Mechanical work in terrestrial locomotion: two basic mechanisms for minimizing energy expenditure

Transcription

1 Mechanica work in terrestria ocomotion: two basic mechanisms for minimizing energy expenditure GIOVNNI. CVGN, NORMN C. HEGLUND, ND C. RICHRD TYLOR Istituto di Fisioogia Umana de Universitci di Miano, Centro di Studio per a Fisioogia de1 Lavoro Muscoare de1 CNR, 2133 Miano, Itay; and Museum of Comparative Zooogy, Harvard University, Cambridge, Massachusetts 2138 CVGN, GIOVNNI., NORMN C. HEGLUND, ND C. RICHRD TYLOR. Mechanica work in terrestriaz ocomotion: two basic mechanisms for minimizing energy expenditure. m. J. Physio. 233(5): R243-R261, 1977 or m. J. Physio.: Reguatory Integrative Comp. Physio. 2(3): R243-R261, The work done during each step to ift and to reacceerate (in the forward direction) and center of mass has been measured during ocomotion in bipeds (rhea and turkey), quadrupeds (dogs, stump-taied macaques, and ram), and hoppers (kangaroo and springhare). Waking, in a animas (as in man), invoves an aternate transfer between gravitationa-potentia energy and kinetic energy within each stride (as takes pace in a penduum). This transfer is greatest at intermediate waking speeds and can account for up to 7% of the tota energy changes taking pace within a stride, eaving ony 3% to be suppied by musces. No kinetic-gravitationa energy transfer takes pace during running, hopping, and trotting, but energy is conserved by another mechanism: an eastic bounce of the body. Gaoping animas utiize a combination of these two energy-conserving mechanisms. During running, trotting, hopping, and gaoping, 1) the power per unit weight required to maintain the forward speed of the center of mass is amost the same in a the species studied; 2) the power per unit weight required to ift the center of mass is amost independent of speed; and 3) the sum of these two powers is amost a inear function of speed. easticity; waking; running; hopping; trotting; gaoping; eficiency ON FIRST CONSIDERTION, the methods which vertebrates have utiized to move aong the earth s surface seem both diverse and compex. Some use two egs whie others use four. They wak, ambe, trot, pace, canter, gaop, and hop (19). Bioogists have concentrated on describing differences between ocomotory types, and detaied descriptions exist for the foot-fa patterns and the anatomic features associated with the different modes of ocomotion (16, 17, 19). In this paper we have tried to find mechanisms which are common to these different modes of terrestria ocomotion. We seected animas to represent what appear to be three very difftirent types of ocomotion: bipeda birds which wak and run (turkey and rhea); quadrupeda mammas which wak, trot, and gaop (dog, monkey, and ram); and bipeda mammas which hop (kangaroo and springhare). We measured the forces appied by the animas to R243 the ground as they moved at different speeds and used different gaits. These force measurements were used to cacuate the mechanica energy which must be provided by the ocomotory system to move the anima s center of mass forward reative to the ground. We considered ony the situation after an anima had acceerated and reached a constant average speed. Locomotion at a constant average speed consists of a series of cyces (steps or strides), during which both the gravitationa potentia energy and the kinetic energy of the center of mass osciate between maximum and mini- mum vaues as the center of mass rises, fas, acceerates, and deceerates. Forces must be appied to the ground to raise and reacceerate the center of mass (12, 15), and mechanica work is performed (work = force x dispacement). This has been caed the externa work output Wezt of ocomotion (7), and the rate at which this work is performed is then externa power output ( Wed. west is ony part of the tota work done by the anima during ocomotion. The musces aso perform work to change the kinetic energy of the imbs reative to the center of mass. This has been caed the interna work output Wint of ocomotion; we did not measure Wint in this study, but it has been measured for man (5), quai (o), and kangaroos (2) by other investigators. Contracting musces use chemica energy to suppy some, but not necessariy a, or even most, of Wert. most no additiona energy woud be required to maintain a constant forward speed of the center of mass, if the decrements of the gravitationa potentia energy (as the center of mass decreases in height during each stride) and the decrements of kinetic energy (as the anima sows during each stride) coud be stored and/or used to reacceerate and raise the center of mass during another part of the stride. There are two mechanisms for aternatey storing and recovering energy within each stride: 1) an exchange between gravitationa potentia energy and kinetic energy, as occurs in a swinging penduum; and 2) an exchange between mechanica energy stored in musce s eastic eements and recovered as both kinetic and gravitationa energy, as in a bouncing ba. Both of these storage-recovery mechanisms have been found to be important for minimizing the chemica energy input required for Wex in human ocomotion: the penduum in the wak and eastic storage in the run (6-8, 13, 2). We wanted to know if these mechanisms are utiized in Downoaded from by on November 7, 216

