ARCHI: A redundant mechanism for machining with unlimited rotation capacities

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1 ACHI: A edundan mechanism fo machining wih unlimied oaion capaciies Fedeic Maque, ebasien Ku, Olivie Compan and Fançois ieo LIMM - M 556 C / M 6, ue Ada Monpellie cede 5, Fance <maque, u, compan, pieo>@limm.f Absac - This pape pesens a 3 dof edundan paallel mechanism, ACHI, dedicaed o machining, as a sub-pa of a 5-ais hbid machine. We descibe he edundan paallel mechanism design, is models, and discuss was fo is conol. ecenl began o sud alenae soluions based on he pinciple of Moion-haing whee boh he ool and he wo-piece ae moved, and whee a leas a pa of he moion is due o paallel chains (: ismaic join, : nivesal, : pheical, : evolue. Inoducion The idea of paallel obos was fis developed b Gough [] and ewa [] in he 5's and 6's wih he idea of "heapods" (6 dofs [3][4]. The evoluion inoduced b Clavel and his Dela sucue [5] in he lae 8's, opened a new aea wih machines able o each eemel high acceleaions and which ae commonl used fo pic-andplace; ecenl, we have used he Dela pinciple fo machining applicaions and paicipaed o he ceaion of ane [6] (a 3-ais dilling-aping-boing machine-ool wih linea dives, and a 3.5~5. g s acceleaion capabili. Consideing he machining of comple shape objecs, he soluion ofen poposed b paallel mechanisms is damaicall diffeen fom he soluion in use in indus; on one hand, mos indusial machines (Figue ae based on a 5-dof seial chain whose wea poin is he so-called head, i.e. he las wo oaing joins (he e issues ae he lac of siffness, he limied speed, and he difficul o ceae compac designs; on he ohe hand, mos paallelmechanism-based machines el on 6-dof aangemen. Diffeen pacical designs have poven he efficienc of paallel mechanisms in his pe of applicaions in ems of speed, accuac and siffness (see [7] and HeaM machine ool [8] poposed b Tooda fo eample. Appoaches using less acuaos (ha is, five acuaos, fo five machining ais have been poposed ecenl o educe cos and complei (Figue, bu he usuall suffe fom he same limiaion ha full paallel chains: a limied iling angle. Hbid seial / paallel o paallel / seial sucues (eos Ticep obo, eoul nivesi Eclipse Machine [9][], D Technologies pin have o be menioned as soluions fo 5-ais machining: on one hand, he paallel / seial aangemen of Ticep offes a good dnamic behavio hans o is paallel sub-pa, bu i sill suffes fom he naual limiaions of a seial head ; on he ohe hand, pin achiecue guaanees a good behavio of he head bu is neveheless limied in ems of iling angle. We Tool Figue. Classical (seial 5-ais machine vs full-paallel 6-dof machine (hee, HeaM pinciple, Tooda Tool Tool Figue. 5-ais, 5-dof wih passive chain Tool Figue 3. Hbid aallel-eial (Ticep, eos o eial-aallel (pin, D Technologies We designed a new paallel edundan sucue, ACHI, as a pa of a 5-ais equipmen fo machining. We descibe Tool Coesponding auho

2 hee is basic design, inemaics and dnamics models, and discuss conol saegies. Moion-haing : ACHI basic design We have alead inoduced one possible design, based on he so-called H4 sucues ([], [], [3], whee 4 dof ae dedicaed o he ool moion, and dof is a wo-piece level. Wih he ACHI pojec, we eend he idea of moion-shaing beween ool and wo-piece a sep fuhe, and we popose o give he ool a complee plana moion ( anslaions, oaion, and le he wo-piece be moved along he emaining anslaion ais and abou he emaining oaion ais. Keeping in mind he needs fo high siffness in machining, we developed fis ACHI concep wih linea dives and hus we focussed ou effos on plana paallel mechanisms diven b linea moos. lana paallel mechanisms have been inensivel sudied (see [4] and i uns ou i is no possible o obain a eall lage ange of moion in oaion wih classical aangemens. This is obviousl because of he eisence of singula posiions, and among hem, moe specificall ove-mobili singula posiions whee he machine siffness is zeo. The machine we developed is designed o allow unlimied oaion capabili hans o a edundan paallel inemaic mechanism. Figue 4 is he ACHI aangemen gaph : fou linea dives ae fied on he base, and hen lined o he nacelle (caing he spindle hans o - o - chains 3 ; on he wo-piece side, wo joins ae aanged in a seial wa. Figue 5 shows a schemaic view of such a mechanism, which can be seen as wo (paallel ams moving in a (, plane and lined ogehe via a igid bod and wo pivos. Those wo ams collaboae o ovecome singulaiies. Indeed, when he nacelle oaes fom o π, each chain can be in a singula posiion when he leg is aligned wih he nacelle: in his case, he singula chain canno poduce an oque abou z ais. Howeve, when a chain is singula, he hee ohe chains ae no singula and he complee mechanical sucue sas conollable. z ingulai Figue 6. Acuaion edundanc o ovecome singulaiies uch mechanisms offe in addiion anohe advanage: he ae eemel eas o build and assemble. We have designed ou fis ACHI poope as a simple es-bed (no focus an aesheic able o offe good pefomances: we use fou LineaDives moos, a FAEMAT spindle and few off-heshelve mechanical componens. Figue 7. A CAD view of a pacical design Tool Wo iece Figue 8. A picue of he fis ACHI poope 4 Figue 4. ACHI basic design 3 Kinemaic and dnamic modeling Figue 9 descibes he geomeical paamees. z Figue 5. A possible configuaion fo ACHI This concep alead eiss fo some seial machine-ools, and has been called he lef-hand/igh-hand paadigm in he oboics communi. 3 The - chains can be eplaced b - chains; he can also be eplaced b - chains if a pefec alignmen of he joins can be obained. 4 Fo moe phoos, see:

