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RESEARCH ARTICLE

Angular momentum synergies during walking

Thomas Robert Æ Æ Bradford C. Bennett Æ Æ

Shawn D. Russell Æ Æ Christopher A. Zirker Æ Æ

Mark F. Abel

Received: 8 December 2008/Accepted: 10 June 2009/Published online: 4 July 2009

? Springer-Verlag 2009

Abstract

during treadmill walking. Specifically, we used the

uncontrolled manifold hypothesis framework to quantify

the segmental angular momenta (SAM) synergies that

stabilize (i.e., reduce the across trials variability) the whole

body angular momentum (WBAM). Seven male subjects

were asked to walk over a treadmill at their comfortable

walking speed. A 17-segment model, fitted to the subject’s

anthropometry, was used to reconstruct their kinematics

and to compute the SAM and WBAM in three dimensions.

A principal component analysis was used to represent the

17 SAM by the magnitudes of the first five principal

components. An index of synergy (DV) was used to

quantify the co-variations of these principal components

with respect to their effect on the WBAM. Positive values

of DV were observed in the sagittal plane during the swing

phase. They reflected the synergies among the SAM that

stabilized (i.e., made reproducible from stride to stride) the

WBAM. Negative values of DV were observed in both

frontal and sagittal plane during the double support phase.

They were interpreted as ‘‘anti-synergies’’, i.e., a particular

organization of the SAM used to adjust the WBAM. Based

We studied the coordination of body segments

on these results, we demonstrated that the WBAM is a

variable whose value is regulated by the CNS during

walking activities, and that the nature of the WBAM

control changed between swing phase and double support

phase. These results can be linked with humanoid gait

controls presently employed in robotics.

Keywords

Motor control ? Biomechanics

Synergy ? Walking ? Angular momentum ?

Introduction

Bipedal walking is one of the most common and repetitive

activities in our everyday life. However, as a high dimen-

sional, nonlinear problem, it appears very complex from a

control point of view. Many studies have focused on the

possible reduction of the dimensionality of the problem,

notably by studying the possibility of grouping the indi-

vidual variables based on their co-variation during the gait

cycle. The underlying hypothesis is that, instead of con-

trolling each individual variable, the CNS would have to

act on a fewer number of groups of individual variables,

often referred to as modes, primitives or synergies (note

that the term synergy has another specific meaning (Latash

et al. 2007; Latash 2008) that will be adopted in this study).

For walking activities, the co-variation of the individual

variables has been extensively studied at the kinematics

level. Notably, it has been showed that the leg segment

rotations in the sagittal plane (the elevation angles) can be

reduced to two variables only (Lacquaniti et al. 1999, 2002;

Hicheur et al. 2006; Ivanenko et al. 2008). These results

were extended to other variables, including the first-order

dynamic variables (the angular momenta), and considering

whole body motions in the 3D space. Recently, it was

T. Robert (&)

Universite ´ de Lyon, 69622, Lyon, France;

INRETS, UMR_T9406, Laboratoire de Biome ´canique

et Me ´canique des Chocs, Bron;

Universite ´ Lyon 1, Villeurbanne, France

e-mail: thomas.robert@inrets.fr

T. Robert ? B. C. Bennett ? S. D. Russell ?

C. A. Zirker ? M. F. Abel

Department of Orthopaedic Surgery, Kluge Children’s

Rehabilitation Center, University of Virginia,

Charlottesville, VA 22903, USA

123

Exp Brain Res (2009) 197:185–197

DOI 10.1007/s00221-009-1904-4

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shown that, due to their co-variation, the angular momenta

of the individual segments considered can be grouped into

a small number of primitives (Popovic et al. 2004; Herr and

Popovic 2008; Russell 2008), whose composition is

invariant of the speed (Bennett et al. 2008). These studies

also indicate that the composition of these primitives ten-

ded to keep the whole body angular momentum (WBAM)

small.

Although interesting, these studies did not focus on how

the controller acts on these groups of individual variables.

An interesting way to study this question is the notion of

multi-segmental synergies and the framework of the

uncontrolled manifold (UCM) hypothesis (Scholz and

Scho ¨ner 1999; reviewed in Latash et al. 2002, 2007).

Within this framework, the control is viewed as a two-level

hierarchy: at the lower level, the individual variables are

grouped into elemental variables (EV) based on their

co-variation. Although the origin of such grouping is still

debated (does it originate from the CNS or is it a conse-

quence of biomechanical constraints?), the result is that the

controller can act at the upper level on a fewer number of

independent EV. At this upper level, the neural controller

forms in the space of EV a subspace, named UCM,

corresponding to a desired value (time profile) of an

important performance variable (PV). By confining the EV

to that subspace, the controller guarantees a constant value

of the PV. Moreover, if most of the trial-to-trial variance is

confined to the UCM, a conclusion can be drawn on a

synergy among the EV stabilizing the PV (i.e., decreasing

its variability across trials). Note that in this framework, the

term stabilization does not refer to the classical mechanical

concept of stability, but to reproducibility of a performance

value (or of its time profile) over successive trials.

The UCM framework has been extensively used to study

synergies for several type of activities: postural tasks

(Krishnamoorthy et al. 2003; Danna-dos-Santos et al. 2007;

Scholz et al. 2007; Robert et al. 2008), multi-finger force

and moment production (Kang et al. 2004; Latash et al.

2004; Zhang et al. 2006; Olafsdottir et al. 2008), pointing

and reaching (Tseng et al. 2002; Domkin et al. 2005) or

more exotic tasks like pistol shooting (Scholz et al. 2000).

