# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 5 Phase 2: HEO Based SSO Objects Flyby
### Multi-Objective Evolutionary Optimization Algorithm  
The parameters of the NSGA-II algorithm in the CRTBP system are shown in Table 11.  
Figure 11: Calculation procedure of Phase 2.  
The optimization parameters in the CRTBP system as shown in Table 12. The optimization objectives are as follows:  
\[\begin{split} f_{1}(\mathbf{x})=&||\Delta\mathbf{V}||\\ f_{2}(\mathbf{x})=& dt_{1}+dt_{2}\end{split} \tag{22}\]  
The lower and upper bounds of the optimization parameters are as follows:  
\[\begin{split}\mathbf{x}_{\min}&=[0,0^{\circ},0^{\circ},0. 5\text{TU},0.5\text{TU}]\\ \mathbf{x}_{\max}&=[\zeta_{4},2\pi,2\pi,1.35\text{TU},1. 35\text{TU}]\end{split} \tag{23}\]  
The inequality constraints are as follows:  
\[\begin{split} c_{1}&=-(h_{p-moon}-\widetilde{h}_{1}) \leq 0\\ c_{2}&=-(h_{p-earth}-\widetilde{h}_{2})\leq 0\\ c_{3}&=-(||\delta\mathbf{r}||-\zeta_{5})\leq 0\\ c_{4}&=||\delta\mathbf{r}||-\zeta_{6}\leq 0\\ c_{5}&=||\delta\mathbf{r}||-\zeta_{7}\leq 0\end{split} \tag{24}\]  
If the spacecraft would enter the Moon's SOI, the inequality constraints should contain the constraint of lunar gravity so that the inequality constraints would be as follows:  
\[\begin{split} c_{1}&=-(h_{p-moon}-\widetilde{h}_{1}) \leq 0\\ c_{2}&=h_{p-moon}-h_{SOI}\leq 0\\ c_{3}&=-(h_{p-earth}-\widetilde{h}_{2})\leq 0\\ c_{4}&=-(||\delta\mathbf{r}||-\zeta_{5})\leq 0\\ c_{5}&=||\delta\mathbf{r}||-\zeta_{6}\leq 0\\ c_{6}&=||\delta\mathbf{r}||-\zeta_{7}\leq 0\end{split} \tag{25}\]  
As discussed before, the orbit with the smallest \(||\Delta\mathbf{V}||\) in the two-objective Pareto-optimal frontier is chosen as the transfer orbit because of the fact that the differences in the time of flight of all solutions in the Pareto-optimal frontier are not significant. Next, the semi-analytical optimization method is proposed to solve the transfer orbit with the minimum impulse.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Population Size**} & \multirow{2}{*}{\begin{tabular}{c} **Maximum** \\ **Iterations** \\ \end{tabular} } & \begin{tabular}{c} **Simulates Binary** \\ **Crossover** \\ **Parameters \(\mathbf{\eta_{c}}\)** \\ \end{tabular} & \begin{tabular}{c} **Polynomial** \\ **Mutation** \\ **Parameter \(\mathbf{\eta_{m}}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Probability of** \\ **Mutation \(\mathbf{p_{m}}\)** \\ \end{tabular} &
\begin{tabular}{c} **Probability of** \\ **Crossover \(\mathbf{p_{c}}\)** \\ \end{tabular} \\ \hline
1500 & 150 & 20 & 100 & 0.2 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Parameters of NSGA2 algorithm.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Meaning** \\ \hline \(||\Delta\mathbf{V}||\) & The modulus of the impulse \(\Delta\mathbf{V}\). \\ \(\alpha\) is the angle between \(\Delta\mathbf{V}\) and the projection of \(\Delta\mathbf{V}\) in the \(\hat{\mathbf{x}}-\hat{\mathbf{y}}\) plane, \\ and \(\beta\) is the angle between the projection of \(\Delta\mathbf{V}\) in the \(\hat{\mathbf{x}}-\hat{\mathbf{y}}\) plane and \(\hat{\mathbf{x}}\). \\ \(dt_{1}\) & Time from the flyby point of the previous revolution to the moment of \\  & maneuver in the current revolution. \\ \(dt_{2}\) & Time from the moment of maneuver to fly by the target debris. \\ \hline \hline \end{tabular}
\end{table}
Table 12: Optimization parameters of the NSGA2 algorithm.