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INTERIM SCIENTIFIC REPORT

on

DEVELOPMENT OF AN EVALUATION TECHNIQUEFOR STRAPDOWN GUIDANCE SYSTEI_S

Submitted to

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

ELECTRONICS RESEARCH CENTER

Cambridge, Massachusetts

February 13, 1968

Contract No. NAS 12-550

BATTELLE _MOR [AL INSTITUTE

Colt,rebus Laboratories

505 King Avenue

Columbus, Ohio 43201

https://ntrs.nasa.gov/search.jsp?R=19680021618 2020-07-15T19:11:35+00:00Z

FOREWORD

This interim scientific report presents the results of a nine month studyconducted by Battelle Memorial Institute, ColumbusLaboratories, for NASA/Electronics Research Center, Contrsct NAS12-550.

The objective of this study was to develop anevaluation technique forstrapdown guidance system performance utilizing performance parameters generatedby both testing and analysis of the inertial sensors and guidance computer and

by system simulation. This objective was accomplished by as follows:

(i) A penalty function was defined which expresses a functional

relationship between the system parameters describing

reliability, power, weight, and accuracy.

(2) Techniques were developed to estimate the system parameters

used in the penalty function.

(3) Digital computer programs were developed implementing the

first and second items. These programs are used as the

basis for (a) system performance evaluation, (b) system

design aids, and (c) system trade studies.

The computer programs developed were exercised on a Jupiter

flyby mission performed with a specific vehicle strapdown

guidance system.

(5) The digital computer program decks, test cases, and

documentation were delivered and an ERC employee instructed

in their use.

..... vu_uLi_ presents a summary of the study results, detailed technical

discussion, recommendations, and conelusions.

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TABLE OF CONTENTS

INTRODUCTION ..............................

Study Objectives ..........................

Summary of Study Elements .....................

Major Assumptions for Exercising the Programs ...........

Mission and Spacecraft ....................

Launch Vehicle ........................

Candidate Components .....................

SUMMARY .................................

Penalty Functions .........................

System Parameters Estimation ....................

Computer Programs .........................

Mission Characteristics ......................

Launch Vehicle Characteristics ............ ....

Trajectory Characteristics .................

Spacecraft .........................

Guidance System A

Mission Results Using System A ...................

TECHNICAL DISCUSSION ..........................

Development of Performance Indices .................

Applicability of SSGS Cost Effectiveness Models ........

Penalty Function Analysis ...................

Penalty Mode i ........................

Pen_aIty Mode 2 ........................

Penalty Mode 3 ........................

Development of System Parameter Estimation Techniques .......

IMU We ight ..........................

Computer Weight ........................

System Power .........................

IMU Power ...........................

Computer Power ........................

Estimated Power ........................

Guidance Power Subsystems ...................

Reliability ..........................

Accuracy Analysis .......................

Injection Errors .......................

IMU Error Mode is . ......................

Computer Error Models .....................

State Transition Matrices ...................

Midcourse Correction Determination ..............

Zeroing All Position Errors ...................

Zeroing One Component of Error ................

Uncertainty in Midcourse Correction ..............

Digital Computer Programs .....................

Program Data .........................

Optimizing Algorithm .....................

Search Housekeeping ......................

Application of Study Techniques to a Jupiter Flyby Mission .....

Data Required ........................

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TABLE OF CONTENTS (Continued)

Pa_e

Boost Trajectory and SEAP Data ................ 76

State Transition Matrices Data ................ 81

Midcourse Propulsion System Data ............... 81

Guidance System Electrical Power Source Data ......... 83

Candidate Component Hardware Data ............... 84

Typical Output Listing .................... 87

IMU Mechanical Weight Estimation Results ........... 92

System B ........................... 93

System B Components ...................... 94

System B Results ....................... 95

Summary of Systems Exercised ................. 95

Parameter Sweeps ....................... 99

CONCLUSIONS ............................... 109

RECOMMENDATIONS ............................. 109

REFERENCES ............................... iii

APPENDIX A

SIMULATION OF T}_ 260(3.7)/SIVB/CENTAUR I/KICK LAUNCHING

A 5,410 LB PAYLOAD ONTO A JUPITER FLYBY TRAJECTORY .......... A- l

INTRODUCTION ............................ A- i

GROUND RULES .............................. A- I

DESCRIPTION OF VEHICLE ......................... A- i

ANALYSIS ............................... A- 3

CONCLUSIONS ............................... A- 5

REFERENCES ............................... A- 9

APPENDIX B

SYNOPSIS OF STANDARDIZED SPACE GUIDANCE SYSTEM

COST-EFFECTIVENESS MODELS ....................... B- 1

Applicability of SSGS Models ..................... B- 8REFERENCES ............................... B-IO

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APPENDIX C

APPROXIMATING THE PROBABILITY DISTRIBUTION OF A

VECTOR WITH NORMAL COMPONENTS AND ZERO MEAN.' ............. C- i

APPENDIX D

CONCEPTUAL IMU DESIGNS ......................... D- I

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APPENDIX E

Pa_e

SINGLE-DEGREE-OF-FREEDOM GYROSCOPE ERROR MODEL ............. E- 1

INTRODUCTION .............................. E- i

DEFINITION OF TERMS ........................... E- i

DERIVATION ............................... E- 2

REFERENCES ............................... E-18

APPENDIX F

TORQUED PENDULUM ACCELEROMETER ERROR MODEL ............... F- 1

INTRODUCTION .............................. F- !

DERIVATION ............................... F- i

DEFINITION OF TERMS ......................... F-15

APPENDIX G

NEW TECHNOLOGY ............................. G- 1

TABLE

TABLE

TABLE III.

TABLE IV.

TABLE V.

TABLE VI.

TABLE VII.

TABLE VIII.

TABLE

TABLE

TABLE XI.

TABLE XII.

TABLE XIII.

TABLE XIV.

TABLE

TABLE XV I.

LIST OF TABLES

I. THREE PENALTY FUNCTIONS FOR EVALUATION OF

STRAPDOWN GUIDANCE SYSTEMS ............... 4

II. SELECTED CHARACTERISTICS OF 260(3.7)/SlVB/

CENTAUR 1/KICK LAUNCH VEHICLE ............ 7

CHARACTERISTICS OW 1972 JUPITER FLYBY MISSION

INTERPLANETARY TRAJECTORY ............... 8

COMMON INPUT DATA FOR SSGS COST-EFFECTIVENESS MODELS. 20

GUIDANCE SYSTEM PARAMETERS ................ 21

MISSION AND SPACECRAFT PARAMETERS ............ 22

PENALTY FUNCTION DEFINITION ............... 22

INTERMEDIATE QUANTITIES USED IN CALCULATINGT_ PENALTY FUNCTIONS ................. 23

IX. STRAPDOWN IMU COMPONENTS AND WEIGHT ........... 33

X. STRAPDOWN IGS COMPUTER COMPONENTS AND

PRELIMINARY WEIGHT FACTORS ............... 40

WEIGHT ESTIMATE FOR 8,192 WORD SRT COMPUTER ....... 41

STRAPDOWN IGS COMPUTER MODULES AND POWER REQUIRED .... 44

RELIABILITY ESTIM/kTION .................. 50

SINGLE-DEGREE-OF-FREEDOM GYROSCOPE ERROR

SOURCES TO BE ADDED TO SEAP .............. 52

XV. TORQUED PENDULUM ACCELEROMETER ERROR SOURCES

TO BE ADDED TO SEAP ................. 54

COMPUTER ERROR MODEL ................... 58

TABLE XVII.TABLE XVIII.TABLE XIX.

TABLE XX.TABLE XXI.TABLE XXII.TABLE XXIII.TABLE XXIV.TABLE XXV.TABLE XXVI.TABLE A-I.

TABLE B-I.TABLE B-II.TABLE B-Ill.

TABLE B-IV.

TABLE B-V.

TABLE B-VI.

TABLE B-VII.

TABLE B-VIII.

TABLE C-I.

TABLE C-II.

TABLE E-I°

TABLE F-I.


Pa_e

MIDCOURSE CORRECTION DEFINITIONS ............. 60

EXAMPLE OF SEARCH ALGORITHM ............... 68

COMPARISON OF THE SEARCH ALGORITHM TO

FINITE ENUMERATION ................... 70

TRAJECTORY COEFFICIENTS FOR 260(3.7)/SIVB/CI/K ...... 77

MIDCOURSE PROPULSION SYSTEM VALUES ............ 83

ASSUMED RTG POWER SOURCE VALUES ............. 84

IMU WEIGHT ESTIMATION RESULTS FOR SYSTEM A ........ 93

SUMMING ACCELEROMETER SENSITIVITIES .......... 94

ERROR SOURCE SENSITIVITIES FOR SYSTEM B ......... 95

ADDITIONAL PENALTY RESULTS ................ 99

SELECTED CHARACTERISTICS OF 260(3.7)/SIVB/

CENTAUR 1/KICK LAUNCH VEHICLE ............. A- 3

IBM EFFECTIVENESS ANALYSIS INPUT DATA .......... B- i

IBM "COST-EFFECTIVENESS" ANALYSIS COST INPUT DATA . . . B° 2

SPERRY EFFECTIVENESS ANALYSIS INPUT DATA ........ B- 2

SPERRY "COST-EFFECTIVENSS" ANALYSIS COST INPUT DATA . . . B- 3

AUTONETICS AND PLANNING RESEARCH CORPORATIONS

EFFECTIVENESS ANALYSIS INPUT DATA ........... B- 5

COST DATA INPUT TO PLANNING RESEARCH CORPORATION MODEL. B- 6

TRW MODEL INPUT ..................... B- 7

COMMON INPUT DATA FOR SSGS COST-EFFECTIVENESS MODELS.. B- 9

COMPUTATION OF a AND d FOR EQUATING MOMENTS ....... C- 2

CUMULATIVE PROBABILITY DISTRIBUTION FUNCTION

[Pr(IX] _ _-'C_ ) ] OF A VECTOR WITH NORMAL,

ZERO MEAN, COMPONENTS ................. C- 4

SINGLE DEGREE OF FREEDOM GYROSCOPES ANGULAR

RATE ERROR SOURCES ................... E-14

SUMMARY OF TORQUED PENDULUM ACCELEROMETER ERROR SOURCES . F-13

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LIST OF FIGURES

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

I. PENALTY, MODE 3, REPORT ON SYSTEM A ............ I0

2. TABLE OF PENALTY, MODE 3, VARIATION WITH SYSTEM AACCELEROMETER BIAS SWEEP ................ 16

3. PLOTS OF PENALTY, MODE 3, VARIATION WITH SYSTEM A

ACCELEROMETER BIAS SWEEPS ................ 17

4. PLOTS OF PENALTY, MODE 3, VARIATION WITH SYSTEM A

GYRO FIXED DRIFT SWEEPS ................. 18

5. PLOT OF PENALTY, MODE 3, (SYSTEM A) VARIATION

WITH NUMBER OF COMPUTER BITS SWEEP ........... 19

6. CUMULATIVE PROBABILITY DISTRIBUTION FUNCTION

[Pr ([X[ _T_)] OF A VECTOR WITH NORMAL,

ZERO MEAN, COMPONENTS ................. 25

7. CALCULATION OF PENALTY, MODE 1 ............. 28

8. CALCULATION OF PENALTY, MODE 2 .............. 30

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FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

ABUSE,

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

A-I•

A-4•


9. CALCULATION OF PENALTY, MODE 3 .............. 32

i0. SKETCH OF CYLINDRICAL COMPONENT IN BLOCK ......... 35

ii. SKETCH OF RECTANGULAR COMPONENT IN BLOCK ......... 35

12. TOP VIEW OF IMU BASE AND SAB, HORIZONTAL DESIGN I..... 37

13. WE IBULL TABLE ...................... 48

14. THE WEIBULL DISTRIBUTION ................ 49

15. LOCAL VERTICAL COORDINATE SYSTEM ............ 51

16. FORMULATION OF COMPUTER ERROR MODEL .......... 55

17. MINIMUM FUEL DELTA-V TO ZERO ONE COMPONENT OF

TARGET ERROR ...................... 64

18. ILLUSTRATION OF A SEARCH FOR A MODE 3 OPTIMUM SYSTEM . . . 71

19. PENALTY, MODE 3, REPORT ON AN OPTIMUM SYSTEM ..... 72

20. ERROR COEFFICIENT SENSITIVITIES FROM SEAP ......... 78

21. INERTIAL FRAME ..................... 81

22. DEFINITION OF ROTATION ANGLE ............... 81

23. N-BODY TRAJECTORY SUMMARY ................ 82

24. CANDIDATE HARDWARE DATA .................. 85

25. PENALTY, MODE 2, REPORT ON SYSTEM A ........... 88

26. PENALTY, MODE 3, VARIATION WITH SYSTEM B

ACCELEROMETER BIAS SWEEP ................ 96

27• PLOT OF PENALTY, MODE 3, VARIATION WITH SYSTEM B

ACCELEROMETER SCALE FACTOR SWEEP ............ 97

28. PLOT OF PENALTY, MODE 3, VARIATION WITH SYSTEM B

BYROFIXEDDRiftSWEEP•••. • _.__ ....... 9829. PLOT OF PENALTY, MODE 2, (SYSTEM A)'VARIATION

WITH TIME TO MIDCOURSE SWEEP .............. 101

30. PLOT OF PENALTY, MODE 3, (SYSTEM A) VARIATIONWITH TIME TO MIDCOURSE SWEEP ............. 102

31. PLOT OF PENALTY, MODE 3, (SYSTEM A) VARIATIONWITH PROBABILITY OF GUIDANCE FAILURE SWEEP ...... 103

32. PLOT OF PENALTY, MODE 3, (SYSTEM A) VARIATIONWITH OPERATING TEMPERATURE SWEEP ........ 104

33. PLOTS OF PENALTY, MODE 3, (SYSTEM A) VARIATION

WITH GYRO EXCITATION POWER SWEEPS ............ 10..5

34. PLOTS OF PENALTY, MODE 3, (SYSTEM A) VARIATIONWITH GYRO WEIGHT SWEEPS ................. 106


WITH IMU COMPONENT SEPARATION SWEEP ......... 107


WITH COMPUTATION FREQUENCY SWEEP ........... 108ASSUMED AERODYNAMIC CHARACTERISTICS OF

260(3 7)/SIVB/CI/K A- 2

TIME HISTORY OF 260(3.7) STAGE TRAJECTORY PARAMETERS . . . A- 4

SECOND STAGE TRAJECTORY FOR SIVB/CI/K PLUS

5,410 POUND PAYLOAD ON JUPITER FLYBY MISSION ...... A- 6

THIRD STAGE TRAJECTORY FOR CI/K PLUS 5,410 POUND

PAYLOAD ON JUPITER FLYBY MISSION ........... A- 7

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ease

FIGURE A-5.

FIGURE C-l.

FIGURE C-2.

FIGURE C-3.

FIGURE C-4.

FIGURE C-5.

FIGURE C-6.

FIGURE D-I.

FIGURE D-2.

FIGURE D-3.

FIGURE D-4.

FIGURE D-5.

FIGURE D-6.

FIGURE D-7.

F IGURE D- 8.

FIGURE D-9.

FIGURE D- i0.

FIGURE D-II.

FIGURE D-12.

FIGURE D-13.

FIGURE D-14.

FIGURE D-15.

FOURTH STAGE TRAJECTORY FOR K PLUS 5,410 POUND

PAYLOAD ON JUPITER FLYBY MISSION ........... A- 8

PROBABILITY VS MAG/MV FOR MONTE CARLO SIMULATION

COMPARED TO CHI-SQUARED DISTRIBUTIONS ......... C- 5


COMPARED TO CHI-SQUARED DISTRIBUTIONS .......... C- 6




COMPARED TO CHI-SQUARED DISTRUBUTIONS ......... C- 8




COMPARED TO CHI-SQUARED DISTRIBUTIONS ......... C-IO

TOP VIEW OF IMU BASE AND SAB, HORIZONTAL DESIGN I..... D- 2

TOP VIEW OF IMU BASE AND SAB, HORIZONTAL DESIGN 2..... D- 3

TOP VIEW OF IMU BASE AND SAB,

HORIZONTAL DESIGN 3 PENDULOUS ONLY ........... D- 4


HORIZONTAL DESIGN 4 PENDULOUS ONLY ........... D- 5


HORIZONTAL DESIGN 5 PENDULOUS ONLY ....... D- 6

TOP VIEW OF IMU BASE AND SAB,HORIZONTAL DESIGN 6 PENDULOUS ONLY ........... D- 7

TOP VIEW OF IMU BASE AND SAB, HORIZONTAL DESIGN 7.... D- 8

TOP VIEW OF IMU BASE AND SAB, HORIZONTAL DESIGN 8 ..... D- 9

TOP VIEW OF IMU BASE AND SAB, HORIZONTAL DESIGN 9 .... D-10

TOP VIEW OF IMU BASE AND SAB, VERTICAL DESIGN i...... D-II


VERTICAL DESIGN 2 PENDULOUS ONLY .......... D-12


VERTICAL DESIGN 3 PENDULOUS ONLY ........... D-13


VERTICAL DESIGN 4 PENDULOUS ONLY ....... . . . D-14

TOP VIEW OF IMU BASE AND SAB, VERTICAL DESIGN 5 ...... D-15

TOP VIEW OF IMU BASE AND SAB, VERTICAL DESIGN 6 ...... D-16

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LIST OF SYMBOLS AND DEFINITIONS

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C 3

D T

D V

Ig

KDc

KDV

Kpl

Kp2

MTBF

MTTF

Hyperbolic excess velocity squared

Target miss degrees-of-freedom

Midcourse correction velocity degrees-of-freedom

Midcourse correction system specific impulse times gravity

Midcourse correction system constant weight

Midcourse correction system tankage factor

Power source constant weight (ibs)

Power source variable weight (ibs/watt)

Mean time between failures

Mean time to failure

PFG

PFR

PFT

PFTR

P___-

WDV

W F

WGS

WICP

WNG

Wp

Probability of mission failure attributable to the guidance

Probability of failure attributable to hardware reliability

Probability of failure attributable to target miss

Probability of failure attributable to hardware reliability or target

target miss

-j - _

The square root of the trace of the midcourse correction velocity

covariance matrix

The square root of the trace of the target miss covariance matrix

Midcourse correction system weight

Midcourse correction system fuel weight

Combined guidance system weight

Inertial measurement unit + computer + electrical energy source weight

Nonguidance spacecraft weight

Electrical energy source weight

ix

W T

VO9

AV

_T

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LIST OF SYMBOLS AND DEFINITIONS (Continued)

Total spacecraft weight

Hyperbolic excess velocity

Midcourse correction velocity

Spacecraft midcourse correction mass ratio

Allowed target miss distance divided by the square root of the trace

of the target miss covariance matrix

Spacecraft delta-V capability divided by the square root of the trace

of the midcourse correction velocity covariance matrix

DEVELOPMENTOFANEVALUATIONTECHNIQUEFORSTRAPDOWNGUIDANCESYSTEMS

By Ellis F. Hitt and F. G. Rea

BATTELLEMEMORIALINSTITUTEColumbusLaboratoriesColumbus,Ohio 43201

INTRODUCTION

This report presents the results of a nine-month study on "Developmentof an Evaluation Technique for Strapdown Guidance Systems" for future NASAunmannedinterplanetary probe missions, conducted by Battelle MemorialInstitute for NASA/Electronics Research Center. This volume presents a summaryof the study results, detailed technical discussion, recommendations, andconclusions.

Study Objectives

The overall objective of this study was to develop an evaluation techniquefor automated interplanetary probe mission strapdown guidance systems. Thistechnique was to be designed to provide a measure or index of guidance systemperformance for this class of missions. The technique was to utilize systemparameters describing _h_ _1_h_1_y, power, weight, _na _,,_=_, T_

addition, techniques were to be developed to estimate the system parameters

using analytical methods as well as available test data.

Summary of Study Elements

To accomplish the study objectives, the project was carried out in the

following five major steps:

(i) A system performance index or penalty function was defined which

expresses a functional relation between system parameters describing

the reliability, power, weight, and accuracy. In performance of

this task, consideration was given to existing "cost-effectiveness"

mode Is.

(2) Techniques were developed to estimate the system parameters used in

the system performance index. These techniques use analytical

methods as well as available test data as the basis of the estimates.

(3) Digital computer programs were developed which:

(4)

(5)

(a) Calculate the system penalty given the system parameters,

or select a quasi-optimum system configuration

from various subsystem alternatives.

(b) Calculate estimates of the system parameters for use in

determining the system penalty.

A boost trajectory for the conceptual 260(3.7)/SIVB/Centaur I/Kick

launch vehicle was generated for a Jupiter flyby mission. The

interplanetary trajectory and state transition matrices were

generated with the Lewis Research Center n-body program. The ERC

Strapdown Error Analysis Program (SElkP) was used with the boost

trajectory data to generate the inertial sensor error sensitivity

coefficients. The computer programs developed under the previous

step were exercised on the Jupiter flyby mission for a NASA/ERC

specified guidance system.

The digital computer program decks, test cases, and documentation

were delivered and a NASA/ERC employee was instructed in the use of

these programs.

Major Assumptions for Exercising the Programs

It was necessary to assume specific mission, launch vehicle, spacecraft,

inertial sensors, strapdown guidance computers, and other data required for

exercising the computer programs developed under this study.

Mission and Spacecraft.--The mission used in exercising of the computer

programs was a Jupiter flyby mission with launch to occur during 1972. This

mission, or variations, thereof has been studied previously (References 1,2,

3, and 4) .

The midcourse propulsion system was assumed to be a gas-pressure-regulated

monopropellant hydrazine unit using a Shell 405-type catalyst (Reference i).

Guidance system electrical power was assumed to be supplied by a RTG

(radioisotope thermoelectric generator)(References i and 2).

Perfect update of velocity, position, and attitude were assumed to occur

during the parking orbit. Hence, the boost analysis considers errors during

the burn from the parking orbit to the final velocity as the only contributors

to injection errors.

Prior to the midcourse, correction, an attitude update was assumed with a

I0 arc second uncertainty a_out each of the principal axes. Error in the mid-

course correction magnitude was assumed to be solely attributable to the roll

axis accelerometer.

Launch Vehicle.--For the purpose of exercising the computer programs, the

conceptual 260(3.7)/SIVB Centaur 1/Kick launch vehicle was assumed to be

suitable for this mission. Data on this conceptual vehicle are contained in the

subsequent section of this report describing the boost trajectory. Guidance

and navigation of this vehicle were assumed to be under control of the strapdown

guidance system located above the Kick stage. It was assumed that the weight

of the strapdown guidance system was included in the final payload weight

injected on the escape trajectory.

Candidate Components.--The major portion of the data assumed for candidate

gyroscopes and accelerometers was based on manufacturers' data and government

agency test reports. Reliabilities for the inertial sensors and computers were

assumed for purposes of exercising the computer programs, and thus, results

_nvolving these data should not be considered conclusive.

Weight and power for the conceptual SRT computer (an assumed computer for

the specified system A) were estimated using simple techniques described in a

subsequent section of this report.

SUMMARY

An evaluation technique for strapdown guidance systems designed for

unmanned interplanetary missions has been developed. This technique provides a

............ jo_=,,, _=L_uLu,=,U= UL, U_UL_ n_ission being examined, in a(l(lltlon, the

technique is useful in evaluation of competitive systems, as an aid in pre-

liminary design of conceptual systems, and for determination of research needed

to improve system performance.

Cost-effectiveness for the problem is defined with cost equal to the

weight of the guidance system and effectiveness equal to the probability of

the guidance system operating correctly. Weight is defined to be the sum of

rhp weights n_ _h= _._oI measurc_e_nt unit (T_T_ .......... _'A .........

correction propulsion system, and guidance electrical energy source. The

effectiveness probability is defined to be the appropriate combination of the

probabilities of failure due to guidance system unreliability, failure due to

insufficient midcourse fuel, and failure due to missing the target by more than

a prescribed distance. Using this cost-(or weight-)effectiveness model, several

penalty functions have been developed. These may be broken into two categories:

those which hold effectiveness or probability constant and consider weight as

the penalty and those which hold weight constant and consider the ineffective-

ness or probability of failure as a penalty. This eliminates dollar cost as

a parameter but combines power, reliability, and accuracy by considering their

effects on total guidance system weight. These penalty functions provide

meaningful and useful information.

Penalty Functions

Three different penalty functions were developed. The need for tl_ree

penalty function_q (indicated by three modes) became apparent when "real world"

launch vehicle and mission considerations were investigated. The three modesare as follows:

ModeI: The probability of mission failure due to lack of guidancereliability and accuracy (PFG) is a user specified constant.

Another user specified constant is all nonguidance weight (WNG).

The penalty function is the combined guidance system weight (WGS)

which consists of the sum of guidance system hardware weight plus

electrical power source weight (WICP) and midcourse propulsion

system weight (WDv). An increase in the combined guidance system

weight necessary to assure a given influence, by the guidance

system, on probability of mission success is reflected in an

increased launch weight (WT).

Mode 2 : The total launch weight, equal to the sum of the nonguidance

weight plus the combined guidance system weight, is a user speci-

fied constant. In addition, the nonguidance weight is specified

as is the combined guidance system weight. Any decrease in

guidance system hardware or power source weight is offset with an

increase in midcourse propulsion system weight or vice-versa.

