Salto Alexander

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Leg Design and Jumping Technique for Humans, other Vertebrates and InsectsAuthor(s): R. McN. AlexanderSource: Philosophical Transactions: Biological Sciences, Vol. 347, No. 1321 (Feb. 28, 1995), pp.235-248Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/55945

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Leg design and jumping technique for humans, othervertebrates and insectsR. McN. ALEXANDERDepartmentf Pure andAppliedBiology, University f Leeds,LeedsLS2 9JT, U.K.

SUMMARYHumans, bushbabies, frogs, locusts, fleas and other animals jump by rapidly extending a pair of legs.Mathematical models are used to investigate the effect muscle properties, leg design and jumpingtechnique have on jump height. Jump height increases with increased isometric force exerted by legmuscles, their maximum shortening speeds and their series compliances. When ground forces are smallmultiples of body mass (as for humans), countermovement and catapult jumps are about equally high,and both are much better than squat jumps. Vertebrates have not evolved catapult mechanisms and usecountermovement jumps instead. When ground forces are large multiples of body mass, catapult jumps(as used by locusts and fleas) are much higher than the other styles ofjump could be. Increasing leg massreduces jump height, but the proximal-to-distal distribution of leg mass has only a minor effect. Longerlegs make higherjumps possible and additional leg segments, such as the elongated tarsi of bushbabies andfrogs, increase jump height even if overall leg length remains unchanged. The effects of muscle momentarms that change as the leg extends, and of legs designed to work over different ranges of joint angle, areinvestigated.

1. INTRODUCTIONA wide variety of animals, including humans, makestanding jumps by rapidly extending a pair of legs.Species that have been studied include fleas (Bennet-Clark & Lucey 1967), locusts (Bennet-Clark 1975),frogs (Calow & Alexander 1973; Hirano & Rome1984; Lutz & Rome 1994) bushbabies and otherprosimians (Guinther 1985; Gtinther et al. 1991) andhumans (Bobbert & van Ingen Schenau 1988; Pandyet al. 1990; Dowling & Varmos 1993). Recognizedadaptations for jumping include long, muscular legs,sometimes with additional segments formed by elonga-tion of tarsal bones (frogs, bushbabies) or by mobilityof the sacro-iliac joint (frogs) (Emerson 1985). At leasttwo techniques are used to improve jumping per-formance by taking advantage of elastic elements inseries with the muscles. Humans make a countermove-ment, bending the legs immediately before extendingthem. Komi & Bosco (1978) have shown that thisenables them to jump higher than they otherwisecould. Jumping insects use catapult mechanisms,storing elastic strain energy and then releasing itsuddenly to power the jump (Bennet-Clark 1976).Bennet-Clark & Lucey (1967) showed that the jumpsof small insects require much higher power outputs perunit mass than any known muscle can provide.Catapult mechanisms enable work done relativelyslowly by muscles to be released much more rapidly attake-off.The aim of this paper is to improve our under-standing of leg design in jumping animals and of thetechniques used for standing jumps. How does jumpperformance depend on muscle properties, on thedistribution of mass in the legs and on the number ofPhil. Trans. R. Soc. Lond. B (1995) 347, 235-248Printed in Great Britain

leg segments? To what angle should the joints bend, inpreparation for a jump? Can an advantage be gainedby having muscles whose moment arms change, as thejoint extends? In what circumstances can a counter-movement improve a jump and when will a catapultmechanism be more effective?These questions will be tackled by mathematicalmodelling. A model will be described that is generalenough to be applied to jumpers of all sizes and taxa;from fleas to humans. Muscle properties, other aspectsof leg design and jumping technique will be varied andthe effects on jump height determined. Only verticaljumps will be considered.2. THEORY(a) Model with two leg segmentsThe model used for most of the calculations is shown infigure 1a. It jumps by extending its legs. The jump ispowered by knee extensor muscles which exert equalmoments about the two knees: the properties of thesemuscles are described in ?2c. The model starts fromrest and its symmetry ensures that the jump is vertical.Each leg consists of two segments, each of length s.The point of contact of each foot with the ground isvertically below the corresponding hip, and at time tthe angle of each knee is 20. The hips are at height yfrom the ground and the knees are x lateral to them.Thusy = 2s sin0,y = 2sB cos0,x = s' os 0,x = - s' sin 0 = - y' tan0,and from equation 20 = (' sec0)/2s.

(1)(2)(3)(4)(5)

? 1995 The Royal Society35


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236 R. McN. Alexander Leg design andjumping technique(c)

force, Fm, F,, iso2-

- 0.5 0 0.5 1.0rate of shortening ( - a / amax

Figure 1. Models of bipedal jumperswith: (a) two segments; and (b) three segmentsin each leg. (c)The force-velocityrelationshipfor the models' muscles,describedby equations (22). The force isexpressedas a multiple of the isometricforce (i.e. as F/Fm iso) and the rate of shortening as a multiple of the maximum shortening speed (as -?d/max).We will obtain an equation of motion by consideringenergy balance. At time t, the knee muscles are

exerting moments T about the knees, each of which isextending at a rate 20. The rate at which the kneemuscles are doing work must equal the sum of the ratesof increase of the potential energy P and kinetic energyK of the model4T0 =P+K. (6)

The trunk has mass m1, the two thighs together m2and the two lower legs together m3. Each leg segmentis a uniform rod, so its centre of mass is midway alongits length. Thus these centres of mass are at heights3y/4, y/4, and the potential energy of the model isP = (gy/4) (4m, + 3m2+ m3), (7)(g is the gravitational acceleration). The equationimplies that the centre of mass of the trunk is at theheight of the hip joints, but this assumption has noeffect on the analysis because we will be using thederivative of the potential energy, rather than thepotential energy itself.The trunk moves vertically with velocity y. Thecentres of mass of the thighs and lower legs havevertical components of velocity 3//4,y//4 and hori-zontal components +?/2, ?+/2. Each thigh hasmoment of inertia m2 2/24 about its centre of mass, andangular velocity + 0: and each lower leg has moment ofinertia m3s2/24 and the same angular velocity. Thusthe total kinetic energy of the model isK = (2/32) (16m +9m2+m3) + (2/8) (m2+ m3)

+ (s202/24) (m2+ 3),=(2/24) [12m + 7m2+m3+tan20(m2+m3)], (8)

(using equations (4) and (5) to substitute for x and 6and remembering that sec2 0 = I + tan2 0)It will be convenient to writem4= m2+ m3, (9)

m5= 4m1+ 3m2+ m3m =- 12m + 7m2 + m,so that equations (7) and (8) becomeP = gym5/4,K = (y2/24) (m6+ 4tan2 0).By differentiating with respect to timeP = g9m5/4,K= (yy/12) (m6+ m4 tan2 o)

+ (g2/24) (2m46 an 0 sec2 0).By substituting equations (14) and (15) in (6),using (5) to eliminate 62 Tsec 0/s = gm5/4+ (y/ 12) (m+ m4 an2 0)

+ (mi4 2/24s) tan 0sec3 0,y = (48 Tsec 0-6m5 gs- m4 2 tan 0 sec3 0)

/[2s(m6 + m4tan2 0)].

(10)(11)

(12)(13)

(14)(15)and

(16)This equation is used to calculate the motion of themodel during take-off. The height y of the hips isobtained by numerical integration and from it the kneeangle 20 by using equation (1). The force Fg on theground is the sum of the weight of the body and theforces needed to give the segments their verticalcomponents of acceleration(17)g= mg+m5 /4,

where m is the total mass of the bodym = m +m2+m3.

When y > 2s or Fg< 0, the feet have left the ground.At the instant when they leave the ground, y = yoff andy = Soff' The centre of mass of the model is thenrising at a rate m5yOff/4m(this is P/mg, equation (14)).A projectile fired vertically with velocity v rises to aheight v2/2g, so the centre of mass will rise bym5yoff/32m2g after the feet have left the ground. If thePhil. Trans. R. Soc. Lond. B (1995)

(18)

(a) (b)


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Leg design andjumping technique R. McN. Alexander 237joints remained fixed at the angles they had when thefeet left the ground, the hips would rise to a heightYoff m5yoff/32m2g at the peak of the jump. However,they will rise a little higher if the legs becomecompletely straight, moving the centres of mass ofthighs and lower legs distances (2s-yoff)/4 and3(2s-yoff)/4 further below the hips. Thus the dif-ference of height between the hips and the centre ofmass of the whole body is increased by(m2+3m3) (2s-yo,,)/4m. The height of the jump,defined as the height of the hipjoints above the groundat the peak of the jump, is thush = yoff + (mnyo2f/32m2g) + (m2+ 3m) (2s-yOff)/4m. (19)(b) Model with three leg segments

Some calculations will be presented for the modelshown in figure 1b, which has three segments in eachleg instead of two. These are a thigh of length s/2, ashank of length s and a metatarsal segment of lengths/2. The distributionof massalong the length of the legis the same as for the previous model; thus the thighand metatarsal segment are uniform rods of massesm2/4,m3/4 and the shank consistsof two uniformrods,each of length s/2, joined end to end; the proximalhalfof the shank has mass m2/4 and the distal half m3/4.The twojoints in each leg are constrained always tohave equal angles, perhaps by a parallel rule mech-anism (not represented n the diagram). The muscularmoment T may all be applied at one of the joints, inwhich case a moment is transmittedto the other by thelinking mechanism. Alternatively, moments totallingTmay be applied to the twojoints by separatemuscles.The mathematical analysis is the same, in either case.The equation of motion can be obtained by a similarargument to the one presentedfor the model with onlytwo leg segments, in ?2a. The more concise argumentthat follows leads to the same conclusion.Mass in this model is distributed over height inpreciselythe same way as in the previousmodel, so atany given hip height y the potential energiesof the twomodels are equal. Also, at any given hip velocity y thevertical components of velocity of particles in cor-responding positions in the two models are equal:therefore, the kinetic energies associated with verticalcomponents of velocity are equal. However, particlesin the legs of this model are on average only half as farfrom the vertical line from hip to foot, as in the othermodel. Therefore the transverse displacements thatoccur as the leg straightens are halved, transversecomponentsof velocity are halved and kinetic energiesassociated with transversecomponents of velocity areonly one quarter as much as in the previousmodel. Inequation (13), thesekinetic energiesare representedbythe term in m4. It follows that we can obtain theequation of motion for the model with three segmentsin each leg by dividing by four those termsin equation(16) which include m4. Equivalently, we can multiplyby four the terms on the right which do not include m4y = (192TsecO-24m5gs-m4y2 tan 0 sec3 0)/ [2s(4m6 + ,4tan2 0)]. (20)

(c) Muscle propertiesThe extensor muscle which powers the jump consistsof a contractile element in series with an elasticelement. Any change 80 in the half-angle of the knee

requires a change 2r 60 in the overall length of themuscle, where r is the moment arm of the muscle aboutthe joint. This is the sum of length changes 8a in thecontractile element and 8b in the elastic element.2r' 80 = 6a+ b. (21)

The contractile element has force-velocity propertiesexpected to be realistic for striated muscle. Morespecifically, the force Fm that the muscle exerts isrelated to the rate of change of length a of thecontractile elementfor -a < 0

Fm= Fm,so[ .8 - 0.8(ma - )/(max + 23a)], (22a)for 0 dmax Fm= 0, (22c)(see figure I c). Here Fmiso is the force exerted inisometric contraction and amax is the maximum rate ofshortening of the contractile element. Equation (22b)is Hill's (1938) equation for muscle shortening, withsome signs changed because shortening is a negativelength change. Similarly, equation (22a) is Otten's(1987) equation for stretching of active muscle. Inthese equations, the constant describing the curvatureof the force-velocity relationship (a/Po in Woledge et al.1985; k in Otten 1987) is given the value 0.33. Thisvalue is typical for fast skeletal muscle (Woledge et al.1985).The elastic element is a linear spring of complianceC, which undergoes extension b when force Fmacts onitFm = bC. (23)At every stage in take-off, the forces given by equations(22) and (23) must be equal.We will see, in section 5 (a), how forcible stretchingof the muscle in a countermovement can enable it toexert increased force in a subsequent contraction. Thisresults from interaction of the series compliance withthe force-velocity properties of the contractile ele-ments. No attempt is made in the model to reproducean additional effect of an initial stretch, 'potentiation'of the contractile machinery itself. This effect seemsrelatively unimportant (Ettema et al. 1990).The moment arm r is related to the angle 20 of thejoint by the equationr = ro[l + (6k/() (0-1/3)]. (24)When k = 0 (as in most of the calculations that will bepresented) the moment arm has a constant value ro.When it has other values, the moment arm changeslinearly from ( - k) rowhen 0 = r/6 to (1 + k) rowhen0 = n/2. Note that the mean moment arm, over thisrange, is always ro. In most of the simulations presentedin this paper, including all those in which r varies, theminimum value of 0 is 71/6. When the leg is fullyextended, 0 = n/2.

Phil. Trans.R. Soc.Lond.B (1995)


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238 R. McN. Alexander Leg design andjumping techniqueThe moment T exerted by the extensor muscleabout the joint is

T= Fmr.Thus the unit of time is (s/g) . The followingparameters will be used to describe muscle properties:

(25) the isometric force parameter

3. COMPUTATIONThe equations given above were incorporated in aprogram run on a desk-top computer. This simulatedjumps starting from rest, following the model's move-ments by numerical integration until the feet left theground. Equation (16) or (20), as appropriate, wasused to calculate the changes of velocity y and hipheight y during each time increment. Hence joint anglewas calculated using equation (1). The increment ofjoint angle was partitioned between the contractile andseries elastic elements of the muscle by stipulating thatthe muscle forces given by equations (22) and (23)must remain equal. The force so obtained was used asthe muscle force for the next time increment.

Integration ceased when the force on the ground(equation (17)) fell to zero. The height of the jump wasthen calculated, using equation (19). Halving the timeincrements in a sample of runs altered jump heights byless than 1 o.Simulations were performed for three jumpingtechniques. In the following description, 20min is theminimum angle to which the knee bends in preparationfor the jump.1. Squat jumps. The simulation starts at rest, withknee angle 20min. Initially force Fm s zero and the kneeis prevented from bending further by a passive stop asoccurs, for example, when a person is squatting withthe posterior surface of the thigh resting on the calf.The muscle is activated and contracts, stretching theseries elastic elements and building up a moment. Nomovement occurs until the acceleration given byequation (16) or (20) becomes positive, at whichinstant the legs start extending.2. Catapultumps. Again, the model is initially at restwith knee angle 20min. The muscle is active, exerting itsisometric force Fm iso, but the joint is prevented fromextending by some other means. For example, the kneeextensor muscle in locusts develops tension prior to ajump, while extension is prevented by an antagonist(Bennet-Clark 1975). The joint is suddenly released (inlocusts, the antagonist relaxes very rapidly) and take-off starts.3. Countermovementjumps. The model is initially atrest with the legs straight and the muscle inactive. Itfalls for a while under gravity according to equation(16) or (20), with Fm fixed at zero. At some stageduring its fall, the muscle is suddenly activated. Amoment is developed at the knee, decelerating the falland then accelerating the model upwards to take off.The time at which the muscles are activated is variedin successive trails to find by trial and error the timerequired to make the minimum knee angle reached inthe simulation equal the chosen value 20min.The results obtained in this study will be presentedin dimensionless form using body mass m as the unit ofmass, leg segment length s as the unit of length andgravitational acceleration g as the unit of acceleration.

