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Research ArticleDynamic Output Feedback Based ActiveDecentralized Fault-Tolerant Control for ReconfigurableManipulator with Concurrent Failures
Yuanchun Li1 Fan Zhou1 and Bo Zhao12
1Department of Control Engineering Changchun University of Technology Changchun 130012 China2State Key Laboratory of Management and Control for Complex Systems Institute of AutomationChinese Academy of Sciences Beijing 100190 China
Correspondence should be addressed to Bo Zhao zhaob09mailsjlueducn
Received 10 July 2014 Revised 27 November 2014 Accepted 29 November 2014
Academic Editor Gerhard-WilhelmWeber
Copyright copy 2015 Yuanchun Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The goal of this paper is to describe an active decentralized fault-tolerant control (ADFTC) strategy based on dynamic outputfeedback for reconfigurable manipulators with concurrent actuator and sensor failures Consider each joint module of thereconfigurable manipulator as a subsystem and treat the fault as the unknown input of the subsystem Firstly by virtue oflinear matrix inequality (LMI) technique the decentralized proportional-integral observer (DPIO) is designed to estimate andcompensate the sensor fault online hereafter the compensated systemmodel could be derivedThen the actuator fault is estimatedsimilarly by another DPIO using LMI as well and the sufficient condition of the existence of 119867
infinfault-tolerant controller in the
dynamic output feedback is presented for the compensated system model Furthermore the dynamic output feedback controlleris presented based on the estimation of actuator fault to realize active fault-tolerant control Finally two 3-DOF reconfigurablemanipulators with different configurations are employed to verify the effectiveness of the proposed scheme in simulationThemainadvantages of the proposed scheme lie in that it can handle the concurrent faults act on the actuator and sensor on the same jointmodule as well as there is no requirement of fault detection and isolation process moreover it is more feasible to the modularityof the reconfigurable manipulator
1 Introduction
The rapid development of robotics leads the reconfigurablemanipulators to be variously applied to the potential unstruc-tured environments especially in the fields where humancannot intervene directly such as the space station nuclearpower plant and battle field However once the faultappeared in the system it might deteriorate the performanceor cause the loss of the system functionality even stabil-ity As a result there is an increasing demand for safetyreliability and performance of reconfigurable manipulatorsystems Therefore it is an urgent requirement to designcontrol systems which can tolerate the occurrence of failuresduring the operation in order to guarantee the stability andfunctionality and maintain the acceptable performance aswell
Generally speaking two strategies namely the passivefault-tolerant control (PFTC) and the active fault-tolerantcontrol (AFTC) were carried out to achieve the aim at FTC[1] For the PFTC the control structure and parameters havebeen redesigned to go against the occurring of failures Thismeans that the FTC was fixed to tolerate a certain set of faultswithout any change in the controller Du et al [2] obtained thefault information by estimating the outputs of the actuatorsand then compared them with the corresponding prescribedcontrol inputs hereafter the FTC was developed by choosinga safe-park point Jiang et al [3] presented the sliding modeFTC method in the view of the nonlinear flexible spacecraftflywheel failure while in fact it was difficult to obtain theminimum value of the spacecraft flywheel fault Brambillaet al [4] adopted an optimal second-order sliding modecontrol method to design observer-control law by using
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 528086 14 pageshttpdxdoiorg1011552015528086
2 Mathematical Problems in Engineering
the unknown input observer and generalized observer toanalyze residuals but this method can only detect a singlecomponent failure In [5] a decentralized tuning PID outputfeedback controller was utilized to ensure the stability of largeflexible space structures (LFSS) suffered sensor and actuatorfailures Moreover a common solution in the PFTC whensome severe failures are taken into account does not alwaysexist In addition it usually presented a low performanceeven though it exists On the other hand the AFTC maychange the control structure andor parameters to overcomethe bad effect on the whole control systems aroused by thefault Even when necessary it needs to introduce a detectionand estimation module to detect and estimate when thefault occurs Hereafter a supervisory controller should bereconstructed based on the estimated information in the caseof the occurrence of severe faults [6 7] such that it canguarantee the faulty systemrsquos stability and provide acceptablecontrol performance In [8ndash12] only single fault is handledwith AFTC but the concurrent failures on actuator andsensor always occur in actual fact In this regard Rotondo etal [13] used virtual actuator and sensor to correct the actualactuator and sensor faults which achieved the objective ofFTC based on the dynamical controller reconfiguration Samiand Patton [14] proposed a new architecture based on acombination of actuator and sensor Takagi-Sugeno (T-S)proportional state estimators augmented with proportionaland integral feedback (PPI) fault estimators together witha T-S dynamic output feedback control for time-varyingreference tracking
In recent literatures some effort has been made forthe reconfigurable manipulators in fault Yuan et al [15]introduced an energy efficiency monitor approach to detectthe fault where the operation failure was reflected by theefficiency decline of mechanical systemThemeasurement ofeach joint torque is used not only to control the running statebut also to reflect the output capacity The method based ontorque measurement is independent of the whole dynamicmodel of robot systems Ahmad et al [16] presented a dis-tributed fault detection method which can gain the forecasterror through comparing the joint torque signal and torqueestimation being filtered Zhao and Li [17] were concernedwith the active fault-tolerant control problem for reconfig-urablemanipulator actuator based on local joint informationThis scheme processed a simple control structure as well asthe fault could be isolated and tolerated in subsystem and itcan be easily applied to different configurations without anyparameters modification However only a single fault in theactuator or sensor was taken into account to be handled inthe aforementioned methods which limited the availabilityin practice
There are severalmethods successfully used in controllingreconfigurable manipulator In centralized control approachLi et al [18] utilized the elastic parameters of joint modulewhich were identified by fuzzy logic to build finite elementmodel of reconfigurable robot then based on the BP neuralnetwork and genetic algorithm the vibration control methodwas proposed based on the finite element model Sun et al[19] divided 4-DOF module into posture coupled subsys-tems and position feedback subsystem and simultaneously
decomposedworkspace into the above two subspaces to solvethe inverse position problem through forecasting methodBiglarbegian et al [20] presented Type-2 TSK fuzzy logiccontrol method aimed at the reconfigurable manipulatorswith uncertain dynamic parameters This control structurehad a complex and fixed control structure and lackedflexibility thus it was difficult to be implemented to thereconfigurable manipulator when its configuration changedThe other one was distributed control method Muller etal [21] simplified the hardware of control system to ensurethe flexibility of system reconstruction and coordinated tooperate all modular robots through independent centralcontrol Zhu and Lamarche [22] described the system as aset of subsystems through virtual decomposition and thenused the exchange information amongmodules to design thesubsystemrsquos controller The distributed control method canreduce computational complexity and has a more harmo-nious and flexible structure compared to those in centralizedcontrol it makes the system compatibility not only betterbut also more suitable to the concept of modularizationThisdistributed control could conduct more thorough coordinatecontrol however the time delay in communication can resultin imprecise control performance To reduce the difficultyin controller design the decentralized control strategy isdeveloped in a large-scale system In fact the main propertyof the reconfigurable manipulator system lies in differentconfigurations and different degree of freedom Therefore itis more suitable to take a jointmodule as a subsystem and thedecentralized control method can satisfy its main propertyKirchoff and Melek [23] designed a PID robust controllerbased on independent joint information for industrial robotLi [24] introduced a dispersion saturated type of robustcontrol method only considering the single joint dynamicsafter the system was decoupled and treated the influence ofother jointsrsquo dynamics as external disturbanceThe controllerdesign in decentralized control approach utilizes only localinformation thus it is more suitable for the system with anuncertain degree of freedom and different configurations
