1. POLITECNICO DI MILANO
Facoltà di Ingegneria Industriale e dell’Informazione
Corso di Laurea Magistrale in
Ingegneria Biomedica
MUSCLE COORDINATION DURING RECTILINEAR
AND CURVILINEAR LOCOMOTION: THE ROLE OF
MUSCLE SYNERGIES
Relatore: Prof.ssa Simona FERRANTE
Correlatore: Noelia CHÍA BEJARANO
Tesi di Laurea di:
Walter BACCINELLI
Matricola 798742
Anno Accademico 2014 – 2015

3. I
ABSTRACT
INTRODUCTION
The human body is a highly complex system, and its large number of degrees of freedom
allows a task to be accomplished by the combination of different activation patterns. The
way the central nervous system (CNS) deals with such a complexity is still under debate,
and many theories have been proposed to explain this mechanism [1]. A recent theory has
suggested that the CNS could manage the problem of the high number of degrees of
freedom in a modular manner, controlling groups of muscles instead of activating them
singularly [2]. This way, the flexible combinations of few “muscle synergies”, that can be
defined as a pattern of relative muscle activations, would be used to produce a wide range
of motor behaviours, and the dimensionality of the system would be greatly reduced and
simplified.
The muscle synergy concept has been formalized in a mathematical model (equation 1) [3].
Assuming that the activation level function u(t) of all the m muscles during a motor task is
the variable to be controlled by the system, it is possible to decompose it as a linear
combination of k vectors w ∈ ℝm, with one-dimensional time-varying coefficients a(t) and
a certain level of reconstruction error, ɛ(t).
𝒖(𝑡) = ∑ 𝑎𝑖(𝑡)𝒘𝑖 + 𝜀 (𝑡)𝑘
𝑖=𝑖 (1)
In this paradigm, each vector of weights w specifies the relative level of activation of the
muscles in the i-th module, whereas the activation profiles a(t) determine the temporal
evolution of the synergy thorough the motor task. The most common method used to
evaluate the muscle synergies consists in searching for regularities in a dataset of muscle
activations acquired with superficial EMG. This search is performed by applying the Non-
negative Matrix Factorization (NMF) to the EMG envelopes, allowing the decomposition
of the data into a matrix of muscles weights and a matrix of activation profiles that
represent the muscle synergies.
This type of analysis can be applied to numerous fields with cyclic biomechanics, such as
human locomotion. Many authors have investigated the synergistic control performed by

4. II
the CNS during such a task, and four to five synergies, each corresponding to a specific
biomechanical subtask, have been found to be sufficient to explain the muscle activity
during walking [4]–[8].
Goal of the work
It has been demonstrated that locomotion is particularly unstable in the medio-lateral
plane, and thus a thorough understanding on how the CNS controls the lower-limb and
trunk muscles in order to assure accurate propulsion and balance during curvilinear waking
is required. This particular walking condition becomes very important to be studied
because it seems very demanding both in elders and neurological subjects having an
impaired motor control. Thus, the present work aims at the study muscle coordination
during walking along rectilinear and curvilinear trajectories, and the investigation the
differences between these tasks. To that end, muscle synergies have been extracted from
surface EMG signals and the different modules have been evaluated and compared across
walking conditions. Additionally, this novel analysis approach has been complemented
with a more traditional study of the activation of the single muscles to give a more
comprehensive interpretation to the gained results.
This study was conducted in collaboration with the clinic FSM Veruno, in the framework
of the PRIN project A quantitative multifactorial approach to the assessment and
prevention of falls in the elderly (grant no.: 2010R277FT).
METHODS
Thirteen healthy, non-impaired subjects were asked to walk at their self-selected speed
along a rectilinear trajectory and a circular trajectory, moving clockwise and counter-
clockwise. Thus, three conditions of walking were considered: walking along a rectilinear
trajectory (RT), walking on a curvilinear trajectory with the dominant leg moving on the
inner part of the trajectory (IT), and walking on a curvilinear trajectory with the dominant
leg moving on the external part of the trajectory (ET). During the trials, the surface EMG
signals from the following 15 muscles of the dominant leg and trunk were recorded:
erector spinae (ES), abdominal external oblique (AO), gluteus maximus (GM), tensor
fasciae latae (TFL), rectus femoris (RF), vastus medialis (VM), vastus lateralis (VL),

5. III
adductor magnus (AM), medial hamstring (HM), lateral hamstrings (HL), medial
gastrocnemius (MG), lateral gastrocnemius (LG), soleus (SO), peroneus longus (PE), and
tibialis anterior (TA).
The kinematic data were acquired by means of inertial and magnetic sensors placed on
both the shanks, allowing the detection of the moments where an initial contact (IC), end
contact (EC) and a mid-stance moment (MS) occurred for each leg. The ICs were
considered as the first event of the gait cycle, and they were used to divide the data into
strides, defined as the interval between two consecutive ICs of the same leg. The detected
events allowed the calculation of the cadence values, the subdivision of the strides into the
three gait phases of double support, initial and terminal swing for each leg, and the
computation of their duration.
The EMG signals were band-pass filtered at 20-400 Hz with a 3rd
order Butterworth filter.
Afterwards, the envelope of the recordings was obtained by rectifying and low-pass
filtering the previous signals at 5 Hz with a 3rd
-order Butterworth filter. The envelopes
were, then, segmented into single strides on the basis of the IC events, and each stride was
normalized in time by interpolating the signal into 100 points, and normalized in amplitude
with the median of the peak values calculated across strides for each walking condition
independently. Twenty strides were extracted for each subject and walking condition, and
this was the dataset considered for the further analysis.
The quality of each muscle was evaluated by checking the inter-individual variability
trough the calculation of the variance ratio. On the other hand, the number of synergies
was evaluated independently for each subject and walking condition, by conducting
separate NMF analysis with the number of synergies ranging from 1 to 15 and computing
each time the Variance Accounted For (VAF). The minimum number of synergies required
to correctly reconstruct the EMG signal was chosen as the lowest number that guaranteed
that each muscle was reconstructed with a VAF higher than 90%, or the minimum number
whose VAF did not increased more than 5% when adding an extra synergy [4].
After the number of synergies was set, the same number was extracted from the EMG
signals of all subjects. The synergies obtained were confronted across subjects by means of
similarity (calculated as the normalized scalar product of the weight vectors of two

6. IV
synergies), matching the pairs with maximum similarity. The mean values of weights and
activation profiles were calculated for each group of synergies, and a set of mean synergies
was obtained for each condition.
The muscle synergies were compared across conditions, after coupling them using the
same similarity criterion of intra-subject matching, obtaining several quantitative
parameters that characterized the comparison between each pair of conditions. The muscles
weights were compared with the similarity, whereas the activation profiles were analysed
by applying the circular cross-correlation between the profiles, and computing the
maximum correlation coefficient (rmax) and the time-lag (shift needed to get rmax).
A further comparison of the synergies across conditions was performed with a cross-
validation analysis that consisted in obtaining the reconstructed EMG signals from the
mean synergies of each condition, and comparing them to the original signals of the other
conditions, by means of the VAF.
Moreover, activation profiles where extracted from each condition of each subject by
applying NMF to the EMG signals, holding fixed the weights matrix W. In particular, two
separate analysis were conducted: in the first analysis the weights matrix fixed was the W
matrix obtained from RT condition of each subject independently; in the second analysis
the fixed matrix was the same for all subjects, and was equal to the mean of the W
matrixes extracted from RT condition of all the subjects.
A single-muscle analysis was also performed on the activation profiles. In particular, the
bursts of the muscle activations were characterized by their onset and offset, which were
extracted and compared across conditions. Additionally, the amplitude of the activity,
calculated as the area under the bursts, was normalized with its corresponding area in the
RT condition and was compared across the conditions.
A preliminary test was conducted on a pathological subject, who performed the same trials
than the healthy subjects. A subset of 8 EMG signals were recorded from GM, RF, VM,
MH, LH, MG, SO, TA muscle of each leg, and the kinematic data were recorded from
inertial and magnetic sensors mounted on both shanks. The data processing and analysis
was performed with the same methodology used for the data from healthy subjects.

7. V
RESULTS
The cadence resulted significantly decreased during curvilinear walking with respect to the
rectilinear walking, with a value lower in ET than in IT. The stance phase duration resulted
significantly longer in IT than in RT, and the duration of the double support phase
presented higher values during curvilinear walking with respect to the rectilinear walking.
From the analysis of VR it emerged that the AO and AM muscles had too much variability,
and were thus excluded from the further analysis.
Five muscle synergies were extracted from all the subjects and conditions, since this was
the minimum required for the correct reconstruction of the EMG signals of all the walking
conditions. The mean synergies obtained for each condition are reported in figure 1.
Figure 1 - Mean synergies for all subjects and walking conditions. The bars represent the weights of each
muscle within each synergy. The continuous lines represent the activation profiles over a gait cycle for each
synergy and condition

8. VI
Synergies from 1 to 4 appeared to be similar across the conditions and were compatible
with the results reported in literature in terms of muscle weights, activation profiles and
biomechanical functions. Synergy 5 did not represent a specific biomechanical function.
These mean muscle synergies were compared between each pair of condition (see table 1),
showing a high similarity both in terms of muscle weights and activation profiles.
The cross-validation analysis proved that the synergies of each condition could properly
reconstruct the EMG of all the conditions, since the VAF values were high for all the
reconstructions, obtaining a minimum of 74% [9].
RT vs IT RT vs ET ET vs IT
Synergy similarity rmax time-lag similarity rmax time-lag similarity rmax time-lag
1 0.99 0.99 1 0.97 0.98 -1 0.98 0.99 -2
2 0.98 0.98 2 1.00 0.99 1 0.98 1.00 -1
3 0.97 0.94 -1 0.96 0.97 1 0.92 0.91 50
4 0.91 0.89 -1 0.98 0.98 1 0.89 0.95 2
5 0.89 0.91 2 0.89 0.93 1 0.85 0.95 1
Table 1 - Similarity, rmax and time-lag calculated for each couple of walking conditions. The computation of
time-lag and rmax was performed with the second term of the comparison shifting in time, thus a positive
value of time-lag means the second term is delayed in time with respect to the first. The time-lag is reported
as % of gait cycle.
Figure 2 - Onset and offset for each muscle and walking condition. The horizontal bars represent the period
of the gait cycle during which the muscle presented a peak. Two peaks were identified for ES and RF,
represented as ES 1/2 and RF 1/2. The horizontal bars represent the first and third quartile values for the
onset and offset of each muscle. The pairs of values that resulted statistically different are highlighted with
red asterisk.