(26)m,.iso Fm so o/gs,the shortening speed parameterdmax= (amax/o) (S/g),and the compliance parameterC= CFm,iso/ro.

(27)

(28)4. VALUES FOR PARAMETERSThe models presented in this paper are designed tothrow light on jumping by animals ranging fromhumans to small insects. Our choice of parameters willbe guided principally by data for humans (body massapproximately 70 kg), bushbabies (Galago senegalensisand moholi, 0.3 kg) and locusts (Schistocercagregaria,2 g).The total mass of the two thighs is 20 ?/ of body massboth in humans and in Galago (Winter 1990; Grand1977). The mass of the two lower legs and feet is 12 ?/of body mass in humans and 10% in Galago (samesources). A reasonably realistic model of jumpingmammals can therefore be obtained by taking m=0.7m, m2= 0.2m and m3=0.1m; these segment masseshave been used except where it is stated otherwise.Note, however, that at least some insects have relativelylighter legs. The two femora of Schistocerca otal only140% of body mass and the two tibiae and tarsi only3 ? (Bennet-Clark 1975).The minimum knee angle, in the countermovementprior to jumping, is about 75? in humans (Bobbert &Van Ingen Schenau 1988) and 30? in Galago (Gunther1985). An intermediate value of 60? will be used as theminimum knee angle (20m,,) in this study, except whenthe effects of varying this angle are being investigated.Note that the chosen angle is much too large to berealistic for Schistocerca,which bends the knee almost to0? in preparation for jumping (Heitler 1977).Peak ground forces in standing jumps are generally2-3 times body mass for humans (Dowling & Vamos1993), up to 13 times body weight in Galago (Gunther1985), about 18 times body weight in Schistocerca(Bennet-Clark 1975) and up to at least 135 times bodyweight in fleas (Bennet-Clark & Lucey 1967). Isometricforce parameters Fm iso of one, five and 25, respectively,will be used in simulations designed to represent jumpsby humans, bushbabies and insects. When the musclesexert these forces at knee angles (20) of 60?, the groundforces (exerted by the two feet together) are 2.3, 12 and58 times body weight, respectively. The peak forcesexerted in simulated jumps may be somewhat more orless than these values, depending on the jumpingtechnique (see figure 2).

Maximum shortening speeds of muscles are usuallyexpressed in terms of muscle fascicle lengths per second.To select realistic values of dmax for investigation, wemust first estimate the resting length of the musclefascicles. In most of the simulations, the knee willextend from a minimum angle of 60? so its workingrange, from the minimum to full extension, is 120? orPhil. Trans.R. Soc. Lond.B (1995)


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Leg design andjumping technique R. McN. Alexander 2392.1 radians. A muscle with a moment arm ro, movingthe joint through that angular range, must shorten by2.1 r0. Studies of rabbit leg muscles (Dimery 1985) andbird wing muscles (Cutts 1986) indicate that theworking range of length of muscles is commonly aboutone quarter of the resting length, so a muscle requiredto shorten by 2.1 ro can be expected to have fasciclesabout 8.4ro long. For a muscle shortening at e lengthsper second, the shortening speed can thus be estimatedas 8.4roe.In this paper, muscle shortening speeds are repre-sented by the dimensionless parameter (a/ro) (s/g)which, by the argument of the previous paragraph,equals 8.4U(s/g)2. Leg segment length s would be about450 mm for humans, 66 mm for Galago (Grand 1977)and 25 mm for Schistocerca(Bennet-Clark 1975). Thecorresponding values of (s/g)2 are 0.21 s for humans,0.08 s for Galago and 0.05 s for Schistocerca.

The maximum shortening speed for fast fibres fromhuman deltoid muscles is 4.9 lengths per second (less intrained swimmers; Fitts et al. 1989), and it seems likelythat knee extensor muscles would be about equally fast.I have no data for bushbaby muscle, but bushbabiesare similar in mass to rats in which a fast leg muscle(extensor digitorum longus) has a maximum short-ening speed of about 15 lengths per second (Woledge etal. 1985). These data give shortening speed parameters(8.4e(s/g)2, see above) of nine for humans and ten forbushbabies. However, Tihanyi et al. (1982) give themaximum shortening speed of human knee muscles interms of the angular velocity of the knee, as 18 rad persecond for subjects with predominantly fast fibres. Thisis dmax/ro, implying that the shortening speed par-ameter is only 18 x 0.21 = 4. These data indicate thatthe parameter is likely to be in the range 4-10, forhumans and bushbabies. Most of the graphs in thispaper show data for a shortening speed parameter of 8,but the effects of variations in the range 2-32 havebeen investigated.

Orthopteran wing muscles have maximum short-ening speeds up to 16 lengths per second (Josephson1984). If locust knee extensor muscles were as fast asthis, their shortening speed parameter would be8.4 x 16 x 0.05 = 7, about the same as for humans andbushbabies. The shortening speed seems actually to bemuch lower than this, about 1.8 lengths per second(Bennet-Clark 1975), giving a shortening speed par-ameter of only 0.8. It seems probable that theparameter is also small for smaller insects. For a flea,segment length s, would be of the order of 1 mm,making (s/g)2 about 0.01 s. The fastest muscles thathave been investigated have shortening speeds of about25 lengths per second (Woledge et al. 1985). Even withmuscles as fast as this, the shortening speed parameterof a flea would be only 8.4 x 10 x 0.01 = 1.7. For someof the insect simulations, the parameter will be given avalue of 1.We must take account of the force the muscle canexert, in selecting values for series compliance. Sarco-meres are stretched about 1.5 / by their isometricforce (Huxley & Simmons 1971). Thus the lowestlikely value for the series compliance of a muscle oflength 8.4r, (see above) is 0.015 x8.4rO/Fm, so=

0.13ro/F, io. At the other extreme, we may imagine amuscle whose tendon stretched elastically by 2.1 ro (ourestimate of the muscle's working range of length) whenthe muscle exerted its isometric force. In that case thecompliance would be 2.1r/Fmi,,o. Results will bepresented for compliance parameters CFm,s,/r rangingfrom 0.125 to 2.The strain energy stored by elastic structures prior tothe jump in locusts exceeds the energy of the jump,implying that the compliance parameter is a little morethan 2 (Bennet-Clark 1975). Realistic values formammals are harder to estimate. Isometric muscleforces impose stresses around 50 MPa on highly-stressed tendons such as the human gastrocnemius(Ker et al. 1988), stretching them by about 4 % of theirlength (see figure 1.6 in Alexander 1988). Suppose thatthe total length of tendon or aponeurosis in series witheach muscle fascicle is twice the length of the fascicle,a value within the commonly found range for pennatemuscles. Then if the length of the fascicles is 8.4ro (asalready estimated) the extension of the tendon is2 x 8.4 x 0.04ro = 0.7ro and the compliance parameteris 0.7.

5. RESULTS AND INTERPRETATIONThis section presents results and tries to explain inwords why the mathematical models behave as theydo. Discussion of the light thrown by the models on thedesign and jumping techniques of real animals isdeferred to the next section.

(a) Predicted forcesFigure 2 shows sample simulations, one for each ofthe three jumping techniques. The chosen isometricforce (1.Omgs/ro) gives peak ground forces in the range2-3 mg which is typical of human jumping.Figure 2 a shows a squat jump. Initially, the force onthe ground equals body mass and the muscle isinactive. At time, t = 0, the muscle is activated and itscontractile element starts to shorten, stretching theseries elastic element and building up tension. By t =0. 14(s/g)2 it is exerting enough moment to startextending the knee. The muscle force continues to riseas the series elastic elements are stretched further: butnever reaches the isometric value because the con-tractile elements are shortening throughout take-off.Later, the muscle force falls because the contractileelements' rate of shortening is increasing, allowing theseries elastic elements to recoil. Eventually, the muscleis exerting too little moment to maintain the angularacceleration of the leg segments that would be neededto keep the feet on the ground while the trunkcontinues to rise. The feet leave the ground at a kneeangle (in this simulation) of 146?.Figure 2b represents a catapult jump. In this casethe muscle is active from the start, exerting its isometricforce, and the series elastic element is correspondinglystretched. The knee is prevented from extending untilt = 0, when it is suddenly released. The force on the

ground rises abruptly and the body accelerates to take-off. As the knee extends, the series elastic elementsPhil. Trans.R. Soc. Lond.B (1995)


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(a)anglegroundmuscleforce force(b) (c)

% groundI \ force1I^ \

I muscle iI force

angle" , ,

0 10 -1 0 1time time timeFigure2. Examplesof simulatedjumps: (a) representsa squatjump; (b) a catapult jump; and (c) a countermovementjump. The force exerted on the ground (expressed as F/mg), the force exerted by the knee extensor muscles(expressedas Fm/Fm,o) and the knee angle (20) are plotted against the time parameter t(g/s)05. These arejumps bythe model with two-segment legs (see figure 1a) with muscles exerting human-like forces (isometricforce parameterFn o= 1.0). The shortening speed parameter d,mais 8, the compliance parameter C is 1.0 and segment masses arem1= 0.7m, m2 = 0.2m, m3= 0.1m.

(b) (c)16

-3C)

14I 2

.1 2 - 2.2

i . 0 .. ---0.125 0.25 0.5

16uc 8.5.S 4E

._ 2x' 2 0 1 - I I .,1 2 0.125 0.25 0.5 1

compliance compliance

4-' 8 )X20\

1510

2 I'aiI I l2 0.125 0.25 0.5

complianceFigure 3. Contour plots showing the dependence of jump height on the maximum shortening speed of the musclesand their seriescompliance: (a) refers to countermovementjumps with human-likeforces (isometricforce parameterF iso= 1); (b) to countermovementjumps with bushbaby-like forces (F iso= 5); and (c) to catapult jumps withinsect-likeforces (Fm,,o= 25). The axes show the shorteningspeed parameteramax nd the compliance parameter C,and the contours give relative jump height his. The legs have two segments, and segment masses are m, = 0.7m,m2= 0.2m and m3= 0.lm, throughout.

recoil, the contractile elements shorten progressivelyfaster and the muscle force falls.Figure 2c represents a countermovement jump.Initially the leg is straight; the muscles are inactive butthe feet rest on the ground exerting a force whichdiminishes as the body falls under gravity. The legbends until, at t = 0 (when the knee angle, in thisexample, is 96?), the muscles are activated. Tensionbuilds up and the fall is decelerated until at t=0.8(s/g)2 (in this example), when the knee angle is 60?,

the fall is halted and the body begins to rise again.Immediately prior to this the muscle was beingstretched and the force in it had risen a little above theisometric value (to 1.08Fm,is). In the very early stagesof knee extension, the force is still above the isometricvalue: the contractile elements are still being stretchedbut the series elastic elements are shortening faster (by

elastic recoil) and the muscle, as a whole, is shortening- extending the knee. Only when the elastic recoil hasproceeded far enough for the muscle force to dropbelow F, iso do the contractile elements begin toshorten.

(b) Effects of muscle propertiesFigure 3 shows how the height of ajump depends onthe maximum shortening speed of the muscles and the

series compliance: (a) for human-like ground forces;(b) for bushbaby-like ground forces; and (c) for insect-like ground forces. The simulations are of counter-movement jumps in figure 3 a, b and a catapult jump infigure 3 c, in accordance with the jumping techniquesused by humans, bushbabies and insects, respectively.Comparison of figure 3 a and 3 b shows, as expected,Phil. Trans.R. Soc. Lond.B (1995)

240 R. McN. Alexander Leg design andjumping technique

1800 -

2

120 -

- 1.0

- 0.5

60? n .

(a)

2

1

i


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Legdesignand umping echniqueR. McN. Alexander 241

// catapult

countermovt.

- --' ~ sqjuat

I I II I .t.. .1$1ta10tI I O t t,0.125 0.25 0.5 1 2 0.125 0.25compliance

0.5compliance

1 2 0' ,,["0.125 0.2: 0.5

compliance1 2

Figure 4. A comparison of threejumping techniques. In each graph relativejump height h/s is plotted against thecompliance parameter C, for: (continuous line) squat jumps; (short dashes) countermovement jumps; and (longdashes) catapult jumps. Isometric forces are (a), human-like (Fm = 1); (b), bushbaby-like (Fmo = 5); and (c),insect-like (Em,iso 25). In every case the shortening speed parameter imaxs 8 and segment masses are m1= 0.7m;m2= 0.2m; and m3 = 0.lm. The legs have two segments.that larger muscle forces give higher jumps. Both thesegraphs refer to countermovement jumps, but isometricforce (expressed as a multiple of body mass) is larger inb than in a, and jump heights (as multiples of leglength) are also larger. Examination of the contours oneach graph shows that for constant isometric force,faster muscles and higher compliances given higherjumps. Faster muscles can exert more force, at givenrates of shortening. Series elastic elements can shortenby elastic recoil at unlimited rates. Also, series elasticelements make it possible for muscle forces that aregreater than the isometric force, developed during acountermovement, to persist into the early stages of legextension. This was explained in ?5a.By how much might increased compliance beexpected to improve jump height? In many cases, thepeak force exerted by the muscles during takeoff isclose to their isometric force F, iso (see figure 2). Thisforce, acting on compliance C, stores strain energy-F 2iC in each leg, a total of Fz C. By equations(26) and (28) this equals Fmiso Cmgs, enough to raisethe animal's centre of mass by F.,so Cs. Thus if all thestored strain energy were converted to gravitationalpotential energy in the jump, an increase in C from0.125 to 2 (the range investigated in figure 3) wouldimprove jump height by 1.9s when Fm,iso = 1; by 9.4swhen Fmso= 5; and by 47s when F so = 25. Theimprovements predicted by the model are substantiallyless than this, as can be seen by comparing jumpheights for compliance parameters of 0.125 and 2, forany chosen value of the shortening speed parameter, infigure 3a, b or c. Reasons for this include peak forcesbeing less than isometric for squat jumps (see figure 2 a)and some countermovement jumps; and to someenergy being required to give kinetic energy to the legs(see ?5d).In countermovement jumps (see figure 3a, b) jumpheight is more sensitive to the speed of the muscles than

to the series compliance: an increase of (say) 10 % inmuscle shortening speed generally increases jumpheight more than a 10%/ increase in compliance. Incatapult jumps (see figure 3 c), however, jump height ismore sensitive to compliance than to the speed of themuscles, except when compliance is very low.The catapult jumps of figure 3c involve much largerisometric forces (relative to body mass) than do thecountermovement jumps of figures 3a, b. The state-ments of the previous paragraph nevertheless remaintrue, when comparisons are made between counter-movement and catapult jumps with equal isometricforces.

