Efficient collision checking in sampling-based motion planning via safety certificates.

Collision checking is considered to be the most expensive computational bottleneck in sampling-based motion planning algorithms. We introduce a simple procedure that theoretically eliminates this bottleneck and significantly reduces collision-checking time in practice in several test scenarios. Whenever a point is collision checked in the normal expensive way, we store a lower bound on that point's distance to the nearest obstacle. The latter is called a ¿safety certificate¿ and defines a region of the search space that is guaranteed to be collision-free. New points may forgo collision checking whenever they are located within a safety certificate of an old point. Testing the latter condition is accomplished during the nearest-neighbor search that is already part of most sampling-based motion planning algorithms. As more and more points are sampled, safety certificates asymptotically cover the search space and the amortized complexity of normal, expensive collision checking becomes negligible with respect to the overall runtime of sampling-based motion planning algorithms. Indeed, the expected fraction of points requiring a normal collision check approaches zero, in the limit, as the total number of points approaches infinity. A number of extensions to the basic idea are presented. Experiments with a number of proof-of-concept implementations demonstrate that using safety certificates can improve the performance of sampling-based motion planning algorithms in practice.