IDENTIFICATION AND COMPENSATION OF GEOMETRIC AND ELASTIC ERRORS IN LARGE MANIPULATORS: APPLICATION TO A HIGH ACCURACY MEDICAL ROBOT

Transcription

1 Pocdgs of DC998: 998 ASM Dsg gg chcal Cofc Sptmb Atlata GA DC98/MCH-5947 IDNIFICAION AND COMPNSAION OF GOMRIC AND LASIC RRORS IN LARG MANIPULAORS: APPLICAION O A HIGH ACCURACY MDICAL ROBO Maco A. Mggolao Massachustts Isttut of chology Dpatmt of Mchacal gg 77 Massachustts Av Cambdg MA 39 USA l Fax Costatos Mavods Rutgs Uvsty Dpt. of Mchacal ad Aospac gg 98 Btt Road Pscataway NJ 8854 USA l Fax Stv Dubowsky Massachustts Isttut of chology Dpatmt of Mchacal gg 77 Massachustts Av Cambdg MA 39 USA l Fax ABSRAC A mthod s pstd to dtfy th souc of dffcto postog os lag mapulatos usg xpmtally masud data. Both os du to maufactug tolacs ad oth gomtc os ad lastc stuctual dfomatos a dtfd. hs o soucs a usd to pdct ad compsat fo th d-pot os as a fucto of cofguato ad masud focs. h mthod s appld to a w lag hgh accuacy mdcal obot. xpmtal sults show that th mthod s abl to ffctvly coct fo th os th systm. INRODUCION Lag obot mapulatos a dd fld svc ad mdcal applcatos to pfom hgh accuacy tasks. xampls a mapulatos that pfom dcotamato tasks ucla sts spac mapulatos such as th Spcal Pupos Dxtous Mapulato (SPDM) ad mapulatos fo mdcal tatmt (Haml t al. 997; Vallacout ad Gossl 994; Flaz 996). I ths applcatos a lag obotc systm may d to hav vy f pcso. Its accuacy spcfcatos may b vy small factos of ts sz. Achvg such hgh accuacy s dffcult bcaus of th mapulato s sz ad ts d to cay latvly havy payloads. Futh may tasks such as spac applcatos qu systms to b lght wght so that stuctual dfomato os may bcom latvly lag. Fo such systms gomtc os du to machg tolacs ad os du to lastc dfomato cat sgfcat dffcto os. Du to task costats t s oft ot possbl to us dct d-ffcto ssg a closd-loop cotol schm to mpov th systm accuacy. hfo th s a d fo modl basd o dtfcato ad compsato tchqus. Whl classcal calbato mthods ca achv such compsato fo som systms thy caot coct th os lag systms wth sgfcat lastc dfomatos bcaus thy do ot xplctly cosd th ffcts of task focs ad stuctual complac. H a mthod s dvlopd that cosds both dfomato ad mo classcal gomtc os a ufd ma. hs mthod s appld h to a w mpotat mdcal applcato of lag mapulato systms. h mapulato s usd as a hgh accuacy obotc patt postog systm a adato thapy sach faclty ow bg costuctd at th Massachustts Gal Hosptal (MGH) th Nothast Poto hapy Ct (NPC) (Flaz t al. 995; Flaz t al. 996). h obotc patt postog systm (PPS) placs a patt a hgh gy poto bam dlvd fom a otatg gaty stuctu (s Fgu ). h PPS s a sx dg of fdom mapulato that covs a lag wokspac of mo tha 4m adus whl cayg patts wghg as Copyght 998 by ASM

2 much as 3 lbs. Patts a fally mmoblzd o th couch attachd to th PPS d-ffcto. h PPS combd wth th otatg gaty that cas th poto bamabls th bam to t th patt fom ay dcto whl avodg th gaty stuctu. Hc pogammabl flxblty offd by obotc tchology s dd. Fgu : Schmatc of th PPS ad th Gaty h qud absolut postog accuacy of th PPS s ±.5 mm. hs accuacy s ctcal as lag os may b dagous to th patt (Rabowtz t al. 985). h qud accuacy s oughly -4 of th omal dmso of systm wokspac. hs s a gat latv accuacy tha may