The Structure and Dimensional Design of a Reconfigurable PKM

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1 Internatonal Journal of Advanced Robotc Systems ARTICLE The Structure and Dmensonal Desgn of a Reconfgurable PKM Regular Paper Xaoqang Tang,*, Dengfeng Sun 2 and Zhufeng Shao The State Key Laboratory of Trbology, Department of Precson Instruments and Mechanology, Tsnghua Unversty,Bejng, P.R. of Chna 2 School of Aeronautcs and Astronautcs Engneerng, Purdue Unversty, West Lafayette, IN,USA * Correspondng author E-mal:tang-xq@mal.tsnghua.edu.cn Receved 4Oct 202; Accepted 26Oct 202 DOI: 0.772/ Tanget al.; lcensee InTech. Ths s an open access artcle dstrbuted under terms of Creatve Commons Attrbuton Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded orgnal work s properly cted. Abstract Parallel Knematc Machnes (PKMs) have many advantages and have been wdely used n machne ndustry. Benefttng from ts modular structure, a PKM s more reconfgurable than tradtonal seral machnes. In ths paper, a new type of drvng strut module and nnovatve jonts are desgned for Reconfgurable Parallel Knematc Machne (RPKM). The new drvng strut module can be changed from lnear drve mode to telescopc drve mode easly, and new sphercal jont and unversal jont are desgned to acheve a large rotaton angle. The nverse knematcs problems n relaton to 6 DOF RPKM are analysed, and Workspace Volume Index (WVI) and Global CondtonIndex (GCI) are adopted to desgn RPKM. Accordng to WVI and GCI analyss of selected parameters for two types of 6 DOF PKM, dmensonal parameters of RPKM are desgned. In end, new type of RPKM prototype s bult, wth whch a wax pattern s machned. Keywords Structure Desgn, Dmensonal Desgn, Reconfgurable PKM. Introducton In recent years, ever changng manufacturng envronment caused by ntensve competton on a global scale and rapd changes n process technology has led to requrement for an ncreasngly effcent producton system. In order to acheve ths, a reconfgurable manufacturng system charactered by modularty, ntegrablty, customaton, convertblty and dagnosablty, whch can easly be reconfgured and respond to unpredctable market changes cost effectvely, s hghly preferred. The earlest large scale research on reconfgurable manufacture systems reported n lterature s Flexble Manufacture Cell (FMC) project launched n 977 by MITI n Japan. Or reconfguraton projects, such as MOSYN, KERNEL and MOTION, focused on modular machne tools. After foundaton of Engneerng Research Center of Reconfgurable Machnng Systems (ERC/RMS) at Unversty of Mchgan n 996, researchers began to develop ory and technology of reconfgurable manufacturng systems systematcally[ 3]. A Parallel Knematc Machne (PKM) has many advantages and has been wdely used n machne ndustry. Thanks to ts modular structure, a PKM s more reconfgurable than tradtonal seral machne. Wth modular desgn, several modules of PKMs can be ntegrated nto a module set so that modules can be reassembled rapdly for dfferent applcatons. As a result, recent trend n reconfgurable manufacturng Xaoqang Int J Adv Tang, Robotc Dengfeng Sy, 203, Sun and Vol. Zhufeng 0, 267:203 Shao: The Structure and Dmensonal Desgn of a Reconfgurable PKM

