IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23,NO.
4,JULY 2015 1465Observer-BasedDecentralizedFaultDetectionand
IsolationStrategyforNetworkedMultirobotSystemsFilippo
Arrichiello,Alessandro Marino, and Francesco PierriAbstractIn this
paper, we present a distributed
faultdetectionandisolation(FDI)strategyforateamof
networkedrobotsthatbuildsonadistributedcontrollerobserverschema.Remarkablydifferent
fromotherworks inliterature, thepro-posed FDI approach makes each
robot of the team able to detectandisolate faults occurringonother
robots, evenif theyarenot direct neighbors. Bymeansof alocal
observer, eachrobotcan estimate the overallstate of the team and it
can use such anestimate to compute its local control input to
achieve global tasks.Thesameinformationusedbythelocal
observersisalsousedto compute residual vectors, whose aim is to
allowthe detectionand the isolation of actuator faults occurring on
any robot of theteam. Adaptivethresholds are
derivedbasedonthedynamicsof theresidual vectors
byconsideringthepresenceof nonzeroinitial observer estimation
errors, and noise terms affecting statemeasurement andmodel
dynamics. Theapproachis validatedviabothnumerical
simulationsandexperimentsinvolvingfourKheperaIIImobilerobots.IndexTermsDistributedmultirobotsystems,
faultdetectionandisolation(FDI), networkedrobots.I.
INTRODUCTIONNETWORKEDrobots have been object of
widespreadresearchinthelatest years [18]. Thisis
motivatedbytheirwideapplicationdomain,aswellasbytheirexibility,potential
robustness to faults, and capacity to accomplish
com-plextasksalternativelyimpossibleforasingleunit. Despitetheir
clear advantages, networked robots pose
challengingproblemsduetotheinteractionamongcontrol, communica-tion,
and perception.When dealing with the control of networked robots,a
decentralized control strategy is a desirable featureand,
sometimes, the only one possible; indeed, in
mostchallengingapplications, acentral control unit represents
aweakpoint of the systemandthe communicationwithallManuscript
received July16, 2014; accepted November 14, 2014. Dateof
publication January 8, 2015; date of current version June 12,
2015.Manuscript received in nal formNovember 24, 2014. This work
wassupported by the Italian Government within the Fund for
Investment inBasic ResearchFuturoin Ricerca2008 through the NECTAR
Project underGrantRBFR08QWUV. RecommendedbyAssociateEditorX.
Zhang.F. Arrichiello is with the University of Cassino and Southern
Lazio,Cassino03043,Italy(e-mail:[email protected]).A.MarinoiswiththeUniversityofSalerno,
Salerno84084,Italy(e-mail:[email protected]).F.PierriiswiththeUniversityofBasilicata,
Potenza85100,Italy(e-mail:[email protected]).Color
versions of oneor moreof thegures inthis paper
areavailableonlineathttp://ieeexplore.ieee.org.DigitalObjectIdentier10.1109/TCST.2014.2377175therobotscouldbeimpossibleinthecaseof
vehicleswithshort-communication-range capabilities. Thus, most
recentresearch in this eld focuses on the development
ofdecentralizedcontrol strategies, whereeachrobot
computesitsownmotioncontrolonlybasedonlocal informationandon
information from a subset of its teammates, usually namedneighbors
[7]. Graph theoretical methods that allow the systemto extract its
analytical properties on the basis of the connec-tivity properties
of the communication network formed by theagents are often used for
the purpose [19]. In this context, oneof the most popular problems
is the consensus, which consistsin reaching an agreement regarding
a specic variable,
exoge-nousordependingonthestateofthesinglerobots, withoutusing
all-to-all communication. Recent results in this eld
aresummarizedin[25]. Asexamples, consensusalgorithmsareinvestigated
with an emphasis on robustness, time-delays, andperformancebounds
in[22] and[26]. Differentlyfromtheconsensus, theproblemof
distributedcontrol of multirobotsystemsisaimedatachievingaglobal
task(e.g.,controllingthe geometrical centroid and formation of the
team) using
onlylocalinformationandinteractions.Awideoverviewonsucha problem
can be found in [7] and [19]. Belta and Kumar [6]propose a
partially decentralized algorithmto control thenetwork centroid,
variance, and orientation of the system.In[12] and[32],
thedesignedapproachbuildson[29]andusesadistributedestimatoroftheactual
collectivebehavior(global task) expressed in terms of formation
statisticsto make it reach a desired constant value.
Decentralizedestimationandcontrol
havebeenalsoinvestigatedin[28]inthe framework of linear-state
feedback control.The intrinsic redundancy of networked robots may
allow foraccomplishing the assigned mission also in the case of
failureofoneormoreteammates. Nevertheless,
whendealingwithdistributedcontrolofmultirobotsystemsaimedat
achievingaglobal task, this feature, without theexplicit
adoptionofasuitablefault detectionandisolation(FDI)schema,
istheonlypotentialandthefaultofonerobotmayjeopardizetheproper
execution of the task. Indeed, almost all the
approachespresentedintheliteraturedonot exploit theredundancyofsuch
systems,as no fault handling strategy is designed.Several FDI
approaches have beenpresentedin the lastdecades for single-unit
systems; the interested reader mayconsult therecent surveyin[15]
andthereferencesthereintoget a wideoverviewonthemainapproaches.
