appliedsciences
ArticleEstimating the Margin of Gait Stability in Healthy Elderly Using
the Triaxial Kinematic Motion of a Single Body FeatureZiqi Liu 1,† , Shogo Okamoto 1, *,† , Tomohito Kuroda 1,† and Yasuhiro Akiyama 2
1

2

*
†

Department of Computer Science, Faculty of Systems Design, Tokyo Metropolitan University,
Tokyo 192-0397, Japan; mistyliu1997@outlook.com (Z.L.)
Faculty of Textile Science and Technology, Shinshu University, Matsumoto 390-8621, Japan;
akiyama_yasuhiro@shinshu-u.ac.jp
Correspondence: okamotos@tmu.ac.jp
These authors contributed equally to this work.

Abstract: Gait stability indices are crucial for identifying individuals at risk of falling while walking.
The margin of stability is one such index, known for its good construct validity. Generally, themeasurement of this stability index requires a motion capture system, rendering it inaccessible
for everyday use. This study proposes an alternative approach by estimating the index throughtime-series data of triaxial kinematic motion from a single body feature. We analyzed an open gait
database comprising data from 60 participants aged over 60 to identify the most accurate bodyfeature for estimating the margin of stability. The margin of stability values were estimated by using
principal motion analysis, with the time series of the triaxial translational velocities of a body feature
as predictors. Among the 10 body feature points, the sacral crest provided the highest accuracy,with the correlation coefficients between observation and estimation being 0.56 and 0.54 for the
mediolateral and anterior directions, respectively. Although these values need to be further improved,
these findings pave the way for developing an accessible system to estimate fall risks.
Keywords: gait; margin of stability (MoS); time series; inertial measurement unit; fall; postural stability

Citation: Liu, Z.; Okamoto, S.;
Kuroda, T.; Akiyama, Y. Estimating
the Margin of Gait Stability in Healthy
Elderly Using the Triaxial KinematicMotion of a Single Body Feature. Appl.
Sci. 2024, 14, 3067. https://doi.org/
10.3390/app14073067
Academic Editor: Claudio Belvedere
Received: 29 February 2024
Revised: 1 April 2024
Accepted: 2 April 2024
Published: 5 April 2024

Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms andconditions of the Creative Commons
Attribution (CC BY) license (https://creativecommons.org/licenses/by/

1. Introduction
A variety of gait stability indices have been investigated to identify individuals at ahigh risk of falling [1,2]. A common objective of these indices is to facilitate the identificationof such individuals, enabling the provision of appropriate interventions to mitigate fallrisks [3]. In the pursuit of accessible measures for assessing fall risks, the potential of inertial
measurement units (IMUs) for gait analysis has garnered significant interest [4–10]. For
instance, Kim et al. [4] employed an IMU sensor embedded in a shoe to calculate basic gaitparameters such as stride and walking speed. Jebelli et al. [6] distinguished between the
maximum Lyapunov exponents of cyclic gait patterns, measured by using IMU sensors
attached to the lower thigh, in individuals bearing different loads. Yang et al. [9] developed aprototype of a real-time warning system for falls using an IMU sensor placed inside a jacket.
The margin of stability (MoS) [11,12] is a gait stability index recognized for its strong
construct validity. As outlined in Section 2.2, it is derived from the kinematic stabilityof an inverted pendulum model, representing a simplified walking figure. This index isinstrumental in assessing the risk of falling due to postural instability during both normal and perturbed walking in healthy individuals as well as those with pathological gait
conditions [13]. Although the MoS can be defined for all directions [14], it is commonly
calculated for the anterior and mediolateral directions. Thus far, numerous studies havefocused on the MoS [15–22]. For instance, Ohtsu et al. [15] explored various representativevalues of the MoS, such as mean and integral values, during a gait cycle among healthy
young adults to establish a reference base. Harro et al. [16] found that individuals with

4.0/).

