Position Control of 2-Link SCARA Robot by using Internal Model Control

Transcription

1 Mmors of th Faculty of Er, Okayama Uvrsty, Vol, pp 9-, Jauary 9 Posto Cotrol of -Lk SCARA Robot by us Itral Modl Cotrol Shya AKAMASU Dvso of Elctroc ad Iformato Systm Er Graduat School of Natural Scc ad choloy Okayama Uvrsty --, sushma-naka Okayama, 7-8 Masam KONISHI ad Ju IMAI Dvso of Idustral Iovato Scc Graduat School of Natural Scc ad choloy Okayama Uvrsty --, sushma-naka Okayama, 7-8 (Rcvd Novmbr, 8) I ths papr, th cotrolld tart s th SCARA robot wth two lks, ad th objct s f cotrol of th arm had posto of th robot o atta th objct, Itral Modl Cotrol () s troducd A olar quatos ar for robot dyamcs formulatd by solv Lara quato, ad s larzd to ds cotrol systm by h cotrollr of s dsd or sythsstd as th vrs systm of th larzd modl, ad fltr modl s slctd Also, rfrc fltr s troducd to mak th mprovmt of prformac h rsult of cotrol prformac by s compard wth that of umrcally, accuracy ad cohrcy ar cofrmd INRODUCION I ral, SCARA(Slctv Complac Assmbly Robot Arm) robot, s a typ of horzotal drv, ad s usd as th arramt of parts to a prtd wr board ad a product assmbly, ad s usually cotrolld by compsator to atta a xact mov I cotroll th SCARA robot, th arm had s movd by motors ad th systm s a MIMO (Multpl-Iput Multpl-Output) systm, so ach lk s rqurd ts accuracy ad cohrcy I ral, as of cotrollr ar dtrmd throuh tral ad rror procsss by sklld xprts, v thouh th adjustmt of as follow to altratos of modl rrors ad frctos ar dffcult I ths study, th cotrolld tart s th SCARA robot wth two lks, ad th arm had of th robot s trd to b cotrolld by us (Itral Modl E-mal:akamatsu@ctrlcokayama-uacjp Cotrol),6 I ral, cotrollr of modl s mad by th vrs of th tart systm modl, ad by us th modl cotrol systm prformac s dvlopd h dffrc th output of th tart ad th modl s fdbackd ad th output of th tart s ood prformac by rflct th dffrc to th cotrollr A olar quato s formulatd solv Lara quato ad th t s larzd,, I ths rasrch, th cotrollr s costructd by vrs systm wthout solv quatos for cotrollr 6 fltr s a slctd modl, but prformac s poor oly by strutt fltr o dvlop th prformac, a rfrc fltr s costructd hrouh ths mprovmt of systm, th adjustmt a bcoms to b asly dtrmd Ds s cofrmd th ffcts of th rfrc fltr for drct drv motor by th xprmt 7 h cotrol prformac by s compard wth that of umrcally, th accuracy ad th cohrcy ar cofrmd hs work s subjctd to copyrht All rhts ar rsrvd by ths author/authors 9

2 Shya AKAMASU t al MEMFACENGOKAUNI Vol SCARA ROBO MODEL I ths scto, a olar quato about th SCARA robot wth two lks s formulatd by solv Lara quato, ad s larzd to ds cotrol systm by Lara Equato of Moto for SCARA Robot If t assums that th arm s homoous, th dm = ρdx ad th dsty s ρ = m /l h momt of rta aroud vrtcal axs to lk, I, ad momt of rta aroud vrtcal axs to lk,i, throuh that lk d ar ad as follows I = l l r dm = ρ l l x dx = ρl (7) I = I m l = m l m l = m l (8) Smlarly, I s calculatd by y θ lk I = m l (9) lk θ motor l Cosdr momt of rta of ach motor, th ktc ry fucto s v by l = m l θ I θ I θ J θ J θ motor F I SCARA robot modl F shows a modl of SCARA robot wth two lks I ths modl, paramtrs ar dfd as show abl, whr th ctr of ravty s v at th ctr of a lk to smplfy th calculato, l s a dstac btw lk d to th ravty pot θ s a al btw y axs ad lk abl Paramtrs m mass of lk k l lth of lk m I momt of rta of lk km f vscous frctoal coffct of shaft of th motor km /s J momt of rta of motor km From F, coordat of th ctr of ravty for lks ar xprssd as follows x = l s θ () y = l cos θ () x = l s θ l s θ () y = l cos θ l cos θ () Also, coordat of potso of arm ar as follows x = l s θ l s θ () y = l cos θ l cos θ (6) x m {l θ l θ l l cos (θ θ ) θ θ } () h pottal ry fucto U s lctd bcaus SCARA robot movs oly o x y pla So, th pottal ry fucto s U = () Bsds, th dsspatv ry fucto D s xprssd by cosdr frcto f as follows D = f θ f ( θ θ ) () By ass ths ry fuctos to Lara quato (), ralzd coordat θ s by quato () ad ralzd forc s v by quato (), th olar quato s formulatd as quato (6) d dt θ d U dt θ U D θ θ θ = τ () θ θ θ, () = τ τ () J (θ) θ D (θ) θ F θ = (6) whr, J J cos (θ θ ) J (θ) = J cos (θ θ ) J J s (θ θ ) D (θ) = J s (θ θ ) f f f F = (7) f f

