Utility Workspace of 3-5R Translational Parallel Mechanism

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1 Utlty Workspe of 3-5R rnsltonl Prllel Mehnsm Mstk nbe Deprtment of Mehnl Senes nd Engneerng okyo Insttute of ehnology okyo, Jpn Symsul Hud Deprtment of Mehnl Senes nd Engneerng okyo Insttute of ehnology okyo, Jpn Yuko ked Deprtment of Mehnl Senes nd Engneerng okyo Insttute of ehnology okyo, Jpn Abstrt In the present pper, we dsussed bout knmt desgn of 3-5R trnsltonl prllel mehnsm wth lrge utlty workspe. We defned the utlty workspe s losed re, from ny pont to other ponts n whh the mehnsm n move wthout sufferng from sngulrty nd workspe boundry. A omputtonl lgorthm of the volume of the utlty workspe ws proposed. We dsussed bout the utlty workspe of 3-5R trnsltonl prllel mehnsms hvng three types of hn wth onsderton of tuton nd onstrnt sngulrtes. We obtned 3-RUU trnsltonl prllel mehnsm wth lrge utlty workspe. Keywords Robots, Knemt desgn, rnsltonl prllel mehnsm, Utlty workspe, Sngulrty I. INRODUCION A prllel mehnsm whh hs three degrees of freedom nd performs pure trnsltonl output moton wthout hngng ts orentton s lled trnsltonl prllel mehnsm. A trnsltonl prllel mehnsm hs potentl ppltons n pk nd ple, mhnng, oordnte mesurement, nd so on. A trnsltonl prllel mehnsm s omposed of bse, pltform nd multple onnetng hns rrnged n prllel between the bse nd pltform. In reent yers, mny reserhers hve nterests n trnsltonl prllel mehnsms nd mnpultors. Knemt ondton of the onnetng hn to obtn pure trnsltonl moton of the pltform hs been nvestgted n [3]-[5]. Vrous knemt strutures of trnsltonl prllel mehnsm, 3-URC type [8], 3-PRRR type [9][19][20], nd 3- UPU type [10], et., hve been nvestgted. Here, C, P, R nd U represent ylndrl, prsmt, revolute nd unversl jonts, respetvely. Mehnsms wth onnetng hn tht onstrns rottonl moton of the pltform nd does not hve ny tve jont, lled supportng leg, hve been proposed n [6][7]. Workspe of 3-5R prllel mehnsm hs been nvestgted n [2]. Optmzton of the mehnsm onsderng ts workspe usng the globl ondtonng ndex hs been done n [11]-[14]. rnsltonl prllel mehnsms hve been ppled to hest ompresson mhne n the proess of rdopulmonry resustton [15] nd mro mnpultors [16][17]. Sngulrty of lower-dof prllel mehnsm hs been nvestgted n [22]-[24]. Even though there re ppltons of trnsltonl prllel mehnsm s desrbed bove, there exst mny problems n ts desgn. One of the problems s the dvson of rehble workspe nto some ndependent sub-workspes due to the exstene of sngulr ponts beuse the re n whh mehnsm n tully work s lmted wthn onseutve re whh does not nlude sngulrty. One the mehnsm s ssembled, t n move nsde sub-workspe tht s surrounded by sngulrty surfes nd workspe boundres. he lrgest re mong the sub-workspes s lled utlty workspe n the present pper. Key ssue of the mehnsm desgn s to obtn mehnsm wth lrger utlty workspe by optmzng ts struture nd ts dmensons. In the lst two dedes, sngulr ponts of prllel mehnsms wth sx degrees of freedom hve been extensvely nvestgted, nd mny works relted to mehnsm desgn wth onsderton of utlty workspe hve been done. For the prllel mehnsm wth sx degrees of freedom, tuton sngulrty (forwrd knemt sngulrty) should be mnly onsdered. However, for lowerdof prllel mehnsms suh s trnsltonl prllel mehnsm, onstrnt sngulrty s well s tuton sngulrty should be smultneously onsdered n ther desgn. In the present pper, we dsuss bout the knemt desgn of trnsltonl prllel mehnsm whh hs three onnetng hns wth fve revolute jonts wth onsderton of utlty workspe tkng tuton nd onstrnt sngulrtes nto onsderton. A tehnque to ompute the volume of utlty workspe from dgtzed dt of the rehble workspe s lso presented. hrough numerl exmples, we show struture nd dmensons of trnsltonl prllel mehnsm hvng lrge utlty workspe. Fnlly, prototype s brefly ntrodued /08 /$ IEEE RAM 2008

