Neural Network Global Sliding Mode PID Control for Robot Manipulators

Transcription

1 Neural Newor Global Sldg Mode PID Corol for Robo Mapulaors. C. Kuo, Member, IAENG ad Y. J. Huag, Member, IAENG Absrac hs paper preses a eural ewor global PID-sldg mode corol mehod for he racg corol of robo mapulaors wh bouded uceraes. A cera sldg mode coroller wh PID sldg fuco s developed. I hs coroller, he swchg ga s ued by a RBF eural ewor o he reachable codo of sldg mode. hus, he effec of chaerg ca be allevaed. Moreover, global sldg mode s realzed by desgg a expoeal dyamc sldg fuco. Mahemacal proof of he sably ad covergece of he corol sysem s gve. Smulao resuls demosrae ha he chaerg ad he seady sae errors are elmaed ad sasfacory raecory racg s acheved. Idex erms Neural ewor, Robo, Robusess, Sldg mode corol. I. INRODUCION A well ow approach o he corol of ucera sysems by olear feedbac laws s he sldg mode corol []-[]. Sldg mode coroller desg provdes a sysemac approach o he problem of maag sably he face of modelg mprecso ad uceray. However, chaerg problem s a maor drawbac of sldg mode corol. he boudary layer s used o avod chaerg pheomea [4]. he cos of hs echology s a reduco he accuracy of he racg performace [5, 6]. I geeral, sldg mode corol has wo phases he corol process. Oe s he reachg mode, ad he oher s he sldg mode. he former s he phase of al saes oward he sldg surface. Whe he sysem raecory says o he sldg surface, he corol sysem ca reec uceraes ad dsurbaces. However, robus racg s guaraeed oly afer he sysem saes reach he sldg surface, ad herefore robusess s o guaraeed durg he reachg phase. I order o overcome hs problem, some researches have bee proposed, such as global sldg mode corol [7-9]. Here, global sldg mode s realzed by desgg a expoeal dyamc sldg fuco. I []-[], he sldg mode corol wh PID sldg surface for robo mapulaors were preseed. Mauscrp receved March, 7. hs wor was suppored par by Naoal Scece Coucl, awa, for facally supporg hs wor uder Corac NSC95--E--.. C. Kuo s wh he Deparme of Elecrcal Egeerg, Chg Yu Uversy, Chugl, awa (e-mal: c@mal.cyu.edu.w). Y. J. Huag s wh he Deparme of Elecrcal Egeerg, Yua-Ze Uversy, Chugl, awa (e-mal: eeyh@saur.yzu.edu.w). Smulao resuls demosraed ha PID sldg surface provded faser respose ha ha of radoal PD-mafold coroller. I hs paper, a robus eural ewor global sldg mode PID-coroller s proposed o corol a robo mapulaor wh parameer varaos ad exeral dsurbaces. he chaerg pheomeo s elmaed by subsug a sgle-pu radal-bass-fuco (RBF) eural ewor. he weghs of hdde layers of he eural ewor are o-le updaed o compesae he sysem uceraes. Moreover, a heorecal proof of he sably ad he covergece of he proposed scheme are provded. II. ROBO MANIPULAOR MODEL A. Dyamcs of he Robo Mapulaor Cosder a -l robo mapulaor, whch aes o accou he frco forces ad dsurbaces, wh he equao of moo gve by [], C( ) + G( + = τ, () + d where q R s he o agular poso vecor of he robo mapulaor; τ R s he appled o orques; R s he era marx; C( ) R s he effec of Corols ad cerfugal forces; G( R R d s he gravaoal orques; ad s he vecor of geeralzed pu due o dsurbaces. B. Properes of he Robo Mapulaor Propery. he era marx M( s symmerc ad posve defe ad sasfes m I mi, q R, () where m ad m are posve cosa, ad I R dey marx. Propery. he Corols ad cerfugal marx sasfes s he C ( ) C( ) ζ q, R, () c where ζ c s a posve cosa, ad ( ) s he Eucldea

