Nearly time-optimal paths for a ground vehicle

Abstract

It is well known that the sufficient family of time-optimal paths for both Dubins' as well as Reeds-Shepp's car models consist of the concatenation of circular arcs with maximum curvature and straight line segments, all tangentially connected. These time-optimal solutions suffer from some drawbacks. Their discontinuous curvature profile, together with the wear and impairment on the control equipment that the bang-bang solutions induce, calls for "smoother" and more supple reference paths to follow. Avoiding the bang-bang solutions also enhances the robustness with respect to any possible uncertainties. In this paper, our main tool for generating these nearly time-optimal, but nevertheless continuous-curvature paths, is to use the Pontryagin Maximum Principle (PMP) and make an appropriate choice of the Lagrangian function. Despite some rewarding simulation results, this concept turns out to be numerically divergent at some instances. Upon a more careful investigation, it can be concluded that the problem at hand is nearly singular. This is seen by applying the PMP to Dubins' car and studying the corresponding two point boundary value problem, which turn out to be singular. This is thus a counterexample to the widespread belief that all the information about the motion of a mobile platform lies in the initial values of the auxiliary variables associated with the PMP.

Optimization of Vehicular Trajectories under Gaussian Noise Disturbances

“…Satisfying either minimum time or minimum distance, the results in [6] showed that trajectory tracing for just one vehicle when looking for a specific heading can be useful when dealing with general robots, but not as much when talking about vehicles, which essentially would expect to avoid an accident, regardless of the final heading. In [7] the authors proposed an approach to solve the problem of tracing the trajectory for a car between two fixed points. This type of problem is a BVP (Boundary Value Problem) in which the main concern is to study the trajectory between two fixed points while optimizing the time employed to complete this trajectory according to some physical constraints in terms of maximum acceleration.…”

Optimal Planar Turns Under Acceleration Constraints

“…The Hamiltonian function (13), being independent of the position x, is invariant under the group of translations (x; v; x ; v ) 7 ! (x + p; v; x ; v ), p 2 .…”

“…The optimal paths were found to be concatenations of line segments, circular arcs of maximum curvature and arcs of clothoids having maximum derivative of the curvature. Other variations of the Dubins' problem considered include the use of a velocity model to replace the assumption of constant speed [12] which yields a suboptimal path, and the use of a cost function for penalizing rapid variations in the acceleration [13].…”

Planar length-optimal paths under acceleration constraints

“…The optimal paths were found to be concatenations of line segments, circular arcs of minimum turn radius and arcs of clothoids having maximum derivative of the curvature. Other variations of the Dubins' problem considered include the use of a velocity model to replace the assumption of constant speed [14] which yields a sub-optimal path, and the use of a cost function for penalizing rapid variations in the acceleration [15].…”

“…Hence, in this paper we consider the path planning problem of taking a particle moving in a two-dimensional plane from a given initial position and velocity to a specified terminal heading along a length-optimal path subject to a magnitude constraint on the acceleration. As a departure from the work of [1]- [5], [10], [15], we do not require the speed of the particle to be constant. While the constant speed case is pertinent to aircraft, our problem is relevant to a terrestrial vehicle moving on a smooth floor that provides limited friction, or to a spacecraft moving in free space under the action of a gimballed thruster of limited capacity.…”