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Terrestrial locomotion is a complex phenomenon that is often linked to the survival of an individual and of an animal species. Mathematical models seek to express in quantitative terms how animals move, but this is challenging because the ways in which the nervous and musculoskeletal systems interact to produce body movement is not completely understood. Models with many variables tend to lack biological interpretability and describe the motion of an animal with too many independent degrees of freedom. Instead, reductionist models aim to describe the essential features of a gait with the smallest number of variables, often concentrating on the center of mass dynamics. In particular, spring–mass models have been successful in extracting and describing important characteristics of running. In this paper, we consider the spring loaded inverted pendulum model under the regime of constant angular velocity, small compression, and small angle swept during stance. We provide conditions for the asymptotic stability of periodic trajectories for the full range of parameters. The hypothesis of linear angular dynamics during stance is successfully tested on publicly available human data of individuals running on a treadmill at different velocities. Our analysis highlights a novel bifurcation phenomenon for varying Froude number: there are periodic trajectories of the spring loaded inverted pendulum model that are stable only in a restricted range of Froude numbers, while they become unstable for smaller or larger Froude numbers.
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This content will become publicly available on October 1, 2026
Sharp conditions for the existence and multiplicity of smooth periodic solutions to a hybrid dynamical system for human running
In this paper, we provide necessary and sufficient conditions for the existence and multiplicity of periodic smooth solutions to a hybrid system of differential equations that describes animal running at constant angular velocity during stance. We show that solutions to this system exist if and only if the angular velocity during stance is lower than the angular velocity of the corresponding linear harmonic oscillator and satisfies a non-resonance condition. The proof of our main theorem relies on a combination of tools from oscillation theory for ODEs, intersection theory for real plane conics and the analysis of Poincaré maps. Furthermore, we demonstrate the validity of our result with numerical simulations for combinations of parameters corresponding to periodic, resonant, and unbounded solutions. Finally, we show that our model fits accurately publicly available data on human running on different conditions, including comfortable and fast running on a treadmill and overground.
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- Award ID(s):
- 2152789
- PAR ID:
- 10646279
- Publisher / Repository:
- 2025 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited
- Date Published:
- Journal Name:
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 481
- Issue:
- 2324
- ISSN:
- 1364-5021
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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