2 systems has promoted research and development of a novel type of machne tool called a Reconfgurable Parallel Knematc Machne (RPKM) [4 ], whch s based on parallel manpulator. More recently, research on RPKMs has made some substantve progress, but some problems that should be solved for ts furr applcaton stll exst [6 7]. Frstly, accordng to drve mode, PKM can be manly classfed nto two knds: telescopc drve PKM, whose struts are extendable wth one end mounted on framework or base platform and or on movng platform; lnear drve PKM, whose struts have constant length wth one end mounted on a slder and or on movng platform. Because both se PKMs have r pros and cons, y are desgned and appled n respectve cases: lnear drve PKM s nclned to be adopted n machnng long shaped parts along ts domnant movement drecton, and telescopc drve PKM s sutable to machne more complcated shapes or than those of a longshape [8]. It would refore be deal f RPKM can reale se two types of PKMs wth same modular set. Moreover, se two types of PKMs need to use unversal jonts and sphercal jonts. A unversal jont can work wth 2 DOFs rotaton, and a sphercal jont wth 3 DOFs rotaton. Wth classc manufacturng process, t s dffcult to fabrcate a hgh precson sphercal jont wth a large rotaton angle. To solve ths problem, an ntegrated jont wth er two DOFs or three DOFs should be desgned. The dmensonal desgn s anor mportant and challengng problem for RPKMs. Two major ssues need to be addressed n optmal desgn of a PKM: performance evaluaton and synss. After desgn of a RPKM, t s necessary to evaluate ts man characterstcs. The second problem s to determne dmensons (lnk lengths) of RPKM, to make t most sutable for specfc task. The latter problem s one of most dffcult ssues to solve n feld of RPKM. As s well known, performances of parallel knematc machnes manly depend upon dmensons. Therefore, process of mechansm dmensonal desgn for a manpulator s largely based on usage of crtera, such as workspace, dexterty, payload, condtonng ndex and stffness [9 ]. Approaches n prevous work dealng wth dmensonal desgn of PKMs can be classfed nto two categores. One category refers to local optmal desgn [ 3], focusng on fndng a famly of dmensonal parameters that generate an sotropy confguraton, where platform presents desrable characterstcs n terms of knematcs accuracy, nose rejecton and sngularty avodance. However, due to possble strut nterference, ths famly of desgn parameters would not be useful n desgn where desrable orentaton capablty of movng platform s also requred. The or category refers to global optmal desgn [4,], whch attempts to fnd a set of desgn parameters to mnme a weghted cost functon n terms of specfed dexterous performance and workspace volume. The reason for mplementng global optmal desgn would be that knematcs performance could not be guaranteed smply by conductng local optmal desgn. However, above two dmensonal desgn methods are only used for telescopc drve PKM or lnear drve PKM respectvely. In ths paper, we wll try to use global optmal desgn method to desgn RPKM. In ths paper, a new type of drvng struts moduleand jonts are desgned for RPKM. The nverse knematcs problems wth regard to two types of 6 DOF parallel manpulators are descrbed and a global optmal desgn ndex s used to desgn RPKM, whch can satsfy requrement of RPKM and reale workspace and dexterty capablty. In end, we show two wax patterns mlled by RPKM. 2. Structure desgn The structure of reconfgurable PKMs s smlar to that of some ordnary PKMs. They consst of common parts lke: a base platform, a movng platform on whch endng executor s nstalled, and some parallel actuated extendble legs or movable legs between moble and base platforms. Unversal jonts and sphercal jonts are mounted on er movng platform or base platform to connect platforms and legs, and to provde mult poston and versatle atttude. The key modular desgns of reconfgurable PKMs nclude strut modules and jont modules. 