Instrumentation
A purposely built instrumented mat was used to measure the beginning (tb) and end (te) of a stride determined by the consecutive contacts of the same foot with the mat, and relevant stride duration (T) was then computed as T = te – tb. Adhesive 5 mm wide copper stripes were attached parallel to each other at a 3 mm distance along a 4 m length linoleum mat. Alternative stripes were connected to an electric circuit so that, when short circuited, a signal was generated. Two independent circuits were constructed for right and left foot. Subjects wore custom designed socks, the bottom part of which was covered with conductive material. The accuracy of the mat was assessed by comparing its data to those simultaneously acquired with a strain gauge force plate (Bertec Corporation, Ohio, USA, sample frequency = 120 samples/s) while a subject stepped on it. The first sample at which the vertical force was greater than its mean value plus two standard deviations recorded for 1 s while the force plate was unloaded, was chosen as indicator of the foot contact. The differences found between the time events detected with the mat and with the force plate were computed for ten different trials, and were always lower than 0.025 s.
A nine-camera VICON® system (Oxford Metrics, Oxford, UK) was used to reconstruct the 3D positions, relative to the stereophotogrammetric set of axes, of 19 markers placed on the body of the subjects. The markers were placed on the head (three markers attached on an elastic band), trunk (spinal process of the seventh cervical vertebra, acromion processes), pelvis (anterior superior iliac spines, midpoint between the posterior superior iliac spines), and lower limbs (greater trochanters, femoral lateral epicondyles, lateral malleoli, calcanei, and second metatarsal heads). From now on, the cluster composed by all the above listed markers will be referred to as whole body (WB) cluster. Two sub-clusters will also be considered: the lower body (LB) cluster, including the 13 markers located on pelvis and lower limbs, and the upper body (UB) cluster, including the 9 markers located on head, trunk and pelvis. While defining the latter cluster, it was decided not to include upper limb markers because of the low sensitivity of the overall gait pattern to the movement of the upper limbs, which, for this reason, may tend to be more aperiodic than that of the rest of the body. In addition, most gait analysis protocols do not include these segments.
Stereophotogrammetric and mat data were simultaneously collected at a sampling frequency of 120 samples/s.
Data analysis
Through a rigid transformation, 3-D marker position data were represented relative to a laboratory set of axes, the X axis of which was aligned with the analysed subject mean speed of progression, and the Y axis was vertical. This data was filtered through a low-pass fourth-order Butterworth filter with a cut off frequency of 8 Hz [
11] and was used to describe the variations of the mechanical state of the subjects' whole body, and of its upper and lower parts.
Each cluster was considered as an ensemble of particles with equal mass and was represented, in each sampled instant of time during movement and relative to the laboratory frame, by the global position vector (
gp) of its centre of mass and by the orientation matrix (
gR) of an arbitrarily chosen set of local axes. To this purpose, the singular value decomposition technique was used [
12]. The position vectors of the markers in the local frame is referred to as
lp. Using this information, energy-like quantities were calculated and used to describe the instantaneous "mechanical state" variation of each cluster and, in turn, of each related body system. Such variations were calculated relative to the reference instant of time t
b.
The vertical coordinate h(t) of the marker cluster centre of mass was considered to represent a gravitational potential energy-like quantity G(t). Its variation was calculated as:
ΔG(t) = h(t) - h(tb). (1)
The first derivative of the centre of mass position vector was estimated via a three-point central difference differentiation method. The modulus of the instantaneous velocity thus obtained (v(t)) was used to calculate a linear kinetic energy-like quantity K(t). Its variation was given by:
ΔK(t) = v2 (t) - v2 (tb). (2)
The instantaneous angular velocity (
ω(t)) of the cluster was computed from the orientation matrix
gR [
13]. The modulus of
ω(t) was used to calculate a rotational kinetic energy-like quantity R(t). Its variation was given by:
ΔR(t) = ω2 (t) - ω2 (tb). (3)
Besides height and velocities variations, during movement the clusters may undergo a variation in orientation and a deformation, both of which were described by elastic potential energy-like quantities. The orientation variation of a cluster between time t
b and time t may be thought to correspond to a rotation of the local set of axes about the corresponding finite helical axis against an elastic torsional constraint. From the orientation matrices of the cluster at times t
b and t, the relevant rotation vector (
θ(t)) was calculated [
14]. The following torsional elastic potential energy-like quantity was, thus, determined:
ΔT(t) = ||θ(t)||2. (4)
Similarly, the variation of the markers local position vectors between time tb and time t allowed for the calculation of another elastic potential energy-like quantity associated with marker cluster deformation:
where N is the number of markers of the relevant cluster.
