An adaptive actuator failure compensation scheme for two linked 2WD mobile robots

Transcription

1 Journal of Physics: Conference Series PAPER OPEN ACCESS An adaptive actuator failure compensation scheme for two linked 2WD mobile robots To cite this article: Yajie Ma et al 27 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - An Integrable Discrete Generalized Nonlinear Schrödinger Equation and Its Reductions Li Hong-Min, Li Yu-Qi and Chen Yong - Lattice gases with a point source P L Krapivsky and Darko Stefanovic - On the determination of observables in d scattering F Pereira and E Ferreira Recent citations - Actuator fault estimation and fault tolerant control in three physically-linked 2WD mobile robots Daniel A. Pereira et al This content was downloaded from IP address on 9//29 at 9:26

2 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 International Conference on Recent Trends in Physics 26 (ICRTP26) Journal of Physics: Conference Series 755 (26) doi:.88/ /755// An adaptive actuator failure compensation scheme for two linked 2WD mobile robots Yajie Ma, Ayad Al-Dujaili, Vincent Cocquempot and Maan El Badaoui El Najjar University of Lille, CNRS, Centrale Lille, UMR 989-CRIStAL-Centre de Recherche en Informatique, Signal et Automatique de Lille, F-59 Lille, France Abstract. This paper develops a new adaptive compensation control scheme for two linked mobile robots with actuator failurs. A configuration with two linked two-wheel drive (2WD) mobile robots is proposed, and the modelling of its kinematics and dynamics are given. An adaptive failure compensation scheme is developed to compensate actuator failures, consisting of a kinematic controller and a multi-design integration based dynamic controller. The kinematic controller is a virtual one, and based on which, multiple adaptive dynamic control signals are designed which covers all possible failure cases. By combing these dynamic control signals, the dynamic controller is designed, which ensures system stability and asymptotic tracking properties. Simulation results verify the effectiveness of the proposed adaptive failure compensation scheme.. Introduction Due to simplicity, efficiency and flexibility, two-wheel drive (2WD) mobile robots are widely used []. In some harsh conditions, resulting from a terrorist attack, a nuclear accident or nature disasters, 2WD rescue robots carrying instruments are employed to help rescuers. These adverse conditions may increase the probability of actuator (wheel motor) failures which will leads to the loss of the failed robots and the important instruments. To deal with this situation, several physical links may be used to connect the robots. For example, if one robot fails, then another robot carrying a mechanical arm can be sent to link the failed robot and to help it continue moving. This robot architecture provides actuator and sensor redundancies which improve the fault tolerance of the system. This paper is concerned with the fault tolerant control design for such a configuration with two linked 2WD robots. There are a lot of research results on the motion control of 2WD mobile robots [2, 3] but without the consideration of actuator faults. For fault tolerant control, various effective design methods are proposed for different applications [4, 5]. As for the application to wheeled robots, a sensor fault accommodation scheme is presented in [6]; a fault-tolerant control design method is given in [7] for four-wheel drive (4WD) mobile robots; and a hybrid adaptive fault tolerant control scheme is proposed in [8] to accommodate partial faults and degradation for 2WD robots. Unlike 4WD robots, a 2WD robot has no redundant actuator, so that if one motor is lost, then the 2WD robot becomes uncontrollable. To our knowledge, there is no research result on the fault tolerant control of 2WD robots with the motors (actuators) being totally faulty. To deal with such actuator failures, physical Content from this work may be used under the terms of the Creative Commons Attribution 3. licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd

3 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 links are used to connect several 2WD mobile robots as shown in Fig. for two robots. For this configuration, if one or two motors are lost, the remaining motors can continue moving the two robots. In this paper, a multi-design integration based adaptive control scheme is developed. The main contributions are as follows. (i) The kinematic and dynamic models of the two 2WD mobile robots with fixed link are proposed. (ii) A new adaptive actuator failure compensation scheme is developed using a multi-design integration method, which ensures system stability and asymptotic tracking properties, despite the presence of actuator failures including simultaneous multi-failures. Y Robot 2b Robot 2 a 2 l a 2l C P y P 2 C 2 2 r 2b 2 d 2r 2r O x 2r 2 X Figure : Two linked 2WD mobile robots The rest of this paper is as follows. In Section 2, system modelling is given and the actuator failure compensation problem is formulated. In Section 3, a multi-design integration based adaptive actuator failure compensation scheme is developed. In Section 4, a simulation study is presented to demonstrate the effectiveness of the proposed adaptive control scheme. Conclusions follow in Section System Modeling and Problem Formulation In this section, the system models are given for the two linked 2WD mobile robots as shown in Fig., and its actuator failure compensation problem is formulated. As shown in Fig., the orientation of robot 2 is consistent with the one of the physical link, but the orientation of robot is independent, and for each robot: the front wheel is passive and the two rear wheels are actuated; P i (i =, 2) is the center between two actuated wheels, C i is the center of mass, a i is the distance between P i and C i, b i is half of the distance between two actuated wheels, r i is the radius of wheels, θ i is the orientation of the robot, and τ il and τ ir are the control torques applied to the left and right actuated wheels, respectively. Moveover, d is the distance between P and P 2, OXY is the inertial frame, and (x, y) denotes the position of P 2 in frame OXY. Kinematic model. Let q = [x, y, θ 2, θ ] T and η = [v 2, ω ] T, where v 2 is the linear velocity of robot 2, and ω is the angular velocity of robot. The kinematic model of the two linked 2WD mobile robots in Fig. is given by q = S(q)η. () 2

4 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 where S(q) = [ cos θ2 sin θ 2 d tan(θ θ 2 ) Moreover, the system constraints may be expressed as ] T. (2) A(q) q =, (3) where A(q) = [ sin θ cos θ d cos(θ θ 2 ) sin θ 2 cos θ 2 ] (4) is the system constraint matrix. Then, we have the following property: S T (q)a T (q) =. (5) Dynamic model. The dynamic equation of these two linked 2WD robots is obtained by using the Lagrange method as: M(q) q + E(q, q) = B(q)τ + A T (q)λ, (6) where M(q) R 4 4 is the inertia matrix that is symmetric positive definite, E(q, q) R 4 is the centripetal and coriolis vector, B(q) R 4 4 is the input injection matrix, τ = [τ r, τ l, τ 2r, τ 2l ] T is the vector of control torques, and λ R 2 is the vector of constraint forces. The matrices and vectors in (6) are given by M(q) = m + m 2 (a 2 m 2 + dm ) sin θ 2 a m sin θ m + m 2 (a 2 m 2 + dm ) cos θ 2 a m cos θ (a 2 m 2 + dm ) sin θ 2 (a 2 m 2 + dm ) cos θ 2 m 2 a m d 2 + I m2 a dm cos(θ θ 2 ) a m sin θ a m cos θ a dm cos(θ θ 2 ) m a 2 + I m E(q, q) = B(q) = a m θ2 cos θ (a 2 m 2 + dm ) θ 2 2 cos θ 2 a m θ2 sin θ (a 2 m 2 + dm ) θ 2 2 sin θ 2 a dm θ2 sin(θ θ 2 ) a dm θ2 2 sin(θ θ 2 ) cos θ r sin θ cos θ r sin θ cos θ 2 r 2 sin θ 2 cos θ 2 r 2 sin θ 2 r r r 2 r 2 d sin(θ θ 2 ) d sin(θ θ 2 ) b 2 r r r 2 b 2 r 2 b r b r,,, (7) where m and m 2 are the masses of the robot and robot 2, and I m and I m2 are the corresponding inertia parameters. Substituting () into (6), and multiplying by S T (q), A T (q)λ is eliminated with (5). Then equation (6) becomes where η = [v 2, ω ] T, and (q) η + 2 (q)η + Ē(q, q) = B(q)τ, (8) (q) = S T (q)m(q)s(q), 2 (q) = S T (q)m(q)ṡ(q), Ē(q, q) = S T (q)e(q, q), B(q) = S T (q)b(q). (9) 3

