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Neuroscience Letters 412 (2007) 148–153

Velocity/curvature relations along a single turnin human locomotion

Anne-Helene Olivier ∗, Armel CretualLPBEM, UFRAPS, Universite Rennes 2-ENS Cachan, Avenue Charles Tillon, CS24414 35044 Rennes, France

Received 12 June 2006; received in revised form 23 October 2006; accepted 2 November 2006

bstract

Neuroscientific approaches have provided an important invariant linking kinematics and geometry in locomotion: a power law controls theelation between radius of curvature and velocity of the trajectory followed. However, these trajectories are predefined and cyclic. Consequently,hey cannot be considered as fully natural. We investigate whether this relationship still exists in one unconstrained turn, which can be compared ton everyday life movement. Two different approaches were developed: an intra-individual one along each turn of each trial and an inter-individualne based on a specific instant for which a subject’s trajectory goes through its maximal curvature. Eleven subjects performed turns at three gaitpeeds (natural, slow, fast). The intra-individual approach did not lead to any power law between velocity and curvature along one single trial.

otwithstanding, the inter-individual approach showed a power law between the whole couples “minimal radius of curvature/associated velocity”.hus, the speed/curvature relation is more a “long term” motor control law linked to the turning task goal rather than a “short term” one dealingith trajectory following all the time of the motion.2006 Elsevier Ireland Ltd. All rights reserved.

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eywords: Locomotion; Turning behaviour; Centre of the turn; Power law; Kin

traight-ahead walking has been extensively studied in stan-ardized conditions, such as on a treadmill. Walking under suchonditions remains far from everyday life path, and locomotions a complex function strongly dependent on the desired desti-ation and the presence of obstacles. Indeed, turning is a partf the basic library of motor synergies [3] and Orendurff et al.12] denote that “the biomechanics of changing direction whilealking has been largely neglected despite its obvious relevancy

o functional mobility”. Nevertheless, several studies, which cane divided into two sets, have considered curved trajectories.

In the first set, one (sometimes two) turn has been studied toighlight comportemental laws which define motor invariantsor locomotor trajectories generation. One particular behaviours the control of head and body orientation while steering3,4,7,9]. Head deviation toward the future direction anticipateshe body shift motion with the “go where we look” strategy,

hatever the conditions of vision or speed. This ensures torepare a stable reference frame [5,14] by finding out about theuture direction. The onset of head deviation is initiated at a

∗ Corresponding author. Tel.: +33 2 99141775; fax: +33 2 99141774.E-mail address: anne-helene.olivier@uhb.fr (A.-H. Olivier).

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onstant distance rather than a constant time from the corner14]. Some others studies are interested in the foot placementtrategy [6,13]. Finally, in the computer graphics domain,rogan and Johnson [1] developed a human locomotion modelhich was created with an a priori knowledge on the environ-ent (i.e. obstacles) thanks to pedestrians video observation in

eal conditions (offices).In the second set, human behaviour while steering has been

tudied on cyclic trajectories which were often pre-defined.n these studies [8,16], the aim was to explore the power lawetween velocity and curvature previously demonstrated in handotion [10,18]. This law states that there exists a linear relation

etween velocity and radius of curvature in the logarithmicpace. More precisely, velocity is proportional to the third rootf the radius of curvature. To avoid confusion, let us recall thathis law can be seen as a relation between curvature (C) andngular speed (Ω) or between radius of curvature (R) and linearpeed (V). As, first, R = 1/C and then V = RΩ, a “two-third”/C relation is equivalent to a “one-third” V/R one. Such a

elation has been demonstrated for head motion while a subjectalks along an elliptic path drawn on the floor [16]. Even if theeural origins of this law are still discussed, Vieilledent et al.16] explained it by the role of central nervous system which

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waWrrFor all trials, we considered the exact instant when this point wasreached as the Key Instant of the Turn. Then, by drawing from theCoT, the parallels to the two axes of the turn, we get a quadrant.The beginning and the end of the turn were, respectively, defined

A.-H. Olivier, A. Cretual / Neuro

omputes locomotion strategies according to the shape of theath to follow. In this sense, Hicheur et al. [8] recently suggestedhat this law is more general and that the actual power value ishape dependent.

This power law is a fundamental result since it provides aotion invariant and helps to better understand the processes

overning human locomotion. Nevertheless, this power law hasnly been demonstrated for cyclic and constrained locomotorrajectories. Such trajectories are never encountered in everydayife. In the first set of studies, working on natural paths, theelation between velocity and geometry has never been studied.onsequently, the first purpose of our study is to check whether

uch a relationship between kinematics and geometry still existslong a single turn which is an everyday life motion and inducespeed and curvature variations.