dustal mapulatos. I addto FM studs ad xpmtal sults show that th chagg ad havy payload (btw ad 3 pouds) cats d-ffcto os du to lastc dfomatos of th od of 6-8 mm. Cosdabl sach has b pfomd th modlbasd o compsato of mapulatos also calld obot calbato (Roth Moog Rava 987; Hollbach 988; Moog Roth ad Dls 99; Zhuag ad Roth 996; Hollbach ad Wampl 997). A majo compot of ths pocss s th dvlopmt of mapulato o modls (Wu 984; Mma ad Gupta 993) som of thm cosdg th ffcts of mapulato jot os whl oths focusg o th ffcts of lk dmsoal os (Waldo ad Kuma 979; Vachav ad Magab 987). Som o modls hav b dvlopd spcfcally fo us th calbato of mapulatos (Bodck ad Cpa988; Zhuag Roth ad Hamao 99; Zhuag Wag ad Roth 993) whl som sachs hav studd mthods to fd th optmal cofguatos to duc th mapulato os by calbato (Zhuag Wag ad Roth 994; Zhuag Wu ad Huag 996; Bom ad Mq 99). Sval calbato tchqus hav b usd to mpov obot accuacy though softwa ath tha chagg th mchacal stuctu (Roth Moog ad Rava 986) cludg op ad closd-loop mthods (Whty Lozsk ad Rouk 986; Hayat so ad Rosto 988; vtt ad L 988) as wll as scw-axs masumt mthods (Hollbach ad Wampl 996) somtms combd wth local calbato (vtt ad L 995). Soluto mthods fo th dtfcato of th mapulato s ukow paamts hav b studd fo ths modl-basd calbato pocsss (Dubowsky Maatuk ad Pa 975; Zhuag ad Roth 993). Most calbato mthods hav b appld to dustal o laboatoy obots achvg good accuacy wh gomtc os a domat. Howv th xstg calbato mthods do ot xplctly compsat fo lastc os du to th wch at th dffcto. A calbato mthod that cosds th wght dpdcy of th os was dvlopd (Dout Mavods ad Dubowsky 998) but t ds a lastc modl of th systm. I ths pap a mthod that compsats fo th posto ad otato os causd by gomtc ad lastc os lag mapulatos s pstd. h mthod xplctly cosds th wght dpdcy of th os. A o modl dvlopd by Mavods t al.(997) ad a st of xpmtally masud postos ad otatos of th obot d-ffcto ad masumts of th payload wch a usd to calculat th obot galzd os wthout dg a mapulato lastc modl. H galzd a calld th os that chaactz th latv posto ad otato of fams dfd at th mapulato lks. hy a foud fom masud data as a fucto of th cofguato of th systm ad th task focs. Kowg ths galzd os th mapulato d-ffcto posto ad otato os a calculatd ad usd at ay cofguato to coct th obot cofguato to compsat fo ths os. h mthod tats all os of th mapulato such as gomtc ad lastc os a ufd ma. h mthod s appld to Patt Postog Systm. A foc/toqu sso has b addd to th systm to masu th wch appld by th patt s wght. It s xpmtally show to b abl to duc th ht 5-7mm to lss tha th qud accuacy of.5 mm. MODL BASD RROR COMPNSAION h a may possbl soucs of os a mapulato. hs os a fd to as "physcal os" to dstgush thm fom "galzd os" whch a dfd lat. h ma soucs of physcal os a mapulato a: Mchacal systm os: hs os a sultg fom machg ad assmbly tolacs of th vaous mapulato mchacal compots. Copyght 998 by ASM

