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2 Robotics and Autonomous Systems 55 (27) Fault-tolerant robot manipulators based on output-feedback H controllers A.A.G. Siqueira a M.H. Terra b C. Buosi b a University of São Paulo at São Carlos Mechanical Engineering Department São Carlos SP Brazil b University of São Paulo at São Carlos Electrical Engineering Department São Carlos SP Brazil Received 22 February 26; received in revised form 26 April 27; accepted 28 April 27 Available online 17 May 27 Abstract This paper develops two fault-tolerant control strategies for robot manipulators. The first is based on linear parameter-varying systems and the second on Markovian jump linear systems. Firstly it is shown that with the LPV approach post-fault stability is guaranteed only if the robot stops completely after a fault detection. Then with an underactuated configuration the manipulator can be controlled appropriately. Secondly it is shown that with the fault-tolerant system based on Markovian jump linear systems stability is guaranteed after a fault is detected even with the robot still moving. This approach incorporates all manipulator configurations in a unified model. Both strategies have been implemented based on output-feedback H controllers which are the main focus of this paper. Experimental results illustrate the performance of each controller. c 27 Elsevier B.V. All rights reserved. Keywords: Fault-tolerant system; H control; Markovian jump linear system; Linear parameter-varying system; Robot manipulator 1. Introduction Parametric uncertainties and exogenous disturbances increase the difficulty of reference tracking control for robot manipulators. H control strategies for robot manipulators based on state-feedback control have been used to minimize the disturbance effects in system performance 1. However the velocity signal considered as state generally is not available and can be obtained indirectly from a position measurement. This procedure can introduce noises and delays which decrease tracking control efficiency. An output-feedback controller can be used in order to avoid these problems. Two design techniques for output-feedback gain-scheduling controllers with a guaranteed H performance are proposed in 2 for linear parametervarying (LPV) systems. In this paper the second design technique named as Projected Characterization is applied to an actual robot manipulator in its Quasi-LPV representation which means the parameters matrix of the model depends on the state. Fault-tolerant systems for robot manipulators have been developed by several authors; see for instance 3 5 and Corresponding author. Fax: addresses: siqueira@sc.usp.br (A.A.G. Siqueira) terra@sel.eesc.usp.br (M.H. Terra) cbuosi@sel.eesc.usp.br (C. Buosi). references therein. Free torque failures in robot manipulators where the torque supply in the motor breaks down suddenly can make these systems uncontrollable. Furthermore if the robot is working in hazardous or unstructured environments where repairs are not allowed the requested movement must be completed according to the fault configuration. When a free torque failure occurs the fully actuated manipulator changes to an underactuated configuration. However when the manipulator changes after a fault occurrence from a fully actuated to an underactuated configuration the system stability is not guaranteed with the deterministic outputfeedback controllers proposed in 2. To use these controllers in a fault-tolerant robot system it is necessary to stop completely the movement of all joints after the fault detection restarting it from zero velocity. To avoid the necessity of stopping the robot when a fault occurs Markov theory is used in this paper to characterize abrupt changes in the operation points of the robotic manipulator. A model is developed based on linear systems subject to abrupt variations namely Markovian jump linear systems (MJLS) 67. In order to formulate this model the manipulator dynamic is linearized around operation points and a Markovian model is developed to encompass the changes of the operation points and the transition rate between fault configurations 18. With the proposed model that represents /$ - see front matter c 27 Elsevier B.V. All rights reserved. doi:1.116/j.robot