However,veryfewapproaches have beendesignedfor
decentralized1063-65362015IEEE. Personaluseispermitted,
butrepublication/redistributionrequiresIEEE
permission.Seehttp://www.ieee.org/publications_standards/publications/rights/index.html
for moreinformation.1466 IEEE TRANSACTIONS ONCONTROL SYSTEMS
TECHNOLOGY, VOL. 23,NO.4,JULY 2015multirobot systems
andmainlymakeuseof acentral unit.A centralized FDI scheme for
networked systems is presentedin[31], whereeachsubsystemtransmits
informationaboutactuatorsandsensorstoacentralizedfault
detectionstationthatdetectsandisolatesfaultsoverthenetwork.In[20],thediagnosis
problemis formulatedin terms of an isolabilityindex for a given
family of fault signatures, and both acentralizedanddecentralized
architectures aredeveloped andcompared; suchapproachhas
beenextendedin[21] tothecaseof large environmental disturbances.In
the last years, some researchers focused their attention onthefault
diagnosis of distributed systems,proposing
differentdecentralizedapproaches. Bankof adaptiveobservers,
usingonly measurements and information from neighboring
subsys-tems, areusedin[11], [33], and[34] todetect andisolatefaults
in interconnected subsystems; for each node in the largescale
system, a fault detectionestimator is designedusinglocal
measurements and information communicated frominterconnected
subsystems. The concept of overlapping systemis exploitedin [30],
where the overall systemis assumedto be linear, with
zero-meanstochastic processes as inputandmeasurement noise,
andpotentiallyaffectedbyadditiveprocess faults; local observers
aredesignedtodetect faultson nonoverlapping parts of the system,
and a consensus-like strategy is adopted for FDI of the
overlappingparts.In [24], an observer-basedschema, where each
subsystemexchanges stateinformation
withalltheothersubsystems,isproposedfor thedetectionof
faultsaffectingboththelocaldynamics and the interconnected
subsystems. Unknown inputobserversareproposedin[27] for theFDI of
networks ofinterconnected systems controlled with a decentralized
controllaw. Feng et al. [10] deal with the FDI problem for
networkedsystemsincaseof communicationfailuresintermsof timedelays,
randompacket losses, and nonlinear
perturbations.Adistributedfaultdetectionlteringapproachforaclassofnonlinear
systems is presented in [16], where each subsystemis monitoredbya
local fault detectionscheme; eachnoderequires the input and output
measurements of the subsystemthat it ismonitoring,aswell
asthemeasurementsofall theinterconnected subsystems that are
inuencing that subsystem.Reference [8] presents a FDI approach for
distributed and het-erogeneousnetworkedsystems,
whereeachagentcandetectandisolatenot onlyitsownfaultsbut
alsothefaultsof itsnearest neighbors.Intheabovecitedpapers, a
necessaryconditionfor thedistributed detection of the fault of a
robot by a healthy one isthat thetworobotsareeither
abletodirectlycommunicateor sense each other. Obviously, this is a
limitation in thecasetherobotsmotioncontrol
lawdependsonthestateofall therobots intheteamandnot
onlyonthestateof itsdirect neighbors, as the formation control
problem consideredin this paper. Thus, here we want to develop a
distributed FDIalgorithmfornetworkedrobots, whichallowseachrobot
oftheteamtodetect faultsoccurringonanyotherrobot,
evenifnotdirectlyconnected; thiscanbeusedinthecasewherethecontrol
input ofeachrobot dependsontheoverall stateof the systemor on
anestimate of it.Antonelli etal. [2], [3]presented a distributed
strategy to control the centroid and therelative formation of a
team of robots based on the use of localobservers that allow each
robot to estimate the collective stateof the system using only
information from the robot itself andfrom its direct neighbors.
Building on the results presented inthe above cited papers, here we
introduce a FDI strategy that,byaproper modicationof
thestate-observer, allows eachrobot to detect additive input faults
affecting any robot of theteam.TheFDI schemaproposedinthis paper is
basedonthefollowing steps.1) Each robot runs a local observer, a
modication ofwhat is presentedin[2] and[3], that uses
onlylocalinformation and a suitable vector received from
neighborrobots to estimate the overall systemstate (i.e.,
thepositions of all therobots).2) Eachrobot computes a
motioncontrol lawthat
usesalocallyavailableestimateoftheoverallsystemstatetotrackdesiredtime-varying
trajectories ofglobal taskfunctions (team centroid and formation
inour case).3) Usingthesameinformation(local
andreceivedfromneighbors) gainedfor theobservercontroller
schema,each robot computes residual vectors relative to
theteammates tomonitor their healthy state.4) The residual dynamics
are analytically investigated bothin the presence and in the
absence of faults. Thus,todetect andisolatefaults of anyother robot
intheteam, theresidual
vectorsarecomparedwithadaptivethresholdsdesignedonthebasisofresidualdynamics,the
presence of likely nonzero initial observer estimationerrors,
measurement noise, and model uncertainties.Apreliminaryversion of
this paper has been presentedin [5]. Here, we include further
analytical details; we considerthe case of noisy state measurements
and model uncertaintiesthat require a newderivation of the adaptive
thresholds;we improve the simulative validation considering the
caseof occurrence of multiple faults in the presence of
noisymeasurement and model uncertainties; nally, we provide
theexperimental results witha teamof networked robots.This paper is
organized as follows. Section II introduces thesystemmodeling and
the decentralized observercontrollerscheme. Section III presents
the fault detection schemadesigned on the basis of the developed
state observer.Sections IV and V, respectively, present the results
ofnumerical simulationsandexperimental validationinvolvingfour
Khepera III robots required to track desired time-varyingcentroid
and formation reference trajectories. Finally,inSectionVI,
wederivetheconclusionanddiscuss futureworks.II. DECENTRALIZED
OBSERVERCONTROLLER SCHEMEA.Robot Dynamics and Communication
ModelingLet us consider asystemcomposedby Nrobots, wherethei
throbotsstateis denotedbyxiIRn. Eachrobot ischaracterized by
thefollowing dynamics: xi = ui +i +i(t , xi) (1)ARRICHIELLO et al.:
OBSERVER-BASED DECENTRALIZED FDISTRATEGY 1467where ui IRnis the
input vector, i IRnis an additive
faulttermthatiszeroinhealthyconditions,and(t , xi) IRnisan
uncertainty term supposed tobe boundedi(t , xi) t i = 1, 2, . . . ,
N. (2)The collective stateisgiven byx = [xT1, . . . , xTN]T IRNnand
the collective dynamics is,then, expressed as x = u + + (3)where
u=[uT1, . . . , uTN]T IRNnis the collective inputvector, = [T1, . .
