i/i meeeemmmmeemme potential of the geostrr …a compu1)lter simulation was performed to demonstrate...
TRANSCRIPT
RD-R154 899 A SIMULATION OF THE NAVIGATION AND ORIENTRTION i/iPOTENTIAL OF THE GEOSTRR RECEIVER(U) NAVAL SURFACEWEAPONS CENTER DRHLGREN VR B R HERMANN OCT 83
UNCLRSSIFIED NSWC/TR-83-377 F/G 17/7 NLmEEEEmmmmEEmmEmmmmmmEEEEmmE
L4 0111112.2
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MICROCOPY RESOLUTION TEST CHARTNAT IONAL BUREAU OF STANDARDS 1963 A
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UNCLASSIFIED%Eu'IYCLASSIFICATION OF THIS PAGE (When Data Entered)
DOCUENTAION AGEREAD INSTRUCTIONSREPORT DOUETTO AEBEFORE COMPLETING FORMIREPORT NUMBER .. 2. G VT ACC SSON . EiPiENT'S CATALOG NUMBER
NSWC TR 83-3774. TITLE (and Subtitle) S. TYPE OF REPORT G PERIOD COVERED
A SIMULATION OF THE NAVIGATION FinalAND ORIENTATION POTENTIAL OF THE
GEOSTR RECIVERS. PERFORMING ORG. REPORT NUMBER
7. AUTHOR(&) G. CONTRACT OR GRANT NUMBER(#)
Bruce R. Hermann
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT PROJECT, TASKAREA & WORK UNIT NUMBERS
Naval Surface Weapons Center (K 13)Dahigren, VA 22448 63701 B
11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE
Defense Mapping Agency Headquarters October 19836500 Brookes Lane 13. NUMBER OF PAGES
Washington, DC 20315 3914. MONITORING AGENCY NAME 6 ADORESS(lf different from Controlling Office) IS. SECURITY CLASS. (of this report)
UNCLASSIFIED15a. DECLASSI FICATION/ DOWNGRADING
SCHEDULE
16. DISTRIBUTION STATEMENT (of this Report)Accession For
Approved for public release; distribution unlimited. DNTIC TA f
1.ainaouxicodJustificat jo
17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, if different from Report) Dsrbt n
Availatbility Codes
lB8 SUPPLEMENTARY NOTES 0" ~i1~/iDis ael
19, K EY WORDS (Continue on reverse side It necessary and Identify by block numbe)l pea
Global Positioning System (GPS) .Bias ResidualThree-Dimensional Navigation Fixes Navigation ResidualIGEOSTAR Navigation Processor Orientation ResidualReial.
20 ABSTRACT (Continue on reverse side If necessary and Identify by block number)
A compu1)Lter simulation was performed to demonstrate that thle signals from the Global
Plositioning System (GPS) satellites are able to provide a user With su~fficient information to]navigate and orient a moving vehicle. A particular application of interest is thle orientation of~ a
0 SloWjy m1ovinlg Survey ship. Orientation accuracies on the order of 0.1 mirad and positionswXithin 20 in RSS are desired./
DD I JAND73 1473 EDITION OF I NOV 65 IS OBSOLETE UNCL.ASSIFIED'N 0 02LF-0 4.607SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)
L
NSWC TR 83-377
'aq
FOREWORD
The Global Positioning System (GPS) is being developed primarily for the purpose of providing a userwith accurate three-dimensional navigation fixes anywhere near the surface of the earth. This system has manyunique characteristics that make it useful for other applications as well. One of these is the capability ofproviding the user with orientation information if he is able to receive signals from several antennassimultaneously.
This report describes a preliminary investigation of the potential for simultaneous, real time, navigationand orientation solutions using time-multiplexed GPS receiver technology such as the Texas Instruments (TI)GEOSTAR. Computer-generated data was used to simulate the gross characteristics of the GEOSTAR. Thisdata was then processed with a least-squares algorithm to provide the desired solutions.
Support for this simulation was provided by the Eastern Space and Missile Center, Patrick Air ForceBase, Cocoa Beach, Florida.
This report was reviewed and approved by R. W. Hill, Head, Space Flight Sciences Branch; and CarltonW. Duke, Jr., Head, Space and Surface Systems Division.
Released b:
T. A. CLARE, HEADStrategic Systems Department
07
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CONTENTS
Page
IM PLEM ENTATIO N ............................................................................ 1CONDITIONS OF THE SIMULATION .......................................................... 2
EST IM A T O R .................................................................................... 3
SIMULATION......................... .. . ................ ................... 4
R E S U LT S ....................................................................................... 5
D ISC U SSIO N ................................................................................... 7
C O N C L U SIO N .................................................................................. 8
REFERENCES ................................................................ 8
APPENDIXES
A: BATCH LEAST SQUARES FORMULATION ............................................ 19
B: COORDINATE TRANSFORMATIONS ....................................... 25
D IST R IBU TIO N ................................................................................ (1)
I'.
A. * , -.
. ....
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FIG'RES
Figure Page
I SATELLITE SELECTION FOR SIMULATION ......................................... 4
A-I PARTIALS OF T,, WITH RESPECT TO COORDINATE STATES i...................... 23
A-2 PARTIALS OF T. WITH RESPECT TO ORIENTATION ANGLES .................... 24
. B-I ANTENNAS AND SATELLITES IN THE ECEF COORDINATE SYSTEM ............. 26
B-2 TRANSFORMATION Tvb: BASELINE SYSTEM IN VEHICLE SYSTEM ............... 27
B-3 TRANSFORMATION T.,: VEHICLE SYSTEM IN LOCAL VERTICAL ................ 28
B-4 TRANSFORMATION Tu: LOCAL VERTICAL IN ECEF SYSTEM ..................... 29
TABLES
Table Page
I INITIAL CONDITIONS OF THE SIMULATIONS ...................................... 52 AVERAGE, RMS, AND STANDARD DEVIATION OF THE RESIDUALS OVER
THE ENTIRE 720-S OBSERVATION SPAN ......................................... 6
vI
. vi
6;" -..-.- -. ... ,;'. .- .'. -.-. - -- " .; .% '.'. -: 2 --. €,/' -2. . .i - -,-.. - - . o
7.
