## 4 Performance Evaluation
### TRN Simulation Results  
Figure 18 shows latitude and longitude error graphs for TRN simulation case #1 and case #4. These graphs show 12 th simulation result among the 50 times MC simulations. Each graph shows the position error of batch processing. The position fix was performed every 40 s, and the position error at that time was indicated in each graph. The performance difference between case #1 and case #4 is not clearly seen in Figure 18a,c,d. On the other hand, the position error of case #4 was always larger than the position error of case #1 as shown in Figure 18b. We argued that the horizontal error of \(p_{k}\) causes the horizontal error of TRN in Section 2.2. In Figure 16, since \(\Delta p_{k,e}\) was larger than \(\Delta p_{k,n}\), the longitude error was relatively larger than the latitude error in the simulation result of the conventional method.  
Figure 19 compares latitude and longitude RMS errors obtained from the 50 times MC TRN simulation. Likewise, the longitude error of case #1 is smaller than that of case #2-#4 in the entire simulation of the conventional method as shown in Figure 19b. However, the performance difference between cases #1 to #4 of the proposed method is not clearly seen, as shown in Figure 19c,d.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline  & **Case \#1** & **Case \#2** & **Case \#3** & **Case \#4** \\ \hline IRA output used in simulation & All Data & \(\Delta p_{k}\)\textgreater{} 5 m & \(\Delta p_{k}\)\textgreater{} 10 m & \(\Delta p_{k}\)\textgreater{} 15 m \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation cases.  
Figure 18: TRN simulation result graphs: (**a**) latitude error for Conventional method; (**b**) longitude error for Conventional method; (**c**) latitude error for Proposed method; (**d**) longitude error for Proposed method.  
Table 4 shows the average latitude and longitude RMS errors for each TRN simulation case. The average RMS errors were calculated using the RMS errors at the time of the TRN position fix in Figure 19. In other words, data in the propagation period without the TRN position fix was excluded when calculating the average RMS errors. In the simulation result of the conventional method, the larger \(\Delta p_{k}\), the larger position RMS error. In the simulation result of the proposed method, the performance difference according to \(\Delta p_{k}\) is not clearly seen. In the simulation case #1, there wasn't a significant difference of position error between two methods. In the simulation cases 2-4, the position errors of the proposed method were smaller than that of the conventional method. From the results of the TRN simulation, it can be concluded that the proposed method calculates the closest point more accurately than the conventional method. The TRN performance is degraded by calculating the closest point using the conventional method at high altitude where the wind blowing from the side is strong. However, the proposed method does not degrade the TRN performance because it calculates the accurate position of the closest point even in high altitude and windy environments.  
Figure 19: Monte Carlo simulation result graphs: (**a**) latitude RMS error for Conventional method; (**b**) longitude RMS error for Conventional method; (**c**) latitude RMS error for Proposed method; (**d**) longitude RMS error for Proposed method.