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Parkinson’s disease exhibited higher MoS values than a healthy control group, indicating a difference in walking strategy between groups at high and low fall risks. Such a
conservative strategy has typically been observed in the elderly [17,23,24]. Additionally,
the response of postural stability to unexpected treadmill movements has been analyzed
using the MoS [19,22]. Sivakumaran et al. [25] linked the variability of gait parameters with
MoS variability, proposing that individuals adjust spatiotemporal gait patterns to preserve
postural stability. Thus, the MoS serves as a reliable indicator of postural stability during
walking under various conditions.
The computation of the MoS necessitates the simultaneous measurement of multiple
body points, typically requiring an optical or IMU-based [10] motion capture system,
limiting MoS recordings to laboratory settings. However, if the MoS is estimated, using
devices such as IMUs embedded in smartphones, this can increase awareness amongpeople about their gait instability and falling risk. Although the computation of the MoS
demands positions from multiple body points, the synergy in human motion suggests thatmeasuring fewer points might suffice for estimating the MoS. Iwasaki et al. [26] estimatedthe mediolateral MoS from the triaxial velocities of the center of body mass in healthyelderly participants, achieving a correlation coefficient of 0.55 between the observed andestimated values. The estimation accuracy can potentially be enhanced by including theangular velocities of body segments [27]. While these studies [26,27] affirm the feasibilityof estimating the MoS using a single IMU, the question of estimate accuracy persists. This
concern arises because accurately computing the MoS requires kinematic information fromat least three body features: the center of body mass and both feet [11,12]. Therefore,
this study identified conditions that can improve the estimation accuracy, specificallydetermining the most suitable body points for predicting the MoS.
We compared 10 body feature points in terms of the accuracy of the anterior andmediolateral MoSs. To this end, a principal motion analysis (PMA) [26,28] has been
conducted using triaxial translational velocities. This study builds upon our previous
study [29], which compared three body points, including seven additional feature points,
using nuanced statistical methods, specifically cross-validation. While the sacral crest
was identified as one of the most promising feature points in our earlier work [29], thisstudy explores the potential of discovering other suitable points. Although this study relies
on data recorded by an optical motion capture system, identifying suitable body points
advances us toward accessible MoS measurements using an IMU.
2. Materials and Methods
2.1. Data Used for Analysis
We employed an open database from the National Institute of Advanced Industrial
Science and Technology [30], where the motion data of gait are recorded for 300 healthy
Japanese individuals with the ethical approval of the local review committee. Thus far, thesame database has been employed in many studies, e.g., [17,31–33]. For our analysis, weselected data from 60 individuals, comprising 30 males and 30 females randomly chosen,
all above the age of 60 years. The means and standard deviations of their height andweight were 159.62 ± 7.58 cm and 60.57 ± 9.33 kg, respectively. They walked in a 10 m
straight line in bare feet with a mean (±standard deviation) stride length and step width of
1.26 ± 0.089 m and 0.15 ± 0.027 m, respectively [17]. Each sample included two succeeding
steps, and five samples starting with left heel contact were used for each individual, with
300 samples (60 people × 5 samples) in total. As elaborated in Section 2.4, our analysis
focuses on the correlation coefficient between the observed and estimated MoS values.
With a hypothetical correlation coefficient of 0.5 [26] and setting α and β values at 0.05 and
0.20, respectively, a power analysis [34] indicates a required sample size of 28.
The database recorded the three-dimensional coordinates of various body featuresusing an optical motion capture system at 200 Hz. The time-series coordinates weresmoothed using a Butterworth low-pass filter with a cutoff frequency of 6 Hz.

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As shown in Figure 1, we selected the ten feature points for the analysis. They included
the vertex of the skull, C7, T10, sacral crest, xiphisternum, right shoulder (acrominon), rightelbow (lateral epicondyle), right great trochanter, right knee (femoral lateral epicondyle),
and right toe (center of second and third metatarsals). The criteria of selection were as
follows. First, the coordinates close to the center of body mass and toe were selected because
they were used to compute the MoS. Points close to the center of body mass, i.e., the sacral
crest, T10, and xiphisternum, are available in the database. Second, as the points between
the center of body mass and toe, the great trochanter and knee were covered. Commercial
smartwatches are usually equipped with IMUs; hence, the forearm is an easy-to-accessmeasurement point, and we selected the elbow. Unfortunately, the records for the wrist
were missing for several individuals; hence, we did not select the wrists. We also included
the vertex of the skull considering that an IMU can be fixed to a cap. The C7 and shoulder
were selected mainly for reference, although these points are not practical body features for
fixing IMUs. For the shoulder, elbow, knee, and toe, the laterality, that is, the right or left
side, did not substantially influence the final results.
Vertex of skull

C7
T10
Sacralcrest

Shoulder(acrominon)
Xiphisternum(front)Elbow(lateral epicondyle)Greattrochanter
Knee (femoral
lateral epicondyle)Toe (center of
second and third
metatarsals)
Figure 1. Feature points examined in the study. Open circles indicate the feature points on thefrontal side.

