aircrafts (e.g. photogrammetric cameras) in the cm-range (REIN et al. 1989, REIN 1990), it became obvious to make some trials to use GPS also for the determination of aircraft position / velocity / acceleration for gravity field reconnaissance work. Flight experiments were carried out by (BROZENA et al. 1989) and IAPG (REIN et al. 1990). First results seem to be very promising, indicating that in near future accuracies of < 0.5 mGal over wavelengths < 1 km might be obtained. Thus, the airborne gravity recovery technique using an sea­air gravity meter and a GPS receiver in an aircraft will become a powerful tool in the determination of the earth's gravity field in the near future.

This paper summarizes the 1989 IAPG experiments in airborne gravimetry and verifies that the determined disturbing vertical accelerations might be even as good as 0.5 mGal when comparing to the results from an independent laser altimeter or even better compared with a second independent GPS solution.

2 Accuracy Requirements

Free-air gravity anomalies can be derived from airborne gravimetry using the standard expression as a function of position (TORGE 1989):

~g(h) = g(h*) - az(h*) + 6gF . (h* - h) + 6gE(h*) - -y(h*)

with the components

~g(ip, A, h) g(h*) az(h*) 6gF ·(h*-h) 6gE(h*) -Y(ip, h*)

... free-air anomaly in (ip, A, h)

... observed acceleration at height h*

... vertical disturbing acceleration caused by aircraft motion, wind gusts, ...

... free-air correction using 6gF :::::: -0.3086 mGal/m

... Eotvos correction

... normal gravity.

(1)

As all quantities are initially time series with independent GPS-time-tag, eq. (1) could also be formulated using the parameter time (FRITSCH, ROESER 1988):

478

Llg(t) = get + r) - az(t) + OgF(t) . Llh(t) + OgE(t) - -yet) (2)

using

get) = go(to) + seLla - 0 . Llt) ... reading of the gravity meter at time t ... relaxation time r

go(to) s Lla o

... initial gravity value at the airport at start time to

... scale factor converting gravity meter readings to mGal

... difference of the readings between start time to and time t

... drift of the gravity meter ( < 0.3 mGalfmonth)

The components az(t), ogF(t)Llh(t), OgE(t) and -yet) have to be filtered adequately to get + r).

Due to the heavy damping of the system the response of the gravity meter to some input comes up r seconds later, so all gravity readings in fact belong to some time instant in the past.

In the sequel accuracy estimates for an anticipated one mGal gravity anomaly accuracy are given for position, velocity, and accelerations, resp.

Obviously the least significant demand on accuracy for airborne gravimetry is for position. As the horizontal gravity gradient is not very large « 2 mGal/km) (REIN 1981) and we can only be interested in averages over a certain time or wavelength (e.g. 1 mGal over 3 km wavelength) the resulting statement is: location accuracy is not the crucial point. Concerning height accuracy a rough estimate can be made by inspection of the free-air gravity gradient OgF ~ -0.3086 mGal/m, leading to a required accuracy for the heights of better than 3 meters.

From own experiences (position determination of a photogrammetric camera during flight using GPS and comparison of the coordinates of the perspective center with those derived by photogrammetry (see BAUSTERT et al. 1989» we can state that cm-accuracy in aircraft position is achievable using GPS carrier phases in differential mode. Of course, state-of-the-art algorithms and software have to be used to end up with such accuracies. TOPAS, developed at the IAPG by H. Landau is such a software. The algorithms incorporated in the software are partially documented in (LANDAU 1988), and a description of the software (running on VAX and PC 286, 386, 486) can be found in (LANDAU 1990).

Another contribution to eq. (1), called the Eotvos correction OgE, can be modeled in terms of the velocity vector, in detail by the magnitude of the horizontal components of the vector and the azimuth of the aircraft motion.

In spherical coordinates the Eotvos correction reads (TORGE 1989):

(3)

with

We ••• earth rotation rate v ... magnitude of the aircraft's horizontal velocity a ... azimuth of the flight path R ... mean radius of the spherical earth.

Using average values for We and R, and introducing the velocity in units of [km/h] leads to the expresssion

OgE = 4.0 . v . sin a cos/;' + 0.0012· v2 [10- 5m. sec- 2]. (4)

479

Further, assuming mid-latitude (<p = 45°), an aircraft velocity of v = 180 km/h, east-west flight (a = 270°), the following rule-of-thumb can be derived:

UtJ [km/h] ~ 1 3' .0'698 [mGal]. (5)

The anticipated one milligal accuracy for U69E results in UtJ ~ 1/3 [km/h], or UtJ ~ 9 [cm/sec].

Based on the analysis of static GPS data and from investigating redundant observations (5 or more satellites with multichannel receivers, e.g. Ashtech LD XII), it can be stated that the necessary accuracy of better than 9 cm/sec for the velocity can be reached in any case.

One milligal accuracy for the determination of the aircraft's vertical motion is impossible at an epoch-to-epoch basis. Thus only averaging over a certain time (in other terms: low-pass filtering) in combination with high-precision algorithms for the differentiation of heights or velocities will lead to the anticipated accuracy. More details will be discussed in section 6 by intercomparison of disturbing vertical accelerations derived from GPS and altimeter data.

3 Hardware Set-Up for Airborne Gravimetry

Designing a system for the determination of airborne gravity consists of two major parts:

• selection of hardware components necessary to form an operational system to gain data during flight, and .

• development or supply of the necessary software (data logging, processing software, ... )

This section summarizes the hardware side which was already described in more detail in (HEIN et al. 1990). Figure 1 shows that our system consists of 5 main hardware subsystems:

1. Bodenseewerk Sea-Air Gravimeter KSS 31 The gravity meter's heart is the axially symmetric gravity sensor GSS 30 which has a straight­line construction principle, i.e., it has only one degree of freedom which avoids cross-coupling. Together with the zero-reading mechanization this leads to a strongly linear measurement behavior over a large range (FRITSCH, KEWITSCH 1987).

The sensor of the gravity meter is mounted on a stabilized platform, the gyrotable KT 31. The erection of the sensitive axis of the sensor into the true vertical is aided using heading and velocity, i.e. the full velocity vector of the aircraft's path. These quantities are derived from the navigation means of the aircraft (GPS, inertial navigation system) and fed into the control electronics of the gravity meter.

2. Two Independent GPS Receivers This means, of course, that both receivers in the aircraft have partners of the same type on the ground because of the differential mode of processing GPS data. One receiver type was the multiplexing TI 4100 dual-frequency P-code receiver with the ability of tracking up to 4 satellites and sampling at an interval of 1.2 seconds. The second GPS receiver equipment was a 5-channel Sercel TR5SB C/ A-code receiver with sampling interval of 0.6 seconds.

These receivers were replaced by two Ashtech LD XII two-frequency devices in recent flight experiments.

3. Laser Altimeter Optech Rangefinder 501 SX This device was mounted as an independent control for the behavior ofGPS determined height and its derivatives. Although resolution (10 cm) and accuracy (20 cm) of the laser altimeter are limited with respect to the potential of the GPS data, the higher sampling rate (25 Hz) was a source of additional information.

480

4. Inertial Navigation System (INS) Honeywell LaserNav In order to reduce the laser altimeter data to the true vertical the roll and pitch angles of the INS are used. This was necessary due to the fact that the laser altimeter was mounted fix with respect to the aircraft and thus the slant range had to be reduced for the aircraft attitude.

5. Data Processing and Registrations Primarily all the different data acquired by the subsystems described above have to be related together via precise time-tagging. Our fundamental time frame is GPS time. All standard data were collected, time-tagged and then stored on magnetic tape. In addition raw GPS data were recorded on disk using a standard IBM compatible 80368 laptop. Also the filtered gravity meter output was time-tagged and stored on the hard-disk of a PC.

One key within our system is the recording of the unfiltered raw sensor signals which are A/D converted, time-tagged and stored on hard-disk. The post-processing of those data provides a large variety of possibilities for digital filtering.

We are aware that such a system is convenient only for the purposes of research, but it was our intention to set-up a low-cost and fully operating system rather than optimize hardware components.

4 Software Developments for G PS and Digital Filtering

Of fundamental importance for the determination of disturbing vertical acceleration is a sophisticated GPS processing software together with digital filter software. IAPG has developed such a system for the processing of GPS data which is called TOPAS and has proven to be a powerful tool for the determination of positions both in static and kinematic mode. Results are documented in (HEIN et al. 1989, BAUSTERT et al. 1989 and REIN 1990).

The software for digital filtering is currently developed by the first author and consists of several modules. Details will be found in a forthcoming publication (HEHL 1991). Here are some brief explanations of the digital filtering:

The input/output relation between input sequence x[n] and output sequence y[n] is given by the difference equation

N-l

y[n] = L h[i] . x[n - .1 ;=0

which describes a non-recursive filter of length N with the filter coefficients h[i].

Introducing the convolution operation

N-l N-l

y[n] = L h[i] . x[n - 21 = L h[n - i] . x[21 = x[n] * h[n] ;=0 ;=0

(6)

(7)

we see that the filter coefficients in some way manipulate the input sequence x[n] to get the output sequence y[n]. Once the filter coefficients are known (after a filter design step), in principle eqn. (7) can be used to calculate the filtered data y[n]. Especially for long data sequences and large N it is of advantage to use fast convolution techniques.

Computing the z-tmnsform of the filter coefficients (z E C)

N-l

H(z) = L h[21z-; (8) ;=0

481

at the unit circle z = eiO (Q = normalized frequency) results in the transfer-function of the filter which is complex in general. Of special importance are linear-phase filters which are exactly only available in post-processing.

As an introduction to digital signal processing the reader is referred to standard literature such as (OPPENHEIM, SCHAFER 1975, RABINER, GOLD 1975).

5 Test Flights

In August 1989 a series of test flights was carried out using the equipment described in section 3. All the components were installed in a research aircraft Dornier DO 128. During ten flight hours data were recorded. Two flights were made over land in the northern part of Germany and two flights over the North Sea. It was always anticipated to fly at a nominal height of 150 m above ground and at a velocity of 180 km/h (corresponding to 50 m/sec). For this report only the data from two flights were used.

The paths of both flights are plotted in Figs. 2 and 3.

6 Results

Processing of the data followed a common scheme. First of all the differential GPS solution was computed using the TOPAS software and resulting in positions for the Sercel data and in positions and velocities for the TI 4100. All the GPS data are implicitly related to a geocentric reference frame, WGS 84. After transforming the data into a local-level system only the vertical channel was basis for further processing.

The filtering of the data employs a cascade of filters. First of all the data are low-pass filtered to remove the short periodic variations. The following differentiation step is one of the critical parts because differentiation tends to amplify or roughen the higher frequency parts in the spectrum of the data. Therefore much care has to be given to numerical differentiation. Final step is to filter the data commensurately with the gravity meter's output because only at this stage the data are of use for the computation of gravity anomalies, see eqn. (1).

For the first day (flight over land) figure 4 shows more than one hour of data passed through the upper filter scheme. The resulting disturbing accelerations have in average a variation of ~ 15 to 20 mGal. These data were derived from the Sercel GPS heights. In the same way TI 4100 heights were processed. Figure 4 gives the accelerations derived from Sercel data for one hour and figure 5 shows the differences between the Sercel and the TI solution for about 20 minutes duration. It turned out that the middle part of the data was unusable for comparison due to the bad TI data accuracy. Clearly the agreement of better than 0.5 mGal can be seen.

For the second flight over the North Sea we could also use the laser altimeter data. Figure 6 shows the differences between the raw Sercel heights and the raw (i.e. only reduced for the aircraft attitude) laser altimeter heights. The differences are in the range of 20 ... 30 cm and thus belong to the same class of accuracy than those reported by (BROZENA et al. 1989).

This flight was made under ideal meteorological circumstances flying at 150 m above the North Sea and therefore the disturbing accelerations due to the aircraft motion have a much smaller amplitude. The differences between laser and Sercel derived accelerations can be seen in figure 7.

However the differences between the Sercel GPS solution and the filtered altimeter accelerations are slightly worse than those between Sercel and TI from flight no. 1.

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7 Conclusions and Further Prospects

It could be shown that vertical disturbing accelerations can be derived from GPS using methods of digital filtering with an accuracy of 0.5 mGal and for ideal conditions even better. These data serve as one of the essential parts in the determination of gravity anomalies from airborne sampled data with a sea-air gravity meter. Further work will consist of fully processing also the gravity meter output and combining all components (i.e. Eotvos effect, free-air reduction, ... ) to get gravity anomalies.

One could even think of processing not only the vertical channel in the local level frame but also the horizontal accelerations in order to get the full gravity vector which is termed vector gravimetry, c.f. (COLOMBO 1990).

8 References

Baustert, G., G.W. Hein, H. Landau (1989): Recent Results in High Precision Kinematic Po­sitioning, Presented at the Workshop on GPS for Geodesy and Geodynamics, Luxemburg, November 1989

Brozena, J., G.L. Mader, M.F. Peters (1989): Interferometric Global Positioning System: Three-Dimensional Positioning Source for Airborne Gravimetry, Journal of Geophysical Re­search, Vol. 94, No. B9, pp. 12,153-12,162

Colombo, O. (1990): Scalar and Vector Airborne Gravimetry with the Help of Precise Kinematic GPS, Presented at the 1990 Spring Meeting of the American Geophysical Society, Baltimore, Maryland, May 29 - June 1, 1990

Eissfeller, B., H. Landau, G.W. Hein (1985): Results of a Gravity Survey with the Honeywell Geo-Spin II in the Testnetwork 'Werdenfelser Land (FRG)', Proc. Inertial Technology for Surveying and Geodesy, Banff, Canada, pp. 203-212

Fritsch, J., P. Kewitsch (1987): Bericht tiber die Eichung des Seegravimeters KSS 31 Nr. 22 auf der Europaischen Gravimeter-Eichlinie zwischen Hannover und Catania vom 29.9. bis 10.10.85, Intern. Rep. Arch. BGR, 101.215,22 pp.

Fritsch, J., A. Roeser (1988): On the Accuracy of Gravity Measurements at Sea, Geol. Jb., E 42, pp. 195-208

Hehl, K. (1991): Bestimmung von Beschleunigungen auf einem bewegten Trager durch GPS und digit ale Filterung, Schriftenreihe Studiengang Vermessungswesen, Universitiit der Bundeswehr Miinchen, in preparation

Hein, G.W. (1981): Untersuchungen zur terrestrischen Schweregradiometrie, Deutsche Geodiitische Kommission, Reihe C, Heft 264, Miinchen

Hein, G.W. (1990): Kinematic Differential GPS-Positioning: Applications in Airborne Photo­grammetry and Gravimetry, Lecture Notes, Udine

Hein, G.W., G. Baustert, B. Eissfeller, H. Landau (1989): High Precision Kinematic GPS Differential Positioning and Integration of GPS with a Ring Laser Strapdown Inertial System, Navigation: Journal of the Institute of Navigation, Vol. 36, No.1, pp. 77-98

483

Hein, G.W., K. Hehl, H. Landau, M. Ertel (1990): Experiments for an Integrated Precise Air­borne Navigation and Gravity Recovery System, Proc. ofthe IEEE PLANS 1990 Meeting, Las Vegas, pp. 279-285

Jekeli, Ch. (1988): The Gravity Gradiometer Survey System (GGSS), EOS, Transactions of the American Geophysical Union, Vol. 69, No.8, pp. 105, 116-117

Landau, H. (1988): Zur Nutzung des Global Positioning Systems in Geodiisie und Geodynamik: Modellbildung, Softwareentwicklung und Analyse, Schriftenreihe Studiengang Vermessungswe­sen, Universitiit der Bundeswehr Miinchen, Heft No. 36

Landau, H. (1990): TOPAS - The Ultimate GPS Processing System, Software Reference Manual

Oppenheim A.V., R.W. Schafer (1975): Digital Signal Processing, Prentice-Hall, Englewood Cliffs, New Jersey

Rabiner R.L., B. Gold (1975): Theory and Application of Digital Signal Processing, Prentice­Hall, Englewood Cliffs, New Jersey

Torge, W. (1989): Gravimetry, de Gruyter, New York

GPS

Other Sensors

Antennae

TI 4100

INTEGRATED AIRBORNE NAVIGATION AND GRAVITY RECOVERY SYSTEM

~ ... 1 __ G_r_~_~s_'1_~t_e_r_.....J

Optech Ronge­finder

501 58

lAM Vt.lE Bus Processor

PDP 11/73 Hon-rwell lAserNav

Databus

Central Unit

ZE 31

c,n Gyro-.a.ctron1cs table Kt 3. KT 31

Sensor Sensor Subsystem GSS 30 GE 30

Power-supply

Figure 1: Hardware Components of the IAPG Gravity Recovery System

484

o 25 50 75 100 125 km

Figure 2: Path of Flight Over Land, Day 219

Braunschweig 6: 33

~----------------------~~ 56°

o 30 60 90 150 km

54°40

54°

53°20'

remen

52°40 .

~----~~-----'------~----~Or---~l~lto 70 SO gO 10

Figure 3: Path of Flight Over Sea, Day 220

485

~,---------------------------------------------------------~

CD

~ ..... CO (!)

.5 CD I

~ N I

1131 0.0 11399.6 1146 9.2 1154 .B 1162 B.4 116 .0 Time [sec]

07:25:50 07:38: 40 07: 51: 29 08: 04: 19 08: 17: 08 08:29: 58

Figure 4: Disturbing Accelerations derived from Sercel data, Day 219

10

0

(I)

0

...

.; ~ ..... co (!)

.5 ... 0 I

"! 0 I

115 3.0 U5B 3.6 1160 4.2 1163 4.B 11655.4 11676.0 Time. [sec]

08:06: 23 08: 10:24 08: 14:24 08: 18:25 08:22: 25 08:26: 26

Figure 5: Differences in Disturbing Accelerations (Sercel minus TI), Day 219

486

0 ~ 0

CD ('f)

0

N -,; :§

N -,; I

CD

~ 0 I

.8 .0 Time .[sec]

07: 05: 10 07:07: 27 07: 09: 45 07: 12: 02 07: 14:20 07: 16:37

Figure 6: Differences in Heights (Laser minus Sercel), Day 220

~ 0

.., 0

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0 I

~ 0 I

1983 0.0 1984 7.4 1988 9.6 1989.7.0 Time

07:05: 10 07:07: 27 07: 09: 45 07: 14: 20 07: 16: 37

Figure 7: Differences in Disturbing Accelerations (Laser minus Sercel), Day 220

487

GPS AND AIRBORNE GRAVIMETRY: RECENT PROGRESS AND FUTURE PLANS

John M. Brozena Naval Research Laboratory

Code 5110 Washington D. C. 20375-5000 USA

INTRODUCTION

The logistical difficulties and expense of measuring gravity over the surface of the earth or ocean provides the motivation for developing techniques for airborne data acquisition. To meet this need, the U.S. Naval Research Laboratory (NRL) has had an active research program in airborne gravimetry since 1979. This paper provides a brief history, current status and proposed future directions of the program.

The general method of aerogravity employed by NRL consists of correcting measurements from a vertically stabilized gravimeter for vertical accelerations caused by aircraft motion. This is accomplished by independently determining non-gravitational accelerations from an extremely precise time series of 3-d air­craft positions or velocity and then by subtracting them from the total vertical acceleration measured by a gravimeter. The correction for the vertical attenuation of gravity with altitude and the normal gravity are also determined from the air­craft position data. Noise in all measurement sources is attenuated by low-pass filtering. The amount of filtering required and the speed of the aircraft determine the shortest gravity wavelengths that can be measured.

The vertical component of the gravity anomaly in an ellipsoidal local-vertical frame of reference is given by [Rustler et al., 1970]

where 8gu

Au Vu , Vn , Ve Vu </> ,\

,\

vertical anomalous gravity, vertical non-gravitational acceleration, vertical, north and east velocity, time derivative of Vu , geodetic latitude, geodetic longitude, time derivative of ,\ = it; ( a + h) cos</>, . Vl-e2sint/>

</>,

488

Wn (~ + O)cos4> , 0 earth rotation rate, a - earth semimajor axis, e - earth eccentricity, h - height above ellipsoid, ,(4)) - reference ellipsoid gravity CIa - free-air correction.

