## 4 Simulation and Comparison
### Simulation Results  
The simulation was performed in an 880 \(\times\) 880 (pixels) space, as shown in Figure 9a, and the radius of a robot is 20 pixels. In the first simulation, there are three robots, and each robot is moving from its starting point to its target point, as shown in Figure 6. The path priority of these robots is that robot-1 is the first, robot-2 is the second, and robot-3 is the third. Let \(R_{j}=4.5L_{j}\), \(\Theta_{j}=45^{\circ}\), and \(L_{j}=L_{n}\). The requirement is that once a robot reaches its target point, it must return to its original starting point immediately and start the next round trip. A collision counter is set in the upper left corner in the simulation, as shown in Figure 9a. The count increased by one means that the collision between two robots happens once. Therefore, in this experiment, the counter remaining at zero means all collisions are avoided. One more thing to be noted here, in mode 2 of simulation, we set \(k_{\alpha}=0.01\) because, considering the hardware limitation or uneven surface of the floor in the real experiment, it is impossible to have \(\alpha\)= 0 exactly at the sampling time; therefore, we used \(\alpha\)= 0.03 in the simulation and the sampling time of the simulation in this study was set at 38 ms. The trajectories of the first trip to the target points of robot-1, robot-2, and robot-3 are shown in Figure 9a-c, respectively. Figure 9d shows the trajectories of each robot moving between the starting point and the target more than 50 times (25 round trips). In Figure 9a, the path of robot-1 crosses the paths of robot-2 and robot-3, whereas the path of robot-2 does not cross the path of robot-3. In Figure 10a, the original path (green line) of robot-2 is blocked by robot-1 at a certain time, then Figure 10b-d shows that robot-1 is regarded as a moving obstacle by robot-2 and the real trajectory of robot-2 is altered to avoid hitting robot-1 on its way. A detailed description is presented in the caption of Figure 10. The same situation occurs for robot-3. A detailed descrip  
Figure 9: **(a)** The first trajectory (red line) of robot-1 arrives at the target point; **(b)** the first trajectory (gray line) of robot-2 arrives at the target point; **(c)** the first trajectory (yellow line) of robot-3 arrives at the target point; **(d)** the trajectories of all robots running between the starting point and the target point more than 50 times, respectively. caption of Figure 11a-d, but it occurs much later than for robot-2. It is noted that robot-2 and robot-3 are of lower path priority than robot-1, so that their paths have turning points near the cross points with robot-1's path to avoid robot-1, respectively. It is concluded that robot-1 has the highest path priority, and its path plan focuses only on avoiding the fixed obstacles and disregards the collisions with robot-2 or robot-3. Thus, the trajectory of robot-1 is almost unchanged on its round trips (see Figure 9a). No collision occurs, so the collision number counter shows \"0\". A video of this simulation result can be seen at [[https://youtu.be/rNPp7](https://youtu.be/rNPp7) bCDbUg]([https://youtu.be/rNPp7](https://youtu.be/rNPp7) bCDbUg) (Accessed on: 16 September 2021). Additional simulations were conducted and are shown in Figure 12, where the starting points of all robots are evenly distributed at the top of the space and their target points are at the bottom corresponding to the respective starting points. In addition, each number above the green circle (robot) in Figure 12 represents the robot's number. In Figure 12a, there are three fixed obstacles in the middle of the space, and the robots can navigate smoothly from the starting point to the target point without any collision. In Figure 12b, there are more obstacles in the space. Nevertheless, each robot can still reach its target point successfully, although the paths of robot-2 and robot-5 have some curved parts. In Figure 12c,d, there are a concave polygon obstacle and two specifically shaped obstacles, respectively, in the space.