Precise Orbit Determination of Precise Orbit Determination of GPS Satellites using Carrier GPS Satellites using Carrier
Phase MeasurementPhase Measurement
Technical ConsultantTechnical Consultant Tomoji TAKASUTomoji TAKASUKasai Design Office Ltd.Kasai Design Office Ltd. Shoji KASAIShoji KASAI
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 with GPS Receiver
http://gpspp.sakura.ne.jp
LCphwrel
pcvrpcvsLC
LLLC
NTdTdtcCC
)(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
),()(
)( ttr
tttr
GMttt
ttt
s
Tss
sE
rrs
s
rr11Φ
rr
)()(
)()()( ttt
tttt s
s
sssLC
rr
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
TAIUTCttUTCUTtrUTGMSTGMST
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
http://gpspp.sakura.ne.jp
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.PCV
APC : 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
http://gpspp.sakura.ne.jp
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
rGM
aaaar
r 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
10-120 10000 20000 30000
Height (km)
GEOGPS
GM
J2,0
J2,2 MoonSun
SRPSolid TideRelativity
J6,6
J18,18
Air Drag
http://gpspp.sakura.ne.jp
GPS SRP ModelGPS SRP Model
• Precise GPS Specific SRP ModelROCK 4/42CODE RPRGSPM04.b
• Eclipse ModelCylindric, Penumbra/Umbra
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
xxxx
),()(),()(
)),(ˆ()(ˆ)(ˆ
111
11
)()()()))(ˆ(()(ˆ)(ˆ
))(()( 1
kkkk
kkkkk
kT
kkkT
kkk
PHKIPxhzKxxRHPHHPK
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 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 : Previous Week PPP results• 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 ParametersParameters Number
of 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
http://gpspp.sakura.ne.jp
IGS Final Orbit/ClockIGS Final Orbit/Clock
• ~300 World-Wide GPS Stations• Weighted Average of 8 Analysis Centers C
ODE, ESA, GFZ, JPL, NRCan, MIT, NOAA, SOPAC
• Various Analysis S/W Packages• Formal Accuracy
Orbit : < 5cm, Clock : < 0.1nsec
http://gpspp.sakura.ne.jp
Estimated Orbit AccuracyEstimated Orbit Accuracy
3D Radial Along-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
GPS
01
GPS
03
GPS
04
GPS
05
GPS
06
GPS
07
GPS
08
GPS
09
GPS
10
GPS
11
GPS
13
GPS
14
GPS
15
GPS
16
GPS
17
GPS
18
GPS
19
GPS
20
GPS
21
GPS
22
GPS
23
GPS
24
GPS
25
GPS
26
GPS
27
GPS
28
GPS
29
GPS
30
GPS
31
Ave
rage
Posi
tion
RM
S Er
ror (
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
Posi
tion
Erro
r 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: 0.0175mPosi
tion
Erro
r 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
GPS
01
GPS
03
GPS
04
GPS
05
GPS
06
GPS
07
GPS
08
GPS
09
GPS
10
GPS
11
GPS
13
GPS
14
GPS
15
GPS
16
GPS
17
GPS
18
GPS
19
GPS
20
GPS
21
GPS
22
GPS
23
GPS
24
GPS
25
GPS
26
GPS
27
GPS
28
GPS
29
GPS
30
GPS
31
Ave
rage
Clo
ck B
ias
RM
S Er
ror (
nsec
)
Clock Error wrt IGS Final
http://gpspp.sakura.ne.jp
Estimated Clock (Example)Estimated Clock (Example)
3.6623.6643.6663.6683.67x105
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 E
rror
(ns)
MEAN: 0.073ns RMS: 0.127ns
PRN01 Clock/Clock Error 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
0.2010.202
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
UT1
-UTC
(sec
)
IGS Final
http://gpspp.sakura.ne.jp
Station Geometry DependencyStation Geometry Dependency
8 Stations 12 Stations 24 Stations
12.3cm 7.7cm 5.7cm0.25ns 0.20ns 0.16ns
Estimated Orbit/Clock RMS Error
Average of 29 Satellites, wrt IGS Final
http://gpspp.sakura.ne.jp
ConclusionsConclusions
• Orbit/Clock Determination of GPS Satellites using Carrier Phase Measurement
• Accuracy wrt IGS FinalOrbit : 5.2cm (3D RMS)Clock : 0.14ns (Bias RMS)
• Future Work : Real-Time Application
Details : See http://gpspp.sakura.ne.jp
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