A vertically oriented accelerometer undergoing arbitrary three-dimensional mo­tion over a curved rotating earth produces a total acceleration output, At, where

(2)

The non-gravitational vertical acceleration, Au, in Equations (1) and (2) may be approximated by

(3)

where

(4)

is a truncated expansion [Harlan, 1968] of the horizontal velocity dependent terms in Equation (1) normally referred to as the Eotvos correction. This expansion is accurate to a fraction of a mGal for altitudes and speeds of less than a few thousands of meters and a few hundred meters/sec respectively.

The vertical component of the gravity anomaly is then given by

(5)

Table 1 lists the rms accuracies needed over the pass-band of interest for the solution of Equation (1) to limit overall system error to 1-2 mGals.

Table 1. Parameters and accuracies for airborne gravimetry.

parameter term accuracy horizontal position normal gravity 100 m horizontal velocity EOtvos correction .1 m/s altitude vertical attenuation 2m altitude Vu .01-.15 m· vertical acceleration At 1 mGal accelerometer verticality At < 5 arc-min··

* - depending on spectral form of error and aircraft velocity [Bower et al., 1990]. ** - [LaCoste, 1967]

489

EVOLUTION OF THE NRL AGMS

In the NRL airborne gravity measurement system (AGMS) the total vertical accel­eration, At, has generally been measured by a LaCoste & Romberg model S air-sea gravimeter. The model S is capable of producing shipboard surveys accurate at the 1 mGallevel.

The AGMS was originally designed for use over open water since the primary source of vertical positioning was a sensitive radar altimeter and the ocean provides a reference surface for the radar heights [Brozena et al., 1986]. Over the ocean we approximate Vu , the time derivative of the radial component of the aircraft velocity, by the second derivative of the radar heights above the sea surface. The sea surface departs from the ellipsoid by the geoidal height + dynamic oceanographic heights (e.g. tide and air pressure induced sea-level changes). The approximation for Vu is reasonable because the second derivative process strongly attenuates long­wavelength features in this sum while accelerations due to short wavelength features of the ocean surface are attenuated by the low-pass filtering required in airborne gravimetry.

The first NRL deployment of the gravimeter and radar aboard a P3-0rion aircraft occurred in 1980. Horizontal positions were determined from LORAN­C. Initial test flights off the coast of California produced several tracks with rms comparisons to ground truth of between 10 and 15 mGals.

In 1981 we replaced the LORAN with an early prototype P-code GPS receiver, a Texas Instruments X-set, for horizontal positioning. A test over the Wallops Island gravity test range off the east coast of the United States yielded average rms errors of <6 mGals and several profiles with <2.5 mGals rms deviation from the ground truth data [Brozena, 1984].

The AGMS was augmented with a sensitive, boom-mounted pressure altimeter in 1984 to allow limited operation over land. At P-3 speeds (350-450 km/hr) the pressure altimetry is quite noisy and provides valid altitudes for only relatively short periods due to pressure variations in the atmosphere. However, it is suitable for patching in between segments of radar altimetry and for extending tracks some distance inland. A test area over eastern North Carolina consisting of roughly half water and half land provided numerous opportunities for updating the pressure altimeter using the radar over water. A one degree square was covered by a nominal 9x9 km grid of tracks in less than 13 hours of flying at 365 km/hour. After gridding, the airborne data was compared with gridded ground truth data yielding a 2.8 mGal rms difference [Brozena and Peters, 1988] some of which may be attributed to errors of ommission in the ground truth data.

Actual operational surveying with the pressure augmented AGMS began in 1987 with the US-Argentina-Chile (USAC) aerogeophysics program over the ice­covered seas surrounding Antarctica. Like the open ocean surface, sea-ice proved to be a suitable reference surface for the radar altimeter providing that the ice is floating and not grounded [Brozena et al., in press]. The large range of the NRL

490

P-3 aircraft allowed the aircraft to be based in Tierra del Fuego, some 1200 km from the survey sites. In spite of the enormous transits, data blocks totaling nearly 700,000 km2 and 400,000 km2 were collected over the Weddell and Bellingshausen Seas respectively with average track spacings of 35 km. The entire field program was accomplished in two, one month deployments. RMS crossover miss-ties of both surveys were <2.5 mGals with nominal aircraft speed and altitude of 460 km/hr and 600m respectively. Along track wavelength resolution was 15-25 km.

In 1987 NRL began a cooperative program with the National Geodetic Survey to add dynamic GPS interferometry to the AGMS as a source of 3-d positioning. This addition extends the system's operational capabilities to over-land regions by removing the requirement for a radar reference surface for the determination of altitude and vertical acceleration.

A series of flights compared radar altitudes over water to the heights derived from differential carrier-phase GPS measurements. Two-frequency carrier-phase data were acquired from a pair of TI-4100 GPS installed aboard the NRL P3 aircraft and at a fixed base station. Baseline lengths of data tracks varied from 20 to 120 km. It became apparent that the TI-4100 receivers were prone to cycle slips under aircraft dynamics. Wherever possible the cycle slips were fixed and the data reduced by the double difference method [Mader, 1986]. One flight that had only a few cycle slips provided the best comparison. Unfiltered RMS height differences on four data tracks from this flight varied from 16-23 cm (Figure 1a). Much of this difference can be attributed to oceanographic and geoid heights contained in the radar measurements but not the GPS positions and to uncorrected geometric effects and coordinate system differences [Brozena et al., 1989]. After minimal filtering these differences were reduced to less than 3 cm RMS. Vertical accelerations were calculated from the heights after filtering with the same scheme used to produce the Antarctic gravity data sets. An RMS difference of 2.0 mGals was achieved on one 60 km long track (Figure 1b).

While this flight series proved that interferometric GPS could be sufficiently accurate to provide corrections for airborne gravimetry, it also showed the difficulty of maintaining continuous phase-lock in an aircraft with the TI-4100. Over half of the flight data was lost to unrecoverable cycle slips. To address this problem a second flight series was flown in December, 1989 in collaboration with Aero Service using a pair of their dual-frequency MINIMAC GPS receivers to collect carrier-phase data. These receivers are less prone to cycle slips on each individual channel than the TI-4100. They are also capable of receiving up to eight satellites simultaneously which reduces the consequences of any single channel cycle slips which occur. Raw inertial data were also acquired from a Litton LTN-72 INS. Data tracks were primarily over water but included some over-land segments and averaged 700 km in length. The goals of the experiment were to test the robustness of the new GPS receivers to loss of phase-lock, determine error growth with baseline length, to determine the length of cycle slip which can be bridged using raw inertial

491

.o .-------------------------------~

. ,00 0 ~ ~60':'-'-'-~,'~O ~,.~o '"'"'"'=',.'7"0 ~300~-':'360~->:.27"'0 ..c.u.: •• ::-,O ~>.O ~ TIME (SECONDS 1 t 20

10,-----------------------------, • ~

A . 'OO~U:.LLO ~1~~u..J"c..JO ~ • • 7'O ~3007'-'-'~3'7'O ~.~:-u-'":: •• ~O ~

n.o€ I SEC"""'SI

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.,

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· 20

B o 120 2 4 0

TIME (SECONDS)

Figure 1: a. Comparison of unfiltered GPS and radar altitudes. Two meters have been added to the GPS data for visibility. h. Comparison of filtered vertical accelerations calculated from the heights in a.

data, to test the error in vertical acceleration resulting from integer phase ambiguity errors and the growth of this error with time and to determine topographic and ocean geoid profiles by differencing the GPS and radar altitudes. Data analysis is still underway but initial results look encouraging.

CURRENT PROGRAM STATUS

The AGMS program at NRL is currently being splitinto a large and small-aircraft programs. The large-aircraft branch will concentrate on utilizing the speed, range and payload capability of the P-3 aircraft for the generation of medium resolution (>20 km) very large scale gravity surveys for geodetic and geophysical studies. NRL and the US Naval Oceanographic Office (NAVOCEANO) have joined forces to pursue this goal. The small-aircraft program is concentrating on the develop­ment and use of a subset of the AGMS that will be installed on a twin Otter for the production of high resolution (5-10 km wavelengths), detailed, continental geophysical studies over limited areas.

NRL/NAVOCEANO Joint Program

While there is still a great deal of research and development necessary to improve the accuracy, wavelength resolution and percentage of data recovery in the AGMS, the system can produce good quality data today. NRL has begun a joint program with NAVOCEANO to continue system development and to simultaneously con­duct large-scale regional gravity surveys. This program will link NRL's expertise in airborne gravimetry with NAVOCEANO's surveying infrastructure which includes

492

several P-3 aircraft, most of the existing Bell BGM-3 gravimeters and numerous highly trained and capable personnel for airborne data acquisition and reduction. NRL will be responsible for continuing systems R&D, the testing of developments aboard the NRL P-3 and the use of collected data for basic scientific research. NAVOCEANO will be responsible for large-scale surveys aboard their P-3 aircraft and for US Government uses of the data. We plan to fly roughly 400 hours of AGMS operations 1991. By 1992 we hope to devote 700-800 P-3 flight hours per year (",,300,000 line-km) to surveying in difficult logistical environments such as Greenland, the polar ice-cap and coastal regions.

In order to begin surveying now, we plan to use the brute force method of over­coming data acquisition problems. The large aircraft system will initially include three gravimeters, an INS equipped with raw data outputs and four GPS receivers on three antennas as well as the sensitive radar and pressure altimeters. Sheer data redundancy will help to overcome problems such as cycle slips, gravimeter off-leveling and hardware failures. Multiple sensors will also allow the concurrent testing and evaluation of different sensor types while producing useful survey data.

The first deployment of the large-aircraft system will be in October of this year, surveying over southern Florida (Figure 2a). Participants will include NRL, NAVOCEANO, USGS, NOAA and the Geological Survey of Canada. A Bell BGM-3 and L & R models S and SL gravimeters will be installed on the aircraft. Airborne GPS receivers will include 2 MINIMACs, a Trimble 4000 and a TI-4100. GPS base stations will be located at Key West and Miami.

Besides the gravity data we will also be collecting radar altimetry from the AGMS. These data will be differenced with the GPS altitudes to test the topo­graphic profiling accuracy of the system.

Small-aircraft System

The small-aircraft system is based on a BGM-3 gravimeter, a MINIMAC and TI-4100 GPS receiver, LTN-90 inertial, boom-mounted pressure altimeter and a lower precision radar altimeter. The stable platform of the BGM-3 has been modified to make it more suitable for use in an aircraft [Brozena et al., 1986]. All data will be logged by a PC based data acquisition system. The system is light and compact enough to be installed in a twin Otter along with the Ohio State University ice radio-echo sounding radar and the USGS towed-bird magnetometer system. There will not be much sensor redundancy and more lines will probably require reflying due to lost data than for the large-aircraft system. This situation will improve with time as more GPS satellites become available and as the system is refined.

The low-speed of the Otter should allow the resolution of anomalies with wave­lengths of a few km. The system is being deployed to Antarctica in this November to begin a three year NSF sponsored campaign to acquire gravity, magnetics and ice thickness in Antarctica between the Transantarctic Mountains and Marie Byrd Land. The goal is to collect a 220 x 330 km block with a 5 x 5 km grid spacing

493

\

AGMS - OCTOBER 1990 AIRBORNE GRAVITY AND MAGNETICS SURVEY

EMPLOYING GPS INTERFEROMETRY PARTICIPANTS: NRl. NOAA (NGS). USGS. NAVOCEANO. GSC

" JACKSONVILLE F.Z.

" " ""-

"'- ......... BAHAMAS F Z.

""-....

c

Sfr N SOON

SOOW

l00w

70'W

lOON lOON

2frW

6frW

B 6001'1

400w 300w

Figure 2: a. Oct, 1990 P-3 survey of southern Florida. h. Proposed 1991-1992 survey of Greenland. c. Proposed 1992-1993 high Arctic survey.

494

during each field season. We hope to make the geophysical aircraft instrumenta­tion available in the future to researchers in various fields such as geodesy, geology, geophysics and glaciology.

FUTURE AGMS DEVELOPMENTS

Although there are similarities in measurement and function, there has been a dichotomy between mechanical platform INS units and gravimeters. Generally, the gravimeter has one, very hiqh quality, accelerometer mounted on a relatively simple and primitive stable platform, while, except for expensive, special purpose units, the INS has two or three medium quality accelerometers mounted on a high quality platform. Future improvements to the AGMS depend on removing this dichotomy.

The stable platform of the L &R gravimeter, in common with that of the Bell BGM-3, orients to the time-averaged local apparent vertical, rather than the normal to the ellipsoid. The platforms are space-stablized by reference to a pair of gyros. This points the platform in a direction which slowly drifts with time due to gyro errors. The direction of space-stabilization is forced to be the local time-averaged apparent vertical by torquing the table to null the average output of a pair of orthogonal horizontal accelerometers. This accounts for both gyro drift and the rotation of the local-vertical frame as the aircraft moves through space.

In the static case, the table aligns the meter with the gravity vector, thus measuring the total magnitude rather than the vertical component of gravity. This is different than the situation described by Equation (1) but, causes negligible errors since, everywhere on the earth, the vertical component of gravity is within a fraction of a mGal of the total magnitude of the vector. However, the orientation is further perturbed in the dynamic case by the horizontal accelerations produced by motion over the surface of the earth.

An aircraft moving over the earth experiences horizontal accelerations which are sensed by the platform accelerometers. The table is then torqued to a new average direction which is the time-averaged resultant of the gravity vector plus the horizontal acceleration vector. The gravimeter will therefore be off-level. It will remain off-level until the horizontal accelerations, averaged over the damping period of the platform, return to zero. This can cause problems in nominal straight and level flight as well as turns since aircraft continually oscillate to some degree in all three axes, and significant Coriolis accelerations are experienced except under unusual circumstances.

The L&R platform addresses this problem by using long damping periods (4, 18 or 84 min) which, however, only make the leveling problem worse during turns as the horizontal acceleration due to the turn is continued long past the turn by the long period averaging. The accelerometer input must therefore be turned off during turns to prevent the platform from being driven off-level and the accelerometer filters from being perturbed by this impulse. Without accelerometer input the

495

platform is stabilized only by the gyros, with no vertical reference, and slowly drifts off-level. The Bell solution is to use a short damping period so that the platform will erect quickly after turns. Unfortunately the platform period, while suitable for ships, is close to aircraft horizontal periods, causing nearly continuous leveling problems.

Both the L&R and the Bell meters could be greatly improved by the incorpo­ration of navigation information into the table leveling circuitry. Real-time hor­izontal accelerations could be computed from GPS and used to compensate the accelerometer / gyro feedback loop for this perturbation. This is reportedly how the Bodenseewerke KSS-31 gravimeter functions (G. Hein, pers. comm.). However, this meter is not widely available.

It might be preferable to install a high quality vertical accelerometer in a local­level INS rather than correcting the flaws in the ~table platform of a commercially available gravimeter. This has the obvious advantage of removing one fairly large, heavy and expensive piece of equipment from an airborne system. For example, it appears that the commercially available Bell model XI accelerometer with its tem­perature control oven should fit in the slot provided for the unused vertical channel of a LTN-72 INS. If an INS is to be utilized as a gravimeter it would be desirable to implement a real-time, closed loop, integrated GPS~INS system to reduce the plat­form orientation error. This would be vital if vector gravimetry is contemplated. Horizontal accelerometers upgrades would probably also be required.

We are pursuing both paths at NRL . We are currently engaged in designing a digital control system for the BGM-3 platform which will accept navigation cor­rections. We are also beginning a program to build an integrated GPS/INS system using high quality accelerometers mounted in a set of surplus LTN-51 gimbals.

FUTURE SURVEY PLANS

Our primary focus with the NRL/NAVOCEANO large-aircraft system will be the acquisition of data in remote, poorly surveyed regions. One of the first priorities will be Greenland. Figure 2b shows the tracks for a proposed gravity/magnetics/ice topography survey of Greenland. The survey is designed to provide half-degree mean anomaly values over the entire island. If the proposed survey is funded we will begin operations in conjunction with NOAA and the Danish Geodetic Institute in the summer of 1991. The survey will use 500-600 hours of aircraft time. This would be accomplished in two months in first season and three during the following summer. ERS-1 should become operational during this period and some of the north-south tracks may be modified to fly along sub-satellite tracks. Ice profiles obtained by subtracting the aircraft radar altitudes from the absolute GPS derived altitudes would serve as a direct calibration/validation check on the ERS-1 data over the ice-sheet. We may also add a laser altimeter to the AGMS as a check on the surface penetration depth of the radar altimeter. The radar can easily be shifted to match the frequency of the ERS-1 unit.

496

Another high priority for the large-aircraft system is the north polar ice-cap. We are tentatively negotiating with the Office of Naval Research to begin a large-scale survey of this region starting in 1992. The stippled area of Figure 2c represents the proposed area. The arcs indicate the area that could be reached by the NRL P-3 operated from Andenes, Norway, Thule, Greenland and northern Alaska. The NAVOCEANO P-3 has significantly better range (up to 5500 km/flight). Four to six months of flying could obtain half-degree mean anomalies over the entire ice-cap. Along track resolution would be about twice as good.

We are currently negotiating other possible future surveys including the tropical rain forests of Brazil, the mountainous island arc of the Aleutians and further work in Antarctica. Over the next decade the use of fast, long-range aircraft for airborne gravity surveying should provide major reductions in gaps in regional gravity coverage of the earth. For areas and tasks that require detailed high­resolution studies, small-aircraft systems will be the method of choice.

REFERENCES

Bower, D.R., Kouba, J. and Beach, R.J. (1990). The spectrum of GPS measure­ment errors and the accuracy of airborne gravity measurements, Geophysics, 55, 1101-1104.

Brozena, J.M. (1984). A Preliminary Analysis of the NRL Airborne Gravimetry System, Geophysics, 49, 1060-1069.

Brozena, J.M., Eskinzes, J.G. and Clamons J.D. (1986). Hardware Design for a Fixed-Wing Airborne Gravity Measurement System, Naval Research Labor­atory Report 9000 1986.

Brozena, J.M. and Peters, M.F. (1988). An Airborne Gravity Study of Eastern North Carolina, Geophysics, 53,245-253.

Brozena, J.M., Mader, G.L. and Peters, M.F. (1989). Interferometric Global Positioning System: Three-Dimensional Positioning Source for Airborne Gravimetry, J. Geophys. Res., 94, 12153-12162.

Brozena, J.M., LaBrecque, J.L., Peters, M.F., Bell, R. and Raymond C. (1990). Airborne gravity measurements over sea-ice: the western Weddell Sea, Geophys. Res. Lett., in press.

Harlan, R.B. (1968). Eotvos corrections for airborne gravimetry, J. Geophys. Res, 73, 4675-4679.

LaCoste, L.J.B. (1967). Measurement of gravity at sea and in the air, Rev. Geophys., 5,477-526.

Mader, G.L. (1986). Dynamic positioni~g using GPS carrier-phase measure-ments, Manuscr. Geod., 11, 272-277. .

Rusler, G.W., Tidwell, N.H. and Brown, B.H. (1970). Design configuration and error analysis study of an airborne gravimetric system, Honeywell Tech Rep. 1!606-FR.

497

ACCURACY OF GRAVITY VECTOR RECOVERY

WITH THE LTN 90-100 RLG STRAPDOWN SYSTEM

ABSTRACT

Elfriede Knickmeyer Department of Surveying Engineering

The University of Calgary 2500 University Drive N.W.

