The Biomechanics of an Overarm Throwing Task: a Simulation Model Examination of Optimal Timing of Muscle Activations

J. theor. Biol. (2001) 211, 39}53 doi:10.1006/jtbi.2001.2332, available online at http://www.idealibrary.com on

The Biomechanics of an Overarm Throwing Task: a Simulation Model Examination of Optimal Timing of Muscle Activations AASIM G. CHOWDHARY*

AND

JOHN H. CHALLIS-?

*Applied Physiology Research ;nit, Biomechanics ¸aboratory, ¹he ;niversity of Birmingham, Birmingham B15 2¹¹, ;.K. and -Biomechanics ¸aboratory, 39, Recreation Building, ¹he Pennsylvania State ;niversity, ;niversity Park, PA 16802-3408, ;.S.A. (Received on 26 June 2000, Accepted in revised form on 4 April 2001)

A series of overarm throws, constrained to the parasagittal plane, were simulated using a muscle model actuated two-segment model representing the forearm and hand plus projectile. The parameters de"ning the modeled muscles and the anthropometry of the two-segment models were speci"c to the two young male subjects. All simulations commenced from a position of full elbow #exion and full wrist extension. The study was designed to elucidate the optimal inter-muscular coordination strategies for throwing projectiles to achieve maximum range, as well as maximum projectile kinetic energy for a variety of projectile masses. A proximal to distal (PD) sequence of muscle activations was seen in many of the simulated throws but not all. Under certain conditions moment reversal produced a longer throw and greater projectile energy, and deactivation of the muscles resulted in increased projectile energy. Therefore, simple timing of muscle activation does not fully describe the patterns of muscle recruitment which can produce optimal throws. The models of the two subjects required di!erent timings of muscle activations, and for some of the tasks used di!erent coordination patterns. Optimal strategies were found to vary with the mass of the projectile, the anthropometry and the muscle characteristics of the subjects modeled. The tasks examined were relatively simple, but basic rules for coordinating these tasks were not evident.  2001 Academic Press

1. Introduction Human throwing is a task for which the performance criterion is generally well de"ned, for example throw as far as possible. Several authors have attempted to gain insight into various aspects of the inter-muscular coordination required to maximize performance in throwing tasks (e.g. Putnam, 1993; Feltner & Dapena, 1986; Alexander, 1991; Herring & Chapman, 1992). The techniques utilized range from direct dynamic ?Author to whom correspondence should be addressed. E-mail: [email protected] 0022}5193/01/130039#15 $35.00/0

simulations to the analysis of kinetics and/or kinematics recorded during simple throwing tasks. A number of authors have suggested that a proximal to distal (PD) sequencing of muscle activation in throwing is necessary to achieve maximal projectile speed or distance (e.g. Gowitzke & Milner, 1980; Feltner & Dapena, 1986; Alexander, 1991; Herring & Chapman, 1992; Putnam, 1993). Most experimental studies employ a small range of projectile masses and assume that the subject is utilizing an optimal strategy, this is di$cult to verify experimentally. An experimental investigation of the e!ects of musculo-skeletal characteristics on optimal  2001 Academic Press

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A. G. CHOWDHARY AND J. H. CHALLIS

strategies is further confounded by the di$culty of exerting independent control over the parameters of interest. For example, experimentally it is di$cult for a subject to change a well learnt movement pattern, or for that subject to increase by a precise amount the strength of one muscle group only. It is for such reasons that recent simulation studies of throwing have been performed (e.g. Alexander, 1991; Herring & Chapman, 1992). Simulation studies also have their limitations arising from the assumptions made when formulating the models. Using a model, Alexander (1991) demonstrated that for three di!erent types of throws a PD sequence of muscle activations was optimal when maximizing projectile energy at release. De Lussanet & Alexander (1997) showed that if biarticular muscles are included in the model then for certain throws a PD sequence is not optimal. In both of these studies the timing of activation of a pair of muscles was systematically varied to "nd optimal performance. With only two variables it is possible to "nd the optimal solution by simply plotting the data. If there are more than two variables such a graphical approach is less feasible. To allow for more subtle variations in muscle activation patterns, which means more than two variables must be optimized, in the present study a numerical approach was adapted to "nd optimal performance. This approach permitted two additional mechanisms to be explored: muscle deactivation and moment reversal. During the throws once a muscle group has been activated it may be advantageous to later deactivate the muscle group. For example, deactivation of a proximal muscle group may permit the distal muscle group to act for a longer period of time and therefore increase the work done on the projectile. Moment reversal requires that the moment at a joint changes direction, for example changing from an elbow extending moment to an elbow #exing moment. Such a mechanism may, for example, permit the transfer of energy within the system resulting in the angular velocity of the wrist joint increasing. Both of these mechanisms require the computation of additional timing parameters for each modeled muscle group. In throwing to hit a target the importance of the timing of muscle activations has been

highlighted using models of the neuro-musculoskeletal system (e.g. Calvin, 1983; Chowdhary & Challis, 1999). This study employed subject speci"c mathematical models of the neuromusculo-skeletal system to examine throws for maximum projectile energy or maximum distance thrown. It was the intention of this study to further scrutinize the general validity of the PD sequencing of muscle activations during a simple throwing task, and to provide an understanding of how throwing performance is dependent upon the interaction of the model's subsystems by analysing the e!ects of changes made to them. These changes involve alterations in muscle characteristics, system inertial properties, and the timing of muscle activations. 2. Methods 2.1. THE MODEL

