Ci a ion: Mo a, B.; Basu ko, J.;
Sabahi, I.; Le u iondo, U.; Albizu i, J.
S ain Vi ual Sensing o S uc u al
Heal h Moni o ing unde Va iable
Loads. Senso s 2023,23, 4706.
h ps://doi.o g/10.3390/s23104706
Academic Edi o : Aldo Mina do
Recei ed: 4 Ap il 2023
Re ised: 8 May 2023
Accep ed: 9 May 2023
Published: 12 May 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
senso s
A icle
S ain Vi ual Sensing o S uc u al Heal h Moni o ing unde
Va iable Loads
Ba omeu Mo a 1,2,* , Jon Basu ko 1, Iman Sabahi 3, U ko Le u iondo 1and Joseba Albizu i 2
1Ike lan Technology Resea ch Cen e, Basque Resea ch and Technology Alliance (BRTA),
20500 A asa e-Mond agon, Spain; [email p o ec ed] (J.B.); [email p o ec ed] (U.L.)
2Facul y o Enginee ing in Bilbao, Uni e si y o he Basque Coun y (UPV/EHU), 48013 Bilbao, Spain;
[email p o ec ed]
3KU Leu en, Depa men o Mechanical Enginee ing, B-3001 Leu en, Belgium; [email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac :
Vi ual sensing is he p ocess o using a ailable da a om eal senso s in combina ion wi h
a model o he sys em o ob ain es ima ed da a om unmeasu ed poin s. In his a icle, di e en
s ain i ual sensing algo i hms a e es ed using eal senso da a, unde unmeasu ed di e en o ces
applied in di e en di ec ions. S ochas ic algo i hms (Kalman il e and augmen ed Kalman il e )
and de e minis ic algo i hms (leas -squa es s ain es ima ion) a e es ed wi h di e en inpu senso
con igu a ions. A wind u bine p o o ype is used o apply he i ual sensing algo i hms and e alua e
he ob ained es ima ions. An ine ial shake is ins alled on he op o he p o o ype, wi h a o a ional
base, o gene a e di e en ex e nal o ces in di e en di ec ions. The esul s ob ained in he pe o med
es s a e analyzed o de e mine he mos e icien senso con igu a ions capable o ob aining accu a e
es ima es. Resul s show ha i is possible o ob ain accu a e s ain es ima ions a unmeasu ed poin s
o a s uc u e unde an unknown loading condi ion, using measu ed s ain da a om a se o poin s
and a su icien ly accu a e FE model as inpu and applying he augmen ed Kalman il e o he
leas -squa es s ain es ima ion in combina ion wi h modal unca ion and expansion echniques.
Keywo ds:
s uc u al heal h moni o ing; i ual sensing; Kalman il e ; augmen ed Kalman il e ;
leas squa es es ima ion; s ain i ual senso
1. In oduc ion
S uc u e heal h moni o ing (SHM) in ol es moni o ing s uc u es o de e mine hei
cu en condi ion. The use o SHM sys ems inc eases he sa e y o s uc u al acili ies
and allows he op imiza ion o he main enance ac ions, p edic ing he emaining use ul
li e o c i ical componen s and de ec ing anomalies ha may indica e he p esence o
damage [
1
]. SHM sys ems equi e measu ed da a om he s uc u e using senso s, bu i is
no always possible o ins all all he necessa y senso s a all he poin s o in e es , ei he
o echnical o economic easons. Vi ual sensing (VS) allows ob aining measu es om a
sys em, no di ec ly om physical senso s, bu using da a in e ence om o he senso s [
2
].
The use o s. in SHM sys ems esul s o in e es akes place when i is necessa y o ob ain
measu emen da a a poin s whe e i is no echnically easible o loca e a eal senso , o
when i is necessa y o ob ain measu emen s a a la ge numbe o loca ions, equi ing a
senso ne wo k ha is oo ex ensi e [
3
]. In conclusion, he use o s. o e s echnical and
economic ad an ages.
VS echniques can be classi ied in o wo main g oups: da a-d i en echniques and
model-based echniques [
4
]. Model-based echniques equi e a physics-based model ca-
pable o eplica ing he beha io o he moni o ed sys em. The model-based me hods
can be u he classi ied in o wo g oups: s ochas ic, whe e he sys em unce ain ies a e
conside ed; and de e minis ic, whe e he sys em unce ain ies a e no conside ed [5,6].
Senso s 2023,23, 4706. h ps://doi.o g/10.3390/s23104706 h ps://www.mdpi.com/jou nal/senso s
Senso s 2023,23, 4706 2 o 19
Neu al ne wo ks (NN) a e commonly used in he da a-d i en s. app oaches. The NN
a e a i icial in elligence algo i hms ha consis o complex ne wo ks o nodes (neu ons)
adjus ed using aining da a, which ela es he p o ided inpu da a wi h he desi ed ou pu s.
A i icial neu al ne wo ks (ANNs) [
7
] and Con olu ional neu al ne wo ks (CNNs) [
8
] ha e
been used o s. applica ions. ANNs a e simple , because inpu s a e p ocessed only in
he o wa d di ec ion, while CNNs a e mo e complex because hey use mul iple ypes o
laye s o p ocessing he p o ided inpu da a.
