## 4 Micro-Vibration Suppression and Image Stabilization Principle
### The First-Order Vibration Suppression  
As shown in Figure 8a, the first-order vibration suppression is composed of three BVILs. Most dynamic characteristics of the BVIL, modeled in 2D, can be carried over to 3D directly. To simplify the analysis, a 2D model of the BVIL is built to analyze the dynamic response of the first-order vibration suppression in this paper. As shown in Figure 8b, the connection between the optical payload and satellite platform is achieved by BVILs with S-joints and U-joints. Each BVIL has a stiffness \(k\), damping \(c\), and total length \(L\). The angle between the two legs is \(90^{\circ}\). Moreover, the distance between the upper platform, carrying an equivalent payload \(m_{pe}\), and the U-joint, connecting with the lower base, is \(L\). The distance between the center of the upper connecting rod, with mass \(m_{u}\), and the U-joint of the lower base is \(L_{u}\). The distance between the center of the lower connecting rod, with mass \(m_{b}\), and the U-joint of the lower base is \(L_{b}\). The moment inertia of the rod groups, connected to the HPSOP and satellite platform, are \(I_{u}\) and \(I_{b}\), respectively. To facilitate the dynamic analysis, a local coordinate system is built on the support center as shown in Figure 8b. Notably, the angle between the disturbance \(F_{d}\) and the \(x\)-axis is \(\alpha\). In particular, the dynamic equation of the BVIL when \(\alpha=0^{\circ}\) is shown in Equation (8):  
\[\left\{ \begin{array}{c}m_{dl}\ddot{x}-F_{l}-F_{r}-F_{d}=0\\ I_{dr}\ddot{F}_{r}+L(m_{pe}+m_{u})\ddot{x}-L(F_{l}+F_{d})=0\end{array} \right. \tag{8}\]  
where \(F_{l}\) and \(F_{r}\) are the reaction forces of the left and right legs, respectively. \(x\) denotes the displacement of the BVIL equivalent mass along the \(x\)-axis of the local coordinate system; however, in conjunction with the small angle approximation, \(y=0\). \(m_{dl}\), which can be  
Figure 8: First-stage vibration supervision platform composed of the BVIL: (**a**) overall layout; (**b**) BVIL represented in 2D. Figure 7: LOS jitter suppression and image stabilization. expanded to \(m_{dl}=m_{pe}+m_{u}+\frac{L_{u}}{L}m_{u}+\frac{L_{b}}{L}m_{b}\), is the mass of the entire BVIL equivalent to the left leg. \(I_{dr}\), which can be expanded to \(I_{dr}=I_{u}+I_{b}+m_{u}L_{u}^{2}+m_{b}L_{b}^{2}\), indicates the moment of inertia of the right leg about the U-joint. Since the rotation angle of the vibration isolation leg under micro-vibration is not large, \(\bar{x}\) can be approximately equal to the product of \(L\) and \(\bar{\beta}_{r}\). Then, Equation (9) can be obtained by combining Equation (8) and the reaction force of the left leg. \[\left\{ \begin{array}{l}F_{l}=-kx-c\hat{x}\\ F_{r}=\gamma(F_{l}+F_{d})\end{array} \right. \tag{9}\]  
where \(\gamma=\frac{m_{dl}l^{2}}{I_{dr}+(m_{pe}+m_{u})L^{2}}-1\). Furthermore, a more meaningful analysis can be found by performing the Laplace transform on Equations (8) and (9). It is as follows:  
\[\left\{ \begin{array}{l}TF_{a}=\frac{-(\gamma+1)(cs+k)}{m_{d}s^{2}+(\gamma+1 )(cs+k)}\\ TF_{s}=\frac{\gamma m_{d}s^{2}}{m_{d}s^{2}+(\gamma+1)(cs+k)}\end{array} \right. \tag{10}\]  
where \(TF_{a}\) and \(TF_{s}\) are the axial and tangential reaction forces of BVIL, respectively. To objectively reflect the mechanical response of BVIL under different disturbances, the transfer function, Equation (11), of any angle \(\alpha\) can be derived from Equation (10). \[\left\{ \begin{array}{l}TF_{l}=\left[\begin{array}{cc}TF_{a}\cos\alpha&TF_{ s}\sin\alpha\\ TF_{r}=\left[\begin{array}{cc}TF_{s}\cos\alpha&TF_{a}\sin\alpha\\ \end{array}\right]^{T}\end{array}\right]^{T}\end{array} \right. \tag{11}\]  
Compared with the optical payload, the mass of BVIL can be negligible. Each isolation leg can be regarded as a 1D vibration isolator without mass (\(m_{u}=m_{b}=I_{u}=I_{b}=0\)). Then, the results \(\gamma=0\) and \(TF_{s}=0\) can be obtained.