Auton Robot (2008) 25: 2535DOI 10.1007/s10514-007-9075-2

On the nonholonomic nature of human locomotion

Gustavo Arechavaleta Jean-Paul Laumond Halim Hicheur Alain Berthoz

Received: 23 October 2006 / Accepted: 3 December 2007 / Published online: 8 January 2008 Springer Science+Business Media, LLC 2008

Abstract In the kinematic realm, wheeled robots determin-ing characteristic lies in its nonholonomic constraint. In-deed, the wheels of the robot unequivocally force the robotvehicle to move tangentially to its main axis. Here we testthe hypothesis that human locomotion can also be partly de-scribed by such a nonholonomic system. This hypothesis isinspired by the trivial observation that humans do not walksideways: some constraints of different natures (anatomical,mechanical. . .) may restrict the way humans generate loco-motor trajectories. To model these constraints, we proposea simple differential system satisfying the so called rollingwithout sliding constraint. We validated the proposed modelby comparing simulated trajectories with actual (recorded)trajectories obtained from goal-oriented locomotion of hu-man subjects. These subjects had to start from a pre-definedposition and direction in space and cross over to a distantporch so that both initial and final positions and directionswere controlled. A comparative analysis was successfullyundertaken by making use of numerical methods to computethe control inputs from actual trajectories. To achieve this,three body segments were used as local reference frames:

G. Arechavaleta () J.-P. LaumondLAAS-CNRS, Toulouse University, 31077 Toulouse, Francee-mail: garechav@laas.fr

J.-P. Laumonde-mail: jpl@laas.fr

H. Hicheur A. BerthozLPPA CNRS-College de France, 11 place Marcelin Berthelot,75005 Paris, France

H. Hicheure-mail: halim.hicheur@college-de-france.fr

A. Berthoze-mail: alain.berthoz@college-de-france.fr

head, pelvis and torso. The best simulations were obtainedusing the last body segment. We therefore suggest an anal-ogy between the steering wheels and the torso segment,meaning that for the control of locomotion, the trunk behav-ior is constrained in a nonholonomic manner. Our approachallowed us to successfully predict 87 percent of trajectoriesrecorded in seven subjects and might be particularly relevantfor future pluridisciplinary research programs dealing withmodeling of biological locomotor behaviors.

Keywords Human locomotion Nonholonomic mobilerobots Trajectory formation

1 Introduction

The generation and control of locomotion field has mainlystudied rhythmic and coordinated motions of the body andlimbs trajectories. It turns out that this synchronization ofmotions between the body and limbs reduce the dimensionof the motor space to be explored. In Robotics, it is wellknown practice to benefit from the system redundancy toperform some tasks (see for instance Siciliano and Slotine1991; Khatib 1987; Yoshikawa 1984 and for an overviewsee Nakamura 1991). The human body is a highly redun-dant system. The human motor learns by discovering mo-tion patterns that reduce the dimension of the motor space(Bernstein 1967). In other words, the human motor tends tobuild a control space with lower dimensions than the motorspace. The challenge of modern computational neuroscienceis the following: to propose control space models that can begeneric enough to account for large classes of tasks (Wolpertand Ghahramani 2000).

26 Auton Robot (2008) 25: 2535

Goal-oriented locomotion has been investigated with re-spect to how different sensory inputs are dynamically in-tegrated. This has facilitated the elaboration of locomotorcommands that allow reaching a desired body position inspace (Berthoz and Viaud-Delmon 1999). Visual, vestibularand proprioceptive inputs have been analyzed during bothnormal and blindfolded locomotion to study how humanscould continuously control their trajectories (see Glasaueret al. 2002 and for a review, see Hicheur et al. 2005a). Re-cently the hypothesis that common principles govern thegeneration (or planning) of hand and whole body trajecto-ries has been tested (Vieilledent et al. 2001; Hicheur et al.2005b). In particular, a strong coupling between path geom-etry (curvature profile) and body kinematics (walking speed)is observed with some quantitative differences between twotypes of movements (Hicheur et al. 2005b). These experi-mental observations have been discussed within the frame-work of the simplifying control strategies that may governthe steering of locomotion in humans. However, these stud-ies are limited to pre-defined paths. In our research, we in-vestigate human forward locomotion in a less restrictive sit-uation: only beginning and end are known, but not the pathto reach the goal (see Hicheur et al. 2007).

