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Journal of Biomechanics 39 (2006) 170–176

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Optimization model predictions for postural coordination modes

Luc Martina,�, Violaine Cahoueta, Myriam Ferrya, Florent Fouqueb

aLaboratoire Sport et Performance Motrice EA 597, UFRAPS Universite Joseph Fourier, 38041 Grenoble cedex 9, FrancebLaboratoire Motricite-Plasticite INSERM/ERIT-M 207, UFRSTAPS, Universite de Bourgogne BP 27877, 21078 Dijon, France

Accepted 13 October 2004

Abstract

This paper examines the ability of the dynamic optimization model to predict changes between in-phase and anti-phase postural

modes of coordination and to evaluate influence of two particular environmental and intentional constraints on postural strategy.

The task studied was based on an experimental paradigm that consisted in tracking a target motion with the head. An original

optimal procedure was developed for cyclic problems to calculate hip and ankle angular trajectories during postural sway with a

minimum torque change criterion. Optimization results give a good description of the sudden bifurcation phase between in-phase

and anti-phase postural coordination modes in visual target tracking. Transition frequency and predicted effects of environmental

and intentional constraints are also in line with experimental observations described in existing literature. In particular, these

investigations pointed out that postural planning process can be related to the minimization of a dynamic cost criterion with an

equilibrium constraint. In conclusion, the optimization technique is well suited for the prediction of postural modes of coordination

and seems to offer many opportunities for better comprehension of neuromuscular movement control.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Dynamic optimization; Postural coordination; Trajectory planning; Minimum torque change criterion

1. Introduction

A classic assumption on movement planning is thathuman beings perform a motion according to certainoptimal criteria, i.e. movement control can be related toa problem of cost function minimization. Over the years,several researchers have used such optimization princi-ples for different biomechanical motion predictions, forexample, arm pointing (Uno et al., 1989; Alexander,1997; Wada et al., 2001), manual lifting (Lin et al., 1999;Chang et al., 2001), pedaling (Kautz and Hull, 1995;Raasch et al., 1997) and walking (Anderson and Pandy,2001). There are various approaches to investigatinghuman planning strategies by which trajectories areselected depending on the choice of a cost functional.Some authors have suggested a planning strategy at a

e front matter r 2004 Elsevier Ltd. All rights reserved.

iomech.2004.10.039

ing author. Tel.: +333 76 63 50 83;

44 69.

ess: luc.martin@ujf-grenoble.fr (L. Martin).

kinematic level with the minimum jerk model (Flash andHogan, 1985). A more reliable approach, when compar-ing with experimental data, is to relate movementstrategies to intrinsic parameters with a human bodydynamics-based criterion. For example Uno et al. (1989)suggested that arm-pointing movements were organizedsuch that the time integral of the square sum ofmuscular joint moment changes is minimal. Many otherauthors focused on joint moments since they are oftenassociated with physical stress or metabolic energy(Kautz and Hull, 1995; Lin et al., 1999; Alexander,1997). These models are objectively and experimentallyexaminable because of their quantitative predictions butnone have been used to examine postural strategy instanding.Human stance requires the control of different body

segments that can be organized in many ways. Nashnerand McCollum (1985) have described two preferentialpostural strategies: hip and ankle strategies. In the firststrategy, the body oscillates around the ankle joint

ARTICLE IN PRESS

.

O1

CoP Fx

FyO1

M1

d

XCoPXb

Xa

O2

yρ

x 1θ

2θ

G1

G2

(a) (b)

G0

Fig. 1. Two-dimensional human body system (a) and mechanical

characteristics of the feet (b).

