Localization for a System of Colliding Robots

We study the

localization problem

in the ring: a collection of

n

anonymous mobile robots are deployed in a continuous ring of perimeter one. All robots start moving at the same time along the ring with arbitrary velocity, starting in clockwise or counterclockwise direction around the ring. The robots bounce against each other when they meet. The task of each robot is to find out, in finite time, the initial position and the initial velocity of every deployed robot. The only way that robots perceive the information about the environment is by colliding with their neighbors; any type of communication among robots is not possible.

We assume the principle of momentum conservation as well as the conservation of energy, so robots exchange velocities when they collide. The capabilities of each robot are limited to: observing the times of its collisions, being aware of its velocity at any time, and processing the collected information. Robots have no control of their walks or velocities. Robots’ walks depend on their initial positions, velocities, and the sequence of collisions. They are not equipped with any visibility mechanism.

The localization problem for bouncing robots has been studied previously in [1,2] in which robots are assumed to move at the same velocity. The configuration of initial positions of robots and their speeds is considered

feasible

, if there is a finite time, after which every robot starting at this configuration knows initial positions and velocities of all other robots. Authors of [1] conjectured that if robots have arbitrary velocities, the problem might be solvable, if the momentum conservation and energy preservation principles are assumed.

In this paper we prove that the conjecture in [1] is false. We show that the feasibility of any configuration and the required time for solving it under such stronger constraints depend only on the collection of velocities of the robots. More specifically, if

v

0

,

v

1

,…,

v

n

 − 1

are the velocities of a given robot configuration

$\mathcal{S}$

, we prove that

$\mathcal{S}$

is feasible if and only if

$v_i\neq \bar{v}$

for all 0 ≤ 

i

 ≤ 

n

 − 1, where

$\bar{v} = \frac{v_0+\ldots+v_{n-1}}{n}$

. To figure out the initial positions of all robots no more than

$\frac{2}{min_{0\leq i\leq n-1} |v_i-\bar{v}|}$

time is required.