## 4 Simulation Experiments and Discussion
### Experiment 2: Analysis of the Reflector TDE Impacts on ABP Accuracy
#### 4.4.1 Analysis for Thermal Misalignment Errors  
First, we used the \"control variate method\" to analyze the BPE caused by each thermal misalignment error (i.e., when analyzing the impact of a certain thermal misalignment error on the BPE, all other TDEs were set to zero). To explore the relationship between the thermally induced BPE, thermal misalignment error, and ECF scan azimuth, the BPE was calculated for every \(1^{\circ}\) of the ECF scan azimuth as the thermal misalignment error was set to \(\pm 10\)\(\upmu\)rad, \(\pm 50\)\(\upmu\)rad, \(\pm 100\)\(\upmu\)rad, \(\pm 500\)\(\upmu\)rad, and \(\pm 1000\)\(\upmu\)rad (note that counterclockwise rotations are positive while clockwise rotations are negative). Here, we chose some representative experimental results, which are shown in Figures 6 and 7. 1. It can be seen from Figure 6 that once \(\varphi_{q}\) was specified, the impact of \(\varphi_{\text{PSR},ma}\) on the BPE increased approximately linearly with the increase in \(|\varphi_{\text{PSR},ma}|\). 2. A careful look at Figure 6a shows that, for a specified level of \(\varphi_{\text{PSR},ma}\), the resulting BPE curve showed a significant sinusoidal wave pattern over a scan period of the ECF, with the minimum appearing near \(\varphi_{q}=0^{\circ}\) and the maximum near \(\varphi_{q}=180^{\circ}\). Furthermore, it could be found that the sinusoidal trend in the BPE curve became much more evident as \(\varphi_{\text{PSR},ma}\) increased. The reason for this phenomenon was that the rate at which the BPE increased changed with \(\varphi_{q}\) (manifested as a dispersion of the BPE growth rates at different values of \(\varphi_{q}\) in Figure 6b). As \(\varphi_{\text{PSR},ma}\) varied from \(\pm 10\)\(\upmu\)rad to \(\pm 1000\)\(\upmu\)rad, the dispersion of the BPE growth rates at different \(\varphi_{q}\) values was widened; thus, the fluctuation of the BPE curve became quite pronounced. 3. Quantitatively speaking, the fluctuation range (i.e., the difference between the maximum BPE and the minimum BPE) of the BPEs was only \(\sim\)0.1 \(\upmu\)rad with \(\varphi_{\text{PSR},ma}=\pm 10\)\(\upmu\)rad; however, when \(\varphi_{\text{PSR},ma}=\pm 1000\)\(\upmu\)rad, the fluctuation range of the BPEs could reach \(\sim\)12 \(\upmu\)rad. Figure 6: _Cont._  
Based on the above experimental results, we could draw the following conclusions:  
1. According to Figure 7a, the BPE caused by the roll misalignment angle \(\varphi_{\text{MR},ma}\) of the MR was primarily related to \(\varphi_{\text{MR},ma}\) itself and did not change with the scan azimuth \(\phi_{\text{q}}\) of the ECF. 2. From Figure 7b, once \(\phi_{q}\) was specified, the \(\varphi_{\text{MR},ma}\)-induced BPE increased approximately linearly with the increase in \(|\varphi_{\text{MR},ma}|\). 3. A careful look at Figure 7a shows that there was not much change in the shape of the BPE curve as \(\varphi_{\text{MR},ma}\) varied from \(\pm 10\) urad to \(\pm 1000\) urad, which was mainly because of the consistency of the BPE growth rates at different values of \(\phi_{q}\) (see Figure 7b). 4. Quantitatively speaking, the fluctuation range of the BPEs was only \(\sim\)0.02 urad with \(\varphi_{\text{MR},ma}=\pm 10\) urad, and when \(\varphi_{\text{MR},ma}=\pm 1000\) urad, the fluctuation range of the BPEs was still only \(\sim\)2 urad. Based on independent impact assessment for each thermal misalignment error, we could provide impact rankings for these errors. For each thermal misalignment error (which was set to \(\pm 10\) urad, \(\pm 50\) urad, \(\pm 100\) urad, \(\pm 500\) urad, and \(\pm 1000\) urad 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 8. Particularly, when the thermal misalignment error level reached the maximum of \(\pm 1000\) urad. The maximum impact of each thermal misalignment error on the BPE is shown in Table 2.  
Based on the experimental results obtained in this subsection, the following conclusions could be drawn:  
1.