The landing–take-off asymmetry of running was thought to derive from,or at least to be consistent with, the physiological property of muscle to resist stretching (after landing) with a force greater than it can develop during shortening (before take-off). In old age, muscular force is reduced,but the deficit in force is less during stretching than during shortening. The greater loss in concentric versus eccentric strength with aging led us to hypothesize that older versus younger adults would increase the landing–take-off asymmetry in running. To test this hypothesis, we measured the within-step changes in mechanical energy of the centre of mass of the body in old and young subjects. The difference between the peaks in kinetic energy attained during the fall and during the lift of the centre of mass is greater in the old subjects. The difference between the time to lift and accelerate the centre of mass (positive work) and to absorb the same amount of energy during the downward displacement (negative work) is also greater in the old subjects. Both these findings imply a difference in force between stretching and shortening during the bounce, which is greater in the old subjects than in the young subjects. This is qualitatively consistent with the more asymmetric force–velocity relation found in aged muscle and supports, even if does not prove, the hypothesis that the landing–take-off asymmetry in running derives from the different response of muscle to stretching and shortening.

Experiments on isolated muscle specimens(Phillips et al., 1991; Ochala et al., 2006) and in vivo on humans (Vandervoort et al., 1990; Porter et al.,1997; Pousson et al.,2001; Klass et al.,2005) showed that in old age muscular force is reduced less during stretching (negative work) than during shortening (positive work). The effect of this change of muscle contractile properties with age on the mechanics of locomotion is unknown.

At each running step, the muscle-tendon units are stretched after landing and shorten before take-off. The peak in kinetic energy attained before landing, to be absorbed by the muscle-tendon units during the negative work phase, is higher than the peak in kinetic energy restored before take-off during the positive work phase. On the other hand, the duration of the positive work phase is greater than the duration of the negative work phase. This landing–take-off asymmetry is consistent with an average force exerted during stretching (after landing) greater than that developed during shortening (before take-off). It has been argued that the landing–take-off asymmetry is a consequence of the force–velocity relation of muscle, and the greater the length change of muscle relative to that of tendon within the muscle-tendon units the larger the landing–take-off asymmetry (Cavagna,2006).

We hypothesized that the increased discrepancy in old age between the greater force resisting stretching and the lower force developed during shortening, if operational during running, would increase the landing–take-off asymmetry of the apparent bounce of the body relative to that of young subjects. To test this hypothesis, we measured the mechanical energy changes of the centre of mass during the negative and positive work phases of the bounce of the body in old and young subjects running on the level at different speeds.

Subjects and experimental procedure

The experiments were conducted on eight healthy old subjects (men, age 73.6±5.5 years, height 1.72±0.06 m, weight 71.1±9.2 kg)and eight healthy young subjects (6 men and 2 women, age 20.8±1.6 years, height 1.76±0.08 m, weight 63.4±10.0 kg) (means ±s.d.). Three subjects from each group were trained (running several kilometres at least once a week) while five were sedentary. The results reported in this study are an average of the data obtained with all subjects of each group regardless of their training. In agreement with the finding that the age-related effects on muscle-tendon units are similar in runners and non-active subjects (Karamanidis and Arampatzis, 2005), similar results were obtained here by comparing the three old trained subjects with the three male young untrained subjects(data not shown). Informed written consent was obtained from each subject. The experiments were carried out in accordance with the Declaration of Helsinki.

Subjects ran back and forth along a 50 m corridor that had built into it,at the level of the floor, a 4 m×0.5 m force platform sensitive to the force exerted by feet in the forward and vertical directions. A total of 124 runs by the elderly subjects at a speed of 3 to 13–17 km h–1, and 229 runs by the young subjects at a speed of 3 to 17–21 km h–1, were analyzed.

