Optimal path planning under full state and map knowledge is often accomplished using some variant of Dijkstra's algorithm, despite the fact that it represents the path domain as a discrete graph rather than as a continuous space. In this paper we compare Dijkstra's discrete algorithm with a variant (often called the fast marching method) which more accurately treats the underlying continuous space. Analytically, both generate a value function free of local minima, so that optimal path generation merely requires gradient descent. We also investigate the use of optimality metrics other than Euclidean distance for both algorithms. These different norms better represent optimal paths for some types of problems, as demonstrated by planning optimal collision-free paths for a multiple robot scenario. When considering approximations consistent with the underlying state space, our conclusion is that fast marching places fewer constraints upon grid connectivity, and that it achieves better accuracy than Dijkstra's discrete algorithm in many but not all cases

We present a semi-parametric control policy representation and use it to solve a series of nonholonomic control problems with input state spaces of up to 7 dimensions. A nearest-neighbor control policy is represented by a set of nodes that induce a Voronoi partitioning of the input space. The Voronoi cells then define local control actions. Direct policy search is applied to optimize the node locations and actions. The selective addition of nodes allows for progressive refinement of the control representation. We demonstrate this approach on the challenging problem of learning to steer cars and trucks-with-trailers around winding tracks with sharp corners. We consider the steering of both forwards and backwards-moving vehicles with only local sensory information. The steering behaviors for these nonholonomic systems are shown to generalize well to tracks not seen in training.