Base of Support, Step Length and Stride Width Estimation during Walking Using an Inertial and Infrared Wearable System

Abstract

The analysis of the stability of human gait may be effectively performed when estimates of the base of support are available. The base of support area is defined by the relative position of the feet when they are in contact with the ground and it is closely related to additional parameters such as step length and stride width. These parameters may be determined in the laboratory using either a stereophotogrammetric system or an instrumented mat. Unfortunately, their estimation in the real world is still an unaccomplished goal. This study aims at proposing a novel, compact wearable system, including a magneto-inertial measurement unit and two time-of-flight proximity sensors, suitable for the estimation of the base of support parameters. The wearable system was tested and validated on thirteen healthy adults walking at three self-selected speeds (slow, comfortable, and fast). Results were compared with the concurrent stereophotogrammetric data, used as the gold standard. The root mean square errors for the step length, stride width and base of support area varied from slow to high speed between 10–46 mm, 14–18 mm, and 39–52 cm², respectively. The mean overlap of the base of support area as obtained with the wearable system and with the stereophotogrammetric system ranged between 70% and 89%. Thus, this study suggested that the proposed wearable solution is a valid tool for the estimation of the base of support parameters out of the laboratory.

1. Introduction

The role of the base of support (BoS) is crucial in the investigation of the dynamic stability during walking [1]. With reference to normal gait, the right BoS is defined as the area enclosed by the outer edges of the footprints with the right foot ahead the left one [2], and it is associated with the right step length and the right stride width (Figure 1). Given the distance between two consecutive footprints, step length and stride width are defined as the projections of the above-mentioned distance, respectively, on the direction of progression (identified by two consecutive homolateral footprints) and on the line perpendicular to it [3]. Several studies have investigated the correlations between BoS-related parameters and margin of stability, balance, and risk of falling both in normal and pathological gait [4,5,6,7,8].

The BoS can be obtained in the laboratory from the direct measures of the feet position of a stereophotogrammetric system (SP) or of an instrumented mat [9,10,11]. Much more difficult is to obtain an accurate description of the BoS out of the laboratory and in the real world.

To date, magneto-inertial measurement units (MIMUs) attached to the feet are the most effective wearable technology for out-of-laboratory gait analysis [12].

MIMU-based methods have been successfully used for the estimation of spatio-temporal parameters such as stride duration and length [13,14]; however, the use of inertial sensing technology alone cannot provide information about the relative position of the two feet, and consequently, BoS-related parameters.

In the literature, different ancillary technologies have been proposed to overcome the intrinsic MIMU’s limitation by integrating other types of sensors enabling the estimation of the relative feet position.

The most direct solution to continuously measure the inter-foot distance is to integrate MIMUs with either ultrasound sensors [15,16,17] or foot-worn cameras [18,19]. These systems can provide accurate foot trajectories, but they are quite cumbersome and therefore not suitable for both clinical and real-world applications.

To reduce the system size, Trojaniello et al. [20] proposed a system integrating MIMUs and a light intensity infrared proximity sensor which embeds a transmitter and a receiver in the same chip. However, infrared proximity sensors do not provide the inter-foot distance, but rather the distance between the infrared emitter and any reflecting target. In addition, their performance may vary due to changes in the environmental conditions, such as reflectance and color of the target surface [21].

A further improvement was obtained by using infrared time-of-flight proximity sensors, which guarantee a good accuracy regardless of the environmental conditions [21]. By instrumenting a single foot with the latter system, Bertuletti et al. [22] developed a method for step detection and relevant inter-foot distance estimation.

In summary, among the various solutions proposed in the literature, some methods limited the analysis to the estimation of inter-foot distance [20,22], some others focused on the reconstruction of feet trajectories for pedestrian navigation purposes [15,18,19], and finally, methods based on ultrasound technology provided step length, stride width [16], and margin of stability [17], but not the BoS area.

In the present study, we propose an original wearable system which integrates miniaturized infrared time-of-flight sensors with inertial sensing for the estimation of BoS parameters. Its main advantage with respect to existing solutions is that only one foot is instrumented, thus improving wearability and portability. The position of the non-instrumented foot during stance is estimated from the distance data recorded during the swing phase of the instrumented foot.

The validation of the proposed system was carried out on thirteen healthy subjects walking at three self-selected speeds (slow, comfortable, and fast) and results were compared in terms of BoS area, step length, and stride width to those obtained from SP, considered as the gold standard.

