An onboard navigation system was developed to aid the design and evaluation of algorithms used in autonomous satellite navigation with Global Positioning System (GPS) data. The performance of the algorithms designed for a GPS Receiver/Processor Assembly (R/PA) intended for LANDSAT-D was investigated during the development phases of the GPS (four to six satellites in the constellation). This evaluation emphasized the effects on the orbit determination accuracy of the expected user clock errors, GPS satellite visibility, force model approximations, and state and covariance propagation approximations. Results are presented giving the sensitivity of orbit determination accuracy to these constraints

Dunham, J. B.

English

NASA Technical Reports Server

AUTONOMOUS SATELLITE ORBIT DETERMINATION DURING THE
DEVELOPMENT PHASES OF THE GLOBAL POSITIONING SYSTEM*
Joan B. Dunham
Computer Sciences Corporation
ABSTRACT
An onboard navigation system was developed to aid the design and evaluation of algorithms used in
autonomous satellite navigation with Global Positioning System (GPS) data. The performance of
the algorithms designed for a GPS Receiver/Processor Assembly (R/PA) intended for Landsat-D was
investigated during the development phases of the GPS (four to six satellites in the constellation).
This evaluation emphasized the effects on the orbit determination accuracy of the expected user
clock errors, GPS satellite visibility, force model approximations, and state and covariance propaga-
tion approximations. Results are presented giving the sensitivity of orbit determination accuracy to
these constraints.
*Work performed under National Aeronautics and Space Administration Contract NAS 5-24300
137
https://ntrs.nasa.gov/search.jsp?R=19800019894 2020-03-21T16:58:33+00:00Z
IN TRODUCTION
The Navstar Global Positioning System (GPS) is a Department of Defense pro-
gram that will provide navigation information to properly equipped users. A
constellation of up to 24 satellites in 12-hour orbits will broadcast coded signals
from which the user's position can be determined. The application of GPS to
onboard satellite navigation has been previously discussed (References i, 2).
As part of the evaluation of the feasibility of autonomous satellite orbit deter-
mination using GPS, an experimental GPS Receiver/Processor Assembly (R/PA)
will be placed on Landsat-D, and the resultant orbital solution will be compared
to that obtained using more conventional ground-based techniques. This experi-
ment will be conducted during the early phases of GPS, during which there will
be four to six GPS satellites available.
The R/PA desig-n proposed for spacecraft applications (Reference 3) consists
of a dual-channel receiver and a Digital Equipment Corporation (DEC) LSI-II
processor. The R/PA measures pseudorange and delta pseudorange observa-
tions from the GPS signals, estimates the corresponding observations using
the GPS navigation message, and uses the observation residuals in a UDU T
formulation of the extended Kalman filter (EKF) to determine the user space-
craft's position, velocity, clock bias and bias rate, and satellite drag coefficient.
Simulation studies are in progress to determine the accuracy attainable with this
use of the GPS data, to identify and evaluate the primary sources of error, ,_md
to examine the algorithms in the proposed R/PA. As an aid to such studies, an
onboard navigation package simulator (ONPAC) was developed on a DEC
PDP-II/70 computer, which has computational accuracy similar to that of the
LSI-II. The simulator is designed for both prernission planning and real-tirne
analysis as well as evaluation of the GPS receiver algorithms.
The simulator is being used in this study to determine the factors affecting the
optimum performance of the onboard processor that are mission independent.
138
Some of these results are Presented here. The topics studied include data
editing, residual smoothing, fading of the filtermemory, clock modeling, state
process noise covariance modeling, and GPS selection.
As an aid to the use and evaluation of GPS pseudorange and delta pseudorange
observations, the capabilities to simulate and use these observation types were
built into the Research and Development Goddard Trajectory Determination Sys-
tem (R&D GTDS). GPS observation can be simulated with R&D GTDS for both
ONPAC and R&D GTDS use.
