M1 Pendular Locomotion

Topic

Pendular locomotion describes walking in mammals: during each step, the center of mass oscillates over the limb in contact with the ground—acting like the mass of an inverted pendulum—and exchanges potential and kinetic energy in opposite directions throughout the cycle, such that when one increases, the other decreases.

This exchange between potential and kinetic energy is never perfect; however, at the optimal walking speed, pendular energy recovery reaches between sixty and seventy percent of the total energy exchanged during each step, and it is precisely this recovery that minimizes the metabolic cost of walking at that speed.

This optimal speed for energy exchange is governed by a dimensionless number that combines movement speed, gravitational acceleration, and the length of the supporting limb: walking, as a mode of locomotion, is only possible while this number remains below a critical value close to one-half, and the speed at which the exchange of potential and kinetic energy is most efficient corresponds to a value of this number close to one-quarter.

This same dimensionless number also predicts the point at which a mammal switches from walking to running: mammals of vastly different body sizes switch from one gait to the other upon reaching approximately the same value of this number—whether the animal is a mouse or an elephant—a prediction that has been experimentally verified across this vast range of body sizes.

The natural frequency at which the animal takes each step is determined, just as with a simple pendulum, by the length of its supporting limb: the longer the limb, the slower its natural oscillation—much like a longer pendulum swings more slowly than a shorter one—meaning that step frequency decreases in proportion to the square root of the limb length. Throughout this same pendulum-like cycle, the vertical component of the ground reaction force follows a profile with two peaks per step—one shortly after foot strike and another just before toe-off. These two peaks correspond to the moments in the cycle when the center of mass is at the lowest point of its arc; at this point, the body must bear not only its own weight but also the additional force required to decelerate and subsequently lift the center of mass again.

The metabolic cost of walking—expressed as energy expended per unit of mass and distance traveled—is lowest at the animal's naturally preferred walking speed, and it increases whether the animal walks slower or faster than that preferred speed.

Within this pendulum-like mode of locomotion, maintaining lateral stability poses the greatest challenge for bipedal mammals (humans and certain primates) because the system is inherently unstable in the frontal (side-to-side) plane, even when the forward exchange of potential and kinetic energy is functioning correctly. Maintaining lateral balance requires continuous adjustments to both foot placement and step width; since the metabolic cost of making these adjustments rises with the square of the speed of movement, walking fast is disproportionately more costly—in terms of lateral stability—than walking slowly.

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