5 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 Note that (q) is also symmetric positive definite as well as M(q). Actuator failure model. The considered actuator failure is that some motors totally lose power or are stuck which will introduce additional frictions. This type of actuator failure for one motor is modeled as τ j (t) = ū j, t t j () where j = r, l, 2r, 2l, ū j is the friction value that is unknown but constant, and ū j = means that the motor loses its power but can rotate freely and ū j > means that the motor is stuck or can not rotate freely caused by some frictions in the bearing, and t j is the unknown failure occurring time instant. Consider that all actuators may be faulty. The control torque in (8) becomes τ(t) = σ(t)u(t) + (I 4 σ(t))ū, () where τ = [τ r, τ l, τ 2r, τ 2l ] T is the vector of control torques generated by the wheel motors, u = [u r, u l, u 2r, u 2l ] T is the control signal vector to be designed, ū = [ū r, ū l, ū 2r, ū 2l ] T is the vector of constant frictions, I 4 is the identity matrix, and σ = diag{σ r, σ l, σ 2r, σ 2l } is the uncertain failure pattern matrix with {, if the jth motor fails, σ j (t) =, otherwise. (2) Actuation redundancy. For the two linked 2WD mobile robots in Fig., the following actuation redundancy condition needs to be satisfied: rank( Bσ) = 2. (3) for all possible failure pattern matrices σ. This condition means that there are enough actuated motors to control η = [v 2, ω ] T. Remark With B(q) in (9), the compensable failure cases satisfying this redundancy condition are: ) fault free case with four actuated wheels; 2) one actuator fails with three remaining actuated wheels; 3) two actuators fail with two remaining actuated wheels. However, for case 3), if the two failed actuators are in robot 2, then the system is similar with a tractor-trailer system [9], and the failures may be tolerated; if each robot has one faulty actuator, then the failures are also compensable; but if the two failed actuators are in robot, then the failures are not compensable, because in this case, rank( Bσ) = with σ = diag{,,, } and ω is uncontrollable. Control objective. The control objective is to develop an actuator failure compensation scheme for two linked 2WD mobile robots in Fig. to asymptotically track a reference trajectory, that is, to design a control signal u(t) to guarantee that all closed-loop system signals are bounded, and lim t (x(t) x r (t)) =, lim t (y(t) y r (t)) = and lim t (θ 2 (t) θ r (t)) = for system (3) and (8), in the presence of some actuator failures modeled as ()-(2) that satisfies the actuation redundancy condition in (3), where x r, y r, θ r are the reference trajectories. Remark 2 In Fig., the position of P is determined by x, y and θ 2, in this sense, the control objective contains the position for the whole system. In addition, this objective is implemented by the control of ω with the dynamic equation (8), as θ = ω, we can see that the orientation angle θ is an intermediate variable to be controlled. 4

6 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 A virtual robot is employed to generate the reference trajectories as follows: ẋ r = cos θ r, ẏ r = sin θ r, θr = ω r. (4) where and ω r are the linear velocity and angular velocity. By choosing appropriate, ω r and initial values x r (), y r () and θ r (), the reference trajectories x r, y r and θ r are determined. In this paper, we consider the tracking problem of a two-robot system. Then the following assumption is given for the reference trajectories. Assumption : The reference trajectories x r, y r and θ r, the velocities and ω r, and their derivatives are continuous and uniformly bounded, moreover. Design issues. The structure of the proposed actuator failure compensation scheme is shown in Fig. 2. To design such a control scheme, the following technical issues need to be solved: x y r r r v c c u u x y v Figure 2: Actuator failure compensation control scheme. to design a kinematic control law η c = [v 2c, ω c ] T, such that when it is applied, the desired control performance can be ensured; to design a dynamic control law u using the designed kinematic control law, to achieve the control objective; to handle the failure uncertainties, for which, a multi-design integration based adaptive method will be employed; to evaluate the control performance. 3. Multi-Design Integration based Adaptive Failure Compensation Scheme In this section, a multi-design integration based adaptive actuator failure compensation scheme is developed including a kinematic controller and a dynamic controller. 3.. Kinematic Controller Design 3... Kinematic control law Define the output tracking error vector as ẽ = [ẽ x, ẽ y, ẽ θ ] T = [x x r, y y r, θ 2 θ r ] T, (5) and a transformation matrix as T e = cos θ r sin θ r sin θ r cos θ r. (6) 5

