# Distributed Weighted Coverage for Multi-Robot Systems in Non-Convex Environment
## Scenario 2 Multi-robot motion control based on BUVC under constant density function  
In this scenario, the initial positions of the robots are the same as those in Scenario 1, and the coverage motion control of multiple robots under the density function shown in Figure 9d is verified. Figure 9a shows the distribution of the Voronoi cells corresponding to the robot position when the number of iterations \(k=0\). The black discs represent the current positions and dimensions of the robots. As the sizes of the red circles increase, both the positioning uncertainties and positioning accuracies of the robot increase. The regions represent the Voronoi cells of the robots, and the dark blue areas between the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \(\mathbf{p_{1}}\) & \(\mathbf{p_{2}}\) & \(\mathbf{p_{3}}\) & \(\mathbf{p_{4}}\) & \(\mathbf{p_{5}}\) & \(\mathbf{p_{6}}\) & \(\mathbf{p_{7}}\) & \(\mathbf{p_{8}}\) & \(\mathbf{p_{9}}\) \\ \hline Coordinate \(p(k=0)\) & (0.4, 1) & (1, 0.5) & (2, 1) & (2.5, 2) & (3, 0.5) & (1, 4) & (1, 3) & (3.7, 3) & (3, 4) \\ \hline Radius \(r_{i}\) & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \\ \hline Positioning uncertainty \(\lambda_{i}\) & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial positions and corresponding parameters of the nine robots. Figure 8: CVT distribution and related data of multi-robot under constant density function. Voronoi cells represent the buffer zones between the cells. Figure 9b shows the position distribution and trajectory diagram of the robot task environment when the number of iterations \(k=33\). Figure 9c represents the final configuration of multi-robot coverage under the constant density function. Figure 9c,g reveal that the buffer area of the Voronoi cell of a robot increases as the robot's positioning uncertainty increases. This is consistent with the characteristics of the BUVC. This means that the boundary between the Voronoi cells is biased toward the side with less positioning uncertainty and divides the cells asymmetrically. The buffer zones of the cells represent robots that cannot be treated as single points. Figure 9e indicates that the cost function of the multi-robots finally reaches the minimum value, that is, the optimal solution. Figure 9f shows that the position error of each robot tends to zero after 123 iterations. At this time, the optimal distribution of the robots in the task environment is obtained, and the iteration is stopped. Figure 9h shows that during the multi-robot movement, all the robots are at safe distances from each other. Scenario 3. Multi-robot motion control of BUVC based on communication distance constraint under constant density function  
In this scenario, the initial positions of the robots are the same as those in Scenarios 1 and 2. Herein, we verify the coverage motion control of multiple robots under the density function shown in Figure 10d. Figure 10a represents the initial position of the robot. Herein, the Voronoi cells corresponding to the robots are all constrained to 0.5 times the communication range. Figure 10b shows the BUVC division in the robot task environment when the number of iterations \(k=80\). Figure 10c shows the final configuration of 10 robots under a constant density function, demonstrating an asymmetric distribution. Figure 10c,g indicate 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. This is consistent with the BUVC characteristics and in stark contrast with the final configuration of the BUVC without the communication range constraints, as shown in Figure 9c. Figure 10e,f show that both the cost function and position error in the multi-robot operation tend to zero. Figure 10h shows that in the process of multi-robot motion, the robots are all at safe distances from each other.