A model was developed to simulate an overarm throwing task involving the forearm and hand, con"ned to the parasagittal plane, and actuated by the elbow extensors and wrist #exors. The model is essentially the same as the model of Chowdhary & Challis (1999), but is used here to examine di!erent tasks, which are outlined in the section on the model controller. The details of the model are presented under the following subheadings: model equations of motion, model of muscles, model controller, and model parameters. The model is described in more detail in Chowdhary & Challis (1999). 2.2. MODEL EQUATIONS OF MOTION

The model consisted of two segments, the forearm and hand, with two joints, the elbow and wrist (see Fig. 1). The hand was considered to hold the ball until the instant of release. For this system the equations of motion represent a coupled set of second-order ordinary di!erential equations, which can be written as qK "M(q)\ (M !v (q, qR )!G(q)), (

(1)

where qK is the vector of angular accelerations, M(q) the inertia matrix, which is a function of the masses, moments of inertia, lengths (¸1, ¸2), and center of mass locations (r1, r2) of the segments,

41

COORDINATION OF HUMAN THROWING

TABLE 1 ¹he inertial properties of the body segments for the two subjects Subject 1 Subject 2 Subject height (m) Subject mass (kg)

FIG. 1. A two-degree-of-freedom link-segment model of the arm. Center of mass location ( ); projectile ( ); q is the  orientation of the forearm, q is the orientation of the hand,  g is the gravitational "eld vector, ¸1 is the length of the forearm, ¸2 is the length of the hand, r1 is the distance from the elbow joint center to the forearm's center of mass, and r2 is the distance from wrist joint center to the hand's center of mass.

M the vector of muscle model generated joint ( moments, v (q, qR ) the vector of centrifugal and Coriolis terms, which is a function of the masses, lengths, and center of mass locations of the segments and G(q) is the vector of gravity terms, which is a function of the masses, lengths, and center of mass locations of the segments. The initial kinematics of the system were that the arm was stationary, with full elbow #exion and wrist extension. The equations of motion of the system, given the moments acting at the joints, were integrated numerically using a "fthorder adaptive step-size Runge}Kutta algorithm (Press et al., 1992). The e!ects of air resistance were assumed to be negligible for the throws and therefore not included in the simulation model. Throw distance was measured along the ground from an origin in line with the elbow joint center, which was assumed to be 1 m above ground level. The inertial parameters of the model segments were determined by modeling them as a series of geometric solids (Hatze, 1980), with the density of these solids taken from Chandler et al. (1975). The dimensions of these shapes were obtained by taking measurements on two experimental subjects, whose resulting inertial parameters are presented in Table 1.

1.74 77.55

1.76 83.65

Forearm Length (¸1) (m) Location of center of mass (r1) (m) Mass (kg) Moment of inertia (kg m)

0.271 0.114 1.316 0.0790

0.270 0.113 1.117 0.0701

Hand Length (¸2) (m) Location of center of mass (r2) (m) Mass (kg) Moment of inertia (kg m)

0.17 0.056 0.483 0.0036

0.15 0.044 0.353 0.0027

Note*locations of center of mass are given from proximal end of segments, moments of inertia are about an axis through the segmental center of mass.

2.3. MODEL OF MUSCLES

Muscle}tendon models (MTMs) representing the net action of the elbow extensors and wrist #exors produced the moments at the joints. The output of each muscle model is a function of the models' active state, moment-angle, and moment angular velocity properties M "A(u, t) M(h) f (hQ ), (

(2)

where M is the moment at a joint produced by ( the MTM, A the active state of the MTM, u the control signal to the muscle, t the time, M(h) the maximum isometric moment the MTM can produce at a joint angle h and f (hQ ) is the normalized moment}angular velocity function at joint angular velocity hQ . The MTMs were allowed to be in one of two states, active (u"1) or inactive (u"0), with a time delay enforced between the change from either of these states to the MTM achieving the desired active state. The moment}angle relationship for each joint, accounting for both the force}length properties and the joint angle} moment arm relationship of the muscles, was modeled using a polynomial of varying order.

