Mechanical and energetic consequences of rolling foot shape in human walking

Abstract

During human walking, the center of pressure under the foot progresses forward smoothly during each step, creating a wheel-like motion between the leg and the ground. This rolling motion might appear to aid walking economy, but the mechanisms that may lead to such a benefit are unclear, as the leg is not literally a wheel. We propose that there is indeed a benefit, but less from rolling than from smoother transitions between pendulum-like stance legs. The velocity of the body center of mass (COM) must be redirected in that transition, and a longer foot reduces the work required for the redirection. Here we develop a dynamic walking model that predicts different effects from altering foot length as opposed to foot radius, and test it by attaching rigid, arc-like foot bottoms to humans walking with fixed ankles. The model suggests that smooth rolling is relatively insensitive to arc radius, whereas work for the step-to-step transition decreases approximately quadratically with foot length. We measured the separate effects of arc-foot length and radius on COM velocity fluctuations, work performed by the legs and metabolic cost. Experimental data (N=8) show that foot length indeed has much greater effect on both the mechanical work of the step-to-step transition (23% variation, P=0.04) and the overall energetic cost of walking (6%, P=0.03) than foot radius (no significant effect, P>0.05). We found the minimum metabolic energy cost for an arc foot length of approximately 29% of leg length, roughly comparable to human foot length. Our results suggest that the foot's apparently wheel-like action derives less benefit from rolling per se than from reduced work to redirect the body COM.

Keywords: arc foot; biomechanics; fixed ankle; foot length; locomotion; metabolic energy; rigid ankle; rocker bottom foot; rollover shape; round foot.

Fig. 1.

The effect of varying radius and length of an arc-shaped foot, in a simple walking model. (A) The model has radius ρ and length l, and performs positive work with a push-off impulse P and negative work with a collision impulse C at the step-to-step transition, to redirect the COM velocity. The amount of redirection (represented by angle δ) is determined by the distance between the points of ground contact for the two arcs. More work is needed to perform greater amounts of redirection. (B) Varying foot length, keeping radius fixed, greatly affects the amount by which the velocity must be redirected. (C) In contrast, varying arc radius, keeping foot length fixed, has substantial effect on the COM trajectory, but no effect on redirection. The velocity change due to push-off and collision are shown as Δv_PO and Δv_CO, respectively. The step length, kept constant, determines the angle between the legs, 2α. Keeping step length constant across conditions, the redirection angle is at a minimum (δ*) when the feet are long enough to roll without pivoting on the heel and toe. The moment arm of GRF about the knee, typically small in human walking, is affected strongly by foot length but only slightly by radius; this effect may be partly responsible for the increased cost caused by long feet of large radius (Adamczyk et al., 2006).

Fig. 2.

Dynamic walking model predictions for step-to-step transition work as a function of foot radius and length. (A) Step-to-step transition work per step (vertical axis) depends on both arc radius and length, the latter having a much greater effect. (B) Work decreases nearly quadratically as a function of foot length, as demonstrated for various foot radii. (C) In contrast, work per step varies little with foot radius, as shown for various foot lengths. The approximate range of experimental conditions is denoted by the shaded area on the horizontal axes. Filled circles in A show the parameters studied experimentally (see Fig. 3). In B and C, the boundary of the parameter space for the current model shows results that agree closely with a previous model, in which length and radius varied together (Adamczyk et al., 2006).

Fig. 3.

Apparatus used to impose an arc-like shape to the foot bottom, while keeping the ankles fixed (see also Adamczyk et al., 2006). Arcs were constructed in five lengths and three radii of curvature.

Fig. 4.

Average ground reaction force data over a step cycle, for varying (A) foot length keeping radius fixed and (B) foot radius keeping length fixed. Curves are averaged across all (N=8) subjects. A step was defined as starting at one heelstrike and ending at opposite heelstrike.

Fig. 5.

(A,B) Vertical versus forward components of center of mass (COM) velocity data, termed COM hodographs, for arc feet of varying (A) length and (B) radius. (C,D) COM work rate versus step cycle for varying (C) foot length and (D) foot radius. Curves are shown for both legs (labeled as leading and trailing leg during step-to-step transition). All curves are averaged across subjects (N=8).

Fig. 6.

Experimental results for varying foot length (left column) and foot radius (right column). (A,B) Average redirection angle of COM velocity (δ_COM) decreased significantly (*P<0.05) as function of (A) foot length (P=0.03), but not (B) foot radius (P=0.08). Angular redirection of COM velocity is a strong predictor of COM work (Adamczyk and Kuo, 2006; Adamczyk et al., 2009). (C,D) Average COM work rate changed as predicted by the model, decreasing significantly across (C) foot length (P=0.04 for linear term) but not (D) foot radius (P=0.33). Average COM work was considerably lower than normal in all cases. (E,F) Net metabolic rate also changed significantly with (E) foot length but not (F) foot radius. Quadratic curve fits (solid lines) yield minima at l=0.285 for foot length (P=0.03) and ρ=0.553 for foot radius (P=0.31). All trials from all subjects (N=8, filled symbols) are shown along with curve fits (solid dark lines) to equations predicted by the model.

Fig. 7.

Comparison of best-fit quadratic curves for (A) COM work rate and (B) metabolic rate versus foot length (solid black lines) and foot radius (dotted cyan lines) from the present study and from a previous experiment varying length and radius together (dashed red lines) (Adamczyk et al., 2006). The close match between the current ‘varying length’ curve and the previous result emphasizes that mechanical and metabolic cost changes attributable to arc foot shape are primarily due to changes in foot length. The ‘varying radius’ curve from the present study departs substantially from the previous result, suggesting that foot radius per se was not responsible for the changes observed in that study (Adamczyk et al., 2006). Note that length and radius share a common axis in the previous study (dashed red lines) because geometric design constraints made the two parameters numerically equal within a reasonable small-angle approximation.

Fig. A1.

Sagittal joint mechanics (angle, moment, power output) across variations in foot length. Curves are averaged across all eight subjects. Values are nondimensionalized on the left-hand axes; values on the right-hand axes units are normalized to body mass.

Fig. A2.

Sagittal plane joint mechanics (angle, moment, power output) across variations in foot radius. Curves are averaged across all eight subjects. Note the trend toward increasing knee moment in late stance, which may reflect the altered timing of GRF center of pressure advancement due to changing foot radius. Values are nondimensionalized on the left-hand axes; values on the right-hand axes units are normalized to body mass.