Satellite Orbit and Ephemeris Determination using Inter Satellite Links

ISL Observation ModelInter Satellite LinksThe range rate, i.e. the velocity along the line of sight vector between two points can bewritten as:x xy yz zLD−21−21−2=1 ⋅(xD1− xD2)+ ⋅(yD1− yD2)+ ⋅(zD1− z2)LLLDEq. 2.2-2with point index 1 being the (first) satellite and point two being either a known location on theearth’s surface or a second satellite.Forming the partials with respect to the satellites state yields∂LD x1− xEq. 2.2-32=∂xDL∂LDxD1− xD2LD⋅ ( x1− x2)= −= 02∂xL LThe partial with respect to the position and velocity in y- and z-direction can be obtained in asimilar manner. From the two equations above it can be see that the range rate equation isalready linear. We can therefore write the linear measurement equation for a range rateobservable in the case of an inter satellite link asL− L=Eq. 2.2-4=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⋅ ∆zNote that the range rate measurement is independent of the clock state of the satellite.2Page 6R. Wolf