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First published online March 14, 2005
Journal of Experimental Biology 208, 979-991 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01498

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Constrained optimization in human walking: cost minimization and gait plasticity

John E. A. Bertram

Department of Nutrition, Food and Exercise Sciences, Florida State University, Tallahassee, FL 32306, USA

View larger version (16K): [in a new window]   Fig. 1. (A) The relation between speed and step frequency in human treadmill walking (data from a mixed gender group of college-age students participating in an undergraduate biomechanics class). A consistent relationship is evident and is usually explained as indicating the most economical means of walking. If this is the case, then it should not matter if speed is determined, as in treadmill walking, or frequency is determined, as walking to a metronome beat (arrows). (B) Different constraints on walking actually result in different speed-step frequency relations (data from one subject). Red, solid line: speed constrained walking (treadmill); blue, broken line: frequency constrained walking (metronome); green, dotted line: step length constrained walking (floor markers).

View larger version (31K): [in a new window]   Fig. 2. The constrained optimization hypothesis. Light blue lines indicate constant cost contours describing the metabolic cost surface for walking, where each contour indicates a higher cost than the one inside (central contour surrounds the minimum cost region). These are generalized contours, roughly following measurements of Molen et al. (1972). For each constraint (speed, frequency or step length), walking cost will be minimized if the individual selects the other gait parameters that occur where the line of determination just touches a cost contour; at any other point the cost will be greater for that constraint. Constraints indicated by line colors and form as in Fig. 1B.

View larger version (52K): [in a new window]   Fig. 3. An illustration of the process for determining the metabolic cost contours and generating the optimization predictions for walking. The pooled data for the subject group (490 original points) are shown. (A) The 49 metabolic cost measurements do not provide an adequate characterization of the cost surface and they generate contours without predictive value (B). Here and in C the cost is plotted as Economy, the inverse of cost, to allow better visualization of the surface. (C) The data were smoothed and the region between measured data points was reconstructed by fitting with a plate using a defined number of Fourier terms as the model. (D) From the smoothed surface, realistic contours are derived. (E) Error resulting from the fitting process can be monitored. (F) Predictions of speed and step frequency according to the constrained optimization hypothesis (as in Fig. 2) can be derived from the smoothed contours. Line colors and form for each constraint match those of previous figures.

View larger version (43K): [in a new window]   Fig. 4. Examples demonstrating the effect of the surface modeling control parameters on the surface, contours and error for the pooled data of this study. The two control parameters are matrix stiffness and number of nodes. In each series of plots the control settings listed at the left are used to generate each of the plots in line to the right. Matrix stiffness refers to the resistance of the smoothing function to allow deviations from the measured data (at the 49 measured points on which the surface is based). The higher this value, the greater influence local points have on the plate form. This can be seen by comparing A and D, where the number of nodes are kept constant and the influence of high stiffness matrix is evident. The number of nodes refers to the elements included in the truncated double Fourier sine series that generates the curvilinear model of the surface. In this case, the greater number of nodes, the more complex the surface shape can be. The influence of this parameter can be seen by comparing surfaces D and G. The analyses performed in the present study used the parameter settings indicated by plots D-F. These maintained much of the complexity of the measured data but linked those points in a more realistic manner than simplistic planar surfaces (Fig. 3A,B). This is indicated by the modest error values (difference between surface model and measured data) for the parameter settings used (error axis resolution in F is an order of magnitude greater than in C or I). For comparison of the optimization predicted for each parameter setting, the optimizations are provided (as in Fig. 3F). General predictions are consistent even over the wide range of control parameters illustrated, but the excessive smoothing of the extreme options removes all subtle features of the surface and increases overall error.

View larger version (23K): [in a new window]   Fig. 5. Results of spontaneous gait selection parameters (data points) compared to minimum metabolic cost as predicted by the constrained optimization hypothesis. In each case the predictions are shown as a range around the tangent point indicating cost minimization for that constraint, where red indicates ±1%, orange ±5%, yellow ±10% and gray bounded by black ±15%. Mean for each constraint is indicated by a solid black circle with error bars showing ± 1 S.D. The data points for speed and step length constraints are fit with a power function (solid black line) with the constraint as the determinant variable (see text for details). For the frequency constraint (B) regressions of those subjects that chose the higher speed (long dashes) and lower speed (short dashes) options are also indicated. For all frequency constraint regressions a simple quadratic was used. Metabolic cost contours used to derive the constrained optimization predictions are shown in blue for reference. Data points are plotted in colors associated with the particular constraint applied, as in previous figures.

View larger version (16K): [in a new window]   Fig. 6. Speed-frequency data for individual subjects as they responded to the applied constraints. Subject labels are consistent between plots within each gender and match the labels indicated in the insets of A and B. These data are those plotted in Fig. 5, but have been reproduced in this format to allow tracking and comparison of individual responses.