Studying an unconstrained single turn should provide a newpproach to investigate the origin of the velocity/curvature rela-ionship, which could be linked to motor control at two differentevels: a “short term” one, the instantaneous trajectory follow-ng, but also a “long term” one, the global turning task. Indeed,locomotor task could be either completed by following a pre-

omputed trajectory (the main objective of the control being stepy step path planning) or by minimising the difference betweenhe original position and orientation and the desired ones (therajectory is then only the result of this minimisation control).

As we cannot observe any power law in straight-ahead motionefore and after the turn, it is primordial to clearly identify theime interval when the subject is actually turning. Indeed, theadius of curvature, despite body periodic lateral oscillations,ends to infinity in particular when oscillation goes from one sideo the other (there is a point of inflexion, i.e. an infinite radiusf curvature) whereas speed obviously remains bounded. There-ore, we first proposed to define the Key Instant of the Turn (KIT)orresponding to the instant where curvature is maximal. Then,wo different ways were considered to explore the power lawlong a single turn. The first one was an intra-individual analy-is of the potential relation between velocity and curvature alongime along the turn, following a procedure inspired by the onesed by Vieilledent et al. [16]. The second one, inter-individualnd original in this study, was to focus on the characteristicoint reached at the KIT. The aim was to investigate if a relationetween velocity and this maximal curvature exists, not alongime, but when considering the set of all those characteristicoints for any trial of any subject.

Eleven subjects, five men and six women, volunteered for thisxperiment. They were 29 years old (±11) (mean ± S.D.) and.71 m tall (±0.07). They had no known vestibular or neurolog-cal pathology, which would affect their locomotion. Subjectsave written and informed consent and the study conformed tohe Declaration of Helsinki.

Subjects, barefoot, performed 90◦ corner trajectories at threendividual gait speeds: natural, slow and fast. For each speed,hey performed this simple motion four times: twice on the left

nd twice on the right. What will be now called a trial is thusne turn at one speed. Between two trials, subjects did fully stopor at least 5 s to ensure a non-periodic motion. Trajectory wasot predefined on the floor. Only three markers were put down

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ce Letters 412 (2007) 148–153 149

n the floor to simulate a corner around which subjects had toalk. The aim was to focus on locomotion behaviour for one

hange of direction.Three-dimensional kinematic data were recorded thanks

o the optoelectronic motion capture device Vicon MXOxford Metrics, Oxford, UK). Twelve high-resolution cameras4 megapixels), at a sampling rate of 120 Hz, formed a circleround the scene. Twenty-eight reflective markers were attachedo the subject skin on the following anatomical landmarks:terno-clavicular joint, xiphoid process, 7th cervical vertebra,0th thoracic vertebra and for both hemi-bodies, occipital androntal bones, gleno-humeral joint, lateral humeral epicondyle,lnar styloid process, radial styloid process, anterior iliospinale,osterior iliospinale, lateral tibiale, lateral malleolus, heel, headf the 2nd metatarsus.

Reconstruction and labellisation were performed using ViconQ software (Oxford Metrics) and computations using Matlab.5 (Mathworks, Natick, MA). The high quality of acquisitionsstatic error on marker position is less than 0.1 mm) allowedvoiding using any filtering.

We calculated the body centre of mass (CoM) thanks to Zat-iorsky et al.’s anthropometric table [19]. We calculated the headosition, seen as the mean of the four head markers position.ven if we did not expect differences with the CoM, we chosetudying the head to compare our power law results with previ-us studies working on this reference point. Three-dimensionalelocity and acceleration were also computed using a classicalingle or double derivative of the position.

In order to specifically study the adaptation of locomotionhile turning, we had to precisely determine when the subjects

ctually turn, in order to avoid straight parts in their trajectory.e focused on the very characteristic cusp of the evolute of the

adius of curvature (Fig. 1). This cusp matched with the minimaladius of curvature, hereafter called Centre of the Turn (CoT).

ig. 1. Evolution of the centre of curvature of the centre of mass trajectory. KITs the Key Instant of the Turn and TIT is the Time Interval of the Turn.

150 A.-H. Olivier, A. Cretual / Neuroscience Letters 412 (2007) 148–153

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ig. 2. Intra-individual and inter-individual approaches which deals, respectivelylong the turn and with the minimal radius of curvature and its associated veloc

y the entrance in this quadrant and the exit. Thus, we get theime Interval of the Turn (TIT).

To assess whether a power law still exists along one turn, wean seek for a linear relation in the logarithmic space betweenhe radius of curvature and its associated velocity [17]:

= KRβ ⇔ log(V ) = β log(R) + log(K) (1)

here V is the velocity in m/s, K the velocity gain factor, Rhe radius of curvature in m and β is the value of the power.uch a linear relation can be quantified using a correlation oretermination coefficient.

