Geometric Motion Estimation and Control for Robotic-Assisted Beating-Heart Surgery

Transcription

1 Geometi Motion Estimation and Contol fo Rooti-Assisted Beating-Heat Sugey Vinent Duindam and Sanka Sasty Depatment of EECS, Univesity of Califonia, Bekeley, CA 97, USA Astat One of te potential enefits of ooti systems in adia sugey is tat tei use an inease te nume of possile off-pump eating eat) oonay atey ypass gafting poedues. Rooti systems an atively synonize te motion of sugial tools to te motion of te sufae of te eat, wit te sugeon speifying only te elative motion of te tool wit espet to te eat. Auate pedition of te motion of te eat sufae is oviously of uial impotane fo te safety and oustness of su a system. Tis pape pesents a novel appoa to pedit te motion of te sufae of a eating eat. We sow ow ECG and espiatoy infomation an e used to extat two peiodi omponents fom te quasi-peiodi motion of te eat sufae. Contay to most existing liteatue, we onside te full geometi motion, inluding otation due to espiation. We ten sow ow to omine te peiodi omponents to auately pedit futue motion of te eat sufae, and ow tis infomation an e used to design an expliit ontolle tat asymptotially stailizes te elative motion of te sugial tool to a desied elative distane and oientation. Ψ d diapagm eat Ψ Ψ Ψ I. INTRODUCTION Rooti assisted sugey is eoming ineasingly ommon, and advaned ooti systems ae moe widely availale in ospitals. One of te advantages of ooti tools is te potential to failitate moe advaned minimally invasive sugey poedues, wi ave te enefits of sote eovey time and less invasive saing. Altoug uent ooti tools ae usually pogammed to only follow te sugeon s movements, tei tue potential lies in te possiility to ontol tem in a moe ative way. One of te appliations in wi tis ould make a diffeene is in adia opeations su as oonay atey ypass gafting CABG). Cuently, many adia poedues equie te eat to e stopped and a eat-lung maine to e used) to allow sugey on a motionless sufae. Howeve, te use of a eat-lung maine an ave damaging effets fo te patient and sould teefoe e used as little as possile. Atively ontolled ooti tools ould e employed instead, and e synonized to move at a fixed elative distane and oientation wit espet to te eat sufae see 1,, 3 fo moe details on tis idea). Toug tis elative stailization, togete wit stailized visual feedak, 5 fom te opeating aea to te sugeon, a sugeon only needs to povide te desied elative motion of te tool wit espet to te eat, tus possily eliminating te need to stop te eat. Fig. 1. Semati system epesentation and definition of vaious oodinate fames. Te motion of a ooti tool wit attaed fame Ψ ) is to e synonized to te motion of a etain small aea on te eat sufae fame Ψ ). Te motion of te eat sufae is aused y a omination of motion of te diapagm due to eating fame Ψ d ) and te eating motion of te eat itself. One of te most uial and in some sense most diffiult) aspets of su a motion synonization system is te auate measuing and pedition of te motion of te eat sufae. Fist, it equies auate sensos to measue te motion of te sufae, and seveal diffeent tenologies ave een poposed in liteatue, inluding ig-speed ameas and piezoeleti ystals 7. But seondly, and tis is te topi of te pesent pape, te motion of te eat sufae is quasi-peiodi, wi means it is te omination of two peiodi motions, as illustated in Fig. 1. One peiodi motion is aused y te diapagm, wi moves at te fequeny of espiation, and te ote peiodi motion is aused y te eating of te eat itself. Te appoa in liteatue to pedit tis quasi-peiodi motion is to assume tat it is simply te sum of two peiodi omponents, and ten to sepaate tese two omponents using e.g. a low-pass filte, an adaptive amoni filte ank 1, o diet time seies analysis. As sown in Setion III, oweve, tis impliitly assumes tat te espi-

