ORIGINAL PAPER
Nagoya J. Med. Sci. 85. 211–222, 2023
doi:10.18999/nagjms.85.2.211

Relationship between gait stability indices andgait parameters comprising joint angles based onwalking data from 288 people
Takashi Inagaki1, Yasuhiro Akiyama2, Shogo Okamoto3,
Takuya Mayumi2 and Yoji Yamada2
Department of Mechanical and Aerospace Engineering, Nagoya University Graduate School of Engineering,
Nagoya, Japan2
Department of Mechanical Systems Engineering, Nagoya University Graduate School of Engineering,
Nagoya, Japan3
Department of Computer Sciences, Graduate School of Systems Design, Tokyo Metropolitan University, Hino,
Japan
1

ABSTRACT
Stability during walking is essential because falling accidents may lead to severe injuries. In this study,
we calculated the margin of stability (MoS) and the maximum Lyapunov exponent (l s), which are two
major stability indices for walking, using a gait database representing 300 healthy people. Previously, therelationships between these indices and other gait parameters, including joint angles, have not been investigated in such a large subject pool. Therefore, we determined the relationships between these stability indicesand the gait parameters by calculating correlation coefficients and performing multiple regression analysis.
The results indicated that MoS is dominated by walking speed in the forward direction and associatedwith various joint angles in the lateral direction. Conversely, no relationships were identified between ls
and the gait parameters. Although both MoS and ls are considered as measures of gait stability, they are
independent. The results of this study suggest that MoS and ls represent different aspects of gait motion.
Keywords: local dynamic stability, margin of stability, gait analysis, gait stability
Abbreviations:
MoS: margin of stability
MoSf: MoS in the forward direction
MoSl: MoS in the lateral direction
ls: maximum Lyapunov exponent
XCoM: extrapolated center of mass position
HC: heel contact
TO: toe off
COG: center of gravity
BoS: base of support
This is an Open Access article distributed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
License. To view the details of this license, please visit (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Received: October 21, 2021; accepted: April 12, 2022
Corresponding Author: Yasuhiro Akiyama, PhD
Faculty of Textile Science and Technology, Shinshu University, 3-15-1 Tokida, Ueda 386-8567, Japan
Tel: +81-268-21-5426, E-mail: akiyama_yasuhiro@shinshu-u.ac.jp211

Takashi Inagaki et al

INTRODUCTION
Accidents during walking caused by gait instability, particularly among the elderly, can be amajor risk factor and cause severe injuries. Therefore, the analysis of gait stability is important
for assessing the risk of falls. Various indices have been proposed to evaluate walking stability.1
The maximum Lyapunov exponent (l s)2 is a major index for evaluating local dynamic gaitstability. It quantifies the stability of a person’s gait by constructing a limit cycle based on the
gait parameters and by calculating the trajectory divergence. Conversely, the margin of stability(MoS) is an index that determines walking stability based on the kinetic margin of the supporting
area, which prevents falling.3,4
Several studies have investigated various factors that can affect gait stability. Kodesh et al
examined gait symmetry and Jordan et al investigated stride-to-stride fluctuations by measuring
walking speed.5,6 Additionally, Espy et al examined the influence of walking speed and step
length on gait stability and fall risk.7 The relationships between gait parameters and gait stability indices, namely, MoS and ls, have also been studied. Hak et al have reported that slower
walking speeds enhance MoS in the forward direction (MoSf) and that a greater stride frequency
increases MoS in the lateral direction (MoSl).8 Additionally, several studies have investigated therelationships between ls and walking speed,9-12 stride length, and stride frequency.8 However, thestability indexes for gait have not been examined while considering more comprehensive gaitparameters such as joint angles during walking. In particular, the relationship between ls and
MoS is unclear, despite these two indices being prominent measures of gait stability.
In this study, we investigated the relationships between ls, MoS, and basic gait parameters,13,14
such as walking speed, step length, and stride frequency, along with joint angles. We used datafrom AIST Gait Database 2019,15 containing gait data from 300 healthy people. Although many
gait indices evaluate gait using different parameters, the motion synergy among body links
required for gait motion may lead to certain regularities among indices, including MoS and ls.
Therefore, we aimed to identify statistical relationships between these two indices. Furthermore,because the relationships between gait parameters, including joint angles, and gait stability indiceshave not been investigated in detail, we aimed to identify the parameters that determine or are
correlated with the aforementioned gait stability indices.