. , TN]TIRNnis thecollectivefaultvector, and =[T1, . . . , TN]T
IRNnis the collectivedynamic uncertainty term.It is supposed that
each robot has access to a noisymeasurexi,mof its own statexi,m =
xi +i(4)where iIRnis theadditivenoise, assumedtobenormbounded by
apositive scalari i = 1, 2, . . . , N. (5)The collective noisy
measure ofthe systemstateisxm = x + (6)where = [T1, . . . , TN]T
IRNnis the collective noise vector.The informationexchangeamongthe
robots is modeledreferring to graph theory approaches [9], [13],
[23]; thus,anetworkofrobotsisdescribedbyagraph G(E,
V)charac-terizedbyitstopology, i.e.,theset Voftheindexes
labelingtheNvertices (nodes), the set of edges (arcs) E = V V,
andthe(N N)adjacency matrixA = {ai j} : aii = 0, ai j =_1 if ( j, i
) E0 otherwisethat is, ai j=1 if there exists an arc fromvertex j
tovertex i . It is assumed that the i th robot receives
informationonlyfromits neighbors Ni={ j V : ( j, i ) E}, anditdoes
not knowthe topologyof the overall communicationgraph.
Thecardinalityof Niisthein-degreeofnodei , i.e.,di = |Ni| = Nj =1
ai j. Moreover, the cardinality of the set
ofnodesreceivinginformationfromnodei representstheout-degree of
node i , i.e., Di = Nk=1 aki.If all the communication links among
the robotsare bidirectional, the graph is called undirected(i.e.,
(i, j ) E ( j, i ) E) andtheadjacencymatrixissymmetric; otherwise,
the graph is called directed. A directedgraph is called strongly
connected if any two distinct nodes ofthe graph can be connected
via a directed path, i.e., a path thatfollows the direction of the
edges of the graph. An undirectedgraph is called connected if there
is an undirected pathbetweeneverypair of distinct nodes. Anodeof
adirectedgraph is balanced if its in-degree and its out-degree are
equal;a directed graph is called balancedif each node of the
graphis balanced (i.e., di= Di,i ). Any undirectedgraph
isbalanced.Moreover, thegraphtopology
canbeassumedeitherxedorswitchingintime(e.g.,
communicationlinksmayappearor disappear over the
time).Thecommunication topology canalsobecharacterized bythe (N
N)Laplacian matrix dened asL = {li j} : lii =N
j =1, j =iai j, li j = ai j, i =j.TheLaplacianexhibitsat least
azeroeigenvaluehavingasthecorrespondingright eigenvectorthe N
1vectorof allones 1N. Hence, rank(L) N1 and L1N = 0N, where
0Nisthe(N 1)nullvector. Forabalanceddirectedgraph(and,thus,
foranundirectedgraph), 1Nisalsoaleft eigenvectorof L, i.e., 1TN
L=0TN. All the eigenvalues of matrix Laresuchthat theirreal part
isequal toorgreaterthanzero[i.e., Re(i) 0,i ]; moreover, if the
graph is stronglyconnectedrank(L) =N 1; rank(L) 0 is a scalar gain;
ym=xm _tt0 u()d,andiy=i x _tt0i u(,i x)d, where t0is the initial
timeinstant; andi u =i u(t ,i x) istheestimateof
thecollectiveinputelaboratedbythei throbotonthebasisofitsestimateof
the collective statei x, and of the control
lawdetailedinSectionII-C. Thei thinput uiin(1)isextractedfromi
uthanks to theselection matrixiui = ii u. (8)It is worth noticing
that iy depends only on local informationavailabletoroboti ,
andthateachobserverisupdatedusingonly the estimatesjy received from
direct neighbors.1468 IEEE TRANSACTIONS ONCONTROL SYSTEMS
TECHNOLOGY, VOL. 23,NO.4,JULY 2015Thus,jy IRNnis the only
information that is required to beexchanged among neighbors.The
collective estimation dynamics is x
= koL
y
+ko
y
+ko
+ u
(9)where L
= LINn, with denoting the Kronecker productoperator and
= diag{[1, . . . , N]} IRN2nN2n x
= [1 xT, . . . , N xT]TIRN2ny
= [1yT, . . . , NyT]TIRN2n u
= [1 uT(t ,1 x), . . . , N uT(t , N x)]T IRN2n
= 1N IRN2ny
= 1N y y
IRN2n(10)withy = x _tt0 u()d.C.ControlObjective and the Feedback
ControlLawThecontrol objectiveandthefeedbackcontrol
lawcon-sideredinthis paper are inheritedfrom[2] and[3]; here,we
recall their essential concepts tomake this paper self-contained,
while interested readers may want to directly refertothe
citedpapers for further details.The control objective is to make
the teamcentroidandthe relative formation follows desired
time-varying references.Tothisaim,
thetwotasksarerepresentedviathefollowingtaskfunctions1) The
centroid of thesystem1(x) =1NN
i=1xi=J1x (11)whereJ1 IRnNnis theJacobian ofthe task.2)
Theformationofthesystem, expressedasanassignedset of relative
displacement between therobots2(x) = [(x2 x1)T, . . . , (xN
xN1)T]T=J2x(12)where J2 IR(N1)nNnis theJacobianof thetaskwhose
expression can befound in [3].It is worth combining both the tasks
in a single vector IRNn =_12_ =_ J1J2_x =Jx, =J x (13)it can be
easily shown that matrixJ IRNnNnis not singularand then
invertible.Eachrobot, usingitscollectiveestimatei x,
computesanestimateof the collective input via thefeedback control
lawi u =i u(t ,i x) =J_ d + kci (i x)_(14)wherei (i x) =_i 1(i x)i
2(i x)_ =_1,d i1(i x)2,d i2(i x)_is theestimate ofthe taskerror = [
1(x)T 2(x)T]T= [1,d 1(x)T, 2,d 2(x)T]Twhere i,dis the desired value
of the task variable i(i = 1, 2) andJis the pseudoinverse of
matrixJ. The inputvector uitorobot i is extracted from(14)
according to (8).D.Convergence Proofof
theObserverControllerSchemaInthefollowing,weprove
thatwiththepresentedversionof observer (modied with respect to the
one presented in [2]),in the absence of faults, measurement noise
and model uncer-tainty, x
= 1N x x
exponentially converges to the origin.Tothisaim,
theexponentialconvergence totheoriginof y
is rst proven.Theorem 2.1: In the presence of a strongly
connecteddirected communication graph (or connected undirected
graph)andintheabsenceoffaults(i = 0, i = 1, 2, . . . ,
N),mea-surement noise(i = 0, i = 1, 2, . . . , N, and y =ym)
andmodel uncertainty (i = 0, i = 1, 2, . . . , N), y
is globallyexponentiallyconvergent totheoriginwiththeupdatelawin
(9) for any ko> 0. Furthermore, in the presence of
boundedmeasurement noise and model uncertainty, y
is globallyuniformly ultimately bounded.Proof: Theproof comes
from the factthat systemin(9)can berearranged asy
= x
u
= koL
y
+ko
y
+ ko
. (15)Then, invirtue of the lastrelationship in(10)L
y
= L
(1N y y
) = L
y
+ L
1N y = L
y
(16)where the propertyL
(1N y) = (L INn)(1N y) = (L1N) (INn y) = 0has been exploited.
