A Manipulated Deformable Object as an Underactuated Mechanical System

Transcription

1 Maipulatd Dfoabl Obct as a Udactuatd Mchaical Syst Hbt G. a Kostas. Kyiaopoulos bstact. Dfoabl obcts ud aipulatio ca b odld usig fiit lts. h sultig odl is i fact a udactuatd chaical syst. h cosucs of ay typ of costaits vald by th odlig pocdu is plod. h study of dfoabl obct odls withi th fawo of udactuatd chaical systs idicatd th istc of scod od oholooic costaits. Fo th idtificatio of this id of costaits th authos hav dvlopd coputatioally fficit sigificatly sipl athatical tools. hi thodology is illustatd tstd by a apl.. Itoductio I obot aipulatio th obct that was big hld was titioally cosidd igid. his assuptio o th obct's atu siplifid sigificatly th pobl. h appoach was ustifid fo aly vy poit of viw. It allowd sachs to focus o th w syst big dvlopd aly th obot ustd its viot to facilitat aalysis: o obstacls pst cla absolutly ow viot igid obcts. s pobls ivolvd i obot aipulatio fid thi solutio with ti gat ffot by th obotics couity all of ths assuptios a gually livd. Obstacl avoidac has b cosidd aptatio thods plod sso ifoatio ta ito accout. It is about ti w did sothig about th assuptio cocig th obct big igid.

2 I what follows w shall discuss so poptis of a odl obtaid fo dfoabl obcts big aipulatd by obotic as. h otivatio fo ivstigatig dfoabl atials ca fo th study of a ultipl obil aipulato syst which hls a dfoabl obct. Wh focusig o th obct issu w fist h to choos a appopiat odl to dscib its bhavio. Pvious appoachs to dfoabl obct hlig focusd aily o foulatig gal cotiuous dyaic uatios fo th obct. h ai was to costuct a appopiat odl to hac coputatio o allow fo th applicatio of ctai cotol statgis. Su t al. followd zopoulos' hybid appoach to dfoabl obcts. his appoach oigiatd fo th study of Coput idd Dsig coput gaphics. It is chaactizd by th dcopositio of th dfoabl obct to a fc copot a dfoatio copot. h fo psts th oigial shap of th obct th latt th chag i its shap as a sult of th applid lo. h positio of ay poit i th body ca b th dtid by a suppositio of th two copots. Kosug t al. 4 usd fiit lts. Loig coditios icludd oly bdig of a sht tal aid th pobl of cotollig th static dfoatio of th plat wh hld by a dual aipulatio syst. h static assuptio siplifis cosidably th dyaics of th obct. Wu t al. 5 poitd out that a flibl obct is actually a distibutd paat syst appoiatd it by cosidig a lupd paat odl. hy also assud uit asoably that th obct is ot vy soft which as that it udgos oly sall dflctios duig its itactio with th obot. cas of a flibl tal plat is cosidd. o costuct th lupd paat syst thy ploitd th goty of th syst of th obot gaspig th tal plat. It is ot cla howv how this ca b tdd to th cas of oth obcts with abitay shap. Yuawa t al. 6 ivstigatd a vibatig flibl obct odld it usig odl ductio thoy. h ai was to aliz positio cotol whil suppssig th vibatio of th obct. Vibatio is assud to ta plac i a two disioal spac. h odlig appoach fo th obct bgis with th distibutd paat odl ds up with a fiit disioal odl. I ou appoach to odlig dfoabl obcts w dcidd to us fiit lts basd o th lastodyaic uatios otivatd aily by a giig attitud tows th pobl i h. his stp has a dcisiv ipact o th atu of th pobl w would th hav to fac. ustio aiss cocig th stuctual poptis of th odl thus obtaid; how dos that bhav fo th cotol poit of viw; what a th liitatios what ca o hop to achiv. alig about achivt w would btt dti what ds to b do fist. I ou study of th ultipl obil aipulato syst w placd phasis o th dfoabl obct sic this dfis at a gat dg th whol syst's spcificatios uits. Evythig has to b do with a y o th obct: should it b glass it ust ot ba should it b pap cloth o lath it ust ot ta apat should it b wood it ust ot factu should it b a tal plat it ust ot bd too uch so o. W thfo dd to pla th otio of th obct

