# Fault Reconstruction for a Giant Satellite Swarm Based on Hybrid Multi-Objective Optimization
## 4 Simulation and Analysis
### Fault Reconstruction Simulation  
According to the model assumptions in Section 2.1, the faulty satellite distribution is represented as a vector \(\mathbf{S_{F}}\); the orbital radius of the virtual center of the satellite swarm is \(R_{\mathrm{s}}\), and Earth's gravitational constant is \(\mu\). The reconstruction model parameters are shown in Table 1. The three-axis coordinate parameters of the faulty satellite in the relative coordinate system are shown in Table 2. The simulated running orbit is nearly circular, so the angular velocities are approximately equal. The initial values of the two moving satellites in the relative system are shown in Table 3.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(Num\) & 61 \\ \(a\) & 30 m \\ \(S_{F}\) & [10, 18, 33, 40, 44, 53] \\ \(\mu\) & \(3.986\times 10^{14}\)\(\mathrm{m^{3}/s^{2}}\) \\ \(R_{\mathrm{s}}\) & \(6958\times 10^{3}\)\(\mathrm{m}\) \\ \(t_{f}\) & 100 s \\ \hline \hline \end{tabular}
\end{table}
Table 1: Model parameters.  
Figure 6: Flowchart of the hybrid multi-objective genetic algorithm.  
The genetic algorithm population size is set to 100, and the maximum number of generations is set to 100. The simulation of the optimal fault reconstruction strategy calculated using the algorithm in Chapter 3 is shown in Figure 7.  
From the scatter plot in Figure 6, the overall fitness value fluctuates and decreases. The fluctuation is due to the inconsistency of the direction of the optimization indicators in the iterative process, resulting in local optima. However, as the number of generations increases, the global optimum is gradually located. Therefore, after 100 iterations, the curve shows convergence. The histogram indicates that the optimal reconstruction scheme has been obtained through the genetic algorithm: \(\mathbf{X}=[22,59,6,23,43,29]\), i.e., the healthy satellites numbered 22, 59, 6, 23, 43, and 29 correspond to the reconstructed faulty satellites 10, 18, 33, 40, 44, and 53. This reconfiguration scheme optimizes the health state, energy consumption, and fuel consumption balance of the entire satellite swarm.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Module** & \(\mathbf{S_{F}}\) & \(\mathbf{X/m}\) & \(\mathbf{Y/m}\) & \(\mathbf{Z/m}\) \\ \hline
1 & \(S_{10}\) & 0 & 30 & \(51.9615\) \\
2 & \(S_{18}\) & 0 & 30 & \(-51.9615\) \\
3 & \(S_{33}\) & 0 & \(-15\) & \(-77.9423\) \\
4 & \(S_{40}\) & 0 & 90 & \(51.9615\) \\
5 & \(S_{44}\) & 0 & 0 & \(103.9230\) \\
6 & \(S_{53}\) & 0 & \(-75\) & \(-77.9423\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Initial position of the faulty satellite in the relative coordinate system.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Module** & \(\mathbf{S_{F}}\) & \(\mathbf{X/m}\) & \(\mathbf{Y/m}\) & \(\mathbf{Z/m}\) & \(\dot{\mathbf{X}/(m/s)}\) & \(\dot{\mathbf{Y}/(m/s)}\) & \(\dot{\mathbf{Z}/(m/s)}\) \\ \hline
1 & \(x_{1}\) & \(x_{x1}-0\) & \(y_{x1}-30\) & \(z_{x1}-51.9615\) & 0 & 0 & 0 \\
2 & \(x_{2}\) & \(x_{x2}-0\) & \(y_{x2}-30\) & \(z_{x2}+51.9615\) & 0 & 0 & 0 \\
3 & \(x_{3}\) & \(x_{x3}-0\) & \(y_{x3}+15\) & \(z_{x3}+77.9423\) & 0 & 0 & 0 \\
4 & \(x_{4}\) & \(x_{x4}-0\) & \(y_{x4}-90\) & \(z_{x4}-51.9615\) & 0 & 0 & 0 \\
5 & \(x_{5}\) & \(x_{x5}-0\) & \(y_{x5}-0\) & \(z_{x5}-103.9230\) & 0 & 0 & 0 \\
6 & \(x_{6}\) & \(x_{x6}-0\) & \(y_{x6}+75\) & \(z_{x6}+77.9423\) & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Initial values of the two satellite reconstructions in the relative coordinate system.  
Figure 7: Fault reconstruction results: (**a**) Adaptation change; (**b**) Optimal reconstruction scheme.