The probability of mission failure due to lack of reliability or

accuracy is the penalty function.

Mode 3: The third mode involves user specified total launch weight and proba-

bility of mission failure due to lack of guidance reliability and

accuracy. The combined guidance system weight is the penalty

function. In this mode the nonguidance weight (useful payload) is

the difference between the launch weight and combined guidance

system weight. Thus, for increasing WGS , WNG is reduced.

The three penalty functions are shown in Table I for comparison.

TABLE I

THREE PENALTY FUNCTIONS FOR

EVALUATION OF STRAPDOWN GUIDANCE SYSTEMS*

Mode WT WNG WGS PFG Remarks

1 V F P F

2 F F F P

3 F V P F

* V_ Variable with System

F_ Constant

P& Penalty Function

Fixed Nonguidance Weight and Probabilityof Guidance Failure

Fixed Total Weight and Guidance System

We igh t

Fixed Total Weight and Probability of

Guidance Failure

II!I

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For each of the modes, the minimum value of the penalty function defines

the best system.

Sensitivity of each penalty function with respect to specific system hard-

ware parameters is expressed as the percent change in penalty per percent change

in data. These sensitivities allow easy determination of the system parameters

and components which affect the penalty function most directly (large sensi-

tivity magnitude). The algebraic sign indicates which direction the penalty

changes for an increase in the system parameter. Further explanation of the

penalty functions and sensitivities is contained in the technical discussion

section of this report.

System Parameters Estimation

The strapdown inertial measurement unit (IMU)(also referred to as the

inertial sensing unit (ISU)) weight is estimated by evaluating analytic

expressions approximating the design of an IMU. Nine designs, of which any

six are evaluated in the computer subroutine MECHDS, have been described

analytically for the case where the base of the IMU is mounted horizontally

in the vehicle. Similarly, six designs have been developed for the case where

the base of the IMU is mounted vertically in the vehicle. The current designs

and their analytic expressions are described further in the technical discussion

section of this report. Other designs for which analytical expressions can be

IIII

The strapdown inertial guidance system (IGS) computer weight can be esti-

mated should such an estimate be required. The recommended means of doing this

requires input of specific values for the weights of each module of the

computer. This assumes the computer would be assembled from trays containing

these modules. The weight factors contained in the technical discussion section

of this report were estimated based upon a computer organized in a modular

of the computer case. Since this technique is quite simple and the computer

requirements vary little from one strapdown IMU to another for the same inter-

planetary mission, a computer subroutine was not written to calculate the IGS

computer weight.

IIIII

Estimation of the system power requirements is dependent upon the thermal

design of the IMU as well as the IGS computer. The primary concern in thermal

control of the IMU should be to minimize the total system power required. The

IMU power is estimated by solution of heat transfer equations presented in the

technical discussion following this summary. Since many possible tradeoffs

exist in thermal control of the IMU, this should by no means be considered as

other than a tool for estimating the IMU power.

Computer prime power can be estimated by summation of the estimated power

required by the input/output (I/0), memory, and processor modules and division

of this summation by the efficiency of the proposed computer power supply.

Since the I/0 module power depends upon the electrical interfaces of the IMU,

telemetry, etc. with the computer, precise power estimates cannot be made until

this information is available. Therefore, a subroutine to calculate computerpowerwas not written.

System reliability is predicted assuming there are no redundant components.Several possible reliability descriptions were examined. The Weibull distribu-tion was chosen to allow more realistic estimation of failure probabilities with-out the large amounts of data needed for more complex distributions such as theMarkovian. The Weibull distribution contains the more commonlyused exponentialmodel if the exponent = is set equal to one. The meantime to failure (MTTF)and

are input data for the candidate gyroscopes, acce]erometers, and computers.The mean times between failure (MTBF)are input data for the guidance electricalpower source and IMUelectronics. The probability of failure of the midcoursepropulsion system to ignite is also input data. Using these data, the programcalculates the reliability of the strapdown guidance system.

IIIiII

Computer Programs

The calculation of the three penalty functions and the necessary estimation

of the system parameters have been coded into a deck of FORTRAN subroutines.

The subroutines, with a short, simple, main program calculate the necessary

system parameters and evaluate them according to the specified penalty function.

Data needed to run the program are divided into four categories. The first

three involve data describing the mission and spacecraft and include: (I) injec-

tion error sensitivities as computed by the Strapdown Error Analysis Program

(SEAP); (2) state transition matrices generated by the n-body program; and

(3) data describing mission values, IMU design values, and spacecraft subsystems.

The fourth category is data describing candidate components (accelerometers,

gyroscopes, and computers) and includes: component (i) weight, (2) dimensions,

(3) excitation power, (4) reliability, and (5) error coefficients. Computer

data required is similar to that for gyros and accelerometers except that the

number of bits used to store each element of the attitude matrix, attitude

update integration frequency, and integration scheme (rectangular, Runge-Kutta

second order, or Runge-Kutta fourth order) are used in the error determination.

The program and its subroutines operate in one of two ways. A specific

set of candidate components may be evaluated, or, by a searching technique

(similar to steepest descent), a combination of the candidate components may be

found which yields the minimum penalty. Energy source and midcourse correction

subsystems as well as the IMU block, base, and cover design are modelled

internally. Options are provided to allow specification of the type of mid-

course correction (zeroing all position or any one component of miss at the

target), mechanical orientation (horizontal or vertical), mechanical configura-

tion (one of the stored designs or the one yielding minimum weight), midcourse

correction accuracy (specified or computed), and an option which indicates if

the gyros and accelerometers must be identical or may be non-identical along

the three axes when the searching technique is selected. After selection of

the desired options and a set of candidate components, either arbitrarily or

by the searching technique, all other system characteristics are calculated and

the penalty obtained. The program output consists of a four-page report listing

IIIIIIIIIIIII

!II

the selected components and their data, mission and subsystem characteristics,

and the penalty function value. In addition, normalized derivatives or sensi-

tivities of the penalty to each piece of data is printed. This allows the

designer to evaluate the relative importance of the data on the penalty function

value. Furthermore, outputs in the form of tables or curves may be obtained to

show the behavior of the penalty function as any piece of hardware or mission

data is swept over a wide range of values.

IIIII

Mission Characteristics

Launch Vehicle Characteristics.--The conceptual 260(3.7)/SlVB/Centaur I/

Kick launch vehicle was assumed to be suitable for the Jupiter flyby mission. A

version of this vehicle has been studied extensively (References 5 and 6) by

Battelle Memorial Institute, NASA Launch Vehicle Planning Project under Contract

NASw-II46. Selected characteristics of this vehicle are given in Table II. The

260(3.7) first stage is a solid propellant, single engine stage and is described

in detail in Reference 7. The SIVB second stage is the same as the SIVB utilized

by the Saturn IB and Saturn V launch vehicles. The Centaur I third stage is bas

based on design evolution and concentrated development effort to employ hydrogen-

flourine propellants (Reference 8). The "Kick" fourth stage is a small high

energy stage originally proposed for use with the SLV3C and SLV3C/Centaur

(Reference 8).

TABLE II

SELECTED CHARACTERISTICS OF 260(3.7)/SIVB/CENTAUR I/KICK LAUNCH VEHICLE

Stage 260(3.7) SIVB Centaur I Kick

IiII

Initial Total

Gross Weight (ib)*

Final Total

Gross Weight (ib)*

Vacuum Thrust (Ib)

Propellant Weight

Flow (Ib/sec)

Exit Area (ft 2)

4,019,302 315,303"** 54,202 12,410

720,302 85,303 17,602 6,460

6,430,000** 205,000 31,000 7,500

24,300** 482 68.20 16.47

376 35.8 16.58 4.14

* All weights include 5,410 ib payload.

** Initial value only for time history, see Appendix A, Figure A-2.

*** Includes 5,600 ib shroud which is ejected 26 seconds after SIVB ignition.

This launch vehicle was simulated using the Battelle three-degree-of-

freedom computer program. The simulated boost trajectory was used in the

guidance error analysis. Further discussion of the launch vehicle simulation

and boost trajectory are contained in Appendix A of this report.

Traiector Y Characteristics.--The boost trajectory was assumed to start

with a vertical rise from Cape Kennedy until a relative velocity of 150 ft/sec

was attained. The vehicle was subjected to an instantaneous deflection of the

flight path through a selected kick angle of 1.83 degrees from the vertical

along an initial azimuth of 90 degrees from true north. The vehicle then flew

a gravity turn until first stage burnout. A linear time dependent pitch

steering profile was used to steer the second stage into a nearly circular

i00 nm parking orbit. It was assumed that the SIVB stage was restarted at the

correct time to begin the final injection phase of the launch trajectory.

Characteristics of the interplanetary trajectory are tabulated in Table III.

The Lewis Research Center n-body computer program was modified to calculate

n-body state transition matrices. The state transition matrices used in exer-

cising of these computer programs were obtained by running the Lewis n-body pro-

gram for the trajectory summarized in Table III. Further discussion of the inter-

planetary trajectory and state transition matrices is contained in a subsequent

section of this report.

III

TABLE III

CHARACTERISTICS OF 1972 JUPITER FLYBY MISSION INTERPLANETARY TRAJECTORY

Launch date

Arrival date

Time of flight

Departure asymptote (from Earth)

V=

C3Dec iina tion

Angle to Sun-Earth line

Approach asymptote (to Jupiter)V=

Declination to plane of Jupiter

Angle to Jupiter-Sun line

Interplanetary Orbit

True anomaly at launch

True anomaly at arrival

Heliocentric central angle

Inclination to ecliptic

Per ihe Iion

Aphe iion*

Eccentricity**

Earth-Jupiter distance at encounter

.. r ' ' . ,'; ,

* Does not pass aphelion on way to Jupiter.

** Varies due to n-body effects.

March 4, 1972

April 18, 1973

410 days

12.195 km/sec

147.718 km2/sec 2

-25.3 deg

93.0 deg

17.53 km/sec

i. 64 deg

153.2 deg

-6 deg

129.3 deg

135.3 deg

-0.57 deg

0.989 AU

650 AU

0.972 - 0.976

7.67989 x i0 II meters

IIIIIIIII

I!!IIII

Spacecraft.--The spacecraft weight derived from the reference trajectory

is 5410 pounds. No assumptions about the spacecraft scientific payload or

structure were made since these have been studied previously (References i, 2,

and 4). The midcourse propulsion system was assumed to be a monopropellant

hydrazine unit using a Shell 405-type catalyst. The guidance system electrical

power source was assumed to be part of the spacecraft total weight as was the

midcourse propulsion system. The strapdown guidance system onboard the space-

craft was assumed to provide the primary guidance and control functions for all

launch vehicle stages as well as the midcourse correction.

Guidance System A.--The strapdown guidance system being assembled

in breadboard form at ERC was used in the exercising of the computer programs

developed under this contract. This system is made up of the Honeywell GG 334A

gyroscope (three units orthogonally mounted) and the Arma D4E accelerometer

(three units orthogonally mounted) and is referred to as system A. Estimates of

the weight of the conceptual IMU containing these instruments have been made.

In addition, estimates of the characteristics of a conceptual computer, referred

to as SRT, were made for purposes of exercising of the computer programs.

IIIIiIlIIII

Mission Results Using System A

The computer programs developed as part of this study were exercised on

the Jupiter flyby mission for the specified strapdown guidance system described

iu L,= _u_=s_ns secti n. _g .... I= presents the p_n_1_y for the mode being

evaluated (mode 3 penalty is weight), lists descriptive information about the

launch vehicle and mission, and presents the hardware data for each instrument

used in the IMU. In addition, data describing the conceptual SRT computer

are presented. The notation RUK - 2 denotes that a Runge-Kutta second order

algorithm was used to update the direction cosine matrix.

Figure ib presents the results of the penalty function and the estimation

techniques incorporated in the program. For example_ all horizontal designs

were evaluated and design number 4 was found to be optimum (yielding minimum

weight). The dimensions of the case of the IMU containing the inertial

sensors are presented for that design. In addition, the calculated weights

of the block, base, cover, and insulation are presented. The e_ctronics

weight was a specified value. The components weight is the summation of the

weights of the inertial sensors and computer. The total estimated weight for

the system A IMU and computer is calculated to be 69.46 pounds for IMU design 4.

The thermal analysis presented indicates that the thermal conductance for

the IMU was calculated to be 2.17 watts/°F on the basis that no heater power or

cooling was to be required at the maximum ambient temperature. The peak watts

listed are the total power (IMU plus computer) required at the minimum ambient

temperature for the calculated thermal conductance. The average watts listed

are the total power required at the average ambient temperature. The energy

source weight is the weight of the radioisotope thermoelectric generator (RTG)

needed to supply the average power required. This assumes that the peak power

is required for a short time and would be drawn from a small secondary energysource as well as the RTG.

o-i

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' I

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tl

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g

Figure Ib also presents the results of the navigation error analysis at

various times along the trajectory. For instance, the velocity errors at

injection are given in down range (DR), cross range (CR), and out of plane (OP)

components. The root mean square (RMS) of these components is 11.03 ft/sec at

injection with the degree of freedom (DOF) of 1.984. The injection errors

propagated to the target with no midcourse correction are seen to result in a

CR position error of 11.6450 x 108 ft. If a single perfect midcourse delta-

velocity correction is made to null the CR position error, the correction delta-

velocity required has an RMS value of 9.6276 ft/sec. If an imperfect midcourse

is made with the relative accuracy shown, the CR position error at the target is

seen to be 84.3593 x 104 ft. Data describing the midcourse propulsion system

are also presented. It can be seen that the delta-V capability is 16.45111

ft/sec and the expected burn time for the 9.6276 ft/sec velocity correction is

32.35 sec. The total midcourse propulsion system weight was calculated to be33.287 lb.

The penalty analysis for this mode, mode 3, is presented. The probability

of failure due to hardware reliability was calculated to be 0.013. For the

specified (fixed) probability of system failure, 0.i000000, and the probability

of failure due to hardware reliability, it is seen that the probability of fail-

ure due to not having sufficient fuel was calculated to be 0.088. The total

weight of the spacecraft and guidance system was specified (fixed) at 5,410 lb.

The penalty for this mode is shown to be a guidance system weight of 222.04 lb.

It should be recalled that this weight is the summation of the IMU + computer +

energy source + midcourse propulsion system weights. The resultant non-guidance

weight is seen to be 5,187.96 lb.

Figure ic lists the mission and spacecraft data used in the computation of

the results presented in Figure lb. A short phrase describing the item of data

and the value used in the computation is followed by the sensitivity of the

penalty to that value. This sensitivity, as noted in Figure ic, is the percent

change in the penalty to a one percent change in the data value. For example,

a one percent change in ID 23, the IMU electronics power, would result in a

0.233 percent change in the penalty for this case. A negative value of sensi-

tivity indicates that the penalty will decrease if the data value is increased

and vice versa.

Sensitivity of the penalty to inertial sensor and computer data is pre-

sented in Figure Id. A zero value of sensitivity indicates that a one percent

change in the data value presented in Figure la for the item of interest

results in no change in the penalty out to the number of significant figures

printed_ or that the data value was zero. The sensitivity to the number of bits

of the computer and integration scheme is not the sensitivity to a percent

change in the data value but is the percent change in penalty due to the change

noted for these items. For example, an increase of one bit results in a 0.00199

percent decrease in the penalty. It is interesting to note that the pitch gyro

fixed restraint drift uncertainty, R, affects the penalty much more than any of

the other error coefficients of the inertial sensors. The sensitivity analysis

presented in Figure id indicates that hardware design effort might be better

directed toward decreasing the weight of the computer by a percent than

decreasing the pitch gyro fixed restraint drift by a percent since the penalty

decrease would be nearly five times greater.

II1

IIIIIIiIIIIII

12

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13

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IIIIIIIIII

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I

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IIIIII

Figure 2 presents a tabular listing of penalty for a range of values of K0,bias, for the yaw, pitch, and roll accelerometers of system A. The value

of the accelerometer bias was swept simultaneously for the three accelerometers.

It can be seen that a one order of magnitude decrease from the original value

made very little improvement in the penalty. An order of magnitude increase

from the original value also made little difference although an increase of two

orders of magnitude over the original value resulted in saturation of the

penalty due to weight of the midcourse fuel required to correct the position

error caused by the accelerometer bias coefficient listed.

Figure 3 presents four parameter plots, _ne of which is a plot of the

tabular data presented in Figure 2. The computer plots the penalty for the mode

being used (in this case mode 3 whose penalty is weight) versus the value of the

parameter being swept. In this figure, a logarithmic sweep of the parameter was

used. Plot I in Figure 3 is a plot of the data swept, K_, for all three accel-

erometers simultaneously. This plot corresponds to the _ata in Figure 2.

Plot 2 depicts the effect on the penalty if K n of the yaw accelerometer only is

swept. Plot 3 presents similar information i_ the pitch acceierometer only is

swept while plot 4 presents the penalty if the roll accelerometer only is swept

while the other data values remain constant. From this plot it can be con- ,

cluded, for example, that the bias of the pitch accelerometer could increase

from 6.7 _g to 0.33 millig with no discernible increase in the penalty. It

should be kept in mind that this is an example dependent upon the data used in

the calculations. This example does illustrate the usefulness of the parameter

sweeping and plotting routines of th_ computer program developed under thi_

study.

Figure 4 presents parameter plots in which the fixed restraint drift, R,

of the system A gyroscopes was swept logarithmically for penalty mode 3.

Plot i presents the penalty versus R when all three gyroscopes are swept

simultaneously. Plot 2 presents a sweep of the yaw gyro only, plot 3 a sweep of

the pitch gyro only, and plot 4 a sweep of the roll gyro only. The normalvalue of R is 0.1°/hr. It can be concluded that for these data, little increase

in the value of R for the pitch gyroscope can be tolerated as seen by the

approximately 6 pound increase in the penalty when the value of R is increasedfrom 0.i to 0.199°/hr. The value of R for the roll gyrocope can be increased

from 0.i to 2.5°/hr before a discernible increase in the penalty occurs.

Information such as this should prove useful in the design and selection of

components for strapdown guidance systems.

Figure 5 presents a plot of the penalty versus the number of bits in a

computer word for the conceptual SRT computer using a Runge-Kutta second order

algorithm (RUK-2). A linear sweep of the parameter was used. It can be con-

cluded that a word length greater than 27 bits does not result in an improve-

ment in the penalty for mode 3. Use of a 23 bit word length would result in a

78 Ib increase in the midcourse propulsion system fuel and tankage over that

needed to perform the midcourse velocity correction required using a 27 bit or

longer word. This is due to the increase in navigation errors the computer

contributes in updating the direction cosine matrix and solving the guidance

equations.

15

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19

The results presented in this summary section were meant to illustrate

some of the features of the computer programs. Other results from the exer-

cising of the computer programs developed under this study are presented in

the technical discussion section of this report.

TECHNICAL DISCUSSION

Development of Performance Indices

Before starting to develop a performance index, four existing "cost-

effectiveness" models were reviewed and evaluated. These models were developed

for the USAF Space Systems Division (now Space and Missile Systems Organization)

"Standardized Space Guidance System (SSGS) Study" and are discussed in

Appendix B of this report. This appendix contains a review of each of the four

models, lists the input data required, discusses the fundamental figure of merit

utilized by each model,and presents an evaluation of the applicability of

these models. This evaluation is summarized below.

Applicability of SSGS Cost Effectiveness Models.--Each of the "cost-

effectiveness" models developed for evaluating competing guidance systems for

the Standardized Space Guidance System (SSGS) may be useful if given the same

requirement, i.e., determine for the national space mission model the SSGS

which is most cost-effective. It can be stated that this was not the problem

for which NASA/ERC was seeking a solution. NASA/ERC was concerned with develop-

ment of a performance index which will trade off the system parameters of

accuracy, weight, power, and reliability, and thereby serve as a design aid for

selection of component subsystems of a strapdown guidance system.

All of the SSGS models require some of the same input data. Table IV

lists some of the common input data. Some of the models required weight and

power of the guidance system (inertial platform, computer, and aids) and others

did not.

TABLE IV

COMMON INPUT DATA FOR SSGS COST-EFFECTIVENESS MODELS

(i)(ii)(iii)(iv)(v)(vi)

(vii)

(viii)

(ix)

Mission model

Navigation accuracy requirement of each mission

Probability of mission success

Candidate guidance system error analysis

Candidate guidance System reliability

Nonrecurring cost for launch vehicle and all

other nonguidance items

Recurring cost for launch vehicle and all other

nonguidance items

Nonrecurring guidance cost

Recurring guidance cost

IIIIIIIIIIIIIIIIII

20

IIIIII

All of the models are heavily cost oriented. The cost data required is

generally subject to doubt or not available. Two of the models make use of a

figure of merit which can be expressed as dollars/pound of payload. For

missions with quite heavy payloads, the figure of merit would appear much less

sensitive to an increase in guidance system weight than a mission utilSzing the

same guidance system with a low payload weight. Two of the models are verymuch total mission cost oriented.

Greater length could be added to this report in providing additional

rationale for non-use of the four SSGS models. It was apparent that a less

complex and different type of cost function or performance index was required

to aid in the design of strapdown guidance systems, be of use in evaluating

competitive systems, and serve as a tool in the determination of research

needed to improve system performance. It was decided that any model developed

should be much less cost oriented than the SSGS models. The following section

presents the models developed by Battelle.

III

Penalty Function Analysis.--The effectiveness of a _.................

specific spacecraft on a specific mission is evaluated by one of three penalty

functions. The guidance system is described by the six system parameters shownin Table V.

TABLE V

GUIDANCE SYSTEM PARAMETERS

Symbol Definition

IiIIIII

PFR

DV

RT

DT

WICP

Probability of a failure due to hardware

reliability.

Root-mean-square midcourse delta-V, the square-

root of the trace of the -_ ....... A_I+=__

covariance matrix.

The degrees-of-freedom of the midcourse delta_

covariance.

Root-mean-square target miss, the square root of

the trace of the target miss covariance matrix.

Degrees-of-freedom of the target miss covariance.

Sum of inertial measurement unit, computer, and

electrical energy source weights.

The penalty functions are calculated from the above system parameters and the

mission and spacecraft parameters shown in Table VI.

21

TABLEVIMISSIONANDSPACECRAFTPARAMETERS

Symbol

PFG Probability of mission failure attributable to theguidance system.

WNG

WT

XMISS

Nonguidance spacecraft weight.

Total spacecraft weight.

Allowed miss distance at target.

Midcourse correction system tankage factor (systemweight/fuel weight).

!!!

KDC

Ig

Midcourse correction system constant weight.

Specific impulse of the midcourse correction system

times gravity.

The three penalty functions are defined in Table VII.

TABLE VII

PENALTY FUNCTION DEFINITION

Penalty PFG WNGMode

1 F F

2 P F

3 F V

WGS W T

P V

F F

P F

III

F = Fixed, used as a requirement

P = Variable, used as the penalty

V = Variable, used for information only

Penalty function, mode I, assumes that a certain probability of mission

failure attributable to guidance (PFG) is reasonable and that nonguidance

22

IIII[II

spacecraft weight (WNG) is fixed. The guidance system weight is calculated and

used as the penalty function.

Penalty function, mode 2, assumes nonguidance, guidance system, and total

spacecraft weights are constants with the probability of mission failure

attributable to guidance (PFG) variable and used as the penalty function.

Penalty function, mode 3, assumes that a certain probability of mission

failure attributable to guidance (PFG) is reasonable and that total spacecraft

weight (WT) is fixed. The guidance system weight (WGS) is variable and used as

the penalty function.

Calculation of the penalty function under any of the three modes will

involve calculating the intermediate quantities defined in Table VIII.

TABLE VIII

INTERMEDIATE QUANTITIES USED INCALCULATING THE PENALTY FUNCTIONS

IIIIiIIII

Symbol Definition

_FV

PFT

PFTR

W F

WDV

AV

_v

_T

Probability of excessive target miss

Probability of failure due to reliability or target

miss

Weight of midcourse fuel

Total weight of midcourse system

Midcourse delta-V capability

Spacecraft mass ratio

Midcourse delta-V capability divided by the square

root of the trace of the delta-V covariance matrix

(number of traces)

Allowed target miss distance divided by the square

root of the trace of target miss covariance matrix

(number of traces)

A detailed discussion of the steps used to calculate each penalty mode is

given below°

23

Penalty Mode l.--Probability of missing the target (PFT) is calculated from

the system parameters describing accuracy at the target by

PFT = Prob(@T, DT)

III

with

_T = XMIss/RT

The function Prob(_, D) is the probability distribution of the magnitude

of a vector with normal, zero-mean, components as discussed in Appendix C.

A table of this distribution is shown in Figure 6.

The combined probability of missing the target or failing due to relia-

bility is obtained from

PFTR = PFR + PFT - PFRPFT

IIII

With the probability of failure due to target miss or reliability known,

the probability of failure due to insufficient fuel which will result in the

specified PFG is calculated by

PFG - PFTR

PFV = i - PFTR

III

Note that if PFTR exceeds PFG, PFV does not exist. In other words, if the

probability of failure due to reliability or target miss is greater than PFG,

even a perfect system (zero probability of insufficient fuel) will not meet

the required guidance system failure probability.

