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 A₁,..., 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 Ω(nkα(k)) in the worst case, where n is the total complexity of the individual Minkowski sums P₁,...,P_k. We also derive an efficient randomized algorithm that constructs this configuration space in expected time O(nk log k log n).