42

A. G. CHOWDHARY AND J. H. CHALLIS

FIG. 2. (a) The wrist #exor moment}angle relationship for the two subjects. (b) The elbow extensor moment}angle relationship for the two subjects. (c) The wrist #exor normalized moment}angular velocity relationship for the two subjects. (d) The elbow extensor normalized moment}angular velocity relationship for the two subjects. Note that the data for subject 1 ), and subject 2 by the ( ). is represented by the (

The normalized moment}angular velocity relationship was described based on the classic force}velocity equation of Hill (1938). Further details of the MTMs are provided in Chowdhary & Challis (1999). Subject-speci"c muscle model parameters were determined for two experimental subjects using a procedure similar to that described in Challis & Kerwin (1994), and for this model described in detail in Chowdhary & Challis (1999). In brief, it was possible via a series of maximal tasks to determine the parameters describing the muscles of the experimental subjects. For these tasks an optimization algorithm, Go!e et al. (1994), was used to iteratively adjust model parameters until they provided a good match between subject produced joint moments and motion, and model

joint moments and motion. The subjects provided informed consent prior to participation in the experimental protocols, which had been approved by the Institutional Review Board. Fig. 2 shows the models of the moment}angle curves and moment}angular velocity curves for the two subjects. Note that the two subjects could achieve di!erent joint angles for full elbow #exion and wrist extension. The shape of the moment}angle curves for the wrist #exors and elbow extensors were similar to those presented by Delp et al. (1996) and Kulig et al. (1984), respectively. The curves for the moment}angular velocities for the two joints are re#ective of the data presented by Wilkie (1950) and Winters and Stark (1985). The activation times are quantitatively similar to those presented in Winters & Stark (1985).

COORDINATION OF HUMAN THROWING

For a set of simulations which employed moment reversal it was necessary to model the elbow #exors. Due to the nature of the movement this muscle group acted eccentrically. It was not possible to use the procedure described in Chowdhary & Challis (1999) to parameterize this aspect of the model. Sensitivity analysis showed that simulation results were not sensitive to the properties of the model of the elbow #exors. Therefore, the models of the elbow #exors were given the same properties for each subject as the model of their extensors. The normalized moment}angular velocity curve for eccentric contractions was assumed to rise linearly to a level of 1.3 times the maximal isometric moment the MTMs could produce as the velocity of elbow joint extension decreased from zero to !3 rad s\. At angular velocities less than !3 rad s\ the value of the curve remained at a plateau of 1.3 times the maximum isometric moment. These characteristics were based on the data available in the literature for the eccentric behavior of the elbow #exors in vivo (e.g. Winters & Stark, 1985; Dowling et al., 1995). The validity of this component of the model is examined in the Discussion. 2.4. MODEL CONTROLLER

The motion of the model was controlled by specifying the activations to the MTMs. The tasks were maximal, and it was assumed that the MTMs were either activated or not, and if they were activated the neural control signal to the MTM was attempting to make it maximally active. The controller had to specify the times of activation, time of ball release, and for some simulations time of deactivation of each of the MTMs. The model was used to examine a number of di!erent tasks. For each task an objective function was written which mathematically described the task to be performed. The simulated annealing algorithm of Go!e et al. (1994) was used to "nd the time instants of MTM activations and ball release which maximize the objective functions. The search space was limited so physiologically infeasible joint con"gurations were not examined. Each task was designed with the hope of providing some insight into the biomechanics and motor control of throwing.

43

The "rst series of simulations investigated throwing for maximal distance, a task common in many sporting activities. The objective function was ;"xR ) tof"distance thrown, NPMH

(3)

where ; is the objective function to be maximized, xR the horizontal velocity of the projecNPMH tile at release and tof the projectile's time of #ight, which is given by yR #(yR  #2 g release height NPMH , (4) tof" NPMH g where yR is the vertical velocity of the projectile NPMH at release, release height the height above ground that projectile is released and g the acceleration due to gravity (9.81 m s\). The simulations were conducted with projectile masses varying from 50 g to 1.5 kg (0.05, 0.1, 0.2, 0.35, 0.5, 0.65, 0.8, 1, 1.25, and 1.5 kg). The instants of muscle activation and ball release had to be found which maximized the objective function for each of the projectile masses. This procedure was also repeated after increasing the output of the maximum isometric moment}angle equations for the elbow extensors and wrist #exors by 10%. The in#uence of deactivating the elbow extensors on performance was also examined, therefore four timing instants had to be found for each of the projectile masses for the models of both the subjects. The second series of simulations investigated throwing which imparts the greatest amount of kinetic energy to the projectile, the objective function was ;" m3 (xR  #yR  ), NPMH NPMH 

(5)

where m3 is the projectile mass. These simulations were performed for the same range of projectile masses as the "rst objective function, and with the 10% increase in output of the maximum isometric moment}angle equations for the elbow extensors and wrist #exors. The in#uence on performance of deactivating the

44

A. G. CHOWDHARY AND J. H. CHALLIS

elbow extensors was also examined. This was the same performance criteria used by Alexander (1991), and permits comparisons with his results. These simulations are also amenable to the examination of the relationship between the work done by the muscles and the energy of the projectile at release. The third series of simulations was designed to illustrate any possible advantages of employing moment reversal about the proximal joint in the performance of throwing for maximum distance or energy. Therefore, to achieve moment reversal it was necessary to specify the time of activation of the elbow extensors and wrist #exors, the time of deactivation of the elbow extensors, the time of activation of the elbow #exors, and the time of ball release. The same objective functions were used as for the other simulations but the search was for more control variables. Increasing the number of control variables increased the search space for the simulated annealing algorithm, so simulations for this task were run for a smaller number of projectile masses (0.05, 0.2, 0.5, 0.8, and 1.5 kg). The controller speci"es a sequence of activations and deactivations which generate muscular moments at the joints. Given the moments at the joints the ordinary di!erential equations described in eqn (1) are integrated forwards in time, producing for each integration step the elbow and wrist angles and their angular velocities. This gives a new set of conditions for the muscle model, which subject to these new conditions produces moments at the joints, and the sequence is repeated until the controller speci"ed time of release. The outputs from the simulations are either the distance thrown or the kinetic energy of the ball, and the timings of the MTM activations. For some of the simulations the net positive work performed by the MTMs was computed from