The following formula gives us the radius of curvature:

(t) = V 3

det((dOM/dt), (d2OM/dt2))(2)

here R(t) is the radius of curvature in m along time, V theorizontal velocity in m/s, OM is the position of the point M.

We used this formula to explore the power law consideringrst the CoM and then the head. The search was divided into twoifferent approaches: an intra-individual approach (which wasven an intra-trial approach) and an inter-individual one. Finally,

eeking for a potential power law, whatever the approach used,an be performed only if at least one parameter has an a prioriarge enough variation. This minimal variation condition allowshen to verify if the second parameter evolves concomitantlyith this first one.

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the evolution of the radius of curvature (R) and the velocity (V) of the trajectory

The first approach was intra-individual and all along the TITf each trial, i.e. the curved path of the turn. It consisted inooking up the relation between the functions R(t) and V(t) (seeig. 2). It therefore led to 132 relations, i.e. one relation per trial4 turns at 3 speeds for any of the 11 subjects which actuallyave 132 trials).

The second approach was inter-individual and only usedne value per trial. For each trial, we considered the coupleade up of minimal radius of curvature (Rmin) and its associ-

ted velocity (V(Rmin)), meaning the couple velocity/curvatureeached at the KIT defined above. We collapsed the data overhe 132 couples in order to seek for one potential relationshipetween all couples (Rmin; V(Rmin)) for all subjects and all trialssee Fig. 2).

Data are presented as mean and standard deviation (±S.D.).ormality distribution was assessed using the Kolmogorov–mirnov test. The strength of the relationships was determinedsing the determination coefficient (R2) given by the Pearsonorrelation test. Statistics within the intra-individual approachere performed on the results of each trial.First of all, let us underline that during the TIT, the situation

as explicitly a turn because mean maximal radius of curvature,hat was 1.49 m (±0.6), is rather small. Moreover, Rmin was

.52 m (±0.21), ensuring large enough variations of the radiusf curvature to allow us to seek for a potential power law. Theime window covered by the TIT was 0.88 s (±0.22) with a rangef 0.45–1.5 s.

A.-H. Olivier, A. Cretual / Neuroscience Letters 412 (2007) 148–153 151

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ig. 3. Intra-individual logarithmic relation for the centre of mass between radiuith a slow velocity (a), a right turn (b) and a left turn (c) with a fast velocity an

For all situations, we did not find any reproducible power lawhenomena between velocity and radius of curvature along oneingle 90◦ turn neither for the CoM nor for the head. Followingwide range of values (0.01–0.99), mean R2 was equal, for

he CoM, to 0.63 (±0.31). Mean R2 was lower for the head0.43 (±0.33)). Four examples on the CoM trajectory, with R2,espectively, equal to 0.99, 0.65, 0.33 and 0.08, illustrate thereat variability of the results in Fig. 3. Moreover, the potentialower values linked to these relations covered a wide intervalrom −0.02 to 0.52, underlying the unreproducible aspect of theelocity/curvature relation within the intra-individual approach.

First, the range of KIT speeds, from 0.35 to 1.80 m/s, fulfilledhe minimal variation condition.

Contrary to the intra-individual approach, the inter-individualne led to very strong linear correlations (Fig. 4a and b)etween Rmin and V(Rmin) in the logarithmic space for the CoMR2 = 0.84, p < 0.0001) and for the head (R2 = 0.81, p < 0.0001) inll situations with a, respectively, power value of 0.69 and 0.65.

No reproducible power law relation between velocity andadius of curvature along one turn was found within the intra-ndividual approach. A first explanation could be that the TIT

ight be too wide including parts of straight-ahead walking.owever, we noticed that the mean maximal radius of curvatureas 1.49 m (±0.6). This dismisses this first hypothesis.

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Then, even though from time to time R can reach a highalue within the intra-individual approach, it could be seen aspower law illusion. This seems to be linked with the mathe-atical continuity of the two variables considered. Indeed, such

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Fig. 4. Inter-individual logarithmic relation between velocity and rad

urvature (RCoM) and velocity (VCoM) for four trials corresponding to a left turnght turn at natural speed (d).

ontinuity a priori imposes to work with 1D curves instead ofets of 2D points. Thereby, any 1D curve observed on a shorteriod of time, which is the case here, can be close to a straightine that is its own tangent.

