Experiments in Fuzzy Control of a Class of Servo Systems for Mobile Robots

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1 Experiment in Fuzzy Control of a Cla of Servo Sytem for Mobile Robot Stefan reitl, Radu-Emil recup Dept. of Automation and Applied nf., olitehnica Univerity of Timioara Bd. V. arvan 2, RO Timioara, Romania tefan.preitl@aut.upt.ro, radu.precup@aut.upt.ro Abtract: The paper deal with experiment with Mamdani -fuzzy controller (-FC) to control a cla of integral plant pecific to ervo ytem playing the role of actuator for control ytem (CS) dedicated to mobile robot. n the firt phae there are deigned linear controller tuned in term of the Extended Symmetrical Optimum method to enure the deired CS performance indice with repect to the tep modification of the et-point and of three poible type of load diturbance input. Then, there i preented an attractive development method for the -FC baed on the linear cae reult and on the modal equivalence principle. An example concerning the peed control of a nonlinear ervo ytem with variable load, accompanied by real-time experimental reult, validate the -FC and the development method a low-cot olution. Keyword: Servo Sytem, -fuzzy Controller, Mobile Robot NTRODUCTON Nowaday although there have been achieved great progree, mobile robot which are capable of performing variou and complex ta in an autonomou and intelligent way, have not yet been able to conquer wide range of application. n thi context, it i very important to develop high performance controller to cope with the three navigation problem [1], tracing control (tracing a reference trajectory), path following and point tabilization. The tracing control problem can be further divided in local and global tracing problem [2]. The majority of controller developed for nonholonomic mobile robot i baed on either inematic [3, 4], or dynamic model [5, 6]. One model that exploit the dynamic of the actuator, of the meauring device and of the control equipment a part of the control ytem (CS) tructure ha been propoed in [7], and the implified model of the controlled plant (C) i characterized by the tranfer function (t.f.) (): ( ) = /[ (1 + TΣ )], (1) where i the controlled plant gain and T Σ i the mall time contant or the time contant correponding to the um of paraitic time contant. Since the t.f. in (1) repreent a implified linearized model of C in nonlinear ervo ytem, the parameter and T Σ are time-variable within certain limit, o controlling the plant (1) i a challenging problem when very good CS performance indice are required in regulation and

2 tracing. The CS tructure in the linear cae i preented in Fig. 1, where: r reference input, F() t.f. of the reference filter, r 1 filtered reference input, y controlled output, u control ignal, e = r 1 y control error, d {d 1, d 2, d 3 } general diturbance input of threee type. Figure 1 Linear control ytem tructure For thi CS tructure and the conidered plant, in the linear cae the ue of controller with the t.f. (2): H C( ) = [ c / ](1 + Tc ), (2) with the gain c and the integration time contant T c, tuned in term of Keler Symmetrical Optimum method, can enure acceptable CS performance indice [8]. But, in ome practical application the CS performance indice overhoot σ 1, ettling time t, rie time t r and phae margin φ m prove to be rather unacceptable due to the large enitivity with repect to the modification of the plant gain accompanied by a poible alleviation of φ m. Thi hortcoming become eriou when T Σ correpond to the um of paraitic time contant generally taing only an approximate value. A imple and efficient way to tune the parameter of the linear controller (2) controlling the plant (1) i repreented by the ESO method [9], characterized by only one deign parameter, β. One olution to enure CS performance enhancement i repreented by fuzzy control. Due to the two-degree-of-freedom CS tructure, the -fuzzy controller (- FC) preented in the paper enure very good CS performance indice with repect to the two input, r and d. The paper i organized a follow. The following Section i dedicated to the preentation of the new development method baed on the tranfer of reult from the linear cae (in term of the ESO method) to the fuzzy cae baed on the modal equivalence principle [10]. Then, there are preented in Section real-time experimental reult in a cae tudy correponding to nonlinear DC drive ervo ytem. The cae tudy prove very good CS performance in both regulation and tracing, and validate the fuzzy control olution a low cot olution in mobile robot control. The concluion are drawn in the end of the paper. DEVELOMENT METHOD FOR MAMDAN -FUZZY CONTROLLERS The -FC replace the linear controller with the tranfer function C() in the CS tructure preented in Fig. 1. The -FC i obtained by fuzzifying the linear controller to enure the aim of low-cot, and it repreent a dicrete-time controller involving a baic fuzzy controller (B- FC, without dynamic), with dynamic being added by the numerical differentiation of the control error e expreed a the increment of control error, Δe =e e 1, and by the numerical integration of the increment of control ignal, Δu. The tructure of the -FC i preented in Fig. 2. The fuzzification i olved in term of the regularly ditributed input and output memberhip function

3 illutrated in Fig. 3. Other ditribution and hape of memberhip function can modify in deired way the controller nonlinearitie. Fig. 3 highlight the three (trictly poitive) parameter of the -FC to be tuned by the development method preented in the equel, B e, B Δe and B Δu. The inference engine in B-FC employ Mamdani MAX-MN compoitional rule of inference aited by the rule bae preented in Table 1, and the defuzzificaton i done in term of the centre of gravity method for ingleton. the modification of CS performance indice (σ 1, t ˆ r = tr / T normalized rie Σ time, t ˆ = t / T normalized ettling Σ time defined in the unit tep modification of r, φ m ) according to deigner option and a compromie to thee indice can be reached uing the diagram preented in Fig. 4, valid in the ituation without F(). However, the preence of F() improve further the CS performance indice. Figure 2 -fuzzy controller tructure Figure 4 Control ytem performance indice veru β Table 1 Deciion table of B-FC Δe e NB NS ZE S B B ZE S M B B S NS ZE S M B ZE NM NS ZE S M NS NB NM NS ZE S NB NB NB NM NS ZE Figure 3 Memberhip function hape and parameter To develop the -FC it i neceary to do the linear cae development focued on the ESO method characterized, a mentioned in Section, by only one deign parameter, β. The adequate choice of the parameter β within the domain 1 < β < 20, enure The tuning condition, pecific to the ESO method, can be expreed in term of (3): 2 c = 1/( β βtσ ), Tc = βt. (3) Σ To apply the ESO method in cae of fuzzy CS it i neceary to dicretize the continuou-time linear controller (2) reulting in the digital controller:

