Fractional Order Controller for Two-degree of Freedom Polar Robot

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13 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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F
r
actional Order Controller
for

Two
-
degree of
F
reedom
P
olar
R
obot


H. Delavari
*
,
R. Ghaderi
*
,

A. Ranjbar N
.

1
**
,
S. Momani
**
*


*Noushirvani University of Technology, Faculty of Electrical and Computer Engineering,


P.O. Box 47135
-
484, Babol, Iran ,(
hdela
vary@gmail.com

),

**
Golestan University, Gorgan P.O. Box

386
, Iran

(
a.ranjbar@nit.ac.ir

)

*
*
*
Department of Mathematics, Mutah University,
P.O. Box: 7, Al
-
Karak, Jordan

Abstract:

A

robust fractional order

cont
rol
ler

is
designed

for a

robotic manipulator
.

T
he controller
design
ation

will be carried out by two strategies
. Primarily,

a sliding surface
-
based linear compensation
networks PD
,

will be designed
. Then,
a

fractional form of
the controller
PD

will be flowingly

developed
.
A fuzzy
control

is
another expansion to
reduce
the
chattering phenomenon in the proposed sliding mode
controller
s. A
genetic algorithm

identifies parameters of the

fuzzy sliding mode controller
s.

Simulation
study has been

carried out to evaluate the performance of the proposed

controller and to compare
the

performance with
respect
the
conventional sliding mode controller.

Keywords
:

Fractional Control, Robotic Manipulators
,
Sliding Mode Control (SMC)
, Fuzzy control
,
G
enetic

algorithm
.





1

Corresponding Author:

a.ranjbar@nit.ac.ir

, Tel: +98 911 112 0971, Fax: +98 111 32 34 201

1. INTRODUCTION

A

robot motion tracking control is one of the challenging
problems due to the highly coupled nonlinear and time
varying dynamic.

Moreover, there always exists uncertainty
in the model which causes

undesired

performance
and
dif
ficult to control.
Robust control is a powerful tool to
control complex systems
.

Several

kinds of control schemes

h
ave
already
been proposed in the field of robotic control
over

the past decades.

Using
feedback linearization
technique
(
Park
et

al.,
2000
)
,
c
ompensates some of
coupling
nonlinearities in the dynamic
. The technique
transform
s

the
nonlinear system into a linear one. Then, the controller is
designed on the basis of the linear and decoupled plant.
Although
, a

global feedback linearization is
theor
e
tically
possible
, a practical insight is restricted. U
ncertainties
also
arise from imprecise knowledge of the kinematics and
dynamics, and
due to

joint and link flexibility, actuator
dynamics, friction, sensor noise, and unknown loads.
A
sliding mode contr
ol
is almost an
effective robust approach
.
Unfortunately, the method
causes a
chattering phenomenon,
due to
high frequency switching of the control signal
.

This
causes

or damage actuators or excites

un
-
modeled

high
frequency property of the system.

To supp
ress the chattering
several techniques have been investigated in the literatures
.
A
Chattering
-
free fuzzy sliding
-
mode control strategy
(
Yau
et

al.,

2006)
, a

fuzzy tuning approach to sliding mode control

with application to

robotic manipulators

(
Ha

et

al.,

1999
)

are

presented.

Based

on a general form
of
a sliding manifold, two
design

strategies for a
da
ptive sliding mode control are
also
presented
(
Su

et

al.
,

1993
)

for

nonlinear robot manipulator.

However
, primarily PD surface sliding mode controller
will
be

proposed. Thereafter a fractional

order

surface
PD

sliding
mode controller
will be designed
. To reduce the chattering
phenomenon a fuzzy logic control
(FSMC)
is used
.

Parameters of FSMC will be determined by a Genetic based
optimization

procedure.

This paper is organized as follows:

in section 2 fundamentals
of fractional calculus is studied.

Section
3

presents

m
a
nipulator dynamic p
roperties
.
Classical sliding mode
controller and fractional

order

surface sliding mode controller

are

disc
ussed in Section
4
.

Fuzzy controller
and the genetic
algorithm are

studied in section
5
.
Finally, concluding

remarks are drawn in Section
6
.

