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International Journal of Innovative Computing, Information and Control Volume 7, Number 2, February 2011

c ICIC International ⃝2011 ISSN 1349-4198 pp. 965–976

REAL-TIME DECENTRALIZED NEURAL BACKSTEPPING CONTROL: APPLICATION TO A TWO DOF ROBOT MANIPULATOR R. Garcia-Hernandez1,3,∗ , E. N. Sanchez1 , E. Bayro-Corrochano1 M. A. Llama2 and Jose A. Ruz-Hernandez3 1

Departamento de Ingenieria Electrica CINVESTAV Unidad Guadalajara Avenida Cientifica 1145, Zapopan, Jalisco, Mexico { rhernand; sanchez; edb }@gdl.cinvestav.mx 2

Division de Estudios de Posgrado Instituto Tecnologico de la Laguna Apartado Postal 49, Adm. 1, C.P. 27001, Torreon, Coahuila, Mexico [email protected] 3

Facultad de Ingenieria Universidad Autonoma del Carmen Avenida 56, No. 4, Cd. del Carmen, Campeche, Mexico [email protected] ∗ Corresponding author: [email protected]

Received October 2009; revised May 2010 Abstract. This paper presents a discrete-time decentralized control scheme for trajectory tracking of a two degrees of freedom (DOF) robot manipulator. A high order neural network (HONN) is used to approximate a decentralized control law designed by the backstepping technique as applied to a block strict feedback form (BSFF). The neural network learning is performed on-line by Kalman filtering. The controllers are designed for each joint using only local angular position and velocity measurements. These simple local joint controllers allow trajectory tracking with reduced computations. The proposed scheme is implemented in real-time to control a two DOF robot manipulator. Keywords: Decentralized control, High-order neural networks, Extended Kalman filter, Backstepping

1. Introduction. Recently, control of robot manipulators has become a significant research area for diﬀerent applications due to the relevancy that they have acquired in performing tasks classified as dangerous, or which require higher accuracy. In this context, diﬀerent control schemes have been proposed to guarantee eﬃcient trajectory tracking and stability [1,2]. Fast advances in computational technology oﬀer diﬀerent possibilities for implementing control algorithms within the approach of a centralized control design [3]. However, there is a great challenge to obtain an eﬃcient control for these systems, due to their highly nonlinear complex dynamics, with strong interconnections, parameters diﬃcult to be measured and unmodeled dynamics. Considering only the most important terms on the mathematical model, control algorithms with great number of mathematical operations are required, which aﬀect real-time implementation feasibility. In [4], the authors presented a control approach based on open-loop optimization using a genealogical decision tree (GDT), which can be used for solving both tracking and regulation. A novel task-space robust control approach with suitable tracking performance under imperfect transformation was presented in [5]; the proposed control law is 965

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derived based on the direct method of Lyapunov to guarantee stability in presence of both structured and unstructured uncertainties. On the other hand, within the area of control systems theory, for more than three decades, an alternative approach has been developed considering a global system as a set of interconnected subsystems, for which it is possible to design independent controllers, considering only local variables to each subsystem: the decentralized control [6,7]. This type of control has been applied in robotics, mainly in cooperative multiple mobile robots and robot manipulators, where it is natural to consider each mobile robot or each manipulator as a subsystem of the whole system. For robot manipulators, each joint and the respective link is considered as a subsystem in order to develop local controllers, which consider only local angular position and angular velocity measurements, and compensate interconnection eﬀects, usually assumed as disturbances. The resulting controllers are easily implemented for real-time applications [8]. In [9], a decentralized control for robot manipulators was developed, decoupling the dynamic model of the manipulator in a set of linear subsystems with uncertainties and simulations for a two joints robot are included. In [10], a decentralized control for robot manipulators was reported; it was based on the estimation of each joint dynamics, using feedforward neural networks. A decentralized control scheme, on the basis of a recurrent neural identifier and the block control structure was shown in [11]. This approach was tested only via simulation, with a two degrees of freedom robot manipulator, and with an Articulated Nimble Adaptable Trunk (ANAT) manipulator, with seven degrees of freedom. In [12], the authors proposed a robust adaptive decentralized control algorithm for trajectory tracking of robot manipulators. The controller, which consists of a PD (Proportional plus Derivative) feedback part and a dynamic compensation part, is designed based on the Lyapunov methodology. In recent literature about adaptive and robust control, numerous approaches have been proposed for the design of nonlinear control systems. Among these, adaptive backstepping constitutes a major design methodology [13]. The idea behind the backstepping approach is that some appropriate functions of state variables are selected recursively as virtual control inputs for lower dimension subsystems of the overall system. Each backstepping stage results in a new virtual control design from the preceding design stages; when the procedure ends, a feedback design for the true control input results, which achieves the original design objective. This paper presents a backstepping decentralized approach in order to design a suitable controller for each subsystem. Afterwards, each local controller is approximated by a high order neural network (HONN) [14]. The neural network learning is performed on-line by means of an Extended Kalman Filter (EKF) [15], and the controllers are designed for each joint, using only local angular position and velocity measurements. These simple local joint controllers allow trajectory tracking with reduced computations. We present the realtime implementation of the proposed scheme to control a two DOF robot manipulator, without a priori knowledge of the plant parameters. It is worth to note that a preliminary version of this paper was presented in [16]. 2. Discrete-Time Decentralized Systems. Let consider a class of discrete-time nonlinear perturbed and interconnected system which can be presented in the block strict feedback form (BSFF) [13] consisting of r blocks

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( ) ( ) χ1i (k + 1) = fi1 χ1i + gi1 χ1i χ2i + Γ1iℓ ( ) ( ) χ2i (k + 1) = fi2 χ1i , χ2i + gi2 χ1i , χ2i χ3i + Γ2iℓ .. . ( ) ( ) r χr−1 (k + 1) = fir−1 χ1i , χ2i , . . . , χr−1 + gir−1 χ1i , χ2i , . . . , χr−1 χi + Γr−1 i i i iℓ ( ) ( ) r r r r χi (k + 1) = fi χi + gi χi ui + Γiℓ (1) [ ]T [ ]T 2T where χi ∈ ℜni , χi = χ1T . . . χrT and χji ∈ ℜnij ×1 , χji = χji1 χji2 . . . χjil , i χi i i = 1, . . . , N ; j = 1, . . . , r; l = 1, . . . , nij ; N is the number of subsystems, ui ∈ ℜmi is the ∑ input vector, the rank of gij = nij , rj=1 nij = ni , ∀χji ∈ Dχj ⊂ ℜnij . We assume that i

fij , gij and Γji are smooth and bounded functions, fij (0) = 0 and gij (0) = 0. The integers ni1 ≤ ni2 ≤ · · · ≤ nij ≤ mi define the diﬀerent subsystem structures. The interconnection terms are given by Γ1iℓ

=

N ∑

( ) γiℓ1 χ1ℓ

ℓ=1, ℓ̸=i

Γ2iℓ

=

N ∑

( ) γiℓ2 χ1ℓ , χ2ℓ

ℓ=1, ℓ̸=i

.. .