2. Materials and Methods

2.1. System Overview

The proposed system included an MIMU and two distance sensors (DS) connected to the inertial module via cable [22,23]. All sensors were embedded in a custom 3D-printed rigid support with known geometry, fixed to the medial side of a shoe through two thin straps, based on the experimental setup proposed in a previous study [22] (Figure 2).

The MIMU (mod. LSMDSO and mod. LIS2MDL, STMicroelectronics, Switzerland; 3D accelerometer: range ±16 g; 3D gyroscope: range ±2000 dps; 3D magnetometer: range ±50 Gauss; sampling frequency: 100 Hz) was calibrated following the methods proposed by [24,25].

The DS (mod. VL6180X, STMicroelectronics, Switzerland; distance: range 0–200 mm; sampling frequency: 50 Hz) reduced the sensor’s dimensions (4.8 mm × 2.8 mm × 1.0 mm), combined the receiver and the transmitter in the same chip, and guaranteed low power consumption (~2–5 mA) and an accuracy independent from environmental conditions. The infrared time-of-flight technology provides an estimate of the distance to the target by measuring the phase shift between the radiated and the reflected infrared waves. DSs were calibrated with a custom 3D-printed cylinder which imposed a known distance between the sensor and the target, so that the offset could be estimated and removed. Distance data were linearly interpolated and resampled at 100 Hz.

Recorded magneto-inertial and distance data were stored onboard and the communication between the laptop and the MIMUs was based on the Bluetooth low energy technology.

2.2. Description of the Method for Base of Support Estimation

By exploiting gait cyclicity, the procedure for the estimation of the BoS is presented with reference to a generic gait cycle of the instrumented foot, which was arbitrarily chosen to be the right one. Thus, hereby we refer to the right BoS as instrumented BoS. The estimation of the instrumented BoS area, step length, and stride width requires one to determine position and orientation of two right and one left footprints with respect to the same global coordinate system (CS_G) (Figure 3). To this purpose, the following actions were implemented: (1) identification of the gait cycle interval of the instrumented foot, (2) estimation of the position and orientation (pose) of the instrumented foot, (3) identification of the footprints of the instrumented foot, (4) identification of the footprint of the non-instrumented foot, and (5) computation of BoS-related parameters (Figure 4).

2.2.1. Identification of the Gait Cycle Interval of the Instrumented Foot

The gait cycle interval was defined as the interval of time between two consecutive flat-foot instants, t₀ and t_f, of the instrumented foot with the lowest kinetic energy. The stance phases were identified through peak detection on the approximated foot mediolateral angular velocities and anteroposterior accelerations [13,26]. The flat-foot instants were searched within the relevant stance phase of the instrumented foot using a parametric zero-velocity detector based on angular velocities [27,28].

2.2.2. Estimation of the Instrumented Foot Pose in the Global Coordinate System CS_G

The coordinate system fixed with the instrumented foot (CS_S) was made to coincide with the coordinate system embedded with the sensor axes of the MIMU (Figure 2). The orientation of CS_S with respect to the Earth magnetic-gravity reference system (CS_E), expressed by the rotation matrix ${}^E\mathit{R}_S(t)$, was obtained from the recorded magneto-inertial data using an optimized complementary filter [28,29,30,31].

The foot acceleration in CS_E, ${}^E\mathit{a}(t)$, was computed by removing the gravity contribution to the recorded accelerations:

$${}^E\mathit{a}(t)={}^E\mathit{R}_S·{}^S\mathit{f}(t)-\mathit{g}$$

(1)

where ${}^S\mathit{f}(t)$ is the specific force measured by the accelerometer in CS_S and g is the gravity vector.

A global coordinate system CS_G was made to coincide with CS_S at the beginning of the gait cycle (t₀) after realignment with the gravity, so that the y-axis of CS_G was perpendicular to the ground (Figure 3). The origin O_G was set to the ground plane.

Then, ${}^E\mathit{a}(t)$ was expressed in CS_G:

$${}^G\mathit{a}(t)={}^G\mathit{R}_E·{}^E\mathit{a}(t)$$

(2)

where ${}^G\mathit{R}_E$ is the rotation matrix from CS_E to CS_G.

