Anti-Swing Fuzzy Controller Applied in a 3D Crane System

Transcription

1 ETASR -, Tehnolog & Applied Siene Researh Vol. 2, o. 2, 2012, Anti-Sing Fuzz Controller Applied in a 3D Crane Sstem Dragan Antić dragan.anti@ Zoran Jovanović zoran.jovanovi@ Staniša Perić stanisa.peri@ Saša Nikolić sasa.s.nikoli@ Marko Milojković marko.milojkovi@e lfak.ni.a.rs Miloš Milošević R&D Center ALFATEC Abstrat It is ell knon that fuzz logi an be used in the ontrol of omplex sstems desribed b highl nonlinear mathematial models. Hoever, the main diffiult in the design of a fuzz ontroller omes ith the adjustment of the ontroller s parameters that are usuall determined b human experts knoledge or trial and error methods. In this paper, e desribe an implementation of fuzz logi in order to redue osillations during the positioning of a 3D rane sstem. The fuzz ontroller s struture is quite simple, requiring onl to input variables. The proposed fuzz ontroller has been applied to an experimental laborator frameork and results sho that osillations are signifiantl redued. Keords- fuzz ontroller; anti-sing ontrol; 3D rane sstem I. INTRODUCTION The rane sstem is an eletromehanial sstem, hih is used to lift and loer some materials and to move them from one plae to another. It is mainl used for lifting heav things and moving loads beond the normal apabilit of a man. There are several tpes of rane sstems, but in our interest is in the 3D rane sstem, here the paload attahed to the end of the lift-line an move freel in three dimensions. The 3D rane mathematial model is rather ompliated, and it is often simplified in some a. The exat mathematial models for rane sstems are ver important for ontroller design. Man attempts ere made to introdue simplified models or to use some ontrol tehniques to ope ith sstem unertainness. In [1], several 3D rane mathematial models, depending on the number of variables, are desribed. The authors gave (i) a omplete nonlinear model ith onstant pendulum length and to ontrol fores, (ii) a omplete nonlinear model ith varing pendulum length and three ontrol fores, and (iii) a simplified model ith tree ontrol fores. On the other hand, hen the frition and gravit are unknon, neural netorks (NNs) an be used [2]. NNs are used for approximation of the hole nonlinearit of robot dnamis, but the an be adjusted for 3D rane sstem. Moreover, the authors in [3] propose an adaptive PD ontrol ith gravit ompensation for the ase here parameters in gravitational torque vetor are unknon. It is ell-knon that some ontrol tehniques do not require an exatl knon mathematial model of the ontrolled plant. Having in mind this, the robust methods are seemed to be the right hoie in the ontrol of 3D rane sstem. The main objetive in the ontrol of 3D rane sstem is to position the paload in the desired loation ithout osillations. There are man papers dealing ith this ontrol problem. In [4], boundar onditions are used in the time optimal ontrol. In [5], the authors improve the previous paper and the optimal ontrol is alulated so that the orresponding trajetor satisfies the speified boundar onditions and the osillations during the transfer are minimized. Hoever, some requirements of time optimization must be negleted if e ant to inrease robustness. In [6], it is proposed to use zero angular veloit at the target point. Gain sheduling, in order to inrease traking aura, is proposed in [7]. Authors in [8], interpreted the rane as a large orkspae serving robot, and developed several observers. Intelligent ontrol algorithms, suh as fuzz, sliding mode, neural, genetis et., have a lot of advantages related to the interpolative reasoning approah, but also have some restritions due to their omplexit. In [9], authors investigate the intelligent ontrol algorithms (PID, fuzz) applied in 3D rane model ontrol. In [10], a netork-based self-tuning ontroller is presented, based on using a multilaer pereptron, as a self-tuner for the ontroller. The given method is verified b simulations in different onditions. The implementation of geneti algorithms in determining parameters of a 3D rane sstem model is given in [11]. The validation of the model and parameters are performed b digital simulation and omparison ith the experimental results. The method proposed had satisfator ontrol performane under a ide range of operating onditions and redued osillations during the positioning of paload art. A sliding mode anti-sing ontrol for overhead ranes is proposed in [12]. The proposed ontrol realizes a tpial anti-sing trajetor ontrol, alloing highspeed load hoisting motion and suffiient damping of load sing. In [13], a ne fuzz anti-sing ontrol sheme for a three-dimensional overhead rane is proposed. The proposed ontrol onsists of a position servo ontrol and a fuzz logi ontrol. The validation of the proposed ontrol is done b

