Research ArticleA Novel Nonlinear Fault Tolerant Control forManipulator under Actuator Fault

Jing Zhao 12 Sen Jiang1 Fei Xie 23 Zhen He 4 and Jian Fu 5

1 Jiangsu Engineering Laboratory for Internet of ings and Intelligent Robots Nanjing University of Posts and TelecommunicationsNanjing 210023 China

2Jiangsu Key Laboratory of 3D Printing Equipment and Manufacturing Nanjing Normal University Nanjing 210023 China3Nanjing Institute of Intelligent High-end Equipment Industry Limited Company Nanjing 210042 China4School of Automation Nanjing University of Aeronautics and Astronautics Nanjing 210016 China5College of Energy and Power Engineering Nanjing University of Science and Technology Nanjing 210094 China

Correspondence should be addressed to Fei Xie xiefeinjnueducn

Received 2 June 2018 Revised 29 August 2018 Accepted 9 September 2018 Published 14 October 2018

Academic Editor Xue-Jun Xie

Copyright copy 2018 Jing Zhao et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A fault tolerant control (FTC) scheme based on adaptive sliding mode control technique is proposed for manipulator with actuatorfault Firstly the dynamic model of manipulator is introduced and its actuator faulty model is established Secondly a fault tolerantcontroller is designed in which both the parameters of actuator fault and external disturbance are estimated and updated by onlineadaptive technology Finally taking a two-joint manipulator as example simulation results show that the proposed fault tolerantcontrol scheme is effective in tolerating actuator fault meanwhile it has strong robustness for external disturbance

1 Introduction

With the rapid development of modern science technologymanipulator has emerged as an important area of researchand more manipulators are applied in our life to reduce theburden of work In [1 2] two cleaning robots are designedto help people complete household cleaning tasks betterBesides some tasks cannot be completed by a single manip-ulator but two more cooperating manipulators are requiredThus a control method of dual manipulators is proposed toreplace the human workers to assemble and grasp objects[3 4] Reference [5] addressed a decentralized controllerwith constrained error variable and a radial basis functionnetwork for spacemanipulator Besides the above applicationmanipulator also plays an important role in dangerousenvironment where people can not directly participate Inouter space nobody can stay much long Therefore anadvanced mechanical arm system is needed to perform someexploratory and experimental tasks especially extravehicularmission such as space assembly spacecraft maintenancesatellite interaction and outer space exploration Thus itgreatly reduces the risks of the astronauts going out of the

cabin and improves the efficiency and safety of space mission[6] In recent years many researchers have done a greatdeal of products on manipulators The American space robotremote service system and the lightweight modular spacemanipulator system developed by German and Canadiangiant robotic arm have been successfully launched followingthe rocket and completed the task What is more there arealso some other space manipulators serving in InternationalSpace Station including Dextre SSRMS and ERA [7ndash9]

In the current industrial application manipulator hasbecome increasingly important [10] therefore the stabilityand reliability of manipulator system are crucial factors[11ndash15] Sophisticated and dangerous work such as weldingand space tasks that require high precision are assignedto robots The robotic manipulator is a typical complexunderactuated system with redundancy multivariate highlynonlinearities and coupling On the one hand as frictioncoefficient between joints always changes over time andexternal disturbance is uncertain [16] fault may occur inmanipulator In particular in dangerous environment faultmay occur more easily such as hard environment conditionsparticle radiation electromagnetic interference [17] and

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 5198615 11 pageshttpsdoiorg10115520185198615

2 Mathematical Problems in Engineering

low temperature consequently the performance will greatlydecrease even leading mission to fail On the other handartificial repair is nearly impossible in outer space In con-clusion manipulator is needed to tolerate fault and continuethe given operation task Consequently fault tolerant controlis vital in security assurance for manipulator FTC techniquealso applies on UAV team cooperative control distributedcontrol mobile wireless networks and communications [18]The performance of feedback control system depends onactuators sensors and data acquisitioninterface compo-nents Faulty components will lead to the deterioration ofthe overall system stability which has been a safety problemin control system [19] At present there have been a largenumber of FTC schemes In the FTC literature differentapproaches have been reported such as robust FTCpresentedin [20] adaptive FTC designed in [21 22] nonlinear FTCproposed in [23] and sliding mode FTC proposed in [24 25]However there are not many FTC schemes for manipulatorIn [26] a novel finite time FTC based on adaptive neuralnetwork nonsingular fast terminal sliding mode is addressedfor uncertain robot manipulators with actuator faults andit is verified that the system possesses strong robustness nosingularity less chattering and fast finite time convergence bysimulation [27ndash33] In [34] a robust LQRLQI FTCmethod isdeveloped for a 2DOF unmanned bicycle robot with actuatorfault Different from [26] [34 35] proposed a decentralizedFTC for reconfigurable manipulator with sensor fault

Compared with the above-mentioned methods slidingmode control (SMC) has attractive advantages of efficientcharacteristics thanks to its insensibility to matched uncer-tainties and disturbances [36] Therefore SMC is adopted todesign the fault tolerant controller in this paper Howeverchattering of the sliding mode control signal has become themajor issue to its actual applications which can lead to thedeterioration of the overall system In order to solve suchproblem a novel sliding mode control technology dynamicsliding mode control is designed [37] In FTC methodobservers are usually designed to estimate disturbances andfault information for example in [38] a fault diagnosisvia higher order sliding mode observers is proposed formanipulator system different from [39 40] observers are notconsidered in this paper in which the proposed controller ismuch easier to realize in practical application

The main theoretical contributions of this paper can bebriefly outlined as follows

(1) Compared with the design of traditional slidingsurface the novel dynamic sliding mode controllerproposed in this paper can reduce the chatteringeffectively

(2) In comparison with traditional method to handleunknown system parameters and disturbances adap-tive algorithm is adopted to update parameters onlineit is not necessary to obtain the accurate value ofdisturbances

(3) In comparison with the conventional FTC methodthere is no need to design a fault diagnosis anddetection observer and the unknown fault can beestimated by the proposed adaptive algorithm

The rest of this paper is organized as follows In Section 2the dynamic model of space manipulator and its actuatorfault model are introduced Then a fault tolerant controller isdesigned in which parameters of actuator fault and externaldisturbance are estimated and updated by online adaptivemethod in Section 3 Finally simulation results demonstratethe proposed fault tolerant controller is able to tolerateactuator fault as well as the strong robustness for externaldisturbance

2 Mathematic Model

21 Dynamic model An articular robot system is a typicalredundant multivariable nonlinear complex dynamic cou-pling system [39]Whendealingwithmanipulator we usuallyadopt its simplified model as follows due to the extremecomplexity of its general dynamic model manipulator

119867(119902) 119902 + 119862 (119902 119902) 119902 + 119866 (119902) + 119865 ( 119902) = 120591 + 120591119889 (1)

where 119902 isin R119899 119902 isin R119899 119902 isin R119899 represent joint positionvelocity and acceleration vector respectively here positionrefers to joint angle119867(119902) isin R119899times119899 denotes symmetric positivedefinite inertia matrix 119862(119902 119902) isin R119899times119899 represents Coriolisforce and centrifugal force matrix 119866(119902) isin R119899 denotesgravity torque vector 119865( 119902) isin R119899 is friction torque vector120591119889 isin R119899 denotes external disturbance and model parameteruncertainties torque vector 120591 isin R119899 denotes control torquevector

The above manipulator system (1) has the followingproperty which is beneficial in subsequent controller design

Property (119902)minus2119862(119902 119902) is a skew symmetric matrix [41] ieΓ119879((119902) minus 2119862(119902 119902))Γ = 0 forallΓ isin R119899

22 Fault Model Instead of locked-joint fault loss of effec-tiveness considered in this paper means the free-swingingfault that is fault joint swings freely without constraints Onthe contrary locked-joint fault means that fault joint is lockedat its current position and cannot move any more The moveof manipulator system depends on the rotation of motorwhich ranges from 0 degree to 300 degreeThe free-swingingfault here that may be caused by a hardware or software faultin a manipulator can lead to the loss of torque (or force) on ajoint [42] Then the free-swinging fault model is establishedas follows

119867(119902) 119902 + 119862 (119902 119902) 119902 + 119866 (119902) + 119865 ( 119902) = 120591119891 + 120591119889 (2)

where 120591119891 = 119864120591 119864 = diag 119890119894 119894 = 1 2 119899 and 119890119894 isin (0 1)represents actuator loss of effectiveness factor which refers tothe free-swinging fault in this paper and the proposed FTC ofmanipulator is mainly designed under such case When 119890119894 =1 the 119894119905ℎ joint is normal without fault when 119890119894 = 0 the 119894119905ℎjoint is with lock in place that means that the joint is lockedat its current position which is not considered in this paperDefine Δ119864(119905) = 119868 minus119864 = diag 1 minus 119890119894(119905) where I is the identitymatrix and Δ119864(119905) = 1 minusmin119890119894(119905) then system (2) can betransformed as

119867 119902 = 120591 minus Δ119864 (119905) 120591 + 120591119889 + 119892 (119902 119902) (3)

Mathematical Problems in Engineering 3

where 119892(119902 119902) = minus119862(119902 119902) 119902 minus 119866(119902) minus 119865( 119902) For conveniencedefine 119867 ≜ 119867(119902) 119862 ≜ 119862(119902 119902) 119866 ≜ 119866(119902) 119865 ≜ 119865( 119902) and119892 ≜ 119892(119902 119902)3 Fault Tolerant Controller Design

31 Problem Statement In this paper we are absorbed ininvestigating an FTC method for manipulator with actuatorfault The trajectory tracking problem of fault system (2)is considered The control objective can be described asthat for a manipulator control system with actuator faultan FTC method is proposed to ensure that the closed-loopsystem is stable ie when 119905 997888rarr infin 119902 997888rarr 119902119889 where 119902119889denotes the desired position signal For this purpose an FTCmethod based on adaptive dynamic sliding mode technologyis proposed Firstly an assumption is given as follows

Assumption 1 The external disturbance 120591119889 is assumed to benorm-bounded 10038171003817100381710038171205911198891003817100381710038171003817 le 119870 (4)

where119870 is an unknown positive constant and sdot represents119871infin norm in this paper

32 Controller Design As mentioned in the above sectionthe desired position signal is defined as 119902119889 so the trackingerror is 119911 = 119902minus119902119889Then the conventional slidingmode surfaceis selected as

119878 = + 119897119911 (5)

where 119897 isin R119899times119899 is a positive matrix further

119878 = + 119897 = 119902 minus 119902119889 + 119897 ( 119902 minus 119902119889) (6)

Next the dynamic sliding mode surface is selected as

119869 = 119878 + 120594 (7)

where 120594 can be considered as the error between 119869 and 119878 andits derivative of time is 120594 = 120588111987805sgn (119878) minus 120588211986905sgn (119869)where sgn(119878) = [sgn (1198781) sgn (1198782) sgn (119878119899)]119879 sgn (119869) =[sgn (1198691) sgn (1198692) sgn (119869119899)]119879 1205881 is the sliding mode gainof conventional sliding mode surface 119878 and 1205882 is the slidingmode gain of dynamic sliding mode surface 119869 They are twopositive constants satisfying 1205881 gt 0 1205882 gt 0 1205881 = 1205882 and formore details please refer to Remark 6

Theorem 2 Considering manipulator system with free-swinging fault and external disturbance (2) under Assump-tion 1 a fault tolerant control input based on adaptive slidingmode controller (8) and adaptive law (9) can guarantee asymp-totic output tracking of manipulator control system in bothcases of no fault and fault which guarantees the boundednessof all the closed-loop signals and asymptotic output tracking

120591 = minus119892 minus 119867119891 minus 119862119869 minus 120601sgn (119869) minus sgn (119869) (8)

119870 = 120583 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (9)

where is the estimated value of 119870 120576 is a positive constant119891 = minus 119902119889 + 119897( 119902 minus 119902119889) + 120594 120601 = (119887(1 minus 119887))[119892 + 119867119891 + 119862119869 +sgn (119869) + 120576] and 119887 = Δ119864 To satisfy the stability of thesystem achieve a good tracking effectiveness and estimate thevalue of disturbances (9) is obtained to put an integral actionin the definition of (8)

Proof Define a Lyapunov function as follows

119881 = 12119869119879119867119869 + 121205832 (10)

where denotes the estimated error of disturbance that is = 119870 minus and 120583 is a positive constant The time derivativeof 119881 is obtained

= 12119869119879119869 + 119869119879119867 119869 minus 1120583 119870= 12119869119879 ( minus 2119862) 119869 + 119869119879119862119869 + 119869119879119867 119869 minus 1120583 119870= 119869119879 (119862119869 + 119867 119869) minus 1120583 119870= 119869119879 (119862119869 + 119867 119902 + 119867119891) minus 1120583 119870

(11)

Substituting (3) into the above equation one can obtain thefollowing

= 119869119879 (119862119869 + 120591 minus 119864 (119905) 120591 + 120591119889 + 119892 + 119867119891) minus 1120583 119870 (12)

Applying controller (8) and adaptive law (9) into the aboveequation one can obtain

= 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869) minus sgn (119869) + 120591119889]minus 1120583 119870

= 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869)] minus 119869119879sgn (119869) + 119869119879120591119889minus 1120583 119870

le 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869)] minus 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 + 119870119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816minus 1120583 119870

= 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869)] + ( 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1120583 119870)

(13)

4 Mathematical Problems in Engineering

Further simplify

= 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869)]= 119869119879 119864 (119905) [119892 + 119867119891 + 119862119869 + 120601sgn (119869) + sgn (119869)]minus 119869119879120601sgn (119869)

le 119887 119869 [1003817100381710038171003817119892 + 119867119891 + 1198621198691003817100381710038171003817 + sgn (119869)] + 119887120601119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816minus 120601 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816le 119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 [1003817100381710038171003817119892 + 119867119891 + 1198621198691003817100381710038171003817 + sgn (119869)]

minus 120601 (1 minus 119887) 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 = minus119887120576 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 le 0

(14)

and thus the stability of the closed-loop system is verified

Remark 3 In practical application the smallest valuableof actuator fault min119890119894(119905) is usually unknown thereforeit is necessary to design an adaptive law to estimate faultinformation To solve this problem an adaptive fault tolerantcontroller will be designed in the following inwhich adaptivescheme is adopted to estimate both the actuator fault andexternal disturbance

When dealing with the faulty term define 119887 = Δ119864(119905)120585 = 1(1 minus 119887) and then adaptive algorithm is adoptedto estimate 120585 which can compensate for the existent faultRegarding the disturbance based on Assumption 1 isused to compensate for the existent disturbance by adaptivetechnique

Theorem 4 Based on eorem 2 considering manipulatorsystem with free-swinging fault and external disturbance (2)under Assumption 1 the minimum boundary value of actuatorfault min119890119894(119905) is unknown and an FTC input based onadaptive sliding mode controller (15) and adaptive laws (16)-(17) can guarantee asymptotic output tracking of manipulatorcontrol system in both cases ie no fault and fault whichguarantees the boundedness of all the closed-loop signals andasymptotic output tracking

120591 = minus119892 minus 119867119891 minus 119862119869 minus 120574 (119905) sgn (119869) minus sgn (119869) (15)

119870 = 120583 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (16)

120585 = 120573120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (17)

where is the estimated value of 119870 120575 = 119892 + 119867119891 + 119862119869 + + 120576 and 120576 is a positive constant 120574(119905) = minus120575 + 120585120575 and 120585 is theestimated value of 120585

Proof In order to prove the stability of the overall system theLyapunov function is selected as

119881 = 1198811 + 1198812 + 1198813 (18)

where 1198811 1198812 1198813 represent three Lyapunov functions whichwill be elaborated in the following 3 steps further

= 1 + 2 + 3 (19)

The process of proof is divided into three steps

Step 1 (adaptive law analysis) To obtain the adaptive laws offault and disturbance the Lyapunov function is chosen as

1198811 = 12119869119879119867119869 + 121205832 + 1 minus 1198872120573 1205852 (20)

where denotes the estimated error of disturbance 120585 denotesthe estimated error of fault = 119870 minus 120585 = 120585 minus 120585 and 120583and 120573 are both positive constants The time derivative of1198811 isobtained

1 = 12119869119879119869 + 119869119879119867 119869 minus 1120583 119870 minus 1 minus 119887120573 120585 120585= 12119869119879 ( minus 2119862) 119869 + 119869119879119862119869 + 119869119879119867 119869 minus 1120583 119870minus 1 minus 119887120573 120585 120585

= 119869119879 (119862119869 + 119867 119902 + 119867119891) minus 1120583 119870 minus 1 minus 119887120573 120585 120585

(21)

Substituting (3) into the above equation one obtains thefollowing

1 = 119869119879 (119862119869 + 120591 minus 119864 (119905) 120591 + 120591119889 + 119892 + 119867119891) minus 1120583 119870minus 1 minus 119887120573 120585 120585

(22)

Applying controller (15) into the above equation yields thefollowing

1 = 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869) minus sgn (119869) + 120591119889]minus 1120583 119870 minus 1 minus 119887120573 120585 120585

= 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869)] + 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816minus 1120583 119870 minus 1 minus 119887120573 120585 120585

= 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869)]+ ( 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1120583 119870) minus 1 minus 119887120573 120585 120585

(23)

Mathematical Problems in Engineering 5

Substituting adaptive law (16) into the above equation yieldsthe following

1 le 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869)] minus 1 minus 119887120573 120585 120585= minus120574 (119905) 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 + 119869119879 119864 (119905) [119892 + 119867119891 + 119862119869 + 120574 (119905) sgn (119869) + sgn (119869)]

le (1 minus 120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585+ 119887 1003817100381710038171003817100381711986911987910038171003817100381710038171003817 [1003817100381710038171003817119892 + 119867119891 + 1198621198691003817100381710038171003817 + 120574 (119905) + ]

le (1 minus 120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 + 119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (minus120576 + 120585120575)

= (1 minus 120585 + 119887120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816= (1 minus 119887) (120585 minus 120585) 120575 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816= (1 minus 119887) 120585(120575 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1120573 120585) minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816

(24)

As a result applying adaptive law (17) into the above equationone can obtain the following

1 = minus120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 le 0 (25)

Step 2 (reach time analysis) Before analysis a lemma isproposed as follows to obtain the convergence time

Lemma 5e dynamic sliding mode function 119869 exists and canconverge within finite time 119905119869 please see [43] for more details

Aer time 119905119869 the surface 119869 = 0 from (7) one can obtainthe following

119878 = minus 120594 = minus1205881 11987805 sgn (119878) (26)

Further a Lyapunov function is selected as follows to obtain theconvergence time of system

1198812 = 12119878119879119878 (27)

Further

2 = 119878119879 119878 = 119878119879 (minus1205881 11987805 sgn (119878)) le minus1205881 11987815 le 0 (28)

To solve the above differential equation the following equationcan be obtained by (27)

1198812 = 12 1198782 (29)

Further

119878 = 100381610038161003816100381621198812100381610038161003816100381612 (30)

Substituting (30) to (28) yields the following

2 le minus1205881 100381610038161003816100381621198812100381610038161003816100381634 (31)

us

119881minus342 1198891198812 le minus2341205881119889119905int119905119905119869

119881minus342 1198891198812le int119905119905119869

minus23412058811198891199054 [1198812 (119905)14 minus 1198812 (119905119869)14]le minus2341205881 (119905 minus 119905119869)

(32)

As time 119905 reaches 119905119878 the conventional slidingmode surface 119878willconverge to zero ie when 119905 = 119905119878 119878 = 0 that is1198812(119905) = 1198812(119905119878)thus

minus41198812 (119905119869)14 le minus2341205881 (119905119878 minus 119905119869) (33)

us one can obtain the following

119905119878 le 41198812 (119905119869)14 + 23412058811199051198692341205881 (34)

Consequently sliding mode surface 119878 will converge to zerowithin finite time

119905119878 le 41198812 (119905119869)14 + 23412058811199051198692341205881 (35)

Now it is verified that conventional sliding mode surface 119878and dynamic sliding mode surface 119869 can both converge to zerowithin finite time and dynamic surface converges faster thanconventional surface ie lim119905ge119905119869(119869119878) = 0 As 119905 997888rarr 119905119878 isreached 119878 = 0 substituting this into (5) one can obtain thefollowing

= minus119897119911 (36)

Step 3 (tracking error analysis) To prove the convergenceof the tracking error 119911 the Lyapunov function is defined asfollows

1198813 = 12119911119879119911 (37)

Thus

3 = 119911119879 le minus119897 1199112 le 0 (38)

and therefore tracking error 119911 is convergent which meansthat when 119905 997888rarr infin 119902 997888rarr 119902119889 119911 997888rarr 0 Therefore accordingto the above three steps it is easy to be seen that lt 0 in(19) which means that the overall system can be stable withthe proposed controller The proof is completed

6 Mathematical Problems in Engineering

Remark 5 Practically due to the hysteresis of nonlinear andswitching 119869 cannot converge to zero accurately within afinite time therefore the adaptive parameters and 120585 of theestimated values for 119870 and 120585 may increase boundlessly Inthe other words and 120585 obtained by the proposed adaptivealgorithm will be not accurate which may increase towardsinfinity In practical engineering it is difficult to apply (16)-(17) directly Consequently to solve such a problem deadzone technique is used [44] and adaptive laws (16) and (17)are modified as

119870 = 120583 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 10038161003816100381610038161198691198941003816100381610038161003816 ge 12058210 10038161003816100381610038161198691198941003816100381610038161003816 lt 1205821

(39)

120585 = 120573120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 10038161003816100381610038161198691198941003816100381610038161003816 ge 12058220 10038161003816100381610038161198691198941003816100381610038161003816 lt 1205822

(40)

where 1205821 and 1205821 are both small positive constants

Remark 6 The dynamic sliding function 119869 reaches andremains on the sliding surface 119869 = 0 before the conventionalsliding function 119878 gets to the sliding surface 119878 = 0 if and onlyif 1205881 1205882 satisfy [45]

|119869 (0)| le (12058821205881)2 |119878 (0)| (41)

