The classification of homotopy classes of bounded curvature paths

Abstract

Abstract. A bounded curvature path is a continuously differentiable piecewise C 2 path with a bounded absolute curvature that connects two points in the tangent bundle of a surface. In this work, we analyze the homotopy classes of bounded curvature paths for points in the tangent bundle of the Euclidean plane. We show the existence of connected components of bounded curvature paths that do not correspond to those under (regular) homotopies obtaining the first results in the theory outside optimality. An application to robotics is presented. PreludeGiven a class of curves satisfying some constraints, understanding when there is a deformation connecting two curves in the class, where all the intermediate curves also are in the class, strongly relies on the defining conditions. When considering continuity any two plane curves (both closed or with different endpoints) are homotopic one into the other. Whitney in 1937 observed that under regular homotopies (homotopy through immersions) not always two planar closed curves lie in the same connected component [26]. In fact, there are as many regular homotopy classes of plane curves as integers. When considering curves with different endpoints the concept of homotopies through immersions do not lead to results different from those obtained when only continuity is considered. Dubins in 1957 introduced the concept of bounded curvature path when characterizing bounded curvature paths of minimal length [9] 1 .Let (x, X), (y, Y ) ∈ T R 2 be elements in the tangent bundle of the Euclidean plane. A planar bounded curvature path is a C 1 and piecewise C 2 path starting at x, finishing at y; with tangent vectors at these points X and Y respectively and having absolute curvature bounded by κ = 1 r > 0. Here r is the minimum allowed radius of curvature. The piecewise C 2 property comes naturally due to the nature of the length minimisers [9].A substantial part of the complexity of the theory of bounded curvature paths is described in the following observation. In general, length minimisers are used to establish a distance function between points in a manifold. This approach may not be considered for spaces of bounded curvature paths since in many cases the length variation between length minimisers of arbitrarily close endpoints or directions is discontinuous. Closely related is the fact that for most cases the length minimisers from (x, X) to (y, Y ) and from (y, Y ) to (x, X) have different length contradicting

On the Homology of the Space of Curves Immersed in the Sphere With Curvature Constrained to a Prescribed Interval

“…We use an idea similar and inspired by Birkhoff curve shortening (see [10]). Similar ideas and related studies by others may be found in [18], [9], [8], [5], [6] and [7]. Given a curve γ ∈L ρ 0 (Q), let L 0 be the length of γ, we construct a new curve that is shorter than or has the same length as the original curve, by the following process: we separate the curve in small sections, each section, except the first and last (that have length ≤ l), have the fixed length l, with l ≤ π sin ρ 0 .…”

“…From Theorem 3.2 and Proposition 3.7 for ρ =ρ, P = Q i and Q = Q i+1 , the length-minimizing curve is of type CSC. The continuity can be proven by using the same argument as J. Ayala and H. Rubinstein's argument in [8] for the plane case. The idea is to define a region Ω that depends continuously on Q i and Q i+1 .…”

“…Thus this implies Equations (4-9) and (4)(5)(6)(7)(8)(9)(10). Again using the fact that three consecutive zeroes cannot happen for x k for k ∈ {0, 1, 2, .…”

“…The research on the topological aspects of spaces of curves has not been restricted exclusively to sphere S n . For curves on 2-dimensional Euclidean plane, here we mention the articles: [34], [11], [12], [27], [28], [6], [5], [8], [9]. We also mention [3] for RP 2 , [29] for two dimensional hyperbolic space with constant curvature −1, and [30] and [19] for general Riemannian manifolds, respectively.…”

“…If both γ 1 and γ 3 are degenerate, we call γ of type S, so on. This kind of nomenclature is commonly used on studies of Dubins' curves (see, for example, [9]). …”