On soft predicates in subdivision motion planning

We propose to design new algorithms for motion planning problems using the well-known Domain Subdivision paradigm, coupled with "soft" predicates. Unlike the traditional exact predicates in computational geometry, our primitives are only exact in the limit. We introduce the notion of resolution-exact algorithms in motion planning: such an algorithm has an "accuracy" constant K > 1, and takes an arbitrary input "resolution" parameter ε > 0 such that: if there is a path with clearance Kε, it will output a path with clearance ε/K; if there are no paths with clearance ε/K, it reports "NO PATH". Besides the focus on soft predicates, our framework also admits a variety of global search strategies including forms of the A∗ search and probabilistic search. Our algorithms are theoretically sound, practical, easy to implement, without implementation gaps, and have adaptive complexity. Our deterministic and probabilistic strategies avoid the Halting Problem of current probabilistically complete algorithms. We develop the first provably resolution-exact algorithms for motion-planning problems in SE(2)=R²×S¹. To validate this approach, we implement our algorithms and the experiments demonstrate the efficiency of our approach, even compared to probabilistic algorithms.