Singularity Surfaces and Maximal Singularity-Free Boxes in the Joint Space of Planar 3-RPR Parallel Manipulators

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1 Sngulrt Surfces nd Mml Sngulrt-Free Boes n the Jont Spce of Plnr 3-RPR Prllel Mnpultors Mzen ZEIN Phlppe WENGER Dmen CHABLAT IRCCN, Insttut de recherche en communctons et cbernéthque de Nntes 1, rue de l Noe, 443 Nntes Abstrct In ths pper, method to compute jont spce sngulrt surfces of 3-RPR plnr prllel mnpultors s frst presented. Then, procedure to determne mml jont spce sngulrt-free boes s ntroduced. Numercl emples re gven n order to llustrte grphcll the results. Ths stud s of hgh nterest for plnnng trjectores n the jont spce of 3-RPR prllel mnpultors nd for mnpultors desgn s t m consttute tool for choosng pproprte jont lmts nd thus for szng the lnk lengths of the mnpultor. Kewords: 3-RPR prllel mnpultors, sngulrt, sngulrt-free zones, jont spce, jont lmts. I. Introducton Most prllel mnpultors hve sngulrtes tht lmt the moton of the movng pltform. The most dngerous ones re the sngulrtes ssocted wth the drect knemtcs, where two ssembl-modes colesce. Indeed, pprochng such sngulrt results n lrge ctutor torques or forces, nd n loss of stffness. Hence, these sngulrtes re undesrble. There ests three mn ws of copng wth sngulrtes, whch hve ther own merts. A frst pproch conssts n elmntng the sngulrtes t the desgn stge b properl determnng the knemtc rchtecture, the geometrc prmeters nd the jont lmts [4,4,7]. Ths pproch s sfe but dffcult to ppl n generl nd restrcts the desgn possbltes. A second pproch s the determnton of the sngulrt-free regons n the workspce [5,15-17,0,4]. Ths soluton does not nvolve pror desgn restrctons but, becuse of the complet of the sngulrt surfces, t m be dffcult to determne defntel sfe regons. Fnll, thrd w conssts n plnnng sngulrt-free trjectores n the mnpultor workspce [,6,19]. Wth ths soluton one s lso fced wth the complet of the sngulrt equtons but lrger zones of the workspce m be eploted. In ths pper, we choose to use the second pproch b determnng mml jont spce sngulrt-free boes. Ths pproch wll help us determne pproprte jont lmts nd lnk dmensons. Plnr prllel mnpultors nd prtculrl mnpultors wth three etensble leg rods, referred to s 3-RPR, hve receved lot of ttenton becuse the hve nterestng potentl pplctons n plnr moton sstems [9,1]. As shown n [18], moreover, the stud of the 3-RPR plnr mnpultor m help better understnd the knemtc behvor of ts more comple sptl counterprt, the 6- dof octhedrl mnpultor, whch hs lso trngulr bse nd pltforms. The sngulrtes of these mnpultors hve been most often represented n ther workspce [13,14,18] but more rrel n ther jont spce [18,4,6]. Hunt nd Prmrose showed tht 3-RPR plnr mnpultor could hve up to 6 ssembl-modes [1]. Mcree nd Dnel nlzed the jont spce sngulrtes through slces to epln non-sngulr chngng trjectores [188], nd Zen et l nlzed the topolog of these slces n [6]. It ws shown n [18,4,5] tht, to chnge ts ssembl mode wthout meetng sngulrt, 3-RPR mnpultor should encrcle cusp pont n ts jont spce. In ths pper, method to compute nd to represent jont spce sngulrtes of 3-RPR plnr prllel mnpultors s frst proposed. A procedure s then provded to determne mml jont spce sngulrt-free boes. Ths work s of hgh nterest for the determnton of pproprte jont lmts nd for plnnng trjectores n the jont spce. II. Mnpultors under stud The mnpultors under stud re 3-DOF plnr prllel mnpultors wth three etensble leg rods (Fg.1). These mnpultors hve been frequentl studed nd hve nterestng potentl pplctons n plnr moton sstems. The geometrc prmeters re the three sdes of the movng pltform d 1, d, d 3 nd the poston of the bse revolute jont centers defned b A 1, A nd A 3. The reference frme s centered t A 1 nd the -s psses through A. Thus, A 1 = (0, 0), A = (A, 0) nd A 3 = (A 3, A 3 ). The prmeter s functon of d 1, d nd d 3. The jont spce Q s defned b the vectors of the lengths of the three ctuted etensble lnks T q. 1 3

