Proceedings of the 2020 IEEE/SICE
International Symposium on System Integration
Honolulu, Hawaii, USA, January 12-15, 2020

Principal motion ellipsoids:
Gait variability index based on principal motion analysis
Tomoyuki Iwasaki, Shogo Okamoto, Yasuhiro Akiyama, and Yoji Yamada
Abstract— We developed a gait variability index on the basisof principal motion analysis (PMA). PMA decomposes multivariate time-series data into independent bases. The resulting
variability index evaluates how individual gaits vary along thesebases. Hence, it allows us to analyze how and to what extent
the gait motions of humans vary, unlike previous popular
indices, which aimed to quantify the level of gait variability.
We computed this variability index for lower limb joints andthe velocity of the center of mass during walking. These indicesexhibited higher correlation coefﬁcients with MeanSD, which isa typical gait variability index. The meanings of the independentbases was successfully interpreted, suggesting the validity andusability of the new gait variability index.

I. INTRODUCTION
There is a large variability in each individuals’ walking,and this gait variability can be related to the risk of fall [1].
England et al. and Bruijn et al. discussed gait variability
in terms of walking speed [2], [3]. Owings et al. focusedon the variability of the step length during walking [4].
There are some studies that discussed gait variability within
individuals not only during straight walking, but also during
curving motion [5], [6]. Thus, many researchers have alreadystudied gait variability. Indices to represent gait variability
are invaluable in generalizing these research works andenhancing commercial applications.
MeanSD [7] and the maximum Lyapunov exponent [8] are
the most popular indices to evaluate gait variability. MeanSD
is the mean value of standard deviation of parameters such as
joint angles. The maximum Lyapunov exponent quantiﬁes1
the average logarithmic rate of the divergence of a systemafter a perturbation, and it is often calculated by using theinformation from an accelerometer or internal measurement
unit attached to the waist. If the values of these indices arelarge, the gait variability within individuals is large. However,given that these indices are represented by single scalar quantities, they are not suitable to discuss how walking motions
vary, although they are certainly suitable for discussing thelevel of gait variability.
Human motions are sometimes analyzed by principal
motion analysis (PMA) [9]. This method decomposes humanmotions into bases called principal motions without losing
the information of interlocked multiple degrees of freedom of
redundant systems. This method has mainly been applied to*This work was in part supported by AMED (Japan Agency for Medical Research and Development 19he2002003h0302) and JSPS KAKENHI(26750121 and 19K21584).
All are with the Department of Mechanical Systems Engineering, Nagoya
University, Nagoya, Japan
1 The maximum Lyapunov exponent is also used as a gait stability index.

978-1-7281-6667-4/20/$31.00 ©2020 IEEE

generate human and robot motions to reduce the dimensionsof parameter spaces [9]–[13]. PMA has also been adapted
to classify and identify individual gaits [14] and sit-to-standmotions [15].
This study aims to develop a new index for gait variability
that is inherently represented by multivariate time series.
For this purpose, we analyze the gait variability within
individuals by applying PMA on walking data measured bya camera-based motion capture system. PMA allows us toanalyze the variability that occurs during the walking phase.
Therefore, our index represents both the level and quality ofvariability, i.e., represents walking motion varies how and atwhich phase within the gait cycle. We compute the variability
indices using the joint angles or the velocities of the centerof mass (CoM) during walking and verify the validity of
the gait variability index. For this purpose, MeanSD and ourindex based on PMA are compared.
The experiment in this study was conducted with theapproval of the Institutional Review Board of the Schoolof Engineering, Nagoya University (#18–2).
II. EXPERIMENT OF WALKING MOTION
A. Experimental method and participants
A walking experiment was conducted on 12 adults (height:
172.5 ± 3.8 cm, weight: 63.6 ± 7.2 kg), having no neurological and musculoskeletal abnormalities. Written consent
was obtained prior to the experiment. The participants were
instructed to repeat straight walk, within a distance ofapproximately 5 m, with their faces turned to the front.
B. Motion data analysis
We extracted a motion–deﬁned gait cycle from continuous
walking by considering data from the left heel contact to thenext left heel contact. The gait cycle was normalized to 0–
100% to align the data length of each trial (Fig. 1). The
measurement space was deﬁned in the reference coordinate
system, taking the horizontal direction as the x-axis (rightfor positive), the anterior posterior direction as the y-axis(anterior for positive), and the vertical direction as the z-axis(upper for positive) according to the right-handed system.
The values of joint angles were positive in the direction of
ﬂexing and negative in the direction of extending from thebasic standing posture. We analyzed hip ﬂexion angles, knee
ﬂexion angles, and ankle dorsiﬂexion angles for the left andright legs. We deﬁned the hip ﬂexion angle as the angle
between the vertical line passing through the hip joint andthe straight line connecting the hip joint and the knee joint.
The joint angles were calculated by ﬁtting the motion of the489

