Area Convergence of Monoculus Robots With Additional Capabilities

Abstract

Abstract This paper considers the area convergence problem, which requires a group of robots to gather in a small area not defined a priori. While it is known that robots can gather at a point if they can precisely measure distances, we, in this paper, show that without any agreement on the coordinate system, it is impossible for robots to converge to an area if they cannot measure distances or angles. We denote these robots without the ability to measure distances or angles as monoculus robots. We present a counterexample showing that monoculus robots fail in area convergence even with the capability of measuring angles. However, monoculus robots with a weak notion of distance or minimal agreement on the coordinate system are sufficient to achieve area convergence. In particular, we present area convergence algorithms in asynchronous model for such monoculus robots with one of the two following simple additional capabilities: (1) locality detection ($\mathcal{L}\mathcal{D}$), a notion of distance or (2) orthogonal line agreement ($\mathcal{O}\mathcal{L}\mathcal{A}$), a notion of direction. We discuss extensions corresponding to multiple dimensions and the termination. Additionally, we validate our findings using simulation and show the robustness of our algorithms in the presence of errors in observation or movement.