dq # M d(q !q #n), =" M CJ@ CVR  UPGQR DJV   (6) where = the total positive work done by the elbow extensors and wrist #exors, M CJ@ CVR the moment produced by elbow extensors, M the moment produced by wrist #exors UPGQR DJV

and q , q are the orientation of the segments (see   Fig. 1) 2.5. MODEL EVALUATION

To evaluate the model the kinematics of the subjects throwing a tennis ball (mass 0.05 kg) for maximum possible distance were compared with the kinematics of the models simulations of these throws. For subject 1 the model predicted a maximum throw of 8.42 m, the actual maximum distance thrown by this subject was 8.38 m. For subject 2 the model predicted a maximum throw of 6.42 m, and the actual maximum distance thrown by the subject was 6.31 m. Therefore, although the model was a simpli"cation of the complex human musculo-skeletal system, it accurately represents movements of the type to be simulated in this study. Further details of the model evaluations are provided in Chowdhary & Challis (1999). 3. Results 3.1. INTRODUCTION

The results of the simulations are presented in the following sub-sections: throwing to maximize distance achieved; throwing to maximize projectile kinetic energy; and throwing employing moment reversal. For the model simulations the delay to onset of wrist #exor activation is given in both actual time and in dimensionless form. The actual delay is the time elapsed after the activation of the elbow extensors until the wrist #exors are activated, a negative value indicates activation of wrist #exors prior to the activation of elbow extensors. The dimensionless or relative delay is the actual delay divided by the time from the initiation of movement to the time at which the projectile was released. A value of zero for the relative delay would indicate simultaneous activation of wrist #exors and elbow extensors, a negative value would indicate activation of wrist #exors prior to the activation of elbow extensors, while a value of one would indicate that the wrist #exors were not recruited. 3.2. THROWING TO MAXIMIZE DISTANCE

In throws for maximum distance a PD sequencing of the muscle activations was observed

45

COORDINATION OF HUMAN THROWING

FIG. 3. The e!ects of projectile mass on actual and relative delays when throwing for maximum distance. (a) and (b) show the actual and relative delays for subject 1, and (c) and (d) show the actual and relative delays for subject 2. Note that the ( ) represents the results for the original model parameter set, and the ( ) the results when the potential output of the moment-angle relationship has been increased by 10%.

for the models of both subjects for the range of projectile masses investigated (Fig. 3). The optimal delays between wrist #exor activation and elbow extensor activation were di!erent for each subject. However, as the projectile mass was increased, the optimal delay also increased at a steady rate for both subjects. The relative delay followed similar patterns for both subjects, as the mass was increased there was an initial decrease in optimal relative delay for both subjects followed by a continuous increase. Increasing the maximum isometric moment of the MTMs by 10% resulted in shorter actual delays for both subjects' at all projectile masses. The relative delays followed a similar pattern after the strength increase as they had before, but they were not the same. There was no enhanced performance for the models of either subject if deactivation of the elbow extensor muscles was permitted.

3.3. THROWING TO MAXIMIZE PROJECTILE KINETIC ENERGY

Unlike the simulation results for throwing for maximum distance the sequencing of muscle activations to maximize projectile energy did not follow the same pattern for both subjects (see Fig. 4). To maximize projectile energy for subject 1 the muscle activations did not always require a PD sequencing. For all projectile masses investigated above 0.35 kg the model representing subject 1 found a negative delay to be optimal. The shape of the curves showing the e!ect of projectile mass on optimal delays was not at all similar to the curves expressing the same relationships for subject 2. For subject 1, for lower projectile masses an increase in muscle strength produced slightly smaller delays, for some of the larger masses the optimal delay was slightly greater. This can in part be explained by the fact

46

A. G. CHOWDHARY AND J. H. CHALLIS

FIG. 4. The e!ects of projectile mass on actual and relative delays when throwing for maximum projectile energy at release. (a) and (b) show the actual and relative delays for subject 1, and (c) and (d) show the actual and relative delays for subject 2. Note that the ( ) represents the results for the original model parameter set, and the ( ) the results when the potential output of the moment-angle relationship has been increased by 10%.

