## 3 Dynamics Modeling and Impact Effect Analysis  
The structure of a space robot with RSEA and target spacecraft systems is shown in Figure 3. The space robot consists of a rigid base \(B_{0}\), rigid links \(B_{i}\) (\(i\) =1,2), and rigid target spacecraft \(B_{3}\). We build the inertial coordinate system \(XOY\), while at the same time, the local coordinate system \(x_{i}O_{i}y_{i}\) (\(i\) = 0,1,2) of each component \(B_{i}\) (\(i\) = 1,2) is established; \(O_{0}\) is the rotation center of the base, \(O_{i}\) is the rotation center of \(B_{i}\) (\(i\) =1,2); \(m_{0}\) is the mass of the base, \(m_{s}\) is the mass of the non-cooperative spacecraft, \(m_{i}\) is the mass of \(B_{i}\) (\(i\) = 1,2). \(I_{0}\) is inertial moment of the base with respect to its mass center, \(I_{s}\) is the inertial moment of the non-cooperative spacecraft with respect to its mass center, \(I_{i}\) (\(i\) = 1,2) is the inertial moment of \(B_{i}\) (\(i\) = 1,2) with respect to their mass center. \(I_{0}\) represents the distance from point \(O_{0}\) to \(O_{1}\), \(I_{i}\) (\(i\) = 1,2) represents length of \(B_{i}\) along the \(x_{i}\) axis. \(d_{i}\) (\(i\) = 1,2) is the distance from the mass center of \(B_{i}\) to \(O_{i}\). \(I_{im}\) (\(i\) = 1,2) is inertial moment of the \(i\)-h actuator. \(k_{im}\) (\(i\) = 1,2) is the spring stiffness of the RSEA device. \(\mathbf{r}_{c}\) is the position vector of the mass center of the entire system in inertial coordinate system (\(XOY\)). \(\mathbf{r}_{i}\) (\(i\) = 1,2) is position vector of the mass center of \(B_{i}\) in the inertial coordinate system (\(XOY\)). Regarding the target spacecraft as a homogeneous rigid body, its dynamic equation can be obtained by the Newton-Euler method:  
\[\mathbf{D}_{\rm s}\ddot{\mathbf{q}}_{\rm s}=\mathbf{J}_{\rm s}^{\rm T}\mathbf{r}^{\prime} \tag{1}\]  
where \(\mathbf{q}_{\rm s}=[x_{\rm s},y_{\rm s},\theta_{\rm s}]^{\rm T}\) the generalized coordinates of the target spacecraft; \(x_{\rm s}\), \(y_{\rm s}\) are the position vectors of the mass center of \(B_{3}\), \(\theta_{\rm s}\) is the attitude angle of the spacecraft system. \(\mathbf{D}(\mathbf{q})\in\mathbf{\mathbb{R}}^{3\times 3}\) are the inertia positive definite matrices, \(\mathbf{J}_{\rm s}\in\mathbf{\mathbb{R}}^{3\times 3}\) is its impact contact point corresponding to the motion Jacobi matrix. \(\mathbf{F}^{\prime}\in\mathbf{\mathbb{R}}^{3\times 1}\) is the force acting on the spacecraft. According to the position vector relation in Figure 2, the position vectors of the mass center of \(B_{i}(i=0,1,2)\) in the pre-contact phase are:  
\[\left\{ \begin{array}{c}\mathbf{r}_{0}=[x_{a},y_{a}]^{\rm T}\\ \mathbf{r}_{1}=\mathbf{r}_{0}+l_{0}\mathbf{e}_{0}+d_{1}\mathbf{e}_{1}\\ \mathbf{r}_{2}=\mathbf{r}_{0}+l_{0}\mathbf{e}_{0}+l_{1}\mathbf{e}_{1}+d_{2}\mathbf{e}_{2}\end{array} \right. \tag{2}\]  
where \(x_{a}\), \(y_{a}\) are the position vector of the mass center of base \(B_{0}\), \(\mathbf{e}_{i}(i=0,1,2)\) is the unit vector along the \(x_{i}\) axis in the \(x_{i}O_{i}y_{i}\) frame. Figure 3: Space robot with RSEA and target spacecraft systems. Differentiating Equation (2) with respect to time \(t\), then the total kinetic energy of the space robot with RSEA is:  
\[T=\sum_{i=0}^{2}\biggl{(}\frac{1}{2}\,m_{i}\dot{r}_{i}^{\text{T}}\dot{r}_{i}+ \frac{1}{2}I_{i}\omega_{i}^{\text{T}}\mathbf{\omega}_{i})+\sum_{j=1}^{2}\frac{1}{2}I _{jm}\mathbf{\omega}_{jm}^{\text{T}}\mathbf{\omega}_{jm} \tag{3}\]  
where \(\mathbf{\omega}_{i}(i=0,1,2)\) is the angular velocity of the rotation center \(O_{i}\), \(\mathbf{\omega}_{jm}(j=1,2)\) is the angular velocity of the actuator. Neglecting the micro-gravity in space, the potential energy of the system only comes from the RSEA device, so the total potential energy of the system is:  
\[U=\sum_{i=1}^{2}\biggl{\{}\frac{3}{2}k_{int}\Bigl{[}(\Delta x_{iL})^{2}+( \Delta x_{iR})^{2}\Bigr{]}\biggr{\}} \tag{4}\]  
where \(\Delta x_{iL}=x(\alpha_{i})\), \(\Delta x_{iR}=-x(\alpha_{i})\), \(x(\alpha_{i})=R\sin(\alpha_{i})\).