3 q q3 A A 3 A A 4 L L L L d C B 34 θ q q4 & & = & 34 & 34 v = G & d.sinθ & d.cosθ & d.sinθ θ& d.cosθ B 3. osiion elaionship Figue 9. ACHI paamees Veco = [ q q q ] q 3 q4 denoes dives posiions and = [,, θ ] is he nacelle configuaion, descibed b he posiion of poin C and he oienaion of he nacelle. If coodinaes of poins B and B 34 ae especivel (, and ( 34, 34, we have: ½ = d.cosθ = d.sinθ Invese posiion elaionship 34 = d.cosθ 34 = d.sinθ. q = s q = s q3 = 34 s q 4 = 34 s whee: s = L² ², s = L² 34 ² ½ Fowad posiion elaionship 5 Depending on which se of 3 chains among he 4 is seleced, diffeen compuaions can be made, which ae simila o: =.( q q3 34 =.( q q4 M = L².( q3 q² 4 34 = L².( q4 q ² 4 The posiion of C, and he nacelle angle, ae given b: 34, 34, an( 34 = = θ = Veloci elaionship The elaion beween velociies of poins (B, B 34 and C can be wien in a mai fom: 5 Bee esuls could be obained b esoing o ieaive numeical scheme The veloci elaionship can be wien hans o he classical pope: V(B A B = V(A A B, =, V(B 34 A B34 = V(A A B34, =3,4. ( denoes he do poduc These equaions can be gouped in a mai fom such as: Whee: J q J v q q = q = This can be epessed as: If J q whee: J v v = J q q & q 34 q3 34 q4 34 q3 J v G & = J q q& o J & = J q q & is no singula, his leads o: J m = s s 34 s 34 s q & = J m & q4 d.(sinθ.cosθ s d.(sinθ.cosθ s 34 d.( sinθ.cosθ s 34 d.( sinθ.cosθ s mehods, whee he Caesian posiion veco is esimaed as: n = n J( n,qn.[ qd qn ]

4 3.3 ingulai issue The Jacobian condiioning inde of a mechanism pemis o evaluae is isoop. In ou case, we compaed he Jacobian of he edundan machine o he Jacobians of he 4 paial mechanisms composed of onl 3 ams (i.e. non-edundan mechanisms: J 4, J 34, J 3, J 34. The fis seies of cuves depics he behavio of each se of chains: singulaiies occu cleal when he condiion numbe ends o infini. On he cona, he edundan complee machine offes good condiioning fo boh maices, guaaneeing ha no singulai occus (of couse, unde-mobili singulai sill eiss fo his mechanism, bu he ae eas o manage as fo mos paallel mechanisms. e3 [ Jm. Md. Jm Mn]. && [ Jm. Md. J& m.jm ]. & fn Mn. g Jm. fd = This elaion can also be epessed on he classical fom: ~ ~ f n = A(. && H(, & 4 Conol saegies simulaion esuls Thans o all he pevious models, we have buil a simulao of ACHI including of couse dnamic effecs, and he possibili of inoducing eos on diffeen paamees. We pesen hee simulaion esuls fo a pical moion combining a anslaion and a oaion fom π / 4 o π / ; duing ha moion, he chain o 4 cosses a singulai. Cond(J34 Cond(J3 hea.5 Cond(J4 Cond(J (m/ad d /a m X (.5 Cond(J Cond(Jq ime (s q4.5 q - angle ( deg. Figue. Jacobian condiioning of edundan and non-edundan mechanisms fo = m, = -.4 m and θ [ -,] degees (L = m, d =. m q (m (m Q.5.5 q q3 3.4 Dnamic elaionship In his pa ams dnamics is negleced. Le s conside f d and f n as, especivel, dive s and nacelle s foces. The basic elaion of dnamics can be wien: Jm.( fd.q& Mn. g fn = Jm.Md.q&& Mn.& wih: M n : Mai conaining nacelle s weighs M d : Mai conaining dives weighs : dives ficion coefficien g: gavi veco. Moeove, he acuaos acceleaion is: q& = J m. && J& m. & and he model becomes: ime (s Figue. A pical es ajeco (Caesian and join moions 4. Independen join space conol Mos (if no all indusial machine-ools conol ssems ae based on independen (linea join conol loops; wih acuaion edundanc, an dimensional eo in inemaic models leads o non-convegence poblems as soon as an inegal effec is included in he conol loops. Figue shows such a poblem wih a % eo on one leg lengh. Ou idea is hen o sud an alenae appoach (dnamic Caesian scheme wih diffeen conol saegies.

