# Integrated Design of Moon-to-Earth Transfer Trajectory Considering Re-Entry Constraints
## 2 Dynamical Model and Frame Construction  
In this paper, the selection of transfer opportunity and the design of precise transfer trajectory are carried out under the simplified Earth-Moon ephemeris model and the high-precision dynamic model, respectively.  
The precise transfer trajectory design should be carried out in a high-precision dynamic model, which is established as  
\[\dot{\mathbf{r}}_{e}=-\mu_{E}\cdot\frac{\mathbf{r}_{e}}{\left\|\mathbf{r}_{e}\right\|^{3}}+ \mu_{M}\cdot\left(\frac{\mathbf{r}_{M}-\mathbf{r}_{e}}{\left\|\mathbf{r}_{M}-\mathbf{r}_{e} \right\|^{3}}-\frac{\mathbf{r}_{M}}{\left\|\mathbf{r}_{M}\right\|^{3}}\right)+\mu_{S} \cdot\left(\frac{\mathbf{r}_{S}-\mathbf{r}_{e}}{\left\|\mathbf{r}_{S}-\mathbf{r}_{e}\right\|^{3 }}-\frac{\mathbf{r}_{S}}{\left\|\mathbf{r}_{S}\right\|^{3}}\right)+\mathbf{a}_{NE}+\mathbf{a}_ {NM}+\mathbf{a}_{srp} \tag{1}\]  
where \(\dot{\mathbf{r}}_{e}\) is the acceleration of the probe in the Earth-Moon gravitational model; \(\mu_{E}\), \(\mu_{M}\), and \(\mu_{S}\) are the gravitational constants of the Earth, the Moon, and the Sun, respectively; \(\mathbf{r}_{e}\) is the position vector of the probe in the Earth-centered inertial (ECI) coordinate frame; \(\mathbf{r}_{M}\) and \(\mathbf{r}_{S}\) are the position vectors of the Moon and the Sun in ECI, respectively; \(\mathbf{a}_{NE}\), \(\mathbf{a}_{NM}\), and \(\mathbf{a}_{srp}\) are the Earth's non-spherical gravitational perturbation, the Moon's non-spherical gravitational perturbation, and the solar radiation pressure perturbation, respectively. Since the gravitational effect of the probe during the Moon-to-Earth transfer process mainly comes from the Earth and the Moon, the simplified dynamics considering the gravities of the Earth and the Moon is adopted when determining the feasible return trajectory and opportunity, which greatly improves the efficiency while ensuring the reliability of the design.  
To describe the state of the probe under the constraints of the re-entry condition, a re-entry coordinate frame (RF) \(OXYZ\) fixed to the center of the probe, which is shown in Figure 1, is established. The rectangle is the position of the re-entry point marked as \(P\) with an altitude of \(H\). The \(Y\)-axis of the frame points to the zenith, and the \(X\)-axis is vertical to the \(Y\)-axis and points to the flight direction in the re-entry plane. The \(Z\)-axis forms a right-hand coordinate frame with the \(X\)-axis and \(Y\)-axis.  
The re-entry point is noted as \(P\), which is marked with a triangle in Figure 1. The longitude and latitude of the re-entry point are \(\lambda_{P},\varphi_{P}\), respectively. The angle between the direction of the velocity at the re-entry point on the local sphere (noted as \(\dot{\mathbf{r}}\) ) and the local north direction is defined as the flight azimuth \(Q\).  
The rotation matrix of the re-entry coordinate frame to the Earth-centered Earth-fixed (ECEF) coordinate frame can be obtained by Euler rotation as  
\[\mathbf{M}_{EF}=\left[\begin{array}{ccc}-\cos Q\cos\lambda_{P}\sin\varphi_{P}- \sin Q\sin\lambda_{P}&\cos\lambda_{P}\cos\varphi_{E}&\sin Q\cos\lambda_{P} \sin\varphi_{P}-\cos Q\sin\lambda_{P}\\ -\cos Q\sin\lambda_{P}\sin\varphi_{P}+\sin Q\cos\lambda_{P}&\sin\lambda_{P} \cos\varphi_{E}&\sin Q\sin\lambda_{P}\sin\varphi_{P}+\cos Q\cos\lambda_{P}\\ \cos Q\cos\varphi_{P}&\sin\varphi_{P}&-\sin Q\cos\varphi_{P}\end{array}\right] \tag{2}\]