Mobility Determination of Displacement Set Fully Parallel Platforms.

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1 12th IFToMM World Congress, Besnçon, June 18-21, 2007 Moblty Determnton of Dsplcement Set Fully Prllel Pltforms. José M. Rco Λ, J. Jesús Cervntes y, Jun Roch z Gerrdo I. Pérez x Alendro Tdeo. FIMEE, Unversdd de Gunuto. I. T. de l Lgun. ITESI. Slmnc, Gto. Méxco. Torreón, Coh. Méxco. Irputo, Gto. Méxco. Abstrct Ths contrbuton present comprehensve pproch for the moblty determnton of fully prllel pltforms where the dsplcements of the movng pltform wth respect to to the fxed pltform do not form subgroup of the Euclden group, SE(3). These pltforms re denomnted dsplcement set fully prllel pltforms for short. These types of prllel pltforms, except for the prllel pltforms of Tnev s type, hve not been prevously nlyzed, regrdng ther moblty, n prevous contrbutons. Keywords:Moblty, prllel pltforms, lower moblty pltforms. I. Introducton. In pr of recent contrbutons, the moblty determnton of fully prllel pltforms ws nlyzed employng group theory ppled to the Euclden group, SE(3), [1] nd the Le lgebr, se(3), of the Euclden group, SE(3), [2]. In these contrbutons, except for one cse, tht of prllel pltforms of Tnev s type, only prllel pltforms where the set of possble dsplcements of the movng pltform wth respect to the fxed pltform form subgroup of the Euclden group 1, were studed. The reson s tht mny reserchers, ncludng ourselves, beleved tht lmost ll pltforms tht do not stsfy ths condton re not - menble of structured pproch for determnng ther moblty. However, n the lst fve yers, mny reserchers such s Hung nd hs coworkers [3-8], Kong nd Gosseln [9-14] nd Crrcto nd Prent-Cstell [15-17] were ble to fnd gret number of prllel pltforms where the set of dsplcements of the movng pltform wth respect to the fxed pltform do not form subgroup of the Euclden group, SE(3). These pltforms re denomnted s dsplcement set fully prllel pltforms for short. The preferred pproch for fndng these prllel pltforms s bsed n n nlyss of recprocl screws. Snce the recprocl screws re more relted to the constrnt nlyss of knemtc chns thn the moblty of the chns, ths fct posed Λ E-ml: rco@slmnc.ugto.mx y E-ml: ecer@slmnc.ugto.mx z E-ml: rcpelus@hotml.com x E-ml: gerrdo p s@hotml.com E-ml: ltdeo@tes.edu.mx 1 In the lnguge of the Le lgebr, se(3), ths condton cn be stted s requrng tht ll the possble velocty sttes of the movng pltform wth respect to the fxed pltform form sublgebr. metholodogy problem for the uthors of ths contrbuton, becuse we hve dvnced the de tht the moblty of fully prllel pltform n prtculr, nd knemtc chns n generl, cn be determned by nlyzng only the subset nd subgroups of the Euclden group, SE(3), or lterntvely nlyzng only the subspces nd sublgebrs of the Le lgebr, se(3), of the Euclden group, SE(3). It should be remnded tht recprocl screws re elements of se(3) Λ, the dul spce of the Le lgebr, se(3), of the Euclden group, SE(3). Recently, the uthors of ths contrbuton were ble to fnd systemtc pproch for snteszng fully prlell pltforms where the dsplcements of the movng pltform wth respect to the fxed pltform my or my not form subgroup of the Euclden group, SE(3). In the cse of dsplcement set fully prllel pltforms, the pproch s bsed n n nlyss of the compostons of the dfferent subgroups of the Euclden group, SE(3), nd t does not requre ny nlyss of recprocl