2 ation omponent does not ontain a otation, wi may not e auate enoug fo ig-peision appliations 9. In addition, te metods only onside te motion of points, disading otational infomation, even toug te oientation of te eat sufae needs to e known in ode to popely oient te sugial tool. Te pupose of tis pape is to pesent a geneal metod to sepaate a quasi-peiodi igid 3-D motion into its two peiodi omponents, and to use te esult in te design of an asymptotially synonizing ontolle. We assume te fequenies of te two motion omponents to e known, fo example ased on an ECG signal and infomation fom te meanial ventilato used duing sugey. Te estimated omponents ae ten omined to pedit futue motion of te eat sufae. Tis infomation, in tun, is used in te design of an expliit ontolle tat stailizes te elative motion of a sugial tool to a desied distane and oientation wit espet to te eat sufae. Afte te neessay matematial peliminaies in Setion II, we disuss in Setion III tat te total motion of te eat sufae is a nonlinea) geometi penomenon, and sow ow te peiodi espiatoy and adia motions an e extated. We ten pesent a ontol law Setion IV) tat uses te pedited motion to asymptotially stailize te motion of te sugial tool to a desied distane and oientation wit espet to te eat sufae. We illustate te esults in simulations. II. MATHEMATICAL PRELIMINARIES We fist iefly summaize te teoy and notation used to epesent igid 3-D motions and veloities. We efe to 1 and 11 fo a moe extensive akgound. Te elative onfiguation of a igt-anded oodinate fame Ψ a wit espet to a igt-anded fame Ψ, meaning tei elative position and oientation, an e desied y an element of te Lie goup SE3), te Speial Eulidean goup. Numeially, tis onfiguation an e expessed y a time-vaying omogeneous matix H a t) of te fom H a R a t) := t) p a t) 1) 1 wee R at) is a 3 3 otogonal matix wit detra ) = 1 tat desies te elative oientation, and p a t) is a 3 1 veto tat desies te elative position of te oigins. A otation an not only e desied y an appopiate 3 3 matix, ut also y so-alled unit quatenions, wi an e tougt of as a set of fou eal numes q,q) := q,q 1,q,q 3 ) R ) satisfying q,q) q, q) = 1,), wit te nonommutative) Gassman podut defined as q,q) p,p) := q p q T p,q p + p q + ˆpq) 3) wit ˆx := x fo x R 3. Unit quatenion epesentations an e onveted to matix epesentations and ak) toug elatively staigt-fowad omputations. Te elative veloity of two fames, meaning ot te linea and angula veloity, an e onisely epesented y an element of se3), te Lie algea of SE3). Numeially, it an e expessed as a 1 veto T,a o matix ˆT,a of te fom T,a ω,a,a := v,a,a ˆω o ˆT := v,a ) wit ω,a,v,a R 3 te elative angula and linea veloity. Te veto T,a is alled a twist and desies te elative veloity of fame Ψ wit espet to Ψ a, expessed in fame Ψ. It is elated to te elative onfiguation 1) as ˆT,a = H aḣa H 5) wit Ḣ denoting te time-deivative of H. Finally, te following two elations ae used in Setion IV. T d,a = Ad H d T,a R d wit Ad H d := ˆp d R d R d d ) d,d ˆω AdH d = adt d,d Ad dt H d wit ad T d,d := ˆv d,d ˆω d,d III. QUASI-PERIODIC MOTION PREDICTION IN SE3) Te motion of te eat sufae is aused y te omined dynamis of espiation mainly affeting te eat toug motion of te diapagm) and adia musle ontations. Te full dynamis ae vey omplex and ad to pedit, and equie many patient-speifi paametes to e fitted 1. To simplify te analysis and allow eal-time pedition, we do not take te ause of te motion into onsideation, and only onside te kinematis of a loal aea of inteest on te eat sufae. Its motion is assumed to e a omination of two peiodi motions: one due to espiation wit peiod T d, wit d fo diapagm ), and one due to eat eating wit peiod T, wit fo eat ). Te omination of two peiodi motions is alled a quasi-peiodi motion. Sine te two peiods T d and T ae geneally not intege multiples of ea ote, te peiod of te total motion te time afte wi it epeats itself) an e aitaily long. As te fequenies of te two peiodi motions ae diffeent typially T d s and T 1 s), a low-pass filte may seem a elatively simple solution to sepaate te espiatoy omponent. Howeve, tis assumes tat te espiation is a puely sinusoidal motion witout ige amonis, and moeove, tat te two signals mix linealy. As sown in Setion III-A, tis seond assumption does not old wen otation is taken into aount. In addition, a ausal) low-pass filte tat leanly distinguises etween two signals wit T d and T in tis lose ange will geneally suffe fom a lage pase sift tat an ause staility and pefomane polems in feedak loops. Fotunately, in tis appliation, we ave a easonaly good estimate of te pase 1 and fequeny of te two motion 1 Te peise definition of pase is not impotant in tis ontext; we an oose any nume tat indiates te pat of te yle as a funtion of time. Fo example, it is suffiient to oose a linea intepolation fom zeo to one etween evey two onseutive QRS-omplexes in te ECG yle.

