Satellite Orbit and Ephemeris Determination using Inter Satellite Links

State EstimationInter Satellite LinksΦ (linearized) transition matrixThe transition matrix can either be derived from the linearized dynamic matrix or bynumerical derivation of the Jacobian (see chapter 3.3). The transition matrix is not onlyneeded for state prediction, but also for propagation of the covariance matrix P. In fact, the"noise shaping" function of the transition matrix is essential, if states which can not bedirectly observed are included in the state vector, e.g. velocity is estimated from rangemeasurements. According to [GEL-88], the covariance propagation can be written as~PkT Tk− 1⋅ ⋅ Φk−1+( Q )= Φ Pˆ diagEq. 3.5-2k−1withP covariance matrixQ process noiseIf measurements are available, the predicted covariance matrix and state vector can beupdated. The updated state is then obtained byxˆ ( z − H ⋅ )~ kxk+ K~k⋅k kxk= Eq. 3.5-3withz measurementK Kalman gain matrixand the updated covariance matrix bywithPˆ = ( I − K kH k) P ~Eq. 3.5-4kI Identity matrixThe Kalman gain matrix can be interpreted as a weighting matrix of the innovation introducedby the measurement z. It depends on the apriori covariance and the measurement noise andcan be computed from the following equation.withRHKk= P ~k⋅ HTk( ~ T−H ⋅ P ⋅ H + diag( R )) 1measurement noise(linearized) observation matrixkkkEq. 3.5-53.5.2 Filtering to EpochIt is possible to operate the Kalman filter as a batch estimator. The filter algorithms are thesame as for the real time filter, except there is no process noise, state transient or covariancepropagation within the processed batch interval. Instead of the real time measurement matrixH, the measurement matrixPage 16R. Wolf