Kinematic Error Correction for Minimally Invasive Surgical Robots

Transcription

1 Knematc Error Correcton for Mnmally Invasve Surgcal Robots Ryan A. Beasley an Robert D. Howe Dvson of Engneerng an Apple Scences Harvar Unversty Cambrge, USA {rbeasley, Perre E. Dupont Department of Aerospace an Mechancal Engneerng Boston Unversty Boston, USA Abstract Robots are useful tools n mnmally nvasve surgery, provng benefts such as reucton n han tremor, navgaton, an workspace scalng. Unfortunately, mnmally nvasve confguratons result n two lkely sources of knematc error: port splacement an nstrument shaft flexon. For a quasstatc system, a measure s presente that relates the errors n the robot Jacoban to the angular fference between esre motons an actual motons. Smulatons an expermental ata emonstrate ths measure for a laboratory system. One potental use for the presente measure s, for boune errors, etermnng whether the system wll monotoncally converge for all ntal an esre postons n the workspace. In aton, the measure s useful for path plannng, etermnng less error-prone paths. Keywors-surgcal robot, mage guance, knematc error I. INTRODUCTION Mnmally nvasve surgery (MIS) uses long nstruments nserte through small ports n the patent's boy. Ths technque greatly reuces patent trauma but ncreases surgcal ffculty ue to ecrease exterty an restrcton of vsual, tactle, an proproceptve feeback. Robot assstance can help offset such complcatons by provng nstrument localzaton, mage guance [1], navgatonal tools (e.g., vrtual fxtures [, 3]), ampenng of han tremor [4], an workspace scalng [5]. Unfortunately, laboratory an clncal trals emonstrate that errors n the robotc system result from uncertan knematcs, unmoele ynamcs, an unmoele forces [, 6]. Internal mammary artery takeown s an example task that exhbts such errors. To reach the surgcal ste near the chest wall, the nstrument shaft apples sgnfcant torque to the port. The moments an forces create urng the operaton cause the patent s rbs to flex, the port to move, an the nstrument shaft to ben. These knematc errors mpar postonng of the robot an cause evatons from the esre motons. Reucng the postonng precson reuces the navgatonal benefts of the robot. In the worst case, a slave robot may move n rectons other than those of the master robot, ecreasng patent safety. Much work has been one on reucng knematc error. Offlne calbraton calculates exact knematc parameters, but oes not account for onlne errors ue to nteractons wth unstructure envronments [7, 8]. Estmatng the benng of the nstrument shaft s problematc ue to unmoele forces at the port an wthn the patent. Research nto flexble robotcs has focuse on ynamc flexblty [9, 1]. Such research assumes known statc knematcs, whch s not the case when there s statc eformaton aganst the patent. Measurng the poston of the tp of the nstrument shaft usng cameras or magnetc poston trackers s another approach. Such measurements elmnate the problem of errors n the forwar knematcs, but o not rectly solve the problem of errors n the nverse knematcs. One metho for usng that poston nformaton to reuce the effects of errors n nverse knematcs s by onlne estmaton of the local Jacoban [11, 1]. Unfortunately, sgnfcant moton n a sngle recton can lea to errors n these estmates. Another metho of usng poston measurements to correct for knematc errors s to use a controller base on the nomnal Jacoban. Cheah, et al., erve a relatonshp between error n a transpose Jacoban controller an the controller gans that result n asymptotc convergence to the commane poston. Whle that result s useful, for surgcal robotcs t s mportant to have more nformaton, partcularly concernng moton of the robot before convergence. For robots usng an nverse Jacoban controller, ths paper presents a measure of the postonng error n the controller base on the error n the Jacoban. That measure can be use to etermne whether the system s monotoncally convergent, an to estmate the mum angular fference between esre motons an actual motons as prouce by the controller. The mum angular fference can then be use to etermne how closely the system woul follow commane motons. Smulatons an expermental ata are use to llustrate the measure for a laboratory robot system. Practcal uses for the measure are then scusse such as prectng monotonc convergence, path plannng, an robot esgn. II. ANALYSIS OF CONTROLLER An eal robotc system moves to the commane poston an oes not stray from the most rect path to that poston. The frst trat translates to asymptotc convergence an the secon trat s relate to monotonc convergence. For esgn an safety reasons, t s mportant to have a measure of the effect that errors n the controller have on these propertes of the system. The measure evelope here uses a quas-statc assumpton to calculate both the Cartesan error length after a Ths work was supporte by the Natonal Scence Founaton uner grant EEC