3 Dflctos: lastc dfomatos of th mmbs of th mapulato ud load ca sult lag d-ffcto os spcally log ach mapulato systms. Masumt ad Cotol: Masumt actuato ad cotol os that occu th cotol systms wll cat dffcto postog os. h soluto of cods ad stpp motos a xampls of ths typ of o. Jot os: hs os clud bag u-out otatg jots al cuvatu la jots ad backlash mapulato jots ad actuato ga box. I most cass physcal os a latvly small. Howv th ffct at th d-ffcto ca b lag. Futhos ca b dstgushd to patabl ad adom os (Slocum 99). Rpatabl os a os whos umcal valu ad sg a costat fo a gv mapulato cofguato. A xampl of a patabl o s a assmbly o. Radom os a os whos umcal valu o sg chags updctably. At ach mapulato cofguato th xact magtud ad dcto of adom os caot b uquly dtmd but oly spcfd ov a ag of valus. Radom os caot b compsatd usg classcal calbato tchqus. A xampl of a adom o s th o that occus du to backlash of a actuato ga ta. Classcal kmatc calbato ad cocto ca oly dal wth patabl os. It wll b show xpmtally Scto 4 that ths os domat th pfomac of th PPS. o dscb th kmatcs of a mapulato th dfto of fc fams at th mapulato basd-ffcto ad at ach of th jots that a chaactzd by th Davt ad Hatbg paamts a dfd (Cag 989). h posto ad otato of a fc fam F wth spct to th pvous fc fam F - s dfd wth a 4x4 matx A that has th gal fom: R A = () h R tm s a 3x3 otato matx composd of th dcto coss of fam F wth spct to fam F - ad s a 3x vcto of th coodats of ct O of fam F F -. h lmts of matcs A dpd o th gomtc paamts of th mapulato ad th mapulato jot vaabls q. Physcal os chag th gomtc popts of a mapulato. As a sult th fams dfd at th mapulato jots a slghtly dsplacd fom th xpctd dal locatos. h posto ad otato of a fam F wth spct to ts dal locato F s pstd by a 4x4 homogous matx. h otato pat of matx s th sult of th poduct of th coscutv otatos s p aoud th Y Z ad X axs spctvly. (hs a th ul agls of F wth spct to F ). h subscpts s ad p pst sp (yaw) oll ad ptchspctvly. h taslatoal pat of matx s composd of th 3 coodats x y ad z of pot O F. h 6 paamts x y z s ad p a calld h "galzd o paamts. Fo a th dg of fdom mapulato th a 6 galzd os whch ca b wtt vcto fom as = [... x y z s p ] wth agg fom to. Sc th physcal os a small th galzd os x y z s ad p a also small so a fst od appoxmato ca b appld to th tgoomtc fuctos ad poducts. Matx aft th fst od appoxmato has th fom: - s x -p y = -s p z h galzd os ca b calculatd fom th physcal os lk by lk ad thy dpd o th systm gomty th systm wght f thy cota lastc os ad th systm jot vaabls. h d-ffcto posto ad otato o s dfd as th 6x vcto that psts th dffc btw th al posto ad otato of th d-ffcto ad th dal o dsd o: () = X X (3) wh X ad X a th 6x vctos composd of th th postos ad th otatos of th d-ffcto fc fam (F ) th tal fc systm (F ) fo th al ad dal casspctvly. Wh th galzd os a cosdd th modl th mapulato loop closu quato taks th fom: A (qs) = A A...A (4) wh A s a 4x4 homogous matx of th typ show quato () that dscbs th posto ad otato of th d-ffcto fam F 6 wth spct to th tal fc fam F as a fucto of th cofguato paamts q th vcto of th galzd os ad th vcto of th stuctual paamts s. h th compots of th vcto ad th th agls of th otato matx R a th sx coodats of vcto X that ca b wtt a gal fom: X = f (q s) (5) wh f s a vcto o-la fucto of q ad s. 3 Copyght 998 by ASM

