Docstoc

robotic manipulator

Document Sample
robotic manipulator Powered By Docstoc
					           Fractional Order Controller for Two-degree of Freedom Polar Robot
                                                   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 ,(hdelavary@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 controller is designed for a robotic manipulator. The controller
            designation 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
            controllers. A genetic algorithm identifies parameters of the fuzzy sliding mode controllers. 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,
            Genetic algorithm.
                                                              

1. INTRODUCTION                                                   However, primarily PD surface sliding mode controller will
                                                                  be proposed. Thereafter a fractional order surface PD sliding
A robot motion tracking control is one of the challenging         mode controller will be designed. To reduce the chattering
problems due to the highly coupled nonlinear and time             phenomenon a fuzzy logic control (FSMC) is used.
varying dynamic. Moreover, there always exists uncertainty        Parameters of FSMC will be determined by a Genetic based
in the model which causes undesired performance and               optimization procedure.
difficult to control. Robust control is a powerful tool to
control complex systems. Several kinds of control schemes         This paper is organized as follows: in section 2 fundamentals
have already been proposed in the field of robotic control        of fractional calculus is studied. Section 3 presents
over the past decades. Using feedback linearization technique     manipulator dynamic properties. Classical sliding mode
(Park et al.,2000), compensates some of coupling                  controller and fractional order surface sliding mode controller
nonlinearities in the dynamic. The technique transforms the       are discussed in Section 4. Fuzzy controller and the genetic
nonlinear system into a linear one. Then, the controller is       algorithm are studied in section 5. Finally, concluding
designed on the basis of the linear and decoupled plant.          remarks are drawn in Section 6. Simulation results illustrate
Although, a global feedback linearization is theoretically        the effectiveness of the proposed controllers.
possible, a practical insight is restricted. Uncertainties also
                                                                  2. FUNDAMENTALS OF FRACTIONAL CALCULUS
arise from imprecise knowledge of the kinematics and
dynamics, and due to joint and link flexibility, actuator         Fractional calculus is an old mathematical topic since 17th
dynamics, friction, sensor noise, and unknown loads. A            century. The fractional integral-differential operators
sliding mode control is almost an effective robust approach.      (fractional calculus) are a generalization of integration and
Unfortunately, the method causes a chattering phenomenon,         derivation to non-integer order (fractional) operators. The
due to high frequency switching of the control signal. This       idea of fractional calculus has been known since the
causes or damage actuators or excites un-modeled high             development of the regular calculus, with the first reference
frequency property of the system. To suppress the chattering      probably being associated with Leibniz and L’Hospital in
several techniques have been investigated in the literatures. A   1695. In recent years, numerous studies and applications of
Chattering-free fuzzy sliding-mode control strategy (Yau et       fractional-order systems in many areas of science and
al., 2006), a fuzzy tuning approach to sliding mode control       engineering have been presented. An algorithm to determine
with application to robotic manipulators (Ha et al., 1999) are    parameters of an active sliding mode controller in
presented. Based on a general form of a sliding manifold, two     synchronizing different chaotic systems have been studied
design strategies for adaptive sliding mode control are also      (Tavazoei et al., 2007). Implementation of fractional order
presented (Su et al., 1993) for nonlinear robot manipulator.      algorithms in the position/force hybrid control of robotic
                                                                  manipulators is studied (Ferreira et al., 2003). Signal
                                                                  propagation and fractional-order dynamics during evolution

