Accurate sampling-based algorithms for surface extraction and motion planning

摘要

Boolean operations, Minkowski sum evaluation, configuration space computation, and motion planning are fundamental problems in solid modeling and robotics. Their applications include computer-aided design, numerically-controlled machining, tolerance verification, packing, assembly planning, and dynamic simulation: Prior algorithms for solving these problems can be classified into exact and approximate approaches. The exact approaches are difficult to implement and are prone to robustness problems. Current approximate approaches may not solve these problems accurately. Our work aims to bridge this gap between exact and approximate approaches. We present a sampling-based approach to solve these geometric problems. Our approach relies on computing a volumetric grid in space using a sampling condition. If the grid satisfies the sampling condition, our algorithm can provide geometric and topological guarantees on the output. We classify the geometric problems into two classes. The first class includes surface extraction problems such as Boolean operations, Minkowski sum evaluation, and configuration space computation. We compute an approximate boundary of the final solid defined using these geometric operations. Our algorithm computes an approximation that is guaranteed to be topologically equivalent to the exact surface and bounds the approximation error using two-sided Hausdorff error. We demonstrate the performance of our approach for the following applications: Boolean operations on complex polyhedral models and low degree algebraic primitives, model simplification and remeshing of polygonal models, Minkowski sums and offsets of complex polyhedral models, and configuration spare computation for low degrees of freedom objects. The second class of problems is motion planning of rigid or articulated robots translating or rotating among stationary obstacles. We present an algorithm for complete motion planning, i.e., finding a path if one exists and reporting a failure otherwise. Our algorithm performs deterministic sampling to compute a roadmap that captures the connectivity of free space. We demonstrate the performance of our algorithm on challenging environments with narrow passages and no collision-free paths.