Fault-Tolerant Control for Physically Linked Two 2WD Mobile Robots with Actuator Faults*

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

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IFAC PapersOnLine 50-1 (2017) 13563–13568 Fault-Tolerant Fault-Tolerant Control Control for for Physically Physically Fault-Tolerant Control for Physically Linked Two 2WD Mobile Robots with Fault-Tolerant Control for Physically Linked with Linked Two Two 2WD 2WD Mobile Mobile Robots Robots with  Actuator Faults  Linked Two 2WD Mobile Robots with Actuator Faults  Actuator Faults  Actuator Faults Yajie Ma ∗,∗∗ Vincent Cocquempot ∗

∗,∗∗ ∗ ∗ Yajie Ma ∗,∗∗ Vincent Cocquempot ∗∗∗ ∗ Yajie Ma Maan El Badaoui ElVincent Najjar ∗∗Cocquempot Andrey Polyakov Yajie Ma ∗,∗∗ Vincent Cocquempot ∗∗∗ ∗,∗∗ ∗ ∗ ∗∗∗ Maan El Badaoui El Najjar Andrey Polyakov ∗∗∗ Yajie Ma Vincent ∗Cocquempot Maan El Badaoui El Najjar Andrey Polyakov Gang Zheng Maan El Badaoui El Najjar ∗ Andrey Polyakov ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ Zheng Maan El BadaouiGang El Najjar Andrey Polyakov ∗∗∗ Gang Gang Zheng Zheng ∗∗∗ Gang Zheng ∗ ∗ Univ. Lille, CNRS, UMR 9189-CRIStAL-Centre de Recherche en ∗ Univ. Lille, CNRS, UMR 9189-CRIStAL-Centre de Recherche en ∗ Univ. Lille, CNRS, 9189-CRIStAL-Centre de en Informatique, Signal etUMR Automatique de Lille, F-59000 Lille, France UMR 9189-CRIStAL-Centre de Recherche Recherche en ∗ Univ. Lille, CNRS, Informatique, Signal et Automatique de Lille, F-59000 Lille, France Univ. Lille, CNRS, 9189-CRIStAL-Centre de Recherche en Informatique, Signal et Automatique de Lille, (e-mail: [email protected]). Informatique, Signal etUMR Automatique de Lille, Lille, F-59000 F-59000 Lille, France France (e-mail: [email protected]). ∗∗ Informatique, Signal et Automatique de Lille, F-59000 Lille, France (e-mail: [email protected]). College of Automation Engineering, Nanjing University of (e-mail: [email protected]). ∗∗ ∗∗ of Automation Engineering, Nanjing University [email protected]). ∗∗ College College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China of College of (e-mail: Automation Engineering, Nanjing University of ∗∗ Aeronautics and Astronautics, Nanjing 210016, China ∗∗∗ College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Non-A, INRIA Lille-Nord Europe, 40 Avenue Halley, 59650 Aeronautics and Astronautics, Nanjing 210016, China ∗∗∗ ∗∗∗ INRIA Lille-Nord Europe, 40 Avenue Halley, 59650 Aeronautics and Astronautics, Nanjing 210016, China ∗∗∗ Non-A, Non-A, INRIA Lille-Nord Europe, 40 Avenue Halley, 59650 Villeneuve Lille-Nord d’Ascq, Europe, France 40 Avenue Halley, 59650 ∗∗∗ Non-A, INRIA Villeneuve Non-A, INRIA Villeneuve Lille-Nord d’Ascq, Europe, France 40 Avenue Halley, 59650 d’Ascq, France Villeneuve d’Ascq, France Villeneuve d’Ascq, France Abstract: In this paper, a fault-tolerant control scheme for two physically linked two wheel Abstract: this fault-tolerant scheme two linked two Abstract: Inmobile this paper, paper, a with fault-tolerant control scheme for for actuator two physically physically linked two wheel wheel drive (2WD)In robotsaa multiple control and simultaneous faults is proposed. The Abstract: In this paper, fault-tolerant control scheme for two physically linked two wheel drive (2WD) mobile robots with multiple and simultaneous actuator faults is proposed. The Abstract: In this paper, a fault-tolerant control scheme for two physically linked two based wheel drive (2WD) (2WD) mobile robots witharemultiple multiple and simultaneous simultaneous actuator faults isdesigned, proposed. The kinematic andmobile dynamic models first presented. A kinematic controlfaults law isis drive robots with and actuator proposed. The kinematic and dynamic models are first presented. A kinematic control law is designed, based drive (2WD) mobile robots with multiple and simultaneous actuator faults is proposed. The kinematic and dynamic models are first presented. A kinematic control law is designed, based on which, and a dynamic investigated. applying the proposed kinematic dynamiccontrol modelslaw areisfirst presented.By A kinematic control law isdynamic designed,control based on which, aadesired dynamic control law is investigated. By applying the proposed dynamic control kinematic and dynamic models are first presented. A kinematic control law is designed, based on which, dynamic control law is investigated. By applying the proposed dynamic control signal, the system stability and tracking properties are ensured. Simulation results are on which, a dynamic control law is investigated. By applying the proposed dynamic control signal, the desired stability tracking are ensured. Simulation results are on which, dynamic control law isand investigated. By applying the proposed dynamic control signal, the system stability and tracking properties are ensured. Simulation results are finally presented tosystem verify the effectiveness of the properties developed fault-tolerant control scheme. signal, the adesired desired system stability and tracking properties are ensured. Simulation results are finally presented to verify the effectiveness of the developed fault-tolerant control scheme. signal, the desired system stability and tracking properties are ensured. Simulation results finally finally presented presented to to verify verify the the effectiveness effectiveness of of the the developed developed fault-tolerant fault-tolerant control control scheme. scheme. are © 2017, presented IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. finally to verify Federation the effectiveness of the developed fault-tolerant control scheme. Keywords: Actuator fault, fault-tolerant control, mobile robot, physical link, two wheel drive, Keywords: Actuator fault, fault-tolerant control, mobile robot, physical link, two wheel drive, Keywords: Keywords: Actuator Actuator fault, fault, fault-tolerant fault-tolerant control, control, mobile mobile robot, robot, physical physical link, link, two two wheel wheel drive, drive, Keywords: Actuator fault, fault-tolerant control, mobile robot, physical link, two wheel drive, 1. INTRODUCTION There are many effective fault diagnosis and fault-tolerant 1. INTRODUCTION INTRODUCTION There are many effective diagnosis and fault-tolerant 1. There many fault diagnosis and control methods, see Yu fault and Jiang (2015); Blanke et al. 1. INTRODUCTION There are are many effective effective fault diagnosis and fault-tolerant fault-tolerant control methods, see Yu and Jiang (2015); Blanke et al. 1. INTRODUCTION There are many effective fault diagnosis and see (2015); Blanke et (2006); Ding (2008). Asand for Jiang the application to wheel control methods, methods, see Yu Yu and Jiang (2015); fault-tolerant Blanke et al. al. Due to simplicity and efficiency, two-wheel drive (2WD) control (2006); Ding (2008). As for the application to wheel Due to simplicity and efficiency, two-wheel drive (2WD) control methods, see Yu and Jiang (2015); Blanke et al.a Ding (2008). As application to drive et al. (2014) proposes Due simplicity and two-wheel (2WD) (2006);mobile Ding robots, (2008). Fourlas As for for the the application to wheel wheel mobile are widely used as described Dixon et al. (2006); Due to to robots simplicity and efficiency, efficiency, two-wheelindrive drive (2WD) drive mobile robots, Fourlas et al. (2014) proposes mobile robots are widely used as described in Dixon et al. (2006); Ding (2008). As for the application to wheel Due to simplicity and efficiency, two-wheel drive (2WD) drive mobile robots, Fourlas et al. (2014) proposes a diagnosis method; and Sarkar (2007)a mobile