# Distributed Weighted Coverage for Multi-Robot Systems in Non-Convex Environment
## 1 Introduction
Thus, the algorithm convergence is slow. Finally, because the calculation of the geodesic distance is expensive, the robot can fall to extreme points because of obstacles. To address robot collision problems that may occur within the geodesic distance, [PERSON] [24] combined the classical Lloyd algorithm with the local path planning TangentBug to calculate robotic motion near obstacles and corners based on the definition of geodesic distance. In addition, the concept of a virtual generator is proposed to address centroid unreachability problems to map the centroids of accessible regions. The convergence and optimality of the algorithm are validated. [PERSON] [25] and [PERSON] [26] proposed a visibility-based definition of a Voronoi diagram based on the decay of sensor capabilities due to boundaries or obstacles. This definition extends the continuous-time Lloyd method for convex domains and modifies the conditions related to the visibility of Voronoi boundaries. [PERSON] used the projection of a centroid to solve problems wherein centroids are not located in Voronoi cells and introduced smoothness constraints during deployment of robots to avoid accidents at sharp turns. As can be observed in the aforementioned contributions on non-convexity, algorithms with simple CVTs, CVTs and the combination of a differential homogeneous embryo, geodesic lines, and artificial potential fields cannot be easily adapted to complex environments but can only adapt to the cluster controls in simpler environments with poor stabilities. Moreover, a greater computational effort is required to adapt algorithms containing geodesic lines and differential homogeneous embryos to non-convex regions. Robots with different capabilities are applied in search and rescue, patrol, surveillance, and other fields. The control problem of multi-robots with different capabilities has been an important topic in robot research. In addition, the uncertainty of the sensors used in robot positioning also increases the concern about the problem of imprecise positioning. [PERSON] [27] aimed to adapt a group of robots to study the performance changes among different robots while enabling them to complete collaborative tasks online. He proposed a distributed approach to the online learning of the relative performance changes among robots and automatically compensated the learning approach by assigning smaller environ ment shares to weaker robots and larger environment shares to stronger robots. Based on this, the algorithm proposed by [PERSON] [28] uses an energy-saving coverage optimal control scheme and a Hopfield neural network to create a power Voronoi diagram. The weights of each agent are calculated according to the different driver performances to independently achieve the optimal coverage configuration. [PERSON] [29; 30] aimed at the area coverage problem of mobile robot networks with inaccurate agent positioning. The convex space was divided according to the guaranteed Voronoi (GV) principle, and the responsibility area of each robot corresponded to its GV cell, bounded by hyperbolic arcs. The scheme based on gradient rise was used for area coverage and guarantees the Voronoi tessellation of the monitored area. In this study, an effective multi-robot coverage control method is developed to coordinate a group of mobile robots to effectively cover the region of interest in the environment at a minimum cost. Primarily aiming at solving the inherent positioning error problem in a positioning system and the inability of a robot to be mass-pointed, this study introduces the positioning uncertainty factor and robot radius. This can buffer uncertainty Voronoi cells for robot construction and create larger cells for robots with large positioning uncertainties, while the introduction of buffer strips ensures that robots close to Voronoi boundaries do not collide. Subsequently, for a non-convex environment, both discrete and constrained Voronoi partitions are used to solve the non-convex limitations of CVTs.