Calgary, Alta., CANADA T2L 2}5

The paper discusses whether the LTN 90-100, which belongs to the 1 Nm/h class, could be used to validate the concept of vector gravimetry by a combination of INS and GPS. For positioning the LTN 90-100/GPS combination has already been successfully employed. To determine to what accuracy the LTN 90-100 could provide the anomalous gravity vector tests on the short and long term stability of the accelerometer and gyro biases were performed. A temperature model was necessary to correct for large residual temperature dependent accelerometer drifts. The effect of the sensor biases on gravity field parameters was estimated by a numerical integration of stationary data. The results for a 1 hour period with 2 min. averaging time indicate errors in the gravity disturbance, deflection, and height anomaly of 5-10 mgal, 7 arcsec, and 1 m, respectively. These errors are too large to use the system for concept validation of vector gravimetry by INS/GPS.

1. INTRODUCTION

Huddle showed (1978) that in order to determine deflections of the vertical at the sub-arcsecond level, an accuracy of the gyro biases of 0.0010 /h is required. This asks for systems in the 0.1 Nm/h (nautical miles per hour) class (Savage, 1984). Since no such system was available for an extended period of time, it should be determined to what accuracy the Litton LTN 90-100 ring laser gyro strapdown inertial system, which is owned by the Department of Surveying Engineering, could provide the anomalous gravity vector. This system belongs to the 1 Nm/h class and has already been successfully combined with GPS for positioning. The analysis was based on

498

raw output data. When judging on the results, one has to keep in mind that the specific instrument tested is subject to harder conditions than typical aircraft navigation systems are: it has been extensively used on rough gravel roads; for lack of power the accelerometers could not always be caged during transportation. Therefore, other systems of the same type might perform better.

2. BIAS STABILITY

The objective was to verify the bias specifications given by the manufacturer and to estimate their effect on gravity field parameters. The specifications (Litton, 1984) are given in Table 1.

Accelerometer A-4

net bias accuracy 500 Jlms-2 10', over 1 year

bias short term stability 50 Jlms-2 10'

Laser Gyro LGG-8028B

bias repeatability I O.OlO/h I all cases

Table 1: LTN 90-100 Sensor Specifications

2.1 Medium Term Bias Stability

The behaviour of the sensor biases was observed over a period of two months, during which the biases were regularly determined by tumbles, using a Cardan frame (Knickmeyer, 1989). The results are plotted for the x­accelerometer and x-gyro in Figs. la and lb. They show that the accelerometer bias variation over longer periods is well within the specifications, whereas the gyro bias variation is considerably larger than specified. For precise surveys the biases therefore have to be determined immediately before the run.

499

,....., -140 C"I

I en [ -160 ---a- Accel bias

.......... en td -180 .....

.0 -Q) u -200 u <:

-220

-240

-260 120 130 140 150 160 170 180 190

Julian Days

Fig. la: Accelerometer Bias Variation over 2 Months

'7il -0.18 .....---------------------, ........

~ -0.20 ..........

~ -0.22

~ l? -0.24

-0.26 Gyro bias

-0.28 L...-_.......&. __ -'--__ .L..-_.......&. __ -'--__ .L..-_--'

120 130 140 150 160 170 180 190 Julian Days

Fig. Ib: Gyro Bias Variation over 2 Months

500

2.2 Short Term Bias Stability

The short term bias stability was assessed during a test of ten hours duration. The biases were determined by tumbles at 24 epochs. The system was not turned off in between. Since the body accelerations showed a very large drift, caused by a change in temperature due to self-heating, a residual temperature correction had to be applied:

f1·(t) = fi(tref) + CJ i" (t - tref) + c2 i" (t - tref)2 + c3 i' (t - tref)3 , , , (1)

where

f i measured specific force in the i-channel

t actual temperature tref reference temperature, e.g. 25 °c.

The temperature change due to self-heating was about 20 °C over the duration of the test. The correlation of the unreduced accelerations and the recorded temperatures was found to be 0.87. No residual temperature correction was applied to the gyro output. The results are shown for the x-accelerometer and x-gyro in Figs. 2a and 2b .

....... N 0 I rJ)

§. -50 ~

rJ) (Ij .... ~ -100 ...... Q) u u « -150

-200

-250

-300 10 12

Fig.2a:

•

14 16 18

without temp. corr. with temp. corr .

20 22 24 Time

Short Term Bias Behaviour x-i\ccelerometer Bias

501

v; -0.1 r-----------------------, "-

~ ~ ....... ~ -0.2 ..... ~

~ l?

-0.3

without temp.corr.

-0.4 1--_---" __ --1..-__ ....1.-__ 1--_---" __ --1..-_----'

10 12 14 16 18 20 22 24

Fig.2b: Short Term Bias Behaviour x-Gyro Bias

Time

Nine months after the last factory calibration the tests showed very large absolute biases but small bias variations. For the accelerometers the absolute biases and the bias variations are smaller that specified. For the gyros the absolute biases are much larger that specified and the variation is about twice the specified value.

3. COVARIANCE FUNCTION FOR RESIDUAL BIASES

If empirical covariance functions could be determined for the gyro and accelerometer biases, the parameters of these functions could then be used in a simulation program to assess the achievable accuracy for the gravity field quantities. When using the raw body rates, the result is a highly oscillating covariance function since the raw output contains periodicities stemming from the dithering of the laser gyros. Removing these effects by a Butterworth filter was investigated by Wong (1988). Since the Butterworth-filter introduced a bias, filtering in the spectral domain has been considered here. For this purpose a spectral analysis of the raw output was performed at different epochs within one set of stationary data. Some of the results are shown in Figs. 3 and 4. The temperature at the epoch of Fig. 3 was about 50 °c, at the epoch of Fig. 4 it was about Soc. The time difference between the two exposures was 1 h 45 m .

502

~

';b 10000 S ;:1.. 8000 .......... en <d ...... 6000 f-~

...... OJ u 4000 f-U

< I

>- 2000 r-d 00

0 <d

~ 0

l . .1.

I I

5 10 15 25 30 20 35

Frequency [Hz]

~

en .......... 100 u

OJ en u

80 I-< <d .......... en <d 60 f-...... ~ 0 40 I-< >-

C) I 20 x

~ 0 I I I I

<d • I • • ~ 0 5 10 15 20 25 30 35

Frequency [Hz]

Fig. 3: Sensor Spectra at Epoch 1

At epoch 1 the peaks at 8.92 Hz, 18.58 Hz, and 29.44 Hz show up in all channels and in most cases have the highest power. It is therefore assumed that these are the aliased dither frequencies. As the sampling rate is 64 Hz and the dither frequencies range from 380 Hz to 420 Hz (Wong, 1988), their frequencies would be 392.92 Hz, 402.58 Hz, and 413.44 Hz. In some of the channels, e.g. the x-gyro, a number of additional peaks show, which cannot be clearly identified. They could be harmonics of the dither frequencies, response of the mounting block to the dithers, signals from the cooling fan, or something else. For a moving system it would be even more difficult to

503

....-.

c;b 10000 S ;:::1.. 8000 r ........ en n:I ..... 6000 r ~

...... Q) u 4000 u « I

I

>. 2000 r:: 00

0 n:I

~ 0 5

I I

10 15 20 25 30 35

Frequency [Hz]

....-. en

" 100 u Q) en u 80 r M n:I ........ en n:I 60 r-..... ~ 0 40 r-~ l? X 20 r

J ~ 0 I .

n:I I I I I

~ 0 5 10 15 20 25 30 35

Frequency [Hz]

Fig. 4: Sensor Spectra at Epoch 2 = Epoch 1 + 1 h 45 m

distinguish between effects due to the dithers and other effects. A comparison of Figs. 3 and 4 shows that the whole spectrum changes with time and/or temperature. The dither frequencies of the two epochs differ by up to 3 Hz and also their amplitudes are different.

Considering that the effects of the dithers cannot be clearly identified and separated from other effects and that they are not constant in time, it is not recommended to try eliminating them either in the time or in the spectral domain.

504

In order to obtain a covariance function that is not dominated by these high-frequency, high-amplitude oscillations, the covariance was computed from 10 second averages. Beforehand, the above-mentioned temperature correction was applied. Also, a constant bias and drift was removed. This corresponds to the procedure of gravity surveys, where a bias determination is performed at the beginning and at the end of the survey. The covariance function obtained (Fig. 5) is still oscillating. The amplitude shows a large peak at t = 0 and then rapidly levels off to a more or less constant value, which it keeps for at least two hours.

~

0.08 ~ * -ell 0.06 ....... u Q) ell 0.04 u ... <IS -....... 0.02 0 ~

0.00 c..? I X :> -0.02 0 U

-0.04 0 10 20 30

Lag [min]

Fig. 5: Covariance Function of x-Gyro from lOs Averages; C(O) = (0.26 arcsec/s)2

The functions are of the form

C(t) = ( a + b . e- c' t) . cos( rot ) (3)

and the corresponding process could be modelled by a state space model. Since the appearing frequencies are not exactly known and not constant, the much simpler model of a first-order Gauss-Markov process seems to be justified. It should consist of two components: one with a short correlation length of about 10 seconds and one with a longer correlation length of the order of a few hours. The specific parameters cannot be precisely determined from the raw data, but should be determined from the integrated data, e.g. by performing calibration runs, see e.g. Wong (1988).

505

4. INFLUENCE OF BIASES ON GRAVITY FIELD PARAMETERS, ESTIMATED FROM STATIONARY DATA

An estimate for the lower limit of the effect of sensor errors on the gravity field parameters can be obtained directly from the reduced output, since the accelerometer and gyro biases are the main error sources. For an averaging time of 2 minutes, which at a speed of 60 km/h corresponds to a resolution of 2 km, the behaviour of the residual biases is shown for the x­channel in Figs. 6 and 7.

The effect on the gravity disturbance is directly

(4)

The effect of the biases on deflection and height anomaly can be approximated by

(5)

t t2 t

B~~(t) = - v . J Jbg(t1) dtl dt2 +;. Jba(t) dt (6)

(Knickmeyer and Schwarz, 1989), where the coupling between different channels has been neglected.

,....... N

I til

§ ....... ...... ~ u < I x

100

50

0

-50

-100 0 60 120

Time [min]

Fig. 6: x-Acceleration Error (2 min. Means)

506

The deviations in the accelerometer bias reach 60 and - 80 Jlms-2 at t-l0 min and t-22 min, respectively. Similar "dents" occur in most of the analyzed data sets and it can be expected that this is even more the case for a moving vehicle where the sensors are subject to disturbing accelerations.

,......, 0.06 ,..------------------.., CIl

........

~ .......

0.04

0.02

60 120

Time [min]

Fig. 7: Angular Velocity Error in x (2 min. Means)

The effect of the accelerometer bias on gravity disturbance and deflection would be

BL\g - 50 - 80 JlIDs-2

B~ - 1 - 2 ".

The effect on the height anomaly was obtained by numerical integration to be

BL\~ - 3 cm.

The effect of the gyro bias on the gravity disturbance is negligible. The effect of gyro bias variations on the deflection and height anomaly are plotted in Figs. 8 and 9. The average bias and drift have been removed. The results largely depend on how well the average bias and drift over the whole run can be recovered from bias determinations at the beginning and at the end, and are influenced by the random walk. For a one hour run the effect on the deflection would be about 10" and the effect on the height anomaly about 1.8 m. By smoothing, using known gravity field parameters at the end of the

507

traverse, this could be reduced to about 1/-{2 times the value at half the travelling time, yielding

o~ -7"

o.1~ - 1.3 m.

These estimates do not include effects due to scale factor errors, misalignments, or disturbing accelerations, and their influence on the sensor behaviour, which would add to the overall error. On the other hand, large sensor errors as the ones shown would be partially observed by update observations.

40 ,........, u

30 OJ fJ) U J-< ~ 20 ........ J-<

8 10 J-< ~

td 0 OJ 0

-10 0 60 120

Time [min]

Fig. 8: Effect of Gyro Bias Variation on Deflection

10

8 ,........, 6 S ........ )JI 4 s:: .....

J-< 2 0 J-< J-<

0 ~

-2

0 60 120

Time [min]

Fig. 9: Effect of Gyro Bias Variation on Height Anomaly

508

5. CONCLUSIONS

The performance of the LTN 90-100 can be considerably improved by determinining the biases immediately before the run and by applying a residual temperature correction. The covariance functions of the biases cannot be determined from raw data due to time varying periodicities.

From analyzing the bias behaviour it can be concluded even without field tests that the LTN 90-100 is not suited for gravity vector recovery. As far as the accelerometers are concerned, this is mainly due to drifts. If these could be avoided or observed through suitable measurement procedures, the determination of the magnitude of gravity might be possible to a few tens of }lms-2. Such measurement procedures could be step methods, crossovers, or rotation of the sensor assembly. The accuracy of the gyros is sufficient for the determination of the magnitude of gravity, but not for the determination of deflections of the vertical or height anomalies. In order to obtain useful results for the gravity vector, a system with better gyros, especially with a smaller random walk, should be used.

Acknowledgement

The contribution of Mr. Radim Zizka, who performed the tumbles f~r the bias calibrations, is thankfully acknowledged. Financial support for thiS research was obtained from an NSERC grant of Dr. K.P. Schwarz.

REFERENCES

Huddle, J.R. (1978): "Theory and Performance for Position and Gravity Survey with an Inertial System." J. Guidance and Control, Vol. I, No.3, May­June 1978, pp.183-188.

Knickmeyer, E.H. (1989): "Calibration, Handling, and Use of a Cardan-Frame with the Litton LTN 90-100 Inertial Reference System." Pub. # 30011, Dep. Surveying Engineering, University of Calgary, Calgary, Canada.

Knickmeyer, E.T. and K.P. Schwarz(1989): "Geoid Determination using GPS­aided Inertial Systems." Proc. Symp. 104, General Meeting lAG , Edinburgh, Springer, New York.

Litton (1984): "Technical Description of LTN 90-100 Inertial Reference System". Document # 500406, Aero Product Division, Litton Systems California, Canoga Park, U.S.A.

Savage, P.G. (1984): "Advances in Strapdown Sensors." AGARD Lecture Series, No. 133, NATO, Neuilly Sur Seine.

Wong, R.V.C. (1988): "Development of a RLG Strap down Inertial Survey System". PhD Thesis, Publication # 20027, Dep. of Surveying Engineering, University of Calgary, Calgary, Canada.

509

A METHOD TO DETERMINE INCREMENTS OF VERTICAL DEFLECTIONS

ABSTRACT

O.S. Salytchev and A.B. Bykovsky Bauman Moscow State Technical University

USSR

The anomalous gravity vector is determined by Inertial Survey Systems (ISS) using measurements of the velocity errors in ZUPT points. These errors are caused by the simultaneous action of the instrument errors of the ISS (accelerometer error, gyroscopic drift, etc.), the misalignment errors and the gravitational model errors due to the anomalous gravity field. The instrumental errors are functions of time, while the anomalous gravity vector is a function of position. The change of the gravity anomalies within the correlation interval can be approximated by a polynomial in coordinate increments. The problem of determining the anomalous gravity vector is essentially the problem of estimating the constant random coefficients of the polynomial using measurements of the velocity errors at ZUPT points. The estimation accuracy depends on the variation in vehicle speed between ZUPT points. Simulation results indicate a fairly accurate approximation of the anomalous gravity vector by polynomials with estimated coefficients.

INTRODUCTION

The anomalous gravity field introduces errors in the commonly used Earth gravitational model which in turn causes errors reaching hundreds of metres in the coordinates determined by an ISS. The determination of the anomalous gravity vector makes it possible to increase ISS accuracy as well as to solve another important problem in geodesy - the determination of the gravitational model of the Earth.

The anomalous gravity field is caused by inhomogeneities in the Earth's density distribution. It is a function of position only. However, the estimation of the anomalies from velocity measurements at ZUPT points uses a time error model of the ISS and time models of the external disturbances. The transition to the time models results in a random process representation which is the output of a shaping filter excited by white noise.

For example, deflections of the vertical are modelled rather well with the aid of a shaping filter having the transfer function (Kriegsman and Mahar, 1986)

og 1 - = 2 Wg L+ 2~~ + 1

W()2 W()

(1)

510

where <>g . .. filter output, w g . .. filter input,

roo . .. natural oscillation frequency of vertical deflections in absence of damping.

The value roo depends on the vehicle speed V which is assumed to be constant:

kV roo=T ~

where k . .. maximum harmonic number in disturbance compensation model, R . .. Earth radius.

Simple time models using filters of first order are employed in the anomaly estimation algorithms (Forsberg, 1985)

. 1 <>g(t) = - - <>g(t) + Wg(t) (3)

'tg

where 'tg . .. correlation time.

The correlation time of the anomaly is related to the correlation length Sg in the spatial domain by

~ ~=V ~

where V = const. Equation (2) is a function of the vehicle's velocity, which must be maintained at a chosen

constant level throughout the motion period. This condition complicates the transformation of the gravity anomaly model from the spatial to the time domain.

The correlation times in the ISS sensor error model (gyroscopic drifts, accelerometer errors) and in the gravity anomaly model have similar magnitude. This reduces the observability of the ZUPT velocity errors when the estimation algorithms (e.g. Kalman fIlter) are used directly in realtime.

Considering that the gravity anomalies are functions of position, measurement of multiple independent returns, reverse as well as cyclical, is recommended to ensure the observability of the gravity anomalies (Benson, 1985). For this reason, we suggest the use of a spatial gravity anomaly model in the ISS error equations together with a real-time estimation algorithm.

SPATIAL DEFLECTION MODEL FOR ISS ERROR EQUATIONS

Components of the anomalous gravity vector may be described within the correlation length

Sg by a number of smooth functions which have the trajectory increment AS as argument. This conclusion has been drawn from the study of experimental results (Forsberg et. al., 1985) and modelling results where deflections of the vertical were taken as functions of coordinates (Groten et. al., 1985).

Figure 1 shows the vertical deflections, <>g, over the trajectory of length, S. The results are obtained using the shaping fIlter (1) and modelling the velocity as V=constant.

Let us now find the connection between the shaping fIlter equation (1) in the time domain and the representation of the vertical deflections in the space domain. Passing from the operator to the time notation, we have:

511

S [km]

-3

-6

-9

- 12

- 15 1\

og,og [mgal]

Fig. 1. Deflections of the vertical.

(5)

Here, the deflections of the vertical, og(t), are a random process in the time domain caused by white noise Wg(t). The vehicle speed V is assumed to be constant.

The change in vertical deflections, a random low-frequency process, may be approximated by a series of base functions, if the deflections are considered as a function of position which is the usual approach. This series could be a polynomial in powers of coordinate increments in the approximation intervals. In this case, the coefficients are random constants.

For example, the deflections of the vertical may be approximated in the interval Se [Sj,Sj+l] by the polynomial

o S ss (S-Sj)2 ogj(s) = ogj + ogj (S-Sj) + ogj 2 (6)

o where ogj deflection of the vertical at the beginning of the j-interval,

<;:s <;:ss ug, u g . .. first and second derivatives of the deflections in the space domain.

The following equations hold for a polynomial of form (3):

aog o S ss ~ = ogj V + ogj V (S-Sj) at a20gj <;:ss 2

012 = ugjV

(7)

(8)

Again, the vehicle speed V is assumed to be constant to assure consistency between the time and space domain models of the vertical deflections Bg.

Let us rewrite equation (5) by taking into account equations (6) to (8):

512

ss k 8 SS k2 8 88 (S-S·)2 k2 agj + 2~~ R {agj + agj (S-Sj)} + R2 (ag9 + agj (S-Sj) + agj 2 J } = R2 Wg(t)

Since it is assumed that V=const, a unique transfonnation between the white noise Wg(t) in the time domain and the white noise Wg(s) in the space domain exists. As a result, the following equation for the spatial shaping filter of the vertical deflections is obtained:

d2agj J:' dagj '2 ~ , 2 + 2~~ + ~ ugj = CO() Wg(s)

dS2 dS (9)

where agj = agj(s) and coO = ~ . The coO corresponds to the natural oscillation frequency of the spatial model of vertical deflections in the absence of damping. It depends on the accuracy of the gravity anomaly model compensating for the disturbance. It also determines the transmission band of the shaping filter. The reciprocal of roO can be considered the correlation length Sg of the random spatial process of the vertical deflections. Within this correlation length it is possible to approximate the function ag(s) to any degree of accuracy by base functions. A

compensation model of moderate accuracy is given by k=700 and Sg == 9 km, which is close in magnitude to Sg = 10 km (Forsberg, 1985).