3 786 A.A.G. Siqueira et al. / Robotics and Autonomous Systems 55 (27) all manipulator configurations in a unified way the outputfeedback H controller for MJLS proposed in 9 is used to guarantee stability after the occurrence of a sequence of faults. This paper is organized as follows: in Section 2 the Quasi-LPV representations of fully actuated and underactuated robot manipulators are presented with experimental results using a deterministic output-feedback H controller; in Section 3 the fault-tolerant manipulator model and the control system based on output-feedback H controller for MJLS are presented and two fault sequences for the UArm II robot are evaluated to demonstrate the effectiveness of this approach. 2. Quasi-LPV representation of the manipulator 2.1. Fully actuated manipulator The dynamic equations of a robot manipulator can be found by Lagrange theory as τ = M(q) q + C(q q) q + F q + G(q) (1) where q R n is the joint position vector M(q) R n n is the symmetric positive definite inertia matrix C(q q) R n n is the Coriolis and centripetal matrix F R n n is the diagonal matrix of frictional torque coefficients G(q) R n is the gravitational torque vector and τ R n is the applied torque vector. A parametric uncertainty can be introduced dividing the parameter matrices M(q) C(q q) F and G(q) into a nominal and a perturbed part where M (q) C (q q) F and G (q) are the nominal matrices and M(q) C(q q) F and G(q) are the parametric uncertainties. A finite energy exogenous disturbance τ d R n can also be introduced resulting in τ + δ(q q q τ d ) = M (q) q + C (q q) q + F q + G (q) with δ(q q q τ d ) = ( M(q) q + C(q q) q + F q + G(q) τ d ). The state is defined as x = q T q T T where q and q are the positions and the velocities of the manipulator joints respectively. The Quasi-LPV representation of a fully actuated manipulator is given by ẋ = A(q q)x + B(q)u + B(q)δ(q q q τ d ) (3) with A(q q) = M 1 B(q) = (q) u = τ G (q). M 1 (q) (C (q q) + F ) I n n (2) 2.2. Underactuated manipulator Underactuated robot manipulators are mechanical systems with fewer actuators than degrees of freedom. For this reason the control of passive joints is made considering the dynamic coupling between them and the active joints. Here the manipulator is considered with n joints in which n p are passive and n a are active joints. From 1 no more than n a joints of the manipulator can be controlled at every instant when breaks are used in the passive joints. Let n u be the number of passive joints that have not already reached their set point in a given instant. If n u n a n a passive joints are controlled and grouped in the vector q u R n a the remaining passive joints if any are kept locked and the active joints are grouped in the vector q a R n a. If n u < n a the n u passive joints are controlled applying torques in n a active joints. In this case q u R n u and q a R n a. The strategy is to control all passive joints until they reach the desired position considering the conditions exposed above and then turn on the brakes. After that all the active joints are controlled by themselves as a fully actuated manipulator. The dynamic Eq. (2) can be partitioned as τa + δa Maa M = au δ u + M ua M uu qa q u + Faa qa + F uu q u Caa C au C ua C uu qa q u Ga (4) G u where the indices a and u represent the active and free (breaks not actioned) passive joints respectively. Factoring out the vector q a in the second line of (4) and substituting in the first one results in τ a + δ(q q q τ d ) = M (q) q u + C (q q) q u + F (q) q u with M (q) = M au M aa M 1 ua M uu C (q q) = C au M aa M 1 ua C uu D (q q) = C aa M aa M 1 ua C ua + F aa F ( q) = M aa M 1 ua F uu G (q) = G a M aa M 1 ua G u δ(q q q τ d ) = δ a M aa M 1 ua δ u + D (q q) q a + G (q) (5) where all matrices and vectors have appropriate dimensions depending on the numbers of active n a and free passive joints n u. The state is defined as x u = q u T T. qt u Hence a Quasi-LPV representation of the underactuated manipulator can be defined as follows ẋ u = A(q q)x u + B(q)u + B(q)δ(q q q τ d ) (6) with A(q q) = M 1 (q) ( C (q q) + F (q) ) I B(q) = M 1 (q) u = τ a D (q q)( q a G (q)).

