Health-aware and fault-tolerant control of an octorotor UAV system based on actuator reliability

In . J. Appl. Ma h. Compu . Sci., 2020, Vol. 30, No. 1, 47–59 DOI: 10.34768/amcs-2020-0004 HEALTH–AWARE AND FAULT–TOLERANT CONTROL OF AN OCTOROTOR UAVSYSTEMBASEDONACTUATORRELIABILITY JEAN CARLO SALAZARa,*,ADRIÁN SANJUANa,FATIHA NEJJARIa,RAMON SARRATE a aPoly echnic Uni e si y o Ca alonia (UPC) Resea ch Cen e o Supe ision, Sa e y and Au oma ic Con ol (CS2AC) 10, Rambla San Neb idi, Te assa, Spain e-mail: {jean.salaza ,ad ian.sanjuan}@upc.edu, { a iha.nejja i, amon.sa a e}@upc.edu A majo goal in mode n ligh con ol sys ems is he need o imp o ing eliabili y. This wo k p esen s a heal h-awa e and aul - ole an con ol app oach o an oc o o o UAV ha allows dis ibu ing he con ol e o among he a ailable ac ua o s based on hei heal h in o ma ion. Howe e , i is wo h men ioning ha , in he case o ac ua o aul occu ence, a eliabili y imp o emen can come in o con lic wi h UAV con ollabili y. The e o e, sys em eliabili y sensi i i y is ede ined and modi ied o p e en uncon ollable si ua ions du ing he UAV’s mission. The p io i y gi en o each ac ua o is ela ed o i s impo ance in sys em eliabili y. Mo eo e , he p oposed app oach can econ igu e he con olle o compensa e ac ua o aul s and imp o e he o e all sys em eliabili y o delay main enance asks. Keywo ds: p ognos ics and heal h managemen , heal h-awa e con ol, aul - ole an con ol, eliabili y analysis, oc o o o , UAV. 1. In oduc ion Unmanned ae ial ehicles (UAVs) a e well-sui ed o a wide ange o mission scena ios, such as sea ch, escue, supe ision and inspec ion, among o he s. Redundancy in senso s, ac ua o s and all o he essen ial componen s o UAVs plays a i al ole in inc easing ligh sa e y and mission accomplishmen in he case o deg ada ion o aul / ailu e occu ence in hose componen s. In pa icula , he use o a mul icop e wi h eigh ac ua o s (oc ocop e ), ins ead o using a quadcop e , makes he UAV capable o main aining no mal ligh and accomplishing he mission despi e he occu ence o ailu es in one o mo e o o s/p opelle s. On he one hand, se e al econ igu a ion con ol echniques, such as gain-scheduled PID (Milhim e al., 2010) o sliding mode con ol (SMC) (Alwi and Edwa ds, 2006; Me heb e al., 2015) ha e been applied o compensa e he aul e ec . On he o he hand, echniques which hide he aul om he con olle poin o iew ha e also been used in aul - ole an con ol (FTC) when he sys em has ac ua o edundancy. In he wo k o Ro ondo *Co esponding au ho e al. (2015), an FTC in o a obus linea pa ame e a ying (LPV) poly opic amewo k was p oposed. A e iew o FTC echniques o quad o o s is p esen ed by Zhang e al. (2013). In he wo k o Cen e al. (2015), an ac i e and passi e FTC scheme based on aul es ima ion o a quad o o ac ua o is p oposed. Ano he popula echnique is con ol alloca ion (Johansen and Fossen, 2013). Howe e , o imp o e sys em pe o mance du ing UAVs mission ope a ions, i could be mo e app op ia e o a oid aul occu ence han ole a e hem. In his sense, a new pa adigm in which he use o bo h con ol and eliabili y heo ies has eme ged in e ms o heal h-awa e con ol (HAC). The eliabili y o UAVs and hei componen s agains aul s and ailu es is one o he mos impo an objec i es o he sa e y o c i ical sys ems. The aim o an HAC sys em is o modi y he con ol ac ions based on sys em eliabili y in o ma ion p o ided by a p ope online p ognos ics ool e en in he p esence o ac ua o /senso aul s. This leads o an inc ease in he ope a ion ime o he sys em (Salaza e al., 2015; 2016; Khelassi e al., 2011). © 2020 J.C. Salaza e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion-NonComme cial-NoDe i s license (h p://c ea i ecommons.o g/licenses/by-nc-nd/3.0/). 