Let us consider the use of the gravity anomaly model in the ISS error equations. For simplicity, we will restrict ourselves to a single-channel ISS because the application of the method to the general case involves no difficulties.

The ISS error equations are

Il. V = - g C\> + &t + Il.g . Il.V

C\> = T+£ . 1 £ = --£+we

'te . 1

aa = --&t+wa 'ta

where Il. V . .. ISS velocity error,

C\> • •• misalignment error of gyro platfonn,

£ . " gyro drift rate,

't . .. correlation time.

(10)

Usually, the time model (3) is used to describe ago Then, the first two error equations become

.. , ... '} 1 . Il.V = -\lIS"Il.V-g£--aa+ag(t)+Wa (11)

'ta

where O>s = ~ . .• Schuler frequency.

Taking into consideration the newly introduced spatial model, equation (7) can be expressed as

513

t • S ss f ogj(t) = ogj V(t) + ogj V(t) V(t) dt

to

where as

V(t) =-at

... vehicle velocity in interval Se [Sj,Sj+l].

S ss Assume that ogj = const. and ogj = const.. Equation (11) can be expressed in matrix form:

x = A x + B(t) f + G Wa (12)

where x = [AV , AV]T

f = Ej + ~, ogj, u [ oa ° S s:SgSJo]T

gta

A =[_~ ~J

B(t) = [ 0 -g

G = [?]

o

V(t) V(t) jV(t)dt ]

Within a time interval of 10 to 12 minutes, or two to three ZUPT points, it is possible to assume that

oa o

Ej + ~ = const. gta

The solution of equation (12) may then be written as the sum of the solutions for the corresponding homogeneous and nonhomogeneous equations. The homogeneous equation

Xh = AXh

with Xh(to) = [AV(to) , AV(to)]T

gives

Xh = F(t-to) Xh(to)

where F(t-IQ) = [ cosCOs(t-to)

-COssinCOs(t-to)

The nonhomogeneous equation

Xnh = A Xnh + B(t) f + G Wa

with Xnh(to) = 0

sincos(t-to)

COs

cosCOs(t-to)

514

l

gives t

Xnh(t) = D(t) f + f F(t-t) G wit) dt (13) to

where D(t) = A D(t) + B(t) (14) and D(to) = 0 . Thus, for the interval SE [Sj,Sj+1l the ISS errors are given by the vector

x(t) = Xh(t) + Xnh(t) (15)

Equation (12) models the errors of an ISS horizontal channel rather accurately since the cross-coupling at time t may be neglected in the interval SE [Sj,Sj+l].

ESTIMATION OF VERTICAL DEFLECTION INCREMENTS

It has been assumed that the ISS vehicle alternate between 3 to 4 minutes of motion and 0.5 to 1 minute of rest for velocity error measurements. Figure 2 shows the vehicle movement over time. Measurements z(t) are made at the time intervals t'l<t<tl and t'2<t<t2 and can be expressed by

z(t) = H x(t) + v(t) (16)

where H = [1 , 0] and v(t). .. measurement noise.

stop period

'III ~'III

motion period

Fig. 2. Vehicle motion over time.

t' t 1 1

t' t 2 2

Using equations (13) and (15) this formulation can be rearranged to give t

z*(t) = z(t) - H F(t-to) Xh(to) = H D(t) f + H f F(t-t) G wa(t) dt + v(t) to

= H D(t) f + v*(t) (17)

Time t has to be counted from to in the calculations of F(t-to) and D(t). The measurement z*(t) depends on the drift rate Ej and the accelerometer error Baj which are assumed to be constant over the time interval to<t<t3. It also depends on the increment of the vertical

s ss deflections determined by the polynomial coefficents Bgj and B gj which are constant over the j-section of the trajectory covered during time 1. The ISS instrument and deflection errors of the previous time intervals, t<to, do not have an effect on z*(t).

The initial conditions

515

~h(to) = [AV(to) , A ~(to)]T are evaluated by the method of least squares using the observation equations

z(t) = AV(t) + v(t)

with to < t < to . Then, AV(t) is written as a polynomial

. (t-tO)2 AV(t) = ao + at{t-to) + a2 -2- (18)

and the initial condition vector is given by A • (t-tQ)2

AV(to) = ~ + tt{t-to) + t2 2

AV(to) = tl + t2 (t-tQ) . s ss

The polynomial coefficients agj and a gj are the components of the vector f and are detennined by least squares:

ANN f = [i~1 OT(tu IfT H O(ti) ]-1 i~1 OT(ti) IfT Z*(ti)

where N=4 or 6 for 2 or 3 ZUPT stops, respectively, starting with a reset at time to. The measurement z*(t) is smoothed by least squares in each ~PT interval t'l<t<tl and t'2<t<t2 using the model of equation (18).

In the time interval Atj=t3-to the vehicle covers the distance ASj and the increment of the vertical deflections is detennined from the equation

t ~ (S· 1 S·)2 Aa~j = agj (Sj+l-Sj) + agj J+i ) (19)

for each j-sequence of intervals between two or three ZUPT points. At the fmal point t3 of the j-sequence the new initial conditions are estimated in the same

way as for time to. Time is then counted from the instant t=t3 for which 0(t3)=O and . t

F(t-t3)=I. When the vehIcle moves on, new measurements z*(t) are collected, the agj+l ~

and a g j+ 1 are determined and the increment Aa~j+ 1 for the 0+ 1 )-section of the trajectory is calculated.

The total increment of the vertical deflection during the entire vehicle movement is equal to

Aa~E = ~Aa~j J

This method of estimating vertical deflection increments considers rectilinear motion of the object only. For arbitrary vehicle motion it is necessary to determine the increments for the South-North and West-East directions separately. In this case, the velocity V(t) in

equation (12) and the trajectory increment ASj in equation (19) denote linear velocity and trajectory increment along the respective axes.

516

ACCURACY OF ESTIMATED DEFLECTION INCREMENTS AS A FUNCTION OF VEHICLE MOTION

The method suggested for the estimation of deflections of the vertical is essentially a least squares determination of the vector f referring to the measurements (16). Let us now analyze the observability of the components of the vector f in terms of the vehicle speed. To simplify the analysis, we restrict ourselves to the approximation of the deflections in the interval Se [Sj,Sj+l] by a first order polynomial

o s ogj(s) = ogj + ogj (S-Sj) (20)

Then

f = [Ej + oaj , ogj]T gta

B(t) = [_~ V~t)]. Let us analyze the possibility of estimating the vector f for the system

f = 0

z*(t) = H D(t) + v*(t)

using the stochastic observability matrix

p(t) = [i DT(t) HT R-I H D(t) dt]-I

where R = M[v*2] = O"~ •

(21)

(22)

(23)

It is assumed that a priori information on f is not available. The system (21) and (22) is unifonnly and completely observable if the integral in equation (23) is positive and limited

s by [to,t]. The quantitative measure for the obseniability of ogj is the estimated error variance P22(t) . •

Full observability of f may be obtained for V(t)~. For the in-flight alignment of aircraft navigation systems this condition can be met by an acceleration mode or a meandering manoeuvre. The moving ground vehicle cannot attain sufficient axial acceleration and the meandering manoeuvre may not be possible due to terrain conditions. We will show that the estimation of the vector f is also possible when the vehicle moves at various constant speeds in the intervals between ZUPT points.

Suppose that the vehicle moves at a speed of VI =const. during the time interval to<t<ti and at a speed ofV2=const. during the time interval tl<t<ti. Define the motion interval as At=ti-to=ti-tl and the stopping time for ZUPT as AT=tl-ti , then the stochastic

observability matrix in the interval [to,to+2At] becomes

[to+2At ]-1

P(At) = J DT('t) HT R-l H D('t) dt (24)

s Hence, the estimated error variance, ogj. is determined from

517

s It is obvious that best observability of ogj is achieved when the values for V 1 and V 2 differ in value and sign. Thus, the best estimation accuracy is obtained when the reverse runs are done with the same absolute velocity but opposite direction for each ZUPT interval. However, the reverse run drastically increases the time needed for the survey.

The requirement for the same absolute values of velocity between ZUPT points can be explained by returning to the spatial model for the deflections, see equation (9). For the medium accuracy compensation model, the correlation length is

1 R Sg = - = -:::: 9km , k-

~ It is possible to approximate the vertical deflections within the limits of the entire correlation length rather accurately by a higher order polynomial in coordinate increments. In such a case, the following condition will be satisfied ~ Vi i1ti < Sg

where Vi vehicle velocity between ZUPT points within the limits of the correlation length,

i1ti time between ZUPTs within the limits of the correlation length.

An approximation accuracy of 5% or better may be obtained with a simpler model if the interval is reduced to Sj+l-Sj == 0.2·Sg . The order of the approximation polynomial should be selected according to the required accuracy and the speed with which the survey has to be conducted.

NUMERICAL RESULTS

We used the proposed estimation method for the determination of vertical deflection increments for a vehicle moving at piecewise constant velocities between ZUPT points. The deflections of the vertical were obtained with the shaping filter (1) and the results are shown in Figure 1. The ISS errors are modelled according to equation (10) and it is assumed that 'te = 120 min, 'ta = 30 min, aWE = 0.001 0/h, and awa = 5·1Q-5g (Benson,

1985). The vertical deflections Og are approximated by a first-order polynomial (20). The vertical deflections of curve #1 were obtained using vehicle speeds of VI = 12 km/h

and V 2 = 48 km/h over a time interval of i1t = 3 min. The deflection derivatives for the ~

corresponding coordinate increments, ogj, j=I,2,3, are estimated for the points S2, S4, and S6. An interpolation curve is plotted according to equation (19).

The curve #2 in Figure 1 corresponds to vehicle speeds of VI = 24 km/h and V 2 = 36 km/h between ZUPT points. The decrease of the ratio C= V 2fV 1 results in a decrease of the estimation accuracy O~j .

Curve #3 gives the results for vehicle speeds of VI = 24 km/h and V 2 = 96 km/h over time intervals of i1t = 3 min. It is evident that the use of a first-order polynomial for the approximation of vertical deflections at high speeds generates a considerable estimation error.

518

CONCLUSIONS

This paper proposes methods for the approximation of deflections of the vertical by a series of base functions in terms of coordinate increments. The transition to the space domain makes it possible to obtain a physically valid model of the random process for the deflections independent of the vehicle velocity. This facilitates improved estimation accuracy and makes near real-time estimation possible. The choice of different velocities between ZUPT points allows the estimation of the deflections of the vertical with sufficient accuracy.

REFERENCES

Benson, D.O. (1985). Scanning, tracking and mapping state variable estimation algorithm, IEEE Transaction on Aerospace and Electronic Systems AES-21.

Forsberg, R. (1986). Gravity-induced position errors in inertial surveying, Proc. Third Int. Symp. Inertial Technology for Surveying and Geodesy, Banff, Canada.

Forsberg, R, Vassiliou, A.A., Schwarz, K.P., and Wong, R.V.C. (1986). Inertial gravimetry - a comparison of Kalman fIltering/smoothing and post-mission adjustment techniques, Proc. Third Int. Symp. Inertial Technology for Surveying and Geodesy, Banff, Canada.

Groten, E., Hausch, W., and Keller, D. (1986). Study of the influence of specific gravity anomaly and deflection fields in inertial surveying, Proc. Third Int. Symp. Inertial Technology for Surveying and Geodesy, Banff, Canada.

Kriegsman, B.A. and Mahar, K.B. (1986). Gravity-model errors in mobile inertial navigation systems, Journal of Guidance, Control, and Dynamics, No.3.

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SESSION6b

GRAVITY AND ATTITUDE APPLICATIONS

CHAIRMAN G.HEIN

UNIVERSITY FAF MUNICH NEUBIBERG, GERMANY

GRAVITY FIELD MODELLING FOR INS

N guyin chi Thong Department of Geodetic Science, University of Stuttgart

KeplerstraBe 11, D-1000 Stuttgart 1 Federal Republic of Germany

ABSTRACT

In all inertial navigation computations approximative formulae for the normal grav­ity are used. The gravity-induced position errors for an unaided INS are suppressed conventionally by increasing the vehicle speed (shortening the ZUPT-interval) or by modelling the local gravity disturbance, which is frequently limited in practical appli­cations.

In the paper the exact computational formulae for the normal gravity will be de­scribed by using spheroidal coordinates. The error process of strapdown-INS will be presented in the earth-fixed Conventional Terrestrial Frame, that has considerable advantages. A convenient method to improve the precision of the positioning will be outlined by increasing the frequency of the gravity model and simulation results for a timespan of the Schuler period are demonstrated.

1. INTRODUCTION

Inertial navigation computations are usually carried out in the spherical or geodetic coordinates. The geodetic coordinate surfaces fit to an oblate spheroid designed for the normal figure of the earth. The normal gravity, which is based on the grav­ity of a normal earth spheroid producing an equipotential surface on its surface, is used as the first approximation to the actual gravity field in all inertial navigation computations. The normal gravity formula represented in the geodetic coordinates mentioned above, however, is an infinite series that complicates numerical computa­tions in real-time. For practical applications the standard normal gravity formulae, which are derived by using linear and quadratic approximations with respect to the geometrical and gravity-flattening as well as the ellipsoidal height (see for instance Heiskanen &Moritz, 1967; Mittermayer, 1969), are frequently used. Other approxi­mation based on truncating the spherical series of the normal gravity and represented

523

in the earth-fixed Cartesian frame (for strapdown applications) has been recently derived by Schwarz and Wei (1990). In any case both the spherical and geodetic coordinates fit to the normal gravity only approximatively. Exact computational for­mulae for the normal gravity are treated in Chapter 2.

When using strapdown-INS, the navigation computations are effectively carried out in the earth-fixed Conventional Terrestrial Frame (CTF) (see Wei & Schwarz, 1990). The corresponding error process is represented in Chapter 3.

The gravity-induced position errors caused by gravity disturbance may be signif­icant, especially for surveys in mountainous areas. The conventional methods to eliminate the gravity-induced position errors are to increase the velocity (shorten the ZUPT-interval) or model the local gravity disturbance (see for instance Forsberg, 1985, Groten et ai, 1987).· Their applications are, however, limited to areas having smooth topography or dense coverage of local gravity data. Furthermore, increas­ing the vehicle speed may produce high dynamic uncertainties, especially when using strapdown-INS. The last chapter treats a convenient method to improve the precision of the positioning and demonstrates simulation results.

2. EXACT NORMAL GRAVITY FOR NAVIGATION COMPUTATIONS

In order to get an exact (non-infinite) representation of the normal gravity field we choose the spheroidal coordinates originating from the confocal quadric oblate spheroids, hyperboloids of revolution and half-planes (Heiskanen & Moritz, 1967). Various Representations of such coordinates have been studied in Thong & Grafarend (1989), from which the variant (CP,A,17):

[; 1 = E [~:~ :~ ::~ 1 Z smcp smh17

(1)

seems to be efficient for practical analyses. Here we have parametrized the coordinate surfaces by (cp, A, 17) with the confocal distance E. The Cartesian coordinates in the earth-fixed Conventional Terrestrial Frame (CTF) f are denoted by (X, Y, Z).

The normal gravity potential can be determined as the solution of Dirichlet '8

boundary value problem for the space outside the normal earth spheroid. Using the spheroidal coordinates (A, cp, 17) mentioned above, the normal gravity potential U can be written in a closed analytical form (c. f. Heiskanen & Moritz, 1967,p. 67):

where

524

G M is the gravitaional constant multiplied by the mass of the Earth Qnm are real-valued Legendre functions of the second kind

(see Thong & Grafarend, 1989) n is the angular velocity of the earth's rotation a is the semi-major axis of the normal earth spheroid 770 is the spheroidal coordinate of the normal earth spheroid Pnm are the Legendre functions of the first kind.

Now we introduce three coefficients

bi = n2E2/2 b2 = a2/[(E2 + 3b2 )arccot(sinh 770) - 3Eb] b3 = 2GM/(n2 E3) - b2/3

where b is the semi-minor axis of the normal earth spheroid. After some rearrange­ments we obtain a simple formula (without any approximation) for the normal gravity potential:

U(CP,'\,77) = bi {(I + b2P + p2) + (b3 + b2p2)arccot(p)- (3) - ((1 + 3b2P + p2) - b2(1 + 3p2) arccot(p) )sin2 cp}

with sinh 77 being now denoted by p. The first derivatives of the normal gravity potential in the spheroidal coordinates are:

8U "f'P := 8cp = -bi

8U "f .- -­

'1 .- 877 - -~I (b3 - b2(1 - 4 sin2 cp) + 2p(3b2p sin2 cp_ cos 77 -(1 + p2) (cos2 cp + b2(1 + 3 sin2 cp )arccot(p))))

8U "f>.. := 8'\ = o.

(4)

Now let us define the spheroidal-south-pointing local-level frame e to have its ori­gin at the station considered and its axes (Xs, YE , Zv) with orthonormal bases (es, eE, ev)as following:

-8(X, Y, Z)/8cp es := 118(X, Y, Z)/8cpll

8(X, Y, Z)/8,\ eE := 118(X, Y, Z)/8,\11

8(X, Y, Z)/877 ev := 118(X, Y, Z)/87711

'spheroidal south'

'spheroidal east' (5)

'spheroidal vertical'.

In the frame e the normal gravity vector can be exactly written by the simple trans­formation (c.f. Thong, 1989)

(6)

525

where Reu denotes the Jacobian matrix of (<p, A, 11) on (Xs, YE , Zv):

R .- 8( <p, A, 11) _ 1 d· (1 1 1) (7) eu .- 8(Xs, YE , Zv) - E lag J p2 + sin2 <p' cos <p cosh 11' J p2 + sin2 <p •

When representing the normal gravity vector in the CTF f, we use the similar transformation (c.f. Thong, 1989)

If := (fx"y"zl = R fele (8)

with the Jacobian matrix of (Xs, YE , Zv) on (X, Y, Z):

8(X Y; Z) [ sin ~ cos A - sin A cos ~ cos A 1 R s, E, V . . \ \ . \

fe:= 8(X, Y, Z) = sm <p Sl~ /\ cos /\ cos: s:.n /\ - cos <p 0 sm <p

(9)

where we have denoted 'i; for arctan (tan <p coth 11). Substituting formulas (6), (7) and (9) into (8), results in

1 ([ - tan <p 1 [ tanh 11 l) [X 1 If = E2( 2 sin2) l<p - tan <p + 11/ tanh 11 * Y p + <p cot <p coth 11 Z

(10)

where "*" denotes the Hadamard product (i.e. entrywise multiplication). By platform-mechanization, the navigation computer receives the specific force fe

from the accelerometers on the platform being commanded into the spheroidal local­level frame e. In addition, the normal gravity (6) is used and the navigation equations

ve= -(2RJeW}f + w!e) x Ve + Ie + fe

Ve = Reu(P,~, iJ)T (11 )

are numerically solved. Subsequently the total angular velocity w~e of the platform with respect to the inertial frame g as the sum of the earth rate w~f = RJeW}f (see Schroder et ai, 1988) and the transport rate

(12)

= (- ~ cos 'i;, -~, ~ sin 'i;)T

is calculated for feedback. However, the navigation computations of strapdown-INS can be carried out suitably in the CTF f (see Wei & Schwarz, 1990). In this case, the normal gravity formulae (8) are used.

