Portable Gait Lab: Tracking Relative Distances of Feet and CoM Using Three IMUs

Ambulatory estimation of gait and balance parameters requires knowledge of relative feet and centre of mass (CoM) positions. Inertial measurement units (IMU) placed on each foot, and on the pelvis are useful in tracking these segments over time, but cannot track the relative distances between these segments. Further, drift due to strapdown inertial navigation results in erroneous relative estimates of feet and CoM positions after a few steps. In this study, we track the relative distances using the assumptions of the Centroidal Moment Pivot (CMP) theory. An Extended Kalman filter approach was used to fuse information from different sources: strapdown inertial navigation, commonly used constraints such as zero velocity updates, and relative segment distances from the CMP assumption; to eventually track relative feet and CoM positions. These estimates were expressed in a reference frame defined by the heading of each step. The validity of this approach was tested on variable gait. The step lengths and step widths were estimated with an average absolute error of 4.6±1.5 cm and 3.8±1.5 cm respectively when compared against the reference VICON©. Additionally, we validated the relative distances of the feet and the CoM, and further, show that the approach proves useful in identifying asymmetric gait patterns. We conclude that a three IMU approach is feasible as a portable gait lab for ambulatory measurement of foot and CoM positions in daily life.

A minimal inertial measurement units (IMU) only approach has great benefits as an ambulatory sensing setup; they are highly portable and can be miniaturised. They are suitable for measuring kinematics of any rigid body segment they are attached to using strapdown inertial navigation. However, this introduces errors in the position estimates. Further, IMUs do not track relative distances from nearby body segments. This results in erroneous estimates of relative distances of segments over time. There are different solutions to solve this issue of drift between two segments. For instance, Xsens TM offers a full body suit of IMUs that employ a biomechanical chain [2] to track movement of associated body segments. This requires setting up a lot of IMUs, and initializing a few biomechanical parameters. Alternatively, specially designed systems such as the ForceShoe TM [3], and SWING [4] use time of flight information from ultrasound or infrared respectively, to measure the relative distance between the feet. Unfortunately, the ForceShoe TM is quite thick and heavy, and not suited for daily wear [1]. Also, additional sensors require extra calibration and synchronisation steps for reliable monitoring.
A minimal three IMU setup, where one IMU is placed on each foot and one on the pelvis is an ideal setup for ambulatory monitoring. However, this may be insufficient for measuring relative foot and pelvis positions, owing to the issue of drift described earlier. Some studies solve this by using mathematical constraints that prevents drift between the feet [5], [6]. However, these may not reflect the true foot positions during continuous tracking, or in cases of an asymmetric gait. Other studies use biomechanical constraints related to the pattern of gait. Bancroft and Lachapelle [7] use information about stride length, and a difference in vector between foot positions to reduce the drift. Zhao et al. [8] use a derivation of step length from information about limb sway for this purpose. In both cases, approximations have been made regarding a general pattern of gait cycle. Sy et al. [9] show that using an extended set of biomechanical constraints can help estimate kinematics from a reduced sensor setup, but they assume a fixed pelvis, and do not comment on relative segment distances.
The theory of Centroidal Moment Pivot (CMP) could serve as a realistic biomechanical principle that relates the movement of the CoM with the stance foot. Assuming an inverted pendulum model of gait, the theory states that for normal, level-ground human walking, the moments around the CoM is zero [10], [11]. This implies that the whole body ground reaction force (GRF) and a vector connecting the virtual CMP point and CoM are parallel [12]. This gives us a relation between the virtual CMP point and CoM, and that of the GRF [12]. For an IMU based approach, the virtual CMP point can be assumed to be the same as foot position. Additionally, estimations of the shear GRF and the height of the CoM are required [12]. Using the pelvis IMU, the 3D GRF can be estimated [13], and the height of the CoM can be tracked by adapting existing methods in literature [14], [15].
The goal of this study is to track the relative positions of the feet and CoM using only three IMUs. For this, first, the foot trajectories were estimated from the foot IMUs using strapdown integration, improved by zero velocity and zero height updates [3]. Then, the 3D instantaneous estimates of GRF [13], CoM velocity [16], and height [14] were estimated from the pelvis IMU. 3D GRF and CoM height were used to solve the CMP theory, and derive relative positions between the feet and CoM [12]. These positions were used to reduce the drift between the feet and CoM. The proposed algorithm was tested for variable gait patterns. The resulting kinematics were expressed in a body-centric frame of reference or current step frame [13], and compared with reference systems.

II. METHODS
Here, the methods used to track the kinematics of feet and CoM are further explained. First, in Section II-A, we briefly explain the reference frames [13] used in this study. Next, Section II-B describes the IMU models used. In Section II-C, the design of the EKF for the tracking is described. Section II-D describes the measurement systems used, and Sections II-E and II-F describe the participant group, and the experimental protocol used to obtain measurements and validate this study respectively. Finally, Section II-G explains how the results were analysed.

A. Reference Frames Used
A changing reference frame [13] was used to express the kinematics in this study. This allows us to provide a body centric frame of expression, as opposed to an arbitrary global frame used commonly. A detailed description of the different frames used and the transformations between them are given in Mohamed Refai et al. [13]. Here, we summarize them briefly.
The changing reference frame was based on the direction of steps being made and will be referred to as the current step frame, denoted as ψ cs . In Fig. 1, an example of ψ cs for the step k made by the right leg is shown as the light green shaded triangle. The frame was defined with the X axis along the heading of the step (bold dotted line in Fig. 1) and Z along the vertical. As the IMU measures in its sensor frames, ψ s , it has to be transfomed to the ψ cs per step.
First, a sensor to segment calibration was performed to the respective segment (seg) frames ψ seg [13]. The segments included the pelvis (p), left foot (fl), and the right foot (fr). Then, the change in orientation of the segments during a step k was first expressed in the current step frame ψ cs(k−1) of the previous step k − 1. The change in orientation was estimated  [13]. The pink foot is the left foot, and the thin line is the CoM trajectory during walking. The changing current step frame ψ cs(k) is defined for each step k. Segments of interest were the feet (fl) and (fr), and pelvis (p).
using an error extended Kalman filter [13]. At the end of the step k, using the change in position of the swing foot the ψ cs(k) was estimated. This procedure was iterated for each step.
In short, four frames of reference were used in this study: sensor frame (ψ s ), segment frames (pelvis ψ p , right foot ψ f l and left foot ψ f l ), a current step frame defined by the previous step (ψ cs(k−1) ), and the current step k (ψ cs(k) ).

B. Inertial Measurement Unit Model
The 3D accelerometer and 3D rate gyroscope present in the IMU provides the acceleration and angular velocities in the sensor frame ψ s respectively, and can be modelled as: Here, a is the linear acceleration of the sensor, g is gravity, and e A is Gaussian white noise. Also, ω is the angular velocity, b is a slowly varying offset, and e G is Gaussian noise. Both (1) and (2) are discrete time equations and are expressed for a given time instance i . Fig. 2 shows a brief overview of the steps involved. There are a few working assumptions we need to consider. We assumed an inverted pendulum model of gait where all mass is concentrated at the CoM that is located within the pelvis [11], [15]. Thus, the GRF accelerates the CoM and opposes gravity. These assumptions should hold as long as the subject walks normally, and doesn't fall or negotiate large obstacles. Additionally, the feet are the only contact with the external world, and no additional load is carried by the body. Given these assumptions, the accelerations measured by the IMU at the pelvis is similar to the accelerations at the CoM. Thereby, the pelvis segment ( p) will be hereto referred as the CoM (C).

