Optimum timing of muscle activation for simple models of throwing

J. theor. Biol. (1991) 150, 349-372

Optimum Timing of Muscle Activation for Simple Models of Throwing R. M c N . ALEXANDER

Department of l~re and Applied Biology, University of Leeds, Leeds LS2 9JT, U.K.t (Received on 26 November 1990, Accepted on 23 January 1991) In diverse throwing activities, muscles contract in sequence, starting with those furthest from the hand. This paper uses simple mathematical models, each with just two muscles, to investigate the consequences of this sequential contraction. One model was suggested by shot putting, another by underarm throwing and the third by overarm throwing, but all are much simpler than real human movements. In each case there is an optimum delay between activation of the more proximal muscle and of the more distal one, that maximizes the speed at which the missile leaves the hand. If the delay is shorter than optimal, the throw is completed sooner and less time is available for contraction of the proximal muscle: it may shorten faster, exerting less torque, or through less than its full range of movement, and so do less work. If it is longer than optimal, less time is available for contraction of the distal muscle, which therefore does less work. The optimal delay is in some cases longer than would maximize total work because the delay influences the proportion of the work that appears as kinetic energy of the missile. 1. Introduction Throwing an object involves the use of many muscles which move many joints, generally starting with those furthest from the hand. In an overarm throw as used in baseball and cricket, body segments reach peak angular velocities in the sequence pelvis, upper trunk and upper arm, forearm and finally hand (Atwater, 1979). In a javelin throw, hip, shoulder and elbow successively reach maximum speeds and then decelerate before the hand reaches its peak speed and releases the javelin (Menzel, 1986). In a shot put, knee, hip, shoulder and wrist successively pass through speed maxima (Zatsiorsky et al., 1981). The authors cited in this paragraph have argued that the sequence of movement allows energy and momentum to be transferred from segment to segment and eventually to the missile, increasing the speed o f the throw. This paper uses mathematical models to investigate more rigorously the consequences of sequential muscle activation in throwing. The aim is to discover basic principles that apply to throwing in general, but a wide variety o f techniques of throwing are used in sports and other activities. Three models with markedly different geometries will therefore be presented. One was suggested by the action o f shot putting, the second by underarm throwing as used by softball pitchers and the third by overarm throwing as used in cricket and baseball. t The work described in this paper was carried out at the Department of Biology, St Francis Xavier University, Antigonish, N.S., Canada. 349

0022-5193/91/110349+24 $03.00/0

(~ 1991 Academic Press Limited

350

R.

MeN.

ALEXANDER

All are very much simpler than the movements that suggested them. Running throws (e.g. javelin throwing) are not considered. A simple model of throwing has previously been described, in a short paper by Herring & Chapman (1988). The geometry of their model is different from all those described here: it represents an overarm throw as a planar movement. It has three muscles which exert joint torques that are independent of angular velocity, while they are active. 2. Features Common to the Three Models

Each model has just two muscles, each of which is either inactive or fully active at any given time. One muscle is activated at zero time and the other after a delay. Both remain active until the missile leaves the hand. At any time t while it is active, each muscle exerts a torque T about a joint which has angle 0 and angular velocity 0. The torque is related to the angular velocity by a form of Hill's equation (see Woledge et al., 1985). if but if

0<0 . . . . 0 _>0,,ax,

T=To(Omax-O)/(Omax+30)"I T = 0,

(1)

where To is the torque exerted by the muscle in an isometric contraction and 0,~ax the angular velocity corresponding to its maximum (unloaded) rate of shortening. To will generally be taken to be a constant for each muscle, independent of joint angle. However, measurements of the maximum isometric torques that can be exerted at human joints show marked dependence on joint angle, in some cases (Kulig et al., 1984). A few results will therefore be presented for versions of the models in which To is a function of 0. The factor 3 in eqn (1) has been selected as typical for fast mammalian muscle. It represents the reciprocal of the parameter called a / P o by Woledge et al. (1985). The movements of the models start from rest, but real throwing is often preceded by a preparatory movement which may stretch active muscles, immediately before they shorten. This should increase the torques that can be exerted during shortening (Cavagna et aL, 1968). An attempt will be made to take account of this by using appropriately high values for To. Equation (1) ignores tendon compliance, which substantially modifies the effects of the human gastrocnemius muscle (Ker et al., 1987) but seems likely to be less important for the knee, shoulder and elbow muscles represented in the models. The gastrocnemius exerts much higher stresses on its tendon than do most other muscles (Ker et al., 1988), stretching it to unusually large strains. While torques are falling and angular velocities rising, as they generally are during the simulated throws performed by the models, tendon compliance would generally result in higher angular velocities at given torque: it would have much the same effect as an increase in 0max. The movements of the joints of the models, and of the missiles, have been calculated from the equations presented in this paper by numerical integration. Some of the integrations were repeated with time increments five times smaller than

SIMPLE MODELS OF THROWING

351

those otherwise used: this altered the calculated missile velocity by less than 0.2%. Results will be presented in dimensionless form so as to be applicable to athletes o f any size. Masses will be expressed as multiples o f body mass M, lengths as multiples of leg segment length s (Fig. 1) and accelerations as multiples of the gravitational acceleration g. A typical man might have a mass of 70 kg, a stature of 1.8 m and leg segments 0.44 m long (estimated from the segment lengths given by Dyson, 1977). For such a man, our dimensionless unit of time ( s / g ) ~/2 would be 0-21 see; our unit o f torque (Mgs) would be 300 Nm; our unit of angular velocity ( g / s ) ~/2 would be 4.7 rad sec-~; and our unit of speed (sg) 1/2 would be 2.1 m sec -1.