2. Strut modules desgn [6] Accordng to drvng mode, PKMs can manly be catalogued nto two knds: telescopc drve PKMs, whose struts are extendable wth one end mounted on framework or base platform and or on movng platform; lnear drve PKMs, whose struts have constant length wth one end mounted on a slder and or on movng platform. In fgure, telescopc drve mode s represented: a drvng motor () s mounted at one end of telescopc drve module. The motor drves ball screw (2), and work pece module (3) moves along axs of ball screw (2). A rod fx module (4) s connected to work pece module along drecton of screw n lnear drve module, n a constant length rod module () and a sphercal jont (6) are mounted n turn. The length of rod module () s represented by l k, and maxmal move dstance of telescopc drve mode s denoted by l. 2 Int J Adv Robotc Sy, 203, Vol. 0, 267:203

3 In fgure 2, lnear drve mode s represented as follows: a drvng motor() s mounted at one end of lnear drve module, motor drves ball screw(2), and work pece module(3) moves along axs of ball screw(2). A unversal jont (7) s mounted on work to pece module, n a rod fx module (4) s connected constant length rod module () and a sphercal jont (6) are mounted n turn. The length of rod module () s represented by l k, and maxmal move dstance s represented by l. rotaton axes only, sphercal jont wll be changed nto unversal jont wth two DOFs. C-axs A-axs C-axs Accordng to desgn, telescopc drve mode can be changed nto lnear drve mode by mountng constant length rod module ()at anor place on work pece module (3). The assemblng process s very easy and can be fnshed n a short perod of tme. Fgure 3. Sphercal jont The rotaton angle of C axs s 90 and rotaton angle of A axs s 60, so general rotaton angle of new type sphercal jont and unversal jont s 60. C-axs A-axs Fgure.Telescopc drve mode Fgure 2.Lnear drve mode. 2.2 Jont module desgn [7] In a RPKM, a unversal jont has two DOFs, and a sphercal jont has three DOFs. Wth classc manufacturng process, t s dffcult to fabrcate a hgh precson sphercal jont. Moreover, general rotaton angle of tradtonal sphercal jont s no more than 20. To resolve ths ssue, ntegrated jonts possessng er two DOFs or three DOFs are desgned, as shown n fgure 3. The new type of sphercal jont ncludes three rotaton axes, whch are C axs, A axs and C axs wth three DOFs. As shown n fgure 4, whenwe use two Fgure 4. Unversal jont 2.3 Two types of 6 DOF PKMs Both lnear drve PKM and telescopc drve PKM have r pros and cons. Therefore, y are generally desgned to be appled n respectve cases: lnear drve PKM s nclned to be adopted n machnng long and telescopc drve PKM s sutable to machne more complcated shapes or than those of a longshape. As shown n fgures and 6, we assemble two types of PKMs wth same module shaped parts along ts lnear movement drecton, set. Xaoqang Tang, Dengfeng Sun and Zhufeng Shao: The Structure and Dmensonal Desgn of a Reconfgurable PKM 3

4 Fgure.The telescopc drve PKM a. The telescopcc drve PKM Fgure 6. The lnear drve PKM 3. Dmensonal desgn As soon as we have structure desgn of RPKM, most mportant step s too decde dmensons of RPKM. It s of prmary mportance to develop a useful desgn method that can reveal relatonshps between crtera and lnk dmensons of machne. Frstly, knematcs model of RPKM s bult and crtera ndexes are formulated. Then we get dmensonal results through analysngg Workspace Volume Index (WVI) and Global Condtonng Index (GCI) [4]. 