1
    Corresponding Author: a.ranjbar@nit.ac.ir , Tel: +98 911 112 0971, Fax: +98 111 32 34 201
of a genetic algorithm, to generate a robot manipulator                where q, q, q  R n , q joint variable n-vector and τ n-vector of
trajectory are also addressed in (Pires et al., 2003). In
(Calderón et al., 2006) several alternative methods for the            generalized forces. M (q)  R nn is a symmetric and positive
control of power electronic buck converters applying                   definite inertia matrix, C (q, q)q is Coriolis/centripetal
Fractional Order Control (FOC) are presented. The control of           vector, and G(q) is the gravity vector. In general, a robot
a special class of Single Input Single Output (SISO) switched
                                                                       manipulator always presents uncertainties such as frictions
fractional order systems (SFOS) is addressed in (Sira-
                                                                       and disturbances. The control has a duty to overcome these
Ramirez et al., 2006). This is done from viewpoints of
Generalized Proportional Integral (GPI) feedback control               problems.
approach and a sliding mode based Σ − Δ modulation
implementation of an average model based designed                      2.1. Two-degree of freedom polar robot manipulator
feedback controller. (Silva et al., 2004) studied the
performance of integer and fractional order controllers in a           A two-degree of freedom polar robot manipulator has one
hexapod robot where joints at the legs having viscous friction         rotational and sliding joint in the (x,y) plane. Neglecting the
and flexibility. Experiments reveal the fractional-order               gravity force and normalizing the mass and length of the arm,
 PD controller implementation is superior to the integer-             a mathematical model of two-degree of freedom polar robot
order PD algorithm, from the point of view of robustness.              can be expressed as follows:
The fractional-order differentiator can be denoted by a
                                                                       x1 (t )  x2 (t )                                                                (4)
general fundamental operator a Dt as a generalization of the
                                                                                   x1 (t )  M ( x1 (t )  a)  x (t )  
                                                                                 
                                                                                                                      2
                                                                                                                            
differential and integral operators, which is defined as               x2 (t )                                             (   m)
                                                                                                                      4

follows (Calderón et al., 2006):                                                 u1 (t )  d1 (t ) 
                                                                                                                           
                                                                                                                            
                                                                      x3 (t )  x4 (t )
          d
                           , R ( )  0                                                                                       J1  J 2         
          dt                                                                    2   x1 (t )  M ( x1 (t )  a)  
                                                                                                                              2                 
        
        1                  , R( )  0                               x4 (t )                                                x1 (t )        
                                                                                  x2 (t ) x4 (t )  u2 (t )  d 2 (t ) 
a Dt                                                             (1)
         t                                                                                                                   M ( x (t )  a) 2 
                                                                                                                                                 
           
                                                                                                                                       1
                  
          ( d )            , R( )  0
         a
                                                                      where  is the mass of motional link, M is the payload, J1 and
where  is the fractional order which can be a complex                 J2 are moments of inertia of the motional link with respect to
number, constant a is related to initial conditions. There are         the vertical axis through c and o respectively. dk (t ) is an
two commonly used definitions for general fractional                   unknown but bounded external disturbance:
differentiation and integration, i.e., the Grünwald–Letnikov
(GL) and the Riemann Liouville (RL). The GL definition is              dk (t )  Duk , k  1, 2                                                        (5)
as:
                                                                       Robot manipulator joints are driven according to the
                          (t  ) h                           (2)    following desired trajectory:
                                                
