Precise Orbit Determination of Precise Orbit Determination of GPS Satellites using Carrier GPS Satellites using Carrier
Phase MeasurementsPhase Measurements
Technical ConsultantTechnical Consultant Tomoji TAKASUTomoji TAKASU
Kasai Design Office Ltd.Kasai Design Office Ltd. Shoji KASAIShoji KASAI
The 15th Workshop on JAXA Astrodynamics and Flight Mechanics, 2005The 15th Workshop on JAXA Astrodynamics and Flight Mechanics, 2005
http://gpspp.sakura.ne.jp
MotivationMotivation
• Precise Orbit/Clock Determination is Key Technology for Accurate Positioning with GPS/GNSS
• Demands for Real-Time / Near Real-Time Precise GPS Orbit/Clock
• Technique can be Applied to Precise Orbit Determination for Satellite equipped with GPS Receiver
http://gpspp.sakura.ne.jp
LCphwrel
pcvrpcvsLC
LLLC
NTdTdtc
CC
)(
21 2211
Measurement ModelMeasurement Model
Undifferenced (Zero-Differenced)Ionosphere-Free Linear Combination of Carrier Phase Measurements (L1,L2)
)/(2)/(1 22
21
22
22
21
21 fffCfffC
http://gpspp.sakura.ne.jp
Geometric DistanceGeometric Distance
)'''()(
)(
apcrsdisprT
apcss
dtt
cdtt
ΔΔrU
Δr
Light-Time Equation
sr
'' sdispr Δr
'apcrΔ
apcsΔ
http://gpspp.sakura.ne.jp
Time SynchronizationTime Synchronization
ttttr
GMttt s
sEss )()(2
1)()( 23
vrr
2233333
)(
)()(3
)(2),(
)(
)(t
tr
tt
tr
GMttt
t
tts
Tss
sE
rrs
s
rr
11Φr
r
)(
)(
)()()( t
tt
tttt s
s
sssLC
r
r
rrr
T
antrsdisprT
antss
s
dtttt
tt
)'''()()(
)(
ΔΔrUΔr
r
)/( cdtt
http://gpspp.sakura.ne.jp
ECI to ECEF TransformationECI to ECEF Transformation
)()()()()()( ttGASTyxt zpxpy PNRRRU
• Precession/Nutation Model :IAU1976/1980, IERS1996, IAU2000A
• Earth Rotation Parameters
)(sec19
))1(()10(
cos
TAIUTCtt
UTCUTtrUTGMSTGMST
GMSTGAST
GPSTUTC
UTCh
http://gpspp.sakura.ne.jp
Tropospheric Delay ModelTropospheric Delay Model
))(1)((
)(
ZHDZTDgGgGElM
ZHDElMT
NNEEwet
dry
)coscotsincot( AzElgAzElg NE
• ZTD : Zenith Total Delay
• GE ,GN : Tropospheric Gradient
• Mdry,Mwet : Mapping Function (NMF etc)
)00028.02cos00266.01/(0022768.0 0 HPZHD
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Satellite Antenna Offset/PCVSatellite Antenna Offset/PCV
PRN22 Nadir Angle - Postfit Residuals (m)
• Satellite Antenna Offset (m)Block II/IIA : (0.279,0,1.023), IIR : (0,0,0)
• Satellite Antenna PCV
)( pcvspcvssataoecisatapcs ΔEΔ
http://gpspp.sakura.ne.jp
Receiver Antenna Offset/PCVReceiver Antenna Offset/PCV
• Eccentricity : Marker to ARP
• Antenna PCV : IGS_01.PCVAPC : ARP to L1/L2 Phase CenterPCV : Elevation Dependent Variation
)(2)(1
21'
21
21
ElΔCElΔCΔ
CC
pcvrpcvrpcvr
apcrapcrecceceflocalapcr
ΔΔΔEΔ
APC
ARP
Marker
http://gpspp.sakura.ne.jp
Site DisplacementsSite Displacements
• Solid Earth Tides
• Ocean Loading
• Polar Tides
• Atmospheric Loading, Snow, ...