S ochas ic s. algo i hms ha e been commonly used o he es ima ion o s a es. One
o he mos known s ochas ic es ima ion algo i hms is he Kalman il e (KF), a physics
model-based algo i hm p oposed by R. Kalman in 1960 [
9
]. The KF uses a s a e-space model
o he sys em o make s a e p edic ions (mean and co a iance) based on in o ma ion om
he p e ious s a es. The KF is hen a Bayesian es ima o [
10
]. Inpu da a om eal senso s
a e used o co ec he p edic ions and o upda e he algo i hm pa ame e s. Some examples
o he use o he KF o s ain es ima ions a e ound in [
11
,
12
]. An implemen a ion o he KF
wi h an augmen ed s a e-space model (which es ima es he inpu s oge he wi h he s a es
o he sys em) was i s p oposed in 1969 by B. F iedland, o pe o m a s a e es ima ion
wi h unknown inpu s [
13
]. In 2010, E. Lou ens e al. used he KF wi h he augmen ed
s a e-space o dynamic o ce iden i ica ion, and he ollowing yea hey consolida ed he
so-called augmen ed Kalman il e (AKF), which has been used in la e publica ions [
14
].
O he a ian s o he KF o nonlinea sys ems ha e been p oposed: o example, he Ex ended
Kalman il e (EKF) [15], ha pe o ms a linea iza ion o he es ima ed mean and co a iance
o each ime s ep, o he Unscen ed Kalman il e (UKF) [
16
], ha a oids linea iza ion by
applying an unscen ed ans o ma ion o he es ima ed mean and co a iance. An al e na i e
o he Kalman il e is he Pa icle il e (PF) [
17
], which a e also s ochas ic Bayesian es ima o s.
Fo each ime s ep, he PF gene a es mul iple andom es ima ions (pa icles) using Mon e
Ca lo simula ions. A weigh is assigned o each pa icle, and he closes pa icles o he
obse a ion measu emen s a e mo e weigh ed o he ollowing ime-s eps.
De e minis ic s. algo i hms ha e been used o o ce and s ain es ima ion [
3
,
18
,
19
].
In his a icle, he leas -squa es s a e es ima ion (LSSE) is used. This me hod uses a Moo e-
Pen ose pseudoin e se (a gene aliza ion o he ma ix in e se which allows o ob ain he
pseudo-in e se ma ix o a non-squa ed ma ix [
20
]) o ob ain he leas -squa es solu ion
o he unknown s ains [
21
]. Unlike he p obabilis ic me hods, such as he p e iously
desc ibed Kalman il e s, he LSSE does no use he in o ma ion o p e ious s a es and does
no upda e i s in e nal pa ame e s o imp o e he es ima ion.
In his a icle, bo h s ochas ic and de e minis ic model-based algo i hms a e es ed.
The classic KF and he AKF a e used as examples o s ochas ic me hods and he LSSE is used
as example o de e minis ic me hod. The AKF has been chosen because i is speci ically
designed o wo k wi hou in o ma ion o he ex e nal o ces (which is o g ea in e es o
he wo k de eloped in his a icle), while he KF has been chosen o compa e i wi h he
AKF unde he unknown ex e nal o ces condi ion. On he o he hand, he LSSE has been
chosen due o i s simplici y. EKF and UKF ha e been disca ded because i is no in ended
o wo k wi h nonlinea models; meanwhile, he PF has been disca ded due o i s much
highe compu a ional complexi y. NN algo i hms a e no used in his wo k because i is
in ended o a oid p o iding subs an ial amoun s o aining da a.
In ecen yea s, Kalman il e and a ian s [
22
,
23
] and de e minis ic algo i hms [
24
,
25
]
ha e been used in SHM sys ems applied o s uc u al acili ies, such as wind u bines o
b idges. In [
22
], he AKF is applied in a wind u bine o es ima e he s a e o he s uc u e
and he ex e nal wind o ces, using he u bine speed and he gene a o o que, oge he
wi h accele ome e da a, as inpu senso s. In [
23
], he KF algo i hm is used o damage
de ec ion in ai c a ames and b idges, using accele ome e s as inpu senso s. In [
24
],
modal expansion is used o s ess and s ain es ima ion in an o sho e s uc u e p o o ype,
using accele ome e s and s ain gauges as inpu senso s. In [
25
], modal expansion is used
o s ain es ima ion in a monopile o sho e wind u bine, using accele ome e s and s ain
gauges as inpu senso s oo.
Senso s 2023,23, 4706 3 o 19
In o de o wo k wi h s ain measu emen s using he men ioned algo i hms, as well as
o ob ain s ain es ima es om hem, i is necessa y o use he modal expansion/ educ ion
me hod [
18
,
26
]. This me hod allows a numbe o s ain measu emen s o be ela ed o
displacemen s in a model, and ice e sa. Model educ ion me hods a e also used o ob ain
ligh e models om complex FE models, capable o being used by he s. algo i hms. S ain
es ima ion is o in e es due o i s ela ionship wi h a igue: by es ima ing he s ain a
c i ical poin s, he emaining use ul li e o a s uc u e due o he accumula ed a igue can
be es ima ed.
The main con ibu ion o his a icle is o es di e en s. algo i hms (s ochas ic and
de e minis ic) using eal da a ob ained om a wide a ie y o expe imen al es s, ob ained
om an o sho e wind u bine scaled p o o ype. Fo each selec ed algo i hm, di e en inpu
senso con igu a ions ha e been es ed unde di e en ypes o ex e nal o ces applied in
di e en di ec ions (using an elec omagne ic shake ins alled on a o a ing base on he op
o he p o o ype), simula ing a iable loads on he p o o ype. The s. algo i hms a e es ed
wi hou measu ing he applied o ces, inc easing he di icul y o he s udy.
