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Design Novel Nonlinear Controller Applied to Robot Manipulator: Design New

Feedback Linearization Fuzzy Controller With Minimum Rule Base Tuning

Method

Article in International Journal of Robotics and Automation · February 2012

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Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 1

Design Novel Nonlinear Controller Applied to Robot

Manipulator: Design New Feedback Linearization Fuzzy

Controller With Minimum Rule Base Tuning Method

Farzin Piltan [email protected]

Industrial Electrical and Electronic

Engineering SanatkadeheSabze

Pasargad. CO (S.S.P. Co), NO:16

,PO.Code 71347-66773, Fourth floor

Dena Apr , Seven Tir Ave , Shiraz , Iran

Mohammad Keshavarz [email protected]

Industrial Electrical and Electronic

Engineering SanatkadeheSabze

Pasargad. CO (S.S.P. Co), NO:16

,PO.Code 71347-66773, Fourth floor

Dena Apr , Seven Tir Ave , Shiraz , Iran

Ali Badri [email protected]

Industrial Electrical and Electronic

Engineering SanatkadeheSabze

Pasargad. CO (S.S.P. Co), NO:16

,PO.Code 71347-66773, Fourth floor

Dena Apr , Seven Tir Ave , Shiraz , Iran

Arash Zargari [email protected]

Industrial Electrical and Electronic

Engineering SanatkadeheSabze

Pasargad. CO (S.S.P. Co), NO:16

,PO.Code 71347-66773, Fourth floor

Dena Apr , Seven Tir Ave , Shiraz , Iran

Abstract

In this paper, fuzzy adaptive base tuning feedback linearization fuzzy methodology to adaption

gain is introduced. The system performance in feedback linearization controller and feedback

linearization fuzzy controller are sensitive to the main controller coefficient. Therefore, compute

the optimum value of main controller coefficient for a system is the main important challenge work.

This problem has solved by adjusting main fuzzy controller continuously in real-time. In this way,

the overall system performance has improved with respect to the classical feedback linearization

controller and feedback linearization fuzzy controller. Adaptive feedback linearization fuzzy

controller solved external disturbance as well as mathematical nonlinear equivalent part by

applied fuzzy supervisory method in feedback linearization fuzzy controller. The addition of an

adaptive law to a feedback linearization fuzzy controller to online tune the parameters of the fuzzy

rules in use will ensure a moderate computational load. Refer to this research; tuning

methodology can online adjust coefficient parts of the fuzzy rules. Since this algorithm for is

specifically applied to a robot manipulator.

Keywords: Feedback Linearization Controller, Fuzzy Logic Methodology, Feedback Linearization

Fuzzy Controller, Adaptive Methodology, Fuzzy Adaptive Feedback Linearization Fuzzy

Methodology.

Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 2

1. INTRODUCTION, BACKGROUND AND MOTIVATION

One of the important challenges in control algorithms is a linear behavior controller design for

nonlinear systems. When system works with different parameters and hard nonlinearities this

technique is very useful in order to be implemented easily but it has some limitations such as

working near the system operating point[2]. Some of robot manipulators which work in industrial

processes are controlled by linear PID controllers, but the design of linear controller for robot

manipulators is extremely difficult because they are nonlinear, uncertain and MIMO[1, 6]. To

reduce above challenges the nonlinear robust controllers is used to systems control. One of the

powerful nonlinear robust controllers is feedback linearization controller (FLIC), this controller

has been analyzed by many researchers [7].This controller is used in wide range areas such as

in robotics, in control process, in aerospace applications and in power converters. Even though,

this controller is used in wide range areas but, pure FLIC has challenged in uncertain (structured

and unstructured) system. A multitude of nonlinear control laws have been developed called

“computed-torque” or “inverse dynamic” controller in the robotics literature [1-3]. These

controllers incorporate the inverse dynamic model of robot manipulators to construct. The

computed-torque controllers have their root in feedback linearization control methodology [4, 9].

