modelling, simulation and optimisation of a human vertical jump

10
Journal of Biomechanics 32 (1999) 521 530 Modelling, simulation and optimisation of a human vertical jump T. Spa¨gele!, A. Kistner!, A. Gollhofer",* ! Institute A of Mechanics, University of Stuttgart, 70550 Stuttgart, Germany " Department of Sport Science, University of Stuttgart, 70569 Stuttgart, Germany Received in final form 21 September 1998 Abstract This paper describes an efficient biomechanical model of the human lower limb with the aim of simulating a real human jump movement consisting of an upword propulsion, a flying and a landing phase. A multiphase optimal control technique is used to solve the muscle force sharing problem. To understand how intermuscular control coordinates limb muscle excitations, the human body is reduced to a single lower limb consisting of three rigid bodies. The biomechanical system is activated by nine muscletendon actuators representing the basic properties of muscles during force generation. For the calculation of the minimal muscle excitations of the jump movement, the trajectory of the hip joint is given as a rheonomic constraint and the contact forces (ground reaction forces) are determined by force plates. Based on the designed musculoskeletal model and on the differential equations of the multibody system, muscle excitations and muscle forces necessary for a vertical jump movement are calculated. The validity of the system is assessed comparing the calculated muscle excitations with the registered surface electromyogramm (EMG) of the muscles. The achieved results indicate a close relationship between the predicted and the measured parameters. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Vertical jump movement; Human movement simulation; Optimal control; Dynamic optimisation 1. Introduction The determination of muscle forces during movement is not only essential for a complex analysis of internal loads acting on bones and joints, it also contributes to a deeper understanding of the underlying neural control. Due to the difficulty of measuring muscle forces directly within the living system by means of invasive techniques and due to the circumstance of the mechanically redund- ant arrangement of the actuators, static optimisation (Crowninshield, 1978; Hardt, 1987; Patriarco et al., 1981) or dynamic optimisation (Davy and Audu, 1987; Pandy et al., 1990; Spa¨gele, 1995) have been used to estimate muscle forces during movement. To produce motion of the body, the muscles must be activated by signals from the central nervous system (CNS) via electrical impulses. The electrophysiological excitations produce muscle for- ces, which are related to joint torques by the moment arms of those muscles. These torques cause angular ac- celerations, which are necessary for the calculation of the multibody movement that can be described by gene- *Corresponding author. ralised coordinates of the mechanical system. For com- plex movement simulations, the human system may be described by a model incorporating two distinct sections. The first part describes the behaviour of the actuators during force generation. The second part includes the musculoskeletal system with the mechanical properties of the limbs and the generation of joint torques as a func- tion of the muscle forces. With the muscle excitations as results of an optimal control algorithm (Pandy et al., 1992, 1995) and an appropriate model of the musculo- skeletal system, it is possible to simulate complex human movements and to resolve neural excitations that are responsible for the measured movement pattern. In order to use a multiphase optimal control approach for a satis- factory solution to the simulation problem, biomechani- cal models are necessary which are both accurate and highly efficient. Previous simulations (e.g. Pandy et al., 1990) investigated only the propulsion phase of the jump with the jump height as the performance criterion. The aim of this study is to simulate a measured complex jump movement consisting of an upward propulsion, a flying and a landing phase. The underlying performance cri- terion contains kinematic-based task criteria and neuro- muscular penalty criteria. 0021-9290/99/$ see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 1 4 5 - 6

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Journal of Biomechanics 32 (1999) 521—530

Modelling, simulation and optimisation of a human vertical jump

T. Spagele!, A. Kistner!, A. Gollhofer",*

! Institute A of Mechanics, University of Stuttgart, 70550 Stuttgart, Germany" Department of Sport Science, University of Stuttgart, 70569 Stuttgart, Germany

