Time-optimal trajectories with bounded curvature in anisotropic media

Abstract

This paper characterizes time-optimal trajectories in anisotropic (direction-dependent) environments where path curvatures are bounded by the inverse of the minimum-turning radius of a mobile agent. Such problems are often faced in the navigation of aerial, ground and naval vehicles when a mobile agent cannot instantaneously change its heading angle. The work presented is a generalization of the Dubins car problem, which considers the fastest paths with bounded curvature while assuming constant speed and minimum-turning radius. We relax this assumption and discuss fastest-path finding problems for the generalized direction-dependent speed and minimum-turning radius functions, to account for the effects of waves, winds and slope of the terrain on the agent's motions. We establish that there exists an optimal path such that it is a portion of a path of the type CSCSC where C denotes a sharpest-turn curve and S a straight-line segment. We further analyze a special case wherein the speed polar plot is convex, and show that in that case there exists an optimal path with the same structure as for the Dubins problem: CCC or CSC. An algorithm that implements our results for the convex speed polar plot is also presented.

The classification of homotopy classes of bounded curvature paths

“…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].…”

Constrained Global Path Optimization for Articulated Steering Vehicles

“…Representation of the performance index depends on a user's interests, such as energy consumption [18], travel time [19], [20], smoothness (evaluated by the integral of the curvaturesquared function) [21], [5], [17], [3], or safety from collisions (evaluated by the integral of the proximity-to-obstacle function) [17], [3]. Kelly and Nagy [21] optimize parametric smoothness motions using cubic polynomial curvature functions of the path lengths.…”

Restricted Orientation Dubins Path With Application to Sailboats

“…The Dubins' vehicles with constrained orientations have been studied. Primary work comes from [9], where the boat studied has a constrained rate of turning. The shortest path therein was proven to be of the type, or a subset of CSCSC in an anisotropic medium, or more specifically with a convex polar speed diagram.…”

“…The problem considered here is similar to [9], but the velocity considered here is binary rather than continuous: the velocity will be null in the range of the restricted orientation and full speed otherwise. While [9] gives a more generic and complete solution to constructing a Dubins path for all vehicles with a continuous convex speed polar plot, this paper proposes a simple geometric algorithm for a more specific Dubins car with a non-continuous binary speed and a constant turning radius. By restricting the domains of orientation, different problems arise such as the feasibility of the path, which are to be discussed here.…”