526

3. STRAP DOWN-ERROR PROCESS IN THE CTF

The specific force-equations of a strapdown-INS can be simply written in the CTF as follows:

(13)

where n~1 denotes the cartan-matrix of the earth rate w~/, /d the specific force in the sensor-frame d and r J the gravity vector in the CTF f. The direction-cosine matrix RJd is determined from the equations

. . R old OglR R ogd RJd= - JdHd = HI Id - JdHd' (14)

When setting a state vector as the set of three orientation angulars (say Ex, Ey and Ez ) and the velocity and position:

(15)

the dynamic process (13) and (14) can be described by the ordinary differential equa­tion

x= h(r J(x), x(t), l(t)). (16)

Here the 9 x 1 vector function h defines the instantaneous behaviour of the system dynamics in the real gravity r J of the earth and under the impact of the (given)

cont,rol variables I := (( w~d)T, II) T. However, due to imperfections of the sensors, the

navigation computer receives only a (random) observation '7 (t) of l(t). In addition we use the normal gravity 'YI as a deterministic approximation to the real gravity r J such that the navigation computations may give only a random approximation x (t) of the state vector for which the consistency condition

~ (t) = h(-yJ(x), x (t), 7 (t)) (17)

holds. Using the linearization with respect to a random approximation (see Schaffrin, 1985)

and the usual pertubation of the gravity we arrive at the stochastic differential equa­tions

Ox (t) = F('Y, t)ox(t) + G(t)w(t) (18)

for the strapdown-error process ox(t) := x(t)- x (t) with

[ Rid O(3X3) O(3X3) 1 (9 c:. 9):= O(3X3) Rid I(3x3)

O(3X3) O(3X3) O(3X3)

(19)

denoting the coefficient matrix of the sensor-errors 01 := 1- 7 and the gravity dis­turbance O'YI := r J - 'YI

527

w := (biT, ~;ff. (20)

The simple form of the dynamic matrix

0 0 0 0 0 0 0 0 0 -0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

'" "'" 0 Iz - Iy 0 -20 0 Uxx Uxy Uxz "'" '" F .- - Iz 0 Ix 20 0 0 Uyx Uyy Uyz

(9 x 9) "'" '"

(21)

Iy -Ix 0 0 0 0 Uzx Uzy Uzz 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0

points out advantages of using the earth-fixed CTF. The matrix U'j of the (second order) Eotvos-tensor of the normal gravity is symmetric: Uij = U ji , therefore it is sufficient to know six elements

vech(Uj) = (Uxx, UyX , Uzx, Uyy, UZy, UZZ)T. (22)

The computation of vech(U'j) from U:: (second derivatives of the normal potential with respect to the spheroidal coordinates) and ;u can be carried out in two steps. Firstly we compute the second derivatives of U with respect to the frame e by the transformation

vech( U~/) = S:u;u + vech( Reu U: R;J

where

S:u := [vech(Se<p), vech(SeA), vech(Se7j)]

with

EPa [Sea]:= 8/38;; Va E {cp,A,1]}, V/3,; E {Xs,YE,Zv}

(23)

(24)

(25)

denoting the Hessian matrix of a on (Xs, YE, Zv). In the second step we use a simple transformation

vech(Uj) = vech(RleU~' RJe) (26)

to get U'j. For more details we refer to Thong (1989).

When assuming ideal (perfect) sensors a (t) = I (t» and zero initial condition (x (0) = x(O», (17) is reduced to the deterministic differential equations

~XI = 2(0}/f~ XI +Uj~XI + ~;I (27)

of the gravity-induced postition errors ~XI caused by the gravity disturbance ~;I.

528

4. GRAVITY MODEL IMPROVEMENTS AND SIMULATION RESULTS

The use of the normal gravity in inertial navigation computations, when applying unaided INS for geodetic positioning, may cause significant gravity-induced posi­tion errors, especially for surveys in mountaineous areas. The conventional methods to restrict the gravity-induced position errors are (see for instance Forsberg, 1985, Groten et al., 1987) to increase the velocity (shorten ZUPT-interva.l) or model the local gravity disturbance. However, in practical a.pplications they may be limited. Increasing the velocity produces high dynamic unce7"tainties, especially when using strapdown-INS. Modelling local gravity disturbance requires a reasonably dense cov­erage of local gravity data that are available only in certain areas.

Another convenient method to eliminate the gravity-induced position errors is to im,prove the gravity model used in inertial navigation computations. The normal gravity, that contains the spheroidal harmonic gravitation of the second degree, is only a lotO frequency part of the earth's gravity field. The order of the gravity dis­turbance, however, decreases by increasing the spheroidal harmonic degree nmax of the gravitation model

of the earth (see jig. 1), Here the funct.ions Q~m a.nd enm denote the real-valued stabilized Legendre functions of the second kind arid the spheroidal sU7face functions, respectively (c. f. Thong & Gmfa1"end, 1989).

mgal

100

so

o o 200 300 360

. nmax

Figure 1 Order of the gravity disturbance

529

Such a spheroidal harmonic model has been derived upto the (nmax = 360) high frequency (see ibid), that may give precise INS-positioning without limitations men­tioned by conventional methods. The computation time may be shortened by

• representing the gravity model in the form of a digital model

• using parallel-minicomputer (near future) and fast computation techniques.

In the following simulation we study numerically, how the gravity-induced position errors depend on the frequency of the gravity model used in the INS. The simulated trajectory for the timespan of the Schuler period was ta.ken from Schroder et al. (1988). Gravity models and their second order tensor were computed on the parallel­computer Gray-2using vectorization techniques described by Thong (1989). Gravity disturbances were taken into account according to Jig. 1. A low frequency model (nmax = 36) produces gravity-induced position errors of 200-800 7n level (fig. 2). If medium frequency models are used, the position errors are substantial1y small. Figure 3 shows the position erorrs caused by using a model with harmonic degree nmax = 180. Especially high frequency models give high precisions of the position­ing. The errors caused by using a model with nm<:tx = 360 (fig. 4) are situated at the level of 2-12 7n. All simulation results are demonstrated in the spheroidal local horizontal system and show the same Schuler period of the error behaviour.

m

800

400

o

40 60 80 100 mm

Figure 2 Gravity induced position errors caused by using a low frequency gravity model (nmax == 36)

530

m

100

50

o o 20 40 60 80 100 min

Figure 3 Gravity-induced position errors caused by using a medium fre­quency gravity model (nmax = 180)

10

5

o o 20 40 60 100 mm

Figure 4- Gravity-induced position errors caused by using a high frequency gravity model (nmax = 360)

5. CONCLUSIONS

In this paper exact computational formulae for the normal gravity, that is used in

531

all navigation computations, have been derived. The strapdown-error process has been reformulated in the earth-fixed Conventional Terrestrial Frame CTF, that offers advantages not only for navigation computations, but also for the error analyses and integration with GPS.

A convenient method to eliminate the gravity-induced position errors has been described by improving the gravity model, such that the practical limitations of the conventional methods can be avoided. For areas where conventional methods can be realized, combinations of all possible methods may give higher precision of the positioning.

REFERENCES

Britting, K. R. (1971). Inertial Navigation Systems Analysis, Whiley-Interscience, New York, 1971.

Forsberg, R. (1985). Gravity-Induced Position Errors in Inertial Surveying. In K. P. Schwarz (ed.) Inertial Technology for Surveying and Geodesy, Banff, 1985.

Groten, E.; W. Hausch and D. Keller (1987). Some special considerations on gravity induced effects in inertial geodesy, manuscripta geodaetica 12 (1987) pp. 16-27.

Heiskanen, W. and H. Moritz (1967). Physical geodesy, Freeman Co., San Francisco 1967.

Mittermayer, E. (1969). Numerical formulas for geodetic reference system 1967, Bolletino di Geofisica, XI (1969) pp. 96-107.

Schaffrin, B. (1985). A Note on Linear Prediction within a Gau6-Markov Model Lin­earized with Respect to a Random Approximation. Dept. of Math. Sci./Statist. Report A-138 (1985) pp. 285-300, University of Tampere, Finland.

Schroder, D; N. C. Thong: S. Wiegner; E. Grafarend and B. Schaffrin (1988). A comparative study of local level and strapdown inertial systems, manuscripta geodaetica 13 (1988) pp. 224-248.

Schwarz, K. P. and M. Wei (1990). Efficient numerical formulas for computation of normal gravity in a Cartesian frame, manuscripta geodaetica 15 (1990) pp. 228-234.

Thong, N. C. (1989). Simulation of gradiometry using the spheridoidal harmonic model of the gravitational field, manuscripta geodaetica 14 (1989) pp. 404-417.

Thong, N. C. and E. Grafarend (1989). A spheroidal harmonic model of the terrestrial gravitational field, manuscripta geodaetica 14 (1989) pp. 285-304.

Wei, M. and K. P. Schwarz (1990). A strapdown inertial algorithm using an earth­fixed Cartesian frame, Navigation: J. of the Institute of Navigation, Vol. 37, No.2, 1990.

532

SMOOTHING AND DESMOOTHING IN THE FOURIER APPROACH TO SPHERICAL COEFFICIENT DETERMINATION

Paul A. Zucker Applied Physics Laboratory

Johns Hopkins University, Laurel, Maryland 20723 USA

ABSTRACT

In order to analyze and evaluate errors in inertial surveys, it is helpful to have good regional and global gravity field models. Such models, given in terms of spherical harmonic coefficients, can also be expressed in terms of Fourier coefficients which can be related to the local field models. Conversely, spherical harmonic models can be constructed based upon a Fourier fit to the regional and global data. When the original data are mean values over grid squares, both the Fourier approach and the customary approach using discretized orthogonal projection integrals need to be modified to account for the smoothing of the higher frequency part of the signal. In this paper, both approaches will be discussed, and it will be shown how each attacks the smoothing/ desmoothing problem. For the Fourier case, the desmoothing factors (depending on order as well as degree) are derived. They are seen to have a simple form which can be rigorously incorporated into the coefficients which transform from the Fourier to the spherical domain. Other useful features of the Fourier approach are summarized and are shown to persist in the presence of the desmoothing factors.

INTRODUCTION

There is a close relationship between inertial surveying and gravity field determina­tion. If the outputs of a set of inertial instruments are known along with their position history, the gravitational accelerations can be determined. Conversely, if the gravity field is known along with inertial instrument outputs, then the position history can be found. The analysis of regional datasets is important from both points of view: computing vertical deflections to compensate errors in the inertial systems and finding estimates for the high frequency behavior of the anomalous gravity.

One of the problems in analyzing regionalized or localized sets of gravity data is the aliasing of the low frequency gravity components. In order to reduce this aliasing, it is essential to subtract off the contributions of a reference field. Since the reference fields available (for example, Rapp and Cruz, 1986) are given in terms of a high

533

degree and order expansion of the anomalous potential using spherical harmonics, the evaluation of such a field (or its horizontal derivatives to obtain vertical deflections) on a set of grid points is a complex procedure requiring specialized software. If the reference field were given in terms of a two-dimensional global Fourier series, then the widely available fast Fourier transform (FFf) procedure could be utilized. Furthermore, once the reference values have been subtracted, Fourier techniques are well-suited for the analysis of the regional data (Schwarz, Sideris, and Forsberg, 1990).

This paper will review and describe how global gravity fields can be constructed in the Fourier domain. It will also address how existing spherical harmonic expansions can be converted to the Fourier domain. There are some subtleties in the process involving the smoothing and desmoothing resulting from the use of mean anomalies. These issues will be examined, and it will be shown that the desmoothing in the Fourier domain takes on a simple and familiar form which fits neatly into the procedure of coefficient estimation.

REPRESENTATION OF GLOBAL ANOMALOUS GRAVITY

The two basic approaches for determining the global field are reviewed and critiqued below. The simple case where a gravity anomaly value is supplied at each grid point is discussed first. The advantages of using mean anomalies and the corresponding modifications to each method are discussed in the following Section.

The gravity anomaly at the surface of the earth can be approximated by a spherical harmonic expansion out to a maximum degree, L, and order, N, in terms of normalized Legendre functions and a set of coefficients, G nm:

L N

.1g(8,l) = L L G_ P _(cos8) eim1 (1) ,.=0 m=-N

where geocentric coordinates are used for the colatitude (B), and the longitude (l). In terms of the spherical coefficients, those for the anomalous potential, Tnm, are easily obtained from the anomaly coefficients:

R G_ T = ~ _ n-1

where R is an equivalent spherical radius for the earth.

Integral Projection Method

The series in Equation (1) can be inverted to obtain the coefficients G nm by using the orthogonality of the Legendre functions:

GMt = _1 J.:1g(8,l) P Mt(cos8) e-im1 sinB dB dl. (3) 41r

If the anomaly were known at all points on the surface, this would be an exact

534

method. Since the values for the anomaly ..:1g are assumed to be given on a latitude­longitude grid with spacing 180/N degrees, this integral is then approximated by the sum:

N-l 2N-l 1C ~ 1 ~ - -i .. ..!. • G_ = - L.J - L.J ..:1g(8; • ..t.) P _ (cos8;) e "J 8m8;

2N;z() 2N ,z() J (4)

with a slight modification for the contribution from the poles. This approach was described and implemented by Colombo (1981), and it was used by Rapp and Cruz (1986) in their determination of the coefficients for the potential out to degree and order 360.

Since this approximation procedure is a direct evaluation of the projection in Equation (3), it seems quite sound. It clearly goes to the appropriate limit as the data grid becomes denser. Furthermore, it does not use a matrix inversion and therefore appears well conditioned. This advantage results from doing the continuous projection before going to the discrete domain. If the Legendre functions in Equation (1) were discretized first, then a matrix inversion would be needed. Such a method, requiring the inversion, could be used in conjunction with an optimal estimation scheme in the presence of measurement noise.

Conversely, it might be said that the projection method of Equation (4) suffers the drawback that it cannot easily be extended to include measurement noise. Furthermore it has the computational burden of evaluating the Legendre functions for each degree and order at each grid point.

Fourier Projection Method

An alternate approach to this problem was first described by Dilts (1985). Instead of discretizing the Legendre function or the inversion integral, the Legendre functions are expanded in Fourier series:

II

P 1/IIf (cos8) = L B!. ei48 • (5) q=-n

The gridded anomaly values are then represented by their discrete Fourier series: N N

..:1g(8j1..t,) = L L Aqm (1 - Y26,qlN) (1 - Y2~'m'N) e i48j e, .. l }. (6) q--N .. =-N

The Kronecker ~ factors compensate for double counting at the extremes of the summations since terms for both Nand -N are included. For this representation it is useful to use a double covering of the earth (colatitude ranging from zero to 360 degrees) where each physical point has the same anomaly value each time it is encountered and where the value at each pole is repeated 2N times.

After the substitution of both of these expansions into Equation (1), the longitude dependence is easily projected out using the discrete Fourier inversion. At this point the coefficients satisfy:

535

N L • E AqIII (1 - Vz6 IfIN) elf" = E E B!. G,.". elf'B,. (7) r~ ~ ~~

Further use of the discrete Fourier inversion (multiplying by an exponential and summing over the latitude grid) gives the defining relation for the unknown G nm in terms of the knownAqm:

L

A = r G [Bf + Bf +2N + Bf -2N + • • • ] (8) qIII L ,."."", ,.".,.. . • -0

Since the B coefficients are zero when the q index is greater in magnitude than n, the additional terms in the right hand bracket are present only for L greater than or equal to N.

The orthogonality relation for the B coefficients can be obtained by the Fourier expansion of the Legendre orthogonality relation. The result can be expressed (see Dilts, 1985, for example):

z! = r Bf ' Vz [1 + (- l)f+f'] (9) ,.". L ,.. 1 12 f'--. - (q + q )

where the Z coefficients then satisfy:

f--· -f~- Z!,. e-1q8 = ; P.,. (cos8) Isin81. (10)

It needs to be remembered that unlike the case for the B coefficients, the Z coefficients are not zero when the magnitude of q exceeds the degree n prime. Operating with these coefficients on Equation (8) gives:

N L

E Z:'. AqIR (1 - Vz6 IfIN) = E G,.". MJIIJ' r~ .~

N M , = r Zf (1 - Vz6 ) [Bf + Bf+2N + Bf -2N + ••• ] (11) .. L .'. If IN ,.. ,.".,.. .

f--N

The matrix M is computable but may not be easily invertible. This expression can be generalized to include noise as well or used as a measurement equation in a Kalman filter (see Zucker, 1989, for example). A special case does exist whenL is chosen less than or equal to N for even values of the order, m, and less than or equal to N-1 for odd values of m. In this particular instance, the matrix M above equals the identity and the coefficients are easily determined to be:

N

G,.". = E z!. AqIR (1 - V2 6 If IN) . (12) f--N

This special result came about as a result of the orthogonality relation in Equation (10) coupled with the fact that n is less than or equal to N. The limits when q equals plus or minus N are handled by the Kronecker delta factor (for m even) and by the exclusion of L = N for m odd.

The Fourier projection technique has the advantages that fast Fourier transforms

536

can be used in the computations, that only exponential functions need to be computed at grid points, and that a matrix inversion is avoided with a judicious choice of maximum degree, L. Estimators of the Kalman type can be utilized based upon adding noise to Equation (8) or to Equation (11). The difficulties appear to be in the computation of the B and the Z coefficients, but these each can be found using recursion relations and need to be computed only once.

Comparison of the Two Methods

At first glance, it may appear that the two procedures discussed above could generate the same numerical results, particularly for the special choice of L = N discussed above. In fact, these two approaches are not equivalent. This can be seen by expanding the Legendre polynomial and sine functions of Equation (4) in terms of exponentials using Equation (10). Performing the indicated Fourier sums then leads to:

eo

(13) q=--

Comparison with Equation (12) shows inequivalence since the limits on the sum over q extend to plus and minus infinity. Although the A coefficients are periodic in q with period 2N, the Z coefficients are not periodic in q.

Both approaches require the computation of a large set of fixed coefficients. The integral method needs the values of all the Legendre functions at all the grid points while the Fourier method needs the comparably-sized set of Band Z coefficients. Evaluation of the spherical harmonic expansion at the grid points is also accom­plished using these same sets of fixed values and the spherical coefficients (such as the Gnm for the anomaly or the Tnm for the potential).

The special choice of L=N (for even m) and N-l (for odd m) does not only give the particularly simple solution of Equation (12), but it also corresponds to the Nyquist frequency at the equator. There is some information at higher degrees due to the closer spacing of grid points at higher latitudes, but this information does not provide for unique estimates of the lower orders within the higher degrees. For a grid spacing of 30 arc minutes, the value of N is equal to 360, and thus the special case discussed for the Fourier method does indeed match the limiting degree and order in the fit by Rapp and Cruz (1986).

SMOOTHING AND DESMOOTHING

The situation discussed in the preceding Section, where point anomaly values were available at each grid point, is an idealized case. In practice, the observations are not all at grid points, there are varying numbers of observations in each grid square, and there is high frequency noise present with the observations. These problems are treated by forming a mean anomaly for each grid square. This process of averaging

537

over the grid square suppresses the high frequency noise which would cause undesirable aliasing in the spherical coefficients. It also suppresses the contributions of components in the true gravity field which are at higher frequency than the limiting degree used for the fit. Unfortunately, the grid square averaging also partially suppresses frequencies of interest, leaving only the zeroth degree term unaffected. Since it is possible to compute the effect of the suppression for each harmonic term, the resulting coefficient (estimated from the mean anomaly) can be compensated for it. While this desmoothing process compensates for the smoothing of the coefficients included in the fit, it does not completely undo the (more complete) suppression of the higher frequency noise and field components.

Integral Projection Method

The integral projection method incorporates the mean anomalies by moving the mean value outside the integral in Equation (3) for each grid square:

N 2N-l

Grun = _1_ L L .1g(Oi,lj) f P tint (cosO) e-im1 sinO dO dl (14) 4nQn;=O j=O D"

where the factor Qn is the desmoothing term discusse.d below. From this expression it is seen that the values of the Legendre functions integrated over each grid square are needed. In order to compute precise desmoothing expressions, the following steps need to be followed. First, Equation (1) is integrated over each grid square to obtain mean anomalies in terms of a reference set of spherical coefficients. Next, Equation (14) is used to estimate the spherical coefficients from the mean anomalies. The resulting expression for the estimated coefficients in terms of the reference coefficients shows the effects of the smoothing. This expression involves a matrix transformation mixing different degrees and orders. Precise desmoothing would require the inversion of this linear relationship. Rapp and Cruz (1986) did not attempt such a complex and intricate computation. Instead, they approximated the desmoothing process by a factor which depends only upon the degree, n. The choice of this factor is based upon consideration of the integral of a Legendre polynomial over a spherical cap the size of a grid square at the equator. Phenomenological fitting is also done, and the factor chosen for degree n less than N /3 is the square of the factor for higher degrees (up to N). Thus it is seen that a large number of approximations were employed in the implementation of the desmoothing.

