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Mechanical work for step-to-step transitions is a major determinant of the metabolic cost of human walking

J. Maxwell Donelan¹, Rodger Kram² and Arthur D. Kuo^3,*

¹ Department of Integrative Biology, University of California, Berkeley, CA 94720-3140, USA
² Department of Kinesiology and Applied Physiology, University of Colorado, Boulder, CO 80309-0354, USA
³ Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA

View larger version (23K): [in a new window]   Fig. 1. (A) The simplest two-dimensional passive dynamic walking model has two degrees of freedom, stance leg angle and swing leg angle, and is restricted to motion in the sagittal plane. Mass is concentrated in points located at the pelvis (M) and feet (m), making it possible to compute step-to-step transition costs analytically (Garcia et al., 1998; Kuo, 2002) (B) The anthropomorphic three-dimensional passive dynamic walking model (Kuo, 1999) extends this model in two ways. First, it employs a torsional hip spring acting between the limbs, making it possible to explore the mechanics of walking at different step lengths or frequencies (after Kuo, 2002). Second, it includes an extra degree of freedom allowing for lateral motion and finite step widths. Step width is adjusted by changing the splay angle, ß. The model has three degrees of freedom (stance, swing and roll angles).

View larger version (25K): [in a new window]   Fig. 2. (A) The direction of the center of mass velocity, vcm, is perpendicular to the stance limb during the single support inverted pendulum phase of the simplest two-dimensional passive dynamic walker. (B) Each transition to a new stance limb requires redirection of the center of mass velocity, from vcm(-) to vcm(+) (with the superscripts `-' and `+' denoting the instances immediately before and after impact, respectively), accomplished by an impulsive heel strike, S, acting along the leading limb. S also causes an instantaneous reduction in the magnitude of the center of mass velocity through negative work by the leading limb with (shaded square). To walk at steady speed, an equal amount of positive work is required (see Kuo, 2002; Donelan et al., 2002). The magnitude of Wtrans(-)·, and thus the step-to-step transition cost, depends on vcm(-) and the angle between the legs, 2 (Equation 1). (C) When step frequency is kept fixed, vcm(-) and 2 are proportional to step length, l, so that Wtrans(-) increases with l4 (denoted by the differences in area of the shaded squares.

View larger version (19K): [in a new window]   Fig. 3. (A) Walking models predict that the rate of external mechanical work dissipated in collisions is proportional to the fourth power of step length l (keeping step frequency fixed; Kuo, 2002). The simplest two-dimensional (Fig. 1A) and anthropomorphic three-dimensional passive dynamic walking models (Fig. 1B) both give similar predictions. (B) The anthropomorphic model predicts that leg motion also contributes to external work rate, with a term proportional to the square of step length. Step length is expressed as a fraction of leg length, L. Mechanical work rate shown is made dimensionless by dividing by , where M is body mass and g is the gravitational acceleration. C,D, constants. See Materials and methods for details.

View larger version (29K): [in a new window]   Fig. 4. Average ground reaction forces from two force plates during a single step of walking at three different step lengths (N=9). (A) Leading leg force plate, (B) trailing leg force plate. (C) Center of mass velocity for a single step, computed from ground reaction forces. The dot product of the ground reaction forces and center of mass velocity yields average external mechanical power produced by (D) the leading leg and (E) the trailing leg during a single step at different step lengths. As a result of the changes in ground reaction forces and center of mass velocities, the external mechanical power generated within a step increased at longer step lengths. Grey lines denote double support. Medio—lateral forces and velocities (not shown) are relatively small and change little with step length. They are, however, included in all calculations. l*, preferred step length.

View larger version (21K): [in a new window]   Fig. 5. (A) Increases in negative external mechanical work rate (black circles) and (B) net metabolic rate (black circles) were both dominated by the fourth power of step length l. External mechanical work rate (A) is compared against a nonlinear regression from Equation 3 (black line), and metabolic power (B) is compared against a regression from Equation 4 (black line). Note that traditional combined limbs measures of total negative external mechanical work rate (grey circles in A) underestimated the external work rate generated by the individual limbs. Values shown are means ± S.D., N=9. l*, preferred step length; C,C', D,D', constants. See Materials and methods for details.

View larger version (23K): [in a new window]   Fig. 6. Correlation between mechanical step-to-step transition costs and metabolic costs for varying step lengths. One estimate for step-to-step transition costs (open circles) is found from negative external mechanical work rate; a least-squares linear regression of these data exhibits linearity (broken line, r2=0.89). This is probably an overestimate because it fails to attribute some of the increases in power to leg motion. A lower bound on step-to-step transition costs (solid circles) is found by correcting for the contribution due to leg motion, Clegl2 (Equation 3); a least-squares linear regression of these data also exhibits linearity (solid line, r2=0.79). These data are for step lengths l* in the range (0.6-1.4l*).

View larger version (37K): [in a new window]   Fig. 7. (A) Walking models predict that the mechanical energy dissipated in collisions is a function of step length l and step width w (expressed as fractions of leg length, L). (B,C) Slices through the surface of A. (B) Collision costs increase with step width squared when walking with a fixed, substantial step length, as tested previously (Kuo, 1999; Donelan et al., 2001). (C) Collision costs increase with step length to the fourth power when walking at a fixed step width, as tested here. These predictions were constructed using a simple 3-D walking model (Fig. 1B; Donelan et al., 2001), walking at different step lengths or widths but constrained to walking with a fixed step frequency. C,D, constants. See text for details

View larger version (27K): [in a new window]   Fig. A1. (Appendix). Conceptual diagram of mathematical cancellation between different contributors to external mechanical power. Simultaneous positive and negative external mechanical power (broken line) is produced during double support due to propulsion and collision, respectively. But collision power in humans also extends beyond double support, and propulsion power also precedes double support. External power due to leg motion (dotted line) overlaps and mathematically cancels these quantities in measurements of external power during single support, even though they do not physically interact. The result is that total external mechanical power (solid line) tends to underestimate the positive and negative power associated with both step-to-step transitions and leg motion.

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