A New Approach to Time-Optimal Path Parameterization Based on Reachability Analysis

Abstract

Time-Optimal Path Parameterization (TOPP) is a well-studied problem in robotics and has a wide range of applications. There are two main families of methods to address TOPP: Numerical Integration (NI) and Convex Optimization (CO). NIbased methods are fast but difficult to implement and suffer from robustness issues, while CO-based approaches are more robust but at the same time significantly slower. Here we propose a new approach to TOPP based on Reachability Analysis (RA). The key insight is to recursively compute reachable and controllable sets at discretized positions on the path by solving small Linear Programs (LPs). The resulting algorithm is faster than NI-based methods and as robust as CO-based ones (100% success rate), as confirmed by extensive numerical evaluations. Moreover, the proposed approach offers unique additional benefits: Admissible Velocity Propagation and robustness to parametric uncertainty can be derived from it in a simple and natural way.

Asymptotic Optimality of a Time Optimal Path Parametrization Algorithm

“…This paper shows the algorithm outlined in [1] is in fact optimal in the limit as the distance between consecutive discretization points tends to zero. The main contribution of this paper is twofold.…”

“…They then jointly recover approximations of the optimal squared speed profile at every point [5] [6]. Although these methods are both numerically stable and converge to optimal solutions, their time complexity is considered to be high for many practical real-time robotics applications [1]. Similar to the convex-optimization-based approaches, algorithms developed in [1], [7]- [9] approximate the optimal squared speed at a set of discretization points.…”

“…Although these methods are both numerically stable and converge to optimal solutions, their time complexity is considered to be high for many practical real-time robotics applications [1]. Similar to the convex-optimization-based approaches, algorithms developed in [1], [7]- [9] approximate the optimal squared speed at a set of discretization points. In addition to being numerically stable, they are computationally efficient [1], which they achieve by exploiting additional structure possessed by time optimal path parametrization problems.…”

“…In addition to being numerically stable, they are computationally efficient [1], which they achieve by exploiting additional structure possessed by time optimal path parametrization problems. However, their optimality has been established for only a strict subset of problems solved optimally by convex optimization approaches [1].…”

“…In Section IV, we present our first main result that characterizes the optimal solution. We recall the algorithm given in [1] in Section V, and we present our second main result that proves the asymptotic optimality of this algorithm in Section VI.…”