(c) Comparison of jumping techniquesFigure 4 shows results of simulations of the threejumping techniques, with the isometric muscle forceschosen to represent: (a) human jumping; (b) bushbabyjumping; and (c) insect jumping. Results are shown ineach case for a range of series compliances, for onemaximum shortening speed.With zero series compliance, the three techniqueswould give jumps of identical height for the followingreasons. In a squat jump, muscle force would riseinstantaneously to the isometric value when the musclewas activated; and in a countermovement jump,muscle force would fall to the isometric value at the

instant when the knee ceased bending and started toextend. Thus knee extension would start in every casewith 0 = Omin, = 0 and m = Fm,isoAs compliance increases, all three techniques givehigher jumps but squat jumping is less successful thanthe others because muscle force is less than the isometricvalue when knee extension starts. The relative merits of

catapult and countermovement jumping depend onthe isometric force. In the human simulations the twotechniques give similar jump heights (see figure 4a).

Phil. Trans.R. Soc. Lond.B (1995)

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242 R. McN. Alexander Leg design and umping technique(a)

3.5

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30

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2.0 . ' I 0 I i I 1 0 1 X 10.125 0.25 0.5 1 2 0.125 0.25 0.5 1 2 0.125 0.25 0.5 1 2compliance compliance compliance

Figure 5. The effects onjump height of the mass and structureof the legs. Relativejump height h/s is plotted againstthe compliance parameter C. Mass distributions are: masslesslegs: m1= m, m2= 0, m3= 0; insect-like legs: m1=0.83m, m2= 0.14m, m3= 0.03m; mammal-like legs and 3-segment legs m1= 0.7m, m2= 0.2m, m3= 0.lm; heavy feetm1= 0.7m, m2= 0.lm, m3= 0.2m.Three segment legs are as shown in figure 1b; others as in figure 1a. (a) shows theheights of countermovement umps with human-likeforces ( ,iso= 1.); (b), countermovement umps with bushbaby-like forces (Fmiso = 5); and (c), catapult jumps with insect-like forces (m,Jso = 25). The shortening speed parameterdmax is 8 in all cases.

With moderate compliances, countermovement jumpsare a little higher than catapult jumps because themaximum muscle forces are greater than isometric (seefigure 2 c). If the compliance is very high, however, thepotential energy lost in the body's fall in a counter-movement is not enough to build up so much force inthe series elastic elements, and catapult jumps arehigher.In the bushbaby and insect simulations (see figure4b, c), isometric muscle force is not attained incountermovement jumps except when the series com-pliance is very low. Consequently, catapult jumps arehigher than countermovement jumps over a widerange of compliances. In the insect case (see figure 4c)a countermovement gives very little advantage over asquat jump.A simple calculation will give a rough indication ofthe circumstances in which a countermovement jumpcan be expected to be higher than a catapult jump. Insimulations like those offigure 4, in which the minimumknee angle is 600, the trunk falls a distance s in acountermovement which starts with the legs straight.The leg segments fall smaller distances, so the potentialenergy lost in the fall is a little less than mgs. When amuscle is exerting its isometric force, strain energyCFU iso is stored in its series elastic element. For thepotential energy lost to supply enough strain energy toraise the force in the series elastic elements of bothmuscles to Fm,smgs > CFm,iso,C < 1/F, iso. (29)

The right-hand side of this inequality is 1.0, 0.2 and0.04 for the human, bushbaby and insect simulations,respectively. These are the maximum values of thecompliance parameter C at which countermovementjumps might be expected to be higher than catapultjumps. However, it should be noted that some of thestrain energy may be supplied as work done by the

muscles, especially if the maximum shortening speed ishigh.(d) Mass distribution in the legs

Figure 5 compares jumps by animals with differentdistributions of mass in their legs. As in previousfigures, the isometric forces have been chosen torepresent: (a) humans; (b) bushbabies; and (c) insects.In each case, the highest jumps were achieved whenthe legs were given no mass. Mass in the legs reducesthe height of the jump because some of the work doneby the muscles is required to provide internal kineticenergy (energy associated with movement of parts ofthe body relative to the centre of mass). Unlike theexternal kinetic energy (associated with movement ofthe centre of mass), this energy does not becomepotential energy as the animal rises to the peak of thejump, so does not contribute to the jump's height.Some of the internal kinetic energy is associated withdifferences in the vertical component of velocity attake-off, between the leg segments and the trunk (seediscussion of the effect of foot mass on jumping,Alexander 1988). The rest is due to the horizontalcomponents of velocity given to parts of the legs, as thelegs straighten in take-off.The total mass of the legs seems more importantthan the distribution of mass within the legs. Amammal-like mass distribution, with the thighs twiceas heavy as the lower leg ('mammal-like', figure 5a)gives only slightly higher jumps than when the massesof thighs and lower legs are reversed ('heavy feet').This seems to be due to the part of the internal kineticenergy at take-off due to transverse components ofvelocity being larger than the part due to differences invertical velocity. The former part is the same for bothmass distributions (for given trunk velocity) but thelatter part is greater when the lower leg is the heaviersegment.



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Legdesignand umping echnique R. McN. Alexander 243

10leg length2 _

leg length2

1 *5 L _51'. 505

00.12... I .- I 0 I25 0.25 0.5 1 2 0.125 0.25 0.5

.. 0 ' I '1 2 0.125 0.25 0.5compliance compliance compliance

Figure 6. The effect of leg length on jump height. In each graph, relative jump height h/s, is plotted against thecompliance parameter C for three different relative leg lengths s'/s: (a) refers to countermovement jumps withhuman-like forces ( = 1); (b) to countermovement jumps with bushbaby-like forces (Fm,iso 5); and (c) tocatapult jumps with insect-like forces (F ,iso = 25). Other details are as in figure 4, and as explained in the text.

The leg segments of jumping insects such as locust?are much smaller fractions of body mass than are thoseof humans and bushbabies (see ?4). Figure 5c showsthat even their mass reduces jump height appreciably,in comparison with hypothetical massless legs.(e) Number ofjoints

Figure 5a, b also shows results for a model withthree-segment legs (figure 1b). This jumps higher thanthe mammal-like two-segment model although it hasthe same distribution of mass along the legs. Thereason is that the joints of the three-segment leg areinitially closer to the vertical line through the hip.Therefore, the transverse displacements and transversevelocities that occur, as the leg straightens, are smallerfor the three-segment leg. It was shown in thederivation of equation (20) that at the same verticalvelocity y, the kinetic energy associated with transverseleg movement is only one quarter as much for three-segment legs, as for two-segment legs.(f) Leg Length

To discover the effect of changing leg length we willcompare animals with equal masses of leg muscle: thatimplies those with equal values of Fmiso ro, as F, iso isproportional to the cross-sectional area of the muscleand (as explained in ?4) r, can be expected to beproportional to muscle fibre length. The leg muscles, ofthe animals to be compared, will be capable ofshortening at equal numbers of lengths per second:hence, as explained in ?4, they have equal values of&max/ro. Thus we will compare a standard animal with

leg segments of length s, with an isometric forceparameter F, iso and a shortening speed parameterdmax; with a modified animal with legs of length s', withan isometric force parameter Fm,io (s/s') and a short-ening speed parameter damax(s'/s) (see equations (26)and (27)). The compliance parameter (see equation(28)) is not affected by the change of leg length. Jump

height will be expressed as a multiple of the standardleg length (i.e. as h/s). Because leg muscles are generallymore massive than the leg skeleton, we will ignore anyincrease of leg skeleton mass that may be madenecessary by increased leg length.Results are shown in figure 6, calculated for isometricforces representing : (a) humans; (b) bushbabies; and(c) insects. In every case, longer legs give higherjumps.This is partly because longer-legged animals start ajump with the centre of mass higher above the ground.When the feet are on the ground with the knees bent at60? (the starting angle in every case, in figure 6) thehips and centre of mass are a height 1.0s above theground when relative leg length s'/s is 1.0, but 2.0sabove the ground when s'/s = 2.0. In addition, longerlegs enable the animal to accelerate over a greaterdistance, so the muscles do not have to shorten in soshort a time, to accelerate the animal to given speed.Their rate of shortening can be lower so they can exertmore force (see figure 1c) and do more work.Notice that for catapult jumps simulating those ofinsects (figure 6c), leg length has little effect on jumpheight when compliance is high. The reasons are thatjump heights are large multiples of leg length, so theinitial height of the centre of mass from the ground isrelatively unimportant; and the work done by elasticrecoil is the same, whether the recoil is fast or slow.

(g) Moment armsSuppose a given volume of muscle of given propertiesis required to operate a joint. Anatomical consider-ations may make it convenient to have a long-fibredmuscle with a large moment arm, or a short-fibredmuscle with a short moment arm. But if fibre length ismade proportional to moment arm, these two musclearrangements will have precisely the same mechanicaleffect: they can exert the same moment and move the

joint at the same angular velocity (Alexander 1981).Because muscle volume is assumed constant, longerfibres imply a smaller physiological cross-sectionalPhil. Trans.R. Soc. Lond.B (1995)

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244 R. McN. Alexander Leg design andjumping technique

50 -

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C" ) max = 2, massless

max --

30 -

20 -

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I I I.0 -0.5 0 0.5 -1.0 I I I-0.5 0 0.5

momentrm actor moment rmactor momentrm actorFigure 7. The effect of changing moment arms on jump height. Relative jump height h/s is plotted against themoment arm parameter k (equation 24): (a) shows the heights of countermovementjumps with human-like forces(F'm so= 1) for threedifferentvalues of the compliance parameterC; (b)shows the heightsof countermovementjumpswith bushbaby-like forces (Fm so= 5), for the same three values of the compliance parameter; and (c) shows theheights of catapult jumps with insect-likeforces (Fm i,o = 25) for two values of the shortening speed parameter. In (a)and (b) the shortening speed parameter is 8. In (c) the compliance parameter is 2. The legs have two segments.Segment mass are mI = 0.7m, m2 = 0.2m, m3= O.lmexcept in the case of the broken line in (c), for which m1= m,m2 = m3 = 0.

(a) (b)1.00.8 -

0.6 -0.4 -

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Imoment rmfactor .5 /1

! I . I, I

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0.2 -0

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I/I II III I I

0.2u02 v0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0muscleength muscleength muscleength

Figure8. Graphsof muscleforce (Fm/Fm,o) against musclelength, forselected umps with muscles of zero compliance.Muscle length is expressedas a fraction of the working range, so that it is zero when the leg is fully extended and 1.0when the knee is bent to its minimum angle. (a) and (b) compare jumps with different moment arm factors k, forjumps with bushbaby-like forces (F, iso = 5) and (a) slow muscles, dmax 4 and (b) fast muscles dmax 16. (c)compares jumps with different starting angles 20min for jumps with human-like forces (Fm,s = 1) and a shorteningspeed parameter of 8. All graphs refer to two-segment legs with ml = 0.7m, m2= 0.2m, m3= 0. m.

area. For this reason, we will not investigate the effectof changing the mean moment arm ro.There may, however, be an advantage in having amoment arm that changes, as the joint extends. This isachieved in the models by giving the factor k (equation(24)) a non-zero value. When k is positive the momentarm increases as the joint extends. The mean momentarm, over the range of knee angles from 60? to 180?,equals ro for all values of k.Figure 7a shows that for countermovement jumpswith isometric forces representative of humans, thelowest values of k give the highest jumps. Figure 7cshows that the same is true of catapult jumps withinsect-like forces. Figure 7b, however, shows that forcountermovement jumps with bushbaby-like groundforces, the highest jumps may be obtained with thehighest values of k (for the higher compliances in this

figure) or with intermediate values (for low compli-ances). In these simulations, the muscles were given amaximum shortening speed which is believed to berealistic for small mammals such as bushbabies (see?4). Simulations with faster muscles gave the highestjumps for the lowest values of k, as in figure 7a.To explain these confusing results we must considerboth the force-velocity properties of the muscles andthe influence of leg mass. To see the effects of theforce-velocity properties clearly, we will comparejumps with zero series compliance. As already ex-plained (see ?5c) such jumps are identical whether asquat, catapult or countermovement technique is used.In figure 8a, b, muscle force is plotted against musclelength for jumps with different values of k. In all cases,the muscle initially exerts its isometric force, but theforce falls as the muscle shortens at an increasing rate.

Phil. Trans.R. Soc. Lond.B (1995)

(a) (b)10 -.0

' 2.5 :

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Legdesignand umping echnique R. McN. Alexander 245(c) amax= 2, masslessI - - _ _10 50

40

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Aa M.max 8maxamax=2

20 -

10

0 o0 300 60? 90? 1200 300 60? 90? 1200 300 60? 90? 12(startingngle startingngle startingngleFigure9. The effectsof starting angle onjump height, when the musclesare adapted to the starting angle as explainedin the text. Relative jump height hfs is plotted against starting angle 2mi,, for countermovement jumps with:(a) human-like; (b) bushbaby-like forces; and (c) catapult jumps with insect-like forces. Other details are as forfigure 7.

At first it falls faster for negative values of k. This isbecause negative values of k give moment arms whichare initially high, requiring the muscle to shorten fasterfor any given angular velocity of the joint. Later,however, muscle force falls more slowly for negativevalues of k (for which the moment arm is decreasing)than for positive values (for which it is increasing).Consequently, the graphs for negative and positivevalues of k cross, and the negative values give thehigher muscle forces in the later stages of take-off.The areas under the graphs in figure 8 represent thework done by the muscles. In figure 8a the maximumshortening speed of the muscle is low and the muscledoes 8 ?/ less work for k =-- 1 than for k = + 0.5. Infigure 8 bhowever, a faster muscle does 18 % moreworkfor k = - 1 than for k = + 0.5. These simulations usedbushbaby-like muscle forces, as did the simulations infigure 6b in which the maximum shortening speed ofthe muscles has an intermediate value.

The optimum value of k in countermovement jumpsdepends mainly on the force-velocity properties of themuscles, though series compliance also has an effect, asfigure 7 b shows. The heights of catapult jumps withinsect-like muscle forces and high series compliancesdepend very little on the force-velocity properties of themuscles (see figure 3 c), and in such cases we must lookfor a different explanation of the dependence of jumpheight on k.An explanation is suggested by a comparison infigure 7c, between the continuous lines (for legs withmass) and the broken one (for legs of zero mass). Theformer show lower jump heights for the reason given in?5 d; some of the work done by the muscles is requiredto provide internal kinetic energy associated withmovement of leg segments relative to the centre ofmass. Another difference between the continuous andbroken lines is that the former show jump heightdecreasing as k increases, but the latter shows heightsthat are almost independent of k. The reason that legmass makes jump height dependent on k is that a major

part of the internal kinetic energy at take-off (repre-sented by the term m4tan20, in equation (13)) isproportionalto tan20 which approachesinfinity as theleg straightens. Consequently, less work is needed toaccelerate the body to a given speed if it reaches thisspeed while the legs are still considerablybent, than ifit does not reach it until the legs are almost straight.Anegative value of k makes the moment arm initiallyhigh, enabling the elasticrecoilof the seriescomplianceto do most of its work early in the process of legextension.(h) Joint angles

So far we have assumed that the minimum kneeangle occurring in the jump (20,,i) is 60?; the workingrange, from this to full extension, is 120?.We will nowask whether there would be an advantage in workingover a different range, for example from 0? to fullextension (a range of 180?) or from 120? to fullextension (a range of 60?). Assume that the volume ofthe muscle and the propertiesof its constituent fibresare constant. Then to adapt the muscle to work over arange of 180? (forexample) instead of 120?,the lengthof its fascicles should be multiplied by 1.5 and theirphysiological cross-sectional area by 0.67. Its maxi-mum shortening rate would then be 1.5 times, and itsisometric force 0.67 times, the values for 120? range.More generally, if the range is to be multiplied by afactor n, amax is multiplied by n and F,misos divided byn.Also, if the serieselastic element is to be stretched bythe same fraction of muscle fascicle length, when themuscle exertsits isometricforce,the complianceC mustbe multiplied by n2.These adjustmentswere made in the calculationsforfigure 9, which shows jump heights for differentminimumknee angles. Jump height is greaterfor lowerminimumknee angles except in figure 9a, which showsintermediate angles giving the highest jumps.Figure 8c will help us to understand these results.


