Robut adaptv uro-fuzzy cotrollr for hybrd poto/forc cotrol of robot mapulator cotact wth ukow vromt Arah Faa ad Mohammad Farrokh * Ira Uvrty of Scc ad chology, Dpartmt of Elctrcal Egrg, Narmak, hra 6844, Ira l.: +982 7808022; Fa: +982 7454055; E-mal: farrokh@ut.ac.r * Corrpodg author Abtract - I th papr, a adaptv cotrol mthod for hybrd poto/forc cotrol of robot mapulator, bad o uro-fuzzy modlg, prtd. Sc th forc cotrol volv applyg crta amout of forc o th urfac of a objct, t mportat to codr th frcto forc btw th d-ffctor ad th urfac to accout. I ordr to compat th frcto forc, a robut ad adaptv uro-fuzzy compator wll b dgd ad corporatd to th clo-loop ytm. Morovr, to dtrm tff coffct of th urfac, a o-l tmator wll b dgd for mor prc computato of th drd forc. Du to th adaptv uro-fuzzy modlg, th propod cotrollr dpdt of th robot dyamc, c th fr paramtr of th uro-fuzzy cotrollr ar adaptvly updatd to cop wth th chag th ytm ad th vromt. A a rult, th trackg rror, both for poto ad forc, wll alway rma mall. Alo, th tablty of th cotrollr guaratd, c th adaptato law bad o Lyapuov thory. I addto to that, th covrgc of th adaptv paramtr wll b provd th papr. h mulato rult how good prformac of th propod cotrollr a compard wth othr covtoal cotrol chm for robot mapulator uch a computd torqu mthod. Kyword: robot hybrd forc/poto cotrol adaptv cotrol uro-fuzzy cotrol urfac frcto compator. Itroducto Nuro-fuzzy cotrollr hav b uccfully appld th pat two dcad by may rarchr to olar ytm. h motvato for th utlzato tm from two fudamtal proprt of uro-fuzzy ytm: ) th ablty to corporat th kowldg of prt to a cotrollr a fuzzy IF-HEN rul, whch mak th dg of th cotrollr dpdt of th ytm dyamc or at lat thr o d for act mathmatcal modl of th ytm, ad 2) th uqu proprty of th ural twork, whch calld trag, to adapt th fr paramtr of th cotrollr to th chag th ytm paramtr ad th vromt, whch mak th cotrollr mor robut. Wh th d-ffctor of a robot mapulator movg fr pac, th ma goal to cotrol th arm of robot uch that t follow th drd trajctory. O th othr had, wh robot mut hold ad mov a objct, or wh th d-ffctor mut com cotact wth a urfac wth frcto ad ukow tff, th thr mut b forc cotrol addto to th poto cotrol of th d-ffctor. h tak wll kow a th poto/forc cotrol of robot mapulator. wo forc cotrol mthod hav b tvly tudd by may rarchr th pat two dcad: ) th hybrd poto/forc cotrol mthod [] ad 2) th mpdac cotrol mthod [2]. I both mthod a prc cotrol chm cary ordr to mov th d-ffctor o th rght path, rtg th corrct amout of forc o th objct. If th forc ot cotrolld wth crta dgr of prco, thr thr wll b o cotact btw th d-ffctor ad th objct, or th objct ad th arm of th robot wll b damagd. Hc, th poto/forc cotrol of robot mapulator rqur accurat cotrollr, pcally wh th tff of th urfac ukow advac, whch commo mot applcato uch a grdg, poudg, polhg, dburrg ad twtg.
I rct yar, tllgt mthod ad algorthm, lk ural twork ad fuzzy logc hav b mployd for poto/forc cotrol of mapulator. Hu ad Fu [3], [4] ad [5], hav propod a adaptv fuzzy hybrd poto/forc cotrollr, whch th adaptv paramtr ar th ctr of th mmbrhp fucto th tally part of th fuzzy IF-HEN rul, but th frcto forc ha ot b tak to accout. Kguch ad Fukuda [6] ad [7], ad Kguch t al. [8] hav alo ud a uro-fuzzy mthod for poto/forc cotrol, but wth a mpl ad tatc modl for th frcto btw th d-ffctor ad th urfac. h frcto compato ha b prformd wth o uro, who output multpld to a wght, whch th coffct of th tatc frcto of th urfac. h wght ot trad adaptvly. Alo, Kguch ad Fukuda [9] hav ud th uro-fuzzy mthod to cotrol a 2DoF robot mapulator, aumg that th tff coffct of th urfac kow ad th frcto forc glgbl. h propod cotrollr [0] by Xao t al. ha th advatag of hadlg th poto/forc cotrol a ucalbratd vromt. But, th cotrollr d a optcal or ad a vual ytm to obrv th codto of th urfac. I may applcato, uch a grdg ad dburrg, du to th dut ad udrd partcl th vromt, th output of th vual ytm ca b oy ad mprc, hc hamprg th tak. I th papr a adaptv uro-fuzzy cotrollr wth a adaptv uro-fuzzy frcto compator propod for hybrd poto/forc cotrol of robot mapulator. hak to th adaptv urofuzzy modlg, both for th cotrollr ad th frcto compator, th propod mthod dpdt of th robot dyamc a wll a th codto of th vromt. h du to th fact that th fr paramtr of th uro-fuzzy cotrollr a wll a th uro-fuzzy compator ar adaptvly updatd to th chag th ytm ad th vromt. h ma advatag of th propod cotrol mthod that th adaptato law bad o th Lyapuov tablty thory, whch guarat th tablty of th cotrollr. Although th mulato ar prformd o a robot mapulator wth thr dgr of frdom wth rvolut jot, th propod cotrollr ca b tdd to robot mapulator wth mor dgr of frdom ad dffrt kd of jot. Morovr, th tructur of th cotrollr ad th compator vry mpl, makg t a vry fat ad approprat mthod for dffrt applcato of robot mapulator. h mulato rult how good prformac of th propod cotrollr a compard wth othr covtoal cotrol mthod uch a computd torqu mthod. 