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First published online December 14, 2006
Journal of Experimental Biology 210, iv (2007)
Copyright © 2007 The Company of Biologists Limited
doi: 10.1242/jeb.02635
Outside JEB |
MODELLING WALKING WITH TWO HUMPS
University of Maryland, College Park
tytell{at}umd.edu
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Since the seventeenth century, researchers have divided running and walking into two different mechanical models. Running requires a springy leg that compresses under the body's mass. This simple `spring-mass' model of running does surprisingly well in capturing the important dynamics of running. In the walking model, by contrast, the body is said to vault over a stiff leg, like an upside-down pendulum. The pivot is where the foot touches the ground. However, this `inverted pendulum' model of walking, despite its nearly universal application, misses important features of how people walk. First, it predicts that we should oscillate up and down much more than we do. Second, and more importantly, it predicts that the foot's force on the ground should rise smoothly as the body vaults over the leg, then drop off producing a force profile with one hump. Yet, experimental results show a two-humped profile - one force peak at the beginning of stance, when the foot is on the ground, and one at the end.
Hartmut Geyer and his colleagues at Friedrich-Schiller University in Jena, Germany, set out to develop a simple model of walking that would replicate the two-humped force trace seen in experiments. Their key insight, stunning as it may seem, is that walking requires two legs. Both the spring-mass and the inverted pendulum models describe only one leg during the time it's touching the ground. During walking, though, there are time periods when both feet are on the ground.
So the team put together a simple model with two legs, not one. More complex models had indicated that springy legs might be important not only for running, but for walking, also. Because of this, the group simulated each leg by a massless spring with a fixed resting length that swings around a point mass representing the body. To specify when the legs contact the ground, they set one simple condition - that they must touch down at a defined angle. This condition defines the body's location relative to the foot when the foot contacts the ground. Once a leg-spring touches down, it begins to compress under the body's weight. As the body moves forward, the spring begins to lengthen again; when it returns to its resting length, the leg comes off the ground and stops influencing the body until it touches down again during the next step. Modelling both legs in this way results in periods when both feet are on the ground and the body is supported by two legs.
The researchers were then left with three unspecified parameters in their model: the touch-down angle, the leg's stiffness, and an initial energy which kick-starts the system. When they tuned the parameter values, they found a plethora of different stable gaits. As they hoped, the model can walk with a two-humped force trace. But, in fact, it shows three different versions of two-humped walking - two high-speed walking gaits, and a slower version - plus gaits with three-, four- and five-humped force profiles. Conveniently, at high initial energies, the model also converges to a one-humped gait that turns out to be the standard spring-mass model of running.
In the end, Geyer's two-legged spring-mass model provides a simple, unified framework for describing the mechanics of both walking and running, as well as some weird patterns that humans don't seem to use. One model for both gaits not only eliminates the need for the over-simplified inverted pendulum model of walking, but also provides a template for exploring the relative efficiencies of walking and running and the dynamics of transitions between gaits.
References
Geyer, H., Seyfarth, A. and Blickhan, R. (2006). Compliant leg behaviour explains basic dynamics of walking and running. Proc. R. Soc. Lond. B. doi:10.1098/rspb.2006.3637 .
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