This paper tries to address an ADFTC for reconfigurablemanipulator with concurrent failures This idea focuses onthe observer design for isolating and estimating the actuatorand sensor faults for the purpose of fault compensation Itdecomposes the entire system into a set of interconnectedsubsystems for developing decentralized control architectureADPIO is designed through using LMI technique to estimateand compensate the sensor fault online and the compen-sated system model is derived Similarly another DPIO isestablished with the sufficient condition of the existence of119867infinfault-tolerant controller and presented in the presence of
the dynamic output feedback Simultaneously the ADFTC isrealized by the estimation of the faults based on the dynamicoutput feedback Finally simulation results show the stabilityand accuracy in the tracking system with simultaneouslyacting actuators and sensors faults
The main advantages of the proposed approach lie inthe following (i) Only local information is used to designthe ADFTC for reconfigurable manipulator with the conceptof decentralized control which can tolerate the concurrentfaults acting in actuator and sensor in an independent joint
Mathematical Problems in Engineering 3
module (ii) LMI technique is used in the design procedure ofDPIOs and dynamic output feedback controller simplifyingthe control structure and making the proof process of systemstability easier on the condition of ensuring the systemstability (iii) There is no requirement of FDI unit here soit saves the reconfiguration time which is necessary in theconventional AFTC (iv) Compared to the existing resultsthe dynamic output feedback is utilized as the state feedbackin the proposed scheme meanwhile it could balance thecontradictions between the irreplaceable state feedback andthe difficulty in physical realization
This paper is presented in the following order Section 2describes nonlinear interconnected subsystem dynamicmodel of the reconfigurable manipulator including thesystems with fault or without fault Section 3 enters into adescription of the observers followed by two subsectionswhich illustrate the stability and performance designconditions for (i) sensor fault estimate observer and (ii)the actuator fault estimate observer In Section 4 thedynamic output feedback controller is designed and itillustrates the stability and performance design conditionsIn Section 5 the effectiveness of the proposed ADFTCmethod is verified by the simulation results of two 3-DOFreconfigurable manipulators with different configurationsSome conclusions are drawn in Section 6
2 Problem Description
For the development of decentralized control consider theentire reconfigurable manipulator with 119899-DOF as a set ofnonlinear interconnected subsystems which are composedof a general joint module And the subsystem 119868 in thereconfigurable manipulator system can be presented by thefollowing state equation [25]
positive-definite functionThe control objective is to design an active decentralized
fault-tolerant controller in order to guarantee the wholeclosed-loop system stability in the case of the system sufferingconcurrent actuator and sensor faults In other words theproposed fault-tolerant control scheme should make theoutputs of the entire system follow the desired trajectorieseven though concurrent faults occur
31 Sensor Fault Observer Design In this subsection adecentralized proportional-integral observer is designed forthe faulty dynamic model (3) in order to estimate the sensorfault
Assumption 1 The desired trajectories 119902119889119894 119902119889
119894 and 119902
119889
119894are
bounded
Assumption 2 The subsystem actuator fault function119891119894119886(119902119894 119902119894 119894) and the sensor fault function 119891
where 120578119894119891and 120578119894119892are positive constants
Note that a challenge in implementing the decentralizedcontrol is to compensate the coupling torque caused bythe interconnected joint modules In such a scenario thefollowing assumption is presented
Assumption 3 The interconnection term ℎ119894(119902 119902 119902) is
bounded by [25]
1003816100381610038161003816ℎ119894 (119902119902 119902)1003816100381610038161003816 le
119899
sum
119895=1
119889119894119895119864119895 (10)
with 119889119894119895ge 0 and 119864
119895= 1 + 119890
119879
119894119875119894119861119894 + 119890
119879
1198941198751198941198611198942
Similarly another RBF neural network term 119894(119890119879
119894119875119894119861119894
119894119901) is introduced to compensate the effect of interconnec-
119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894) 119870119894V119862119894119863119894
]
119873119894119904= [
119861119894
0
119870119894V119862119894119861119894 119868
] 119911119894119904= [
119898119904
119891119894119904
]
(19)
Lemma4 (see [28]) In the given system the eigenvalues of thesystem are located in a LMI region in the complex plane definedby 119863(119902 119903) which is defined by merging different eigenvaluesconstraints to produce a119863(119902 119903)LMI region inwhich 119902 and 119903 arethe radius and center of the disc region If there exist symmetric
Mathematical Problems in Engineering 5
positive-definite matrices 119875 and 119876 and matrices 119870119894119901 119870119894119897 and
119870119894V as well as the corresponding LMI such that
performance is guaran-teed with an attenuation level 120574
Theorem 5 Based on Lemma 4 given 120574 gt 0 and error systemmodel (18) if there exist symmetric positive-definite matrices 119875and 119876 and matrices 119870
119894119901 119870119894119897 and 119870
119894V as well as matrix LMIsuch that (20) holds then system (18) is robust asymptoticallystable and satisfies the119867
Inequality (22) can be obtained Therefore the observersatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 5
32 Actuator Fault Observer Design This subsection designsthe actuator fault estimator alongwith the observer driven bythe corrected (sensor fault compensated) output and controlsignalsTherefore the system given in (3) can be converted to
Theorem 6 Also based on Lemma 4 given 120574 gt 0 and errorsystem model (31) if there exist symmetric positive-definitematrices 119875 and 119876 and matrices 119870
119875119886 119870119894119886 and 119870V119886 as well as
hold then system (31) is robust asymptotically stable andsatisfies the119867
infinperformance indicator as follows10038171003817100381710038171198901198941198861003817100381710038171003817
2
le 12057410038171003817100381710038171199111198941198861003817100381710038171003817
2
+ 119881 (0) (37)
where 1198901198941198862
= int1199051
0
(119890119894119886
119879
119890119894119886)119889119905 119911
1198941198862
= int1199051
0
(119911119894119886
119879
119911119894119886)119889119905 and 119868
119890
119868119904 and 119868
1are unit matrices
Proof The proof procedure ofTheorem 6 is similar to that ofTheorem 5 here it is omitted
4 Active Decentralized Fault-TolerantController Design
In this section the ADFTC based on dynamic output feed-back is designed to ensure the stability and tracking accuracyof a reconfigurable manipulator with acting actuator andsensor faults concurrently
Considering the faulty subsystem dynamic model (29)the decentralized fault-tolerant controller is designed asfollows
Similarly based on Lemma 4 the existing conditionof dynamic output feedback robust control is given inTheorem 7
Theorem 7 Given 120574 gt 0 and fault subsystems dynamic model(29) if there exist symmetric positive-definite matrices119875 and119876and matrices 119860
119894119888 119861119894119888 119862119894119888 and 119863
119894119897119909119894119897) 119889119905 le int
1199051
0
120574119889119879
119894119889119894119889119905 + 119881 (0) (49)
Inequality (43) can be obtained Therefore the systemsatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 7
5 Simulation Results
To verify the effectiveness of the proposed ADFTC strategyin this subsection two 3-DOF reconfigurable manipulatorswith different configurations shown in Figure 1 are employedto illustrate the simulation results
The initial position and velocity are set as 1199021(0) = 119902
2(0) =
1199023(0) = 1 and 119902
1(0) = 119902
2(0) = 119902
3(0) = 0 respectively
By solving the LMI conditions given in 1 2 and 3 thedynamic output feedback controller and observer gains are
1198601119888=
[[[[[
[
17261 29781 44521 63562
11781 21535 33562 53070
minus10682 minus15344 45082 51247
minus09344 minus14606 31247 41075
]]]]]
]
1198602119888=
[[[[[
[
10261 28531 40951 59062
11141 20031 31639 50250
minus10009 minus13614 41163 49428
minus08314 minus12171 27319 40014
]]]]]
]
8 Mathematical Problems in Engineering
I[1]
I[2]
I[2]
I[2]
I[2]
I[2]
O[2]
O[2]
O[2]O[2]
O[2]O[2]
L[0]
L[0]
L[1]L[0]
L[0]
L[1]
L[3]
L[3]
T
T
BB
Configuration bConfiguration a
Figure 1 Configurations of 3-DOF reconfigurable manipulators
Figure 2 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
10 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 3 p
ositi
on (r
ad)
minus1
minus05
minus08
minus06
minus04
minus02
minus06
minus04
minus02
Figure 3 The tolerant tracking performance of configuration 119886
The actuator fault observer gains are calculated as
1198701119901=
[[[[[[
[
271343 minus224461 157691 08612
40364 01558 177803 11852
141557 117100 82266 minus129699
