Sequential parametrized motion planning and its complexity, II

Abstract

This is a continuation of our recent paper [6] in which we developed the theory of sequential parametrized motion planning. A sequential parametrized motion planning algorithm produced a motion of the system which is required to visit a prescribed sequence of states, in a certain order, at specified moments of time. In [6] we analysed the sequential parametrized topological complexity of the Fadell - Neuwirth fibration which is relevant to the problem of moving multiple robots avoiding collisions with other robots and with obstacles in the Euclidean space. In [6] we found the sequential parametrised topological complexity of the Fadell - Neuwirth bundle for the case of the Euclidean space Rd of odd dimension as well as the case d=2. In the present paper we give the complete answer for an arbitrary d≥2 even. Moreover, we present an explicit motion planning algorithm for controlling multiple robots in Rd having the minimal possible topological complexity; this algorithm is applicable to any number n of robots and any number m≥2 of obstacles.