2. Mapulator Dyamc ad th Modl of Cotact Surfac 2- Mapulator Dyamc h dyamc quato of a robot mapulator ca b prd Carta pac a []: M ( +C(,)+G )&& & ( ) +D() & =f f f () whr C(,) D() vctor to mov th d-ffctor, th poto vctor of th d-ffctor, M( ) th rta matr, & th ctrfugal ad corol forc vctor, G() th gravty forc vctor, & th vctor for jot frcto forc of th robot arm, f th rqurd forc f th appld forc vctor to th urfac by th d- ffctor, ad f th forc vctor for th urfac frcto Carta coordat. Equato () ca alo b wrtt a M ( +C )&& (,)+G & & ( ) +D() & =f f f (2) m whr C m. Fg. how th chmatc dagram of a 3DoF robot mapulator, cotg of wat, houldr ad lbow. h poto of th d-ffctor th Carta coordat ca b calculatd trm of jot agl a
( ( )) ( ( )) θ ( θ θ ) = l coθ + l co θ + θ coθ 2 2 3 2 3 = l coθ + l co θ + θ θ 2 2 2 3 2 3 = l + l + 3 2 2 3 2 3 (3) 3 l 3 θ 3 l 2 l θ θ 2 2 Fg. - Schmatc dagram of a 3DoF robot mapulator Carta coordat Dffrtatg Eq. (3) w.r.t. tm ad wrtg vctor-matr form, w hav & = J( θθ ) & (4) whr J( θ ) calld th Jaccoba matr, whch ca b ud to fd th rlatohp btw th jot-torqu vctor ad th d-ffctor forc vctor a f=j ( θ) τ (5) Alo, th followg quato hold btw th matrc ad vctor Carta coordat ad jot-agl coordat, ug th Jaccoba matr M() =J M( θ) J ( C(θ, θ) & +G(θ) +D(θ)-MJ & JJ & ) & & & C(, ) +G() +D() =J whr M(θ), C(θ,θ) &, G(θ), ad D(θ) & ar th quvalt of M ( ), C(,) &, G( ), ad D() &, jot-agl coordat, rpctvly, ad ca b prd for a 3DoF robot mapulator a 2 2 2 2 2 2 2 2 mllcc + mlc + ml + ml c + ml c 0 0 3 3 2 2 23 3 2 2 3 3 23 2 2 2 3 3 3 2 2 2 2 M(θ) = 0 mllc + ml + ml + ml mllc + ml 3 3 2 3 3 3 3 2 2 2 3 2 3 3 3 3 3 3 2 3 2 2 0 mllc + ml ml 3 3 2 3 3 3 3 3 2 3 3 C(θ, θ) & = θ& θ& + ω θ + θ + θ& + θ& + θ& + θ& θ + θ& θ 3 3 2 2 2 [ ml ( )2( ) mll ( c ( ) c ) ml 2 ml 2 ] 3 3 2 3 2 3 3 3 2 2 2 23 2 3 2 23 3 2 2 2 2 2 2 2 2 2 2 2 2 2 mll θ& (2 θ& + θ& ) + m ( l θ& 2( θ + θ ) + ll θ& (2 θ + θ ) + l θ& 2 θ ) + mlθ& 2 2θ 3 3 2 3 2 3 3 3 3 2 3 3 2 2 3 2 2 2 2 2 2 2 3 6 mll && + m l & + + ll & & + & + ll & c 2 2 3 2 2 2 θθ 3 3 2 3 2 3 3 ( θ 3 2( θ θ 2 3 ) θ 3 2 2 ( θ θ 2 3 ) θ 3 3 2 2 23 ) (6)
0 m ( lc + lc ) + mlc 3 3 23 2 2 2 2 2 G(θ) = 2 2, 3 3 23 mlc 2 D(θ) & 2θ& + g( θ& ) = 2θ + g( θ ) 2 2 & & 2θ& + g( θ& ) 3 3 whr l, m,ad θ ( =,2,3) ar th lgth, wght ad agular poto of jot, rpctvly; ad c = coθ, = θ, c = co( θ + θ ), ad = ( θ + θ ). j j j j (7) 2-2 h Charactrtc of th Cotact Surfac I ordr to charactrz th urfac, whch com cotact wth th d-ffctor, two mathmatcal modl ar codrd hr. h frt o th drcto of th forc prpdcular to th urfac ad th cod o th drcto of movmt, tagt to th urfac. h prpdcular modl dcrb th tff dgr of th urfac, ad ud for forc cotrol. h tagt modl pr th frcto proprt of th urfac ad ud for forc a wll a poto cotrol of th d-ffctor. 2-2- h Prpdcular Modl h mathmatcal modl th prpdcular drcto of th cotact urfac a prg modl (Fg. 2) [8], [3], [4], [5]. If th tral appld forc to th urfac how wth f, th f = k( ) (8) whr k th tff coffct of th urfac, N/m, ad ad ar th poto of th d-ffctor ad th urfac, rpctvly (Fg. 2). f k tff Fg. 2- h prpdcular modl of th urfac Accordg to Fg. 2, f th tral forc f appld to th urfac wth tff coffct k, th hap of th urfac wll b dformd form of a hol. h hghr th tff of th urfac, th mallr th dpth of th ptrato wll b. For tac, 0 N forc appld to a urfac wth k = 0000 N/m ca ptrat 0.00 m. I th papr t aumd that th urfac of th objct l o 2 pla, hc = 0. 2-2-2 h agt Modl Wh a objct movg o a urfac, thr wll b frcto forc btw th objct ad th urfac. h forc du to th appld prpdcular forc to th urfac ad th oppot drcto of th movmt. Fg. 3 how th affctg forc appld to a objct, movg o a urfac. I th Fg., f τ ad f τ 2 ar movg forc alog ad 2 a, rpctvly. h frcto forc f ad f 2 alway act th oppot drcto of th movg forc o th urfac.