Finally, the foot trajectory, represented by the position of the origin O_S of CS_S, ${}^G\mathit{p}_{OS}(t)$, was computed by double integrating ${}^G\mathit{a}(t)$ between t₀ and t_f:

$${}^G\mathit{p}_{OS}(t)={}^G\mathit{p}_{OS}(t_0)+\underset{t_0}{\overset{t}{\iint }}{}^G\mathit{a}\left(\tau \right)d\tau \hspace{1em}t\in [t_0,t_f]$$

(3)

To reduce the drift, an optimal filtering of accelerations and a direct and reverse integration technique for the velocity estimation were implemented [28,32]. The initial boundary conditions of velocity and displacement were set to zero.

The foot orientation in time with respect to CS_G was described by the rotation matrix from CS_S to CS_G, ${}^G\mathit{R}_S\left(t\right)$, computed as follows:

$${}^G\mathit{R}_S\left(t\right)={}^G\mathit{R}_E·{}^E\mathit{R}_S(t)\hspace{1em}t\in [t_0,t_f]$$

(4)

The foot pose in time with respect to CS_G was described by the transformation matrix from CS_S to CS_G, ${}^G\mathit{T}_S\left(t\right)$, computed as follows:

$${}^G\mathit{T}_S\left(t\right)=\left[\begin{array}{cc}{}^G\mathit{R}_S\left(t\right)& {}^G\mathit{p}_{OS}\left(t\right)\\ 0& 1\end{array}\right]\hspace{1em}t\in \left[t_0,t_f\right]$$

(5)

where ${}^G\mathit{p}_{OS}\left(t\right)$ is the translation vector of the foot position (i.e., origin O_S) from the origin O_G.

2.2.3. Identification of the Footprints of the Instrumented Foot

The footprint was defined by the outer borders of the contact region between the foot insole and the ground plane during the relevant flat-foot instant. For simplicity, the footprint was approximated to a rectangle with vertices V_i (i = 1, 2, 3, 4), and length L and width W equal to the measured subject’s shoe sizes.

The time-invariant position of each vertex V_i of the rectangle in the CS_S, ${}^S\mathit{p}_{V_i}$ (i = 1, 2, 3, 4), was identified based on L, W, and distance b as obtained during a calibration procedure (Figure 5).

The positions of each vertex V_i (i = 1, 2, 3, 4) expressed in CS_G, ${}^G\mathit{p}_{V_i}(t)$ (i = 1, 2, 3, 4), at t₀ and t_f can be computed as follows:

$$\left[\begin{array}{c}{}^G\mathit{p}_{V_i}\left(t_0\right)\\ 1\end{array}\right]={}^G\mathit{T}_S\left(t_0\right)·\left[\begin{array}{c}{}^S\mathit{p}_{V_i}\\ 1\end{array}\right]\hspace{1em}i=1,2,3,4$$

(6)

$$\left[\begin{array}{c}{}^G\mathit{p}_{V_i}(t_f)\\ 1\end{array}\right]={}^G\mathit{T}_S\left(t_f\right)·\left[\begin{array}{c}{}^S\mathit{p}_{V_i}\\ 1\end{array}\right]\hspace{1em}i=1,2,3,4$$

(7)

Footprints are then defined by the x_G-z_G components of ${}^G\mathit{p}_{V_i}$ (i = 1, 2, 3, 4) at t₀ and t_f.

2.2.4. Identification of the Footprint of the Non-Instrumented Foot

The footprint of the non-instrumented foot was determined based on the knowledge of the instrumented foot pose and the data recorded by the two DS. In fact, during the swing phase of the instrumented foot, occurring during the stance phase of the non-instrumented foot, the two feet face each other, and the DS readings are different from zero and equal to the distance between medial shoe surfaces.

Let t₁ be the timing of the first reading different from zero obtained from the DS attached to the front portion of the instrumented foot (DS_f) and equal to the distance $d_f\left(t_1\right)$ between DS_f and the point $F(t_1)$ of the medial side of the non-instrumented foot. Let t₂ be the timing of the last reading different from zero of DS attached to the rear portion of the instrumented foot (DS_r) and equal to the distance $d_r\left(t_2\right)$ between DS_r and the point $R(t_2)$ of the medial side of the non-instrumented foot (Figure 6).

Let t* be a generic instant of time between t₁ and t₂, during which the non-instrumented foot was still, when both DS_f and DS_r could measure non-zero values equal to distances $d_f\left(t^{*}\right)$ and $d_r\left(t^{*}\right)$ between DS_f and DS_r and the points $F(t^{*})$ and $R(t^{*})$ of the medial side of the non-instrumented foot, respectively (Figure 6).

