## 4 Approach
### Iterative Sampling in the Neighborhood
We regard a single generation with a total number of attempts greater than 1000 and time consumed more than 1 s as a failure. The result is shown in Table 1, the success rate and time consumed in all cases are acceptable, which indicates that the method is effective. We then perform 100 runs for each algorithm in each case. _NumOfAlter_ in Algorithm 4 is 100, and the threshold for determining the appropriate end-effector path is 0.1. The stepsize of the mobile base is 0.01, and the stepsize of the manipulator is 0.1. For the whole path planning process, a single path planning taking more than 120 s is regarded as a failure. Table 2 shows the success rates of the different algorithms in different cases. In all cases, the proposed HSPP-pt method exhibits the best success rates. In contrast, RRT, IK, SP-1, and SP-2 fail to find a path in the given time in any case. For the RRT and IK methods, the main cause of failure is that they require a large amount of time, whil \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Success Rate** & \begin{tabular}{c} **Mean \# of** \\ **Samples** \\ \end{tabular} & \
\begin{tabular}{c} **Mean Time** \\ **Consumed [s]** \\ \end{tabular} &
\begin{tabular}{c} **Median \# of** \\ **Samples** \\ \end{tabular} \\ \hline C1G1 & 98.6\% & 159.64 & 0.48 & 115 \\ C1G2 & 99.8\% & 128.99 & 0.14 & 93 \\ C2G3 & 100\% & 92.25 & 0.31 & 63 \\ \hline \hline \end{tabular}

Except for cases and success rates, failed generations are not included.

\end{table} 
Table 1: Results of the SCG method. Figure 8: Simulation scenarios. (**a**) A complex 3D scenario. (**b**) A tunnel-like 3D scenario. methods, the main cause of failure is that they cannot find a collision-free path near the target configuration. For the HSPP method, different case difficulties do not significantly affect the success rate, whereas for the HSPP-t method, which only uses \(cost_{target}\), different settings affect its success rate due to the fact that it may fall into a local optimum solution at a particular configuration. Table 3 compares the average completion times of different algorithms. HSPP-t achieves faster search speeds in scenario \(C1\), while HSPP-pt performs better in scenario \(C2\). It can also be seen that, similar to the case of success rates, the time consumption of the HSPP-pt method is less affected by the different case settings and consumes a more even amount of time across cases, whereas the time consumption of the HSPP-t method is more affected. The time consumed by each part of the algorithm is illustrated in Figure 9. The SCG part consumes very little time and is negligible in relation to the total time consumed. The time consumed by the sampling in neighborhood for HSPP-t is much more than that of HSPP-pt, which means that \(cost_{target}\) improves the performance of the process. Although the HSPP-pt method increases the time consumed by adding a reference path planning part, it is comparable to the total time consumed by the HSPP-t method, and the HSPP-pt method has better stability and higher success rate. In addition, the HSPP-pt method can further reduce the time consumed by optimizing the reference path planning. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & **RRT** & **IK** & **SP-1** & **SP-2** & **HSPP-t** & **HSPP-pt** \\ \hline C1S1G1 & 0\% & 0\% & 0\% & 0\% & 80\% & 97\% \\ C1S1G2 & 0\% & 0\% & 0\% & 0\% & 73\% & 98\% \\ C1S2G1 & 0\% & 0\% & 0\% & 0\% & 79\% & 96\% \\ C1S2G2 & 0\% & 0\% & 0\% & 0\% & 72\% & 94\% \\ C2S1G3 & 0\% & 0\% & 0\% & 0\% & 81\% & 99\% \\ C2S2G3 & 0\% & 0\% & 0\% & 0\% & 80\% & 98\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of success rates.