69872 57800 40606 minus64019
]]]]]]
]
1198702119901=
[[[[[[
[
258643 minus204813 125191 07132
40374 01098 137013 10352
13977 117100 80136 minus119819
58924 51320 39836 minus53419
]]]]]]
]
1198703119901=
[[[[[[
[
221463 minus204001 117251 0702112
40364 09958 129803 09562
138937 117100 82266 minus129699
58924 50972 39001 minus52919
]]]]]]
]
1198701V = [65289 31970 52071 65190]
1198702V = [60963 30023 51325 62351]
1198703V = [59235 28971 50701 61180]
1198701119897= [50231 35576 75657 64152]
1198702119897= [49421 30893 71097 62321]
1198703119897= [48511 29416 65725 60757]
(52)
Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578
119894119891= 0002
120578119894119892= 0002 and 120578
119894119901= 500 and the 119867
infinperformance
indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added
to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
the unknown input observer and generalized observer toanalyze residuals but this method can only detect a singlecomponent failure In [5] a decentralized tuning PID outputfeedback controller was utilized to ensure the stability of largeflexible space structures (LFSS) suffered sensor and actuatorfailures Moreover a common solution in the PFTC whensome severe failures are taken into account does not alwaysexist In addition it usually presented a low performanceeven though it exists On the other hand the AFTC maychange the control structure andor parameters to overcomethe bad effect on the whole control systems aroused by thefault Even when necessary it needs to introduce a detectionand estimation module to detect and estimate when thefault occurs Hereafter a supervisory controller should bereconstructed based on the estimated information in the caseof the occurrence of severe faults [6 7] such that it canguarantee the faulty systemrsquos stability and provide acceptablecontrol performance In [8ndash12] only single fault is handledwith AFTC but the concurrent failures on actuator andsensor always occur in actual fact In this regard Rotondo etal [13] used virtual actuator and sensor to correct the actualactuator and sensor faults which achieved the objective ofFTC based on the dynamical controller reconfiguration Samiand Patton [14] proposed a new architecture based on acombination of actuator and sensor Takagi-Sugeno (T-S)proportional state estimators augmented with proportionaland integral feedback (PPI) fault estimators together witha T-S dynamic output feedback control for time-varyingreference tracking
In recent literatures some effort has been made forthe reconfigurable manipulators in fault Yuan et al [15]introduced an energy efficiency monitor approach to detectthe fault where the operation failure was reflected by theefficiency decline of mechanical systemThemeasurement ofeach joint torque is used not only to control the running statebut also to reflect the output capacity The method based ontorque measurement is independent of the whole dynamicmodel of robot systems Ahmad et al [16] presented a dis-tributed fault detection method which can gain the forecasterror through comparing the joint torque signal and torqueestimation being filtered Zhao and Li [17] were concernedwith the active fault-tolerant control problem for reconfig-urablemanipulator actuator based on local joint informationThis scheme processed a simple control structure as well asthe fault could be isolated and tolerated in subsystem and itcan be easily applied to different configurations without anyparameters modification However only a single fault in theactuator or sensor was taken into account to be handled inthe aforementioned methods which limited the availabilityin practice
There are severalmethods successfully used in controllingreconfigurable manipulator In centralized control approachLi et al [18] utilized the elastic parameters of joint modulewhich were identified by fuzzy logic to build finite elementmodel of reconfigurable robot then based on the BP neuralnetwork and genetic algorithm the vibration control methodwas proposed based on the finite element model Sun et al[19] divided 4-DOF module into posture coupled subsys-tems and position feedback subsystem and simultaneously
decomposedworkspace into the above two subspaces to solvethe inverse position problem through forecasting methodBiglarbegian et al [20] presented Type-2 TSK fuzzy logiccontrol method aimed at the reconfigurable manipulatorswith uncertain dynamic parameters This control structurehad a complex and fixed control structure and lackedflexibility thus it was difficult to be implemented to thereconfigurable manipulator when its configuration changedThe other one was distributed control method Muller etal [21] simplified the hardware of control system to ensurethe flexibility of system reconstruction and coordinated tooperate all modular robots through independent centralcontrol Zhu and Lamarche [22] described the system as aset of subsystems through virtual decomposition and thenused the exchange information amongmodules to design thesubsystemrsquos controller The distributed control method canreduce computational complexity and has a more harmo-nious and flexible structure compared to those in centralizedcontrol it makes the system compatibility not only betterbut also more suitable to the concept of modularizationThisdistributed control could conduct more thorough coordinatecontrol however the time delay in communication can resultin imprecise control performance To reduce the difficultyin controller design the decentralized control strategy isdeveloped in a large-scale system In fact the main propertyof the reconfigurable manipulator system lies in differentconfigurations and different degree of freedom Therefore itis more suitable to take a jointmodule as a subsystem and thedecentralized control method can satisfy its main propertyKirchoff and Melek [23] designed a PID robust controllerbased on independent joint information for industrial robotLi [24] introduced a dispersion saturated type of robustcontrol method only considering the single joint dynamicsafter the system was decoupled and treated the influence ofother jointsrsquo dynamics as external disturbanceThe controllerdesign in decentralized control approach utilizes only localinformation thus it is more suitable for the system with anuncertain degree of freedom and different configurations
This paper tries to address an ADFTC for reconfigurablemanipulator with concurrent failures This idea focuses onthe observer design for isolating and estimating the actuatorand sensor faults for the purpose of fault compensation Itdecomposes the entire system into a set of interconnectedsubsystems for developing decentralized control architectureADPIO is designed through using LMI technique to estimateand compensate the sensor fault online and the compen-sated system model is derived Similarly another DPIO isestablished with the sufficient condition of the existence of119867infinfault-tolerant controller and presented in the presence of
the dynamic output feedback Simultaneously the ADFTC isrealized by the estimation of the faults based on the dynamicoutput feedback Finally simulation results show the stabilityand accuracy in the tracking system with simultaneouslyacting actuators and sensors faults
The main advantages of the proposed approach lie inthe following (i) Only local information is used to designthe ADFTC for reconfigurable manipulator with the conceptof decentralized control which can tolerate the concurrentfaults acting in actuator and sensor in an independent joint
Mathematical Problems in Engineering 3
module (ii) LMI technique is used in the design procedure ofDPIOs and dynamic output feedback controller simplifyingthe control structure and making the proof process of systemstability easier on the condition of ensuring the systemstability (iii) There is no requirement of FDI unit here soit saves the reconfiguration time which is necessary in theconventional AFTC (iv) Compared to the existing resultsthe dynamic output feedback is utilized as the state feedbackin the proposed scheme meanwhile it could balance thecontradictions between the irreplaceable state feedback andthe difficulty in physical realization
This paper is presented in the following order Section 2describes nonlinear interconnected subsystem dynamicmodel of the reconfigurable manipulator including thesystems with fault or without fault Section 3 enters into adescription of the observers followed by two subsectionswhich illustrate the stability and performance designconditions for (i) sensor fault estimate observer and (ii)the actuator fault estimate observer In Section 4 thedynamic output feedback controller is designed and itillustrates the stability and performance design conditionsIn Section 5 the effectiveness of the proposed ADFTCmethod is verified by the simulation results of two 3-DOFreconfigurable manipulators with different configurationsSome conclusions are drawn in Section 6
2 Problem Description
For the development of decentralized control consider theentire reconfigurable manipulator with 119899-DOF as a set ofnonlinear interconnected subsystems which are composedof a general joint module And the subsystem 119868 in thereconfigurable manipulator system can be presented by thefollowing state equation [25]
positive-definite functionThe control objective is to design an active decentralized