48 J.C. Salaza e al. The con ol o mul i o o sys ems and pa icula ly he oc o o o is a opic o in e es and a challenging p oblem due o hei unde -ac ua ed na u e and nonlinea dynamics. Se e al con ol echniques ha e been used, such as model p edic i e con ol (MPC) (Ra o e al., 2010; Liu e al., 2012; Abdolhosseini e al., 2013), PID (Rinaldi e al., 2014), o LQR (Ma ks e al., 2012; Adî and S oica, 2012). Thanks o i s ac ua o edundancy, he oc o o o has he po en ial o imp o e sa e y and eliabili y. Se e al app oaches ha e been p oposed and used o manage he edundancy o his kind o sys ems by dis ibu ing he con ol e o among a se o o o s. One app oach o achie e his goal consis s in using an op imal con ol design o shape in one s ep he closed-loop dynamics and ac ua o con ol dis ibu ion. Ano he echnique is sepa a ing he con ol ask om he e o dis ibu ion one. In his case, a sepa a e con ol alloca ion module is in oduced in he closed loop o dis ibu e he con ol e o among he ac ua o s (Khelassi e al., 2011; Du ham, 1993; Bodson, 2002; Salaza e al., 2015; 2017). The con ol alloca ion app oach allows he aul y sys em o be accommoda ed wi hou he need o modi y he con olle . This wo k p esen s he ad an ages o componen and sys em eliabili y in eg a ion in o he con ol by a con ol alloca ion scheme. The p oposed app oach is applied o a mul i o o UAV sys em. Such kind o sys ems has ac ua o s edundancy which allows he design o con olle s ha can op imize he dis ibu ion o he con ol e o in such a way ha he eliabili y o he sys em is p ese ed o e en ex ended. The objec i e is o combine a de e minis ic pa ela ed o sys em dynamics and a s ochas ic pa ela ed o sys em eliabili y. The esul ing scheme p o ides con ol pe o mance and p ese es sys em eliabili y in aul y and non- aul y scena ios. The p oposed app oach can also be used o schedule main enance asks based on he eliabili y o he ac ua o s. This abili y is e y use ul in he case o sys ems whose main enance ac ions ep esen s a cos ly p ocess ei he by hei down imecos o hei economic cos in componen s, o labo cos , among o he s. The case s udy is an oc o o o UAV sys em ha has eigh p opelle s in he con igu a ion I (Fig. 1). Fou p opelle s can o a e in a clockwise di ec ion, while he emaining ones can o a e an i-clockwise. The oc o o o is mo ed by changing he o o speeds. Fo example, by inc easing o dec easing oge he he speeds o he eigh p opelle s, a e ical mo ion is achie ed. Changing only he speeds o he p opelle s si ua ed a he opposi e loca ions p oduces ei he oll o pi ch o a ion, coupled wi h he co esponding la e al mo ion. Finally, yaw o a ion esul s om he di e ence in he coun e - o que be ween each pai o p opelle s. Addi ionally, he oc o o o has ac ua o edundancy ha can o e a deg ee o o e -ac ua ion as i can wo k wi h a leas ou p opelle s o ming a quad o o s uc u e. The pape is o ganized as ollows. The dynamic modeling o he oc o o o UAV sys em is de ailed in Sec ion 2. In Sec ion 3, he con olle design o he UAV sys em is p esen ed. Sec ion 4 add esses he p oblem o con ol e o edis ibu ion. In Sec ion 5, eliabili y modeling is p esen ed. The HAC scheme is p esen ed in Sec ion 6. The simula ion esul s a e p esen ed in Sec ion 7 and, inally, some conclusions a e gi en. 2. Oc o o o dynamics To desc ibe he dynamics o a mul i o o , i is necessa y o de ine wo ames in which i will ope a e: he ine ial ame and he body ame. The o me {I}is s a ic and ep esen s he e e ence o he mul i o o while he la e {B}is de ined by he o ien a ion o he mul i o o and i is si ua ed in i s cen e o mass. The wo ames a e ela ed by he o a ion ma ix (1). RI B ans o ms a ec o in body e e ence o a ec o in ine ial e e ence. In his case, he Eule angles, namely, he oll angle (φ), he pi ch angle (θ) and he yaw angle (ψ), a e used o model his o a ion ollowing he sequence z−y−x: RI B=⎡ ⎣ c(ψ)c(θ)c(ψ)s(θ)s(φ)−s(ψ)c(φ) s(ψ)c(θ)s(ψ)s(θ)s(φ)+c(ψ)c(φ) −s(θ)c(θ)s(φ) c(ψ)s(θ)c(φ)+s(ψ)s(φ) s(ψ)s(θ)c(φ)−c(ψ)s(φ) c(θ)c(φ)⎤ ⎦,(1) whe e s(·)and c(·)deno e sin(·)and cos(·), espec i ely. A simila no a ion will be used o an(·)which will be deno ed by (·). The dynamics o a mul i o o can be de ined using he New on