3. Model 1: Throwing by Pushing 3.1. THE MODEL

This model (Fig. 1) was suggested by shot putting, but neither the body nor the missile has any horizontal component of velocity: both move vertically upwards. The knee and elbow are initially bent, and are extended to project the missile. Their extensors are the only muscles represented, so no torques act at the ankle, hip or shoulder, which are at all times vertically below the centre of mass o f the missile. The trunk has mass M and is rigid. The masses of limb segments are neglected, except that the mass o f the hand is included in the mass mbali of the missile. The relative lengths of segments (shown in Fig. 1) are reasonably realistic (Dyson, 1977). At time t, the angles of knee and elbow are Ok and 0~. While the foot is on the ground and the missile in the hand, the heights of hip and missile are given by Yhip s[0"2+2 sin (0k/2)]

(2)

Yba, = S[1"4+2 sin (0k/2) + 1"6 sin ( 0 J 2 ) ]

(3)

:

The corresponding velocities can be obtained by differentiation Yhip ----"$/~k COS ( 0 k / 2 )

(4)

YbaH= S[0k c o s ( 0 k / 2 ) + 0 " 8 0e COS (0¢/2),

(5)

where 0k, 0~ are the angular velocities of the joints. While the muscles are active they exert torques Tk, T~ which are related to these angular velocities by eqn (1). Consequently, the leg exerts a force Fh~p at the hip and the arm exerts a force Fb~n on the missile. Fhip = Tk/[s COS (0k/2)]

(6)

Fba,, = T,/[O.8s cos ( 0 J 2 ) ]

(7)

The accelerations of trunk and ball are Phip = ( F . i . - Fba,O / M - g

(8)

Yba~,=/;'ball/rnbaH - g.

(9)

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T

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12

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0

_

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0-6 Dimensionless deloy

0-4

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Missile moss=0.25 /14

~.0'\0~ ~'~

0.8

I

--'~~..~

(b)

.0

FIG. I. (a) Model 1. The missile is pushed upwards by extension of the knee and elbow. (b) A graph of the energy with which the missile leaves the hand, plotted against the delay between activation of the knee and of the elbow muscles, for three different missile masses. Energy is expressed as Eball/mb,lngs and delay as At/(s/g) l/z. The properties of the muscle and the initial angles of the joints are as given in column (a) of Table 2. The broken line shows the effect of making isometric torque a function of joint angle, as explained in the text.

f

1_

t-2s

_

°~..~ 0"8s

(a)

14

rn

Z

Lra

Z

t~

/[

;¢

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353

These equations apply when both the knee and the elbow muscles are active, but we will consider throws in which the knee muscle is activated first. While only it is active, the elbow remains strongly bent with hand and missile resting on the shoulder, and

fihip

= fiball = Fhip/(M + mball) -- g.

(10)

At zero time, while the model is stationary, the knee muscle is activated and eqn (10) starts to apply. After some delay At the elbow muscle is also activated and eqns (8) and (9) start to apply. The foot may leave the ground if the angular velocity o f the knee reaches its maximum value [0 . . . . eqn (1)], because the knee muscle can shorten no faster. Thereafter, Fhip is zero. The missile leaves the hand when the angular velocity of the elbow reaches its maximum value. Its total (kinetic plus potential) energy at that time is calculated from the equation E b a . = m b a l l ( ) ) 2 a . / 2 + Yballg).

(11 )

The initial knee angle will generally be taken to be 75 ° and the initial elbow angle 30 ° (as when the back of the hand rests on the shoulder). The isometric torques that will be used in most o f the calculations have been chosen to obtain peak forces similar to those that act in shot putting. During the stage of the put in which the acceleration o f the shot is high, the right leg of a right-handed shot putter has its foot on the ground and behaves much like the leg o f the model. It exerts a peak force on the ground o f about three times body weight, and the hand exerts a peak force on the shot of about 0-3 times body weight (Zatsiorsky et al., 1981). Similar forces are obtained for the model by making the isometric torques o f knee and elbow 2.5Mgs and 0.7Mgs, respectively. The same knee torque was used in a model o f jumping (Alexander, 1990). For a typical 70 kg man it represents a torque of about 750 Nm, which is probably higher than the torques exerted by the knee muscles o f athletes, even after rapid stretching. The model needs a larger torque at the knee than a real athlete, to obtain the same ground force, because it keeps this force in line with the hip and ankle instead of allowing it to act further forward, through the ball of the foot. Alexander (1990) supposed that the peak angular velocity attained by the knee in standing jumps might be about half the unloaded angular velocity [0 . . . . eqn (1)], which was accordingly estimated as 8(g/s) 1/2. We will use this value in most of our calculations, both for the knee and for the elbow. Results will also be represented for other values, of all the parameters discussed in this paragraph. 3.2. RESULTS FOR M O D E L 1

Figure l(b) shows the energy of the missile as it leaves the hand, plotted against the delay between activation of the knee muscle and o f the elbow muscle. Results are presented for three different missile masses of which the intermediate one (0-1 times body mass) is roughly equivalent to the shot used in athletics (4-0 kg for women and 7.3 kg for men). For each mass there is an optimum delay which maximizes the energy imparted to the missile.

354

R. McN. ALEXANDER

These results were calculated using the standard values for parameters (selected in section 3.1), which were intended to be realistic. The energies imparted to the missiles may nevertheless seem small. For example, the maximum energy of 10.3 mballgs attained by a missile of mass 0 . 1 M represents about 320 joules (J), for a 7.3 kg shot thrown by an adult man. An athlete putting the shot 16 m, however, releases it with velocity 12 m sec -1 at a height of about 2.2 m (Balireich & Kuhlow, 1986); its energy, calculated from eqn (11), is then 680J. The discrepancy arises because the model imitates only part of the action of shot putting. It moves vertically upwards with its trunk rigid and accelerates the shot over a distance o f only 2s, 0.9 m for a typical man. In contrast, a real shot putter moves across the circle as well as upward, extending his back as well as his limbs, and the shot may be accelerated over a distance of as much as 2.4 m (Dyson, 1977). If the same force is exerted over a larger distance, more work is done and the shot gains more energy. An athlete imitating the movements of the model would probably not perform much better than it. We will study the behaviour of the model to find out why there is an optimum delay, for any given missile mass. Figure 2 shows how it moves when the delay is [(a) and (d)] less than the optimum; [(b) and (e)] close to the optimum; and [(c) and (f)] greater than the optimum. The unloaded angular velocities 0. . . . above which the muscles can exert no torque, a r e 8 ( g / s ) 1/2 in these simulations. The missile is released when the elbow reaches this angular velocity, at which time it is fairly well-extended (to 130° or more). If the delay is not too short the knee reaches its unloaded angular velocity, and the foot leaves the ground, before the missile leaves the hand [Fig. 2(b) and (e), (c) and (f)]. In these cases it extends to more than 160 ° while its muscles are exerting torque. If the delay is very short, however, the missile leaves the hand while the knee is still quite strongly bent [t29 ° in Fig. 2(a)]. The knee muscles move the joint through rather a small angle, and so can do rather little work. The faster the trunk is accelerating while the elbows extend, the longer the hand can maintain contact with the missile. Consequently, the shorter the delay, the longer the extension time for the arm (from elbow muscle activation to missile release). Figure 2 and Table 1 show this. The Table also shows that when the delay is shorter the elbow muscles do more work, because they contract more slowly and so [by eqn (1)] exert more torque. Not only does the acceleration of the trunk influence the work that can be done by the arms: that o f the missile influences the work that can be done by the legs. Table 1 shows that the extension time for the legs and the work they do are greatest when the delay has intermediate values. (Leg extension time is measured from zero time to foot off or missile release.) The leg extension time is longest and the work done by the legs is greatest, when the delay is just long enough to ensure that the missile does not leave the hand until the feet have left the ground [0.4(s/g) I/2, Table 1]. The total work done by legs and arms is greatest when the delay has almost the same value, but more energy is given to the missile if the delay is rather longer [about 0.6(s/g)I/2]. The reasons seems to be that more energy is transmitted from

