On Polygonal Paths with Bounded Discrete-Curvature: The Inflection-Free Case

A shortest path joining two specified endpoint configurations that is constrained to have mean curvature at most ς on every non-zero length sub-path is called a ς-geodesic. A seminal result in non-holonomic motion planning is that (in the absence of obstacles) a 1-geodesic consists of either (i) a (unit-radius) circular arc followed by a straight segment followed by another circular arc, or (ii) a sequence of three circular arcs the second of which has length at least π [Dubins, 1957]. Dubins’ original proof uses advanced calculus; Dubins’ result was subsequently rederived using control theory techniques [Sussmann and Tang, 1991], [Boissonnat, Cérézo, and Leblond, 1994], and generalized to include reversals [Reeds and Shepp, 1990]. We introduce and study a discrete analogue of curvature-constrained motion. Our overall goal is to show that shortest polygonal paths of bounded “discrete-curvature” have the same structure as ς-geodesics, and to show that properties of ς-geodesics follow from their discrete analogues as a limiting case, thereby providing a new, and arguably simpler, “discrete” proof of the Dubins characterization. Our focus, in this paper, is on paths that have non-negative mean curvature everywhere; in other words, paths that are free of inflections, points where the curvature changes sign. Such paths are interesting in their own right (for example, they include an additional form, not part of Dubins’ characterization), but they also provide a slightly simpler context to introduce all of the tools that will be needed to address the general case in which inflections are permitted.