Platform records analysis

The mechanical energy of the centre of mass of the body(Fig. 1) was determined from the ground reaction forces as previously described(Cavagna, 1975). Only motion in a sagittal plane was considered when calculating the mechanical energy of the centre of mass. Rotational kinetic energy of the body and lateral translational energy were ignored. The only instruction given to each subject was to run normally, trying to reach and maintain a constant step-average speed over the section of the corridor where the platform was placed. The average running speed was measured by two photocells placed 1–3 m apart(depending on speed) along the side of the platform. The characteristics of the force platform were as previously described(Cavagna, 1975). The experimental procedure consisted of measuring the force exerted on the ground in the sagittal plane during running at different speeds. A microcomputer acquired data at a rate of 500 Hz per channel from (i) the platform signal proportional to the force exerted in the forward direction, (ii) the platform signal proportional to the force exerted in the vertical direction, and (iii)the signal from the photocells. Data acquisition and analysis were made via a dedicated DAQ board and custom LabView software (version 7.1,National Instruments, Austin, TX, USA). The platform signal from the unloaded force platform was measured immediately before each run and subtracted from the platform records of the vertical force, Fv, and fore–aft force, Ff, in order to account for a possible drift of the base line. Only the subset of the Fvand Ff records obtained between photocells crossing was used for subsequent analysis. The vertical and forward velocities(Vv and Vf, respectively) of the center of mass were obtained as follows.

The average vertical force measured by the force platform, v,plate, in a time interval, nfτ, corresponding to an integer number, nf, of steps periods, τ (selected between peaks or valleys of the force records), must equal the body weight measured with a balance, v,scale.

v,plate was measured after each run as the area below the Fv record in the time interval nfτ divided by nfτ. We analyzed records where v,plate/v,scale=1.005±0.02(N=124) for the group of old subjects and 1.002±0.01(N=229) for the group of young subjects.

v,plate was then subtracted from the Fv array and the result(Fvv,plate)/Mb(where Mb is the mass of the body) was integrated to obtain the record of vertical velocity changes for the time interval between photocells crossing.

One or more regular steps were subsequently chosen for analysis between two peaks or valleys of the record of Vv changes corresponding to a time interval nvτ where nv is an integer number of steps. The regularity of the steps was determined by the difference between positive and negative increments in the Vv and Vf changes divided by the sum of the increments. In the old group this ratio was 3.83±3.73%(vertical) and 10.95±9.2% (forward) (N=124), whereas in the young group it was 3.55±3.29% (vertical) and 12.98±11.63%(forward) (N=229). During running on the level, the upward and downward vertical displacements of the centre of mass of the body are on average equal over nv steps, i.e. the average Vv must be nil. On this basis, the area below the Vv changes (ΔVv) record,corresponding to the nvτ interval selected above, was divided by nvτ and the result subtracted from the whole ΔVv record between photocells crossing to obtain the instantaneous positive (upward) and negative (downward) values of Vv.

The ΔVf record was determined by integration of the Ff/Mb array during the time interval between photocells crossing, Δtphoto. The area below this ΔVf record was then divided byΔ tphoto and the result subtracted from the same record to locate the average running speed on the tracing. The average running speed, measured as photocells distance/Δtphoto, was then summed to the resulting array to obtain the instantaneous values of Vf.

The instantaneous vertical velocity Vv(t) was used to calculate the instantaneous kinetic energy of vertical motion Ekv(t)=0.5MbVv(t)2and, by integration, the vertical displacement of the centre of mass, Sv(t), with the corresponding gravitational potential energy Ep(t)=MbgSv(t)(where g is the acceleration of gravity). The kinetic energy of forward motion was calculated as Ekf(t)=0.5MbVf(t)2,the total translational kinetic energy of the centre of mass in the sagittal plane as Ek(t)=Ekf(t)+Ekv(t),and the translational mechanical energy of the centre of mass in the sagittal plane as Ecm(t)=Ekv(t)+Ekf(t)+Ep(t). Since, as mentioned above, selection was initially made between peaks (or valleys) of the ΔVv, the records were expanded to include the previous valley (or peak) of Ep(t)until a clear picture of the step(s) was obtained(Fig. 1).

Algorithms were made to calculate the work done during the selected steps between Ep valleys (or peaks): Wv, Wkf and Wext were calculated from the amplitudes of valleys and peaks, and the initial and final values in the Ep(t), Ekf(t) and Ecm(t) records. Positive values of the energy changes gave positive work, negative values gave negative work. In a perfect steady run on the level the ratio between the absolute values of positive and negative work done in nv steps should be equal to one. Experimental values were as follows: in the old subjects group(N=124):

\(W_{\mathrm{v}}^{+}{/}W_{\mathrm{v}}^{-}\)
=0.99±0.08,
\(W_{\mathrm{kf}}^{+}{/}W_{\mathrm{kf}}^{-}\)
=1.01±0.13,
\(W_{\mathrm{ext}}^{+}{/}W_{\mathrm{ext}}^{-}\)
=0.99±0.07;in the young subjects group (N=229):
\(W_{\mathrm{v}}^{+}{/}W_{\mathrm{v}}^{-}\)
=0.99±0.07,
\(W_{\mathrm{kf}}^{+}{/}W_{\mathrm{kf}}^{-}\)
=1.03±0.14,
\(W_{\mathrm{ext}}^{+}{/}W_{\mathrm{ext}}^{-}\)
=1.00±0.07. These means refer to steps where 0.6<W+/W<1.5.