In our current study, we analyze the spatial and temporalfeatures of the locomotor trajectories when human subjectsperform natural displacements. We test the following sim-ple statement saying that the natural way for walking is toput one foot in front of the other and to repeat again theseactions. This basic statement is not so trivial. Indeed, Infront of means that the direction of the motion is given bythe direction of the body: it implies a coupling between thedirection of the body and the tangent to the trajectory. This isa differential non integrable coupling known as being non-holonomic.1 This research has been conducted in an effort toconceive a first control system that accounts for human lo-comotor trajectories. We follow a methodology based on anaccessibility domain geometric study of forward locomotortrajectories.

We first consider the 3-dimensional space defined bythe body position (x, y) and direction . Within this space,the non integrable 2-dimensional distribution gathers all theconfigurations (x, y, ). A basis of the distribution is doneby two control vector fields supporting the linear and theangular velocities respectively. Both linear and angular ve-locities are the only two controls that define the shape of thepaths in the 3-dimensional manifold. It means that it is pos-sible to integrate the locomotor trajectories knowing thesetwo control inputs to simulate trajectories. In particular, asmentioned before, the model we study should be valid for all

1Nonholonomy is a classical concept from mechanics which has beenvery fruitful in mobile robotics in the past twenty years (Bellaiche andRisler 1996).

possible intentional goals reachable by a forward walk. Weexclude from the study the goals located behind the startingposition and the goals requiring side walk steps. Neverthe-less, any goal in an empty space, even one located behindthe starting position, may be reachable by a forward walk.However, this is not the natural way to do so. A humanwould not intentionally walk all around the room to reach apoint that is right behind them. This important assumptionis related to the accessibility space of a control system. Herewe reasonably assume that the accessibility domain of theforward locomotion is a kind of a 3-dimensional cone ap-proximated by the accessibility domain we consider in theprotocol.2

In addition to the considerations relative to the geomet-ric aspects of the trajectories, some motor aspects need tobe mentioned here. Indeed, it can be argued that geometricconfigurations of human bodies are constrained, at the jointlevel, by anatomical parameters that limit a given rotationof a body segment within a certain space. For example, ab-duction/adduction movements of a given leg cannot covera wide range of spatial configurations as it can be the casefor the shoulders segment. Ground reaction forces also actfirst at the legs level and constraint indirectly the center ofmass trajectory. Such a mechanical point of view has beeninvestigated in biomechanics for the study of the human lo-comotion (see for instance Winter 2004), in computer ani-mation (see for instance Multon et al. 1999) and in roboticsfor the study of the humanoid robots locomotion (see for in-stance the pioneering work, Raibert 1986 or the more recentworked out example of HRP robot, Hirukawa et al. 2005).

Our approach differs from the previous ones since wedo not consider sensory inputs or the complex mechani-cal system that models the human body. Our point of viewis complementary and more macroscopic than the standardbiomechanics approaches. Our study is devoted to analysethe steering of locomotion at the trajectory planning level.We focus on the shape of the locomotor trajectories in thesimple 3-dimensional space defined by the position and thedirection of the body. As a consequence of this neurophysio-logical perspective, appropriate experimental protocols havebeen defined to exhibit the behavior under study. Then, wehave formalized the knowledge acquired by experimentationin terms of mathematical models already used for mobile ro-bots (Li and Canny 1993; Laumond et al. 1998).

The differential model we propose could serve as a bridgebetween the researches performed in the human physiologyand the mathematical background developed on the non-holonomic systems in mobile robotics. This point of viewconstitutes the first contribution of the paper.

