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First published online October 5, 2006
Journal of Experimental Biology 209, 3953-3963 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02455

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The advantages of a rolling foot in human walking

Peter G. Adamczyk^*, Steven H. Collins and Arthur D. Kuo

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA

View larger version (19K): [in a new window]   Fig. 1. A simple model demonstrates how a rolling foot can affect walking energetics. (A) Modeling the legs as pendulums supporting the body center of mass (COM), a step can be produced by passive limb dynamics with no energy input (McGeer, 1990a). Work is required, however, in the step-to-step transition to redirect the COM velocity. This can be accomplished with positive push-off work performed by the trailing leg, and negative collision work by the leading leg (Kuo, 2002). These leg actions redirect the pre-transition COM velocity vpre to a post-transition velocity vpost. For point feet, the net directional change in velocity is equal to the angle between the legs, 2. (B) A model with arc feet applies collision at the heel of the leading leg, and push-off at the toe of the trailing leg. This reduces the directional change in COM velocity and therefore the step-to-step transition work. (C) COM velocity change may be understood geometrically. The pre-transition velocity vpre is directed perpendicular to the line from the trailing leg's rolling point of ground contact to the COM. Push-off, directed along this line (angle /2 from vertical), causes a change in velocity (vmid=vpre+vpush-off). A periodic gait is achieved if push-off and collision velocity changes (vpush-off and vcollision, respectively) are of the same magnitude, so that vpost is equal in magnitude to vpre but directed according to rolling of the leading leg. Work is proportional to the square of each velocity change. As the arc foot radius (, defined as a fraction of leg length L) increases, less step-to-step transition work is needed. There is no redirection of COM velocity for a radius equal to leg length, =1.

View larger version (11K): [in a new window]   Fig. 2. Work performed on COM as a function of foot radius of curvature , from various dynamic walking models. Models are powered by push-off to walk on level ground: the `Simplest Model' (SM) with point mass pelvis and feet (Kuo, 2001), the `Anthropomorphic Model' (AM) with human-like mass distribution (Kuo, 2001), the `Forward-foot Model' (FM) with feet facing forward from the legs, and the `Kneed Model' (KM) with knees and forward feet (after McGeer, 1990b). All simulations generally predict decreasing step-to-step transitions with increasing arc foot radius, roughly in proportion to (1–)2 as in Eqn 7. However, FM and KM have a slight upward trend for larger values of , due to different foot geometry and introduction of knees. The SM is used as a prediction for experimental results. Over the range of arc radii studied experimentally, all other models match the trend of Eqn 7 reasonably well, with r2 ranging 0.940–0.998.

View larger version (10K): [in a new window]   Fig. 3. Apparatus used to rigidly restrict human ankle motion and control rolling characteristics of the foot. (A) Subjects wore a boot and arc apparatus bilaterally, each consisting of a rigid walking boot modified to accept wooden arc shapes of varying radius. (B) `Arc foot' shapes of varying arc radius (defined as fraction of leg length) were rigidly attached with pyramidal prosthesis adapters. Arcs ranged in radius 0.02–0.40 m in absolute dimensions, and each subtended a sufficient range of angles to ensure continuous rolling ground contact throughout the stance phase. Arcs had matched mass of 1.1±0.1 kg, and boots had mass 0.85 or 1.05 kg, depending on size.

View larger version (18K): [in a new window]   Fig. 4. Vertical ground reaction forces (GRF) versus time over one step, measured during walking with arcs of different radius and in normal shoes. Larger arc radii resulted in smoother collisions during the step-to-step transition. Small arc radii resulted in very large initial peaks in ground reaction force. With larger arcs this peak decreased to below its magnitude in normal walking, but it always occurred earlier in the step. Walking on arcs resulted in shorter double-support times, decreasing with smaller radii. Arc radius had little effect on the second peak in vertical force. Data shown are averaged over all subjects and plotted over the mean step period. A step begins at heelstrike and ends at opposite heelstrike, with double support occurring over the first 0.10–0.15 s. BW, body weight. Blue and green traces indicate forces under the trailing and leading legs, respectively.

View larger version (12K): [in a new window]   Fig. 5. The angular direction change COM in COM velocity decreased with increasing arc foot radius . COM direction change was estimated as the angle between the steepest upward and downward velocities of the COM in the sagittal plane (defined below; compare with Fig. 1B). The relationship between COM and is described well by the linear fit of Eqn 9, r2=0.89. Different coloured symbols indicate different subjects.

View larger version (17K): [in a new window]   Fig. 6. Instantaneous COM mechanical work rate for each leg over one complete step, measured with arcs of different radii. The trailing leg performed positive work and the leading leg negative work to redirect the COM during the step-to-step transition. Leading leg negative work rate was highest in magnitude for small-radius arcs. Work rate magnitudes decreased with increasing arc radius for the leading leg during double support, and through most of single support. Average rate of negative work was computed by integrating the magnitude of negative regions of instantaneous work rate (shaded areas for 0.40 m arc) and dividing by step period. Data shown are averaged from all subjects and plotted over the mean step period. Blue and green traces indicate work rate of the trailing and leading leg, respectively.

View larger version (11K): [in a new window]   Fig. 7. The average rate at which negative work is performed on the COM (, see shaded areas in Fig. 6) fell with increasing arc foot radius . The Simplest Model Fit of Eqn 10 predicted the trend well (r2=0.95). The magnitude of work rate was greater for small arcs than for normal walking (broken line), and lower for arcs of approximately >0.2. Less work is needed to redirect the COM velocity with larger arcs, due to a smaller directional change during the step-to-step transition. The work rate observed with the smallest arcs was 2.37 times that for the largest arcs. Different coloured symbols indicate different subjects.

View larger version (10K): [in a new window]   Fig. 8. Net metabolic rate met exhibited a U-shaped curve as a function of arc radius . For small radii, metabolic rate decreased with much as predicted by the Simplest Model (Fig. 2). However, metabolic rate reached a minimum at =0.30 according to the Empirical Fit of Eqn 11, r2=0.86, and began to increase with larger . The energetic cost of walking was 59% higher than the minimum for the smallest arcs, and higher for all arc radii compared to normal walking (broken line). Different coloured symbols indicate different subjects.

View larger version (9K): [in a new window]   Fig. 9. Comparison of net metabolic rate met with expected cost based on step-to-step transition work. Assuming a peak efficiency of 25%, the observed work performed on the COM (Fig. 7) would be expected to yield a strictly decreasing metabolic rate with increasing . Subtracting the expected cost from observed yields a residual cost not explained by the Simplest Model. The residual cost is substantial for arcs of smallest and largest radii. (Region a) The high cost for small radii may be caused by the effort of balancing on a small contact patch through large collisions in the step-to-step transition. (Region b) The cost for large radii may be associated with stabilizing the knee joint against a hyperextension moment caused by the ground reaction force late in single support.