Another potential explanation could be that such a power lawnly exists in periodic motions. Consequently, a post hoc exper-mentation has been performed, strongly inspired by Vieilledentt al.’s study [16]. Our eleven subjects had to turn five times, inhe clockwise direction, around four markers forming a rectan-le on the floor. The only difference with this latter study layn the fact that the trajectory was not explicitly drawn whichet subjects free to define it. The same calculations than in thentra-individual approach were used for the head and CoM tra-ectories. During this situation, we did notice any significantower law neither for the CoM (R2 = 0.47 (±0.07)) nor for theead (R2 = 0.41 (±0.07)). The clockwise direction was arbi-rarily chosen, as it has no effect on the phenomena stemmingrom power law [15]. Hicheur et al. [8] state that the power lawxists when walking along a predefined trajectory, even thoughhe power value is shape dependent. Not finding any veloc-ty/curvature relationship within our intra-individual approach,here subjects’ trajectory is let free, allows us to conclude thatsimple turning task is not completed by, first, performing a fullath planning and then, afterwards, following the trajectory that

tems from this planning.

We must also take into account the inverted pendulum mecha-ism. The CoM vaults over the stance leg, arcing firstly upwardsnd secondly downwards over this leg [2]. Then, between touch-

ius of curvature for the centre of mass (a) and for the head (b).

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52 A.-H. Olivier, A. Cretual / Neuro

own and mid-stance, the forward velocity decreases and, duringhe second half of the stance phase, it increases. It involveselocity oscillations over one step cycle which influence,oncomitantly with the turn, the walking speed. This intra-cycleariation of speed can be considered as a low frequency noiseot synchronised with the speed evolution due to the turningask. In Vieilledent et al. [16] and Hicheur et al. [8] studies, ashe motion is much longer, the effect of the low frequency noises reduced. However, in our post hoc experiment our subjectsalked for similar durations as theirs. As we still did not findvelocity/curvature relation in that case, we can conclude that

he pre-definition of the trajectory is essential to raise such aelation.

Within the inter-individual approach, a strong correlationinked all couples (Rmin; V(Rmin)). This result is very interestingnsofar as all subjects, who followed trajectories closely similarrom each other’s, with individual velocities, were governedy a common law of movement. It can be seen in fact as theefinition of a “mandatory waypoint” for the extremum of theelocity/radius curves for each intra-individual trial.

One can wonder whether this result is only a mathemati-al illusion. First, Schaal and Sternad incriminated the low-passmoothing of continuous variables [15]. In our study, such aypothesis cannot be hold. Indeed, the signal we were workingn, i.e. the set of all couples (Rmin; V(Rmin)), was not smoothedsing any low-pass filter and, moreover, is a set of indepen-ent and discrete variables. Finally, Maoz et al. [11] showedhat any white noise signal can lead to a pseudo-one-third poweraw, which would correspond, following our formulation, to a

value close to 0.3. We here obtained a β value around 0.7.herefore, this hypothesis should also be rejected.

The intra-cycle speed variation due to pendulum mechanismould also explain the inter-individual law. Indeed, there may inact exist a velocity/curvature relation at any phase of the stepycle, due to mechanical reasons. So, we have identified heeltrikes which surround the KIT within each trial. It actuallyppears that the KIT distribution is normal with a mean equalo 30% of the step cycle. It matches with the midstance phasehich provides a stable support. However, after dividing step

ycle in 10% sections and seeking for inter-individual poweraws at these instants, R2 values are always lower than whate get at KIT (maximal R2 value is 0.77 at 30% of step cycle).oreover, when moving away from 30%, R2 dramatically

alls (e.g. R2 = 0.04 at 90% of step cycle). Therefore, there is,rst, a high relationship at KIT and then, as KIT is alwaysround 30% of step cycle, we get a reasonable R2 around thishase.

Therefore, the inter-individual relationship we get allows uso conclude that completing a turning task leads to a simplifi-ation of the motor control by coupling velocity and curvature.owever, results within the intra-individual approach show that

his coupling is not ensured all the time over the locomotor path.he intra-individual relation found in previous studies [8,16]

ould be linked with the fully imposed trajectory. This involvedvery specific visual feedback which information is used tostep-by-step control of the orientation along the turn. In our

tudy, no specific information on the shape of the trajectory

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ce Letters 412 (2007) 148–153

s given. Then subjects might change the way the turn is con-rolled. Therefore, the coupling between velocity and curvatureithin such a turning task would take place in a “long term”

ontrol level. The task completion is ensured by minimising therror between the original and the desired point in the orien-ation/position space. This minimisation process is based on aigher level control law that remains to be found. The trajectoryapped out by the subject in the orientation/position space is

nly the result of that minimisation process.To complete our results, further experiments are necessary

o investigate the influence of the turning angle on β, which ishape dependent [8]. If so, it will then emphasize our conclusionhat the power law observed within the inter-individual approachs linked to the “long term” control of the turning task, reachingdesired goal in the body orientation space.

cknowledgements

This study has been partially supported by the Regionalouncil of Brittany, the General Council of Ille-et-Vilaine andennes Metropole.

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