and Eule equa ions (Mahony e al., 2012) desc ibing he ansla ion and o a ion o he igid body: ˙ ξI= I,(2) ˙ I=1 m( I),(3) ˙ ηI=WηωB,(4) ˙ ωB=1 J(τB−ωB×JωB),(5) whe e I=[ x y z]Tis he linea speed ec o in he ine ial ame, ξI=[xyz]Tis he posi ion ec o in he ine ial ame, η=[φθψ]Tis he o ien a ion ec o , ωB=[pq ]Tis he body angula eloci y ec o , mis he mass o he ehicle, Jis he ine ia enso , I and τB ep esen he ex e nal o ces and o ques applied o he UAV and Wη, which ep esen s he ans o ma ion ma ix o angula eloci ies om he body ame o he Heal h-awa e and aul - ole an con ol o an oc o o o UAV sys em ... 49 ine ial ame, gi en by (Blakelock, 1991) Wη=⎡ ⎢ ⎢ ⎣ 1s(φ) (θ)c(φ) (θ) 0c(φ)−s(φ) 0s(φ) c(θ) c(φ) c(θ) ⎤ ⎥ ⎥ ⎦ .(6) The ex e nal o ces in e ac ing wi h he ehicle a e he li o he o o s (T), he ansla ional d ag and he g a i y (g). The ex e nal o ques a e he o ques gene a ed by he mo o s (τx,τyand τz) and he o a ional d ag. The model has been de eloped unde he ollowing assump ions (F eddi e al., 2011): • The s uc u e o he UAV is symme ical. • The body and p opelle s a e igid. • The ee s eam ai eloci y is ze o. • The mo o dynamics can be neglec ed since hey a e ela i ely as . • The blade lexibili y can be neglec ed conside ing ha i is ela i ely small. • The ine ia enso o he oc o o o body is diagonal J= diag(Jxx,J yy,J zz). • The ine ia o he oc o o o body is much la ge han ha o he p opelle (i includes he o a ing pa s o he o o ), so JJp op. • T ansla ional and o a ional d ags a e negligible. The mul i o o UAV model is ob ained by expanding Eqns. (2)–(5), doing he ans o ma ions om he body o he ine ial ame (1), and applying he p e ious assump ions: ˙xI= x,(7) ˙yI= y,(8) ˙zI= z,(9) ˙ x=1 m[c(φ)s(θ)c(ψ)+s(φ)s(ψ)]T, (10) ˙ y=1 m[c(φ)s(θ)s(ψ)−s(φ)c(ψ)]T, (11) ˙ z=1 m[c(φ)c(θ)]T−g, (12) ˙ φ=p+s(φ) (θ)q+c(φ) (θ) , (13) ˙ θ=c(φ)q−s(φ) , (14) ˙ ψ=s(φ) c(θ)q+c(φ) c(θ) , (15) ˙p=1 Jxx [−(Jzz −Jyy)q −JpqΩp+τx],(16) ˙q=1 Jyy [(Jzz −Jxx)p +JppΩp+τy],(17) ˙ =1 Jzz [−(Jyy −Jxx)pq +τz],(18) Fig. 1. Oc o o o PPNNPPNN s uc u e. Table 1. Pa ame e alues. Pa ame e Symbol Value Body ine ia Jxx =Jyy 150 ·10−3[kg·m2] Body ine ia Jzz 400 ·10−3[kg·m2] P opelle ine ia Jp104 ·10−6[kg·m2] Mass m8[kg] A m leng h l0.4[m] Th us ac o kb54.2·10−6[N·s2] D ag ac o kd813 ·10−9[m] whe e Jpis he ine ia momen o he mo o ( o a ing pa s) and he p opelle a ound he z-axis. Then, o he PPNNPPNN oc o o o s uc u e, as he one p esen ed in Fig. 1, whe e Pand Nde ine posi i e and nega i e eac i e mo o o ques, espec i ely, ( ep esen ed as a ows in Fig. 1), Ωp=−|Ω1|−|Ω2|+|Ω3|+|Ω4| −|Ω5|−|Ω6|+|Ω7|+|Ω8|,(19) whe e Ωiis he angula eloci y o he i- h mo o . I is assumed ha he h us o ce o a p opelle is p opo ional o he angula eloci y squa ed, such ha uTi=kbΩ2 i, whe e uTiis he h us gene a ed by mo o iand kbis he us ac o . The pa ame e alues which de ine he oc o o o model a e p esen ed in Table 1. Equa ions (7)–(18) desc ibe he model o a gene ic mul i o o s uc u e whe e he con ol ac ion is gi en by u =[Tτ xτyτz]T. Gene ally, u is e e ed o as he i ual con ol ac ion due o he ac ha no physical ac ua o s co espond o he o ces and momen ums which i desc ibes. In his wo k, he con ol alloca ion block ans o ms and edis ibu es he con ol e o (u ) gene a ed by he inne con ol loop in o he h us ha each mo o should deli e (uT) as p esen ed in Sec ion 4. 3. Con olle s uc u e In his wo k, he con ol o he UAV is a cascade s uc u e (Fig. 2), whe e wo LQR con olle s, inne and ou e , de ine he dynamics o he oc o o o , and he 50 J.C. Salaza e al. Fig. 2. Con ol scheme. con ol alloca ion dis ibu es he con ol e o among he ac ua o s. The linea ized model needed o he design o he LQR con olle is ob ained by applying he Taylo se ies app oxima ion a he ho e posi ion, which co esponds o he si ua ion whe e he x-yplanes o bo h ames ({I} and {B}(Fig. 1) a e pa allel and he mo o s gene a e a li ing o ce equal o he weigh o he oc o o o . The s a e and inpu a e conside ed o be x=[xyzφθψ x y zpq ]Tand u =[Tτ xτyτz]T, espec i ely. The LQR con olle is designed ollowing he me hodology p oposed by Oga a (1995). 