Kinematics of a dedicated 6DOF Robot for Tele-echography

Transcription

1 Knematcs of a dedcated DOF Robot for ele-echography L Al Basst G Posson P Veyres Laboratory of Vson and Robotcs Unversty of Orleans 1800 Bourges France lbasst@bourgesunv-orleansfr Abstract hs paper presents the knematcs structure of an ultrasound probe holder nnovatve robot It s a sx degree-of-freedom dedcated structure remotely controlled by a medcal expert to perform a tele-echography examnaton It has been desgned wth respect to users requrements fxed by medcal experts from varous clncal felds It s used to poston the probe on the patent s skn, for every needed posture Knematcs and dynamc models of the robot have been establshed, sngularty problem has been studed, geometrc parameters optmaton has been nvestgated hs robot has been desgned n the frame of the European project Otelo 1 Introducton ele-robotcs and tele-operaton have currently a very mportant role to play n the medcal feld he am of our project s to develop a dedcated tele-operated robotc system allowng people lvng n solated areas, far from medcal experts, or not able to be transported to a medcal center, to take advantage of a medcal ultrasound department he overall system conssts of: A lght robot, holdng an ultrasound probe, located at the patent ste and operated by a paramedc nonexpert n echography; A haptc nterface, located at the expert ste, allowng the medcal doctor to control the robot and to receve ultrasound mages and force feedback; A communcaton lnk (satellte or terrestral) allowng nformaton exchange between the two stes (eg ambent mages, ultrasound mages and robot control data) he laboratory of vson and robotc bult two prevous prototypes of tele-echography systems he frst system, usng DOF prototype manpulator, has been desgned, mplemented and tested between two stes: Bourges n France and Katmandu n Nepal n 1998 [5] hs frst prototype has consttuted a valdaton of the teleechography concept used n the followng projects In 1999, an ndustral prototype for tele-chography has been studed n collaboraton wth Snters company [8] hs second prototype uses a 4 DOF probe holder robot It has been manufactured and clncally valdated n 00 [] Otelo * (moble ele-echography usng an ultra-lght robot) project started n September 001 [7] In ths project, a new probe holder robot s beng desgned hs mproved robot should better respond to the medcal requrements In ths paper we present the knematcs aspects of the mechancal desgn of ths new dedcated robot Robot Desgn he Moton of the probe held by a medcal doctor, durng varous ultrasound nvestgatons: cardac, abdomnal and renal, has been studed he choce of robot knematcs structure s based on probe movement specfcatons establshed by future users of the system: medcal doctors 1 Probe Movement Specfcatons Probe movement ams to scan the organ nvestgated by the plane of the emtted ultrasound waves and to search for the best ntersecton, between ths plane and the organ, to obtan the needed nformaton of the ultrasound mage, see Fg 1 Based on the probe movement study, the followng specfcatons were ssued: * Project supported by the European Commsson 9 partners: unverstes, ndustrals and hosptals

2 Fgure 1 Area of probe movement Intersecton of the ultrasound plane wth the nvestgated organ and the resultng ultrasound mage Once the probe s postoned on the area of nterest, a contnuous contact must be kept between the tp of the probe and the skn he doctor mantans the same contact pont between the probe and the skn durng the organ nvestgaton hs contact pont s chosen wth accuracy, based on anatomcal references o reach the best angle of ncdence, the probe axs s moved nsde a concal space and rotated around ts axs he apex of ths cone s the contact pont, see Fg1 Durng most of the nvestgatons, the nclnaton of the probe axs, from the normal drecton of the skn, s most of the tme smaller than 5 Although n some cases and for some organs nvestgatons, the probe axs s nclned up to 0 from the normal drecton of the skn Robot Knematcs Structure he robot structure, desgned to obtan the movement of the probe prevously descrbed, s a degree-offreedom seral type wth three rotatons around concurrent axes and three translatons, as depcted n Fg Furthermore, ths dedcated slave probe holder has to be lght enough to be manpulated and moved wth ease on the patent s skn when t s requested by the dstant expert he robot wll be postoned on the patent s skn and operated by a paramedc or a person wth lmted medcal experence he frst two prsmatc pars, P1 and P, allow a precse Fgure Robot knematcs structure postonng of the probe on the area of nterest, the range for each of these translaton movements s 5cm he revolute pars R1 and R have concurrent axes; they allow the postonng and the orentaton of the probe axs nsde a concal space wth half angle apex of s chosen from the nterval [ / 8, / ] to obtan the best knematcs performance of the robot, as shown n secton 5 he probe s orentated, mantanng a contnuous contact wth the same pont of the skn; ths pont s the ntersecton pont of the two revoluton axes Revolute par R allows the rotaton of the probe around ts symmetrc axs and enables the expert to access a longtudnal or transverse vew of a gven organ he axs of R s merged nto the symmetrc axs of the probe he axes of R1, R and R are concurrent and the same angle s defned between R1-R and between R-R he sole role of the last prsmatc par P s to modfy the contact force between the probe and the skn, t sldes the probe along ts symmetrc axs wthn a cm range Robot work space: o represent the work space of our Fgure Accessble orentaton space