(2)

Γr−1 = iℓ

N ∑

( ) γiℓr−1 χ1ℓ , χ2ℓ , . . . , χr−1 ℓ

ℓ=1, ℓ̸=i

Γriℓ

=

N ∑

( ) γiℓr χℓ

ℓ=1, ℓ̸=i

where χℓ represents the state vector of the ℓ-th subsystem with 1 ≤ ℓ ≤ N and ℓ ̸= i. Interconnection terms (2) reflect the interaction between the i-th subsystem and the other ones. 3. High-Order Neural Networks. 3.1. Discrete-time HONN’s. Considering the HONN described by ϕ(w, z) = w⊤ S(z) ⊤ ⊤ S(z) = [s⊤ 1 (z), s2 (z), · · · , sm (z)] [ ]⊤ ∏ ∏ si (z) = [s(zj )]dj (i1 ) · · · [s(zj )]dj (im ) j∈I1

(3)

j∈Im

i = 1, 2, · · · , L where z = [z1 , z2 , · · · , zp ]⊤ ∈ Ωz ⊂ ℜp , p is a positive integer which denotes the number of external inputs, L denotes the neural network node number, ϕ ∈ ℜm , {I1 , I2 , · · · , IL } is a collection of not ordered subsets of {1, 2, · · · , p}, S(z) ∈ ℜL×m , dj (ij ) is a nonnegative

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integer, w ∈ ℜL is an adjustable synaptic weight vector, and s(zj ) is chosen as the hyperbolic tangent function: ezj − e−zj (4) s(zj ) = zj e + e−zj For a desired function u∗ ∈ ℜm , assume that there exists an ideal weight vector w∗ ∈ ℜL such that the smooth function vector u∗ (z) can be approximated by an ideal NN on a compact subset Ωz ⊂ ℜq u∗ (z) = w∗⊤ S(z) + ϵz (5) m where ϵz ⊂ ℜ is the bounded NN approximation error vector; note that ∥ϵz ∥ can be reduced by increasing the number of the adjustable weights. The ideal weight vector w∗ is an artificial quantity required only for analytical purposes [14,17]. In general, it is assumed that there exists an unknown but constant weight vector w∗ , whose estimate is w ∈ ℜL . Hence, it is possible to define: w(k) ˜ = w(k) − w∗

(6)

as the estimation error. 3.2. EKF training algorithm. It is known that Kalman filtering (KF) estimates the state of a linear system with additive state and output white noises [18]. For KF-based NN training, the network weights become the states to be estimated. In this case, the error between the NN output and the measured plant output can be considered as additive white noise. Due to the fact that NN mapping is nonlinear, an EKF-type is required. The training goal is to find the optimal weight values wij (k) which minimize the prediction error. We use an EKF-based training algorithm described by: K(k) = P (k)H(k)[R(k) + H ⊤ (k)P (k)H(k)]−1 w(k + 1) = w(k) + ηK(k)[x(k) − xˆ(k)]

(7)

⊤

P (k + 1) = P (k) − K(k)H (k)P (k) + Q(k) where P ∈ ℜL×L is the prediction error covariance matrix, w ∈ ℜL is the weight (state) vector, η is a design parameter such that 0 ≤ η ≤ 1, L is the respective number of NN weights, x ∈ ℜm is the measured plant state, xˆ ∈ ℜm is the NN output, K ∈ ℜL×m is the Kalman gain matrix, Q ∈ ℜL×L is the state noise associated covariance matrix, R ∈ ℜm×m is the measurement noise associated covariance matrix, and H ∈ ℜL×m is a matrix, for which each entry (Hij ) is the derivative of one of the NN output (ˆ xi ), with respect to one NN weight (wj ), as follows [ ] ∂ xˆi (k) Hij (k) = (8) ∂wj (k) where i = 1, . . . , m and j = 1, . . . , L. Usually P and Q are initialized as diagonal matrices, with entries P (0) and Q(0), respectively. It is important to remark that H(k), K(k) and P (k) for the EKF are bounded [19]. 4. Controller Design. The model of many practical nonlinear systems can be expressed in (or transformed into) a special state-space form named the block strict feedback form (BSFF) [13] as follows: ) ) ( ( xji (k + 1) = fij xji (k) + gij xji (k) xj+1 (k) + dji (k) i xri (k + 1) = fir (xi (k)) + gir (xi (k))ui (k) + dri (k) y(k) = x1i (k),

j = 1, 2, . . . , r − 1

(9)

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j r⊤ ⊤ 1⊤ 2⊤ where xi (k) = [x1⊤ i (k), . . . , xi (k)] are the state variables, xi (k) = [xi (k), xi (k), . . . , j n ⊤ xj⊤ i (k)] , xi (k) ∈ ℜ , i = 1, . . . , N , r ≥ 2, r is the number of blocks, N is the number of subsystems, ui (k) ∈ ℜm are the system inputs, y(k) ∈ ℜm is the system output, dji ∈ ℜni is the bounded unknown disturbance vector which includes all the eﬀects of the other j j connected subsystems; then, there exist a constant di such that ∥dji (k)∥ ≤ di , for 0 < k < ∞. fij (·) and gij (·) are unknown smooth nonlinear functions. If we consider the original system (9) as a one-step ahead predictor, we can transform it into an equivalent maximum r-step ahead one, which can predict the future states x1i (k +r), x2i (k +r −1), . . . , xri (k +1); then, the causality contradiction is avoided when the controller is constructed based on the maximum r-step ahead prediction by backstepping [14,20]; then, system (9) can be rewritten as

x1i (k + r) = Fi1 (x1i (k)) + G1i (x1i (k))x2i (k + r − 1) + d1i (k + r) .. . r−1 xi (k + 2) = Fir−1 (xr−1 (k)) + Gr−1 (xr−1 (k)) i i i r−1 r xi (k + 1) + di (k + 2) r xi (k + 1) = Fir (xi (k)) + Gri (xi (k))ui (k) + dri (k) y(k) = x1i (k)

(10)

where Fij (·) and Gji (·) are unknown smooth functions of fij (xji (k)) and gij (xji (k)) respectively. To easy the analysis, let us define 1 ≤ j ≤ r − 1. The objective is to design a control ui (k) to force the system output χi (k) to follow a desired trajectory xid (k). Once (10) is defined, we apply the well-known backstepping technique [13]. For system (10), we can define the desired virtual controls (αj∗ (k), j = 1, . . . , r − 1) and the ideal practical control (u∗ (k)) as follows: ( ) αi1∗ (k) , x2i (k) = φ1i x1i (k), xid (k + r) ( ) αi2∗ (k) , x3i (k) = φ2i x2i (k), αi1∗ (k) .. . ) ( r−1 αir−1∗ (k) , xri (k) = φr−1 xi (k), αir−2∗ (k) i ( ) u∗i (k) = φri xi (k), αir−1∗ (k)

(11)

χi (k) = x1i (k) φji (·) with 1 ≤ j ≤ r are nonlinear smooth functions. It is obvious that the desired virtual controls αi∗ (k) and the ideal control u∗i (k) will drive the output χi (k) to track the desired signal xid (k) only if the exact system model is known and there are no unknown disturbances. However, in practical applications, these two conditions cannot be satisfied. In the following, neural networks will be used to approximate the desired virtual controls, as well as the desired practical control, when the conditions established above are not satisfied. As in [14], we construct the virtual and practical controls via backstepping without the causality contradiction [18]. Let us approximate the virtual controls and practical control by the following HONN: αij (k) = wij⊤ S j (zij (k)) ui (k) = wir⊤ S r (zir (k)),

j = 1, · · · , r − 1

(12)

970

with

R. GARCIA-HERNANDEZ ET AL.

[ ]⊤ zi1 (k) = x1i (k), x1id (k + r) [ ]⊤ zij (k) = xji (k), αij−1 (k) , j = 1, · · · , r − 1 [ ]⊤ zir (k) = xi (k), αir−1 (k)

where wij ∈ ℜLj are the estimates of ideal constant weights wij∗ and S j ∈ ℜLj ×nj with j = 1, . . . , r. Define the estimation error as w˜ij (k) = wij (k) − wij∗ .