Hence, by deningL
= L
+
,(15) becomesy
= ko L
y
+ko
. (17)Bynoting that from(3), y = +,it holdsy
= 1N y y
=
+
ko L
y
ko
(18)with
=1N and
=1N . In[2], it wasprovedthat matrix L
is Hurwitz provided that the communicationgraphis
stronglyconnected. Thus, intheabsenceof faults(
= 0), measurement noise (
= 0), and model uncertainty(
= 0), (18) proves the theorem ko> 0.Inthepresenceofbounded
measurement noiseandmodeluncertainty, thetermko
+
in(9) canbeviewedasanonvanishingboundedperturbationwhoseupper
boundisgiven byko
+
ko +
N(ko +N). (19)The exponential stability of the origin of the
nominalsystem ensures that the solutions of the perturbedsystem are
globally uniformly ultimately bounded(see[17, Lemma 9.2,
p.347]).ARRICHIELLO et al.: OBSERVER-BASED DECENTRALIZED
FDISTRATEGY 1469Theorem 2.2: In the presence of a strongly
connecteddirected communication graph (or connected undirected
graph)andintheabsenceoffaults(i = 0, i = 1, 2, . . . ,
N),mea-surement noise (i.e., i = 0, i = 1, 2, . . . , N), and
modeluncertainty (i = 0, i = 1, 2, . . . , N), with the update
lawin(9), thestackedvector of thecollectivestate estimationerrors
x
is exponentially convergent to the origin. Moreover,inthe
presenceof boundedmeasurement noise andmodeluncertainty, x
is globally uniformly ultimately bounded.Proof:
SeetheAppendix.Remark 2.1: In [3], it was proven that the
exponentialstability of the observer leads also to the exponential
stabilityofthetaskerrors l(l =1, 2)withthecontrollawin(14).Thus,
the proof relative to the convergence of l(l = 1, 2) isomitted.III.
FAULT DETECTIONAND ISOLATION SCHEMATo detect the occurrence of a
fault, let us dene thefollowing residual vector for the i
throbot:ir =
j Ni(jy iy) + i( ym yi). (20)Itisworthnotingthat thecomputation
oftheabove quantitydoes not require additional information exchange
since itmakes use of the same terms used by the local
stateobserverin(7).The vectorir can be seen as a stacked vector,
i.e.,ir =[irT1,irT2, . . . ,irTN]T IRNn, where each componentirk
IRnrepresents theresidual computed by robot i relativeto robot k.
As it will be explained in the following,
thevectorirkallowstheroboti tomonitorthehealthystateofrobot k.The
following lemma represents the key point of thedesigned
FDIstrategy.Lemma 3.1: A fault occurring on the robot k at time
tf> 0 = [0T, . . . , Tk , . . . ,
0T]T(21)affectsonlytheresidualcomponentsirk( i = 1, 2, . . . ,
N)and not residualir j(i = 1, 2, . . . , Nandj = k).Proof: Based on
(20), the stacked vectorr
= [1rT, . . . , NrT]TIRN2n, i.e., thecollectiveresidualvector,
can be expressed asr
= L
y
+
(22)andfromTheorem2.1, it is straightforwardtoderive thatin the
absence of faults, measurement noise, and modeluncertainty, such a
term converges exponentially to zero, whilein thepresence of
bounded noise and model uncertainty, it isbounded as well.From
(22), the vector collecting all the residuals relative totherobot
k, namely, r
k = [1rTk , . . . , NrTk]TIRNn, canbeexpressed asr
k =diag{k,k, . . . , k}r
=
k(L
y
+
)= L
ky
k + k (23)wherethevector y
k = [1yTk , . . . , NyTk]TIRNncollectstheestimation errorsiykof
the observers and L
k = L In+k.It can be shown that it holdsy
k = diag{k,k, . . . , k} y
=
k y
. (24)Byconsidering (18),thedynamics of y
kisy
k =
ky
= ko
k L
y
ko
k
+
k
+
k
= ko L
k y
k ko
k +1N k +1N k(25)where
k
=1N k(
k
=1N k). Equation(25)shows that y
kis only affected by the fault on robot k(k) andnot by faults on
other robots (jwithj = k). Since L
khas allits eigenvalues in the right half plane when the
communicationgraph is strongly connected (this can be proven
analogously tothe case of L
), (25) is asymptotically stable and its solution,starting from
thetimet = 0,isy
k(t ) = eko L
kty
k(0)+_t0eko L
k(t )(1N k() +1N k() ko
k())d. (26)Consequently, residual vectorirkcan beexpressed asirk
= ir
k = i L
ky
k + i
k =ihk +ifk(27)whereihk(healthy) isthepartoftheresidual
notdependingonthefault, whileifk(faulty)isthepartdependingonk,whose
expressions from(26) and(27) areihk = i
k + iL
k_eko L
kty
k(0) +_t0eko L
k(t )(1N k() ko
k())d_(28)ifk = i L
k_t0eko L
k(t )(1N k())d (29)respectively.Therefore, from(27)(29),the
following can beargued.1) The residualsirkfor all i = 1, 2, . . . ,
N (i.e., theresiduals referredtothefaultyrobot) areaffectedbythe
fault kviathe termifkin(29).2) All theresidualsirlfor all i = 1, 2,
. . . , N, and for alll = k(i.e., all the residuals referred to
arobot differentto the faulty one) are insensitive to the fault on
robot k.This completes theproof.Remark 3.1: Itisworthremarking
thatsincetheresidualsaredecoupledinsuchawaythat thefault kaffects
onlythe residualsirk(i =1, 2,. . . , N), the proposedschemeis
effectivealsointhepresenceof multiplefaultsaffectingdifferent