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PLOTS
- Plot Page
. I NAVIGATION RESIDUALS, EARTH-FIXED X (ESTIMATE-TRUTH);
2-M ANTENNA BASELINES ................................................................ 10
2 NAVIGATION RESIDUALS, EARTH-FIXED Y (ESTIMATE-TRUTH);
2-M ANTENNA BASELINES ................................................................ 10
3 NAVIGATION RESIDUALS, EARTH-FIXED Z (ESTIMATE-TRUTH);
2-M ANTENNA BASELINES ................................................................ I I
4 ORIENTATION RESIDUALS, LOCAL VERTICAL ROLL (ESTIMATE-TRUTH);
2-M ANTENNA BASELINES ................................................................ I I
5 ORIENTATION RESIDUALS, LOCAL VERTICAL PITCH (ESTIMATE-TRUTH);
2-M ANTENNA BASELINES ................................................................ 12
6 ORIENTATION RESIDUALS, LOCAL VERTICAL YAW(ESTIMATE-TRUTH);
2-M ANTENNA BASELINES ................................................................ 12
7 LOCAL CLOCK OFFSET RESIDUALS (ESTIMATE-TRUTH);
2-M A N T EN N A BA SELIN ES ................................................................... 13
8 ANTENNA 2 BIAS RESIDUAL (ESTIMATE-TRUTH); 2-M ANTENNA BASELINES .... 13
9 ANTENNA 3 BIAS RESIDUAL (ESTIMATE-TRUTH); 2-M ANTENNA BASELINES .... 14
10 THIS PLOT NUMBER IS INTENTIONALLY OMITTED
I I NAVIGATION RESIDUALS, EARTH-FIXED X (ESTIMATE-TRUTH);
20-M ANTENNA BASELINES ............................................................... 14
12 NAVIGATION RESIDUALS, EARTH-FIXED Y (ESTIMATE-TRUTH);
20-M ANTENNA BASELINES ............................................................. 15
13 NAVIGATION RESIDUALS, EARTH-FIXED Z (ESTIMATE-TRUTH);
20-M ANTENNA BASELINES ........................ 15'
14 ORIENTATION RESIDUALS, LOCAL VERTICAL ROLL (ESTIMATE-TRUTH);
20-M A N T EN N A BA SELIN ES .................................................................. 16
15 ORIENTATION RESIDUALS, LOCAL VERTICAL PITCH (ESTIMATE-TRUTH);
20-M A N T EN N A BA SEL IN ES .................................................................. 16
16 ORIENTATION RESIDUALS, L.OCAL VERTICAL YAW (ESTIMATE-TRUTH);
20-M A N T EN N A BA SE LIN ES .................................................................. 17
17 LOCAL. CI.OCK OFFSET RESIDUAI.S (ESTIMATE-TRUTH):20-M ANTENNA BASEI.INES .. 17 ."
S A S E I.................................................................. 1
18 ANTENNA 2 BIAS RESIDUAL (ESTIMATE-TRUTH); 20-M ANTENNA BASELINES 18 19 ANTENNA 3 BIAS RESIDUAl .. (ESTIMATE-TRUTH); 20-M ANTENNA BASEL.INES .. 18"-'
b vii I''4q
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IMPLEMENTATION
Three-dimensional orientation requires at least three independent antennas to define a plane. Four non-coplanar antennas could be used to define two planes and provide redundancy. Orientation solutions of theplanes containing the antennas are related to the vehicle and, therefore, allow the orientation of the vehicle tobe determined. The vehicle position can be obtained from pseudoranges in a manner similar to the conven-tional navigation solution if four GPS satellites are in view'. Four satellites will also add strength to theorientation solution. For these reasons, and algorithm simplicity, this simulation requires that four satellitesalways be in view. Operational versions of this software will need to be able to handle cases where fewer thanfour satellites are available, but this complication will be deferred pending the results of the current simulation.
The TI GEOSTAR 2 receiver has eight software trackers available. These are intended to be used to trackfour satellites on both L band frequencies simultaneously. The two frequency measurements provide the meansto correct for the propagation effects due to ionospheric refraction. If the receiver were operated on a singlefrequency, there would, in principle, be enough trackers available to allow four satellites to be tracked by twoantennas. Assuming that the antennas are not separated by more than a few tens of meters, the loss of L2 would
-" not affect the orientation solution appreciably. However, the position solution would be adversely affected; for.- this reason alone, two-frequency operations will probably be required.
The desirability of two-frequency observations from four satellites implies that a complete GEOSTARreceiver needs to be dedicated to each antenna. Since at least three antennas are required, three GEOSTARsfeeding a single processing computer will be necessary. This is probably an extravagant use of hardware, but itmay be necessary to realize the precision desired.
An alternative might take advantage of the similarity of the signal dynamics received from each antenna.rhe GEOSTAR has a fundamental clocking interval (T) of 20 ms duration. All receiver operations are someinteger fraction or multiple of T. Typically, the GEOSTAR dwells for T/2 ms on each satellite and T/4 on eachfrequency of a particular satellite. Thus, it has completed an observation cycle (two frequencies and foursatellites) after 2T ms'. If the antennas were connected to a single GEOSTAR via an RF switch that operated insynchronism with the GEOSTAR cycle, then a new antenna could be switched on line every 2T ms. Since thesignal dynamics will be similar from each antenna, reacquisition of the same four satellites via the new antennausing the GEOSTAR receiver software should not be difficult. Thus, it might be possible to keep three or fourauxiliary sets of tracking loops (one per antenna) running in the navigation processor, each set being updatedh% the rccciser processor or software. Update intervals of the trackers in the GEOSTAR navigation processorwould then be 2NT ms, where N is the number of antennas being utilized.
Seeral variations of this scheme may be possible. The trackers in the navigation processor might be
duplicates of the receiver processor trackers, except that they would be updated less frequently. They could alsob difference trackers: again taking advantage of the similarity of signal dynamics. In this case, one antenna
Iq
", . .. . . .... . . .... . ........... •.•
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would be the reference and its tracker would follow the full-signal dynamics. The other trackers would justtrack the difference between the dynamics received by its antenna and the reference. These difference trackerswould not need frequent updates and so the more critical reference trackers could be updated in alternate2T-ms intervals. The antenna switching sequence might be ABACADAB... where antenna A is the designatedreference. In this operating mode, the GEOSTAR navigation processor would be busy tracking signals and.consequently, an outboard computer would be required to perform the real-time position and orientationsolutions.
aIn either mode of operation. each tracker will be updated at a regular but different interval. Consequently.
it will be necessary to propagate the current state of each tracker to a common future time. The error involvedin this operation is small in low-dynamic applications, and it has the important benefit of providing simultane-ous observations from all trackers at each requested observation time. The simulation uses this property and,therefore, requires that all observations occur simultaneously at a time tag obtained from the local clock.