SIMPLE

MODELS

OF THROWING

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R. M c N . A L E X A N D E R TABLE 1

Work done by the muscles of Model 1, in simulations with parameter values as in Fig. 2. Time and energy are expressed as multiples of ( s / g ) I/2 and M g s , respectively. Work was calculated as S T. dO Delay Extension time for legs Work done by legs Extension time for arms Work done by arms Total work Energy gained by missile

0.2 1.01 1.37 0.81 0.41 1.78 0.63

0.4 1.15 1.68 0-79 0-40 2-08 0.70

0-6 1-13 1.67 0.75 0.37 2.04 0.72

1.0 1-10 1.61 0-64 0-31 1-92 0.67

t r u n k to m i s s i l e w h e n t h e a r m e x e r t s a g i v e n i m p u l s e , i f t h e t r u n k is m o v i n g faster. I f a s m a l l i m p u l s e P a c t e d o n t h e t r u n k w h i l e it w a s m o v i n g w i t h v e l o c i t y )~hip this v e l o c i t y w o u l d b e r e d u c e d b y P / M a n d its k i n e t i c e n e r g y w o u l d b e r e d u c e d b y a b o u t P3~hip. T h i s q u a n t i t y o f e n e r g y , w h i c h is p r o p o r t i o n a l t o t h e v e l o c i t y , w o u l d be transferred to the missile. F i g u r e l ( b ) s h o w s t h a t t h e g r e a t e r t h e m a s s o f t h e m i s s i l e , t h e less t h e o p t i m u m d e l a y . T h e r e a s o n is t h a t a h e a v i e r m i s s i l e c a n n o t b e g i v e n s u c h l a r g e a c c e l e r a t i o n s , so m o r e t i m e is n e e d e d to e x t e n d t h e e l b o w . T a b l e 2 s h o w s t h e effects o f c h a n g i n g other parameters. Stronger or faster muscles (with greater isometric torques or TABLE 2

The effects of different parameter values on the performance of Model 1, throwing a missile of mass O. 1M. In column a, the parameters have their standard values, and in each of the other columns one parameter ( b o l d type) has been changed. In each case the optimum delay that maximizes the energy given to the missile is shown, and the corresponding energy. Relative delay means the delay divided by the time at which the missile leaves the hand. Quantities are expressed in dimensionless form, as explained in section 2 a

b

c

d

e

f

g

Isometric torque at knee elbow

2.5 0-7

5 0-7

2.5 1.5

2.5 0.7

2.5 0-7

2.5 0.7

2.5 0.7

Unloaded angular velocity of knee elbow

8 8

8 8

8 8

8

8 8

8 8

Initial angle of knee elbow

75° 30 °

75° 30°

75 ° 30 °

75 ° 30 °

75 ° 30 °

115° 30 °

Optimal delay

0.62

0.30

0-77

0-60

0.66

0.05

0"80

Optimal relative delay

0-46

0.28

0.61

0-44

0.51

0-06

0.63

9.4

9"1

Energy gained by missile

10.3

11-9

12.5

20 8

11.7

20

14.0

75° 75 °

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unloaded angular velocities) increase the energy that can be given to the missile (columns b-e). Larger initial joint angles reduce the energy because the joints extend through smaller angular ranges, so the muscles do less work (columns f and g). Stronger knee muscles (column b) reduce the optimum delay because they extend the knees faster, and faster knee muscles (column d) have a similar but much smaller effect. Stronger or faster elbow muscles increase the optimum delay because they extend the elbow faster (columns c and e). A larger initial knee angle reduces the optimum delay and a larger initial elbow angle increases it (columns f and g), because the joints take less time to extend through their reduced angular ranges. The results presented so far assume isometric torque to be independent of joint angle, which is unrealistic: the knee extensors actually exert isometric torques that vary with knee angle, with a maximum at about 120°, and some data for elbow extensors show maximum isometric torque at about 90 ° (Kulig et al., 1984). The broken line in Fig. l(b) shows the data for a missile of mass 0 . 1 M recalculated, with the isometric torques multiplied by sin (Ok -- zr/6) for the knee and sin 0r for the elbow to imitate the observed dependences o f torque on joint angle. This particular change reduces the energy that can be given to the missile because it reduces torques at most joint angles, but alters the optimum delay very little.

4. Model 2: Underarm Throwing 4.1. T H E

MODEL

This model imitates underarm throwing as used in bowling (Plagenhoef, 1971: fig. 8.13), but with the trunk stationary. Softball pitching is also performed underarm but involves more complex movements (Hay, 1973). As in Model 1, the arm is represented by two segments each of length 0-8s [Fig. 3(a)]. It swings forward in a vertical plane, moved by the flexor muscles of the shoulder and elbow. The upper arm has mass m,a and the forearm mass mrs. These are treated for simplicity as point masses at the centres o f mass of the segments, at distances r,~ and rr~ from the shoulder and elbow, respectively. A mass mball at the distal end o f the arm represents the ball and hand together. The shoulder is located at the origin o f Cartesian co-ordinates and the arm moves in the XY plane. At time t the upper arm and forearm make angles 0,a, 0fa with the horizontal. The equation of motion of the model will be obtained by the method o f Lagrange. Gravitational potential energy changes will be ignored because they can be assumed to be small in comparison to the kinetic energy changes. The kinetic energy of the upper arm is

Ea~=½m.~r~O2a.

(12)

The co-ordinates o f the centre of mass o f the forearm are xra = - 0 " 8 s cos 0ua - rra cos 0f~

(13)

yf~ = -0"8s sin 0 , ~ - rf~ sin 0f~.

(14)

-Y

"E

f

r> 0

<

o

~B

I0

J

I 0-2

I 0"6 Dimensionless deloy

! 0-4

! O. 8

(b)

1-0

FIG. 3. (a) Model 2. The missile is thrown underarm by flexion and rotation of the shoulder. (b) A graph of the speed with which the ball leaves the hand, against the delay between activation of the shoulder flexor and of the elbow flexor. Speeds and delays are expressed in dimensionless form as explained in section 2. Parameters are as listed in column (a) of Table 4. The broken line shows the effect of making isometric shoulder torque a function of joint angle, as explained in the text.

/ ~ ma f

(a)

L~ OO

SIMPLE

MODELS

OF

THROWING

359

so its components of velocity are xra = 0"8s0.a sin 0.a+

rfa0fa sin

0ra

Yfa = --0"8S0.a COS 0u~-- rr~t~ra cos 0ra,

(15) (16)

and the kinetic energy of the forearm is Efa

1 .2 -2 = irnra(Xta+yr~) 1 2"2 =~mra[0"64S 0ua+ l'6srra0ua0ra COS (0t~-- 0ua) + rra0f~]. 2 "2

(17)

Similarly, the kinetic energy of the ball is Eb~n = ~mban[0.64Sl 2 0ua_{_" 2 1.28s20.~ 0c~ cos (0ra- 0ua) + 0"64s20~a]

(18)

The total kinetic energy of the system is obtained by adding eqns (12), (17) and (18). E = E ~ + Era+ Ebal~ 1 "2 " " = ~[k~O.~+2kzO.~Ora cos (0f~- 0~) + k30f2a,

where kl = r~amua + 0"64s2(rnfa + mball) k2 = 0" 8s( rrdnr~ + 0'8smbaI0

(19)

k3 = r~mra + 0"64s2mba,.

The work performed by the muscles is

w= I Ts.dO°+f

Ter.d(Ofa-Oua).

(20)

where Ts. Ter are the torques exerted by the muscles at shoulder and elbow. (The subscript ef is used to distinguish this torque exerted by a flexor of the elbow, from the torque exerted by an extensor of the elbow in Model 1.) The equations of Lagrange are thus d (00~u~) O E = T ~ - T ~ f dt - O0ua

(21)

dt \aOrJ -0-~r~ = Tee.

(22)

By inserting E from eqn (19) and performing the differentiations we obtain k~O,.+kzOr~cos(Or~-O,a)-k202asin(Ora-O.,)= T~-T~r

(23)

k2 b',. cos (0r, - 0,a) + k30"ra+ k202- sin (0r. - 0.,) = T,r.

(24)

These equations were solved to obtain 0"u~and 0"ra explicitly and the movements of the model were determined by numerical integration. The angular velocities of the joints, which were required for calculating the torques [eqn (1)]. were 0~a for the shoulder and (0f.-O.a) for the elbow. Simulated throws were started with the arm stationary and aligned with the X-axis, with the elbow straight (0ua = 0ra = 0). The shoulder muscle was activated at zero time and the elbow muscle after a delay At. While only the shoulder muscle was

360

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ALEXANDER

active the elbow remained straight, prevented from hyperextending by its bony structure. Throughout this stage angles 0ua and 0fa remained equal, and their derivatives were also equal, so eqns (23) and (24) simplified to

Oua(k~ + 2kt+ k3) = Ts.

(25)

After the elbow muscles had been activated, the more complex equations applied. The aim of the calculations was to find out how fast the ball could be thrown (in any direction) for any chosen delay and set of parameters. Each simulation was therefore continued either until the kinetic energy of the ball [given by eqn (18)] reached a maximum or until the shoulder angle Ou~ reached 180 °, whichever happened sooner. It seems unlikely that the shoulder would be flexed through more than 180° in any real throwing action. The final speed o f the ball was calculated by taking the square root of the expression in square brackets in eqn (18). Dyson (1977) gives the masses of upper arm, forearm and hand as 3.3, 2.1 and 0-85% o f body mass. The mass mb,H in the model is that o f the hand and ball together and was taken to be 1-1% o f body mass, to include the mass of a cricket ball or baseball (about 0.15 kg). Values for the constants in eqn (19) were calculated from these masses, together with the positions of the centres of mass of the segments given by Dyson (1977): kl = 0"0258Ms 2, k2 = 0-0118Ms 2 and k3 = 0-0086Ms 2. A few simulations were also performed with larger values of these constants, corresponding to a (hand plus ball) mass of 5% body mass. "Wilkie (1950) and de Koning et al. (1985) investigated the force-velocity properties o f the flexor muscles o f the elbow. Their best subjects achieved maximum isometric torques of about 90 Nm (about 0.32Mgs) and unloaded angular velocities of about 18 rad sec -~ [about 3.8(g/s)I/2]. It seems likely that larger torques could be exerted immediately after the stretch imposed by the preparatory movements for a throw (Cavagna et al., 1968). Also, series compliance enhances the shortening speeds o f muscles in actions in which forces fall from initial high values, as in the model throws. We will therefore take 0.7Mgs and 8(g/s) 1/2 as our standard values for the isometric torque and unloaded angular velocity of the elbow. These values for the flexor happen to be identical to those used for the extensor in Model 1. The measurements of shoulder flexor strength reported by Kulig et al. (1984) are given as forces rather than torques, but if the measurements were made as illustrated by Clarke (1966), the isometric torques for men were around 90 Nm. In the absence o f better information, the same standard values of isometric torque and unloaded angular velocity will be used for the shoulder flexors as for the elbow flexors. Some results will also be presented for other values of all the muscle parameters. 4.2. R E S U L T S F O R M O D E L

2

Figure 3(b) shows the speeds at which the ball leaves the hand in throws with different delays, when the muscle parameters have their standard values. As for Model 1 there is an optimum delay which maximizes the speed. The maximum speed of 8"7(sg) 1/2 represents about 18 m sec -~ for an adult man. This is much less than the speeds of 30 m sec -~ or more at which softballs are often pitched (Hay,