3 so that it ca b safly b tasfd fo o poit to aoth without placig cssiv lo o th obots which could tip ov o ach a sigula positio i thi ffot to accoplish thi issio. hat is why w h to s what th odl w chos fo th obct could do. Fist w idtifid a udactuatd syst 7. Fiit lt pocdus sult to a odl which has a fw poits o ods wh lo is dictly td a ub of itdiat poits which a o o lss iflucd by that lo. S fo th cotol pspctiv th ods wh lo is applid a dictly actuatd whil th st a statically dyaically coupld to th. hfo th is a ub of dgs of fdo which a actuatd oths which a ot. his is oughly th ida of a udactuatd syst: udactuatd systs hav fw cotol iputs tha dgs of fdo. Viwig th aipulatd obct as a udactuatd syst w ca cosidably bo th otio of a dfoabl obct to th cas of systs which a ot cotiuous: chais stuctus with passiv oits flibl lis ollig cotacts tc. his allows a galizatio of ou appoach to obct hlig puts a gat vaity of systs i a uifid fawo of study. I th aalysis of udactuatd systs it will soo bco cla that a pat of thi dyaics which oths pf to call th zo dyaics ca b thought of as a st of dyaic costaits. I this cott w f to ths uatios as itisic costaits sic thy st dictly fo th atu of th syst a pst glss of th assuptios o ay a fo th aipulatd obct. pat fo ths pactic ca foc oth liitatios such as th avoidac of factu o cssiv loig which ca sult to plastic o-lastic dfoatio. hs tal liitatios ca lat to atial stgth liitatios coditios fo obstacl avoidac duig th otio of th obct so foth. Noally th tal costaits ca b pssd i th fo of a st of iualitis.. h Udactuatd Syst Stictly spaig a dfoabl obct has a ifiit ub of dgs of fdo. his is bcaus vy poit i th body ca b cosidd as a idividual dg of fdo. h dp o gos i a icoscopic scal th bigg th ub gows. lthough athaticias would fl uit cofotabl with th otio of ifiity i th pobl this would pobably distss ost gis who a assigd to th tas of poposig a alistic solutio. Egis would thfo ty to dcas this ub to a fiit o a a toff btw coputatioal coplity accuacy of solutio. I th pocss of this siplificatio i od to dal with th pobl usig istig wll ow appovd tools a atual appoach is to disctiz th obct: cosid it as a syst of idtical itcoctd atial lts that ach cotibuts by a sall potio to th syst's ovall bhavio. his is actually th ida udlyig th fiit lts appoach. h td of disctizatio dpds o ay factos. O is th idividual chaactistics of th atial: obcts that a faily igid do ot ui uch dis-

