## 5 Positioning and Orbit Determination of Space Targets
### Gaussian Minimum Mean Square Error Correction Positioning Algorithm  
The optimal solution of the nonlinear static equation can be determined with the Gaussian minimum mean square error differential correction algorithm [26]. The loss equation determined by the Wahba minimum mean square error is as follows:  
\[J\ =\ \frac{1}{2}\sum_{i\ =\ 1}^{N}\left(\widetilde{b}_{i}-h_{i}(\hat{x}) \right)^{T}W_{i}\left(\widetilde{b}_{i}-h_{i}(\hat{x})\right) \tag{19}\]  
The optimization of the loss equation can be attributed to the minimization of the redundant mean square error with the weight. Where, \(i\) is a single measurement label, and \(N\) is the total number of available measurements. \(W\ =\ W^{T}<0\) is a weight coefficient matrix to assess the relative importance of each measurement. Estimated by the minimum variance estimation criterion, the optimal weight coefficient matrix \(W\) is the inverse matrix of the measurement covariance matrix \(R^{-1}\). The Gaussian minimum mean square error differential correction algorithm uses an iterative approximation to find accurate minimum mean square estimates. Assuming the estimate of the current state is \(\hat{x}_{k}\), then the next target state estimates \(\hat{x}_{k+1}\) can be expressed as follows:  
\[\hat{x}_{k+1}\ =\ \hat{x}_{k}+\Delta x \tag{20}\]  
If \(\Delta x\) is small enough, \(h(\hat{x})\) can be linearized with [PERSON]'s first-order expansion linearization. \[h_{i}(\hat{x})\approx h_{i}(\hat{x}_{k})+H_{i,k}\Delta x \tag{21}\]  
where \(H_{i,k}\) is the Jacobian matrix of the measurement model. \[H_{i,k}\ =\ \left[\frac{\partial h_{i}}{\partial x}\right]_{\hat{x}_{k}} \tag{22}\]  
Figure 11: Multiple star trackers observation schematics. Then the redundancy error of the state estimation can be expressed as follows:  
\[ \begin{array}{l}\Delta b_{i,k+1}\ =\ \widetilde{b}_{i}-h_{i}(\hat{x}_{k+1})\\ \approx\ \widetilde{b}_{i}-h_{i}(\hat{x}_{k})-H_{i,k}\Delta x\\ =\ \Delta b_{i,k}-H_{i,k}\Delta x\end{array}  \tag{23}\]  
The measurement vector and the prediction vector can be expressed as follows:  
\[\widetilde{Y}\ =\ \left[ \begin{array}{c}\widetilde{b}_{1}\\ \widetilde{b}_{2}\\ \vdots\\ \widetilde{b}_{N}\end{array} \right],\hat{Y}_{k}\ =\ \left[ \begin{array}{c}h_{1}(\hat{x}_{k})\\ h_{2}(\hat{x}_{k})\\ \vdots\\ h_{N}(\hat{x}_{k})\end{array} \right] \tag{24}\]  
Then the measurement error and sensitivity matrix can be expressed as follows:  
\[\Delta Y_{k}\ =\ \widetilde{Y}-\hat{Y}_{k}\ =\ \left[ \begin{array}{c} \widetilde{b}_{1}-h_{1}(\hat{x}_{k})\\ \widetilde{b}_{2}-h_{2}(\hat{x}_{k})\\ \vdots\\ \widetilde{b}_{N}-h_{N}(\hat{x}_{k})\end{array} \right],A_{k}\ =\ \left[ \begin{array}{c}H_{1}\\ H_{2}\\ \vdots\\ H_{N}\end{array} \right] \tag{25}\]  
To minimize the loss Equation of (19), that is, the optimal correction amount is searched to minimize the linearized prediction redundancy mean square error:  
\[J_{k+1}\approx\frac{1}{2}(\Delta Y_{k}-A_{k}\Delta x)^{T}W(\Delta Y_{k}-A_{k} \Delta x) \tag{26}\]  
Solve the above formula, the optimal approximate solution is  
\[\Delta x\ =\ P_{k}A_{k}^{T}W\Delta Y_{k} \tag{27}\]  
where \(P_{k}\) is the covariance matrix,  
\[P_{k}\ =\ \left(A_{k}^{T}WA_{k}\right)^{-1} \tag{28}\]  
Therefore, \(\hat{x}_{k}\) of Formula (20) will be updated to be \(\hat{x}_{k+1}\). An initial estimation state \(\hat{x}_{0}\) is required to start the entire Gaussian minimum mean square error differential correction algorithm, and an iterative condition is required to stop the entire algorithm.