Th( broadcast ephemeris and satellite clock correction from each satellite are used to compute the earth-fixed coordinates. These ephemeris parameters are assumed by the simulation to be correct. There are nocompensating parameters in the state vector. However, the local clock offset is unknown and is a parameter inthe state vector. Also unknown are antenna biases. These states model the differences in cable length betweenthe various antennas a'nd the receiver. These differential cable delays need not be calibrated as long as they canbe modeled by a linear polynomial of the form
p(t) = a,, + aI(t - to)
However, it is assumed that there are no unexpected discontinuities that might influence the coefficients ao anda, during the period of observation. There is the possibility that certain other effects (e.g., interchannel biases inmultihardware-channel receivers) may behave like antenna biases over the short term, but may not be properlymodeled by p(t) over the long term. In the GEOSTAR, there is only one hardware channel and whenever a newsatellite is acquired. it is presumed that the pha-e observations from the new satellite are consistent with the oldand not randomied b% an arbitrary phase offset. If experience demonstrates that this condition cannot be met.then a neu antenna bias will have to be defined whenever a different satellite is selected.
CONDITIONS OF THE SIMULATION
-1 he obserxation error has been selected on the basis of the information in the TI Design Analysis ReportIII' and t o-frequenc% operation. The G(EOSTAR observation errors follow:
Pseudorange: L, = 0.687 m RSSL_, = 0.768 mn RSS
lonospherically Corrected Pseudorange = 2.11 mt
Phase: (P Code Tracking): L, = 0.147 cm RSS1., = 0.199 cm RSS
lonospherically Corrected Phase 0.48 cm
OP 6.48 o I + 2.39 o ,
2
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All simulated observations are free from tropospheric refraction and relativity effects; however, flicker noise
has been added to model the expected random variations due to the local frequency standard. The simulation
assumptions follow:
1. Four satellites always available2. Observation errors: Pseudorange a 2 m,
Phase a = 0.005 m3. All observations simultaneous4. Satellite ephemeris and time corrections contain no error
5. Antenna baseline length and orientation with respect to vehicle unknown
6. Antenna biases modeled by bias and drift
7. Tropospheric refraction and relativity neglected
8. Receiver tracker phase does not randomize when switching satellites
The GPS satellite constellation used in the simulation was the 18-satellite, 6-plane system. The right
ascension of the ascending nodes of the constellation are separated by 60 deg, with the satellites within a plane
separated by 120 deg. The phasing of satellites in adjacent planes is offset by 40 deg.5 The orbital period is the
standard 11. 966 hr, eccentricity was selected as 0.001, and the inclination of each plane was 55 deg. Figure I
shows the look angles from the receiver to the five satellites in view during the simulation. Of these five
satellites. 7. 10, 15, and 17 produced the best Geometric Dilution of Precision (GDOP) and were the four for
which synthetic data was generated.
ESTIMATOR
The data processing was performed by a least-squares batch estimator using both the pseudorange and
phase data from all antennas at each observation time. Two classes of states are employed in the solution. The
first class includes all time-dependent states: the vehicle earth-fixed coordinates (x, y, and z), the vehicle
orientation with respect to the local vertical (roll, pitch, and yaw), and the local clock offset. These states are
estimated solely from the data obtained at a single observation time and are independent of previous results in
all respects with the exception of the correlations that exist between the first and second class of states.
The second class of states include the antenna biases that are assumed to be independent of time. This
assumption allows all previous data to be employed at each observation time of these states. The accumulation
of data is accomplished by eliminating, in a formal elimination procedure at each observation time, the first
class of states. The results of the elinination are then saved and accumulated through the entire simulation. The
matrix manipulations leading to the solution are described in Appendix A.
Since the vehicle position and orientation are members of the state vector and the data is collected by
seeral antennas with positions that arc not states, it is required that the location of the antennas in the vehicle
coordinate system be accurately known eforehand. The assumption requires that these locations can be
surveyed with a precision better than the phase measurement accuracy so that they do not contribute to the
orientation errors. A description of the %i rious coordinate transformations that are required is presented in
Appendix B.
3. - - -S.1
0 NSWC TR 83-377 j
270 90
GDOP = 2.8
17 .
180
SAT NODE m + co AZIMUTH ELEVATION
5 90 160 241.9 56.47 150 80 319.6 30.4
10 210 120 55.0 23.615 270 40 60.6 71.717 330 320 187.1 18.9
FIGURE 1. SATELLITE SELECTION FOR SIMULATION
SIMULATION
Simulated pseudorange and phase data were generated for these antennas at 6-s intervals for a span of 720s. Vehicle motion was at a constant velocity of 2.54 m/s and a constant heading of 127.8 deg. Sinusoidal roll,pitch, and yaw signals of different periods and amplitudes were introduced as perturbations. The heading wasadded directly to the yaw signal. The local clock was offset from GPS time by 50 ps with a drift of I ns/s.Added to this offset was the flicker noise process that models the random variations of the local crystalfrequency standard". Antenna biases were also introduced on all antennas except the reference. The biases forthe reference are assumed to be included in the local clock offsets. All the conditions of the simulation aresummarized below:
l)vnamic Cases: Time Step 6 s, Time Span 720 s
Baseline System Translation and Rotation: Vh, = -0.61 01 = -45Vh, = + 10.36 0, = -5V ,, = + 20.73
4
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True States: !.ongitude = 2822 + 0.000018 ° t"
Latitude = 28' - 0.0000140 tHeight = 0 m
Roll = 5- * sin ( 2n- * 0.01) t
Pitch = 10- * sin (2 7 * 0.03) tYax% = 8- * sin (2rr * 0.02) tClock Bias 50 ps + I ns s + flickerAntenna 2 = 2.99-79 mAntenna 3 = 5.9957 mAntenna 4 = -2.9979 m
Flicker: = I: 1'7 1 00, 7 , 0b: r10 = 10'
The results from two cases will be presented. Both use all the same conditions, except that the antennabaseline lengths are different by a factor of 10. In both cases, the antennas are located in a coordinate s.stemoffset from the vehicle b\ the .ector V, and rotated in azimuth by 0t and in elevation by 02. The antennalocations (meters) in this rotated system are as follows:
Case 1: 0, 0. 0 Case 2: 0, 0, 02. 0, 2 20, 0, 200, 2, 2 0, 20, 20
RESULTS
The initial conditions given the estimator, vs the true conditions, are tabulated in Table 1. The given con-ditions were offset from the truth by 2 deg in longitude and 2 deg in latitude. The heading or yaw angle was inerror by 37.8 deg. These offsets were included to demonstrate the rapid convergence to the true solution as isillustrated in the residual plots. Both the navigation solution and the orientation solution converge within thefirst few observation intervals.