2 ETASR -, Tehnolog & Applied Siene Researh Vol. 2, o. 2, 2012, experiments ith a three-dimensional prototpe overhead rane. As e mentioned before the main ontrol goal is to redue the sing of the paload hile moving it to the desired position as fast as possible. Hoever, the sing of the paload depends on the aeleration of the trolle, so minimizing the operation time and minimizing the paload sing are partiall onfliting requirements. Having this in mind, one partiular method for limiting vibration in flexible sstems b shaping the input to the sstem is presented in [14]. As differene to the most previous input shaping strategies, this method does not require a preise sstem model or length numerial omputation. In [15], the author proposes one method for a high performane anti-sing ontrol of overhead ranes, here the motion planning problem is solved as a kinemati problem. In [16], the authors developed a nonlinear model for an overhead rane and then that model is used for anti-sing ontrol development. The feedbak ontrol is used to speif the rane speed at ever moment. In [17], a nonlinear traking ontroller ith to loops (outer and inner), is designed. Loops are used to stabilize the internal osillator dnamis. Similar to the most of the previousl desribed papers, in this paper e developed an anti-sing ontrol method in order to redue the osillations during the positioning of a 3D rane sstem. The differene beteen the desired and atual position and the derivative of that differene are used as the input variables for the fuzz ontroller. The main advantage of this ontroller is that its struture is quite simple in omparison to some other strutures mentioned above. The paper is organized as follos. Setion II starts ith the desription of the 3D rane experimental frameork used for the validation of the proposed ontrol algorithm, and ontinues ith the mathematial model of the 3D rane sstem. The ontroller design is presented in setion III. In Setion IV, the proposed ontrol method is applied to the rane sstem and experimental results are disussed. Conluding remarks are given in Setion V. II. 3D CRANE SYSTEM A. Laborator Experimental Frameork Desription Pratial 3D rane frameork, used in experiments performed in this paper, is presented in Figure 1 [1]. This frameork, made b Inteo, is plaed in the laborator of Control Sstems in Fault of Eletroni, Nis, Serbia. 3D rane represents a nonlinear eletromehanial sstem having a omplex dnami behavior and reating hallenging ontrol problems. The 3D rane sstem onsists of a paload hanging on a pendulum-like lift-line ound b a motor mounted on a art. The paload attahed to the end of the lift-line an move freel in three dimensions: in x diretion (both the rail and the art an move horizontal), in diretion (the art an move horizontal along the rail) and in z diretion (the paload is lifted and loered in this diretion). Fig. 1. 3D Crane Sstem made b Inteo. There is one DC motor, at the end of eah axis. Furthermore, there are five idential measuring enoders measuring five state variables: the art oordinates on the horizontal plane (used for ontroller design), the lift-line length (in our ase used as onstant), and to deviation angles of the paload. The enoders measure movements ith a high resolution equal to 4096 pulses per rotation. The deviation of the load is measured ith a high aura equal to rad. The poer interfae amplifies the ontrol signals hih are transmitted from the PC to the d motors. It also onverts the enoders pulse signals to the digital 16-bit number. The PC equipped ith the RT-DAC/PCI multipurpose digital I/O board ommuniates ith the poer interfae board. The hole logi neessar to ativate and read the enoder signals and to generate the appropriate sequene of pulses of pulse idth modulation (PWM) to ontrol the d motors is onfigured in the Xilinx hip of the RT-DAC/PCI board. All funtions of the board are aessed from the 3D rane toolbox hih operates diretl in the MATLAB /Simulink environment. The user has rapid aess to all basi funtions of the 3D rane ontrol sstem from the Main Control Windo in MATLAB. It inludes tests, drivers, models and appliation examples. The main driver is loated in the RTWT Devie Driver olumn. The driver is a softare go-beteen for the real rane MATLAB environment and the RT-DAC/PCI aquisition board. The driver has three PWM inputs (d motor ontrols) for the X, Y and Z axes. There are eight outputs of the driver: X position, Y position, Z position, to angles and additionall three safet sithes. Aording to a preprogrammed logi the internal XILINX program of the RT- DAC/PCI board an use the sithes to stop the DC motors. B. 3D Crane Mathematial Model Graphial representation of the 3D rane sstem is depited in Figure 2. Despite the model simplifiation, it preserves the fundamental harateristis of the real sstem. The fores ating on this sstem and relevant variables, needed for the model development, are presented. An important element in the onstrution of the mathematial model is the appropriate hoie of the sstem of oordinates. Despite the fat that the