robots.The following Lemma for the detection and isolation of
thefaultkcanbe stated.Lemma 3.2: Thefault k, affectingthekthrobot,
canbedetected and isolatedby the robot i if_t > tf : irk(t )
>ik(t )l (1, 2, . . . , N), l =kt >0, irl(t ) il(t)(30)where
tf>0is the instant at whichthe fault starts andij(t ) (i, j )
are suitable thresholds dened in the following.1470 IEEE
TRANSACTIONS ONCONTROL SYSTEMS TECHNOLOGY, VOL. 23,NO.4,JULY
2015Moreover,thefollowingsufcient detectabilityconditionfortherobot
i holds:t > tf : ifk 2ik(t ). (31)Proof:
Condition(30)iseasilyderivedbythefact thatonlytheresidualirkis
affectedbythefault k, as statedin Lemma 3.1, while the other
residuals computed by robot iare insensitive to it. In the presence
of nonzero initial observerestimation errors, measurement noise,and
model uncertainty,the residuals in (20) can be different from zero
as well, even inthe absence of faults. To avoid the occurrence of
false alarms,adaptive thresholds can be dened on the basis of the
residualstructureandtheboundin(5), thenthedecisionabout
theoccurrenceofafault ismadewhenaresidual
exceedssuchthreshold.Concerning the generic thresholdik, it is
calculated basedon (28), considering the upper bound of the
residualihk(i.e., inthe absence of faults)
ihk i
k +
i L
keko L
kty
k(0) +
i L
k_t0eko L
k(t )(1N k() ko
k())d i
k +
i L
keko L
kty
k(0)+
i L
k_t0eko L
k(t )_ko +N_diky
k(0)et+
i L
k_ko +N_(1 et) (32)where ik = 1 if i = k and ik = 0 otherwise,
and the property__eko L
kt__ et(33)with, > 0has been exploited being ko L
kHurwitz [14].Finally, since
i L
k i(L In) +i
k
n i (L In) +i
k
n di +ik(34)with dibeing the dimension of Ni(i.e., the in-degree
ofnode i ), (32) becomes
ihk _ndi+ik__y
k(0)et+_ko +N_(1 et)_+ik =ik(t ). (35)As far as the
detectability condition is concerned, in thepresence of fault,
based on (27), the following chain ofinequalities holds:
irk = ifk +ihk ifk ihk ifk ik(t ).(36)Since the fault is
detected by the robot i if irk(t ) >ik(t ),from (36) a sufcient
detectability condition can be dened as t > tf : __ifk__ik(t )
ik(t ) (37)from which condition (31) is straightforwardly
derived.Remark 3.2: Threshold in(32) requires areliable estimateof
y
k(0), , and.The constant y
k(0) can be estimatedon the basis of approximate information
about the initialconditions ofthesystem(e.g.,therobots
startfromaknownbounded area).Themorepreciseistheknowledge about
theinitialconditionsofthesystem, themorechancetodetectafault
intheinitial phaseoftheexperiment. IftheLaplacianmatrix of the
systemis known to the robots, parametersandcan be computed, as
shown in [14], as =_M(H)m(H) =1M(H)where M() (m()) represents the
maximum(minimum)eigenvalue of the symmetric positive denite matrixH
IRNnNnsolution of theLyapunov equationko L
kTH +koH L
k = 2INn.If the Laplacianmatrixis not knowntothe robots of
theteam, the matrix L
kcan be estimated by considering the worstcasescenario,
correspondingtothe maximumvalueof thesteady-state threshold. From
(32), this case corresponds to themaximumvalueof /for all
thepossiblecommunicationgraphs. However, tothebest of our
knowledge, thereisnoresearch result on which is the worst case,
thus, as the possibleconnectivity graphs withnitenumber ofrobots
areniteaswell, it is possible to run an ofine exhaustive search of
theseparameters.Remark 3.3: Concerning thecomputational
load,themainburden for each robot is represented by the observer in
(7) that,in turn, requires the calculation of the estimated control
inputin(14). Suchcalculationsaremandatoryfor theestimationof the
overall state of the systemand, then, for
detectingandisolatingfaults evenrelativetonot
neighboringrobots.However,
thisloadisfullycompatiblewithstandardroboticboards even in the
unlikely case of robot teams with hundredsof robots. The experiment
in Section Vproves howthedesignedapproachworks using a setup with
quite limitedcomputational capabilities.IV. NUMERICAL SIMULATIONSIn
this section, we present the results of a numericalsimulation
analysis performedto validate the effectivenessof the
controllerobserver and FDI strategies presented
intheprevioussections.
Inparticular,weconsideredateamofvemobilerobots, eachof
themimplementingthecontrolalgorithmin(14)andtheobserverin(7),
wherethecontrolandobserver gains wereset, respectively, tokc
=1.3andko = 2.Thenetworktopology isassumedasarigiddirectedgraph, as
reported in Fig.1.Therobots
werecommandedtokeepaconstantlylinearformation with a relative
distance of 0.4 m, while the team cen-troid was commanded tofollow
asinusoidal path. UniformlyARRICHIELLO et al.: OBSERVER-BASED
DECENTRALIZED FDISTRATEGY 1471Fig.1.
Simulationcasestudy.Topologyofthecommunicationnetwork.Fig. 2.
Simulation case study. Paths of the robots during the mission.