hr ar may modl avalabl th ltratur to rprt frcto forc. Boa ad Idr [6] hav how that a comprhv mathmatcal modl for frcto forc f ca b gv trm of appld prpdcular forc f a v 2 v f 0 β β = γ γ γ 2 ( ) + + f g( v) (9) whr v th pd of th objct, ( γ 0 + γ) f th tcto frcto, ( γ 0 + γ 2) f th dyamc frcto at hghr pd, ( γ = α f, = 0,,2), β how th dampg rat of frcto at lowr pd ad alo a dcato of gatv lop th tcto rgo (Fg. 4), ad β 2 pcf th acdg rat of frcto at hghr pd. Fg. 4 how th urfac frcto forc vru th pd of th objct [7], [8] ad [9]. I th Fg., thr a gatv lop at lowr pd, whch calld th Strbck ffct. If th ffct ot compatd, thr could b tablty aroud th org of th frcto forc-pd charactrtc [6]. I poto/forc cotrol of robot mapulator, th oppog urfac frcto forc mut b compatd appropratly; othrw t could crat dffcult cotrol proc, pcally poto cotrol, ad yt could lad to vry larg tady-tat rror or v ca bcom utabl. 3 f Strbck ffct Effct of vcou frcto hrhold of Coulomb frcto f f S 2 v f S2 f τ M f τ 2 Fg. 3. Appld forc to a objct movg o a urfac Stcto (tatc) frcto Fg. 4. h complt modl for frcto wth Strbck ffct 2-3 Surfac Coffct Etmato Uually, th urfac frcto coffct a ukow paramtr ad hould b tmatd wth good accuracy; othrw t could lad to larg rror forc appld to th urfac or v tablty. Sc th frcto ca chag wth tm, th bt way to tmat t to mploy a ol mthod [24]. h rlatohp btw th appld forc to th urfac ad th amout of th dpth of th ptrato ca b prd a (Fg. 2): f k = (0) ( ) Hr, f ca b maurd by a forc or locatd at th d-ffctor ad ( ) ca b calculatd by traformg th agular jot poto to th Carta pac. Sc th obtad k ot th actual valu of th urfac coffct, thrfor, th tmatd valu of th paramtr how wth k ˆ Fg. 5. I th papr t aumd that th d-ffctor alway cotact wth th urfac ad th forc maurmt at th d-ffctor accurat ough, avodg zro tmato of th urfac tff coffct.
3- Adaptv Nuro-Fuzzy Cotrollr h goal th papr to cotrol poto ad forc of a robot mapulator wth 3DoF, wth th d-ffctor movg o a urfac th 2 pla (Fg. ), whl applyg forc th gatv drcto of th 3 a (.. prpdcular to th 2 pla). h cotrol acto prformd th Carta pac = [ 2 3], but practc th robot movmt th jot pac θ = [ θ θ2 θ 3]. hrfor, th Carta pac ud for cotrollr dg, ad th ug τ =Jf, whr J th Jaccoba matr, th cotrol forc f ca b covrtd to th cotrol torqu τ to apply to th robot arm. Hc, th tak of th cotrollr to mov th d-ffctor alog th d d d 2 d 3. Sc th forc appld to th urfac alog th 3 drd path = [ ] ( ) 3 3 a, th f = k (0) Alo, c w aum that th urfac of th objct l o th 2 pla, th 3 = 0 ad w hav [ f k ] 2 / hrfor, th drd path ad drd forc ca b dfd a = () = [ f k ], = [ ] d d d 2 d/ f d 0 0 f d (2) h rror btw th drd ad th actual vctor for th poto, vlocty ad acclrato of th d-ffctor fd f = d = d d 2 2 k, & = & d &, && = && d && (3) Now, lt df ldg vctor a =& + Λ (4) Λ = dag λ,..., λ a dagoal ad potv dft matr [20]. Dffrtatg whr { } (4) w.r.t. tm ad calculatg && trm of ug (2), w hav M(-Λ)=M & & && d +Cm& d -C m(-λ)+g +D +f +f -f (5) or M=M( & && d +Λ)+C & m( & d +Λ)-Cm+G +D +f +f -f (6) h, lctg th cotrol law a f=m( && +Λ)+C & (& +Λ)+G +D +f +f +K (7) whr d m d K a potv dft matr, ad ubttutg (7) to (6) yld M+(C & m +K)=0 (8) Eq. (8) how that ug cotrol law (7), th clod-loop quato a frt ordr ytm. Morovr, c th coffct (8) ar ot zro, th ldg vctor mut approach zro ordr to atfy (8). But th ma obtacl ralzg th cotrol law that matrc G,C, m,m ad D (7) ar uually udfd ad ukow. Som of th trm of th cotrol law (7) ca b wrtt a [2] Y(,,,)φ & & =M( && +Λ)+C & (& +Λ)+G +D (9) d d d m d r r whr φ a vctor of th ukow paramtr ad Y(,,,) & & d d a rgro matr ad kow. Vctor φ ca b codrd a th adaptv paramtr, whch hould b adjutd uch a way to guarat th tablty ad th mmum rror of th clod loop ytm. Eq. (9), whch th olar part of cotrol law (7), ca b wrtt ug robut cotrol mthod a [5] Y(,,,)φ & & =h(,) = M( && +Λ)+C & (& +Λ)+G +D (20) d d d m d
