## 4 Performance Comparison of Observability Indices
### Methods for Performance Comparison of Observability Indices  
In order to compare the performance of five observability indices in the kinematic calibration of six-hardpoint positioning mechanisms, a numerical simulation method for performance comparison was designed. The method is based on the measurement pose selection algorithm in Section 3.2, the numerical simulation method in Section 4.1, and the accuracy evaluation index in Section 4.2. The performance of the observability index is compared through the accuracy of many simulations of calibration under various conditions. In simulation, the theoretical kinematic parameters of the six-hardpoint positioning mechanism are shown in Table 2.  
In order to make the conclusions more universal, various measurement conditions were set in the simulation to comprehensively evaluate the performance of observability indices in the calibration of six-hardpoint positioning mechanisms. The measurement conditions refer to the number of measurement poses \(n\) and the level of measurement noise. The number of poses should meet the condition of \(n\geq 7\). In addition, in order to ensure the efficiency of simulation, \(n\) cannot be too large. Finally, the range of \(n\) is determined to be 10 to 30. The level of measurement noise mainly depends on the measurement method. Length gauges, laser trackers, linear variable differential transformers (LVDT), measuring  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(i\) & \(\textbf{x}_{Ai}\) & \(\textbf{y}_{Ai}\) & \(\textbf{z}_{Ai}\) & \(\textbf{x}_{Bi}\) & \(\textbf{y}_{Bi}\) & \(\textbf{z}_{Bi}\) & \(\textbf{h}_{i}\) \\ \hline
1 & 421.1 & \(-112.8\) & 0 & 388.9 & \(-388.9\) & 0 & 388 \\
2 & 421.1 & 112.8 & 0 & 388.9 & 388.9 & 0 & 388 \\
3 & \(-112.8\) & 421.1 & 0 & 142.4 & 531.3 & 0 & 388 \\
4 & \(-3 arms, etc., can be used to measure the pose of the primary mirror [29; 30; 31; 32], and different pose resolving methods are required. Different equipment and resolving method can lead to different measurement noise. The final determination of the measurement noise range is \(\pm\)3-\(\pm\)100 \(\upmu\)m. In summary, simulations of calibration is performed under the four measurement conditions shown in Table 3. Compare the performance of five observability indices by comparing their calibration accuracy under each condition. In order to make the conclusion more universal, 100 sets of kinematic parameter errors were randomly set. The parameter errors present a normal distribution with a standard deviation of 0.5 mm. The process and method of performance comparison are as follows:  
* Generate 100 sets of kinematic parameter errors \(\mathbf{C_{0|}}\) (j = 1, 2,..., 100), with 42 errors per set. Set 10,000 primary mirror poses evenly distributed in the workspace as the poses to be selected. LOOP 1: For each measurement condition, do:
* The measurement noise matrix is generated based on the measurement condition, and the size of the matrix is \(n\)\(\times\) 6; LOOP 2: For each observability index, do:
* The improved DETMAX algorithm is used to select \(n\) poses that maximize observability index, to obtain the optimal measurement poses. LOOP 3: For each \(\mathbf{C_{0|}}\) (j = 1, 2,..., 100), do:
* Calculate the actual kinematics parameter \(\mathbf{C^{\prime}}\) corresponding to \(\mathbf{C_{0|}}\). * Execute the calibration procedure in Section 4.1 to identify kinematic parameter errors, to obtain the corrected kinematic parameter \(\mathbf{C^{\prime}}\), and establish the corrected kinematic model K\({}^{\prime\prime}\). * Calculate the integrated error \(\mathrm{d}\mathrm{X}\mathrm{z}_{\mathrm{j}}\) (j = 1, 2,..., 100) for this calibration. END LOOP 3
* Calculate the mean value \(\mathrm{d}\mathrm{X}\mathrm{z}_{\mathrm{mean}}\) of the integrated error \(\mathrm{d}\mathrm{X}\mathrm{z}_{\mathrm{j}}\) (j = 1, 2,..., 100).