# RRT*-Fuzzy Dynamic Window Approach (RRT*-FDWA) for Collision-Free Path Planning
## 5 Experimental Results
### Experimental Comparison  
In this paper, we used a two-wheel differential speed robot for the simulation. The maneuverability degree, \(\delta_{m}\), of this mobile robot can, therefore, be obtained as two, and according to Equation (18), \(rank\{C_{a}(\beta_{a})\}\) is the number of maneuverable wheels. \[\delta_{m}=3-rank[C_{a}(\beta_{a})] \tag{18}\]  
In dynamic obstacle avoidance, robot behavior includes deceleration or acceleration to avoid obstacles when it encounters them, but such behaviors may cause damage to safety. In this paper, the behavior of avoiding obstacles is proposed as the standard deviation formula for speed, the smaller value of which represents higher safety, as shown in Equation (19), where \(v_{i}\) indicates the line speed at the current moment. The integrated speed difference formula presents the maneuvering magnitude, which indicates the robot's evasion ability when encountering obstacles, and the higher its value, the better the evasion ability, as shown by Equation (20). \[\sigma_{w}^{2}=\sqrt{\frac{\sum_{i=1}^{n}\left(v_{i}-v\right)^{2}}{n}} \tag{19}\]  
\[\delta_{r}=\sigma_{w}^{2}\delta_{m}/\theta_{d}\times 100\% \tag{20}\]  
Figure 9 shows the RRT*-FDWA path planning process; the solid red circles indicate the current position and solid blue circles indicate \(p_{new}\). Hollow circles indicate the previous moment position and the planned route and dashed hollow circles indicate the moving obstacle at the previous moment. Figure 9: RRT*-FDWA path planning diagram. (**a**) Position planning diagram at the moment \(t=1\). (**b**) Position planning diagram at the moment \(t=2\). (**c**) Position planning diagram at the moment \(t=5\). (**d**) Position planning diagram at the moment \(t=7\). Figure 10 shows the RRT*-FDWA experimental path planning diagram, and the experimental results show that the algorithm is effective in practical applications. The RRT*-FDWA algorithm was packaged into the path planner in the ROS navigation package before experimental validation. In the figure, the basketball and the white mobile robots are treated as moving obstacles and the experimental environment was the same as the simulation experiment. In the real experimental process, there was an error in the path planning process due to the robot control drive factor. Figure 10: _Cont._  
The RRT*-FDWA algorithm proposed in this paper was compared to the hybrid A*--an improved DWA algorithm--the hybrid algorithm being SOTA. Figure 11 shows the final completed path planning graph for both algorithms, the red dot [1,1] is the starting point and the green dot [9,9] is the target point; it can be seen that the A*-DWA algorithm still falls into local minima, while the RRT*-FDWA algorithm can complete the path planning more smoothly. It can be seen from Table 3 that the \(\sigma_{w}^{2}\) of the RRT*-FDWA algorithm is smaller and \(\delta_{r}\) is 12% higher than those of the A*-DWA algorithm, and it takes less time to complete. This indicates that the RRT*-FDWA algorithm is safer, fits better, and plans faster. Figure 11: Algorithm comparison procedure. \begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{\sigma_{w}^{2}}\) & **Time (s)** & \(\delta_{r}\) \\ \hline A*-DWA & 0.548 & 11.93 & 79\% \\ RRT*-FDWA & 0.424 & 9.62 & 91\% \\ \hline \end{tabular}
\end{table} 
Table 3: Algorithm efficiency test comparison. Figure 10: RRT*-FDWA real environment path planning map. (**a**) The robot position map at the moment \(t=0\). (**b**) The robot position map at the moment \(t=3\). (**c**) The robot position map at the moment \(t=5\). (**d**) The robot position map at the moment \(t=7\). (**e**) The robot position map at the moment \(t=9\). (**f**) The robot position map at the moment \(t=13\). (**g**) The robot position map at the moment \(t=16\). (**h**) The robot position map at the moment \(t=18\).