Innovative Kinematics and Control to Improve Robot Spatial Resolution

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The 00 IEEE/RSJ Inenaional Confeence on Inelligen Robos and Sysems Ocobe 8-, 00, Taipei, Taiwan Innovaive Kinemaics and Conol o Impove Robo Spaial Resoluion Jean-Fançois Behé, Membe, IEEE Absac The

1 The 00 IEEE/RSJ Inenaional Confeence on Inelligen Robos and Sysems Ocobe 8-, 00, Taipei, Taiwan Innovaive Kinemaics and Conol o Impove Robo Spaial Resoluion Jean-Fançois Behé, Membe, IEEE Absac The pape pesens innovaive kinemaics and conol of a plana edundan obo designed o impove spaial esoluion by a faco 5. This esul is obained in a esiced aea of he wokspace using posiion infomaion fom exenal sensos. This innovaion esuls fom a cleae undesanding of he facos ha influence he obo micomeic behavio: axes conol esoluion geneaes a se of aainable poins in he obo wokspace and diffeen spaial esoluion paens appea when inoducing edundancy depending on he final axes chosen o coec he posiion. C I. INTRODUCTION HOOSING and designing he adequae obo o pefom a ask is no easy. Manufacues povide some pefomance cieia such as payload, wokspace dimensions, acceleaion and epeaabiliy. These cieia ae deailed in ISO983 [] o ANSI R [] bu hese noms descibe numeous ohe cieia which ae no ofen available fo indusial obos. Fo insance, if we waned o choose he igh obo o pefom minue assembly asks like pick and place in eleconic assembly, which cieia should be used? Should we conside he pose epeaabiliy, he pose accuacy, he obo spaial esoluion? Should we choose a seial o a paallel obo? Opimal obo design is a key issue in oboics and a clea insigh is given in [3]. Numeous mehodologies exis o opimize obo kinemaics. A lage body of hem poposes using manipulabiliy o dexeiy indexes [4]. Ohes ae moe concened wih dynamic [5]-[7] o elasosaic pefomance [8], [9]. The pos and cons of hese diffeen mehodologies ae well-known fo seial obos [0]. Now ecen eseach deals wih accuacy analysis in paallel oboics []. One of hese papes [] is a compaison of he accuacy of a paallel and a seial obo and he mehodology is based on he sensiiviy. Ohe cieia can be used fo his kind of compaison fo example in elaion o he sochasic ellipsoid heoy based on he angula covaiance maix [3]. In his pape, we descibe innovaive seial obo kinemaics ha can gealy impove pecision in a limied aea of he wokspace. This achiecue was no discoveed using a adiional appoach because oboicians usually avoid edundancy and singulaiies. In fac, new popeies of Jean-Fançois Behé is wih he GREAH laboaoy, fom Le Have Univesiy, Fance; jean-fancois.behe@univ-lehave.f edundancy on spaial esoluion ae hee pesened and used in an innovaive conol scheme o impove local pecision. In secion II, we descibe he adiional appoach o local esoluion modeling based on axis esoluion. In secion III, we sudy he SCARA obo case and descibe in deail he song vaiaions of he spaial esoluion in he wokspace. In secion IV, we inoduce edundancy and poin ou wo ineesing popeies. A he same ime, we show how inenal and exenal sensos can be used in a conol saegy paving he way o diffeen spaial esoluions. In secion V, we pesen a edundan kinemaic obo sucue impoving spaial esoluion in a limied aea of he wokspace. In secion VI, we explain how his achiecue is efficien o impove pecision on he micomeic scale. II. SPATIAL RESOLUTION DEFINITION The pecision of a oaional o linea axis is chaaceized by diffeen indices displayed in fig.. The conol esoluion is he smalles diffeence beween wo diffeen ages = Ti+ Ti. The pecision eo can be descibed by wo diffeen indices : The epeaabiliy index Rep which esimaes he disance eo aound he posiion mean; The exaciude index Ex which is he disance beween he posiion mean and he eal age. Fig. Resoluion, epeaabiliy, exaciude. When he conol is pefec and he age T i is o be eached, he final posiion lies in he ineval Ii = [ + T; ] i + Ti +. Some auhos conside ha he posiion disibuion is consan ove he ineval I i. We sudied saisically his disibuion and showed ha in /0/$ IEEE 3495