This a icle is o ganized as ollows: in Sec ion 2, he modeling p ocesses, he i ual
sensing algo i hms used, and he use case a e desc ibed. In Sec ion 3, he ob ained expe i-
men al esul s a e shown. In Sec ion 4, he esul s a e discussed and in Sec ion 5, he inal
conclusions a e p esen ed.
2. Ma e ials and Me hods
2.1. Sys em Modeling
In his subsec ion, he heo e ical bases used on sys em modeling a e desc ibed: mass-
dampe -sp ing equa ion, s a e-space o mula ion, and model disc e iza ion. The selec ed
model educ ion me hod, modal unca ion, is also desc ibed.
2.1.1. Fini e Elemen Model
A Fini e Elemen (FE) linea model o he moni o ed s uc u e is c ea ed. Geome y,
cons uc ion de ails and bounda y condi ions mus be aken in o accoun du ing he model
c ea ion. Ma hema ically, a FE model is de ined by he mass-dampe -sp ing second o de
di e en ial Equa ion (MCK) Equa ion (1), ha is able o desc ibe he dynamical beha io o
he model o e ime.
M..
q( )+CD
.
q( )+Kq( )= ( )(1)
Wi h n being he numbe o deg ees o eedom (DoFs) o he model,
q
( ) is he
displacemen ec o (wi h n
×
1 dimension),
M
,
CD
and
K
a e he s i ness, damping and
mass ma ices, espec i ely (wi h n
×
ndimension), and
( ) is he ex e nal o ces ec o
(wi h n×1 dimension).
The FE models o complex s uc u es con ain a la ge numbe o deg ees o eedom
(DoF), which implies ha a big p ocessing capaci y and la ge amoun s o ime a e needed
o wo k wi h hem. To emedy his issue, educ ion me hods need o be applied.
2.1.2. Model Reduc ion
By applying model educ ion me hods o a ull FE model, i is possible o ob ain models
wi h a much smalle numbe o DoFs, which a e much ligh e in e ms o compu a ion. I is
a necessi y when i is in ended o wo k wi h FE models ha ep esen complex s uc u es
(usually made o housands o e en millions o DoFs) and i is equi ed o pe o m a high
numbe o calcula ions o e ime ( o example, a ansien simula ion) [
27
]. The educed
models can ep oduce he dynamic beha io o he s uc u e in limi ed anges o use.
Se e al model educ ion me hods can be ound in he bibliog aphy. Some examples a e
he Guyan s a ic condensa ion [
28
], he imp o ed educed sys em (IRS) [
29
], he C aig-
Bamp on componen mode syn hesis [
30
] and he modal unca ion [
31
]. In his a icle, he
modal unca ion is selec ed as model educ ion me hod because i is a me hod ha allows
o main ain a g ea p ecision om he ull model, wi hin a de ined ange o use, and due o
i s simplici y o applica ion [32].
Senso s 2023,23, 4706 4 o 19
To in oduce he modal unca ion me hod, i s i mus be explained ha a dynamic
model can be desc ibed h ough i s mode shapes, using he mode-shapes ma ix (
Φ
). Each
column o
Φ
co esponds o an eigen ec o (
ϕi
), associa ed o an eigen alue (
λi
). The
squa e oo o e e y eigen alue co esponds o a na u al equency o he sys em (
ωi
). The
Φ
- ans o ma ion implies a change o domain o he model, om he physical domain
(wi h ca esian base) o he modal domain.
Φ
can be ob ained sol ing he undamped
Equa ion (2), disca ding he i ial solu ion
Φ
=
0
.
Φ
is conside ed mass-no malized when
exp ession (3) is sa is ied.
(K−λM)Φ=K−ω2MΦ=0(2)
ΦTMΦ=I(3)
In i s ull o m,
Φ
con ains as many mode shapes as DoFs o he ull model, bu i is
possible o educe he model emo ing he modes ou o he equency ange o in e es
(modal unca ion). Fo a knumbe o modes o in e es ,
Φ
is educed o
ΦK
(4), wi h i s
dimension educed o n×k.
Φ(n,k)=[ϕ1,ϕ2. . . ϕk](4)
Th ough he
Φ
- ans o ma ion, he dynamic Equa ion (1) can be ans o med in o
he gene alized dynamical Equa ion (5), whe e
z
( ) is he ec o o modal displacemen s
(also known as gene alized displacemen s), ob ained wi h he ans o ma ion
q
( ) =
ΦK
z
( ). Equa ion (5) can also be exp essed as (6),
ΦKTMΦK
being an iden i y ma ix,
Σ
a
diagonal ma ix con aining he damping a ios (
ξ
) associa ed wi h each equency, and
Ω he diagonal ma ix wi h he na u al equencies o he model (ω).
ΦkTMΦk
..
z( )+ΦkTCDΦk
.
z( )+ΦkTKΦkz( )=ΦkT ( )(5)
..
z( )+2ΣΩ .
z( )+Ω2z( )=ΦkT ( )(6)
2.1.3. S a e-Space Model
A MCK model can be desc ibed as a s a e-space sys em (7), ha consis s o wo
equa ions: he s a e Equa ion (abo e) and he ou pu Equa ion (below).
x
is he s a e ec o ,
wi h 2n
×
1 dimension. As shown in (8), he s a e ec o con ains he displacemen s and he
eloci ies o each DoF.
u
is he inpu ec o , and wi h n
×
1 dimension, con ains he possible
ex e nal inpu o each DoF.
