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Pendular energy transduction within the step in human walking

G. A. Cavagna^1,*, P. A. Willems², M. A. Legramandi¹ and N. C. Heglund²

¹ Istituto di Fisiologia Umana, Università degli Studi di Milano, 20133 Milan, Italy
² Unité de Réadaptation, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

View larger version (21K): [in a new window]   Fig. 8. The pendular recovery of mechanical energy, defined by equation 1 (Rstep, open circles) and by equation 6 (Rint, open squares), and the external work done per unit distance (W+ext, filled circles) plotted as a function of speed during unloaded walking for the 11 European subjects of this study. Values are means ± S.D. (N is given by the numbers near the filled circles) for data grouped into the following intervals along the abscissa: <2, 2 to <2.5,...., 8.5 to <9, >9 km h-1. Lines are fitted using a third-order polynomial fit (r2=0.98, KaleidaGraph 3.5). The crosses and the dotted line show how Rstep changes when the kinetic energy of vertical motion of the centre of mass, Ekv, is taken into account in the calculation of W+v and W+k (see text). Note that Wext attains a minimum at a speed lower than the speed at which the pendular recovery attains a maximum.

View larger version (18K): [in a new window]   Fig. 2. Simulation: effect of the phase shift . The fraction of the mechanical energy recovered through the pendular transduction in the simulation, calculated as Rstep (dotted line, Equation 4) or as Rint (solid line, Equation 6) is plotted as a function of the phase shift between the curves Ep=-sinx and Ek=sin(x-) illustrated in Fig. 1. The two vertical continuous lines encompass the values of attained during all speeds of walking: decreases from approximately 45° at the lowest speeds to approximately -45° at highest speeds (Cavagna et al., 1983). The two vertical broken lines encompass the values of (10°-20°) observed in this study (Table 1). Note that RstepRint over the entire range of measured during walking.

View larger version (16K): [in a new window]   Fig. 7. Recordings such as those depicted in Fig. 5, showing an extreme case in which loading results in tpk- being reduced to zero and in an increase in the instantaneous recovery of mechanical energy r(t) during the descent of the centre of mass to a level equal to that attained during the lift. This leads to very high values of pendular recovery during the step (African woman, Luo, 88.9 kg, 4.5 km h-1, loaded with 19.1 kg). For further details, see legend to Fig. 5.

View larger version (33K): [in a new window]   Fig. 1. Simulation of the transduction between kinetic and potential energy of the centre of mass when the maximum in kinetic energy is set to lag behind the minimum in potential energy by a value of =10° (A) and =20° (B), which covers the range of mean experimental values measured in this study during walking (see values of in Table 1). is the phase shift between the maximum of the kinetic energy Ek and the minimum of the potential energy Ep. Upper panels: the total energy of the centre of mass of the body (Ecg, thin continuous line) is simulated as the sum of two sine waves representing its potential energy (Ep=-sinx; dotted lines) and kinetic energy [Ek=sin(x-10°) in A, and Ek=sin(x-20°) in B: broken lines) during a step cycle, expressed in degrees. The fraction of the mechanical energy recovered at each instant by the pendular transduction within the cycle, r(x) (thick lines and right-hand ordinates), is calculated according to Equation 5 from the relative changes in the Ek, Ep and Ecg curves. r(x) is zero when the changes in the Ek and Ep curves have the same sign, and attains unity when the Ecg curve is at a maximum or at a minimum. Lower panels: the area under the r(x) curve divided by 360°, defined as , attains the value Rint(360°)=Rint at the end of each cycle. Time-averaged Rint is less than Rstep, calculated according to Equation 1 from the total amplitude reached by the Ep, Ek and Ecg curves during the cycle. The relationship between Rint and Rstep for different values of is shown in Fig. 2.

View larger version (16K): [in a new window]   Fig. 4. Unloaded walking. (A) Average curves of the instantaneous recovery of mechanical energy r(t) for the African women (thick continuous line, mean of 32 steps by four subjects), the European women (thin continuous line, mean of 17 steps by five subjects) and all European subjects (males and females, broken line, average of 32 steps on ten subjects). The time-average of the standard deviation of the mean was less than 25% of r(t) during the lift of the centre of mass (ttr,up in Fig. 3) and less than 35% during its descent (ttr,down). (B) The area under the average r(t) curve divided by the step period attains a value of Rint()=Rint at the end of the step which is equal for all groups of subjects. The corresponding mean values of the parameters for all European subjects and for the African women are given in Table 1 (unloaded).

View larger version (16K): [in a new window]   Fig. 6. Loaded walking. (A) Average curves of the instantaneous recovery of mechanical energy r(t) for the African women (thick continuous line, mean of 32 steps by four subjects), the European women (thin continuous line, mean of 14 steps by five subjects) and all European subjects (males and females, dotted line, mean of 32 steps by ten subjects). A comparison with Fig. 4 shows that, in all subjects, loading decreases tpk- and increases r(t) at the beginning of the descent of the centre of mass (ttr,down), but that this results in a net increase in Rint in the African women only (final value attained by the thick line at the end of the step in B). The corresponding mean values of the parameters for all European subjects and the African women are given in Table 1 (loaded). For further details, see legend to Fig. 4.

View larger version (24K): [in a new window]   Fig. 3. Typical experimental recordings of unloaded walking. The fraction of the mechanical energy recovered during unloaded walking at each instant of the step cycle [r(t), thick lines] is superimposed on the mechanical energy changes of the centre of mass (Ep, gravitational potential energy; Ek, kinetic energy; and Ecg=Ep+Ek, where Ecg is the total mechanical energy of the centre of mass, thin lines). Typical record obtained from (A) a European subject (male, 66.2 kg, 4.86 km h-1) and (B) an African woman (Kikuyu, 83.3 kg, 4.85 km h-1). The vertical broken lines on the Ek and Ep curves delimit the periods when the instantaneous recovery of mechanical energy r(t) is zero, indicated on the figure as tpk+ when Ep and Ek increase simultaneously, and as tpk- when Ep and Ek decrease simultaneously. The periods during which energy transduction between Ep and Ek occurs are indicated as ttr,up during the lift of the centre of mass and ttr,down during the descent of the centre of mass. As in the simulation, Rstep is greater than Rint but, in contrast to the simulation, the r(t) curves recorded during the rise and fall of the centre of mass during walking are not symmetrical see (Fig. 1).

View larger version (24K): [in a new window]   Fig. 5. Typical experimental recordings of loaded walking for (A) a European subject (male, 65.6 kg, 3.71 km h-1, loaded with 19.3 kg) and (B) an African woman (Luo, 83.5 kg, 3.95 km h-1, loaded with 19.5 kg). For further details, see legend to Fig. 3. Note that in the African subject, loading results in a reduction in tpk- and in an increase in r(t) at the beginning of the descent of the centre of mass (ttr,down).