Singular value decomposition-based gait characterization

Abstract

Background

Human gait varies based on personal characteristics, the existence of walking problems, or variability of gait parameters. Identifying the sources of variations is significant in detecting walking problems, designing orthotic/prosthetic products, etc.

Research questions

What are the types of variations in joint angles and ground reaction forces? How do age, sex, height, weight, and walking speed affect the distribution of the modes?

Methods

In this study, temporal variations in the joint angles and ground reaction forces were obtained using singular value decomposition. Then, the relationships among age, sex, height, weight, walking speed, and the coefficients obtained by singular value decomposition were investigated using Pearson’s correlation coefficient matrix.

Results

The first mode of joint angles and ground reaction forces represent the overall characteristics; the first six modes of joint angles and the first two modes of ground reaction forces express 99.9% of the gait parameter space. We concluded that the walking speed dramatically affects joint kinematics and ground reaction forces. In addition, the effects of age, gender, height, and weight on the joint kinematics and ground reaction forces were also found, but with less contribution.

Significance

The temporal behavior of the joint angles and ground reaction forces was expressed using a few coefficients due to singular value decomposition. The effects of age, sex, weight, height, and walking speed on the modes were found. The proposed method can be applied to understand the progression of an abnormality, and design orthotic/prosthetic products etc. in future studies.

Gait analysis; Singular value decomposition; Gait variations; Temporal variations.

1. Introduction

Understanding human locomotion to detect the development of gait abnormalities, design shoes, or orthotic products that better fit individuals and produce more comfortable prostheses, is quite important; however, each gait shows some differences. Personal parameters (e.g., age [1], gender [2], walking speed [3]), and the existence of an abnormality [4, 5] may cause certain variations in human gait. The identification of the source of variation facilitates the diagnosis and treatment of gait abnormalities. Thus, determining the source of variation related to the personal parameters or abnormalities is considerably important. In addition, the variability of the gait parameters needs to be considered and the range of normal gait parameters should be identified.

Several techniques have been used to investigate the relationship among gait variations, personal parameters, and abnormalities. Previous studies have shown that walking speed significantly affects various gait parameters [3, 6]. Age is another parameter that affects joint kinematics, predominantly by reducing the range of joint motion [7]. Ground reaction forces (GRFs) exhibited some variations in the second peak of vertical forces, as well as braking and propulsive force during level walking [8]. Kobayashi et al. reported that gender and age affect joint kinematics using principal component analysis (PCA), and age-independent gender differences were captured in one of the principal components [9]. The effects of weight on ground reaction forces were investigated by comparing obese and normal-weight adults, and the study showed that a higher vertical loading rate and reduction of knee flexion excursion were observed in obese subjects at a standardized speed [10]. Another study showed that walking speed, age, gender, and body mass index can predict joint kinematics based on several key features, determined by considering curvature parameters such as the minimum/maximum point [11]. However, expressing the variations of joint angles and GRF in a more comprehensive and simpler manner instead of using parameters such as maximum/minimum points and rate of change, and factoring what kind of variations exist in temporal parameters allows us to better understand normal walking.

There are several methods to investigate biomechanical gait parameters such as joint angles and GRFs. Fourier Series analysis was used to model the normative knee joint angle data [12]. Knee joint angle was converted to a weighted summation of sines and cosines during one gait cycle. The variations of gait parameters were investigated in terms of sines and cosines. Wavelet analysis was also a commonly used method to investigate the locomotion of human. The effect of carrying load on the lumbar motion was examined by using wavelet decomposition [13]. Although Fourier Series analysis and wavelet analysis were useful decomposition techniques, the variation of each coefficient does not have a direct meaning. PCA and singular value decomposition (SVD) detects the variations which were expressed naturally. PCA was also used as a statistical method to understand the variation of gait parameter. The effects of sex and age on the kinematics and kinetics of the lower extremities were investigated by using PCA [14]. SVD is a matrix decomposition method [15] that creates orthogonal basis vectors to explain the row-vise and column-vise characteristics of a phenomenon. SVD creates a set of basis vectors to describe the behavior of the temporal gait parameters.

