# Summary of Lunar Constellation Navigation and Orbit Determination Technology
## 3 Overview of Single-Satellite Navigation and Orbit Determination Methods
### Astronomical Navigation Technology  
The exploration mission faces problems such as extended information transmission time, complex environment, and many unknown factors in space, which has higher demands on the autonomous navigation of the probe. Radio navigation methods based on ground station measurement and control are widely used in space exploration-related missions. However, due to problems such as transmission rate, transmission delay, and cumulative errors caused by orbital dynamics recursion, it is difficult to achieve real-time high-precision navigation with single ground-based deep space radio navigation. Therefore, with the development of exploration missions, navigation methods that combine more astronomical information such as images and spectra have become a research hotspot. According to the differences in measurement methods, astronomical autonomous navigation methods are divided into three categories: angle measurement navigation, ranging navigation and velocity measurement navigation. 1. Astronomical angle measurement navigation uses optical images of target celestial bodies captured by navigation cameras and known celestial ephemeris, calculates the direction information of the probe relative to the navigation source using image-processing techniques, and finally combines navigation filter methods to achieve autonomous navigation, obtaining information such as the position, velocity and attitude of the probe in a certain coordinate system [21; 22]. The astronomical angle measurement navigation method is widely used in deep space exploration missions, and its schematic diagram is shown in Figure 2. The height difference principle proposed by the French navigator [PERSON] is the basis of astronomical angle measurement navigation. The earliest astronomical navigation method was designed based on this principle [23]. In the early stage of development, astronomical angle measurement navigation methods were mainly used as an auxiliary to terrestrial radio navigation methods. Subsequently, this technology has been continuously developed and improved, and has been successfully applied in several exploration missions [24; 25; 26; 27; 28]. Since the accuracy of astronomical angle measurement  
Figure 1: Schematic diagram of radio interferometry. navigation is greatly affected by distance, when the probe is far away from the navigation target, a small error in the angle estimate may lead to a large error in the probe position estimation after calculation. In actual exploration missions, this technology is generally used in the approach or capture phase when the probe is closer to the target celestial body. In the heliocentric inertial coordinate system, the geometric relationship is [29]:  
\[ \begin{array}{l}\gamma=\arccos(I_{ps}I_{pm})\\ \theta=\arccos(I_{sm}I_{pm})\end{array}  \tag{6}\]  
Among them, \(I_{ps}\) is the sight vector of the sun relative to the probe, and \(I_{pm}\) is the sight vector of the target celestial body relative to the probe, which can be obtained through the measurement of the navigation sensor. \(I_{sm}\) is the direction vector from the sun to the target celestial body, which can be obtained by means of ephemeris calculation. The position of the probe in the inertial frame is:  
\[\mathbf{r}=-\frac{\|r_{sm}\|\sin\theta}{\sin\gamma}I_{ps}=-\|r_{sm}\|\sqrt{\frac{ 1-\left(\cos\theta\right)^{2}}{1-\left(\cos\gamma\right)^{2}}}I_{ps} \tag{7}\]  
That is:  
\[\mathbf{r}=-\|r_{sm}\|\sqrt{\frac{1-\left(I_{sm}I_{pm}\right)^{2}}{1-\left(I_{ps} I_{pm}\right)^{2}}}I_{ps} \tag{8}\]  
Among them, \(I_{ps}\) and \(I_{pm}\) can be obtained through navigation sensors and solar sensors, and their observation models are:  
\[ \begin{array}{l}\widetilde{I_{ps}}=I_{ps}+\mathbf{V}_{ps}\\ \widetilde{I_{pm}}=I_{pm}+\mathbf{V}_{pm}\end{array}  \tag{9}\]  
Among them, \(\mathbf{V}_{ps}\) and \(\mathbf{V}_{pm}\) are the sight vector observation noises. Taking the \(\mathbf{r}\) observed value as \(\mathbf{Z}\), the observation equation is:  
\[ \begin{array}{l}\mathbf{Z}\\ \end{array}