2 Exstng robot controller Poston tracker x x x Ĵ 1 c k k c Jont level control τ Arm & nstrument moton an the angle between esre an actual motons. Possble applcatons n control of orentaton have been left for future work. A. Controller an Assumptons An mportant use of surgcal robots s for mage-gue surgery. The surgeon watches mages from cameras an preoperatve scans whle manpulatng the master robot. The motons of the master are cope by the slave robot, whch hols a surgcal nstrument. In MIS, nstruments have long shafts that pass through a small port n the patent's boy. Unfortunately, knematc errors n the slave robot result n ncorrect mage guance, navgatonal as (e.g., vrtual fxtures), an robot motons. Drectly measurng the robot poston wth cameras or a magnetc tracker can fx the problems wth mage guance, but the actual motons of the robot woul stll not match the commane motons. Focusng on MIS robots nfluences the choce of feeback controllers. Many surgcal robots have a bult-n jont-level controller. That controller s runnng wth a hgh servo rate an has crtcally ampe or over-ampe behavor, wth propertes set at ntal esgn. Atonally, the jont-level controller has regulatory approval an has been use n thousans of surgcal cases, factors that nhbt mofcatons. Often the knematcs for the master an slave robots are ssmlar, so an nverse Jacoban controller s practcal for the outer loop controller. The nverse Jacoban controller requres the current poston of the robot, whch can be measure va camera or magnetc poston sensor. Combnng these nto one system (Fg. 1), the esre poston for the robot can be set an the robot wll move to that poston, barrng knematc errors. Whle contacts between the nstrument shaft an the envronment can ntrouce errors, these contacts have the beneft of reucng the effects of system ynamcs. The contacts ncrease the ampng of the system, leang to a quasstatc stuaton. Crtcally ampe jont-level control also leas to a quas-statc stuaton ue to ts mnmum settlng tme an no overshoot. Wth these features, a quas-statc assumpton hols as long as the commane angular changes are small enough to be complete before the next upate of the nverse Jacoban controller. To satsfy ths constrant, the moton scalng gan of the controller s ecrease untl the system behaves n a quas-statc manner. Fgure 1. Block agram of controller. B. Dervaton of Error Measures For an nverse Jacoban controller wth an mperfect n Jacoban, the commane change n jont angles, c where n s the egrees of freeom for the robot, s calculate as ˆ kj 1 x x kjˆ 1 x, (1) c 3 3 where x s the current Cartesan poston, x s the 3 esre Cartesan poston, x s the esre moton n Cartesan space, Ĵ s an estmate of J, the true system Jacoban, an k s a moton scalng gan. For nfntesmal steps an an eal quas-statc system, the resultng moton s ˆ 1 x J kjj x. () c Therefore the new poston after completng the moton s ˆ 1 x xx kjj x x, (3) new an the error after completng the moton s ˆ 1 x x x x xkjj x. (4) A mappng, G, k new new, s erve from x to xnew. ˆ 1 x G, k x I kjj x. (5) new To analyze convergence, the nuce Euclean norm of G, k can be efne n the stanar way. where G G k Gx G, k G, (6) x x enotes the mum sngular value of G. If, 1,, the Cartesan error length s always smaller after a step than before the step. Therefore, the Cartesan error length monotoncally ecreases to zero wth subsequent steps, makng the controller monotoncally an asymptotcally convergent. Let v be the rght sngular vector that correspons to the mum sngular value, G. Note that v s the esre recton that results n the largest x. Also note, nether new G, k nor v have any applcaton to the orentaton of the robot. Whle G, k s useful, a potentally more mportant measure s the angle,, between the esre moton, x, an the actual moton, x (Fg. ). If s small for all possble motons across the workspace, then for a teleoperate system

