**4. Discussion**

The present method for quantifying regularity in pseudo-periodic human movements was a direct evolution and a generalization of methods based on the autocorrelation analysis to study trunk oscillatory movements during locomotion [28,29]. The proposed method's novelty is in the application of the autocorrelation analysis to the module of acceleration and to the concurrent measurement of acceleration on four different anatomical points.

The autocorrelation analysis was applied to the module of acceleration, derived from measurements by a triaxial sensor, which was periodic to the extent that single components are periodic. This novel aspect was expected to imply a larger robustness than single-component methods since the latter require an accurate placement for sensor orientation and location (sensor axes have to be aligned with anatomical or functional axes, and this may be particularly prone to operator errors in experiments involving persons with abnormal anatomical features). On the contrary, the acceleration module is independent from sensor orientation, but relies only upon sensor location. As to the computational aspects, the autocorrelation was applied to a time-finite windowed signal after having standardized it (subtracted average and divided by standard deviation), implying a unitary value for RI indicating perfect regularity. The average removal (equivalent to the autocovariance approach explicitly adopted by Moe-Nilssen [29]) is relevant, since an offset does corrupt, proportionally to the offset relevance/amplitude, the resulting autocorrelation coefficient and, thus, the regularity index. Therefore, this approach increases the robustness of the method.

The proposed sensor positions were not limited to location on the sagittal symmetry plane of the trunk–pelvis as in previous studies, but other body locations, particularly on either upper and lower limbs, were considered. This was possible since a periodic movement implies periodic patterns of the acceleration measured on any moving body location. A multi-sensor arrangement as already considered by Rispens et al. [36]; however, they only compared a regularity assessment of different locations on the trunk–pelvis segment, omitting the limbs. Obviously, locations outside the body's sagittal symmetry plane cannot support the analysis of symmetry according to Reference [29]; however, the multi-point approach here proposed has the advantage of disclosing how different body parts contribute to the movement regularity, thus supporting understanding and potentially targeting treatment or training on specific body segments.

The proposed autocorrelation analysis produced an autocorrelation time profile (see Figure 1E) whose average on a one-minute-long track was considered here as a regularity index (RI). However, analysis on longer recordings produced longer regularity profiles, which could be further studied for the occurrence of trends, potentially unveiling effects of fatigue on regularity. Schutte already dealt with this aspect by simply analyzing two spot assessments before and after a prolonged fatiguing task [34]. Moreover, the multi-sensor approach may further support a fatigue analysis allowing the identification of which anatomical functional part acts as a fatigue trigger.

The experimental data were obtained from one-minute-long locomotion trials at constant velocity on a treadmill, whereas previous studies considered recordings of walking for one minute or less [17,28,29]. The subjects were randomly requested to perform the sequence from slower walking to faster running or the reverse-order sequence, and no bias resulted in regularity index.

The experimental dataset also allowed, for all three components of acceleration singularly considered, to compute stride regularity according to the previous method [29]. For the sake of simplicity, the highest and the lowest outcome indexes as computed from single components were identified and then compared with the regularity indexes computed on the acceleration module pattern. The results show that the average difference between the highest single-component-based index and the module-based index was very little for the considered dataset, about 1% in relative terms, though statistically significant. However, no substantial bias was observed, since about half of the trials displayed a higher regularity index for the module-based index. Nonetheless, the option for a single-component regularity analysis, which showed validity in healthy subjects and in the study of selected locomotor disturbances [17,21,29], requires that this component be identified before the experiment takes place, which is a decision that may suffer from errors particularly when motor disturbances affect the performing subject. Ultimately, the proposed module-based analysis on a specific anatomical location [38] does not require that the researcher accurately align the sensor axes along predefined directions, which is particularly difficult when skeletal deformities affect the performing subject. Obviously, the adoption of triaxial accelerometers does not exclude the possibility to apply both the module-based approach and the single-component one.

Interesting findings emerged also from the analysis of period duration, a useful secondary outcome of the autocorrelation method. Firstly, it was confirmed that the estimate was very robust across sensors and across methods in accordance with published studies [39], and results were comparable with published normative data [40]. Secondly, the method provided a tracking of the fundamental period value along with the monitoring time; such information is supportive of studies concerning the gait variability [10]. Moreover, since the sensors may be placed on most parts of the body and adopted to any pseudo-periodic movements, this method appears to be of general applicability, while gait-specific methods for gait event identification cannot be extended to other motor tasks.

The application of the method was intended to disclose if different body segments showed different regularity during locomotion, and to analyze if locomotion strategy and locomotion speed influenced movement regularity.

Since the experiments involved the synchronized measurement of acceleration from four sensors, we were able to compare the autocorrelation coefficients observed on different anatomical sites; the highest regularity was found for C7 and the ankle, while the lowest (but still very high in absolute terms) was found for the wrist. The interpretation of C7's higher regularity may be related to biomechanical aspects (segments with larger mass and inertia, such as the trunk, show more regular movements, in the same manner that flywheels are expected to), to motor-control specific features (C7 movements are more regular because one of the objectives of locomotion is to keep the head, and the sensory organs inside it, the most stable and regular [35]) and to the balancing role played by limbs (particularly upper limbs) in keeping the body stability [41]. The reason for the lower regularity of upper limbs may include the possibility of accomplishing other episodic motor tasks, asynchronous with walking [42,43].

The effect of gait speed was assessed by comparing trials at different speeds but with the same strategy; a speed effect was observed for walking trials only, thus confirming that increasing speed is associated with increased regularity [20]. However, we observed that a faster locomotion resulted in the regularity index being closer to its maximum possible unitary value; therefore, it is reasonable to hypothesize the emergence of a ceiling effect at faster gait speeds, which could explain the missing detection of a speed effect for the running trials.

Interestingly, when comparing walking and running trials, the trunk (i.e., C7 and the pelvis) and, even more so, the upper arms tended to be more regular in running, according to the speed effect already evidenced, while lower limbs (ankle) were less regular. Such a decrease in regularity in lower limbs may be due to a technical artefact; the feet contacts on the ground are characterized, during running, by large impulsive impact forces which may transmit irregular oscillation on the nearby musculoskeletal structures and, therefore, also on the accelerometer tighten to the ankle [44].

As to the motor strategy, no between-strategy difference in regularity was observed at one single matched velocity close to the transition speed (RIW3 = RIR3 in Figure 4C), thus confirming previous studies concerning the analysis of spatio-temporal parameters [23]. While walking and running performances showed values of regularity index very close each other and close to the unitary upper limit value, a perusal of the whole dataset showed how RI never got below 0.9 when running, while 10 out of 100 RI values when walking were below 0.9, ranging down to approximately 0.7. Interestingly, all those ten values referred to regularity assessed by the wrist sensor. A possible explanation is that the role of upper limbs during running is more strictly related to locomotion than during walking, since they must counterbalance larger inertial forces which occur in the body [45]. Conversely, it can be expected that, during walking, the upper limbs keep the possibility to also perform some locomotor-unrelated tasks, characterized by less regular features [43].