A measure of the system mechanical state variation, in any observed interval of time, could be obtained through the sum of the absolute values of the above-defined energy-like quantities. However, since such quantities have arbitrary dimensions, their values are incomparable and should hence be normalised. The maximum amplitude of one (arbitrarily chosen) of the energy-like quantities could be considered as a reference normalisation factor for the others. In such way, the variation of the mechanical state of the system can be calculated according to the following weighed sum:
where kG, kK, kR, kT, and kD are weighing constants. These constants, for the i-th trial, are arbitrarily calculated considering (for example) the maximum amplitude of gravitational potential energy-like quantity as the reference normalisation factor (i.e. setting kG = 1):
However, if different trials are to be compared, a fixed reference value of the constants should be chosen for all of them. Since no reference values were available to this purpose, previously (unpublished) available kinematic gait data, recorded at natural speed from a similarly aged group of 15 healthy subjects adopting the same instrumentation and marker set as the ones in the present study, were used. The values of the constants were computed as in (7a-7e) for each trial, and their mean values (kG = 1.00, kK = 0.04, kR = 0.27, kT = 0.05, kD = 0.70) were used in the rest of the study.
The variation of the state of the system during a gait cycle, starting from a foot contact (tb) and normalised with respect to its maximum value, was assessed by means of the index:
The sought aperiodicity index was computed as:
J
min
= min (J(t)). (9)
The larger the
J
min
value, the further the analysed gait is from a periodic process. The time instant for which J(t) =
J
min
is proposed as an estimate of the end of the period (
e), which can be used to determine the pseudo-period
=
e - t
b. The value assumed by J(t) at t
e, i.e. that measured by the mat at the end of the stride, will be referred to as
Je.
To assess the sensitivity of the index J
min
to the values of the constants, a set of 100 different combinations of values was generated by randomly varying them in the ranges defined by their corresponding mean values plus or minus one standard deviation, computed over the above described 75 trials (kG = 1.00, kK = 0.04 ± 0.03, kR = 0.27 ± 0.15, kT = 0.05 ± 0.03, kD = 0.70 ± 0.19).
As previously mentioned, gait aperiodicity depends on step length, cadence, and width, which can differently affect cluster kinematics: step length variation can be expected to mostly affect h(t), and partly v(t) and lp(t); step cadence variation mostly affects v(t); step width variation mostly affects h(t) and lp(t). Thus, it can be hypothesised that within the same stride, the quantities ΔG(t) and ΔD(t) are the most sensitive to changes in step length and width, and the index ΔK(t) to changes in step length and cadence. Moreover, the two terms ΔR(t) and ΔT(t) are expected to be negligible when walking straight. In such case, the equations (6), (8) and (9) can be replaced by the following:
(1)
min
= min (
(t)). (9a)
The above described hypotheses were verified by means of ad-hoc constrained tests. One subject (male, 23 years, 1.70 m, 70 kg) was asked to follow the auditory input of a metronome to modulate step cadence (C), and the visual input of markers on the floor to control step length (L) and width (W) while walking along the 8-m pathway. L, C, and W were first kept unconstrained, and then made to vary, one at a time, from step to step (ΔL = 0.4 m, ΔC = 1 step/s, ΔW = 0.2 m).