3 x the slave robot wll accurately follow the motons of the master. Let be the mum absolute value of over all esre motons, for a specfc jont confguraton,. From lnear algebra, Gv σ u, (7) where v are the rght sngular vectors of G, u are the left sngular vectors of G, an σ are the corresponng sngular values. So pars of nput an output vectors can be efne where the nput vectors, v, are unt length, m m m v v, 1 u u, (8) 1 1 G 1 where m=m(x). Then T 1 cos v v u v v u. (9), but Equaton (9) calculates the exact value of requres searchng over a sphere of raus 1, nvokng computatonal requrements tolerable for offlne calculatons but that may be too great for real-tme use. Instea, an upper boun on can be etermne geometrcally from G, k (Fg. ). Note that x new must le wthn a sphere of raus G, k x from x. The conservatve estmate, ˆ,k, of s constructe by assumng the actual moton has the largest possble. Then the actual moton s tangental to the surface of the sphere about x, puttng a rght angle between the actual moton an the raus of the sphere. The resultng rght trangle has the esre moton as ts hypotenuse. An upper boun on s therefore ˆ x ˆ, k sn x x new Fgure. Geometry of G, k x x, 1. 1 x esre G G G (worst angle case) G k x ˆ., 1 sn, 1 (1) Let p be the recton of esre moton that results n. Note that there s no smple relatonshp between p an v. v s erve from G, k, a measure of the mum length of postonal error, unlke, whch rectly measures the angular error. III. EXAMPLES To emonstrate the varous measures an varables, a smplfe MIS robot system was smulate quas-statcally usng the controller n Fg. 1. Knematc errors ue to splacement of the nstrument port were ntrouce, an, G, k, ˆ, p, an v were calculate across the workspace of the system. Ths source of error was chosen for ts ease of mplementaton an straghtforwar mplcatons. Atonally, the expermental setup (Fg. 3), usng the controller n Fg. 1, generate trajectores from varous ntal an esre postons n the presence of knematc errors, an these trajectores were compare wth trajectores calculate va smulaton. A. Computer Smulaton The smulaton consste of a planar two-lnk robot attache to an nstrument shaft. Fg. 4 splays the workspace an confguraton for both the smulaton an experment. The workspace of the robot s wrst s broken nto two classfcatons epenng on whether the nstrument shaft coul reach the port wth the robot at that poston. The lnks were each 1 mm an the nstrument shaft was 3 mm. The home poston was wth the frst lnk horzontal (parallel to the z-axs) an the secon lnk vertcal (parallel to the y-axs). The orgn was at the wrst of the robot n ts home poston. The nstrument shaft passe through a port that was place at (y=, z=16). We assume that the frst jont s constrane to angular values between 1. an 1.9 raans as measure from the home poston. The secon jont s constrane to angular values between 1. an.7 raans as measure from the home poston. Atonally, the jonts are constrane such that the sum of the angles for both jonts s between.8 an.3 raans. These jont constrants are smlar to the physcal constrants of the robot use n the expermental setup, escrbe n the next secton. For each pont n the tp workspace, we calculate the error metrcs wth a port postonng error of + mm n y an +1 mm n z, an assumng quas-statc steps (Fg. 5). Ths port error s somewhat larger than measure n our anmal trals (Zeus surgcal robot system, Computer Moton Inc., Goleta, Calforna), but serves to llustrate behavor of the error metrcs. The crcle s the locaton of the port an the x s the estmate locaton of the port use by the controller. Fg. 5a plots the nuce Euclean norm, G,1. G,1 1 for a large porton of the workspace; therefore the system s not guarantee to be monotoncally convergent over the entre workspace. However, G, k.5 n the mle of the workspace, so small esre motons there shoul be monotoncally convergent. In general, as the controller gan, k, ecreases from unt gan to zero, the commane motons G, k constrcts to values closer shrnk n length an so to one. As k ncreases from unt gan, G, k ncreases proportonally, ue to the larger motons. The regons sgnfe by the ponts L, M, an N n Fg. 5 were use n the expermental tests splaye n Fg. 8.