2 As As 1 A 3 q 3 A 3 q 3 B 3 As B 3 d 3 d q d 1 B B 1 d 3 d d 1 B B 1 1 q 1 q q 1 q 1 III. A 1 A Fg. 1. A 3-RPR prllel mnpultor Knemtcs of 3-RPR prllel mnpultors The relton between the jont spce Q nd the workspce W cn be epressed s sstem of non-lner lgebrc equtons, whch cn be wrtten s: F(, q ) 0 (1) where nd q re respectvel the vectors of the workspce nd jont spce vrbles. Dfferenttng equton (1) wth respect to tme leds to the veloct model: At + Bq 0 () where w T t c (w s the sclr ngulr veloct nd c s the two-dmensonl veloct vector of the opertonl pont B 1 of the pltform f we used the frst workspce prmeters), A nd B re 33 Jcobn mtrces whch re confgurton dependent, nd T jont veloct vector. IV. mnpultors q s the 1 3 Jont spce sngulrtes of 3-RPR prllel The sngulrtes of 3-DOF plnr prllel mnpultors hve been etensvel studed (see for emple [3,8,14,18,]). The were defned n the workspce (,, nd to the uthor s knowledge, there est smll number of works delng wth the sngulr confgurtons n the mnpultors jont spce ( 1,, 3 ). In prllel sngulrt, mtr A s sngulr. To derve the sngulrt equtons, t s usul to epnd the determnnt of A. We use rther geometrc pproch tht does not nvolve complcted lgebrc clculus. The 3- RPR prllel mnpultor s n sngulr confgurton whenever the es of ts three legs re concurrent or prllel [11] (Fg. ). Fg.. A 1 A A 3-RPR prllel mnpultor on sngulr confgurton. In order to derve ths geometrc condton, we derve the condton for the three leg es to ntersect t common pont (possbl t nfnt). We frst wrte the equtons of the three leg es: (As 1) : cos( q1) sn( q1) (As ) : cos( q) ( A )sn( q) (As 3) : cos( q3) ( A3 )sn( q3) A3 cos( q3) Elmntng nd elds the followng sngulrt equton n the tsk prmeters (q 1, q, q 3 ): A s s A s A c s (4) where s sn q, c cosq nd sj sn q q j. It s possble to epress Eq. (4) s functon of the jont spce prmeters 1, nd 3 b usng the constrnt equtons of the 3-RPR mnpultor. However, the resultng equton would be too complcted to eld rel nsghts, nd dffcult to hndle. Our pproch to compute the sngulr confgurtons n the jont spce conssts n reducng the dmenson of the problem b frst consderng two-dmensonl slces of the confgurton spce b fng the frst leg rod length 1. The sngulr surfces n the full jont spce re then clculted b stckng the slces. Step 1: We rewrte Eq. (4) s functon of 1, nd q 1 usng the constrnt equtons of the mnpultor. A c 1 c1 d1 cos 0 s 1 s1 d1 sn 0 A3 3 c3 1 c1 d3 cos 0 A3 3 s3 1 s1 d3 sn 0 The frst (respectvel lst) two equtons mke t possble to epress (respectvel 3 ) s functon of 1, nd q 1. Then, c nd s (respectvel c 3 nd s 3 ) re clculted s functon of 1, nd q 1 from the frst (respectvel lst) (3) (5)

