# Dynamics Analysis of Space Netted Pocket System Capturing Non-Cooperative Target
## 3 Dynamics Modeling of Space Netted Pocket
### Dynamics Model of the Closing Ropes  
In this subsection, the dynamics model of the closing ropes is established based on the ANCF flexible cable dynamics model [17]. As shown in Figure 2, the variable length flexible cable element model is presented, where \(v_{1}\) is the mass flow velocity at the element boundary node. Figure 1: Structure composition and size of the netted pocket system. With the assumption of the mass flow at the boundary, the Lagrange method cannot be applied in the case of varying element length. [PERSON] principle [41] states that the sum of virtual work done by inertial force and acting force on arbitrary virtual displacement is 0, by which the dynamics equations Equation (8) is obtained. \[\sum_{i}(\mathbf{F}_{i}-m\ddot{\mathbf{r}_{i}})\cdot\delta\mathbf{r}_{i}=0 \tag{8}\]  
where \(\delta\mathbf{r}_{i}\) is the material mass \(i\) in any moment to meet the constraints of the virtual displacement. The force on the rope element consist of the elastic force \(\mathbf{F}_{\text{E}}\) and the external force \(\mathbf{F}_{t}\), then the dynamics equation of the cable element can be written as:  
\[\int\limits_{0(t)}^{L(t)}\delta^{j}\mathbf{r}^{\text{T}}\Big{(}\mathbf{F}_{\text{f}}+ \mathbf{F}_{\text{E}}-\rho^{j}\ddot{\mathbf{r}}\Big{)}\text{d}l=0 \tag{9}\]  
Due to the material mass of the cable element being variable, the upper and lower limits of the above integral changes with time. Considering the axial and bending deformation of the cable, neglecting the rotational motion of the cable, the virtual work of each part can be expressed as:  
\[ \begin{array}{ll}\int\limits_{0(t)}^{L(t)}\delta^{j}\mathbf{r}^{\text{T}}(\mathbf{F} _{\text{f}})\text{d}l&=\delta^{j}\mathbf{q}^{\text{T}}\int\limits_{0(t)}^{L(t)} \mathbf{S}^{\text{T}}(l)\mathbf{f}(l,t)\text{d}l\\ &=\delta^{j}\mathbf{q}^{\text{T}}\frac{\text{d}l}{\text{d}\xi}\int\limits_{-1}^{1 }\mathbf{S}^{\text{T}}(\xi)\mathbf{f}(\xi,t)\text{d}\xi\\ \int\limits_{0(t)}^{L(t)}\delta^{j}\mathbf{r}^{\text{T}}(\mathbf{F}_{\text{E}})\text{ d}l=-\delta^{j}\mathbf{q}^{\text{T}}\frac{\text{d}l}{\text{d}\xi}\int\limits_{-1}^{1 }\left(\left(\frac{\dot{\alpha}\dot{\alpha}_{0}}{\partial^{j}\mathbf{q}}\right)^{ \text{T}}\cdot\right.\\ EA\left(i\dot{\epsilon}_{0}+\dot{c^{j}\dot{\epsilon}_{0}}\right)+\left(\frac{ \dot{\beta}\dot{\kappa}}{\partial\dot{\alpha}}\right)^{\text{T}}\cdot EJ \left(i\kappa+\dot{c^{j}\kappa}\right)\right)\text{d}\xi\\ \int\limits_{0(t)}^{L(t)}\left(\delta^{j}\mathbf{r}\right)^{\text{T}}\left(-\rho^{ j}A\ddot{\mathbf{r}}\right)\text{d}x=-\delta^{j}\mathbf{q}^{\text{T}}\frac{\text{d}l}{ \text{d}\xi}\int\limits_{-1}^{1}\rho A\mathbf{S}^{\text{T}}\ddot{\mathbf{r}}\text{d} \xi\end{array}  \tag{11}\]  
where \(\rho^{j}_{0}\kappa\) is the axial strain and curvature, \(E\) is elasticity modulus, \(A\) is the cross-sectional area and \(J\) is the moment of inertia of the flexible cable element. \(c\) is the damping coefficient and \(\frac{\text{d}l}{\text{d}\xi}=\frac{L(t)}{2}\). Combining Equations (10)-(12), the dynamics equation of the variable flexible cable element can be obtained as:  
\[ \begin{array}{ll}{}^{j}\mathbf{M}^{j}\ddot{\mathbf{q}}+{}^{j}\mathbf{M}^{j}_{\text{v}} \dot{\mathbf{q}}+{}^{j}\mathbf{M}^{j}_{\text{q}}q+{}^{j}\mathbf{Q}=0\end{array}  \tag{13}\]  
where  
\[ \begin{array}{ll}{}^{j}\mathbf{M}&=\frac{L(t)}{2}\int\limits_{-1}^{1}\rho\mathbf{S}^ {\text{T}}\mathbf{S}\text{d}\xi^{j}\mathbf{M}_{\text{v}}=\frac{L(t)}{2}\int\limits_{- 1}^{1}\rho\mathbf{S}^{\text{T}}\dot{\mathbf{S}}\text{d}\xi^{j}\mathbf{M}_{\text{q}}\\ &=\frac{l(t)}{2}\int\limits_{-1}^{1}\rho\mathbf{S}^{\text{T}}\ddot{\mathbf{S}}\text{d} \xi\\ \end{array}  \tag{14}\]  
\[