2 orm. Propery. he gravy erm s bouded as G(, q R, (4) where g b s a ow posve fuco of q. g b Propery 4. Usg a proper defo of he marx C ( ), he M & ( C( ) s sew-symmerc ad sasfes x [ M& ( C( )] x =, x R. (5) III. NEURAL NEWORK GLOBAL SLIDING MODE CONROLLER A. Defo of Sldg Fuco Le he racg error vecor be q d e = q q d, e R, (6) where s he desred raecory. he sldg fuco s defed as σ ( ) = e& + Λe + Λ ed β ( ), (7) where Λ ad Λ are cosa posve defe dagoal marces. Now we defe β () as β () = σ () exp( α), where α > ad σ () s he al value of sldg fuco. he choce of β () should sasfy: () β ( ) = e& () + Λ e (), () β ( ) as, ad () & β () exs ad s bouded. Noably, he fuco β () drves sysem saes ay sae space drecly o he sldg mode whou a reachg phase. I oher words, he sysem saes are ally locaed he sldg mode. If sysem saes are maaed o he surface for >, he e approaches zero ad q q d. he follow sldg codo wll be used o develop he corol law, d [ σ σ ] < d. (8) Equao (8) meas ha he dsace o he sldg surface decreases o zero eveually alog wh all sysem raecores. hus, he sysem saes are drve o he sldg surface o whch sldg mode aes place. B. Defo of Corol Ipu Le he subscrp o sad for he omal value, ad symbol Δ sad for he ucera value,.e., M = M o + ΔM, C = Co + ΔC, G = Go + ΔG. Assumpo. he uceraes of he -l robo mapulaor () ca be lumped as Δf, Δ f = ΔM ( Λe& + Λe & d ) ΔC( d Λe Λ ed) + & β ΔG d. (9) he corol pu of coveoal sldg mode corol cosss of a dscouous swchg corol par, ad causes he chaerg problem. Here we propose a sgle-pu RBF eural ewor o fd a suable ga marx o replace he swchg corol pu. Defe he sldg mode corol law as follows, based o equvale corol, τ = M o( Λe& + Λe & + & d β ) + Co( d Λe Λ ed) + Aσ K. () G o where A = dag[ a a L a ], a s posve cosa, () [ L ] K =, = W Φ σ ). () ( he ga marx K s obaed by sgle-pu RBF eural ewor. he symbol W s he m vecor of oupu layer weghs, m s he umber of odes hdde layer, ad m [ φ φ L φ ] Φ ( σ ) = s he m vecor of oupus of hdde layer odes. hey ca be chose as Gaussa-ype fuco, φ ( σ ) = γ exp( σ μ / υ ), () where μ ad υ are he ceer ad varace of he h bass fuco of he h RBF eural ewor. he ga γ s a posve cosa. Defe s he deal value of, so ha W d W = W Φ ( σ ) s he opmal compesao for Δ f d, where Δ f s he h row of Δ f. Accordg o he propery of uversal approxmao of RBF eural ewor, here exss δ > ad he followg codo sasfed, Δ f W Φ ( σ ) δ, (4) where δ s posve ad ca be chose small. C. Sably Proof d heorem. Cosder a -l robo mapulaor, as descrbed (), whch coas uow bu bouded uceraes. If () s corolled applyg he corol pu () o (), he he corol sysem () s globally sable. Proof. Choose he Lyapuov fuco caddae as V ( ) = σ ( ) σ ( ). (5)

3 he, V& ( ) = σ & σ + σ M& ( σ. (6) Usg Propery 4 ad Assumpo ad subsug corol law () o () ad Gaussa-ype fuco (), he Eq. (6) becomes V & () = a σ + σ [ Δf W Φ (σ )] = = = = a σ + σ Δf W Φ (σ ). (7) Form he propery of uversal approxmao of RBF eural ewor, assume d d Δ f W Φ ( σ ) δ ρ σ, (8) where < ρ <. he, he secod erm o he rgh sde of (7) sasfes herefore, oe ca ge d d ( σ ) ρσ σ Δ f W Φ. (7) a = V& ( ) σ + ρ σ. (8) = a Sce a s a posve cosa ad > ρ s chose, s clear ha V & ( ). (9) Equao (9) guaraees he decay of he eergy of σ as log as σ. hus, he overall sysem s sable. IV. EXAMPLE AND SIMULAION RESULS Cosder a wo-l robo mapulaor [4] as show Fg.. he parameer marces are as follows: = ) ) + s( q) s( q)( + C( ) = s( q) G( = g ( + ) ) + g 4 5 g + q ) where g s he gravaoal accelerao ad 6 = ml c + ml + I = ml c + I 6 + q, (), (), (), (), (4) = m l l, (5) c = m l, (6) 4 c, (7) 5 = ml = m l. (8) 6 c Assume ha he parameers of he uloaded robo are gve by able. he desred raecores are q d q = q d d.6.6exp( 8).8 exp( 8) =. (9).6.6exp( 8).8 exp( 8) Regardg a uow load carred by he robo as par of he secod l, he parameers m, lc ad I chage o m O + Δm, lco + Δlc, ad I O + ΔI, respecvely. Suppose ha he varao of parameers les he ervals: Δm, Δl c. 5, ad ΔI. 5. he exeral dsurbace s assumed o be d. + cos(.) =. ().5 +.7s(.) I order o acheve ha he desred respose of each o of he mapulaor beg a secod-order crcally damped respose, we choose dampg rao o be ad aural frequecy o be rad/sec. herefore, he sldg fuco 6 69 cosas are Λ = ad Λ =. he 6 69 corol pu s chose as () o (). he marx A s 5 A =. he gas of Gaussa-ype fuco are 5 γ = 8 ad γ =. he smulao resuls are show Fgs. o 5. Fgs. ad show ha boh ad q coverge o he desred q raecores. From Fgs. 4 ad 5, s obvous ha chaerg of he corol pu s elmaed by applyg he proposed mehod. V. CONCLUSION I hs paper, a robus eural ewor global sldg mode PID-coroller s proposed o corol a robo mapulaor wh parameer varaos ad exeral dsurbaces. I classcal sldg mode corol, he corol pu ga s chose o be lager ha he boud of he uceraes, whch meas he coroller has o have a pror owledge of he uceraes. he proposed mehod ca ole updae he ceer ad varace of he bass fuco of he RBF eural ewor, ad compesae he uceraes. he commo problem of pu chaerg s also elmaed ad hece he corol pu s smooh. he oher advaage of he proposed mehod s ha possesses sldg mode characerscs all he me whou a reachg phase.