are used as in drive mobile fault robots, Fourlas et al.Ji(2014) proposes a (2001). Especially, in some conditions, such et asal. a model-based mobile robots robots are widely widely usedharsh as described described in Dixon Dixon et al. model-based fault diagnosis method; Ji and Sarkar (2007) (2001). Especially, in some harsh conditions, such as a drive mobile robots, Fourlas et al. (2014) proposes mobile robots are widely used as described in Dixon et al. model-based fault diagnosis method; Ji and Sarkar (2007) presents a sensor fault accommodation scheme; Koh et al. (2001). Especially, in some harsh conditions, such as a fault diagnosis method; Ji and Sarkar (2007)a terrorist attack, a nuclear or nature such disasters, (2001). Especially, in some accident harsh conditions, as a model-based aa sensor Koh et al. terrorist attack, nuclear accident or nature nature disasters, model-based faultfault diagnosis method; Ji scheme; and (2001). Especially, inrobots some accident harsh conditions, as a presents presents fault accommodation scheme; Koh et (2012) Rotondo etaccommodation al. (2014) give two Sarkar fault-tolerant terrorist attack, aaa nuclear or disasters, presentsand a sensor sensor fault accommodation scheme; Koh(2007) et al. al. 2WD rescue mobile are employed. For such example, terrorist attack, nuclear accident or nature disasters, (2012) and Rotondo et al. (2014) give two fault-tolerant 2WD rescue mobile robots are employed. For example, presents a sensor fault accommodation scheme; Koh et al. terrorist attack, a nuclear accident or nature disasters, (2012) and Rotondo et al. (2014) give two fault-tolerant designs for four-wheel drivegive (4WD) robots; 2WD rescue mobile robots employed. example, (2012) and Rotondo et al. (2014) twomobile fault-tolerant some instruments, 2WD robots rescue carry mobileautomated robots are aremanipulation employed. For For example, control control designs for four-wheel drive (4WD) mobile robots; some robots carry automated manipulation instruments, (2012) and Rotondo et al. (2014) give two fault-tolerant 2WD rescue mobile robots are employed. For example, control designs for four-wheel drive (4WD) mobile robots; and Ji et al. (2003) proposes a hybrid adaptive fault tolersome robots carry automated manipulation instruments, environment sensors or rescuemanipulation tools, and some robots control designs for four-wheel drive (4WD) mobile robots; some robots carry automated instruments, and Ji etdesigns al.method (2003) proposes hybrid fault tolerenvironment sensors or rescue rescue tools, and some robots control for four-wheel drive (4WD) mobile some robots automated manipulation instruments, Ji proposes hybrid adaptive fault tolerant control for 2WD aaarobots toadaptive accommodate parenvironment sensors or tools, and some robots and Ji et et al. al. (2003) (2003) proposes hybrid adaptive faultrobots; tolerare used to carry collaboratively carry some heavy and big and environment sensors or rescue tools, and some robots ant control method for 2WD robots to accommodate parare used to collaboratively carry some heavy and big and Ji et al. (2003) proposes a hybrid adaptive fault tolerenvironment sensors or rescue tools, and some robots ant control method for 2WD robots to accommodate partial faults and degradation. However, these fault tolerant are used to collaboratively carry some heavy and big ant control method for 2WD robots to accommodate parinstruments or an injured. However, the adverse conditions are used to collaboratively carry some heavy and big tial faults and degradation. However, these fault tolerant instruments or an injured. However, the adverse conditions ant control method for 2WD robots to accommodate parare used to collaboratively carry some heavy and big tial faults and degradation. However, these fault tolerant for 4WD mobile robots have instruments an However, adverse conditions tial faultsare anddeveloped degradation. However, these faultthat tolerant may result inor (wheel motor)the faults leading to lose methods instruments oractuator an injured. injured. However, the adverse conditions methods are developed for 4WD mobile robots that have may result in actuator (wheel motor) faults leading to lose tial faults and degradation. However, these fault tolerant instruments or an injured. However, the adverse conditions methods are developed for 4WD mobile robots that have redundant actuators or without considering loss of control may result in actuator (wheel motor) faults leading to lose are developed for 4WD mobile robots that have failed robots the important instruments or the injured may result inand actuator (wheel motor) faults leading to lose methods redundant actuators or without considering loss of control failed robots and the robot important instruments oractuator, the injured injured are developed for mobile robots have may inAand actuator (wheel faults leading to lose redundant actuators or considering loss of control failures for 2WD robots. For4WD linked multiple mobile failed robots the instruments or redundant actuators or without without considering loss that of robots, control they carry. 2WD hasmotor) no redundant it methods failedresult robots and the important important instruments or the the injured failures for 2WD robots. For linked multiple mobile robots, they carry. A 2WD robot has no redundant actuator, it redundant actuators or without considering loss of control failed robots and the important instruments or the injured 2WD robots. linked Samson (1995) kinematic control mobile methodrobots, withthey carry. 2WD actuator, failures for for 2WDpresents robots. aFor For linked multiple multiple mobile robots, will if no oneredundant wheel is inoperative. they become carry. A Auncontrollable 2WD robot robot has has no redundant actuator, it it failures (1995) aaFor kinematic control method withwill become uncontrollable if will oneredundant wheel is is inoperative. inoperative. failures for 2WDpresents robots.dynamics; linked Khalaji multiple mobile robots, they carry. Auncontrollable 2WD robot has actuator, it Samson Samson (1995) presents control method without considering system Moosavian will if one wheel Samson (1995) presents a kinematic kinematic controland method withConsequently, the failed robots that will become become uncontrollable if no one become wheel isobstacles inoperative. out considering system dynamics; Khalaji and Moosavian Consequently, the failed robots will become obstacles that Samson (1995) presents a kinematic control method withwill become uncontrollable if one wheel is inoperative. out considering system dynamics; Khalaji and Moosavian (2014) designs a kinematic control and a dynamic control Consequently, the failed robots will become obstacles that considering system dynamics; Khalaji and Moosavian makes the mission more robots difficultwill andbecome less efficient. Consequently, the failed obstacles that out (2014) designs aasystem kinematic control and aa dynamic control makes the mission mission more robots difficultwill andbecome less efficient. efficient. out considering dynamics; Khalaji and Moosavian Consequently, the failed obstacles that (2014) designs kinematic control and dynamic control for a tractor with one trailer; Yuan et al. (2015) develops makes the more difficult and less (2014) designs a kinematic control and a dynamic control makes the mission moretolerance difficult and for aa tractor one trailer; Yuan et al. (2015) develops To improve the fault of less 2WDefficient. robots, some (2014) designswith a kinematic control a dynamic control makes the mission moretolerance