3 Jauary 9 Posto Cotrol of -Lk SCARA Robot by us Itral Modl Cotrol I quato (7), J ar trms about mass, lth ad momt of rta xcpt al matrx J D J = (m m )l I h (8) J = m l l (9) J = J () J = m l I h () Drvato of th Larzd Equato h ad quato s a olar quato, so t s t abl to drv trasfr fucto hrfor, th olar quato s larzd ad us t th cotrollr s dsd for systm Also, systm, larzd trasfr fucto s usd to ds cotrol systm as ts modl P At frst, torqu s cosdrd A pottal ry s zro SCARA robot, so th torqu to mata thw ravty at qulbrum s ot cssary = () Nxt, ach qulbrum posto s cosdrd Hr, th quato (6) ot smlarts btw matrx J ad D, all θ s cossts of θ θ If al of two lks s sam as follows, θ = θ () th sd of troomtrcal fucto θ θ bcoms th troomtrcal fucto s always or hat s, wh two lks ar paralll, t s abl to larz wthout cosdr larzd als By putt th paramtr as follow X = θ θ () th lar quato wth ordr s ad Ẋ = AX BU () Y = CX (6) J = J J J J, F = F, E = (8) h trasfr fucto matrx P s ad by follow quato k (s a), k (s b) P (s) = s(s α)(s β) k (s b), k (s c) whr, all pols ad zros ar (9) α β a b c > () DESIGNING CONROL SYSEM I ths scto, cotrol systm s dsd -stps h frst s ds cotrollr, scod s fltr systm, ad th last s th rfrc fltr Systm Cotrollr h frst stp s to ds cotrollr Q(s) F shows costructo of systm I systm, th U Q P P ~ F systm cotollr s dtrmd by put typ U ad tart systm P as follows Q = P M W {W Y P A U M} U M () whr, E A = J ( ) F B = J ( ) C = ( ) (7) Hr, { } M dscrbs mmum phas fucto ad { } A dscrbs allpass fucto A symbol {} dscrbs omtt fracto that clud pols aftr partal fracto xpaso Howvr, ral cotrollr s usd vrs systm of modl P Ivrc matrx of modl s P J s(s c) J s(s b) (s) = () J s(s b) J s(s a)