The positions of the points $F(t^{*})$ and $R\left(t^{*}\right)$ with respect to CS_S, ${}^S\mathit{p}_F\left(t^{*}\right)$ and ${}^S\mathit{p}_R\left(t^{*}\right)$, were computed as follows:

$${}^S\mathit{p}_F\left(t^{*}\right)=\left[\begin{array}{c}-c\\ 0\\ d_f\left(t^{*}\right)\end{array}\right]\hspace{1em}{t}^{*}\in \left[t_1,t_2\right]$$

(8)

$${}^S\mathit{p}_R\left(t^{*}\right)=\left[\begin{array}{c}c\\ 0\\ d_r\left(t^{*}\right)\end{array}\right]\hspace{1em}{t}^{*}\in \left[t_1,t_2\right]$$

(9)

where $d_f(t^{*})$ and $d_r(t^{*})$ are the distances recorded at t* by DS_f and DS_r, respectively, and c is the constant distance measured between each DS and O_S (Figure 7).

Then, the detected points were expressed with respect to CS_G:

$$\left[\begin{array}{c}{}^G\mathit{p}_F\left(t\right)\\ 1\end{array}\right]={}^G\mathit{T}_S·\left[\begin{array}{c}{}^S\mathit{p}_F\left(t\right)\\ 1\end{array}\right]\hspace{1em}t\in [t_1,t_2]$$

(10)

$$\left[\begin{array}{c}{}^G\mathit{p}_R\left(t\right)\\ 1\end{array}\right]={}^G\mathit{T}_S·\left[\begin{array}{c}{}^S\mathit{p}_R\left(t\right)\\ 1\end{array}\right]\hspace{1em}t\in [t_1,t_2]$$

(11)

The projection of points $F(t)$ and $R(t)$ to the ground plane (x_G-z_G), ${}^{G}p_{F_x}$, ${}^{G}p_{F_z},{}^{G}p_{R_x}$, and ${}^{G}p_{R_z}$, were then stored in the matrix M:

$$\mathit{M}=\left[\begin{array}{cc}{}^{G}p_{F_x}\left(t_1\right)& {}^{G}p_{F_z}\left(t_1\right)\\ {}^{G}p_{F_x}\left(t_1+∆t\right)& {}^{G}p_{F_z}\left(t_1+∆t\right)\\ {}^{G}p_{R_x}\left(t_1+∆t\right)& {}^{G}p_{R_z}\left(t_1+∆t\right)\\ .& .\\ .& .\\ .& .\\ {}^{G}p_{R_x}\left(t_2\right)& {}^{G}p_{R_z}\left(t_2\right)\end{array}\right]$$

(12)

where M is a Nx2 matrix with N equal to the total number of the recorded points of the non-instrumented foot.

The line l ∈ x_G-z_G, representing the medial side of the approximated footprint of the non-instrumented shoe, was obtained through least square fitting using the points contained in the matrix M. In addition, the centroid B of the points was also computed. The local coordinate system of the non-instrumented foot, CS_M, was then defined with the x-axis aligned with line l and centered in B (Figure 8).

The time-invariant positions of vertices $V_i$ (i = 5, 6, 7, 8) of the rectangle approximating the footprint with respect to CS_M, ${}^M\mathit{p}_{V_i}$ (i = 5, 6, 7, 8), were defined as follows:

$${}^M\mathit{p}_{V_5}\left[\begin{array}{c}\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0\end{array}\right], {}^M\mathit{p}_{V_6}\left[\begin{array}{c}\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ W\end{array}\right], {}^M\mathit{p}_{V_7}\left[\begin{array}{c}-\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ W\end{array}\right], {}^M\mathit{p}_{V_8}\left[\begin{array}{c}-\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0\end{array}\right]$$

(13)

where L and W are the shoe length and width, respectively.