3 omponents: te espiatoy pase and fequeny an e otained fom te meanial ventilato, and te adia pase and fequeny ae deteted y an ECG monito. We an exploit tis pio knowledge to extat te two peiodi omponents fom measuements of te omined motion. A. Heat motion as a goup podut As te fist step to aateize te motion of te eat sufae, we define wat we mean exatly y omination of two motions. Sine ot te espiatoy motion and te adia motion geneally ontain tanslation as well as otation, we an wite te elative displaement of te eat sufae as a goup podut of two omogeneous maties Hd and H d, as illustated in Fig. 1. Te matix H d desies te elative motion of te diapagm Ψ d wit espet to an inetial fame Ψ, and H d desies te elative motion of te aea of inteest Ψ on te eat sufae wit espet to Ψ d. Ea omponent is assumed to ave a ougly onstant) peiod ut unknown sape, i.e. H dt + T d ) = H dt) H d t + T ) = H d t) ) Te omination of te two motions is te podut of te two maties, i.e. H = HdH d R = d R d p d + R d pd 7) 1 It is impotant to note tat, even if we ae only inteested in te elative positions, te omination of te positions is not a simple linea sum of te vetos p d and pd, ut also nonlinealy depends on te otation of te diapagm Rd. As sown in esea on adia image egistation 9, even toug te otation of te diapagm is geneally only a few degees, it an still ause sustantial deviations if not popely taken into aount. Fo tat eason, we onside in tis pape ot otation and tanslation omponents. Te oientation of te eat sufae in itself is also a useful signal to pedit, sine geneally we want to stailize ot te elative position and te elative oientation of te sugial tool wit espet to te eat sufae. Note tat te pysial loation of te intemediate fame Ψ d is not lealy defined, wi esults in amiguities. Moe peisely, given any omination Hd,Hd ) tat satisfies 7), te omination Hd H, H 1 H d ) also satisfies 7), fo any H of te fom 1). Tis amiguity an e esolved y oosing a onvenient initial position and oientation fo Ψ d, e.g. su tat p d ) = and R d ) = I. B. Filteing peiodi omponents Given a geneal quasi-peiodi signal xt), omposed of two peiodi omponents x 1 t) and x t). Suppose we ave suffiient measuements of xt) and of te two pases φ 1 t) and φ t) of te omposing signals, ten we an estimate te peiodi omponents x 1 φ 1 ) and x φ ) temselves as funtions of tei espetive pases. Te appoa we take is illustated in Fig.. We paameteize ea peiodi signal x φ 1 φ t t t B11B1 B1k B1nB11 φ 1 ompute optimal a ij B1 B Fig.. Semati epesentation of te algoitm. Based on measuements of a quasi-peiodi signal x and te pase tajetoies of its two peiodi omponents, and given a oie of asis funtions B ij, te optimal paametes a ij ae omputed. Te esulting epesentation of te omponents x i φ i ) an e used to pedit futue xt). as a sum of asis elements B ij φ i ) and wite te total signal xt) at time t as xt) = x 1 φ 1 t)) x φ t)) ǫt) ) ) ) = a 1k B 1k φ 1 t)) a m B m φ t)) ǫt) k wee denotes te omposition of te two peiodi signals, ǫt) is a measuement eo, and a ij ae te weigts fo te asis elements. Given suffiient measuements, we an estimate te paametes a ij su tat ǫt) is minimized, e.g. in te least-squaes sense. Depending on te oie of asis funtions B ij, te oeffiients may not e uniquely detemined. Fo example, te two peiodi omponents an only e detemined up to a onstant: if denotes te egula sum opeato, ten adding a onstant to one of te omponents and sutating te same onstant fom te