Only a few recent studies have applied this analysis to

walking activities. Cusumano et al. (2008), by using a very

similar approach named the Goal Equivalent Manifolds,

quantified the synergy between stride length and stride

duration in order to keep the walking speed constant. Black

et al. (2007) used the UCM framework to study multi-

segmental synergies at one instant of the gait. They showed

the existence of multi-segmental kinematical synergies,

stabilizing both the CoM and the head position at the heel

contact. However, they did not provide any insight into the

temporal evolution of these synergies.

To our knowledge, despite the interesting results

suggesting that the WBAM is a controlled variable (see

previous paragraph), there is no study focusing on inter-

segmental synergies at the angular momentum level.

Moreover, although it is known that gait cycle is composed

of different phases (double support and swing phase) dur-

ing which the system is in very different configurations

(e.g., open loop vs. close loop), there are no studies

focusing on the evolution of the synergies along the gait

cycle.

In this study, we explored the angular momentum syn-

ergies during walking and the time evolution of these

synergies along the stride. We hypothesized that: (H1)

there will be a synergy among the segmental angular

momenta (SAM) stabilizing the WBAM; (H2) the strength

of these synergies will change within a stride, especially

between the gait phases.

Methods

Experiment

Seven male subjects participated in this experiment. Their

mean age was 31 years old (between 19 and 55 years),

mean height 1.80 m (between 1.75 and 1.94 m) and mean

weight 76.1 kg (between 60.7 and 104.5 kg). None of them

had any known neurophysiological or peripheral disorders.

All of them had previously experienced treadmill walking.

All subjects gave their informed consent, according to the

University of Virginia’s Human Investigation Committee.

The tests were conducted in the Motion Analysis and

Motor Performance Laboratory at the University of

Virginia. A full body marker set of 38 markers was attached

to the subjects after anthropometric measurements were

taken. Three-dimensional kinematic data were collected

using an eight camera Vicon Motion Analysis System

sampling at 120 Hz. In addition, subjects wore two compact

Oxycon Mobile modules and a face mask to measure the

metabolic cost. These data were not used in this study.

The experiment began with 1 min of acclimation on the

treadmill. The subject-specific comfortable walking speed

(CWS) was found as being the average of each subject’s

self-determined ascending and descending CWS. After a

rest period of several minutes for subjects to demonstrate

steady-state resting metabolic cost, the subjects were asked

to walk over the treadmill at their own CWS until a new

metabolic steady state was reached (about 10 min). During

this period, the 3D kinematics were recorded for at least

five series of ten consecutive strides. Strides were defined

from left heel strike to left heel strike. The heel strikes were

automatically identified using the kinematics of the heel

186 Exp Brain Res (2009) 197:185–197

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markers (Zeni et al. 2008). Each stride was then time

normalized over a 100% time window. In a first approxi-

mation, the two double support phases of the stride cycle

were defined as the two periods of 15% of the gait cycle

following the heel strikes (1 to 15 and 51 to 65%). The

remaining parts of the cycle were considered as the swing

phases.

Data reduction

The kinematic data were then processed using MSC.

Adams, with the LifeMod plug-in developed by Biome-

chanics Research Group (San Clemente, CA, USA). A

17-segment, 16-joint, subject-specific model was created

for each subject. The 17 model segments were the head,

neck, upper torso, central torso, lower torso, upper arms

(2), lower arms (2), hands (2), upper legs (2), lower legs (2)

and feet (2). The 16 joints were specified as tri-axis hinge

joint arrangements, except for the elbow and wrist joints

(two axes) and the knee joints (flexion–extension only).

The model was adjusted to the subjects using the generator

of body data (GeBOD) database (Cheng et al. 1994): body

segment length and inertia parameters (mass, center of

mass and inertia) were scaled based on the subject’s age

mass and weight. For each stride, the inverse kinematics

problem was solved using a global optimization procedure

in order to determine the joint motions based on the

markers’ kinematics. A visual screening of the data was

performed to ensure that all strides submitted to further

analysis were correctly reconstructed. At least 45 strides

per subjects were kept.

From the reconstructed kinematics and the segment

inertia, the individual SAM relative to the whole body

center of mass were computed for the three dimensions of

the laboratory reference frame (which axes were assimi-

lated to subject’s frontal, vertical and sagittal axes). Each

segment revolved about its own CoM and moved relative

to the whole body CoM. The angular momentum of each

segment was computed as the sum of two terms: the local

angular momentum and a transfer term (see Eq. 1). For the

ith segment, this can be expressed as:

Li

!¼ Li;local

with Libeing the SAM of the ith segment, ICoM,ithe inertia

tensor of the segment, xi

ri

!and vi

CoM to the whole body CoM, and mithe mass of the

segment.

For these computations, the angular momentum of the

body segments about the long axes of the segments was

neglected, as these terms were found to be very small. The

?? ?!þ Li;transfer

????!¼ ICoM;i? xi

?!the angular velocity vector,

?!þ ri

!? mivi

!

ð1Þ

!the relative position and velocity of the segment

WBAM (LWB) was computed as the sum of the SAM (see

Eq. 2):

LWB

? ?!¼

For comparison across subjects, the three components of

the angular momenta were normalized by the mass, height

and CWS of the subject (Herr and Popovic 2008; Bennett

et al. 2008). Note that these normalized angular momenta

are unitless.

The angular momentum being a vectorial quantity, the

following analyses were performed independently for each

of the three dimensions of the reference space.

