## 4 Dynamic Analysis of the Docking Process  
### Contact Impact Analysis of the Docking Process  
#### 4.1.1 Modeling Contact Torque  
During the docking process, attitude correction of the passive docking mechanism was implemented with the contact force and torque exerted by the locking jaw of the active docking mechanism. Therefore, it was necessary to analyze the force and torque exerted on the passive docking mechanism during the docking process.  
When attitude errors existed between the active and passive docking mechanisms, the locking jaw was first contacted with the V-shaped groove surface of the passive docking mechanism in the docking process, as illustrated in Figure 7.  
As shown in Figure 7a, during the docking process, there was a certain angular deviation between the active and passive motion simulators and the origin coordinate system. The hook claw completed the hugging action and adjusted the attitude deviation under the action of the three V-shaped guide grooves, and finally reached the bottom of the V-groove to complete the docking operation. In this analysis, the collision types were divided into two cases: (1) when there was a y-direction yaw angle, it was positioned as a positive collision; (2) when there was a roll angle or pitch angle, it was defined as an oblique collision. In Figure 7, \(F_{1}\), \(F_{2}\), and \(F_{3}\) represented the contact force exerted on the side contact point of the V-shaped groove by the three locking jaws; given this, the attitude of the passive docking mechanism was continuously changed and adjusted as the locking jaw closed in response to the contact force. Here, the trajectory of the contact point between the top of the locking jaw and the V-shaped groove surface forms an arc on the groove surface. The contact impacting force of each locking jaw can therefore be decomposed into the circumferential force perpendicular to the groove surface, the force directed toward the groove center along the groove surface, and the force directed toward the bottom surface along the groove surface.  
Given that A was the initial contact point and C was the contact point between the locking jaw and the V-shaped groove when the docking was completed, then trajectory AC was the displacement vector of the jaw during the entire docking process. When the docking mechanism rotated at an angle of \(\Delta\psi\) around the \(z\)-axis in response to the contact force, there were  
\[\begin{array}{l}M_{Z}=\sum\limits_{i=1}^{3}F_{iv}d_{i}\\ \text{in the formula}\\ d_{i}=(l_{OA_{i}}-v_{0}I)\cos(\beta^{\prime}-\psi+\Delta\psi)\\ l_{OA_{i}}=\frac{a\sin\psi}{\tan(\beta^{\prime}-\psi)}+a\cos\psi\\ F_{iv}=F_{i}\sin(\beta^{\prime}-\psi+\Delta\psi)\end{array} \tag{11}\]  
Here, \(M_{z}\) is the rotational torque applied on the \(z\)-axis; \(F_{iv}\) is the rotational force; \(d_{i}\) is the torque. As can be seen from Equation (11), when the hook moves to the position B shown in Figure 7b, the calculation of the contact force was determined by the equivalent  
Figure 7: Illustrating the contact force between the active and passive docking mechanisms (a) analysis of forces; (b) trajectory of locking jaw.  
stiffness and damping characteristics, and the hook moment was obtained. The calculation of the impact force below provided the basis.  
Note that the same method can be used to determine expressions for \(Mx\) and \(My\), which represented the rotational torques that the \(x\)- and \(y\)-axes of the passive docking mechanism exert in response to the locking jaw during the docking process.