2 ealiy i was Gaussian [4]. Fom his i is possible o model epeaabiliy wih sochasic ellipsoids. This appoach is highly ineesing because i is possible o sudy all epeaabiliy influence facos. Fo example, we poved ha fo wo 6 axis seial obo sudied, he load influence was fa less impoan han he wokspace locaion influence [5], [6]. This was no he fis wok on his subjec. Saisical wok has been poposed o quanify he influence of some facos on epeaabiliy [7], [8]. Rieme and Edan wee ineesed in wokspace locaion influence [9]. Offodile and Ugwu have sudied load and speed influence [0]. When conolling he obo fom he angula space, he angula esoluions of each acuao ae combined o ceae he spaial esoluion paens. Fo a given age Θ= ( θ, θ, θ3, θ4, θ5, θ6) whee he esoluion of each acuao is R = (,, 3, 4, 5, 6), he final angula posiion will be: R Θ± = ( θ±, θ ±, θ3 ±, θ4 ±, θ5 ±, θ6 ± ) We sudied he pefomances of one Samsung Faaman FARA and one Kuka IR384 obos wih especive angula esoluion of 0.0 and Cuenly, we ae woking on an EPSON RS450. I is possible o conol his obo by specifying he numbe of pulses. If he sevo-conol is pefec wih nil eos, he spaial esoluion is hen a is bes. Le he spaial esoluion SR be he se of poins of he X = xyz,, coesponding o he age final obo posiion ( ) Θ wih he unceainy R. I can be compued in wo diffeen ways. In he fis mehod 6 poins ae compued in he Caesian space using he ineval analysis and he fowad kinemaics funcion X = F( θ ) ansfoming join coodinaes Θ ino wokspace coodinaes X = ( xyz,, ). The spaial esoluion SR will be a convex polyhedon 6 obained fom he convex envelop of he poins R F Θ± : ( ) { ( R )} SR = Conv F Θ±. The second mehod consiss in diffeeniaing he fowad kinemaics funcion X = F( θ ) and using he Jacobian funcion mapping he join velociy veco o he Caesian velociy veco in he linea ansfomaion dx = J ( θ ) dθ. This elaionship can also be undesood as he link beween small angula and Caesian vaiaions. The spaial esoluion SR will be a convex polyhedon obained fom 6 R he convex envelop of he poins ± J ( Θ ) : { } R SR = Conv ± J ( Θ ) Whaeve he mehod, he esuls will be he same because he esoluion R is usually vey small compaed o he angula value Θ so he fis ode appoximaion is vey pecise. The wo mehods ae only diffeen because he fis one consides he se of poins and he ohe one, he poin vaiaions. In boh cases, he following esuls in secion III ae idenical. In geneal, i is difficul o evaluae he evoluion of size, volume, oienaion of hese diffeen polyheda. This could explain why spaial esoluion has been someimes eplaced by epeaabiliy, hoping ha he epeaabiliy sphee could appoximae he polyhedon. Bu he epeaabiliy index will no give any clue abou spaial disibuion []. Using his adiional appoach whee he esoluion is consan in angula space, he vaiaions of he spaial esoluion in he wokspace depend only on he lenghs of he links. The final unceainy ineval a he end of a oaing am is equal o he poduc of he link lengh by he angula esoluion. If he angula esoluion is consan, he longe he am, he lage he unceainy fo he exemiy s final posiion. This phenomenon is known as leve-am amplificaion eo. As obos ae mosly buil wih oaional joins, he spaial esoluion depends on he obo posue and locaion which deemine he leve-am lengh fo each acuao. This funcion is non-linea and difficul o sudy in he geneal case. III. SPATIAL RESOLUTION VARIATIONS FOR THE SCARA To illusae he song vaiaions of spaial esoluion, le us conside a SCARA obo wih he usual fou dof and he same angula esoluion on he s and nd axis. The hid axis is a oaion aound he obo endpoin. I does no inevene fo spaial esoluion bu only fo oienaion esoluion. The fouh axis coesponding o he Z- anslaion is easy o ake ino accoun adding veical veices o he paens dawn on he plane. So o illusae he esoluion vaiaions, he epesenaion will be dawn on he hoizonal plane consideing only he fis and second axes. In his paicula case, he spaial esoluion is a paallelogam. Fo he hee diffeen locaions in he SCARA wokspace displayed on fig., we daw he mesh of he diffeen paallelogams coesponding o he closes ages wih he same scale. Fig. The hee diffeen chosen locaions of he SCARA. 3496

