J Intell Robot Syst DOI 10.1007/s10846-014-0162-2

Robust Adaptive Trajectory Tracking Sliding Mode Control Based on Neural Networks for Cleaning and Detecting Robot Manipulators Cuong Van Pham · Yao Nan Wang

Received: 6 June 2014 / Accepted: 14 November 2014 © Springer Science+Business Media Dordrecht 2014

Abstract This paper proposes an robust adaptive control method based on Radial Basis Function Neural networks (RBFNN) to investigate the joint position control for periodic motion and predefined trajectory tracking control of two link Cleaning and Detecting Robot Manipulators (CDRM). The proposed control scheme uses a three layer RBFNN to approximate nonlinear robot dynamics. The RBF network is one of the most popular intelligent approaches which has shown a great promise in this sort of problems because of simple network structure and its faster learning capacity. When the RBF networks are used to approximate a nonlinear dynamic system, the control system is stable. In addition, Sliding mode control (SMC) is a well known nonlinear control strategy because of its robustness. A robust term function is selected as an auxiliary controller to guarantee the stability and robustness under various envirorments, such as the mass variation, the external disturbances and modeling uncertainties. The adaptation laws for the weights C. Van Pham () · Y. N. Wang College of Electrical and Information Engineering, Hunan University, Hunan Changsha, China e-mail: [email protected] Y. N. Wang e-mail: [email protected] C. Van Pham College of Electrical Technical Technology, Hanoi University of Industry, Hanoi, Vietnam

of the RBFNN are adjusted using the Lyapunov stability theorem, the global stability and robustness of the entire control system are guaranteed, and the tracking errors converge to the required precison, and position is proved. Finally, experiments performed on a twolink CDRM in electric power substation are provided in comparison with proportional differential (PD) and adaptive Fuzzy (AF) control to demonstrate superior tracking precision and robustness of the proposed control methodology. Keywords Sliding mode control · Neural networks · Robust adaptive control · Robot manipulators

1 Introduction Robot manipulators are multivariable nonlinear systems and they suffer from various uncertainties in their dynamics, which deteriorate the system performance and stability, such as external disturbance, nonlinear friction, highly time-varying, and payload variation. Therefore, achieving high performance in trajectory tracking is a very challenging task. To overcome these problems, various control methods have been proposed, including adaptive control, intelligent control, sliding mode control and variable structure control, etc. [1–10]. Over the last few decades, the applications of intelligent control such as fuzzy control and neural control to the position control

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of robotic manipulator have received considerable attention for the approximation of nonlinear system. It has been proven that fuzzy and neural network can approximate any nonlinear function to any desired accuracy because of the universal approximation theorem. Recently, much research has been done on using neural networks to provided online learning algorithms and deal with unmodeled unknown dynamics in robot model. In [11], a new neuro fuzzy representation is proposed for the filtered error approximation based control of robotic manipulators. In [12], an approach and a systematic design methodology are presented to adaptive motion control based on NN for high performance robot manipulators. In [13], an approach is proposed for nonlinear dynamical system to model reference adaptive control based on neural networks. In [14], the authors proposed neural adaptive control for sigle master multiple slaves teleoperation to enforce motion coordination of mobile manipulators so as to guarantee the desired trajectories tracking whereas the the tracking error remains bounded. In [15], an adaptive neural network compensator is proposed for control system to improve the control performance without hardware modification. The proposed adaptive NN compensator is very useful when the conventional controller can not properly handle large disturbances and parameter changes. The singular perturbation analysis is employed to investigate the stability and robustness properties of dynamical NN identifier. Various cases that lead to modeling errors are taken into consideration and prove stability and convergence in [16]. In [17], a robust neural network output feedback control scheme that includes a novel neural network observer is presented for the motion control of robot manipulator. In [18], the authors proposed a neural- based control with robust force motion tracking performance for constrained robot manipulators. In [19], the proposed MPC approach can handle a formulated QP problem using a neurodynamic optimination approach. The applied neural networks can make the formulated constrained QP converging to the exact optimal values. In [20], adaptive neural networks are used to approximate the unknow model of the robot and adapt interactions between the robot and the patient to deal with the system uncertainties and improve the system robustnees.