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THROWING

361

1973) but is probably reasonably realistic for a throw in which the trunk is kept stationary. Figure 4 shows how the model moves, with various delays. When the elbow flexors are first activated, the angular velocity of the shoulder falls as that of the elbow rises: the torque that accelerates the forearm decelerates the upper arm. However, the angular velocity o f the shoulder soon starts rising again, and rises very rapidly, while the angular velocity of the elbow falls. The angular velocity of the shoulder tends to increase to maintain angular momentum as the bending of the elbow reduces the moment of inertia o f the arm as a whole. The angular velocity of the elbow falls because its flexors cannot shorten so fast against the increasing moment, due to the centripetal forces needed to maintain the curved paths of the forearm and hand. Figure 4 and Table 3 show that the longer the delay, the less time is available for contraction of the elbow muscles, the smaller the angle through which they move the elbow, and the less work they do. Thus, long delays tend to be disadvantageous. However, as for Model 1, the time for which the shoulder muscles are active and the work that they do are greatest when the delay is fairly long [0.9(s/g) ~/2 in the case illustrated]. The total work done by the shoulder and elbow muscles together is greatest when the delay is 0.5(s/g) ~/2 but the ball is given a little more kinetic energy at the optimum delay of 0.8(s/g) ~/2. Table 4 shows the effects of changing the parameters of the model. Stronger or faster muscles (columns b-e) allow the ball to be thrown faster, and a heavier bali (column f) cannot be thrown so fast. The optimum delay is reduced if the shoulder muscle is made stronger (column b) or faster (column d) but only enough to keep the relative delay almost constant. Stronger elbow muscles (column c) have very little effect on optimum delay but faster ones (column e), increase both the absolute and the relative values. The heavier missile (column f) requires a greater absolute delay, but a smaller relative one. For all the combinations of parameter values that were tried, the shoulder reached 180 °, the limit of its assumed rate of flexion, in throws with optimum delay. However, in the cases represented by columns c and e of Table 4 (unusually strong or fast elbow muscles) the speed o f the ball reached a maximum before the shoulder angle reached 180 °, if the delay was made sufficiently small. In all the trials of Model 2 reported so far, the isometric torques were independent of joint angle. However, the torque that the flexors of the shoulder can actually exert seems to be less at larger angles of flexion (Kulig et al., 1984). The broken line in Fig. 3(b) shows the effect of multiplying the isometric torque at the shoulder by (1 - 0J2~r), thus making it fall to half over the range 0-180 °. Elbow flexor torque seems to vary less with joint angle, and was left independent of joint angle in the model.

5. Model 3: Overarm Throwing 5.1. THE MODEL Model 3 resembles the overarm style generally used to throw a baseball (Atwater, 1979: fig. 2), a cricket ball or an American football (Plagenhoef, 1971: fig. 8.3). It

0

4

0"5

0"5

I'0

(d

I'0

/

0

0

- ~ 1

I

I

1.0

7,

.J

/

(e)

/cl

0-5 1.0 Dimensionlesstime

0"5

I ~ [/ ,

60 I--

°If o~oV/

oI

0

I

/,

/,

2 --

6t

60

120

180

0-5

0.5

(f)

1.0

e~bOW

I

1.0

f

(c]

FIG. 4. Graphs of (a)-(c) joint angles and (d)-(f) angular velocities, plotted against time, for Model 2 with parameters as in column a of Table 4. The delays were (a) and (d) 0.6(s/g)l/2; (b) and (e) 0.8(s/g) ~/2, close to the optimal value; and (c) and (f) 1.0(s/g) 1/2. Angular velocities and times are expressed in dimensionless form as explained in section 2.

C)

c:

o

"E

8

"G 6 o

6O

'~ 120

t80

X > Z

>

Z

/[

SIMPLE

MODELS

OF THROWING

363

TABLE 3

Work done by the muscles, of Model 2, in throws with parameter values as in Fig. 4. Time and energy are expressed as multiples of (s/g) 1/2 and Mgs, respectively Delay Shoulder flexion time Work done by shoulder muscles Elbow flexion time Work done by elbow muscles Total work Kinetic energy of ball and hand

0 0.87 0-32 0.87 0,46 0.78 0.31

0.6 1.08 0-57 0.48 0-28 0.85 0.40

0-8 1.12 0.62 0.32 0,19 0-81 0.41

1.0 1.12 0-62 0.12 0-06 0.68 0.40

TABLE 4

The effects of different parameter values on the behaviour of Model 2. Relative delay means the delay divided by the time at which the missile leaves the hand. Quantities are expressed in dimensionless form as explained in section 2 a

b

c

d

e

f

Isometric torque at shoulder elbow

0.7 0.7

1-5 0-7

0.7 1.5

0.7 0.7

0.7 0.7

0-7 0.7

Unloaded angular velocity of shoulder elbow

8 8

8 8

8 8

20 8

Mass of hand and ball

0'011

0'011

0'011

0'011

0"011

0.050

Optimum delay

0'81

0'62

0"80

0"64

0-97

1"02

Optimum relative delay

0-72

0'72

0"71

Speed of ball for optimum delay

8-7

10-2

9-5

0"70 10"7

8 20

8 8

0'85

0"63

9"7

6"1

Bold = Indicates where one parameter has been changed.

differs from normal overarm throwing, however, in that the trunk is kept stationary and the elbow is kept bent at 90 ° throughout the movement. The elbow is extended at the end of the movement in real throws, but this does not seem to contribute very much to the speed of the throw (Vaughn, 1985). The shoulder is located at the origin of three-dimensional Cartesian co-ordinates [Fig. 5(a)]. The upper arm OP moves in the XY plane making (at time t) an angle On with the X-axis. (The subscript fl indicates flexion.) The elbow remains bent at 90 ° but the humerus rotates about its long axis so that the forearm makes a variable angle 0rot with PQ, the tangent to the path of the elbow. Two shoulder muscles, a flexor and a rotator, exert torques Tn and Tro, which depend on the appropriate angular velocities according to eqr~ (1). These muscles are activated in turn, and 0n and 0rot both increase in the course of the the throw. The upper arm has mass mua, the forearm mra and the hand and ball together mball, each located at the appropriate centre o f mass as in Model 2.