7 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 Then, a new error vector is defined as e = [e x, e y, e θ ] T = T e ẽ. (7) Since T e is nonsingular with det[t e ] =, if lim t e(t) =, then lim t ẽ(t) =. It follows that Introduce the diffeomorphism: ė x = ω r e y + v 2 cos e θ, (8) ė y = ω r e x + v 2 sin e θ, (9) ė θ = v 2 d tan(θ θ 2 ) ω r. (2) z = e x, (2) z 2 = e y, (22) z 3 = tan e θ, (23) an additional signal: and an input transformation: z 4 = tan(θ θ 2 ) d cos 3 e θ α = [ α ] = α 2 Then, the derivatives of z, z 2, z 3 and z 4 are From (24) and (25), we can obtain ω r cos 2 e θ + e y, (24) [ ] v2 cos e θ. (25) ż 4 ż = ω r z 2 + α, (26) ż 2 = ω r z + ( + α )z 3, (27) ż 3 = (z 4 z 2 ) + α (z 4 z 2 + ω r ( + z v 3)), 2 r (28) ż 4 = α 2. (29) α = T α η + f α, (3) where T α = [ ] [ ] Tα T α2 fα, f T α2 T α =, (3) α22 f α2 with T α = cos e θ, T α2 =, T α2 = 3 tan2 (θ θ 2 ) sin e θ d 2 cos 4 e θ T α22 = d cos 3 e θ cos 2 (θ θ 2 ), f α =, tan(θ θ 2 ) d 2 cos 3 e θ cos 2 (θ θ 2 ) 2ω r tan(θ θ 2 ) sin e θ d cos 3 e θ + sin e θ, f α2 = 3ω r tan(θ θ 2 ) sin e θ d cos 4 e θ ω r cos 2 + ω r e θ vr 2 cos 2 + 2ω2 r sin e θ e θ cos 3 ω r e x. e θ 6

8 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 Define a virtual kinematic control signal as and design it as α c = T α η c + f α, (32) α c = k (z + z 3 (z 4 + ω r ( + z v 3)), 2 r (33) α c2 = k 2 z 3 k 3 z 4, (34) where k >, k 2 > and k 3 > are chosen to be constant. Then, with (32), the kinematic control law is Performance analysis Define the velocity tracking error as Then, we have η c = T α (α c f α ). (35) η e = η η c. (36) α e = [α e, α e2 ] T = α α c = T α η e. (37) For the preliminary analysis of the designed kinematic control law, a positive-definite function is chosen as With (26)-(29), the time derivative of V is V = 2 (z2 + z z k 2 z 2 4). (38) V =(z + z 3 (z 4 + ω r ( + z 2 3)))α c + z 3 z 4 + z 4 k 2 α c2 + (z + z 3 (z 4 + ω r ( + z 2 3)))α e + z 4 k 2 α e2. (39) Letting f η = [z + z 3 (z 4 + ωr ( + z3 2)), z 4 k 2 ] T and substituting (33), (34) and (37) into (39), we have V = k (z + z 3 (z 4 + ω r ( + z 2 3))) 2 k 3 k 2 z f T η T α η e. (4) If there is no f T η T α η e, then V is nonpositive, which means the system is stable. To eliminate it and ensure desired system performance, a dynamic controller is designed in the next section Dynamic Controller Design Multi-design integration Substituting () into (8), we have the dynamic equations with actuator failures as follows: η = = 2 η 2 η Ē + Ē + Bσu + Bσu + B(I 4 σ)ū Bū f, (4) where ū f = (I 4 σ)ū, and u = [u r, u l, u 2r, u 2l ] T is the applied control signal to be designed. 7

