# Navigation Resource Allocation Algorithm for LEO Constellations Based on Dynamic Programming
## 2 Model Analysis of LEO Constellation Navigation Resource Allocation
### Constraints and Objective Functions  
From the above analysis, it can be seen that the LEO constellation navigation enhancement system has both navigation signal transmission and SBM functions, and its resource allocation algorithm must comprehensively consider the navigation signal transmission and SBM performance. For navigation signal transmission performance, the optimization goal is the global coverage of the constellation. First, you need to select the ground stations used for constellation coverage assessment. If there is \(P\) a total ground station, the ground station can be expressed as \(SN_{r}\), \(r=1,2,\cdots,P\). Suppose there are \(N\) LEO satellites in a LEO constellation; then, the LEO satellite can be expressed as \(SL_{i}\), \(i=1,2,\cdots,N\). If a LEO satellite \(SL_{i}\) and a ground station are visible \(SN_{r}\) at a certain moment \(t\), then there is a visible link between them, defined as \(V(t,SL_{i},SN_{r})\). Express the complete return period of a LEO constellation as \(T_{1}\). When the satellite constellation is determined, its trajectory is periodic and fixed. Therefore, the visibility of ground stations to LEO satellites within the time period \(T_{1}\) encompasses the visibility of ground stations to LEO satellites at any given time. All visible links between LEO satellites and ground stations within the time period \(T_{1}\) constitute the visible link set of the navigation signal broadcasting network, defined as \(VS_{1}\), and can be expressed as  
\[VS_{1}=\left\{V_{1}(t,SL_{i},SN_{r})\right\}_{t\in T_{1},i\in[1,2,\cdots,N],r \in[1,2,\cdots,P]} \tag{1}\]  
Some links are selected from the visible link set \(VS_{1}\) as navigation signal transmission links. All links used for navigation signal broadcasting within the time period \(T_{1}\) constitute the activated link set of the navigation signal broadcasting network, which is defined as \(WS_{1}\), and can be expressed as  
\[WS_{1}=\left\{V_{2}(t,SL_{i},SN_{r})\right\}_{t\in T_{1},i\in[1,2,\cdots,N],r \in[1,2,\cdots,P]} \tag{2}\]  
The constraints of the navigation signal transmission network are as follows:  
(1) The navigation signal transmission link must first be a visible link; that is, the activated link set of the navigation signal transmission network is a subset of the visible link set, expressed as  
\[WS_{1}\subseteq VS_{1} \tag{3}\]  
(2) The different requirements for navigation signal enhancement impose varying demands on global constellation coverage. For instance, achieving independent four-satellite positioning similar to the GNSS system requires global coverage of at least quadruple. Doppler positioning, similar to the Iridium system, only requires single global coverage. Joint positioning with GNSS satellite signals does not have explicit requirements for global coverage. To reduce computational complexity, this paper analyzes the constellation with the goal of achieving single global coverage as an example. In this condition, there is at least one navigation signal transmission link for any time \(t\) and any ground station \(SN_{k}\), which can be expressed as  
\[\big{|}\big{\{}V\big{(}t_{0},SL_{i_{0}},SN_{r_{0}}\big{)}\big{\}}\big{|}\geq 1, \quad V\big{(}t_{0},SL_{i_{0}},SN_{r_{0}}\big{)}\in WS_{1},\forall r_{0}\in[1,2,\cdots,P],\forall t_{0}\in T_{1} \tag{4}\]  
where \(|\cdot|\) represents the number of elements in the set. The set of LEO satellites \(US_{1}\) used for navigation signal transmission is expressed as  
\[US_{1}=\big{\{}SL_{i_{0}}\big{\}},\ \ i_{0}\in[1,2,\cdots,N] \tag{5}\]  
For SBM performance, the optimization goal is the coverage of GNSS constellations. Assuming that there are \(M\) GNSS satellites, then the GNSS satellites can be expressed as \(SG_{j}\), \(j=1,2,\cdots,M\). If a LEO satellite \(SL_{i}\) and a GNSS satellite \(SG_{j}\) are visible at a certain moment \(t\), then there is a visible link between them, defined as \(V\big{(}t,SL_{i},SG_{j}\big{)}\).