Precise Orbit Determination of Precise Orbit Determination of GPS Satellites using Carrier GPS Satellites using Carrier

Phase MeasurementPhase Measurement

Technical ConsultantTechnical Consultant Tomoji TAKASUTomoji TAKASU

Kasai 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

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

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.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

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

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 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

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 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 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

http://gpspp.sakura.ne.jp

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

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

• 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