2.2. Computation of Margin of Stability
As shown in Figure 2, the MoS is computed using the position vectors of the centerof mass (xcom ) and the edge point in the base of the support area (xbos ) in the transverse
plane. In this study, xcom is the centroid of the right and left anterior superior iliac spines
and sacral crest, and xbos is the position of the toe. The distance between these two vectors
measures the margin against falling under a still condition. To express the margin during
walking, the MoS adopts the concept of an extended center of mass, that is, the position
vector of the center of mass prospected in the future [11,12]. The position vector of theextended center of mass is defined asxxcom = xcom +

ẋcom
,
ω

(1)

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where ω is the natural angular frequency of an inverted pendulum with a support lengthof l. The height of the center of body mass corresponds to l. ω is calculated using l and thegravitational acceleration g:
r
ω=

g
.
l

(2)

ୠ୭௦
ୡ୭୫
Right side Σ଴

ୟ୬୲

x
y
Walking direction ୡ୭୫


ୠ୭௦

୶ୡ୭୫

୪ୟ୲  0୶ୡ୭୫
Figure 2. Margin of stability. Top view of a walking person.

The MoS values for the anterior direction (y) and mediolateral direction (x) are computed separately. The anterior MoS is defined as
(y)

(y)

mant = xbos − xxcom .

(3)

Notably, x(y) indicates the y component of vector x. Figure 3a shows an example of achange in the extended center of mass (xxcom ) and the endpoint of the base of support (xbos )during a gait cycle. As a representative value, the most critical moment, which is just after
the heel contact, is referred to. Hence, the smaller value between the MoS values at 0% (firstheel contact) and ∼50% (second heel contact) is used as the representative MoS value inthis study. The anterior MoS value can either be positive or negative. Moreover, the greater
the value, the more stable the posture. In the example in the figure, xxcom precedes xbos ,and the anterior MoS is negative.
(a)

(b)
࢞୶ୡ୭୫

0.2

Position in x-axis (m)Position in y-axis (m)

0.5
mant
0

࢞ୠ୭ୱ
െ0.5

െ1
0

20

40
60
Gait cycle (%)

80

mlat
0
mlat

െ0.2
0

100

20

40
60
Gait cycle (%)

80

100

Figure 3. Margin of stability and changes in the extended center of mass and base of support.
(a) Y-direction (anterior) and (b) X-direction (mediolateral).

The mediolateral MoS is defined as
(x)

(x)

mlat = | xbos − xxcom |.

(4)

The mediolateral MoS assumes positive values during normal walking, indicating that theposture remains stable in the mediolateral direction. The value is at a minimum at either∼10% or ∼60% gait cycle, and the smallest value between them is employed as the mostcritical value in this study.

Appl. Sci. 2024, 14, 3067

5 of 162.3. Principal Motion Analysis to Estimate MoS Values from Kinematic Data
We employ PMA [28] to estimate the MoS values from the triaxial translational velocities of a body feature. PMA encompasses a suite of linear time-series data analysis
techniques. It involves decomposing a multidimensional time-series data sample into a
set of independent bases, which are also multidimensional time series. The unsupervised
variant, known as PMA0, extends the principal component analysis to time-series dataand has been applied in the analysis of human motion [35–39]. For this study, we used asupervised variant, PMA1 [28], which is a time-series extension of partial least-squares
regression analysis. In PMA1, a time-series data sample is represented as a linear combination of multiple independent time-series bases. These bases are selected to maximize
the covariance between the objective variable and principal motion scores, which quantify
the extent to which each sample incorporates the bases. Herein, we introduce PMA1 by
first explaining partial least-squares regression analysis, followed by the adaptation of its
explanatory variables to time-series data.
Let x ∈ R p×1 and y be the explanatory vector and objective variable, respectively. xi
and yi are the values from the ith sample (i = 1, . . . , n). These variables are centered suchthat their mean values are zero: x = 0 and y = 0. The explanatory vectors of all the samples
are displayed in a matrix (X ∈ Rn× p ):
 Tx1
 .. 
X =  . .

(5)

xTn

Similarly, the objective values of all the samples are stored in a vector y ∈ Rn×1 :y = [ y1 . . . y n ]T .

(6)

X and y are approximated as a linear summation of a′ (∈ N) principal components:a′

X = ∑ ta l aT + E

(7)

a =1
a′

y = ∑ ba ta + e,

(8)

a =1

where ta ∈ Rn×1 and l a ∈ R p×1 are the scores and loadings of the ath principal component(a = 1, . . . , a′ ). E and e are the error matrix and vectors. Let Xa and ya be
Xa = Xa−1 − ta−1 l aT−1

(9)ya = ya −1 − b a −1 t a −1 .

(10)Here, X1 = X and y1 = y. ta and l a are computed as
XaT ya| XaT ya |tal a = XaT T
ta ta

ta = Xa

ba =

taT
ya ,
taT ta

(11)
(12)
(13)

where | ◦ | is the L2 norm. a′ is the number of principal components to approximate
the original data and is determined by cross-validation. In this study, a′ = 3, which is
close to the typical value of two or three for approximating the gait motions of healthypeople [26,38–40].

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To expand the above-mentioned partial least-squares regression to time-series data, x
is replaced with a time-series vector. In the case of triaxial kinematic data for gait analysis,
each sample includes the translational velocities v along the x, y, and z-directions for
one gait cycle, that is, two succeeding steps. The time-series data are discretized by eachpercentage of the gait cycle. We then acquirevx = [v x,0 , . . . , v x,100 ]T(14)

vy = [vy,0 , . . . , vy,100 ]

T

(15)

vz = [vz,0 , . . . , vz,100 ]

T

(16)

x = [vxT vyT vzT ]T ,(17)

where v{ x,y,z},j is the velocity at j (∈ {0, 1, . . . , 100}) percentage of the gait cycle.
2.4. Comparison of Ten Body Features
To compare the 10 body features, we calculated PMA1 for both the anterior and mediolateral directions, with the critical MoS values as the objective variables. The explanatory
variables were the triaxial velocities of each of the 10 body features described in Section 2.1.
We randomly divided 300 samples from 60 individuals into 10 groups, conducting 10-foldcross-validation. Subsequently, we calculated the root mean square errors and correlation coefficients between the estimated MoS values and corresponding observations as
performance metrics. Then, the body feature with the highest correlation coefficient was
statistically compared to the others. We transformed the correlation coefficients into z
values using Fisher’s z transformation [41] and performed paired t-tests for comparison.
MATLAB (2023b, Mathworks Inc., Natick, MA, USA) was used for these computations andcomparisons. For multiple comparisons, Bonferroni correction of factor nine was adopted
to adjust the significance level.
To assess the impact of random bias in the 10-fold cross-validation, we repeated theanalysis with different random divisions of samples 20 times. In 18 out of 20 iterations, thesame body feature demonstrated the highest performance metrics for both the anterior andmediolateral directions, indicating the robustness of our analysis.
3. Results
Figure 4 shows the mean triaxial velocities of the 10 feature points used for the analysis.
Figure 5 shows the means and standard errors of the correlation coefficients between
the observation and prediction of mediolateral and anterior MoS. The values were calculated from the 10 repetitions of cross-validation. Additionally, the figure displays the means
and standard errors of the root mean squared errors.
For the mediolateral direction, the sacral crest, knee, and shoulder exhibited thegreatest correlation coefficients with r = 0.563, 0.560, and 0.558, respectively. These three
features were followed by C7 and T10, with r = 0.508 and 0.489, respectively. The lowest
correlation coefficient was recorded at the great trochanter and toe with r = 0.330 and 0.250,
respectively. The correlation coefficients are significantly different among the 10 featurepoints (F (9, 90) = 150.4, p < 0.001). The coefficients of the sacral crest that exhibited thehighest value were significantly different from those of the seven features: C7 (t(9) = 7.11,
p = 2.52 × 10−4 ), T10 (t(9) = 4.91, p = 3.76 × 10−3 ), elbow (t(9) = 12.31, p = 2.79 × 10−6 ),xiphisternum (t(9) = 17.12, p = 1.60 × 10−7 ), skull (t(9) = 8.75, p = 4.86 × 10−5 ), great
trochanter (t(9) = 14.85, p = 5.53 × 10−7 ), and toe (t(9) = 22.18, p = 1.64 × 10−8 ).