9 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 Let σ (k), k =, 2,..., N denote the kth possible failure pattern matrix satisfying actuation redundancy condition (3), where N is the number of all possible failure pattern matrices that are under consideration. For each σ (k), a corresponding dynamic control signal u (k) will be designed such that the desired system performance can be ensured if u = u (k) and σ = σ (k). Then, to cover all possible failure patterns, a nominal multi-design integration based dynamic control signal is constructed as where u = χ (k) u (k) = k= diag{χ (k), χ (k) χ (k), χ (k) }u (k), (42) k= χ (k) = { if σ = σ(k) otherwise. (43) Dynamic control law Since the actual failure pattern σ is uncertain, χ (k) is also uncertain for k =, 2,..., N. To deal with such uncertainties, in this paper, the dynamic control law is designed as u = ˆχ (k) u (k), (44) k= where ˆχ (k) = diag{ˆχ (k)r, ˆχ (k)l, ˆχ (k)2r, ˆχ (k)2l } is the estimate matrix that will be described in the following. With (4), the time derivative of the velocity tracking error η e = η η c in (36) is η e = 2 η Ē + Bσu + Bū f η c. (45) Let ˆū f denote the estimate of ū f. Then, the dynamic control signal u (k) = [u (k)r, u (k)l, u (k)2r, u (k)2l ] T is designed as u (k) = ( Bσ (k) ) + ( k 4 η e T T α f η + 2 η + Ē Bˆū f + η c ) (46) for σ (k), k =, 2,..., N, where k 4 > is a chosen constant, and ( inverse matrix satisfying Bσ(k)( Bσ (k) ) + = I 2. Bσ (k) ) + is a generalized Remark 3 Some elements of some u (k) may be zero. For example, if σ (k) = diag{,,, } and ( Bσ (k) ) + = σ (k) ( B) T ( Bσ (k) ( B) T ), then u (k)r =. In this sense, the estimate matrix ˆχ (k) in (44) can be simplified for some σ (k). For instance, for σ (k) = diag{,,, }, ˆχ (k) = diag{, ˆχ (k)l, ˆχ (k)2r, ˆχ (k)2l }; and for σ (k) = diag{,,, }, ˆχ (k) = diag{, ˆχ (k)l, ˆχ (k)2r, }. This can reduce the number of estimates. Define the estimation errors as ū f = ū f ˆū f, χ (k)r = χ (k) ˆχ (k)r, χ (k)l = χ (k) ˆχ (k)l, χ (k)2r = χ (k) ˆχ (k)2r, χ (k)2l = χ (k) ˆχ (k)2l, (47) 8

10 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 for k =, 2,..., N. Then, substituting (42), (44) and (46) into (45), we have η e = 2 η Ē + Bσu + = k 4 η e Tα T f η + B ū f ( B) c σ r χ (k)r u (k)r ( k= ( k= B) c3 σ 2r χ (k)2r u (k)2r k= ( k= Bū f η c + B) c2 σ l χ (k)l u (k)l Bσ(u u ) B) c4 σ 2l χ (k)2l u (k)2l, (48) where ( B) ci (i =, 2, 3, 4) denotes the ith column vector of matrix B Adaptive laws To construct the control signal u (k) in (46), the adaptive law of ˆū f is chosen as ˆū f = Γ f ( B) T η e, (49) where Γ f = Γ T f R4 4 is the positive-definite adaptation gain matrix that is chosen to be constant. To construct the dynamic control law u in (44), the adaptive laws of ˆχ (k)r, ˆχ (k)l, ˆχ (k)2r and ˆχ (k)2l are chosen as ˆχ (k)r = γ kr u (k)r η T e ( ˆχ (k)l = γ kl u (k)l η T e ( ˆχ (k)2r = γ k2r u (k)2r η T e ( ˆχ (k)2l = γ k2l u (k)2l η T e ( B) c + f kr, B) c2 + f kl, B) c3 + f k2r, B) c4 + f k2l, (5) for k =, 2,..., N, where γ kr >, γ kl >, γ k2r > and γ k2l > are chosen to be constant, and f kr, f kl, f k2r and f k2l are the projection signals. Let p kr = γ kr u (k)r ηe T ( B) c, p kl = γ kl u (k)l ηe T ( B) c2, p k2r = γ k2r u (k)2r ηe T ( B) c3 and p k2l = γ k2l u (k)2l ηe T ( B) c4. The projection signals are given as follows:, if ˆχ (k)j (, ), or if ˆχ f kj = (k)j = and p kj, or (5) if ˆχ (k)j = and p kj, p kj, otherwise, for j = r, l, 2r, 2l. Lemma The adaptive laws in (5) with the projection signals in (5) for k =, 2,..., N guarantee that: i) ˆχ (k)r, ˆχ (k)l, ˆχ (k)2r, ˆχ (k)2l [, ]; and ii) χ (k)r f kr, χ (k)l f kl, χ (k)2r f k2r and χ (k)2l f k2l. Proof: The proofs for the four adaptive laws are similar. Here the details for ˆχ (k)r are given as an example. Choose the initial estimate as ˆχ (k)r () [, ]. The projection signal in (5) with j = r ensures ˆχ (k)r (t) [, ], and (χ (k)r ˆχ (k)r )f kr = χ (k)r f kr (52) 9

11 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 that is analyzed for the following three cases: (i) if ˆχ (k)r = and p kr <, then χ (k)r ˆχ (k)r and f kr = p kr >, which means (52) is ensured; (ii) if ˆχ (k)r = and p kr >, then χ (k)r ˆχ (k)r and f kr = p kr <, which also means (52) is ensured; and (iii) otherwise, we have f kr =, then (52) is ensured. Similarly, we can also obtain the same properties for the other three adaptive laws. The proof is completed System performance Choose the global Lyapunov function candidate as V 2 =V + 2 ηt e η e + 2 ū T f Γ f ū f k= Then, the derivative of V 2 is σ 2r γ k2r χ2 (k)2r + 2 k= k= σ r γ kr χ2 (k)r + 2 k= σ l γ kl χ2 (k)l σ 2l γ k2l χ2 (k)2l. (53) V 2 k (z + z 3 (z 4 + ω r ( + z 2 3))) 2 k 3 k 2 z 2 4 k 4 η T e η e, (54) which indicates: z, z 2, z 3, z 4, η e, z + z 3 (z 4 + ωr ( + z3 2)) L and all estimates are bounded, and z 4, η e, z + z 3 (z 4 + ωr ( + z3 2)) L2. It follows from (23) and (24) that cos e θ and tan(θ θ 2 ) L meaning cos(θ θ 2 ). Then, with (8)-(35) and (4)-(46), we have: T α is nonsingular and bounded, and f α, α c, α e, α, η c, η, ż, ż 2, ż 3, ż 4, α c, η c, u (k), η, η e, α e, α L, which also implies that the derivative of z + z 3 (z 4 + ωr ( + z3 2 )) is bounded. According to Barbalat s lemma, we can conclude that all closed-loop signals are bounded, and lim t (z + z 3 (z 4 + ωr ( + z3 2))) =, lim t z 4 = and lim t η e =, which also means lim t α e =, lim t α c =, and lim t α = with (37) and (33). From (29), z 4 = α 2 = α c2 + α e2 L with α c2, α e2 L, which means that ż 4 is t uniformly continuous, together with lim t ż4(τ)dτ = z 4 ( ) z 4 () = z 4 (), we have lim t ż 4 = lim t α 2 = lim t (α c2 + α e2 ) = according to Barbalat s lemma. Then, with lim t α e =, lim t z 4 =, α c2 = k 2 z 3 k 3 z 4 and, we have lim t α c2 = and lim t z 3 =, it follows that lim t z = with lim t (z + z 3 (z 4 + ωr ( + z3 2 ))) =. On the other hand, from ż 3 = (z 4 z 2 ) + α (z 4 z 2 + ωr ( + z3 2)) in (28), we have z 3 L. Similarly, lim t ż 3 = is ensured according to Barbalat s lemma. Then, we obtain lim t z 2 = with lim t z 4 =, lim t α = and. Finally, we can conclude that: all closed-loop signals are bounded, and lim t z i (t) = (i =, 2, 3, 4) and lim t (η(t) η c (t)) =, which also means lim t (x(t) x r (t)) =, lim t (y(t) y r (t)) = and lim t (θ 2 (t) θ r (t)) = with the diffeomorphism in (2)-(23) and the transformation in (7). In summary, we have the following theorem. Theorem The developed multi-design integration based adaptive actuator failure compensation control scheme, constituted by the kinematic control law in (35), and dynamic control law in (44) with multiple control signals in (46) and updated by the projection adaptive laws in (49) and (5), applied to two linked 2WD mobile robots modeled as () and (8), guarantees that all closed-loop signals are bounded, and lim t (x(t) x r (t)) =, lim t (y(t) y r (t)) = and lim t (θ 2 (t) θ r (t)) =, despite the presence of actuator failures modeled as ()-(2).