9. VII
The single-muscle analysis revealed that changes in the onset and offset were present
among conditions in some muscles (see figure 2).
In particular, a delayed offset was reported during IT for HM, and also for muscles with an
action on the hip joint (GM, TFL, RF) during ET. On the other hand, an anticipated
activation was present for muscles that act on the knee and ankle joints (MG, LG, SO, PE)
during the curvilinear trajectories. Moreover, a statistically-significant effect of the
walking condition on the amplitude of the muscle activity for the GM, both the bursts of
RF, HM, HL, MG, LG and SO muscles was reported.
PRELIMINARY RESULTS ON A PATHOLOGICAL SUBJECT
The enrolled patient had suffered an ischemic stroke in 2004 and was characterized by
relevant weakness of the knee flexion, plantar dorsiflexion and foot eversion on the paretic
leg. The cadence values for the curvilinear trajectory were lower than those of the
rectilinear trajectory, and also lower than those of healthy subjects. The computation of the
stance ratio between the non-paretic and the paretic limb revealed a longer stance phase
duration of the non-paretic limb during curvilinear trajectory.
The analysis of synergy dimensionality revealed that 3 synergies were required for the
paretic limb, whereas 4 synergies were required for the non-paretic limb. The synergies
extracted are reported in figure 3 and figure 4.
Figure 3 - Synergies of the paretic limb of the pathological subject for all the conditions.

10. VIII
Synergy 2 and synergy 3 of the paretic limb were similar to the synergies 2 and 3 of the
healthy subjects, whereas synergy 1 could be decoded as the merging of the synergies 1
and 4 of healthy subjects.
The comparison of the synergies between conditions (table 2) revealed that the weights
were similar across conditions (similarity always greater than 0.75 [9]), whereas the
activation profiles exhibited relevant time shifts.
The synergies of the non-paretic limb did not resulted consistent across the conditions. The
changes across conditions are particularly evident in synergy 1.
Figure 4 - Synergies of the non-paretic limb of the pathological subject for all the conditions.
Table 2 - Similarity, rmax and time-lag calculated for each couple of conditions for the paretic leg. The
computation of time lag and rmax was performed with the second term of the comparison shifting in time, thus
a positive value of time-lag means the second term is delayed in time. The time-lag is reported as % of gait
cycle.
PARETIC LIMB
RT vs IT RT vs ET ET vs IT
Synergy similarity rmax time-lag similarity rmax time-lag similarity rmax time-lag
1 0.96 0.97 -45 0.99 0.97 -27 0.97 0.99 -9
2 0.90 0.90 -49 0.96 0.99 21 0.97 0.94 40
3 0.97 0.92 42 0.99 0.97 3 0.98 0.97 39

11. IX
The comparison of the synergies between conditions (table 3) revealed that the weights
were not similar for synergy 1, whereas the activation profiles resulted to be consistent
across conditions.
DISCUSSION
The results proved that the adaptations in terms of cadence and stance duration were
implemented to maintain the balance during the curvilinear walking.
The analysis of the synergies revealed that in healthy subjects 5 synergies explained
correctly the muscle activity for all the walking conditions. Such synergies were invariant
with the walking trajectory, and each of them was responsible of a specific biomechanical
subtask. Specifically, synergy 1 performed the load acceptance and provided body support
during the stance phase. Synergy 2 provided body support during the terminal stance
phase, initiated the swing phase and was responsible of the propulsion of the leg. Synergy
3 performed the leg flexion and ground clearance during the swing phase. Synergy 4 was
responsible of leg deceleration and preparation to the approach of the leg to the ground
during terminal swing phase. The same biomechanical functions were reported in previous
literature [2, 4, 10].
The invariance of the synergies led to hypothesize, in accordance with the neural
interpretation of the synergies reported in literature, that the same coordination strategy
implemented by the CNS is adopted to perform rectilinear and curvilinear walking.
From the single-muscle analysis it emerged that changes in the timing and amplitude of the
activity were implemented for some muscles in order to maintain balance during the
curvilinear trajectory. Thus, whereas at the CNS level the motor coordination strategy is
NON-PARETIC LIMB
RT vs IT RT vs ET ET vs IT
Synergy Similarity rmax time-lag similarity rmax time-lag similarity rmax time-lag
1 0.67 0.89 1 0.70 0.94 -2 0.87 0.97 3
2 0.99 0.97 -6 0.98 0.96 -2 0.96 0.97 -1
3 0.95 0.93 -10 0.74 0.86 -12 0.88 0.89 -1
4 0.88 0.79 -1 0.75 0.80 -27 0.95 0.95 -2
Table 3 - Similarity, rmax and time-lag calculated for each couple of conditions for the non-paretic leg. The
computation of time lag and rmax was performed with the second term of the comparison shifting in time,
thus a positive value of time-lag means the second term is delayed in time. The time-lag is reported as % of
gait cycle.

12. X
invariant, modulations of single muscles’ activity are necessary to implement different
walking trajectories.
The analysis in the pathological subject revealed a diminished complexity of locomotor
output in the paretic limb during rectilinear walking, in accordance with a previous work
reported in literature [4]. The same reduced number of synergies was also obtained in the
other two walking conditions, but with a different structure of the synergies, indicating a
pathological motor control. It was also reported a change of the synergies of the non-
paretic limb with the walking condition, maybe caused by the adaptation of the
coordination mechanism to compensate for the loss of strength of the paretic limb.
Thus, in the present study, evidences of the existence of a muscle coordination strategy
invariant between rectilinear and curvilinear locomotion has been reported, whereas
changes at single-muscle level were found to be adopted to implement locomotion on
curvilinear trajectories. Moreover, a novel method has been used to assess the quality of
motor coordination of pathological subjects. Such a method could help the understanding
of the disruption of the central mechanisms after stroke and give useful information for the
choice of the most effective rehabilitation solutions for recovery of physiological
locomotion.

13. XI
SOMMARIO
INTRODUZIONE
Il corpo umano è un sistema altamente complesso, e il suo elevato numero di gradi di
libertà permette di compiere un’attività attraverso la combinazione di diversi schemi di
attivazione. Il modo in cui il sistema nervoso centrale (CNS) gestisce una tale complessità
rimane ancora oggetto di dibattito, e sono state proposte diverse teorie per spiegare tale
meccanismo [1]. Una teoria recente ha suggerito che il CNS possa gestire il problema
dell’elevato numero di gradi di libertà in maniera modulare, controllando i muscoli in
gruppi anziché singolarmente [2]. In questo modo la combinazione di poche “sinergie
muscolari”, che possono essere definite come un pattern di attivazioni muscolari relative,
potrebbe essere usata per produrre un ampio spettro di comportamenti motori, e la
dimensionalità del sistema potrebbe essere molto ridotta e semplificata.
Il concetto di sinergia muscolare è stato formalizzato in un modello matematico (equazione
1) [3]. Considerando la funzione u(t) del livello di attivazione di tutti gli m muscoli durante
un’attività motoria la variabile controllata dal sistema, è possibile scomporre tale variabile
in una combinazione lineare di k vettori w ∈ ℝm con coefficienti monodimensionali
tempo-varianti a(t) e un certo errore di ricostruzione ɛ(t).
𝒖(𝑡) = ∑ 𝑎𝑖(𝑡)𝒘𝑖 + 𝜀 (𝑡)𝑘
𝑖=𝑖 (1)
In questo paradigma, ogni vettore w specifica il livello di attivazione nell’i-esimo modulo,
mentre i profili di attivazione a(t) determinano l’evoluzione temporale della sinergia
durante il compito motorio. Il metodo più comunemente usato per valutare le sinergie
muscolari consiste nella ricerca di regolarità in un dataset di attivazioni muscolari acquisito
tramite EMG di superficie. Tale ricerca è svolta applicando agli inviluppi dell’EMG la
Non-negative Matrix Factorization (NMF), che consente di scomporre i dati in una
matrice di pesi muscolari e una matrice di profili di attivazione che rappresentano le
sinergie muscolari.
Uno dei campi la cui biomeccanica ciclica può trarre vantaggio dallo studio delle sinergie
muscolari è la locomozione umana. Molti autori hanno studiato il controllo sinergistico

14. XII
compiuto dal CNS durante tale attività, ed è stato trovato che quattro o cinque sinergie,
ognuna corrispondente a uno specifica funzione biomeccanica, sono sufficienti a spiegare
l’attività muscolare durante il cammino [4]-[8].
Obiettivo del lavoro
È stato dimostrato che la locomozione è particolarmente instabile sul piano medio-laterale
ed è quindi richiesta una comprensione scrupolosa di come il CNS controlli i muscoli
dell’arto inferiore e del tronco per assicurare una propulsione e un equilibrio adeguati
durante il cammino curvilineo. Lo studio di questa particolare condizione di cammino
diventa molto importante poiché sembra molto difficoltoso per soggetti anziani e
neurologici con un controllo motorio compromesso. Dunque, il presente lavoro ha come
obiettivo lo studio della coordinazione muscolare durante il cammino lungo una traiettoria
rettilinea e curvilinea, e l’indagine delle differenze tra i due compiti. A questo scopo, le
sinergie muscolari sono state estratte dall’EMG di superficie e i diversi moduli sono stati
valutati e comparati tra le diverse condizioni di cammino. Per di più, questo nuovo
approccio analitico è stato affiancato da uno studio complementare più tradizionale
dell’attivazione dei singoli muscoli per fornire un’interpretazione più comprensiva dei
risultati ottenuti.
Il presente studio è stato condotto in collaborazione con la clinica FSM Veruno,
nell’ambito del progetto PRIN A quantitative multifactorial approach to the assessment
and prevention of falls in the elderly (grant no.: 2010R277FT).
METODI
A tredici soggetti sani, con mobilità non ridotta, è stato chiesto di camminare a una
velocità da loro scelta lungo una traiettoria rettilinea e una traiettoria circolare,
percorrendola in senso orario e in senso antiorario. Tre condizioni di cammino sono state,
dunque, considerate: cammino lungo la traiettoria circolare (RT), cammino lungo la
traiettoria curvilinea con la gamba dominante a percorrere la parte interna della traiettoria
(IT), cammino lungo la traiettoria curvilinea con la gamba dominante a percorrere la parte
esterna della traiettoria (ET). Durante le prove il segnale EMG di superfice è stato
registrato dai seguenti 15 muscoli della gamba dominante e del tronco: erettore della

