Hip Actuations Can be Used to Control Bifurcations and Chaos in a Passive Dynamic Walking Model

[+] Author and Article Information
Max J. Kurz1

Integrative Physiology Laboratory, Department of Health and Human Performance,  University of Houston, 3855 Holman St., 104S Garrison, Houston, TX 77204-6015mkurz@uh.edu

Nicholas Stergiou

HPER Biomechanics Laboratory, School of HPER,  University of Nebraska at Omaha, Omaha, NE 68182


Corresponding author.

J Biomech Eng 129(2), 216-222 (Dec 01, 2006) (7 pages) doi:10.1115/1.2486008 History: Received February 27, 2004; Revised December 01, 2006

We explored how hip joint actuation can be used to control locomotive bifurcations and chaos in a passive dynamic walking model that negotiated a slightly sloped surface (γ<0.019rad). With no hip actuation, our passive dynamic walking model was capable of producing a chaotic locomotive pattern when the ramp angle was 0.01839rad<γ<0.0190rad. Systematic alterations in hip actuation resulted in rapid transition to any locomotive pattern available in the chaotic attractor and induced stability at ramp angles that were previously considered unstable. Our results detail how chaos can be used as a control scheme for locomotion.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

The passive dynamic walking model where ϕ is the angle of the swing leg, θ is the angle of the stance leg, γ is the angle of inclination of the supporting surface, k is the stiffness of the hip torsional spring that is acting on the swing leg, m is the mass of the foot, M is the large mass of the combined pelvis and torso, and g is gravity. Both legs are of length l.

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Figure 2

Bifurcation map of the model's locomotive patterns for the respective ramp angles. A similar bifurcation map was found by Garcia (18). Chaotic locomotive patterns are present in the bifurcation map between the ramp angles of 0.01839rad<γ<0.0190rad. Without hip joint actuation, no stable locomotive patterns were found beyond 0.0190rad.

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Figure 3

Poincaré maps for (a) γ=0.0180rad, (b) γ=0.01823rad, (c) γ=0.0183rad, (d) strange attractor at γ=0.0189rad. Axes of the respective graphs are presented in seconds.

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Figure 4

Percentage of global false nearest neighbors for the time series intervals of the passive dynamic walking model at γ=0.0189rad with a time lag of 1

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Figure 5

Poincaré sections for the model while walking at γ=0.01823rad and (a) k=0s−2, (b) k=0.002s−2, (c) k=0.01s−2, (d) k=0.06s−2. Increases in k result in the emergence of lower order gaits. Axes of the respective graphs are presented in seconds.

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Figure 6

A series of steps from the passive dynamic walker at a ramp angle of 0.0189rad. No hip actuation was supplied prior to the 200th step and the model walked with a chaotic gait pattern. After the 200th step, a hip actuation of 0.06s−2 was applied which promoted the gait pattern to rapidly transition to a period-1 gait.



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