Fuzzy Sliding Mode Controller with Neural Network Applications to Mobile Robot

Transcription

1 Fuzzy Sdng Mode Controer wth Neura Network Appcatons to Mobe Robot Lon-Chen Hung and Hung-Yuan Chung* Department of Eectrca Engneerng, Natona Centra Unversty, Chung-L, Tao-Yuan, 3, Tawan, R.O.C TEL: ext 4475 Fax: *E-ma: Abstract: - Ths paper deas wth the desgn of sgned fuzzy sdng mode contro wth neura network (SFSMNN) systems by usng the approach of sdng mode contro. The fuzzy sdng mode contro (FSMC) n the system guarantee the state can reach the user-defned surface n fnte tme and then t sdes nto the orgn aong the surface. The key dea s to appy parameters of the membershp functon of the fuzzy rue base are sef-tunng wth neura network (NN). The nta fuzzy membershp functon can be gven by human experts, and t s tuned automatcay by the tunng scheme so as to emnate the chatterng effect of the system. Next, we w appy the SFSMNN controer to a nonnear mobe robot system. At ast, experment resuts show the feasbty of the method that used to verfy the effectves and robustness. Key-Words: - fuzzy, sdng mode, neura network, mobe robot Introducton In cassca contro theory, the controer s desgned based on the mathematca mode of the system. However, the mathematca mode of the system s hard to know. So t s not easy to desgn a mode-based controer to contro these compex and -defned system. Fortunatey, human knowedge can do we n deang wth ths knd of probem. It s we known that fuzzy ogc provdes a systematc procedure to transform a knowedge base nto a practca contro strategy. Partcuary, fuzzy contro shows ts powerfu capabty n compex systems wth -defned processes, or systems ack of knowedge of ther dynamcs. However, the desgn of fuzzy controers s tme-consumng and experment-orented and requres the knowedge acquston, defnton of the controer structure, defnton of rues, and other controer parameters. So, one mportant ssue n fuzzy ogc systems s how to reduce the number of nvoved rues and ther correspondng computaton requrements [,]. It s mportant the fuzzy membershp functons are updated teratvey and automatcay, because a change n fuzzy membershp functons may ater the performance of the fuzzy controer sgnfcanty. Severa agorthms of tunng of the fuzzy membershp functons have been proposed n [3,4]. These agorthms have an mportant characterstc of goba tunng of the fuzzy membershp functons and that most of them need off-ne preprocessng. The fuzzy technques have been wdey apped to the systems wth uncertantes. Recenty, researchers have utzed the fuzzy technques together wth the sdng mode contro for many engneerng contro systems [5-7]. In [5], the sdng mode contro schemes wth fuzzy system are proposed wth the uncertan system functon s approxmated by the fuzzy system. The fuzzy contro rues based on the sdng functon and the dervatve of the sdng functon are constructed n [6]. The compexty of the fuzzy SMC s ncreased qucky as the number of sdng functons ncreases. The technoogy that combnes or fuses the neura-network theory wth the fuzzy reasonng s beng watched wth keen nterest []. In ths paper, an SFSMNN poston contro s nvestgated, n whch the SFSMNN s utzed to estmate the bound of uncertantes rea-tme for the poston contro and trackng contro system. The nputs of the SFSMNN are the swtchng surface, and the output of the SFSMNN s the estmated bound of uncertantes. If the uncertantes are absent, once the swtch surface s reached ntay, a very sma postve estmated vaue of bounded of uncertantes woud be suffcent to keep the trajectory on the swtchng surface. We aso appy t to controng a mobe robot drven by two ndependent whees. The mathematca mode of mobe robot system s not competey known. But, to measure some physca parameters, e.g., vscous frcton factor and moment of nerta around the center of gravty for the robot, so that the SFSMNN whch s not based on the mathematca mode s recommendabe for ths subject.