L. Martin et al. / Journal of Biomechanics 39 (2006) 170–176 171

whereas in the second, postural equilibrium is regulatedby tilting the torso forward or backward. This definitionis confusing because it is widely accepted that hipstrategy may include ankle rotation and several studieshave reported that hip rotation was often apparent inankle strategy (Horack and Nashner, 1986; Nashner andMcCollum, 1985). These observations suggested that itwas not the involvement of these different joints thatdetermined the type of postural organization adopted,but rather the way in which the movements of thedifferent joints were coordinated. Accordingly, Bardyet al. (2002,1999) analyzed the relative phase (Frel)between hips and ankles in standing participants whowere stressed to track a visual target with the head.Participants were found to switch suddenly from in-phase mode (the ankles and the hips moving simulta-neously in the same direction, Frel close to 01) to anti-phase mode (the ankles and hips rotating simultaneouslyin opposite directions, Frel close to 1801) as a result ofconstraints acting on the system. Newell (1986) distin-guished three categories of constraints which influencedobserved coordination modes: environmental con-straints (i.e. support surface properties), intrinsic con-straints (i.e. height of the center of mass, length of thefeet) and intentional constraints (or task constraint, i.e.the instruction to track target motion).The aims of this paper are to investigate the ability of

dynamic optimization model to predict in-phase andanti-phase postural modes of coordination and toevaluate influence of two particular environmental(length of support base) and intentional (amplitude ofhead displacement) constraints on postural strategy.The studied task was based on the experimentalparadigm widely used in literature (Bardy et al., 2002;Oullier et al., 2002; Marin et al., 1999a) consisting intracking a target motion with the head. An originaloptimal procedure was developed for cyclic problems tocalculate hip and ankle angular trajectories duringpostural sway with a minimum torque change criterion.

2. Method

2.1. Biomechanical model

During postural sway, the human body was modeledas a two-dimensional system comprising three rigidsegments representing the feet (segment 0), the lowerlimbs (segment 1) and the upper part of the body(segment 2) which were linked by two articulations,ankle and hip joints (Fig. 1) modeled as frictionlesshinges. The feet were assumed to remain at rest, instatic contact with the floor. Assuming that theligaments and bone-on-bone contact forces were negli-gible, the inverse dynamics of the body-system wasused to expressed muscular net joint moments at ankle

and hip joints as a function of segmental kinematics ateach time t:

M1 ¼ ðI1 þ I2 þ l21m2 þ 2m2l1r2 cos y2Þ€y1

þ ðI2 þ m2l1r2 cos y2Þ€y2 � ðm2l1r2 sin y2Þ

� ð_y2

2 þ 2_y1 _y2Þ þ gðm1r1 cos y1

þ m2ðl1 cos y1 þ r2 cosðy1 þ y2ÞÞÞ; ð1Þ

M2 ¼ ðI2 þ m2l1r2 cos y2Þ€y1 þ I2 €y2 þ ðm2l1r2 sin y2Þ_y2

1

þ gðm2r2 cosðy1 þ y2ÞÞ; ð2Þ

where, for i ¼ 1; 2 : Mi was the net muscular torque atjoint i, yi; _yi and €yi were respectively angular position,velocity and acceleration of joint i as defined in Fig. 1.For each segment i ¼ 1; 2 : li and mi were, respectively,the length and the mass, Gi the center of mass, ri thedistance between Gi and joint i and Ii, the moment ofinertia with respect to the axis ðOi; zÞ:The maintenance of balance during postural sway

depends on the horizontal anterior–posterior position ofthe center of pressure (CoP), which can be expressed as afunction of the body kinematics. The static equilibriumof the feet segment led to

XCoP ¼ðM1 � F xd þ m0r0gÞ

F y

; (3)

where Fx and Fy were respectively horizontal andvertical ground reaction force components.As regards the body pluri-articulated system, Euler’s

equations could relate Fx and Fy to the rate of change ofthe respective horizontal and vertical linear momenta of

ARTICLE IN PRESSL. Martin et al. / Journal of Biomechanics 39 (2006) 170–176172

the whole system at each time t (Cahouet et al., 2002):

