Why not walk faster?

Abstract

Bipedal walking following inverted pendulum mechanics is constrained by two requirements: sufficient kinetic energy for the vault over midstance and sufficient gravity to provide the centripetal acceleration required for the arc of the body about the stance foot. While the acceleration condition identifies a maximum walking speed at a Froude number of 1, empirical observation indicates favoured walk-run transition speeds at a Froude number around 0.5 for birds, humans and humans under manipulated gravity conditions. In this study, I demonstrate that the risk of 'take-off' is greatest at the extremes of stance. This is because before and after kinetic energy is converted to potential, velocities (and so required centripetal accelerations) are highest, while concurrently the component of gravity acting in line with the leg is least. Limitations to the range of walking velocity and stride angle are explored. At walking speeds approaching a Froude number of 1, take-off is only avoidable with very small steps. With realistic limitations on swing-leg frequency, a novel explanation for the walk-run transition at a Froude number of 0.5 is shown.

Figure 1

Stance during the ‘compass gait’ model of walking. The point mass (black circle) vaults over the massless, rigid leg of length l, following an arc about the leg's connection with the ground. The components of acceleration acting in line with the leg are indicated for two positions: (a) midstance, or ϕ=0, when the velocity of the mass is at a minimum for the step (V_min) and gravity acts directly in line with the leg and (b) at the extreme of leg angle ϕ=−Φ or Φ, when the velocity of the mass is highest for the step (V_max) and a smaller component of gravity acts along the leg. The conditions for successful inverted pendulum walking are considered broken if there is insufficient energy for the vault over midstance or if the centripetal acceleration required to keep the leg arcing around the foot cannot be provided by gravity.

Figure 2

Limiting parameters for walking as an inverted pendulum with sufficient energy for the vault past midstance (i) but without ‘take-off’ (ii). (a) Values appropriate for a human walking on Earth (l=1 m, g=9.81 m s⁻²). High walking speeds can only be achieved with low step lengths and high step frequencies. An approximate upper limit to comfortable walking speed for men is indicated by a circle. In non-dimensionalized form (b), the step frequency $\hat{f}$ is the step frequency normalized by the frequency of a pendulum of length l swinging over small angles. The rectangle bounds the range of values relating to the walk–run transition for seven species of ground-dwelling bird ranging from 0.045–90 kg (where Φ is taken as half of the stance leg excursion angle published in Gatesy & Biewener (1991)). The largest possible step angle, 48.2°, is indicated but is achievable at only one velocity. Lower leg angles are required for higher walking speeds; however, this requires higher step frequencies. The combination of avoidance of take-off and a limitation to swing-leg frequency provides realistic limitations to inverted pendulum style walking speeds and may account for the consistent preferred speed of the walk–run transition in bipeds of F_r=0.5 (vertical dotted lines).