Satellite Orbit and Ephemeris Determination using Inter Satellite Links

State EstimationInter Satellite Links⎡ X ⎤1⎢ ⎥⎢X2X = ⎥⎢ ... ⎥⎢ ⎥⎢⎣X n ⎥⎦Eq. 3.2-3With at least 7 states, which have to be considered per satellite, it can easily be seen that thestate of a complete constellation gets very large. This leads for instance to a state vectormagnitude of 126 states for a constellation of 18 space vehicles. Although many small filters(one per each satellite) would result in a smaller computational burden, it is absolutelynecessary to process all satellites in one large filter, because the state estimates of thesatellites get correlated due to the inter satellite links.3.3 State Transition and Transition MatrixThe system of linear differential equationsxD = F ⋅ xEq. 3.3-1is not very well suited for the implementation of a discrete estimation process in a digitalcomputer. The discrete formulation of the Kalman filter for example requires the statetransient to be expressed by a simple vector-matrix-operationxk( tk, tk−1) ⋅ xk−1= Φ Eq. 3.3-2with Φ(t k ,t k-1 ) being the transition matrix from the epoch t k-1 to the epoch t k . In a more generalway, Eq. 3.3-2 can be expressed as ∂xkEq. 3.3-3xk= ⋅ xk−1∂xk−1with the transition matrix Φ(t k ,t k-1 ) being interpreted as the Jacobian∂x∂xkk−1⎡ ∂x⎢∂xk⎢⎢ ∂y⎢= ⎢∂xk⎢ ∂z⎢⎢∂xk⎢⎢⎣...k−1k−1k−1∂x∂y∂y∂y∂z∂yk−1k−1k−1...kkk∂x∂z∂y∂z∂z∂zk−1k−1k−1...kkk⎤...⎥⎥⎥... ⎥⎥⎥... ⎥⎥⎥... ⎥⎦Eq. 3.3-4Page 10R. Wolf