difficult and less efficient. for with one trailer; Yuan et al. (2015) aforkinematic control without system a tractor tractor with onescheme trailer; Yuanand et considering al. (2015) develops develops To improve the fault of 2WD robots, some To improve the fault tolerance of 2WD some kinematic control scheme without considering system physical linksthe canfault be used to connect themrobots, as shown in aafor a tractor with one trailer; Yuan et al. (2015) develops To improve tolerance of 2WD robots, some kinematic control scheme without considering system dynamics for a double-steering two linked 2WD mobile a kinematic control scheme without considering system physical links can be used to connect them as shown in To improve the fault tolerance of 2WD robots, some physical links can be used to connect them as shown in dynamics for a double-steering two linked 2WD mobile Fig. 1 for two 2WD robots with a physical link. Such adynamics kinematic control scheme without considering physical links can be used to connect them as shown in robots. dynamics for a double-steering two linked 2WD mobile However, these schemes are proposed without confor a double-steering two linked 2WD system mobile Fig. 1 for two 2WD robots with a physical link. Such physical links can be used to connect them as shown in Fig. robots with physical Such However, these schemes are proposed without conarchitecture has 2WD actuator and sensor dynamics for a double-steering two linked 2WD mobile Fig. 11 for for two two 2WD robots with aaredundancies. physical link. link.If some Such robots. robots. However, these schemes are proposed without considering actuator faults. robots. However, these schemes are proposed without conarchitecture has actuator and sensor redundancies. If some Fig. 1 for with physical the link.If Such robots. architecture has actuator and sensor some sidering actuator faults. of them fail,two the 2WD othersrobots can help. Foraredundancies. example, system However, these schemes are proposed without conarchitecture has actuator and sensor redundancies. If some sidering actuator faults. sidering actuator faults. of them fail, the others can help. For example, the system architecture actuator and sensor redundancies. If some This paper proposes a fault-tolerant control scheme for two of them fail, the others can For example, the system 2WD robots inhelp. Fig. 1 has four wheel sidering actuator faults. of two themlinked fail, has the others can help. For example, themotors, system This paperlinked proposes aa fault-tolerant scheme for two of two linked 2WD robots inhelp. Fig. has four operative wheel motors, them fail, the others For example, themotors, system control scheme two physically 2WD mobile robotscontrol with actuator faults, of two linked 2WD robots in 111 has four This paper paper proposes proposes a fault-tolerant fault-tolerant control scheme for for two if or two of them arecan lost, the remaining ones This of one two linked 2WD robots in Fig. Fig. has four wheel wheel motors, physically linked 2WD mobile robots with actuator faults, if one or two of them are lost, the remaining operative ones This paper proposes a fault-tolerant control scheme for two of two linked 2WD robots in Fig. 1 has four wheel motors, 2WD mobile with actuator which are linked uncertain, is torobots say we don’t knowfaults, when if or them the ones physically linked 2WDthat mobile robots with actuator faults, can continue moving the lost, two robots. On theoperative other hand, if one one or two two of of them are are lost, the remaining remaining operative onesa physically which are uncertain, that is to say we don’t know when can continue moving the two robots. On the other hand, a physically linked 2WD mobile robots with actuator faults, if one or two of them are lost, the remaining operative ones which are uncertain, that is to say we don’t know when actuator(s)that faults can moving robots. On hand, whichwhich are uncertain, is totake(s) say weplace. don’tThese know faults when system of robots thatthe aretwo used to collaboratively can continue continue moving the two robots. On the the other othercarry hand,anaa and which actuator(s) faults take(s) These system of robots robots that aretwo used to collaboratively collaboratively carry ana and which are the uncertain, is to say weplace. don’tTo know when can continue the robots. theasother hand,an which actuator(s) faults take(s) place. These faults will applied that control gain uncertain. dealfaults with system of are used to carry and make which actuator(s) faults take(s) place. These faults instrument ormoving an that injured also beOn seen a physically system of robots that are can used to collaboratively carry an and will make applied control gain uncertain. To deal with instrument or an an that injured can also be seen seen as as aa physically physically and whichthe actuator(s) faultsdefinite take(s) place.gain These faults system of robots are used to collaboratively carry an will make the applied control gain uncertain. To deal with this problem, a new positive control matrix is instrument or injured can also be will make the applied control gain uncertain. To deal with linked multi-robot system. instrument or an injured can also be seen as a physically this problem, a new positive definite control gain matrix is linked multi-robot system. will make the applied control gain uncertain. To deal with instrument or an injured can also be seen as a physically this problem, a new positive definite control gain matrix is constructed, and the lower boundary of its eigenvalues linked multi-robot system. this problem, a new positive definite control gain matrix is linked multi-robot system.  This work was partially supported by the Regional project SUCR´ constructed, and the lower boundary of its eigenvalues is e this problem, a new positive definite control gain matrix linked multi-robot system. constructed, and the lower boundary of its eigenvalues is  used in the control design. The main contributions of this constructed, and the lower boundary of its eigenvalues is This work was partially supported by the Regional project SUCR´ e   u (Sˆ ret´ e de fonctionnement et r´ e silience pour la gestion et le contrˆ o le This work was partially supported by the Regional project SUCR´ e used in the control design. The main contributions of this constructed, and the lower boundary of its eigenvalues is This work was partially supported by the Regional project SUCR´ e used in the control design. The main contributions of this paper are as follows. used in the control design. The main contributions of this (Sˆ u ret´ e work de fonctionnement fonctionnement et r´ r´ esilience silience pour la gestion gestion etHomme(s)le contrˆ contrˆ ole lee  u This was partially supported by the Regional project SUCR´ coop´ e ratifs des syst` e mes sociotechniques: Coop´ e ration (Sˆ ret´ e de et e pour la et le o paper follows. (Sˆ uret´ e de fonctionnement et r´ esilience pour la gestion et le contrˆ ole used inare theas control design. The main contributions of this paper are as follows. coop´ e ratifs des syst` e mes sociotechniques: Coop´ e ration Homme(s)paper are as follows. (Sˆ uret´ e deenfonctionnement et esilience la gestion etHomme(s)le contrˆ ole Robot(s) milieu hostile) of r´ the Hautspour deCoop´ France region. coop´ eeratifs des syst` eemes eeration coop´ ratifs des syst` mes sociotechniques: sociotechniques: Coop´ ration Homme(s)paper are as follows. Robot(s) endes milieu hostile) of the Hauts deCoop´ France region. coop´ eratifs syst` emes sociotechniques: eration Homme(s)-