Fy ¼ ðm1r1 þ m2l1Þ cos y1 €y1 þ m2r2 cosðy1 þ y2Þð€y1 þ €y2Þ

� ðm1r1 þ m2l1Þ sin y1 _y2

1

� m2r2 sinðy1 þ y2Þð_y1 þ _y2Þ2

þ ðm0 þ m1 þ m2Þg; ð4Þ

Fx ¼ � ðm1r1 þ m2l1Þ sin y1 €y1 � m2r2 sinðy1 þ y2Þ

ð€y1 þ €y2Þ � ðm1r1 þ m2l1Þ cos y1 _y2

1

� m2r2 cosðy1 þ y2Þð_y1 þ _y2Þ2: ð5Þ

2.2. Optimal calculation for hip and ankle angular

trajectories

2.2.1. Minimum torque change criterion

The optimization model assumed that the humanbody performed postural sway by minimizing timederivatives of net joint moments. Considering a stea-dy-state cyclic two-joint movement, postural activitywas formulated into a mathematical optimizationproblem with an objective function specifying theminimization of the sum of net torque changes:

C ¼1

2

ZTc

0

dM21

dtþdM2

2

dt

� �dt; (6)

where Tc was the period of the movement correspondingto the period of the target motion. The task related tothe studied movement was two-fold: first, the head hadto move in phase with the target (frequency ¼ fT) andwith the same amplitude aT and secondly, the body-system had to maintain balance. This could be expressedby adding constraints in the optimization search. Thehead amplitude displacement was imposed to equaltarget one

l1 cos y1Tc

2

� �� �� cos y1ð0Þð Þ

� �

þ l2 cos y1Tc

2

� �þ y2

Tc

2

� �� �� cos y1ð0Þ þ y2ð0Þð Þ

� �

¼ aT : ð7Þ

The other constraint imposed in the optimization searchwas the maintenance of balance during oscillation. Tokeep the CoP antero-posterior position (3) of the subjectwithin the base of support and feet flat on the floor, theconstraint could be expressed as

X apXCoPðtÞpXb; (8)

where Xa and Xb were the boundary positions on theCoP in respectively backward and forward directionswith respect to the ankle joint to prevent toe-off orheel-off.

The associated non-linear constrained optimizationproblem consisted in finding cyclic joint displacementsyi¼1;2 such that criterion (6) was minimized andconstraints (7) and (8) were satisfied.

2.2.2. Time discretization and angular trajectory models

Discretizing the cycle time scale of the angulartrajectories, the infinite dimensional of the minimizationproblem (6) was converted into a finite one:

Cn ¼Xn

i¼1

dM21

dtðtiÞ þ

dM22

dtðtiÞ

� �Dt; (9)

where n was the number of discrete time ti and Dt wasthe time discretization step such that

Dt ¼Tc

ðn � 1Þ:

Considering that angular displacements in steady-statepostural sway were cyclic time functions, they could bedescribed by a N-harmonic Fourier series:

yiðtÞ ¼a0i

2þ

XN

k¼1

aki cosðko0tÞ þ bki sinðko0tÞð Þ; (10)

where aki and bki were amplitude coefficients and thefundamental frequency o0 ¼ 2p=Tc:Consequently, the optimization problem could be

reformulated using (10) as follows: find the 2(2N+1)coefficients of the Fourier series representing y1 andy2; such that Cnða01; . . . ; aN1; b11; . . . ;bN1; a02; . . . ; aN2;b12; . . . ;bN2Þ was minimal and (7) and (8) wererespected. This non-linear constrained optimizationwas solved with the sequential quadratic programming(SQP) method (Boggs and Tolle, 1996) using Matlab(Math Works Natick, MA).

2.2.3. Input data values

Specific input data values were chosen to performsimulations of postural sway during head tracking task.Typical male proportions were used to determineanthropometrical parameters (Paı and Patton, 1997).According to experimental observations of posturalsway movements (Marin et al., 1999a), the amplitudespectra of hip and ankle position data present very littleactivity above the second harmonic. Consequently, thenumber of harmonics in the Fourier series was set toN ¼ 3 in the optimization search. All simulations wereperformed with the same initial solution in theoptimization search, where Fourier series’ coefficients(10) were set to zero. The periods of target oscillationswere varied with an increment of 0.5 s between 5 s and1 s (corresponding to frequencies of between 0.2 and1.0Hz) to examine hip and ankle coordination evolu-tion. Focusing on the transition phase, the correspond-ing bifurcation frequency calculation was refined with aprecision of 0.01Hz.