4 ctizatio; flibl obcts d ay lts to sufficitly dscib thi shap. oth facto is loig coditios: fo th sa obct if a lag lo is to b applid th disctizatio gid should b ds sic ach lt would b calld to udta a lag displact. If th asu of this displact is cssiv th th assuptios o which th appoiatio is basd could cas to hold th sults ay ot b valid ay o. Fo a giv atial appoiatly ow loig coditios a ptty good disctizatio gid ca b costuctd. h fiit lt thod yilds a dyaic odl i which so dgs of fdo a div dictly as a sult of tal lo th st coply with th displact of th fo. h dgs of fdo that a dictly cotolld cospod to th ods of th gid at th locatio wh th aipulatos gasp th obct. ll oth ods a displacd i accodac to th loig coditios iposd by th div ods. h whol syst ca b thfo classifid as udactuatd 8 sic th ub of dgs of fdo big dictly cotolld is lss tha th total ub of dgs of fdo i th syst. h class of udactuatd chaical systs is vy bo. It icluds systs with passiv oits flibl li costuctios chai chaiss obil obots ay oths. hs systs a pculia i th ss that thi Lagagia dyaics ay cotai udsiabl poptis such as o-iiu phas zo dyaics oholooic costaits tc. Viwig th aipulatd obct as a udactuatd syst allows fo th ivstigatio of w aipulatio tass icludig systs of igid bodis o cobiatios of igid dfoabl obcts. h obct big hld ca ow b a tuss a fa a plat a chai o a abitay shapd th disioal body th atial cosidd ca b pactically aythig that huas thslvs ca aipulat. I Lagagia dyaics th uatios of otio fo a chaical syst with dgs of fdo ca b divd as d d t L L i i i i F wh a th galizd coodiats chos fo th syst: Q It is wll ow that ths uatios ca ta th ati uatio fo: M C K u wh M is th itia ati which is sytic positiv dfiit C is a ati latd to Coiolis ctifugal focs K is th vcto fod by th lastic gavitatioal ts u is th iput to th syst. Fo th cas of udactuatd systs u with <. h difft disio btw th spac of cotol iputs th spac of galizd coodiats otivats th patitio of th latt ito a actuatd a uactuatd subspac. Without loss of gality w ca aag th ts i th vcto as follows:

5 Q cospods to th uactuatd coodiats Q wh to th actuatd o cotolld coodiats. I th light of th abov patitio uatio ca b witt 8 a b c c b u wh th ati b is assud osigula. Euatio a is a st of - scod od ucotolld difftial uatios which ca b cosidd as dyaic costaits. So pf to itpt ths uatios as th zo dyaics howv this is ot foally coct sic o output has b spcifid fo this syst. I this cott w will a ths uatios itisic costaits to distiguish th fo ay tal iposd costaits that ca lat to atial stgth liitatios o obstacl avoidac uits. atual ustio to as is what id of costaits ths a. Ca ths difftial uatios b itgatd oc? If so th o has a st of pssios latig th galizd vlocitis with th galizd coodiats i.. th will b o acclatio ts pst. Do ths sultig uatios pit a scod fial itgatio? Should such itgatio b pittd th sult would b a st of algbaic uatios latig th galizd coodiats. I this fial cas thigs a ath sipl: h algbaic uatios thus obtaid dfi a - hypsufac o which th syst ust liv. his is th cas of holooic costaits. Fo a obct big aipulatd this as that th obots hav o cotol whatsov ov - dgs of fdo of th syst. hs dgs a copltly spcifid by th aiig. hfo thy cas to b dgs of fdo: thy a ust so fully dpdt vaiabls that ca b ually igod i th dyaic odl. his could idicat a igid bhavio hibitd by a pat of th obct. I fact such a situatio is ipobabl sic v igid atial hibits so sot of lastic bhavio at last i thoy. It ay ais howv if duig th iitial odlig pocdu so physical costaits hav b ovlood th odl cos lat o to big th ito vidc. his iplis that th is a way to ipov th odl duc th coplity of th pobl. If th uatios ca b itgatd oly oc o ds up with a st of fist od oholooic costaits. h a oly galizd coodiats thi fist od ti divativs pst. Now thigs a o coplicatd: ths latios iply that th a liitatios i th spac of vlocitis that do ot ifluc dictly th spac of positios. h ifluc of ths vlocity liitatios is uit subtl. hy ipos sigificat liitatios i cotol. I sipl ts thy iply that o ay b abl to st th syst fo o plac to aoth but th path it ust follow is ot cssaily th shot i so spct. sipl apl is paig a ca: th is a stig suc you hav to follow o caot siply tu th ca i ay dsiabl dictio. If this is th cas th fo th tas of plaig tactois fo th obct o has to sot to spcial havy athatical tools.

