Mass/ Inertia and Joint Friction Minimization for a Low force Five dof Haptic Device *

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1 Mass/ Inerta and Jont Frcton Mnmzaton for a Low force Fve dof Haptc Devce * Kostas Vachos and Evangeos Papadopouos Donssos N. Mtropouos Department of Mechanca Engneerng Schoo of Medcne Natona echnca Unversty of Athens Natona Kapodstran Unversty of Athens Athens, Greece 1157 Athens, Greece {kostasw, egpapado}@centra.ntua.gr dmp@otenet.gr Abstract hs paper presents a desgn methodoogy, whch ams at the mnmzaton of the mass, nerta and jont frcton for a ow force fve dof haptc devce. he haptc devce s optmzed aong a typca path wth proper toerances, rather than at some workspace operatng pont. he devce, part of a tranng medca smuator for uroogca operatons, conssts of a two dof, 5 bar nkage and a three dof spherca jont. he requrement for reabe reproducton of ow torques and forces ead to the need for mnmzaton of devce nduced parastc forces and torques. he mutobjectve optmzaton empoyed s based on two objectve functons that ncude mass/ nerta propertes and jont frcton. Knematca and operatona constrants are taken nto account. he resutng optmzed mechansm s substantay mproved wth respect to an exstng devce. Keywords Force feedback, mutobjectve optmzaton, haptc devces, tranng medca smuators I. INRODUCION he use of smuators s now an accepted too n the tranng of surgeons, [1]. Athough t s st eary for defntve concusons, t seems that there are many advantages n the use of smuators. Smuator-based tranng s ess expensve and resuts n effcent and customzabe tranng n compex operatons, []. ranng on patents can resut n serous damages and awsuts whe tranng on anmas becomes an undesrabe aternatve for ethca and economca reasons. Furthermore the anatomy of an anma s not aways cose enough to that of the human. Aso, the exstence of a tranng smuator ncreases the avaabty of the tranng envronment, aows an easer evauaton of the performance of the tranee, and can be used to ntroduce varous operaton scenaros or stuatons. Reastc medca smuators consst of a graphca envronment, whch reproduces the vsua nformaton that the surgeon obtans durng an operaton, and a haptc devce, whch reproduces the haptc nformaton. oday, one can dstngush two trends n the deveopment of medca smuators. he frst s characterzed by the use of generapurpose haptc devces, ke the Phantom or the Freedom 7, [,, 5]. he second trend s characterzed by the use of haptc devces desgned for a specfc operaton, [6, 7, 8, 9]. he fathfu reproducton of the rea forces and torques that the surgeon fees durng an operaton s of great mportance. hs means that the mechansm shoud be transparent as much as possbe. Optmzaton technques have been aready used n mprovng the performance of mechansms and manpuators. he nerta and acceeraton characterstcs of manpuators have been dscussed n [10]. Optmzaton technques are used to determne the smaest nerta propertes and the maxmum achevabe acceeraton of the end-effector n every drecton over the workspace. A goba sotropy ndex has been proposed to quantfy a confguraton ndependent sotropy of a robot s Jacoban or mass matrx, [11]. hs ndex was used to compare the performance of three manpuators, ncudng two parae patform robots and a hybrd robot, [1]. A two degree-offreedom (dof) haptc devce was optmzed wth respect to workspace, ntruson, nerta, response and structura propertes, [1]. he archtecture of a parae redundant mechansm has been optmzed from a knematc vewpont, [1]. he dexterty, unformty and actuator forces have been nvestgated as potenta objectve functons. Authors prevous work has presented the desgn of a fve dof haptc nterface, whch was party optmzed wth respect to ts condton number and perceved nerta under severa knematc constrants, [9]. hs paper presents a mutobjectve optmzaton methodoogy resutng n an mproved ow force fve bar haptc devce, whch s a part of a tranng medca smuator for uroogca operatons. Unke other haptc devces n whch the reproducton of arge forces or torques s of prme mportance, here t s desred to bud a devce that can reproduce fathfuy very sma forces and torques, such as those appearng n uroogca operatons. o ths end, a major effort s paced n desgnng the mechansm such that t s characterzed by ow mass and nerta and sma jont frcton. wo objectve functons are defned, the frst focusng at mass/ nerta optmzaton and the second at jont frcton. he optma desgn s acheved for a typca endoscope path, aowng at the same tme sma devatons from t. he methodoogy resuts n an optmum mechansm geometry and an optmum ocaton of the endoscope path wth respect to the haptc devce base. he proposed optmzaton methodoogy s sutabe for any mechansm that shoud be optmzed aong a gven path. he paper descrbes n deta the objectve functons empoyed, the optmzaton constrants and the * Support of ths work by the PENED and PRAXE programs of the Heenc Genera Secretarat for Research and echnoogy s acknowedged.

