Fault Tolerant Control for Robot Manipulators Using Neural Network ...

Fault Tolerant Control for Robot Manipulators Using Neural Network and Second-Order Sliding Mode Observer Mien Van1 and Hee-Jun Kang2,* 1

Graduate School of Electrical Engineering, University of Ulsan, 680-749, South Korea [email protected] 2 School of Electrical Engineering, University of Ulsan, 680-749, South Korea [email protected]

Abstract. This paper investigates an algorithm for fault tolerant control of uncertain robot manipulator with only joint position measurement using neural network and second-order sliding mode observer. First, a neural network (NN) observer is designed to estimate the modeling uncertainties. Based on the obtained uncertainty estimation, a second-order sliding mode observer is then designed for two purposes: 1) Providing the velocity estimation, 2) providing the fault information that is used for fault detection, isolation and identification. Finally, a fault tolerant control scheme is proposed for compensating the effect of uncertainties and faults based on the fault estimation information. Computer simulation results on a PUMA560 industrial robot are shown to verify the effectiveness of the proposed strategy. Keywords: Fault diagnosis, Neural network, Second-order sliding mode observer, Robot dynamics.

1

Introduction

Various fault diagnosis approaches for nonlinear systems as well as robotic system have been studied during the last three decades. The model-based analytical redundancy based fault detection and isolation have been widely investigated in the literature [1]. By using neural network (NN) learning, a robust fault diagnosis schemes have been proposed [2-3]. Other observer based technique which is so-called sliding mode (SM) observer have been developed, in which the equivalent output injection (EOI) signal is utilized to detect and reconstruct the presence of the uncertainties and faults [4]. In some applications of sliding mode, chattering is the major drawbacks of the SM approach in the practical realization. The most widely used in practical applications to eliminate chattering are using higher-order sliding mode such as second-order sliding mode (SOSM) [5-6, 10]. After a fault has been detected and isolated, in some applications of robotic system it is required that the fault must be self-corrected, to guarantee that the robot can *

Corresponding author.

D.-S. Huang et al. (Eds.): ICIC 2013, LNCS 7995, pp. 526–535, 2013. © Springer-Verlag Berlin Heidelberg 2013

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continue working. It is referred as fault tolerant control (FTC). In general, FTC can be divided into two main approaches [7]: 1) passively, one controller is used for both normal case and fault case without the need to detect the presence of the fault [8]. This approach however, requires the partial knowledge of possible system faults. It is limited in real application; 2) actively, FTC is designed based on fault diagnosis information [2-3, 9-10]. In the actively FTC, fault diagnosis (FD) is the first step to provide the fault information. Based on the obtained fault information, a FTC is then designed to compensate for the effect of the faults in the system by online controller reconfiguration. With the correct fault information, performance of an active FTC system is more effective than that of a passive FTC system and hence is more desirable for practical applications. For robotic system, a number of FTC approaches have been proposed based on active strategy [2-3, 9-10]. However, these approaches did not compensate for the modeling uncertainty. Hence, the tracking performance is decreased. In addition, these schemes were designed based on the assumption that the joint velocity measurements are available. It is limited in real application. To overcome this obstacle, exact velocity observer has proposed based on super-twisting SOSM observer [6, 13]. In light of the remarkable benefits, in this paper, an active FTC scheme is proposed for uncertain robot manipulators using only position measurement. In the proposed scheme, both uncertainty and fault are considered and compensated. The FTC scheme is constructed based on the fault estimation which is obtained from a fault diagnosis observer scheme. The main contribution of this paper is as follows: 1) a second-order sliding mode observer is designed to obtain both velocity and fault estimation, 2) a neural network observer is used to estimate the uncertainties, 3) an active FTC scheme is designed based on the velocity estimation, uncertainty estimation and fault estimation for compensating the effects of the uncertainties and faults. Simulation results using a PUMA560 robot arm are used to verify the effectiveness of the proposed algorithm.