METHODS
DatasetWe used data from the AIST Gait Database 2019,15 which consists of 10 gait trials withapproximately 1.5 strides for 300 healthy adults, wherein the left and right strides are included
evenly. Measurements were obtained using a motion capture system (Vicon MX, Oxford, UK)and a floor ground reaction force plate (AMTI, Watertown, MA, USA). The sampling frequency
was set to 200 Hz, and 55 to 59 reflective markers were attached to different major body parts
to obtain positional information. We analyzed data from 288 healthy adult men and women(141 men and 147 women, age range of 20 to 78 years, average age of 49.7 years). The datafrom 12 subjects were excluded from our analysis as a result of incomplete stride information.
Therefore, a total of 2880 trials were analyzed.
Because several studies have reported that gait characteristics are dependent on sex and age,16-18
we considered various attributes of gait motion in the subjects of this study.
The midpoints of both the anterior superior iliac spine marker and sacral marker were considered as centers of mass. Additionally, heel and toe markers were used to determine foot contact
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Relationships of gait stability indicesand calculate the position of the base of support (BoS). Various gait parameters were consideredin our analysis. The data were analyzed using the software MATLAB R2020b (MathWorks, Inc,
Natick, MA, USA).
Gait parameters
Table 1 presents the gait parameters considered in this study. The posture angle of each bodylink was determined based on the marker positions. In this study, in addition to the parameters
identified in our previous study,19 stride frequency was newly identified as the inverse of singlestride time, which is calculated as walking speed divided by stride length.
Table 1

1

Gait parameters
Step length (m)23Step width (m)MFC (m)4

Thigh tilt (HC) (°)

5
6
7
8
9
10

Thigh tilt (TO) (°)
Knee angle (HC) (°)
Knee angle (TO) (°)
Shank tilt (HC) (°)
Shank tilt (TO) (°)
Foot angle (°)

11

Tilt of upper body (°)

12

Ratio of COG position (lateral)

13
14

Ratio of COG position (forward)
Walking speed (m/s)

15Stride frequency (Hz)List of gait parameters

Definitions
Distance between the heel markers of the right and left
legs in the forward direction at HC
Lateral distance between the right and left heels at HC
Height of the sole when the foot becomes parallel to theground during the swing phase
Angle between the thigh and horizontal axis in the sagittal
plane
Thigh tilt at TO
Angle between the thigh and shank at HC
Knee angle at TO
Angle between the shank and horizontal direction at HC
Shank tilt at TO
Angle between the vector from the distal end to theproximal end of foot in the horizontal direction at TO
Inclination of the line connecting the midpoint between C7
and the upper body margin of the sternum
Ratio of the COG positions and the area of the BoS in
the lateral direction at HC
Ratio of the COG position in the forward direction at HC
Average value of the velocity of the center of mass in theforward direction
Inverse of one stride time calculated as walking speed/
stride length

HC: heel contact
TO: toe off
MFC: minimum foot clearance
COG: center of gravity

Margin of stability
The MoS evaluates the gait stability using an inverted pendulum model.3,4 Initially, theextrapolated center of mass position (XCoM) is calculated using the position and velocity of thecenter of mass, which are denoted as CoM and VCoM, respectively.
(1)

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Here, l denotes the height of the center of mass, and g is the acceleration due to gravity.
Additionally, we calculated the BoS based on the position of the toe marker. The margin between
BoS and XCoM defines MoS, which is obtained as follows:(2)
A system is mechanically stable when MoS > 0 and unstable when MoS < 0. Figure 1
presents the MoS in the stable state. MoSf is calculated as the distance between the position of
the toe marker and XCoM, as shown in Equation (2), where the toe marker of the forward footis considered as BoS. MoSl is calculated as the distance between the position of the toe marker
and XCoM in the lateral direction. In this study, the smallest margin at the beginning of the
double-stance phase, which appears after heel contact (HC), was selected as the representative
MoS value in the forward and lateral directions of each stride, respectively.

