Page - 102 - in Contributions to GRACE Gravity Field Recovery - Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations

Here,RNOMCRF is the rotation to the CRF from the nominal orientation of the spacecraft as described in section 4.3, with the KBR antenna pointing to the other spacecraft. This rotary is computed from the satellite orbits and the calibrated APC coordinates and is assumed to be error-free. The remaining rotation R˜SRFNOM from the SRF to the nominal orientation is the inverse of the small-angle Euler rotation sequence described in eqs. (6.2.4) and (6.2.5): R˜SRFNOM= R˜ T α (8.1.10) This is the rotation for which the covariance matrix was determined in the sensor fusion algorithm of section 6.2. Substituting these rotations in eq. (8.1.8) gives ∆ρsAOC=c T SRF ( RNOMCRF R˜ SRF NOM )T eCRF =cTSRF ( R˜SRFNOM )T( RNOMCRF )T eCRF =cTSRFR˜ NOM SRF R CRF NOMeCRF . (8.1.11) The rightmost product in eq. (8.1.11),RCRFNOMeCRF, is the satellite baseline expressed in the nominal orientation of the spacecraft. There is no need to explicitly calculate this vector, though, as by definition it is exactly the direction of the APC vector expressed in the SRF, as illustrated in fig. 8.2. In other words, the satellite baseline in the nominal orientation is RCRFNOMeCRF= cSRF ‖cSRF‖ . (8.1.12) Inserting into eq. (8.1.11) then gives ∆ρsAOC= 1 ‖cSRF‖ ·c T SRFR˜ NOM SRF cSRF , (8.1.13) and finally, using eq. (8.1.10) ∆ρsAOC= 1 ‖cSRF‖ ·c T SRFR˜αcSRF . (8.1.14) The partial derivative of the AOC w.r.t. the small angle rotationsα is then, via the chain rule ∂∆ρsAOC ∂α = ∂∆ρsAOC ∂vec ( R˜α )∂ vec(R˜α) ∂α . (8.1.15) With the APC vectorcnow understood to be given in the SRF, and using the ordering for matrix derivatives as described in eq. (2.1.7), the first partial is ∂∆ρsAOC ∂vec ( R˜α )= 1‖c‖ ·cT⊗cT = 1 ‖c‖ · [ cxcx cxcy cxcz cycx cycy cycz czcx czcy czcz ] . (8.1.16) Chapter8 Star Camera Observations and Uncertainties102