(13)

Then, the corresponding weights updating laws using an EKF, are defined as wij (k + 1) = wij (k) + ηij Kij (k)eji (k)

(14)

Kij (k) = Pij (k)Hij (k)Mij (k) [ ]−1 Mij (k) = Rij (k) + Hij⊤ (k)Pij (k)Hij (k)

(15)

with

Pij (k + 1) = Pij (k) − Kij (k)Hij⊤ (k)Pij (k) + Qji (k) ] [ j ∂ υˆi (k) j Hi (k) = ∂wij (k)

(16)

and eji (k) = υij (k) − υˆij (k) (17) j j nj nj where υi (k) ∈ ℜ is the desired signal and υˆi (k) ∈ ℜ is the HONN function approximation defined, respectively as follows υi1 (k) = x1id (k) υi2 (k) = x2i (k) .. .

(18)

υir (k) = xri (k) and υˆi1 (k) = χ1i (k) υˆi2 (k) = αi1 (k) .. .

(19)

υˆir (k) = αir−1 (k) eji (k) denotes the error at each step as e1i (k) = x1id (k) − χ1i (k) e2i (k) = x2i (k) − αi1 (k) .. .

(20)

eri (k) = xri (k) − αir−1 (k). The whole proposed neural backstepping control scheme is shown in Figure 1. 5. Two DOF Robot Manipulator Application. This section presents the real-time application for a two DOF robot manipulator.

REAL-TIME DECENTRALIZED NEURAL BACKSTEPPING CONTROL

References

xid (k )

ei (k )

Robot Manipulator Link 1

Neural Network 1 Neural Network 2

ui (k )

Link 2

Neural Network N

NN Backstepping Controller

971

c i (k )

Link N

wi (k ) EKF

Figure 1. Decentralized neural backstepping control scheme 5.1. Robot manipulator description. In order to evaluate, via real-time implementation, the performance of the proposed control algorithm, we use a two DOF robot manipulator moving in the vertical plane as presented in Figure 2. The considered robot manipulator consists of two rigid links; high-torque brushless direct-drive servos are used to drive the joints without gear reduction. These joints present reduced backlash and significantly lower joint friction as compared to the actuators with gear drives. The motors used in the experimental arm are DM1200-A and DM1015-B from Parker Compumotor, for the shoulder and elbow joints, respectively. Angular information is obtained from incremental encoders located on the motors, which have a resolution of 1,024,000 pulses/rev for the first motor and 655,300 for the second one (accuracy 0.0069◦ for both motors), and the angular velocity information is computed via numerical diﬀerentiation of the angular position signal.

Figure 2. Robot manipulator

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5.2. Control objective. Define the following states: [ ] [ ] 1 2 χ (k) χ (k) 1 1 x1 (k) = ; x2 (k) = ; χ12 (k) χ22 (k) [ ] x11d (k) xd (k) = ; χ(k) = x1 (k) x12d (k)

[ u(k) =

] u1 (k) u2 (k)

; (21)

where χ11 (k) and χ12 (k) are the angular positions, χ21 (k) and χ22 (k) are the angular velocities, x11d (k) and x12d (k) are the desired trajectory signals, u1 (k) and u2 (k) represent the applied torque to joints 1 and 2, respectively. The control objective is to drive the output χ(k) to track the reference xd (k). Using (21), the discrete-time two DOF robot manipulator model can be represented in the BSFF, consisting of two blocks (r = 2) for each joint x1i (k + 1) = fi1 (x1i (k)) + gi1 (x1i (k))x2i (k) x2i (k + 1) = fi2 (x2i (k)) + gi2 (x2i (k))ui (k)

(22)

⊤

where x2i (k) = [x1i (k) x2i (k)] , i = 1, 2 subsystems, fi1 (x1i (k)), gi1 (x1i (k)), fi2 (x2i (k)) and gi2 (x2i (k)) are assumed to be unknown. To this end, we use a HONN to approximate the desired virtual controls and the ideal practical control described as αi1∗ (k) , x2i (k) = φ1i (x1i (k), x1id (k + 2)) u∗i (k) = φ1i (x1i (k), x2i (k), αi1∗ (k))

(23)

χi (k) = x1i (k). The HONN proposed for this application is defined as follows: αi1∗ (k) = wi1⊤ S 1 (zi1 (k)) ui (k) = wi2⊤ S 2 (zi2 (k))

(24)

with zi1 (k) = [x1i (k), x1id (k + 2)] zi2 (k) = [x1i (k), x2i (k), αi1 (k)].

(25)

The weights are updated using the EKF (14) – (20) with i = 1, 2 and e1i (k) = x1id (k) − x1i (k) e2i (k) = x2i (k) − αi1 (k).

(26)

The training is performed on-line using a series-parallel configuration. All the NN states are initialized in a random way. The covariance matrices are initialized as diagonal, and the nonzero elements are: Pi1 = Pi2 = 100, Q1i = Q2i = 1000 and Ri1 = Ri2 = 1 × 1012 , respectively. These values are heuristically selected. Comment 1. In fact, the system model must be expressed in the block strict feedback form (BSFF) [13] before to start the designing. The electromechanical system considered in this paper is already in this form. According to the actuators manufacturer, the direct-drive motors are able to supply torques within the following bounds: |u1 | ≤ τ1max = 150 [Nm] |u2 | ≤ τ2max = 15 [Nm].

REAL-TIME DECENTRALIZED NEURAL BACKSTEPPING CONTROL

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5.3. Real-time results. For the experiments we select the following discrete-time trajectories 3

3

3

3

x11d (k) = b1 (1 − ed1 kT ) + c1 (1 − ed1 kT ) sin(ω1 kT )[rad] x12d (k) = b2 (1 − ed2 kT ) + c2 (1 − ed2 kT ) sin(ω2 kT )[rad] where b1 = π/4, c1 = π/18, d1 = −2.0 and ω1 = 5 [rad/s] are parameters of the desired position trajectory for the first joint, whereas b2 = π/3, c2 = 25π/36, d2 = −1.8 and ω2 = 1.0 [rad/s] are parameters of the desired position trajectory for the second joint. The sampling time is selected as T = 2.5 milliseconds. These trajectories incorporate a sinusoidal term to evaluate the performance for relatively fast periodic signals, where the non-linearities of the robot dynamics are really important; additionally, they include a term, which smoothly grows for maintaining the robot in an operation state without saturating actuators. The actuator saturation limits are 150 [Nm] and 15 [Nm], respectively. The trajectory tracking results for decentralized neural backstepping (DNBS) control scheme are presented in Figures 3 and 4. For the real system, the initial conditions are the same that those of the reference system, both restricted to be equal to zero according to the experimental prototype architecture; therefore transient errors do not appear. The tracking error performance can be verified for joints 1 and 2 in Figure 5. The applied torques to each joint are shown in Figure 6.