whr th orm of th poto vctor, th ldg vctor dfd (4), ad h(,) a ukow vctor who lmt ca b dfd a whr r η (, ) ad cotrol law (7) bcom h(,)=w η (, ) (2) * * w th optmal valu of r w. Hc, quato (6) ad th * * * 2 2 2 3 3 3 m M= & w η (, ) w η (, ) w η (, ) -C +f +f -f (22) * * * f= w η (, ) w2 η 2(, 2) w3 η 3(, 3) +f +f +K (23) I fact, th olar part of th rct quato ha b gv a th product of η(,) ad w, whr w = [ w w2 w 3 ] th adaptv paramtr of th cotrollr ad mut b adaptvly trad a o-l to mmz th trackg rror, to cop wth th chag th ytm paramtr ad th vromt, ad to mata th tablty of th clod-loop ytm. Sc th propod cotrollr th papr of uro-fuzzy typ, thrfor, w codrd a th wght vctor of th uro-fuzzy twork, whch trad o-l ad adaptvly to prform th abov thr tak. Alo, aothr adaptv uro-fuzzy tmator, who dg wll b plad th t cto, wll compat th frcto forc. h block dagram of th propod mthod th papr ha b how Fg. 5. h cotrollr block cot of thr ubcotrollr, o for ach jot of th robot. hr ar alo o block for tmato ad compato of urfac frcto forc ad aothr block for o-l tmato of tff coffct of th urfac. Sc th coffct uually ot oly ukow but alo varyg, t mportat to tmat t wth good accuracy. Alo, t hould b mtod that or, locatd at th jot of th robot, maur th agl ad th agular vlocty of jot. hrfor, th quatt d to b covrtd to poto ad vlocty of th d-ffctor Carta pac ug Eq. (3) ad (4). Each ubcotrollr ha two put, (.. th orm of th poto vctor Carta coordat) ad (th th compot, =, 2, 3, of th ldg vctor ). h tral cocto of ach ubcotrollr ar how Fg. 6. h fuzzy rul of thr ubcotrollr ca b dfd a 2 2 2 R j j :IF A j ad A j HEN y B j j (24) j j2 whr A ad A ( =,2,3, j =, K, l, j2 =, K, l2) ar th j th ad th j th 2 fuzzy t jj2 dfd for th fuzzy put varabl ad, th th ubcotrollr, rpctvly, ad B ar th fuzzy t for th tally part of th fuzzy rul. Ug th Mamda product frc g, glto fuzzfr, ad ctr avrag dfuzzfr, th output of th th ubcotrollr ca b calculatd a f Nf (, ) = w η = l l 2 j= j2= l l 2 j= j2= µ ( ) µ ( ) j j A A 2 j j A A 2 µ ( ) µ ( ) y jj2 jj 2 jj2 whr y th ctr of th mmbrhp fucto dfd for fuzzy t B (24). h adaptv paramtr of th ubcotrollr ar th ctr of th mmbrhp fucto jj2 jj dfd for fuzzy t B (.. y 2 ), whch ar th lmt of w th adaptv cotrol law (23). hrfor th thr ubcotrollr mut b trad uch a way that th olar fucto w η (, ) ( =, 2, 3) ar tmatd wth good accuracy. I othr word, w d a adaptato law for chagg th fr paramtr of th ubcotrollr (.. (25) jj y 2 ) a o-l faho ubjct
to: ) th tablty of th clo-loop ytm guaratd at all tm, 2) th trackg rror for poto ad forc rma boudd ad mall, v for larg chag th ytm paramtr ad th vromt. h adaptato law bad o Lyapuov tablty thory ad wll b drvd cto 5. I th t cto, a adaptv uro-fuzzy compator wll b dvlopd for urfac frcto. Adaptato Law for Cotrollr f d δ d d 2 d 3 h Prpdcular Modl of th Surfac d/dt + Σ Eq. (4) + & Σ S 2 3 Cot. Cot. 2 Cot. 3 f f 2 f 3 Adaptv Nuro-Fuzzy Cotrollr f f + + Σ + f f f f =f c J Surfac Frcto t θ & θ f Robot f Cotact Pot Btw th Edffctor ad th Surfac Objct Forc Sor k ˆ Etmator for Stff Coffct of th Cotact Surfac 3 S Adaptato Law for Surfac Frcto Drct Kmatc Adaptv Nuro- Fuzzy Compator for Surfac Frcto & J Fg. 5- Block dagram of th propod adaptv uro-fuzzy cotrollr for hybrd poto/forc cotrol of a 3DoF robot mapulator alog wth frcto forc compator ad th tmator for tff coffct of th cotact urfac Iput Layr Fuzzfr Layr Fuzzy Rul Layr Π Dfuzzfr Layr Output Layr Π Π w r Π Π Π Σ / Π f Nf Π Π Π Fg. 6- h uro-fuzzy tructur of th adaptv hybrd poto/forc cotrol
