On the Structure of the Time-Optimal Path Parameterization Problem with Third-Order Constraints

TL;DR

This paper explores the structure of the time-optimal path parameterization problem under third-order constraints, introduces key difficulties, and proposes a new algorithm, TOPP3, to address these challenges in robotics.

Contribution

It provides the first detailed analysis of third-order constrained TOPP and introduces the TOPP3 algorithm to improve solution computation.

Findings

Identifies key difficulties in connecting optimal profiles and handling singularities.

Proposes the TOPP3 algorithm to address these issues.

Marks progress towards efficient solutions for third-order constrained TOPP.

Abstract

Finding the Time-Optimal Parameterization of a Path (TOPP) subject to second-order constraints (e.g. acceleration, torque, contact stability, etc.) is an important and well-studied problem in robotics. In comparison, TOPP subject to third-order constraints (e.g. jerk, torque rate, etc.) has received far less attention and remains largely open. In this paper, we investigate the structure of the TOPP problem with third-order constraints. In particular, we identify two major difficulties: (i) how to smoothly connect optimal profiles, and (ii) how to address singularities, which stop profile integration prematurely. We propose a new algorithm, TOPP3, which addresses these two difficulties and thereby constitutes an important milestone towards an efficient computational solution to TOPP with third-order constraints.