Artificial Neural Network Based Prediction Model of the Sliding Mode Control in Coordinating Two Robot Manipulators Parvaneh Esmaili1 and Habibollah Haron2 1

Department of Computer Science, Faculty of Computing, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia [email protected] 2 Department of Computer Science, Faculty of Computing, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia [email protected]

Abstract. The design of a decentralized controlling law in the coordinated transportation area of an object by multiple robot manipulators employing implicit communication between them is a specific alternative in synchronization problems. A decentralized controller is presented in this work which is combination of the sliding mode control and artificial neural network which guarantees robustness in the system. Implicit communication among robot manipulators considers the light weight beam angle in this controller. A multi layer feed forward neural network based prediction model is presented not only to improve trajectory tracking of multiple robots but also to solve the chattering phenomena in the sliding mode control. The simulation results show the effectiveness of the proposed controller on two cooperative PUMA 560 robot manipulators. Keywords: Decentralized control, neural network, cooperative robot manipulators, handles an object, synchronization.

1

Introduction

Recently, in cooperative robot manipulator systems which can accurately control the system in the presence of structured and unstructured uncertainties have become a challenging issue. Several types of controllers have been categorized into different groups. Cooperative robot controlling methods are categorized into one of two methods: Centralized and Decentralized. Centralized controlling methods are approaches which consider all robot arms and the handling object as one closed chain system. Conversely decentralized methods almost entirely use implicit behavior to communicate instead of explicit information which helps robots to act individually. In addition, there are several kinds of controlling methods within both decentralized and centralized classification methods such as kinematics, dynamics, master slave, sliding mode, adaptive robust methods and adaptive neural network based sliding mode control. Several researchers have considered kinematics and N.T. Nguyen et al. (Eds.): ACIIDS 2014, Part I, LNAI 8397, pp. 474–483, 2014. © Springer International Publishing Switzerland 2014

Artificial Neural Network Based Prediction Model of the Sliding Mode Control

475

dynamics method for control, but in cooperative robot system, due to complexity problems, these methods were not sufficient [1] [2] [3]. Some studies used various combinations with force control and position/force control [4-5] [6] [7]. Component based methods are considered as [8-9]. Impedance control was proposed [10] to achieve accurate system control and there are several works based on impedance controller in a multiple manipulator structure [11]. However, impedance control method was not sufficient to handle the dynamic properties of the real world environment such as stiffness, damping and inertia. A variable structure system with sliding mode control was first suggested in the 1950’s, and after 1970 the sliding mode control became commonly used because it consistently showed good performance for nonlinear systems in multi input multi output systems and has been usefully implemented in discrete time systems as well. It is robust controller and is also insensitive in the presence of parameter changes and external disturbances. However, the sliding mode control has two disadvantages. The first is chattering phenomena which is high output frequency of the controller. The second is that the calculation of known sections of nonlinear system as equivalent control is difficult [12] when solving these kinds of problems. Some researchers have been investigated on some soft computing methods which have been adapted from artificial intelligence such as fuzzy logic, neural network, evolutionary computing and so on [12][13] [14][15]. A hybrid sliding mode control with neural network has been investigated by several researchers to develop one robot manipulator which was initially proposed [13] and investigated by some researchers [12] [14] [16]. In a similar manner, some works have focused only on the dual arm robot as a cooperative system based on the sliding mode with adaptive neural network controlling method. Also, in these kinds of systems, the system is considered a closed chain kinematic or dynamics equation. However, the ability of act separately faces problems as they must use a centralized controlling method which is not sufficient and accurate enough for multiple robotic systems in the presence of structured and unstructured uncertainties in the real world. The aim of this work is to design a decentralized control law by using implicit communication, the robot manipulators and the object transfer simultaneously with the same constant yet limited velocity. First, the sliding mode control schema for PUMA 560 is then implemented. The artificial neural network based prediction model for the dynamic section of PUMA 560 is presented. Finally the proposed controller in this work is extended for two decentralized robot manipulators and the results of the controller are shown as follows.

