ANALYTICAL DYNAMIC MODEL AND EXPERIMENTAL ROBUST AND OPTIMAL CONTROL OF SHAPE-MEMORY-ALLOY BUNDLE ACTUATORS

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1 ANALYICAL DYNAMIC MODEL AND EXPERIMENAL ROBUS AND OPIMAL CONROL OF SHAPE-MEMORY-ALLOY BUNDLE ACUAORS Chuho Joseph Lee R & D Plig Geerl Moors M/C: Mou Ro Box 9 Wrre MI 489 Cosios Mvroiis Roboics Mechroics Lborory Deprme o Mechicl Aerospce Egieerig Rugers Uiversiy he Se Uiversiy o New Jersey 98 Bre Ro Piscwy NJ ABSRAC I his pper he lyicl ymic moel erivio he robus opiml posiio corol o Shpe Memory Alloy (SMA) bule cuors usig he LQR H echiques re presee. SMA bule cuors compose o muliple SMA wires plce i prllel hve bee recely propose s mes o cosierbly icrese he liig cpbiliies o SMA cuors. Robus opiml lier corollers coul provie he esire robuse i he perormce o hese olier highly sesiive cuors combie wih he simpliciy o hese corol schemes. he ovel coribuios o he prese reserch re: ) he erivio o geeric lierize ime-ivri lyicl sysem moel or SMA Bule cuors h is use i he esig o he LQR H bse corollers; b) he evelopme o ew improve esimor i iscree-ime H opiml corol esig bse o he Klm Filer preicor orm or use i he corol o SMA bule cuors; c) he experimel suy o wo corol esig mehos usig se-spce moels LQR H Opiml Desig i iscree-ime omi usig experimel SMA bule cuor cosisig o 48 Flexiol SMA wires ble o pply up o lbs. (44 N). As emosre i he experimes he esige corollers provie siscory resuls i ccurcy sbiliy spee.. Iroucio I my pplicios o roboic mechicl sysems such s spce explorio meicl operios eerime iusry miliry sks here is icresig ee or evelopig smll size lighweigh evices h will be ble o pply lrge orces evelop high spees chieve lrge isplcemes be highly eergy eicie. Avcig such roboic sysems i pr requires usig ew cuors sice clicl orms o cuors such s DC moors hyrulics or peumics re hevy cumbersome. Uilizig vce cuors bse o smr merils c mke poible he evelopme o iovive roboic sysems h woul sisy my o he requiremes se bove. I his reserch he key mehoology i rsiclly reucig he weigh size o roboic sysems is he use o Shpe Memory Alloy (SMA) wires s cuors o he robo jois. SMA wires such s Nickel-iium (Ni-i) wires hve he propery o shoreig whe hee hus re ble o pply orces. his pheomeo clle he Shpe Memory Eec (SME) occurs whe he meril is hee bove ceri rsiio emperure chgig is crysllie phse rom mresie o useie. Heig hus cuio o SMA wire is esily ccomplishe by pplyig volge rop cro he wire cusig curre o low hrough he meril resulig i joule heig. Ese o cuio is o he oly vge o SMA cuors. Oher vges re heir icreibly smll size weigh heir high orce o weigh rio heir low cos heir oisele operio. heir limiios iclue relively smll bwih low eergy eiciecy. Despie o hese limiios SMAs hve oe o he highes pylo o weigh rios mog "smr meril" bse cuors. hereore SMAs re oe o he ew "smr merils" h he prese ime c be use i pplicios h require smll size lrge orces rom he cuors. While esig moelig ymics o SMA cuors hve bee suie exesively very lile work hs bee oe i he re o corol. his is very iicul problem o solve or hree mi resos: ) SMA cuors prese complex herml-elecriclmechicl ymics h re iicul o moel; b) ue o heir emperure epeecy SMA cuors re very sesiive i emperure chges; c) ue o he lexible chrcerisics o SMA cuors subsil vibrios c be excie whe hese cuors re use o power he jois o roboic sysems. Corollers or SMA cuors ee o be robus i sysem eviromel chges moelig errors. hey lso ee o hve vibrio suppreio chrcerisics. I iio he corollers ee o be ble o hle boh posiio orce corol sks be simple i implemeio. Achievig ccure robus perormce o SMA cuors is very impor sice i will llow heir use i my impor pplicios. Hshimoo ke Sgw Chib S [] pplie PD corol scheme o he SMA wires use s cuors o bipe wlkig robo. Iku sukmoo Hirose [] use cive PID corol o segmee cive eoscope me wih SMA sprigs. roisoie Biu Drio [] pplie PI corol o SMA cuors wih iiol herml sesor. Mill Wg [4] use very simple Seior Reserch Egieer ASME Member Aocie Proeor ASME Member Correspoig Auhor Copyrigh by ASME

