Higher-order Adaptive Iterative Control for Uncertain Robot Manipulators *

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1 Proceeigs of he 6 h Chiese Corol Coferece July Zhagiaie Hua Chia Higher-orer Aapive Ieraive Corol for Ucerai Robo aipulaors * Qua Qua Wag Xihua Cai Kaiyua School of Auoaio Sciece a Elecrical Egieerig Beihag Uiversiy Beiig 83 P. R. Chia E-ail: qq_buaa@asee.buaa.eu.c Absrac: his paper preses higher-orer aapive ieraive learig corol for raecory rackig of ucerai robo aipulaors. he propose corol schees have bee give rigorous proof of covergece uer soe assupios. he schees are base upo he use of a proporioal-erivaive (PD) feeback srucure for which a ieraive er is ae o cope wih he ukow paraeers a isurbaces. Higher-orer aapive ieraive learig corol has poeial o give a beer covergece perforace ha he firs-orer schee algorihs because of usig pas syse corol iforaio fro ore ha oe pas ieraive cycle. he effeciveess of he propose eho is show hrough uerical siulaio resuls. Key Wors: Aapive ieraive corol Robo aipulaors INRODUCION he corol of robo aipulaors has arace a grea eal of aeios ue o heir coplex yaics a wie applicaios i iusrial syses. he robo aipulaors are geerally use i repeiive asks (e.g. auooive aufacurig iusries). herefore i is ieresig o ake avaage of he fac ha he referece raecory is repeae over a give operaio ie. Recely here is a ieres i cobiig aapive corol a ILC echiques ogeher o solve rackig probles of robo aipulaors. he ieraive aapive coroller cosiss of hree uis: feeback corol ui oliear copesaio a learig corol ui soeies he las wo ers ui are copose as oliear copesaio which has ieraive characer. We usually use PD coroller as feeback corol ui. I fac a PD coroller wih precise graviy copesaio is able o globally asypoically sabilie he oi posiios of rigi robo aipulaors a a give se-poi []. he graviy copesaio er coul reuce he corol loa fro he feeback corol ers a keep he reasoable sall. Bu he precise copesaio is o easy o saisfy i pracical because he graviy ers geerally epe o he ukow a possibly ie-varyig payloas aipulae by he robo urig a give ask. Wihou copesaig for he graviy forces he PD corol schee leas o a seay-sae error which ca eveually be reuce by icreasig he proporioal a erivaive gais or by iroucig a iegral acio. he rawback of he high-gai feeback soluio is relae o he fac ha i ay saurae he oi acuaors or/a excie high-frequecy oes. O he oher ha wih he PID corol schee oly local asypoic sabiliy was prove uer soe relaively coplex coiios uil he iroucio of he passiviy propery for robo aipulaors which allowe o esig globally asypoically sabiliig PID corollers wihou graviy copesaio []. * his work is suppore by he Naioal Naural Sciece Fouaio of Chia (Gra No: 64746) a he 863 Prograe of Chia (Gra No: 6AAZ74). 85 I s well kow ha higher-orer schee algorihs will iprove he rasie learig behavior i he ieraio oai. oreover os of he propose higher-orer schee algorihs are show o be ore robus for he easuree oise ha firs-orer schee algorihs. I is also ieresig o oe ha a he 5h IFAC Worl Cogress o Auoaio a Corol a special sessio was evoe o higher-orer ieraive corol; e.g. [3]-[4]. Furherore i is iscusse i [] ha lower orer ILC algorihs are show o ouperfor (spee of covergece) HO-ILC i he sese of ie weighe or. However he resuls i [5] are oly applicable are oly applicable o sigle-sigle sigle-oupu (SISO) liear ie-ivaria (LI) syses i absece of easuree errors. I [6] i is show ha HO-ILC coul be use o reuce he variace of he effec of easuree oise. I [7] [8] he oliear copesaio is esiae i he ieraio oai wih firs-orer schee algorihs. So we ca expec ha he higher-orer aapive ieraive corol is capable of givig beer perforace ha he raiioal aapive ieraive corol. I his paper we shall cosier he proble of higher-orer aapive ieraive corol by raecory rackig of ucerai roboic aipulaors. Copare wih exisig resuls he ai coribuios of his paper are: ) Higher-orer schee algorihs are firsly applie o aapive ieraive corol a have bee give rigorous proof of covergece. ) he evelope approach is show ha i ca ehace he covergece rae copare wih raiioal aapive ieraive corol via uerical siulaio resuls. PROBLE FORULAION Usig he Lagrage forulaio he equaios of oio of a egrees-of-freeo rigi aipulaor ay be expresse by D( q( ) ) q( ) B( q( ) q ( ) ) q ( ) () F q = ( ()) () a