2

Modeling of the System

The decentralized control law uses the implicit information between robots. As implicit information for each robot manipulator, the edge angle of the object, reveals the objects position and the fixed end- effectors on the object by using set points which are mentioned in Fig.1. By using the dynamic equation, the motion of the robot manipulator will be derived. As implicit information, by changing the angle of the object and related end-effector, the other robot can understand information about another robot. For this reason, robot manipulators work by implicit communication

476

P. Esmaili and H. Haaron

with respect to each otherr. The purpose of this work is to design a decentraliized artificial neural network based b prediction model of sliding mode control law by using implicit communicaation, the robot manipulators and the object transfer simultaneously with the sam me constant limited velocity.

n of the two cooperative robot manipulators to handle object Fig. 1. The construction

Dynamic equation of PU UMA 560 robot based on Lagrange – Euler [17] formulla is as follows in Eq. (1).

.. .

(1)

Where M is a [6 6] masss matrix, B is the Carioles- Coefficient which is [6 15], C is Centrifugal- Coefficient which is [6 6] matrix, G is Gravity term [6 1] maatrix and is Torque [6 1] maatrix. The measured parameters of PUMA 560 are driiven using the Denavitt- Hartenb berg [18]. Therefore, , ,…, is the vector off the angle of the 6-DOF robot manipulator and , ,…, , , ,…, is the velocity and acceleraation of the robot manipulator. In general, Eq. (2) is uused for robot manipulator dynam mics. ,

,

2

This work attempts to develop d a decentralized robust sliding mode control w with neural network for two coop perative manipulators in a cooperative manner.

3

The Controller Design D

3.1

The Sliding Mode Control C

The sliding mode controlleer will be designed based on the dynamic equation of the cooperative system. There are a two steps needed for the sliding mode controller [12]]. In the first step, a sliding surfface for the cooperative system is chosen. The second sstep

Artificial Neural Network Based Prediction Model of the Sliding Mode Control

477

involves the equivalent controller. In this work, to omit the chattering phenomenon, the sliding mode controller uses the neural network based prediction model. Some researchers used soft computing methods [13] [19]. In general, the dynamic second order nonlinear model of multi input multi output system can be explained in Eq. (3) [19]. ,

(3)

Where x= [ , , … , is the vector of the generalized coordinates such as position , ,…, , , ,…, is the of the n DOF mechanical system and velocity and acceleration of the robot manipulators. In addition, a bounded nonlinear vector is a function of the state vector to present the nonlinear term of the system and is a bounded positive definite nonlinear function over the entire state space known as the control gain and is the control input of the system. Therefore, the proposed dynamic equation for the robot manipulator to handle an object as mentioned above is mapped into Eq. (4) as follows. ,

, ,

, ,

, , ,

, (4)

The control inputs are as follows: (5)

In the SMC design, tracking control problems can be investigated by holding the system trajectory on the sliding surface. To switch the function design in the first step, the PD type sliding surface is chosen. (6)

, are Where is a 6 1 vector, is a constant. So, tracking error vector and the rate of tracking error vector. The second step of the controller design is to define the control law with variable parameters. The following control law is considered as: (7)

Where and terms are defined as follows. The term is the equivalent control term which is proposed for the approximately known section of the system in the presence of perturbations. This section makes the derivative of the sliding surface equal to zero in order to remain on the sliding surface that has been previously mentioned as the low frequency control law. The term is the corrective control

478

P. Esmaili and H. Haron

term which is proposed to compensate the derivatives from the sliding surface known as the high frequency control law. The term helps the system to set on the sliding surface. ,

So, by using

,

(8)

, there is: (9)

To keep the system on the sliding surface requires that is as follows as expressed in Eq. 10. So,

0 and

0.