2 II. 3-5R RANSLAIONAL SPAIAL PARALLEL MECHANISM A. Mehnsm Confgurton Fgure 1 shows 3-5R trnsltonl sptl prllel mehnsm. It hs three onnetng hns, nd they re symmetrlly loted between the pltform nd the bse. Eh onnetng hn onssts of fve revolute jonts. When three of the fve revolute jonts re prllel nd resultnt two re prllel n eh onnetng hn, the mehnsm performs pure trnsltonl pltform moton [5]. In the mehnsm shown n Fg.1, the frst, seond nd ffth revolute jonts re prllel, nd the thrd nd fourth re prllel. hs type of onnetng hn s denoted s R R RRR by usng br or dot on R to lerly ndte geometrl reltonshp mong fve jont xes. An R R RRR hn onstrns rottonl moton of the pltform round n xs perpendulr to jont xes 1 nd 3. Beuse there re three onnetng hns, the pltform s rottonl moton s ompletely onstrned s fr s onstrnts by three onnetng hns re ndependent. B. Coordnte Systems A oordnte system O-XYZ s tthed on the bse. he frst revolute jont of eh onnetng hn s symmetrlly loted on rle whh s perpendulr to the Y-xs. A oordnte system A x y z s loted t A on the bse s shown n Fg. 2, so tht x z plne s prllel to the XZ plne nd z s the xs of rotton of the frst revolute jont, where denotes the onnetng hn s number ( = 1, 2,3 ). C. Knemt Constnts nd Vrbles Followng the defnton of the D-H prmeters, knemt onstnts to desrbe the reltonshp between djent jonts re defned s shown n Fg. 2(d). o desrbe the loton of the revolute jonts on the bse nd the pltform, knemt onstnts rb, rp nd γ re used s shown n Fgs. 2(b) nd 2(). In the present pper, γ 1 = 0D, γ 2 = 120D, γ 3 = 240D, α12 = α 34 = 0, α 23 = α 45 = 90D re ommonly used unless spefed. Eh revolute jont on the bse s onsdered s tve jont. Angulr dsplements of the tve jonts re denoted s θ = [θ1,1 θ1,2 θ1,3 ] s shown n Fg. 2(b). he dreton of the j-th(j=1,2, 5) jont n the -th onnetng hn s denoted s w j,, nd the poston of the j-th jont of the -th onnetng hn s denoted s rj,. III. SINGULARIY ANALYSIS here exst four types of sngulrty n prllel mehnsm. wo of them re tuton sngulrty nd onstrnt sngulrty t whh the mehnsm nnot keep ts onfgurton even when ll tutors re loked. hese types of sngulrty re seen only n losed-loop mehnsm. he other two types of sngulrty re one tht lotes t the workspe boundry nd the other t whh one of the onnetng hns hs lol moblty. In the desgn of prllel Fgure R trnsltonl sptl prllel mehnsm hvng R R RRR onnetng hns () shemt dgrm () loton of onnetng hn (b) dsplement of tve jont (d) DH prmeters Fgure 2. Coordnte systems nd knemt prmeters mehnsm, the former two sngulrtes should be refully onsdered. In ths seton, method to fnd onstrnt nd tuton sngulrtes of 3-5R prllel mehnsm wth R R RRR onnetng hns s desrbed. A. Veloty Reltonshp Beuse 3-5R prllel mehnsm s sptl prllel mehnsm wth three dof, onstrnt equton s well s nput-output veloty reltonshp should be onsdered to nvestgte nstntneous hrtersts of the mehnsm. Usng 6 6 overll Jobn mtrx J defned n [1], the veloty reltonshp of the mehnsm s desrbed s ω θ J =, (1) v 0 where ω nd v re ngulr veloty vetor, trnsltonl veloty vetor of the pltform, nd θ nd 0 re nput veloty vetor nd three dmensonl zero vetor, respetvely. he upper three slr equtons of Eq. (1) desrbe the reltonshp between nput nd output velotes, nd the lower the onstrnt equton. he overll Jobn mtrx n be wrtten usng 3 3 mtres J, J b nd J s

