## 4 Experimental Results

Table 1: Performance comparison between Expanded Douglas–Peucker (EDP) algorithm and Douglas–Peucker (DP) algorithm for various \(\epsilon\) values (The number of obstacles and vertices of each obstacle were fixed as 15, and \(\theta_{rot}\) for EDP was fixed as 30\({}^{\circ}\)). EDP is completely zero and better than DP since the result of the EDP can cover the entire area of the original obstacle regions. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{\(\epsilon\) (m)} & \multicolumn{2}{c}{**Average Number of Vertices**} \\ \cline{2-3}  & **EDP** & **DP** \\ \hline
0.05 & 31.08 & 15.92 \\
0.08 & 27.87 & 11.92 \\
0.11 & 25.89 & 10.19 \\
0.14 & 24.68 & 9.18 \\
0.17 & 23.67 & 8.33 \\
0.20 & 22.85 & 7.66 \\
0.23 & 22.13 & 7.32 \\
0.26 & 21.55 & 6.83 \\
0.29 & 20.97 & 6.56 \\
0.32 & 20.15 & 6.20 \\
0.35 & 19.32 & 5.91 \\
0.38 & 18.97 & 5.58 \\
0.41 & 18.63 & 5.40 \\
0.44 & 18.06 & 5.06 \\
0.47 & 17.65 & 5.00 \\
0.50 & 17.16 & 4.78 \\ \hline AVG & 21.91 & 7.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average number of vertices created by EDP and DP (The numbers of obstacles and vertices of each obstacle were fixed as 15, and \(\theta_{rot}\) for EDP was fixed as 30\({}^{\circ}\)). Figure 7: Performance comparison between EDP and DP for various \(\varepsilon\) values: (**a**) Ratio of the inner space of the approximated polygons outside the obstacle region (IA); (**b**) Ratio of the outer space of the approximated polygons inside the obstacle region (OA); (**c**) Sum of IA and OA (SA); (**d**) Average number of vertices created by EDP and DP (The numbers of obstacles and vertices of each obstacle were fixed as 15, and \(\theta_{rot}\) for EDP was fixed as 30\({}^{\circ}\)). Table 3 compares the performance of the modified OAECD algorithm and VCD algorithm. Here, NV is the total number of vertices in the connectivity graph, VV is the number of vertices that composes the generated path by the A* algorithm, and PL is the path length. From Table 3, the path length by the modified OAECD algorithm is 60.7% shorter than the path length by VCD algorithm on average. Figure 8 shows the comparison of VCD with DP and OAECD with EDP to show the feasibility of the modified OAECD algorithm for maps with static curvilinear obstacles. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**(NO, NV)**} & \multicolumn{3}{c}{**OAECD**} & \multicolumn{3}{c}{**VCD**} \\ \cline{2-7}  & **NV (ea)** & **VV (ea)** & **PL (m)** & **NV (ea)** & **VV (ea)** & **PL (m)** \\ \hline (1,10) & 7.60 & 4.30 & 28.43 & 12.00 & 7.00 & 29.27 \\ (3,10) & 19.80 & 6.00 & 29.16 & 36.60 & 16.50 & 33.43 \\ (5,10) & 31.70 & 10.20 & 29.85 & 61.80 & 24.70 & 37.84 \\ (7,10) & 44.10 & 10.60 & 30.37 & 88.20 & 29.90 & 45.89 \\ (9,10) & 56.80 & 11.90 & 29.68 & 111.80 & 34.60 & 47.90 \\ (11,10) & 68.50 & 13.40 & 30.88 & 139.00 & 36.80 & 53.64 \\ (13,10) & 79.20 & 14.90 & 30.62 & 164.70 & 43.80 & 56.38 \\ (15,10) & 92.50 & 17.60 & 30.92 & 192.00 & 45.70 & 48.31 \\ (15,3) & 40.70 & 9.80 & 28.88 & 76.60 & 28.90 & 57.03 \\ (15,5) & 54.90 & 11.10 & 29.72 & 108.00 & 29.80 & 51.94 \\ (15,7) & 69.20 & 13.80 & 29.83 & 141.20 & 37.70 & 59.94 \\ (15,9) & 84.70 & 15.70 & 30.48 & 174.60 & 43.50 & 54.79 \\ (15,11) & 100.60 & 17.60 & 30.82 & 207.80 & 43.00 & 53.86 \\ (15,13) & 116.10 & 21.40 & 31.26 & 249.50 & 46.90 & 58.88 \\ (15,15) & 130.60 & 20.30 & 31.32 & 279.20 & 43.30 & 55.28 \\ \hline AVG & 66.47 & 13.24 & 30.15 & 136.20 & 34.14 & 49.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison between modified Opposite Angle-Based Exact Cell Decomposition (OAECD) and Vertical Cell Decomposition (VCD) (The numbers of obstacles and vertices of each obstacle were fixed as 15). Figure 8: Comparison between OAECD with EDP and VCD with DP (\(\varepsilon\) = 0.05 m): (a) Path planning using EDP and OAECD; (b) Path planning using DP and VCD (The numbers of obstacles and vertices of each obstacle were fixed as 15, and \(\theta_{\text{ref}}\) for EDP was fixed as 30\({}^{\circ}\)).