screws. Moreover, ths pproch reveled tht dsplcement set fully prllel pltforms, re lso menble of structured pproch for determnng ther moblty. Ths s the purpose of ths present contrbuton. Nmely, to determne the moblty of fully prllel mnpultors where there re t lest two legs such tht, the possble dsplcements of the movng pltform wth respect to the fxed pltform, when they re only connected by ths, s not equl to the correspondng composte group 2, H m=f. Dsplcement set fully prllel pltforms where clssfed n Tble I. Clssfcton of prllel mnpultors bsed on ther moblty of [1] s II-3A, nd they were denomnted Prdoxcl Pltforms. The results ndcted n ths contrbuton represent step forwrd to the gol of clssfyng nd determnng the moblty of the dfferent types of fully prllel mnpultors. leg, known s the mechncl lson, II. The Dfferent Types of Dsplcement Set Fully Prllel Pltforms. In Rco et l [18] t ws shown tht there re two dfferent types of dsplcement set fully prllel pltforms. These types re 1. Type S1. In these prllel pltforms the set of ll dsplcements of the movng pltform wth respect to the fxed 2 Of course, t mght be possble to cst these results n the lnguge of the Le lgebr, se(3), of the Euclden group, SE(3). 1

2 12th IFToMM World Congress, Besnçon, June 18-21, 2007 pltform does not form subgroup of the Euclden group, SE(3). Furthermore, ths set s the ntersecton of the subsets generted by the serl connector chns, where ll the subsets re equl, see fgure 1. H m=f = H m=f =1; 2; :::; k: Employng dgrm of Venn-Euler, ths cse s depcted n fgure 3. Fg. 1. Intersecton of Equl Subsets,, for Dsplcement Set Fully Prllel Pltforms, when k =4. 2. Type S2. In these prllel pltforms the set of ll dsplcements of the movng pltform wth respect to the fxed pltform does not form subgroup of the Euclden group, SE(3). Furthermore, ths set s the ntersecton of the subsets generted by the serl connector chns, where not ll the subsets re equl 3. Some but not ll of these subsets my be subgroups, see fgure 2. Fg. 3. Venn-Euler Dgrm for Trvl Pltform of Tnev s Type If the Tnev s type pltform s exceptonl, then n generl H m=f 6= H m=f =1; 2; :::; k: Employng dgrm of Venn-Euler, ths cse s depcted n fgure 4. Fg. 2. Intersecton of Unequl Subsets,, for Dsplcement Set Fully Prllel Pltforms, when k =4. It should be noted the type S2 contns s specl cses the prllel pltform trvl of Tnev s type nd exceptonl of Tnev s type, see [1,2]. In the cse of ll prllel pltforms of Tnev s type, ll k serl connector chns or legs except one stsfy the condton tht nd = H m=f =1; 2; :::; k 1; k 6= H m=f k : If the Tnev s type pltform s trvl, then 3 Otherwse, the ntersecton would be subgroup nd the pltform would not be dsplcement set fully prllel pltform. Fg. 4. Venn-Euler Dgrm for n Exceptonl Pltform of Tnev s Type. The necessry nd suffcent condtons for these prllel pltforms of Tnev s type to be movble s, well s, ther moblty determnton ws nlyzed n Rco et l [1,2]. III. Moblty Condtons nd Moblty Determnton for Dsplcement Set Fully Prllel Pltforms of Type S1. In ths secton, t wll be presented the necessry nd suffcent condtons for dsplcement set fully prllel plt- 2