fault-tolerant controller in order to guarantee the wholeclosed-loop system stability in the case of the system sufferingconcurrent actuator and sensor faults In other words theproposed fault-tolerant control scheme should make theoutputs of the entire system follow the desired trajectorieseven though concurrent faults occur
31 Sensor Fault Observer Design In this subsection adecentralized proportional-integral observer is designed forthe faulty dynamic model (3) in order to estimate the sensorfault
Assumption 1 The desired trajectories 119902119889119894 119902119889
119894 and 119902
119889
119894are
bounded
Assumption 2 The subsystem actuator fault function119891119894119886(119902119894 119902119894 119894) and the sensor fault function 119891
where 120578119894119891and 120578119894119892are positive constants
Note that a challenge in implementing the decentralizedcontrol is to compensate the coupling torque caused bythe interconnected joint modules In such a scenario thefollowing assumption is presented
Assumption 3 The interconnection term ℎ119894(119902 119902 119902) is
bounded by [25]
1003816100381610038161003816ℎ119894 (119902119902 119902)1003816100381610038161003816 le
119899
sum
119895=1
119889119894119895119864119895 (10)
with 119889119894119895ge 0 and 119864
119895= 1 + 119890
119879
119894119875119894119861119894 + 119890
119879
1198941198751198941198611198942
Similarly another RBF neural network term 119894(119890119879
119894119875119894119861119894
119894119901) is introduced to compensate the effect of interconnec-
119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894) 119870119894V119862119894119863119894
]
119873119894119904= [
119861119894
0
119870119894V119862119894119861119894 119868
] 119911119894119904= [
119898119904
119891119894119904
]
(19)
Lemma4 (see [28]) In the given system the eigenvalues of thesystem are located in a LMI region in the complex plane definedby 119863(119902 119903) which is defined by merging different eigenvaluesconstraints to produce a119863(119902 119903)LMI region inwhich 119902 and 119903 arethe radius and center of the disc region If there exist symmetric
Mathematical Problems in Engineering 5
positive-definite matrices 119875 and 119876 and matrices 119870119894119901 119870119894119897 and
119870119894V as well as the corresponding LMI such that
performance is guaran-teed with an attenuation level 120574
Theorem 5 Based on Lemma 4 given 120574 gt 0 and error systemmodel (18) if there exist symmetric positive-definite matrices 119875and 119876 and matrices 119870
119894119901 119870119894119897 and 119870
119894V as well as matrix LMIsuch that (20) holds then system (18) is robust asymptoticallystable and satisfies the119867
Inequality (22) can be obtained Therefore the observersatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 5
32 Actuator Fault Observer Design This subsection designsthe actuator fault estimator alongwith the observer driven bythe corrected (sensor fault compensated) output and controlsignalsTherefore the system given in (3) can be converted to
Theorem 6 Also based on Lemma 4 given 120574 gt 0 and errorsystem model (31) if there exist symmetric positive-definitematrices 119875 and 119876 and matrices 119870
119875119886 119870119894119886 and 119870V119886 as well as
hold then system (31) is robust asymptotically stable andsatisfies the119867
infinperformance indicator as follows10038171003817100381710038171198901198941198861003817100381710038171003817
2
le 12057410038171003817100381710038171199111198941198861003817100381710038171003817
2
+ 119881 (0) (37)
where 1198901198941198862
= int1199051
0
(119890119894119886
119879
119890119894119886)119889119905 119911
1198941198862
= int1199051
0
(119911119894119886
119879
119911119894119886)119889119905 and 119868
119890
119868119904 and 119868
1are unit matrices
Proof The proof procedure ofTheorem 6 is similar to that ofTheorem 5 here it is omitted
4 Active Decentralized Fault-TolerantController Design
In this section the ADFTC based on dynamic output feed-back is designed to ensure the stability and tracking accuracyof a reconfigurable manipulator with acting actuator andsensor faults concurrently
Considering the faulty subsystem dynamic model (29)the decentralized fault-tolerant controller is designed asfollows
Similarly based on Lemma 4 the existing conditionof dynamic output feedback robust control is given inTheorem 7
Theorem 7 Given 120574 gt 0 and fault subsystems dynamic model(29) if there exist symmetric positive-definite matrices119875 and119876and matrices 119860
119894119888 119861119894119888 119862119894119888 and 119863
119894119897119909119894119897) 119889119905 le int
1199051
0
120574119889119879
119894119889119894119889119905 + 119881 (0) (49)
Inequality (43) can be obtained Therefore the systemsatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 7
5 Simulation Results
To verify the effectiveness of the proposed ADFTC strategyin this subsection two 3-DOF reconfigurable manipulatorswith different configurations shown in Figure 1 are employedto illustrate the simulation results
The initial position and velocity are set as 1199021(0) = 119902
2(0) =
1199023(0) = 1 and 119902
1(0) = 119902
2(0) = 119902
3(0) = 0 respectively
By solving the LMI conditions given in 1 2 and 3 thedynamic output feedback controller and observer gains are
1198601119888=
[[[[[
[
17261 29781 44521 63562
11781 21535 33562 53070
minus10682 minus15344 45082 51247
minus09344 minus14606 31247 41075
]]]]]
]
1198602119888=
[[[[[
[
10261 28531 40951 59062
11141 20031 31639 50250
minus10009 minus13614 41163 49428
minus08314 minus12171 27319 40014
]]]]]
]
8 Mathematical Problems in Engineering
I[1]
I[2]
I[2]
I[2]
I[2]
I[2]
O[2]
O[2]
O[2]O[2]
O[2]O[2]
L[0]
L[0]
L[1]L[0]
L[0]
L[1]
L[3]
L[3]
T
T
BB
Configuration bConfiguration a
Figure 1 Configurations of 3-DOF reconfigurable manipulators
Figure 2 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
10 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 3 p
ositi
on (r
ad)
minus1
minus05
minus08
minus06
minus04
minus02
minus06
minus04
minus02
Figure 3 The tolerant tracking performance of configuration 119886
The actuator fault observer gains are calculated as
1198701119901=
[[[[[[
[
271343 minus224461 157691 08612
40364 01558 177803 11852
141557 117100 82266 minus129699
69872 57800 40606 minus64019
]]]]]]
]
1198702119901=
[[[[[[
[
258643 minus204813 125191 07132
40374 01098 137013 10352
13977 117100 80136 minus119819
58924 51320 39836 minus53419
]]]]]]
]
1198703119901=
[[[[[[
[
221463 minus204001 117251 0702112
40364 09958 129803 09562
138937 117100 82266 minus129699
58924 50972 39001 minus52919
]]]]]]
]
1198701V = [65289 31970 52071 65190]
1198702V = [60963 30023 51325 62351]
1198703V = [59235 28971 50701 61180]
1198701119897= [50231 35576 75657 64152]
1198702119897= [49421 30893 71097 62321]
1198703119897= [48511 29416 65725 60757]
(52)
Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578
119894119891= 0002
120578119894119892= 0002 and 120578
119894119901= 500 and the 119867
infinperformance
indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added
to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
module (ii) LMI technique is used in the design procedure ofDPIOs and dynamic output feedback controller simplifyingthe control structure and making the proof process of systemstability easier on the condition of ensuring the systemstability (iii) There is no requirement of FDI unit here soit saves the reconfiguration time which is necessary in theconventional AFTC (iv) Compared to the existing resultsthe dynamic output feedback is utilized as the state feedbackin the proposed scheme meanwhile it could balance thecontradictions between the irreplaceable state feedback andthe difficulty in physical realization
This paper is presented in the following order Section 2describes nonlinear interconnected subsystem dynamicmodel of the reconfigurable manipulator including thesystems with fault or without fault Section 3 enters into adescription of the observers followed by two subsectionswhich illustrate the stability and performance designconditions for (i) sensor fault estimate observer and (ii)the actuator fault estimate observer In Section 4 thedynamic output feedback controller is designed and itillustrates the stability and performance design conditionsIn Section 5 the effectiveness of the proposed ADFTCmethod is verified by the simulation results of two 3-DOFreconfigurable manipulators with different configurationsSome conclusions are drawn in Section 6
2 Problem Description
For the development of decentralized control consider theentire reconfigurable manipulator with 119899-DOF as a set ofnonlinear interconnected subsystems which are composedof a general joint module And the subsystem 119868 in thereconfigurable manipulator system can be presented by thefollowing state equation [25]
positive-definite functionThe control objective is to design an active decentralized
fault-tolerant controller in order to guarantee the wholeclosed-loop system stability in the case of the system sufferingconcurrent actuator and sensor faults In other words theproposed fault-tolerant control scheme should make theoutputs of the entire system follow the desired trajectorieseven though concurrent faults occur
31 Sensor Fault Observer Design In this subsection adecentralized proportional-integral observer is designed forthe faulty dynamic model (3) in order to estimate the sensorfault