Robot(s) Robot(s) en en milieu milieu hostile) hostile) of of the the Hauts Hauts de de France France region. region. Robot(s) en milieu hostile) of the Hauts de France region. Copyright © 2017 IFAC 14105 Copyright 2017 14105 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017, 2017 IFAC IFAC 14105 Copyright © 2017 IFAC 14105 Peer review©under of International Federation of Automatic Control. Copyright 2017 responsibility IFAC 14105 10.1016/j.ifacol.2017.08.2367

Proceedings of the 20th IFAC World Congress 13564 Yajie Ma et al. / IFAC PapersOnLine 50-1 (2017) 13563–13568 Toulouse, France, July 9-14, 2017



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T

.

(2)

2.2 Dynamic Model

 

1 cos θ2 sin θ2 tan(θ1 − θ2 ) 0 d 0 0 0 1

The system constraints is expressed as Ma et al. (2016): A(q)q˙ = 0, (3) where   sin θ1 − cos θ1 −d cos(θ1 − θ2 ) 0 (4) A(q) = sin θ2 − cos θ2 0 0 is the system constraint matrix. Then, we have (5) S T (q)AT (q) = 0.

 





S(q) =





By using the Lagrange method, the dynamics of two physically linked 2WD robots is given by Ma et al. (2016):





M (q)¨ q + E(q, q) ˙ = B(q)τ + AT (q)λ,

Fig. 1. Two physically linked 2WD mobile robots. (1) The configuration of physically linked 2WD mobile robots is proposed to deal with actuator faults. (2) A fault-tolerant control scheme is developed for two 2WD mobile robots with a physical link, which ensures desired system stability and tracking properties, despite the presence of simultaneous multi-faults, including loss of actuation effectiveness and loss of control. The rest of this paper is organized as follows. In Section 2, the kinematic and dynamic models are presented for two 2WD robots with physical link, and its fault-tolerant control problem is formulated. In Section 3, a fault-tolerant control scheme is developed, consisting of a kinematic control law and a dynamic control law. In Section 4, a simulation study is given to verify the effectiveness of the proposed fault-tolerant control scheme. Conclusions follow in Section 5.