   
a Dt   f (t )  lim
               h 0
                      1
                      
                      h                (1) j   f (t  jh)
                                               j
                                                                        xd 1  0.5sin( t /10) m,                      (6)
                             j 0                                      xd 3   sin( t /10) rad
where . is a flooring-operator while the RL definition is
                                                                       During simulation studies, parameters are chosen as:
given by:
                                                                       M=1.5kg,   1kg , J1=J2=1 kgm2, a=1m, and initial
                                    t                           (2)
                                             f ( )
                                     (t  )
                1      dn                                                                                             [q1 (0), q1 (0), q 2 (0), q 2 (0)]T =
a Dt f (t )                                      n 1
                                                           d          conditions                  are
              (n   ) dt n                                           [-0.2, -0.25, 0.36, 0.98]T .  d1 (t )  0.3 cos(4 t ) N   and
                                    a
For (n  1    n) and ( x) is the well known Euler’s                d2 (t )  0.5cos(4 t ) Nm are also considered as disturbance.
Gamma function. Dynamic of manipulator will be discussed
in next section.                                                       4. SLIDING MODE                           CONTROL             FOR        ROBOT
                                                                       MANIPULATORS
3. MANIPULATOR DYNAMICS
                                                                       Sliding Mode control is a robust nonlinear Lyapunov-based
A manipulator is defined as an open kinematics, chain of               control algorithm in which an nth order nonlinear and
rigid links. Each degree of freedom of manipulator is                  uncertain system is transformed to a 1st order system. The
powered by independent torques. In the absence of friction or          sliding mode design approach consists of two steps. The first
other disturbances, dynamic of an n-link rigid robotic                 involves the design of a switching function, S  0 , such that
manipulator system can be described by the following second            the sliding motion satisfies the design specifications. Second
order nonlinear vector differential equation:                          one is concerned with selection of a control law which will
                                                                       enforce the sliding mode. This section explains the design
M (q)q  C(q, q)q  G(q)                                      (3)    procedure of sliding mode controller for a robot manipulator
using alternative techniques based on Fractional Order                          where  is a positive constant. To guarantee the stability the
Control (FOC). Two alternative sliding surfaces are presented                   energy of S should decay towards zero. All trajectories are
in order to achieve a good performance. First, sliding surfaces                 seen improved and approach to the sliding surface in a finite
based on linear compensation networks PD is presented.                          time and will stay on the surface for all future times. S (t ) is
Then, the fractional form of these networks, PD is used in
                                                                                called the sliding surface. Once the behaviour of the system is
order to obtain the sliding surfaces.
                                                                                settled on the surface, is called the sliding mode ( S  0 ) is
The dynamic in (4) can be described by a coupled second-                        happened.Taking the time derivative from both sides of
order nonlinear system of the form                                              equation in (12), obtains:

x2k 1 (t )  x2k (t )                                                  (7)     Sk (e(t ))  k e2k 1 (t )  e2k (t ),         k  0                        (15)
x2k (t )  f k ( x, t )  bk ( x)uk (t ),   k  1, 2,..., n                     Replacing (13) in (7) yields:
x0  x(t0 )
                                                                                e2k 1(t )  e2k (t )  xd 2k (t )  xd 2k 1(t )                             (16)
where t  R ,                                                                   e2k (t )  f k ( x, t )  bk ( x)uk (t )  xd 2k (t ),   k  1,2,..., n
                                                                                Substituting (16) into (15) results:
x(t )  [ x1 (t ) x2 (t ) x3 (t )... x2n (t )]T  R 2n                  (8)
                                                                                Sk (e(t ))  k (e2k (t )  xd 2k (t )  xd 2k 1 (t ))                      (17)
x (t ) is state vector, u(t )  R is a control action vector,
                                                                                 f k ( x, t )  bk ( x)uk (t )  xd 2k (t ),      k  1, 2,..., n
 x0  x(t0 ) is the arbitrary initial conditions given at initial
time t0 , bk ( x) and f k ( x, t ) , k  1,2,..., n are the control gains and   Using (13) and then forcing S k  0 one can obtain the input
the nonlinear dynamics of the robot respectively. The desired                   control signal as:
state variables are defined as:                                                                         k ( x2 k (t )  xd 2 k 1 (t ))                 (18)
                                                                                uk (t )  bk 1 ( x, t )                                                  
xd (t )  [ xd1 (t ) xd 2 (t ) xd 3 (t )... xd 2n (t )]T  R 2n         (9)                              f k ( x, t )  xd 2 k (t )  K k sgn( S k (t )) 
                                                                                where K k is a switching feedback control gain and might be
The tracking error e(t )  R can be defined as:
                            2n
                                                                                any positive number, and
e(t )  x(t )  xd (t )                                                (10)                      1 if Sk (t )  0                        (19)
                                                                                                 