Then the positions of footprint vertices $V_i$ (i = 5, 6, 7, 8) with respect to CS_G, ${}^G\mathit{p}_{V_i}$(i = 5, 6, 7, 8), were computed:

$$\left[\begin{array}{c}{}^G\mathit{p}_{V_i}(t)\\ 1\end{array}\right]=\left[\begin{array}{cc}{}^G\mathit{R}_M& {}^G\mathit{p}_B(t)\\ 0& 1\end{array}\right]·\left[\begin{array}{c}{}^M\mathit{p}_{V_i}\\ 1\end{array}\right]\hspace{1em}i=5,6,7,8\hspace{1em}t\in [t_1,t_2]$$

(14)

where ${}^G\mathit{p}_B$ is the translation vector between $B$ and O_G, and ${}^G\mathit{R}_M$ is the time-invariant rotation matrix between CS_M and CS_G, defined as

$${}^G\mathit{R}_M=\left[\begin{array}{cc}{x}_G·x_M& z_G·x_M\\ x_G·y_M& z_G·y_M\end{array}\right]$$

(15)

Since the non-instrumented foot is still during t₁-t₂, its footprint can be defined from the estimated positions of the footprint vertices ${}^G\mathit{p}_{V_i}(t)$ (i = 5, 6, 7, 8) with t within t₁ and t₂.

To improve the robustness of the fitting procedure, a data cleaning procedure on the detected points of the medial side of the non-instrumented foot was applied for outlier removal. For further details see Appendix A.

2.2.5. Estimation of the Base of Support Parameters

The position of the footprints of both instrumented and non-instrumented feet were described by the relevant rectangle centroids (C_I and C_NI). The direction of progression of the instrumented foot was identified by the line between the footprint centroids of the instrumented foot, C_I, between t₀ and t_f.

Then, the following parameters chosen for the description of the BoS were extracted (Figure 9):

• BoS area was defined as the largest area among the ones identified by the footprints vertices of opposite feet including the entire footprint regions. Outer edges of the BoS should not intersect the footprint regions. Rectangle footprints areas were easily calculated, while the irregular area between them was achieved with Bretschneider formula for irregular polygons. The instrumented BoS area ended with a contact with the ground of the instrumented foot and it was defined by footprints #2 and #3 (Figure 9).
• Step length (coinciding with the BoS length) was identified as the displacement along the direction of progression between a footprint centroid position and the consecutive centroid position of the opposite footprint [3]. Thus, the instrumented step length was defined along the direction of progression of the instrumented foot between C_NI(t₁ − t₂) and C_I(t_f).
• Stride width (coinciding with the BoS width) was determined as the perpendicular distance between a footprint centroid and the direction of progression of the opposite foot [3,4]. Thus, the instrumented stride width was defined by C_NI(t₁ − t₂) and the direction of progression of the instrumented foot.

Considering a generic gait cycle of the non-instrumented foot, all the non-instrumented BoS-related parameters could be similarly estimated. Thus, with reference to two consecutive gait cycles, the BoS area, step length, and stride width of both instrumented and non-instrumented sides were extracted (Figure 9).

For the sake of clearness, the non-instrumented BoS area ended with a contact with the ground of the non-instrumented foot and it was defined by footprints #1 and #2. The non-instrumented step length was defined along the direction of progression of the non-instrumented foot between C_I(t₀) and C_NI(t₁ − t₂). The non-instrumented stride width was defined by C_I(t₀) and the direction of progression of the non-instrumented foot.

2.3. Reference Base of Support Parameters Estimation Based on Stereophotogrammetry Data

For validation purposes, an SP was used to obtain gold standard estimates of the BoS. A total of 18 retro-reflective markers were attached to both feet (m₁–m₁₈) (Figure 10).

The positions of virtual markers m₉–m₁₅ on the instrumented foot were calibrated during a preliminary standing trial with respect to the rigid marker cluster defined by m₂–m₄, according to the CAST procedure to avoid visibility issues [33]. Similarly, the positions of virtual markers m₁₆–m₁₈ on the non-instrumented foot were calibrated with respect to a rigid marker cluster defined by m₆–m₈ to avoid visibility issues and undesired infrared wave reflections during walking trials. Then, virtual markers m₉–m₁₈ were removed before recording the walking trials, and their virtual trajectories were reconstructed [33].

Reference initial and final contact instants were estimated using the method proposed by O’Connor et al. [34,35] from the trajectories of the midpoint (Mid) between the heel and toe markers (m₁–m₂ for instrumented foot and m₅–m₆ for non-instrumented foot). The flat-foot instants of both feet were determined based on a parametric zero-velocity detector applied to the norm of the velocity of the foot midpoint [27,28].