6 h last cas is wh th uatios caot b itgatd at all. his iplis th psc of scod od oholooic costaits. h dp ipact of such costaits has ot b vy wll udstood yt. Such systs could b h to cotol as i th cas of fist od oholooic costaits but sotis th systs ca v b cotolld with lia ti-ivaiat cotolls 8. his is typical fo systs that hav a lastic/gavity pottial t. h udlyig chaactistics of scod od oholooic costaits will b claifid as sach cotius. It is thfo ipotat to ivstigat th typ of th costaits iposd o th syst sic ths could dti to a ctai dg th cotollability poptis of th syst o iply ways i which th odl ca b ipovd.. Collocatd Liaizatio Ctai fos of syst dsciptio hac aalysis. otabl o is th oal fo which will b usd i subsut aalysis. his fo vals a stuctu that is paticulaly usful i idtifyig th typ of costaits iposd o th syst. May of th pvious sults cocig oholooic costaits hav usd this typ of dsciptio. h oal fo is obtaid by a st tchiu of fdbac liaizatio. h syst a-b ca always b patially fdbac liaizd with spct to th actuatd dgs of fdo. his is a ipotat popty of such systs 8. h pocss of patial fdbac liaizatio fo this class of systs is also ow as collocatd liaizatio. It ca asily b udstood that th ati i uatio a is sua osigula. h latt is guaatd by th positiv dfiitss of th itia ati M. If a is solvd fo w obtai wh c his ca b usd to substitut i b so that it bcos: c b u c c c Usig ow th liaizig fdbac v c u b wh v is th w cotol vcto th syst a b ca ta th fo v 4a 4b

7 wh th ts a dfid as follows c. Stat Spac Dsciptio h stat spac uatios of th udactuatd syst ca b asily obtaid by uatios 4 by sttig: his way uatios 4 ca ta th fo: 4 v 4 v 4 with cofiguatio vcto 4 5a 5b 5c 5d M. h abov uatios hav a uactuatd lia pat 5a 5b a actuatd olia pat 5c 5d. hy will b th statig poit fo th aalysis of th costait uatios th stabilizatio poptis that will follow. W chos to a th cofiguatio vaiabls bcaus th patitio of th oigial cofiguatio spac th aagt of vaiabls ca asily caus cofusio.. Costait Classificatio. Pliiais I th sul w us so athatical cocpts fo th fild of difftial goty. Fo th who is ot failia w ths ts w will attpt a shot ifoal itoductio. hos failia with th ts ay sip this sctio. h dfiitios giv ifoally blow a by o as coplt o accptabl athatically thy sv oly to allow a o-spcialist to follow ituitivly ou appoach. h itstd ca f to 9. aifold is a locally Euclida spac. Locally as that o caot us th sa costuctios ot v th sa coodiat syst to ov fo a poit to ay oth poit. sph is a aifold which is oly locally Euclida. W cosid th ath's sufac as a pla bcaus i ou scal it ss so pactically it

8 is a cllt appoiatio. Evyo howv has a cla pictu of th sphical shap of Eath dos ot pct to oi th oth th south pol with a staight li without that li cossig th sufac. h plaa appoiatio ca oly hold locally. tagt vcto is a vcto tagt to th aifold's sufac attachd to a poit o th aifold. h tagt vcto is always fd to i coctio to th poit at which it is attachd. ll tagt vctos at a poit fo th tagt spac at that poit. vcto fild is a appig that assigs to ach poit o th aifold a tagt vcto o that aifold. h vlocity of a paticl ovig o th aifold is a vcto fild. Vcto filds a closly latd to difftiatio. hat is why w us th sybols to dot th bas vctos i th tagt spac. h ight h sid of th stat uatios of th syst ca dfi a st of vcto filds g i : g v g p v p Havig o o o vcto filds o a aifold you ca assig at ach poit o o o tagt vctos. hs tagt vcto ay spa a tagt subspac at this poit. h collctio of all tagt subspacs gatd by th tagt vctos of th vcto filds fos a distibutio. Sval opatios ca b dfid o vcto filds. O of th is th Li bact which sbls a out poduct opatio. h poduct of th Li bact opatio btw two vcto filds is aoth vcto fild. Ud th Li bact opatios th vcto filds ca fo a algba aly th Li algba. With vy distibutio spad by so vcto filds a algba is associatd. Studyig th athatical poptis of this algba o ca daw sigificat coclusios cocig th cotollability poptis of a syst th vcto filds of which gat th associatd distibutio..4 Dfiitios Costait Idtificatio Noholooic costaits ca b idtifid with th us of so tools fo difftial goty. Cosid uatio 5. W stat th followig dfiitio optd fo Dfiitio : Cosid th syst 5 dfi th followig vcto filds: i i i spa b th distibutio gatd by th. Cosid th accssibility algba C ~ of th distibutio i.. th sallst subalgba that lt { } t

