Repetitive Gait of Passive Bipedal Mechanisms in a Three-Dimensional Environment

[+] Author and Article Information
Harry Dankowicz

Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061e-mail: danko@vt.edu

Jesper Adolfsson, Arne B. Nordmark

Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

J Biomech Eng 123(1), 40-46 (Oct 16, 2000) (7 pages) doi:10.1115/1.1338121 History: Received January 18, 1999; Revised October 16, 2000
Copyright © 2001 by ASME
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Bifurcation diagrams of five different model mechanisms under variations in slope showing stable (solid) and unstable (dashed) periodic solutions. All solution branches in the compliant models terminate in foot scuffing. The limping-gait branches in the constrained models are born in pitchfork bifurcations from the main branch (the other branch is not shown). The numbers included in the top five panels indicate the average gait speed. The bottom panel is a blow-up of the stable region in the fifth panel showing the birth of quasi-periodic (dotted) and eventually chaotic (cloud) solutions at slope 0.0795.
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The amplitude of lateral oscillations of a branch of periodic solutions under a one-parameter variation of the ground compliance (here the damping is proportional to the square root of the stiffness). The inlays indicate the location of the toe points during the gait. Note, in particular, the switch in the right-most inlay at 0.5 s between the inner and outer toe, and at 0.58 s between the inner toe on one foot and the outer toe on the other foot.
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Geometry and mass distribution of the planar walker. The toe circles represent the cylindrical feet used in the simulations. Note that the heels have no physical significance in our model.
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A typical gait cycle of the planar walker illustrated by six characteristic events
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The path from a constrained walker with cylindrical feet to an unconstrained mechanism with point contacts
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Geometric make-up and mass properties of the three-dimensional mechanism
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A simulated gait sequence of the planar walker including the trajectories of the hip, a knee, and a toe. Note that the time interval between successive images of the walker slightly exceeds a full stride time.
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A simulated gait sequence of the unconstrained mechanism including the trajectory of the center of the hip over eleven consecutive steps. The percentages correspond to the time of the event relative to one stride. Stride time = 1.62 s, stride length = 0.7 m. The largest modulus of any eigenvalue of the linearized dynamics in the vicinity of this periodic orbit is 0.9685, i.e., weakly asymptotically stable.
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The time histories of the hip and knee angles of the left leg during one stride of the gait in Fig. 6. Note that after the knee impact, the knee angle is less than the hip angle, corresponding to a “locked” knee.
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Trajectory of the center of mass of the mechanism projected on the inclined plane and the toe-point locations. The numbered events along the continuous trajectory indicate contact being established or interrupted by the corresponding toes. The center of mass spends 0.054 s outside the lateral support line.




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