                                                                                 sgn( Sk (t ))  0 if Sk (t )  0
The nonlinear dynamics fk ( x, t ), k  1,2,..., n are not known                                 1 if S (t )  0
                                                                                                             k
exactly, but are estimated as fˆk ( x, t ) with an error bounded by             Substitution equation (18) in to (17), results:
a known function f k ( x, t ) :                                                  Sk (t )   K k sgn( Sk (t ))                             (20)

       ˆ                                                               (11)     A simulation result for this controller has been shown in Fig.
 f k  f k  f k ( x), k  1,2,..., n                                           1. Chattering phenomena has occurred when the state hits the
                                                                                sliding surface Fig. 1(c), (d). After reaching time the actual
The control objective is to get the states x (t ) to track the                  trajectory response x1(t) is almost identical to the desired
specific states xd (t ) . It is of the goal to drive the tracking               command xd1(t), the same results is noticed for x3(t)and xd3(t).
error asymptotically to zero for any arbitrary initial
conditions and uncertainties.                                                   4.2 Fractional order PD surface Sliding Mode Controller

4.1 Classical PD surface Sliding Mode Controller                                The following Fractional PD sliding surface is proposed
                                                                                 Sk (e(t ))  k e2k 1 (t )  D e2k 1 (t ), k  0                        (21)
A typical sliding surface can be expressed as:                                  It can be rewritten as:
Sk (e(t ))  k e2k 1 (t )  e2k (t ),     k  0                     (12)      Sk (e(t ))  k e2k 1 (t )  D 1 (e2k 1 (t ))                           (22)
                                                                                Substituting (16) into (22) results:
where k is a positive constant, and
                                                                                 Sk (e(t ))  k e2k 1 (t )  D 1 (e2k (t )  xd 2k (t )  xd 2k 1 (t )) (23)
e2k (t )  x2k (t )  xd 2k (t )                                       (13)
e2k 1 (t )  x2k 1 (t )  xd 2k 1 (t )                                       Taking the time derivative from both sides of (23), results:
                                                                                Sk (e(t ))  k e2k 1 (t )  D 1 (e2k (t )  xd 2k (t )  xd 2k 1 (t )) (24)
The control objective can now be achieved by choosing the
control input such that the sliding surface satisfy the
following sufficient condition:                                                 Again substituting (16) into (24) and then forcing S k  0 , the

                                                                       (14)     control signal results:
1 d 2
     Si  i Si                                                                                         D1 (k e2 k 1 (t ))  f k ( x, t )            (25)
                                                                                           
2 dt                                                                            uk (t )  bk 1 ( x, t )                                           
                                                                                                         xd 2 k 1 (t )  K k sgn( S k (t ))
                                                                                                                                                  
                                                                                                                                                   
Then using (16) it can simplify (25) to:                                                                                                    2


                        D1 (k [ x2 k (t )  xd 2 k 1 (t )]  f k ( x, t )  (26)                                                     1

uk (t )  bk 1 ( x, t )                                                                                                                   0

                          xd 2 k 1 (t )  K k sgn( Sk (t ))
                                                                                
                                                                                 




                                                                                                                           u1 (t),N.
                                                                                                                                        -1

                                                                                                                                        -2

                                                                                                                                        -3
Similar to the previous section substitution equation (26) in
                                                                                                                                        -4
to (24), results:
                                                                                                                                        -5
 S (t )   K sgn( S (t ))
  k                                                    k
                                                           (27)
                                                               k
                                                                                                                                          0         2       4       6       8        10
                                                                                                                                                                            Time(second)
                                                                                                                                                                                               12        14        16        18        20


When the control law uk (t ) is chosen as (26), chattering                                                                                                                      (c)
phenomena will occur as soon as the state hits the sliding                                                                         8

surface because of discontinuity in signum function.                                                                               6

To reduce the chattering a saturation function is used instead                                                                     4




                                                                                                                     u2 (t),N.m.
of the signum function. Hence, the alternative control signal                                                                      2

in (18) becomes as:                                                                                                                0


