## 3 Dynamics Modeling and Impact Effect Analysis
\(x(\alpha_{i})\) is the deformation of the spring on the block of the RSEA device, \(\alpha_{i}\) is the angular difference between the sweeping arm and the input disk. Based on Equations (3) and (4), and combing with the Lagrange equations, the dynamic equations of the space robot of pre-capture phase are as follows  
\[\left\{ \begin{array}{c}\mathbf{D}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q} })\dot{\mathbf{q}}=\mathbf{\tau}_{c}+\mathbf{J}^{\text{T}}\mathbf{F}\\ \mathbf{I}_{m}\ddot{\mathbf{\theta}}_{m}+\mathbf{K}(\mathbf{\theta}_{m}-\mathbf{\theta})=\mathbf{\tau} _{m}\\ \mathbf{K}(\mathbf{\theta}_{m}-\mathbf{\theta})=\mathbf{\tau}_{\theta}\end{array} \right. \tag{5}\]  
where \(\mathbf{q}=[x_{a},y_{a},\theta_{0},\theta_{1},\theta_{2}]^{\text{T}}\) are the generalized coordinates of the system, \(\theta_{0}\) is the attitude angle displacement of the base, \(\theta_{i}(i=1,2)\) is the attitude angle displacement of the \(i\)-th link, \(\theta_{im}(i=1,2)\) is the attitude angle displacement of the \(i\)-th actuator. \(\mathbf{D}(\mathbf{q})\in\mathbb{R}^{5\times 5}\) are the inertia positive definite matrices, \(\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}\in\mathbb{R}^{5\times 1}\) is the Coriolis/centrifugal matrix. \(\mathbf{\theta}_{m}=[\theta_{1m},\theta_{2m}]^{\text{T}}\), \(\mathbf{\theta}=[\theta_{1},\theta_{2}]^{\text{T}}\). \(\mathbf{\tau}_{c}=[\mathbf{\tau}_{a}^{\text{T}},\tau_{0},\mathbf{\tau}_{\theta}^{\text{T} }]^{\text{T}}\), \(\mathbf{\tau}_{a}=[0,0]^{\text{T}}\) is the position control torque of the base, \(\tau_{0}\) is the attitude control torque of base. \(\mathbf{\tau}_{m}=[\tau_{1m},\tau_{2m}]^{\text{T}}\) is the joint torque/force delivered by actuators. \(\mathbf{I}_{m}=\text{diag}(I_{1m},I_{2m}),\mathbf{K}=\text{diag}(k_{1},k_{2})\) is the equivalent stiffness of joints, and its calculation formula is given in Equation (46). \(\mathbf{J}\in\mathbb{R}^{3\times 5}\) is its end-effector impact contact point corresponding motion Jacobi matrix, \(\mathbf{F}\in\mathbb{R}^{3\times 1}\) is the force acting on the end-effector. In the capturing operation phase, the space robot contacts and collides with the target spacecraft, and the interaction force at the end is satisfied:  
\[\mathbf{F}=-\mathbf{F}^{\prime} \tag{6}\]  
Based on Equation (6), and combining it with Equations (1) and (5), we can obtain that:  
\[\mathbf{D}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}=\mathbf{\tau} _{c}-\mathbf{J}^{\text{T}}(\mathbf{J}_{\text{s}}^{\text{T}})^{-1}\mathbf{D}_{\text{s}} \ddot{\mathbf{q}}_{\text{s}} \tag{7}\]  
The actuators will be turned off during the capture phase, which is \(\mathbf{\tau}_{c}=0_{5\times 1}\). Integrating Equation (7) over the momentary period of collision [13]:  
\[\mathbf{D}(\mathbf{q})(\dot{\mathbf{q}}(t_{0}+\Delta t)-\dot{\mathbf{q}}(t_{0}))+\mathbf{J}^{ \text{T}}(\mathbf{J}_{\text{s}}^{\text{T}})^{-1}\mathbf{D}_{\text{s}}(\dot{\mathbf{q}}_{ \text{s}}(t_{0}+\Delta t)-\mathbf{q}_{\text{s}}(t_{0}))=0 \tag{8}\]  
The space robot and spacecraft satisfy the velocity constraint in the post-capture phase. Based on this, the following generalized velocity of the post-capture hybrid system can be obtained:  
\[\dot{\mathbf{q}}(t_{0}+\Delta t)=\mathbf{N}^{-1}[\mathbf{D}(\mathbf{q})\dot{\mathbf{q}}(t_{0})+\bm {J}^{\text{T}}(\mathbf{J}_{\text{s}}^{\text{T}})^{-1}\mathbf{D}_{\text{s}}\dot{\mathbf{q}} (t_{0})] \tag{9}\]  
where \(\mathbf{N}=\mathbf{D}(\mathbf{q})+\mathbf{J}^{\text{T}}(\mathbf{J}_{\text{s}}^{\text{T}})^{-1}\mathbf{D }_{\text{s}}\mathbf{J}_{\text{s}}^{-1}\mathbf{I}\). Integrating first item of Equation (5), we have:  
\[\mathbf{D}(\mathbf{q})(\dot{\mathbf{q}}(t_{0}+\Delta t)-\dot{\mathbf{q}}(t_{0}))=\mathbf{J}^{\rm T}\bm {P} \tag{10}\]  
where \(\mathbf{P}=\int_{t_{0}}^{t_{0}+\Delta t}\mathrm{F}\mathrm{d}t\) is the impact impulse during the capture phase.