2 R244 CVGN, HEGLUND, ND TYLOR some reguar way by a terrestria vertebrates, and if so, how. MTERILS ND METHODS nimas We tried to achieve as great a diversity of ocomotory modes as possibe in our seection of animas. We chose a arge bird, the rhea (Rhea americana, one anima weighing 22.5 kg) and a smaer bird, the wid turkey (Meeagris gaopavo, two animas each weighing 7 kg). Locomotion in these two bipeds coud be compared with that of man, who had aready been carefuy studied with the same apparatus (9). Bipeda ocomotion in birds ooks quite different from that of humans, since the anke of the bird occupies a simiar position reative to the ground as the knee of humans. Thus the eg of a bird appears to bend in the opposite direction from a eg of a human. For our quadrupeda animas we seected one which was a highy speciaized and efficient runner and appeared to move gracefuy and effortessy at a speeds: the dog (Canis famiiaris, a sma terrier weighing 5 kg and a arge mongre weighing 17.5 kg). We compared the dog with another quadruped which was highy speciaized to use its egs for nonocomotory tasks, and which appeared to move awkwardy and inefficienty at a speeds (i.e., with its egs faiing and swinging in a directions): a monkey, the stump-taied macaque (Macaca speciosa, two maes, each weighing 3.6 kg). We seected the ram (Ovis musimon, two animas weighing 6 and 85 kg) because it was the argest quadruped we coud convenienty find that was sti within the capacity of our force pates. We seected, two hoppers which were very far apart phyogeneticay: the kangaroo ( MegaZeia rufa, one femae weighing 2 kg and one mae weighing 21 kg), which is a marsupia, and the springhare (Pedetes cafer, one anima weighing 2.5 kg), which is a rodent. This suggests that their satatory modes of ocomotion have evoved independenty. Procedures The horizonta and vertica components of the resutant force appied by the anima to the ground were measured by means of a force patform. The atera movements were disregarded. The animas were not restrained in any way when they ran across the force patform. We caed, coaxed, or chased them (more or ess vehementy) to achieve a range of speeds. The force patform was inserted with its surface at the same eve as the foor, about 3 m from the beginning of a corridor. Thus the animas had penty of space to reach a constant speed over the patform, even during a fast run. Since we wanted to study ocomotion at a constant average speed, we accepted ony those trias where the acceerations taking pace during one or more compete strides differed from the deceerations by ess than 25%. The force patform was 4 m ong and.5 m wide and this ength aowed us to record the force exerted by the feet against the ground over severa strides. The horizonta force and the vertica force minus the body weight were integrated eectronicay to determine the instantaneous veocity of the center of mass of the body to yied the instantaneous kinetic energy. The change in potentia energy was cacuated by integrating vertica veocity as a function of time to yied vertica dispacement, and mutipying this by body weight. The tota mechanica energy as a function of time was obtained by adding the instantaneous kinetic and potentia energies. The positive externa mechanica work was obtained by adding the increments in tota mechanica energy over an integra number of strides. Both the mechanica detais of our force patform and the procedures invoved in using force patforms to measure externa mechanica work have been thoroughy described in a recent artice by Cavagna (3). For the convenience of the reader, we have incuded a ist of the symbos which we use throughout the paper with their definitions in Tabe 1. RESULTS Waking We found a remarkabe simiarity in the force and veocity records from a of our experimenta animas when they waked across our force pates (Fig. 1). In fact it was often not possibe to identify which record came from which anima without knowing the absoute vaues of the forces invoved. great dea of information (e.g., the patterns of thrust from individua feet) can be obtained from these records, but we have omitted this detaied information in this paper. Instead, we have concentrated on trying to understand genera mechanisms which animas utiize to keep their center of mass moving forward at a constant average speed. Waking in a of our experimenta animas invoved an aternate exchange between the kinetic energy and the gravitationa potentia energy of the center of mass within each stride. In this respect, waking is simiar to a swinging penduum or an egg roing end-over-end (7). The mech anica energy which the musces have to provide to keep the center of mass moving forward is decreased by the amount of this exchange which depends on three factors: I) the phase reationship between the changes in kinetic energy and gravitationa potentia energy within the stride; 2) the reative magnitude of the two; and 3) the degree of symmetry between the two (i.e., how cosey they approximate mirror images of each other). If the kinetic and gravitationa potentia energy changes are 18 out of phase, if their magnitudes are equa, and if the changes are symmetrica, then the musces wi not need to provide any additiona energy to keep the center of mass moving, provided of course friction and wind resistance are negected. The amount of muscuar work required to keep the center of mass of a waking anima moving at a constant speed wi depend on the deviation of the measured kinetic and potentia energy changes from this optima condition. Phase reationship between kinetic and gravitationa potentia energy. The phase reationship between the Downoaded from by on November 7, 216

3 h-). h-). MECHNISMS OF TERRESTRIL LOCOMOTION R245 --iif Ee Ekf Eku EP E tot f Ff F, i K K L Lc L dc Lc p s* t act tc tdec 47 average deceeration forward of the center of mass taking pace at each step as a consequence of the impact with the ground, -if = -bvf/tdec where - Vf is a decrement of Vf and tee is the time during which this decrement takes pace eastic energy stored during the negative work phase of each step and deivered during the positive work phase kinetic energy of forward motion of the center of mass kinetic energy of vertica motion of the center of mass gravitationa potentia energy of the center of mass tota mechanica energy of the center of mass, Etot = Ekf + E, + Ekv step or stride frequency horizonta component of the resutant force exerted by the feet against the ground vertica component of the resutant force exerted by the feet ainst t&e ground acceeration of gravity sope of a inear reationship, starting from the origin, cacuated for the function -tif = f(vf); i.e., K = -afqf (t,-jt,) t& = K ength of the step or stride, L = 6,-T forward dispacement of the center of mass taking pace during each step or stride whie the body is in contact with the ground, L, = vf* tc forward dispacement of the center of mass taking pace at each step of waking when both feet contact the ground, L& = vf forward dispacement of the center of mass taking pace at each step of waking when one foot ony is in contact with the ground, Lx = vf*t, body mass body weight sum of the upward dispacements of the center of mass taking pace during a cyce of movement T; S, = W,/P fraction of the period T during which the forward speed Vf increases fraction of the period T during which the feet contact the ground fraction of the period 7 during which the forward speed Vf decreases fraction of the period T during which the body is off the ground rate of oxygen consumption, above the rest rate, which we changes in kinetic and gravitationa potentia energy can be easiy seen from the records of instantaneous kinetic and potentia energy of the center of mass over a series of strides in Fig. 2 (computed from the force and veocity data presented in Fig. 1). The top ine of each experiment in Fig. 2 represents the kinetic energy due to the veocity of the center of mass in the horizonta direction (E&, the two midde ines (which amost competey overap) are the gravitationa potentia energy (E,) and the sum of the gravitationa potentia energy pus the kinetic energy due to the veocity of the center of mass in the vertica direction (E, + E,J. The kinetic energy in the vertica direction (Ekv) is so sma compared to E, that two separate ines representing E, and E, + Ekv can ony be distinguished in a few of the records (e.g., the rhea waking at 3.7 km* h-). The records of the turkey and the rhea ceary show that at sow waking speeds E, is at its maximum when E, is at its minimum, i.e., they are amost competey out of phase; as the animas wak faster, E, and E, become more and more in phase with one another (compare the experiments where the turkey waks at 3.4 km. h- vs. 4.6 km h- or where the rhea have converted to metaboic power input using the energetic equivaent of 4.8 Ca/m O2 V -f instantaneous speed of forward motion of the center of mass Vf average speed of ocomotion, vf = L/T W ext positive work done at each step to increase the mechanica energy of the center of mass; Wed is the sum of the increments of Etot during 7; this is caed externa work Wf positive work done at each step to increase the forward speed of the center of mass; Wf is the sum of the increments of Ekf during 7 WV positive work done at each step to ift the center of mass; W, is the sum of the increments of E, during 7 Knt the positive work done during each step to acceerate parts ~ K? of the body reative to the center of mass. This is caed interna work rate at which Ee is deivered during T; We = E,/T rir,, = W,,JT: this mechanica power is smaer, about haf or ess, than the average power deveoped by the musces during contraction; in fact not ony positive work but aso negative work is done during 7; in addition 7 may incude a Wf ~ WV R?t = W& fight period t, = Wf/T; other indications as for West = W&; other indications as for West wmetab rate at which the musces consume chemica energy for ocomotion; Wmetd = (Wmetd during ocomotion) - (Wmtti at I rest) WI work done per unit distance and per unit of body mass; w 4 Y ange between the vertica and a ine connecting the hip joint with the point of contact on the ground (Fig. 3) period of a repeating change in forward veocity and height of the center of mass; in waking, running, and trotting 7 is the period of the step; in gaoping 7 is the period of the stride overa efficiency of positive work production, i.e., the ratio between the tota mechanica power output (We, + W& and the net energy expenditure per unit time (Wmet,); the atter incudes the cost of negative work = wextrir,,,; Y < Y waks at 2.2 km* h- vs. 5.1 km* h-). The bottom ine in Fig. 2 (E,,,) is the instantaneous sum of Ekf and (E, + E&. When the two curves are MO out of phase, the osciations of Etot are ess than those of E, or (E, + Ekv) individuay by the amount of the exchange between the two (see Fig. 2, man waking at 3.9 km When the Ekf and the (E, -+ Ekv) curves are amost competey in phase, itte exchange is possibe, and the increments of Etot approach the sum of the increments of the two curves (see Fig. 2, ram waking at 4.6 km Other experiments in Fig. 2 show a range of phase reationships between these two extremes. Magnitude of kinetic and gravitationa potentia energy changes. In a simpe penduum the change in gravitationa potentia energy (E,) wi equa the changes in kinetic energy in the forward (&J and vertica directions (E,J at each instant in time 3 P = hekf + mkv (1) In the simpest inverted penduum system which has been used to characterize waking - the stiff-egged wak of exander ( )-Ek, woud be ost from the system in each step when the front foot hits the ground (Fig. Downoaded from by on November 7, 216