TABLE I. INITIAl. CONDITIONS OF THE SIMULATION
True Initial Conditions Given Initial Conditions
I.ongitudc (deg) ....... 282 280i.atitude (deg) ......... 28 26Height (m ) ........... 0 0R oll (dg) ............ 1) 0Pitch ,(lCg) ( 0
N im\ t(Ic) .. . .. . . 1 90
I ()Cal (C ock (P ) ..... So 0
Antenna 2 (M") ...... I) 0
n\rte a ,) . .20 0
Antcnnia 4 0n,) I) 0
. . . . . .. •. "
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The performance of the estimator is evaluated by comparing the estimated solution for each state with the
true value at each observation time. The difference (estimate-truth) and the standard deviations of' variousstates are plotted. The average values of the residuals, the RMS of the residuals over the entire 720-s span. and
the standard deviation at the end of the simulation are given in T-able 2. Plots 1-9 illustrate the residuals from
Case 1 and Plots 11-19 are residuals from Case 2. The plots are semilog with GPS time the linear horizontal
axis and the residuals plotted on the vertical log axis. The standard deviations at each time are plotted as heavyisolid lines and, in most cases, bracket the residual values that are plotted as "o" symbols.
TABLE 2. AVERAGE, RMS, AND STANDARD DEVIATION OF THE RESIDUALSOVER THE ENTIRE 720-S OBSERVATION SPAN
CASE 1: 3 ANTENNAS, 2-M BASELINES iState Average RMS S. D.
x (M) 0.096 1.345 1.315\(in) - 0.181 2.052 2.392
/(Mn) 0.029 1.324 1.289roll (rad) - 0.000262 0.002004 0.002063pitch (rad) 0.000336 0.002474 0.001934
yaw (rad) - 0.000114 0.002374 0.002671clock (in) - 0.040 1.354 1.533
CASE 2: 3 ANTENNAS, 20-M BASELINES
X (in) 0.090 1.340 1.3 14
o. (in) - 0.185 2.052 2.392
/(in) 0.026 1.321 1.289roll (rad) +0.000035 0.000386 0.000206pitch (rad) 0.000245 0.000739 0.000193%a%% (rad) +0.000076 0.000610 0.000267clock (in) - 0,044 !.354 1.533
If the estimator has done its 'oh properly, the residual plots of' all Class I states should contain no
s\ stemnatic signtals or trends. If the estimator has modeled the physical system correctly, the residual,, should hereprc~erwat xc_ of the noise processes that are inherent in the measurement and reflected In the states. C~lass 21xliii e',. the aniten na hiases. are correlated from ohserx atwin to ohseix at ion d tIC to the accumilulat ionl Process.-( 1nseuuejontr%. the (lass 2 residuals ,hoxk eonsiderathle cmurclat:oul as, thex hunlt for the /clo residual cx ci.
Plots, 1-1 and 1 1-13 sho%% the liax itionr yi i ll 111i t %, w iw a I lie, chanie inl haxelinle IcuethI has little
effetct uiponi the naxigation solution or upon thica k4 'ttxet WI)tx - and I-),. I hl, resuilt l" e\pected. sinlce
callii he performed xxlf it 1 xciie n fljl 11, 1 1I. ia iill(ar h0%%n 1 u Plots, 4-0 and 14-16.
1lie 11Imo ient xkth the longer h&,txe1iwi (( ixi 2. ii lxi I lie:xtlindanii (lations of the threeo
micitittions approach 0.0012 tadI in (a I ill! 100 jkl '.I~ I c a' " 2 8 e plo\ Itie the "i"lssII lit loll' of til'
. . . -,,r.. . -. W
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The construction of this A matrix is pursued in the next section of this appendix.
Batch least squares proceeds by rewriting equation (A-I) and then premultipling h%\ A\
AA , = ( - )
A'WAA. = A'W (0 - C) X.- 2)
The matrix W is a \,eight matrix that weighs the two data classes appropriately. This expression can hcsimplified by letting
B =- A WAand
E o-C
Next, the B matrix is partitioned to allow for two classes of states: Class I is represented by the states A*,, and(lass 2 by the states z.h,. The partitioning is as follows:
Br,, Bbh Es
Substitution of equation (A-3) into equation (A-2) produces the following two simultaneous matrix equations:
BiAx + B,, Ax h E, (A-4)
B,',, Ax, + Bhh L\X£ =E (A-5)
(lass I statcs arc eliminated by solking equation (A-4) for A:
,-X.= 1 (E, x
S,'uhstiUti,, ,I tl, Cslet nhl ) equation (A-5) produces the matrix equation
B B' B B:,) Z, E- B B . (A-6)
I hi-. cq a 'I] i; pr- ne i t ,id the ( lass I states ,l.,. I equation (A-6) is summed over all obscr\ations up toihe p fr. all Oi c' iniornation acqtluired ahout the Class 2 states.
|' . 1" 3 [ I V' 13 13 ... I'*I< 13' 13
I h'se ac ntel11lllcd nilitfeCS Larn thcn he ced along xxth the data fll the most recent ohscr\atioi al I. to
ctiniac the currnt state I his final form is as to lmo s
21)
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APPENDIX A jBATCH LE,.SE SQUARES FORMULATION
MATRIX MANIPI'IATIONS
The obserxations used in this simulation consist of (1) pseudoranges from each GPS satellite to each
antenna and (2) phase differences between each antenna and the reference antenna from each of the satellites. Ifthere are n, satellites and n, antennas, the number of observations at each time step tk is
n, (2n,- I)
At each time step, there kkill exist an estimate of each observatio. based upon previous information. It the
actual observations are represented by the column vector Ok and tie estimate of the observations by Ck. then
the difference provides information about the error in the estimate. The parameters of interest are the states Xk.
