Asymptotic Optimality in Sampling-based Motion Planning

Although one of the fundamental problems in robotics, the motion planning problem is inherently hard from a computational point of view. In particular, the piano movers’ problem [1], [2] is known to be PSPACE-hard, which implies that any algorithm aimed to solve this problem (with completeness guarantees) is expected not to scale well with increasing number of dimensions of the configuration space. In fact, the computation time required by well-known complete algorithms scale exponentially with the dimensionality of the configuration space in the worst case [3], which makes them impractical, e.g., in motion planning problems for robotic arms with several joints. The optimal motion planning problem is known to be significantly harder computationally when compared to finding just a feasible solution, even when the dimensionality of the configuration space is fixed. In particular, it is known that the optimal motion planning problem is NP-hard for a point mass moving among polygonal obstacles in a three dimensional configuration space [4]. Early research on motion planning during 1980s has involved optimal motion planning algorithms only with limited success [5], [6], [7]. However, especially after the emergence of a deeper understanding of the computational complexity of the problem during the late 1980s, the community shifted towards designing algorithms that can quickly find feasible solutions, usually with no optimality guarantees.