Asymptotic Optimality of a Time Optimal Path Parametrization Algorithm

Abstract

Time Optimal Path Parametrization is the problem of minimizing the time interval during which an actuation constrained agent can traverse a given path. Recently, an efficient linear-time algorithm for solving this problem was proposed [1]. However, its optimality was proved for only a strict subclass of problems solved optimally by more computationally intensive approaches based on convex programming. In this paper, we prove that the same linear-time algorithm is asymptotically optimal for all problems solved optimally by convex optimization approaches. We also characterize the optimum of the Time Optimal Path Parametrization Problem, which may be of independent interest.

Joint Feature Selection and Time Optimal Path Parametrization for High Speed Vision-Aided Navigation

“…To analyze proposed algorithms, we draw upon several results in [22] that tell us that for any fixed selection of landmarks, the set of feasible square speed profiles is closed under: (a) pointwise suprema and infima; and (b) convex combinations. We note that result (a) implies that the time optimal profile for tracking a chosen subset of landmarks is the pointwise maximum of the set of such profiles.…”

Perception-aware time optimal path parameterization for quadrotors

“…The convexity of the problem implies that at an arbitrary discretization point s i , the sets of square speeds that can be extended to feasible profiles on [s i , S end ] and [0, s i ] are both intervals. The algorithm leverages this insight to recover an asymptotically optimal solution [20] consisting of upper endpoints of such intervals, obtained incrementally in a pair of sweeps through (s i ) n i=0 , as detailed in stage one of Algorithms 1 and 2.…”