Assumption 1 The desired trajectories 119902119889119894 119902119889
119894 and 119902
119889
119894are
bounded
Assumption 2 The subsystem actuator fault function119891119894119886(119902119894 119902119894 119894) and the sensor fault function 119891
where 120578119894119891and 120578119894119892are positive constants
Note that a challenge in implementing the decentralizedcontrol is to compensate the coupling torque caused bythe interconnected joint modules In such a scenario thefollowing assumption is presented
Assumption 3 The interconnection term ℎ119894(119902 119902 119902) is
bounded by [25]
1003816100381610038161003816ℎ119894 (119902119902 119902)1003816100381610038161003816 le
119899
sum
119895=1
119889119894119895119864119895 (10)
with 119889119894119895ge 0 and 119864
119895= 1 + 119890
119879
119894119875119894119861119894 + 119890
119879
1198941198751198941198611198942
Similarly another RBF neural network term 119894(119890119879
119894119875119894119861119894
119894119901) is introduced to compensate the effect of interconnec-
119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894) 119870119894V119862119894119863119894
]
119873119894119904= [
119861119894
0
119870119894V119862119894119861119894 119868
] 119911119894119904= [
119898119904
119891119894119904
]
(19)
Lemma4 (see [28]) In the given system the eigenvalues of thesystem are located in a LMI region in the complex plane definedby 119863(119902 119903) which is defined by merging different eigenvaluesconstraints to produce a119863(119902 119903)LMI region inwhich 119902 and 119903 arethe radius and center of the disc region If there exist symmetric
Mathematical Problems in Engineering 5
positive-definite matrices 119875 and 119876 and matrices 119870119894119901 119870119894119897 and
119870119894V as well as the corresponding LMI such that
performance is guaran-teed with an attenuation level 120574
Theorem 5 Based on Lemma 4 given 120574 gt 0 and error systemmodel (18) if there exist symmetric positive-definite matrices 119875and 119876 and matrices 119870
119894119901 119870119894119897 and 119870
119894V as well as matrix LMIsuch that (20) holds then system (18) is robust asymptoticallystable and satisfies the119867
Inequality (22) can be obtained Therefore the observersatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 5
32 Actuator Fault Observer Design This subsection designsthe actuator fault estimator alongwith the observer driven bythe corrected (sensor fault compensated) output and controlsignalsTherefore the system given in (3) can be converted to
Theorem 6 Also based on Lemma 4 given 120574 gt 0 and errorsystem model (31) if there exist symmetric positive-definitematrices 119875 and 119876 and matrices 119870
119875119886 119870119894119886 and 119870V119886 as well as
hold then system (31) is robust asymptotically stable andsatisfies the119867
infinperformance indicator as follows10038171003817100381710038171198901198941198861003817100381710038171003817
2
le 12057410038171003817100381710038171199111198941198861003817100381710038171003817
2
+ 119881 (0) (37)
where 1198901198941198862
= int1199051
0
(119890119894119886
119879
119890119894119886)119889119905 119911
1198941198862
= int1199051
0
(119911119894119886
119879
119911119894119886)119889119905 and 119868
119890
119868119904 and 119868
1are unit matrices
Proof The proof procedure ofTheorem 6 is similar to that ofTheorem 5 here it is omitted
4 Active Decentralized Fault-TolerantController Design
In this section the ADFTC based on dynamic output feed-back is designed to ensure the stability and tracking accuracyof a reconfigurable manipulator with acting actuator andsensor faults concurrently
Considering the faulty subsystem dynamic model (29)the decentralized fault-tolerant controller is designed asfollows
Similarly based on Lemma 4 the existing conditionof dynamic output feedback robust control is given inTheorem 7
Theorem 7 Given 120574 gt 0 and fault subsystems dynamic model(29) if there exist symmetric positive-definite matrices119875 and119876and matrices 119860
119894119888 119861119894119888 119862119894119888 and 119863
119894119897119909119894119897) 119889119905 le int
1199051
0
120574119889119879
119894119889119894119889119905 + 119881 (0) (49)
Inequality (43) can be obtained Therefore the systemsatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 7
5 Simulation Results
To verify the effectiveness of the proposed ADFTC strategyin this subsection two 3-DOF reconfigurable manipulatorswith different configurations shown in Figure 1 are employedto illustrate the simulation results
The initial position and velocity are set as 1199021(0) = 119902
2(0) =
1199023(0) = 1 and 119902
1(0) = 119902
2(0) = 119902
3(0) = 0 respectively
By solving the LMI conditions given in 1 2 and 3 thedynamic output feedback controller and observer gains are
1198601119888=
[[[[[
[
17261 29781 44521 63562
11781 21535 33562 53070
minus10682 minus15344 45082 51247
minus09344 minus14606 31247 41075
]]]]]
]
1198602119888=
[[[[[
[
10261 28531 40951 59062
11141 20031 31639 50250
minus10009 minus13614 41163 49428
minus08314 minus12171 27319 40014
]]]]]
]
8 Mathematical Problems in Engineering
I[1]
I[2]
I[2]
I[2]
I[2]
I[2]
O[2]
O[2]
O[2]O[2]
O[2]O[2]
L[0]
L[0]
L[1]L[0]
L[0]
L[1]
L[3]
L[3]
T
T
BB
Configuration bConfiguration a
Figure 1 Configurations of 3-DOF reconfigurable manipulators
Figure 2 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
10 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 3 p
ositi
on (r
ad)
minus1
minus05
minus08
minus06
minus04
minus02
minus06
minus04
minus02
Figure 3 The tolerant tracking performance of configuration 119886
The actuator fault observer gains are calculated as
1198701119901=
[[[[[[
[
271343 minus224461 157691 08612
40364 01558 177803 11852
141557 117100 82266 minus129699
69872 57800 40606 minus64019
]]]]]]
]
1198702119901=
[[[[[[
[
258643 minus204813 125191 07132
40374 01098 137013 10352
13977 117100 80136 minus119819
58924 51320 39836 minus53419
]]]]]]
]
1198703119901=
[[[[[[
[
221463 minus204001 117251 0702112
40364 09958 129803 09562
138937 117100 82266 minus129699
58924 50972 39001 minus52919
]]]]]]
]
1198701V = [65289 31970 52071 65190]
1198702V = [60963 30023 51325 62351]
1198703V = [59235 28971 50701 61180]
1198701119897= [50231 35576 75657 64152]
1198702119897= [49421 30893 71097 62321]
1198703119897= [48511 29416 65725 60757]
(52)
Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578
119894119891= 0002
120578119894119892= 0002 and 120578
119894119901= 500 and the 119867
infinperformance
indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added
to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
where 120578119894119891and 120578119894119892are positive constants
Note that a challenge in implementing the decentralizedcontrol is to compensate the coupling torque caused bythe interconnected joint modules In such a scenario thefollowing assumption is presented
Assumption 3 The interconnection term ℎ119894(119902 119902 119902) is
bounded by [25]
1003816100381610038161003816ℎ119894 (119902119902 119902)1003816100381610038161003816 le
119899
sum
119895=1
119889119894119895119864119895 (10)
with 119889119894119895ge 0 and 119864
119895= 1 + 119890
119879
119894119875119894119861119894 + 119890
119879
1198941198751198941198611198942
Similarly another RBF neural network term 119894(119890119879
119894119875119894119861119894
119894119901) is introduced to compensate the effect of interconnec-
119870119894119897119862119894+ 119870119894V119862119894 (119860119894 minus 119870119894119901119862119894) 119870119894V119862119894119863119894
]
119873119894119904= [
119861119894
0
119870119894V119862119894119861119894 119868
] 119911119894119904= [
119898119904
119891119894119904
]
(19)
Lemma4 (see [28]) In the given system the eigenvalues of thesystem are located in a LMI region in the complex plane definedby 119863(119902 119903) which is defined by merging different eigenvaluesconstraints to produce a119863(119902 119903)LMI region inwhich 119902 and 119903 arethe radius and center of the disc region If there exist symmetric
Mathematical Problems in Engineering 5
positive-definite matrices 119875 and 119876 and matrices 119870119894119901 119870119894119897 and
119870119894V as well as the corresponding LMI such that
performance is guaran-teed with an attenuation level 120574
Theorem 5 Based on Lemma 4 given 120574 gt 0 and error systemmodel (18) if there exist symmetric positive-definite matrices 119875and 119876 and matrices 119870
119894119901 119870119894119897 and 119870
119894V as well as matrix LMIsuch that (20) holds then system (18) is robust asymptoticallystable and satisfies the119867
Inequality (22) can be obtained Therefore the observersatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 5
32 Actuator Fault Observer Design This subsection designsthe actuator fault estimator alongwith the observer driven bythe corrected (sensor fault compensated) output and controlsignalsTherefore the system given in (3) can be converted to
Theorem 6 Also based on Lemma 4 given 120574 gt 0 and errorsystem model (31) if there exist symmetric positive-definitematrices 119875 and 119876 and matrices 119870
119875119886 119870119894119886 and 119870V119886 as well as
hold then system (31) is robust asymptotically stable andsatisfies the119867
