Classification of the Dubins set

Given two points in a plane, each with a prescribed direction of motion in it, the question being asked is to find the shortest smooth path of bounded curvature that joins them. The classical 1957 result by Dubins gives a sufficient set of paths (each consisting of circular arcs and straight line segments) which always contains the shortest path. The latter is then found by explicitly computing all paths on the list and then comparing them. This may become a problem in applications where computation time is critical, such as in real-time robot motion planning. Instead, the logical classification scheme considered in this work allows one to extract the shortest path from the Dubins set directly, without explicitly calculating the candidate paths. The approach is demonstrated on one of two possible cases that appear here — when the distance between the two points is relatively large (the case with short distances can be treated similarly). Besides computational savings, this result sheds light on the nature of factors affecting the length of paths in the Dubins problem, and is useful for further extensions, e.g. for finding the shortest path between a point and a manifold in the corresponding configuration space.