3 two equtons of (5) nd ther epressons re nput n Eq. (4), whch, now, depend onl on L 1, nd q 1. Step : We f vlue for 1, so Eq. (4) depends now onl on nd q 1. B vrng or q 1, we compute the roots of the equton, to obtn the sngulr confgurtons ( s, q 1s ) for fed 1s. Step 3: For ever sngulr confgurton computed n the spce (, q 1 ) n the second step of the pproch, we clculte the correspondng vlues s nd 3s usng the equton sstem (6). We hve thus the sngulr confgurtons curves n slce of the jont spce (, 3 ) wth 1 fed. A 1 cos( q1) d1 cos( ) 1 sn( q1) d1 sn( ) A3 1 cos( q1) d3 cos( ) 3 A3 1 sn( q1) d3 sn( ) Fgure 3 shows slce of the jont spce sngulr confgurtons for 1 =17 obtned for the sme 3-RPR mnpultor used n [14,18,1]. We refer onl to ths mnpultor n ths pper n order to llustrte our work. The geometrc prmeters of ths mnpultor re reclled below n n rbtrr length unt: A 1 =(0, 0) d 1 =17.04 A =(15.91, 0) d =16.54 A 3 =(0, 10) d 3 =0.84 (6) obtned n step 4 nto CAD softwre, nd we hve meshed them together. Obvousl, there s contnut between the sngulrtes slces, one cn clm, wthout n doubt, tht there s sngulrt between the dfferent slces. The surfce depcted n Fg. 4 s of nterest:. for plnnng trjectores n the jont spce becuse t shows clerl the jont spce regons tht re free of sngulrtes.. for mnpultor desgn, becuse t consttutes tool for defnng the vlues of the jont lmts such tht the jont spce s sngulrt-free bo Fg. 4. Jont spce sngulrt surfces of the 3-RPR mnpultor studed when 1 vres from 0 to 50. Fg. 3. Sngulr confgurtons n (, 3) for 1=17. Step 4: We compute the jont spce sngulrt slces for number of 1 vlues, to do ths we hve to repet steps nd 3 whle vrng 1. Fnll, we collect ll the computed slces n one fle to obtn the sngulrtes n the jont spce ( 1,, 3 ). Fgure 4 represents the sngulrtes n the jont spce of the mnpultor studed when 1 vres from 0 to 50. To obtn ths surfce, we hve mported the solutons V. Mml jont spce sngulrt-free boes In contet of desgn nd/or trjector plnnng, n mportnt problem s to fnd sngulrt-free zones n the jont spce. In ths secton, we ntroduce new procedure to determne mml sngulrt-free boes n the jont spce of 3- RPR mnpultors. These sngulrt-free boes wll help us f the mnpultor jonts lmts. Two numercl emples re provded to llustrte the effectveness of the procedure. A. Procedure Step 1: We choose n ntl jont spce confgurton Q 0 ( 10, 0, 30 ). Ths confgurton cn be chosen ccordng to severl consdertons, for emple choosng Q 0 s the mge through the nverse knemtcs of prescrbed workspce center, or choosng t drectl n the jont spce s the center of lrge sngulrt-free zone. Step : We clculte the lrgest sngulrt-free cube centered t Q 0 ( 10, 0, 30 ).

4 To do ths, we clculte the nfnt norm dstnce d, lso known s Chebshev dstnce, between the center pont Q 0 ( 10, 0, 30 ) nd ech of the jont spce sngulr ponts Q s ( 1s, s, 3s ) computed n Secton IV: d m,, (7) 10 1s 0 s 30 3s nd we keep the mnml dstnce d mn found over ll, becuse we re serchng for the dstnce between the closest jont spce sngulrt confgurton Q s from the center pont Q 0. The length of the sngulrt-free cube edge wll be: d (8) mn Step 3: The choce of the ntl center pont Q 0 ( 10, 0, 30 ) does not led to n optmzed soluton, n other words vrng lghtl the center pont poston m led to lrgest sngulrt-free cube. Thus, the poston of the ntl pont must be optmzed, whch we hve done usng Hooke nd Jeeves optmzton scheme [10]. Note tht the soluton found s locl optmum. Step 4: The cube found n step 3 touches the closest sngulr confgurton to the center pont. In order tht the cube does not touch the sngulrtes surfce nd for more securt we subtrct smll securt vlue s from the dstnce d mn. Such vlue cn be relted to lws of commnd to stop the moton when the jont veloct s mmum. Fg Jont spce sngulrt surfces nd mml jont spce sngulrt-free cube centered t Q(41.65, 4.875, 44.15). Fgures 6 shows the mges through the drect knemtcs of the mml jont spce sngulrt-free cube, whch re two seprte sngulrt-free components, ech of them beng locted n n spect of the workspce [4]. The projectons