(10)

To overcome the dynamic uncertainty in the controlling system caused by system uncertainty, external disturbances, friction and parameter variation, the approximation of is termed [19]. It should be mentioned that an approximate of an artificial neural network based prediction model has been designed. So, the equivalent control is as follows: ∆

,

(11)

Where, ∆ is the difference between real and approximated f function. Thus, the F is a positive function which is bounded the values of f and . The approximation in this work uses a predictive neural network model to find accurate values for dynamic uncertainty in the control input to achieve good trajectory tracking in the presence of uncertainties. Discontinues control unit (corrector) is used to achieve good trajectory tracking performance in very fast switching. The term: here is considered a function to compensate the derivations from the sliding surface and reach to sliding surface. ,

1

0

-1

0

(12)

To stabilize such systems, the differential equation solutions of describing dynamic systems can be achieved by Lyapunov stability theory [19] which is concerned with the stability of solutions near the equilibrium point. It also lends qualitative results to the stability equations which may be useful in designing stabilizing controllers of nonlinear dynamical systems. By using the Lyapunov stability method, the parameter boundaries of sliding mode controllers to stabilize the proposed system can be determined. Positive definite Lyapunov function is as follows: (13)

Thus, for stability the Lyapunov function V(x) should be positive and the derivative of the Lyapunov function should be negative.

Artificial Neural Network Based Prediction Model of the Sliding Mode Control .

479

0

14

As mentioned before

: | |

By choosing

(15)

as follows: | |

(16)

The interest of using artificial neural network arises from non linearity in the system which caused disorder in the performance of the system. One of the common and most applicable neural networks is multi layer neural network. The important key of using multi layer neural network is that this type of neural network can solve continuous non linearity function (mapping between input and output) of the system [13] [16]. 3.2

The Artificial Neural Network Based Prediction Model

The predictive model based neural network is implemented by Matlab 2013 software which uses a neural network model of the nonlinear plant to predict plant performance. The controller also computes the control input to increase plant performance. The Multi layer feed forward neural network is used in this model. Because of the 6 inputs and outputs in the robot manipulator, 6 separate but same prediction models for each robot manipulator angle are considered here. The 10 hidden layers are used in all 6 neural network layers. Therefore, it uses both Tangsig and Pureline activation functions as seen in Fig 2. The controller can act in two steps which are Identification and Predictive control. The first one determines the neural network plant model to identify of the system. The second step is to use the controller to predict future system performance in terms of predictive control. In system Identification, the prediction model is used to train the plants neural network which uses the prediction error between the plant and neural network outputs as the neural networks training signals. Based on the previous inputs and outputs, future outputs of the plant will be predicted by the neural network. For training, multilayer back propagation neural network algorithm is used. A multi layer network is an artificial neural network model that includes multiple layers of fully connected nodes to map the input data into the desired outputs. To learn multilayer perception, the model applies a learning algorithm such as back propagation algorithm. Back propagation or backward error propagation is a training algorithm that consists of two steps: propagation and weight update in the propagation step activities, feed forward propagation in terms of training input patterns are used. The backward propagation of output activities propagation is then used to create the rule of gradient descent learning for weight updating of hidden

480

P. Esmaili and H. Haron

and output layers (neurons). In the second step, to achieve the gradient of the weights, the outputs data set and input activation are multiplied. Thus, the weights are set in the opposite direction. The second step in designing the prediction model based on neural network is predictive control. The prediction section includes the neural network and the optimization part which corrects the data coming from the neural network for the plant.

Fig. 2. The NN model of prediction model

So, the proposed controller for two robot manipulators is extended as follows as in Fig.2. The linkage between two robots as implicit communication is object’s angle.

Fig. 3. The Decentralized proposed controller

4

Simulation Results

Multi layer feed forward neural network with 10 hidden layers (which is based on the Baum-Haussler rule [20]) is used for f function approximation. As there are six robot manipulator angles in PUMA 560, neural networks also use six inputs and outputs. To all six neural network prediction boxes is applied 0.05 control weighting ( ), 0.001 search parameter ( , 2 iteration per sample time, 10 cost horizon (N2), 2 cost Horizon (Nu), 0.2 (Sec) sampling interval, 2 delayed plant input, 2 delayed plant output and 1000 training epochs.

Artificial Neural Netwo ork Based Prediction Model of the Sliding Mode Control

481

Table 1. Results of regressiion, training state and Neural network with validation check=66 Angle No.