4 R246 CVGN, HEGLUND, ND TYLOR VF(m/scc).5 u const. -~ I, -.5 E F,(h) ' -4 E V, ( m/see ).5 const E F,(kg) 1 c 2ooL :,- _. 1 c - -1 t 2 ' 4 E WLK -I b' -.4 E i,--j /L./ - 4-L--'~-~~ E *c?upt/-.f 8 4 I,.. c a5 - RHE 3.7 k-i?hr.2 c. - - c.- ' F F -.5L -.2' FF(kd ~/",,'..,.", ~,' (_,," ;, "._,,,.'",,." /i+,,,., I,/f/,,/ '.,",/x. r.",-,/ir," /? 'G J -4 E -1 E i -8 E v,(+cc) const. I ~I~_ "..;',,._, '.-,,,, c. - E _- _ 1 _._ I' _.' - (., - ~/- /' ;'. _ I E E km~~~ ~-- o.5 we Q8 i E r 1 4 E -.._I :,.,<.//' ;,,' r-' I,/,'.(,_. E - I (/i I,.", ~",1' - E..._/* FJ"ih/" ,' i /.4 1 a ,-,,/ #, 1. /? ~,. i..._ -._.._.._ (.. _ :-., I,,. - c. - E E E -j VF(m/sec) E F,(kg) 4 V, ( m/set ) const. F,(kg) const. - c.-.., Q8 ' _---.- _ x' w 1 2-d 8 'd % t"* 4 E 2 E i FIG. 1. Experimenta records obtained during waking. Each set of records indicates from bottom to top: vertica force exerted by the feet against the ground (F,), osciations of vertica component of veocity of the center of gravity (V,.), forward component of force exerted on the ground (F,) and osciations of forward component of veocity of the center of gravrty (V,). Positrve vaues of F, mdicate a backward push of the foot against the ground during this push the 1 2 set _ forward veocity V, increases; the opposite occurs when F, is negative. Veocity tracings begin and end when the anima crossed two photoces over the patform. Meaning of integration constant is described by Cavagna (3). For turkey and rhea, records are given for different speeds of waking to show modifications of the tracings with speed. Note simiarity of records between different animas and man. c Downoaded from by on November 7, 216 3), and the transfer of kinetic energy into gravitationa potentia energy coud ony take pace between E,, and E,. exander has concuded that most of the energy used during this stiff-egged waking woud go into repacing the Ek,. ost during each step. It woud therefore be advantageous to keep E,,, sma. One way to do this is to keep the ampitude of the swing sma. For exampe, in Fig. 3 (top) E,, = E,,, when v, is 45 and LU?,~ > E,,. when C,C ~45. It can be seen ceary in Fig. 2 that waking animas keep Ek,. sma, in fact it never exceeds 6% of Ekf. Not ony is E,,. sma during waking, but it appears aso to be converted directy into E,, by the appication of a force norma to the direction of the veocity of the center of mass as it fas forward. The extension of the back foot which occurs during waking (9) may appy

5 MECHNISMS OF TERRESTRIL LOCOMOTION R247 WLK MONKEV 26 km/b MN 3.9 km/hr RHE 2.2 km/b T Q2ca TURKEV TURKEV 3.4 km/hr 4.6 km/m RHE RHE 3.7 km/hr 5.1 km/hr RM 4.6 km/hr Downoaded from TIME FIG. 2. Experimenta records of mechanica energy changes of center of mass of body during waking. Curves were cacuated from records of Fig. 1; they were drawn directy using the potter output of a computer. In each set of tracings, the upper curve refers to kinetic energy of forward motion & = 4 m Vt; the midde curve to the sum of gravitationa potentia energy E, and kinetic energy just such a norma force (Fig. 3, bottom). so the center of mass continues to fa afier the front foot hits the ground during part of the period of doube support. If the two feet act to appy a resutant force norma to the direction of the veocity of the center of mass during this time, then there coud aso be a transfer ofekv into E,, Our records of the potentia energy changes of the center of mass before and during the period of doube contact show that E, decreases graduay as the center of mass approaches its owest point (as in Fig. 3, bottom); this is consistent with a transfer ofek, into Ekp How do the reative magnitudes of E, and Z& compare during waking in bipeda birds and quadrupeda mammas? Summing the increments of (E, + Eke) over a stride gives the positive work done against gravity (WV) and dividing by the stride period gives the average power (I&,>. Summing the increments of Ekf gives the positive work done to acceerate the body F-- SCCF of vertica motion Ekv = 4 m V,Z; the bottom curve to tota energy E tot = Ekf + E, + Eke. Curves E, and (E, + Ekf) are aso given (thin ine), but often they cannot be distinguished from (E, + Ekv) and Eto curves, since Ekv is very sma in waking. Note that at ow and average speeds E, and Ekf change in opposition of phase as in a penduum. forward (W,> and dividing by the stride period gives the average forward power (W&. Mass specific power, 8$,rn and Wfm, is potted as a function of waking speed in Fig. 4. Using these graphs it is easy to compare the reative magnitudes of the positive energy changes in E, and E,,-, and to see how they vary as a function of waking speed. In both birds (rhea and turkey) W, = Wf at the sowest waking speeds, and Wf > W, at the highest waking speeds as has been found in man (9). There is a simiar trend in the ram, but at its highest waking speeds W, = WV. In the dogs Wf = W, over the entire range of waking speeds and in the monkey W, > Wf over the entire range of speeds. The point which emerges from the measurements of W, and W, is that their reative magnitudes are often simiar during waking: Therefore, from the viewpoint of reative magnitude aone, it is possibe to have a significant transfer of energy between kinetic and gravitationa potentia by on November 7, 216

6 h-) FIG. 3. Schematic representation of inverted penduum system in waking. Top diagram represents stiff-egged wak of exander (1) where ony one foot makes contact with the ground at a time. Step ength, therefore, equas forward dispacement of the center of mass whie one foot is in contact with the ground (L,). EkF is ost during each stride when the foot first makes contact with the ground, whie Ekf is transferred into E,. Ekv can be kept sma by keeping sma the ange cp between a ine connecting the hip joint with the ground and a ine drawn perpendicuar to the ground. coser approximation to penduum system used by waking animas is given in bottom diagram. The rear foot pushes upward before it eaves the ground, exerting a force norma to the veocity of the center of mass. This resuts in a conversion of Ekv directy into EkP This norma force continues during the period when both feet make contact with the ground as a resutant of the forces exerted by both feet. The center of mass moves forward during the period of doube contact by an amount Ldc, and the step ength L equas Ldc + L,. energy within the stride in a of the animas. How important is the exchange between kinetic and gravitationa potentia energy within each stride? s mentioned at the beginning of this discussion, phase ange and the reative ampitude of kinetic and potentia energy changes are ony two of the factors which determine the competeness of the exchange between kinetic and gravitationa potentia energy; symmetry is aso important. The two changes potted as a function of time must be mirror images of one another if the exchange is to be compete. If there is an increase in potentia energy without a simutaneous and equa decrease in kinetic energy, then contracting musces (or stored eastic energy) must provide the missing energy. Likewise, if there is a decrease in potentia energy without a corresponding increase in kinetic energy, the energy must ether be ost as heat, or stored in eastic eements. Perhaps the simpest way to quantify the net effect of a three parameters is to compare the magnitude of the power output required to maintain a constant waking steed if there were no exchange (I Wt + I W,,) with CVGN, HEGLUND, ND TYLOR the amount of power actuay expended (W&. W,,, can be cacuated from EtO (Fig. 2) by summing a the increments in + over a stride and dividing by stride period, just as W, and W, were cacuated from E,, and (E, + Ekv). This was done and W,,, is potted as a function of waking speed in the second (from the top) set of graphs in Fig. 4. The magnitude of the exchange between gravitationa potentia energy and kinetic energy is then equa to the tota power, which woud be required. if there were no exchange (1 Wfi + 1 WU I) minus W ext. This difference can be expressed as a percentage of the tota power required with no exchange % recovery = P& + I%( - Kxt x 1oo I %I + I Kq I If the exchange were compete, then W,,, woud be zero, and the recovery woud be 1%. Percentage recovery is potted as a function of waking speed in the third (from the top) set of graphs in Fig. 4. The figure shows ceary that there was a arge exchange between gravitationa potentia energy and kinetic energy during waking in a of our animas. This exchange reached a maximum at intermediate waking speeds in bipeds: 7% in the rhea at 3-4 km* h-, and 7% in turkeys at 3-4 km. h-. The exchange decined at sower and faster waking speeds. This is identica to the situation which Cavagna et a. (9) found in man, and their curve has been potted as a dotted ine in the graph of percent recovery for the turkey for comparison. Percent recovery in the quadrupeda ram aso reached a maximum (35%) at intermediate waking speeds (3-4 km and decreased at sower and faster speeds. In dogs and monkeys, percent recovery reached vaues of 5%, but it did not change in a reguar way with speed. The externa work required to move one kiometer has been cacuated by dividing We,, by speed and is potted in the bottom set of graphs in Fig. 4. It appears that there is an optima waking speed for both birds and the ram, where West to move a given distance is minima and the exchange between gravitationa potentia energy and kinetic energy is maxima. This is aso the case for man (9) as can be seen from the dotted ine in the turkey graph. Running, Trotting, and Hopping Just as we had found a remarkabe simiarity in the force and veocity records obtained during waking (Fig. I), we aso found a striking simiarity in the force and veocity records of humans, birds, dogs, monkeys, rams, kangaroos, and springhares as they ran, trotted, or hopped across the force pates (Fig. 5). Running, trotting, and hopping animas a used their musces to push upward and forward simutaneousy during one part of the stride and to break their fa and sow their forward speed simutaneousy in another part of the stride. There was usuay, but not necessariy, an aeria phase. The reative proportion of the time spent in the air was argest in the hoppers (up to 75% of the step period), intermediate in runners (up to 3% of the step period), and smaest in the trotters (O-15% of the step period). It is easy to identify this aeria phase on the (2) Downoaded from by on November 7, 216