here, x̄ represents the mean value of the motions of all theparticipants.
From the eigenvector expansion of covariance matrix
X T X, we obtain
X ∼ Y V T,
(4)
(1)

Fig. 1. Gait cycle normalized to 0–100%. Deﬁned from the left heel contact(0%) to the next left heel contact (100%).

(s′ )where Y = (y1 , ..., yk′ )T is a score matrix that represents
how much information of each principal motion is included
by each observed motion and V = (v1 , ..., vq , ..., vr ) is aprincipal motion matrix composed of r eigenvectors corre′ ′sponding to the r largest eigenvalues; vq (∈ Rs k ×1 ) is
the q-th principal motion vector whose eigenvalue is theq-th largest. Any motion can be represented by a linear
combination of the principal motion vectors weighted bytheir corresponding scores:
(s)

(s)

xk ∼ V yk .

(5)markers to the human model of SIMM (Mulsculographics
Inc., U.S.) by using the least-squares method. We also
analyzed the velocities of the center of mass (CoM) in the
direction of the three axes. These velocities of the CoM wereobtained by differentiating the position of the CoM of thewhole body calculated using Zatsiorsky’s method [16].

2) Extraction of the repetition error within individuals:
The variability of the walking motion is evaluated by considering the repetition errors within individuals. A matrix′ ′of repetition errors within individuals B (∈ Rs k ×pu ) is
computed asIII. GAIT VARIABILITY INDICES

B = (x1 − x̄(1) , ..., xk − x̄(s) , ..., xk′ − x̄(s ) )T , (6)A. Gait variability indices based on principal motion analysis

where x̄(s) represents the mean of the motion of experimental participant s . A score matrix Yb and a principal motion
matrix Vb of this matrix B are obtained from the eigenvector
expansion of covariance matrix B T B as follows:

1) Principal Motion Analysis (PMA): Park et al. [9]extended principal component analysis (PCA) to make itapplicable to time-series data and achieved a representationof human motions with a reduced number of variables. PCA
is a multivariate analysis method that identiﬁes the combinations of correlated variables, which are called principal
components. These components, which are calculated fromthe same data pool, are linearly independent of each other.
PMA interlocks the variables along the temporal dimension
as well as among multiple variables. The combination ofinterlocked variables in the temporal direction is called theprincipal motion. PMA is a linear analysis method, and it ispossible to interpret the meanings of the obtained principal
motions.
Suppose that a human motion at a certain instant isrepresented by p variables. For variable i (i = 1, ..., p) at kth trial (k = 1, ..., k ′ ), the time-series data vector θik , which
consist of discretized u moments, is given asθik = (θik1 , ..., θikl , ..., θiku )T .

(1)

Using this, we create an extended column vector xk consisting of p variables as
xk = (θ1k T , ..., θik T , ..., θpk T )T .

(2)

(s)

Note that xk represents a time-series column vector of theexperimental participant s (s = 1, ..., s′ ) at trial k. If weobtain k ′ sets of walking motion from each s′ participants
as motion samples, the time-series data of all the motions′ ′are represented by a matrix X (∈ Rs k ×pu ) as follows:
(1)

(s)

(s′ )

X = (x1 − x̄, ..., xk − x̄, ..., xk′ − x̄)T

(3)

(s′ )

(s)

(1)

B ∼ Yb VbT .

′(7)

We use the size of the error ellipsoid calculated individually from the distribution of the principal motion scores Yb
as a gait variability index. We let this error ellipsoid be the(s)principal motion ellipsoid (PM ellipsoid). Suppose that am
represents the length of the major axis m (m = 1, ..., m′ ) of
the PM ellipsoid of participant s with m′ being the number(s)of principal motions calculated from B. Also, am is thesquare root of the eigenvalue of the covariance matrix of thescores of participant s. We deﬁne the size of the PM ellipsoidc(s) by using the length of each axis as
′

c

(s)

=

m
∑

(s)
am
.

(8)

i=1

Fig. 2 shows the deﬁnition of the size of PM ellipsoidfor participant s when using the ﬁrst and second principal
motions. In the latter example, we used three principal
motions.
B. MeanSD
Many indices evaluating variability are based on the standard deviation of the analyzed variables [1]. In this study, we
explore MeanSD as a standard index of gait variability that
employs standard deviation. MeanSD is an mean value ofstandard deviation based on the gait cycle. It has been used
so far for the joint angle of the lower limbs [7], [17], [18]and for velocity [3], [17], [19], [20] and acceleration [7],490Fig. 2. Deﬁnition of the size of PM ellipsoid for participant s. An error
ellipsoid (red) is calculated from the distributed samples (black dots).

[21] obtained from an accelerometer attached to the waist.
For a participant, SD(u) represents the standard deviation at
the moment u on a gait cycle of a variable. Then, MeanSD,
which represents gait variability on the whole gait cycle ofthe variable, is deﬁned as
MeanSD = ⟨SD(u)⟩, u ∈ {0 − 100% gait cycle}(9)

where ⟨ ⟩ represents the mean value of all the temporal
points [7]. We computed MeanSD for joint angles andvelocities of the CoM. We calculated each MeanSD of six
angles for the joint angles and velocities of three directions
for the velocity of the CoM, respectively, and we deﬁned
their mean value as MeanSD of the joint angle and thevelocity of the CoM.
IV. RESULTS
In this study, we analyzed the 1st–3rd principal motions.
Fig. 3 shows the relationship between MeanSD and the sizeof the PM ellipsoid c computed from the six joint angles(top) and the three velocity components (x, y, and z) of theCoM (bottom). The correlation coefﬁcient for MeanSD andc for the joint angles is 0.96 and the correlation coefﬁcientfor MeanSD and c for the velocity of the CoM is 0.95. These
results indicate that MeanSD and the size of the PM ellipsoid
are similar indices for the degree of gait variability.
V. DISCUSSION
A. Interpreting principal motions for joint angles
We interpret how the gait variability within individuals
exposes by interpreting each principal motion. Fig. 4 shows
the time-series change of the loads of the 1st–3rd principal
motion. Given that the load represents gait variability within
individuals, if the load is 0, the variability is small, and if theload is positive or negative, the joint angle varies. If the load
at a particular time instant is positive, the joint angle ﬂexes
more than the mean of the individual. The contribution ratioof each principal motion is as follows: for the 1st principal
motion it is 29.3%, for the 2nd principal motion it is 11.1%,
and for the 3rd principal motion it is 8.0%.
Top of Fig. 4, which represents the 1st principal motion,shows that the hip and knee joints tend to ﬂex (positive),

Fig. 3. The relationship between MeanSD and the size of the PM ellipsoid
c computed from the six joint angles (top) and the three velocity components(x, y, and z) of the CoM (bottom).

