Yoyo Dynamics: Sequence of Collisions Captured by a Restitution Effect

Abstract

This paper focuses on modeling the dynamics of a yoyo—a popular toy, which presents a challenging task for robotic control. A dynamic two-degree-of-freedom (DOF) model of the yoyo with a unilateral constraint is first proposed. The flight of the yoyo is naturally split into four phases, namely, a free-motion phase, a constrained-motion phase, a bottom phase, and a transition phase. It is shown that the energy loss is mainly due to a sequence of collisions that occur between the yoyo and the string during the bottom and transition phases. The dynamics in the bottom and transition phases is simplified by a single impulsive effect with an equivalent restitution coefficient, which depends only on the mass, inertia and axle radius of the yoyo. The resulting one-DOF model captures the energy loss while greatly simplifying the dynamic model, and thus facilitating the analysis and design of yoyo control. Both models are verified by experiments conducted on three different yoyos.