In addition, the RBFNN, one of the most popular intelligent computation approaches, has an inherent learning ability and can approximate a nonlinear continuous function. The RBF network is a kind of neural network, and it is very useful to control the dynamic systems [21–25]. The RBF network adaptation can effectively improve the control performance against large uncertainty of the system. The adaptation law is derived using the Lyapunov method, so that the stability of the entire system and the convergence of the weight adaptation are guaranteed. In [24], the adaptive controller using RBF network is proposed to derive elements of inertia matrix, Coriolis matrix, and gravitational force vector for the dynamics of robot manipulators in task space. By designing the control law in the task space, force control can be easily formulated. In [26], an adaptive neurocontroller based on the RBF network is proposed to deal with tracking control problems for robot manipulators. The adaptation laws for the RBF network are derived to adjust the weights of Gaussian function in real time. In this case, the function approximation error and the neural tuning weights had to be bounded. In [27], the RBFSMC controller is proposed to eliminate the chattering and control an SMA actuator. In [28], an adaptive output feedback control scheme which used RBF network is presented to compensate adaptively for output tracking of continuous time nonlinear systems. The RBF network is also used to adjust adaptively the gain of the sliding mode to eliminate the effects of dynamical uncertainties and guarantee asymptotic error convergence in [29]. The Sliding mode control theory for a class of nonlinear systems has been successfully studied by many researchers, because of its robustness [30–33]. The SMC generates smooth switching between the adaptive and robust modes from integration of advantages of robust and intelligent control. The SMC is designed to reduce the effects of the approximation errors, and it uses again which is large enough to compensate the bounded uncertainties and guarantees stability and passive of nonlinear systems. In [34], the SMC is used only the measurable output estimation error, and the unknown input can be estimated from the measurable sling surface. The gain design is based on HGO that guarantees exponential convergence of the estimation error. In [35], multiple parameters model based

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siding mode technique is presented to improve the tracking performance for a certain class of nonlinear mechanical system. The proposed method can be used to remove the chattering and infinitely fast switching control problem from the robust control strategy. In [36], a control scheme is proposed which can not only overcome the chattering problem caused by sliding mode control, but also determine the rules and parameters for fuzzy logic control. In [37], adaptive algorithms which use adaptation laws for tuning both the SMC gain and the thickness of the boundary layer has been proposed to reject a discontinuous control input, and to improve the tracking performance. Here, the tracking performance can be improved, good for rejection of control chattering phenomenon, and fairly large parameter variation and disturbances can be handled. In this paper, we propose a robust adaptive trajectory tracking SMC based on RBFNN for two link CDRM in electric power substation to achieve the high precision position tracking under various environments. The SMC is designed to be robust to disturbance with a guarantee of the stability of the system. The weights of RBFNN are updated online by using Lyapunov stability theory. And we also use Lyapunov stability theory to prove that the proposed scheme is not only ensure the stability of the control system, but also has the robustness to the unmodeled dynamics. When compared with the existing results in the literatures [25, 28] and [30], our proposed controller is more flexible, inverse dynamical model evaluation is not required, and the time consuming training process is not necessary. Furthermore, based on simulation and experiment results, the disadvantages which are the chattering phenomenon, tracking errors from the discontinuous control efforts of SMC, are improved. The remainder of this paper is organized as follows. The dynamic of CDRM is presented in Section 2. Section 3 describes the structure of RBFNN. Section 4 presents design of robust adaptive RBFNN controller with SMC robust term. Section 5 provides experimental results of two-link CDRM in electric power substation. Finally, in Section 6, concluding remarks are given.

2 Dynamic of Robot Manipulators We consider the dynamics of an n- link CDRM that is shown in Fig. 1 can be expressed in the Lagrange as follows: M(q)q¨ + C(q, q) ˙ q˙ + G(q) = τ

(1)

where (q, q, ˙ q) ¨ ∈ R n×1 are the vectors of joint position, velocity and acceleration, respectively. M(q) ∈ R n×n is the symmetric inertial Matrix. C(q, q) ˙ ∈ n×n R is the vector of Coriolis and Centripetal forces. G(q) ∈ R n×1 is a vector containing Gravity forces and torques, and τ ∈ R n×1 is the control input torque vector. For the purpose of designing controller, the dynamics of the robot model (1) has the following properties: Property 1: The inertial matrix M(q) is a positive symmetric matrix and bounded: m1 x2 ≤ x T M(q)x ≤ m2 x2 , ∀x ∈ R n×1

(2)

With m1 and m2 are known positive constants and they depend on the mass of the robot manipulators.

