2021 IEEE/SICE International Symposium on System Integrations (SII)
Iwaki, Fukushima, Japan. January 11 - 14, 2021

Kinematic gait stability index highly correlated withthe margin of stability: Concept and interim report
Tomoyuki Iwasaki, Shogo Okamoto, Yasuhiro Akiyama, Takashi Inagaki, and Yoji Yamada
Hak et al. compared the short-term Lyapunov exponents
with the minimum MoS values in the forward direction andthe medio-lateral direction and reported that they exhibit
different properties [4], [10]. Vieira et al. compared theshort-term Lyapunov exponents in the three axial directions
with the minimum MoS values in the forward and mediolateral directions during the walking of fatigued people. They
found no substantial correlation among them [11]. Yang etal. compared the feasible stability region with the minimum
MoS values, Floquet multiplier, and Lyapunov exponent for
the fall prediction and concluded that they exhibited different
properties [12].
One of their goals was to ﬁnd kinematic stability indicesthat are correlated with the kinetic stability indices. If thiscorrelation can be found, the kinetic stability indices canbe replaced with easy-to-access kinematic ones. This is
beneﬁcial especially for commercial applications. However,as mentioned above, earlier studies have suggested that there
is no substantial correlation between popular kinetic andkinematic stability indices. In the present study, we employ
an approach that is different from those of previous studies.
We construct a new kinematic index that is correlated witha popular kinetic index.
In this work, we construct a new kinematic gait variability index that is highly correlated with a popular kinetic
stability index i.e., the MoS. For this, we extend partial
least squares regression (PLS) [13], which is a supervised
multivariate analysis, and apply it to a subset of the gait
motion database [14], which is measured by a camera-based
motion capture system. This kinematic index uses the timeseries of velocity at the center of human body mass that canbe easily measured by an inertial measurement instrument
such as the one built-in phones.

Abstract— Gait stability indices that are easy to measure andcompute are necessary to enhance their commercial applications. We constructed a new kinematic gait variability index
that is highly correlated with a popular kinetic stability index,
i.e., the margin of stability (MoS). The new stability index wascomputed by using the velocity of the center of the humanbody mass, which can be easily measured by equipment suchas inertial measurement instruments. A time-series extensionof partial least squares regression was applied on the velocity
series to construct the principal motions. These motions wereindependent of each other and were correlated with MoS. The
linear combination of these principal motions exhibited a highcorrelation coefﬁcient of 0.82 with the minimum values of MoS,
suggesting that the proposed index can be used as an easy-tomeasure alternative of the MoS.

I. I NTRODUCTION
Falling down while walking is a major concern for theelderly. Thus far, many studies on gait stability and stable
walking have been conducted. For example, a previous
study found that the stability of gait depends on some gaitparameters such as the gait speed [1], [2] and step length [3],[4]. Some studies focused on how humans avoid falling after
tripping or coming in contact with obstacles [5], [6]. In
particular, indices to represent gait stability are invaluable
for judging the risk of falling and for enhancing commercial
applications such as walking aids [7].
There are two types of gait stability indices: kinetic andkinematic. Kinetic indices are computed based on the lawof physical dynamics, and they directly represent the risk or
safety margin of falls; however, their computation requires
the measurement of the positions or angles of multiple humanbody parts. Further, some indices require the measurement ofground reaction forces. Hence, kinetic indices are suitable forlaboratory environments. The margin of stability (MoS) [8]is a representative kinetic gait stability index. In contrast,
kinematic indices are usually computed by using kinematic
information of fewer body parts. Such information canbe easily acquired through inertial measurements. Further,because they do not require the measurement of forces,
the kinematic stability indices are relatively easy to obtain;
however, they may not be directly related to kinetic stability.
For example, the maximum Lyapnov exponent [9] is often
used.
Considering these properties of the two types of stability
indices, many studies have compared them. For example,

II. G AIT STABILITY INDEX : MARGIN OF STABILITY
In this study, we evaluate the MoS values of participants
by using the extrapolated center of mass (XCoM) [8]. For
this computation, as shown in Fig. 1, the human body wasapproximated to be an inverted pendulum with a constantlength of l. The y-axis is the forward direction, and the zaxis is the upward direction. The coordinate vector of XCoM
(xcom ) was computed by the velocity vector of the centerof mass (CoM) (vcom ), and it represents the range of the
CoM movement at each instant. MoS is the distance vector
between the XCoM and the end of the base of support (BoS).
The larger the MoS value, the greater the margin against
falls, because falls are induced when the CoM is outside theend of BoS. When the MoS value is negative, a human falls*This work was in part supported by AMED (Japan Agency for Medical Research and Development 19he2002003h0302) and JSPS KAKENHI(19K21584).
All are with the Department of Mechanical Systems Engineering, Nagoya
University, Nagoya, Japan.

