Optimal control approach to Olympic weightlifting exercise: Minimal model of the snatch pull

TL;DR

The paper tackles modeling the explosive vertical barbell motion in Olympic weightlifting using optimal control. It formulates a minimal one-dimensional vertical dynamics system with two phase-specific objectives that penalize the applied force and its rate of change, yielding analytic trajectories under acceleration- or jerk-optimization. The results show that both optimization schemes reproduce the observed barbell trajectory $z(t)$ with distinct force and velocity profiles, revealing how impulse and rate of force development (RFD) interplay across the first and second pulls. This framework provides an analytic basis for understanding weightlifting mechanics and can inform technique analysis and training design, with potential extensions to multi-body models and other lifts.

Abstract

We theoretically investigate the biomechanical aspects of Olympic weightlifting within the framework of optimal control theory. The squared force and the rate of force development (RFD) defined by the time derivative of the force are taken into account in the evaluation functions of the first and second pull phases of the snatch motion. Focusing on the vertical trajectory of the barbell, we develop a minimal model to describe the mechanical characteristics of the weightlifting exercise. The calculated barbell trajectory agrees well with the experimental data obtained by video analysis. Our study would be useful for the further development of mathematical models for weightlifting motions and related exercises.

Figures (2)

• Figure 1: Observed vertical position of the bar during the snatch in Ref. nagao2022validation. At the beginning, an athlete pulls the bar relatively slowly to prepare for a strong drive at an appropriate posture. This phase is called "first pull." In turn, at the appropriate posture which might be close to that of the vertical jump, an athlete use the whole muscle strength to induce a strong force on the bar, called "second pull." After these pulling phases, an athlete tries to catch the bar.
• Figure 2: (a) Vertical position $z(t)$, (b) velocity $v(t)$, and (c) force $F(t)$ during the snatch motion. The gray circles show the experimental data in Ref. nagao2022validation. The solid and dashed curves represent the acceleration optimization model (A-opt.) and the jerk optimization model (J-opt.), respectively. The chain-dot curve at $t\geq t_{2,{\rm f}}=0.90\,{\rm s}$ corresponds to the free fall motion with $F(t)=0$ in the catch phase.