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

Area of Science:

• Biomechanics
• Sports Science
• Human Movement Analysis

Background:

• Force-velocity (Fv) and force-position (Fp) models are established for single-joint movements.
• Characterizing multi-joint movements like vertical jumps requires integrated modeling approaches.
• Optimizing athletic performance necessitates accurate models of force production during dynamic actions.

Purpose of the Study:

• To evaluate the efficacy of combined Fv and Fp models in characterizing force production during the vertical jump push-off phase.
• To compare the performance of six different Fv,p models integrating Anderson, Hill, and Linear Fv models with Cosine and Quadratic Fp models.
• To determine the most suitable model for practical application in sports performance analysis.

Main Methods:

• Fifteen trained CrossFit athletes performed maximal countermovement jumps under varied loads and push-off depths.
• Ground reaction forces were recorded using force plates during jump execution.
• Six integrated force-velocity-position (Fv,p) models were fitted to the experimental data.

Main Results:

• All six Fv,p models demonstrated high goodness-of-fit (r²: 0.885–0.886) and low RMSE (262.6–266.5 N).
• No significant differences in fitting or descriptive capacity were found among the Anderson, Hill, and Linear Fv models.
• The Quadratic Fp model, when combined with the Linear Fv model, showed potential in reducing maximal force (Fmax) deviations in specific cases.

Conclusions:

• Combined Fv,p models effectively characterize force production during the vertical jump push-off.
• A simple, three-parameter heuristic model offers biomechanically and physiologically relevant parameters for applied sports science.
• This modeling approach provides practical insights for optimizing multi-joint movement performance in athletes.