and the ankle joints tend to plantarﬂex (negative) in theﬁrst half of the swing phase whereas the knee joints tend
to extend (negative) in the second half of the swing phase.
It can be considered that this motion represents the advanceof the phase because this motion seems to be differentiatedthe average walking motion. Fig. 5 shows the knee ﬂexion
angles of gait samples with maximum and minimum scoresof the 1st principal motion for one participant. Gait samples
with maximum scores of the 1st principal motion shifted thegait phase earlier than minimum ones. Therefore, we caninterpret that the 1st principal motion is the principal motion
related to the speed of shifting the gait phase.
The walking motion with positive 1st principal motion
score shifts to the free leg quickly. Conversely, the walking
motion with negative score shifts to the free leg slowly. The
variance of the gait sample along the 1st principal motion isgreater than the variance along the other main motions, andthe 1st principal motion indicates main variance of walking
motions for many participants. Variations in the length of
the standing and swing phases for this motion as describedabove account for a large proportion in gait variation.
Middle of Fig. 4, which represents the 2nd principal
motion, shows that the hip and knee joint of the right leg tend
to ﬂex (positive), and the ankle joint of the right leg tends491

Fig. 5. Gait samples with maximum (blue line) and minimum (black line)scores of the 1st principal motion. Solid line: left knee angle. Dotted line:
right knee angle.

B. Interpreting principal motions for the velocity of the CoM

Fig. 4. Loads of each principal motion computed from joint angles. Top:
1st principal motion. Middle: 2nd principal motion. Bottom: 3rd principal
motion.

to plantarﬂex (negative) in the ﬁrst half of the gait phase
whereas the knee joint of the left leg tends to ﬂex (positive)in the second half of the gait phase. This motion means thatthe left leg ﬂexes signiﬁcantly more than the average motionin the latter half of the swing phase, and as compensation for
it, the hip and knee joints ﬂex and the ankle joint plantarﬂex
at the moment of the right toe gets off. In short, this motionmeans that the left step length is smaller than the right step
length. Therefore, the 2nd principal motion represents leftright asymmetry.
Bottom of Fig. 4, which represents the 3rd principal
motion, shows that the ankle joints of both of legs tend to
dorsiﬂex (positive), and the knee joints of both of legs tend
to ﬂex (positive) just before heel contact. This motion has
long double stance phase and this is common for walking
with a small step length. Therefore, the 3rd principal motion
represents the step length.

Fig. 6 shows the time-series change of the loads of the 1st–
3rd principal motion. As discussed above, the load represents
gait variability within individuals. If the load at a particulartime instant is positive, it indicates that the velocity of
the CoM is larger than the mean of the individual. The
contribution ratio of each principal motion is as follows: forthe 1st principal motion it is 40.3%, for the 2nd principal
motion it is 7.8%, and for the 3rd principal motion it is
5.9%.
Top of Fig. 6, which represents the 1st principal motion,shows that the velocity of the CoM in the anterior direction(y axis) is large (positive) throughout a gait cycle. Therefore,
we can interpret that the 1st principal motion is related
to the walking velocity towards anterior direction. The 1stprincipal motion represents the main variance of walking
motions for many participants. Hence, for the velocity of theCoM, variations in the walking velocity towards the anterior
direction account for a large proportion of gait variation.
Middle of Fig. 6, which represents the 2nd principal
motion, shows that the velocity of the CoM in the horizontal(x axis) and anterior (y axis) directions is large (positive) in
the left stance phase, whereas the velocity of the CoM in thehorizontal (x axis) and anterior (y axis) directions is small(negative) in the right stance phase. The right step lengthis larger than the left step length and the CoM is close to
the swing leg. Fig. 7 shows the angles of gait samples withmaximum and minimum scores of the 2nd principal motionof one participant. In the walking sample with the large 2nd
principal motion score, the left knee angle is relatively small
compared with the right. Therefore, the second principal
motion represents left-right asymmetry.
Given that the 3rd principal motion is the motion thatdifferentiated the average walking motion, as the bottom of
Fig. 6 shows, it can be considered that this motion represents
the advance of the phase. Therefore, we can interpret thatthe 3rd principal motion is related to the speed of gait phaseshifting.