4 2.3. Output-feedback H LPV control To apply the control techniques presented in 2 the robot manipulator needs to be represented according to equation ẋ = A(θ)x + B 1 (θ)w + B 2 (θ)u z = C 1 (θ)x + D 11 (θ)w + D 12 (θ)u y = C 2 (θ)x + D 21 (θ)w where θ = ρ 1 (t)... ρ N (t) T belongs to a convex space P and ρ i (t) i = 1... N are the time-varying parameters satisfying ρ i (t) ν i with ν i i = 1... N the bounds of the parameter variation rates. Consider as system disturbances the desired position q d and the combined torque disturbance δ that is: w = δ T (q d ) T T. The system outputs z are the position error q d q and the control input u. The control output is the position error y = q d q since we only have the position measured directly. Note that for the underactuated case instead of using ( q q) as the state one must use ( q u q u ). Hence the robot system can be described by (7) with A(θ) = A(q q) B 1 (θ) = B(q) B 2 (θ) = B(q) I C 1 (θ) = C 2 (θ) = I I D 11 (θ) = D 12 (θ) = I T D 21 (θ) = I D 22 (θ) = where matrices A(q q) and B(q) are obtained from (3) for the fully actuated case and (6) for the underactuated case. In 2 two H control techniques for LPV systems are presented. The one named as Projected Characterization which uses the projection lemma 11 to reduce the number of unknown variables was applied to the manipulator in its Quasi-LPV representation (7). The controller dynamics is defined as ẋk = u AK (θ θ) B K (θ θ) C K (θ θ) D K (θ θ) xk y A.A.G. Siqueira et al. / Robotics and Autonomous Systems 55 (27) (7). (8) To obtain the controller one must solve the following set of linear matrix inequalities (LMI) for X (θ) and Y (θ) minimizing γ N T Ẋ + X A + A T X X B 1 C X 1 T B1 T I X γ I DT 11 B 1 D 11 γ I NX < (9) I N T Ẏ + Y A T + AY Y C Y 1 T B 1 C 1 Y γ I D 11 I B1 T D11 T γ I NY < (1) I X I > (11) I Y where N X and N Y designate any bases of the null spaces of C 2 D 21 and B2 T DT 12 respectively. Note that the matrices depend on θ; this dependency was omitted for convenience. After finding X and Y the LPV controller can be designed by the following sequential scheme Compute D K solution to σ max (D 11 + D 12 D K D 21 ) < γ (12) and set D cl := D 11 + D 12 D K D 21. Compute B K and Ĉ K solutions to the linear matrix equations D 21 D21 T γ I Dcl T B T C 2 K = B 1 T X D cl γ I C 1 + D 12 D K C 2 (13) D12 T D 12 γ I D cl Dcl T γ I = Compute ĈK B T 2 C 1 Y (B 1 + B 2 D K D 21 ) T. (14) Â K = (A + B 2 D K C 2 ) T + X B 1 + B K D 21 (C D 12 D K C 2 ) T γ I D T cl D cl γ I (B1 + B 2 D K D 21 ) T. (15) C 1 Y + D 12 Ĉ K Solve for N M the factorization problem I XY = N M T. Finally compute A K B K and C K as A K = N 1 (XẎ + Â K X (A B 2 D K C 2 )Y B K C 2 Y X B 2 Ĉ K )M T B K = N 1 ( B K X B 2 D K ) C K = (Ĉ K D K C 2 Y )M T. The LMI problem defined by (9) (11) is infinitedimensional since the parameter vector θ varies continuously. To solve this problem one can divide the parameter space P in several points. The variables X (θ) and Y (θ) will be a solution if both satisfy the LMIs on all points. There is no systematic rule that defines how X (θ) and Y (θ) vary upon θ although this problem is usually solved using basis functions to describe the unknown matrices which lead them to be written as Z(θ) = M f i (θ)z i i=1 where { f i (θ)} i=1 M are differentiable functions of θ. See 2 for more details on the controller synthesis.

