A Simple Model of Bipedal Walking Predicts the Preferred Speed–Step Length Relationship

[+] Author and Article Information
Arthur D. Kuo

Dept. of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109-2125e-mail: artkuo@umich.edu

J Biomech Eng 123(3), 264-269 (Jan 11, 2001) (6 pages) doi:10.1115/1.1372322 History: Received September 22, 1999; Revised January 11, 2001
Copyright © 2001 by ASME
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Grahic Jump Location
Predicted speed-step length relationships for the Anthropomorphic Model (AM), compared with the empirical curve E∼v0.42 (thick solid line). Each prediction is based on a metabolic cost per distance E composed of toe-off work plus a hypothesized cost (listed in legend) for tuning the swing leg. The relative weight of these costs, c, was adjusted to make the predictions intersect the empirical curve at about v=0.35. The best prediction was made by the cost of force/time (force divided by duration of step). Swing work and peak force tend to modulate step frequency rather than length. The cost of impulse (integral of force over time) favors shorter step lengths at high speeds, opposite to observed behavior.
Grahic Jump Location
(a) Predicted contours of metabolic cost of transport, E, versus speed, v, and step length, s, assuming a cost proportional to a weighted sum of mechanical work performed at toe off and the force/time associated with tuning the swing leg of the Anthropomorphic Model (AM). Constant energy contours shown (thin solid lines) are in constant increments. For any given speed, E is minimized at a step length close to s=v0.42 (thick solid line). There is also a sharp increase in E with increasing step frequency (dotted lines denote contours of constant step frequency, as in Fig. 3). Units of bottom and left sides are dimensionless; units on top and right sides are for a model with leg length 1 m. (b) Relative contribution to the cost of transport between the toe-off impulse, Etoe, and the cost of force/time needed to actuate the swing leg, Eswing for preferred gaits. At least 50 percent of the overall cost of transport is due to the propulsive toe-off work.
Grahic Jump Location
Metabolic energy costs as a function of speed, v, and step length, s. Data are from 1, replotted in the manner of 2. Also shown is the preferred speed-stride length relationship s∼v0.42 (thick solid line) reported by 4. Energy contours are shown as a percentage of a nominal gait, with the cost of standing subtracted. Note that there is a sharp increase in energy consumption if speed is increased while keeping step length constant. At fast speeds and short step lengths, energy consumption appears to be dominated by step frequency, plotted as dotted line contours (τ is the dimensionless step period). Top and right axes are in actual SI units. Bottom and left axes are in dimensionless units: Step length is normalized by leg length l=0.98 m, and speed is normalized by gl and is equal to the square root of a Froude number 16. Dimensionless step period τ is normalized by l/g.
Grahic Jump Location
Three models of walking: (a) The Idealized Simplest Model (ISM) consists of point mass pelvis and feet connected by massless legs, after Garcia et al. 20. It employs a linearized analytic solution to the equations of motion. (b) The Simplest Model (SM) is identical to the ISM except that the equations of motion are solved numerically. (c) The Anthropomorphic Model (AM) is similar to that of McGeer 18, with legs with more realistic inertial parameters as well as curved feet to improve efficiency. All three models are powered on level ground by an impulsive push P along the stance leg applied at toe-off, as well as a springlike hip torque between the legs 22. The SM is used to test the linearizing assumptions of the ISM, while the AM tests the idealized inertial parameters.
Grahic Jump Location
Mechanical work performed: (a) by toe-off impulses on the stance leg, and (b) by hip torques on the swing leg, as a function of speed v and step length, s, using the Anthropomorphic Model. (a) Lines of constant (dimensionless) toe-off mechanical work per distance, Wtoe (solid lines, with energy levels as labeled), increase with both speed and step length. For a given toe-off impulse, the effect of increasing the natural frequency τ of the swing leg is to increase speed at slightly shorter step lengths. Walking becomes less costly because the hip spring decreases collision losses. Without a cost assigned to tuning the swing leg, there is no obvious relationship between mechanical work and the preferred speed-step length relationship (thick solid line from Fig. 1). Shaded regions denote unstable gaits. (b) Swing leg mechanical work per distance, Wswing, increases sharply with step frequency (1/τ, denoted by dotted lines). No linear combination of (a) and (b) predicts the preferred speed-step length relationship, indicating that other factors may contribute to cost of transport.




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