4- Adaptv Nuro-Fuzzy Compator for Surfac Frcto Wh th tp of th d-ffctor mov o th urfac of a objct, th urfac frcto forc affct th prformac of th cotrollr ad dtr th corrct proc of th robot mapulator. Hc, t mportat to compat th frcto proprly; othrw thr wll b larg tady-tat rror. Morovr, th ytm mght bcom utabl for urfac wth larg frcto coffct. h frcto forc affctd by th movmt of th d-ffctor o th urfac ad caot b drctly maurd. h modl gv cto 2 hav ucrtat ad thr paramtr ar ukow practc. Morovr, th modl may ot prcly df th urfac frcto. hrfor, ordr to compat th frcto forc, o mut fd a good tmat of th forc. Svral mthod hav b propod ltratur for urfac frcto forc compato. O mthod to tak advatag of hgh ga cotrollr. h mthod compat th rqurd forc to om dgr, but hgh ga ca cau problm lk utablty ad lack of provdg ough forc by rvomotor. Amrat t al. hav propod a robut compator, whch compat frcto to om good dgr, but du to adaptablty atur of th mthod, t caot work wll wh frcto forc crad. Alo, It ha ot b mtod how to dtrm th paramtr of th compator [2]. h othr mthod to u torqu or vry jot of th robot to maur th actual torqu ad comparg thm wth th drd torqu (output of th cotrollr). h dffrc th amout of torqu coumd for urfac frcto [22]. h mthod d torqu or vry jot of th robot. h propod mthod th papr to tmat th urfac frcto forc ug a adaptv uro-fuzzy tmator. h fr paramtr of th tmator ar updatd adaptvly ug th trackg rror utl th frcto forc compltly compatd. Sc th urfac frcto forc drctly proportoal to th appld forc to th urfac ad ha a olar rlatohp wth th vlocty of th d-ffctor (Fg. 4), th approprat put to th propod tmator hould b th vlocty of jot Carta pac. h, ordr to calculat th rqurd forc for frcto compato, th output of th uro-fuzzy compator wll b multpld to th prpdcular forc, appld to th urfac. hrfor, thr compator mut b dgd, o for vry drcto. h fuzzy rul of ubcompator ar dfd a j j j R :IF & A HEN y B (26) whr & ( =,2,3) th th compot of th vlocty vctor of th d-ffctor Carta coordat, A ad B j ( =,2,3, j =, K, l) ar th jth mmbrhp fucto of th put ad j th output for th th ubcompator, rpctvly, ad y th output of th th ubcompator, whch mut b multpld to f ad addd to th output of th uro-fuzzy cotrollr obtad form th quato th prvou cto. Ug glto fuzzfr, product frc g, ad ctr avrag dfuzzfr, th output of th th uro-fuzzy ubcompator cab b prd a y = l j = l µ j = A j µ A ( & ) j ( & ) j whr y th ctr of th jth mmbrhp fucto of th HEN part of th fuzzy rul (26) ad alo th fr paramtr of th uro-fuzzy compator. Hc, h compatd forc for urfac frcto th th drcto f = y f (28) ˆ whr f ˆ th tmato of th th compot for urfac frcto ad f th prpdcular forc, maurd by forc or, ad appld to th urfac of th objct. Eq. (27) ca b wrtt th followg vctor otato form y j (27)
whr y = θ ξ ( & ) (29) l θ = y L y a vctor cotag th fr paramtr of th th uro-fuzzy ubcompator, ξ ( & ) = [ µ ( & ) L µ ( & )] µ ( & ), ad l th umbr of fuzzy rul. l l j A A A j = hrfor, th tmatd forc for urfac frcto compato ˆ ˆ [ ( ) ( ) 0] f =f(& θ) = θξ & f θξ & f (30) l 2 2 2 whr θ= 0 θ θ 2. h thrd compot of th frcto forc zro, bcau thr ar o drd movmt th drcto. Hc, th cotrol law gv Eq. (23) ca b wrtt a * f=w η +f +f( ˆ & θ)+k (3) * * * * whr w = w w2 w 3. hrfor, th fr paramtr th cotrollr, w, ad th compator, θ, mut b appropratly adjutd durg oprato of robot to guarat tablty of th clod-loop ytm ad to ur mall trackg rror for poto ad forc a wll. Nt cto gv th propod algorthm ad th tablty aaly of that, whch ar th ma cotrbuto of th papr. 5- Adaptato Algorthm for Cotrollr ad Compator hr ar two t of paramtr, w ad θ, whch mut b trad durg oprato of robot to * * rach thr optmal valu w ad θ, rpctvly. hrfor, th rror for ubcotrollr ad ubcompator ca b dfd a * ψ =w -w, =,2,3 (32) * φ =θ -θ A a rult, thr ar thr t of rror th whol ytm, amly ψ = [ ψ ψ2 ψ 3 ], φ= [ φ φ2 φ 3], ad = ( & d & ) + Λ( d ), whch ca b corporatd to o rror fucto quadratc form, whch our caddat for th Lyapuov fucto V = M + ψ Γ ψ + φφ φ (33) 2 2 2 whr =3 for a 3DoF robot, { } = = r r M th am a Eq. (), ad Γ = dag {, K, } γ γ ad l l Φ = dag φ, K, φ ar potv dft matrc, whch r ad l ar th umbr of r fuzzy rul th th ubcotrollr ad th ubcompator, rpctvly. Dffrtatg (33) w.r.t. tm yld V& = M+ & M & + ψ Γ ψ& + φφ φ& (34) 2 = = h, ug Eq. (22) gv ( * * * V& = [ w η (, ) w2 η 2(, 2) w3 η 3(, 3) ] +f +f -f) + (M& -2C m ) 2 (35) ψ Γ ψ& φφ φ& = = + + Sc M& -2Cm a kw ymmtrc matr, w hav (M& -2C m )=0 (36) Subttutg f from (3) to (35) rult r