that the original delays were negative for these masses, so stronger wrist #exors would result in the wrist reaching its limit of #exion earlier if the delay was maintained at the same level or reduced. This would reduce the relative contribution of the elbow extensors to the throw. However, not all the negative delays are increased in magnitude which means that maintaining the same relative work contribution for the involved muscle groups is not always the most appropriate strategy. A co-ordination pattern which reduces the net work that may be done by the muscles, but results in an advantageous redistribution of angular velocities, can increase the energy transferred to the projectile. In the simulations for subject 2 the optimal approach was always one in which a PD sequencing was preferred (see Fig. 4). Increasing the strength of the MTMs by 10% resulted in lower actual and relative delays. This is to be expected because as the proximal muscles are now

stronger, it will take less time for the elbow joint to reach its limit of extension. Therefore, the delay to recruitment of the wrist #exors must be decreased if their contribution to the throw is not to be reduced. Analyses of the e!ects of varying PD delay times for the subjects modeled showed that the optimal delays for maximizing projectile energy were indeed close to the delays which allowed the muscles to do the most work (for an example see Fig. 5). The delays for maximizing muscle work and those for maximizing projectile energy were not always in complete coincidence, more energy was transferred to the projectile at delays slightly longer than those which maximized the work output of the muscles. When deactivation of the elbow extensors MTM was allowed for the model of subject 1 there was increased projectile energy for all masses tested greater than or equal to 0.8 kg (Table 2). Deactivation of the proximal muscle group

47

COORDINATION OF HUMAN THROWING

before projectile release may enable the distal muscle groups to perform for a slightly longer period of time in a manner which increases the net work done by the muscles or; it may result in a more e!ective transfer of energy to the projectile. Analysis of the work done by the muscles

demonstrated that the increase in projectile energy was due to a more e!ective transfer of the work done by the muscles to the system. The actual work done by the muscles was reduced a little by deactivating the elbow extensors (see Table 2) but the energy transferred to the projectile was increased, albeit by a very small amount (on average a little greater than 0.5%). The simulations for subject 2 demonstrated that no bene"t in performance could be obtained for this subject's model by deactivation of MTMs. 3.4. MOMENT REVERSAL THROWS

FIG. 5. E!ect of delay on work done by muscle groups and energy transferred to projectile of mass 0.1 kg during throws to maximize projectile energy. (a) for subject 1, and (b) for subject 2.

The only throw where moment reversal (MR) increased throwing distance was for subject 2 with the lightest projectile (mass 0.05 kg). The optimal activation sequencing required that the elbow extensors be de-recruited before recruitment of the elbow #exors, and the elbow #exors were activated after the wrist #exors (Table 3). Inspection of the joint angular velocities for the simulations of subject 2 demonstrate that at the time of projectile release, moment reversal resulted in a much greater wrist joint angular velocity (50.0 rad s\ with MR, 24.3 rad s\ without MR) while the elbow joint angular velocity was very close to zero when MR was employed. The angular velocity of the wrist joint rose to much higher values later in the simulation, reaching almost 90 rad s\, while the angular velocity of the elbow fell to approximately !8 rad s\. This meant that the projectile velocity was also greatly increased later in the simulation, but by this time

TABLE 2 ¹he e+ects of elbow extensor deactivation on throws to maximize projectile kinetic energy for subject 1 Projectile mass (kg)

0.8 0.8* 1.0 1.0* 1.25 1.25* 1.5 1.5*

Work done by elbow extensors (J)

Work done by wrist #exors (J)

Total positive work done by muscles (J)

Projectile energy (J)

36.39 34.79 39.41 37.56 42.43 40.87 45.05 44.84

14.19 14.35 14.40 14.68 15.12 15.51 15.16 15.45

50.58 49.14 53.81 52.24 57.55 56.38 60.21 59.29

30.34 31.17 33.24 34.03 35.17 36.02 35.35 36.10

*Indicates that deactivation was employed.

48

A. G. CHOWDHARY AND J. H. CHALLIS

TABLE 3 Distance thrown, timing of muscle activations and release for subject 2 throwing a 0.05 kg projectile, either with or without moment reversal (MR) Throw type

Distance thrown (m)

Wrist #exors activation time (s)

Elbow extensors deactivation time (s)

Elbow #exors activation time (s)

Release time (s)

6.83 6.42

0.0547 0.0610

0.0628 *

0.0773 *

0.0933 0.0892

Moment reversal No moment reversal

TABLE 4 E+ect of MR (moment reversal) on work done during throws which impart maximum kinetic energy to the projectile. Note that only the data for those throws where projectile energy was greater for the MR throw compared with the throws with no MR are presented Projectile mass (kg)

Projectile energy with MR (J)

Projectile energy no MR (J)

Total positive work no MR (J)

Total positive work with MR (J)

Total work (positive#negative) with MR (J)