ote omponent does not ange tei omposition, and similaly wen is multipliation. Ote amiguities will ou if te asis funtions ae osen in an unfotunate way tat makes tem fit in ot peiodi omponents. We now apply tis geneal appoa sepaately to te otation and tanslation omponents of te quasi-peiodi motion H t), as given y 7). We stat wit otation and oose to use a epesentation in unit quatenion oodinates, as desied in Setion II. Unit quatenions povide a good alane etween minimizing te dimension of te signal spae fou fo quatenions) and simplifying te expession fo omining two otations defined y te Gassman podut fo quatenions). Altoug oosing exponential oodinates 13 would fute edue te dimension of te signal spae to tee, witing te podut of two otations in tems of tei exponential oodinates podues a quite involved expession, wi is mu ade to optimize tan te simple i-linea expession fo quatenions: Bk m Qt) = ǫt) Q φ t)) Q d φ d t)) 9) in wi Qt) is te quatenion expession fo te measued total otation at time t, Q d and Q ae te quatenion expessions fo te otations Rd and Rd, espetively, ǫt) desies φ B1 x 1 x φ 1 φ

4 te measuement eo lose to te 1, ) quatenion), and is te Gassman podut defined in 3). We paameteize ea omponent in 9) as in ) y oosing a set of asis funtions B ij fo te quatenions Q d φ d ) and Q φ ). Wit tis oie and suffiient measuements Qt), te next step is to estimate te oeffiients a ij. An exta ompliation aises ee eause of te onstaint tat Q d and Q sould ave unit lengt at all times, and ene tei fou omponents annot e osen aitaily. Tis an e solved y inluding onstaints in te optimization poess, o, moe pagmatially toug not matematially sound, to just feely optimize ove all quatenions and nomalize te esult aftewads. A seond aspet is te oie of meti fo te optimization, i.e. ow to judge wat ǫ in 9) is minimal. Te simplest solution is to minimize Q 1 Q d Q ) 1,) in te least squaes sense, o even just Q Q d Q. Bot ae not pope metis on SO3) ut ae easy to optimize and give good esults in simulations. One te otation omponents, in patiula Rd, ave een detemined, we an wite te estimation polem ) fo te tanslational omponents as follows. p t ) = R dt )p d φ t )) + p dφ d t )) + ǫt ) 1) wit p and R d known and pd and p d to e estimated. Tis elation is unonstained and linea in te unknowns and an e solved e.g. y a standad linea least squaes omputation. Fo ot otation and tanslation omponents, seveal design oies ave to e made. Fist, te nume of asis funtions B ij φ i ) and tei sapes need to e detemined. Seondly, a meti on te data spae needs to e osen, i.e. a mapping tat weigs te elative influene of olde and newe samples in te estimation of te oeffiients a ij and ene in te pedition of te futue signal. If te signal is igly peiodi, ten olde samples sould e weiged as well as new samples, su tat any andom noise is aveaged and tus edued. If te sape of te signal anges signifiantly ove seveal peiods, ten only te newe samples sould e taken into aount. Studying expeimental data fom vaious patients sould elp make tese oies. One te oeffiients a ij fo otation and tanslation ave een detemined and ene te sapes of te two peiodi omponents of H ae known, futue values of H an e pedited y omining te values of te two omponents Hd φ d) and H dφ ), evaluated at tei pedited pases. Te pedition of te pases φ d and φ an e a simple extapolation at te uent espiation and eat eat ate, espetively. Tese ates ae stailized y a meanial ventilato and y mediation