A
and
B
a e he s a e and inpu ma ices, espec i ely. As i
seen in (9) and (10), he dimensions o hese ma ices a e 2n
×
2nand 2n
×
n, espec i ely.
The elemen s o he ou pu equa ion, he ou pu ec o yand he ou pu and eed h ough
ma ices
C
and
D
, change acco ding o he desi ed ou pu a iables. S a e-space no a ion is
equi ed o implemen he model in Kalman il e s and a ian s.
.
x=Ax +Bu
y=Cx +Du (7)
x=q
.
q(8)
A=0 I
−M−1K−M−1CD(9)
B=0
M−1(10)
To use he s a e-space model in a disc e e- ime app oach, he Aand Bma ices mus be
disc e ized.
Ad
(11) and
Bd
(12) a e he disc e ized e sions o he s a e-space model ma ices.
Ad=eA∆ (11)
Senso s 2023,23, 4706 5 o 19
Bd=A−1(Ad−I)B(12)
2.2. Vi ual Sensing Algo i hms
In his subsec ion, he selec ed s. algo i hms in his a icle a e desc ibed: he Kalman
il e , he Augmen ed Kalman il e and he leas -squa es s ains es ima ion. The obse -
abili y condi ions o each algo i hm a e also desc ibed.
2.2.1. Kalman Fil e
The KF is a Bayesian ecu si e algo i hm used o es ima e he hidden s a es o a sys em.
A s a e-space model o he sys em is used o make p edic ions o he s a es, and in o ma ion
coming om a limi ed numbe o eal senso s is used o co ec he p edic ions.
The KF is an algo i hm o s ochas ic na u e ha manages gaussian unce ain ies
associa ed wi h he used model and wi h he measu emen s.
Q
(13) is he co a iance ma ix
o he model (wi h 2n
×
2n dimension) and
R
(14) is he co a iance ma ix o he inpu
senso s (wi h
×
dimension, being he numbe o inpu senso s). Assuming ha he
s a es and he measu emen s a e no co ela ed wi h each o he , he ma ices
Q
and
R
a e simpli ied o diagonal ma ices, whe e each alue o he diagonal co esponds o he
unce ain y associa ed wi h each s a e (q) and wi h each senso inpu ( ), espec i ely. The
Qma ix mus be disc e ized when used in a disc e e- ime Kalman il e (15).
Q=diag(q1, q2. . . , q2n)(13)
R=diag( 1, 2. . . , )(14)
Qd=(AdQAdT∆ (15)
In absence o ex e nal o ce measu emen s, he KF is implemen ed as ollows: s a es
p edic ion (16), co a iance p edic ion (17), Kalman gain de e mina ion (18), s a es p edic-
ion upda e (19) and co a iance p edic ion upda e (20).
x =Ax −1(16)
P =AP −1AT+Q(17)
K =P HTHP HT+R−1(18)
x upda ed =x +K (z −Hx )(19)
P upda ed =P −K HP (20)
The inco po a ion o he eal senso measu emen s in o he il e (desc ibed in s a es
p edic ion upda e s ep) is pe o med wi h he measu emen ma ix (
H
). This ma ix ela es
each measu emen wi h hei co esponding s a es. I has
×
2ndimension, being 2n he
numbe o s a es o he sys em and he numbe o inpu senso s.
Wi h a measu emen da a ec o
z
( ) con aining xnumbe o s ain gauges and ynum-
be o accele ome e s, he
H
ma ix is buil as seen in (21). To ela e he s ain gauge da a o
he modal s a es, he modal s ains a e ob ained om he FE model. These can be ob ained
om a modal analysis o he FE model o he s uc u e, compiling he s ain alue (
ε
)
ob ained in each gauge (1 o x) o each mode (1 o n). To ela e he accele ome e da a o
he modal s a es, he co esponding ows o modal M,Cand Kma ices a e used.
Because o he ex e nal o ce measu emen s a e no a ailable, no ela ion be ween
he o ce and accele a ion measu emen s is implemen ed (in he case ha ex e nal o ce
measu emen s we e a ailable, hese would be ela ed o he accele a ion measu emen s
h ough a
J
ma ix (22)). Because o his, he unce ain y o accele ome e measu emen s is
expec ed o be g ea e .
Senso s 2023,23, 4706 6 o 19
H=
ε1,1· · · ε1,n
.
.
.....
.
.
εx,1· · · εx,n
0· · · 0
.
.
.....
.
.
0· · · 0
−M1,1
−1K1,1· · · −M1,n
−1K1,n
.
.
.....
.
.
−My,1
−1Ky,1· · · −My,n−1Ky,n
−M1,1
−1C1,1· · · −M1,n
−1C1,n
.
.
.....
.
.
−My,1
−1Cy,1· · · −My,n−1Cy,n
(21)
J=
0· · · 0
.
.
.....
.
.
0· · · 0
M1,1
−1· · · M1,n
−1
.
.
.....
.
.
My,1
−1· · · My,n−1
(22)
In a KF, obse abili y can be de ined as he capaci y o he algo i hm o ob ain enough
in o ma ion om he eal sys em ( h ough he inpu senso s and he obse a ion ma ix)
o be able o es ima e all he s a es. To de e mine i a KF is obse able, he obse abili y
ma ix
O
(23) is calcula ed using he anspose o
A
. Only i he ank o
O
is equal o 2n
( he numbe o s a es o he model) is he KF is ully obse able.