3 12th IFToMM World Congress, Besnçon, June 18-21, 2007 forms of type S1 to be movble. In ddton, ths secton ncludes the determnton of the degrees of freedom or moblty of ths type of prllel pltforms. The result s specl cse of proposton 5 n Rco et l [1]. Proposton 1. Moblty condtons for dsplcement set fully prllel pltforms of type S1. Consder fully prllel pltform wth k serl connector chns or legs such tht nd 6= H m=f =1; 2;:::;k; (1) = 8 ; 2f1; 2;:::;kg: (2) The condtons (1) nd (2) ndcte tht the prllel pltforms s ndeed dsplcement set fully prllel pltforms of type S1, see fgure 1. Therefore = k =1 Fnlly, f = neghborhood 4, N e ρ 8 2f1; 2;:::;kg: = ; 8 2 =1; 2;:::;k, contns, of the dentty mppng e of the Euclden group, SE(3), then, the pltform es movble. Furthermore, the moblty or degree of freedom of the prllel pltform;.e. the degree of freedom of the possble dsplcements of the movng pltform wth respect to the fxed pltform, s the dmenson of the subset. Proof: Consder prllel pltform tht stsfes the condtons of proposton 1, nd let ψ 2 N e ρ trry. Snce N e ρ = 8 2f1; 2;:::;kg be rb- Then for ny serl connector chn, or leg, of the prllel pltform there exst dsplcements ψ r f;f+1, ψr f +1;f +2,..., ψ r m 2;m 1, ψ r m 1;m such tht ψ = ψ r f;f+1 ψr f +1;f +2 ψr m 2;m 1 ψr m 1;m 8 r 2f1; 2;:::;kg (3) where ψ 2 N e ρ. Snce ths result s vld for ny ψ 2 N e ρ nd N e s n open subset, t follows tht the dsplcement set fully prllel pltforms type S1 re movble. Remrk 1: It cn be proved, see Appendx A, tht, except for the trvl subgroup feg <SE(3), ll the remnng, proper, subgroups of the Euclden group re open sets tht contn the dentty mppng. Thus,the exstence of the 4 A neghborhood Ne of of the dentty mppng e 2 SE(3) s n open subset of tht contns e 2 SE(3). neghborhood round the dentty N e s ensured. Furthermore, the product of two proper subgroups of the Euclden group lso hs negborhhood N e round the dentty. Therefore, f = 8 2 f1; 2;:::;kg s the product of two subgroups of the Euclden group, such s those ndcted n Tble I, the condtons of proposton 1 of re fullflled. IV. Moblty Condtons nd Moblty Determnton for Dsplcement Set Fully Prllel Mnpultors of Type S2. In ths secton, t wll be presented the necessry nd suffcent condtons for dsplcement set fully prllel pltforms of Type S2 to be movble. In ddton, ths secton ncludes the determnton of the moblty of ths type of prllel pltforms. The result s lso specl cse of proposton 5 n Rco et l [1]. Proposton 2. Moblty condtons for dsplcement set fully prllel pltforms of type S2. Consder fully prllel pltform wth k serl connector chns or legs such tht S s proper subset of f1,2,...,kg, 5 wth t lest two elements, such tht = H m=f 8 2f1; 2;:::;kg=S 6= H m=f 8 2 S; (4) nd, n generl 6= for some ; 2f1; 2;:::;kg: (5) Therefore = k =1 nd n generl 6= for some 2f1; 2;:::; kg: (6) It should be noted tht S cn not be the empty set, otherwse, the set of dsplcements of the movng pltform wth respect to the fxed pltform do form subgroup of the Euclden group, SE(3), nd the prllel pltform s not dsplcement set fully prllel mnpultors. Then condtons (4) nd (5) ndcte tht the prllel pltform s ndeed dsplcement set fully prllel pltforms of type S2, see fgure 2. contns neghborhood of the dentty mppng e 2 SE(3); then, the pltform s movble. Furthermore, the moblty or degree of freedom of the possble dsplcements of the movng pltform wth respect to the Fnlly, f fxed pltform, s the dmenson of subset. Proof: Consder prllel pltform tht stsfes the condtons of propostons 2, nd let ψ 2 N e ρ trry. Snce be rb- 5 The notton A=B ndctes the set of ll elements tht belong to A nd do not belong to B. 3