where M (q) ∈ R is the inertia matrix which is symmetric positive definite, E(q, q) ˙ ∈ R4 is the centripetal and coriolis vector, B(q) ∈ R4×4 is the input injection matrix, τ = [τ1r , τ1l , τ2r , τ2l ]T is the control torque vector, λ ∈ R2 is the constraint force vector, and  0 m 1 + m2 0 m 1 + m2  M (q) =  −(a2 m2 + dm1 ) sin θ2 (a2 m2 + dm1 ) cos θ2 −a1 m1 sin θ1 a1 m1 cos θ1  −(a2 m2 + dm1 ) sin θ2 −a1 m1 sin θ1 (a2 m2 + dm1 ) cos θ2 a1 m1 cos θ1  , m2 a22 + m1 d2 + Im2 a1 dm1 cos(θ1 − θ2 )  m1 a21 + Im1 a1 dm1 cos(θ1 − θ2 )   −a1 m1 θ˙12 cos θ1 − (a2 m2 + dm1 )θ˙22 cos θ2  −a1 m1 θ˙2 sin θ1 − (a2 m2 + dm1 )θ˙2 sin θ2  2 1 , E(q, q) ˙ =   −a1 dm1 θ˙12 sin(θ1 − θ2 ) 2 ˙ a1 dm1 θ2 sin(θ1 − θ2 )  cos θ cos θ cos θ  cos θ 1

2. PROBLEM FORMULATION For the configuration of the two robots in Fig. 1, the orientation of robot 1 is independent, but the one of robot 2 is consistent with the link. For each (ith, i = 1, 2) robot: the two rear wheels are actuated while the front wheel is passive; Pi is the center between two rear wheels which denotes the position of the robot, Ci is the mass center, ai denotes the distance between Pi and Ci , bi denotes the half distance of two actuated wheels, ri is the radius of wheels, θi is the orientation, and τil and τir are the control torques of the left and right actuated wheels, respectively. Moreover, d denotes the distance between P1 and P2 , and (x, y) denotes the position of P2 in the inertial frame OXY .

The kinematic model of the two physically linked 2WD robots in Fig. 1 is given by Ma et al. (2016): q˙ = S(q)η, (1)

1

2

2

 r1 r1 r2 r2   sin θ1 sin θ1 sin θ2 sin θ2     r1 r1 r2 r2  . B(q) =   d sin(θ1 − θ2 ) d sin(θ1 − θ2 ) b2 b2    −  r1 r1 r2 r2    b1 b1 − 0 0 r1 r1 where m1 and m2 are the robot masses, and Im1 and Im2 are the inertia parameters. Substituting the derivative of (1) into (6), and multiplying by S T (q), we have ¯ 2 (q)η + E(q, ¯ q) ¯ ¯ 1 (q)η˙ + M ˙ = B(q)τ, (7) M with (5), where ¯ 1 (q) = S T (q)M (q)S(q), M ˙ ¯ 2 (q) = S T (q)M (q)S(q), M T ¯ E(q, q) ˙ = S (q)E(q, q), ˙

2.1 Kinematic Model

T

(6)

4×4

¯ B(q) = S T (q)B(q).

(8)

¯ 1 (q) is also symmetric positive definite. Note that M 2.3 Fault Model and Actuation Redundancy Condition

T

where q = [x, y, θ2 , θ1 ] , η = [v2 , ω1 ] , v2 is the linear velocity of Robot 2, ω1 is the angular velocity of Robot 1 and

The considered actuator faults are modeled as τ (t) = σ(t)u(t),

14106

(9)

Proceedings of the 20th IFAC World Congress Yajie Ma et al. / IFAC PapersOnLine 50-1 (2017) 13563–13568 Toulouse, France, July 9-14, 2017

where u = [u1r , u1l , u2r , u2l ]T is the control signal vector that needs to be designed, σ = diag{σ1r , σ1l , σ2r , σ2l } is the uncertain control effectiveness matrix with σj ∈ [0, 1] for j = 1r, 1l, 2r, 2l, and situation σj = 1 denotes fault free, situation σj = 0 denotes the total loss of control failure and situation 0 < σj < 1 denotes the loss of partial effectiveness fault.

13565

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



Consider the dynamic equation (7) with η = [v2 , ω1 ]T . To control the two physically linked 2WD robots in Fig. 1, the following redundancy condition is needed: ¯ =2 rank(Bσ) (10) for all possible fault cases.

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 

       



Fig. 2. Block diagram of the fault-tolerant control scheme.

2.4 Control Objective and Design Issue In this paper, the control objective is to develop a faulttolerant control scheme for two physically linked 2WD mobile robots to track a desired reference path, despite the presence of actuator faults. In other words, the control objective is to design a u(t) to guarantee that all closedloop signals in (1) and (7) are bounded and limt→∞ (x(t)− xr (t)) = 0, limt→∞ (y(t) − yr (t)) = 0 and limt→∞ (θ2 (t) − θr (t)) = 0, despite the presence of uncertain σ(t) in (9) satisfying the actuation redundancy condition (10), where xr , yr , θr are desired reference trajectories. Note that the position of P1 for robot 1 is determined by the position (x, y) of P2 and the orientation θ2 of robot 2. So that this control objective contains the position of the whole system. On the other hand, from the dynamic system (7), it can be seen that the control objective is implemented by the control of ω1 that is related with θ1 . In this sense, θ1 is an intermediate variable to be controlled. The desired reference trajectories are generated by a virtual robot as follows: x˙ r = vr cos θr , (11) y˙ r = vr sin θr , (12) (13) θ˙r = ωr , where vr and ωr are the reference linear velocity and angular velocity, respectively. By choosing appropriate xr (0), yr (0), θr (0), vr and ωr , the trajectories xr , yr and θr are computed. Assumption 1. The desired xr , yr , θr , vr , ωr and their derivatives are continuous and uniformly bounded, moreover vr = 0. In this paper, the structure of the fault-tolerant control scheme is shown in Fig. 2. To implement it, the following issues need to be solved: • design a kinematic control law consisting of v2c and ω1c to ensure desired performance of kinematic system (1); • using the kinematic control law, design the faulttolerant control law u(t) to ensure system stability and tracking properties for the whole system, in the presence of actuator faults with uncertain σ(t); • evaluate the control performance. 3. FAULT COMPENSATION SCHEME

3.1 Kinematic Control Design Define the output tracking error as e˜ = [˜ ex , e˜y , e˜θ ]T = [x − xr , y − yr , θ2 − θr ]T .

