## 6 Experimental Verification of the Mirror Alignment Method
### Numerical Experiment  
Theoretical validity of the three-rotation and two-rotation alignment method was assessed by numerical experiments. The detail and steps of this numerical experiment as follows:  
(1) Ideal parameters of the mirror unit. The outermost ring mirror unit in the 38 kW XEM-Dish system (Figure 1) was selected as Part1 and then the support position of bolt 3 was changed as Part2 for the numerical simulation. Due to the numerical experiment did not involve measuring the coordinate of the feature points, so the ball-hinge centers A, B, and C were taken as feature point T\({}_{1}\), T\({}_{2}\), and T\({}_{3}\), respectively. In ideal facet, the coordinates of the ball-hinge center and feature point, as well as the axis vector \(\mathbf{N}_{0}\) and position vector \(\mathbf{Q}\) of the vertex in Part 1 and 2 as shown in Table 1 (in coordinate system O-_xyz_). (2) Installation error of the mirror unit. In the numerical experiments for Parts 1 and 2, the installation error of the mirror unit was introduced by ThreeR-OneT motion in order BCA and translation vector \(\mathbf{M}_{1}=[15.0,12.0,5.0]\) (in Section 4). That is the Parts 1 and 2 through ThreeR-OneT motion from ideal facet to get error condition (i.e., facet 1). In the ThreeR-OneTwith order BCA, the error combination of rotation angles \(\theta_{\text{B}}\), \(\theta_{\text{C}}\) and \(\theta_{\text{A}}\) is given by:  
\[\left[\theta_{\text{B}},\theta_{\text{C}},\theta_{\text{A}}\right]= \left[(i-1)\frac{2\varphi}{n-1}-\varphi,\ (j-1)\frac{2\varphi}{n-1}-\varphi,\ (k-1)\frac{2 \varphi}{n-1}-\varphi\right] \tag{27}\]  
where,\(\varphi\) is the maximum rotation angle; \(n\) is the equal divided number; \(i,j\) and \(k\) are all integers within the range of 1-nam the initial values of them are 1. In this paper, \(\varphi=52.40\) mrad and \(n=4\). The combination process of the angles \(\theta_{\text{B}}\), \(\theta_{\text{C}}\), and \(\theta_{\text{A}}\) is that if \(k\) from 1 to \(n\) cycle once, \(j=j+1\); if \(j\) from 1 to \(n\) cycle once, \(i=i+1\). The combination number was \(N=(i-1)\times n^{2}+(j-1)\times n+k\), \(N\in 1,2,\cdots,n^{3}\). For example, if \(N=21\) presents \(i=2,j=2\), and \(k=1\), i.e., \(\theta_{\text{B}}=-8.73\) mrad, \(\theta_{\text{C}}=-8.73\) mrad, and \(\theta_{\text{A}}=-26.20\) mrad. (3) Numerical simulation for mirror-facet alignment. The equivalent of facet 1 is adopted as the ABC rotation order (in Section 4) and the mirror-facet alignment adopts the CBA rotation order (in Section 5). The numerical results of the facet alignment in Parts 1 and 2 are shown in Figure 8.  
The error angle \(\theta\) of axis vector range is 0.001\(\sim\)0.080 mrad and 0.003\(\sim\)0.45 mrad (the error range was 2.950\(\sim\)61.190 mrad before alignment), by adopting the three-rotation and two-rotation alignment methods, respectively (Figure 8a,c). The results show that the alignment method has higher theoretical accuracy, which has verified their validity. The vertex position error of mirror unit can be greatly reduced by both methods but cannot be completely eliminated (Figure 8b,d). In Part 1, the error distance \(L\) was in the range 16.45\(\sim\)73.00 mm and 19.22\(\sim\)44.29 mm for the two-rotation and three-rotation methods, respectively (\(L\) ranged from 22.41 to 506.86 mm before alignment) (Figure 8b). To further reduce the vertex position error, the facet position of the mirror unit must be controlled before the initial installation. In this way, the vertex position error of the mirror unit can be effective controlled by real-time alignment using two visual measurement methods, which will be discussed in Section 7.