*4.1. Simulation Experiments*

The simulation experimental platform was configured with MATLAB 2019b, 64-bit Windows 10, processor Inter(R) Core (TM) i7-10700F CPU @ 2.90 GHz, and 16 GB of memory. The experimental simulation area was a 100 × 100 × 100 cube area with the starting position [5, 5, 5] and the target position [95, 95, 95]. The experimental environment was designated as a simple and complex one according to the number of obstacles [22]. Figures 4 and 5 show the simulation results of various algorithms in simple and complex environments.

**Figure 4.** Paths planned by various algorithms in a simple environment. The red cube indicates the starting position and the green cube indicates the target position. The red sphere indicates the obstacles in the environment, the green dot indicates the points obtained by sampling during the path planning process, the red solid line indicates the branches of the random tree, and the black solid line indicates the feasible paths obtained. (**a**) RRT algorithm; (**b**) RRT\* algorithm; (**c**) RRT-Connect algorithm; (**d**) PPD-RRT algorithm; (**e**) PPRO-RRT algorithm.

(**e**)

**Figure 5.** Paths planned by various algorithms in a complex environment. The red cube indicates the starting position and the green cube indicates the target position. The red sphere indicates the obstacles in the environment, the green dot indicates the points obtained by sampling during the path planning process, the red solid line indicates the branches of the random tree, and the black solid line indicates the feasible paths obtained. (**a**) RRT algorithm; (**b**) RRT\* algorithm; (**c**) RRT-Connect algorithm; (**d**) PPD-RRT algorithm; (**e**) PPRO-RRT algorithm.

The simulation results in Figures 4 and 5 showed that there was a significant reduction in the number of sampling points because the PPRO-RRT algorithm incorporated the idea of PPD and RO.

#### *4.2. Results and Analysis*

In order to avoid the influence of the randomness of the path planning algorithm on the experimental results, each method was tested for 5000 iterations in the same environment, and the upper limit of the number of single iterations was set to 1000. The proposed algorithm was compared with other algorithms in terms of the success rate, the number of path points, the path planning time, and the path length. The results of each experiment are shown in Tables 1–4.


**Table 1.** The success rate of various algorithms in two kinds of experimental environments.

Table 1 indicates the success rate of various path planning algorithms in two kinds of experimental environments. The experimental results showed that the success rates of the PPD-RRT and PPRO-RRT path planning algorithms were significantly higher than those of the other algorithms. The success rates of the PPD-RRT and PPRO-RRT algorithms were 99.80% and 100.00% in simple environments and 99.48% and 99.99% in complex environments, respectively. The success rate improvement was significant compared with other algorithms. The PPRO-RRT algorithm achieved success rate increases of 94.88% and 96.05% compared with the RRT algorithm; 94.32% and 95.37% compared with the RRT\* algorithm; and 50.30% and 48.11% compared with the RRT-Connect algorithm. With a set number of iterations of 1000, the PPRO-RRT algorithm greatly enhances the success rate of the path planning algorithm.

**Table 2.** The number of random tree points of various algorithms in two kinds of experimental environments.


Table 2 indicates the number of random tree points obtained by various path planning algorithms. For each of the algorithms, the number of random tree points hardly varied with the complexity of the environment. Compared with the RRT algorithm, the number of random tree points of the PPD-RRT algorithm was reduced by approximately 81.59% and 81.76% in two kinds of experimental environments, and after adding the RO idea, the number of points was reduced by about 93.25% and 93.32%. Compared to the RRT\* and RRT-Connect algorithms, the number of random tree points of the PPRO-RRT algorithm was reduced by 93.23% and 87.42% in the simple environments and 93.23% and 87.55% in the complex environments, respectively.

**Table 3.** The path planning time(s) of various algorithms in two kinds of experimental environments.


Table 3 indicates the path planning time of different algorithms in two kinds of experimental environments. For each of the proposed algorithms, the path planning time was approximately reduced by one order of magnitude. In the simple experimental environment, the path planning time of the PPD-RRT and PPRO-RRT algorithms was 90% and 92.92% shorter than that of the RRT algorithm, respectively. In the complex experimental environment, the path planning time of the PPD-RRT and PPRO-RRT algorithms was 88.66% and 91.49% shorter than that consumed by the RRT algorithm, respectively. As the RRT\* algorithm and the RRT-Connect algorithm both require a longer path planning time than the RRT algorithm, the PPD-RRT algorithm and the PPRO-RRT algorithm were far superior to the RRT\* algorithm and the RRT-Connect algorithm in terms of path planning time. Compared with the RRT\* and RRT-Connect algorithms, the path planning time of the PPRO-RRT algorithm was reduced by 97.59% and 94.65% in the simple environments and 97.05% and 92.84% in the complex environments, respectively.


**Table 4.** The path length [23] of various algorithms in two kinds of experimental environments.

Table 4 indicates the path length of the various algorithms in the two kinds of experimental environments. Compared with the RRT algorithm, the path length of the PPD-RRT algorithm became longer in two kinds of experimental environments due to the restricted sampling area. After adding the RO idea, the path length of the PPRO-RRT algorithm was reduced by 7.89% and 10.01% in simple environments and complex environments. While the path length of the PPRO-RRT algorithm was longer than that of the RRT\* algorithm, the difference was not significant and was far superior to the RRT\* algorithm in other aspects. Compared with the RRT-Connect algorithm, the path length of the PPRO-RRT algorithm was reduced by 11.84% and 15.81% in simple environments and complex environments.

From the above analysis, the PPRO-RRT algorithm can converge at a small number of iterations, and the success rate reaches close to 100% with a set upper limit of 1000 iterations. The PPRO-RRT algorithm saves the process of searching parent points, which greatly saves the path planning time; the real-time judgment of whether new points should be retained reduces the generation of more redundant points. We can conclude that the PPRO-RRT algorithm has significant advantages over the RRT algorithm in terms of success rate, number of points, planning time, and path length.