Aerial time, brake-push durations and vertical displacement during contact

Since the mechanical energy of the centre of mass is constant when the body is airborne (air resistance is neglected), the aerial time was calculated as the time interval during which the derivative dEcm(t)/dt=0. This time interval was measured using two reference levels set by the user above and below the section of the record where dEcm(t)/dt≈0(Cavagna, 2006). The brake duration, tbrake, i.e. the time during which external negative work is done, was calculated as the time interval during which the dEcm(t)/dt record was below the reference level. The push duration, tpush, i.e. the time during which external positive work is done, was calculated as the time interval during which the dEcm(t)/dtrecord was above the reference level. Due to the noise of the dEcm(t)/dt record, the aerial time was in some cases overestimated (7%), and tbrake and/or tpush were in some cases underestimated (5–7%)(Cavagna, 2006). Similarly, the downward and upward displacements of the centre of mass during contact, Sc,down and Sc,up(Fig. 2), were measured from the descending and ascending portions, respectively, of the Ep(t) curve during the time interval where dEcm(t)/dt was lower or greater,respectively, than the two reference levels.

Vertical displacement below and above the equilibrium points

The vertical force, Fv, applied by the foot on the ground is:
\[\ F_{\mathrm{v}}=\mathrm{body}{\ }\mathrm{weight}+M_{\mathrm{b}}\mathbf{a}_{\mathrm{v}},\]
(1)
where av is the vertical acceleration of the centre of mass,i.e. the time derivative of its vertical velocity, Vv. When the Vv and Ekv(=0.5MbVv2) are at a maximum, the derivative is nil, av=0, and as a consequence Fv=body weight. The locations of the Ekv peaks attained during the step(Fig. 1) were therefore used to determine the instants where the vertical force equals body weight. The locations and the amplitudes of Ep simultaneous with the peaks of Ekv were used to determine the part of the vertical displacement taking place below the equilibrium points during the downward deceleration of the centre of mass, Sce,down, and during the upward acceleration of the centre of mass, Sce,up (Fig. 3). The differences Sv,downSce,down=Sae,downand Sv,upSce,up=Sae,upgave the vertical displacement when the centre of mass of the body accelerates downwards and decelerates upwards, respectively(Fig. 3).

Within the step EpEktransduction

The time course of the transduction taking place within the step between gravitational potential energy Ep and kinetic energy of the centre of mass Ek was determined from the absolute value of the changes, both positive and negative increments, of Ep, Ek and Ecm in short time intervals within the step cycle(Cavagna et al., 2002):
\[\ r(t)=1-|{\Delta}E_{\mathrm{cm}}(t)|{/}(|{\Delta}E_{\mathrm{p}}(t)|+|{\Delta}E_{\mathrm{k}}(t)|).\]
(2)
The EpEk transduction is complete [r(t)=1] during the aerial phase (ballistic lift and fall) when no external work is done by the muscular force. However the EpEk transduction also occurs during contact, when the body is partially supported by the foot on the ground[0<r(t)<1]. The EpEk transduction is nil[r(t)=0] in two phases of the step, α and β,where Ek increases or decreases, respectively,simultaneously with the gravitational potential energy Ep(Cavagna et al., 2002). Note that positive and negative external work is done both in the phases αand β of the step [r(t)=0] and in the phases of the step where a transduction occurs between Ek and Ep [0<r(t)<1].

The cumulative value of energy recovery, Rint(t), resulting from the instantaneous EkEp transduction, was measured from the area below the r(t) record divided by the step period: Rint(t)=[∫ t0r(u)du]/τ. At the end of the step Rint(τ)=Rint(Cavagna et al., 2002).