3. Knematcs model Knematcs models of se two types of PKM are developed as shown n fgure 7. The vertces of movng platform are denoted as platform jonts A (, 2, 3,..., 6 ), and vertces of base plate are denoted as B (, 2, 3,,...,6 ). A fxed global reference system : o xy s located at centre of crcle B...B 6 wth axs vertcal to base plate and y axs n concdence wth OB. The crcumcrcle radus of ponts B...B6 s denoted as R B. Anor reference frame, defnng top frame : oʹ xy, s located at centre of crcle A...A 6. The axs s perpendcular to movng platform and y axs parallel to vector O A. The crcumcrcle radus of ponts A...A 6 s denoted as RA. Vector AB s denoted as L (, 2, 3,..., 6 ), and lnk length for each leg s denoted as l, where AB l, (, 2,...,6 ). Becausee structure s symmetrcal, B 2 / 3 B and A 2 / 3 A. b. The lnear drve PKM Fgure 7. The geometrc archtecture of 6 DOF parallel manpulators. The objectve of nverse knematcs soluton s to defne a mappngg from pose of movng platform n Cartesan space to set of actuated leg nputs n order to acheve am pose. For ths analyss, pose of movng platform s assumed to be known. The poston s gvenn by poston vector Oʹ and orentaton s gven by a matrxx R. Then t s shown Oʹ x y The rotaton of coordnate system relatve to can be expressed by rotaton matrx c c c ss s c c sc s s R s c s ss c c s sc c s s cs cc The vectors B n base platform coordnate system can be wrtten as: ʹ T () (2) 4 Int J Adv Robotc Sy, 203, Vol. 0, 267:203

5 B R cos(( ) / 3),R sn(( ) / 3),,(,3,) B B RB cos(( 2) / 3 B), B RB sn(( 2) / 3 B), ( 2,4,6) where 0,...6, for PKM wth telescopc drves. The vectors A n movng platform coordnate system can be wrtten as: A R cos(( ) / 3),R sn(( ) / 3),0,(,3,) A A RA cos(( 2) / 3 A), A, RA sn(( 2) / 3 A),0 ( 2,4,6) Therefore, vectors A can be expressed n base platform coordnate system as: (3) (4) A R A O, (,2,...,6) () The relatonshp between vectors A, B and strut vectors has followng form accordng to geometrc constrant of archtecture: B A L l, (,2,...,6) (6) where l l k,,2,...,6 f PKM s equpped wth lnear drves. The nverse knematcs problem of machne can be solved by Eq. (6), where unknown tems are er l (,2,...,6 ) for PKM wth telescopc drves, or (,2,...,6 ) for PKM wth lnear drves. Hence, for a gven mechansm and prescrbed poston and orentaton of movng platform, requred actuator nputs can be drectly computed from Eq. (6). Eq. (6) can be dfferentated wth respect to tme to obtan velocty equaton. Ths leads to an equaton as: J p J q (9) p where q s velocty vector of movng platform defned n Cartesan coordnate systemas q T q x,y,,,, (0) and p s vector of nput veloctes. As mentoned earler, re are two dfferent types of drve modes. When PKM s equpped wth telescoped legs, p can be expressed as T p [l,l,l,l,l,l ] () whle when PKM s actuated wth slder legs, ts expresson s T p [,,,,, ] (2) Matrces J p and J q are 6 6 forward and nverse Jacoban matrces of mechansm, and can be expressed as Jp Jp2 0 Jp3 J p J (3) p4 0 J p J where w s unt vector of J I, J q,(=,2,...,6) ; q p6 6 6 w ( v w) w2 ( v2 w2) w ( v w ) J q (4) w4 ( v4 w4) w ( v w) w ( v w ) v R A. L and ʹ When PKM s actuated wth telescoped legs, J q,(,2,...,6) and J q I ; f PKM s equpped wth slder legs, Jq ( w ), where ( w ) s thrd element of vector w wth respect to axs coordnate. If J q 0, equaton can be wrtten as: 3.2 Workspace and ndex p q p J J q J q () The workspace of a PKM s work regon whch tool can reach, and s an mportant ndcaton of PKM s performance. In ths secton, Workspace Volume Index (WVI) s used to represent workspace volume of RPKM. A numerc method s appled to compute PKM s reachable ponts to ntegrate workspace surface. Accordng to earler research of Huang [3], same optmal desgn result for workspace can be obtaned by settng rotaton angles of movng platform to ero or or s. To smplfy analyss, three rotaton angles of movng platform are all set to ero n ths calculaton. If we are gven ntal as shown n Table, workspaces of two knds of PKMs are shown n fgure 8 and r WVIs are.x0 8 mm 3 and.2x0 7 mm 3 respectvely. Xaoqang Tang, Dengfeng Sun and Zhufeng Shao: The Structure and Dmensonal Desgn of a Reconfgurable PKM