364

R.

MeN.

ALEXANDER

The equation of motion of the system will be obtained by Lagrange's method. The kinetic energy of the upper arm is 2 "2 Eua -1 ~muar ua On .

(26)

To calculate the kinetic energy of the forearm note that PQ = rfa cos 0rot, and is perpendicular to OP. Thus, the co-ordinates of the centre of mass of the forearm are Xra = 0"8s cos 0n + rra cos 0rot sin 0n

(27)

Yra = 0"8s sin On - rra cos 0to, cos On.

(28)

zra = rrsin 0rut

(29)

The components o f its velocity are Xfa = -0"8s0n sin On+ rr~0r c o s 0rot C O S

rra0~ot sin 0~otsin 0n

(30)

)ra = 0"8s0n cos 0n+ rra0n cos 0rot sin 0n+ rra0rot sin 0rot cos On

(31)

-~ra= rra0rot COS 0rot.

(32)

0 fl - -

Its kinetic energy is 1 .2 .2 Eta = imfa(Xra -t- Yfa + -~2a) 1

2 "2

2

"2

2

"2

= ~mra(0.64s On+ rraOn cos ~ 0,ot+ l'6srr,0~ot0n sin 0~ot+ rra0~o~)

(33)

and the kinetic energy o f the ball Eba~ can be obtained similarly. The kinetic energy o f the whole system is E = Eua + Era + Eball 1 "2 " " • =~[k~On+2k20~otOn sin O~ot+ k3( 0"2r o t + 02 C O S 2 0rot)]

(34)

where k~, k2 and k3 are the same constants as in eqn (19). The work performed by the muscle is

W = I Tn.dOn+ I Trot.d0rot ,

(35)

so the equations o f Lagrange are d-t \ ~ n / - O 0 n

= Tn

(36)

and

d ( ae ]

ae = to,.

(37)

By inserting E from eqn (34) and performing the differentiations we obtain "2 (kl+k3cos2 Orot)Ofl+kzsinOrot.Orot+k2cOSOrot.Orot-k3sin2Orot. OrotOfl=Tn

k2sin 0rot. 0n+k3 ~/rot+0.5k3 sin 20ro t. 02__ Trot,

(38) (39)

SIMPLE

MODELS

OF

THROWING

365

where the angular accelerations 0"n and 0"rot can be obtained explicitly. The movements o f the model were determined by numerical integration. Simulated throws were started with On = 0 and at some chosen value o f 0rot, usually 30 °. [ H a y (1973) points out that lateral rotation of the humerus in a baseball pitch goes to an extreme that could not be held with the arm stationary.] The flexor muscle was activated at zero time and the rotator after an interval At. While only the flexor was active the angle of rotation remained constant and eqn (38) simplified to (k~ + k3 cos 2 0rot)0"n = Tn,

(40)

but after the rotator had been activated eqns (38) and (39) applied. The aim was to discover how fast the ball could be thrown in any reasonable direction, for any chosen delay and set of parameters. Each simulation was continued until the angle of flexion reached 180 ° (which was taken to be the anatomical limit o f flexion of the shoulder with the humerus abducted as for overarm throwing) or the angle of rotation reached 120 °. The latter limit was set because throws are generally required to be horizontal or tilted upward. It would be difficult to tilt the flexion plane [the XY plane in Fig. 5(a)] enough to prevent a throw from a larger angle o f rotation from being directed downward. In every case, the speed o f the ball continued to increase up to the instant when one or other joint reached its limit, and the ball was released. The same values were used for the constants kl, k2 and k3 as in Model 2. The isometric torques and unloaded angular velocities of both muscles were generally taken to be 0.7Mgs and 8(g/s) I/2, respectively, but other values were also tried. These values were mere guesses, but give plausible throwing speeds. 5.2. R E S U L T S F O R M O D E L

3

Figure 5(b) shows the speeds at which Model 3 throws the ball, when the muscle parameters have their standard values. As for the other models, there is an o p t i m u m delay that gives the fastest throw. The m a x i m u m speed of 8"4(sg) 1/2 would be about 17 m sec -~ for an adult man. This is much less than the speeds of about 35 m sec -~ achieved in overarm throws by skilled baseball players (Atwater, 1979) but seems reasonable for a throw in which the trunk is kept stationary. T o y o s h i m a et al. (1974) found that men who could throw 0-1 kg balls at 28 m sec -~ when allowed to move freely, could throw at only 15 m sec -~ when the whole trunk was immobilized. Figure 6 shows the movements of the model, in throws with various delays. The angular velocity of flexion falls a little when the rotator is first activated, but soon rises again. It may make a second fall if the angle of rotation passes 90 °, because the m o m e n t o f inertia o f the arm as a whole about the flexion axis then starts to increase. Figure 6 and Table 5 show that when the delay is short, rotation reaches its limit o f 120 ° while flexion is still far short of its 180 ° limit. Therefore, the flexor does less work than when the delay is longer. I f the delay is long, however, there is little time for the rotator to contract before the limit of flexion is reached, little rotation occurs

\

J

E3

c

< 0

6

-

I 0,6

Dimension|ess detoy

,I 0-4

I 0"2

,x,

I 0-8

\ I-0

FIG. 5. (a) Model 3. The missile is thrown overarm by flexion and rotation of the shoulder. (b) A graph of the speed at which the bali leaves the hand, against the delay between activation of the shoulder flexor and of the rotator. Speeds and delays are expressed in dimensionless form as explained in section 2. Parameters are as listed in column a of Table 6. The broken line shows the effect of making isometric shoulder torque a function of joint angle, as explained in the text.

