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

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.

💡 Research Summary

The paper addresses a long‑standing difficulty in optimal motion planning for symmetric trajectories whose boundary conditions are zero and whose derivatives up to an arbitrary order are bounded. Classical treatments of double‑S (or S‑curve) motions treat each order separately and rely on a cascade of case distinctions (e.g., 4‑phase, 5‑phase, 6‑phase, 7‑phase profiles) to derive the relationship between the travel distance, the peak values of velocity, acceleration, jerk, etc., and the total motion time. The authors propose a unified analytical framework that eliminates all such case‑by‑case analysis.

First, the authors formalize the problem. A one‑dimensional trajectory x(t) is defined on the interval