2Drawing the exact frontiers of the forward locomotion accessibilitydomain is typically a topic for future work open by this study.

Auton Robot (2008) 25: 2535 27

One of the most popular nonholonomic system is the uni-cycle. This mechanical system rolls without sliding. Motionplanning and control for rolling vehicles is an active researcharea in mobile robotics (Li and Canny 1993; Laumond et al.1998). The controls of a vehicle are usually the linear ve-locity (via the accelerator and the brake) and the angular ve-locity (via the steering wheel). A second aspect of this workis relative to the examination of the motion of the differentsegments constituting the pluri articulated human body, andtheir potential respective roles for the steering of human lo-comotion.

We then designed an original experimental protocolbased on the generation of locomotor trajectories in a freespace (in the absence of obstacles). These intentional tra-jectories were freely chosen by subjects and were only con-strained both in terms of position and direction in space atthe starting and end of movement execution. No constraintswere given as to what path to take or how fast to approachthe final destination point.

In order to cover as much as possible the 3-dimensionalaccessibility region, we sampled the domain by defining dif-ferent positions on a 2-dimensional grid (within a 5 m by 9 mrectangle) and 12 directions each. The starting position is al-ways the same. More than 1500 trajectories were recordedwith a motion capturing optoelectronic device which pro-vided the position of 34 different body markers. This is thedata basis used for statistical analysis. We found that morethan 87 percent of the recorded trajectories are approxi-mated with an error (in terms of distance) less than 10 cm.

These results, obtained using the torso markers, suggestthat the trunk motion during human locomotion tends to

obey the same rules as the ones governing the motion ofsimple nonholonomic systems.

2 Apparatus and protocol

2.1 Subjects and materials

To examine the geometric properties of human locomotorpaths, actual trajectories were recorded in a large gymna-sium in seven normal healthy males who volunteered forparticipation in the experiments. Their ages, heights andweights ranged from 26 to 29 years, from 1.75 to 1.80 me-ters, and from 68 to 80 kilograms respectively.

We used motion capture technology to record the tra-jectories of body movements. Subjects were equipped with39 light reflective markers located on their head and bod-ies. The 3D positions of the light reflective markers wererecorded using an optoelectronic Vicon motion device sys-tem (Vicon V8, Oxford metrics) composed of 24 cameras.The sampling frequency of the markers was 120 Hz. It isimportant to mention that we do not apply any kind of filterto raw data in our analysis (see Fig. 1).

Only nine markers have been directly used for this analy-sis. Three reflective markers were fixed on a helmet (200 g).The helmet was placed so that the midpoint between the twofirst markers was aligned with the head yaw rotation (naso-occipital) axis. We also used other two reflective markerslocated at each shoulder and finally four markers located onthe bony prominences of the pelvis.

In order to specify the position of the subject on the planewe established a relationship between the laboratorys fixed

Fig. 1 Some examples of actual trajectories followed by the torso with the same final direction. a, b, c and d show all actual trajectories wherethe final direction is 330 deg., 120 deg., 270 deg. and 90 deg. respectively

28 Auton Robot (2008) 25: 2535

Fig. 2 The porch and the room used in the experiments. We sampleda region of the gymnasium with 480 points defined by 40 positionson floor (within a 5 m by 9 m rectangle) and 12 directions each. Thestarting position was always the same while the goal was randomlyselected. One subject performed all the 480 trajectories while other 6performed only a subset of them chosen at random.

reference frame and the trajectorys reference frame whichcan be computed using either head, torso or pelvis markersas we explain in Sect. 2.3. Hence, the configuration A of thesubject is described as a 3-vector (xa, ya, a).

2.2 Protocol

The aim of the experimental protocol was to validatewhether subjects, in a free environment, perform stereotypedtrajectories in terms of geometric and kinematic attributes.

