Census of bounded curvature paths

Abstract

A bounded curvature path is a continuously differentiable piece-wise C2 path with bounded absolute curvature connecting two points in the tangent bundle of a surface. These paths have been widely considered in computer science and engineering since the bound on curvature models the trajectory of the motion of robots under turning circle constraints. Analyzing global properties of spaces of bounded curvature paths is not a simple matter since the length variation between length minimizers of arbitrary close endpoints or directions is in many cases discontinuous. In this note, we develop a simple technology allowing us to partition the space of spaces of bounded curvature paths into one-parameter families. These families of spaces are classified in terms of the type of connected components their elements have (homotopy classes, isotopy classes, or isolated points) as we vary a parameter defined in the reals. Consequently, we answer a question raised by Dubins (Pac J Math 11(2):471481, 1961).

História e cultura: negros do Rosário de Pombal

“…That is, in general, the length of the Dubins path from (x, X) to (y, Y ) is different from (y, Y ) to (x, X). Length minimising bounded curvature paths may not be unique, refer to [14] for details about these claims. Definition 4.11.…”

“…And, due to symmetry γ is also of minimal length under these constraints. We use the methods in [14] to compute the following values. Since the length of each ccc path is aproximatelly 6, 0325 we have that Length(γ) ≈ 2 × 6, 0325 and Length(β) ≈ 8, 2831 both of thickness 1 we conclude that: Rop(γ ∪ β) ≈ 20.3481.…”