• IERS Conventions 1996/2003
)(' polaroceansolideceflocalsdisp ΔΔΔEΔ
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Relativity/Phase Wind-up EffectRelativity/Phase Wind-up Effect
• Relativistic Effect to Satellite Clock
• Phase Wind-up EffectRCP phase affected by Relative Rotation between Satellite/Receiver Antenna
)'/'(cos)21( 121 DDDD CCΔphw
sr
sr rrrrkykxkkxDykxkkxD /)(')'('')(
cΔ ssrel /2 vr
http://gpspp.sakura.ne.jp
Satellite Orbit ModelSatellite Orbit Model
• Geopotential : JGM-3 + Tidal Effects
• 3rd Body : Moon/Sun, JPL DE405
• Solar Radiation Pressure : GPS SRP Model
• Relativity Effect
relsrprdbodygeop
sEs
r
GMaaaa
rr 33
http://gpspp.sakura.ne.jp
Perturbations of Satellite Orbit Perturbations of Satellite Orbit A
ccel
erat
ion
(m/s
2 )
1
10-2
10-4
10-6
10-8
10-10
10-12
0 10000 20000 30000Height (km)
GEOGPS
GM
J2,0
J2,2 Moon
Sun
SRPSolid TideRelativity
J6,6
J18,18
Air Drag
http://gpspp.sakura.ne.jp
GPS SRP ModelGPS SRP Model
• Precise GPS Specific SRP Model :ROCK 4/42CODE RPRGSPM04.b
• Eclipse Model :Cylindric, Penumbra/Umbra
• Ambiguous in Eclipse and Post-Eclipse Maneuver Period
http://gpspp.sakura.ne.jp
Other State Transition ModelsOther State Transition Models
• Satellite Clock : 1st order Gauss-Marcov
• Receiver Clock : White-Noise
• Tropos. Parameters : Random-Walk
• Earth Rotation Parameters : Random-Walk
• Carrier Phase Bias :Fixed in a Arc, Reinitialized on Cycle-Slip
http://gpspp.sakura.ne.jp
Estimation FilterEstimation Filter
kT
kkkkkk
t
tkkk
tttt
dk
k
QΦPΦP
xfxx
),()(),()(
)),(ˆ()(ˆ)(ˆ
111
11
)()()(
)))(ˆ(()(ˆ)(ˆ
))(()( 1
kkkk
kkkkk
kT
kkkT
kkk
PHKIP
xhzKxx
RHPHHPK
Extended Kalman Filter
kz
http://gpspp.sakura.ne.jp
Backward Filter/SmootherBackward Filter/Smoother
)(ˆ)()(ˆ)()()()(ˆ 11111 ttttttTt bbb xPxPPPx
10/2 0:00 10/2 12:00 10/3 0:00 10/3 12:00 10/4 0:00 10/4 12:00 10/5 0:00
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
3D P
ositi
on E
rror
(m
)
Forward (RMS:66.9cm)Backward (RMS:4.0cm)Smoothed (RMS:3.6cm)
PRN01 Orbit Estimation Error (wrt IGS Final)
http://gpspp.sakura.ne.jp
Quality ControlQuality Control
• Prefit Residuals Test
• Postfit Residuals Test
• Outlier Exclusion and Re-Filtering
• Statistical Test of Outlier Rate, Prefit/Postfit Residuals RMS and Valid Data Outage
• Bad Satellite/Station Exclusion and Rollback/Re-Estimation
http://gpspp.sakura.ne.jp
Evaluation of AccuracyEvaluation of Accuracy
• GT ver.0.5.5
• GPS Week 1291 (2004/10/3-10/9)
• GPS 29 Satellites / IGS 40 Stations
• Initial Orbit/Clock : Broadcast Ephemeris
• Station Pos : Fixed to Previous Week PPP
• 3Pass Filter/Smoother
• 24H+Overlap24H, Interval : 300sec
http://gpspp.sakura.ne.jp
GpsTools (GT) ver.0.5.5GpsTools (GT) ver.0.5.5
http://gpspp.sakura.ne.jp
IGS 40 StationsIGS 40 Stations
150 W 120 W 90 W 60 W 30 W 0 30 E 60 E 90 E 120 E 150 E 180 E
60 S
30 S
0
30 N
60 N
ALGO
AMC2
AREQASC1
AUCK
BAHR
CAS1
CEDU
COCO
DAV1
DRAO
FAIR
HOB2
IISC
IRKT
KERG
KOUR
LHAS
LPGS
MAC1
MAG0
MAS1MATE
MKEA
NRC1
NRIL
ONSA
PERT
PIE1
PIMO
POL2
REUN
RIOG
STJO
THTI
TROM
TSKBUSNO VILL
YELL
http://gpspp.sakura.ne.jp
Estimated ParametersEstimated Parameters
ParametersNumberof States
Sats/Stas Total
Satellite Position/Velocity 6 29 174Satellite SRP Parameters 4 29 116Satellite Clock 2 29 58Station Tropos. ZTD 1 40 40Station Tropos. Gradient 5 40 200Station Receiver Clock 1 40 40Carrier Phase Bias 1 29 x 40 1160Earth Rotation Parameters 3 1 3Total Parameters per Epoch 1788
