## 5 Simulation Analysis
### Simulation Results  
#### 5.2.1 Without Altimeter Scenario  
The single-satellite navigation and positioning results in the scenario without an altimeter are shown in Figure 6. We also give the navigation and positioning results of the LEO 3 satellites + LS algorithm as a reference. Figure 6 shows that, without the aid of an altimeter, the positioning error, velocity error, and navigation trajectory error of LEO 1 satellite + INS are smaller than those in INS alone. To quantitatively analyze and evaluate the positioning error of the LEO 1 satellite + INS, we chose the mean value and standard deviation (STD) to describe the accuracy performance of the algorithm. Tables 2-4 show the statistical data of the positioning results of the LEO 1 satellite + INS and INS navigation and the standard LEO 3 satellites + LS algorithm. Among them, NPE stands for north position error, NVE stands for north velocity error, and the meaning of the other acronyms can be deduced by analogy. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Orbit Parameter Type** & **Value** & **Error Parameter Type**[28; 32] & **Value** \\ \hline Period (min) & 110 & Signal in space error (m) & 0.02 \\ Altitude (km) & 1000 & Zenith ionosphere error (m) & 0.02 \\ Orbit radius (km) & 7515 & Zenith troposphere error (m) & 0.002 \\ Inclination (\({}^{\circ}\)) & 99.5 & Code tracking error (m) & 0.1 \\ Number of orbit surfaces & \textgreater{}6 & Range rate tracking error (m/s) & 0.002 \\ Number of satellites per orbit & 12 & Pseudorange measurement noise (m) & 0.025 \\ Total number of satellites & 117 & Pseudorange rate measurement noise (m/s) & 0.001 \\ User terminal elevation angle (\({}^{\circ}\)) & 20 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The main parameters of the Telesat system. Figure 6: Single-satellite navigation and positioning results in the scenario without an altimeter. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Error** & **Algorithm** & **Mean** & **STD** \\ \hline \multirow{3}{*}{NPE (m)} & INS & \(-47.6103\) & \(53.1964\) \\  & LEO 3 satellites & \(0.0134\) & \(0.1354\) \\  & INS+LEO 1 satellite & \(-32.6619\) & \(21.4059\) \\ \hline \multirow{3}{*}{EPE (m)} & INS & \(67.7172\) & \(93.1811\) \\  & LEO 3 satellites & \(0.0276\) & \(0.1080\) \\  & INS+LEO 1 satellite & \(-41.9907\) & \(46.4180\) \\ \hline \multirow{3}{*}{DPE (m)} & INS & \(-104.1891\) & \(93.3793\) \\  & LEO 3 satellites & \(0.0021\) & \(0.1105\) \\ \cline{1-1}  & INS+LEO 1 satellite & \(-70.4270\) & \(79.3741\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Position error statistics. From Tables 2-4, compared with INS navigation, we find the following:  
1. In terms of the position error, the position error accuracy of the INS + LEO 1 satellite in the north direction is increased by 31.40%, and the STD is increased by 59.76%; the position error and STD accuracy in the east direction are increased by 37.99% and 50.19%, respectively; and the position error accuracy and STD of the down direction are improved by 32.40% and 15.00%, respectively;
2. The accuracy of the INS + LEO 1 satellite in the north direction is improved by 57.34%, and the STD is increased by 70.58%; the velocity error and STD accuracy in the east direction are increased by 28.31% and 37.63%, respectively; and the velocity error accuracy in the earth direction is increased by 33.17%, while the STD is reduced by 12.71%;
3. In the final trajectory error, the mean error accuracy of the longitude error is improved by 10.59% and the STD is increased by 38.12%. Since IMU is constantly correcting the navigation error, there are some fluctuations in the latitude error. This result can be seen in Figure 6c. Therefore, the mean and STD of the latitude decrease significantly, but the average accuracy of the altitude error is improved by 32.40% and the STD is increased by 15.00%.