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246 R. McN. Alexander Leg design and umping techniqueLike the other parts of this figure it shows muscle forceplotted against length, for a muscle with no seriescompliance. The greater the minimum angle, the fastermuscle stress falls as the body accelerates and the lesswork does the muscle do. (Remember that work isrepresented by the areas under the graphs.)Our assumptions imply that the muscle has the samevolume in every case and is, in principle, capable ofperforming the same amount of work in a contraction.However, when the minimum angle is larger, thedistance over which the body has to be accelerated totake-off speed is less, so it has to be accelerated in lesstime to reach the same speed. Consequently, themuscles shorten faster and can exert less force, whenthe minimum angle is high.The argument so far suggests that the lowestminimum angles should give the highest jumps.However, figure 9 a shows that for jumps with human-like isometric forces a minimum angle of 40? gives ahigher jump than does one of 0?. The muscles do 22 %less work, but the resulting jump is higher. Theexplanation is that with a minimum angle of 0? thehips start at ground level, but if the angle is 400 theystart at a height 2s sin 40 = 0.68s.This effect can only be significant if jump height isquite small, compared to leg length. The difference ofstarting height is too small to counteract the advantageof a very low minimum angle, in the simulations withbushbaby-like muscle forces (see figure 9b).There is another advantage of low minimum angleswhich has limited importance in the simulations ofmammal jumps but predominates injumps with insect-like muscle forces (see figure 8c). This is that the lowerthe minimum angle, the more of the muscle's work canbe done while the leg is still quite strongly bent, and thelower the proportion of this work that is lost as internalkinetic energy. The argument in ?5f, relating to theterm m4tan20 in equation (13), applies again here.Simulations with insect-like muscle forces and legs ofzero mass give jump heights almost independent ofminimum angle.6. DISCUSSIONThe models presented in this paper are highlysimplified. Their anatomies resemble those of realanimals only in broad outline. Many simplifyingassumptions have been made: for example, thatmuscles are fully activated instantaneously and that, ifthere are several extensor muscles, they are activatedsimultaneously. These assumptions were avoided byPandy et al. (1990) in an optimal control model ofhuman jumping. There is much uncertainty about thevalues of muscle properties such as maximum short-ening speed and series compliance, which would berealistic for any particular species. The results never-theless may help us to understand the principles ofjumping.They show us that different jumping techniques areappropriate for animals exerting forces that aredifferent multiples of body mass. As a general rule,larger animals exert forces that are smaller multiples ofbody weight (Alexander 1985): humans making

standing jumps exert forces on the ground of 2-3 timesbody weight, and fleas over 100 times body weight (see?4). However, frogs exert maximum forces of onlyabout 3.5 times body weight (Hirano & Rome 1984).Despite the difference in size between them andhumans, the maximum forces they exert are not muchgreater, relative to body weight.Figure 4a tells us that with human-like muscleforces, countermovement and catapult jumps arehigher than squat jumps. Humans use the counter-movement technique and healthy young men jumpabout 5 cm higher with a countermovement than theycan in a squat jump (Komi & Bosco 1978). For them,leg segment length s is about 45 cm, so the advantagein jump height that a countermovement gives is 0. s.This matches the advantage given by the simulationsfor a compliance parameter of 1.0. It was argued in ?4(admittedly on sparse evidence) that a complianceparameter of 0.7 might be realistic for mammals.Figure 4 b shows that for animals exerting bushbaby-like forces, catapult jumps could be much higher thancountermovement jumps, which in turn can be higherthan squat jumps. Mammals seem not to have evolvedcatapult mechanisms, so the options available to themare countermovement jumping and squat jumping.The larger prosimians make a countermovement beforejumping, as is shown by forces falling below body massin records of jumps by Lemurcatta (2.4 kg) and Galagogarnetti (0.8 kg: see figure 6 of Gunther etal. 1991). Thesmall bushbaby Galagomoholisometimes makes a smallhop which may function as a countermovement, beforejumping (Ginther et al. 1991).Figure 4c shows that catapult jumping is by far themost effective jumping technique for animals exertinginsect-like ground forces. A variety of catapults haveevolved in insects including the resilin springs of fleas(Bennet-Clark & Lucey 1967) and the apodemes andsemilunar processes of locusts (Bennet-Clark 1975).The compliances of these catapults are high, enough inthe locust for their elastic recoil to move the kneethrough its whole angular range (Bennet-Clark 1975).Figure 3c shows that for catapult jumps with such highcompliances, the maximum shortening speed of themuscle makes little difference to the height of the jump.The knee extensor muscles of locusts seem to be slow,with maximum shortening speeds of only 2 lengths persecond (Bennet-Clark 1975). It is probably inevitablethat they should be fairly slow, as their sarcomeres arelong, with 5.5 jum-thick filaments. They exert highisometric stresses, of about 0.7 MPa. Muscles with longthick filaments can exert high stresses because largenumbers of cross-bridges connect each thick filament toa neighbouring thin one, but they tend to be slowbecause high cross-bridge cycling rates are needed tomake the muscle contract at any given strain rate(Ruigg 1968). The long sarcomeres of locust kneeextensor muscle allow it to exert high stresses, enablinga given volume of muscle to do a large quantity of workas it shortens to deform the catapult springs. The goodeffect of this on jump performance must far outweighthe small disadvantage of the muscles' being slow.Figure 5 shows that jump height is reduced by heavylegs. However, if the jump is powered by a leg muscle

Phil. Trans. R. Soc. Lond. B (1995)


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Leg design andjumping technique R. McN. Alexander 247(as in locusts) that muscle must be large to power astrong jump, and the leg cannot be very light.Comparison of the simulations with mammal-like legproportions in figure 5a with those with thigh andlower leg masses reversed shows that the total mass ofthe leg influences jumping ability more than thedistribution of the mass. This is in contrast to running,for which it is particularly important that the distalparts of the limbs should be light, to minimize thekinetic energy required for each forward or backwardleg swing (Fedak et al. 1982). Jumping vertebrates donot have the very light feet found in ungulates.The finding that legs with more than two segmentsmake higher jumps possible is striking (see figure5a, b). It seems to throw light on the evolution of theelongated tarsal bones of bushbabies and frogs (illu-strated in Rogers 1986) which, in effect, add a segmentto the lower leg. The long ilia of frogs, with theirmoveable iliosacral joints, add a further functionalsegment to frog legs.When humans jump, the forefoot presses on theground and the heel rises off the ground, so themetatarsals form an additional short leg segment. Theimportance of this was stressed by Bobbert & vanIngen Schenau (1988).Figure 6 shows that longer legs can be expected tomake higher jumps possible. Accordingly, manyjumping vertebrates have longer legs than relatedanimals of similar mass, that do not jump (Emerson1985). Figure 6c suggests that for small insects usingcatapult mechanisms, the advantage of long legs mightbe small. Locusts have remarkably long hind legs but(as Dr R. F. Ker has pointed out to me) flea beetles(Phyllotreta) do not.Figure 7a, c indicates that, to be most effective forjumping, the moment arms of the knee extensormuscles of humans and insects should decrease as thejoint extends. The reverse is found both in humans andin locusts. The kinematics of the human knee iscomplicated by the effect of the patella (Bishop 1977),but Lindahl & Movin's (1967) graphs of quadricepselongation against knee angle show that the muscles'effective moment arm is 80 %? reater when the knee isnear full extension than when it is bent to 90?. Theexplanation here may be that the leg is principallyadapted for walking and running, in which maximumforces act when the knee is much straighter than whenbent in preparation for a jump.The moment arm about the knee of the extensortibiae muscle of locusts is very small when the knee isfully flexed and increases as the knee extends (Bennet-Clark 1975). This is an essential feature of the catapultmechanism because it enables the small flexor tibiaemuscle to hold the knee flexed while tension builds upin the extensor. The requirements of the catapultrelease mechanism apparently override the advantage(indicated by figure 7 c) of a moment arm that decreasesas the joint extends.Lutz & Rome (1994) found that the semimembra-nosus muscles of frogs shorten at a constant rateduring take-off for a jump, presumably a consequenceof moment arms that fall as the leg extends.Figure 9 suggests that humans should not bend their