Simulation results illustrate
the effectiveness of the proposed controller
s
.

2.

FUNDAMENTALS OF FRACTIONAL CALCULU
S

Fractional calculus is an old mathematical topic since 17th
century.
The fractional
integral
-
differential
operators
(fractional calculus) are

a generalization of integration and
derivation to non
-
integer order (fractional) operators. The
idea of fraction
al calculus has been known since the
development of the regular calculus, with the first reference
probably being associated with Leibniz and L’Hospital in
1695.

In recent years, numerous studies and applications of
fractional
-
order systems in many areas o
f science and
engineering have been presented.
An algorithm to determine
parameters
of
an

active sliding mode controller in
synchronizing different chaotic systems ha
ve

been
studied
(
Tavazoei

et al.
, 2007)
.

I
mplementation of fractional order
algorithms in
the position/force hybrid control of robotic
manipulators
is studied

(Ferreira
et al.
, 2003)
.
S
ignal
propagation and f
ractional
-
order dynamics during

evolution






of a genetic algorithm,
to

generat
e
a robot manipulator
trajectory are
also
addressed in
(
Pires

et al.
, 2003)
.
In
(
Calderón

et al.,
2006)

several alternative methods for the
control of power electronic buck converters applying
F
ractional
O
rder
C
ontrol (FOC) are presented
.
The control

of
a special class of Single Input Single

Output (SISO) switched
fr
actional order systems (SFOS)
is
addressed

in (
Sira
-
Ramirez

et al.
,
2006).

This is done
from viewpoints of
Generalized Proportional Integral (GPI) feedback control
approach and a sliding mode based Σ − Δ modulation
implementation of an average model based
designed
feedback controller
.
(
S
ilva

et al
.,
200
4
)

studie
d
the
performance of integer and fractional order controllers in a
hexapod robot
where

joints at the legs having viscous friction
and flexibility.
E
xperiments reveal the fractional
-
order
PD

controller implementation is superior to the integer
-
order PD algorithm, from the point of view of robustness.

The fractional
-
order differentiator can be denoted by a
general fundamental operator
a t
D


as a generalization of the

differential and integral operators, which is defined as
follows
(
Calder
ó
n
et al.,

2006)
:

,( ) 0
1,( ) 0
( ),( ) 0
a t
t
a
d
R
dt
D R
d R






 







 









(1)

w
here

is the fractional order
which
can be a complex
number, constant
a

is related to initia
l conditions
. There are
two commonly used definitions for general fractional
differentiation and integration, i.e., the Gr
ü
nwald

Letnikov
(GL) and the Riemann Liouville (RL). The GL definition is
as:



( )
0
0
1
( ) lim ( 1) ( )
t h
j
a t
h
j
D f t f t jh
j
h







 
  
 
 


(
2
)

w
here


.

is a flooring
-
operator while the RL definition is
given by
:

1
1 ( )
( )
( )
( )
t
n
a t
n n
a
d f
D f t d
n
dt t






 

 



(
2
)

For
( 1 )
n n

  
and
( )
x

is the well known Euler’s
Gamma function.

Dynamic of

manipulator
will be

discussed
in next section.



3
.
M
ANIP
ULATOR DYNAMIC
S

A manipulator is defined as an open
kinematics
,

chain of
rigid links
.


Each degree of freedom of manipulator is
powered by independent torques.

In the absence of friction or
other disturbances, dynamic of an
n
-
link rigid robotic
manipulator

system can be described by the following second
order nonlinear vector differential equation:

( ) (,) ( )
M q q C q q q G q

  

(
3
)

w
here

,,
n
q q q R

,

q

joint variable
n
-
vector and τ
n
-
vector of
generalized forces.
( )
n n
M q R


is
a
symmetric and positive
definite

inertia matrix
,
(,)
C q q q

is
Coriolis/centripetal
vector, and
( )
G q

is the

gravity vector. In general, a robot
manipul
ator always present
s

uncertainties such as frictions
and disturbances.
The control
has a duty to
overcome these
problems.

