In this paper, we compute a conservative approximation of the path-connected components of the free space of a rigid object in a 2D workspace in order to solve two closely related problems: to determine whether there exists a collision-free path between two given configurations, and to verify whether an object can escape arbitrarily far from its initial configuration -- i.e., whether the object is caged. Furthermore, we consider two quantitative characteristics of the free space: the volume of path-connected components and the width of narrow passages. To address these problems, we decompose the configuration space into a set of two-dimensional slices, approximate them as two-dimensional alpha-complexes, and then study the relations between them. This significantly reduces the computational complexity compared to a direct approximation of the free space. We implement our algorithm and run experiments in a three-dimensional configuration space of a simple object showing runtime of less than 2 seconds.
In this paper, we address the problem determining the connectivity of a robot's free configuration space. Our method iteratively builds a constructive proof that two configurations lie in disjoint components of the free configuration space. Our algorithm first generates samples that correspond to configurations for which the robot is in collision with an obstacle. These samples are then weighted by their generalized penetration distance, and used to construct alpha shapes. The alpha shape defines a collection of simplices that are fully contained within the configuration space obstacle region. These simplices can be used to quickly solve connectivity queries, which in turn can be used to define termination conditions for sampling-based planners. Such planners, while typically either resolution complete or probabilistically complete, are not able to determine when a path does not exist, and therefore would otherwise rely on heuristics to determine when the search for a free path should be abandoned. An implementation of the algorithm is provided for the case of a 3D Euclidean configuration space, and a proof of correctness is provided.