Actuator Fault Compensation in Three Linked 2WD Mobile Robots Using Multiple Dynamic Controllers

The International Federation of Automatic Proceedings of the 20th World The International Federation of Congress Automatic Control Control Proceedings of the 20th World Congress Toulouse, France, July 2017 Proceedings of the 20th9-14, World The International Federation of Congress Automatic Control Toulouse, France, July 9-14, 2017 The International Federation of Automatic Control Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 in (2017) 13556–13562 Actuator Three Linked Actuator Fault Fault Compensation Compensation in Three Linked 2WD 2WD Mobile Mobile Robots Robots Using Using Actuator Fault Compensation in Three Linked 2WD Mobile Robots Using Actuator Fault Compensation in Three Linked 2WD Mobile Robots Using Multiple Dynamic Controllers Multiplein Dynamic Controllers Actuator Fault Compensation Three Linked 2WD Mobile Robots Using Multiple Dynamic Controllers Multiple Dynamic Controllers Multiple Dynamic Controllers AL-DUJAILI AL-DUJAILI Ayad, Ayad, Ma Ma Yajie, Yajie, EL EL BADAOUI BADAOUI EL EL NAJJAR NAJJAR Maan Maan and and COCQUEMPOT COCQUEMPOT Vincent Vincent

AL-DUJAILI Ayad, Ma Yajie, EL BADAOUI EL NAJJAR Maan and COCQUEMPOT Vincent AL-DUJAILI Ayad, Ma Yajie, EL BADAOUI EL NAJJAR Maan and COCQUEMPOT Vincent AL-DUJAILI Ayad, Ma Yajie, EL BADAOUI EL NAJJAR Maan and COCQUEMPOT Vincent University of centre in in Computer Computer Science, Science, Signal Signal and and Automatic Automatic Control University of Lille, Lille, CRIStAL CRIStAL (Recherch (Recherch centre Control of of Lille), Lille), University of Lille, CRIStAL (Recherch centre in Computer Science, Signal and Automatic Control of Lille), (UMR 9189), 59655, Villeneuve d’Ascq, France (UMR 9189), 59655, Villeneuve d’Ascq, France University of Lille, CRIStAL (Recherch centre in Computer Science, Signal and Automatic Control of Lille), University of Lille, CRIStAL (Recherch centre in Computer Science, Signal and Automatic Control of Lille), (UMR 9189), 59655, Villeneuve d’Ascq, France (UMR 9189), 59655, Villeneuve d’Ascq, France (UMR 9189), 59655, Villeneuve d’Ascq, France Abstract: The paper presents a new compensation control scheme to control three linked linked wheeled wheeled mobile mobile Abstract: The paper presents a new compensation control scheme to control three Abstract: The paper presents a new compensation control scheme to control three linked wheeled mobile robots with actuator failures. First, a configuration of three mobile robots with fixed link is described, and robots withThe actuator failures. First, a compensation configuration of three scheme mobile robots withthree fixedlinked link iswheeled described, and Abstract: paper presents aa new control to control mobile Abstract: The paper presentsare new compensation control scheme tobased control three linked wheeled mobile robots with actuator failures. First, a configuration of three mobile robots with fixed link is described, its kinematics and dynamics modelled. Then, a multiple-model control scheme is developed to its kinematics and dynamics are modelled. Then, aofmultiple-model basedwith control scheme isdescribed, developedand to robots with actuator failures. First, a configuration three mobile robots fixed link is and robots with actuator failures. First, a configuration ofmultiple-model three mobile robots with fixed link controllers isisdescribed, and its kinematics and dynamics are modelled. Then, a based control scheme developed to compensate for actuator failures, consisting of a kinematic controller, multiple dynamic and compensate forand actuator failures, consistingThen, of a kinematic controller, multiple dynamic controllers andtoaa its kinematics dynamics are modelled. aa multiple-model based control scheme is developed its kinematics and dynamics areThe modelled. Then, multiple-model based control scheme isthe developed toa compensate for actuator failures, consisting of a kinematic controller, multiple dynamic controllers and control switching mechanism. kinematic controller is based on the transformation of considered control switching mechanism. The kinematicofcontroller is based on the transformation of the considered compensate for actuator failures, consisting aa kinematic controller, multiple dynamic controllers and aa compensate for actuator failures, consisting of kinematic controller, multiple dynamic controllers and control switching mechanism. The kinematic controller is based on the transformation of the kinematics into system, after the technique is the kinematic kinematics into aa chained-form chained-form system, after that that the recursive recursive technique is used used to to derive derive theconsidered kinematic control switching mechanism. The kinematic controller is based on the transformation of the considered control switching mechanism. The kinematic controller is based on the transformation of the considered kinematics aa chained-form system, after that the recursive is used to derive the kinematic control law, based on dynamic controllers are designed all failure control law,into based on which, which, multiple multiple dynamic controllers are technique designed considering considering all possible possible failure kinematics into chained-form system, after that the recursive technique is used to derive the kinematic kinematics into a chained-form system, after that the recursive technique is used to derive the kinematic control law, based on which, multiple dynamic controllers are designed considering all possible cases. From these dynamic controllers, an appropriate one to the control signal cases. From these dynamic controllers, an appropriate one is is selected selected to generate generate the applied applied control failure signal control law, based on which, multiple dynamic controllers are designed considering all possible failure control law, these based on which, multiple using dynamic controllers are designed considering all possible failure cases. dynamic controllers, an appropriate is selected to generate the applied control signal by the control switching mechanism multiple reconstruction dynamic signals to desired by theFrom control switching mechanism using multiple one reconstruction dynamic signals to ensure ensure desired cases. From these dynamic controllers, an appropriate one is selected to generate the applied control signal cases. From these dynamic controllers, an appropriate one is selected todynamic generate the applied control signal by the control switching mechanism using multiple reconstruction signals to ensure desired system performance. Simulated fault scenarios are presented to illustrate the performance of the proposed system performance. Simulated fault scenarios are presented to illustrate the performance of the proposed by the control switching mechanism using multiple reconstruction dynamic signals to ensure desired by the control switching mechanism using multiple reconstruction dynamic signals to ensure desired system performance. Simulated fault scenarios are presented to illustrate the performance of the proposed approach. approach. system performance. system performance. Simulated Simulated fault fault scenarios scenarios are are presented presented to to illustrate illustrate the the performance performance of of the the proposed proposed approach. approach. Keywords: Fault tolerant control, Mobile robot, Fixed link, Failure compensation. © 2017, IFAC (International Federation of Automatic Control) Hostingcompensation. by Elsevier Ltd. All rights reserved. Keywords: Fault tolerant control, Mobile robot, Fixed link, Failure approach. Keywords: Fault control, Mobile robot, Fixed link, Failure compensation. Keywords: Fault tolerant tolerant compensation. Keywords:1. tolerant control, control, Mobile Mobile robot, robot, Fixed Fixed link, link, Failure Failure compensation. y 1.Fault INTRODUCTION Robot 1 y INTRODUCTION Robot 1 2b 2b y 1. INTRODUCTION Robot 1 a  2b y 1. INTRODUCTION a Robot 1  The demand for Wheeled Mobile Robots (WMR) is increased y 1. INTRODUCTION The demand for Wheeled Mobile Robots (WMR) is increased Robot 2 Robot 1 2b a Robot 2  2b  a a The demand for Wheeled Wheeled Mobile Robots useful (WMR)to isoperate increased  in many many places, which are are particularly useful tois operate in a Robot 2 in places, which particularly in The demand for Mobile Robots (WMR) increased a   Robot 2 C The demand for Wheeled Mobile Robots (WMR) isnuclear increased  a  in places, are to operate in C harsh environments resulting from for useful instance or a Robot 2 P harsh environments resulting from for instance  in many many places, which which are particularly particularly useful to nuclear operate or in P C  a in many places, which are particularly useful to operate in harsh environments resulting from for instance nuclear or chemical accident. In Inresulting such harsh harsh environments where the the  P C chemical accident. such where harsh environments environments from environments for instance instance nuclear nuclear or  P P C  harsh resulting from for or  P chemical such harsh P  C robots are sent, probability of wheel failures is C robots areaccident. sent, the theIn probability of environments wheel motor motor where failuresthe is chemical accident. In such harsh environments where the  2r  P chemical accident. In such harsh environments where the Robot 3 C  2r  robots are sent, the probability of wheel motor failures is P Robot 3 increased. Such failure will result in stopping the robot and 𝑑𝑑 increased. will resultofinwheel stopping the failures robot and C robots are are Such sent, failure the probability probability motor is  2r P 𝑑𝑑11 𝑎𝑎 Robot 3 𝜃𝜃 C robots sent, the of wheel motor failures is 3 2r 𝑎𝑎 Robot 3 𝜃𝜃3 increased. Such failure will result in stopping the robot and will make the instruments it carries unusable. One solution to  𝑑𝑑 1 will make the instruments it carries unusable. One solution to 2r  increased. Such failure will result in stopping the robot and Robot 3 𝑑𝑑 𝜏𝜏3𝑙𝑙𝑎𝑎 1 𝜃𝜃3 𝜏𝜏3𝑙𝑙 increased. Such failure will result in stopping the solution robot and 𝑑𝑑1 𝑎𝑎 will make the instruments it carries unusable. One to  2r 𝜃𝜃3 overcome this situation is to physically link several robots, overcome thisinstruments situation isittocarries physically link One several robots, will make the unusable. solution to  2r 𝜃𝜃3 𝜏𝜏3𝑙𝑙𝑎𝑎 𝐶𝐶 𝑃𝑃3 will make the instruments ittocarries unusable. One solution to  𝜏𝜏 𝐶𝐶 overcome this is robots, 𝑃𝑃3 where some of them instruments and others to 2r y 𝜏𝜏3𝑙𝑙 where some of situation them carry carry the instrumentslink andseveral others help help to overcome this situation is the to physically physically link several robots, y 3𝑙𝑙 2r 𝐶𝐶 𝑃𝑃3 overcome this situation is to physically link several robots, 2r where some of them carry the instruments and others help to 𝐶𝐶 y move, in normal situation or in presence of faulty robots [Ma 𝑃𝑃3 move, in normal situation or in presence of faulty robots [Ma where some of them carry the instruments and others help to y 𝐶𝐶 2𝑏𝑏 𝑃𝑃 3 𝜏𝜏 3𝑟𝑟𝑟𝑟 2𝑏𝑏 where some of them carry the instruments and others help to y 𝜏𝜏 move, in normal situation or in presence of faulty robots [Ma 3 et al., 2016]. 2016]. Thissituation paper is is concerned concerned with of fault tolerant control et al., This paper with fault tolerant control move, in normal normal or in in presence presence faulty robots [Ma 2𝑏𝑏 𝜏𝜏 3𝑟𝑟 22𝑟𝑟𝑟𝑟 2𝑏𝑏 move, in situation or of faulty robots [Ma et al., 2016]. This paper is concerned fault tolerant control 𝜏𝜏 3𝑟𝑟 design for such such configuration with with three linked two-wheel xx O X 2𝑏𝑏 design for configuration with three linked two-wheel et al., 2016]. This paper is concerned with fault tolerant control O 𝜏𝜏 3𝑟𝑟 2𝑟𝑟 X et al., mobile 2016]. This paper is concerned with fault tolerant control 2𝑟𝑟 design for such configuration with three linked two-wheel drive robots. x O Figure X 1: Three Linked 2WD Mobile Robot. 𝑟𝑟 drive mobile robots. 