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