2 overa procedure. Fnay, optmzaton resuts are presented and dscussed. II. DESCRIPION OF HE HAPIC DEVICE As mentoned above, the haptc devce s used n a tranng smuator for uroogca operatons. Durng a uroogca operaton on a mae patent, the surgeon nserts a ong cyndrca endoscope unt ts endpont reaches the patent s badder. Durng nserton, the endoscope foows a path such as the typca one shown n Fg. 1. he surgeon moves the tp of the endoscope from the nserton pont A to the fna pont C, va an ntermedate pont B, see Fg. 1. At pont B, the endoscope orentaton changes wthout transaton, so as to agn the entre urethra and contnue the nserton phase wthout traumas. he correspondng endoscope confguratons abeed by a, b, c, d, are shown n Fg. 1. paced at the base. he transmsson system s mpemented usng tendon drves wth capstans, [9]. he forces that the surgeon fees durng the operaton are sma but of great mportance, because they provde the necessary feedback for the successfu accompshment of the operaton. In order to reproduce these sma forces, the haptc mechansm must have ow mass and nerta, ow frcton, no backash, and be absoutey backdrveabe, [9]. he upper mts of forces and torques durng a uroogca operaton were measured n coaboraton wth specast surgeons and are shown n abe II. However, the mnmum forces and torques that are actuay fet durng an operaton can be a fracton of these mts. Fgure. he frst verson of the haptc devce ABLE II. FORCE/ORQUE UPPER LIMIS DURING AN OPERAION Fgure 1. he Path When the tp of the endoscope reaches the badder (pont C n Fg. 1), the surgeon nserts through the endoscope a mechansm wth a scssor-ke hande and begns the second phase. hs phase s the man operaton n whch tssue remova occurs. Durng ths phase, the movements of the endoscope are many rotatona. he actua knematc requrements that defne the mnmum workspace of the haptc nterface are shown n abe I. hese were found by actua observatons of typca uroogca operatons. ABLE I. HAPIC DEVICE WORKSPACE REQUIREMENS Moton Lengths ransaton aong the X axs 0.1 m ransaton aong the Y axs 0.1 m ransaton aong the Z axs 0.0 m Rotaton about the X axs ±180 Rotaton about the Y axs ±0 Rotaton about the Z axs ±0 Observatons durng our prevous work showed that a haptc mechansm wth two transatona and three rotatona dof s needed, [9]. he mechansm, shown n Fg., conssts of a two dof, 5 bar nkage and a three dof spherca jont. o reduce mechansm movng mass and nerta, a actuators are Moton Force aong the X axs Force aong the Y axs orque about the X axs orque about the Y axs orque about the Z axs Lengths.50 N.50 N mnm mnm mnm In the frst verson of the haptc devce, see Fg., the nk engths of the fve bar mechansm were optmzed n order to mnmze the condton number of the mechansm aong a path, under knematca and structura constrants. he path was fxed n space reatve to the mechansm base. abe III presents the optmzed confguraton. ABLE III. PREVIOUS WORK RESULS Lnk Length = 0.15 m m 0.15 m 0. m III. OPIMIZAION DESIGN he probem that we present here s more compex. A the nk engths are free to be optmzed. he path that the mechansm shoud foow s free n space reatve to the base. he optmzaton resuts are these unknown parameters under severa knematca and mpementaton constrants.

