Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Inter Satellite Links⎡ 1⎢ 2σ1⎢⎢ 0⎢W = ⎢⎢ 0⎢⎢ ...⎢⎢ 0⎣012σ210...00012σ3...0............0000012σn1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦with σ i ² being the variance of the i-th measurement.State EstimationEq. 3.4-113.4.2 Introducing apriori Statistic InformationSometimes, a good a-priori estimate of some states or the complete state vector, together witha related accuracy value (variance) is available. One way would be, to introduce the aprioriknowledge of the known state variables as pseudo observations, and therefore to augment themeasurement vector.If an estimate of the complete state vector is available, typically from the last iteration in aniterative process, Eq. 3.4-2 can be, according to [BIR-77] rewritten toxLS=T −1T( Λ + H H) ⋅ ( Λ ⋅ x + H ⋅ z) Eq. 3.4-120with Λ being the so called apriori information matrix. The information matrix is the inverse ofthe covariance matrix. Especially in the case of bad observation geometry together with goodpredictability (high orbit altitudes) this method can be used very successful.3.5 Kalman Filtering3.5.1 Real Time EstimationIf real time state estimation is desired, the state estimator can be implemented as a linearizedor extended Kalman filter. In the following, only a brief overview of the Kalman filteralgorithm is given. More detailed information can be found in literature, e.g. [GEL-88]. TheKalman filter estimates the state vector x of dynamic system, described by a system of firstorder linear differential equations contained in the transition matrix Φ.With the linearized equations of motion, the transition or prediction of the error state can bewritten aswithx~ x k= Φ ⋅ xˆEq. 3.5-1k−1state vectork−1R. Wolf Page 15