(14)

Then the following error transformation is employed: e = [ex , ey , eθ ]T = Te e˜, where Te =



 cos θr sin θr 0 − sin θr cos θr 0 . 0 0 1

(15)

(16)

is the transformation matrix. Note that: Te is nonsingular with det[Te ] = 1, so that if limt→∞ e(t) = 0, then limt→∞ e˜(t) = 0. The time derivative of e is e˙ x = ωr ey + v2 cos eθ − vr , e˙ y = −ωr ex + v2 sin eθ , v2 tan(θ1 − θ2 ) − ωr . e˙ θ = d

(17) (18) (19)

Introduce a diffeomorphism: z 1 = ex , z 2 = ey , z3 = tan eθ ,

(20) (21) (22)

an additional signal: tan(θ1 − θ2 ) ωr − + ey , z4 = d cos3 eθ vr cos2 eθ and an input transformation:     v2 cos eθ − vr α1 = . α= α2 z˙4

(23)

(24)

The nonzero properties of cos eθ and cos(θ1 − θ2 ), will be analyzed in the system performance analysis section. Then, the time derivatives of z1 , z2 , z3 and z4 are z˙1 = ωr z2 + α1 , z˙2 = −ωr z1 + (vr + α1 )z3 , z˙3 = vr (z4 − z2 ) + α1 (z4 − z2 +

(25) (26) ωr (1 + z32 )), vr

z˙4 = α2 .

(27) (28)

With (23) and (24), α is calculated as

In this section, a fault-tolerant control scheme is developed for two physically linked 2WD robots in Fig. 1.

α = Tα η + fα , where Tα ∈ R

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2×2

with Tα11 = cos eθ , Tα12 = 0,

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Proceedings of the 20th IFAC World Congress 13566 Yajie Ma et al. / IFAC PapersOnLine 50-1 (2017) 13563–13568 Toulouse, France, July 9-14, 2017

¯ B ¯T M ¯ −T is positive definite. For ¯ −1 Bσ (10), the matrix M 1 1 the control design the following assumption is given.

3 tan2 (θ1 − θ2 ) sin eθ tan(θ1 − θ2 ) − 2 2 4 d cos eθ d cos3 eθ cos2 (θ1 − θ2 ) 2ωr tan(θ1 − θ2 ) sin eθ − + sin eθ , dvr cos3 eθ 1 , Tα22 = 3 d cos eθ cos2 (θ1 − θ2 ) and fα ∈ R2 with fα1 = −vr , 3ωr tan(θ1 − θ2 ) sin eθ ω˙ r fα2 = − − 4 d cos eθ vr cos2 eθ 2 ωr v˙ r 2ω sin eθ + 2 + r 3 − ωr e x . vr cos2 eθ vr cos eθ Tα21 =

Assumption 2. For the eigenvalues λ1 , λ2 of the new control ¯ B ¯T M ¯ −T , there is a known constant ¯ −1 Bσ gain matrix M 1 1 λmin > 0 such that λi > λmin , i = 1, 2. Then, the virtual control signal v is designed as 1 v= (−k4  ηe  −  fηT Tα  λmin ¯ −1 E ¯ −1 M ¯ 2η  −  M ¯  −  η˙ c ), (42) −M 1 1 where k4 > 0 is a chosen constant, and  ·  denotes the 2-norm.

Let (30) ηc = [v2c , ω1c ]T denote the kinematic control law and αc = Tα ηc + fα . (31) be a virtual kinematic control law. Now we design αc as ωr (1 + z32 )), (32) αc1 = −k1 (z1 + z3 (z4 + vr (33) αc2 = −k2 vr z3 − k3 z4 , where k1 > 0, k2 > 0 and k3 > 0 are chosen to be constant. Then we have the kinematic control law as ηc = Tα−1 (αc − fα ). (34) For the preliminary analysis, we define the velocity tracking error as ηe = η − ηc . (35) Then, we have αe = α − αc = Tα ηe . (36) Choose a positive-definite function as 1 1 (37) V1 = (z12 + z22 + z32 + z42 ). 2 k2 The time derivative of V1 is ωr k3 V˙ 1 = − k1 (z1 + z3 (z4 + (1 + z32 )))2 − z42 vr k2 (38) + fηT Tα ηe , where fη = [z1 + z3 (z4 + ωvrr (1 + z32 )), kz42 ]T . If there is no fηT Tα ηe , then V˙ 1 is nonpositive, which means that the system is stable. To implement this and ensure desired system performance, a dynamic controller will be designed. 3.2 Dynamic Control Design With (7) and (9), the dynamics of linked two 2WD robots under actuator faults is ¯ −1 M ¯ −1 E ¯ 2η − M ¯+M ¯ −1 Bσu, ¯ η˙ = −M (39) 1 1 1

where u = [u1r , u1l , u2r , u2l ]T is the control signal to be designed. Then, with (35), we have ¯ −1 M ¯ −1 E ¯ 2η − M ¯+M ¯ −1 Bσu ¯ − η˙ c . η˙ e = −M (40) 1 1 1