O=
ATH0
ATH1
.
.
.
ATH2n−1
(23)
2.2.2. Augmen ed Kalman Fil e
The AKF is a a ian o he KF in which he ex e nal o ces o e he sys em a e
conside ed addi ional s a es o he model. Thanks o his ea u e, his il e does no need
he ex e nal o ce applied on he moni o ed sys em as inpu . The AKF uses an augmen ed
s a e-space model o he sys em ha combines he
A
and
B
ma ices o he s a e-space
model in a single ma ix
A*
(24) wi h (2n+n
)
×
(2n+n
) dimension (n
being he numbe
o expec ed ex e nal o ces), and an augmen ed ec o o s a es
x
* (25) ha combines he
displacemen s, hei i s de i a i es and he ex e nal inpu o ces ( esul ing in a 2n+n
dimension). The disc e iza ion o A* is shown in (26).
A*=A B
0 0(24)
x*=
q
.
q
u
(25)
Ad*=AdBd
0 I (26)
The unknown inpu is modeled as a ze o-mean andom walk model, so he co a iance
ma ix o he model
Q
mus be augmen ed o (2n+n
)
×
(2n+n
) dimension by adding a
e m ela ed o he unce ain y associa ed o he ex e nal o ces (27).
Q*=Qd0
0 Qu(27)
An augmen ed obse a ion ma ix
H*
(28) mus be de ined by combining he obse a ion
ma ix
H
(21) and he inpu obse a ion ma ix
J
(22), esul ing in a ma ix o
×
(2n+n
)
dimension.
Senso s 2023,23, 4706 7 o 19
H*=H J(28)
In he AKF, obse abili y has he same meaning as in he classical KF. To de e mine i
an AKF is obse able, he obse abili y ma ix
O*
(29) mus be calcula ed. Only i he ank
o O* is equal o 2n +n is he AKF is ully obse able.
O*=
A*TH*0
A*TH*1
.
.
.
A*TH*2n−1
(29)
2.2.3. Leas -Squa es S ain Es ima ion (LSSE)
The LSSE is a de e minis ic i ual sensing algo i hm ha uses a ma ix gene alized
in e sion o ob ain he leas squa es solu ion o he unknown s ains. The Moo e-Pen ose
pseudoin e se [
20
] and he Modal Expansion [
24
] a e used o his pu pose. This me hod
allows ob aining s ain es ima es a unmeasu ed poin s bo h in he p esence and absence
o dynamic e ec s.
The linea equa ion is s a ed by ela ing he measu ed s ain and he modal displace-
men s o he sys em (30). In a linea sys em, displacemen s
x
( ) and measu ed s ains
zi
( )
a e linea ly ela ed h ough he modal s ain ma ix
Gi
(wi h g
×
mdimension, gbeing he
numbe o s ain measu emen s, and m he numbe o modal displacemen s).
zi( )=Gix( )(30)
Using he same s a emen , s ain i ual measu emen s
z s
( ) can be ob ained om he
modal displacemen s, h ough he modal s ain ma ix
G s
(wi h o
×
mdimension, obeing
he numbe o i ual s ain senso s, and m he numbe o modal displacemen s) (31).
z s( )=G sx( )(31)
Using he pseudoin e se o
Gi
, bo h s a emen s can be combined o ob ain s ain
i ual measu emen s om a se o eal s ain measu emen s (32).
z s( )=G sGi+z( )(32)
I he numbe o s ain measu emen s gis equal o he numbe o modal displacemen s
m, he s a emen (31) is de e mined, and he solu ion is ound by he LSSE. I gis highe han
m, he s a emen is o e de e mined. I , on he con a y, gis lowe han m, he s a emen (31)
is unde de e mined. In bo h cases, he LSSE gi es a bes - i app oxima ion o he solu ion.
To p o ide a good app oxima ion o he solu ion, he condi ion numbe o he ma ix
Gi
mus be close o 1. I he condi ion numbe o
Gi
is high, he s a emen (31) is ill condi ioned
and signi ican e o s can be expec ed in he solu ion.
2.3. Vi ual Sensing Implemen a ion
The selec ed s. algo i hms a e es ed on a use case de ined in Sec ion 2.4. Fi s , an FE
model o he use case is c ea ed. This model is used o choose he loca ion o he senso s
(s ain gauges and accele ome e s) in he eal p o o ype. Measu emen da a ob ained om
he senso s is i s used o adjus and alida e he model, and hen o eed he s. algo i hms.
The ob ained es ima ions a e compa ed o he equi alen measu emen da a o e alua e
he pe o mance o he s. algo i hms unde he di e en condi ions. The en i e p ocess is
summa ized in he lowcha shown in Figu e 1.
Senso s 2023,23, 4706 8 o 19
Senso s 2023, 23, x FOR PEER REVIEW 8 o 20
e alua e he pe o mance o he s. algo i hms unde he di e en condi ions. The en i e
p ocess is summa ized in he lowcha shown in Figu e 1.
Figu e 1. Flowcha o he p ocess ollowed o implemen and es s. algo i hms.
2.4. Use Case
In his subsec ion, he use case, he ins alled senso s, and he modelling p ocess a e
desc ibed.