In the experiment, subjects walked from the same initialconfiguration Ainit where the initial direction approximatelyorthogonal to the horizontal axis of the laboratory to a ran-domly selected final configuration Af inal represented by thedoorway. The target consisted in a porch which could be ro-tated around a fixed point to indicate the desired final direc-tion (see Fig. 2). The subjects were instructed to cross oversuch porch (from Ainit to Af inal) without any spatial con-straints relative to the path they might take. Subjects wereallowed to choose their natural walking speed in order toperform the task. They were not asked to stop walking afterentering the door because such instruction could influencedtheir behavior few steps before reaching the porch.

To be more precise, when the subjects were asked togo through a distant doorway without any instructions onspeed or accuracy, they had several possibilities for planningand executing the sequence of movements allowing them toreach the goal. The only constraints were the initial posi-tion and direction that were always the same and the final

Fig. 3 It shows all the final configurations considered for the first sub-ject

position and direction given by the doorway (see Fig. 3).Surprisingly, we observed that in such simple goal-orientedtask the subjects reproduced similar trajectories.

The final direction varied from to in intervals of6 at each final position. In order to exclude the positiveand negative acceleration effects at the beginning and at theend of trajectories, the first and the last steps of the sub-jects trajectories are not considered in this study. The sub-jects started to walk straight ahead one meter before the ini-tial configuration Ainit and stopped two meters after passingthrough the porch. Thus, the initial and final linear velocitywere never zero.

The experiment was carried out in seven sessions. Thefirst subject was asked to perform 480 different trajectoriesin two sessions (corresponding to 40 positions 12 direc-tions).

The six other subjects were asked to perform 180 dif-ferent trajectories during the next six sessions. Each sub-ject performed 3 trials for a given configuration of the porchAf inal . Therefore, they walked 180 trajectories with only 60different final configurations. It means that they have done asubset of the recorded trajectories executed by the first sub-ject.

The length of the trajectories performed by the first sub-ject ranged between 2 and 10 meters. The length of trajecto-ries performed across the other six subjects and trials rangedfrom 1.96 to 2.12 meters for the nearest targets and from6.48 to 6.50 meters for the farthest targets.

2.3 Global, head, torso and pelvis coordinate frames

While walking, the body generated trajectories in the spacerelative to the laboratorys reference frame LRF. To describethe movement of the body, a local reference frame was de-fined (see Fig. 4). Three body coordinate frames were used

Auton Robot (2008) 25: 2535 29

Fig. 4 Definition of the local frames and their trajectories pro-jected on the ground. All of them correspond to the same mo-tion. a shows a 3D reconstruction of human body from themarkers. b shows the trajectory followed by the head refer-

ence frame and its directions. c shows the trajectory followedby the torso reference frame and its directions. d shows thetrajectory followed by the pelvis reference frame and its direc-tions

for the head RFH , the torso RFT and the pelvis RFP respec-tively. The origins of RFH , RFT and RFP and their direc-tions have been determined from the markers coordinates.

To calculate the origin xH ,yH of RFH , we used the mark-ers located on the back and the forehead. The direction Hof RFH is easily identified according to the segment whoseendpoints are the back and the forehead markers. Therefore,the desired direction is merely the rigid body transformationof RFH onto LRF.

The midpoint of the shoulder markers and the directionorthogonal to the shoulder axis corresponding to the originxT , yT and the direction T of RFT respectively. Finally,to find the origin xP , yP and the direction P of RFP , fourmarkers are used, left and right-front, left and right-back.These markers are located on the bony prominences of thepelvis.

2.4 Data processing

Numerical computation is performed to obtain the walkingvelocity profile. Each recorded trajectory is represented as asequence of discrete points on the plane. We computed thelinear v and angular velocities at each point such that

x(t) x(t +t) x(t t)2t

,

y(t) y(t +t) y(t t)2t

, (1)

v(t)x2(t)+ y2(t),

(t) (t +t) (t t)2t

(2)

where x(t), y(t) and (t) are the configuration parametersof the body along the trajectory. Therefore, these parametersdescribe the motion of any of the three RFH , RFT or RFPlocal frames. We computed the desired tangential direction(t) along the path as

(t) tan1(y(t)

x(t)

). (3)

It should be pointed out that (t) has been calculatedfrom the markers while (t) is computed from the sequenceof discrete points x(t), y(t). We used (2) to obtain the in-stantaneous variation of (t) replacing (t) and (t) with (t) and (t) respectively.