2 Proporiol corol o veriy he SMA sysem moel hey pe iscu he sysem sbiliy. he corol gis re ue eiher olie or hrough simulios wih ril error meho. he rwbck o lier P PI or PID corol is h he coroller my perorm well i he rge where he corol gis re ue bu eeriores rmiclly oce ousie he rge. Vrious pive corol lgorihms hve bee propose or use i SMA cuors. Dickiso use pive corol o compese irecly he hyseresis o SMA whe hese cuors re use i vibrio suppreio pplicios [] [6]. Webb Wilso Lgous Reiiois [7] ook io ccou i heir pive corol lgorihm he esimio o herml chges bewee he SMA cuors he evirome clcule he compesig ipu usig esblishe SMA moel. he rwbck o his meho is h he moel he clculios re very complice hus icresig he bure or o-lie compuio i experimel implemeios. Aoher pive/olier corol o SMA is Pulse Wih Moulio (PWM) [] [8] [9] []. Gr Hywr ([] []) pplie vrible srucure/sliig moe corol mehos uer PWM o perorm orce corol o SMA cuor me or roboic eye. he corol gis were vrie o-lie. ebbe Schroeer Buler [] suie se-spce mulivrible corol i lrge lexible smr srucures cue by SMA wires. he corollers were esige usig eiher Eigevlue pole plceme or LQR mehos. he moel o he lexible srucure wih orce ipus lexible isplceme ws suie pplie i he coroller esig bu he moel h escribes he orce/curre relio bewee he SMA wires he srucure ws esblishe experimelly by ope-loop ipu/oupu igrm ws o urher suie. I his pper he lyicl ymic moel erivio he robus opiml posiio corol o SMA bule cuors usig he LQR H echiques re suie. SMA bule cuors compose o muliple SMA wires plce i prllel hve bee recely propose s mes o cosierbly icrese he liig cpbiliies o SMA cuors [4-6]. he esig o SMA bule cuors hs bee suie i [4] [] he o-lier ope loop ymics hve bee suie i [6]. Iiil corol suies usig PID corollers ehce wih expoeil ipu precoiioig ucio hve bee perorme i [4]. Due o he ihere o-lier behvior o SMA cuors heir high sesiiviy o chges i eviromel operig coiios he PID bse corollers coul o perorm well whe iere or vryig posiio ipus re use. Robus opiml lier corollers coul provie he esire robuse combie wih he simpliciy o hese corol schemes. he ovel coribuios o he prese reserch re: ) he erivio o geeric lierize ime-ivri lyicl sysem moel or SMA Bule cuors h is use i he esig o he LQR H bse corollers; b) he evelopme o ew improve esimor i iscree-ime H opiml corol esig bse o he Klm Filer preicor orm or use i he corol o SMA bule cuors; c) he experimel suy o wo corol esig mehos usig se-spce moels LQR H Opiml Desig i iscree-ime omi usig experimel SMA bule cuor cosisig o 48 Flexiol SMA wires ble o pply up o lbs. (44 N). As emosre i he experimes he esige corollers provie siscory resuls i ccurcy sbiliy spee.. Sysem Moel o Shpe Memory Alloy Wires Shpe memory lloys cosis o group o mellic merils h emosre he biliy o reur o some previously eie shpe or size whe subjece o he pproprie herml proceure. hese pheome re eie s he Shpe Memory Eecs (SME) which occur ue o emperure sre epee shi i he meril s crysllie srucure bewee wo iere phses clle Mresie Auseie. Mresie he low emperure phse is relively so wheres Auseie he high emperure phse is relively hr. he chge h occurs wihi SMA s crysllie srucure urig he SME is o hermoymiclly reversible proce. As resul emperure hyseresis occurs. his emperure hyseresis bewee Mresie rio ξ m (or Auseie rio ξ ) emperures A S A F M S M F rsles irecly io hyseresis i he sri/emperure relioship. he hyseresis behvior mkes i chllegig o evelop moelig corol schemes or SMA cuors. For give SMA he hyseresis is epee o he composiio o he lloy he mucurig procees. Mos Shpe Memory Alloys (such s ickel-iium (Ni-i)) hve hyseresis loop wih o 7.8 o 67.8 C ( o ºF) wih he excepio o some wie hyseresis lloys use or joiig pplicios such s coupligs. Aiiol iormio o he priciple o operio o shpe memory lloys c be ou i [7- ]. Developig mhemicl moel h cpures he behviors o Shpe Memory Alloy s i uergoes emperure sre phse chges is complice chllegig problem. Reserchers coiue o suy wh re he bes wys o moel corol cuors h use his uique mily o merils. As iscue erlier i is he sigiic hyseresis loop h cuses he problems. hereore his secio ocuses o evelopig ew moel which will be use o esig eebck corollers. his moel embles iere eures o severl exisig moels or SMA cuors. he mehos o esblish he mhemicl moels or SMA cuors c be sepre io wo relms: ) Usig experimel o i he relios bewee orces eormio ipus (volge or emperure chges) such s [8] []; b) Iroucig vrible ξ he Mresie (or