Satellite Orbit and Ephemeris Determination using Inter Satellite Links

State EstimationInter Satellite Linksand∂F = f Dynamic matrix of the residual or error state∂x x=xˆh( x()t ) = h xˆ ( t) ∂h ( ) + ( x − xˆ ) + ...∂x x=xˆwith∂H = h Measurement matrix of the residual or error state∂x x=xˆThe error state is the difference between thr real and the nominal system stateEq. 3.1-6∆ x = x − xEq. 3.1-7kkrealkno min alwhere the nominal state is obtained by integrating the nonlinear equations, i.e. numericallyintegrating the equations of motion.t∫x = f (x(t), t) dtknominaltkk−1In a similar manner, the residual observations can be derivedEq. 3.1-8with∆ z = z − zEq. 3.1-9kkrealkno min alzkno min al= h(x(t), t)Eq. 3.1-10no min al3.2 State VectorThe state vector at least contains the position errors, i.e. the difference between nominal andreal position. If the clock offset can not be measured directly e.g. by two way measurements,it has to be estimated together with the orbit errors. This implies that for each epoch at leastfour measurement are available to estimate the instantaneous position. However, in orbitdetermination there are frequently less observations than states per measurement epoch. Forexample, a GPS satellite is (nearly) never tracked by more than two ground stationssimultaneously. To allow the accumulation of measurements over a longer orbit arc, thevelocity errors have to be estimated as well.Thus, the minimum state vector consists of the following elementsPage 8R. Wolf