## 2 Principle and Structural Design of the Active Support System for a 4m SiC Primary Mirror
### Pneumatic Surface Control Actuators and Hardpoints  
The pneumatic surface control actuators correct the surface shape of the primary mirror and support its weight. It is structurally divided into single- and two-axis actuators. Single-axis actuators provide only axial force, whereas two-axis actuators provide both axial and radial forces.  
A total of 54 pneumatic surface control actuators were installed at 54 support points on the back of the primary mirror. The positions of the support points were obtained through a finite element analysis and optimization, and their positions and numbers are shown in Figure 2. The origin of the coordinate system shown in Figure 2 coincided with the vertex of the reflecting surface of the primary mirror. The y-axis was parallel to the elevation axis, the unmarked z-axis was parallel to the optical axis and pointed to the observation target, and the x-axis direction was determined using the right-hand rule. Among the 30 two-axis actuators, 26 were arranged parallel to the x-axis and could output a radial force in the x-direction to balance the radial component of gravity. The two-axis actuators 19, 24, 25, and 30 were arranged in the y-axis direction and could output a radial force in the y-direction. The structures of the single- and two-axis actuators were as shown in Figure 1.  
The actuator was powered with a cylinder. The stroke of the cylinder was \(\pm\)20 mm and the output force range was \(\pm\)0.85 kN. The range of the output force of the single-axis actuator was the same as that of the cylinder. The range of the axial force output by the two-axis actuator was \(\pm\)1.45 kN, and the range of radial force was \(\pm\)0.85 kN. Each cylinder was connected in series with two hinges, enabling the pneumatic actuator to meet the displacement of \(\pm\)20 mm and rotation of \(\pm\)0.21\({}^{\circ}\) of the primary mirror. The range of motion of the mirror was obtained through a kinematic analysis based on the Monte Carlo method. Each cylinder was connected in series with an S-type tension and pressure sensor that could achieve closed-loop control of the output force.  
Six hardpoints with the same structure formed a six-hardpoint positioning mechanism, which was used to define the relative pose between the primary mirror and mirror cell. The numbering of the hardpoints is illustrated in Figure 2. The hardpoints can be understood as linear displacement actuators with a high axial stiffness, as shown in Figure 1. The  
Figure 2: Layout and numbering of pneumatic surface control actuators and hardpoints.  
six-hardpoint positioning mechanism adjusted the pose of the primary mirror by changing the length of the hardpoints. Maintaining the length of the hardpoints unchanged could preserve the pose of the primary mirror. The flexible hinges in the hardpoints provided the necessary degrees of freedom, and a breakaway mechanism provided an overload protection function. A spoke-type tension and pressure sensor measured the force on the hardpoints. The axial stiffness of the hardpoints, 33.044 N/\(\upmu\)m, could provide a one-order natural frequency of over 25 Hz for the system [28]. The displacement accuracy of the hardpoints depended on the displacement accuracy of the actuator. The displacement stroke of the hardpoints was \(\pm\)10 mm, and the displacement resolution was 0.33 \(\upmu\)m. The repeatability accuracy was 7.026 \(\upmu\)m.