A family of skeletons for motion planning and geometric reasoning applications

The task of planning a path between two spatial configurations of an artifact moving among obstacles is an important problem in practically all geometrically intensive applications. Despite the ubiquity of the problem, the existing approaches make specific limiting assumptions about the geometry and mobility of the obstacles, or those of the environment in which the motion of the artifact takes place. We present a strategy to construct a family of paths, or roadmaps, for two- and three-dimensional solids moving in an evolving environment that can undergo drastic topological changes. Our approach is based on a potent paradigm for constructing geometric skeletons that relies on constructive representations of shapes with R-functions that operate on real-valued half-spaces as logic operations. We describe a family of skeletons that have the same homotopy as that of the environment and contains the medial axis as a special case. Of importance, our skeletons can be designed so that they are “attracted to” or “repulsed by” prescribed spatial sites of the environment. Moreover, the R-function formulation suggests the new concept of a medial zone, which can be thought of as a “thick” skeleton with significant applications for motion planning and other geometric reasoning applications. Our approach can handle problems in which the environment is not fully known a priori, and intrinsically supports local and parallel skeleton computations for domains with rigid or evolving boundaries. Furthermore, our path planning algorithm can be implemented in any commercial geometric kernel, and has attractive computational properties. The capability of the proposed technique are explored through several examples designed to simulate highly dynamic environments.