# Precision Analysis and Error Compensation of a Telescope Truss Structure Based on Robotics
## 3 Aberration-Compensation Method
### Compensation of the PM Surface Error  
For large-aperture telescopes, there are two main reasons for the misalignment of the PM and SM:  
1. The misalignment of the SM is caused by the deformation of the truss obtained in Table 2. In addition, the deviation of the SM is due to the large span in the height direction of the truss coupled with its large mass.
2. The position deviation of the primary mirror is caused by the deformation of the support structure and the surface error of the primary mirror. The influence of the surface error is much greater than the error caused by the support component.  
In practice, as the azimuth and pitch angles change, the gravitational field experienced by the PM also changes continuously, resulting in a PM surface error. This is problematic because the PM is difficult to adjust in actual engineering. Therefore, this study uses the finite element method to analyze the surface error of the PM and attempts to use the adjustment of the SM to compensate for the primary aberration to the system caused by the PM [22; 23; 24; 25; 26].  
The PM model was established using finite element analysis software. A PM with a diameter of 2120 mm was used as the analysis object. The radius of curvature fitted by the mirror was 6000 mm, and the edge thickness of the mirror body was 260 mm. The axial support of the main mirror adopts an 18-point whiffletree support structure, and the lateral support adopts a tangential rod support method. The constraints adopted by the main mirror model are shown in Figure 10.  
By changing the loaded gravity to simulate the pitching movement of the PM during operation, the simulation was performed with a resolution of 5\({}^{\circ}\), tracking the movement process of the optical axis from vertical to horizontal. The surface error has been shown in Figure 11.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Z4** & **Z5** & **Z6** & **Z7** & **Z8** & **Z9** \\ \hline F (0, 0) & 0.2039 & 0.195 & 0.2343 & 0.0403 & 0.0417 & 0.0023 \\ F (0, \(-\)0.1) & 0.1991 & 0.170 & 0.2257 & 0.0309 & 0.0332 & 0.0021 \\ F (0, 0.1) & 0.2032 & 0.206 & 0.2329 & 0.0414 & 0.0370 & 0.0051 \\ \hline \hline \endThe original coordinate node value and the node deformation of the PM surface can be obtained using the finite element analysis software. The surface fitting program was compiled by Matlab, with the least square method used to fit the mirror error and the Zernike polynomial decomposition to obtain the RMS value of the mirror shape and the decomposed Zernike polynomial coefficients [27]. The extracted Zernike fringe coefficients of the PM were introduced into the optical system using the Zemax software, and the simulation finally obtained the wavefront RMS and the 4 th to 9 th Zernike coefficients (Z4-Z9), as shown in Figure 12.  
The simulated system wavefront error is calculated using the high-order sensitivity matrix method. The adjustment of each angle can be obtained through the Zernike coefficients of each angle and the high-order sensitivity matrix, and consequently, the amount of compensation required via adjusting the SM. The adjusted Z4-Z9 fringe Zernike coefficients are shown in Figure 13.  
Zernike polynomials were used to fit the low-order surface errors of the PM caused by gravity, and analyze the ability of the system to compensate for the surface errors associated with the optical components after adjusting the SM. This leads to the following conclusion: when the misalignment is caused by the relative positions of the PM and SM, the high-order sensitivity matrix method can compensate for the error in the optical system effectively. The RMSEs in F (0,0) are 0.2039, 0.195, 0.2343, 0.0403, 0.0417, and 0.0023, respectively. The sensitivity matrix can compensate for the astigmatism and coma caused by the PM surface error, but the compensation effect is limited.  
Figure 11: Surface error of the PM under the gravity field when (**a**) the optical axis is vertical and (**b**) the optical axis is horizontal.  
Figure 12: Z4–Z9 change curves in the gravity field.