2 design for such configuration with three linked two-wheel Figure 1: Three Linked 2WD Mobile Robot. x O X design for such configuration with three linked two-wheel x 1: Three Linked 2WD Mobile Robot.X O Figure drive mobile robots. There are a lot of research results on fault diagnosis and fault system overactuated. drive mobile robots. Figure 1: Three Linked 2WD Mobile Robot. Theremobile are a lot of research results on fault diagnosis and fault system overactuated. drive robots. Figure 1: Three Linked 2WD Mobile Robot. There aa lot research results on and fault system overactuated. tolerant different applications [Tao tolerant control for different applications [Tao et et al., al., 2004], For this thisoveractuated. configuration, if if one, one, two, two, three, three, even even four four in in some some There are arecontrol lot of offor research results on fault fault diagnosis diagnosis and2004], fault system For configuration, There are a lot of research results on fault diagnosis and fault system overactuated. tolerant for different applications et al., 2004], [Shen etcontrol al., 2013]. 2013]. For the the applications applications of[Tao mobile robots, For this configuration, configuration, ifsix one, two, three, three, evenare four in some some [Shen et al., For of mobile robots, aa For specific cases, of of the the if sixone, actuated wheels arefour faulty, the tolerant control for different applications [Tao et al., 2004], specific cases, actuated wheels faulty, the this two, even in tolerant control for different applications [Tao etinal., 2004],a For this configuration, if one, two, three, even four in some [Shen et al., 2013]. For the applications of mobile robots, fault tolerant control design method is presented [Rotondo specific cases, of the six actuated wheels are faulty, the fault tolerant control design method is presented in [Rotondo remaining actuated wheels can continue moving the three [Shen et al., 2013]. For the applications of mobile robots, a remaining actuated wheels can continue moving the three specific cases, of the six actuated wheels are faulty, the [Shen et al.,for 2013]. For thedrive applications of mobile robots, fault control design method is presented in [Rotondo cases, of isthe six actuated wheels are faulty, the et al.,tolerant 2014] four-wheel mobile robot; neural networka specific remaining actuated wheels can continue moving the three et al., 2014] for four-wheel drive mobile robot; neural network robots. This paper focused on developing an actuator failure fault tolerant control design method is presented in [Rotondo robots. Thisactuated paper is focused on developing an actuator failure remaining wheels can continue moving the three fault tolerant control designdrive method is presented in [Rotondo et al., 2014] for four-wheel mobile robot; neural network remaining actuated wheels can continue moving the three is used in [Skoundrianos E. and Tzafestas S., 2004] to robots. This paper paper is focused focused on developing developing an robots. actuator failure failure is used in for [Skoundrianos E. and Tzafestas S., 2004] to robots. compensation scheme for mobile et al., al., 2014] four-wheel drive drive mobile robot; neural neural network compensation scheme for three-linked three-linked mobile robots. This is on an actuator et 2014] for four-wheel mobile robot; network is used in [Skoundrianos E. and Tzafestas S., 2004] to robots. This paper is focused on developing an robots. actuator failure diagnose faults in motors for drive compensation scheme for three-linked mobile diagnose faults in wheels wheels and and motors for two-wheel two-wheel drive is used in [Skoundrianos E. and Tzafestas S., 2004] to The paper paper is is organized organized as follows. follows. In Section Section 2, the the kinematics kinematics scheme three-linked mobile is used infaults [Skoundrianos E. and Tzafestas S.,is 2004] to compensation The as In diagnose in two-wheel drive scheme for for three-linked mobile2,robots. robots. mobile hybrid fault adaptive mobile robot; and hybrid and fault motors adaptivefor control is designed designed diagnoserobot; faultsand in aawheels wheels and motors forcontrol two-wheel drive compensation The paper is organized as follows. In Section 2, the kinematics and dynamics of the system and its actuator failure diagnose faults in wheels and motors for two-wheel drive and dynamics of the system and its actuator failure The paper is organized as follows. In Section 2, the kinematics mobile robot; and aa hybrid fault control designed to accommodate partial faults and degradation for two-wheel The paper is organized as follows. In Section 2, the kinematics to accommodate partial faults andadaptive degradation foris two-wheel mobile robot; and hybrid fault adaptive control is designed and dynamics of the system and its actuator failure compensation problem are formulated. Section 3 is dedicated mobile robot; and a hybrid fault adaptive control is designed compensation problem are formulated. Section 3 is dedicated and dynamics of the system and its actuator failure to accommodate partial degradation for two-wheel drive mobile in et 2003]. and dynamics ofthethecontroller system and the its control actuator failure drive mobile robot robot in [Ji [Ji faults et al., al., and 2003]. to accommodate partial faults and degradation for two-wheel compensation problem are formulated. Section 3 is dedicated to the design of and switching to accommodate partial faults and degradation for two-wheel compensation to the design problem of the controller and the control switching are formulated. Section 33 is dedicated drive mobile robot in [Ji et al., 2003]. problem are formulated. Section is switching dedicated There are several several types of WMR, one often often used used is is the the twotwo- compensation drive mobile robot in [Ji et al., 2003]. to the design of the controller and the control There are types of WMR, one mechanism. Section 4 discusses the results from detailed drive mobile robot in [Ji et al., 2003]. mechanism. Section 4controller discusses and the the results fromswitching detailed to the of control There are of one used the to the design design of the the 4and controller and the control switching wheel (2WD) mobile robot simplicity, mechanism. Section discusses thethe results from detailed wheel drive (2WD)types mobile robot due due tooften simplicity, efficiency computer simulations illustrates proposed theoretical There drive are several several types of WMR, WMR, oneto often used is isefficiency the twotwo- mechanism. computer simulations and illustrates the proposed theoretical Section 4 discusses the results from detailed There are several types of WMR, one often used is the twowheel drive (2WD) mobile robot due simplicity, mechanism. Section 4and discusses thethe results from detailed and flexibility. The problem of 2WD mobile robot computer simulations illustrates proposed theoretical and flexibility. The control control problem ofto 2WD mobileefficiency robot has has computer developments. Conclusions are given in Section 5. wheel drive (2WD) mobile robot due to simplicity, efficiency developments. Conclusions are given the in Section 5. theoretical simulations and illustrates proposed wheel drive (2WD) mobile robotindue to2WD simplicity, efficiency and flexibility. The control problem of mobile robot has computer simulations and illustrates the proposed theoretical attracted considerable attention industry and research, and developments. Conclusions Conclusions are are given given in in Section Section 5. 5. attracted considerable attention in industry research, and flexibility. flexibility. The control control problem of 2WD 2WDand mobile robot and has developments. and The problem of mobile robot has 2. MODELLING PROBLEM attracted considerable attention in industry and research, Conclusions are givenAND in Section 5. various control schemes have developed [Fukao et al., 2. SYSTEM SYSTEM MODELLING AND PROBLEM various control schemes have been been developed [Fukao et and al., developments. attracted considerable attention in industry and research, and 2. SYSTEM SYSTEM MODELLING MODELLING AND PROBLEM PROBLEM attracted considerable attention in industry and [Fukao research, and FORMULATION various control schemes have been developed et al., 2000], [Huang et al., 2014] but without considering actuator FORMULATION 2. AND 2000], al., 2014] butbeen without considering actuator various[Huang control etschemes schemes have developed [Fukao et al., al., 2. SYSTEM MODELLING AND PROBLEM FORMULATION various control have been developed [Fukao et 2000], [Huang et al., 2014] but without considering actuator faults. Unlike 4WD mobile robots which have redundant FORMULATION faults. Unlike 4WD mobile robots which have redundant 2000], [Huang et al., 2014] but without considering actuator In this section, the actuator failure compensation problem is is FORMULATION In this section, the actuator failure compensation problem 2000], [Huang et al., 2014] butrobots without considering actuator faults. Unlike 4WD mobile have redundant actuators, 2WD robot mobile has nowhich redundant actuator, so In this section, section, the actuator failure compensation compensation problem is actuators, aa 2WD robot mobile has no redundant actuator, so faults. Unlike 4WD mobile robots which have redundant formulated for three differentially driven wheeled mobile formulated for three differentially driven wheeled mobile In this the actuator failure problem is faults. Unlike 4WD mobile robots which haveis redundant actuators, aa 2WD robot mobile has no redundant actuator, so that it becomes becomes uncontrollable when one motor lost. One One In this with section, the actuator failure compensation problem is formulated for three differentially driven wheeled mobile that it uncontrollable when one motor is lost. actuators, 2WD robot mobile has no redundant actuator, so robots fixed link. As shown in Fig. 1, the orientations of robots with fixed link. differentially As shown in Fig. 1, the orientations of formulated for three driven wheeled mobile actuators, a 2WD robot mobile hassuch no redundant actuator, so that it becomes motor is lost. possible solution to actuator fault is to formulated for three differentially driven wheeled mobile robots with with fixed link. As shown in Fig. 1, the orientations of possible solutionuncontrollable to overcome overcomewhen suchone actuator fault isOne to robots that it becomes uncontrollable when one motor is lost. One 2 and 3 are consistent with the physical links, but the 2 andfixed 3 arelink. consistent withinthe physical links, but the As shown Fig. 1, of that it becomes uncontrollable when onewhich motor isfault lost.isOne possible to such physically connect mobile robots with fixed As shown Fig.physical 1, the the orientations orientations of robots 2 and and 3robot arelink. consistent withinthe the links, but but the the physically connect several several mobile robots, robots, which makes makes the orientation of 1 is independent. possible solution solution to overcome overcome such actuator actuator fault the is to to robots orientation of robot 1 is independent. 2 3 are consistent with physical links, possible solution to overcome such actuator fault is to physically connect several mobile robots, which makes the robots 2 and 3 are consistent with the physical links, but the orientation of of robot 11 is is independent. physically connect connect several several mobile mobile robots, robots, which which makes makes the the orientation physically orientation of robot robot 1 is independent. independent. 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Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Al-Dujaili Ayad et al. / IFAC PapersOnLine 50-1 (2017) 13556–13562

For the 𝑖𝑖th (𝑖𝑖 = 1, 2,3) robot in Fig. 1: 𝑃𝑃𝑖𝑖 is the center between two actuated wheels, 𝐶𝐶𝑖𝑖 is the center of mass, 𝑎𝑎𝑖𝑖 is the distance between 𝑃𝑃𝑖𝑖 and 𝐶𝐶𝑖𝑖 , 𝑏𝑏𝑖𝑖 is half of the distance between two actuated wheels, 𝑟𝑟𝑖𝑖 is the radius of wheels, 𝜃𝜃𝑖𝑖 is the orientation of the robot, and 𝜏𝜏𝑖𝑖𝑖𝑖 and 𝜏𝜏𝑖𝑖𝑖𝑖 are the control torques applied to the left and right actuated wheels, respectively. In addition, 𝑑𝑑𝑖𝑖 is the distance between 𝑃𝑃1 , 𝑃𝑃2 , and 𝑃𝑃3 , frame 𝑂𝑂𝑂𝑂𝑂𝑂 is the inertial frame, (𝑥𝑥, 𝑦𝑦) denotes the coordinates of point 𝑃𝑃3 that is the middle point of Robot 3 rear wheels. The configuration of the system can be expressed with the generalized coordinate vector 𝑞𝑞 = [𝑥𝑥, 𝑦𝑦, 𝜃𝜃3 , 𝜃𝜃2 , 𝜃𝜃1 ] 𝑇𝑇 . 2.1 Kinematic Modelling

