## 3 Image-Motion Compensation Technique of the Secondary Mirror Motion
### Aberration Field Constant Technique
#### 3.2.1 Secondary Mirror Motion Induces First-Order Effects  
In order to elucidate the operational principle of the secondary mirror in compensating for image motion, it is essential to establish a model of the misalignment optical system and investigate the interrelationship between the deviation vector of the secondary mirror and the misalignment of the secondary mirror. The offset of the image point position of the optical system caused by the movement of the secondary mirror can be determined by tracing the optical axis rays to the image plane [14; 15], as illustrated in Figure 3.  
According to the theory of the off-axis vector aberration, the offset of the image point position of the optical system caused by the motion of the secondary mirror can be obtained by Equation (3):  
\[\overset{\rightarrow}{H}_{IMG}=\sqrt{\overset{\rightarrow}{H}_{x\_IMG}^{2}+ \overset{\rightarrow}{H}_{y\_IMG}^{2}}, \tag{3}\]  
where \(\overset{\rightarrow}{H}_{IMG}\) is the offset of the image point position of the optical system, \(\overset{\rightarrow}{H}_{x\_IMG}\) is the vector components of the image point position offset in the \(x\)-axes, and \(\overset{\rightarrow}{H}_{y\_IMG}\) is the vector components of the image point position offset in the \(y\)-axes. In Equation (3), the vector components of the image point position offset in the X and \(y\)-axes of the image plane can be expressed as follows:  
\[\left[\overset{\rightarrow}{H}_{x\_IMG}\right]=\frac{\beta L(1-m)}{(f^{\prime }_{\text{SYS}})\gamma\overline{y}_{IMG}}\left[ \begin{matrix}E_{SM\_x}-\frac{2 \gamma(f^{\prime}_{\text{SYS}})}{1-m}\varphi_{SM\_y}\\ E_{SM\_y}+\frac{2\gamma(f^{\prime}_{\text{SYS}})}{1-m}\varphi_{SM\_x}\end{matrix}  \right], \tag{4}\]  
\[L=\gamma f^{\prime}_{\text{SYS}}, \tag{5}\]  
\[m\equiv-\frac{f^{\prime}_{\text{SYS}}}{f^{\prime}_{\text{PM}}}, \tag{6}\]  
Figure 3: Interrelationship between the deviation vector and the misalignment of the secondary mirror. where \(\beta\) is the magnification of the correction lens, \(\gamma\) is the blocking ratio, \(L\) is the distance from the aspherical vertex of the secondary mirror to the Gaussian image plane, \(f^{\prime}_{SYS}\) is the focal length of the telescope system, \(f^{\prime}_{PM}\) is the focal length of the primary mirror, \(\overline{y}_{IMG}\) is the height of the primary optical axis of the image coordinate system, \(E_{SM\_x}/E_{SM\_y}\) is the eccentricity of the secondary mirror, \(\varphi_{SM\_x}/\varphi_{SM\_y}\) is the tilt of the secondary mirror, and \(m\) is the ratio of \(f^{\prime}_{SYS}\) to \(f^{\prime}_{PM}\). In order to facilitate calculation, Equation (4) is further derived to obtain:  
\[ \begin{bmatrix}\stackrel{{\rightarrow}}{{H}}_{x\_IMG}\\ \stackrel{{\rightarrow}}{{H}}_{y\_IMG}\end{bmatrix} =\frac{\beta} {\overline{y}_{IMG}} \begin{bmatrix}(1-m)E_{SM\_x}-(2L)\varphi_{SM\_y}\\ (1-m)E_{SM\_y}+(2L)\varphi_{SM\_x}\end{bmatrix} , \tag{7}\]  
In addition, the center of rotation of the secondary mirror is set as the coma-free pivot point. When the tilt of the secondary mirror is small, according to the geometric relationship shown in Figure 4, the eccentricity of the secondary mirror can be simplified as Equation (8). \[\left[ \begin{array}{c}E_{SM\_x}\\ E_{SM\_y}\end{array} \right]= \begin{bmatrix}L^{cfp}_{SM}\cdot\sin\begin{pmatrix} 180\varphi_{SM\_y}\\ \pi\end{pmatrix}\\ L^{cfp}_{SM}\cdot\sin\begin{pmatrix}180\varphi_{SM\_x}\\ \pi\end{pmatrix}\end{bmatrix} \approx \begin{bmatrix}L^{cfp}_{SM}\cdot\varphi_{ SM\_y}\\ L^{cfp}_{SM}\cdot\varphi_{SM\_x}\end{bmatrix} , \tag{8}\]  
where \(L^{cfp}_{SM}\) is the distance from the aspherical vertex of the secondary mirror to the center of curvature of the secondary mirror. By substituting Equation (8) into Equation (7), we obtain:  
\[ \begin{bmatrix}\stackrel{{\rightarrow}}{{H}}_{x\_IMG}\\ \stackrel{{\rightarrow}}{{H}}_{y\_IMG}\end{bmatrix} =\frac{\beta} {\overline{y}_{IMG}} \begin{bmatrix}\left((1-m)L^{cfp}_{SM}-2L\right)\varphi_{ SM\_y}\\ \left((1-m)L^{cfp}_{SM}+2L\right)\varphi_{SM\_x}\end{bmatrix}