7 I / MECHNISMS OF TERRESTRIL LOCOMOTION Fu49 BIPEDS QUDRUPEDS --MONKEYS --RMS I 5 25 I I I I I I I I 8 t WLR*QLL()p e-- wm -t --- i 1 i t I I I I I I 1 I I 1 I I I I I I I -- 1 f 1 + Downoaded from N /... w -...,_ *,, H I I 1 I I I I I I 1 I 1 1 I I I VERGE SPEED OF WLKING (km/hr) FIG. 4. Top set of graphs gives power required to ift center of km), which reaches a minimum when recovery of mechanica energy mass W, and to reacceerate center of mass in the forward direction is maximum; the same is true for man (dotted Lines). Verticay W1 during each step of waking at different speeds. Second (from the crossed symbos refer to a wak-run transition, obiquey crossed top) set of graphs gives tota externa power Wert as a function of symbos to an unidentified gait. The two turkeys performed different waking speed. Third set of graphs gives percent recovery of mechan- W, but the same W,; this is true aso for the two monkeys; the ica energy (Eq. 2) resuting from an exchange between gravita- monkey doing more work against gravity in waking aways shired tiona potentia energy E, and kinetic energy of forward motion EkP to a trot, the other to a gaop. Bottom set of graphs gives externa work per unit distance (W,,,/ by on November 7, 216 force and veocity records of Fig. 5, since both F, and F, are zero when the anima is not touching the pate. so V, remains constant and V, changes at a rate of 9.8 me ss2 due to the acceeration of gravity. Phase reationship and exchange between kinetic and gravitationa potentia energy within each stride. Energy changes due to the motion in the vertica (E, + Eke) and forward (E,,) directions were aways amost competey in phase during running, trotting, and hopping (Fig. 6); therefore there was itte possibiity for an exchange between the two. Wett = 1 Wfi + 1 IV,1 and percent recovery cacuated using Eq. 2 was neary zero (Fig. 7). E,, becomes quite arge in parts of the stride during running (much greater than during waking). It reaches maximum vaues twice: at the instant the anima eaves the ground (easiy seen in Fig. 6 as the point where (E, + Ekv) becomes constant) and at the instant the anima ands (seen in Fig. 6 as the point where (E, + Ek,,) starts to decrease again). fter the anima eaves the ground, Ekv decreases as the center of mass rises, unti it is competey converted into E, when the center of mass reaches its maximum height. t this instant the vertica veocity is zero and the E, and (E, + Ek,,) curves in Fig. 6 coincide. Then the anima begins to fa as E, is converted into E,,; this then decreases as

8 R25 CVGN, HEGLUND, ND TYLOR RUN = HOP = TROT VF(m/sec) Q5r const. - t - const. - t -.5L -.4L FF(kg) or -4 E ', -2 E Vv(m/sec 1 2 ~_ 2, const. I, - const E -2 E -.4r' const. -,, x -%, I: -.4 t.8 E const. - _ - r - -a*, F&kg) 4 E."L,?.,i,. i- Ii b 2 P -4 (_ -2, r E L : FIG. 5. Experimenta force and veconst ocity records obtained during running -.2 t (man, turkey, and rhea), hopping (kangaroo and spring hare), and trot- 8r.?,.I( ting (monkey, dog, and rami. Two difa ---i, * / - ferent speeds are given for the kanga- * : : i -8-1 roo to show increase of force, particuary of the forward component, with -()'.5 E *' - *. - p ' speed. Right-hand ordinates give vertica acceeration of center of mass as const. o- L ^ L..."_( a mutipe of gravity; note that this is greatest in hopping (particuary in springhare, 2.5 kg body wt), intermediate in running, and minima in trot- 1 I 2.:g ting. Other indications as in Fig * " $2 Downoaded from by on November 7, 216 the anima breaks its fa unti it reaches zero when the center of mass is at its owest point. Since E,,. is zero when (E, + E,,.) is both at its maximum and at its minimum vaues, the increments in (E, + IX,,,) per step equa the increments in E,. Magnitude of kinetic and gravitationa potentia energy changes. Perhaps the most unexpected and intriguing finding of this study was that the mass specific power output for reacceerating the center of mass (in the forward direction) was neary the same for bipeda running, quadrupeda trotting, and hopping when pot- ted as a function of speed ( WJrn in Fig. 8). In a of the animas wf/rn seemed to approach zero at zero running speed and increased more and more steepy with increasing speed. The functions reating wfrn and speed for birds, springhares, kangaroos, dogs, monkeys, and rams are amost competey superimposabe. Even the data which Cavagna et a. (9) obtained from humans in running (dotted ine in kangaroo graph in Fig. 8) are hardy distinguishabe from the data from our diverse assortment of animas. The mass specific power output for raising the center

9 MECHNISMS OF TERRESTRIL LOCOMOTION R251 RUN = HOP = TROT MN 14.3 km/hr KNGROO 1.2 km/hr MONKEY 5.7 km/hr T 2Sca rnekf ti EP r 25ca E TOT g-r EP* EKF TURKEY 13.5 km/hr KNGROO 24.4 km/hr SP. HRE 15.5 km/hr T 2ca 1 DOG 1.4 km/hr /mn RM 6. km/hr FIG. 6. Experimenta records of mechanica energy changes of center of mass of the body whie running, hopping, and trotting. Curves were cacuated from tracings of Fig. 5. Their meaning is described in Fig. 2. Note that gravitationa potentia energy E, and kinetic energy of forward motion Ekf change in phase in running as in hopping and trotting. Other expanations are given in text. Downoaded from RHE 16.9 km/hr Y Q-7 4 $7-j7j T 5ca 1 by on November 7, 216 TIME -1 set----+ of mass within each stride (&/m in the top set of graphs in Fig. 8) was amost independent of running speed in a of the animas. Since the vertica dispacement of the center of mass S, generay decreased with increasing speed (Fig. 9) and w Jm = S,*g f = const (3) (where g is the acceeration of gravity) an increase in stride frequency f must compensate for the decrease of S,. &,/m was greater in hopping than in running or trotting as a resut of the greater vertica dispacement of their center of mass. S, does not seem to depend on body mass aone, nor on the overa dimensions of the body; e.g., in the springhare (2.5 kg), S, is greater than in the ram (85 kg). It seems more ikey that S, depends on the pecuiar characteristics of the eastic system on which the body bounces at each step. For exampe when the kangaroo deceerates downward and forward, its feet are fexed so that the chies tendons are stretched storing eastic energy. In order to reease this eastic energy an extension of the feet is necessary, but this impies an appreciabe upward dispacement of the body of the order of magnitude of the dimensions of the foot of the kangaroo (Fig. 9). In other words, to

