Asymptotic Solution of a Boundary Value Problem for a Spring–Mass Model of Legged Locomotion

Running is the basic mode of fast locomotion for legged animals. One of the most successful mathematical descriptions of this gait is the so-called spring–mass model constructed upon an inverted elastic pendulum. In the description of the grounded phase of the step, an interesting boundary value pro...

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Bibliographic Details
Published in:Journal of nonlinear science 2020-12, Vol.30 (6), p.2971-2988
Main Authors: Okrasińska-Płociniczak, Hanna, Płociniczak, Łukasz
Format: Article
Language:English
Subjects:
Analysis
Angle of attack
Asymptotic methods
Asymptotic series
Boundary value problems
Classical Mechanics
Economic Theory/Quantitative Economics/Mathematical Methods
Gait
Locomotion
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Pendulums
Stiffness
Theoretical
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Summary:Running is the basic mode of fast locomotion for legged animals. One of the most successful mathematical descriptions of this gait is the so-called spring–mass model constructed upon an inverted elastic pendulum. In the description of the grounded phase of the step, an interesting boundary value problem arises where one has to determine the leg stiffness. In this paper, we find asymptotic expansions of the stiffness. These are conducted perturbatively: once with respect to small angles of attack, and once for large velocities. Our findings are in agreement with previous results and numerical simulations. In particular, we show that the leg stiffness is inversely proportional to the square of the attack angle for its small values, and proportional to the velocity for large speeds. We give exact asymptotic formulas to several orders and conclude the paper with a numerical verification.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-020-09641-w