Collisionless Gathering of Robots with an Extent

Gathering

n

mobile robots in one single point in the Euclidean plane is a widely studied problem from the area of robot formation problems. Classically, the robots are assumed to have no physical extent, and they are able to share a position with other robots. We drop these assumptions and investigate a similar problem for robots with (a spherical) extent: the goal is to gather the robots as close together as possible. More exactly, we want the robots to form a sphere with minimum radius around a predefined point. We propose an algorithm for this problem which synchronously moves the robots towards the center of the sphere unless they block each other. In this case, if possible, the robots spin around the center of the sphere. We analyze this algorithm experimentally in the plane. If

R

is the distance of the farthest robot to the center of the sphere, the simulations indicate a runtime which is linear in

n

and

R

. Additionally, we prove a theoretic upper bound for the runtime of

O

(

nR

) for a discrete version of the problem. Simulations also suggest a runtime of

O

(

n

 + 

R

) for the discrete version.