and since C, is computed from estimates of these states at each time step, the error in the estimate provides
information to ,-orrect the states. The formulation that adjusts the states to match the observations begins by
expanding Ck by use of Ta\ lor's Series around the current state estimate Xk. The general expression for the
expansion followks:
('.I C"( I anCk ..-. _ ?', ( . x - .. ... x,+ . .. + (.aC-X k)"-
2' X, n! ax,
In ordcr to emplo\ linear cstiman ,i theor\. this e pression must be approximated by truncating the series
atcr the term ioMa InThc th list patnal dcnan\c. I hen, C, is moxed to the left hand side of the equality and',,hk i] '201d105m' Is, the luneat appi,*\Imtioil~.
x.(A-)
>' ,1 [hd! de~atc,, C. X. is generall\ known as the "A" matrix:
. X( )N I axt
I t c r cr ce ent,, the numer of states in the solution and k is the time step. The difference between the
-" C , A,. CstI nated ,tatc x-kill he designated b\ the \ector /A k, wkhere
- I
19
• " - -.:- ". -::::":":::::: :::::::::::::::::: :: : .:":::: : : : = :- -;2::-: : 2: 2 :2. 5 7
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10-1 - 10.1
10-2- 10-2
E0
-j
o 0
w
-10-1 -2o-
0 100 200 300 400 500 600 700 800
GPS TIME (s)
PLOT 18. ANTENNA 2 BIAS RESIDUALS (ESTIMATE-TRUTH);
20-rn ANTENNA BASELINES
10-1- 10-1
10-2 10-2
.q 1 4 10 1 4
_1o 2 1 -
-1I 1 -10
0 100 200 300 400 500 600 700 800
GPS TIME (s)
PL.OT 19. ANTENNA 3 BIAS RESIDUAI.S (ISTINIATE-TRUTII);20-rnANTENNA BASElINFS
rflV- '.r77,
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10-1 - 10-1
10-2 10-1
00 000 0o 0 0
w~ 0 0 0 0
0 0 0 0 0
_1-1 3 0-ia-
-10-1.- I -10-2
0 100 200 300 400 500 600 700 800
GPS TIME()
PLOT 16. ORIENTATION RESIDUALS, LOCAL VERTICAL YAW (ESTIMATE-TRUTH);20-rn ANTENNA BASELINES
101 - 101
0 000 000 0
100 00 0000 000 U 0 0 00 100
000 o0 0 0 0 0 000
o0 0 0 0 0 0001- 0 .100
00
0 01
o0 0 0 0
0 10 030 0 50 600 700 00
o0r 0~IN n 0EI.0N00 0 0) 0 0
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10-1' 10-1
102 10-2
10-3-
o -oi0 0
0I 000-100 0 20 30)0 40 50 00 700 00
00-i - 10-4
100 200
ix 0~
'a -10- -1-- _____________-3________mI-10-2 -10-2
0 100 200 300 400 500 600 700 800
GPS TIME ()
PLOT 1. ORIENiTATION RE:SIDUAL~S, LOCAL. VERTIC'AL ROLLI (ES1IVIATE-TRUTH);201-m ANTEFNNA BASELINES10-1I
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101 101
0 0 00
00 0 0 00 0 0 0000 00
0 000 00 0 0 0 0 00
100 00 0 00 0 010
00000 000
0 0
10-'-E 0 0 00
FA 0 0wu 0
0 00 0 0
00 0
00 00 0d 0 0 10
0 100 200 0 40 50 00 70 0
GPS TIME (a)
PLOT 12. NAVIGATION RESIDUALS, EARTH-FIXED Y (ESTIMATE-TRUTH);20-rn ANTENNA BASELINES
01. -10'
00 000 0( 0 0c 0
100 vo 0 0 v 0 0 0- 0 100o0 0 0 0 00
00
0 1 0' 0
(A0 0 0
0 0 00
10 00 0 000 0
0 0 00 00 0 000 0 0
-101 -101
0 100 200 300 400 500 600 700 800
GPS TIME (s)
PLOT1 13. NAVICATION RESIDU ALS, EARTH-FIXED Z (ESTIMATE-TRUTH);20-rn ANTENNA BASELINES
15
NSWC TR 83-377
10-2 10-1
10-1 lo-2
-CD
i0-
10-3 0 C)10-3
Oo 0 0 0 0 0
0 00 0
0 0
-10-2
0 0 10 200, 30 00 So0 0 0 o
000 0 0 0 00 0 c
.10 0 0 0 0 -10
0 &0 000
0E 000 00~~0 o 0 000
0 0 00 340 50 0 00
104
0 1 13---- 0.- .. . ~ . - - . - -
NSWC TR 83-377
101 -101
00
09 0 0-o 0
10 0 0 0 00 0
0 0 00 000 00 000 00000100 0
0 0 00 0
'U 0-. l0
U
~100 0 00 :o 0 100 00
0 0 0 0D00 0 0
0 00 0 0 0 0 00 0
-10' - _ 101
60 100 200 300 400 500 600 700 600
GPS TIME (9)
PLOT 7. LOCAL CLOCK OFFSET RESIDUALS, (ESTIMATE-TRUTH);2-rn ANTENNA BASELINES
10.
100
*~~~ 0 0(900 0
0U 0
* 10-4 -0 n10-4
-10-3. _ 10-3
0 100 200 300 400 500 600 700 go0
* GPS TIME()
PLOT 8. ANTENNA 2 BIAS RESIDUAL., (ESTIMATE-TRUTH);2-rn ANTENNA BASEL.INES
13
NSWC TR 83-377
10.1 lo-
10-2 10-20 0
0 0 0 0 00 CD
10i-3 % 0 0 cp 0 00000000 0b00 0 0 00
00 0 0-. 0
~3 i00
Wu 00 00 0
0 00 0 0
00
0 c0o 0 0 0 00 0
_10-1 -101
0 100 200 300 400 500 600 700 800
GPS TIME (s)
PLOT 5. ORIENTATION RESIDUALS, LOCAL VERTICAL PITCH (ESTIMATE-TRUTH);2-rn ANTENNA BASELINES
10-1 10-1
10-2 10.2
.~ia0 0
00
* - -)o 1-10 0 0
00 0 0
a 10-40 1-
00
-1- 0 00 0 0 0 0 -1-
0-- - - - - - - -- - - - - - - . o--. - -0 -- --0 *--"0.