3 he schematc vew of the dof 5 bar mechansm, whch s gong to be optmzed, and of a typca path, n a random ocaton, to foow s ustrated n Fg.. he symbos are expaned n abe IV. ABLE IV. SYMBOL EXPLANAION OF FIGURE Symbo 1 Expanaton = Length of nk 1 Length of nk Length of nk Length of nk Length of nk Length of nk q q 1 Ange between nk 1 and X axs Ange between nk and X axs Fgure. Schematc vew of the 5-bar nkage and of a random ocated typca path to foow. he optmzaton goa s to fnd the mechansm nk engths and the ocaton of the path ABC wth respect to the base pont O, so that mass, nerta and frcton at the jonts of the 5 bar mechansm are mnmzed. Because of the nature of the surgca operaton, the path ABC es aways on the XY pane. Aso, because the patent assumes a constant and predetermned poston wth respect to the vertca, the same appes to the orentaton of path ABC. Because of these observatons, the reatve ocaton of path ABC wth respect to the haptc nterface base pont O can be descrbed by two parameters that ocate one of ts pont wth respect to O. Next, two objectve functons f 1, and f are defned that can be used as the objectve functon f n a mutobjectve optmzaton approach. A. he frst mnmzaton 1) he frst objectve functon f 1 he dagona terms of the mass matrx of the 5 bar mechansm depend on the ength, the mass and the nerta of the nks. We assume that the nks are cynders wth mass m () = ρ ( b a) (1) where ρ s the densty, b s the outer dameter, a s the nner dameter and s the ength of the nk. he nk nerta s I( ) = m( )( a + b + )/1 () Mass and nerta are functons of nk engths. Hence, the frst objectve functon can be the sum of the nk engths. he mnmzaton of ths sum resuts n the mnmzaton of mechansm mass and nerta. f1 = () A more exact optmzaton of the mechansm mass and nerta propertes woud requre the mnmzaton of the mass matrx as seen from the endoscope sde, M, [9], but ths woud make the procedure too compcated. Instead we chose the above functon, Eq. (), whch s very smpe and resuts n an optmum wth a very good approxmaton. In addton to the above objectve, one has to take nto account severa confctng knematca and mpementaton constrants. ) Constrants for the frst objectve functon An mportant constrant s that the mechansm must be arge enough to foow typca endoscope paths, such as the one shown n Fg. 1. he foowng nequaty par descrbes ths constrant for a ponts aong the path, ( ) ( ) ( ) x s y s ( ) where x( s), ( ) + + () y s are the coordnates of the mechansm tp aong the path. he mechansm shoud be we condtoned at a confguratons. It can be shown that the mechansm condton number s optmum when = 1 and q q1 = /, whe t ncreases when 1 and q q. he above gves us the foowng constrant, 1 / 1 e 1+ e (5) 1 1 e q q1 + e (6) where e1, e ndcates how strcty the constrant s. It s mportant that Eq. (6) hods more strcty at the end poston of the path, pont C n Fg.1, where the man operaton takes pace. hs ntroduces the next constrant, e q, C q1, C + e (7) where e > e and the subscrpt n qc,, = 1, denotes the vaues of anges q at pont C. Another requrement resut from mpementaton constrants,.e. to avod coson between nk and the endoscope, the ange Ψ = q that s formed by nk and the endoscope, see Fg., has to be bounded accordng to