2 Where eoes he ie a he oegaive ieger Z eoes he operaio or ieraio uber. he sigals q R q R q R are he oi posiio oi velociy a oi acceleraio vecors respecively a he ieraio. D( q ) R is he ieria arix B( q q ) q R cerifugal forces. F ( q ) is a vecor resulig fro Coriolis a is he vecor resulig fro he graviaioal forces. R is he corol ipu vecor coaiig he orques a forces o be applie a each oi. a R is he vecor coaiig he uoele yaics a oher ukow exeral isurbaces. Assuig ha he oi posiios a he oi velociies are available for feeback our obecive is o esig guaraeeig he boueess of a corol law q ( ) [ ] a Z a he covergece of q ( ) o he esire referece raecory [ ] whe es o ifiiy. q for all We assue ha all he syse paraeers are ukow a we ake he followig reasoable assupios: (A) he referece raecory a is firs a seco q q q as well as ie-erivaive aely he isurbace ( ) are boue [ ] a a Z. (A)he reseig coiio is saisfie i.e. q = q q = q.we will also ee he followig properies which are coo o robo aipulaors. (P) D( q ) R is syeric boue a posiive efiie. D q B q q is skew syeric (P) he arix ( ) hece ( D( q) B( q q) ) = R (P3) B( q q) ξ( ) F( q) = Ψ ( q q) p( ) ( ) Ψ ( q q ) R a ξ ( ). where respecively p R vecor over [ ]. (P4) If q < q < is efie as [7]. he ( ) is a ukow coiuous B q q <. 3 HIGHER-ODER ADAPIVE CONROL DEVELOPEN Defie he filer rackig error i he ieraio oai a he h cycle as Where q = q q () = () α () i i q q () is he rackig error. he filer rackig error i he ieraio oai uses pas syse corol iforaio fro ore ha oe pas ieraive cycle. We will eglec ie soeies for brief i he ex. Lea If ρ = αi < a li a i = < Z he li q a q < Z Proof ( ) eoes he k h row of k. [ ] q q = α k k i ik We obai q k ( ) αi q ik ( ) k ( ) If li he [ ] ( ) li. α < sowecaco- he coefficies of αi ee i clue li q ( ) for [ ] k obai li q k Z k is a kow arix a vecor i i. We ca easily q α q ρ q where q = ax ( q ) q ρ ( ) also we have q q q = ax q q ρ ρ q q Expaig he righ of above iequaliy uil q < ρ he q ρ q ρ where = ax ( ) Z a <. So we ca coclue q < Z. heore Cosier () wih properies (P P3) uer he followig corol law: = Q Q Φp (3) wih p = p ΓΦ (4) Where pk = q k = k Z Q R Q R a Γ R are syeric posiive efiie. he arix Φ = Ψ q q sg where ξ = q α q i ( ) (P3) a p ( p ) i i = he coefficies of αi ee ρ = αi < he q < q < < for all i = Z a li q = li q =. Reark ( q q) Ψ coais he iforaio of forer cycle eawhile also akes avaages of forepasse iforaio. So hese schee algorihs are calle higher-orer aapive ieraive corol. 86