According to the unit sliding mode control method in Gutman (1979) and Utkin (1992), the control signal in this paper is designed as ¯T M ¯ −T ηe v, (41) u=B 1  ηe  where v ∈ R is a virtual control signal to be designed. Note that: with the actuation redundancy condition in

3.3 Performance Analysis Choose the global Lyapunov function candidate as 1 V2 = V1 + ηeT ηe . (43) 2 With (38), (40) and (41), the time derivative of V2 is ωr k3 V˙ 2 = − k1 (z1 + z3 (z4 + (1 + z32 )))2 − z42 vr k2 ¯ −1 M ¯ −1 E ¯ 2η − M ¯ + fηT Tα ηe + ηeT (−M 1 1 η e ¯ −1 Bσ ¯ B ¯T M ¯ −T +M v − η˙ c ) 1 1  ηe  ωr k3 (1 + z32 )))2 − z42 ≤ − k1 (z1 + z3 (z4 + vr k2 ¯ −1 M ¯ −1 E ¯ 2η  +  M ¯ +  ηe  ( fηT Tα  +  M 1 1 η e ¯ −1 Bσ ¯ B ¯T M ¯ −T +  η˙ c ) + ηeT M v. (44) 1 1  ηe  With v ≤ 0 in (42), we have ¯ −1 Bσ ¯ B ¯T M ¯ −T ηe v ≤ λmin  ηe 2 v. ηeT M (45) 1 1 Substituting (42) and (45) into (44) yields ωr k3 V˙ 2 ≤ − k1 (z1 + z3 (z4 + (1 + z32 )))2 − z42 vr k2 − k4  ηe 2 , (46) ∞ which indicates that z1 , z2 , z3 , z4 , ηe ∈ L which also means cos eθ and cos(θ1 − θ2 ) is not zero in (22)-(23), and z1 + z3 (z4 + ωvrr (1 + z32 )), z4 , ηe ∈ L2 . Together with (17)(34) and (39)-(41), we can further obtain that all closedloop signals are bounded, and the time derivatives of z1 + z3 (z4 + ωvrr (1 + z32 )), z4 , ηe are bounded. Then, according to Barbalat’s lemma, we have limt→∞ (z1 + z3 (z4 + ωvrr (1 + z32 ))) = 0, limt→∞ z4 = 0 and limt→∞ ηe = 0, which also implies limt→∞ αe = 0, limt→∞ α1 = 0 and limt→∞ αc1 = 0 with (36) and (32). Further more, from (23), we can obtain that z¨4 is bounded, it means that z˙4 is uniformly t continuous, then with limt→∞ 0 z˙4 (τ )dτ = −z4 (0), we have limt→∞ z˙4 = limt→∞ α2 = limt→∞ (αc2 + αe2 ) = 0 according to Barbalat’s lemma, which also means that limt→∞ αc2 = 0, limt→∞ z3 = 0, and limt→∞ z1 = 0. Similarly, from (27), we can obtain that z¨3 is bounded. then limt→∞ z˙3 = 0 is ensured according to Barbalat’s lemma. It follows that limt→∞ z2 = 0. Finally, the developed control scheme can guarantee that all closed-loop signals are bounded, and limt→∞ zi (t) = 0, i = 1, 2, 3, 4, limt→∞ (η(t) − ηc (t)) = 0, which implies limt→∞ (x(t) −

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xr (t)) = 0, limt→∞ (y(t) − yr (t)) = 0 and limt→∞ (θ2 (t) − θr (t)) = 0, despite the presence of actuator faults.

6

In summary, the following result is presented.

5

Theorem 1. The developed fault tolerant control scheme, constituting the kinematic control law in (32) and (33) and the dynamic control law in (41) and (42) applied to two linked 2WD mobile robots modeled as (1) and (7), guarantees that all closed-loop signals are bounded and limt→∞ (x(t) − xr (t)) = 0, limt→∞ (y(t) − yr (t)) = 0, limt→∞ (θ2 (t)−θr (t)) = 0, despite the presence of actuator faults modeled as (9).

4

Remark 1. The control signal v(t) in (41) may be chattering due to the use of ηηee  . To deal with this problem, the e common approximation function ηeη+ with  > 0 can ηe be used to replace ηe  , which makes the control signal be continuous. However, this approximation function can guarantee that the system tracking error converges to a threshold near zero but not go to zero (Tao (2003)), and the convergence value can be made small by an appropriate choice of small . 

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robot 2 reference robot 1

Y (m)

3 2 1 0 −1 −2 −6

−5

−4

−3

−2

X (m)

−1

0

1

2

3

Fig. 3. Robot paths: Robot 2 (solid), Reference (dashed), Robot 1 (dot-dashed). 1 x−xr (m)

4. SIMULATION STUDY To verify the effectiveness of the developed fault tolerant control scheme for linked two 2WD mobile robots shown in Fig. 1, the following simulation study is presented.

0 −0.5 0

20

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1 y−yr (m)

In this simulation, the two 2WD robots are chosen as the one in Fukao et al. (2000), and the physical parameters are: a1 = a2 = 0.3 m, b1 = b2 = 0.75 m, r1 = r2 = 0.15 m, m1 = m2 = 30 kg, Im1 = Im2 = 15.625 kg m2 . The length of the link is assumed to be d = 1.7 m. The desired trajectory is calculated by (11)-(13) with vr = 0.5 m/s, ωr = 0.2 rad/s, xr (0) = yr (0) = 0 and θr (0) = 45 deg.

