As the robot manipulators are highly nonlinear, time varying and Multiple Input Multiple Output (MIMO) systems, one of the most important challenges in the field of robotics is robot manipulators control with acceptable performance. In this research paper, a simple and computationally efficient Fuzzy Logic Controller is designed based on the Fuzzy Lyapunov Synthesis (FLS) for the position control of PUMA-560 robot manipulator. The proposed methodology enables the designer to systematically derive the rule base thereby guarantees the stability of the controller. The methodology is model free and does not require any information about the system nonlinearities, uncertainties, time varying parameters, etc. The performance of any fuzzy logic controller (FLC) is greatly dependent on its inference rules. The closed-loop control performance and stability are enhanced if more rules are added to the rule base of the FLC. However, a large set of rules requires more on-line computational time and more parameters need to be adjusted. Here, a Fuzzy Logic Controller is first designed and then the controller based on FLS is designed and simulated with a minimum rule base. Finally the simulation results of the proposed controller are compared with that of the normal Fuzzy Logic Controller and PD controlled Computed Torque Controller (PD-CTC). Results show that the proposed controller outperformed the other controllers.

1. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
DESIGN OF LYAPUNOV BASED FUZZY LOGIC
CONTROLLER FOR PUMA-560 ROBOT
MANIPULATOR
1
1
Ch Ravi kumar 2DV Pushpalatha 3K R Sudha 4KA Gopala Rao
Research Scholar, Department of Electrical Engineering, Andhra University, India
2
Professor, Department of EEE, GRIET, Hyderabad, India
3
Professor, Department of Electrical Engineering, Andhra University, India
4
Professor, Department of EEE, GVPCE, Visakhapatnam
ABSTRACT
As the robot manipulators are highly nonlinear, time varying and Multiple Input Multiple Output (MIMO)
systems, one of the most important challenges in the field of robotics is robot manipulators control with
acceptable performance. In this research paper, a simple and computationally efficient Fuzzy Logic
Controller is designed based on the Fuzzy Lyapunov Synthesis (FLS) for the position control of PUMA-560
robot manipulator. The proposed methodology enables the designer to systematically derive the rule base
thereby guarantees the stability of the controller. The methodology is model free and does not require any
information about the system nonlinearities, uncertainties, time varying parameters, etc. The performance
of any fuzzy logic controller (FLC) is greatly dependent on its inference rules. The closed-loop control
performance and stability are enhanced if more rules are added to the rule base of the FLC. However, a
large set of rules requires more on-line computational time and more parameters need to be adjusted.
Here, a Fuzzy Logic Controller is first designed and then the controller based on FLS is designed and
simulated with a minimum rule base. Finally the simulation results of the proposed controller are
compared with that of the normal Fuzzy Logic Controller and PD controlled Computed Torque Controller
(PD-CTC). Results show that the proposed controller outperformed the other controllers.
KEYWORDS
PUMA-560 robot manipulator, Fuzzy controller, Fuzzy Lyapunov Synthesis.
1. INTRODUCTION
Robotic arm is an important class in the robot anatomy such as manipulator of PUMA-560 robot.
These arms are widely used for mechanical handling, welding, assembling, painting, grinding and
other industrial applications. These applications may require path planning, trajectory generation
and control design. All these factors make the study of robot manipulators, interesting.
Conventional methods of controlling a nonlinear system are based on models, especially in the
field of robot control. Many controllers like LQG, Hα [1] and input shaping as well as singular
perturbation, feedback linearization, manifolds and output redefinition techniques have been used
for controlling purpose if the exact model of the system is known. Many robotic control schemes
can be considered as special cases of model-based control called computed torque approach. The
basic concept of computed torque is to linearize a nonlinear system, and then to apply linear
control theory. But these controllers suffer from the lack of an exact simple-enough model of the
system and this calls for the use of the intelligent controllers. And for these above mentioned
controllers stability also became hard task. Fuzzy inference systems [2] have been proven to be
powerful tools to deal with the nonlinear systems on the basis of fuzzy rules [3], particularly those
DOI : 10.5121/ijfls.2014.4101
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2. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
possessing a high degree of uncertainty and nonlinearity. Thus considerable research
development has been achieved in the use of fuzzy inference system for the control of a robot
manipulator that suffers from structured and unstructured uncertainties such as load variation,
friction, and external disturbances etc. To avoid these difficulties and ensuring the stability of the
system, a model free technique Fuzzy Lyapunov Synthesis (FLS) is proposed to control the
PUMA-560 robot in this paper.
