Proceedings of the 2021 IEEE International
Conference on Intelligence and Safety for Robotics
Nagoya, Japan, March 4-6, 2021

Walking motions with high margin-of-stability values
Tomoyuki Iwasaki, Shogo Okamoto, Yasuhiro Akiyama, Takashi Inagaki, and Yoji Yamada

Abstract— Many researchers studied on gait parameters,such as step width and speed, correlated with gait stability
indices. In contrast, to understand stable gait motions against
fall, we computed the time-series of joint angles that are
correlated with a popular kinetic gait stability index: marginof stability (MoS). A time-series extension of the partial leastsquare regression was applied on the joint angles to construct
the principal motions, i.e., motion elements included in gait
motions. These principal motions were linearly independent ofeach other and correlated with MoS. The linear combination of
three principal motions produced a quantity that was correlatedwith the minimal MoS values having a coefﬁcient of 0.87.
Interpretation of the obtained principal motions suggests thatthe stable gait is of the small takeoff and step lengths.

I. I NTRODUCTION
Falling while walking can lead to serious disabilities suchas fractures and sprains for the elderly; hence, the reductionof such incidents is a major area of focus to aid the aging
society. The clariﬁcation of the gait motions with a high riskof falling will be useful for risk assessment, reduction, andpractical applications such as the evaluation and developmentof walking aids. Thus far, there have been several studies
on gait stability. England et al. reported that highly stablegaits are observed at slower walking speeds inferring fromthe correlation between the gait speeds and the maximum
Lyapunov exponents calculated from the variability of joint
angles [1]. However, if the gait speed is slower than thesuitable speed, walking becomes rather unstable [2], [3].
Elderly people tend to exhibit unstable walking motion
because of the variability of step length [4]. Thus, it has beenreported that gait stability depends on some gait parameterssuch as gait speed and stride length. Some studies also
focused on how humans avoid falling when they stumble
or come in contact with obstacles [5], [6].
Margin of stability (MoS) [7], [8] is a popular kinetic index
to represent gait stability. MoS represents the kinetic margin
against falling. Thus far, many studies have analyzed therelationship between various gait parameters and MoS. For
example, the MoS in the medio-lateral direction increases
at high cadence [7], [9]. Similarly, the MoS in the forward
direction increases when the step length is large and the gait
speed is small [10]–[12]. The objectives of these studies are
to identify the gait parameters that are correlated with MoS.
These gait parameters including step length, cadence, andgait speed are scalar values; however, gait is a redundant*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.

978-0-7381-4674-4/21/$31.00 ©2021 IEEE

multi-degree-of-freedom system and time-series involving
the whole body. Gait motions that exhibit large MoS values
and are less likely to result in a fall need to be investigatedusing the time-series of multiple human-limb joints.
In the earlier studies, researchers tried to ﬁnd gait parameters that are highly correlated with the kinetic stability index.
Differing from them, we identify multi-joint time-series
during walking that are highly correlated with MoS by using
statistical approaches. This difference in the approach may
substantially divide earlier studies and ours. For this purpose,
we apply partial least squares regression (PLS) [13], which isa supervised multivariate analysis method, to principal motion analysis (PMA) [14], [15], which is an analysis method
for multi-variate time-series. This approach is expected toresult in a new motion parameter highly correlated with
MoS because PLS produces compound quantities for which
the correlation coefﬁcients with MoS are maximized. By
combining PLS and PMA, we can calculate scores that arehighly correlated with the MoS, and we can understand themeaning of them by interpreting the principal motions. We
apply this method to a subset of the gait motion database [16]measured by a camera-based motion capture system.
II. G AIT STABILITY INDEX : MARGIN OF STABILITYMoS is estimated by using the extrapolated center of mass
(XCoM) [7], which assumes a walking human body as aninverted pendulum of height l as shown in Fig. 1. The yaxis is positive in the forward direction, and the z-axis ispositive in the upward direction. The coordinate vector of
XCoM (xcom ), which is computed by the velocity vector of
the center of mass (CoM), (vcom ) represents the range of the
CoM movement at each instant. The position vector and thevelocity vector of CoM at a certain instant are represented
as com and vcom , respectively. xcom is computed as√lxcom = com + vcom
(1)gwhere g is the gravitational acceleration. MoS (mos ) is
computed by using the coordinate vector of the end of
the base of support (BoS), (bos ) and x′com , which is the
projection of xcom on the ground as
mos = bos − x′com .

