Fast reconfiguration of robot swarms with uniform control signals

Abstract

This paper investigates a simplified model of robot motion planning where particles on a board respond to global signals, causing them to move uniformly in a particular direction. We consider two types of movement commands: (1) Steps, which cause particles to move one unit distance in the given direction, and (2) Tilts, which cause particles to move maximally in the given direction. Under the overarching theme of reconfiguring robot swarms, we look at the problem of assembling general shapes both within systems that exclusively use step commands and systems that exclusively use tilt commands. We derive upper and lower bounds on the worst-case number of movements needed to reconfigure a general purpose board into a target shape. Under step transformations, we show a set of obstacles that can reconfigure n robots from any size-n shape to construct any other size-n shape with optimal $\varTheta (n)$ steps, which improves on previous techniques taking $O(n^2)$ steps. We then provide a board configuration that, under tilt transformations, can construct any size-n shape (given “helper particles”) in optimal $\varTheta (n)$ tilts, which also improves upon the previous best known time of $O(n^2)$ tilts.