X

17

i¼1

Li

!:

ð2Þ

Principal component analysis

Prior to the UCM analysis, a principal component analysis

(PCA) was performed on the SAM. For each subject and

each of the three dimensions independently, the SAM

matrices (Nstrides*101 rows, Nstrides[45; 17 columns,

one for each segment of the model) were subjected to PCA,

using standard Matlab?7.3 function ‘‘princomp.m’’ based

on the covariance matrix. For each subject and each

dimensions, it resulted in 17 principal components (PC) or

orthonormal vectors. The five first PCs in term of per-

centage of variance explained were kept for further anal-

ysis. This number was defined so that at least 95% of the

variance was explained for every subject and every plane.

As this study does not focus on the PCs’ compositions, no

rotation methods (e.g., Varimax) were applied. As a result

of this analysis, at each instant, the 17 SAM are represented

by the five magnitudes of the PCs:

L ¼ PC ? M

where L is the vector of SAM Li(17 9 1), PC the matrix

made of the five principal components’ orthonormal vec-

tors (17 9 5), constant across frames and trials, and M is

the vector of the magnitudes of each of the five PCs

(5 9 1).

Two interesting characteristics of this PCA are: (1) it

guarantees that the EV obtained are independent (orthog-

onal), which is a requirement for the UCM analysis; (2) it

is an interesting way to ‘‘filter’’ the EV, i.e., to remove the

local artifacts that are present in the last PCs (Daffertshofer

et al. 2004). As this study focused on the inter-trial vari-

ability, more PCs were required than for studying the

average trend of the gait (Daffertshofer et al. 2004). The

number of PCs considered in this study (five) was therefore

slightly larger than that used in previous studies on angular

momentum during walking (usually three PCs) (Herr and

Popovic 2008; Bennett et al. 2008; Russell 2008). Note that

a difficulty would have arisen if the PCA was based on the

ð3Þ

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correlation

(accounting for only a small amount of variance) would

have been comparable to magnitudes of the first PCs (in

terms of variance explained for), leading to an artificial

inflation of the UCM variance (variance that does not

influence the PV, see next paragraph) and thus to an

overestimation of the index of synergy (Latash et al. 2007).

In this study, the use of the covariance matrices prevented

this artifact.

matrices:magnitudesof marginalPCs

Link between elemental variables and performance

variable

Within the UCM framework, we need to investigate how

the stride-to-stride variations of the SAM (Li) affected the

WBAM (LWB), i.e., were the SAM variations confined to a

particular subspace producing a constant WBAM value?

The PV considered in this study was the WBAM (LWB),

and the EV were the magnitudes of the five PCs (vectors

M). The relation between EV and PV can be deducted from

the linear relation between LWBand the SAM Li(see Eq. 2

above). Considering Eqs. 2 and 3, the relation between EV

and PV can be written using a Jacobian matrix J as follows:

LWB¼ J ? M

where J is a Jacobian matrix, here degenerated to a 5 9 1

vector.

The component of J can be computed as:

ð4Þ

Jj¼

X

17

i¼1

PCij;

j ¼ 1 ? 5:

ð5Þ

UCM analysis

Within the UCM analysis, the trial-to-trial (here stride to

stride) variance in the EV is spread into two components:

one, which lies in a particular subspace (the UCM) and

does not affect the PV, and the other, which lies in the

orthogonal subspace. These components, normalized by the

dimensionality of their respective subspace, are then

compared to quantify the strength of the synergy (Latash

et al. 2007).

The following steps were performed for each subject

and each of the three planes (details of this analysis can be

found in many previous studies):

1.Definition of the UCM subspace: in this case, the

relation between EV and PV is linear and expressed by

a Jacobian matrix (see preceding paragraph). The

UCM space is, by definition, the null space of this

Jacobian matrix. In this study, its dimension is four

(dimensionality of the EV space minus dimensionality

of the PV space).

2.Computation of the stride-to-stride changes in the EV:

for every frame, the PC magnitudes were averaged

across strides. For each stride and every frame, the

stride-to-stride changes in the EV were then computed

as the mean free changes of the PCs’ magnitudes.

Projection of these mean free vectors onto the UCM

subspace and the orthogonal subspace;

Computation of the total trial-to-trial variance (VTOT)

and of the components of this variance in each of the

two subspaces (VUCM and VORT), normalized per

degree of freedom of the respective subspace (5, 4

and 1, respectively);

Computation of the index of synergy DV (Eq. 6). Note

that, by definition, DV is bounded between -5 (all the

variance in the orthogonal subspace) and ?1.25 (all

the variance in the UCM subspace):

DV ¼VUCM? VORT

VTOT

3.

4.

5.

:

ð6Þ

Detrended fluctuation analysis

The UCM analysis focuses on the variability of the EV

across strides, considering each stride as an independent

sample. On the contrary, a detrended fluctuation analysis

(DFA) was used to investigate the variability of the EV

along the stride, considering the succession of strides as a

whole. This analysis (Peng et al. 1992) allows investigating

the structure of the long-range correlations in nonlinear

time series and has been used to analyze different types of

physiological signals: heart beats (Peng et al. 1995), tap-

ping (Kadota et al. 2004), COP trajectories (Lin et al. 2008,

Duarte and Sternad 2008) and gait parameters (Cusumano

et al. 2008, Jordan et al. 2009). Briefly, the steps of the

DFA are the following: (1) the considered time series is

integrated and divided into boxes of equal size; (2) for each

box, the local trend is removed; (3) the RMS fluctuation of

the whole integrated and detrended time series is com-

puted; (4) the operation is repeated for different size of

boxes; (5) the linear relationship that best fits the log of the

fluctuation versus the log of the box size is estimated;

(6) the output of the DFA, commonly referred to as the

scaling exponent a, is the slope of this linear relationship.

