## 4 Experimental Results
In terms of appearance, the one to which the post triangular processing of midpoint interpolation method (\(\varepsilon\): 10 px) was applied seems to have a smoother slope compared to those of the other algorithms, and the path length with the method (\(\varepsilon\): 10 px) seems to be the shortest. The path planning results (average values after repeating the trial 100 times) are shown in Table 4 for Map 3 among the specified environment maps for each algorithm. By applying the post triangular processing of midpoint interpolation method (\(\varepsilon\): 10 px), the path length becomes 82% (505/613(%)) compared to the RRT, which is the shortest of all  
Figure 11: Experimental results of Map 3: (**a**) RRT; (**b**) Proposed method (\(\varepsilon\): 50 px) applied; (**c**) Proposed method (\(\varepsilon\): 30 px) applied; (**d**) Proposed method (\(\varepsilon\): 10 px) applied. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Performance** & **RRT** & \begin{tabular}{c} **Proposed Method** \\ **(\(\varepsilon\): 50 px)** \\ \end{tabular} & \
\begin{tabular}{c} **Proposed Method** \\ **(\(\varepsilon\): 30 px)** \\ \end{tabular} &
\begin{tabular}{c} **Proposed Method** \\ **(\(\varepsilon\): 10 px)** \\ \end{tabular} \\ \hline Path length (px) & 986 (100\%) & 780 (7\%) & 752 (76\%) & 730 (74\%) \\ Planning time (ms) & 10 (100\%) & 10 (100\%) & 11 (110\%) & 11 (110\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental results of Map 2 (numbers in parentheses (averages of repeating trial 100 times) are relative ratios to RRT results (values less than 1 are counted as 1)). the algorithms, and the planning time is the most similar to that of the method (\(\epsilon\): 10 px), as it takes 16% more time than the RRT algorithm. However, the absolute time difference is 2 ms, and since it is an environment that requires less planning time, the difference appears to be large when the method is applied. The path planning results are shown in Figure 12 for Map 4 among the specified environment maps for each algorithm. In terms of appearance, the one to which the post triangular processing of the midpoint interpolation method (\(\epsilon\): 30 px) was applied seems to have a smoother slope compared to those of the other algorithms, and the path length with the method (\(\epsilon\): 10 px) seems to be the shortest. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Performance** & **RRT** & \begin{tabular}{c} **Proposed Method** \\ **(\(\epsilon\): 50 px)** \\ \end{tabular} & \
\begin{tabular}{c} **Proposed Method** \\ **(\(\epsilon\): 30 px)** \\ \end{tabular} &
\begin{tabular}{c} **Proposed Method** \\ **(\(\epsilon\): 10 px)** \\ \end{tabular} \\ \hline Path length (px) & 613 (100\%) & 535 (87\%) & 512 (83\%) & 505 (82\%) \\ Planning time (ms) & 6 (100\%) & 7 (116\%) & 8 (133\%) & 8 (133\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Experimental results of Map 3 (numbers in parentheses (averages of repeating 100 times) are relative ratios to the RRT results (values less than 1 are counted as 1)). Figure 12: Experimental results of Map 4: (**a**) RRT; (**b**) Proposed method (\(\epsilon\): 50 px) applied; (**c**) Proposed method (\(\epsilon\): 30 px) applied; (**d**) Proposed method (\(\epsilon\): 10 px) applied. The path planning results (average values after repeating the trial 100 times) are shown in Table 5 for Map 4 among the specified environment maps for each algorithm. By applying the post triangular processing of the midpoint interpolation method (\(\varepsilon\): 10 px), the path length becomes 77% (1190/1536(%)) compared to the RRT, which is the shortest of all the algorithms, and all the planning times are similar. Consequently, the post triangular processing of the midpoint interpolation method (\(\varepsilon\): 10 px) performed well in the path length aspect for all maps, demonstrating that the proposed method is efficient in terms of optimality.