                         k ( x2 k (t )  xd 2 k 1 (t ))                      (28)                                           -2
 uk (t )  bk 1 ( x, t )                                                         
                          f k ( x, t )  xd 2 k (t )  K k s at ( S k (t )  k ) 
                                                                                                                                   -4

                                                                                                                                        0       2       4       6       8       10        12        14        16        18        20

Fractional surface sliding mode control (26) will be                                                                                                                    Time(second)
                                                                                                                                                 (d)
                         D1 (k [ x2k (t )  xd 2k 1 (t )]                                           (29)
                                                                                                                  Fig.1. PD Sliding Mode Control with signum function (a):
           
uk (t )  bk 1 ( x, t )  f k ( x, t )  xd 2k 1 (t )                                                            Tracking response of joint No.1 (b): Tracking response of
                         K s at ( S (t )  )                                                                      joint No.2 (c): Control signal u1(t) (d): Control signal u2(t)
                         k             k         k               
where                                                                                                                                  5. FUZZY CONTROLLER
             sgn(  ) if    1
             
                                                                                                            (30)   The signum function in (18) and (26) can cause the chattering
sat (  )  
                     if    1                                                                                 effect. The saturation function of the control law in (28) is
             
                                                                                                                   replaced by a fuzzy controller. The same procedure is done
In (28) and (29)  k is width of boundary layer and K k is a                                                       for the control law in (29). The combination of fuzzy control
positive switching gain. The saturation function can reduce                                                        strategy with SMC becomes a feasible approach to preserve
the chattering, but to have a satisfactory compromise between                                                      advantages of these two approaches. A fuzzy sliding surface
small chattering and good tracking precision in presence of                                                        will be introduced to develop the control law in (28) (and of
parameter uncertainties, a fuzzy logic control is proposed                                                         course 29). The IF-THEN rules of fuzzy sliding mode
(Delavari et al., 2007a,b , Yau et al., 2006). Fuzzy Sliding                                                       controller are described as:
Mode Controller (FSMC) can also be used as in next section.                                                        R1:If  is NB then K is PB
                                                                                                                   R2: If  is NM then K is PM
                                                                                                                   R3: If  is NS then K is PS
                                             0.4
      xd1 (t),x1 (t),e1 (t),m.




                                             0.2

                                              0                                                                    R4: If  is ZE then K is ZE                              (31)
                                      -0.2
                                                                                                 x d1(t)           R5: If  is PS then K is NS
                                                                                                                   R6: If  is PM then K is NM
                                      -0.4                                                       x 1(t)
                                      -0.6                                                       e1(t)

                                      -0.8                                                                         R7: If  is PB then K is NB
                                                   0       2   4   6   8     10   12   14   16   18         20
                                                                       Time(second)                                where NB, NM, NS, ZE, PS, PM, PB are the linguistic terms
                                                                           (a)                                     of antecedent fuzzy set. They mean Negative Big, Negative
                                              4                                                                    Medium, Negative Small, Zero, Positive Medium Positive
                                                                                                  x d3(t)
                                                                                                                   Small and Positive Big, respectively. A fuzzy membership
                xd3 (t),x3 (t),e3 (t),rad.




