## 4 Dynamic Analysis of the Docking Process
### Dynamic Model and Simulation Analysis of the Docking Process  
The active and passive docking mechanisms are respectively installed on the active and passive motion simulator through the flange and supporting rod. Here, the passive motion simulator is the follow-up system of the passive docking mechanism during the docking process. Furthermore, the rotation and displacement are generated with the change in attitude of the passive docking mechanism in response to the contact force of the locking jaw. During the docking process, the rotational motion of the mandrel balance assembly, spherical joint, and the end-effector take the spherical center O\({}_{2}\) as the center of rotation. Therefore, the rotation equation of the rolling system is  
\[\overset{\rightharpoonup}{M_{O_{2}}}=\overset{\rightharpoonup}{M_{S}}-\overset{ \rightharpoonup}{M_{Z}}-\overset{\rightharpoonup}{M_{P}} \tag{14}\]  
\[T_{2}=\frac{1}{2}A\,P^{2}+\frac{1}{2}Br^{2}+\frac{1}{2}Hq^{2} \tag{15}\]  
Here, \(P\), \(r\), and \(q\) represent the velocities of spherical joint turns around the \(X_{2}\), \(Y_{2}\), and \(Z_{2}\) axes, respectively. In addition, the Nissen equations corresponding to components of the spherical joint angle are  
\[ \begin{cases}\frac{\partial T_{2}}{\partial\dot{y}}-2\frac{\partial T_{2}}{ \partial\dot{y}}=M_{\psi}\\ \frac{\partial T_{2}}{\partial\dot{y}}-2\frac{\partial T_{2}}{\partial\dot{y }}=M_{\theta}\\ \frac{\partial T_{2}}{\partial\dot{\varphi}}-2\frac{\partial T_{2}}{\partial \dot{\varphi}}=M_{\varphi}\end{cases}  \tag{16}\]  
where \(M_{\psi}\), \(M_{\theta}\), and \(M_{\varphi}\) are the components of \(M_{\text{O}_{2}}\) for each coordinate axis, respectively. The dynamic differential equation of the rotational motion of the spherical joint during the docking process is  
\[\left\{ \begin{array}{l}(A\sin^{2}\theta+J_{Z}+J_{P})\ddot{\psi}=(H-2A\sin \theta)\dot{y}\dot{\theta}\cos\theta+H\dot{\varphi}\dot{\theta}\sin\theta+M_{ \psi}-M_{\varphi}\cos\theta\\ A\ddot{\theta}=(A-H)\dot{\psi}^{2}\sin\theta\cos\theta-H\dot{\varphi}\dot{ \psi}\sin\theta+M_{\theta}\\ AH\ddot{\varphi}\sin^{2}\theta=\dot{\psi}\dot{\theta}(AH\sin\theta+AH\sin \theta\cos^{2}\theta-H^{2}\cos^{2}\theta)-H^{2}\dot{\varphi}\dot{\theta}\sin \theta\cos\theta\\ +(A\sin^{2}\theta+B\cos^{2}\theta)M_{\varphi}-HM_{\psi}\cos\theta\end{array} \right. \tag{17}\]Equation (17) was obtained by Nielsen transformation of Equation (15), where \(J_{Z}\), \(J_{P}\) are the rotational inertia of spherical joint and the end-effector, respectively. From Formula (17), it can be known that the above equations described the dynamic characteristics of the rolling device during the docking process, which consisted of a spherical joint, a mandrel balancing assembly, and an end-effector. Here, the effect of the impact on the test platform was analyzed by combining with the dynamic simulation software. The effect of the impacting force on the test platform during the docking process was reflected in the change in performance of specific components in response to the impacting force. These components include the spherical joint, the mandrel, and so on. We conducted a co-simulation of the test platform and the docking mechanism by using ADAMS, a dynamic analysis software system. Simulate the real material properties on the simulation material property settings. Ask the partner to learn that high-strength lightweight materials such as aluminum alloy and titanium alloy are used, and set their properties as material properties in the simulation. Through the simulation, we obtained the variation of the impacting force generated during the docking process as a reference for further analysis of related components. According to the theorem of impulse, when an impact occurs, the impacting force can be expressed as  
\[F_{I}=\frac{mv}{t} \tag{18}\]  
where \(m\) is the mass of moving objects, \(v\) is the relative velocity, and \(t\) is the impacting time. Here, the impacting velocity and time both depend on the initial conditions of the docking and the retracting speed of the locking jaw.