Genetic Tuning of Fuzzy Logic Controller for a Flexible-Link Manipulator

Transcription

1 Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator Lnda Zhxa Sh Mohamed B. Traba Department of Mechancal Unversty of Nevada, Las Vegas Department of Mechancal Engneerng Las Vegas, NV Based on Sh, L. Z., and M. Traba, Genetc Tunng of Fuzzy Logc Controller for a Flexble Lnk Manpulator, Proceedngs of the Internatonal Conference on Mathematcs and Engneerng Technques n Medcne and Bologcal Scences (METMBS 00), Las Vegas, Nevada, June 000, CSREA Press, pp Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 1

2 Overvew Fuzzy control presents a practcal alternatve to conventonal controllers that s both computatonally effcent and robust. Whle fuzzy logc control has been successfully mplemented n many applcatons, several questons, concernng t, reman however unanswered. For example, desgn of fuzzy controllers usually depends on ntuton or experence, whch works well f the desgner s famlar wth the system. Fuzzy controller for flexble manpulators does not ft well wthn ths category. Whle the rules for such controllers may be ntutve, selectng membershp functons for nput and output varables of the manpulator can be challengng. Wrong choces may lead to sluggsh response, excessve vbraton, or nstablty. Ths paper uses a dstrbuted controller composed of two PD-lke fuzzy logc controllers to control angle and lnk vbratons of ths manpulator. It presents an approach for tunng the membershp functons of the varables of these controllers usng a genetc algorthm. The results of these controllers are analyzed to smplfy the structure of controllers. Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator

3 Lterature Revew A control law that conssts of a fuzzy logc controller plus a nonlnear effects negotator for a flexble manpulator arm was derved n [Ln, Y. and Lee, T., 1993]. The nonlnear effects negotator was used to enhance the controller's ablty n dealng wth the model s uncertantes. An error response plane method was proposed to obtan the fuzzy control rules. The optmal overlap of membershp functons that results n the best performance of the manpulator was estmated. A fuzzy control strategy to control the rgd body and the frst flexural mode of vbraton n a sngle lnk robotc arm by desgnng a separate controller for each was descrbed n [Kubca, E. and Wang, D., 1993]. Controllers for manpulators wth flexble lnks were mplemented by usng a class of fuzzy nference systems, [Arcnegas et al., 1993], where fuzzy nference systems were shown structurally smlar to those of neural networks consstng of a collecton of locally tuned unts. Ths observaton was used to generate the fuzzy rules functon by a method smlar to the problem of effcently placng the centers of the receptve felds of unts n neural networks. A fuzzy model reference learnng controller for a manpulator wth flexble lnks was developed n [Moudgal et al., 1995]. The proposed approach n that paper can synthesze a rule-base for a fuzzy controller and tune t to adapt to varatons n the payload. Fuzzy logc was used to suppress vbraton of a flexble arm, [Yoo et al., 1995] usng a lngustc model of the system. A new structure for a dstrbuted fuzzy logc controller and a method for tunng the membershp functons of ths controller whle reducng ts dmensonalty were proposed n [Traba, 1998]. Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 3

4 Objectves Ths paper s addressng several ssues related to the desgn of a fuzzy logc controller for a flexble-lnk manpulator. The objectve of ths controller s to drve the manpulator through a desred trajectory wthout exctng vbraton. The desgn of a fuzzy logc controller deals wth these ssues: a. Identfyng the varables of the controller. b. Identfyng the structure of the controller. c. Choosng fuzzy nference rules that the controller uses. d. Estmatng the number of the membershp functons that descrbe each of ts varables. e. Determnng the shape and locaton of each of these membershp functons. f. Evaluatng the performance of the controller to determne f any of the above elements, such as the number of membershp functons that descrbe a varable, should be modfed. Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 4

