A New Perspective on Double-S Curve Motions of Higher Order and Optimal Motion Planning

TL;DR

This paper introduces a universal equation for symmetric trajectories with zero boundary conditions, applicable to various motion phases, and proposes an algorithm for optimal motion planning that simplifies complex calculations.

Contribution

It provides a novel, general equation for symmetric trajectories of arbitrary order, and an efficient algorithm for time-optimal motion planning without case distinctions.

Findings

Equation holds regardless of the number of motion phases

Algorithm reduces time minimization to solving a system of equations

Application examples demonstrate effectiveness in minimum time, velocity, and acceleration motions

Abstract

This paper presents and proves an equation for the time horizon of symmetric trajectories with zero boundary conditions and bounded derivatives of arbitrary order. This equation holds regardless of the number of phases comprising the associated motion. This avoids case distinctions in calculations. Application examples of motions with minimum time, minimum velocity, and minimum acceleration are discussed. Furthermore, an algorithm is derived that reduces the time minimization problem to solving a system of equations. This algorithm avoids nested case distinctions and complex optimizations.