A fuzzy logic controllerfora two link functional manipulator

International Journal of Computer Networks & Communications (IJCNC) Vol.6, No.6, November 2014
A FUZZY LOGIC CONTROLLERFORA TWO-LINK
FUNCTIONAL MANIPULATOR
Sherif Kamel Hussein1,Mahmoud Hanafy Saleh2
1Department of Communications and Electronics October University for Modern
Sciences and Arts Giza - Egypt
2Electrical Communication and Electronics Systems Engineering Department,
Canadian International College-CIC
ABSTRACT
This paper presents a new approach for designing a Fuzzy Logic Controller "FLC"for a dynamically
multivariable nonlinear coupling system. The conventional controller with constant gains for different
operating points may not be sufficient to guarantee satisfactory performance for Robot manipulator. The
Fuzzy Logic Controller utilizes the error and the change of error as fuzzy linguistic inputs to regulate the
system performance. The proposed controller have been developed to simulate the dynamic behavior of A
Two-Link Functional Manipulator. The new controller uses only the available information of the input-output
for controlling the position and velocity of the robot axes of the motion of the end effectors
KEYWORDS
Fuzzy Logic Control “FLC”,Degree of Freedom "DOF", MATLAB Simulink
1. INTRODUCTION
Robotic manipulators are a major component in the manufacturing industry. They are used for
many reasons including speed, accuracy, and repeatability. Increasingly, robotic manipulators are
finding their way into our everyday life. In fact in almost every product we encounter a robotic
manipulator has played a part in its production. The equations of motion for the two arms are
described by nonlinear differentialequations. Because closed-form solutions are not available, the
equations of motion are numericallystudied using a numerical method. Special interest is devoted
to determine the motion of the two arms to yield a desired xy-position of the robot hand.
The robot is a multi-functional manipulator designed to move materials, parts, tools, or
specialized devices, through variable programmed motions for the performance of a variety of
tasks. The industrial robot is a programmable mechanical manipulator, capable of moving along
several directions, equipped at its end with a work device called the end effectors or tool and
capable of performing factory ordinarily done by human beings. The term robot is used for a
manipulator that has a built-in control system and is capable of stand-alone operation. The robot
is one of the most important machines for industrial automations; flexible multifunctional robotic
manipulators can be applied to dangerous environments or routine labor as substitutes for the
workers.
Robotic manipulators are multivariable nonlinear coupling dynamic systems. Since, the robotic
manipulators have complicated nonlinear mathematical models. Control systems based on the
system model are difficult to design [1-6]. Controlling the position and velocity of the robot axes
DOI : 10.5121/ijcnc.2014.6608 109

of motion generates the motion of the end effectors. An axes of motion in robotics means a
degree of freedom in which the robot can move, The degrees of freedom, or DOF, is a very
important term to understand. Each degree of freedom is a joint on the arm, a place where it can
bend or rotate or translate. You can typically identify the number of degrees of freedom by the
number of actuators on the robot arm. Now this is very important - when building a robot arm
you want as few degrees of freedom allowed for your application!!! Why? Because each degree
requires a motor, often an encoder, and exponentially complicated algorithms and cost.
Thus, an n-degree of freedom manipulator contains n-joints, or in more general terms, n-axes of
motion. The axes of motion of the robot arm can be either rotary or linear. A rotary axis is
designed in kinematics as a revolute pair, which is simple hinge without axial sliding. It is
usually driven by an electric motor, which is coupled to the axis either directly or through a chain
or a gear system.
Control of an industrial robot includes nonlinearities, uncertainties and external perturbations that
should be considered in the design of control laws. Afuzzy controller is used for monitoring and
control the input scaling factors of the fuzzy controller according to the actual tracking position
error and the actual tracking velocity error [6,7].
This paper presents the dynamics of the robot manipulators and presents a fuzzy control for a
multi-variable nonlinear system to provide robust control characteristics. We derive the
equations of motion for a general open-chain manipulator and, using the structure present in the
dynamics, construct control laws for asymptotic tracking of a desired trajectory. In deriving the
dynamics, we will make explicit use of twists for representing the kinematics of the manipulator
and explore the role that thekinematics play in the equations of motion. We assume some
familiarity with dynamics and control of physical systems[8-10].
This paper is organized as follows; the second section describes the manipulator dynamic. The
third section is devoted to discuss a fuzzy control scheme. The fourth section is concerned with
the design of the proposed fuzzy control scheme. Finally the last section with the analysis of
simulation, and the conclusions.
110
2.MANIPULATOR DYNAMICS
There are two problems in robot kinematics; the first problem is referred to as the forward
kinematics problem, while the second problem is the inverse kinematics (arm solution) problem.
Robot arm dynamics deals with the mathematical formulations of the equations of robot arm
motion. The dynamic equations of motion of a manipulator are a set of mathematical equations
describing the dynamic behavior of the manipulator [3-5].
Our study of manipulators has focused on kinematics consideration only. There are two problems
related to the dynamics of a manipulator that we wish to solve. In the first problem, we are given
a trajectory point, the position, velocity, and the acceleration; we wish to find the required joint
torque. This formulation of dynamics is useful for the problem of controlling the manipulator.
The second problem is to calculate how the mechanism will move under application of a set of
joint torques that is given a torque to calculate the resulting motion of manipulator, the position,
the velocity, and the acceleration, this useful for simulation. The motion control problem consists
of obtaining dynamic models of the manipulator, and using these models to determine control
laws to achieve the desired system response and performance. The dynamic model of multi-link

robot arm can be described by [3,4,11]. The simplified model of a two-link manipulator system is
shown in Fig.1.
111
M2, L2 , 2
M1, L1 , 1
x
y
g
1
2
Fig. 1: A simplified model of a two--link manipulator
The system consists of two masses connected by weightless bars. The bars have length L1and L2.
The masses are denoted by M1and M2, respectively.Let 1and 2denote the angles in which the
first bar rotates around the origin and the second barrotates about the endpoint of the first bar,
respectively.
2.1 Kinetic Energy
The kinetic energy is based on the x-yaxis labeled on the first link. The velocities are only going
to occur in the x-yplane. The velocity vi, i = 1,2is the magnitude of the xyvelocity of the center of
mass of each link. We have no velocities in the zaxis because this problem had the joints fixed in
a barring that only moves in the x-yplane.The equation for the kinetic energy can be written as
KE=1/2m1[(dx1/dt)2+dy/dt)2]+1/2m2[(dx2/dt)2+(dy2/dt)2] … (1)
Where m1 is the mass of the link 1, m2 is the mass of the link 2,.
2.2 Potential Energy
We have to understand the potential energy due to gravity of each arm. Thepotential energy of the
arm-link is
PEi ()=mi g hi() … (2)
= m1 g l1 sin(1) + m2 g(l1 sin (1) + l2 sin(2))
where hiis the height of the center of mass of the iarm, gis the acceleration due to gravityconstant,
andl1is the length of the link 1, 1 is the angle of the link connection 1, 2 is the angle of the link
connection 2, l2is the length of the link 2,
The definitions for kinetic energy and potential energy can be considered by Lagrange Dynamics,
we form the Lagrangian which is defined as
–
… (3)
ℒ =
Substituting the values for the kinetic and potential energies in for KEand PEwe get :
ℒ =
1/2m1[(dx1/dt)2+dy/dt)2]+1/2m2[(dx2/dt)2+ dy2/dt)2] + m1 g l1 sin(1) + m2 g(l1 sin (1) + l2
sin(2)) (4)

International Journal of Computer Networks Communications (IJCNC) Vol.6, No.6, November 2014
112
The equations for the x-position and the y-position of 1are given by
1 = l1cos1 … (5)
And
1 = l1 sin 1 … (6)
Similarly, the equations for the x-position and the y-position of 2are given by
2 = l1cos1 + l2cos2 … (7)
and
2 = l1 sin 1 + l2 sin 2 … (8)
Next, we define the velocity of 1as
1 =[(d1/ dt)2 + (d1/dt)2] …(9)
Similarly, the velocity of 2is defined as
2 = [(d2/ dt)2 + (d2/dt)2] … (10)
The dynamic model of the robot arm excluding the dynamic of the joint motors, backlash, and
gear friction can be obtained from lagrange-euler or Newton-euler approach. It is often
convenient to express the dynamic equations of a manipulator in a single equation
T=A() q + B(, q) +C() …(11)
Where;
T A generalized vector of joint torques
, q A generalized joint coordinate angle and acceleration vectors
A() The n*n mass matrix
B(, q) The n*1 vector of centrifugal and coriolis terms
C() The n*1 n-vector of gravity terms
Each element of A(), and C() are complex function which depend on the angle , the position
of all the joints of the manipulator, and each element of B(, q) is a complex function of both the
angle , and the rate of change of the angle .
The dynamic model of multi-link robot arm can be described by:
T1=A11q
1+A12q
2+A122q2
2+A112q
1q
2+D1q
1+A1 … (12)
T2 = A21q
1 + A22q
2 + A211q2
2 + D2q
2 + A2 … (13)
Where:
A1 = m1 g s1sin 1 + m2g(l1sin 1+ s2 sin(1+2))
A2 = m2 g s2 sin2