5 A Structure for a PD-Lke Dstrbuted Fuzzy Controller Fuzzy logc control has an ntutve nature, whch may work well n controllng smple systems. It may be dffcult to desgn a fuzzy logc controller for flexble manpulators, Fgure 1, whose equatons of moton are coupled. A controller that combnes both jont angle and tp dsplacement, as well as ther tme dervatves, n a sngle controller wll have large number of rules that may be dffcult to determne. It wll be also computatonally ntensve. y T(t) x P v m t,j t x J m L 0 θ Fgure 1 Schematc of a Flexble Manpulator Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 5

6 To fully explot the potental of fuzzy logc, the control acton s dstrbuted between two fuzzy controllers: jont angle controller and tp controller, Fgure. Both controllers use the error as nput. Error s defned as the dfference between the desred value of a varable and ts actual one. The output of both controllers s torque. Jont angle controller has two nputs: jont angular dsplacement and velocty errors, e θ and e dθ respectvely, and one output: the torque needed to correct these errors: T θ. Smlarly, the tp controller has two nputs: tp dsplacement and velocty errors, e tp and e dtp respectvely, and one output: the torque needed to correct these errors: T tp. The sum of the output of these two controllers s fed to the manpulator. v(l) _ + 0 d(v(l))/dt _ + 0 T Flexble Manpulator θ _ + θ d dθ/dt _ + (dθ/dt) d e dθ T θ Jont Angle Fuzzy Controller e θ + + e dtp T tp Tp Fuzzy Controller e tp Fgure Structure of Dstrbuted PD-Lke Fuzzy Logc Controller for a Flexble Manpulator Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 6

7 Membershp Functons The queston of how many membershp functons are suffcent to descrbe a fuzzy varable s dffcult to readly answer. In ths paper, an ntal guess of seven membershp functons s used to descrbe each of the sx varables: negatve bg (NB), negatve medum (NM), negatve small (NS), zero (Z), postve small (PS), postve medum (PM), and postve bg (PB). Subsequent sectons n ths paper wll help determne f ths number s suffcent, excessve, or nadequate. Dscusson at ths paper s lmted to Gaussan curve membershp functons whose form s, ( z c ) σ, e ( z σ, c) µ = (1) Gaussan curve membershp functon has the advantage of beng descrbed usng only two parameters: c, the center of the functon and, σ, curve shape parameter. Selectng the membershp functons and ranges of varables s not as ntutve as determnng the rules of the controllers. Detals of the proposed algorthms for selectng membershp functons and ranges of varables for optmal performance are shown n the next sectons. Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 7

8 Fuzzy Rules The rules for both controllers are based on ntuton and observatons of nertal system. The goal of the jont angle fuzzy controller s to mantan the manpulator along a desred trajectory. Its nputs are selected to produce output smlar to that of a typcal PD controller. The rules of ths controller, Table 1, are selected to avod overshoot or laggng wth respect to the desred jont trajectory. Table 1 Rules for the Jont Angle Fuzzy Controller e θ NB NM NS Z PS PM PB e dθ NB NB NB NB NM NM NS Z NM NB NB NM NM NS Z PS NS NB NM NM NS Z PS PM Z NM NM NS Z PS PM PM PS NM NS Z PS PM PM PB PM NS Z PS PM PM PB PB PB Z PS PM PM PB PB PB Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 8

9 On the other hand, the rules of the tp fuzzy controller, Table, are selected such that f the tp s approachng the correct poston or f velocty error belongs to the zero functon, the controller wll produce no torque. These rules attempt to use the stran energy of the lnk to dampen the arm vbraton. Ths controller produces torque only when the tp s movng away from the desred target poston. Table Rules for the Tp Fuzzy Controller e tp NB NM NS Z PS PM PB e dtp NB PB PB PB PM PS Z Z NM PB PM PS PS PS Z Z NS PM PS PS PS PS Z Z Z Z Z Z Z Z Z Z PS Z Z NS NS NS NS NM PM Z Z NS NS NS NM NB PB Z Z NS NM NB NB NB Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 9

