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,...,Ak 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 $P_i=A_i\oplus (-B)$, 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 P1,...,Pk. We also derive an efficient randomized algorithm that constructs this configuration space in expected time O(nk log k log n).