# Spatial Grid-Based Position Calculation Method for Satellite-Ground Communication Links
## 2 Materials and Methods
### Calculation of Satellite-Ground Visibility  
The satellite information transmission system generally consists of a space segment and a ground control segment, and information transmission can be carried out only after the establishment of a satellite-ground link. In the process of satellite motion, the satellite is in dynamic motion relative to the terminal, which poses difficulties for real-time calculation of the relative position of the satellite and the ground. In this section, we will establish a satellite grid computational model based on the GeoSOT model to solve the satellite selection problem in satellite-ground communication under the maximum service time. The spatial reception area coverage of the ground terminal can also be expressed by using the ground coverage grid framework of the satellite. Each satellite in the satellite constellation moves according to a certain orbit, and satellite trajectories can be calculated based on satellite orbit dynamics. Therefore, by mapping the satellite motion trajectory information with the sensor coverage area information in the GeoSOT grid coding system, we can perform grid modeling (Figure 3). The satellite motion trajectory is mainly composed of a series of ordered spatial 3D points. Under the grid coordinate system, we map the satellite's latitude and longitude positions into a series of grid codes by determining the size of a specific grid. The motion speed of a low-orbiting satellite is about 7.5 km/s, so depending on the satellite response time, a GeoSOT grid with 9-15 levels is generally used for modeling. Grid modeling for satellite beams, terminals, etc., is mainly used to calculate their coverage areas (Figure 2). By determining the boundary information of the coverage area, we can dissect the original coverage area and generate a collection of grids at the base level. Figure 3: Grid-based modeling of satellite trajectories. This can then be aggregated according to the actual requirements to generate multi-scale grid codes and establish a one-to-many mapping relationship with temporal information. In the model proposed in this paper, we use the distance of the satellite relative to the center of the terminal receiving area to determine whether the satellite is in the communicable range of the ground terminal. For a deterministic satellite, we know the orbital altitude, and the space coverage area of the ground terminal is a closed inverted conic surface body. As shown in Figure 4, the bottom of the inverted conic surface body can be projected onto the surface of the Earth, forming a closed surface region \(S\). \(S\) is a circular area of radius r centered on the ground measurement and control station, and the circular area is the coverage area of the ground terminal for orbit height \(H\). Gridding the coverage area \(S\), we have:  
\[C_{u,H}\rightarrow\{grid_{1},\;grid_{2},\ldots,grid_{n}\} \tag{3}\]  
Further onwards,  
\[\left\{ \begin{aligned} C_{u,H}\to G_{u,H}\\ G_{u,H}=\{gcode_{i}|C_{u,Hj}-gcode_{i},\;C_{u,Hj}\subset C_{u,H}\} \end{aligned} \right. \tag{4}\]  
Equation (4) is the form of the grid code description of the coverage area of the ground terminal for orbit height \(H\). The primary condition for establishing a stable satellite-ground link is to be within the visibility range of the terminal, followed by the state of motion of the satellite equivalent to the center of the terminal. Because the visibility of the ground terminal and the satellite is mainly related to the angle of the satellite relative to the terminal, the distance will hardly affect the visibility of the satellite signal, but will affect the strength of the signal. Additionally, within the same altitude layer, the distance L between the satellite and the center point is the key factor for measuring whether the satellite is within the acceptance range. Therefore, converting the position analysis of the satellite and the location station to  
Figure 4: Simplified schematic of ground terminal satellite coverage area.