9 cotais th vcto filds { }. Lt C ~ b th accssibility distibutio gatd by th accssibility algba. If di C t t M th ~ th syst 5 is calld copltly scod od oholooic. Wh th coditio fo th disio of th accssibility algba is satisfid fo th udactuatd syst 5 th it posssss scod od oholooic costaits. his iplis that th costait uatios b caot b itgatd to poduc a distibutio fo th syst. O should cafully distiguish btw th itgatio with spct to ti th itgatio w a discussig. h latt fs to Fobius itgatio i.. fidig sooth fuctios λ that solv a patial difftial uatio λ i i. If such a itgatio is possibl th it as that th dyaic uatios stict th syst to dvlop o a disioal distibutio. If o th oth h th costaits a scod od oholooic th disio of th stat spac ais. Whth ths costaits ifluc dcisivly th cotollability poptis of th syst is aoth issu which will b discussd lat. Suppos ow that th costaits a ot scod od oholooic a itgatd oc. h sultig uatios will ow lat th galizd coodiats with thi fist ti divativs. h sultig uatios ca th b pssd i th fo: 6 h aihilatos of th ows of ati fo aoth ati S fo which S h istc of S iplis a latioship of th fo S 7 h colus of S a vcto filds which dfi a distibutio. Cosid th accssibility algba that is gatd by thos vcto filds th sall subalgba that cotais thos vcto filds. If th accssibility algba has a disio ual to th th syst is copltly fist od oholooic by Fobius tho. his as that th syst is stictd to volv o a tagt budl of disio which ca b ily show with th st aalysis that will follow. Noholooic costaits allow th syst to ach ay dsid positio howv thy locally liit th dictios o which th syst ca ov. h liitatio dos ot apply to cofiguatio vaiabls but ath to vlocitis: at a spcific cofiguatio th syst ay ot b abl to dvlop vlocity at ctai dictios. h ub ca b thought of as th w disio of th stat spac. ft difftiatio uatio 7 bcos S S his ca b substitutd i to yild

10 M S S C K u ultiplyig by S fo th ight { S M S S C S S } S K S u S M S 8 wh th dpdc o has b doppd. Euatio 8 is ow disioal giv th ducd dyaic odl of th syst. ft so algbaic aipulatio aagig of ts uatio 8 ca b witt as D G u fo th w ducd od odl of th syst 9 S D G u which aft aagig th ts ca b bought to th stat spac fo 5. Suppos ow that uatios 6 ca still b itgatd. h sult of this itgatio is a st of - holooic uatios pscibig - dgs of fdo i ts of th aiig : g h iplicit fuctio tho povids th cssay coditios ud which th abov uatio ca b solvd fo to yild: h h abov latio ca b usd to liiat fo uatio b so as to obtai a ducd od odl. Mˆ C ˆ K ˆ ˆ u Q oth cas is wh th dyaic costaits that fo uatio a a a collctio of scod od oholooic fist od oholooic v holooic. I this occasio th pocdu outlid fo ach cas should b followd fo th pat that falls withi ach catgoy. his could b uit toublso but it is th oly way to cla out th sc obtai a cosistt iiu od odl. 4. Cotollability Issus Wh dyaic costaits a holooic th th syst otio is stictd to a - disioal aifold. It caot ach ay positio outsid this aifold. Poits that do ot blog to that aifold a siply uachabl. I th cas of copltly fist od oholooic costaits th cofiguatio spac is ot cofid. h syst is ow to b accssibl. Du howv to th dift t D i uatio 9 accssibility fo th syst dos ot iply cotol-

