# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 2 Problem Statement and Equations of Motion
### Flyby Mission Description  
The mission scenario is shown in Figure 2. The steps of the flyby mission are illustrated in Table 2 and Figure 3. As shown in Figure 2, the spacecraft is located in DRO at the initial moment \(\overline{\tau}\). At \(\tau_{0}\), by imposing \(\Delta V_{0}\), the spacecraft leaves DRO and approaches the Moon. At \(\tau_{1}\), the spacecraft applies \(\Delta V_{1}\) in lunar sphere of influence (SOI) to take a powered lunar gravity assist. Then, the spacecraft enters the Moon-Earth transfer orbit. At \(\tau_{2}\), the spacecraft near-coplanar flies by a piece of space debris near the perigee. After that, the spacecraft flies towards the apogee and applies an impulse near the apogee. By adjusting the orbit and the phase, the spacecraft can near-coplanar fly by the next piece of space debris in the highly elliptical orbit (HEO). By iterating Step 4-Step 5 in Table 2, the multiple-object flyby mission is accomplished.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Target Orbits** & \(1\,\Delta V_{\text{total}}\,|\,1\) & **Types of Transfer** \\ \hline DRO & 3.1934 km/s [45] & \multirow{2}{*}{Lunar gravity assist and weak stability boundary} \\ GEO & 4.2457 km/s [45] & Direct two impulses \\ PLO & 3.8 km/s [46] & Direct two impulses \\ NRHO & 4 km/s [46] & Direct two impulses \\ L\({}_{1}\) halo orbit & 3.61 km/s [47] & Two impulses and stable manifold \\ L\({}_{2}\) halo orbit & 3.36 km/s [48] & Two impulses and stable manifold \\ \hline \hline \end{tabular}
\end{table}
Table 1: Velocity increment budgets for transferring from LEO to different target orbits.  
Figure 2: Mission scenario. (**a**) The center of mass rotating frame; (**b**) the J2000 ECI coordinate.  
This paper divides the multiple-object near-coplanar flyby mission into two phases. As shown in Figure 2b, Phase 1 is the segment in which the spacecraft transfers from DRO to Earth and implements the first flyby, i.e., Step 1-Step 3 in Table 2. Phase 2 is the segment in which the spacecraft near-coplanar flies by all the target debris, i.e., Step 4-Step 5 in Table 2.  
The requirements of the flyby mission are shown in Table 3. In Table 3, \(\zeta_{1}=31\) days, \(\zeta_{2}=0.4\) km/s, \(\zeta_{3}=0.2\) km/s, \(\zeta_{4}=0.1\) km/s, \(\zeta_{5}=50\) km, \(\zeta_{6}=100\) km, \(\zeta_{7}=4\) km/s, \(\widetilde{h}_{1}=100\) km, and \(\widetilde{h}_{2}=400\) km. Note that the errors of measurement and controlling are not considered. The two impulses of Phase 1 are applied on DRO (\(\Delta V_{0}\)) and the perilune (\(\Delta V_{1}\)), respectively. The impulse \(\Delta V\) of Phase 2 is applied in HEO. \(h_{p-moon}\geq\widetilde{h}_{1}=100\) km means that the PLF cannot impact the Moon. \(h_{p-earth}\geq\widetilde{h}_{2}=400\) km means that the spacecraft cannot enter the Earth's atmosphere. When the spacecraft with an apogee radius close to the Moon coplanar flies by a space debris with an orbital height of 800 km, the relative velocity is approximately 3 km/s. Thus, this paper requires the modulus of the relative velocity when the spacecraft flies by the target debris to be less than or equal to 4 km/s.