Appl. Sci. 2024, 14, 3067

1

0

(e)
X-direction
Y-direction
Z-direction

1050
Gait cycle (%)
Average motion of elbow (m/s)

(g)

100

50
Gait cycle (%)

X-direction
Y-direction
Z-direction

2

1

0
0

50
Gait cycle (%)

X-direction
Y-direction
Z-direction

2

100

3

X-direction
Y-direction
Z-direction

2

1

0
0

50
Gait cycle (%)

100

(f)
3

(h)

3

1

0
0

50
Gait cycle (%)

100

(i)
Average motion of knee (m/s)

00Average motion of great troch. (m/s)

2

1

0

(d)
3

2

100

Average motion of xiphisternum (m/s)

50
Gait cycle (%)

Average motion of sacral crest (m/s)

0X-direction
Y-direction
Z-direction

100

3

3

X-direction
Y-direction
Z-direction

2

1

0
0

50
Gait cycle (%)

100

X-direction
Y-direction
Z-direction

2

1

0
0

50
Gait cycle (%)

100

(j)
3

X-direction
Y-direction
Z-direction

2

1

0
0

50
Gait cycle (%)

100

3
Average motion of toe (m/s)

2

(c)

3

Average motion of shoulder (m/s)

X-direction
Y-direction
Z-direction

Average motion of C7 (m/s)(b)

3

Average motion of T10 (m/s)

Average motion of skull (m/s)(a)

7 of 16

X-direction
Y-direction
Z-direction

2

1

0
0

50
Gait cycle (%)

100

Figure 4. Mean translational velocities of 10 feature points during a gait cycle. (a) Skull, (b) T10,(c) right shoulder, (d) sacral crest, (e) xiphisternum, (f) C7, (g) right elbow, (h) right great trochanter,(i) right knee, and (j) right toe.

Appl. Sci. 2024, 14, 3067

8 of 16(a) Mediolateral

***
***
***
**

0.8

1.6

***

1.2

0.6

0.8

0.4

Toe

Trochanter

Knee

Skull

Xiphisternum

Shoulder

0

Elbow

0

T10

0.4

C7

0.2

Sacral crest

Correlation coefficient

1.0

Root mean squared error (cm)

***
***

(b) Anterior
***
***

*
***
***

0.8

1.6

***

1.2

0.6

0.8

0.4

Toe

Trochanter

Knee

Skull

Xiphisternum

Shoulder

0

Elbow

0

T10

0.4

C7

0.2

Sacral crest

Correlation coefficient

1.0

Root mean squared error (cm)*****Figure 5. Means and standard errors of correlation coefficients and root mean squared errors
between estimated and observed MoS values. (a) Mediolateral and (b) anterior MoS values.
*, ** and *** indicate significant difference at p < 0.05, 0.01, and 0.001, respectively, by paired t-tests
with Bonferroni correction of factor nine.

For the anterior direction, the sacral crest exhibited the highest correlation coefficient
between the observation and estimation, with r = 0.542. The sacral crest was followed by
the shoulder and knee with r = 0.523 and 0.518, respectively. The lowest coefficient was
exhibited by the toe (r = 0.301). The coefficients were significantly different among the
10 body features (F (9, 90) = 83.2, p < 0.001). The correlation coefficient for the sacral crest
was significantly greater than those for C7 (t(9) = 8.57, p = 5.72 × 10−5 ), T10 (t(9) = 7.28,
p = 2.10 × 10−4 ), elbow (t(9) = 20.32, p = 3.55 × 10−8 ), shoulder (t(9) = 3.43, p = 0.034),
xiphisternum (t(9) = 10.97, p = 7.41 × 10−6 ), skull (t(9) = 5.58, p = 0.0015), great
trochanter (t(9) = 23.68, p < 0.001), and toe (t(9) = 18.59, p < 0.001).

