The Optimal Design of Three Degree-of-Freedom Parallel Mechanisms for Machining Applications

Transcription

1 Submtted to ICAR D. Chablat h. Wenger F. Majou he Optmal Desgn of hree Degree-of-Freedom arallel Mechansms for Machnng Applcatons Damen Chablat - hlppe Wenger Félx Majou Insttut de Recherche en Communcatons et Cybernétque de Nantes, rue de la Noë, 44 Nantes, France Damen.Chablat@rccyn.ec-nantes.fr Abstract he subject of ths paper s the optmal desgn of a parallel mechansm ntended for three-axs machnng applcatons. arallel mechansms are nterestng alternatve desgns n ths context but most of them are desgned for three- or sx-axs machnng applcatons. In the last case, the poston and the orentaton of the tool are coupled and the shape of the workspace s complex. he am of ths paper s to use a smple parallel mechansm wth two-degree-of-freedom (dof) for translatonal motons and to add one leg to have one-dof rotatonal moton. he knematcs and sngular confguratons are studed as well as an optmzaton method. he three-degree-of-freedom mechansms analyzed n ths paper can be extended to four-axs machnes by addng a fourth axs n seres wth the frst two. Key Words: arallel Machne ool, Isotropc Desgn, and Sngularty. Introducton arallel knematc machnes (KM) are commonly clamed to offer several advantages over ther seral counterparts, lke hgh structural rgdty, hgh dynamc capactes and hgh accuracy []. hus, KM are nterestng alternatve desgns for hgh-speed machnng applcatons. he frst ndustral applcaton of KMs was the Gough platform, desgned n 957 to test tyres []. KMs have then been used for many years n flght smulators and robotc applcatons [] because of ther low movng mass and hgh dynamc performances []. hs s why parallel knematc machne tools attract the nterest of most researchers and companes. Snce the frst prototype presented n 994 durng the IMS n Chcago by Gddng&Lews (the VARIAX), many other prototypes have appeared. o desgn a parallel mechansm, two mportant problems must be solved. he frst one s the dentfcaton of sngular confguratons, whch can be located nsde the workspace. For a sx-dof parallel mechansm, lke the Gough-Stewart platform, the locaton of the sngular confguratons s very dffcult to characterze and can change under small varatons n the desgn parameters []. he second problem s the non-homogenety of the performance ndces (condton number, stffness...) throughout the workspace []. o the authors' knowledge, only one parallel mechansm s sotropc throughout the workspace [4] but the legs are subject to bendng. Moreover, ths concept s lmted to three-dof mechansms and cannot be extended to four or fve-dof parallel mechansms. Numerous papers deal wth the desgn of parallel mechansms [4,5]. However, there s a lack of four- or fve-dof parallel mechansms, whch are especally requred for machnng applcatons [6]. o decrease the cost of ndustralzaton of new KM and to reduce the problems of desgn, a modular strategy can be appled. he translatonal and rotatonal motons can be dvded nto two separated parts to produce a mechansm where the drect knematc problem s decoupled. hs smplfcaton yelds also some smplfcatons n the defnton of the sngular confguratons. he organzaton of ths paper s as follows. Next secton presents desgn problems of parallel mechansms. he knematc descrpton and sngularty analyss of the parallel mechansm used, are reported n sectons. and.. Sectons. and.4 are devoted to desgn and the optmzaton. IRCCyN: UMR CNRS 6596, École Centrale de Nantes, Unversté de Nantes, École des Mnes de Nantes /6