12 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 4. Simulation Study To verify the effectiveness of the developed multi-design integration based adaptive actuator failure compensation scheme for two linked 2WD robots shown in Fig., the following simulation study is presented. 4.. Simulation Conditions In this simulation, the physical parameters of each robot are a = a 2 =.3 m, b = b 2 =.75 m, r = r 2 =.5 m, m = m 2 = 3 kg, I m = I m2 = kg m 2. The length of the link is assumed to be d =.7 m. To verify the system tracking property, a circle reference trajectory is considered. To generate such reference a trajectory, the velocities and ω r are chosen as: =.5 m/s and ω r =.5 rad/s. Then, x r, y r and θ r are generated by (4) with x r () = y r () = and θ r () = 45 deg. In order to verify the failure compensation effectiveness of the developed adaptive control scheme, the following failure cases are simulated: no failure, σ () = diag{,,, }, t < s. τ r fails, σ (2) = diag{,,, }, τ r =, s t < 2s; τ r, τ 2l fail, σ (3) = diag{,,, }, τ r =, τ 2l = 2 Nm, 2s t < 3s; τ 2l fails, σ (4) = diag{,,, }, τ 2l = 2 Nm, 3s t < 4s; τ 2r, τ 2l fail, σ (5) = diag{,,, }, τ 2r = Nm, τ 2l = 2 Nm, t 4s. There are 5 failure pattern matrices which satisfy the actuation redundancy condition in (3), covering the cases of fault free, one actuator fails, both two actuators of robot 2 fail, and one actuator of each robot fails. The initial conditions are chosen as: x() =, y() = m, θ 2 () = deg, θ () =, v 2 () =, and ω () =. The adaptation gains are chosen as: Γ f = I 4, and γ kr = γ kl = γ k2r = γ k2l = for k =, 2, 3, 4, 5. The control gains are chosen as: k =, k 2 =, k 3 =.2 and k 4 = Simulation Results Robot 2 Reference Robot τ r (Nm) τ l (Nm) Y (m) τ 2r (Nm) τ 2l (Nm) X (m) t (sec) Figure 3: Robot trajectories in (X, Y ) plane. Figure 4: Control torques. Fig. 3 shows the positions of the robot 2, the reference robot and the robot. Fig. 4 shows the control torques generated by the four wheels, which are consistent with the faulty cases in simulation conditions. Fig. 5 shows the tracking errors. Fig. 6 shows the orientation error

13 3th European Workshop on Advanced Control and Diagnosis (ACD 26) IOP Conf. Series: Journal of Physics: Conf. Series 783 (27) 22 doi:.88/ /783//22 x x r (m) y y r (m) θ 2 θ r (deg) t (sec) θ θ 2 (deg) t (sec) Figure 5: Tracking errors. Figure 6: Orientation error of two robots. between two robots. From them, we can see that the desired system stability and asymptotic tracking properties are ensured by the developed multi-design integration based adaptive failure compensation scheme, despite the presence of some actuator failures. 5. Conclusions This paper developed a multi-design integration based adaptive actuator failure compensation scheme for two linked 2WD robots. The kinematics and dynamics of this robot configuration were modeled, for which, the proposed adaptive failure compensation scheme was designed. The developed control scheme ensures desired system stability and asymptotic tracking properties, which was verified by simulation results. Extending the proposed method for n(n > 2) linked 2WD mobile robots is our interest in the 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 [] Dixon W, D. Dawson, Zergeroglu E and Behal A. 2 Adaptive tracking control of a wheeled mobile robot via an uncalibrated camera system IEEE T. Syst., Man, Cybern., Cybern , [2] Fierro R and Lewis F 998 Control of a nonholonomic mobile robot using neural networks IEEE T. Neural Network [3] Huang J, Wen C, Wang W and Jiang Z 24 Adaptive output feedback tracking control of a nonholonomic mobile robot Automatica [4] Zhang Y and Jiang J, 28 Bibliographical review on reconfigurable fault-tolerant control systems Annu. Rev. Control [5] Blanke M, Kinnaert M, Lunze J and Staroswiecki M 26 Diagnosis and Fault-Tolerant Control Springer- Verlag Berlin Heidelberg [6] Ji M and Sarkar N 27 Supervisory fault adaptive control of a mobile robot and its application in sensor-fault accommodation IEEE T. Robotics [7] Rotondo D, Puig V, Nejjari F and Romera J 24 A fault-hiding approach for the switching Quasi-LPV fault-tolerant control of a four-wheeled omnidirectional mobile robot IEEE T. Ind. Electron [8] Ji M, Zhang Z, Biswas G and Sarkar N 23 Hybrid fault adaptive control of a wheeled mobile robot IEEE-ASME T. Mech [9] Khalaji A and Moosavian S 24 Robust adaptive controller for a trackor-trailer mobile robot IEEE-ASME T. Mech