Fig. 1 The cleaning and detecting robot manipulator

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˙ Property 2: M(q) − 2C(q, q) ˙ is skew symmetric matrix, ∀x: ˙ x T [M(q) − 2C(q, q)]x ˙ =0 Property 3: follows:

(3)

C(q, q) ˙ q˙ and G(q) are bounded as

C(q, q) ˙ q ˙ ≤ Ck q ˙ 2 , G(q) ≤ Gk ,

We assume the output values of ideal RBFNN are MR (q), GR (q, q), ˙ GR (q), and calculated as: T M(q) = MR (q) + εM = WM ∗ hM (q) + εM

C(q, q) ˙ = CR (q, q) ˙ + εC = WCT ∗ hC (q, q) ˙ + εC(6) G(q) = GR (q) + εG = WGT ∗ hG (q) + εG

(4)

where Ck , Gk , are positive constants.

3 The Structure of RBF Neural Networks RBF neural networks are a local mapping networks, which have a few neurons respond to local area of the input space and determine the output of RBF networks. RBF networks can approximate any single value continuous function with arbitrary precision by the enough number of neurons in hidden layer. The configuration of RBFNN is described in Fig. 2.

(5)

(7)

where WM , WC , and WG are ideal optimum weight value of RBF; hM , hC , hG are outputs of hidden layer, εM , εC and εG are modeling error of M(q), C(q, q) ˙ and G(q), respectively, and are assumed to be bounded. The estimated values of MR (q), CR (q, q), ˙ and GR (q) can be expressed by RBF as follows: T Mˆ R (q) = Wˆ M ∗ hM (q)  n  T = Wˆ Mi ∗ hMi (q)

(8)

Cˆ R (q, q) ˙ = Wˆ CT ∗ hC (q, q) ˙  n  T = Wˆ Ci ∗ hCi (q, q) ˙

(9)

ˆ R (q) = Wˆ GT ∗ hG (q) G  n  T = Wˆ Gi ∗ hGi (q)

(10)

i=1

i=1

i=1

where Wˆ M , Wˆ C , and Wˆ C are estimates of WM , WC , and WC respectively, and n is the number of hidden nodes.

4 Design of Robust Adaptive RBF Neural Networks Controller

Fig. 2 Structure of RBF neural networks

In this section, our objective of robot manipulator control is to determine the control torque τ , so that the tracking error between the desired joint position qd and the joint position vector q can converged to zero when t → ∞. Thus, the unknown functions of robot manipulator control system are estimated, and the stability of control system can be guaranteed. The architecture of the robot manipulator control system is shown in Fig. 3.

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Fig. 3 Architecture of the CDRM control system

Define a tracking error vector e(t) and the sliding mode function s(t) as the following equations:

For the dynamics of an n-link robot manipulator (1), the adaptive control law is proposed as:

e(t) = qd − q

τ = fˆ(x) + τs + τP I

s(t) = e˙ + λe

(11) (12)

where λ = diag(λ1 , λ2 , .., λn) is the positive constant gain matrix. It is typical to define an error metric s(t) to be a performance measure. When the sliding surface s(t) = 0, the sliding mode is governed by the following differential equation according to the theory of sliding mode control: e˙ = −λe. The behavior of the system on the sliding surface is determined by the structure of the matric λ. When the error metric s(t) is smaller, the performance is better. Therefore, Eq. 1 can be rewritten as follows: M s˙ +Cs = M q¨d +(Mλ+C)q˙d +Cλqd +G(q)−Mλq−τ (13)

where, τP I = Kp s + KI 0 sdt, diag{Kp1 , Kp2 , . . . , Kpn } is the positive definite matrix, τs is a SMC robust term that is used to suppress the effects of uncertainties and approximation errors, and fˆx is the approximation of the adaptive function fx and is defined as fˆx = Mˆ R q¨d + ˆ R (q) − Mˆ R λqˆ − (Mˆ R λ + Cˆ R )q˙d + Cˆ R λqd + G Cˆ R λq. Follow above analysis, we propose a SMC robust term τs as: τs = Ks sgn(s)

(16)

where Ks = diag{Ks1 , Ks2 , . . . , Ksn }, and Ks > ε. Substuting Eq. 15 into Eq. 14, we have: M(q)˙s + C(q, q)s ˙ = fˆ(x) − τP I − τ s + ε

since Eqs. 5-7, 13 becomes M s˙ + Cs = f (x) − τ + ε

(15) τ

(17)