and geneally only vay quite slowly. Fig. 3 sows a simulation of te extation of te peiodi diapagm motion wit peiod s) and te peiodi adia motion wit peiod 1.1 s) fom an atifiial omined motion signal, ontaminated wit zeo-mean Gaussian noise wit a standad deviation of.1. We used equally spaed tiangula windows as te asis funtions fo ea omponent. Fom te measuements noisy signals in te top gaps) etween and seonds, te oeffiients a ij wee optimized omined motion diapagm motion adia motion omined motion diapagm motion adia motion istoy pedition a) Rotation omponents. ) Tanslation omponents. istoy pedition Fig. 3. Extation of diapagm peiod s) and adia peiod 1.1 s) motions fom noisy measuements of te omined motion. Te figues sow te tee degees of feedom fo otation and tanslation, wit x solid), y dased), and z dotted) omponents. Reonstuted signals ae sown on top of measued noisy signals. and te peiodi signals eonstuted lean signals in te ottom gaps, ovelaid on te noisy eal peiodi soue omponents, wi ae unknown to te algoitm). Using te otained paametes and te asis funtions, te omined signal an e eonstuted t < s) and its futue values t > s) an e auately pedited. Note tat Fig. 3a) sows tee omponents fo ea otation exponential oodinates) altoug te optimization was pefomed using fou omponents quatenions). IV. EXPLICIT ASYMPTOTIC CONTROL FOR RELATIVE MOTION SYNCHRONIZATION Afte omputing a pedition of te motion of te eat sufae, te next step is to synonize te motion of te

5 sugial tool to te motion of te aea of inteest. Tis polem is usually ast as a tee-dimensional position taking polem and often solved using a fom of adaptive) model peditive ontol 1,. In tis setion, we genealize te polem to inlude not only position, ut also oientation infomation, and we pesent an altenative expliit modelased ontolle. A. Coie of ojetive funtion Te ojetive of te ontolle is to stailize te elative onfiguation of te oot wit espet to a etain aea on te eat sufae. Tis elative onfiguation is desied y a omogeneous matix of te fom H = R p 1 =: x y z p 1 11) wit index fo oot. To quantify te ontol ojetive, we oose an eo funtion JH ) as follows JH ) := J tan + J ot := 1 k pp z ) T p z ) + k x 1 e T x x ) + k y 1 e T y y ) + k z 1 e T z z ) 1) wit te desied elative distane etween te tool and te eat, E := e x e y e z a otation matix desiing te desied oientation of te tool fame wit e z =,,1)), and k p,k x,k y,k z > onstant paametes. Te funtion J as a minimum equal to zeo only if p = z and R = E, i.e. te oigin of Ψ is along te sufae nomal away fom te oigin of Ψ, and te axes of te tool fame ae popely aligned. Te eo funtion ineases fo distanes diffeent fom and fo deviations fom te desied oientation speified y E. Note tat wen k x = k y = k z, J ot is popotional to 3 taee T R) = 1 osθ), wit θ te angle of otation away fom E. One of te advantages of te ost funtion 1) is tat its time-deivative along te tajetoies of te system takes a patiulaly simple fom, as sown elow. J = k p p z ) T v ˆpω + ˆ z ω + k x e T x ˆ x ω + k y e T y ˆ y ω = kx ê x x + k y ê y y + k z ê z z k p p z ) ) + k z e T z ˆ z ω T ω v =: dj) T T 13) Wee we used te fat tat, fom 5) and 11), we ave ˆω Ḣ = x ˆω y ˆω z ˆω p + v 1) and tat ˆp ˆ z )p z ) =. Equation 13) is used in te following setions, as is te expession fo te timedeivative of dj given elow. dj) = kx ê xˆ x ω k p v k y ê yˆ y ω ˆpω + ˆ z ω kx ê = xˆ x k y ê yˆ y k z ê zˆ z k p ˆp ˆ z ) k p I k z ê zˆ z ω ) T 15) B. Asymptotially