Statistics

The data collected as a function of running speed were grouped into classes of 1 km h–1 intervals as follows: 3 to <4 km h–1, 4 to <5 km h–1..., 20 to <21 km h–1. The data points in Figs 2 and 4 represent the mean ±s.d. in each of the above speed intervals and the figures near the symbols in Fig. 2 give the number of items in the mean. A paired samples t-test was used to determine when the means, within a subject group with the same number of items at a given speed interval, are significantly different(Table 1). When comparing the means of different variables between the two subject groups having different numbers of items, an independent-samples t-test was used(Table 2). The t-tests were performed using SPSS for Windows version 11.0.1 (SPSS, Chicago, IL,USA).

The apparent landing-take-off asymmetry

Fig. 2 shows the fractions of the vertical displacement of the centre of mass taking place when the foot is in contact with the ground, plotted separately during the lift(Sc,up/Sv,up, open circles) and during the descent (Sc,down/Sv,down, filled circles).

When the aerial phase is nil, at the lowest running speeds (e.g. Fig. 1A,B), the whole vertical displacement takes place with the foot in contact with the ground and Sc/Sv=1. With increasing speed, an aerial phase of progressively greater extent takes place during the step. It follows that the fraction of the vertical displacement in contact with the ground decreases. The decrement is less during the lift than during the fall,i.e. Sc,up/Sv,up is greater than Sc,down/Sv,down(Fig. 2). In other words, the height of the centre of mass at the instant of take-off is greater than its height at the instant of touchdown: i.e. the ballistic lift is smaller than the ballistic fall (light-blue segments of Ep in Fig. 3)(Cavagna, 2006).

This landing–take-off asymmetry is present in both subject groups,but is larger in the old subjects than in the young subjects, mainly due to a smaller reduction with speed of Sc,up/Sv,up: a larger fraction of the lift of the centre of mass takes place in contact with the ground in the old subjects.

The effective landing–take-off asymmetry

The landing–take-off asymmetry described above bears no relation to loading and unloading of the spring-mass system during the bounce of the centre of mass at each running step(Cavagna, 2006). Indeed, as mentioned above, landing and take-off may not occur at all during low-speed running. It is obvious that, in this extreme case, the time of contact gives no information on the loading of the elastic system. Even in the presence of an aerial phase, the time of contact exceeds the time during which the spring-mass system is loaded beyond its equilibrium position, where the vertical force equals body weight(Blickhan, 1989).

It is therefore more appropriate to consider `effective landing' as the instant where the vertical force increases above body weight (rather than the instant where the foot contacts the ground) and `effective take-off' as the instant where the vertical force drops below body weight (rather than the instant where the foot leaves the ground). Loading of the elastic system with a force greater than body weight (downward deceleration and upward acceleration) takes place during the lower part (Sce) of the vertical oscillation of the centre of mass, and unloading (upward deceleration and downward acceleration) during its upper part(Sae) (Cavagna et al.,1988).

The changes in gravitational potential energy, Ep,translational kinetic energy, Ek=Ekf+Ekv, and their transduction Rint(t) are therefore depicted in Fig. 3 during loading (red)and unloading (blue) of the system relative to its equilibrium position,regardless of the contact time and the aerial phase. These records have been analyzed in detail (Cavagna,2006) and are only briefly described here to assess the different landing–take-off asymmetry in old and young subjects.

During the downward acceleration (Sae,down, Fig. 3, blue), the support of the body on the ground is low with the consequence that a large transduction of Ep into Ek occurs both in the presence and the absence of an aerial phase. The amount of this transduction is given by the increment Rint,down of the Rint(t) curve. As a consequence of this transduction, the kinetic energy Ek attains its highest peak in the running step, just prior to the effective landing (start of downward deceleration).

During the downward deceleration (Sce,down, Fig. 3, red), negative external work is done to decrease Ep and Eksimultaneously, as indicated by the horizontal tract of the Rint(t) curve showing that no transduction occurs between Ep and Ek: this is the βfraction of the step.