infinperformance indicator as follows10038171003817100381710038171198901198941198861003817100381710038171003817
2
le 12057410038171003817100381710038171199111198941198861003817100381710038171003817
2
+ 119881 (0) (37)
where 1198901198941198862
= int1199051
0
(119890119894119886
119879
119890119894119886)119889119905 119911
1198941198862
= int1199051
0
(119911119894119886
119879
119911119894119886)119889119905 and 119868
119890
119868119904 and 119868
1are unit matrices
Proof The proof procedure ofTheorem 6 is similar to that ofTheorem 5 here it is omitted
4 Active Decentralized Fault-TolerantController Design
In this section the ADFTC based on dynamic output feed-back is designed to ensure the stability and tracking accuracyof a reconfigurable manipulator with acting actuator andsensor faults concurrently
Considering the faulty subsystem dynamic model (29)the decentralized fault-tolerant controller is designed asfollows
Similarly based on Lemma 4 the existing conditionof dynamic output feedback robust control is given inTheorem 7
Theorem 7 Given 120574 gt 0 and fault subsystems dynamic model(29) if there exist symmetric positive-definite matrices119875 and119876and matrices 119860
119894119888 119861119894119888 119862119894119888 and 119863
119894119897119909119894119897) 119889119905 le int
1199051
0
120574119889119879
119894119889119894119889119905 + 119881 (0) (49)
Inequality (43) can be obtained Therefore the systemsatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 7
5 Simulation Results
To verify the effectiveness of the proposed ADFTC strategyin this subsection two 3-DOF reconfigurable manipulatorswith different configurations shown in Figure 1 are employedto illustrate the simulation results
The initial position and velocity are set as 1199021(0) = 119902
2(0) =
1199023(0) = 1 and 119902
1(0) = 119902
2(0) = 119902
3(0) = 0 respectively
By solving the LMI conditions given in 1 2 and 3 thedynamic output feedback controller and observer gains are
1198601119888=
[[[[[
[
17261 29781 44521 63562
11781 21535 33562 53070
minus10682 minus15344 45082 51247
minus09344 minus14606 31247 41075
]]]]]
]
1198602119888=
[[[[[
[
10261 28531 40951 59062
11141 20031 31639 50250
minus10009 minus13614 41163 49428
minus08314 minus12171 27319 40014
]]]]]
]
8 Mathematical Problems in Engineering
I[1]
I[2]
I[2]
I[2]
I[2]
I[2]
O[2]
O[2]
O[2]O[2]
O[2]O[2]
L[0]
L[0]
L[1]L[0]
L[0]
L[1]
L[3]
L[3]
T
T
BB
Configuration bConfiguration a
Figure 1 Configurations of 3-DOF reconfigurable manipulators
Figure 2 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
10 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 3 p
ositi
on (r
ad)
minus1
minus05
minus08
minus06
minus04
minus02
minus06
minus04
minus02
Figure 3 The tolerant tracking performance of configuration 119886
The actuator fault observer gains are calculated as
1198701119901=
[[[[[[
[
271343 minus224461 157691 08612
40364 01558 177803 11852
141557 117100 82266 minus129699
69872 57800 40606 minus64019
]]]]]]
]
1198702119901=
[[[[[[
[
258643 minus204813 125191 07132
40374 01098 137013 10352
13977 117100 80136 minus119819
58924 51320 39836 minus53419
]]]]]]
]
1198703119901=
[[[[[[
[
221463 minus204001 117251 0702112
40364 09958 129803 09562
138937 117100 82266 minus129699
58924 50972 39001 minus52919
]]]]]]
]
1198701V = [65289 31970 52071 65190]
1198702V = [60963 30023 51325 62351]
1198703V = [59235 28971 50701 61180]
1198701119897= [50231 35576 75657 64152]
1198702119897= [49421 30893 71097 62321]
1198703119897= [48511 29416 65725 60757]
(52)
Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578
119894119891= 0002
120578119894119892= 0002 and 120578
119894119901= 500 and the 119867
infinperformance
indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added
to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
performance is guaran-teed with an attenuation level 120574
Theorem 5 Based on Lemma 4 given 120574 gt 0 and error systemmodel (18) if there exist symmetric positive-definite matrices 119875and 119876 and matrices 119870
119894119901 119870119894119897 and 119870
119894V as well as matrix LMIsuch that (20) holds then system (18) is robust asymptoticallystable and satisfies the119867
Inequality (22) can be obtained Therefore the observersatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 5
32 Actuator Fault Observer Design This subsection designsthe actuator fault estimator alongwith the observer driven bythe corrected (sensor fault compensated) output and controlsignalsTherefore the system given in (3) can be converted to
Theorem 6 Also based on Lemma 4 given 120574 gt 0 and errorsystem model (31) if there exist symmetric positive-definitematrices 119875 and 119876 and matrices 119870
119875119886 119870119894119886 and 119870V119886 as well as
hold then system (31) is robust asymptotically stable andsatisfies the119867
infinperformance indicator as follows10038171003817100381710038171198901198941198861003817100381710038171003817
2
le 12057410038171003817100381710038171199111198941198861003817100381710038171003817
2
+ 119881 (0) (37)
where 1198901198941198862
= int1199051
0
(119890119894119886
119879
119890119894119886)119889119905 119911
1198941198862
= int1199051
0
(119911119894119886
119879
119911119894119886)119889119905 and 119868
119890
119868119904 and 119868
1are unit matrices
Proof The proof procedure ofTheorem 6 is similar to that ofTheorem 5 here it is omitted
4 Active Decentralized Fault-TolerantController Design
In this section the ADFTC based on dynamic output feed-back is designed to ensure the stability and tracking accuracyof a reconfigurable manipulator with acting actuator andsensor faults concurrently
Considering the faulty subsystem dynamic model (29)the decentralized fault-tolerant controller is designed asfollows
Similarly based on Lemma 4 the existing conditionof dynamic output feedback robust control is given inTheorem 7
Theorem 7 Given 120574 gt 0 and fault subsystems dynamic model(29) if there exist symmetric positive-definite matrices119875 and119876and matrices 119860
119894119888 119861119894119888 119862119894119888 and 119863
119894119897119909119894119897) 119889119905 le int
1199051
0
120574119889119879
119894119889119894119889119905 + 119881 (0) (49)
Inequality (43) can be obtained Therefore the systemsatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 7
5 Simulation Results
To verify the effectiveness of the proposed ADFTC strategyin this subsection two 3-DOF reconfigurable manipulatorswith different configurations shown in Figure 1 are employedto illustrate the simulation results
The initial position and velocity are set as 1199021(0) = 119902
2(0) =
1199023(0) = 1 and 119902
1(0) = 119902
2(0) = 119902
3(0) = 0 respectively
By solving the LMI conditions given in 1 2 and 3 thedynamic output feedback controller and observer gains are
1198601119888=
[[[[[
[
17261 29781 44521 63562
11781 21535 33562 53070
minus10682 minus15344 45082 51247
minus09344 minus14606 31247 41075
]]]]]
]
1198602119888=
[[[[[
[
10261 28531 40951 59062
11141 20031 31639 50250
minus10009 minus13614 41163 49428
minus08314 minus12171 27319 40014
]]]]]
]
8 Mathematical Problems in Engineering
I[1]
I[2]
I[2]
I[2]
I[2]
I[2]
O[2]
O[2]
O[2]O[2]
O[2]O[2]
L[0]
L[0]
L[1]L[0]
L[0]
L[1]
L[3]
L[3]
T
T
BB
Configuration bConfiguration a
Figure 1 Configurations of 3-DOF reconfigurable manipulators
Figure 2 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
10 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 3 p
ositi
on (r
ad)
minus1
minus05
minus08
minus06
minus04
minus02
minus06
minus04
minus02
Figure 3 The tolerant tracking performance of configuration 119886
The actuator fault observer gains are calculated as
1198701119901=
[[[[[[
[
271343 minus224461 157691 08612
40364 01558 177803 11852
141557 117100 82266 minus129699
69872 57800 40606 minus64019
]]]]]]
]
1198702119901=
[[[[[[
[
258643 minus204813 125191 07132
40374 01098 137013 10352
13977 117100 80136 minus119819
58924 51320 39836 minus53419
]]]]]]
]
1198703119901=
[[[[[[
[
221463 minus204001 117251 0702112
40364 09958 129803 09562
138937 117100 82266 minus129699
58924 50972 39001 minus52919
]]]]]]
]
1198701V = [65289 31970 52071 65190]
1198702V = [60963 30023 51325 62351]
1198703V = [59235 28971 50701 61180]
1198701119897= [50231 35576 75657 64152]
1198702119897= [49421 30893 71097 62321]
1198703119897= [48511 29416 65725 60757]
(52)
Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578
119894119891= 0002
120578119894119892= 0002 and 120578
119894119901= 500 and the 119867
infinperformance
indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added
to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
Theorem 6 Also based on Lemma 4 given 120574 gt 0 and errorsystem model (31) if there exist symmetric positive-definitematrices 119875 and 119876 and matrices 119870
119875119886 119870119894119886 and 119870V119886 as well as
hold then system (31) is robust asymptotically stable andsatisfies the119867
infinperformance indicator as follows10038171003817100381710038171198901198941198861003817100381710038171003817
2
le 12057410038171003817100381710038171199111198941198861003817100381710038171003817
2
+ 119881 (0) (37)
where 1198901198941198862
= int1199051
0
(119890119894119886
119879
119890119894119886)119889119905 119911
1198941198862
= int1199051
0
(119911119894119886
119879
119911119894119886)119889119905 and 119868
119890
119868119904 and 119868
1are unit matrices