5 788 A.A.G. Siqueira et al. / Robotics and Autonomous Systems 55 (27) Fig. 1. Underactuated arm II. Remark 1. Note that to obtain the best performance of this controller the choice of ν i (in order to guarantee ρ i (t) ν i ) should be checked a posteriori Experimental results The output-feedback H LPV controller described in the previous section was applied to the experimental underactuated manipulator UArm II (Underactuated Arm II) Fig. 1 which is a special-purpose planar manipulator whose joints can be configured to act as passive (P) or active (A). More information on the manipulator can be found in Fully actuated control For the fully actuated control design the selected parameters which are part of the state vector are θ(x) = q 2 q 3 T. The parameter space P is defined as θ The parameter variation rate is bounded by θ 9 9 /s. X (θ) and Y (θ) were defined as follows X (θ) := X f 1 (θ) Y (θ) := Y f 1 (θ) + Y 1 f 2 (θ) + Y 2 f 3 (θ) (16) where f 1 (θ) = 1 f 2 (θ) = sin(q 2 ) + cos(q 2 ) f 3 (θ) = sin(q 3 ) + cos(q 3 ). The parameter space was divided in L = 5 for each parameter and the best level of attenuation found was γ = Underactuated control The underactuated configuration used in the experiment presented in the next section is the APA configuration i.e. joint 2 is passive and joints 1 and 3 are active. For this configuration two control phases hence two controllers are necessary to control all joints to the set point. In the first control phase the passive joint 2 is controlled by the dynamic coupling with the active joint 1 that is q u = q 2 q a = q 1 and joint 3 is kept locked; in the second one the active joints 1 and 3 are controlled. For the first control phase of the experiment the parameter θ selected is the state representing the position of joint 2 θ(x) = q 2. The parameter space P is defined as θ 3 3. The parameter variation rate is bounded by θ 9 /s. X (θ) and Y (θ) are defined as in (16) with f 1 (θ) = 1 f 2 (θ) = sin(q 2 ) f 3 (θ) = cos(q 2 ). The parameter space was divided in L = 5 and the best level Fig. 2. Joint position Quasi-LPV control not using brakes in the reconfiguration. of attenuation found was γ = For the second control phase the selected parameters that are part of the state vector were θ(x) = q 1 q 3 T. The parameter space P is defined as θ The parameter variation rate is bounded by θ 9 /s 9 /s. X (θ) and Y (θ) are the same as in (16) with f 1 (θ) = 1 f 2 (θ) = sin(q 1 ) + cos(q 1 ) f 3 (θ) = sin(q 3 ) + cos(q 3 ). The parameter space was divided in L = 5 for each parameter and the best level of attenuation found was γ = AAA APA fault sequence The controllers designed in Sections and do not guarantee that the joints will reach the set point if a free torque fault occur in the second joint changing suddenly the configuration from AAA to APA. To verify this behavior one experiment was performed considering initially the manipulator in the fully actuated configuration AAA with initial position q() = T and desired final position q(t ) = T. When the joint positions reached approximately 15 for all joints at t f = 2.5 s a free torque fault was introduced in the second joint. Hence the controller changes from the fully actuated configuration to the underactuated one maintaining the manipulator movement. Here we assume that a fault detection system indicates the fault instantaneously. As can be seen in Fig. 2 the LPV controllers were not able to react efficiently to the fault occurrence resulting in an unstable system. An alternative procedure is to use brakes during the control reconfiguration which is equivalent to using each controller considering their design specifications. For the UArm II when the brakes are turned on with the joints still in movement there are some oscillations in the joint positions which take at least 1 s to vanish; see Fig. 3. In this case all joints are locked for t l = 1 s between the fault detection and the beginning of the APA configuration control phase. One disadvantage of this procedure is that some components can be damaged mainly when the robot is performing high-speed motions. The results of this experiment can be seen in Fig. 3. The next objective is to design a control strategy that eliminates the necessity of stopping the

