Optimal feedback velocity profile generation for a vehicle with given acceleration limits: A level set implementation

Abstract

In this paper, we use level set methods to numerically generate the minimum-time optimal velocity profiles for a vehicle with given acceleration limits driving along a specified path. The proposed approach solves the HamiltonJacobi-Bellman (HJB) equation associated with the given optimal control problem by computing the level sets of the value function. Once the optimal cost is found, the optimal feedback control can be computed online thus generating the velocity profile quickly. The results are compared to a semi-analytic approach that was developed recently for the same problem by the last two authors. I. INTRODUCTIONIn a series of recent papers the last two authors of this paper have developed a semi-analytic solution to the problem of optimal velocity generation of a point-mass vehicle on a prescribed path with a given, elliptical acceleration en-provides an alternative approach for the trajectory optimization problem of ground vehicles, which, typically, is dealt with in the literature using numerical methods [4], [5], [6]. These numerical optimization approaches are computationally intensive, and they cannot be readily applied in cases where the environment changes unpredictably. As a result, they are not suitable for real-time path-planning optimization. Motivated by the requirement to reduce the on-board computational cost, references [1], [2], [3] have exploited the potential of solving the trajectory planning problem (or at least part of this problem) analytically or semi-analytically. Indeed, for a point mass model moving on a given path, the problem is essentially one-dimensional (using the path arc length as the independent variable) with double-integrator dynamics. Hence, it can be readily solved. However, several complications arise from additional state and control constraints that make the problem more intricate than the standard time-optimal problem of the double-integrator [7]. Although "intuitive" solutions for this problem were known for sometime [8], [9], [10] (see also [11], [12], [13]), nonetheless, the work in [1], [3] was the first to offer a rigorous proof of optimality using optimal control theory. The necessary optimality conditions were explicitly derived, allowing one to determine the number and type of control switchings. In particular, the switching points can be found semi-analytically instead of numerically (as it was done in [11], [12] and [13]).