# Distributed Weighted Coverage for Multi-Robot Systems in Non-Convex Environment
## Scenario 2 Multi-robot motion control based on BUVC under constant density function
Figure 9: Multi-robot motion control based on BUVC under constant function. **Scenario 4. Multi-robot motion control of BUVC based on communication distance constraint under Gaussian density function. **  
In this scenario, the initial positions of the robots are the same as those in Scenarios 1, 2, and 3. The coverage motion control of the multiple robots under the Gaussian density function, shown in Figure 11d, is verified. The density function is obtained using Equation (24). Figure 11a shows the initial positions of the robots. Figure 11c shows the final configuration of the nine robots under the Gaussian density function, demonstrating an asymmetric distribution. Figure 11c,g reveal that the area of the Voronoi cell of a robot increases as the robot's positioning uncertainty increases. Moreover, the Voronoi buffer between the robots is biased toward the side containing robots with low positioning uncertainties, and the distribution is asymmetric, which is consistent with the BUVC characteristics. \[\sigma(x,y)=e^{-\delta(a(x-x_{c})^{2}+b(y-y_{c})^{2})^{2}} \tag{24}\]  
Figure 11e,f show that the cost function and position error of the multi-robot operation tend to zero. Figure 11h indicates that during the multi-robot motion, the robots are all at safe distances from each other. **Scenario 5. Multi-robot motion control of BUVC based on communication distance constraint under Gaussian density function. **  
In this scenario, the initial positions of the robots are the same as those in Scenarios 1, 2, 3, and 4. However, the presence of an obstacle in the upper right corner of the area under consideration transforms the area into a non-convex environment. The coverage motion control of the multi-robots under the Gaussian density function shown in Figure 12d is verified. The density function is obtained using Equation (24). Figure 12a shows the initial position of the robots. Figure 12c shows the final configuration of the nine robots under the Gaussian density function, showing an asymmetric distribution. Figure 12c,g show that the area of the Voronoi cell of a robot increases as the robot's positioning uncertainty increases. Moreover, the Voronoi buffer between the robots is also biased toward the side containing robots with low positioning uncertainties, that is, the distribution is asymmetric. This is consistent with the BUVC characteristics. Figure 12e,f show that both the cost function and  
Figure 10: BUVC distribution and related data for multi-robot with communication distance constraints under constant density function. Gaussian density function, which shows an asymmetric distribution. Figure 13c,g show that the area of the Voronoi cell of a robot increases as the robot's positioning uncertainty increases. Moreover, the Voronoi buffer between the robots is also biased toward the side containing robots with low positioning uncertainties, that is, the distribution is asymmetric. This is consistent with the BUVC characteristics. Figure 13e,f reveal that both the cost function and position error in the multi-robot operation tend to zero. Figure 13h shows that during the multi-robot movement, all the robots are at safe distances from each other. The results of the multi-robot coverage simulations in the six scenarios are compared and analyzed. The comparison results are presented in Table 2. Notably, the number of iterations in Scenario 2 is less than that in Scenario 1, under stable conditions. This is because the existence of the buffer zone between the robots under the BUVC relatively reduces the area under consideration. This means that the area where the robots should operate is reduced, so the convergence speed is accelerated, which also reflects the advantages of the BUVC. However, the number of iterations and average position errors when the system is stable in both scenarios are the same. This shows that the method proposed in this study does not reduce the position accuracies of the robots when the positioning errors and actual sizes of the robots are considered.