3 Jb J J = J 0, (2) 3 where 03 represents the three dmensonl zero mtrx. hese mtres re derved n III.D. B. Constrnt Sngulrty he onstrnt equton of 3-5R trnsltonl prllel mehnsm s wrtten s J ω = 0. (3) When the rnk of the mtrx J s less thn three, onstrnt mposed on the pltform by three onnetng hns re dependent. hen, rottonl moton of the pltform n not be onstrned. hs s the onstrnt sngulrty. herefore, the onstrnt sngulrty s defned by det J = 0. (4) C. Atuton Sngulrty If the rnk of the mtrx J s three, the mehnsm n perform three-dof pure trnsltonl pltform moton. In ths stuton, the followng reltonshp between nput nd output velotes holds J v = θ. (5) When the rnk of the mtrx J s less thn three, trnsltonl moton of the pltform n not be onstrned even f ll the nput jonts re loked. hs s the tuton sngulrty. herefore, the tuton sngulrty s defned by det J = 0. (6) D. Dervton of Mtres J nd J Frstly, the mtrx J s derved usng the reprol srew theory. Jont srew S j, (sx dmensonl vetor) of the j-th jont n the -th onnetng hn s denoted s j= 1 ( ) S j, = s s r, j, t, j, = j, j, j, w r w, where s r, j, nd s t, j, re three dmensonl vetors. Consder srew tht s reprol to ll jont srews of eh onnetng hn. Suh reprol srew s denoted s SR, = s Rf, s Rm,, where s Rf, nd s Rm, re three dmensonl vetors nd represent onstrnt fore nd moment mposed on the pltform by the -th onnetng hn. Sne the vrtul power by onstrnt fore by onnetng hn nd veloty of the pltform generted by eh jont veloty of onnetng hn should be zero, SR, s obtned by solvng the followng equton, SR, Sj, = srf, st, j, + srm, sr, j, = 0( j = 1,2,,5), (7) where the opertor represents the reprol produt. he output veloty V = ω O vo s wrtten s 5 V = S j, θ j,. (8) Applyng the reprol produt wth S R, to both sdes of Eq. (8) nd substtutng the reltonshp n Eq. (7) nto ths equton, the followng reltonshp s obtned. SR, V = 0 (9) Substtutng the ondtons where w1, = w2, = w 5, nd w3, = w 4, nto Eq. (7), S R, for 3-5R prllel mehnsm wth R RRR R onnetng hns s obtned s w1, w3, srf, = 0, srm, =. (10) w1, w3, Usng Eqs. (9) nd (10), the mtrx J s obtned s srm,1 ( w1,1 w3,1) / w1,1 w 3,1 J = srm,2 = ( w1,2 w3,2 ) / w1,2 w3,2. (11) srm,3 ( w1,3 w3,3 ) / w1,3 w3,3 Next, the mtrx J s derved. A srew SRA, = s RAf, s RAm, whh stsfes the followng equtons s onsdered. SRA, Sk, = 0, k = 2,3,4,5 (12) SRA, SR, = 0 Here the opertor represents the nner produt. For 3-5R prllel mehnsm wth R RRR R onnetng hns, the srew s obtned s {( ) } {( ) } sraf, = r2, r5, w1, r3, r 4, w3,, (13) sram, = mw1, + nw3, where ( F w1, w3, F ) ( F w1, w3, F) m=, n = 2 2 ( w1, w3, ) 1 ( w1, w3, ) 1. F = s r w, F = s r w ( ) ( ) RAf, 2, 1, RAf, 3, 3, Applyng the reprol produt wth S RA, to both sdes of Eq. (8) nd substtutng the reltonshp n Eq. (13) nto ths equton, the followng reltonshp s obtned. SRA, V = SRA, S 1, θ1, (14) From Eq. (14), the Jobn mtrx J s obtned s sraf,1 ( SRA,1 S1,1 ) J = sraf,2 ( SRA,2 S1,2). (15) RAf,3 ( RA,3 1,3 ) s S S E. Identfton of Sngulr Surfes nd Utlty Workspe As desrbed n [21] nd other ltertures, sngulr ponts exst on surfes nsde the rehble workspe. If the sgns of determnnt of Jobn mtrx J re dfferent between two seprte ponts, there must be pont to be det J = 0 : sngulr pont. Bsed on ths de, sngulrty surfes onsderng onstrnt nd tuton sngulrtes re obtned s follows. (1) Defne ltte ponts n the XYZ spe by eqully dvdng ts spe.