(14)

where f (x) is defined as f (x) = MR q¨d + (MR λ + CR )q˙d + CR λqd + GR (q) − MR λq˙ − CR λq, and ε = εM (q¨d + λq˙d − λq) ˙ + εC (q˙d + λqd − λq) + εG

where f˜(x) = f (x) − fˆ(x) = M˜ R q¨d + (M˜ R λ + C˜ R )q˙d ˜ R (q) − M˜ R λq˙ − C˜ R λq + C˜ R λqd + G (18)

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By applying the adaptive control law Eq. 15 to the dynamic Eq. 1, and using the sliding mode control ⎧ ⎪ ˙ i ⎨ Wˆ Mi = (Mi ∗ hMi )(q¨d + λq˙d )si − λ(Mi ∗ hMi )qs ˆ WCi = (Ci ∗ hCi )(q˙d + λqd )si − λ(Ci ∗ hCi )qs ˙ i ⎪ ⎩ Wˆ Gi = (Gi ∗ hGi )si

where Mi , Ci , and Gi , are symmetric positive definite constant matrices. The stability of the closed-loop system in Fig. 3 which is described by Eq. 15 is established in the following theorem.

robust term function Eq. 16, the online RBF neural networks adaptive update laws are designed as:

i = 1, 2, . . . , n

(19)

According property 2, Eq. 21 becomes: V˙ (t) = s T



t M s˙ + Cs + KI sdt 0

n n T T ˜ ˜ + W˜ Mi −1 W˜ Ci −1 Mi WMi + Ci WCi i=1

Theorem: Consider an n-link robot manipulator represented by Eq. 1. If the RBF neural networks adaptive update laws are designed as Eq. 19, and the SMC robust term τs is given by Eq. 16, and the adaptive control law designed in Eq. 15, then the tracking error and the convergence of all the system parameters can be assured and approached to zero. According to the Lyapunov stability analysis, if the Lyapunov function is positive definite and its derivative is negative semidefinite, then the control system is stable. Therefore, to guarantee the stability of the total control system, we consider the Lyapunov function candidate as follows:

 t

T  t

1 T V (t) = s Ms + sdt KI sdt 2 0 0 n +

W˜ T −1 W˜ Mi + i=1 Mi Mi

+

n i=1



W˜ T −1 W˜ Ci i=1 Ci Ci (20)

where and W˜ M = WM − Wˆ M ; W˜ C = WC − Wˆ C ; W˜ G = WG − Wˆ G . Defferentiating V (t) along to time, the following equation can be obtained as: 

t 1 ˙ V˙ (t) = s T M s˙ + Ms + KI sdt 2 0 n n T T ˜ ˜ + W˜ Mi −1 W˜ Ci −1 Mi WMi + Ci WCi i=1

n +

i=1

i=1

T ˜ W˜ Gi −1 Gi WGi

i=1

i=1

T ˜ W˜ Gi −1 Gi WGi

(22)

Submitting Eqs. 17 and 18 into Eq. 22, yields V˙ (t) = −s T Kp s − s T Ks sgn(s) + s T ε    +s T M˜ R q¨d + M˜ R λ + C˜ R q˙d + C˜ R λqd  ˜ R (q) − M˜ R λq˙ − C˜ R λq +G n + W˜ T −1 W˜ Mi i=1 Mi Mi n + W˜ T −1 W˜ Ci i=1 Ci Ci n T ˜ + W˜ Gi −1 Gi WGi i=1

V˙ (t) = −s Kp s − s T Ks sgn(s) + s T ε    T +s T W˜ M ∗ hM (q¨d + λq˙d    T −λ W˜ M ∗ hM q˙    +s T W˜ CT ∗ hC (q˙d + λqd    −λ W˜ CT ∗ hC q   +s T W˜ GT ∗ hG n + W˜ T −1 W˜ Mi i=1 Mi Mi n + W˜ T −1 W˜ Ci i=1 Ci Ci n T ˜ + W˜ Gi −1 Gi WGi T

n

T ˜ W˜ Gi −1 Gi WGi

n +

(21)

i=1

(23)

