## 4 Simulation Experiment and Results
### Case 1: Simple Scenario
#### 4.2.1 Simulation Results  
The calculation results of the five algorithms for the Case 1 simple scenario are shown in Table 3. In Table 3, \(t_{cpu}\) is the CPU running time for different algorithms computation. As can be seen, the PRM algorithm requires the shortest computation time, which is only 1.206 s. The PH-RPM proposed in this paper can also be efficiently carried out in simple scenarios for paths to carry out planning, and the computation time required is only 1.884 s, which is slightly longer than that of the PRM but is still shorter than that of other path planning methods. \(t_{f}\) is the time of the cabin pushing from the starting point to the endpoint under kinematic constraints, where the A\({}^{\star}\), RRT\({}^{\star}\), and PRM algorithms do not take the characteristics such as the velocity of the PMCU pushing into account in the planning, so the corresponding results are null. Compared with the conventional RPM algorithm, the PH-RPM algorithm solves the simple scenario with 154.43 s for the pushing, which is 20.45 s less than the time solved by the conventional RPM algorithm. Regarding the path planning results of different algorithms, we need to combine the path length and path diagram for comparison, as shown in Figures 7 and 8. As shown in Figure 7, the RPM achieves the shortest path with a length of only 68.51 m. The path length \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Setting** \\ \hline \((x_{0},y_{0})\) & Initial position of PMCU & (106.2, 44.8) \\ \(\theta_{0}\) & Angle at initial position & \(-\frac{\pi}{2}\) \\ \(v_{0}\) & Velocity at the initial position & 0 \\ \(t_{0}\) & Time to leave the initial position & 0 \\ \([v_{min},v_{max}]\) & The range of velocity changes \((\mathrm{m/s})\) & \([-0.5,\,0.5]\) \\ \([\theta_{min},\theta_{max}]\) & The range of angle changes \((\mathrm{rad})\) & \([-\pi,\,\pi]\) \\ \([u_{1 min},u_{1 max}]\) & The range of acceleration changes & \([-0.1,\,0.1]\) \\ \([u_{2 min},u_{2 max}]\) & The range of angular acceleration & \([-\frac{\pi}{4},\,\frac{\pi}{4}]\) \\ \([x_{min},x_{max}]\) & Horizontal boundary & \([0,180]\) \\ \([y_{min},y_{max}]\) & Vertical boundary & \([0,50]\) \\ \(N\) & Number of collocation points & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter description. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & \(t_{cpu}/s\) & \(t_{f}/s\) & \(Lm\) \\ \hline A\({}^{\star}\) & 2.921 & - & 70.74 \\ RRT\({}^{\star}\) & 7.679 & - & 82.99 \\ PRM & 1.206 & - & 84.12 \\ RPM & 6.160 & 174.88 & 68.51 \\ PH-RPM & 1.884 & 154.43 & 70.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of simple scenarios planned by different algorithms. planned by the PH-RPM proposed in this paper is 70.25 m, which is slightly longer than RPM but still shorter than A*, RRT*, and PRM. Although the traditional RPM algorithm finds the shortest path length, it can be seen in the rectangular areas (A) and (B) in Figure 8. This is because the RPM only satisfies the process constraints at the collation points, but the paths between the collocation points may still not satisfy the path constraints, resulting in collisions. Although the path length of the PH-RPM method is slightly longer than that of the RPM, it can be seen from Figure 8 that no collision occurs along the entire length of the path, which ensures the path is passable. The other algorithms (A*, RRT*, and PRM) are also able to plan the resulting paths, but the lengths of the generated paths are all longer than PH-RPM, and the kinematic constraints are not fully taken into account in the planning process. The variation curves of the velocity and orientation angle of the cabin pushing process by the PH-RPM algorithm and the RPM algorithm are shown in Figure 9. From an overall perspective, the variation of velocity and orientation angle of the results planned by the PH-RPM are within the set constraints, and the rate of change is acceptable.