legs as much as bushbabies and insects, when preparingto jump. This is observed; humans bend the knee toabout 75? (Bobbert & van Ingen Schenau 1988),Galago to 30? (Guinther 1985) and locusts almost to 0?(Heitler 1977). I am not inclined to attach muchsignificance to this correspondence between theory andobservation because human legs are not principallyadapted for jumping.Jumping is a relatively simple process, performed insimilar ways by a wide variety of animals. An objective(to jump as high or as far as possible) is easily defined.These features make jumping a peculiarly attractivesubject for investigations such as this one, which hasexplored the effects of muscle properties, leg design andtechnique on jumping performance.REFERENCESAlexander, R.McN. 1981 Mechanics of skeleton andtendons. In Handbookf physiology thenervousystem(ed.V. B. Brooks), edn 2, vol. 2, pp. 17-42. Bethesda, Mary-land: American Physiological Society.Alexander, R.McN. 1985 The maximum forces exerted byanimals. J. exp.Biol. 115, 231-238.Alexander, R.McN. 1988 Elastic mechanismsn animalmovement.ambridge University Press.Bennet-Clark,H.C. 1975 The energetics of the jump of thelocust Schistocercaregaria.J. exp.Biol. 63, 53-83.Bennet-Clark,H.C. 1976 Energy storageinjumping insects.In The nsectntegumented. H. R. Hepburn), pp. 421-443.Amsterdam: Elsevier.Bennet-Clark,H.C. & Lucey, E.C.A. 1967 The jump of the

flea: a study of the energetics and a model of themechanism. J. exp.Biol. 47, 59-76.Bishop, R.E.D. 1977 On the mechanics of the human knee.Engng.Med. 6 (2), 46-52.Bobbert, M.F. & van Ingen Schenau, G.J. 1988 Co-ordination in verticaljumping. J. Biomechan.1, 249-262.Calow, L.J. & Alexander, R.McN. 1973 A mechanicalanalysis of a hind leg of a frog (Ranatemporaria).. Zool.171, 293-321.Cutts, A. 1986 Sarcomere length changes in the wingmuscles during the wing beat cycle of two bird species.J. Zool.A209, 183-185.Dimery, NJ. 1985 Muscle and sarcomere lengths in thehind leg of a rabbit (Oryctolagus uniculus)during agalloping stride. J. Zool.A205, 373-383.Dowling, JJ. & Vamos, L. 1993 Identification of kineticand temporalfactors related to verticaljump performance.J. appl.Biomechan., 95-110.Emerson, S.B. 1985 Jumping and leaping. In Functionalvertebrateorphologyed. M. Hildebrand, D. M. Bramble &K. F. Liem), pp. 58-72. Cambridge, Massachussetts:Belknap Press.Ettema, GJ.C., Huijing, P.A., van Ingen Schenau, G.J. & deHaan, A. 1990 Effects of prestretch at the onset ofstimulation on mechanical work output of rate medialgastrocnemius muscle-tendon complex. J. exp.Biol. 152,333-351.Fedak, M.A., Heglund, N.C. & Taylor, C.R. 1982Energetics and mechanics of terrestrial locomotion. II.Kinetic energy changes of the limbs and body as a functionof speed and body size in birds and mammals. J. exp.Biol.97, 23-40.Fitts, R.H., Costill, D.L. & Gardetto, P.R. 1989 Effect ofswim exercise training on human muscle fiber function.J. appl.Physiol.66, 465-475.



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248 R. McN. Alexander Leg design andjumping techniqueGrand, T.I. 1977 Body weight: its relation to tissuecomposition, segment distribution and motor function. I.Interspecific comparisons. Am. J. phys. Anthrop.47,211-240.Gunther, M.M. 1985 Biomechanische Voraussetzungenbeim Absprung des Senegalgalagos. Z. Morph.Anthrop.5,

287-306.Guinther,M.M., Ishida, H., Kumakura, H. & Nakano, Y.1991 The jump as a fast mode of locomotion in arborealand terrestrialbiotopes. Z. Morph.Anthrop. 8, 341-372.Heitler, W.J. 1977 The locust jump. III. Structuralspecializations of the metathoracic tibiae. J. exp.Biol. 67,29-36.Hill, A.V. 1938 The heat of shortening and the dynamicconstants of muscle. Proc.R. Soc. LondB 126, 136-195.Hirano, M. & Rome, L.C. 1984 Jumping performance offrogs (Rana pipiens) s a function of muscle temperature.J. exp.Biol. 108, 429-439.Huxley, A.F. & Simmons, R.M. 1971 Mechanical proper-ties of the crossbridges of frog striated muscle. J. Physiol.,Lond.218, 59P-60P.Josephson, R.K. 1984 Contraction dynamics of flight andstridulatorymuscles of tettigoniid insects. J. exp.Biol. 108,77-96.Ker, R.F., Alexander, R.McN. & Bennett, M.B. 1988 Why

are mammalian tendons so thick? J. Zool., Lond. 216,309-324.Komi, P.V. & Bosco, C. 1978 Utilization of stored elasticenergy in leg extensor muscles by men and women. Med.Scienta.Sports10, 261-265.Lindahl, O. & Movin, A. 1967 The mechanics of extensionof the kneejoint. Actaorthop. candinav.8, 226-234.Lutz, G.J. & Rome, L.C. 1994 Builtforjumping: the designof the frog muscular system. Science,Wash.263, 370-372.Otten, E. 1987 A myocyberneticmodel of thejaw system ofthe rat. J. Neurosci.Meth.21, 287-302.Pandy, M.G., Zajac, F.E., Sim, E. & Levine, W.S. 1990 Anoptimal control model for maximum-height humanjump-ing. J. Biomechan.3, 1185-1198.Rogers, E. 1986 Looking t vertebrates. arlow: Longman.Ruegg,J.C. 1968 Contractilemechanismsof smooth muscle.Symp.Soc.exp.Biol. 22, 45-66.Tihanyi, J., Apor, P. & Fekete, Gy. 1982 Force-velocity-power characteristicsand fibercompositionin human kneeextensor muscles. Eur.J. appl.Physiol.48, 331-343.Winter, D.A. 1990 Biomechanicsnd motorcontrolof humanmovement,dn. 2, New York: Wiley.Woledge, R.C., Curtin, N.A. & Homsher, E. 1985 Energeticaspects f muscle ontraction. ondon: Academic Press.Received 6 May 1994; accepted8 August1994


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