The main feature in kinematic models of wheeled mobile robots is the presence of nonholonomic constraints due to the rolling without slipping conditions between the wheels and the ground. The system constraints take the matrix form as follows where

𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃3 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃2 𝐴𝐴(𝑞𝑞) = [ 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃1

𝐴𝐴(𝑞𝑞)𝑞𝑞̇ = 0

− 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃3 − 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃2 − 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃1

(1)

0 −𝑑𝑑2 𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2 − 𝜃𝜃3 ) − 𝑑𝑑2 𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1 − 𝜃𝜃3 )

0 0 −𝑑𝑑1 𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1 − 𝜃𝜃2 )

0 0] 0

(2)

(3)

where 𝜂𝜂 = [𝑣𝑣3 𝜔𝜔1 ]𝑇𝑇 , 𝑣𝑣3 is the linear velocity of robot 3, and 𝜔𝜔1 the rotational velocity of robot 1. and the matrix 𝑆𝑆(𝑞𝑞) 𝑆𝑆(𝑞𝑞) = [

𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃3

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𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃2 − 𝜃𝜃3 ) 0

𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃1 − 𝜃𝜃2 ) / 𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2 − 𝜃𝜃3 ) 0 0

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(4)

Moreover, we can show the following, which describes the natural orthogonal complement property

2.2 Dynamic Modelling

𝑆𝑆(𝑞𝑞)𝑇𝑇 𝐴𝐴(𝑞𝑞)𝑇𝑇 = 0

(5)

The dynamic model of three mobile robots with fixed links can be obtained using the Lagrange method in matrix form as 𝑀𝑀(𝑞𝑞)𝑞𝑞̈ + 𝐶𝐶(𝑞𝑞, 𝑞𝑞̇ )𝑞𝑞̇ = 𝐵𝐵(𝑞𝑞)𝜏𝜏 + 𝐴𝐴(𝑞𝑞)𝑇𝑇 𝜆𝜆

(6)

where 𝑀𝑀(𝑞𝑞) ∈ ℜ5×5 is the system inertia matrix, 𝐶𝐶(𝑞𝑞, 𝑞𝑞̇ ) ∈ ℜ5×5 is the centripetal and Coriolis forces vector, 𝐵𝐵(𝑞𝑞) ∈ ℜ5×6 is the input transformation matrix, 𝜏𝜏 ∈ ℜ6×1 is the vector of control torques. 𝐴𝐴(𝑞𝑞) ∈ ℜ3×5 is the system constraint matrix, 𝜆𝜆 ∈ ℜ3×1 Lagrange multipliers vector. In order to remove Lagrange multipliers, substituting from the derivative of (3) into (6), and multiplying by 𝑆𝑆(𝑞𝑞)𝑇𝑇 , the system dynamic equation takes the following form ̅ (𝑞𝑞)𝜂𝜂̇ + 𝐶𝐶̅ (𝑞𝑞, 𝑞𝑞̇ )𝜂𝜂 = 𝐵𝐵̅ (𝑞𝑞)𝜏𝜏 𝑀𝑀

Multiple Model Control Scheme Three robots with fixed link

Control Switching Mechanism Switching signal Multiple dynamic Controllers Reference robot 𝑥𝑥𝑑𝑑 , 𝑦𝑦𝑑𝑑 , 𝜃𝜃3𝑑𝑑 , 𝜃𝜃2𝑑𝑑 , 𝜃𝜃1𝑑𝑑

𝑣𝑣3𝑐𝑐 , 𝜔𝜔1𝑐𝑐

Kinematic Controller

Actuator failures 𝑢𝑢

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 Dynamics

𝑥𝑥, 𝑦𝑦, 𝜃𝜃3 , 𝜃𝜃2 , 𝜃𝜃1

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Kinematics

System output

Figure 2: Block Diagram of the Multiple Model Actuator Failure Compensation Control Scheme. where ̅ (𝑞𝑞) = 𝑆𝑆(𝑞𝑞)𝑇𝑇 𝑀𝑀(𝑞𝑞)𝑆𝑆(𝑞𝑞), 𝑀𝑀 ̅ 𝐶𝐶 (𝑞𝑞, 𝑞𝑞̇ ) = 𝑆𝑆(𝑞𝑞)𝑇𝑇 𝑀𝑀(𝑞𝑞)𝑆𝑆̇(𝑞𝑞) + 𝑆𝑆(𝑞𝑞)𝑇𝑇 𝐶𝐶(𝑞𝑞, 𝑞𝑞̇ )𝑆𝑆(𝑞𝑞), (8) 𝐵𝐵̅ (𝑞𝑞) = 𝑆𝑆(𝑞𝑞)𝑇𝑇 𝐵𝐵(𝑞𝑞)

2.3 Actuator failure model

Actuator failures may occur in some motors of the six differentially driven wheels of the three robots system. An actuator failure results in losing the motor power which makes it impossible to apply the control torque corresponding to the control signal. This type of actuator failure is modelled as

The kinematic equations of three mobile robots with fixed links as shown in Fig. 1 are given by 𝑞𝑞̇ = 𝑆𝑆(𝑞𝑞)𝜂𝜂

13557

(7)

𝜏𝜏(𝑡𝑡) = 𝜎𝜎(𝑡𝑡)𝑢𝑢(𝑡𝑡)

(9)

where τ = [𝜏𝜏1𝑟𝑟 , 𝜏𝜏1𝑙𝑙 , 𝜏𝜏2𝑟𝑟 , 𝜏𝜏2𝑙𝑙 , 𝜏𝜏3𝑟𝑟 , 𝜏𝜏3𝑙𝑙 ]T is the torque vector generated by the motors, 𝑢𝑢 = [𝑢𝑢1r , 𝑢𝑢1l , 𝑢𝑢2r , 𝑢𝑢2l , 𝑢𝑢3r , 𝑢𝑢3l ]T is the control signal vector to be designed, and 𝜎𝜎(𝑡𝑡) = diag{𝜎𝜎1𝑟𝑟 , 𝜎𝜎1𝑙𝑙 , 𝜎𝜎2𝑟𝑟 , 𝜎𝜎2𝑙𝑙 , 𝜎𝜎3𝑟𝑟 , 𝜎𝜎3𝑙𝑙 } is the uncertain failure pattern matrix such that 0, if the jth motor fails 𝜎𝜎𝑗𝑗 (𝑡𝑡) = { (10) 1, otherwise Assumption 1: for the failure compensation design, the following actuation redundancy condition needs to be satisfied 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝐵𝐵̅ 𝜎𝜎) = 2

(11)

for all possible failure matrix 𝜎𝜎. This condition means that there are enough operative motors to control 𝜂𝜂 = [𝑣𝑣3 𝜔𝜔1 ]𝑇𝑇 for the three robots with any motor failure. 2.4 Control Objective

Our goal is to design feedback control signal 𝑢𝑢(𝑡𝑡) for the system modelled by equations (3) and (7), to guarantee the overall closed-loop system stability and asymptotic tracking in the presence of actuator failures. 2.5 Design issue A multiple model control scheme covering all possible cases of 𝜎𝜎(𝑡𝑡) is proposed to achieve the control objective. The structure of this control scheme is shown in Fig. 2. Our failure compensation control design consists of three steps Step 1: For the kinematic equations in (3), the linear velocity 𝑣𝑣3 and the angular velocity 𝜔𝜔1 can be seen as intermediate control signals. So, we first design a kinematic control law 𝜂𝜂𝑐𝑐 = [𝑣𝑣3𝑐𝑐 𝜔𝜔1𝑐𝑐 ]𝑇𝑇 such that when it is applied, the desired system performance can be ensured. Step 2: Then, the multiple controllers are designed, each of which is designed using one possible failure pattern matrix. If

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the failure pattern which is used in the controller is consistent with the actual one, then the applied control signal can ensure 𝜂𝜂 ⟶ 𝜂𝜂𝑐𝑐 and desired system performance. Step 3: Finally, the control switching mechanism is used to select the appropriate controller in order to generate the applied control signal 𝑢𝑢(𝑡𝑡). 3. MULTIPLE MODEL ACTUATOR FAILURE COMPENSATION SCHEME

3.1 Kinematic Controller Design The design of the kinematic controller starts from converting the kinematic model into the chained form. After that a recursive technique is used which appears to be an extension of the popular integrator backstepping idea to derive the kinematic control law for trajectory tracking. According to [Sørdalen O., 1993], the kinematic model of three robots with fixed links is transformed into chained form equations using the following change of coordinates 𝜉𝜉1 = 𝑥𝑥,

+(1/(𝑑𝑑22 𝑐𝑐𝑐𝑐𝑐𝑐 4 𝜃𝜃3 )) 𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃2 − 𝜃𝜃3 ) [3 𝑡𝑡𝑡𝑡𝑡𝑡 𝜃𝜃3 𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃2

(12)

Together with the following input transformations 𝛼𝛼1 = 𝑣𝑣3 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃3 , 𝛼𝛼2 = (1/𝑑𝑑1 𝑑𝑑2 )𝑠𝑠𝑠𝑠𝑠𝑠 4 𝜃𝜃3 𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃1 − 𝜃𝜃2 )𝑠𝑠𝑠𝑠𝑠𝑠 4 (𝜃𝜃2

−𝜃𝜃3 )[(1/𝑑𝑑1 )(3𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃2 − 𝜃𝜃3 )𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃1 − 𝜃𝜃2 )

+𝑠𝑠𝑠𝑠𝑠𝑠 2 (𝜃𝜃1 − 𝜃𝜃2 )) + (1/𝑑𝑑2 )(6𝑡𝑡𝑡𝑡𝑡𝑡𝜃𝜃3 𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃2 −2𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃2 − 𝜃𝜃3 )𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃2 − 𝜃𝜃3 ) − 𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃2

−𝜃𝜃3 )]𝑣𝑣3 + (1/𝑑𝑑1 𝑑𝑑2 )𝑠𝑠𝑠𝑠𝑠𝑠 4 𝜃𝜃3 𝑠𝑠𝑠𝑠𝑠𝑠 2 (𝜃𝜃1 −𝜃𝜃2 )𝑠𝑠𝑠𝑠𝑠𝑠 3 (𝜃𝜃2 − 𝜃𝜃3 )𝜔𝜔1

(13)

