A New Concept of Modular Parallel Mechanism for Machining Applications

Transcription

1 Submtted to ICRA A New Concept of Modular arallel Mechansm for Machnng Applcatons Damen Chablat and hlppe Wenger 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 desgn of a new concept of modular parallel mechansms for three, four or fve-axs machnng applcatons. Most parallel mechansms 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 translaton motons and to add one or two legs to add one or two-dofs for rotaton motons. he knematcs and sngular confguratons are studed for each mechansm. 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 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 as a functon of small varatons of the desgn parameters []. he second problem s the nonhomogenety of ts performance ndces (condton number, stffness...) throughout the workspace []. o the authors' knowledge, only one parallel mechansm s sotropc throughout the workspace [4] but ts stffness s nsuffcent to be used n machnng applcatons because ts legs are subject to bendng. Unfortunately, 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 needed 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 translaton and rotaton 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 secton. Secton 4 s devoted to desgn of two new archtectures of parallel mechansms wth one or two dofs of rotaton. IRCCyN: UMR CNRS 6596, Ecole Centrale de Nantes, Unversté de Nantes, Ecole des Mnes de Nantes /6

2 Submtted to ICRA 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 gven 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 Gough-Stewart platform, a mechansm wth sx-dof, the problem s very dffcult []. Wth a modular archtecture, when poston and orentaton of the moble platform are decoupled, the determnaton of 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 [9] (servce angle, dexterous workspace and manpulablty ). 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. Another problem s that to 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 /6

3 Submtted to ICRA encoder resoluton s 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]. Unfortunately, ths concept s qute dffcult to apply when the tool frame can produce rotaton and translaton motons. Indeed, n ths case, the Jacoban matrx s not homogeneous [9]. he frst way to solve ths problem s ts normalzaton by computng ts characterstc length [9]. he second one s to change the form of the Jacoban matrx. he frst part of the mechansm for translatonal moton s optmzed usng homogeneous matrx. hen, the part dedcated to rotaton moton can be optmzed usng the method ntroduced by []. Knematcs of mechansms for translaton motons. Fgure shows a KM wth two dofs. 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. Fgure : arallel mechansm wth two-dof 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 and 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. o produce the thrd translatonal moton, t s possble to place orthogonally a thrd prsmatc jont. hs one can be located as n the case of Fgure 4. Another soluton s to use the Orthoglde mechansm, an sotropc three-dof mechansm []. he choce between these two solutons depends on the man applcaton of the mllng machne. For example, for aeronautcal peces, the soluton of fg. 4 s more approprate because long and fne peces are bult. Conversely, for rapd prototypng of compact parts, we can choose the Orthoglde. Fgure 4: Hybrd mechansm wth three-dof 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 ( ) b a (a) p a ( b ) (b) a where a, b, a and b represent the poston vectors of the ponts A, B and B, respectvely. Moreover, the veloctes a and a of A and A are gven by a e and a e, respectvely. For an sotropc confguraton to exst where the velocty amplfcaton factors are equal to one, we must have e. e [] (Fgure 5). wo square useful workspaces can be used. he frst one has horzontal and vertcal sdes. he second one has oblque sdes but ts sze s hgher. Fgure 5: Cartesan workspace and sotropc confguraton /6

4 Submtted to ICRA We would lke to elmnate the two passve jont rates and 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 (a) ) ( b e a) p ( b a) (b) Equatons (a-b) can be cast n vector form, namely Ap B, 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 s defned as the vector of actuated jont rates and p s the velocty of pont,.e., x and p y When A and B are not sngular, we obtan the relatons, p J wth J A B arallel sngulartes occur whenever the lnes ( AB ) and AB ) are colnear,.e. when k, for k =,,... ( Seral sngulartes occur whenever e b a or e b a. In [], the range lmts are defned to avod these two sngulartes n usng sutable bounds on the velocty factor amplfcaton. 4 Knematcs of mechansms for translaton and rotaton motons he am of ths secton s to defne the knematcs of two mechansms wth one and two dofs of rotaton, respectvely. 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 dofs defned n the prevous secton. he new jont admts one or two-dof accordng to the prescrbed tasks. 4. Knematcs of a spatal parallel mechansm wth one-dof of rotaton 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 6 shows the mechansm obtaned wth two translatonal dofs and one rotatonal dof. Fgure 6: 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 be referred to as the vector of actuated jont rates and p as the velocty vector of pont, [ ] 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, p a ( b a) (a) p a ( b a) (b) p a ( j k) ( b a ) j( p b ) (c) where a and b represent the poston vectors of the ponts A and B for,,, respectvely. Moreover, the veloctes a, a and a of A and A are gven by a e, a e and a e, respectvely. We want to elmnate the passve jont rates and from Eqs. (a-c), whch we do upon dot-multplyng Eqs. (a-c) by b a, ( ) b a p ( b a ) e (4a) ( b a ) p ( b a ) e (4b) ( b a) p ( b a) e ( b a ) j( p b ) (4c) 4/6