A new Fuzzy rule base is derived for stabilizing the system and minimizing the error. The basic
idea to choose Lyapunov function and derive the Fuzzy rules is to make its derivative negative.
For this, the knowledge of output relative degree of the system model is only sufficient. The basic
assumption of the fuzzy Lyapunov synthesis is that, for a Lyapunov function V (x), if the
linguistic value of (x) is Negative, then (x) < 0, so the stability can be guaranteed.
Prior to this work, the mathematical model of PUMA-560 robot manipulator is implemented in
Simulink, the normal three input Fuzzy controller is applied to the model [4], the results are
compared with the results obtained by PD controlled Computed Torque Controller (PD-CTC).
The rest of the paper is organized as follows: The mathematical model of PUMA-560 robot
manipulator is presented in section 2, The Fuzzy Lyapunov Synthesis (FLS) is presented in
section 3, the FLS Control law is explained in section 4, and Simulation results are presented in
section 5.
2. THE MATHEMATICAL MODEL OF PUMA-560 ROBOT MANIPULATOR
The mathematical model Armstrong [5] [6] [7] of PUMA-560 robot manipulator [8] model is
derived by using Lagrange’s equation considering only three links among the total six links, such
that
.
-------eq. (1)
Where
q: nx1 position vector ,
A(q): nxn inertia matrix of the manipulator,
G(q): nx1 vector of gravity terms
: nx1 vector of torques
B(q): nxn(n-1)/2 matrix of Coriolis torques
C(q): nxn matrix of Centrifugal torques
: n vector of acceleration
and
are notation for n(n-1)/2 vector of velocity products and the n-vector of squared
velocities respectively.
Where
The above model of the robot arm is derived by generating the kinetic energy matrix and gravity
vector symbolic elements by performing the summation of Lagrange’s nonlinear formulation [9].
These elements are simplified by combining inertia constants that multiply common variable
expressions. The Coriolis and centrifugal matrix elements are then calculated in terms of partial
derivatives of kinetic energy, and then reduced using four relations that hold the partial
derivatives.
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3. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
Where
a11=Im1+I1+I3CC2+I7SS23+I10SC23+I11SC2+I20(SS5(SS23(I21+CC4)–1)–2SC23C4SC5)
+I21SS23CC4+2{I5C2S23+I12C2C23+I15(SS23C5+SC23C4S5)+I16C2(S23C5+C23C4S5)+I18S4S
5+I22(SC23C5+CC23C4S5);
a12=I4S2+I8C23+I9C2+I13S23–I15C23S4S5+I16S2S4S5+I18(S23C4S5–C23C5)+I19S23SC4 +
I20S4(S23C4CC5+C23SC5)+I22S23S4S5;
a13=I8C23+I13S23–I15C23S4S5+I19S23SC4+I18(S23C4S5–C23C5)+I22S23S4S5+I20S4(S23C4
CC5+C23SC5);
a22=Im12+I2+I6+I20SS4SS5+I21SS4+2(I3S3+I12C3+I15C5+I16(S3C5+C3C4S5)+I22C4S5;
a23=I5S3+I6+I12C3+I16(S3C5+C3C4S5+I20SS4SS5+I21SS4+2{I15C5+I22C4S5};
a33=Im5 I6+I20SS4+SS5+I21SS4+2{I15C5+I22C4S5};
a34=-I15S4S5+I20S4SC5; a35=I15C4C5+I17C4+I22S5;
a36=I23S4S5; a44=Im4+I14–I20SS5; a46=I23C5;a55=Im5+I17; a66=Im6+I23;
b112=2{-I3SC2+I5C223+I7SC23–I12S223+I15(2SC23C5+(1-2SS23)C4S5+I16(C223C5S223C4S5)+I21SC23CC4+I20(1+CC4)SC23SS5-(1-2SS23)C4SC5+I22{(1–2SS23)C52SC23)C4S5)}+I10(1-2SS23)+I11(1-2SS2)