This transformation is a local diffeomorphism around configurations for which the angle of each robot and the angles between each adjacent robot are not equal to (𝜋𝜋/2). In these new coordinates, the kinematic model of the system has the following chained form 𝜉𝜉1̇ = 𝛼𝛼1 , 𝜉𝜉2̇ = 𝛼𝛼2 , 𝜉𝜉3̇ = 𝜉𝜉2 𝛼𝛼1 , 𝜉𝜉4̇ = 𝜉𝜉3 𝛼𝛼1 ,

𝜉𝜉5̇ = 𝜉𝜉4 𝛼𝛼1

(14)

where 𝜉𝜉 = (𝜉𝜉1 , 𝜉𝜉2 , … , 𝜉𝜉5 ) is the state and 𝛼𝛼1 , 𝛼𝛼2 are the two control inputs. The desired trajectory 𝜉𝜉𝑑𝑑 = (𝜉𝜉1𝑑𝑑 , 𝜉𝜉2𝑑𝑑 , … , 𝜉𝜉5𝑑𝑑 ) can be generated after the change of coordinates of the system in equation (12) by the following equations ̇ = 𝛼𝛼2𝑑𝑑 , 𝜉𝜉3𝑑𝑑 ̇ = 𝜉𝜉2𝑑𝑑 𝛼𝛼1𝑑𝑑 , ̇ = 𝛼𝛼1𝑑𝑑 , 𝜉𝜉2𝑑𝑑 𝜉𝜉1𝑑𝑑 ̇ = 𝜉𝜉3𝑑𝑑 𝛼𝛼1𝑑𝑑 , 𝜉𝜉4𝑑𝑑

̇ = 𝜉𝜉4𝑑𝑑 𝛼𝛼1𝑑𝑑 𝜉𝜉5𝑑𝑑

(16) The object is to find the kinematic control law such that the system in equation (14) track the desired trajectory in equation (15). Toward this end, the following tracking control problems will be addressed in this paper. Definition 1: The tracking control problem will be said to be semi globally solvable [Jiang Z. and Nijmeijer H., 1999] for system (14) if, given any compact set 𝑆𝑆 ∈ ℝ5 containing the origin, we can design appropriate continuous time-varying state feedback controllers of the form 𝛼𝛼1 = 𝜇𝜇1 (𝜉𝜉) and 𝛼𝛼2 = 𝜇𝜇2 (𝜉𝜉), such that for any initial tracking error 𝜉𝜉𝑒𝑒 (0) = 𝜉𝜉(0) − 𝜉𝜉𝑑𝑑 (0) in 𝑆𝑆, all the signals of the closed loop system (16), 𝛼𝛼1 and 𝛼𝛼2 are uniformly bounded over [0, ∞). Furthermore (17) 𝑙𝑙𝑙𝑙𝑙𝑙 |𝜉𝜉(𝑡𝑡) − 𝜉𝜉𝑑𝑑 (𝑡𝑡)| = 0 3.1.1 Kinematic control law

−𝜃𝜃3 ) − 𝑠𝑠𝑠𝑠𝑠𝑠 2 (𝜃𝜃2 − 𝜃𝜃3 )], 𝜉𝜉3 = 𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃2 − 𝜃𝜃3 )/𝑑𝑑2 𝑐𝑐𝑐𝑐𝑐𝑐 3 𝜃𝜃3 , 𝜉𝜉5 = 𝑦𝑦

̇ = 𝜉𝜉3𝑒𝑒 𝛼𝛼1𝑑𝑑 + 𝜉𝜉3 (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 ) 𝜉𝜉4𝑒𝑒 ̇ = 𝜉𝜉4𝑒𝑒 𝛼𝛼1𝑑𝑑 + 𝜉𝜉4 (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 ) 𝜉𝜉5𝑒𝑒

𝑡𝑡⟶+∞

𝜉𝜉2 = [𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃1 − 𝜃𝜃2 ) /(𝑑𝑑1 𝑑𝑑2 𝑐𝑐𝑐𝑐𝑐𝑐 4 𝜃𝜃3 𝑐𝑐𝑐𝑐𝑐𝑐 3 (𝜃𝜃2 − 𝜃𝜃3 ))] 𝜉𝜉4 = 𝑡𝑡𝑡𝑡𝑡𝑡𝜃𝜃3 ,

̇ = 𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 𝜉𝜉1𝑒𝑒 ̇ = 𝛼𝛼2 − 𝛼𝛼2𝑑𝑑 𝜉𝜉2𝑒𝑒 ̇ = 𝜉𝜉2𝑒𝑒 𝛼𝛼1𝑑𝑑 + 𝜉𝜉2 (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 ) 𝜉𝜉3𝑒𝑒

The main purpose of this section is to design the kinematic control law 𝛼𝛼1 and 𝛼𝛼2 cited above for trajectory tracking of the treated system. First, an error dynamics is considered for the tracking problem. Second, a change of coordinate is proposed and system (16) is rearranged into a triangular form as follows 𝑧𝑧1 = 𝜉𝜉5𝑒𝑒 − [𝜉𝜉4𝑒𝑒 + 𝜉𝜉4𝑑𝑑 ]𝜉𝜉1𝑒𝑒 𝑧𝑧2 = 𝜉𝜉4𝑒𝑒 − [𝜉𝜉3𝑒𝑒 + 𝜉𝜉3𝑑𝑑 ]𝜉𝜉1𝑒𝑒 𝑧𝑧3 = 𝜉𝜉3𝑒𝑒 − [𝜉𝜉2𝑒𝑒 + 𝜉𝜉2𝑑𝑑 ]𝜉𝜉1𝑒𝑒 𝑧𝑧4 = 𝜉𝜉2𝑒𝑒 (18) 𝑧𝑧5 = 𝜉𝜉1𝑒𝑒 The tracking error dynamics is transformed into the following system 𝑧𝑧̇1 = 𝛼𝛼1𝑑𝑑 𝑧𝑧2 − 𝜉𝜉3 (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 )𝑧𝑧5 𝑧𝑧̇2 = 𝛼𝛼1𝑑𝑑 𝑧𝑧3 − 𝜉𝜉2 (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 )𝑧𝑧5 𝑧𝑧̇3 = 𝛼𝛼1𝑑𝑑 𝑧𝑧4 − 𝛼𝛼2 𝑧𝑧5 𝑧𝑧̇4 = 𝛼𝛼2 − 𝛼𝛼2𝑑𝑑 𝑧𝑧̇5 = 𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 (19) The algorithm which is used in this paper was proposed by [Jiang Z. and Nijmeijer H., 1999]. Let us formulate the backstepping design scheme for the system (19). In this aim, we need to consider an intermediate virtual control functions 𝛽𝛽𝑖𝑖 (1 ≤ 𝑖𝑖 ≤ 3), which would make the subsystem stabilizable. Step 1: Treating the 𝑧𝑧1 subsystem of system (19) (20) 𝑧𝑧̇1 = 𝛼𝛼1𝑑𝑑 𝑧𝑧2 − 𝜉𝜉3 (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 )𝑧𝑧5 Suppose the variable 𝑧𝑧2 is a virtual control input and the variables 𝛼𝛼1𝑑𝑑 and 𝑧𝑧5 are time-varying function. Set 𝑧𝑧̅1 = 𝑧𝑧1 . Along the solutions of system (19), the time derivatives of the positive definite and proper function

where 𝛼𝛼𝑑𝑑 = (𝛼𝛼1𝑑𝑑 , 𝛼𝛼2𝑑𝑑 ) is the reference control inputs. For later use, denote the tracking error as 𝜉𝜉𝑒𝑒 = 𝜉𝜉 − 𝜉𝜉𝑑𝑑 . It is directly established that the 𝜉𝜉𝑒𝑒 dynamics errors satisfy the following differential equations

𝑉𝑉1 (𝑧𝑧̅1 ) = 𝑧𝑧̅12 1

(15)

2

𝑉𝑉1̇ = 𝛼𝛼1𝑑𝑑 𝑧𝑧̅1 𝑧𝑧2 − 𝑧𝑧̅1 𝜉𝜉3 (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 )𝑧𝑧5 Choosing 𝛽𝛽1 (𝑧𝑧1 ) as follows 𝛽𝛽1 (𝑧𝑧1 )=𝑧𝑧2 + 𝑧𝑧̅2

satisfies

14100

(21) (22) (23)

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With the 𝛽𝛽1 (𝑧𝑧1 ) = 0, is a stabilizing function for the system (20) whenever 𝑧𝑧5 = 0. It follows, equation (22) becomes (24) 𝑉𝑉1̇ = 𝛼𝛼1𝑑𝑑 𝑧𝑧̅1 𝑧𝑧̅2 − 𝑧𝑧̅1 𝜉𝜉3 (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 )𝑧𝑧5 Step 2: We apply the same procedure at step 1 for the (𝑧𝑧1 , 𝑧𝑧2 ) subsystem of system (19) with 𝑧𝑧3 viewed as a virtual control input. To achieve this goal, let us choose the function 1 (25) 𝑉𝑉2 (𝑧𝑧̅1 , 𝑧𝑧̅2 ) = 𝑉𝑉1 (𝑧𝑧̅1 ) + 𝑧𝑧̅22 2 Differentiating the function 𝑉𝑉2 along the solutions of system (19) yields 𝜕𝜕𝛽𝛽 𝑉𝑉2̇ = − [𝑧𝑧̅1 𝜉𝜉3 + 𝑧𝑧̅2 𝜉𝜉2 − 𝑧𝑧̅2 1 𝜉𝜉3 ] (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 )𝑧𝑧5 𝜕𝜕𝑧𝑧1 𝜕𝜕𝛽𝛽1

+𝛼𝛼1𝑑𝑑 𝑧𝑧̅2 [𝑧𝑧̅1 +𝑧𝑧3 −

Choosing

𝑧𝑧 ] 𝜕𝜕𝑧𝑧1 2

𝛽𝛽2 (𝑧𝑧1 , 𝑧𝑧2 ) = −𝑧𝑧̅1 +

(26)

𝑧𝑧 𝜕𝜕𝑧𝑧1 2

𝜕𝜕𝛽𝛽1

(27)

𝑧𝑧̅3 = 𝑧𝑧3 − 𝛽𝛽2 (𝑧𝑧1 , 𝑧𝑧2 )

(28)

It follows, equation (26) becomes 𝜕𝜕𝛽𝛽 𝑉𝑉2̇ = − [𝑧𝑧̅1 𝜉𝜉3 + 𝑧𝑧̅2 𝜉𝜉2 − 𝑧𝑧̅2 1 𝜉𝜉3 ] (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 )𝑧𝑧5 𝜕𝜕𝑧𝑧1

+𝛼𝛼1𝑑𝑑 𝑧𝑧̅2 𝑧𝑧̅3

1

satisfies

2

(29)

(30)