4 ACKNOWLEDGMEN he auhors would le o ha Naoal Scece Coucl, awa, ROC, for supporg hs wor uder Gras NSC95--E--. REFERENCES [] V. I. U, Sldg Mode Corol ad Opmzao. New Yor: Sprger-Verlag, 99. [] J. Y. Hug, W. Gao, ad J. C. Hug, Varable srucure corol: a survey, IEEE ras. Id. Elecr., vol. 4, pp. -, 99. [] K. D. Youg, V. I. U, ad Ü. Özgüer, A corol egeer's gude o sldg mode corol, IEEE ras. Corol Sys. ech., vol. 7, pp. 8-4, 999. [4] J. J. E. Sloe ad W. L, Appled Nolear Corol. New Jersy: Prece-Hall, 99. [5] P. Gua, X. J. Lu, ad J. Z. Lu, Adapve fuzzy sldg mode corol for flexble saelle, Egeerg Appl. Ar Iell., vol. 8, pp , [6] [7] [8] [9] [] [] [] [] 5. Y. Fag,. W. S. Chow, X. D. L, Usg of a recurre eural ewor dscree sldg-mode corol, Proc. Is. Elecr. Eg. Corol heory Appl., vol. 46, pp. 84-9, 999. H. S. Cho, Y. H. Par, Y. Cho, ad M. Lee, Global sldg-mode corol: mproved desg for a brushless DC moor, IEEE Corol Sys. Mag., vol., pp. 7-5,. R. J. Maz, H. de Basa, ad F. D. Bach, VSS global performace mproveme based o AW cocep, Auomaca, vol. 4, pp. 99-, 5. Z. Wag, J. Zhag, Z. Che, ad Y. He, Neural ewor_based o adapve dscree-me global sldg mode corol scheme, Iellge Corol ad Auomao. Lecure Noes Corol ad Iformao Scece, New Yor: Sprger-Verlag, Vol. 44, 6. Y. Sepaeo, Y. Gao, ad C. Y. Su, Varable srucure corol of robos wh PID sldg surfaces, I. J. Robus Nolear ad Corol, vol. 8, pp. 79-9, 998. V. Parra-Vega, S. Armoo, Y. H. Lu, G. Hrzger, ad P. Aella, Dyamc sldg PID corol for racg of robo mapulaors: heory ad expermes, IEEE ras. Rob. ad Auo., vol. 9, pp ,. E. M. Jafarov, M. N. A. Parlac, ad Y. Isefaopulos, A ew varable srucure PID-coroller desg for robo mapulaors, IEEE ras. Corol Sys. ech., vol., pp. -, 5. M. W. Spog, O he robus corol of robo mapulaors, IEEE ras. Auo. Corol, vol. 7, pp , 99. [4] K. K. Shyu, P. H. Chu, ad L. J. Shag, Corol of rgd robo mapulaors va combao of adapve sldg mode corol ad compesaed verse dyamcs approach, IEE Porc-Corol heory Appl., vol. 4, 8-88, 996. able. Parameers of he robo mapulaor m m o l l o l c l co I I o / 5/ Jo agle (Rad) Jo agle (Rad) m q l l c l Fg.. wo-l robo mapulaor q q d me(s) m q Fg.. he respose of q ad desred pah q d q q d me(s) Fg.. he respose of q ad desred pah q d l c

5 5 orque of L (N - M) me(s) Fg.4. he corol pu τ 8 6 orque of L (N - M) me(s) Fg.5. he corol pu τ