978-1-7281-7657-4/21/$31.00 ©2021 IEEE347

Fig. 1. Deﬁnition of MoS along the y-axis. x′com is the projection of
xcom on the ground. mos is calculated by using bos and x′com .

Fig. 2. Gait cycle and minimum MoS value. (a) Gait cycle (deﬁned fromthe left-heel contact (0%) to the next left-heel contact (100%)) normalized
to 0–100%. Modiﬁed from [17]. (b) Example of the variation in MoS along
the y-axis (mos,x ) with the gait cycle.

unless a new leading foot is grounded [15], [16]. The position
vector and velocity vector of CoM at a certain instant are
represented as com and vcom , respectively. xcom is computedby using the natural frequency of the inverted pendulum ω
as
vcom
(1)
xcom = com +ωHere, ω is calculated as
√
g
(2)
ω=
l

coordinate system. These CoM velocities were obtained bydifferentiating the position of the CoM of the whole body.
Fig. 2 (b) represents the variation in MoS along the y-axis
along a gait cycle.
IV. C OMPUTATION OF KINEMATIC GAIT STABILITY
INDEX

By using kinematic or motion data, we constructed thegait stability index that is highly correlated with the marginof stability. For this, we used PLS, which is a kind of
multivariate analysis [13]. PLS determines a quantity througha linear combination of the explanatory variables. The correlation coefﬁcient between the quantity and an objective
variable is maximized. There exist multiple such quantities
that are independent of each other. We applied PLS on
the analysis of principal motions [17], [18], for which the
explanatory variables were the multivariate time-series. In
the present study, the time-series of the three-axial velocityof the center of human body were the explanatory variables.
Then, multiple principal motions that are highly correlatedwith the minimum MoS values were determined.
For the direction j (j = x, y, z) at the k-th trial (k =1, ..., k ′ ), vcom,j,k was the time-series vector of the velocities
of com , comprising discretized 101 moments (one value
every percent of gait cycle):

where g is the acceleration of gravity, and l is the height of
the CoM at an upstanding posture. MoS (mos ) is computedby using the coordinate vector of the end of the BoS (bos )and x′com , which is the projection of xcom on the ground as
mos = bos − x′com .

(3)

We deal with the MoS in the forward direction (y-axis) and
use the minimum MoS value during a gait cycle that isobserved just before heel contact, as shown in Fig. 2.
III. DATA FROM GAIT MOTION DATABASE
In this study, we analyzed a subset of the gait motion
database [14]. This subset included ten adult participants(age: 67.8 ± 2.5 (mean and standard deviation) years,
height: 160.1 ± 8.1 cm, weight: 59.3 ± 7.7 kg) without
any neurological and musculoskeletal abnormalities. They
walked on a 10-meter straight course at their comfortablegait speeds. The motions of their whole body were recordedby using an optical motion capture system. The details of
measurement are provided in [14].
We analyzed 53 gait trials; a trial is deﬁned as a gait
cycle from left-heel contact to the next left-heel contact
while walking. The gait cycle was normalized to 0–100% foraligning the data length of each trial (Fig. 2 (a)). We analyzed
the velocities of the CoM in the three-axial directions byusing the x- and z-axes as well as the y-axis, which is themedio-lateral direction (left for positive) in the referencevcom,j,k = (vcom,j,k,1 , vcom,j,k,2 , ..., vcom,j,k,101 ).

(4)

k ′ is the number of gait samples and k ′ = 53 in the present
study. By using this, we constructed an extended column
vector ak (∈ R303×1 ) asak = (vcom,x,k , vcom,y,k , vcom,z,k )T .

(5)

The time-series data of all the motions are represented by′
the matrix Aobs (∈ Rk ×303 ) as follows:
Aobs = (a1 , ..., ak , ..., ak′ )T .
348

(6)

TABLE I
C ORRELATION COEFFICIENTS BETWEEN mos,min AND PRINCIPAL
MOTION SCORES .
First principal motion

0.71

Second principal motion

0.36

Third principal motion

0.19

In the following procedure, the main formulas of the PLS
are as follows:n∑(7)
Aobs =
ti pT
i + EA

Fig. 3.