                                                                                                  x 3(t)
                                              2
                                                                                                  e3(t)
                                                                                                                   function for each fuzzy term should be a proper design factor
                                                                                                                   in the fuzzy control problem (Delavari et al., 2007a,b). A
                                              0
                                                                                                                   general form is used to describe these fuzzy rules as:
                                                                                                                   Ri: If  is Ai , then K is Bi                            (32)
                                              -2
                                                                                                                   where Ai has a triangle membership function (depicted in
                                              -4
                                                   0       2   4   6   8     10   12   14   16   18         20
                                                                                                                   Fig.2.) and Bi is a fuzzy singleton. The modified controller
                                                                       Time(second)                                invites an idea to restrict the width of boundary layer  k ,
                                                                           (b)                                     which uses a continuous function to smoothen the control
                                                                                                                   action. Therefore, the problem of the discontinuousness of
                                                                                                                   the signum 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 chattering phenomenon are two                    that employing the PD FSMC can impressively improve the
important factors that influence the performance of the               tracking performance and provides a faster tracking response
proposed controller. The width of boundary layer  k ,                with minimum reaching phase time in comparison with the
influences the chattering magnitude of the control signal,            conventional controller Figs.1(a)–(b) and PDFSMC
whilst the gain K k , will influence speed of synchronization.        Figs.3.(a)–(b). In addition, all root mean square errors of
The reaching time can be reduced via a suitable selection of          employing the proposed PD FSMC are minimized
parameter K k ,  k . GA is used to search for a best fit for         comparing with these by employing the PDFSMC.
                                                                      From table 1 can be seen that the reaching times (Rt1, Rt2)
these parameters in (28) and (29). The tracking error and the
                                                                      and root mean square errors (E1rms, E3rms) of employing
chattering of the controlled response are chosen as a
performance index to select the parameters.                           the PD FSMC are less than PDFSMC.
                                                                      Finally simulations results assure the validity of the proposed
                                                                      controller to enhance the tracking performance of a nonlinear
           NB        NM NS ZE PS PM                          PB       system and prove the robustness and effectiveness of the
                                                                       PD FSMC against model parameter uncertainty.

                                                                                                               0.4




                                                                         xd1 (t),x1 (t),e1 (t),m.
                                                                                                                0.2

                 2 -Φ/4 0 Φ/4  2                                                                           0

                                                                                                         -0.2                                                   x d1(t)
                                      z
                                      (a)                                                                -0.4                                                   x 1(t)
                                                                                                                                                                e1(t)
                                                                                                         -0.6

                NB     NM    NS     ZE      PS      PM   PB                                              -0.8
                                                                                                                      0   2   4   6   8     10   12   14   16   18         20
                                                                                                                                      Time(second)
                                                                                                                                          (a)
                                                                  K                                              4

               k k 2  k / 4 0            k /4 k 2     k                                                                                                       x d3(t)
                                                                                   xd3 (t),x3 (t),e3 (t),rad.




                                                                                                                                                                 x 3(t)
                                                                                                                 2
                           (b)                                                                                                                                   e3(t)

Fig. 2. (a): The input membership function of the FSMC (b):                                                      0
The output membership FSMC
                                                                                                                 -2


The cost function is defined in such a way that the selected                                                     -4
parameters to minimize the error to provide a less chattering                                                         0   2   4   6   8     10   12   14   16   18         20
                                                                                                                                      Time(second)
at the same time. The cost function is defined as follows:
                                                                                                                                          (b)

                                                           (33)
    [W1( Si )  W2 (ek )]dt , i  1,2 k  1,2,3,4
        2       2
                                                                                                                 4
                                                                                                                                                                 x d3(t)
                                                                                   xd3 (t),x3 (t),e3 (t),rad.




where ek is defined in (13), and W1 and W2 are the weighting                                                     2
                                                                                                                                                                 x 3(t)
                                                                                                                                                                 e3(t)
factors. Parameters of GA based FSMC with the above
control rules- are specified as follows:                                                                         0

Population size = 70,             Crossover probability = 0.75,
Generations = 50,                 Mutation probability = 0.03                                                    -2


 K k belongs to [0, 10]             k belongs to [0, 2].
                                                                                                                 -4
                                                                                                                      0   2   4   6   8     10   12   14   16   18         20
Theses are chosen from the author experience without losing                                                                           Time(second)
the generality. Let W1=2 and W2=1, an optimal parameters of                                                                               (c)
the FSMC are obtained with GA, K1=3.3205, K2=6.1032 and                                                          2
Φ=0.2421, and the others parameters are chosen                                                                   1

as 1  10, 2  10,   0.8 .                                                                                   0
                                                                                   u1 (t),N.