2.1.
Two
-
degree of freedom polar robot manipulator

A

two
-
degree of freedom polar robot manipulator

has one
rotational and sliding join
t in the (
x,y
) plane.
N
eglecting the
gravity force and normalizing
the
mass and length of the arm,
a

mathematical model of two
-
degree of freedom polar robot
can be expressed as follows:









1 2
2
1 1 4
2
1 1
3 4
1 2
1 1
2
4 1
2 4 2 2
2
1
( ) ( )
( ) ( ( ) ) ( )
( ) ( )
( ) ( )
( ) ( )
2 ( ) ( ( ) )
( ) ( )
( ) ( ) ( ) ( )
( ( ) )
x t x t
x t M x t a x t
x t m
u t d t
x t x t
J J
x t M x t a
x t x t
x t x t u t d t
M x t a





 
  
 
 
 

 
 

 
 
 
  
 
 
 
 
 
  
 
 
 

 

(
4
)

w
here

is the ma
ss of motional link,
M

is the payload,
J
1

and
J
2

are moments of inertia of the motional link with respect to
the vertical axis through
c

and
o

respectively
.

( )
k
d t
is
an

unknown b
ut bounded external disturbance:

( ),1,2
k uk
d t D k
 

(
5
)

R
obot manipulator joints are driven according to the
following
desired trajectory
:



1
3
0.5sin(/10),
sin(/10)
d
d
x t m
x t rad

 



(
6
)

During
simulation studies,

parameters are chosen as:

M
=1
.5
kg
,
1
kg


, J
1
=J
2
=1 kgm
2
, a=1m,

and initial
conditions
a
re
1 1 2 2
[ (0),(0),(0),(0)]
T
q q q q
=
[-0.2,-0.25, 0.36,0.98]
T
.
1
( ) 0.3 cos(4 )
d t t N



and
2
( ) 0.5cos(4 )
d t t Nm


are also considered as disturbance
.


4
.

S
LIDING MODE CONTROL FOR ROBOT
MANIPULATORS

Sliding Mode control is a robust nonlinear Lyapunov
-
based
control algorithm in which an
n
th order nonlinear and
uncertain system is transformed to a 1st order system.
The
sliding mode design approach consists of

two steps. The first
involves the design of a

switching function,
0
S

,
such

t
hat
the sliding motion

satisfies the design specifications.
S
econd
one is

concerned with selection of a control law which

will
enforce the sliding mode
.
This section explains the design
procedure
of
s
liding

mode

controller

for a
robot manipulator







using alt
ernative techniques based on Fractional Order
Control (FOC). Two alternative
s
liding

surfaces are presented
in order to achieve a good
performance
. First, sliding surfaces
based on linear compensation networks PD is presented.
Then, the fractional form of
these networks,
PD

is used in
order to obtain the sliding surfaces.

The dynamic
in

(
4
)

can be described by a coupled second
-
order nonlinear system of the form

2 1 2
2
0 0
( ) ( )
( ) (,) ( ) ( ),1,2,...,
( )
k k
k k k k
x t x t
x t f x t b x u t k n
x x t


  


(
7
)

w
here

t R

,

2
1 2 3 2
( ) [ ( ) ( ) ( )...( )]
T n
n
x t x t x t x t x t R
 

(
8
)

( )
x t

is state vector,
( )
u t R

is
a

control action vector,
0 0
( )
x x t

is the

arbitrary initial conditions given at initial
time
0
t
,
( )
k
b x
and
(,)
k
f x t
,
1,2,...,
k n

are the control gains and
the nonlinear dynamics of the robot respectively.

The desired
state variables are defined as:

2
1 2 3 2
( ) [ ( ) ( ) ( )...( )]
T n
d d d d d n
x t x t x t x t x t R
 

(
9
)

The tracking error

2
( )
n
e t R

can be de
fined as:

( ) ( ) ( )
d
e t x t x t
 

(
10
)

The nonlinear dynamics
(,),1,2,...,
k
f x t k n


are not known
exactly, but are estimated as
ˆ
(,)
k
f x t
with an error bounded by
a known
function
(,)
k
f x t
:

ˆ
( ),1,2,...,
k k k
f f f x k n
  

(
11
)

Th
e control objective is to get the states
( )
x t
to track the
specific states
( )
d
x t
.
I
t is
of the goal
to drive the tracking
error asymptotically to zero for any arbitrary initial
conditions and uncertainties.

