## 3 Active Optics Correction of the 4m SiC Primary Mirror
### Pose Correction of the 4m SiC Primary Mirror
The stationary primary mirror was in a state of force equilibrium, and the resultant force and moment of the primary mirror were 0. The resultant force and moment of the hardpoints on the primary mirror were calculated based on the force on each hardpoint. Subsequently, by applying the same force and moment to the primary mirror through the pneumatic actuators, the primary mirror could still be in a state of force equilibrium, but the force on the hardpoints was offset. This could eliminate the influence of the force on the hardpoints on the surface shape and compensate for the force exerted on the hardpoints. First, the resultant force of the hardpoints on the primary mirror was calculated. Let the force on the hardpoint be \(F_{Dj}\), with a positive value for compression and a negative value for tension, where \(j=1,2,\ldots,6\). Projecting the force on the hardpoints toward the coordinate axis gave  
\[F_{DXj}=F_{Dj}\cos\alpha_{D} \tag{14}\]  
Figure 11: Primary mirror surface defects caused by force on hardpoints (external conditions: gravity; elevation angle: 15\({}^{\circ}\); order of the bending modes: 40; the rms after surface correction: 53.9 nm). \[F_{DYj}=F_{Dj}\cos\beta_{D} \tag{15}\]  
\[F_{DZj}=F_{Dj}\cos\gamma_{D} \tag{16}\]  
where \(F_{DXj}\), \(F_{DYj}\), and \(F_{DZj}\) are the projections of \(F_{Dj}\) along the x-, y-, and z-axes, respectively. \(\alpha_{D}\), \(\beta_{D}\), and \(\gamma_{D}\) are the angles between \(F_{Dj}\) and the positive directions of the x-, y-, and z-axes, respectively. \(F_{DXj}\), \(F_{DYj}\), and \(F_{DZj}\) were summed to obtain the three components of the resultant force of the hardpoints on the primary mirror:  
\[F_{X}=\sum_{j=1}^{6}F_{DXj} \tag{17}\]  
\[F_{Y}=\sum_{j=1}^{6}F_{DYj} \tag{18}\]  
Figure 10:0  
Then, the three components of the resultant moment of the hardpoints on the primary mirror could be calculated as  
Figure 10:1  
Figure 10:2  
Figure 10:3  
where \(x_{j}\) and \(y_{j}\) are the x and y coordinates of the upper support point of the \(j\)-th hardpoint, respectively. To compensate for the force on the hardpoints, we used the pneumatic actuators to apply a force and moment equal to the resultant force and moment of the hardpoints to the primary mirror. That is, the resultant force and moment of the hardpoints were distributed over the pneumatic actuators. Considering the installation direction of the two-axis pneumatic actuator, a specific actuator could only be used to output a force in a specific direction. The force \(F_{X}\) in the x-direction was distributed over actuators 1-18, 20-23, and 26-29. The output force \(F_{FX}\) of each actuator was \(F_{X}/26\). The force \(F_{Y}\) in the y-direction was distributed over actuators 19, 24, 25, and 30. The output force \(F_{FY}\) was \(F_{Y}/4\). The force \(F_{Z}\) in the z-direction was distributed over actuators 1-30, and the output force \(F_{FZ}\) was \(F_{Z}/30\). In terms of the moment distribution, the moments \(M_{X}\) and \(M_{Y}\) were generated by the z-direction force output by actuators 1-30. To generate \(M_{X}\), the required output force for each actuator was  
Figure 10:4  
where  
Figure 10:5  
where \(y_{i}\) is the y-direction coordinate of support point \(i\). To generate \(M_{Y}\), the output force required for the actuator was  
Figure 10:6where  
Figure 10:7  
where \(x_{i}\) is the x-direction coordinate of support point \(i\). The moment \(M_{\rm Z}\) was parallel to the z-axis and was generated by the y-direction radial force output by actuators 19, 24, 25, and 30. The required output force was  
Figure 10:8  
where  
Figure 10:9  
In summary, to compensate for the force on the hardpoints, the combined force and moment of the hardpoints on the primary mirror were calculated. Then, the equivalent forces and moments were output through the pneumatic actuators to compensate for the force on the hardpoints. The forces output by the pneumatic actuators are listed in Figure 9:0