Proof The proof procedure ofTheorem 6 is similar to that ofTheorem 5 here it is omitted
4 Active Decentralized Fault-TolerantController Design
In this section the ADFTC based on dynamic output feed-back is designed to ensure the stability and tracking accuracyof a reconfigurable manipulator with acting actuator andsensor faults concurrently
Considering the faulty subsystem dynamic model (29)the decentralized fault-tolerant controller is designed asfollows
Similarly based on Lemma 4 the existing conditionof dynamic output feedback robust control is given inTheorem 7
Theorem 7 Given 120574 gt 0 and fault subsystems dynamic model(29) if there exist symmetric positive-definite matrices119875 and119876and matrices 119860
119894119888 119861119894119888 119862119894119888 and 119863
119894119897119909119894119897) 119889119905 le int
1199051
0
120574119889119879
119894119889119894119889119905 + 119881 (0) (49)
Inequality (43) can be obtained Therefore the systemsatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 7
5 Simulation Results
To verify the effectiveness of the proposed ADFTC strategyin this subsection two 3-DOF reconfigurable manipulatorswith different configurations shown in Figure 1 are employedto illustrate the simulation results
The initial position and velocity are set as 1199021(0) = 119902
2(0) =
1199023(0) = 1 and 119902
1(0) = 119902
2(0) = 119902
3(0) = 0 respectively
By solving the LMI conditions given in 1 2 and 3 thedynamic output feedback controller and observer gains are
1198601119888=
[[[[[
[
17261 29781 44521 63562
11781 21535 33562 53070
minus10682 minus15344 45082 51247
minus09344 minus14606 31247 41075
]]]]]
]
1198602119888=
[[[[[
[
10261 28531 40951 59062
11141 20031 31639 50250
minus10009 minus13614 41163 49428
minus08314 minus12171 27319 40014
]]]]]
]
8 Mathematical Problems in Engineering
I[1]
I[2]
I[2]
I[2]
I[2]
I[2]
O[2]
O[2]
O[2]O[2]
O[2]O[2]
L[0]
L[0]
L[1]L[0]
L[0]
L[1]
L[3]
L[3]
T
T
BB
Configuration bConfiguration a
Figure 1 Configurations of 3-DOF reconfigurable manipulators
Figure 2 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
10 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 3 p
ositi
on (r
ad)
minus1
minus05
minus08
minus06
minus04
minus02
minus06
minus04
minus02
Figure 3 The tolerant tracking performance of configuration 119886
The actuator fault observer gains are calculated as
1198701119901=
[[[[[[
[
271343 minus224461 157691 08612
40364 01558 177803 11852
141557 117100 82266 minus129699
69872 57800 40606 minus64019
]]]]]]
]
1198702119901=
[[[[[[
[
258643 minus204813 125191 07132
40374 01098 137013 10352
13977 117100 80136 minus119819
58924 51320 39836 minus53419
]]]]]]
]
1198703119901=
[[[[[[
[
221463 minus204001 117251 0702112
40364 09958 129803 09562
138937 117100 82266 minus129699
58924 50972 39001 minus52919
]]]]]]
]
1198701V = [65289 31970 52071 65190]
1198702V = [60963 30023 51325 62351]
1198703V = [59235 28971 50701 61180]
1198701119897= [50231 35576 75657 64152]
1198702119897= [49421 30893 71097 62321]
1198703119897= [48511 29416 65725 60757]
(52)
Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578
119894119891= 0002
120578119894119892= 0002 and 120578
119894119901= 500 and the 119867
infinperformance
indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added
to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
119894119897119909119894119897) 119889119905 le int
1199051
0
120574119889119879
119894119889119894119889119905 + 119881 (0) (49)
Inequality (43) can be obtained Therefore the systemsatisfies the119867
infinperformance indicator and this completes the
proof of Theorem 7
5 Simulation Results
To verify the effectiveness of the proposed ADFTC strategyin this subsection two 3-DOF reconfigurable manipulatorswith different configurations shown in Figure 1 are employedto illustrate the simulation results
The initial position and velocity are set as 1199021(0) = 119902
2(0) =
1199023(0) = 1 and 119902
1(0) = 119902
2(0) = 119902
3(0) = 0 respectively
By solving the LMI conditions given in 1 2 and 3 thedynamic output feedback controller and observer gains are
1198601119888=
[[[[[
[
17261 29781 44521 63562
11781 21535 33562 53070
minus10682 minus15344 45082 51247
minus09344 minus14606 31247 41075
]]]]]
]
1198602119888=
[[[[[
[
10261 28531 40951 59062
11141 20031 31639 50250
minus10009 minus13614 41163 49428
minus08314 minus12171 27319 40014
]]]]]
]
8 Mathematical Problems in Engineering
I[1]
I[2]
I[2]
I[2]
I[2]
I[2]
O[2]
O[2]
O[2]O[2]
O[2]O[2]
L[0]
L[0]
L[1]L[0]
L[0]
L[1]
L[3]
L[3]
T
T
BB
Configuration bConfiguration a
Figure 1 Configurations of 3-DOF reconfigurable manipulators
Figure 2 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
10 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 3 p
ositi
on (r
ad)
minus1
minus05
minus08
minus06
minus04
minus02
minus06
minus04
minus02
Figure 3 The tolerant tracking performance of configuration 119886
The actuator fault observer gains are calculated as
1198701119901=
[[[[[[
[
271343 minus224461 157691 08612
40364 01558 177803 11852
141557 117100 82266 minus129699
69872 57800 40606 minus64019
]]]]]]
]
1198702119901=
[[[[[[
[
258643 minus204813 125191 07132
40374 01098 137013 10352
13977 117100 80136 minus119819
58924 51320 39836 minus53419
]]]]]]
]
1198703119901=
[[[[[[
[
221463 minus204001 117251 0702112
40364 09958 129803 09562
138937 117100 82266 minus129699
58924 50972 39001 minus52919
]]]]]]
]
1198701V = [65289 31970 52071 65190]
1198702V = [60963 30023 51325 62351]
1198703V = [59235 28971 50701 61180]
1198701119897= [50231 35576 75657 64152]
1198702119897= [49421 30893 71097 62321]
1198703119897= [48511 29416 65725 60757]
(52)
Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578
119894119891= 0002
120578119894119892= 0002 and 120578
119894119901= 500 and the 119867
infinperformance
indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added
to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
Figure 2 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
10 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 3 p
ositi
on (r
ad)
minus1
minus05
minus08
minus06
minus04
minus02
minus06
minus04
minus02
Figure 3 The tolerant tracking performance of configuration 119886
The actuator fault observer gains are calculated as
1198701119901=
[[[[[[
[
271343 minus224461 157691 08612
40364 01558 177803 11852
141557 117100 82266 minus129699
69872 57800 40606 minus64019
]]]]]]
]
1198702119901=
[[[[[[
[
258643 minus204813 125191 07132
40374 01098 137013 10352
13977 117100 80136 minus119819
58924 51320 39836 minus53419
]]]]]]
]
1198703119901=
[[[[[[
[
221463 minus204001 117251 0702112
40364 09958 129803 09562
138937 117100 82266 minus129699
58924 50972 39001 minus52919
]]]]]]
]
1198701V = [65289 31970 52071 65190]
1198702V = [60963 30023 51325 62351]
1198703V = [59235 28971 50701 61180]
1198701119897= [50231 35576 75657 64152]
1198702119897= [49421 30893 71097 62321]
1198703119897= [48511 29416 65725 60757]
(52)
Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578
119894119891= 0002
120578119894119892= 0002 and 120578
119894119901= 500 and the 119867
infinperformance
indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added
to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
Figure 2 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
10 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 3 p
ositi
on (r
ad)
minus1
minus05
minus08
minus06
minus04
minus02
minus06
minus04
minus02
Figure 3 The tolerant tracking performance of configuration 119886
The actuator fault observer gains are calculated as
1198701119901=
[[[[[[
[
271343 minus224461 157691 08612
40364 01558 177803 11852
141557 117100 82266 minus129699
69872 57800 40606 minus64019
]]]]]]
]
1198702119901=
[[[[[[
[
258643 minus204813 125191 07132
40374 01098 137013 10352
13977 117100 80136 minus119819
58924 51320 39836 minus53419
]]]]]]
]
1198703119901=
[[[[[[
[
221463 minus204001 117251 0702112
40364 09958 129803 09562
138937 117100 82266 minus129699
58924 50972 39001 minus52919
]]]]]]
]
1198701V = [65289 31970 52071 65190]
1198702V = [60963 30023 51325 62351]
1198703V = [59235 28971 50701 61180]
1198701119897= [50231 35576 75657 64152]
1198702119897= [49421 30893 71097 62321]
1198703119897= [48511 29416 65725 60757]
(52)
Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578
119894119891= 0002
120578119894119892= 0002 and 120578
119894119901= 500 and the 119867
infinperformance
indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added
to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
Figure 3 The tolerant tracking performance of configuration 119886
The actuator fault observer gains are calculated as
1198701119901=
[[[[[[
[
271343 minus224461 157691 08612
40364 01558 177803 11852
141557 117100 82266 minus129699
69872 57800 40606 minus64019
]]]]]]
]
1198702119901=
[[[[[[
[
258643 minus204813 125191 07132
40374 01098 137013 10352
13977 117100 80136 minus119819
58924 51320 39836 minus53419
]]]]]]
]
1198703119901=
[[[[[[
[
221463 minus204001 117251 0702112
40364 09958 129803 09562
138937 117100 82266 minus129699
58924 50972 39001 minus52919
]]]]]]
]
1198701V = [65289 31970 52071 65190]
1198702V = [60963 30023 51325 62351]
1198703V = [59235 28971 50701 61180]
1198701119897= [50231 35576 75657 64152]
1198702119897= [49421 30893 71097 62321]
1198703119897= [48511 29416 65725 60757]
(52)
Here the control law (39) is applied to the whole controlsystem and the control parameters are selected as 120578