2 The rest of the paper s dvded nto fve sectons. In Secton, the snge fuzzy sdng mode controer wth neura network s presented. In Secton 3, the mode of a mobe robot s descrbed. In Secton 4, the proposed controer s used to contro a mobe robot system. Fnay, we concude wth Secton 5. Sgned Fuzzy Sdng Mode Controer wth Neura Network. Sgned Dstance Fuzzy Logc Contro In ths secton, the dea of [8] named the sgned dstance s used, and the feasbty of the present approach w be demonstrated. The swtchng ne s defned by: s: x + c x= () Frst, we ntroduce a new varabe caed the sgned dstance. Let P( xx, ) be the ntersecton pont of the swtchng ne and the ne perpendcuar to the swtchng ne from an operatng pont Q( x, x ), as ustrated n Fg.. Next, ds s evauated. The dstance between P( xx, ) and Q( x, x ) can be gven by the foowng expresson: d s = x + cx + c () The sgned dstance d s s defned for an arbtrary pont Q( x, x ) as foows: d s x + cx = sgn( s) + c x+ c x s = = (3) + c + c where for s > sgn( s) = (4) - for s < For the swtchng ne s chosen as s = cx + x (5) By takng the tme dervatve of both sdes of (5), we can obtan s = cx + x = cx + f( x) + b( x ) u+ d (6) Then, mutpyng both sdes of the above equaton by gves ss = scx + sf( x) + sb( x) u+ sd (7) Here, we assume that b ( x) >. In (6), t s seen that ncreases as ncreases and vce versa. Equaton (7) provdes the nformaton that f s>, the decreasng u w make ss decrease and that f s<, the ncreasng u w make ss decrease. Hence, the fuzzy rue tabe can be estabshed on a one-dmensona space of as shown n Tabe nstead of a two-dmensona space of x and x. The contro acton can be determned by d s ony. Hence, we can easy add or modfy rues for fne contro. For mpementaton, the membershp functon of the nput s tranguar-shaped whch are n the form of x m μ() x = max[,], where m s the center of σ membershp functon and σ s the wdth. The membershp functon of the output s sngeton. The fuzzy sdng mode wth rue base s as Fg. and the Structure of the SFSMNN contro s as Fg. 3.. Tunng Sgned Dstance Fuzzy Logc Contro wth Neura network We appy neura network to tunng sgned dstance fuzzy ogc controer. The neura network based fuzzy ogc controer (NN-FLC) s a mutayer network. It combnes the merts of fuzzy controer and neura network. It ntegrates the basc eements and functons of a tradtona fuzzy ogc controer nto a connected structure that possesses the earnng abty. The archtecture of NN-FLC s show n Fg. 4. The NN-FLC contans fve ayers.the frst ayer s nput ayer and the ffth ayer s output ayer. In the second ayer, nodes compute the degree of compatbty from the frst ayer, and connect to reatve nodes of next ayer. Nodes at the thrd ayer are rue nodes whch ndcate the connecton wth fourth ayer by fred rues. Nodes at the fourth ayer are term nodes whch s smar to the second ayer, representng the membershp functon of the consequence abe. The nks between every ayer represent the weght. The second and the ffth ayer are fu connected.the arrow of nk ndcate the propagaton drecton of the sgna. There are two knd of functon requred n each node. One s summaton of a nputs from the node of the prevous ayer. The functon can be expressed as foows: () () () () () () Node = I( u, u,, u ; w, w,, w ) (8) where nput n n u w represents the nput sgna from prevous ayer, represents the -th nk weght of the -th ayer, and n represent the number of the connected nodes. The other functon s transfer functon whch denotes the output of ths node. The functon can be wrtten as

3 Nodeoutput = O( I) (9) The detas of ths network are descrbed as foows: Layer : Ths ayer just transmts the vaue of the nput sgna to the next ayer. So the functon can be wrtten as I = u n and O= I () We note that the nk weght w =. Layer : In ths ayer, we use tranguar membershp functon to fuzzfy the nput sgnas that propagate from the prevous ayer. The functon s u mj I = max(,) and O= I () σ mj j Where and σ j are the convex vertex and the wdth of the tranguar-shape functon of the j-th term of the -th nput ngustc varabe, hence the nk weght w can be represented as m. j Layer 3: Nodes at ths ayer perform the fuzzy AND operaton and ndcate where the output connected by matchng rue. The functon can be denoted (3) (3) (3) I = mn( u, u,, un ) and O = I () 3 We note that the nk weght w =. Layer 4: Nodes at ths ayer sum up the contrbutons whch have the same consequence from Layer 3 and perform fuzzy OR operaton. The functon can be denoted as n 4 = I = w I u and O= and the nk weght 4 =. j (3) Layer 5: Ths ayer executes the defuzzfcaton. One seects the COA (center of area) method. The functon can be denoted as n 5 I I = ( m) u and O = 5 = (4) u Where m s the vaue of each sngeton fuzzy output functon of the -th term of the output ngustc varabe, hence the nk weght can be represented as 5 w = m. The back-propagaton tranng agorthm s apped to ths fuzzy system. We usuay check the earnng performance by the performance ndex (cost functon) and mnmze the error between desred and current output. The functon can be wrtten as E= ( yt ( )- yd ( t) ) (5) In genera, we mnmze the functon by the gradent descent method whch s used to update the weght of the neura network. That s, the modfed vaue can be represented as: E Δw w (6) E wt ( + ) = wt ( ) + η( ) w (7) where η s the earnng rate. For smpfyng the tunng probem, we derve the tunng agorthm to Layer fve ony. In Layer fve, the adaptve rue of the vaue m s derved as E E O E O I = = m O m O I m (8) From (5) E = [ yd ( t) y( t) ] O (9) O = I u () I m = u () Then the membershp functon can be wrtten as: u mt ( + ) = mt ( ) + η( yd( t) yt ( )) u () Next, by measurng the chatterng of the system, we ntroduce the chatterng ndex [], β s defned as foows: 3 β = α (3) = f SkTSk ( ) ( ) T> α = (4) f SkTSk ( ) ( ) T< where T s the sampng tme. Because y d s unknown, we repace ( y d - y) by ( β ), and then () can be represented as u m( t+ ) = m( t) ηβ (5) u 3 Mode of a Mobe Robot Let the mobe robot wth two ndependent drvng whees be rgd movng on the pane. It s assumed that the absoute coordnate system O-XY s fxed on the pane as shown n Fg. 6. Then, the dynamc property of the mobe robot s gven by the foowng equaton of moton []: I = Dr D (6) M = Dr D (7) For the rght and eft whees, the dynamc property of the drvng system becomes

4 Iω θ + cθ = ku rd, = r, (8) where each parameter and varabe are defned by () I : moment of nerta around the e.g. of robot; () M : mass of robot; (3) Dr, D : eft and rght drvng forces; (4) : dstance between eft or rght whee and the e.g. of robot; (5) : azmuth of robot; (6) : veocty of robot; (7) I ω : moment of nerta of whee; (8) c : vscous frcton factor; (9) k : drvng gan factor; () r : radus of whee; () θ : rotatona ange of whee; and () u : drvng nput. The geometrca reatonshps among varabes,, θ are gven by r r = + θ r = (9) θ (3) From these equatons, defnng the state varabe for the robot as x = [ ] T. 4 Experment of the Mob Robot System The veocty and azmuth of the robot are controed by manpuatng the torques for the eft and the rght whees. That s, the contro system consdered here s of mut-nput/mut-output. In the SFSMNN wth e, e, e, and e as nputs, and wth ur and u as reasonng outputs. From the property of a mobe robot wth two ndependent drvng whees, the foowng reatons are consdered: ur = u + u (3) u = u u (3) where u s the torque requred for controng the robot s veocty by usng the measurement of the robot s veocty and u s the torque requred for controng the robot s azmuth by usng the measurement of the robot s azmuth. If u s generated from the SFSMNN wth the veocty error e and ts rate as the nputs and u s generated from the SFSMNN wth the azmuth error e and ts rate as nputs, then the earnng controer for the veocty and azmuth of a robot. The expermenta mobe robot system used n ths paper s shown n Fg. 7. It conssts of a vehce wth two drvng front whees mounted on the same axs. The moton and orentaton are acheved by ndependent actuators, e.g., each front whee s drven by a dc motor. The dspacement output of each motor was by a crude ncrementa encoder on the motor shaft. No pre-fterng was used to fter the nosy feedback sgnas. Veocty sgnas of the servo were obtaned by dfferentatng the dspacement outputs. The trackng controer was mpemented n a PC 486. A -bt resouton D/A card and a 6-bt encoder care were used to transfer the contro and feedback sgnas. Fg. 8 and Fg. shows the response of ange and the state trajectory. 5 Concusons In ths paper, the SFSMNN contro system has been proposed to sove the poston and trackng probem for a wheeed mobe base has been addressed. In order to cope wth the unavodabe presence of uncertantes n the dynamca mode and to guarantee the mpementabty of the controer, the proposed contro aw has been desgned usng snge fuzzy sdng mode wth neura network contro technques. The asymptotc boundedness of the trackng errors has been theoretcay proved. It s shown that the proposed method s feasbe and effectve. References: [] C.L. Tan, A sef-tunng fuzzy ogc controer based on neura networks structure, M.S thess, Department of Eectrca Engneerng, Natona Centra Unversty, 995. [] C.T. Ln, and C.S.G. Lee, Neura Fuzzy System: A Neuro-Fuzzy Synergsm to Integent Systems, Prentce-Ha Interantona, Inc., 996. [3] K. Erbatur and O. Kaynak, Use of adaptve fuzzy systems n parameter tunng of sdng-mode controers, IEEE/ASME Trans. on Mechatroncs, Vo, 6, No. 4,, pp [4] D. Park, A. Kande and G. Langhoz, Genetc-based new fuzzy reasonng modes wth appcaton to fuzzy contro, IEEE Trans. on Syst., Man, Cybern., Vo. 4, 994, pp [5] R. Pam, Robust contro by fuzzy sdng mode, Automatc, Vo. 3, No. 9, 994, pp