Auseie) rio which is show i he relio o emperure chge i he hyseresis eec. he moel o Shu Lgous Hughes We [] ler pe i Dickiso [] [6] he moel o Iku sukmoo Hirose [] re i his cegory. he ew lierize moel evelope i his pper which is i he seco cegory bove iclues hree relios: ) he emperure chge isie he SMA ue o he elecric power ipu he he covecio wih he mbie evirome; b) he relio bewee he emperure o he SMA he Auseie rio (ξ ); c) he relio bewee he Mresie rio he mechicl properies which is mily bse o he sub-lyer moel by Iku e l. [] wih he iio o impor eures rom he eormio moel by Shu e l. []. HEA CONDUCION he emperure o he SMA wire is cosiere o be iepee o sri Mresie/Auseie rcio o be oly ucio o he he rser. he geerl soli he coucio equio is erive by pplyig he coservio o m he irs lw o hermoymics Fourier s lw [4]. Usig he elecricl chrge s he ipu power he equio is wrie s: ρc = i( k ) h( ) + Pe () where: ρ is he meril esiy; c is he speciic he coeicie; (xyz) is he emperure s ucio o ime () o geomery ˆ ˆ (xyz); is he grie vecor i + j + k ; h represes he he rser coeicie which epes o he surrouig luis/gs; is he mbie emperure; Pe is he elecricl power where V Pe = = i R R. V represes volge; i elecric curre; R elecric resisce. Copyrigh by ASME ˆ

3 I he irecio x is se o be log hi SMA wire he irecios y z re o he cro secio (umig circulr oe) he vriios he pril erivives y z i boh y z irecios re egligible becuse he cro secio is very smll compre wih he vriio x i x irecio. I iio he rsie emperure respose o he wire he coucio i he irecio x c be cosiere o be much ser h he he covecio. hereore Equio () c be simpliie io he ollowig orm [4]: ρcvv = ha( ) + Pe () where Vv A re he ol volume surce re o he SMA wire. MARENSIE/AUSENIE RAIO Iku e l. [] esblishe he vrible sub-lyer moel. I c be ume h here re wo mi phses (Mresie Auseie) exisig i he SMA. A exr phse (Rhoboherl phse or R- phse) exi lso i bewee bu mos o he ime i c be eglece [4]. his is umpio h is ope i his pper. he wo mi phses c be cosiere s beig wo prllel sub-lyers coece eve hough he phses re isribue romly i he bulk mel. he mcroscopic view o he wo lyers cig s composie meril wih wo iere properies is show i Figure. Whe he Mresie/Auseie rio chges he cro seciol res he volumes o he wo phses chge ccorigly. FIGURE : Vrible Sub-lyer Moel o SMA. I he igure bove ξ m ξ represe Mresie Auseie rios σ m σ σ eoe srees o Mresie Auseie ol sre respecively. he relio o he wo rios re simply: ξm + ξ = () he Mresie rio ξ m ecreses rom oe o zero whe he emperure rises rom he m-phse sr emperure M s o he iish emperure M. O he oher h he Auseie rio ξ icreses rom o whe he emperure rises. All he relios c be wrie eiher i erms o ξ m or ξ he Auseie rio is use i his work. I Iku e l. [] he Auseie rio ξ (or Mresie rio ξ m ) is expoeil ucio o emperure bu his rio c be simpliie urher. Usig he hyseresis loop he rio ξ s ucio o emperure c be pproxime i wo lier relios: i heig proce A (4) i coolig proce ξ ξ s As = A A A As As s M s M = M M M s M M s SRESS/SRAIN RELAION I oe-imesiol eormio Shu e l. [] pplie he geerlize Hooke s lw which hs he orm: (6) e E E[ ( )] where: σ ε e ε ε re he uixil sre elsic sri ol sri rsormio sri respecively; is he curre emperure is he mbie emperure (or he emperure wihou y () heig/coolig proce); E is he Youg s Moulus; α is he coeicie o herml expsio. Boh E α re ume o be ucios o ξ urig he rsormio proce hve he relio: A M A E = E + ξ ( E E ); A M A α = α + ξ( α α ) (7) where E A α A E M α M re he meril properies pure Auseie Mresie ses respecively. he re o chge o he rsormio sri is ume o be proporiol o he re o chge o he Mresie rcio: ε =Λξ (8) where Λ i oe imesio is eie o be: H( σ ) ξ > H( σ ) ξ > σ σ (9) Λ= H( ε ) ξ < H( ε ) ξ < ε ε Equio (9) provies he irecios i which he rsormio sris evelop. Prmeer H is he mximum xil rsormio sri (ε ). From his equio i c be show h i he SMA is uer esio oly Λ is posiive cos equl o H or boh ξ> ξ<. he ol sri ε is eie s: LSMA LSMA ε = + H L SMA () i which L SMA is he curre legh o he SMA wire L is he SMA origil legh o he wire. Accorig o he oe-imesiol eormio moel i Shu e l. [] he ol sri ε o he eormio cois he elsic sri ε e he rsormio sri ε he herml expsio. he herml expsio c be eglece becuse i is - orer smller h ε. hereore he ol