10 Tunng of a Dstrbuted Fuzzy Controller Usng Genetc Algorthm The performance of a fuzzy controller depends on the range of ts nput and output varables and shape of the membershp functons that descrbe ths functon. An automated method to tune the membershp functons of a fuzzy controller s presented. The algorthm chooses the ranges and the shapes of the membershp functons of ts varables usng genetc programmng. Ths algorthm reduces the need to havng a requred pror knowledge of the controlled system that s currently necessary for desgnng a fuzzy controller. The performance ndex chosen for the flexble manpulator s, nt e e tp dtp e + e θ dθ = 1 L L PI = () nt e e dtp dtp 1 ( e e ) + = dθ dθ 1 L L where, nt s the total samplng tme dvded by samplng nterval. The frst term n performance ndex equaton represents a measure for the manpulator errors whle the second term represents a measure for the manpulator vbraton. Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 10

11 In the absence of gravty, t s far to assume that membershp functons are symmetrcal. The problem s then modeled as havng forty-two varables (genes) that correspond to the shapes (Z, PS, PM, and PB) and centers of the membershp functons (PS, PM, and PB) of:e θ, e dθ, T θ, e tp, e dtp, and T tp respectvely. Each varable s represented by real numbers. The problem s subject to spacng constrants n the form of: g nms 1 nms + 1 ( k 1 ) + j = c c k, j k, j 1 > 0 n + 1 ms (3) where + 1 j nms 1 k 7 These constrants ensure that the center of PS (PM) membershp functon les to the left of that of PM (PB) membershp functon can be ncorporated n the performance ndex usng the bracket functon. The modfed functon s, mnmze, m FC = PI + Ω Ω = R * g ( x) f g 0 Ω = 0 f g > 0 where, R s a penalty term of a hgher magntude than the value of PI. = 1 (4) Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 11

12 Genetc Algorthm The ntal populaton of 00 chromosomes s randomly generated. The algorthm selects ffty percent of the populaton wth the best ftness value as parents, as well as members, of next generaton. The rest of the new populaton s generated by crossng over two randomly chosen parents usng the weghted average operator: v ' = a v + (1 a) v j v ' = a v + (1 a) v where, a s a randomly generated number from [0, 1]. A mutaton rate of 0.01 s selected. At each generaton, the number of mutated strngs s equal to, Mutate_Number = Mutaton_rate * Populaton_sze* Number_of_strngs The postons of the mutated strngs are ncluded n an array of random nteger numbers that are selected from the array: [1,,, Populaton_sze* Number_of_strngs]. The values of these strngs are randomly generated. The process contnues for a maxmum number of 00 generatons. j j Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 1

13 Tunng the fuzzy logc controller can be represented by the block dagram of Fgure 3. The system has three blocks: (1)Plant: The system that wll be controlled. It receves controller nputs and produces sensors outputs. ()Fuzzy Controller Traner: Ths traner uses a genetc algorthm to evaluate the system performance ndex. It suggests modfcatons of the membershp functons to mnmze ths performance ndex. (3)Fuzzy Controller: Fuzzy controller produces the nputs for the plant. Fuzzy Controller Inputs Plant Performance Index Outputs c's & σ's Fuzzy Controller Traner Fgure 3 Block Dagram of the Algorthm used for Tunng Membershp Functons of Fuzzy Controllers Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 13

14 Smulaton Example A computer program has been developed to smulate the dynamcs of a sngle-lnk flexble manpulator. The physcal parameters and mechancal propertes of the manpulator used n ths smulaton are lsted n Table 3. Four elements of equal length are used to model the beam. The ntal angle of the manpulator s equal to zero whle desred fnal angle s equal to one radan. The number of samples, nt, s equal to one thousand samples over the smulaton perod of ten seconds. Table 3 Physcal Parameters and Mechancal Propertes of the Smulated Flexble Manpulator Parameter Value Lnk length, L 1.0 m Densty, ρ 0.1 kg/m Bendng stffness, EI.0 Nm Moment of nerta of the hub, J m 0.05 Kgm Radus of the hub, L m Tp mass, m t 1.0 kg Tp mass moment of nerta, J t 10-5 Kgm Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 14