An overview of the steps involved in simulation and use of GPS data are sum-
marized in Figure i. Both truth model information and simulated data are
passed to the ONPAC program. There, the orbit estimation is done, and the
estimated trajectory is compared to the truth model.
DATA SIMULATION
The force model used in generating the true user ephemeris can be selected
from the options available to the R&D GTDS EPHEM program (Reference 4).
These include geopotential harmonic coefficients (up to 21-by-21), drag, solar
radiation pressure, and perturbations from the Sun, the Moon, and the other
planets.
Data simulation options for parameters affecting the data accuracy are listed
in Figure 2. Itshould be noted that GPS satellitescan be scheduled for specific
subsets of the total simulation time span. If a GPS satelliteis scheduled for
observations only during periods when itis not visible, itis "scheduled out" of
the data set. The default inclinationwill also be a modifiable option in the
future.
The information passed to ONPAC is summarized in Figure 3. Because all the
computations in ONPAC are done in the Earth-centered Earth-fixed (ECEF)
coordinate system, this information is in ECEF coordinates.
139
TRUE USER
R& It/USERSAT L-|LITEFORCE H EPHEMERIS EPHEMERIS_(TAPE i )
IMODEL OPT IONSI GENERATION '606££'__#,.;,.._,,--
T _ DATA .l _Y"_ GPSI SIMULATION DATA I EPHEMER[S_(TAPE 2/
D [ OPTIONS _SIMULATIONI GPS "\'_ '-JS IOBSERVATIONS_
........
OPTIONS /_,,I I / ESTIMATED A
k _ :_ SIMULA#jON I ,,_USER EPHEM- { 1
ur.,ar_ I _ ERIS AND I /
f _ _ R/PA . \CLOCK OFFSET V
P __,, EVALUAT,ON_'-
A T_
C _ IS°LuT_°_F'- --
 PLOTS)
ACCURACY OF
GPS R/PA
ESTIMATES
L
Figure 1. Overview of Analysis Approach
140
o USERCLOCKERRORrlODELOPTIONS:
NO ERROR
OOADRATIC
RANDOMWALK
o OBSERVATIONMEASUREMENTRRORS
o GPSCLOCKERRORMODELOPTIONS:
No ERROR
CONSTANT BIAS, UNCORRELATED
CONSTANT BIAS, CORRELATED (I.E., ALLOPS
CLOCKS HAVE THE SA;4E ERROR}
O GPS CONF[OURATION OPTIONS:
1 TO 2q GPSs (DEFAULTS: 5 IN PHASE ], _2 IN
PHASE [[, 2q IN RHASEIi[)
3 ORBIT PLANES
ALL ORBITS CIRCULAR; INCL[NATION = 63 DEGREES;
12-HOUR PERIODS
o GPSDATASPACINGOPTIONS:
_tI (PROMo TO _o)
At 2 (FROM_PSn TOGPSn_i)
At 3 [FROM LAST GPS It4CCNSTELLATION TO GPS _1)
o GPSSELECT[0N:
ALL OBSERVABLE
GEOMETRIC DILUTION OF PRECISION (GDOP)
EACHGPSMAYBE SCHEDULED FOR SUBSET(S}
OF THE TOTAL SIMULATION TIME SPAN
o GPSEPHEMERISERROROPTIONS;
NONE
RANDOM CONSTANTS FOR RADIAL AND CROSS-TRACK
(H,C); LINEARLY INCREASING ALONG-TRACK (L) TO
A RANDOMLY SELECTED MAXIMUM:
-- UNCORRELATED
-- 0RBIT-W[SE CORRELATED
-~ TOTALLY CORRELATED
SINUSOIDAL:
-- INPUT IS H,C,L AMPLITUDES, PERIOD _P},
AND ALONG-TRACK RATE L[)
-- DIFFERENT mNASE OF=SET FOR EACH _PS,
COMPUTATION BASED ON _IUMBER OF _msB
IN THE CONFIGURATION
Figure 2. R&D GTDS GPS Data Simulation Options
141
FOREACHOBSERVATION:
rOBS = TIME OF OBSERVATION (INCLUDING USER CLOCK OFFSET)