Received in final form 21 September 1998

Abstract

This paper describes an efficient biomechanical model of the human lower limb with the aim of simulating a real human jumpmovement consisting of an upword propulsion, a flying and a landing phase. A multiphase optimal control technique is used to solvethe muscle force sharing problem. To understand how intermuscular control coordinates limb muscle excitations, the human body isreduced to a single lower limb consisting of three rigid bodies. The biomechanical system is activated by nine muscle—tendon actuatorsrepresenting the basic properties of muscles during force generation. For the calculation of the minimal muscle excitations of the jumpmovement, the trajectory of the hip joint is given as a rheonomic constraint and the contact forces (ground reaction forces) aredetermined by force plates. Based on the designed musculoskeletal model and on the differential equations of the multibody system,muscle excitations and muscle forces necessary for a vertical jump movement are calculated. The validity of the system is assessedcomparing the calculated muscle excitations with the registered surface electromyogramm (EMG) of the muscles. The achieved resultsindicate a close relationship between the predicted and the measured parameters. ( 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Vertical jump movement; Human movement simulation; Optimal control; Dynamic optimisation

1. Introduction

The determination of muscle forces during movementis not only essential for a complex analysis of internalloads acting on bones and joints, it also contributes toa deeper understanding of the underlying neural control.Due to the difficulty of measuring muscle forces directlywithin the living system by means of invasive techniquesand due to the circumstance of the mechanically redund-ant arrangement of the actuators, static optimisation(Crowninshield, 1978; Hardt, 1987; Patriarco et al., 1981)or dynamic optimisation (Davy and Audu, 1987; Pandyet al., 1990; Spagele, 1995) have been used to estimatemuscle forces during movement. To produce motion ofthe body, the muscles must be activated by signals fromthe central nervous system (CNS) via electrical impulses.The electrophysiological excitations produce muscle for-ces, which are related to joint torques by the momentarms of those muscles. These torques cause angular ac-celerations, which are necessary for the calculation of themultibody movement that can be described by gene-

*Corresponding author.

ralised coordinates of the mechanical system. For com-plex movement simulations, the human system may bedescribed by a model incorporating two distinct sections.The first part describes the behaviour of the actuatorsduring force generation. The second part includes themusculoskeletal system with the mechanical properties ofthe limbs and the generation of joint torques as a func-tion of the muscle forces. With the muscle excitations asresults of an optimal control algorithm (Pandy et al.,1992, 1995) and an appropriate model of the musculo-skeletal system, it is possible to simulate complex humanmovements and to resolve neural excitations that areresponsible for the measured movement pattern. In orderto use a multiphase optimal control approach for a satis-factory solution to the simulation problem, biomechani-cal models are necessary which are both accurate andhighly efficient. Previous simulations (e.g. Pandy et al.,1990) investigated only the propulsion phase of the jumpwith the jump height as the performance criterion. Theaim of this study is to simulate a measured complex jumpmovement consisting of an upward propulsion, a flyingand a landing phase. The underlying performance cri-terion contains kinematic-based task criteria and neuro-muscular penalty criteria.

0021-9290/99/$ — see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 1 4 5 - 6

Fig. 1. Block diagram showing the calculation of the vector fM of the muscle forces as a function of the normalised muscle excitations u, muscle lengthslM and muscle velocities vM.

2. Methods

2.1. Muscle model

The muscles are the active elements controlled by thecentral nervous system. With their capacity of activeshortening they can apply forces to the skeleton. Onemajor problem in modelling large-scale motions is to findappropriate descriptions of the actuators with accuratemathematical models. In the literature, several very de-tailed and complex muscle models are described whichcan roughly be divided into two classes (Audu and Davy,1985). The molecular models describe the processes offorce generation on the sarcomer level; they range incomplexity from single-molecular models (Huxley, 1957,1974) to more complex approaches with many specificparameters (Hill et al., 1975; Zahalak and Ma, 1990).Despite their apparent beneficial effects on the repres-entation of the biophysical processes, these models, how-ever, do not appear to be practical for simulations oflarge-scale movements (Winters and Stark, 1987).

