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ScienceDirect

Procedia Technology 19 (2015) 607 - 614

8th International Conference Interdisciplinarity in Engineering, INTER-ENG 2014,9-10 October

2014, Tirgu Mures, Romania

Multi-input multi-output fuzzy logic controller for complex system: Application on two-links manipulator

Fatima Zahra Baghlia'*, Larbi El bakkalia, Yassine Lakhala

"Modeling and Simulation of Mechanical Systems Laboratory, Abdelmalek Essaadi University, Faculty of Sciences, BP.2121, M'hannech,

93002, Tetouan, Morocco

Abstract

The robot manipulator is a mechanical system multi-articulated, in which each articulation is driven individually by an electric actuator, it is the most robot used in industry, this system need an efficient control strategy, such as Fuzzy Logic law control, by means each articulation is controlled independently, this kind of control present a lot of inconvenient, such as error of each articulation, isn't taken account into others.

In this present work, we present a Multi-Input Multi-Output Fuzzy controller (MIMO) to ensure the articulation robot control strategy, the results obtained present satisfactory and shows clearly the efficiency of the present Fuzzy-MIMO controller. © 2015 The Authors.PublishedbyElsevierLtd.This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of "Petru Maior" University of Tirgu Mures, Faculty of Engineering

Keywords:Robot Manipulator; Dynamic Modeling; Fuzzy Logic; Control, MIMO.

1. Introduction

Research on the dynamic modeling and control of the arms manipulators has received increased attention since the last years due to their advantages.

A robot manipulator is a high-speed process, that is highly nonlinear, dynamically coupled and often it is not adequate to use linear servo control, if accurate performance in high bandwidth operations is desired. Many efforts have been made in developing control scheme to achieve the precise tracking control of robot manipulators.

* Corresponding author. Tel.: +212644989050; fax: +212539994500. E-mail address:baghli.fatimazahra@gmail.com

2212-0173 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of "Petru Maior" University of Tirgu Mures, Faculty of Engineering doi: 10.1016/j.protcy.2015.02.086

Knowledge based control, expert control and intelligent control are somewhat synonymous and fuzzy control is a particular type of intelligent control. Fuzzy logic control has a great potential since it is able to compensate for the uncertain nonlinear dynamics, using the programming capability of human control behavior. The main features of fuzzy control is that a control knowledge base, is available within the controller and control actions are generated by applying existing conditions or data to the knowledge base, making us of inference mechanism [2,4]. Also, the knowledge base and inference mechanism can handle no crisp, incomplete information; the knowledge itself will improve and evolve through learning and past experience [2].

Fuzzy logic control does not require a conventional model of the process, whereas most conventional techniques require either an analytical model or an experimental model. Fuzzy logic control is particularly suitable for complex and ill-defined process in which analytical modeling is difficult due to the fact that the process is not completely known and experimental model identification is not feasible because the required inputs and output of the process may not be measurable.

The majority of process industries are nonlinear, multi-input multi-output (MIMO) systems. The control of these systems is met with a number of difficulties due to process interactions, dead time and process nonlinearities. The difference between MIMO systems control and Single-Input Single-Output (SISO) systems control is based on an estimation and compensation of the process interaction among each degree of freedom. It is obvious that the difficulty of MIMO systems control is how to overcome the coupling effects among each degree of freedom. To obtain good performance, coupling effect cannot be neglected. Hence SISO system control scheme is not easy to implement on complicated MIMO systems. In addition, the control rules and controller computation will grow exponentially with respect to a number of considered variables. Therefore, intelligent control strategy is gradually drawing attention [5].

In this work, after the system modeling, simulation and control robot manipulator using two articulations for motion using MatLab/Simulink software were carried, when the proposed MIMO Fuzzy controlled is used to improve the articulation robot stability. Two types of control Fuzzy-SISO and Fuzzy-MIMO were studied; analysed and comparative studies were made.

The reminder paper was structured as follow: the robot modeling is presented in second part of this paper, in the third part of this paper, the Fuzzy-MIMO is detailed, the results obtained present the efficiency and the robustness of the proposed control with good performances compared with Fuzzy- SISO.