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Since Eqs. 8-10, we have       ⎧    T   T T ∗h ˜ T ∗ hM q˙ = n ⎪ W˜ Mi ∗ hMi (q¨d + λq˙d ) si − λ ni=1 W˜ Mi s T W˜ M ∗ hMi qs ˙ i M (q¨d + λq˙d ) − λ W ⎪ i=1 M ⎪ ⎨        n  T n  T T T T W˜ C ∗ hC (q˙d + λqd ) − λ W˜ C ∗ hC q = i=1 W˜ Ci ∗ hCi (q˙d + λqd ) si − λ i=1 W˜ Ci ∗ hCi qsi s ⎪    ⎪ ⎪   T ⎩ s T W˜ GT ∗ hG (q) = ni=1 W˜ Gi ∗ hGi si

Now, Eq. 23 can be rewritten as

i=1 n  

−λ +

T W˜ Mi

0

Because V(0) is a bounded function, and V (t) is non-increasing and bounded, we have t limt →∞ s T Kp sdt < ∞ (29)



∗ hMi qs ˙ i

 T W˜ Ci ∗ hCi (q˙d + λqd ) si

−λ

0

Accordingto Barbalat’s Lemma, it can be shown t that limt →∞ 0 s T Kp sdt = 0. Since V(0) and Kp are positive constants, it follows that s ∈ Ln2 . Consequently, e ∈ Ln2 ∩ Ln∞ , e is continuous and converged to zero when t → ∞, and e˙ ∈ Ln2 . From Eq. 26, it follows that V (t) ∈ L∞ , it is mean that W˜ Mi ; W˜ Ci ; W˜ Gi are bounded, and W˜ Mi ; W˜ Ci ; W˜ Gi are bounded. Therefore, we can conclude that (˙s , τs , τ ) ∈ Ln∞ . Using the fact that s ∈ Ln2 and s˙ ∈ Ln2 , thus s → 0

 T W˜ Ci ∗ hCi qsi

i=1 n    T + W˜ Gi ∗ hGi (q) si i=1

+ + +

n i=1 n i=1 n i=1

T ˜ W˜ Mi −1 Mi WMi T ˜ W˜ Ci −1 Ci WCi T ˜ W˜ Gi −1 Gi WGi

0

(28)

i=1 n  

i=1 n  

Equation 27 can be rewritten as: ∞ ∞ T s Kp sdt ≤ − V˙ (t)dt = V (0) − V (∞)



V˙ (t) = −s T Kp s − s T Ks sgn(s) + s T ε n    T + W˜ Mi ∗ hMi (q¨d + λq˙d ) si

(25)

Applying adaptive update laws (19) into (25), and follows substitutions W˜ Mi = −W˜ Mi , W˜ Ci = −W˜ Ci , and W˜ Gi = −W˜ Gi , and considering Ks > ε, the derivative V˙ (t) can be bounde V˙ (t) = −s T Kp s − s T Ks sgn(s) + s T ε ≤ −s T Kp s Then V˙ (t) ≤ 0

(26)

Hence, all parameters of the adaptive control system are bounded with τ > 0, and all initial conditions are bounded at t = 0, 0 ≤ V (0) ≤ ∞ is ensured. Furthermore, integrating V˙ (t) with respect to time as follows: ∞ ∞ V˙ (t)dt ≤ − s T Kp sdt (27) 0

0

(24)

Fig. 4 Two link cleaning and detecting robot manipulator

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when t → ∞, hence e˙ → 0 when t → ∞, both the global stability of the system and the tracking errors are guaranteed and converged to zero when t → ∞ by the adapting control law (15). The proof is completed.

technology, which is shown in Fig. 1, is employed to verify the effectiveness of the proposed control scheme. We consider, the dynamic equation of the two-link CDRM in electric power substation model, is shown in Fig. 4, can be described by using Lagrang method.

5 Simulation and Experiment Results



5.1 Simulation Results In this section, for illustrative purposes, a two-link cleaning and detecting robot manipulator in electric power substation in our Lab of intelligent automation

M11 (q2 ) M12 (q2 ) M21 (q2 ) M22 (q2 )



q¨1 q¨2





  C11 (q2 ) C12 (q2 ) q˙1 C21 (q2 ) C22 (q2 ) q˙2     τ G1 (q1 , q2 ) = 1 + G2 (q1 , q2 ) τ2 +

