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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: [email protected]
J.-P. Laumonde-mail: [email protected]
H. Hicheur A. BerthozLPPA CNRS-College de France, 11 place
Marcelin Berthelot,75005 Paris, France
H. Hicheure-mail: [email protected]
A. Berthoze-mail: [email protected]
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).
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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.
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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
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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
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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.
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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
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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.
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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
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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
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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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