3.1. Cascade s uc u e con olle . The inne and ou e LQR linea ized models a e de ailed below. The inne loop ocuses on he o ien a ion and he al i ude o he UAV and is de ined as ˙ ein( )=Ainein( )+BinΔu ( ), ˙ ein( )=04×4I4×4 04×404×4ein( )+04×4 βin Δu ( ),(20) whe e βin =diag(1/m, 1/Jxx,1/Jyy,1/Jzz),I4×4is he iden i y ma ix, and ein is he inne s a e ec o : ein =X e in −Xin =[ezeφeθeψe zepeqe ]T,(21) whe e X e in deno es he al i ude, he o ien a ion, he linea eloci y (z-axis) and he angula eloci y se -poin s, and Δu is he inne inpu ec o deno ed as Δu =[ΔTΔτxΔτyΔτz]T,(22) wi h Δu =u −u =u −[mg 000] T,andu is he eed- o wa d i ual inpu ha keeps he sys em a he equilib ium poin (ho e posi ion) in open loop. The ou e loop ocuses on he posi ion o he UAV (x-yplane). I s model is as ollows: ˙ eo( )=Aoeo( )+BoΔuo( ), ˙ eo( )=02×2I2×2 02×202×2eo( )+⎡ ⎢ ⎢ ⎣ 00 00 0g −g0 ⎤ ⎥ ⎥ ⎦ Δuo( ), (23) whe e eois he ou e s a e ec o deno ed as eo=X e o−Xo=exeye xe yT,(24) whe e X e ois he posi ion and linea eloci y se -poin (x-yplane), and Δuois he ou e inpu ec o deno ed as Δuo=[ΔφΔθ]T=[φ e θ e ]T.(25) 4. Con ol alloca ion loop The con ol dis ibu ion ask will be p o ided by he con ol alloca ion block. I consis s in dis ibu ing he con ol e o gene a ed in he inne loop (u ) among he ac ua o s (uT) a ailable in he sys em (Johansen and Fossen, 2013). Taking in o accoun ha he i ual con ol ac ions a e always in he a ainable con ol se (ACS) (Schneide e al., 2012) o he ac ua o s, he mos ex ended solu ion is o use he gene alized in e se, which is desc ibed as ollows: min uT∈R 1 2(uT)TW(uT), subjec o u =BCAuT. (26) I BCA has ull ank, hen his weigh ed leas squa es p oblem has he ollowing explici solu ion (Johansen and Fossen, 2013): uT=Win BT CA(BCAWin BT CA)−1u ,(27) whe e Win =W−1,andBCA is a unc ion o he s uc u al ma ix o he PPNNPPNN oc o o o (Bs ): u =BCAuT=Bs ΞuT,(28) whe e Ξ=diag(ξ1,ξ 2, ... ξ 8)deno es he con ol e ec i eness ma ix (ξi=1de ines he nominal beha io and a comple e ailu e co esponds o ξi=0)andBs is he ma ix ha con ains he ela ion be ween he h us gene a ed by he ac ua o s and he i ual ac ions, Bs =⎡ ⎢ ⎢ ⎣ 1111 0−ls(45◦)−l−ls(45◦) −l−lc(45◦)0+lc(45◦) +kd/kb+kd/kb−kd/kb−kd/kb Heal h-awa e and aul - ole an con ol o an oc o o o UAV sys em ... 51 111 1 0ls(45◦)lls(45◦) llc(45◦)0−lc(45◦) +kd/kb+kd/kb−kd/kb−kd/kb ⎤ ⎥ ⎥ ⎦ ,(29) whe e kdis he mo o d ag ac o and lis he dis ance be ween he cen e o mass o he o o cen e . No e ha , he highe he diagonal componen o he ma ix Win , he g ea e he co esponding ac ua o usage, and ice- e sa. In his wo k, he con ol inpu s a e compu ed based on he edis ibu ed pseudo-in e se (RPI) (Johansen and Fossen, 2013) and on he cascaded gene alized in e se (CGI) (Bo dingnon and Du ham, 1995) h ough Algo i hm 1. I in ol es he pa i ion o he i ual con ol ec o (u )in onsub ec o s (S eps 1–2): u = n  i=1 u i,(30) whe e u iis a sub ec o ha con ains a g oup o i ual ac ions wi h compa able cha ac e is ics o ele ance. In his wo k, hese a iables will be clus e ed depending on how hey a ec o he ac ua o deg ada ion. Nex , a ma ix Win iis de ined o each u iapplying gi en c i e ia (S ep 4). Then, he gene alized in e se (31) is applied o compu e uTi(S eps 5–7): uTi=Win iBT CA(BCAWin iBCAT)−1u i,(31) and uT(S ep 8): uT= n  i=1 uTi.(32) Algo i hm 1 s ops once uTis be ween he limi s o he ac ua o (S ep 9): uT<uT<uT.(33) I his is no sa is ied, e u n o S ep 4 and ede ine he weigh s. No e ha , con a y o wha is done in he RPI me hodology whe e sa u a ed ac ua o s a e neglec ed, in he p oposed app oach, i he condi ion (33) is no sa is ied, he weigh s mus be ede ined ollowing al e na i e c i e ia. Also, as long as a solu ion exis s o (26), he ollowing choice will p o ide a easible solu ion: Win i=I,∀i∈[1,n], wi h I∈R8×8being he iden i y ma ix. The e o e, he p oposed con ol alloca ion scheme ies o dis ibu e he con ol ac ions depending on he sys em eliabili y (Win i) and he ac ua o aul s (Ξ). Algo i hm 1. Dis ibu ion o he con ol e o . 