119894119891= 0002
120578119894119892= 0002 and 120578
119894119901= 500 and the 119867
infinperformance
indicator is defined as 120574 = 10First considering configuration 119886 fault signals are added
to the actuators of joint 1 the position sensor of joint 2 andboth velocity sensor and actuator of joint 3 at 119905 = 3 s 119905 = 5 s119905 = 6 s and 119905 = 3 s respectively
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
Without sensor fault compensation trajectoryDesired trajectory
minus04
minus02
(f)
Figure 4 (a) Actuator fault estimation of joint 1 (b) sensor fault estimation of joint 2 (c) tracking performance with uncompensated sensorfault of joint 2 (d) actuator fault estimation of joint 3 (e) sensor fault estimation of joint 3 (f) tracking performance with uncompensatedsensor fault of joint 3
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
08
1
Time (s)
Join
t 1 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
05
1
Time (s)
Join
t 2 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
0 1 2 3 4 5 6 7 8 9 10
0
02
04
06
Time (s)
Join
t 3 p
ositi
on (r
ad)
Desired trajectoryActual trajectory
minus06
minus04
minus02
minus1
minus05
minus04
minus02
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
Figure 5 The tolerant tracking performance of configuration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
5 sin (1199021) 1199021 119905 gt 3
1198912119904=
0 119905 le 5
minus03 cos (1199022) 119905 gt 5
1198913119886=
0 119905 le 3
15 119905 gt 3
1198913119904=
0 119905 le 6
minus031199022 119905 gt 6
(53)
As illustrated in Figure 2 from Figures 2(a) 2(b) 2(d)and 2(e) it can be obtained that the sensor faults and actuatorfaults can be detected online in real time when the faultsoccur via DPIO and the actual trajectory cannot track thedesired trajectory due to without sensor fault compensationfrom Figures 2(c) and 2(f) Figure 3 shows the tolerant
tracking performance of configuration 119886 in which it canbe seen that the actual trajectories can follow the desiredtrajectories in each joint
To further test the effectiveness of the proposed schemeunder different configurations the same scheme applies toconfiguration 119887
The fault functions are as follows
1198911119886=
0 119905 le 3
2 sin (1199021) 119905 gt 3
1198912119904=
0 119905 le 5
3 sin (119905) 119905 gt 5
1198913119886=
0 119905 le 6
20 119905 gt 6
1198913119904=
0 119905 le 4
minus051199022 119905 gt 4
(54)
Mathematical Problems in Engineering 13
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
The simulation results are shown as in Figures 4 and 5 thatillustrate that the proposed active decentralized fault-tolerantcontrol can be applied to different configurations of reconfig-urable manipulator without any parameters modification
6 Conclusion
The paper develops a novel tolerant control strategy forreconfigurable manipulator with simultaneous actuator andsensor faults based on dynamic output feedback controlUsing LMI technology the DPIO is designed to estimate andcompensate the actuator and sensor faults on line and theestimator of actuator fault is put into the designed dynamicoutput feedback controller to realize active fault-tolerantcontrol The method obviates the disturbance of time delayfor system by cancelling the need for the use of a faultdiagnosis and isolation (FDI) unit and the controller not onlycan make fault system robust stability but also can meet therequirement of119867
infinperformance indicatorsThe effectiveness
of the proposed scheme is verified under the conditions ofdifferent configurations without modifying any parameter
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants 61374051 and 60974010 andthe Scientific and Technological Development Plan Project inJilin Province of China under Grant 20150520112JH
References
[1] D Rupp G Ducard E Shafai and H P Geering ldquoExtendedmultiple model adaptive estimation for the detection of sensorand actuator faultsrdquo in Proceedings of the 44th IEEE Conferenceon Decision and Control and the European Control Conference(CDC-ECC rsquo05) pp 3079ndash3084 December 2005
[2] M Du J Nease and P Mhaskar ldquoAn integrated fault diagnosisand safe-parking framework for fault-tolerant control of non-linear systemsrdquo International Journal of Robust and NonlinearControl vol 22 no 1 pp 105ndash122 2012
[3] Y Jiang Q Hu and G Ma ldquoAdaptive backstepping fault-tolerant control for flexible spacecraft with unknown boundeddisturbances and actuator failuresrdquo ISATransactions vol 49 no1 pp 57ndash69 2010
[4] D Brambilla L M Capisani A Ferrara and P Pisu ldquoSecondorder sliding mode observers for fault detection of robotmanipulatorsrdquo in Proceedings of the 47th IEEE Conferenceon Decision and Control (CDC rsquo08) pp 2949ndash2954 CancunMexico December 2008
[5] S T Huang E J Davison and R Kwong ldquoDecentralizedrobust servomechanism problem for large flexible space struc-tures under sensor and actuator failuresrdquo IEEE Transactions onAutomatic Control vol 57 no 12 pp 3219ndash3224 2012
[6] J Jiang and Y Zhang ldquoAccepting performance degradationin fault-tolerant control system designrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 284ndash292 2006
[7] L Liu Y Shen E H Dowell and C Zhu ldquoA general 119867infin
fault tolerant control and management for a linear system withactuator faultsrdquo Automatica vol 48 no 8 pp 1676ndash1682 2012
[8] H H Niemann ldquoA model-based approach to fault-tolerantcontrolrdquo International Journal of Applied Mathematics andComputer Science vol 22 no 1 pp 67ndash86 2012
[9] J C D Silva A Saxena E Balaban and K Goebel ldquoAknowledge-based system approach for sensor fault modelingdetection andmitigationrdquo Expert Systems with Applications vol39 no 12 pp 10977ndash10989 2012
[10] M Petkovic M R Rapaic Z D Jelicic and A Pisano ldquoOn-lineadaptive clustering for process monitoring and fault detectionrdquoExpert SystemswithApplications vol 39 no 11 pp 10226ndash102352012
[11] Y Xu S Tong and Y Li ldquoAdaptive fuzzy fault-tolerant controlof static var compensator based on dynamic surface controltechniquerdquo Nonlinear Dynamics vol 73 no 3 pp 2013ndash20232013
[12] S J Yoo ldquoActuator fault detection and adaptive accommodationcontrol of flexible-joint robotsrdquo IET Control Theory and Appli-cations vol 6 no 10 pp 1497ndash1507 2012
[13] D Rotondo F Nejjari and V Puig ldquoA virtual actuator andsensor approach for fault tolerant control of LPV systemsrdquoJournal of Process Control vol 24 no 3 pp 203ndash222 2014
[14] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[15] J Yuan G Liu and B Wu ldquoPower efficiency estimation-basedhealth monitoring and fault detection of modular and reconfig-urable robotrdquo IEEE Transactions on Industrial Electronics vol58 no 10 pp 4880ndash4887 2011
[16] S AhmadH Zhang andG Liu ldquoDistributed fault detection formodular and reconfigurable robots with joint torque sensing aprediction error based approachrdquo Mechatronics vol 23 no 6pp 607ndash616 2013
[17] B Zhao and Y Li ldquoLocal joint information based activefault tolerant control for reconfigurablemanipulatorrdquoNonlinearDynamics vol 77 no 3 pp 859ndash876 2014
[18] Y Li Y Liu X Liu and Z Peng ldquoParameter identificationand vibration control in modular manipulatorsrdquo IEEEASMETransactions on Mechatronics vol 9 no 4 pp 700ndash705 2004
[19] T Sun Y M Song Y G Li and J Zhang ldquoWorkspacedecomposition based dimensional synthesis of a novel hybridreconfigurable robotrdquo Journal of Mechanisms and Robotics vol2 no 3 Article ID 031009 8 pages 2010
[20] M Biglarbegian W W Melek and J M Mendel ldquoDesign ofnovel interval type-2 fuzzy controllers for modular and recon-figurable robots theory and experimentsrdquo IEEETransactions onIndustrial Electronics vol 58 no 4 pp 1371ndash1384 2011
[21] RMuller M Esser M Jansen and B Corves ldquoModular controlsystem for reconfigurable robot applicationsrdquo in Proceedings ofthe IEEE International Symposium on Assembly and Manufac-turing pp 1ndash5 May 2011
[22] W-H Zhu and T Lamarche ldquoModular robot manipulatorsbased on virtual decomposition controlrdquo in Proceedings of theIEEE International Conference on Robotics and Automation(ICRA rsquo07) pp 2235ndash2240 Rome Italy April 2007
14 Mathematical Problems in Engineering
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007
[23] S Kirchoff and W W Melek ldquoA saturation-type robust con-troller for modular manipulators armsrdquo Mechatronics vol 17no 4-5 pp 175ndash190 2007
[24] Z LiDevelopment and Control of aModular and ReconfigurableRobot with Harmonic Drive Transmission System University ofWaterloo Waterloo Canada 2007
[25] B Zhao and Y Li ldquoMultisensor fault identification schemebased on decentralized sliding mode observers applied toreconfigurable manipulatorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 327916 9 pages 2013
[26] R N Murray Z X Li and S S Sastry A MathematicalIntroduction to Robotic Manipulation CRC Press Boca RatonFla USA 1994
[27] M Sami and R J Patton ldquoActive fault tolerant control for non-linear systems with simultaneous actuator and sensor faultsrdquoInternational Journal of Control Automation and Systems vol11 no 6 pp 1149ndash1161 2013
[28] L Bai Z Tian and S Shi ldquoRobust fault detection for a class ofnonlinear time-delay systemsrdquo Journal of the Franklin Institutevol 344 no 6 pp 873ndash888 2007