4
.1 Cla
ssical

PD

surface

Sliding

Mode Controller

A typical sliding surface can be
expressed

as:

2 1 2
( ( )) ( ) ( ),0
k k k k k
S e t e t e t
 

  

(
1
2
)

w
here
k


is a positive constant, and

2 2 2
2 1 2 1 2 1
( ) ( ) ( )
( ) ( ) ( )
k k d k
k k d k
e t x t x t
e t x t x t
  
 
 

(
13
)

The control objective can now be achie
ved by choosing the
control input
such

that
the
sliding surface satisf
y

the
following sufficient condition
:

2
1
2
i i i
d
S S
dt

 

(
14
)

w
here


is a positive constant.
To guarantee the stability
the
energy of
S

should decay
towards
zero.

All trajectories are
seen improved and
approach

to the sliding surface in
a
finite
time and
will
stay

on the surface for all future times.
( )
S t
is
called the sliding surface. Once the
behaviour

of the system is
settled on the surface, is called
the
sliding
mode (
0
S

)

is
happened
.
Taking the time derivative from both sides of
equation in (
1
2
), obtains:

2 1 2
( ( )) ( ) ( ),0
k k k k k
S e t e t e t
 

  

(
15
)

Replacing

(
13
)

in (
7
)
yields
:

2 1 2 2 2 1
2 2
( ) ( ) ( ) ( )
( ) (,) ( ) ( ) ( ),1,2,...,
k k d k d k
k k k k d k
e t e t x t x t
e t f x t b x u t x t k n
 
  
   

(
16
)

S
ubstituting (
16
) into (1
5
)

results
:

2 2 2 1
2
( ( )) ( ( ) ( ) ( ))
(,) ( ) ( ) ( ),1,2,...,
k k k d k d k
k k k d k
S e t e t x t x t
f x t b x u t x t k n


   
  

(
17
)

Using (
13
) and

then f
orcing
0
S
k

one can obtain the input

control signal as:

2 2 1
1
2
( ( ) ( ))
( ) (,)
(,) ( ) sgn( ( ))
k k d k
k k
k d k k k
x t x t
u t b x t
f x t x t K S t



  
 

 
 
 

(
18
)

w
here
k
K
is a switching feedback c
ontrol gain and might be
any positive number
, and

1 if ( ) 0
sgn( ( )) 0 if ( ) 0
1 if ( ) 0
k
k k
k
S t
S t S t
S t
 


 


 


(
19
)

Substitution equation (
18
) in to (
17
),

results:

( ) sgn( ( ))
k k k
S t K S t
 

(
2
0
)

A

simulation result for this controller has been shown in Fig.
1.

Chattering

phenomena
has

occur
re
d

when

the state hits the
sliding surface

Fig. 1
(c)
, (
d)
.

After reaching

time the actual
trajectory
response x
1
(t) is almost identical to the
desired

command x
d
1
(t), the same results is
noticed

for x
3
(t)and x
d
3
(t).


4
.
2

Fractiona
l
order
PD


surface

Sliding

Mode Controller

The
following
Fractional

PD

sliding surface

is proposed

2 1 2 1
( ( )) ( ) ( ),0
k k k k k
S e t e t D e t

 
 
  

(
2
1
)

It can be rewritten as
:

1
2 1 2 1
( ( )) ( ) ( ( ))
k k k k
S e t e t D e t



 
 

(
2
2
)

Substituting

(
16
) into (
22
)

results
:

1
2 1 2 2 2 1
( ( )) ( ) ( ( ) ( ) ( ))
k k k k d k d k
S e t e t D e t x t x t



 
   

(
2
3
)

Taking the time derivative from both sides of (
23
), results
:

1
2 1 2 2 2 1
( ( )) ( ) ( ( ) ( ) ( ))
k k k k d k d k
S e t e t D e t x t x t



 
   

(
2
4
)

Again substituting (
16
) into (
24
)

and then forcing
0
S
k

, the
control signal results
:

1
1
2 1
2 1
( ( )) (,)
( ) (,)
( ) sgn( ( ))
k k k
k k
d k k k
D e t f x t
u t b x t
x t K S t






 
  

 

 
 

(
2
5
)







Then

using (
16
) it can simplify
(25) to
:

1
1
2 2 1
2 1
( [ ( ) ( )] (,)
( ) (,)
( ) sgn( ( ))
k k d k k
k k
d k k k
D x t x t f x t
u t b x t
x t K S t






 
  

 
 
 
 

(
2
6
)

Similar to the previous section
s
ubstitution equation (
26
) in
to (
2
4
), results:

( ) sgn( ( ))
k k k
S t K S t
 

(
2
7
)

When

the control law
( )
k
u t
is chosen as (
2
6
), chatterin
g
phenomena will occur as soon as the state hits the sliding
surface

because of discontinuity in
signum

function
.

To reduce the chattering a
saturation

function is used instead
of the
signum

function. Hence, the alternative control signal
in (
18
) becomes
as:

2 2 1
1
2
( ( ) ( ))
( ) (,)
(,) ( ) s ( ( ) )
k k d k
k k
k d k k k k
x t x t
u t b x t
f x t x t K at S t



  
 

 
  
 

(
2
8
)

F
ractional surface sliding mode control
(
26
)
will be

1
2 2 1
1
2 1
( [ ( ) ( )]
( ) (,) (,) ( )
s ( ( ) )
k k d k
k k k d k
k k k
D x t x t
u t b x t f x t x t
K at S t






 
  
 
  
 
 

 


(2
9)

w
here

sgn( ) if 1
( )
if 1
sat



    

 

   



(
30
)

In
(2
8
) and (2
9
)

k

is width of boundary layer and
k
K
is a
positive switch
ing

gain.

The s
aturation function can reduce
the chattering, but to have a
satisfactory

compromise between
small chattering and good tracking precision in presence of
parameter uncertainties, a fuzzy logic control is prop
osed

(
Delavari

et al.,

200
7a,b ,

Yau
et al.,
2006)
.
Fuzzy Sliding
Mode Controller (FSMC)
can
also
be
used as in next section.


0
2
4
6
8
10
12
14
16
18
20
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Time(second)
x
d1
(t),x
1
(t),e
1
(t),m.


x
d1
(t)
x
1
(t)
e
1
(t)

(a)

0
2
4
6
8
10
12
14
16
18
20
-4
-2
0
2
4
Time(second)
x
d3
(t),x
3
(t),e
3
(t),rad.


x
d3
(t)
x
3
(t)
e
3
(t)

(b)

0
2
4
6
8
10
12
14
16
18
20
-5
-4
-3
-2
-1
0
1
2
u
1
(t),N.
Time(second)

(c)

0
2
4
6
8
10
12
14
16
18
20
-4
-2
0
2
4
6
8
u
2
(t),N.m.
Time(second)

(d)

Fig.1.
PD Sliding Mode Control with
sign
um

function

(a):

T
racking response of joint

No.
1

(b):

T
racking re
sponse of
joint

No.
2

(c):

Control
signal
u
1
(t)

(
d
):

Control
signal
u
2
(t)



5
.
FUZZY CONTROLLER

The
sig
num

function
in (1
8
) and (2
6
)

can cause the chattering
effect.
The
saturation

function of the control law in (
2
8
) is
replaced by a fuzzy controller. The s
ame procedure is done
for the control law
in (2
9
).
The combination of fuzzy control
strategy

with SMC becomes

a feasible approach to preserve
advantages of these two approaches. A fuzzy sliding surface
will be
introduced to develop
the
control law
in
(2
8
)
(and of
course 2
9
).