6 Fig. 3. Joint position Quasi-LPV control using brakes in the reconfiguration. joints between the fully actuated and underactuated control phases. 3. Manipulator robots as Markovian model In this section MJLS are adopted to model free-joint faults of robot manipulators. The dynamic model of an underactuated manipulator (5) can be represented as τ a = M (q) q u + b (q q) + δ(q q q) (17) with b (q q) = C (q q) q u + F (q) q u + D (q q) q a + G (q). The fully actuated manipulator (2) can be represented by (17) with q u = q M (q) = M (q) b (q q) = b (q q) = C (q q) q + F q + G (q) and δ(q q q) = δ(q q q). A proportional controller can be introduced in the form τ a = K P x + u in order to pre-compensate model imprecisions. The linearization of (17) around an operation point with position q and velocity q is given by ẋ = Ax + Ew + Bu z = C 1 x + D 1 u y = C 2 x + D 2 w A.A.G. Siqueira et al. / Robotics and Autonomous Systems 55 (27) (18) with the equations given in Box I where q d is the desired trajectory α and β are constants defined by the designer and used to adjust the controllers and w = δ T. The model presented in this section describes the changes between the linearization points of the plant (18) and the probability of a fault occurrence for the three-link manipulator I A = ( M 1 ) q (q)b (q q) + M 1 (q)k P M 1 (q) ( b (q q) ) q αi E = B = M 1 (q) C 1 = D 1 = β I I q d q C 2 = D 2 = x = q d q Box I. UArm II. The workspace of each joint is divided into two sectors of 1 each. For each sector a linearization point is defined 5 for the first sector and 15 for the second one. All the possible combinations to position the three joints q 1 q 2 q 3 in these two points are used to map the manipulator workspace. Then eight linearization points with the velocities set to zero are found. For a three-link manipulator robot seven possible fault configurations can occur: AAP APA PAA APP PAP PPA and PPP where A represents active joints and P represents passive joints. Here it is considered that two or more failures cannot occur simultaneously. The fault configurations AAP APA and PAA have n a = 2 then two control phases are necessary to control all joints to the set point. The first control phase is denoted by the configuration name followed by the subscript u (the passive joint is unlocked); and the second control phase is followed by the subscript l (locked). The fault configurations APP PAP and PPA have n a = 1 then three control phases are necessary to control all joints to the set point. The first control phase is denoted by the configuration name followed by the subscript u 1 ; the second control phase is followed by the subscript u 2 ; and the third control phase is followed by the subscript l. The Markovian states are the manipulator dynamic model linearized properly according to (18) in the eight points for all control phases of all configurations Fig. 4. Remark 2. Here the transition rate matrix Λ is used instead of the well-known transition probability matrix P adopted in the fault-tolerant model based on state-feedback control for MJLS presented in 18. The difference between them is that while the sum of the row elements of P is one the sum of the row elements of Λ is zero. Λ describes the transition rate among the Markovian states which is positive when the system jumps to a different state and negative when it remains in the same state; see 13 for more details. The transition rate matrices Λ f Λ s Λ and Λ 1 describe respectively the rates of a fault occurrence the passive joint being controlled to reach the set point the defective joint to be repaired and the manipulator to stay in the configuration PPP. Remark 3. The generalization of the Markovian model developed in this section for free-joint failures can be performed for n-link robot manipulators. The total number of

7 79 A.A.G. Siqueira et al. / Robotics and Autonomous Systems 55 (27) Fig. 4. Fault-tolerant model. Markovian states can be computed by the following formula ( ) n 1 T M S = 1 + n lp (n cpi n f ci ) + 1 (19) i=1 n! where n f ci = i!