The idea is to design a nonlinear feedback (maybe calculated using the inverse dynamic model

of the robot manipulator to be controlled) which cancels the nonlinearities of an actual robot

manipulator. In this manner the closed-loop system becomes exactly linear or partly linear

depending on the accuracy of the dynamic model, and then a linear controller such as PD and

PID can be applied to control the robot manipulator. The main potential difficulty encountered in

implementation of the computed-torque control methodology described above is that the

dynamic model of the robot manipulator to be controlled is often not known accuratly. For

instance, then the ideal performance (i.e., the exact linearization) of the computed-torque control

has been proposed in [18, 23]. In the adaptive computed-torque contol methodology, it is

assumed that the structure of the robot manipulator dynamics is perrfectly known but physcical

parameters such as links mass, links inertia and friction coefficient are unknown. Therefore, the

liearity in the parameters property of robot manipulator dynamic model presented in next part are

exploited either to identify unnown parameters or modify the partially known parameters in

order to account for the model uncertainty. The two important requirements of adaptive FLIC

methodology are: the parameters must be updated such that the estimated inertia matrix

remains positive definite and bounded at all times, which means the lower and upper bounds of

ineritia parameters must be known a priori; and the measurment of acceleration is need in order

to realize the update law [7]. Furthermore, due to the fact that parameters errors are not the sole

source of imperfect decoupling and linearization of the robot manipulator dynamics, thus this

control methodology cannot provie robustness against external disturbances and unstructured

uncertainties [8]. Another difficulty that may be encountered in the implementation of FLIC is that

the entire dynamic model (the inertia matrix and the vector of Coriolis, centrifugal, and

gravitational terms) of the robot manipulator, i.e., all terms of equation in robot manipulator, must

be computed on-line and in the control law, since control is now based on the nonlinear feedback

of the current system state. For a robot manipulator with many joints and links, for example for a

6-DOF serial robot manipulator (Stewart-Gough platform) these computations can be

complicated and time consuming. The problem of computation burden can even be increased

when the adaptive FLIC is used. This is due to extra computation needed to update the

parameters in each sample time. Two methods can be found in the literature to deal with the

problem of computation burden described above. One method to deal with the problem of

computation burden is to use feedforward computed-torque control in which the tourqe vector is

computed on the basis of the desired trajectory of the joints (i.e., desired joints positions,

velocities and accelerations) and FLIC the nonlinear coupling effects. As opposed to feedback

FLIC, in the feedforward method it is possible to pre-compute all the terms of the dynamic model

off-line and reduce the computation burden to a large extent [3-9]. The second method to deal

with the problem of heavy computation burden in the FLIC is to develop a computationally

efficient dynamic model. The feedback linearization-based (computed-torqe/inverse dynamic)

control methodologies rely on the knowledge of the robot manipulator dynamic model and its

parameters. In the case of imperfect dynamic model the closed-loop dynamics will no longer be

Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 3

decoupled and linearized, for detailed information the reader is referred to [4, 18, 23].

Furthermore, in the feedback linearization-based control methodology, the control law may

cancel some beneficial nonlinearity such as friction [18, 23].

2. ROBOT MANIPULATOR DYNAMICS, OBJECTIVES, PROBLEM

STATEMENTS AND FEEDBACK LINEARIZATION FORMULATION

Robot manipulator dynamic formulation: The equation of an n-DOF robot manipulator

governed by the following equation [1, 3, 15-29]:

(1)

Where τ is actuation torque, M (q) is a symmetric and positive define inertia matrix, is the

vector of nonlinearity term. This robot manipulator dynamic equation can also be written in a

following form:

(2)

Where B(q) is the matrix of coriolios torques, C(q) is the matrix of centrifugal torques, and G(q) is

the vector of gravity force. The dynamic terms in equation (2) are only manipulator position. This is

a decoupled system with simple second order linear differential dynamics. In other words, the

component influences, with a double integrator relationship, only the joint variable ,

independently of the motion of the other joints. Therefore, the angular acceleration is found as to

be [3, 10-29]:

(3)

Feedback Linearization Control: This technique is very attractive from a control point of view.

