On translational motion planning of a convex polyhedron in 3-space

Let B be a convex polyhedron translating in 3-space amidst k convex polyhedral obstacles A(1),...,A(k) with pairwise disjoint interiors. The free configuration space (space of all collision-free placements) of B can be represented as the complement of the union of the Minkowski sums P-i = A(i) + (-B), for i = 1,...,k. We show that the combinatorial complexity of the free configuration space of B is O(nk log k), and that it can be Omega(nk alpha(k)) in the worst case, where n is the total complexity of the individual Minkowski sums P-1,...,P-k. We also derive an efficient randomized algorithm that constructs this configuration space in expected time O(nk log k log n).