## 3 Simulation and Analysis
### Simulation of the 6-DOF Manipulator  
In this section, the above algorithms are applied to a 6-DOF serial manipulator. As presented in Figure 8, the manipulator was surrounded by obstacles (blue solid blocks) mainly distributed in the upper and lower parts of the manipulator workspace. These two parts restricted the path of the manipulator through a tunnel. Each algorithm was implemented and applied to the manipulator to complete the task of passing through the tunnel from the initial configuration to the target configuration. Because the success rate of using uniform sampling for planning in high-dimensional spaces was almost zero, all the algorithms in this experiment adopted the sampling strategy with a 10% goal bias to maintain the consistency of the conditions. The step size \(\epsilon\) was set to \(2^{\circ}\), and the maximum number of failures n was set to 2000. \(k=15,\ c=2\) were set in the NC-RRT and CSA-RRT algorithms, and the average time obtained after 50 runs of each algorithm is presented in Table 4. It could be seen that compared with RRT with the 10% goal bias, the algorithm proposed in this paper significantly enhanced the efficiency.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & \begin{tabular}{c} **Average** \\ **Computational** \\ **Time (s)** \\ \end{tabular} & \begin{tabular}{c} **Average** \\ **Number of Nodes** \\ **in the Tree** \\ \end{tabular} & \begin{tabular}{c} **Average** \\ **Number of Collision** \\ **Detections** \\ \end{tabular} & \
\begin{tabular}{c} **Average** \\ **Path Length** \\ \end{tabular} &
\begin{tabular}{c} **The Success** \\ **Rate of the** \\ **Algorithm** \\ \end{tabular} \\ \hline Basic-RRT & 0.463 & 844.520 & 38,026 & 954.369 & 96\% \\ CSA-RRT & 0.058 & 92.900 & 4860.1 & 915.982 & 100\% \\ NC-RRT & 0.055 & 82.860 & 3941.7 & 977.667 & 100\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation results of the three algorithms in the cluttered environment. \((k=1,\ c=2)\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & \begin{tabular}{c} **Average** \\ **Computational** \\ **Time (s)** \\ \end{tabular} & \begin{tabular}{c} **Average** \\ **Number of Nodes** \\ **in the Tree** \\ \end{tabular} & \begin{tabular}{c} **Average** \\ **Number of Collision** \\ **Detections** \\ \end{tabular} & \
\begin{tabular}{c} **Average** \\ **Path Length** \\ \end{tabular} &
\begin{tabular}{c} **The Success** \\ **Rate of the** \\ **Algorithm** \\ \end{tabular} \\ \hline Basic-RRT & 0.471 & 928.680 & 35,821 & 865.236 & 84\% \\ CSA-RRT & 0.118 & 122.909 & 9705.3 & 805.751 & 88\% \\ NC-RRT & 0.159 & 126.817 & 9214.6 & 992.595 & 88\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation results of the three algorithms in the trapped environment. \((k=3,\ c=2)\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & \begin{tabular}{c} **Average** \\ **Computational** \\ **Time (s)** \\ \end{tabular} & \begin{tabular}{c} **Average** \\ **Number of Nodes** \\ **in the Tree** \\ \end{tabular} & \begin{tabular}{c} **Average** \\ **Number of Collision** \\ **Detections** \\ \end{tabular} & \
\begin{tabular}{c} **Average** \\ **Path Length** \\ \end{tabular} &
\begin{tabular}{c} **The Success** \\ **Rate of the** \\ **Algorithm** \\ \end{tabular} \\ \hline Basic-RRT & 0.550 & 1145.600 & 40,159 & 871.825 & 78\% \\ CSA-RRT & 0.106 & 172.732 & 8867.2 & 868.609 & 82\% \\ NC-RRT & 0.117 & 126.245 & 6746.4 & 942.615 & 98\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation results of the three algorithms in the narrow environment. \((k=1,\ c=2)\).