Page - 103 - 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

GRACE-A GRACE-B zSRF xSRF eNOM cA Figure 8.2: Orientation of spacecraft in nominal attitude with APC vector The second partial depends on the chosen Euler sequence used to represent the small- angle rotation R˜α. After analytically forming the complete rotary as described in eq. (6.2.5), partials w.r.t. to roll, pitch, and yaw are taken. For the Euler sequence from eq. (6.2.5) used here, these can for example be found in Diebel, 2006, section 8.11. For the individual Euler rotations, the derivatives are ∂R˜α ∂roll =    0 spitchcrollcyaw−srollsyaw spitchsrollcyaw+syawcroll0 −spitchsyawcroll−srollcyaw −spitchsrollsyaw+crollcyaw 0 −cpitchcroll −srollcpitch    , (8.1.17) ∂R˜α ∂pitch =    −spitchcyaw srollcpitchcyaw −cpitchcrollcyawspitchsyaw −srollsyawcpitch syawcpitchcroll cpitch spitchsroll −spitchcroll    , (8.1.18) ∂R˜α ∂yaw =    −syawcpitch −spitchsrollsyaw+crollcyaw spitchsyawcroll+srollcyaw−cpitchcyaw −spitchsrollcyaw−syawcroll spitchcrollcyaw−srollsyaw 0 0 0    , (8.1.19) with sroll= sin(roll), croll= cos(roll), and equivalently for pitch and yaw. Using the correct ordering, the full matrix-vector derivative is ∂ vec ( R˜α ) ∂α =                  0 −spitchcyaw −syawcpitch 0 spitchsyaw −cpitchcyaw 0 cpitch 0 spitchcrollcyaw−srollsyaw srollcpitchcyaw −spitchsrollsyaw+crollcyaw −spitchsyawcroll−srollcyaw −srollsyawcpitch −spitchsrollcyaw−syawcroll −cpitchcroll spitchsroll 0 spitchsrollcyaw+syawcroll −cpitchcrollcyaw spitchsyawcroll+srollcyaw −spitchsrollsyaw+crollcyaw syawcpitchcroll spitchcrollcyaw−srollsyaw −srollcpitch −spitchcroll 0                   . (8.1.20) 8.1 The Antenna Offset Correction in the ll-SST Observation Equation 103