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Put a bounce in your step

September 11, 2005 By Michael Hopkin This article courtesy of Nature News.

Maths model shows why we move the way we do.

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As Monty Python's John Cleese famously complained, silly walks just don't get the support they deserve. But there is probably a good reason for that: a mathematical model shows that the traditional styles of walking and running really are the most efficient ways to get around.

That's despite the fact that both walking and running involve bouncing up and down, says Andy Ruina of Cornell University in Ithaca, New York, who carried out the analysis with his colleague Manoj Srinivasan.

The leading theory of locomotion holds that humans and other animals should move in a way that minimizes the amount of energy they expend. So Ruina and Srinivasan were intrigued to find out why all this jiggling is more efficient than slinking along as smoothly as possible, like a waiter carrying a tray of soup.

Ground level

It shows the most energy-efficient way to move along is to bounce up and down.
Andy Ruina
Cornell University
The pair created a mathematical model that reduces locomotion to its fundamentals: a certain mass, representing the body, that must be moved around on two struts, the legs, that can absorb forces and expend energy to shift the mass.

They used the model to deduce the styles that require the least effort to move a mass. As they report in their study, to be published by Nature1, walking is best at slow speeds, and running is the most efficient mode when moving in top gear.

Although other models have investigated the efficiency of walking and running before, this is the first to work out the best means of locomotion from first principles. "It shows that the most energy-efficient way to move along is to bounce up and down," says Ruina.

The model also predicted a third efficient way of moving, somewhere between running and walking. The style, dubbed 'pendular running', is not generally used in the real world. Ruina describes it as looking like an 80-year-old trying to run.

"It's like a lumbering run, maybe for a fat or out-of-shape person," he adds.

Upended monkeys

When walking, each leg acts as a pendulum, swinging the body's mass from leg to leg like an upside-down version of a monkey swinging from branch to branch. As a result, each footfall generates two force peaks: one as the foot lands, and another as it pushes off again.

Walking is most efficient at speeds where the body is moving slowly enough to be passed smoothly from one pendulum to the other. This explains why walking is still efficient even when though the body bobs up and down - this is simply a product of the legs' natural pendular swing.

When running, the foot acts more like a ball, bouncing the body back up into the air in the same instant that it lands. Each footfall therefore generates a single force peak. At speeds where the body is going too fast to be transferred smoothly from pendulum to pendulum, the most efficient method is to bounce it along in a series of free-falling flights.

The third way

Pendular running falls halfway between the two. As in running, the two feet are never on the ground at the same time. But each foot stays on the ground long enough to cause the double force peak characteristic of walking, Ruina explains.

Perhaps the reason most of us don't use this in the real world is simply because we don't need to, he adds.

But some people may benefit from the odd gait. Ruina notes that his model doesn't take into account our legs' natural springiness, which would make bounce-running even more efficient. But some people, such as those who are very heavy, might lack this springiness. For them, a lolloping half-run might be the most efficient way to travel.


  1. Srinivasan M. & Ruina A. Nature, doi: 10.1038/nature04113 (2005).


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