0.5 0 −0.5 0 50

θ2−θr (deg)

The fault scenario is considered as: σ = diag{1, 1, 1, 1} for 0 ≤ t < 10s; σ = diag{0, 1, 1, 1} for 10s ≤ t < 30s; σ = diag{0, 0.5, 1, 0.6} for 30s ≤ t < 50s; σ = diag{0, 0.5, 1, 0} for 50s ≤ t < 70s; σ = diag{1, 0.5, 1, 0} for 70s ≤ t < 90s; σ = diag{1, 0.5, 0, 0} for 90s ≤ t < 110s; σ = diag{0.3, 0.5, 0, 0} for t ≥ 110s.

0.5

0 −50 0

Fig. 4. Output tracking errors e˜ = [x − xr , y − yr , θ2 − θr ]T .

The initials values are chosen as: x(0) = 0, y(0) = 1 m, θ2 (0) = 10 deg, θ1 (0) = 0, v2 (0) = 0, and ω1 (0) = 0. The control parameters are chosen as: k1 = 1, k2 = 1, k3 = 0.2, k4 = 0.05, λmin = 0.001 and  = 0.1, and the guideline in choosing them can be found in Ma et al. (2015).

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Fig. 3 shows the positions of the reference robot (dashed), robot 1 (dot-dash) and robot 2 (solid). Fig. 4 shows the system tracking errors. Fig. 5 shows the orientation error between the two robots. Fig. 6 and Fig. 7 show the control torques (solid) and designed control signal (dashed) generated by the the wheels in two robots, respectively. From them, we can see that the system stability is ensured despite the presence of actuator faults. On the other hand although the system tracking errors do e not go to zero because ηηee  is replaced by ηeη+ in the control signal (41), they are in small regions near zero by choosing a small  = 0.1. On the other hand, because the link is fixed with robot 2, the two robot can not track the same circular reference path, which makes the orientation error between the two robots are not zero.

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Fig. 5. Orientation error between two robots: θ1 − θ2 .

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5 τ1r (Nm)

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Fig. 6. Control inputs in robot 1. 1 τ2r

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Fig. 7. Control inputs in robot 2. 5. CONCLUSION This paper developed a fault-tolerant control scheme for two physically linked 2WD mobile robots with multiple and simultaneous actuator faults. To improve the fault tolerance of 2WD mobile robots, some physical links are employed as shown in Fig. 1 for two linked robots. The kinematics and dynamics were modeled. Using such models,the kinematic and dynamic control laws were designed, which ensured desired system stability and tracking properties. For the control design, a new positive definite control gain matrix was constructed, and the lower boundary of its eigenvalues is used. The effectiveness of the proposed faulttolerant control scheme was demonstrated by simulation results. Research on the fault-tolerant control for n(n > 2) linked mobile robots is our future interest. REFERENCES Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M. (2006). Diagnosis and Fault-Tolerant Control. SpringerVerlag, Berlin Heidelberg.

Ding, S. (2008). Model-based Fault Diagnosis Techniques: Design Schemes, Algorithms, and Tools. SpringerVerlag, Berlin Heidelberg. Dixon, W., Dawson, D., Zergeroglu, E., and Behal, A. (2001). Adaptive tracking control of a wheeled mobile robot via an uncalibrated camera system. IEEE Transactions on System, Mam, and Cybernetics-Part B: Cybernetics, 31, 341–352. Fourlas, G., Karkanis, S., Kostas, G., and Kyriakopoulos, K. (2014). Model based actuator fault diagnosis for a mobile robot. In IEEE International Conference on Industrial Technology, 79–84. Busan, Korea. Fukao, T., Nakagawa, H., and Adachi, N. (2000). Adaptive tracking control of a nonholonomic mobile robot. IEEE Transactions on Robotics and Automation, 16, 609–615. Gutman, S. (1979). Uncertain dynamic systems - a lyapunov min-max approach. IEEE Transactions on Automatic Control, AC-24, 437–449. Ji, M. and Sarkar, N. (2007). Supervisory fault adaptive control of a mobile robot and its application in sensorfault accommodation. IEEE Transactions on Robotics, 23, 174–178. 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– 233. Khalaji, A. and Moosavian, S. (2014). Robust adaptive control for a trackor-trailer mobile robot. IEEE/ASME Transactions on Mechatronics, 19, 943–953. Koh, M., Noton, M., and Khoo, S. (2012). Robust faulttolerant leader-follower control of four-wheel-steering mobile robots using terminal sliding mode. Australian Journal of Electrical and Electronics Engineering, 9, 247–254. Ma, Y., Cocquempot, V., Najjar, M., and Jiang, B. (2016). Multipl-model based actuator fault compensation for two linked 2wd mobile robots. In Proceedings of the 3rd Conference on Control and Fault-Tolerant Systems, 33–38. Barcelona, Spain. Ma, Y., Tao, G., Jiang, B., Liu, H.H., and Jiang, B. (2015). Multiple-model based adaptive compensation of actuation sign uncertainty using an error transformation. In American Control Conference, 165–170. Chicago, USA. Rotondo, D., Puig, V., Nejjari, F., and Romera, J. (2014). A fault-hiding approach for the switching quasi-lpv fault-tolerant control of a four-wheeled omnidirectional mobile robot. IEEE Transactions on Industrial Electronics, 62, 3932–3944. Samson, C. (1995). Control of chained systems application to path following and time-varying point-stabilization of mobile robots. IEEE Transactions on Automatic Control, 41, 64–77. Tao, G. (2003). Adaptive Control Design and Analysis. John Wiley & Sons, New Jersey. Utkin, V. (1992). Sliding Modes in Control and Optimization. Springer-Verlag, Berlin Heidelberg. Yu, X. and Jiang, J. (2015). A survey of fault-tolerant controllers based on safety-related issues. Annual Reviews in Control, 39, 46–57. Yuan, J., Sun, F., and Huang, Y. (2015). Trajectory generation and tracking control for double-steering tractortrailer mobile robots with on-axle hitching. IEEE Transactions on Industrial Electronics, 62, 7665–7677.

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