4 Δu = K Δe + K e =, (4) = K ( Δe + α e ) where the parameter K, K and α can be calculated, for example, in term of (5) in cae of Tutin dicretization method: K = C [1 T /(2T c )], (5) K = C T / Tc, α = K / K = 2T /(2T T ), with T ampling period. The development method for the conidered cla of fuzzy CS with - FC controlling ervo ytem of type (1) conit of the phae A) and B): A) The phae of linear controller development, referred to a the linear cae, with the tep A1) A3): A1) Expre the implified mathematical model of the ervo ytem in term of the t.f. in (1) with the parameter and T Σ. A2) Chooe the initial value of the deign parameter β taing into account the deired / impoed CS performance indice and the diagram preented in Fig. 4, deign the reference filter, the implet one having the t.f. F() in the linear cae: F( ) = 1/(1 + β TΣ ), (6) and dicretize the reference filter. A3) Set the value of T in accordance with the requirement of quaicontinuou digital control and calculate the parameter in (5) correponding to (4). B) The phae of fuzzy controller development, referred to a the fuzzy cae, with the modal equivalence principle reulting in (7): B Δ e = α Be, BΔu = K B, (7) e where the free parameter B e repreent the deigner option. The choice of thi parameter can be done to enure the tability of the c fuzzy CS [11, 12]. The enitivity analyi with repect to the parametric variation of C i recommended alo in [13] taing into account the plant model in (1) a linearized implified model and one of the aim of fuzzy control i to control complex plant. EXERMENTAL RESULTS To validate the development method preented in Section it i conidered a cae tudy with the C characterized in it linearized implified form by the t.f. in (1). The experimental etup correpond to the peed control of a nonlinear laboratory DC drive (AMRA DR300). The DC motor i loaded uing a current controlled DC generator, mounted on the ame haft, and the drive ha built-in analog current controller for both DC machine having rated peed equal to 3000 rpm, rated power equal to 30 W, and rated current equal to 2 A. The peed control of the DC motor i digitally implemented uing an A/D-D/A converter card. The peed enor are a tacho generator and an additional incremental rotary encoder mounted at the free drive haft. The picture and the chematic diagram of the hardware tation are preented in Fig. 5. The mathematical model of C can be well approximated by the tranfer function () in (1) with the nominal value of the parameter =4900 and T Σ = Then, the propoed development method i applied, tarting with the choice of the deign parameter, β = 6. The parameter of the linear controller obtain the value c = and T c = The -FC tuning parameter obtain the value B e = 0.3, B Δe = 0.03 and B Δu =

5 art of the real-time experimental reult i preented in Fig. 6 and Fig. 7 for the CS with the original linear controller and with the developed - FC, repectively. The experimental cenario concern the triangular-type modification (Fig. 6a and Fig. 7a), the inuoidal variation of r (Fig. 6b and Fig. 7b), and a 5 period of 10% d 3 - type rated load in all four ituation. Figure 5 icture and chematic diagram of hardware tation Figure 6 Speed repone of control ytem with linear controller Concluion The paper preent a development method for a cla of Mamdani - FC dedicated to ervo ytem a actuator in mobile robot application.the method can be applied with minor problem in cae of D-, D-fuzzy controller and of other complex fuzzy controller tructure [14, 15, 16]. The cae tudy preented in the paper, with real-time experimental reult, highlight the CS performance enhancement enured by the propoed -FC in comparion with one.

6 Future reearch will be focued on the automatic model-free development of fuzzy controller. Acnowledgement The upport temming from two CNCSS grant and from the cooperation between Budapet Tech and olitehnica Univerity of Timioara a part of the Hungarian- Romanian nter-governmental S & T Cooperation rogram i appreciated. Figure 7 Speed repone of control ytem with -fuzzy controller Reference [1] R. Fierro, F. L. Lewi: Control of a nonholonomic mobile robot uing neural networ, in EEE Tran. NN, Vol. 9, No. 4, 1998, pp [2] Z.-. Jiang, H. Nijmeijer: Tracing control of mobile robot: a cae tudy in bactepping, in Automatica, Vol. 33, No. 7, 1997, pp [3] C. Canuda de Wit, O. Sordalen: Exponential tabilization of mobile robot with nonholonomic contraint, in EEE Tranaction on Automatic Control, Vol. 13, No. 11, 1992, pp [4] J.-M. Yang, J.-H. Kim, Sliding mode control for trajectory tracing of nonholonomic wheeled mobile robot, in EEE Tranaction on Robotic and Automation, Vol. 15, No. 3, 1999, pp [5] J.-M. Yang, J.-H. Kim: Sliding mode motion control of nonholonomic mobile robot, in EEE Control Sytem Magazine, Vol. 19, No. 2, 1999, pp [6] Z.-. Jiang: Robut exponential regulation of nonholonomic ytem with uncertaintie, in Automatica, Vol. 36, No. 2, 2000, pp [7] R.-E. recup, S. reitl, C. Szabo,. Korondi,. Szeme: On Some Low-Cot Tracing Controller for Mobile Robot, in Control and ntelligent Sytem, Vol. 33, No. 1, 2005, pp. 1-12

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