\

(

/

8

(b)

t~

i

0

0.5

I 0.5

S

ii"

60

120

I .o

(d)

I-O

o

60

J

12C

IBC

Dimensionless

0-5

/f

0.5

I

time

•0

I.O

(e)

(b)

0

6

0

60

120

180

~

k

I 0.5

\

0.5

I

~

~"

rototion , , , ~ /

1.0

I

(f)

1.0

I

(c)

FIG. 6. Graphs of (a)-(c) joint angles and (d)-(f) angular velocities, plotted against time, for Model 3 with parameters as in column a of Table 6. The delays were (a) and (d) 0.2(s/g)~/2; (b) and (e) 0.5(s/g) 1/2 close to the optimum; and (c) and (f) 0.8(s/g) ~/2. Angular velocities and times are expressed in dimensionless form as explained in section 2,

-£

<3

>

8 o oJ

c

i

+of (o)

t~ --d

z C3

O

:z

"11

o

t" tn

o ~7

t"

~r

o~

368

R. McN. ALEXANDER TABLE 5

Work clone by the muscles of Model 3, in throws with parameter values as in Fig. 6. Time and energy are expressed as multiples of ( s / g ) 1/2 and Mgs, respectively Delay Flexion time Work done by flexor muscle Rotation time Work done by rotator muscle Total work Kinetic energy of ball and hand

0 0-43 0-22 0-43 0.23 0.45

0.2 0.60 0.34 0-40

0-5 0.91 0.50 0.41 0.21 0.71 0.39

0.21 0.55 0-31

0.25

0.8 0.89 0-46 0-09 0.05 0-51 0.22

a n d the r o t a t o r does less work t h a n for shorter delays. At a n i n t e r m e d i a t e delay

[0.51(s/g) 1/2 w h e n the p a r a m e t e r s have their s t a n d a r d values] b o t h m o v e m e n t s reach their limits s i m u l t a n e o u s l y . The total work d o n e b y the two muscles is t h e n greater, as is the speed a t t a i n e d by the ball. T a b l e 6 shows the effects o f v a r y i n g the p a r a m e t e r s . I n every case the two angles reach their limits s i m u l t a n e o u s l y , w h e n the delay is optimal. S t r o n g e r or faster m u s c l e s m a k e faster t h r o w s possible ( c o l u m n s b - e ) b u t a larger initial a n g l e o f r o t a t i o n (i.e. less lateral r o t a t i o n o f the h u m e r u s ) or a h e a v i e r missile r e d u c e s p e e d ( c o l u m n s f a n d g). O p t i m u m delay, b o t h a b s o l u t e a n d relative, is r e d u c e d w h e n the flexor m u s c l e s are s t r o n g e r ( c o l u m n b) or faster ( c o l u m n d) or the missile is h e a v i e r ( c o l u m n g). It is i n c r e a s e d w h e n the r o t a t o r muscles are s t r o n g e r ( c o l u m n c), or faster ( c o l u m n e), or w h e n the initial angle o f r o t a t i o n is larger ( c o l u m n f). TABLE 6

The effects of different parameter values on the behaviour of Model 3. Relative delay means the delay divided by the time at which the ball leaves the hand. Quantities are expressed in dimensionless form as explained in section 2 a

b

c

d

e

f

g

Isometric flexor torque rotator torque

0-7 0.7

I-5 0-7

0.7 1-5

0-7 0-7

0.7 0.7

0-7 0-7

0-7 0-7

Unloaded angular velocity of flexion rotation

8 8

8 8

8 8

20 8

8 20

8 8

8 8

30

30

30

30

30

60

30

Initial angle of rotation Mass of ball and hand

0,011

0-011

0.011

0.011

0.011

0.011

0-050

Optimum delay

0.51

0.25

0.63

0-21

0.67

0.63

0.46

Optimum relative delay

0.55

0.35

0.67

0.30

0.71

0.70

0-38

Speed of ball for optimum delay

8.4

9.5

8.9

7.9

6.1

Bold = Indicates where one parameter has been changed.

10.3

10.1

SIMPLE

MODELS

OF

THROWING

369

The broken line in Fig. 5(b) shows the effect of making the isometric torque of shoulder flexion a function of the angle of flexion: it was multiplied by (1 - 0n/2~), as was done also to obtain the broken line for Model 2, in Fig. 3(b). 6. Discussion

Each of the three models has two muscles, one of which has its main effect on more distal parts of the body than the other. In Model 1 the arm muscle (the more distal in its effect) accelerates only the hand and the missile upward, but the leg muscle (proximal) accelerates the whole body. In Model 2 the elbow muscle (distal) accelerates the forearm and hand forward, but the shoulder muscle (proximal) accelerates the whole arm. In Model 3 both muscles act at the shoulder but the rotator may be regarded as the more distal in its effect because it accelerates only the forearm and hand forward (its effect on the upper arm is merely to rotate it about its long axis), whereas the flexor accelerates the whole arm. In each case the model predicts an optimum delay between activation of the "proximal" muscle and activation of the "distal" one. These simple models, each with just two muscles, help to explain why the numerous muscles involved in real throwing actions tend to be activated in sequence from proximal to distal (Atwater, 1979; Zatsiorsky et al., 1981). It seems possible to explain why the models predict optimum delays, by considering the times available for the contractions of the muscles. Tables 1, 3 and 5 show for all the models that the later distal muscles are activated, the shorter the duration of their activity: they move faster through their range of movement (Model 1) or move the joint through a smaller angular range (Models 2 and 3), and in either case do less work. The Tables also show for all three models that increasing the delay up to a certain limit increases the time available for contraction of the proximal muscle, before it reaches the limit of its motion a n d / o r the missile is released. This implies that it contracts more slowly (exerting more torque) or over a greater angular range, and so does more work. Thus increasing the delay (up to a certain limit) increases the work done by the proximal muscles and decreases that done by the distal ones. The total work done on body and missile is maximized at some intermediate delay. This is not necessarily the optimum, however, since the requirement is to impart as much energy as possible to the missile alone. The optimum delay for Models 1 and 2 is longer than the delay that maximizes total work because the longer delay increases the proportion of the work that becomes kinetic energy of the missile. The point has been made repeatedly, that if a muscle moves a joint through the same angular range in less time it exerts less torque [according to Hill's equation, eqn (1)] and so does less work. It may be asked whether the existence of an optimum delay depends on torques being less when muscles shorten faster. To find the answer, the relationship between torque and angular velocity was virtually eliminated from all three models by giving both muscles exceedingly high unloaded angular velocities, o f 109(g/s)i/2, while giving the standard values to all the other parameters. All three models still gave optimal delays. Model 1 gave a very fiat optimum in the range of

