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 ,(firstname.lastname@example.org ), ** Golestan University, Gorgan P.O. Box 386, Iran (email@example.com ) *** 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: firstname.lastname@example.org , 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 nn 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. 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