Survivability of Swarms of Bouncing Robots

Abstract

Bouncing robots are mobile agents with limited sensing capabilities adjusting their movements upon collisions either with other robots or obstacles in the environment. They behave like elastic particles sliding on a cycle or a segment. When two of them collide, they instantaneously update their velocities according to the laws of classical mechanics for elastic collisions. They have no control on their movements which are determined only by their masses, velocities, and upcoming sequence of collisions.

We suppose that a robot arriving for the second time to its initial position dies instantaneously. We study the survivability of collections of swarms of bouncing robots. More exactly, we are looking for subsets of swarms such that after some initial bounces which may result in some robots dying, the surviving subset of the swarm continues its bouncing movement, with no robot reaching its initial position.

For the case of robots of equal masses and speeds we prove that all robots bouncing in the segment must always die while there are configurations of robots on the cycle with surviving subsets. We show the smallest such configuration containing four robots with two survivors. We show that any collection of less than four robots must always die. On the other hand, we show that $|{\mathcal S}_{}^+-{\mathcal S}_{}^-|$ robots always die where ${\mathcal S}_{}^+$ (and ${\mathcal S}_{}^-$ ) is the number of robots starting their movements in clockwise (respectively counterclockwise) direction in swarm ${\mathcal S}_{}$.

When robots bouncing on a cycle or a segment have arbitrary masses we show that at least one robot must always die. Further, we show that in either environment it is possible to construct swarms with n − 1 survivors. We prove, however, that the survivors in the segment must remain static (i.e, immobile) indefinitely, while in the case of the cycle it is possible to have surviving collections with strictly positive kinetic energy.

Our proofs use results on dynamics of colliding particles. As far as we know, this is the first time that these particular techniques have been used in order to analyze the behavior of mobile robots from a theoretical perspective.