4 Sc th galzd os a small ca b calculatd by th followg la quato : = J (6) wh J s th 6x6 Jacoba matx of th fucto f wth spct to th lmts of th galzd o vcto. h lmts of J a dfd as: J f [ ] [ j] = [ j] h valu of ags fom to 6 ad j ags fom to 6. I gal J dpds o th systm cofguato gomty ad wght f th a lastc dflctos th systm. Mo fomato o th dvlopmt of th o modl ca b foud Mavods t al. (997). If th galzd os a kow th th dffcto posto ad otato o ca b calculatd usg quato (6). Fgu shows how a o modl of th typ of quato (6) ca b usd a o compsato algothm. h mthod to obta s xplad Scto 3. Dsd d-ffcto posto X Ivs kmatcs wth o os Offl Masumts Jot vaabls wth o o cocto q q o Modl Idtfcato Pocss d-ffcto o J - w + Jot vaabl cocto q Wch fom th d-ffcto Fgu : o Compsato Schm + (7) Jot vaabls wth o cocto q IDNIFICAION OF H GNRALIZD RRORS h fst stp th mthod s to calculat th galzd os fom off-l masumt data. h dtfcato mthod to calculat s basd o th assumpto that som compots of vcto ca b obtad xpmtally at a ft umb of dfft mapulato cofguatos. Howv sc posto coodats a much as to masu pactc tha otatos may cass oly th th posto coodats of a masudqug th twc th umb of masumts fo th calculato. Assumg that all 6 compots of ca b masud fo a th dg of fdom mapulato ts 6 galzd os ca b calculatd by fully masug vcto at dfft cofguatos ad th wtg quato (6) tms: =... J J = J ( q w) ( q w ) = J... ( q w ) (8) wh s th 6 x vcto fomd by all masud vctos at th dfft cofguatos ad J s th 6 x 6 total Jacoba matx fomd by th o Jacoba matcs at th cofguatos. If matx J s o-sgula ad th galzd os do ot dpd o th cofguato th s obtad smply by vtg J : = J If J s sgula th claly quato (9) caot b appld. hs ca occu f som of th galzd os sult d-ffcto os sam dcto. By masug ths d-ffcto o t s ot possbl to dstgush th amout of th o cotbutd by ach galzd o. hs codto usually occus bcaus of th xstc of spcal gomtc codtos btw th mapulato jot axs such as paalll o othogoal axs o th xstc of psmatc jots (Hayat t al. 988). Patal masumt of vcto such as masug oly th posto but ot th otato of th d-ffcto ca also lad to a sgula J. I ths cas oly la combatos of galzd os ca b calculatd. Mathmatcally th sgulaty of J s xpssd wth a la dpdcy of th colums of J. quato (9) also caot b succssfully appld wh som of th galzd os dpd o th mapulato cofguato amly (q). Fo xampl th galzd os catd by dflctos dpd o th cofguato. h pocdu to fd fo a sgula J o fo = (q) s dscbd blow. Rducto of J to a No-Sgula Matx If th colums of J a ducd to a la dpdt st by goupg th galzd os that cospod to la dpdt colums th J ca b mad o-sgula. If λ s a gvalu of J ad c th cospodg gvcto th: (9) J c = λ c () If J s sgula sval of ts gvalus a zo. Lt c = [ ] c t c... c b th th gvcto that cospods to a ull gvalu wh s qual to 6 whch s th maxmum dmso of J ad th supscpt "t" dots th taspos of a vcto. h quato () s wtt as: J c = [ J ; J ;...; J ] c = () J. c + J. c J. c = Fom quato () t ca b s that th coodats of th gvcto c a th coffcts of la dpdt colums of J. Assumg that total J has gvctos cospodg to a ull gvalu ths gvctos ca fom a matx C: 4 Copyght 998 by ASM

