## 5 Simulation Analysis
### Discussion  
According to the above simulation and analysis results, we will present a brief analysis and discussion, as follows:  
1. Compared with INS navigation, the usability of our algorithm is obviously better than INS in the scenario without an altimeter and in the scenario with an altimeter,  
\begin{table}
\begin{tabular}{c c c c} \hline
**Error** & **Scene** & **Mean** & **STD** \\ \hline \hline \multirow{2}{*}{NPE (m)} & altimeter without deviation & \(-6.6661\) & 7.8891 \\  & reference [20] & 247.8372 & 534.3183 \\ \hline \hline \multirow{2}{*}{EPE (m)} & altimeter without deviation & \(-9.5780\) & 17.4749 \\  & reference [20] & 254.5632 & 485.8885 \\ \hline \hline \multirow{2}{*}{DPE (m)} & altimeter without deviation & \(-8.1980\times 10^{-4}\) & 0.0182 \\  & reference [20] & 679.7182 & 1424.0361 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Statistics of position error.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Error** & **Scene** & **Mean** & **STD** \\ \hline \hline \multirow{2}{*}{Lon (\({}^{\circ}\))} & altimeter without deviation & \(-5.9788\times 10^{-5}\) & 7.0759 \(\times 10^{-5}\) \\  & reference [19] & 0.1025 & 0.0004 \\ \hline \hline \multirow{2}{*}{Lat (\({}^{\circ}\))} & altimeter without deviation & \(-1.3456\times 10^{-4}\) & 2.4540 \(\times 10^{-4}\) \\  & reference [19] & 0.0082 & 0.0011 \\ \hline \hline \multirow{2}{*}{Alt (m)} & altimeter without deviation & \(8.1980\times 10^{-4}\) & 0.0182 \\  & reference [19] & \(-0.7636\) & 15.1179 \\ \hline \hline \end{tabular}
\end{table}
Table 12: Statistics of the trajectory.  
but in the scenario without an altimeter, the variance is relatively large, and there is the problem of INS divergence;
2. To overcome the problem of INS divergence, we have introduced an altimeter to improve this problem. We can find that, in the scene of an unbiased altimeter, our algorithm can significantly improve the position and velocity accuracy in all directions and the final trajectory accuracy, especially the altitude accuracy. In addition, even if the altimeter exhibits deviation, the positioning accuracy can reach a cm level, and the divergence of INS can be effectively overcome;
3. Compared with the scene without an altimeter, for the algorithm in the scene of an altimeter, the position accuracy, velocity accuracy, and final trajectory accuracy are significantly improved, which shows that in the specific engineering application, the introduction of an altimeter will improve the robustness of the whole navigation system, especially the barometric altimeter with smaller altitude deviation;
4. Compared with the MEO constellation, the navigation trajectory of LEO constellation is closer to the real trajectory of aircraft, and it can effectively overcome the divergence problem of INS, so the LEO constellation is more suitable for the preferred constellation of the PNTRC scheme;
5. Compared with the traditional algorithm, each index accuracy has improved by more than 95%, which is enough to explain the progressiveness of our algorithm, because it can achieve such a high positioning accuracy without continuous observation, and there is no positioning delay, so it satisfies the real-time, high-precision positioning needs in challenging environments.