Algorithmic motion planning in robotics: Coordinated motion of several disks amidst polygonal obstacles

Given k ≥ 2 circular disks D1, D2...Dkin a 2-dimensional region bounded by a finite set of polygonal obstacles, an algorithm to determine the existence of a collision free coordinated motion of the disks is given. The time complexity is shown to be O(n^k} where n is the number of edges composing the walls. This algorithm is believed to be optimal.