# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 4 Phase 1: Two Impulses DRO-Earth Transfer Orbit Design  
As shown in Figure 5a, this section discusses how to use a two-impulse maneuver [7] to transfer from DRO to the Earth and fly by a target debris near the perigee. The objective of this section is to find an orbit that minimizes the total impulse while satisfying the flyby requirements. As a rule of thumb, in the typical Earth-Moon three-body problem, a key factor in this transfer orbit design problem is to fully utilize the lunar gravity assist.  
This section proposes the spacecraft's flight procedure as follows: at \(\tau_{0}\), the spacecraft applies a departure impulse \(\Delta V_{0}\) in DRO to transfer to the Moon. At \(\tau_{1}\), the spacecraft applies the impulse \(\Delta V_{1}\) when it arrives at the perilune. At \(\tau_{2}\), the spacecraft flies to the Earth and flies by the SSO target debris near the perigee. The transfer time of the spacecraft from DRO to the perilune is \(\Delta t_{0}=\tau_{1}-\tau_{0}\). The transfer time from the perilune to flyby is \(\Delta t_{1}=\tau_{2}-\tau_{1}\).  
Typical DRO-Earth transfer methods include the internal transfer, which is mainly accomplished based on the CRTBP system, and the external transfer, in which the spacecraft  
Figure 4: Orbits of target debris, J2000 ECI coordinate.  
Figure 5: (a) Two impulses DRO–Earth transfer orbit in the Earth–Moon Rotating Frame; (b) illustration of \(\Delta V_{0}\) in the Earth–Moon rotating frame.  
files to the Sun-Earth gravitational equilibrium region and achieves a low-energy Moon-Earth transfer using the weak stability boundary theory. Note that the external transfer method requires a much longer transfer time (100 days to 130 days), which does not conform to the requirements shown in Table 3. So, this paper chooses the internal transfer approach.  
Taking the perilune as a node, the complete DRO-Earth transfer and flyby orbit is composed of two segments: (1) The segment from DRO to the perilune, denoted by \(\mathrm{G}_{0}\), as shown in Figure 5a. (2) The segment from the perilune to the perigee, denoted by \(\mathrm{G}_{1}\). The two maneuvers are denoted as \(\mathrm{\Gamma}_{0}(\mathbf{X}_{0})\) and \(\mathrm{\Gamma}_{1}(\mathbf{X}_{1})\). The notation \(\left\{\mathrm{G}_{0}(\mathrm{\Gamma}_{0})+\mathrm{G}_{1}(\mathrm{\Gamma}_{1 })\right\}\) means the trajectory of the spacecraft is composed of \(\mathrm{G}_{0}\) and \(\mathrm{G}_{1}\), with two orbital maneuvers \(\mathrm{\Gamma}_{0}(\mathbf{X}_{0})\) and \(\mathrm{\Gamma}_{1}(\mathbf{X}_{1})\). The notation \(\left\{\mathrm{G}_{0}(\mathrm{\Gamma}_{0})+\mathrm{G}_{1}(0)\right\}\) means the trajectory of the spacecraft is composed of \(\mathrm{G}_{0}\) and \(\mathrm{G}_{1}\), with only one maneuver \(\mathrm{\Gamma}_{0}(\mathbf{X}_{0})\).  
The plan is constructed as follows: First, the DRO-Earth transfer orbit with a single impulse \(\left\{\mathrm{G}_{0}(\mathrm{\Gamma}_{0})+\mathrm{G}_{1}(0)\right\}\) is to be investigated. Then, based on the results of \(\left\{\mathrm{G}_{0}(\mathrm{\Gamma}_{0})+\mathrm{G}_{1}(0)\right\}\), the DRO-Earth transfer orbit with two impulses \(\left\{\mathrm{G}_{0}(\mathrm{\Gamma}_{0})+\mathrm{G}_{1}(\mathrm{\Gamma}_{1 })\right\}\) is designed. The parameters of \(\left\{\mathrm{G}_{0}(\mathrm{\Gamma}_{0})+\mathrm{G}_{1}(\mathrm{\Gamma}_{1 })\right\}\) are to be optimized to solve the local optimization problem of the first flyby.