Epoch No.

1

17

Gradient 483.6533

Mu 0.1

2

54

155.4274

0.001

3

47

1.2332

0.1

4

40

0.096128

0.0001

5

21

0.016548

0.001

6

497

9.9687e-11

1e-13

Table 2. The regression resullts for all six angles of PUMA in neural network with trainning performance Angle No.

Training

validation

Test

All regression

Training Performance

1

0.954662

0.88318

0.71784

0.83915

55.4822

2

0.99968

0.98219

0.97627

0.98763

22.5304

3

0.99982

0.96905

0.96917

0.95569

9.306

4

0.99992

0.88538

0.7119

0.88382

0.47042

5

0.99442

0.94824

0.74094

0.9355

0.32236

The performance of thee proposed controller just for one robot is compared w with classic sliding mode controller at fig. 4. The x axis of the each plot shows the anglee of each joint of robot and the y axis of the plots shows time(second). The dashed linne is classic sliding mode controlller and the regular line is proposed controller.

Fig. 4. The results of thee decentralized controller for two cooperative robots

482

P. Esmaili and H. Haron

As can see in fig.4 the proposed controller has regular motion in the sine input for angle of robot joint’s. The purpose of the controller is to achieve minimum tracking error which obtains as follows in Fig. 4. Then, the output of the proposed scheme for two robot manipulators which is mentioned at fig.3 is presented at fig. 5(a). for evaluation the proposed controller at fig.5(b) the classic sliding mode controller for two robot manipulators is extended which is revealed the proposed scheme has as regular and reliable motion in the environment to handle an object in the desired trajectory.

(a)

(b)

Fig. 5. The results of controller for two cooperative robots (a) the proposed controller and (b) classic sliding mode controller

5

Conclusion

A decentralized control law is proposed to allow two PUMA560 robot manipulators to cooperate when handling objects. This controller is robust with no chattering controller because of the combination of the sliding mode controller with neural network based prediction model. The angle of the lightweight beam is used as implicit communication in a cooperative manner to solve the synchronization problem. The simulation results guarantee the validity of the proposed controller.