3 Comparison between head, torso and pelvis directions

The purpose of this section is to analyze the time course ofthe body turning behavior of the three different direction pa-rameters H (t), T (t) and P (t). These parameters corre-spond to the rotation of different body segments with respectto the trajectory. This quantitative and qualitative analysis isdone to determine which of them better approximates (t).

To accomplish such evaluation, we performed some testsand measurements for the different reference frames RFH ,RFT and RFP : the direction of the head, torso and pelvisversus (t) while steering along a path.

30 Auton Robot (2008) 25: 2535

Fig. 5 Head, pelvis, torso and tangential direction profiles. All of themcorrespond to the same motion. a shows the head direction profile withrespect to the tangential direction. b shows the pelvis direction profile

with respect to the tangential direction. c shows the torso and the tan-gential directions. d shows the torso direction shifted 16 s backward andthe tangential direction

3.1 Head direction profile

Defining RFH as the local coordinate frame it is noted thatH (t) points most of the time towards the direction of thetarget as it is illustrated in Fig. 4b. Furthermore, there aresome cases where H (t) is pointing to the opposite half-plane with respect to (t). For instance, analyzing the be-havior of H (t) and (t) in the trajectory of Fig. 4b, it canbe shown that H (t) and (t) follow a similar trace until0.55 s and just after that both directions start to diverge until1.4 s (see Fig. 5a).

3.2 Torso direction profile

Choosing RFT rather than RFH , we observed that for everytrajectory the curves traced by T (t) and (t) had a simi-

lar form. However, comparing T (t) and (t) in time, it isnoted that T (t) is shifted between 14 and

18 s backward (see

Fig. 5c and Fig. 5d). It means that the torso as well as thehead anticipates the direction relative to the current walkingdirection. In other terms

T (t + ) (t) (4)where represents the time shifted backwards.

3.3 Pelvis direction profile

Examining P (t) relative to (t) while steering along a path,we observed that P (t) oscillates with amplitude close to15 degrees even along a curve (see Fig. 5b). These instanta-neous variations reflect the significant influence of the gait

Auton Robot (2008) 25: 2535 31

cycle at each step. It could be possible to fit the curves ofP (t) in agreement with the shape of (t) by filtering P (t)using a fourth-order low-pass filter algorithm with a cut-offfrequency of 0.5 Hz. However, we did not apply any kind offilter to experimental data.

4 A control model of locomotion

4.1 Introduction

Human beings usually walk forward and the direction oftheir body is tangent to the trajectories they perform (ne-glecting fluctuations induced by steps alternation). This cou-pling between the direction and the position (x, y) of thebody can be summarized by the following differential equa-tion: tan = y

x. It is known that this differential equation

defines a non integrable 2-dimensional distribution in the 3-dimensional manifold R2 S1 gathering all the configura-tions (x, y, ): the coupling between the direction and theposition is said to be a nonholonomic constraint. A basis ofthe distribution is done by the two following vector fields:

cos sin

0

and

001

(5)

supporting the linear velocity and the angular one respec-tively. Both linear and angular velocities appear as the onlytwo controls that perfectly define the shape of the pathsin the 3-dimensional manifold R2 S1. Checking whethera differential coupling is integrable or not is done by theFrobenius theorem, a classical tool from differential geom-etry (Varadarajan 1984). The study of nonholonomic sys-tems generates works in the community of pure mathemat-ics (e.g., Bellaiche and Risler 1996), control theory (e.g., Liand Canny 1993) and robotics (e.g., Laumond et al. 1998).

The purpose here is to answer the following question:what is the body frame that better accounts for the nonholo-nomic nature of the human locomotion?

