# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 2 Problem Statement and Equations of Motion  
### General Scenario  
Envisioning the active removal of space debris, this paper investigates the mission design problem of a multiple-object flyby such that the detailed information of the object can be obtained by the on-board inspection payload. Debris in SSO are chosen as the target database because SSO is one of the busiest orbits around the Earth. Figure 1 shows the mission scenario. Initially, the spacecraft is in the parking orbit. At point P\({}_{1}\), the spacecraft undertakes the departure maneuver to leave the parking orbit. At point P\({}_{2}\), the spacecraft flies by one piece of target debris. After the first flyby, the spacecraft flies to point P\({}_{3}\). When the spacecraft reaches P\({}_{3}\), it applies the second maneuver to adjust its orbit. As a result, the spacecraft flies by another piece of target debris at point P\({}_{4}\). In this way, the spacecraft performs a multiple debris flyby mission through several maneuvers. Points P\({}_{1}\) and P\({}_{3}\) are calculated and optimized in Sections 4 and 5. The motion of the spacecraft is in cislunar space and described in the circular restricted three-body system. The orbit of the spacecraft is presented in the Earth-Moon rotating coordinate, i.e., the center of mass rotating frame \(\{C-\hat{\pi},\hat{y},\hat{z}\}\), where \(C\) is the center of mass of the Earth-Moon system, the \(\hat{\pi}-\hat{y}\) plane is the orbital plane of the Moon, \(\hat{\pi}\) points from the Earth to the Moon, and \(\hat{z}\) is the normal of the \(\hat{\pi}-\hat{y}\) plane. The spacecraft's equation of motion in the Earth-Moon rotating coordinate is as follows [42]:  
\[\ddot{\boldsymbol{r}}+2\left[ \begin{array}{c}-\dot{y}\\ \dot{x}\\ 0\end{array} \right]=-(\frac{\partial W}{\partial\boldsymbol{r}})^{\text{T}} \tag{1}\]  
where \(\boldsymbol{r}=[x,y,z]^{\text{T}}\) is the spacecraft's position in the Earth-Moon system. \(\dot{\boldsymbol{r}}=[\dot{x},\dot{y},\dot{z}]^{\text{T}}\) is the spacecraft's velocity and \(\ddot{\boldsymbol{r}}\) is the spacecraft's acceleration. \[ \begin{array}{l}W=\frac{1}{2}(x^{2}+y^{2})+U(r_{1},r_{2})\\ U(r_{1},r_{2})=(1-\mu)/r_{1}+\mu/r_{2}\\ \mu=\frac{M_{2}}{M_{1}+M_{2}}\\ J_{t}=2W-||\boldsymbol{v}||^{2}\end{array}  \tag{2}\]  
Figure 1: Mission scenario. (**a**) The panorama of flyby mission; (**b**) the enlarged view of flyby. where \(r_{1}\) and \(r_{2}\) is the distance from the spacecraft to the Earth and the Moon, respectively. \(J_{t}\) is the Jacobi constant. \(M_{1}\) is the mass of the Earth and \(M_{2}\) is mass of the Moon. \[ \begin{split} r_{1}&=\sqrt{\left(x+\mu\right)^{2}+y^{ 2}+z^{2}}\\ r_{2}&=\sqrt{\left(x+\mu-1\right)^{2}+y^{2}+z^{2}} \end{split}  \tag{3}\]  
In the circular restricted three-body problem (CRTBP), the relevant common parameters are as follows [43]:  
\[ \begin{split}\mu&=0.0121506683\\ \text{TU}&=104.8585095\text{ h}\\ \text{DU}&=388405\text{ km}\\ \text{VU}&=3665.892272\text{ km/h}\end{split}  \tag{4}\]  
where TU, DU, and VU are the normalized units of time, distance, and the modulus of velocity in the Earth-Moon rotating coordinate. In addition, the ephemeris model should be used if the eccentric orbit of the Moon and the perturbations due to the gravity of the Sun and other celestial bodies are considered. The orbits in CR3 BP cannot be directly used in the ephemeris model, since the CRTBP is a low-fidelity dynamical model. It is necessary to correct and optimize the orbits obtained from CRTBP in the ephemeris model [44]. This could be analyzed in future research. In this paper, the SSO space debris is affected by the \(J_{2}\) perturbation.