4 Y 4 Wrst workspace: too far from ncson Robot lnks X Incson Z Instrument Frst lnk 1 Secon lnk Base Val wrst workspace Port ncson Wrst Wrst Tp workspace Fgure 3. Expermental setup wth axes. Fg. 5b shows the estmate, ˆ,k wth unt gan. The effects of changng k on ˆ are to rve t to ether 9 egrees for a k ecreasng to zero or 18 egrees for a k ncreasng to nfnty, as requre by (1) an the effects of k on G, k. Fg. 5c shows the mum absolute angular error,. Ths plot s unaffecte by changes n the controller s moton scalng gan, k. The latter two plots emonstrate that ˆ s a conservatve estmate of. In fact, the contour lne at the rght of the workspace, beyon whch ˆ s greater than egrees, approxmately correspons to where s greater than egrees. Examnng, the actual moton s wthn 9 egrees of the esre moton for most of the workspace. In all three plots, larger values occurre when ether the robot or the shaft tp were near the port poston, locatons at whch the effects of the port s poston error on the controller are greatest. Fg. 6a s a vector plot of v across the workspace, splayng the rectons of esre moton that mze x new. Ths plot shows the esre moton rectons that result n the largest Cartesan error lengths after the actual motons. Of nterest are the locatons where nearby vector lnes are perpencular to each other, because at those locatons small changes of poston cause large changes n v, such as n the lower left corner an aroun the port. Except n the lower left corner, the v for postons along the ege of the workspace are all approxmately perpencular to the bounary of the workspace. Fg. 6b contans the same lne segments scale by G, k, whch shows that even though v s behavng nterestngly n the center of the workspace, the errors there are not sgnfcant. Fg. 6c s a vector plot of p, the rectons of esre motons that result n. These vectors are smlar to the ones n Fg. 6a, but the fferences are rather notceable n the bottom left, an along the ege of the workspace. Also, p seems to change more smoothly than v, aroun the left-center of the workspace. Fg. 6 s the same plot scale by. Comparng Fg. 6 to Fg. 6b emphaszes the fference between v an p where errors n moton are largest, near the workspace bounary furthest from the port. To show the effect of fferent error rectons n the port estmate, Fg. 7 contans smulaton ata for varous port - Instrument shaft - 4 Fgure 4. Dagram of wrst an tp workspaces, wth lnks an shaft as lnes an port as a crcle. postonng errors. The errors are 1 mm n the +z, -z, +y an - y rectons, respectvely. Plots of an ˆ,1 are shown. ˆ s agan shown to be a conservatve estmate of. Port poston errors n the z-recton cause greater error at the bottom of the workspace, whereas port errors n the y- recton cause more error on the rght se of the workspace. Lookng more closely, postve y errors n the port poston cause the hgh error regon to shft ownwar compare to negatve y errors n the port poston. The tren s postons along the lne of port error o not have errors n moton. B. Physcal Experment The expermental setup (Fg. 3) ncorporate a Phantom robot (moel 1.5, Sensable Technologes, Inc., Woburn, MA). For ths setup, the roll jont was remove from the wrst an all postons were kept n the y-z plane by actvely rvng the base jont to a set value. Lmtng moton to the vertcal plane allows easy vsualzaton of results. The home poston of the Phantom s wth the frst lnk horzontal an the secon lnk vertcal. The orgn of graph axes s at the poston of the tp of the robot n ts home poston. The controller n Fg. 1 was use to control the robot. The jont level controller was PID wth the same upate rate as the nverse Jacoban control. The varous gans were chosen for a slow, over-ampe moton to any poston wthn the workspace. A hollow cylner 3 mm n length an mae of stanless steel was attache to the en of the Phantom to smulate a surgcal nstrument. The Jacoban n the nverse Jacoban controller relates the change n the poston of the tp of ths shaft to the change n the robot jont angles. Ths shaft passe through a moel of a port constrant. The port constrant consste of a low frcton acetyl ball jont. The ball freely rotate but coul not translate. A cylnrcal hole n the ball allowe the shaft to pass through. For ths work, the port constrant was place halfway along the shaft n the z recton,

5 G 1 1 ˆ.8 L M N 5 5 L M N L M N A B C Fgure 5. (A) G, (B) ˆ (egrees), an (C) (egrees) for port postonng error of (, 1) mm. The crcle shows the true port locaton. The x s the estmate port locaton A 1 B C D FIGURE 6. (A) Drecton of esre motons that result n mum Cartesan error length, v ; (B) v scale by G ; (C) Drecton of esre motons that result n mum absolute angular error, p ; (D) p scale by. The crcle shows the true ncson locaton. The x s the estmate ncson locaton.