4 12th IFToMM World Congress, Besnçon, June 18-21, 2007 N e ρ 8 2f1; 2;:::;kg: tht s ndcted n [18], the composton s not commuttve; nmely, the resultng subset depends on the order of the nvolved subgroups. Then, for ny serl connector chn, or leg, of the prllel pltform there exst dsplcements ψ r f;f+1, ψr f +1;f +2,..., ψ r m 2;m 1, ψ r m 1;m such tht ψ = ψ r f;f+1 ψ r f +1;f +2 ψ r m 2;m 1 ψ r m 1;m 8 r 2f1; 2;:::;kg; (7) where ψ 2 N e ρ. Snce ths result s vld for ny ψ 2 N e ρ nd N e s n open subset, t follows tht the dsplcement set fully prllel pltform type S2 re movble. Remrk 2: The remrk 1 lso pples to ths cse. However, n ddton, t must be noted tht the ntersecton of the respectve neghborhoods N e, correspondng to ech one of the for 2f1; 2;:::;kg s the requred neghborhood N e correspondng to, see Appendx A. V. Subgroups of the Euclden Group Whose Composton s not Subgroup. The results obtned n the prevous sectons depend on the exstence of subgroups of the Euclden group whose composton s not subgroup. The results ndcted n Tble I were obtned by Fnghell nd Gllett, [19], nd recently completed nd employed n the knemtc synthess of prllel pltforms, [18]. The notton of the subgroups follows tht of Herve s, [20], nd t s s follows 1. R P;^u, the subgroup of rottons round fxed xs tht psses through pont P nd ts drecton s gven by the unt vector ^u. 2. T^u, the subgroup of lner trnsltons long the drecton gven by the unt vector ^u. 3. T^u?, the subgroup of plne trnsltons n ll the drectons perpendculr to the unt vector ^u. 4. S O, the subgroup of rottons round the fxed pont gven by O. 5. G^u, the subgroup of dsplcements n plne perpendculr to the unt vector ^u. 6. X^u, the subgroup of Schönfles dsplcements ssocted wth the unt vector ^u Furthermore, t should be reclled tht f H 1 nd H 2 re two subgroups of group G;.e. H 1 ;H 2» G, the product of subgroups H 1 H 2 G s defned s H 1 H 2 fg 2 G g = h 1 h 2 ;h 1 2 H 1 ;h 2 2 H 2 g: The tble llustrtes ll possble combntons of nonhelcodl subgroups of the Euclden group whose composton s not subgroup, together wth the geometrc condtons requred to ttn the result. It should be remrked Tble I. Clssfcton of Non-Helcodl Subgroups of the Euclden Group Whose Composton s not Subgroup. No. Subgroups Comp. Subgroup Subset nd Cond nd Dmenson Dmenson ton 1 R P;^u1 R Q;^u2 SE(3) 6 T 2 R 2 2, d 2 R P;^u1 R Q;^u2 G^v 3 T 2 R 1 2 b 3 R P;^u1 R Q;^u2 S O 3 T 2 R 2 2 c 4 R P;^u1 T^u2 X^v 4 T 2 R R P;^u1 T^u2 G^v 3 T 2 R 1 2 d 6 R P;^u1 C Q;^u2 SE(3) 6 T 3 R 2 3, c, d 7 R P;^u1 C Q;^u2 X^v 4 T 3 R 1 3 b 8 R P;^u1 T^u? 2 X^v 4 T 3 R 1 3, d 9 R P;^u1 S O SE(3) 6 T 3 R 3 4 e 10 R P;^u1 G^u2 SE(3) 6 T 3 R 2 4, d 11 R P;^u1 X^u2 SE(3) 6 T 3 R 2 5 *,d* 12 T^u1 C P;^u2 X^v 4 T 3 R 1 3, d 13 T^u1 S O SE(3) 6 T 3 R 3 4 g 14 C P;^u1 C Q;^u2 SE(3) 6 T 3 R 2 4, c, d 15 C P;^u1 C Q;^u2 X^v 4 T 2 R 1 3 b* 16 C P;^u1 T^u? 2 X^v 4 T 3 R 1 3 d 17 C P;^u1 G^u2 SE(3) 6 T 3 R 2 5 * 18 C P;^u1 G^u2 SE(3) 6 T 3 R 2 4 d 19 C P;^u1 S O SE(3) 6 T 3 R 3 5 e 20 C P;^u1 S O SE(3) 6 T 3 R 3 4 f 21 C P;^u1 X^u2 SE(3) 6 T 3 R 2 5 *,d* 22 T^u? S O 1 SE(3) 6 T 3 R 3 5 g 23 G^u1 S O SE(3) 6 T 3 R 3 5 g 24 G^u1 G^u2 SE(3) 6 T 3 R 2 5 *,d* 25 G^u1 X^u2 SE(3) 6 T 3 R 2 5 *,d* 26 S O1 S O2 SE(3) 6 T 3 R 3 5 h 27 X^u1 X^u2 SE(3) 6 T 3 R 2 5 *,d* Condton. The unt vectors tht defnes the drectons of the subgroups re nether prllel nor perpendculr. If both subgroups defne xes, they do not ntersect. Condton b. The