3 Reark ξ = q α q is a kow vecor a he i i h cycle so (P3) ca be use. Proof: Applyig he (P3) io () he we have D q q B q q q F q ( ) a ( ( ) ξ ) Ψ ( ) B q q F q = q q p Le ξ = q α q hewewillhave i i D q q B q q q q D q q ( ) α Ψ ( ) i i a = q q p Subsiuig (3) io (6) we he obai D q q B q q D q q (5) (6) ( ) ( ) a Q Q Ψ ( q q) p sg( ) p = p p. If we efie U = D( q) = (7) where αiq i q a he (7) becoes D( q ) = U Ψ ( q q ) p ( B( q q ) Q ) (8) Q sg ( ) As a oivaio o geerae a learig algorih le s efie a Lyapuov fucio caiae V( ) as D( q) Q.he he erivaive V alog he error raecory is of V D q D q Q = = ( Ψ ( ) ( ( ) ) Q sg ( ) ) D Q = U Ψ ( q q) p ( B( q q ) Q ) D sg( ) U q q p B q q Q Usig (P) he above equaio becoes V = U Ψ q q p Q sg ( ) Iegraig boh sie of he above equaio we obai V = V ( ) ( ) Ψ U q q p Q sg Usig (A) (P) a Reark 3 we will ge U = D( q) αiq i q a ( ()) ( ) ( Φ ) Where V V p Q (9) p = p p.le us cosier he followig Lyapuov like coposie eergy fucio: W() = V( () ) p Γ p W is give by W() W () = V( () ) p Γ p V ( () ) p Γ p () = V V he ifferece of ( ()) () Δp Γ Δ p Δp Γ p Where Δ p = p p ( p ) ( p ) p p = = Applyig (4) io above equaio p = p ΓΦ he Δ p = p p = ΓΦ W( ) W ( ) = V( ( ) ) V ( ( ) ) ΓΦ Γ ΓΦ ΓΦ Γ p = V( () ) V ( () ) Φ ΓΦ Φ p Usig (9) we obai W W () ( ) ( Φ ) V V p Q Φ ΓΦ Φ p = V ( () ) V( ) ( ) Q Φ ΓΦ Because qk = qk ( ) k Z he ( ) V V = usig(a).wecaobai W() W () ( ) Q Φ ΓΦ Because Q Φ ΓΦ W W. > he he followig proof is siilar o [7] [8] we coul obai < < < for all k Z a li = li =. Fially we ca easily ge li q = li q = a for all Z applyig Lea. Reark 3: If q i < we ca coclue usig q D ( B q F a ) q < 87

4 (A) (P) (P4) a q < q < <. αiq i < ca be go ex ie. he followig proof coul use iucio. A he begiig q i = < Reark 4: Leα > = Q Q Φp α α where αiq i p = p ΓΦ α =. So we cawrie he corol law as = Q Q Φ p i i p = p ΓΦ where = α q.i he case of firs-orer aapive ieraive corol he coefficies areα = α = = α =.he coefficies ee 4 SIULAION α i α < he siulaio of wo egrees of freeo plaar aipulaor wih revolue ois are sae as [7].he iffere par is he arix Φ where φ = q q α q q φ = q q q si i ( i i = αiq si q φ q q αiq si q i = i I orer o copare he wo schees we choose = a α =.7 for firs-orer aapive ieraive corol. α =.7 α =. α =. for higher-orer aapive ieraive corol a α α α α <.he resuls show i Fig. a Fig.. Fig. Sup-or of he rackig error of lik versus he uber of ieraios for firs-orer a higher-orer aapive ieraive corol Nuerical siulaio resuls have show ha higher-orer aapive ieraive learig corol give a beer covergece perforace ha he firs-orer schee algorihs because of usig pas syse corol iforaio fro ore ha oe pas ieraive cycle. I pracical applicaios he aipulaor ois are equippe wih icreeal ecoers o easure he oi posiios. We ca use ocausal FIR filer o filer oise of q q backgrou a he ieraio a he pracice is easier a he perforace is beer ha ha of filerig oise of q real-ie. he eho coul reuce he variace of he effec of easuree oise. 5 CONCLUSION his paper exes he exisig aapive ieraive corol o higher-orer aapive ieraive corol. I has bee give he proof of covergece a eawhile ol ha how o se he coefficies i he higher-orer aapive ieraive corol law o esure he covergece. he eho has bee propose for he posiio rackig proble of rigi robo aipulaors wih ukow paraeers a subec o exeral isurbaces. he evelope approach is show ha i ca ehace he covergece rae copare wih raiioal aapive ieraive corol via uerical siulaio resuls. Bu i is ore coplex a which orally requires uch copuaioal power a eory for he geeraio of real-ie corol ipu. Foruaely he CPU power a eory becoe less of proble hese ays as fas processors becoe available a low price. oreover ay copuaioally-efficie algorihs are available which ca be use o iplee he aapive learig corol srucure. Fig. Sup-or of he rackig error of lik versus he uber of ieraios for firs-orer a higher-orer aapive ieraive corol 88 REFERENCES [] Arioo S Corol heory of No-liear echaical Syses[]. Oxfor: Oxfor Sciece Publicaios 996. [] oore K L Che Y. O he oooic covergece of higher orer ieraive learig upae laws. Proc. 5 h IFAC rieial Worl Cogress. [3] Pha Q Loga R W. Higher-orer ieraive learig corol by pole placee a oise filerig. Proc. 5 h IFAC rieial Worl Cogress. [4] Al-owai S Lewi P Rogers E. Higher orer ILC ver-

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