2020 IEEE 2nd Global Conference on Life Sciences and Technologies (LifeTech 2020)

Comparison of gait variability and stability indices
Takashi Inagaki, Shogo Okamoto, Tomoyuki Iwasaki, Yasuhiro Akiyama, and Yoji Yamada
Department of Mechanical Systems Engineering, Nagoya University, Japan

Abstract—Indices of variability and stability are essential
in evaluating the motions of the human gait. Owing to theinconsistent results of earlier studies, the relationships among
popular indices, i.e., maximum Lyapunov exponent (λ), MeanSD,and long-range correlation (α), are inconclusive. In this study,
we examined the correlations among these indices using thevelocity of the center of body mass for 10 participants walking
on a treadmill. In terms of the velocity along the gravitational
direction, MeanSD exhibited a negative correlation with theshort-term λS and the long-term λL when the walking speed was
4.0 km/h. Although a positive correlation between MeanSD and
λ is expected based on the meanings of these indices, the present
and earlier studies collectively suggest that there is no positive
correlation between MeanSD and λ. In the same direction, apositive correlation was found between λS and α when thewalking speed was 3.5 km/h. MeanSD and α tended to exhibit
a negative correlation for some conditions, which is reasonablebecause MeanSD and α indicate the level of variability and thelong-term consistency of walking, respectively.
Index Terms—gait variability, human locomotion, long-range
correlation, maximum Lyapunov exponent, MeanSD

I. I NTRODUCTION
To create systems that assist human locomotion, it is essential to quantify the variability and stability of walking. Manyindices have been proposed and compared thus far.
MeanSD is a popular index for gait variability [1], [2]. It
is the mean of the standard deviation of walking parameters.
Another index is the long-range correlation (α). It evaluates thesystem’s error-tolerance and resistance to external perturbationand quantiﬁes the long-term consistency of the system. The
maximum Lyapunov exponent (λ) is an index for gait stability.
It quantiﬁes the reaction of the system against an inﬁnitesimal
perturbation [1], [3].
Although the relationship between λ and MeanSD has
been discussed in the past, their relationship remains unclear.
Dingwell et al. [3] did not ﬁnd signiﬁcant correlations between
λ and MeanSD. Terrier et al. [4] reported a negative correlation between the long-term maximum Lyapunov exponent(λL ) and MeanSD for gait motions on a treadmill. Brujinet al. [1] reported a positive correlation between the shortterm maximum Lyapunov exponent (λS ) and MeanSD. Thesestudies are inconsistent in terms of the relationships between
λ and MeanSD. For α and λ, Terrier et al. [4] reported that
there was no correlation between them.
In this study, we examined and discussed the correlations
among maximum Lyapunov exponent, MeanSD, and longrange correlation using the data of 10 healthy males walking
on a treadmill [5].

978-1-7281-7063-3/20/$31.00 ©2020 IEEE

II. C OMPUTATION OF GAIT INDICES
A. Maximum Lyapunov exponent
The maximum Lyaounov exponent was computed as follows [3], [4], [6]. State vector S(t) was constructed fromdiscrete time series q(t), which was the velocity of the centerof body mass in the present study, as S(t) = [q(t), q(t +τ ), . . . , q(t + (d − 1)τ )] where τ is the time delay and dis the dimension of S(t). S(t) forms a limit cycle in thestate space over t. Euclidean distances between neighboring
trajectories of the limit cycle were calculated. Subsequently,
they were averaged over all discrete time points to obtain
the average logarithmic rate of divergence. The slope of theaverage logarithmic rate of divergence is represented by λ. Inthis study, we computed the slope using the ﬁrst 0.5–1 stride
for the short-term (λS ) and fourth to tenth strides for the longterm (λL ). We set τ to be 10% gait cycle and d = 5.
B. Long-range correlation
The long-range correlation was computed as follows [4],[7]. The original time series was ﬁrst integrated to be anothertime series y(k) of length N as described in [4] and [7].
Subsequently, y(k) was divided into boxes with equal lengthof n. In each box, a regression line yn (k) was ﬁtted to y(k).
y(k) was then detrended by subtracting the local trend yn (k)in each box. The average ﬂuctuation F (n) for box size n wascalculated byvuN
u1 X[y(k) − yn (k)](1)
F (n) = tN
k=1