5 C = [ c ; c ;...; c ] () Matx C psts a bass of a la spac composd of vctos that wh multpld wth J sult zo. Aft pfomg la combatos of ows of matx C t s obtad a ducd ow chlo fom (Lo 994). I ths fom matx C s composd of may zo lmts ad hc t s vy asy to dstgush th la dpdt colums of J. By spcto of th lmts of ach colum of matx C th sts of la dpdt colums of J a dtfd. Fom ach st o colum s kpt J th oths a dltd ad th galzd os that cospod to ths la dpdt colums fom la combatos usg th coffcts of th colum of matx C. A xampl s gv xt to llustat ths pocdu. Assum that o colum of matx C aft ts ducto to ow chlo fom has all ts lmts qual to zo xcpt th lmts of th th ad j th ows that a qual to ad - spctvly. Fom quato () t ca b dducd that th ad jth colums of J a qual: j t j [ J... J... J... J ] [ ] = J = J (3) Fo xampl usg quato (8) t ca b s that th galzd os ad j cospodg to th ad j th colums of J ca b goupd to o w o j whch s th sum ad o of th colums th th o j th of J ca b lmatd: J = J = J J j j j (4) J ( + ) J J J h sam pocdu ca b appld to vy colum of matx C ad thus J ca b ducd to a o-sgula matx J ad vcto s ducd to vcto composd of la combatos of th lmts of. Hc quato (8) s wtt as: = J (5) h la dpdcs of colums of J a th sam btw th colums of matx J usd quato (6). hfo quato (6) ca b wtt as: = J (6) wh J s th ducd o Jacoba matx of th mapulato aft la dpdt colums hav b lmatd. quato (6) s th w o modl of th systm ad ca b usd th o compsato schm dscbd Fgu. Polyomal Appoxmato of th Galzd os I gal th lmts of vcto a ot costat but dpd o th systm cofguato payload wght o oth o-gomtc paamts such as tmpatu. A xampl a th galzd os du to dflctos: thy dpd o = both th systm cofguato ad payload wght amly (qw). So vcto caot b calculatd by vtg quato (5) bcaus s ot th sam at all dfft cofguatos wh s masud. I ths cas th th lmt of vcto must b dfd as a fucto of q ad w. Fo smplcty of calculato ths fuctos a appoxmatd by polyomal ss xpasos of th fom: j ( j) a a a b = ( q q... q w ) (7) wh q q... q a th mapulato jot paamts w m s (j) a lmt of th wch vcto fom th d-ffcto ad a th polyomal coffcts. hotcally th s a ft umb of tms quato (7). Howv fo a dsd accuacy of th mthod oly a fw tms a usd ad th coffcts d to b calculatd. Fom th dfto of th galzd os th os assocatd wth th th lk dpd oly o th paamts of th th jot. If lastc dflctos of lk a cosdd th th galzd os catd by ths dflctos would dpd o th wght wch w appld at th th lk. Fo a sal mapulato ths wch s du to th wght of th payload ad to th cofguato of th lks aft th th. Hc th wch w dpds oly o th jot paamts q +...q. hus th umb of tms th poducts of quato (7) ca b ducd. (j) I quato (7) th coffcts a costat paamts ad bcom th w ukows of th poblm. (j) quato (7) s substtutd to (5) all coffcts a goupd to o vcto ad th pat of quato (7) that s kow s copoatd to matx J ad foms a w matx J. h quato (5) bcoms: = J (8) Vcto s calculatd by vtg quato (8). h mmum umb of cofguatos wh s masud dpds o th umb of tms usd quato (7) to appoxmat. o cas th accuacy of vso of matx J mo masumts th dd a mad ad a last ma squa pocdu s usd to vt quato (8): t t $ = (J J ) J m (9) h mthod of dtfyg th galzd os s summazd Fgu 3. APPLICAION O H PAIN POSIIONING SYSM h PPS s a sx dg of fdom obot mapulato (s Fgu 4) bult by Gal Atomcs (Gal Atomcs 995; Flaz t al. 996). h fst th jots a psmatc wth maxmum tavl of 5cm 56cm ad 47cm fo th latal (X) vtcal (Y) ad logtudal (Z) axsspctvly. h 5 Copyght 998 by ASM

