# Precision Analysis and Error Compensation of a Telescope Truss Structure Based on Robotics
## 3 Aberration-Compensation Method
### Principle of Second-Order Sensitivity Matrix  
When the optical system is in a misaligned state, the i-th fringe Zernike coefficient, \(Z_{i}\), representing the wavefront aberration of the exit pupil of the j-th field of view can be expressed by the Taylor expansion as follows:  
\[Z_{i}^{(j)}(U)=Z_{i}^{(j)}(N+\Delta M)=\sum_{p=0}^{m}\frac{1}{p!}(\sum_{q=1}^{ n}\Delta m_{q}\frac{\partial}{\partial x_{q}})^{p}Z_{i}^{(j)}(N)+R_{N,m}( \Delta M), \tag{18}\]  
where \(N=(x_{01},x_{02},\cdots,x_{0n})\) is the design state of the n adjustment degrees of freedom of the optical system, \(\Delta M=(\Delta m_{1},\Delta m_{2},\cdots,\Delta m_{n})\) is the misalignment amount on the n adjustment degrees of freedom, and \(R_{N,m}(\Delta M)\) is the expansion remainder. Figure 8: (**a**) Both variables have a linear relationship with a certain aberration coefficient, such as \(Z\gamma\) vs. (\(\Delta x\), \(\Delta\beta\)). (**b**) One of the variables shows a linear relationship, the other shows a significant non-linear relationship, such as \(Z\)S vs. (\(\Delta x\), \(\Delta\beta\)). (**c**) Both variables show a non-linear relationship, such as \(Z\)S vs. (\(\Delta\alpha\), \(\Delta\beta\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(x\) (\(mm\)) & \(y\) (\(mm\)) & \(z\) (\(mm\)) & \(\Delta\alpha\) (\({}^{\circ}\)) & \(\Delta\beta\) (\({}^{\circ}\)) \\ \hline \(\pm 2\) & \(\pm 2\) & \(\pm 2\) & \(\pm 1\) & \(\pm 1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Secondary mirror misalignment range. Figure 7: Optical path of the telescope. Then, the second-order expansion of Equation (12) can be expressed as:  
\[ \begin{array}{c}Z_{i}^{(j)}(N+\Delta M)=Z_{i}^{(j)}(N)+(\sum\limits_{q=1}^{n} \Delta m_{q}\frac{\partial}{\partial x_{q}})Z_{i}^{(j)}(N)+\\ \frac{1}{2}(\sum\limits_{q=1}^{n}\Delta m_{q}\frac{\partial}{\partial x_{q}}) ^{2}Z_{i}^{(j)}(N)+R_{N,2}(\Delta M)\approx Z_{i}^{(j)}(N)+\\ \sum\limits_{q=1}^{n}\Delta m_{q}\frac{\partial}{\partial x_{q}}+\frac{1}{2} \sum\limits_{q=1}^{n}\Delta m_{q}^{2}\frac{\partial^{2}Z_{i}^{(j)}(N)}{ \partial x_{q}}+\frac{1}{2}\sum\limits_{\begin{array}{c}p,q=1\\ p\
eq q\end{array}}^{n}\Delta m_{p}\Delta m_{q}\frac{\partial Z_{i}^{(j)}(N)}{ \partial x_{p}}\frac{\partial Z_{i}^{(j)}(N)}{\partial x_{q}}.\end{array}  \tag{19}\]  
Using the ray-tracing method, the derivative quantity in Equation (13) can be replaced by a linear fitting value:  
\[ \begin{array}{c}Z_{i}^{(j)}(N+\Delta M)=Z_{i}^{(j)}(N)+\sum\limits_{q=1}^{n }\Delta m_{q}b_{q}+\\ \frac{1}{2}\sum\limits_{q=1}^{n}\Delta m_{q}^{2}a_{qq}+\frac{1}{2}\sum\limits _{\begin{array}{c}p,q=1\\ p\
eq q\end{array}}^{n}\Delta m_{p}\Delta m_{q}a_{pq},\\ p\
eq q\end{array}  \tag{20}\]  
which can be expressed in the following matrix form:  
\[Z_{i}^{(j)}(N+\Delta M)=\Delta M\cdot\left(A_{i}^{(j)}\right)_{n\times n}\cdot \Delta M^{\rm T}+\Delta M\cdot\left(B_{i}^{(j)}\right)_{n\times 1}+Z_{i}^{(j)}(N), \tag{21}\]  
in which  
\[\left(A_{i}^{(j)}\right)_{n\times n}=\left[ \begin{array}{ccc}a_{i}^{(j)}{}_ {11}&\cdots&a_{i}^{(j)}{}_{1n}\\ \vdots&\ddots&\vdots\\ a_{i}^{(j)}{}_{n1}&\cdots&a_{i}^{(j)}{}_{nn}\end{array} \right],a_{i}^{(j)}{}_ {pq}=a_{i}^{(j)}{}_{qp},\left(B_{i}^{(j)}\right)_{n\times 1}=\left[ \begin{array}{c}b_{i}^{(j)}{}_ {1}\\ \vdots\\ b_{i}^{(j)}{}_{n}\end{array} \right].\]  
The optical design software is used to analyze the random misalignment example of the optical system model with a sample volume of k, i.e., each \(Z_{i}^{(j)}(N+\Delta M)\) can be obtained via ray tracing by each \(\Delta M\), after which we can establish the linear equation system for the quadratic coefficient and the linear coefficient with k equations:  
\[\left[ \begin{array}{c}Z_{i}^{(j)}(N+\Delta M_{1})-Z_{i}^{(j)}(N)\\ \vdots\\ Z_{i}^{(j)}(N+\Delta M_{k})-Z_{i}^{(j)}(N)\end{array} \right]=M_{k\times(\frac {n^{2}+3n}{2})}P_{(\frac{n^{2}+3n}{2})\times 1}. \tag{22}\]  
The minimal norm least squares solution of Equation (16) is  
\[P_{(\frac{n^{2}+3n}{2})\times 1}=\textit{pinv}\left(M_{k\times(\frac{n^{2}+3n}{2 })}\right)\times\left[