During the upward acceleration (Sce,up, Fig. 3, red), positive external work is done to increase Ep and Eksimultaneously, as indicated by the lower horizontal tract of the Rint(t) curve showing that no transduction occurs between Ep and Ek: this is the αfraction of the step. At the end of the α fraction the kinetic energy Ek attains its maximum during the lift, which is lower than that attained during the fall. The difference between the two Ek peaks is on average greater in the old subject. Note that whereas β is almost totally contained within Sce,down, α continues after the end of Sce,up, within the upward deceleration of the centre of mass (Sae,up, Fig. 3, blue). In other words, the muscular push is still lifting and accelerating the body forwards even though the vertical force has dropped below body weight. The intrusion of α into Sae,up is larger in the old subject.

During the upward deceleration (Sae,up, Fig. 3, blue), the transduction of Ek into Ep given by the increment Rint,up of the Rint(t) curve is confined to the last part of the lift because, as described above, Ep and Ek increase simultaneously for an appreciable part of Sae,up. Indeed, the transduction of Ek into Ep is almost nil in the oldest subject of Fig. 3. In particular, the ratio Rint,down/Rint,up is much larger in the old subject.

The different features of the rebound of the body described above translate into different durations of positive and negative work (red/blue bars in Fig. 1). These are plotted in Fig. 4 as a function of speed. It can be seen that, in both old and young subjects, the duration of positive work tpush is greater than the duration of negative work tbrake up to ∼13 km h–1. At higher speeds, both durations fall below 0.1 s and become similar. In the common speed range, the ratio tpush/tbrake is greater in the old subjects than in the young subjects(Table 2).

The records in Figs 1, 3 and 4 show that the landing–take-off asymmetry is greater in the old than in the young subjects. The results supporting this conclusion are summarized in Table 2 where a comparison is made between old subjects, young subjects and the symmetric rebound of an elastic system.

The physiological meaning of Rint,down/Rint,up being greater in the elderly

Rint(t) is the cumulative value of energy recovery resulting from the instantaneous transduction r(t)between gravitational potential energy Ep and translational kinetic energy Ek=Ekf+Ekv during the step cycle (Cavagna, 2006). In running, the kinetic energy of forward motion Ekfincreases and decreases essentially in phase with the potential energy Ep (Cavagna et al.,1964), with the consequence that the EkfEp transduction is negligible. It follows that in running the EkEp transduction takes place essentially between gravitational potential energy Ep and kinetic energy of vertical motion Ekv. This transduction is obviously complete during the aerial phase when the support of the body on the ground is nil [r(t)=1], but it also occurs during contact when the body is only partially supported by the foot on the ground in the upper part of the trajectory of the centre of mass[0<r(t)<1] [see the increment of the Rint(t) curve in Fig. 3A,B where the aerial phase is nil].

During running, therefore, Rint gives a quantitative measure of the `lack of support' of the body on the ground over the whole step cycle, including both the aerial phase and the ground contact phase. Rint would attain unity in a hypothetical `step' made up completely of an aerial phase.

In the elderly, Rint,old=0.28±0.05, which is about 70% of the value attained by the young subjects: Rint,young=0.38±0.06(P=3.66×10–44). This gives a measure of the greater support on the ground during the step in the old subjects relative to the young subjects. A lower flight time in the elderly has already been reported during running at ∼10 km h–1(Karamanidis and Arampatzis,2005).

The EkEp transduction has a different meaning during the descent Rint,down and during the lift Rint,up of the centre of mass(Fig. 3).

Rint,down, i.e. the Ep into Ek transduction during the fall, can be viewed as a mechanism exploiting gravity to passively increase the vertical downward velocity and, as a consequence, the kinetic energy. Rint,down precedes the negative work phase of the step. This has two physiological effects. (i) It provides mechanical energy to be stored within the muscle-tendon units during the subsequent brake, but (ii) it requires a force to decelerate the body downwards which, in the old subjects,may be insufficient and/or may decrease the safety of their muscular–skeletal system(Karamanidis and Arampatzis,2005). A large value of Rint,down relies on an adequate muscular force to be exerted during subsequent stretching. In the elderly, Rint,down,old=0.18±0.02, which is about 85% of the value attained by the young subjects: Rint,down,young=0.21±0.03(P=3.97×10–25).

Rint,up, i.e. the Ek into Ep transduction during the lift, follows the positive work phase of the step and is greater the greater the push-average power developed before take-off. In fact, the greater the push, the greater the increment in kinetic energy of vertical motion and therefore its subsequent decrement when the centre of mass is lifted during the phase of partial support and the aerial phase. The push-average power depends in turn on the capability to (i)recover elastically the mechanical energy stored during the preceding negative work phase, and (ii) add work done during shortening by the contractile component. A large Rint,up therefore relies on an adequate muscular force to be exerted during shortening by the muscle-tendon units. In the elderly, Rint,up,old=0.10±0.04, which is about 60% of the value attained by the young subjects: Rint,up,young=0.17±0.04(P=1.34×10–45).