15. XIII
colonna vertebrale (ES), addominale obliquo esterno (AO), gluteo massimo (GM), tensore
della fascia lata (TFL), retto femorale (RF), vasto mediale (VM), vasto laterale (VL),
grande adduttore (AM), ischiocrurale mediale (HM), ischiocrurale laterale (HL),
gastrocnemio mediale (MG), gastrocnemio laterale (LG), soleo (SO), peroneo lungo (PE),
e tibiale anteriore (TA).
I dati sulla cinematica sono stati acquisiti tramite sensori inerziali e magnetici posizionati
su entrambi gli stinchi, per permettere il rilevamento dei momenti di avvenimento di un
contatto iniziale (IC), un contatto finale (EC) e un momento di metà fase di appoggio (MS)
per entrambe le gambe. Gli IC sono stati considerati come il primo evento del ciclo del
passo, e sono stati usati per dividere i dati in passi, definiti come gli intervalli tra due IC
consecutivi della stessa gamba. Gli eventi rilevati hanno permesso il calcolo dei valori di
cadenza, la suddivisione di ogni passo nelle tre fasi di doppio appoggio, fase di volo
iniziale e finale per entrambe le gambe, e il calcolo della loro durata.
Il segnale EMG è stato filtrato in passa-banda a 20-400 Hz con un filtro Butterworth del 3°
ordine. Dopodiché, l’inviluppo delle registrazioni è stato ottenuto rettificando e filtrando in
passa-basso a 5 Hz i segnali precedenti con un filtro Butterworth del 3° ordine. Gli
inviluppi, poi, sono stati divisi in singoli passi sulla base degli eventi di IC, e ogni passo è
stato normalizzato in tempo tramite l’interpolazione del segnale su 100 punti, e
normalizzato in ampiezza con la mediana dei valori di picco calcolati sui passi per ogni
condizione di cammino indipendentemente. Sono stati estratti venti passi per ogni soggetto
e per ogni condizione di cammino, e questo ha costituito il dataset considerato per le
analisi successive.
La qualità di ogni muscolo è stata valutata tramite il controllo della variabilità inter-
individuale attraverso il calcolo del variance ratio (VR). Dall’altro lato, il numero di
sinergie è stato valutato indipendentemente per ogni soggetto e condizione di cammino,
effettuando analisi NMF separate con il numero di sinergie che variava da 1 a 15 e
calcolando ogni volta la varianza spiegata (VAF). Il numero minimo di sinergie è stato
scelto come il numero più basso che garantisse, per ogni muscolo, una VAF maggiore del
90%, o il numero minimo per cui la VAF non crescesse di più del 5% aggiungendo una
nuova sinergia [4].

16. XIV
Dopo che il numero di sinergie è stato stabilito, lo stesso numero di sinergie è stato estratto
dall’EMG di tutti i soggetti. Le sinergie ottenute sono state confrontate tra i soggetti
attraverso la similarity (calcolata come il prodotto scalare normalizzato tra i vettori dei
peso di due sinergie), abbinando le coppie con massima similarity. I valori medi dei pesi e
dei profili di attivazione sono stati calcolati per ogni gruppo di sinergie, ed è stata ottenuta
una serie di sinergie medie per ogni condizione.
Le sinergie muscolari sono state confrontate tra le condizione, dopo averle accoppiate
usando lo stesso criterio basato sulla similarity per l’abbinamento intra-soggettivo,
ottenendo alcuni parametri quantitativi che hanno caratterizzato il confronto tra ogni
coppia di condizioni. I pesi muscolari sono stati confrontati tramite la similarity, mentre i
profili di attivazione sono stati analizzati applicando la cross-correlazione circolare tra
profili, e calcolando il coefficiente di correlazione massimo (rmax) e il time-lag (lo
spostamento necessario per ottenere rmax).
Un ulteriore confronto delle sinergie tra le condizioni è stato svolto con un’analisi di cross-
validazione, che è consistita nell’ottenimento dei segnali EMG ricostruiti dalle sinergie
medie di ogni condizione, e nel confronto di questi con i segnali originali delle altre
condizioni, tramite la VAF.
Inoltre, per ogni soggetto e per ogni condizione, sono stati estratti i profili d’attivazione
attraverso l’applicazione della NMF ai segnali EMG tenendo fissa la matrice W dei pesi. In
particolare sono state condotte due analisi separate: nella prima la matrice fissata era la
matrice W ottenuta dalla condizione RT, per ogni soggetto indipendentemente; nella
seconda analisi la matrice fissata era la stessa per tutti i soggetti, ed era pari alla media
delle matrici W estratte dalla condizione RT d tutti i soggetti.
Anche un’analisi singolo-muscolo è stata svolta sui profili di attivazione. In particolare, i
picchi delle attivazioni muscolari sono stati caratterizzati tramite i loro onset e offset, che
sono stati estratti e confrontati tra le condizioni. Per di più, l’ampiezza dell’attività,
calcolata come l’area sotto i picchi, è stata normalizzata con l’area corrispondente nella
condizione RT ed è stata confrontata tra le condizioni.
Un test preliminare è stato svolto su un soggetto patologico, che ha svolto le stesse prove
dei soggetti sani. Un sottogruppo di 8 segnali EMG è stato registrato dai muscoli GM, RF,

17. XV
VM, MH, LH, MG, SO, TA per entrambe le gambe, e I dati cinematici sono stati registrati
da sensori montati su entrambi gli stinchi. L’elaborazione e l’analisi dei dati sono state
svolte con la stessa metodologia applicate ai dati dei soggetti sani.
RISULTATI
La cadenza è risultata significativamente diminuita durante il cammino rettilineo rispetto al
cammino curvilineo, con un valore minore per ET ed IT. La durata della fase di appoggio è
risultata significativamente più lunga per IT che per RT, e la durata della fase di doppio
appoggio ha riportato valori maggiori durante il cammino curvilineo rispetto al cammino
rettilineo.
Dall’analisi del VR è emerso che i muscoli AO ed AM presentavano troppa variabilità, e
sono, dunque, stati esclusi dalle successive analisi.
Cinque sinergie muscolari sono state estratte per ogni soggetto e condizione, poiché questo
era il numero minimo richiesto per la corretta ricostruzione dei segnali EMG di tutte le
Figura 1 – Sinergie medie per tutti i soggetti e le condizioni di cammino. Le barre rappresentano i pesi di
ogni muscolo all’interno di ogni sinergia. Le linee continue rappresentano i profili di attivazione su un
ciclo del passo per ogni sinergia e condizione.

18. XVI
condizioni di cammino. Le sinergie medie ottenute per ogni condizione sono riportate in
figura 1.
Le sinergie da 1 a 4 sono apparse simili tra le condizioni ed erano in linea con i risultati
riportati in letteratura in termini di pesi muscolari, profili di attivazione e funzione
biomeccanica.
Queste sinergie medie sono state confrontate tra ogni coppia di condizioni (tabella 1),
mostrando un’elevata similarity in termini di pesi muscolari e profili di attivazione.
L’analisi di cross-validazione ha provato che le sinergie di ogni condizione potevano
ricostruire in maniera adatta l’EMG di tutte le condizioni, poiché i valori di VAF erano alti
per tutte le ricostruzioni, con un valore minimo ottenuto del 74% [9].
RT vs IT RT vs ET ET vs IT
Synergy similarity rmax time-lag similarity rmax time-lag similarity rmax time-lag
1 0.99 0.99 1 0.97 0.98 -1 0.98 0.99 -2
2 0.98 0.98 2 1.00 0.99 1 0.98 1.00 -1
3 0.97 0.94 -1 0.96 0.97 1 0.92 0.91 50
4 0.91 0.89 -1 0.98 0.98 1 0.89 0.95 2
5 0.89 0.91 2 0.89 0.93 1 0.85 0.95 1
Tabella 1 - Similarity, rmax e time-lag calcolati per ogni coppia di condizione di cammino,. Il confronto di
time-lag e rmax è stato svolto spostando il secondo termine di confronto, quinid un valore positive di
time_lag significa che il secondo termine è ritardato nel tempo rispetto al primo. Il time-lag è riportato
come % del ciclo del passo.
Figura 2 – Onset ed offset per ogni muscolo e condizione di cammino. Le barre orizzontali rappresentano i
periodi nel ciclo del passo durante i quali il muscolo presentava un picco. Per ES e RF sono stati identificati
2 picchi. ES 1 ed ES 2 rappresentano il primo e il secondo picco di ES rispettivamente. RF 1 e RF 2
rappresentano il primo e il secondo picco di RF rispettivamente. Le barre orizzontali rappresentano il primo
e terzo quartile per l’onset e l’offset di ogni muscolo. Le coppie statisticamente diverse sono contrassegnate
con un asterisco rosso.

19. XVII
L’analisi singolo-muscolo he mostrato che tra le condizioni erano presenti cambiamenti
negli onset e offset di qualche muscolo (figura 2).
In particolare, è stato trovato un ritardo nell’offset di HM durante IT, e anche per i muscolo
con azione sull’articolazione dell’anca (GM, TFL, RF) durante ET. Dall’altra parte, un
anticipo dell’attivazione era presente per i muscoli agenti sull’articolazione del ginocchio e
della caviglia (MG, LG, SO, PE) durante la traiettoria curvilinea. Inoltre, è stato trovato un
effetto statisticamente significativo della condizione di cammino sull’ampiezza dell’attività
muscolare per i muscoli GM, entrambi i picchi di RF, HM, HL, MG, LG e SO.
RISULTATI PRELIMINARI SU DI UN SOGGETTO PATOLOGICO
Il paziente reclutato ha subito un ictus ischemico nel 2004 ed era caratterizzato da una
rilevante debolezza della flessione del ginocchio, della dorsiflessione plantare e
dell’eversione del piede nella gamba paretica. I valori di cadenza per la traiettoria
curvilinea erano più bassi che quelli per la traiettoria rettilinea, e anche minori di quelli dei
soggetti sani. Il calcolo dello stance ratio tra l’arto non paretico e l’arto paretico ha
mostrato una fase d’appoggio più lunga nell’arto non paretico nel cammino curvilineo.
L’analisi della dimensionalità delle sinergie ha mostrato che 3 sinergie erano richiesto per
l’arto paretico, mentre 4 sinergie erano richieste per l’arto non paretico. Le sinergie estratte
sono riportane nella figura 3 e nella figura 4.
Figure 3 - Synergies of the paretic limb of pathological subject for all the conditions.

20. XVIII
La sinergia 2 e la sinergia 3 dell’arto paretico erano simili alle sinergie 2 e 3 dei soggetti
sani, mentre la sinergia 1 potrebbe essere decodificata come l’unione delle sinergie 1 e 4
dei soggetti sani.
Il confronto delle sinergie tra condizioni (tabella 2) ha mostrato che i pesi erano simili tra
le condizioni (similarity sempre maggiore di 0.75 [9]), mentre i profili di attivaziine
mostravano uno sfasamento temporale rilevante
Le sinergie dell’arto non paretico non sono risultate consistenti attraverso le condizioni. I
cambiamenti attraverso le condizioni sono particolarmente evidenti nella sinergia 1.
Figure 4 - Sinergie dell'arto non paretico del soggetto patologico per tutte le condizioni.
PARETIC LIMB
RT vs IT RT vs ET ET vs IT
Synergy similarity rmax time-lag similarity rmax time-lag similarity rmax time-lag
1 0.96 0.97 -45 0.99 0.97 -27 0.97 0.99 -9
2 0.90 0.90 -49 0.96 0.99 21 0.97 0.94 40
3 0.97 0.92 42 0.99 0.97 3 0.98 0.97 39
Tabella 2 - Similarity, rmax e time-lag calcolati per ogni coppia di condizione per la gamba paretica. Il
confronto di time-lag e rmax è stato svolto spostando il secondo termine di confronto, quindi un valore
positive di time-lag significa che il secondo termine è ritardato nel tempo rispetto al primo. Il time-lag è
riportato come % del ciclo del passo.

