# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 5 Phase 2: HEO Based SSO Objects Flyby
### Semi-Analytical Optimization Method
Step 5: The optimized solution with the smallest \(||\Delta V||\) is selected as the design result for the spacecraft's orbit. We output the following variables: the parameters of the maneuver \([||\Delta V||,\alpha,\beta,dt_{1},dt_{2}]^{\mathsf{T}}\), where \(dt_{1}\) is the time from the previous flyby moment to the application of the impulse in the current orbit, and \(dt_{1}=\tau_{e}-\tau_{3}\) and \(dt_{2}=\tau_{4}-\tau_{e}\). The calculation procedure of the semi-analytical optimization method is summarized as follows: in Step 1, the states of the spacecraft in the CRTBP system are converted into the two-body system. Then, in Step 2-Step 3, the orbital parameters of the spacecraft and the moment of the maneuver that are able to fly by a target piece of debris in the two-body system are obtained. In Step 4, these parameters are used as the initial values and are optimized in the CRTBP system. In Step 5, the optimized solution with the smallest \(||\Delta V||\) is chosen to be the flyby orbit. The calculation procedure of Phase 2 is shown in Figure 12. \(\tau_{3}\) is the previous flyby moment. \(\mathrm{D}_{i}\) is the serial number of the debris. angle\({}_{i}\) is the angle between the spacecraft and \(\mathrm{D}_{i}\) debris at \(\tau_{4}=\tau_{3}+10.5\) day. flyby\({}_{i}\) is the number of times that \(\mathrm{D}_{i}\) debris was flown by. The colors of the block diagram in Figure 12 correspond to Figure 11. The green parts indicate that the NSGA-II algorithm is utilized for optimization. For each flyby target, the transfer orbit is optimized using the NSGA-II algorithm up to two times when the Pareto-optimal results are not obtained. When a piece of debris cannot be flown by using the NSGA-II algorithm, another piece of debris will be chosen as the flyby target until all three pieces of debris have been used as flyby targets and Pareto optimization results have still not been obtained. The yellow part means that the semi-analytical optimization method is used and the purple diagram means that the spacecraft flies without flying by any pieces of debris. Figure 13 shows the calculation procedure when the spacecraft orbits the Earth without flying by any target. \(T_{e}\) is the orbit period of the spacecraft. As shown in Figure 13, when the spacecraft flies without flying by any target and the height of perigee and the inclination  
Figure 12: Design of multiple transfer orbits. of the spacecraft satisfy the conditions in Equation (40), the spacecraft does not perform orbit maneuver, i.e., \(||\Delta\mathbf{V}||=0\). \[ \begin{split}\widetilde{h}_{2}&\leq h_{p- earth}\leq\widetilde{h}_{3}\\ \widetilde{i}_{1}&\leq i_{1}\leq\widetilde{i}_{2} \end{split}  \tag{40}\]  
where \(i_{1}\) is the orbital inclination of the spacecraft; \(\widetilde{i}_{1}=80^{\circ}\) and \(\widetilde{i}_{2}=110^{\circ}\). The reason that Equation (40) is not satisfied is that the spacecraft is influenced by the lunar gravity. There are two general cases. One is that an impulse should be applied to prevent the spacecraft from entering the Earth's atmosphere when the height of the perigee is less than 400 km because of the fact that the lunar flyby is in front of the Moon. The other one is that when the height of the perigee is increased because of the fact that the lunar flyby is behind the Moon and the spacecraft is far away from the nearest debris, the spacecraft will continue to fly several revolutions without maneuver rather than changing the orbit with a large impulse to comply with the conditions of debris flyby. When an impulse should be applied, the optimization parameters are as Table 12 shows. The objective function is as follows:  
\[f(\boldsymbol{x})=||\Delta\mathbf{V}|| \tag{41}\]  
The lower and upper boundaries of the optimization parameters are as follows:  
\[ \begin{split}\boldsymbol{x}_{\text{min}}&=[0,0^{ \circ},0^{\circ},0,0]\\ \boldsymbol{x}_{\text{max}}&=[\zeta_{4},2\pi,2\pi,3 \text{ TU},3\text{ TU}]\end{split}  \tag{42}\]  
The inequality constraints are as follows:  
\[