The value of this scaling exponent allows estimating the

long-range correlations in the time series, while limiting

artifacts due to the potential trends and nonstationarity in

the time series (Peng et al. 1995; Chen et al. 2002). More

specifically, a = 0.5 corresponds to a white noise signal;

0\a\0.5 indicates an anti-persistent signal (the smaller

the a, the more it is anti-persistent), i.e., a deviation in one

direction is more likely to be followed by a deviation in the

other direction, which is consistent with an error-correcting

signal; 0.5\a\1 indicates a persistent signal (the larger

the a, the more it is persistent), i.e., a deviation in one

188Exp Brain Res (2009) 197:185–197

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direction is more likely to be followed by a deviation in the

same direction.

For each subject and each of the three planes, this

analysis was run on the variability of the EV (the mean free

changes of the PCs’ magnitudes) that lies on the orthogonal

subspace, later referred to as ‘‘orthogonal variability’’.

Note that the variability lying in the UCM subspace, being

a four-dimensional time series, was not investigated in this

study. For every frame and every stride, the vectors of the

mean free changes of the PCs’ magnitudes were projected

onto the one-dimensional vector representing the orthogo-

nal subspace (see preceding paragraph). The resulting time

series was submitted to the DFA using the algorithm pro-

posed by Peng et al. (1995).

Statistics

The different components of variances (VTOT, VUCMand

VORT) are always positive and the index of synergy DV is

bounded (-5\DV\1.25). Therefore, the distributions of

these variables do not follow a normal distribution. To

overcome this bias, and be able to apply classical statistical

analyses, mathematical transformations were applied. The

components of variances were log-transformed and DV was

transformed using a Fisher’s z-transformation adapted to

the boundaries of DV (Robert et al. 2008):

?

Note that these transformed data were used for the

statistical analyses only. Figures and results presented in

the paper refer to the original data. Statistical analyses were

performed for each of the three planes.

To investigate the evolution of the two components of

variance, VUCM and VORT, between double support and

swing phases, the log-transform of the two components of

variance (see previous paragraph), averaged for the double

support and swing phases, were submitted to a one-way

ANOVA with factor Phase (double support and swing).

In order to check if DV was, on average, positive or

negative, T tests were run on the z-transformed values of

DV averaged across the stride cycle. Bonferroni corrections

were applied to correct the fact that the three tests were run

(one per plane).

The evolution of DV between phases (double support

and swing) was investigated as follows: for each of the

three planes, DV was averaged for both the double stance

and swing phase and a one-way ANOVA with factor Phase

(double support and swing) was run on the z-transformed

values.

In order to check if DV was significantly positive or

negative for each of the two phases, T tests with Bonferroni

corrections were run on the averaged values of the

zDV¼ 0:5 ? Log

5 þ DV

1:25 ? DV

?

:

ð7Þ

z-transformed DV for the two phases and the three planes

and were applied.

Finally, to verify that the scaling exponents a of the

DFA were significantly different from zero, three one-

sample T tests (one per plane) were run and Bonferroni

corrections were applied.

Results

Angular momentum

The angular momenta obtained in this study were typical of

those previously described (Herr and Popovic 2008;

Russell et al. 2008; Russell 2008; Bennett et al. 2008).

Figure 1 shows the time profile of the normalized WBAM

for a typical subject. One can observe the small stride-to-

stride variability, especially in the sagittal plane.

For each subject and each plane, the matrix of the SAM

was submitted to PCA. Of the 17 PCs, the first five in terms

of variance explained were kept for further analysis. On

average, these five PCs accounted for 96, 98 and 99% of

the total variance for the frontal, horizontal and sagittal

plane, respectively. Figure 2 shows, for a typical subject,

the composition of the first five PCs for each plane and the

Frontal

0

e-3

-0.02

0

0.01

Horizontal

-0.01

0.05

Sagittal

mean std=0.8*10e-3

Normalized whole body angular momen tum (no units)

-0.05

0

mean std=2.8*10e-3

% of the stride

010 20 30405060 708090 100

0.02

mean std=1.7*10

Fig. 1 Normalized whole body angular momentum (WBAM) aver-

aged across strides (± one standard deviation) for a typical subject

and the three planes. The mean standard deviation is displayed on the

figures. Note the different scales

Exp Brain Res (2009) 197:185–197 189

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percentages of variance explained by each. The composi-

tion of the first PCs were similar to those previously

described (Herr and Popovic 2008; Bennett et al. 2008).

The last two PCs, accounting for much less variance, were

variable across subjects.

Variance

As a result of the PCA, the 17 SAMs were represented by

the magnitudes of the five PCs (see ‘‘Methods’’). At each

time frame, these magnitudes were averaged across strides

and these averages were removed from the individual stride

magnitudes. The total variance in the space of EV, VTOT,

was computed from these mean free stride-to-stride chan-

ges of the PCs’ magnitudes and normalized by the

dimension of the EV space (5). Figure 3 shows the evo-

lution of the total variance in the EV space along the gait

cycle. Note the increase of variance around the heel strikes.

The peaks of variance arose slightly after the heel strike for

the frontal and horizontal plane (6% after the heel strike on

average), while it was simultaneous with the heel strike for

the sagittal plane.

The total variance in the elemental space was projected

onto the two complementary subspaces (the UCM and its

orthogonal space) and normalized by the dimension of each

subspace (four and one for the UCM and the orthogonal

space, respectively). It resulted in two components, VUCM

and VORT. Recall that VUCMdoes not affect the PV (the

WBAM). Figure 4 shows the evolution of both variance

components along the gait cycle, averaged across subjects.

