## 3 Error Modeling and Analysis
### Influence of Assembly Error on Pointing Accuracy  
Another factor affecting CE is assembly error, which mainly contains reflector and axis assembly errors. The reflectors in Figure 1 can be divided into three kinds. The first kinds are the FSM and PAM; these two mirrors will give feedback on the rotation angle information and correct the rotation error based on the feedback signal. The second kind of mirror is Mirrors 1 and 2, which are only used to fold the beam direction. The third kinds of mirrors are the azimuth and elevation folding mirrors. These two mirrors will introduce a CE if they are not well assembled before being utilized, which may cause axis bounce because the CE is measured in a static state; hence, the axis bounce is not considered here. Furthermore, suppose the azimuth axis and elevation axis are not strictly perpendicular (or parallel) to the reference coordinate system after assembly. In that case, it will lead to another rotation error in the folding mirror relative to the reference coordinate (because the axis and folding mirrors are rigidly connected). Based on the above analysis, it is necessary to simulate the influence of an assembly error on the beam-pointing direction. Figure 4: Beam spot energy distribution of CCR under different geometrical errors (**a**) without error, (**b**) only dihedral angle error, (**c**) only planeness error, and (**d**) both errors. According to Coordinate o-xyz in Figure 1, take the \(M_{1}\) as the target, and assuming that the rotation error angle induced by assembly error around \(x\), \(y\), and \(z\) are \(\alpha_{\text{MI}}\), \(\beta_{\text{MI}}\) and \(\gamma_{\text{MI}}\), then the rotation matrix of \(M1\) around \(x\), \(y\), and \(z\) are  
\[R_{M1}^{x}= \begin{bmatrix}1&0&0\\ 0&\cos\alpha_{\text{MI}}&\sin\alpha_{M1}\\ 0&-\sin\alpha_{M1}&\cos\alpha_{M1}\\ \end{bmatrix} ,R_{M1}^{y}= \begin{bmatrix}\cos\beta_{M1}&0&-\sin\beta_{M1}\\ 0&1&0\\ \sin\beta_{M1}&0&\cos\beta_{M1}\\ \end{bmatrix} ,R_{M1}^{z}= \begin{bmatrix}\cos\gamma_{\text{MI}}&\sin\gamma_{ \text{MI}}&0\\ -\sin\gamma_{M1}&\cos\gamma_{M1}&0\\ 0&0&1\\ \end{bmatrix} \]  
According to Figure 1, the normal of M1 is \(N_{\text{MI}}=[\surd 2/2,-\surd 2/2,0]\); therefore, its new normal after rotation is  
\[{N^{\prime}}_{M1}=R_{M1}^{x}R_{M1}^{y}R_{M1}^{z}N_{M1}. \tag{11}\]  
According to Equation (3), we can drive out the reflection matrix of \(M_{1}\) as follows:  
\[{A^{\prime}}_{M1}=E-2{N^{\prime}}_{M1}{({N^{\prime}}_{M1})}^{T}=E-2{R_{M1}^{x} }R_{M1}^{y}R_{M1}^{z}N_{M1}{\left({R_{M1}^{x}R_{M1}^{y}R_{M1}^{z}N_{M1}}\right) }^{T}. \tag{12}\]  
From Figure 1, we can determine the outgoing beam reflection order, which is PAM\(\rightarrow\)M1, \(\rightarrow\)FSM\(\rightarrow\)Az FM\(\rightarrow\)El FM, and the reflection order of the incoming beam is El FM\(\rightarrow\)Az FM\(\rightarrow\)FSM\(\rightarrow\)M2. At last, for the beam incidence on the surface of the detector, assuming that the vector of the laser from the laser source is \(A_{out}\) = [1, 0, 0], then the outgoing beam vector \({A^{\prime}}_{out}\) can be written as  
\[{A^{\prime}}_{out}=A_{El}A_{Az}A_{FSM}A_{M1}A_{PAM}\cdot A_{\text{out}}. \tag{13}\]  
Assuming that the incoming beam vector is \(A_{in}\) = [0, 0, -1], then similar to Equation (13), the beam vector \({A^{\prime}}_{in}\) in the front of the detector after it passes through the system can be derived. \[{A^{\prime}}_{in}=A_{M2}A_{FSM}A_{Az}A_{El}\cdot A_{in} \tag{14}\]  
The vector \({A^{\prime\prime}}_{out}\) is [0, 0, 1]. When \(A_{out}\) passes through the APT system, which is without error, and assuming that \({A^{\prime\prime}}_{in}\) is the vector when A\({}_{in}\) passes through the system without error, \({A^{\prime\prime}}_{in}\) can be written as [0, 1, 0]. We can obtain the error angle through the vector angle formula depending on the vector angle formula.