Crossesare the initial positions, diamonds the nal ones, andcircles
intermediatecongurations.distributedrandomnoisewithabounded norm( =
0.05m)was addedtothestatemeasurements, as in(4), as well
asuniformly distributed random uncertainty with a bounded
norm(=0.05m/s) was addedtotheprocessinput signals, asin(1).During
the motion, we induce two successive input failurestotwoof
therobots(robot 2androbot 3); specically,
weappliedanadditiveconstantinputterm3 = [0.3, 0.3]Tfort
>20storobot 3; andweassumethat robot 2stopsfort >
30(thismeansnullinput command and2(t ) = u2(t )for t > 30s).The
information initially available to each robot is assumedto be
limited to its position (xi(t0)) and to the number of
robotsintheteam(N). Eachlocal observeri xistheninitializedasall the
robots were in the same position (i xj(t0) = xi(t0), j ),i.e.,i
x(t0) =1N xi(t0). Therobotsareinitiallydisplacedinside aclosed
region of 2m 2m (centered in [1.5, 2] m)following a uniformly
random distribution. Assuming that
thesizeoftheareainwhichtherobotsareinitiallydisplacedisknown, the
previous initialization choices allow for evaluatingthe maximum
initial values of the residual errors and for usingthem inthe
residual adaptive thresholds computation in(32).Fig. 2 shows the
paths of the robots during the mission andthe desired path of
theteam centroid, while Fig.3 shows theinput to the robots. From
the gures, it is possible to notice thatduetotheoccurrence
ofthetwofaults,robot 2stopsduringthecourseof themission(null input
for t >30s), whilerobot 3escapes fromthedesiredlinear
formation(additiveinput term for t > 20s).Fig.3.
Simulationcasestudy.Realinputui(t ) =ii u(i )totherobots.Fig. 4.
Simulationcase study. Plots of the state estimation( x
), centroid,andformationtaskerrors( 1and 2).Fig.5.
Simulationcasestudy.Solidline:normoftheresidualcomponentsrelativetorobot
1(anonfaultyrobot) ascomputedbythedifferent
robots.Dashedline:thresholdusedforthefaultdetection.Fig. 4 shows
the plots of the norms of the estimationandtaskfunctionserrors,
andit ispossibletonotethat
alltheerrorsapproachaneighborhoodoftheoriginbeforetheoccurrence of
the rst fault. Figs. 57 show the norms of theresidual components
(solidlines) of thedifferent observersrelativetorobots1, 2, and3,
respectively;suchguresalsoshow the adaptive threshold values used
for the fault detection1472 IEEE TRANSACTIONS ONCONTROL SYSTEMS
TECHNOLOGY, VOL. 23,NO.4,JULY 2015Fig.6.
Simulationcasestudy.Solidline:normoftheresidualcomponentsrelative
torobot 2(the faulty robot withnull input termfor t >30)
ascomputedbythedifferent robots. Dashedline: thresholdusedfor
thefaultdetection. Faultoccurredatt = 30s.Fig.7.
Simulationcasestudy.Solidline:normoftheresidualcomponentsrelativeto
robot 3 (the faulty robot with constant additive constant input
termfort >
20)ascomputedbythedifferentrobots.Dashedline:thresholdusedforthefaultdetection.
Faultoccurredatt = 20s.(dashedlines). Indetail, Fig. 5showsthat
theresidualsir1(i =1, 2, . . . , N), i.e.,
theresidualscomputedbytherobotsandrelatedtoanonfaulty robot (robot
1inthiscase)donotovercome the corresponding thresholdi1. The same
happensfortheresiduals oftheother nonfaulty robots
andtheirplotsarehereomittedforbrevity. Figs. 6and7showthat all
therobots areabletodetectthefaultsofrobots 2and3,even ifnot
directly connected tothem, as shown inFig. 1.V. EXPERIMENTAL
RESULTSInthissection, wepresent theresultsof
anexperimentalanalysisperformedwithamultirobot
systemtovalidatetheeffectiveness of the FDI strategy presented
above. In particular,weusedateamoffourKheperaIIIrobots(Fig. 8)that
aresmall-sized (12 cm diameter) differential drive mobile
robots.Each robot is equippedwith a HokuyoURG-04LX-UG01Fig. 8.
Experimentalcase study. Pictureof the four Khepera III robots
usedfor
theexperiments.laserrangenderandadoptsthesoftwaremoduledevelopedin[4]
toperformlocalizationinindoor environmentsbasedon extended Kalman
lter and on the Hough transform. Sucha module allows for estimating
the position of the robotswithrespect toaknownmapoftheenvironment.
Basedonseveral trialsrelativetothelocalization
moduleusingknownreferencepointsandassignedpaths,
thelocalizationerrorofeachrobot isoftheorderof0.05m.
Inaddition,eachrobotisequippedwithaIEEE802.11wirelesscardthat
weusedto built a wireless ad hoc network to allow direct
informationexchangeamongtherobots. Sincethecommunicationrangeof the
wireless links is much larger than the size of theexperiments area,
the topology of the communication graph iscomplete (all-to-all
communication); however, to avoid havingacomplete communication
graph, theadjacency matrixes areassigned a priori so as to reduce
the number of communicatingneighbors.Theusedrobots aredifferential
driverobots that donothave a single integrator dynamics but a
unicycle-like kinematicmodel as xi =_ xi yi_
=_cos(i)sin(i)_viwhereiistherobotorientationsuchthat i =i,
viisthelinear velocityandiistheangularvelocity. Thedesignedcontrol
strategyproduces adesiredvelocityvector uithat,in general, cannot
be executed by robot i because of itsnonholonomic constraint. For
this reason, a low-levelcontroller has been implemented on board of
the
robots,whichisinchargeofgeneratingangularandlinearvelocitycommands
to make the robots track the assigned velocitycommandui. The
effects of the low-level control andtheestimate of the localization
error module are used in anempirical procedure toset
theFDIthresholds.Observer and control gains were set to ko = 2 and
kc = 1.3in (7) and (14), respectively. The network topology is
assumedas acircular undirected graph.The information initially
available to each robot is assumedtobelimitedtoits position(xi(t0))
andtothenumber ofrobots in the team (N). Thus, as in the simulation
case study,the initial state estimation is set equal to i
x(t0)|t0=0 = 1Nxi,i.e., each robot initializes the estimate of the
collectivestate assuming that all the other robots have its
sameinitial position (that is the only variable directly
measurable).Therobots areinitiallydisplacedinsideaclosedregionof2 m
2 m. As for the simulation case study, this informationARRICHIELLO
et al.: OBSERVER-BASED DECENTRALIZED FDISTRATEGY 1473Fig.9.