1: Gi en u 2: Clus e u in o nsub ec o s (u i)(30) 3: epea 4: De ine Win i ollowing gi en c i e ia i 5: o i=1 o ndo 6: Compu e uTias (31) 7: end o 8: Compu e uTas (32) 9: un il {uT>uTand uT<uT} 5. Reliabili y modeling Reliabili y is de ined as he p obabili y ha componen s, uni s, equipmen , and sys ems will pe o m sa is ac o ily o a speci ied pe iod o ime unde speci ied ope a ing condi ions and en i onmen s (Ge sbakh, 2001). In pa icula , he eliabili y o he i- h componen o a sys em can be exp essed as Ri( )=R0 ie− 0λi(τ)dτ,∀i=1,...,m, (34) whe e R0 iis he ini ial eliabili y a he mission ini ial ime and λi( )is he ailu e a e o he i- h ac ua o , espec i ely. Se e al de ini ions o he ailu e a e can be ound in he li e a u e. In his wo k, a de ini ion based on he p opo ional haza d p oposed by Cox (1972) is used, λi( )=λ0 i·gi(, ϑ),∀i=1,...,m, (35) whe e λ0 i ep esen s he nominal ailu e a e o he i- h componen and gi(, ϑ)is a load unc ion also known as a co a ia e, which ep esen s he e ec o s ess on he componen ailu e a e as a unc ion o he employed load () and a componen pa ame e (ϑ). Di e en de ini ions o gi(, ϑ)exis in he li e a u e. Fo example, in he wo k o Khelassi e al. (2011), a load unc ion based on he oo -mean-squa e o he employed con ol inpu un il he end o he mission ( ) was p oposed. This load unc ion is used o dis ibu e he con ol e o s be ween he edundan ac ua o s, and he con ol ac ion is calcula ed using a eliable s a e eedback con olle . Salaza e al. (2017) de ined he co a ia e unc ion as he cumula i e con ol e o du ing he usage li e o he i- h ac ua o . In his wo k, he co a ia e o he i- h ac ua o is exp essed h ough unc ion gi(uno m i( ),˙uno m i( )) as gi(·)=1+βi 0 uno m id +γi 0 ˙uno m i( )d , (36) wi h gi(·)is de ined as a cumula i e unc ion o he no malized con ol e o and i s no malized de i a i e o he i- h ac ua o om he beginning o he mission up o 52 J.C. Salaza e al. he cu en ime ,andβiand γia e cons an pa ame e s ha desc ibe he con ibu ion o each e m. No malized alues belong o he in e al [0,1], wi h 0 ep esen ing no con ibu ion and 1 he maximum con ibu ion o ac ua o deg ada ion. Then, eplacing (36) in (35) yields λi( )=λ0 i1+βi 0 uno m id +γi 0 ˙uno m i( )d . (37) In con as o ou p e ious wo k (Salaza e al., 2017), he ailu e a e (37) conside edhe e includes a e m ela ed o con ol a ia ions which ep esen s he ac ua o deg ada ion due o a igue. This de ini ion implies ha ac ua o s a e unde a eliabili y decay due o he baseline ailu e a e which is inc eased when he ac ua o s a e used. 5.1. Sys em eliabili y. The o e all sys em eliabili y can be compu ed by means o i s s uc u e unc ion. I exp esses he s a e o he sys em in e ms o he s a e o i s componen s, allowing he compu ing o he sys em eliabili y. This sys em s uc u e could be se ial, pa allel, o a mix o bo h. In complex s uc u es (i.e., a b idge s uc u e), he s uc u e unc ion can be compu ed ollowing he pi o al decomposi ion me hod (Ge sbakh, 2001) o based on he se o minimal pa h se s o minimal cu se s. Al e na i ely, sys em eliabili y can be modeled using a dynamic Bayesian ne wo k (DBN) (Salaza e al., 2015). In his wo k, oc o o o mo o s a e he only componen s ha a e conside ed in he sys em eliabili y compu a ion. This compu a ion is based on de e mining he minimal pa h se s. Al hough he oc o o o sys em has eigh ac ua o s ( i ∀i∈[1,8]), con ol pe o mance is gua an eed as long as, a leas , ce ain quad o o con igu a ions a e ope a ional (Sanjuan e al., 2019) (see Fig. 4). Based on hem, he ollowing minimal pa h se s a e de ined: ζ1:{ 1, 3, 5, 7},ζ 2:{ 2, 4, 6, 8}, ζ3:{ 2, 3, 6, 7},ζ 4:{ 1, 4, 5, 8}.