In this study, we aimed to find the temporal variations of joint angles and GRFs, investigate the relationship between personal parameters and temporal variations, and demonstrate normative values depending on personal parameters. Singular value decomposition of matrices comprising joint angles and GRFs reveals average behavior and variations. Owing to the orthogonality of each singular mode obtained by SVD, any type of joint angle and GRF data can be expressed as a weighted summation of each singular mode. Further investigations were conducted to understand the effects of gender, age, height, weight, and walking speed on the coefficients. A large motion capture dataset helped understand the extent to which the gait variation for each gait pattern was related to personal parameters.

2. Methods

In this study, SVD-based gait characterization was proposed. First, the AIST gait database 2019 [16] was used to obtain 3D motion capture (20 cameras, VICON MX, sampled at 200 Hz) and ground reaction force (GRF) data (seven force plates, AMTI, sampled at 1000 Hz). Then, marker positions and GRFs data were filtered using a Butterworth filter with 6 and 10 Hz cut-off frequencies.

Data from 225 subjects were selected from the database. Subjects with ages lower than 18 years and subjects who have missing markers or GRF data were not included. The participants of the experiments were Asian, and the distribution of the subjects' gender, age, height, weight, and walking speeds are given in Table 1. The subjects were instructed to enter the motion capture room and walk straight at their preferred walking speed. After leaving the room, the subjects were instructed to turn around and walk straight again. Each subject performed ten trials and a total of 2250 walking trials were obtained. The experimental protocol was approved by the local institutional review board (Environment and Safety Headquarters, Safety Management Division, AIST) and written consent was obtained from the participants of experiments.

Table 1.

Mean values and standard deviations of the subjects' age, height, weight, and speed.

	Total (225 Subjects)	Male (108 Subjects)	Female (117 Subjects)
AGE	50.03 ± 18.71	50.47 ± 19.60	49.63 ± 17.98
HEIGHT	162.33 ± 8.52 cm	168.06 ± 6.72 cm	157.04 ± 6.35 cm
WEIGHT	59.43 ± 10.41 kg	65.68 ± 9.13 kg	53.66 ± 7.95 kg
SPEED	1.33 ± 0.16 m/s	1.30 ± 0.15 m/s	1.36 ± 0.16 m/s

The human body was modeled as eight rigid bodies (head-arms-trunk, pelvis, two thighs, two shanks, and two feet) and seven revolute joints in the sagittal plane. The gait cycle was determined from the first ground contact to consecutive contact of the same foot. Joint angles were calculated using the experimental marker data. The vector size of each trial’s joint angle and GRFs showed differences because of the different durations of the gait cycle. Therefore, joint angles and GRFs were resampled at 60 data points. In addition, the mean values of each joint angle data were subtracted from the corresponding data, so that each joint angle data had a zero mean value to reduce possible errors due to marker placements.

Singular value decomposition was used to understand the overall human dynamics and what kind of variations were mostly performed. Singular value decomposition is a useful matrix decomposition method that divides a matrix, $\mathbf{Q}$, into two orthogonal matrices, $\mathbf{U}$ and $\mathbf{V}$, and one rectangular matrix, $\mathbf{\Sigma }$, whose only diagonal entries were filled, as shown in Eq. (1).

$${\mathbf{Q}}_{\left(\mathrm{n}\times \mathrm{m}\right)}={\mathbf{U}}_{\left(\mathrm{n}\times \mathrm{n}\right)}{\mathbf{\Sigma }}_{\left(\mathrm{n}\times \mathrm{m}\right)}{\mathbf{V}}_{\left(\mathrm{m}\times \mathrm{m}\right)}^{\mathrm{T}}$$ (1)

Column vectors, ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$, were created for each gait parameter in all trials, as shown in Eqs. (2) and (3). The joint angle vector, ${\mathbf{q}}_1$, combines the resampled 60 data points of the ankle (${Ankle}_1\dots {Ankle}_{60}$), knee (${Knee}_1\dots {Knee}_{60}$), hip (${Hip}_1\dots {Hip}_{60}$), and pelvis-trunk (${Pl-Tr}_1\dots {Pl-Tr}_{60}$) joint angle data, and is a column vector of length 240. The GRF vector, ${\mathbf{q}}_2$, contains the resampled 60 data points of the vertical and horizontal GRFs, and is a column vector of length 120. Then, matrices, ${\mathbf{Q}}_1$ and ${\mathbf{Q}}_2$, were created for joint angles and GRFs by combining the column vectors, ${\mathbf{q}}_1^{\mathrm{j}}$ and ${\mathbf{q}}_2^{\mathrm{j}}$, obtained for all 2250 walking trials as given in Eqs. (4) and (5). The column vectors, ${\mathbf{q}}_1^{\mathrm{j}}$ and ${\mathbf{q}}_2^{\mathrm{j}}$, were defined as joint angles and GRFs vectors of the j^th walking trial.