370

R. M c N . A L E X A N D E R

delays that allowed the foot to leave the ground before the missile left the hand, but not before the arm muscle had been activated. Models 2 and 3 gave quite sharp optima at delays just long enough to allow shoulder flexion to proceed through its full angular range of 180°. A proximal to distal sequence of muscle activation is also required for optimal throws by Herring & Chapman's (1988) model of throwing, in which joint torques are independent of angular velocity. It seems to be widely believed that the significance of the sequence of muscle action in throwing, is that it allows energy and momentum to be transferred from segment to segment (see, e.g. Atwater, 1979.) Similarly in a whip, a travelling wave transmits energy and momentum from segment to segment, and the light distal end reaches a much higher peak speed than did the heavier proximal end. The models presented in this paper confirm that considerations of transfer of energy and momentum have a role in determining the optimum sequence of muscle action, making the optimum delays for Models 1 and 2 longer than would maximize total work. However, they also make it clear that the delay affects the time available for the contraction of each muscle, and so the range or speed of its contraction and the work that it does. The work done by the muscles equals the mechanical energy gained by the body and missile together. Optimal sequencing of throwing movements requires a compromise between maximizing this energy and maximizing the proportion of it that goes to the missile. In all the examples that have been presented, the optimal throw is obtained when the proximal muscles are activated before the distal ones. This seems likely to be the case for most real throws, but it is possible to envisage situations in which distal muscles should be activated first. For example, Model 1 gives negative optimal delays (arm muscles activated before leg muscles) when it throws sufficiently large masses. Figure l(b) shows a small positive optimum delay [O.l(s/g) 1/2] when a mass of 0.25M is thrown, but if this is increased to 0.3M the optimum delay becomes -0.2( s/ g) 1/2. The muscles in the models are always fully active or completely inactive. It seems necessary to consider the possibility that an optimal throw might require one of the muscles to be only partially active, for at least part of the time. It would exert less torque than if it were fully activated and would do less work, but this might increase the time available for the action of the other muscle which might thereby be enabled to do more work. Incomplete activation for the whole period of activity is easily simulated in the models by reducing the isometric torque for one of the muscles. The speed given to the missile in throws with very small delays can be increased a little in all three models, by reducing the isometric torque exerted by the distal muscle, but it does not seem to be possible to improve in this way the throw attainable with optimum delay. Optimal throwing probably requires bang-bang control of muscle activation.

REFERENCES ALEXANDER, R. MCN. (1990). Optimum take-of/techniques for high and long jumps. Phil. Trans. R. Soc. Land. B. 329, 3-10.

SIMPLE

MODELS OF THROWING

371

ATWATER, A. E. (1979). Biomechanics of overarm throwing movements and of throwing injuries. Exercise Sport Sci. Reo. 7, 43-85. BALLREICH, R. & KUHLOW, A. (1986). Biomechanik des Kugelstosses. In: Biomechanik der Sportarten (Ballreich, R. & Kuhlow, A., eds) pp. 89-109. Stuttgart: Enke. CAVAGNA, G. A., DUSMAN, B. & MARGARIA, R. (1968). Positive work done by a previously stretched muscle. J. appl. Physiol. 24, 21-32. CLARKE, H. H. (1966). Muscular Strength and Endurance in Man. Englewood Cliffs: Prentice-Hall. DE KONING, F. L., BINKHORST, R. A. VOS, J. A. & VAN'T HOF, M. A. (1985). The force-velocity relationship of arm flexion in untrained males and females and arm-trained athletes. Eur. J. Appl. Physiol. 54, 89-94. DVSON, G. H. G. (1977). The Mechanics of Athletics 7th edn. New York: Holmes & Meier. HAY, J. G. (1973). The Biomechanics of Sports Techniques. Englewood Cliffs: Prentice-Hall. H ERRING, R. M. & CHAPMAN, m. E. (1988). Computer simulations of throwing: optimization of endpoint velocity and projectile displacement. Proceedings 5th Biennial Conference and Symposium, Canadian Society of Biomechanics. pp. 76-77. University of Ottawa. KER, R. F., ALEXANDER, R. MCN. & BENNETT, M. B. (1988). Why are mammalian tendons so thick? J. ZooL, Lond. 216, 309-324. KER, R. F., BENNETT, M. B., BIBBY, S. R., KESTER, R. C. & ALEXANDER, R. MCN. (1987). The spring in the arch of the human foot. Nature. Lond. 325, 147-149. KULIG, K., ANDREWS, J. G. & HAY, J. G. (1984). Human strength curves. Exercise Sport Sci. Rev. 12, 417-466. MENZEL, H. J. (1986). Biomechanik des Speerwurfs. In: Biomechanik der Sportarten (Ballreich, R. & Kuhlow, A., eds) pp. 110-120. Stuttgart: Enke. PLAGENHOEF, S. (1971). Patterns of Human Motion, A Cinematographic Analysis. Englewood Cliffs: Prentice-Hall. TOYASHIMA,S., HOSHIKAWA,T., MIYASHITA,M. & OGURI, T. (1974). Contribution of the body parts to throwing performance. In: Biomechanics IV (Nelson, R. C. & Morehouse, C. A., eds) pp. 169-174. Baltimore: University Park Press. VAUGHN, R. E. (1985). An algorithm for determining arm action during overarm baseball pitches. In: Biomechanics IXB (Winter, D. A., Norman, R. W., Wells, R. P., Hayes, K. C. & Patla, A. E., eds) pp. 510-515. Champaign: Human Kinetics Publishers. WILKIE, D. R. (1950). The relation between force and velocity in human muscle. J. Physiol., Lond. 110, 249-280. WOLEDGE, R. C., CURT1N, N. A. & HOMSHER, E. (1985). Energetic aspects of Muscle Contraction. London: Academic Press. ZATSIORSKY, V. M., LANKA, G. E. & SHALMANOV, A. A. (1981). Biomechanical analysis of shot putting technique. Exercise Sport Sci. Reo. 9, 353-389.

APPENDIX A Subscripts ball e ef fa fl hip k max rot s ua 0 1, 2, 3

relating to the missile and the hand holding it elbow elbow flexor forearm flexion hip knee maximum rotation shoulder upper arm isometric subscripts to distinguish constants from each other

372

R.

MeN.

ALEXANDER

APPENDIX B

Symbols E F g k M m P r 8

T t

W x, y, z 0

energy force gravitational acceleration constants defined by eqn (19) body mass mass of a segment of the body, or of a missile impulse distance of the centre o f mass o f a segment from the proximal joint length of a leg segment torque time work Cartesian co-ordinates angle