On calculation of optimal paths with constrained curvature: the case of long paths

Given two points in the plane, each with the prescribed direction of motion, the question being asked is to find the shortest smooth path of bounded curvature that joins them. The classical result by Dubins (1957) that is commonly used gives a sufficient set of paths which is guaranteed to contain the shortest path; the latter is then found by explicitly calculating every path in the set. In this paper we show that in the case when the distance between the two points is above some minimum, the solution sought can be found via a simple classification scheme. Besides computational savings (essential, for example, in real-time motion planning), this result sheds light on the nature of factors affecting the length of paths in the Dubins's problem.