T(tk) = USER CLOCK OFFSET = rOB S- tk
TD(tk) = USER CLOCK DRIFT AT tk
POBS = "OBSERVED" OBSERVATION (PSEUDORANGE)
= TRUE OBSERVATION (PSEUDORANGE)
APOBS = "OBSERVED" OBSERVATION (DELTA PSEUDORANGE)
_p = TRUE OBSERVATION (DELTA PSEUDORANGE)
-_GPS = GPS POSITION AND VELOCITY VECTORS IN ECEF
_GPS'
COORDINATES, INCLUDING THE EFFECT OF GPS
EPHEMERIS ERRORS
r, V = TRUE USER POSITION AND VELOCITY VECTORS IN
ECEFCOORDINATES
GPSSATELLITE IDENTIFICATION
TRUTHMODELINFORMATIONPROVIDEDURINGDATAGAPS
Figure 3. Simulated Data Produced for ONPAC From R&D GTDS
142
ONPAC ESTIMATION
The ONPAC estimation is done with the UDU T form of the EKF, as described
in the mathematical specifications (Reference 5). The estimation process is
briefly described in Figure 4. The a priori state and covariance matrix can be
either the values from the last observation processed or the input values. The
integration of the satellite equations of motion is done with a modified Euler
integrator. The state transition matrix is computed with a Taylor series ap-
proximation. Studies have demonstrated that these propagation techniques
have sufficient accuracy for nearly circular orbits for the filter as long as the
propagation stepsize is held small, i.e., less than i0 seconds.
The ONPAC state vector is given in Figure 5. The clock bias and bias rate are
estimated in position and velocity coordinates as the clock offset and drift times
the speed of light. The user can select all nine members as the solve-for state,
drop the drag and estimate only eight parameters, or drop the drag and clock
drift and estimate seven parameters.
Parameters that can be varied in the ONPAC program are listed in Table i.
The force model options for the user satellite are the Earth geopotentia[ up to
the 5-by-5 harmonics, rotation terms, and drag. The state transition matrix
is computed with a Taylor series approximation and has only a two-body geo-
potential contribution plus rotation and drag terms. The effect of even further
limitations to this force model can be studied, as can the effect of the integra-
tor stepsize. A tunable parameter study can be done with variations of the
process noise parameters, the fading memory smoothing factor, maximum
values for the memory factor and the residual, and the obser_ration measure-
ment noise.
RESULTS
As part of the autonomous orbit determination evaluation, an experimental
R/PA will be placed on Landsat-D. The proposed Landsat-D orbit was used in
studies of the orbit determination.
I43
START^ ^ _(WITH A PRIORI XK_I, UK_I DK_I
'I XK-I = STATE ESTIMATE AT tK_l
Y(tOBS)____ PREDICT tK PK-I= UK-I DK-I O_-l'
,; COVARIANCE
I Y(toB S) = OBSERVATION
COMPUTE XK,
_(tKJtK_1)i,@ K _(_K_tK_I)=STATETRANSITION
MATRIX
'_' QK = STATE PROCESS NOISE
COMPUTE COVARIANCE MATRIX
PK' UK' DK SGPS(TK) = GPS POSITION
GK = COMPUTED OBSERVATION
_GPS(fK HK, O-C HK OBSERVATION PARTIAL
I DERIVATIVES
OUTPUT
XK..
ACCEPTED .