4.2 Model

To measure the error of the approximation expressed by (4),we defined RFT as the local reference frame of the body toperform numerical integration using the following controlsystem:

xTyTT

=

cosTsinT

0

u1 +

001

u2. (6)

The control inputs u1 and u2 are the linear and angu-lar velocities respectively. The nonholonomic constraint ex-

Fig. 6 Bicycle model

pressed by the (7) force the control system to move tangen-tially to its main axis.

yT cosT xT sinT = 0. (7)A parametric interpretation of the anticipation effect (de-

lay of 16 s) can be considered within the control system. It canbe seen as a bicycle as shown in Fig. 6. Such kind of systemhas been used extensively in robotics (Laumond et al. 1998).The equations describing the motion of the bicycle are givenby:

xTyT

T

=

cos sin tanTL

0

u1 +

0001

u2 (8)

where (xT , yT ) denotes the position of the bicycle relativeto some inertial frame, the angle of the bicycle relative tothe horizontal axis, T the angle of the front wheel relativeto the bicycle, u1 the driving speed, u2 the steering rate andL is the length of the link between the back and the frontwheel. For our purpose, L characterizes the anticipation ef-fect represented previously by . We made a simple transfor-mation from a delay of 16 s to a distance of 16.6 cm. Sincethe model is subject to rolling constraints, (7) must hold atevery point along any achievable trajectory.

5 Results

We observe that the trajectories performed by all subjects aresimilar both in geometric and kinematic terms. To validateour hypothesis, we perform several experiments on differentsystems, including a unicycle on the plane subject to rollingconstraint (6), and also, a bicycle system (6) in order to takeinto account the anticipation effect.

32 Auton Robot (2008) 25: 2535

Fig. 7 Representative examples of comparisons between recorded(thin) and integrated (bold) locomotor trajectories. a shows the behav-ior of the recorded and integrated trajectories by translating the finalposition in the vertical axis with a fixed final direction. b shows thebehavior of the recorded and integrated trajectories by translating thefinal position in the horizontal axis with a fixed final direction

In this section we describe the comparisons conductedto quantify, in our models, the instantaneous error of therecorded trajectory with respect to the simulated trajectory.We have defined the average and the maximal error betweenboth trajectories.

It is important to emphasize that all the actual trajecto-ries have not been filtered. The data from each trial of eachsubject is analyzed separately (i.e., no averaging over trialsis performed). Thus, for each trajectory represented by a se-quence of discrete points on the plane, we use (1) and (2)to extract the linear u1(t) and angular u2(t) velocities. Wethen obtain the control inputs of the recorded locomotor tra-jectory expressed by RFT .

Having the control inputs, we integrate the differentialsystem (see 6). Figure 7 shows some examples of the behav-ior of the recorded and integrated trajectories by translatingthe final position over both: the vertical and the horizontalaxes with a fixed final direction. Figure 8 show some exam-ples of recorded and integrated trajectories for a fixed finalposition. The final direction varies in intervals of 6 .

To measure how well the model approximates locomotortrajectories, we compute the difference between both trajec-tories at instant t . To do that, we define the trajectory errorTE such as

TE(t)=(xi(t) xa(t))2 + (yi(t) ya(t))2 (9)

where (xi(t), yi(t)) and (xa(t), ya(t)) are the positions at in-stant t of the integrated and actual trajectories respectively.

Fig. 8 Representative examples of comparisons between recorded(thin) and integrated (bold) locomotor trajectories. a and b show therecorded and integrated trajectories for a fixed final position. The finaldirection varies in intervals of 6

Then, as in Quang-Cuong et al. (2007), we compute the av-eraged and the maximal trajectory errors

ATE =

t[0,T ]TE(t)dt

MTE = maxt[0,T ]

TE(t).(10)

These two quantities indicate the similarity between theintegrated and the actual trajectories. Thus, small values ofATE and MTE mean that the similarity degree is high be-tween both trajectories.