Appl. Sci. 2024, 14, 3067

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Figure 6 shows the MoS values estimated by cross-validation. Each of these two
figures includes 300 points, estimated using 10 different models, with 30 points for each.
(b) Anterior MoS

6

Estimated anterior MoS (cm)

Estimated mediolateral MoS (cm)(a) Mediolateral MoS

4

2

0
2
4
0
6
Obeserved mediolateral MoS (cm)

10

0

െ10
10
െ10
0
Obeserved anterior MoS (cm)Figure 6. Observed and estimated MoS. (a) Mediolateral direction and (b) anterior direction. MoS
values were estimated by 10-fold cross-validation.

Figure 7 shows the mean triaxial velocities of the sacral crest and the first to thirdprincipal motions to estimate the mediolateral MoS. As discussed in Section 2.3, each gait
sample is represented as a linear combination of these three principal motions with theprincipal motion scores (ta ) indicating the contribution of each motion. Notably, all samples
were normalized to have x = 0, and the principal motions highlight the deviations fromthe mean motions. The estimation of the mediolateral MoS for the sacral crest involves thesum of the three principal motion scores:
mlat = 0.160t1 + 0.0488t2 + 0.0457t3 + mlat ,(18)

where mlat is the mean mediolateral MoS value across all samples. As shown in Figure 7b,
the characteristic movement of the first principal motion is along the x-axis. On average,
as shown in Figure 7a, the x-direction velocity reaches its minimum and maximum atapproximately 0% and 50% of the gait cycle, respectively. Conversely, the load distributionof the first principal motion demonstrates almost opposite characteristics, with maximum
and minimum values at roughly 0% and 50% of the gait cycle, respectively. This patternsuggests that individuals with higher first scores exhibit less variation in the x-direction
velocity. This observation is further supported by Figure 8a, where gait samples withthe highest and lowest first scores are depicted with solid and dotted lines, respectively.
Notably, the higher the score, the higher the MoS value and postural stability. The extended
center of mass becomes close to the center of the body with low speeds in the x-direction,
implying a high postural stability. Therefore, individuals with high first scores and minimal
fluctuation in the x-directional speed tend to have large mediolateral MoS values. Figure 7cshows that the dominant feature of the second principal motion is along the z-axis, lagging
behind the mean motion by approximately 10%. This delay suggests that individuals withelevated second scores experience delayed peaks in the z-directional velocities, as shownin Figure 8b. Similarly, Figures 7d and 8c show that the defining characteristic of the thirdprincipal motion is a reduced walking speed (y-directional velocity), with individuals
exhibiting large third scores when walking slowly, probably because of the short strides, as
indicated by the reduced vertical movement.

Appl. Sci. 2024, 14, 3067

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Figure 9 shows the mean translational velocities of the sacral crest and the first to thirdprincipal motions for estimating the anterior MoS. The anterior MoS for the sacral crest
was estimated by summing the following three principal motion scores:
mant = 0.0792t1 + 0.0480t2 + 0.0432t3 + mant ,(19)

where mant represents the mean anterior MoS value across all samples. Figure 10 compares
motion samples with the highest (represented by solid curves) and smallest (represented
by dotted curves) scores for each principal motion. Figure 9b shows that the prominent
feature of the first principal motion is small velocities in the y-direction, with velocities
in the x- and z-directions approximately in the opposite phase to the average motions.
Consequently, as shown in Figure 10a, participants with a large first score walk slowly withsmall velocity amplitudes in the mediolateral (x) and vertical (z) directions, indicating a gaitwith small strides. The second principal motion is characterized by a lag in the z-direction
velocity, as shown in Figures 9c and 10b. A motion sample with a large second score showsa delay in the z-direction phase. Figures 9d and 10c show noticeable differences in motions
between large and small third scores, particularly in the x- and z-directions. Comparedwith the sample with a small score, the sample with a large score advances in phase in thez-direction. Additionally, the sample with a large score exhibits minimal change in thex-directional velocity.
(b) First principal motion for mediolateral MoS
0.2

Motion load (-)

Mean velocity (m/s)(a) Mean velocity

1
0.5
0
0

50
Gait cycle (%)100(c) Second principal motion for mediolateral MoS

െ0.2

50
Gait cycle (%)

100

Motion load (-)

0.2

0

െ0.2

0

(d) Third principal motion for mediolateral MoS

0.2

Motion load (-)

0

0

50
Gait cycle (%)
X-direction (right)

100

0

െ0.2

0

50
Gait cycle (%)

Y-direction (forward)

Z-direction (up)100

Figure 7. Mean translational velocities of the sacral crest and principal motions to estimate themediolateral MoS. (a) Mean translational velocities. (b) First principal motion. (c) Second principal
motion. (d) Third principal motion.