w η (, ) w η (, ) V& = w η (, ) w η (, ) + f f(& θ) K + ψ Γ ψ& + φφ φ& (37) * * ˆ 2 2 2 2 2 2 * = = w3 η 3(, 3) w3η 3(, 3) Now, lt th rror btw th tmatd frcto forc ˆ * f(& θ ) whr ε R. h rror ca alo b wrtt a ˆ * ad t actual valu f() & b ε=f()-f( & & θ ) (38) ˆ * f(& θ ) f( ˆ & θ) = [ φξ ( & ) f φξ ( & ) f 0] (39) 2 2 2 hrfor, combg Eq. (38) ad (39) yld ˆ [ f()-f( & & θ)= φξ ( & ) f φξ ( & ) f 0] +ε (40) 2 2 2 Ug (40) (37) gv * w η (, ) wη (, ) * V& = w2 η 2(, 2) w2η 2(, 2) +ε K + ψ Γ ψ& + φφ φ& * = = 3 3 3 3 3 w η (, ) w η (, 3) Now, lt or = = = - K+ ε+ ψ Γ ψ& + ψ η (, ) + φφ φ& + φξ ( & ) f ψ Γ ψ& + ψ η (, ) = 0 & φξ & f φφ φ + ( ) = 0 ( & ) ( & & f ) ψ Γ ψ + η (, ) = 0 φ Φ φ + ξ ( ) = 0 =, K, =, K, whch yld ψ & = Γη(, ) =, K, (44) φ& = f Φξ ( & ) * * Sc ψ & =w& -w& ad φ =θ& -θ& * &, ad alo c at th optmal pot w & = 0 ad θ & * = 0, t ca b cocludd w & = Γη(, ) =, K, (45) θ& = f Φξ ( & ) Itgratg th quato gv th adaptv algorthm for updatg th fr paramtr of th uro-fuzzy cotrollr ad th uro-fuzzy compator t2 w ( t ) = w ( t ) + Γη(, ) dt 2 t t2 θ ( t ) = θ ( t ) + f Φξ( & ) dt 2 t =, K, Ug th quato, th paramtr ar updatd utl thy rach thr optmal valu ordr to brg th poto ad forc rror to mma. O mportat pot that quato (46) ar dpdt of th robot dyamc ad th urfac frcto. I th followg, t wll b provd that adaptato law gv (46) ca guarat th tablty of th clod loop ytm udr crta codto. (4) (42) (43) (46) t. horm : If (46) hold t ad f ε (38) boudd, th th clod loop ytm tabl
Proof: Sc (46) drctly dductd from (44), hc, by ubttutg (44) to (4) t yld Now, c Γ Γ = = V& = - K + ε + ψ Γ Γη + ψ η + f φφ Φξ + f φξ (47) Γ ad Φ ( =, K, ) ar dagoal matrc, th thr vr t ad = I ad Φ Φ whch yld = I. hrfor = = V& = - K + ε + ψ η + ψ η + f φξ + φξ (48) & (49) V = -K+ ε Sc ε boudd, th matr K (7) ca b cho a a dagoal ad potv dft matr wth larg lmt a compard to th lmt of ε uch that K > ε (50) hrfor, V & < 0 ad th clod loop ytm tabl, whch coclud th proof. Corollary : h adaptato law, gv (46) ha th followg proprt w( tk+ ) < w( tk) =, K, (5) θ( tk+ ) < θ( tk) tk, k =,2, K whr w( tk+ ) = w( tk+ ) w( tk) =, K, (52) θ( tk+ ) = θ( tk+ ) θ( tk) tk, k =,2, K Proof: Wthout lo of gralty, lt tk = 0 ad t k = t. Accordg to thorm, th adaptato law (46) yld a tabl clod loop ytm t. Hc, accordg to th ldg cotrol thory (Slot, 99) th ldg vctor =& + Λ (4) mut dcra aymptotcally. hat lm () t 0 (53) t Coqutly, th tgral trm of (46) go to zro Hc, a k Or quvaltly whch mut b tru oly f h complt th proof. t 0 0 lm Γη(, ) dt 0 t t lm f Φξ ( & ) dt 0 t w ( t ) w ( t ) k+ k θ ( t ) θ ( t ) k+ k lm w ( t ) 0 k k+ lm θ ( t ) 0 k k+ w( tk+ ) < w( tk) =, K, θ( tk+ ) < θ( tk) tk, k =,2, K Rmark: I proof of horm, t wa aumd that th lmt of dagoal matr Kar larg ough uch that qualty (50) hold for dffrt valu of ε. h mght crat om dffcult th cotrol proc, pcally wh th drd forc too larg, whch mght crat larg (54) (55) (56) (57)