unt vectors tht defne the drectons of the subgroups re prllel. If both subgroups defne xes, they re not coxls. Condton c. The xes re ntersectng but they re nether prllel nor perpendculr. Condton d. The unt vectors tht defne the drectons of the subgroups re perpendculr. If both subgroups defne xes, they do not ntersect. Condton e. The xs does not pss through the pont ssocted to the subgroup. Condton f. The xs does psses through the pont ssocted to the subgroup. Condton g. There s no one condton between drecton nd pont ssocted to the subgroup. Therefore, ny locton of the pont provdes the sme result. Condton h. The ponts ssocted to the subgroup re dfferent. Remrk. * Ths symbol ndctes tht the trnsltonl nd rottonl dsplcements re not relted. If the symbol * does not pper, the number of reltons, between the trnsltonl nd rottonl dsplcements, s gven by the sum of trnsltonl nd rottonl dsplcements mnus the subset dmenson. VI. Exmple of dsplcement set fully prllel pltform of type S1. Fgure 5 presents n exmple of dsplcement set fully prllel pltform of type S1. In ths prllel mnpultor, the movng pltform, represented by the trngulr plte, s connected wth the fxed pltform, represented by the trngulr frme, by three serl connector chns or legs. Ech 4

5 12th IFToMM World Congress, Besnçon, June 18-21, 2007 serl connector chn s formed by two prsmtc prs, wth perpendculr drectons, followed by three revolute prs. The three revolute prs ntersect t pont O. The frst two prs of ech serl connector chn form plnr trnsltons subgroup. Ths plnr trnsltons subgroup llows generl trnsltons n the plne of the fxed pltform. The remnng three revolutes, n ech serl connector chn, form the sphercl subgroup whose fxed pont s O, nmely S O. Ths wy, the subset of llowed dsplcements of ech serl connector chns, nd therefore of the movng pltform wth respect to the fxed pltform, s formed by n rbtrry trnslton long the plne of the fxed pltform followed by n rbtrry rotton round pont O. Ths pont O moves n the trnslton plne. Ths set cn be obtned by the composton of the subgroups ndcted n lne 22 n Tble I. Furthermore, the moblty of the prllel pltform s fve nd there s no pssve degrees of freedom. followed by one-dmensonl sptl trnslton. Furthermore, the moblty of the prllel pltform s two. Fg. 6. Dsplcement set fully prllel pltform of type S2, where the requred dsplcements set form only subset of the Euclden group SE(3), nd not ll the serl connector chns hve the sme requred dsplcements set. Fg. 5. Dsplcement set fully prllel pltform of type S1, where the requred dsplcements set form only subset of the Euclden group SE(3), nd ll the serl connector chns hve the sme requred dsplcements set. VII. Exmple of dsplcement set fully prllel pltform of type S2. Fgure 6 presents n exmple of dsplcement set fully prllel pltform of type S2. In ths prllel mnpultor, the movng pltform, represented by the upper cudrlterl plte s connected wth the fxed pltform, represented by the lower cudrlterl plte, by three serl connector chns. The frst two serl connector chns provde n llowed dsplcement set formed by rbtrry rottons round pont O 1 (pont O 2 ) followed by trnsltons long dfferent plne tngent to sphere wth center pont O 1 (pont O 2 ). These subsets re generted by the composton of the subgroups ndcted n lne 22 of Tble I, wth the opposte order; nmely S O T^u? 1. The thrd serl connector chn genertes the Schönfles subgroup wth n