Since Rint,down is 15% less in the elderly than in the young subjects whereas Rint,up is 40% less, the ratio Rint,down/Rint,up is appreciably greater in the old subjects than in the young subjects(Table 2). As mentioned above,a large Rint,down requires a large force to be exerted during the following negative work phase (stretching), whereas a large Rint,up requires a large force to be exerted during the preceding positive work phase (shortening). The finding that Rint,down is less affected by age than Rint,up suggests that the deficit in force during stretching is less than the deficit in force during shortening, which is a characteristics of aged muscle(Vandervoort et al., 1990; Porter et al., 1997; Pousson et al., 2001; Klass et al., 2005). The relatively greater Rint,down and lower Rint,up in the elderly translate into a larger difference between peaks in kinetic energy attained during the fall Ek,mx,down and during the lift Ek,mx,up (Table 2).

The physiological meaning of tpush/tbrake being greater in the elderly

Fig. 4 and Table 1 show that the positive work duration tpush is greater than the negative work duration tbrake up to ∼13 km h–1. This is true in both subject groups, but the ratio tpush/tbrake is on average greater in the old subjects than in the young subjects(Table 2).

During running on the level at a constant step-average speed, the momentum lost during negative work, braketbrake,equals the momentum gained during positive work, pushtpush. When tpush>tbrake(Fig. 4) then push<brake,i.e. the average force during positive work is less than the average force during negative work, as expected from the force–velocity relation of muscle (Cavagna, 2006).

The present findings show that tpush/tbrake is on average greater in the old subjects (Table 2),i.e. that push/brakeis less in the old subjects than in the young subjects. This indicates a lower force during shortening relative to stretching in old age, which is qualitatively consistent with the more asymmetric force–velocity relation of aged muscle (Vandervoort et al., 1990; Porter et al.,1997; Pousson et al.,2001; Klass et al.,2005).

At running speeds greater than about 14 km h–1, negative and positive work durations fall below 0.1 s and approach each other, seen more clearly in the young than in the old subjects. An explanation for this finding has previously been proposed(Cavagna, 2006) and is briefly given below.

At each running step the muscle-tendon units are subjected to a stretch–shorten cycle as the body bounces off the ground. Muscle-tendon units are composed of two structures in series having a very different response to stretch and recoil. While tendons have a similar stretch–shorten relation due to their small hysteresis(Ker, 1981; Alexander, 2002), muscle exerts a larger force during stretching then during shortening, depending on its force–velocity relation. This fact results in a different response of the muscle-tendon units to a stretch–shortening cycle depending on the relative length change of muscle and tendon during the cycle(Cavagna, 2006).

The lengthening of muscle relative to the lengthening of tendon depends on the stiffness of muscle relative to that of tendon; the stiffness of muscle,in turn, depends on its activation, i.e. on the force exerted by its contractile component. If the muscle is relaxed, i.e. the force is nil, the whole of the lengthening will be taken up by muscle. If the force is low, as at low speeds of locomotion [e.g. fig. 3 of Biewener (Biewener,1998)], an appreciable fraction of the length change will be taken up by muscle. In this case, the average force developed during stretching is expected to exceed that developed during shortening according to the force–velocity relation of muscle, and the characteristics of the bounce would deviate from those of an elastic body. If, on the other hand, the force is high, as at high speeds of locomotion [e.g. fig. 3 of Biewener(Biewener, 1998)] and the muscle is kept isometric, as some studies suggest for running(Kram and Taylor, 1990; Roberts et al., 1997; Biewener et al., 1998), most of the length change will be taken up by tendons and the characteristics of the bounce will approach that of an elastic body.

These observations are in agreement with the finding that at low and intermediate running speeds the positive work duration is greater than negative work duration (Fig. 4), indicating that the average force during positive work is less than the average force during negative work as expected from the force–velocity relation of muscle. At speeds greater than 14 km h–1, tpushtbrake,indicating that pushbrake,as expected from an elastic system. This suggests that when the duration of the rebound, tpush+tbrake, falls below 0.2 s (Fig. 4) the length change of the muscle-tendon units is taken up almost completely by tendons and other undamped elastic elements, with muscle in a quasi-isometric contraction. Fig. 4 shows that this apparent elastic behavior at high speeds is more evident in the young subjects than in the elderly.