2 Submtted to ICAR D. Chablat h. Wenger F. Majou About parallel knematc machnes. General remarks In a KM, the tool s connected to the base through several knematc chans or legs that are mounted n parallel. he legs are generally made of telescopc struts wth fxed foot ponts (Fgure a), or fxed length struts wth moveable foot ponts (Fgure b). the vertcal drecton s null and the force amplfcaton factor s nfnte. Fgure b shows a parallel sngularty. he velocty amplfcaton factor s nfnte along the vertcal drecton and the force amplfcaton factor s close to zero. Note that a hgh velocty amplfcaton factor s not necessarly desrable because the actuator encoder resoluton s amplfed and thus the accuracy s lower. Fgure a: A bpod KM Fgure b: A bglde KM For machnng applcatons, the second archtecture s more approprate because the masses n moton are lower. he lnear jonts can be actuated by means of lnear motors or by conventonal rotary motors wth ball screws. A classfcaton of the legs sutable to produce motons for parallel knematc machnes s provded by [6] wth ther degrees of freedom and constrants. he connecton of dentcal or dfferent knematc legs permts the authors to defne two-, three-, four- and fve-dof parallel mechansms. However, t s not possble to remove one leg from a four-dof to produce a three-dof mechansm because no modular approach s used.. Sngulartes he sngular confguratons (also called sngulartes) of a KM may appear nsde the workspace or at ts boundares. here are two types of sngulartes [7]. A confguraton where a fnte tool velocty requres nfnte jont rates s called a seral sngularty. hese confguratons are located at the boundary of the workspace. A confguraton where the tool cannot resst any effort and n turn, becomes uncontrollable s called a parallel sngularty. arallel sngulartes are partcularly undesrable because they nduce the followng problems () a hgh ncrease n forces n jonts and lnks, that may damage the structure, and () a decrease of the mechansm stffness that can lead to uncontrolled motons of the tool though actuated jonts are locked. Fgures a and b show the sngulartes for the bglde mechansm of Fg. b. In Fg. a, we have a seral sngularty. he velocty amplfcaton factor along Fgure a: A seral sngularty Fgure b: A parallel sngularty he determnaton of the sngular confguratons for twodof mechansms s very smple; conversely, for a sx-dof mechansm lke the Gough-Stewart platform, a mechansm wth sx-dof, the problem s very dffcult []. Wth a modular archtecture, when the poston and the orentaton of the moble platform are decoupled, the determnaton of the sngulartes s easer.. Knetostatc performance of parallel mechansm Varous performance ndces have been devsed to assess the knetostatc performances of seral and parallel mechansms. he lterature on performance ndces s extremely rch to ft n the lmts of ths paper (servce angle, dexterous workspace and manpulablty ) [9]. he man problem of these performance ndces s that they do not take nto account the locaton of the tool frame. However, the Jacoban determnant depends on ths locaton [9] and ths locaton depends on the tool used. o the authors' knowledge there s no parallel mechansm, sutable for machnng, for whch the knetostatc performance ndces are constant throughout the workspace (lke the condton number or the stffness ). For a seral three-axs machne tool, a moton of an actuated jont yelds the same moton of the tool (the transmsson factors are equal to one). For a parallel machne, these motons are generally not equvalent. When the mechansm s close to a parallel sngularty, a small jont rate can generate a large velocty of the tool. hs means that the postonng accuracy of the tool s lower n some drectons for some confguratons close to parallel sngulartes because the encoder resoluton s /6

3 Submtted to ICAR D. Chablat h. Wenger F. Majou amplfed. In addton, a hgh velocty amplfcaton factor n one drecton s equvalent to a loss of stffness n ths drecton. he manpulablty ellpsod of the Jacoban matrx of robotc manpulators was defned two decades ago [8]. he JJ - egenvalues square roots, γ and γ, are the lengths of the sem-axes of the ellpse that defne the two velocty amplfcaton factors between the actuated jonts veloctes and the velocty vector t& (λ = /γ and λ = /γ ). For parallel mechansms wth only pure translaton or pure rotaton motons, the varatons of these factors nsde the Cartesan workspace can be lmted by the followng constrants λ mn λ λ max Unfortunately, ths concept s qute dffcult to apply when the tool frame