Mathematical modeling of a force-velocity-position relationship in vertical jump

Abstract

This study aimed to evaluate whether combining force-velocity Fv and force-position Fp models, originally developed for single-joint movements, could effectively characterize force production during the push-off phase of a vertical jump. Six force-velocity-position Fv,p models, integrating three Fv (Anderson, Hill, and Linear) and two Fp (Cosine and Quadratic) models were assessed. Fifteen trained CrossFit athletes performed maximal countermovement jumps under varying loads and push-off depths with ground reaction forces recorded via force plates. All six models demonstrated high goodness-of-fit, with r² ranging from 0.885 to 0.886 and RMSE values ranging from 262.6 to 266.5 N, effectively capturing key experimental data characteristics. No significant differences in fitting or descriptive capacity were observed among Anderson, Hill, and Linear models, reflecting the near-linear behavior of the force-velocity relationships in vertical jump. Nevertheless, the Linear model offers simplicity and interpretability by focusing on key physiological parameters (e.g., maximal force, maximal velocity, and optimal position) commonly used in applied sports contexts. The Cosine and Quadratic models showed no significant impact on overall fit quality, although significant differences in optimal vertical position (p_opt) and theoretical maximal force (F_max) were observed. When paired with the Linear model, the Quadratic model slightly reduced F_max deviations in participants with slightly curvilinear force-velocity relationships. This study highlights the strength of a simple three-parameter heuristic model, whose parameters are biomechanically and physiologically relevant, in describing the force production as a function of position and velocity. This combination of simplicity and interpretability represents a significant step forward in the modeling of multi-joint movements, offering practical insights for sport performance optimization.

Keywords: Biomechanical modeling; Countermovement jump; Force–velocity-position relationship; Maximal force production; Multi-joint.