^{1,a)}

### Abstract

Many aspects of steady human locomotion are thought to be constrained by a tendency to minimize the expenditure of metabolic cost. This paper has three parts related to the theme of energetic optimality: (1) a brief review of energetic optimality in legged locomotion, (2) an examination of the notion of optimal locomotion speed, and (3) an analysis of walking on moving walkways, such as those found in some airports. First, I describe two possible connotations of the term “optimal locomotion speed:” that which minimizes the *total* metabolic cost per unit distance and that which minimizes the *net* cost per unit distance (total minus resting cost). Minimizing the total cost per distance gives the maximum range speed and is a much better predictor of the speeds at which people and horses prefer to walk naturally. Minimizing the net cost per distance is equivalent to minimizing the total daily energy intake given an idealized modern lifestyle that requires one to walk a given distance every day—but it is not a good predictor of animals’ walking speeds. Next, I critique the notion that there is no energy-optimal speed for running, making use of some recent experiments and a review of past literature. Finally, I consider the problem of predicting the speeds at which people walk on moving walkways—such as those found in some airports. I present two substantially different theories to make predictions. The first theory, minimizing total energy per distance, predicts that for a range of low walkway speeds, the optimal absolute speed of travel will be greater—but the speed relative to the walkway smaller—than the optimal walking speed on stationary ground. At higher walkway speeds, this theory predicts that the person will stand still. The second theory is based on the assumption that the human optimally reconciles the sensory conflict between the forward speed that the eye sees and the walking speed that the legs feel and tries to equate the best estimate of the forward speed to the naturally preferred speed. This sensory conflict theory also predicts that people would walk slower than usual relative to the walkway yet move faster than usual relative to the ground. These predictions agree qualitatively with available experimental observations, but there are quantitative differences.

A popular hypothesis in legged locomotion is that we move in a manner that roughly minimizes the energy cost. A brief account of the evidence for and against this hypothesis is presented. A specific instance of possible energy minimization is the preference of specific walking speeds by humans when walking in a manner that is most comfortable. What energy-like quantity should humans minimize? Should humans minimize the total energy cost per unit distance walked or should they minimize the energy cost per distance over and above what they expend while resting? Some adaptationist evolutionary arguments and experimental evidence favor the minimization of the total cost. Unlike for walking, there is a common perception that running has no optimal speed. We critique this perception, building on the discussion on optimal walking speeds and drawing from past literature. Finally, the problem of walking on moving walkways found in some airports is considered. Two theories are presented: one based on energy minimization and another based on the idea that the brain may get confused because the walking speed the eyes see is different from the walking speed the legs feel. For low walkway speeds, both theories predict that people would walk slower than usual relative to the walkway yet move faster than usual relative to the ground. At higher walkway speeds, the energy minimization theory predicts that people would ride the walkway standing still.

This work was supported by NSF Grant No. EF-0425878 (Frontiers in Biological Research), awarded to Philip Holmes at Princeton University. I thank Philip Holmes, John Milton, and an anonymous reviewer for thoughtful comments on an earlier version of this manuscript.

I. INTRODUCTION

II. BACKGROUND: A BRIEF REVIEW OF ENERGY OPTIMALITY

III. OPTIMAL LOCOMOTION SPEEDS

A. Preferred speeds

B. Optimal walking speed

C. Adaptationist accounts

D. Optimal running speed

IV. WALKING ON MOVING WALKWAYS: TWO THEORIES

A. Theory 1: Energy minimization

B. Theory 2: Resolving sensory conflicts optimally

C. Comparison with observations

V. DISCUSSION

VI. CONCLUSION

## Figures

The total metabolic cost per unit distance and the net metabolic cost per unit distance (total cost minus the resting cost). The speed that minimizes the total cost is substantially higher than the net cost.

The total metabolic cost per unit distance and the net metabolic cost per unit distance (total cost minus the resting cost). The speed that minimizes the total cost is substantially higher than the net cost.

Walking on a moving walkway, minimizing the cost per unit distance traveled. (a) Costs for walking at the optimal walking speed and cost for standing still on the walkway. [(b) and (c)] Walking at the optimal speed [Eqs. (5) and (11)] is better than standing still on the walkway at low walkway speeds; standing still is better at higher walkway speeds. The transition for human walking is at about 0.75 m/s. This figure uses values for and from Sec. III B.

Walking on a moving walkway, minimizing the cost per unit distance traveled. (a) Costs for walking at the optimal walking speed and cost for standing still on the walkway. [(b) and (c)] Walking at the optimal speed [Eqs. (5) and (11)] is better than standing still on the walkway at low walkway speeds; standing still is better at higher walkway speeds. The transition for human walking is at about 0.75 m/s. This figure uses values for and from Sec. III B.

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