5 A A 3 A A 4 ( ( 5-5 B C B 34 θ f n - f f, m z ime (s Figue. Moos foces do no convege in case of dimensional eo 4. Geneal appoach: Dnamic Caesian scheme To ovecome he poblems due o dimensional eos, we popose o use onl Caesian conol schemes; moeove, since paallel mechanisms ae inended o offe high speeds and high acceleaions, we conside ha aing ino accoun dnamics is mandao fo bee pefomances. Howeve, edundanc issue has been consideed oo: he non-unici of dives foces coesponding o a given eenal foce ma be addessed in vaious was. d & d & d K p = h ( q K w ~ v A ( - m J Figue 3. Dnamic Caesian conol sucue 4.3 -am obo lie conol We have eplained in secion ha ACHI could be seen as a wo-am obo caing a solid objec. Thus appealing conol saegies ma be simila o hose poposed fo woam obos; fo eample, Dauchez [5] poposed o define inenal foces as foces acing inside he caied objec. In a simila wa, we can define conol saegies whee he inenal foce acing in he nacelle is consideed Wihou gavi effec - The simples wa o conside such inenal foce is o egad saics onl: hen he inenal foce ( inside he nacelle is defined as he diffeence beween he saic foces, f and f, poduced b each individual obo (i.e. each -dof sub-pa, pojeced on he nacelle diecion (given b veco n in Figue 4. An appealing conol saeg is hen o se his inenal foce o zeo: q f n dq / d f d compuaion f d OBOT ~ H This condiion leads o: wih: Figue 4. Foces on he mechanism [ f f f f ] [ f f m ] J f. = Moeove, J f = d.sinθ cosθ f d = [ J.J q ] d.cosθ sinθ Then dives foces ae given b: d.sinθ cosθ z d.cosθ sinθ.[ f f f ] f.[ f f m ] f d = [ J. J q ]. J f z 4.3. Wih gavi effec In his case, he supplemena condiion is: ( f f. n = M n. G.sinθ 4.4 Minimizing a given cieion As dim ( Ke( J m =, he equaion J m. f d = f has an infini of soluions [6][7]: f d = J m. f [ I J m. J m ]. z, 4 z In fac, [ I J m. J m ]. z belongs o he null-space [8] and epesens inenal foces in he mechanism: [ I J m. J m ]. z Ke( J m Then, dives foces compuaion can be ealized b minimizing a given cieion [9][][] as he -nom o he infinie nom of dives foces. ( f f n =

6 4.4. seudo-invese based conol The soluion coesponding o z =, i.e. he esul of a pseudo-invesion, minimizes f d (if z =, he inenal foces ae equal o zeo Infini-nom based conol Minimizaion of he infinie nom of f d can be pefomed if we choose z =. gad( f d,. I coesponds o a minimizaion of he maimal dive foce. ( ( ime (s imulaion esuls Figue 5 and Figue 6 descibe especivel dives foces and he nom of he inenal foce in he case of a -am conol. If we neglec gavi effecs, he saic inenal foce is quie impoan wheeas if we don neglec hese effecs he saic inenal foce is ve small. Fo he conol using he pseudo-invese (Figue 7, he esul is bee han he -am conol because dives foces componen in he null-space is equal o zeo. Figue 8 shows ha he minimizaion of he infinie nom of dives foces is moe efficien han he simple pseudo-invese, bu he cuve epesening dives foces pesens undesiable "disconinuiies". ehaps a good soluion could consis in he minimizaion of a p nom, p > (compomise beween he nom and he infinie nom. If he wo-am conol seems o be less efficien han he conol using he minimizaion of a given cieion, howeve i is ve simple o implemen and i needs few calculaions. om (Fe ( e o n ime (s Figue 6. Foces wih wo-am lie conol, wih gavi ( ( ime (s ( ( -5 - om (Fe ( e o n ime (s ime (s Figue 7. seudo-invese based conol 7 om (Fe e o n ime (s Figue 5. Moos foces and nom of he inenal foces wih a woam lie conol, wihou gavi