$${\mathbf{q}}_1={\left[\begin{array}{cccccccccccc}Ankl{e}_1& \dots & Ankl{e}_{60}& Kne{e}_1& \dots & Kne{e}_{60}& {Hip}_1& \dots & {Hip}_{60}& {PlTr}_1& \dots & {PlTr}_{60}\end{array}\right]}^T$$ (2)

$${\mathbf{q}}_2={\left[\begin{array}{cccccc}GR{F}_{Vertical-1}& \dots & GR{F}_{Vertical-60}& GR{F}_{Horizontal-1}& \dots & GR{F}_{Horizontal-60}\end{array}\right]}^T$$ (3)

$${\mathbf{Q}}_{\mathbf{1}}=\left[\begin{array}{ccccc}{\mathbf{q}}_1^1& {\mathbf{q}}_1^2& {\mathbf{q}}_1^{\dots }& {\mathbf{q}}_1^{2249}& {\mathbf{q}}_1^{2250}\end{array}\right]$$ (4)

$${\mathbf{Q}}_{\mathbf{2}}=\left[\begin{array}{ccccc}{\mathbf{q}}_2^1& {\mathbf{q}}_2^2& {\mathbf{q}}_2^{\dots }& {\mathbf{q}}_2^{2249}& {\mathbf{q}}_2^{2250}\end{array}\right]$$ (5)

Singular value decomposition of the matrix ${\mathbf{Q}}_{\mathrm{i}}$ generates the matrices of ${\mathbf{U}}_{\mathrm{i}}$, ${\mathbf{\Sigma }}_{\mathrm{i}}$ and ${\mathbf{V}}_{\mathrm{i}}$, as shown in Eq. (1). The column-vise characteristics in matrix ${\mathbf{Q}}_{\mathrm{i}}$ are stored in the orthogonal matrix ${\mathbf{U}}_{\mathrm{i}}$. The columns of ${\mathbf{U}}_{\mathrm{i}}$ are the modes of gait parameters. The first column of the ${\mathbf{U}}_{\mathrm{i}}$ shows average gait characteristics, and the following columns exhibit variations that mostly occurred in order. The coefficient matrix was defined as the multiplication of ${\mathbf{\Sigma }}_{\mathrm{i}}$ and ${\mathbf{V}}_{\mathrm{i}}^{\mathrm{T}}$. The coefficients' mean values and standard deviations were obtained for each gait parameter mode. Then, mean coefficient vectors, ${\mathbf{\mu }}_1$ and ${\mathbf{\mu }}_2$, and standard deviation vectors, ${\mathbf{\sigma }}_1$ and ${\mathbf{\sigma }}_2$, were created for the modes of joint angles, ${\mathbf{U}}_1$, and GRFs, ${\mathbf{U}}_2$. Even though a large number of modes exists, the effect of modes decreases in ascending order. The effect of the ith mode was calculated as the ratio of the square of ith singular value, $s_i$, to the sum of squares of all singular values. Hence, the cumulative effect of the first k modes is given by Eq. (6). l is the number of singular values and is equal to min(m,n).

$${\epsilon }_{\mathrm{k}}=\frac{\sum _{i=1}^k{s}_i^2}{\sum _{j=1}^l{s}_j^2}$$ (6)

Personal parameters were defined using gender, age, height, weight, and walking speed. Data processing and statistical analyses were conducted using MATLAB software (MATLAB R2020a, The MathWorks, Inc., Massachusetts, USA). The effects of gender, age, weight, height, and walking speed on the coefficients were investigated using Pearson’s correlation coefficients. Pearson’s correlation coefficients, r, vary between −1 and 1. The value of r indicates the existence of strong liner relations between the parameters. The value of $r$ which is closer to −1 or 1 implies that a strong relationship between parameters exists negatively or positively. Statistical significance was set at p < 0.05.