b113=2{I5C2C23+I7SC23-I12C2S23+I15(2SC23C5+(1-2SS23)C4S5)+I16C2(C23C5S23C4S5)+I21SC23CC4+I20{(1+CC4)SC23SS5-(1-2SS23)C4SC5)+I22{(1–2SS23)C52SC23)C4S5)}+I10(1-2SS23)
b114=2{-I15SC23S4S5–I16C2C23S4S5+I18C4S5–I20(SS23SS5SC4–SC23S4+SC5)+
I22CC23S4S5–I21SS23SC4);
b115=2{I20(SC5(CC4(1-CC23)CC23)-SC23C4(1-2SS5))–I15(SS23S5–SC23C4C5)I16C2(S23S5-C23C4C5)+I18S4C5+I22{CC23C4C5-SC23S5)}
b123=2{-I8S23+I13C23+I15S23S4S5+I18(C23C4S5+S23C5)+I19C23SC4+I20S4(C23C4CC5–
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4. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
S23SC5)+I22C23S4S5};
b124=- I182S23S4S5+I19S23(1-(2SS4)+I20S23(1-2SS4CC5)–I14S23;
b125=-I17C23S4+I182(S23C4C5+C23S5)+I20S4(C23(1-2SS5)–S5S23C4*2SC5;
b126=-I23(S23C5C23C4S5); b134= b124; b135 = b125; b136 = b126;
b145=2{I15S23C4C5+I16C2C4C5+I18C23S4C5+I22C23C4C5)+I17S23C4-I20(S23S4(12SS5)+2C23SC5);
b146=I23S23S4S5; b156=-I23(C23S4S5+S23C4C5);
b214=I14S23+I19S23(1-(2SS4))+2{-I15C23C4S5+I16S4C4S5+I20(S23(CC5CC4–
0.5)+C23C4SC5)+I22S23C4S5};
b215=2{-I15C23S4C5+I22S23S4C5+I16S2S4C5}I17C23S4+I20C23S4(1-2SS5)–2S23SC4SC5);
b216=-b126; b223 =2{-I12S3I3C3+I16(C3C5–S3C4S5)}
b224=2{-I16C3S4S5+I20+SC4SS5+I21SC4–I22S4S5}
b225=2{-I15S5I16(C3C4C5–S3S5)+I20SS4SC5+I22C4C5};
b254= b224; b235 =b225; b256=I23S4C5;
b245=2{-I15S4C5–I16S3S4C5-I17S4+I20S4(1-2SS5}; b246=I23C4S5;
b314=2{-I15C23C4S5+I22S23C4S5+I20(S23{CC5+CC4–0.5)+C23C4SC5)}+I14S23+I19S23(1(2SS4));
b315=2{-I15C23S4C5+I22S23S4C5-I17C23S4+I20S4{C23(1-2SS5)-2S23C4SC5);
b316=-b136; b324=2{I20SC4SS5+I21SC4–I22S4S5)
b325=2{-I15S5+I20SS4SC5+I22C4C5;
b334=b324; b335=b325; b346=b246; b356=b256; b412=- b214; b413=-b314; b416=- b146; b423=- b324;
b345=I152S4C5–I17S4+I20S4(1-2SS);
b415=I20{S23C4(1-2SS5)+2C23SC5)-I17S23C4};
b425=I17S4+I20S4(1-2SS5); b426=- b246; b435=
b425; b436=- b346; b512=- b215; b515=- b515; b514=b415; b516=- b156; b525=-b325; b524=- b425; b526=- b256; b534= b524; b536=- b356; b546=- b456; b612=b126;
b613=b136; b614=b146; b613=b156; b624=b246; b625=b256; b634=b624; b635=b625; b645=- b456.
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5. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
Matrix C is
Where
c12=I4C2–I8S23–I9S2+I13C23+I15S23S4S4+I16C2S4S5+I18(C23C4S5+S23C5)+I19C23SC4+
I20S4(C23CC5-S23SC5)+I22C23S4S5;
c13=0.5b123; c21=-0.5b112; c23=0.5b223; c24=-I15C4S5–I16S3C4S5+I20C4SC5;
c31=-0.5b113; c32=c23; c34=-I15C4S5+I20C4SC5;c35=-I15C4S5+I22C5;
c41=-0.5b114; c42=0.5b224; c43=0.5b423; c31=-0.5b115; c52=-0.5b225; c53=0.5b523; c34=-0.5b145.
And matrix G is:
Where
g2=g1C2+g2S23+g3S2+g4C23+g5(S23C5+C23C4S5)
g3=g2S23+g4C23+g5(S23C5+C23C4S5); g5=g5(C23S5+S23C4C5);
Where
are the inertial constants,
are the gravitational constants. And
we have abbreviated the trigonometric functions by writing S2 to mean sin( ), C23 to mean
cos(
) and CS4 to mean cos( )*sin( ).
With all the above parameters the dynamic model equation (1) can be written as follows.
-------- eq. (2)
Where
-------- eq. (3)
Selecting the Proportional-Derivative (PD) [10] feedback results in the PD-Computed Torque
Controller (PD-CTC) Nguyen [11], this forms equation (4).