can produce both rotatonal and translatonal motons. In ths case, ndeed the Jacoban matrx s not homogeneous [9]. A frst way to solve ths problem s ts normalzaton by computng ts characterstc length [9-]. he second approach s to lmt the values of all terms of the Jacoban matrx to avod sngular confguraton and to assocate these values to a physcal measurement (See secton.4). Knematcs of mechansms studed. Knematcs of a parallel mechansm for translatonal motons he am of ths secton s to defne the knematcs and the sngular confguraton of a two-dof translatonal mechansm (Fgure ), whch can be extended to threeaxs machnes by addng a thrd axs n seres wth the frst two. he output body s connected to the lnear jonts through a set of two parallelograms of equal lengths L = AB, so that t can move only n translaton. he two legs are a dentcal chans, where and a stand for rsmatc and arallelogram jonts, respectvely. hs mechansm can be optmzed to have a workspace whose shape s close to a square workspace and the velocty amplfcaton factors are bounded []. he jont varables ρ and ρ are assocated wth the two prsmatc jonts. he output varables are the Cartesan coordnates of the tool center pont = [ x y]. o control the orentaton of the reference frame attached to, two parallelograms can be used, whch also ncrease the rgdty of the structure, Fgure. A e B θ e y θ x j z Fgure : arallel mechansm wth two-dof o produce the thrd translatonal moton, t s possble to place orthogonally a thrd prsmatc jont. he velocty p& of pont can be expressed n two dfferent ways. By traversng the closed loop ( A B A B ) n two possble drectons, we obtan p& = a& ( ) + θ& b a (a) p& = a& + θ& ( b a ) (b) where a, b, a and b represent the poston vectors of the ponts A, B, A and B, respectvely. Moreover, the veloctes a& and a& of A and A are gven by a & = eρ& and a & = eρ&, respectvely. For an sotropc confguraton to exst where the velocty amplfcaton factors are equal to one, we must have e. e = [] (Fgure 5). We would lke to elmnate the two passve jont rates θ & and θ & from Eqs. (a-b), whch we do upon dot-multply the former by ( ) b a and the latter by ( b ) a, thus obtanng ( b a ) p& = ( b a e ρ& (a) ) ( b a ) p& = ( b a ) eρ& (b) Equatons (a-b) can be cast n vector form, namely A p& = Bρ&, wth A and B denoted, respectvely, as the parallel and seral Jacoban matrces, ( b a ) ( b a) e A B ( b a ) ( b a ) e where ρ& s defned as the vector of actuated jont rates and p& s the velocty of pont,.e., ρ & = [ & ρ & ρ ] and p& = [ x& y& ] When A and B are not sngular, we obtan the relatons, p & = Jρ& wth J = A B arallel sngulartes occur whenever the lnes AB and AB are colnear,.e. when θ θ = kπ, for k =,,... Seral sngulartes occur whenever e b a or e b a. o avod these two sngulartes, the range lmts are defned n usng sutable bounds on the velocty factor amplfcaton (See secton.). /6

4 Submtted to ICAR D. Chablat h. Wenger F. Majou. Knematcs of a spatal parallel mechansm wth one-dof of rotaton he am of ths secton s to defne the knematcs of a smple mechansm wth two-dof of translaton and onedof of rotaton. o be modular, the drect knematc problem must be decoupled between poston and orentaton equatons. A decoupled verson of the Gough- Stewart latform exsts but t s very dffcult to buld because three sphercal jonts must concde []. hus, t cannot be used to perform mllng applcatons. he man dea of the proposed archtecture s to attach a new body wth the tool frame to the moble platform of the two-dof mechansm defned n the prevous secton. he new jont admts one or two-dofs accordng to the prescrbed tasks. o add one-dof on the mechansm defned n secton, we ntroduce one revolute jont between the prevous moble platform and the tool frame. Only one leg s necessary to hold the tool frame n poston. Fgure 4 shows the mechansm obtaned wth two translatonal dofs and one rotatonal dof. α e A k e θ B B Fgure 4: arallel mechansm wth two-dof of translaton and one-dof of rotaton he archtecture of the leg added s UU where and U stand for rsmatc and Unversal