2

Problem Statement

Consider the robot dynamics is described by

q = M −1 (q)[τ − Vm (q, q )q − G (q) − Δ '(q, q , t )] + β (t − T f )φ (q, q ,τ )

(1)

where q ∈ℜn is the state vector, τ ∈ ℜn is the torque produced by the actuators,

M (q) ∈ ℜn×n is the inertia matrix. Vm (q, q ) ∈ ℜn is the Coriolis and centripetal torque, G (q) ∈ ℜn is the term of gravitational torque, Δ′(q, q , t ) ∈ ℜn

is an

uncertainty term. The term φ (q, q ,τ ) ∈ ℜ is a vector which represents the changes in the system dynamics due to the occurrence of a fault, β (t − T f ) = diag{β1 (t − T f ), β 2 (t − T f ),... , β n (t − T f )} ∈ ℜn represents the time n

profile of the faults, and T f is the time of occurrence of the faults, that is

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0 −ϕi 1 − e (t − T f )

β i (t − T f ) = 

if t < T f if t ≥ T f

(2)

where ϕi > 0 represent the unknown fault evolution rate. Small ϕi values characterize slowly developing forms, also called incipient fault. For very large values of ϕi , the profile of βi approaches a step function that models abrupt faults. When

ϕi → ∞ , the βi become a step function so that incipient fault becomes an abrupt faults. To simply the subsequent design and analysis, Eq. (1) can be rewritten as q = M −1 (q )[τ − H (q, q )] + Δ (q, q , t ) + β (t − T f )φ (q, q ,τ )

(3)

where H (q, q ) = V (q, q ) + G (q) and Δ(q, q , t ) = − M −1 (q)Δ '(q, q , t ) . In this paper, we investigate a FTC scheme to handle the effects of faults of uncertainty robot manipulator rely on the following assumptions. Assumption 1: The states of the robot system are bounded for all time. Assumption 2: The modeling uncertainty is bounded

Δ(q, q , t ) < Δ

(4)

where Δ is a known constant. Assumption 3: The unknown fault function is bounded

φ (q, q , t ) < φ

(5)

where φ is a known constant. By introducing x1 = q ∈ ℜn , x2 = q ∈ ℜn , the robot dynamics expressed in eq. (3) can be written in state space form as: x1 = x2 x2 = f ( x1 , x2 , u ) + Δ ( x1 , x2 , t ) + β (t − T f )φ ( x1 , x2 , u )

(6)

y = x1 where u = τ ,

f ( x1 , x2 , u ) = M −1 (q)[τ − H (q, q )] denotes the nominal of robot

dynamics.

3

Design of Uncertainty Observer

The uncertainty Δ ( x1 , x2 , t ) can be described by a neural network approximation as: Δ ( x1 , x2 , t ) = W σ (Vx (t )) + δ

(7)

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where W , V are the ideal weights of three layer neural network, x (t ) = [ x1 , x2 , u ] is the neural network input, σ is sigmoid activation function and δ is approximation error. In order to estimate the uncertainties, a neural network based observer is designed as

ω = A(ω − x2 ) + f ( x1 , ω , u ) + U (t )

(8)

where A = diag {−k1 , −k2 ,..., −kn } is stable matrix with ki > 0 , ω denotes the state of the observer. U (t ) is the observer input which is used to estimate the uncertainties and it can be selected by ˆ (t )) U (t ) = Wˆ σ (Vx

(9)

where Wˆ and Vˆ are the estimation of the ideal weights W and V , respectively. 1 By chosen a cost function as J = (ω )T (ω ) , ω = x2 − ω , the update law based on 2 the back-propagation plus an e-modification terms is designed as [12]:

∂J  − ρ 2 ω Wˆ Wˆ = − ρ1 ∂Wˆ

(10)

∂J  − ρ 4 ω Vˆ Vˆ = − ρ3 ∂Vˆ

(11)

where ρ1 , ρ3 > 0 are the learning rates and ρ2 , ρ4 > 0 are small positive numbers. Substituting Eq. (6) into Eq. (8) when the robot works in normal operation mode, the observer error is defined as:

ω = Aω + f ( x1 , x2 , u ) − f ( x1 , ω , u ) + Δ( x1 , x2 , t ) − U (t )

(12)