(n i)! is the number of possible fault configurations for i faults n cpi = ceil( n i n ) is the number of control phases for a configuration with i faults (ceil(x) rounds x to the nearest integer towards infinity) and n lp = (n γ ) n is the number of linearization points (n γ is the number of sectors where the linearizations will be performed) Output-feedback H control for MJLS The output-feedback H control for MJLS briefly presented in this section was originally presented in 9. Consider the collections of real matrices A = (A 1... A N ) dim(a i ) = n n E = (E 1... E N ) dim(e i ) = n m B = (B 1... B N ) dim(b i ) = n r C 1 = (C C 1N ) dim(c 1i ) = p n D 1 = (D D 1N ) dim(d 1i ) = p r C 2 = (C C 2N ) dim(c 2i ) = q n and D 2 = (D D 2N ) dim(d 2i ) = q m i = 1... N. Let us consider a continuous-time homogeneous Markov chain Θ = {Θ(t) : t > } with transition probability Pr(θ t+ t = j θ t = i) defined as Pr(Θ(t + t) = j Θ(t) = i) { λi = j (t) + o(δ) if i j 1 + λ ii (t) + o(δ) if i = j where > and λ i j (t) is the transition rate of the Markovian state i to j (i j) and λ ii (t) = N j=1 j i λ i j (t). The probability distribution of the Markov chain at the initial time is given by µ = (µ 1... µ N ) in such a way that Pr(Θ() = i) = µ i. The Markovian jump linear system is given by ẋ(t) = A θ(t) x(t) + E θ(t) w(t) + B θ(t) u(t) z(t) = C 1θ(t) x(t) + D 1θ(t) u(t) y(t) = C 2θ(t) x(t) + D 2θ(t) w(t) t (2) with w L 2 ( T ) E( x 2 ) < Θ() µ where x = {x(t) t } z = {z(t) t } e y = {y(t) t } are respectively the state the controlled output and the measured output of (2). Thus whenever Θ(t) = i S one has A θ(t) = A i E θ(t) = E i B θ(t) = B i C 1θ(t) = C 1i D 1θ(t) = D 1i C 2θ(t) = C 2i and D 2θ(t) = D 2i. The dynamic controller is given by v(t) = A cθ(t) v(t) + B cθ(t) y(t) u(t) = C cθ(t) v(t) t (21) where A c = (A c1... A cn ) dim(a ci ) = n n B c = (B c1... B cn ) dim(b ci ) = n q and C c = (C c1... C cn ) dim(c ci ) = p n. The output-feedback H problem for MJLS is to find a controller (A c B c C c ) such that the H norm of the

8 AT i X i + X i A i + L i C 2i + C2i T LT i + C1i T C 1i + A.A.G. Siqueira et al. / Robotics and Autonomous Systems 55 (27) N λ i j X j j=i X i E i + L i D 2i < Ei T X i + D2i T LT i γ 2 I A i Y i + Y i Ai T + B i F i + Fi T Bi T + λ ii Y i + γ 2 E i Ei T Y i C1i T + FT i D T 1i R i (Y ) C 1i Y i + D 1i F i I < Ri T (Y ) S i(y ) Yi I > I X i Box II. closed-loop system is smaller than γ. To find this controller the LMI system given in Box II must be solved with R i (Y )= λ1i Y i... λ (i 1)i Y i λ (i+1)i Y i... λ Ni Y i S i (Y )= diag(y 1... Y i 1 Y i+1... Y N ). Once the solution is obtained the controller is designed with C ci = F i Y 1 i B ci = (Y 1 i X i ) 1 L i A ci = (Y 1 i where X i ) 1 M i Y 1 i (22) M i = A T i X i A i Y i X i B i F i L i C 2i Y i C T 1i (C 1iY i + D 1i F i ) γ 2 (X i E i + L i D 2i )E T i 3.2. Experimental results N j=1 λ i j Y 1 j Y i AAA APA fault sequence For the experimental implementation it is considered the fault sequence where a free-joint fault occurs in joint 2 named AAA APA fault sequence represented in the fault-tolerant model by the numbers 1 2 and 3; see Fig. 4. The vector of controlled joints q c is chosen as q c = q 2 q 3 T for the control phase APA u and q c = q 1 q 3 T for APA l. There exist 24 Markovian states for this fault sequence; see Table 1. Following the approach presented in 13 it is necessary to group the transition rates between the Markovian states in a transition rate matrix Λ of dimension The matrix Λ is partitioned in nine submatrices of dimension 8 8 Λ = Λ AAA Λ f Λ Λ Λ APAu Λ s Λ Λ s Λ APAl. (23) The submatrix Λ AAA shows the relations between linearization points of configuration AAA and the diagonal submatrix Λ f determines the probabilities of a fault occurring. After the fault occurrence the system changes to the second line of Λ where Λ APAu defines the relations between the linearization points in the control phase APA u Λ shows that the defective Table 1 AAA APA Markovian states and linearization points AAA APA u APA l q 1 q 2 q 3 q 1 q 2 q joint cannot be repaired and the matrix Λ s represents how the transition rate of the system goes to the control phase APA l. In the third line of Λ Λ APAl defines the relations between the linearization points in the set APA l Λ s represents the possibility of the system returning to the control phase APA u and Λ represents again the impossibility of the defective joint being repaired. The transition rate matrix Λ is selected as (23) with Λ AAA(i j) =.9 Λ APAu(i j) =.8 Λ APAl(i j) =.8 for i j Λ AAA(ii) =.73 Λ APAu(ii) =.76 Λ APAl(ii) =.76 for i = j Λ f =.1I 8 Λ s =.2I 8 Λ =. The output-feedback H control presented in 9 is implemented according to the proposed fault-tolerant model aforementioned in the planar three-link robot manipulator UArm II. The experiments were performed for an initial position q() = T and for a desired final position q(t ) = T. The initial configuration is AAA with a linearization point starting in 1; see Table 1. To validate the fault-tolerant control proposed a fault was introduced at t f = 1.5 s. The fault-detection system proposed in 14 was implemented considering the filtered torque prediction error approach. When the fault is detected at t d = 1.54 s the Markov chain changes from the configuration AAA to the control phase APA u keeping the related linearization point. Torque disturbances were also