In order to highlight the respective evolution of the two

components of the variance between double support and

swing phase, the values of VUCMand VORTwere averaged

across subject and over time for the two phases. Results are

displayed in Fig. 5. For each plane, a one-way ANOVA

with factor Phase (double support and swing) was run on

the log-transformed of the two components. In the frontal

plane, it showed a significant decrease between double

support and swing phase for VORTonly ([F1,12= 15.89,

P\0.05] and [F1,12= 2.69, P[0.05] for VORT and

VUCM, respectively). No significant differences were found

for the horizontal plane ([F1,12= 3.94, P[0.05] and

[F1,12= 3.86, P[0.05] for VUCMand VORT, respectively).

For the sagittal plane, an opposite evolution of the two

components was found: VUCM increased while VORT

decreased between the double support and the swing phase

([F1,14= 5.13, P\0.05] and [F1,12= 83.95, P\0.01]

for VUCMand VORT, respectively).

Index of synergy

The index of synergy DV was computed as the difference

between the two components of variance (VUCMand VORT),

normalized by the total variance. Positive values of this

index mean that most of the variance lies in the UCM

space, i.e., the variance of the EV is organized in such a

way that it mostly does not affect the PV. Note that due to

the normalizations, DV ranges between -5 (all the variance

is in the orthogonal space) and ?1.25 (all the variance lies

in the UCM space).

1

1

0

1

PC1

1

1

0

1

1

1

0

1

67%

l a tno z i roH l a tnorF

Sagittal

%28%78

-1

-1

1

0

0

PC2

-1

-1

1

0

0

-1

-1

1

0

0

16%6% 14%

6%

4%

3%

1%

3%

1%

-1

1

1

0

PC3

PC4

-1

1

1

0

-1

1

1

0

-1

1

-1

0

PC5

oot

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nk

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rso

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LowTors

rso

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% 1 . 0%1%3

LFoo

RFoo

RLarm

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LFoo

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LFoo

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Fig. 2 Composition of the first five principal components (PC) for a typical subject and the three planes. The amount of variance explained by

each PC is displayed on the graphs. These PCs are typical from those already described in previous studies

190Exp Brain Res (2009) 197:185–197

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Figure 6 shows the time profile of DV along the stride

cycle averaged across subjects. Overall, DV was positive

for the sagittal plane, but not different from zero for the

frontal and horizontal plane. A T test with Bonferroni

corrections was run for the three planes on the averaged

values of the z-transformed DV across the stride cycle. It

confirmed that DV was not different from zero for the

frontal horizontal planes ([T(6) = -1.261, P[0.05] and

[T(6) = 1.815, P[0.05], respectively), but significantly

positive for the sagittal plane [T(6) = 4.207, P\0.01].

However, large disparities between subjects were observed

in the horizontal plane (e.g. Fig. 6). Three of the seven

subjects showed an overall strong synergy (DV averaged

across the stride for these three subjects was 0.46 ± 0.08),

while four others did not (DV averaged across the stride for

these four subjects was 0.00 ± 0.21).

In general, we see that the typical time profile of DV has

maxima during swing phase and minima during double

stance. This is especially striking for the sagittal plane, and,

to a lesser extent, for the frontal plane. To support this

observation, DV was averaged for both the double stance

and swing phase. Figure 7 shows these results averaged

across subjects. For each of the three planes, a one-way

ANOVA with factor Phase (double support and swing) was

run on the z-transformed values. It confirmed that there

were statistically significant increases of DV between

double support and swing phases for both the frontal and

thesagittalplane([F1,12= 10.02,

[F1,12= 62.89, P\0.01], respectively), but no significant

differencesforthehorizontal

P[0.05]. T test with Bonferroni corrections were run on

the averaged values of the z-transformed DV for the two

phases and the three planes. For the frontal plane,

DV was significantly negative for the double support phase

andnon-different fromzero

([T(6) = -3.59, P\0.05] and [T(6) = 0.52, P[0.05],

P\0.01]and

plane[F1,12= 0.29,

fortheswingphase

0.6

Frontal

0.2

0.4

0.15

0

Horizontal

0

0.05

2

3

4

Sagittal

% of the stride

Total Variance in the E..V. space * 10^6(No Units)

0

1

0102030 405060 708090100

0.1

Fig. 3 Total variance in the elemental variable averaged across

subjects (± one standard deviation) for the three planes. Due to the

normalization of the angular momentum (see text for details), the

variance had no units. For a better readability, average of the data

across 5% times windows are displayed. Note the increase of variance

around the heel strike. Note also that the peak of variance arose earlier

for the sagittal planes than for the two other planes

0.4

0.5

0.6

Frontal

VUCM

VORT

0.1

0.2

0.3

0

0.1

0.15

Horizontal

0

0.05

3

4

Sagittal

Variance * 10^6 (no units)

0

1

2

% of the stride

010

20

30405060708090 100

Fig. 4 Time profiles of the two components of the variance in the

elemental space (VUCM, black bars, and VORT, white bars) averaged

across subjects (± one standard deviation) for the three planes

Exp Brain Res (2009) 197:185–197191

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respectively). No significant results were found for the

horizontal plane ([T(6) = 1.35, P[0.05] and [T(6) =

1.88, P[0.05] for the double support and swing phase,

respectively). For the sagittal plane, DV was significantly

negativeduringthedouble

P\0.01] and significantly positive during the swing phase

[T(6) = 8.61, P\0.01].

Large disparities between subjects, observed in the

horizontal plane (e.g., Fig. 6), highlights this point: three

subjects showed an overall strong synergy (DV averaged

across the stride for these three subjects was 0.46 ± 0.08),

while four others did not (DV averaged across the stride for

these four subjects was 0.00 ± 0.21).

support[T(6) = -2.91,

Detrended fluctuation analysis

A DFA was performed on the nonlinear time series of the

variability of the PC magnitude that lies on the orthogonal

subspace (‘‘orthogonal variability’’). Figure 8 displays the

box plots of scaling exponent a obtained from this analysis.