Nomenclatures

NB Negative Big

NM Negative Medium

NS Negative Small

PB Positive Big

PM Positive Medium

PS Positive Small

ZE Zero

FLC Fuzzy Logic Controller

FLC MIMO Multi input Multi output Fuzzy Logic Controller

SFs Scaling Factor

MFC Main Fuzzy Controller

CFC Coupling Fuzzy Controller

2. Physical Model

Robot Manipulators are familiar examples of trajectory controllable mechanical systems. However, their nonlinear dynamics present a challenging control problem, since traditional linear control approaches do not easily apply.

The manipulator used for simulation is a two revolute joined robot, whose position can be described by a (2x1) vector q of joint angles and whose actuator inputs consists of a (2x1) torque vector t applied to the manipulator joints as illustrated in Fig.l

The dynamical equation of manipulator robot of 2 solids articulated between us is given by the following matrix equation [6]:

z = M(q)q + C(q,q) + G(q) (1

where:

ris the (2 x 1) vector ofjoint torque, M(q) is the (2 x 2) symmetric positive definite manipulator inertia matrix, C(q, q) present the (2 x 1) vector of Coriolis and centrifugal forces, G{q) is the (2 x 1) vector of gravitational references and q,q ,¿1 are: position, velocity and acceleration of each articulations.

The elements of the inertia matrix M(q) in the terms of the parameters of the robot manipulator are given by:

Mn(q) = /1 +12 + mxl2cl + m2 l2c2 + m2 /12 + lm2 ljc 2 c2 M12 (q) = M21 (q) = I2 + m2 l2c2 + lrn2 ljc 2 c2 M22(q) = 2 /2 + w2 /c22

The matrix elements (q,g)(/, j = 1,2) centrifugal and Coriolis force are:

Qlfo ?) = "^2/l/c2^2^2 Cn(q, q ) = "^2 /1/c 2 s2( qx + g2)

Ql(tf, tf) = ^2/l/c2^2 C22( ^) = 0

Fig.l. Structure of manipulator robot of two degree of freedom

where:

Finally the elements of the vector of gravitational torques G (q) are given by:

Gl(q) = (m, + m2)glclcl + m2 glc2 c12 G2(q) = m2 glc2 c12

3. MIMO Fuzzy Controller Design

Generally, the principal design elements of the Fuzzy Logic control system are as follows: Fuzzification, Inference mechanism and Deffuzification [4]. The first one comprises the process of transforming crisp values into grades of membership for linguistic terms of fuzzy sets.

The fuzzy inference system has been considered the min-max method (Mamdani), where the implication has been assumed to min and the aggregation has been considered to max. In addition, the Deffuzification method has been considered to the centroid method and the Deffuzification is the conversion of a fuzzy quantity represented by a membership function into a precise or crisp value. This fuzzy controller is to be designed to automate how a human expert who is successful at this task would control the system. First, the expert tells us (the designers of the fuzzy controller) what information she or he will use as inputs to the decision-making process. The Fig.2 shows the FLC structure.

Fig.2.Fuzzy logic controller structure

where: FUZZIFICA is the Fuzzification process and DEFFUZIFF is the Deffuzification process

In this paper the MIMO fuzzy control strategy is used to multi-machines system position control. The block diagram of the MIMO fuzzy control scheme is shown in Fig.3. The design procedure of the fuzzy control strategy consists of main fuzzy controller (MFC) and coupling fuzzy controller (CFC). The concept of adding coupling controller is to compensate the coupling effects of system dynamics among each degree of freedom.

Fig.3. Block diagram of the MIMO Fuzzy Control schema

An ordinary fuzzy controller that usually operates with system output error and error change was chosen as the main controller to control each degree of freedom of the MIMO systems. Here, the input variables of the conventional fuzzy controller for among each degree of freedom of a MIMO system were defined individually

ei (k) = q',{k) - q, (k)

Ae; (k) = {k)- e, (k -1) (2)

where ei (k) is the position error of the i'k degree, Ae{ (k) is used for change in error of the i'k degree, q* {k) is the reference input of the i'k degree and qt {k) represents the i'k position output of each degree of freedom of this MIMO system at the k'k sample. The relationship between the scaling factors (Ge, G0e, Gr)and the input and output variables of the FLC are:

eiN = Ge X e.