113
A11=m1 s1
1 + s2
2 + 2 l1s2cos2)
2 + m2 (l2
A12= m2 (s2
2 + l1s2cos2)
A122=-m2l1s2sin2
A112= - 2m2 l1s2sin2
A21= A12 .A22=m2 s2
2 + jm . A211=m2 l1s2sin2
Where :
m1 is the mass of the link1
m2 is the mass of the link2
l1 is the length of the link 1
1 is the angle of the link connection 1
2 is the angle of the link connection 2
l12 is the length of the link2
S1 is the center of gravity of the link 1
S2 is the center of gravity of the link 2
Jm is the moment of inertia
The motion equations of a manipulator are coupled,and nonlinear second-order ordinary
differential equations.
3.A FUZZY LOGIC CONTROLLER FLC
Fig. (2) Shows aFuzzy Logic Controllerusually takes the form of an iteratively adjusting model. In
such a system, input values are normalized and converted to fuzzy representations, the fuzzy rule
base isexecuted to produce a consequent fuzzy region for each solution variable, and the consequent
regions are defuzzified to find the expected value of each solution variable [8,11].
Input
Output
Fuzzy Rule-Base-2
Fuzzifier Defuzzifier
Fuzzy Inference Engine
Fuzzy Set in U Fuzzy Set in V
Fig.(2). A Fuzzy Logic Controller
On the other hand, a Fuzzy Logic Controlleradjusts its control surface in accord with parameters,
and not only adjusts to time, or process phased conditions, but also changes the supporting system
control, [7,10].
4.PROPOSED FUZZY LOGIC CONTROLLER
In such a system input values are normalized and converted to fuzzy representations, the model’s
rule base is executed to produce a consequent fuzzy region for each solution variable, and the
consequent regions are defuzzified to find the expected value of each solution variable. On the
other hand, a Fuzzy Logic Controlleradjusts its control surface in accord with parameter, the

system can be made monitoring and controlling by adding a facility for changing the
normalization of the universe of discourse.
The proposed rules depend on the following concepts [9-11] :
• The fuzzy controller maintains the output value, when the output value is set value and
114
the steady state error changes is zero
• Depending on the magnitude and signs ofposition error and velocity error changes, the
output value will return to the set value
The error “e” and the error change “ e” are defined as a difference between the set point value
and the current output value
e(k) = r (k) – c (k)
e(k) = e(k) – e(k-1) … (14)
That is,
r (k) = r (k-1)
This assumption is also satisfied in most cases:
Case (1)
e(k) ˂0 and e(k) 0
r(k) c(k)
and c(k) c(k)
Case (2)
e(k) 0 and e(k) 0
r(k) c(k)and r(k) c(k
Where
• r(k) is the reference of the fuzzy logic controller at k-th sampling interval
• c(k) is the fuzzy logic controller signal at k-th sampling interval
• e(k) is the error signal
• e(k) is the error change signal
Rule-Base
e
e
N Z P
N P N Z
Z N Z P
P Z P N

After the inputs have been fuzzified, the necessary action, i.e. output required is determined from
the following linguistics rue:
115
IF e is N AND e is N
Then u is P
IF e is N AND e is Z
Then u is N
IF e is N AND e is P
Then u is Z
IF e is Z AND e is N
Then u is N
IF e is Z AND e is Z
Then u is Z
IF e is Z AND e is P
Then u is P
IF e is P AND e is N
Then u is Z
IF e is P AND e is Z
Then u is P
IF e is P AND e is P
Then u is N
The proposed programs have been developed to simulate the dynamic behavior of the robotic
system. The new controller uses only the available information of the input-output. The proposed
Fuzzy Logic Controllercan obtain the good control performance.
The computer simulation have demonstrated the effectiveness of the proposed controller in
improving drastically proposed controller be used to cope with the possible variation in system
parameters. Simulation results using MATLAB SIMULINK show the transient response and the
same time have removed any error in the resulting scheme.
5. SIMULATION RESULTS
Numerical simulations using the dynamic model of a three DOF planar rigid robot manipulator
with uncertainties show the effectiveness of the approach in set point tracking problems.
Simulation studies on a pole balancing robot and a multilink robot manipulator demonstrate the
effectiveness and robustness of the proposed approach.
In the following, the parameters of a robotic model are given, each of the physical parameters
used in the simulation, where l is the length of a link, s is the center of gravity, m is the mass, D is
the coefficient of viscous friction, and j is the moment of inertia.[3,4,11]:

