## 5 Mirror Facet Alignment Method
### Two-Rotation Alignment Method  
The two-rotation alignment method only adjusts two bolts and the third is not adjusted (i.e., the adjustment value is 0 mm). This could improve the efficiency of mirror facet alignment. The mathematical model of the two-alignment method is presented in this section. In many cases, the distance between any two ball-hinge centers in the mirror unit is much greater than the adjustment value of the bolts, so the rotation angles \(\theta_{A}\), \(\theta_{B}\), and \(\theta_{C}\) are micro-amounts in Three-ROneT motion (Figure 4). For example, \(\theta_{C}\approx d_{c}^{t}/\|\vec{\mathbf{C}_{1}}\|\leq 0.06\) rad due to \(\|\vec{\mathbf{C}_{1}}\|>>d_{c}^{t}\) when \(\|\vec{\mathbf{C}_{1}}\|\geq 500\) mm and \(d_{c}^{t}\leq 30\) mm. When angle \(\beta\in[0,\ \ 0.18]\) rad, the maximum relative error \(\max((\beta-\sin\beta)/\beta)\) is less than 0.6% satisfied by \(\sin\beta\approx\beta\); therefore, it is assumed that \(\sin\theta_{C}\approx\theta_{C}\), \(\sin\theta_{B}\approx\theta_{B}\), and \(\sin\theta_{A}\approx\theta_{A}\), as well as \(\mathbf{e_{A_{1}^{i}C}}\approx\mathbf{e_{AC}}\) and \(\mathbf{e_{B_{1}^{i}A_{1}^{i}}}\approx\mathbf{e_{BA}}\). Based on these assumptions, abandoning the second order, and allowing the micro-amounts (such as the \(\theta_{A}\cdot\theta_{B}\) is second order microamount) to be negligible, the total rotation matrix \(\mathbf{H}^{\text{ABC}}\) (Equation (15)) can be simplified in order to derive the linearization of the total rotation matrix \(\mathbf{H}_{1}^{\text{ABC}}\) as:  
\[\mathbf{H}_{1}^{\text{ABC}}=\left[ \begin{array}{ccc}1.0&z_{1}\cdot\theta_{A }+z_{2}\cdot\theta_{B}+z_{3}\cdot\theta_{C}&-y_{1}\cdot\theta_{A}-y_{2}\cdot \theta_{B}-y_{3}\cdot\theta_{C}\\ -z_{1}\cdot\theta_{A}-z_{2}\cdot\theta_{B}-z_{3}\cdot\theta_{C}&1.0&x_{1}\cdot \theta_{A}+x_{2}\cdot\theta_{B}+x_{3}\cdot\theta_{C}\\ y_{1}\cdot\theta_{A}+y_{2}\cdot\theta_{B}+y_{3}\cdot\theta_{C}&-x_{1}\cdot \theta_{A}-x_{2}\cdot\theta_{B}-x_{3}\cdot\theta_{C}&1.0\end{array} \right] \tag{24}\]  
where \(\left[x_{1},y_{1},z_{1}\right]=\mathbf{e_{CB}}\), \(\left[x_{2},y_{2},z_{2}\right]=\mathbf{e_{AC}}\), \(\left[x_{3},y_{3},z_{3}\right]=\mathbf{e_{BA}}\). In the same way, the linearization of the total rotation matrix of different permutations (ACB, BAC, CAB, or CBA) could also be derived, and it was found that the linearization of each total rotation matrix are the same as \(\mathbf{H}_{1}^{\text{ABC}}\) such that \(\mathbf{H}_{1}^{\text{ABC}}=\mathbf{H}_{1}^{\text{ACB}}=\mathbf{H}_{1}^{\text {BCD}}=\mathbf{H}_{1}^{\text{BDC}}=\mathbf{H}_{1}^{\text{CAB}}=\mathbf{H}_{1}^ {\text{CBA}}\). Figure 7: Flowchart of the three-rotation alignment method for aligning the axis vector of single mirror unit in the dish concentrator. This shows that when the rotation angle \(\theta_{A}\)\(\sim\)\(\theta_{C}\) is small, the facet parameter of the mirror unit does not affect the rotation order in ThreeR-OneT motion. Therefore, the equivalent of facet 1 is always adopted as the ABC rotation order (Section 4). The mirror facet alignment adopts the CBA rotation order (Section 5) in this study. Using matrix \(\mathbf{H}_{1}^{\text{ABC}}\) to replace \(\mathbf{H}^{\text{ABC}}\), the axis vector \(\mathbf{N}_{\text{t}}^{2}\) can be deduced and is given by:  
\[\left\{ \begin{array}{l}\mathbf{N}_{\text{t}}^{2}(x)=y_{1}\cdot\theta_{A}+y_{ 2}\cdot\theta_{\text{B}}+y_{3}\cdot\theta_{\text{C}}\\ \mathbf{N}_{\text{t}}^{2}(y)=-x_{1}\cdot\theta_{A}-x_{2}\cdot\theta_{B}-x_{3} \cdot\theta_{C}\\ \mathbf{N}_{\text{t}}^{2}(z)=1.0\end{array} \right. \tag{25}\]  
Then, according to the \(x\)-axis and \(y\)-axis component of the vector \(\mathbf{N}_{\text{t}}^{2}\) and vector \(\mathbf{N}_{\text{t}}\) are equal respectively for determining the relationship between the angles \(\theta_{A}\)\(\sim\)\(\theta_{C}\) as:  
\[\left\{