QUICK SEARCH:   [advanced] Author: Keyword(s):
Home     Help     Feedback     Subscriptions     Archive     Search     Table of Contents

First published online August 31, 2004
Journal of Experimental Biology 207, 3545-3558 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.01177

This Article Summary Full Text Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Griffin, T. M. Articles by Farley, C. T. Search for Related Content PubMed PubMed Citation Articles by Griffin, T. M. Articles by Farley, C. T.

Biomechanics of quadrupedal walking: how do four-legged animals achieve inverted pendulum-like movements?

Timothy M. Griffin^1,*, Russell P. Main² and Claire T. Farley³

¹ Orthopaedic Bioengineering Laboratory, Department of Surgery, Duke University Medical Center, Durham, NC 27710, USA
² Concord Field Station, Museum of Comparative Zoology, Department of Organismic and Evolutionary Biology, Harvard University, Bedford, MA 01730, USA
³ Locomotion Laboratory, Department of Integrative Physiology, University of Colorado, Boulder, CO 80309, USA

View larger version (23K): [in a new window]   Fig. 1. A hypothetical diagram of quadrupedal walking. The fore quarters and hind quarters are represented as independent inverted pendulums. If the mass distribution is equal between the fore and hind quarters and the limbs cycle at evenly spaced time intervals, the pendular movements of the fore quarters and hind quarters offset each other. When the fore quarters are highest (i.e. gravitational potential energy is maximum), the hind quarters are lowest. Similarly, when the fore quarters are moving fastest (i.e. maximum kinetic energy), the hind quarters are moving slowest. As a result, the gravitational potential energy (Ep) and kinetic energy (Ek) are constant throughout the stride. Bars indicate foot–ground contact times, and the footfall order is left hind (LH), left fore (LF), right hind (RH) and right fore (RF) limb. COM, center of mass.

View larger version (16K): [in a new window]   Fig. 2. Gravitational potential energy (Ep), kinetic energy (Ek) and total mechanical energy (Ecom) of the center of mass versus time for a dog walking at four different speeds. Ep and Ek generally fluctuated out of phase so the fluctuations in Ecom were smaller than either one. Bars indicate foot–ground contact times. Data are for typical trials for one stride beginning with the left hind limb ground contact for a 30 kg dog. LH, left hind limb; LF, left fore limb; RH, right hind limb; RF, right fore limb.

View larger version (9K): [in a new window]   Fig. 3. Inverted pendulum mechanics of the center of mass for dogs walking at a range of speeds. (A) Recovery of mechanical energy via the inverted pendulum mechanism (recovery=–117.5u2+213.6u–27.1, where u is speed; r2=0.38). (B) Mass-specific work performed on the center of mass per distance traveled (Wcom=0.407u2–0.718u+0.440; r2=0.39). (C) Mass-specific mechanical work per unit distance to lift (Ep; filled circles) and accelerate (Ek; open circles) the center of mass (Ep=0.088u2–0.236u+0.332, r2=0.88; Ek=–0.153u2+0.412u–0.032, r2=0.88). (D) Phase difference between the fluctuations in Ep and Ek (phase=–75.5u2+62.7u+196.4; r2=0.44). Values are means ± S.E.M. for all of the dogs. Lines are least-squares regressions.

View larger version (19K): [in a new window]   Fig. 4. Vertical displacements of the fore quarters, hind quarters and center of mass versus time for one typical walking stride at 0.8 m s–1. The relative magnitudes and the timing of the actual displacement data (A) corresponded to the compass gait prediction (B). The similarities suggest that the fore and hind quarters actually vault over their support limbs like independent bipeds. The sharp transition points in the compass gait prediction (B) correspond to an instantaneous transfer from left to right limbs at the middle of double support. However, in a dog (A), this transition is smooth because it occurs over the entire period of double support. The dog's leg length and contact time were 0.54 m and 0.696 s, respectively, for the fore limbs and 0.45 m and 0.629 s, respectively, for the hind limbs. (C) The dog's actual footfall pattern from A; LH, left hind limb; LF, left fore limb; RH, right hind limb; RF, right fore limb.

View larger version (39K): [in a new window]   Fig. 5. Vertical displacement of the fore pendulum, hind pendulum and system center of mass versus time for specific examples from the two-pendulum model of quadrupedal walking. The model consists of two identical independent pendulums representing a quadruped's fore and hind quarters with a variable phase relationship () and mass distribution (Mf). is the percentage of stride time that the fore pendulum displacement lags the hind pendulum displacement (analogous to the phase between a dog's ipsilateral fore and hind limbs), and Mf is the ratio of fore pendulum mass to the combined mass of both pendulums (equivalent to the fraction of body weight supported by the fore limbs). The time interval shown is equivalent to a complete dog stride. zcom is the magnitude of the center of mass vertical displacement relative to the pendulum displacement, and is the time interval between the peak center of mass vertical displacement and the peak hind pendulum vertical displacement, expressed as a percentage of time between peak hind and fore pendulum displacements. A represents the pendulum phase and mass distribution combination actually used by the dogs, and D represents the phase and mass distribution shown in the hypothetical diagram of Fig. 1. B and C represent intermediate patterns.