Fig. 1 Schematic of XCoM, BoS, and MoS in the stable state
The foot is grounded in the direction the body is moving. The minimum value of MoS in the beginning of the
double-stance phase is used as the representative value of MoS.
CoM: center of mass
XCoM: extended center of mass
BoS: base of support
MoS: margin of stability

Maximum Lyapunov exponent
The ls evaluates local dynamic stability, which quantifies the divergence of small perturbations
in a state space.1,2,9,20 An m-dimensional state space S with time delay t is reconstructed using
the center of mass velocity v(t) based on the Takens theorem21 as follows:(3)
The values of the embedding dimension m and time delay t must be appropriately selected for
Equation (3). Therefore, we calculated the optimal m using the false nearest neighbor algorithm.22
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Relationships of gait stability indicesIn this algorithm, the false nearest neighbor of a cyclic signal is determined as the point at
which the distance from the closest point of the neighboring cycle changes significantly whenm changes. Typically, m is identified as the minimum value at which the false nearest neighbors
converge. We obtained average values of m = 3 and m = 5 for all participants in the forward
and lateral directions, respectively.
Additionally, the time delay t was calculated using average mutual information.23 Typically, t
is determined as the minimum value at which the mutual information exhibits a local minimum.
For each t, we determined the amount of mutual information between the original time series
data and the time series data delayed by t. Consequently, average values of 13 and 6 time points
for all participants were calculated in the forward and lateral directions, respectively. Therefore,
the state space was reconstructed using the derived values of m and t, and divergence wascalculated using Rosenstein’s algorithm.24
The average logarithmic rate of divergence was obtained by computing the Euclidean distances
between neighboring trajectories in the state space. The slope of the obtained divergence curveis the maximum Lyapunov exponent. In this study, a stride of 0.5 to 1.0 steps was consideredas the short-term maximum Lyapunov exponent, which is denoted as ls. Although the optimal
stride length was not identified for computing ls,25 we selected this short stride length based on
its applicability to real-time scenarios. A positive ls indicates that a system is unstable.
Statistical analysis
The correlation coefficients between the gait stability indices and the gait parameters were
calculated. We performed multiple regression analysis considering the two types of stability
indices and gait parameters as the objective and explanatory variables, respectively. Additionally,
the parameters were standardized (z-score) based on the original data distribution to facilitate theinterpretation of regression coefficients.

RESULTS
Correlation coefficients between gait parameters and stability indices
Table 2 lists the value of the correlation coefficients between the two calculated stability
indices and each gait parameter. As indicated in Table 2, no meaningful correlation exists between
ls and the gait parameters. Additionally, the correlation coefficients obtained between ls and MoS
are nearly negligible at r = –0.02 and r = 0.03 in the forward and lateral directions, respectively.
However, the correlation coefficient between the walking speed and MoSf is negative (r =–0.69). Additionally, a negative correlation (r = –0.42) can be observed between the step lengthand MoSf. Furthermore, weak correlations were identified between several joint angles and
MoSl. The correlation coefficients between MoSl and the step width are small and positive (r= 0.25). Additionally, the correlation coefficients between MoSl and the thigh tilt at HC andbetween MoSl and the tilt of the upper body are small and negative at r = –0.24 and r =–0.27, respectively. Negative correlation coefficients were also observed between MoSf and thefoot angle and MoSf and the ratio of the center of gravity (COG) position (forward) at r =–0.33 and r = –0.25, respectively. Furthermore, the correlation coefficients between MoSf andthe thigh tilt at toe off (TO) and MoSf and the shank tilt at TO are positive and small at r =0.26 and r = 0.27, respectively.
Regression analysis of gait parameters and stability indicesFigure 2 presents the coefficient values and their 95% confidence intervals obtained through
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Table 2Computed Pearson’s correlation coefficients between gait parameters and MoS and λs

Variables

mean

std

r
MoSl

MoSf

λsl

λsf

Step length (m)