4 Simulation

In order to verify the validity of control method proposedin this paper we applied it to two-joint manipulator modelIn this section four cases with considering signal fault anddouble joints faults respectively will be simulated to presentthe tracking effectiveness of manipulator

41 Simulation Cases

Case 1 Link 1 and link 2 are both healthy without fault ie119864 = 119868 where 119868 is the identity matrix

Case 2 Link 1 is healthy and actuator fault occurs in link 2 at10s ie

1198901 = 11198902 =

1 119905 lt 1011990402 119905 ge 10119904

(42)

Case 3 Actuator fault occurs in link 1 at 10s and link 2 ishealthy ie

1198901 = 1 119905 lt 1011990406 119905 ge 10119904

1198902 = 1(43)

Case 4 Actuator fault occurs in link 1 and link 2 at 10s ie

1198901 = 1 119905 lt 1011990406 119905 ge 10119904

1198902 = 1 119905 lt 1011990402 119905 ge 10119904

(44)

Simulation Parameters The parameters of simulation aredesigned as follows

119867(119902) = [366 + 174 cos 1199022 076 + 087 cos 1199022076 + 087 cos 1199022 076 ]

119866 (119902) = [304119892 cos 1199021 + 087119892 cos (1199021 + 1199022)087119892 cos (1199021 + 1199022) ]

120591119889 = [02 sin 119902102 sin 1199022]

(45)

The initial state of manipulator system is selected as 1199020 =(0 0)119879119903119886119889119904 1199020 = (0 0)119879119903119886119889119904Tracked objective trajectory is 1199021198891 = 07 sin (120587119905)119903119886119889 1199021198892 =

sin (120587119905)119903119886119889 respectivelyTo make the overall system stable equipped with strong

robustness and fault tolerance the parameters of the pro-posed controller are selected as 120576 = 1 and the learning gainsof adaptive laws are adopted as 120583 = 05 120573 = 01 To realizefast convergence of the sliding mode surface sliding modeparameters are designed as 119897 = diag 0001 0001 1205881 = 031205882 = 5 1205821 = 1205822 = 005Simulation Results and Analysis In the simulation accordingto the designed control law (15) and adaptive laws (39) (40)the time of fault tolerance in each case is shown in Table 1 andthe corresponding simulation results are depicted in Figures1ndash8 which show time responses of link position trackingvelocity tracking

Figure 1 shows the tracking responses in Case 1 FromFigure 1 we can see that the position and speed signal oflink 1 can track the corresponding desired signal within5s Meanwhile Figure 1 shows that tracking trajectories ofposition and speed of link 2 between actual signal and desiredsignal can converge to zero in 5s and then reach stabilityFrom Figure 5 it is easy to be seen that the control torquecan converge within 4 seconds without actuator fault in thiscase

Figure 2 depicts the time responses of trajectory trackingin Case 2 FromFigure 2 it can be easily seen that the positionand speed tracking errors of link 1 between actual signal anddesired signal can converge to zero in 5s without actuatorfault Also Figure 2 shows that tracking error of two linkscan converge to zero in 5s after actuator fault occurs in link2 at 10s system can rapidly deal with the fault and realizetrajectory tracking within 5 seconds From Figure 6 we cansee that when actuator fault occurs in link 2 the control

Mathematical Problems in Engineering 7

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

resp

onse

of l

ink

(rad

)

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Figure 1 Tracking responses of links 1 and 2 in Case 1

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minusminus

minus

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

resp

onse

of l

ink

(rad

)

Figure 2 Tracking responses of links 1 and 2 in Case 2

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

trac

king

of l

ink

(rad

)

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Figure 3 Tracking responses of links 1 and 2 in Case 3

8 Mathematical Problems in Engineering

Table 1 Time of fault tolerance

CASESPosition

tracking oflink1(s)

Speedtracking oflink1(s)

Controltorque oflink1(s)

Positiontracking oflink2(s)

Speedtracking oflink2(s)

Controltorque oflink2(s)

1 2 5 5 33 7 7 3 4 7 7 3 5 5 3

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minus

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

trac

king

of l

ink

(rad

)

Figure 4 Tracking responses of links 1 and 2 in Case 4

minus

minus

minus

Link 1Link 2

Time (s)

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 5 Time responses of control torque in Case 1

torque of link 2 can handle the fault within 3 seconds but alittle chattering phenomenon appears

The tracking responses in Case 3 are shown in Figure 3from which it can be easily seen that link position can dealwith fault by itself and basically track the desired positionsignal in 7s when actuator fault appears in link 1 at 10sCompared with other cases the error between actual signaland desired signal is a little high but is still in an acceptablerange Figure 3 also provides the actual speed of link 1 withactuator fault being able to track expected signal within 7sFrom Figure 3 we can respectively see position and speed oflink 2 with no fault being able to easily realize the trajectory

tracking in 5s Figure 7 shows the time response of controltorque from which we can see that the amplitude of link 1increases and control torque can reconvergewithin 3 seconds

Figure 4 provides the trajectory tracking responses of twolinks in this case The tracking response of link 1 is shown inFigure 4 from which it is easy to see that when actuator oflink 1 failed at 10s position signal and speed signal of link 1can deal with fault and basically track the desired signals in 7seconds Although there exists a certain error in both positionand speed tracking the error is acceptable in programmingFigure 4 also depicts the tracking response of link 2 fromwhich we can see that it takes 5 seconds for position and

Mathematical Problems in Engineering 9

Link 1Link 2

Time (s)

minus

minus

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 6 Time responses of control torque in Case 2

Link 1Link 2

Time (s)

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 7 Time responses of control torque in Case 3

Link 1Link 2

Time (s)

minus

minus

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 8 Time responses of control torque in Case 4

speed signal to track the corresponding desired signals whenactuator fault occurs in link 2 at 10s Figure 8 depicts the timeresponse of control torque in this case from which it can beeasily seen that the amplitudes of 2 links increase and controltorque can reconverge within 3 seconds but a little chatteringphenomenon appears

From Table 1 and Figures 1ndash8 we can see that when bothactuators are healthy without fault the overall system canrealize trajectory tracking within 5 seconds when actuatorfault occurs in link 2 tracking responses of link 2 canreconverge to desired signal within 5 seconds when actuator

fault occurs in link 1 tracking responses of link 1 canreconverge to desired signal within 7 seconds when actuatorfault occurs in two links tracking responses of links 1 and2 can reconverge to desired signal within 7 and 5 secondsrespectivelyTherefore the trajectory tracking effectiveness ofthe FTC method proposed in this paper is verified

5 Conclusions

In this paper a novel FTC scheme based on adaptive slidingmode method is investigated for manipulator with actuator

10 Mathematical Problems in Engineering

fault and external disturbance Firstly the general dynamicmodel of space manipulator is introduced and further itsactuator faulty model is established Secondly an adaptivefault tolerant controller is designed for manipulator withactuator fault Parameters of fault and disturbance are esti-mated and updated by online adaptive method Finally theproposed controller is applied to two-joint manipulator fromthe simulation results it is shown that the controller proposedin this paper can not only realize a good trajectory trackingbut also tolerate actuator fault and present strong robustnessfor external disturbance while effectively reducing the chat-tering phenomenon of sliding mode control

Data Availability

The data supporting the conclusions of our manuscript aresome open access articles that have been properly citedand the readers can easily obtain these articles to verify theconclusions replicate the analysis and conduct secondaryanalysesTherefore we did not create a publicly available datarepository

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by National Nature Science Founda-tion (under Grant 61601228 61403210 and 61603191) NaturalScience Foundation of Jiangsu (BK20161021) the Open Pro-gram of Jiangsu Key Laboratory of 3D Printing Equipmentand Manufacturing (BM2013006) Item number 3DL201607and the National Key Research and Development Program ofChina (Grant No 2017YFB1103200)

References

[1] D Zhao S Li and Q Zhu ldquoA new TSMC prototype robustnonlinear task space control of a 6 DOF parallel roboticmanipulatorrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1189ndash1197 2010

[2] YTaniseK Taniguchi S YamazakiMKamata YYamada andT Nakamura ldquoDevelopment of an air duct cleaning robot forhousing based on peristaltic crawlingmotionrdquo in Proceedings ofthe 2017 IEEE International Conference on Advanced IntelligentMechatronics AIM 2017 pp 1267ndash1272 Germany July 2017

[3] A S Al-Yahmadi J Abdo and T C Hsia ldquoModeling andcontrol of two manipulators handling a flexible objectrdquo Journalof e Franklin Institute vol 344 no 5 pp 349ndash361 2007

[4] Buren G Sun and R Hu ldquoFlexible force control on robot armfor spacecraft assemblyrdquo Spacecra Environment Engineeringvol 465 no 21 pp 147ndash152 2014

[5] L A Tuan Y H Joo P X Duong et al ldquoParameter estimatorintegrated-sliding mode control of dual arm robotsrdquo Interna-tional Journal of Control Automation amp Systems vol 15 no 6pp 2754ndash2763 2017

[6] Y Xu H Sun Q Jia et al ldquoControl of space robotic manipula-tors with faultsrdquo Mechanical Science amp Technology vol 25 no12 pp 1503-1423 2006

[7] E Coleshill L Oshinowo R Rembala B Bina D Rey andS Sindelar ldquoDextre improving maintenance operations on theInternational Space StationrdquoActaAstronautica vol 64 no 9-10pp 869ndash874 2009

[8] S B Nokleby ldquoSingularity analysis of the canadarmrdquo Mecha-nism ampMachine eory vol 42 no 4 pp 442ndash454 2007

[9] R Boumans and C Heemskerk ldquoThe European Robotic Armfor the International Space Stationrdquo Robotics and AutonomousSystems vol 23 no 1-2 pp 17ndash27 1998

[10] I Duleba and M Opalka ldquoA comparison of Jacobian-basedmethods of inverse kinematics for serial robot manipulatorsrdquoInternational Journal of Applied Mathematics amp ComputerScience vol 23 pp 373ndash382 2013

[11] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquoUkrainianMathematical Journal vol 69 no 8 pp 1220ndash1233 2017

[12] W Liu J Cui and J Xin ldquoA block-centered finite differencemethod for an unsteady asymptotic coupled model in fracturedmedia aquifer systemrdquo Journal of Computational and AppliedMathematics vol 337 pp 319ndash340 2018

[13] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[14] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 1ndash21 2017

[15] L Gao D Wang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[16] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[17] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[18] Z Cen H Noura and Y A Younes ldquoSystematic fault tolerantcontrol based on adaptiveThau observer estimation for quadro-tor UAVsrdquo International Journal of Applied Mathematics andComputer Science vol 25 no 1 pp 159ndash174 2015

[19] M Bonf P Castaldi NMimmo et al ldquoActive fault tolerant con-trol of nonlinear systems the cart-pole examplerdquo Internationaljournal of applied mathematics amp computer science vol 21 no3 pp 441ndash445 2011

[20] M Z Gao G P Cai and N Ying ldquoRobust adaptive fault-tolerant H8 control of re-entry vehicle considering actuator andsensor faults based on trajectory optimizationrdquo InternationalJournal of Control Automation amp Systems vol 14 no 1 pp 198ndash210 2016

[21] C Yang T Teng B Xu et al ldquoGlobal adaptive trackingcontrol of robotmanipulators usingneural networkswith finite-time learning convergencerdquo International Journal of ControlAutomation amp Systems vol 11 pp 1ndash9 2017

[22] Z Yu F Wang and L Liu ldquoAn adaptive fault-tolerant robustcontroller with fault diagnosis for submarinersquos vertical move-mentrdquo International Journal of Control Automation and Sys-tems vol 13 no 6 pp 1337ndash1345 2015

Mathematical Problems in Engineering 11

[23] L Li Y Yang S X Ding Y Zhang and S Zhai ldquoOn fault-tolerant control configurations for a class of nonlinear systemsrdquoJournal of e Franklin Institute vol 352 no 4 pp 1397ndash14162015

[24] P Yang R Guo X Pan et al ldquoStudy on the sliding mode faulttolerant predictive control based on multi agent particle swarmoptimizationrdquo International Journal of Control Automation ampSystems vol 3 pp 1ndash9 2017

[25] J Yang F Zhu X Wang and X Bu ldquoRobust sliding-modeobserver-based sensor fault estimation actuator fault detectionand isolation for uncertain nonlinear systemsrdquo InternationalJournal of Control Automation and Systems vol 13 no 5 pp1037ndash1046 2015

[26] M Van S S Ge and H Ren ldquoFinite time fault tolerantcontrol for robot manipulators using time delay estimation andcontinuous nonsingular fast terminal sliding mode controlrdquoIEEE Transactions on Cybernetics 2016

[27] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[28] S Liu J Wang Y Zhou and M Feckan ldquoIterative learningcontrol with pulse compensation for fractional differentialsystemsrdquoMathematica Slovaca vol 68 no 3 pp 563ndash574 2018

[29] C Ma T Li and J Zhang ldquoConsensus control for leader-following multi-agent systems with measurement noisesrdquo Jour-nal of Systems Science and Complexity vol 23 no 1 pp 35ndash492010

[30] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[31] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[32] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[33] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[34] A Owczarkowski and D Horla ldquoRobust LQR and LQI controlwith actuator failure of a 2DOF unmanned bicycle robotstabilized by an inertial wheelrdquo International Journal of AppliedMathematics and Computer Science vol 26 no 2 pp 325ndash3342016

[35] D Y Li and L Y Chun ldquoSensor fault identification anddecentralized fault-tolerant control of reconfigurable manip-ulator based on sliding mode observerrdquo in Proceedings ofthe 2012 International Conference on Control Engineering andCommunication Technology ICCECT 2012 pp 137ndash140 ChinaDecember 2012

[36] A Ferreira de Loza J Cieslak D Henry A Zolghadri andL M Fridman ldquoOutput tracking of systems subjected toperturbations and a class of actuator faults based on HOSMobservation and identificationrdquo Automatica vol 59 pp 200ndash205 2015

[37] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[38] L M Capisani A Ferrara A Ferreira De Loza and L MFridman ldquoManipulator fault diagnosis via higher order sliding-mode observersrdquo IEEE Transactions on Industrial Electronicsvol 59 no 10 pp 3979ndash3986 2012

[39] A Yarza V Santibanez and JMoreno-Valenzuela ldquoAn adaptiveoutput feedbackmotion tracking controller for robotmanipula-tors uniform global asymptotic stability and experimentationrdquoInternational Journal of Applied Mathematics and ComputerScience vol 23 no 3 pp 599ndash611 2013

[40] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[41] Q L Hu L Xu and A H Zhang ldquoAdaptive backsteppingtrajectory tracking control of robotmanipulatorrdquo Journal ofeFranklin Institute vol 349 no 3 pp 1087ndash1105 2012

[42] J-H Shin and J-J Lee ldquoFault Detection And Robust FaultRecovery Control for Robot Manipulators with Actuator Fail-uresrdquo in Proceedings of the 1999 IEEE International Conferenceon Robotics and Automation ICRA99 vol 2 pp 861ndash866 May1999

[43] J K Liu SlidingMode Control Design andMATLAB SimulationeBasiceory andDesignMethod TsinghuaUniversity Press2005

[44] S Mondal and C Mahanta ldquoAdaptive second order terminalsliding mode controller for robotic manipulatorsrdquo Journal ofe Franklin Institute vol 351 no 4 pp 2356ndash2377 2014

[45] A J Koshkouei K J Burnham and A S I Zinober ldquoDynamicsliding mode control designrdquo IEE Proceedings - Control eoryand Applications vol 152 no 4 pp 392ndash396 2005

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Mathematical Problems in Engineering 3

where 119892(119902 119902) = minus119862(119902 119902) 119902 minus 119866(119902) minus 119865( 119902) For conveniencedefine 119867 ≜ 119867(119902) 119862 ≜ 119862(119902 119902) 119866 ≜ 119866(119902) 119865 ≜ 119865( 119902) and119892 ≜ 119892(119902 119902)3 Fault Tolerant Controller Design

31 Problem Statement In this paper we are absorbed ininvestigating an FTC method for manipulator with actuatorfault The trajectory tracking problem of fault system (2)is considered The control objective can be described asthat for a manipulator control system with actuator faultan FTC method is proposed to ensure that the closed-loopsystem is stable ie when 119905 997888rarr infin 119902 997888rarr 119902119889 where 119902119889denotes the desired position signal For this purpose an FTCmethod based on adaptive dynamic sliding mode technologyis proposed Firstly an assumption is given as follows

Assumption 1 The external disturbance 120591119889 is assumed to benorm-bounded 10038171003817100381710038171205911198891003817100381710038171003817 le 119870 (4)

where119870 is an unknown positive constant and sdot represents119871infin norm in this paper

32 Controller Design As mentioned in the above sectionthe desired position signal is defined as 119902119889 so the trackingerror is 119911 = 119902minus119902119889Then the conventional slidingmode surfaceis selected as

119878 = + 119897119911 (5)

where 119897 isin R119899times119899 is a positive matrix further

119878 = + 119897 = 119902 minus 119902119889 + 119897 ( 119902 minus 119902119889) (6)

Next the dynamic sliding mode surface is selected as

119869 = 119878 + 120594 (7)

where 120594 can be considered as the error between 119869 and 119878 andits derivative of time is 120594 = 120588111987805sgn (119878) minus 120588211986905sgn (119869)where sgn(119878) = [sgn (1198781) sgn (1198782) sgn (119878119899)]119879 sgn (119869) =[sgn (1198691) sgn (1198692) sgn (119869119899)]119879 1205881 is the sliding mode gainof conventional sliding mode surface 119878 and 1205882 is the slidingmode gain of dynamic sliding mode surface 119869 They are twopositive constants satisfying 1205881 gt 0 1205882 gt 0 1205881 = 1205882 and formore details please refer to Remark 6

Theorem 2 Considering manipulator system with free-swinging fault and external disturbance (2) under Assump-tion 1 a fault tolerant control input based on adaptive slidingmode controller (8) and adaptive law (9) can guarantee asymp-totic output tracking of manipulator control system in bothcases of no fault and fault which guarantees the boundednessof all the closed-loop signals and asymptotic output tracking

120591 = minus119892 minus 119867119891 minus 119862119869 minus 120601sgn (119869) minus sgn (119869) (8)

119870 = 120583 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (9)

where is the estimated value of 119870 120576 is a positive constant119891 = minus 119902119889 + 119897( 119902 minus 119902119889) + 120594 120601 = (119887(1 minus 119887))[119892 + 119867119891 + 119862119869 +sgn (119869) + 120576] and 119887 = Δ119864 To satisfy the stability of thesystem achieve a good tracking effectiveness and estimate thevalue of disturbances (9) is obtained to put an integral actionin the definition of (8)

Proof Define a Lyapunov function as follows

119881 = 12119869119879119867119869 + 121205832 (10)

where denotes the estimated error of disturbance that is = 119870 minus and 120583 is a positive constant The time derivativeof 119881 is obtained

= 12119869119879119869 + 119869119879119867 119869 minus 1120583 119870= 12119869119879 ( minus 2119862) 119869 + 119869119879119862119869 + 119869119879119867 119869 minus 1120583 119870= 119869119879 (119862119869 + 119867 119869) minus 1120583 119870= 119869119879 (119862119869 + 119867 119902 + 119867119891) minus 1120583 119870

(11)

Substituting (3) into the above equation one can obtain thefollowing

= 119869119879 (119862119869 + 120591 minus 119864 (119905) 120591 + 120591119889 + 119892 + 119867119891) minus 1120583 119870 (12)

Applying controller (8) and adaptive law (9) into the aboveequation one can obtain

= 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869) minus sgn (119869) + 120591119889]minus 1120583 119870

= 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869)] minus 119869119879sgn (119869) + 119869119879120591119889minus 1120583 119870

le 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869)] minus 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 + 119870119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816minus 1120583 119870

= 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869)] + ( 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1120583 119870)

(13)

4 Mathematical Problems in Engineering

Further simplify

= 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869)]= 119869119879 119864 (119905) [119892 + 119867119891 + 119862119869 + 120601sgn (119869) + sgn (119869)]minus 119869119879120601sgn (119869)

le 119887 119869 [1003817100381710038171003817119892 + 119867119891 + 1198621198691003817100381710038171003817 + sgn (119869)] + 119887120601119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816minus 120601 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816le 119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 [1003817100381710038171003817119892 + 119867119891 + 1198621198691003817100381710038171003817 + sgn (119869)]

minus 120601 (1 minus 119887) 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 = minus119887120576 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 le 0

(14)

and thus the stability of the closed-loop system is verified

Remark 3 In practical application the smallest valuableof actuator fault min119890119894(119905) is usually unknown thereforeit is necessary to design an adaptive law to estimate faultinformation To solve this problem an adaptive fault tolerantcontroller will be designed in the following inwhich adaptivescheme is adopted to estimate both the actuator fault andexternal disturbance

When dealing with the faulty term define 119887 = Δ119864(119905)120585 = 1(1 minus 119887) and then adaptive algorithm is adoptedto estimate 120585 which can compensate for the existent faultRegarding the disturbance based on Assumption 1 isused to compensate for the existent disturbance by adaptivetechnique

Theorem 4 Based on eorem 2 considering manipulatorsystem with free-swinging fault and external disturbance (2)under Assumption 1 the minimum boundary value of actuatorfault min119890119894(119905) is unknown and an FTC input based onadaptive sliding mode controller (15) and adaptive laws (16)-(17) can guarantee asymptotic output tracking of manipulatorcontrol system in both cases ie no fault and fault whichguarantees the boundedness of all the closed-loop signals andasymptotic output tracking

120591 = minus119892 minus 119867119891 minus 119862119869 minus 120574 (119905) sgn (119869) minus sgn (119869) (15)

119870 = 120583 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (16)

120585 = 120573120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (17)

where is the estimated value of 119870 120575 = 119892 + 119867119891 + 119862119869 + + 120576 and 120576 is a positive constant 120574(119905) = minus120575 + 120585120575 and 120585 is theestimated value of 120585