6 Dmensonal Varable R A (mm) Descrpton R B (mm) l (mm) Radus of movng platform The maxmal move range of strut The strutt mounted angle on A ( o ) movng platform(fgure 7) The strutt mounted angle on B ( o ) base platform(fgure 7) PKM wth telescopc legs l mn (mm) Mnmal length of telescopc legs 00 l max = l mn + l Maxmal length of telescopc (mm) legs 700 PKM wth lnear legs Maxmal heght of work mn (mm) pece module max = mn + Mnmal heght of work l (mm) pece module l k (mm) Length of rod module 600 Table. The dmensonal varables Intal Value Radus of base platform ( / 3 ) 30( / 6 ) The sngularty s k / where (=,2,,6) are sngularty s of Jacoban matrx J. Accordng to characterstc equaton of sngularty det(e J J =0 T 6 ( J J ), (= 6) 6 T ) (6) (7) (8) Now, Local Condtonng Index (LCI) s used to evaluatng performance of RPKM, whch s: LCI / k 6 / (9) It has been acknowledged that a larger of LCI ndcates a better moton dexterty of movng platform. As shown n fgure 9, gven dmensonal varables n Table, change trends of LCI can be shownn one X Y plane of workspace. a. The telescopc drve PKM a. The telescopcc drve PKM b. The lnear drve PKM Fgure 8. RPKM workspace surface 3.3 Condtonal number and GCI ndex The condton number of Jacoban matrx J can be used to evaluate dexterty [8]. The condton number expresson s wrtten as: b. The lnear drve PKM Fgure 9.LCI of RPKM However, LCI can only evaluate dexterty of X Y plane for PKM workspace, and we adopt Globall 6 Int J Adv Robotc Sy, 203, Vol. 0, 267:203

7 Condton Index (GCI) [4] to evaluate dexterty of whole workspace, a promsng approach defned as GCI / k dw dw (20) W where dw s a dfferental workspace volume, and k s condton number of Jacoban matrx at a gven movng platform pose, whch s defned as rato of maxmum and mnmum sngularty s of manpulator Jacoban matrx. It s well known that a larger of GCI reveals a better moton dexterty of movng platform. W 3.4 Workspace and GCI analyss From Table, we can know that ten dmensonal varables should be desgned. It wll be very dffcult to analyse all varables at one tme. So dscussed desgn varables are defned as A, 2 B, 3 R A / l, 4 R B / l, l mn / l and 6 l k / l, where 3, 4, and 6 are all dmensonless varables. Based on workspace and formula of GCI, effects of desgn varables on dexterty and workspace can be dscussed and evaluated. The ball screw length lmtaton gves l 200mm. Or desgn parameters that are not dscussed below are same as specfed n Table. b. The lnear drve PKM Fgure 0. Workspace volumes of RPKM wth dfferent & 2 As shown n fgure, workspace volumes ncrease wth 3 ncreasng and 4 decreasng. If we only want to get largest workspace volume, selectng largest of 3 and smallest of 4 wll be a good choce Workspace effect analyss Fgure 0 shows WVI curve when and 2 are gven dfferent s. It ndcates that workspace expands as ncreases. When 2 30, maxmal workspace volume can be obtaned. Because proposed RPKM structures are symmetrcal, of should not be larger than 60, so we wll let 60 and 2 30 for followng analyss. a. The telescopc drve PKM a. The telescopc drve PKM b. The lnear drve PKM Fgure. Workspace volumes of RPKM wth dfferent 3 and 4 Xaoqang Tang, Dengfeng Sun and Zhufeng Shao: The Structure and Dmensonal Desgn of a Reconfgurable PKM 7