The positions of the instrumented and non-instrumented footprint vertices with respect to the SP coordinate system (CS_SP), ${}^{SP}\mathit{p}_{V_i}$ (i = 1,…,8), were defined during flat-foot instants from the positions of m₃ and m₁₃-m₁₅ (instrumented foot), and m₇ and m₁₆–m₁₈ (non-instrumented foot).

To compare BoS parameters as estimated by the wearable system and the SP, the footprints estimated by the SP were expressed in the same global coordinate system of the MIMU (CS_G). To this purpose, the marker cluster m₉–m₁₂, rigidly connected with the rigid support, was used to define a marker-based local coordinate system coinciding with CSs and to obtain the MIMU orientation with respect to CS_SP. By knowing the orientation of the MIMU in both CS_G and CS_SP, it was then possible to obtain the transformation matrix between CS_G and CS_SP. Then, the positions of the footprints vertices could be expressed with respect to CS_G (${}^G\mathit{p}_{V_i}$ (i = 1,…,8)) and the BoS parameters were identified following the definitions adopted for the estimates from wearable system data (see Figure 9).

2.4. Experimental Data Collection

The right foot was instrumented with the wearable system, including an MIMU and two DSs, as shown in Figure 11. The trajectories of retro-reflective markers were recorded by a 12-camera SP system (mod. Vero, Vicon, UK). The wearable system and SP were synchronized using a hardware solution.

Thirteen healthy volunteers (gender: 7F, 6M; age: 25.6 ± 1.8 y.o.) were enrolled. Before starting the experiments, a ten-minute MIMU warm-up was performed to limit the temperature effects on the sensor readings. After a 10-second standing acquisition, the participants were asked to walk along a 5 m straight path on level ground 10 times at 3 different self-selected speeds (slow, comfortable, and fast).

Experiments conformed to the standards set in the Declaration of Helsinki. This protocol was reviewed and approved by the Ethics Committee of the University Hospital of Cagliari (Prot. PG/2021/1195) and participants provided their written informed consent to participate in this study.

2.5. Method Performance Assessment and Statistical Analysis

For each subject, the three tested speeds (slow, comfortable, and fast) were analyzed separately. Right and left estimates were averaged assuming the symmetry of healthy gait.

For each single estimate of each BoS parameter (step length, stride width, and BoS area), errors were computed as the difference between values provided by the wearable system and the reference SP. Subsequently, for each speed level, mean errors (ME), root mean square errors (RMSE), and mean absolute percentage errors (MAE%) were calculated over the 10 trial repetitions and over subjects.

In addition, mean values of the estimated parameters (MV) were calculated over the 10 trial repetitions and over subjects separately for each speed.

The agreement between wearable system and SP estimates was quantified by performing a Bland–Altman analysis with limits of agreement at 95% (LoA) and by calculating the Pearson correlation coefficient (r_xy).

Errors affecting the BoS estimation were evaluated in terms of errors on the estimation of the area positioning (BoS overlap%) (Equation (16)) and errors on footprint positioning (footprint shift) (Equation (17)), as shown in Figure 12:

• BoS overlap%:

$$BoS overlap\%=\frac{BoS overlap}{BoS Are{a}_{SP}}\times 100$$

(16)

where $BoS overlap$ is the area shared by BoS from the wearable system and from SP (BoS Area_SP). Thus, a perfect overlap would be equal to 100%.

• Footprint shift was defined as the distance between the footprints’ centroids of the same foot calculated with the wearable system and SP:

$$Footprint shift=\stackrel{-}{C_{WS}{C}_{SP}}$$

(17)

where $C_{WS}$ and $C_{SP}$ are the footprint centroids calculated with the wearable system and SP, respectively. Footprint shifts were analyzed on the ground floor (plane x_G-z_G), thus footprint shift_x and shift_z were computed.

The analysis was conducted separately for the instrumented and non-instrumented side. We expected that the footprint positioning of the non-instrumented foot would be affected by larger errors than the instrumented one, since its position was only determined by the detection of its medial side by the DS.

Preliminary Shapiro–Wilk tests of normality on the different types of error distributions were carried out to select the most appropriate subsequent statistical analysis.

To assess the effect of speed on BoS parameters estimation (normal distributions), a one-way repeated measure analysis of variance (ANOVA) was performed. Conversely, to assess the effect of speed and differences between instrumented/non-instrumented sides on BoS overlap% and footprint shift (non-normal distributions), a 3 × 2 Friedman test was used. The significance for all statistical tests was determined at p < 0.05 and a Bonferroni adjustment was used to determine statistical significance.