4 Shya AKAMASU t al MEMFACENGOKAUNI Vol Accord to systm, codto for Q has ot ustabl pols trasfr fucto h quato () has oly zros, but t has ot pol So, quato () satsfs th codto for cotrollr hrfor, cotrollr s xprssd as follow U F r Q P P ~ P ~ Y Q(s) P (s) () Fltr h scod stp s th ds of fltr F (s) For robustss Q has to b aumtd by low-pass fltr F (s) fltr s formd to rmovd rfluc of os ad rfluc of modl rror I prcpl both th structur ad th paramtrs of f(s) should b dtrmd such that btw prformac ad robustss bcom a optmal comproms o smplfy th ds w fx th fltr structur ad sarch ovr a small umbr of fltr paramtrs, usually just o, to obta dsrd robustss charactrstcs F (s) s xprssd as follows f (s) F (s) = () f (s) ad trasfr fucto f(s) s slctd as follow f (s) = λ s (λ s ) =, () whr,, s th ordr of fltr, s slctd lar ouh to mak cotrollr Q propr, ad λ s th adjustabl paramtr Q(s) = QF (6) If λ s a lar umbr, cotrol systm s rachd mor robustss, but f λ s small umbr, cotrol systm bcoms poor prformac Bcaus th quato () all lmts ar cosst of two ordr, th cotrollr bcoms propr = = (7) I ds of th cotrollr for MIMO systm s dtrmd by r-outr factorzato, so th cotrollr s uquss Howvr, ths papr th cotrollr s dtrmd by vrs of modl wthout routr factorzato, bcaus of vrtbl modl Rfrc Fltr By abov two stps, th ds of cotrol systm s compltd I thrd stp, rfrc fltr F r (s) s addd F systm wth rfrc fltr to mprov prformac F h rfrc fltr s low-pass fltr ad s formd as follows f r (s) F r (s) = (8) f r (s) hr, trasfr fucto f rl (s) s f r (s) = r s (9) whr, rl s th adjustabl paramtr ad s dtrmd as follws f r (s) = r s = λ s () ths form domator s th sam as umrator of quato (), ad by ths form prformac s xpctd I F, by troducto F R trasfr fucto G(s) s xpssd as follows G(s) = QP F F R QF (P P ) () I quato (), f t assums that th modl s prfct about tart systm, th quato () bcom G(s) = F F R f f r (s) = f f r (s) () ths assumpto s ot ralstc, but t prformd a approxmato to costatato rfrc advata of fltr by ths assumpto G(s) s cossts of fltr ad rfrc fltr quato (), th t s ad as follow G(s) = (λ s ) (λ s ) () I quato (), ths systm fally dpds o oly λ, bcaus s dtrmd by modl hrfor th structur s smplcty ad b abl to adjust asly by us systm wth rfrc fltr

5 Jauary 9 Posto Cotrol of -Lk SCARA Robot by us Itral Modl Cotrol NUMERICAL SIMULAION Numrcal smulato s prformd by us th valus of th paramtrs abl abl valus of paramtrs smulato paramtr valu m 8 k m 6 k l 7 m l 6 m f km /s f km /s h km h km Effct of fltr h smulato rsult s show Furs ad F shows th rsult that of systm wth fltr ad wthout fltr h rsult of systm wthout fltr ovrshoots, but th rsult of systm wth fltr for prformac was mprovd F As a rsult prformac bcoms ood by addto fltr F ( =,, λ =, λ = ) F shows th rsult wh tm costat of fltr was chad From abov l, R < λ =, R = λ =, R > λ = 6, I F, ood prformac s cofrmd by R = λ l a tmsc > λ R = λ R < λ F Effcts of rffrc fltr abl valus of paramtrs cotrollr R λ K p K K d u u Comparso of wth Ga paramtrs of ad cotrollr ar usd th valus abl I abl, u xprssd th cotrollr of th motor As stp put rfrc al put sc, θ = d,θ = d h rsult of comparso to cotrol s show F 6 9 F 6,7 show th rsult for rfrc al ad F 8,9 show th arm posto I F 6, th rsult of θ by us cotrollr s vbrat about d, but th rsult of θ by rmas d, ad prformac of quck sttl tm s ad l a tmsc wth rfrcf ltr wthout rfrcf ltr d θ F Effct of rffrc fltr tmsc F 6 Comparso for lk al θ

6 Shya AKAMASU t al MEMFACENGOKAUNI Vol 6 d θ m y tmsc tmsc F 7 Comparso for lk al θ F 9 Comparso for arm posto y m x tmsc F 8 Comparso for arm posto x CONCLUSIONS systm wth rfrc fltr s usd to atta ood prformac ds th cotrollr of th SCARA robot wth two lks Ad th th cotrol s dtrmd asly by smpl mar h cotrol prformac by s compard wth that of umrcally, ad th suprorty of s cofrmd REFERENCES KYoda, subouch ad HOhkuma : h frst robot crato ds, Koudasya Sctfc() Evahlos Zafrou ad Mafrd Morar : Robust Procss Cotrol Prmtc-Hall(998) Mta ad KOsuka : Itroducto to robot cotrol r, Coroa Publsh(989), 8- SKawamura : Itroducto to robot cotrol r, Ohmsha(997), 7-9 Fujshro : Nw dto Physcs, okyokyoakusha(99), - 6 MOhshma : Procss cotrol systm, Coroa Publsh(), - 7 SAkamatsu : Ds cotrol systm for Drct Drv motor by us Itral Modl Cotrol, Okayama Uv(6)