8 As shown n fgure 2, desgn varable only has mpact on workspace volumes of telescopc drve PKM, whle desgn varable 6 only affects workspace volumes of lnear drve PKM. Workspace ncreases wth ncreasng and 4 decreasng. Workspace volumes ncrease as 6 ncreases and 4 decreases. If we only want to get largest workspace, selectng largest s of and 6 wll be a good choce. However, ncreasng s of and 6 wll decrease stffness of machne. a. The telescopc drve PKM a. The telescopc drve PKM b. The lnear drve PKM Fgure 3.GCI of RPKM wth dfferent 3 and 4 b. The lnear drve PKM Fgure2. Workspace volumes of RPKM wth dfferent 4, and 6 a. The telescopc drve PKM GCI effect analyss As shown n fgure 3, we can get maxmal GCI as long as of 3 s close to one, and s of GCI ncrease as 4 decreases. As shown n fgure 4, desgn varable of only affects GCI of telescopc drve PKM, whle desgn varable of 6 only affects lnear drve PKM. The maxmal of GCI can be obtaned as of and 6 are near 2.. The s of GCI ncrease when ncreases. 4 b. The lnear drve PKM Fgure4.GCI of RPKM wth dfferent 4, and 6 8 Int J Adv Robotc Sy, 203, Vol. 0, 267:203

9 3. Dmensonal varables desgn The dmensonal varables effects on WVI and GCIare shown n Table 2. Dmensonal varables Gettng larger 60 WVI Gettng larger GCI Dmensonal varables 4 Gettng larger The smallest WVI The largest Gettng larger GCI Table 2. The dmensonal varables effects on WVI and GCI Accordng to defntonn of WVI and GCI, larger s of WVI and GCI ndcate a larger workspace volume and a better moton dexterty of movng platform. If we only analyse workspace, desgn varables should be selected as 60, 2 30, 3., 4., 3. and 6 4. However, GCI s very mportant n PKM s performance, so we should consder GCI frst, and n endeavour to obtan a larger workspace. Therefore, accordng to GCI effect analyss results, desgn varables are selected as 3, 4 2., 2. and Accordng to fgures, 2, 3 and 4, when 3.2, 4 2, 3 and 6 3, RPKM wll get a larger workspace and reasonable GCI relatvely. Then dmensonal varables can be calculated by rato of desgn parameters,,2,....,6. Fnally, dmensonal varables of RPKM can be obtaned and shown n Table 3. Dmensonal varables RA Value 20 Dmensonal varables l max Value 800 RB 2 30 Table 3.The dmensonal parameter of RPKM prototype The largest Near 2. Near 2. l mn max The largest Near 6 The largest A ( o ) B ( o ) l mn l k A ( o ) B ( o ) () A new type of strut s desgned to reale er telescopc drve or lnear drve wthout any or modules, where telescopc drve mode can be changed nto lnear drve mode easly and n a short perod of tme. (2) A compound jont wth three rotaton freedoms s desgned to be a sphercal jont, based on whch a unversal jont can also be obtaned when one rotaton s locked. Compared wth tradtonal jonts, general rotaton angle of new type of sphercal jont or unversal jont s threee tmes largerr and can reach 60. (3) The ndces of WVI and GCI are used to evaluate and desgn varables of 6 DOF RPKM. Accordng to WVI and GCI analyss, dmensonless varables are obtaned, and n prototype s bult. Mllng experments proved that new type of RPKM can mplement machnng functon. However, only two 6 DOF PKMs are dscussed n ths paper. In future work, we wll desgn and research or RPKMswth same drve mode and evaluaton ndex. a. The telescopcc drve PKM Based on Table 3, a 6 DOF RPKM prototype was manufactured n 200, as shown nfgure.in fgure 6(a), telescopc drve PKM was machnng one wax pattern, and n fgure 6(b), lnear drve PKM prototype was machnng anor wax pattern, whch proved feasblty of 6 DOF RPKM prototype. 4. Conclusonn In ths paper, a 6 DOF RPKM s proposed, whch can be assembled nto er telescopc drve PKM or lnear drve PKM wth same module set. Conclusons can be drawn as follows: b. The lnear drve PKM Fgure. The 6 DOF RPKM prototype Xaoqang Tang, Dengfeng Sun and Zhufeng Shao: The Structure and Dmensonal Desgn of a Reconfgurable PKM 9