(2)

As the CoM is outside the BoS and far from its end, the bodyis unstable and unless a leading step supports the body, a falloccurs [10], [17]. In this study, we only deal with the MoS
in the forward direction (y-axis). Generally, the minimum
MoS value in the forward direction is observed just before
the heel contacts during a gait cycle.

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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 . Modiﬁed
from [14].

Fig. 2. Gait cycle and MoS value. (a) Gait cycle (deﬁned from the left-heelcontact (0%) to the next left-heel contact (100%)) normalized to 0–100%.
Modiﬁed from [14]. (b) Example of the variation in MoS along the y-axis(mos,y ) with the gait cycle. Modiﬁed from [14].

III. G AIT MOTION DATAThis study used a part of the gait database [16], which
included ten elderly participants (age: 67.8±2.5 (mean andstandard deviation) years, height: 160.1±8.1 cm, weight:
59.3±7.7 kg) who declared no walking disabilities. They
were instructed to walk on a 10 m straight course attached
a force measurement sensor at their comfortable gait speeds.
An optical motion capture system was used to record the
motions of their entire body. The details of the measurement
are provided in [16].
We analyzed 50 gait samples in total, i.e., 5 samples foreach participant. Each gait sample included two steps startingand ending at left-heel contacts and was normalized to 0–
100% (Fig. 2 (a)). The measurement space was deﬁned in thereference coordinate system with the medio-lateral direction
on the x-axis (right for positive), the front-back direction on
the y-axis (forward for positive), and the vertical direction on
the z-axis (upper for positive). Each joint angle is represented
as a relative angle of the distal segment seen from theproximal segment. In this study, the direction of the joint
ﬂexing and extending from the basic standing posture on thesagittal plane was set as positive and negative, respectively.
We analyzed hip ﬂexion, knee ﬂexion, and ankle dorsiﬂexion
angles considering both the legs. An exemplar variationin MoS along the forward direction along a gait cycle isrepresented in Fig. 2 (b). The minimum MoS value during a
gait cycle was recorded immediately before the heel contact
at ∼50% or ∼100%.

some principal motions whose scores are highly correlatedwith the minimum MoS value. That is, the minimum value of
MoS was estimated by a linear synthesis of multiple principal
motions that are independent of each other. The principal
motion was the time-series of the angles of the six lowerlimb joints. The minimum value of MoS was standardized(z-score) among all trials. Joint angles were standardized
across all trials after discretizing every 1% of the gait cycle.
For the joint i (i ∈ 1, ..., 6) at the k-th trial (k ∈ 1, ..., k ′ ),let ji,k be the time-series vector of the joint angles of 101
discretized moments (one value every percent of gait cycle):ji,k = (ji,k,1 , ji,k,2 , ..., ji,k,101 ).

(3)

By using this, we constructed an extended column vector ak(∈ R606×1 ) asak = (j1,k , j2,k , ..., j6,k )T .

(4)

The time-series data of all the motions are represented by′
the matrix A (∈ Rk ×606 ) as follows:
A = (a1 , ..., ak , ..., ak′ )T .

(5)

Here, k ′ is the number of gait samples and k ′ = 50 in thepresent study.
The model formulas of PLS in this problem are as follows:A=

IV. C ALCULATION OF THE GAIT MOTION CORRELATED

n
∑

ti pT
i + EA(6)

i=1

WITH MARGIN OF STABILITY

We identiﬁed the gait motions correlated with MoS by
using the motion data of the joint angles during walking. For
this purpose, we applied PLS [13] to PMA [14], [15]. PLS
determines a quantity as the correlation with the objective
variable which is maximized by a linear synthesis of the
explanatory variables. Using this method, we can calculate

mos,min =n∑qi ti + eb

(7)

i=1
′

Here, mos,min (∈ Rk ×1 ) represents the vector of the minimum MoS values obtained from each trial. n is the number′of principal motions, ti (∈ Rk ×1 ) is the score vector for thei-th principal motion, pi (∈ R606×1 ) is the loading of the i-th

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

0.68

Second principal motion

0.44

Third principal motion

0.37

Forth principal motion

0.25

principal motion, and qi is the regression coefﬁcient for thei-th principal motion. EA and eb are the residuals of A andmos,min , respectively. First, using these formulas, t1 wasdetermined to maximize the covariance of t1 and mos,min .
p1 and q1 are calculated so that the sum of the squares of theelements of EA and eb are minimized, respectively. Then,t2 was determined to maximize the covariance of t2 andmos,min − q1 t1 . The third and subsequent principal motions
are also determined in this way. In each principal motion,
the motion with a high score is a stable motion with a large
MoS. Adopting up to n-th principal motion, the minimum
value of MoS is estimated as follows;m̄os,min =n∑q i ti .

