## 3 Selecting Optimal Measurement Poses Based on Observability Index
### Algorithm for Selecting Measurement Pose  
In order to select a group of poses that maximize a certain observability index from many poses, we need a selection algorithm. We considered the DETMAX algorithm [32], which was proposed and used to maximize the determinant of the covariance matrix. The algorithm uses the method of exchanging design points in the initial set and candidate point set multiple times to improve the performance of the initial set, and finally obtains the best solution. The algorithm is widely used in the design of experiments, and the principle of this algorithm is as follows:  
1. Set the candidate point set to \(\mathbf{p}\) and select the initial solution \(\mathbf{s}\).
2. Select an element from \(\mathbf{p}\) and add it to \(\mathbf{s}\) to maximize the performance of \(\mathbf{s}\).
3. Remove an element from \(\mathbf{s}\) and return it to \(\mathbf{p}\) to maximize the performance of \(\mathbf{s}\).
4. Repeat steps 2 and 3, and the termination condition is that the added element and the removed element are the same. This algorithm will improve the performance of \(\mathbf{s}\).  
After applying this algorithm to measurement pose selection, it is found that the algorithm has extremely fast convergence speed, and the observability index of the selected poses is indeed higher than the index of the initial poses. However, two problems of this algorithm were found. The first is that it is easy to converge to a local optimal solution, and the second is that the obtained results are greatly affected by initial values. Therefore, this article uses tabu search to improve the DETMAX algorithm, and the improved algorithm adds a tabu list to store a certain number of poses that have been temporarily eliminated during the iteration process. The above improvements effectively avoid the two problems of DETMAX. The process of the improved DETMAX algorithm is as follows:  
1. Assume that there are 10,000 primary mirror poses to be selected. Let the set of 10,000 primary mirror poses be \(\Omega_{0}\). Extract \(n\) poses from \(\Omega_{0}\) as the initial pose set \(\zeta_{\mathrm{n}}\), and extract \(m\) poses as the initial tabu list \(\eta_{\mathrm{m}}\). Then the set \(\Omega_{0}\) becomes set \(\Omega\), and there are 10,000-\(n\)-\(m\) poses in \(\Omega\).
2. Extract a pose from \(\Omega\) and add it to \(\zeta_{\mathrm{n}}\) to form \(\zeta_{\mathrm{n+1}}\), and calculate the observability index of \(\zeta_{\mathrm{n+1}}\). Perform this step for all 10,000-\(n\)-\(m\) poses in \(\Omega\), retain the pose that maximizes the observability index, and the \(\zeta_{\mathrm{n+1}}\) corresponding to this pose. There are \(n+1\) poses in \(\zeta_{\mathrm{n+1}}\). Then the set \(\Omega\) becomes \(\Omega-\) and there are 9999-n-m poses in \(\Omega-\).
3. Remove a pose from \(\zeta_{\mathrm{n+1}}\), to obtain \(\zeta_{\mathrm{n}}\). Calculate the observability index of \(\zeta_{\mathrm{n}}\). Perform this step for all \(n+1\) poses in \(\zeta_{\mathrm{n+1}}\), retain the \(\zeta_{\mathrm{n}}\) that maximizes the observability index. Record the pose that was removed from \(\zeta_{\mathrm{n+1}}\) in the last position of the tabu list \(\eta_{\mathrm{m}}\). Then, remove the first pose of the \(\eta_{\mathrm{m}}\), and add the pose to \(\Omega-\) to form \(\Omega\). Now, there are \(n\) poses in \(\zeta_{\mathrm{n}}\).  
4. Return to step 2 and continue to execute until the maximum number of iterations is reached. Finally, the \(n\) poses in set \(\zeta_{n}\) are the optimal measurement pose that maximize the observability index. In order to compare the performance of the improved DETMAX algorithm with the traditional DETMAX algorithm, two algorithms were used to select the poses, and the results are shown in Table 1. The results of observability indices of the selected poses indicate that the improved DETMAX algorithm has obtained bigger observability indices, therefore, the improved DETMAX algorithm has better performance.