ovrhoot. Morovr, wh th tat of th ytm ar clo to thr drd valu, but th dffrc btw th drd forc ad th actual forc too hgh (.. wh th d-ffctor vry clo to th drd pot but ha t touch t yt) th lmt of matr K mut b trmly hgh, whch could lad to larg chattrg aroud th ldg urfac. o ovrcom th problm, th followg robut cotrol law ha b ud th papr (Sug, 988): wη(, ) f= ˆ w2η2(, 2) +f +f( & θ ) +K+ Qg( ) (58) 3 3( 3) w η, whr Q = dag { q,..., q } a potv dft matr. Now, th drvatv of th Lyapuov fucto bcom V & = K+ ε Qg( ) (59) Lt df th lmt of matr Q to b q ε, =,..., (60) h, V & < 0, ad V & = 0 oly f = 0, whch ma that ug cotrol law (58), th clod-loop ytm aymptotcally tabl. Morovr, th cotrol law ca avod larg chattrg aroud th ldg urfac. 6- Smulato h mmbrhp fucto for th put ad th output varabl of ubcotrollr ar how Fg. 7 ad 8, rpctvly. h adaptv paramtr for th cotrollr (.. th lmt of vctor jj w ) ar th ctr of th mmbrhp fucto how Fg. 8 (.. y 2 (25)). Alo, th fuzzy rul for ubcotrollr hav b gv abl. hrfor, ach ubcotrollr ha oly 9 rul wth oly 9 adaptv paramtr, whch mut b trad adaptvly to mmz th trackg rror. h mmbrhp fucto for th put ad th output varabl of th ubcompator for urfac frcto ar how Fg. 9 ad 0, rpctvly. h adaptv paramtr of th compator (.. th lmt of vctorθ) ar th ctr of th mmbrhp fucto how Fg. 0 (.. y j (27)). Alo, th fuzzy rul for ubcompator hav b gv abl 2. A th tabl how, th compator ha oly 7 rul wth oly 7 adaptv paramtr, whch mut b updatd adaptvly for a bttr tmato of th drd urfac frcto. Fg. 7- h mmbrhp fucto for put varabl ad for th ubcotrollr. P=Potv, PM=Potv Mdum, PB=Potv Bg, N=Ngatv, Z=Zro, P=Potv.
Fg. 8- h mmbrhp fucto for th output varabl ubcotrollr Fg. 9- h mmbrhp fucto for th put varabl & ad & 2 of th urfac-frcto compator. NB=Ngatv Bg, NM=Ngatv Mdum, N=Ngatv, Z=Zro, P=Potv, PM=Potv Mdum, PB=Potv Bg. Fg. 0- h mmbrhp fucto for th output varabl of th urfac-frcto compator abl : Fuzzy rul for th ubcotrollr N P PM PB M9 M8 M7 Z M6 M5 M4 P M3 M2 M
abl 2: Fuzzy rul for th ubcompator - IF & NB, HEN y NB 2- IF & NM, HEN y NM 3- IF & N, HEN y N 4- IF & Z, HEN y Z 5- IF & P, HEN y P 6- IF & PM, HEN y PM 7- IF & PB, HEN y PB Although th dgd cotrollr th papr adaptv, but for fatr covrgc, th tal valu of th adaptv paramtr (th ctr of mmbrhp fucto for th tally part of th fuzzy rul) ar dfd ug pror kowldg of th ytm, whch o of may advatag of fuzzy ytm. h 3DoF robot mapulator, ud mulato, ha th followg cotat: l = 0.5m, l = 0.8m, l = 0.3m 2 3 m = 2kg, m2 = kg, m3 = kg Alo, th followg paramtr hav b cho for th uro-fuzzy cotrollr ad th uro-fuzzy compator: K=dag{30 30 30}, Λ =dag{0.4 0.4 0.4} Γ=dag{50 50 50}, Φ=dag{50 50 50}, Q =dag{0 0 0} h drd path a crcl 2 pla wth a radu of o mtr ad th ctr of th org. h d-ffctor mut rt 0 N forc o th urfac whl followg th drd path. h tal valu of th adaptv paramtr of th cotrollr ha b cho qually a follow (although t worth to mto that th prformac of th dvlopd mthod th papr dpdt of th tal valu of th adaptv paramtr): w (0) = w (0) = w (0) = 4 3 2 0 2 3 4 2 3 [ ] I ordr to how th adaptato ablty of th propod cotrollr ad th compator, mulato hav b prformd for dffrt valu of tff coffct of th urfac, urfac frcto, jot frcto ad arm ma of th robot, ad maurmt o. Fg. to 8 how th mulato rult. h chag th paramtr ar 00% of thr tal valu occurrg at t = 5. Fg. 8 how th rult for addg 0% maurmt o poto ad forc or. A th mulato rult how, th propod cotrollr prfctly capabl of cotrollg th poto ad th appld forc of th d-ffctor to th drd valu v wth larg amout of chag ytm paramtr ad th vromt. o compar th prformac of th propod cotrollr wth aothr mthod, mulato hav b prformd for th am robot wth computd torqu mthod, whch dyamc dpdt ad oadaptv []. h mulato rult hav b how Fg. 9 to 25. A th fgur how, wh thr ar o chag th ytm paramtr or th vromt, th cotrollr ca prform a cllt tak wth almot o tady tat rror. But, c th computd torqu mthod dyamc dpdt, ay chag th robot paramtr ca crat larg rror, pcally for 00% cra th arm ma t bcom utabl (Fg. 2). Morovr, du to ucompatd urfac-frcto forc, ay varato urfac frcto may v rult parato of th dffctor from th urfac (Fg. 23). I addto, Fg. 25 how that computd torqu mthod vry ucptbl to maurmt o.
