metric self-similarity, o c curs. It is able to unveil hid-
den time-harmonic and self-similar structures along
the new direc tion toward a fractal human walking de-
composition. The a bove index
2
, has also innovatively
involved a term relying on a new experimental con-
jecture that relies on the position of the foot relative
to the tibia while op ening new a nalysis and diagno-
sis perspectives on the internal a nalysis of the do u-
ble support phase. The resulting theoretical approach
thus moves along the direction of using temporal gait
analyses to complement, in c linical or general per-
formance evaluations the classical gait analyses in-
cluding motion analysis, dynam ic electromyography,
force plate recordings, e nergy cost measurements or
energetics, measurement of stride characteristics.
Now, even though the Φ-bonacci gait number is
able to comprehensively capture the m ost reliable and
objective (quantitive) outco me measures of recursiv-
ity, asymmetry, consistency, and self-similarity (har-
monicity) of the gait cycle, however, up to this stage,
only two different simplified versions of such an in-
dex have been tested: i) in (El Ar a yshi et al., 2022),
concerning the distinction between patients affected
by Ataxia Telangiectasia and th eir healthy counter-
parts, no internal analysis of the double support phase
has be en performed; in (Verrelli et al., 2021), con-
cerning patients with highly a symmetric deficits (such
as patients with hemiparetic stroke and patients with
an alteration in gait ratio not always being acco m-
panied by motor asymmetries [such as patients with
quite symmetric symptoms due to Parkinson’s Dis-
ease]), data concerning the adjoint gaits are neglected.
In this paper, we thus illustrate, for the first time
in the literature, not only the effectiveness of th e com-
plete version of the aforementioned index in discrim-
inating healthy subjects from pathological ones, but
also its responsiveness in quantifying patients’ im-
provements coming from rehabilitation. To this aim,
we have recruited a cohort of patients with BPPV,
i.e., a peripheral vestibular disorder leading to b al-
ance difficulties and increased fall risks (Zhang et al.,
2021). Such BPPV patients suffer from transient ver-
tigo and nystagmus, leading to balance impairm e nts
and incre a sed fall risk , so their treatment typically in-
volves a canalith reposition maneuver, practiced by
expert physicians, and requires, at least, two weeks
to have an appreciable effect. These featu res promote
them as g ood candida te s to be tested, before and after
the repositioning maneuver, in order to show that the
Φ-bonacci gait number, in its complete version, repre-
sents a mea ningful index, capable of explicitly quan-
2
Even though it can be naturally extended to even as-
sess gait index variability along past walking gaits, this is
how ever out of the focus of this paper.
tifying and detecting the recovery level and improve-
ments due to rehabilitation. Experimen ta l results con-
firm such a conjecture.
2 MATERIALS AND METHODS
This section r ecalls the concept of com posite gait cy-
cle and the notions of recursivity, harmonicity, sym-
metry, and double support consistency as d efined in
(Verrelli et al., 2021 ). It also reports the mathemat-
ical expression of the Φ-bonacci ga it numbe r, in its
complete and simplified versions. Methods are then
described, along with the experimental setup and the
data acquisition modality. Finally, the main features
of the participants are introduced and the results com-
ing from the statistical analysis are reported.
2.1 Φ-Bonacci Sequence-Based Indices
Consider a walking gait (Iosa et al., 2013; Verrelli et
al., 2021) and let: GC stand for gait cycle; HS stand
for heel-strike; T O stand for toe-off; r and l stand for
right and left, respectively; ad j stand for adjoint; ST
stand for stance; SW stand for swing; DS stand for
double support. In Figure 1, a comprehensive mode l
of the composite gait cycle in (Verrelli et al., 2021),
which involves two specific couples of overlap ping
gait cycles, namely the left and right gait cycles (GCl
and GCr) and the adjoint right a nd left gait cycles
(GCr
ad j
and GCl
ad j
), is shown. For the sake of clar-
ity, STr, STl, SWr, and SWl represent the right and
left stance phase durations and the right and left swing
phase d urations, respec tively. Moreover, the dura-
tions DSr and DSl of the r ight and left double support
phase satisfy DSr = DSx + DSy, DSl = DSy + DSz,
with DSx, DSy, DSz being graphically defin e d in Fig-
ure 1. Accordingly, the equal partition of the double
support sub-phases, i.e., DSx = DSy (and DSw = DSy,
DSy = DSz in Figure 1) involves the concept of dou-
ble support consistency. The same duration s for the
adjoint right and left gait cycles, denoted by STradj,
STladj, SWradj, SWladj, DSradj, and DSladj, are re-
ported in Figure 1. Now, Verrelli et al. (2021) have
innovatively c haracterized the aforementione d com-
posite gait cycle by means of a new mathematical and
meaningful index, namely the Φ-bonacci gait number,
which relies, in its self-similar kernel, on generalized
finite-length Fibona cci sequences, exploiting the ro le
of the golden ratio φ. Spe cifically, the complete ver-
sion of such an ind ex, here called Y
φ
and r eported in
(10) of (Verrelli et al., 2021), has relied on a new ex-
perimental conjecture concerning an extended fractal
walking decomposition paying attention to the posi-