## 3 Active Optics Correction of the 4m SiC Primary Mirror
### Pose Correction of the 4m SiC Primary Mirror  
The six-hardpoint positioning mechanism consisted of six hardpoints of the same structure, which can be understood as linear displacement actuators with a high axial stiffness. When the length of the hardpoints remained unchanged, the rigid displacement of the primary mirror was limited, and the pose of the primary mirror remained unchanged. The pose of the primary mirror could be controlled by controlling the length of the hardpoints, thereby actively adjusting the pose of the primary mirror. The structural diagram of the six-hardpoint positioning mechanism is shown in Figure 7, where Figure 7a shows the initial pose state of the mechanism and Figure 7b shows the state of the mechanism after a pose adjustment [29]. When considered as a rigid body, the primary mirror had six degrees of freedom, with \(q=[x\ y\ z\ \alpha\ \beta\ \gamma]^{\mathsf{T}}\) representing the pose of the primary mirror, where \(q\) could be decomposed into a position vector \(t=[x\ y\ z]^{\mathsf{T}}\) and attitude vector \(s=[\alpha\ \beta\ \gamma]^{\mathsf{T}}\). The relationship between the pose of the primary mirror and the length of the hardpoints was  
\[I_{j}=t+Ra_{j}-b_{j} \tag{12}\]  
where \(j=1,2,\ldots,6\) corresponds to the six hardpoints. \(I_{i}\) is the vector that represents the direction of the hardpoint, and the norm of the vector was the length of the hardpoint. \(t\) is the coordinate of origin O\({}_{1}\) of the coordinate system O\({}_{1}\)-x\({}_{1}\)y\({}_{1}\)z\({}_{1}\) in the coordinate system O-xyz. \(a_{j}\) is the coordinate of the upper support point A\({}_{j}\) in the coordinate system O\({}_{1}\)-x\({}_{1}\)y\({}_{1}\)z\({}_{1}\), and \(b_{j}\) is the coordinate of the lower support point B\({}_{j}\) in the coordinate system O-xyz. \(R\) is the rotation matrix:  
\[R= \begin{bmatrix}\cos\gamma\cos\beta&\cos\gamma\sin\beta\sin\alpha-\sin \gamma\cos\alpha&\cos\gamma\sin\beta\cos\alpha+\sin\gamma\sin\alpha\\ \sin\gamma\cos\beta&\sin\gamma\sin\beta\sin\alpha+\cos\gamma\cos\alpha&\sin \gamma\sin\beta\cos\alpha-\cos\gamma\sin\alpha\\ -\sin\beta&\cos\beta\sin\alpha&\cos\beta\cos\alpha\end{bmatrix}  \tag{13}\]  
Figure 6: Relationship between the amplitude of bending-mode force \(F_{Bi}\) and the order \(i\) of the bending mode. where \(\alpha\), \(\beta\), and \(\gamma\) are the rotation angles of the primary mirror around the coordinate taxes \(\mathrm{x_{2}}\), \(\mathrm{y_{2}}\), and \(\mathrm{z_{2}}\), respectively, and are the components of the attitude vector \(\mathbf{s}=[\alpha\ \beta\ \gamma]^{\mathrm{T}}\). Equation (12) is called the inverse kinematics model of the six-hardpoint positioning mechanism, and it established the relationship between the pose of the primary mirror and the length and direction of the hardpoints. Based on the above theory, we could use the six-hardpoint positioning mechanism to compensate for and correct the primary mirror pose error caused by the mirror cell deformation. During a telescope observation, the mirror cell undergoes deformation due to its own gravity and the reaction force of the pneumatic actuators, which can lead to an error in the pose of the primary mirror. By changing the length of the hardpoints, the pose of the primary mirror can be corrected. Considering the case of elevation angles of \(45^{\circ}\) and \(90^{\circ}\) as examples, the corresponding deformation of the mirror cell is shown in Figure 8. The deformation of the mirror cell caused a displacement at the lower support point of the hardpoint, as shown in Table 1. The displacement of the lower support point resulted in an error in the pose of the primary mirror. The pose errors of the primary mirror at different elevation angles were calculated, as shown in Figure 9.  
Figure 8: Deformation diagram of the mirror cell. (**a**) Elevation angle of \(45^{\circ}\); (**b**) Elevation angle of \(90^{\circ}\). Figure 7: Structural diagram of the six-hardpoint positioning mechanism.