By similar way with Ref. [10], it is easy to demonstrate that the estimation error ( ω ) and the resulting parameter errors ( W = W − Wˆ , V = V − Vˆ ) are stability and being bounded under the observer scheme expressed in Eq. (8). After the NN weights vector Wˆ , Vˆ approach to the optimal vectors W , V under the update laws expressed in Eqs. (10) and (11) in normally working condition of robot system. At this time, the NN weights are fixed and noted as W 0 ,V0 . Up to now, the uncertainty can be expressed as: Δ ( x1 , x2 , t ) = W 0σ (V 0 x (t )) + ε

(13)

and the uncertainty estimation expressed in eq. (7) can be replaced by obtained NN: Δˆ ( x1 , x2 , t ) = W 0σ (V 0 x (t )) where Δˆ ( x1 , x2 , t ) denotes the estimation of the uncertainty Δ ( x1 , x2 , t ) .

(14)

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Velocity and Fault Diagnosis Observer Scheme

Based on [6], the second-order sliding mode based state observer can be described by 1/ 2 xˆ1 = xˆ2 + λ x1 sign( x1 ) xˆ2 = f ( x1 , xˆ2 , u ) + Δˆ ( x1 , x2 , t ) + α sign( x1 )

(15)

where xˆ1 and xˆ2 are states estimation of x1 and x2 with initial condition xˆ1 = x1 , xˆ2 = 0 . x1 = x1 − xˆ1 is the state estimation error. Substituting Eq. (6) into Eq. (15), we obtain the states estimation error: x1 = x2 − λ x1 sign( x1 ) x2 = d ( x1 , x2 , xˆ2 , u ) + ε + φ ( x1 , x2 , u ) − α s ign( x1 ) 1/2

where d ( x1 , x2 , xˆ2 , u ) = f ( x1 , x2 , u ) − f ( x1 , xˆ2 , u ) , ε = Δ( x1 , x2 , t ) −

(16) Δˆ ( x1 , x2 , t ) is

the uncertainty estimation error, it is bounded by ε < ε , ε is a known constant. If we defined

F ( x1 , x2 , xˆ2 , u, t ) = d ( x1 , x2 , xˆ2 , u ) + ε + φ ( x1 , x2 , u ) , based on the

assumptions 1-3, there exist a constant f + such that: F ( x1 , x2 , xˆ2 , u, t ) < f +

(17)

Based on the Lyapunov approach in Ref. [13], Theorem 1 gives the convergence of the estimation error to zero in finite time: Theorem 1: Suppose that condition (17) holds for a system as in Eq. (16); if the sliding mode gains of the observer scheme in Eq. (16) are chosen as

λ >0 +

α >3f +2

f+

2

(18)

λ

Then the observer scheme is stable, and the states of the observer in Eq. (15) ( xˆ1 , xˆ2 ) converges to the true state ( x1 , x2 ) in Eq. (6) in finite time. Proof: by similar way with the Ref. [13], we can verify that the observer states converge to the true states in finite time.

5

Fault Tolerant Control Scheme

5.1

Fault Reconstruction

Consider the estimation error as expressed in Eq. (16) converges to zero. Eq. (16) can be written as

Fault Tolerant Control for Robot Manipulators Using NN and SOSM Observer

x1 ≡ 0 x2 = ε + φ ( x1 , x2 , u ) − α s ign( x1 ) ≡ 0

531

(19)

Notice that d ( x1 , x2 , xˆ2 , u ) = f ( x1 , x2 , u ) − f ( x1 , xˆ2 , u ) = 0 due to the estimation states converge to the true states ( xˆ2 = x2 ). Then the equivalent output injection (EOI) is defined as zeq = α sign( x1 ) = ε + φ ( x1 , x2 , u )

(20)

Theoretically, the equivalent output injection is the result of an infinite switching frequency of the discontinuous terms α sign( x1 ) that is so-called chattering. To eliminate this high frequency chattering, we use a low pass filter has the form a zeq (t ) + zeq (t ) = zeq (t )