5 Submtted to ICRA Equatons (4a-c) can be cast n vector form, namely, t J wth J A B and t [ x y ] 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. 4. Knematcs of a spatal mechansm wth two-dof of rotaton o add two dofs on the mechansm defned n secton, we ntroduce a unversal jont, between the prevous moble platform and the tool frame. wo legs are necessary to hold the tool frame n poston and we use the same archttecture than for the prevous mechansm,.e. the UU mechansm. Fgure 7 depcts a parallel mechansm wth two translatonal dofs and two rotatonal dofs. Let be referred to as the vector of actuated jont rates and p as the velocty vector of pont, and p [ x y ] 4 p can be wrtten n four dfferent ways by traversng the three chans A B, p a ( b a) (5a) p a ( b a) (5b) p a ( k) ( b a) ( j ) ( p b) (5c) p a ( k) ( b a ) ( j ) ( p b ) (5d) where a and b represent the poston vectors of the ponts A and B, respectvely, for,,,4. Moreover, the veloctes a, a, a and a 4 of A and A 4 are gven by a e, a e, a e and a e, respectvely Fgure 7: arallel mechansm wth two-dof of translaton and two-dof of rotaton We want to elmnate the passve jont rates and from Eqs. (6a-d), whch we do upon dot-multplyng Eqs. (5a-d) by b a, ( ) b a p ( b a ) e (7a) ( b a ) p ( b a ) e (7b) ( b a) p ( b a) e ( b a ) ( j ) ( p b ) ( b4 a4) p ( b4 a4) e 44 ( b a ) ( j ) ( p b ) (7c) (7d) Equatons (8a-d) can be cast n vector form, namely, t J wth J A B and t x y where A and B are the parallel and seral Jacoban matrces, respectvely, ( b a) ( b a) A ( b a) ( b a) j ( p b) ( b a) ( p b) ( b4 a4 ) ( b4 a4 ) j ( p b4 ) ( b4 a4 ) ( p b4) and ( b a) e ( b a ) e B ( b a) e ( b4 a4) e4 When two legs are added, the same sngular confguratons as the prevous mechansm occur, a parallel sngularty occurs when the lnes ( A B ), ( AB 4 4) and (C) are coplanar, and a seral sngularty occurs 5/6

6 Submtted to ICRA when ( b a) e,.e., a b e or ( b4 a4) e4,.e., a4 b4 e Dscusson he problem of the optmal desgn of the two mechansms defned n the prevous sectons s not addressed n ths paper. In future works, for both mechansms, t s possble to defne a confguraton for whch the Jacoban matrx s sotropc. hs result permts us to defne the locaton of the prsmatc jonts and to defne the condton length to normalze the Jacoban matrx. A method to defne the range lmts s explaned n [], va a dstance to the sotropc confguraton. 5 Conclusons In ths paper, a new class of modular mechansms s ntroduced wth two, three, four and fve dofs. All the actuated jonts are prsmatc jonts, whch can be actuated by means of lnear motors or by conventonal rotary motors wth ball screws. he topology of the legs used to add one or two dof s the same. 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 avod by proper desgn. In the future, for the modular archtecture, the lengths of the legs as well as ther postons wll be optmzed, to take nto account the velocty amplfcaton factors. 6 Acknowledgments hs research was partally supported by the CNRS (roject ROBEA Machne à Archtecture complexe ). he authors would lke to thank Mr. Caro for hs valuable remarks on ths paper. 7 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. [] 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, arallel robots, Kluwer Academc ubl., Dordrecht, 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 closed-loop 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, Second Edton, Sprnger-Verlag, New York,. [] Chablat, D. and Angeles, J., On the Knetostatc Optmzaton of Revolute-Coupled lanar Manpulators, Journal 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,. [] Chablat, D. and Wenger, h, Desgn of a hree- Axs Isotropc arallel Manpulator for Machnng Applcatons: he Orthoglde, Workshop on Fundamental Issues and Future Research Drectons for arallel Mechansms and Manpulators, October -4, Québec,. [] Khall, W. and Murarec, D., Knematc Analyss and Sngular confguratons of a class of parallel robots, Mathematcs and Computer n smulaton, pp. 77-9, /6