Artificial Neural Network Based Prediction Model of the Sliding Mode Control

483

References 1. Callegari, M., Suardi, A.: Hybrid kinematic machines for cooperative assembly tasks. In: Proceedings of the International Workshop: Multiagent Robotic Systems: Trends and Industrial Applications, Padova, Italy, pp. 223–228 (2003) 2. Tarn, T.J., Bejczy, A.K., Yun, X.: Design Of Dynamic Control Of two Cooperative Robot Arms: Closed Chain Formulation. In: Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 2004, pp. 7–13 (1987) 3. Zhao, Y.S., Ren, J.Y., Huang, Z.: Dynamic loads coordination for multiple cooperating robot manipulators. Mech. Mach. Theory 35, 985–995 (2000) 4. Chen, G.: Study on Dynamic Hybrid Position/Force Control of Two Coordinated Manipulators. Adv. Mater. Res. 468, 1224–1230 (2012) 5. Chiaverini, S., Sciavicco, L.: The Parallel Approach to ForcePosition Control of Robotic Manipulators. IEEE Trans. Robot. Autom. 9, 361–373 (1993) 6. Ishida, T.: Force control in coordination of two arms. In: Proc. 5th Int. Conf. on Artificial Intelligence, pp. 717–722 (1977) 7. Tarn, T.J., Bejczy, A.K., Yun, X.: Co-ordinated control of two robot arms. In: Proc. IEEE Int. Conf. on Robotics and Automation, pp. 1193–1202 (1986) 8. Jawawi, D., Deris, S., Mamat, R.: Software Reuse for Mobile Robot Applications Through Analysis Patterns. Int. Arab J. Inf. Technol. 4, 220–228 (2007) 9. Jawawi, D.N.A., Deris, S., Mamat, R.: Early-Life Cycle Reuse Approach for ComponentBased Software of Autonomous Mobile Robot System. In: 2008 Ninth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing, pp. 263–268 (2008) 10. Hogan, N.: Impedance control: An Approach to Manipulation, Parts I-Theory, II Implementation, III-Applications. ASME J. Dyn. Syst. Meas. Control 107, 1–24 (1985) 11. Esakki, B.: Modeling and Robust Control of Two Collaborative Robot Manipulators (2011) 12. Yagiz, N., Hacioglu, Y., Arslan, Y.Z.: Load transportation by dual arm robot using sliding mode control. J. Mech. Sci. Technol. 24, 1177–1184 (2010) 13. Panwar, V., Kumar, N., Sukavanam, N., Borm, J.-H.: Adaptive neural controller for cooperative multiple robot manipulator system manipulating a single rigid object. Applied Soft Computing 12, 216–227 (2012) 14. Hacioglu, Y., Arslan, Y.Z., Yagiz, N.: MIMO fuzzy sliding mode controlled dual arm robot in load transportation. J. Franklin. Inst. 348, 1886–1902 (2011) 15. Moosavian, S.A.A., Papadopoulos, E.: Cooperative Object Manipulation with Contact Impact Using Multiple Impedance Control. 8, 314–327 (2010) 16. Tsai, C.-C., Cheng, M.-B., Lin, S.-C.: Robust Tracking Control for A Wheeled Mobile Manipulator With Dual Arms Using Hybrid Sliding-Mode Neural Network. Asian Journal of Control 9(4), 377–389 (2007) 17. Lewis, F.L., Abdallah, C.T., Dawson, D.M.: Control of robot manipulators. Macmillan Pub. Co. (1993) 18. Armstrong, B., Khatib, O., Burdick, J.: The Explicit Dynamic Model and Inertial Parameters of the PUMA 566 Am, pp. 510–518. IEEE (1986) 19. Zeinali, M., Notash, L.: Adaptive sliding mode control with uncertainty estimator for robot manipulators. Mech. Mach. Theory 45, 80–90 (2010) 20. Baum, E.B., Haussler, D.: What size net gives valid generalization? Neural Computation 1, 151–160 (1989)

Department of Computer Science, Faculty of Computing, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia [email protected] 2 Department of Computer Science, Faculty of Computing, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia [email protected]

Abstract. The design of a decentralized controlling law in the coordinated transportation area of an object by multiple robot manipulators employing implicit communication between them is a specific alternative in synchronization problems. A decentralized controller is presented in this work which is combination of the sliding mode control and artificial neural network which guarantees robustness in the system. Implicit communication among robot manipulators considers the light weight beam angle in this controller. A multi layer feed forward neural network based prediction model is presented not only to improve trajectory tracking of multiple robots but also to solve the chattering phenomena in the sliding mode control. The simulation results show the effectiveness of the proposed controller on two cooperative PUMA 560 robot manipulators. Keywords: Decentralized control, neural network, cooperative robot manipulators, handles an object, synchronization.

1

Introduction

Recently, in cooperative robot manipulator systems which can accurately control the system in the presence of structured and unstructured uncertainties have become a challenging issue. Several types of controllers have been categorized into different groups. Cooperative robot controlling methods are categorized into one of two methods: Centralized and Decentralized. Centralized controlling methods are approaches which consider all robot arms and the handling object as one closed chain system. Conversely decentralized methods almost entirely use implicit behavior to communicate instead of explicit information which helps robots to act individually. In addition, there are several kinds of controlling methods within both decentralized and centralized classification methods such as kinematics, dynamics, master slave, sliding mode, adaptive robust methods and adaptive neural network based sliding mode control. Several researchers have considered kinematics and N.T. Nguyen et al. (Eds.): ACIIDS 2014, Part I, LNAI 8397, pp. 474–483, 2014. © Springer International Publishing Switzerland 2014

