The cost of bounded curvature

We study the motion-planning problem for a car-like robot whose turning radius is bounded from below by one and which is allowed to move in the forward direction only (Dubins car). For two robot configurations @s,@s^', let @?(@s,@s^') be the shortest bounded-curvature path from @s to @s^'. For d>=0, let @?(d) be the supremum of @?(@s,@s^'), over all pairs (@s,@s^') that are at Euclidean distance d. We study the function dub(d)[email protected]?(d)-d, which expresses the difference between the bounded-curvature path length and the Euclidean distance of its endpoints. We show that dub(d) decreases monotonically from dub(0)[email protected]/3 to dub(d^@?)[email protected], and is constant for d>=d^@?. Here d^@?~1.5874. We describe pairs of configurations that exhibit the worst-case of dub(d) for every distance d.

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