Page - 82 - 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, the original initial statey0 appears. The difference between the true motion and the reference motion is given by the Encke vectors∆r¨,∆r˙, and∆r. Using eqs. (7.2.2) to (7.2.4) for the true motion and eqs. (7.2.6) to (7.2.8) for the reference motion they are ∆r¨(τ)= r¨(τ)− r¨ref(τ)=∆f(τ) , (7.2.9) ∆r˙(τ)= r˙(τ)− r˙ref(τ)=∆r˙0+T ∫ τ 0 ∆f(τ′)dτ′ , (7.2.10) ∆r(τ)=r(τ)−rref(τ)=∆r0+∆r˙0(τT)+T2 ∫ τ 0 (τ−τ′)∆f(τ′)dτ′ . (7.2.11) The vectors∆r˙0 = r˙0− r˙ref,0 and∆r0 = r0−rref,0 are the differential initial state betweenthereferencemotionandthetruemotion.This formulation isnowverysimilar to the original integration problem treated in chapter 5, with only two differences to be found: First, the original initial values of the true motiony0 are replaced by the differential initial values ∆y0= [ ∆r0 ∆r˙0 ] . (7.2.12) Second, the full forcef(τ) is replaced by the disturbing force∆f(τ). This system can be solved with only minor adjustments to the algorithm presented in chapter 5. The complete steps are as follows: 1. Select a reference forcef0 with an associated analytically determinable reference trajectory. Computerref and r˙ref for the entire orbit arc. Compute the disturbing forces∆f at the approximate positionre according to eq. (7.2.1). 2. Following eqs. (5.2.12) and (5.2.13), compute the integrated Encke position and velocity ∆rinte =Kr∆r¨e . (7.2.13) 3. With Φ¯r as in eq. (5.2.7), solve the system re−rref−∆rinte = Φ¯r∆y0 (7.2.14) to compute an estimate of the differential state∆yˆ0. 4. In analogy to eq. (5.2.25), the estimated coordinate difference to the true position is ∆re=[I−KrT]−1 [ Φ¯r∆yˆ0+∆r int e +rref−re ] . (7.2.15) 5. ComputeΦr,Φr˙, andΦr¨ according to eqs. (5.2.30), (5.2.33) and (5.2.36). 6. Following eq. (5.2.37), use∆re to correct the accelerations due to the disturbing forces ∆r¨c=∆r¨e+T∆re . (7.2.16) Chapter7 Numerical Optimization in Orbit Integration82