6 at poston (y= mm, z=16 mm). Fg. 4 shows the workspace of the robot s wrst an the workspace of the tp of the shaft. The controller use an estmate port poston of (, 17). Ths port error was the only knematc error n the experment. Snce the true port poston was known, the nstrument tp poston (x n Fg. 1) was calculate usng forwar knematcs. For the experment, ntal tp postons were place at (y,z): (, 3), (, 375), an (, 415), labele L-N n Fg. 5. Those postons were chosen to le n areas of ncreasng error. Eght stop postons were then chosen for each start poston at stances of ± mm along the y-axs, z-axs an agonal rectons. Expermental poston ata were gathere as the system move from each start poston to the corresponng eght esre postons. Fg. 8 contans plots of the expermental trajectores escrbe for the esre ponts L, M, an N, respectvely. Smulate paths overlay the plots n ashe lnes, calculate as quas-statc steps wth a controller gan of.1. Though none of these results show the robot movng away from the esre poston before convergng, such a path s possble. The smulate path for the esre poston at (435, ) actually oes move away from the esre poston just before convergng. IV. DISCUSSION Robots can be useful tools n mnmally nvasve surgery, ncreasng stuatonal awareness, reucng han tremor, an provng tools to reuce the surgeon's mental workloa. Unfortunately, experence shows that port moton an flexng of the nstrument shaft can result n localzaton an moton errors, mtgatng the helpfulness of the robot. Such errors can reuce or elmnate the usefulness of navgatonal as an mage guance, as well as cause the robot to move the nstrument n unesrable rectons. A measure, ˆ, was propose for mnmally nvasve surgcal robots wth nverse Jacoban controllers. Gven the true Jacoban an the estmate Jacoban, ˆ conservatvely estmates the mum absolute angle,, between esre moton an actual moton. Ths measure assumes a quas-statc system, an assumpton justfe by low gan on the nverse Jacoban controller, a hgh-spee jont-level controller, an the presence of ampng. The smulaton emonstrates the measures of the moton error across the workspace. Comparng Fg. 5b an 5c, ˆ s larger than at all postons, behavng as a conservatve estmate. The contours of ˆ are much steeper than those of. Ths results n ˆ closely estmatng aroun the mle of the workspace, but beng qute large at postons close to, or far from, the port. In the purest sense, ˆ an G, k are not practcally usable metrcs snce they requre knowlege of the true Jacoban of the system. One practcal use oes exst when the error sources on the knematcs are boune. In ths case, calculatng G, k for the worst cases shows whether the robot wll converge wth a monotoncally ecreasng error length for all possble cases, an allows the observaton of subsets of the workspace where the behavor of the system s most esrable. Smlarly, the measure can be use as a esgn A1 A A3 A ˆ ˆ tool to etermne the necessary tolerances of varous system components. Usng the system n ths paper as an example, f postons along y= are the most mportant, then Fg. 7 shows that errors n the y poston of the port are less tolerable than errors n the z poston. Another practcal result les n usng angular errors to prect error sources. Usng an external poston measurng system to track the tp poston, a plot of the measure errors n angle,, throughout the workspace coul be compose. By comparng those measurements to the varous plots of ˆ ˆ B1 B B3 B (eg) (eg) Fgure 7. (A) ˆ (egrees). (B) (egrees). Each wth 1 mm port postonng error n a fferent recton: (1) +z. () z. (3) +y. (4) y. A crcle shows the true ncson locaton. An x shows the estmate ncson locaton. 9 (eg) (eg)