The length of link, l1= l2, = 0.5 m, the center of gravity ,s1 = s2 = 0.25 m, the mass m1 = m2 = 0.5
kg, the coefficient of viscous frictionD1 = D2 = 0.1 N.m / rad/s, the moment of inertia J = 0.1 kg
m2
Fig.(4)shows the system response including the tracking positions and velocities using the
proposed Fuzzy Logic Controllertechnique.
As it is expected, the Fuzzy Logic Controllerhas a minimum steady state error. In such controller
, input values are normalized and converted to fuzzy representations, the model’s rule base is
executed to produce a consequent fuzzy region for each solution variable, and the consequent
regions are defuzzified to find the expected value of each solution variable This technique should
be independent of either the model structure or the model parameters.
116
Time in Sec
Fig.(4-a) The desired and simulated tracking Position of the link-1
Fig.(4-b) The desired and simulated tracking Position of the link-2

117
Time in Sec.
Fig.(4-c) The desired and simulated tracking Velocity of the link-1
Fig.(4-d) The desired and simulated tracking Velocity of the link-2
6- CONCLUSION
We present an introduction to a proposed fuzzy logic control for realization of a linguistic
controller for a multi-functional manipulator, which designed to move materials from point–to-another
point. Therefore, the main objective of the fuzzy logic control scheme is to replace an
expert human operator with a fuzzy rule-based control system. There is an analogous form of in
mathematics, where we solved a complicated problem in the complete plant. The Fuzzy Logic
Controlleris faster and more accurate. The results validate that the robot dynamic response is free
speed. The paper presents a fuzzy logic control strategy to ensure excellent study and guarantees
the operation of interconnected power system. Simulation results show that the control
performance can be obtained. Finally, we can conclude that the analysis of the operational
characteristics resulted in key findings enabling a further derivation of control algorithms and
examination of the fuzzy logic controller under dynamic operating conditions.
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Applications of Artificial Intelligence (TAAI), 2013 Conference .pp: 90 - 96
[3] Zand, R.M.; Shouraki, S.B. Designing a fuzzylogic controller for a quadruped robot using human expertise
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mobile robot in an indoor environment Control System graduate Research Colloquium (ICSGRC), 2011
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[5] Jalali, L. ; Ghafarian, H. Maintenance of
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AUTHORS
Multi-Area Load Frequency Control
Chapter, Alexandria, Egypt, September
– 2012.
Sherifkamel kamel Hussein Hassan Ratib :
Graduated from the faculty of engineering in
1989Communications and Electronics Department ,Helwan University. He received his
DiplomaMSc,and Doctorate in Computer Science ,Major Information Technology
andNetworking. He has been working in many private and a
governmental universities
insideand outside Egypt for almost 13 years .He shared in the development of many
industrialcourses .His research interest is GSM Based Control and Macro mobility based
on Mobile IP.
Mahmoud HanafySaleh ySaleh has received the M.Sc. degree in Automatic Control
- Electrical
Engineering and PhD degree in Automatic Control from Faculty of Engineering, Helwan
University (Egypt). He is currently
an Assistant Professor in Electrical Communication
and Electronics s Systems Engineering Department,Canadian International College-College
CIC. Dr
Mahmoud Hanafy has worked in the areas of Fuzzy logic, Neural Network, System
Dynamics, Intelligent Control Logic Control, Physics of Electrical Materials, Electronic
Circuit-Analog-Digital, Electric Circuit Analysis DC
Conversion Solar energy-Photovoltaic, and Electric Power System analysis Interconnected Power System.
His research interests include: Control System Analysis, Systems Eng
Control, Neural network and fuzzy logic controllers, Neuro
Computer Simulation, Statistical Analysis.
gital, DC-AC, Power electronic, Analysis of Electric Energy
Engineering, System Dynamics, Process
Neuro-Fuzzy systems, Mathematical Modeling and
118
. 1986.
bernetics, Cybernetics,
EEE ”ineering,

A fuzzy logic controllerfora two link functional manipulator

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