(21)

where a is the filter time constant. After the filtration, we have zeq = zeq + γ

(22)

where every elements of zeq is the filtered version of zeq and γ is the difference caused by filtration process. Nevertheless, as it is shown in [13]:

lim zeq (t ) = zeq (t ),

a→0

a ∈ℜ

Now, once the information of the equivalent output injection is available, it is possible to reconstruct the fault function by mean of the following expression zeq = φˆ( x1 , x2 , u ) + εˆ

(23)

where φˆ( x1 , x2 , u ) , εˆ are the estimation of φ (q, q ,τ ) and ε , respectively. Because φ ( x1 , x2 , u )  ε , hence the fault estimation can be approximated by z ≈ φˆ( x , x , u ) . eq

5.2

1

2

Fault Tolerant Control Scheme

The key problem of the FTC scheme is online reconfiguration of the controller based on the obtained fault estimation. The FTC architecture is illustrated in Fig. 1 that consists of three parts: 1) the CTC where the real joint velocity measurement is replaced by the estimated velocity; 2) the estimated uncertainties compensating; 3) the estimated fault compensating.

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Fig. 1. Structure of fault tolerant control scheme

The fault tolerant control scheme is design as

τ = τ0 +τu +τ f

(24)

where τ 0 = M ( x1 )(qd + KV (qd − xˆ2 ) + K P (qd − x1 )) + H ( x1 , xˆ2 ) , τ u and τ f

are

used to compensate for the modeling uncertainties and faults, respectively. They are defined as follows:

τ u = − M ( x1 )W0σ (V0 x (t ))

(25)

τ f = − M ( x1 ) zeq

(26)

and

6

Simulation Results

In order to verify the effectiveness of the proposed algorithm, its overall procedure is simulated for a PUMA560 robot where the first three joints are used. Its explicit dynamic model and its parameter are given in Ref. [14]. The uncertainties that include friction, small joint backlash are given by  q1 + sin(3q1)    Fu = 1.1q2 + 0.8sin(2q2 )  0.8q3 + 1.1sin(q3 )   

(27)

In this simulation, the sliding gains are chosen α = 12 , β = 8 . To estimate the uncertainties, a three layer neural network has 20 neurons in hidden layer, the tuning law was given in eqs. (10) and (11) with ρ1 = ρ3 = 5 and ρ2 = ρ4 = 0.03 . First in normal operation, Fig. 2 shows the uncertainty estimation performance of the neural network observer. We see that the neural network observer obtain good uncertainty approximation.

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To verify the performance of the FTC, we assumed a fault φ = [0, 6, 4 sin(t )]T occurs in the second and third joints at t = 10 s . The time histories of joint angles are given in Fig. 3 without fault estimation compensating ( τ = τ 0 + τ u ). The fault signal is reconstructed from the EOI is given by zeq = α i sign( x1i ) using a low pass filter with

a = 0.002 , the result is shown in Fig. 4. From this figure, we see that the fault are correctly detected, isolated and identified. The mismatch between the EOI of sliding mode and the fault function inevitably come from the filtration and model uncertainty estimation error. Fig. 5 shows the performance of the fault tolerant control scheme under the presence of the fault φ . Comparison of Fig. 5 with Fig. 3 shows that the tracking performance is improved due to its model self-correction capability. uncertainty2 NN-output2

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uncertainty1 NN-output1

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Fig. 3. Joint angles of the robot manipulator when fault φ occurs without fault estimation compensating

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Fig. 5. Joint angles of the robot manipulator when fault φ occurs with fault estimation compensating

7

Conclusions

An active fault tolerant control scheme for uncertainty robot manipulator using only position measurement has been presented in this paper. Here, the uncertainty observer for estimation of modeling uncertainties and the second-order sliding mode observer for the fault diagnosis are investigated to show their effectiveness. In addition, a fault tolerant control based on a modification of computed torque control is suggested for better tracking performance in the system. The results of computer simulations for a 3-DOF PUMA560 robot verify the effectiveness of the proposed strategy. Acknowledgements. The authors would like to express financial supports from Korean Ministry of Knowledge Economy both under Human Resources Development Program for Convergence Robot Specialists and under Robot Industry Core Technology Project.

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