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The cost of leg forces in bipedal locomotion: a simple optimization study.

Rebula JR, Kuo AD - PLoS ONE (2015)

Bottom Line: We therefore tested a generalized force cost that can penalize force amplitude or its n-th time derivative, raised to the p-th power (or p-norm), across a variety of combinations for n and p.A simple model shows that this generalized force cost only produces smoother, human-like forces if it penalizes the rate rather than amplitude of force production, and only in combination with a work cost.Humans might find it preferable to avoid rapid force production, which may be mechanically and physiologically costly.

View Article: PubMed Central - PubMed

Affiliation: University of Michigan, Ann Arbor, Michigan, USA.

ABSTRACT
Simple optimization models show that bipedal locomotion may largely be governed by the mechanical work performed by the legs, minimization of which can automatically discover walking and running gaits. Work minimization can reproduce broad aspects of human ground reaction forces, such as a double-peaked profile for walking and a single peak for running, but the predicted peaks are unrealistically high and impulsive compared to the much smoother forces produced by humans. The smoothness might be explained better by a cost for the force rather than work produced by the legs, but it is unclear what features of force might be most relevant. We therefore tested a generalized force cost that can penalize force amplitude or its n-th time derivative, raised to the p-th power (or p-norm), across a variety of combinations for n and p. A simple model shows that this generalized force cost only produces smoother, human-like forces if it penalizes the rate rather than amplitude of force production, and only in combination with a work cost. Such a combined objective reproduces the characteristic profiles of human walking (R² = 0.96) and running (R² = 0.92), more so than minimization of either work or force amplitude alone (R² = -0.79 and R² = 0.22, respectively, for walking). Humans might find it preferable to avoid rapid force production, which may be mechanically and physiologically costly.

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Comparison of work-minimizing model of locomotion with human walking and running gaits, in terms of body center of mass (COM) trajectories and vertical ground reaction forces vs. time.The model’s COM trajectories feature a sharp, instantaneous redirection due to ideal impulsive vertical ground reaction forces (asterisks denote infinitely high peaks, shown truncated) in both walking and running. In contrast, human gait has much smoother COM trajectories and more rounded vertical leg forces with finite peaks.
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pone.0117384.g001: Comparison of work-minimizing model of locomotion with human walking and running gaits, in terms of body center of mass (COM) trajectories and vertical ground reaction forces vs. time.The model’s COM trajectories feature a sharp, instantaneous redirection due to ideal impulsive vertical ground reaction forces (asterisks denote infinitely high peaks, shown truncated) in both walking and running. In contrast, human gait has much smoother COM trajectories and more rounded vertical leg forces with finite peaks.

Mentions: One cost in locomotion is for the production of mechanical work. During walking, work is needed to redirect the body center of mass (COM) between pendulum-like steps [5], with a roughly proportional metabolic cost [6]. A simple two dimensional walking model [7] shows that the redirection is performed most economically by applying perfectly impulsive (i.e., instantaneously brief but with high amplitude) forces along the two legs, pushing off with the back leg just before the front leg impacts the ground. A more general model relaxes the pendulum assumption and allows for arbitrary leg forces, yet still discovers the same walking gait using work minimization alone [8]. For higher speeds, the same optimization also discovers running gaits, where impulsive leg forces are again favored, except separated by aerial phases. These optimizations suggest that the strategy for reducing mechanical work would be to attempt to walk and run with rigid legs (Fig. 1).


The cost of leg forces in bipedal locomotion: a simple optimization study.

Rebula JR, Kuo AD - PLoS ONE (2015)

Comparison of work-minimizing model of locomotion with human walking and running gaits, in terms of body center of mass (COM) trajectories and vertical ground reaction forces vs. time.The model’s COM trajectories feature a sharp, instantaneous redirection due to ideal impulsive vertical ground reaction forces (asterisks denote infinitely high peaks, shown truncated) in both walking and running. In contrast, human gait has much smoother COM trajectories and more rounded vertical leg forces with finite peaks.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4338056&req=5

pone.0117384.g001: Comparison of work-minimizing model of locomotion with human walking and running gaits, in terms of body center of mass (COM) trajectories and vertical ground reaction forces vs. time.The model’s COM trajectories feature a sharp, instantaneous redirection due to ideal impulsive vertical ground reaction forces (asterisks denote infinitely high peaks, shown truncated) in both walking and running. In contrast, human gait has much smoother COM trajectories and more rounded vertical leg forces with finite peaks.
Mentions: One cost in locomotion is for the production of mechanical work. During walking, work is needed to redirect the body center of mass (COM) between pendulum-like steps [5], with a roughly proportional metabolic cost [6]. A simple two dimensional walking model [7] shows that the redirection is performed most economically by applying perfectly impulsive (i.e., instantaneously brief but with high amplitude) forces along the two legs, pushing off with the back leg just before the front leg impacts the ground. A more general model relaxes the pendulum assumption and allows for arbitrary leg forces, yet still discovers the same walking gait using work minimization alone [8]. For higher speeds, the same optimization also discovers running gaits, where impulsive leg forces are again favored, except separated by aerial phases. These optimizations suggest that the strategy for reducing mechanical work would be to attempt to walk and run with rigid legs (Fig. 1).

Bottom Line: We therefore tested a generalized force cost that can penalize force amplitude or its n-th time derivative, raised to the p-th power (or p-norm), across a variety of combinations for n and p.A simple model shows that this generalized force cost only produces smoother, human-like forces if it penalizes the rate rather than amplitude of force production, and only in combination with a work cost.Humans might find it preferable to avoid rapid force production, which may be mechanically and physiologically costly.

View Article: PubMed Central - PubMed

Affiliation: University of Michigan, Ann Arbor, Michigan, USA.

ABSTRACT
Simple optimization models show that bipedal locomotion may largely be governed by the mechanical work performed by the legs, minimization of which can automatically discover walking and running gaits. Work minimization can reproduce broad aspects of human ground reaction forces, such as a double-peaked profile for walking and a single peak for running, but the predicted peaks are unrealistically high and impulsive compared to the much smoother forces produced by humans. The smoothness might be explained better by a cost for the force rather than work produced by the legs, but it is unclear what features of force might be most relevant. We therefore tested a generalized force cost that can penalize force amplitude or its n-th time derivative, raised to the p-th power (or p-norm), across a variety of combinations for n and p. A simple model shows that this generalized force cost only produces smoother, human-like forces if it penalizes the rate rather than amplitude of force production, and only in combination with a work cost. Such a combined objective reproduces the characteristic profiles of human walking (R² = 0.96) and running (R² = 0.92), more so than minimization of either work or force amplitude alone (R² = -0.79 and R² = 0.22, respectively, for walking). Humans might find it preferable to avoid rapid force production, which may be mechanically and physiologically costly.

Show MeSH