One can remark that they are always less than 0.5, indi-

cating that the orthogonal variability time series are anti-

persistent. One-sample T tests performed confirmed that

a was statistically different from 0.5 for each of the

three planes ([T(6) = -11.44, P\0.01], [T(5) = -6.14,

P = 0.01], [T(6) = -14.72, P\0.01]: the frontal, hori-

zontal and sagittal plane, respectively).

Discussion

In ‘‘Introduction’’ of this study, we made two hypotheses:

(H1) there will be a synergy among the SAM, stabilizing

the WBAM; (H2) the strength of these synergies will

change during a stride, especially between the gait phases.

The results obtained confirmed both of these hypotheses:

we did observe a particular organization of the individual

segmental angular momentum (SAM) with regard to their

Double Support

Swing

1.5

2

^6 (n.u.)

ariance * 10^

Va

FrontalHorizontal Sagittal

*

*

*

0.4

0.5

0.08

0.1

0

0.5

1

0

0.1

0.2

0.3

0

0.02

0.04

0.06

VUCM

VORT

VUCM

VORT

VUCM

VORT

Fig. 5 The two components of the variance in the elemental space

(VUCMand VORT) averaged for the two phases (double support black

bars and swing phase white bars) and across subjects (± one standard

deviation) for the three planes. The stars and arrows indicate

significant differences (P\0.05) between double support and swing

phase. For the frontal plane, both components decrease during the

swing phase compared to the double support phase, although this was

only significant for the VORTcomponent. For the horizontal plane, no

significant differences were found. Note the opposite evolution of the

two components in the sagittal plane (increase of VUCMand decrease

of VORTbetween double support and swing phase)

Frontal

0.5

1

-1.5

-1

-0.5

0

∆V

Index of Synergy

Horizontal

Sagittal

% of the stride

0

10 2030405060708090 100

0.5

1

-1.5

-1

-0.5

0

0.5

1

-1.5

-1

-0.5

0

Fig. 6 Time profile of the index of synergy DV averaged across

subjects (± one standard deviation) for the three planes. For a better

readability, average of the data across 5% times windows are

displayed. Note the decrease in DV during the double support phase,

especially for the sagittal plane

192Exp Brain Res (2009) 197:185–197

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effect on the WBAM, and the strength and nature of these

synergies varied between the gait phases. These results are

discussed below.

The whole body angular momentum is a controlled

variable

The authors previously suggested that the WBAM was

highly regulated during gait (Popovic et al. 2004; Herr and

Popovic 2008). However, by this, the authors meant that the

WBAM remained small during the gait cycle, despite sub-

stantial individual SAM. They notably demonstrated that

individual SAM can be grouped into a small number of

primitives, whose particular composition leads to small

valuesofWBAM.Thepresent studynotonlyconfirms these

results, but goes a step further: it also investigates how the

primitives were used to regulate the WBAM time profile.

In this study, it was clearly observed that the individual

SAM were grouped into a small number of independent

primitives. The origin of such grouping remains unclear.

As stated in ‘‘Introduction’’, this issue (the origin of the

grouping of individual variables into smaller number of

primitives) is still debated in the literature. Although it is

usually attributed to the CNS (e.g., this is strongly suggested

in Robert and Latash 2008 and Asaka et al. (2008) based on

the changes in the composition of the primitives during the

reaction to an external postural perturbation and during the

learning process of a challenging task), mechanics is sus-

pected to be involved in some instances (e.g., Hicheur et al.

2006 for the co-variation of the lower limb elevation angles

during walking activities). Nevertheless, wherever this

grouping originates from, its major consequence is the

dimensionality reduction of the control problem to be

solved by the CNS.

The index of synergy DV quantifies the co-variation of

the EV (the five first principal components representing the

SAM in this study) across repetitive trials to limit the

variability of an important PV (here the WBAM). In

the way it is computed, values different from zero indicate

that the EV are organized in a specific way that lead to a

decrease (DV positive) or an increase (DV negative) in the

variability of the PV. These two cases will be discussed in

more detail in ‘‘Synergies and ‘‘anti-synergies’’’’. How-

ever, in both cases, DV different from zero indicates that

the motor abundance is organized in a non-trivial way with

regard to its effect on the PV (remember that for a random

process, the variance would be spread randomly between

the two subspaces resulting in a DV statistically not dif-

ferent from zero). One can thus claim that, in both cases,

the PV is of particular importance for the controller.

In this study, we found that DV was significantly dif-

ferent from zero for two of the three planes. This is a clear

indication that, during walking activities, the WBAM is a

particular variable that the CNS tends to regulate via the

individual SAM. This result constitutes a support to the

numerous studies on humanoid gait control, which

Double Support

Swing

Frontal

Horizontal

Sagittal

*

0.5

1

*

*

*

*

-1

-0.5

0

Index of Synergy ∆V

-1.5

0.5

1

-1

-0.5

0

-1.5

0.5

1

-1

-0.5

0

-1.5

Fig. 7 Index of synergy DV for the double support phase (black bars)

and swing phase (white bars), averaged across subjects (± one

standard deviation) for the three planes. The stars over the bars

indicate that DV is significantly different from zero (P\0.05). The

stars over the arrows indicate a statistically significant change of DV

(P\0.05) between the double support and swing phases. See text for

details. Note the change of sign for DV in the sagittal plane

1

0.5

0

Scaling exponent α

Frontal

Horizontal

Sagittal

Fig. 8 Scaling coefficient a for the three planes, and results of the

detrended fluctuation analysis performed on the ‘‘orthogonal vari-

ability’’ time series (see text for details). Note that in the three cases,

the values of a are smaller than 0.5. One clear outlier (represented by

the cross) was found for the horizontal plane and was not considered

in the statistical analysis

Exp Brain Res (2009) 197:185–197193

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considered the WBAM to be a controlled variable (Mitobe

et al. 2004; Goswami and Kallem 2005). Details about this

control will be discussed in the following sections.

