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

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.


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]. 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 MI-MUs 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 interfoot 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. MIMU-based methods have been successfully used for the estimation of spatiotemporal 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 noninstrumented 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.

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). 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.

Figure 2.
Wearable system: a magneto-inertial unit and two infrared time-of-flight distance sensors are cabled (in blue) and fixed to a custom 3D-printed rigid support attached to the medial side of a shoe. The size of the wearable system (sensors and rigid support) is 155 mm × 42 mm × 22 mm and the overall weight is ~40 g. The MIMU coordinate system (CSS) is depicted in white.

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 (CSG) (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).

Figure 2.
Wearable system: a magneto-inertial unit and two infrared time-of-flight distance sensors are cabled (in blue) and fixed to a custom 3D-printed rigid support attached to the medial side of a shoe. The size of the wearable system (sensors and rigid support) is 155 mm × 42 mm × 22 mm and the overall weight is~40 g. The MIMU coordinate system (CS S ) is depicted in white.
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.

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).

Identification of the Gait Cycle Interval of the Instrumented Foot
The gait cycle interval was defined as the interval of time between two consecutive flatfoot instants, t 0 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].

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

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 tf, 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] The coordinate system fixed with the instrumented foot (CSS) was made to coincide with the coordinate system embedded with the sensor axes of the MIMU (Figure 2). The orientation of CSS with respect to the Earth magnetic-gravity reference system (CSE),

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 tf, 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] The coordinate system fixed with the instrumented foot (CSS) was made to coincide with the coordinate system embedded with the sensor axes of the MIMU (Figure 2). The orientation of CSS with respect to the Earth magnetic-gravity reference system (CSE), The foot acceleration in CS E , E a(t), was computed by removing the gravity contribution to the recorded accelerations: where S 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 (t 0 ) 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 a(t) was expressed in CS G : where G 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 p OS (t), was computed by double integrating G a(t) between t 0 and t f : 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 R S (t), computed as follows: The foot pose in time with respect to CS G was described by the transformation matrix from CS S to CS G , G T S (t), computed as follows: where G p OS (t) is the translation vector of the foot position (i.e., origin O S ) from the origin O G .

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 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 , at t 0 and t f can be computed as follows: Footprints are then defined by the x G -z G components of G p V i (i = 1, 2, 3, 4) at t 0 and t f .

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 (DSf) and equal to the distance ( ) between DSf and the point F( ) 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 (DSr) and equal to the distance ( ) between DSr and the point R( ) of the medial side of the non-instrumented foot ( Figure 6). The instrumented foot (light grey) is in swing phase and faces the non-instrumented foot (dark grey), which is still during its stance phase. Between t1 and t2, the front and rear distance sensors (DSf and DSr) record distances df(t) and dr(t) by detecting points F(t) (green squares) and R(t) (green dots) of the medial side of the non-instrumented foot.
Let t* be a generic instant of time between t1 and t2, during which the non-instrumented foot was still, when both DSf and DSr could measure non-zero values equal to distances ( * ) and ( * ) between DSf and DSr and the points F( * ) and R( * ) of the medial side of the non-instrumented foot, respectively ( Figure 6).
were measured with a ruler. Position vectors of the vertices in CS S were expressed by means of simple geometrical rules (e.g., . Then, the position vector of the vertices of the footprint G p V i (t) at t 0 and t f are expressed in the global coordinate system CS G .

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 t 1 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 (t 1 ) between DS f and the point F(t 1 ) of the medial side of the non-instrumented foot. Let t 2 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 (t 2 ) between DS r and the point R(t 2 ) of the medial side of the non-instrumented foot ( Figure 6).