Proof In order to prove the stability of the overall system theLyapunov function is selected as

119881 = 1198811 + 1198812 + 1198813 (18)

where 1198811 1198812 1198813 represent three Lyapunov functions whichwill be elaborated in the following 3 steps further

= 1 + 2 + 3 (19)

The process of proof is divided into three steps

Step 1 (adaptive law analysis) To obtain the adaptive laws offault and disturbance the Lyapunov function is chosen as

1198811 = 12119869119879119867119869 + 121205832 + 1 minus 1198872120573 1205852 (20)

where denotes the estimated error of disturbance 120585 denotesthe estimated error of fault = 119870 minus 120585 = 120585 minus 120585 and 120583and 120573 are both positive constants The time derivative of1198811 isobtained

1 = 12119869119879119869 + 119869119879119867 119869 minus 1120583 119870 minus 1 minus 119887120573 120585 120585= 12119869119879 ( minus 2119862) 119869 + 119869119879119862119869 + 119869119879119867 119869 minus 1120583 119870minus 1 minus 119887120573 120585 120585

= 119869119879 (119862119869 + 119867 119902 + 119867119891) minus 1120583 119870 minus 1 minus 119887120573 120585 120585

(21)

Substituting (3) into the above equation one obtains thefollowing

1 = 119869119879 (119862119869 + 120591 minus 119864 (119905) 120591 + 120591119889 + 119892 + 119867119891) minus 1120583 119870minus 1 minus 119887120573 120585 120585

(22)

Applying controller (15) into the above equation yields thefollowing

1 = 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869) minus sgn (119869) + 120591119889]minus 1120583 119870 minus 1 minus 119887120573 120585 120585

= 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869)] + 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816minus 1120583 119870 minus 1 minus 119887120573 120585 120585

= 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869)]+ ( 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1120583 119870) minus 1 minus 119887120573 120585 120585

(23)

Mathematical Problems in Engineering 5

Substituting adaptive law (16) into the above equation yieldsthe following

1 le 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869)] minus 1 minus 119887120573 120585 120585= minus120574 (119905) 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 + 119869119879 119864 (119905) [119892 + 119867119891 + 119862119869 + 120574 (119905) sgn (119869) + sgn (119869)]

le (1 minus 120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585+ 119887 1003817100381710038171003817100381711986911987910038171003817100381710038171003817 [1003817100381710038171003817119892 + 119867119891 + 1198621198691003817100381710038171003817 + 120574 (119905) + ]

le (1 minus 120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 + 119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (minus120576 + 120585120575)

= (1 minus 120585 + 119887120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816= (1 minus 119887) (120585 minus 120585) 120575 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816= (1 minus 119887) 120585(120575 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1120573 120585) minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816

(24)

As a result applying adaptive law (17) into the above equationone can obtain the following

1 = minus120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 le 0 (25)

Step 2 (reach time analysis) Before analysis a lemma isproposed as follows to obtain the convergence time

Lemma 5e dynamic sliding mode function 119869 exists and canconverge within finite time 119905119869 please see [43] for more details

Aer time 119905119869 the surface 119869 = 0 from (7) one can obtainthe following

119878 = minus 120594 = minus1205881 11987805 sgn (119878) (26)

Further a Lyapunov function is selected as follows to obtain theconvergence time of system

1198812 = 12119878119879119878 (27)

Further

2 = 119878119879 119878 = 119878119879 (minus1205881 11987805 sgn (119878)) le minus1205881 11987815 le 0 (28)

To solve the above differential equation the following equationcan be obtained by (27)

1198812 = 12 1198782 (29)

Further

119878 = 100381610038161003816100381621198812100381610038161003816100381612 (30)

Substituting (30) to (28) yields the following

2 le minus1205881 100381610038161003816100381621198812100381610038161003816100381634 (31)

us

119881minus342 1198891198812 le minus2341205881119889119905int119905119905119869

119881minus342 1198891198812le int119905119905119869

minus23412058811198891199054 [1198812 (119905)14 minus 1198812 (119905119869)14]le minus2341205881 (119905 minus 119905119869)

(32)

As time 119905 reaches 119905119878 the conventional slidingmode surface 119878willconverge to zero ie when 119905 = 119905119878 119878 = 0 that is1198812(119905) = 1198812(119905119878)thus

minus41198812 (119905119869)14 le minus2341205881 (119905119878 minus 119905119869) (33)

us one can obtain the following

119905119878 le 41198812 (119905119869)14 + 23412058811199051198692341205881 (34)

Consequently sliding mode surface 119878 will converge to zerowithin finite time

119905119878 le 41198812 (119905119869)14 + 23412058811199051198692341205881 (35)

Now it is verified that conventional sliding mode surface 119878and dynamic sliding mode surface 119869 can both converge to zerowithin finite time and dynamic surface converges faster thanconventional surface ie lim119905ge119905119869(119869119878) = 0 As 119905 997888rarr 119905119878 isreached 119878 = 0 substituting this into (5) one can obtain thefollowing

= minus119897119911 (36)

Step 3 (tracking error analysis) To prove the convergenceof the tracking error 119911 the Lyapunov function is defined asfollows

1198813 = 12119911119879119911 (37)

Thus

3 = 119911119879 le minus119897 1199112 le 0 (38)

and therefore tracking error 119911 is convergent which meansthat when 119905 997888rarr infin 119902 997888rarr 119902119889 119911 997888rarr 0 Therefore accordingto the above three steps it is easy to be seen that lt 0 in(19) which means that the overall system can be stable withthe proposed controller The proof is completed

6 Mathematical Problems in Engineering

Remark 5 Practically due to the hysteresis of nonlinear andswitching 119869 cannot converge to zero accurately within afinite time therefore the adaptive parameters and 120585 of theestimated values for 119870 and 120585 may increase boundlessly Inthe other words and 120585 obtained by the proposed adaptivealgorithm will be not accurate which may increase towardsinfinity In practical engineering it is difficult to apply (16)-(17) directly Consequently to solve such a problem deadzone technique is used [44] and adaptive laws (16) and (17)are modified as

119870 = 120583 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 10038161003816100381610038161198691198941003816100381610038161003816 ge 12058210 10038161003816100381610038161198691198941003816100381610038161003816 lt 1205821

(39)

120585 = 120573120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 10038161003816100381610038161198691198941003816100381610038161003816 ge 12058220 10038161003816100381610038161198691198941003816100381610038161003816 lt 1205822

(40)

where 1205821 and 1205821 are both small positive constants

Remark 6 The dynamic sliding function 119869 reaches andremains on the sliding surface 119869 = 0 before the conventionalsliding function 119878 gets to the sliding surface 119878 = 0 if and onlyif 1205881 1205882 satisfy [45]

|119869 (0)| le (12058821205881)2 |119878 (0)| (41)

4 Simulation

In order to verify the validity of control method proposedin this paper we applied it to two-joint manipulator modelIn this section four cases with considering signal fault anddouble joints faults respectively will be simulated to presentthe tracking effectiveness of manipulator

41 Simulation Cases

Case 1 Link 1 and link 2 are both healthy without fault ie119864 = 119868 where 119868 is the identity matrix

Case 2 Link 1 is healthy and actuator fault occurs in link 2 at10s ie

1198901 = 11198902 =

1 119905 lt 1011990402 119905 ge 10119904

(42)

Case 3 Actuator fault occurs in link 1 at 10s and link 2 ishealthy ie

1198901 = 1 119905 lt 1011990406 119905 ge 10119904

1198902 = 1(43)

Case 4 Actuator fault occurs in link 1 and link 2 at 10s ie

1198901 = 1 119905 lt 1011990406 119905 ge 10119904

1198902 = 1 119905 lt 1011990402 119905 ge 10119904

(44)

Simulation Parameters The parameters of simulation aredesigned as follows

119867(119902) = [366 + 174 cos 1199022 076 + 087 cos 1199022076 + 087 cos 1199022 076 ]

119866 (119902) = [304119892 cos 1199021 + 087119892 cos (1199021 + 1199022)087119892 cos (1199021 + 1199022) ]

120591119889 = [02 sin 119902102 sin 1199022]

(45)

The initial state of manipulator system is selected as 1199020 =(0 0)119879119903119886119889119904 1199020 = (0 0)119879119903119886119889119904Tracked objective trajectory is 1199021198891 = 07 sin (120587119905)119903119886119889 1199021198892 =

sin (120587119905)119903119886119889 respectivelyTo make the overall system stable equipped with strong

robustness and fault tolerance the parameters of the pro-posed controller are selected as 120576 = 1 and the learning gainsof adaptive laws are adopted as 120583 = 05 120573 = 01 To realizefast convergence of the sliding mode surface sliding modeparameters are designed as 119897 = diag 0001 0001 1205881 = 031205882 = 5 1205821 = 1205822 = 005Simulation Results and Analysis In the simulation accordingto the designed control law (15) and adaptive laws (39) (40)the time of fault tolerance in each case is shown in Table 1 andthe corresponding simulation results are depicted in Figures1ndash8 which show time responses of link position trackingvelocity tracking

Figure 1 shows the tracking responses in Case 1 FromFigure 1 we can see that the position and speed signal oflink 1 can track the corresponding desired signal within5s Meanwhile Figure 1 shows that tracking trajectories ofposition and speed of link 2 between actual signal and desiredsignal can converge to zero in 5s and then reach stabilityFrom Figure 5 it is easy to be seen that the control torquecan converge within 4 seconds without actuator fault in thiscase

Figure 2 depicts the time responses of trajectory trackingin Case 2 FromFigure 2 it can be easily seen that the positionand speed tracking errors of link 1 between actual signal anddesired signal can converge to zero in 5s without actuatorfault Also Figure 2 shows that tracking error of two linkscan converge to zero in 5s after actuator fault occurs in link2 at 10s system can rapidly deal with the fault and realizetrajectory tracking within 5 seconds From Figure 6 we cansee that when actuator fault occurs in link 2 the control

Mathematical Problems in Engineering 7

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

resp

onse

of l

ink

(rad

)

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Figure 1 Tracking responses of links 1 and 2 in Case 1

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minusminus

minus

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

resp

onse

of l

ink

(rad

)

Figure 2 Tracking responses of links 1 and 2 in Case 2

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

trac

king

of l

ink

(rad

)

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Figure 3 Tracking responses of links 1 and 2 in Case 3

8 Mathematical Problems in Engineering

Table 1 Time of fault tolerance

CASESPosition

tracking oflink1(s)

Speedtracking oflink1(s)

Controltorque oflink1(s)

Positiontracking oflink2(s)

Speedtracking oflink2(s)

Controltorque oflink2(s)

1 2 5 5 33 7 7 3 4 7 7 3 5 5 3

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minus

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

trac

king

of l

ink

(rad

)

Figure 4 Tracking responses of links 1 and 2 in Case 4

minus

minus

minus

Link 1Link 2

Time (s)

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 5 Time responses of control torque in Case 1

torque of link 2 can handle the fault within 3 seconds but alittle chattering phenomenon appears

The tracking responses in Case 3 are shown in Figure 3from which it can be easily seen that link position can dealwith fault by itself and basically track the desired positionsignal in 7s when actuator fault appears in link 1 at 10sCompared with other cases the error between actual signaland desired signal is a little high but is still in an acceptablerange Figure 3 also provides the actual speed of link 1 withactuator fault being able to track expected signal within 7sFrom Figure 3 we can respectively see position and speed oflink 2 with no fault being able to easily realize the trajectory

tracking in 5s Figure 7 shows the time response of controltorque from which we can see that the amplitude of link 1increases and control torque can reconvergewithin 3 seconds

Figure 4 provides the trajectory tracking responses of twolinks in this case The tracking response of link 1 is shown inFigure 4 from which it is easy to see that when actuator oflink 1 failed at 10s position signal and speed signal of link 1can deal with fault and basically track the desired signals in 7seconds Although there exists a certain error in both positionand speed tracking the error is acceptable in programmingFigure 4 also depicts the tracking response of link 2 fromwhich we can see that it takes 5 seconds for position and

Mathematical Problems in Engineering 9

Link 1Link 2

Time (s)

minus

minus

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 6 Time responses of control torque in Case 2

Link 1Link 2

Time (s)

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 7 Time responses of control torque in Case 3

Link 1Link 2

Time (s)

minus

minus

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 8 Time responses of control torque in Case 4

speed signal to track the corresponding desired signals whenactuator fault occurs in link 2 at 10s Figure 8 depicts the timeresponse of control torque in this case from which it can beeasily seen that the amplitudes of 2 links increase and controltorque can reconverge within 3 seconds but a little chatteringphenomenon appears

From Table 1 and Figures 1ndash8 we can see that when bothactuators are healthy without fault the overall system canrealize trajectory tracking within 5 seconds when actuatorfault occurs in link 2 tracking responses of link 2 canreconverge to desired signal within 5 seconds when actuator

fault occurs in link 1 tracking responses of link 1 canreconverge to desired signal within 7 seconds when actuatorfault occurs in two links tracking responses of links 1 and2 can reconverge to desired signal within 7 and 5 secondsrespectivelyTherefore the trajectory tracking effectiveness ofthe FTC method proposed in this paper is verified

5 Conclusions

In this paper a novel FTC scheme based on adaptive slidingmode method is investigated for manipulator with actuator

10 Mathematical Problems in Engineering

fault and external disturbance Firstly the general dynamicmodel of space manipulator is introduced and further itsactuator faulty model is established Secondly an adaptivefault tolerant controller is designed for manipulator withactuator fault Parameters of fault and disturbance are esti-mated and updated by online adaptive method Finally theproposed controller is applied to two-joint manipulator fromthe simulation results it is shown that the controller proposedin this paper can not only realize a good trajectory trackingbut also tolerate actuator fault and present strong robustnessfor external disturbance while effectively reducing the chat-tering phenomenon of sliding mode control

Data Availability

The data supporting the conclusions of our manuscript aresome open access articles that have been properly citedand the readers can easily obtain these articles to verify theconclusions replicate the analysis and conduct secondaryanalysesTherefore we did not create a publicly available datarepository

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by National Nature Science Founda-tion (under Grant 61601228 61403210 and 61603191) NaturalScience Foundation of Jiangsu (BK20161021) the Open Pro-gram of Jiangsu Key Laboratory of 3D Printing Equipmentand Manufacturing (BM2013006) Item number 3DL201607and the National Key Research and Development Program ofChina (Grant No 2017YFB1103200)

References

[1] D Zhao S Li and Q Zhu ldquoA new TSMC prototype robustnonlinear task space control of a 6 DOF parallel roboticmanipulatorrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1189ndash1197 2010

[2] YTaniseK Taniguchi S YamazakiMKamata YYamada andT Nakamura ldquoDevelopment of an air duct cleaning robot forhousing based on peristaltic crawlingmotionrdquo in Proceedings ofthe 2017 IEEE International Conference on Advanced IntelligentMechatronics AIM 2017 pp 1267ndash1272 Germany July 2017

[3] A S Al-Yahmadi J Abdo and T C Hsia ldquoModeling andcontrol of two manipulators handling a flexible objectrdquo Journalof e Franklin Institute vol 344 no 5 pp 349ndash361 2007

[4] Buren G Sun and R Hu ldquoFlexible force control on robot armfor spacecraft assemblyrdquo Spacecra Environment Engineeringvol 465 no 21 pp 147ndash152 2014

[5] L A Tuan Y H Joo P X Duong et al ldquoParameter estimatorintegrated-sliding mode control of dual arm robotsrdquo Interna-tional Journal of Control Automation amp Systems vol 15 no 6pp 2754ndash2763 2017

[6] Y Xu H Sun Q Jia et al ldquoControl of space robotic manipula-tors with faultsrdquo Mechanical Science amp Technology vol 25 no12 pp 1503-1423 2006

[7] E Coleshill L Oshinowo R Rembala B Bina D Rey andS Sindelar ldquoDextre improving maintenance operations on theInternational Space StationrdquoActaAstronautica vol 64 no 9-10pp 869ndash874 2009

[8] S B Nokleby ldquoSingularity analysis of the canadarmrdquo Mecha-nism ampMachine eory vol 42 no 4 pp 442ndash454 2007

[9] R Boumans and C Heemskerk ldquoThe European Robotic Armfor the International Space Stationrdquo Robotics and AutonomousSystems vol 23 no 1-2 pp 17ndash27 1998

[10] I Duleba and M Opalka ldquoA comparison of Jacobian-basedmethods of inverse kinematics for serial robot manipulatorsrdquoInternational Journal of Applied Mathematics amp ComputerScience vol 23 pp 373ndash382 2013

[11] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquoUkrainianMathematical Journal vol 69 no 8 pp 1220ndash1233 2017

[12] W Liu J Cui and J Xin ldquoA block-centered finite differencemethod for an unsteady asymptotic coupled model in fracturedmedia aquifer systemrdquo Journal of Computational and AppliedMathematics vol 337 pp 319ndash340 2018

[13] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[14] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 1ndash21 2017

[15] L Gao D Wang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[16] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[17] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[18] Z Cen H Noura and Y A Younes ldquoSystematic fault tolerantcontrol based on adaptiveThau observer estimation for quadro-tor UAVsrdquo International Journal of Applied Mathematics andComputer Science vol 25 no 1 pp 159ndash174 2015

[19] M Bonf P Castaldi NMimmo et al ldquoActive fault tolerant con-trol of nonlinear systems the cart-pole examplerdquo Internationaljournal of applied mathematics amp computer science vol 21 no3 pp 441ndash445 2011

[20] M Z Gao G P Cai and N Ying ldquoRobust adaptive fault-tolerant H8 control of re-entry vehicle considering actuator andsensor faults based on trajectory optimizationrdquo InternationalJournal of Control Automation amp Systems vol 14 no 1 pp 198ndash210 2016

[21] C Yang T Teng B Xu et al ldquoGlobal adaptive trackingcontrol of robotmanipulators usingneural networkswith finite-time learning convergencerdquo International Journal of ControlAutomation amp Systems vol 11 pp 1ndash9 2017

[22] Z Yu F Wang and L Liu ldquoAn adaptive fault-tolerant robustcontroller with fault diagnosis for submarinersquos vertical move-mentrdquo International Journal of Control Automation and Sys-tems vol 13 no 6 pp 1337ndash1345 2015

Mathematical Problems in Engineering 11

[23] L Li Y Yang S X Ding Y Zhang and S Zhai ldquoOn fault-tolerant control configurations for a class of nonlinear systemsrdquoJournal of e Franklin Institute vol 352 no 4 pp 1397ndash14162015

[24] P Yang R Guo X Pan et al ldquoStudy on the sliding mode faulttolerant predictive control based on multi agent particle swarmoptimizationrdquo International Journal of Control Automation ampSystems vol 3 pp 1ndash9 2017

[25] J Yang F Zhu X Wang and X Bu ldquoRobust sliding-modeobserver-based sensor fault estimation actuator fault detectionand isolation for uncertain nonlinear systemsrdquo InternationalJournal of Control Automation and Systems vol 13 no 5 pp1037ndash1046 2015

[26] M Van S S Ge and H Ren ldquoFinite time fault tolerantcontrol for robot manipulators using time delay estimation andcontinuous nonsingular fast terminal sliding mode controlrdquoIEEE Transactions on Cybernetics 2016

[27] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[28] S Liu J Wang Y Zhou and M Feckan ldquoIterative learningcontrol with pulse compensation for fractional differentialsystemsrdquoMathematica Slovaca vol 68 no 3 pp 563ndash574 2018

[29] C Ma T Li and J Zhang ldquoConsensus control for leader-following multi-agent systems with measurement noisesrdquo Jour-nal of Systems Science and Complexity vol 23 no 1 pp 35ndash492010

[30] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[31] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[32] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[33] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[34] A Owczarkowski and D Horla ldquoRobust LQR and LQI controlwith actuator failure of a 2DOF unmanned bicycle robotstabilized by an inertial wheelrdquo International Journal of AppliedMathematics and Computer Science vol 26 no 2 pp 325ndash3342016

[35] D Y Li and L Y Chun ldquoSensor fault identification anddecentralized fault-tolerant control of reconfigurable manip-ulator based on sliding mode observerrdquo in Proceedings ofthe 2012 International Conference on Control Engineering andCommunication Technology ICCECT 2012 pp 137ndash140 ChinaDecember 2012

[36] A Ferreira de Loza J Cieslak D Henry A Zolghadri andL M Fridman ldquoOutput tracking of systems subjected toperturbations and a class of actuator faults based on HOSMobservation and identificationrdquo Automatica vol 59 pp 200ndash205 2015

[37] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[38] L M Capisani A Ferrara A Ferreira De Loza and L MFridman ldquoManipulator fault diagnosis via higher order sliding-mode observersrdquo IEEE Transactions on Industrial Electronicsvol 59 no 10 pp 3979ndash3986 2012

[39] A Yarza V Santibanez and JMoreno-Valenzuela ldquoAn adaptiveoutput feedbackmotion tracking controller for robotmanipula-tors uniform global asymptotic stability and experimentationrdquoInternational Journal of Applied Mathematics and ComputerScience vol 23 no 3 pp 599ndash611 2013

[40] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[41] Q L Hu L Xu and A H Zhang ldquoAdaptive backsteppingtrajectory tracking control of robotmanipulatorrdquo Journal ofeFranklin Institute vol 349 no 3 pp 1087ndash1105 2012

[42] J-H Shin and J-J Lee ldquoFault Detection And Robust FaultRecovery Control for Robot Manipulators with Actuator Fail-uresrdquo in Proceedings of the 1999 IEEE International Conferenceon Robotics and Automation ICRA99 vol 2 pp 861ndash866 May1999

[43] J K Liu SlidingMode Control Design andMATLAB SimulationeBasiceory andDesignMethod TsinghuaUniversity Press2005

[44] S Mondal and C Mahanta ldquoAdaptive second order terminalsliding mode controller for robotic manipulatorsrdquo Journal ofe Franklin Institute vol 351 no 4 pp 2356ndash2377 2014