4, NSWC TR 83-377
10' - 10'
100 10
0o 0 0 0 0 00 0~ 0 0 0 0 00
00 ,0 0 0 000000
0 0 0000
_j~1 1
10-1- 10-1
-J 0w 0 (
0 0 0
0 00 0 0
0-0 000 00
0 100 200 300 400 500 600 700 800
* GPS TIME ()
* - PlOTLO 3.. A;ATION RESIDUALS,, EOAVRT-IXAL ROL (ESTIMATE-TRTH);2-r ANTENNA BASELINES
10-1- 10-
1011-
.o 0 0 0 0 0
NSWC TR 83-377
10' 101
10 01
* 10.1
0 00
101 - v_ 101
0 0000 00 0 0
0 0 0 0 000 0 0 0
100 0 0 0 0 00 0
0E0 0 0
-1 00 0
0
rn0 00U 0 0 0
.10 000.0x0 0 0 0
*~ 000 0 CJ,0 ~ 00
0 0 0 0
-101 -10,0 100 200 300 400 500 600 700 800
* OPS TIME (s)
PLOT 1. NAVIGATION RESIDUALS. EARTrH-FIXED X (ESTIMATE -TRU TH),2-rn ANTENNA BASELINES
10, 11,
0 0
NSWC TR 83-377
4. Design Analysis Report IH For The T14100 GEOSTAR Receiver, 23 December 1982.
5. P. Kruh, The NA VSTA R Global Positioning System Six Plane 18-Satellite Constellation, IEEE NationalTelecommunication Conference. New Orleans, 1981.
6. G.L. Mealy and D.R. Vander Steop, Time Standard Error Modeling With Applications to SatelliteNavigation. Proc. 29th Annual Symposium on Frequency Control, May 1975, pp. 417-424.
9
-0.-.
- .??]
NSWC TR 83-377
model fits the cable behavior. Unpredictable phase variations due to kinks or thermal effects of sunlight andshadow are not modeled and must be examined separately.
Clearly, a larger baseline prt "ices higher precision orientation capability for a given measurement accu-racy. However, as noted above, longer baselines are prone to introduce unmodeled effects that may make thepotential improvement unrealizable. This simulation demonstrates that the three-antenna array nominallyseparated by 20-m approaches 0.2 mrad resolution in the three angles for the given 0.5-cm phase measurementaccuracy. As expected, the 2-m array is a factor of 10 worse. In general, the angular precision can besummarized by the following relationship:
,, A 4to - csc 4'Ib
where A , = angular precision (rad)A I phase measurement error (m)b = baseline length (m)4, = angle between the baseline vector and the vector to the satellite
Use of several satellites and two or more baselines will tend to minimize the effect of csc4 in this equation.Thus, A /,b more nearly represents the standard deviation observed for roll, pitch, and yaw in this simulation.
CONCLUSION
All troublesome propagation effects have been neglected in this simulation and, consequently, the resultsrepresent a best case. Under these optimistic conditions, the simulation shows that milliradian angular solutionsare feasible using the GPS and a suitable receiver such as the TI GEOSTAR. Submilliradian accuracies are alsopossible. but will require a distance between antennas of about 20-m. It is doubtful that additional improve-ments in phase measurement accuracy will buy a proportional amount of improvement in angular resolution.Effort should instead be placed toward improving the models of propagation effects, working to remove themechanical instabilities of the antenna baseline un a moving vehicle, and controlling the thermal variations inlong signal cables. Clever mechanical design and more sophisticated software will be needed to make 0. 1-mradorientations routine.
REFERENCES
I. R.J. Milliken and C.J. Zoller, "Principle of Operation of NAVSTAR and System Characteristics", NAV-IGiATION. Vol 25, t2. 1978.
2. P.W. Ward, An Advanced NA VSTA R GPS Multiplex Receiver, IEEE Position location and NavigationSymposium. Atlantic City, N.J., 1980.
3. C.R. Johnson, P.W. Ward. M.D. Turner, and S.D. Roemerman. "Applications of a Multiplexed GPSUser Set", NAVIGATION. Vol 28. #4, 1981.
8
. . . . . . . . . .. .N
NSWC TR 83-377
simulation, longer baselines would continue to improve the angular resolution. However, practical considera-
tions, discussed in the next section. would work in the opposite direction.
The antenna biases, initially unknown, gradually converge toward a small residual as data is accumulated.
The bias residuals appear in Plots 8-9 and 18-19. As discussed previously, the knowledge of these statesimproves due to the accumulation of information over the span of observations. The effect of this improvement
is reflected in the orientation residuals and can be seen particularly in the case of pitch (Plot 15). During the first2 min of observations, the pitch residual is larger than it is later when the antenna bias residuals are reduced. It
can also be seen in Plot 5, but the larger residuals make the effect less obvious. In both cases, the pitchcomponent shows the effect more than roll or yaw because the correlation coefficients of pitch with respect to
the antenna biases have the same sign. The correlation coefficients of roll and yaw with respect to the antenna
biases have opposite sign and, consequently, tend to cancel each other when the antenna bias residuals are inthe same sense. .
DISCUSSION
This simulation demonstrates that the intrinsic measurement accuracy of the GPS signals readily meets the
20-m three-dimensional navigation error requirement of the application. However, there are many sources of
measurement error, not included in the simulation, that are capable of biasing the result and its effects are
difficult to detect.
Accurate prediction of the satellite trajectory over a long enough period of time with sufficient accuracy to
make the broadcast ephemeris suitable for high-precision navigation and point positioning will continue to be a
challenge. Improvement of our understanding of the forces acting on the satellites and an increase in the
number of tracking stations may combine to decrease the size of the broadcast ephemeris error in the future. IClosely related to the ephemeris error is the prediction of satellite time. Satellite clock corrections are includedin the broadcast ephemeris and need to be considered as part of the ephemeris prediction problem.