More tractable approaches, single-element models, de-scribe the dynamical properties of force generation of themuscles. They proved to be practical, especially in nu-merical applications (Spagele, 1998). The muscle forces ofthe musculoskeletal system are represented by the vectorfM. This vector is a function of the normalised muscleexcitations u, the muscle lengths lM and the muscle vel-ocities vM (Fig. 1). The muscle—tendon element (MTE)contains the parameters of the activation dynamics andthe contraction mechanics of the muscle. The activationdynamics is considered to be related to the release ofcalcium ions from the sarcoplasmatic reticulum de-scribed by the vector c. Their subsequent function form-ing crossbridges to enable contraction, can be expressedby the active states a of the muscles (Audu and Davy,1985; Winters, 1990). In parallel to the MTE, a dampingelement (DE) and a parallel elastic element (PEE)

describe the passive behaviour of the muscle. The PEEcan be represented by a exponential spring, whereas theDE is modelled as a linear damper. The resultant forcesfM in the muscles comprise the forces of the three ele-ments. They are the sum of the active muscle forces fMTE,the passive muscle forces fPEE and fDE:

fM (u, lM,, vM)"fMTE (a(u), lM,, vM)#fPEE( lM )#fDE(vM).(1)

Based on the activation dynamics and the normalisedmuscle excitations u it is possible to calculate the vectora of the active states (Fig. 1). The muscle mechanicsdescribing the dependencies of active muscle forces onthe muscle lengths lM and muscle velocities vM contain theHill force—velocity relationship and the tension—lengthrelationship (Winters, 1990). The active muscle forceFMTEi

3fMTE produced by the ith muscle group is cal-culated as the product of the maximum isometric forceF*40.!9i

and the factors fADi

(ui), f

TLi(lMi) and f

FVi(vM

i), which

depend on excitation and mechanical conditions of forcegeneration (Spagele, 1998) of the active muscle force.

FMTEi

(ui, lM

i, vM

i)"F*40

.!9ifADi

(ui) f

TLi(lMi) f

FVi(vM

i). (2)

The muscle model contains the basic active and passivecharacteristics of force generation (Fig. 1). It can be usedin particular to calculate the muscle forces for the simula-tion and optimisation of human movements.

2.2. Multibody model

Mechanical systems and objects such as vehicles, ro-bots or even human bodies, can be modelled precisely asmultibody systems. Their mechanical properties can becharacterised by motion as a rigid body and they can beconstrained by elements such as joints, bearings andsupports. Force elements may be realised by springs,dampers or motors (Schiehlen, 1991). Mechanical prin-ciples (e.g. principle of d’Alembert in Lagrange’s version)

522 T. Spagele et al. / Journal of Biomechanics 32 (1999) 521—530

Fig. 2. Block diagram showing the calculation of the generalised coordinates q, the muscle lengths lM, and the muscle velocities vM as a function of themuscle forces fM.

allow the equations of motion for a holonomicmultibody system to be derived (Pfeiffer, 1989; Schiehlen,1986) under these conditions. Subsequent equations canbe arranged in a specific form and results for a planarmultibody motion according to

M(q)q#gc(q, q5 , t)"gc(q, q5 , t), (3)

where M(q) is the matrix of inertia

M(q)"nB+i/1

(miJT

TiJTi#HzSiJT

RiJRi), (4)

gc(q, q5 , t) the vector of generalised gyroscopic forces

gc(q, q5 , t)"nB+i/1

(miJT

Tia6 si#Hz

SiJT

RiaNi), (5)

and ge(q, q5 , t) the vector of generalised applied forces

ge(q, q5 , t)"nB+i/1

(JTTif e

i#JTRiTe,z

Si). (6)

For the calculation of Eqs. (3)—(6), the Jacobian matricesJTi

(translation) and JRi

(rotation) as well as the segmentmasses m

i, the moments of inertia Hz

Si, the local acceler-

ations a6Si

and angular accelerations aNi, the applied forces

f%iand the torques T e,z

Si acting on the centre of masses arenecessary for all n

Bmodelled rigid bodies. The equations

of motion (3) are constructive parts of the model of themusculoskeletal system (Fig. 2) to determine the general-ised coordinates q, the muscle lengths lM and the musclevelocities vM as a function of the muscle forces fM. Theanatomical arrangement of the muscles is essential for

the evaluation of the driving forces and the joint torquesacting on the single-body segments.

For the muscle force generation, there exists a subtleinteraction between the required normalised excitations,the muscle lengths and the muscle velocities. With know-ledge of the muscle forces and their moment arms, it ispossible to calculate the active joint torques t! contribu-ting with the passive joint torques t1 to the joint torquest responsible for the active movement of the multibodysystem (Fig. 2). The functions of the moment arms d, thepassive joint torques t1, the muscle lengths lM and musclevelocities vM depend on the generalised coordinates q andtheir first time derivatives q5 . They are given in detail inSpagele (1998).

2.3. Complete state space model

The kinetics of the skeleton as well as the excita-tion—contraction dynamics of the muscles are determinedby a set 2f#n

Mdifferential equations of first order

x5 "f (x, u, t), x(t0)"x0, x3IR2f`nM, (7)

where f is the degree of freedom of the mechanicalsystem and n

Mis the number of muscles modelled in the

musculoskeletal system. For state space notation, thevector q of the generalised coordinates is described by x

1,

the vector q5 of the generalised velocities by x7, and the

normalised calcium ion concentrations c of the musclesby the vector xc . The state vector x describing the com-plete biomechanical behaviour of the human body is

T. Spagele et al. / Journal of Biomechanics 32 (1999) 521—530 523

represented by first-order differential equations (ODE):

x5 "

x51

x57

x5 c

"

xv

M!1(xp)(ge(xp, xu, xc, t)!gc(xp, xv, t))

C(u, xc)(8)

where the vector C contains the excitation functions ofthe modelled muscles and describes the characteristicproperties of all muscles during activation. With the statevector x of the musculoskeletal system and the dynamicoptimisation approach, it is possible to compute theoptimal controls u* responsible for a measured humanmovement with minimal cost functional. For a pre-scribed type of movement, the optimisation problemrequires the control functions, the model design para-meters and the phase separation times of the distinctmovement phases to minimise a scalar cost functional I,which may be described by a combination of Mayerterms / and Lagrange terms L (Jansch et al., 1989).

I"/0#m+j/1A/j#P

j

tj~1

LjdtB. (9)

Additionally to differential equations (8) the solution isrequired to satisfy separable multipoint boundary condi-tions at the initial time of the movement as well as at thephase time points, the pure parameter constraints andthe path constraints of the multiphase optimal controlproblem. The subdivision of the entire movement intom-independent phases requires certain conditions at theirrespective phase transients, which describe the connec-tions of the single phases.

3. Application

As an application, a vertical one-legged jump exerciseis investigated to simulate human movement. This jumpmovement can be divided into an upward propulsion,a flying and a landing phase. In order to restrict thecomplexity of the musculoskeletal system, the activelower limb is described separately (Fig. 3). This approachrequires both knowledge of the three-dimensional kinet-ics of the body (Fig. 4) and the ground reaction forces (seeAppendix, Fig. 8). It allows, however, a detailed descrip-tion of the reduced biomechanical system using morecomprehensive musculoskeletal models. For simulationof the vertical jump, the reduced model consists ofa single lower limb represented by three rigid bodies. Thesegments of the thigh, the shank and the foot are joinedtogether by frictionless joints. They are allowed to bemoved in the sagital plane by pure rotation around thehip, knee and ankle joint. The biomechanical system ofthe leg is activated by nine muscle—tendon actuators(Fig. 3). The force generation of each muscle group isdescribed by the three-element muscle model.