1

0.9

Joint position (rad)

0.8

0.7

0.6

0.5

0.4

0.3

− x11d

0.2

− − χ11

0.1

0

0

2

4

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8

10

12

14

16

18

20

Time (s)

Figure 3. Trajectory tracking for joint 1 x11d (k) (solid line) and χ11 (k) (dashed line)

It is easy to see that both control signals are always inside the prescribed limits given by the actuators manufacturer; that is, their absolute values are smaller than the bounds τ1max and τ2max , respectively. Time evolution of the position error e1i reflects that the control system performance is very good. The performance criterion considered in this paper is the mean square error

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3.5 3

− x12d

2.5

− − χ12

Joint position (rad)

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2

0

2

4

6

8

10

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Time (s)

Figure 4. Trajectory tracking for joint 2 x12d (k) (solid line) and χ12 (k) (dashed line)

Error (rad)

0.1

e11

0.05

0

−0.05

−0.1

0

2

4

6

8

10

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Time (s) 0.2

Error (rad)

e12 0.1

0

−0.1

−0.2

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Time (s)

Figure 5. Tracking errors for joints 1 and 2 (MSE) value of the position error calculated as v u n u1 ∑ 1 MSE[ei ] = t ∥e1 ∥2 T t k=0 i

(27)

where T is the sampling time and t = 20 seconds. The respective mean square errors for the proposed scheme are included in Table 1. According to the mean square errors presented above, the proposed scheme based on the backstepping technique presents a good performance for trajectory tracking and reduced computational complexity.

REAL-TIME DECENTRALIZED NEURAL BACKSTEPPING CONTROL

Torque (Nm)

100

975

τ1

50

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−10

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Time (s)

Figure 6. Applied torques to joints 1 and 2 Table 1. Mean square error for the real joint positions Control algorithm MSE[e11 (k)] MSE[e12 (k)] DNBS

4.1611e-5

0.0013

6. Conclusions. This paper presents a decentralized neural control scheme based on the backstepping technique. The control law for each joint is approximated by a high order neural network. The training of each neural network is performed on-line using an extended Kalman filter. Real-time results confirm the eﬀectiveness of the proposed scheme for trajectory tracking when is applied to a two DOF robot manipulator. Acknowledgment. Authors thank CONACYT-Mexico for the financial support during this research through projects 57801Y, 82084 and 86940. The first author thanks to Universidad Autonoma del Carmen (UNACAR) and the Programa de Mejoramiento del Profesorado (PROMEP) for supporting this research. The fourth author thanks to DGEST-Mexico for supporting this research through project 2166.09-P. They also thank the anonymous reviewers for helping to improve this paper. REFERENCES [1] E. N. Sanchez and L. J. Ricalde, Trajectory tracking via adaptive recurrent neural control with input saturation, Proc. of International Joint Conference on Neural Networks, Portland, OR, pp.359-364, 2003. [2] V. Santiba˜ nez, R. Kelly and M. A. Llama, A novel global asymptotic stable set-point fuzzy controller with bounded torques for robot manipulators, IEEE Transactions on Fuzzy Systems, vol.13, no.3, pp.362-372, 2005. [3] R. Gourdeau, Object-oriented programming for robotic manipulator simulation, IEEE Robotics and Automation, vol.4, no.3, pp.21-29, 1997. [4] K. Najim, E. Ikonen and E. Gomez-Ramirez, Trajectory tracking control based on a genealogical decision tree controller for robot manipulators, International Journal of Innovative Computing, Information and Control, vol.4, no.1, pp.53-62, 2008.

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[5] M. M. Fateh and M. R. Soltanpour, Robust task-space control of robot manipulators under imperfect transformation of control space, International Journal of Innovative Computing, Information and Control, vol.5, no.11(A), pp.3949-3960, 2009. [6] Z. P. Jiang, New results in decentralized adaptive nonlinear control with output feedback, Proc. of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, pp.4772-4777, 1999. [7] S. Huang, K. K. Tan and T. H. Lee, Decentralized control design for large-scale systems with strong interconnections using neural networks, IEEE Transactions on Automatic Control, vol.48, no.5, pp.805-810, 2003. [8] M. Liu, Decentralized control of robot manipulators: Nonlinear and adaptive approaches, IEEE Transactions on Automatic Control, vol.44, no.2, pp.357-363, 1999. [9] M. L. Ni and M. J. Er, Decentralized control of robot manipulators with coupling and uncertainties, Proc. of the American Control Conference, Chicago, IL, pp.3326-3330, 2000. [10] R. Safaric and J. Rodic, Decentralized neural-network sliding-mode robot controller, Proc. of the 26th Annual Conference on the IEEE Industrial Electronics Society, Nagoya, Japan, pp.906-911, 2000. [11] E. N. Sanchez, A. Gaytan and M. Saad, Decentralized neural identification and control for robotics manipulators, Proc. of the IEEE International Symposium on Intelligent Control, Munich, Germany, pp.1614-1619, 2006. [12] L. C. Fu, Robust adaptive decentralized control robot manipulators, IEEE Transactions on Automatic Control, vol.37, no.1, pp.106-110, 1992. [13] M. Krstic, I. Kanellakopoulos and P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, 1995. [14] S. S. Ge, J. Zhang and T. H. Lee, Adaptive neural network control for a class of MIMO nonlinear systems with disturbances in discrete-time, IEEE Transactions on Systems, Man, and Cybernetics Part B, vol.34, no.4, pp.1630-1645, 2004. [15] A. Y. Alanis, E. N. Sanchez and A. G. Loukianov, Discrete-time adaptive backstepping nonlinear control via high-order neural networks, IEEE Transactions on Neural Networks, vol.18, no.4, pp.1185-1195, 2007. [16] R. Garcia-Hernandez, E. N. Sanchez, V. Santiba˜ nez, M. A. Llama and E. Bayro-Corrochano, Realtime decentralized neural backstepping controller for a robot manipulator, Proc. of International Symposium on Intelligent Control, Saint Petersburg, Russia, pp.701-706, 2009. [17] G. A. Rovithakis and M. A. Christodoulou, Adaptive Control with Recurrent High-Order Neural Networks, Springer, London, 2000. [18] S. Haykin, Kalman Filtering and Neural Networks, John Wiley & Sons, New York, 2001. [19] Y. Song and J. W. Grizzle, The extended Kalman filter as local asymptotic observer for discrete-time nonlinear systems, Journal of Mathematical Systems, Estimation and Control, vol.5, no.1, pp.59-78, 1995. [20] F. Chen and H. Khalil, Adaptive control of a class of nonlinear discrete-time systems using neural networks, IEEE Transactions on Automatic Control, vol.40, no.5, pp.791-801, 1995. [21] E. N. Sanchez, A. Y. Alanis and A. G. Loukianov, Discrete-Time High Order Neural Control Trained with Kalman Filtering, Springer-Verlag, Berlin, 2008.