where



   M11 (q2 ) M12 (q2 ) (m1 + m2 )l12 + m2 l22 + 2m2 l1 l2 cos(q2 ) m2l22 + m2 l1 l2 cos(q2 ) M(q) = = M21 (q2 ) M22 (q2 ) m2l22 + m2 l1 l2 cos(q2 ) m2l22     C11 (q2 ) C12 (q2 ) −m2 l1 l2 sin(q2)2˙ 2 −m2 l1 l2 (q˙1 + q˙2 ) sin(q2 ) C(q, q) ˙ = = C21 (q2 ) C22 (q2 ) m2 l1 l2 sin(q2 )q˙1 0     G1 (q1 , q2 ) (m1 + m2 )l1 g cos(q2 ) + m2 l2 g cos(q1 + q2 ) G(q) = = G2 (q1 , q2 ) m2 l2 g cos(q1 + q2 )

in which, m1 and m2 are links masses; l1 and l2 are links lengths; g = 10(m/s 2 ) is acceleration of gravity. The parameters of two link cleaning and detecting robot manipulator in electric power substation are given by Table 1. The object is to design control input in order to force joint variables q = [q1 q2 ]T to track desired trajectories as time goes to infinity. Here, the desired position trajectories of the two link cleaning and detecting robot manipulator in electric power substation are chosen by qd = [qd1 qd1 ]T = [0.5 sin(πt) sin(πt)]T , and initial positions of joints are q0 = [0.1 − 0.1]T , and initial velocities of joints are q˙0 = [0.0 0.0]T . Let e = qd − q = [e1 e2 ]T be the tracking errors. The element of the gain matrix in the adaptive control law (19) is: Mi = 5; Ci = 5.

Table 1 Parameters of CDRM Link

Mass (kg)

Length (m)

Link 1 Link 2

2 1

0.8 1

In the following passage, our proposed control scheme is applied to the CDRM in comparison with the adaptive Fuzzy (AF) [4] and the proportional differential (PD) control, where the output of the PD controller can be expressed to be ypd = Kα e(t) + Kβ e˙(t), and the PD gains were selected to be Kα = diag[200, 250]; Kββ = diag[20, 20]. The simulation results of joint position responses, tracking errors and control torques in following the desired trajectories for joint 1 and joint 2 are shown in Fig. 5, when there is not any disturbance (d(t) =0). From these simulation results, we can see that the proposed system converges to the desired trajectory more quickly and achieves tracking performance better than both the case with PD and AF control. Therefore, the use of proposed scheme with adaptation weights can effectively improve the performance of the closed- loop system compared with the existing results. It seems that the robust tracking performance of the proposed control scheme is more excellent and effective than the PD and AF control. The parameter values used in the adaptive control system are chosen for the convenience of simulations as follows λ = diag[5, 5]; Kp = diag[50, 50]; KI = diag[100, 100]; Ks = diag[0.1, 0.1].

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Fig. 5 Simulated position responses, tracking errors, and control efforts of the proposed system, AF and PD control

5.2 Experiment Results In this section, we have also applied the proposed strategy to control cleaning and detecting robot

manipulator which are movable in the condenser water chamber in our Lab of intelligent automation technology (Fig. 1). The CDRM system is an intelligent robotic system combined with the auxiliary equipment

High-pressure water jet Condensers Link 2

Remote/Monitoring control system Link 1

Cameras

Driven wheel Manual control device

Fig. 6 Mechanical structure of the CDRM

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for condenser cleaning and detecting. This system is designed to guarantee the CDRM can self operate in various conditions under requirement for safety. The cleaning and detecting are implemented by the support of cameras, sensors system and a high pressure water jet. The mechanical structure of the CDRM is shown in Fig. 6. In this experiment, electrical control system of the CDRM which is expressed in Fig. 7 is based on the distributed control structure adopting multi-parallel processing system. The internal centre of the main controller is connected to a number of coprocessor module through CAN bus. The co-processor module generates internal network of the robot pose structure. The design makes the robot possess the abilities of high speed information processing and real time communication. The Cleaning and detecting robot can be controlled with both manual and automatic method. The manual method is performed by worker, and the robot activities are controlled and supervised via the remote control system and cameras. With the automatic method, the robot operations are automatic