[45] A J Koshkouei K J Burnham and A S I Zinober ldquoDynamicsliding mode control designrdquo IEE Proceedings - Control eoryand Applications vol 152 no 4 pp 392ndash396 2005

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

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4 Mathematical Problems in Engineering

Further simplify

= 119869119879 [minus 119864 (119905) 120591 minus 120601sgn (119869)]= 119869119879 119864 (119905) [119892 + 119867119891 + 119862119869 + 120601sgn (119869) + sgn (119869)]minus 119869119879120601sgn (119869)

le 119887 119869 [1003817100381710038171003817119892 + 119867119891 + 1198621198691003817100381710038171003817 + sgn (119869)] + 119887120601119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816minus 120601 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816le 119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 [1003817100381710038171003817119892 + 119867119891 + 1198621198691003817100381710038171003817 + sgn (119869)]

minus 120601 (1 minus 119887) 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 = minus119887120576 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 le 0

(14)

and thus the stability of the closed-loop system is verified

Remark 3 In practical application the smallest valuableof actuator fault min119890119894(119905) is usually unknown thereforeit is necessary to design an adaptive law to estimate faultinformation To solve this problem an adaptive fault tolerantcontroller will be designed in the following inwhich adaptivescheme is adopted to estimate both the actuator fault andexternal disturbance

When dealing with the faulty term define 119887 = Δ119864(119905)120585 = 1(1 minus 119887) and then adaptive algorithm is adoptedto estimate 120585 which can compensate for the existent faultRegarding the disturbance based on Assumption 1 isused to compensate for the existent disturbance by adaptivetechnique

Theorem 4 Based on eorem 2 considering manipulatorsystem with free-swinging fault and external disturbance (2)under Assumption 1 the minimum boundary value of actuatorfault min119890119894(119905) is unknown and an FTC input based onadaptive sliding mode controller (15) and adaptive laws (16)-(17) can guarantee asymptotic output tracking of manipulatorcontrol system in both cases ie no fault and fault whichguarantees the boundedness of all the closed-loop signals andasymptotic output tracking

120591 = minus119892 minus 119867119891 minus 119862119869 minus 120574 (119905) sgn (119869) minus sgn (119869) (15)

119870 = 120583 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (16)

120585 = 120573120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (17)

where is the estimated value of 119870 120575 = 119892 + 119867119891 + 119862119869 + + 120576 and 120576 is a positive constant 120574(119905) = minus120575 + 120585120575 and 120585 is theestimated value of 120585

Proof In order to prove the stability of the overall system theLyapunov function is selected as

119881 = 1198811 + 1198812 + 1198813 (18)

where 1198811 1198812 1198813 represent three Lyapunov functions whichwill be elaborated in the following 3 steps further

= 1 + 2 + 3 (19)

The process of proof is divided into three steps

Step 1 (adaptive law analysis) To obtain the adaptive laws offault and disturbance the Lyapunov function is chosen as

1198811 = 12119869119879119867119869 + 121205832 + 1 minus 1198872120573 1205852 (20)

where denotes the estimated error of disturbance 120585 denotesthe estimated error of fault = 119870 minus 120585 = 120585 minus 120585 and 120583and 120573 are both positive constants The time derivative of1198811 isobtained

1 = 12119869119879119869 + 119869119879119867 119869 minus 1120583 119870 minus 1 minus 119887120573 120585 120585= 12119869119879 ( minus 2119862) 119869 + 119869119879119862119869 + 119869119879119867 119869 minus 1120583 119870minus 1 minus 119887120573 120585 120585

= 119869119879 (119862119869 + 119867 119902 + 119867119891) minus 1120583 119870 minus 1 minus 119887120573 120585 120585

(21)

Substituting (3) into the above equation one obtains thefollowing

1 = 119869119879 (119862119869 + 120591 minus 119864 (119905) 120591 + 120591119889 + 119892 + 119867119891) minus 1120583 119870minus 1 minus 119887120573 120585 120585

(22)

Applying controller (15) into the above equation yields thefollowing

1 = 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869) minus sgn (119869) + 120591119889]minus 1120583 119870 minus 1 minus 119887120573 120585 120585

= 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869)] + 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816minus 1120583 119870 minus 1 minus 119887120573 120585 120585

= 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869)]+ ( 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1120583 119870) minus 1 minus 119887120573 120585 120585

(23)

Mathematical Problems in Engineering 5

Substituting adaptive law (16) into the above equation yieldsthe following

1 le 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869)] minus 1 minus 119887120573 120585 120585= minus120574 (119905) 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 + 119869119879 119864 (119905) [119892 + 119867119891 + 119862119869 + 120574 (119905) sgn (119869) + sgn (119869)]

le (1 minus 120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585+ 119887 1003817100381710038171003817100381711986911987910038171003817100381710038171003817 [1003817100381710038171003817119892 + 119867119891 + 1198621198691003817100381710038171003817 + 120574 (119905) + ]

le (1 minus 120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 + 119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (minus120576 + 120585120575)

= (1 minus 120585 + 119887120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816= (1 minus 119887) (120585 minus 120585) 120575 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816= (1 minus 119887) 120585(120575 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1120573 120585) minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816

(24)

As a result applying adaptive law (17) into the above equationone can obtain the following

1 = minus120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 le 0 (25)

Step 2 (reach time analysis) Before analysis a lemma isproposed as follows to obtain the convergence time

Lemma 5e dynamic sliding mode function 119869 exists and canconverge within finite time 119905119869 please see [43] for more details

Aer time 119905119869 the surface 119869 = 0 from (7) one can obtainthe following

119878 = minus 120594 = minus1205881 11987805 sgn (119878) (26)

Further a Lyapunov function is selected as follows to obtain theconvergence time of system

1198812 = 12119878119879119878 (27)

Further

2 = 119878119879 119878 = 119878119879 (minus1205881 11987805 sgn (119878)) le minus1205881 11987815 le 0 (28)

To solve the above differential equation the following equationcan be obtained by (27)

1198812 = 12 1198782 (29)

Further

119878 = 100381610038161003816100381621198812100381610038161003816100381612 (30)

Substituting (30) to (28) yields the following

2 le minus1205881 100381610038161003816100381621198812100381610038161003816100381634 (31)

us

119881minus342 1198891198812 le minus2341205881119889119905int119905119905119869

119881minus342 1198891198812le int119905119905119869

minus23412058811198891199054 [1198812 (119905)14 minus 1198812 (119905119869)14]le minus2341205881 (119905 minus 119905119869)

(32)

As time 119905 reaches 119905119878 the conventional slidingmode surface 119878willconverge to zero ie when 119905 = 119905119878 119878 = 0 that is1198812(119905) = 1198812(119905119878)thus

minus41198812 (119905119869)14 le minus2341205881 (119905119878 minus 119905119869) (33)

us one can obtain the following

119905119878 le 41198812 (119905119869)14 + 23412058811199051198692341205881 (34)

Consequently sliding mode surface 119878 will converge to zerowithin finite time

119905119878 le 41198812 (119905119869)14 + 23412058811199051198692341205881 (35)

Now it is verified that conventional sliding mode surface 119878and dynamic sliding mode surface 119869 can both converge to zerowithin finite time and dynamic surface converges faster thanconventional surface ie lim119905ge119905119869(119869119878) = 0 As 119905 997888rarr 119905119878 isreached 119878 = 0 substituting this into (5) one can obtain thefollowing

= minus119897119911 (36)

Step 3 (tracking error analysis) To prove the convergenceof the tracking error 119911 the Lyapunov function is defined asfollows

1198813 = 12119911119879119911 (37)

Thus

3 = 119911119879 le minus119897 1199112 le 0 (38)

and therefore tracking error 119911 is convergent which meansthat when 119905 997888rarr infin 119902 997888rarr 119902119889 119911 997888rarr 0 Therefore accordingto the above three steps it is easy to be seen that lt 0 in(19) which means that the overall system can be stable withthe proposed controller The proof is completed

6 Mathematical Problems in Engineering

Remark 5 Practically due to the hysteresis of nonlinear andswitching 119869 cannot converge to zero accurately within afinite time therefore the adaptive parameters and 120585 of theestimated values for 119870 and 120585 may increase boundlessly Inthe other words and 120585 obtained by the proposed adaptivealgorithm will be not accurate which may increase towardsinfinity In practical engineering it is difficult to apply (16)-(17) directly Consequently to solve such a problem deadzone technique is used [44] and adaptive laws (16) and (17)are modified as

119870 = 120583 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 10038161003816100381610038161198691198941003816100381610038161003816 ge 12058210 10038161003816100381610038161198691198941003816100381610038161003816 lt 1205821

(39)

120585 = 120573120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 10038161003816100381610038161198691198941003816100381610038161003816 ge 12058220 10038161003816100381610038161198691198941003816100381610038161003816 lt 1205822

(40)

where 1205821 and 1205821 are both small positive constants

Remark 6 The dynamic sliding function 119869 reaches andremains on the sliding surface 119869 = 0 before the conventionalsliding function 119878 gets to the sliding surface 119878 = 0 if and onlyif 1205881 1205882 satisfy [45]

|119869 (0)| le (12058821205881)2 |119878 (0)| (41)

4 Simulation

In order to verify the validity of control method proposedin this paper we applied it to two-joint manipulator modelIn this section four cases with considering signal fault anddouble joints faults respectively will be simulated to presentthe tracking effectiveness of manipulator

41 Simulation Cases

Case 1 Link 1 and link 2 are both healthy without fault ie119864 = 119868 where 119868 is the identity matrix

Case 2 Link 1 is healthy and actuator fault occurs in link 2 at10s ie

1198901 = 11198902 =

1 119905 lt 1011990402 119905 ge 10119904

(42)

Case 3 Actuator fault occurs in link 1 at 10s and link 2 ishealthy ie

1198901 = 1 119905 lt 1011990406 119905 ge 10119904

1198902 = 1(43)

Case 4 Actuator fault occurs in link 1 and link 2 at 10s ie

1198901 = 1 119905 lt 1011990406 119905 ge 10119904

1198902 = 1 119905 lt 1011990402 119905 ge 10119904

(44)

Simulation Parameters The parameters of simulation aredesigned as follows

119867(119902) = [366 + 174 cos 1199022 076 + 087 cos 1199022076 + 087 cos 1199022 076 ]

119866 (119902) = [304119892 cos 1199021 + 087119892 cos (1199021 + 1199022)087119892 cos (1199021 + 1199022) ]

120591119889 = [02 sin 119902102 sin 1199022]

(45)

The initial state of manipulator system is selected as 1199020 =(0 0)119879119903119886119889119904 1199020 = (0 0)119879119903119886119889119904Tracked objective trajectory is 1199021198891 = 07 sin (120587119905)119903119886119889 1199021198892 =

sin (120587119905)119903119886119889 respectivelyTo make the overall system stable equipped with strong

robustness and fault tolerance the parameters of the pro-posed controller are selected as 120576 = 1 and the learning gainsof adaptive laws are adopted as 120583 = 05 120573 = 01 To realizefast convergence of the sliding mode surface sliding modeparameters are designed as 119897 = diag 0001 0001 1205881 = 031205882 = 5 1205821 = 1205822 = 005Simulation Results and Analysis In the simulation accordingto the designed control law (15) and adaptive laws (39) (40)the time of fault tolerance in each case is shown in Table 1 andthe corresponding simulation results are depicted in Figures1ndash8 which show time responses of link position trackingvelocity tracking

Figure 1 shows the tracking responses in Case 1 FromFigure 1 we can see that the position and speed signal oflink 1 can track the corresponding desired signal within5s Meanwhile Figure 1 shows that tracking trajectories ofposition and speed of link 2 between actual signal and desiredsignal can converge to zero in 5s and then reach stabilityFrom Figure 5 it is easy to be seen that the control torquecan converge within 4 seconds without actuator fault in thiscase

Figure 2 depicts the time responses of trajectory trackingin Case 2 FromFigure 2 it can be easily seen that the positionand speed tracking errors of link 1 between actual signal anddesired signal can converge to zero in 5s without actuatorfault Also Figure 2 shows that tracking error of two linkscan converge to zero in 5s after actuator fault occurs in link2 at 10s system can rapidly deal with the fault and realizetrajectory tracking within 5 seconds From Figure 6 we cansee that when actuator fault occurs in link 2 the control

Mathematical Problems in Engineering 7

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

resp

onse

of l

ink

(rad

)

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Figure 1 Tracking responses of links 1 and 2 in Case 1

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minusminus

minus

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

resp

onse

of l

ink

(rad

)

Figure 2 Tracking responses of links 1 and 2 in Case 2

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

trac

king

of l

ink

(rad

)

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Figure 3 Tracking responses of links 1 and 2 in Case 3

8 Mathematical Problems in Engineering

Table 1 Time of fault tolerance

CASESPosition

tracking oflink1(s)

Speedtracking oflink1(s)

Controltorque oflink1(s)

Positiontracking oflink2(s)

Speedtracking oflink2(s)

Controltorque oflink2(s)

1 2 5 5 33 7 7 3 4 7 7 3 5 5 3

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minus

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

trac

king

of l

ink

(rad

)

Figure 4 Tracking responses of links 1 and 2 in Case 4

minus

minus

minus

Link 1Link 2

Time (s)

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 5 Time responses of control torque in Case 1

torque of link 2 can handle the fault within 3 seconds but alittle chattering phenomenon appears

The tracking responses in Case 3 are shown in Figure 3from which it can be easily seen that link position can dealwith fault by itself and basically track the desired positionsignal in 7s when actuator fault appears in link 1 at 10sCompared with other cases the error between actual signaland desired signal is a little high but is still in an acceptablerange Figure 3 also provides the actual speed of link 1 withactuator fault being able to track expected signal within 7sFrom Figure 3 we can respectively see position and speed oflink 2 with no fault being able to easily realize the trajectory

tracking in 5s Figure 7 shows the time response of controltorque from which we can see that the amplitude of link 1increases and control torque can reconvergewithin 3 seconds

Figure 4 provides the trajectory tracking responses of twolinks in this case The tracking response of link 1 is shown inFigure 4 from which it is easy to see that when actuator oflink 1 failed at 10s position signal and speed signal of link 1can deal with fault and basically track the desired signals in 7seconds Although there exists a certain error in both positionand speed tracking the error is acceptable in programmingFigure 4 also depicts the tracking response of link 2 fromwhich we can see that it takes 5 seconds for position and

Mathematical Problems in Engineering 9

Link 1Link 2

Time (s)

minus

minus

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 6 Time responses of control torque in Case 2

Link 1Link 2

Time (s)

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 7 Time responses of control torque in Case 3

Link 1Link 2

Time (s)

minus

minus

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 8 Time responses of control torque in Case 4

speed signal to track the corresponding desired signals whenactuator fault occurs in link 2 at 10s Figure 8 depicts the timeresponse of control torque in this case from which it can beeasily seen that the amplitudes of 2 links increase and controltorque can reconverge within 3 seconds but a little chatteringphenomenon appears

From Table 1 and Figures 1ndash8 we can see that when bothactuators are healthy without fault the overall system canrealize trajectory tracking within 5 seconds when actuatorfault occurs in link 2 tracking responses of link 2 canreconverge to desired signal within 5 seconds when actuator

fault occurs in link 1 tracking responses of link 1 canreconverge to desired signal within 7 seconds when actuatorfault occurs in two links tracking responses of links 1 and2 can reconverge to desired signal within 7 and 5 secondsrespectivelyTherefore the trajectory tracking effectiveness ofthe FTC method proposed in this paper is verified

5 Conclusions

In this paper a novel FTC scheme based on adaptive slidingmode method is investigated for manipulator with actuator

10 Mathematical Problems in Engineering

fault and external disturbance Firstly the general dynamicmodel of space manipulator is introduced and further itsactuator faulty model is established Secondly an adaptivefault tolerant controller is designed for manipulator withactuator fault Parameters of fault and disturbance are esti-mated and updated by online adaptive method Finally theproposed controller is applied to two-joint manipulator fromthe simulation results it is shown that the controller proposedin this paper can not only realize a good trajectory trackingbut also tolerate actuator fault and present strong robustnessfor external disturbance while effectively reducing the chat-tering phenomenon of sliding mode control

Data Availability

The data supporting the conclusions of our manuscript aresome open access articles that have been properly citedand the readers can easily obtain these articles to verify theconclusions replicate the analysis and conduct secondaryanalysesTherefore we did not create a publicly available datarepository

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by National Nature Science Founda-tion (under Grant 61601228 61403210 and 61603191) NaturalScience Foundation of Jiangsu (BK20161021) the Open Pro-gram of Jiangsu Key Laboratory of 3D Printing Equipmentand Manufacturing (BM2013006) Item number 3DL201607and the National Key Research and Development Program ofChina (Grant No 2017YFB1103200)

References

[1] D Zhao S Li and Q Zhu ldquoA new TSMC prototype robustnonlinear task space control of a 6 DOF parallel roboticmanipulatorrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1189ndash1197 2010

[2] YTaniseK Taniguchi S YamazakiMKamata YYamada andT Nakamura ldquoDevelopment of an air duct cleaning robot forhousing based on peristaltic crawlingmotionrdquo in Proceedings ofthe 2017 IEEE International Conference on Advanced IntelligentMechatronics AIM 2017 pp 1267ndash1272 Germany July 2017

[3] A S Al-Yahmadi J Abdo and T C Hsia ldquoModeling andcontrol of two manipulators handling a flexible objectrdquo Journalof e Franklin Institute vol 344 no 5 pp 349ndash361 2007

[4] Buren G Sun and R Hu ldquoFlexible force control on robot armfor spacecraft assemblyrdquo Spacecra Environment Engineeringvol 465 no 21 pp 147ndash152 2014

[5] L A Tuan Y H Joo P X Duong et al ldquoParameter estimatorintegrated-sliding mode control of dual arm robotsrdquo Interna-tional Journal of Control Automation amp Systems vol 15 no 6pp 2754ndash2763 2017

[6] Y Xu H Sun Q Jia et al ldquoControl of space robotic manipula-tors with faultsrdquo Mechanical Science amp Technology vol 25 no12 pp 1503-1423 2006

[7] E Coleshill L Oshinowo R Rembala B Bina D Rey andS Sindelar ldquoDextre improving maintenance operations on theInternational Space StationrdquoActaAstronautica vol 64 no 9-10pp 869ndash874 2009

[8] S B Nokleby ldquoSingularity analysis of the canadarmrdquo Mecha-nism ampMachine eory vol 42 no 4 pp 442ndash454 2007

[9] R Boumans and C Heemskerk ldquoThe European Robotic Armfor the International Space Stationrdquo Robotics and AutonomousSystems vol 23 no 1-2 pp 17ndash27 1998

[10] I Duleba and M Opalka ldquoA comparison of Jacobian-basedmethods of inverse kinematics for serial robot manipulatorsrdquoInternational Journal of Applied Mathematics amp ComputerScience vol 23 pp 373ndash382 2013

[11] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquoUkrainianMathematical Journal vol 69 no 8 pp 1220ndash1233 2017

[12] W Liu J Cui and J Xin ldquoA block-centered finite differencemethod for an unsteady asymptotic coupled model in fracturedmedia aquifer systemrdquo Journal of Computational and AppliedMathematics vol 337 pp 319ndash340 2018

[13] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[14] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 1ndash21 2017

[15] L Gao D Wang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[16] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[17] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[18] Z Cen H Noura and Y A Younes ldquoSystematic fault tolerantcontrol based on adaptiveThau observer estimation for quadro-tor UAVsrdquo International Journal of Applied Mathematics andComputer Science vol 25 no 1 pp 159ndash174 2015

[19] M Bonf P Castaldi NMimmo et al ldquoActive fault tolerant con-trol of nonlinear systems the cart-pole examplerdquo Internationaljournal of applied mathematics amp computer science vol 21 no3 pp 441ndash445 2011

[20] M Z Gao G P Cai and N Ying ldquoRobust adaptive fault-tolerant H8 control of re-entry vehicle considering actuator andsensor faults based on trajectory optimizationrdquo InternationalJournal of Control Automation amp Systems vol 14 no 1 pp 198ndash210 2016

[21] C Yang T Teng B Xu et al ldquoGlobal adaptive trackingcontrol of robotmanipulators usingneural networkswith finite-time learning convergencerdquo International Journal of ControlAutomation amp Systems vol 11 pp 1ndash9 2017

[22] Z Yu F Wang and L Liu ldquoAn adaptive fault-tolerant robustcontroller with fault diagnosis for submarinersquos vertical move-mentrdquo International Journal of Control Automation and Sys-tems vol 13 no 6 pp 1337ndash1345 2015

Mathematical Problems in Engineering 11

[23] L Li Y Yang S X Ding Y Zhang and S Zhai ldquoOn fault-tolerant control configurations for a class of nonlinear systemsrdquoJournal of e Franklin Institute vol 352 no 4 pp 1397ndash14162015

[24] P Yang R Guo X Pan et al ldquoStudy on the sliding mode faulttolerant predictive control based on multi agent particle swarmoptimizationrdquo International Journal of Control Automation ampSystems vol 3 pp 1ndash9 2017

[25] J Yang F Zhu X Wang and X Bu ldquoRobust sliding-modeobserver-based sensor fault estimation actuator fault detectionand isolation for uncertain nonlinear systemsrdquo InternationalJournal of Control Automation and Systems vol 13 no 5 pp1037ndash1046 2015

[26] M Van S S Ge and H Ren ldquoFinite time fault tolerantcontrol for robot manipulators using time delay estimation andcontinuous nonsingular fast terminal sliding mode controlrdquoIEEE Transactions on Cybernetics 2016

[27] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[28] S Liu J Wang Y Zhou and M Feckan ldquoIterative learningcontrol with pulse compensation for fractional differentialsystemsrdquoMathematica Slovaca vol 68 no 3 pp 563ndash574 2018