The IF
-
THEN rules of fuzzy sliding mode
controller
are

described as:

R1:If


is NB then
K

is PB

R2: If


is NM then
K

is PM

R3: If


is NS then
K

is PS

R4: If



is ZE then
K

is ZE

R5: If


is PS then
K

is NS

R6: If


is PM then
K

is NM

R7: If


is PB then
K

is NB

(
31
)

w
here

NB, NM,

NS, ZE, PS, PM, PB are the linguistic terms
of antecedent fuzzy set. They mean Negative Big, Negative
Medium, Negative Small, Zero, Positive Medium Positive
Small and Positive Big, respectively.
A

fuzzy membership
function for each fuzzy term should be a
proper design factor
in the fuzzy control problem

(
Delavari

et al.,

200
7a,b
)
.

A
general form is used to describe these fuzzy rules as:

R
i
: If


is
A
i

, then
K

is
B
i

(
32
)

w
here
A
i

has a triangle membership function
(depicted in
Fi
g.
2
.) and
B
i

is a fuzzy singleton. The modifie
d controller
invites an idea to restrict the width of boundary layer
k

,
which uses a continuous function to smoothen the control
action. Therefore, the problem of the discontinuousness
of
the
sign
um

function can be treated, and the chattering
phenomena will be decreased. From the control point of
view, the parameters of structures should be automatically
modified by evaluating the results of fuzzy control in
(
31
).







The hitting time and ch
attering phenomenon are two
important factors that influence the performance of
the
proposed controller. The width of boundary layer
k

,
influences the chattering magnitude of the control signal,
whilst the
gain
k
K
, will influence speed of synchronization.
The reaching time can be reduced via a suitable selection of
parameter
k
K
,
k

. GA is used to search for a best fit for
these parameters
in (2
8
) and (2
9
).

The t
racking error and the
chattering of the controlled response are chosen as a
performance index to select the parameters.


2



0
z
2


PB
PM
ZE
NM
NB

PS
NS
Φ
/
4
-
Φ
/
4

(a)

K
k
0
k

2
k
2
k

/4
k
/4
k


PB

PM

PS
ZE

NS

NB

NM




(b)

Fig.
2
. (a):

The

input
membership function of the F
SMC
(b):
The output membership
FSMC



The
cost

function is defined in such a way that the selected
parameters
to minimize the error

to provide a l
ess chattering
at the same time.

The
cost

function is
defined as follows:

2 2
1 2
[ ( ) ( )],1,2 1,2,3,4
i k
W S W e dt i k
  


(
33
)

where
k
e
is

defined in (
13
), and

W
1

and
W
2

are the weight
ing

factors.

P
arameters of
GA based
FSMC with the above
control rules
-

are specified as follows:

Population size =
7
0,

Crossover probability =
0.
75
,

Generations =
5
0,

Mutation probability =
0.0
3

k
K
belongs to [
0,
1
0
]

k

belongs to [
0, 2
].

Theses are chosen from the author experience

without losing
the generality
.

Let
W
1
=
2

and
W
2
=1
,
an

optimal paramete
rs of
the FSMC are obtained with GA,
K
1
=
3.3205,

K
2
=
6.1032

and
Φ=0.2
42
1
,

and
the others parameters
are

chosen

as
1 2
10,10,0.8
  
  
.

The simulation results of employing Genetic based
F
uzzy
PD


Sliding Mode Control

(
PD

FSMC)

and Genetic based
F
uzzy
PD

Sliding Mode Control

(
PD
F
SMC
)
with
+20%

variations

in system parameters the system responses have
been shown in Fig
.3 and
Fig
.4 respectively.

A
fast

tracking
response is observed by employing the proposed
PD

FSMC

in comparison with the response obtained by

employing

the
PD
F
SMC
.

In addition, it can be seen that employing the
proposed
PD

FSMC

provides a smooth control action. The
chattering of
u
1
(t)

and

u
2
(t)
;
are

shown
minimized
in Figs.
4
(
c
) and (
d
); respectively.

From Figs.
4
(a)

(
b
), it

is observed
that employing

the
PD

FSMC

can impressively improve the
tracking performance and provides a

faster tracking response
with minimum reaching phase time in comparison with

the
conventional controller

Figs.
1
(a)

(
b
)

and
PD
F
SMC

Fi
gs.
3
.
(a)

(
b
)
.

In addition
,

all
root mean square errors of
employing

the proposed
PD

FSMC

are minimized
comparing with these by employing

the
PD
F
SMC.

From table 1 can be seen that t
he
r
eaching times

(Rt1,

Rt2)
and
root mean square errors

(E1rms, E3rms)

of employing
the
PD

FSMC

are

less than

PD
F
SMC
.

Finally simulations results assure the validity of the proposed
controller to

enhance the tracking performance of a nonlinear
system and prove the robustness

and effectivenes
s of the
PD

FSMC

against model parameter uncertainty.

0
2
4
6
8
10
12
14
16
18
20
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Time(second)
x
d1
(t),x
1
(t),e
1
(t),m.


x
d1
(t)
x
1
(t)
e
1
(t)

(a)

0
2
4
6
8
10
12
14
16
18
20
-4
-2
0
2
4
Time(second)
x
d3
(t),x
3
(t),e
3
(t),rad.


x
d3
(t)
x
3
(t)
e
3
(t)

(
b
)

0
2
4
6
8
10
12
14
16
18
20
-4
-2
0
2
4
Time(second)
x
d3
(t),x
3
(t),e
3
(t),rad.


x
d3
(t)
x
3
(t)
e
3
(t)

(
c
)

0
2
4
6
8
10
12
14
16
18
20
-5
-4
-3
-2
-1
0
1
2
u
1
(t),N.
Time(second)

(
c
)







0
2
4
6
8
10
12
14
16
18
20
-4
-2
0
2
4
6
8
u
2
(t),N.m.
Time(second)

(
d
)

Fig.3.
PD
F
SMC

with
20%

variation in
parameters of the
system
,

(
a):

T
racking response of joint1
(b):

T
racking
response of joint2

(c):

Control
signal
u
1
(t
)


(d):

Control
signal
u
2
(t)

0
2
4
6
8
10
12
14
16
18
20
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Time(second)
x
d1
(t),x
1
(t),e
1
(t),m.


x
d1
(t)
x
1
(t)
e
1
(t)

(
a
)

0
2
4
6
8
10
12
14
16
18
20
-4
-2
0
2
4
Time(second)
x
d3
(t),x
3
(t),e
3
(t),rad.


x
d3
(t)
x
3
(t)
e
3
(t)

(
b
)

0
2
4
6
8
10
12
14
16
18
20
-5
-4
-3
-2
-1
0
1
2
u
1
(t),N.
Time(second)

(
c
)

0
2
4
6
8
10
12
14
16
18
20
-4
-2
0
2
4
6
u
2
(t),N.m.
Time(second)

(
d
)

Fig.
4
.
PD

FSMC

with
20%

variation in
parameters of the
system
,

(a):

T
racking response of joint1
(b):

T
racking
response of joint2

(c):

Control
signal
u
1
(t)

(d):

Control
signal
u
2
(t)

Ta
ble
1
.
Results of controllers performances

with
20%

variation in
parameters of the system

Controller

Rt1

Rt2

E1rms

E3rms

PD

FSMC

0.5101

0.8121

0.1522

0.3483

PD
F
SMC

0.8431

0.9452

0.3228

0.6513



6
.
CONCLUSION

In this paper, a controller based on
fractional order
surface
sliding mode control is
proposed
.

A fuzzy logic
controller

is incorporated with a chattering index to tune

adaptively
the switching gain of the sliding mode controller

is also
other improvement
over the last controller. This is done
in
order to shorten the

duration of reaching phase and to
minimize chattering of the control action of
sliding mode
control
.

The performance

of
the
proposed controller with
uncertainties and disturbance ha
s

been inves
tigated. The

sliding mode controller

performance will be improved
when the sliding surface is chosen fractional. More
improvement has
also
been achieved when the
signum

function is replaced with a fuzzy controller. The work has
been progressed to find best

fit parameters of the fuzzy
controller through
a
genetic based technique.
The proposed
controller has been applied

to a trajectory
tracking of a
polar manipulator

with uncertainties of its parameters
.

The proposed controller assure
the

validity, effective
ness

and
the

superiority to conve
ntional sliding mode controller
in the sense of a

much faster trajectory tracking time,
smoothing the control actions and robustness

against model
parameter uncertainties

and disturbances.



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,

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D.C.,

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