Artificial Neural Network Based Prediction Model of the Sliding Mode Control

475

dynamics method for control, but in cooperative robot system, due to complexity problems, these methods were not sufficient [1] [2] [3]. Some studies used various combinations with force control and position/force control [4-5] [6] [7]. Component based methods are considered as [8-9]. Impedance control was proposed [10] to achieve accurate system control and there are several works based on impedance controller in a multiple manipulator structure [11]. However, impedance control method was not sufficient to handle the dynamic properties of the real world environment such as stiffness, damping and inertia. A variable structure system with sliding mode control was first suggested in the 1950’s, and after 1970 the sliding mode control became commonly used because it consistently showed good performance for nonlinear systems in multi input multi output systems and has been usefully implemented in discrete time systems as well. It is robust controller and is also insensitive in the presence of parameter changes and external disturbances. However, the sliding mode control has two disadvantages. The first is chattering phenomena which is high output frequency of the controller. The second is that the calculation of known sections of nonlinear system as equivalent control is difficult [12] when solving these kinds of problems. Some researchers have been investigated on some soft computing methods which have been adapted from artificial intelligence such as fuzzy logic, neural network, evolutionary computing and so on [12][13] [14][15]. A hybrid sliding mode control with neural network has been investigated by several researchers to develop one robot manipulator which was initially proposed [13] and investigated by some researchers [12] [14] [16]. In a similar manner, some works have focused only on the dual arm robot as a cooperative system based on the sliding mode with adaptive neural network controlling method. Also, in these kinds of systems, the system is considered a closed chain kinematic or dynamics equation. However, the ability of act separately faces problems as they must use a centralized controlling method which is not sufficient and accurate enough for multiple robotic systems in the presence of structured and unstructured uncertainties in the real world. The aim of this work is to design a decentralized control law by using implicit communication, the robot manipulators and the object transfer simultaneously with the same constant yet limited velocity. First, the sliding mode control schema for PUMA 560 is then implemented. The artificial neural network based prediction model for the dynamic section of PUMA 560 is presented. Finally the proposed controller in this work is extended for two decentralized robot manipulators and the results of the controller are shown as follows.

2

Modeling of the System

The decentralized control law uses the implicit information between robots. As implicit information for each robot manipulator, the edge angle of the object, reveals the objects position and the fixed end- effectors on the object by using set points which are mentioned in Fig.1. By using the dynamic equation, the motion of the robot manipulator will be derived. As implicit information, by changing the angle of the object and related end-effector, the other robot can understand information about another robot. For this reason, robot manipulators work by implicit communication

476

P. Esmaili and H. Haaron

with respect to each otherr. The purpose of this work is to design a decentraliized artificial neural network based b prediction model of sliding mode control law by using implicit communicaation, the robot manipulators and the object transfer simultaneously with the sam me constant limited velocity.

n of the two cooperative robot manipulators to handle object Fig. 1. The construction

Dynamic equation of PU UMA 560 robot based on Lagrange – Euler [17] formulla is as follows in Eq. (1).

.. .

(1)

Where M is a [6 6] masss matrix, B is the Carioles- Coefficient which is [6 15], C is Centrifugal- Coefficient which is [6 6] matrix, G is Gravity term [6 1] maatrix and is Torque [6 1] maatrix. The measured parameters of PUMA 560 are driiven using the Denavitt- Hartenb berg [18]. Therefore, , ,…, is the vector off the angle of the 6-DOF robot manipulator and , ,…, , , ,…, is the velocity and acceleraation of the robot manipulator. In general, Eq. (2) is uused for robot manipulator dynam mics. ,

,

2

This work attempts to develop d a decentralized robust sliding mode control w with neural network for two coop perative manipulators in a cooperative manner.

3

The Controller Design D

3.1

The Sliding Mode Control C

The sliding mode controlleer will be designed based on the dynamic equation of the cooperative system. There are a two steps needed for the sliding mode controller [12]]. In the first step, a sliding surfface for the cooperative system is chosen. The second sstep

Artificial Neural Network Based Prediction Model of the Sliding Mode Control

477

involves the equivalent controller. In this work, to omit the chattering phenomenon, the sliding mode controller uses the neural network based prediction model. Some researchers used soft computing methods [13] [19]. In general, the dynamic second order nonlinear model of multi input multi output system can be explained in Eq. (3) [19]. ,

(3)