The paallelogam paens ae displayed on fig.3 and lead o he following conclusions: The suface, size and oienaion of he spaial esoluion paallelogams ae vey diffeen accoding o he wokspace locaion.

3 The paallelogam paens ae displayed on fig.3 and lead o he following conclusions: The suface, size and oienaion of he spaial esoluion paallelogams ae vey diffeen accoding o he wokspace locaion. The spaial esoluion is much fine in he cene of he wokspace coesponding o he s locaion, compaed o he ohe locaions because he suface of he paallelogam is smalle. I means ha he final posiion will be in a smalle aea and ha i is possible o eposiion moe pecisely he obo changing slighly he age. The posiion can be minuely coeced in he X-diecion bu as he ecangle sides ae in a /0 aio, he esuling Y-posiion unceainy is lage. Moeove, in his case, displacemens ae decoupled in he X and Y diecions. Fo he 3d locaion, he spaial esoluion is bee in he Y diecion han in he X diecion. The main conclusion is ha spaial esoluion is bee nea he singulaiies of he obo. Fo he s locaion, his bes esoluion diecion coesponds o he move induced by he fis axis oaion. The lengh of his move is obained muliplying he disance beween he s axis and he locaion by he fis axis angula esoluion. So i is eally he disance of he leve-am ha is he main paamee o ake ino accoun fo esoluion impovemen. Roughly speaking, he inoducion of edundancy poduces boh a densificaion and paen modificaions elaed o he moving axes. A. Densifying Caesian space To illusae his popey, le us daw he spaial esoluion polygons fo wo plana obos, SCARA and SCARA3. SCARA is he SCARA descibed in he peceding secion. SCARA3 is a plana edundan obo wih hee degees of mobiliy as shown in fig.4. The hee links of he SCARA3 have he same lengh. Le J SC3 be he Jacobian maix of he SCARA3. Fig.4 Geomey of SCARA3 obo and posue Fig.3 Paallelogam of spaial esoluion fo he hee locaions. IV. IMPACT OF REDUNDANCY ON SPATIAL RESOLUTION Redundancy is ofen used in oboics o give moe dexeiy o he obo. When some aeas ae difficul o aain because of obsacles o when some asks need special oienaions ha ae no compaible wih a 6 dof obo, inoducing edundancy can be a soluion. Unfounaely he obo conol is hen less easy. Wha is he impac of edundancy on spaial esoluion? Le = = 3 = be he esoluion of angula conol. The spaial esoluion polygons ae hexagons coesponding o he convex envelop of he 3 poins: { J SC3 [ ±, ±, ± 3] } One of his hexagon displayed on fig.5 is geomeically buil fom he vaiaions ± u =± J SC3 [,0,0] ± u =± J SC3 [ 0,,0 ] ± u =± J 0,0, [ ] 3 SC3 3 The disposiion of some close polygons ae shown on fig.6 coesponding o 3 anslaions of u, u,u 3. I illusaes he fac ha a poin in a given aea does no coespond o a unique age. This is quie obvious because he obo is edundan bu his has impoan consequences fo he micomeic posiioning pefomance. On he ohe hand, i does no mean i is easie o conol he obo o each he desied aea. The choice of he igh age is sill a difficul poblem. Bu when one of he hee axes is locked, acivaing an elecomechanical bake fo insance, hen he polygon mesh epesening he diffeen eposiioning aea associaes again one poin o one unique age. 3497

is fine han he hexagon of fig.5. The squae on he lef side of fig.7 coesponds o he gey squae of fig.5. A his poin, we ae sill awae ha he final polygon suface is sill he same as he move is pefomed using he hee axes and he global spaial esoluion has no been impoved ye.