In conclusion, a muscular force greater during stretching than during shortening with a large energy loss during the stretch–shorten cycle,which are both consequences of the force–velocity relation of muscle,may explain, at least qualitatively, the ensemble of the deviations from the elastic model during the bounce of the body(Table 2). Assuming that the old subjects we tested exhibit the modification of the force–velocity relation characteristic of their age, the larger deviation from the elastic model found in the old subjects may be due to: (i) the greater difference in force between stretching and shortening described in the aged muscle, and (ii)the lower force developed by their muscles, implying a relatively larger length change of muscle relative to tendons within the muscle–tendon units.

LIST OF SYMBOLS AND ABBREVIATIONS

     
  • av

    vertical acceleration of the centre of mass

  •  
  • Ecm

    mechanical energy of the centre of mass

  •  
  • Ek

    kinetic energy of the centre of mass in the sagittal plane

  •  
  • Ek,mx,down

    kinetic energy peak attained during the descent of the centre of mass

  •  
  • Ek,mx,up

    kinetic energy peak attained during the lift of the centre of mass

  •  
  • Ekf

    kinetic energy of forward motion of the centre of mass

  •  
  • Ekv

    kinetic energy of vertical motion of the centre of mass

  •  
  • Ep

    gravitational potential energy of the centre of mass

  •  
  • brake

    average force during negative external work

  •  
  • Ff

    fore–aft force

  •  
  • push

    average force during positive external work

  •  
  • Fv

    vertical force

  •  
  • g

    acceleration of gravity

  •  
  • Mb

    body mass

  •  
  • r(t)

    instantaneous recovery of mechanical energy calculated from the absolute value of the changes, both increments and decrements, in Ep, Ek and Ecmduring each time interval in the step period(Eqn 2)

  •  
  • Rint

    cumulative energy recovery attained at the end of the step due to EkEp transduction

  •  
  • Rint,down

    fraction of Rint attained during the downward displacement of the centre of mass

  •  
  • Rint,up

    fraction of Rint attained during the lift of the centre of mass

  •  
  • Sae,up

    upward displacement of the centre of mass taking place during its upward deceleration

  •  
  • Sae,down

    downward displacement of the centre of mass taking place during its downward acceleration

  •  
  • Sc,down

    downward displacement of the centre of mass during contact

  •  
  • Sc,up

    upward displacement of the centre of mass during contact

  •  
  • Sce,down

    downward displacement of the centre of mass from the equilibrium position where Fv=Mbgto the lowest point of its trajectory

  •  
  • Sce,up

    upward displacement of the centre of mass from the lowest point of its trajectory to the equilibrium position where Fv=Mbg

  •  
  • Sv,down

    downward displacement of the centre of mass

  •  
  • Sv,up

    upward displacement of the centre of mass

  •  
  • tbrake

    duration of negative external work production

  •  
  • tpush

    duration of positive external work production

  •  
  • Vf

    forward velocity

  •  
  • Vv

    vertical velocity

  •  
  • \(W_{\mathrm{ext}}^{+}\)

    positive work done at each step to increase the mechanical energy of the centre of mass:

    \(W_{\mathrm{ext}}^{+}\)
    is the sum of the positive increments of Ecm during the periodτ

  •  
  • \(W_{\mathrm{kf}}^{+}\)

    positive work done at each step to increase the forward speed of the centre of mass:

    \(W_{\mathrm{kf}}^{+}\)
    is the sum of the positive increments of Ekf during the periodτ

  •  
  • \(W_{\mathrm{v}}^{+}\)

    positive work done at each step to sustain the gravitational potential energy changes of the centre of mass:

    \(W_{\mathrm{v}}^{+}\)
    is the sum of the positive increments of Ep during the period τ

  •  
  • α, β

    step phases during which Ep and Ek increase (α) or decrease (β)simultaneously

  •  
  • Δtphoto

    time interval between photocells crossing

  •  
  • τ

    step period, i.e. period of repeating change in forward and vertical velocity of the centre of mass

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