## 5 Computational Results  
To verify feasibility and effectiveness of planning with the proposed \(\mathbb{G}^{3}\)-continuous path primitive, it was investigated how it impacts computation times for the most crucial primitive operations in path planning, such as collision checking, checking of curvature-constraint satisfaction (\(\kappa\) checking), and extension procedure computation. Similarly, we tested the computational performance of our approach in path-planning scenarios S1-S3, shown in Figures 7-9. The proposed extend procedure was integrated with the RRT\({}^{\star}\) motion-planning algorithm (see Reference [23] for details). All simulations were performed with the following parameters: \(L=5.7\) m, \(b=2.5\) m, \(a=5.9\) m, \(\beta_{m}=0.96\) rad. We used motion cost \(J\triangleq\bar{I}+\int_{0}^{s_{fin}}\left(\frac{  
The computation times obtained in MATLAB are presented in Tables 2 and 3. One can observe that the \(\mathbb{G}^{3}\) path continuity of our approach comes with the price of increased computational cost in comparison to the classic RS paths with curvature discontinuities. This was expected, since one has to specifically account for additional transition segments during collision checking, and also check a larger amount of paths due to availability of additional combinations of transition segments and circle arcs. However, we note that the computational cost for the more general \(\eta^{3}\)-splines, which are also \(\mathbb{G}^{3}\)-continuous, is significantly higher than in our approach. This observation holds even if one assumes that numerical checking of curvature constraints and collision checking can be performed in parallel. Unsurprisingly, since collision-checking procedures can account for over 90% of planning time, those computational performance tendencies propagate to computation times of a full path-planning procedure for example Scenario S2. Such results suggest that our approach can represent a viable alternative when the planning of \(\mathbb{G}^{3}\)-continuous paths is necessary due to task-specific constraints or the mechanical construction of the robot. Figures 7 and 8 illustrate the results of path planning with the proposed \(\mathbb{G}^{3}\)-continuous extend procedure for parking Scenarios S1 and S2, whereas in Figure 9 we show a challenging lane-change-like maneuver in Scenario S3. During Scenario S3, only forward robot motion was assumed admissible. Obstacles are shown in black, whereas the planned path is in blue. The magenta rectangles correspond to robot footprints at the endpoints of path segments comprising \(\mathbb{G}^{3}\)-continuous path primitives, whereas the green and red rectangles illustrate the initial and final robot configuration, respectively. It can be seen that planning is successfully performed in a severely constrained environment despite the conservative approximation utilized in our collision-checking algorithm. Curvature profiles and \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Path Primitive** & **Collision Checking (\(\upmu\)s)** & \(\kappa\)**Checking (\(\upmu\)s)** & **Extend Procedure (\(\upmu\)s)** \\ \hline Reeds-Shepp & 24 & 0 & 27 \\ \(\eta^{3}\)-splines & 1397 & 1264 & 63 \\ \(\mathbb{G}^{3}\)**-continuous path primitive** & 124 & 0 & 67 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average computation times of primitive operations for 1000 random scenarios. \begin{table}
\begin{tabular}{c c} \hline \hline
**Path Primitive** & **Planning Time of S2 (S)** \\ \hline Reeds-Shepp & 27 \\ \(\eta^{3}\)-splines & 86 \\ \(\mathbb{G}^{3}\)**-continuous path primitive** & 39 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average computation times in MATLAB for 10 planning trials in Scenario S2. Figure 9: Planned path and curvature profile for Scenario S3 corresponding to a lane-change-like maneuver. curvature arc-length derivative profiles are continuous; however, in some cases, e.g., in Scenario S3, relatively low \(\mu\) values corresponding to relatively high curvature arc-length derivative values have been planned.