ARTICLE IN PRESS

1.48

1.5

1.52

1.54

Ank

le a

ngle

(rad

)

0.1

p an

gle

(rad

)

0

0.04

0.08

0.12

Hea

d po

sitio

n (m

)

L. Martin et al. / Journal of Biomechanics 39 (2006) 170–176 173

The influence of environmental (length of supportbase) and intentional (amplitude of head displacement)constraints was investigated using three different ampli-tudes of target motion (0.07, 0.1, 0.13m) and threedistinct lengths of support base (0.07, 0.1, 0.13m).For each trial, angular displacement and net muscular

power were estimated at the hip and ankle joints. Netjoint powers, calculated at each joint using scalarproduct between muscular net joint moment andangular velocity, demonstrated concentric muscularaction when positive and eccentric muscular actionwhen negative. Head motion, body center of gravity(CoG) and CoP displacements were also evaluated foreach trial.

-0.1

Hi

0.5 Hz 0.3 Hz

0

0.05

0.1

0 50 100 150 200

Cycles (%)

Hor

izon

tal p

ositi

on (m

)

CP 0.5 Hz CG 0.5Hza CP 0.3Hz CG 0.3 Hz

(a)

(c)

(b)-10

-5

0

5

10

Net

join

t pow

er (

W)

ankle 0.5 Hz hip 0.5 Hz ankle 0.3 Hz hip 0.3 Hz

Fig. 2. Two typical trials (aT ¼ 0:1m) of in-phase (0.3Hz) and anti-phase (0.5Hz) coordination modes: horizontal head position and ankle

and hip angles (a), net joint power at hip and ankle joints (b) and CoG

and CoP horizontal displacements (c) are plotted as a function of the

cycle proportion.

3. Results

Two distinct stable modes of coordination appearedin the predicted joint trajectories with increasingoscillation frequencies. Typical optimal results, corre-sponding to in-phase (f ¼ 0:3Hz) and anti-phase(f ¼ 0:5Hz) coordination patterns are illustrated inFig. 2. In in-phase mode coordination, hip and anklejoints flexed in the first part of the cycle and extendedsimultaneously in the second one. Conversely, the anti-phase mode was characterized by hip flexion and ankleextension in the first part of the cycle and by hipextension and ankle flexion in the last one. Net jointpower time histories (Fig. 2b) exhibited one eccentricand one concentric muscular action at ankle joint, andtwo eccentric and two concentric muscular actionsappeared at hip articulation regardless of the trial. Atthe ankle joint, the appearance chronology of distinctmuscular actions was inverted between in-phase andanti-phase coordination modes, while it remained un-changed at the hip joint. Peak-to-peak CoG displace-ment along the antero-posterior axis (Fig. 2c) wasgreater at low frequency (0.3Hz), and conversely,CoP oscillated with greater amplitude at the higherfrequency (0.5Hz). For both trials, CoG displacementspresented lower amplitude than CoP displacements.The mean antero-posterior value of CoG positionobserved at 0.3Hz was located in front of the onepredicted at 0.5Hz.The bifurcation frequency corresponding to a sudden

transition between coordination modes was estimated at0.41Hz (Fig. 3b). In-phase coordination mode corre-sponding to relative phase equal to zero were observedfor low target motion frequencies (0.2–0.4Hz). An anti-phase pattern of coordination was obtained for higherfrequencies (0.42–1Hz), where the coordination of theankle was 1801 out of phase with respect to the hip joint.The simulation trend exhibited angular displacementamplitudes at ankle joint that were more reducedthan at the hip, regardless of frequency (Fig. 3a).