4 1.rad q 5.1rad (8) whch forms another optmzaton constrant. It s possbe, durng the tranng procedure, that the smuator tranees make mstakes,.e. they may devate from the dea path. In ths case, the haptc devce must have the capabty not ony to foow these wrong paths, but aso to mantan an optmum functonaty. In coaboraton wth specasts we determned that the possbe erroneous dstance s about e= 0.01m from the typca path. herefore, t s mportant to ocate not path ABC wth respect to O but a whoe famy of paths that e wthn the bounds defned by the possbe devaton e. In other words we fnd not the optmum ocaton of the typca path, but the optmum ocaton of an area about the typca path. hs requrement eads to addtona constrants. he mechansm has to foow the perturbed paths, ( ) x ( s) y ( s) ( ) + + (9) ( ) ( ) ( ) x+ s y+ s ( ) where the subscrpt n x ( s), y ( s) and x ( s), y ( s) + + (10) + + denotes the mnmum and maxmum perturbaton about the dea path respectvey. he mechansm shoud be we condtoned even n the perturbed path, e q+ q1 + + e (11) e5 q q1 + e5 where the subscrpt n q, +, = 1, and q,, = 1, denotes the vaues of the anges at the maxmum and mnmum wrong ocatons respectvey. Fnay the ast optmzaton constrants for the perturbed paths s due to the same mpementaton constrants as n Eq. (8) 1.0rad q rad (1) 1.0rad q 5.585rad Equatons () to (1) form the set of optmzaton constrants for the frst objectve functon. If we ook carefuy at the objectve functon and the optmzaton constrants, we see that the ength of nk ( ) s not affected by the varous constrants. hs means that the optmzaton procedure w aways yed as optmum ength of nk the predefned ower bound. Assumng a sma, then mass and nerta s reduced. However to transmt gven forces or torques, the nterna forces at the jont of the mechansm ncrease resutng n a hgh eve of frcton n the mechansm. hs effect s made worse gven the fact that the mechansm ncudes tendon drves, whch have some non-zero preoad. On the other hand, f s made arge, then frcton s reduced, but the nerta and mass due to s ncreased. hese observatons ntroduce the second mnmzaton. B. he second mnmzaton 1) he second objectve functon f We wsh to mnmze the frcton that appears at the jonts of the 5 bar mechansm. he frcton s drecty proportona to the nterna constrant forces at the jonts accordng to the reaton M = Rr, see Fg., where r s the radus of the jont axe and R represents the frcton force at the jont. In order to fnd these forces we construct the free body dagram of each nk of the mechansm. Fgure. Mechansm nterna forces and actuator torques. he actuator forces and torques are shown n Fg.. F s an externa force, F j s the constrant force from nk to nk j, 1, are the apped torques by the actuators to nks, to keep the mechansm n baance aganst F, m s the mass of nk and g s the gravtatona acceeraton. For each nk force and moment baance equatons must hod,.e. F j = 0 (1a) M Gj = 0 (1b) he ower part of Fg. shows a magnfcaton of the axe and bearng ncudng the actuated forces and torques at the base of the 5 bar mechansm where nk s couped wth the correspondng actuator. 1 s the torque, from the actuator to the nk, F 0 s the constrant force from the base to nk, r s the radus of the axe, R represents the frcton force at the jont and M s the frcton that appears. Equatons (1) ead to a set of tweve equatons wth tweve unknowns. Sovng ths set yeds the mechansm constrant forces and the actuator torques. Fg. 5 shows the souton of the set as functon of at a specfc pont, namey the end of the path, or pont C n Fg. 1. We see that the nterna forces decrease as ength ncreases. In the same fgure, one can aso observe that a mnmum exsts for the sum of the torques 1,. hs s due to the fact that as decreases, the nterna forces and therefore the frcton at the jonts ncreases. hs aso