7 1-1 L M N Fgure 8. Trajectores followe n experment (sol lnes) an smulaton (ashe lnes), to the esre postons (re ots) each mm away from the corresponng start postons (black ots) at (L): (3, ), (M): (375, ), (N): (415, ). generate by assumng fferent sources of error, smlartes coul suggest whch sources of error are sgnfcant. Onlne calculaton of ˆ,k coul use a pror knowlege to enhance onlne estmaton of the Jacoban. By assumng bouns for sources of sgnfcant error, worst-case Jacobans can be calculate for the errors at each of those bouns. The Jacoban can then be estmate urng onlne motons, allowng calculatons of ˆ,k between the estmate Jacoban an each of the worst-case Jacobans. The mum of those angles proves an upper boun on the actual angular error. The esre motons, v, plotte n Fg. 6a, are smlar to the esre motons, p, n Fg. 6c. Therefore, the esre motons that result n the largest length of Cartesan error for a unt controller gan are smlar to the motons that actually result n the largest angular error. The fferences occur wherever changng the moton scalng gan of the nverse Jacoban controller coul ecrease the Cartesan error. Path plannng s another practcal use for ths work. If the lkely sources of error are known, v can easly be calculate for those error sources. A path planner coul then trae off between nrect an rect paths to the esre fnal poston, where the nrect paths may have less error than the rect paths. The expermental paths n Fg. 8 approxmately follow the smulate paths, provng some expermental valaton for the smulaton. The most obvous fferences are for the start poston at (415, ), the start poston closest to the ege of the workspace. For ths start poston, three of the expermental paths o not closely follow the smulate paths, those for the esre postons at (49, 14), (41, -14), an (435, ). Another fference between expermental an smulate paths n Fg. 8 s that the angular error between esre moton an actual moton s sometmes greater for the expermental paths than for the smulate paths. The combnaton of three sources of error coul explan these screpances. Frst, frcton at the port constrant mpees moton ansotropcally base on velocty. Secon, lack of gravty compensaton n the jont-level control results n unmoele forces on the system that vary over the workspace. Thr, ynamc effects coul nfluence the expermental system, though ths seems unlkely snce the controller gans were set for very slow moton. Future stues wll apply ths work to shaft flexon as a source of knematc error. Subsequently, the two sources of error, port moton an nstrument flexon, wll be combne. The result wll be bouns on those error sources such that the system wll stll converge wth a monotoncally ecreasng postonal error length. An atonal possblty s to nvestgate the relatonshp between the measures presente n ths paper an the orentaton of the robot by usng the full Jacoban nstea of just the poston porton of the Jacoban. REFERENCES [1] T. M. Peters, Image-gue surgery: from X-rays to vrtual realty, Computer Methos n Bomechancs an Bomecal Engneerng, vol. 4, pp. 7-57,. [] S. S. Park, R. D. Howe, an D. F. Torchana, Vrtual fxtures for robotassste mnmally-nvasve carac surgery, Mecal Image Computng an Computer-Assste Interventon 1, W.J. Nessen an M.A. Vergever, Es. Berln: Sprnger-Verlag, Berln, 1, pp [3] J. J. Abbott an A. M. Okamura, Vrtual fxture archtectures for telemanpulaton, IEEE Intl. Conf. on Robotcs an Automaton, pp , 3. [4] L. Panat, C. Doarn, an R. Merrell, Applcatons of robotcs n surgery, Chrurga Bucurest, vol. 97, no. 6, pp ,. [5] G. Guthart an K. Salsbury, Jr., The IntutveTM telesurgery system: overvew an applcaton, IEEE Intl. Conf. on Robotcs an Automaton, pp ,. [6] R. A. Beasley an R. D. Howe, Tactle trackng of arteres n robotc surgery, IEEE Intl. Conf. on Robotcs an Automaton, Vol. 4, pp ,. [7] J. M. Hollerbach an C. W. Wampler, The calbraton nex an taxonomy for robot knematc calbraton methos, Intl. J. of Robotcs Research, vol. 15, no. 6, pp , [8] Z. Roth, B. W. Moorng, B. Ravan, An overvew of robot calbraton, IEEE Trans. of Robotcs an Automaton, vol 3, no 5, pp , [9] D. M. Rovner an R. H. Cannon, Jr., Experments towar on-lne entfcaton an control of a very flexble one-lnk manpulator, Intl. J. of Robotcs Research, vol. 6, no. 4, pp. 3-19, 1987 [1] D.-S. Kwon an W. J. Book, A tme-oman nverse ynamc trackng control of a sngle-lnk flexble manpulator, J. of Dynamc Systems, Measurement, an Control, vol. 116, pp. 193-, [11] M. Jägersan, Vsual servong usng trust regon methos an estmaton of the full couple vsual-motor Jacoban IASTED Applcatons of Robotcs an Control, pp , [1] K. Hosoa an M. Asaa, Versatle vsual servong wthout knowlege of true Jacoban, Intl. Conf. on Intellgent Robots an Systems, pp , [13] C. C. Cheah, M. Hrano, S. Kawamura, an S. Armoto, Approxmate Jacoban control for robots wth uncertan knematcs an ynamcs, IEEE Trans. Robotcs an Automaton, vol 19, no. 4, pp. 69-7, 3.