Fourier Projection Method

In the Fourier domain, a similar logical procedure is followed. Based on a reference expansion of the field, mean anomalies are computed. In the Fourier representation, the grid cell averaging involves integration over exponentials, for example:

538

N fB, + ~ iqB d8 _ (2N • qTr) iq8, - 1re --sm-e. Tr 8, - 2N qTr 2N

(15)

A similar relation holds for the longitude averaging. The expression in brackets (sometimes called a sinc function) is also interpreted as the well-known Fourier transform of a discrete window.

When the grid cell averages are used in the Fourier projection method discussed above, a generalization of Equation (8) results which explicitly shows the effects of the smoothing:

L

a = L G (2N sin mTr) qm n=O Mr mTr 2N

x [ (2N sin qTr) Bq + ( 2N sin qTr 2N Mr (q + 2N)Tr

(16)

In this expression the G nm are the desired spherical coefficients for the anomaly field, and the aqm are the Fourier coefficients computed from the mean anomaly values. From thIS expression it is clear that a new set of coefficients bq nm could be defined by incorporating the sinc factors with the B coefficients. Correspondingly, a set of z coefficients could be defined by dividing the Z coefficients by the sinc factors. Then the solution for the Gnm would have the same form as Equations (11) or (12) using the modified coefficients and the aqm.

In the special case of interest (maximum degree N for even m and N-l for odd m), an even simpler approach is possible. Due to the lack of the extra terms on the right­hand side of Equation (16), it is possible to combine the sinc factors with the Fourier coefficients aqm:

a A = qm (17)

qlft (2N sin mTr)(2N sin qTr) . mTr 2N qTr 2N

Under these circumstances, the Aqm can be interpreted as point anomaly coefficients obtained from the mean anomaly coefficients by desmoothing in the Fourier domain. Equation (12) then gives the solution for the spherical coefficients Gnm• By placing the desmoothing factors on the A coefficients, the Band Z coefficients remain unchanged.

Thus, it is seen that the use of mean anomalies is treated quite simply by the Fourier projection method. The sinc factors obtained are well-known and easy to compute, and no additional manipulations involving Legendre functions are required. Furthermore, the procedure described above is rigorous and not approximate. This contrasts with the integral projection method, which required evaluation of integrals of Legendre functions and an approximate set of desmoothing factors.

539

IMPLICATIONS FOR REGIONAL ANALYSIS

In order to illustrate the use of the global fields for removing and introducing low frequency information in regional computations, the example of determining vertical deflections from locally measured anomalies will be discussed. Three approaches will be discussed and critiqued: full use of the integral projection method, full use of the Fourier projection method, and use of a hybrid approach.

Integral Projection Method

The spherical coefficients from a fit such as Rapp and Cruz (1986) are used. Since the regional grid is finer than the global grid for that fit, it is not sufficient to use a file of Legendre functions evaluated at the global grid points. Specialized software needs to be used in evaluating the series at the local grid points. Once the anomalies from the global fit have been subtracted at the local grid points, then the local computations (e.g. for the vertical deflections) are performed. Then global results need to be added back in. In the case of vertical deflections, the spherical series has to be differentiated, recursion relations used, corresponding coefficients determined, and values at the grid points determined.

Fourier Projection Method

The Fourier coefficients for the global anomaly can be determined from the spherical coefficients using Equations (2) and (8). Then the values at the local grid points can be determined via Equation (6). Vertical deflection values can be obtained by finding the Fourier coefficients for the potential and differentiating the Fourier expansion (analogous to Equation (6». For both the differentiation and the evaluation at the grid points, it is the exponential functions rather than the Legendre functions which are needed. As a result, the computations are greatly simplified. In order to go from the spherical to the Fourier coefficients using Equation (8), the set of B coefficients is needed.

Hybrid Method

It is also possible to start with a set of spherical coefficients which were computed using the integral projection method and then transform to the Fourier domain. Equations (2) and (8) could be used, for example, on the Rapp and Cruz (1986) coefficients. Since a more precise desmoothing is possible in the Fourier domain, it would be preferable first to remove the desmoothing factors used in the spherical coefficients. After the corresponding Fourier coefficients were determined, the Fourier desmoothing factors could be incorporated as in Equation (17). The rest of the computation is as in the Fourier method discussed above. Thus an improved

540

treatment of desmoothing is possible coupled with the computational advantages of using exponential functions rather than Legendre functions.

The B coefficients appearing in Equation (8) can be determined using recursion relations. If the values of the Legendre functions are available at all the global grid points, then these coefficients can be obtained by performing a discrete Fourier transform using Equation (5). (A discrete Fourier transform based upon Equation (10) can be used to compute the Z coefficients if needed.)

CONCLUSIONS

Both the integral and Fourier projection techniques for fitting global spherical harmonic expansions have been reviewed and discussed. Particular attention was paid to the smoothing and desmoothing issues introduced when mean anomalies are used as the initial data. From this analysis, it appeared that the Fourier approach could handle the desmoothing in a simpler and more precise fashion.

The manner in which the global expansions could be used as reference fields for localized computations was also discussed. The Fourier approach was shown to be useful for this purpose since all the computations involve the evaluation of exponential functions instead of Legendre functions. In either case, a set of fIXed coefficients is needed for evaluation of the spherical harmonic series at the global grid points. For the integral approach this set contains the values of the Legendre functions at the global grid points; for the Fourier approach this set constitutes the B coefficients of Equation (5), which is the discrete Fourier transform of the preceding set. Evaluation of the series at the localized grid points is considerably simpler in the Fourier representation. Since computations involving exponential func­tions are familiar and efficient (especially when fast Fourier transforms can be used), employment of the Fourier techniques (or the hybrid technique discussed above) could lead to greater use of high degree global expansions for the reference field.

Acknowledgment. This work was supported by U. S. Navy contract NOOO39-89-C-5301.

REFERENCES

Colombo, O. L. (1981). Numerical methods for harmonic analysis on the sphere, Department of Geodetic Science Report 310, Ohio State University, Columbus.

Dilts, G. A (1985). Computation of spherical harmonic coefficients via FFf's, 1 of Computational Physics 57, 439-453.

Rapp, R. H. and Cruz, J. Y. (1986). Spherical harmonic expansions of the Earth's gravitational potential to degree 360 using 30 minute mean anomalies, Depart­ment of Geodetic Science Report 376, Ohio State University, Columbus.

541

Schwarz, K P., Sideris, M. G., and Forsberg, R. (1990). The use of FFf techniques in physical geodesy, Geophysical 1 Int. 100, 485-514.

Zucker, P. A. (1989). Precise solution for a finite set of spherical coefficients from equiangular gridded data, in Progress in the determination of the Earth s gravity field, Department of Geodetic Science Report 397, 35-38, Ohio State University, Columbus.

542

AIRBORNE GRAVITY SURVEYING AN EFFECTIVE EXPLORATION TOOL

William Gumert, Victor Graterol, Gerald Washcalus, John Kratochwill

May 1990

Numerous papers have been given, (see references), to describe the system used by Carson Services, Inc. in their collection of airborne gravity and magnetic data. As many studies have been made to determine the system accuracies by both private oil companies and government agencies. The final outcome is that Carson has been contracted to collect airborne gravity data in eighty, (80) , prospective areas around the world and that these data have been used to evaluate geological structures with petroleum potential and to guide the collection of more expensive exploration information including the collection of seismic data.

Several areas will be discussed to demonstrate the advantages and capabilities of this system.

In early 1983 a helicopter gravity and magnetic survey was flown over the santa Maria Basin located along the southern Pacific coast of California. (Fig. 1) The survey was flown as a 1.5 mile, (2.78 kilometers), grid at 2500 feet, (762 meters), above sea level with a ground speed of 50 knots, (90 kilometers/hour). The observed data had an intersection mistie array with a standard deviation of 2.2 milligals. After systematic first order correction, the approximately 1400 intersections have a standard deviation of 1.4 milligals.

This area is a mature petroleum exploration region. Several existing oil fields are producing, (Fig. 2), and there have been many geophysical exploration efforts using all disciplines. Questions remained about the geological setting, primarily in connecting the known onshore structures with the known offshore trends. The land-water interface needed to be surveyed and the airborne system was the only acceptable technique.

Reports from oil companies who used these airborne data concluded that the airborne gravity data were better than the existing marine gravity data and a report from a government consultant comparing the airborne data to existing land and marine data give a precision of 1-2 milligal.

543

A third degree residual bouguer gravity map was produced to best define the anomalous structures, (Fig. 2). This map delineates a sharp gravity low that is straddled by the existing oil production and certainly suggests the areas for further exploration. Clearly defined is the change of structural trend from major onshore faults previously mapped to the offshore pattern that cuts obliquely across these known trends.

From late 1987 to the end of 1988 three survey areas were flown in Venezuela for the three operating companies located there. One area was in Eastern Venezuela in the state of Monagas.

The main purposes of this survey were the following:

- To investigate in a rapid area of difficult access. of low elevation flooded, Orinoco River Delta.

way and at very low cost an The area consists mainly

swampy and wild land in the

- To correlate the results of this survey with the Venezuelan structural patterns. These structures are presently under very intense exploration, especially west of the city of Maturin, Monagas state.

- To select, by the means of the potential field methods of gravity and magnetics, areas of good possibilities for the accumulation of oil and gas deposits to be surveyed systematically in detail to define possible drilling sites.

This survey in Monagas was flown by helicopter at 50 knots, (90 kilometers/hour), with a 2 x 4 kilometer grid. 10587 kilometers of data were collected in January and February. Standard interpretation techniques were modified to address the elevation and filter effects on the computed bouguer gravity values that were mapped.

In the Maturin area a series of regional and residual Bouguer gravity maps were produced with detailed input from company geologists and the correct version was selected to best match the intended targets. The Bouguer data, (Fig. 3), represents one of the largest continental negative gravity anomalies in the world. This feature was created by a series of overthrusts from the collision of the Caribbean plate with the Venezuelan precambrian shield. Magnetic data, (Fig. 4), was also collected and mapped. Shadow graphs were created from both data sets, (Fig. 5) and (Fig. 6), and a qualitative structural pattern interpretation was made. These shadow graphs are produced by regridding the data set at a fine grid interval usually 500 meters, applying the proper smoothing function, and manipulating the grid in the software. Various vertical angles above the horizontal and a complete 360 degree rotation can quickly find the proper combination in the interactive software to define the faults and structural trends.

544

The general and detailed structural trends are easily defined in a quick time frame, (Fig. 7).

wi th this data and the geologists input, models of the geological structures can be made and modified to match the observed information, (Fig. 8).

From this data set and two other surveys carried out in other geological provinces of Venezuela the president of PDVSA, Juan Chacin Guzman, pointed out, at the inauguration of the Fourth Geophysical Congress in Caracas, 1988, the esteem that the oil industry has for the new geophysical techniques.

Chacin emphasized that gravimetric survey methods have simplified and perfected the task in such a way that a "many-year survey" now takes only a few months to complete. To quote only one case: the information recently obtained in a two month aerogravimetric survey of the areas Eastern Venezuela, Guarumen and North Andes Front would not have been possible with the old techniques in less than two decades and would not have had the obtained accuracy.

A modeled anomaly with the attenuation due to height, filter function and aircraft speed is shown in Figure 9. The structure in the example is 2 kilometers wide with a density contrast of .33. The effects at 50 knots, (90 kilometers/ hour), and 100 knots, (180 kilometers/hour), are shown. The maximum filter length is a digital low pass filter representing an R-C electrical circuit with a 60 second time constant. At the lower speed almost 80% of the anomaly is passed while the high speed only 30% is passed and the anomaly approaches the noise level of the technique.

One of the difficulties in using a fixed wing airborne gravity system is the inability to maintain a constant and/or consistent ground speed. Much aeronautical engineering effort has been expended to control this problem including extensive modification to the airframe control surfaces and to the autopilot electronics. Even with these measures the slowest practical speed approaches 100 knots, (180 kilometers/hour), and the speed in a survey grid pattern can vary from 70 to 140 knots, (125 to 250 kilometers/ hour). with this diagram the effects of these speed variations on the gravity data can be appreciated. with the helicopter system this is not a problem and ground speed variations can be controlled to within 2 knots, (3.6 kilometers/hour).

with the increased availability of GPS satellites the practical use of this system becomes a reality. Carson began collecting GPS data in 1984 during a survey of the Bahamas. At this time several lines of gravity data were collected using GPS and ground based Motorola transponders. Eotvos corrections with a 2 milligal accuracy were made when good satellite geometry was available. Limited satellite coverage has not allowed effective use of this system until 1990. This year Carson surveyed 5 geothermal areas at

545

a 750 meter spacing in Indonesia, and has flown surveys in Malaysia, Papua, New Guinea, and Burma. In Malaysia both Motorola Falcon 484 ground based transponder and 8 channel Golden Eagle GPS receiver were used.

These data were collected before selective availability was turned on. Differences in X, Y, & Z coordinates between GPS and the Motorola Falcon varied from 1 to 10 meters with most larger errors deriving from systematic shifts in the coordinates. Eotvos corrections computed from both data sets and compared showed differences of .3 to near .9 milligal standard deviation.

To conclude, (Fig. 10), a new gravity meter package will be tested in October-November of this year. This system incorporates the standard Lacoste & Romberg zero length spring sensor installed in a molded carbon fiber frame and meter box reducing a three axis system from 700 pounds, (320 kilos), to less than 200 pounds, (90 kilos) • A board level electronics package has been designed, constructed and tested at Carson's completely instrumented gravity meter laboratory in Austin, Texas.

This system was designed solely for the airborne environment and will allow extra fuel capacity for longer missions. Its board level components allows easy operation, maintenance, and calibration.

REFERENCES

Carson, F. and Gumert, W., Airborne Gravity Surveying Method, Canadian Patent No.1 256 537, Issued 27 June 1989.

Carson, F. and Gumert, W., Airborne Gravity Surveying, Canadian Patent No. 1 204 158, Issued 06 May 1986.

Carson, F. and Gumert, W., Airborne Gravity Surveying, united States Patent No. 4,435,981, Issued 13 May 1984.

Carson, F. and Gumert, W., Airborne Gravity surveying, French Patent No. 83 16899, Issued 24 October 1983.

Carson, F. and Gumert, W., Airborne Gravity surveying Method, French Patent No. 86 01237, Issued 29 January 1986.

Carson, F. and Gumert, W., Airborne Gravity surveying, United Kingdom Patent No. 2148016B, Issued 22 May 1985.

Durboraw, Dr. I. Newton, III, Test Results of a Differential GPS Receiver in a Dynamic Environment, International Congress Federation of International Geodesy, Toronto.

Gumert, William, (1974). Helicopter Gravity System for Geophysical Exploration, Society of Exploration Geophysicists 44th International Meeting, Dallas.

Gumert, William, (1977). Helicopter Gravity Survey Over the Outer continental Shelf Off the Virginia Coast, Society of Exploration Geophysicists 47th Annual International Meeting, Calgary.

546

Gumert, William, (1985), Advantages of the Continuous Profiling Airborne Gravity Surveys, International Meeting on Potential Fields in Rugged Topography, switzerland. ,

Gumert, W. and Cobb G., (1970). Helicopter Gravity Measuring system, Symposium on Dynamic Gravimetry, Texas.

Gumert, W. and Navazio, F., (1980). Airborne Gravity - 1980's, Pennsylvania.

Gumert, w. and Navazio, F., (1982). Airborne Gravity of the 1980s, PennWell Publishing Co.

Gumert, W. and Stancato, T., (1985). Current Status of Airborne Gravity Surveys, SEG Bookmart, 55th Annual International Meeting, Washington, D.C.

Hicks, Forrest L., (1967). Data Reduction of Airborne Gravity Data, Massachusetts.

Krabill, W. and Martin, C., (1987). Aircraft Positioning Using Global Positioning System Carrier Phase Data, Vol. 34, No.1.

LaCoste, Lucien, (1967). Measurement of Gravity at Sea and in the Air, Reviews of Geophysics, Vol. 5, No.4.

Porter, W., Kruczynski, L., Abby, D. and Weston, E., (1985). Global Positioning System Differential Navigation Tests at the Yuma Proving Ground, Journal of the Institute of Navigation, Vol. 32, No.2. Printed U.S.A.

Rowe, R., Salvermoser, F. and Beruff, R., (1983). Developmental Efforts to Improve the Accuracies of Geodetic and Geophysical Surveys, Venezuela.

~7

LOCATION MAP OF Figure 1. SANTA MARIA BASIN SURVEY AREAS

Figure 2.

548

Figure 3. Bouguer gravity, eastern Venezuela.

Figure 4. Total magnetic intensity, eastern Venezuela.

549

Figure 5. Structural interpretation, Bouguer gravity.

Figure 6. Structural interpretation, total mag­netic intensity.

550

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552

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5

AN EVALUATION OF ALIGNMENT PROCEDURES FOR STRAPDOWN INERTIAL SYSTEMS

ABSTRACT

K.P. Schwarz and Ziwen Liu Department of Surveying Engineering

The University of Calgary 2500 University Drive N.W.

Calgary, Alberta, Canada, T2N IN4

The alignment of strap down inertial systems is typically achieved by a real-time estimation process which uses a reduced Kalman filter model for coarse alignment and a more complete model for fine alignment. In surveying applications with strapdown systems, there is usually no need for real-time results and the alignment procedure can be optimized for the complete data set taken during the alignment period. This gives much greater freedom in the choice of the estimation model used for alignment. The paper compares different approaches and evaluates the results with respect to external accurate azimuth information.

1. INTRODUCTION

In navigation applications of inertial systems, alignments are performed in real time. This often goes hand in hand with a requirement to minimize the alignment time. Studies of alignment procedures therefore have the objective of balancing accuracy and time requirements. Usually this amounts to achieving a specified accuracy in as short a time as possible. Highest possible accuracy of the alignment is therefore usually not an issue.

In survey applications of strap down inertial technology, real-time computations are often not needed and high alignment accuracy is more important than short alignment time. Thus, post-mission alignment methods are an alternative in this case. The paper investigates the question whether or not the accuracy of the initial values can be improved by making use of refined analysis techniques. Good initial alignment values are especially important in inertial positioning and inertial gravimetry.

In other applications, especially in a number of the emerging industrial applications, the stability of the alignment is an important requirement. A typical case is machinery alignment where shafts or rollers have to be aligned parallel in space with very little room to do the measurements. The use of an inertial reference unit has been proposed because it allows the transport of a reference direction from one point to the next without awkward forward and backsights. In this case, it is very important that the reference direction remains stable during transport and during the actual measuring periods. Occasional determination of linear gyro drift seems to be feasible by other methods.

Post-mission alignment has a number of advantages compared to real-time alignment. It widens the range of methods that can be used to estimate the alignment parameters. It allows the analysis of residuals and thus makes it possible to identify the steady state measurement phase. Finally, by comparing different types of update measurements for the same data set,

553

conclusions can be drawn on the observability of certain states in an operational environment and on optimal data combinations for specific applications.

Post-mission methods that can be applied besides Kalman filtering are spectral methods for data screening and elimination of unwanted peaks, improvement of the initial values by repeated Kalman filtering, optimal smoothing, batch least-squares adjustment of the whole data set, and non-statistical methods of eliminating transient drifts. Since this fIrst series of tests was exploratory in nature and time was limited, not all methods have been used. However, the results presented should give a good indication of their suitability for this task.