10 Ft252 CVGN, HEGLUND, ND TYLOR 6 c I - RMS I I I I I I I I I I I o- SPRING HRE 4 RHE i z g 4 c) g DOGS t f-1 TURKEYS i VERGE SPEED OF LOCOMOTION (km/hr) FIG. 7. Recovery of mechanica energy (Eq. 2) in waking (open kinetic energy as in a penduum (eastic energy is not taken into symbos), running (cosed symbos) and gaoping (haf-open sym- account). This recovery is maxima in waking, intermediate in bos) as a function of speed. Recovery indicates extent of mechani- gaop, and minima in running (run = hop = trot). ca energy reutiization through the shift between potentia and utiize the eastic energy stored by his eastic system, pare the mechanica power output of the anima with the kangaroo must (jump over his feet. its chemica power input and define the efficiency (y) In running, trotting, and hopping the mass specific with which an anima converts stored chemica energy power for externa work (We,Jm) is into positive mechanica work as Westm = IWjmI + (WJmI (4) since there is no exchange between E, and Ekf. &Jrn increased ineary in a the animas with increasing speed as in man (Fig. 8). The equations for these ines, cacuated using the method of east squares, are given in Tabe 2. The high correation coefficients for a inear reationship indicate that the curviinear reationship for &/m must be compensated for by a curviinear reationship for &,/m as has been described for man (9). Thus WJ m is probaby not independent of speed but our data are too scattered to define this reationship precisey. Is there storage and recovery of mechanica energy in eastic eements? Measuring the magnitude of the exchange between kinetic energy and gravitationa potentia energy is easiy done and its importance can be quantified as percent recovery using Eq. 2. Determining the magnitude of an exchange between kinetic and/or gravitationa potentia energy and eastic potentia energy (Ee) is much more difficut. There is a way, however, to approach the question of eastic storage using our force pate measurements (8). We can com- = wett + wint Y * (5) W metab where IVest is externa power output, PVint is interna power output, and VVmetab is the chemica power input into the musces for ocomotion ( VSImetab during ocomotion, -YSmetab at rest). The maximum efficiency with which musces can convert chemica energy into positive mechanica work is about.25. If the efficiency of ocomotion at steady state exceeds.25, then the expanation has to be storage and recovery of energy using musce s eastic eements. Negecting the cost of negative work, the efficiency with which the contractie machinery of musce transforms chemica energy into positive work becomes (wead + ~int - we Ym =. W metab where VVe is the power output deivered free of cost during the recoi of stretched eastic eements (i.e., the energy stored during the deceeration, when the musce performs negative work, and reeased during the sub- Downoaded from by on November 7, 216

11 .. i FIG. 8. bove: mass specific power for raising center of mass against gravity within each stride WV/m (top set of graphs); mass specific power for reacceerating center of mass in the forward direction after the deceeration due to impact with the ground wf/rn (midde set of graphs); and mass specific power for tota externa work, w&m, are potted as a function of running speed. Curves through I&/m data were cacuated using Eq. 12. Straight ines through w&m data were cacuated using eastsquares method (equations are given in Tabe 2). Dotted ines are data from man (9). Some vaues of I@ /ti for rhea and springhare (open circes) have been cacuated from the equa-. tion W+2 = vf*vf*f, using data such as those in Fig. 5. other vaues (cosed and haf cosed symbos) were cacuated as described in (3). Beow: mass specific power data, given above, were divided by the average speed v to obtain work done per unit distance, W( m L). I 1 I BIPEDS I I I I I HOPPERS QUDRUPEDS - I I ---I RHE 1 _ WV TURKEYS 1 m 5 - -I a 1 m 5 1 I 1 I 1 SPRING HRE 1 I I I I I KNGROOS / DOGS I I 1 I I MONKFYS 1 I I I I I I I I WV ix RHE. t I I RMS I I I I I I I I.I I I I I I I I I I I I I I I 1 TURKEYS-- 4 smw wud -. DOGS 1 MONKEYs RMS Downoaded from by on November 7, 216 I I 1 I 1 WEXT f t I I VERGE SPEED OF LOCOMOTIQN (km/hr)

12 h-; h-. h-; CVGN, HEGLUND, ND TYLOR KNGROOS I I I 1 SPRING HRE 25 m #... Downoaded from VERGE SPEED OF LOCOMOTION (km/hr) FIG. 9. Vertica dispacement of center of mass, taking pace potted as a function of the speed of ocomotion. See text for during each step of waking (open symbos), running (cosed sym- interpretation of this figure. bos), and during each stride of gaoping (haf-open symbos) is sequent acceeration, when the musce performs posi- We have measurements of Wer and Wmeab for our tive work). animas, but not of Wint. However, if we cacuate the minimum vaue for We can then be cacuated efficiency of doing externa work (7 ) from our data using the maxima vaue for ym of.25.. W W 4 = ( wezt + wint> -.25 wmetab (7) (8) y'=i@= metab (W, is defined ony when We > ) and if y >.25 this ceary indicates a arge amount It must be emphasized that this vaue for W; is ony a of storage of energy in eastic eements, since W, must minimum vaue since 1) musces usuay operate at then account for both (We, -.25 Wmet&) and a of efficiencies beow.25 (e.g., isometric contractions have the Wint according to E+ 7. an efficiency of zero); and 2) even in the optima y has been cacuated as a function of speed for a conditions of contraction, Ym in Eq. 6 cannot be.25 of the animas and is potted in Fig. 1. It increases because Wmetab incudes some chemica energy which from about.1 to.15 as turkeys increase running is used by active musces when they are stretched by speed from 5 to 2 km. h-; from.25 to.27 as dogs externa forces and perform negative work. Therefore, increase trotting speed from 5 to 15 km from about the approach of using y to study eastic storage can zero to as monkeys increase trotting or gaopte us that eastic storage may exist if y is greater ing speed from 5 to 25 km from.16 to.25 as than.25. It can aso give us a minimum vaue for springhares increase their hopping speed from 1 to 25 this storage, but it cannot te us that eastic storage km= h-; and from.24 to.76 as kangaroos increase does not exist when y is ess than.25. their hopping speed from 1 to 3 km by on November 7, 216