492

Fig. 8. Error ellipsoids of participants C, G, and I obtained from principal
motion scores. The size and the direction of major axes of the error ellipsoids
differ among participants.

C. Relationship of PM ellipsoid and gait variability

Fig. 6. Loads of each principal motion computed from the velocity of the
CoM. Top: 1st principal motion. Middle: 2nd principal motion. Bottom: 3rdprincipal motion. CoM X, Y, and Z means the velocity of the CoM in x, y,
and z directions, respectively.

We analyzed the variations in walking within individuals
by using PM ellipsoid as an index of gait variability. The
1st principal motion of joint angles and the 3rd motion ofvelocities of the CoM represent the speed of gait phaseshifting, whereas the 2nd principal motion of joint angles
and velocities of the CoM represent left-right asymmetry.
In Fig. 8, the PM ellipsoids for joint angles obtained fromthe experiment are shown on the 1st and 2nd principal motion
plane. Participant G was the participant whose PM ellipsoidwas smallest whereas participant C and I exhibited the largest
PM ellipsoids. The direction of the major axis of the PM
ellipsoid of participant C was mostly along the 1st principal
motion whereas that of participant I was mostly along the
2nd principal motion. This indicates that the gait variabilityof participant C is mainly related to the speed of gait phaseshifting and the gait variability of participant I is mainly
related to left-right asymmetry. The size and direction ofmajor axes of the PM ellipsoids differed among participants.
Thus, each principal motion represents how walking motions
vary within individuals. MeanSD and Lyapunov exponent
are scalar values to express the size of walking variation.
PM ellipsoids also express how gait motions vary. From thisviewpoint, PM ellipsoids can be used as a general index toobtain walking variations.
VI. C ONCLUSION

Fig. 7. Gait samples with maximum (blue line) and minimum (black line)scores of the 2nd principal motion. Solid line: left knee angle. Dotted line:
right knee angle.

We proposed the size of the PM ellipsoid as a new
gait variability index. We compared MeanSD, which is a
representative gait variability evaluation index that has beencommonly used in the literature, with the size of the PM
ellipsoid obtained from the principal motion score. As aresult, there was a high correlation between the two indiceswhen we considered both joint angles of lower limbs andvelocities of the CoM. The variability index based on PMA
is a useful index because after understanding how walking
motions vary, we can interpret the meanings of the principal
motions.

