Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Inter Satellite LinksAutonomous Onboard ProcessingUp to that point, the covariances of the satellites are assumed to be uncorrelated.⎡pP =⎢⎢0⎢⎣011p02200 ⎤0⎥⎥p ⎥33 ⎦Eq. 7.2-2Now, satellite number one is transmitting an inter satellite ranging signal which is received bysatellites number two and tree. Therefore, the measurement equation system is written as⎡z⎢⎣z1312⎤⎥⎦k⎡h= ⎢⎣h1121h022h13⎤0⎥⎦k⎡x⋅⎢⎢x⎢⎣x123⎤⎥⎥⎥⎦k−1,⎡r13R = ⎢⎣ 00 ⎤r⎥12 ⎦Eq. 7.2-3with R being the covariance matrix of the (uncorrelated) measurements. The indices for themeasurements z and variances r represent the link direction, i.e. z 13 means "link from satelliteone to satellite three". Let us now only look at the equation concerning the Kalman gainmatrix,K = PHTT −( HPH + R) 1Eq. 7.2-4which is a 3 x 2 matrix. The element K jk contains the effect of the k th measurement on the j thstate. Performing the equation using our presumptions above leads to a lengthy expression.Here, we only concentrate on a few elements. K 32 contains the effect of the measurementbetween sat1 and sat2 (measured by sat2) on the state of sat3. The expression is none-zero andrequires all partial matrices to be evaluated.( h h h p p ) det(Inv)K32 −11 13 21 11 33⋅with= Eq. 7.2-51det(Inv) = Eq. 7.2-6h + + + + + + +2 22 22 2222211h22p11p22h13h21p11p33h13h22p22p33h11p11r12h13p33r12h21p11r13h22p22r13r12r13The problem is that the measurement sat1-sat2 is not available at sat3. The Kalman gain onthe state of sat2 evaluates to22( h p + h p + r ) det(Inv)2K22= h22p22 11 11 13 33 13⋅Eq. 7.2-7if all three satellites are processed in one filter.Let us assume now that we split the filter and process the measurement sat1-sat2 and sat1-sat3independently in two separate filters.R. Wolf Page 157