With respect to the rheonomic constraints (measuredhip point acceleration) of the hip joint, the mechanicalsystem of the lower limb possesses f"3 degrees of free-dom for the plane motion. The kinetics of the multibodysystem can be described by the independent absoluteangles q

1, q

2and q

3of the thigh, shank and foot, repre-

sented by the vector q of the generalised coordinates

q"[q1

q2

q3]T. (10)

For the description of the entire biomechanical system ofthe jump movement, 15 first-order differentials are neces-sary incorporating six equations of motion and nineactivation equations of the modelled muscle groups.

In the approach described in this manuscript, the opti-mal control problem is defined to find the muscle excita-tions u minimising the scalar cost functional I . This costfunctional contains not only the weighted differencesbetween the measured and calculated kinematic data ofthe leg motion, but also the weighted sum of all neuro-muscular values of muscle excitations

I"I,*/

#I/%630

"

3+j/1

wj1P

tj

tj~1

((u!u/0.

)2#(t!t/0.

)2

#(m!m/0.

)2) dt#3+j/1

wj2P

tj

tj~1 A9+i/1

u2i Bdt. (11)

The equations of motion for the jump movement are setup applying the principle of d’Alembert in Lagrange’sversion. They can be specified by three non-linear differ-entials of second order (see Appendix A). The biologicalpart of the system consists of the muscle model and thearrangement of the nine actuators within the musculo-skeletal system. Together with the muscle excitationsu these are inputs to the moments t acting at the joints,which contain the active t! and the passive t1 joint mo-ments responsible for the body motion.

t"t!#t1"

¹1

¹2

¹3

"

¹!1#¹1

1¹!

2#¹1

2¹!

3#¹1

3

. (12)

The active joint moments t! are the sum of the productsresulting from the muscle forces fM"[FM

1,2, FM

9]T and

their respective moment arms d. For the jump motion,the active joint moments at the hip, knee and ankle jointcan be as follows:

ta"

FM1du

1#FM

2du

2!FM

4du

4!FM

5du

5FM2dt2#FM

3dt3!FM

5dt5!FM

6dt6!FM

7dt7

FM9dm9!FM

7dm7!FM

8dm8

, (13)

where d0

idescribes the moment arm of the ith muscle

group acting at the joint with the relative angle 03Mu, t,mN. The moment arms d of the muscles, the passive mo-ments t1 of the system and the muscle kinetics lM and

524 T. Spagele et al. / Journal of Biomechanics 32 (1999) 521—530

Fig. 3. Schematic representation of the measured human vertical jump with one leg and the musculoskeletal model of the lower limb activated by ninemuscle groups.

vM are functions of the relative angles. The detailed func-tions and the anatomic parameters of the subjects investi-gated in this study are given in Spagele (1998). The anglesu, t and m (Fig. 4) describe the relative position of thesegments and can be calculated from the absolute coordi-nates q of the mechanical system

u"q1, t"q

1!q

2, m"q

3!q

2#

n2. (14)

4. Results and discussion

The aim of the study was to simulate a complex jumpmovement consisting of an upward propulsion, a flyingand a landing phase. The used performance criterion

contained kinematic based as well as neuromuscularpenalty criteria (Zajac and Winters, 1990). Previoussimulations (e.g. Pandy et al., 1990) of jumping haveconcentrated only on the propulsion phase with the jumpheight as the performance criterion. The basic result ofthe dynamic optimisation is shown in Fig. 5. The differ-ences between calculated and measured trajectories areconsiderably small, according to the definition of the costfunctional.