In his saegy, he final polygon esoluion depending of he las pai of axes, is smalle han he global one and i is possible o choose a bee age o impove he final posiion.

Modifying polygon esoluion paens We ae now consideing a new saegy consising of conolling he SCARA3 obo using only wo axes insead of hee. The emaining axis is locked.

4 is fine han he hexagon of fig.5. The squae on he lef side of fig.7 coesponds o he gey squae of fig.5. A his poin, we ae sill awae ha he final polygon suface is sill he same as he move is pefomed using he hee axes and he global spaial esoluion has no been impoved ye. Hee is he new saegy: a fis ough posiioning is pefomed using he hee axes and a he end of he move, he minue final posiion coecion is done using only wo of he hee axes. In his saegy, he final polygon esoluion depending of he las pai of axes, is smalle han he global one and i is possible o choose a bee age o impove he final posiion. This is he saegy ha will be used in he nex secions. Fig.5 Hexagon fo spaial esoluion of he SCARA3. Fig.7 Spaial esoluion paens fo s and 3d axes conol (lef) o nd and 3d axes final conol (igh). Fig.6 The mesh of spaial esoluion hexagons fo SCARA3 B. Modifying polygon esoluion paens We ae now consideing a new saegy consising of conolling he SCARA3 obo using only wo axes insead of hee. The emaining axis is locked. The obo is no longe edundan and a new polygon mesh can be buil. Fo insance, fo he posue of fig.4, he paen will change depending on he pai of axes chosen o dive he obo. Fig.7 illusaes his pinciple. If he final conol uses only he s and 3 d axes, hen he polygons ae paallelogams in an ohogonal layou. If he conol uses only nd and 3 d axes, hen he oienaion of he paallelogam is quie diffeen. So i is clea ha he choice of he final pai of axes has an impoan influence on he sucue of he final paallelogam mesh. I is hen possible o choose he axes o be conolled in he las posiioning sep accoding o he pefomance cieion, hee spaial esoluion. I will be easie o coec he posiion using he s and 3 d axes because he induced displacemens ae ohogonal. The final unceainy will be wice lage in he X-diecion if he nd and 3 d axes ae chosen fo he conol, bu he final eo will be he same in he Y-diecion. Bu in boh cases, he final spaial esoluion V. INNOVATIVE KINEMATICS AND CONTROL TO IMPROVE SPATIAL RESOLUTION We ae now consideing he pevious esuls o impove he spaial esoluion fo a specific locaion P I poin of inees in he wokspace. A minue posiioning is demanded nea his poin, so he final spaial esoluion mus be fine in wo ohogonal diecions. This could be done using wo join aiculaions whose oaion cenes ae as close as possible o P I. I is fo insance s and nd axes of fig.8. Then o enlage he wokspace, i is necessay o add a hid axis. Consequenly, he obo is now edundan, and able o gasp an objec in a wide aea. The conol saegy consiss of wo seps: in a fis sep, he obo endpoin is bough close o he desied age P I using he hee axes. Then he 3 d axis is locked. The posiion eo is esimaed fom exenal senso infomaion and he new age is compued. In he second sep, he obo comes close o P I using only he s and nd axes. Fo he sucue displayed in fig.8, le he lenghs of he links be l =, l = 5, l 3 = 4.5 (dm). Fo he angula posiion ( θ = 35, θ = 90, θ3 = 73 ), he locaion of he obo endpoin is PI ( 0.697, 0.058) (dm) vey close o he he s and nd axes. The paallelogam mesh esuling fom he s and nd axes final conol in he viciniy of he poin of inees is ohogonal and consiss of sho-sided squaes displayed on Fig.9. The scale of he dawing is obained fo 4 a esoluion of.0 ad. Fig.0 is a compaison of his squae mesh wih he hexagon of spaial esoluion when he 3 axes ae used. I hen becomes obvious ha i is easie o 3498