0.66

0.06

–0.05**–0.42**

0.01

–0.01

Step width (m)

0.07

0.03

0.25**–0.04*

0.09**

0.04*

MFC (m)

0.02

0.01

0.14**–0.13**

–0.02

0.00

Thigh tilt (HC) (°)

117.5

3.42

–0.24**–0.16**

–0.01

0.04

Knee angle (HC) (°)

8.3

4.14

0.10**

0.07**

0.02–0.05**

Shank tilt (HC) (°)

109.9

2.56

–0.12**–0.11**

0.01

–0.03

Foot angle (°)

69.1

8.99

–0.17**–0.33**

–0.01

0.04*

Tilt of upper body (°)

17.1

7.38

–0.27**

–0.02

0.00

0.01

Ratio of COG position (lateral)

0.50

0.13

–0.03

–0.02

0.02

0.00

Ratio of COG position (forward)

0.65

0.03

0.11**–0.25**

–0.01

–0.02

Thigh tilt (TO) (°)

80.5

4.10

0.09**

0.26**

0.00

0.01

Knee angle (TO) (°)

41.1

5.03

–0.18**–0.07*

0.03

–0.03

Shank tilt (TO) (°)

38.4

2.80

–0.01

0.27**

0.05*–0.07**Walking speed (m/s)

1.34

0.15

–0.11**–0.69**

0.00

0.03

Stride frequency (Hz)

1.07

0.07

0.01

0.01

–0.02

–0.00

Mean and standard deviation values are also presented. _l and _f denote the lateral and forward directions,
respectively. The p-values are indicated as follows: *>0.05, **>0.01.
HC: heel contact
TO: toe off
MFC: minimum foot clearance
COG: center of gravity
MoS: margin of stability
l s: maximum Lyapunov exponent

the multiple regression analysis of the gait parameters and stability indices. As shown in thisfigure, MoSf is negatively affected by the walking speed, shank tilt at TO, and ratio of the COG
position (forward). Additionally, the step length has a positive effect on MoSf. In the lateral
direction, MoSl is negatively affected by the knee angle and the thigh tilt at TO. In contrast, theshank tilt at the TO exhibits a positive effect on MoSl. Additionally, MoSl is positively affectedby the step width, step length, and minimum foot clearance (MFC). The large R2 value (R2 =0.55) of MoSf indicates that it can be predicted to a certain extent using these gait parameters.
Additionally, a relatively large R2 value can be observed in the case of MoSl with R2 = 0.30.
Conversely, the value of R2 for ls is extremely small and the gait parameters do not appear toaffect the value of ls. Although several gait parameters exhibited significant effects, as shown in
Fig. 2(d), wide confidence intervals were observed with significant differences between individual
subjects and between trials. Additionally, the observed trends were common among the young,middle-aged, and elderly groups. We examined the inter-correlations among the gait parameters to
identify multicollinearity in the regression analysis, but none of the gait parameters were highly
correlated with each other (r > 0.8).

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Relationships of gait stability indicesFig. 2 Partial correlation coefficients of multiple regression analysis
Two stability indices MoS and ls are modeled using gait parameters. Error bars represent 95% confidence intervals.
R2 values are also indicated. The p-value is indicated on the left side (*>0.05).
HC: heel contact
TO: toe off
MFC: minimum foot clearance
COG: center of gravity
MoS: margin of stability
l s: maximum Lyapunov exponent