3 dedcated robot, we dstngush between the space of accessble postons and the space of accessble orentatons Here we are nterested by the accessble orentaton space Projectng the rotaton axes on a sphere centered on the axes ntersecton pont, we obtan a planar representaton from whch ts work space s easer determned, see Fg Robot Models Wth a prevous 4 DOF prototype [] developed n our laboratory and based on a smlar mechancal structure, we compared two solutons for robot control: one used nverse geometrc model and the other used nverse knematcs model Knematcs model was more smple to mplement than the geometrc one However, due to a larger samplng tme n the knematcs soluton, whch provded some nstabltes n the robot movement, we chose to mplement the geometrc one For ths DOF prototype, and to nvestgate the best path followng, we have to establsh the three models to be used for control laws and compare the result, before the fnal mplementaton on the robot hese models needed for the robot control are establshed, e, geometrc, knematcs and dynamc drect and nverse models [4], [], we present here the knematcs and the dynamc models 1 Knematcs Model It gves the relaton between the probe velocty n the ω V, ω represents the rotaton velocty of the probe and V the translaton velocty of the end pont of the probe, and the jonts velocty, e the dervatve of the generaled coordnates Cartesan space q 1 d d d 1 hs relaton s lnear, then the drect knematcs model could be wrtten as: ω J( q V where J ( s the Jacoban matrx In our case, J ( has the followng form: A 0 J wth A, B, C are matrces and B C 0 s a ero matrx We notce that the rank of the jacoban matrx s equal to the number of the jonts: For some confguratons, the rank of J ( s smaller than : these correspond to the sngular confguratons where det( J ) 0 hs aspect s developed n secton 4 Dynamc Model hs model gves the relaton between the actuators moments & forces and the artcular postons, veloctes and acceleratons Here we are nterested by the nverse dynamc model whch s expressed as: Γ f( q, q, q, Fe ) where Γ s the actuators moments & forces 1 matrx and F e the external moments & forces appled by the probe on the patent s skn Lagrange formulaton s used to establsh ths model he dynamc model s obtaned and s wrtten as Where: Γ M( q C( q, q Q( Γ f Γ e M ( s the mass matrx, C q, q ( represents Corols and centrfugal moment & forces, Q s the gravty moments & forces 1 matrx, Γ s a 1 matrx represents the effects of jonts frcton forces on the movement and Γ e represents the effects of external moments & forces on the movement Here we only present the shape of the mass matrx, obtaned drectly from the expresson of knetc M ( 1 energy E q M( q M ( s a symmetrc matrx M1 M M( : M M k l mkl R ( M L I G M kl k 1 :, mkl mlk kl k l mkl R ( M L M l l k 1 1,,, : and l mkl ( M L k l1,, M Sym 1 M 0 M 1 M R M R M f