Observability of specific states is obviously dependent on the type of update measurements used. Alignment in a stable environment allows to introduce updates for zero velocity, for zero position change, and for zero direction change. A comparison of the effectiveness of these update measurements for the estimation of specific states will therefore be included in the study.

The paper uses four sets of alignment data on the same station to fInd answers to some of the above questions. Each set consists of about one hour of data taken at a 64 Hz rate with the Litton LTN-90-100 unit owned by the University of Calgary. An external astronomically determined azimuth is available as an absolute reference on this station. It has been related to the internal azimuth of the inertial unit by a mirror-collimator system and calibration of the x­axis of the body frame. Although the measurement procedures and mathematical techniques are applied to data of a specific system, the results should be applicable with small modifications to any navigation-grade inertial strapdown unit.

2. ESTIMATION MODELS

All methods used in the following are based on the same mathematical model. It consists of the linearized solution of the first-order system of differential equations

x = Fx + Gu (1)

with initial values

x(to) = c

and update measurements of the form

y = Hx (2)

Assuming a linearized, time-invariant system, the solution of (1) can be written in the standard form

tk X(tk) = <l>(tk,tk-l) X(tk-l) + f <l>(tk,t) G(t) u(t) dt

lk-l (3)

where <l>(tk,tk-l) is the transition matrix between time tk-l and tk. In the following, the state vector will be of the form

x = {E,or,ov,b,d}T (4)

554

where

E are attitude errors

Br are position errors

Bv are velocity errors

b are accelerometer biases

d are gyro drifts.

All states are defined as corrections to an initial estimate and all subvectors are of dimension 3.

The three methods discussed in the following estimate an optimal x by using the measurements (2) together with the model structure (3). The differences between the methods are due to either a different definition of optimality or different assumptions on the statistical processes involved. In addition, preftltering of the data will be used in certain cases to eliminate unwanted effects. This can either take place in the spectral domain and thus amounts to data cleaning, or can be built into the time-domain filter and then amounts to a change of the optimality criterion.

Kalman filtering

This is the standard real-time method giving an optimal estimate for a specific time tk based on all previous measurements and their accuracy. The unknown alignment parameters at time tk are estimated by using the standard algorithm

x(+) = x(-) + K{y - Hx(-)}

K = P (_)HT {HP(-) HT + R }-l } (5)

where (-) and (+) denote estimates before and after update, K is the Kalman gain matrix, and P and R are the covariance matrices of the system noise and the measurement noise. The time subscript tk has been omitted to simplify notation. For a detailed derivation, see for instance Gelb (1974).

Theoretically, results of this method will agree with other methods only at the end of the alignment period but not at any intermediate points because it process a subset of the data used with the other methods. Practically, convergence of the Kalman filter estimates towards the results of the other methods may be achieved earlier and is a good indicator of having reached the steady-state phase.

Filtering with improved initial values and covariance estimates

This method is in principle a double run of the Kalman filter on the same data set. The idea is simply to improve the initial values and covariance estimates of the alignment which are known to be very poor for the first run. The procedure is quite simple. After processing a specific data set through the Kalman filter, the state vector estimate at the end of the run and its covariance matrix are taken as initial values for another run with the same data set. By starting out with values close to the true ones, nonlinearities in the filter will be minimized and the state variations in the second run should be a good indicator for the system accuracy during the alignment phase.

555

Filtering and optimal smoothing

This method improves the Kalman filter results by taking into account the effect of all measurements - before, at and after time tk+ 1. It can therefore only be applied post mission. The Rauch-Tung-Striebel algorithm is often used. It has the form

For a discussion, see again Gelb (1974).

Batch least-squares adjustment

In this case, equation (2) is rewritten as

} (6)

(7)

where Xn is typically the state vector at the end of the alignment period. By rewriting equation (7) as

y = DXn (8)

where y is now the vector of all update measurements and D contains design and transition matrices at all update periods, the least-squares solution for Xn can be written as

(9)

where the xn above a letter denotes an estimate. If the covariance matrices of the system noise and the measurement noise are known, say P and R, then the weighted least-squares solution is obtained from

(10)

Note that the components of R and P are different for different update periods. In formulas (9) and (10), the state vector at the end of the period is estimated from all update measurements. It can then be transferred to intermediate points by using

(11)

The choice of the end point is arbitrary. If instead an intermediate point is chosen, the same procedure applies.

Results from equation (10) are theoretically equivalent to results of filtering and optimal smoothing if the same covariance assumptions are made. The method has, however, some operational advantages, specifically in this application without vehicle dynamics.

556

Spectral data cleaning

The data obtained from the inertial system contain a number of large spikes in the spectrum which are most likely due to aliasing caused by the dither frequencies. By transforming the data to the frequency domain, removing the spikes and transforming back to the time domain, clean data are obtained which theoretically should give a better result when used with any of the above methods. The spectral approach is discussed in Czompo (1990) and reference is made to his paper for all details.

3. DATA ANALYSIS AND RESULTS

Experimental setup

The tests were done on the roof of the Engineering Building of the University of Calgary. The choice of this location was mainly determined by the fact that one of the observation pillars installed there has a well-determined astronomical azimuth to a radio beacon a few kilometres away. This azimuth was used as a reference for the internal azimuth of the INS.

To relate the internal azimuth to the reference azimuth, several steps were necessary. First, the inertial system had to be equipped with a mirror in a plane approximately normal to the x-axis of the inertial system. It was then put into a Cardan frame which allowed free rotational motion of the system. The Cardan frame and the mirror arrangement are described in Knickmeyer (1989).

Second, a precise theodolite with a collimator was set up a few meters away with sights to the INS, the radio beacon and the pillar. Directions to these three targets were determined several times and the azimuth of the direction normal to the mirror plane was derived from it A sunshot azimuth was taken independently and the azimuth of the mirror normal was also derived in this way. The two determinations agreed within 10 arc seconds.

Finally, the mirror misalignment, i.e. the directional bias between the direction normal to the mirror plane and the internal x-axis of the INS, was determined by a method described in Knickmeyer (1989). It involves a series of rotations about the systems axes and uses the inertial system output for bias determination. The mirror misalignment was found to be about 35 arc minutes.

By correcting the mirror azimuth by this amount, the azimuth of the x-axis was found to be 355 34' 28". Taking all the error sources into account, the standard deviation of this determination is probably not better than 30 arc seconds.

The INS was kept fixed in this position and five independent alignments were done, each lasting about one hour. All data were recorded at a rate of 64 Hz. Unfortunately one data set was subsequently lost in a computer breakdown, so that only four sets were available for the data analysis described in the following.

Processing characteristics

Since the data sets represent a 'no dynamics' case, processing can be simplified by reducing the number of data points used. This can be done by averaging the raw data output over a certain time period, say 10 seconds. This speeds up the mechanization computations considerably. To test the effectiveness of this method, a comparison was made between the

557

standard method using a 64 Hz rate for the mechanization equations and the averaging method using 10 second means of the raw data for the mechanization. Results were similar but in general the use of the high data rate gave slightly better results. It was therefore decided to use the 64 Hz data rate for this test series and to return to the problem of an optimal averaging period later on.

The update rate used in these tests was ten seconds. It gives a detailed picture of the azimuth change as a function of time and this was the main reason for choosing such a high rate. It is not required from an operational point of view but it allows to plot the deviations of the estimated azimuth from the reference azimuth as a continuous curve. Updates were made using the conditions of zero velocity, zero position change and zero direction change. The initial variances of the state vector elements, the spectral densities for the system noise and the variances for the update measurement noise are given in Table 1.

Parameter Initial Variance Spectral Density

EE, EN 40 arcsec. 0.1 arcsec2/sec.

EU 1 degree 0.1 arcsec2/sec.

o<j>,oA.,oh 0.1 m 0

&i>,o~,oh 0.003 m/s 0.25 • 10-6 m2/s3

dx, dy, dz 0.01 deg./h 1.38 • 10-9 deg2/h3

bx, by, bz lOmgal 1.38 • 10-3 mgal2/s

zero velocity update 0.0005 m/s

zero position update 0.01 m

zero angular velocity update 1. • 10-6 deg./h

Table 1: Statistical assumptions

Influence of initial values

All data sets were first processed through the standard Kalman filter using an azimuth error of 1 degree as standard error after the coarse alignment. A typical result of this procedure is displayed as solid line (a) in Figure 1. It shows the deviation of the azimuth estimate from the reference azimuth which is shown as zero line. As expected, this first run shows large variations during the first half hour, even though convergence seems to occur after about 10 minutes or about 60 updates. After half an hour, the CUlve becomes smoother and varies only within 1 arcminute. It appears that any data point from there on gives an estimate of the azimuth which is good to about 1 arc minute. This result does not significantly improve by a longer observation time. It is similar for all other runs.

The estimated state vector at the end of this first run and its covariance matrix were then taken as initial values and the data set was processed again. The result is shown as broken line (b) in Figure 1. This second run is much smoother and does not show the large initial variations. It seems to be quite stable over periods of several minutes but still shows some distinct long term variations. Again, this result is similar for all runs. It indicates that a good approximation is most likely possible with a smaller data set if a number of iterations can be made in post mission.

558

3

2

.- 1 C ·s

0 (J

'"' ~ '--'

.c ... -1 =' E ~ -2

-3 85 1085 2085 3085 4085

T ime (sec.)

Figure 1: Effect of initial values (curve (a): poor initial values; curve (b): good initial values)

To investigate this question, only about twelve minutes of the same data set were taken and were processed repeatedly through the standard Kalman filter, improving the initial value for the azimuth and its covariance in each run. The result is shown in Figure 2, where the numbers at the curves indicate their Kalman filter iteration. The result is as expected.

3

2

.-d 1 ·s (J

0 I.. ~

, ...... . . -._._._._. 4 __ . -._.-._._0_. ....... yHt>I' • • .-..,~. .. -. .... ... .......... ..... .. ,... ..... _ .... ... ~.

'-" 3 .c ... =' - 1 E ~ .

'N -<

-2

-3 85 185 285 385 485 585 685

Time (sec.)

Figure 2: Azimuth iteration for a short data span

559

After a good approximation of the reference azimuth has been reached at the end of the fIrst Kalman fIlter run, the subsequent iterations show a very smooth and stable behaviour. The accuracy of the result is, however, very much dependent on the accuracy of the estimate at the end of the fIrst run. If it has a bias, it is unlikely to be eliminated in subsequent runs. The stability of the results stays the same,however. This means that if a bias is of no concern, i.e. if variation about a given reference direction is the important parameter, these results can be still quite useful.

The following conclusions can therefore be drawn from these experiments. Convergence and accuracy of the alignment depend heavily on the initial azimuth estimate and its covariance. For real-time methods these initial values are usually poor and an improvement is only possible by collecting additional data. For post-mission processing, iterative procedures can be applied, where the end results of one processing step are used as initial values for the next step. In this way, shorter data spans can be used and convergence can be achieved after a few iterations. In general, accuracies of estimates are within a standard deviation of about 30 arc seconds. This accuracy can be reached after about 30 to 40 minutes of real-time Kalman filtering and after about 10 minutes of iterative post-mission processing.

Influence of update measurements

Figure 3 shows the effect of different types of update measurements on the same data set. In all cases the update period is ten seconds and the same initial values have been used for all runs. They are typical values obtained after processing the data through the Kalman fllter once. Again, the deviation of the azimuth estimate from the reference azimuth is shown as a function of time.

3 ~ __________________________________________ -,

2

-1

-2

85 585 1085 1585 2085 2585 3085 3585 Time (sec.)

Figure 3: Effect of different update measurements (1: zero velocity; 2: zero position change; 3: zero direction change)

It is clear from the graph that the results obtained from zero velocity updates and zero position change updates are very close. This is an expected result because the effect of the two

560

updates on the azimuth is essentially the same for a stationary system. The third update, enforcing a zero direction change, shows a very different pattern. The resulting curve is smoother than the others but shows a more systematic deviation from the reference azimuth. The same characteristical behaviour occurs in all four runs. It appears that this updating procedure eliminates residual gyro drifts very well but affects the estimation of the misalignments in some way.

A summary of results from all four runs is given in Table 2. It shows that the mean of the four runs is in all cases within 10 to 16 arc seconds of the reference azimuth, well within the standard deviation computed by standard least squares error propagation.

Update Estimated Azimuth Standard Deviation

Zero velocity 3550 34'12" ± 26 arcsec.

Zero position 3550 34'11" ± 26 arcsec.

Zero angular velocity 3550 34'18" ± 30 arcsec.

Reference 3550 34'28" ± 30 arcsec.

Table 2: Mean values from different update procedures

Stability of estimates

It has been mentioned before that stability of the azimuth determination over time is more important in some industrial applications than the accuracy of the azimuth itself. Figure 4 shows results for the four runs using zero velocity updates. In this case, the same initial value

3 ~------------------------------------,

2

-

-2 c

85 585 1085 1585 2085 2585 3085 3585 Time (sec.)

Figure 4: Comparison of four runs

561

and covariance matrix have been used for all four runs, assuming that the data have been processed one in the usual way. Variations over ten minute intervals are usually not larger than 10 to 20 arc seconds. Over the total period of one hour, variations can be as large as 1 arc minute.

In some of the emerging applications, the typical operational procedure would require setups of 10 minutes at individual stations, during which time the internal azimuth would be used as a reference, followed by a short transport time to the next station. Table 3 indicates what could be expected under such circumstances. It shows the azimuth difference with respect to the control value for 10 minute means. Statistical information in terms of mean and its standard deviation is given for each run, and for the mean across runs. The table shows clearly that the variations between 10 minute means of individual runs are much smaller than the variations between runs. This indicates that the mean of a run may be biased but that the variations around that bias are small. Thus, if the stability of a reference line is more important than the accuracy of the North direction, then the system tested will reliably maintain such a reference direction for 10 minute means within a standard deviation of 10 to 15 arc seconds.

Time Data Set

Interval A B C D (Minutes)

0- 10 -39" 0" _76" 42"

10- 20 -26" 0" -50" 19"

20- 30 -39" 5" -37" 19"

30 - 40 -36" _3" -37" 10"

40- 50 _43" 7" -20" 22"

50- 60 -37" 3" _44" 22"

60 -70 -30" 5"

Mean -36" 2" _44" 20"

RMS for ±6" ±4" ± 19 ± 12

RMS about total mean: ±29"

RMS about ref. azimuth: ±32"

Table 3: 10 minutes means for different runs

Spectral data cleaning

A few tests were made with a data set where the spikes in the spectrum had been removed before processing the data through the Kalman filter. This method of spectral data cleaning is described in more detail in Czompo (1990). A typical result is shown in Figure 5 where the solid curve shows the result for the raw data, while the broken curve shows the result for the cleaned data. The two curves display a parallel shift but show otherwise exactly the same behaviour. The explanation of this peculiarity is most likely that the spectral procedure introduces a small difference in the mean value which causes the parallel shift. The spikes that have been removed from the data have sufficiently high frequencies and do therefore not show

562

up in the curve. Since factory calibration of the system is usually done without spike removal, the calibrated gyro drift should be changed to reflect the change in the mean value.

5

4

3

..-.. 2 c ·5 1 (.,I

0 s... ~ '-'

.::: - 1 -::s E -2 ·N < -3

-4

-5 85 1085 2085 3085 4085

Time (sec.)

Figure 5: Comparison of filtering results for raw (a) and spectrally cleaned data (b)

4. CONCLUSIONS

Surveying applications of inertial strapdown technology do usually not require real-time processing but place high demands on accurate initial azimuth determination and stability of the internal azimuth reference. The paper explores the consequences of these requirements, i.e. it looks at alignment from a post-mission processing point of view. Such an approach gives more flexibility in the choice of estimation procedures and allows the analysis of residuals. The paper formulates a common framework for the different methods and presents a first comparison of real-time and post-mission techniques on the same data set.

Comparisons were done using Kalman filtering as the real-time method and iterative Kalman filtering as the post-mission method. Results show that the accuracy of the estimated azimuth and its stability is strongly dependent on the quality of the initial values. Since the quality of these values is usually poor in real-time methods, because the initial azimuth is dependent on the coarse alignment, a data span of 30 to 40 minutes is needed to get results at the noise level of the system. In post-mission methods this accuracy can be reached for a data span of only 10 to 15 minutes. The standard error with respect to an external reference azimuth is about 30 arc seconds. The largest part of this error is due to a constant bias which changes randomly from one alignment to the next. Its size is typically 20 to 25 arc seconds.

The stability of the estimated azimuth is therefore much better than indicated by the overall standard deviation. Again, post-mission results are in general more stable than real-time results. Variations about the mean value of the run are small, typically 10 to 15 arc seconds.

563

The overall accuracy of the result does not depend very much on the type of update measurement used, however, there are differences when azimuth estimation is viewed as a time function. While results obtained from zero velocity and zero position change updates are virtually identical, this is not true for zero direction change updates. The more systematic behaviour in the latter case requires further study.

The use of spectral methods to eliminate unwanted spikes in the spectrum and thus provide a clean set of data, did not improve the overall accuracy of the results. It introduced a small bias in the data which can be easily accommodated by changing the calibrated constant gyro drift by the same amount.

The results reported here should be considered as preliminary. Further tests are planned and different estimation methods will be used to arrive at optimal operational procedures for different applications.

ACKNOWLEDGEMENTS

Financial support for this research was obtained from an NSERC Cooperative Grant, entitled The Industrial Alignment Project. D. Lapucha, H.E. Martell and J. Czompo are acknowledged for their assistance in setting up the tests and in providing support during the data processing phase.

References

Czompo, S. (1990). Use of spectral methods in strapdown ISS data processing. This volume.

Gelb, A. (ed. 1974). Applied Optimal Estimation. M.LT. Press, Cambridge, Massachussets.

Knickmeyer, E.H. (1989). Calibration, handling and use of a Cardan-frame with the Litton LTN 90-100 inertial reference system. UCSE Report 30011, University of Calgary.

564

APPLICATION OF STRAPDOWN INERTIAL SURVEYOR IN DETERMINATION OF HOIST SKIP AND MINE SHAFT

TRAJECTORY

H.E. Martell, M. Wei, K.P. Schwarz Department of Surveying Engineering

The University of Calgary, Calgary, Alberta T2N IN4

W. Griffin, A. Peterson Department of Mining and Mettulurgical Engineering

Department of Civil Engineering The University of Alberta, Edmonton, Alberta T6G 2G7

1 . Introduction

In March of 1990, the University of Calgary was approached by Mr. Wayne Griffin and Mr. Art Peterson of the University of Alberta with a unique proposal for the utilization of our Litton LTN-90-100 strapdown inertial surveyor. This involved detennination of the trajectory of the two production skips which carry potash to the surface of a one kilometre deep mine shaft at Central Canada Potash in Colonsy, Saskachewan. In the following we will describe the problems encountered by the mining company, the procedure used in the survey, the computations involved in reconstructing the trajectory parameters for curvature, attitude detennination and displacement of the elevator conveyances from the nominal plumb line, as well as the results obtained.

2 . The Mine Site Problem

The production skips, at the mine in question, operate in a 4.88 m. diameter shaft of ap­proximately 1045 m. in depth. One skip hauls about 27 tons of potash to the surface, while the other travels in the opposite direction to the bottom. The skips are lowered on four ropes and held in place by another four wire ropes, two on each side of each car. The optimal speed of the skips, for production purposes, has been detennined to be 16 m.s·-l by mine engineers. Unfortunately, problems develop at this velocity.

From the beginning of the trajectory, it was noticed, by mine site staff, that the skips had a tendency to twist in a contra-rotating fashion, with the back ends of each skip converging toward one another and the front ends tending towards the wall of the shaft. Near sea-level the two conveyances pass one another. Owing to increased air turbulence, among other factors , the skips experience their worst rotational behaviour at this point. Figure 1 illustrates the general behaviour of the skips, in plan view, at approximately mid­shaft. This shows, in the background, a nominal clearance of about 1/2 metre between the two skips. However at mid-level the processed data indicates the skips may be separated by much less. Torsional behaviour abates slowly as the skips continue their ascent and de­scent until they halt at the top and come into fIXed metal guides at the bottom. Note that the diagram below only shows displacement due to rotation. Smoothed position differences in x and y indicate that pure translations are in the <10 cm. range - something that is quite fortunate for mine management.