DISCUSSION
Effects of gait parameters on MoS
Forward direction. Based on the results of multiple regression analysis presented in Fig.
2(b), we conclude that the effect of the walking speed on MoSf is dominant and that a negative
relationship exists between them. Furthermore, the values of the correlation coefficients between
the walking speed and MoSf are large. Compared to the walking speed, the other parameters
exhibit weaker effects on the MoSf value. This trend matches the results of a previous study,8which reported a positive relationship between the walking speed and the MoS in the backward
direction. This is because the XCoM moves forward as the walking speed increases, which
reduces MoSf.
Additionally, the parameter that determines the BoS, namely, the partial regression coefficientof the step length, is positive. Furthermore, a negative correlation and effect are identified
between the ratio of the COG position and MoSf based on the narrowing of the margin as the
COG position moves forward.
Additionally, the partial regression coefficients of the shank tilt, knee angle, and thigh tilt
significantly affect MoSf. This can be attributed to the greater values of the knee angle, thightilt, and step length increasing BoS, which in turn increases MoSf. Additionally, we analyzed
the effects of the joint angles on MoSf by considering the HC parameters. We observed that theeffects of the joint angles detected at the TO on MoSf were similar to those detected during
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HC. This represents the effect of the kicking motion at TO on the stability of the stance phase.
Lateral direction. As shown in Fig. 2(a), MoSl is significantly affected by the knee angle,
thigh tilt, and shank tilt at TO, which implies that lateral stability relies heavily on the joint
angles at TO. In contrast, the contributions of the step width and ratio of the COG position(lateral), which affect MoSl directly, are relatively small. This implies that the joint angles arestronger indicators of lateral gait stability than the gait parameters. Furthermore, the effects of
the joint angles at the TO are significantly greater than those observed at HC. This is probably
because the kicking motion at TO is strongly reflected in MoSl, similar to the forward direction.
Furthermore, we found that the walking speed and the stride frequency did not affect MoSl
and that stability improved when the step length was increased. A previous study8 has reported
that walking speed and step length are not associated with lateral stability but are positively
related to stride frequency. Another study has reported that faster walking speed helps maintainlateral gait stability.26 The reported relationships between these three gait parameters, namely,
the stride frequency, walking speed, and step length, and lateral gait stability are inconsistent.
However, our findings verify that the joint angles in the sagittal plane contribute to lateral gaitstability more significantly than the aforementioned basic gait parameters. Furthermore, existingstudies have considered the balance between the forward, backward, and mediolateral directions
independently. However, as shown in Fig. 2(a), our results suggest that the kicking motion in
the sagittal plane (ie, tilt angles at TO, knee angle (at TO), and foot angle) affects stability inthe mediolateral direction.
Effects of gait parameters on ls
The coefficients of determination for the gait parameters were extremely low, indicatingthat they did not influence the value of ls significantly. Although our results revealed a weak
positive trend between the walking speed and ls, the confidence intervals were large based on
the individuality and the variance between trials. In other words, the results indicate that ls is
not directly associated with the gait parameters. However, previous studies have reported that ls
can be used to distinguish the gait sets of the young and elderly,27 and is associated with fall
history.28 These results indicate that although ls may reflect the properties of gait motion, it is
not directly associated with basic gait parameters.
Furthermore, variations in the data used to calculate ls complicate the comparison of the results
reported in related studies.29 For example, Hak et al used a combination of three-axis walking
speeds to calculate ls and reported a negative relationship between ls and the walking speed.8
In contrast, Dingwell et al used body acceleration and reported a quadratic positive relationshipbetween ls and the walking speed.9 Furthermore, England et al reported that ls calculated using
joint angles exhibited a positive correlation with the walking speed.10 Another study identified
a negative relationship between ls calculated using time series velocity data and the walking
speed in the forward direction.12
Additionally, the number of strides considered in this study is small compared to the numbers
used in previous studies.25 Although the ls value calculated from a small stride number can beapplied in real-time scenarios, the results of this study suggest that it is not associated with the
MoS and the gait parameters. However, the l s values calculated using a large stride number
may be more relevant, but this requires further investigation.
Relationship between ls and MoS
The correlation coefficient between l s and MoS in the forward direction is small, and no
relationship is identified between the two indices. Furthermore, no correlation exists between ls
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Relationships of gait stability indicesand MoS in the lateral direction. Therefore, we can conclude that MoS and ls represent different
aspects of walking and that no direct relationship exists between the two indices. Additionally,
the differences between MoS and ls in terms of their responses to gait parameters suggest thatthey are independent of each other. However, a greater number of strides may affect these results
based on the sensitivity of ls to stride length.
Effects of outliers in our analysis
To investigate the effects of outliers on the MoS and ls, Spearman’s rank correlation coefficients,
which are more robust to outliers than Pearson’s correlation, were calculated. The results are shown in
Table 3. According to Tables 2 and 3, the differences between Spearman’s correlations and Pearson’s
correlations are small, except for the relationships between the stride frequency and the MoS andls. Considering the small standard deviation of the stride frequency, it is nonlinearly related to thestability indices.