The central idea of FLIC is feedback linearization so, originally this algorithm is called feedback

linearization controller. It has assumed that the desired motion trajectory for the

manipulator , as determined, by a path planner. Defines the tracking error as:

(4)

Where e(t) is error of the plant, is desired input variable, that in our system is desired

displacement, is actual displacement. If an alternative linear state-space equation in the

form can be defined as

(5)

With and this is known as the Brunousky canonical form. By

equation (4) and (5) the Brunousky canonical form can be written in terms of the state

as [1]:

(6)

With

(7)

Then compute the required arm torques using inverse of equation (7), is;

(8)

This is a nonlinear feedback control law that guarantees tracking of desired trajectory. Selecting

proportional-plus-derivative (PD) feedback for U(t) results in the PD-computed torque controller

[7-9, 18-23];

(9) | and the resulting linear error dynamics are |

(10)

According to the linear system theory, convergence of the tracking error to zero is guaranteed [6].

Where and are the controller gains. The result schemes is shown in Figure 1, in which two

feedback loops, namely, inner loop and outer loop, which an inner loop is a compensate loop and

an outer loop is a tracking error loop. However, mostly parameter is all unknown. So the

Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 4

control cannot be implementation because non linear parameters cannot be determined. In the

following section computed torque like controller will be introduced to overcome the problems.

The application of proportional-plus-derivative (PD) FLIC to control of PUMA 560 robot

manipulator introduced in this part. PUMA 56o robot manipulator is a nonlinear and uncertain

system which needs to have powerful nonlinear robust controller such as computed torque

controller.

Suppose that in (9) the nonlinearity term defined by the following term

(11)

Therefore the equation of PD-CTC for control of PUMA 560 robot manipulator is written as the

equation of (12);

(12)

FIGURE 1: Block diagram of PD-computed torque controller (PD-CTC)

Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 5

The controller based on a formulation (12) is to robot dynamics therefore it has problems

in uncertain conditions.

Problem Statement: feedback linearization controller is used in wide range areas such as in

robotics, in control process, in aerospace applications and in power converters because it has an

acceptable control performance and solve some main challenging topics in control such as

resistivity to the external disturbance. Even though, this controller is used in wide range areas

but, pure FLIC has the following disadvantage: the main potential difficulty encountered in

implementation of the computed-torque control methodology described above is that the dynamic

model of the robot manipulator to be controlled is often not known accuratly. On the other hand,

pure fuzzy logic controller (FLC) works in many areas, it cannot guarantee the basic requirement

of stability and acceptable performance[8]. Although both FLIC and FLC have been applied

successfully in many applications but they also have some limitations. Proposed method focuses

on substitution fuzzy logic system applied to main controller to compensate the uncertainty in

nonlinear dynamic equivalent equation to implement easily and avoid mathematical model base

controller. To reduce the effect of uncertainty in proposed method, adaptive method is applied in

feedback linearization fuzzy controller in robot manipulator.

Objectives: The main goal is to design a position controller for robot manipulator with acceptable

performances (e.g., trajectory performance, torque performance, disturbance rejection, steady

state error and RMS error). Robot manipulator has nonlinear dynamic and uncertain parameters

consequently; following objectives have been pursuit in the mentioned study.

• To design and implement a position feedback linearization fuzzy controller in order to

solve the uncertainty in nonlinear parameters problems in the pure feedback linearization

control.

• To develop a position adaptive feedback linearization fuzzy controller in order to solve the

disturbance rejection.

3. METHODOLOGY: DESIGN A NOVEL ADAPTIVE FEEDBACK

LINEARIZATION FUZZY ESTIMATION CONTROLLER

First step, Design feedback linearization fuzzy controller: In recent years, artificial intelligence

theory has been used in robotic systems. Neural network, fuzzy logic, and neuro-fuzzy are

combined with nonlinear methods and used in nonlinear, time variant, and uncertainty plant (e.g.,

robot manipulator). This controller can be used to control of nonlinear, uncertain, and noisy

systems. This method is free of some model-based techniques that used in classical controllers.

