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

Autor:	Łukasz Płociniczak, Hanna Okrasińska-Płociniczak
Rok vydání:	2020
Předmět:	Applied Mathematics Mathematical analysis General Engineering Pendulum Phase (waves) Stiffness Numerical verification 01 natural sciences Square (algebra) Computer Science::Robotics Gait (human) Spring (device) Modeling and Simulation 0103 physical sciences medicine Boundary value problem medicine.symptom 010306 general physics 010301 acoustics Mathematics
Zdroj:	Journal of Nonlinear Science. 30:2971-2988
ISSN:	1432-1467 0938-8974
DOI:	10.1007/s00332-020-09641-w
Popis:	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.
Databáze:	OpenAIRE
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