C. Fusion Filter to Track Relative Feet and CoM Positions
The method of Skog et al. [17] was used to estimate the foot contact instances for each foot. As the IMUs were synchronized in time, double stance instances can be estimated. Though, distinct gait events can be estimated from foot IMUs [18], for sake of simplicity, a step was defined to take place between halfway of a double stance until halfway of the subsequent double stance.  An extended Kalman filter (EKF) was used to track the velocity and position of the three segments; both feet and CoM, in the current step frame ψ cs . The filter notations used are tabulated in Table I. The state vector of the EKF was denoted as x and its covariance matrix as P. The states as shown in (3) included the 3D position p, and 3D velocity v of each segment, with the superscript denoting the corresponding segment.
The following text expands on the overview shown in Fig. 2. It describes the a-priori estimate determined using strapdown integration of the accelerations measured at each segment. Then, the special biomechanical constraints applied to each segment are shown. The implementation of the CMP assumptions to reduce the drift between the three segments is also described. 1) Strapdown Inertial Navigation: This is the prediction step of the EKF. The accelerations at each segment was expressed in the ψ cs(k) as [13]: The R cs(k),cs(k−1) k , and R cs(k−1),seg i were estimated using the methods described in Section II-A [13]. Here, i denotes the samples during the current step k. The position and velocity can be estimated as [3]: Here, T is the time step. The Kalman filter prediction equation was given as [19]: where F = I 3 T 0 3 I 3 , and u = T 2 2 ·â cs i T ·â cs i (8) and the covariance matrix was predicted usinĝ where, Q is the process noise covariance matrix.
2) Measurement Update: The measurement updates used to reduced drift in the estimation of position and velocity of the feet and CoM are as follows. A summary of when these updates are applied is shown in Fig. 2.
• Zero Velocity Update (ZV): As shown in Fig. 2, this is applied when the foot is in contact with the ground [17], as the velocity of the feet are assumed to be zero. The measurement z zv is provided as follows, and applied only to the feet.
and, H In the above equations, H transforms the state vector to a measurement predictionẑ, and z denotes the actual measurement [19]. e zv denotes the error associated with this measurement. The same notations are used in the following equations. • Zero Height Update (ZH): During the same foot contact instances as above, if we assume walking over a flat surface, we have information regarding the height of the foot from the floor.
and, H f l The measurement matrix H zh has only one row as it transforms only the Z axis of each foot. Here, p f Z ,init is the initial height and the superscript f denotes either the left ( f l) or right foot ( f r). This update was only applied to the feet. • CoM Velocity (CV): The ZV update ensures that the velocity of the feet do not drift due to integration errors. However, the CoM is constantly moving. Therefore, an estimate of the CoM velocity was derived by fusing two complementary sources of information [11], [16].
A high frequency information (v C h f ) was derived from an optimally filtered direct and reverse strapdown integration [14] of the CoM accelerations using a cut off of 0.6 Hz. Then, low frequency information (v C l f ) of the CoM velocity was derived from low pass filtering the average of the foot velocities using the same cut off used for v C h f . The two sources were fused to get estimates of the instantaneous 3D CoM velocity [16]: • CoM Height (CH): The height of the CoM (p C Z ) was also estimated using a complementary filter method [11]. An optimally filtered direct and reverse strapdown integration [14] of vertical CoM velocity was used to obtain the changes in CoM height during gait using a cut off of 0.3 Hz to obtain the p C Z ,h f . Then, as the subject does not crouch or jump while walking, the height of the CoM should oscillate around an offset. Assuming an average walking CoM height as 98% of the height during quiet standing (p C Z ,init ) showed least errors when validating this method. The average height and the p C Z ,h f were fused to get an estimate of CoM height p C Z during walking.
The CMP update was used to restrict the feet and CoM from drifting apart or towards each other. Fig. 2 summarizes the steps involved; the CMP update was used to first estimate the horizontal position of stance foot with respect to the movement of the CoM. After correcting discrete changes at the start of each swing phase, the CoM trajectory was updated for the corrected foot positions. These steps are explained further below.
(14) is a mathematical expression of the CMP theory [10]. When the 3D components of GRF are known, the distance between the CoM (p C ax ) and a virtual CMP point (cmp f ax ) under the stance foot can be estimated [13]. Here, ax denotes either X or Y axes, and F is the 3D GRF in a specific axis. The 3D components of GRF estimated from the CoM accelerations [13] were used to estimate the ratio F ax /F Z . The p C Z or height of the CoM was already estimated using the update CH in (13). Note that there are a few assumptions regarding (14). First, we assumed that the virtual CMP position (cmp During right swing: The measurement matrices H f r cmr and H f l cml transform only the X and Y axes of the foot positions in the state vector. This update corrects the drift in relative positions between the CoM and stance foot during swing phase. However, this may cause a discrete jump in relative foot distances at the start of the swing phase. To have a smooth change in relative foot distances between subsequent steps, knowledge of the relative foot distances at the end of the preceeding step was used to update the relative foot distances at the beginning of the subsequent swing phase.
Start of left swing: In (16a)  This correction of relative foot distances requires a final update of the CoM position following the CMP theory. (14) was adapted to obtain the CoM from foot estimates as follows.
During left swing, cmp f ax represents the right foot, and vice-versa for the right swing. During these instances, the CoM position was improved using the following measurement update: transforms the X and Y axes of the CoM positions to the measurement. The measurement updates were applied to the EKF using the standard equations. The Kalman gain was estimated using (19), the state matrix was updated with (20), and the error covariance matrix was updated using (21).