Subject 1 0.05 0.20 0.50

5.77 17.19 24.22

3.69 13.57 23.23

29.34 36.18 42.42

21.55 25.02 34.26

27.97 30.01 37.41

Subject 2 0.05 0.20

5.02 10.34

2.55 8.98

12.15 23.97

9.23 21.94

13.67 24.89

the angle of release which could be produced directed the projectile towards the ground therefore giving a much shorter throw. The models of the subjects were used to evaluate whether MR could increase projectile energy compared with simulations without MR (Table 4). These simulations were run for "ve ball masses only (0.05, 0.2, 0.5, 0.8, and 1.5 kg). Subject 1 increased the energy given to the projectile using MR for three projectile masses (0.05, 0.2, and 0.5 kg), compared with optimal performances without MR. The greatest increases were seen for the smaller projectile masses. As the positive work done by the muscles was reduced in all cases, it is evident that the increases in projectile energies were achieved by a redistribution of joint angular velocities. For subject 1, even when the negative work done by the elbow #exors was accounted for, the net muscle work done when

using MR was less than the net positive work done without MR. In simulations for subject 2 MR was only able to increase the energy transferred to two of the projectile masses tested (0.05 and 0.2 kg). An energy analysis showed that when moment reversal was utilized the task was performed with greater e$ciency (Table 4). The net positive work done by the muscles was reduced in all cases. Work is done by the elbow #exors acting eccentrically about the elbow joint, this is negative work, with the elbow #exors acting to brake the extensor motion of the elbow. 4. Discussion A proximal to distal (PD) sequence of muscle activations was seen in many of the simulated throws but not all. Under certain conditions

COORDINATION OF HUMAN THROWING

moment reversal produced a longer throw and greater projectile energy, and deactivation of the muscles resulted in increased projectile energy. Therefore, simple timing of muscle activation does not fully describe the patterns of muscle recruitment which can produce optimal throws. The results also demonstrate that changing both the mass of the projectile and the strength of the subjects required new timings of muscle activations to achieve optimal performance. 4.1. OPTIMAL THROWS

In throws for maximum distance a PD sequencing of the muscle activations was observed, with di!erent optimal delays between wrist #exor activation and elbow extensor activation for each subject. This is not surprising as the subjects had di!erent anthropometric and muscle properties. In particular, subject 1 had stronger elbow extensors than subject 2, this enabled this muscle group to do more work, which thus resulted in increased performance. With increasing mass of the projectile optimal delays increased, which must in part be explained by the greater mass projectiles resulting in longer movement times. If the strength of the muscles was changed, then the timings had to be changed, even though the strength increase was the same for both modeled muscle groups. These results indicate the subtlety of the timing required for long throws in terms of muscle activation, and that changes to the system require new timings. Di!erent subjects require di!erent timings even when expressed as a relative delay. In contrast, for some of the projectile masses, throws to maximize ball energy did not employ a PD sequence. When throwing to maximize projectile energy the optimal delay for maximizing projectile speed is close to the delay which enables the net work done by the proximal and distal muscles to be maximized. Too short a delay means that the proximal joint does not reach the limit of its extension thereby reducing the amount of work the larger proximal muscles can contribute to the throw. Too long a delay results in the distal muscles not being able to contribute su$cient work during the throw, consequently reducing the total work the muscles contribute to the task. A similar argument was made by

49

Alexander (1991). In throws for maximum distance, angle and height of release in#uence performance, whereas in throws for maximum projectile energy these two factors are not important. Maybe it is the removal of these constraints which leads to optimal performance throws not requiring a PD sequencing. For some throws to maximize projectile kinetic energy performance was enhanced due to moment reversal. Moment reversal about the elbow joint resulted in a reduction of its angular velocity, but as a consequence of the transfer of energy within the system, the angular velocity of the wrist joint increased. As the combined inertia of the hand and projectile increased, the less the wrist angular velocity increased for a given reduction in elbow joint angular velocity. The results show that for heavier projectiles MR becomes ine!ective in the transfer of energy to the projectile. The simulations suggest that subject 1 is capable of utilizing MR to his advantage for greater projectile masses than subject 2. This is because the elbow extensors of subject 1 can produce a greater moment than those of subject 2, and therefore do more work (see Fig. 5). These properties enabled the elbow to reach higher angular velocities before MR was employed. Alexander (1991) demonstrated for his models of throwing that a PD activation sequence was always employed to produce maximum projectile kinetic energy at release, except for a case with a very high projectile mass. In this study it was shown that a PD sequence is not always optimal. This arose for the customized model parameters of one subject, illustrating that coordination patterns may be very di!erent for individuals with di!erent muscle and anthropometric properties. The study also showed that projectile energy can further be maximized if certain muscles are deactivated, or if moment reversal occurs. De Lussanet & Alexander (1997) demonstrated that if the task was to maximize projectile release velocity, a performance criteria equivalent to maximizing projectile kinetic energy, the presence of biarticular muscles could cause coordination patterns which maximized performance which did not have a PD activation sequence. The present study also showed that a PD sequence does not always produce maximum projectile energy, showing that this phenomenon

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does not only exist because of the presence of biarticular muscles. In the studies of Alexander (1991), De Lussanet & Alexander (1997), and the present study, when maximizing projectile kinetic energy the balls velocity vector at release is the only important factor. The objective function used in this study which examined maximum distance thrown considered three factors, as the distance the ball travels is a nonlinear function of height, angle, and velocity of release. For this objective function distance thrown was always maximized with a PD sequence. However, once activated with this PD sequence, for one condition it was then advantageous to perform moment reversal. Throws performed athletically are normally more tightly constrained than those in this study which maximized projectile energy; therefore, the simulated throws for maximum distance are perhaps more realistic, and as such emphasize the optimality of a PD sequence of muscle activations. 4.2. MODEL ASSUMPTIONS

