## 5 Simulation Analysis
### Method Comparison and Analysis  
To further verify the effectiveness, our method is compared with the centralized method [6] and the soft constraint-based DMPC method [17].  
The starts and goals are the same as in Table 1. Meanwhile, the robot operating environment and parameter settings are the same as in Section 5.1. The covariance matrix is set to \(N_{noise}=diag(0.1\text{ m},\ 0.1\text{ m},\ 0.5\text{ deg})\). Fifteen independent repeated simulations are carried out. The different methods are evaluated in terms of task success rate, safety, operational efficiency, and computational efficiency. Results are shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Value of the Covariance Matrix** & \(0.5*N_{m,k}^{noise}\) & \(N_{m,k}^{noise}\) & \(2*N_{m,k}^{noise}\) \\ \hline Minimum distance from obstacles & 0.38 m & 0.40 m & 0.49 m \\ Minimum distance between robots & 0.45 m & 0.50 m & 0.56 m \\ Task success rate & 100\% & 100\% & 100\% \\ Average velocity & 0.50 m/s & 0.45 m/s & 0.44 m/s \\ Average standard deviation of velocity & 0.21 m/s & 0.20 m/s & 0.19 m/s \\ Task time & 18.70 s & 21.25 s & 22.10 s \\ Calculation time per horizon & 0.18 s & 0.20 s & 0.21 s \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison under different level disturbances.  
Figure 9: Comparison of minimum distance under different level disturbances, (**a**) the minimum distance between robots, (**b**) the minimum distance between robot and obstacle.  
It is important to note that task time refers to the time spent on the mission in the simulation while calculation time refers to the time spent by the CPU in the real world.  
In terms of task success rate, our method and centralized method achieve a 100% task success rate. There is a failure when the soft constraint-based DMPC is utilized. The task success rate of soft constraint-based DMPC is 93.3%. Thus, the soft constraint-based DMPC is more sensitive to disturbances and our method is more robust.  
In terms of safety, the successful results of all methods are compared. Figure 10a presents the minimum distance between robots. In addition, Figure 8b illustrates the minimum distance between robot and obstacle. The minimum distance between robot and obstacle for our method, the centralized method, and the soft constraint-based DMPC are 0.40 m, 0.38 m, and 0.40 m, respectively. Moreover, the minimum distances between the robots for our method, the centralized method, and the soft constraint-based DMPC are 0.50 m, 0.51 m, and 0.44 m, respectively.  
In terms of operational efficiency, the average velocity of our method is 15% higher than that of the centralized method, and the task time of our method is 6.80% less than that of the centralized method. Moreover, the average velocity of our method is 25% higher than that of the soft constraint-based DMPC, and the task time of our method is 12.1% less than that of the soft constraint-based DMPC.  
In terms of computational efficiency, the calculation time per horizon of our method is longer than that of the soft constraint-based DMPC. However, the difference is not significant. Moreover, it can be seen that our method is much more computationally efficient than the centralized method.