The main reasons to use fuzzy logic technology are able to give approximate recommended

solution for unclear and complicated systems to easy understanding and flexible. Fuzzy logic

provides a method which is able to model a controller for nonlinear plant with a set of IF-THEN

rules, or it can identify the control actions and describe them by using fuzzy rules. Besides

applying fuzzy logic in the main controller of a control loop, it can be used to design adaptive

control, tuning parameters, working in a parallel with the classical and non classical control

method. However the application area for fuzzy control is really wide, the basic form for all

command types of controllers consists of;

• Input fuzzification (binary-to-fuzzy[B/F]conversion)

• Fuzzy rule base (knowledge base)

• Inference engine

• Output defuzzification (fuzzy-to-binary[F/B]conversion).

Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 6

As a summary the design of fuzzy logic controller based on Mamdani’s fuzzy inference method

has four steps, namely, fuzzification, fuzzy rule base and rule evaluation, aggregation of the rule

output (fuzzy inference system), and deffuzzification [10-15, 29].

Fuzzification: the first step in fuzzification is determine inputs and outputs which, it has one input

( ) and one output ( ). The input is which measures the summation of linear loop and

nonlinear loop in main controller. The second step is chosen an appropriate membership function

for inputs and output which, for simplicity in implementation and also to have an acceptable

performance the researcher is selected the triangular membership function. The third step is

chosen the correct labels for each fuzzy set which, in this research namely as linguistic variable.

The linguistic variables for input ( ) are; Negative Big (NB), Negative Medium (NM), Negative

Small (NS), Zero (Z), Positive Small (PS), Positive Medium (PM), Positive Big (PB), and it is

quantized in to thirteen levels represented by: -1, -0.83, -0.66, -0.5, -0.33, -0.16, 0, 0.16, 0.33, 0.5,

0.66, 0.83, 1 and the linguistic variables to find the output are; Large Left (LL), Medium Left (ML),

Small Left (SL), Zero (Z), Small Right (SR), Medium Right (MR), Large Right (LR) and it is

quantized in to thirteen levels represented by: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

Fuzzy Rule Base and Rule Evaluation: the first step in rule base and evaluation is provide a

least structured method to derive the fuzzy rule base which, expert experience and control

engineering knowledge is used because this method is the least structure of the other one and the

researcher derivation the fuzzy rule base from the knowledge of system operate and/or the

classical controller. Design the rule base of fuzzy inference system can play important role to

design the best performance of fuzzy sliding mode controller, that to calculate the fuzzy rule base

the researcher is used to heuristic method which, it is based on the behavior of the control of robot

manipulator suppose that the fuzzy rules in this controller is;

F.R1: IF is NB, THEN is LL. (13)

The complete rule base for this controller is shown in Table 1. Rule evaluation focuses on

operation in the antecedent of the fuzzy rules in fuzzy sliding mode controller. This part is used

fuzzy operation in antecedent part which operation is used.

Aggregation of the Rule Output (Fuzzy Inference): Max-Min aggregation is used to this work

which the calculation is defined as follows;

(14)

Deffuzzification: The last step to design fuzzy inference in our fuzzy sliding mode controller is

defuzzification. This part is used to transform fuzzy set to crisp set, therefore the input for

defuzzification is the aggregate output and the output of it is a crisp number. In this design the

Center of gravity method is used and calculated by the following equation;

(15)

This table has 7 cells, and used to describe the dynamics behavior of fuzzy controller.

NB NM NS Z PS PM PB

LL ML SL Z SR MR LR

TABLE 1: Rule table

Figure 2 is shown the feedback linearization fuzzy controller based on fuzzy logic controller and

minimum rule base.

Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 7

FIGURE2: Block Diagram of Feedback Linearization Fuzzy Controller with Minimum Rule Base

Second Step; Design Fuzzy Adaptive Feedback Linearization Fuzzy Controller With

Minimum Rules: All conventional controller have common difficulty, they need to find several

parameters. Tuning feedback linearization fuzzy method can tune automatically the scale

parameters using artificial intelligence method. To keep the structure of the controller as simple as

possible and to avoid heavy computation, a two inputs Mamdani fuzzy supervisor tuner is

selected. In this method the tuneable controller tunes the PD coefficient feedback linearization

controller using gain updating factors.