Experimentalcasestudy.Robotpathsfollowedduringthemission.Fig. 10.
Experimental casestudy. Normof thestateestimationerror x
(thickline)andofitsindividual components(thinlines).allows for
computing a conservative guess of y
k(0)in(32).Intheconsideredcase, theteamof
robotsiscommandedtoexecuteaswitchingformationcontrol
missioninalargeindoor environment (20moflength).
Specically,therobotswerecommandedtomovethecentroidalongastraightline,followingatrapezoidal
velocityprole, whiletheformationshape switches fromlinear
tocircular. Duringthe motion,we induce a temporaryinput failure to
one of the robotsby applying a null input command (4(t ) =u4(t )
for80 < t < 110 s), i.e., the robot suddenly stops (80 t
110)and recovers (t > 110).Fig.9showsthepaths oftherobots during
themissionaswell as afewintermediatepositions. Fromthegure, it
ispossiblenoticingthatrobot 4stopsbetween80< t < 110sdue to
the fault occurrence but it recovers the formation whenthefault is
over (t > 110 s).Fig. 10shows thetimehistoryof thenormof
thestateestimationerror x
andofitsindividual components; con-sideringthat x
isacumulativevectorofdimensionN2n(that, inthespeciccase, is
equal to32), it is possibletonote that at steady state and in the
absence of faults, themeanestimationerror of eachrobot is of
fewcentimeters.Thus, theestimationerrordecreasesuntil
theoccurrenceofthefaultonrobot 4(att =
80s)increaseswhenthefaultispresent (80 t 110 s) and decreases when
the fault is over(t > 110 s).Fig. 11 shows the centroid and
formationtask functionerrors ( 1and 2). In the absence of faults,
the task functionerrors remain limitedandclosetozero.
Thesmallerrors canbe justiedby thelocalization accuracy.Figs. 12
and 13 show the norms of the residual components(solid line) of the
different observers relative to therobots 1 and 4, respectively,
and also showthe adaptivethresholdvalues usedfor the fault
detection(dashedline).Fig. 13showsthat all therobotsareabletodetect
thefaultof robot 4, evenif thereis not direct communicationwithFig.
11. Experimental case study. Centroid ( 1) and formation ( 2)
trackingerrors.Fig. 12. Experimental case study. Solid line: norm
of the residual componentsir1(i =1, 2,. . . ,N) as computedbythe
different robots relative toanonfaultyrobot (robot 1inthis case).
Dashedline: thresholdusedfor thefaultdetection.Fig. 13.
Experimental case study. Solid line: norm of the residual
componentsir4 (i = 1, 2, . . . , N) as computed by the different
robots relative to the
faultyrobot(robot4).Dashedline:thresholdusedforthefaultdetection.thefaultyrobot;
moreover, it shows that theresidual
termsgobackbelowthecorrespondingthresholdswhenthefaultis over.
Finally, Fig. 12shows that thecomponents of the1474 IEEE
TRANSACTIONS ONCONTROL SYSTEMS TECHNOLOGY, VOL. 23,NO.4,JULY
2015residuals relatedtoanotfaultyvehicle(robot 1inthiscase)are not
affected by the fault occurrence. This feature is,
indeed,obtainedthankstotheintroductionof thevariable yintheobserver
in(7). Thesamehappensfor theresiduals of theother not faulty
robots; theirplot is here omitted for brevity.Avideo of the
experiment can be found in[1].VI. CONCLUSIONIn this paper, we
presented a distributed FDI strategyfor networked robots jointly
working with a distributedobservercontroller schema.
Basedonthedesignedobserver,a proper vector of residuals is
introduced, and fault
detectabil-ityandisolabilityconditionsareanalyticallyderived.Differ-ently
from previous works in literature, healthy robots are abletodetect
andisolatethefaultyones eveninthecasetheyarenot directlyconnected.
Theapproachhasbeenvalidatedvia numerical simulation considering the
case of multiplefaults, aswell aswiththeexperimental
resultswithateamofnetworkedrobotsconsideringtemporaryfaults.
Asfutureworks, the extension of the diagnosis approach to more
generalcontrol inputs andtothecaseof switchingtopologies
willbeinvestigated;moreover,afault accommodationschemaisunder
investigation.APPENDIXPROOFOF THEOREM 2.2The systemin(9) can be
writtenas x
= ko L
y
+ u
. (38)Since x
=1N x =1N (u + ) =u
+
, thesystemin(38) canbe written as x
= ko L
y
+ u
+
= u
(t , x
) + d (39)with u
= 1N u u
= [1 uT2 uT, . . . , N uT]T IRN2nandd = ko L
y
+
. Provided that the conditions inTheorem 2.1 and in (2) and (5)
hold, the above system can beseen as a dynamical systems withdas a
bounded disturbanceterm.As a consequence, the error x
in (39) exponentially reachestheorigin provided the system x
= u
(t , x
) + d (40)is exponentially stable. It is, thus, necessary to nd
an expres-sionof u
asafunctionof x
inthecaseofthecontrollawin(14).At this end, it holdsi u =u(t ) i
u(t ) =N
k=1
kk u(t ) i u(t )= kc_J1Ji x N
k=1
k J1Jk x_(41)and thus by collecting the differences u(t ) i u(t
) fori = 1, 2, . . . , N, it is u
= 1N u u
= kc_____________1 xN
k=1
kk x__2 x N
k=1
kk x_..._N x N
k=1
kk x___. (42)Let us dene thefollowing matrix:
= 1N [1,2, . . . , N] IRN2nN2n; (43)it is1N u u
= kc(IN2n
) x
. (44)The properties listed below will be exploited in
thefollowingProperty1:
(1N v) = (1N v).Property2: As the matrix(IN2n
)is idempotentrank(IN2n
) =N2n rank(
) =N2n Nn.Property3: Matrix(IN2n
) is symmetric and it hasN2n Nneigenvaluesequal
to1andNneigenvaluesequalto0.
