## 3 Results
### Optimization Results and Analysis
#### 3.1.3 Optimization Design of LEO Tilt Constellation  
The LEO hybrid constellation adopts a layered design. On the one hand, considering the overall optimization will make the design space larger but will also increase the computation load exponentially. On the other hand, considering that the different hierarchical constellations undertake different navigation tasks, step-by-step optimization can be the first to serve the more populated mid-latitude regions and then to meet the polar service area. The optimization process can consider the constellation configuration, expandability, and redundancy.  
The constellation parameters were configured with the minimum observation elevation \(\gamma=5^{\circ}\), grid \(5^{\circ}\times 5^{\circ}\), and a global navigation service area. The selection of the orbital plane and the number of satellites in the tilt constellation is based on predetermined conditions, while the optimization parameter upper and lower limits are as follows:  
\[\left\{\begin{array}{l}X_{low}=\{1100,45,10,8\}\\ X_{up}=\{1200,60,20,15\}\\ 60^{\circ}\\ s.t.\quad PDOP^{\textit{unlarkor}_{LEO}}_{-60^{\circ}}\leq 2.0\\ \end{array}\right. \tag{12}\]  
By fine-tuning the algorithm parameters, a population size of 50 and a maximum evolutionary generation of 500 were selected. The hypervolume (HV) was employed as a metric for diversity assessment [26], where larger HV values indicate superior solution sets. The optimization results and the HV evaluation index are depicted in Figure 7.  
Figure 7a utilizes the PDOP and total number of satellites in the LEO tilt constellation as the objective function, with the average PDOP of the BDS-3 constellation between \(60^{\circ}\) north and south latitude serving as the constraint function. The average PDOP constraint function is approximately 2.0. For a Walker 216/18/1 constellation configuration, the average PDOP is around 2.0. As the total number of constellation satellites increases, for a Walker 280/20/1 configuration, the average PDOP decreases to about 1.5; however, this also leads to increased costs associated with constellation deployment, operation, and control. Considering performance, cost, configuration, and scalability factors, Walker 216/18/1 was chosen as the LEO tilt constellation configuration.  
Figure 6: Average number of visible satellites.  
Constellation performance refers to the PDOP, i.e., cost simplified as the total number of satellites in the constellation. The constellation configuration and scalability need to consider the selection of the orbital plane and the number of satellites per orbital plane in the constellation optimal design. Scalability is expressed, as the sub-constellation should be expanded from the previous sub-constellation and has the ability to expand to the next sub-constellation. In Figure 7b, the evaluation index HV converges in the 300 th generation, signifying the feasibility of the optimization results.  
Table 1 presents five optimization results of the LEO tilt constellation scheme. The number of coverage reps across different schemes is approximately 11, with a variation in PDOP of around 0.17. Taking into account the altitude and inclination of the constellation comprehensively, Scheme 3, i.e., Walker 216/18/1:150/50 (\(N=216\), \(P=18\), \(F=1\), \(h=1150\) km, \(i=50\)), was chosen as the deployment and construction scheme for the LEO tilt constellation. Other solutions served as backup options.