sri c be wrie s: e ε = ε + ε () he rsormio sri is ucio o he Mresie rio s i is show i Equio (8). Wihou losig geerliy he re-chge relio (8) c be rewrie s: ε =Λ ξm =Λ( ξ) () I he proce is oly uer esio Λ is equl o H where H is he mximum xil rsormio sri (presri) i is equl o pproximely %. he elsic sri c be rele o sre usig Equio (6). I c be represee i he Auseie rio s: e M A e σ = Eε = ( E ξm + E ξ) ε () M A M e = [ E + ( E E ) ξ ] ε. Discree-ime Robus Opiml Corol Desig I his pper he iscree ime Lier Quric Regulor (LQR) [] he H robus opiml corol [6] esig echiques will be use or he corol o SMA cuors. I his secio boh mehos re presee briely. A ovel eure presee here is he evelopme o improve esimor i iscree-ime H opiml corol esig bse o he Klm Filer preicor orm. Boh LQR H mehos re usig he seprio heory o eermie he se eebck corol gi mrix K esimor gi mrix L iepeely. he iscree-ime eebck corol loop is escribe usig he iscreize lier ymicl sysem i Equio (4) x = Ax + + Bu y = Cx + Du (4) where i geerl x is he (p ) se vecor; u is he (r ) ipu vecor; y is (q ) oupu vecor; A is (p p) mrix; B is (p r) mrix; C is (q p) mrix; D is (q r) mrix. he subscrip represes ech ime sep. Copyrigh by ASME

4 A ech ime sep he corol ipu u is clcule bse o he ull se vecor x he corol mrix K (r p): u = Kx () Becuse he ull se spce vecor x co be irecly rom he sesors ise ech ime sep he esime se spce vecor x ˆ is clcule usig he y he esimor mrix L (p q): xˆ = A xˆ + B u + L( y ( C ( A xˆ + B u ) + D u )) (6) + + I ymic sysem is require o ollow reerece or ph y he coroller esig becomes servo problem. A sey se ( or ) he sysem oupu y is equl o he reerece y. Give he sysem equios i iscree-ime omi wih reerece y he sysem Equio (4) sey se c be rewrie s: x = A x + Bu y = Cx + Du (7) where he coss x u re he se vecor ipu vecor sey se respecively. I is esire o i x u wih y give rom Equio (7): x A I B (8) u = C D y where I is he ieiy mrix. I orer o mke he sey-se Equio (7) he origil sysem Equio (4) o oe regulor sysem Equio ew se o sysem equios ech ierio is esblishe by subrcig he wo equios (4) (7). Wih he iroucio o ew se vecor x ew ipu vecor u ew oupu vecor y he sysem o equios becomes: x + = Ax + Bu y = Cx + Du (9) where x = x x u = u u y = y y LINEAR QUADRAIC REGULAOR AND KALMAN ESIMAOR MEHOD Boh LQR Klm esimor require opimizio or which he Discree-ime Algebric Rici equios re compose []. I he sysem i Equio (4) (A B ) is corollble he esig crierio or he bes corol gi K is o miimize errors rom oupus (or se vribles) eergy cosumpio o ipus. he corol gi K is erive s: K = ( B X B + ( D D + R) ) ( B X A + D QC ) () K K where boh Q R re weighig mrices. he ocie Discree Algebric Ricci Equio (DARE) is: AX K KAK XK ( AX K KBK + SK)( BX K KBK + RK) ( BX K KAK + SK) + QK = AK = A BK = B SK = CQD RK = D D + R QK = CQC () I he sysem (A C ) is observble he Klm esimor gi L c be clcule s [] L = X LC ( CX LC + DwRwmDw) () Where R wp R wm re he covrices o proce me oises ccorigly. Wih he DARE: AX A X AXC CXC + DR D CX A + BR B = () ( ) L L L L w wm w L w wp w H OPIMAL ROBUS CONROL DESIGN MEHOD While H opiml robus corol esig i coiuous ime sysem hs bee use requely is implemeio i iscree ime sysems hs bee give le eio. he implemeio o H corol o iscree ime sysem is presee i his subsecio he ull erivio c be ou i Lee [7]. he robuse o he coroller is eermie by kig io ccou he isurbces o he sysem. he isurbces o ipus w /or oupus v re weighe iegre io he sysem o equios coiuous-ime omi. he sysem is iscreize is wrie s: x = Ax + B w + + Bu z= C x+ D w+ D u y= Cx+ Dw+ Du (4) w= { w } v where z is he oupu crierio eie by users vecor w represes ll he weighe isurbces. For H opiml robus corol meho he H orm (orm i Hry spce) is use o orm he cos ucio or opimizio. he H orm o Lier ime Ivri sysems i iscree ime omi is eie s i [6]: * * * * G() z = rce { D D+ B LoB } = rce { DD + CL cc } A B () Gz () = RH C D where L o L c re he corollbiliy observbiliy Grmis respecively. Similrly o he LQR/Klm Esimor meho he seprio heory is pplie so h opiml gis bse o ull-se eebck opiml gis bse o Klm se esimio re clcule seprely he combie ogeher wihou losig sbiliy. he objecive ucios or boh gi mrices K L re ormule usig he H orm rom isurbce w o se vecor oupu z. I he sysem mrices (A