Correlation between mos,min and m̄os,min .

i=1

mos,min =n∑qi ti + eb

(8)of the principal motions by inspecting their time-series
characteristics. Fig. 4 shows the time-series change for theload of the ﬁrst–third principal motions. As the motion isnormalized, when the load is positive, the velocity of the
CoM is larger than the mean value of all trials at a certain
instant.
The top of Fig. 4, which is the ﬁrst principal motion,shows that the velocity of the CoM along the y-axis shifted
toward the stance leg from before heel contact to the ﬁrst
half of the stance phase, indicating a relatively larger lateralmotion. Along the y-axis, the CoM velocity tends to be slow
(negative), especially during the double support phase (0–10%, 50–60% of gait cycle). Thus, the ﬁrst principal motion
is a characteristic motion of slow walking in the double
stance phase. In other words, the slower the motion during
the double-support phase, the greater the minimum MoS, andthe gait is judged to be more stable. Just before the double
support phase, the velocity along the y-axis is large, and the
support base is small. Therefore, XCoM moves considerably
ahead of BoS, making the walking motion unstable, and the
MoS value is usually minimum during this phase. Therefore,
it is reasonably speculated that the gait becomes more stableby employing these strategies.
The middle of Fig. 4, which is the second principal motion,
represents that the CoM motion along the x-axis is opposite
to the normal CoM motion because the velocity of the CoM
tends to be toward the swing leg from before heel contact
to the ﬁrst half of the stance phase. The y-axial velocity
is large (positive) during the single support phase, which is
opposite to the normal motion of the CoM. In contrast, themotion of the CoM along the z-axis is the enhancement of
its typical motion as it exhibits the maximum value at toe
off and the minimum value at heel contact. Collectively, thesecond principal motion represents walking motion with asmall change in the velocity of the CoM along the x- andy-axes.
The bottom of Fig. 4, which is the third principal motion,
indicates that the velocity of the CoM along the x-axis issimilar to the reversal of the ﬁrst principal motion. The
velocity of the CoM along the y-axis is slow (negative).
The velocity of the CoM along the z-axis is large (positive)in the single support phase, and it is maximum before heel

i=1′

Here, mos,min (∈ Rk ×1 ) represents the vector of theminimum MoS values obtained from each trial. n is the′number of principal motions, ti (∈ Rk ×1 ) is the score
vector for the i-th principal motion, pi (∈ R303×1 ) is theloading of the i-th principal motion, and qi is the regression
coefﬁcient for the i-th principal motion. EA and eb are
the residuals of A and mos,min , respectively. Based onthese formulas, ﬁrst, t1 was determined by making thecovariance of t1 and mos,min the largest. p1 and q1 arethen computed to minimize the residuals in (7) and (8). t2
was then determined by making the covariance of t2 andmos,min − q1 t1 the largest. Similarly for the third, fourth,and other principal motions. Next, the loading and coefﬁcientof each principal motion were calculated. After adopting
the appropriate number of principal motions, the objective
variable was estimated as follow:n∑q i ti .
(9)m̄os,min =i=1

V. R ESULT
Table I lists the correlation coefﬁcients between mos,min
and the principal motion scores. Based on the magnitudeof the correlation coefﬁcients, we included up to the thirdprincipal motions in our analysis. Strong to moderate, i.e.,
0.71–0.19, linear relationships were observed between thescores and the minimum MoS. The kinematic gait stability
index was computed by the linear synthesis of these threeprincipal motions:m̄os,min = 0.10t1 + 0.05t2 + 0.03t3 .

(10)

Fig. 3 shows the scatter plot of mos,min and m̄os,min . Theircorrelation coefﬁcient was as high as 0.82. The constructedindex is highly correlated with the minimum value of MoS,
unlike previous stability indices.
VI. D ISCUSSION
As aforementioned, the newly constructed kinematic index
exhibited concurrent validity in terms of the correlation withthe minimum MoS values. The semantic validity of theindex is also important. Here, we interpret the meanings349