UPDATE I _0 BACK Tb
_K,OK,§K,tK START
,i ]
IF THE NO OF "-_- A /
' OUTPUT
GPSs IS _ 4, G
ADE MEMORY / K
NOTE: ALL OPERATIONS ARE DONE
IN DOUBLE PRECISION
Figure 4. ONPAC Estimation Algorithm
144
X _ CARTESIAN POSITION COMPONENTS
y (/ IN EARTH-CENTERED_ EARTH-FIXED
z ) (ECEF)COORDINATES
B CLOCK BIAS
X = _
-- , _ CARTESIAN VELOCITY COMPONENTSY
I INECEFCOORDINATESi /
CLOCK BIAS RATE
D DRAG COEFFICIENT
CD Acx
D =
2M
WHERE
MCX = CROSS-SECTIONAL AREA OF THE
SATELLITE
M = MASS OF THE SATELLITE
CD CONSTANT COEFFICIENT
Fig-are5. ONPAC StateVector
145
Table 1. User Options in ONPAC
TYPE OPTION ADJUSTABLEPARAMETERS
ACCELERATION IIODEL
GEOPOTENTIAL - DEGREE & ORDER, 2-BODY TO 5X5
ATMOSPHERIC DRAG 0N/OFF TIME CONSTANT, Td
EI<FALGORITHM
RESIDUAL TEST FOR ACCEPTANCE ON/OFF PMAX
FADING MEMORY 0N/OFF 'P2
PROCESS NOISE 0N/OFF a2, qa, qb, qd
SOLVE-FOR PARAMETERS INCLUDE/EXCLUDE -
DRAG PARAMETER
OBSERVATION MEASUREMENT NOISE - O-p2' (YA_
STATE TRANSITION MATRIX - ORDER OF APPROXIMATION TOAt 3
INTEGRATOR - STEP SIZE
USER CLOCK - TIME CONSTANT, If
Initial conditions for Landsat-D and the GPS satellites are given in Figure 6.
The GPS satellites constitute the default Phase I configuration. Since the launch
of Landsat-D is expected during the early phases of the GPS, efforts were con-
centrated on orbit determination using the Phase I and subsets of the Phase I
configuration.
Sample results are presented with four different sets of simulated data for
October i, 1980, 0 hours to 6 hours Universal Time (UT). The data simulation
options used in common for these data sets are given in Figure 7.
The ONPAC options used for four sample cases used with these data sets are
given in Table 2. In addition, all runs were done using a 5-by-5 geopotential,
drag in the force model, a state transition matrix approximated to zkt3, and
a 3-second stepsize. The level of process noise used was found from tunable
paYameter studies to give the best results during periods of poor visibility.
Figure 8 shows the root-sum-square (RSS) position error for the baseline case
and the Phase I visibility over the 6 hours of the data span. During periods when
the fading memory is used and four or more GPS satellites are in view, the
RSS position error ls less than i0 meters. The curve has a "fiat bottomed"
appearance found to be characteristic of the cases when fading memory is used.
The studies discussed here have shown that the best results occur when the
fading memory is tuned to the periods of good GPS visibility and the process
noise covariance to periods of poor GPS visibility. The fading memory multi-
plies the covariance matrix and inflates the entire matrix, whereas the process
noise is additive to certain terms of the covarianee. The fading memory swamps
the effect of the process noise when they are used together.