3 ETASR -, Tehnolog & Applied Siene Researh Vol. 2, o. 2, 2012, Cartesian sstem is simple in interpretation, it is not onvenient for the desription of the rotational motion dnamis, so e hoose the spherial sstem. As e an see from Figure 2, the position of the paload is desribed b to angles α and β. The shortoming of the spherial sstem of oordinates is that for ever point on the -axis, the orresponding value of β is not uniquel determined. Hoever, this is not valid in the ase of the real rane sstems, so it an be negleted. The position of the paload is desribed b the folloing equations: TABLE I. PARAMETERS OF 3D CRANE MODEL Smbol Desription R length of the lift-line α angle beteen the -axis and the lift line β m m angle beteen the negative diretion on the z-axis and the projetion of the lift line onto the xz plane mass of the paload mass of the art x = x + R osα, (1) = + R sinα sinβ, (2) z = Rsinα osβ, (3) m s x,, z S F x F mass of the moving rail oordinates of the paload reation fore in the lift-line ating on the art fore driving the rail ith art fore driving the art along the rail here x represents the distane of the rail ith the art from the enter of the onstrution frame, and is the distane of the art from the enter of the rail. The parameters of the 3D rane model are given in the Table I. In similar manner, the dnamis of the rane an be obtained as: mɺɺ x = S, (4) x mɺɺ x = S, (5) mɺɺ x = S m g, (6) z mɺɺ x = F T + S, (7) ( ) x x x m + m ɺɺ x = F T + S, (8) s here S x, S and S z are the omponents of the vetor S. T x, T frition fores S = S osα, (9) x S = S sinα sinβ, (10) S = S sinα osβ. (11) z The mathematial model of 3D rane sstem is ompletel determined b these equations. This model is used as the ontrolled plant for the ontroller design, presented in the next Setion. III. CONTROLER DESIGN In [18], e onsidered the implementation of geneti algorithms in the determination of the parameters of PI ontrollers. In this setion e propose an anti sing fuzz ontroller, hih gives better results ompared to [18]. A ontrol struture, based on fuzz ontrollers, is developed for the previousl desribed experimental frameork. The main fuzz ontroller s struture is shon in Figure 3. As it an be seen, it onsists of three proesses: fuzzifiation, fuzz onluding, based on fuzz rules, and deffuzifiation. Let us define the error for x position of the paload as: e = x x (12) x des here the x des represents desired and the x is atual, x position of the paload, respetivel. Differentiating (12) results in: Fig. 2. 3D rane sstem: oordinates and fores. d d eɺ = x ( ex) ( xdes x) dt = dt (13) In a similar manner, the error e and the error first derivative eɺ for the position of the paload an be defined. These four

4 ETASR -, Tehnolog & Applied Siene Researh Vol. 2, o. 2, 2012, variables are used as inputs for the to anti-sing ontrollers, one for the x position and one for the position of the paload, as shon in Figure 4. The oordinate z of the paload is onsidered to be a onstant. The ontroller outputs are named u x and u, and the represent the PWM inputs (DC motor ontrols) for the x- and - axis, respetivel. Eah of the assoiated fuzz variables for e, eɺ and u has seven linguisti terms. This terms are desribed b equall distributed Gaussian membership funtions for e and eɺ and triangular ones for u. We hoose a qualitative ategor for eah of them as NL (negative large), NM (negative medium), NS (negative small), ZE (zero), PS (positive small), PM (positive medium) and PL (positive large). The rule base for the antising ontroller is presented in Table II, and it has the same form for x- and -axis, so e give onl one. As a defuzzifiation method aording to the ontrol problem speifiit, the enter of gravit (COG) method as hosen. Fig. 3. Standard struture of fuzz ontroller. TABLE II. eɺ e ANTI-SWING FUZZY CONTROLLER RULE BASE L M S ZE PS PM PL PL ZE NS NM NL NL NL NL PM PS ZE NS NM NL NL NL PS PM PS ZE NS NM NL NL ZE PL PM PS ZE NS NM NL S PL PL PM PS ZE NS NM M PL PL PL PM PS ZE NS L PL PL PL PL PM PS ZE IV. EXPERIMENTAL RESULTS Pratial verifiation of the proposed anti-sing fuzz ontroller is performed on the experimental 3D rane setup presented in Figure 1. As mentioned earlier, this is the experimental environment in hih the users verif their 3D ontrol algorithms in real time using