included more participants, and a greater number of participants should be analyzed in the future. Our approach will
be applied along the medio-lateral direction as well as thefrontal-back direction. Finally, the general applicability of
the index should be tested via cross-validation.
R EFERENCES[1] S. A. England and K. P. Granata, “The inﬂuence of gait speed on
local dynamic stability of walking,” Gait & Posture, vol. 25, no. 2,
pp. 172–178, 2007.
[2] 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.
[3] T. M. Owings and M. D. Grabiner, “Variability of step kinematics inyoung and older adults,” Gait & Posture, vol. 20, no. 1, pp. 26–29,
2004.
[4] L. Hak, H. Houdijk, P. J. Beek, and J. H. van Dieën, “Steps to take toenhance gait stability: the effect of stride frequency, stride length, andwalking speed on local dynamic stability and margins of stability,”
Plos One, vol. 8, no. 12, 2013.
[5] Y. Akiyama, Y. Fukui, S. Okamoto, and Y. Yamada, “Effects of
exoskeletal gait assistance on the recovery motion following tripping,”
Plos One, vol. 15, no. 2, p. e0229150, 2020.
[6] Y. Akiyama, K. Mitsuoka, S. Okamoto, and Y. Yamada, “Experimental
analysis of the fall mitigation motion caused by tripping based on the
motion observation until shortly before ground contact,” Journal of
Biomechanical Science and Engineering, vol. 14, no. 1, pp. 18–00 510,
2019.
[7] S. Bruijn, O. Meijer, P. Beek, and J. ｖ an Dieen, “Assessing thestability of human locomotion: a review of current measures,” Journalof the Royal Society Interface, vol. 10, no. 83, p. 20120999, 2013.
[8] A. Hof, M. Gazendam, and W. Sinke, “The condition for dynamic
stability,” Journal of Biomechanics, vol. 38, no. 1, pp. 1–8, 2005.
[9] J. B. Dingwell and J. P. Cusumano, “Nonlinear time series analysis ofnormal and pathological human walking,” Interdisciplinary Journal of
Nonlinear Science, vol. 10, no. 4, pp. 848–863, 2000.
[10] L. Hak, H. Houdijk, F. Steenbrink, A. Mert, P. van der Wurff,
P. J. Beek, and J. H. van Dieën, “Speeding up or slowing down?:
Gait adaptations to preserve gait stability in response to balance
perturbations,” Gait & posture, vol. 36, no. 2, pp. 260–264, 2012.
[11] M. F. Vieira, G. S. d. S. e Souza, G. C. Lehnen, F. B. Rodrigues, andA. O. Andrade, “Effects of general fatigue induced by incremental
maximal exercise test on gait stability and variability of healthy young
subjects,” Journal of Electromyography and Kinesiology, vol. 30, pp.
161–167, 2016.
[12] F. Yang and Y.-C. Pai, “Can stability really predict an impending sliprelated fall among older adults?” Journal of Biomechanics, vol. 47,
no. 16, pp. 3876–3881, 2014.
[13] S. Wold, M. Sjöström, and L. Eriksson, “PLS-regression: a basic toolof chemometrics,” Chemometrics and Intelligent Laboratory Systems,
vol. 58, no. 2, pp. 109–130, 2001.
[14] Y. Kobayashi, N. Hida, K. Nakajima, M. Fujimoto, and M. Mochimaru, “2019: Aist gait database,” 2019.
[15] D. D. Espy, F. Yang, T. Bhatt, and Y.-C. Pai, “Independent inﬂuence ofgait speed and step length on stability and fall risk,” Gait & Posture,
vol. 32, no. 3, pp. 378–382, 2010.
[16] Y.-C. Pai and J. Patton, “Center of mass velocity-position predictions
for balance control,” Journal of Biomechanics, vol. 30, no. 4, pp. 347–
354, 1997.
[17] T. Iwasaki, S. Okamoto, Y. Akiyama, and Y. Yamada, “Principal
motion ellipsoids: Gait variability index based on principal motion
analysis,” in Proceedings of IEEE/SICE International Symposium on
System Integration, 2020, pp. 489–494.
[18] ——, “Generalized principal motion analysis: classiﬁcation of sitto-stand motions,” in Proceedings of IEEE Global Conference on
Consumer Electronics, 2019, pp. 679–681.

Fig. 4.
Substance each principal motion. Top: ﬁrst principal motion.
Middle: second principal motion. Bottom: third principal motion. A positive
load at an instant indicates that the velocity is greater than the averagevelocities among all trials.

contact. From these characteristics, the third principal motion
may be related to the small step length. In fact, Pai andPatton [16] demonstrated that a decrease in the step length,
for an unchanged walking speed, led to an increase in MoS
in the backward direction.
VII. CONCLUSION
We constructed a new kinematic gait stability index thatis highly correlated with a popular kinetic stability index
i.e., the MoS. This index uses the velocity of the centerof the human body and is designed to be correlated with
the MoS by using a method of principal motion analysis.
Unlike previous kinematic indices, this index exhibited ahigh correlation coefﬁcient of 0.82 with the MoS. Thissuggests that the index can be used as an easy-to-measurealternative of the MoS. In this study, the MoS was computedby using the velocities of the CoM measured by a camerabased motion capture system, but the same results can be
expected by using IMU.
Although we analyzed the data of ten participants in thepresent study, the gait motion database we used originally350