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:f.arrichiello@unicas.it).A.MarinoiswiththeUniversityofSalerno, Salerno84084,Italy(e-mail:almarino@unisa.it).F.PierriiswiththeUniversityofBasilicata, Potenza85100,Italy(e-mail:francesco.pierri@unibas.it).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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Johansson,Distributedfault detectionfor interconnected second-order systems,Automatica, vol.47,no.12,pp.27572764, 2011.[28] R.S.SmithandF.Y.Hadaegh, Closed-loopdynamicsofcooperativevehicleformationswithparallel estimatorsandcommunication,IEEETrans.Autom.Control, vol.52,no.8,pp.14041414, Aug.2007.[29] D.P. Spanos,R.Olfati-Saber, andR.M.Murray,Dynamicconsensusonmobilenetworks,inProc.IFAC WorldCongr.,2005.[30] S. Stankovic, N. Ilic, Z. Djurovic, M. Stankovic, andK. H. Johansson, Consensus based overlapping decentralized faultdetection andisolation, inProc. Conf. Control Fault-Tolerant Syst.,Oct.2010,pp.570575.[31] Y.Wang,H. Ye,S.X. Ding,Y.Cheng,P. Zhang,andG. Wang,Faultdetectionof networkedcontrol systemswithlimitedcommunication,Int. J.Control, vol.82,no.7,pp.13441356, 2009.[32] P. Yang, R. A. Freeman, andK. M. Lynch, Multi-agent coordinationbydecentralizedestimationandcontrol,IEEETrans. Autom. Control,vol.53,no.11,pp.24802496, Dec.2008.[33] X. Zhang, Decentralized fault detection for a class of large-scalenonlinearuncertainsystems,inProc. Amer. Control Conf., Baltimore,MD,USA, Jun./Jul.2010,pp.56505655.[34] X. Zhang and Q. Zhang, Distributed fault diagnosis in a class ofinterconnected 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.