## 3 Dynamic Modeling and Impact Analysis
\tag{5}\]  
where \(\mathbf{M}_{\rm r}\in\mathbf{R}^{(n+3)\times(n+3)}\) is the symmetric positive definite inertia matrix of space robot, \(\mathbf{H}_{\rm r}\in\mathbf{R}^{(n+3)\times(n+3)}\) is the matrix containing Coriolis force and centrifugal force, \(\mathbf{D}_{\rm L}\in\mathbf{R}^{(n+3)\times(n+3)}\) is the augmented equivalent damper coefficient matrix of manipulator, \(\mathbf{D}_{\rm mc}\in\mathbf{R}^{n\times n}\) is the equivalent damper coefficient matrix of motor, \(\mathbf{D}_{\rm tc}\in\mathbf{R}^{n\times n}\) is the damper coefficient matrix in SDD. \(\mathbf{q}_{\rm r}=[x_{0},y_{0},\theta_{0},\mathbf{q}_{\rm c}^{\rm T}]^{\rm T}\) is the generalized coordinate of space robot system, and \(\mathbf{q}_{\rm c}=[\theta_{1},\theta_{2},\cdots,\theta_{\rm n}]^{\rm T}\), \(\mathbf{q}_{\rm m}=[\theta_{\rm m1},\theta_{\rm m2},\cdots,\theta_{\rm mn}]^{\rm T}\). \(\mathbf{\tau}_{\rm r}=[\mathbf{\tau}_{\rm r}^{\rm T},\mathbf{\tau}_{0},\mathbf{\tau}_{\rm c} ^{\rm T}]^{\rm T}\), \(\mathbf{\tau}_{\rm B}=[0,0]^{\rm T}\), \(\mathbf{\tau}_{0}\) is the input torque of carrier, \(\mathbf{\tau}_{\rm c}=[\tau_{1},\tau_{2},\cdots,\tau_{\rm n}]^{\rm T}\) is the joint input torque, \(\mathbf{\tau}_{\rm mc}=[\tau_{\rm m1},\tau_{\rm m2},\cdots,\tau_{\rm mn}]^{\rm T}\) is the motor output torque.\(\mathbf{I}_{\rm m}=[I_{\rm m1},I_{\rm m2},\cdots,l_{\rm mn}]^{\rm T}\) is the inertia matrix of motor rotor, \(\mathbf{K}_{\rm s}=[k_{\rm s1},k_{\rm s2},\cdots,k_{\rm sn}]^{\rm T}\) is the stiffness matrix of spring. \(\mathbf{J}_{\rm r}\in\mathbf{R}^{3\times(n+3)}\) is the Jacobian matrix of space robot, \(\mathbf{F}_{\rm P}\in\mathbf{R}^{3\times 1}\) is the force acting on the end-effector of space robot. The model of satellite system before capture is obtained by the Newton-Euler equation:  
\[\mathbf{M}_{\rm s}\ddot{\mathbf{q}}_{\rm s}=\mathbf{J}_{\rm s}^{\rm T}\mathbf{F}_{\rm P^{ \prime}} \tag{6}\]  
where \(\mathbf{M}_{\rm s}\in\mathbf{R}^{3\times 3}\) is the symmetric positive definite inertia matrix of satellite, \(\mathbf{q}_{\rm s}=[x_{\rm s},y_{\rm s},\theta_{\rm s}]^{\rm T}\) is the generalized coordinate of satellite system, \(x_{\rm s}\) and \(y_{\rm s}\) are the satellite centroid coordinates, \(\mathbf{J}_{\rm s}\in\mathbf{R}^{3\times 3}\) is the Jacobian matrix for satellite,\(\mathbf{F}_{\rm P^{\prime}}\in\mathbf{R}^{3\times 1}\) is the force acting on the satellite, and \(\mathbf{F}_{\rm P}+\mathbf{F}_{\rm P^{\prime}}=0\). After the capture operation, the velocity on end-effector of space robot and satellite meets the following requirements:  
\[\mathbf{S}_{\rm P}(t_{0}+\Delta t)=\mathbf{S}_{\rm P^{\prime}}(t_{0}+\Delta t) \tag{7}\]  
where \(\Delta t\) is the length of collision time, and \(\mathbf{S}_{\rm P}=[\dot{x}_{\rm P},\dot{y}_{\rm P^{\prime}},\omega_{2}]^{\rm T}\), \(\mathbf{S}_{\rm P^{\prime}}=[\dot{x}_{\rm P^{\prime}},\dot{y}_{\rm P^{\prime}}, \dot{\theta}_{\rm s}]^{\rm T}\). Through Equation (7), the acceleration of satellite after collision:  
\[\ddot{\mathbf{q}}_{\rm s}(t_{0}+\Delta t)=\mathbf{J}_{\rm s}^{-1}\mathbf{I}_{\rm r}\ddot{ \mathbf{q}}_{\rm r}(t_{0}+\Delta t)+\mathbf{J}_{\rm s}^{-1}(\mathbf{J}_{\rm r}-\dot{\mathbf{J} }_{\rm s}\mathbf{J}_{\rm r}^{-1}\mathbf{J}_{\rm r})\dot{\mathbf{q}}_{\rm r}(t_{0}+\Delta t) \tag{8}\]  
Combining the law of momentum conservation with Equations (5) and (6), we can get the following equation:  
\[\left\{ \begin{array}{c}\mathbf{M}_{\rm r}\big{[}\dot{\mathbf{q}}_{\rm r}(t_{0}+ \Delta t)-\dot{\mathbf{q}}_{\rm r}(t_{0})\big{]}=\mathbf{J}_{\rm r}^{\rm T}\mathbf{f}_{ \rm P}\\ \mathbf{M}_{\rm s}\big{[}\dot{\mathbf{q}}_{\rm s}(t_{0}+\Delta t)-\dot{\mathbf{q}}_{\rm s}(t _{0})\big{]}=\mathbf{J}_{\rm s}^{\rm T}\mathbf{f}_{\rm P^{\prime}}\end{array} \right. \tag{9}\]  
where \(\mathbf{f}_{\rm P}=\int_{t_{0}}^{t_{0}+\Delta t}\mathbf{F}_{\rm P}dt\) and \(\mathbf{f}_{\rm P^{\prime}}=\int_{t_{0}}^{t_{0}+\Delta t}\mathbf{F}_{\rm P^{\prime}}dt\) are the impact impulse, and \(\mathbf{f}_{\rm P}+\mathbf{f}_{\rm P^{\prime}}=0\).