(38) Gi en sminimal pa h se s, i is possible o compu e he s uc u e unc ion as Φ=1− s  j=1 ⎛ ⎝1− i∈ζj Xi⎞ ⎠,(39) whe e Xiis a andom bina y a iable ep esen ing he s a e o he i- h o o (up o down). To compu e he eliabili y o he sys em, he s uc u e unc ion exp ession mus be p e iously expanded. Then, he expanded exp ession should be simpli ied aking in o accoun ha , due o i s bina y na u e, any powe o Xiis equal o Xi. The sys em eliabili y exp ession (RS) is ob ained om Fig. 3. Sys em eliabili y block diag am. Fig. 4. Oc o o o con ollable con igu a ions. he esul ing s uc u e unc ion exp ession by eplacing Xi wi h he componen eliabili y Ri. Figu e 3 ep esen s he eliabili y block diag am o he oc o o o sys em based on i s minimal pa h se s (ζi). The esul ing exp ession o he compu a ion o i s sys em eliabili y is RS=R1R3R5R7+R1R4R5R8+R2R3R6R7 +R2R4R6R8−R1R2R3R5R6R7 −R1R2R4R5R6R8−R1R3R4R5R7R8 −R2R3R4R6R7R8+R1R2R3R4R5R6R7R8. (40) 5.2. Reliabili y impo ance measu e. The sensi i i y o sys em eliabili y o he i- h ac ua o eliabili y (Bi nbaum’s measu e) (Bi nbaum, 1969) is de ined as IBi( )= ∂RS( ) ∂Ri( )=RS(1i, )−RS(0i, ),(41) whe e RS(1i, )deno es he sys em eliabili y when he i- h ac ua o is pe ec ly eliable, and RS(0i, )deno es he sys em eliabili y when he i- h ac ua o is aul y. This index indica es how sensi i e sys em eliabili y is agains changes in pa icula ac ua o eliabili y. This measu e will be la e used as a c i e ion o de ine he Win (27) o he con ol alloca ion p ocedu e. Heal h-awa e and aul - ole an con ol o an oc o o o UAV sys em ... 53 6. Heal h-awa e con ol me hodology The heal h-awa e con ol me hodology uses he weigh ma ix Win (27) o inco po a e he heal h s a e o he ac ua o s, ep esen ed by hei eliabili y le el and hei impo ance o sys em eliabili y, as a c i e ion o edis ibu e he con ol e o in he con ol alloca ion module. In ac , his me hodology can be applied o sa is y wo di e en objec i es du ing he mission. On he one hand, we ha e an objec i e aimed a maximizing he sys em li e ime. In e ec , enhancing sys em eliabili y a mission ime will lead o an inc ease in sys em li e ime. On he o he hand, he e is an objec i e aimed a maximizing indi idual o o li e ime. In ac , inc emen ing o o eliabili y will enhance o o li e ime. In his case, i is assumed ha p edic i e main enance is pe o med on he sys em when any o o eliabili y dec eases below a p ede ined eliabili y h eshold R h. Usually, ac ua o s a e b ushless o o s, he e o e main enance could be jus a isual inspec ion and a check o he as ening o he p opelle s. The e o e, di e en cases ha e been gene a ed o maximize each o hese policies as discussed below. Nominal case. This case ep esen s he beha io o he sys em when eliabili y is no aken in o accoun and, consequen ly, all o o s ha e he same impo ance. The e o e, he weigh s (31) o he con ol alloca ion block a e Win =I. Case A. The objec i e is o maximize he o e all sys em eliabili y a mission ime (Rs( )). In his case, he weigh s (31) o he con ol alloca ion block a e Win = diag(1 −IB),(42) whe e IBis a measu e o he impo ance o each ac ua o o he sys em eliabili y (41). The e o e, i is expec ed ha applying a smalle load on he o o s wi h highe IB will imp o e sys em eliabili y a mission ime. Case B. The objec i e is o maximize he ime a which he indi idual o o eliabili y dec eases below R h. The policy consis s in educing he load applied o he ac ua o s ha a e mo e deg aded. Thus, he weigh s will be gi en by Win =diag(win i),∀i∈[1,8] ,(43) whe e win i= 60σ(25(Ri−0.95)) + 5Ri, wi h σbeing he sigmoid unc ion. No e ha he objec i e o (43) is o inc ease he di e ence be ween he weigh s when hei alues a e nea o 1. As he deg ada ion o he ac ua o s (37) is a unc ion o he magni ude and a ia ions in he con ol inpu , ui and ˙ui, weigh ing hem equally wi h a uniqueWin ma ix migh lead o a subop imal beha io o each c i e ion. Mo eo e , he UAV ac ua o s ha e wo asks. The i s oneis o gene a ea h us ha compensa es he weigh o he ehicle such ha he ho e posi ion can be achie ed and gua an eed. The second ask is o ollow he e e ence ajec o y by gene a ing an imbalance be ween he o ces p oduced by he o o s. Taking his ac in o accoun , he ollowing pa i ion o he i ual con ol ac ion is p oposed acco ding o (30): u =u 1+u 2,(44) whe e u 1=[T000] Tis he cons an load ha he ac ua o s should gene a e and u 2=[0τxτyτz]Tis he a iable load which allows modi ying i s o ien a ion, and posi ion. Taking in o accoun he di e en c i e ia p oposed, he weigh s o u 1a e de ined as (42) o Case A and (43) o Case B. S onge es ic ions a e imposed on u 2because i is assumed ha his i ual ac ion causes a la ge deg ada ion han u 1. Algo i hm 1 has been applied o gene a e he h us o he o o s when di e en weigh s o he i ual con ol inpu s a e in ol ed. Then, he selec ed c i e ia o Win ia e as ollows. Case A*. As in Case A, he objec i e is o maximize sys em eliabili y. The e o e, he assignmen done in (42) is used o he i s pa o he i ual con ol inpu (u 1), and o he second pa (u 2) he assignmen is Win 2= diag((1−IB)5).(45) Case B*. As in Case B, he objec i e is o maximize he main enance ime mo he ac ua o s. Fo he i s pa o he i ual con ol ac ion (u 1), he assigmen in (43) is used, and o he second pa (u 2) he assigmen is Win 2=diag(win 2,i)∀i∈[1,8] ,(46) whe e win 2,i = 300σ(30(Ri−0.8)) + 5Ri. 7. Simula ion esul s 7.1. Simula ion se up. The eliabili y beha io o he PPNNPPNN oc o o o is s udied du ing a co n ield ae ial Fig. 5. UAV e e ence ajec o y. The a ows ep esen he o i- en a ion o he oc o o o . 54 J.C. Salaza e al. supe ision in ol ing di e en c i e ia. The co n ield has an a ea o 3000m2which is o e lown by he ai c a a an al i ude o 10m ollowing he g id pa h and e u ning o he s a ing poin as p esen ed in Fig. 5. The oc o o o is egula ed by a cascade LQR con ol s uc u e (de ailed in Sec ion 3) and he simula ion, whose pa ame e s a e p esen ed in Table 3, is done using he non-linea model o he PPNNPPNN oc o o o . To illus a e he di e en policies p oposed, highe o o ailu e a e alues ha e been conside ed, so ha hey a e scaled o he mission ime. I mus be highligh ed ha a o o eliabili y h eshold (R h)o 0.6hasbeen conside ed. I is assumed ha he deg ada ion o he ac ua o inc eases when i s eliabili y (Ri) is lowe han ha eliabili y h eshold (i.e., pa ame e βiin (37) is 10 imes highe ). In ini ial condi ions o he sys em, Ac ua o 8 s a s wi h a lowe eliabili y (Ro 8). Thus, his componen will de ine he sys em beha io . No e ha he ansi ions o he con ol e o a e gene a ed by he ajec o y e e ence ha he UAV mus ollow and he pe o mance o he con olle . The chosen dynamics o he con olle a e e y as in o de o inc ease he deg ada ion p oduced by he a igue o he o o s. The objec i es p esen ed in Sec ion 6 ha e been s udied in bo h he nominalcondi ions o he oc o o o and when a o o ails a some poin du ing he mission. 7.2. Faul - ee scena io. Cases A and B a e analyzed when all ac ua o s a e wo king p ope ly. Scena io 1 (Cases A and B). The simula ion esul s a e summa ized in Table 2, whe e Rs( )is sys em eliabili y a he end o he mission and mis he ime when an ac ua o equi es a main enance (when Ri=R h). The dash symbol means ha he main enance ime is highe han he end o mission ime ( m> ). The eliabili y o each componen (Ri) is p esen ed in Fig. 8, whe eas he sys em eliabili y (Rs) is shown in Fig. 6. The h us gene a ed by he ac ua o s unde he di e en c i e ia is shown in Fig. 12. Applying he gene alized in e se (27), i can be seen ha sys em eliabili y is sligh ly be e in Case A, whe e he sensi i i y is used as weigh s (42), compa ed o he nominal case. Also, in compa ison wi h he nominal case, in Case B he eliabili y o he mos c i ical ac ua o (Ro o 8) is main ained 100 seconds longe ; be o e ha , a main enance ask should be pe o med. Scena io2(CasesA*andB*). The ob ained esul s applying he p oposed app oach wi h Algo i hm 1 a e p esen ed in Table 2. Wi h he p oposed me hodology (Case A*) he eliabili y o he sys em is imp o