**1. Introduction**

Human functional periodic movements include locomotion, which is usually described in relation to its fundamental period [1], and upper limb activities, particularly those related to working tasks, sports, or art performances [2]. However, it is also possible to observe non-functional periodic movements such as tremor [3] or some cyclic gestures related to dystonic syndromes [4]. The periodicity of human locomotion is normally automatic and not necessarily consciously planned, controlled, and performed since specific neural structures, the central pattern generators, are in charge of controlling locomotor movements [5]. Moreover, inherent passive biomechanical and inertial characteristics of the different body parts naturally support the occurrence of passive pendulum-like periodic movements [6].

While, strictly speaking, a periodic phenomenon/signal repeats itself every predefined period of time *T*, a human cyclic movement and, particularly, kinematic and dynamic variables related to it are only approximately periodic; small variations occur both in the time domain and in the physical domain of the phenomena (*y*), according to

$$y(t + T + \Delta T) = y(t) + \Delta y,\tag{1}$$

where *y* is a generic gait analysis variable (with the exception of point trajectories, but including their derivatives), *T* is the fundamental period, and *t* is the time variable. Smaller values of Δ*T* and Δ*y* indicate the movement (described by a variable y) being closer to a periodic phenomenon. Therefore, since Δ*T* and Δ*y* only tend to zero, it is more proper to refer to the "pseudo-periodicity" of human movements [7]. In the scientific literature, such a unique aspect was assessed by different quantitative methods and related numerical indexes, concerning variability [8–13], stability [13,14], and regularity [12,15–21].

The variability of a pseudo-periodic movement may be increased due to endogenous physiological factors, like dual-task interference [13], or by pathological factors, such as those associated with neurodegenerative diseases [10]. Therefore, the quantification of the alteration of movement periodicity, i.e., the regularity of variables related to the performance, may represent a pathological marker to be considered in the clinical decision-making process [22], but only if confounding factors are taken into consideration.

Referring to gait, many studies addressed the variability of gait-related temporal parameters both in healthy individuals, to explore the possible influence of age [15] and different gait strategies (i.e., walking and running) [23], and in specific pathologic conditions, e.g., in frail elderly people [24], in people with neurological diseases, either degenerative [10] or focal [25], and in persons suffering from orthopedic diseases [12]. Interestingly, gait analysis methodologies based on wearable sensors [26] can efficaciously support quantitative assessment of variability and regularity; acceleration and angular velocity of anatomical parts, which are directly measured by inertial units, are characterized by pseudo-periodic patterns according to the pseudo-periodic nature of the considered locomotor act [27].

The study of locomotion regularity is generally based on an established time-domain approach involving autocorrelation analysis; movement regularity is assumed as the degree of similarity between two consecutive patterns of variables or signals characterizing the same cyclic movement [28,29]. Particularly, the measured acceleration undergoes an autocorrelation analysis [17,28] or an autocovariance analysis [29], whose outcome functions are characterized by a peak in correspondence of the fundamental period of the signal itself; the greater the peak Y-value is, the more regularly the signal pattern repeats. This latter number, thus, represents a regularity index. This method was successfully applied to the assessment of various samples of people, including elderly [19,30] and persons with locomotor disturbances [17,21,31–33].

Many factors, particularly gait speed [9,20,27] and gait strategy, i.e., walking or running [23], but also cognitive loads [20] and shoe type [16], were identified as modulating gait regularity. Moreover, when tracking regularity during long-term monitoring, an effect of fatigue on regularity was shown [34]. Furthermore, in some specific cases such as a musical performance, the regularity can be voluntarily modulated [2].

As to the sensor position, few studies considered more than one sensor placed on the trunk or pelvis to study head stabilization [35] or to assess across-sensor agreement [36]. All the cited articles about regularity reported analysis on single components of acceleration, thus requiring the identification of the meaningful acceleration components and the related accurate sensor alignment with the considered anatomical plane. Moreover, only sensors located on the trunk or pelvis were considered, and were, thus, unable to report regularity for limb movements.

The present study's aims were as follows:

(A) To define a method to assess regularity by generalizing and further developing the already proposed method based on autocorrelation analysis [29]; innovative aspects include (1) the autocorrelation analysis being applied to the module of acceleration (i.e., norm of the acceleration vector) and not to one acceleration component, thus removing errors due to sensor misalignment; (2) a multi-sensor approach which enables comparatively studying the regularity of anatomical parts, particularly of upper and lower limbs;

(B) To perform experiments and analyses on healthy subjects to quantify the effect of factors "strategy" (walking vs. running), "speed" (four speeds considered) [37], and "sensor location" (synchronized sensors located on pelvis, the seventh cervical vertebra (C7), and lower and upper limbs) on the regularity of human pseudo-periodic locomotor movements.

This study outcome may support future studies on human locomotion regularity by providing a robust methodology and reference data. Moreover, the application of the method to the study of non-functional periodic movements, such as tremors, is straightforward.