stailizing ontol law Using te eo funtion 1), we popose te following expliit ontol law fo te oot tat moves te sugial tool. We do not onside speifi oot dynamis at tis point and only speify te desied inetial aeleation T, of its end effeto fame Ψ. Fo a patiula oot, tis desied aeleation sould e aieved y a suitale lowe level ontolle. Te poposed desied aeleation is te following. T, ) des = T, Ad H K 1 dj) + ad T, T, Ad H K T + K 1 dj ) 1) wit K 1,K > two symmeti positive-definite maties, and H, T,, and T, te estimated onfiguation, veloity, and aeleation of te fame Ψ at te aea of inteest on te eat sufae. Intuitively, tis ontolle dives T to K 1 dj y a gain K ), wi makes te onfiguation move along te dietion of steepest desent of J defined using K 1 as a meti). To pove asymptoti staility of te equiliium J = fo te losed loop system, we onside te following andidate Lyapunov funtion. V := κ 1 J + 1 T + K 1 dj ) T K 1 T + K 1 dj ) wit < κ 1 < σ min K 1 ), i.e. κ 1 is stitly less tan fou times te smallest singula value of K 1 wi is stitly positive sine K 1 > ). Assuming te oot aieves pefet taking of T, ) des, we an ompute te time-deivative of V along system tajetoies as follows. V = κ 1 dj) T T = κ 1 dj) T T + T T + K 1 dj) T K 1 T + K 1 dj) T T + K 1 dj ) + K 1 dj) = T + K 1 κ ) T 1 I)dJ T + K 1 κ ) 1 I)dJ κ 1 dj) T K 1 κ ) 1 I dj 17) Tis expession is non-positive sine we ose κ 1 su tat K 1 κ1 is stitly positive definite. Tus, V = only wen T = and dj =. Fom 13), we see tat dj = wen te following two equations ae satisfied. = p z 1) = k x ê x x + k y ê y y + k z ê z z 19) Te fist equation as only one solution, p = z. Te seond equation, togete wit te onstaint tat x, y, z ) defines an otonomal fame, gives ise to seveal possile solutions. Only te solution R = E is stale, all ote solutions ae unstale and oiented at least 9 degees away fom E, ene tey an e safely ignoed fo patial puposes. By La Salle s Piniple 1, tis poves tat te poposed ontol law 1) semi-gloally asymptotially stailizes te system to te desied state. Fig. sows a simulation of te poposed ontolle applied to te ideal oot system T, = T, ) des )

6 1 V J J tan J ot ange in e x,e y ) ange in a) Time tajetoies of te Lyapunov and eo funtions. ) Snapsots as viewed fom te tool fame Ψ along te negative z dietion plus sign denotes te ente of te pitue). Fig.. Simulation of te ontol law applied to an ideal oot. wit all ontol gains set to unity. Stating fom some initial onfiguation and veloity, and wit H t) an aitay motion simila to Fig. 3, te position and oientation of te oot tool onvege to te desied elative distane and oientation. At t = 7, te desied oientation of te tool, speified y E, is anged y 9 degees aound te vetial axis te viewing dietion). At t = 1, te desied distane is edued. In ot ases, te system eoves fom te step ange in te efeene signals and onveges asymptotially to te new equiliium. V. CONCLUSIONS AND FUTURE WORK Tis pape pesents a novel metod to extat peiodi 3-D motion signals fom noisy measuements of a quasi-peiodi motion and knowledge of te pase of te omposing signals. Te total quasi-peiodi motion of a egion on te sufae of a eating eat is a nonlinea omination of espiatoy and adia motions, if otation of te diapagm due to espiation is taken into aount. Using a quatenion epesentation, te nonlinea signal estimation polem an e fomulated as a easonaly simple optimization polem, fom wi te peiodi espiatoy and adia motions an e extated and