Satellite Orbit and Ephemeris Determination using Inter Satellite Links
Satellite Orbit and Ephemeris Determination using Inter Satellite Links Satellite Orbit and Ephemeris Determination using Inter Satellite Links
ISL Observation ModelInter Satellite Links2 ISL OBSERVATION MODELThe majority of the observation used in the orbit determination of satellites orbiting the earth,are radio frequency (pseudo-) range and Doppler measurements. Angle measurements, i.e.azimuth and elevation provide insufficient accuracy for precise orbit determination. Laserranging measurements, which are the most precise measurements available today, are stronglysubjected to weather conditions. Thus, they are used mainly for calibration purposes. Theobservations considered in this text, are therefore only one and two-way range and range rate(Doppler) measurement.2.1 Derivation of the Range EquationThe pseudo range between two points is the difference between two clock readings, the clockat the sender and the clock at the receiver. If the clocks are coarsely synchronized, the largestpart of the measured clock difference will be due to the signal travelling at the speed of light,thus representing the geometric distance.( TSat− TGroundSat 2) ⋅ c =+ c ⋅ ( δTSat− δTGround/ Sat 2) + δiono+ δTropo+ δMultipathεnoiseL =/Eq. 2.1-1= ρ +GeometricwithLandPseudo rangec Speed of lightδT Sat Deviation of satellite clock from system timeδT Groundδ IonoDeviation of ground receiver clock from system timeIonospheric delayδ Tropo Troposheric delayδ MultipathMultipath errorε noise Thermal noiseρGeometric=22( x − x ) + ( y − y ) + ( z − ) 21 2 1 2 1z2being the geometric distance between the two points.Eq. 2.1-2To obtain a linear measurement equation, the partials with respect to the unknown parametershave to be formed. Assuming that all other error contributions except the satellite clock can bemeasured or modelled, and therefore removed, we can write the linearized observationequation as a function of the three position errors and the satellite clock error. Remainingerrors e.g. due to mismodelling are added to the measurement noise.Page 4R. Wolf
Inter Satellite LinksISL Observation ModelFor a ground measurement, only the partials with respect to the satellites states are formed.The position of the ground station is assumed to be exact. The range equation for examplewould yieldL − L0=xSat− xL0GS⋅ ∆x+ySat− yL0GSz⋅ ∆y+Sat− zL0GS⋅ ∆z+ c ⋅δTwithL 0 Predicted pseudo range computed from nominal trajectorySatEq. 2.1-3For inter satellite links, the partial of the range equation with respect to both satellites stateswould have to be formed. Above equation would transform toL − L =Eq. 2.1-4=xx−0Sat,1Sat,1− xLL0− x0Sat,2Sat,2⋅ ∆x⋅ ∆x12+−yySat,1Sat,1− yLL0− y0Sat,2Sat,2⋅ ∆y1⋅ ∆y2z+z−Sat,1Sat,1− zLL0− z0Sat,2Sat,2⋅ ∆z1⋅ ∆z2+ c ⋅δT− c ⋅δTSat1Sat 2As can be seen from the equation above, an ISL observation impacts the state variables ofboth, the measuring and the target satellite.2.2 Derivation of the Range Rate EquationA radio signal being emitted from a moving sender is subjected to shift in the receivedfrequency, called the Doppler shift. This frequency shift is proportional to the velocity alongthe line of sight.ffTransmit⎛ LD⎞⎜1−c⎟⎝ ⎠or∆fRe ceive =Transmit=fLD⋅cEq. 2.2-1Normally, the frequency shift can not be directly measured, but has to be derived from thephase rate, (or the so called integrated Doppler count) instead. In the context of orbitdetermination, we are not interested in the frequency shift itself, but in the range rate whichcaused the shift. Fortunately, the phase rate can be directly scaled to a delta-range bymultiplying with the carrier wave length. A division through the integration time yields therange rate, the value we are interested in. A drawback of a range rate derived from integratedDoppler counts is that it is an averaged instead of an instantaneous value. But for shortintegration times, this fact can be neglected.From geometric considerations, or by forming the derivative of the range equation withrespect to time, we obtain the measurement equation for a range rate observable.R. Wolf Page 5
- Page 1 and 2: Satellite Orbit and EphemerisDeterm
- Page 3 and 4: Inter Satellite LinksAbstractGlobal
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- Page 7 and 8: Inter Satellite Links6.2.2 IGSO WAL
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- Page 13 and 14: Inter Satellite LinksList of Tables
- Page 15 and 16: Inter Satellite LinksJ2Earths Oblat
- Page 17 and 18: Inter Satellite LinksIntroduction1
- Page 19: Inter Satellite LinksIntroductionGE
- Page 23 and 24: Inter Satellite LinksState Estimati
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<strong>Inter</strong> <strong>Satellite</strong> <strong>Links</strong>ISL Observation ModelFor a ground measurement, only the partials with respect to the satellites states are formed.The position of the ground station is assumed to be exact. The range equation for examplewould yieldL − L0=xSat− xL0GS⋅ ∆x+ySat− yL0GSz⋅ ∆y+Sat− zL0GS⋅ ∆z+ c ⋅δTwithL 0 Predicted pseudo range computed from nominal trajectorySatEq. 2.1-3For inter satellite links, the partial of the range equation with respect to both satellites stateswould have to be formed. Above equation would transform toL − L =Eq. 2.1-4=xx−0Sat,1Sat,1− xLL0− x0Sat,2Sat,2⋅ ∆x⋅ ∆x12+−yySat,1Sat,1− yLL0− y0Sat,2Sat,2⋅ ∆y1⋅ ∆y2z+z−Sat,1Sat,1− zLL0− z0Sat,2Sat,2⋅ ∆z1⋅ ∆z2+ c ⋅δT− c ⋅δTSat1Sat 2As can be seen from the equation above, an ISL observation impacts the state variables ofboth, the measuring <strong>and</strong> the target satellite.2.2 Derivation of the Range Rate EquationA radio signal being emitted from a moving sender is subjected to shift in the receivedfrequency, called the Doppler shift. This frequency shift is proportional to the velocity alongthe line of sight.ffTransmit⎛ LD⎞⎜1−c⎟⎝ ⎠or∆fRe ceive =Transmit=fLD⋅cEq. 2.2-1Normally, the frequency shift can not be directly measured, but has to be derived from thephase rate, (or the so called integrated Doppler count) instead. In the context of orbitdetermination, we are not interested in the frequency shift itself, but in the range rate whichcaused the shift. Fortunately, the phase rate can be directly scaled to a delta-range bymultiplying with the carrier wave length. A division through the integration time yields therange rate, the value we are interested in. A drawback of a range rate derived from integratedDoppler counts is that it is an averaged instead of an instantaneous value. But for shortintegration times, this fact can be neglected.From geometric considerations, or by forming the derivative of the range equation withrespect to time, we obtain the measurement equation for a range rate observable.R. Wolf Page 5
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