MATLAB and Simulink toolboxes. The experimental results of the 3D rane ith the proposed anti-sing fuzz ontroller are ompared ith the results obtained b the implementation of a PI ontroller [18]. The square ave is used as a referene signal. This tpe of signal mathes the real situation, here the art ith paload is driven from the home position to the desired position and bak to the home position. The square signal has amplitude of 0.8m and frequen of 0.1Hz. It is assumed that the rane starts from home position (x, )=(0, 0). Experimental time as set to 25s. The results of the paload positions after using anti-sing fuzz ontrollers are shon in Figure 5. It is shon that the proposed ontrol algorithm gives quite better results in omparison to those in [18]. Hoever, due to osillations, atual position slightl exeeds the desired position. On the other hand, it traks desired position quite ell. Fig. 4. Simulation blok diagram of the 3D rane ontrol. V. CONCLUSION This paper deals ith the problem of reduing osillations during the positioning of the paload. The 3D rane sstem is desribed b a nonlinear model, so using some robust methods in the ontrol seems to be the right hoie. In this paper, e present an anti-sing fuzz ontroller for 3D rane positioning. The differene beteen desired and atual position and the first derivative of that differene are used as inputs for the ontroller. The short desription of experimental frameork and its mathematial model are given first, and then, experimental setup is emploed for the pratial verifiation of the proposed ontrol algorithm. Results sho that the method desribed in the paper shos good sstem aura. Fig. 5. Experimental results for x and positions. ACKNOWLEDGMENT This paper as realized as a part of the projets TR and III 44006, finaned b the Ministr of Eduation and Siene of the Republi of Serbia for the period

5 ETASR -, Tehnolog & Applied Siene Researh Vol. 2, o. 2, 2012, REFERENCES [1] Inteo, 3D Crane Sstem-User s Manual, available at.inteo.om.pl, [2] F. L. Leis, T. Parisini, Neural netork feedbak ontrol ith guaranteed stabilit, Int. J. Control, Vol. 70, pp , [3] M. Takegaki, S. Arimoto, A ne feedbak method for dnami ontrol of manipulator, ASME J. Dnami Sst. Measurement, and Contr., Vol. 103, pp , [4] J. W. Beeston, Closed-loop time optimal ontrol of a suspended paload-a design stud, 4th IFAC World Congress, Warsa, Poland, pp , [5] Y. Sakaa, Y. Shindo, Optimal ontrol of ontainer ranes, Automatia, Vol. 18, No. 3, pp , [6] M. W. Noakes, J. F. Jansen, Generalized input for damped-vibration ontrol of suspended paloads, Journal of Robotis and Autonomous Sstems, Vol. 10, No. 2, pp , [7] G. Corriga, A. Giua, G. Usai, An impliit gain-sheduling ontroller for ranes, IEEE Trans. Control Sstems Tehnolog, Vol. 6, No. 1, pp , [8] O. Saodn, H. Ashemann, S. Lahres, An automated gantr rane as a large orkspae robot, Control Pratie, Vol. 10, No. 12, pp , [9] A. Bara, S. Dale, Z. T. Nag, Comparative real-time experimental stud ase for ontrol algorithms, from implementation point of vie, Proeedings of the 8 th WSEAS International Conferene on Sstem Siene and Simulation in, Wisonsin, USA, pp , [10] J. A.Méndez, L. Aosta, L. Moreno, S. Torres, G.N.Marihal, An appliation of a neural self-tuning ontroller to an overhead rane, Neural Computing and Appliations, Vol. 8, No. 2, pp , [11] D. Antić, S. Nikolić, S. Perić, M. Milojković, M. Milošević, Geneti algorithms applied in positioning 3D Crane Sstem, 55 th Conferene ETRAN, Banja Vrućia (Teslić), Bosnia and Herzegovina, pp. AU , [12] H. -H. Lee, Y. Liang, D. Segura, A sliding-mode antising trajetor ontrol for overhead ranes ith high-speed load hoisting, Journal of Dnami Sstems, Measurement, and Control, Vol. 128, No. 4, pp , [13] S. K. Cho, H. H. Lee, A fuzz-logi antising ontroller for three dimensional overhead ranes, ISA Trans., Vol. 41, No. 2, pp , [14] W. Singhose, W. Seering, N. Singer, Residual vibration redution using vetor diagrams to generate shaped inputs, Journal of Dnami Sstems, Measurement, and Control, Vol. 116, No. 2, pp , [15] H. H. Lee, A ne motion-planning sheme for overhead ranes ith high-speed hoisting, Journal of Dnami Sstems, Measurement, and Control, Vol. 126, No. 2, pp , [16] K. A. Moustafa, A. M. Ebeid, Nonlinear modeling and ontrol of overhead rane load sa, Journal of Dnami Sstems, Measurement, and Control, Vol. 110, No. 3, pp , [17] J. Yu, F.L. Leis, T. Huang, Nonlinear feedbak ontrol of a gantr rane, Amerian Control Conferene, Seattle, USA, pp , [18] Z. Jovanović, D. Antić, Z. Stajić, M. Milošević, S. Nikolić, S. Perić, Geneti algorithms applied in parameters determination of the 3D rane model, FACTA UNIVERSITATIS Series: Automati Control and Robotis, vol. 10, no. 1, pp , 2011.