9 792 A.A.G. Siqueira et al. / Robotics and Autonomous Systems 55 (27) PAA l q c = q 1 for PAP u1 q c = q 3 for PAP u2 and q c = q 2 for PAP l. There exist 48 Markovian states for this fault sequence; see Table 2. The matrix Λ is partitioned in 36 submatrices of dimension 8 8 Λ = Λ AAA Λ f Λ Λ f Λ Λ Λ Λ PAAu Λ s Λ f Λ Λ Λ Λ s Λ PAAl Λ Λ f Λ Λ Λ Λ Λ PAPu1 Λ s Λ Λ Λ Λ Λ s Λ PAPu2 Λ s Λ Λ Λ Λ s Λ s Λ PAPl. (24) The matrix Λ is defined following the same arguments presented in Section as Λ AAA(i j) =.8 Λ PAAu(i j) =.7 Λ PAAl(i j) =.7 Λ PAPu1(i j) =.8 Λ PAPu2(i j) =.6 Λ PAPl(i j) =.6 for i j Λ AAA(ii) =.76 Λ PAAu(ii) =.79 Λ PAAl(ii) =.79 Λ PAPu1(ii) =.76 Λ PAPu2(ii) =.82 Λ PAPl(ii) =.82 for i = j Λ f =.1I 8 Λ s =.2I 8 Λ =. Fig. 5. Joint position. introduced to verify the robustness of the controllers. For these experiments the proportional controllers were selected heuristically as 2.25 K PAAA = 2. K PAPAu = K PAPAl = The controllers were computed considering α = 5 and β = 1 for all configurations; see (18). The best value of γ found was 1.5. The experimental results joint positions and Markovian states for the output-feedback H controller are shown in Fig. 5. Even after the fault the system kept the stability with the manipulator in movement during the control reconfiguration AAA PAA PAP fault sequence A more complete experiment was implemented considering the AAA PAA PAP fault sequence represented in the faulttolerant model by the numbers and 8; see Fig. 4. With this second experiment the guidelines to apply this procedure to manipulators with n-links and m-faults is better understood. The vector of controlled joints q c is chosen as q c = q 1 q 3 T for the control phase PAA u q c = q 2 q 3 T for This experiment was performed for an initial position q() = T and for a desired final position q(t ) = T. The initial configuration is AAA with the linearization point starting in 1; see Table 2. The fault-detection system used in the AAA APA fault sequence was also adopted here to determine the fault occurrence. The first fault was introduced at t f 1 = 2.5 s and detected at t d1 = 2.58 s changing the Markovian state from AAA to PAA u maintaining its linearization point. The second fault was introduced at t f 2 = 3 s and detected at t d2 = 3.35 s before joint 1 has reached its set point resulting in the Markovian state jumping from PAA u to PAP u1. After joint 1 reached its set point at t r1 = 4.73 s the Markovian state changed to PAP u2 and finally at t r3 = 6.75 s the state changed to PAP l. Torque disturbances and an additional payload of.5 kg were introduced in order to check the controller robustness. The proportional controllers selected were 2.25 K PAAA = 2. K PPAAu = 1.8 K PPAPu1 = 5 K PPAPu2 = 3 K PPAPl = The controllers were computed considering α = 1 and β = 1 for all configurations; see (18). The best value of γ found was 1. The experimental results joint positions and Markovian states for the output-feedback H controller are shown in Fig. 6. The system kept the stability in a more critical situation than the one presented in Section The use of a brake while the system operates in the Markovian states PAP u1

10 A.A.G. Siqueira et al. / Robotics and Autonomous Systems 55 (27) Table 2 AAA PAA PAP Markovian states and linearization points AAA PAA u PAA l PAP u1 PAP u2 PAP l q 1 q 2 q 3 q 1 q 2 q information is required to implement these control strategies. The first one based on an LPV output feedback controller is subbested for use when the robot does not operate at high velocities. The second one based on an output feedback controller for MJLS can be applied in a more complex situation because it is not necessary to stop the robot when it is subjected to faults. By virtue of the MJLS approach being designed as depending on a set of linear models three extra variables were introduced (K p α and β) in order to pre-compensate the effects of the MJLS controller. Despite pre-compensations being common in linear modelbased controllers this represents a weak point of this controller. For the LPV controller it is necessary to choose basic functions in order to solve the LMI and the designer can start with functions related to the robot model usual for this kind of problem. In this paper only free torque failures are considered. Fault tolerant systems for different fault types and concurrent faults are under investigation and will be presented in future works. Acknowledgement This work is supported by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo-Brazil) under grants 1/ and 5/ Fig. 6. Joint position. is necessary due to the fact that n u > n a and there is no way of controlling two passive joints at the same time using only one active joint; see Section 2.2. The advantage in this case is that the robot remains in movement. 