futue eat sufae motions an e pedited. Te pedited motion signal an ten e used in te design of an expliit model ased ontolle tat asymptotially synonizes te motion of a sugial tool to te motion of a egion on te eat sufae. Tis ontolle povides an altenative to te impliit) model-ased ontol algoitms known fom liteatue. In futue wok, we plan to extend te motion sepaation algoitm to a eusive vesion tat an quikly eompute te optimal motion paametes wen a few new samples aive. We will also fute investigate wat sensos and sensing teniques an most elialy and oustly extat te eat motion, wi is te input fo te pesented algoitm. Wen measuements of te full igid motion of te eat sufae ae availale, we an veify te signal estimation and ontol appoa pesented in tis pape on eal expeimental data, and tune te vaious paametes in te algoitms. ACKNOWLEDGMENTS Tis esea is sponsoed toug a Ruion gant fom te Netelands Oganization fo Sientifi Resea NWO). REFERENCES 1 R. Ginoux, J. A. Gangloff, M. F. de Matelin, L. Sole, M. M. A. Sanez, and J. Maesaux, Ative filteing of pysiologial motion in ootized sugey using peditive ontol, IEEE Tansations on Rootis, vol. 1, no. 1, pp. 7 79, Feuay 5. O. Beek and M. C. Çavuşoğlu, Peditive ontol algoitms using iologial signals fo ative elative motion anelling in ooti assisted eat sugey, in Poeedings of te IEEE Intenational Confeene on Rootis and Automation, May, pp T. J. Otmaie, Motion ompensation in minimally invasive ooti sugey, P.D. dissetation, Tenial Univesity of Muni, Ma 3. M. Göge and G. Hizinge, Image stailisation of te eating eat y loal linea intepolation, Medial Imaging : Visualization, Image-Guided Poedues, and Display, vol. 11, pp. 7 5, Ma. 5 D. Stoyanov, M. ElHewl, B. P. Lo, A. J. Cung, F. Bello, and G. Z. Yang, Cuent issues in potoealisti endeig fo vitual and augmented eality in minimally invasive sugey, in Poeedings of te IEEE Intenational Confeene on Infomation Visualization, 3, pp R. Ginoux, J. A. Gangloff, M. F. de Matelin, L. Sole, M. M. A. Sanez, and J. Maesaux, Beating eat taking in ooti sugey using 5 Hz visual sevoing, model peditive ontol and an adaptive oseve, in Poeedings of te IEEE Confeene on Rootis and Automation, Apil, pp M. B. Ratliffe, K. B. Gupta, J. T. Steie, E. B. Savage, D. K. Bogen, and L. H. Edmunds, Use of sonomiomety and multidimensional saling to detemine te tee-dimensional oodinates of multiple adia loations: Feasiility and initial implementation, IEEE Tansations on Biomedial Engineeing, vol., no., pp , June T. Otmaie, M. Göge, D. H. Boem, V. Falk, and G. Hizinge, Motion estimation in eating eat sugey, IEEE Tansations on Biomedial Engineeing, vol. 5, no. 1, pp , Otoe 5. 9 G. Sete, C. Oztuk, J. R. Resa, and E. R. MVeig, Respiatoy motion of te eat fom fee eating oonay angiogams, IEEE Tansations on Medial Imaging, vol. 3, no., pp. 1 15, August. 1 R. M. Muay, Z. Li, and S. S. Sasty, A Matematial Intodution to Rooti Manipulation. CRC Pess, S. Stamigioli, Modeling and IPC Contol of Inteative Meanial Systems A Coodinate-fee Appoa. Spinge-Velag, 1. 1 R. Haddad, P. Claysse, M. Okisz, P. Coisille, D. Revel, and I. E. Magnin, Funtional Imaging and Modeling of te Heat, se. Letue Notes in Compute Siene. Spinge, 5, vol. 35,. A Realisti Antopomopi Numeial Model of te Beating Heat, pp J. M. Selig, Geometi Fundamentals of Rootis, nd ed. Spinge- Velag, 5. 1 S. S. Sasty, Nonlinea Systems: Analysis, Staility, and Contol, se. Intedisiplinay Applied Matematis. Spinge, 1999.