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R EFERENCES[1] S. Bruijn, O. Meijer, P. Beek, and J. Van Dieen, “Assessing the stabilityof human locomotion: a review of current measures,” Journal of theRoyal Society Interface, vol. 10, no. 83, p. 20120999, 2013.
[2] S. A. England and K. P. Granata, “The inﬂuence of gait speed on local
dynamic stability of walking,” Gait and Posture, vol. 25, no. 2, pp.
172–178, 2007.
[3] S. M. Bruijn, J. H. van Dieen, O. G. Meijer, and P. J. Beek, “Is slow
walking more stable?” Journal of Biomechanics, vol. 42, no. 10, pp.
1506–1512, 2009.
[4] T. M. Owings and M. D. Grabiner, “Variability of step kinematics inyoung and older adults,” Gait and Posture, vol. 20, no. 1, pp. 26–29,
2004.
[5] Y. Akiyama, H. Toda, T. Ogura, S. Okamoto, and Y. Yamada,“Classiﬁcation and analysis of the natural corner curving motion ofhumans based on gait motion,” Gait and Posture, vol. 60, pp. 15–21,
2018.
[6] Y. Akiyama, S. Okamoto, H. Toda, T. Ogura, and Y. Yamada, “Gaitmotion for naturally curving variously shaped corners,” Advanced
Robotics, vol. 32, no. 2, pp. 77–88, 2018.
[7] J. Dingwell, J. P. Cusumano, P. Cavanagh, and D. Sternad, “Local dynamic stability versus kinematic variability of continuous overgroundand treadmill walking,” Journal of Biomechanical Engineering, vol.
123, no. 1, pp. 27–32, 2001.
[8] J. B. Dingwell and J. P. Cusumano, “Nonlinear time series analysis ofnormal and pathological human walking,” An Interdisciplinary Journalof Nonlinear Science, vol. 10, no. 4, pp. 848–863, 2000.
[9] K. J. Frank C. Park, “Movement primitives and principal component
analysis,” in On Advances in Robot Kinematics, 2004, pp. 97–106.
[10] N. Yamada, S. Okamoto, H. Okumura, Y. Yamada, K. Isogai, and
K. Kawakami, “Similarities and differences in manual stretching ofphysical therapists for equinovarus,” in 2014 IEEE/SICE International
Symposium on System Integration. IEEE, 2014, pp. 490–495.
[11] K. Mishima, S. Kanata, H. Nakanishi, T. Sawaragi, and Y. Horiguchi,“Extraction of similarities and differences in human behavior using
singular value decomposition,” IFAC Proceedings Volumes, vol. 43,
no. 13, pp. 436–441, 2010.
[12] H. Kang and F. C. Park, “Humanoid motion optimization via nonlinear
dimension reduction,” in Proceedings of IEEE International Conference on Robotics and Automation, 2012, pp. 1444–1449.
[13] S. Morishima, K. Ayusawa, E. Yoshida, and G. Venture, “Wholebody motion retargeting using constrained smoothing and functional
principle component analysis,” in Proceedings of IEEE-RAS 16th
International Conference on Humanoid Robots, 2016, pp. 294–299.
[14] T. Akiduki, K. Kawamura, Z. Zhang, and H. Takahashi, “Extraction
and classiﬁcation of human gait features from acceleration data,” International Journal of Innovative Computing, Information and Control,
vol. 14, no. 4, pp. 1361–1370, 2018.
[15] T. Iwasaki, S. Okamoto, Y. Akiyama, and Y. Yamada, “Generalized
principal motion analysis: classiﬁcation of sit-to-stand motions,” inProceedings of IEEE-GCCE 8th International Conference on Consumer Electronics, 2019, pp. 679–681.
[16] P. De Leva, “Adjustments to Zatsiorsky-Seluyanov’s segment inertia
parameters,” Journal of Biomechanics, vol. 29, no. 9, pp. 1223–1230,
1996.
[17] H. G. Kang and J. B. Dingwell, “Separating the effects of age andwalking speed on gait variability,” Gait and Posture, vol. 27, no. 4,
pp. 572–577, 2008.
[18] D. H. Gates, B. J. Darter, J. B. Dingwell, and J. M. Wilken, “Comparison of walking overground and in a computer assisted rehabilitation
environment (caren) in individuals with and without transtibial amputation,” Journal of NeuroEngineering and Rehabilitation, vol. 9, no. 1,
p. 81, 2012.
[19] J. B. Dingwell and L. C. Marin, “Kinematic variability and local
dynamic stability of upper body motions when walking at different
speeds,” Journal of Biomechanics, vol. 39, no. 3, pp. 444–452, 2006.
[20] M. Punt, S. M. Bruijn, H. Wittink, and J. H. van Dieen, “Effect ofarm swing strategy on local dynamic stability of human gait,” Gait
and Posture, vol. 41, no. 2, pp. 504–509, 2015.
[21] P. Terrier and O. Deriaz, “Kinematic variability, fractal dynamics andlocal dynamic stability of treadmill walking,” Journal of NeuroEngineering and Rehabilitation, vol. 8, no. 1, p. 12, 2011.

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