13 MECHNISMS OF TERRESTRIL LOCOMOTION R255 TBLE 2. Externa power per unit of body mass deiver externa power (thanks to externa forces) ony ( W&m> reative to average forward speed (Yr> when an anima is in contact with the ground. During each step (or stride) there is a time when the anima Bipeds is in contact with the ground (t,) and a time when it Rhea Kvtm = Tf r =.93 is in the air ( tv) Turkeys %sm = = Yr VERGE SPEED OF LOCOMOTION (km/iv) r =.952 FIG. 1. Externa power output, west, tota energy expenditure cacuated from the oxygen consumption, wrnptab (data for birds from Fedak et a. (14), data for kangaroo from Dawson and Tayor (), and data for springhares and dogs from unpubished measurements by Tayor) and efficiency of doing externa work (7 = werjwmeab) are given as a function of speed. Dotted horizonta ine indicates upper imit of.25 for efficiency of transformation of chemica energy into mechanica work by the contractie machinery of musce. High y vaues attained by some animas suggest a substantia recovery of mechanica energy through musce easticity v, r =.98 7 = t, + t,! (9) Hoppers where 7 is the period of a repeating change in forward Springhare K?rrm = yf r =.924 veocity and height of the center of mass. In both Kangaroo Kttm = Vf r =.973 running and trotting there was a symmetrica change Quadrupeds in forward veocity and height at each step: in this Dog, 5 kg, trot Ksrm = I$ r =.945 case 7 is the period of the step. When the anima Dog, 17.5 kg, trot &,,/m = t_, r=.988 changed from a trot to a gaop there were no onger Dog, gaop Kwm = Yr r =.966 two symmetrica steps in each stride and 7 became the Monkey, trot %stm = Yr r =.987 Monkey, gaop KTtm = y/ r =.936 period of a stride. Vaues of t, and t, were measured Rams, trot %tm = Vf r =.622 from the force records and are potted in Fig. 11. The vaue of t, decreased markedy with increasing speed &,/m is measured in Ca/kg- min and Q, is measured in km/ h. * From Cavagna et a. (9). in a of our animas whie t, either increased or remained constant. The increase in t, in Fig. 11 when animas changed from a trot to a gaop is the resut of 7 changing from the period of a step to the period of a stride. The distance an anima moves forward whie it is in contact with the ground can be measured fairy accuratey from t, and & L c = t;p, (1) because the osciations in the speed around P, are sma. Two curves are potted in Fig. 11 using Eq. 1 for two vaues of L, which just bracket the observed vaues for t,. These two vaues for L, approximate minimum and maximum vaues and show how L, changes with speed. s speed increases, either t, must decrease and/or L, must increase. For a given W&step, the power must increase as t, decreases. However, the power coud be kept constant with increasing speed, if L, increases proportionatey to P, Some animas increase L, appreciaby as they move faster; however t, decreases in a animas, approaching a minimum vaue of about.1 s (aso observed in man by Cavagna et a. (9)). The anima deceerates and fas (i.e., performs neg- ative work) during the first part of the time of contact ( tdec), and it acceerates and ifts its center of mass during the second part (t,,,) tc = tdec + t,cc (11) We can reasonaby concude from the vaues of y Since t, decreases and W,,, increases with increasing that dogs, monkeys, kangaroos, and springhares rey speed, the externa power which must be absorbed and on power recovered from eastic eements. This can be deivered by the musces and tendons must increase. very arge; for exampe, when a kangaroo hops at 3 Both the externa power which has to be dissipated km- h-, We amounts to a minimum of two-thirds of and/or stored whie the anima deceerates (W&/t&c) the measured W,,, pus a the Wint. and the externa power which has to be recovered and/ It is interesting to note that y often increases with or suppied by the musces whie it reacceerates ( W,,,I increasing speed, suggesting that eastic storage and t,,,) have been cacuated (Tabe 3). In the turkeys recovery become reativey more important at higher Wextt dec is more than 6% greater than Wezt/tacc. This speeds. This is in agreement with the findings on man woud be expected from the force-veocity reationship (5, 6) and on isoated musce (4). of musce: arger forces are deveoped when the musces How does the externa power absorbed and deivered are stretched (during deceeration) than when they when the anima is in contact with the ground change shorten (during acceeration). Thus musces wi rewith speed? The musces and tendons can absorb and quire ess time to deceerate the body than to reacceer- Downoaded from by on November 7, 216

14 t tdec/tc, and vf) Fu56 CVGN, HEGLUND, ND TYLOR 4. I I RiE ) TbRKiYS 1 smk I-~RE 1 i KPING~~~SI LJ r.4 t 8 L, a. k,t Ii&T... L, = Q5m,.7 m 1, y* LCF1;4 m 17.5kg-DOG 2Sm +4 m :,; :.6m tc t... : :, GLLOP a tv Oo Ooo O Off f ;t t Y, 8 &*-... c ooo*-. CC OV kg- DOG Lc=.6m 1.1 m 1, 4,, =,2m a k, y io.3; Y * =e :, 3 GLLOP : &,&&,,. 8, TROT L ds * a hi O-LOO I 1 1 I ~CQ % LC Lc=.5m.85 m MONKEY TROT =.3 m,.4 m? 2, t, k k ho a others t,=o I I I I I I -io VERGE FIG. 11. Time in which the feet contact the ground (t,, cosed symbos ) and in which the body is off the ground (t,, open symbos) during each step of running (circzes and squares) and each stride of gaoping (trianges) are given as a function of speed. Interrupted ate it. However, if the energy were absorbed in and deivered by passive springs (e.g. tendons) then about the same amount of time woud be required for deceeration and acceeration, so that W,,t,, = Werttacc. This is what we found in the kangaroos (Tabe 3). It is therefore reasonabe to concude that the contractie component of musce is more important in deivering positive work in the turkey than it is in the kangaroo and that passive eastic eements pay a more important roe in the kangaroo than in the rhea. This is consistent with the arger efficiency vaues (y ) which we found for the kangaroo compared with the other animas (Fig. o), indicating independenty a arger storage and recovery of eastic energy per unit time, We. The difference between Wdec and IV,,, shows that another hopper, the springhare, does not use passive SPEED OF LOCOMOTION (km/hr) ines were constructed assuming that forward dispacement when the body is in contact with the ground (L, = t, is independent of speed and equa to the indicated vaue. eastic eements to an unusua extent; this is in agreement with the ower efficiency vaues y that we found for this anima (Fig. 1). Why do animas use neary the same mass specific power to reacceerate in the forward direction at a given speed? Cavagna et a. (9) derived an equation for the mass specific power output due to the forward speed changes in man. vt w/ fm=k tv +-Tf L (12) where K = K K is the sope of a inear reationship found between the average deceeration forward of the center of mass during each step (/> and the average speed of ocomotion (VJ. Downoaded from by on November 7, 216