5 coec he final posiion using only s and nd axes in he final sep. posiioning is pa of he obo sucue iself, hidden in is geomeical chaaceisics. Fig.8 Innovaive plana edundan sucue SCARA3 This minue spaial esoluion is sill ue if he final poin moves away fom P I bu says in he viciniy of P I. The mesh is slighly changed in oienaion and he squae becomes a paallelogam, bu he dimensions of he paallelogam emain small as long as he final poin says in a disk ceneed on P I wih a adius lowe han 0.35 fo insance. This poin P I is named poin of inees and he viciniy of PI is an ineesing aea fo local pecise eposiioning. The ineesing aea suface may seem o be quie small bu in ealiy, ohe s and nd axis configuaions poduce moe ineesing sufaces and he oal ineesing aea is a ing ceneed aound he s axis and wih 0.35 dm inenal and.05 dm exenal adius. This can be a disadvanage in ceain applicaions bu i can be useful in some specific asks fo insance in some eleconic manufacuing opeaions whee he assembly zone is ofen small and pecision is limied by he camea o binocula visual field, and he pieces ae gasped elsewhee in he wokspace. The pecision fo he gasping in his example is no as high as fo he final placemen on he PCB. This impovemen in spaial esoluion can be oughly esimaed fom he aio of he hid link lengh above he fis link lengh, eaching a faco 5 a leas in his example. This innovaive kinemaics and conol saegy have o be compaed o a adiional appoach o obain minue spaial esoluion. In fac, unil now, he answe consised in associaing a mico and a maco posiione, in wha was called a mico-maco appoach. One soluion is fo ool held by he obo o have a micoposiione fo minue spaial esoluion. In his case, he obo pefoms he ough posiioning and he ool he minue posiioning. The ohe soluion is o pu he ool on he obo and he pa on a able. The minue spaial esoluion is obained by small movemens of he able. In boh cases, 4 acuaos ae necessay fo he pocess. The innovaive soluion descibed in his pape needs only 3 acuaos. So he oal cos is lowe. In fac, he mico Fig.9 spaial esoluion paallelogam esuling fom diffeen final conol (in micomees) Fig.0 compaison of he spaial esoluion esuling fom diffeen final conol ; hexagon fo he 3 axes and squae fo s and nd axes. VI. EXTERNAL SENSORS AND CONTROL STRATEGY In he peceding secion, we explained he saegy. A he end of he ough posiioning, in he las sep, we need o esimae he posiion coecion befoe he minue las move. This las movemen mus be esimaed fom measues by exenal sensos. Fo example, hese sensos can be a camea wih high esoluion. As he final zone fo minue posiioning is small compaed o he oal wokspace aea, i is possible o choose a camea wih high esoluion and a naow visual field, fo insance 4-micon esoluion wih a visual field of one cm. Two impoan poblems may appea and uin ou effos if no ackled seiously. The fis one deals wih he los moion zone descibed in fig.. I is someimes impossible o move he obo endpoin diecly by a small incemen, because of ficion. 3499