11 lability. his dos ot a that th syst is ot cotollabl; it siply as that th is o dfiit clu that it is. Fo oholooic systs with dift th is o availabl gal cssay sufficit sult fo stablishig coplt cotollability. O has to sot to oth fos of cotollability such as stog accssibility sall-ti-local cotollability. Fo th latt th ist oly sufficit coditios but oc it has b stablishd o ca us th aifold of uilibiu poits of th dift vcto fild to ach a abitay sall ighbohood of th dsid cofiguatio. difficult poit is that fist od oholooic systs a ot stabilizabl via cotiuous ti-ivaiat stat fdbac. I suay this is a situatio that o should wish to avoid. If howv th syst 5 is povd to b scod-od oholooic it has b povd that it is autoatically stogly accssibl. Moov th is v a chac fo sooth fdbac stabilizatio povidd that a sufficit coditio fo o-istc of a sooth stabilizig cotol law is ot satisfid: ho : ssu that i M fo i -. Lt lt dot a uilibiu solutio. h th scod od oholooic syst 5 is ot asyptotically stabilizabl to usig ti-ivaiat cotiuous static o dyaic stat fdbac law. If o th oth h i M i th th syst could phaps b stabilizabl by cotiuous cotol law. 4.5 h Fiit Elt Modl Lts tu ow to th fiit lt odl w hav dvlopd fo th obct. h fiit lt aalysis sults to a dyaic odl which is lia : M C K u Moov th chaactistic atics M C K a idpdt of th od coodiats. h fiit lt thod povids a aigful way of liaizig th oigial dyaic uatios of th dfoabl obct. Ud spcific coditios fo slctig th itpolatio fuctios withi th lt th lia odl ca b povd to b uivalt to th iitial olia difftial uatio so that o of th ifoatio cotaid i th oigial uatios is lost i th pocss all ods a pstd by th appoiatd odl. h stat uatios divd fo hav th fo v h stat uatios ca b fod as

12 v v W giv th followig La: La : Fo th syst it holds:. { }. fo > th th colu of is th th bas vcto of. h vctos that fo th vcto fild ca b pssd as follows: with Poof:. Dfi t. Fo th Li bact it is. Fo w hav. callig Dfiitio calculatig fo th syst w ca s that. Howv is idpdt of so that both a zo. It is asy to s that ay Li bact.

13 .. It ca b show by iductio: W fist vify fo. It is. h i bact is show to b: wh I. Mati I is th idtity ati. Now fo. y th stuctu of th fiit lt odl it ca b s that dos ot cotai uatic ts i o i. hfo is idpdt of. O th oth h is also costat sic is idpdt of. hus. Calculatig th sua of this ati o ca vify that which ultiplid by yilds:

14 his ca b witt uivaltly which povs ou sult fo. Suggst ow that it holds fo. h fo uivaltly I which coplts th poof. W a ow y to pst ou ai sult: Popositio : Dfi th suc of atics:

15 I I wh a. If fo so th tdd ati I I has a th th syst is scod od oholooic. Poof: h poof follows fo th Dfiitio th pvious La. h sis dfid abov is dictly associatd with th accssibility distibutio. Idd th vcto filds dfi o Phillip Hall basis fo th accssibility distibutio. It is asily show that vcto filds a lialy idpdt. his is obvious fo by thi dfiitio. O th oth h I s it ca b s by ispctio th vcto filds a lialy idpdt gat a disioal distibutio. Icludig o vcto filds i th st th disio of th distibutio gows. his suc of distibutios i i i G G G G is a filtatio. Each i G is spad by vcto filds of th pvious o plus so vcto filds fod by taig i- Li bacts. If fo so th a of th tdd ati is is as that it cotais lialy idpdt colus. h colus of th ati howv a actly th vcto filds that o would calculat fo th P. Hall

16 basis of th accssibility algba. Choosig lialy idpdt colus o has idpdt vcto filds that spa th accssibility algba of th syst. y Dfiitio is scod od oholooic. Gally as it will b show i th apls systs divd by fiit lts a usually scod od oholooic. h pocss of dtiig th istc of scod od oholooic costaits i systs with possibly hudd dgs of fdo though th covtioal way of sig fo a Phillip Hall basis bcos awfully cubso. O th oth h th algoith ust dscibd povids a idiat way of ivstigatio sic o ca asily autoat th abov pocdu. ig scod od oholooic th fiit lt odl is stogly accssibl. O ca asily vify howv that it dos ot satisfy th sufficit coditio fo sall ti local cotollability pstd i 4. Howv this dos ot ul out th possibility that th syst ay b sall ti locally cotollabl. I fact sic th fiit lt odl is lia a sipl ati calculatio ca show whth that syst is cotollabl. I that cas th stabilizig fdbac law is ot oly cotiuous but also lia! his cos i accodac to ho sic th fiit fiit lt lt F F O O Figu : dfoabl obct ud aial lo coditio fo oistc of a sooth fdbac law satisfid. is ot 5. Eapl - od Ud ial Lo Cosid a ba ud aial lo Figu. h ba is dividd ito two fiit lts. h syst has th dgs of fdo two of which a dictly cotolld. h lt chaactistic atics a: E M C K 6 6 wh! is th dsity of th atial is th coss sctio is th lgth of th lt E th lasticity odulus. h uatios fo th two lts a

17 assbld aft aagig th ts to distiguish th actuatd pat fo th uactuatd th coplt uatios hav th followig fo: F F E E E E E E E E pplyig th liaizig fdbac th abov uatios bco v v v v wh 6 6 E E E s it ca b vifid th lia syst is cotollabl which as that it ca b div with cotiuous lia fdbac law. Idd coditio dos ot hold. St 4 Usig Popositio w ca pov that th syst is scod od oholooic. Notic how th colus i th suc of atics dfid i Popositio giv actly th vcto filds which w to b calculatd fo th P. Hall basis. Lts calculat fist th vcto filds that a fod with up to o lvl of bactig: cotiuig with o lvl of bactig:

18 Cotiuig to high lvls yilds: E E E E Sic vcto filds a idpdt th syst is scod od oholooic. Now otic how ths vcto filds appa plicitly i th tdd ati of Popositio :

19 E E E E L E E L I this cas th patial divativs of a calculatd oly oc at th bgiig of th pocdu whas th dict calculatio of th Li bacts of th vcto filds uis th calculatio of th acobias of th vcto filds ivolvd.

20 6. Matial Costaits h dgs of fdo of th aipulatd obct ay b subct to costaits. hs costaits ca ais fo atial stgth liitatios /o obstacl avoidac uits. Whil th latt apply dictly o th obct dgs of fdo th fo a usually pssd i th fo wh is th stss tso of th stuctu is th aiu issibl stss spcifid fo th paticula atial obct. With so algbaic aipulatio th abov latio ca b taslatd i ts of th fiit lt odl od displacts. If o calls th wll ow pssios that lat th dfoatio th displact U th stss D U E alog with th lt itpolatio fuctios U N th atial costaits ca b pssd i ts of th dgs of fdo. hs costaits ca b icludd i with th us of Kuh-uc ultiplis: M C K u. I this cas th ultiplis caot b liiatd bcaus thy cospod to iuality coditios which do ot duc th disio of th stat spac. h costait ts i th abov uatios ca b icludd though i th lastic gavity focs ts as follows: c c. bu Wh th stss coditios a satisfid vaishs th uatios dscib th otio of th ucostait syst. It should ot b attptd howv to dti th istc of oholooic costaits usig th abov uatios. O th o h wh costaits a spctd th uatios coicid with. If th costaits a violatd it ay b too lat to ai cotollability. O th oth h atial costaits do ot cospod to th atual bhavio of th syst but a ath tally iposd coditios which a statd i od to cofi dfoatios avoid phaps factu. hy a piaily usd fo optiizatio puposs. t ay cas th sults of th pcdig sctios ay o log hold if th iclusio of th atial costaits dstoys th lia stuctu of th lastic t.