This paper focuses on a description of how the bio-mechanical system controls the single muscle groups inorder to perform the movement with minimal muscleexcitations. The normalised muscle excitations of themodelled musculoskeletal system correspond to the ele-ments of the control vector u of the optimal controlalgorithm. The results of the simulated vertical jump

T. Spagele et al. / Journal of Biomechanics 32 (1999) 521—530 525

Fig. 4. Schematic representation of the relative angles u, t and m actingon the lower limb.

movement are summarised in Fig. 6. The calculated timetrajectories of the minimal muscle excitation functionsare presented for each muscle group separately. Thesecomputed optimal controls can be verified and validatedby comparison with the actual muscle action potentialsregistered with surface electromyography (Fig. 7).

The entire jump has been divided into three distinctphases and this is also reflected in the determined muscleexcitations. In the first part of the jump, a plantar move-ment of the foot must be performed for the accelerationof the body in vertical direction. This motion is producedprimarily by the gastrocnemius (muscle group 7) and thesoleus (muscle group 8). Therefore, strong excitations inthese two groups with maximal excitation rates at theend of the first phase are characteristic (Fig. 6). As tibialisanterior (muscle group 9) counteracts the plantar motionof the foot this muscle does not seem to be activated

Fig. 5. Comparison of measured (dotted line) and calculated (solid line)trajectory of the hip angle, the knee and the ankle angle of the verticaljump.

during the first part of the jump. During vertical acceler-ation, extension of the shank is produced by simulta-neous action of the rectus femoris (muscle group 2) andthe vastus group (muscle group 3). This is indicated byrather similar trajectories characterised by increasing ex-citation rates.

During the first phase, extension of the thigh can beproduced by the gluteus group (muscle group 4) and thehamstring group (muscle group 5). The hamstrings and thebiceps femoris c. b. (muscle group 6) counteract the re-quired motion of the shank. They are therefore not con-trolled during the entire jump movement. The dynamicsof the thigh is primarily generated by the gluteus group, asshown by an increasing excitation during the first jumpphase. During flight phase only few muscles show con-siderable activation amplitudes. The largest torque actsin the knee joint at the beginning of the landing phase.Therefore, intense activation of the second (rectusfemoris) and third (vastus) muscle group must be ex-pected for deceleration of the body after landing. This isalso achieved by the ability of the muscles to develophigh forces during eccentric muscle actions. Due to theproperties of the contractile apparatus in enhancing theforce output during active lengthening the calculated

526 T. Spagele et al. / Journal of Biomechanics 32 (1999) 521—530

Fig. 6. Trajectories of the calculated control values u of the nine modelled muscle groups necessary for the vertical jump movement with minimalmuscle excitations.

time histories of the muscle excitations are reduced in thedeceleration phase as compared with the amplitudes cal-culated for the vertical acceleration of the body. Evenpreactivation can be observed in the plantar flexors in theshort time interval before landing. Functionally preac-tivation is necessary to stiffen the joint complexes beforemechanical loading (Gollhofer et al., 1992).

The human musculoskeletal system is characterised bynon-linear skeletal and excitation—contraction dynamics,high dimensionality of the state vector and a great num-ber of control functions. Therefore, dynamic optimisa-tion problems of human movement simulation are very

time expensive. The calculation time of the presentedjump movement depends on the initial conditions as wellas on the determination of the biomechanical parameterswhich must be frequently adapted to the problem duringthe simulation process.

The optimisation of a single upward propulsion hasalready been published in the literature (Pandy et al.,1990). In the present study, however, a multi-phasemotion consisting of an upward propulsion, a flying anda landing phase is simulated. To solve the describedoptimisation problem, it is necessary to register the con-tact forces and the kinetics of the jump movement. The

T. Spagele et al. / Journal of Biomechanics 32 (1999) 521—530 527

Fig. 7. Experimental EMG activities of the leg muscle groups measured during the jump movement.

presented biomechanical model has been validated withempirical data from the excitation of the entire muscularchain of the lower limb during the multi-phasejump performance, including the flying and landingphase.