The simulation results of employing Genetic based Fuzzy                                                          -1


 PD Sliding Mode Control ( PD FSMC) and Genetic based
                                                                                                                 -2

                                                                                                                 -3
Fuzzy PD Sliding Mode Control (PDFSMC) with +20%                                                                 -4
variations in system parameters the system responses have                                                        -5
been shown in Fig.3 and Fig.4 respectively. A fast tracking                                                        0      2   4   6   8     10   12
                                                                                                                                      Time(second)
                                                                                                                                                      14   16   18         20


response is observed by employing the proposed PD FSMC                                                                                   (c)
in comparison with the response obtained by employing the
PDFSMC. In addition, it can be seen that employing the
proposed PD FSMC provides a smooth control action. The
chattering of u1(t) and u2(t); are shown minimized in Figs.
4(c) and (d); respectively. From Figs. 4(a)–(b), it is observed
                                                                                                                                                                       Table 1. Results of controllers performances with 20%
                                                 8

                                                 6
                                                                                                                                                                                variation in parameters of the system
                                                                                                                                                                       Controller       Rt1         Rt2      E1rms        E3rms
           u2 (t),N.m.




                                                 4


                                                                                                                                                                       PD FSMC       0.5101      0.8121     0.1522       0.3483
                                                 2

                                                 0

                                             -2
                                                                                                                                                                       PDFSMC         0.8431      0.9452     0.3228       0.6513
                                             -4

                                                         0               2       4       6       8       10        12        14        16        18         20
                                                                                                 Time(second)                                                                               6. CONCLUSION
                               (d)                                                                                                                                     In this paper, a controller based on fractional order surface
Fig.3. PDFSMC with 20% variation in parameters of the                                                                                                                  sliding mode control is proposed. A fuzzy logic controller
system, (a): Tracking response of joint1 (b): Tracking                                                                                                                 is incorporated with a chattering index to tune adaptively
response of joint2 (c): Control signal u1(t) (d): Control signal                                                                                                       the switching gain of the sliding mode controller is also
u2(t)                                                                                                                                                                  other improvement over the last controller. This is done in
                                                                                                                                                                       order to shorten the duration of reaching phase and to
                                      0.4
                                                                                                                                                                       minimize chattering of the control action of sliding mode
 xd1 (t),x1 (t),e1 (t),m.




                                      0.2
                                                                                                                                                                       control. The performance of the proposed controller with
                                                 0
                                                                                                                                                  x d1(t)
                                                                                                                                                                       uncertainties and disturbance has been investigated. The
                                  -0.2
                                                                                                                                                  x 1(t)
                                                                                                                                                                       sliding mode controller performance will be improved
                                  -0.4

                                  -0.6
                                                                                                                                                  e1(t)                when the sliding surface is chosen fractional. More
                                  -0.8
                                                                                                                                                                       improvement has also been achieved when the signum
                                                         0               2       4       6       8       10
                                                                                                 Time(second)
                                                                                                                   12        14        16        18         20
                                                                                                                                                                       function is replaced with a fuzzy controller. The work has
                                                                                                         (a)                                                           been progressed to find best fit parameters of the fuzzy
                                                                                                                                                                       controller through a genetic based technique. The proposed
                                                             4
                                                                                                                                                       x d3(t)         controller has been applied to a trajectory tracking of a
                                                                                                                                                                       polar manipulator with uncertainties of its parameters.
                            xd3 (t),x3 (t),e3 (t),rad.