21. XIX
Il confronto delle sinergie tra condizioni (tabella 3) ha rivelato che i pesi non erano simili
per la sinergia 1, mentre i profili di attivazione sono risultati consistenti attraverso le
condizioni
DISCUSSIONE
I risultati hanno dimostrato che sono stati implementati degli adattamenti in termini di
cadenza e durata della fase d’appoggio per mantenere l’equilibrio durante il cammino
curvilineo.
L’analisi delle sinergie ha mostrato che nei soggetti sani 5 sinergie spiegavano
correttamente l’attività muscolare per tutte le condizioni di cammino. Tali sinergie erano
invarianti rispetto alla traiettoria del cammino, e ognuna di esse era responsabile di una
specifica funzione biomeccanica. Nello specifico, la sinergia 1 effettuava l’accettazione del
carico e forniva sostegno al corpo durante la fase d’appoggio. La sinergia 2 forniva
sostegno al corpo durante la parte terminale della fase d’appoggio ed era responsabile della
propulsione della gamba. La sinergia 3 effettuava la flessione della gamba e il
sollevamento del piede dal suolo durante la fase di volo. La sinergia 4 era responsabile
della decelerazione della gamba e della preparazione al contatto dell’arto con il suolo
durante la parte terminale della fase di volo. Le stesse funzioni biomeccaniche sono state
riportate nella letteratura precedente [2, 4, 10].
L’invarianza delle sinergie ha portato a ipotizzare, in accorto che l’interpretazione in
chiave neurale delle sinergie riportata in letteratura, che la stessa strategia di coordinazione
implementata dal CNS venga adottata per effettuare il cammino rettilineo e curvilineo.
Dall’analisi singolo-muscolo è emerso che venivano implementati cambiamenti in termini
temporali e d’ampiezza dell’attività per alcuni muscoli al fine di mantenere l’equilibrio
NON-PARETIC LIMB
RT vs IT RT vs ET ET vs IT
Synergy Similarity rmax time-lag similarity rmax time-lag similarity rmax time-lag
1 0.67 0.89 1 0.70 0.94 -2 0.87 0.97 3
2 0.99 0.97 -6 0.98 0.96 -2 0.96 0.97 -1
3 0.95 0.93 -10 0.74 0.86 -12 0.88 0.89 -1
4 0.88 0.79 -1 0.75 0.80 -27 0.95 0.95 -2
Tabella 3 - Similarity, rmax e time-lag calcolati per ogni coppia di condizione per la gamba non paretica. Il
confronto di time-lag e rmax è stato svolto spostando il secondo termine di confronto, quindi un valore
positive di time-lag significa che il secondo termine è ritardato nel tempo rispetto al primo. Il time-lag è
riportato come % del ciclo del passo.

22. XX
durante la traiettoria curvilinea. Dunque, mentre a livello del CNS la strategia motoria di
coordinamento è invariante, sono necessarie delle modulazione dell’attività dei singoli
muscoli per effettuare diverse traiettorie di cammino.
L’analisi del soggetto patologico ha rivelato un diminuzione della complessità dell’output
locomotorio nell’arto paretico durante il cammino rettilinea, in accordo con un lavoro
precedente riportato in letteratura [4]. Lo stesso ridotto numero di sinergie è stato ottenuto
anche nelle altre due condizioni di cammino, ma con una struttura differente delle sinergie,
indice di un controllo motorio patologico. È stato anche riportato un cambiamento delle
sinergie dell’arto non paretico con la condizione di cammino, forse causato da un
adattamento del meccanismo di coordinazione per compensare la perdita di forza dell’arto
paretico.
Quindi, nel presente studio, sono state riportate prove dell’esistenza di una strategia di
coordinazione muscolare invariante tra la locomozione rettilinea e curvilinea, mentre è
stato trovato che vengono adottati dei cambiamenti a livello del singolo muscolo per
implementare la locomozione lungo traiettorie curvilinee. Inoltre, è stato utilizzato un
nuovo metodo per verificare la qualità della coordinazione motoria di soggetti patologici.
Tale metodo potrebbe aiutare la comprensione del disturbo post ictus dei meccanismi
centrali e fornire informazioni utili per la scelta delle soluzioni riabilitative più efficaci per
il recupero della locomozione fisiologica.

23. TABLE OF CONTENTS
1 Introduction ...................................................................................................................... 1
1.1 Motor Control .............................................................................................................. 1
1.2 Mechanics Of Locomotion ......................................................................................... 3
1.3 Muscle Synergies ........................................................................................................ 6
1.3.1 Algorithms For Synergies Extraction.................................................................... 9
1.3.2 Synergies In Walking ......................................................................................... 12
1.4 Goal Of The Work .................................................................................................... 15
2 Methods ........................................................................................................................... 17
2.1 Treadmill Trials ........................................................................................................ 17
2.2 Overground Trials ..................................................................................................... 20
2.3 Data Analysis ............................................................................................................ 22
2.3.1 Kinematics .......................................................................................................... 22
2.3.2 EMG Data .......................................................................................................... 24
2.3.2.1 Synergies In Treadmill Trials ..................................................................... 26
2.3.2.2 Synergies In Overground Trials .................................................................. 27
2.3.2.3 Single-Muscle Analysis In Overground Trials ........................................... 33
2.4 Statistical Analysis .................................................................................................... 34
2.5 Pathological Subject .................................................................................................. 34
3 Results.............................................................................................................................. 36
3.1 Kinematics Analysis ................................................................................................. 36
3.2 EMG Analysis ........................................................................................................... 38
3.2.1 Preprocessing Results ......................................................................................... 38
3.2.2 EMG Synergies Extraction ................................................................................ 42
2.3.2.1 Treadmill Trials .......................................................................................... 42
2.3.2.2 Overground Trials ....................................................................................... 44
3.2.3 Single-Muscle Analysis In Overground Trials .................................................. 53

24. 3.3 Pathological Subject .................................................................................................. 57
4 Discussion ........................................................................................................................ 66
4.1 Kinematics ................................................................................................................ 67
4.2 EMG Data ................................................................................................................. 67
4.3 Pathological Subject .................................................................................................. 72
4.4 Limitations And Future Sights .................................................................................. 73
4.5 Conclusions ............................................................................................................... 74
5 Bibliography.................................................................................................................... 76

25. 1
1. INTRODUCTION
1.1 MOTOR CONTROL
The human body is a highly complex system, in which many physiological elements are
coordinated throughout the body to correctly perform countless different tasks. The large
number of degrees of freedom of the human body (e.g., joints, muscles, motor units,
neurons…) allows a task to be accomplished by the combination of different activation
patterns, thus the way the central nervous system (CNS) deals with such a complexity is
not a trivial matter, and its full understand is still under debate [11].
To generate a voluntary movement, the CNS transforms the sensorial and cognitive
information that receives as inputs into the motor-command outputs. In particular, once the
motor task is planned, the system has first to convert this cognitive information into the
space coordinates, in terms of joints angles and segment position of the effector (kinematic
transformation), and then has to determine the right muscle forces to apply to every joint to
reach such coordinates (dynamic transformation). Information on the position of the
effector are finally reported to the CNS by the sensory system, so that a more precise
control can be performed in a closed loop [12, 13].
This type of system must have a high level of complexity, since it must function under the
effect of delays, noise, nonlinearities, and internal and external changes. In fact, there are
delays in the afferent and efferent pathways of the CNS communication, which make the
sensorial information unusable to guide at least the first part of the movement. In addition,
intrinsic neural noise on sensorial and motor control signals limits the possibility of the
motor system to perform simultaneously fast and accurate movements. Another aspect to
consider is that the motor control cannot be assumed to work in a linear environment, since
there are both external non linearities (like gravity) and internal non linearities (like the
joint range of motion). Finally, there are continuous changes both in the musculoskeletal
system, due to the growing processes, and in the external environment that induces non-
stationarity in the relationship between motor commands and movement.
The study of the neuro-physiological mechanisms underlying movement production has a
long history and a lot of investigators since Berstein [14] tried to understand whether and

26. 2
how the CNS can solve such complexity relying only on the combination of a much
smaller set of variables able to actuate the whole set of voluntary movements.
Even if, in line of principle, the CNS could control independently each muscle and degree
of freedom to produce a desired movement, all the mentioned complexities would make
such a control strategy very difficult to be implemented outside a limited number of cases
[2]. Amongst the different motor-control hypotheses that have been proposed to simplify
this complex task, a recent theory has suggested that the CNS could manage the problem of
the high number of degrees of freedom in a modular fashion, controlling groups of muscles
instead of activate them singularly. This way, the dimensionality of the system would be
greatly reduced, and the control would be simplified [2].
From the anatomical point of view, the motor control is distributed in hierarchically
organized areas of the CNS, each containing circuits that can regulate a specific part of the
motor response (Figure ).
The highest level of motor control is the cerebral cortex. The primary motor cortex and
premotor areas are responsible for the planning and the coordination of complex sequences
of movements, but its activity also codes for elementary characteristics of the voluntary
movements such as the force that muscles have to develop, through the changes in the fire
rate of its neurons, and the direction of the movement.
The next level is in the brain stem, which has two systems: the medial descending system,
that contributes to the control of posture by integrating sensory information, and the lateral
descending system, that controls distal limb muscles. The lowest level of the organization
is the spinal cord: it contains the neuronal circuits that mediate a variety of reflexes and
rhythmic automatism such as locomotion. All motor commands, finally, converge on
motor neurons, located in the ventral grey matter of the spinal cord and in the brain stem.
There are evidences that neural circuitry responsible for the modular control of muscle
coordination could be located right at spinal level [15]. Each motor neuron send its axon to
a muscle, where it spreads out and innerves a variable number of muscular fibers [13]. The
union of a motor neuron and the innerved fibers is called motor unit. When a motor neuron
transmit an activation signal, the innervated fibers contract.

27. 3
1.2 MECHANICS OF LOCOMOTION
One of the most important motor tasks in daily life is locomotion. It is fundamental for
healthy subjects, allowing them to move among different places, and it constitutes a crucial
problem in subjects with impairment. In fact, in subjects with neurological diseases (e.g.,
post-stroke hemiplegia, which compromises the correct use of one side of the body), one of
the most invalidating consequences is the loss of the ability to correctly walk. Thus,
locomotion in both healthy and pathological condition is an essential field of study, since a
better knowledge of the neuromuscular mechanism underlying this motor task can lead to
Figure 1 - Areas involved in the motor control and connection between them. Figure from [13].