-------- eq. (4)
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6. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
From the equation (4), we can separate the mathematical model into linear and nonlinear parts.
The schematic diagram of a 6DOF PUMA-560 robot manipulator is shown in Fig. 1.
Figure 1. The 6DOF PUMA-560 robot manipulator
3. FUZZY LYAPUNIV SYNTHESIS (FLS)
The FLS, Margaliot [12] is fuzzy model-free approach based on selecting a Lyapunov function
candidate [13] [14] and make its derivative negative by designing fuzzy control rules [15]. The
FLS was applied to some minimum phase single input single output plants where some fuzzy
rules describing the linguistic relation between the input and the output and the output relative
degree were known [16]. And in this process there is no direct insight to the system dynamic and
structural properties was given there, so that the applicability of the FLS and the study of stability
analysis were less tractable. Here, the reviewed formulation is to show the essentials of the theory
and its control rules [12] are clearly modified to illustrate applicability of the FLS to the
nonminimum phase systems and improve the system performance. Consider the nonlinear system
p
, u
n
, y
-------eq. (5)
y = h(x)
The control objective is that the error e = y – yd goes to zero asymptotically where yd is the
desired reference trajectory. First we had chosen a positive definite function V as a lyapunov
function candidate and then design u to make its time-derivative as negative along the system
trajectory, i.e.
,
--------eq. (6)
If the knowledge about (5) is limited to fuzzy descriptions of the proposed system, (6) may be
used again, but here this time as a linguistic inequality yielding u in terms of fuzzy mamdani IFTHEN rules. This methodology is called Fuzzy Lyapunov Synthesis. The control strategy of FLS
is based on designing u to make negative.
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7. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
4. FLS CONTROL LAW
The Lyapunov function candidate [17] can be selected in several approaches based on some
specific controlling objectives. The following are some possible approaches in them.
1. Select a Lyapunov function candidate in order to guarantee the system stability and to meet the
performance measures. And this approach often becomes more complex.
2. The Lyapunov function candidate parameters are tuned accordingly by observing the
performance measures and the internal stability of system dynamics so that system stability [18]
is guaranteed.
3. Add all control terms to the main system controller, such that each control term that satisfies
the performance measures pertaining to the internal stability.
In this section, to derive the control rules two Lyapunov function candidates are proposed. Even
though these two Lyapunov functions are simple easy functions of output error e, with their
integral and derivative, these also give good insight to the problem. These Lyapunov functions
and corresponding derived FLS control rules are modified and reviewed to improve the
performance and to stabilize the internal dynamics of the system. Let us consider the first
Lyapunov function candidate and its time derivative are
The FLS control rules are derived assuming ë is proportional to u with e and ė. And the premise
variables are summarized in the Table 1. Using
and, can be rewritten as:
-------- eq. (7)
The FLS control rules framed based on equation (7) do not generate effective control signals
when the steady state error is large. Under these conditions, the second Lyapunov function can be
chosen which gives effective control. The equation is:
2
)
-----eq. (8)
The related control rules corresponding equation (8) are given below in the Table 1.
Table 1. Fuzzy Lyapunov control rules
+
+
-
+
-
+
-
+
+
-
-
+
+
-
-
+
U
-
+
+
+
-
-
-
-
nb
nm
z
ns
ps
z
pm
pb
Where
nb- negative big, nm- negative medium, z- zero, ns- negative small, ps- positive small, pmpositive medium, pb- positive big.
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8. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
5. RESULTS
The Fuzzy Lyapunov Synthesis (FLS) controller was designed by means of selecting appropriate
rule base such that the system stability was also guaranteed and tested to step and ramp inputs. At
first the mathematical model of PUMA-560 robot manipulator was implemented in simulation. In
this simulation the first, second and third joints are moved from home to final position with and
without uncertainties. The simulation was implemented in MATLAB/SIMULINK environment.
The results obtained for Fuzzy Lyapunov Synthesis (FLS) controller were compared with the
results obtained for PD controlled Computed Torque Controller (PD-CTC) Nguyen [4] and
normal Fuzzy controller. From the figures from Fig. 2 to Fig. 12 it was observed that the FLS
controller gave better results in all cases when compared with PD-CTC and Fuzzy controllers.
