# Optimization Algorithm for 3D Smooth Path of Robotic Arm in Dynamic Obstacle Environments
## 4 Experiment Results
Based on the comprehensive analysis of the above experiments, it can be concluded that the strategy of combining adaptive angle guidance for the search tree with the back-to-source update algorithm enhances the goal-directedness of expanding new nodes in the random tree, reduces the trajectory length and path convergence time, and also optimizes the number of turning points in the trajectory after Catnull-Rom trajectory optimization. The verification method employed in this subsection's experiments involves a comparison between two scenarios. Scenario 1 features only static obstacles, while Scenario 2 adds additional dynamic obstacles to the static obstacles of Scenario 1. This setup is used to validate the effectiveness of local path replanning in both scenarios. Furthermore, to verify the robustness of the proposed algorithm against different speeds of dynamic obstacles, multiple experiments were conducted with the speed of dynamic obstacles as the controlled variable. Scenario 1 with static obstacles and Scenario 2 with dynamic obstacles are illustrated in Figures 11 and 12, respectively. In the dynamic obstacle scenario, the blue obstacles are in motion, while the rest are static. The dynamic adjustment of the weight factor \(\alpha\) is given as follows: \(\alpha\) values for nodes 1 to 4 are 0.214, 0.578, 0.682, and 0.315, respectively. \(Q_{\text{start}}\) and \(Q_{\text{end}}\) are located outside the protected area. Since the second node is closer to an obstacle, the \(\alpha\) value is greater than 0.5 (\(\alpha>0.5\)), indicating that the path expands more towards the RRT. This weakens the influence of the artificial potential field (APF), thereby affecting the path search direction. In contrast, the fourth node is farther away from obstacles, resulting in an \(\alpha\) value less than 0.5 (\(\alpha<0.5\)), which means that the path expands more towards the APF. Consequently, the influence of the APF is enhanced, guiding the path search direction towards areas with a higher probability of convergence. To verify the capability of local path replanning in a dynamic obstacle scenario, Figure 13 illustrates the real-time path adjustment when moving obstacles (blue) block the original path. The \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & \
\begin{tabular}{c} **Mean** \\ **Convergence Time** \\ \end{tabular} &
\begin{tabular}{c} **Average Number of** \\ **Turning Points** \\ \end{tabular} \\ \hline RRT* & 0.41579 & 1.267 \\ RRT-Connect & 0.20172 & 0.800 \\ Lazy-RRT & 0.27613 & 1.167 \\ Random Tree Guided Path & 0.19621 & 0.133 \\ Back-to-Source Update Algorithm & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of algorithm convergence time and path turning points. Figure 10: Comparison of motion paths of various algorithms. local path replanning effectively avoids new obstacles, making the path smoother and safer, demonstrating the algorithm's adaptability and robustness in complex environments. To verify the robustness of the hybrid local path replanning algorithm combining rapidly exploring random tree (RRT) and artificial potential field (APF) proposed in this study under dynamic environments, it is necessary to evaluate its performance against dynamic obstacles of different speeds, positions, and quantities. First, the performance of the replanning algorithm in response to obstacles moving at different speeds is statistically analyzed. The path replanning times for dynamic obstacles with velocities of 5 cm/s, 10 cm/s, and 20 cm/s are 2.43 ms, 3.76 ms, and 4.59 ms, respectively. Then, following the experiment in Scenario 2 with only one dynamic obstacle, we proceed to verify the case with two dynamic obstacles, as shown in Figure 14. The left  
Figure 11: Static obstacle scene and trajectory. Figure 12: Initial path in dynamic obstacle scene. Figure 13: Local path replanning in dynamic obstacle scene. image is a detailed view of \(Q_{\text{start}}\) and \(Q_{\text{end}}\) in the local path replanning algorithm, while the right image shows the final replanned path.