of these two components onto the (,) plne re plotted n gr n Fgure 6. The mnpultor jont lmts correspondng to the cube found cn be esl computed s follows: d s mn 0 mn m 0 mn d s wth =1,,3 (9) B. Applcton of the procedure In ths secton, two emples re provded n order llustrte the pplcton of the procedure. Emple 1: For the sme mnpultor studed, we consder the center pont Q 0 (35,5,45) n the jont spce. Ths pont ws chosen n the center of lrge sngulrt-free zone n the jont spce. B computng the Chebshev dstnces between ech jont spce sngulr pont Q s computed n secton IV, nd Q 0, the mnml dstnce obtned s d mn =5.3, so the edge length of the sngulrt-free cube s = B runnng the optmzton lgorthm, we fnd mml vlue d mn =7.175 for center pont Q(41.65, 4.875, 44.15). We subtrct securt vlue of 0.1 from d mn whch becomes d mn = Fgure 5 shows the jont spce sngulrt surfce nd the mml jont spce sngulrt-free cube centered t Q(41.65, 4.875, 44.15) for the mnpultor studed. Fg. 6. Imges b drect knemtcs of the jont spce sngulrt-free cube (n blck), nd ther projecton on the (,) plne (n gr). Fgures 7 shows the two workspce components nd the workspce sngulrtes of the 3-RPR mnpultor studed, the sngulrtes re plotted n color.

5 two seprte sngulrt-free components, one n ech spect of the workspce. The projectons of these two components onto the (,) plne re plotted n gr n Fgure 9. Ths fgure s dspled wth the sme vewng ngle s n Fgure 6 to show the dfference between the two emples. Fg. 7. Sngulrt-free components wth workspce sngulrtes. Emple : For the sme mnpultor, we consder n ths emple the center pont Q 0 (30,50,35) n the jont spce. B computng the Chebshev dstnces between ech jont spce sngulr pont Q s computed n Secton IV nd Q 0, the mnml dstnce obtned s d mn =4, so the edge length of the sngulrt-free cube s = 8. B runnng the optmzton lgorthm, we fnd mml vlue d mn =5.794 for center pont Q(38.15, 50, 33). We subtrct securt vlue of 0.1 from d mn, whch becomes d mn = Fgure 8 shows the jont spce sngulrt surfce nd the mml jont spce sngulrt-free cube centered t Q(38.15, 50, 33) for the mnpultor studed. Fg Jont spce sngulrt curves nd mml jont spce sngulrt-free cube centered t Q(38.15, 50, 33). Fgures 9 dspls the mges b the drect knemtcs of the mml jont spce sngulrt-free cube, whch re Fg. 9. C. Future works Imges b drect knemtcs of the jont spce sngulrt-free cube (n blck), nd ther projecton on the (,) plne (n gr). We cn see n fgures 6 nd 9 tht the components n the workspce do not hve regulr forms. Becuse ths stud s crred out n the jont spce onl, t cnnot tke nto ccount n of the propertes, n the workspce, of the mge b drect knemtcs of the sngulrt-free cube found. Ths work wll be etended b tkng nto ccount the lrgest regulr volume (cube, clnder ) nsde the workspce components mges of the sngulrt-free cube. The de wll then be to optmze the locton of the ntl pont Q 0 ( 10, 0, 30 ) such tht the mge of the mml sngulrt-free cube n the workspce genertes regulr volume of mml sze. VI. Concluson A procedure for computng jont spce sngulrtes of 3- RPR prllel mnpultors hs been presented frstl n ths pper. Secondl, procedure for the determnton of mml jont spce sngulrt-free boes hs been provded. These two procedures re of nterest for plnnng trjectores n the jont spce, nd for mnpultors desgn becuse the provde tool for choosng the vlues of the jont lmts. Future work wll optmze the choce of the cube center pont Q 0 n the jont spce n order to mmze the volumes of the workspce components mges of the sngulrt-free cube.

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