# Autonomous Navigation Based on the Earth-Shadow Observation near the Sun-Earth L2 Point
## 2 Design of Navigation System
### The Earth-Shadow Measurement Model
Meanwhile, we can obtain the distance between the projection center of the Earth and the Sun, which is given as  
\[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]6  
As shown in Figure 6, according to the analysis of the spatial motion of the Sun, the Earth, and the spacecraft, it can be concluded that there are three different occlusion types in the whole cycle. The following sections will discuss the three different circumstances, respectively. \[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]7 Measurement model of Earth shadow. As shown in Figure 6a, when \(AF\geq AB+FG\),  
\[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]8  
the Earth cannot block the Sun. Therefore, it can be obtained that the measurement function \(H\) is equal to zero. As shown in Figure 6b, when \(AF\leq FG-AB\), the Earth's projection is included in the Sun's projection. Therefore, the measurement function \(H\) is given in the following formula:  
\[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]9  
As shown in Figure 6c, when \(FG-AB\leq AF\leq FG+AB\),  
Figure 3:0  
and the Earth's projection and the Sun's projection have overlapping parts. Thus, the measurement function \(H\) is given as  
Figure 3:1  
where \(S_{overlap}\) represents the overlapping areas of the Earth's projection and the Sun's projection, which can be calculated by adding the area of Fan \(AH_{1}H_{2}\) and Fan \(FH_{1}H_{2}\), and by subtracting the area of Quadrilateral \(AH_{1}FH_{2}\), given by  
Figure 3:2  
where \(S_{AH_{1}H_{2}}\) represents the area of Fan \(AH_{1}H_{2}\), \(S_{FH_{1}H_{2}}\) represents the area of the Fan \(FH_{1}H_{2}\), and \(S_{AH_{1}FH_{2}}\) represents the area of the Quadrilateral \(AH_{1}FH_{2}\). In order to simplify the calculation, it is noted that \(P=r_{e}+r_{s}+AF\). According to the Law of Cosines,  
Figure 3:3 Three types of Earth-shadow. (**a**) Occlusion situation 1. (**b**) Occlusion situation 2. (**c**) Occlusion situation 3. \[ \begin{split}\cos\angle H_{1}AF=\frac{r_{s}^{2}-\left(r_{e}^{2}+\|AF \|^{2}\right)}{2r_{e}\|AF\|}\\ \angle H_{1}AF=\cos^{-1}\left(\frac{r_{s}^{2}-\left(r_{e}^{2}+\|AF \|^{2}\right)}{2r_{e}\|AF\|}\right)\\ S_{AH_{1}H_{2}}=\frac{1}{2}\left(\cos^{-1}\left(\frac{r_{s}^{2}- \left(r_{e}^{2}+\|AF\|^{2}\right)}{2r_{e}\|AF\|}\right)r_{e}^{2}\right)\\ SA_{iH_{1}FH_{2}}=2\sqrt{P(P-\|AF\|)(P-r_{e})(P-r_{s})}\\ S_{FH_{1}H_{2}}=\frac{1}{2}\left(\cos^{-1}\left(\frac{r_{e}^{2}- \left(r_{s}^{2}+\|AF\|^{2}\right)}{2r_{s}\|AF\|}\right)r_{s}^{2}\right),\end{split}  \tag{19}\]  
where \(r_{e}\) is the radius of the Earth, \(r_{s}\) denotes the radius of the Sun, and \(\|AF\|\) is the distance between point A and point F.  
Therefore, the measurement function \(H\) is as follows:  
\[ \begin{split} H=&\frac{1}{4\pi r_{s}{}^{2}}\left( \cos^{-1}\left(\frac{r_{s}^{2}-\left(r_{e}{}^{2}+\|AF\|^{2}\right)}{2r_{e}\| AF\|}\right)r_{e}{}^{2}\right)\\ &+\frac{1}{4\pi r_{s}{}^{2}}\left(\cos^{-1}\left(\frac{r_{e}{}^{ 2}-\left(r_{s}{}^{2}+\|AF\|^{2}\right)}{2r_{s}\|AF\|}\right)r_{s}^{2}\right)\\ &-\frac{\sqrt{P(P-\|AF\|)(P-r_{e})(P-r_{s})}}{\pi r_{s}{}^{2}}. \end{split}  \tag{20}\]  
It can be seen that the expression of the measurement function \(H\) only contains one variable \(\|AF\|\). According to the analysis of the geometric relationship, \(\|AF\|\) is closely related to the position of the spacecraft. Depending on the above hypothesis, this section is to study the dynamics of the CRTBP in the two-dimensional plane. It is assumed that the position vector of the Earth, the Sun, and the spacecraft are \(\vec{E}=E\big{(}E_{x},E_{y}\big{)}\), \(\vec{S}=S\big{(}S_{x},S_{y}\big{)}\), and \(\vec{r}=r\big{(}r_{x},r_{y}\big{)}\), respectively. The position coordinates of the Earth and the Sun can be obtained through the ephemeris. Thus, they can be regarded as known quantities. The following formulas are the position vector \(\vec{r}=r\big{(}r_{x},r_{y}\big{)}\) and the measurement function of the spacecraft.