# Summary of Lunar Constellation Navigation and Orbit Determination Technology
## 3 Overview of Single-Satellite Navigation and Orbit Determination Methods
### Astronomical Navigation Technology
 = -\|r_{sm}\|\sqrt{\frac{1-\left(I_{sm}\widetilde{I_{pm}}\right)^{2}}{ 1-\left(\widetilde{I_{pm}}\widetilde{I_{pm}}\right)^{2}}}\widetilde{I_{ps}} \widetilde{I_{ps}}\] \[= -\|r_{sm}\|\sqrt{\frac{1-\left[I_{sm}\left(I_{pm}+\mathbf{V}_{pm} \right)\right]^{2}}{1-\left[\left(I_{ps}+\mathbf{V}_{ps}\right)\left(I_{ps}+\mathbf{V }_{ps}\right)\right]^{2}}}\left(\mathbf{I}_{ps}+\mathbf{V}_{ps}\right) \tag{10}\]  
Figure 2: Schematic diagram of angle measurement navigation. For the above equation, Taylor expansion at \(I_{ps}\) and \(I_{pm}\) gives:  
\[\mathbf{Z}=\mathbf{r}+\left(\frac{\partial\mathbf{r}}{\partial I_{ps}}\mathbf{V}_{ps}+\frac{ \partial\mathbf{r}}{\partial I_{pm}}\mathbf{V}_{pm}\right)+o(\bullet) \tag{11}\]  
Ignoring the higher-order small quantities in the observed quantities, then:  
\[\mathbf{Z}_{t}=\mathbf{r}+\mathbf{V}(t) \tag{12}\]  
Among them, the observation error is calculated as follows:  
\[\mathbf{V}(t)=\frac{\partial\mathbf{r}}{\partial I_{ps}}\mathbf{V}_{ps}+\frac{\partial\bm {r}}{\partial I_{pm}}\mathbf{V}_{pm} \tag{13}\]  
Among them, \(\mathbf{V}_{ps},\mathbf{V}_{pm}\) are measurement noises, which are generally treated as Gaussian white noise. 2. Astronomical ranging navigation  
Astronomical ranging navigation is an autonomous spacecraft navigation method based on X-ray pulsar observations. X-ray pulsars will emit strong and stable periodic X-ray radiation. The difference between the time when the same pulse signal reaches the spacecraft X-ray detector and the time when it reaches the center of mass of the solar system predicted by the phase time model is used as the observation quantity to construct X-Ray pulsar navigation measurement equations. Combined with the speed of light, the plane where the spacecraft is located is obtained, which is the projected length of the distance from the spacecraft to the center of mass of the solar system in the direction of the pulsar, as shown in Figure 3. By detecting three pulsars at the same time, a geometric relationship can be constructed to calculate the space position of the spacecraft and achieve autonomous navigation. The radio pulse signal navigation proposed by [PERSON] is the basis of astronomical ranging navigation [30]. The earliest autonomous navigation method of X-ray pulsars was designed based on this principle [31]. In the early stage of development, astronomical ranging navigation remained at the theoretical level. It was not until 2017 that NASA announced the implementation of the first space demonstration of X-ray pulsar navigation [32]. A series of subsequent tests also proved that this technology has the advantages of a stable navigation source signal, strong anti-interference and long application distance [33; 34; 35; 36]. The key to follow-up research is the establishment of a pulsar library and effective on-orbit exploration technology of pulse signals. 3. Astronomical velocity navigation  
Astronomical velocity measurement navigation uses the solar system and extrasolar stars as navigation sources. It obtains relative motion speed information by observing  
Figure 3: Schematic diagram of ranging navigation. the astronomical spectral frequency shift caused by the motion of the probe relative to the navigation source, and further calculates the velocity vector of the probe in space [37]. The schematic diagram is shown in Figure 4, whereby \(v_{s}\), \(v_{t}\) are the velocity vectors of the star and the probe in the reference coordinate system. \(l_{ts}\) is the azimuth vector of the star relative to the probe. \(v_{ts}\) is the radial velocity of the probe relative to the star. \(v_{ts}\) can be derived from the spectral frequency shift amount \(v_{ts}=c\bullet\frac{\Delta f}{f_{r}}=l_{ts}^{T}(v_{s}-v_{t})\) and \(f_{r}\) is the reference spectral frequency. The basis of astronomical velocity measurement and navigation is the feasibility analysis of using the visible spectrum band and radio spectrum band of stars for velocity measurement proposed by [PERSON] [38]. [PERSON] et al.