a. The first experimental case

Visual controller

CAN bus

Main controller Manual control

Automatic control

Control system

Wheels controller Servo valve

Fig. 7 Electrical control structure of CDRM

Fluid motor

Robot arm controller Servo valve

Fluid motor

Liquid controller Liquid pumps

Underwater camera

Detector of condenser smudginess

Underwater source

Flux sensor of steam

Detector of water status

Pressure sensor of steam

Temperature of cooling water

Cooling water sensor

Pressure and follow sensors

Underwater sonar

Arm encoder

Loca lization encoder

Controller of position

controlled and supervised. Firstly, the CDRM control system will check all conditions to guarantee robot can work. After that, the cleaning and detecting robot will automatically detect exact condenser positions of the tube. Next, the CDRM control system will automatically determine the optimization of pressure washing, cleaning time or cleaning cycles by supporting cameras and sensors system as cooling water sensor, flux sensor, pressure and follow sensor. Then, the high pressure water jet which is on robot will be used to clean by high pressure liquid pumbs. After the cycle of cleaning, the CDRM system will continuously detect condenser smudginess positions of the tube. In this study, to test the applicability and the performance of the proposed technique, two different experimental cases are done to consider the performance under various environments which are the parameter variation and the change of the external disturbance.

J Intell Robot Syst Table 2 Parameters of CDRM Link

Mass (kg)

Length (m)

Link 1 Link 2

3 2

0.8 1

ˆ ˆ Fig. 9. From Fig. 9, we can see that both M(q), G(q) ˆ and C(q, q) ˙ do not converge to M(q), G(q) and C(q, q), ˙ respectively. This is mean that the desired trajectory is not exciting persistenly, which happens often in real application. b. The second experimental case

In this case, we assume that the masses of two links are changed as in Table 2 with the same desired trajectories and the others parameters as in the simulation case. Figure 8 shows the experimental results of joint position responses, tracking errors and control torques. From these results, it is easy to see that the responses and the tracking error norm of the proposed control scheme are quite better than both PD and AF control method. Moreover, Fig. 8 imply that our control torques are less and smooth than the PD control and method in [4]. It seems that the robust tracking performance of our proposed control scheme is better than the PD and AF control [4] under parameter variation. The variation of the elements of M(q), G(q), C(q, q) ˙ and the estimation of ˆ ˆ ˆ them, respectively, M(q), G(q), C(q, q) ˙ are shown in

In this case, we assume that the robot is tracking a trajectory and suddenly the external disturbance is injected into control system. This happened after the first 5s of the experimental time, and all other parameters are chosen as in the simulation case. The shapes of the external disturbance are expressed as follows: d(t) = [50 sin(10t)

50 sin(10t)]T

The experimental responses of joint position, tracking error and control torque are shown in Fig. 10. From this experimental results, we can find that, the control performance and robustness of the proposed controller under external disturbance are better than PD and AF controller. The performance of our proposed

Fig. 8 The experimental results of position responses, tracking errors, and control efforts for the first case

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Fig. 9 Estimated norms of M(q), G(q), C(q, q˙ ) of the adaptive RBF neural networks in the first case

approach is only slightly affected, while the performance of PD approach is seriously affected. The variation of the elements of M(q), G(q), C(q, q) ˙ and the estimated norms of them, respectively, ˆ ˆ ˆ M(q), G(q), C(q, q) ˙ are shown in Fig. 11. From

ˆ ˆ ˆ Fig. 11, we can see that both M(q), G(q) and C(q, q) ˙ do not converge to M(q), G(q) and C(q, q) ˙ respectively. This is mean that the desired trajectory is not exciting persistently, which happens often in real application.

Fig. 10 The experimental results of position responses, tracking errors, and control efforts for the second case

J Intell Robot Syst

Fig. 11 Estimated norms of M(q), G(q), C(q, q˙ ) of the adaptive RBF neural networks in the second case

6 Conclusion

References

In this paper, a robust adaptive trajectory tracking SMC based on RBFNN is proposed for two link CDRM in electric power substation to achieve the high precision position tracking under various environments. Based on the Lyapunov stability theorem, the estimation convergence, the stability robustness, the uniformly ultimate boundedness and the tracking performance of the proposed control system can be guaranteed. Simulation and experiment results were presented on a two link CDRM and comparisons were made with the performance of PD control and AF control. Finally, as demonstrated in the illustrated simulation and experiment results, the proposed control scheme in this approach is not only reduce the chattering phenomenon, but also can achieve the high precision position tracking and good robustness in the trajectory tracking control of two link CDRM in electric power substation under various environments over the existing results.

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Acknowledgments This work was supported by National Natural Science Foundation of China (Grant nos. 61175075) National Hightech Research Research and Development Projects (Grant nos. 2012AA112312, Grant nos. 2012AA11004). The authors would like to thank the editor and the reviewers for their invaluable suggestions , which greatly improved the quality for this paper dramatically.

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