Lihat lebih banyak...
c ICIC International ⃝2011 ISSN 1349-4198 pp. 965–976

REAL-TIME DECENTRALIZED NEURAL BACKSTEPPING CONTROL: APPLICATION TO A TWO DOF ROBOT MANIPULATOR R. Garcia-Hernandez1,3,∗ , E. N. Sanchez1 , E. Bayro-Corrochano1 M. A. Llama2 and Jose A. Ruz-Hernandez3 1

Departamento de Ingenieria Electrica CINVESTAV Unidad Guadalajara Avenida Cientifica 1145, Zapopan, Jalisco, Mexico { rhernand; sanchez; edb }@gdl.cinvestav.mx 2

Division de Estudios de Posgrado Instituto Tecnologico de la Laguna Apartado Postal 49, Adm. 1, C.P. 27001, Torreon, Coahuila, Mexico [email protected] 3

Facultad de Ingenieria Universidad Autonoma del Carmen Avenida 56, No. 4, Cd. del Carmen, Campeche, Mexico [email protected] ∗ Corresponding author: [email protected]

Received October 2009; revised May 2010 Abstract. This paper presents a discrete-time decentralized control scheme for trajectory tracking of a two degrees of freedom (DOF) robot manipulator. A high order neural network (HONN) is used to approximate a decentralized control law designed by the backstepping technique as applied to a block strict feedback form (BSFF). The neural network learning is performed on-line by Kalman filtering. The controllers are designed for each joint using only local angular position and velocity measurements. These simple local joint controllers allow trajectory tracking with reduced computations. The proposed scheme is implemented in real-time to control a two DOF robot manipulator. Keywords: Decentralized control, High-order neural networks, Extended Kalman filter, Backstepping

1. Introduction. Recently, control of robot manipulators has become a significant research area for diﬀerent applications due to the relevancy that they have acquired in performing tasks classified as dangerous, or which require higher accuracy. In this context, diﬀerent control schemes have been proposed to guarantee eﬃcient trajectory tracking and stability [1,2]. Fast advances in computational technology oﬀer diﬀerent possibilities for implementing control algorithms within the approach of a centralized control design [3]. However, there is a great challenge to obtain an eﬃcient control for these systems, due to their highly nonlinear complex dynamics, with strong interconnections, parameters diﬃcult to be measured and unmodeled dynamics. Considering only the most important terms on the mathematical model, control algorithms with great number of mathematical operations are required, which aﬀect real-time implementation feasibility. In [4], the authors presented a control approach based on open-loop optimization using a genealogical decision tree (GDT), which can be used for solving both tracking and regulation. A novel task-space robust control approach with suitable tracking performance under imperfect transformation was presented in [5]; the proposed control law is 965

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derived based on the direct method of Lyapunov to guarantee stability in presence of both structured and unstructured uncertainties. On the other hand, within the area of control systems theory, for more than three decades, an alternative approach has been developed considering a global system as a set of interconnected subsystems, for which it is possible to design independent controllers, considering only local variables to each subsystem: the decentralized control [6,7]. This type of control has been applied in robotics, mainly in cooperative multiple mobile robots and robot manipulators, where it is natural to consider each mobile robot or each manipulator as a subsystem of the whole system. For robot manipulators, each joint and the respective link is considered as a subsystem in order to develop local controllers, which consider only local angular position and angular velocity measurements, and compensate interconnection eﬀects, usually assumed as disturbances. The resulting controllers are easily implemented for real-time applications [8]. In [9], a decentralized control for robot manipulators was developed, decoupling the dynamic model of the manipulator in a set of linear subsystems with uncertainties and simulations for a two joints robot are included. In [10], a decentralized control for robot manipulators was reported; it was based on the estimation of each joint dynamics, using feedforward neural networks. A decentralized control scheme, on the basis of a recurrent neural identifier and the block control structure was shown in [11]. This approach was tested only via simulation, with a two degrees of freedom robot manipulator, and with an Articulated Nimble Adaptable Trunk (ANAT) manipulator, with seven degrees of freedom. In [12], the authors proposed a robust adaptive decentralized control algorithm for trajectory tracking of robot manipulators. The controller, which consists of a PD (Proportional plus Derivative) feedback part and a dynamic compensation part, is designed based on the Lyapunov methodology. In recent literature about adaptive and robust control, numerous approaches have been proposed for the design of nonlinear control systems. Among these, adaptive backstepping constitutes a major design methodology [13]. The idea behind the backstepping approach is that some appropriate functions of state variables are selected recursively as virtual control inputs for lower dimension subsystems of the overall system. Each backstepping stage results in a new virtual control design from the preceding design stages; when the procedure ends, a feedback design for the true control input results, which achieves the original design objective. This paper presents a backstepping decentralized approach in order to design a suitable controller for each subsystem. Afterwards, each local controller is approximated by a high order neural network (HONN) [14]. The neural network learning is performed on-line by means of an Extended Kalman Filter (EKF) [15], and the controllers are designed for each joint, using only local angular position and velocity measurements. These simple local joint controllers allow trajectory tracking with reduced computations. We present the realtime implementation of the proposed scheme to control a two DOF robot manipulator, without a priori knowledge of the plant parameters. It is worth to note that a preliminary version of this paper was presented in [16]. 2. Discrete-Time Decentralized Systems. Let consider a class of discrete-time nonlinear perturbed and interconnected system which can be presented in the block strict feedback form (BSFF) [13] consisting of r blocks

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( ) ( ) χ1i (k + 1) = fi1 χ1i + gi1 χ1i χ2i + Γ1iℓ ( ) ( ) χ2i (k + 1) = fi2 χ1i , χ2i + gi2 χ1i , χ2i χ3i + Γ2iℓ .. . ( ) ( ) r χr−1 (k + 1) = fir−1 χ1i , χ2i , . . . , χr−1 + gir−1 χ1i , χ2i , . . . , χr−1 χi + Γr−1 i i i iℓ ( ) ( ) r r r r χi (k + 1) = fi χi + gi χi ui + Γiℓ (1) [ ]T [ ]T 2T where χi ∈ ℜni , χi = χ1T . . . χrT and χji ∈ ℜnij ×1 , χji = χji1 χji2 . . . χjil , i χi i i = 1, . . . , N ; j = 1, . . . , r; l = 1, . . . , nij ; N is the number of subsystems, ui ∈ ℜmi is the ∑ input vector, the rank of gij = nij , rj=1 nij = ni , ∀χji ∈ Dχj ⊂ ℜnij . We assume that i

fij , gij and Γji are smooth and bounded functions, fij (0) = 0 and gij (0) = 0. The integers ni1 ≤ ni2 ≤ · · · ≤ nij ≤ mi define the diﬀerent subsystem structures. The interconnection terms are given by Γ1iℓ

=

N ∑

( ) γiℓ1 χ1ℓ

ℓ=1, ℓ̸=i

Γ2iℓ

=

N ∑

( ) γiℓ2 χ1ℓ , χ2ℓ

ℓ=1, ℓ̸=i

.. .