Appl. Sci. 2024, 14, 3067

11 of 16(a) First principal motion (mediolateral MoS)(b) Second principal motion (mediolateral MoS)(c) Third principal motion (mediolateral MoS)

1

0

100

Gait cycle (%)
X-direction0

Y-direction

0100

Gait cycle (%)

100

Gait cycle (%)

Motion sample with the highest score

Z-direction

Motion sample with the smallest score(a) Mean velocity(b) First principal motion for anterior MoS

Motion load (-)

Figure 8. Examples of gait motions with the highest (rigid curves) and smallest (dotted curves) scoresfor each principal motion to estimate the mediolateral MoS.

Mean velocity (m/s)

1
0.5

0

0
0

50
Gait cycle (%)

െ0.1

100

(c) Second principal motion for anterior MoS
0.1

0

െ0.1

0

50
Gait cycle (%)100(d) Third principal motion for anterior MoS
0.2

Motion load (-)

0

Motion load (-)

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2

0

െ0.2
0

50
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0100
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Figure 9. Mean translational velocities of the sacral crest and principal motions to estimate the anterior
MoS. (a) Mean translational velocities (same as Figure 7a). (b) First principal motion. (c) Second
principal motion. (d) Third principal motion.

Appl. Sci. 2024, 14, 3067

12 of 16(a) First principal motion (anterior MoS)(b) Second principal motion (anterior MoS)(c) Third principal motion (anterior MoS)

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100
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1000Gait cycle (%)

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Motion sample with the highest score
Motion sample with the smallest score

Figure 10. Examples of gait motions with the highest (rigid curves) and smallest (dotted curves)scores for each principal motion to estimate the anterior MoS.

4. Discussion
As shown in Section 3, the kinematic information from the sacral crest and knee
can most accurately estimate both the anterior and mediolateral MoSs. The sacral crest
was anticipated to be a suitable estimation point owing to its proximity to the center ofbody mass (xcom ), which is used for MoS calculations. Further, the positions of the feetcan be accurately predicted using the motion of the center of mass [42,43], implying that
only the kinematic information near the center of mass is satisfactory to predict the MoSs.
Following the sacral crest, the xiphisternum, T10 vertebra, and great trochanter are thefeature points closest to the center of body mass. However, the estimation accuracies forthe xiphisternum and T10 did not match that of the sacral crest. This discrepancy could
be attributed to the separation of the xiphisternum and T10 from the center of body massby several spinal and coastal bones, potentially weakening their motion correlation withthe sacral crest. The great trochanter, despite its proximity to the center of body mass,
proved to be the least suitable point for MoS estimation, similar to the toe. This may be
because the femur, prominently featuring the great trochanter, moves independently fromthe pelvis, which comprises the sacral crest, rendering it less relevant for MoS estimations.
Contrarily, the femoral lateral epicondyle, another prominent feature of the femur, achieved
estimation accuracies comparable to the sacral crest. A possible explanation for this is thatthe great trochanter is located near the instant center of rotation of the femur [44], where its
translational motion is minimal and hence less contributory to MoS estimation.
The knee, along with the sacral crest, was identified as a suitable point for estimating
the MoS. This suitability may stem from the position of the knee, which captures motion
information from both the center of body mass and toe, as it is located between them. In
healthy individuals, the knee is almost fully extended at the moment of heel contact [45],rendering its motion information valuable for predicting foot position. The position markerof the knee, attached to the femoral lateral epicondyle, does not capture information regarding the abduction and adduction movements of the knee joint. However, unexpectedly, theknee also proved effective for estimating the mediolateral MoS. This finding necessitatesfurther investigation to elucidate the underlying mechanisms in future studies.
The toe was unsuitable, although it is used for computing the MoS. This is potentiallybecause the toe does not include significant kinematic information during the stance phase.
As shown in Figure 4j, the right toe rarely moves for approximately 40% of the gait cycle.
Notably, the typical stance phase occupies 60% of the gait cycle [46,47], but the toe continues
to move immediately after the heel contact. Further, during the swing phase, the ankle
joint can move freely in the air or at least without the restriction caused by the contact
with the ground, although such motions are used to determine the toe clearance [48–50].