This procedure has been executed for 1,560 trajectoriesperformed by seven subjects. The length of the trajectoriesranged between 2 and 10 meters. The walking speed of thesubjects was equal to 1.26 0.3 meters/seconds (m/s). It isinteresting to note that the model approximates 87 percentof trajectories with ATE < 10 cm and MTE < 20 cm.

For the subset of recorded trajectories executed by sixsubjects (i.e., for 60 targets), we compute the trajectory de-viation TD between actual trajectories corresponding to thesame task such as

TD(t)= 1

N

Ni=1

((xi(t) x(t))2 + (yi(t) y(t))2) (11)

where (x(t), y(t)) is the mean trajectory and N is the num-ber of actual trajectories. The averaged and maximal devia-tions between the actual and the mean trajectories are givenby ATD and MTD respectively. To perform the analysis, weclassified the subset of 60 targets in terms of the trajectory

Auton Robot (2008) 25: 2535 33

Fig. 9 Classification of actualtrajectories (grey) for a giventarget in terms of the curvatureinduced by the final direction.The mean trajectory (black) iscomputed for all targets.a shows an example of HC:High Curvature. b shows anexample of MC: MediumCurvature. c shows an exampleof LC: Low Curvature. d showsan example of S: Straight

curvature induced by the final direction: HC (High Curva-ture), MC (Medium Curvature), LC (Low Curvature) and S(Straight) (see Fig. 9).

The accuracy of the model is also supported by the factthat ATE and MTE are always lower than ATD and MTD(see Fig. 10). In other words, the integrated trajectory is al-ways inside the area defined by the trial-to-trial variabilityof actual trajectories. Consequently, T (t + ) satisfies themodel. To validate the model (8), we perform the same pro-cedure that we have done for the unicycle system (6).

6 Conclusion

This model shows that human locomotion can be approxi-mated by the motion of a nonholonomic system. Indeed, we

were able to approximate more than 87 percent of the 1560trajectories recorded in 7 subjects during walking tasks witha < 10 cm accuracy. Thus, nonholonomic constraints, sim-ilar to that described in wheeled robots, seem to be at workduring human locomotion. Nevertheless, choosing differentbody reference frames yields different results. We obtainedthe best results using the shoulders segment. It appears thatyaw oscillations induced by step alternation affect the head,torso or pelvis movements differently in such a way thatonly the shoulders midpoint trajectory provided good fit ourmodels predictions. Further investigation is required to ac-count for these differences.

The present model will be the starting point of the nextstage of our work where we plan to provide further evidenceand details about how nonholonomic constraints are exertedduring the generation of human locomotor trajectories. Our

34 Auton Robot (2008) 25: 2535

Fig. 10 The accuracy of themodel is also supported by thefact that the integrated trajectoryis closer to the correspondingactual trajectory than thetrial-by-trial variability of actualtrajectories. a shows thecomparison between theaveraged trajectory errors (ATE)and the averaged trajectorydeviations (ATD). b shows thecomparison between themaximal trajectory errors(MTE) and the maximaltrajectory deviations (MTD)

current model does not explain the geometric shape of thelocomotion trajectories. Why in some cases (see Fig. 4) weare turning first on the right to finally reach a goal whoseposition is on the left of the starting configuration? Such adifficult question is related to optimal control theory (e.g.Sussmann 1990) already successfully applied to mobile ro-botics (e.g. Laumond et al. 1998). The application of thesetools to the understanding of human locomotion opens anoriginal promising route which is currently under develop-ment.

Acknowledgements The authors are grateful to Stephane Dalberafor his help with the Vicon motion device system. G. Arechavaletabenefits from a SFERE-CONACyT grant. Work partially funded by theEuropean Community Projects FP5 IST 2001-39250 Movie for LAAS-CNRS and by a Human Frontiers grant for LPPA.

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Gustavo Arechavaleta received the M.S andthe PhD degree in computer science from theMonterrey Institute of Technology, Mexico andthe Toulouse University, France in 2003 and2007, respectively. He worked on motion plan-ning for robots and virtual characters. His cur-rent work seeks to understand the computationalprinciples of movement neuroscience via theoptimal control and robotics tools.