The model had a number of assumptions, the primary ones are presented along with a justi"cation for making them. Rather than model separately each of the muscles crossing a joint they were lumped together into a single equivalent muscle. This is a common approach for musculo-skeletal modeling (e.g. Alexander, 1991; Wilkie, 1950; Winters & Stark, 1985). If the task had been to elucidate the contributions of individual muscles to the task such an approach would not have been appropriate, but here the net output at a joint was the only concern. The moments produced at the joints caused by the passive structures crossing the joint and joint friction were considered insigni"cant compared with that produced by the muscles. Data examining the passive moments at the elbow (e.g. Hayes & Hatze, 1977) and wrist (e.g. Delp et al., 1996) support this assumption. The model ignored the potential contributions of bi-articular muscles to the throws. There is evidence to suggest that bi-articular muscles can act to transport energy from the proximal muscle groups to the distal joints and consequently enhance performance (van Ingen Schenau, 1994).

The two major wrist #exors are also elbow #exors, and two of the major wrist extensors are also elbow extensors. This generates a complicated situation for the activity analysed, in that when the elbow extensors were active an extensor moment would have been generated at the wrist joint, and when the wrist #exors were active an elbow #exor moment generated. Examination of the moment arms and cross-sectional areas of these wrist #exors and extensors suggests that they produce moments equal in magnitude but opposite in direction at the elbow joint (Amis et al., 1979). Therefore, our analysis is in part justi"ed in ignoring their action, certainly the evaluations of the model seem to justify this assumption. A more complex model would have had to be employed to account for the bi-articular muscles, which would have made both the identi"cation of model parameters, and the interpretation of the model results more di$cult. The subjects have elastic elements, the tendons, in series with the contractile elements of their muscles. It is feasible that storage and subsequent utilization of elastic energy could enhance performance during throwing. The simulations did ignore this aspect, but there are a number of reasons why this factor may not be as relevant for the throws examined here as it maybe for other activities. The energy stored in a tendon depends on the force applied on it, its sti!ness, and the tendon's resting length (Alexander, 1988). The "rst two factors depend on the ratio of a muscle's cross-sectional area to its tendon cross-sectional area (Challis, 2000). Compared, for example, with the gastrocnemius the muscles of the upper body are less suited to the storage of elastic energy (Ker et al., 1988). In jumping, elastic energy storage does not signi"cantly a!ect performance (Bobbert et al., 1996), whereas it can during running (Alexander, 1988). The di!erence between these two tasks is the load acting on the tendons to cause them to stretch, it is much greater in running. During the throws examined here the balls were not of su$cient mass to generate signi"cant amounts of stretch in the tendons. Other models of human throwing have ignored the contribution of series elasticity (e.g. Alexander, 1991), or have concluded that it does not signi"cantly contribute to performance (de Lussanet & Alexander, 1997). The model of the moment}angle

COORDINATION OF HUMAN THROWING

relationship of the muscles used in this study implicitly included aspects of the series elasticity of the tendons, and given the good evaluations of the model series elasticity has probably been adequately accounted for in the simulations presented. The model of activation dynamics was simple, it assumed that the muscles were either becoming maximally active or inactive. The time delays in the model of activation meant that once a muscle was activated or deactivated, it did not immediately become maximally active or inactive (bang}bang). The assumption is that during the tasks examined the timing of the activations of the muscles is crucial, and that once activated performance is maximized by maximally activating the muscles. For high-speed athletic activities muscle models are often assumed to become instantaneously maximally active (e.g. throwing*Alexander, 1991), or there is a rapid rise in activation until the muscles become maximally active (e.g. jumping*Pandy et al., 1990). Most of the MTM parameters were experimentally determined, but some were not estimated in this way. Speci"cally, the model parameters associated with elbow #exor activity and the time constant associated with MTM deactivation (¹dac) were estimated from the literature. Throws employing moment reversal would be dependent on all of these model parameters. The model predicted that moment reversal could increase throw performance with a 0.05 kg ball from 6.42 to 6.83 m for subject 2. To explore the validity of these model components, the subject was given feedback and instructed to perform the throw with moment reversal in a similar way to the model. Under these conditions the subject threw 6.88 m, close to the model determined value. The subjects elbow trajectory had a percentage root mean square di!erence of 1.76% from the models trajectory. These results provide additional vindication of the validity of the model for simulating these throwing tasks. The search algorithm may have found a local maximum not global maximum for some of the simulations. The search algorithm required a starting point at which to start its search. By seeding the algorithm with multiple starting points, over a sequence of runs, it was possible to con"rm that the optimal solutions were the global maxima.

