## 4 Simulations  
The numerical simulations are conducted to evaluate the proposed path-smoothing algorithm. In the simulations, two sequences of waypoints are considered, as shown in Tables 1 and 2. The turning radius of the vehicle is set to \(R=30\) m, which corresponds to the maximum curvature constraint of \(\kappa_{max}=1/30\). In the first waypoint sequence, the waypoints are far away relative to the turning radius, but the second is not. The approaching direction to the final waypoint of both sequences is assumed to be the \(-y\) direction. Considering (24) or (25), when designing a Dubins reference path for Bezier curve approximation, the split turning angle is chosen as \(\theta_{s}=30^{0}\) and the base radius of the angular arc as \(R_{base}=34.872\) m to meet the maximum curvature constraint.  
The Dubins path and Bezier curve approximation path are generated from a series of waypoints in Table 1, as shown in Figure 14, and the horizontal and vertical plan views are shown in Figure 15. From the figures, it can be seen that both paths pass through all waypoints marked with 'o'.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Coordinate**} & \multicolumn{6}{c}{**Waypoints**} \\ \cline{2-7}  & **\#1** & **\#2** & **\#3** & **\#4** & **\#5** & **\#6** \\ \hline \(x(m)\) & 200 & 200 & 0 & 0 & 200 & 0 \\ \(y(m)\) & 0 & 200 & 200 & 400 & 400 & 0 \\ \(z(m)\) & 100 & 100 & 100 & 80 & 60 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Waypoints I.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Coordinate**} & \multicolumn{6}{c}{**Waypoints**} \\ \cline{2-7}  & **\#1** & **\#2** & **\#3** & **\#4** & **\#5** & **\#6** \\ \hline \(x(m)\) & 20 & 20 & 0 & 0 & 20 & 0 \\ \(y(m)\) & 0 & 20 & 20 & 40 & 40 & 0 \\ \(z(m)\) & 100 & 100 & 100 & 80 & 60 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Waypoints II.  
Figure 14: Three-dimensional Dubins path and Bezier approximation.  
Note that the Dubins path and Dubins reference path are different. The Dubins reference path is for designing Bezier curve approximation, and it is created by adjusting the circular arc radius as \(R_{base}\), whereas the Dubins path is designed with radius \(R\). Therefore, the trajectory difference seen in the figures should not be interpreted as trajectory estimation error.  
The Dubins path consists of the following sequence: RSL, LSR, LSR, LSR, and RSL, and has a length of 1351.5 m. The length of the Bezier curve approximation is 1371.0 m, which is a 19.5 m increase. However, the curvature of the Dubins path is discontinuous at the transition points, since the path directly transitions from line segments to circular arcs. In contrast, the Bezier curve approximation is a continuous curvature path satisfying maximum curvature constraints, as shown in Figure 16. The horizontal dashed line in the figure indicates the maximum curvature limit. The Bezier curve approximation is, therefore, a feasible path.  
Figure 16: Curvature.  
Figure 15: Horizontal and vertical plan views.  
Figure 17 is a partially enlarged view of Figure 16. It shows that the circular arc segment of the Dubins path is approximated by several Bezier curves to meet the maximum curvature requirement.  
For the sequence of waypoints in Table 2, the Dubins path and its Bezier curve approximation are shown in Figures 18 and 19. The Dubins path and its approximated Bezier paths look complicated because the waypoints are relatively close to the turning radius. However, both paths pass through all waypoints. The Dubins path consists of the following sequence: LSL, RSR, RSL, RSL, and RSL, and has a length of 1042.6 m. The length of the Bezier curve approximation is 1196.8 m, which is a 154.2 m increase.  
Figure 20 is the curvature plot of both paths, showing that only the Bezier curve approximation has continuous curvature and satisfies the maximum curvature requirement.