3) Reinitialising for
Step k+1: After the state vector has been updated using the prediction and measurement updates, the trajectory of the segments are known for the current step k. The ψ cs(k) was adjusted using methods described in Section II-A using the improved estimates of the foot positions. For the next step k + 1, accelerations are transformed to the current step frame ψ cs(k+1) using (4), and the steps described in Sections II-C were reiterated. 4) Initialisation and Noise: Before applying the EKF, the states for each segment and their covariance noises have to be initialised. The right foot was assumed to be the origin. The initial locations of the CoM, and the left foot were measured from VICON c . All initial velocities v seg were set to zero, and the initial noise was set to arbitrary values. The process and measurement noises shown in Table II were estimated from sensor specifications, and then fine tuned by optimizing the error between estimated and reference values.

D. Measurement System
Three Xsens TM MTw IMUs formed the minimal setup and can be visualized in Mohamed Refai et al. [13]; one IMU was mounted on the pelvis, and one on each foot. The pelvis IMU was placed below the midway point between the line connecting the left and right posterior superior iliac spine. The foot IMUs were placed on the midfoot region. The MT Manager (version 4.8) software was used to read the data from the IMU wirelessly, which was sampled at 100 Hz.
Two reference systems were used. The ForceShoe TM was used as wearable reference for the estimation of forces required in (14). The ForceShoe TM consists of a 6-DoF Force and Torque sensor, and an IMU under each toe and heel of both feet [20]. It has been validated against force plates (AMTI ® ) for measurement of contact forces [11]. A VICON c motion capture system (Oxford Metrics PLC.) was used as the reference system for validating the velocities and positions estimated using the state vector. Markers were placed on the following locations on both the left and right limbs: anterior superior iliac spine, posterior iliac spine, the second and fifth metatarsal, and heel. One marker was also placed on each IMU. The data from VICON c and ForceShoe TM were sampled at 100 Hz. The data was then low pass filtered at 10Hz with a zero-phase second order butterworth filter.
Foot contact was estimated when the magnitude of forces measured by the ForceShoe TM was below a set threshold of 30 N on each foot. Foot positions were derived from the marker on the IMU. The CoM position obtained from VICON c was assumed to lie at the centroid of the four pelvis markers. The feet and CoM positions were differentiated and low pass filtered with a second order zero phase Butterworth filter of cut off 10 Hz to obtain the respective velocities.
The measurements by both reference systems were transformed to the ψ cs frame that was determined using the VICON c foot positions. To synchronize the two reference systems with that of the Xsens TM MTw IMUs, the subjects were asked to raise their right leg before starting the experimental protocol. The magnitude of angular velocities measured with the Xsens TM , as well as the IMUs in the ForceShoe TM were used to synchronize these systems. The change in right foot position was used to synchronize the VICON c with the other two systems. A manual check was performed in order to verify if all the signals were properly synchronized.