Fig. 3.

Correlation between mos,min and m̄os,min .

(8)

i=1

V. R ESULTS
Correlation coefﬁcients between mos,min and the principal motion scores are summarized in Table I. The principal motions exhibited strong to moderate relationships,i.e., 0.68–0.25, with the minimum MoS. In our analysis,
we considered motions up to the third principal motion
considering the magnitude of the correlation coefﬁcients. We
approximated the gait motions by a linear combination of thethree principal motions;m̄os,min = 0.08t1 + 0.04t2 + 0.06t3 .

(9)As shown in Fig. 3, which is the scatter plot of mos,min
and m̄os,min , the linear synthesis of the three principal
motions was highly correlated with the minimum value of
MoS. Further, Fig. 4 shows the distribution of the ﬁrst andsecond principal motion scores. Clusters were formed foreach participant, indicating that the principal motions well
represent the gait characteristics of each individuals.
VI. D ISCUSSION
Herein, we interpret the implications of the principal
motions based on their time-series proﬁles. Fig. 5 shows
the contents of the ﬁrst–third principal motion vectors that
represent the time-series of the standardized joint angles.
When the motion load is positive, the joint ﬂexes more thanthe mean value of all trials at a certain instant because themotion is standardized.
Fig. 5 (a) shows the ﬁrst principal motion. The hip andknee joints of both legs tend to extend (negative) throughthe entire phase. The right ankle joint tends to dorsiﬂex(positive) between ∼90% (immediately before the left-heelcontact) to ∼50% (at the right heel contact). Similarly, theFig. 4. Scatter plot of the ﬁrst and second principal motion scores. Trialsof the same participant are denoted by the same symbols.

left ankle joint tends to dorsiﬂex (positive) between ∼40%
(immediately before the right heel contact) to ∼100% (atthe left-heel contact) of the gait cycle. These characteristics
suggest that the ﬁrst principal motion expresses the gait withweak takeoffs, i.e., weak kicking motions at toe-off. As thesecharacteristics are prominent, the minimum MoS values arehigh and the gait motions are more stable.
Note that people with the small takeoff walk slowly.
Previous studies have shown that when the walking speed
is slow, the gait stability is large in terms of the short-term
Lyapunov exponent [1], [18], [19], which is consistent withthe ﬁrst principal motion. However, in contrast, it is reported
that slow walking directly increases the risk of fall [20] andlowers clinical balance scales [21]. The effects of gait speed
on stability are not fully consistent among earlier studies.
As shown in Fig. 5 (b), in the second principal motion, thehip joints tend to ﬂex (positive) through the entire gait cycle.
The knee joints tend to ﬂex (positive) in the stance leg. This
motion represents a small step length with bending knees
and stooping hip. As mentioned in Section 1, with smaller
stride lengths, the MoS values in the forward direction are
greater [10]–[12]. As such, the second principal motion
reasonably pertains to the MoS value.
As in Fig. 5 (c), in the third principal motion, the minimum

226these principal motions can be kinetically stable, referring toearlier studies on stable walking. Although we analyzed thedata of ten participants in the present study, the original gait
motion database included more participants. To obtain better
results, the data of a greater number of participants should
be included in future studies. Further, in future studies, ourapproach will be applied along the medio-lateral direction as
well as the frontal-back direction.
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Fig. 5. Substance of each principal motion. (a): ﬁrst principal motion. (b):second principal motion. (c): third principal motion. A positive value at an
instant indicates that the angle is greater than the average angle among alltrials.

angle for the left hip and maximum angle for the right hip are
observed at left-heel contact (∼0%). Similarly, the maximum
angle for the left hip joint and the minimum angle for theright hip joint are observed at right heel contact (∼100%).
This motion indicates small angular changes in the hip joints.
The knee joints in both legs tend to extend (negative). The
ankle joint tends to planterﬂex (negative) in the stance leg,and dorsiﬂex (positive) in the swing leg. Therefore, thismotion may represent a small step length without stooping.
VII. C ONCLUSION
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