## 5 Evaluation and Discussion of the Kinematic Calibration Method Using Optimal Measurement Pose  
In order to evaluate the proposed kinematics calibration method of six-hardpoint positioning mechanisms using optimal measurement pose, the simulations of calibrations using the proposed method and traditional calibration method were performed under the same conditions. The same conditions refer to the same kinematic parameter errors and the same measurement noise. To eliminate chance, a new set of initial configuration parameters were used, as shown in Table 5. The set kinematic parameter error is shown in Table 6. The randomly generated parameter error assumes a normal distribution with a standard deviation of 0.5 mm. The measurement noise of the pose of the primary mirror is also normally distributed, with a standard deviation of 0.001 mm. The theoretical kinematic parameters and kinematic parameter errors are superimposed to simulate the actual kinematic parameters, and the measurement noise is superimposed on the actual pose of the primary mirror to obtain the measurement value of the pose. The number \(n\) of the measurement poses is 20. Figure 5: Histogram superimposed PDF curve of integrated error of simulations of calibration. (**a**) Measurement condition I; (**b**) Measurement condition II; (**c**) Measurement condition III; (**d**) Measurement condition IV. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(i\) & \(x_{Ai}\) & \(y_{Ai}\) & \(z_{Ai}\) & \(x_{Bi}\) & \(y_{Bi}\) & \(z_{Bi}\) & \(h_{i}\) \\ \hline
1 & 1299.170 & \(-\)348.112 & 0 & 1164.388 & \(-\)977.037 & 0 & 800 \\
2 & 1299.170 & 348.112 & 0 & 1164.388 & 977.037 & 0 & 800 \\
3 & \(-\)348.112 & 1299.170 & 0 & 263.945 & 1496.908 & 0 & 800 \\
4 & \(-\)951.059 & 951.059 & 0 & \(-\)1428.333 & 519.871 & 0 & 800 \\
5 & \(-\)951.059 & \(-\)951.059 & 0 & \(-\)1428.333 & \(-\)519.871 & 0 & 800 \\
6 & \(-\)348.112 & \(-\)1299.170 & 0 & 263.945 & \(-\)1496.908 & 0 & 800 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: New theoretical kinematic parameters (unit: mm). The proposed calibration method requires selecting the optimal measurement poses based on the observability index \(O_{1}\). Therefore, the improved DETMAX algorithm was used to select 20 poses from 2000 poses to be selected, with the criteria of maximizing \(O_{1}\). The 2000 poses are evenly distributed in the workspace. In the algorithm, the size of the tabu list is set to 100, resulting in 20 optimal measurement poses. The observability index of the optimal measurement poses was \(O_{1}\) = 0.03571. After 219 iterations, the result was obtained. The variation of \(O_{1}\) during the selection process is shown in Figure 6. When using the traditional DETMAX algorithm, the result is obtained by only 21 iterations, and the obtained value of \(O_{1}\) is smaller than the value obtained by the improved DETMAX algorithm. The improved DETMAX algorithm improves the problem of convergence to the local optimal solution. In order to ensure the effectiveness of the improved DETMAX algorithm, 1000 sets of poses were randomly selected from 2000 poses, and each set has 20 poses. The observability index \(O_{1}\) corresponding to each set was calculated and the results were plotted as a histogram, as shown in Figure 7. The maximum value of \(O_{1}\) corresponding to these sets is 0.02199, and the average value is 0.01713. The results show that the value of \(O_{1}\) has been maximized, and the improved DETMAX algorithm performs well.