(2)

Γr−1 = iℓ

N ∑

( ) γiℓr−1 χ1ℓ , χ2ℓ , . . . , χr−1 ℓ

ℓ=1, ℓ̸=i

Γriℓ

=

N ∑

( ) γiℓr χℓ

ℓ=1, ℓ̸=i

where χℓ represents the state vector of the ℓ-th subsystem with 1 ≤ ℓ ≤ N and ℓ ̸= i. Interconnection terms (2) reflect the interaction between the i-th subsystem and the other ones. 3. High-Order Neural Networks. 3.1. Discrete-time HONN’s. Considering the HONN described by ϕ(w, z) = w⊤ S(z) ⊤ ⊤ S(z) = [s⊤ 1 (z), s2 (z), · · · , sm (z)] [ ]⊤ ∏ ∏ si (z) = [s(zj )]dj (i1 ) · · · [s(zj )]dj (im ) j∈I1

(3)

j∈Im

i = 1, 2, · · · , L where z = [z1 , z2 , · · · , zp ]⊤ ∈ Ωz ⊂ ℜp , p is a positive integer which denotes the number of external inputs, L denotes the neural network node number, ϕ ∈ ℜm , {I1 , I2 , · · · , IL } is a collection of not ordered subsets of {1, 2, · · · , p}, S(z) ∈ ℜL×m , dj (ij ) is a nonnegative

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integer, w ∈ ℜL is an adjustable synaptic weight vector, and s(zj ) is chosen as the hyperbolic tangent function: ezj − e−zj (4) s(zj ) = zj e + e−zj For a desired function u∗ ∈ ℜm , assume that there exists an ideal weight vector w∗ ∈ ℜL such that the smooth function vector u∗ (z) can be approximated by an ideal NN on a compact subset Ωz ⊂ ℜq u∗ (z) = w∗⊤ S(z) + ϵz (5) m where ϵz ⊂ ℜ is the bounded NN approximation error vector; note that ∥ϵz ∥ can be reduced by increasing the number of the adjustable weights. The ideal weight vector w∗ is an artificial quantity required only for analytical purposes [14,17]. In general, it is assumed that there exists an unknown but constant weight vector w∗ , whose estimate is w ∈ ℜL . Hence, it is possible to define: w(k) ˜ = w(k) − w∗

(6)

as the estimation error. 3.2. EKF training algorithm. It is known that Kalman filtering (KF) estimates the state of a linear system with additive state and output white noises [18]. For KF-based NN training, the network weights become the states to be estimated. In this case, the error between the NN output and the measured plant output can be considered as additive white noise. Due to the fact that NN mapping is nonlinear, an EKF-type is required. The training goal is to find the optimal weight values wij (k) which minimize the prediction error. We use an EKF-based training algorithm described by: K(k) = P (k)H(k)[R(k) + H ⊤ (k)P (k)H(k)]−1 w(k + 1) = w(k) + ηK(k)[x(k) − xˆ(k)]

(7)

⊤

P (k + 1) = P (k) − K(k)H (k)P (k) + Q(k) where P ∈ ℜL×L is the prediction error covariance matrix, w ∈ ℜL is the weight (state) vector, η is a design parameter such that 0 ≤ η ≤ 1, L is the respective number of NN weights, x ∈ ℜm is the measured plant state, xˆ ∈ ℜm is the NN output, K ∈ ℜL×m is the Kalman gain matrix, Q ∈ ℜL×L is the state noise associated covariance matrix, R ∈ ℜm×m is the measurement noise associated covariance matrix, and H ∈ ℜL×m is a matrix, for which each entry (Hij ) is the derivative of one of the NN output (ˆ xi ), with respect to one NN weight (wj ), as follows [ ] ∂ xˆi (k) Hij (k) = (8) ∂wj (k) where i = 1, . . . , m and j = 1, . . . , L. Usually P and Q are initialized as diagonal matrices, with entries P (0) and Q(0), respectively. It is important to remark that H(k), K(k) and P (k) for the EKF are bounded [19]. 4. Controller Design. The model of many practical nonlinear systems can be expressed in (or transformed into) a special state-space form named the block strict feedback form (BSFF) [13] as follows: ) ) ( ( xji (k + 1) = fij xji (k) + gij xji (k) xj+1 (k) + dji (k) i xri (k + 1) = fir (xi (k)) + gir (xi (k))ui (k) + dri (k) y(k) = x1i (k),

j = 1, 2, . . . , r − 1

(9)

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j r⊤ ⊤ 1⊤ 2⊤ where xi (k) = [x1⊤ i (k), . . . , xi (k)] are the state variables, xi (k) = [xi (k), xi (k), . . . , j n ⊤ xj⊤ i (k)] , xi (k) ∈ ℜ , i = 1, . . . , N , r ≥ 2, r is the number of blocks, N is the number of subsystems, ui (k) ∈ ℜm are the system inputs, y(k) ∈ ℜm is the system output, dji ∈ ℜni is the bounded unknown disturbance vector which includes all the eﬀects of the other j j connected subsystems; then, there exist a constant di such that ∥dji (k)∥ ≤ di , for 0 < k < ∞. fij (·) and gij (·) are unknown smooth nonlinear functions. If we consider the original system (9) as a one-step ahead predictor, we can transform it into an equivalent maximum r-step ahead one, which can predict the future states x1i (k +r), x2i (k +r −1), . . . , xri (k +1); then, the causality contradiction is avoided when the controller is constructed based on the maximum r-step ahead prediction by backstepping [14,20]; then, system (9) can be rewritten as

x1i (k + r) = Fi1 (x1i (k)) + G1i (x1i (k))x2i (k + r − 1) + d1i (k + r) .. . r−1 xi (k + 2) = Fir−1 (xr−1 (k)) + Gr−1 (xr−1 (k)) i i i r−1 r xi (k + 1) + di (k + 2) r xi (k + 1) = Fir (xi (k)) + Gri (xi (k))ui (k) + dri (k) y(k) = x1i (k)

(10)

where Fij (·) and Gji (·) are unknown smooth functions of fij (xji (k)) and gij (xji (k)) respectively. To easy the analysis, let us define 1 ≤ j ≤ r − 1. The objective is to design a control ui (k) to force the system output χi (k) to follow a desired trajectory xid (k). Once (10) is defined, we apply the well-known backstepping technique [13]. For system (10), we can define the desired virtual controls (αj∗ (k), j = 1, . . . , r − 1) and the ideal practical control (u∗ (k)) as follows: ( ) αi1∗ (k) , x2i (k) = φ1i x1i (k), xid (k + r) ( ) αi2∗ (k) , x3i (k) = φ2i x2i (k), αi1∗ (k) .. . ) ( r−1 αir−1∗ (k) , xri (k) = φr−1 xi (k), αir−2∗ (k) i ( ) u∗i (k) = φri xi (k), αir−1∗ (k)

(11)

χi (k) = x1i (k) φji (·) with 1 ≤ j ≤ r are nonlinear smooth functions. It is obvious that the desired virtual controls αi∗ (k) and the ideal control u∗i (k) will drive the output χi (k) to track the desired signal xid (k) only if the exact system model is known and there are no unknown disturbances. However, in practical applications, these two conditions cannot be satisfied. In the following, neural networks will be used to approximate the desired virtual controls, as well as the desired practical control, when the conditions established above are not satisfied. As in [14], we construct the virtual and practical controls via backstepping without the causality contradiction [18]. Let us approximate the virtual controls and practical control by the following HONN: αij (k) = wij⊤ S j (zij (k)) ui (k) = wir⊤ S r (zir (k)),

j = 1, · · · , r − 1

(12)

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with

R. GARCIA-HERNANDEZ ET AL.

[ ]⊤ zi1 (k) = x1i (k), x1id (k + r) [ ]⊤ zij (k) = xji (k), αij−1 (k) , j = 1, · · · , r − 1 [ ]⊤ zir (k) = xi (k), αir−1 (k)

where wij ∈ ℜLj are the estimates of ideal constant weights wij∗ and S j ∈ ℜLj ×nj with j = 1, . . . , r. Define the estimation error as w˜ij (k) = wij (k) − wij∗ .