3. Results

The singular value decomposition of a gait parameter matrix yields the modes and coefficients. In SVD, the first mode has the maximum effect, and the contribution of the modes decreases gradually. First, the mean joint angles and GRFs were calculated by the multiplying the modes of matrices, ${\mathbf{U}}_1$ and ${\mathbf{U}}_2$, and mean coefficient vectors, ${\mathbf{\mu }}_1$ and ${\mathbf{\mu }}_2$. Then, the effect of ith mode was obtained by increasing the ith coefficient by five times of the standard deviation of the corresponding coefficient. The effects of the first, second, and third modes on the joint angles and GRFs are shown in Figure 1(a-b).

Figure 1.

Effects of first three modes on joint angles (a) and GRFs (b). ${\mathbf{\mu }}_1$ and ${\mathbf{\mu }}_2$ are mean coefficients vectors of joint angles and GRFs. Then, the first, second and third coefficients were increased by 5 times of the standard deviation of each mode. ${\mathbf{\sigma }}_{1,\mathrm{i}}$ and ${\mathbf{\sigma }}_{2,\mathrm{i}}$ are the standard deviations of the ith coefficients of joint angle and GRFs modes, respectively.

Although the first three modes of joint angles and GRFs were significant and had frequently occurring variations, the cumulative effect of the modes should be considered to determine the effect of each mode. For this purpose, the cumulative effects of each mode were calculated. The cumulative effects of the first 20 modes for the joint angles and GRFs are shown in Figure 2(a-b). The cumulative effects of the first six modes of joint angles and the first two modes of GRF exceeded 99% of the motion. The mean values, ${\mathbf{\mu }}_1$ and ${\mathbf{\mu }}_2$, and standard deviations, ${\mathbf{\sigma }}_1$ and ${\mathbf{\sigma }}_2$, of the coefficients of the modes were obtained and are shown in Figure 2(c-d) for the first ten modes of joint angles and GRFs. All modes except the first mode have almost zero mean value. The standard deviation was the largest in the second mode and decreased in the following modes.

Figure 2.

Cumulative effects of the first 20 modes of joint angles (a) and GRFs (b)were given in percentage. The coefficients of the first ten modes of joint angles (c) and GRFs (d) were given as mean values, ${\mathbf{\mu }}_1$ and ${\mathbf{\mu }}_2$, (line plot) and standard deviations, ${\mathbf{\sigma }}_1$ and ${\mathbf{\sigma }}_2$ (error bars).

Then, the relationship between the personal parameters and the coefficients obtained using SVD was investigated using Pearson’s correlation coefficient. Tables 2 and 3 show the correlation coefficients and p-values for the first ten modes of the joint angles and GRF, respectively. If the absolute value of r is higher than 0.5, there is a strong relationship between the personal parameter and the coefficient of the corresponding mode. The r-value ranging from 0.2 to 0.5 was considered a moderate relation and an r-value less than 0.2 was considered negligible.

Table 2.

r-value and p-value for the relation between personal parameters and joint angles. Bolded r-values show that |r|>0.2 and p < 0.05.

Mode	R-Value					P-Value
	Gender	Age	Height	Weight	Speed	Gender	Age	Height	Weight	Speed
1	-0.162	-0.109	-0.116	-0.149	0.452	0.000	0.000	0.000	0.000	0.000
2	0.130	-0.044	0.107	0.062	-0.534	0.000	0.035	0.000	0.003	0.000
3	-0.169	0.040	-0.099	-0.264	0.221	0.000	0.060	0.000	0.000	0.000
4	0.167	0.031	0.131	0.161	-0.358	0.000	0.144	0.000	0.000	0.000
5	0.074	0.117	-0.015	-0.023	0.190	0.000	0.000	0.465	0.281	0.000
6	0.052	0.128	-0.039	0.007	-0.049	0.013	0.000	0.067	0.751	0.020
7	-0.013	0.205	-0.118	0.051	0.093	0.546	0.000	0.000	0.016	0.000
8	-0.081	0.162	-0.135	-0.074	0.079	0.000	0.000	0.000	0.000	0.000
9	-0.301	0.111	-0.249	-0.345	-0.203	0.000	0.000	0.000	0.000	0.000
10	-0.375	0.139	-0.326	-0.192	0.005	0.000	0.000	0.000	0.000	0.818

Table 3.

r-value and p-value for the relation between personal parameters and GRFs. Bolded r-values show that |r|>0.2 and p < 0.05.

