## 4 Simulation Experiments and Discussion
### Experiment 2: Analysis of the Reflector TDE Impacts on ABP Accuracy
#### 4.4.2 Analysis for Thermal Translation Errors  
Next, by using the \"control variate method\" as previously mentioned, we analyzed the independent impact of each thermal translation error on the BPE. To explore the relationship between the thermally induced BPE, thermal translation error, and ECF scan azimuth, the BPE was calculated for every \(1^{\circ}\) of the ECF scan azimuth as the thermal translation error was set to \(\pm\)0.2 mm, \(\pm\)0.4 mm, \(\pm\)0.6 mm, \(\pm\)0.8 mm, and \(\pm\)1.0 mm. As an example, the independent impact of the pitch axial translation \(y_{\text{MR},ma}\) of the MR on the BPE was shown in Figure 9.  
Based on the above experimental results, similar conclusions could be drawn:  
1. According to Figure 9, the BPE caused by the pitch axial translation \(y_{\text{MR},ma}\) of the MR was mainly related to \(y_{\text{MR},ma}\) itself and showed very little change with the scan azimuth \(\phi_{q}\) of the ECF. 2. From Figure 9b, once \(\phi_{q}\) was specified, the \(y_{\text{MR},ma}\)-induced BPE increased approximately linearly with the increase in \(|y_{\text{MR},ma}|\). 3. A careful look at Figure 7a shows that the fluctuation range of the BPEs hardly changed as \(y_{\text{MR},ma}\) varied from \(\pm\)0.2 mm to \(\pm\)1.0 mm, which was mainly because of the consistency of the BPE growth rates at different values of \(\phi_{q}\) (see Figure 9b). 3. Quantitatively speaking, the fluctuation range of the BPEs was only \(\sim\)0.4 \(\upmu\)rad with \(y_{\text{MR},ma}=\pm\)0.2 mm, and when \(y_{\text{MR},ma}=\pm\)1.0 mm, the fluctuation range of the BPEs was still only \(\sim\)2 \(\upmu\)rad. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Thermal** & \multirow{2}{*}{\(\boldsymbol{\varphi_{\text{SSR},ma}}\)} & \multirow{2}{*}{\(\boldsymbol{\theta_{\text{SSR},ma}}\)} & \multirow{2}{*}{\(\boldsymbol{\psi_{\text{SSR},ma}}\)} & \multirow{2}{*}{\(\boldsymbol{\psi_{\text{SSR},ma}}\)} & \multirow{2}{*}{\(\boldsymbol{\varphi_{\text{FSR}  
Based on independent impact assessment for each thermal translation error, we could provide impact rankings for these errors. For each thermal translation error (which was set to \(\pm 0.2\) mm, \(\pm 0.4\) mm, \(\pm 0.6\) mm, \(\pm 0.8\) mm, and \(\pm 1.0\) mm as shown previously), the RMSE of the ABPs over a scan period of the ECF was calculated using Equation (21), and the results are shown in Figure 10. Particularly, when the thermal translation error level reached the maximum of \(\pm 1.0\) mm, the maximum impact of each thermal translation error on BPE was as shown in Table 3.  
Figure 9: The impact of the pitch axial translation \(y_{\text{MR,ma}}\). of the MR on the BPE. (**a**) The variation of the BPE with respect to the ECF scan azimuth at some fixed levels of \(y_{\text{MR,ma}}\). (**b**) The variation of the BPE with respect to \(y_{\text{MR,ma}}\) at some fixed ECF scan azimuths. Based on the experimental results obtained in this subsection, the following conclusions could be drawn:  
1. Among the thermal translation errors, \(x_{\text{SSR},ma}\) and \(y_{\text{SSR},ma}\) had no impact on the thermally induced BPE. For the remaining thermal translation errors, their impacts on the BPE had the same order of magnitude. Among them, \(y_{\text{MR},ma}\) had the highest impact on the BPE; \(y_{\text{PSR},ma}\), \(z_{\text{PSR},ma}\), and \(z_{\text{MR},ma}\) also had relatively higher impacts; \(x_{\text{MR},ma}\) and \(x_{\text{PSR},ma}\) had relatively lower impacts; and \(z_{\text{SSR},ma}\) had the lowest impact. Specifically, the impact of \(y_{\text{MR},ma}\) was about four times larger than that of \(z_{\text{SSR},ma}\). 2. Once the scan azimuth of the ECF (i.e., \(\phi_{q}\)) was specified, there was an approximately positive linear relationship between the thermal translation error and the resulting BPE.