## 4 Numerical Simulation
### GMM-Based Artificial Potential Function Simulation  
Suppose the main body of the target spacecraft is a cube of 8 cubic meters, which has two solar panels with a length of 6 m and a width of 2 m, and two antennas with a length of 2 m, as shown in Figure 3. By using equal division method, obtain 40,000 feature points data. Then, combining the K-means and EM method, calculate the key parameters of GMM. The initial parameters are listed in Table 1 and the calculation results are listed in Table 2. Because of the uniform sampling points and the regular shape of target spacecraft, the initial key parameters in Table 1 are very similar to the calculation results in Table 2.  
Figure 3: Complex shape of target spacecraft. Based on the calculation results, the GMM-based reconstructed point-cloud and potential distribution are illustrated in Figure 4. Figure 4a gives the reconstructed point-cloud with additional red contours for better appearance. Figure 4b-d are the projection of GMM potential distribution in \(x\)-\(O\)-\(y\), \(x\)-\(O\)-\(z\), and \(y\)-\(O\)-\(z\) planes, respectively, where the brighter color means the higher potential. It is obvious that the GMM can give an accurate depiction of the complex shape of spacecraft with an analytical expression in Equation (10). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(j\) & \(\pi_{j}\) & \(\mu_{j}\) & \(\Sigma_{j}\) & \\ \hline
1 & \(8.302\times 10^{-2}\) & \(\left[ \begin{array}{cccc}1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ \end{array} \right]^{T}\) & diag\(\left\{ \begin{array}{cccc}0&0.3301&0.3301\\ 0&0.3301\\ 0&1&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ -1&0&0\\ \end{array} \right]^{T}\) & diag\(\left\{ \begin{array}{cccc}0&0.3300&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300\\ \end{array} \right\}\) \\
12 & 1.252\times 10^{-1}\) & \(\left[ \begin{array}{cccc}5.5&0&0\\ 0&0\\ -5.5&0&0\\ 0&0\\ \end{array} \right]^{T}\) & diag\(\left\{ \begin{array}{cccc}0&0.7550&0&0.3300\\ 0&0.3300\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ \end{array} \right\}\) \\
12 & 1.252\times 10^{-1}\) & \(\left[ \begin{array}{cccc}5.5&0&0\\ 0&0\\ \end{array} \right]^{T}\) & diag\(\left\{ \begin{array}{cccc}0.7500&0&0.3300\\ 0.7500&0&0.3300\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ \end{array} \right\}\) \\
12 & 1.252\times 10^{-1}\) & \(\left[ \begin{array}{cccc}5.5&0&0\\ 0&0\\ \end{array} \right]^{T}\) & diag\(\left\{\begin{array}{cccc}0.7500&0&0.3300\\ 0.7500&0&0.3300\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&0\\ 0&0.3300&1\\ 0&0.3382&1\times 10^{-5}\\ 0&0.3380&1\times 10^{-5}\\ 0&0.3382&1\times 10^{-5}\\ 0&0.3380&1\times 10^Assume that the desired position is \(\mathbf{r}_{f}=\left[ \begin{array}{cccc}-2,&2,&2\end{array} \right]^{T}\), and \(\mathbf{M}=\text{diag}\Big{\{} \begin{array}{cccc}2&2&2\end{array} \Big{\}}\), \(\mathbf{P}=\text{diag}\Big{\{} \begin{array}{cccc}2&2&2\end{array} \Big{\}}\). Then, the simulation of GMM-based artificial potential function in Equation (17) is obtained, as shown in Figure 5. The three figures show the potential of GMM-based APF in \(x\)-\(y\), \(x\)-\(z\) and \(y\)-\(z\) 2D coordinates, respectively. It is obvious that the surface of main body and two solar panels have high potential value, as well as the two antennas. The desired position has the globally lowest potential value, equaling to zero, which means that the spatial positions away from the desired position have higher potential value. So that the service spacecraft, moving in the potential space determined by GMM-based APF, will be attracted to the desired position, as well as avoiding collision with the target spacecraft.