For the simulation of human movements, it is neces-sary to use efficient and attractive models of the humanmuscular system. The developed muscle model with itsthree-elements is especially attractive for the computa-tional investigation of large-scale movements. The valid-ity of the simulations can be assessed by comparing thecalculated muscle excitations with originally registeredsurface EMG of the muscles of interest. The achievedresults indicate a close relationship between the predictedand the measured parameters.

Appendix A

A.1. Multibody dynamics

The equations of motion describing the mechanicalbehaviour of the skeleton are given below. The differen-tial equations of the multibody system can be generallywritten as

M(q)q#g#(q, q5 , t)"g%(q, q5 ,t).

The inertia matrix M of the system

M"

M11

M12

M13

M12

M22

M23

M13

M23

M33

,

528 T. Spagele et al. / Journal of Biomechanics 32 (1999) 521—530

M11"H

S1#m

1z21#(m

2#m

3) l21,

M12"(m

2z2#m

3l2) l1cos(q

1!q

2),

M13"m

3l1(z

3cos(q

1!q

3)#z

4sin(q

1!q

3)),

M22"H

S2#m

2z22#m

3l22,

M23"m

3l2(z

3cos(q

2!q

3)#z

4sin(q

2!q

3)),

M33"H

S3#m

3(z2

3#z2

4).

The vector g# of the generalised gyroscopic forces of thesystemg#"

(m2z2#m

3l2)l1sin(q

1!q

2)qR 2

2#m

3l1(z

3sin(q

1!q

3)!z

4

cos(q1!q

3))qR 2

3#(m

1z1#(m

2#m

3)l1)(cos(q

1)a

y(t)#sin(q

1)a

z(t))

(m2z2#m

3l2)l1sin(q

2!q

1)qR 2

1#m

3l2(z

3sin(q

2!q

3)!z

4

cos(q2!q

3))qR 2

3#(m

2z2#m

3l2)(cos(q

2)a

y(t)#sin(q

2)a

z(t))

m3l1(z

3sin(q

3!q

1)#z

4cos(q

3!q

1))qR 2

1

#m3l2(z

3sin(q

3!q

2)#z

4cos(q

3!q

2))qR 2

2#m

3((z

3cos(q

3)!z

4sin(q

3))a

y(t)#(z

3sin(q

3)#z

4cos(q

3))a

z(t))

.

The vector g% of the generalised external forces of thesystem

g%"

Fy(t)l

1cos(q

1)#((F

z(t)!(m

2#m

3)g)l

1!m

1gz

1) sin(q

1)#¹

1!¹

2

Fy(t)l

2cos(q

2)#((F

z(t)!m

3g)l

2!m

2gz

2) sin(q

2)#¹

2!¹

3

Fy(t)(l

3cos(q

3)!(L(t) sin(q

3))#F

z(t)(l

3sin(q

3)#L(t) cos(q

3))

!m3g(z

3sin(q

3)#z

4cos(q

3))#¹

3

.

The mechanical definitions are:ay(t) measured horizontal acceleration of the hip

pointaz(t) measured vertical acceleration of the hip point

Fy(t) measured horizontal ground reaction force

Fz(t) measured vertical ground reaction force

g gravity constantH

Simoment of inertia of segment i about its centreof mass

li

length of segment i¸(t) measured distance from the ankle joint to the

ground reaction forcesm

imass of the segment i

qi

generalised coordinates of the mechanical sys-tem

t time¹

itorque acting at the joint i

zi

distance from distal point to centre of mass ofsegment i

Fig. 8. Measured trajectories of the ground reaction forces Fy(t) and

Fz(t) of the vertical jump movement and the distance ¸(t) from the ankle

joint to the ground reaction forces.

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Hardt D.E., 1978. Determining muscle forces in the leg during normalhuman walking — an application and evaluation of optimizationmethods. Journal of Biomedical Engineering 100, 72—78.

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