Satellite Orbit and Ephemeris Determination using Inter Satellite Links

State EstimationInter Satellite Links∂z∂z∂x∂xH ′k = = ⋅ = H ⋅∂x∂x∂x∂x000Eq. 3.4-5The partials of the actual state x with respect to the state at epoch x 0 are expressed by theJacobian, and therefore the transition matrix Φ.∂xx = ⋅ x 0 = Φ∂x0( t, t 0 ) ⋅ x 0Eq. 3.4-6Thus we can write for an arbitrary instant of time t kH′ = H ⋅ ΦEq. 3.4-7kkkThe transition matrix from epoch to a time t k can be computed successive from the precedingtransition matrices, only the transient from the previous point to the instant has to becomputedΦ( t , t ) = Φ( t , t ) ⋅ Φ( t , t ) ⋅ .... ⋅ Φ( t , t )k = Φ k 0 k k− 1 k−1k−21 0Eq. 3.4-8If the transition matrix F is computed from a linearized dynamic matrix F, the time interval(t k ,t 0 ) has to be relatively short. For longer batch lengths one would use the numericallyderived Jacobian (see Eq. 3.3-10, Eq. 3.3-11).The measurement equation system containing measurements of a certain time interval isobtained by forming the appropriate observation matrices H' k . For example, if themeasurements of four observation times t 0 – t 3 are used to determine the state at t 0 , theobservation model would look the following way⎡z⎢z⎢⎢z⎢⎣z3210( t 3,t 0 ) ⎤( t 2 , t 0 )⎥⎥ ⋅ x 0( t , t ) ⎥⎤ ⎡H 3 ⋅ ΦEq. 3.4-9⎥ ⎢⎢H 2 ⋅ Φ⎥ =⎥ ⎢ H1⋅ Φ 1 0⎥ ⎢⎥⎦ ⎣ H 0 ⎦3.4.1 Weighted Least SquaresUsually, not all measurements z are made with the same accuracy. Thus, Eq. 3.4-2 has berewritten asx =T−1T( H ⋅ W ⋅ H) ⋅ H ⋅ W ⋅ zEq. 3.4-10to account for the weights of the individual measurements. For uncorrelated measurements,the weighting matrix W is simplyPage 14R. Wolf