We also plotted the frequency distributions and skewness of the MoS and ls, respectively Fig.
3. The frequency distributions are approximately normally distributed, and the skewness value isinconsequential. Therefore, the effects of outliers are assumed to be small.

Table 3

Computed Spearman’s correlation coefficients between gait parameters and MoS and λs

Variables

mean

std

r
MoSl

MoSf

λsl

λsf

Step length (m)

0.66

0.06

–0.03*–0.55**–0.01–0.01*

Step width (m)

0.07

0.03

0.23**–0.02*

0.10**

0.04*

MFC (m)

0.02

0.01

0.13**–0.17**–0.02*

0.01

Thigh tilt (HC) (°)

117.5

3.42

–0.23**–0.19**–0.01

0.05**

Knee angle (HC) (°)

8.3

4.14

0.12**

0.08**

0.01*–0.07**

Shank tilt (HC) (°)

109.9

2.56

–0.10**–0.15**

0.01–0.02*

Foot angle (°)

69.1

8.99

–0.16**–0.42**

–0.01

0.04*

Tilt of upper body (°)

17.1

7.38

–0.29**–0.03*

–0.01

0.01*

Ratio of COG position (lateral)

0.50

0.13

–0.11**–0.10**–0.07**

0.03*

Ratio of COG position (forward)

0.65

0.03

0.12**–0.29**–0.01–0.03*

Thigh tilt (TO) (°)

80.5

4.10

0.08**

0.32**

0.02*

0.01

Knee angle (TO) (°)

41.1

5.03

–0.17**–0.07**

0.02*–0.03*

Shank tilt (TO) (°)

38.4

2.80

–0.02*

0.32**

0.04*–0.08**Walking speed (m/s)

1.34

0.15

–0.08**–0.84**

0.01

0.04*

Stride frequency (Hz)

1.07

0.07

0.10**–0.63**–0.01*

0.07**

Mean and standard deviation values are also presented. _l and _f denote the lateral and forward directions,
respectively. The p-values are indicated as follows: *>0.05, **>0.01.
HC: heel contact
TO: toe off
MFC: minimum foot clearance,
COG: center of gravity
MoS: margin of stability
λs: maximum Lyapunov exponent
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Fig. 3 Histograms of MoS and λs
Skewness of each frequency distribution is also shown.
MoS: margin of stability
λs: maximum Lyapunov exponent

Limitations
Elderly individuals sometimes have underlying conditions that come with aging, and suchdiseases may affect gait ability.30 It is important to investigate the relationship between these
diseases and gait characteristics. However, participant histories of disorders were not includedin the database used in this study. Therefore, we could not analyze the effects of underlying
conditions on the gait characteristics in this study.

CONCLUSIONSThis study aimed to identify the relationships between the gait stability indices MoS and ls
and various gait parameters using 10 trials of walking data from 288 subjects. We investigatedthe effects of various parameters on the values of the MoS and ls based on the calculation of thecorrelation coefficients and multiple regression analysis. The relative contributions of individual
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Relationships of gait stability indicesgait parameters such as walking speed, walking frequency, step length, and several joint angles
to the values of MoS and ls were determined.
The results indicate that no meaningful correlations exist between ls and MoS. Based on our
analysis, we conclude that MoS is dominated by the negative effect of the walking speed andis closely associated with various joint angles, whereas no relationships exist between ls andmost of the gait parameters. Although these two indices are used as measures of gait stability,
the results of our study indicate that they can be used to examine different aspects of gait.
Further investigation of the relationships between MoS and various gait parameters (ie, effects
of symptoms or lower-limb injuries) will facilitate the development of a measure that quantifies
the risk of falls and gait stability. This index is expected to be useful as an indicator for gait
rehabilitation.

CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest.

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