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IGS Final Orbit/ClockIGS Final Orbit/Clock
• ~300 World-Wide GPS Stations
• Weighted Average of 8 Analysis Centers CODE, ESA, GFZ, JPL, NRCan, MIT, NOAA, SOPAC
• Various Analysis S/W Packages
• Formal AccuracyOrbit : < 5cm, Clock : < 0.1nsec
http://gpspp.sakura.ne.jp
Estimated Orbit AccuracyEstimated Orbit Accuracy
3D RadialAlong-Track
Cross-Track
PositionRMSError
5.18cm 1.97cm 3.43cm 3.12cm
Average of 29 Satellites, wrt IGS Final
http://gpspp.sakura.ne.jp
Orbit Accuracy by SatellitesOrbit Accuracy by Satellites
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
GP
S01
G
PS
03
GP
S04
G
PS
05
GP
S06
G
PS
07
GP
S08
G
PS
09
GP
S10
G
PS
11
GP
S13
G
PS
14
GP
S15
G
PS
16
GP
S17
G
PS
18
GP
S19
G
PS
20
GP
S21
G
PS
22
GP
S23
G
PS
24
GP
S25
G
PS
26
GP
S27
G
PS
28
GP
S29
G
PS
30
GP
S31
A
vera
ge
Pos
ition
RM
S E
rror
(m
)
3DRadialAlong-TrackCross-Track
Orbit Error wrt IGS Final
http://gpspp.sakura.ne.jp
Estimated Orbit (Example)Estimated Orbit (Example)
0
0.1
0.2
0.3
0.4RMS: 0.0385m
Pos
itio
n E
rror
3D
(m
)
10/3 10/4 10/5 10/6 10/7 10/8 10/9
-0.2
-0.1
0
0.1
0.2
RMS R: 0.0121m A: 0.0321m C: 0.0175mPos
itio
n E
rror
R/A
/C (
m)
RadialAlong-TrackCross-Track
PRN01 Orbit Error wrt IGS Final
http://gpspp.sakura.ne.jp
Estimated Clock AccuracyEstimated Clock Accuracy
With BIAS Without BIAS
Clock BiasRMS Error 0.135ns 0.099ns
Average of 29 Satellites, wrt IGS Final
http://gpspp.sakura.ne.jp
Clock Accuracy by SatellitesClock Accuracy by Satellites
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
GP
S01
G
PS
03
GP
S04
G
PS
05
GP
S06
G
PS
07
GP
S08
G
PS
09
GP
S10
G
PS
11
GP
S13
G
PS
14
GP
S15
G
PS
16
GP
S17
G
PS
18
GP
S19
G
PS
20
GP
S21
G
PS
22
GP
S23
G
PS
24
GP
S25
G
PS
26
GP
S27
G
PS
28
GP
S29
G
PS
30
GP
S31
A
vera
ge
Clo
ck B
ias
RM
S E
rror
(ns
ec)
Clock Error wrt IGS Final
http://gpspp.sakura.ne.jp
Estimated Clock (Example)Estimated Clock (Example)
3.662
3.664
3.666
3.668
3.67x10
5
Clo
ck B
ias
(ns)
10/3 10/4 10/5 10/6 10/7 10/8 10/9-1
-0.5
0
0.5
1
Clo
ck B
ias
Err
or (
ns)
MEAN: 0.073ns RMS: 0.127ns
PRN01 Clock/Clock Error wrt IGS Final
http://gpspp.sakura.ne.jp
Comparison with OthersComparison with OthersRMS Error
Orbit/Clock Analysis S/WOrbit Clock
GT 3pass GpsTools ver.0.5.5 5.2cm 0.14nsCODE Bernese ver.5.0 3.3cm 0.11nsNRCan GIPSY/OASIS-II ver.2.6 3.8cm 0.10nsESOC BAHN,GPSOBS,BATUSI 14.2cm 0.13nsGFZ EPOS.P.V2 3.1cm 0.09nsJPL GIPSY/OASIS-II 5.5cm 0.15nsMIT GAMIT ver.10.02,GLOBK ver.5.08 5.3cm 0.31ns
NOAA page5 9.9cm -
IGSAC
SOPAC GAMIT ver.9.72,GLOBK ver.4.17 7.5cm -Average of 29 Satellites, wrt IGS Final
http://gpspp.sakura.ne.jp
Estimated ZTD (Example)Estimated ZTD (Example)ALGO Tropospheric ZTD
http://gpspp.sakura.ne.jp
Estimated ERPEstimated ERP
0.2
0.201
0.202
0.203
Xp
(")
0.415
0.42
0.425
Yp
(")
10/3 10/4 10/5 10/6 10/7 10/8 10/9-0.4576-0.4574-0.4572-0.457
-0.4568-0.4566
UT
1-U
TC
(se
c)
IGS Final
http://gpspp.sakura.ne.jp
Station Geometry DependencyStation Geometry Dependency
8 Stations 12 Stations 24 Stations
12.3cm 7.7cm 5.7cm
0.25ns 0.20ns 0.16ns
Estimated Orbit/Clock RMS Error
Average of 29 Satellites, wrt IGS Final
http://gpspp.sakura.ne.jp
ConclusionsConclusions
• Precise Orbit/Clock Determination of GPS Satellites by Carrier Phase Measurements
• Accuracy wrt IGS FinalOrbit : 5.2cm (3D RMS)Clock : 0.14ns (Bias RMS)
• Future Work :Real-Time, LEO Sats. Orbit, ...Details : See http://gpspp.sakura.ne.jp
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