Jean-Paul Laumond is Directeur de Rechercheat LAAS-CNRS (group Gepetto) in Toulouse,France. He received the M.S. degree in Math-ematics, the Ph.D. In Robotics and the Ha-bilitation from the University Paul Sabatier atToulouse in 1976, 1984 and 1989 respectively.In Fall 1990 he has been invited senior sci-entist from Stanford University. He has beena member of the French Comit National dela Recherche Scientifique from 1991 to 1995.

He has been coordinator of two European Esprit projects PROMo-tion (19921995) and MOLOG (19992002), both dedicated to robotmotion planning technology. In 2001 and 2002 he created and man-aged Kineo CAM, a spin-off company from LAAS-CNRS devotedto develop and market motion planning technology. Kineo CAM wasawarded the French Research Ministery prize for innovation and en-terprise in 2000 and the IEEE-IFR prize for Innovation and Entrepre-neurship in Robotics and Automation in 2005. His current researchis devoted to human motion studies along three perspectives: artifi-cial motion for humanoid robots, virtual motion for digital actors andmannequins, and natural motions of human beings. He teaches Robot-ics at ENSTA and Ecole Normale Suprieure in Paris. He has editedthree books. He has published more than 100 papers in internationaljournals and conferences in Computer Science, Automatic Control and

Robotics. He is 2006-7 IEEE Distinguished Lecturer, IEEE Fellow andmember of the IEEE RAS AdCom. He is currently co-director of JRL-France.

Halim Hicheur is a research fellow at the College de France. He holdsa PhD in physiology and biomechanics of motion at the Paris 6 univer-sity (2006), within the Brain-Cognition-Behaviour doctoral school.His research orientations are related to the principles governing thegeneration and the control of locomotion in humans at both the plan-ning and motor implementation levels. He is interested in both the ex-perimental analysis of the locomotor behaviour using kinematic, elec-tromyographic and videooculographic measurements and in the mod-elling of the principles underlying the generation and the control of thelocomotor behaviour. He actively collaborates with roboticists, mathe-maticians, computer scientists and mechanicists in order to provide anintegrative view for the understanding of human locomotion.

Alain Berthoz is Professor at the College deFrance in Paris. He holds the chair of Physiol-ogy of Perception and Action. Trained as an En-gineer in one of the Grandes Ecoles in France.He graduated in Psychology and Neurophysiol-ogy. Full time researcher in the Centre Nationalde la Recherche Scientifique for twenty years hecreated the Laboratory of Physiology of Percep-tion and Action (LPPA) specialised in the studyof the neuronal mechanisms of multisensory in-

tegration, gaze control, posture and equilibrium and locomotion. In re-cent years his work has focussed on the neural basis of spatial mem-ory and navigation in humans and rats. He has co-directed thesis onthis topic with roboticians and has therefore experience in cooperationwith roboticians. He has more than 200 publications in Internationaljournals (recently one in Science and one in Nature Neuroscience onvisual vestibular integration and sensory conflicts). He has extensiveexperience in cooperative work in European consortium (Coordinatorof an ESPRIT project with 7 countries, of a BIOMED project). He isthe co-Director of a European laboratory with Italy. Member of theFrench Academy of Sciences, of American academy of Art and Sci-ence and of theBelgium Royal Academy of Medicine he has recentlyreceived the Grand Prix of the French Academy of Sciences for hiswork on multisensory integration. He is author of a book published in2000 by Harvard University Press (The brains sense of movement).

On the nonholonomic nature of human locomotionAbstractIntroductionApparatus and protocolSubjects and materialsProtocolGlobal, head, torso and pelvis coordinate framesData processing

Comparison between head, torso and pelvis directionsHead direction profileTorso direction profilePelvis direction profile

A control model of locomotionIntroductionModel

ResultsConclusionAcknowledgementsReferences

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