Pattern Formation Problem for Synchronous Mobile Robots in the Three Dimensional Euclidean Space

We consider a swarm of autonomous mobile robots each of which is anonymous and oblivious (memory-less), and synchronously executes the same algorithm. The pattern formation problem requires the robots to from a given target pattern from an initial configuration. We investigate the pattern formation problem for oblivious fully-synchronous (FSYNC) robots moving in the three dimensional Euclidean space (3D-space) and characterize the problem by showing a necessary and sufficient condition for the robots to form a given target pattern $F$ from an initial configuration $P$. We define the symmetricity $\varrho(P)$ of positions of robots in 3D-space as the set of rotation groups formed by rotation axes that do not contain any robot, in other words, the rotation axes that the robots can never eliminate. We show the following necessary and sufficient condition for the pattern formation problem which is a natural extension of existing results of the pattern formation problem in 2D-space: The oblivious FSYNC robots in 3D-space can form a target pattern $F$ from an initial configuration $P$ if and only if $\varrho(P) \subseteq \varrho(F)$.