[29] C Ma T Li and J Zhang ldquoConsensus control for leader-following multi-agent systems with measurement noisesrdquo Jour-nal of Systems Science and Complexity vol 23 no 1 pp 35ndash492010

[30] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[31] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[32] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[33] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[34] A Owczarkowski and D Horla ldquoRobust LQR and LQI controlwith actuator failure of a 2DOF unmanned bicycle robotstabilized by an inertial wheelrdquo International Journal of AppliedMathematics and Computer Science vol 26 no 2 pp 325ndash3342016

[35] D Y Li and L Y Chun ldquoSensor fault identification anddecentralized fault-tolerant control of reconfigurable manip-ulator based on sliding mode observerrdquo in Proceedings ofthe 2012 International Conference on Control Engineering andCommunication Technology ICCECT 2012 pp 137ndash140 ChinaDecember 2012

[36] A Ferreira de Loza J Cieslak D Henry A Zolghadri andL M Fridman ldquoOutput tracking of systems subjected toperturbations and a class of actuator faults based on HOSMobservation and identificationrdquo Automatica vol 59 pp 200ndash205 2015

[37] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[38] L M Capisani A Ferrara A Ferreira De Loza and L MFridman ldquoManipulator fault diagnosis via higher order sliding-mode observersrdquo IEEE Transactions on Industrial Electronicsvol 59 no 10 pp 3979ndash3986 2012

[39] A Yarza V Santibanez and JMoreno-Valenzuela ldquoAn adaptiveoutput feedbackmotion tracking controller for robotmanipula-tors uniform global asymptotic stability and experimentationrdquoInternational Journal of Applied Mathematics and ComputerScience vol 23 no 3 pp 599ndash611 2013

[40] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[41] Q L Hu L Xu and A H Zhang ldquoAdaptive backsteppingtrajectory tracking control of robotmanipulatorrdquo Journal ofeFranklin Institute vol 349 no 3 pp 1087ndash1105 2012

[42] J-H Shin and J-J Lee ldquoFault Detection And Robust FaultRecovery Control for Robot Manipulators with Actuator Fail-uresrdquo in Proceedings of the 1999 IEEE International Conferenceon Robotics and Automation ICRA99 vol 2 pp 861ndash866 May1999

[43] J K Liu SlidingMode Control Design andMATLAB SimulationeBasiceory andDesignMethod TsinghuaUniversity Press2005

[44] S Mondal and C Mahanta ldquoAdaptive second order terminalsliding mode controller for robotic manipulatorsrdquo Journal ofe Franklin Institute vol 351 no 4 pp 2356ndash2377 2014

[45] A J Koshkouei K J Burnham and A S I Zinober ldquoDynamicsliding mode control designrdquo IEE Proceedings - Control eoryand Applications vol 152 no 4 pp 392ndash396 2005

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Mathematical Problems in Engineering 5

Substituting adaptive law (16) into the above equation yieldsthe following

1 le 119869119879 [minus 119864 (119905) 120591 minus 120574 (119905) sgn (119869)] minus 1 minus 119887120573 120585 120585= minus120574 (119905) 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 + 119869119879 119864 (119905) [119892 + 119867119891 + 119862119869 + 120574 (119905) sgn (119869) + sgn (119869)]

le (1 minus 120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585+ 119887 1003817100381710038171003817100381711986911987910038171003817100381710038171003817 [1003817100381710038171003817119892 + 119867119891 + 1198621198691003817100381710038171003817 + 120574 (119905) + ]

le (1 minus 120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 + 119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 (minus120576 + 120585120575)

= (1 minus 120585 + 119887120585) 120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816= (1 minus 119887) (120585 minus 120585) 120575 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1 minus 119887120573 120585 120585 minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816= (1 minus 119887) 120585(120575 119899sum

119894=1

10038161003816100381610038161198691198941003816100381610038161003816 minus 1120573 120585) minus 120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816

(24)

As a result applying adaptive law (17) into the above equationone can obtain the following

1 = minus120576119887 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 le 0 (25)

Step 2 (reach time analysis) Before analysis a lemma isproposed as follows to obtain the convergence time

Lemma 5e dynamic sliding mode function 119869 exists and canconverge within finite time 119905119869 please see [43] for more details

Aer time 119905119869 the surface 119869 = 0 from (7) one can obtainthe following

119878 = minus 120594 = minus1205881 11987805 sgn (119878) (26)

Further a Lyapunov function is selected as follows to obtain theconvergence time of system

1198812 = 12119878119879119878 (27)

Further

2 = 119878119879 119878 = 119878119879 (minus1205881 11987805 sgn (119878)) le minus1205881 11987815 le 0 (28)

To solve the above differential equation the following equationcan be obtained by (27)

1198812 = 12 1198782 (29)

Further

119878 = 100381610038161003816100381621198812100381610038161003816100381612 (30)

Substituting (30) to (28) yields the following

2 le minus1205881 100381610038161003816100381621198812100381610038161003816100381634 (31)

us

119881minus342 1198891198812 le minus2341205881119889119905int119905119905119869

119881minus342 1198891198812le int119905119905119869

minus23412058811198891199054 [1198812 (119905)14 minus 1198812 (119905119869)14]le minus2341205881 (119905 minus 119905119869)

(32)

As time 119905 reaches 119905119878 the conventional slidingmode surface 119878willconverge to zero ie when 119905 = 119905119878 119878 = 0 that is1198812(119905) = 1198812(119905119878)thus

minus41198812 (119905119869)14 le minus2341205881 (119905119878 minus 119905119869) (33)

us one can obtain the following

119905119878 le 41198812 (119905119869)14 + 23412058811199051198692341205881 (34)

Consequently sliding mode surface 119878 will converge to zerowithin finite time

119905119878 le 41198812 (119905119869)14 + 23412058811199051198692341205881 (35)

Now it is verified that conventional sliding mode surface 119878and dynamic sliding mode surface 119869 can both converge to zerowithin finite time and dynamic surface converges faster thanconventional surface ie lim119905ge119905119869(119869119878) = 0 As 119905 997888rarr 119905119878 isreached 119878 = 0 substituting this into (5) one can obtain thefollowing

= minus119897119911 (36)

Step 3 (tracking error analysis) To prove the convergenceof the tracking error 119911 the Lyapunov function is defined asfollows

1198813 = 12119911119879119911 (37)

Thus

3 = 119911119879 le minus119897 1199112 le 0 (38)

and therefore tracking error 119911 is convergent which meansthat when 119905 997888rarr infin 119902 997888rarr 119902119889 119911 997888rarr 0 Therefore accordingto the above three steps it is easy to be seen that lt 0 in(19) which means that the overall system can be stable withthe proposed controller The proof is completed

6 Mathematical Problems in Engineering

Remark 5 Practically due to the hysteresis of nonlinear andswitching 119869 cannot converge to zero accurately within afinite time therefore the adaptive parameters and 120585 of theestimated values for 119870 and 120585 may increase boundlessly Inthe other words and 120585 obtained by the proposed adaptivealgorithm will be not accurate which may increase towardsinfinity In practical engineering it is difficult to apply (16)-(17) directly Consequently to solve such a problem deadzone technique is used [44] and adaptive laws (16) and (17)are modified as

119870 = 120583 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 10038161003816100381610038161198691198941003816100381610038161003816 ge 12058210 10038161003816100381610038161198691198941003816100381610038161003816 lt 1205821

(39)

120585 = 120573120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 10038161003816100381610038161198691198941003816100381610038161003816 ge 12058220 10038161003816100381610038161198691198941003816100381610038161003816 lt 1205822

(40)

where 1205821 and 1205821 are both small positive constants

Remark 6 The dynamic sliding function 119869 reaches andremains on the sliding surface 119869 = 0 before the conventionalsliding function 119878 gets to the sliding surface 119878 = 0 if and onlyif 1205881 1205882 satisfy [45]

|119869 (0)| le (12058821205881)2 |119878 (0)| (41)

4 Simulation

In order to verify the validity of control method proposedin this paper we applied it to two-joint manipulator modelIn this section four cases with considering signal fault anddouble joints faults respectively will be simulated to presentthe tracking effectiveness of manipulator

41 Simulation Cases

Case 1 Link 1 and link 2 are both healthy without fault ie119864 = 119868 where 119868 is the identity matrix

Case 2 Link 1 is healthy and actuator fault occurs in link 2 at10s ie

1198901 = 11198902 =

1 119905 lt 1011990402 119905 ge 10119904

(42)

Case 3 Actuator fault occurs in link 1 at 10s and link 2 ishealthy ie

1198901 = 1 119905 lt 1011990406 119905 ge 10119904

1198902 = 1(43)

Case 4 Actuator fault occurs in link 1 and link 2 at 10s ie

1198901 = 1 119905 lt 1011990406 119905 ge 10119904

1198902 = 1 119905 lt 1011990402 119905 ge 10119904

(44)

Simulation Parameters The parameters of simulation aredesigned as follows

119867(119902) = [366 + 174 cos 1199022 076 + 087 cos 1199022076 + 087 cos 1199022 076 ]

119866 (119902) = [304119892 cos 1199021 + 087119892 cos (1199021 + 1199022)087119892 cos (1199021 + 1199022) ]

120591119889 = [02 sin 119902102 sin 1199022]

(45)

The initial state of manipulator system is selected as 1199020 =(0 0)119879119903119886119889119904 1199020 = (0 0)119879119903119886119889119904Tracked objective trajectory is 1199021198891 = 07 sin (120587119905)119903119886119889 1199021198892 =

sin (120587119905)119903119886119889 respectivelyTo make the overall system stable equipped with strong

robustness and fault tolerance the parameters of the pro-posed controller are selected as 120576 = 1 and the learning gainsof adaptive laws are adopted as 120583 = 05 120573 = 01 To realizefast convergence of the sliding mode surface sliding modeparameters are designed as 119897 = diag 0001 0001 1205881 = 031205882 = 5 1205821 = 1205822 = 005Simulation Results and Analysis In the simulation accordingto the designed control law (15) and adaptive laws (39) (40)the time of fault tolerance in each case is shown in Table 1 andthe corresponding simulation results are depicted in Figures1ndash8 which show time responses of link position trackingvelocity tracking

Figure 1 shows the tracking responses in Case 1 FromFigure 1 we can see that the position and speed signal oflink 1 can track the corresponding desired signal within5s Meanwhile Figure 1 shows that tracking trajectories ofposition and speed of link 2 between actual signal and desiredsignal can converge to zero in 5s and then reach stabilityFrom Figure 5 it is easy to be seen that the control torquecan converge within 4 seconds without actuator fault in thiscase

Figure 2 depicts the time responses of trajectory trackingin Case 2 FromFigure 2 it can be easily seen that the positionand speed tracking errors of link 1 between actual signal anddesired signal can converge to zero in 5s without actuatorfault Also Figure 2 shows that tracking error of two linkscan converge to zero in 5s after actuator fault occurs in link2 at 10s system can rapidly deal with the fault and realizetrajectory tracking within 5 seconds From Figure 6 we cansee that when actuator fault occurs in link 2 the control

Mathematical Problems in Engineering 7

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

resp

onse

of l

ink

(rad

)

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Figure 1 Tracking responses of links 1 and 2 in Case 1

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minusminus

minus

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

resp

onse

of l

ink

(rad

)

Figure 2 Tracking responses of links 1 and 2 in Case 2

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

trac

king

of l

ink

(rad

)

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Figure 3 Tracking responses of links 1 and 2 in Case 3

8 Mathematical Problems in Engineering

Table 1 Time of fault tolerance

CASESPosition

tracking oflink1(s)

Speedtracking oflink1(s)

Controltorque oflink1(s)

Positiontracking oflink2(s)

Speedtracking oflink2(s)

Controltorque oflink2(s)

1 2 5 5 33 7 7 3 4 7 7 3 5 5 3

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minus

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

trac

king

of l

ink

(rad

)

Figure 4 Tracking responses of links 1 and 2 in Case 4

minus

minus

minus

Link 1Link 2

Time (s)

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 5 Time responses of control torque in Case 1

torque of link 2 can handle the fault within 3 seconds but alittle chattering phenomenon appears

The tracking responses in Case 3 are shown in Figure 3from which it can be easily seen that link position can dealwith fault by itself and basically track the desired positionsignal in 7s when actuator fault appears in link 1 at 10sCompared with other cases the error between actual signaland desired signal is a little high but is still in an acceptablerange Figure 3 also provides the actual speed of link 1 withactuator fault being able to track expected signal within 7sFrom Figure 3 we can respectively see position and speed oflink 2 with no fault being able to easily realize the trajectory

tracking in 5s Figure 7 shows the time response of controltorque from which we can see that the amplitude of link 1increases and control torque can reconvergewithin 3 seconds

Figure 4 provides the trajectory tracking responses of twolinks in this case The tracking response of link 1 is shown inFigure 4 from which it is easy to see that when actuator oflink 1 failed at 10s position signal and speed signal of link 1can deal with fault and basically track the desired signals in 7seconds Although there exists a certain error in both positionand speed tracking the error is acceptable in programmingFigure 4 also depicts the tracking response of link 2 fromwhich we can see that it takes 5 seconds for position and

Mathematical Problems in Engineering 9

Link 1Link 2

Time (s)

minus

minus

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 6 Time responses of control torque in Case 2

Link 1Link 2

Time (s)

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 7 Time responses of control torque in Case 3

Link 1Link 2

Time (s)

minus

minus

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 8 Time responses of control torque in Case 4

speed signal to track the corresponding desired signals whenactuator fault occurs in link 2 at 10s Figure 8 depicts the timeresponse of control torque in this case from which it can beeasily seen that the amplitudes of 2 links increase and controltorque can reconverge within 3 seconds but a little chatteringphenomenon appears

From Table 1 and Figures 1ndash8 we can see that when bothactuators are healthy without fault the overall system canrealize trajectory tracking within 5 seconds when actuatorfault occurs in link 2 tracking responses of link 2 canreconverge to desired signal within 5 seconds when actuator

fault occurs in link 1 tracking responses of link 1 canreconverge to desired signal within 7 seconds when actuatorfault occurs in two links tracking responses of links 1 and2 can reconverge to desired signal within 7 and 5 secondsrespectivelyTherefore the trajectory tracking effectiveness ofthe FTC method proposed in this paper is verified

5 Conclusions

In this paper a novel FTC scheme based on adaptive slidingmode method is investigated for manipulator with actuator

10 Mathematical Problems in Engineering

fault and external disturbance Firstly the general dynamicmodel of space manipulator is introduced and further itsactuator faulty model is established Secondly an adaptivefault tolerant controller is designed for manipulator withactuator fault Parameters of fault and disturbance are esti-mated and updated by online adaptive method Finally theproposed controller is applied to two-joint manipulator fromthe simulation results it is shown that the controller proposedin this paper can not only realize a good trajectory trackingbut also tolerate actuator fault and present strong robustnessfor external disturbance while effectively reducing the chat-tering phenomenon of sliding mode control

Data Availability

The data supporting the conclusions of our manuscript aresome open access articles that have been properly citedand the readers can easily obtain these articles to verify theconclusions replicate the analysis and conduct secondaryanalysesTherefore we did not create a publicly available datarepository

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by National Nature Science Founda-tion (under Grant 61601228 61403210 and 61603191) NaturalScience Foundation of Jiangsu (BK20161021) the Open Pro-gram of Jiangsu Key Laboratory of 3D Printing Equipmentand Manufacturing (BM2013006) Item number 3DL201607and the National Key Research and Development Program ofChina (Grant No 2017YFB1103200)

References

[1] D Zhao S Li and Q Zhu ldquoA new TSMC prototype robustnonlinear task space control of a 6 DOF parallel roboticmanipulatorrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1189ndash1197 2010

[2] YTaniseK Taniguchi S YamazakiMKamata YYamada andT Nakamura ldquoDevelopment of an air duct cleaning robot forhousing based on peristaltic crawlingmotionrdquo in Proceedings ofthe 2017 IEEE International Conference on Advanced IntelligentMechatronics AIM 2017 pp 1267ndash1272 Germany July 2017

[3] A S Al-Yahmadi J Abdo and T C Hsia ldquoModeling andcontrol of two manipulators handling a flexible objectrdquo Journalof e Franklin Institute vol 344 no 5 pp 349ndash361 2007

[4] Buren G Sun and R Hu ldquoFlexible force control on robot armfor spacecraft assemblyrdquo Spacecra Environment Engineeringvol 465 no 21 pp 147ndash152 2014

[5] L A Tuan Y H Joo P X Duong et al ldquoParameter estimatorintegrated-sliding mode control of dual arm robotsrdquo Interna-tional Journal of Control Automation amp Systems vol 15 no 6pp 2754ndash2763 2017

[6] Y Xu H Sun Q Jia et al ldquoControl of space robotic manipula-tors with faultsrdquo Mechanical Science amp Technology vol 25 no12 pp 1503-1423 2006

[7] E Coleshill L Oshinowo R Rembala B Bina D Rey andS Sindelar ldquoDextre improving maintenance operations on theInternational Space StationrdquoActaAstronautica vol 64 no 9-10pp 869ndash874 2009

[8] S B Nokleby ldquoSingularity analysis of the canadarmrdquo Mecha-nism ampMachine eory vol 42 no 4 pp 442ndash454 2007

[9] R Boumans and C Heemskerk ldquoThe European Robotic Armfor the International Space Stationrdquo Robotics and AutonomousSystems vol 23 no 1-2 pp 17ndash27 1998

[10] I Duleba and M Opalka ldquoA comparison of Jacobian-basedmethods of inverse kinematics for serial robot manipulatorsrdquoInternational Journal of Applied Mathematics amp ComputerScience vol 23 pp 373ndash382 2013

[11] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquoUkrainianMathematical Journal vol 69 no 8 pp 1220ndash1233 2017

[12] W Liu J Cui and J Xin ldquoA block-centered finite differencemethod for an unsteady asymptotic coupled model in fracturedmedia aquifer systemrdquo Journal of Computational and AppliedMathematics vol 337 pp 319ndash340 2018

[13] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[14] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 1ndash21 2017

[15] L Gao D Wang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[16] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[17] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[18] Z Cen H Noura and Y A Younes ldquoSystematic fault tolerantcontrol based on adaptiveThau observer estimation for quadro-tor UAVsrdquo International Journal of Applied Mathematics andComputer Science vol 25 no 1 pp 159ndash174 2015

[19] M Bonf P Castaldi NMimmo et al ldquoActive fault tolerant con-trol of nonlinear systems the cart-pole examplerdquo Internationaljournal of applied mathematics amp computer science vol 21 no3 pp 441ndash445 2011

[20] M Z Gao G P Cai and N Ying ldquoRobust adaptive fault-tolerant H8 control of re-entry vehicle considering actuator andsensor faults based on trajectory optimizationrdquo InternationalJournal of Control Automation amp Systems vol 14 no 1 pp 198ndash210 2016

[21] C Yang T Teng B Xu et al ldquoGlobal adaptive trackingcontrol of robotmanipulators usingneural networkswith finite-time learning convergencerdquo International Journal of ControlAutomation amp Systems vol 11 pp 1ndash9 2017

[22] Z Yu F Wang and L Liu ldquoAn adaptive fault-tolerant robustcontroller with fault diagnosis for submarinersquos vertical move-mentrdquo International Journal of Control Automation and Sys-tems vol 13 no 6 pp 1337ndash1345 2015

Mathematical Problems in Engineering 11

[23] L Li Y Yang S X Ding Y Zhang and S Zhai ldquoOn fault-tolerant control configurations for a class of nonlinear systemsrdquoJournal of e Franklin Institute vol 352 no 4 pp 1397ndash14162015

[24] P Yang R Guo X Pan et al ldquoStudy on the sliding mode faulttolerant predictive control based on multi agent particle swarmoptimizationrdquo International Journal of Control Automation ampSystems vol 3 pp 1ndash9 2017

[25] J Yang F Zhu X Wang and X Bu ldquoRobust sliding-modeobserver-based sensor fault estimation actuator fault detectionand isolation for uncertain nonlinear systemsrdquo InternationalJournal of Control Automation and Systems vol 13 no 5 pp1037ndash1046 2015

[26] M Van S S Ge and H Ren ldquoFinite time fault tolerantcontrol for robot manipulators using time delay estimation andcontinuous nonsingular fast terminal sliding mode controlrdquoIEEE Transactions on Cybernetics 2016

[27] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[28] S Liu J Wang Y Zhou and M Feckan ldquoIterative learningcontrol with pulse compensation for fractional differentialsystemsrdquoMathematica Slovaca vol 68 no 3 pp 563ndash574 2018

[29] C Ma T Li and J Zhang ldquoConsensus control for leader-following multi-agent systems with measurement noisesrdquo Jour-nal of Systems Science and Complexity vol 23 no 1 pp 35ndash492010

[30] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[31] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[32] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[33] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[34] A Owczarkowski and D Horla ldquoRobust LQR and LQI controlwith actuator failure of a 2DOF unmanned bicycle robotstabilized by an inertial wheelrdquo International Journal of AppliedMathematics and Computer Science vol 26 no 2 pp 325ndash3342016

[35] D Y Li and L Y Chun ldquoSensor fault identification anddecentralized fault-tolerant control of reconfigurable manip-ulator based on sliding mode observerrdquo in Proceedings ofthe 2012 International Conference on Control Engineering andCommunication Technology ICCECT 2012 pp 137ndash140 ChinaDecember 2012

[36] A Ferreira de Loza J Cieslak D Henry A Zolghadri andL M Fridman ldquoOutput tracking of systems subjected toperturbations and a class of actuator faults based on HOSMobservation and identificationrdquo Automatica vol 59 pp 200ndash205 2015

[37] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[38] L M Capisani A Ferrara A Ferreira De Loza and L MFridman ldquoManipulator fault diagnosis via higher order sliding-mode observersrdquo IEEE Transactions on Industrial Electronicsvol 59 no 10 pp 3979ndash3986 2012