2.4.1. P o o ype Desc ip ion
The use case is a scaled wind u bine owe p o o ype ins alled on a jacke - ype s uc-
u e, which is ixed o he g ound (Figu e 2). The main speci ica ions o he p o o ype can
be seen in Table 1. An elec omagne ic ine ial shake , conside ed as pa o he sys em, is
placed on op o he p o o ype a ached o a o a ing pla o m o exci e he s uc u e in
di e en di ec ions (Figu e 3) and equency componen s (<25 Hz). The speci ica ions o
he shake used can also be seen in Table 1.
Figu e 1. Flowcha o he p ocess ollowed o implemen and es s. algo i hms.
2.4. Use Case
In his subsec ion, he use case, he ins alled senso s, and he modelling p ocess
a e desc ibed.
2.4.1. P o o ype Desc ip ion
The use case is a scaled wind u bine owe p o o ype ins alled on a jacke - ype
s uc u e, which is ixed o he g ound (Figu e 2). The main speci ica ions o he p o o ype
can be seen in Table 1. An elec omagne ic ine ial shake , conside ed as pa o he sys em,
is placed on op o he p o o ype a ached o a o a ing pla o m o exci e he s uc u e in
di e en di ec ions (Figu e 3) and equency componen s (<25 Hz). The speci ica ions o
he shake used can also be seen in Table 1.
Senso s 2023, 23, x FOR PEER REVIEW 8 o 20
e alua e he pe o mance o he s. algo i hms unde he di e en condi ions. The en i e
p ocess is summa ized in he lowcha shown in Figu e 1.
Figu e 1. Flowcha o he p ocess ollowed o implemen and es s. algo i hms.
2.4. Use Case
In his subsec ion, he use case, he ins alled senso s, and he modelling p ocess a e
desc ibed.
2.4.1. P o o ype Desc ip ion
The use case is a scaled wind u bine owe p o o ype ins alled on a jacke - ype s uc-
u e, which is ixed o he g ound (Figu e 2). The main speci ica ions o he p o o ype can
be seen in Table 1. An elec omagne ic ine ial shake , conside ed as pa o he sys em, is
placed on op o he p o o ype a ached o a o a ing pla o m o exci e he s uc u e in
di e en di ec ions (Figu e 3) and equency componen s (<25 Hz). The speci ica ions o
he shake used can also be seen in Table 1.
Figu e 2. Gene al iew o he p o o ype and he conc e e ounda ion.
Senso s 2023,23, 4706 9 o 19
Senso s 2023, 23, x FOR PEER REVIEW 9 o 20
Figu e 2. Gene al iew o he p o o ype and he conc e e ounda ion.
Figu e 3. Ine ial shake Da a Physics IV47 a ached on op o he p o o ype.
Table 1. Main specs o he p o o ype and he shake .
Fea u e Value
Towe + nacelle weigh 42 kg
Jacke weigh 13.5 kg
Shake + suppo weigh 27 kg
To al weigh
82.5 kg
Towe heigh 1790 mm
Jacke heigh 1300 mm
To al heigh 3090 mm
Ma e ial S eel
Suppo s Fixed o a conc e e base
Shake model Da a Physics IV47
Ine ial mass 14.5 kg
Max sinus o ce (peak) 250 N
To al shake mass 21 kg
Shake main mode eq. 20 Hz
2.4.2. FE Model and Model Reduc ion
A FE model o he p o o ype is buil based on he 3D CAD o he s uc u e. The shake
and i s suppo , including he bushing, a e simpli ied o an equi alen poin mass loca ed
a he mass cen e o he eplaced componen s and a ached o he s uc u e. The beha io
o he bushing has been es ed in he equency ange o in e es (0 o 25Hz), e i ying i s
linea i y. The pla o m has been designed o keep he mass cen e o he o a ing compo-
nen s in he o a ing axis, so he sys em can be conside ed in a ian . The bol ed join s
p esen in he p o o ype a e also simpli ied using bonded con ac s. Due o he owe and
nacelle o he wind u bine p o o ype being hin s eel p o ile componen s, shell- ype ele-
men s ha e been used o educe he o al numbe o elemen s in he mesh. In he jacke
suppo s uc u e, solid elemen s ha e been used. The p ima y ea u es o he FE model
can be seen in Table 2, and he FE model o he p o o ype can be seen in Figu e 4.
Figu e 3. Ine ial shake Da a Physics IV47 a ached on op o he p o o ype.
Table 1. Main specs o he p o o ype and he shake .
Fea u e Value
Towe + nacelle weigh 42 kg
Jacke weigh 13.5 kg
Shake + suppo weigh 27 kg
To al weigh 82.5 kg
Towe heigh 1790 mm
Jacke heigh 1300 mm
To al heigh 3090 mm
Ma e ial S eel
Suppo s Fixed o a conc e e base
Shake model Da a Physics IV47
Ine ial mass 14.5 kg
Max sinus o ce (peak) 250 N
To al shake mass 21 kg
Shake main mode eq. 20 Hz
2.4.2. FE Model and Model Reduc ion
A FE model o he p o o ype is buil based on he 3D CAD o he s uc u e. The
shake and i s suppo , including he bushing, a e simpli ied o an equi alen poin mass
loca ed a he mass cen e o he eplaced componen s and a ached o he s uc u e. The
beha io o he bushing has been es ed in he equency ange o in e es (0 o 25Hz),
e i ying i s linea i y. The pla o m has been designed o keep he mass cen e o he
o a ing componen s in he o a ing axis, so he sys em can be conside ed in a ian . The
bol ed join s p esen in he p o o ype a e also simpli ied using bonded con ac s. Due o
he owe and nacelle o he wind u bine p o o ype being hin s eel p o ile componen s,
shell- ype elemen s ha e been used o educe he o al numbe o elemen s in he mesh. In
he jacke suppo s uc u e, solid elemen s ha e been used. The p ima y ea u es o he FE
model can be seen in Table 2, and he FE model o he p o o ype can be seen in Figu e 4.
Senso s 2023,23, 4706 16 o 19
Table 14. E alua ion o he esul s shown in Figu e 8.
F XY Noise X-1-90 Y-1-90
KF 19.8/97.4 13.7/99.1
AKF 20.0/97.4 13.9/99.0
LSSE 18.2/97.3 15.9/98.9
Table 15. E alua ion o he esul s shown in Figu e 9.
F VAR 5 Hz X-1-90 Y-1-90
KF 12.3/92.8 8.6/99.0
AKF 19.5/99.0 4.9/98.9
LSSE 16.8/99.3 4.1/98.7
Table 16. E alua ion o he esul s shown in Figu e 10.
F X HIT X-1-90 Y-1-90
KF 11.6/99.5 2.1/93.7
AKF 11.3/99.5 1.8/93.3
LSSE 11.6/99.6 3.2/92.4
Senso s 2023, 23, x FOR PEER REVIEW 16 o 20
sys em beha io , as is p edic ed by senso loca ion me hods (such as he Modal Kine ic
Ene gy me hod, men ioned in Sec ion 2.4.4).
Acco ding o he ob ained esul s in he i ual senso s when o ces in X di ec ion a e
applied, he es ima es ob ained in he i ual senso s a e gene ally be e han when o ces
in he Y di ec ion o in combined di ec ions a e applied. This di e ence can be explained
by he ce ain lack o p ecision in he beha io o he FE model ( om which he educed
model has been ob ained) in he Y-di ec ion bending and he Z-di ec ion o sion. O he
senso con igu a ions es ed in he di e en s. algo i hms, he bes -pe o ming one is
con igu a ion 3. This con igu a ion uses he 3X, 3Y, 4X, 4Y, 5X and 5Y gauges as inpu
senso s. Con igu a ion 8, ha uses he same inpu gauges bu adding an accele ome e ,
also pe o ms well when is used wi h he AKF.
F om he da a ob ained om he expe imen s ca ied ou in his a icle, i can be con-
cluded ha , in e ms o obus ness, he LSSE is p e e able because, unlike Kalman il e s,
i does no depend on uning pa ame e s. Among he Kalman il e s, he AKF can be con-
side ed mo e obus han he KF because, unde condi ions o unmeasu ed o ces, i has
a s able pe o mance when accele ome e s a e used as inpu senso s.
Some examples o he ob ained esul s applying di e en inpu o ces, using con ig-
u a ion 3 (inpu gauges X-2-90, Y-2-90, X-3-90, Y-3-90, X-4-90, Y-4-90, X-5-90, Y-5-90), a e
p o ided in Figu es 6–10. The alues co esponding o hese esul s a e summa ized in
Tables 12–16. The i ual senso s a e he gauges 1X ( i s column o he ables) and 1Y
(second column o he ables). The es ima ions ob ained wi h he KF, AKF and LSSE a e
compa ed wi h eal s ain da a a he same loca ion (indica ed as REF). The le alues in
he ables co espond o he pe cen age e o o he es ima ion, and he igh alues co -
espond o he PCC e o .
Figu e 6. Resul s ob ained applying a 5Hz sinusoidal o ce in X di ec ion.
Figu e 6. Resul s ob ained applying a 5Hz sinusoidal o ce in X di ec ion.
Senso s 2023, 23, x FOR PEER REVIEW 17 o 20
Figu e 7. Resul s ob ained applying a 15 Hz sinusoidal o ce in Y di ec ion.
Figu e 8. Resul s ob ained applying a ze o-mean whi e noise o ce in 45° di ec ion.
Figu e 9. Resul s ob ained applying a 5 Hz sinus o ce in a iable di ec ion along ime.
Figu e 7. Resul s ob ained applying a 15 Hz sinusoidal o ce in Y di ec ion.
Senso s 2023,23, 4706 17 o 19
Senso s 2023, 23, x FOR PEER REVIEW 17 o 20
Figu e 7. Resul s ob ained applying a 15 Hz sinusoidal o ce in Y di ec ion.
Figu e 8. Resul s ob ained applying a ze o-mean whi e noise o ce in 45° di ec ion.
Figu e 9. Resul s ob ained applying a 5 Hz sinus o ce in a iable di ec ion along ime.
Figu e 8. Resul s ob ained applying a ze o-mean whi e noise o ce in 45◦di ec ion.
Senso s 2023, 23, x FOR PEER REVIEW 17 o 20
Figu e 7. Resul s ob ained applying a 15 Hz sinusoidal o ce in Y di ec ion.
Figu e 8. Resul s ob ained applying a ze o-mean whi e noise o ce in 45° di ec ion.
Figu e 9. Resul s ob ained applying a 5 Hz sinus o ce in a iable di ec ion along ime.
Figu e 9. Resul s ob ained applying a 5 Hz sinus o ce in a iable di ec ion along ime.
Senso s 2023, 23, x FOR PEER REVIEW 18 o 20
Figu e 10. Resul s ob ained applying a hamme impac in X di ec ion.
