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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