565

-z e E

00 N

m Figure 1: Approximate relative displacement of the skips due to z-axis rotation only

3 . On-site Surveying

Initial surveys were carried out by Central Canada Potash (CCP) simply by placing metal rods on the exterior of the elevator conveyances and observing which rods were broken off by contact with the shaft walls. A later approach involved the use of a video camera mounted to the top of one skip. This seemed to confirm that the skips twisted out of their guide ropes in a mirror image of each other, and further, that increased speed resulted in more exaggerated behaviour on the part of each skip, especially at mid shaft. It was as­sumed that the conveyance descending against the upcoming airflow, from ventilation, was contributing most of the relative rotation between the vehicles, since this conveyance would be fighting the on-coming air mass.

What was not known was the degree of rotation and the amount of relative dis­placement of each skip at discrete points in the shaft. Nor was it certain that the aerody­namics of the skips was owing to the assumption stated above. For these reasons, the con­sultants to CCP, Professors Griffin and Peterson, decided to employ the U. of C's strap­down inertial unit in order to quantify the trajectory behaviour of the mine elevator cars. It was indicated to the U. of A. consultants, prior to the survey, that at a conservative esti­mate, with the LTN-90-100 inertial surveyor we could expect a drift of several hundred arc seconds in azimuth and several tens of arc seconds in level misalignments, for a survey of several hours duration. Again, referring to figure 1, it can be seen that such misalign­ments, if translated into errors in displacements from the origin of the ISS to the outside corners of the skips, are negligible for the purpose required. No estimates were given for the ability of the unit to resolve displacements due to pure translational behaviour for a number of reasons, the main one being that we had no idea what sort of problems would have to be confronted on-site. However, as the twisting of the skips was identified as the major problem to be resolved it was felt that the survey would provide useful data. In ad­dition, as we will show later the performance of the ISS was adequate in position dis­placement, as well.

566

The L TN -90-100 was fixed in place at approximately the location of the origin of the axes as shown in figure 1. The unit travelled, by itself, up and down for six runs in each skip. The runs were performed at various speeds ranging from 5 ms-1 to 16 ms-l. The cars were loaded for three runs and empty for the remainder. The duration of each survey was in the order of 5000 seconds. Some of the problems encountered were quite interesting, and given the quality of the results, attest to the robust nature of the Litton strapdown unit.

The first problem was lack of a stable platform for the ISS to align itself on, since the skips sway considerably at the mine shaft surface, and even at the bottom, in fixed guides, there is an alarming amount of vibration present. Initially, a ten minute alignment was performed outside the skips and the ISS carried inside and fixed to the skip. To be safe, four minutes of ZUPTS were also done at both top and bottom of the shaft, so that in effect an alignment could be performed virtually anywhere in the data set. Eventually, during processing, the bottom of the shaft showed the most promise for alignment pur­poses and the data sets were processed from there. Another problem, of course, was the unknown nature of the gravity field at different levels of the shaft. Unfortunately, the gravity meter which we attempted to utilize could, in no way, be levelled in this unstable environment, and the usual approximations had to be made for the discontinuity at the sur­face and below. A final problem, was the lack of a precise known geodetic position for the plumb line of each skip, and furthermore, an uncertainty in the actual elevation of the top and bottom of the mine shaft, this being known only to the nearest several metres. The relative position of the skips being the real consideration here, it was felt that the problem of absolute position could safely be ignored.

4. Data Processing

It was decided to follow several strategies in processing the data. Firstly, we employed the raw body rates in an attempt to find some overall curvature pattern in the trajectory of the skips. It was not expected that the shaft differed significantly from the vertical. Instead we had theorized that the path of the skips might take the shape of a very shallow arc, with the conveyances tending toward the wall of the shaft. Given the small signal present in the horizontal accelerations, it was felt that there would be little possibility of recovering curva­ture parameters, but these were included in post-mission procedures for the sake of com­pleteness.

The data was also filtered post-mission in a 15 state Kalman filter. Velocity, posi­tion and attitude information was output at one second intervals. Finally, the horizontal position data was smoothed using both optimal and empirical (linear) techniques, for pur­poses of comparison. The x and y position data were then differenced with respect to the nominal plumb line of each skip, in order to obtain some estimate of the translational movement of the vehicles. It should be made clear that we put little emphasis on the accu­racy of the smoothed position data, assuming that it served to give only a general idea of the skip trajectories. On the other hand, it was felt that the quality of the attitude informa­tion was more than sufficient to track the rotational behaviour of the skips, and then com­pute the horizontal displacements due to rotation about the z-axis of the skips, to an accu­racy in the order of several cm ..

4.1 Curvature Parameters

Figure 2 illustrates one possible configuration of the path of the skips. The assumption here was that the trajectory was a shallow circular curve, with the z body axis of the ISS tangent to the curve. In the figure below, it is also assumed that the x-axis of the body

567

frame is normal to the curve, and for the simple curve specified, in the direction of the ra­dius.

z body azis_

centre of circle

Figure 2: Tangential and normal components of an assumed circular trajectory

With the assumption of simple curvature, several different algorithms can be employed to recover curvature or alternately, the magnitude of the radial vector. Curvature can be de­fined, in terms of the arc length parameter s, as (Stoker, 1967)

de K=-d s· (1)

Examining angle e in figure 1, or alternately its rate of change '0(, the above simply means that curvature can be expressed as the rate of change of the tangent vector to a curve, with respect to the arc length. This rate of change can be expressed in vector notation as having magnitude K in the direction perpendicular to the tangent, that is, in the direction of the principal normal. We have

dt (s) de ds = ds n = K n . (2)

This allows several possibilites, from the point of view of strapdown navigation. The first involves the use of centrifugal force. The radius of curvature, r, which is nothing but the inverse of the curvature, K can be computed from the following relations.

The first differential of the position vector with respect to time is the velocity vector,

dx dx ds v = dt = ds dt . (3)

The magnitude of velocity I v I is given by the change in arc length with time, ds/dt and from e.g. Chorlton, 1983 direction is given by the unit vector dx/ds. This unit vector is nothing but the tangent vector, t in arc length parametrization, hence velocity can be ex­pressed as

v = v t . (4)

Acceleration can be computed from the time rate of change of equation (3) as

dv. t dt=vt+v , (5)

568

or . dt ds

a=vt+v dsdt ' (6)

which is recognized as

a = v t + v2 1( n . (7)

Under the assumptions made in figure 1, the first term of equation (6) is nothing but tan­gential acceleration sensed in the z-direction, after compensation for normal gravity, while the second term is acceleration sensed in the axes orthogonal to the tangent i.e. the direction of the principal normal, in the case of simple curvature. This radial acceleration can be, in this case, in either the x or y axes, in general. For purposes of simplification, if one as­sumes the skip is drifting to the north or south, which is in fact the most likely scenario, the scalar value

I v2 a = k v2 = - , (8) x p

where al indicates body sensed acceleration rotated into the local level frame, and p is the radius 01 curvature. In principal then, it is possible to reconstruct the curvature parameter, or alternately the radius of curvature, from inertially derived acceleration and velocity measurements, that is,

v2 r=- (9)

al x

Another possiblity for curvature derivation, in strapdown surveying, is through uti­lization of angular rates. If the trajectory of a vehicle is assumed to be on a simple 'arc, then from, e.g. Wells, 1967, the tangential and normal components described above are equiva­lent to radial and transverse components in a polar coordinate frame. In the ease of a sim­ple curve, the radial quantity is the one of interest. Referring to figure 1, once more, we have acceleration in the direction of the radial vector as

a =.j: - r e2 r '

(10)

where for a curve of constant radius

~ = - r e2 . (11)

Looking at figure 1, it can be seen that the change in the direction of the tangent vector i.e. the z-axis of the ISS, is directly related to a change in the magnitude of the angle, S sub­tending the arc. If the inertial unit were travelling along a highway, for instance, we would clearly have

S = azimutht+l - azimuth t

or in this case, where the x-axis is assumed radial

S = roll t+ 1 - roll t

It would seem reasonable that in the limit

S· - droll - dt

569

(12)

(13)

(14)

The quantity, drolVdt can be obtained in a very natural way from the output of a strapdown system. The roll rate is, (Wong, 1989)

droll . CIt = 8 = cozsin cp + ~cos cp

where co~ and co~ are measured body rates. To fIrst approximation one can write

. b 8 =co x '

if the body frame is fairly close to the level frame, or

1· b a = -r 8 2 ~ -r co 2, x x

so that the radius of curvature can be computed, to fIrst order, as

~ p=-­

cob 2 . x

(15)

(16)

(17)

(18)

Equations (9) and (18) can then be used to independently compute the radius of curvature of a vehicle travelling on a path which is assumed to be a simple arc.

4.2 Filtering

The data was also filtered to obtain six degrees of freedom information, namely the atti­tudes, in terms of roll, pitch, and azimuth, as well as position, in curvilinear coordinates. In addition, the velocities in local level frame were computed The LTN-90-100 outputs body rate information at 64 hz., while the quantities above were fIltered at 1 hz.

The ISS was updated at the top and bottom of each skip, during each run, for both zero velocities and position. A defInite swinging motion is apparent in the data during the so-called zero velocity updates, however we had no choice but to live with this. Another cause for concern was the long periods between updates - up to seven minutes for the slower runs at 5 m. s-l.

where

The 15 state fIlter employed in post-mission estimates the following states:

ee, en, eup are misalignments in local level frame, oA, 8cp, oh are position errors in curvilinear coordinates, gA, o·cp, oh are velocity errors, and

(19)

dx dy dz and bx by bzare exponentially correlated terms representing gyro drift and accelerometer bias respectively.

The prediction and fIlter equations are well known. For the specifIc implementation utilized here the reader is referred to Wong (1989).

4.3 Smoothing Equations

570

The position data was smoothed with both an optimal and empirical smoother. It was not felt that the coordinate data was all that reliable, especially for runs of seven minutes dura­tion without update, so both smoothers were employed for the sake of comparison.

The first smoother used makes an optimal estimate of the states Xk at time tk by making use of all the available measurement information both before and after time tk. The Rauch-Tung-Strieble formulation (Wong,1982), written below gives an optimal estimate of the states in a recursive form. The formulas are summarized below:

,...",..." ,..." -xk/m = xk/k + "\ (xk+I/m - xk/k )

P kIm = Pk/k + "\ ( Pk+I/m - Pk+I/k)

with

P t -1 Ak = k/k <l>k+l/k Pk+1/k .

(20)

(21)

(22)

In the formulas above tk/m indicates the final epoch in the survey, while tk/k is the present update epoch being smoothed. If one wishes to optimally estimate any state between up­dates, then (20) to (21) can be utilized by replacing (20) with

,...",..." - ,..."

x./k = x./k + A (x. I/k 1 - X. 1/k)' 11 11++ 1+ (23)

with

A. = P./k <l>~ 1/· P: I I/· ., (24) 111+11+1

where i is the present epoch and k the kth update. The equations above indicate that the misclosure between the predicted and filtered results can be used to improve the predicted results at previous epochs, if the covariance matrix P describes the noise behaviour of the system in a reasonable way.

Another method of back smoothing is empirical smoothing,. which relies on collo­cation to distribute the observed position misclosure at the beginning and end of a run, in a time-wise linear fashion. This method is much less dependent on the choice and propaga­tion of variances during the mission, but also much less effective if the error behaviour displays non-linearities. This was not the case here, especially in the axes of interest i.e. north and east, and so was used as a check on the results obtained from the optimal smoother. The model is of the form (Schwarz et aI, 1984),

0<1> = q I1A + <2 11<1> + cs I1A I1t

OA = -q 11<1> + '3 I1A + c4 11<1> - cs 11<1>I1t .

(25)

(26)

The coordinate corrections, 0<1> and OA can be computed for any epoch, t if the coefficients Ci are estimated from the least squares algorithm

c = C Dt (D C Dt )-1 I ss ss ' (27)

where I is the observed misclosure between predicted and known coordinates at the end of the traverse, D is the design matrix associated with (25) and (26), and Css is the covariance matrix of the unknowns Ci. For a more complete discussion see the article referred to above.

571

5. Results

The quantities which were originally of most interest were the radius of curvature parame­ter, the attitudes, especially azimuth, and the smoothed displacement information. It was not regarded as likely that the curvature parameters could actually be recovered.

For a simple arc, with a maximum deviation from the chord - the plumb line in this case - of 30 cm., at most, there is very little signal in either the horizontal accelerometers or in the x and y gyro channels. Examination of equations (9) and (18) shows immediately that strength of signal is imperative if curvature is to be computed with any degree of accu­racy.and alternately, if there is little in the way of horizontal acceleration or angular rates, it can be concluded that curvature is not present. For instance, in equation (9), it is obvious that for small accelerations p becomes very large, indicating straight line behaviour. Similarly, for equation (18) if the angular rate is at noise level then the radius of curvature again becomes a number approaching infmity.

In fact, no consistent curvature behaviour was observed here. The radii of curva­ture at any given instant were generally in the tens of thousands of metres, and very fre­quently changed sign. The output of the curvature calculations were then indicative of a vehicle which probably followed a sinusoidal path, deviating so little from a straight line that an INS of this accuracy could not resolve with any precision the trajectory parameters in terms of curvature. This result was hardly unexpected, given the nature of the vehicle path and the noise-like output of the x and y channels. A far more profitable exercise was in computation of the output of the azimuth channel.

It is obvious here that the twisting behaviour of the skips would be directly ob­served by looking simply at the difference in azimuths from epoch to epoch and at various

elevations in the mine shaft. In addition, a check on the azimuth drift could be made simply by outputting the attitude information at the bottom of the shaft, where the skips come into

fixed metal guides. At this point the skips are relatively stable and one could expect that the azimuth would show a repeatibility for the LTN -90-100 of +/- several hundred arc seconds

if the system was performing in a reasonable fashion. This was in fact the case, and as well the ISS recorded very similar rotational patterns for each skip on each run, the only difference from run to run being the magnitude of the twist in the skips. This seemed to depend very much on the velocity, the rotation increasing with increased speed. Figure 3

illustrates the behaviour of the skips as they ascend and descend in tandem.

l:J south skip down -84 • north skip up

-86 ........ 00 Q) -88

"0 ......, ..c: -90 .... ::l S

-92 ..... N <

-94

-96 -600 -400 -200 0 200 400 600

Height (m)

Figure 3: Worst Case Trajectory of the Contra-Rotating Skips

572

Figure 1 is a plan view of the skips, at about mid-shaft, based on the azimuth information as plotted above, in figure 3. Given the 15 cm. clearance we have calculated, based on ro­tational displacement at the outside edges of the skips, there is very litde room for transla­tional movements, superimposed on these. Figures 4 and 5 give some idea of the horizon­tal displacements we have estimated. These are taken as position differences from the nominal plumb line.

....... E

'-"

Q) ..... C'tS ~ ....

"'0 J-o 0 0 u

....... E

'-"

Q) ..... C'tS ~ ....

"'0 J-o 0 0 U

1.2

1.0 • III

0.8 • 0.6

0.4

0.2

0.0

-0.2 0

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

-1.2 0

filtring results optimal smoothing emprical smoothing

.....

100 Time (s)

100 Time (5)

....

200 300

200 300

Figures 4 and 5: East and north displacements from the plumb line

Figure 3 illustrates one unexpected result of the survey. From the outset, it was thought that the main twisting effect was coming from the skip which was travelling down, against the upward flow of the ventilated air. While this was true for the south skip, in fact the north skip displays worst case behaviour while on the way up, with the air current. The conclusion was drawn that aerodynamics is not the only problem facing the engineers at CPP.

573

6. Conclusions

The use of a strapdown inertial surveyor has been described, for the purpose of recon­structing the trajectory of two mine shaft conveyances which demonstrate a tendency to ro­tate towards one another as they ascend and descend. Azimuth output was very consistent, and of high enough quality to estimate skip displacements, due to rotations, to the order of several cm. The ISS not only produced a great deal of useful information on the behaviour of the mine skips but demonstrated a great deal of robustness, in an environment of high vibration and sway. We feel that the strapdown system has proven that it can be success­fully implemented, for this, and other positioning and trajectory tasks in the mining indus­try.

Acknowledgements

We would like to extend our thanks to the staff at Central Canada Potash for their help and cooperation. In addition, we would like to acknowledge Dr. E. Knickmeyer for his origi­nal ideas on the mine shaft and curvature problems.

References

Chorton, F. (1983): "Textbook of Dynamics". Ellis Horwood Limited, Chichester, England.

Gelb, A. (ed.) (1974): "Applied Optimal Estimation." The M.LT. Press, Cambridge, Mass.

Knickmeyer, E.H., K.P. Schwarz, and P.J.G. Teunissen (1988): "Strapdown - ein Tragheitsnavigationskonzept fUr Ingenieur-anwendungen ". In: Schnadelbach, K./E. Ebner (eds): "Ingenieurvermessung 88", Diimmler, Bonn.

Schwarz, K.P., D.A.G. Arden, J.H. English (1984): "Comparison of Ajustment and Smoothing Methods for Inertial Networks". University of Calgary Publication 30006, Calgary, Canada.

Schwarz, K.P., E.H. Knickmeyer, and H. Martell (1990): "The Use of Strapdown Technology in Surveying." CISM Journal ACSGC Vol. 44, No.1, Spring 1990, pp.29-37.

Stoker, J.J. (1969): "Differential Geometry". Wiley-Interscience, New York.

Wong, R.V.C. (1982): "A Kalman Filter-Smoother for an Inertial Survey System of Local Level Type." MSc Thesis, Publ. # 20001, Dep. of Surveying Eng., University of Calgary, Calgary.

Wong, R.V.C. (1988): "Development of a RLG Strapdown Inertial Survey System". PhD Thesis, Publ. # 20027, Dep. of Surveying Eng., University of Calgary, Calgary.

574

Ashkenazi, 140, 329 Bartha, 27 Becker,154, 363 Bolcsvolgyi-Bim, 37 Brozena, 488 Bykovsky, 238, 510 Camberlein, 134 Cannon, 443 Cocard, 341 Colombo, 463 Czompo, 228 de la Fuente, 85 Doizi,393 Donna, 95 Dong, 261 Doucet, 178 Doufexopoulou, 27 Eissfeller, 47 Ertel,477 Euler, 285 Ferguson, 319 Forsberg, 351 Fritsch, 477 Geier, 309 Geiger, 341 Georgiadou, 178 Graterol, 543 Griffin, 565 Gumert,543

Author Index

Hadfield, 119 Haines, 382 Hanna, 140 Hatch, 168, 299 Hehl, 477 Hein, 477 Hill,433 Huddle, 125 Jacob, 405 Keller, 154 Kewitsch,477 Kleusberg, 178 Knickmeyer, Ernst, 105 Knickmeyer, Elfriede, 105,498

Knight, 168 Korakitis, 27 Kratochwill, 543 Lachapelle, 17, 273 Langley, 178 Landau, 477 Lapucha, 201, 372 Lidberg, 363 Liu,553 Loomis, 309 Lu, 273 Mamon,95 Martell, 201, 565 Mazzanti, 85

575

Moore, 329 Napier, 140 Norling, 70 Peterson, 565 Ren, 261 Rohrich, 154 Ross, 433 Salzmann, 218, 251 Salytchev, 238, 510 Schwarz, 3, 443, 553, 565 Schroder, 37 Schaffrin, 285 Senese, 433 Sideris, 251 Smith, 59 Soltz, 95 Sun, 453 Teunissen, 191, 251 Thong, 523 van Willigen, 423 Veatch,319 Wan, 261 Washcalus, 543 Wei, 201, 565 Westrop, 329 Weyrauch, 59 Zhang, 453 Zucker, 533