[39] A Yarza V Santibanez and JMoreno-Valenzuela ldquoAn adaptiveoutput feedbackmotion tracking controller for robotmanipula-tors uniform global asymptotic stability and experimentationrdquoInternational Journal of Applied Mathematics and ComputerScience vol 23 no 3 pp 599ndash611 2013

[40] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[41] Q L Hu L Xu and A H Zhang ldquoAdaptive backsteppingtrajectory tracking control of robotmanipulatorrdquo Journal ofeFranklin Institute vol 349 no 3 pp 1087ndash1105 2012

[42] J-H Shin and J-J Lee ldquoFault Detection And Robust FaultRecovery Control for Robot Manipulators with Actuator Fail-uresrdquo in Proceedings of the 1999 IEEE International Conferenceon Robotics and Automation ICRA99 vol 2 pp 861ndash866 May1999

[43] J K Liu SlidingMode Control Design andMATLAB SimulationeBasiceory andDesignMethod TsinghuaUniversity Press2005

[44] S Mondal and C Mahanta ldquoAdaptive second order terminalsliding mode controller for robotic manipulatorsrdquo Journal ofe Franklin Institute vol 351 no 4 pp 2356ndash2377 2014

[45] A J Koshkouei K J Burnham and A S I Zinober ldquoDynamicsliding mode control designrdquo IEE Proceedings - Control eoryand Applications vol 152 no 4 pp 392ndash396 2005

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

Remark 5 Practically due to the hysteresis of nonlinear andswitching 119869 cannot converge to zero accurately within afinite time therefore the adaptive parameters and 120585 of theestimated values for 119870 and 120585 may increase boundlessly Inthe other words and 120585 obtained by the proposed adaptivealgorithm will be not accurate which may increase towardsinfinity In practical engineering it is difficult to apply (16)-(17) directly Consequently to solve such a problem deadzone technique is used [44] and adaptive laws (16) and (17)are modified as

119870 = 120583 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 10038161003816100381610038161198691198941003816100381610038161003816 ge 12058210 10038161003816100381610038161198691198941003816100381610038161003816 lt 1205821

(39)

120585 = 120573120575 119899sum119894=1

10038161003816100381610038161198691198941003816100381610038161003816 10038161003816100381610038161198691198941003816100381610038161003816 ge 12058220 10038161003816100381610038161198691198941003816100381610038161003816 lt 1205822

(40)

where 1205821 and 1205821 are both small positive constants

Remark 6 The dynamic sliding function 119869 reaches andremains on the sliding surface 119869 = 0 before the conventionalsliding function 119878 gets to the sliding surface 119878 = 0 if and onlyif 1205881 1205882 satisfy [45]

|119869 (0)| le (12058821205881)2 |119878 (0)| (41)

4 Simulation

In order to verify the validity of control method proposedin this paper we applied it to two-joint manipulator modelIn this section four cases with considering signal fault anddouble joints faults respectively will be simulated to presentthe tracking effectiveness of manipulator

41 Simulation Cases

Case 1 Link 1 and link 2 are both healthy without fault ie119864 = 119868 where 119868 is the identity matrix

Case 2 Link 1 is healthy and actuator fault occurs in link 2 at10s ie

1198901 = 11198902 =

1 119905 lt 1011990402 119905 ge 10119904

(42)

Case 3 Actuator fault occurs in link 1 at 10s and link 2 ishealthy ie

1198901 = 1 119905 lt 1011990406 119905 ge 10119904

1198902 = 1(43)

Case 4 Actuator fault occurs in link 1 and link 2 at 10s ie

1198901 = 1 119905 lt 1011990406 119905 ge 10119904

1198902 = 1 119905 lt 1011990402 119905 ge 10119904

(44)

Simulation Parameters The parameters of simulation aredesigned as follows

119867(119902) = [366 + 174 cos 1199022 076 + 087 cos 1199022076 + 087 cos 1199022 076 ]

119866 (119902) = [304119892 cos 1199021 + 087119892 cos (1199021 + 1199022)087119892 cos (1199021 + 1199022) ]

120591119889 = [02 sin 119902102 sin 1199022]

(45)

The initial state of manipulator system is selected as 1199020 =(0 0)119879119903119886119889119904 1199020 = (0 0)119879119903119886119889119904Tracked objective trajectory is 1199021198891 = 07 sin (120587119905)119903119886119889 1199021198892 =

sin (120587119905)119903119886119889 respectivelyTo make the overall system stable equipped with strong

robustness and fault tolerance the parameters of the pro-posed controller are selected as 120576 = 1 and the learning gainsof adaptive laws are adopted as 120583 = 05 120573 = 01 To realizefast convergence of the sliding mode surface sliding modeparameters are designed as 119897 = diag 0001 0001 1205881 = 031205882 = 5 1205821 = 1205822 = 005Simulation Results and Analysis In the simulation accordingto the designed control law (15) and adaptive laws (39) (40)the time of fault tolerance in each case is shown in Table 1 andthe corresponding simulation results are depicted in Figures1ndash8 which show time responses of link position trackingvelocity tracking

Figure 1 shows the tracking responses in Case 1 FromFigure 1 we can see that the position and speed signal oflink 1 can track the corresponding desired signal within5s Meanwhile Figure 1 shows that tracking trajectories ofposition and speed of link 2 between actual signal and desiredsignal can converge to zero in 5s and then reach stabilityFrom Figure 5 it is easy to be seen that the control torquecan converge within 4 seconds without actuator fault in thiscase

Figure 2 depicts the time responses of trajectory trackingin Case 2 FromFigure 2 it can be easily seen that the positionand speed tracking errors of link 1 between actual signal anddesired signal can converge to zero in 5s without actuatorfault Also Figure 2 shows that tracking error of two linkscan converge to zero in 5s after actuator fault occurs in link2 at 10s system can rapidly deal with the fault and realizetrajectory tracking within 5 seconds From Figure 6 we cansee that when actuator fault occurs in link 2 the control

Mathematical Problems in Engineering 7

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

resp

onse

of l

ink

(rad

)

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Figure 1 Tracking responses of links 1 and 2 in Case 1

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minusminus

minus

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

resp

onse

of l

ink

(rad

)

Figure 2 Tracking responses of links 1 and 2 in Case 2

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

trac

king

of l

ink

(rad

)

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Figure 3 Tracking responses of links 1 and 2 in Case 3

8 Mathematical Problems in Engineering

Table 1 Time of fault tolerance

CASESPosition

tracking oflink1(s)

Speedtracking oflink1(s)

Controltorque oflink1(s)

Positiontracking oflink2(s)

Speedtracking oflink2(s)

Controltorque oflink2(s)

1 2 5 5 33 7 7 3 4 7 7 3 5 5 3

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minus

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

trac

king

of l

ink

(rad

)

Figure 4 Tracking responses of links 1 and 2 in Case 4

minus

minus

minus

Link 1Link 2

Time (s)

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 5 Time responses of control torque in Case 1

torque of link 2 can handle the fault within 3 seconds but alittle chattering phenomenon appears

The tracking responses in Case 3 are shown in Figure 3from which it can be easily seen that link position can dealwith fault by itself and basically track the desired positionsignal in 7s when actuator fault appears in link 1 at 10sCompared with other cases the error between actual signaland desired signal is a little high but is still in an acceptablerange Figure 3 also provides the actual speed of link 1 withactuator fault being able to track expected signal within 7sFrom Figure 3 we can respectively see position and speed oflink 2 with no fault being able to easily realize the trajectory

tracking in 5s Figure 7 shows the time response of controltorque from which we can see that the amplitude of link 1increases and control torque can reconvergewithin 3 seconds

Figure 4 provides the trajectory tracking responses of twolinks in this case The tracking response of link 1 is shown inFigure 4 from which it is easy to see that when actuator oflink 1 failed at 10s position signal and speed signal of link 1can deal with fault and basically track the desired signals in 7seconds Although there exists a certain error in both positionand speed tracking the error is acceptable in programmingFigure 4 also depicts the tracking response of link 2 fromwhich we can see that it takes 5 seconds for position and

Mathematical Problems in Engineering 9

Link 1Link 2

Time (s)

minus

minus

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 6 Time responses of control torque in Case 2

Link 1Link 2

Time (s)

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 7 Time responses of control torque in Case 3

Link 1Link 2

Time (s)

minus

minus

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 8 Time responses of control torque in Case 4

speed signal to track the corresponding desired signals whenactuator fault occurs in link 2 at 10s Figure 8 depicts the timeresponse of control torque in this case from which it can beeasily seen that the amplitudes of 2 links increase and controltorque can reconverge within 3 seconds but a little chatteringphenomenon appears

From Table 1 and Figures 1ndash8 we can see that when bothactuators are healthy without fault the overall system canrealize trajectory tracking within 5 seconds when actuatorfault occurs in link 2 tracking responses of link 2 canreconverge to desired signal within 5 seconds when actuator

fault occurs in link 1 tracking responses of link 1 canreconverge to desired signal within 7 seconds when actuatorfault occurs in two links tracking responses of links 1 and2 can reconverge to desired signal within 7 and 5 secondsrespectivelyTherefore the trajectory tracking effectiveness ofthe FTC method proposed in this paper is verified

5 Conclusions

In this paper a novel FTC scheme based on adaptive slidingmode method is investigated for manipulator with actuator

10 Mathematical Problems in Engineering

fault and external disturbance Firstly the general dynamicmodel of space manipulator is introduced and further itsactuator faulty model is established Secondly an adaptivefault tolerant controller is designed for manipulator withactuator fault Parameters of fault and disturbance are esti-mated and updated by online adaptive method Finally theproposed controller is applied to two-joint manipulator fromthe simulation results it is shown that the controller proposedin this paper can not only realize a good trajectory trackingbut also tolerate actuator fault and present strong robustnessfor external disturbance while effectively reducing the chat-tering phenomenon of sliding mode control

Data Availability

The data supporting the conclusions of our manuscript aresome open access articles that have been properly citedand the readers can easily obtain these articles to verify theconclusions replicate the analysis and conduct secondaryanalysesTherefore we did not create a publicly available datarepository

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by National Nature Science Founda-tion (under Grant 61601228 61403210 and 61603191) NaturalScience Foundation of Jiangsu (BK20161021) the Open Pro-gram of Jiangsu Key Laboratory of 3D Printing Equipmentand Manufacturing (BM2013006) Item number 3DL201607and the National Key Research and Development Program ofChina (Grant No 2017YFB1103200)

References

[1] D Zhao S Li and Q Zhu ldquoA new TSMC prototype robustnonlinear task space control of a 6 DOF parallel roboticmanipulatorrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1189ndash1197 2010

[2] YTaniseK Taniguchi S YamazakiMKamata YYamada andT Nakamura ldquoDevelopment of an air duct cleaning robot forhousing based on peristaltic crawlingmotionrdquo in Proceedings ofthe 2017 IEEE International Conference on Advanced IntelligentMechatronics AIM 2017 pp 1267ndash1272 Germany July 2017

[3] A S Al-Yahmadi J Abdo and T C Hsia ldquoModeling andcontrol of two manipulators handling a flexible objectrdquo Journalof e Franklin Institute vol 344 no 5 pp 349ndash361 2007

[4] Buren G Sun and R Hu ldquoFlexible force control on robot armfor spacecraft assemblyrdquo Spacecra Environment Engineeringvol 465 no 21 pp 147ndash152 2014

[5] L A Tuan Y H Joo P X Duong et al ldquoParameter estimatorintegrated-sliding mode control of dual arm robotsrdquo Interna-tional Journal of Control Automation amp Systems vol 15 no 6pp 2754ndash2763 2017

[6] Y Xu H Sun Q Jia et al ldquoControl of space robotic manipula-tors with faultsrdquo Mechanical Science amp Technology vol 25 no12 pp 1503-1423 2006

[7] E Coleshill L Oshinowo R Rembala B Bina D Rey andS Sindelar ldquoDextre improving maintenance operations on theInternational Space StationrdquoActaAstronautica vol 64 no 9-10pp 869ndash874 2009

[8] S B Nokleby ldquoSingularity analysis of the canadarmrdquo Mecha-nism ampMachine eory vol 42 no 4 pp 442ndash454 2007

[9] R Boumans and C Heemskerk ldquoThe European Robotic Armfor the International Space Stationrdquo Robotics and AutonomousSystems vol 23 no 1-2 pp 17ndash27 1998

[10] I Duleba and M Opalka ldquoA comparison of Jacobian-basedmethods of inverse kinematics for serial robot manipulatorsrdquoInternational Journal of Applied Mathematics amp ComputerScience vol 23 pp 373ndash382 2013

[11] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquoUkrainianMathematical Journal vol 69 no 8 pp 1220ndash1233 2017

[12] W Liu J Cui and J Xin ldquoA block-centered finite differencemethod for an unsteady asymptotic coupled model in fracturedmedia aquifer systemrdquo Journal of Computational and AppliedMathematics vol 337 pp 319ndash340 2018

[13] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[14] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 1ndash21 2017

[15] L Gao D Wang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[16] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[17] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[18] Z Cen H Noura and Y A Younes ldquoSystematic fault tolerantcontrol based on adaptiveThau observer estimation for quadro-tor UAVsrdquo International Journal of Applied Mathematics andComputer Science vol 25 no 1 pp 159ndash174 2015

[19] M Bonf P Castaldi NMimmo et al ldquoActive fault tolerant con-trol of nonlinear systems the cart-pole examplerdquo Internationaljournal of applied mathematics amp computer science vol 21 no3 pp 441ndash445 2011

[20] M Z Gao G P Cai and N Ying ldquoRobust adaptive fault-tolerant H8 control of re-entry vehicle considering actuator andsensor faults based on trajectory optimizationrdquo InternationalJournal of Control Automation amp Systems vol 14 no 1 pp 198ndash210 2016

[21] C Yang T Teng B Xu et al ldquoGlobal adaptive trackingcontrol of robotmanipulators usingneural networkswith finite-time learning convergencerdquo International Journal of ControlAutomation amp Systems vol 11 pp 1ndash9 2017

[22] Z Yu F Wang and L Liu ldquoAn adaptive fault-tolerant robustcontroller with fault diagnosis for submarinersquos vertical move-mentrdquo International Journal of Control Automation and Sys-tems vol 13 no 6 pp 1337ndash1345 2015

Mathematical Problems in Engineering 11

[23] L Li Y Yang S X Ding Y Zhang and S Zhai ldquoOn fault-tolerant control configurations for a class of nonlinear systemsrdquoJournal of e Franklin Institute vol 352 no 4 pp 1397ndash14162015

[24] P Yang R Guo X Pan et al ldquoStudy on the sliding mode faulttolerant predictive control based on multi agent particle swarmoptimizationrdquo International Journal of Control Automation ampSystems vol 3 pp 1ndash9 2017

[25] J Yang F Zhu X Wang and X Bu ldquoRobust sliding-modeobserver-based sensor fault estimation actuator fault detectionand isolation for uncertain nonlinear systemsrdquo InternationalJournal of Control Automation and Systems vol 13 no 5 pp1037ndash1046 2015

[26] M Van S S Ge and H Ren ldquoFinite time fault tolerantcontrol for robot manipulators using time delay estimation andcontinuous nonsingular fast terminal sliding mode controlrdquoIEEE Transactions on Cybernetics 2016

[27] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[28] S Liu J Wang Y Zhou and M Feckan ldquoIterative learningcontrol with pulse compensation for fractional differentialsystemsrdquoMathematica Slovaca vol 68 no 3 pp 563ndash574 2018

[29] C Ma T Li and J Zhang ldquoConsensus control for leader-following multi-agent systems with measurement noisesrdquo Jour-nal of Systems Science and Complexity vol 23 no 1 pp 35ndash492010

[30] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[31] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[32] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[33] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[34] A Owczarkowski and D Horla ldquoRobust LQR and LQI controlwith actuator failure of a 2DOF unmanned bicycle robotstabilized by an inertial wheelrdquo International Journal of AppliedMathematics and Computer Science vol 26 no 2 pp 325ndash3342016

[35] D Y Li and L Y Chun ldquoSensor fault identification anddecentralized fault-tolerant control of reconfigurable manip-ulator based on sliding mode observerrdquo in Proceedings ofthe 2012 International Conference on Control Engineering andCommunication Technology ICCECT 2012 pp 137ndash140 ChinaDecember 2012

[36] A Ferreira de Loza J Cieslak D Henry A Zolghadri andL M Fridman ldquoOutput tracking of systems subjected toperturbations and a class of actuator faults based on HOSMobservation and identificationrdquo Automatica vol 59 pp 200ndash205 2015

[37] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[38] L M Capisani A Ferrara A Ferreira De Loza and L MFridman ldquoManipulator fault diagnosis via higher order sliding-mode observersrdquo IEEE Transactions on Industrial Electronicsvol 59 no 10 pp 3979ndash3986 2012

[39] A Yarza V Santibanez and JMoreno-Valenzuela ldquoAn adaptiveoutput feedbackmotion tracking controller for robotmanipula-tors uniform global asymptotic stability and experimentationrdquoInternational Journal of Applied Mathematics and ComputerScience vol 23 no 3 pp 599ndash611 2013

[40] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[41] Q L Hu L Xu and A H Zhang ldquoAdaptive backsteppingtrajectory tracking control of robotmanipulatorrdquo Journal ofeFranklin Institute vol 349 no 3 pp 1087ndash1105 2012

[42] J-H Shin and J-J Lee ldquoFault Detection And Robust FaultRecovery Control for Robot Manipulators with Actuator Fail-uresrdquo in Proceedings of the 1999 IEEE International Conferenceon Robotics and Automation ICRA99 vol 2 pp 861ndash866 May1999

[43] J K Liu SlidingMode Control Design andMATLAB SimulationeBasiceory andDesignMethod TsinghuaUniversity Press2005

[44] S Mondal and C Mahanta ldquoAdaptive second order terminalsliding mode controller for robotic manipulatorsrdquo Journal ofe Franklin Institute vol 351 no 4 pp 2356ndash2377 2014

[45] A J Koshkouei K J Burnham and A S I Zinober ldquoDynamicsliding mode control designrdquo IEE Proceedings - Control eoryand Applications vol 152 no 4 pp 392ndash396 2005

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

resp

onse

of l

ink

(rad

)

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Figure 1 Tracking responses of links 1 and 2 in Case 1

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minusminus

minus

minusSpee

d re

spon

se o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

resp

onse

of l

ink

(rad

)

Figure 2 Tracking responses of links 1 and 2 in Case 2

minus

minusPosit

ion

resp

onse

of l

ink

1 (r

ad)

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minusPosit

ion

trac

king

of l

ink

(rad

)

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Figure 3 Tracking responses of links 1 and 2 in Case 3

8 Mathematical Problems in Engineering

Table 1 Time of fault tolerance

CASESPosition

tracking oflink1(s)

Speedtracking oflink1(s)

Controltorque oflink1(s)

Positiontracking oflink2(s)

Speedtracking oflink2(s)

Controltorque oflink2(s)

1 2 5 5 33 7 7 3 4 7 7 3 5 5 3

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minus

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

trac

king

of l

ink

(rad

)

Figure 4 Tracking responses of links 1 and 2 in Case 4

minus

minus

minus

Link 1Link 2

Time (s)

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 5 Time responses of control torque in Case 1

torque of link 2 can handle the fault within 3 seconds but alittle chattering phenomenon appears

The tracking responses in Case 3 are shown in Figure 3from which it can be easily seen that link position can dealwith fault by itself and basically track the desired positionsignal in 7s when actuator fault appears in link 1 at 10sCompared with other cases the error between actual signaland desired signal is a little high but is still in an acceptablerange Figure 3 also provides the actual speed of link 1 withactuator fault being able to track expected signal within 7sFrom Figure 3 we can respectively see position and speed oflink 2 with no fault being able to easily realize the trajectory

tracking in 5s Figure 7 shows the time response of controltorque from which we can see that the amplitude of link 1increases and control torque can reconvergewithin 3 seconds

Figure 4 provides the trajectory tracking responses of twolinks in this case The tracking response of link 1 is shown inFigure 4 from which it is easy to see that when actuator oflink 1 failed at 10s position signal and speed signal of link 1can deal with fault and basically track the desired signals in 7seconds Although there exists a certain error in both positionand speed tracking the error is acceptable in programmingFigure 4 also depicts the tracking response of link 2 fromwhich we can see that it takes 5 seconds for position and

Mathematical Problems in Engineering 9

Link 1Link 2

Time (s)

minus

minus

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 6 Time responses of control torque in Case 2

Link 1Link 2

Time (s)

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 7 Time responses of control torque in Case 3

Link 1Link 2

Time (s)

minus

minus

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 8 Time responses of control torque in Case 4

speed signal to track the corresponding desired signals whenactuator fault occurs in link 2 at 10s Figure 8 depicts the timeresponse of control torque in this case from which it can beeasily seen that the amplitudes of 2 links increase and controltorque can reconverge within 3 seconds but a little chatteringphenomenon appears

From Table 1 and Figures 1ndash8 we can see that when bothactuators are healthy without fault the overall system canrealize trajectory tracking within 5 seconds when actuatorfault occurs in link 2 tracking responses of link 2 canreconverge to desired signal within 5 seconds when actuator

fault occurs in link 1 tracking responses of link 1 canreconverge to desired signal within 7 seconds when actuatorfault occurs in two links tracking responses of links 1 and2 can reconverge to desired signal within 7 and 5 secondsrespectivelyTherefore the trajectory tracking effectiveness ofthe FTC method proposed in this paper is verified

5 Conclusions

In this paper a novel FTC scheme based on adaptive slidingmode method is investigated for manipulator with actuator

10 Mathematical Problems in Engineering

fault and external disturbance Firstly the general dynamicmodel of space manipulator is introduced and further itsactuator faulty model is established Secondly an adaptivefault tolerant controller is designed for manipulator withactuator fault Parameters of fault and disturbance are esti-mated and updated by online adaptive method Finally theproposed controller is applied to two-joint manipulator fromthe simulation results it is shown that the controller proposedin this paper can not only realize a good trajectory trackingbut also tolerate actuator fault and present strong robustnessfor external disturbance while effectively reducing the chat-tering phenomenon of sliding mode control