𝜕𝜕𝛽𝛽 𝜕𝜕𝛽𝛽 𝑉𝑉3̇ = − [𝑧𝑧̅1 𝜉𝜉3 + 𝑧𝑧̅2 𝜉𝜉2 − (𝑧𝑧̅2 1 𝜉𝜉3 + 𝑧𝑧̅3 2 𝜉𝜉3

+𝑧𝑧̅3

𝜉𝜉 )] (𝛼𝛼1 𝜕𝜕𝑧𝑧2 2

𝜕𝜕𝛽𝛽2

𝜕𝜕𝑧𝑧1

+𝛼𝛼1𝑑𝑑 𝑧𝑧̅3 [𝑧𝑧̅2 + 𝑧𝑧4 − (

Choosing

𝜕𝜕𝑧𝑧1

− 𝛼𝛼1𝑑𝑑 )𝑧𝑧5 − 𝑧𝑧̅3 𝑧𝑧5 𝛼𝛼2 𝑧𝑧2 +

𝜕𝜕𝛽𝛽2 𝜕𝜕𝑧𝑧1

𝛽𝛽3 (𝑧𝑧1 , 𝑧𝑧2 , 𝑧𝑧3 ) = −𝑧𝑧̅2 +

𝑧𝑧 )] 𝜕𝜕𝑧𝑧2 3

𝜕𝜕𝛽𝛽2

𝑧𝑧2 +

𝜕𝜕𝛽𝛽2 𝜕𝜕𝑧𝑧1

𝑧𝑧̅4 = 𝑧𝑧4 − 𝛽𝛽3 (𝑧𝑧1 , 𝑧𝑧2 , 𝑧𝑧3 )

𝑧𝑧 𝜕𝜕𝑧𝑧2 3

𝜕𝜕𝛽𝛽2

+𝑧𝑧̅3

𝜉𝜉 𝜕𝜕𝑧𝑧1 3

𝜕𝜕𝛽𝛽2

+ 𝑧𝑧̅3

𝜉𝜉 )] (𝛼𝛼1 𝜕𝜕𝑧𝑧2 2

𝜕𝜕𝛽𝛽2

− 𝛼𝛼1𝑑𝑑 )𝑧𝑧5

(31)

(33) 𝜕𝜕𝑧𝑧1

(34)

Step 4: Let us consider the (𝑧𝑧1 , 𝑧𝑧2 , 𝑧𝑧3 , 𝑧𝑧4 ) subsystem of (19) and let us choose the function 1 1 𝑉𝑉4 (𝑧𝑧̅1 , 𝑧𝑧̅2 , 𝑧𝑧̅3 , 𝑧𝑧̅4 ) = 𝑉𝑉3 (𝑧𝑧̅1 , 𝑧𝑧̅2 , 𝑧𝑧̅3 ) + 𝑧𝑧̅42 = ∑4𝑖𝑖=1 𝑧𝑧̅𝑖𝑖2 (35) 2

2

According to (19) and (34), the time derivative of 𝑉𝑉4 along the solutions of (19) satisfies 𝜕𝜕𝛽𝛽 𝜕𝜕𝛽𝛽 𝑉𝑉4̇ = 𝑧𝑧̅4 [𝛼𝛼1𝑑𝑑 𝑧𝑧̅3 + 𝛼𝛼2 − 𝛼𝛼2𝑑𝑑 − ( 3 𝛼𝛼1𝑑𝑑 𝑧𝑧2 + 3 𝛼𝛼1𝑑𝑑 𝑧𝑧3 +

𝜕𝜕𝛽𝛽3 𝜕𝜕𝑧𝑧3

𝛼𝛼1𝑑𝑑 𝑧𝑧4 )] − [𝑧𝑧̅3 − 𝑧𝑧̅4

+𝑧𝑧̅2 𝜉𝜉2 − (𝑧𝑧̅2 − (𝑧𝑧̅4

𝜉𝜉 𝜕𝜕𝑧𝑧1 3

𝜕𝜕𝛽𝛽3

𝜉𝜉 𝜕𝜕𝑧𝑧1 3

𝜕𝜕𝛽𝛽1

+ 𝑧𝑧̅4

+ 𝑧𝑧̅3

𝜕𝜕𝑧𝑧1 𝜕𝜕𝛽𝛽3

𝜕𝜕𝑧𝑧3 𝜕𝜕𝛽𝛽2

] 𝛼𝛼2 𝑧𝑧5 − [𝑧𝑧̅1 𝜉𝜉3

𝜉𝜉 )] (𝛼𝛼1 𝜕𝜕𝑧𝑧2 2

𝜕𝜕𝛽𝛽3

𝜕𝜕𝑧𝑧2

𝜉𝜉 𝜕𝜕𝑧𝑧1 3

+ 𝑧𝑧̅3

𝜉𝜉 ) 𝜕𝜕𝑧𝑧2 2

𝜕𝜕𝛽𝛽2

− 𝛼𝛼1𝑑𝑑 )𝑧𝑧5

Applying the following control law 𝜕𝜕𝛽𝛽 𝛼𝛼2 = 𝛼𝛼2𝑑𝑑 − 𝑐𝑐4 𝑧𝑧̅4 − 𝛼𝛼1𝑑𝑑 𝑧𝑧̅3 + 3 𝛼𝛼1𝑑𝑑 𝑧𝑧2 +

𝜕𝜕𝛽𝛽3 𝜕𝜕𝑧𝑧2

with 𝑐𝑐4 > 0, we obtain:

𝛼𝛼1𝑑𝑑 𝑧𝑧3 +

𝜕𝜕𝛽𝛽3 𝜕𝜕𝑧𝑧3

𝜕𝜕𝑧𝑧1

𝛼𝛼1𝑑𝑑 𝑧𝑧4

𝜕𝜕𝑧𝑧3

where ∆1 (𝑧𝑧1 , 𝑧𝑧2 , 𝑧𝑧3 , 𝑧𝑧4 , 𝜉𝜉̃𝑑𝑑 ) is a smooth function given by ∆1 = [𝑧𝑧̅1 𝜉𝜉3 + 𝑧𝑧̅2 𝜉𝜉2 − (𝑧𝑧̅2 +𝑧𝑧̅3

𝜉𝜉 ) 𝜕𝜕𝑧𝑧2 2

𝜕𝜕𝛽𝛽2

−

𝜕𝜕𝛽𝛽 𝜉𝜉 + 𝑧𝑧̅3 2 𝜉𝜉3 𝜕𝜕𝑧𝑧1 3 𝜕𝜕𝑧𝑧1 𝜕𝜕𝛽𝛽3 𝜕𝜕𝛽𝛽3 (𝑧𝑧̅4 𝜉𝜉 + 𝑧𝑧̅4 𝜉𝜉 )] 𝜕𝜕𝑧𝑧 3 𝜕𝜕𝑧𝑧 2 𝜕𝜕𝛽𝛽1 1

2

(39)

Step 5: In the final step, the objective is to design 𝛼𝛼1 such that 𝑧𝑧̅1 , 𝑧𝑧̅2 , 𝑧𝑧̅3 , 𝑧𝑧̅4 and 𝑧𝑧5 converge to zero. Toward this end, consider the positive definite and proper function 𝑉𝑉5 which serves as Lyapunov function candidate for the complete system (19) 1 1 1 1 𝜅𝜅 (40) 𝑉𝑉5 (𝑧𝑧̅) = 𝑧𝑧̅12 + 𝑧𝑧̅22 + 𝑧𝑧̅32 + 𝑧𝑧̅42 + 𝑧𝑧52 2

2

2

2

2

where 𝜅𝜅 > 0 is a design parameter. Differentiating 𝑉𝑉5 along the solutions of (19) gives 𝑉𝑉5̇ = −𝑐𝑐4 𝑧𝑧̅42 + 𝑧𝑧5 [(𝜅𝜅 − ∆1 )(𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 ) −(𝑧𝑧̅3 − 𝑧𝑧̅4

𝜕𝜕𝛽𝛽3 𝜕𝜕𝑧𝑧3

)𝛼𝛼2 ]

𝛼𝛼1 = 𝛼𝛼1𝑑𝑑 + (𝜅𝜅 − ∆1 )−1 [−𝑐𝑐5 𝑧𝑧5 + (𝑧𝑧̅3 − 𝑧𝑧̅4

with 𝑐𝑐5 > 0, we obtain the 𝑉𝑉5̇ (𝑧𝑧̅) = −𝑐𝑐4 𝑧𝑧̅42 − 𝑐𝑐5 𝑧𝑧52

𝜕𝜕𝛽𝛽3 𝜕𝜕𝑧𝑧3

)𝛼𝛼2 ]

(41) (42)

(43) ̇ One can remark that 𝑉𝑉5 is negative which proves the stability of the controlled system (19). The described five steps permit us to find kinematic control laws 𝛼𝛼1 and 𝛼𝛼2 which there are assumed as signals for dynamic controller in the next section. 3.1.2 Performance analysis

(32)

It follows, equation (31) becomes 𝜕𝜕𝛽𝛽 𝑉𝑉3̇ = 𝑧𝑧̅4 𝑧𝑧̅3 𝛼𝛼1𝑑𝑑 − 𝑧𝑧̅3 𝑧𝑧5 𝛼𝛼2 − [𝑧𝑧̅1 𝜉𝜉3 + 𝑧𝑧̅2 𝜉𝜉2 − (𝑧𝑧̅2 1 𝜉𝜉3

𝜕𝜕𝛽𝛽 𝑉𝑉4̇ = −𝑐𝑐4 𝑧𝑧̅42 − (𝑧𝑧̅3 − 𝑧𝑧̅4 3) 𝛼𝛼2 𝑧𝑧5 − ∆1 (𝛼𝛼1 − 𝛼𝛼1𝑑𝑑 )𝑧𝑧5 (38)

By choosing the following control law

Step 3: At the same way on preceding step, we consider now the (𝑧𝑧1 , 𝑧𝑧2 , 𝑧𝑧3 ) subsystem of system (19), the time derivative of the following function 𝑉𝑉3 (𝑧𝑧̅1 , 𝑧𝑧̅2 , 𝑧𝑧̅3 ) = 𝑉𝑉2 (𝑧𝑧̅1 , 𝑧𝑧̅2 ) + 𝑧𝑧̅32

13559

(36)

(37)

In this section, we prove the tracking control problem for system (14) Proposition 1: Assume that 𝜉𝜉𝑖𝑖𝑖𝑖 (2 ≤ 𝑖𝑖 ≤ 4), 𝛼𝛼𝑑𝑑 , and 𝛼𝛼̇ 1𝑑𝑑 are bounded over [0, ∞). Then, the tracking control problem is semi globally solvable for system (14). This means that using coordinates transformation (18) and applying the design procedure described in the above section to system (19), given any compact neighborhood 𝑆𝑆 of the origin in ℝ5 , we can find a sufficiently large 𝜅𝜅 > 0 so that, for any initial tracking conditions 𝜉𝜉𝑒𝑒 (0) in 𝑆𝑆, all the solutions of the closed loop system, (16), (37), and (42) are uniformly bounded. Furthermore, if 𝛼𝛼1𝑑𝑑 (𝑡𝑡) does not converge to zero, we have 𝑙𝑙𝑙𝑙𝑙𝑙 |𝜉𝜉𝑒𝑒 (𝑡𝑡)| = 0. 𝑡𝑡⟶+∞