ItisadirectconsequenceofProperty2andofthefact that (IN2n
)isidempotent(idempotentmatrices admits only 0and 1as
eigenvalues).Applying Property 1,it is straightforward to show
thatkc(IN2n
) x
= kc(IN2n
) x
.In sum, thesystem in(40) becomes x
= kc(IN2n
) x
+ d. (45)FromProperty(3),
itfollowsthatthedynamicmatrixoftheabove system is symmetric
negative semidenite. Thus, in theabsenceof measurement andprocess
noise, thesolutiontothe above systemis x
(t ) = ekc( IN2n
)t x
(t0) (46)where x
(t0)isthecollectionoftheestimationerrorsattheinitial time
instant.As anal step,it is x
= limt + x
(t ) =
x
(t0) (47)wheretheproperty limt +eAt=I AforanyidempotentmatrixA
was exploited. In sum x
=
x
(t0) = (1N [1,2, . . . , N]) x
(t0)= 1N ([1,2, . . . , N] x
(t0)). (48)Equation(48)clearlystatesthati x =j x, i, j ,
andthus, alltheestimatesi
xexponentiallyconvergetoacommonvalue.However, thereisnoevidencethat
thisvalueisx. Toshowthat the consensus value is,indeed,x, it is
worth noting that,as aconsequence of y
= 0at steady statex i x =_tt0ud _tt0i ud i (49)ARRICHIELLO et
al.: OBSERVER-BASED DECENTRALIZED FDISTRATEGY 1475andsince i(u i u)
=i(_tt0 ud _tt0i ud) =0, i ,it holds
i(x i x) = 0 (50)i.e., each vehicle is able to estimate its own
state. Thestraightforward consequence of (48) and (50) is thati
xreaches x, i = 1, 2, . . . , N. Based on Theorem 2.1 and boundsin
(2) and (5), in the presence of bounded measurement noiseand model
uncertainty, the disturb d is a nonvanishing boundedinput; thus,
x
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nonlinear uncertain systems, Int. J. Control, vol.
85,no.11,pp.16441662,2012.Filippo Arrichiello was borninNaples,
Italy, in1979.He
receivedtheLaureadegreeinmechanicalengineeringfromtheUniversityofNaples,
Naples,in 2003, and the Ph.D. degree in electrical and
infor-mationengineeringfromtheUniversityofCassinoandSouthernLazio,Cassino,Italy,
in 2007.He joined as a Visiting Ph.D. Student withtheCentreof
ExcellenceandtheCentreof Ships andOceanStructures,
NorwegianUniversityofScienceand Technology, Trondheim, Norway, in
2005. From2008to2011, hespent sevenmonthsasaVisitingResearcher
withtheRoboticEmbeddedSystems Laboratory,
UniversityofSouthernCalifornia, LosAngeles, CA, USA. Heis
currentlyanAssociateProfessor of Control Engineering with the
University of Cassino and SouthernLazio, where he held a
post-doctoralpositionand was an Assistant Professorfrom 2006 to
2014. He has authored over 50 papers published in
internationaljournals andconferences proceedings intheeldof
robotics. His researchinterests include industrial andmobile
robotics with a specic interest
inmultirobotsystemsandmarinerobotics.Dr.
ArrichiellohasbeenamemberoftheIEEERoboticsandAutomationSocietysince2004.1476
IEEE TRANSACTIONS ONCONTROL SYSTEMS TECHNOLOGY, VOL. 23,NO.4,JULY
2015AlessandroMarinowasborninPotenza, Italy, in1982.
HereceivedtheLaureadegree incomputerengineeringfrom theUniversityof
NaplesFedericoII, Naples, Italy, in 2006, and the Ph.D. degreein
industrial and innovation engineering
fromtheUniversityofBasilicata, Potenza, in
2010.Heheldapost-doctoral positionwiththeUniver-sityofCassino,
Cassino, Italy, from2010to2011.In 2008, he was a VisitingScholar
with the Depart-ment of Computer Science, University of
Tennessee,Knoxville, TN, USA, and a Visiting Researcher
withtheInstitutoSuperiorTcnico,Lisboa,Portugal,in 2011.Since2011,
hehasbeenanAssistant Professor withtheUniversityof Salerno,
Salerno, Italy.Hehas authoredseveral conferencepapers andjournal
papers. Hiscurrentresearch interests include cooperativerobot
manipulation, trajectory
planning,controlofmultirobotsystems,andmarinerobotics.Dr.
Marinoiscurrentlyamember oftheIEEERobotics
andAutomationSocietyandtheIEEEControl SystemSociety. Since2013,
hehasbeenanAssociate Editor of the IEEE INTERNATIONAL CONFERENCE ON
ROBOTICSANDAUTOMATION. He is also on the ProgramCommittee of the
IEEEInternational
ConferenceonRoboticsandBiomimetics.FrancescoPierri
receivedtheLaurea(cumlaude)degree in mechanical engineering and
thePh.D. degree in environmental engineering
fromtheUniversityofBasilicata, Potenza, Italy,
in2003and2007,respectively.He was aVisitingScholar
withtheDepartmentof Automatic Control, Lund University,
Lund,Sweden, in 2006. Since 2008, he has been anAssistantProfessor
with the Schoolof Engineering,University of Basilicata.He has
co-authored over 35journal andconferencepapers
andabookentitledControl andMonitoringof Chemical
BatchReactors(Springer, 2011). Hiscurrent researchinterests
includeplanningandcontrol of aerial
robotsandfaultdiagnosisandfault-tolerant
controlfornonlinearsystems.Dr. Pierri has been a member of the
IEEERobotics and AutomationSocietyandtheIEEEControl
SystemSocietysince2004. Since2013, hehasbeenanAssociateEditor of
theInternational Journal of Robotics andAutomation. In2012, 2013,
and2014, hewas anAssociate Editor of
theIEEEINTERNATIONALCONFERENCEONROBOTICSANDAUTOMATION.He is on the
ProgramCommitteeof the 2014 IEEE InternationalConferenceon
RoboticsandBiomimetics.