## 5 Mirror Facet Alignment Method
### Three-Rotation Alignment Method
\tag{18}\]  
According to matrix \(\mathbf{H}^{\mathrm{ABC}}\), vector \(\mathbf{N}_{\mathrm{t}}^{1}\) has strong nonlinearity, so that the parallel problem of vector \(\mathbf{N}_{\mathrm{t}}^{1}\) and \(\mathbf{N}_{\mathrm{t}}\) is equivalent to the minimum optimization problem of their intersection angle. The optimization objective can be defined as:  
\[\min\ f(\theta_{A},\theta_{B},\theta_{C})=\arccos(\left|\frac{\mathbf{N}_{ \mathrm{t}}^{1}\cdot\mathbf{N}_{\mathrm{t}}}{\|\mathbf{N}_{\mathrm{t}}^{1}\| \cdot\|\mathbf{N}_{\mathrm{t}}\|}\right|) \tag{19}\]  
which can be calculated by the Genetic Algorithm method. More specifically, using the Genetic Algorithm Tool in Matlab (R2012a) [32], Equation (19) was optimized for determining the rotation angle \(\theta_{A}\), \(\theta_{B}\), and \(\theta_{C}\). To improve the optimization accuracy of the results, Equation (19) is optimized \(m\) times (\(m=45\) in this paper) to get \(m\) sets of angle \(\theta_{A}\), \(\theta_{B}\), and \(\theta_{C}\). Finally, a set of angles \(\theta_{A}\), \(\theta_{B}\), and \(\theta_{C}\) is selected to make the error angle \(\theta\) as small as possible for the particular facet alignment parameters of the mirror unit. Under the actual operation, the facet alignment of the mirror unit is adjusted using bolts 3, 2, and 1, and it is necessary to turn the rotation angles \(\theta_{C}\), \(\theta_{B}\), and \(\theta_{A}\) to the adjust values of bolt 3, 2, and 1, respectively. These adjustment values are determined in relation to the support-adjust structure. Taking the adjustment of bolt 3 as an example (Figure 6a), bolt 3 can be adjusted by rotating the mirror unit around axis \(\mathbf{A}_{\mathrm{t}}\mathbf{B}_{\mathrm{t}}\) with rotation angle is \(\theta_{C}\) while bolts 1 and 2 remain unchanged. Bolt 3 will be deflected in the connection hole during the adjustment process as it is close to the triangle. The triangle geometry satisfies following relationship:  
\[\left(L_{C}^{AL}\right)^{2}+\left(L_{C}^{\mathrm{t}}\right)^{2}-2L_{C}^{AL}L_{ C}^{\mathrm{t}}\cos\beta_{c}=\left(d_{C}^{\mathrm{t}}\right)^{2} \tag{20}\]  
where, \(L_{C}^{\mathrm{t}}\) is the support length of bolt 3 before adjustment, \(L_{C}^{AL}\) is length of bolt 3 after adjustment, \(d_{C}^{\mathrm{t}}\) is distance moved by the ball-hinge center \(C\), where \(d_{C}^{\mathrm{t}}=\|\mathbf{C}_{\mathrm{t}}-\mathbf{C}_{\mathrm{AL}}\|\), and \(\beta_{c}\) is the intersection angle before and after adjusting bolt 3. \(\beta_{c}\leq\beta_{\mathrm{max}}\), where \(\beta_{\mathrm{max}}\) is the maximum intersection angle before and after the adjustment of bolt 3 (Figure 6b) and can be calculated as Equation (22). In Figure 6b, the relationship between the diameter \(d_{2}\)of the bolt, thickness\(L_{1}\) of connection plate, and diameter \(d_{1}\) of the connection hole can be described by:  
\[d_{1}=L_{2}+L_{3}=\frac{d_{2}}{\cos\beta_{1}}+L_{1}\tan\beta_{1} \tag{21}\]  
where \(L_{2}\) is the hypotenuse length of the red triangle (left hand side), \(L_{3}\) is right-angle side length of the green triangle (right hand side). Hence, the angle \(\beta_{\mathrm{max}}\) can be derived as:  
\[\beta_{\mathrm{max}}=2\beta_{1}=2(\arccos(\frac{d_{2}}{\sqrt{\left(d_{1}\right) ^{2}+\left(L_{1}\right)^{2}}})-\arccos(\frac{d_{1}}{\sqrt{\left(d_{1}\right)^{2}+ \left(L_{1}\right)^{2}}})) \tag{22}\]Normally, the diameter \(d_{2}\) of the bolt is smaller (2.0-4.0 mm) than the diameter \(d_{1}\) of the connection hole, so that \(\beta_{\max}\) will also be small. Due to \(\beta_{c}\leq\beta_{\max}\) and assuming \(\cos\beta_{c}\approx 1.0\), Equation (20) can be rewritten as \(d_{\text{C}}^{\text{t}}=\left|L_{\text{C}}^{AL}-L_{\text{C}}^{\text{t}}\right|\). The adjustment value \(d_{\text{C}}\) of bolt 3 is the difference between \(L_{\text{C}}^{\text{t}}\) and \(L_{\text{C}}^{AL}\) therefore \(|d_{\text{C}}|=d_{c}^{\text{t}}=\left|L_{\text{C}}^{AL}-L_{\text{C}}^{\text{t}}\right|\).