E. Participants
Six healthy subjects were recruited for the study. The average and standard deviation of the height, weight, and age was 1.7±0.1 m, 74.1±10 kg, and 25.6±2.8 years respectively. Leg length was measured from the greater trochanter to the ground [21] and was 0.9 ± 0.04 m. All participants signed an informed consent before the experiment. The study was conducted in accordance with the Declaration of Helsinki, and the protocol was approved by the Ethical Committee of the faculty. The inclusion criteria included subjects with no history of impaired gait or leg injury. One subject was female and all of their shoe sizes were 40 (European Size Chart).

F. Experimental Protocol
The ForceShoe TM was calibrated using the MT Manager software, and the VICON c was calibrated using standard procedures. Fig. 3 summarizes the experimental protocol. The subjects began by standing still for a few seconds with their feet placed parallel, and were asked to bend the trunk forward. This was used to calibrate the pelvis segment frame in Section II-A [13]. The subjects were then asked to perform a set of walking tasks, each repeated four times: 1) Normal Walk (NW): The subject was asked to walk at their preferred walking speed for 5 m. in an asymmetric manner. The instruction given was to induce a stiff left knee and abduct the hip as much as possible, and also have a shorter step on the right side.

G. Analysis of Results
In the following text, the minimal IMU sensing setup, along with the algorithms explained in Section II-C will be referred to as Portable Gait Lab (PGL). A zero-phase Butterworth low pass filter of order 4 and cut off 3 Hz was used to filter noise from the estimated kinematics.
The estimated forces and kinematics were compared against the reference systems, ForceShoe TM and VICON c respectively. First, the errors in estimating the ratio of forces ( F ax F Z ) in (14) from the pelvis IMU [13] was studied. The errors were expressed as the root mean square of the differences normalised by the range of the reference values in both X (rRat X ) and Y (rRat Y ) axis. Then, the root mean square of the differences in estimating CoM height (RCoM Z ) was studied. The error margins rRat X Z , rRat Y Z , and RCoM Z are required to understand the errors associated with (14), and eventually, the relative distance estimates.
The root mean square of the errors (Right X , Right Y , Le f t X , Le f t Y , CoM X , and CoM Y ) in estimating the horizontal positions of feet and CoM were then analysed for each step. The vertical foot clearance comparison has been neglected in this study, as it is not novel [22]. Then, the average 2D horizontal Euclidean distance (E D) between the feet at the end of each step for all walking tasks was measured and compared against the VICON c reference. Following this, spatial gait parameters, such as the Step Lengths (SL), and Step Widths (SW ) were estimated [23]. A metric CoM width (CW ) was derived by estimating the average 2D Euclidean distance between the stance foot and CoM trajectory for each step. This provided an average relative distance between either foot and CoM. Correlation and Bland-Altman plots were used to compare the SL, SW, and CW derived from the PGL with the reference VICON c . Finally, the feasibility of the PGL in differentiating between symmetrical and asymmetrical walking was studied by comparing the differences between left and right steps in two walking tasks, the NW and AW.