3. Results

The number of strides analyzed for each tested walking speed and the average speed values are shown in

Table 1. Number of analyzed strides and averaged walking speed.

	Slow	Comfortable	Fast
Stride number	790	778	355
Walking speed (m/s)	0.90 ± 0.14	1.17 ± 0.14	1.51 ± 0.20

.

For each BoS parameter, a description of the method performance is presented in

Table 2. Results of base of support parameters: mean values (MV) and differences with respect to stereophotogrammetric system. ME = mean error, RMSE = root mean squared error, MAE = mean absolute error, r_xy = Pearson correlation coefficient, LoA = limits of agreement at 95%.

		Slow	Comfortable	Fast
Step length	MV ± SD (mm)	618 ± 43	687 ± 41	753 ± 65
	ME ± SD (mm)	1 ± 11	−1 ± 11	−24 ± 41
	RMSE (mm)	10	10	46
	MAE (%)	1.37	1.23	4.12
	$r_{xy}$	0.979	0.968	0.802
	LoA (mm)	−20 to 22	−22 to 20	−100 to 57
Stride width	MV ± SD (mm)	123 ± 23	124 ± 25	124 ± 36
	ME ± SD (mm)	0 ± 15	0 ± 17	0 ± 19
	RMSE (mm)	14	16	18
	MAE (%)	9.44	10.8	11.45
	$r_{xy}$	0.835	0.801	0.857
	LoA (mm)	−29 to 30	−32 to 34	−37 to 36
Base of support area	MV ± SD (cm2)	1246 ± 148	1238 ± 160	1434 ± 218
	ME ± SD (cm2)	17 ± 36	21 ± 42	22 ± 50
	RMSE (cm2)	39	45	52
	MAE (%)	2.53	2.78	2.74
	$r_{xy}$	0.972	0.967	0.976
	LoA (cm2)	−54 to 88	−61 to 103	−75 to 119

.

For each parameter, Bland–Altman plots are reported in Figure 13.

According to the Shapiro–Wilk tests, mean errors of BoS parameters (step length, stride width, and BoS area) were normally distributed. No significant differences across walking speeds were found (p > 0.05).

The instrumented BoS overlap% at slow, comfortable, and fast speed was equal to 75.7 ± 4.3%, 73.9 ± 5.4%, and 70.7 ± 7.3%, respectively. While the non-instrumented BoS overlap% at slow, comfortable, and fast speed was equal to 89.1 ± 3.5%, 88.8 ± 4.3%, and 88.7 ± 3.3%, respectively. According to the Shapiro–Wilk test, BoS overlap% values were not normally distributed. Statistically significant differences were found between instrumented and non-instrumented BoS overlap% values for each speed (p < 0.05). A significant difference was found across speeds in the BoS overlap% only for the instrumented side between slow and fast speeds (p < 0.05).

Results showed that footprint shift values for the non-instrumented foot were larger than those found for the instrumented foot for each walking speed (Figure 14). According to the Shapiro–Wilk test, footprint shift values were not normally distributed. Statistically significant differences were found between instrumented and non-instrumented footprint shift_x values for fast speed (p < 0.05). A significant difference was found across speeds in the footprint shift_x only for the non-instrumented foot between slow and fast speeds (p < 0.05). No significant differences were found pairwise comparing results of footprint shift_z values (p > 0.05).

4. Discussion

In this study, an original method for the stride-by-stride estimation of the BoS and related parameters, such as right and left step length, stride width, and BoS area, is presented, and validated on gait data recorded on 13 young healthy adults at different speeds.

The unique feature of the proposed method is that it requires to instrument one foot only, thus improving system wearability with respect to previous solutions requiring equipping both feet with an emitter and a receiver [15,16,18,19].

The validity of the estimated BoS parameters was assessed on more than 1900 strides against a gold standard (SP) and strong to very strong correlations were found for all parameters at every walking speed (0.8 $<r_{xy}<$ 0.98).

The RMSE values affecting stride width estimates varied from slow (~0.90 m/s) to high speed (~1.51 m/s) between 14 and 18 mm (MAE% = 9.4% to 11.5%), whereas the RMSE values affecting the BoS area varied between 39 and 52 cm² (MAE% = 2.5% to 2.8%). No significant differences on the estimation errors were found for different walking speeds (p > 0.05).