= GAe xAe;

= GAt xArm (3)

Selection of suitable values for Ge, Gae, and GT are made based on the knowledge about the process to be controlled and sometimes through trial and error to achieve the best possible control performance. This is so because, unlike conventional no fuzzy controllers to date, there is no well-defined method for good setting of scaling factor's for FLC's. The SFs are the significant parameters of FLC because changing the SFs changes the normalized universe of discourse, the domains, and the membership functions of input or output variables of FLC. All membership functions (MFs) for controller inputs (e;, Ae;) and incremental change in controller output (Ar; )are

defined on the common normalized domain [-1, +l] .We use symmetric triangles (except the two MFs at the extreme ends) as shown in Fig. 4.

Membership functions for controller inputs and controller output

Fig.4. Membership functions of ejjAe.andAr,

This is the most natural and unbiased choice for MFs. By way of the above design process, the actual control input torque for the main fuzzy controller can be written as:

(*) - (k-1) + Ar;(k)

The fuzzy rule-base is shown in Table.l, and these rules can be written in the format of IF-THEN

Table. 1. Rules base

NB NS z PS PB

NB NB NB NS NS z

NS NB NS NS z PS

Z NS NS z PS PS

PS NS Z PS PS PB

PB Z PS PS PB PB

The fuzzy control rules of the coupling fuzzy controller are similar to the main fuzzy controller. The output of the coupling fuzzy controller is chosen directly as the coupling control input torque. The main reason is that there is a different coupling effect for each sampling interval and it does not have an accumulating feature. The coupling effect is incorporated into the main fuzzy controller for each step to improve system performance and robustness. Fig.4 illustrates the structure of MIMO fuzzy control schema. Therefore, the total control input torque of the MIMO fuzzy controller is represented as:

(*) - (*) + T;_ (*) (i * I) (5)

where Tj(k) expresses the system control input torque of the i'k degree of a main fuzzy controller. T(k)i^j represents the coupling effect control of the Ith degree relative to the i'k degree of the coupling fuzzy controller.

Fig.5. Structure of MIMO fuzzy control schema

4. Simulation results

SISO control based on FLC model and intelligent control based on MIMO FLC model were tested to sinus response trajectory. This simulation applied to two degrees of freedom robot arm was implemented in Matlab/Simulink as shown in Fig.6 and Fig.7. Position error is compared in these controllers.

02-► fk

-►02

Fig. 6. Arm manipulator robot FLC Control.

Fig. 7. Arm manipulator robot FLC-MIMO Control

In Fig. 8-9 are shown error performance, by comparing position error for the first and second link; FLC's error is higher than FLC-MIMO.

We can summaries all the obtained results in the Table 2.

FLCl-Position erroe of joint 1

FLC MIMO- Position error of first link

J °'5

Position 1 error

> Time [secj \

1.6 1.4 1.2

.S 0'6

'% 0.4

- Positionl error

Time [sec]

Fig.8.(a) FLC1 position error for the first link; (b) FLC-MIMO position error for the first link.

FLC2- Position error of joint 2

i------1

■ Position 2 error

- 1------r -

i"ime [secf

0.1 0 -0.1

t -°-2

.2 -0-3

^ -0.4 -0.5 -0.6

FLC MIMO-Position error of second link

position 2

Time [sec]

Fig.9.(a) FLC1 position error for the second link; (b) FLC-MIMO position error for the second link.

Table.2. FLC and FLC MIMO Results

Controller FLC FLC-MIMO

Links Linkl Link2 Linkl Link2

Position error[rad] 0.0033 0.0010 0.0025 0.0004

overshoot [%] 5% 0% 0% 0%

5. Conclusion

In this present work, an arm manipulator robot using two degree of freedom was controlled using two types of controls strategies, SISO control based on FLC model and MIMO model based on Multi-Inputs Multi-Outputs (FLC) controller and (FLC-MIMO), this last one present maximum control structure of our control model, it give more and more efficiency for the robot model with more position stability and good dynamical performances, with no overshoot. So industrials would take into account the efficiency of the developing control model for the futures two freedom robot design considerations.

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