5 ncreases the torques needed to compensate for frcton. herefore the sum of the torques woud be a sutabe objectve functon to mnmze. IV. OPIMIZAION RESULS For the optmzaton we used the Matab optmzaton toobox and the functon fgoaattan. hs functon s based on a Sequenta Quadratc Programmng (SQP) method wth a few modfcatons, [15]. he functon soves the goa attanment probem, whch s one formuaton for mnmzng a mutobjectve optmzaton probem. he startng guess s gven n abe V. he ower and upper bounds are shown n abe VI. ABLE V. SARING GUESS A HE SOLUION Fgure 5. Constrant forces and torques at the end of the path (pont C) as ncreases from 0.001m to 0.m. Optmzed varabes Lnk 1, 1 Lnk, Lnk, Lnk, X poston of pont C Y poston of pont C Startng Guess 0.01 m 0.01 m 0.01 m 0.0 m 0.0 m 0.0 m It s aso desrabe to fnd the optmum ength not for a specfc confguraton but aong the whoe path. herefore we dvde the path n n segments. 1, are vectors wth the actuator torques at the n segments of the path. = [,1( s1 ),...,, n( s n)] (1) where, = 1,. We defne the second objectve functon as the sum of the nfnty norms of the vectors 1,. f = w1norm( 1,nf) + wnorm(,nf) (15) where, Norm(,nf) = max( (, k ( sk) () (16) and w1, w are the approprate weghts n order to wegh the contrbuton of the two actuators equay. k = 1,..., n. ) Constrants for the second objectve functon he frst constrant s due to mpementaton mtatons,.e. must have a mnmum ength, 0.06m (17) Next, we wsh to bound the nterna constrant forces at the jonts of the mechansm to a acceptabe mt. We construct vectors F wth the nterna constrant forces F at the n segments of the path, F = [ F,1( s1 ),..., F, n( s n)] (18) where, = 1,..., 1 and then we construct matrx F, whch contans the F. he second optmzaton constrant s the foowng, Norm( F,nf) g (19) where, g represents the maxmum predefned acceptabe frcton mt. ABLE VI. LOWER AND UPPER BOUNDS Optmzed varabes Lower bounds Upper bounds Lnk 1, m 0.15 m Lnk, 0.01 m 0.15 m Lnk, 0.01 m 0.15 m Lnk, 0.0 m 0.0 m X poston of pont C m 0.50 m Y poston of pont C m 0.50 m he functon fgoaattan may ony gve oca soutons. herefore the optmzaton search area s dvded n severa subspaces. he best resut of a s chosen as the optmum. he optma resuts are shown n abes VII and VIII and graphcay n Fg.6, where the subscrpt n qs,, = 1, and qe,, = 1, denotes the vaues of the anges at the start and the end of the path n the optmum ocaton. ABLE VII. Lnks Lnk 1, 1 Lnk, Lnk, Lnk, ABLE VIII. LENGHS OPIMIZAION RESULS Lengths m m m m PAH LOCAION OPIMIZAION RESULS Axs Locaton of pont C (Fg. 1) X 0.06 m Y m

6 reducton to the mass matrx dagona terms of the 5 bar mechansm of about 0% s cacuated. VI. CONCLUSIONS A desgn methodoogy whch ams at the mnmzaton of the mass, nerta and jont frcton for a ow force fve dof haptc devce s presented. he haptc devce s optmzed aong a typca path, rather than at some workspace operatng pont. he mnmzed objectve functons ncude mass/ nerta propertes and jont frcton. he optmzaton took nto account severa knematca and operatona constrants that are detaed descrbed. Sgnfcant better resuts are obtaned wth respect to an exstng devce. Fgure 6. Optmzed nk enghts and ocaton of the typca path. Accordng to the optmzaton resuts we have that / = 0.9, that aong the path 1.7rad q q1 1.97rad and that at pont C, q q 1 = 1.7rad and q = 1.rad. he new mechansm, confgured at the end