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First published online January 31, 2006
Journal of Experimental Biology 209, 622-632 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02010

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Constrained optimization in human running

Anne K. Gutmann¹, Brian Jacobi², Michael T. Butcher³ and John E. A. Bertram^4,*

¹ Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 12853, USA
² Department of Biological Sciences, Florida State University, Tallahassee, FL 32306, USA
³ Department of Biological Sciences, University of Calgary
⁴ Department of Cell Biology and Anatomy, Faculty of Medicine, University of Calgary, Calgary, AB T2N 4N1, Canada

View larger version (14K): [in a new window]   Fig. 1. A plot of speed-frequency relations for a single subject running under the three constraint conditions imposed in this study. Red circles show frequencies selected when speed is constrained in treadmill running, blue circles show speeds selected when frequency is constrained in over-ground running to a metronome beat, and green circles indicate the speed-frequency combinations selected when step length is constrained by stepping in registry with ground markers. Each relation was fit with a least-squares linear regression with the constrained parameter as the independent variable, then the relationship determined was converted to speed-frequency for comparison (see text for details). The point of intersection of the v-constrained, f-constrained and d-constrained relationships gives apparent preferred speed and frequency (vp and fp).

View larger version (25K): [in a new window]   Fig. 2. Behavioral data for all subjects with least-squares linear regressions determined as in Fig. 1. Data for v-constrained conditions, red circles; f-constrained conditions, blue circles; d-constrained conditions, green circles. All three slopes are significantly different from one another, P<0.001. v-constrained conditions, f/fp=0.202(v/vp)+0.796; f-constrained conditions, v/vp=1.347(f/fp)-1.3684; d-constrained conditions, f/fp=0.117(d/dp)+1.078.

View larger version (26K): [in a new window]   Fig. 3. Economy (the inverse of cost of transport) as a function of normalized speed (v/vp) and frequency (f/fp). (A) The 3-D surface; (B) a flattened, overhead view of the surface, showing the distribution of the data points more clearly. Economy is used instead of cost of transport simply for visual clarity in depicting the surface shape. The surface is interpolated based on metabolic measurements from Cavanagh (1982) (red triangles), Rouviere (2002) (red circles), and Knuttgen (1961) (red squares), in which running parameters were fully constrained. The v-constrained metabolic measurements from Knuttgen (1961) (x), were considered to be outliers by virtue of the peculiar frequency selection of the subject in comparison to frequencies selected by subjects in other studies. The measurements of Liefeldt (1992) (+), were considered to be outliers because the costs were considerably lower than those reported in other studies under similar conditions. These two sets of outliers were not used to generate the cost surface. The color of the surface is determined by the height of the surface. Dark blue indicates regions of low economy (high cost), bright green indicates regions of high economy (low cost), and blue-green indicates regions of intermediate economy (intermediate cost).

View larger version (29K): [in a new window]   Fig. 4. Predicting optimal behavior by finding the points where constraint lines are tangent to cost contours. Cost contours are shown as black curves. Cost is least in the region bounded by the central curve and greater for curves lying outside each other. Constrained optimization predicts that for any given constraint gait, parameters will be chosen such that cost of transport is minimized. This occurs at the tangent of the constraint line and a cost contour, because any other point on the constraint line lies outside the contour and indicates a greater cost. This method is equivalent to predicting optimal behavior by finding the points where one of the partial derivatives is equal to zero and may be used to verify the optimal behavior predictions shown in Fig. 5. Speed and frequency constraints can be visualised as horizontal and vertical lines, respectively, and step length constraints can be visualised as lines radiating from the origin whose slopes are equal to the specified step lengths - i.e. lines whose equations are of the form v=fd, where d=constant.

View larger version (19K): [in a new window]   Fig. 5. Predicted and measured running gait parameter selection for all subjects. Solid circles indicate measured parameter selection under specific constraint conditions; (A) red circles, v-constrained; (B) blue circles, f-constrained; (C) green circles, d-constrained. Thick black lines indicate least-squares linear regression of the behavioral data, as determined using each constrained parameter as the independent variable. The broken black lines give 95% confidence intervals of the regression. Contours lines indicate equivalent cost of transport with the region of least cost surrounded by the inner contour and cost increasing outward from that. The bold red lines indicate the optimal predicted behavior (zero slope/minimum cost), the orange area represents the region of Cmin + 0.001 ml O2 kg-1 m-1, and the yellow area that of Cmin + 0.005 ml O2 kg-1 m-1.

View larger version (27K): [in a new window]   Fig. 6. Comparison of optimal minimal cost behavior predictions for constrained frequency running using cost regions calculated for (A) speed-frequency-cost space and (B) frequency-step length-cost space. For comparison, both predictions are displayed on equivalent speed-frequency plots. Thick red lines represent optimal predicted behavior (zero slope), the orange area represents region of minimal cost+0.001 ml O2 kg-1 m-1, and the yellow area that of minimal cost+0.005 ml O2 kg-1 m-1. The general features of the predicted behavior are not affected by method of calculation.

View larger version (19K): [in a new window]   Fig. 7. Comparison of raw (A,C,E) and normalized (B,D,F) gait parameter selection data for all five subjects. Speed and frequency are normalized according to vp and fp, respectively. Normalization reduced inter-subject variability for all constraint conditions, but the reduction of variability is most noticeable for v-constrained (A,B) and f-constrained (C,D) conditions. Subject 1, green triangles; Subject 2, black x; Subject 3, blue +; Subject 4, red squares; Subject 5, blue circles.

View larger version (17K): [in a new window]   Fig. 8. Comparison of optimal behavior predictions generated using (A) cost per distance, (B) cost per time, and (C) cost per step surfaces. Cost contours from each surface are shown as black curves. Colored broken lines represent least-squares regressions of self-selected behavioral data and colored solid lines represent optimal predicted behavior. Red lines are used for v-constrained conditions, blue for f-constrained conditions and green for d-constrained conditions. The cost per time plot predicts v-constrained and d-constrained behavior quite well, but does not predict f-constrained behavior (no solid blue line). The cost per step plot also does not predict f-constrained behavior and predicts that the v-constrained behavior should occur where, instead, we observe f-constrained behavior. Only the cost per distance plot correctly predicts three different self-selected behaviors and places all three curves in the correct regions of v-f space. Therefore, minimization of cost per distance seems to be the best predictor of running behavior.