28. 4
the development of more efficient therapies for the rehabilitation of patients affected by
neurological diseases, and improvement of their quality of life.
In physiological conditions, human walking is bipedal and plantigrade, with two important
features: the body is erect and supported alternatively by the lower limbs, allowing the
contralateral limb to be lifted and pushed forward until it returns to the ground.
Walking is a repetitive process, so it is possible to determinate temporal events in the gait
cycle. Usually the starting point of the cycle is considered the moment when the foot hits
the ground, called Initial Contact (IC). In healthy subjects the IC is performed with the
heel, but in pathological conditions it can be made in other ways, like putting the whole
foot on the ground, or laying the toe first. A stride is defined as the interval between two
subsequent ICs of the same leg, and the distance covered during a stride cycle is called
stride length. The instant in which the foot is lifted from the ground is called End Contact
(EC); the interval between two consecutive ICs (one per each foot) is called step, and the
distance covered during a step is called step length.
The gait cycle can be divided in two main phases, stance and swing, which can in turn be
partitioned into more subphases, corresponding to functionally important moments during
which the body produces different biomechanical outputs in order to deal with the different
subtasks that characterize each phase (Figure 2).
In the stance phase (0-60% of the gait cycle) the ipsilateral foot is in contact with the
ground, either in double (0-10% and 50-60% of the gait cycle) or single support (10-40%
of the gait cycle). The first subphase of the stance is the loading acceptance, where the
ipsilateral limb has to accept the body load and has to dampen the impact of the foot on the
ground; the muscles active in this interval are the foot plantar flexors, knee extensors and
hip extensors. The next subphases are the mid and terminal stance (or pre-swing), during
which the activity of the foot plantar flexors is maximal, since they develop the push-off
necessary for the start of the swing phase.
After the ipsilateral EC, the limb enters the swing phase (60-100% of the gait cycle) until
the IC restarts the gait cycle again. In this phase, the hip, knee and ankle are flexed to
allow foot clearance and guarantee the forward progression of the body. During the

29. 5
terminal part of the swing, the knee flexors perform a deceleration of the limb, to
counteract the inertia and stabilize the body before the IC.
A particular condition of walking is the walking along a curvilinear trajectory. This is an
important task that involves changes in the walking mechanics and the motor strategy, in
order to maintain the body stability over a trajectory with continuous changes of direction.
Such task could be of particular interest for patients affected by neurological diseases,
since it is a daily-life task but it is much more challenging than walking on a rectilinear
trajectory [16].
The differences between rectilinear and curvilinear walking, both in terms of kinematics
and muscle activity, have been studied in literature [17, 18]. In particular, when walking
over a circle, the stance duration of the internal and the external leg of the rotation
increases and decreases, respectively, with respect to the rectilinear walking. In addition,
the foot-pelvis distance changes in curvilinear walking compared with rectilinear walking,
with the inner foot moving closer to the pelvis and the outer foot moving away. The timing
of both legs also differs: whereas in rectilinear walking the heel contact of the contralateral
leg occurred at nearly 50% of the gait cycle of the ipsilateral leg, in curvilinear walking the
IC occurred earlier and later respectively for the inner and outer legs. Finally, no major
Figure 2 - Representation of a stride cycle and the phases of stance, swing and double support of both feet,
with their duration in percentage of the stride cycle. Figure from Molson Medical Informatics Project.

30. 6
changes were found in the muscle-activity pattern of the main trunk and lower-limb
muscles between the two conditions; however, a fine modulation in timing and amplitude
of the basic motor pattern of some muscles was necessary to implement the curvilinear
walking.
1.3 MUSCLE SYNERGIES
One of the fundamental questions in motor control is how the CNS manages the high
complexity and the high number of degrees of freedom of the musculoskeletal system, in
order to perform a task [3]. Experiments on spinalized frogs, freely moving frogs and rats
have provided evidence for a modular organization of the spinal cord [15], where
functional units generate specific motor outputs by imposing a specific pattern of muscle
activation [19]. Therefore, it has been suggested that the CNS uses flexible combinations
of few “muscle synergies” to produce a wide range of motor behaviours.
Surface ElectroMyoGraphy (EMG) is the most common method used to investigate the
neural strategies used by the CNS during the execution of movement [20]. By using
innovative computational methods (see Section 1.3.1), the EMG signals recorded from
several muscles involved in the execution of a motor task can be decomposed into a small
number of muscle synergies. A muscle synergy can be defined as a pattern of relative
levels of muscle activation [21], and, for a given motor task, several muscle synergies can
be activated in varying combinations to produce the motor behaviour. Such hypothesis
suggests that the nervous system uses muscle synergies as a set of heuristic solutions to
transform task-level goals into detailed spatiotemporal patterns of muscle activation,
allowing its higher centers to rapidly encode task-level variables, instead of control low-
level variables (e.g, individual muscle activations).

31. 7
From a computational point of view, the hypothesis that many muscles are controlled by
the modulation of a small set of predefined muscle synergies, implies a significant
reduction of the dimensionality of the motor control problem, and simplifies the problem
of inter and intra variability of subjects [3]. The muscle synergy concept has been
formalized in a mathematical model (equation 1). Assuming that the activation level
function u(t) of all the m muscles during a motor task is the variable to be controlled by the
system, it is possible to decompose it as a linear combination of k vectors w ∈ ℝm, with
one-dimensional time-varying coefficients a(t) and a certain level of reconstruction error,
ɛ (t).
𝒖(𝑡) = ∑ 𝑎𝑖(𝑡)𝒘𝑖 + 𝜀 (𝑡)𝑘
𝑖=𝑖 (1)
Each vector w (vector of weights) specifies the relative level of activation of the muscles in
the i-th module, and constitutes the spatial structure of the synergies. On the other hand,
the coefficient a(t) (activation profile) determines the temporal evolution of the synergy
during the motor task and, thus, constitute its temporal structure [2]. Such a model
provides a dimensionality reduction if the number of synergies is lower than the number of
muscles (k < m), since the variables to be controlled are no more the single muscles, but
the synergies.

32. 8
Figure 3 - Model of muscle synergy. Motor signals are explained as a linear combination of the one-
dimensional activation profiles and the m-dimensional weight vectors. Figure from[3].
The validity of the hypothesis of a synergy-based control of muscle coordination is still an
open debate [1]. In fact, it has been proposed that the synergies might not reflect a modular
neural organization, but rather task or biomechanical constraints, which could reduce the
dimensionality of the motor control problem as well [22]. An additional alternative to the
synergy-based control underlies on the performance optimization criterion that might be
applied internally to complete a task, which has been proved to produce muscle activations
similar to the synergistic muscle modules [1]. Indeed, a neuro-physiological mechanism
able to fully justify the muscles synergy model is still lacking and has to be thoroughly
investigated [23].
It is important to observe that, even if the synergies hypothesis would not represent the
actual neural mechanism of control of muscle coordination of the CNS, it is an important
tool for the study of body biomechanics both in healthy and pathological conditions. An
analysis of electromyographic data based on muscle synergies would reduce the
dimensionality of the variables to be considered, and facilitate the comparison of data from
different subjects and conditions.

33. 9
1.3.1 ALGORITHMS FOR SYNERGIES EXTRACTION
The most common method used to evaluate the muscle synergies consists in searching for
regularities in a dataset of muscle activations acquired by means of EMG signals. This
search is performed by using innovative computational methods based on factorization
algorithms applied to the EMG envelopes.
There are various factorization algorithms that can be used to decompose multiple signals,
each leading to a specific result, making the choice of the algorithm very important for the
correct analysis and interpretation of the results. The algorithms mainly used in synergies
extraction are the Principal Component Analysis (PCA) and the Non-negative Matrix
Factorization (NMF) [24], and to a much letter extent the Independent Component
Analysis (ICA) and the k-means analysis.
Both PCA and NMF are linear decomposition techniques that assume that the set of
measured data can be composed in a linear combination of a smaller number of underlying
elements. Any observation in the given dataset can be thus represented as:
𝑴𝑗 = ∑ 𝑐𝑖𝑗 𝑾𝑖 + 𝜀𝑚
𝑖=1 (2)
Where Mj is a vector representing the measurement of j-th channel of the data, Wi are
vectors that remain invariant across different channels, cij are scalar values that,
multiplying vectors Wi, specify the contribution of the invariant i-th component to the j-th
channel, and ɛ is the error present due to the factorization. It is clear that such a factorial
representation of the data perfectly fits the mathematical model of the synergies previously
described, where Mi is the muscle activation, Wi is the vector of the weights and c is the
activation profile.
PCA is an analytical technique that requires the components to be orthogonal to each other,
so a unique solution is guaranteed to any decomposition, and it can be found through a
straightforward set of computations. Moreover, it is possible to easily choose the adequate

34. 10
number of components needed to explain the given dataset. On the other hand, NMF is a
search algorithm, and therefore it has to start from a set of random components and
iteratively update the solution until the reconstruction accounts for an adequate proportion
of variability of the original dataset. Furthermore, NMF constrains all the components of
the factorization to be nonnegative; this implies that the components do not have to be
orthogonal, but independent. Finally, the NMF algorithm requires the number of
components to be specified in advance, so a criterion based on multiple searches to
determine the right number must be used.
Since these algorithms will be applied to EMG signals to obtain muscle synergies, is
crucial to analyse to which extent the differences in the algorithm outputs have
consequences, from a physiological point of view. In PCA, the vector of weights is
allowed to contain positive or negative numbers representing relative muscle activation
levels. In the context of muscle activation patterns, the equal relationship between positive
and negative activation is inconsistent with the transformation between motorneuron action
potential and muscle activity, since an inhibitory effect would be seen only with active
muscle. Moreover, excitatory pathways are different from inhibitory ones, so it is not
correct to suppose that an excitatory and an inhibitory pattern would be identical.
On the other hand, NMF constraints its components to be either positive or zero, which
corresponds to the physiological behaviour of both the neural and muscle output, since
neurons are either firing or in a rest state, and muscles are active or quiescent. Moreover,
the non-negative constraint forces the components to be combined in a part-based, additive
way to reconstruct the original signal. This is not true for PCA, in which its components
can be also subtracted. The part-based decomposition is similar to the type of neural
representation observed in the visual system and other sensory encoding systems [25].
In order to assess the quality of the output of the decomposition, it is possible to multiply
the components calculated to obtain a reconstruction of the original signal, and then
compare the reconstructed signal with the original one. Such a comparison can be
performed by evaluating the Variability Accounted For (VAF), which is an index of
goodness of fit between the original and the reconstructed signals. The higher the VAF, the
best is the performance of the algorithm.

35. 11
The VAF associated to the EMG of a muscle can be calculated using the following
equation [26], which obtains a ratio between the unexplained variation (sum of squared
errors) and the pooled variation of the data (total sum of squares).
𝑉𝐴𝐹𝑖 = 1 −
∑ (𝐸𝑀𝐺 𝑜(𝑖,𝑗)−𝐸𝑀𝐺 𝑟(𝑖,𝑗))2𝑘
𝑗=1
∑ 𝐸𝑀𝐺 𝑜(𝑖,𝑗)2𝑘
𝑗=1
(3)
Where i indicates the muscle under consideration, j indicates the current sample, k is the
total number of samples, EMGo is the original EMG signal, and EMGr is the signal
reconstructed from the output of the factorization.
The VAF associated to the entire dataset can be calculated as well by applying the
following equation [10], using the same symbol convention used in the equation (3), and m
indicating the total number of muscles.
𝑉𝐴𝐹 = 1 −
∑ ∑ (𝐸𝑀𝐺 𝑜(𝑖,𝑗)−𝐸𝑀𝐺 𝑟(𝑖,𝑗))2𝑘
𝑗=1
𝑚
𝑖=𝑖
∑ ∑ 𝐸𝑀𝐺 𝑜(𝑖,𝑗)2𝑘
𝑗=1
𝑚
𝑖=1
(4)
It has been shown that applying both NMF and PCA to the same set of data, NMF leads to
a higher value of VAF [25]. Therefore, considering all the aspects discussed above, NMF
appears to be the best factorization algorithm to be used for extracting synergies.
Since the NMF algorithm requires the number of components to be specified a-priori, a
method to determine such a number must be used. Different criterions have been applied in
literature, most of them based on the analysis of the VAF. The first step is to apply the
NMF to the dataset multiple times, varying the number of extracted components, and
calculating the VAF for each condition. There are different ways to select the adequate
number of components. One possibility is to plot the value of the VAF as a function of the
number of components extracted (Figure ) and select the correct number of components as
the one in which a change in the slope of the curve is present [27].

36. 12
Figure 4 - Example of the plot of the VAF as function of the number of components extracted for the same
dataset. Figure from [28].
Another option is to choose the minimal number of components in correspondence of
which the value of total VAF is higher than 90% [6]. A constraint can be added to this
criterion, by selecting the number of components that respects the condition described
above and that ensure for each muscle a value of VAF higher than 75% [10].
Other studies have opted for a more-constraining criterion [4], where the number of
extracted components must guarantee a VAF higher than 90% for all muscles, or an
increase lower than 5% when adding a component.
All the described criterions have been applied in the literature, without any clear agreement
on whether or when they should be applied [29].
1.3.2 SYNERGIES IN WALKING

37. 13
One of the fields whose cyclic biomechanics can be studied with muscle synergies is
human locomotion. Various authors have investigated the composition of the muscle
synergies used by the CNS to perform locomotion under different conditions, showing that
the EMG activity of trunk and leg muscles can be adequately reconstructed as a linear
combination of four to five basic synergies [4]–[8].
Figure 5 - Mean weights and activation profiles of the four muscle synergies extracted from 20 healthy
subjects during normal walking at self-selected speed. The 8 muscles considered are tibialis anterior(TA),
soleus (SO), medial gastrocnemius (MG), vastus medialis (VM), rectus femoris (RF), lateral hamstring (LH),
medial hamstring (MH) and gluteus medius (GM). Figure from [4] .
Looking at the composition of the vectors of muscle weights and the shape of the
activation profiles (Figure ), it is possible to associate each of the extracted synergies to a
specific biomechanical function [4]. Specifically, module C1 consists mainly of the

38. 14
activity of the hip extensor and abductor (GM), knee extensors (RF, VM, VL) and hip
flexor (RF). The timing of this module, primarily active in early stance, and the
simultaneous activity of a hip extensor and a hip flexor muscle suggest that the function of
module C1 is to provide body support during weight acceptance.
Module C2 consists mainly ankle plantarflexors (SO, MG) and knee flexor (MG), and was
active during late stance. It likely contributes to body support, forward propulsion, and
swing initiation.
Module C3 primarily consists of activity of ankle dorsiflexor (TA), and hip flexor (RF)
during early stance; it likely provides dorsiflexion during and immediately after heel strike
and early swing, where it contributes to ground clearance of the foot.
Module C4 consists mainly in the activation of knee flexors and hip extensors (MH, LH),
during late swing and early stance and may decelerate the leg at the end of swing and
propel the body during early stance.
The configuration of the described synergies was found to be stable over a variety of
conditions. Analysing the lower-limb muscle synergies for walking at different speeds, it
emerged that the only difference in the results under the different conditions was that, at
higher speeds, the amplitude and definition of the activation-profile peak increased, but the
weights composition and the shape of the profiles were maintained [4].
Other findings suggest that the number of modules and the composition of the weights
vector are the same as the one found during normal walking when the functional task
demands were altered (walking with maximum cadence, maximum step length, maximum
step height). The only difference reported over the different tasks is in the activation
profiles, where a slightly altered timing is present [6].
The consistency of the synergies used in walking has also been proved for walking with
the presence of perturbations. In fact, three out of five modules used in walking in presence
of multidirectional perturbations are the same used in unperturbed walking, showing that
walking and reactive balance may be constructed by a common set of muscle synergies [5].
Since it has been hypothesized that the synergistic control of muscles is implemented in the
CNS by the existence of dedicated neural circuits, it is interesting to analyse how the

39. 15
synergies organisation changes due to neural diseases that compromise neural circuits and
lead to impairments of lower limbs and to unnatural walking. An example of such a
situation is provided by post-stroke subjects: a stroke occurs when the blood supply to the
brain is compromised by an ischemia or a haemorrhage, with a consequent damage to the
area of the brain affected, and the probable impossibility to move one or more limbs of one
side of the body (paretic limbs). Investigations on post-stroke subjects show that different
changes in the synergies organisation occur due to the disease [4], and such changes and
the severity of the impairment of the subject are related. The first change is in the number
of modules needed to reconstruct the EMG; in fact two to four modules are sufficient to
reconstruct the activity of the paretic limb during walking, suggesting that the loss of
motor function is a consequence of an insufficient muscle-activity complexity. These
modules can be usually directly matched to the physiological synergies or be reconstructed
by a combination of them.
1.4 GOAL OF THE WORK
The purpose of the present work is to study muscle activations and muscle synergies in
regular walking (rectilinear trajectory at self-selected speed), but also investigate the
changes that occur during walking on a curvilinear trajectory. This task is more
challenging, since it changes balance conditions with respect to rectilinear walking, and
requires the neuromuscular system to adapt the walking strategy to a different condition. It
is important in patients with neurological diseases, despite resulting easy to accomplish by
healthy subjects, since the slight changes with respect to rectilinear walking can represent a
challenge for pathological subjects. Therefore, understanding how the motor system of
healthy subjects deals with the curvilinear walking could be useful to understand the
pathological condition too.
In order to study both the rectilinear and curvilinear walking conditions and the differences
between them, muscle synergies have been extracted from surface EMG signals applying
the NMF algorithm, and have been evaluated.

40. 16
Until now, some studies have reported the differences in single muscle activity between
rectilinear and curvilinear walking, analysing the effect on the EMG signal of single
muscles, but a study on the overall effect of curvilinear trajectory on synergistic control
strategy used in walking based on the synergies hypothesis has never been published
before. Thus the present study propose a novel approach to the study of curvilinear
walking and the differences between this condition and the rectilinear walking.
Moreover, the acquired dataset had a number of muscles larger than those considered in
the majority of the studies present in literature, and it was combined with different data-
processing methods, in order to find the optimal processing for muscle synergies
extraction.

41. 17
2. METHODS
Two different experimental trials were prepared to acquire and analyse the muscle
synergies on treadmill and overground walking. Twenty healthy, non-impaired subjects
enrolled the trials, not presenting any lower-limb injuries or dependency to walk in the
conditions requested by the experiments. Similar conditions were maintained across
conditions, allowing the age-matched subjects to walk with their own shoes at their self-
selected speed, being monitored with inertial (and magnetic, for the overground trials)
sensors and surface EMG on the main lower-limb muscles during gait. Treadmill trials
were conducted in order to perform a rigorous comparison of the results with literature
data, since the majority of the studies on muscles synergies during locomotion was
conducted on treadmill walking, and numerical data of one of these studies [4] were
available.
The complexity of the experimental setup was increased for the overground trials, in order
to monitor the lower-limb muscles with a higher density.
2.1 TREADMILL TRIALS
Ten healthy, non-impaired subjects (4 men and 6 woman, age: 24.6 ± 2.3 years, height:
1.68 ± 0.10 m, weight: 61.0 ± 11.6 kg) enrolled the treadmill trials.
The activity of the ten main lower-limb muscles of the left leg was recorded with surface
EMG electrodes: gluteus maximus (GM), tensor fasciae latae (TFL) rectus femoris (RF),
vastus medialis (VM), vastus lateralis (VL), medial hamstrings (HM), lateral hamstrings
(HL), medial gastrocnemius (MG), soleus (SO) and tibialis anterior (TA). The skin was
preliminary shaved and cleaned with alcohol, and bipolar electrodes were placed following
the SENIAM recommendations [30]. A multi-channel bio-signals amplifier (Porti 32™,
TMS International) with a sampling frequency of 1024 Hz was used to record the EMG
data. During the trials the subject wore two inertial sensors (MTx, Xsens technologies B.V)
to acquire the kinematics and segment the gait strides. The sensors were sampled at 100

42. 18
Hz, and they were mounted on external part of both shanks, fixed with bi-adhesive tape
and Velcro straps.
The experimental protocol comprised three separate trials of three-minute walking, with an
initial phase of 5 seconds of standing in erect position. Prior to the acquisition, the subjects
were initially asked to walk on the treadmill and choose their most comfortable walking
speed (self-selected speed), which was maintained for the three trials.
1
2
3
4
5
6
7
8
9
10

43. 19
Figure 6 - Setup for treadmill trials. Electrodes placement on: 1) RF, 2) VM, 3) TA, 4) MG, 5) SO, 6) GM, 7)
LH, 8) MH, 9) TFL, 10) VL. And inertial sensors placement on: 10) right shank, 11) left shank.
10
11
Figure 2-Experimental setup for treadmill trials
Figure 7 - Experimental setup for treadmill trials.

44. 20
2.2 OVERGROUND TRIALS
Thirteen healthy, non-impaired subjects (7 men and 6 woman, age: 24.8 ± 1.3 years,
height: 1.73 ± 0.11 m, weight: 60.8 ± 11.4 kg) were recruited for the overground trials.
The same experimental setup described in the previous section was adopted to acquire the
EMG and kinematic data. In addition it was possible to acquire data from the
magnetometers integrated in the inertial sensors, since no electromagnetic interference was
present. Surface EMG data were recorded from 15 muscles of the dominant leg (see figure
8Errore. L'origine riferimento non è stata trovata.): erector spinae (ES), abdominal
external oblique (AO), gluteus maximus (GM), tensor fasciae latae (TFL), rectus femoris
(RF), vastus medialis (VM), adductor magnus (AM), vastus lateralis (VL), medial
hamstring (HM), lateral hamstrings (HL), medial gastrocnemius (MG), lateral
gastrocnemius (LG), soleus (SO), peroneus longus (PE), and tibialis anterior (TA).
The experimental protocol comprised an initial stage of walking along a circular trajectory
with a 1.2 m radius, and a second phase of walking across a 10-meter long rectilinear
trajectory. Both of them were performed at the subject’s self-selected speed. The
curvilinear trials comprised two clockwise trials and two counter-clockwise trials, in which
the subject covered six consecutive times the circumference drawn on the ground, leaving
the dominant leg at the inner and outer part of the trajectory. The rectilinear trials consisted
on walking ten times a 10-meter trajectory. Before each trial, for both the circular and the
rectilinear trajectory, subjects were asked to stand in erect position for 5 seconds.
Three walking conditions were extracted from this dataset: walking on rectilinear
trajectory (RT), walking on curvilinear trajectory with the dominant leg moving on the
inner part of the trajectory (IT), and walking on curvilinear trajectory with the dominant
leg moving on the external part of the trajectory (ET).

45. 21
1
2
3
4
5
6
7
9
8
10
12
11
14
16
17
13
15

46. 22
Figure 8 - Setup for overground trials. Electrodes placement on: 1)ES, 3) AO, 4)TFL, 5) GM, 6) HM, 7) LH,
8) RF, 9) AM, 10) VL, 11) VM, 14) GM, 15) GL, 16) SO, 18) PE, 19) TA. And inertial sensors placement on:
2) chest, 12) thigh of dominant leg, 13) shank of contralateral leg, 17) foot of dominant leg, 20) shank of
dominant leg.
2.3 DATA ANALYSIS
All the data were analysed by means of the software MATLAB® R2014 [31]. The two
conditions (overground and treadmill) had a common initial processing for the kinematics
and EMG data, and then different analysis on the processed signals were performed.
2.3.1 KINEMATICS
The data analysis was limited to the part of the trial where the subject had already reached
a stable gait, removing the initial and final acceleration and deceleration phases. Therefore,
a preliminary analysis of the shanks angular velocity was done, to stablish the initial and
terminal moment of each trial as the points in which the shank angular velocity remained
stable.
The angular velocities of the shanks were then used to detect the moments where an initial
contact (IC), end contact (EC) and a mid-stance events (MS) occurred for each leg [32].
The ICs were considered as the first event of the gait cycle, and they were used to divide
the data into strides, defined as the interval between two consecutive ICs of the same leg.
19
18
20

47. 23
The duration of the gait strides and its inverse, the cadence, were calculated for the
complete trial. The other gait events detected from the two legs allowed a subdivision of
the stride into three gait phases per leg: double support, initial and terminal swing [32]
(figure 9).
Figure 9 - Shank sagittal-plane angular velocity (a) and flexion/extension angle (b) for the left (dashed blue)
and right (solid black) legs. IC, EC and MS defined 6 gait phases. The angle (AN) used in the detection of EC
is also shown. Figure from [32].
The duration of three gait phases was evaluated for the leg whose muscles’ activity was
recorded: stance phase (considered as the interval between an IC and an EC), swing phase
(considered as the interval between an EC and the subsequent IC), and total double
support. Total double support was computed by summing up the duration of the two
double support phases present in the gait cycle, the first going from the IC of the leg
considered to the EC of the contralateral leg, and the second going from IC of the
contralateral leg to the EC of the leg considered.
The duration of each phase was calculated as a percentage of the stride cycle. The resulting
lengths were compared across conditions, as detailed on section 2.4.
The stance ratio index between the stance duration of the two legs was computed as the
ratio between the stance duration of ipsilateral leg and the stance duration of the
contralateral leg, where the stance duration was measured as a percentage of the stride
cycle [33].

48. 24
2.3.2 EMG DATA
In order to remove part of the noise, the raw EMG signals were band-pass filtered at 20-
400 Hz with a 3rd
order Butterworth filter [10]. Successively, the envelope of the
recordings was obtained by rectifying and low-pass filtering the previous signals at 5 Hz
with a 3rd
order Butterworth filter. The cut-off frequency of the low-pass filter was set at 5
Hz in order to maintain the 95% of the total power of the signal [34].
The start and end points previously identified with the analysis of the kinematics data was
used to select the EMG envelopes measured during stable gait; thus only the central part of
the recording of each trial, in which the velocity of the subject was found to be stable, was
maintained for the subsequent analysis.
The EMG envelopes were segmented into single strides on the basis of the IC events
obtained in the previous kinematics analysis. In order to allow a between-stride
comparison, the time scale of each stride was normalized by linearly interpolating the
EMG signals on 100 points [10], [35].
An additional visual analysis was performed to check the quality of the resulting signals
thus avoiding outliers. When artefacts on the profiles were identified, the stride containing
the irregularity was eliminated for all muscles, so that further analysis would not be
affected by signals not related to muscle activity [36].
For each walking condition, the envelopes of all the trials were concatenated so that, for
each subject, a matrix M of size m x n (where m is the number of muscles and n is equal to
the number of strides x 100) was created, containing the signals from all the trials. Then,
the maximum peak of muscle activity was calculated for each stride and muscle, and the
signals of each muscle were normalized, for all strides, with the median of the muscle
peaks [35].

49. 25
In order to guarantee that the results of the further analysis would not be affected by the
different number of strides collected for each subject, 20 strides were selected for each
subject and walking condition, reducing the dimensions of the matrix M to m x 2000. The
selection of these 20 strides was performed by processing, for each subject and condition,
all the possible combinations of 20 consecutives strides and choosing those that provided
the highest analysis quality. The specifics of the processing procedure (muscle synergies),
and the quantitative parameters to measure analysis quality (Variance Accounted For,
VAF) will be explained in section 2.3.2.1. A schematic representation of the EMG
processing is reported in figure 10.
Figure 10 - Flowchart of data processing. The processing applied to raw EMG and acceleration data to get
to matrix M of EMG envelopes is shown.

50. 26
2.3.2.1 SYNERGIES IN TREADMILL TRIALS
The results obtained from the analysis of the treadmill data were compared with those
reported in the literature in a study performed with the same experimental conditions [4].
This study analysed the EMG signals of the same 8 lower-limb muscles acquired from 20
healthy subjects walking on a treadmill at their self-selected speed, extracting, with similar
VAF criteria, four muscle synergies. Therefore, since numerical data of the matrix of
weights (Wclark) and the matrix of activation profiles (Hclark) obtained in [4] were available,
it was, thus, possible to perform a rigorous comparison.
The analysis of treadmill data allowed to implement a methodology used for synergies
extraction and analysis. Since the same methodology was applied to overground data and
treadmill data, if it emerged that the results of the treadmill-data analysis were similar to
the results reported in literature, it would also validate the results of the overground data
analysis, since the same acquisition instrumentation and processing methods were used.
Comparison with literature
The set of muscles recorded was restricted to eight to match those used in the literature
(GM, VM, VL, MH, LH, MG, SO, TA).
For each subject, the NMF algorithm was applied to the M matrix of EMG signals,
imposing a number of synergies equal to 4, in order to compare them with the 4 synergies
extracted in literature. The matrices W of muscle weights, of size 8 x 4, and H of activation
profiles, of size 4 x 2000, were than obtained. Since each column of H contains a number
of concatenated activation profiles equal to the number of strides contained in the matrix
M, the mean over the all the profiles was computed, and the matrix dimensions were
reduced to 4 x 100, where 100 is the length of a gait cycle.
For each synergy, a normalization of both the weights and the activation profile was
performed by dividing all the elements of the vector of weights for its maximum value, and
multiplying all the points of the activation profile for the same value, with the following
equations.
𝑊𝑛𝑜𝑟𝑚(𝑖, 𝑗) = 𝑊(𝑖, 𝑗)/ 𝑚𝑎𝑥
𝑗
𝑊 (5)

51. 27
𝐻 𝑛𝑜𝑟𝑚(𝑗, 𝑘) = 𝐻(𝑗, 𝑘)/ 𝑚𝑎𝑥
𝑗
𝑊 (6)
The similarity (see section 2.3.2.2) was calculated between each column of Wclark and each
column of Wnorm for each subject, and the pairs with maximum similarity were grouped
together. Thus, the synergies of all the subjects were grouped on the base of their similarity
with the synergies of literature.
For each group of synergies, mean values of the vectors of weights and activation profiles
were computed, and the matrices Wtreadmill and Htreadmill, containing such mean values, were
created.
Each synergy obtained from the treadmill data was compared with the corresponding
synergy of the literature data, in terms of weights (similarity), and in terms of activation
profiles (circular cross-correlation, rmax, and time-lag) (see section 2.3.2.2).
2.3.2.2 SYNERGIES IN OVERGROUND TRIALS
Inter-individual variability
The variance ratio (VR) coefficient was calculated for each muscle, subject and walking
condition using the following equation [28]:
𝑉𝑅 =
∑ ∑ (𝑋 𝑖𝑗−𝑋̅ 𝑖)
2
/𝑘(𝑠−1)𝑠
𝑗=1
𝑘
𝑖=1
∑ ∑ (𝑋 𝑖𝑗−𝑋̅)
2
/(𝑘𝑠−1)𝑠
𝑗=1
𝑘
𝑖=1
with 𝑋̅ =
1
𝑘
∑ 𝑋𝑖
̅𝑘
𝑖=1 (7)
where Xij is the i-th bin of the EMG profile of the j-th stride, k is the number of points of
each EMG profile (100, in our case), s is the number of strides over which the VR is
calculated (20, in our case) and Xi
̅ is the mean value of the profile obtained for the i-th bin
calculated across the strides. The VR is a quantitative parameter for the inter-subject
variability, measured as the overall variation of the EMG data with respect to its mean. The
higher the value of VR, the higher the inter-subject variability present in the data.