B ) re corollble he corol gi mrix K o ull-se eebck hs he orm [7]: ( ) K = B XKB + DD ( B XKA + DC) (6) he correspoig Discree-ime Algebric Rici Equio is AKXKAK XK ( AKXKBK + SK)( BKXKBK + RK) ( BKXKAK + SK) + QK = AK = A BK = B SK = CD RK = DD QK = CC (7) he gi mrix L is clcule usig he meho o Klm Filerig ise o he regulr orm escribe i [6]. I he sysem (A C ) is observble he sysem (A C A ) is lso observble hus he soluio o he esimor is guree []. he gi mrix L is clcule s i [7]: L= ( AX( ) ( ) )(( ) ( ) ( )( ) ) L AC + B CB + D CA XL CA + CB + D CB + D (8) he ocie Algebric Rici Equio is: AXA L L L XL ( AXB L L L + SL)( BXB L L L + RL) ( BXA L L L + SL ) + QL = AL = A BL = ( CA) SL = B( CB+ D) R = C B + D C B + D Q = B B (9) ( )( ) L L 4. SMA Bule Acuor Experimel Se-Up he SMA Wire Bule cuor experimel seup show i Figure is use i his pper o pply he robus opiml corollers. I cosi o 48.cm (i.) log µm (.6i.) imeer wires. he wires mkig up he bule re brice rom Nickel-iium (Ni-i) lloy me by Dylloy Ic. re mucure such h hey uergo mximum legh corcio o 8% c pply cosierble mou o orce compre o heir weigh [8]. he bule ws esige o pply mximum o 44N (lbs.) over isce o pproximely.7cm (.i.). he 48 wires were coece mechiclly i prllel bewee wo.6cm (.i.) hick.8cm (.i.) imeer virgi elo e ples. WiReC sowre evelope i our lborory provies eermiisic s imers Hz i his experime o Wiows N plorms [9]. IIR (Iiie Impulse Respose) Buerworh igil ilers re use i WiReC o iler he sesor sigls. he cuo requecy o ech iler is chose equl o Hz becuse he bwih o he SMA wire is lower h h. he Power Supply or bule cuio ws ellbs 48vol (omil) mp DC power source. he 48W supply ws corolle usig cusom esige x operiol mpliier circui which use Burr Brow OPA Power Operiol Ampliier. he 4 Copyrigh by ASME

5 recio orce bewee he lo bule wires ws sese usig rsucer echiques MLP- Mii Low Proile Lo Cell which hs cpciy o 44N (lbs.) ccurcy o ±.44N (±.lbs.). Lier isplceme o he pylo ws sese by Spce Age Corol Series Alog-Oupu Ulr-Smll Submiiure Posiio rsucer which ws see o jusble brcke o he bse o he es rig he exesio cble ws che o he boom o he lo. Is rge o me is o.8cm ( o.i.) wih mximum lieriy error o ±.6% o ull rge. he ol curre pig hrough he bule se ws sese usig Hewle Pckr HP4A Amp Curre Shu which coie precisio.ω resisor. his precisio resisor provie oupu sigl o mv per Amp. he recommee mximum elecric curre pig hrough he SMA wire is 4mA. he 48-wire bule usge ws resrice creully o he ol power cosumpio whe i ws cosruce. Mesure experimelly he resisce o he whole bule vries i he rge rom o Ω he mximum curre wih 8 ses o wires coece elecriclly i prllel is.a. hus he correspoig mximum volge bewee he bule wires shoul o excee V. hereore he mximum corol volge se by he compuer o he bule is limie o V. FIGURE : SMA Bule Experimel Seup. I iio o he lier lo coigurio show i Figure we hve lso evelope moiie versio o he seup h is ble o power revolue joi. Due o spce limiio he escripio experimel resuls obie wih he revolue lo coigurio hve o bee iclue i his pper c be ou i [7].. Dymics Corol o he SMA Bule Acuor Se-Up DYNAMICS OF HE SMA BUNDLE ACUAOR I his sub-secio we prese he moio ymics o he SMA Bule cuor escribe i Secio 4. While oly simpliie versio o he erive moel will be use i he coroller esig we prese i here i is ull orm i cse h will be eee i uure works. I is ume h he emperures o ll 48 bule SMA wires chge homogeeously uer he sme coiios. Equios () (4) () which escribe he he rser eecs he relio o emperure Auseie rio re use irecly i his bule-wire ymics. he relio bewee he ieri lo orces o he bule wires he srees i he wires ees o be esblishe pplie o Equio (). I iio he relio o he legh chge o he bule SMA wires he sri i he wires is lso eee or Equio (). his ymic moel o he SMA bule cuor is urher lierize i orer o be pplie i he coroller esig. he sysem ymics o he SMA bule cuor i he lier lo coigurio c be erive rom he sre/sri relio o Equio (). he sre (σ) pplie o he bule wires is pproxime s: F σ F F = () A 48 where: A is he cro secio re o sigle SMA wire; F is he ol recio orce; F is he recio orce o ech wire he verge o F ou o 48 wires umig h ll he orces re evely isribue o ech wire. he Newo s equio o he ol recio orce F is wrie s: F = mwg+ m wx+ F () i which m w