7 ( om (Fe ( e o n ime (s ime (s Figue 8. Infini-nom based conol. 5 Conclusion. In his pape we descibed Achi, a new 3 dof edundan paallel obo acuaed b fou dives. Afe giving is inemaic and dnamic models, we demonsaed is abili o easil evolve in he whole wospace and o allow unlimied oaions. imulaions based upon diffeen join foces compuaions mehods wee compaed, a poope was buil and he implemenaion of a conol sofwae is unde wa. Then, Achi obo will be included as a pa of an hbid obo dedicaed o machining. [8] ieo F., hibuawa T., Fom Hea o HeaM. In oc. IK'98: Inenaionales aallelinemai-kolloquium, Züich, June 4, 998, pp [9] Kim J. and a F.C., Eclipse A new paallel mechanism poope. osiion pape in oceedings of he Fis Euopean- Ameicana Foum on aallel Kinemaic Machines, Milan, Ial, Augus 3- epembe, 998. [] Kim J., a F.C., Lee J.M., A new paallel mechanism machine ool capable of five-face machining. Annals of he CI, Vol.48, o., pp , 999. [] Compan O. and ieo F., A new 3T- paallel obo, ICA 99, Too, Japan, Ocobe 5-7, 999, pp [] ieo F., Maque F., Compan O., Gil T., H4 paallel obo: modeling, design and pelimina epeimens. IEEE In. Conf. On oboics and Auomaion, eoul, Koea, Ma. [3] hp:// [4] Innoceni C. and aeni-caselli V., Ehausive enumeaion of full paallel inemaic chains. Dnamic sem and Conol, Vol. 55-, pp [5] Dauchez., Delebae X., Joudan., Hbid conol of a woam obo handling fiml a single igid objec, oc. of he nd IFI, aagoza, pain, ovembe 989, pp [6] Koc., chumache W., A mied elasic and igid bod dnamic model of an acuaion edundan paallel obo wih higheducion geas, IEEE In. Conf. On oboics and Auomaion, Apil. [7] Dasgupa B., Muhunjaa T.. Foce edundanc in paallel manipulaos: heoeical and pacical issues. Mechanism and Machine Theo, 33(6: , Augus 998. [8] Koc., chumache W., A paallel - manipulao wih acuaion edundanc fo high-speed and acive siffness applicaions. IEEE In. Conf. On oboics and Auomaion, Leuven, Belgium. Vol. 3:. 95-3, Ma 998. [9] Kuz., Hawad V., Muliple goal inemaic opimizaion of a paallel spheical mechanism wih acuao edundanc, IEEE In. Conf. On oboics and Auomaion, Vol. 8, o Ocobe 99. [] Yoshiawa T., Analsis and conol of obo manipulaos wih edundanc, The in. Jounal of oboics eseach, Vol. 4,985. [] Koinis T. e Millies., A paallel obo-am egional sucue wih acuaional edundanc. Mechanism and Machine Theo, 6(6: , 99. Acnowledgemens: This wo has been paiall suppoed b he Euopean Commission, GOWTH ogam, Conac GD CT efeences: [] Gough V.E., Conibuion o discussion of papes on eseach in auomoive sabili, conol and e pefomance. oc. Auo Div. Ins. Mechanical Enginees, [] ewa D., A plafom wih 6 degees of feedom. oc. of he Ins. of Mech. enginees, 8 (a, 5, pp , 965. [3] Mele J.-., Les obos paallèles, nd Ediion, Hemes, 997. [4] Tönshoff H.K., A ssemaic compaison of paallel inemaics. Kenoe in oceedings of he Fis Foum on aallel Kinemaic Machines, Milan, Ial, Augus 3- epembe, 998. [5] Clavel., ne nouvelle sucue de manipulaeu paallèle pou la oboique légèe, AII, 3(6, pp. 5-59, 989. [6] Compan O., ieo F., Laua F., Fioini C., Modeling and pelimina design issues of a 3-ais paalell machine-ool, KM' : Inenaional Confeence on aallel Kinemaic Machines, Ann Abo, Michigan, epembe 3-5,, pp [7] ieo F., Dauchez. and Founie A., Fas paallel obos. Jounal of oboic sems, 8(6, pp , 99.

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