# Navigation Resource Allocation Algorithm for LEO Constellations Based on Dynamic Programming
## 2 Model Analysis of LEO Constellation Navigation Resource Allocation
### Network Model
#### 2.1.2 LEO Space-Based Monitoring  
Currently, the four major global navigation satellite systems (GNSS), including GPS, GLONASS, GALILEO, and BDS, have been widely used. To provide better services, the four systems all use ground-based augmentation integrity monitoring (GAIM) [5] to monitor the integrity of GNSS navigation signals. In GAIM, there is a need to realize the dense distribution of ground systems around the world to ensure global integrity monitoring, which is very expensive and difficult to achieve. Therefore, the GNSSs also adopt satellite autonomous integrity monitoring (SAIM) [6]. However, since SAIM directly collects and detects radio frequency (RF) signals or intermediate frequency (IF) signals, it is impossible to evaluate the impact of antennas and wireless channels on navigation signals.  
[PERSON] proposed SBM at the Stanford Position Navigation and Timing (PNT) Conference in 2010 [4]. In SBM, after passing through the transmitting antenna and wireless channel, the GNSS signals are received by antennas of other satellites and analyzed by signal monitoring devices of other satellites. SBM greatly simplifies the traditional GAIM, breaks through geographical constraints, and can obtain a high signal-to-noise ratio (SNR). Therefore, SBM has great advantages in signal monitoring. Previously, due to the same orbital altitude of GNSS satellites, it was difficult for GNSS satellites to receive downlink navigation signals from other GNSS satellites. The emergence of the LEO navigation system makes SBM possible.  
SBM in the LEO constellation can instantly monitor existing navigation signals. The optimization goal is to optimize the SBM coverage performance, specifically to use the least number of LEO satellites to achieve uniform coverage of the GNSS system under the condition that the multiple requirements for GNSS satellite coverage are determined. Multiple requirements for GNSS satellite coverage refer to the various performance metrics, such as coverage multiplicity and geometric dilution of precision (GDOP), that need to be met to ensure robust and reliable navigation services.  
The condition that the navigation signals of GNSS satellites can be monitored by LEO satellites is that GNSS satellites are visible to LEO satellites. The spatial geometric position relationship between GNSS satellites and LEO satellites is crucial for determining visibility. This relationship is depicted in Figure 6, which shows how LEO satellites must have an unobstructed line of sight to GNSS satellites for effective monitoring. In the Figure, the GNSS satellite is visible to the LEO satellite L1, but the GNSS satellite and the LEO satellite L2 cannot send and receive signals due to the Earth's obstruction, so there is no visibility between them.  
In summary, the conditions for judging visible GNSS satellites from low-orbit satellites are as follows:  
1. The line-of-sight vectors of low-orbit satellites and GNSS satellites are not blocked by the Earth;  
Figure 6: GNSS satellites and LEO satellites.  
2. The elevation angle of the receiving antenna of the low-orbit satellite is within the coverage range of the beam angle of the transmitting antenna of the GNSS satellite. When the ephemeris and respective antenna elevation angle constraints of the low-orbit constellation and the GNSS constellation are known, the visual situation between the LEO constellation and the GNSS constellation can be calculated through STK 11.6 software simulation or through the analytical algorithm [26]. This analytical algorithm considers satellite ephemeris data and antenna elevation constraints, which can ensure accurate visibility predictions by accounting for the dynamic positions and orientations of both LEO and GNSS satellites. Since satellite motion is periodic and deterministic, the visibility of the GNSS constellation from the LEO constellation at each moment can be calculated in advance.