# Distributed Weighted Coverage for Multi-Robot Systems in Non-Convex Environment
## 8 Conclusions  
In this study, we proposed a buffer uncertain Voronoi cell with communication distance constraints in a non-convex environment to optimize the multi-robot formation control strategy while improving the robustness of the formation. The main contents of the work are summarized as follows:  
1. In the traditional CVT, an actual robot cannot be treated as a single point. To address this limitation, we created a buffer zone between the Voronoi cells by introducing the radius weight of the robot. Subsequently, we shrank the edge of the Voronoi cell to the safe radius of the robot to improve the collision avoidance performance of the robot when it was close to the boundaries. 2. Second, considering the inherent positioning error in the positioning system, a positioning uncertainty factor was introduced as a weighting value. Consequently, the robots had high positioning uncertainty factors. In other words, the size of a Voronoi cell increased with an increase in the value of the position error. The size of the Voronoi cell could be varied in the area under consideration according to the positioning error of the current robot, thereby further improving the collision avoidance performance and robustness of the BUVC division in the process of a multi-robot formation movement. 3. Next, considering the limitation of the non-convex environment of the traditional CVT, this study realized a BUVC partition in a non-convex environment by introducing a discretized Voronoi partition and non-convex constraints. Moreover, the challenge of the unreachable centroid in a non-convex environment was solved by a non-visibility area and the corresponding obstacle avoidance strategy. This enabled the robots to achieve an effective area division for non-convex working areas with obstacles and ensured their working safety during the working process. 4. Given the limited communication distance of the robots in practice, this study constrained the BUVC of each robot in the non-convex environment obtained through the communication distance constraint. Furthermore, we addressed challenges such as robot collision, deterioration of the cost function, and failure to consider the centroid of the cell that may generate errors under the condition of limited communication distance. Finally, the proposed algorithm was improved with regard to the traditional CVT, and the robustness of the proposed algorithm to the formation of robots with communication distance constraints, actual sizes, and positioning errors in a non-convex environment were simulated in six simulation scenarios and two experiments. Thus, the innovation and effectiveness of the proposed algorithm were validated. In future work, we will extend the proposed algorithm to more complex obstacle environments, including 3D environments for multi-robot formation control. Another future work will consist of the adjustment of the Voronoi division of designated areas in time to restore the stable state of a multi-robot system in a short time when a robot fails or a new robot joins the system. Conceptualization, K.C., Y.C. and S.G.; Methodology, K.C. and D.A. ; Software, K.C. and H. D.; Validation, K.C., Y.C., H.D. and D.A. ; Formal analysis, K.C. ; Investigation, K.C. and D.A. ; Data curation, H.D. and D.A. ; Writing--original draft, K.C. ; Writing--review & editing, Y.C. and S.G.; Visualization, H.D. ; Supervision, Y.C. and S.G.; Project administration, S.G.; Funding acquisition, S.G. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China grant number 62103315, the Key R & D Program Project of Shaanxi Province grant number 2022 GY-24, 2022 GY-243 and the Key Industrial Innovation Chain Project of Shaanxi Province grant number 2022 QFY01-16, 2023-ZDLNY-63. Data Availability Statement: Data sharing is not applied.