# Distributed Weighted Coverage for Multi-Robot Systems in Non-Convex Environment
## 3 Buffer Uncertain Voronoi Cell (BUVC)  
For a distributed multi-robot system operating in a task environment, a collision avoidance control strategy should be designed, especially for robots that cannot be presented as points. Although a CVT provides an obstacle avoidance attribute for robots, in  
Figure 1: Schematic diagram of Voronoi cells. practality, they have real sizes. When the robots are simultaneously located at the edge of a Voronoi cell, a collision may occur, even if the robots' centers are located in the cell. To solve the abovementioned problems, this study introduces the concept of a buffer zone and appropriately shrinks the edge of the original Voronoi cell so that the robots' centers are located at the edge of the Voronoi cell and the robots do not collide with each other. The traditional Voronoi cell boundary renders the robots i and j equidistant from their shared Voronoi edges. The buffer zone in this study adds weights based on the original equation, as follows:  
\[v_{i}^{b}=\{q\in Q\ |\ \|\ p_{i}-q\ \|^{2}{\leq}\|\ p_{j}-q\ \|^{2}-\omega_{ij}\}. \tag{4}\]  
\[\omega_{ij}=(r_{i}+r_{j})\ \|\ p_{i}-p_{j}\ \|. \tag{5}\]  
In Equation (5), \(\omega_{ij}\) is the cell weight value between robot \(i\) and its neighbor \(j\), and \(r\) is the radius of the robot. Equations (4) and (5) create a gap between the boundaries of the Voronoi cells as a safe buffer zone to ensure that robots that are close to the edge of the cell do not collide. Notably, each robot can measure its position through a positioning technology or communicate with its neighboring robots. However, owing to errors in the positioning system, the position of each robot cannot be accurately obtained. Therefore, in general, under laboratory conditions, each robot has different positioning uncertainties. Assuming that the upper bound of the positioning uncertainty of each robot \(R_{i}^{u}\) is known, to describe the positioning uncertainty, the positioning uncertainty factor \(\lambda_{i}\in[0,1]\) is introduced to measure the positioning uncertainty (positioning error) of the robot. If \(\lambda_{i}\) is close to 1, the radius of the robot's positioning uncertainty is closer to the upper bound (that is, the positioning error is significant). On the contrary, the lower the robot's positioning uncertainty, the smaller the positioning error. \[\omega_{ij}=[1+\frac{1}{2}(\lambda_{i}-\lambda_{j})](r_{i}+r_{j})\ \|\ p_{i}-p_{j}\ \|-\frac{1}{2}(\lambda_{i}-\lambda_{j})\ \|\ p_{i}-p_{j}\ \|^{2}. \tag{6}\]  
\[v_{i}^{b,u}=\{q\in Q\ |\ \|\ p_{i}-q\ \|^{2}{\leq}\|\ p_{j}-q\ \|^{2}-\omega_{ij}\}. \tag{7}\]  
When Equation (6) is changed to (5), both robots are located at the edge of their respective cells, and the distance between the two robots is \(d_{ij}=(r_{i}+r)\). In Equation (6), when \(\omega_{ij}\) increases, the Voronoi cell of the robot also increases. \[\partial v_{i}^{b,u}=\{q\in Q\ |\ \|\ p_{i}-q\ \|^{2}-\ \|\ p_{j}-q\ \|^{2}{=}\omega_{ij}\}. \tag{8}\]  
According to hyperbolic theories, Equation (8) represents a single branch equation of a hyperbolic equation. The positions of the two robots are located at the focus points of the long axis. The parameter \(v_{i}^{b,u}\) may either be convex or non-convex because \(\omega_{ij}\) may either be positive or negative. (a) When a buffer zone of a Voronoi cell and the positioning uncertainty of a robot are not considered, \(r=0\), \(\lambda_{i}=0\), and \(\omega_{ij}=0\). The distribution of the robots is governed by CVT, as shown in Figure 1(a). The cell edge between the robots is located on the vertical line of the robot and is symmetrically distributed. (b) When the buffer zone of the Voronoi cell is introduced, \(r_{i}\
eq 0\), \(\lambda_{i}\)=0, and \(\omega_{ij}=(r_{i}+r_{j})\ \|\ p_{i}-p_{j}\ \|\). The distribution of the robots is shown in Figure 1(b), wherein the black discs represent the actual robot sizes, and the buffer zone between the robots is located on the vertical bisector between the robots and is symmetrically distributed.