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Ankle motions in the air may not be directly linked to the MoS that is computed referringto the foot position during the stance phase. Collectively, the toe is reasonably judged as an
unsuitable point to estimate the MoS.
This study identifies two body points—the sacral crest and knee—as suitable for
estimating the anterior and mediolateral MoSs based on their kinematic information. When
attaching the IMUs to these points using a cloth bandage, areas near joints where theattachment may loosen during extended periods of walking should be avoided. In practice,
IMUs are typically affixed near the knee, either on the femoral or tibial parts [10,51–53].
Considering these practical considerations for fixation, further research is needed to refine
our understanding and approach.
The accuracy of MoS estimation requires further enhancement, although specificcriteria have not been established in this study. We propose two potential strategies for
improvement. First, considering that MoS values can also be estimated from the angularvelocities of the pelvis [27], integrating angular and translational velocities might enhanceaccuracy. Second, employing multiple IMUs could increase the accuracy. With IMUs now
commonly incorporated into smartphones and smartwatches, it is feasible that individuals
might carry two IMU-equipped devices in their daily lives. Future research will explore
the optimal locations for attaching these two IMUs.
A limitation of this study is our reliance on optical motion capture systems rather than
IMUs to collect body motion data. Although IMUs are recognized as potential alternatives
to optical systems [10,54–56], they may entail a loss of accuracy. A challenge with IMU
usage concerns securing them to the body effectively. Because of their inertia, including
the weight of batteries and communication devices, IMUs may exacerbate noise issues
associated with soft tissue movements beneath skin, in contrast to the lightweight reflective
markers used in optical systems [57]. Another consideration is the calculation of thecenter of mass. We approximated the center of mass of the body using the geometriccenter of three pelvis points because of the lack of inertial data for body segments. Thisapproximation is close to the true center of body mass [58] and is deemed acceptable for
MoS calculations. However, this study specifically associates the center of mass with thesacral crest, potentially leading to an overestimation of its effectiveness in MoS estimation.
To address these issues and validate the feasibility of IMU-based MoS estimations, futurestudies should involve experiments that concurrently use optical motion capture systems
and IMUs. The definitive potential of IMUs for MoS estimation can be confirmed by further
experimental investigations.
5. Conclusions
We investigated suitable body points for estimating the anterior and mediolateral
MoSs from triaxial translational velocities, assuming the use of an IMU, such as thoseinstalled in smartphones. To the best of our knowledge, no previous studies have reported
similar findings. Among the 10 points evaluated, the sacral crest and knee were identified
as the most accurate for MoS estimation. When using the sacral crest, the mean correlation
coefficients between the estimated and observed MoSs were 0.563 for the mediolateral
direction and 0.542 for the anterior direction, respectively. For the knee, these coefficientswere 0.560 and 0.518, respectively. These two features are considered as promising points
to attach IMUs. Future research will explore the feasibility of IMU-based MoS estimation
as a method for accessible MoS measurement in everyday life. Our findings pave the way
for developing an accessible system to estimate fall risks.
Author Contributions: Conceptualization, S.O. and Y.A.; methodology, S.O. and T.K.; software, Z.L.
and T.K.; validation, Z.L. and T.K.; formal analysis, Z.L., T.K. and S.O.; investigation, Z.L. and T.K.;
resources, T.K.; writing—original draft preparation, Z.L., S.O. and T.K.; writing—review and editing,
S.O. and Y.A.; visualization, Z.L. and S.O.; supervision, S.O.; project administration, S.O. All authors
have read and agreed to the published version of the manuscript.
Funding: This study was in part supported by MEXT Kakenhi (21H01572).

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Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: This study is based on a semi-open data [30] that are accessible undercertain conditions. We are not allowed to redistribute the data.
Conflicts of Interest: The authors declare no conflicts of interest.

Abbreviations
The following abbreviations are used in this manuscript:
MoS
IMU
PMA

Margin of stability
Inertial measurement unit
Principal motion analysis

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