Although the control of the WBAM (in the sense DV

was different from zero) was clear in the sagittal plane,

such control was less, or not present, in the two other

planes: DV was statistically not different from zero for both

phases in the horizontal plane, and for the double support

phase in the frontal plane. An explanation could be the

need (or the absence of the need) of an effective control.

The basic hypothesis of the UCM analysis is that the CNS

cares about the value of the PV. If this hypothesis is not

verified, the index of synergy will be close to zero (random

repartition of the variability of the EV between the UCM

and orthogonal manifolds) and no synergies, according to

our definition, can be observed. In this study, subjects were

asked to walk over the belt of the treadmill, without any

specific requirements or feedbacks. Therefore, the only

needs for an active stride-to-stride control were the balance

demands and matching walking speed to the belt velocity

(although this latter need was a loose constraint due to the

length of the belt). This loose need of control could explain

the large disparities observed between subjects in the

horizontal plane: three subjects demonstrated strong syn-

ergies (average DV for these three subjects was 0.46), while

for the four others no synergies could be observed (average

DV was 0.00). It could be interesting to focus on this

question by studying these synergies while walking with

various situations of constraints: over ground walking (less

constrained), walking on a tiny bar (constrained in the ML

plane), etc.

Synergies and ‘‘anti-synergies’’

In this study, both positive and negative values of the index

of synergy were observed. Positive values of DV have been

classically described in literature (see Latash et al. 2007;

Latash 2008 for reviews). They have been interpreted as

multi-segmental synergies among the EV (the PCs repre-

senting the SAM in this study) stabilizing the PV (the

WBAM). In other words, positive values of DV observed in

this study revealed that the CNS organized the variability

of the individual SAM in such a way that it mostly lays in a

particular subspace that produces identical values of

WBAM. By doing so, it ensured a reproducible pattern of

the WBAM across strides while reducing the need to

generate detailed time courses for all EV. This notably

increased the robustness to the perturbation, which can be

critical during walking activities.

Negative values of DV were less classically described

than positive values of DV. By oppositions to the synergies,

they have been referred to as ‘‘anti-synergy’’. It has been

principally observed in situations where a quick shift of the

PV was required (e.g., quick change in finger force pro-

duction) and was understood as a way for the CNS to not

have to fight against its own pre-existent synergies (Wang

et al. 2006; Goodman and Latash 2006; Olafsdottir et al.

2008). In this study, however, such ‘‘anti-synergies’’ were

observed at a specific time of the gait cycle (the double

support phase) not associated with higher rate of change of

the WBAM (see Fig. 1). Another explanation for the role

of these anti-synergies could be the fact that the controller

cares about adjusting (not making it reproducible) the value

of the PV from stride to stride. Two facts tend to support

this. First, periods where negative values of DV were

observed were mainly associated with an increase in VORT

(see Fig. 5). The second fact comes from the results of the

DFA performed on the ‘‘orthogonal variability’’ (the vari-

ability of the EV projected onto the subspace orthogonal to

the UCM, which directly affects the PV). This analysis

gives information on the nature of the correlations in the

analyzed time series. Consistent

Cusumano et al. (2008), it was found that the ‘‘orthogonal

variability’’ is an anti-persistent time series, i.e., a positive

deviation at one frame of the stride is more likely to be

followed by a negative one at the next frame. This type of

anti-persistent time series has been interpreted as the result

of an error-correcting type of control (Peng et al. 1992).

Considering these two facts together (anti-synergies being

associated with an increase in the ‘‘orthogonal variability’’

and this orthogonal variability corresponding to an error-

correcting control), the anti-synergies observed in this

study can be associated with a need of adjusting the PV

(the WBAM).

with resultsfrom

Modulation along the gait cycle

The human gait cycle has been extensively described in

literature. One of the well-known characteristics is that the

cycle can be viewed as two different phases: single support

phase or swing phase, during which only one foot is in

contact with the ground, and double support phase, during

which both feet are in contact with the ground. The system

(wholebody)configurationisverydifferentforeachofthese

phases:notably,itcanbeviewedfromakinematicalpointof

view as an open loop system (swing phase) or a closed loop

system (double support phase). This has consequences in

terms of stability and control. For example, it has been

shown that children with CP tend to increase the duration of

the double support phase as it is more stable (Russell 2008;

Hsueetal.2009).Inthisstudy,weinvestigatedtheevolution

of the synergies along the stride cycle, and, more particu-

larly, the differences between single support and double

support phases. This study represents, to the authors’ best

knowledge, the first description of the temporal evolution of

multi-segmental synergies during walking activities.

194Exp Brain Res (2009) 197:185–197

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Results showed that the strength of the synergy (the

value of DV) changed significantly between double support

and swing phase, for at least two of the three planes. More

specifically, for the frontal plane, the index of synergy DV

was negative during the double support phase and non-

different from zero during the swing phase. For the sagittal

plane, DV switched from negative to positive values. This

indicates a modification of the nature and goal of the

control between double support and swing phase (see

‘‘Synergies and ‘‘anti-synergies’’’’ for a discussion of the

meaning of positive and negative values of DV).