Data Availability

The data supporting the conclusions of our manuscript aresome open access articles that have been properly citedand the readers can easily obtain these articles to verify theconclusions replicate the analysis and conduct secondaryanalysesTherefore we did not create a publicly available datarepository

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by National Nature Science Founda-tion (under Grant 61601228 61403210 and 61603191) NaturalScience Foundation of Jiangsu (BK20161021) the Open Pro-gram of Jiangsu Key Laboratory of 3D Printing Equipmentand Manufacturing (BM2013006) Item number 3DL201607and the National Key Research and Development Program ofChina (Grant No 2017YFB1103200)

References

[1] D Zhao S Li and Q Zhu ldquoA new TSMC prototype robustnonlinear task space control of a 6 DOF parallel roboticmanipulatorrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1189ndash1197 2010

[2] YTaniseK Taniguchi S YamazakiMKamata YYamada andT Nakamura ldquoDevelopment of an air duct cleaning robot forhousing based on peristaltic crawlingmotionrdquo in Proceedings ofthe 2017 IEEE International Conference on Advanced IntelligentMechatronics AIM 2017 pp 1267ndash1272 Germany July 2017

[3] A S Al-Yahmadi J Abdo and T C Hsia ldquoModeling andcontrol of two manipulators handling a flexible objectrdquo Journalof e Franklin Institute vol 344 no 5 pp 349ndash361 2007

[4] Buren G Sun and R Hu ldquoFlexible force control on robot armfor spacecraft assemblyrdquo Spacecra Environment Engineeringvol 465 no 21 pp 147ndash152 2014

[5] L A Tuan Y H Joo P X Duong et al ldquoParameter estimatorintegrated-sliding mode control of dual arm robotsrdquo Interna-tional Journal of Control Automation amp Systems vol 15 no 6pp 2754ndash2763 2017

[6] Y Xu H Sun Q Jia et al ldquoControl of space robotic manipula-tors with faultsrdquo Mechanical Science amp Technology vol 25 no12 pp 1503-1423 2006

[7] E Coleshill L Oshinowo R Rembala B Bina D Rey andS Sindelar ldquoDextre improving maintenance operations on theInternational Space StationrdquoActaAstronautica vol 64 no 9-10pp 869ndash874 2009

[8] S B Nokleby ldquoSingularity analysis of the canadarmrdquo Mecha-nism ampMachine eory vol 42 no 4 pp 442ndash454 2007

[9] R Boumans and C Heemskerk ldquoThe European Robotic Armfor the International Space Stationrdquo Robotics and AutonomousSystems vol 23 no 1-2 pp 17ndash27 1998

[10] I Duleba and M Opalka ldquoA comparison of Jacobian-basedmethods of inverse kinematics for serial robot manipulatorsrdquoInternational Journal of Applied Mathematics amp ComputerScience vol 23 pp 373ndash382 2013

[11] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquoUkrainianMathematical Journal vol 69 no 8 pp 1220ndash1233 2017

[12] W Liu J Cui and J Xin ldquoA block-centered finite differencemethod for an unsteady asymptotic coupled model in fracturedmedia aquifer systemrdquo Journal of Computational and AppliedMathematics vol 337 pp 319ndash340 2018

[13] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[14] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 1ndash21 2017

[15] L Gao D Wang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[16] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[17] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[18] Z Cen H Noura and Y A Younes ldquoSystematic fault tolerantcontrol based on adaptiveThau observer estimation for quadro-tor UAVsrdquo International Journal of Applied Mathematics andComputer Science vol 25 no 1 pp 159ndash174 2015

[19] M Bonf P Castaldi NMimmo et al ldquoActive fault tolerant con-trol of nonlinear systems the cart-pole examplerdquo Internationaljournal of applied mathematics amp computer science vol 21 no3 pp 441ndash445 2011

[20] M Z Gao G P Cai and N Ying ldquoRobust adaptive fault-tolerant H8 control of re-entry vehicle considering actuator andsensor faults based on trajectory optimizationrdquo InternationalJournal of Control Automation amp Systems vol 14 no 1 pp 198ndash210 2016

[21] C Yang T Teng B Xu et al ldquoGlobal adaptive trackingcontrol of robotmanipulators usingneural networkswith finite-time learning convergencerdquo International Journal of ControlAutomation amp Systems vol 11 pp 1ndash9 2017

[22] Z Yu F Wang and L Liu ldquoAn adaptive fault-tolerant robustcontroller with fault diagnosis for submarinersquos vertical move-mentrdquo International Journal of Control Automation and Sys-tems vol 13 no 6 pp 1337ndash1345 2015

Mathematical Problems in Engineering 11

[23] L Li Y Yang S X Ding Y Zhang and S Zhai ldquoOn fault-tolerant control configurations for a class of nonlinear systemsrdquoJournal of e Franklin Institute vol 352 no 4 pp 1397ndash14162015

[24] P Yang R Guo X Pan et al ldquoStudy on the sliding mode faulttolerant predictive control based on multi agent particle swarmoptimizationrdquo International Journal of Control Automation ampSystems vol 3 pp 1ndash9 2017

[25] J Yang F Zhu X Wang and X Bu ldquoRobust sliding-modeobserver-based sensor fault estimation actuator fault detectionand isolation for uncertain nonlinear systemsrdquo InternationalJournal of Control Automation and Systems vol 13 no 5 pp1037ndash1046 2015

[26] M Van S S Ge and H Ren ldquoFinite time fault tolerantcontrol for robot manipulators using time delay estimation andcontinuous nonsingular fast terminal sliding mode controlrdquoIEEE Transactions on Cybernetics 2016

[27] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[28] S Liu J Wang Y Zhou and M Feckan ldquoIterative learningcontrol with pulse compensation for fractional differentialsystemsrdquoMathematica Slovaca vol 68 no 3 pp 563ndash574 2018

[29] C Ma T Li and J Zhang ldquoConsensus control for leader-following multi-agent systems with measurement noisesrdquo Jour-nal of Systems Science and Complexity vol 23 no 1 pp 35ndash492010

[30] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[31] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[32] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[33] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[34] A Owczarkowski and D Horla ldquoRobust LQR and LQI controlwith actuator failure of a 2DOF unmanned bicycle robotstabilized by an inertial wheelrdquo International Journal of AppliedMathematics and Computer Science vol 26 no 2 pp 325ndash3342016

[35] D Y Li and L Y Chun ldquoSensor fault identification anddecentralized fault-tolerant control of reconfigurable manip-ulator based on sliding mode observerrdquo in Proceedings ofthe 2012 International Conference on Control Engineering andCommunication Technology ICCECT 2012 pp 137ndash140 ChinaDecember 2012

[36] A Ferreira de Loza J Cieslak D Henry A Zolghadri andL M Fridman ldquoOutput tracking of systems subjected toperturbations and a class of actuator faults based on HOSMobservation and identificationrdquo Automatica vol 59 pp 200ndash205 2015

[37] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[38] L M Capisani A Ferrara A Ferreira De Loza and L MFridman ldquoManipulator fault diagnosis via higher order sliding-mode observersrdquo IEEE Transactions on Industrial Electronicsvol 59 no 10 pp 3979ndash3986 2012

[39] A Yarza V Santibanez and JMoreno-Valenzuela ldquoAn adaptiveoutput feedbackmotion tracking controller for robotmanipula-tors uniform global asymptotic stability and experimentationrdquoInternational Journal of Applied Mathematics and ComputerScience vol 23 no 3 pp 599ndash611 2013

[40] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[41] Q L Hu L Xu and A H Zhang ldquoAdaptive backsteppingtrajectory tracking control of robotmanipulatorrdquo Journal ofeFranklin Institute vol 349 no 3 pp 1087ndash1105 2012

[42] J-H Shin and J-J Lee ldquoFault Detection And Robust FaultRecovery Control for Robot Manipulators with Actuator Fail-uresrdquo in Proceedings of the 1999 IEEE International Conferenceon Robotics and Automation ICRA99 vol 2 pp 861ndash866 May1999

[43] J K Liu SlidingMode Control Design andMATLAB SimulationeBasiceory andDesignMethod TsinghuaUniversity Press2005

[44] S Mondal and C Mahanta ldquoAdaptive second order terminalsliding mode controller for robotic manipulatorsrdquo Journal ofe Franklin Institute vol 351 no 4 pp 2356ndash2377 2014

[45] A J Koshkouei K J Burnham and A S I Zinober ldquoDynamicsliding mode control designrdquo IEE Proceedings - Control eoryand Applications vol 152 no 4 pp 392ndash396 2005

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

Table 1 Time of fault tolerance

CASESPosition

tracking oflink1(s)

Speedtracking oflink1(s)

Controltorque oflink1(s)

Positiontracking oflink2(s)

Speedtracking oflink2(s)

Controltorque oflink2(s)

1 2 5 5 33 7 7 3 4 7 7 3 5 5 3

minus

minus

Time (s)

minus

minus

minusSpee

d re

spon

se o

f lin

k 1

(rad

s)

Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

Time (s)Actual signalDesired signal

minus

minus

minus

minusSpee

d tr

acki

ng o

f lin

k

(rad

s)

Posit

ion

resp

onse

of l

ink

1 (r

ad)

Posit

ion

trac

king

of l

ink

(rad

)

Figure 4 Tracking responses of links 1 and 2 in Case 4

minus

minus

minus

Link 1Link 2

Time (s)

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 5 Time responses of control torque in Case 1

torque of link 2 can handle the fault within 3 seconds but alittle chattering phenomenon appears

The tracking responses in Case 3 are shown in Figure 3from which it can be easily seen that link position can dealwith fault by itself and basically track the desired positionsignal in 7s when actuator fault appears in link 1 at 10sCompared with other cases the error between actual signaland desired signal is a little high but is still in an acceptablerange Figure 3 also provides the actual speed of link 1 withactuator fault being able to track expected signal within 7sFrom Figure 3 we can respectively see position and speed oflink 2 with no fault being able to easily realize the trajectory

tracking in 5s Figure 7 shows the time response of controltorque from which we can see that the amplitude of link 1increases and control torque can reconvergewithin 3 seconds

Figure 4 provides the trajectory tracking responses of twolinks in this case The tracking response of link 1 is shown inFigure 4 from which it is easy to see that when actuator oflink 1 failed at 10s position signal and speed signal of link 1can deal with fault and basically track the desired signals in 7seconds Although there exists a certain error in both positionand speed tracking the error is acceptable in programmingFigure 4 also depicts the tracking response of link 2 fromwhich we can see that it takes 5 seconds for position and

Mathematical Problems in Engineering 9

Link 1Link 2

Time (s)

minus

minus

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 6 Time responses of control torque in Case 2

Link 1Link 2

Time (s)

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 7 Time responses of control torque in Case 3

Link 1Link 2

Time (s)

minus

minus

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 8 Time responses of control torque in Case 4

speed signal to track the corresponding desired signals whenactuator fault occurs in link 2 at 10s Figure 8 depicts the timeresponse of control torque in this case from which it can beeasily seen that the amplitudes of 2 links increase and controltorque can reconverge within 3 seconds but a little chatteringphenomenon appears

From Table 1 and Figures 1ndash8 we can see that when bothactuators are healthy without fault the overall system canrealize trajectory tracking within 5 seconds when actuatorfault occurs in link 2 tracking responses of link 2 canreconverge to desired signal within 5 seconds when actuator

fault occurs in link 1 tracking responses of link 1 canreconverge to desired signal within 7 seconds when actuatorfault occurs in two links tracking responses of links 1 and2 can reconverge to desired signal within 7 and 5 secondsrespectivelyTherefore the trajectory tracking effectiveness ofthe FTC method proposed in this paper is verified

5 Conclusions

In this paper a novel FTC scheme based on adaptive slidingmode method is investigated for manipulator with actuator

10 Mathematical Problems in Engineering

fault and external disturbance Firstly the general dynamicmodel of space manipulator is introduced and further itsactuator faulty model is established Secondly an adaptivefault tolerant controller is designed for manipulator withactuator fault Parameters of fault and disturbance are esti-mated and updated by online adaptive method Finally theproposed controller is applied to two-joint manipulator fromthe simulation results it is shown that the controller proposedin this paper can not only realize a good trajectory trackingbut also tolerate actuator fault and present strong robustnessfor external disturbance while effectively reducing the chat-tering phenomenon of sliding mode control

Data Availability

The data supporting the conclusions of our manuscript aresome open access articles that have been properly citedand the readers can easily obtain these articles to verify theconclusions replicate the analysis and conduct secondaryanalysesTherefore we did not create a publicly available datarepository

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by National Nature Science Founda-tion (under Grant 61601228 61403210 and 61603191) NaturalScience Foundation of Jiangsu (BK20161021) the Open Pro-gram of Jiangsu Key Laboratory of 3D Printing Equipmentand Manufacturing (BM2013006) Item number 3DL201607and the National Key Research and Development Program ofChina (Grant No 2017YFB1103200)

References

[1] D Zhao S Li and Q Zhu ldquoA new TSMC prototype robustnonlinear task space control of a 6 DOF parallel roboticmanipulatorrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1189ndash1197 2010

[2] YTaniseK Taniguchi S YamazakiMKamata YYamada andT Nakamura ldquoDevelopment of an air duct cleaning robot forhousing based on peristaltic crawlingmotionrdquo in Proceedings ofthe 2017 IEEE International Conference on Advanced IntelligentMechatronics AIM 2017 pp 1267ndash1272 Germany July 2017

[3] A S Al-Yahmadi J Abdo and T C Hsia ldquoModeling andcontrol of two manipulators handling a flexible objectrdquo Journalof e Franklin Institute vol 344 no 5 pp 349ndash361 2007

[4] Buren G Sun and R Hu ldquoFlexible force control on robot armfor spacecraft assemblyrdquo Spacecra Environment Engineeringvol 465 no 21 pp 147ndash152 2014

[5] L A Tuan Y H Joo P X Duong et al ldquoParameter estimatorintegrated-sliding mode control of dual arm robotsrdquo Interna-tional Journal of Control Automation amp Systems vol 15 no 6pp 2754ndash2763 2017

[6] Y Xu H Sun Q Jia et al ldquoControl of space robotic manipula-tors with faultsrdquo Mechanical Science amp Technology vol 25 no12 pp 1503-1423 2006

[7] E Coleshill L Oshinowo R Rembala B Bina D Rey andS Sindelar ldquoDextre improving maintenance operations on theInternational Space StationrdquoActaAstronautica vol 64 no 9-10pp 869ndash874 2009

[8] S B Nokleby ldquoSingularity analysis of the canadarmrdquo Mecha-nism ampMachine eory vol 42 no 4 pp 442ndash454 2007

[9] R Boumans and C Heemskerk ldquoThe European Robotic Armfor the International Space Stationrdquo Robotics and AutonomousSystems vol 23 no 1-2 pp 17ndash27 1998

[10] I Duleba and M Opalka ldquoA comparison of Jacobian-basedmethods of inverse kinematics for serial robot manipulatorsrdquoInternational Journal of Applied Mathematics amp ComputerScience vol 23 pp 373ndash382 2013

[11] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquoUkrainianMathematical Journal vol 69 no 8 pp 1220ndash1233 2017

[12] W Liu J Cui and J Xin ldquoA block-centered finite differencemethod for an unsteady asymptotic coupled model in fracturedmedia aquifer systemrdquo Journal of Computational and AppliedMathematics vol 337 pp 319ndash340 2018

[13] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[14] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 1ndash21 2017

[15] L Gao D Wang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[16] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[17] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[18] Z Cen H Noura and Y A Younes ldquoSystematic fault tolerantcontrol based on adaptiveThau observer estimation for quadro-tor UAVsrdquo International Journal of Applied Mathematics andComputer Science vol 25 no 1 pp 159ndash174 2015

[19] M Bonf P Castaldi NMimmo et al ldquoActive fault tolerant con-trol of nonlinear systems the cart-pole examplerdquo Internationaljournal of applied mathematics amp computer science vol 21 no3 pp 441ndash445 2011

[20] M Z Gao G P Cai and N Ying ldquoRobust adaptive fault-tolerant H8 control of re-entry vehicle considering actuator andsensor faults based on trajectory optimizationrdquo InternationalJournal of Control Automation amp Systems vol 14 no 1 pp 198ndash210 2016

[21] C Yang T Teng B Xu et al ldquoGlobal adaptive trackingcontrol of robotmanipulators usingneural networkswith finite-time learning convergencerdquo International Journal of ControlAutomation amp Systems vol 11 pp 1ndash9 2017

[22] Z Yu F Wang and L Liu ldquoAn adaptive fault-tolerant robustcontroller with fault diagnosis for submarinersquos vertical move-mentrdquo International Journal of Control Automation and Sys-tems vol 13 no 6 pp 1337ndash1345 2015

Mathematical Problems in Engineering 11

[23] L Li Y Yang S X Ding Y Zhang and S Zhai ldquoOn fault-tolerant control configurations for a class of nonlinear systemsrdquoJournal of e Franklin Institute vol 352 no 4 pp 1397ndash14162015

[24] P Yang R Guo X Pan et al ldquoStudy on the sliding mode faulttolerant predictive control based on multi agent particle swarmoptimizationrdquo International Journal of Control Automation ampSystems vol 3 pp 1ndash9 2017

[25] J Yang F Zhu X Wang and X Bu ldquoRobust sliding-modeobserver-based sensor fault estimation actuator fault detectionand isolation for uncertain nonlinear systemsrdquo InternationalJournal of Control Automation and Systems vol 13 no 5 pp1037ndash1046 2015

[26] M Van S S Ge and H Ren ldquoFinite time fault tolerantcontrol for robot manipulators using time delay estimation andcontinuous nonsingular fast terminal sliding mode controlrdquoIEEE Transactions on Cybernetics 2016

[27] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[28] S Liu J Wang Y Zhou and M Feckan ldquoIterative learningcontrol with pulse compensation for fractional differentialsystemsrdquoMathematica Slovaca vol 68 no 3 pp 563ndash574 2018

[29] C Ma T Li and J Zhang ldquoConsensus control for leader-following multi-agent systems with measurement noisesrdquo Jour-nal of Systems Science and Complexity vol 23 no 1 pp 35ndash492010

[30] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[31] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[32] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[33] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[34] A Owczarkowski and D Horla ldquoRobust LQR and LQI controlwith actuator failure of a 2DOF unmanned bicycle robotstabilized by an inertial wheelrdquo International Journal of AppliedMathematics and Computer Science vol 26 no 2 pp 325ndash3342016

[35] D Y Li and L Y Chun ldquoSensor fault identification anddecentralized fault-tolerant control of reconfigurable manip-ulator based on sliding mode observerrdquo in Proceedings ofthe 2012 International Conference on Control Engineering andCommunication Technology ICCECT 2012 pp 137ndash140 ChinaDecember 2012

[36] A Ferreira de Loza J Cieslak D Henry A Zolghadri andL M Fridman ldquoOutput tracking of systems subjected toperturbations and a class of actuator faults based on HOSMobservation and identificationrdquo Automatica vol 59 pp 200ndash205 2015

[37] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[38] L M Capisani A Ferrara A Ferreira De Loza and L MFridman ldquoManipulator fault diagnosis via higher order sliding-mode observersrdquo IEEE Transactions on Industrial Electronicsvol 59 no 10 pp 3979ndash3986 2012

[39] A Yarza V Santibanez and JMoreno-Valenzuela ldquoAn adaptiveoutput feedbackmotion tracking controller for robotmanipula-tors uniform global asymptotic stability and experimentationrdquoInternational Journal of Applied Mathematics and ComputerScience vol 23 no 3 pp 599ndash611 2013

[40] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[41] Q L Hu L Xu and A H Zhang ldquoAdaptive backsteppingtrajectory tracking control of robotmanipulatorrdquo Journal ofeFranklin Institute vol 349 no 3 pp 1087ndash1105 2012

[42] J-H Shin and J-J Lee ldquoFault Detection And Robust FaultRecovery Control for Robot Manipulators with Actuator Fail-uresrdquo in Proceedings of the 1999 IEEE International Conferenceon Robotics and Automation ICRA99 vol 2 pp 861ndash866 May1999

[43] J K Liu SlidingMode Control Design andMATLAB SimulationeBasiceory andDesignMethod TsinghuaUniversity Press2005

[44] S Mondal and C Mahanta ldquoAdaptive second order terminalsliding mode controller for robotic manipulatorsrdquo Journal ofe Franklin Institute vol 351 no 4 pp 2356ndash2377 2014

[45] A J Koshkouei K J Burnham and A S I Zinober ldquoDynamicsliding mode control designrdquo IEE Proceedings - Control eoryand Applications vol 152 no 4 pp 392ndash396 2005

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

Link 1Link 2

Time (s)

minus

minus

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 6 Time responses of control torque in Case 2

Link 1Link 2

Time (s)

minus

minus

minusTim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 7 Time responses of control torque in Case 3

Link 1Link 2

Time (s)

minus

minus

Tim

e res

pons

e of c

ontr

ol to

rque

(Nmiddotm

)

Figure 8 Time responses of control torque in Case 4

speed signal to track the corresponding desired signals whenactuator fault occurs in link 2 at 10s Figure 8 depicts the timeresponse of control torque in this case from which it can beeasily seen that the amplitudes of 2 links increase and controltorque can reconverge within 3 seconds but a little chatteringphenomenon appears

From Table 1 and Figures 1ndash8 we can see that when bothactuators are healthy without fault the overall system canrealize trajectory tracking within 5 seconds when actuatorfault occurs in link 2 tracking responses of link 2 canreconverge to desired signal within 5 seconds when actuator

fault occurs in link 1 tracking responses of link 1 canreconverge to desired signal within 7 seconds when actuatorfault occurs in two links tracking responses of links 1 and2 can reconverge to desired signal within 7 and 5 secondsrespectivelyTherefore the trajectory tracking effectiveness ofthe FTC method proposed in this paper is verified