Ionospheric refraction, tropospheric refraction, and relativity are biases that were not considered in this
simulation. Errors here will particularly effect the navigation solution,since they, like ephemeris errors, can
introduce time-varying radial signals that cannot be easily separated from the true signal. The orientation
solution is not significantly degraded by these types of errors as long as the phase front remains intact across a
dimension comparable to the longest antenna baseline length. However, warping of the phase front due to "
propagation effects so that arrival times at the antennas are perturbed would be of serious concern
Any phenomena that differentially influences the arrival times of the signals from the several antennas at
the receiver in an unexpected manner would produce errors in the orientation solution. Any movement of the "'
antenna phase center as a function of the direction to a satellite must be measured and modeled. The distance
between antenna phase centers (i.e., the respectixe baselines) and the orientation of the antenna arra\ \ith '
respect to the vehicle must all be measured and remain stable to an accuracy better that that desired 1or the
system performance. This requirement tends to favor short baselines and systems where all antennas are rigidly
mounted on a single structure that can then be precisely mounted on the %ehicle.
Pr(,pagation delays of signals from each antenna to the receiver can be accounted for in the antenna
* biases, it the effects arc time-independent. Calibration of these signal cables will not be necessary as long as the
7
NSWC TR 83-377
1, AX, Bphk Ak -I
(A-7) _
B, hk A , + (HB, + Dk-I) Axhk = (Ehk + Fk-) (A-7
Solution at tk proceeds by premultiplving both sides of the unpartitioned matrix equation by B '. ihe result is -
the vector Axk as desired.
CONSTRtTION OF THE A MATRIX
As discussed above, two data classes are used to produce the combined position and orientation solution.
The partial derivatives for each of these will be presented independently. Reference to the diagram in Appendix
B (Figure B-I) will be helpful.
Pseudorange
lhe pseudorange observations can be modeled by the following equation:
rC k+ T +TTtt) i> (A-8)/,k, I, -DIk+ C [To, + To + r,(t, - to))]i> IA 8''-
where subscript j the satellitesubscript i the antenna
The two vectors e, and , (defined in Figure B-1) are the ECEF coordinates of the i th antenna and the
ECEF coordinates of the j th satellite, respectively. The biases, represented by the taus in equation (A-8).arc composites. The local clock bias T, and the reference antenna biases T, and j are grouped together to
make a single Class I state .,. The Class 2 states are the biases and drifts associated with the remaining
antennas plus ,. These effects arise due to cable variations, etc., and are represented by 1-, and T-,
respectively (i > I). The sum of all these effects is the total delay impressed upon the signal from each
antenna. In summary, the biases are as follows:
reference antenna: j, = r + T, + T1 (tk - o)
all other antennas: + T, + r, (tk - to)
Equation (A-8) is the observation equation for the pseudorange data class. It represents Ck. which wasdefined in the first section of this appendix. In order to form the A matrix, the partial derivative of h, is
required with respect to each state variable:
''C ,,2 c7.e... 71, T . To. to, T,, . T,, T . 1.
Ihc partial derivatives with respect to these states are written below, with the subscript m indicating thecoordinate. m :- I. 2. 3.
ackc, +ae,, ae,
21 - , - ,)
21
NSWC TR 83-377
Se., ae~, ae,aPki(ex e,) ay- + (r, - e ,) a + (r,.- e,,) a e-ay(m I e,)i'/ 1 ,- e1 -
a pk, k
-_ -c i> la7,
• apkl,a'k -C(tk t0) >.
The result from Appendix B, equation (B-I) is required to evaluate the partials of the antenna positionswith respect to the states. The partial of equation (B-I) with respect to coordinates follows:
aT,,, aTeuT.F T, + T], T, bae,,m N .e ae,m ",
Since T,,, is independent of Z,, (Figure B-3), the above matrix equation reduces to
a , T,1,2.3ae = ae,,m T, Th bm
The evaluation of aiT- ae,,, is presented in Figure A-l; the form of T,.T,... and T,h are given in Appendix B.
The partial of equation (B-I) with respect to the orientation angles roll, pitch, and yaw follows:
a. = aT + a T,1T,h b,
In this case, aT,, am = 0, and this allows the above matrix equation to be simplified to
ae, aT,, "-T,. T, b, m= 1.2.3D y ',,, aD y ,. .
* The evaluation of a-., ay , is presented in Figure A-2. All these pieces can be put together to form therequired partial derivatives.
Phase Difference
The observation equation for the phase difference data class can be obtained by subtracting the phase predictedat the reference antenna from the phase at all the other antennas:
APk,,. = Pk, Pk , i > I (A-9)
Ihe expansion of the above, using equation (A-B), is
22
4
-7-, 2
4 NSWC TR 83-377
~Pk =Ir- I~l - ~+ c[ r, + ,(tk tu)I >1 (A- 1)
The partial derivatives can be fashioned from the previous results and equations (A-9) and (A-10).
S~p, Sku_ ~k i> Im =1.2. 3Kaem ae"m aeuM
KaLPk, _ Pkj ______
a~r aym yun i> Im 1,I 2. 3
aAPkjt
C(tk t4)) i>a il
~4 CO X 2 sin6 f3 COSX+C2 Cos 6 k1 -aTell C2 C4 sinl fifl fsin + f Cos 0 k2
e,, : ,2 eu1 eLI3
aTe,, 1 r3 R31
C2 1 + f 1 + k k3k1 0r r R Rr
2
ae", e ,e,,2 eu2 r e,
r~-+ kr kk 0
0 C 30 + k,=
0~ k, k2 0R
42[ c1 r el. I el 21 R r2 + e12
FIGURE A-1. PARTIALS OF T-. WITH RESPECT TO COORDINATE STATES i,
23-
NSWC TR 83-377
co , i -COiyin -yIS , sin y.cos y, 0 cos y cos /jsinfy, siny2 sinl 0 :
DT-, sin -psin-/,+ cosy in y2 cosy, 0 sin ycos y,+ cos ysin y2 sin y 01- COS '-,COS Yl 0 -cos 12siny, 0):
ay0 0 00
-sin -y cos -y, s~ in sin -y, sin -y2 sifl y, cos y2 cos yi 0]
ZIT- co y cos y2 sinl y) cos -y, sin Y2 -COS 'YA COS 'Y2 W~S 'y' 0___-sin-y2 sin-y Cosy -Y2sin y,,cos yi 0
ai~0 0 0 0
-sin -y cos -y; co -Sy, sin -y, sin-/, - cos y, cos y2 -sin y, sinl yi + cos p~ sin ^12 cos yi*- cos cos CO -sin y,4 sinl -y-, sifl' - cos y., sin 'Y3 cos y, sin y, + sin y, sin y., cos y', 0
*0 0 0 00 0 0 01
FIGURE A-2. PARTIALS OF T, WITH RESPECT TO ORIENTATION ANGLES
0
K 24 4
7: W - 1V V - P6V "
"°..