52. 28
The VR was also used as an indicator of the quality of the signal. In fact, a high variability
of the signal of a muscle and, thus, a high value of VR could indicate that too much noise
is present in the signal or that there is not a specific pattern of muscle activation, resulting
in both cases in a signal not suitable for a correct analysis. A mean VR was obtained for
each muscle and walking condition, calculated across subjects and trials, and was
compared with a threshold below which the variability was considered not tolerable [37],
fixed at 0.3. The muscles that presented a mean value of VR higher than 0.3 were
eliminated from the successive analysis.
Synergies dimensionality
The extraction of muscle synergies was performed by means of the NMF algorithm [24].
Such a decomposition algorithm, applied to the matrix M, gives the following solution:
M ≈ W x H (8)
where, being syn the number of extracted synergies, W is the 15 x syn matrix of muscle
weights, and H is the syn x 100 matrix of activation profiles. Given the solution of the
algorithm, it is possible to obtain a matrix R of reconstructed signals by multiplying W and
H. Since each column of H contains a number of concatenated activation profiles equal to
the number of strides contained in the matrix M, the mean over the all the profiles was
computed, and the matrix dimensions were reduced to syn x 100, where 100 is the length
of a gait cycle.
Since the NMF algorithm requires the number of synergies to be specified as an input, it
was necessary to first determine the correct number of synergies. Therefore, for each
subject and condition, separate NMF analyses were performed on the matrix M with the
number of synergies ranging from 1 to m (number of muscles contained in M), since, in the
worst-case scenario, a one-on-one correspondence between muscles and synergies would
be found.
The quality of the signals reconstructed with the extracted muscle synergies was
quantitatively assessed with the Variance Accounted For (VAF) metric, which is an index
of the variation that can be explained by the model. The VAF was obtained for both the

53. 29
single muscles (equation 9) and the entire dataset (equation 10) [35], where VAFs(i)
indicates the VAF of the i-th muscle.
𝑉𝐴𝐹𝑠(𝑖) = 1 −
∑ (𝑀(𝑖,𝑗)−𝑅(𝑖,𝑗))2𝑛
𝑗=1
∑ 𝑀(𝑖,𝑗)2𝑛
𝑗=1
(9)
𝑉𝐴𝐹𝑡𝑜𝑡 = 1 −
∑ ∑ (𝑀(𝑖,𝑗)−𝑅(𝑖,𝑗))2𝑛
𝑗=1
𝑚
𝑖=𝑖
∑ ∑ 𝑀(𝑖,𝑗)2𝑛
𝑗=1
𝑚
𝑖=1
(10)
The criterion for the determination of the number of synergies to extract depends on
multiple variables, like the processing that was made on the data, the dimension of the
dataset and the quality of the signals. In this study a criterion based on the analysis of the
quality of single muscles reconstruction was used: the minimum number of synergies
needed to adequately reconstruct the data in M was considered the lowest in
correspondence of which, for each muscle, VAFs was ≥ 90% or was not increased more
than 5% by adding a synergy [4].
The number of muscle synergies was selected by averaging across subjects the number of
synergies required for each condition. If the number of synergies would be found to be
same for all the conditions, an inter-condition comparison would be possible.
Synergies extraction
For each subject and walking condition, five muscle synergies were extracted using the
NMF algorithm on the EMG profiles (M), obtaining the muscle weights (W) and the
activation profiles (H). For each synergy, the weights and activation profiles were
normalized by dividing all the elements of the weight vector by their maximum value, and
multiplying all the points of the activation profiles by the same value [35]. This procedure
allows the comparison between different subjects, without affecting the reconstructed
signal, since the normalization factors are cancelled out when W and H are multiplied [35].
The normalized matrices Wnorm and Hnorm were, thus, calculated following equations (5)
and (6).
The muscle synergies that were extracted from each subject for a specific walking
condition were then confronted against their analogous pairs from other subjects, matching

54. 30
the synergies that corresponded to the same biomechanical function. This similarity
between muscle synergies was quantitatively calculated with the normalized scalar product
between the weights of the two muscle synergies [7]. The similarity ranges from 0 (no
matching between the vectors) to 1 (perfect matching between the vectors). This parameter
allowed a specific synergy to be recognized across subjects, therefore it was calculated for
all the possible combination of synergy pairs between two subjects, grouping together
those with the highest value.
𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 =
𝑣∙𝑤
‖𝑣‖‖𝑤‖
(11)
Where v and w are vectors of length m containing the weights of all muscles for one
synergy.
The mean values of weights and activation profiles were calculated for each group of
synergies, and the matrices Wmean and Hmean, were, thus, obtained. To assess the quality of
the mean synergies and to verify if they were representative of a general inter-subject
trend, for each single walking condition a matrix of reconstructed signals was computed
starting from Wmean and Hmean, and, for each subject, the VAFtot was calculated between the
reconstructed signal and the original signal. The median and interquartile range values of
VAFtot from all the subjects was then computed.
Synergies comparison
The mean synergies were compared across walking conditions. The weights of a specific
synergy were compared across two conditions by computing their similarity, considering
the two synergies quantitatively similar when this metric overcame 0.75 [10]. The
comparison between two activation profiles was performed by means of a circular cross-
correlation, which is a procedure analogous to linear cross-correlation that introduces a
circular shift between the signals [38], particularly suited for periodic signals (like the
activation profile, whose period is the gait cycle). The circular cross-correlations were
computed for all synergies across each pair of walking condition, obtaining for each
comparison the maximum correlation (rmax) and the time-lag, defined as the time shift

55. 31
needed to get rmax [29]. The rmax is an index of the similarity between the shapes of two
profiles, and its values range from 0 (no correlation between the profiles) to 1 (perfect
matching on the profiles). On the other hand, the time-lag gives information on the
temporal shift between two profiles. It ranges from 0 to 100, but, since the signals were
circularly shifted in time, it can be assumed that its actual range goes from -50 to 50,
because the time shift of a profile in one direction is equal to a negative time shift in the
other direction. Therefore, a time-lag equal to 0 means that there is no temporal shift
between the profiles, whereas a value of 50 or -50 means that a profile is anticipated or
delayed of half a period with respect to the other, respectively.
Cross-validation
To further investigate the differences among the synergies extracted from the three walking
conditions, it was tested whether the mean synergies of a specific walking condition could
explain the EMG signals of the other trials. To this aim, for each condition of walking, the
EMG signals were reconstructed using the mean synergies previously obtained. Then, they
were compared with the EMG signals of all subjects for the other conditions, by means of
VAFtot [10], whose median and interquartile rage values were calculated afterwards across
subjects.
Since the comparison of the activation profiles, H, across different conditions would not
result rigorous if they were referred to different spatial structures (W), even if such
structures resulted similar, a cross-validation analysis different from the previously
described was implemented. For each subject independently, the NMF was applied to the
EMG envelopes of each condition, allowing only the matrix H of the activation profiles to
vary, and keeping the matrix W of muscles weights fixed and equal to the matrix W
extracted from the RT condition. The matrix H was normalized following the equation (6),
and the mean activation profiles for each condition were computed afterwards across
subjects. For each subject, the VAFtot was calculated between the original signals and the
signals reconstructed from the extracted synergies for each condition independently, and
the median and interquartile rage values of VAFtot were computed across subjects. A
comparison between the activation profiles was performed between each pair of conditions

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by means of the rmax and the time-lag. This procedure allowed a rigorous comparison of the
timing features of the synergies across different conditions, that presented similar spatial
structures, for a given group of subjects.
Finally, a further comparison of the activation profiles was performed by applying for each
subject independently, the NMF to the EMG envelopes of each condition, allowing only
the matrix H of the activation profiles to vary, and keeping the matrix W of muscles
weights fixed and equal to the mean matrix W obtained across subjects for the RT
condition, normalized following the equation (5). The matrix H was, then, normalized
following equation (6), and the mean activation profiles were computed for each condition
across subjects. For each subject, the VAFtot was calculated between the original signals
and the signals reconstructed from the extracted synergies for each condition
independently, and median and interquartile rage values of VAFtot were computed across
subjects. A comparison between the activation profiles was performed between each pair
of conditions by means of the rmax and the time-lag. This procedure assumes that a
common spatial structure exists for all subjects and, thus, that a set muscles weights can be
created for a group of subjects.
Comparison with treadmill data
The synergies extracted from the treadmill trials were compared to those of the overground
trials, only in the RT condition of the overground trials. The set of muscles of the
overground trials was restricted to match that of the literature (GM, VM, VL, MH, LH,
MG, SO, TA).
The extraction of muscle synergies from the restricted overground dataset was performed
by applying, for each subject, the NMF algorithm to the M matrix of the EMG signals,
imposing a number of synergies equal to 4. Thus, the matrices W of muscle weights, of
size 8 x 4, and H of activation profiles, of size 4 x 2000, were obtained. Since each column
of H contains a number of concatenated activation profiles equal to the number of strides
contained in the matrix M, the mean over the all the profiles was computed, and the matrix
dimensions were reduced to 4 x 100, where 100 is the length of a gait cycle.

57. 33
For each synergy, a normalization of both the weights and the activation profile was
performed by dividing all the elements of the vector of weights for its maximum value, and
multiplying all the points of the activation profile for the same value, following equation
(5) and (6).
The muscle synergies from Wtreadmill and Woverground were paired when the weights obtained
a maximum value of similarity. For each group of synergies, mean values of the vectors of
weights and activation profiles were computed, and the matrices Woverground and Hoverground,
containing such mean values, were created.
Synergies extracted from overground walking condition were compared with the synergies
extracted from the treadmill walking condition and the synergies of literature. The
comparison was performed using the same metrics adopted for the comparison of the
synergies obtained from treadmill walking conditions with the literature data. Thus, each
synergy obtained from the overground data were compared with the corresponding synergy
of the treadmill data and literature data in terms of weights (similarity), and activation
profiles (circular cross-correlation, rmax, and time-lag).
2.3.2.3 SINGLE-MUSCLE ANALYSIS IN OVERGROUND TRIALS
Identification of muscular onset and offset
The onset and offset of each single muscle activations were extracted for all the trials, in
order to compare across conditions the differences in the gait subphases where the muscles
are active. This was done by defining first a threshold of muscle activity, defined for each
subject and muscle independently, as the minimum of the average EMG profile plus the
25% of the difference between minimum and maximum [29]. The onset was considered as
the point where the muscle activity overcame such a threshold, and the offset was
identified as the point where the muscle activity decreased below the threshold. A pair of
onset and an offset were considered acceptable only if the maximum value of the profile
was located between them.
Amplitude of muscle activity

58. 34
The amplitude of muscle activity was calculated, for each subject, muscle and condition
independently, as the integral of the muscle envelope during the time interval between a
pair of an onset and an offset. A normalization was performed by dividing the amplitude of
activity of each muscle by the amplitude of activity of the same muscle during RT [17].
2.4 STATISTICAL ANALYSIS
Kinematics
The median and interquartile ranges of the cadence, phases length and stance index were
computed for each condition, across subjects. The differences of each parameter between
conditions were assessed with the Kruskal-Wallis test, and post-hoc analyses were
performed in the data that resulted statistically different.
EMG data
After having evaluated that the computed parameters were not normally distributed, for
each walking condition, the median values and interquartile of the muscles onsets, offsets
and amplitude of activity were computed across subjects. The Kruskal-Wallis test was
applied to the data to assess if there were differences among conditions, and post-hoc
analyses were performed in the data that resulted statistically different.
2.5 PATHOLOGICAL SUBJECT
The method described in this work for the study of muscular coordination during
rectilinear and curvilinear walking could be useful in the evaluation of locomotion
strategies in subjects with neurological diseases, since a task of daily life is proposed, but
with a more challenging condition, which may require changes in the coordination strategy
in pathological conditions.