Where x= [ , , … , is the vector of the generalized coordinates such as position , ,…, , , ,…, is the of the n DOF mechanical system and velocity and acceleration of the robot manipulators. In addition, a bounded nonlinear vector is a function of the state vector to present the nonlinear term of the system and is a bounded positive definite nonlinear function over the entire state space known as the control gain and is the control input of the system. Therefore, the proposed dynamic equation for the robot manipulator to handle an object as mentioned above is mapped into Eq. (4) as follows. ,

, ,

, ,

, , ,

, (4)

The control inputs are as follows: (5)

In the SMC design, tracking control problems can be investigated by holding the system trajectory on the sliding surface. To switch the function design in the first step, the PD type sliding surface is chosen. (6)

, are Where is a 6 1 vector, is a constant. So, tracking error vector and the rate of tracking error vector. The second step of the controller design is to define the control law with variable parameters. The following control law is considered as: (7)

Where and terms are defined as follows. The term is the equivalent control term which is proposed for the approximately known section of the system in the presence of perturbations. This section makes the derivative of the sliding surface equal to zero in order to remain on the sliding surface that has been previously mentioned as the low frequency control law. The term is the corrective control

478

P. Esmaili and H. Haron

term which is proposed to compensate the derivatives from the sliding surface known as the high frequency control law. The term helps the system to set on the sliding surface. ,

So, by using

,

(8)

, there is: (9)

To keep the system on the sliding surface requires that is as follows as expressed in Eq. 10. So,

0 and

0.

(10)

To overcome the dynamic uncertainty in the controlling system caused by system uncertainty, external disturbances, friction and parameter variation, the approximation of is termed [19]. It should be mentioned that an approximate of an artificial neural network based prediction model has been designed. So, the equivalent control is as follows: ∆

,

(11)

Where, ∆ is the difference between real and approximated f function. Thus, the F is a positive function which is bounded the values of f and . The approximation in this work uses a predictive neural network model to find accurate values for dynamic uncertainty in the control input to achieve good trajectory tracking in the presence of uncertainties. Discontinues control unit (corrector) is used to achieve good trajectory tracking performance in very fast switching. The term: here is considered a function to compensate the derivations from the sliding surface and reach to sliding surface. ,

1

0

-1

0

(12)

To stabilize such systems, the differential equation solutions of describing dynamic systems can be achieved by Lyapunov stability theory [19] which is concerned with the stability of solutions near the equilibrium point. It also lends qualitative results to the stability equations which may be useful in designing stabilizing controllers of nonlinear dynamical systems. By using the Lyapunov stability method, the parameter boundaries of sliding mode controllers to stabilize the proposed system can be determined. Positive definite Lyapunov function is as follows: (13)

Thus, for stability the Lyapunov function V(x) should be positive and the derivative of the Lyapunov function should be negative.

Artificial Neural Network Based Prediction Model of the Sliding Mode Control .

479

0

14

As mentioned before

: | |

By choosing

(15)

as follows: | |

(16)

The interest of using artificial neural network arises from non linearity in the system which caused disorder in the performance of the system. One of the common and most applicable neural networks is multi layer neural network. The important key of using multi layer neural network is that this type of neural network can solve continuous non linearity function (mapping between input and output) of the system [13] [16]. 3.2

The Artificial Neural Network Based Prediction Model

The predictive model based neural network is implemented by Matlab 2013 software which uses a neural network model of the nonlinear plant to predict plant performance. The controller also computes the control input to increase plant performance. The Multi layer feed forward neural network is used in this model. Because of the 6 inputs and outputs in the robot manipulator, 6 separate but same prediction models for each robot manipulator angle are considered here. The 10 hidden layers are used in all 6 neural network layers. Therefore, it uses both Tangsig and Pureline activation functions as seen in Fig 2. The controller can act in two steps which are Identification and Predictive control. The first one determines the neural network plant model to identify of the system. The second step is to use the controller to predict future system performance in terms of predictive control. In system Identification, the prediction model is used to train the plants neural network which uses the prediction error between the plant and neural network outputs as the neural networks training signals. Based on the previous inputs and outputs, future outputs of the plant will be predicted by the neural network. For training, multilayer back propagation neural network algorithm is used. A multi layer network is an artificial neural network model that includes multiple layers of fully connected nodes to map the input data into the desired outputs. To learn multilayer perception, the model applies a learning algorithm such as back propagation algorithm. Back propagation or backward error propagation is a training algorithm that consists of two steps: propagation and weight update in the propagation step activities, feed forward propagation in terms of training input patterns are used. The backward propagation of output activities propagation is then used to create the rule of gradient descent learning for weight updating of hidden

