# Design and Stiffness Optimization of Bionic Docking Mechanism for Space Target Acquisition
## 5 Numerical Solution
### Optimal Stiffness Coefficients Solution  
Based on the PSO algorithm in 4, the dynamic model established in Section 3 is simulated and trained for 100 times under the max collision to obtain min fitness (\(F_{g}\)), and the optimal stiffness of the torsion springs and linear spring is obtained, which is used to realize the stiffness optimization of the bionic docking mechanism. The dynamic parameters of the dynamic model established in Section 3 are shown in Table 2.  
The simulation parameters are as follows: the fitness function of the PSO algorithm is shown in Equation (21). The instantaneous force and torque in Figure 8 are \(F\) = [\(F_{x}\), \(F_{y}\), \(F_{z}\)] = [100, 100, 100] N, \(M\) = [\(Mx\), \(My\), \(Mz\)] = [100, 100, 100] N \(\cdot\) m, The collision time is 0.01 s. The dampers are all in the initial state, and the initial damping is shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(k\)} & \multirow{2}{*}{\(m_{k}\)/kg} & \multicolumn{5}{c}{\(\mathbf{I_{k}/\text{kg}\cdot\mathbf{m}^{2}}\)} & \multirow{2}{*}{\(\mathbf{d_{k}/\text{m}}\)} & \multirow{2}{*}{\(\mathbf{r_{k}/\text{m}}\)} \\ \cline{3-3} \cline{5-7}  & & & \(\mathbf{X}\) & & \(\mathbf{Y}\) & \(\mathbf{Z}\) \\ \hline
1 & 200 & 55 & 55 & 50 & 6.71 & 1.51 \\
2 & 1.9 & 0.03 & 0.005 & 0.03 & 0 & 0 \\
3 & 1.7 & 0.015 & 0.01 & 0.005 & 0 & 0 \\
4 & 1.7 & 0.02 & 0.02 & 0.004 & 0.21 & 0.21 \\
5 & 8 & 0.5 & 0.5 & 0.25 & 0.22 & 0.13 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Model parameters of space–capture mechanism.  
Figure 8: Collision diagram of spatially six–dimensional collision.  
Figure 9 is a broken line diagram of the fitness values of 100 numerical simulations, in which the 64 th experiment has the max fitness and the 82 nd experiment has the min fitness. Figure 10 is a broken line diagram of the stiffness coefficients of the torsion springs and linear spring in 100 numerical simulations. The stiffness of the bionic docking mechanism corresponding to the min fitness and the max fitness are shown in Table 4; Figures 11-14 show the comparison of vibration displacements of various joints under the stiffness of the min fitness and the max fitness in Table 4. As shown in Table 5, within 50 s, under the min fitness, the vibration peak of the Gyro-X joint is \(50.5^{\circ}\), the vibration peak of the Gyro-Y joint is \(29.5^{\circ}\), the vibration peak of the Gyro-Z joint is \(78.9^{\circ}\) and the vibration peak of the Slide-Z joint is \(21.4\) mm, which are all above the control amplitudes in Table 1. Under the max fitness, the vibration peak of the Gyro-X joint is \(41.0^{\circ}\), the vibration peak of the Gyro-Y joint is \(26.2^{\circ}\), the vibration peak of the Gyro-Z joint is \(54.1^{\circ}\) and the vibration peak of the Slide-Z joint is \(16.2\) mm, we can see that only the vibration peak of the Gyro-Y joint is above the control amplitude in Table 1. Therefore, the stiffness coefficients under the min fitness should be the optimal stiffness coefficients of the torsion springs and linear spring, that is, the stiffness optimization of the bionic docking mechanism is achieved.  
According to the above analysis of 100 numerical simulations based on PSO algorithm in Section 4, the optimal stiffness coefficient of the Gyro-X joint can be selected as \(30\ \mathrm{N}\cdot\mathrm{mm}/\mathrm{deg}\), the optimal stiffness coefficient of the Gyro-Y joint is \(193\ \mathrm{N}\cdot\mathrm{mm}/\mathrm{deg}\), the optimal stiffness coefficient of the Gyro-Z joint is \(43\ \mathrm{N}\cdot\mathrm{mm}/\mathrm{deg}\), and the optimal stiffness coefficient of the Slide-Z joint is \(0.85\ \mathrm{N}/\mathrm{mm}\).  
Figure 11: Gyro–X joint vibration displacements under min and max fitness.  
Figure 10: Torsion/Linear spring stiffness coefficients of 100 numerical simulations.