Interpretation of the changes of the nature and goals of

the control is delicate. More particularly, it is difficult from

our results to firmly claim that the change in the control

observed between double support and swing phase cannot

be attributed to mechanical effects (the structure changes

from a complex chain with kinematical loop to a less stable

open chain between these two phases). Nevertheless, an

interesting interpretation is to link the change in control to

the difference of stability during the gait phases. This

relation between mechanical stability and the use of syn-

ergies was already proposed by Black et al. (Black et al.

2007). They observed that children with Down syndrome

exhibited stronger kinematical synergies during walking

than typically developed children. They explained this

finding by the fact that children with Down syndrome take

more profit of their motor abundance than typically

developed children to counteract the fact that they are

inherently less stable than their typically developed peers.

In the present study, we similarly observed that during the

swing phase of the gait, during which the body is in a less

stable state, the subject tends to take profit of the motor

abundance of the segments (positive values of index of

synergy). This is understood as a way to produce repro-

ducible patterns of an important whole body parameter (the

angular momentum in this study) with an increased

robustness against perturbations. On the contrary, during

the more stable double support phase, the segmental

redundancy was used to adjust the value of the whole body

parameters from stride to stride. These results also dem-

onstrate that motor abundance can be used in two different

ways during the same task.

These observations can be linked with studies on gait

control of bipedal robot. Notably, the zero moment point

control (ZMP) is a classical method that has often been

applied to robotics imitating humanoid bipedal gait,

including the Honda humanoid Robot ASIMO (Hirai et al.

1998; Asano et al. 2000; Kajita et al. 2001; Ono and

Takahashi 2001). The principle is to control parameters

such as feet placement or ankle propulsive torque in order

to adjust the WBAM. This control is typically employed

just prior to and during double support. Results of the

present study, showing that the WBAM is a controlled

variable and suggesting that the nature and goal of the

control are different between double support and swing

phase, bring a new support to these methods.

Complementarities between the detrended fluctuation

analysis and the uncontrolled manifold analysis

In this study, two types of analyses were used to investigate

the variability of the EV. The UCM framework was used to

identify the sharing patterns among the EV that ensure a

reproducible value of the PV. The UCM analysis investi-

gates the structure of the variance of the EV and is per-

formed across strides, each stride being considered

independent of the previous one. The DFA was used to

investigate the temporal evolution of the variance of the

EV, and more particularly the nature of the correlations

that might exist in these signals. This analysis is performed

along strides, all strides being considered in a unique time

series. These two analyses, different by nature, bring dif-

ferent and complementary information about the control of

a given variable.

In this study, for example, the DFA showed that the

variance of the EV laying in the subspace orthogonal to the

UCM (VORT) has the same properties as if it was the result

of an error-correcting type of control, i.e., tends to be

maintained close to an objective profile. Note that due to

the properties of the orthogonal subspace (the subspace

where the changes in EV affects the PVs), results of the

DFA obtained on VORTcould be extended to the variance

of the PV. This could be interpreted as a way for the CNS

to maintain the PV close to an objective time profile (time

profile that is likely to be similar from stride to stride).

However, such conclusion would not be redundant with the

results of the UCM analysis. Remember that the aim of this

analysis is to understand how the motor abundance is used

to make the PV reproducible across strides (i.e., are there

synergies that stabilize the PV across strides?). To do so,

the relative difference between the variances within and

orthogonal to the UCM is investigated (see ‘‘Methods’’ for

the definition of the index of synergy DV). Information

about both components of variance is thus needed in order

to draw a conclusion about the presence or absence of

synergies. This point was clearly emphasized by studies

focusing on learning motor synergies (Latash et al. 2007):

diminutions with practice of the amount of variance

orthogonal to the UCM were associated with an increase

(Yang and Scholz 2005; Asaka et al. 2008), decrease

(Domkin et al. 2002) or stagnation (Domkin et al. 2005)

of DV.

Moreover, although synergies have been classically

described in situation where an error-correcting type of

control (feedback control) is suspected, theoretical models

of feedforward control able to reproduce features of

Exp Brain Res (2009) 197:185–197195

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synergies have been proposed (Goodman and Latash

2006). Therefore, one should not infer the type of control

(result of the DFA) from the presence or absence of syn-

ergies (result of the UCM analysis).

Finally, the UCM analysis gives a picture of the struc-

ture of the variance of the EV at a given instant of the task.

When performed at successive instants of a task, it can

bring insight into the evolution of synergies along the task.

This was used in this study, for example, to reveal the

evolution of synergies between the gait phases. This kind

of information is difficult to obtain from a DFA, which

consider the time series as a whole.

Summary

In this study, we showed that the WBAM is a particular

variable for the CNS, which tends to regulate its value

along the gait cycle by exploiting the motor abundance of

the human body. We also showed that this motor abun-

dance was used in different ways during the different

phases of the gait cycle. During the swing phase, in which

the body is in a less stable state, the CNS tends to profit

from the segments’ redundancy to produce reproducible

patterns of WBAM with an increased robustness against

perturbations. This was interpreted as synergies among the

individual SAM, stabilizing the WBAM. On the contrary,

during the more stable double support phase, the segmental

redundancy was used to adjust the value of the WBAM

from stride to stride. This was interpreted as ‘‘anti-syner-

gies’’ among the individual SAM used to adjust the

WBAM.

These results bring insights into the control of bipedal

walking. They could be used in the field of human walking,

including impaired gait and rehabilitation, and has direct

implications in gait control of bipedal robot. The results

could be extended by studying these synergies in walking

with various situations of constraints.

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