During in-phase coordination and for increasing fre-quencies from 0.2 to 0.35Hz, angular displacementamplitude decreased at hip joint, while it slightlyincreased at ankle joint. This trend was inverted at botharticulations for higher frequencies ranging from0.35Hz to the bifurcation frequency. During anti-phasecoordination, raising motion frequency resulted in anincrease in angular displacement amplitude at bothjoints.The model predictions are sensitive to modified

environmental and/or intentional constraints (Fig. 4).In general, observed transition frequency increased withincreasing base of support length, and decreased as head

ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

Ang

ular

am

plitu

de (r

ad)

Ankle Hip

(a)

0

180

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Frequencies (Hz)

Rel

ativ

e ph

ase

(°)

(b)

Fig. 3. Evolution of (a) hip and ankle angular motion amplitude and

(b) relative phase evolutions with head motion frequencies increasing

from 0.2 to 1.0Hz. Bifurcation frequency (0.41Hz) is marked with a

vertical dotted line. These simulations were conducted with parameters

values of base of support and head motion amplitude both set at 0.1m.

0.1

0.3

0.5

0.05 0.07 0.09 0.11 0.13 0.15

Base of support (m)

Bif

urca

tion

freq

uenc

y (H

z)

(0.07 m) (0.1m) (0.13 m)Head motion amplitude

Fig. 4. Bifurcation frequencies plotted as a function of base support

length and head motion amplitude.

L. Martin et al. / Journal of Biomechanics 39 (2006) 170–176174

motion amplitude was extended, except for a single trialcorresponding to a base of support length of 0.13m anda head displacement amplitude of 0.07m.

4. Discussion

The constrained optimization model we have devel-oped, provided realistic predictions of postural swaymovements during head tracking task.Considering the formulation of the optimization

problem, the chosen criterion, which involves a dynamicplanning control (Uno et al., 1989) satisfactorilyreproduced postural experimental observations. Moreprecisely, the use of minimum torque change criterion to

predict cyclic postural sway movements, which are oflow metabolic cost, was adequate. The formulation ofthe criterion suggests that the planning strategy can berelated to some extent, to the minimization of the centralmotor command variations. Therefore, this criterion,combined with a pertinent equilibrium constraint, mayprovide suitable predictions of all postural tasks, whileassociated energy expenditure remains restrained.Traditionally, joint angular velocities and accelera-

tions are expressed as a function of angular positionsusing a finite difference method (Chang et al., 2001).This method is convenient but can lead to unrealisticpredictions because it does not ensure that angularkinematics are continuously differentiable functions.Some alternatives are to use spline curves (Lo andMetaxas, 1999) or polynomial approximations (Wadaet al., 2001; Lin et al., 1999) for angular displacements.Considering that angular displacements in steady-statepostural sway are cyclic time functions, the use ofFourier series to approximate joint angular trajectory isconvenient because it ensures that angular positions andvelocities are continuously differentiable functions.Furthermore, it avoids enforcing supplementary con-straints to ensure periodic displacements and allowsbetter convergence of the optimization process.This study has proved the ability of the optimization

model to predict two different modes of coordination inpostural sway for varying frequencies with a suddentransition phase. We calculated an in-phase coordina-tion between hip and ankle joints for low motionfrequencies and anti-phase coordination for higherfrequencies. These results are consistent with observa-tions related to similar experimental paradigms, i.e.head tracking, found in existing literature (Bardy et al.,2002; Marin et al., 1999a,b; Oullier et al., 2002).Examining the optimization process, it appears thatthe sudden bifurcation emerges from both equilibriumconstraint and cost minimization. For low frequencies(below 0.35Hz), the predicted CoP displacement whichminimizes the cost functional is naturally inside base ofsupport boundaries and equilibrium constraint is notactive in the optimization search. The predicted in-phasecoordination is the less expensive strategy with regard tothe chosen criterion. Increasing oscillation frequencyleads to augmented CoP displacement amplitude, whichimperils balance. Consequently, the equilibrium con-straint becomes active in the minimization search. Inthese conditions, optimal process do not predict the lessexpensive strategy but the one that can ensure equili-brium conditions while minimizing the cost functional.The activation of equilibrium constraint in the optimalsearch also leads to a modification in angular amplitudeevolution at hip and ankle joint, as observed in Fig. 3 forfrequency values comprised between 0.35Hz andbifurcation frequency. For frequencies above 0.41Hz,anti-phase coordination mode appears because, in these

