We demonstrate that an irreducibly simple, uncontrolled, two-dimensional, two-link model, vaguely resembling human legs, can walk down a shallow slope, powered only by gravity. This model is the simplest special case of the passive-dynamic models pioneered by McGeer (1990a). It has two rigid massless legs hinged at the hip, a point-mass at the hip, and infinitesimal point-masses at the feet. The feet have plastic (no-slip, no-bounce) collisions with the slope surface, except during forward swinging, when geometric interference (foot scuffing) is ignored. After nondimensionalizing the governing equations, the model has only one free parameter, the ramp slope γ. This model shows stable walking modes similar to more elaborate models, but allows some use of analytic methods to study its dynamics. The analytic calculations find initial conditions and stability estimates for period-one gait limit cycles. The model exhibits two period-one gait cycles, one of which is stable when 0 < γ < 0.015 rad. With increasing γ stable cycles of higher periods appear, and the walking-like motions apparently become chaotic through a sequence of period doublings. Scaling laws for the model predict that walking speed is proportional to stance angle, stance angle is proportional to γ1/3, and that the gravitational power used is proportional to ν4 where ν is the velocity along the slope.
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April 1998
Technical Papers
The Simplest Walking Model: Stability, Complexity, and Scaling
Mariano Garcia,
Mariano Garcia
Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell University, Ithaca, NY 14853
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Anindya Chatterjee,
Anindya Chatterjee
Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell University, Ithaca, NY 14853
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Andy Ruina,
Andy Ruina
Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell University, Ithaca, NY 14853
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Michael Coleman
Michael Coleman
Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell University, Ithaca, NY 14853
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Mariano Garcia
Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell University, Ithaca, NY 14853
Anindya Chatterjee
Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell University, Ithaca, NY 14853
Andy Ruina
Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell University, Ithaca, NY 14853
Michael Coleman
Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell University, Ithaca, NY 14853
J Biomech Eng. Apr 1998, 120(2): 281-288 (8 pages)
Published Online: April 1, 1998
Article history
Received:
November 12, 1996
Revised:
July 31, 1997
Online:
October 30, 2007
Citation
Garcia, M., Chatterjee, A., Ruina, A., and Coleman, M. (April 1, 1998). "The Simplest Walking Model: Stability, Complexity, and Scaling." ASME. J Biomech Eng. April 1998; 120(2): 281–288. https://doi.org/10.1115/1.2798313
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