6 cofguatos J( q w) J( q w ) J J c = λ c q w J = fo λ =... J ( q w ) C = [ c ;...; c ] J us polyomal fuctos to appoxmat = (qw) masumts J t $ = (J J ) J =... t la ducd ow combatos chlo fom of J $ X $ = J $ Fgu 3: Flow-chat of th Mthod to Idtfy Galzd os C duc J to o-sgula fom $ X( q w s) last th jots a volut jots. h fst jot otats paalll to th vtcal (Y) axs ad ca otat ±9. h last two jots a usd fo small coctos aoud a axs of otato paalll to th Z (oll) ad X (ptch) axs ad hav a maxmum otato agl of ±3. h mapulato "dffcto" s a couch whch suppots th patt a sup posto accommodatg patts up to 88 cm hght ad 3 lbs wght omal opato. h tscto pot of th poto bam wth th gaty axs of otato s calld th systm soct. h couch tatmt volum s dfd by a tatmt aa o th couch of 5cm x 5cm ad a hght of 4cm (s Fgu 4). hs aa covs all possbl locatos of tatmt pots (.. tumo locatos at a patt). h objctv s that th PPS maks ay pot ths volum b cocdt wth th soct at ay otato. Fgu 4: h Patt Postog Systm h jot paamts of th PPS a th dsplacmts d d d 3 of th th psmatc jots ad th otatos θ α β of th th otatoal jots. A 6 axs foc/toqu sso s placd btw th couch ad th last jot. By masug th focs ad momt at ths pot t s possbl to calculat th patt wght ad th coodats of th patt ct of gavty. h systm motos a vy slow ad smooth du to safty qumts. Hc th systm s quas-statc ad ts dyamcs do ot fluc th systm accuacy ad a glctd. h accuacy of th PPS was masud usg a Lca 3D Las ackg Systm (Lca 997). Mo spcfcally th masumts w to valuat th PPS patablty th olaty of ts wght dpdt dflctos th ht ucompsatd PPS accuacy ad th mthod dvlopd abov. h tagts w placd o th couch at th postos P P ad P 3 show Fgu 5. h tagts a locatd about mm abov th couch. h posto accuacy of th masumts s appoxmatly.4mm. Am Z P 3 P X Y O Fam F P Couch NP atmt Aa Fgu 5: Clos Vw of th Couch A fc fam F s fxd to th couch (s Fgu 5). h tscto pot of th pla (P P P 3 ) wth th Y axs of th fxd fc fam s calld O. A fxd fc fam F o s usd to xpss th coodats of all pots. Wh th PPS s at ts hom cofguato (all jot vaabls st qual to zo) th fc fams F ad F o a cocdt. h locato of a tumo o a patt dfd as th Nomal atmt Pot (NP) s spcfd th fam coodat F. Fo th sults pstd blow th NP coodats F a tak as ( 9-84) mm. Fo mo tha 7 cass (at dfft cofguatos of th PPS ad usg dfft wghts) th locato of pots P P ad P 3 fam F o was masud ad th NP coodats fam F o calculatd. Fom th systm kmatc modl wth o os th dal coodats of NP w calculatd ad subtactd fom th xpmtally masud valus to yld th vcto (qw). I ths wok 45 masumts w usd to valuat th basc accuacy of th PPS ad lat usd to valuat th accuacy of th compsato mthod dscbd abov. Fo ths plmay quato two dfft payloads w cosdd: o wth o wght ad aoth wth a 54 lbs wght at th ct of th tatmt aa. h PPS cofguatos usd w goupd to two sts: St a) atmt Volum. h 8 vtcs of th tatmt volum (s Fgu 4) a achd wth th NP wth agl θ 6 Copyght 998 by ASM