(13)

Then, the corresponding weights updating laws using an EKF, are defined as wij (k + 1) = wij (k) + ηij Kij (k)eji (k)

(14)

Kij (k) = Pij (k)Hij (k)Mij (k) [ ]−1 Mij (k) = Rij (k) + Hij⊤ (k)Pij (k)Hij (k)

(15)

with

Pij (k + 1) = Pij (k) − Kij (k)Hij⊤ (k)Pij (k) + Qji (k) ] [ j ∂ υˆi (k) j Hi (k) = ∂wij (k)

(16)

and eji (k) = υij (k) − υˆij (k) (17) j j nj nj where υi (k) ∈ ℜ is the desired signal and υˆi (k) ∈ ℜ is the HONN function approximation defined, respectively as follows υi1 (k) = x1id (k) υi2 (k) = x2i (k) .. .

(18)

υir (k) = xri (k) and υˆi1 (k) = χ1i (k) υˆi2 (k) = αi1 (k) .. .

(19)

υˆir (k) = αir−1 (k) eji (k) denotes the error at each step as e1i (k) = x1id (k) − χ1i (k) e2i (k) = x2i (k) − αi1 (k) .. .

(20)

eri (k) = xri (k) − αir−1 (k). The whole proposed neural backstepping control scheme is shown in Figure 1. 5. Two DOF Robot Manipulator Application. This section presents the real-time application for a two DOF robot manipulator.

REAL-TIME DECENTRALIZED NEURAL BACKSTEPPING CONTROL

References

xid (k )

ei (k )

Robot Manipulator Link 1

Neural Network 1 Neural Network 2

ui (k )

Link 2

Neural Network N

NN Backstepping Controller

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c i (k )

Link N

wi (k ) EKF

Figure 1. Decentralized neural backstepping control scheme 5.1. Robot manipulator description. In order to evaluate, via real-time implementation, the performance of the proposed control algorithm, we use a two DOF robot manipulator moving in the vertical plane as presented in Figure 2. The considered robot manipulator consists of two rigid links; high-torque brushless direct-drive servos are used to drive the joints without gear reduction. These joints present reduced backlash and significantly lower joint friction as compared to the actuators with gear drives. The motors used in the experimental arm are DM1200-A and DM1015-B from Parker Compumotor, for the shoulder and elbow joints, respectively. Angular information is obtained from incremental encoders located on the motors, which have a resolution of 1,024,000 pulses/rev for the first motor and 655,300 for the second one (accuracy 0.0069◦ for both motors), and the angular velocity information is computed via numerical diﬀerentiation of the angular position signal.

Figure 2. Robot manipulator

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5.2. Control objective. Define the following states: [ ] [ ] 1 2 χ (k) χ (k) 1 1 x1 (k) = ; x2 (k) = ; χ12 (k) χ22 (k) [ ] x11d (k) xd (k) = ; χ(k) = x1 (k) x12d (k)

[ u(k) =

] u1 (k) u2 (k)

; (21)

where χ11 (k) and χ12 (k) are the angular positions, χ21 (k) and χ22 (k) are the angular velocities, x11d (k) and x12d (k) are the desired trajectory signals, u1 (k) and u2 (k) represent the applied torque to joints 1 and 2, respectively. The control objective is to drive the output χ(k) to track the reference xd (k). Using (21), the discrete-time two DOF robot manipulator model can be represented in the BSFF, consisting of two blocks (r = 2) for each joint x1i (k + 1) = fi1 (x1i (k)) + gi1 (x1i (k))x2i (k) x2i (k + 1) = fi2 (x2i (k)) + gi2 (x2i (k))ui (k)

(22)

⊤

where x2i (k) = [x1i (k) x2i (k)] , i = 1, 2 subsystems, fi1 (x1i (k)), gi1 (x1i (k)), fi2 (x2i (k)) and gi2 (x2i (k)) are assumed to be unknown. To this end, we use a HONN to approximate the desired virtual controls and the ideal practical control described as αi1∗ (k) , x2i (k) = φ1i (x1i (k), x1id (k + 2)) u∗i (k) = φ1i (x1i (k), x2i (k), αi1∗ (k))

(23)

χi (k) = x1i (k). The HONN proposed for this application is defined as follows: αi1∗ (k) = wi1⊤ S 1 (zi1 (k)) ui (k) = wi2⊤ S 2 (zi2 (k))

(24)

with zi1 (k) = [x1i (k), x1id (k + 2)] zi2 (k) = [x1i (k), x2i (k), αi1 (k)].

(25)

The weights are updated using the EKF (14) – (20) with i = 1, 2 and e1i (k) = x1id (k) − x1i (k) e2i (k) = x2i (k) − αi1 (k).

(26)

The training is performed on-line using a series-parallel configuration. All the NN states are initialized in a random way. The covariance matrices are initialized as diagonal, and the nonzero elements are: Pi1 = Pi2 = 100, Q1i = Q2i = 1000 and Ri1 = Ri2 = 1 × 1012 , respectively. These values are heuristically selected. Comment 1. In fact, the system model must be expressed in the block strict feedback form (BSFF) [13] before to start the designing. The electromechanical system considered in this paper is already in this form. According to the actuators manufacturer, the direct-drive motors are able to supply torques within the following bounds: |u1 | ≤ τ1max = 150 [Nm] |u2 | ≤ τ2max = 15 [Nm].

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5.3. Real-time results. For the experiments we select the following discrete-time trajectories 3

3

3

3

x11d (k) = b1 (1 − ed1 kT ) + c1 (1 − ed1 kT ) sin(ω1 kT )[rad] x12d (k) = b2 (1 − ed2 kT ) + c2 (1 − ed2 kT ) sin(ω2 kT )[rad] where b1 = π/4, c1 = π/18, d1 = −2.0 and ω1 = 5 [rad/s] are parameters of the desired position trajectory for the first joint, whereas b2 = π/3, c2 = 25π/36, d2 = −1.8 and ω2 = 1.0 [rad/s] are parameters of the desired position trajectory for the second joint. The sampling time is selected as T = 2.5 milliseconds. These trajectories incorporate a sinusoidal term to evaluate the performance for relatively fast periodic signals, where the non-linearities of the robot dynamics are really important; additionally, they include a term, which smoothly grows for maintaining the robot in an operation state without saturating actuators. The actuator saturation limits are 150 [Nm] and 15 [Nm], respectively. The trajectory tracking results for decentralized neural backstepping (DNBS) control scheme are presented in Figures 3 and 4. For the real system, the initial conditions are the same that those of the reference system, both restricted to be equal to zero according to the experimental prototype architecture; therefore transient errors do not appear. The tracking error performance can be verified for joints 1 and 2 in Figure 5. The applied torques to each joint are shown in Figure 6.

