# Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit
## 5 Phase 2: HEO Based SSO Objects Flyby
### Semi-Analytical Optimization Method  
This section presents the semi-analytical optimization method to solve the transfer orbit with the minimum velocity increment. Assume that, in the CRTBP system, \(\tau_{3}\) is the previous flyby moment. The spacecraft's position and the velocity at \(\tau_{3}\) in the center of mass rotating frame \(\{C-\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}\}\) are \((\mathbf{r}_{3},\mathbf{v}_{3})\). At \(\tau_{3m}\), the spacecraft applies an impulse \(\Delta V\) to change its orbit. Before changing the spacecraft's orbit at \(\tau_{3m}\), the position and the velocity in the \(\{C-\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}\}\) frame are \((\mathbf{r}_{3m-},\mathbf{v}_{3m-})\), which are denoted as \((\mathbf{R}_{3m-},\mathbf{V}_{3m-})\) in the ECI coordinate \(\{O-\hat{\mathbf{X}},\hat{\mathbf{Y}},\hat{\mathbf{Z}}\}\). The corresponding orbital elements are denoted as \((a_{3m},e_{3m-},i_{3m-},\Omega_{3m-},\omega_{3m-},f_{3m-})\). After changing the spacecraft's orbit at \(\tau_{3m}\), the position and the velocity in the ECI coordinate \(\{O-\hat{\mathbf{X}},\hat{\mathbf{Y}},\hat{\mathbf{Z}}\}\) are denoted as \((R_{3m+},V_{3m+})\), and the corresponding orbital elements are \((a_{3m+},e_{3m+},i_{3m+},\Omega_{3m+},\omega_{3m+},f_{3m+})\). At \(\tau_{3m}\), the states should satisfy the following:  
\[\mathbf{R}_{3m-}=\mathbf{R}_{3m+},\ \Delta\mathbf{V}=\mathbf{V}_{3m+}-\mathbf{V}_{3m-} \tag{26}\]  
\[\mathbf{h}_{3m+}\bot\mathbf{R}_{3m-},\ \mathbf{h}_{3m+}\bot\mathbf{R}_{3m+} \tag{27}\]  
\[\mathbf{h}_{3m+}\cdot\mathbf{R}_{3m-}=0,\ \mathbf{h}_{3m+}\cdot\mathbf{R}_{3m+}=0 \tag{28}\]  
where \(\mathbf{h}_{3m+}\) is the unit angular momentum after changing the spacecraft's orbit at \(\tau_{3m}\). The flyby target is denoted as Target-1. The orbital elements of Target-1 at \(\tau_{3}\) are \((a_{13},e_{13},i_{32},\Omega_{13},\omega_{13},f_{13})\). At \(\tau_{4}\), the spacecraft flies by Target-1. The orbital elements of Target-1 at \(\tau_{4}\) are \((a_{4},e_{4},i_{4},\Omega_{4},\omega_{4},f_{44})\), and the orbital elements of the spacecraft at \(\tau_{4}\) are \((a_{4},e_{4},i_{4},\Omega_{4},\omega_{4},f_{44})\). It is required that the spacecraft will fly by the target debris after changing its orbit. At \(\tau_{4}\), the orbital plane of the spacecraft should satisfy the following:  
\[i_{4}\approx i_{\tau 4},\ \Omega_{4}\approx\Omega_{\tau 4} \tag{29}\]  
The specific calculation procedure for the optimization method is composed of five steps as follows:  
Step 1: Input the spacecraft's states \((\mathbf{r}_{3},\mathbf{v}_{3})\) at \(\tau_{3}\) in the \(\{C-\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}\}\) frame, and the states of Target-1 \((\mathbf{R}_{13},\mathbf{V}_{13})\) at \(\tau_{3}\) in the ECI coordinate \(\{O-\hat{\mathbf{X}},\hat{\mathbf{Y}},\hat{\mathbf{Z}}\}\). Integrate the spacecraft's orbit an orbital period away from \(\tau_{3}\) in the two-body system. The moment that satisfies Equation (28) is denoted as \(\tau_{e}\). The spacecraft's states \(\mathbf{r}_{e},\mathbf{v}_{e}\) at \(\tau_{e}\) in the CRTBP system are recorded. In Equation (28), the equations of \(\mathbf{h}_{3m+}=[h_{3m+x};h_{3m+y};h_{3m+z}]\) are calculated through the following:  
\[\left\{ \begin{array}{l}h_{3m+x}=\sin(\Omega_{3m+})\sqrt{1-h_{3m+z}^{2}}\\ h_{3m+y}=-\cos(\Omega_{3m+})\sqrt{1-h_{3m+z}^{2}}\\ h_{3m+z}=\cos i_{3m+}\end{array} \right. \tag{30}\]  
where \(i_{3m+}=i_{\tau 4},\ \Omega_{3m+}=\Omega_{\tau 4}\). Step 2: \(\tau_{e},\mathbf{r}_{e},\mathbf{v}_{e}\) are converted into the ECI coordinate \(\{O-\hat{\mathbf{X}},\hat{\mathbf{Y}},\hat{\mathbf{Z}}\}\) and then are denoted as \(\tau_{e},\mathbf{R}_{e-},\mathbf{V}_{e-}\). We calculate the spacecraft's orbital elements \((a_{e},e_{e},i_{e},\Omega_{e},\omega_{e},f_{e})\) after it changes its orbit at \(\tau_{e}\). Then, let us implement the ergodic calculation as Equation (31) shows. \[