Error in theta3 of link3 for ramp input without uncertainties
0
PD-CTC
Fuzzy
Ref
Fuzzy Lyap
-1
-2
Error
-3
-4
-5
-6
-7
0
1
2
3
4
5
Time
6
7
8
9
10
Fig. 2 Response of error in angle at link3 for ramp input without uncertainties
Error in theta3 of link3 for ramp input with uncertainties (positive)
0
Reference
PD-CTC
Fuzzy
Fuzzy Lyap
-1
Error
-2
-3
-4
-5
-6
-7
0
1
2
3
4
5
Time
6
7
8
9
10
Fig. 3 Response of error in angle at link3 for ramp input with uncertainties (positive)
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9. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
Error in theta3 of link3 for step input without uncertainties
1.2
1
Reference
PD-CTC
Fuzzy
Fuzzy Lyap
0.8
Error
0.6
0.4
0.2
0
-0.2
0
1
2
3
4
5
Time
6
7
8
9
10
9
10
Fig. 4 Response of error in angle at link3 for step input without uncertainties
E rror in theta3 of link 3 for s tep input with unc ertainties (pos itive)
1.2
1
Referenc e
P D-CTC
Fuz z y
Fuz z y Ly ap
0.8
E rror
0.6
0.4
0.2
0
-0.2
0
1
2
3
4
5
Tim e
6
7
8
Fig. 5 Response of error in angle at link3 for step input with uncertainties (positive)
Error in theta2 of link2 for ramp input without uncertainties
7
6
PD-CTC
Ref
Fuzzy
Fuzzy Lyap
5
Error
4
3
2
1
0
0
1
2
3
4
5
Time
6
7
8
9
10
Fig. 6 Response of error in angle at link2 for ramp input without uncertainties
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10. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
Error in theta2 of link2 for ramp input with uncertainties (positive)
7
Reference
PD-CTC
Fuzzy
Fuzzy Lyap
6
5
Error
4
3
2
1
0
0
1
2
3
4
5
Time
6
7
8
9
10
Fig. 7 Response of error in angle at link2 for ramp input with uncertainties (positive)
E rror in theta2 of link 2 for s tep input without unc ertainties
2
1
E rror
0
Referenc e
P D-CTC
Fuz z y
Fuz z y Ly ap
-1
-2
-3
-4
0
1
2
3
4
5
Tim e
6
7
8
9
10
Fig. 8 Response of error in angle at link2 for step input without uncertainties
Error in theta1 of link1 for ramp input without uncertainties
0.005
0
PD-CTC
Fuzzy
Ref
FUZZY LYAP
-0.005
Error
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
0
1
2
3
4
5
Time
6
7
8
9
10
Fig.9 Response of error in angle at link1 for ramp input without uncertainties
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11. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
Error in theta1 of link1 for ramp input with uncertainties (positive)
0.005
0
Reference
PD-CTC
Fuzzy
Fuzzy Lyap
-0.005
Error
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
0
1
2
3
4
5
Time
6
7
8
9
10
Fig. 10 Response of error in angle at link2 for ramp input with uncertainties (positive)
E rror in theta1 of link 1 for s tep input without unc ertainties
0.01
0
Referenc e
P D-CTC
Fuz z y
Fuz z y Ly ap
-0.01
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
0
1
2
3
4
5
Tim e
6
7
8
9
10
Fig.11 Response of error in angle at link1 for step input without uncertainties
-5
12
Error in theta1 of link1 for step input with uncertainties (positive)
x 10
10
Fuzzy Lyap
Reference
8
6
Error
E rror
-0.02
4
2
0
-2
0
2
4
6
8
10
Time
12
14
16
18
20
Fig. 12 Response of error in angle at link1 for step input with uncertainties (positive)
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12. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014
6. CONCLUSIONS
In this paper, a novel approach for determining the rule base of a Fuzzy lyapunov Synthesis (FLS)
controller is proposed. Starting with minimal knowledge concerning the plant’s behavior, a
lyapunov function candidate V is chosen and determines conditions so that V will indeed be a
lyapunov function. These conditions provide us the rule base of the fuzzy controller.
Our proposed FLS approach combines two works here. On one hand, the plant is model-free in
the sense only minimal fuzzy knowledge is available. On the other hand, it follows the classical
Lyapunov Synthesis method. This combination provides us with a solid analytical basis from
which the rules are obtained and justified. And now the concept of Fuzzy Lyapunov Synthesis
(FLS) is applied to a PUMA-560 robot manipulator, the effectiveness of the control is observed
when compared with PD controlled Computed Torque Controller (PD-CTC) and the normal
Fuzzy controller. Only with the knowledge of output relative degree and some structural
properties of the robot manipulator model the framework of stability and performance analysis of
the system was done.
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