1

0.9

Joint position (rad)

0.8

0.7

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0.3

− x11d

0.2

− − χ11

0.1

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Time (s)

Figure 3. Trajectory tracking for joint 1 x11d (k) (solid line) and χ11 (k) (dashed line)

It is easy to see that both control signals are always inside the prescribed limits given by the actuators manufacturer; that is, their absolute values are smaller than the bounds τ1max and τ2max , respectively. Time evolution of the position error e1i reflects that the control system performance is very good. The performance criterion considered in this paper is the mean square error

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3.5 3

− x12d

2.5

− − χ12

Joint position (rad)

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2

0

2

4

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Figure 4. Trajectory tracking for joint 2 x12d (k) (solid line) and χ12 (k) (dashed line)

Error (rad)

0.1

e11

0.05

0

−0.05

−0.1

0

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4

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8

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e12 0.1

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−0.1

−0.2

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Figure 5. Tracking errors for joints 1 and 2 (MSE) value of the position error calculated as v u n u1 ∑ 1 MSE[ei ] = t ∥e1 ∥2 T t k=0 i

(27)

where T is the sampling time and t = 20 seconds. The respective mean square errors for the proposed scheme are included in Table 1. According to the mean square errors presented above, the proposed scheme based on the backstepping technique presents a good performance for trajectory tracking and reduced computational complexity.

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Torque (Nm)

100

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τ1

50

0

−50

−100

0

2

4

6

8

10

12

14

16

18

20

18

20

Time (s)

Torque (Nm)

10

τ2

5

0

−5

−10

0

2

4

6

8

10

12

14

16

Time (s)

Figure 6. Applied torques to joints 1 and 2 Table 1. Mean square error for the real joint positions Control algorithm MSE[e11 (k)] MSE[e12 (k)] DNBS

4.1611e-5

0.0013

6. Conclusions. This paper presents a decentralized neural control scheme based on the backstepping technique. The control law for each joint is approximated by a high order neural network. The training of each neural network is performed on-line using an extended Kalman filter. Real-time results confirm the eﬀectiveness of the proposed scheme for trajectory tracking when is applied to a two DOF robot manipulator. Acknowledgment. Authors thank CONACYT-Mexico for the financial support during this research through projects 57801Y, 82084 and 86940. The first author thanks to Universidad Autonoma del Carmen (UNACAR) and the Programa de Mejoramiento del Profesorado (PROMEP) for supporting this research. The fourth author thanks to DGEST-Mexico for supporting this research through project 2166.09-P. They also thank the anonymous reviewers for helping to improve this paper. REFERENCES [1] E. N. Sanchez and L. J. Ricalde, Trajectory tracking via adaptive recurrent neural control with input saturation, Proc. of International Joint Conference on Neural Networks, Portland, OR, pp.359-364, 2003. [2] V. Santiba˜ nez, R. Kelly and M. A. Llama, A novel global asymptotic stable set-point fuzzy controller with bounded torques for robot manipulators, IEEE Transactions on Fuzzy Systems, vol.13, no.3, pp.362-372, 2005. [3] R. Gourdeau, Object-oriented programming for robotic manipulator simulation, IEEE Robotics and Automation, vol.4, no.3, pp.21-29, 1997. [4] K. Najim, E. Ikonen and E. Gomez-Ramirez, Trajectory tracking control based on a genealogical decision tree controller for robot manipulators, International Journal of Innovative Computing, Information and Control, vol.4, no.1, pp.53-62, 2008.

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[5] M. M. Fateh and M. R. Soltanpour, Robust task-space control of robot manipulators under imperfect transformation of control space, International Journal of Innovative Computing, Information and Control, vol.5, no.11(A), pp.3949-3960, 2009. [6] Z. P. Jiang, New results in decentralized adaptive nonlinear control with output feedback, Proc. of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, pp.4772-4777, 1999. [7] S. Huang, K. K. Tan and T. H. Lee, Decentralized control design for large-scale systems with strong interconnections using neural networks, IEEE Transactions on Automatic Control, vol.48, no.5, pp.805-810, 2003. [8] M. Liu, Decentralized control of robot manipulators: Nonlinear and adaptive approaches, IEEE Transactions on Automatic Control, vol.44, no.2, pp.357-363, 1999. [9] M. L. Ni and M. J. Er, Decentralized control of robot manipulators with coupling and uncertainties, Proc. of the American Control Conference, Chicago, IL, pp.3326-3330, 2000. [10] R. Safaric and J. Rodic, Decentralized neural-network sliding-mode robot controller, Proc. of the 26th Annual Conference on the IEEE Industrial Electronics Society, Nagoya, Japan, pp.906-911, 2000. [11] E. N. Sanchez, A. Gaytan and M. Saad, Decentralized neural identification and control for robotics manipulators, Proc. of the IEEE International Symposium on Intelligent Control, Munich, Germany, pp.1614-1619, 2006. [12] L. C. Fu, Robust adaptive decentralized control robot manipulators, IEEE Transactions on Automatic Control, vol.37, no.1, pp.106-110, 1992. [13] M. Krstic, I. Kanellakopoulos and P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, 1995. [14] S. S. Ge, J. Zhang and T. H. Lee, Adaptive neural network control for a class of MIMO nonlinear systems with disturbances in discrete-time, IEEE Transactions on Systems, Man, and Cybernetics Part B, vol.34, no.4, pp.1630-1645, 2004. [15] A. Y. Alanis, E. N. Sanchez and A. G. Loukianov, Discrete-time adaptive backstepping nonlinear control via high-order neural networks, IEEE Transactions on Neural Networks, vol.18, no.4, pp.1185-1195, 2007. [16] R. Garcia-Hernandez, E. N. Sanchez, V. Santiba˜ nez, M. A. Llama and E. Bayro-Corrochano, Realtime decentralized neural backstepping controller for a robot manipulator, Proc. of International Symposium on Intelligent Control, Saint Petersburg, Russia, pp.701-706, 2009. [17] G. A. Rovithakis and M. A. Christodoulou, Adaptive Control with Recurrent High-Order Neural Networks, Springer, London, 2000. [18] S. Haykin, Kalman Filtering and Neural Networks, John Wiley & Sons, New York, 2001. [19] Y. Song and J. W. Grizzle, The extended Kalman filter as local asymptotic observer for discrete-time nonlinear systems, Journal of Mathematical Systems, Estimation and Control, vol.5, no.1, pp.59-78, 1995. [20] F. Chen and H. Khalil, Adaptive control of a class of nonlinear discrete-time systems using neural networks, IEEE Transactions on Automatic Control, vol.40, no.5, pp.791-801, 1995. [21] E. N. Sanchez, A. Y. Alanis and A. G. Loukianov, Discrete-Time High Order Neural Control Trained with Kalman Filtering, Springer-Verlag, Berlin, 2008.

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