On soft predicates in subdivision motion planning

Abstract

We propose to design new algorithms for motion planning problems using the wellknown 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 K ε, 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) = R 2 × S 1 . To validate this approach, we implement our algorithms and the experiments demonstrate the efficiency of our approach, even compared to probabilistic algorithms.

Rods and Rings: Soft Subdivision Planner for R^3 x S^2

“…We regard this as a (G1-G2) trade-off. In fact, our implementations here as well as in our previous papers [31,19,34] are such machine implementations. This follows the practice in the robotics community, in order to have a fair comparison against other implementations.…”

“…Remarkably, the single bit of information encoded by NO-PATH is often missing in discussions. The standard definitions of correctness for path planners (resolution completeness and probabilistic completeness) omit this bit [31].…”

“…Exact path planning has many issues including a serious gap between theory and implementability. In [31,32], we introduced a theoretical framework based on subdivision to close this gap. This paper demonstrates for the first time that our framework is able to achieve rigorous state-of-the-art planners in 3D.…”

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

“…Another interesting direction of subdivision algorithms, of more geometric nature, concerns the approximation of algebraic varieties [30,25,5,29,37,21], and the computation of the approximate Voronoi diagrams [39]. There are also quite important applications of these algorithms to the problem of robot motion planning [33].…”

“…A big step in this direction is the introduction of soft tests [34,38] that, roughly speaking, replace hard exact tests, usually comparisons with zero, with approximate computations, and they are exact in the limit. They introduce a new notion of correctness called resolution-exactness.…”