ed om 0.271 o 0.302 compa ed wi h ha o Case A, (Fig. 6). On he o he hand, he main enance ime is inc eased a ound 100 seconds (Case B*) wi h espec o Case B and Table 2. Policy esul s in aul - ee and aul y scena ios ( o o 1 blocked). Case s udy Rs( )Main enance ime ( m[s])Case s udy Rs( )Main enance ime ( m[s]) 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Nominal 0.269 448 - - - 660 - - 279 Nominal 0.069 Faul y 443 - - 626 - - 238 A(W1=W2) 0.271 424 - - - 599 - - 268 A2 (W1=W2) 0.068 Faul y 437 - 610 588 - - 230 A* (W1=W2) 0.302 367 - - 651 493 - - 260 A*2 0.075 Faul y 434 - 550 502 - - 216 B(W1=W2) 0.264 507 641 - - - - - 384 B(W1=W2) 0.061 Faul y 418 - - - - - 337 B* (W1=W2) 0.255 539 586 - - - - - 475 B* 0.062 Faul y 436 668 589 661 - 676 417 Heal h-awa e and aul - ole an con ol o an oc o o o UAV sys em ... 55 Fig. 6. Sys em eliabili y ( aul - ee scena io). Fig. 7. Sys em eliabili y (scena io o a aul in Ro o 1). app oxima ely 200 seconds wi h espec o he nominal case (Fig. 8). 7.3. Faul y scena io. Fo he aul y scena io, i has been conside ed ha he o o 1 becomes aul y a he ime o 120 seconds. I is also assumed ha a aul de ec ion and diagnosis (FDD) module de ec s and es ima es he aul in a de e minis ic ime o 1 second a e i s occu ence. The FDD in o ma ion is hen in oduced in he e ec i eness ma ix Ξ(28) in he con ol alloca ion block. I has been obse ed ha in Case A he oc o o o becomes uncon ollable a e he occu ence o a aul . This is due o he ac ha eliabili y and con ollabili y objec i es a e in con lic in he aul y scena io. Analyzing he minimal pa h se s o he sys em (38), i is possible o de e mine ha when he o o 1 is aul y he pa hs ζ1 and ζ2become una ailable. Hence, he mos impo an o o s a e 2 and 6 because hey belong o he ac i e pa hs. This beha io poses a p oblem o sys em con ollabili y because o o s 1 and 2 o he oc o o o PPNNPPNN canno be aul y o ha e es ic ed use (Win i,i →0)a he same ime. Algo i hm 2 can be ep esen ed as in Fig. 10, whe e wo swi ches (Sw1and Sw2) a e added o he diag am o Fig. 3. The swi ches a e closed in he p esence o Table 3. Simula ion pa ame e s. Pa ame e Symbol Value Inne sampling ime si0.05 [s] Ou e sampling ime so0.25 [s] Mission ime 684 [s] LQR con olle Qinne loop Qin diag([100.0 820.7 820.7 131.3 25.0 8.2 8.2 3.6 ]) Rinne loop Rin I[4×4] Qou e loop QoI[4×4] Rou e loop RoI[2×2] Ro o pa ame e s β;γ;R h 3; 323; 0.6 Ini ial eliabili y R0 idiag([0.83 0.95 0.89 0.97 0.94 1 1 0.72]) Baseline ailu e a e λ0 i1.86 ·10−4[s−1] Ini ial condi ions con olled ou pu s y(0) 0x,0y,0z,0ψ s a es x(0) 0[12×1] Feed- o wa d inpu u [mg, 0,0,0]T a ailu e o hose ac ua o s belonging o hei espec i e mesh. No e ha Algo i hm 2 is used o he pu pose o compu ing he IBimeasu e when an ac ua o s a e DOWN. This measu e allows building Win in Cases A and A*, and hey ha e been deno ed as Cases A2 and A*2, espec i ely. Thei componen eliabili y will s ill be compu ed ollowing (34). No e ha he PPNNPPNN oc o o o UAV becomes uncon ollable i con iguous o o s o a ing in he same di ec ion a e aul y (o no used). Thus, he minimal cu se s o he s uc u e a e C1:{ 1, 2},C 2:{ 3, 4}, C3:{ 5, 6},C 4:{ 7, 8}.(47) The minimal cu se s a e composed o he o o s whose aul s o limi a ion use cause he loss o he o al con ollabili y o he sys em. In ui i ely, obse ing Fig.1, i Ro o s 1 and 2 a e aul y, hen he e will be mo e o o s o a ing clockwise han coun e clockwise. Thus, i is impossible o ind a easible solu ion o he con ol alloca ion p oblem wi hou losing con ollabili y. In he p esence o ac ua o aul s, a elaxa ion o weigh assignmen s should be aken. Hence, an app oach has been de eloped o compu e IBi, aking in o accoun he c i ical componen se s o cu se s (47), in o de o achie e he objec i es o Sec ion 6 wi hou losing con ollabili y. This p oblem is handled as p oposed in Algo i hm 2. Algo i hm 2 aims o assign he same impo ance o all ac ua o s loca ed in he mesh which con ain a aul y