## 4 Micro-Vibration Suppression and Image Stabilization Principle
### The Second-Order Vibration Suppression  
The second-order vibration suppression is a decoupled FSM driven by piezoelectric actuators. As shown in Figure 10a, it is composed of a mirror, a tip-tilt mechanism about two axes, and piezoelectric actuators. In general, the LOS jitter is corrected by rotating the mirror around the X-axis and Y-axis through the tip-tilt mechanism. The rotation of the X-axis is taken as an example, as shown in Figure 10b. The piezoelectric ceramics produce elongation and shortening deformations, respectively. Then, the deformations are amplified by the amplification mechanism and transmitted to the flexible support. Finally, the flexible support pushes and pulls the mirror frame to rotate the mirror around the X-axis. Notably, to guarantee structural rigidity and motion decoupling, the FSM adopts an integrated design. This results in the rigidity being too high to achieve a large rotation angle. To overcome this shortcoming, a two-stage amplification mechanism is proposed in this paper, as shown in Figure 10c. Moreover, the image stabilization process is shown in Figure 11. Affected by micro-vibration disturbances, optical components, such as the primary mirror (PM) and the secondary mirror (SM), undergo displacement relative to the calibration state, resulting in the LOS jitter and image offset as shown in Figure 11a. Then, the FSM is controlled to correct the optical axis to the CCD center, as shown in Figure 11b. The control structure is shown in Figure 11c, where \(R(s)\) and \(Y(s)\) are the LOS jitter and target position, respectively; \(e^{-\tau s}\) expresses the time delay characteristics of CCD; \(C_{p}(s)\) indicates the controller on position; \(G_{FSM}(s)\) is the transfer function of the FSM; and \(D_{Com}(s)\) represents the LOS jitter caused by the jitter on the PM and SM. To obtain the plant transfer function of position \(G_{p}(s)=e^{-\tau s}G_{FSM}(s)\), frequency sweep experiments on FSM are conducted in this paper, and the results after eliminating the low-frequency measurement noise are shown in Figure 12. Then, \(G_{p}(s)\) can be obtained through parameter identification, as shown in Figure 12. According to the fitting curve, the transfer functions around the X-axis and Y-axis can be obtained as follows:  
\[G_{FSM-X}(s)=e^{(-0.0013s)}\frac{N_{FSM-X}}{D_{FSM-X}} \tag{12}\]  
where \(N_{FSM-X}=8.374\times 10^{36}s^{4}+9.86\times 10^{38}s^{3}+9.523\times 10^{4 }s^{2}+5.694\times 10^{45}s+2.674\times 10^{50}\) and \(D_{FSM-Y}=2.589\times 10^{27}s^{7}+4.626\times 10^{30}s^{6}+4.674\times 10^{34}s^{5}+ 6.62\times 10^{37}s^{4}+2.664\times 10^{41}s^{3}+2.882\times 10^{44}s^{2}+4.782 \times 10^{47}s+3.628\times 10^{50}\). \[G_{FSM-Y}(s)=e^{(-0.0014s)}\frac{N_{FSM-Y}}{D_{FSM-Y}} \tag{13}\]  
Figure 10: Composition and working principle of the FSM. (**a**) FSM mechanism. (**b**) Tip–tilt mechanism around the two axes. (**c**) Amplification principle. where \(N_{FSM-Y}=2.245\times 10^{36}s^{4}+1.09\times 10^{39}s^{3}+2.182\times 10^{43}s^{2}+5.146\times 10^{45}s+5.136\times 10^{49}\) and \(D_{FSM-Y}=1.135\times 10^{27}s^{7}+1.86\times 10^{30}s^{6}+1.74\times 10^{34}s^{5}+2.373\times 10^{37}s^{4}+8.66\times 10^{40}s^{3}+9.654\times 10^{43}s^{2}+1.4 \times 10^{47}s+1.24\times 10^{50}\). Then, to satisfy the design requirements of a phase margin greater than 45\({}^{\circ}\) and a gain margin greater than 6 dB, controllers \(C_{P-X}(\mathrm{s})\) and \(C_{P-Y}(\mathrm{s})\) can be designed by the pole-zero cancellation according to \(G_{P-X}(s)\) and \(G_{P-Y}(s)\). The forms of the controllers are as follows:  
\[C_{P-X}(s)=74750\frac{(1+0.32s)}{s(1+110s)} \tag{14}\]  
\[C_{P-Y}(s)=62973\frac{(1+0.99s)}{s(1+130s)} \tag{15}\]  
Furthermore, the disturbance attenuation function \(S_{x}=\frac{1}{1+G_{FSM-X}\cdot C_{P-X}}\) can be obtained by combining Equations (12) and (14), and \(S_{y}=\frac{1}{1+G_{FSM-Y}\cdot C_{P-Y}}\) can be obtained by combining Equations (13) and (15).