7 takg valus fom -9 to 9 wth a stp of 3 fo a total of cofguatos. St b) Idpdt Moto of ach Axs. ach axs s movd dpdtly whl all oth axs a hld at th hom (zo) valus. h stp of moto fo d s 5 mm fo d mm fo d 3 5mm ad fo θ 5 sultg 338 cofguatos. h PPS ucompsatd accuacy combg th two sts s show Fgu 6. h dots pst th postog os of NP. It s claly s that spt of th hgh qualty of th PPS physcal systm ts ucompsatd accuacy s o th od of mm. hs s appoxmatly tms hgh tha th spcfcato. dy (mm) dx (mm) coctd os ucoctd os dx (mm) -5 - dz (mm) dz (mm) dy (mm) dy (mm) dx (mm) dz (mm) Fgu 6: Masud ad Rsdual os Aft Compsato h patablty o s du to th adom systm os ad t caot b compsatd by a modl basd tchqu. It psts th accuacy lmt of ay o compsato algothm ad t also shows how wll a o compsato tchqu pfoms. H th patablty was basd o how wll th systm would tu th NP to cta abtay cofguatos. A total of 7 masumts w tak wth zo payload wght. Fgu 7 shows th dstbutos of th patablty o fo ach axs. h patablty o ca b s to b lss tha.5mm (3σ). hus ths systm wth a spcfcato of.5mm s a good caddat fo a modl basd o cocto mthod. I mplmtg th mthod a gal ola fucto of th wch w ca b usd. o hlp stablshg ths fucto th bhavo of th PPS postog os fo dfft payload wghts was xamd wth masumts mad at th hom (zo) cofguato. h wghts agd fom to 3 lbs stps of appoxmatly 5 lbs. h sults showd that th postog os of th PPS a aly la wth th payload wght. h last squa o s lss tha.mm fo th la ft. Fgu 7: Rpatablty Dstbuto h galzd os a calculatd wth quato (9) usg th cofguatos of st (b) (dpdt moto of ts axs) ad half of th tatmt volum data (st a). Fo a Ptum PC 66MHz th computg tm was lss tha two muts. h PPS s th commadd to go to compsatd pots (s Fgu ) fo th mag cofguatos of st (a). h sdual postog os of th PPS aft compsato fo ths pots a show Fgu 6. h sdual os a closd a sph of.38mm adus whch s small tha th sph of.5mm adus that psts th accuacy spcfcato. h qud umb of data pots fo ths calculato was lss tha 4. h o dstbuto alog ach axs s show Fgu 8. Hc th compsato appoach usd ths pap abls th systm to mt ts spcfcato. populato populato ma=.5 3σ= o alog X (mm) 4 3 ma=-.9 3σ= o alog Z (mm) populato populato ma=-.3 3σ= o alog Y (mm) 3..4 om of o vcto (mm) Fgu 8: Statstcal Rsults at NP ma=.94 3σ=.93 7 Copyght 998 by ASM

8 CONCLUSIONS I ths pap a mthod s pstd to dtfy th postog d-ffcto os of lag mapulatos. h mthod ca dtfy th soucs of th d-ffcto os both gomtc ad lastc os. Pvous calbato tchqus dd t xplctly cosd th wch at th dffcto to compsat fo lastc os. hs mthod cosds th wght dpdcy wthout th d to dvlop a lastc modl of th systm. It s valuatd xpmtally o a hgh accuacy lag mdcal mapulato. h sults showd that th basc accuacy of th mapulato xcdd ts spcfcatos but aft applyg th mthod to compsat fo d-ffcto os th accuacy spcfcatos a mt. ACKNOWLDGMNS h suppot of th NIH va th Nothast Poto hapy Ct fom th Massachustts Gal Hosptal fo ths wok ad th tchcal fomato povdd by Gal Atomcs Sa Dgo CA a ackowldgd. RFRNCS Bom J.H. ad Mq C.H. 99 Dtmato of Optmal Masumt Cofguatos fo Robot Calbato Basd o Obsvablty Masu h Itatoal Joual of Robotcs Rsach (): Bodck P. ad Cpa R. 988 "A Mthod fo Dtmg ad Coctg Robot Posto ad Otato os Du to Maufactug" asactos of th ASM Joual of Mchasms asmssos ad Automato Dsg : 3-. Dout P. Dubowsky S. ad Mavods C. 998 "Compsato of Gomtc ad lastc Dflcto os Lag Mapulatos Basd o xpmtal Masumts: Applcato to a Hgh Accuacy Mdcal Mapulato submttd to th 6th Itatoal Symposum o Advacs Robot Kmatcs Austa. Dubowsky S. Maatuk J. ad Pa N.D. 975 A Paamtc Idtfcato Study of Kmatc os Plaa Mchasms. asactos of th ASM Joual