Toward Optimal Sampling in the Space of Paths

While spatial sampling of points has already received much attention, the motion planning problem can also be viewed as a process which samples the function space of paths. We define a search space to be a set of candidate paths and consider the problem of designing a search space which is most likely to produce a solution given a probabilistic representation of all possible environments. We introduce the concept of relative completeness which is the prior probability, before the environment is specified, of producing a solution path in a bounded amount of computation. We show how this probability is related to the mutual separation of the set of paths searched. The problem of producing an optimal set can be related to the maximum k-facility dispersion problem which is known to be NP-hard. We propose a greedy algorithm for producing a good set of paths and demonstrate that it produces results with both low dispersion and high prior probability of success.