                                                                                                                                                       x 3(t)
                                                             2
                                                                                                                                                       e3(t)
                                                                                                                                                                       The proposed controller assure the validity, effectiveness
                                                             0                                                                                                         and the superiority to conventional sliding mode controller
                                                                                                                                                                       in the sense of a much faster trajectory tracking time,
                                                         -2                                                                                                            smoothing the control actions and robustness against model
                                                                                                                                                                       parameter uncertainties and disturbances.
                                                         -4
                                                                 0           2       4       6       8        10        12        14        16        18         20
                                                                                                     Time(second)
                                                                                                         (b)                                                                                REFERENCES
                                                             2

                                                             1
                                                                                                                                                                      Ha, Q.P., Rye, D.C., Durrant-Whyte, H.F. (1999), Fuzzy
                                                             0
                                                                                                                                                                          moving sliding mode control with application to robotic
                            u1 (t),N.




                                                         -1
                                                                                                                                                                          manipulators, Automatica 35 607-616
                                                         -2
                                                                                                                                                                      Su, Ch. S. , Stepanenko, Y. (1993), Adaptive Sliding Mode
                                                         -3
                                                                                                                                                                          Control of Robot Manipulators with General Sliding
                                                         -4

                                                         -5
                                                                                                                                                                          Manifold, International Conference on Intelligent Robots
                                                           0                 2       4       6       8        10        12        14        16        18         20
                                                                                                                                                                          and Systems, Japan, 1255-1259
                                                                                                     Time(second)
                                                                                                         (c)                                                          Silva, M. F., Machado, J. A. T., and Lopes, A. M. (2004),
                                                                                                                                                                          Fractional Order Control of a Hexapod Robot, Nonlinear
                                                             6
                                                                                                                                                                          Dynamics 38 417–433,
                                                             4
                                                                                                                                                                      Park, K.Ch., Chung, H., and Lee, J. G. (2000), Point
                            u2 (t),N.m.




                                                             2                                                                                                            stabilization of mobile robots via state-space exact
                                                             0                                                                                                            feedback linearization, Robotics and Computer Integrated
                                                         -2
                                                                                                                                                                          Manufacturing 16 353-363
                                                                                                                                                                      Calderón, A.J., Vinagre, B.M. and Feliu, V. (2006),
                                                         -4
                                                                                                                                                                          Fractional order control strategies for power electronic
                                                                 0           2       4       6       8        10        12        14        16        18         20
                                                                                                     Time(second)                                                         buck converters, Signal Processing 86 2803–2819
                               (d)                                                                                                                                    Sira-Ramirez, H. and Feliu-Batlle, V. (2006), On the GPI-
                                                                                                                                                                         sliding mode      control of switched fractional order
Fig.4. PD FSMC with 20% variation in parameters of the
system, (a): Tracking response of joint1 (b): Tracking                                                                                                                    systems, International Workshop on Variable Structure
response of joint2 (c): Control signal u1(t) (d): Control signal                                                                                                          Systems, Italy 310-315.
u2(t)                                                                                                                                                                 Delavari, H., Ranjbar, A. (2007a), Robust Intelligent Control
                                                                                                                                                                           of Coupled Tanks, WSEAS International Conferences,
                                                                                                                                                                           Istanbul 1-6.
 Delavari, H., Ranjbar, A. (2007b), Genetic-based Fuzzy
    Sliding Mode Control of an Interconnected Twin-Tanks,
    IEEE Region 8 EUROCON 2007 conference, Poland,
    714-719.
Fonseca Ferreira, N. M., Tenreiro Machado, J. A. (2003),
   Fractional-Order      Hybrid    Control     of   Robotic
   Manipulators, The 11th International Conference on
   Advanced Robotics, Portugal 393-398
Solteiro Pires, E.J., Tenreiro Machado, J.A., and de Moura
   Oliveira, P.B. (2003), Fractional order dynamics in a GA
   planner, Signal Processing 83 2377 – 2386
Tavazoei M. S. and M. Haeri (2007), Determination of active
   sliding mode controller parameters in synchronizing
   different chaotic systems, Chaos, Solitons and Fractals,
   32 583–591.
Yau, H. T., Ch. Li Chen (2006), Chattering-free fuzzy
    sliding-mode control strategy for uncertain chaotic
    systems, Chaos, Solitons and Fractals, 30, 709–718.