jonts, respectvely [6]. he new prsmatc jont s located orthogonaly to the frst two prsmatc jonts. hs locaton can be easly justfed because on ths confguraton,.e. when b a b a and b a b p, the thrd leg s far away from seral and parallel sngulartes. Let ρ& be referred to as the vector of actuated jont rates and p& as the velocty vector of pont, ρ & = [ & ρ & ρ & ρ] and p& = [ x& y& ] Due to the archtecture of the two-dof mechansm and the locaton of, ts velocty on the z-axs s equal to zero. p& can be wrtten n three dfferent ways by traversng the three chans A B, A B A e j θ θ β z y x p& = a& + & θ ( b a) (a) p& = a& + & θ ( b a) (b) p& = a& + (& θj+ & αk) ( b a) + & βj ( p b) (c) where a and b are the poston vectors of the ponts A and B for =,,, respectvely. Moreover, the veloctes a&, a& and a& of A, A and A are gven by a & = eρ&, a & = eρ& and a& = e & ρ, respectvely. We want to elmnate the passve jont rates θ & and α& from Eqs. (a-c), whch we do upon dot-multplyng Eqs. (a-c) by b a, ( b a) p& = ( b a) e & ρ (4a) ( ) b a p& = ( b a) e & ρ (4b) ( b a) p& = ( b a) e & ρ + ( b a) & β j ( p b) (4c) Equatons (4a-c) can be cast n vector form, namely, t = J&ρ wth J = A B and t = [ x & y& β& ] where A and B are the parallel and seral Jacoban matrces, respectvely, ( b a) A ( b a) ( ) ( ) ( ) b a b a j p b B ( b a ) e ( b a ) e ( b a ) e here are two new sngulartes when one leg s added. he frst one s a parallel sngularty when ( b a) j ( p b ) =,.e., when the lnes ( A B) and ( B ) are colnear, and the second one s a seral sngularty when ( b a) e =,.e., a b e. However, these sngular confguratons are smple and can be avoded by proper lmts on the actuated jonts.. Optmzaton of the useful workspace for translatonal motons wo types of workspaces can be defned, () the Cartesan workspace s the manpulator s workspace defned n the Cartesan space, and () the useful workspace s defned as a subset of the Cartesan workspace. Workspace and sze are prescrbed where some performance ndces are prescrbed. For parallel mechansm, the useful workspace shape should be smlar to the one of classcal seral machne 4/6

5 Submtted to ICAR D. Chablat h. Wenger F. Majou tools, whch s paralleleppedc f the machne has three translatonal degrees of freedom for nstance. So, a square useful workspace s prescrbed here where the velocty amplfcaton factors reman under the prescrbed values. wo square useful workspaces can be used, () he frst one has horzontal and vertcal sdes (Fgure 5a) and () the second one has oblque sdes but ts sze s hgher (Fgure 5b). e e e e A B B A B Cartesan workspace Useful workspace (a) Close to sngularty locus (b) Fgure 5: Cartesan workspace and sotropc confguraton o fnd the best useful workspace (center locus and sze), we can shft the useful workspace along x-axs ( Δ x) and y-axs ( Δ y) (Fgure 6) and the velocty amplfcaton factors are computed for each confguraton. hs method was developed n []. Δu y y Δu Δy 4 Δx (a) x 4 (b) Fgure 6: Lookng for the best useful workspace (center locus and sze) In each case, velocty amplfcaton factor extrema are located along the sdes j: they start from at pont S, then they vary untl they reach prescrbed boundares (/ λ ). hen computaton (whch analytcal expressons λ ) has been obtaned wth Maple along the four sdes of the square and for the same length of the leg equal to one. For the frst mechansm, soluton (a), the sze of the optmal surface s equal to,89 m and for the second mechansm, soluton (b), the sze s equal to,6 m. he result obtaned for the soluton (b) s smaller than for the soluton (a) but s more approprate for the extenson three-axs mechansm of Fgure 4. In effect, we want the axs of rotaton to be parallel to one of the sde of the B x useful workspace. In the next secton, the lengths of the thrd leg wll optmze to acheve ths square useful workspace wthout sngularty..4 Optmzaton of the thrd-axs for rotatonal motons he am of ths secton s to defne the two lengths of the leg, L = B and L = AB such that t s possble to acheve the maxmum range varaton of thrd