## 5 The Complete-Inverse Problem  
In Sections 3 and 4.2, we showed the feasibility of the two sub-problems, respectively. However, the feasibility of the two sub-problems cannot guarantee the feasibility of the complete-inverse problem. In this case, the gravity field and orbit are coupled, and we need to estimate the orbit and gravity field together. In our work, the mutual influence between the two sets of parameters (the gravity field and the orbits) is so strong that it is hard to converge. When the error of the initial estimate is too large, the complete-inverse problem is hard to converge. To solve this problem, we use the adaptive step size to control in the iteration process. As shown in Equation (19). \[\widetilde{x}=\frac{\hat{x}}{1+k|\hat{x}|} \tag{19}\]  
\(\hat{x}=[\hat{x}_{0},\hat{c}]\) is the calculated correction value by the iteration process. It is quite possible that \(\hat{x}\) is too large. Directly correcting the estimate with this calculated value may lead to a worse estimate and eventually the failure of the iteration process. To overcome this problem, we deliberatively decrease the size of the correction value by introducing the step-size control parameter \(k\). \(\hat{x}\) is the actual correction value taken and \(k\) is an empirical constant which usually takes the value of 1. In such a case, when \(\hat{x}\) is large (i.e., the estimate is far from the true value), the size of the actual correction \(|\hat{x}|\) is always smaller than 1. When \(\hat{x}\) is small (i.e., the estimate is close to the true value), \(\hat{x}\approx\hat{x}\). Test 3 shows two examples with good initial estimates. The simulation observation data of them are the same: The initial position is the same as Section 3.2. But the initial estimate of initial position is different. Random error (with a threshold of 10 m) along all three directions is added differently in the two examples. The initial estimate of the gravity field is the same, where \(\mathrm{C}_{20}=-5.20\times 10^{-2}\) and \(\mathrm{C}_{22}=8.30\times 10^{-2}\). For an arc length of 50(T), the result is shown in Figure 6 and Table 3. And Table 4 reflects the change of RMS for two examples before and after the calculation. Figure 5: Contour map of the residual with respect to the gravity-field coefficients \(\mathrm{C}_{20}\) and \(\mathrm{C}_{22}\) (erroneous initial position). Obviously, the results after iteration are much better than the initial estimate. For test 3a, the residuals of observation are less than 0.5 m, and the position errors of estimated orbits are less than 0.8 m. For test 3b, the residuals are less than 0.3 m and the position errors are less than 2 m. In addition, the difference in the results of these two tests is small. The accuracy of the estimated gravity-field coefficients is about 0.02% for test 3a and about 0.1% for test 3b. The RMS of them are similar. Thus, we can say that the orbit determination process has converged. \begin{table}
\begin{tabular}{c c c} \hline \hline
**RMS** & **Test 3.a** & **Test 3.b** \\ \hline
**Initial Estimate** & 6.4053 & 8.0739 \\ \hline
**Result** & 0.1167 & 0.1417 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The change of RMS in test 3. Figure 6: Residual in the observation data (**left**) and norm of the position error of the determined orbit from the true one (**right**) (test 3 with small errors in the initial estimate). However, when the initial estimate is poor, the iteration process converges harder, and the accuracy of the results obviously decreases. In test 4, we randomly added a position error of 50 m in all the three directions of both the spacecraft. The initial estimate of the gravity-field coefficients are \(C_{20}=-5\times 10^{-2}\) and \(C_{22}=9\times 10^{-2}\). Setting the arc length as 50(T), the result is shown in Figure 7 and Table 5.  
Figure 7 and Table 5 indicate that the iteration process has not converged, because some of the final result is even worse than the initial estimates.