III. RESULTS
A few trials were removed from analysis due to issues with the reference setups. Further, it was made sure that each subject had at least three walking trials per walking task. First, an example of the CoM height estimated using the complementary filter approach in (13) is shown in Fig. 4 as the solid blue line. The dotted red line shows the measurement by the VICON c . Here, the subject is performing a WT task and makes the 180 • turn around 25 seconds. Table III shows the errors (rRat X Z , and rRat Y Z ) in estimating the ratio of forces and that of CoM height (RCoM Z ) in (14) for the different   Fig. 5 shows a graphical step-wise comparison of the feet and CoM positions estimated by the PGL with the reference VICON c for the same trial as shown in Fig. 4. In Fig. 5, circles denote estimations by the PGL, and the lines denote the measurements by VICON c . For both systems, the trajectory of the right foot is shown in red, the CoM in green, and the left foot in blue. Each subplot is a top-down view of a step expressed in the current step frame ψ cs . The X axis of each plot corresponds to the X axis of the ψ cs , and similarly for the Y axis. As each step is represented in its own ψ cs frame, they all progress to the right, even during turns. In the first subplot, the left foot moves first. Then, the subject can be seen to make consecutive steps, until step 6, where they prepare for the 180 • turn. The turning step is highlighted with a shaded light red background. Although the PGL shows deviations during the turn when compared to the reference, it converges to the reference values two steps after.
Table IV displays the errors (Right X , Right Y , Le f t X , Le f t Y , CoM X , and CoM Y ) in estimating the horizontal feet and CoM positions for each step. These are an average across all steps in a walking task excluding the turning steps. Turning steps were those that made a 60 • or larger change in direction when compared to the preceding step. Table IV also summarizes the difference in relative distance between the   feet (E D) at the end of each step. Across all walking tasks, this was found to be 8.8 ± 1.0 cm on average.
The estimates of SL, SW, and CW for all tasks except AW are compared against the reference using Fig. 6, 7, and 8 respectively. Some steps had particularly large SL or CW values measured by the VICON c than the average, thereby skewing the distribution as outliers. They were removed based on the interquartile range of the distribution of the VICON c estimates for each parameter (SL, SW, and CW ). Fig. 6, 7, and 8 distinguish between a step with relatively straight heading, denoted as a blue filled square and turning steps denoted by a red filled circle. In each of the three figures 6, 7, and 8, the left subplot shows the agreement between the PGL estimates and reference using correlations. The dotted gray line is the identity line, the solid blue line is the linear fit for the straight steps, and solid red line is the linear fit for the turning steps. The legends in each figure indicate the correlation between the PGL and reference and its significance denoted by a star. The average root mean square of the errors is also shown.
Further, in Fig. 6, 7, and 8, the right subplot shows the Bland-Altman plot. The solid lines denote the median difference between the two systems, with a star to denote no significant difference between the mean of the two systems. The dotted lines denote the 95% limits of agreement (LoA). The LoA for SL were found to be [−25. Finally, the feasibility of PGL in differentiating symmetric from asymmetric gait is shown in Fig. 9 and 10. These figures compare the boxplot distribution of SL on the left and right side. Fig. 9 plots this comparison for the NW task, where the subjects walked symmetrically, and Fig. 10 plots this for the asymmetric AW task. In both figures, the left subplot shows the distributions measured by the reference system, and the right subplot shows that of the PGL. Fig. 10 shows that the   both, the reference and PGL, find significant differences in SL between the right and left side for the AW task.