axs β wthout meetng a sngular confguraton throughout the square workspace wth the sze defned n the prevous secton. he fst step of ths optmzaton s to fnd the locaton of the prsmatc jont. When s on the center of the square workspace, we chose to place the thrd leg furthest away from sngular confguraton,.e. when b a and e are colnear and b a p b (Fgure 7a). σ B A (a) β B A (b) B A (c) e ρ e ρ e ρ L B A γ e ρ (d) Fgure 7: Optmal, seral and parallel confguraton of the thrd leg As t s defned n Secton., the thrd leg s n a sngular confguraton whenever ( b a) j ( p b ) = (Fgure 7b-c) or ( b a) e = (Fgure 7d). In the optmzaton functon, we set: ( b a) e >, b a (4a) ( b a) ( p b) j >, b a p b (4b) wth arcsn(, ) =, 5 hs means that γ = AB s n [,568,5 ] and AB σ = e s n [ 78,5 78,5]. he result of ths optmzaton as a functon of L and L s depcted n Fg. 8. 5/6

6 Submtted to ICAR D. Chablat h. Wenger F. Majou L L =,8L Fgure 8: Range of varaton n degrees of β as a functon of L and L Wth a sutable chose of lengths, t s easy to obtan a range varaton of β hgher than 8. So, ths value can be reduced f we ncrease the constrant defned n Eqs. (4). However, when L =,8L, the range of varaton s optmal. 4 Conclusons In ths paper, a parallel mechansm wth two degrees of poston and one degree of rotaton s studed. All the actuated jonts are fxed prsmatc jonts, whch can be actuated by means of lnear motors or by conventonal rotary motors wth ball screws. Only three types of jonts are used,.e., prsmatc, revolute and unversal jonts. All the sngulartes are characterzed easly because poston and orentaton are decoupled for the drect knematc problem and can be avoded by proper desgn. he lengths of the legs as well as ther postons s optmzed, to take nto account the velocty amplfcaton factors for the translatonal motons and to avod the sngular confguraton for the rotatonal motons. 5 Acknowledgments hs research was partally supported by the CNRS (roject ROBEA Machne à Archtecture complexe ). 6 References [] reb,. and Zrn, O., Smlarty laws of seral and parallel manpulators for machne tools, roc. Int. Semnar on Improvng Machne ool erformances, pp. 5-, Vol., 998. L [] Gough, V.E., Contrbuton to dscusson of papers on research n automoble stablty, control an tyre performance, roc. Auto Dv. Inst. Eng., [] Merlet, J-, he parallel robot, Kluwer Academc ubl., he Netherland,. [4] Kong, K. and Gosseln, C., Generaton of parallel manpulators wth three translatonal degrees of freedom based on screw theory, roc. of CCoMM Symposum on Mechansms, Machnes and Mechantroncs, Sant-Hubert, Montreal,. [5] Hervé, J.M. and Sparacno, F., 99, Structural Synthess of arallel Robots Generatng Spatal ranslaton, roc. of IEEE 5th Int. Conf. on Adv. Robotcs, Vol., pp. 88-8, 99. [6] Gao, F., L, W., Zhao, X., Jn Z. and Zhao. H., New knematc structures for -, -, 4-, and 5-DOF parallel manpulator desgns, Journal of Mechansm and Machne heory, Vol. 7/, pp. 95-4,. [7] Gosseln, C. and Angeles, J., Sngularty analyss of closedloop knematc chans, IEEE ransacton on Robotc and Automaton, Vol. 6, No., June 99. [8] Salsbury, J-K. and Crag, J-J., Artculated Hands: Force Control and Knematc Issues, he Int. J. Robotcs Res., Vol., No., pp. 4-7, 98. [9] Angeles, J., Fundamentals of Robotc Mechancal Systems, nd Edton, Sprnger-Verlag, New York,. [] Chablat, D. and Angeles, J., On the Knetostatc Optmzaton of Revolute-Coupled lanar Manpulators, J. of Mechansm and Machne heory, vol. 7,(4).pp. 5-74,. [] Chablat, D., Wenger, h. and Angeles, J., Concepton Isotropque d'une morphologe parallèle: Applcaton à l'usnage, rd Internatonal Conference On Integrated Desgn and Manufacturng n Mechancal Engneerng, Montreal, Canada, May,. [] Khall, W. and Murarec, D., Knematc Analyss and Sngular confguratons of a class of parallel robots, Mathematcs and Computer n smulaton, pp. 77-9, 996. [] Majou F., Wenger h. et Chablat D., A Novel method for the desgn of -DOF arallel mechansms for machnng applcatons, 8th Int. Symposum on Advances n Robot Knematcs, Kluwer Academc ublshers,,. 6/6