## 3 Dynamic Modeling and Impact Analysis  
The structure of a space robot with SDD and target satellite systems is shown in Figure 3. The \(O_{0}\), \(O_{\rm s}\) and \(O_{i}(i=1,2,\cdot\cdot\cdot,n)\) are the centroids of carrier, satellite, and joints respectively. \(P\) and \(P^{\prime}\) are the acquisition point of space manipulator and the acquisition point of satellite, respectively. \(xOy\) is the inertial coordinate system moving with orbit, \(x_{0}O_{0}y_{0}\) and \(x_{\rm s}O_{\rm s}y_{\rm s}\) are the coordinate systems fixed on the centroids of carrier and satellite, respectively, \(x_{i}O_{i}y_{i}(i=1,2,\cdot\cdot\cdot,n)\) is the coordinate system fixed at the centre of \(i\)th joint. The parameters of space robot and satellite are defined as follows: \(m_{0}\), \(m_{i}(i=1,2,\cdot\cdot\cdot,n)\) and \(m_{\rm s}\) are the mass of space robot carrier, manipulator, and satellite, respectively, \(I_{i}\), \(I_{mi}(i=1,2,\cdot\cdot\cdot,n)\) and \(I_{\rm s}\) are the moment of inertia of manipulator, motor rotor and satellite respectively, \(d_{0}\), \(d_{i}(i=1,2,\cdot\cdot\cdot,n)\) and \(d_{\rm s}\) are the distance from \(O_{0}\) to \(O_{1}\), from the centre of joint to manipulator and from satellite centroid to the end, respectively. \(L_{i}(i=1,2,\cdot\cdot\cdot,n)\) is the length of the manipulator, and \(\theta_{0}\), \(\theta_{i}\), \(\theta_{\rm s}\), and \(\theta_{mi}(i=1,2,\cdot\cdot\cdot,n)\) are the angle of carrier attitude, manipulator, satellite attitude, and motor rotor, respectively. As the system of space robot with SDD (see Figure 3), the total kinetic energy of space robot in the pre-contact phase is as follows:  
\[T_{\rm r}=\frac{1}{2}\sum_{i=0}^{n}\left(m_{i}\dot{r}_{i}^{\rm T}\dot{r}_{i}+I _{i}\omega_{i}^{2}\right)+\frac{1}{2}\sum_{j=1}^{n}\left(I_{mj}\omega_{mj}^{2}\right) \tag{1}\]  
where \(\mathbf{r}_{i}\) is the position vector of manipulator's mass centre, and \(\omega_{i}\) and \(\omega_{mj}\) are the angular velocity of manipulator and motor rotor, respectively. If the microgravity is ignored, the potential energy of space robot comes from the spring in SDD:  
\[U_{\rm r}=\frac{1}{2}\sum_{i=1}^{n}\left[k_{\rm s}(\theta_{mi}-\theta_{i})^{2 }\right] \tag{2}\]  
Due to the addition of SDD in the joints of space robot, the space robot is a nonpotential system, the dissipation function should be added:  
\[\dot{\sigma}_{\rm r}=\sum_{i=1}^{n}\left[D_{mi}\dot{\theta}_{mi}^{2}+D_{ti}( \dot{\theta}_{mi}-\dot{\theta}_{i})^{2}+D_{Li}\dot{\theta}_{i}^{2}\right] \tag{3}\]  
Figure 3: Space robot with SDD and target satellite systems. Reference [36], the Lagrange equation of dissipative system is as follows:  
\[\frac{d}{dt}\bigg{(}\frac{\partial L_{\rm r}}{\partial\dot{q}_{\rm r}}\bigg{)}- \frac{\partial L_{\rm r}}{\partial q_{\rm r}}+\frac{\partial\dot{\theta}_{\rm r }}{\partial\dot{q}_{\rm r}}=\mathbf{Q} \tag{4}\]  
Combined Equations (1)-(4), the model of space robot system before capture is obtained as follows:  
\[\left\{ \begin{array}{c}\mathbf{M}_{\rm r}(\mathbf{q}_{\rm r})\ddot{\mathbf{q}}_{\rm r}+ (\mathbf{H}_{\rm r}(\mathbf{q}_{\rm r},\dot{\mathbf{q}}_{\rm r})+\mathbf{D}_{\rm L})\dot{\mathbf{q} }_{\rm r}=\mathbf{\tau}_{\rm r}+\mathbf{J}_{\rm r}^{\rm T}\mathbf{F}_{\rm P}\\ \mathbf{I}_{\rm m}\ddot{\mathbf{q}}_{\rm m}+\mathbf{D}_{\rm mc}\dot{\mathbf{q}}_{\rm m}+\mathbf{\tau }_{\rm c}=\mathbf{\tau}_{\rm m}\\ \mathbf{K}_{\rm s}(\mathbf{q}_{\rm m}-\mathbf{q}_{\rm c})+\mathbf{D}_{\rm tc}(\dot{\mathbf{q}}_{\rm m }-\dot{\mathbf{q}}_{\rm c})=\mathbf{\tau}_{\rm c}\end{array} \right.