IV. DISCUSSION
This study shows the feasibility of relying on simply a three IMU setup for estimating relative distances of the feet and CoM. Estimating kinetics [13], and kinematics using the principles in this study allows us to use the three IMU setup as a portable gait lab.
There are several biomechanical assumptions considered for this study. In order to apply (14), 3D GRF and height of CoM have to be estimated instantaneously. Refai et al. [13] have already discussed the feasibility of using the PGL in estimating 3D GRF while assuming that the CoM resides within the centroid of the pelvis. Here, the errors associated with the ratios (rRat X Z , and rRat Y Z ) as seen in Table III was 19.3 ± 3.1% of the range of the reference values in the worst case. The average RMS error in estimating the vertical CoM position by Floor-Westerdijk et al. [15] was 3.5 ± 1.3 mm. In our study, it was on average 7 ± 2.4 mm across all walking tasks. Though this is larger than the [15], they only compared the differences in a detrended position for an average stride. We state the average error in estimating the instantaneous CoM height over the complete gait including initiation, turning, and stopping. Knowledge of the force ratios and CoM height allows us to use the CMP constraint using (14) and (17). As the CMP measurement updates were applied only during specific instances of gait, the errors in estimating ratio of forces and CoM height influences the relative positions during these instances. Fig. 5 shows the step-wise comparison of the tracking by the PGL for a subject performing the WT task. Each step is shown in its respective current step frame ψ cs , with the feet and CoM moving from left to right. In some steps, for instance step 3, the trajectory of the stance (right) foot position measured by PGL is slightly different from the VICON c . This could be the rolling of the foot during stance phase, which is measured by VICON c . As the PGL tracks the foot as a fixed point, this rolling is not modelled during stance phase. Thus, we observe a steady medio-lateral position of the stance foot for the PGL estimates. Furthermore, although the virtual CMP point follows the trajectory of the Center of Pressure (CoP) [10], in (14) we assumed that it follows the trajectory of the stance foot tracked by the PGL. These two issues could induce a systematic difference in estimations of foot positions or spatial parameters of gait. The PGL could be further improved with a model for CoP movement [24] or the rolling of the stance foot. Further, we see more discrepancies during the turning steps than other steps. These issues could be because the movement of the CoM deviates further from the centroid of the pelvis when making turns.
Additional biomechanical constraints should be explored to improve estimation of kinematics during turning. Measuring the influence of the rotational inertia of the upper body could improve the assumptions of CMP. Nevertheless, the estimated positions converge to the reference values when the subject continues to make additional steps. Table IV shows that the algorithm has comparable performances across different walking tasks in estimating the absolute position of the feet and CoM. As these errors are an average across all steps in a trial, they indicate the usefulness of PGL in restricting drift between the segments with time. Although marginal, the differences in the orientation of ψ cs defined separately by the PGL and VICON c could influence these errors. The error margins can be put into perspective when comparing it against the average stride length, which was found to be 1.1 ± 0.2 m by VICON c for all tasks except AW. The mean absolute error in estimating stride lengths across all walking tasks was found to be 5.9 ± 1.5 cm in our study as compared to −1.5 ± 4.6 cm as found by Kitagawa and Ogihara [25]. Note that the errors in estimating positions for the AW task are similar to the others, even though this task comprised of an asymmetrical gait with shorter steps on the right. In our validation study [12] on the assumptions of CMP theory, we found that there was an average error of 6.7 ± 0.6 cm between the virtual CMP position and true foot position as measured by VICON c . These influence the errors in the relative distances (E D) between either feet as seen in Table IV. Sy et al. [9] used a set of biomechanical constraints to track the lower limb using a similar three IMU approach with an average error of 5.2 ± 1.4 cm. This was 13.5 ± 0.7 cm for our study. However, in the reference study [9], they assume a fixed pelvis, and measure all segments with respect to it. On the other hand, we track the relative distances between the feet and CoM for variable as well as asymmetric gait. The error margins include all steps in the walking task, thereby showing robustness against drift in relative distances, which becomes larger with time. It may be of interest to include the constraints explored in Sy et al. [9] where they also estimate joint kinematics. Combining Sy et al. [9] with the current study can result in a system that provides complete linear and joint kinematics of the lower limb using a minimal IMU setup.
We have also validated the PGL for estimating spatial parameters: SL, SW, and CW for variable gait. These parameters are dependent on good estimations of relative distances of both feet and CoM. In healthy subjects and hemiparetic populations, the SL variability is close to 2 cm and 3.4 cm respectively, and the SW variability is close to 2 cm and 1.8 cm respectively [26]. Using an ultrasound sensor to measure relative distances, Weenk et al. [3] estimated the SL and SW with an average absolute error of 1.7 ± 2 cm and 1.5 ± 1.5 cm respectively. In our study, using only three IMUs, we found it to be 4.6 ± 1.5 cm and 3.8 ± 1.5 cm respectively. These errors include variable walking, and are slightly larger than clinical variability. Analysing straight line walking tasks such as the 10 metre walk will reduce the impact of variable walking when using the PGL for clinical studies. Furthermore, although the estimates were significantly correlated with the reference system (Fig. 6, 7, and 8), the correlations were found to be moderate for estimations of SL, and CW, and weak for the SW. This suggests that there is merit in using the PGL to study average spatial parameters over a number of trials, although caution must be taken when comparing individual steps. Figures 6, 7, and 8 do not include the spatio-temporal parameters for AW task, as the steps were asymmetric on either side. The spread of the spatio-temporal parameters for straight steps in the correlation subplot in all three figures lies along the identity line. The bias seen in the Bland-Altman plots could be due to systematic differences owing to the several assumptions considered and discussed in this study.
Finally, Fig. 9 and 10 show that the algorithm can distinguish between normal and asymmetric walking patterns, as identified by significant differences in step lengths for the AW task. This indicates that the PGL may be adequately sensitive for differences of clinical importance such as gait asymmetry due to stroke or other conditions.