5 Conclusions

In this paper a novel FTC scheme based on adaptive slidingmode method is investigated for manipulator with actuator

10 Mathematical Problems in Engineering

fault and external disturbance Firstly the general dynamicmodel of space manipulator is introduced and further itsactuator faulty model is established Secondly an adaptivefault tolerant controller is designed for manipulator withactuator fault Parameters of fault and disturbance are esti-mated and updated by online adaptive method Finally theproposed controller is applied to two-joint manipulator fromthe simulation results it is shown that the controller proposedin this paper can not only realize a good trajectory trackingbut also tolerate actuator fault and present strong robustnessfor external disturbance while effectively reducing the chat-tering phenomenon of sliding mode control

Data Availability

The data supporting the conclusions of our manuscript aresome open access articles that have been properly citedand the readers can easily obtain these articles to verify theconclusions replicate the analysis and conduct secondaryanalysesTherefore we did not create a publicly available datarepository

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by National Nature Science Founda-tion (under Grant 61601228 61403210 and 61603191) NaturalScience Foundation of Jiangsu (BK20161021) the Open Pro-gram of Jiangsu Key Laboratory of 3D Printing Equipmentand Manufacturing (BM2013006) Item number 3DL201607and the National Key Research and Development Program ofChina (Grant No 2017YFB1103200)

References

[1] D Zhao S Li and Q Zhu ldquoA new TSMC prototype robustnonlinear task space control of a 6 DOF parallel roboticmanipulatorrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1189ndash1197 2010

[2] YTaniseK Taniguchi S YamazakiMKamata YYamada andT Nakamura ldquoDevelopment of an air duct cleaning robot forhousing based on peristaltic crawlingmotionrdquo in Proceedings ofthe 2017 IEEE International Conference on Advanced IntelligentMechatronics AIM 2017 pp 1267ndash1272 Germany July 2017

[3] A S Al-Yahmadi J Abdo and T C Hsia ldquoModeling andcontrol of two manipulators handling a flexible objectrdquo Journalof e Franklin Institute vol 344 no 5 pp 349ndash361 2007

[4] Buren G Sun and R Hu ldquoFlexible force control on robot armfor spacecraft assemblyrdquo Spacecra Environment Engineeringvol 465 no 21 pp 147ndash152 2014

[5] L A Tuan Y H Joo P X Duong et al ldquoParameter estimatorintegrated-sliding mode control of dual arm robotsrdquo Interna-tional Journal of Control Automation amp Systems vol 15 no 6pp 2754ndash2763 2017

[6] Y Xu H Sun Q Jia et al ldquoControl of space robotic manipula-tors with faultsrdquo Mechanical Science amp Technology vol 25 no12 pp 1503-1423 2006

[7] E Coleshill L Oshinowo R Rembala B Bina D Rey andS Sindelar ldquoDextre improving maintenance operations on theInternational Space StationrdquoActaAstronautica vol 64 no 9-10pp 869ndash874 2009

[8] S B Nokleby ldquoSingularity analysis of the canadarmrdquo Mecha-nism ampMachine eory vol 42 no 4 pp 442ndash454 2007

[9] R Boumans and C Heemskerk ldquoThe European Robotic Armfor the International Space Stationrdquo Robotics and AutonomousSystems vol 23 no 1-2 pp 17ndash27 1998

[10] I Duleba and M Opalka ldquoA comparison of Jacobian-basedmethods of inverse kinematics for serial robot manipulatorsrdquoInternational Journal of Applied Mathematics amp ComputerScience vol 23 pp 373ndash382 2013

[11] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquoUkrainianMathematical Journal vol 69 no 8 pp 1220ndash1233 2017

[12] W Liu J Cui and J Xin ldquoA block-centered finite differencemethod for an unsteady asymptotic coupled model in fracturedmedia aquifer systemrdquo Journal of Computational and AppliedMathematics vol 337 pp 319ndash340 2018

[13] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[14] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 1ndash21 2017

[15] L Gao D Wang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[16] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[17] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[18] Z Cen H Noura and Y A Younes ldquoSystematic fault tolerantcontrol based on adaptiveThau observer estimation for quadro-tor UAVsrdquo International Journal of Applied Mathematics andComputer Science vol 25 no 1 pp 159ndash174 2015

[19] M Bonf P Castaldi NMimmo et al ldquoActive fault tolerant con-trol of nonlinear systems the cart-pole examplerdquo Internationaljournal of applied mathematics amp computer science vol 21 no3 pp 441ndash445 2011

[20] M Z Gao G P Cai and N Ying ldquoRobust adaptive fault-tolerant H8 control of re-entry vehicle considering actuator andsensor faults based on trajectory optimizationrdquo InternationalJournal of Control Automation amp Systems vol 14 no 1 pp 198ndash210 2016

[21] C Yang T Teng B Xu et al ldquoGlobal adaptive trackingcontrol of robotmanipulators usingneural networkswith finite-time learning convergencerdquo International Journal of ControlAutomation amp Systems vol 11 pp 1ndash9 2017

[22] Z Yu F Wang and L Liu ldquoAn adaptive fault-tolerant robustcontroller with fault diagnosis for submarinersquos vertical move-mentrdquo International Journal of Control Automation and Sys-tems vol 13 no 6 pp 1337ndash1345 2015

Mathematical Problems in Engineering 11

[23] L Li Y Yang S X Ding Y Zhang and S Zhai ldquoOn fault-tolerant control configurations for a class of nonlinear systemsrdquoJournal of e Franklin Institute vol 352 no 4 pp 1397ndash14162015

[24] P Yang R Guo X Pan et al ldquoStudy on the sliding mode faulttolerant predictive control based on multi agent particle swarmoptimizationrdquo International Journal of Control Automation ampSystems vol 3 pp 1ndash9 2017

[25] J Yang F Zhu X Wang and X Bu ldquoRobust sliding-modeobserver-based sensor fault estimation actuator fault detectionand isolation for uncertain nonlinear systemsrdquo InternationalJournal of Control Automation and Systems vol 13 no 5 pp1037ndash1046 2015

[26] M Van S S Ge and H Ren ldquoFinite time fault tolerantcontrol for robot manipulators using time delay estimation andcontinuous nonsingular fast terminal sliding mode controlrdquoIEEE Transactions on Cybernetics 2016

[27] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[28] S Liu J Wang Y Zhou and M Feckan ldquoIterative learningcontrol with pulse compensation for fractional differentialsystemsrdquoMathematica Slovaca vol 68 no 3 pp 563ndash574 2018

[29] C Ma T Li and J Zhang ldquoConsensus control for leader-following multi-agent systems with measurement noisesrdquo Jour-nal of Systems Science and Complexity vol 23 no 1 pp 35ndash492010

[30] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[31] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[32] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[33] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[34] A Owczarkowski and D Horla ldquoRobust LQR and LQI controlwith actuator failure of a 2DOF unmanned bicycle robotstabilized by an inertial wheelrdquo International Journal of AppliedMathematics and Computer Science vol 26 no 2 pp 325ndash3342016

[35] D Y Li and L Y Chun ldquoSensor fault identification anddecentralized fault-tolerant control of reconfigurable manip-ulator based on sliding mode observerrdquo in Proceedings ofthe 2012 International Conference on Control Engineering andCommunication Technology ICCECT 2012 pp 137ndash140 ChinaDecember 2012

[36] A Ferreira de Loza J Cieslak D Henry A Zolghadri andL M Fridman ldquoOutput tracking of systems subjected toperturbations and a class of actuator faults based on HOSMobservation and identificationrdquo Automatica vol 59 pp 200ndash205 2015

[37] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[38] L M Capisani A Ferrara A Ferreira De Loza and L MFridman ldquoManipulator fault diagnosis via higher order sliding-mode observersrdquo IEEE Transactions on Industrial Electronicsvol 59 no 10 pp 3979ndash3986 2012

[39] A Yarza V Santibanez and JMoreno-Valenzuela ldquoAn adaptiveoutput feedbackmotion tracking controller for robotmanipula-tors uniform global asymptotic stability and experimentationrdquoInternational Journal of Applied Mathematics and ComputerScience vol 23 no 3 pp 599ndash611 2013

[40] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[41] Q L Hu L Xu and A H Zhang ldquoAdaptive backsteppingtrajectory tracking control of robotmanipulatorrdquo Journal ofeFranklin Institute vol 349 no 3 pp 1087ndash1105 2012

[42] J-H Shin and J-J Lee ldquoFault Detection And Robust FaultRecovery Control for Robot Manipulators with Actuator Fail-uresrdquo in Proceedings of the 1999 IEEE International Conferenceon Robotics and Automation ICRA99 vol 2 pp 861ndash866 May1999

[43] J K Liu SlidingMode Control Design andMATLAB SimulationeBasiceory andDesignMethod TsinghuaUniversity Press2005

[44] S Mondal and C Mahanta ldquoAdaptive second order terminalsliding mode controller for robotic manipulatorsrdquo Journal ofe Franklin Institute vol 351 no 4 pp 2356ndash2377 2014

[45] A J Koshkouei K J Burnham and A S I Zinober ldquoDynamicsliding mode control designrdquo IEE Proceedings - Control eoryand Applications vol 152 no 4 pp 392ndash396 2005

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

fault and external disturbance Firstly the general dynamicmodel of space manipulator is introduced and further itsactuator faulty model is established Secondly an adaptivefault tolerant controller is designed for manipulator withactuator fault Parameters of fault and disturbance are esti-mated and updated by online adaptive method Finally theproposed controller is applied to two-joint manipulator fromthe simulation results it is shown that the controller proposedin this paper can not only realize a good trajectory trackingbut also tolerate actuator fault and present strong robustnessfor external disturbance while effectively reducing the chat-tering phenomenon of sliding mode control

Data Availability

The data supporting the conclusions of our manuscript aresome open access articles that have been properly citedand the readers can easily obtain these articles to verify theconclusions replicate the analysis and conduct secondaryanalysesTherefore we did not create a publicly available datarepository

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by National Nature Science Founda-tion (under Grant 61601228 61403210 and 61603191) NaturalScience Foundation of Jiangsu (BK20161021) the Open Pro-gram of Jiangsu Key Laboratory of 3D Printing Equipmentand Manufacturing (BM2013006) Item number 3DL201607and the National Key Research and Development Program ofChina (Grant No 2017YFB1103200)

References

[1] D Zhao S Li and Q Zhu ldquoA new TSMC prototype robustnonlinear task space control of a 6 DOF parallel roboticmanipulatorrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1189ndash1197 2010

[2] YTaniseK Taniguchi S YamazakiMKamata YYamada andT Nakamura ldquoDevelopment of an air duct cleaning robot forhousing based on peristaltic crawlingmotionrdquo in Proceedings ofthe 2017 IEEE International Conference on Advanced IntelligentMechatronics AIM 2017 pp 1267ndash1272 Germany July 2017

[3] A S Al-Yahmadi J Abdo and T C Hsia ldquoModeling andcontrol of two manipulators handling a flexible objectrdquo Journalof e Franklin Institute vol 344 no 5 pp 349ndash361 2007

[4] Buren G Sun and R Hu ldquoFlexible force control on robot armfor spacecraft assemblyrdquo Spacecra Environment Engineeringvol 465 no 21 pp 147ndash152 2014

[5] L A Tuan Y H Joo P X Duong et al ldquoParameter estimatorintegrated-sliding mode control of dual arm robotsrdquo Interna-tional Journal of Control Automation amp Systems vol 15 no 6pp 2754ndash2763 2017

[6] Y Xu H Sun Q Jia et al ldquoControl of space robotic manipula-tors with faultsrdquo Mechanical Science amp Technology vol 25 no12 pp 1503-1423 2006

[7] E Coleshill L Oshinowo R Rembala B Bina D Rey andS Sindelar ldquoDextre improving maintenance operations on theInternational Space StationrdquoActaAstronautica vol 64 no 9-10pp 869ndash874 2009

[8] S B Nokleby ldquoSingularity analysis of the canadarmrdquo Mecha-nism ampMachine eory vol 42 no 4 pp 442ndash454 2007

[9] R Boumans and C Heemskerk ldquoThe European Robotic Armfor the International Space Stationrdquo Robotics and AutonomousSystems vol 23 no 1-2 pp 17ndash27 1998

[10] I Duleba and M Opalka ldquoA comparison of Jacobian-basedmethods of inverse kinematics for serial robot manipulatorsrdquoInternational Journal of Applied Mathematics amp ComputerScience vol 23 pp 373ndash382 2013

[11] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquoUkrainianMathematical Journal vol 69 no 8 pp 1220ndash1233 2017

[12] W Liu J Cui and J Xin ldquoA block-centered finite differencemethod for an unsteady asymptotic coupled model in fracturedmedia aquifer systemrdquo Journal of Computational and AppliedMathematics vol 337 pp 319ndash340 2018

[13] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[14] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 1ndash21 2017

[15] L Gao D Wang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[16] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[17] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[18] Z Cen H Noura and Y A Younes ldquoSystematic fault tolerantcontrol based on adaptiveThau observer estimation for quadro-tor UAVsrdquo International Journal of Applied Mathematics andComputer Science vol 25 no 1 pp 159ndash174 2015

[19] M Bonf P Castaldi NMimmo et al ldquoActive fault tolerant con-trol of nonlinear systems the cart-pole examplerdquo Internationaljournal of applied mathematics amp computer science vol 21 no3 pp 441ndash445 2011

[20] M Z Gao G P Cai and N Ying ldquoRobust adaptive fault-tolerant H8 control of re-entry vehicle considering actuator andsensor faults based on trajectory optimizationrdquo InternationalJournal of Control Automation amp Systems vol 14 no 1 pp 198ndash210 2016

[21] C Yang T Teng B Xu et al ldquoGlobal adaptive trackingcontrol of robotmanipulators usingneural networkswith finite-time learning convergencerdquo International Journal of ControlAutomation amp Systems vol 11 pp 1ndash9 2017

[22] Z Yu F Wang and L Liu ldquoAn adaptive fault-tolerant robustcontroller with fault diagnosis for submarinersquos vertical move-mentrdquo International Journal of Control Automation and Sys-tems vol 13 no 6 pp 1337ndash1345 2015

Mathematical Problems in Engineering 11

[23] L Li Y Yang S X Ding Y Zhang and S Zhai ldquoOn fault-tolerant control configurations for a class of nonlinear systemsrdquoJournal of e Franklin Institute vol 352 no 4 pp 1397ndash14162015

[24] P Yang R Guo X Pan et al ldquoStudy on the sliding mode faulttolerant predictive control based on multi agent particle swarmoptimizationrdquo International Journal of Control Automation ampSystems vol 3 pp 1ndash9 2017

[25] J Yang F Zhu X Wang and X Bu ldquoRobust sliding-modeobserver-based sensor fault estimation actuator fault detectionand isolation for uncertain nonlinear systemsrdquo InternationalJournal of Control Automation and Systems vol 13 no 5 pp1037ndash1046 2015

[26] M Van S S Ge and H Ren ldquoFinite time fault tolerantcontrol for robot manipulators using time delay estimation andcontinuous nonsingular fast terminal sliding mode controlrdquoIEEE Transactions on Cybernetics 2016

[27] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[28] S Liu J Wang Y Zhou and M Feckan ldquoIterative learningcontrol with pulse compensation for fractional differentialsystemsrdquoMathematica Slovaca vol 68 no 3 pp 563ndash574 2018

[29] C Ma T Li and J Zhang ldquoConsensus control for leader-following multi-agent systems with measurement noisesrdquo Jour-nal of Systems Science and Complexity vol 23 no 1 pp 35ndash492010

[30] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[31] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[32] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[33] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[34] A Owczarkowski and D Horla ldquoRobust LQR and LQI controlwith actuator failure of a 2DOF unmanned bicycle robotstabilized by an inertial wheelrdquo International Journal of AppliedMathematics and Computer Science vol 26 no 2 pp 325ndash3342016

[35] D Y Li and L Y Chun ldquoSensor fault identification anddecentralized fault-tolerant control of reconfigurable manip-ulator based on sliding mode observerrdquo in Proceedings ofthe 2012 International Conference on Control Engineering andCommunication Technology ICCECT 2012 pp 137ndash140 ChinaDecember 2012

[36] A Ferreira de Loza J Cieslak D Henry A Zolghadri andL M Fridman ldquoOutput tracking of systems subjected toperturbations and a class of actuator faults based on HOSMobservation and identificationrdquo Automatica vol 59 pp 200ndash205 2015

[37] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[38] L M Capisani A Ferrara A Ferreira De Loza and L MFridman ldquoManipulator fault diagnosis via higher order sliding-mode observersrdquo IEEE Transactions on Industrial Electronicsvol 59 no 10 pp 3979ndash3986 2012

[39] A Yarza V Santibanez and JMoreno-Valenzuela ldquoAn adaptiveoutput feedbackmotion tracking controller for robotmanipula-tors uniform global asymptotic stability and experimentationrdquoInternational Journal of Applied Mathematics and ComputerScience vol 23 no 3 pp 599ndash611 2013

[40] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[41] Q L Hu L Xu and A H Zhang ldquoAdaptive backsteppingtrajectory tracking control of robotmanipulatorrdquo Journal ofeFranklin Institute vol 349 no 3 pp 1087ndash1105 2012

[42] J-H Shin and J-J Lee ldquoFault Detection And Robust FaultRecovery Control for Robot Manipulators with Actuator Fail-uresrdquo in Proceedings of the 1999 IEEE International Conferenceon Robotics and Automation ICRA99 vol 2 pp 861ndash866 May1999

[43] J K Liu SlidingMode Control Design andMATLAB SimulationeBasiceory andDesignMethod TsinghuaUniversity Press2005

[44] S Mondal and C Mahanta ldquoAdaptive second order terminalsliding mode controller for robotic manipulatorsrdquo Journal ofe Franklin Institute vol 351 no 4 pp 2356ndash2377 2014

[45] A J Koshkouei K J Burnham and A S I Zinober ldquoDynamicsliding mode control designrdquo IEE Proceedings - Control eoryand Applications vol 152 no 4 pp 392ndash396 2005

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

[23] L Li Y Yang S X Ding Y Zhang and S Zhai ldquoOn fault-tolerant control configurations for a class of nonlinear systemsrdquoJournal of e Franklin Institute vol 352 no 4 pp 1397ndash14162015

[24] P Yang R Guo X Pan et al ldquoStudy on the sliding mode faulttolerant predictive control based on multi agent particle swarmoptimizationrdquo International Journal of Control Automation ampSystems vol 3 pp 1ndash9 2017

[25] J Yang F Zhu X Wang and X Bu ldquoRobust sliding-modeobserver-based sensor fault estimation actuator fault detectionand isolation for uncertain nonlinear systemsrdquo InternationalJournal of Control Automation and Systems vol 13 no 5 pp1037ndash1046 2015

[26] M Van S S Ge and H Ren ldquoFinite time fault tolerantcontrol for robot manipulators using time delay estimation andcontinuous nonsingular fast terminal sliding mode controlrdquoIEEE Transactions on Cybernetics 2016

[27] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[28] S Liu J Wang Y Zhou and M Feckan ldquoIterative learningcontrol with pulse compensation for fractional differentialsystemsrdquoMathematica Slovaca vol 68 no 3 pp 563ndash574 2018

[29] C Ma T Li and J Zhang ldquoConsensus control for leader-following multi-agent systems with measurement noisesrdquo Jour-nal of Systems Science and Complexity vol 23 no 1 pp 35ndash492010

[30] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[31] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[32] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[33] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[34] A Owczarkowski and D Horla ldquoRobust LQR and LQI controlwith actuator failure of a 2DOF unmanned bicycle robotstabilized by an inertial wheelrdquo International Journal of AppliedMathematics and Computer Science vol 26 no 2 pp 325ndash3342016

[35] D Y Li and L Y Chun ldquoSensor fault identification anddecentralized fault-tolerant control of reconfigurable manip-ulator based on sliding mode observerrdquo in Proceedings ofthe 2012 International Conference on Control Engineering andCommunication Technology ICCECT 2012 pp 137ndash140 ChinaDecember 2012

[36] A Ferreira de Loza J Cieslak D Henry A Zolghadri andL M Fridman ldquoOutput tracking of systems subjected toperturbations and a class of actuator faults based on HOSMobservation and identificationrdquo Automatica vol 59 pp 200ndash205 2015

[37] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[38] L M Capisani A Ferrara A Ferreira De Loza and L MFridman ldquoManipulator fault diagnosis via higher order sliding-mode observersrdquo IEEE Transactions on Industrial Electronicsvol 59 no 10 pp 3979ndash3986 2012

[39] A Yarza V Santibanez and JMoreno-Valenzuela ldquoAn adaptiveoutput feedbackmotion tracking controller for robotmanipula-tors uniform global asymptotic stability and experimentationrdquoInternational Journal of Applied Mathematics and ComputerScience vol 23 no 3 pp 599ndash611 2013

[40] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[41] Q L Hu L Xu and A H Zhang ldquoAdaptive backsteppingtrajectory tracking control of robotmanipulatorrdquo Journal ofeFranklin Institute vol 349 no 3 pp 1087ndash1105 2012

[42] J-H Shin and J-J Lee ldquoFault Detection And Robust FaultRecovery Control for Robot Manipulators with Actuator Fail-uresrdquo in Proceedings of the 1999 IEEE International Conferenceon Robotics and Automation ICRA99 vol 2 pp 861ndash866 May1999

[43] J K Liu SlidingMode Control Design andMATLAB SimulationeBasiceory andDesignMethod TsinghuaUniversity Press2005

[44] S Mondal and C Mahanta ldquoAdaptive second order terminalsliding mode controller for robotic manipulatorsrdquo Journal ofe Franklin Institute vol 351 no 4 pp 2356ndash2377 2014

[45] A J Koshkouei K J Burnham and A S I Zinober ldquoDynamicsliding mode control designrdquo IEE Proceedings - Control eoryand Applications vol 152 no 4 pp 392ndash396 2005

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom