## 4 Simulation and Comparison
### Comparison Results  
The proposed NSPP was implemented and compared with the other algorithms mentioned in [33]. The comparison was implemented in a circular space, in which four or eight robots are distributed equally on the circle, as shown in Figure 13[33]. Each robot has its own target at the opposite side of the circle, and its initial orientation is toward the target point. For example, the robot at P2 (or P6) will go forward to and stop at position P6 (or P2). Figure 13 is the configuration diagram of the eight robots in which all robots can be randomly assigned from P1 to P8. In the simulation of four robots, only P1, P3, P5, and P7 are needed in the configuration. Other hypotheses in the simulation are as follows. (a) The circle radius is 260 pixels, (b) the radius of a robot is 20 pixels, and (c) the speeds of all robots are the same. Figure 12: **(a)** A simple test with three obstacles, **(b)** nine obstacles interlaced test, **(c)** concave polygon obstacle test, and **(d)** passage-type obstacle test. First, we consider the simulation with four robots. Because of the different priorities and different positions of the four robots, there are six (i.e., 3 \(\times\) 2 \(\times\) 1) configurations and six different simulation results. There are 5040 configurations and 5040 different simulation results for eight robots. We chose six configurations randomly for both simulation cases. For the convenience of comparison, we need to define some notations as follows. In Figure 13, the symbol NSPP(1432) means there are four robots and robot-1 is at position P1, robot-4 is at P3, robot-3 at P5, and robot-2 at P7, initially. In the case of eight robots, the symbol NSPP(14785623) means robot numbers 14,785,623 are initially at P1-P8, respectively. We show the simulation results of a four-robot configuration, and Figure 14a,b shows the shortest and longest average length of four robot's paths, respectively, in which the number under each robot is the robot's trajectory length. In Figure 15a,b, the simulation results are shown for the eight-robot configuration in which NSPP(14632875) has the shortest trajectory and NSPP(18526743) has the longest one. It is well known that the shortest distance algorithm (SDA) [33] and the reciprocal orientation algorithm (ROA) [33] are two benchmark methods in this issue. Let us define  
\[\text{ATL}(\text{average trajectory length})=\frac{\text{sum of all robot trajectory lengths}}{\text{number of robots}} \tag{7}\]  
and  
\[\text{MTL}(\text{maximum trajectory length})=\text{maximum trajectory length in all trajectories of robots} \tag{8}\]  
Figure 13: The configuration diagram of eight robots for the benchmark comparison. The comparison is shown in Figure 16a, where all data of the NSPP in the ATL column are better than the results of both the SDA and ROA; the data of the NSPP are almost better than those of the SDA and ROA in the MTL column except that the NSPP(1342) is worse than SDA. The case of the NSPP(1342) has the maximum trajectory length in robot-3 since robot-1 is waiting for robot-2 and robot-2 is waiting for robot-3 at a certain time, simultaneously. Then robot-3 must make a big turn around robot-1 and robot-2. When the number of robots is eight, it is hard to tell which one is better compared to the MTL in Figure 16b. However, the NSPP performs better than the SDA and ROA in the ATL. It is noted that the data of the ATL and MTL are the same in the ROA and SDA, which means all robot trajectory lengths are the same in their methods, since the methods of ROA and SDA do not give the path-priority order for all robots. Figure 14: The results of four robots. (**a**) The best result of NSPP(1234) and (**b**) the worst result of NSPP(1342). Figure 15: The results of eight robots. (**a**) The best result of NSPP(14632875) and (**b**) the worst result of NSPP(18526743).