of gg fo Idusty vtt L.J. L C.Y. 988 Kmatc Calbato of Mapulatos wth Closd Loop Actuatd Jots Pocdgs of th 988 I Itatoal Cofc of Robotcs ad Automato vtt L.J. L J. 995 Impovd Mapulato Pfomac hough Local D-H Calbato Joual of Robotcs Systms (7): Flaz J. t al. 995 "Ovvw of th MGH-Nothast Poto hapy Ct: Plas ad Pogss" Nucla Istumts ad Mthods Physcs Rsach B 99: Flaz J.t al. 996 Dsg Appoach fo a Hghly Accuat Patt Postog Systm fo NPC. Pocdgs of th POOG XXV ad Hadothapy Symposum Blgum Sptmb. Gal Atomcs 995 Patt Posto Plmay Dsg Documts. Haml W. Malad S. ad Wd. 997 A Modl-Basd Cocpt fo lobotc Cotol of Dcotamato ad Dsmatlmt asks Pocdgs of th 997 I Itatoal Cofc of Robotcs ad Automato Albuququ Nw Mxco Apl. Hayat S. so K. Rosto G. 988 Robot Gomty Calbato Pocdgs of th 988 I Itatoal Cofc of Robotcs ad Automato Hollbach J. 988 "A Suvy of Kmatc Calbato." Robotcs Rvw Khatb O. t al dtos Cambdg MA; MI Pss. Hollbach J.M. Wampl C.W. 996 h Calbato Idx ad axoomy fo Robot Kmatc Calbato Mthods Itatoal Joual of Robotcs Rsach 5(6): Lca 997 Wb Pag of Lca Gosystms Lo S. 994 La algba wth applcatos. Idaapols; Macmlla Collg Publshg Compay. Mavods C. Dubowsky S. Dout P. Htst J. Flaz J. 997 "A Systmatc o Aalyss of Robotc Mapulatos: Applcato to a Hgh Pfomac Mdcal Robot" Pocdgs of th 997 I Itatoal Cofc of Robotcs ad Automato Albuququ Nw Mxco Apl. Mma C. ad Gupta K. 993 "Idtfcato of Posto Idpdt Robot Paamt os Usg Spcal Jacoba Matcs". Itatoal Joual of Robotcs Rsach (3): NPC 996 Wb Pag of th Nothast Poto hapy Ct at th Massachustts Gal Hosptal Rabowtz I. t al 985 "Accuacy of Radato Fld Algmt Clcal Pactc." Itatoal Joual of Radato Ocology Bology ad Physcs : Roth Z.S. Moog B.W. Rava B. 986 A Ovvw of Robot Calbato. I Southco Cofc Olado Floda. 8 Copyght 998 by ASM

9 Slocum A. 99 Pcso Mach Dsg.. glwood Clffs. Vachav R. ad Magab. 987 "A Gal Pocdu to valuat Robot Postog os." h Itatoal Joual of Robotcs Rsach 6(): Vallacout C. ad Gossl G. 994 Compsatg fo th Stuctual Flxblty of th SSRMS wth th SPDM Pocdgs of th Itatoal Advacd Robotcs Pogam Scod Wokshop o Robotcs Spac Caada Spac Agcy Motal Caada. Waldo K. ad Kuma V. 979 "Dvlopmt of a hoy of os fo Mapulatos." Pocdgs of th Ffth Wold Cogss o th hoy of Machs ad Mchasms : Whty D.. Lozsk C.A. Rouk J.M. 986 Idustal Robot Fowad Calbato Mthod ad Rsults. asactos of th ASM Joual of Dyamc Systms Masumt ad Cotol 8: -8. Wu C. 984 "A Kmatc CAD ool fo th Dsg ad Cotol of a Robot Mapulato." h Itatoal Joual of Robotcs Rsach 3(): Zhuag H. Roth Z. ad Hamao F. 99 "A Complt ad Paamtcally Cotuous Kmatc Modl fo Robot Mapulatos". I asacto Robotcs ad Automato 8(4): Zhuag H. Roth Z.S. 993 A La Soluto to th Kmatc Paamt Idtfcato of Robot Mapulatos. I asactos Robotcs ad Automato 9(): Zhuag H Wag L.K. Roth Z.S. 993 o-modl- Basd Robot Calbato Usg a Modfd CPC Modl. Robotcs ad Comput Itgatd Maufactug (4): Gat Bta. Zhuag H. Wag K. ad Roth Z. 994 "Optmal Slcto of Masumt Cofguatos fo Robot Calbato Usg Smulatd Aalg." Pocdgs of th I 994 Itatoal Cofc Robotcs ad Automato : Sa Dgo CA. Zhuag H. Wu J. Huag W. 996 Optmal Plag of Robot Calbato xpmts by Gtc Algothms. Pocdgs of th I 996 Itatoal Cofc Robotcs ad Automato : Mapols Msota. 9 Copyght 998 by ASM