# Integrated Design of Moon-to-Earth Transfer Trajectory Considering Re-Entry Constraints
## 4 Precise Three-Impulse Transfer Trajectory Determination
### Perilune to the Boundary of SOI  
Choose any \(T_{0}\in\Gamma\) in the Moon-to-Earth transfer opportunity; the feasible transfer trajectory with a state of \(X_{i}\) at the re-entry point that reached the vicinity of the Moon can be obtained after the calculation in the previous section. Integrate the initial value \(\mathbf{X}_{i}\) inversely in the high-precision dynamic Equation (1), and the state of the perilune in the ECI is obtained. Transform it to \(\mathbf{X}_{f}=\left[\mathbf{r}_{f},\mathbf{v}_{f}\right]\) in the Moon-centered inertial (MCI) coordinate frame. With orbital angular momentum \(\mathbf{h}\), semi-major axis \(\mathbf{a}\), eccentricity \(\mathbf{e}\), semilatus rectum \(p\), true anomaly \(f\), and eccentric anomaly \(H\) as intermediate variables, calculate the time from perilune \(\tau\), altitude of perilune \(\tau_{pm}\), and inclination of perilune \(i_{m}\). Note the corresponding relationship between the scalars and the vectors as  
\[a=\|\mathbf{a}\|,e=\|\mathbf{e}\|,h=\|\mathbf{h}\|,r_{f}=\left\|\mathbf{r}_{f}\right\|,v_{f}= \left\|\mathbf{v}_{f}\right\| \tag{23}\]  
Then, the time from the perilune, the perilune radius, and the inclination at the perilune can be obtained as [46]  
\[\tau=\sqrt{\frac{a^{3}}{\mu_{M}}}(e\sinh H-H),\quad r_{pm}=a(1-e),i_{m}=\arccos \left(\frac{\mathbf{h}\cdot\mathbf{n}}{h}\right) \tag{24}\]  
where \(\mathbf{n}=[0,0,1]\). The design of the high-precision transfer trajectory in the SOI of the Moon is realized according to the nominal departure parameters at the perilune. Given the radius and inclination of the perilune as \(\tilde{r}_{pm}\) and \(\tilde{i}_{m}\), respectively, the differences between the quantities of the time from the perilune \(\tau\), the actual perilune altitude \(r_{pm}\), and the perilune inclination \(i_{m}\) and the nominal values are chosen as the target constraint variables (the nominal time from the perilune is 0), so the solution of the Moon-to-Earth transfer trajectory can be transformed into the following nonlinear equations as  
\[\left[r_{pm}-r_{pm}^{\prime},i_{m}-i_{m}^{\prime},\tau\right]=0 \tag{25}\]  
After the inverse integration of the initial value \(\mathbf{X}_{i}\) in the high-precision dynamic model, the state of the probe at the boundary of the SOI of the Moon in the ECI is obtained and converted into the form of orbital elements. The correction amounts are selected as the semi-major axis \(a\), the right ascension of the ascending node \(\Omega\), and the argument of the perigee \(w\) at the boundary of the SOI of the Moon, and the influence of the correction amounts on the target constraint variables are analyzed by numerical methods. Respectively giving a small disturbance \(\delta\) to the initial state of each correction amount, the response of the target constraint variable to the perturbation can be obtained after inverse integration into the perilune. Then, the sensitivity matrix \(\mathbf{\mathrm{K}}_{1}\) of the target constraint variables relative to the initial states of the correction amounts can be obtained as  
\[\mathbf{\mathrm{K}}_{1}=\left[ \begin{array}{cc}\frac{\partial r_{pm}}{\partial a }&\frac{\partial r_{pm}}{\partial\Omega}&\frac{\partial r_{pm}}{\partial a \omega}\\ \frac{\partial u}{\partial a}&\frac{\partial u}{\partial\Omega}&\frac{\partial u }{\partial\omega}\\ \frac{\partial r}{\partial a}&\frac{\partial r}{\partial\Omega}&\frac{\partial r }{\partial\omega}\\ \end{array} \right] \tag{26}\]Then, the relationship between the differential of the correction amounts and the error of the target constraint variables is  
\[\left[ \begin{array}{c}\Delta a\\ \Delta\Omega\\ \Delta\omega\end{array} \right]=\mathbf{K}_{1}^{-1}\left[ \begin{array}{c} \delta r_{pm}\\ \delta i_{m}\\ \delta\tau\end{array} \right] \tag{27}\]  
After several iterations, the initial states \(a,\Omega,w\) at the boundary of the SOI of the Moon that satisfy the nonlinear Equation (25) can be obtained.