10 Fgure 6. Mllng wax pattern. Acknowledgments Ths research s sponsored by Natonal Natural Scence Foundaton of Chna (nos. 7802, 09749), and Natonal S&T Major Project of Chna (no. 20ZX040 0). 6. References Mllng by telescopc drve PKM Mllng by lnear drve PKM [] Y. Koren, A. G. Ulsoy, Reconfgurable Manufacturng Systems, Engneerng Research Center for Reconfgurable Machnng Systems(ERC/RMS), Report #, The Unversty of Mchgan, Ann Arbor, Ml, 997. [2] Y. Koren, Computer Control of Manufacturng Systems, McGraw Hll, New York, 983. [3] Farhad, P. Kourosh, A Reconfgurable Robot Wth Lockable Cylndrcall Jonts, IEEEE Transactonss on Robotcs, vol. 2,2009, pp [4] F. Jovane, S. P. Negr, I. Fass, L. Molnar Tosatt, Desgn Issues for Reconfgurablee PKMs, n 2002 Parallel Knematcs Machnes Internatonal Conference, Germany, 2002, pp [] V. Angelo, L. Govann, et al., Performance analyss of a reconfgurable redundant parallel manpulator, n The 2009 ASME/IFToMM Internatonal Conference on Reconfgurable Mechansms and Robots, Kng s Collegee London, London, UK, 2009, pp [6] X. Tang,C. Ln, et al., Structure Desgn and Task Plannng of Reconfgurable Parallel Knematc Machnes System,n The 2009 ASME/IFToMMM Internatonal Conference on Reconfgurable Mechansmss and Robots, Kng s College London, London, UK, 2009, pp [7] G. Patrck, D. G. Raffaele, Plannng for a Novel Reconfgurable Parallel Manpulator wth Lockable Revolute Jonts, n 200 IEEE Internatonal Conferencee on Robotcs and Automaton, Anchorage, Alaska, USA, 200, pp [8] X. Tang, J. Wang, M. Gao, Knematc calbraton of gantry hybrd machne tool based on estmaton errorr and local measurement nformaton, Internatonal Journal of Advanced Manufacturng Technology, vol. 26, 200, pp [9] K. H. Hunt, Structural knematcs of n parallel actuated robot arm, ASME Transacton, AutomatonDesgn, Vol. 4, 983, pp [0] Fattah and G. Kasae, Knematcs and dynamcs of a parallel manpulator wth a new archtecture, Robotca, Vol. 8, 2000, pp [] X. Tang, R. Yao, Dmensonal desgn on sx cable drven parallel manpulator of FAST, Journal of Mechancal Desgn,vol. 33, 20, pp. 2. [2] K. H. Pttens, R.P. Podhordedsk, A famly of Stewartt platforms wth optmal dexterty. Journal of Robotc Systems, vol. 0, 993, pp [3] T. Huang, D. J. Whtehouse, J. Wang, Local dexterty, optmum archtecture and desgn crtera of parallell machne tools. CRIP Annals, vol. 47,998, pp [4] C. Grosseln, J. Angels, A global performance ndex for knematc optmaton of robotc manpulators. Journal of Mechancal Desgn, vol. 3, 99, pp [] C. Grosseln, M. Gullot, The synss of manpulator wth prescrbed workspace, Journal of Mechancal Desgn, vol 3, 99, pp [6] J. Wang, X. Tang, C. Ln, G. Duan, J. Sh, Modular drve strut archtecture for reconfgurable parallell knematcs machne. Chnese Patents, No , [7] X. Tang, J.S. Wang, C. S. Ln, et al., An ntegrated jont for reconfgurable parallel knematcs machne, Chnese Patents, No , [8] J. P. Merlet, Sngular confguratons of parallell manpulators and Grassmann geometry, n Geometry and Robotcs (Lecture Notes n Computer Scence), New York: Sprnger Verlag, vol. 39, 989, pp Int J Adv Robotc Sy, 203, Vol. 0, 267:203