The gradient of logF (n) to log(n) was deﬁned as the longrange correlation (α). α usually ﬁts into 0.5–1.0, and a greater
α value indicates more self-correlated walking. We varied thebox size n from 100 to 1000.
C. MeanSD
MeanSD is an index of gait variability. To compute
MeanSD, the periods of walking were ﬁrst divided into 100%
gait cycles. For each percentage, the standard deviation of thegait parameter, i.e., the velocity of the center of body mass,was calculated. The mean of the standard deviations over allpercentages is the MeanSD [3].
III. WALKING MOTION DATA
Ten participants’ walking motions on a treadmill wererecorded at 100 Hz at both 3.5 km/h and 4.0 km/h [5]. For
each participant, we calculated the indices of Sec. II using thevelocities of the center of body mass. We excluded the ﬁrst ten

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TABLE I: Correlation coefﬁcients between MeanSD, maximum Lyapunov exponent (λL and λS ), and long-range correlation (α). ∗ and +indicate the signiﬁcance levels of p = 0.05 and p = 0.10, respectively.

MeanSD and α
MeanSD and λS
MeanSD and λL
λS and α
λL and α

x
0.09
-0.25
0.32
0.03
0.11

3.5 km/hy−0.40+
-0.32
0.30
0.14
-0.35

z
-0.36
-0.16
0.09
0.75∗
0.33x
−0.59+
0.20
0.11
0.10
0.42

4.0 km/h
yz-0.44-0.03-0.18−0.64∗-0.42−0.72∗
0.37
-0.34
0.40
-0.07

x-0.27
-0.10
0.24
0.06
0.30

Overallyz
-0.40
-0.22
-0.21
-0.21
-0.1
−0.40+
0.32
0.37
0.24
0.09

V. D ISCUSSION

Fig. 1: Walking on treadmill
strides and selected 11th to 50th strides from the recorded datato compute the walking indices. One gait cycle was deﬁned
from the left heel contact (0%) to the next left heel contact(100%). The coordinate system was deﬁned as follows: rightas the x-axis, forward as the y-axis, and up as the z-axis asshown in Fig.1.
IV. R ESULTS
Table I shows the correlation coefﬁcients among the gait
indices. The correlation coefﬁcients were calculated by thewalking speeds, i.e., 3.5 or 4.0 km/h. Further, the values shownin the three right columns were computed by collectively
using the two walking speed conditions. Most values were notstatistically signiﬁcant, in part because of the small numberof participants. For 4.0 km/h along the z-axis, we found anegative correlation (r = −0.64, p = 0.047) between MeanSDand λS and a negative correlation (r = −0.72, p = 0.020)
between MeanSD and λL . For 3.5 km/h along the z-axis, wefound a positive correlation (r = 0.75, p = 0.012) between λS
and α. In addition, we observed that MeanSD and α tended
to exhibit a negative correlation.

MeanSD is an index of the gait variability, and λS is anindex of the instability at the beginning of walking. Thus, it isunintuitive that MeanSD and λS exhibit a negative correlation.
However, in earlier studies, Dingwell et al. [3] and Terrier etal. [4] reported the same results (the overall correlation wasweak and negative). A positive correlation between λL , whichaccumulates relatively long-term information of walking, andMeanSD, which calculates the variability of the entire walk,seems natural. However, such positive relationships have notbeen reported so far [1], [3], [4]. In our study, a weaknegative correlation between them was observed. The presentand related studies suggest that there is no positive correlation between λ and MeanSD. Note that there is no generalexplanation for the relationships between these two popular
gait indices.
A negative correlation between MeanSD and α is understandable, because MeanSD indicates the variability andα indicates the long-term consistency of walking. As thevariability increases (MeanSD increases), the long-range correlation decreases (α decreases). This relationship was also
reported in [4].
A positive correlation was found between α and λS for
3.5 km/h along the z-axis. In an earlier study, Terrier et al. [4]stated “they seem not correlated together.” The computationof α largely depends on the box size, n, and we are unable
to determine the relationship between α and λ without furtherresearch.
VI. C ONCLUSION
We examined correlations among the gait variability andstability indices for treadmill walking, for which there has been
no agreement thus far. Unexpectedly, the maximum Lyapunov
exponent λ and MeanSD exhibited negative or insigniﬁcantcorrelations. The present and earlier studies [3], [4] suggest
at least that there is no positive correlation between them.
Furthermore, for a limited condition, a positive correlation wasobserved between λS and α. MeanSD and α tended to exhibit
negative correlations.
ACKNOWLEDGMENT
This work was supported in part by AMED (Japan Agency
for Medical Research and Development 19he2002003h0302)
and JSPS KAKENHI (26750121 and 19K21584).

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