6 If he obo endpoin is wihin he los moion zone, nohing may happen o he obo could move (fom S o F) bu he ajecoy is unknown which can be vey dangeous in some applicaions. Convesely i is possible o go diecly fom S o F. Fig. Coecive ajecoies o come close o he age So wha could be done when he obo endpoin is S3, in he losmoion aea, bu oo fa fom he desied age? The coec saegy is o epea one poion of a ajecoy and y o change i slighly so ha he final endpoin posiion is vey close o he peceding one. The sa poin of he ajecoy o be epeaed will be named hamonizaion poin (HP), because i is fom his poin ha evey ajecoy is epeaed in nealy he same condiions. Fo insance, saing a S3, coming backwads o HP, coecing he age, aiving a F3, backwads o HP, coecing again he age, aiving a F4 show how his saegy woks. Fo boh aemps, he final age is slighly changed. The second poblem is o decide when o sop coecing he age consideing ha he final posiion is good enough. In geneal, if he posiion eo is infeio o he spaial esoluion, he esul is consideed saisfacoy and he coecion pocess is sopped. VII. CONCLUSIONS Spaial esoluion was hee invesigaed and we have concluded ha i was songly dependen on he obo opology, achiecue, posue and wokspace locaion. Hence an innovaive obo sucue is deailed based on edundan kinemaics and a new conol saegy: a ough posiioning using 3 axes, hen fo he final conol, exa infomaion is colleced via exenal sensos and a minue final move is pefomed using only he wo axes wih he shoe leve-am lenghs. This innovaion opens he field o gea impovemens in he pefomance of seial obo fo minue micomeic posiioning wih pomising applicaions in minue assembly asks, eleconic and micoeleconic assembly indusy, opoeleconics, medical and biological fields REFERENCES [] ISO Manipulaing indusial obos Pefomance cieia and elaed es mehods. ISO983 (998) [] Insiue, A. N. S., Ameican Naional Sandad fo Indusial Robos and Robo Sysems - Poin-o-Poin and Saic Pefomance chaaceisics -Evaluaion, R [3] Angeles, Joge, & Pak, Fanck Spinge handbook of oboics. Spinge. Chap. 0. [4] C.Gosselin, he opimum design of oboic manipulaos using dexeiy indices, Roboics and Auomonous Sysems, Vol.9, N 4, pp.3-6, 99 [5] H. Asada: A geomeical epesenaion of manipulao dynamics and is applicaion o am design, Tans. ASME J. Dyn. Meas. Con. 05(3), 3-35, 983. [6] T. Yoshikawa : dynamic manipulabiliy of obo manipulaos, Poc. IEEE In.Conf. Robo. Auom., pp , 985. [7] P.A. Voglewede, I. Ebe-Uphoff : Measuing closeness o singulaiies fo paallel manipulaos, Poc. IEEE In. conf. obo. Auom., pp , 004. [8] M. giffis, J. Duffy: Global siffness modelling of a class of simple complian couplings, Mechanism Machine Theoy 8, 07-4, 993. [9] S. Howad, M.J. Zefan Kuma: On he 6x6 caesian siffness maix fo he hee-dimensional moions, Mechanism and Machine Theoy 33, , 998. [0] J-P. Mele, Jacobian, manipulabiliy, condiion numbe and accuacy of paallel obos, Jounal of Mechanical design 8, Jan 006, pp [] J. Hesselbach, J. Wege, A. Raaz, O. Becke, Aspecs on he design of high pecision paallel obos, Assembly Auomaion 4 (), [] S. Bio and I.A. Bonev, Ae paallel obos moe accuae han seial obos? CSME Tansacions, 007, vol3 n 4 pp [3] Behé, Jean-Fançois, & Lefebve, Dimii.. Ganula space sucue on a micomeic scale fo indusial obos. pp ICRA07. Roma, 007. [4] J-F. Behé, E. Vasselin, D. Lefebve and B. Dakyo, Deeminaion of he epeaabiliy of a Kuka Robo using he sochasic ellipsoid appoach, in ICRA05, Bacelone, pp [5] J-F. Behé and B. Dakyo, A sochasic ellipsoid appoach o epeaabiliy modelisaion of indusial obos, in IROS0, Lausanne, pp [6] J-F. Behé, E. Vasselin, D. Lefebve and B. Dakyo, Modelling of epeaabiliy phenomena using he sochasic ellipsoid appoach, in Roboica, 006, vol.4, pp [7] Eina Ramsli. Pobabiliy disibuion of epeaabiliy of indusial obos. The inenaional Jounal of Roboics Reseach. 0(3):79-83, 99 [8] Y.Edan, L. Fiedmae A. Mehez and L. Sluski. A heedimensional saisical famewok fo pefomance measuemen of oboic sysems, Roboics and Compue Inegaed Manufacuing, 4, 998, pp [9] Raziel Rieme e Yael Edan. Evaluaion of influence of age locaion on obo epeaabiliy, Roboica, vol.8 (000) pp [0] O.F. Offodile and K. Ugwu, Evaluaing he effec of speed and payload on obo epeaabiliy, Roboics and compue Inegaed Manufacuing 8, 99,pp.7-3 [] J-F. Behé, Ininsic Repeaabiliy: a new index fo epeaabiliy chaaceisaion, in ICRA0, Anchoage, acceped fo publicaion. 3500

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES In igid body kinemaics, we use he elaionships govening he displacemen, velociy and acceleaion, bu mus also accoun fo he oaional moion of he body. Descipion of he moion of igid