is he m o he lo h he cuor lis; g is he grviiol cos; x is he ccelerio o he weigh isplceme; F re he isurbces such s ricio. I shoul be oe h he m o he SMA wires is eglece becuse i is smll compre o he lo. he lier isplceme is he legh chge rom he origil o he il posiio h correspos o he emperure chge rom room emperure o he il emperure er he elecricl chrge is pplie. L is se o represe he origil legh o he bule SMA wires uer o exerl los; ε ε e ε re he ol sri he elsic sri he rsormio sri room emperure. hus he iiil elogio o he wire er he lo is pplie he il elogio o he wire wih loig log wih he emperure chges re presee ccorigly: e Lε= L( ε+ ε) e Lε = L( ε + ε ) () he ol isplceme x s movig upwr (wires corc) i he posiive irecio c be clcule usig Equios () (): x = L( ε ε) () e e = L( ε ε ) + L( H ε ) e e = L( ε ε ) + LHξ Subsiuig Equios () () () io equio () he sysem ymic moel c be wrie s: e mg w + ml w ( Hξ ε ) + F M A M e = [ E + ( E E ) ξ] ε (4) 48A I ew vrible: e e e ε = ε ε is irouce is iiil coiio whe he wires re room emperure is zero. As he sysem is i sic moe room emperure he grviy c be presee s M e mg w = AE ( ε ) () where A is equl o 48A. he sysem ymics (4) c be e rewrie usig he ew sri vrible ε : ε e A M e A M e e F = [ E ( E E )( ) ] H ml ε ε + ε ξ + + w A ξ (6) wih ε e cos. CONROL OF HE SMA BUNDLE ACUAOR I his secio he geerl guielies re se i orer o esig he eebck coroller usig se-spce vrible represeio o he sysem equio. hrough hese guielies he lierize sysem o equios o he SMA bule cuor c be esblishe use i he coroller esig by he LQR or H mehos. Due o he complexiy o he sysem ymics show i he previous secio he coroller or his oe-wy cuio evice will be esige by eglecig he ymics o he lier lo coigurio. his pproximio is ccepble becuse he ouble ime erivive Copyrigh by ASME

6 e o he elsic sri ε i Equio (6) is ume relively smll compre wih oher sigiic erms rom he srees. hereore his equio is reuce io lier sic relios will o be iclue i he moel or coroller esig. he lier sysem o se equios is cosruce usig oly he he coucio Equio (). his irs orer sysem hs ( ) se vecor x ipu u s ollows: x = [( )] u= V where V is he volge pplie o he SMA wires. his coiuousime sysem o equios is: x = Ax+ Bu; ha K (7) A = ρcv B = v ρcvvr i which K is he mpliier gi o he ipu volge. I shoul be oe h he iiil coiio o he bove equio room emperure is zero. he oupu s he isplceme o he pylo/weigh usig he poeiomeer. Becuse he bule cuor is oe-wy cuio sysem oly he heig proce o Equio (4) is ke io ccou. From Equio () i c be ou hrough lier pproximio: x LHξ (8) LH = ( A s) A s A A A s he oupu equio c be wrie s: yo = Cx C = As where: C LH (9) = A A s i which y o is he oupu me or he lier isplceme. Equio (9) is wrie io he orm: y = yo + C (4) = Cx where is ume o be cos; y is he oupu vecor i he lier sysem o equios. Becuse he bule cuor is require o ollow reerece ipus o move he los he posiio corol o he sysem is servo problem s escribe i Secio. he reerece y h is require i Equio (8) correspos o Equio (4) is wrie s: y = y + C (4) o where y o is he esire posiio o he lier isplceme. I shoul be oe h vrible is elimie whe iere vrible y is use i Equio (9) where: y = y y he prmeers o Ni-i SMA wires h re eee o eermie he coeicies o he sysem equios re lise i ble [8] []. ABLE : Prmeers o µm Ni-i SMA Wire rom Flexiol Prmeer (ui) Symbol Vlue Wire Dimeer (µm) r Lier Elecricl Resisce ( -8 Ω/m) R 8-89 Desiy (Kg/m ) ρ 64 He Cpciy (Joul/Kg- C) c (sigle phse) 69 (urig phse chge) He rser Coeicie (Joul/sec.- C) h 8 Auseie Sr emperure ( C) A s 68 Auseie Fiish emperure ( C) A 78 Mresie Sr emperure ( C) M s Mresie Fiish emperure ( C) M 4 Presri (%) H Origil SMA wire Legh (m) L. I shoul be oe h he mbie (room) emperure rou he SMA-wire bule rges roughly rom o C. he cos i Equio (4) c be clcule rom he A s ; hereore is ou o be equl o 4 C. However whe he iiil experime ws couce wih he esire posiio chose s zero he bule move oo much his vlue o. For his reso Equio (4) is moiie o: = M (4) he ew cos is chose experimelly o be equl o C o coie he bule moveme wihi olerble rge. 6. Experimel Resuls LQR/KALMAN-ESIMAOR CONROL A weigh o 4.99Kg (lb) is pplie o he bule cuor s show i Figure. he lier moel o he sysem i Equios (7) (4) is use i he coroller esig. Followig