# Design and Stiffness Optimization of Bionic Docking Mechanism for Space Target Acquisition
## 4 Stiffness Optimization Method
### Stiffness Optimization  
Since the traditional particle swarm algorithm is easy to fall into the local minimum, PSO algorithm with inertia weight is used to optimize the calculation, as shown in (22), in order to search quickly at the initial moment, when close to the best position, slow down the speed to search and enhance the capability of local search. \[ \begin{array}{l}v_{ij}(t+1)=\omega\cdot v_{ij}(t)+c_{1}\cdot\text{rand}_{1} \cdot\left[p_{ij}(t)-z_{ij}(t)\right]\\ +c_{2}\cdot\text{rand}_{2}\cdot\left[p_{gj}(t)-z_{ij}(t)\right]\\ z_{ij}(t+1)=v_{ij}(t+1)+z_{ij}(t)\\ \omega=\omega_{2}-c\cdot\left(\omega_{2}-\omega_{1}\right)\end{array}  \tag{22}\]  
where \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Gyro–X** & **Gyro–Y** & **Gyro–Z** & **Slide–Z** \\ \hline Limited Amplitude & 70.0 & 30.0 & 90.0 & 25.0 \\ Control Amplitude & 50.0 & 20.0 & 70.0 & 17.0 \\ Unit & & deg & & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Limited amplitude and control amplitude of each DOF. Figure 6: Modeling of particle swarm optimization. \(v_{\bar{y}}(t)\) is the flight velocity of the particle;  
\(z_{\bar{y}}(t)\) is the current position of the particle;  
\(p_{\bar{y}}(t)\) is the best location of individual particle;  
\(p_{\bar{y}}(t)\) is the best location of group particles;  
\(c_{1}\) is the cognitive learning coefficient;  
\(c_{2}\) is the social learning coefficient;  
\(c\) is the coefficient;  
\(\omega\in[\omega_{1},\omega_{2}]\) is the inertia weight, positive definite constant;  
\(\text{rand}_{1}\), \(\text{rand}_{2}\in[0,1]\) is the random number. For standard PSO algorithm, the asymptotic convergence condition is as follows  
\[\left\{ \begin{array}{l}\omega-\varphi-1<0\\ (1-\omega+\varphi)2-4\varphi\geq 0\end{array} \right. \tag{23}\]  
where \(\varphi=c_{1}\cdot\text{rand}_{1}+c_{2}\cdot\text{rand}_{2}\). As long as the parameters \(\omega\), \(c_{1}\), and \(c_{2}\) satisfy (23), the particle can asymptotically converge to an optimal state in the end. In this paper, appropriate parameters will be chosen according to (23) to guarantee the stability of proposed method. For objective function (21), when the vibration peak \(|\mathbf{x}_{l}|_{p}\) (\(l\) = 4, 5, 6, 10) along each direction is more than the control amplitude, namely \(F_{g}\) < 1, the optimization has met the desired requirements. At that moment, terminate the particle swarm iteration algorithm and output the optimal stiffness coefficients \(K_{k}\) (\(k\) = 1, 2, 3, 4). The optimization flow chart is shown in Figure 7 and the detailed control algorithm is described as follows:  
Step 1: According to the initial collision momentum and the initial configuration of the bionic docking mechanism, calculate the vibration peak \(|\mathbf{x}_{l}|_{p}\) (\(l\) = 4, 5, 6, 10) at iteration \(i\) according to (17). Step 2: Using \(|\mathbf{x}_{l}|_{p}\) (\(l\) = 4, 5, 6, 10) and the particle swarm algorithm (22), derive the optimal stiffness coefficients through iterative steps (1), (2) and (3). (1) Define particle swarm scale \(m\), and determine the particle dimension \(d\) according to the number of the torsion springs and linear spring. By adjusting the particle swarm scale \(m\), the vibration peak of each DOF can be improved. The stiffness coefficients  
Figure 7: Flow chart of stiffness optimization based on PSO. of the torsion springs and linear spring corresponds to the particle position in particle swarm. Initialize the position of each particle in particle swarm \(z_{ij}\). Substituting \(z_{ij}\) into (17), if \(\left|x_{l}(t)\right|>\left|x_{l}\right|_{L}\), proceed directly to step 3, otherwise, figure out the vibration peak \(\left|x_{l}\right|_{p}\) (\(l\) = 4, 5, 6, 10) of each particle at the iteration \(i\), which is substituted into (21) to compute fitness \(F_{g1}\) and regard current position as \(p_{ij}\) and \(p_{gj}(i)\). (2) Update the current particle position according to (22) and substitute \(z_{ij}\) into (17) and (21) to continue to compute \(F_{g2}\).