Table 12. E alua ion o he esul s shown in Figu e 6.
F X 5 Hz X-1-90 Y-1-90
KF 3.6/99.4 178.6/61.6
AKF 7.7/99.4 187.3/60.2
LSSE 4.1/99.8 197.1/58.3
Table 13. E alua ion o he esul s shown in Figu e 7.
F Y 15 Hz X-1-90 Y-1-90
KF 195.6/95.4 7.3/99.9
AKF 209.3/95.2 7.0/99.9
LSSE 217.3/98.5 6.9/99.9
Table 14. E alua ion o he esul s shown in Figu e 8.
F XY Noise X-1-90 Y-1-90
KF 19.8/97.4 13.7/99.1
AKF 20.0/97.4 13.9/99.0
LSSE 18.2/97.3 15.9/98.9
Table 15. E alua ion o he esul s shown in Figu e 9.
F VAR 5 Hz X-1-90 Y-1-90
KF 12.3/92.8 8.6/99.0
AKF 19.5/99.0 4.9/98.9
LSSE 16.8/99.3 4.1/98.7
Table 16. E alua ion o he esul s shown in Figu e 10.
F X HIT X-1-90 Y-1-90
KF 11.6/99.5 2.1/93.7
AKF 11.3/99.5 1.8/93.3
LSSE 11.6/99.6 3.2/92.4
5. Conclusions
In his a icle, h ee di e en s. algo i hms ha e been es ed o ob ain i ual s ain
es ima ions unde unknown o ces. Twel e di e en senso con igu a ions ha e been
used unde 15 di e en dynamic loads. Two i ual s ain gauges ha e been implemen ed
Figu e 10. Resul s ob ained applying a hamme impac in X di ec ion.
5. Conclusions
In his a icle, h ee di e en s. algo i hms ha e been es ed o ob ain i ual s ain
es ima ions unde unknown o ces. Twel e di e en senso con igu a ions ha e been
Senso s 2023,23, 4706 18 o 19
used unde 15 di e en dynamic loads. Two i ual s ain gauges ha e been implemen ed
in he base o he owe o he p o o ype, and wo eal s ain gauges ha e been used as
e e ence senso s, o compa e he es ima ed da a ob ained om he i ual senso s wi h he
equi alen eal senso da a.
I has been e i ied ha , h ough he modal unca ion, a educed model can be used
o ob ain he esponse o a much mo e complex FE model (in a limi ed ange o equencies)
using a limi ed numbe o modes (which implies a signi ica i e educ ion in he numbe
o DoFs used). The AKF shows i sel o be be e han he classical KF in absence ex e nal
o ce measu emen s, especially when s ain and accele a ion measu emen s a e a ailable.
I only s ain measu emen s a e a ailable, he AKF and he LSSE pe o m simila ly, so, o
ob ain s ain i ual measu emen s, he LSSE may be p e e able due o i s simplici y.
The expe ience and esul s ob ained wi h he expe imen s p esen ed in his a icle can
be use ul when implemen ing s ain i ual senso s. Examples o applica ion can ange
om wind u bines (as in he case o his a icle) o many o he ypes o complex s uc u al
asse s, o example dis inc ypes o o sho e s uc u es, b idges, communica ion owe s o
e en la ge indus ial ames (such as indus ial p esses).
Se e al u u e lines o inqui y may con inue he wo k p esen ed in his a icle. On one
hand, i would be in e es ing o add eal- ime o ce measu emen s o allow compa ison s.
esul s wi h known o ces and wi h unknown o ces. Fo ces can also be es ima ed using s.
algo i hms. On he o he hand, i would also be in e es ing o ins all mo e accele ome e s
o he use case, in o de o be able o es mo e senso con igu a ions in he di e en s.
algo i hms. I would be also in e es ing o ins all gauges wi h an o ien a ion o 45
º
wi h
espec o he owe axis, wi h he aim o measu ing o sional s ain. Fu he mo e, i would
be in e es ing o apply he es ed s. algo i hms in o he use cases o a di e en na u e, o
check i s. es ima ions o compa able quali y can be ob ained in o he ypes o s uc u es.
Au ho Con ibu ions:
B.M. and J.B. concei ed he concep ual ideas and ca ied ou he expe imen al
wo k. I.S. and J.A. helped wi h he me hodology and wi h he imp o emen o he pe o mance o
he s. algo i hms. B.M. buil he necessa y so wa e and managed he expe imen al da a. B.M., J.B.,
I.S., U.L. and J.A. pa icipa ed in he w i ing o he a icle, and I.S., U.L. and J.A. e iewed he inal
manusc ip and made s yle sugges ions. All au ho s ha e ead and ag eed o he published e sion
o he manusc ip .
Funding:
The esea ch p esen ed in his wo k has been ca ied ou by Ike lan Resea ch Cen e ,
a cen e ce i ica ed as “Cen o de Excelencia Ce e a”. This wo k has been unded by CDTI,
dependen on he Spanish Minis e io de Ciencia e Inno ación, h ough he “Ayudas Ce e a pa a
cen os ecnológicos 2019” p og am, p ojec MIRAGED wi h expedien numbe CER-20190001.
Da a A ailabili y S a emen :
The da a p esen ed in his s udy a e a ailable on eques om he
co esponding au ho . The da a a e no publicly a ailable due o p i acy es ic ions.
Con lic s o In e es : The au ho s decla e no con lic o in e es in his wo k.
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