Mode	R-Value					P-Value
	Gender	Age	Height	Weight	Speed	Gender	Age	Height	Weight	Speed
1	-0.125	-0.022	-0.058	-0.171	0.414	0.000	0.291	0.006	0.000	0.000
2	0.141	-0.160	0.157	0.203	-0.707	0.000	0.000	0.000	0.000	0.000
3	-0.115	-0.192	0.023	0.111	0.425	0.000	0.000	0.267	0.000	0.000
4	0.068	-0.082	0.102	-0.065	0.082	0.001	0.000	0.000	0.002	0.000
5	-0.243	-0.152	-0.106	-0.289	0.004	0.000	0.000	0.000	0.000	0.849
6	-0.089	-0.183	-0.014	-0.071	0.086	0.000	0.000	0.498	0.001	0.000
7	-0.192	0.264	-0.202	-0.062	0.103	0.000	0.000	0.000	0.003	0.000
8	-0.117	-0.106	-0.141	-0.076	0.006	0.000	0.000	0.000	0.000	0.761
9	0.011	-0.031	0.015	0.004	0.137	0.606	0.136	0.478	0.862	0.000
10	0.115	0.017	0.030	-0.024	-0.023	0.000	0.408	0.160	0.250	0.270

SVD-based gait characterization allows us to understand walking variations related to personal parameters during normal walking. The coefficients of the first three modes are correlated with the walking speed. The third mode of joint angles and the second mode of the GRFs are also related to the weight. The fifth, ninth, and tenth modes of the joint angles, and the fifth and seventh modes of the GRFs exhibit variations related to personal parameters. The relationships between personal parameters and coefficients are given with the effect of each coefficient in Figure 3.

Figure 3.

Effects of gender on joint angles (a) and GRFs (b), and the effects of age (c) and weight (d) on joint angles. ${\mathbf{\mu }}_1$ and ${\mathbf{\mu }}_2$ are mean coefficient vectors for the joint angles or GRFs. ${\mathbf{\sigma }}_{1,\mathrm{i}}$ and ${\mathbf{\sigma }}_{2,\mathrm{i}}$ are the standard deviations of the ith coefficients of joint angles and GRFs modes. The effect of each mode was visualized by changing the corresponding coefficient by 3 times standard deviation.

4. Discussion

In this study, we aimed to obtain temporal variations in joint angles and GRFs using SVD. The mean values of the coefficients except for the first mode were almost equal to zero. The first mode expressed overall behavior, while the rest of the modes were temporal variations that were orthogonal to each other. The standard deviation of the coefficients was the largest in the second mode because the second mode contained information on the most common gait variation type. The effect and variation of the next modes as well as the standard deviation decreased.

The first mode was almost equal to the mean joint angles during gait and the increase in the first coefficient resulted in an increase in the range of joint motion owing to a zero mean value. The second mode affected ankle joint behavior in the second rocker phase, knee joint behavior during initial contact and weight acceptance, rotational speed of the hip joint, and loading and releasing rate of the GRFs. The third mode of the joint angles corresponded to the changes related to the stance/swing ratio. The region where maximum extension of the hip, knee, and ankle joints occurs shifted left. The first mode of GRF affected the range of the GRF data, similar to the joint angles. The second mode of GRF changed the loading/releasing rate, and the third mode affected the ratio of peaks in the vertical GRF, and maximum braking/propulsive forces. The adaptations caused by the first three modes can be considered as predominantly occurring types of variations.

The changes in the coefficients of each mode help us understand the variations in human gait, such as speed, gender, and age-specific adaptations. In addition, the development of an abnormality or adaptation during rehabilitation can be tracked by the changes in the coefficients. Thus, determining the range of normal values of coefficients and the effect of personal parameters on the coefficients of the modes have serious significance. Walking speed significantly affected the first three modes and was the most effective parameter for determining the behavior of the joint angles and GRFs. The effects of gender, age, height, and weight were evident in a higher number of modes. Chebab et al. also showed similar results in that walking speed mostly affects joint kinematics [6].

The effect of walking speed was observable in the second mode of joint angles and GRFs. Tables 1 and 2 show that walking speed has a negative correlation with the coefficients in the second modes of joint angles and GRFs. As shown in Figure 1, an increase in the walking speed reduces the coefficients in the second mode of joint angles and GRFs, increases the range of joint motions, reduces ankle joint velocity during the second rocker phase, increases knee flexion during the stance phase, and increases the vertical peak GRF and loading rate of GRFs [6, 11].