NSWC TR 83-377
APPENDIX B
COORDINATE TRANSFORMATIONS
''he GPS broadcast ephemeris prox ides the user with the satellite position in the earth-fixed coordinatesystem' . 1o be able to perform the least squares differential correction algorithm described in this report. one
must be able to calculate the recei~cr antenna positions in the same system. The locations of the antennas are
assumed to be surveyed in some convenient local reference frame. The relationship of this frame to the
earth-fixed frame is the subject of this appendix.
The diagram in Figure B-I shows two antennas: one at B, and one at B. One satellite is at P, in the earth
centered earth-fixed (ECEF) coordinate system. The vector T, represents the coordinates of the i th antenna
location, the vector r, represents the j th satellite location, and the baseline vector -, connects the reference
antenna (I = 1) to the i th antenna. These relationships are described mathematically in the figure.
The observations consist of pseudoranges and phase differences. The pseudoranges include the geometric
ranges h. plus the antenna biases r, and the local clock offset To. The phase differences include the segment 1,
plus the difference in antenna offsets r, - Ti. These observations, obtained from three or more antennas and 7
four satellites, are required to produce a solution.."
The antenna locations are defined in a reference frame independent of the vehicle. This reference frame is
the bi. b-. b; frame drawn in Figure B-2. Each antenna has coordinates B, (b, b2,. b,) in this baseline system.
The vehicle reference frame is represented by the vi, v2, vi, coordinates. The baseline system is rigidly attached
to the vehicle system, but is translated by vh and rotated by the angles 01 and 02. The vector Vh and the angles 01and 02 are assumed to be known. The transformation matrix T,, which relates any point in the baseline system
to its equivalent in the vehicle system, is presented in Figure B-2.
In a quiescent state, the vehicle system and the local vertical system would be coincident. However, under !6-
Sdynamic conditions. roll, pitch, and ,aA are permitted; therefore, the vehicle system can rotate about each axis
as illustrated in Figure B-3. These rotations are the orientation angles of the vehicle system with respect to the
local %ertical and arc states of the solution %ector. The transformation matrix T_, which transforms any point
in the xchicle system to its cqtii\alent in the local vertical system, is also presented in Figure B-3. Note that the
first three rows of the fourth columns are /ero. This indicates that there is no translation and so the originsaliays coincide.
A third transformation relates the local vertical to the ECEF coordinate system. Figure B-4 shows that the 2
local vertical system is right handed with the u, axis positive in the radial direction and ul positive eastward. It
. A .1 Van l)icrendonck, S.S. Russell. F.R. Kopit/kc, and M. Birnbaum. "The GPS Navigation Message", NAVIGA-[ lION. Vol 25. t:2. 1978
25
... . . --1
- -- . rr - r - 7
* 1NSWC TR 83-377
. is translated from the origin of the ECEF system by the vector-, and rotated by A and B- TheE, vector joins theorigin of the two coordinate systems and is different from , in Figure B-I, which joins the origin of the ECEF
system to each antenna. The angles X and /3 are, respectively, the longitude and latitude of the vehicle originand can be computed if-,, is known. The elements of the vector d, are state elements, and along with roll, pitch.yaw, and the local clock bias, complete the first class of states described in Appendix A. The transformation
matrix T,.. which relates any point in the local vertical system to its equivalent in the FCEF system. is shown inFigure B4.
"* These three transformations together transform the local antenna coordinates b, into the ECEF coordi-nate system e,:
T, T14 Th b, (B-I)
Pi
r 1
BB
r, r, x + r, N, + r,, z satellites (subscript j)
Se e ,, x + e,, y + e,, antennas (subscript i)
h , (r ,, - c,) x + (r , , - e,,) y + (r,, - el,)
* - (e,, - e,) x + (e, - e, , )y + (e,, -e,l),
h = I,, + h,
S(FIGRE B-I. ANTENNAS AND SATEILITES IN THE EI(EF (OORDINATE SYSTENI
26
; i .. . . " . .. . ...-." . - " - "" .... . . ,, - . , . ' '. - .
NSWC TR 83-377
V3 b3
V3.
b2
b,
V2
' .V c o s 9 1 s i n G 1 c o s O 2 -s n 1 1n G 2 V b l b
v2 -sinG 2 case1 COSG 2 -c0se 1 sinG2 Vb2 * b2
0 sinG
2 C0
2 Vb3 b3
v s 0 0 0 11
FIGURE B-2. TRANSFORMATION Th: BASELINE SYSTEM IN VEHICLE SYSTEM
27.
q
• ". 27
° .. a . *
NSWC TR 83-377
V3 Yy
-V 2
0 U2
Yr
Yp
U1
V I
U1 cosy y CoS>'r -in S~y siflyp s'nyr -sily~, cosyp, cosyy, siflY + sinyy Siflyp COSYr 0 v1
U2 siflyy COSyr + cosy~, siflyp sinYr COSYy, COgyp, siflyy snyr - cosyy siflyp COSyr 0 V
0 1 V
0 0v
FIGURE B-3. TRANSFORMATION T.,: VEHICLE SYSTEM IN LOCAL VERTICAL
0
* - . *d.. . .. . . --- --- --- --- -- ~ !... ~- -- --28
NSWC TR 83-377
e3
e3
3 U
0 e2
e,1
61 -sinA -COSA sln/3 COSA co3, U1
e2 -COSA -siflA sifl/3 sifA cos/3 eU2 U2
e3 0 COSf3 sin/3 eu U
e, 0 0 0 us
FIGURE B-4. TRANSFORMATION T,,,: LOCAL VERTICAL IN ECEF SYSTEM
29
-, . .- 'C- - -h'. . - . -~, - . -7h %1-. - "-4 17% 07 171 -7.7 .7 7 -7 4
NSWC TR 83-377
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NSWC TR 83-377
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