5 [6] Y.R. Hwang and M. Tomzuka, Fuzzy smoothng agorthms for varabe structure systems, IEEE Trans. on Fuzzy Syst., Vo., 994, pp [7] J.C. Lo and Y.H. Kuo, Decouped fuzzy sdng-mode contro, IEEE Trans. on Fuzzy Syst., Vo. 6. No. 3, 998, pp [8] B.J. Cho, S.W. Kwak and B. K. Km, Desgn of a snge-nput fuzzy ogc controer and ts propertes, Fuzzy Sets and Syst., Vo. 6, 999, pp [9] J. Zhang and A. J. Morrs, Fuzzy neura networks for nonnear systems modeng, Contro Theory and Appcatons, IEE Proceedngs, Vo. 4, No. 6, 995, pp [] K. Watanabe, J. Tang, N. M. Koga and S.T. Fukuda, A fuzzy-gaussan neura network and ts appcaton to mobe robot contro, IEEE Trans. on Contro Systems Technoogy, Vo. 4, No., 996, pp [] Q.S. Zhang and J.J. Barre, Robust backsteppng and neura network contro of a ow-quaty nonhoonomc mobe robot, Internatona Journa of Machne Toos and Manufacture, Vo. 39, No. 7, 999, pp r - S Neura Network SFSMC Pant Fg. 3. The structure of the SFSMNN contro. Tabe. The rue of SFSMC tabe. Layer 5 Layer 4 Outputs Concequence abes S NB NS ZE PS PB u PB PS ZE BS NB Y Y' u Y'm Ym y x Layer 3 Rues x Layer Antecedent abes Q ( x, x ) Layer P ( x, x ) x + c x = Inputs X X Xn Fg. 4. The tunng fuzzy rue wth neura network. Fg.. Dervaton of a sgned dstance. μ NB NS ZE PS PB - Fg. Fuzzy varabe of tranguar type. S d d - + e + - e S = e +λ e S = e +λ e SFSMNN SFSMNN u r u Mobe Robot Fg. 5. The structure of the SFSMNN contro for the mobe robot.

6 D Left Whee G I Y D r Rght whee O X Fg. 6. A mobe wth two ndependent drve whees. Fg. 9. The mobe robot trackng ne ( y = -6sn(x/3+π)). Fg. 7. The photo of the expermenta mobe robot. Fg.. The mobe robot trackng ne ( y = -6cos(x/6+π)). Fg. 8. The mobe robot trackng ne ( y = x+).