Page - 45 - 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

from the calibrated KBR antenna phase centre coordinates. It is of note that for the nominal orientation, the KF and the LOSF coincide, so the rotation from the KF to the LOSF is the identity matrix. The nominal orientation can thus be computed analytically, without relying on orientation observations by the spacecraft. The actual orientation of the spacecraft R˜SRFCRF is observed by the SCA. Due to the active steering of the satellites throughout science operations, the difference between these two rotations R˜α= ( RNOMCRF )T R˜SRFCRF = ( RCRFLOSFR LOSF KF R KF SRF ) R˜SRFCRF = ( RCRFLOSFR KF SRF ) R˜SRFCRF (6.2.4) is a small angle rotation, on the order of some few milliradians (Herman et al., 2004). This attitude deviation can be written as an Euler sequence of roll, pitch, and yaw rotations with respect to the axes of the SRF R˜α=Rz,SRF(yaw)Ry,SRF(pitch)Rx,SRF(roll) . (6.2.5) The orientation anglesα for one epochτ are then α(τ)= [ roll(τ) pitch(τ) yaw(τ) ]T . (6.2.6) 6.2.1 Parametrization The original implementation of the SCA/ACC sensor fusion at IfG was parametrized in terms of full rotation quaternions. This approach has some caveats: Estimating the full four quaternion parameters for a three dimensional rotation is an over-parametrization of the problem. The solution must be constrained in such a way that the length of the quaternion is unity. Further, when considering the covariance information of a single epoch’s quaternion, it is clear that this dependency of the quaternion elements leads to not fully invertible covariance matrices. The implementation described here is parametrized in terms of small angle rotations in roll, pitch, and yaw with regard to a reference orientation. This has the advantage that only three parameters must be determined per epoch. No singularities or gimbal lock can occur as, due to the active steering of the spacecraft, the rotations are always small angles. Further, the covariance matrices of each epoch are fully invertible without additional constraints. The parameter vector is x= [ αT1 α T 2 · · · αTN xTcal ]T (6.2.7) with theαi the orientation at each of the N epochs of a variational arc andxcal some calibration parameters. 6.2 Sensor Fusion 45