4 M s the mass of sold, I G s the nerta matrx of sold wrtten n the frame attached to the sold F, L s the cross-product matrx of vector l drected from the orgn of frames F, F4 or F 5 (whch s the same) to the center of mass of sold j, R s the rotaton matrx between F j and F and s the three dmensonal vector We can obtan the sngular confguratons by studyng the rank of the Jacoban matrx whch goes down to 5 n the prevous two cases Fgure 5 shows the varaton of the determnant of the Jacoban matrx, det(j ), as a for dfferent values of, where functon of det( J ) det( A)det( C) S ( S C C ) S 4 Sngularty problem For / 8, /, sngular confguratons are present n two cases: a) he axes of R1, R and R are coplanar: ( 4 ) 5 0, n ths case, the probe axs s: ether perpendcular to the patent s skn and k k 1,,, as depcted n Fg 4a1, or t s nclned from the normal drecton of the skn by, k k 0,,, as depcted n Fg 4a b) he axs of P s parallel to the plane defned by the axes of P1 and P, ) 0, n ths ( 1 case cos( ) 1/ tg, see Fg 4b We note that the prevous equaton has only solutons when / 4 Fgure 5 Graph of determnant J for dfferent value of Sngular confguratons can be avoded by rotatng the probe axs around the normal to the ultrasound plane wthn a small angle he goal of the probe movement, as t s mentoned above, s to fnd the best ntersecton between the ultrasound plane and the nvestgated organ A small rotaton (<5 ) of the probe axs n the ultrasound plane rotates the ultrasound mage wthout modfyng the ntersecton hs small rotaton of the mage, see Fg, can be corrected by mage processng, f necessary, wthout loss of crtcal nformaton hs soluton allow us to avod sngular confguratons but t does not avod us to work near the robot sngular confguratons Fgure 4 Sngular confguratons of the robot Fgure Rght ultrasound mage obtaned from the left mage after a less than 5 rotaton n the ultrasound plane 5 Optmaton of Geometrc Parameters he am of ths optmaton s to establsh the value of the geometrc parameter, the angle, n order to mprove the robot performance near the sngular confguratons he ndex used to evaluate ths performance s the

5 condtonng of the manpulator [1], [] It s the condton number of the dmensonally homogenous Jacoban matrx J r ; It measures the nversblty of the matrx and s gven by the rato of the hghest sngular value of the matrx to the lowest one: k( J r ) max / mn hs numercal values goes from 1, for the sotropc confguratons to nfnty, for the sngular confguratons he prncple role of the prsmatc lnk P s to modfy the value of the contact forces, as we mentoned n subsecton, For ths reason, we chose to apply the optmaton crtera when the value of the dsplacement d s null In ths case, the Jacoban matrx s dmensonally homogenous and s equal to J A 0 0 C r he condtonng of the manpulator s a posturedependent ndex; t depends, for ths robot, on the jonts 1, and on the fxed angle varables o obtan a global ndex for optmng the performance we calculate the sum of the condtonng of the manpulator for all confguratons, except sngulartes he value was chosen to mnme the functon (,, ) 1 d1d f( ) : K he two followng constrants were mposed on : / 4 to avod more sngular confguratons, as show n secton 4; and / 8 to obtan at least a mnmum nclnaton of the probe axs of / 4 from the normal of the skn he result obtaned for s 4 7 Matlab software optmaton tools were used to compute the results for ths study Summary and Conclusons In ths paper we present a new DOF robot dedcated to a tele-echography medcal applcaton From the functonal specfcatons descrbed by the medcal doctors requrements and geometrcal parameters resultng from ths study of optmaton, a dedcated robot was desgned and manufactured Fg7 he next step of our work wll consst of mplementaton and comparson of control methods usng the varous models presented n ths paper, geometrc, knematcs and dynamc he correspondng results should allow us to choose the best effcent soluton n term of path followng A specfc study s n progress to take nto account the sngularty problem and to mnme ts effects on the robot s movements by an adapted control law hs work wll be a technologcal step before the clncal valdaton scheduled n the second trmester of 00 Acknowledgments he project s supported by the European Commsson, contract number IS References Fgure 7 Otelo-1 DOF robot prototype [1] J Angeles, Fundamentals of Robotc Mechancal Systems: heory, Methods and Algorthms Second Edton, Sprnger, 00 [] J Angeles, F Ranjbaran, RV Patel, On the Desgn of the Knematc Structure of Seven-axes Redundant Manpulators for maxmum condtonng, Proc IEEE Int Conf on Robotcs and Automaton, pp , 199 [] F Courreges, N Smth, G Posson, P Veyres, A Gourdon, M Speg, O Mergeaux, Real-tme exhbton of a smulated space tele-echography usng an ultra-lght robot, ISAIRAS, Canada, 001 [4] E Dombre, Analyse et Modélsaton des Robots Manpulateurs, Hermes Scence Publcatons, 001 [5] A Gourdon, Etude et Réalsaton d un Robot Médcal à ros Degrés de Lberté et Axes Concourants: Applcaton à la élé-échographe, hess of the Unversty of Versalles Sant-Quentn, 000 [] W Khall, E Dombre, Modélsaton Identfcaton et Commande des Robots, nd edton, Hermes Scence Publcatons, 1999 [7] Otelo Web ste: [8] Snters company web ste: