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research-article

A detailed look at the SLIP model dynamics: bifurcations, chaotic behaviour, and fractal basins-of-attraction

[+] Author and Article Information
Petr Zaytsev

Robotics and Motion Laboratory, ME, University of Michigan, Ann Arbor, MI, Institute for Nonlinear Mechanics, University of Stuttgart, Stuttgart, Germany
petr.zaytsev@inm.uni-stuttgart.de

Tom Cnops

Robotics and Motion Laboratory, ME, University of Michigan, Ann Arbor, MI
tcnops@umich.edu

David Remy

Robotics and Motion Laboratory, ME, University of Michigan, Ann Arbor, MI, Institute for Nonlinear Mechanics, University of Stuttgart, Stuttgart, Germany
david.remy@inm.uni-stuttgart.de

1Corresponding author.

ASME doi:10.1115/1.4043453 History: Received September 20, 2018; Revised April 04, 2019

Abstract

This paper provides a comprehensive numerical analysis of a simple 2D model of running, the Spring-Loaded Inverted Pendulum (SLIP). The model consists of a point-mass attached to a massless spring leg; the leg angle at touch-down is fixed during the motion. We employ numerical continuation methods combined with extensive simulations to find all periodic motions of this model, determine their stability, and compute the basins-of-attraction of the stable solutions. The result is a detailed and complete analysis of all possible SLIP model behavior which expands upon and unifies a range of prior studies. In particular, we demonstrate and explain the following effects: (i) saddle-node bifurcations which lead to two distinct solution families for a range of energies and touch-down angles; (ii) period-doubling bifurcations which lead to chaotic behavior of the model; and (iii) fractal structures within the basins-of-attraction. In contrast to prior work, these effects are found in a single model with a single set of parameters while taking into account the full non-linear dynamics of the SLIP model.

Copyright (c) 2019 by ASME
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