## 3 Compensation Correction Models for the Off-Axis TMA Telescope
\begin{array}{l}f_{i,1}^{(q)}=f_{0i}+\frac{\partial f_{i}^{(q)}}{\partial(x _{SM})}x_{SM,1}+\frac{\partial f_{i}^{(q)}}{\partial(y_{SM})}y_{SM,1}+\frac{ \partial f_{i}^{(q)}}{\partial(y_{SM})}\theta_{SM,1}+\frac{\partial f_{i}^{(q )}}{\partial(y_{SM})}\phi_{SM,1}+\frac{\partial f_{i}^{(q)}}{\partial(z_{SM}) }z_{SM,1}\\ f_{i,2}^{(q)}=f_{0i}+\frac{\partial f_{i}^{(q)}}{\partial(x_{SM})}x_{SM,2}+ \frac{\partial f_{i}^{(q)}}{\partial(y_{SM})}y_{SM,2}+\frac{\partial f_{i}^{( q)}}{\partial(y_{SM})}\theta_{SM,2}+\frac{\partial f_{i}^{(q)}}{\partial(y_{SM}) }\phi_{SM,2}+\frac{\partial f_{i}^{(q)}}{\partial(z_{SM})}z_{SM,2}\\ \vdots\\ f_{i,N}^{(q)}=f_{0i}+\frac{\partial f_{i}^{(q)}}{\partial(z_{SM})}x_{SM,N}+ \frac{\partial f_{i}^{(q)}}{\partial(y_{SM})}y_{SM,N}+\frac{\partial f_{i}^{( q)}}{\partial(y_{SM})}\theta_{SM,N}+\frac{\partial f_{i}^{(q)}}{\partial(y_{SM}) }\phi_{SM,N}+\frac{\partial f_{i}^{(q)}}{\partial(z_{SM})}z_{SM,N}\\ \end{array} \Bigg{\}}.\] (27) Equation (27) is the mathematical model of the constructed multiple linear regression. 4. Writing Equation (27) in the form of a matrix so as to get \[\alpha=\beta\chi,\] (28) where \[\alpha=\left(f_{i,1}^{(q)}\quad f_{i,2}^{(q)}\quad\cdots\quad f_{i,N}^{(q)} \right)^{T},\] (29) \[\beta= \begin{pmatrix}1&x_{SM,1}&y_{SM,1}&\theta_{SM,1}&\phi_{SM,1}&z_{SM,1 }\\ 1&x_{SM,2}&y_{SM,2}&\theta_{SM,2}&\theta_{SM,2}&z_{SM,2}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 1&x_{SM,N}&y_{SM,N}&\theta_{SM,N}&\phi_{SM,N}&z_{SM,N}\\ \end{pmatrix} ,\] (30) \[\chi=\left(f_{0i}^{(q)}\quad\frac{\partial f_{i}^{(q)}}{\partial(x_{SM})} \quad\frac{\partial f_{i}^{(q)}}{\partial(y_{SM})}\quad\frac{\partial f_{i}^{ (q)}}{\partial(\theta_{SM})}\quad\frac{\partial f_{i}^{(q)}}{\partial(\theta_ {SM})}\quad\frac{\partial f_{i}^{(q)}}{\partial(z_{SM})}\right)^{T}.\] (31)  
Finally, using the least squares method to estimate \(\chi\), we can get  
\[\chi=\left(\beta^{T}\beta\right)^{-1}\beta^{T}\alpha, \tag{32}\]  
where the superscript \(T\) represents the matrix transpose operation; the superscript \(-1\) represents the matrix inversion operation. In the above, a correction model for compensating TM misalignments and PM figure errors is constructed by using SM as the compensator. When using TM as the compensator to construct a correction model for compensating SM misalignments and PM figure errors, it is only necessary to modify Equations (8)-(10) as the following equations:  
\[C_{5,C}^{TM}+C_{5,L}^{TM}+C_{5,O}^{TM}=C_{5}-C_{5,C}^{SM}-C_{5,L}^{SM}-C_{5,O} ^{SM}-(n^{\prime}-n)_{(FIGURE)}C_{5,C}\Bigg{\}}, \tag{33}\] \[C_{6,C}^{TM}+C_{6,L}^{TM}+C_{6,O}^{TM}=C_{6}-C_{6,C}^{SM}-C_{6,L}^{ SM}-C_{6,Q}^{SM}-(n^{\prime}-n)_{(FIGURE)}C_{6,C}\Bigg{\}},\] (34) \[C_{7,C}^{TM}+C_{7,L}^{TM}=C_{7}-C_{7,C}^{SM}-C_{7,L}^{SM}-(n^{ \prime}-n)_{(FIGURE)}C_{7,C}\Bigg{\}},\] \[C_{8,C}^{TM}+C_{8,L}^{TM}=C_{8}-C_{8,C}^{SM}-C_{8,L}^{SM}-(n^{ \prime}-n)_{(FIGURE)}C_{8,C}\Bigg{\}},\] \[C_{9,C}^{TM}=C_{9}-C_{9,C}^{SM}-(n^{\prime}-n)_{(FIGURE)}C_{9,C}. \tag{35}\]The calculation method of the compensation adjustments required for the TM is similar to the calculation method of the SM, and will not be repeated here.