One hundred and sixty of the collected samples were post-processed using Vicon Nexus, and used as reference for IMU-based gait analysis evaluation. To avoid overfitting, we split data into development and validation sets. Approximately 30% of the samples, from uniquely randomly selected users, were included in the development set, and used for algorithm design, debug and optimization. The remaining 70% were used for validation and orientation-invariance proof. The resulting dataset split is shown in Table 1.

Dataset characteristics after splitting.

To compare gait parameters extracted from the IMU with those extracted from Vicon, systems were first synchronized. To this purpose, we used the cross-correlation between acceleration magnitude—obtained from the IMU—and the centroid of the sensor markers—obtained after deriving the centroid trajectory two times; the maximum cross-correlation was used to compensate for the time-shift between data sources, as in [6].

A tolerance of 0.1 s was employed to classify FC events (and its corresponding stride gait parameters) as true positive cases. Only the strides whose FC time was consistent with those obtained with Vicon were considered for comparison. Reference strides without any corresponding IMU-derived candidate were classified as not detected.

The development set was used to tune parameters, and select the most reliable methods for orientation estimation and double integration. As shown in Section 2.4.2, the possible orientation estimation methods were the Madgwick CF, the Euston CF and the gyroscope integration, which could be used in combination with two possible double integration methods: the direct and reverse integration, and the linear dedrifting—both presented in Section 2.4.3. Additionally, the horizontal correction method presented in Section 2.4.3 could or not be employed. Each method—with the exception of gyroscope integration and linear dedrifting—included a set of parameters (also detailed in Section 2.4.2 and Section 2.4.3) that needed to be optimized in view to improve the performance of the gait analysis method. Using a grid-search approach, all possible combination of methods (i.e., orientation estimation and double integration methods) and a set of candidate parameters could be tested, where the number of resultant combinations depended on the number for parameters tested. For this reason, parameter tuning was first performed using a coarse grid of parameters—i.e., a small amount of candidate parameters covering a wider range of values. The most promising combinations of parameters and methods were used to define a finer grid that considered candidate parameters defined in the neighbourhood of the best parameter configurations. Parameters tested within coarse (resulting in 140 combinations) and fine grid-search (resulting in 105 combinations) are shown in Table 2.

Methods and parameters used in coarse and fine grid-search.

To select the best combination of parameters and methods, we calculated the root mean square error (RMSE) between IMU-derived and reference gait parameters. The RMSE of each parameter was normalized by the average of the reference, and then averaged to obtain a single score per configuration. The minimum normalized RMSE defined the most appropriate set of parameters and methods.

We compared gait parameters extracted from the IMU with those extracted from Vicon using the validation set. For each cycle, we estimated the difference between IMU-derived and reference gait parameters. Accuracy (mean of relative and absolute error) and precision (standard deviation of relative and absolute error) were reported for each parameter. Agreement between the two instruments was assessed using 95% limits of agreement, as introduced by Bland Altman [31]. Data were assessed for normal distribution using Shapiro–Wilk tests, to decide for the use of parametric or non-parametric tests. Correlation between instruments was calculated using the correlation coefficients of Pearson (rp)—in case of normal distribution—or Spearman (rs)—when data could not be assumed to be normally distributed. We have also reported RMSE and equivalence tests using an equivalence zone of ±5% of the average of the metric. Equivalence tests were based on Paired T-test (T)—for parametric—or Wilcoxon signed-rank test (W)—for non-parametric.

To validate results in a scenario where only straight walking is considered for gait assessment (as required to assess several gait disorders [32]), we repeated validation tests without including turns. For this purpose, a turning stride was considered a stride where the turning angle (as measured by the reference system) was above 20 degrees (as in [7]).

A significance level (p-value) of 5% was used to evaluate results.

To test for orientation invariance, we simulated multiple rotations of the IMU on the shoes. For that purpose, we sampled uniform random rotations (quaternions), as suggested by Shoemake, K. [33], and used those quaternions to synthetically rotate raw inertial sensor data. To evaluate the performance of the system when IMUs were placed at random rotations, we compare gait parameters extracted from the original sensor orientation with those extracted from a rotated version of the sensor. To quantify differences, we calculated the Root Mean Square Deviation (RMSD), correlation (using Pearson-parametric-or Spearman-non-parametric) and equivalence tests using a stricter equivalence zone of ±1% of the average of the metric. Equivalence tests were based on Paired T-test (T)—for parametric—or Wilcoxon signed-rank test (W)—for non-parametric. To choose an appropriate test, samples were first tested for normal distribution using Shapiro–Wilk; non-parametric tests were chosen in case of non-normal distribution. A significance level (p-value) of 5% was used to evaluate results.

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