Circle packaging with robots in a dynamic environment

This thesis investigates the problem of arranging multiple autonomous mobile robots within a square-shaped environment in such a way that their mutual distances are maximized. Traditionally, this problem can be rephrased in terms of the circle packing problem which studies solutions with a fixed number of circles inside static containers. In contrast, the focus of this thesis is on a dynamic perspective: new circles (robots) can be added sequentially, compelling all circles (robots) to adapt their positions accordingly. In this thesis, we start by examining established circle packing results in two-dimensional domains (e.g., rectangles, circles, and certain special shapes). Next, we propose a theoretical framework for mobile robots that emphasizes synchronization requirements and the constraints arising from limited communication. After that, building on these foundations, we introduce two dynamic packing possible strategies for placing robots within a square: one that use a precomputed static solutions (benefiting from broad visibility), while the other employs relay-based interaction to manage partial connectivity. Finally, we review trade-offs, point to practical applications, and outline future research avenues in scalability, failure resilience, and advanced optimization.