Detecting Geometric Infeasibility

We consider the problem of deciding whether an assembly of polyhedra can partitioned by an arbitrary sequence of translations. Di erent subassemblies may be moved at di erent stages. It is shown that certain D dimensional arrangements of hyperplanes can be searched in the following way: only a single connected component is traversed during the search, and the arrangement is searched as an arrangement of surface patches rather than full hyperplanes. This reduction in search e ort allows for polynomial time bounds in appropriate cases. The described methods give rise to an exact and practical method for translational motion planning with many degrees of freedom. Heuristic and randomized planners cannot return an information about infeasibility of planning problems. Experiments with an implementation of the new methods suggest that translational infeasibility can be detected in practical cases.