## 4 Simulation Results and Discussion
### Path Planning in Static Scenarios  
In this section, the grid map used to model the environmental information is set to 150 m \(\times\) 250 m. The parameters of APF are set as: attractive coefficient \(APF\_a=200\), the maximum attractive force \(APF\_amax=1\), repulsive coefficient \(APF\_r=150\), and repulsion distance \(APF\_po=50m\). The maximum number of iterations is set to 3000, and the step size of \(x_{rand}\) and \(x_{near}\) are both set to 5. Table 1 shows the test environments which contain start position, target position and the obstacles information. Our aim is to find a feasible and collision-free optimal path from a start point to a target point in each obstacles test environment. Figure 5: A comparison of the APF algorithm and APF-IRRT* algorithm in a simulated example. (**a**) shows the simulation result of APF algorithm. (**b**) shows the simulation result of APF-IRRT* algorithm. Figure 6 presents the best path which has the shortest collision-free path length based on the proposed algorithm in each test environment described in Table 1. To understand the advantage of the proposed hybrid algorithm directly, the path lengths of the RRT\({}^{\star}\), IRRT\({}^{\star}\), A*-RRT\({}^{\star}\), P-RRT\({}^{\star}\) and APF-IRRT\({}^{\star}\) algorithms for the above six test environments are shown in Figure 7. The simulation results in Figure 7 shows that the proposed APF-IRRT\({}^{\star}\) algorithm provides better solutions than those obtained using RRT\({}^{\star}\) and IRRT\({}^{\star}\), A*-RRT\({}^{\star}\), and P-RRT\({}^{\star}\) algorithms, especially in more complex environments. For example, in the test environment M5, the RRT\({}^{\star}\) algorithm obtained an average path length of 305.47 m, the IRRT\({}^{\star}\) algorithm obtained 295.47 m, the A*-RRT\({}^{\star}\) algorithm obtained 289.74 m and the P-RRT\({}^{\star}\) algorithm obtained 283.24 m from 20 independent experiments, the APF-IRRT\({}^{\star}\) algorithm obtained an average of 268.01 m. This gives a difference of 37.47 m, 27.46 m, 21.50 m and 15.23 m between the average path length that was obtained by the RRT\({}^{\star}\), IRRT\({}^{\star}\), A*-RRT\({}^{\star}\) and P-RRT\({}^{\star}\) algorithms, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Environment** & **Start Position** & **Target Position** & **Obstacles Position** \\ \hline M1 & (10, 75) & (240, 75) & (125, 75) \\ M2 & (10, 75) & (240, 75) & (125, 52), (125, 75), (125, 98) \\ M3 & (10, 75) & (240, 75) & (100, 52), (160, 75),  
Figure 7 also shows that the standard deviation for the 20 independent experiments of the APF-IRRT* algorithm, compared with other four algorithms, is superior in each test environment. Similarly, taking the test environment M5 as an example, the RRT*, IRRT*, A*-RRT* and P-RRT* algorithms obtained a standard deviation of 9.21, 7.46, 6.45, and 4.60 m, respectively, while the standard deviation of APF-IRRT* algorithm was 3.35 m. This reveals that the path length value of the APF-IRRT* algorithm is closer to the mean value in each experiment, which indicates that the proposed algorithm has lower deviation than the other three algorithms. We can also see from Figure 7 that the more complex the environment is, the more apparent the superiority of the standard deviation of APF-IRRT* algorithm over the other algorithms is. For example, in test environment M1, the RRT*, IRRT*, A*-RRT* and P-RRT* algorithms obtained a standard deviation of 6.54 m, 5.00 m, 4.05 m, and 4.16 m, respectively; the standard deviation of APF-IRRT* algorithm was 3.18 m. This gives a difference of 3.36, 1.82, 0.87, and 0.98 m between the RRT*, IRRT*, A*-RRT*, P-RRT* algorithms and the APF-IRRT* algorithm. While in test environment M5, the differences are 5.86, 4.11, 1.41, and 1.25 m, respectively. It is indicated that compared with the other three algorithms, the APF-IRRT* algorithm has an advantage in complex environments. Figure 6: The best path planning of simulation results for different test environments.