he sr proceure escribe i Secio he LQR/Klm-Esimor coroller is esige usig he weighig mrices s: Q = R =. BR w wpb w =. R v = Due o c h he precisio o he corol perormce is more impor h he ipu surio he vlues o he pely weighig mrices Q R wm which re ocie wih precisio errors re chose much lrger h he vlues o R BR w wpb which re w ocie wih ipu limis. he clcule gi mrices rom Equios () () re: K = 8.6 L =.49 he resuls o he LQR coroller or ui sep ipu (mm) re show i Figure -(). his ui sep es hs bee use o ie-ue he corol gis. he risig ime rom. o.9mm is.sec he sey-se error is wihi % o he reerece ipu. Figure 4- () shows he correspoig corol volge o he cuor he iercio orce bewee he weigh he cuor he error ierece bewee he esime oupu rom he lier moel he rel me rom he poeiomeer. he ipu volge reches he surio limi V whe he moio begis becuse he coroller ries o rive he weigh o he esire posiio s s s poible. A he sey se he volge is miie rou V ue o he he covecio becuse he emperure o he SMA wires shoul be cos o hol he bule wire posiio sey. he iercio orce bewee he bule weigh is quie sey bewee o.4kg hroughou he recilier moio. he error bewee he esime oupu ue o he Klm esimor he rel me is very smll wih sr eviio o.9 - mm. posiio ( mm.) reerece Copyrigh by ASME posiio ( mm. ) reerece FIGURE : Ui Sep Respose o () LQR () H Corolle SMA Bule Acuor he coroller is ese usig series o sep ipus: rom o 4 bck o mm. Figure -() shows he isplceme o he lier bule cuor uer he LQR corol he error bewee he guice sep ipus he posiio o he weigh. Eve

7 hough he SMA bule is oly oe-ireciol (upwr) cuor he coroller sill works very well s he reerece ipus chge rom higher posiio o lower oe. he corol volge is reuce (excly o zero vols) o llow he wires o cool ow so h he weigh sreches he SMA bule owwr uil i reches he esire posiio. ipu volge ( vol.) recig orce ( Kg.) esimio error( mm.) x ipu volge ( vol. ) recig orce ( Kg. ) esimio error( mm. ) x FIGURE 4: Correspoig Volge Force Error o Posiio Esimio o () LQR () H Corol or Ui-Sep Lier Acuio. he repeiivee o he corolle bule is ese usig squre wve ucio o.8hz bewee mm. he ime sp o relxio is. imes lrger h h o corcio o guree h he weigh reches he lower posiio beore he ex cycle srs. he seco imer o he squre wve rjecory i WiReC is se o be Hz s i is escribe i [9]. Figure 6-() shows he resuls or he squre wve ucio. posiio ( mm.) posiio error( mm.) 4 reerece posiio ( mm. ) posiio error( mm. ) reerece FIGURE : Sep Resposes o LQR Corolle Lier Bule Acuor. H CONROL I his secio H coroller is esige pplie o he SMA bule. Becuse he velociy (ccelerio) o he weigh shoul be limie i is cosiere i he opimizio objecive vecor z. hereore he ew cusomize objecive ucio ollowig he escripio i Secio becomes: z = { y x u} he vlues or he weighigs re chose so h he error o he isplceme is pelize prior o he ipu surio. he corol gis K L re: K = 8.69 L =.9 Becuse he moel o he SMA wire bule is irs orer pl he H corol gis c be ue o hve he sme orer s he oes i he LQR corol. posiio ( mm.) posiio error( mm.) reerece po siio ( mm. ) posiio error( mm. ) reerece FIGURE 6: Squre Wve Guice o () LQR () H Lier Bule Corol. he experimes ollow he sme proceure s i he LQR/Klm- Esimor corol. Figure -() Figure 4-() show he ui sep es o he H coroller. he risig ime o he ui sep respose rom. o.9mm is.sec which is he sme s he respose o he LQR coroller. he sey-se error however is beer wihi 4% o he ipu reerece. Also he iercio orce me bewee he SMA wire bule he weigh he ipu volge o he H coroller re very similr o he oes o he LQR corol. he error o esime oupus is very smll gi wih sr eviio o mm. he resposes o he bule or series o sep ipus or squre wve ucio re show i Figure -() Figure 6-() i which he robuse o he coroller is emosre. here re some iereces bewee he PID coroller LQR or H corollers which mke he ler oes superior. I PID corol gis o he corollers re ue oly bse o experimel perormce. I he gis ge oo high he close-loop sysem es o be rgile. However i LQR/H corol he gis re ue wihi he Lypuov sbiliy crieri. Eve hough he eebck gi mrix K c be icrese s he weighig o oupu precisio becomes higher he observer gi mrix L hs limi remis he limi o mer how lrge he weighig icreses. his gurees he sbiliy o he sysem. he oher vge o LQR/H corol is is biliy o