The tenth mode of joint angles and the fifth mode of GRFs showed significant differences depending on gender. The effect of gender was demonstrated by changing the corresponding coefficients of the average coefficient vector. The relevant adaptations with gender are shown in Figure 3(a-b). The tenth coefficient of the joint angle matrix shows gender-related variations in the sagittal plane motion. Female subjects had a higher coefficient in the tenth mode of joint angles. When the tenth coefficient of the joint angle matrix increased, the range of the hip joint increased, and the range of the pelvis-body joint decreased. The gender-related adaptations of GRFs were mostly observable in the fifth mode. However, the fifth mode was affected by gender and weight at a considerable level. Female subjects had a slightly larger coefficient and a higher second peak of GRF.

The effect of aging on the joint angles and GRFs was most evident in the seventh mode for both matrices. The coefficient of the seventh mode of the joint angles increased with aging, as shown in Figure 3(c). Aging effects are observable, particularly in the ankle and knee joints.

The effect of weight on the joint angles can be observed better in the ninth mode of the joint angle matrix, as shown in Table 2. The effect of weight on the ninth coefficient and the joint angles is shown in Figure 3(d). The increase in the subject weight resulted in a decrease in the ninth coefficient of the joint angles. If the ninth coefficient drops, the maximum ankle plantar flexion angle and minimum knee angle also decrease. The most specific variation related to the ninth coefficient occurs at the ankle joint during the late swing to properly adjust foot position and to avoid heel strike. Although the contribution of the weight is the maximum among personal parameters, gender, height, and walking speed also have considerable effects on the ninth mode.

SVD-based gait characterization resulted in a group of vectors defining any complex variation during one gait cycle. The group of vectors, in other words, modes, formed a basis for future studies to investigate the effect of abnormalities, devices, orthoses, prostheses, etc. Compared to Fourier Series analyses or wavelet analyses, the modes obtained by SVD are natural type of variations. Whereas coefficients affect sinusoidals or wavelet functions in Fourier Series analyses and wavelet analyses, the adjustments of the coefficients of modes obtained by SVD have a direct meaning on joint angles and GRFs. Moreover, the distributions of the coefficients were related to personal parameters, and specific patterns related to certain personal parameters were obtained. In addition to variations, SVD can identify the average gait characteristics compared to PCA. Each joint angle data can be rewritten in terms of the modes obtained by SVD. Thus, the progressive adaptation due to a walking abnormality or a device can be investigated in terms of the coefficients of modes.

There are several limitations of the present study that should be considered during the interpretation of the data. All the participants of the experiments walked at their preferred walking speeds; thus, the range of walking speed was limited. Besides, the ethnicity of the participants and whether participants have regular exercise habits have not been included in this research. Moreover, the joint angle matrix was designed in such a way that the mean value of each joint angle is zero to reduce the shifting error due to soft tissue movement, marker placement, etc. Therefore, the numerical value of the joint angle was not interpreted due to having zero mean value.

5. Conclusion

In this study, a large dataset of motion capture data was used to create the joint angles and ground reaction force matrices. Each gait data was converted to the weighted sum of the orthogonal gait patterns using SVD. The relationship between the coefficients (weights) of each gait pattern, and personal parameters was investigated using Pearson’s correlation matrix. We found that the walking speed has considerable effects on the joint angles and ground reaction forces. Moreover, the effects of gender, aging, and weight on joint angles and ground reaction forces were studied. The present study suggests that temporal variations can be obtained by using SVD and the modes of SVD are useful to model the adaptations related to aging, gender, walking speed, etc. In addition, SVD-based gait characterization can be applied in rehabilitation to show the improvement of gait and, design of orthotics and prosthetics for observing the effect of design parameters, etc. in a similar manner.

Declarations

Author contribution statement

Cem Guzelbulut: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Katsuyuki Suzuki: Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data.

Satoshi Shimono: Conceived and designed the experiments; Contributed reagents, materials, analysis tools or data.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

The experimental data was requested from AIST gait database. It can be accessed by the url below. https://unit.aist.go.jp/harc/ExPART/GDB2019_e.html.

Declaration of interest’s statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.