ARTICLE IN PRESSL. Martin et al. / Journal of Biomechanics 39 (2006) 170–176 175

conditions, balance cannot be maintained anymoreusing in-phase strategy.Observation of typical trials of in-phase and anti-

phase coordination modes (Fig. 2b) showed that thefour-component pattern of net muscular power presentat hip joint was maintained in each mode of coordina-tion. As regards the ankle joint, the observed bi-phasicmuscular action pattern was inverted between in-phasemode (eccentric followed by concentric actions) andanti-phase mode (concentric followed by eccentricactions). Consequently, the ankle extensor groupremained agonist for the whole flexion and extensionmotion, regardless of the coordination mode. Theseobservations are strongly dependant on the base ofsupport characteristic values used in the model and haveto be confirmed by further experiments.For all studied target velocities, CoP displacements

remained in phase with respect to CoG movement. Thistrend has already been widely observed in quiet standing(Winter et al., 1998). The optimal process also predictedopposite influence of motion velocity on CoP and CoGdisplacement amplitudes. Peak-to-peak CoG oscilla-tions were reduced as head motion frequency wasincreased (Fig. 2c). This result is in accordance withthose obtained by Pozzo et al. (2002) during whole bodypointing movements for different execution velocities.The sensitivity of the predictive model to modified

environmental and intentional constraints was exploredby manipulating the value of the length of the supportbase and the amplitude of the head motion. In general,increased base support length resulted in increasedbifurcation frequency estimates (Fig. 4). This result is inaccordance with the experimental tendency described byMarin et al. (1999a). The optimal model also predicteddecreasing transition frequency for increased headmotion amplitude (Fig. 4). This tendency is qualitativelyin line with the experimental observations of Marin et al.(1999b); for a given motion frequency, in-phase coordi-nation mode measured with a small target motionamplitude was changed into anti-phase coordinationwith larger amplitude. It can be observed in Fig. 4 thatresults from a single trial, corresponding to a supportlength of 0.13m and head motion amplitude of 0.07m,were inconsistent with these general tendencies. For thistrial, the optimal model predicted exaggerated trunkbending. This does not correspond to a realistic positionthat can be observed in this kind of visual target trackingexperimental paradigm and can be explained by the factthat only horizontal head displacements were constrainedin the optimal search. Adding some constraints in theoptimal process to better express intentional constraintand limit head vertical motion should improve thereproduction of the experimental task.In general, this study brings to light the role of the

CoP position, which seems to be of crucial importancein the postural coordination planning. Indeed, it appears

from our results that this parameter is of primeimportance in the modification of the postural strategy.Moreover, since pressure receptors are located under thefeet, proprioceptive information reflecting CoP positioncan be directly available to the system and, hence,involved in the motor command elaboration.In conclusion, the study has proved that the

optimization technique is well suited for the predictionof postural coordination. The original formulationdeveloped for cyclic angular displacement is efficientand could be successfully applied to the simulation of allperiodic tasks such as pedaling or walking. In general,these investigations pointed out that postural planningprocess can be related to a dynamic cost criterionminimization coupled with an equilibrium constraintexpressed by confining CoP displacement. Particularly,optimal predictions are consistent with the hypothesisthat multi-segment coordination in postural controlemerges from interactions between constraints imposedby the support surface and those imposed by thetask. Possibilities for future studies include the exam-ination of the influence of modified intrinsic constraints(as limited moment strength) on coordination modesand the simulation of other postural tasks, i.e. posturalperturbation effect observations (Runge et al., 1999) oroscillating plate-form paradigm (Buchanan and Horak,1999; Corna et al., 1999; Ko et al., 2001). Theoptimization model approach also seems to offeropportunities to allow a better comprehension ofneuromuscular movement control.

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