## 3 Results
### Optimization Results and Analysis
#### 3.1.4 Optimization Design of LEO Polar Constellation  
Both LEO tilt constellations and LEO polar constellations can be used as independent constellations to provide navigation services to mid-latitudes as well as high latitudes, while LEO hybrid navigation constellations can provide global services. From Equation (8), the optimization range for satellite inclination within a polar constellation is illustrated in Figure 8.  
As depicted in Figure 8, the optimal inclination range for the polar Walker is determined through the LEO tilt constellation setup. Here, \(walker_{1}\) represents the LEO tilt constellation, and \(walker_{2}\) denotes the LEO polar constellation. The simulation analysis considers a total of 300 satellites in \(walker_{1}\) with an inclination of approximately 50\({}^{\circ}\), while for the \(walker_{2}\) polar constellation, there are about 40 to 120 satellites, with an inclination range of around 64\({}^{\circ}\)-90\({}^{\circ}\). When \(S=2\), there exist an infinite number of hybrid Walker configurations with polar Walker configurations. After introducing one Walker configuration, the orbital inclination angle of the other Walker configuration should be appropriately adjusted based on the minimum GDOP of a single Walker configuration. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Scheme 1** & **Scheme 2** & **Scheme 3** & **Scheme 4** & **Scheme 5** \\ \hline Altitude \(h/\)km & 1190 & 1176 & 1150 & 1125 & 1110 \\ Inclination \(i/^{\circ}\) & 45 & 49 & 50 & 52 & 53 \\ Number of coverage & 11.88 & 11.64 & 11.48 & 11.28 & 11.11 \\ PDOP & 2.0072 & 1.8365 & 1.8498 & 1.9084 & 1.9494 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: LEO tilt constellation scheme. Figure 7: Optimization results of LEO tilt constellation: (**a**) optimization results and (**b**) evaluation index. Hence, the optimization parameters for an LEO polar constellation are constrained by upper and lower limits:  
\[\left\{ \begin{array}{l}X_{low}=\{1100,80,5,8\}\\ X_{up}=\{1200,90,12,10\}\\ \end{array} \right. \tag{13}\] \[s.t.\quad PDOD_{mean}^{\text{Hulker}_{LEO+Polar}}\leq 2.0\]  
where the PDOP constraint condition for a single constellation is modified to accommodate a hybrid constellation. The constraint condition is set at 65\({}^{\circ}\)-90\({}^{\circ}\) north and south latitude, with settings consistent with those in the previous section for other scenarios. Additionally, polar constellations featuring different configurations of LEO tilt constellations are presented along with optimization results and HV in Figure 9.  
Based on the configuration of the tilt constellation Walker 216/18/1:1115/50, optimization is performed for the LEO polar constellation. Figure 9(a) utilizes the PDOP of the hybrid constellation and the total number of satellites in the LEO polar constellation as the optimization objectives while imposing a constraint function based on the average PDOP (approximately 2.0) within a latitude range of 65\({}^{\circ}\)-90\({}^{\circ}\) for BDS-3 constellation configuration. The average PDOP for Walker72/8/1 configuration is approximately 1.96, and it increases to about 1.62 with an increase in total satellites for Walker 100/10/1 configuration. Considering performance, cost, constellation configuration, and scalabi  
Figure 8: Inclination optimization range of polar constellation. Figure 9: Optimization results of LEO polar constellation: (**a**) optimization results and (**b**) evaluation index. chosen as the optimal LEO polar constellation configuration. In Figure 9b, the evaluation index HV converges at generation 350, indicating the feasibility of the optimization results. Table 2 presents the optimized LEO polar constellation scheme for an LEO tilt constellation configuration. The number of coverage of the five schemes is approximately 13, with a variation in PDOP of around 0.1491.