A. Limitations and Future Work
The PGL requires reliable estimations of the CoM height and velocity, feet velocities, and 3D GRF before it can track the relative distances. Therefore, a number of assumptions were used regarding the biomechanics of gait. For instance, in (13), we found the average height of CoM during walking by optimizing the errors between the PGL estimate and reference values. Further, we used an inverted pendulum model of walking where the CoM is encompassed within the pelvis segment [15]. There may be larger errors if the CoM deviates further from the centroid of the pelvis, or if the subject crouches or jumps. Errors may also be larger for subjects with an asymmetric body posture due to an impairment or paralysis.
Applying the PGL requires knowledge of initial relative distances of the feet and CoM, which could be input using a tape or other sources. Further, the algorithms require a few steps to calibrate and define the different reference frames, and to initialise the heading of the feet. This could mean that there are some restrictions to be considered when designing a real time application system. The current design of the sensor fusion filter tracks 18 parameters for each iteration that includes the position and velocity of each foot and CoM, and might result in a heavy computational load. However, this should not pose a problem if the processing is performed offline, on a desktop, or a cloud service. Further, it is important that the three IMUs are synchronized well, as the movements are related to each other.
The PGL has not been tested on other aspects of variable gait such as shuffling of the feet, ascending or descending stairs, walking backwards, etc. Although, we have shown that the errors are close to margins of variability, they can be improved, and a follow up study with measurements from a free daily life environment must be designed. The error margins found in the current PGL could be acceptable to derive an overview of gait patterns, track people in daily life, and also to derive balance and stability measures using the relative foot distances. For instance, the BoS and MoS [1] can be derived using this approach. However, employing the PGL to study individual steps must be considered within the provided error margins.
Here, the AW task was used to validate the feasibility of using PGL in asymmetric gait. Nevertheless, a validation study using subjects with impaired and/or asymmetric gait is required, as it might be difficult to detect distinct gait events in impaired gait. Issues with lower limb motor control may result in shuffling patterns, such as freezing of gait as seen in patients with Parkinson's disease. This will influence the estimation of gait events, and therefore the application of CMP updates.

V. CONCLUSION
The feasibility of using a minimal three IMU based setup in measuring relative distances between the feet and CoM is shown for overground variable gait. The average absolute errors in estimating step lengths and step widths were 4.6 ± 1.5 cm and 3.8 ± 1.5 cm respectively. The approach is sensitive in differentiating symmetric and asymmetric gait. Further validation in free walking conditions and subjects with impaired gait patterns must be done.