of the optmzed path (pont C n Fg. 6), s presented n Fg. 7 as CAD drawng. he haptc devce s now under constructon and we expect to be ready for evauaton n two weeks. Fgure 7. he optmzed mechansm n CAD drawng V. DISCUSSION As we can see n Fg. 6 the path can be contaned between two crces ndcatng the maxmum and mnmum dstance of endoscope tp from pont O. he dfference between maxmum and mnmum dstance from pont O tends to be mnmum, takng nto account the varous constrants. In other words, the path ocaton that resuts tends to be as much crcuar as possbe, reatve to pont O. hs s expected because we know that a crcuar path gves mnmum condton number, whch s one of our requrements. Other ocatons, whch gve better resuts reatve to the condton number, may be obtaned from the optmzaton procedure but these ocatons contradct to severa operatona constrants. he optmzaton gves a reducton of the nk engths of about 0% reatve to the prevous mechansm. Furthermore a REFERENCES [1] Laguna, M. P., Hatznger, M., and Rassweer, J., 00, Smuators and endouroogca tranng, Current Opnon n Uroogy 00, vo. 1, pp [] Chen, E., Marcus, B., 1998, Force Feedback for Surgca Smuaton, n Proceedngs of the IEEE, 86, pp [] Sasbury, J. K., Srnvasan, A. M., 1997, Projects n VR. Phantom Based Haptc Interacton wth Vrtua Objects, IEEE Computer Graphcs and Appcatons, pp [] D Augnac, D., Baanuk, R., Lauger, C., 000, A Haptc Interface for a Vrtua Exam of the Human hgh, n Proc. IEEE Int. Conference on Robotcs and Automaton, pp [5] Atsuko,., Koch, H., oyohsa, K., 1998, Vrtua Cuttng wth Force Feedback, n Proceedngs of the Vrtua Reaty Annua Internatona Symposum, pp [6] Baumann, R., et a., 1997, he PantoScope: A Spherca Remote Center of Moton Parae Manpuator for Force Refecton, n Proc. IEEE Int. Conference on Robotcs and Automaton, pp [7] Baur, C., Guzzon, D., Georg, O., 1998, Vrgy, A Vrtua Reaty and Force Feedback Based Endoscopy Surgery Smuator, n Proceedngs Medcne Meets Vrtua Reaty 98, (MMVR 98), pp [8] Kühnapfe, U., et a., 1997, he Karsruhe Endoscopc Surgery raner as an exampe for Vrtua Reaty n Medca Educaton, n Mnmay Invasve herapy and Aed echnooges (MIA), pp. 1-15, Backwe Scence Ltd. [9] Vachos, K., Papadopouos, E., and Mtropouos, D. N., 00, Desgn and mpementaton of a haptc devce for tranng n uroogca operatons, IEEE ransactons on Robotcs and Automaton, vo. 19, no. 5, October 00, pp [10] Khatb, O. and Bowng, A., 1996, Optmzaton of the nerta and acceeraton characterstcs of manpuators, Proc. IEEE Internatona Conference on Robotcs and Automaton (ICRA'96), Mnneapos, Mnnesota, Apr 1996, vo., pp [11] Stocco, L., Sacudean, S. E., and Sassan, F., 1999, Fast constraned goba optmzaton of robot parameters, Robotca, Vo.16, pp [1] Stocco, L. J., Sacudean, S. E., and Sassan, F., 001, Optma knematc desgn of a haptc pen, IEEE/ASME ransactons on Mechatroncs, vo. 6, no.. [1] Hayward, V., Choks, J., Lanvn, G., and Ramsten, C., 199, Desgn and mut objectve optmzaton of a nkage for a haptc nterface, Advances n Robot Knematcs and Computatoned Geometry, A. J. Lenarcc and B. B. Ravan (eds.), Kuwer Academc Pubshers, pp [1] Kurtz, R., Hayward, V., 199, Mutpe goa knematc optmzaton of a parae spherca mechansm wth actuator redundancy, IEEE ransactons on Robotcs and Automaton, vo. 8, no. 5, pp [15] Brayton, R.K., S.W. Drector, G.D. Hachte, and L.Vdga, A New Agorthm for Statstca Crcut Desgn Based on Quas-Newton Methods and Functon Spttng, IEEE ransactons on Crcuts and Systems, Vo. CAS-6, pp , Sept