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The simulation model employed in this investigation of throwing was a simpli"ed one; allowing motion of the forearm and the hand in the parasagittal plane only, actuated by muscle} tendon models representing the net action of the elbow extensors, elbow #exors, and wrist #exors. While this is not the preferred style of throwing for most people it is an easily executable motor task. One of the major bene"ts of simulating a simpli"ed throw of this nature was the ability to acquire accurate subject speci"c parameters describing the models anthropometry and musculo-tendinous actuators. However, any conclusions drawn from this simulation study are speci"c to the tasks simulated. The model evaluations (Section 2.5) vindicate the use of the methods and the assumption employed in this theoretical simulation based investigation of throwing. 4.3. IMPLICATIONS

A PD sequence of muscle activation was present for many of the simulations, but not all. Therefore, a simple coordination pattern for these throws was not evident. Throws for maximum distance used a PD sequence, but throws to maximize projectile energy did not necessarily require a PD sequence. For throws for maximum distance a strategy of moment reversal was optimal for one subject for the lightest projectile examined, while it was optimal for both the subjects when throwing to maximize projectile energy for the lighter projectiles examined. The optimal delays were not entirely coincident with the delays that allowed the muscles to do the most work, as sometimes a slightly longer delay allowed a better transfer of energy to the projectile. Except for the lightest projectiles, as the mass of the projectile increased the time delay between elbow extensor activation and wrist #exor activation increased, for the maximum distance throws. While both actual delays and relative delays followed similar patterns for maximum distance throws for the range of projectile masses used for both modeled subjects, one cannot deduce from this that such a general principle would apply for all subjects who could be modeled, although this remains a possibility. The simulations have

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highlighted that the coordination of throwing di!ers between subjects, if their physical properties vary. The two modeled subjects were similar in stature, yet had di!erent muscle properties. It could be anticipated that other subjects could have even greater di!erences in their muscle properties, which may result in di!erent coordination patterns emerging compared with those observed in the present study. In studies of vertical jumping a PD sequence is seen (Van Ingen Schenau, 1989), but it is di$cult to equate jumping with throwing even though both require the subject to generate high velocities. Two potential reasons for di!erences between jumping and throwing will be highlighted. In a jump it is advantageous to have the center of mass of the jumper as high as possible at the instant of take o! by having the lower limb joints fully extended, therefore jumps have a relatively constrained "nal position. In a throw the orientation of the segments at release is not necessarily constrained in the same way. Another potentially important factor is that in moving proximally to distally in jumping the leg segments have lower inertial properties, the same is the case for the upper limb but the inertial properties of the projectile can mean that the e!ective terminal segment can have greater inertia than more proximal segments. If the reasons for di!erences and similarities in coordination patterns in maximal jumping and throwing could be identi"ed, additional insight could be obtained into the coordination of human movement. 4.4. CONCLUSIONS

While there are general patterns of muscular coordination during the simple throwing tasks examined in this study, they do not hold for all situations. The models used were customized to two subjects, and the models of these two subjects did not always use the same muscular coordination. Increase in projectile mass causes a change in muscular coordination. Under certain conditions it was advantageous to deactivate a muscle group once it had been activated, under other conditions it was advantageous to generate moment reversal. The results of the simulations in this study have highlighted aspects of the coordination of throwing, they also serve to highlight

how coordination patterns can be dependent on a subject's physical properties, a feature not highlighted in studies which perform simulations for a single &&typical'' subject.

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HERRING, R. M. & CHAPMAN, A. E. (1992). E!ects of changes in segmental values and timing of both torque and torque reversal in simulated throws. J. Biomech. 25, 1173}1184. HILL, A. V. (1938). The heat of shortening and dynamic constants of muscle. Proc. R. Soc. ¸ond. B 126, 136}195. KER, R. F., ALEXANDER, R. M. & BENNETT, M. B. (1988). Why are mammalian tendons so thick? J. Zool. ¸ond. 216, 309}324. KULIG, K. ANDREWS, J. G. & HAY, J. G. (1984). Human strength curves. Exerc. Sport Sci. Rev. 12, 417}466. DE LUSSANET, M. H. E. & ALEXANDER, R. MCN. (1997). A simple model for fast planar arm movements; optimising mechanical activation and moment-arms of uniarticular and biarticular arm muscles. J. theor. Biol. 184, 187}201. PANDY, M. G., ZAJAC, F. E., SIM, E. & LEVINE, W. S. (1990). An optimal control model for maximum-height human jumping. J. Biomech. 23, 1185}1198.

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PRESS, W. H., TEUKLOSKY, S. A., VETTERLING, W. T. & FLANNERY, B. P. (1992). Numerical Recipes in FOR¹RAN. ¹he Art of Scienti,c Computing. Cambridge: Cambridge University Press. PUTNAM, C. A. (1993). Sequential motions of body segments in striking and throwing skills: descriptions and explanations. J. Biomech. 26(supp. 1), 125}135. VAN INGEN SCHENAU, G. J. (1989). From rotation to translation: constraints on multi-joint movements and the unique action of bi-articular muscles. Human Movement Sci. 8, 301}337. VAN INGEN SCHENAU, G. J. (1994). Proposed actions of bi-articular muscles and the design of hindlimbs of bi- and quadrupeds. Human Movement Sci. 13, 665}681. WILKIE, D. R. (1950). The relation between force and velocity in human muscle. J. Physiol. 110, 249}280. WINTERS, J. M. & STARK, L. (1985). Analysis of fundamental human movement patterns through the use of in-depth antagonistic muscle models. IEEE ¹rans. Biomed. Eng. 32, 826}839.