480

P. Esmaili and H. Haron

and output layers (neurons). In the second step, to achieve the gradient of the weights, the outputs data set and input activation are multiplied. Thus, the weights are set in the opposite direction. The second step in designing the prediction model based on neural network is predictive control. The prediction section includes the neural network and the optimization part which corrects the data coming from the neural network for the plant.

Fig. 2. The NN model of prediction model

So, the proposed controller for two robot manipulators is extended as follows as in Fig.2. The linkage between two robots as implicit communication is object’s angle.

Fig. 3. The Decentralized proposed controller

4

Simulation Results

Multi layer feed forward neural network with 10 hidden layers (which is based on the Baum-Haussler rule [20]) is used for f function approximation. As there are six robot manipulator angles in PUMA 560, neural networks also use six inputs and outputs. To all six neural network prediction boxes is applied 0.05 control weighting ( ), 0.001 search parameter ( , 2 iteration per sample time, 10 cost horizon (N2), 2 cost Horizon (Nu), 0.2 (Sec) sampling interval, 2 delayed plant input, 2 delayed plant output and 1000 training epochs.

Artificial Neural Netwo ork Based Prediction Model of the Sliding Mode Control

481

Table 1. Results of regressiion, training state and Neural network with validation check=66 Angle No.

Epoch No.

1

17

Gradient 483.6533

Mu 0.1

2

54

155.4274

0.001

3

47

1.2332

0.1

4

40

0.096128

0.0001

5

21

0.016548

0.001

6

497

9.9687e-11

1e-13

Table 2. The regression resullts for all six angles of PUMA in neural network with trainning performance Angle No.

Training

validation

Test

All regression

Training Performance

1

0.954662

0.88318

0.71784

0.83915

55.4822

2

0.99968

0.98219

0.97627

0.98763

22.5304

3

0.99982

0.96905

0.96917

0.95569

9.306

4

0.99992

0.88538

0.7119

0.88382

0.47042

5

0.99442

0.94824

0.74094

0.9355

0.32236

The performance of thee proposed controller just for one robot is compared w with classic sliding mode controller at fig. 4. The x axis of the each plot shows the anglee of each joint of robot and the y axis of the plots shows time(second). The dashed linne is classic sliding mode controlller and the regular line is proposed controller.

Fig. 4. The results of thee decentralized controller for two cooperative robots

482

P. Esmaili and H. Haron

As can see in fig.4 the proposed controller has regular motion in the sine input for angle of robot joint’s. The purpose of the controller is to achieve minimum tracking error which obtains as follows in Fig. 4. Then, the output of the proposed scheme for two robot manipulators which is mentioned at fig.3 is presented at fig. 5(a). for evaluation the proposed controller at fig.5(b) the classic sliding mode controller for two robot manipulators is extended which is revealed the proposed scheme has as regular and reliable motion in the environment to handle an object in the desired trajectory.

(a)

(b)

Fig. 5. The results of controller for two cooperative robots (a) the proposed controller and (b) classic sliding mode controller

5

Conclusion

A decentralized control law is proposed to allow two PUMA560 robot manipulators to cooperate when handling objects. This controller is robust with no chattering controller because of the combination of the sliding mode controller with neural network based prediction model. The angle of the lightweight beam is used as implicit communication in a cooperative manner to solve the synchronization problem. The simulation results guarantee the validity of the proposed controller.

Artificial Neural Network Based Prediction Model of the Sliding Mode Control

483

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