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 (DSf) and equal to the distance ( ) between DSf and the point F( ) 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 (DSr) and equal to the distance ( ) between DSr and the point R( ) of the medial side of the non-instrumented foot ( Figure 6). Figure 6. The instrumented foot (light grey) is in swing phase and faces the non-instrumented foot (dark grey), which is still during its stance phase. Between t1 and t2, the front and rear distance sensors (DSf and DSr) record distances df(t) and dr(t) by detecting points F(t) (green squares) and R(t) (green dots) of the medial side of the non-instrumented foot.
Let t* be a generic instant of time between t1 and t2, during which the non-instrumented foot was still, when both DSf and DSr could measure non-zero values equal to distances ( * ) and ( * ) between DSf and DSr and the points F( * ) and R( * ) of the medial side of the non-instrumented foot, respectively ( Figure 6). The instrumented foot (light grey) is in swing phase and faces the non-instrumented foot (dark grey), which is still during its stance phase. Between t 1 and t 2 , the front and rear distance sensors (DS f and DS r ) record distances d f (t) and d r (t) by detecting points F(t) (green squares) and R(t) (green dots) of the medial side of the non-instrumented foot.
Let t* be a generic instant of time between t 1 and t 2 , 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 (t * ) and d r (t * ) 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(t * ) with respect to CS S , S p F (t * ) and S p R (t * ), were computed as follows: 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).
The positions of the points F( * ) and R( * ) with respect to CSS, ( * ) and ( * ), were computed as follows: where ( * ) and ( * ) are the distances recorded at t* by DSf and DSr, respectively, and c is the constant distance measured between each DS and OS ( Figure 7). Then, the detected points were expressed with respect to CSG: The projection of points F( ) and R( ) to the ground plane (xG-zG), , , , and , were then stored in the matrix M: where M is a Nx2 matrix with N equal to the total number of the recorded points of the non-instrumented foot. Then, the detected points were expressed with respect to CS G : 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: 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). 5,6,7,8), were computed: where is the translation vector between B and OG, and is the time-invariant rotation matrix between CSM and CSG, defined as Since the non-instrumented foot is still during t1-t2, its footprint can be defined from the estimated positions of the footprint vertices ( ) (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. , is calculated by means of geometrical rules exploiting /2. Then, the position vector is computed to describe the point V5 with respect to the global coordinate system CSG.

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 (CI and CNI). The direction of progression of the instrumented foot was identified by the line between the footprint centroids of the instrumented foot, CI, between t0 and tf.
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 p V i (i = 5, 6, 7, 8), were computed: where G p B is the translation vector between B and O G , and G R M is the time-invariant rotation matrix between CS M and CS G , defined as Since the non-instrumented foot is still during t 1 -t 2 , its footprint can be defined from the estimated positions of the footprint vertices G p V i (t) (i = 5, 6, 7, 8) with t within t 1 and t 2 .
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.

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 t 0 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 (t 1 − t 2 ) 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 (t 1 − t 2 ) and the direction of progression of the instrumented foot. 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 CNI(t1 − t2) and CI(tf).

•
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 CNI(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 noninstrumented foot between CI(t0) and CNI(t1 − t2). The non-instrumented stride width was defined by CI(t0) and the direction of progression of the non-instrumented foot.

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 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 noninstrumented foot between C I (t 0 ) and C NI (t 1 − t 2 ). The non-instrumented stride width was defined by C I (t 0 ) and the direction of progression of the non-instrumented foot.

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 (m 1 -m 18 ) ( Figure 10).
Sensors 2023, 23, x FOR PEER REVIEW 10 of 20 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]. Figure 10. Positions of the 18 retro-reflective markers attached for validation purposes. The right foot was instrumented with the wearable system. The virtual markers in red (m9-m18) were used only during the initial standing acquisition for calibration purposes. Mid was the midpoint between heel and toe markers (m1-m2 for instrumented foot; m5-m6 for non-instrumented foot). The marker clusters highlighted in orange (m2-m4 for instrumented foot; m6-m8 for non-instrumented foot) defined foot coordinate systems.
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 (CSSP), (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 (CSG). 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 CSSP. By knowing the orientation of the MIMU in both CSG and CSSP, it was then possible to obtain the transformation matrix between CSG and CSSP. Then, the positions of the footprints vertices could be expressed with respect to CSG ( (i = 1,…,8)) and the BoS parameters were identified following the definitions adopted for the estimates from wearable system data (see Figure 9).

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). The positions of virtual markers m 9 -m 15 on the instrumented foot were calibrated during a preliminary standing trial with respect to the rigid marker cluster defined by m 2 -m 4 , according to the CAST procedure to avoid visibility issues [33]. Similarly, the positions of virtual markers m 16 -m 18 on the non-instrumented foot were calibrated with respect to a rigid marker cluster defined by m 6 -m 8 to avoid visibility issues and undesired infrared wave reflections during walking trials. Then, virtual markers m 9 -m 18 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 (m 1 -m 2 for instrumented foot and m 5 -m 6 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 p V i (i = 1, . . . 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 m 9 -m 12 , 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 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).

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 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.

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 (rxy).
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%: where is the area shared by BoS from the wearable system and from SP (BoS AreaSP). 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: = where and are the footprint centroids calculated with the wearable system and SP, respectively. Footprint shifts were analyzed on the ground floor (plane xG-zG), thus footprint shiftx and shiftz 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 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.