However proposed feedback linearization fuzzy controller has satisfactory performance but

calculate the main controller coefficient by try and error or experience knowledge is very difficult,

particularly when system has uncertainties; fuzzy adaptive feedback linearization fuzzy controller

is recommended.

The adaption low is defined as

(16)

where the is the positive constant and

(17)

As a result proposed method is very stable with a good performance. Figure 3 is shown the block

diagram of proposed fuzzy adaptive applied to feedback linearization fuzzy controller. The fuzzy

system can be defined as below

(18)

where

(19)

where | is adjustable parameter in (18) and | is membership function. |

error base fuzzy controller can be defined as | ||

(20) |

the fuzzy division can be reached the best state when and the error is minimum by

the following formulation

Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 8

(21)

Where is the minimum error, is the minimum approximation error.

The adaptive controller is used to find the minimum errors of .

suppose is defined as follows

(22)

Where

(23)

the adaption low is defined as

(24)

where the is the positive constant.

FIGURE 3: Design fuzzy adaptive feedback linearization fuzzy controllers

4 SIMULATION RESULTS

Pure feedback linearization controller (FLIC) and fuzzy adaptive feedback linearization fuzzy

controller (FAFLIFC) are implemented in Matlab/Simulink environment. Tracking performance and

disturbance rejection are compared.

Tracking Performances

From the simulation for first, second and third trajectory without any disturbance, it was seen that

FLIC and FAFLIFC have the same performance because this system is worked on certain

environment. The FAFLIFC gives significant trajectory good following when compared to pure

fuzzy logic controller. Figure 4 shows tracking performance without any disturbance for FLIC and

FAFLIFC.

Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 9

FIGURE 4: FLIC Vs. FAFLIFC: applied to 3DOF’s robot manipulator

By comparing step response trajectory without disturbance in FLIC and FAFLIFC, it is found that

the FAFLIFC’s overshoot (2.4%) is lower than FLIC’s (14%) and the rise time in FAFLIFC’s (1.2

sec) and FLIC’s (0.8 sec).

Disturbance Rejection

Figure 5 has shown the power disturbance elimination in FLIC and FAFLIFC. The main targets in

these controllers are disturbance rejection as well as the other responses. A band limited white

noise with predefined of 40% the power of input signal is applied to the FLIC and FAFLIFC. It

found fairly fluctuations in FLIC trajectory responses.

Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 10

FIGURE 5: FLIC Vs. FAFLIFC: applied to robot manipulator.

Among above graph relating to trajectory following with external disturbance, FLIC has fairly

fluctuations. By comparing some control parameters such as overshoot and rise time it found that

the FAFLIFC’s overshoot (2.4%) is lower than FLIC’s (60%), although both of them have about

the same rise time.

5 CONCLUSIONS

In this research, fuzzy adaptive base tuning feedback linearization fuzzy methodology to outline

learns of this adaption gain is recommended. Since proof of stability is an important factor in

practice does not hold, the study of stability for robot manipulator with regard to applied artificial

intelligence in robust classical method and adaptive low in practice is considered to be a subject

in this work. The system performance in feedback linearization controller and feedback

linearization fuzzy controller are sensitive to the main controller coefficient. Therefore, compute

the optimum value of main controller coefficient for a system is the main important challenge

work. This problem has solved by adjusting main controller coefficient of the feedback

linearization controller continuously in real-time. In this way, the overall system performance has

improved with respect to the classical feedback linearization controller. As mentioned in previous,

this controller solved external disturbance as well as mathematical nonlinear equivalent part by

applied fuzzy supervisory method in feedback linearization fuzzy controller. By comparing

between fuzzy adaptive feedback linearization fuzzy controller and feedback linearization fuzzy

controller, found that fuzzy adaptive feedback linearization fuzzy controller has steadily stabilised

Farzin Piltan, M. Keshavarz, A. Badri & A. Zargari

International Journal of Robotics and Automation (IJRA), Volume (3) : Issue (1) : 2012 11

in output response but feedback linearization fuzzy controller has slight oscillation in the presence

of uncertainties.

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