21 7. Coclusio dfoabl obct ud aipulatio ca b odld usig fiit lt. his way th distibutd paat odl of th oigial syst is covtd to a fiit disioal udactuatd chaical syst. I this light th class of obcts ud study ca b tdd to iclud systs with cobiatio of igid dfoabl obcts uactuatd oits flibl lis oits ollig cotacts tc. his odlig thod vals a st of dyaic costaits. Dpdig o thi typ sval coclusios about th bhavio of th odl ud cotol statgis ca b daw. Fo ach typ th dfoabl obct odl is tatd accodigly. pat fo ths dyaic costaits howv th could b ditioal os that ca lat to atial stgth liitatios /o obstacl avoidac uits. way to icopoat ths costaits ito th obct odl is dscibd. h study of dfoabl obcts i th fawo of udactuatd chaical systs idicatd th istc of scod od oholooic costaits. Fo th idtificatio of this typ of costaits th ist so havy athatical tools fo olia cotol. h odl divd fo th fiit lt thodology howv is lia o should ot b obligd to us such copl tchius. Fo this pupos a altativ thodology has b dvlopd which dos ot ui spcial athatical sills owldg is also o coputatioally fficit tha th oigial thod. his thod is illustativ with a sipl apl. fcs a H G Kyiaopoulos K 998 Modlig of ultipl obil aipulatos hlig a coo dfoabl obct. oual of obotic Systs 5 : Su D Shi X Liu Y 996 Modlig coopatio of ultipl obil aipulatos hlig a coo dfoabl obct. I: Poc. of th 996 IEEE It. Cofc o obotics utoatio Miapolis Misota pp zopoulos D Flisch K 988 Dfoabl Modls. h Visual Coput 4:6-. 4 Kosug K Saai M Kaitai K Yoshida H Fuuda 995 Maipulatio of a flibl obct by dual aipulatos. I: Poc. of th 995 IEEE It. Cofc o obotics utoatio pp Wu Luo Z Yaaita M Ito K 996 daptiv hybid cotol of aipulatos o uctai flibl obcts. dvacd obotics 5: Yuawa Uchiyaa M Iooa H 996 Stability of Cotol Syst i Hlig of a Flibl Obct by igid obots. I: Poc. of th 996 IEEE It. Cofc o obotics utoatio Miapolis MN pp. -8.

22 7 a H G Kyiaopoulos K 999 alysis of Dfoabl Hlig. I: Poc. of th 999 IEEE It. Cofc o obotics utoatio Dtoit Michiga pp Spog M W 998 Udactuatd chaical systs. I: Siciliao Valavais K P ds 998 Cotol Pobls i obotics utoatio Lctu Nots i Cotol Ifoatio Scic Spig pp oothby M W 986 Itoductio to Difftiabl Maifolds iaia Goty d d. cic Pss Ic. yhaoglu M va d Schaft McClaoch N H Kolaovsy I 996 Nolia cotol of a class of udactuatd systs. I: Pocdigs of th 5 th IEEE Cofc o Dcisio Cotol Kob apa pp Capio G D' da-novl asti G 99 Modlig stat fdbac cotol of oholooic chaical systs. I: Pocdigs of th 99 IEEE Cofc o Dcisio Cotol ighto Egl. Niih H va d Schaft 99 Nolia Dyaical Cotol Systs Spig-Vlag. ao S S 989 h fiit lt thod i Egiig Pgao. 4 D Luca Matto Oiolo G 996 Dyaic obility of dudat obots usig d-ffcto cos. I: Poc. of th 996 It. Cofc o obotics utoatio Miapolis MN pp