4. Conclusion The fault-tolerant control strategies developed in this paper based on output feedback controllers are complementary to the strategies presented in 1. In this case it is not necessary to measure the velocities of the manipulator; only the position References 1 A.A.G. Siqueira M.H. Terra Nonlinear and Markovian H controls of underactuated manipulators IEEE Transactions on Control Systems Technology 12 (6) (24) P. Apkarian R.J. Adams Advanced gain-scheduling techniques for uncertain systems IEEE Transactions on Control Systems Technology 6 (1) (1998) M.L. Visinsky J.R. Cavallaro I.D. Walker Robotic fault detection and fault tolerance: A survey Reliability Engineering & Systems Safety 46 (1994) M.L. Visinsky J.R. Cavallaro I.D. Walker A dynamic fault tolerance framework for remote robots IEEE Transactions on Robotics and Automation 11 (4) (1995) R. Tinós M.H. Terra M. Bergerman A fault tolerance framework for cooperative robotic manipulators Control Engineering Practice 15 (5) (27)

11 794 A.A.G. Siqueira et al. / Robotics and Autonomous Systems 55 (27) Y. Ji H.J. Chizeck X. Feng K.A. Loparo Stability and control of discrete-time jump linear systems Control Theory and Advanced Technology 7 (2) (1991) O.L.V. Costa M.D. Fragoso R.P. Marques Discrete-Time Markov Jump Linear Systems (Probability and its Applications) 1st edition Springer A.A.G. Siqueira M.H. Terra A fault-tolerant manipulator robot based on H 2 H and mixed H 2 /H Markovian controls in: Proceedings of the 24 IEEE International Conference on Control Applications Taipei Taiwan 24 pp D.P. de Farias J.C. Geromel J.B.R. do Val O.L.V. Costa Output feedback control of Markov jump linear systems in continuos-time IEEE Transactions on Automatic Control 45 (5) (2) H. Arai S. Tachi Position control of a manipulator with passive joints using dynamic coupling IEEE Transactions on Robotics and Automation 7 (4) (1991) P. Gahinet P. Apkarian A linear matrix inequality approach to H control International Journal of Robust and Nonlinear Control 4 (1994) A.A.G. Siqueira C. Buosi M.H. Terra Output feedback nonlinear H control of underactuated manipulators in: Proceedings of the IEEE 11th Mediterranean Conference on Control and Automation Rhodes Greece Y. Ji H.J. Chizeck Controllability stability and continuous-time Markovian jump linear quadratic control IEEE Transactions on Automatic Control 35 (7) (199) W.E. Dixon I.D. Walker D.M. Dawson J.P. Hartranft Fault detection for robot manipulators with parametric uncertainty: a prediction-error-based approach IEEE Transactions on Robotics and Automation 16 (6) (2) Adriano Almeida Gonçalves Siqueira is Assistant Professor at the Department of Mechanical Engineering at the University of São Paulo São Carlos Brazil. He received a Bachelor s degree in Mechanical Engineering (1999) and a Ph.D. degree in Electrical Engineering (24) from the University of São Paulo. He was finalist of the Best Student Paper Award of the IEEE Conference on Decision and Control 22 and finalist of the Best Student Paper Award of the IEEE Conference on Control Applications 24. His research interests are underactuated robots cooperative robots robust control nonlinear control and exoskeletons. Marco Henrique Terra is Associate Professor at the Department of Electrical Engineering at the University of São Paulo São Carlos Brazil. He received his Ph.D. in Electrical Engineering in 1995 from USP. His research interests cover filtering estimation and control theories fault detection and isolation problems and robotics. Cleber Buosi received his Bachelor s degree in Electrical Engineering in 23 from the University of São Paulo at São Carlos. He has worked with Chemtech a Siemens company in Brazil. His interests are in advanced control applied in industrial processes multivariable control process information management systems and manufacturing execution systems.