The proof of proposition 1 can be found in [Khalil H., 1996] and [Jiang Z. and Nijmeijer H., 1999] and is omitted due to space limitation. From equation (13), we can obtain 𝛼𝛼 = 𝑇𝑇𝛼𝛼 𝜂𝜂 (44) where 𝑇𝑇𝛼𝛼12 = 0, 𝑇𝑇𝛼𝛼11 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃3 , 4 𝑇𝑇𝛼𝛼21 = (1/𝑑𝑑1 𝑑𝑑2 )𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃3 𝑡𝑡𝑡𝑡𝑛𝑛(𝜃𝜃1 − 𝜃𝜃2 )𝑠𝑠𝑠𝑠𝑠𝑠 4 (𝜃𝜃2 −𝜃𝜃3 )[(1/𝑑𝑑1 )(3𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃2 − 𝜃𝜃3 )𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃1 − 𝜃𝜃2 ) +𝑠𝑠𝑠𝑠𝑠𝑠 2 (𝜃𝜃1 − 𝜃𝜃2 )) + (1/𝑑𝑑2 )(6𝑡𝑡𝑡𝑡𝑡𝑡𝜃𝜃3 𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃2 −𝜃𝜃3 ) − 2𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃2 − 𝜃𝜃3 )𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃2 − 𝜃𝜃3 ) −𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃2 − 𝜃𝜃3 ))], 𝑇𝑇𝛼𝛼22 = (1/𝑑𝑑1 𝑑𝑑2 )𝑠𝑠𝑠𝑠𝑠𝑠 4 𝜃𝜃3 𝑠𝑠𝑠𝑠𝑠𝑠 2 (𝜃𝜃1 − 𝜃𝜃2 )𝑠𝑠𝑠𝑠𝑠𝑠 3 (𝜃𝜃2 − 𝜃𝜃3 )

Define a virtual kinematic control law 14101

Proceedings of the 20th IFAC World Congress 13560 Al-Dujaili Ayad et al. / IFAC PapersOnLine 50-1 (2017) 13556–13562 Toulouse, France, July 9-14, 2017

𝛼𝛼𝑐𝑐 = 𝑇𝑇𝛼𝛼 𝜂𝜂𝑐𝑐 and the velocity tracking error 𝜂𝜂𝑒𝑒 = 𝜂𝜂 − 𝜂𝜂𝑐𝑐 Then we have 𝛼𝛼𝑒𝑒 = 𝛼𝛼 − 𝛼𝛼𝑐𝑐 = 𝑇𝑇𝛼𝛼 𝜂𝜂𝑒𝑒

(45) (46)

Now, we design the virtual kinematic control law 𝛼𝛼𝑐𝑐 as 𝜕𝜕𝛽𝛽 𝛼𝛼2𝑐𝑐 = 𝛼𝛼2𝑑𝑑 − 𝑐𝑐4 𝑧𝑧̅4 − 𝛼𝛼1𝑑𝑑 𝑧𝑧̅3 + 3 𝛼𝛼1𝑑𝑑 𝑧𝑧2 𝛼𝛼1𝑐𝑐

+

𝜕𝜕𝛽𝛽3 𝜕𝜕𝑧𝑧3

𝛼𝛼1𝑑𝑑 𝑧𝑧3 +

𝜕𝜕𝛽𝛽3 𝜕𝜕𝑧𝑧3

𝜕𝜕𝑧𝑧1

𝛼𝛼1𝑑𝑑 𝑧𝑧4

= 𝛼𝛼1𝑑𝑑 + (𝜅𝜅 − ∆1 )−1 [−𝑐𝑐5 𝑧𝑧5 + (𝑧𝑧̅3 − 𝑧𝑧̅4

𝜕𝜕𝛽𝛽3 𝜕𝜕𝑧𝑧3

(47)

(48)

) 𝛼𝛼2𝑐𝑐 ] (49)

where 𝑐𝑐4 > 0 and 𝑐𝑐5 > 0 are chosen to be constant. From equation (45), the kinematic control law is (50) 𝜂𝜂𝑐𝑐 = 𝑇𝑇𝛼𝛼−1 𝛼𝛼𝑐𝑐 Substituting (47), (48) and (49) into equation (41) yields 𝑉𝑉5̇ = −𝑐𝑐4 𝑧𝑧̅42 − 𝑐𝑐5 𝑧𝑧52 + 𝑧𝑧5 (𝑘𝑘 − ∆1 )𝛼𝛼𝑒𝑒1

1, 2, … , 𝑁𝑁, then the boundedness of all closed loop signals is ensured, and lim (𝑥𝑥(𝑡𝑡) − 𝑥𝑥𝑑𝑑 (𝑡𝑡)) = 0, lim (𝑦𝑦(𝑡𝑡) − 𝑦𝑦𝑑𝑑 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

lim (𝜃𝜃3 (𝑡𝑡) − 𝜃𝜃3𝑑𝑑 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

𝑡𝑡⟶∞

lim (𝜃𝜃2 (𝑡𝑡) − 𝜃𝜃2𝑑𝑑 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

lim (𝜃𝜃1 (𝑡𝑡) − 𝜃𝜃1𝑑𝑑 (𝑡𝑡)) = 0 and lim (𝜂𝜂(𝑡𝑡) − 𝜂𝜂𝑐𝑐 (𝑡𝑡)) = 0.

𝑡𝑡⟶∞

𝑡𝑡⟶∞

Proof: Consider 𝜎𝜎 = 𝜎𝜎(𝑎𝑎) and 𝑢𝑢 = 𝑢𝑢(𝑎𝑎) and choose the global Lyapunov function as follows 1 (57) 𝑉𝑉6(𝑎𝑎) = 𝑉𝑉5 + 𝜂𝜂𝑒𝑒𝑇𝑇 𝜂𝜂𝑒𝑒 2

Deriving equation (57) and using (52) and (55) lead to ̇ 𝑉𝑉6(𝑎𝑎) = −𝑐𝑐4 𝑧𝑧̅42 − 𝑐𝑐5 𝑧𝑧52 + 𝑓𝑓𝜂𝜂 𝑇𝑇 𝑇𝑇𝛼𝛼 𝜂𝜂𝑒𝑒 ̅ −1 𝐶𝐶̅ 𝜂𝜂 + 𝑀𝑀 ̅ −1 𝐵𝐵̅ 𝜎𝜎𝑢𝑢 − 𝜂𝜂̇ 𝑐𝑐 ] +𝜂𝜂𝑒𝑒𝑇𝑇 [−𝑀𝑀

(58) Substituting the dynamic control law in (56) with 𝑘𝑘 = 𝑎𝑎 in the equation (58), we have ̇ 𝑉𝑉6(𝑎𝑎) = −𝑐𝑐4 𝑧𝑧̅42 − 𝑐𝑐5 𝑧𝑧52 + −𝑐𝑐6 𝜂𝜂𝑒𝑒𝑇𝑇 𝜂𝜂𝑒𝑒 ≤ 0 (59)

It is clear that from this equation if 𝑓𝑓𝜂𝜂𝑇𝑇 𝑇𝑇𝛼𝛼 𝜂𝜂𝑒𝑒 → 0, then 𝑉𝑉5̇ ≤ 0, which means that the system is stable. To implement this and to ensure desired system performance, a dynamic controller will be designed in the next section.

The equation (59) indicates that: 𝑧𝑧1 , 𝑧𝑧2 , 𝑧𝑧3 , 𝑧𝑧4 , 𝑧𝑧5 , 𝑧𝑧̅1 , 𝑧𝑧̅2 , 𝑧𝑧̅3, 𝑧𝑧̅4, 𝜂𝜂𝑒𝑒 ∈ 𝐿𝐿∞ , and 𝑧𝑧̅4, 𝑧𝑧5 , 𝜂𝜂𝑒𝑒 ∈ 𝐿𝐿2 . It follows from the 2nd, 3rd and 4th equations of (12) that 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃3 ≠ 0, 𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2 − 𝜃𝜃3 ) ≠ 0 and 𝑡𝑡𝑡𝑡𝑡𝑡(𝜃𝜃1 − 𝜃𝜃2 ) ∈ 𝐿𝐿∞ which means 𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1 − 𝜃𝜃2 ) ≠ 0. Then from the equations (16), (19), and also from (45)-(50) and (53)-(56), we obtain the following: 𝑇𝑇𝛼𝛼 is bounded and nonsingular, and 𝛼𝛼𝑐𝑐 , 𝛼𝛼𝑒𝑒 , 𝛼𝛼, 𝜂𝜂𝑐𝑐 , 𝜂𝜂𝑒𝑒 , 𝜂𝜂, 𝑧𝑧̇1 , 𝑧𝑧̇2 , 𝑧𝑧̇3 , 𝑧𝑧̇4, 𝑧𝑧̇5, 𝑧𝑧̅1̇ , 𝑧𝑧̅2̇ , 𝑧𝑧̅3̇ , 𝑧𝑧̅4̇ , 𝛼𝛼̇ 𝑐𝑐 , 𝜂𝜂̇ 𝑐𝑐 , 𝑢𝑢(𝑘𝑘) , 𝜂𝜂̇ , 𝜂𝜂̇ 𝑒𝑒 , 𝛼𝛼̇ , 𝛼𝛼̇ 𝑒𝑒 ∈ 𝐿𝐿∞ . According to Barbalat Lemma, it is concluded that all the closed loop system signals are bounded, and lim 𝑧𝑧̅4 = 0, lim 𝑧𝑧5 = 0 and lim 𝜂𝜂𝑒𝑒 = 0,

3.2 Multiple Dynamic Controller Design

according to the proposition 1, we have

+𝑧𝑧5 (𝑧𝑧̅4

𝜕𝜕𝛽𝛽3 𝜕𝜕𝑧𝑧3

− 𝑧𝑧̅3 ) 𝛼𝛼𝑒𝑒2

𝑇𝑇

(51)

By letting the 𝑓𝑓𝜂𝜂 = [𝑧𝑧5 (𝜅𝜅 − ∆1 ), 𝑧𝑧5 (𝑧𝑧̅4 3 − 𝑧𝑧̅3 )] we get 𝜕𝜕𝑧𝑧3 (52) 𝑉𝑉5̇ = −𝑐𝑐4 𝑧𝑧̅42 − 𝑐𝑐5 𝑧𝑧52 + 𝑓𝑓𝜂𝜂𝑇𝑇 𝑇𝑇𝛼𝛼 𝜂𝜂𝑒𝑒 𝜕𝜕𝛽𝛽

Since the failures are uncertain it is difficult to ensure system stability and asymptotic tracking by using a single control law. To cover all possible failure patterns, we will design one specific control law for each failure pattern. Then, we will establish a control switching mechanism to select an appropriate control law to be the applied one. 3.2.1. Dynamic Control Laws Substituting (9) in (7), we get the following dynamic equation with actuator failures ̅ −1 (−𝐶𝐶̅ 𝜂𝜂 + 𝐵𝐵̅ 𝜎𝜎𝑢𝑢) (53) 𝜂𝜂̇ = 𝑀𝑀 The time derivative of the velocity tracking error is (54) 𝜂𝜂̇ 𝑒𝑒 = 𝜂𝜂̇ − 𝜂𝜂̇ 𝑐𝑐 Substituting (53) in (54), we get the following equation ̅ −1 𝐶𝐶̅ 𝜂𝜂 + 𝑀𝑀 ̅ −1 𝐵𝐵̅ 𝜎𝜎𝑢𝑢 − 𝜂𝜂̇ 𝑐𝑐 𝜂𝜂̇ 𝑒𝑒 = −𝑀𝑀