xs tht psses through ponts O 1 nd O 2. The dsplcement set of the movng pltform wth respect to the fxed pltform s rotton round the xs defned by the lne O 1 O 2 VIII. Concluson. Ths contrbuton hs presented group theoretcl nlyses tht ndcte the necessry condtons for the two types of dsplcement set fully prllel mnpultors to be movble s well s ther moblty, or degree of freedom, determnton. From the methodology pont of vew these nlyses nd results re mportnt becuse they do not requre ny constrnt nlyss nvolvng recprocl screws. IX. Acknowledgements. The utors thnk Concyteg, the stte of Gunuto Councl of Scence nd Technology, for ts support of severl proects ncludng M. Sc. thess completon scholrshp for the three lst uthors nd FOMIX proect. The uthors lso thnk Concyt, the Ntonl Councl Scence nd Technology, for the grdute study scholrshps for the three lst uthors. The results n ths contrbuton re prt of ther M. Sc. theses. Fnlly, the frst uthor thnks Promep, Progrm for the Improvement of Unversty Educton nd Reserch, funded by the Mexcn Mnstry of Educton nd dmnstered by Cosuper, by ts support. References [1] Rco, J.M., Aguler, L.D., Gllrdo, J., Rodrguez, R., Orozco, H. nd Brrer, J.M. A group theoretcl dervton of more generl moblty crteron for prllel mnpultors. Proceedngs of the Insttuton of Mechncl Engneers, Prt C, Journl of Mechncl Engneerng Scence, Vol. 220, pp , [2] Rco, J.M., Aguler, L.D., Gllrdo, J., Rodrguez, R., Orozco, H. nd Brrer, J.M. A more generl moblty crteron for prllel pltform. ASME Journl of Mechncl Desgn, Vol. 128, pp ,

6 12th IFToMM World Congress, Besnçon, June 18-21, 2007 [3] Zho, T. S., D, J. S. nd Hung, Z. Geometrc Anlyss of Overconstrned Prllel Mnpultors wth Three nd Four Degrees of Freedom. JSME Interntonl Journl, Seres C, Vol. 45, pp , [4] L, Q.C., Hung, Z. nd Hervé, J. M. Type Synthess of 3R2T 5- DOF Prllel Mechnsms Usng the Le Group of Dsplcements. IEEE Trnsctons on Robotcs nd Automton, Vol. 20. pp , [5] Hung, Z. nd L, Q.C. Type Synthess of Symmetrcl Lower-moblty Prllel Mechnsms Usng the Constrnt-synthess Method The Interntonl Journl of Robotcs Reserch, Vol. 22,No. 1, pp , [6] L, Q.C. nd Hung, Z. A Fmly of Symmetrcl Lower-Moblty Prllel Mechnsms wth Sphercl nd Prllel Subchns. Journl of Robotc Systems, Vol. 20, pp , [7] L, Q.C. nd Hung, Z. Moblty Anlyss of Novel 3-5R Prllel Mechnsm Fmly. ASME Journl of Mechncl Desgn, Vol. 126, pp , [8] D, J.S., Hung, Z. nd Lpkn, H. Moblty of Overconstrned Prllel Mechnsms. ASME Journl of Mechncl Desgn, Vol. 128, pp , [9] Kong, X. nd Gosseln, C. M. Type Synthess of lner trnsltonl prllel mnpultors. Advnces n Robot Knemtcs- Theory nd Applctons, Lenrčč, J. nd Thoms, F. Eds. Dordrecht: Kluwer Acdemc Publshers, pp , [10] Kong, X. nd Gosseln, C. M. Type Synthess of 3-DOF Trnsltonl Prllel Mnpultors Bsed on Screw Theory. ASME Journl of Mechncl Desgn, Vol. 126, pp , [11] Kong, X. nd Gosseln, C. M. Type Synthess of 3-DOF Sphercl Prllel Mnpultors Bsed on Screw Theory. ASME Journl of Mechncl Desgn, Vol. 126, pp , [12] Kong, X. nd Gosseln, C. M. Type Synthess of 3-DOF PPR- Equvlent Prllel Mnpultors Bsed on Screw Theory nd the Concept of Vrtul Chn. ASME Journl of Mechncl Desgn, Vol. 127, pp , [13] Kong, X. (2003), Type Synthess nd Knemtcs of Generl nd Anlytcl Prllel Mechnsms, Ph. D. Dssertton, Unversté Lvl. [14] Kong, X. nd Gosseln, C. M. Type Synthess of 5-DOF Prllel Mnpultors Bsed on Screw Theory. Journl of Robotc