Essential Properties of Numerical Integration for Time-optimal Trajectory Planning Along a Specified Path

TL;DR

This paper rigorously analyzes and proves key properties of the Numerical Integration method for time-optimal trajectory planning, addressing its limitations under kinematic constraints and providing verification through simulations.

Contribution

It provides the first rigorous mathematical proofs of properties of the NI method and introduces a failure detection algorithm considering kinematic constraints.

Findings

Properties of switch points and acceleration curves are established.

Failure conditions under torque constraints are mathematically derived.

Simulation results verify the theoretical properties.

Abstract

This letter summarizes some known properties and also presents several new properties of the Numerical Integration (NI) method for time-optimal trajectory planning along a specified path. The contribution is that rigorous mathematical proofs of these properties are presented, most of which cannot be found in existing literatures. We first give some properties regarding switch points and accelerating/decelerating curves of the NI method. Then, for the fact that when kinematic constraints are considered, the original version of NI which only considers torque constraints may result in failure of trajectory planning, we give the concrete failure conditions with rigorous mathematical proof. Accordingly, a failure detection algorithm is given in a run-and-test manner. Some simulation results on a unicycle vehicle are provided to verify those presented properties. Note that though those known properties are not discovered first, their mathematical proofs are given first in this letter. The detailed proofs make the theory of NI more complete and help interested readers to gain a thorough understanding of the method.