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%: 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 shi f t = 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.
Sensors 2023, 23, x FOR PEER REVIEW 12 of 20 affected by larger errors than the instrumented one, since its position was only determined by the detection of its medial side by the DS.

Figure 12.
Metrics to describe the base of support estimation. In this example the instrumented base of support areas from the wearable system (black) and stereophotogrammetric system (grey) are illustrated. The base of support overlap (green) is defined as the percentage of the total base of support area shared by sensor-based area and the one obtained from the stereophotogrammetric system. Instrumented and non-instrumented shifts are the distances between footprints' centroids of instrumented (CI) and non-instrumented foot (CNI) calculated with the wearable system and the stereophotogrammetric system.
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.

Results
The number of strides analyzed for each tested walking speed and the average speed values are shown in Table 1. For each BoS parameter, a description of the method performance is presented in Table 2. 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, rxy = Pearson correlation coefficient, LoA = limits of agreement at 95%.

Slow Comfortable Fast
Step length  Metrics to describe the base of support estimation. In this example the instrumented base of support areas from the wearable system (black) and stereophotogrammetric system (grey) are illustrated. The base of support overlap (green) is defined as the percentage of the total base of support area shared by sensor-based area and the one obtained from the stereophotogrammetric system. Instrumented and non-instrumented shifts are the distances between footprints' centroids of instrumented (C I ) and non-instrumented foot (C NI ) calculated with the wearable system and the stereophotogrammetric system.
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.

Results
The number of strides analyzed for each tested walking speed and the average speed values are shown in Table 1. For each BoS parameter, a description of the method performance is presented in Table 2. 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%.
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). 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 over-lap% 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).

Base of support area
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 shiftx values for fast speed (p < 0.05). A significant difference was found across speeds in the footprint shiftx only for the non-instrumented foot between slow and fast speeds (p < 0.05). No significant differences were found pairwise comparing results of footprint shiftz values (p > 0.05).

Figure 14.
Mean plots with standard deviation bars for footprint shiftx with respect to footprint shiftz for the different walking speeds. The footprint shifts of the instrumented foot (black) and non-instrumented foot (grey) are the distances on the ground plane (xG-zG) between sensor-based footprint centroids and the ones obtained with the stereophotogrammetric system. * statistically significant difference at p < 0.05.

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 Figure 14. Mean plots with standard deviation bars for footprint shift x with respect to footprint shift z for the different walking speeds. The footprint shifts of the instrumented foot (black) and non-instrumented foot (grey) are the distances on the ground plane (x G -z G ) between sensor-based footprint centroids and the ones obtained with the stereophotogrammetric system. * statistically significant difference at p < 0.05.

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 cm 2 (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 tõ 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 noninstrumented foot and, consequently, the estimation of BoS parameters.
As expected, errors in the estimation of the position of the footprint of the noninstrumented 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].

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. Funding: This study was partially supported by DoMoMEA grant, Sardegna Ricerche POR FESR 2014/2020. This study is also part of the project NODES which has received funding from the MUR-M4C2 1.5 of PNRR with grant agreement no. ECS00000036. The content of this study represents the views of the authors only and is their sole responsibility; it cannot be considered to reflect the views of the European Commission and/or the MUR. The European Commission and the MUR do not accept any responsibility for use that may be made of the information it contains.

Institutional Review Board Statement:
The study was conducted in accordance with the Declaration of Helsinki and approved by Independent Ethics Committee of University Hospital of Cagliari, Italy (Prot. PG/2021/1195 of 27 July 2021).

Informed Consent Statement:
Informed consent was obtained from all subjects involved in the study.

Data Availability Statement:
The Matlab code used in this project can be provided upon request to the corresponding author. Sensors 2023, 23, x FOR PEER REVIEW 18 of 20 Figure A1. Example of recorded distance data from rear (dots) and front (squares) distance sensors during a swing of the instrumented foot during a walking trial at fast speed (1.8 m/s). Red data correspond to the outliers exceeding the 95% confidence intervals. The dashed horizontal lines determine the thresholds within which the distance data from rear (black) and front (blue) distance sensors are considered reliable. Figure A1. Example of recorded distance data from rear (dots) and front (squares) distance sensors during a swing of the instrumented foot during a walking trial at fast speed (1.8 m/s). Red data correspond to the outliers exceeding the 95% confidence intervals. The dashed horizontal lines determine the thresholds within which the distance data from rear (black) and front (blue) distance sensors are considered reliable.