# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 5 Phase 2: HEO Based SSO Objects Flyby
### Semi-Analytical Optimization Method
\begin{split}& r_{p}\in[6771\ \text{km}:20\ \text{km}:717 \ \text{km}]\\ & r_{a}\in[380000\ \text{km}:500\ \text{km}:390000\ \text{km}]\\ & e_{e}\in[0.8:0.01:0.99]\\ & i_{e}=i_{3m+},\ \Omega_{e}=\Omega_{3m+}\end{split}  \tag{31}\]The argument of latitude \(u_{e}\) is calculated by Equation (32). From Equations (33) and (34), \(\omega_{e},f_{e}\) can be obtained. \[\mathbf{R}_{e}=\left[ \begin{array}{c}R_{ex}\\ R_{ey}\\ R_{ez}\end{array} \right]=\frac{a_{e}(1-e_{e}{}^{2})}{1+e_{e}\cos f_{e}}\left[ \begin{array}{c}\cos\Omega_{3m+}\cos u-\sin\Omega_{3m+}\sin u_{e}\cos i_{3m+ }\\ \sin\Omega_{3m+}\cos u+\cos\Omega_{3m+}\sin u_{e}\cos i_{3m+}\\ \sin u_{e}\sin i_{3m+}\end{array} \right] \tag{32}\]  
\[f_{e}=a_{e}\cos\biggl{(}\biggl{(}\frac{a_{e}(1-e_{e}^{2})}{||\mathbf{R}_{e}||}-1 \biggr{)}/e_{e}\biggr{)} \tag{33}\]  
\[\omega_{e}=u_{e}-f_{e} \tag{34}\]  
Step 3: Each set of \((a_{e},e_{e},i_{e},\Omega_{e},\omega_{e},f_{e})\) is converted into the spacecraft's position and velocity \(\mathbf{R}_{e+},V_{e+}\). The impulse required to change the orbit is \(\Delta V=\mathbf{V}_{e+}-\mathbf{V}_{e-}\). When \(||\Delta\mathbf{V}||\leq 0.3\ \mathrm{km}/\mathrm{s}\), \(\mathbf{R}_{e+},\mathbf{V}_{e+}\) is integrated in the two-body system. The time span of the integration is \([\tau_{e},\tau_{e}+T_{e}]\) and the stepsize is set to 50 s. \(T_{e}=2\pi\sqrt{a_{e}^{3}/\mu_{e}}\). We calculate the following parameters: the integral time \(\tau_{4}\), and the spacecraft's orbit \(\mathbf{R}_{4},\mathbf{V}_{4}\) at \(\tau_{4}\). In the \(J_{2}\) perturbed two-body system, the orbit of Target-1 is \(\mathbf{R}_{te}\), \(\mathbf{V}_{te}\) at \(\tau_{e}\), and \(\mathbf{R}_{te},\mathbf{V}_{te}\) are integrated in the \(J_{2}\) perturbed two-body system. The time span of the integration is \([\tau_{e},\tau_{e}+T_{e}]\) and the stepsize is set to 50 s. Multiple sets of Target-1's orbit \(\mathbf{R}_{4t}\), \(\mathbf{V}_{te}\) are obtained. The relative distance and the relative velocity between the spacecraft and Target-1 are as follows:  
\[\mathbf{R}_{rel}=\mathbf{R}_{4}-\mathbf{R}_{4t},\mathbf{V}_{rel}=\mathbf{V}_{4}-\mathbf{V}_{t4} \tag{35}\]  
For each set of \([\tau_{4},\mathbf{R}_{4},\mathbf{V}_{4},\mathbf{R}_{te},\mathbf{V}_{te}]\) and \(\tau_{e}\), the spacecraft's orbital elements \((a_{e},e_{e},i_{e},\Omega_{e},\omega_{e},f_{e})\) after orbit changes and \(dt=\tau_{4}-\tau_{e}\) are recorded if \(||\mathbf{R}_{rel}||\leq 2000\ \mathrm{km}\), \(||\mathbf{V}_{rel}||\leq 6\ \mathrm{km}/\mathrm{s}\). After the pruning, the orbital parameters of the spacecraft after the maneuver and the moment of the maneuver that is able to fly by a piece of target debris in the two-body system are obtained. Step 4: Each set of \((a_{e},e_{e},i_{e},\Omega_{e},dt)\) is used as the initial values of the optimization parameters in the CRTBP system. The objective function is defined as follows:  
\[f(\mathbf{x})=||\Delta\mathbf{V}|| \tag{36}\]  
The optimization parameters are as follows:  
\[\mathbf{x}=[a_{e},e_{e},i_{e},\Omega_{e},dt]^{\mathrm{T}} \tag{37}\]  
The lower and upper boundaries of the optimization parameters are as follows:  
\[ \begin{array}{l}\mathbf{x}_{\min}=\left[1.9339\times 10^{5}\ \mathrm{km},\ 0.8,\ 80^{\circ},\ 0,\ 0\right]\\ \mathbf{x}_{\max}=\left[2.2879\times 10^{5}\ \mathrm{km},\ 0.99,\ 110^{\circ},\ 360^{\circ},\ 13\ \mathrm{days}\right]\end{array}  \tag{38}\]  
The inequality constraints are as follows:  
\[ \begin{array}{l}c_{1}=-(h_{p-mon}-\widetilde{h}_{1})\leq 0\\ c_{2}=-(h_{p-earth}-\widetilde{h}_{2})\leq 0\\ c_{3}=(h_{p-earth}-\widetilde{h}_{3})\leq 0\\ c_{4}=-(||\delta\mathbf{r}||-\zeta_{5})\leq 0\\ c_{5}=||\delta\mathbf{r}||-\zeta_{6}\leq 0\\ c_{6}=||\delta\mathbf{r}||-\zeta_{7}\leq 0\end{array}  \tag{39}\]  
The optimization frame in the CRTBP system is shown in Equations (35)-(38). From Step 4, the orbital parameters of the spacecraft and the moment of the maneuver obtained in Step 3 are optimized in the CRTBP system.