Eecient Distance Computation Using Best Ellipsoid Fit

Knowledge of the distance between a robot and its surrounding environment is vital for any robotic system. The robot must obtain this information rapidly in order to plan and react in realtime. Our technique rst surrounds the robot links and the obstacles by optimal ellipsoids, and then computes the clearance of the links from the obstacles with a generalized distance function. This approach o ers an attractive alternative to the widely used technique of computing the distance via polyhedral representation of the robot and the obstacles. In particular, our approach o ers a drastic reduction in the complexity of the data structures: each polyhedron, typically represented by a list of its features and their adjacency graph; is replaced by a minimum-volume ellipsoid represented by its center and a symmetric matrix whose dimension is either two or three (the workspace dimension). Moreover, while the computation time of the distance between polyhedra is often a function of their geometrical complexity, computation time in the ellipsoidal case is essentially constant; and becomes even more rapid when it is computed repeatedly along the robot's trajectory. Our method consists of the following two algorithms: The rst computes the optimal ellipsoid surrounding a convex polyhedron. The second is an analytic formula for the free margin about one ellipsoid with respect to another, that is computed as a standard eigenvalue problem. An e cient incremental version of the latter algorithm is then proposed. This system has been implemented and preliminary simulation results are provided throughout the paper.