Systems, Vol. 22, pp , [15] Crrcto, M. nd Prent-Cstell, V. Sngulrty-free fully-sotropc trnsltonl prllel mnpultors. The Interntonl Journl of Robotcs Reserch, Vol. 21, pp , [16] Crrcto, M. nd Prent-Cstell, V. Sngulrty-free fully-sotropc trnsltonl prllel mnpultors. Proceedngs of the ASME DETC02 Desgn Engneerng Techncl Conferences nd Computer nd Informton on Engneerng Conference, Montrel, Cnd Sep. 29- Oct. 2002, Pper DETC2002/MECH [17] Crrcto, M. nd Prent-Cstell, V. A Fmly of 3-DOF Trnsltonl Prllel Mnpultors, ASME Journl of Mechncl Desgn, Vol. 125 pp , [18] Rco, J.M., Cervntes-Sánchez, J. J., Tdeo-Chávez, A., Pérez-Soto, G.I. nd Roch-Chvrrí, J. A comprehensve theory of type synthess of fully prllel pltforms. Proceedngs of IDETC/CIE 2006, September 10-13, Phldelph, USA. ASME DETC [19] Fnghell, P. nd Gllett, C. Metrc Reltons nd Dsplcement Groups n Mechnsm nd Robot Knemtcs, ASME Journl of Mechncl Desgn, vol. 117, pp , [20] Hervé, J.M. Anlyse Structurelle des Mécnsmes pr Groupe des Déplcements, Mechnsm nd Mchne Theory, Vol. 13, pp , [21] Krger, A. nd Novk, J. Spce Knemtcs nd Le Groups. New York: Gordon nd Brech Scence Publshers, Appendx A. Some Results on the Topology of Le Groups. In ths ppendx, some results requred for the moblty determnton of dsplcement set fully prllel mnpultors wll be ncluded, the reference s [21]. Defnton A1 (Le Group). A set G s Le group f 1. G s n lgebrc group. 2. G s n nlytc mnfold. 3. The mppng G G! G, (g 1 ;g 2 )! g 1 g 2 s nlytc. Proposton A1. Let G be subgroup of the generl lner group of order n, GL(n), defned by some system of polynoml equtons. Then G s Le group n the nduced topology. Ths result mples tht, n prtculr, the Euclden group SE(3), tht t hs 4 4 homogeneous mtrx representton, nd ll ther lgebrc subgroups re Le groups. Furthermore, snce Le group s n nlytc mnfold, t s lso topologcl spce; therefore, the Le group, G, s n open set nd there exsts fmly of sets, clled open subsets, such tht 1. The unon of n rbtrry collecton of open subsets s lso open. 2. The ntersecton of fnte number of open subsets s lso open. These open sets nd subsets hve the property tht for ny element g tht belong to the set there s n open neghborhood tht contns the element g nd the neghborhood belongs to the set. These results prove tht the proper subgroups of the Euclden group SE(3) re open sets, Remrk 1. Therefore, t s lwys possble to fnd the requred neghborhood N e round the dentty element, or mppng, e. It s now, necessry to prove tht the product of two proper subgroups of SE(3) lso hs requred neghborhood N e round the dentty element, or mppng, e. Proposton A2. The product of proper subgroups of the Euclden group SE(3) hs neghborhood N e round the dentty mppng. Proof: Let H 1 ;H 2» SE(3). Let N e ρ H 1 be neghborhhod of the dentty element e. Snce e 2 H 2, then N e = N e e ρ H 1 H 2. Thus N e s neghborhood of the dentty element tht belongs to H 1 H 2. Fnlly, t s necessry to prove the exstence of the requred neghborhood n the ntersecton of subsets of SE(3) obtned by the product of subgroups. Proposton A3. Let = H 1 H f1; 2;:::;kg be subsets of SE(3) where H 1 ;H 2» SE(3). Then by proposton A2, ech one of the hs neghborhood N e round the dentty element e. Then N e = k N =1 e s neghborhood of the dentty element e tht belongs to = k =1 Lm=f. Proof: Obvously k N =1 e ρ ; 8 2 s the re- f1; 2;:::;kg. Thus, N e = k N =1 e ρ L m=k qured neghborhood. 6