Plane Formation by Synchronous Mobile Robots in the Three Dimensional Euclidean Space
This addresses a foundational challenge in distributed computing for autonomous robot swarms, providing the first theoretical result for agreement problems in 3D space, though it is incremental relative to existing 2D work.
The paper tackles the plane formation problem for a swarm of synchronous mobile robots in 3D Euclidean space, establishing a necessary and sufficient condition for them to land on a common plane, with the result showing that robots can form a plane from every regular polyhedron except a regular icosahedron but not from most semi-regular polyhedra.
Creating a swarm of mobile computing entities frequently called robots, agents or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, we investigate the plane formation problem that requires a swarm of robots moving in the three dimensional Euclidean space to land on a common plane. The robots are fully synchronous and endowed with visual perception. But they do not have identifiers, nor access to the global coordinate system, nor any means of explicit communication with each other. Though there are plenty of results on the agreement problem for robots in the two dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three dimensional space. This paper presents a necessary and sufficient condition for fully-synchronous robots to solve the plane formation problem that does not depend on obliviousness i.e., the availability of local memory at robots. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can form a plane from every regular polyhedron (except a regular icosahedron), whose symmetry is usually considered to be higher than any semi-regular polyhedrdon.