(55)

Let 𝜎𝜎(𝑘𝑘) , 𝑘𝑘 = 1, 2, … , 𝑁𝑁 denote the 𝑘𝑘𝑘𝑘ℎ possible failure matrix which fulfils the condition (11), where 𝑁𝑁 is the number of all possible failure pattern matrices that are under consideration. Then, the dynamic control law is ̅ −1 𝐵𝐵̅ 𝜎𝜎(𝑘𝑘) ) [−𝑐𝑐6 𝜂𝜂𝑒𝑒 − 𝑇𝑇𝛼𝛼𝑇𝑇 𝑓𝑓𝜂𝜂 + 𝑀𝑀 ̅ −1 𝐶𝐶̅ 𝜂𝜂 + 𝜂𝜂̇ 𝑐𝑐 ] 𝑢𝑢(𝑘𝑘) = (𝑀𝑀 +

(56)

for 𝑘𝑘 = 1, 2, … , 𝑁𝑁, where 𝑐𝑐6 > 0 is chosen to be constant, and ̅ −1 𝐵𝐵̅ 𝜎𝜎(𝑘𝑘) )+ is a generalized inverse matrix. (𝑀𝑀 3.2.2. Performance Analysis

The performance of the designed dynamic control laws is given as follows Lemma 1: If the applied control law matches with the actual failure pattern matrix, i.e., 𝜎𝜎 = 𝜎𝜎(𝑎𝑎) and 𝑢𝑢 = 𝑢𝑢(𝑎𝑎) for 𝑎𝑎 =

𝑡𝑡⟶∞

𝑡𝑡⟶∞

𝑡𝑡⟶∞

which also implies lim 𝛼𝛼𝑒𝑒 = 0. With equation (47) and 𝑡𝑡⟶∞

lim 𝑧𝑧1 = 0,

𝑡𝑡⟶∞

lim 𝑧𝑧2 = 0, lim 𝑧𝑧3 = 0, lim 𝑧𝑧4 = 0, lim 𝑧𝑧5 = 0, lim 𝑧𝑧̅1 =

𝑡𝑡⟶∞

𝑡𝑡⟶∞

𝑡𝑡⟶∞

𝑡𝑡⟶∞

𝑡𝑡⟶∞

0, lim 𝑧𝑧̅2 = 0, lim 𝑧𝑧̅3 = 0, lim 𝑧𝑧̅4 = 0, and |𝛼𝛼1𝑑𝑑 | ≥ 0, then 𝑡𝑡⟶∞

𝑡𝑡⟶∞

𝑡𝑡⟶∞

we can further obtain lim 𝛼𝛼2𝑐𝑐 = 𝛼𝛼2𝑑𝑑 from equation (48), it 𝑡𝑡⟶∞

follows that lim 𝛼𝛼2 = 𝛼𝛼2𝑑𝑑 , and also lim 𝛼𝛼1𝑐𝑐 = 𝛼𝛼1𝑑𝑑 and from 𝑡𝑡⟶∞

𝑡𝑡⟶∞

equation (49), it follows that lim 𝛼𝛼1 = 𝛼𝛼1𝑑𝑑 . 𝑡𝑡⟶∞

Finally, all the closed loop system signals are uniformly bounded, furthermore lim 𝑧𝑧𝑖𝑖 (𝑡𝑡) = 0, (𝑖𝑖 = 1, … 5) and 𝑡𝑡⟶∞

lim 𝜉𝜉𝑖𝑖𝑖𝑖 (𝑡𝑡) = 0, (𝑖𝑖 = 1, … 5) and

𝑡𝑡⟶∞

which also means

𝑦𝑦𝑑𝑑 (𝑡𝑡)) = 0,

lim (𝜂𝜂(𝑡𝑡) − 𝜂𝜂𝑐𝑐 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

lim (𝑥𝑥(𝑡𝑡) − 𝑥𝑥𝑑𝑑 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

lim (𝜃𝜃3 (𝑡𝑡) − 𝜃𝜃3𝑑𝑑 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

𝜃𝜃2𝑑𝑑 (𝑡𝑡)) = 0, lim (𝜃𝜃1 (𝑡𝑡) − 𝜃𝜃1𝑑𝑑 (𝑡𝑡)) = 0. 𝑡𝑡⟶∞

lim (𝑦𝑦(𝑡𝑡) −

𝑡𝑡⟶∞

lim (𝜃𝜃2 (𝑡𝑡) −

𝑡𝑡⟶∞

3.3 Control Switching Mechanism Design To design the control switching mechanism, we first reconstruct the velocity vector 𝜂𝜂 for all possible 𝜎𝜎(𝑘𝑘) , 𝑘𝑘 = 1, 2, … , 𝑁𝑁. Then, multiple cost functions are calculated from the reconstruction errors, and employed to generate the switching signal. 1) Signal reconstruction: Start with the dynamic equation in (53) ̅ −1 (−𝐶𝐶̅ 𝜂𝜂 + 𝐵𝐵̅ 𝜎𝜎𝑢𝑢) (60) 𝜂𝜂̇ = 𝑀𝑀 Choosing a stable filter

1

𝑆𝑆+𝛾𝛾

with 𝛾𝛾 > 0 and adding to both

sides of equation (60), we have 𝑆𝑆

𝑆𝑆+𝛾𝛾

[𝜂𝜂](𝑡𝑡) =

1

𝑆𝑆+𝛾𝛾

̅ −1 𝐶𝐶̅ 𝜂𝜂](𝑡𝑡) + [−𝑀𝑀

where 𝜂𝜂̇ (𝑡𝑡) = 𝑠𝑠[𝜂𝜂](𝑡𝑡) and

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1

𝑆𝑆+𝛾𝛾

1

𝑆𝑆+𝛾𝛾

̅ −1 𝐵𝐵̅ 𝜎𝜎𝑢𝑢 ](𝑡𝑡) [𝑀𝑀

(61)

[𝜒𝜒](𝑡𝑡) denotes the output of the

Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Al-Dujaili Ayad et al. / IFAC PapersOnLine 50-1 (2017) 13556–13562

filter

1

𝑆𝑆+𝛾𝛾

closed-loop signals are bounded and lim (𝑥𝑥(𝑡𝑡) − 𝑥𝑥𝑑𝑑 (𝑡𝑡)) = 0,

with the input 𝜒𝜒(𝑡𝑡).

From equation (61), the following is obtained 𝛾𝛾 1 ̅ −1 𝐶𝐶̅ 𝜂𝜂](𝑡𝑡) [𝜂𝜂](𝑡𝑡) + [−𝑀𝑀 𝜂𝜂(𝑡𝑡) = 𝑆𝑆+𝛾𝛾

+

1

𝑆𝑆+𝛾𝛾

𝑆𝑆+𝛾𝛾

̅ −1 𝐵𝐵̅ 𝜎𝜎𝑢𝑢 ](𝑡𝑡) [𝑀𝑀

lim (𝑦𝑦(𝑡𝑡) − 𝑦𝑦𝑑𝑑 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

lim (𝜃𝜃2 (𝑡𝑡) − 𝜃𝜃2𝑑𝑑 (𝑡𝑡)) = 0,

(62)

for each possible 𝜎𝜎(𝑘𝑘) , 𝑘𝑘 = 1, 2, … , 𝑁𝑁, we reconstruct a signal as follows 𝛾𝛾 1 ̅ −1 𝐶𝐶̅ 𝜂𝜂](𝑡𝑡) [𝜂𝜂](𝑡𝑡) + [−𝑀𝑀 𝜂𝜂̂ (𝑘𝑘) (𝑡𝑡) = 𝑆𝑆+𝛾𝛾 1

+

𝑆𝑆+𝛾𝛾

𝑆𝑆+𝛾𝛾

̅ −1 𝐵𝐵̅ 𝜎𝜎(𝑘𝑘) 𝑢𝑢 ](𝑡𝑡) [𝑀𝑀

(63)

Define the reconstruction error as follows

𝜂𝜂̃(𝑡𝑡) = 𝜂𝜂(𝑡𝑡) − 𝜂𝜂̂ (𝑡𝑡)

(64)

Then, consider that the actual failure pattern matrix is 𝜎𝜎 = 𝜎𝜎(𝑎𝑎) , then with (62) and (63), the matched reconstruction error takes the form of (65) 𝜂𝜂̃(𝑎𝑎) (𝑡𝑡) = 0

and the unmatched reconstruction errors take the form of 𝜂𝜂̃(𝑏𝑏) (𝑡𝑡) =

1

𝑆𝑆+𝛾𝛾

̅ −1 𝐵𝐵̅ 𝜎𝜎(𝑎𝑎) 𝑢𝑢 − 𝑀𝑀 ̅ −1 𝐵𝐵̅ 𝜎𝜎(𝑏𝑏) 𝑢𝑢](𝑡𝑡) [𝑀𝑀

(66)

for 𝑏𝑏 = 1, 2, … , 𝑁𝑁, 𝑏𝑏 ≠ 𝑎𝑎, which maybe not zero. 2) Multiple cost functions: The cost functions, calculated from the reconstruction errors, are chosen as 𝑇𝑇 (𝑡𝑡)𝜂𝜂̃(𝑘𝑘) (𝑡𝑡) (67) 𝐽𝐽(𝑘𝑘) (𝑡𝑡) = 𝜂𝜂̃(𝑘𝑘) for 𝑘𝑘 = 1, 2, … , 𝑁𝑁. 3) Control signal selection: The control switching, is implemented by comparing all the cost functions in (67), and determining the index 𝑘𝑘 corresponding to the minimum one, that is (68) 𝑘𝑘(𝑡𝑡) = 𝑎𝑎𝑎𝑎𝑎𝑎 min 𝐽𝐽(𝑘𝑘) (𝑡𝑡) 𝑘𝑘=1,2,… ,𝑁𝑁

Then the corresponding control law is selected as the applied control signal from (56), that is (69) 𝑢𝑢(𝑡𝑡) = 𝑢𝑢(𝑘𝑘) (𝑡𝑡) 3.4 Overall System Performance Analysis

Theorem 1: The developed multiple-model actuator failure compensation control scheme, constituted by the kinematic control law in (48) and (49), multiple dynamic control laws in (56) and the control switching mechanism implemented by (68) and (69) with multiple reconstructed signals in (63) and multiple cost functions in (67), applied to three differentially driven wheeled mobile robots with fixed link modelled as (3) and (7), guarantees that all closed-loop signals are bounded lim (𝑦𝑦(𝑡𝑡) − 𝑦𝑦𝑑𝑑 (𝑡𝑡)) = 0, and lim (𝑥𝑥(𝑡𝑡) − 𝑥𝑥𝑑𝑑 (𝑡𝑡)) = 0, 𝑡𝑡⟶∞

lim (𝜃𝜃3 (𝑡𝑡) − 𝜃𝜃3𝑑𝑑 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

𝑡𝑡⟶∞

lim (𝜃𝜃2 (𝑡𝑡) − 𝜃𝜃2𝑑𝑑 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

lim (𝜃𝜃1 (𝑡𝑡) − 𝜃𝜃1𝑑𝑑 (𝑡𝑡)) = 0, despite the presence of actuator

𝑡𝑡⟶∞

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failures modelled as (9) and (10) that are uncertain in failure time instants and patterns. Proof: Consider 𝜎𝜎 = 𝜎𝜎(𝑎𝑎) From (65) and (67), we have 𝐽𝐽(𝑎𝑎) =

