Abstract
Concerning the coordination of autonomous mobile robots, the main focus has been on the important class of Pattern Formation problems, where the robots are required to arrange themselves to form a given geometric shape. This class of problems has been extensively studied in the continuous environment (e.g., the Euclidean plane), whereas few results exist when robots move in a discretization of the plane, like infinite grids. In this environment, to form any pattern, the problem of breaking symmetries emerges. Breaking the symmetry by moving some leader robot is not a straightforward task due to the movement restrictions as all the adjacent nodes of the leader may be occupied. It may even happen that before obtaining the requested asymmetric configuration, most of the robots must be moved. Due to the asynchrony of robots, this fact greatly increases the difficulty of the problem. We assume very weak robots moving on any regular tessellation graph as a discretization of the Euclidean plane, and we devise an algorithm Abreak able to solve the Symmetry Breaking problem on both the square and triangular grids. It is important to note that Abreak is proposed so that it can be used as a module for solving more general problems. As a case study, we use Abreak to deal with the Line Formation problem, where n≥3 robots must arrange themselves to occupy n contiguous vertices along a grid line. In this respect, we first provide an algorithm ALF- able to partially solve this problem (it works with configurations in which it is not necessary to break symmetries), and then we show how Abreak and ALF- can be combined to form ALF. We provide a complete characterization of the solvability of the Line Formation problem on the considered topologies by showing that ALF solves the problem in each configuration where this is possible.
Highlights
The coordination of autonomous mobile entities has long been the object of study in several fields, including robotics, control, AI, as well as distributed computing
Each robot, when active, operates in Look-Compute-Move cycles: it determines the positions of the robots in the system (Look), it uses this information to compute a trajectory toward destination point (Compute), and it moves along the computed trajectory towards the destination point (Move)
When robots move in a continuous environment like the Euclidean Plane, they are often viewed as points, and more than one robot can occupy the same location at the same time; when this occurs, we say that there is a multiplicity
Summary
The coordination of autonomous mobile entities has long been the object of study in several fields, including robotics, control, AI, as well as distributed computing. When robots move in a continuous environment like the Euclidean Plane, they are often viewed as points (they are dimensionless), and more than one robot can occupy the same location at the same time; when this occurs, we say that there is a multiplicity It is often assumed (as dictated by impossibility results) that, in combination with the LCM-model, robots are endowed with the so-called multiplicity detection capability This problem is of particular importance and has been extensively studied when robots move in the Euclidean plane In this environment, it has been fully characterized in [17] (for a recent survey, see [23] and references therein). The Arbitrary Pattern Formation for a set of oblivious asynchronous robots on the infinite square grid in the absence of any global coordinate
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