15 Results of Tunng the Flexble Manpulator Fuzzy Controller Usng GA The fgures show the membershp functons for of the member wth the lowest value of the performance ndex at the fnal generaton. The membershps functons have sgnfcant amount of overlap. The results show an overshoot and steady state error of radans for e θ. Tp vbraton dsappears after sx seconds. The torque curve s smooth. Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 15

16 1. GA Controller Reduced GA Controller 1 jont angle (rad) tme (s) Fgure 4 Jont Angle Response for the GA Controllers GA Controller Reduced GA Controller tp dsplacement (m) tme (s) Fgure 5 Tp Pont Dsplacement Response for the GA Controllers GA Controller Reduced GA Controller 0.5 torque (Nm) tme (s) Fgure 6 Appled Torque for the GA Controllers Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 16

17 NB NM NS Z PS PM PB Degree of membershp DE Fgure 7 Membershp Functons of e θ of the GA Jont Angle Fuzzy Controller NBNM NS Z PS PMPB Degree of membershp VE Fgure 8 Membershp Functons of e dθ of the GA Jont Angle Fuzzy Controller NB NM NS Z PS PM PB Degree of membershp torque Fgure 9 Membershp Functons of T θ of the GA Jont Angle Fuzzy Controller Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 17

18 NB NM NS Z PS PM PB Degree of membershp DE Fgure 10 Membershp Functons of e tp of the GA Tp Vbraton Fuzzy Controller NB NMNS Z PSPM PB Degree of membershp VE Fgure 11 Membershp Functons of e dtp of the GA Tp Vbraton Fuzzy Controller NB NM NS Z PS PM PB Degree of membershp torque Fgure 1 Membershp Functons of T tp of the GA Tp Vbraton Fuzzy Controller Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 18

19 Reducng the GA Fuzzy Controller Observatons: Membershp functon Z of e tp overlaps the other functons completely. The value of ths functon s very close to one for the whole range of e tp. Ths membershp functon effectvely neutralzes the effect of e tp on the tp fuzzy controller. Ths controller was therefore reduced by changng t to a sngle nput, e dtp, sngle output, T tp, controller. The rules of the reduced controller are lsted n Table 4. Implementng these changes, The reduced controller effectvely elmnates the steady state error whle mantanng the tp vbraton and the jont torque as before. Table 4 Rules for the Reduced Tp Fuzzy Controller e dtp PB PM PS Z NS NM NM T tp NB NM NS Z PS PM PM Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 19

20 Conclusons Successful mplementaton of fuzzy logc control depends on the avalablty of knowledge of the controlled system. For complex systems, such as flexble manpulators, ths knowledge may be lackng, whch lmts the scope of usng fuzzy controllers. Ths paper uses a dstrbuted controller composed of two fuzzy logc controllers to control jont angle and tp deflecton ndependently. Separate controllers are used snce smple rules can be proposed for each controller when ts acton s consdered ndependently, based on observaton of the dynamcs of the sub-system. The outputs of these two controllers are combned and used as the commanded torque to the manpulator. Genetc algorthm s used to tune the performance of the controller by varyng the ranges and shapes of the membershp functons snce t s dffcult to estmate the approprate values of these varables for every flexble manpulator. The method s satsfactorly appled to a sngle-lnk flexble manpulator. The results show that the ntal number of membershp functons used to descrbe some of the varables can be reduced from what was orgnally proposed. Ths reducton can produce a smpler and faster controller than the orgnal wthout sgnfcant dfference n the performance. The procedures presented n ths paper can be appled to other systems that are dffcult to characterze. Usng hybrd genetc algorthms to ths type of problems s currently under consderaton. Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator 0