Satellite Orbit and Ephemeris Determination using Inter Satellite Links

Inter Satellite LinksState EstimationΦ( t ,t )kk −1⎡ x⎢⎢x⎢⎢⎢ x= ⎢⎢⎢ x⎢⎢ x⎢⎢⎢⎣x,ky,kz,kx,k Dy,k D− x∆x− x∆y− x∆z− x∆xD− x∆yD...0,k0,k0,k0,k0,kyyyyyx,ky,kz,kx,k Dy,k D− y∆x− y∆y− y∆z− y∆xD− y∆yD...0,k0,k0,k0,k0,kzzzzzx,ky,kz,kx,k Dy,k D− z∆x− z∆y− z∆z− z∆xD− z∆yD...0,k0,k0,k0,k0,kxDxDxDxDxDx,ky,kz,kx,k Dy,k D− xD∆x− xD∆y− xD∆z− xD∆xD− xD∆yD...0,k0,k0,k0,k0,k...⎤... ⎥⎥... ⎥⎥⎥... ⎥⎥... ⎥⎥⎥... ⎥⎥... ⎥⎦Eq. 3.3-11In the case of using inter satellite links, a transition matrix for the complete constellation isobtained simply arranging the individual transition matrices as indicated in Eq. 3.3-12.ΦTotal⎡Φ⎢⎢ 0=⎢ ...⎢⎢⎣0Sat,10 ... 0 ⎤Eq. 3.3-12⎥ΦSat,2... 0 ⎥... ... ... ⎥⎥0 ... ΦSat,n ⎥⎦3.4 Least Squares Batch EstimationUsing a linear or linearized relationship between measurement z and state vector xz = H ⋅ xEq. 3.4-1the sum of squares of the residual error gets minimised byx =T −1T( H H) ⋅ H ⋅ zEq. 3.4-2The Matrix H contains the partial derivatives of the measurements with respect to theinstantaneous state. For orbit determination, the measurements of a longer orbit arc have to beconsidered to estimate the state at a certain epoch, so the equation has to be rewrittenz = H ′ ⋅Eq. 3.4-3x 0where the modified measurement matrix H' contains the partial derivatives of themeasurements z with respect to the state vector at epoch x 0 . This transforms Eq. 3.4-2 toxT −1T( H ′ H′) ⋅ H′z0 =⋅The partials of the measurements with respect to the state at epoch are obtained byEq. 3.4-4R. Wolf Page 13