Polyhedral Assembly Partitioning Using Maximally Covered Cells in Arrangements of Convex Polytopes

We study the following problem: Given a collection A of polyhedral parts in 3D, determine whether there exists a subset S of the parts that can be moved as a rigid body by infinitesimal translation and rotation, without colliding with the rest of the parts, A\S. A negative result implies that the object whose constituent parts are the collection A cannot be taken apart with two hands. A positive result, together with the list of movable parts in S and a direction of motion for S, can be used by an assembly sequence planner (it does not, however, give the complete specification of an assembly operation). This problem can be transformed into that of traversing an arrangement of convex polytopes in the space of directions of rigid motions. We identify a special type of cells in that arrangement, which we call the maximally covered cells, and we show that it suffices for the problem at hand to consider a representative point in each of these special cells rather than to compute the entire arrangement. Using this observation, we devise an algorithm which is complete (in the sense that it is guaranteed to find a solution if one exists), simple, and improves significantly over the best previously known solutions. We describe an implementation of our algorithm and report experimental results obtained with this implementation.