## 5 Conclusions  
Firstly, the LEO navigation constellation adopts a segmented optimization method. The tilt constellation and polar constellation are designed to meet the navigation performance requirements for middle latitude and polar regions, respectively. Taking into account satellite constellation altitude, inclination, navigation performance, cost, and scalability, the optimal tilt constellation construction scheme is Walker 216/18/1:150/50. Similarly, the optimal polar constellation construction scheme is Walker 72/8/1:1176/85. The average PDOP for tilt and polar constellations are 2.0 and 1.96, respectively, satisfying latitude constraints. The average PDOP of the designed LEO hybrid navigation constellation is 1.676--outperforming traditional navigation constellations. The performance and configuration of the LEO hybrid navigation constellation meet the design requirements effectively. Furthermore, considering the diverse failure modes of three satellites in the MEO navigation constellation and the extensive manual screening workload, the Monte Carlo method was employed to analyze satellite failures. The findings indicate that in three-satellite failure mode, a single-satellite failure within each orbital plane and the phase of adjacent satellites is relatively close, significantly impacting the overall constellation performance. To identify the most impactful satellite failure mode on the MEO navigation constellation's performance, simulating the model of a single-satellite failure within each orbital plane can reduce computational costs and offer insights for constellation backup. This approach is adaptable to various constellation configurations and four or more satellite failure modes. The staged deployment strategy accounts for the potential degradation of constellation service performance resulting from the failure of MEO navigation satellites. The LEO hybrid navigation constellation is deployed in three stages, each tailored to different mission requirements and aimed at steadily improving overall constellation performance. In the initial stage, 66 satellites are deployed to compensate navigational performance loss following the failure of MEO satellites. Subsequently, in the second stage, deployment involves 156 satellites, with a target PDOP of 1.456 to enhance navigation capabilities in response to more demanding needs. Finally, a total of 288 satellites are deployed in third stage. With an average PDOP of 1.676 achieved by this stage, an independent system capable of operating autonomously is established to provide comprehensive PNT services as a backup navigation system. In conclusion, the LEO hybrid navigation constellation proposed in this study and its deployment strategy exhibit promising potential in terms of constellation navigation  
Figure 16: Staged deployment of PDOP in various scenarios for LEO hybrid navigation constellations. performance, cost effectiveness, and scalability. Future research should explore additional types of constellations, with an expectation that this study will offer valuable insights and practical applications for LEO constellation technology. In practical engineering, as the positioning quality is also related to range measurement quality, it also requires a high-quality ranging payload on LEOs. LEO satellites are developing in the direction of low cost and high integration. Conceptualization--W.X., M.H., and Y.R. ; funding acquisition--M.H. ; investigation--W.X., X.W., and M.Y. ; methodology--W.X. ; writing--original draft, W.X. ; writing--review and editing, M.H., Y.R., X.W., and M.Y. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (Grant No. 61403416). The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.