Overground Gait transitions: Optimality of Walk-Run-Rest mixtures
Human experiments and Mathematical Theory

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Title: Walking, running, and resting under time, distance, and average speed constraints: optimality of walk–run–rest mixtures.

Journal and year: Journal of Royal Society Interface, 10, 20120980, 2013.

Authors: Leroy L. Long and Manoj Srinivasan

Full article: (PDF with supplementary information)

Abstract: On a treadmill, humans switch from walking to running beyond a characteristic transition speed. Here, we study human choice between walking and running in a more ecological (non-treadmill) setting. We asked subjects to travel a given distance overground in a given allowed time duration. During this task, the subjects carried, and could look at, a stopwatch that counted down to zero. As expected, if the total time available were large, humans walk the whole distance. If the time available were small, humans mostly run. For an intermediate total time, humans often use a mixture of walking at a slow speed and running at a higher speed. With analytical and computational optimization, we show that using a walk–run mixture at intermediate speeds and a walk–rest mixture at the lowest average speeds is predicted by metabolic energy minimization, even with costs for transients—a consequence of non-convex energy curves. Thus, sometimes, steady locomotion may not be energy optimal, and not preferred, even in the absence of fatigue. Assuming similar non-convex energy curves, we conjecture that similar walk–run mixtures may be energetically beneficial to children following a parent and animals on long leashes. Humans and other animals might also benefit energetically from alternating between moving forward and standing still on a slow and sufficiently long treadmill.

Major results

For intermediate average speeds (just over 2 m/s), humans use a mixture of walking and running, as shown in the following GPS speed measurements for five subjects.

Humans mixing walking and running.

The following figure shows that for low average speeds, humans mostly use walking, for high average speeds, humans use mostly use running, and for intermediate average speeds, humans use a mixture of walking and running.

Humans mixing walking and running.

Why is a walk-run mixture or a walk-rest mixture optimal? This picture tells the story. IF one is forced to choose between walking or running -- we simply choose whichever has lower cost -- the blue curve (walking cost) or the red curve (running cost). But, if we can mix walking and running, we can do better. For instance, for a range of speeds (yellow band), mixing and running lowers the average to the black straight line -- which is below both blue and red curves. This improvement has to do with the non-convexity of the lower envelope of the walking and running cost curves.

Humans mixing walking and running.

Other Implications, predictions, and conjectures

1. A child following her parent at 1.4 m/s may save energy by a walk-run mixture.

2. Dogs on a long leash at average 1.3 m/s may save energy by walk-trot mixtures

3. Animals and humans on long slow treadmills may stand/coast backward and then scoot forward (walk or run).

4. Average Marathoners (2.3 m/s, 5 hour marathon, 10 minute mile) will save energy by a walk-run mixture.

5. Soccer players (2.1 m/s), and persistence hunters (1.75 m/s) have average speeds close to when a walk-run mixture is optima. While they do have other constraints, do they mix walking and running?

6. If you are given a choice between walk and then rest by sitting versus walk and then rest by lying down, theory predicts that you should walk faster in the second situation!

7. Theory suggests greatest behavioral variability near the gait transition, as observed in experiment.

The appendix includes a number of other mathematical optimization calculations applied to gait transitions on a long treadmill, etc.


ScienceNOW (Science Magazine's website): Are we built to be lazy? By Lizzie Wade.
Wired Magazine. By Lizzie Wade (same as above).
Runner's World. By Scott Douglas.
National Geographic. Why We Walk … and Run … And Walk Again to Get Where We're Going. By Marc Silver.
Prevention Magazine. By Kiera Aaron.

KennisLink (Dutch).

OSU News Release. PhysOrg. by Pam Gorder.
EurekaAlert. Hindustan Times.

Spanish. Chinese. Italian. Korean. (Translations and Adaptations of above)


This work was supported by the Ohio State University.