0 for the matched cost function. But for the unmatched functions 𝐽𝐽(𝑏𝑏) , 𝑏𝑏 = 1, 2, … 𝑁𝑁, 𝑏𝑏 ≠ 𝑎𝑎, these zero properties may not hold due to (66). Since all cost functions are nonnegative, the matched cost function will generically become smaller than the other ones. Then, the matched control law 𝑢𝑢(𝑎𝑎) will be selected as the applied one with (68) and (69). According to Lemma 1, the selected control law can guarantee that all

𝑡𝑡⟶∞

𝑡𝑡⟶∞

lim (𝜃𝜃3 (𝑡𝑡) − 𝜃𝜃3𝑑𝑑 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

lim (𝜃𝜃1 (𝑡𝑡) − 𝜃𝜃1𝑑𝑑 (𝑡𝑡)) = 0,

𝑡𝑡⟶∞

despite the presence of actuator failures. This is the generic (generally true) matched case. On the other hand, if the unmatched control law 𝑢𝑢(𝑏𝑏) , 𝑏𝑏 ≠ 𝑎𝑎 is selected as the applied one meaning 𝐽𝐽(𝑏𝑏) (𝑡𝑡) ≤ 𝐽𝐽(𝑎𝑎) , then there is a time interval [𝑇𝑇1 , 𝑇𝑇2 ] such that 𝐽𝐽(𝑏𝑏) (𝑡𝑡) = 0 for 𝑡𝑡 ∈ [𝑇𝑇1 , 𝑇𝑇2 ], as 𝐽𝐽(𝑎𝑎) = 0. From ̅ −1 𝐵𝐵̅ 𝜎𝜎(𝑎𝑎) 𝑢𝑢(𝑎𝑎) − 𝑀𝑀 ̅ −1 𝐵𝐵̅ 𝜎𝜎(𝑏𝑏) 𝑢𝑢(𝑏𝑏) = 0 (66), 𝐽𝐽(𝑏𝑏) (𝑡𝑡) = 0 means 𝑀𝑀 ̅ −1 𝐵𝐵̅ 𝜎𝜎(𝑏𝑏) 𝑢𝑢(𝑏𝑏) = for 𝑡𝑡 ∈ [𝑇𝑇1 , 𝑇𝑇2 ], together with 𝑀𝑀 𝑇𝑇 −1 −1 ̅ 𝐵𝐵̅ 𝜎𝜎(𝑎𝑎) 𝑢𝑢(𝑎𝑎) = −𝑐𝑐6 𝜂𝜂𝑒𝑒 − 𝑇𝑇𝛼𝛼 𝑓𝑓𝜂𝜂 + 𝑀𝑀 ̅ 𝐶𝐶̅ 𝜂𝜂 +𝜂𝜂̇ 𝑐𝑐 it also means 𝑀𝑀 that: if 𝜎𝜎(𝑎𝑎) but 𝑢𝑢(𝑏𝑏) is selected, then 𝑢𝑢(𝑏𝑏) has the same control effectiveness for the dynamic system in (53) as compared with the matched control law 𝑢𝑢(𝑎𝑎) . 4. SIMULATION STUDIES

In this simulation studies, we assume the three wheeled driven robots are the one used in [Fukao, 2000], then the physical parameters of the three robots are chosen as: 𝑎𝑎1 = 𝑎𝑎2 = 𝑎𝑎3 = 0.3 𝑚𝑚, 𝑏𝑏1 = 𝑏𝑏2 = 𝑏𝑏3 = 0.75 𝑚𝑚, 𝑟𝑟1 = 𝑟𝑟2 = 𝑟𝑟3 = 0.15 𝑚𝑚, 𝑚𝑚1 = 𝑚𝑚2 = 𝑚𝑚3 = 30 𝑘𝑘𝑘𝑘, 𝐼𝐼𝑚𝑚1 = 𝐼𝐼𝑚𝑚2 = 𝐼𝐼𝑚𝑚3 = 15.625 𝑘𝑘𝑘𝑘/ 𝑚𝑚2 . The length of the link between each two robots is assumed to be 𝑑𝑑 = 1.7 𝑚𝑚. The 𝑥𝑥𝑑𝑑 , 𝑦𝑦𝑑𝑑 , 𝜃𝜃3𝑑𝑑 , 𝜃𝜃2𝑑𝑑 , 𝜃𝜃1𝑑𝑑 , 𝑣𝑣3𝑑𝑑 and 𝜔𝜔1𝑑𝑑 , are the reference trajectories with 𝑥𝑥𝑑𝑑 (0) = 𝑦𝑦𝑑𝑑 (0) = 𝜃𝜃3𝑑𝑑 (0) = 𝜃𝜃2𝑑𝑑 (0) = 𝜃𝜃1𝑑𝑑 (0) = 0. In order to verify the fault compensation effectiveness of the developed multiple model scheme, the following failure cases are simulated 0 ≤ t < 20s no failure, σ(1) = diag{1, 1, 1, 1, 1, 1}, σ(2) = diag{0, 1, 1, 1, 1, 1}, 20s ≤ t < 40s 𝜏𝜏1𝑟𝑟 fails, 40s ≤ t < 60s 𝜏𝜏2𝑙𝑙 , 𝜏𝜏3𝑙𝑙 fails, σ(3) = diag{1, 1, 1, 0, 1, 0}, 𝜏𝜏1𝑙𝑙 ,𝜏𝜏2𝑟𝑟 , 𝜏𝜏3𝑟𝑟 fails, σ(4) = diag{1, 0, 0, 1, 0, 1}, 60s ≤ t < 80s 𝜏𝜏1𝑙𝑙 , 𝜏𝜏2𝑟𝑟 , 𝜏𝜏2𝑙𝑙 , 𝜏𝜏3𝑟𝑟 , fails, σ(5) = diag{1, 0, 0, 0, 0, 1}, t ≥ 80s There are 5 failure pattern satisfying the actuation redundancy condition in (11). Then we need 5 control laws in (56), reconstructed signals in (66) and cost functions in (67). The initial conditions are chosen as: 𝑥𝑥(0) = 𝑦𝑦(0) = 0.5, 𝜃𝜃3 (0) = 𝜃𝜃2 (0) = 𝜃𝜃1 (0) = 30 𝑑𝑑𝑒𝑒𝑔𝑔, and the control gains are chosen as: 𝐶𝐶4 = 𝐶𝐶5 = 𝐶𝐶6 = 2 and 𝜅𝜅 = 5. Fig. 3 shows the positions of robot 1, 2, 3 and reference trajectory, and Fig. 4 shows the tracking errors of all the states. From them, we can see that the desired system stability and asymptotic tracking are ensured despite the presence of actuator failures. Fig. 5 shows the control torques generated by the wheels in robot 1, robot 2, and robot 3 respectively, from which we can see that the actuator failures are consistent with the failure cases in simulation conditions. Fig. 6 shows the control switching index, the sequence of which is 1→2→3→4→5. We can see that the control switching index matches with the actual failure pattern index, although there is some wrong switching at some short time intervals after the failure occurring time instants. These wrong control switching do not affect system stability and asymptotic tracking properties. For the short time interval after 20s, in which the control signal has a wrong switching to 3, this case is consistent with the unmatched case in the proof of theorem 1.

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Proceedings of the 20th IFAC World Congress 13562 Al-Dujaili Ayad et al. / IFAC PapersOnLine 50-1 (2017) 13556–13562 Toulouse, France, July 9-14, 2017

Figure 3: Robot trajectories in (x, y) plane.

Figure 4: Tracking errors.

Figure 5: Control torques generated by robot 1, robot 2 and robot 3.

Figure 6: Control switching index. 5. CONCLUSIONS This paper developed an actuator fault compensation scheme for three 2WD mobile robots with fixed link. The kinematics and dynamics of this robot configuration were modelled, and

a multiple model failure compensation scheme was designed. First, a kinematic control law was designed, based on which, multiple dynamic controllers were designed covering all possible failure patterns. Each of these dynamic controllers can ensure desired system performance when it is the applied one and also matches with the actual failure pattern. Then, a control switching mechanism was established by employing multiple cost functions calculated from multiple reconstruction signal errors to select the dynamic controller corresponding to the minimum cost function to generate the applied control signal. Finally, the effectiveness of the proposed failure compensation system was shown by simulation results. Extension of the proposed method to the application of 𝑁𝑁 mobile robots with fixed link is the interest of our future work. ACKNOWLEDGMENT This work was partially supported by the Regional project SUCRé (Sûreté de fonctionnement et résilience pour la gestion et le contrôle coopératifs des systèmes sociotechniques: Coopération Homme(s)-Robot(s) en milieu hostile) of the Hauts de France region. REFERENCES Fukao T., Nakagawa H. and Adachi N. (2000) Adaptive tracking control of a nonholonomic mobile robot IEEE Transactions on Robotics and Automation 16 609-15. Huang J, Wen C., Wang W. and Jiang Z. (2014) Adaptive output feedback tracking control of a nonholonomic mobile robot Automatica 50 821-31. Ji M., Zhang Z., Biswas G. and Sarkar N. (2003) Hybrid fault adaptive control of a wheeled mobile robot IEEE-ASME Transactions on Mechatronics 8 226-33. Jiang Z. and Nijmeijer H. (1999) A recursive technique for tracking control of nonholonomic systems in chained form IEEE Transactions on Automatic Control 44 265-79. Khalil H. (1996) Nonlinear Systems, 2nd ed. Upper Saddle River, NJ: Prentice Hall. Ma Y., Cocquempot V., El Badaoui El Najjar M. and Jiang B. (2016) Multiple-model based actuator fault compensation for two linked 2WD mobile robots SysTol’16: 3rd International Conference on Control and Fault-Tolerant Systems, B arcelo na, Sp ain . Rotondo D., Puig V., Nejjari F. and Romera J. (2014) A faulthiding approach for the switching Quasi-LPV faulttolerant control of a four-wheeled omnidirectional mobile robot IEEE Transactions on Electronic Devices Ind. Electron. 62 3932-44. Skoundrianos E. and Tzafestas S. (2004) Finding fault: fault diagnosis on the wheels of a mobile robot using local model neural networks IEEE Robotics & Automation 11 83-90. Sørdalen O. (1993) Conversion of the kinematics of a car with n trailers into a chained form, IEEE Conference on Robotics and Automation 382–387. Shen Q., Jiang B. and Cocquempot V. (2013) Fuzzy logic system-based adaptive fault tolerant control for near space vehicle attitude dynamics with actuator faults IEEE Transactions on Fuzzy Systems 21 289-300. Tao G., Chen S., Tang, X. and Joshi S. (2004) Adaptive control of systems with actuator failures New York, NY, USA: Springer-Verlag.

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