The delta-V capability required to achieve the required PFV is now calcu-

lated by

III

*V = _(PFv' DV)

and

Av = RV_v

24

IIII!IIIIIIilIIII

(U

,.£:)

E-_

Co

4_

0

m

0

tM

0

t_

0

.IJ

N

m

4-1

,,.4

°,..4

25

where _(P, D) is obtained from function Prob(_, D) by solving for _ knowing P

and D.

The familiar rocket equation,

AV = Ig loge(_) ,

is used to obtain the spacecraft mass ratio

AV/Ig_= e

The spacecraft mass ratio is the initial spacecraft weight divided by the

final spacecraft weight or

I!I!!II

W T

W T - W F

The midcourse fuel weight required is then

WF = WT (_ - i)

IIII

Now

WT = WIC P + WNG + WDV

and since the midcourse correction system weight is estimated by

WDV = WFKDv + KDC

substitution into the equation for WF in terms of WT and _ yields

(_ - I)(WIcP + WNG + KDC)

WF = _ -[_ - i] KDV

IIII

26

The effective weight of the complete ,guidance system is then calcul_ted by

WGs:Wtcp+W+_v+ _c

I!!I

This is the desired penalty function. The above equations are shown in flow

chart form in Figure 7.

Penalty Mode 2.--The probability of failing due to reliability or miss at

the target (PFTR) is calculated as in penalty mode i. The combined guidance

system probability of failure (PF_, the penalty of mode 2, is calculated by

including the probability of insufficient midcourse correction fuel (PFV).

With a total spacecraft weight (WT) and nonguidance spacecraft weight (WNG)the total guidance system weight is fixed,

WGS = WT - WNG

I!Iii

The midcourse system weight is assu_d to be

.. *T T T

wDV _ wGS - "ICP

Thus, the fuel weight is given by

WDV - KDC

WF = KD V

and the spacecraft mass ratio is

II

II

The delta-V capability is found by the rocket equation,

AV = Ig log(_)

27

SYSTEMPARAMETERS

PFR_i PFTR =PFR + PFT-PFRPFT-t

:_PROB(._T, DT)

PFTR

1IPFV .PFG -PFTR- I-PFTR I

RT

Dv

Rvi

WICP

L;_(PFv, DV) I"

l_,v;[,,v=q,vRvF,,v

PFV

](/.L-1)(WICP +WNG-I- K OC)

/.z - (/u.- 1)KDV

_WF

I'WDV=WFKDV + KDC I WDV 1-I

-[ WGS=WICP + WDV IWGS

PENALTY

FIGURE 7. CALCULATION OF PENALTY, MODE 1

28

I

I

I and the probability of insufficient fuel is

I PFV = Pr°b(_v' DV)

I where

IIIIIIII

The combined probability of guidance system failure is found by

PFG = PFTR + PFV " PFTRPFv

which £s the desired penalty, mode 2. The above equations are shown in flow

chart form in Figure 8.

Penalty Mode 3.--Penalty mode 3 is similar to penalty mode 1 in that the

penalty ks the effective weight of the guidance system (WGS). However, the

Lutal spacecraft weight (WT) is held constant under mode 3 unlike mode 1 where

the noaguidance weight was held constant.

The probability of failing due to reliability or miss at the target (PFTR)

is calculated as under mode i. Likewise, the probability of insufficient

fuel is obtained from

||

JIII

e_ _b PFTR

PFV = I - PFTR

which is used to compute the required midcourse delta-V capability by

Av= %%

where the number of traces (_V) is obtained from the statistical distribution

_V = *(PFv' DV) "

II

29

I

SYSTEMPARAMETERS

PFR

I!III

PFTR I

1 i,- T. v_ I

DT

m"

PFTR=PFR + PFT-PFRPF T I

PFT

PROB(_ T,DT) F--j

_qqs T= XMISS_T JDv

PROB(_V, DV) tPFV

I

I

RV

WICP

T_'v I_V=_ V_ /.L _ WT

_WDV =WGS-WICP JWDV-KDC I WFWDV :! WF =

i ii

T !I

FIGURE 8. CALCULATION OF PENALTY, MODE 2|

I

I

I

30I

I

With the delta-V requirement known, the mass ratio is found by

aV/ig_= e

!

ISince the total spacecraft weight is known, the required fuel weight may beobtained directly from

WF = W T .(_ - 1!

IIIIIIiIIIIII

The total effective guidance system weight is obtained by adding the midcourse

system weight to the weight of the inertial measurement unit plus computer pluspower source weight_

WGS = WIC P + WF% V + KDC

The equations for penalty mode 3 are shown in Figure 9.

Development of System Parameter Estimation Techniques

The system parameters which must be estimated for conceptual strapdown

guidance systems are those used in the system performance indices (penalty

functions). These parameters were discussed in the preceding section and are

listed in Table V. This section describes the techniques developed to estimate

the weight of (I) the inertial measurement unit, (2) the computer, and (3) the

power source. In addition, the estimation of the system reliability and the

accuracy analysis are discussed.

IMU Weight.--The strapdown IMU was broken down into the components shown

in Table IX. Data shown is for the United Aircraft Corporate Systems Center

(UACSC) strapdown IMU that was proposed for Centaur (Reference 9) and the

Honeywell, Inc. SIGN III IMU (Reference i0). The electronics, wiring harness,

etc., weigh 9.47 pounds for the UACSC design and 11.7 pounds for the Honeywell

SIGN III IMU. Based upon this information, it was decided that the weight of

the IMU electronics should be a user specified constant. A value of i0 pounds

is believed representative of the state-of-the-art electronics weight for an

IM]J having three gyroscope and three accelerometer rebalance loops. It should

be noted that the majority of the electronics weight is made up by the power

supply, conditioning, and sequencing as shown in Table IX.

The weight of the sensor assembly block (SAB) is estimated as follows.

Block weight is calculated by assuming each component (three gyroscopes and

three accelerometers) is placed in a separate solid block of material as shown

31

SYSTEMPARAMETERS

PFR

DT

RT

DV

J PROB(_/T, DT)q

J 9r= xM,s___iT

"1 RT /

-_ PFTR=PFR +PFT-PFRPFT ! PFTR

PFT "1= PFG- PFTR1- PFTR

RV

WICP

WF,

I WDV=WFKDV+ KDC Ij

J _/(I:'FV, DV) _ PFV-I,/,v II

,,v t

!

WDV

-_ WGS=WlCP + WDV iWGS

PENALTY

FIGURE 9. CALCULATION OF PENALTY, MODE 3

32

TABLEIXSTRAPDOWNIMUCOMPONENTSANDWEIGHT

IiIIIII!IIiI

Component

Gyroscope(s)

Acce lerometer (s)

Gyro(s) Torquing Loop Electronics

Accelerometer(s) Torquing Loop Electronics

Temperature Control Assembly

IMU Power Supply, Conditioning

and Sequencing

Timing (Oscillator and Frequency

Countdown)

Sensor Assembly Block

Base Plate

Cover(s) and Insulation

Calibration and Alignment Optical

Components

Interface Electronics

Wiring Harness

Connectors

Vibration Isolators

Vertical Sensing Element (Option)

Telemetry (Option)

Other

/

t

UACSC

CENTAUR IMU

Weight

4.89

I. 14

0.64

0.61

0.58

4.32

O. 20

7.60

6.70

8.77

0.51

3.12

0.20

Honeywell

SIGN III IMU

Weight

4.95

1.50

1.15

0.56

0.24

8.00

0.16

5.5]

6.10

5.85

O. i0

0.]5

1.20

0.25

0.46

0.05

TOTAL WEIGHT 39.28 36.23

33

in Figure I0 for cylindrical componentsand Figure ii for rectangular components.The volume of the solid block of material is calculated. From this volume issubtracted the volume of the component in the block. The remaining volume ofmaterial is multiplied by the density of the material making up the block tofind the weight of the block holding that component. The weight of the SAB isthe summation of the weight of each of the individual componentblocks.

I!i

For rectangular components, the equation for finding the volume of the

component block can be written as

VBLK = (FL[I] + SEP) x (D[I] + $2) x (W[I] + $2) - W(1) x FL(1) x D(1)

where:

VBLK = Volume of the individual component block

D(I) = Dimension along the I th sensor input axis

FL(I) = Dimension along the I th sensor output axis

W(I) = Dimension perpendicular to D(I) and FL(I) of I th sensor

SEP = Separation between sensors in assembled block

$2 = Twice the separation (SEP) between sensors.

For cylindrical components, the equation for finding the volume of the component

block can be written as

2 _ 2VBLK = (D[I] + $2) x (FL[I] + SEP) - _D(1) x FL(1)

The weight of the block in either case can then be expressed as

WBLK = VBLK x RHOBLK

where :

WBLK = Weight of the individual component block

VBLK = Volume of the individual component block

RHOBLK = Density of the material comprising the block.

For a given set of components, the block weight will always be estimated as the

same value no matter which me.chanical design is used.

The weight of the base, cover, and insulation is estimated as follows.

Analytic expressions are written which describe each of the possible critical

paths along each of the three dimensions which are used in calculation of the

volume of the complete IMU. Each critical path length is calculated and the

longest critical path in each direction establishes the dimensions of the IMU.

These dimensions are used in calculation of the area of the base, sides, and top.

I!IIIiIfiII!a!i

34

D(1)+2x SEP ---_

II1

j FL(1)+ SEP

i II

/

FIGURE i0. SKETCH OF CYLINE)RICAL COI_IPONENT IN BLOCK

--w(I )+2 x SEP-_I

] , FL(I)+SEPI I II I1 I

1 ill' I i

I I I LJ( _

FIGURE ii. SKETCH OF RECTANGULAR COMPONI_NT iN BLOCK

35

The thickness of the base, sides, top, and insulation are used with the appro-priate area to determine the volume of the base, cover, and insulationmaterials. Thesevolumes are then multiplied by the density of the materialused in each item to find the estimated weight of that item.

Figures D-I through D-9 in Appendix D depict nine designs for the casewhere the base of the IMU is mounted horizontally in the vehicle. Theanalytic expressions which are solved to determine the dimensions of the IMUfor a set of three gyroscopes and three accelerometers are presented in eachfigure. Similarly, Figures D-10 through D-15 in Appendix D depict six designsfor the case where the base of the IMU is mounted in a vertical plane in thevehicle. Restrictions such as limitation of that design to pendulous acceler-ometers are noted on the figures.

As an exampleof the procedure used to determine the dimensions of the IMU,horizontal design I shown in Figure 12 can be analyzed as follows. The criticalpaths which must be evaluated to determine the dimension are

I

A = W(1) + D(3) + D(6) + 2 x $2 + 2 x XBSE,

!!!II'|

III

A = W(1) + W(4) + D(6) + 2 x $2 + 2 x XBSE,

A = FL(2) + W(5) + D(3) + 3 x SEP + 2 × XBSE, and

A = FL(2) + W(5) + W(4) + 3 x SEP + 2 x XBSE,

where:

A

w(1)w(4)w(5)D(3)

D(6)

= Dime ns

= Dimens

= Dimens

= D imens

= Dimens

= Dimens

ion along one side of the base

ion perpendicular to the input axis of accelerometer 1

ion perpendicular to the input axis of gyro 4 (roll)

ion perpendicular to the input axis of gyro 5 (yaw)

ion aEong the input axis of accelerometer 3

ion along the input axis of gyro 6 (pitch)

FL(2) = Dimenslon along the output axis of accelerometer 2

$2 = Twice the separation between sensors

XBSE = Clearance between the edge of the SAB and the edge of the base

SEP = Separation between sensors in the assembled block.

The critical paths which must be evaluated to determine the B dimension are

IIIII

B = FL(1) + D(2) + $2 + 2 x XBSE,

B = W(6) + D(5) + 3 x SEP + 2 x XBSE,

36

•I" -/.= 3 -J

_ _. 0

III

jI_J

c.}0 _-

_r

x

qrQ.W

m

!

o Cg _0

,.--4

Z0

0

g

mz

r_A

r_q

b-4

o

o

,--4

37

B = FL(4) + SEP + 2 x XBSE, and

B = W(3) + $2 + 2 x XBSE,

whe re

B = Dimension along other side of the base

FL(1) = Dimension along the output axis of accelerometer I

FL(4) = Dimension along the output axis of gyro 4 (roll)

D(2) = Dimension along the input axis of accelerometer 2

D(5) = Dimension along the input axis of gyro 5 (yaw)

W(6) = Dimension along the spin axis of gyro 6 (pitch).

The critical paths which must be evaluated to determine the C (height) dimension

are

C = D(1) + $2 + XCOV,

C = W(2) + $2 + XCOV,

C = FL(5) + SEP + XCOV,

C = FL(6) + SEP + XCOV, and

C = D(4) + FL(3) + $2 + XCOV,

where

C = Height dimension of IMU

D(1) = Dimension along the input axis of accelerometer 1

D(4) = Dimension along the input axis of gyro 4 (roll)

W(2) = Dimension perpendicular to the input axis of accelerometer 2 _

FL(5) = Dimension along the output axis of gyro 5 (yaw)

FL(6) = Dimension along the output axis of gyro 6 (pitch)

FL(3) = Dimension along the output axis of accelerometer 3

XCOV = Separation between cover and highest point on SAB.

III

The longest critical path in each direction, A, B, and C, is determined fromdimensions of the inertial sensors selected for the IMU and the specified

clearances.

The volume and weight of the base are calculated as follows:

38

VBSE= A x B x TBSE,and

WBSE = VBSE x RHOBSE,

where

VBSE = Volume of the base,

TBSE = Thickness of the base,

WBSE = Weight of the base, and

RHOBSE = Density of the material comprising the base.

The volume and weight of the cover are calculated as follows:

VCOV = (2 x C x [A + B] + A x B) x TCOV, and

WCOV = VCOV x RHOCOV,

where

VCOV = Volume of the cover,

TCOV = Thickness of the cover,

WCOV = Weight of the cover, and

RHOCOV = Density of the material comprising the cover.

The volume and weight of the insulation are calculated as follows:

VINS = (2 x C x [Ax B] + A x B) x TINS, and

WINS = VINS x RHOINS,

where

VINS = Volume of the insulation ,

TINS = Thickness of the insulation,

WINS = Weight of the insulation, and

RHOINS = Density of the insulation material.

The total inertial measurement unit (IMU) weight is estimated by summing

the weight of (i) the inertial sensors in the IMU (three gyroscopes and three

accelerometers), (2) the sensor assembly block (SAB), (3) the base, (4) the

cover, (5) the insulation, and (6) the electronics plus any other weight which

may be considered constant such as vibration isolators. In designing a SAB,

access to the inertial sensors and center of gravity considerations must be

39

kept in mind. The conceptual designs in Appendix D are by no means firm designs

and are for illustrative purposes only.

Computer Weight.--The suggested means of estimating the strapdown inertial

guidance system computer weight was determined by analyzing state-of-the-art

computers such as the UNIVAC 1824, Honeywell, Inc. SIGN III, CDC 5360, and

Nortronics NDC-1051. The computer was broken down into major components and a

weight factor estimated for each of the components. Table X contains the

computer component breakdown and preliminary weight factors.

TABLE X

STRAPDOWN IGS COMPUTER COMPONENTS

AND PRELIMINARY WEIGHT FACTORS

r -- , i " ,

Component Preliminary Weight Factor

Input/Output (I/0) Logic

Modules

4.0 Ib/tray

Input/Out Tray Structure

Memory Module (s) --Memory

Stack and Electronics

1.5 ib/tray

5.0 ib/tray of 4,096 words, 18-24 bit Fe Core

4.0 Ib/tray of 2,048 words, 18-24 bit Fe Core

2.0 Ib/tray of 1,024 words, 18-24 bit Fe Core

Memory Tray Structure

Processor Module(s)--Control

Logic and Arithmetic Register

Processor Tray Structure

Power Supply and Timing

Assembly Module

Power Supply Tray Structure

1.0 Ib/tray

4.0 Ib/tray

1.5 Ib/tray

5.0 Ib/tray assuming 80-120 watts required

1.0 Ib/tray

1III

Computer Base

Computer Cover

Wiring

2.0 Ib

2.0 Ib

2.0 ib/2,048 to 4,096 words

Other

40

lII

The preliminary weight factors were estimated based upon a computer

organized in a modular fashion in which the tray structure of the modules are

assumed to form four sides of the computer case. The trays containing the

modules would be fastened together with the base and cover completing the

computer case. For an 8,192 word computer memory capacity, two memory trays

would be required and possibly an additional one pound allowed for wiring.

It is emphasized that the data in Table X is an estimate and based upon ferritecore memories.

I!I

Using the weight factors presented in Table X, an estimate of the weight

of an 8,192 word computer was made and is shown in Table XI.

TABLE XI

WEIGHT ESTIMATE FOR 8,192 WORD SRT COMPUTER

Component Preliminary Weight Est. (ib)

I/0 Modules 4.0

I/0 Tray Structure 1.5

Memory Module(s) i0.0

Memory Tray Structure 2.0Central Processor A.O

Power Supply 5.0

Power Supply Tray 1.0

Base 2.0

Cove r 2.0

Wiring 3.0

EST. TOTAL WEIGHT 36.0 ib

Since this technique is quite simple and the computer requirements vary

little from one strapdown IFFU to another for the same interplanetary mission,

it was felt that this weight calculation did not require a computer subroutine.

If a particular change in one of the computer modules such as the memory stack

and electronics is to be evaluated, the user should substitute his weight

estimate for that module in place of the weight factor in Table X. Since the

primary objective of this contract was not to develop a computer weight esti-

mating technique, development of a more sophisticated weight estimation

technique based on density of materials, type of memory, number of bits, etc.,

was not investigated further.

System Power.--Estimation of the system power requirements is dependent

upon the thermal design of the IMU as well as the computer. The guidance sys-

tem operating sequence also is a factor in the total system power requirements.

After determining the power requirements, consideration must be given to the

energy source which will supply this power. This section discusses

41

the techniques developed to determine the system power.

IMU Power.--The inertial sensors can be maintained at their designed

operating temperature by two methods. These are (I) use of the heater windings

on each sensor with the appropriate temperature control electronics duplicated

for each instrument, or (2) use of a sensor assembly block (SAB) heater with

its associated temperature control amplifier, bridge circuit, thermistor, etc.

If the former method is used, it is desirable to have a high thermal resistance

between each inertial sensor and the SAB. Use of the latter method requires

high thermal conductance between the inertial sensors and the SAB. Many of the

present-day strapdown guidance systems are heated by the latter method

(References ii and 12).

Heating of the SAB alone with no fine control of the temperature of each

inertial component is likely to cause some degradation in the accuracy of

floated instruments since it is probable no two instruments will float at

neutral buoyancy at exactly the same temperature. This degradation in accuracy

is offset somewhat by a slight reliability gain and weight decrease achieved

by elimination of temperature control electronics for each sensor.

The primary concern in thermal control of the IMU should be to minimize

the total system power required. The power (heat) flow of the IMU can be

expressed by the equation

QR + QC + Qexc = QS + QL

where:

QR-- is the sun,nation of direct and reflected solar radiations, direct

and reflected earth albedo radiation, and direct and reflected

earth thermal radiation on the IMU,

QC

Qexc

QS

QL

Furthermore,

-- is the power to the IMU from the temperature controller,

-- is the power into the IMU from the excitation of the wheels,

signal and torque generators, electronics, etc.,

-- is the rate at which heat is stored in the IMU, and

-- is the lost or dissipated power due to radiation and conduction

from the IMU.

QL = KI(T - Ta)

IIIIII

42

where (T - T a) is the difference between the temperature of the inertial

sensors on the SAB, T, and the ambient temperature, T a. KI, the thermal

conductance between the IMU and the vehicle, can be calculated for QC = 0 when

T a is at its maximum value, T . In this case,a

Qexc

KI -- T - T (watts/° F)amax

IIIIItI

If a material with such a value of thermal conductance could be found and used,

no cooling of the IMU would be required. If the average ambient temperature is

much lower than the maximum ambient and if the maximum ambient occurs for only

a short time period, consideration should be given to setting the thermal

impedance to minimize the total temperature controller power. This involves a

tradeoff between the heat sink weight (thermal impedance set for less than

maximum ambient temperature) and the ext['a power source weight (thermal

impedance set for no cooling at maximum ambient temperature) when the average

ambient temperature is much lower than the maximum. The computer program

containing this IMU power estimation technique allows the user to load a user

selected value for K I or will calculate K I for QC = 0 as in the previous

equation. Also,

Q'I"

QS = K2 d-_

where K 2 is the thermal storage constant and has units of watts/(°F/sec). It

is presently assumed that dT/dt, the rate of change of the sensor operating

temperature with time, is zero. Further discussion of temperature control

problems for inertial components can be found in References 13 and 14. K I and

Data on T a as a function of time along the trajectory are needed to

perform a precise thermal (power) analysis of the IMU. If these data are not

available, an estimate of the average T a can be made by the user as was done

in the exercising of this program. Of course, QR and T a depe1_d upon the loca-

tion of the IbRJ on the vehicle. Since many possible tradeoffs e×ist Jt_ Li_ermal

control of the IMU, this should by no means be considered as other thai_ a tool

for estimating the IFRJ power. The user should keep in mind that Ll_e value of

K I he selects,should he elect not to have QC = 0 at .Ta , is quite imporLat_t.max

In summary, the IMU power is currently estimated by (I) caLcu]atin_ o[

specifying a thermal conductance, KI, (2) assuming an average ambient tempera-

ture, Taa , and (3) assuming an IMU operating temperature, Top. Expressedanalytically, the IHU power is

PIMU = Kl(Top - Ta )ave

43

Computer Power.--The power being supplied to the computer is that power

required to operate the various modules such as input/output plus that dissi-

pated in the power supply. The computer does not usually require maintenance

of a constant temperature. From a thermal viewpoint, the designer is princi-

pally concerned with dissipating enough of the heat to prevent a temperature

rise over the designed operating temperature range of the computer. Since a

precise power and thermal analysis depends upon the design of a specific

computer which depends upon its use, we are principally concerned only with

estimating the power required to operate the computer. Therefore, the logical

approach is to sum the estimates of the power required by each of the modules

making up the computer. These are shown below in Table XII. For example, if

a change from one type of memory to another is made, the power estimates should

be checked and revised as required to reflect the requirements of the new

memory type. Similar practice should be followed for the other modules.

IIIIIII

TABLE XII

STRAPDOWN IGS COMPUTER MODULES AND POWER REQUIRED

ModuleExample of Preliminary

Power Estimate

Input/Output (I/0) Logic Module(s)

Memory Module(s) - Ferrite Core, 8K Stack

and Electronics

Processor Module(s)

20 watts

20 watts

15 watts

Power 55 watts

To find the total computer prime power, divide the summation of the power of

the three functional modules in Table XII by the efficiency of the computer

power supply module. That is:

Computer Prime Power _(I/0 + Memory + Processor) Power= Efficiency Power Supply

For the example above, a typical efficiency of 60% would give:

Computer Prime Power -55

0.6092 watts

I

I

I

I

44

II!I

IIIIIIIIII

IIIII

Since the I/0 module power depends upon the electrical interfaces of the

IMU, telemetry, etc., with the computer, precise power estimates cannot be made

until this information is available. Therefore, a subroutine to calculate

computer power was not written. Instead, the user is required to input actual

or user estimated computer power as data. If the user of these techniques does

estimate the computer power required from the guidance power source, he mustkeep in mind the significance of the I/0 interfaces.

Estimated Power.--The guidance system power required from the power source

is simply tile summation of the IMU power and the computer power. As currently

used in the computer program implementing these techniques, the average system

power is found by assuming that the system operates continuously from launch to

the midcourse correction burn in an average ambient temperature. Expressed

analytically, we have

PIC = PC + Kl(Top - Ta )ave

where:

PIC

PC

K I

= Average power required by the IMU and computer,

= Average p_ver required by the computer,

= Thermal conductance,

Top

Taave

= Operating temperature of the IMU, and

= Average ambient temperature over the trajectory.

Gu]dam-o P+_wor N_h_ys_-=m= .-An ...... ,- +-1........ c_. _ p_W_'r bubsystems

available for production of raw electrical power: chemical fttelm, solar energy,

radioisotope thermal energy, and nuclear fission energy (References 115 through

22). Chemical subsystems (hydrogen-oxygen, hydrazine and solid propellants,

fuel cells, batteries) provide energy limited sources whose performances are

time dependent. Primary batteries, for example, serve quite well For

relatively low power requirements (up to about i00 watts) for aromld one to

three day mission. Chemical dynamic systems using storable propellants (I12-02,hydrazine) can produce higher power than primary batteries but ,_how about

the same time dependency characteristic. Fuel cells using hydrogen and oxygen

as reactants extend the duration period to three or four weeks for power levels

up to about 1-2 KW (e.g., Apollo Command and Service Module). When the energy

conversion unit limits the available power (as opposed to energy source

limitation) the subsystem is categorized as power limited. Solar energy con-

version is an example of a power limited subsystem and has been utilized

through the application of single-crystal silicon cells since 1960 (Tiros).

As of today, the solar cell is the dominant source of power in Ion_-lived space-

craft (i.e., in excess of one year in space) requiring 0.5-1 KIn. Radioisotopes

45

are another example of power limited sources. Power levels of a few kilowatts

lasting in excess of a year are available through proper use of the radioisotope.

Its big advantage over the silicon solar cell is that its output does not vary

with distance to the sun. Nuclear reactors with varying conversion units are

being produced under the SNAP (Space Nuclear Axiliary Power program to provide

higher levels (i.e., i0-I00 KW or more) for durations well over a year.

III

Between the source and the load there is a power conversion device des-

cribed as a cell, dynamic, thermoelectric, or thermionic type. These converters

are applicable to the various sources (except for the chemical direct conver-

sion) so that the choice of system concepts for a given requirement is quite

broad. However, performances of the different combinations are not separated by

thin, well-defined lines found on a plot of power levels versus mission duration.

The problem of this "overlapping" effect is that the choice of a suitable power

subsystem is often far from obvious, and, as one would begin to suspect,

details of the specific application are needed for a satisfactory solution.

When the details of the specific application are known, a suitable power

subsystem can be selected. Factors such as the time of operation of the power

subsystem, its specific power or specific energy, i.e., watts/ib or

watt-hours/ib, and any constant weight factor are used in determination of the

principal item of concern, the guidance system power subsystem weight. It is

this weight which is dependent upon the guidance system power requirements and

is used in the penalty functions as a factor in the total effective guidance

system weight.

An example of the guidance power subsystem weight determination result is

given in a subsequent section of this report. In this example, the selection

of the power subsystem was based upon mission analysis. The weight in this

specific case was determined by solving the following equation:

Wp = Kpl + Kp2PIc

where Kpl is power source constant weight, Kp2 is the reciprocal of power

source specific power density, and PIC is the average power required by the

IMU and.computer.

Reliability.--Several possible reliability descriptions were examined.

The Weibull distribution was chosen to allow more realistic estimation of

failure probabilities without the large amounts of data needed for more complex

distributions such as the Markovian. The Weibull distribution is especially

attractive as it contains the more cormnonly used exponential model as a special

case (References 23 and 24).

The Weibull distribution assumes a hazard function with a failure rate

proportional to a power of time or,

IIIIIIIIIIIIIII

46

Ps (t)/P (t) =-_.t _-Is

which yields the probability of success,

.%t _P (t) = es

which for a = 1 is identical to the exponential distribution.

The expected life or mean time to failure (MTTF) is obtained by

MTTF = S ° tP s(t)dt

o

which for the Weibull distribution gives

III

IIIII!I

I/_,

I_° .ita. -- (lj r_(1 + i/_.)MTTF = , te d tLJ " A." I " -

0

I

where I (x) is the Gamma function. Since MTTF is a more meaningful parameter

than the arbitrary %, the following substitutions will be made.

Let _(,_) = [_(1 + 1/o.),

T = MTTF,

= (_(_.) IT)

P (t) = e (_('_)t/T)C_ ands

PF(t) = i Ps(t).

The form of the equation for Ps(t) is convenient as it contai,_s t_me

normalized to NTTF. The function _(c_) is between approximately 0.88 a,d 1.0

for all _ greater than 1.0. Figure 13 shows t/T for several values of

probability and _.

Figure 14 sllows the effect of increasing _. For _ larger tl,an L.0,

failures are more likely to occur near the _FfTF. In the limit,as _ approaches

Ln_:inity, all units would fail at e_actly their _TTF. Obta[,_i1_g da_a for "

will involve special test procedures. However, as stated above, assumi[_g _

47

c

;D

c)

e:

!

c _

C-

¢_

c_

.I

.J o*..4 o

tJ_0

_,- c)I,.- o

J _...t,4 •

._[

"D o

"l o

_,_. _ _._: _ _.._. : ":: :_.

'_ ,_ ,TJD _ :"1 '.D _ _X, _ ,'_,N '3" IM LJ_ _JO --_ t'M C_:.D

C ............ _ :'M._ fM r"),'_'l -_i 4 Jr) 4": ..IL"

r-.. _ _, ._ ,_ ,D -,"3 LD _" '_ "_0 .4)':0 ,--_ _%l t_'l ."M t,,1

_'_ t_ r_l :£ _ i...- 3_ r... c_l _1..xj ,,..._ e_..1_ ...,,.. o Oh 1D ,o

3E, "JO .£_ "JD £,, "Y, ..7. _ '_t, _ _', _e, _I', _J-. :3_

_. >,_, _. _. >.> '_.>,>.>._ _. >._.

_" _D f"- l_.-,D .I" .t ,'_ m'1_l

_0 000_000_

"22"--2"'.:.

_ _- c', :,q ,D .D._ _D ,,1"f_ _ "_. "M "_- O, ,_ 1=. _ .if"

-- .... Z--_

• • . • • • . • , . e • . • * . • • • • * • • , * . • . • • • • • • • . • • • • • • • • •

oe.

icL

J

|

i

1-

o._J

48

• • . o •

,JM

0

F*-

_B

U,.I

Ud

Z

_LC'

r.=l

a_i-4

i,-,I

I--I

!IIIIIIIIIIIIIIIIIIlII

IIII

eJnl!o_ _o _!l!qoqoJcl

49

equals 1.0 yields the more familiar exponential distribution. This assumption

should be made when better knowledge of _ is not available.

The probability of failure due to reliability for the entire guidance

system is computed with the distributions and operating times shown in

Table XIII.

!II

TABLE Xlll

RELIABILITY ESTIMATION

Component Distribution Operating Time

III

Acce ler ome ters

Gyroscopes

Weibull

Weibull

Time of thrust(s)

Time to midcourse

Computer

Electronics

Energy Source

Weibull

Exponential

Exponential

Time to midcourse

Time to midcourse

Time to midcourse

Midcourse System Fixed probability of failure ..........

Accuracy Analysis.--In the evaluation of any of the three penalty functions,

four system parameters related to accuracy are needed. These are the square

root of the trace and the degrees of freedom for each covariance matrix

describing midcourse delta-V and errors at the target. Additional information

in the form of standard deviations or covariance matrices of other error

quantities may be of interest but are not needed for penalty evaluation.

Nine components of error are of interest.. In general these are split into

a set of six position-velocity errors and a set of three attitude errors. All

errors are described in the local vertical coordinate system defined in

Figure 15. Attitude error components are small angles about each axis.

In_ection _rrors.--The injection errors are obtained by multiplying the

vector representing hardware (accelerometers, gyroscopes, and the computer)

errors by a matrix produced by the Strapdown Error Analysis Program (SEAP)

supplied by NASA/ERC. SEAP obtains the injection error sensitivities for each

source by integrating the navigation error equations with the appropriate

forcing function for the error model.

IMU Error Models.--Appendix E of this report contains a derivation of the

error model for the single-degree-of-freedom gyroscope, and Appendix F contains

a similar derivation for the pendulous accelerometer. Of the error sources

50

VELOCIT_VECTOR

POSITION

PLANE

OP

DOMINANTGRAVITYCENTER

THE THREE ORTHONORMAL COMPONENTS ARE:

DR DOWN RANGE (THE DIRECTION OF VELOCITY)

CR CROSS RANGE (NORMAL TO VELOCITY, IN THEP-V PLANE)

OP OUT-OF-PLANE (COMPLETES THE SET).

FIGURE 15. LOCAL VERTICAL COORDINATE SYSTEM

51

listed in Table E-I of Appendix E, those listed in Table XIV are not presentlycontained in SEAP.

For error analysis we are primarily concerned with the uncertainty ineach of the error coefficients. If the error coefficient is a deterministicvalue that can be compensatedwith zero uncertainty, then it could be con-sidered to makeno contribution to the navigation errors.

TABLEXIVSINGLE-DEGREE-OF-FREEDOMGYROSCOPEERRORSOURCESTOBE ADDEDTOSEAP

!IIII

Symbol Forcing Exc ita tion Nomenc lature

S ioo_H [8;M]

DKSI

DKIS

DKSO

DKIO

DUO

IO___A

H

(IIA ISA)

H

@ or _IA

2

aI

2

as

aoa I

aoa S

aO

wOA

Rebalance loop scale factor - includes

signal and torque generator, loop

transfer function, and wheel speedmodulation errors

Spin axis compliance drift

Input axis compliance drift

Spin axis compliance drift

Input axis compliance drift

Dump term per MIT Instrumentation Lab.

Output axis inertia (OA angular

acceleration error)

Anisoiner tia

_SA 8 or WiA Spin axis cross coupling (kinematic _

rectification)

E Elastic restraint

Coning

Quantization

52

Table XV lists those pendulous accelerometer errors sources in Table F-Iof Appendix F which are not presently contained in SEAP.

System errors can exist due to the effects discussed below:

(1) Gyro loops with different gains and phase shifts (time constants) at

a specific vibration frequency cause errors in the computation

process. If the net drift from this effect is expressed as

wd = _ w (_ - T )• r r,3 'j+l j+2 gj+2 gj+l

j = 1,2,3,1,2 ....

then the effect of instrument quantization, AS, can be evaluated by

substituting the best estimate for the time of occurrence of AS,A8

2w for T This error has been referred to as pseudo coningr. gjJ

(Reference 25), fictitious coning, or commutation error (Reference 26).

IIII

iIIIIII

(2) Angular vibration about the input axis of one gyro is also an angular

vibration about the output axis of another gyro. A pseudo coning

effect results when the computer processes the signals due to this

effect (Reference 25).

(3) Accelerometer and gyro loop with different gain and phase shift

characteristics at the same vibration freqency cause the computational

process to introduce an acceleration error when subjected to a

coherent linear and angular vibration. This error has been called

pseudo sculling. (Reference 25).

(4) The separation of the accelerometer axes in a system produces a

_y_tem _ize effect error which cou]d he partially compensated=

The uncertainty in the computed velocity and position due to this

effect should be considered in the system error analysis.

Computer Error Models.--The errors due to computer parameters (computation

frequency, number of bits, and integration scheme) were studied. Reference 27

formulates the problem as shown in Figure 16. This is the same approach that

is applied to hardware error models in computer programs such as SEAP. The

problem is r_duced to describing instantaneous generated error _ates _ as a

function of U and X where U is the vector o_ sensor inputs and X the state of

the system.

Reference 27 solves for _(t) by defining a matrix function Z by the

differential equation

IdZ(t,s_ _[_I _ 1 [ )] i idt = (t,s_ (t e s); Z(t,s = I (t = s)

53

54

[_

::> r,.)

_o

Z_

r._m

o_t_

o

oZ

o.r.4

.IJ

.,-40

o

o

,-.4o

..o

o

oo

°_N

_., o

0

o

4.1

0

h-I

o,.-¢

o

,._

.oo_

o

oo

o

o

o

o

o

!

v

+

_ _o

L v | ]

"1' _

_J

o

N-4

u

I

_z

ov

+

%ov

(J}

u

c_

I>,"o

4-J

o

_j .M

_ t_ _

,-4 .iJ

c_CY_

ou

IIIlIIIIIIIIIIlIIIIIII

I

I

I

.-4

u(t) ] x(t)>

(a) Ideal System

IIIII

_(t) J

I

>

= _(_,U) + 7(_,U) where ¢ = Generated Errors

_(t)>

(b) Real System

iIIII

,U:

_(t)

• _[e = e(X, + X,U) . e

e(t) >

(c) Error Model

FIGURE 16. FORMULATION OF COMPUTER ERROR MODEL

55

then

t

e(t) = Z(t,o)! _(0)+ IZ(t,s) i ¢(s)dsL ; o' ]

III

where Z(t,s) is the state transition matrix mapping errors at time = s to

t imp = t.

To maintain compatibility with SEAP, the instantaneous error generation

rate _(X,U) must be expanded into the form

¢. : T.q..K.l 13 j

where

_°

l

.th= i generated error rate component,

S.. = SEAP sensitivity, and13

Ko = jth error coefficient.]

The SEAP sensitivity is a function of the instantaneous environment, X, U,

supplied to SEAP as trajectory information, and the error coefficients are

functions of the computer parameters.

Simulation of several integration schemes were made to investigate possible

methods for expressing Sij and Kj for both GP and DDA computers. The formula-tion of Reference 28 is as good as possible while maintaining compatability with

SEAP. Computer errors due to three sources are included in the model:

(i) Constant rate truncation

(2) Round off

(3) Vibration.

Constant rate truncation results in an equivalent instantaneous drift rate

in degrees per hour of

(a) 36.5 w_/f (Rectangular Algorithm)i C

3 2(b) 0.133 w./f (Runge-Kutta - 2nd Order Algorithm)

1 e

(c) 4.62 x 10 -6 w_/f 4 (Runge-Kutta - 4th Order Algorithm)1 C

IIIIII!II

IIII

56

IIIIlI!lIII

IIIII

where fc is the attitude update computation frequency and w i is each component

of body angular rate.

These equations have been separated into the time variation of w to the

proper power as a forcing function on attitude errors in SEAP. This handles

the generation and propagation as described above, on a per unit basis. The

proper SEAP sensitivity is then multiplied by the proper I/fc N as an error

coefficient where N is the power dependent upon the integration scheme as shown

on the previous page.

Round off is expressed as a drift uncertainty in deg/hour of

2.1 x 105 f /2 NBc

where NB is the number of bits in the computer word or words (if multiple

precision is used) used to store the attitude matrix. Since the drift rate is

not a function of time, it is treated as three identical additional error

coefficients to be multiplied by the three drift sensitivities generated by

SEAP for gyro fixed drift.

Vibration is treated in a similar fashion in Reference 28 with the drift

uncertainty for angular vibration given as

-fo

4--_-117_2f-_o _ _2_f-cT) s in2(T)T2 dT(Rectangular Algorithm)

_fo

0.0985 r2f-- e2 _2fc!_-- sin4-T-( ) dT

16_2f Jo _ - \ _ J T 4c

"rrfo

16_2f o T 4c

(Runge-Kutta - 4th Order Algorithm)

where _ is the spectral vibration density passed to the computer by the gyro.

It has been assumed that _(f) is a constant spectrum _ for 0 _ f _ f •o

A summary of the computer error contributions is shown in Table XVI.

I

I 57

I

TABLE XVI

COMPUTER ERROR MODEL

Error Error Coefficient

Type Associated with a Computer

Sensitivity Mapping to Pos.,

Vel.,or Attitude Errors

III

Roundoff X

Roundof f Y

Roundo ff Z

Vibration X

Vibration Y

Vibrat ion Z

Trunc at ion*:

NB = # of bits

2.1 x 10 5 f /2NBc

Equation 14 a, b, or c

llfc

2I/f c

llf 4C

RE

Ry Repeated

RZ Gyro

iRE Fixedb

i

S Ry Drift

Rz Terms

RECT

RUK2 New SEAP

Terms

RUK4

III!III

fc

= Attitude computation frequency

* A single computer is given only _he appropriate integration scheme dependent

truncation coefficient. The other two are zero.

State Transition Matrices.--The state transition matrices necessary to

propagate the velocity and position errors from injection to other points in

the flight were obtained from a modification to an n-body integration program

supplied by NASA/Lewis Research Center. The Lewis n-body program was modified

to produce the state transition matrices by the following technique. After

integration of the nominal trajectory, six additional trajectories are

integrated, each with one of the six initial conditions perturbed by a small

amount. The perturbations were set to 10 -6 times the radius magnitude for

position components and 10-6 times the velocity magnitude for the velocity

components. Matrices are then obtained for many points along the trajectory

by subtracting the nominal trajectory state vector from the perturbed trajectory.

Tij(t) = Xij(t) - Xio(t)

IIIIII

58

I where

!

!!

!

1

!

1

!

!!

|

!

Tij(t ) = an element of the state transition matrix from time = 0to time = t,

Xij(t) = the ith component of the state vector at time t with thejth component of the initial vector X(O), perturbed, and

Xio(t) = the ith component of the nominal state vector at time = t.

Other perturbation ratios ranging from 10-4 to 10 -7 were tried and the

resultant matrices were equivalent out to the sixth decimal place. The per-turbation of i0 -v is small enough to avoid errors due to nonlinearities and

large enough to avoid noise due to finite computer word length. The program

uses two 60 bit words for each state variable when run on a CDC 6400 computer.

A state transition matrix between any two points may then be found by

where

[T(a,b)l = state transition matrix from time = a to time = b,

I_ _I(0,a = state transition matrix from time = 0 to time = a, and

_(O,b)] = state transition matrix from time = 0 to time = b.

Attitude errors are propagated by the identity matrix. However, during

coasting portions of the flight (e.g., injection cutoff to midcourse correction)

additional attitude errors are generated due to gyro fixed drift and computer

errors. Under the assumption of small angle attitude errors, the generated

errors are approximated by multiplying the drift rates by the elapsed time.

The resultant attitude transformation becomes

! ea(tb) = _a(ta) + (tb - ta)

I where

!

!

--9

e (t) = attitude error vector at time t , anda i i

= fixed drift error vector.

!

! 59

!

When handled statistically, the attitude propagation becomes

where

T

iCOVa(ti) = the three by three covarlance of attitude errorsat time = t., and

l

_D = the three by n rr_trix describing n fixed drift gyroscope"( and computer error sources.

Midcourse Correction Determination.--Midcourse correction delta-V may be

determined from the position and velocity errors at midcourse and the state

transition matrix mapping errors at midcourse to errors at the target. Theerrors and matrices of interest are defined in Table XVII.

!l!!!!!

TABLE XVII

MIDCOURSE CORRECTION DEFINITIONS

Symbol Definition

III

[A]

em

et

The 6 by 6 state transition matrix from midcourse to the target

The 6 element error vector prior to midcourse correction

The 6 element error vector at the target

em

xp

The 6 element error vector after midcourse correction

The 3 element position error vector subset of any 6 element error

vector ex

exv

The 3 element velocity subset of ex

The midcourse correction vector

, _ ', =.

Midcourse correction velocity is obtained by determining the required

velocity (for desired target miss) after midcourse (emv) in terms of the erroEs

at midcourse (Em) and computing the difference,

60

IIII

-_ e "_mv mv

Since errors have been assumed linear and zero errors produce zero midcourse

delta-V, the required velocity just after midcourse may be expressed by

I

e [c] -'= emv m

and the correction becomes

_v: [[_]-[oli]! ;_

II!IIi

where

[C] - is the 3 by 6 matrix mapping position-velocity errors prior to

midcourse to required velocity after midcourse,

I - is the 3 by 3 identity matrix, and

0 - is the 3 by 3 zero matrix.

A matrix [F] (3 by 6) may be defined to be the matrix yielding A_ from

errors prior to midcourse by

IF] = [C] - [01I]

The problem is then to obtain [F] in terms of [A] to meet miss requirements at

the target.

IIIII

Two types of target miss requirements hage been studied. TheY are:

(i) Zero all position errors at the target, and

(2) Zero one position or velocity component error at the target.

When making a midcourse correction, three variables may be specified (three

components of delta-V). Thus, three conditions may be met at the target.

Zeroing all position errors specifies three conditions. However, zeroing any

one component at the target specifies only one condition. In other words,

there are an infinite number of possible corrections which will zero one

component of error at the target. In this case, it is desirable to make the

correction that uses the least fuel (minimum magnitude of delta-V).

61

I

I

Zeroin_ All Position Errors.--The state transition matrix, [A], may be I

partitioned into four 3 by 3 submatrices giving I

LetvJ_, _2@Le_vJ • I

For zero position errors at the target

111V

or expressing e i in terms of e ,my mp

I

Thus , e =mp mp

el =mv

I

I

I

Ior

The matrix [F] mapping em

to _V is then

I

I

I[FI = [C] - [01I]

or

yielding the _V in terms of error prior to midcourse.

III

62

I

I

I

I

I

I

I

I

I

I

I

The required midcourse delta-V is then obtained by multiplying the errorsat midcourse by the matrix F.

Zeroing One Component of Error.--The state transition matrix A may be

partitioned into a set of six row vectors, each vector representing a row of A.Thus,

i .

e -_'=i . et I • m

La6

and any component of target error becomes

#

7e = • e

ti i m

To zero the ith component of error at the target

J

70 = • • e

1 m

Breaking the 6 element vector a. into its position and velocity subsets and-'_e I.

substituting for em the error prlor to midcourse and delta-V yields

or

0 = a'. • e_ +a. . e_ +a'. . AV•p mp _v mv _v

I

I

I

0 = a. • e + a. • AV1 m iv

which defines delta-V to lie in a plane as shown in Figure 17. The problem is

now to find the _V with minimum magnitude which is the normal to the plane that

passes through the origin, or

_V

aiv <7 i " _m)

(7iv 7iv)

63

64

FIGURE17.

m_

of A¥ to zero.

o componentof llrgeterror.

Normol to plone

_Minirnum IAVi to zero

o componentof torget error

DR

MINIMUM FUEL DELTA-V TO ZERO ONE COMPONENT OF TARGET ERROR

IIIIIIIIIIIIIIlIIIIIII

which when expressed as a matrix becomes

IIII

AV: [F] em

where an element of F, fjk' is given by

= a a ifjk ii i+3 ,k

(ai42 2 a 2)+ ai5 + i6

with i equal to the index of the component to be zeroed and a:k equal to the

components of a ip (k = i to 3) or air (k = 4 to 6). It shoul_ be noted that

the zeroing of any one target miss component is completely general and any

position or velocity component may be zeroed. However, zeroing of the cross

range (CR) position error is of the greatest interest. At a time of nominal

periapsis the nominal position and velocity errors are perpendicular. The

CR component of error is in the direction of the periapsis vector, and

zeroing this component is equivalent to zeroing the error in periapsismagnitude.

Uncertainty in Midcourse Correction.--If a perfect impulsive delta-V

correction is made according to either scheme discussed above, no target error

occurs. That is, either all three position errors will be zero or the desired

component will be zero. However, the midcourse correction cannot be made

perfectly. Errors will occur in aiming the vehicle for the midcourse burn and

the magnitude of the correction will be in error due to imperfections in its

measurement. The inaccuracy in making the midcourse is calculated by the

techniques described in Reference 29 and shown below.

2 2

" a -_ -_ _Ua ii"ov ': [cov ([cov

where

iCov (e ) = the covariance matrix of the corrected velocity,VC

-_ = the covariance matrix of the midcourse correction delta-V,iC°v. (edv)

U = the relative uncertainty in midcourse correction magnitude,m

Ua

= the uncertainty in midcourse angle,

65

tr = the trace of a matrix, and

I = the identity matrix.

The relative uncertainty in midcourse correction magnitude and angle are either

loaded as data or calculated by the program.

The uncertainty in midcourse magnitude (Um) may be approximated by assumingthe roll axis accelerometer is used to measure the thrust acceleration of the

midcourse correction and terminate the thrusting when the integrated accelera-

tion equals the desired velocity magnitude. The bias, scale factor, and

nonlinearity error terms for the roll axis accelerometer are used in this

calculation as shown below.

2;.-4 I

2 iedv ! 2 2 2U - I-_ 1j2 = (K0/a) + (El) + (K2a)m IAV I

where

a = midcourse correction acceleration =Thrust

Mass and assumed constan%

IIIIIIIII

K0 = bias uncertainty,

K 1 = scale factor uncertainty, and

K 2 = nonlinearity uncertainty.

The midcourse correction angle uncertainty (Ua) may be approximated by

the square root of the trace of the propagated attitude angle covariance at the

time of the midcourse, or Ua may be arbitrarily set to reflect an attitudeerror update.

Errors at the target may be obtained by propagating the errors in making

the midcourse delta-V correction to the target.

IIIlI

Digital Computer Programs

The calculation of the three penalty functions and the necessary estimation

of the system parameters have been coded into a deck of FORTRAN subroutines.

The subroutines, with a short, simple, user written main program may be used to

evaluate an arbitrary guidance system using specified hardware components.

Also, an optimum system (minimum penalty) may be found from a catalog of possible

components.

IIII

66

IIIIIIIIIIIIIIIII

Program Data.--Data needed to run the program consist of the following:

(I) Injection error sensitivities as computed by the Strapdown Error

Analysis Program (SEAP)

(2) A set of state transition matrices as generated by the Lewis n-bodyProgram

(3) Data describing the hardware components (accelerometers, gyroscopes,and computers).

(4) Data describing the mission and spacecraft subsystems.

Accelerometer and gyro data needed are:

(i) We ight

(2) Excitation power

(3) Mean time to failure (MTTF)

(4) Alpha (Weibull constant)

(5) Length

(6) Diameter or height

(7) Width (rectangular components only)

(8) Error coefficient uncertainties

(9) Temperature dependent error coefficients.

Computer data required is similar to that for gyros and acceierometers except

the number of bits used to store each element of the attitude matrix, attitude

update frequency, and integration scheme (rectangular, Runge-Kutta 2nd order,

or Runge-Kutta 4th order) are used in the error model. The spacecraft and

mission parameters needed for penalty function analysis are set internally to

standard values unless the user specifies an alternate value.

A guidance system is defined in the subroutines by indicating the seven

components (three accelerometers, three gyros, and a computer) by seven indices.

The indices refer to the components location in the candidate hardware catalog.

A report such as the one shown in Figure 1 is printed under any of the three

penalty modes.

Additional output in the form of tables or graphs of penalty variation with

any hardware, spacecraft, or mission parameter may be obtained. Examples of

these tables and graphs are shown in Figures 2, 3, and 4.

III

Options are provided to allow specification of the type of midcourse

correction (zeroing all position or any one component of miss at the target),

mechanical orientation (horizontal or vertical), mechanical configuration

67

(one of the stored designs or the one yielding minimumweight), midcoursecorrection accuracy (specified or computed), thermal conductance (specified orcomputed), and an optimization option indicating if the gyros and accelerometersmust be identical or be of differing types in the three axes.

Optimizing Algorithm.--Optimum systems are found by a method similar to a

steepest descent technique. The search algorithm makes a step from one system

to another by making the single substitution that produces the least penalty.

If identical components are required, all three gyros are changed together and

likewise for all three accelerometers. Table XVIII is an example of a hypo-

thetical case of one step in the search algorithm, with three candidate

accelerometers (numbered i-3), three candidate gyros (numbered 21-23), and three

candidate computers (numbered 41-43).

TABLE XVIII

EXAMPLE OF SEARCH ALGORITHM

IIIIIII

Remarks System PenaltyAccelerometers Gyros Computers

Present System i 21

Changing 2 21Acce lerome te rs

41 I00

41 95 Best New

Accelerometer

3 21 41 97

Changing Gyros 1 22 41 96 Best New Gyro

i 23 41 98

Changing i 21 42 96

Computersi 21 43 94 Best New

Computer

New System I 21 43 94

IIIIIII

There is no better system that differs from the original system by only

one substitution. Each step requires many evaluations of the penalty function.

For identical components

Number of Penalty Evaluations = N + N + N ,a g c

III

68

and for mixed components

IIIIIIIIII

Number of Penalty Evaluations = 3N + 3N + N ,a g c

where

N = the number of candidate accelerometers,a

N = the number of candidate gyroscopes, andg

N = the number of candidate computers.C

This is similar to a search in three dimensional space (seven dimensional

if mixed gyros and accelerometers are permitted). However, the dimensions are

arbitrary catalog indices and only di_c_=te _ +.... _v_n_s exist in the hyperspace.

Changing a component represents motion in one spatial direction. However,

since the indices are arbitrary, any number of steps in a direction are of

equal interest. Steepest descent usually involves testing one step in each

direction, then moving in some direction combining the results of the tests.

In this technique all possible steps in each direction are tried, then the

change is made in only the one direction to the new point with the least

penalty. In sunnnary, this algorithm changes only one component, never moves

to an untried location, and never moves to a location with a larger penalty

value. Other search algorithms have been tried but either failed to converge

or resulted in long running times. Finite enumeration (trying all possible

combinations) showed some promise. This approach would find the absolute

optimum in the following number of evaluations of the penalty:

Number of Penalty Evaluations = N N Na gc

for identical components, and

Number of Penalty Evaluations = N3N3Na g c

III

for mixed components. For most searches the algorithm must be applied about

ten times. For equal numbers of candidate accelerometers, gyroscopes, and

computers, Table XIX gives a comparison of the number of penalty evaluations

for the algorithm and finite enumeration. This shows that finite enumeration

results in excessive running time for more than simple problems.

69

TABLEXIXCOMPARISONOF THESEARCHALGORITHM

TOFINITE ENUMERATION

Numberof ComponentsN = N = Na g c

Mixed IdenticalAlgorithm* Enumeration Algorithm* Enumeration

5 350 78,125 150 125

I0 700 i0,000,000 300 1,000

15 1,050 (15)7 450 3,375

* Assumesten iterations to find optimum.

The algorithm has been coded into a separate subroutine (SYSNEW)so it maybeeasily replaced with any other algorithm that finds a new seven element set ofcomponent indices from a given set.

Search Housekeeping.--Another subroutine (SYSOPT) initiates the search,

repetitively calls the algorithm, checks for convergence, and prints a histpry

of the search. The search for a mode 3 optimum system, nominal data, is shown

in Figure 18, and a report on the optimum system is shown in Figure 19.

Actually, two independent searches are made. The first starts with the lowest

numbered components (accelerometers = I, gyroscopes = 21, and computer = 41).

The second starts with the highest numbered components (accelerometers = N ,a

gyroscopes = 20 + N , and computer = 40 + N ). Each search is terminated

by one of three conditions: c

(i) The algorithm returns its present system as the best system.

This indicates a local minima in the penalty.

(2) The algorithm returns a previously tried system. This indicates

the start of an unending limit cycle, impossible with the present

algorithm but included should a new algorithm be tried.

(3) More than one hundred iterations have been made and terminations 1

or 2 have not occurred.

Termination number i is the only one encountered in the exercising of the

program. It represents a local optimum and is used as the optimum for that

search. Terminations 2 and 3 are not local optima. The system tried during

these searches that had the least penalty is used at the end point of the search.

The results of the two searches are compared, and if equal, a message

indicating identical convergence is printed and the result assumed to be

optimum. If the searches have unequal end points, a message is printed, and

the one with least penalty is used as the optimum.

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70

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Unequal end points occurred occasionally in the exercising of the program.

However, in every case one of the endpoints was a system th._t was saturated.

Saturation occurs when the system fails an implied constraint such as one with

a PFR greater than PFG (probability of failure due to reliability greater

than the required probability of total system failure). When a system is

saturated, the penalty function coding arbitrarily adds one million to the

penalty. This number was chosen to be a ridiculously large penalty (spacecraft

weight or probability of failure).

Although not strictly a steepest descent technique, this search routine and

its algorithm are vulnerable to the usual pitfalls of steepest descent. The

most obvious danger is that the local minima is not the global minima. However,

the use of two separate searches and comparison of the end points helps check

for this problem. In any case, it should be remembered that there is no system

that differs by one substitution from the indicated end point with a smaller

penalty.

Application of Study Techniques to a Jupiter Flyby Mission

Tile techniques developed in this study were applied to a Jupiter flyby

mission. The data required for exercising the computer programs implementing

these techniques were compiled or derived in a cooperative effort with NASA/ERC,

personnel of the NASA Launch Vehicle Planning (NLVP) Project at Battelle, and

those Battelle staff members assigned to this study. Much of the hardware data,

particularly the reliability and error coefficient data, is assumed based on

manufacturers' data and are not to be used conclusively.

Data Required.--For the purpose of exercising the computer programs,

trajectory, mission, spacecraft, and candidate component_ data are required.

These are listed and discussed in the following section.

Boost Traiector Y and SEAP Data.--Curves were fitted to the non-zero body

rates and accelerations of the simulated boost trajectory discussed in detail

in Appendix A of this report. These curves were of the form

tn Cao + alt + "'" + an for the body rates and l-t/T + bo + bit + "'" + bn tn for the

accelerations where T is the time when the stage, if it were all fuel, would be

burned up (see Table XX). The coefficients in Table XX for Stages 3 and 4

i.e., the curves expressing the body rates and accelerations as a fun,_tion of

time, were used as the trajectory input data for the Strapdown Error Analysis

Program (SEAP). The error sensitivity coefficients output from SEAP are shown

in Figures 20a, 70b, and 20c for the trajectory used. The first column in the

figures denotes the error coefficient whose sensitivity is listed and the last

symbol in that c_iumn denotes whether the sensitivity is position, xxxP,

velocity, xxxV, _ attitude, xxxA. The columns headed by X, Y, and Z are shown

in Figure 2l. The columns headed by DR (down range along the velocity vector,

V), CR (in the pI_ne determined by the radius and velocity vectors and normal to

DR), and OP (out L_[ plane)are the sensitivities after rotation througll the angle

defined in Figure 22, in this case -29.3 ° .

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FIGLTRE 21. INERTIAL FRAME

CR

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FIGURE 22. DEFINITION OF ROTATION ANGLE

I!1IIII

State Transition Matrices Data.--The Lewis Research Center n-body computer

program was modified to calculate n-body state transition matrices. The state

transition matrices used in exercising of these computer programs were obtained

by running the Lewis n-body program for the trajectory previously sun_narized in

v .... j .... j ..... j .......................

transition matrices were calculated.

As a check on the compatibility of the three trajectory integration

programs (three degree of freedom, SEAP, and Lewis n-body), the program reading

the SEAP and Lewis decks prints the velocity and flight path angle of each

trajectory at booster burnout. This information and the error in the matching

is also presented in Figure 23.

Figure 23 also presents the state transition matrix which propagates

injection errors to the target. This matrix is presented for information only

and serves as a check point in using the program.

Midcourse Propulsion System Data.--The midcourse propulsion system data

required for exercising the computer programs, the fixed weight, specific

impulse, thrust, and tankage constant were extracted from Reference 1. This

system is a constant gas-pressure-regulated monopropellant hydrazine unit

using a Shell 405-type catalyst. It was selected because this type of system

is applicable to accelerometer and burn timer shutoff mechanisms.

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Bipropellant midcourse propulsion systems are competitive with monopro-

pellant systems when the total impulse requirement is 50,000 ib-sec or greater

(Reference 2). Assuming the maximum delta velocity required is 20 ft/sec for

example, the total impulse requirement for the 5,410 ib spacecraft isapprox ima te ly

5410I -t 32.2

x 20 = 336 ib-sec

Since in none of the exercise runs was the delta velocity required as great as

20 ft/sec, a monopropellant system should be used for the midcourse propulsion

system for the payload and mission considered.

The choice between the pressure regulated and blowdown monopropellant

systems (References i and 2) was not considered to be of major importance for

the purpose of exercising the computer programs. Therefore, the midcourse pro-

pulsion system data used in the exercising was based upon the pressure regulated

system. This data is presented in Table XXI. For the purpose of the exercising,

it was assumed that the probability that the midcourse engine would not igniteupon command was 0.001.

TABLE XXI

MIDCOI_SE PROPULSION SYSTEM VALUES

Midcourse System Fixed Weight

Midcourse System Tankage Factor

Specific impulse

Midcourse System Thrust

20.30 Ib

1.096 ib/Ib of fuel

ZJJ See

50 Ib

IIIII

Guidance System Electrical Power Source Data.--The guidance system power

source was assumed to be a radioisotope thermoelectric generator (RTG) for the

purpose of exercising the computer programs. Mission durations restrict power

source consideration to solar, nuclear reactors, or an RTG. Due to the extreme

distance from the sun, solar power sources were ruled out due to size and

weight considerations. Solar thern_l energy intensity at Jupiter's orbit is

approximately 4 percent of that at a near Earth orbit according to Reference i.

This low level prohibits use of any currently envisioned solar energy collection

system. Nuclear reactors have a minimum critical size and weight required to

maintain a controlled nuclear reaction (Reference I). The minimum weight is

currently approximately 250 ibs, and therefore the only practical choice of powerat this time is the RTG.

83

The major componentsof an RTGare: (i) an isotope heat source, (2) thermo-electric converters, and (3) a heat rejection system (Reference 2). A powerconditioning and distribution system is also required. These components weights

can be estimated by assuming the weight to be made up of a fixed weight plus a

specific power factor expressed in ib/watt required. Table XXII lists the

values assumed for the RTG data used in the exercising.

I!!

TABLE XXII

ASSUMED RTG POWER SOURCE VALUES

Fixed Weight, Kpl

Specific Power Factor, Kp2

13.20 Ib

0.345 Ib/watt

III

These values are quite optimistic but suffice for exercising of the program.

It was assumed throughout the power source weight estimation that,the

power source required at the target planet for the experiments could be used in

the earlier phases of the mission for the guidance system. Should one assume

guidance only through the first midcourse, a lower power source weight might

result if a different source were selected. Since the power needed for experi-

ments on a Jupiter flyby mission will require use of an RTG (References i and 2),

it was decided to make use of the same power source to keep the total scientific

experiment payload as large as possible.

Candidate Component Hardware Data.--Figures 24a and 24b present the candi-

date component hardware data used in the exercising of the program. As

previously noted, the MTTF (mean time to failure) data of the components were

assumed for the purpose of exercising the computer programs and are not to be

used conclusively. Data on the uncertainty in the gyroscope and accelerometer

error coefficients per OF (DEG-F in figure) was generally incomplete, so no

values are given. When these data are available, they should be included

since temperature uncertainty effects are included in the accuracy analysis.

A zero value for width of the inertial sensors denotes the sensor is cylindrical.

The SIGN III computer is listed with a 40 bit word length and integration scheme

(INT. SCH.) I, a 20 bit word length and integration scheme 2, and a 20 bit word

length and integration scheme 3. These word lengths and integration schemes

were assumed for purposes of exercising the programs and do not necessarily

correspond to actual hardware. Integration scheme i uses a rectangular

algorithm to update the direction cosine matrix. Integration scheme 2 uses a

Runge-Kutta second order algorithm to update the direction cosine matrix.

Computation frequency (COMP. FREQ. in Figure 24b) is the algorithm integration

frequency.

IIIIIIIII!

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Typical Output Listing.--Figures 25a, 25b, 25c, and 25d make up a typical

four-page report for mode 2 using system A components and nominal mission

data. Similar reports for mode 3 were presented in Figures la through id in

the summary section of this report for system A and Figures 19 a through

19d for an optimum system. Figure 25a presents the penalty mode 2, 0.01309

probability of mission failure due to guidance,lists descriptive information

about the launch vehicle and mission, and presents the hardware data for each

instrument used in the conceptual system A IMU. In addition, data describing the

conceptual SRT computer are presented. The notation RUK-2 denotes that a Runge-

Kutta second order algorithm was used to update the direction cosine matrix.

IMU mechanical data is presented in Figure 25b. Horizontal design number 4

(see Appendix D for sketch of design) was found to be optimum (yields minimum

weight) for the specified instruments. The dimensions of the case of the IMU,

the calculated weights of the block, base, cover, and insulation, and the

_u_t_on of the components (inertial sensors and computer) weights are

presented. The electronics weight was a specified value. The total estimated

weight for the system A IMU and computer is 69.46 Ib for IMU design number 4.

The thermal analysis presented indicates that the thermal conductance for

the IMU was calculated to be 2.17 watts/°F on the basis that no heater power

or cooling was to be required at the maximum ambient temperature, 140 ° F. The

peak watts, 372.7, are the total power (IMU plus computer) required at the

minimum ambient temperature, 0u v F, for the calcu,acea cnerma_ conauc_au_e.

The average watts, 307.5, are the total power required at the average ambient

temperature, 60 ° F, for the calculated thermal conductance. The energy source

weight is the weight of the RTG needed to supply the average power requi_ed.

This assumes that the peak power is required for a short time and would be

drawn from a small secondary energy source as well as the RTG.

Figure 25b also presents the results of the navigation error analysls ac

various times along the trajectory. The velocity errors at injection are

given in down range (DR), cross range (CR), and out of plane (OP) components.

The root mean square (RMS) of these components is 11.03 ft/sec at injection

with the degree of freedom (DOF) of 1.984. The injection errors propagated to

the target with no midcourse correction are seen to result in a CR positionerror of 11.645 x 108 ft. If a single perfect midcourse delta-velocity

correction is made to null the CR position error, the correction delta-velocity

required has an RMS value of 9.6276 ft/sec. If an imperfect midcourse is made

with the relative accuracy shown, 0.07% error in magnitude and angular aiming

error of 0.00291 radians, the CR position error would be 84.3593 x 104 ft.

Data describing the midcourse propulsion system is also presented. Under this

penalty, mode 2, the total guidance weight is fixed, and the total midcourse

propulsion system weight is calculatedto be the difference between the total

guidance weight and the sum of the IMU, computer, and energy 6ource weight.

For this example, the total guidance weight was fixed at 410 Ibs and the

resultant midcourse propulsion system weight after subtracting the weights of

the IMU, computer, and energy source was calculated to be 221.249 ibs. The

fuel weight of 183.34773 ibs gives a delta-velocity capability of 258.68 ft/sec.

From this capability, the calculated probability of failure due to lack of

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sufficient fuel was zero. Therefore, the total penalty, probability of mission

failure due to guidance, is equal to the probability of failure due to hardware

reliability, 0.01309.

Mission and spacecraft data used in the computation of the results pre-

sented in Figure 25b are listed in Figure 25c. A short phrase describing the

item of data and the value used in the computation is followed by the

sensitivity of the penalty to that value. This sensitivity, as noted in

Figure 25c, is the percent change in the penalty to a one percent change in

the data value. For example, a one percent change in ID l, the time to mid-

course correction, would result in a 0.917 percent change in the penalty for

this case. A negative value of sensitivity indicates that the penalty will

decrease if the data value is increased and vice versa. It should be noted

that even though the mission and spacecraft data values listed in Figure 25c

are the same as those listed earlier in Figure lc, the sensitivity of the

penalty to these values is quite different for the two different modes.

Sensitivity of the penalty to inertial sensor and computer data is pre-

sented in Figure 25d. A zero value of sensitivity indicates that a one

percent change in the data value presented in Figure 25d for the item of

interest results in no change in the penalty out to the number of significant

figures printed, or that the data value was zero. The sensitivity to the

number of bits of the computer and integration scheme is not the sensitivity

to a percent change in the data value, but is the percent change in penalty to

the change noted for these items. Under mode 2, the principal data affecting

the penalty are seen to be the mean time to failure, MTTF, and the Weibull

constant, _.

IMU Mechanical Weight Estimation Results.--The system A IMU components,

the GG 334A and the D4E, were used to evaluate horizontal mechanical designs i

through 6 and the six vertical designs illustrated in Appendix D. The same

material densities, component separation, base offset and thickness, cover

clearance and thickness, and insulation thickness were used for each design.

These data values are listed in Figure 25c. Table XXIII lists the results of

this evaluation.

III|I1IIIIIII1III1l14

92

TABLE XXIII

IMU WEIGHT ESTIMATION RESULTS FOR SYSTEM A

IIIIIIII

Item Weight (ibs)

Sensors + Total

Des i_n No. Block Base Cover Insulation Electronics IMU

Horizontal i 8.70 5.05 3.09 1.45 16.00 34.29

Horizontal 2 8.70 4.97 3.06 1.43 16.00 34.16

Horizontal 3 8.70 4.59 2.92 1.37 16.00 33.58

Horizontal 4 8.70 4.55 2.87 1.34 16.00 33.46

Horizontal 5 8.70 5.17 3.14 1.47 16.00 34.48

Horizontal 6 8.70 5.19 3.14 1.47 16.00 34.50

Vertical 1 8.70 4.73 3.33 1.56 16.00 34.32

Vertical 2 8.70 5.49 3.28 1.54 16.00 35.01

Vertical 3 8.70 6.07 2.71 1.27 16.00 34.75

VeLt :--I 4 8._ 6.7= o _ 16 nq _ v_

Vertical 5 8.70 6.07 3.47 1.63 16.00 35.87

Vertical 6 8.70 5.17 3.62 1.70 16.00 35.19

In reality, horizontal designs three tnrougn six are for pendulous accel-

erometers, and therefore, the results are not strictly true since the D4E is

a vibrating string accelerometer. Similarly, vertical designs two through four

are for pendulous accelerometers only. The above results do illustrate that

the mechanical design variations make little difference for these conceptual

designs for the system A components. The total weight variation is seen to be

1.29 ibs from the lightest to the heaviest.

Although horizontal design 4 yields minimum weight for system A,

another design may yield minimum weight for another set of components as shown

previously for an optimum system, Figure 19b.

System B.-- Results for system A, penalty mode 3, nominal

conditions, as shown in Figures i through 5 indicate that the inertial

components have more than sufficient accuracy. When gyro fixed drift (R) and

accelerometer bias (K_ are swept in Figures 2, 3, and 4, the curves show that

increases in these values should not increase the penalty significantly. It

was decided to investigate a system similar to system A with components

identical to the system A components except for increased gyro fixed drift (R),

accelerometer bias (K0), and accelerometer scale factor (KI) error coefficients.

93

The new componentsare labeled Pert-ACC and Pert-GYRO, and the system isreferred to as system B.

System B Components.--The sweeps and sensitivities shown in Figures I, 2,

3, and 4 indicate changes in penalty for changing only one parameter. It

was decided to increase the error coefficients in such a way that the three

become equally important (nearly equal sensitivities) and that the combined

penalty increase be about ten percent by allotting a five percent penalty

change to gyros and five percent to accelerometers.

A sweep of gyro drift for all three gyros (Plot 1 of Figure 4) shows that

a penalty increase of five percent occurs when fixed drift is increased from

0.i to about 0.2 deg/hr. The Pert-GYRO was defined to be identical to the

GG 334-A except for a 0.2 deg/hr fixed drift uncertainty.

A sweep of accelerometer bias for all three accelerometers (Plot I of

Figure 3 and the table in Figure 2) show that bias may be increased from

6.7 x 10 -6 to about 2.2 x 10 -4 g's (a factor of about 39 before penalty is

increased by about five percent. Rather than allocate the entire increase to

K0, however, it was decided to increase both K0 and KI. No sweep was made

for K I. However, its effect may also be estimated from the sensitivities

shown in Figure id.

Since only the sensitivities per single component are shown, the sensi-

tivities for changing all three accelerometers are calculated by adding as

shown in Table XXIV.

iIiW

IIIIII

TABLE XXIV

SUMMING ACCELEROMETER SENSITIVITIES

Error Yaw Pitch Roll TotalSource

K0 0.0000381 0.0000044 0,0000249 0.0000674

K 1 0.0 0.0 0.0000523 0.0000523

Thus, for changing all three accelerometer Ko's and Kl'S , the sensitivityis about the same. The five percent increase was split to a two percent

increase in penalty for both K0 and K I. From the table in Figure 2 a two per-

cent increase (to 226 ibs) in the penalty occurs at about 1.6 x 10-4 g's or a

factor of 30. Since the sensitivities are about the same, it was decided to

increase K1 by a factor of 30 also. The Pert-ACC was defined to be identical

to the D-4E except for 1.68 x 10"4gbias and 3.0 x 10 -4 scale factor

uncertainties.

94

Sj!stem B Results.--The penalty, mode 3, for system B under

nominal conditions is 239.5 ibs, an increase of about 8% over system A.

The sensitivities are shown in Table XXV.

I

!!!!

TABLE XXV

ERROR SOURCE SENSITIVITIES FOR SYSTEM B

ErrorYaw Pitch Roll Total

Source

R 0.01238 0.10839 0.00058 0.12135

K0 0.01782 0.00204 0.08662 0.10648

K I 0.0 0.0 0.03501 0.03501

Ko+K I 0.14149

IIIIIIIII

Plots of the penalty vs the error coefficients are shown in Figures 26, 27,

and 28. Circles have been drawn to indicate the plot points closest to the

values for the system B components. It can be seen that these values represent

a tight condition where any further increase will result in large increases in

the penalty.

This technique of loosening specifications will work equally well on anycomponent data. Care should be used when aDDlvin_ this approach to certain

error coefficients, however. The SEAP coefficients for sensitivity to yaw and

pitch accelerometer scale factor are zero. This would indicate unlimited

increase in these error coefficients and would not change the penalty, or in

fact that these accelerometers are not needed. This is an obvious fallacy in

that, although this nominal trajectory does not have acceleration components in

these body coordinates, an actual flight would due to wind disturbances, etc.

Any trajectory which involves yaw steering or a plane change such as synchronous

equatorial orbital missions would have non-zero sensitivities for these terms

due to the additional accelerations appearing in the nominal trajectories. In

general, however, any term with small sensitivity should be considered one for

which an easing of specifications would not seriously change the penalty.

Surmary of Systems Exercised.--Additional studies of system A,

system B, and optimum systems were made for variations of the nominal

conditions and under different penalty modes. These results are summarized in

Table XXVI.

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TABLE XXVI

ADDITIONAL PENALTY RESULTS

System

(Optimum

Penalty defined

Mode below)

Probability of

Guidance Failure

Time to Midcourse

25 hours 240 hours

2 A Ca icu la ted 0.01309

2 B Ca icula ted 0. 04042

2 Opt imum Ca Iculated 0.011601

3 A O o! 222.04

3 B 0.i 239.47

3 Optimum 0.i 208.802

3 A 0.05 224.89

3 B 0.05 250.96

3 Optimum 0.05 210.554

0.11113

0.13575

0.098241

Fails

Fails

222.083

Fails

Fails

None

IIII

Notes: Fails = Probability of hardware failure exceeds PFG

None = No system was found with sufficient reliability

Optimum Systems: Accelerometers

(i) 2401-005

(2) GG-177

Gyroscopes

/ SYG-1440

/ 18-1RIG-B

(3) GG-177 / SYG-1440

(4) 2401-005 / SYG-1440

Computer

/ SRT RUK-2

/ SIGN-Ill (Full report shown

in Figures 19a-19d)

/ SIGN-Ill

/ SRT-RUK-2

IIII

Parameter Sweeps.--During the exercise of the computer programs selected

parameters were swept, and the results are shown in Figures 29 through 36.

Time to midcourse correction was swept from zero to 240 hours in linear steps

of ten hours as shown in Figure 29 for penalty mode 2 and Figure 30 for penalty

mode 3. For the mission data used, the penalty mode 2 is essentially the proba-

bility of failure due to reliability; thus, the plot of penalty mode 2 vs time

to midcourse increases due to increased gyro and computer operating time.

The similar plot for penalty mode 3 shows very little change between i0 and

200 hours. Above 200 hours the penalty increases rapidly, and above 220 hours

99

the penalty is saturated. This is because the probability of failure due toreliability exceeds the required probability of guidance failure (PFG= 0.i).This can be confirmed by noting that penalty mode 2, essentially PFG, crossesthe 0.i value at 220 hours.

Probability of guidance failure (PFG)was swept in linear steps of 0.002from 0.05 to 0.I0, and the results for penalty mode3 are shown in Figure 31.This plot, with the time of midcourse at the nominal value of 25 hours, showslittle change. However, had the sweepincluded values lower than 0.050, thepenalty would increase greatly whenPFGbecameclose to the probability offailure due to reliability (PFR = 0.013 for 25 hours to midcourse).

A plot of penalty, mode3, vs operating temperature from 150 to 180° Fis shown in Figure 32. Note the increase in penalty for lower operatingtemperature. This is easily understood when the effect of designing thethermal conductance is considered. The thermal conductance is designed todissipate the exciting power through a drop from operating temperature tomaximumambient temperature. The nominal data includes a maximumambienttemperature of 140°_ Therefore, operating temperatures near 140° F resultin very high conductance and a corresponding increase in heater power forportions of the mission when the ambient temperature is below its maximum.In fact, if the operating temperature were equal to the maximumambienttemperature, zero conductance would be used and infinite heater power would beneeded during the colder portions of the mission.

Figure 33 presents four parameter plots of logarithmic sweepsof theparameter versus the penalty for system A. Plot I is a sweepofexcitation power to all three gyros versus the penalty for mode3. Plots 2,3, and 4 coincide and are shownas plot 4. In this case, the excitation powerto each gyro was swept while the other two gyros' power remained at the originalvalue for the GG334A.

Plots of logarithmic sweepsof gyro weight versus the penalty forsystem A are presented in Figure 34. Plot I is a sweepof the weight ofall three gyroscopes simultaneously. Plots 2, 3, and 4 coincide and areshownas plot 4. In this case, the weight of each gyro was swept while theother two gyros' weight remained at the original value for the GG334A.

Figure 35 presents a plot of a linear sweep of IMU componentseparationversus the penalty for system A for mode3. Increasing the componentseparation from the nominal value of 0.25 inch to 1.00 inch increases thepenalty from 222.04 lbs to 256.38 ibs as shown in the figure.

A plot of the penalty versus a logarithmic sweepof the computationfrequency (frequency at which the direction cosine matrix is updated) ispresented in Figure 36 for the system A computer, SRTRUK-2.

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108

I!I!

CONCLUSIONS

An evaluation technique was developed for pure inertial strapdown guidance

systems designed for interplanetary missions. This technique is useful in

evaluating competitive systems, in determination of research needed to improve

system performance, as an aid in preliminary design of conceptual systems,

and provides a measure of system performance on each mission being examined.

•he technique assumes a perfect update of velocity, position, and attitude

occurs prior to the burn from the parking orbit.

I!I!!!III

The system parameters used in the evaluation technique can be estimated

using techniques that are relatively unsophisticated but are of sufficient

accuracy to accomplish the desired result.

These techniques were implemented in computer programs now in use at

NASA/ERC. The computer programs were exercised on a Jupiter flyby mission.

For this mission and the assumptions made, it can be concluded that the accuracy

of the strapdown guidance systems evaluated is more than sufficient to

accomplish the guidance and navigation of the first spacecraft to flyby Jupiter.

If the first midcourse correction is made i0 days after launch and the

required probability of mission failure attributable to guidance is 0.05,, the

reliability (mean time to failure) of the pure inertial strapdown guidance

continuously from launch to midcourse. If the probability of mission failure

attributable to guidance is relaxed to 0.i, the two specified systems still fail

if rnidcourse correction is made I0 days after launch. An optimum system whichwill succeed was found.

Concentrated attention to reduction of system power requirements would

yield a signiticant reauctlon in _ne weigh_ a_EribuL_bi_ Lu _uidau_= fuL L_,i_

specific mission. This might be achieved by development of a lightweight

variable thermal impedance for the IMU.

RECOMMENDATIONS

IIII

I"

/

The techniques developed, the computer programs implementingthese

techniques, and the results from exercising these computer programs on a

Jupiter flyby mission lead to the £oiiowing recon_nendations.

It is recommended that NASA/ERC consider extension of the work which led

to the present parameter estimation techniques to more sophisticated techniques

that could eventually result in computer-aided design of strapdown guidance

sys terns.

For the Jupiter flyby mission studied, a concentrated effort on reduction

of systern power, and hence power source weight, is recommended. The direction

to be followed in this research depends upon cost and other factors such as the

likelihood of success of each research approach to power (or power source

weight) reduction.

109

It is recommendedthat attention be directed toward improving the relia-bility of strapdown guidance systems which might be used for a Jupiter flybymission. This effort should consider not only improvement in the meantimeto failure (MTTF)of the hardware, but also investigate reliability models forstrapdown guidance systems and methods of testing to verify these models.As shown in the exercise cases previously discussed, an improvement in the_TF of the strapdown guidance system hardware must be made if the system is tooperate continuously from launch until midcourse correction ten days later, orthe probability of mission failure attributable to guidance must be relaxed.

If the reliability of the system can be increased, a decrease in accuracycould be considered on acceptable tradeoff. Such a relaxed accuracy system

(system B) was used in the exercise runs.

ii0


I

I

!

I

[

I

I

I

I

(1)

(2)

(3)

(4)

(5)

(6)

REFERENCES

"A Study of Jupiter Flyby Missions, Final Technical Report", General

Dynamics, Fort Worth Division, FZM-4625, May 17, 1966.

"Advanced Planetary Probe Study, Final Technical Report", TRW Systems,

Reports 4547-6004-R000 to 4547-6007-R000, Volumes 1-4, July 27, 1966.

"Final Report for the Study of Simplified Navigation and Guidance Schemes",

Volume I-Technical Analysis, United Aircraft Corporate Systems Center,SCR 290-1, October 31, 1966.

"Application of the Saturn V Launch Vehicle to Unmanned Scientific

Exploration of the Solar System, Jupiter Orbiter/Solar Probe Mission

Study, Advanced Mission Investigations", Northrop Space Laboratories,

TM-29213-6-075, September, 1966.

Porter, R. F., "Preliminary Trajectory Analysis of the 260(0.75) and

260(FL) Stacks", Battelle Memorial Institute, BMI-NLVP-IM-66-64,

Mmy 26, 1966.

Porter, R. F., "Examination of a 'Dog-Leg' Trajectory for the 260(3.7)-

SlVB Launch Vehicle", Battelle Memorial Institute, BMI-NLVP-IM-66-81,

August 15, 1966.

IIII

(7)

(8)

• .

"Saturn IB Improvement Study (Solid First Stage) Phase II, Final Detailed

Report", Douglas Missile and Space Systems Division, Report SM-51896,

Volume II, March 30, 1966.

"Launch Vehicle Estimating Factors for Generating OSSA Prospectus 1968",

Launch Vehicle and Propulsion Programs, Office of Space Science and

Applications, National Aeronautics and Space Administration, December, 1967.

'"l_'in=l R=n,",',"_" "F,',,,* *I_=, P=,.,* .... D_o_ TT p,,-:.-I .... f'}. .... C't..--ly ' ,T_I .... I

Technical", United Aircraft Corporate Systems Center, SCR 264-1, April 15,1966.

(i0) Borland, L. D., Honeywell Aerospace Division, St. Petersburg, Florida,

July, 1967, Note to Ellis F. Hitt, Battelle Memorial Institute.

IIIII

(11)

(12)

(13)

"Final Report for the Centaur Phase II Guidance Systems Study, Volume l-

Technical Appendices", Appendix A, United Aircraft Corporate Systems

Center, SCR 264-1, April i_ la_=._.w, _Jvv.

"Proposal for a Strapdown Inertial Sensing Unit Subassembly, Volume l-

Technical", Honeywell Aerospace Division, St. Petersburg, Florida,

R-ED 24644, July 25, 1967.

McKinley, Howard L, Jr., "An Analysis of Temperature Control Problems for

Inertial Components and Experimental Results", Guidance and Control

Division, Central Inertial Guidance Test Facility, Air Force Missile

Development Center, Technical Report GL-2, June, 1963.

iii

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

Mednick, Ralph, "GE-E-3 Temperature Controller", Instrumentation

Laboratory, Massachusetts Institute of Technology, September, 1960.

Stafford, G. B., "Space Power Subsystem Capabilities", AF Aero Propulsion

Laboratory, WPAFB, Ohio, AIAA Paper No. 65-466, dtd., July, 1965.

Electrical Power Generation Systems for Space Applications, NASA SP-79,

1965.

Shair, Lerner, Joyner, and Evans, "A Review of Batteries and Fuel Cells

for Space Power Systems", Journal of Spacecraft and Rockets, July, 1967,

pp 833-838, Vol 4, No. 7.

"Space Power Systems Engineering", Edited by G. C. Azego and J. E. Taylor,

Academic Press, 1966, AIAA Series - Vol 16.

Corliss, W. R., and Harver, D. G., "Radioisotopic Power Generation",

Prentice-Hall Series, 1964.

"Final Report for the Study of Simplified Navigation and Guidance Schemes",

Vol 1-Technical Analysis, Contract NAS 12-40, UACSC, October 31, 1966.

"Advanced Planetary Probe", Final Technical Report, Vol 2, Spin-Stabilized

Spacecraft for the Basic Mission, TRW Systems, July 27, 1966.

"Application of the Saturn V Launch Vehicle to Unmanned Scientific

Exploration of the Solar System", Contract NAS 8-20082, September, 1966.

Lloyd, David K., and Lipow, Myron, "Reliability: Management, Methods, and

Mathematics", Prentice Hall, 1962, Englewood Cliffs, New Jersey.

Hayre, H. S., Schrader, G. F., Daruvalla, S. R., and Hearn, N. K.,

"Accelerated Life-Testing and Reliability Prediction of Missile Control

Systems", Air Force Missile Development Center, AFMDC-TR-65-25,

August, 1965.

Roantree, J., and Kormanik, N., "A Generalized Design Criterion for

Strapped-Down Inertial Sensor Loops", AIAA/JACC Guidance and Control

Conference, August, 1966.

Farrell, J. L., "Performance of Strapdown Inertial Attitude Reference

Systems", AIAA/JACC Guidance and Control Conference, August, 1966.

Vichnevetsky, Robert, "Error Analysis in the Computer Simulation of

Dynamic Systems: Variational Aspects of the Problem", IEEE Transactions on

Electronic Computers, Vol EC-16, No. 4, August, 1967.

A Study of the Critical Computational Problems Associated with Strapdown

Inertial Navigation Systems, Volumes I and II", United Aircraft Corporate

Systems Center, SCR 328-I and SCR 328-II, February 23, 1967.

Battin, Richard H., "Astronautical Guidance", McGraw Hill Book Company,1964.

IIIII]IIIIIIIIIIIIII

112

IIIIIIIIIlIIIIIIIIIII

APPENDIX A

SIMULATION OF THE 260(3.7)/SIVB/CENTAUR 1/KICK

LAUNCHING A 5,410 LB PAYI,OAD ONTO

A JUPITER FLYBY TRAJECTORY

iI!

APPENDIX A

SIMULATION OF THE 260(3.7)/SIVB/CENTAUR 1/KICK

LAUNCHING A 5,410 LB PAYLOAD ONTO

A JUPITER FLYBY TRAJECTORY

INTRODUCTION

III

This appendix discusses the simulated performance of the 260(3.7)/SIVB/

Centaur 1/Kick launch vehicle launching a 5,410 Ib payload onto a Jupiter flyby

interplanetary trajectory. The simulation was for a due east launch from the

Eastern Test Range (ETR). The SIVB/CI/K plus payload was injected into a 100.2

by I00 nm parking orbit at 420.22 seconds into the SIBV burn. It was assumed

that the SIVB was restarted at the correct time to begin the final injection

phase of the launch trajectory.

IIIII

GROUND RULES

To provide the most meaningful results, the thrust shaping of the first

stage and the kick angle were chosen to meet criteria established by NASA/MSFC

for this type of vehicle. Specifically, these constraints are:

(i) Maximum dynamic pressure < 950 Ib/ft 2

(2) Maximum axial load factor _ 6 g's

(3) Dynamic pressure at first stage burnout _ i0 ib/ft 2

DESCRIPTION OF VEHICLE

Weights, thrust, propellant weight flow, and other selected characteristics

of the vehicle were obtained from References AI and A2. Table A-I contains

pertinent data for this vehicle. Representative aerodynamic data were obtained

from Reference A2 and are presented in Figure A-I.

A-I

p-,.4o

0

I--I

0

t-'4

A-2

TABLE A-I.

SELECTED CHARACTERISTICS OF 260(3.7)/

SIVB/CENTAUR I/KICK LAUNCH VEHICLE

Stage 260(3.7) SIVB Centaur I Kick

Initial Total* 4,019,302 315,303"** 54,202 12,410Gross Weight (ib)

IiI

Final Total*

Gross Weight (ib)

Vacuum Thrust (Ib)

Propellant Weight

Flow (ib/sec)

Exit Area (ft 2)

720,302 85,303 17,602 6,460

6,430,000** 205,000 31,000 7,500

24,300** 482 68.20 16.47

376 35.8 16.58 4.14

* All weights include 5,410 Ib payload

** Initial value only: for time history, see Figure A-2

*** includes 5,600 ib shroud which is ejected 26 seconds after SIVB ignition.

ANALYSIS

IlIIIIII

All simulations were performed using the Battelle three-degree-of-freedom

computer program. The pitch steering program for the SIVB burn was computed

using a linear steering technique (Reference A3).

The first stage trajectory consisted of a vertical rise until a relative

velocity of 150 ft/sec was attained. The vehicle was then subjected to an

instantaneous deflection of the flight path through a selected kick angle of

1.83 degrees from the vertical along an initial azimuth of 90 degrees from true

North. The vehicle then flew a gravity turn until first stage burnout.

Figure A-2 is a time history from liftoff of selected stage and trajectory

parameters during the first stage, 260(3.7), burn. Th_ thrust shaping shown was

necessary to keep the dynamic pressure below 950 ib/ft _. Acceleration along

the roll axis of the vehicle (axial load factor) is quite nonlinear during first

stage burn due to this thrust shaping. Also displayed in Figure A-2 are the

velocity relative to Earth, the flight path angle from the local vertical, and

the vehicle altitude as functions of time from liftoff.

The final conditions at first stage burnout were used in determining

linear pitch steering program for the SIVB stage. The steering used is depicted

in Figure A-3 with the altitude, inertial velocity, flight path angle, and roll

axis acceleration plotted as functions of time from liftoff. At 563 seconds

into the mission (420.22 seconds from first SIVB ignition), the SlVB/CI/K plus

A-3

24

22

0 20

III11IIIIIIIIIIII

FIGURE A-2. TIME HISTORY OF 260(3.7) STAGE TRAJECTORY PARAMETERS

A-4

IIII!

payload was injected into a 100.2 by i00 nm parking orbit. It was assumed that

the SIVB thrust was terminated and the vehicle was then oriented such that the

pitch angle and flight path angle were equal. It was further assumed that the

SIVB was restarted at the correct time to begin the final injection phase of

the launch trajectory as shown in Figure A-3. The SIVB was burned to fuel

depletion in its second burn. At that time, the SIVB was jettisoned and theCentaur I ignited.

Figure A-4 depicts the inertial velocity, roll axis acceleration, and

hyperbolic excess velocity as functions of total burn time from liftoff for the

Centaur I burn. Figure A-5 depicts the same information for the Kick stageburn.

The hyperbolic excess velocity at injection was 39,728 ft/sec. This is

equivalent to a characteristic velocity of 53,732 ft/sec.

CONCLUSIONS

III

Based upon this simulation, it appears that the 260(3.7)/SIVB/CI/K launch

vehicle is capable of launching a 5,410 pound payload onto an interplanetary

trajectory to Jupiter with an expected time of flight of 410 days. This simu-

lation was made assuming the guidance system was part of the payload. If one is

L_ _ L_ _0_ Ltl_ _O_[LL _LULLI _LE_ LL ULLI_ LIL ULI_L LU _U V _ _d _ L_ LLL_

SIVB and 260(3.7), the results of this simulation will not be applicable. This

simulation was performed to generate the boost trajectory data needed for the

Strapdown Error Analysis Program (SEAP).

A-5

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IIIIIIIIIIIIIIIII

A-8

(AI)

(A2)

(A3)

APPENDIX A

REFERENCES

Porter, R. F., "Preliminary Trajectory Analysis of the 260(0.75) and

260(FL) Stacks", Battelle Memorial Institute, BMI-NLVP-IM-66-64,

May 26, 1966.

Porter, R. F., "Examination of a 'Dog-Leg' Trajectory for the 260(3.7)-

SIVB Launch Vehicle", Battelle Memorial Institute, BMI-NLVP-IM-66-81,

August 15, 1966.

Duty, R. L., "Trajectory Determination by Linear Steering", Journal of

Spacecraft and Rockets, Vol 3, No. i, January, 1966.

A-9

IiIIII!!I!I

APPENDIX B

SYNOPSIS OF STANDARDIZED SPACE GUIDANCE

SYSTEM COST-EFFECTIVENESS MODELS

III

APPENDIX B

SYNOPSIS OF STANDARDIZED SPACE GUIDANCE

SYSTEM COST-EFFECTIVENESS MODELS

Prior to defining a performance index for use in evaluating strapdown

guidance systems, existing "cost-effectiveness" models for space guidance

systems were reviewed. This appendix presents a review and evaluation of

four existing models developed by contractors for the USAF Space Systems

Division (now Space and Missile Systems Organization) "Standardized Space

Guidance System (SSGS) Study". The input data required and fundamental figure

of merit of each model is followed by an evaluation of the applicability ofthe four models discussed.

(I) IBM Federal Systems Division, Space Guidance Center, developed a

"cost-effectiveness" model (Reference BI) as part of their work on the

Standardized Space Guidance System (SSGS). Input data required for the effec-

tiveness analysis for each configuration is tabulated in Table B-I.

TABLE B-I

T'I_I_ 1_]171_/"_rTSr_M'_ AMAT V_T¢2 TM'DTTrr l"_Arl_.&

(i)

(iii)

(iv)

(v)

(vi)

Navigation accuracy requirements for eachmission

Guidance error anaiysls for eacn candidaEe

guidance system configuration

Mission time profile broken into specific

periods, "t"

Constant guidance system failure rate for

each period, "t"

Average repair time duration

Probability that a spare is available

(vii) Additional logistic delay time

Table B-II contains a listing of cos. data required for each candidate

guidance system.

B-I

TABLE B-II

IBM "CO_T-EFFECTIVENESS" ANALYSIS

COST INPUT DATA

(i)

(ii)

(iii)

(iv)

Research and Development Cost - all

expenditures for research development,

test, and engineering of all subsystems.

Investment Cost - purchase price per

developed subsystem in quantity buys.

Operating Expense one-time other opera-

ting costs (technical manuals, training,

field engineering, etc.) for each mission

plus spares cost.

All non-guidance costs associated with

mission - includes launch vehicle space-

craft, etc.

!II1ItI

This model requires that a fixed number of missions be accomplished.

Therefore, the user must estimate the number of attempts required per mission.

To determine the "best" guidance system configuration, the user per-

forms the analysis for each configuration and that configuration with the lowest

total cost of ownership is the best for the mission considered.

(II) The model developed by Sperry Gyroscope Company during their SSGS

work (References B2 and B3) also requires that a fixed number of successful

missions be accomplished. Table B-III tabulates some of the input data required

for each candidate system to determine the probability of mission success for

the candidate.

(i)

(ii)

TABLE B-III

SPERRY EFFECTIVENESS ANALYSIS INPUT DATA

Probability of successful operation of

guidance AGE

Probability of successful guidance

countdown - failure rate, repair rate, launch

window duration, time between launch

windows, abort probability

IIII!

B-2

TABLE B-Ill (Continued)

(iii) Probability that systems other than guidance

operate reliably

(iv) Probability of guidance system operating

reliably during ascent

(v) Probability of guidance system operating

reliably during subsequent phases of mission

(vi) Navigation accuracy requirements for each

mission

(vii) Probability that candidate guidance system

configuration satisfies prescribed accuracy

requirement

(viii) Total useful payload weight

(ix) Guidance system weight (burdened for power

Some of the cost input data required for the Sperry model is tabulated in

Table B-IV.

TABLE B-IV

SPERRY "COST-EFFECTIVENESS"

ANALYSIS COST INPUT DATA

IIII

(i)

(ii)

(iii)

Production cost of each candidate guidance

system required for each mission type

Cost of all other systems (launch vehicle

services, etc.) required for each mission

type

Research, development, test, and engineer-

ing cost for each candidate guidance system

B-3

As with the IBM model, the user must estimate the number of attempts

(systems) required to accomplish the mission with a specified probability of

SucceSS .

The basic figure of merit resulting from the manipulation of the input

data is an "assurance cost", A/C. This cost is

N(G + C)

(A/C) = P(MS)

where

= Number of attempts

G = Cost of candidate guidance system

C = Launch vehicle and other system costs

P(MS) = Probability of mission success.

A subordinate measure of guidance system worth which brings into effect weight

is the assurance cost per pound of useful payload. The candidate guidance

system with the lowest assurance cost is the preferred system according to

Sperry.

(III) Autonetics subcontracted development of the cost-effectiveness

model to Planning Research Corporation (PRC). This model (Reference B4) was

also designed on the assumption that a fixed number of missions must be accom-

plished successfully. Accuracy and reliability were treated as the measures of

probability of success of the candidate guidance system. The probability of

mission success was defined as the product of the probability of vehicle-payload

success and the probability of guidance success. The number of launches

required to achieve a specified fixed effectiveness level is found by dividing

the fixed number of successful missions by the probability of mission success.

The parameters weight and power are not considered either explicitly or

implicitly in the PRC cost model. Autonetics did include them by converting

power required to an equivalent power source we;ght which was added to the

guidance system weight. The sum of these weights was converted to cost by a

dollar/ib factor and added to the guidance hardware cost.

Some of the non-cost input data and calculations required for the PRC

developed model are tabulated in Table B-V.

IiIIIIII!lIIII

B-4

TABLE B-V

AUTONETICS AND PLANNING RESEARCH CORPORATIONS

EFFECTIVENESS ANALYSIS INPUT DATA

(i)

(ii)

Estimated Gross National Product

Estimated Air Force Budget

(iii)

(iv)

Forecast Air Force Astronautics Budget

Estimate mission types and number of launches

required for each mission (iterative comparison

with astronautics budget)

I!!

(v)

(vi)

l-_-" -"N% vJ._;

Perform mission analysis to determine navigation

accuracy requirements, guidance reliability

requirements, etc.

Perform navigation error analysis and reliability

for each candidate guidance system on each mission

IIII1I

(viii)

(ix)

(x)

(xi)

(xii)

(xiii)

(xiv)

(xv)

(xvi)

Determine probability of guidance success

Number of R&D flights

Number of operational launches per year

Number of years in the operational program

Number of payloads to be used per year

Number of guidance systems required per year

Number of men to be trained initially

Number of launch vehicles

Number of guidance and control readout stations

IIIII

(xvii)

(xviii)

(xix)

(xx)

Launch vehicle spares factor

Payload spares factor

Guidance spares factor

Number of circuit miles of land line communications

network per station

B-5

Cost data required for the PRCdeveloped model is tabulated in Table B-VI.These costs are (i) research, development, test, and engineering (RDT&E),(2) investment, and (3) operating costs.

TABLEB-VICOSTDATAINPUTTO PLANNINGRESEARCHCORPORATIONMODEL

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

Initial unit cost for launch vehicle hardware

Guidance research, development, test, and

engineering (RTD&E) cost

Initial investment for ground facilities

Payload development

Initial cost of guidance and control readout station

Unit cost for launch vehicles used in flight test

Payload prototype cost

Flight test unit cost of guidance system

Per-launch cost of engineering backup support for

flight test

Per-launch cost for refurbishment of test units and

facilities

B-6

(xi)

(xii)

(xiii)

(xiv)

(xv)

(xvi)

(xvii)

(xviii)

(xix)

(x×)

(xxi)

(xxii)

Investment cost per booster

Cost per payload

Cost per guidance system

Ground environment equipmen£ cost

Cost of payload AGE

Cost of guidance system AGE

Training cost per man of average skill

Pay and allowances

Data reduction and analysis cost per launch

Operational cost per readout station per year

Cost of land conmaunication network

Contractor support cost

IIIIIII!I

In addition, cost data on booster, payload, guidance, and AGE modifications

are inputs for this model. Other miscellaneous cost data are also required but

will not be listed in this report.

Basically, this model searches for the candidate guidance system which

satisfies a specified fixed effectiveness (determined by analysis of reliabil-

ity and system accuracy) and has minimum cost. The program computes cost first

on the basis of the probability of mission success, Psm, equal to the probabil--

ity of vehicle and payload success, Psv. It then computes cost with

III1!II

P =P P ,sm sv sg

where P_ = probability of candidate guidance system success. Psg is theI_ok_I_+,, o_ =_.._._,, _.,_ _ The differ-product of t.e probabilities of re ........ j ........... j .... e ....

ence between the two costs represents the cost of additional launch vehicles,

etc., due to imperfect guidance. Autonetics contends this cost difference could

be spent justifiably for designing and constructing an essentially perfect

guidance system.

(IV) TRW Space Technology Laboratories (now TRW Systems) developed

another cost-effectiveness model in their SSGS study (Reference B5). This

model is based on optimizing the dollar cost added to a family of missions by

the guidance system. Optimization may be made assuming a fixed number of

launches with varying success or variable number of launches for a fixed number

of successes. For a system to be considered, it must satisfy some admissibility

criteria established for performance and cost. A surm_ary of the data required

is listed in Table B-VII.

TABLE B-VII

TRW MODEL INPUT DATA

IIIII

(i)

(ii)

(iii)

(i_

(v)

Number of flights

Average reliability over _i_,_1_gh_._ (._n_,,r__r........ nF

individual reliabilities of system elements)

Change in number of flights due to alternative

systems

Unit cost for one flight including all non-guidance

equipment and operating costs

Change in nonrecurring cost due to alternative

guidance systems

B-7

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

TABLE B-VII (Continued)

Change in guidance equipment weight due to

alternative systems

Change in guidance power due to alternative systems

Launch vehicle unit cost (dollars/ib payload)

Power source cost (dollars/watt)

Power source weight (ib/watt)

Change in unit recurring costs for alternative

guidance systems

Change in performance (accuracy) of alternative

guidance systems

I

!I

i

I

I

I

IThis trade off approach optimizes total space program costs for alternative

guidance systems and requires detailed launch vehicle dollar costs. The funda-

mental measure of competing candidate guidance systems is seen to be dollars/

payload pound. That system with minimum dollars/payload pound would be optimum.

Applicability of SSGS Models

Each of the "cost-effectiveness" models developed for evaluating competing

guidance systems for the SSGS may be useful if given the same requirement, i.e.,

determine for the national space mission model that standardized space guidance

system which is most cost-effective. It can be stated that this was not the

problem for which NASA/ERC was seeking a solution. NASA/ERC was concerned with

development of a performance index which will trade off the system parameters

of accuracy, weight, power, and reliability, and thereby serve as a design aid

for selection of component subsystems of a strapdown guidance system.

Each of the SSGS models require some of the same input data. Table B-VIII

lists some of the common input data.

IIIIIII

B-8

TABLE B-VIII

COMMON INPUT DATA FOR SSGS

COST-EFFECTIVENESS MODELS

(i)

(ii)

Mission model

Navigation accuracy requirement of each mission

(iii)

(iv)

(v)

Probability of mission success

Candidate guidance system error analysis

Candidate guidance system reliability

(vi) Nonrecurring cost for launch vehicle and all other

non-guidance items

(vii) Recurring cost for launch vehicle and all other

non-guidance items

(viii)

(ix)

Nonrecurring guidance cost

Recurring guidance cost

Some of the models required weight and power of the guidance systems. Othersdid not.

IIIII

All of the models are heavily cost oriented. The cost data required is

generally subject to doubt or not available. Two of the models make use of a

figure of merit which can be expressed as dollars/pound of payload. For missions

with quite heavy payloads, the figure of merit would appear much less sensitive

to an increase in guidance system weight than a mission utilizing the same

guidance system with a low payload weight. The IBM and Autonetics models are

very much total mission cost oriented.

Greater length could be added to this appendix in providing additional

rationale for non-use of the four SSGS models. It was apparent that a less

complex and different type of cost function or performance index was required

to aid in the design of a strapdown guidance system. This model should be much

less mission and cost oriented than the SSGS models.

B-9

(B1)

(B2)

(B3)

(B4)

(B5)

APPENDIX B

REFERENCES

"Standardized Space Guidance System (SSGS), Program Definition Studies,

Phase IA Final Report, Volume VI, Cost Effectiveness (U)", SSD-TDR-64-

130(VI), May, 1964 (SECRET), IBM Federal Systems Division, Space

Guidance Center.

"Standardized Space Guidance Study (SSGS), Program Definition Study

Phase IA, Volume 3, Sections 4 and 5 (U)", SSD-TDR-674-131, Vol. 3 of 13,

May, 1964 (SECRET), Sperry Gyroscope Company.

"Standardized Space Guidance Study (SSGS), Program Definition Study -

Phase IA, Volume 9, Annexes D through F (U)", SSD-TDR-64-131, Vol. 9 of

13, May, 1964 (SECRET), Sperry Gyroscope Company.

"Standardized Space Guidance System Phase IA Study, Section 4 (U)", SSD-

TDR-64-129, May 28, 1964 (SECRET), North American Aviation, Autonetics

Division.

"Final Report for Phase IA Standardized Space Guidance System (SSGS),

Volume IIl: System and Subsystem Studies, Section 8 (U)", SSD-TDR-64-132,

May 29, 1964 (SECRET), TRW Space Technology Laboratories.

B-IO

APPENDIX C

APPROXIMATING THE PROBABILITY DISTRIBUTION OF

A VECTOR WITH NORMAL COMPONENTS AND ZERO MEAN

III

APPENDIX C

APPROXIMATING THE PROBABILITY DISTRIBUTION OF

A VECTOR WITH NORMAL COMPONENTS AND ZERO MEAN

III

An N dimensional random vector with normal components and zero mean is

completely described by its N by N covariance matrix. For a vector in three

dimensional space N = 3. Since only the magnitude of the vector is of interest,

two vectors which differ by an orthonormal rotation will have the same

distribution. The covariance matrix is always symmetric and nonsingular with

positive real eigenvalues. This implies that for any vector there is an

equivalent vector, differing by only an orthonormal rotation which has a

diagonal covariance matrix in the form

m

lI 0 0

0 12 0

0 0 13

where the l's are the eigenvalues of the original covariance matrix. The eigen-

values represent the variances of a vector with normal uncorrelated components.

The problem is then to determine the distribution of

J 2 2 2

_IIX I + %2X2 + 13X 3

IIIIII

where 11 > 0, %2 > 0, 13 > 0, and XI, X2, and X3 are independent and all normal

(0,i) random variables. This is a specific case of the general problem of2

determining the distribution of %1s2 + ... %Ks2, where the s i are independent

and s2 _ 2 X ./fi (Chi-Squared of degree fi)- Here 2 = i, fi = i, and K = 3.i

In general, this exact distribution is not known, so an approximation must

be used.* If Ii = %2 = 13 = a, the distribution is that of ax3, or 3ax /3. This

suggests approximating the distribution of IiX2 + 12X2 + 13X33 by the distribution

a_/d for some constant a and degree of freedom d. The standard way of doing

* K. A. Brownlee, Statistical Theory and Methodology in Science and Engineering,

John Wiley and Sons, New York, 1960.

I C-I

I

2this is to determine a and d by equating the first two moments of aXd/d with

those of klX _ + %2X2 + %3X_. For a discussion, see the reference, page 235.

It can be shown that the third moment of a_/d, with a and d determined as

above, is smaller than the third moment of XIX_ + _2X2 + ,X3X_ (if the %i's are

not equal). Thus, it appears that the tail of aX_/d is smaller than that of

%1X2 + 12X2 + _3X_.

As we are particularly interested in the upper 0.99, 0.999, and 0.9999

points of the distribution, it was thought that a better approximation in this

range might be obtained by determining a and d by equating the first and third,

or perhaps second and third, moments of a_/d and _iX2 + k2X2 + %3x2.

Several sets of ki's were chosen, and for each set a Monte Carlo experiment

of 1,000,000 random vectors was performed to approximate the distribution of

XlX2 + X2Xp2 + ,k3X_ . This distribution was then compared to the three a_/d

distributions for a and d chosen by equating the first and second, first and

third, and second and third moments. Figures C-I and C-6 show the Monte Carlo

results compared to distributions based on equating first and second moments,

first and third, and second and third. Figures C-5 and C-6 are special cases

2where %IXI + X2X + k3X is exactly distributed as a /d and equating any twomoments gives the correct a and d and serves as a check on the Monte Carlo

technique.

The necessary m's and d's for each approximation are calculated as shown

in Table C-I.

TABLE C-I

COMPUTATION OF a AND d FOR EQUATING MOMENTS

IllIIII!IIII

Moment a d

C-2

1 and 2 kl + %2 + %3

1 and 3 _i + k2 + k3

2 and 3 2 %23)id(_21 + %2 +

2

(k I + k2 + k3)

2 2 _23)(Xl + %2 +

(_i + 12 + k3)

3/2

3 >33)1/2(X31 + k2 +

(_21+ _2 X_)32 +

3 >,33)2(_31 + %2 +

IIIII

It is not necessary to solve the eigenvalue problem since the necessary sums can

be calculated directly from any covariance matrix as follows:

3 3

Z X = ZC..

i= 1 i i= 1 ii

3 3 3

ZX2= E ZC..

i= I i i=l j=l l]

3 3 3 3

E X3 = _ _ kE!CijCikCjki= 1 i i=l j=l =

where the C.. are the elements of the covariance matrix.lj

T_ble C-II presents the Chi-Squared distribution for non-integer degrees

of freedom ranging from I to 3. The approximation technique may be used by

obtaining a and d from Table C-I and entering them in Table C-II to obtain the

desired probability. It should be noted that for approximations equating

moments I and 2, or I and 3, a is the trace of the covariance matrix.

The significance of d, the degrees of freedom, may be seen by studying

the shapes of the surfaces of equal probability. With d = I the surface is a

line, i.e., all random vectors are constrained to lie on a single line.• _I.... ; .... _ A = 9 2 A_e_ nw A = R = _nh_r_ _ ohr_ned for surfaces of

equal probability.

The state of the art prior to this study has been to use only the first

moment. This assumes the distribution to be normal (d = i) with a standard

deviation equal to the square root of trace of the covariance matrix.* For

larger degrees of freedom, this is a conservative assumption as can be seen by

comparing Figure C-5 to the other curves.

* "AC-9 Guidance System Accuracy Analysis", GDC-BKM 66-028, September 30, 1966,

General Dynamics Corporation, Convair Division.

C-3

0

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ol-i

U,4

o.,....4

0rm

4-)

(D

,,..4

.,-4

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C-4

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C-5

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C-6 II

I

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C-7

!!!I!!IIIIIIIIIII

C-9

I

IIII

IiIIII

I

II

III

C-10

IIIIIIIIIIIIIIIIIIIIII

APPENDIX D

CONCEPTUAL IMU DESIGNS


APPENDIX D

CONCEPTUAL IMU DESIGNS

This appendix contains sketches of fifteen conceptual strapdown IMU designs.

Figures D-I through D-9 depict nine designs for the case where the base of the

IMU is mounted horizontally in the vehicle. The analytic expressions which are

solved to determine the dimensions of the IMU for a set of three gyroscopes and

three accelerometers are presented on Figures D-I to D-6. Similarly, Figures D-IOthrough D-15 depict six designs for the case where the base of the IMU is

mounted in a vertical plane in the vehicle. Restrictions such as limitation of

that design to pendulous accelerometers are noted on the figures.

D-I

}--4

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APPENDIX E

SINGLE-DEGREE-OF-FREEDOM GYROSCOPE ERROR MODEL

I!III!II

APPENDIX E

SINGLE-DEGREE-OF-FREEDOM GYROSCOPE ERROR MODEL

INTRODUCTION

The error model contained in the ERC provided strapdown error analysis

program (SEAP) accounts for only the major steady state error torques of the

single-degree-of-freedom (SDF) gyroscope. Recent papers(Ei-E4)*have shown

that an expansion of the error model to include the effects of vehicle dynamics,

gyroscope dynamics, and rebalance loop characteristics should be considered

since these effects can increase the total error quite significantly. Of equal

importance with the expansion of the SEAP error models is the need for astandard set of ---_ i_ _ +-_uenoLl._ the error coe_yL,,_o_ fficients for each error source.

It is suggested that NASA/ERC give serious consideration to adopting and

requiring use by its contractors of a standard error coefficient nomenclature

and set of symbols.

DEFINITION OF TERMS

M

m

H

Vector sum of all external torques acting on gyro

Vector angular momentum of gyro (referred to an origin at

pivot point and based on inertial velocity)

I £0sp sp

g

Wl ,g

G imba 1

Angular velocity of the gimbal with respect to (wrt)

inertial space

x,y,z Gimbal (moving)axes, where x 111Ay IIOA

z liSA

Ix, ly I ZMoments of inertia of the gimbal about x, y, and z axes

Sp Spin

* Superscript number denotes references.

E-I

00 du:/dt

Aca,g

Angle of gimbal wrt the case

i, j, k

ica' Jca' kca

Unit vectors along moving axes x, y, z

Unit vectors along (fixed case axes X 'Yca' Z )ca ca

Angular velocity of case wrt inertial space

The X component of case wrt inertial spaceca

II!I

i_, G Torque generator applied torque

C DDamping coefficient

a I

aS

a o

Acceleration along Input Axis (IA)

Acceleration along Spin Axis (SA)

Acceleration along Output Axis (OA)

IIIi

DERIVATION

Newton's law in rotational form, applied to a single degree of freedom

gyro (rigid body) may be written as

app I

(])

E-2

where the angular momentum, _, is referred to an origin at the body's cg and is

based on inertial velocity (cg and pivot point are coincident).

app ,gg

g - Gimbal (or float) (2)

= " " + Hwh) (3)+ JglyUJl, + _(IzUOl,g zH(b°dy) iglxWl 'gx gy

where H .= I m . X, y, z are gimbal axes with x corresponding to (_) IA,_. wh-. spasm.

y _ uA, ano z _a. From (3)

g= mg_ixWl,g x) +jg(iy°° I ) +- _ ),gy kg(Iz_l,gz

(4)

(treating Hwh as a constant). Then using (,3) and (4), (2) becomes

IIII

Mnpp

i jgg g

mT = WT. _ 00I.m + _g(IxWl,g.) + jg(lyml,g,) + kg(IzWI,gz)•VX " -y Z j

+ Hwh)1;IxW I I o0 (IzOOl,

'gx y l,gy gz

(5)

or

- [ix IM = _g - (I I ) WI + W ]+app 'gx y z ,gy Wl,g z Hwh l,gy_

7g [lyLl,gy- (Iz -Ix)e I _I'gz 'gx- Hwh Wl,gx] + (6)

kg [Ix_I, gy- (Ix-I) Y

E-3

The torque componentof interest in this case (SDFgyro) is the one along theOA(_ yg), so with x _, IA, y _ OA, and z _ SA

I_appl = IoA'J_l'gy - (IsA - IIA) WI'gzWI'gy - HwhWl''gx " (7)

To convert from gimbal motion with respect to (wrt) inertial space to gimbalmotion wrt the case (ca), use

+ ® = +A (8)= wI 0JWI,g ,ca ca,g I,ca ca,g

Gimbal freedom is about yg (_ OA), so the transformation from moving axes tofixed (case) will be through A as shown in Figure E-I.

ca,g

IIII

X

g,

1___J___,_ Yca ' OA

/ Aca,g

/ A Z , SRA! ca,g ca

_ z , SA

X gca

IA

IIIII

7X

g

FIGURE E-I. GlObAL AND CASE AXES

= I X cos A + 0 - i sin Aca,g Z ca,gca ca

= 0+]y +0Yg ca

(9)

II!I

7Z

= 1 sin A + 0 + i cos Ax ca,g z ca,g

g g g

E-4

or

FW I'gx

I

i I,gy

i r

i wI IL 'gz

Then by treating Aca,g

cos Aca,g

sin Aca,g

-sin Aca,g

0

cos Aca,g

as "small",

ml -- w - wI A'gx l,ca X ,ca Z ca,g

eI -----_ +A,gy l,cay ca,g

l,ca X ]

_l,cay + Aca,gl

' 1el,ca

_ z _j

(lO)

(11)

Wl,g z _ Wl,ca_, Aca,g + Wl,ca 7A

IIII

Before expanding (7) by using (ii), we will include the effects of a small mis-

alignment of the gyro's OA and SA with respect to the IA (i.e., the OA and SA

are each nonorthogonal with respect to the IA). Symbolize these small angles

by Am_. and Am_ ., and let "t" indicate "true" as shown in Figures E-2 and E-3.

The misalignment angles will cause cross-coupling of the components of gimba£

rate since the "y" and "z" components are not orthogonal with respect to "x"

(consider each rate component singly)•

E-5

A

mOA

-,_f-_- J_ Ygt

__ _

_ ZgZ

(oA t)

gt , (SA t)

X _ X

g gt

FIGURE E-2. MISALIGNMENT OF IA WITH RESPECT TO SA AND OA

Wl,g xt

= Wl,g x + (a component of _l,yy

+ a component of _'l,g z)

Aw A + Wl,g z ms Aw I = w I

,gxt 'gx l,gy mOA

(letting sin A _ A ) and similarly,mOA mOA

- Wl, AmOA= + (component of w I ) = Wl,gy gx'gxw I 071

,gy'gYt

and

= W.. + w I o. A'_x ms A'331 ,r I ._gz

e_zt

(12)

(13)

(14)

(15)

I

lIIIIIIIIIIIIIII

E-6

IiI

-x

gt

+A

i ,mOA

I!iI!

f _ Ygt (oAt)

J '_-z7 Yg OAf, .."i• _......... "_Am

t \

_o__/ _',,2 _,g__o_x-.x,x-.x,x-.x,x-.x__ e AI, gx mSA

._. _ -_ mSAV ms A z

gt , (SA t)

x

-t

xg

Using (13), (14), and (15) to rewrite (ii)

(a) e = (el, caxI ,gxt

el,caz Aca,g)" + (_l,cay + Aca,g) A•mOA

+ (_Ol,caz + Wl,ca x A ,g) Aca mSA

(b)

Wl'gy t= (O)l,cay + Aca ,g) - el,ca Z AmOA

(16)

(c)= (el,ca Z + el,ca z A g) + eI Aca, ,caX mSA

E-7

•_.,:o_l'_¢ Lob,

_c_, _

_i.,$%%

%

%

1!

!!I!!!II!!

IMappllt OA = IOA _I,cay + I0A Aca,g + (IIA - IsA)(Wl,ca X Wl,ca Z)

+ HWl, A - HWl, A" HWl,cax + HWl,caz Aca,g Cay mOA ca Z mSA

For an imperfect single-degree-of-freedom (SDF) rate integrating gyroscope

operating in a closed loop

IMappl|| OA-- -CD Aca,g + _RROR + MTG

where:

RROR = Error torques

MTG = Applied torque from gyroscope

CD = Viscous damping coefficient °

L_ UW

MTG = -Aca,g SSG[A;e] SEL[e;i] STG[i;M] = -Aca,g SLoop[A;M]

SSG[A;e]

SEL[e;i]

STG[i;m]

SLoop[A;m]

= Sensitivity of the gyro signal generator, volts/degree

= Gain of the electronics, amperes/volt

= Sensitivity of the torque generator, dyne cm/ampere

= Closed loop scale factor, dyne cm/degree o

The error torques are caused by imperfections in the construction of the

gyroscope. The error sources are principally:

(i) Fixed torque due to flex lead and signal and torque generator

reaction torques.

(2) Elastic restraint torque due to torque gradient proportional to

output angle.

(18)

(19)

(20)

E-9

(3) Rigid pendulosity torques caused by the center of mass of the torque

summing member being displaced from the gyro output axis.

(4) Elastic compliance torques caused by the deflection of the center of

mass of the torque summing member when a specific force is applied.

Treating the compliance torques first, the elastic deflection, A, resolved

along the principal gyro axes, SRA, IA, and OA, shown in Figure E-I can be

written as

IIII

ASR A = m[Kss(Sf)sRA + KsI(Sf) IA + Ko(Sf)oA](21)

AIA = m[KIs(Sf)sRA + Kii(sf) iA + Kio(Sf)oA ]

AOA = m[Kos(Sf)sRA + Koi(Sf) iA + Koo(Sf)oA ]

where K_. = compliance coefficient due to a displacement along the x axis as aA%

result of a specific force along the y a=<is.

The torque about OA resulting from the deflection A, of a unit mass m,

and the specific force, sf, is then given by

(22)

(23)

2M = m f) _ AiA(sf ) ] (24)c [AsRA(S IA SRA "

For an SDF gyr% AOA should make no contribution to torque about the output

axis and is therefore not considered further.

Substituting equations (21) and (22) into (24) one finds after combining

terms that

2 + KsI(s f) 2 (sf) 2M = m [(Kss - KII)(sf)IA(sf)sRA IA KIS SRA

C

+ Kso(Sf)oA (sf) IA - Klo(Sf)oA (sf) SRA] "

(25)

The torques due to mass unbalance along the spin reference and input axes

can be expressed as

E-10

!!!

!!!!

M U = Us(sf)l A - UI(Sf)sRA (26)

where US, U I = moment of mass unbalance about OA along + SRA and + IA.

With the addition of the fixed torque term R, elastic restraint torque

HEAca,g , and a term to account for torque about OA due to the specific force

along the output axis, Uo(sf)oA, one now has the ten terms that make up the

error torques of the SDF gyroscope based upon linear theory. Summing these

terms, we have

MERRO R = R + HEA + Us(sf) - Ul(Sf) + Uo(Sf)ca,g IA SRA OA

+m 2

(Kss - Kll)(sf)iA(Sf)sR A + KsI(Sf)_A RAI( -KIs(Sf)2RA + Kso(Sf)oA(Sf)iA - Kio(Sf)oA(Sf)s o

(27)

In terms of acceleration components (as, al, ao) we have

_RROR = R ÷ HEA + - m2(_S - KII)alAaSAca,g " USalA UIaSA UOaOA +

2 2 2,. 2 ± 2_ .... 2v+ *u _SIalA .....IS_SA ......SO-OS-IA ....IO_OA "

(28)

Substituting for IMappltL 0A in equating (18) we find that

/_-CDI + HEA + R + - UoaoA_ca,g ca,g - USaIA UIaSA

< _ + 2 7Kss KII) alAaSA KSlalA

+m2 I- K a2 + K a a -K a a I

__ IS SA SO OA IA IO OA SAj

_ _ Aca,gSLoop[A;M ]

_IoA_I , ca Y loAA'ca ,g

+ "" _ HWl, ca>

+ (IIA - IsA)(Wi,caxWi,CaZ )

+ HWl,caz Aca,g + HWl,cayAmoA

- H_ I A\ 'CaZ mSA

• (29)

E -ii

In guidance error analysis one is interested in the error in the angularrate measurement, so we must divide all terms by H, the angular momentum,toobtain angular rate. To avoid carrying through coefficients involving l/H, wesuggest the following definitions of which manyare used by the CentralInertial GuidanceTest Facility, Holloman AFB, NewMexico.

III

R U0 m2KDFR = _ DUO = DKI S = IS

H H

2(Kss 2KsoUI = m - KII ) = m

DUI - H DKM H DKSO H

US = m2KsI = m2KloDUS = _-" DKSI DKIO

H H

Now

w I, A = 0' 0:i,cax WlAca = wOA ' ca, g 'Y

_t_l,Cay _OA ' _)I,caZ _SA ' Aca,g = e

III

A = eca,g

Substituting and rearranging terms in Equation (29) we have

E-12

mlA + E8

!DFR - Dusa I + DuIa S - Du0a 0

t + DKMasa I + D 2• KS IaI

!- DKIS a2 + DKSOaOa I - DKlOaOa S ) =

| 1OA• -

i -_ T wOA" _.'' H ISA) WlAWSA

!

CD+

+ SLoop [8;M]_

(30)

As previously stated, we are interested in the error in the angular rate, so it

is appropriate to separate those terms which contribute this error from the rate

we are trying to measure, WlA. Table E-I lists the error term symbols, forcing

exgitation, and some of the nomenclature associated with the terms. In addition,

two errors not in Equation (30), coning and quantization, are lis _-=

We shall assume that the forcing excitations are made up of trajectory

accelerations and angular rates and vibratory excitations. The vibratory excita-

tions are caused by body bending and engine induced vibrations. We assume the

an_ular vibration to be _vib = am sin wt and the linear vibration to be avi b =Aw L cos wt. The linear vibration may be averaged by integration over one period:

TAw2 cos wt dt = Aw2w [sin wt]T = 0

O

Similarly, for the linear vibration squared, we have:

J" TA2 4 2T w cos wt dt -o

A2_ 4 ....... T A2w 3

Tw LilZ wt + 1/4 sin zWCjo - T [1/2 wT]

Therefore, the following is true for the averages over one cycle

2 _ A2w 4

avib 2 ' avi b 0

E-13

r-_

Q)

or._

o

r.z.]

Z

r_

0

o

0

r_

0

o

z

r._

o01_ .,-4

o o

Jo

g

o

e_,.-i ,.-.i

_ o.,_ _ul

,.-4o,-4

..4 _

_.i ,.rj

o

tl-t

_ -_,

.,_ ._ "8

oiJ

p.0

o

4J

:J

.. _ .. _=•_ I:_ _ ..4 14 -,-t I._

•. ._ '_ .._ • _ •

o° .._ _ ,_ _ _ o. _

,po

,.--4

0

II/

N

0

(D

H 0 0

o

_J

o

oQJ

"_ _ °

IIJ o _

.,-4

M m _

_ N N

o _ _CO 0 co

m 0 oo

-,_ _ -,_ ._

._ .,.4 m.

0

m _

_ o

<_

IIIiIIIIIIIIIII

E-14

IIIIIi

It is easily shown that linear vibration contributes to gyroscope angular

rate measurement error through the compliance drifts. Angular vibrations cause

rectified drifts for those terms excited by angular rates.

For terms which involve cross coupling or the product of two accelerations

such as asal, only those frequencies present in both aS and a I will yield anet contribution when averaged over many cycles.

Let aI

AlW = cos (Wit) and aA = AS002 cos (Wst + _IS )

o

LimT -_

ala s = 0 if wI _ wS

A IAs w4

alas = 2 cos _IS if wI = ms

A similar procedure would apply for angular vibration, if it is ass_Ted

that the angular vibrations remain the same during a particular phase of the

flight, then the rectified drift terms can be treated identically to a constant

gyro drift.

SEAP presently contains the error in the torque generator scale factor

(STn). The error in the rebalance loop scale factor also consists of the error

in the signal generator (SGs) and rebalance loop electronic scale factors (SEL).

These two terms' contribution to the loop scale factor error could be treated

separately and then the root sum square of the three terms could be formed to

find the overall effect of the rebalance loop scale factor errors.

An additional error in the measurement of WlA which may be included underscale factor error is the error in the gyro angular momentum, AH, caused by

subjecting the gyro to angular rates about the spin axis. The resultant drift

error as a function of time can be expressed as(Reference E5)

IspAf_s(t) = £ -I[G(s) %A!

H _)sp

where G(s) is of the form:

G(s) =

2s

2

s + k I s + k2

E-15

A constant rate about the spin axis would not cause a steady-state Af_s to occur.

Sinusoidal vibrations occurring about the spin axis and at the same frequency

but phase shifted about the input axis cause rectified drift effects. This

error source, sometimes referred to as spin speed modulation error, can con-

tribute significant drift rates which are forced by the vehicle's angular

vibra t ion o

Addition of the four compliance terms as well as the dump term to SEAP

poses no problem for the trajectory accelerations. The vehicles linear vibra-

tion spectrum must be known to determine the drifts due to effects of the

vibration and the five gyroscope compliance coefficients.

For a constant body rate profile, no drift rate due to OA angular accelera-IOA

tion occurs, but a gimbal angle, _ = --_- _OA, would be interpreted as the

integral of a rate about IA. Therefore, compensation might be required. This

compensation could be performed in either of two ways. (E5) If we assume the

vehicle follows the nominal trajectory, the angular rate about each of the

gyroscopes' output axes could be stored as a function of time in the computer

and compensation accomplished. Alternatively, the rate about a gyroscope

input axis collinear with another gyroscope output axis could be used by the

computer to subtract the incorrect output angle due to this error.

The error due to anisoinertia is usually negligible for the nominal rate

profile. Sinusoidal vibratory motions about the spin and input axes give rise

to a rectified drift. The average value of this drift would be of the form

IIA - ISA_ abe2 cos• H 2

IIIIIIIIIII

where WlA = aw sin _Dt and _SA = be sin (wt + _).

Spin axis cross coupling primarily gives rise to kinematic rectifi¢ _tion

errors. If angular vibrations about IA occur at the same frequency as angular

vibrations about SA, the resultant error has been called spin-input

rectification. If angular vibrations at the same frequency occur about OA and

SA, the resultant error has been called spin-output rectification. In a pulse

torqued gyro, the dead zone of the float can also cause a rectified drift of

4 00a cos @ where 0 4_ = .the form _0 d 2 = _ d sin wt and _SA _a sin (_t + _) (E5)

The elastic restraint of a gyroscope is easily measured and can be

compensated. Therefore, this term can usually be neglected in navigation

error analysis.

Coning has been discussed thoroughly in References E2, E4, E5, and E6.

For a strapdown system mounted on a perfectly rigid body with infinitely tight

gyro rebalance loops and negligible quantization error, no inherent geometric

coning would occur if the data were processed with a perfect computer (infinite

speed) .

E-16

Bending and flexing of the IMU and vehicle structure does give rise to geometricconing errors which are usually neglected. In strapdown, coning motions aboutIA (00SAand _OAare sinusoidal and properly phased, _IA = 0) and OA(_SA and _elAare sinusoidal and properly phased, _OA= 0) cause cross coupling to occurthrough the spin axis cross coupling error term. The average drift rate forgyro geometric coning has been given as (Reference E6)

i= -- 0_ sin_D 2 qoro

where

q = amplitude of angular motion of gyro about its OA,

= qo sin wt,

r = amplitude of angular motion of gyro about its SA, and

= r sin (_t - _).O

Quantization error occurs for pulse rebalanced instruments because the

H J'Toutput of the gyroscope is no longer 0 = CD o _IA dt but is the summation of

Ae pulses. This gives rise to errors in the computed direction cosines. Since

this error is a system associated error, it will not be discussed further in

this appendix.

Sensitivity of the gyroscope to thermal gradients, excitation variation,

and external magnetic fields can be treated by preprocessing the appropriate

then taking the root sum square of the sensitivity terms for each coefficient

and the uncertainty of the error coefficient. For example, let us assume thatO O

the sensitivity of DUS is 0.i ( /hr/g)/ F, the uncertainty in DUS is 0.0iO

°/hr/g and that the temperature can be controlled to 0.5 F. Therefore,

=_0.i_. x 0.5) 2 + (0.01) _ = 0.051°/hr/g.. This value would be used as theDUS

error coefficient in the SEAP analysis.

E-17

(E1)

(E2)

(E3)

(E4)

(E5)

(E6)

APPENDIX E

REFERENCES

Roantree, J., and Kormanik, N., "A Generalized Design Criterion for

Strapped-Down Inertial Sensor Loops", AIAA/JACC Guidance and Control


Farrell, J. L., "Performance of Strapdown Inertial Attitude Reference

Systems", AIAA/JACC Guidance and Control Conference, August, 1966.

Thompson, J. A., and Unger, F. G., "Inertial Sensor Performance in a

Strapped-Down Environment", AIAA/JACC Guidance and Control Conference,

August, 1966.

Alongi, R. E., "Strapdown System Application Studies Related to a Surface

to Surface Missile of i00 to 400 Nautical Miles", AIAA/JACC Guidance and

Control Conference, August, 1966.

Levinson, Emanuel, "Aircraft, Missile, and Space Inertial Navigation",

Chapter 5, Training Notes, Sperry Gyroscope Company, 1963.

"n "Macomber, G. R., and Fernandez, M., "Inertial Guidance Engineerl g

pp 505-507, Prentice Hall, Inc., 1962, Englewood Cliffs, New Jersey.

IIIIIIIIII1I

1

!

E-18

IIIIIIII

II

II

APPENDIX F

TORQUED PENDULUM ACCELEROMETER ERROR MODEL

APPENDIX F

TORQUED PENDULUM ACCELEROMETER ERROR MODEL

INTRODUCTION

The error model contained in SEAP for accelerometers does not include

those errors due to the dynamics of the instrument. A derivation for the

torqued pendulum accelerometer (TPA) has been made to determine those terms

which are not contained in SEAP.

DERIVATION

Applying Newton's second law of motion in rotational form to the rigid

body with mass, m, depicted in Figure F-l, we have

J

P (Pivot)

•'f c (cg).// "

/-

FIGURE F-I. PENDULUM

i "" dH

X = X(l)

where _ is an externally applied force at the center of gravity (cg).

Cext

l - x dH -- _) .

Now,

(2)

F-I

Let Xp, y_, and z denote pendulumaxes (chosen to coincide with the principalaxes so t_at products of inertia vanish), and

(force of pivot on the body due to

mRIp= FP I

inertial acceleration of pivot). Then

Now

x Fp --PC

i jp zP p

Rpcx

ip (Rpc Fp - Rpc Fp )

Y z z y

Rpc Rpc = +3p (Rpc Fp Fp )Y z z x " RpCx z

Fp Fp Fp { --+k (Rpcx y z i P x Fp - Rpc Fp )

Y y x

and

H = i (I w ) + jp (I a_ ) + k (Iz w )p x l,p x y l,Py p l,pz

dH _ - - .

d-_ j = i (Ix ) + jp (lye I ) +p P _I,P ,p Px y

Taking the cross product of -_I,P and H, we have

71 J kP p p l (-) (I

P y

Wl, P x H = WI,P _ Wx I,P I,pzY = +jp (') (Iz

IxWI,P x IyWl,Py IzU'l,pz+k (-) (i

P x

(I z Wl, P ) °g

-I) a_ w

z l,Py l,p z

Ix) a_l,Pz_l,Px

- ly) w a_I,P x I,P Y

F-2

(3)

(4)

(5)

(6)

III

!Using (3), (5), and (6) to expand (2), treating Rpc as a principal axis

! z(Rpc = Rpc = 0), and constraining pendulum motion so that only the OA (y) componentx y

I of torque is of concern, we have

MapPy -_MapPOA (Rpcz FPx ) + ly Wl, P - (Iz " Ix) wl'pz wl'pz (7)

Making a coordinate transformation from pendulum axes (x,y,z) to case axes

(X,Y,Z) by using a positive displacement angle Aca,P (i.e., rotation about +Y)as shown in Figure F-2, gives

ZZ

/

I Y, y, OA/

| u,v_ + Aca,p

! ,x (IA)

I /-xI,

N Aca, p

IIlI

FIGURE F-2. COORDINATE TRANSFORMATION

i = iX cos A + 0 - k sin Ax ca,p z ca,p

jy 0 + jy + 0(8)

k = P_. sin A + 0 + k cos Ay X ca,p z ca,p

F-3

or

cos A 0 -sin Aca,p ca,p

0 i 0

m I i sin A 0 cos A i

'Pz_j ca,p ca,p i_ __ _.J

Let us treat cos A m i and sin A m A . Thenca,p ca,p ca,p

wl,Px _ (Wl,ca X - _l,ca Z Aca,p) ,

=.

ml,Py (Wl,cay

I

u_I ,ca X

p) and (9)+ Aca '

IIIIIIIII

wl,pz _ (Wl,ca Z + Wl,ca X A p)ca,

Including effects of a "small misalignment angle of the pendulous accelerometer's

and An_ shown* arbitrarily as anglesOA and the pendulum axis, PA (i.e., Amo A A'

developed by + rotations) gives

III

Ool,Px = 001,Px - wl,Py A + Wl,pz AmOA mp At

eI = Wl, P - wl,Px A'PYt Y mOA

wl,pz = wl,pz + wl,Px AmpAt

(lO)

IlII

* See Figure E-2, Appendix E, page E-6.

F-4

!

!

!

Using (9) in (i0) gives

--_ A p) - A" (_l,cay) + A )WI'Pxt (WI,cax WI,caz ca, mOA mpA (cUI,caz

I _I --_(Wl,cay + A ) - A (ml,ca X)'PYt ca, p mOA

-- (Wl,ca z + UDI,caX Aca,p)

!el,p

+A )_A (Wl'cax

(II)

Now

°,

Wl, P = (Wl,ca Y + Aca p) - A Wl,ca XY t ' mOA

(12)

I Also,

Fp m (FPx - )I Xp FPz Aca'p

I 8o,

I Fp _ (FPx - Fpz Aca,p) - A (Fp_ + A (Fez)x t mOA mpA

!Fur the rmo re,

R -----Rpcz PCz

t

(13)

F-5

wI,pZt

wl,pz, _l,Px = (O_l,caz + _l,ca Z Aca,p) + AmpA (Wl,cax)t t

p)(Wl,ca X _) Al,ca Z ca, (AmoA Wl Cay + A, mpA

2

Wl, P --_Wl,ca X Wl,caz + el,ca X (Aca,pXt

2

+A ) - _I AmpA ,cab ca, p

L

2

- el,ca Y el,ca z A + AmOA el,ca Z mpA

_I, ca Z) ]

(15)

!!I!!

2 2

R_I,P el,P _ el,ca X el,ca Z + A + _I )zt x t mpA (el,cax ,ca Z

- A (_l,cay Wl,ea z) + A 2 - e_, )mOA ca,p (el,cax ca Z

IIl

and

R F =R (F - F A - F A +F A )

PCzt Pxt PCz PX PZ ca,p PY mOA PZ mPA

(16)

Then, (7) becomes [using (16), (15), and (12)]

E

M = R e|Fpx - F A - F A + FapPoA PCz PZ ca,p PY mOA PZ

°.

AmpA] + IOA el,cay + IOA Aca,p

+ (IIA - IpA) [_l,ca X o01,caZ + AmpA (o01,2

2

'+ e I - A (el,ca Y el,ca Z)caX ,caZ) mOA(17)

( _I

+ Aca,p l,ca X - e 1,ca Z

Equation (17) equates externally applied torques to inertia reaction torques arising

from inertial acceleration of the pivot and angular motion with respect to inertial

space.

F-6

I to be included in the _p terms, (Fpx ___ip x

!

- - (_iAssuming the effects of Earth's gravitational field [g = G - a;l,E x ,E

! ma ) we than have

PX

or

Mdamp + Mtg _+ (1)M = [right hand side of (17)]

" L ]+ Cd A = mR -a + a A + a A - a AIOA Aca,p ca,p PCz PlA PPA ca,p POA mOA PPA mPA

m

x _c ) ],

(18)

"I0 A Wl,caoA

r-

- (IIA- IpA) LWl,CalA Wl,capA2

AmpA (_ + wI )+ ,cal A ,CapA

2 _ 2

+ Aca,p (Wl,CalA Wl,capA) " Wl,capA _l,caoA AmOA] - M t _+ (1) Mg

where :

(1)M - inaccuracy torques

(1)M _ (E)M + (U)M

(E)M - error torques (determinlStlC or average)

(U)M - uncertainty torques.

Let

= [Stg[ • sMtg i;m]s "A

Sm[e:i ] -°[A;e] - ,p

M - torque

tg - torque generator

S - sensitivity

i - current

sm.- signal modifier

sg - signal generator

e - volts

A - angle

ca - case

p - pendulum

(19)

(20)

F-7

Tests which provided an understanding of the error sources within accel-erometers were followed by different and improved designs to meet demandsforlong-term stability, greater accuracy, and wider dynamic ranges. One particularsource of difficulty which tends to limit single-axis performance is output-axis friction. A frictionless restraint on the OA is desired, but is mostdifficult to achieve whencross-axis suspension must be sufficiently stiff.The torqued pendulous accelerometer (TPA) is an example of improved designwhich forced that difficulty into submission.

Nonlinearities of the restraint torque device (torque generator) can provetroublesome, and require high loop gain so that it functions near a nullposition (i.e., the pendulous element displacement angle should be very closeto zero). Resulting error is a function of "g" since it increases as torquecurrent increases.

Precision balance is a design requirement of the TPAsince an uncertaintyin balance contributes to instrument error through the scale factor. Also,scale factor instability, while predictable to someextent, can produce anintolerable uncertainty.

Null instability is another source of instrument error which can limit theaccelerometer's performance. The uncertainty in output resulting from nullshift has long been one of the foremost concerns when greater accuracy wasspecified since it is very difficult to control or predict. Uncertainty inknowing exactly where the instrument axes are (misalignment) with respect toeach other leads to errors in the output when cross-axis acceleration andangular velocity in space occur.

Pendulum inertias about the instrument's input axis and the pendulous-element axis will, if unequal, cause output errors.

The sources of uncertainty in an accelerometer are represented in anempirical equation which assumesthem to be sensitive to powers of acceleration(i.e., zero power, Ist power, 2nd power, etc.). Test results, then, showinglarge and repeatable characteristic values ascribed to definite physical factorswithin the instrument are used to affect an improved design (i.e., through modi-fication or redesign).

In terms of specific force, to which all accelerometers are designed torespond, with

x _ input axis (IA)

y _ output axis (OA)

z _ pendulous axis (PA), andA

(sf) due to acceleration = -a =ForceMass

F-8

IIIIIIIIIII

(Output-Input) _A (E)sfind=K 0 + KI (sf)IA

,

+ K4 (Sf)oA

+ K 2 (sf) 2 3IA + K3 (sf)IA

+ K 5 (Sf)pA

+ K6 (sf) iA (Sf)oA + K 7 (sf)iA (Sf)pA + ...

The accelerometer unit (au) output varies in form with the design of the instru-

ment, as the units of the coefficients vary accordingly. Let the error

coefficients be referred to as "Instrument Output Units", IOU, and be defined as

follows:

K 0

K 1

K 2

K 3

- bias (in IOU's)

- scale factor (in IOU/g)

- second order nonlinearitv effect (in IOU/g 2)

- third order nonlinearity effect (in IOU/g 3)

K4,K 5 - cross-axls acceleration sensitivities (in IOU/g)

K6,K 7 - cross-coupling non-linear effects (in IOU/g 2)

(The model can be expanded to include higher order cross-axis accelera-

tion and cross-coupling nonlinearities if necessary.) The performance function

for an ideal TPA [from (19)] is

ImRocz z

IOA "" + " = _" _d )_-ain_IA - Mapp/Cd

(21)

t

/c dPCZ

= (-a in)

Aca'p(S) FfloA'N 17=I_,5--- _ s +-L,.C d J J

As a pendulum displacement, Aca,p _ develops, it is detected electrically(signal generator), and the szgna_ produced is proportional to the angle's

magnitude. To use this signal for actuating a torquer, it is first modified

(amplified and conditioned) and then becomes the input current to the torque

generator. All are sources of error and uncertainty.

(23)

F-9

-a.in

Let : Ssg[A;e]

Mapp

SSm[e;i]

tg

=AStg SpA[A;i ]

s[_IOA_ s + 1

Cd

s Ltg[i;m]r

Aca,p

sg[A;e] ,i]

(24)

then

itg

(s) _mRpc/S t_ (-a in), -- I II _ n

_---_s + s+ 1pA

(25)

The kinematic equation of motion for the TPA (Equation 17) shows that false or

erroneous contributions to acceleration occur when a pendulous element angle

develops, when misalignment of axes is considered, and when case angular motion

exists. The errors due to case rotation are likely to be exaggerated (compared

with the stabilized platform) for the strapped-down inertial measurement. These

error contributors (some of which may be found insignificant) may be termed as

follows:

(a)(mRpcz)(appAA + a A - a A ) -- these error torques areca,p POA mOA PPA mpA

due to cross-axis (PA and OA) accelerations with the magnitudes being

dependent on pendulous element displacement angle and misalignment of

instrument axes with respect to the input axis (in this case). It

has been shown* that when the forcing acceleration is vibratory, a

non-zero average torque results which falsifies the output. This

affect is called vibropendulous and has the form

cos A(Pendulu m phase lag angle)]

* "Strapdown System Application Studies Related to a Surface to Surface Missile

of i00 to 400 Nautical Miles", by R. E. Alongi, AIAA/JACC Guidance and Control


[

!

IIIII!!i!IiIiI1Ir

F-10

IIIIIIIIllII

I

(b)

(The pendulum axis of the TPA should be orthogonal to the thrust axis

to help minimize this error effect.)

IoAWl,caoA this torque contributes error when OA angular velocity

is changing. For example, a changing roll rate would cause erroneous

outputs in the pitch and yaw accelerometers. Thus, this effect is

usually called output-axis coupling error or OA rotation effect•

- (0v 00 ) -- this is known as the anisoinertia(c) (IIA IpA) l'CalA l'capA

effect which occurs for simultaneous angular rates about the IA and

PA. The product of angular rates causes a non-zero average errorfor sinusoidal motion about these axes•

(d) (IIA - IpA) u_f(W2,CalA + w2- A + 2 _ 2, caBA) mpA (_I, CalA _I, CapA) Aca,p

" (Wl,CalAwl,caoA) AmOA]

These error torques are due to cross-coupling of angular rates.

Their magnitudes depend on pendulous element displacement angle and

misalignment of the instrument axes with respect to the input axis.

They produce errors called rotational cross-coupling.

In addition to the kinematically derived error sources, other error producing

effects are known to be present.

2+ w_ R -- this error in output accelera-

(e) (_l,ca_. ,car^ ) (cg..^u - cgm_^)

tion is due to the separation of the cg of the TPA and the cg of the

vehicle whose acceleration is being measured. Treating this distance

as fixed, angular motion (and vibrations) results in centripetal and

tangential acceleration components• The error depends on the separa-

tion distance, so it is not a characteristic of instrument

imperfection• It is termed a size _ffect.

I (f) (alp?(yo - Y1 ) --a!p?(x 0

!a

- X_) or f PPAI _--_--Aca,p cos A(pendulu m phaseJ

lag angle)

III

The strapdown platform experiences the same spatial motions as the

vehicle, while the computer maintains the space reference coordinates.

Thus, for vehicle acceleration (linear vibration) along a given axis,

the remaining two reference axes are displaced in accordance with the

vibratory input. So far, things are as they should be, but if an

angular vibration (of like frequency) occurs about either of the

linearly displaced axes, the product of these motions results in

rectified acceleration along an axis perpendicular to the two

F-II

linearly displaced axes. This effect is called sculling.it is not attributable to instrument imperfection.)

The complete performance equation becomes

(Note thatIII

o"

IoAA + + kA = mR E-a + a A + a Aca,p CdAca,p ca,p PCz PlA PPA ca,p POA mOA - a_ AAmpA ]

• (ipA F- IoA_i,caoA + - IIA) L°°I,CaiA l,capA

2 2 2

+ A + ml ) + AmpA (Wl'CalA ,CapA ca,p (Wl,CalA

III

2- wI ) (28)

'ca_A I

A I + KO + KI (-alA) + K2 (-alA) 2Wl,capA_l,caoA mOA

3+ K 3 (-alA) + K4 (-aOA) + K 5 (-a)pA + K 6 (-alA)(-aOA)

+ K 7 (-alA) (-apA)

IIII

Table F-I lists the error term symbols, forcing excitation, and some of the

nomenclature associated with the terms.

F-12

0°,.tiJ

0

u

rl ,,-4 -_

_ -,4

'_0

• .4 __ _J _

v

mv

F-13

4J

o(ov

r.O

g0

0

rjr_

z

0

o

I-4

!

r-_

0Hv

_ 1.4

,-4

_ o

o_, o

0

.,.4

,--t

o

p v

c_O0

0

v

0• I-4

v

_J@, -._

40

o

,._ o

_J0

I-4

-,-4 I_1

00 _

12)

01"-4V

_J

>.,4

-,'4

U_

o

X

!

o_-,

<

v

l....

0O

0t-qv

F_

0

g_o

m

o

0

v

v

f_

0

_Jc:

or_

_0

c_

oo

o

<

<H

u_

v

r---

IIIIaIIliIIIIIIII

F-14

IIlIIlIitII

Rpc

Fcext

m

RIp

FP]I

I , ]p, kP P

Rpcx'Rpcy'Rpc z

DEFINITION OF TERMS

Position vector, from pivot to the cg

Vector resultant of all external forces acting through the cg

(center of gravity)

Mass .of the accelerometer unit

The inertial acceleration of the pivot, P

Force of the pivot on the body due to inertial acceleration

of the pivot

Represent unit vectors along pendulum axes, x,y, and z,

respectively

Components of R along pendulum axespc

IlIlIlI

I, ly, Ix z

_I ' Wl,'PX _[_l,py' PZ

Mapp

OA

IA

PA

Vector angular momentum of the body referred to the pivot

point and based on inertial velocity

Moments of inertia of the pendulous body about pendulum

Angular velocity components of the pivot with respect to

inertial space, in pendulum axes

Applied torque

Input axis

Pendulous element axis (or pendulous arm)

Acceleration

ca Case

F-15

DEFINITION OF TER,_q (Continued)

X,Y,Z Case fixed axes

Aca,p

Angle of tile pendu]ous element displacement with respect

to a case reference axis

_l,cax'Wl,cay'_l,ca Z

A

mOA

Am

PA

Components of angular velocity of the case with respect to

inertial space, in case fixed coordinates

Angle of misalignment of the pendulous body's output axis

(with respect to OA reference fixed to the case),

arbitrarily represented as a positive rotation about the

input axis reference

Angle of misalignment of the pendulous element axis (with

respect to the pendulous axis reference fixed to the case),

arbitrarily represented as a position rotation about the

OA reference

IIIII

g

kEC

Vector acceleration due to gravity

Earth's gravitational fie]d vector

Earth's angular velocity with respect to inertial space

Defined as a position vector from Earth's center to location

of pendulous body's cg

IIII

MdampTorque due to viscous damping

MtgTorque developed by the torque generator

(I)M

(E)M

Inaccuracy torque

Error torque (deterministic or average portion of the total

inaccuracy)

(U)M

(I)M _ (E)M + (u)_l

Uncertainty torque (random portion of the total inaccuracy)

6- -- Symbol meaning "de[ined as"

F-16

IIIIIIIIIlIII

IIIIIIIII

S

tg[i;m]

S

Sm[e;i]

S

sg[A;e]

TPA

(s f)

ind

IOU

K0 through K 7

Veh

DEFINITION OF TERMS (Continued)

Sensitivity of the torque generator for current (i) input

and torque (m) output

Sensitivity of the signal modifier for volts input and current

current output

Sensitivity of the signal generator for angle input and

volts output

An "elastic restraint" coefficient defined as the product

(StgSsmSsg)

Torqued Pendulous Accelerometer

Specific force, defined as (-a) _-Force

Mass

Instrument Output Units

Accelerometer error coefficients (see Table C-l)

Vehicle

F-17

IIIIIIIIil!IIIIIIIIlII

APPENDIX G

NEW TECHNOLOGY

IIIII

IIBiBi

Ii.

i.iIIIII

APPENDIX G

NEW TECHNOLOGY

During the period covered, no new concepts were conceived or first reducedto practice.

C_-I

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