Strain Virtual Sensing for Structural Health Monitoring under Variable Loads

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−ω2MΦ=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 HTHP 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 sGi+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”. 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