The RMSE values for the step length estimation were equal to 10 mm (MAE% = 1.2% to 1.4%) at slow and comfortable (~1.17 m/s) speeds. By increasing the walking speed to ~1.51 m/s, the accuracy in the detection of the step length worsened, resulting in an RMSE equal to 46 mm (MAE% = 4.1%). However, despite the larger errors found during fast walking, these differences among speeds were not statistically significant (p > 0.05).

LoA intervals and the visual inspection of the Bland–Altman plots confirmed a slight worsening in the agreement between SP and the wearable system at high speed, which could be associated with a lower number of data points recorded by DS sampling at 50 Hz and the expected lower accuracy in the orientation estimation [36,37].

It can be assumed that the use of DS with higher sampling frequency, allowing an increase in the number of detected points, would improve the identification of the non-instrumented foot and, consequently, the estimation of BoS parameters.

As expected, errors in the estimation of the position of the footprint of the non-instrumented foot were larger than those affecting the instrumented foot (absolute averaged footprint shift_x = 90 mm vs. 56 mm) with a significant difference at fast speed (p < 0.05). This difference is due to the different procedures implemented to estimate the feet position: the non-instrumented foot was identified by scanning its medial side using the DSs, whereas the position of the instrumented foot was determined by double integrating its accelerations.

Similarly, the BoS overlap% values between sensor-based and SP-based BoS areas were significantly larger for the non-instrumented side than the instrumented one (averaged BoS overlap% = 88.9% vs. 73.4%, p < 0.05). In fact, as shown in Figure 9, the estimation of the non-instrumented BoS was only affected by errors in the position of the estimated footprint of the non-instrumented foot (#2). In fact, the position of the previous footprint of the instrumented foot (#1) was made to coincide with the origin of the CS_G, and therefore set equal to zero at every gait cycle. Conversely, the estimation of the instrumented BoS was corrupted by both errors in the position of the non-instrumented foot (#2) and the position of the footprint of the instrumented foot (#3). A possible solution to overcome this problem could be the implementation of a time reversal and inverting the reference initial foot position.

In general, errors affecting the estimation of step length and stride width at comfortable speed were comparable or lower than those reported by Weenk and colleagues (MAE on step length = 17 mm; MAE on stride width = 12 mm) [16].

To the best of authors’ knowledge, there are no previous studies computing the BoS in terms of area during walking using wearable technologies against which to compare our results.

The proposed method comes with some limitations. First, it should be noted that, in contrast with previous studies instrumenting both feet [16,18,19], our method does not provide the trajectory and orientation of the non-instrumented foot over time, but only its footprint position and orientation.

Second, the method performance was validated on normal and almost straight walking, and caution should be paid to extend results to the analysis of curvilinear gait and to patients exhibiting abnormal gait patterns mixed of unpredictable accelerations and decelerations in walking speed with superimposed foot twisting, such as choreiform gait. In real world conditions, the non-instrumented foot might not be detected, or detected with a lower accuracy, during sharp turns, obstacles avoidance, or walking on uneven terrains. In these cases, a solution to increase method robustness would be to attach an additional IMU on the non-instrumented foot in order to track its trajectory during the strides for which inter-foot distance data are missing.

Lastly, it should be acknowledged that, due to the positioning of the wearable system on the medial side of a shoe, the minimum inter-foot distance to avoid the collision between feet must be greater than the system thickness (i.e., 22 mm). We did not encounter any problems when analyzing the gait of healthy subjects and most neurological disorders lead to a wider-based gait [38]. However, it cannot be excluded that this could be an issue in parkinsonian patients typically showing a normal to narrow base of support [39].

5. Conclusions

This study described and validated a wearable system and a novel method for the estimation of BoS parameters such as step length, stride width, and BoS area, providing valuable and accurate information for a complete gait analysis and dynamic stability investigation.

The designed system is lightweight, it only requires instrumenting a single shoe, and it can be suitable for acquisitions in free-living contexts.

Overall, results on healthy subjects were very promising, suggesting that the proposed system and the associated algorithms may provide an accurate and valid solution for the dynamic estimation of BoS parameters, expanding the possibilities to investigate dynamic balance also outside the laboratory setting.

Future work should focus on experimental sessions analyzing more complex locomotion tasks including turning, clinical tests, and simulated daily activities. In addition, the accuracy and the robustness of the proposed system should be investigated on subjects suffering from movement disorders.