# Distributed Weighted Coverage for Multi-Robot Systems in Non-Convex Environment
## 6 Numerical Simulation  
The simulation experiment performed in this study is conducted on MATLAB on the Windows operating system. In MATLAB, the size of the specified area is \(1000\times 1000\). Nine robots can be controlled in this area to complete the coverage under different density functions and environments. The density function of the BUVC can be manually specified in advance to control the robots to form the desired coverage formation.  
The novelty of this study is highlighted by considering the multi-robot motion control based on a CVT and BUVC under a constant density function and comparing it with the traditional CVT. Subsequently, in Scenarios 3 and 4, the communication range is added to the algorithm, and the density function of the robot task area is changed in Scenario 4. While the effectiveness of the algorithm is verified, the robustness under different density functions is reflected. Finally, by considering multi-robot motion control under different obstacle environments in Scenarios 5 and 6, the proposed method can effectively classify different non-convex environments. Thus, robots with actual sizes, positioning errors, and limited communication ranges can effectively cover different task environments.  
**Scenario 1. Multi-robot motion control based on CVT under constant density function.**  
The simulation utilizes nine robots to verify the CVTs of multiple robots under the constant density function shown in Figure (d)d. Figure (a)a represents the initial positions of the robots. Herein, the black points represent the current position of the robot, the red crosses represent the centroid positions of the Voronoi cell, and the colored regions where the robots are located represent the Voronoi cell of the robots. Figure (b)b illustrates the CVT in the robot task environment when the number of iterations \(k=35\). The colored curve represents the trajectory of the robot. Notably, each robot tracks its centroid in its Voronoi cell. Figure (c)c shows the final configuration of all the robots under the constant density function, which is symmetrical; that is, the edge of the Voronoi cell between every two robots is located on the vertical bisector of the robots. Figure (g)g,c reveal that the areas of the nine robots are equal. Moreover, the distribution conforms to the CVT characteristics,that is, the edges of the Voronoi cells are all located on the vertical bisector between the robots, and the areas of the cells are approximately equal. When measuring the workloads of the robots using the size of the work area, the workload of each robot is considered to be approximately equal.  
Figure 8e shows that the cost function of the multi-robots finally reaches the minimum value, that is, the optimal solution. Figure 8f depicts the error of the current position of each robot relative to the position of the centroid in each iteration. Notably, the position error of each robot is zero when the iteration reaches 140, and the cost function of the system is also minimized. When the optimal distribution of multi-robots in the task environment is obtained, the experiment is stopped. According to the distance between each pair of consecutive robots during the process of the multi-robot movement, the robots can be observed to be at safe distances from each other, as shown in Figure 8h.  
All the subsequent simulations in this study use nine robots. Their initial positions and corresponding parameters are listed in Table 1.