Classification of the Dubins set

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

show abstract

“…Applications can be found in computer science [1,5,8,11,15,21], control theory [7,13,16,17,18,20,22,23] and engineering [3,4,6,14,19,25].…”

Section: Preludementioning

confidence: 99%

The classification of homotopy classes of bounded curvature paths

Ayala

Rubinstein

2016

Isr. J. Math.

View full text Add to dashboard Cite

show abstract

“…45 Our purpose in this section is to study the length of optimal paths given different boundary conditions. Namely, the problem of steering a Dubins vehicle from a given initial configuration to a point in the plane will be considered; the difference with the original problem posed by Dubins 5 is that the final heading at the target point is not constrained a priori.…”

Section: B Minimum-length Paths and The Reachable Set For The Dubinsmentioning

confidence: 99%

Stochastic and Dynamic Routing Problems for Multiple Uninhabited Aerial Vehicles

Enright

Savla

Bullo

2009

Journal of Guidance, Control, and Dynamics

View full text Add to dashboard Cite

Consider a routing problem for a team of vehicles in the plane: target points appear randomly over time in a bounded environment and must be visited by one of the vehicles. It is desired to minimize the expected system time for the targets, i.e., the expected time elapsed between the appearance of a target point, and the instant it is visited. In this paper, such a routing problem is considered for a team of Uninhabited Aerial Vehicles (UAVs), modeled as vehicles moving with constant forward speed along paths of bounded curvature. Three algorithms are presented, each designed for a distinct set of operating conditions. Each is proven to provide a system time within a constant factor of the optimal when operating under the appropriate conditions. It is shown that the optimal routing policy depends on problem parameters such as the workload per vehicle and the vehicle density in the environment. Finally, there is discussion of a phase transition between two of the policies as the problem parameters are varied. In particular, for the case in which targets appear sporadically, a dimensionless parameter is identified which completely captures this phase transition and an estimate of the critical value of the parameter is provided. Special Euclidean group in two dimensions (i.e., the group of rigid planar motions) t time, s T the steady-state system time, s v vehicle speed, ms −1

show abstract

“…This work is based on the alternating algorithm of Savla et al (2005) with the considerations of headings introduced by Ny et al (2012). The notations and results of Shkel and Lumelsky (2001) are used to achieve efficient computations of the Dubins curves.…”

Section: Automatic Planningmentioning

confidence: 99%