View larger version (20K): [in a new window]   Fig. 6. Contour plots of the magnitude of the center of mass vertical displacement relative to the pendulum vertical displacement (zcom; A) and the phase relationship between the center of mass and the hind pendulum vertical displacement (; B) for the two-pendulum model of walking. These data are plotted as a function of the relative mass of the fore pendulum, Mf, and the phase difference between the fore and hind pendulums, . Both pendulums have the same amplitude in all cases. The circled letters correspond to the combinations of phase and mass distribution in Fig. 5A–D.

View larger version (13K): [in a new window]   Fig. 7. Limb phase versus speed. Each limb phase was calculated relative to the left hind limb (i.e. left hind limb phase=0%). At all speeds, each fore limb contacted the ground approximately 15% of the stride time after the ipsilateral hind limb. Hildebrand defined this limb phase and footfall pattern as a lateral sequence walk (Hildebrand, 1968, 1976). The order of footfalls was left hind (LH), left fore (LF), right hind (RH) and right fore (RF). Values are means ± S.E.M. for all of the dogs. Error bars are too small to be visible in most cases.

View larger version (21K): [in a new window]   Fig. 8. Individual limb vertical (Fz), fore–aft (Fy) and lateral (Fx) ground reaction force components for the fore and hind limbs versus time for dogs walking at 0.8 m s–1. Forces are expressed as a fraction of body weight (Wb). The fore limb forces were much larger than the hind limb forces. The solid line represents the mean trace for the six dogs, and the broken lines are ±1 S.D. Note that the y-axis scales differ. Positive values correspond to up (Fz), forward (Fy) and medial (Fx).

View larger version (18K): [in a new window]   Fig. 9. Average fore–aft ground reaction forces (Fy) and kinetic energy fluctuations (Eky) for all dogs walking at 0.8 m s–1. (A) The limbs generated propulsive and braking forces simultaneously throughout most of the stride. Consequently, the summed limb fore–aft force was smaller than the individual limb forces. Shaded areas indicate the net propulsive and braking impulses, which determine the velocity fluctuations of the center of mass. Limb phase was 15% of stride time, as observed in dogs. (B) Kinetic energy fluctuations were smaller for the center of mass than for the fore and hind quarters because the nearly out-of-phase fluctuations of the fore and hind quarters partly offset each other. Data assume that (1) the fore and hind quarters were, respectively, 63% and 37% of the total body mass (37.8 kg), (2) the fore and hind quarters each had a mean velocity of 0.8 m s–1 and (3) the velocity fluctuations of the fore and hind quarters were determined completely by their respective fore–aft ground forces. The first two assumptions are reasonable, but the third assumption is likely to be false because forces transmitted via the trunk probably play a role. The fore and hind limbs generate net braking and propulsive forces, respectively, so trunk forces would presumably counteract these net forces. Otherwise, the net propulsive ground reaction force on the hind quarters would cause them to overtake the fore quarters. The trunk is most likely loaded in compression during steady-speed walking because the hind quarters must, on average, push the fore quarters forward, and the fore quarters must, on average, push backwards on the hind quarters over a complete stride. If these trunk interaction forces were accounted for, we would expect the kinetic energy values of the fore and hind quarters to return to their respective initial values at the end of the stride instead of having net changes as shown in B. (C) The dogs' average footfall pattern; LH, left hind limb; LF, left fore limb; RH, right hind limb; RF, right fore limb.

View larger version (21K): [in a new window]   Fig. 10. Decreasing limb phase increased gravitational potential energy (Ep) and fore–aft kinetic energy (Eky) fluctuations of the center of mass relative to those of the fore and hind quarters. As limb phase approached 0%, the vertical and fore–aft movements of the fore and hind quarters were more synchronous. These changes were similar to those predicted by the two-pendulum model. We assumed that (1) the fore and hind quarters were, respectively, 63% and 37% of the total mass (37.8 kg) and (2) limb phase did not affect the ground reaction force. Values were calculated using mean individual limb ground reaction force data for dogs walking at 0.8 m s–1 (Fig. 8) and time-shifting the data to simulate a range of limb phases. The values for 15% limb phase correspond to the example in Fig. 9.