Cases 2 and 3, whose RSS errors are shown in Fig-ure 9, are done with data
sets B and C, which include GPS clock bias errors. The GPS satellites are
selected with the geometric dilution of precision (GDOP) procedure, which,
when six or more GPS satellites are in view, picks for observation only those
147
LANDSAT-DINITIALCONDITIONS
SEMIMAJOR AXIS 7086,901 KILOMETERS
ECCENTRICITY 0,001
INCLINATION 98,181 DEGREES
LONGITUDE OF ASCENDING NODE 354,878 DEGREES
ARGUMENT OF PERIGEE 180,000 DEGREES
MEAN ANOMALY 0,000 DEGREES
PERIOD 98,956 MINUTES
GPSCONFIGURATIONANDINITIALCONDITIONS
PHASEI CONFIGURATION
INCLINATION 63 DEGREES
ECCENTRICITY 0,0
SATELLITES 1, 2, 3:
- LONGITUDE OF ASCENDING NODE 120 DEGREES
- MEAN ANOMALIES i00,140,180 DEGREES
SATELLITES 4, 5, 6:
- LONGITUDE OF ASCENDING NODE 240DEGREES
- MEAN ANOMALIES 60,i00,140 DEGREES
PERIOD !2 HOURS
Figure6. InitialConditions
148
LANDSAT-D FORCE MODEL:
- 8X8 GEOPOTENTIAL
- LUNI-SOLAR PERTURBATIONS
- DRAG
- SOLAR RADIATION PRESSURE
QUADRATIC USER CLOCK ERROR:
- T_ = 3,3360x 10.5 SECONDS
- T2 = 3,475x i0-I0SECONDS/SECOND
- T3 = 5,0x 10-16SECONDS/SECOND2
OBSERVATION SPACING:
- &t] = 0,6 SECOND
I
- _t2 = 6 SECONDS
- &t3 = 6 SECONDS
OBSERVATION STANDARD DEVIATIONS:
- _p = 2,0 METERS
- O-Ap = 1,7CENTIMETERS
OPTIONS VARIED:
- GPS EPHEMERIS ERROR MODEL
- GPSCLOCK BIAS
- GPS CONFIGURATION AND SELECTION
Figure 7. Data Simulation Options Used for Data
Sets A, B, C, and D From R&,DGTDS
149
Table 2. ONPAC _md Data Simulation Options Used in Sample Cases
FADING
CASE MEMORY cTp DATA GPSEPHEMERIS GPSCLOCK SELECTIONPARAMETERS SET ERRORMODEL BIAS
i
BASELINE CASE
TESTCASE1 /_ = 0.2 8.1M A RANDOMLYSELECTED NONE PHASEI,
(3507) P2 = 2.0 (O-H"O-C"O-L)=(5M, 5M_ 10M) ALL
FADING MEMORY/GPS CLOCK BIAS
- TEST CASE 2 /_ : 0.2 7.5 M B SINUSOIDAL CORRELATED PHASE I
(4010) P2 = 1.05 (H,C,L,L)= (5M,5M, TG = 3 Ns GDOP
10M,0.05M/SEC)
P = 24HOURS
TEST CASE 3 NOT USED 7.5 M C SINUSOIDAL UNCORRELATED PHASE I,
(4011) (H,C,L.L)= (5M,5M, CTTG= 3 NS GDOP10M,0.05M/SEC)
P = ]2 HOURS
EFFECT OF FOUR IN CONSTELLATION
TEST CASE 4 /_ = 0.2 8.1 M D SAME AS C ABOVE NONE # 2,3,5,6
(8102) P2 = 2,0 IN PHASE I
ROOT SUM SQUARE OF P O S I T I O N  ERROR 
FOR B A S E L I N E  CASE (DATA SET A)  
RSS OF 
POSITION 
ERROR 
(METERS) 100 
- 
0 
PHASE I - GPS S A T E L L I T E  V I S I B I L I T Y  
Figure 8. Baseline Case and Visibility 
151 
ROOTSUMSQUAREOF POSITIONERROR
FORTESTCASE2 (DATASETB)
200
RSSOF
POSITIONI00-
ERROR
(METERS)
0
0,0 1,5 3,0 4,5 6,0
ROOTSUMSQUAREOF POSITIONERROR
FORTESTCASE3 (DATASETC)
200
RSSOF
POSITION
(METERS)
O'
0,0 1,5 3,0 4,5 6,0
TIME FROM 0 HOURS ON !0/I/80(HOURS)
F:gure 9. Root-Sum-Square Position Errors of Test Cases 2 and 3
152
four with the best geometric distribution. If four or fewer are visible, those
seen are picked for observation.
The effect of the GPS clock bias error is to increase the baseline case RSS
position error during periods of good visibility. The fading memory option used
in case 2 causes the RSS error to drop to the minimum value more quickly than
in case 3 at periods of good visibility and makes the curve plotted flatter than
that in case 3. Study of the correlated versus the uncorrelated GPS clock errors
shows very little difference in their effects.
