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First published online March 31, 2005
Journal of Experimental Biology 208, 1513-1524 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01549
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A new model predicting locomotor cost from limb length via force production

Herman Pontzer

Harvard University, Department of Anthropology, 11 Divinity Avenue, Cambridge, MA 02138, USA



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  		Fig. 1. Movement of the limb through stance. Thick line indicates position of lower limb and COM at heel strike and toe-off. L, limb length; {phi}, excursion angle; U, running speed. For this model: step length is d = 2Lsin{phi}/2; contact time per step is tc = U–1d and tc = U–1 [2Lsin({phi}/2)].

 


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  		Fig. 2. Vertical acceleration of the COM during walking as derived via the LiMb model. During walking the COM follows a sinusoidal trajectory in the sagittal plane resulting in alternating periods of upward and downward acceleration (+ay and –ay) during which the COM is accelerated. Maximum velocity, ±Vmax, is a function of the precise shape of the COM trajectory. Assuming that Vmax=2Vavg during normal walking, the change in velocity (i.e., the mass-specific change in momentum) during one period of acceleration, {Delta}Vy=4L[1–cos({phi}/2)](U–1[Lsin({phi}/2)])–1 and thus {Delta}Vy=4U[1–cos({phi}/2) sin({phi}/2)]–1. Given the duration of acceleration, U–1[Lsin({phi}/2)], this requires an average acceleration {alpha}y=4U[1–cos({phi}/2)]sin({phi}/2)–1(U–1[Lsin({phi}/2)])–1 and thus {alpha}y=4U2L–1[1–cos({phi}/2)]sin({phi}/2)–2. Because sin({phi}/2)–2=([1–cos({phi}/2)][1+cos({phi}/2)])–1, this simplifies to {alpha}y=4U2L–1[1+cos({phi}/2)]–1.

 


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  		Fig. 3. Deriving the horizontal force component of the LiMb model. Diagram shows the position of the limb (Limb) at heel strike and the instantaneous vertical GRF vector (open arrow, GRF). The horizontal force (Fx) required to produce a resultant (Fres) that is directed toward te COM (filled circle) is given by Fx=GRFxtan({phi}/2).

 


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  		Fig. 4. Predicted versus observed COL for running trials. Line indicates LSR (N=27, r2=0.43, P<0.01).

 


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  		Fig. 5. Predicted versus observed COL for walking trials. Line indicates LSR (N=34, r2=0.94, P<0.001).

 


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  		Fig. 6. The contribution of vertical, horizontal and leg-swing forces to total estimated force production during walking and running. Data for one representative subject (Female, 66 kg, hind-limb length: 95 cm.

 


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  		Fig. 7. Predicted versus observed COL for walking and running trials. Solid lines: LSR for walking (open circles) and running (filled circles). Broken line: LSR for all trials combined (N=61, r2=0.91, P<0.001).

 


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  		Fig. 8. Between-subjects variation in k. (A) While the LiMb model predicted over 90% of the variance in COL for each subject, differences in k, measured as the slope of the LSR for each subject, were marked. Data for two subjects are highlighted here against data from all subjects. (B) Predicted versus observed COL, using estimates of k derived for each subject. Line indicates LSR. open circles, walking trials (N=34, r2=0.95, P<0.01); filled circles, running trials (N=27, r2=0.87, P<0.01).

 


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  		Fig. 9. The effect of limb length on vertical force production during walking and running. The magnitude of vertical ground force at a given speed, estimated as (contact time)–1 / speed, decreases with hind-limb length. Group means are calculated from individual trials (walking and running trials combined). P-values are given for each comparison, calculated via Student's one-tailed t-test assuming unequal variance. Error bars indicate the standard error of the mean. Ranges for each hind limb-length category are: short 79–80 cm (N=2 subjects); medium 89–95 cm (N=4); long 103–112 cm (N=3).

 

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© The Company of Biologists Ltd 2005