Alo, Fg. 26 how th chag th paramtr of o of th adaptv uro-fuzzy ubcotrollr, for th ca of 00% cra th arm ma. A th fgur how, th tal valu of th adaptabl wght ar vry clo to th drd valu, c th wght hav b dfd ug th kowldg w hav from th ytm. Alo, a th paramtr of th ytm chag at t=5, th wght quckly adapt thmlv to th chag, matag mall amout of trackg rror both for th poto cotrol a wll a th forc. Fg. 27 dcat that th drvatv of th Lyapuov fucto, dfd Eq. (33), rma gatv for th ca of 00% cra th arm ma of th robot, mag that th ytm tabl at all tm. O mportat pot th mulato rult that although th fuzzy ytm ar uvral appromator ad ar alway aocatd wth om rror, but, du to th adaptato proprty of th cotrollr a wll a th compator, th appromato rror, hrt th fuzzy ytm, caclld out by cotuouly chagg th adaptv paramtr. Although th mulato rult how vry good ad robut prformac for th propod mthod, thr ar om lmtato that hould b addrd: ) th gular pot th robot work pac, whch ha't b codrd th mulato, hould b tak to accout, 2) th cotact urfac wa aumd to b flat ad th -2 pla, whch may ot b th gral ca practc. Evaluato of th propod cotrollr wh coutrg a uv urfac, whch th urfac frcto forc mut b dcompod to thr compot alog thr a, wll b addrd th futur work, 3) t wa aumd that th pd of th d ffctor ca b maurd wth om or. I practc, th or ar uually aocatd wth o, 4) v though, maurmt o wa codrd mulato, tmatg th pd of th d ffctor, tad of maurg t, could bft th practcal applcato of th propod mthod. - - - Drd Surfac-Stff Coffct cra by 00% - - - Drd Fg. - Poto ad forc cotrol wth 00% cra urfac-tff coffct
Fg. 2- h jot torqu for th ca of Fg. 8 - - - Drd - - - Drd Arm ma cra by 00% Fg. 3- Poto ad forc cotrol wth 00% cra arm ma
Fg. 4- h jot torqu for th ca of Fg. 0 Jot frcto cra by 00% - - - Drd - - - Drd Fg. 5- Poto ad forc cotrol wth 00% cra jot frcto - - - Drd Surfac frcto cra by 00% - - - Drd Fg. 6- Poto ad forc cotrol wth 00% cra urfac frcto
- - - Drd - - - Drd Surfac frcto cra by 00% Surfac frcto cra by 00% Fg. 7- Frcto forc ad 2 drcto (actual ad tmatd) - - - Drd - - - Drd Fg. 8- Poto ad forc cotrol wth 0% maurmt o poto ad forc or Surfac frcto cra by 00% - - - Drd - - - Drd Fg. 9- Poto ad forc cotrol wth 00% cra urfac tff coffct (computd torqu mthod)
Fg. 20- h jot torqu for th ca of Fg. 6 - - - Drd - - - Drd Arm ma cra by 00% بازوها Arm ma cra by 00% Fg. 2- Poto ad forc cotrol wth 00% cra arm ma (computd torqu mthod)
Fg. 22- h jot torqu for th ca of Fg. 8 - - - Drd - - - Drd Jot frcto cra by 00% Sparato of th d ffctor from th urfac Fg. 23- Poto ad forc cotrol wth 00% cra urfac frcto (computd torqu mthod)
Fg. 24- h jot torqu for th ca of Fg. 20 - - - Drd - - - Drd Forc ( N ) m (c ) Fg. 25- Poto ad forc cotrol wth 0% maurmt o poto ad forc or (computd torqu mthod)
Fg. 26- Adaptv wght chag of o of th cotrollr durg oprato of th robot, wh 00% cra arm ma occur at t=5. Fg. 27- Drvatv of th Lyapuov fucto, dfd Eq. (33), durg oprato of th robot, wh 00% cra arm ma occur at t=5. 7- Outl of th Futur Work h drcto of th futur work for th papr ca b outld a follow: - Prformg a robut aaly o th dgd cotrollr a wll a th urfac frcto compator. 2- Codrg gular pot th work pac, wh th robot prformg poto/forc cotrol. 3- Evaluato of th propod cotrollr wh coutrg a uv urfac, whch th urfac frcto forc mut b dcompod to thr compot alog thr a. 4- Etmatg th pd of th d ffctor, tad of maurg t, that uually aocatd wth o cratd by th pd or. 8- Cocluo A adaptv uro-fuzzy cotrollr for hybrd poto/forc cotrol of robot mapulator ad a adaptv urfac frcto compator wa prtd th papr. I addto, a o-l tmator wa dgd for adaptv computato of th tff coffct of th urfac. h propod algorthm wr dgd to b dpdt of th robot dyamc, rultg to a cdgly robut clod loop ytm. h bcau, th adaptato law of th fr paramtr bad o th Lyapuov tablty thory. h othr advatag of th propod mthod th mpl tructur of th cotrollr ad th compator. hr ar oly 9 ad 7 fuzzy rul for vry ubcotrollr ad vry ubcompator, rpctvly, makg t a vry fat ad approprat mthod for dffrt applcato of robot mapulator. Morovr, t wa provd that th adaptv paramtr ar
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