Data set D was simulated using only four GPS satellites from Phase h satel-
lites 2 and 3 in one plane and satellites 4 and 5 in another. Figure i0 shows the
visibility and GDOP for this data set. Test case 4, whose RSS error is shown
in Figure ii, was run using this data set. The RSS position error grows to
more than 300 meters during the data gaps, and the user clock is poorly esti-
mated when fewer than four GFS satellites are visible. Given that the error
in the data and the GPS ephemeris is approximately 7 meters, the GDOP from
Figure i0 would predict an error in the position determination of 35 meters or
more when four GPS satellites are visible. The results in test case 4 are in
the 25-through-35-meter range at times of good visibility, within the range pre-
dicted by the GDOP.
CONCLUSIONS
The conclusions of the studies are _ven below.
• The algorithms used are sufficient for accurate orbit determination.
• The errors in orbit estimation are less than those predicted from
the GDOP.
• Accurate orbit determination is possible with only four GPS satel-
lites in the constellation.
• The orbit determination accuracy is limited by the GPS ephemeris
,and clock accuracies.
153
GDOPOF GPSs# 2, 3, 5, 6 FROM
R&D GTDS PHASEI (DATASET D)
i0,0
DILUTION OF
PRECISION 7,5 -
(GDOP) L d XJ
5,0 I , ,
0,0 1,5 3,0 4,5 6,0
GPSVISIBILITYFORDATASETD
6,0
4,0 ....NUMBER
2,0 -
i0,0 l , ,I
0,0 1,5 3,0 4,5 6,0
TIME FROM 0 HouRsON 10/1/80(HOURS)
Figxlre i0. GDOP and Visibilityof Data Set D
154
TRUE-ESTIMATED CLOCK B IAS FOR TEST CASE 4 (DATA SET D) 
ROOT SUM SQUARE OF POSITION ERROR 
FOR TEST CASE 4 (DATA SET D) 
3 0 0  
RSS OF 
POSITION 
ERROR 200  
(METERS) 
1 0 0  
Figure 11. Clock and Position Errors of Test Case 4 
• The fading memory enhances the orbit determination accuracy,
especially when the a priori Imowledge of the user clock offset is
poor.
All computations have been done in double precision; the effect of performing
some operations in single precision has not yet been investigated.
The studies discussed here have shown that the filter is not overly sensitive to
the tunable parameters. The results presented are typical of the results
gathered from a range of parameter values.
ACKNOWLEDGMENTS
The author wishes to thank A. C. Long for her help in this research and in pre-
paring this paper, W. H. Wooden, II and A. J. Fuchs for their support, and
J. A. Carlson for assistance in the GPS data simulation. The author gives
special thanks to E. Smith for her assistance in the documentation of the results.
REFERENCES
i. A.J. Fuchs, W. H. Wooden, and A. C. Long, Autonomous Satellite
Navigation With the Global Positioning Svstem_ (paper presented at XAS/
AIAA Astrodynamics Specialist Conference, Jackson Lake Lodge,
Wyoming, September 1977)
2. W.H. Wooden and J. B. Dunham, Simulation of Autonomous Satellite
Navigation With the Global Positionin[ System_ (paper presented at AI.4_k/
AXS Astrodynamics Conference, Palo Alto, California, August 1978)
3. Magnavox Advanced Products Division,Technical Proposal for the
Receiver/Processor Assembly to be Used inCon)unctionWith the Global
PositioningSystem NavigationSatellites,June 30, 1976
4. Computer Sciences Corporation, Research and Development Ooddard
Traiectory Determination System (R&D GTDS) User's Guide (draft),
August 1978
5. J.B. Dunham, _lathematical Specifications of the Onboard Navigation
Package Simulator (ONPAC), CSC/SD-78/6002, December 1978
156


Autonomous satellite orbit determination during the development phases of the global positioning system

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