15 x 3 v 2 MECHNISMS OF TERRESTRIL LOCOMOTION R257 TBLE 3. Mass specific externa power stored andor dissipated after anima ands whie it deceerates (We&& and mass specific externa power recovered and/or suppied by musces whie anima reacceerates before taking off (W,,,t,,) nima and Rody Mass Bipeds Turkey, 7 kg % Rhea, 22.5 kg , % Hoppers Kangaroo, 2.5 kg % Springhare, 2.5 kg QU&drUpdS Dog, 17.5 kg % % Dqt, 5 kg % To see whether Eq. 12 appies generay to running, trotting, and hopping, we cacuated vaues for the constant K and t,/l, for a our animas. K was cacuated as foows: -& (measured from the force and veocity records) was potted as a function of &, and a ine for the best fit of a inear reationship between -& and Vf (with its origin at zero) was cacuated using the method of east squares. The sope of this ine (K) was mutipied by an average vaue for tdec/tc (taken directy from the force records and incuded in Tabe 4) to give K. In a the animas except the kangaroo, tjl, was independent of speed, and we were abe to cacuate average vaues for this constant (Tabe 4). In the kangaroo tjl, increased ineary with Vf, and we used the equation for this inear reationship 4J -(s- m-) = v (m -s-) (13) L c instead of a constant. The reationships between I&/m and Vf, cacuated using Eq. 12, are potted as soid ines in the second from the top set of graphs on Fig. 8. The cacuated ines are in good agreement with the experimenta data, in spite of the many simpifications which we made when cacuating the constants. Both K and t,l, vary a great dea from anima to anima (Tabe 4), but they change in such a way that the reationship between I&/m and vf remains neary the same in a our animas. nimas which have the argest K (ike the hoppers) aso have the argest t,/l,; thus the effect of a arge K on increasing WJrn is neary exacty compensated for by a arger t,/l,. This suggests that the animas which suffer a big forward deceeration when they and at a given speed (i.e., have a arge K) increase the time which they spend in the air in order to minimize &/m (Fig. 12). However, they have to pay a price for their arge t,l, in a greater WV/m. Gaoping Like the two other modes of ocomotion, gaoping aso invoved a simiar mechanism in a animas, athough this might not be obvious when one first ooks at the compex force and veocity tracings of gaoping animas (Fig. 13). The force tracings are particuary difficut to interpret because of the ringing of the force pate. The artifacts due to these vibrations were excuded in our computations of energy changes (3). TBLE 4. Constants for equation reating I&m and vf (Eq. 12 in text) Bipeds Turkeys Rhea Human Hoppers Kangaroos 1.12 Springhare n = n = n = n= *.4 n= f.33 n = n = n = * n= n=6 Quadrupeds Monkey, trot n= n=8 Dog, 5 kg, trot Dog, 17.5 kg, trot.479 n = n = Dog, gaop.331 n = n= n = n = 35 Ram, trot ~.4 n = 18 n = 18 tdccte and tjlc vaues are means + SD. * In kangaroos t,/l, increases with, as shown in Eq tabe and in Fig. 12 an average vaue was used. n v a E 4.. U C < CI 1 > K(sec-1)' FIG. 12. nimas that suffer a arge forward deceeration when they and at a given running speed (arge K on abscissa) increase the time they spend in air (arge t,l, on ordinate) to minimize V&/ m (Eq. 12). See text. Downoaded from by on November 7, 216

16 R258 CVGN,HEGLUND, ND TYLOR GLLOP VF(m/sec).2 const..,.,_- II kmhr -.2 =,(b) 2 a -2 g V, (m/set).4 const F,ckg) -.4 1c 2( V, (m/set 1.4 F&kg) consi -.1 ( ii Y -! g V~(m/sec ).1 5: cons Fv(kg) C t -.: II 21 VF(m/sec 1. cons -. FF(kg) 4 ZE a -4 IX VV(m/sec). FV(kg) cons ou III 1 :." " -.._ :.:.i I..* _s. I I _/- _ >a.._ L-- 1 -L 1 ', :. i".'. 2 set '2 -a Og 2 FIG. 13. Force and veocity records obtained at different speeds of gaoping. Indications as in Fig. 1. Downoaded from by on November 7, 216 It is much easier to refer to the energy records (Fig. 14) than to the force and veocity records in order to understand gaoping. Motion pictures were taken simutaneousy with the force and veocity measurements to determine the sequence of foot-fa patterns and to correate these with the energy records. In this paper we wi discuss the energy records from dogs to expain the mechanism of gaoping, since essentiay the same mechanism was found in the monkey and ram. t sow speeds, gaoping combines the two separate mechanisms utiized in waking and running. Within a stride there are times when E, is converted into E,, E, is converted into E,, and E, increases simutaneousy with E,. s the animas gaop faster and faster, the exchange between gravitationa potentia energy and kinetic energy becomes smaer and approaches zero. t the highest gaoping speeds the anima bounces first on its hind egs and then on its front egs. Phase reationship and exchange between kinetic and gravitationa potentia energy during each stride. t sow gaoping speeds, E, decreases whie the average E,, increases after the two back feet contact the ground (the numbers 1, 2, 3, and 4 above the E,, ine in Fig. 14 indicate the sequence of foot-fais beginning with the first rear foot). The decrease of E,, simutaneous with the increase ofe, indicates a shift of gravitationa potentia energy into kinetic energy whie the anima is faing forward. When IEkfJ > 1 E,I, an additiona