clcule he sey-se ipus which c s eeorwr corol. PID corol wihou eeorwr co ge he sey-se ipus irecly; hereore higher iegrl gi K i is eee. Bu he higher he iegrl gi is he more sesiive he close-loop sysem is. 7. Coclusios I his pper we suie he lyicl ymic moel erivio he robus opiml posiio corol o SMA bule cuors usig he LQR H echiques. We evelope geeric lierize imeivri sysem moel or SMA Bule cuors h c be use i he esig o he LQR H bse corollers. We lso propose ew improve esimor i iscree-ime H opiml corol esig bse o he Klm Filer preicor orm or use i he corol o SMA bule cuors. We perorme eile experimes o he wo corol esig mehos usig se-spce moels LQR H Opiml Desig i iscree-ime omi usig experimel SMA bule cuor cosisig o 48 Flexiol SMA wires ble o pply up o lbs. (44 N). As emosre i he experimes he esige corollers provie siscory resuls i ccurcy sbiliy spee. 7 Copyrigh by ASME

8 REFERENCES [] Hshimoo M. ke M. Sgw H. Chib I. S K. 98 Shpe Memory Alloy Roboic Acuor Jourl o Roboic Sysems () -. [] Iku K. sukmoo M. Hirose S. 988 Shpe Memory Servo Acuor Sysem wih Elecric Resisce Feebck Is Applicio o Acive Eoscope Proceeigs o he 988 IEEE Ieriol Coerece o Roboics Auomio Philelphi PA USA [] roisoie N. Biu P. Drio P. 997 Corol Experimes o wo SMA bse Micro-Acuors Proceeigs Experimel Roboics (ISER 97) Brcelo Spi. [4] Mill D. R. Wg D. 998 Moelig L Sbiliy o Shpe Memory Alloy Posiio Corol Sysem IEEE rscios o Corol Sysems echology 6(4) [] Dickiso C. A. 997 Feebck Compesio o Shpe memory Alloy Hyseresis Ph. D. hesis Reeler Polyechic Isiue. [6] Dickiso C. A. We J Feebck Corol Usig Shpe Memory Alloy Acuors Jourl o Iellige Meril Sysems Srucures 9(4) 4-. [7] Webb G. Wilso L. Lgous D. Reiiois O. Apive Corol o Shpe Memory Alloy Acuors or Uerwer Biomimeic Applicios AIAA Jourl 8() - 4. [8] Kuribyshi K. 986 A New Acuor o Joi Mechism Usig ini Alloy Wire he Ieriol jourl o Roboics Reserch 4(4) [9] k Y. Ym A. 99 A Rory Acuor Usig Shpe Memory Alloy or Robo Alysis o he Respose wih Lo Proceeigs IROS 99 IEEE/RSJ Ieriol Workshop o Iellige Robos Sysems Ielligece or Mechicl Sysems New York NY USA [] Ghrybeh M. A. Bure G. C. 99 Ivesigio o Shpe Memory Alloy Acuor or Dexrous Force-Feebck Msers Avce Roboics 9() 7-9. [] Gr D. Hywr V. 997 Vrible Srucure Corol o Shpe memory Alloy Acuors IEEE Corol Sysems Mgzie 7() [] Gr D. Hywr V. Cosrie Force Corol o Shpe Memory Alloy Acuors Proceeigs IEEE Ieriol Coerece o Roboics Auomio S Frcisco CA USA 4-. [] ebbe C. Schroeer. Buler R. 99 Moelig Robus Corol o Smr Srucures Proceeigs o SPIE - he Ieriol Sociey or Opicl Egieerig Bellighm WA USA [4] Mosley M. Mvroiis C. Desig Corol o Shpe Memory Alloy Wire Bule Acuor Proceeigs o he ASME Mechisms Roboics Coerece Blimore MD Sepember -. Pper DEC/MECH- 47. [] DeLureis K. Fisch A. Nikiczuk J. Mvroiis C. "Opiml Desig o Shpe Memory Alloy Wire Bule Acuors" Proceeigs - IEEE Ieriol Coerece o Roboics Auomio Wshigo D.C. [6] Mosley M. Mvroiis C. Experimel No-Lier Dymics o Shpe Memory Alloy Wire Bule Acuor" Jourl o Dymic Sysems Mesureme Corol rscios o he ASME () -. [7] oki Corporio 987 Biomel Guiebook okyo Jp. [8] Wrm. 99 Acuor Desig Usig Shpe Memory Alloys E. [9]. W. Duerig (Eior) Egieerig Aspecs o Shpe Memory Alloys Buerworh-Heiem 99. [] Mvroiis C. "Developme o Avce Acuors Usig Shpe Memory Alloys Elecrorheologicl Fluis" Reserch or No-Desrucive Evluio Vol. pp. -. [] k K. 986 A hermomechicl Skech o Shpe Memory Eec: Oe-imesiol esile Behvior Res Mechic 8-6. [] Shu S. G. Lgous D. C. Hughes D. We J Moelig o Flexible Bem Acue by Shpe memory Alloy Wires Smr Meril Srucures [] Iku K. sukmoo M. Hirose S. 99 Mhemicl Moel Experimel Veriicio o Shpe Memory Alloy or Desigig Micro Acuor Proceeigs - IEEE Micro Elecro Mechicl Sysems Nr Jp -8. [4] Arpci V. S. 99 Coucio He rser Abrige E. Gi Pre. [] Og K.997 Moer Corol Egieerig r E. Preice-Hll Ic. [6] Zhou K. Doyle J. C. Glover K. 996 Robus Opiml Corol Preice-Hll Ic. [7] Lee C. J. Discree-ime Opiml Robus LQR/H Corol o Roboic Sysems PhD Dierio Rugers he Se Uiversiy o New Jersey. [8] Dylloy Ic. 998 echicl Chrcerisics o Flexiol Acuor Wires brochure Irvie CA USA. [9] Lee C. J. Mvroiis C. WiReC v.: Rel-ime Corol Sowre or Wiows N is Applicios Proceeigs o he Americ Corol Coerece Chicgo IL Copyrigh by ASME