4 Fgure 3. Combnton of sgns of mtres J nd J n the rehble workspe (Z=0) ( r = r 1, = = = ) p b = = (2) For ll ses when eh ltte pont s gven s the poston of the pltform, nverse dsplement nlyss s done. (3) When rel soluton s obtned n (2), Jobn mtres J nd J re lulted. he sgns of these mtres re then obtned. All the ponts where nverse dsplement soluton exsts re lssfed nto four sets by the ombnton of the sgns of J nd J. (4) A sngulrty surfe n be dentfed between two sets of ponts obtned n (3). (5) Beuse one set of ponts my be dstrbuted to two or more res, utlty workspe should be dentfed onsderng the oseutveness of ponts n one set. Fgure 3 shows numerl exmple of dstrbuton of the ombnton of sgns of the Jobn mtres for rb = rp = 1, 12 = 23 = 34 = 45 = In Fg. 3, one set of ponts wth the sme sgns ombnton s pnted by sme densty. From ths result, we n dentfy sngulrty surfes s the boundry between res of dfferent denstes ()-(v). However, utlty workspe n not be evluted from ths result. We need to develop n lgorthm to hek the onseutveness of ponts n one regon wth the sme ombnton of sgns. IV. COMPUAIONAL ALGORIHM OF UILIY WORKSPACE In the prevous seton, the rehble workspe of 3-5R prllel mehnsm ws dvded nto four regons ordng to the ombnton of sgns of determnnts of Jobn mtres J nd J. However, s desrbed, suh dvded regon my exst seprtely. So, utlty workspe, from ny pont to ny other ponts of whh mehnsm n move wthout sufferng from sngulrty nd workspe boundry, should be dentfed onsderng the onseutveness of ponts n one regon. In ths seton, omputtonl lgorthm to dentfy utlty workspe of mehnsm hekng the onseutveness of ponts s proposed. Fgure 4. A vsul explnton of the steps (4) nd (5) to ompute the utlty workspe for plnr se he proposed lgorthm s summrzed s follows. Here, ths lgorthm s bsed on the lulton result mentoned n the prevous seton. (1) Lbel ll ponts nsde the rehble workspe ordng to the ombnton of sgns of determnnts of the Jobn mtres J nd J. (2) Choose one of the res, t every pont n whh the lbel s the sme, nd whh hs lbeled ponts. If there s no re, go to the step (7). (3) Choose one pont nsde ths re. hs pont s lled bse pont. If there s no pont, go to the step (2). (4) Chek the lbel of ll the djent ponts of the bse pont. Obtn n ntl re by ombnng ll ponts round the bse pont, whh hve the sme lbel, wth the bse pont. hs re s lled sub-workspe. (5) Chek the lbel of ll djent ponts of the ponts tht were ombned wth the ntl sub-workspe n (4), nd subworkspe s updted ordng to the lbel followng the sme tehnque n (4). Repet ths step untl the subworkspe s not updted. (6) he sub-workspe s onsdered s losed re where the mehnsm n move onseutvely, nd s regstered. Delete ll ponts whh belong to ths sub-workspe from the re, nd go to the step (3). (7) Determne the utlty workspe s the re hvng the mxmum volume mong the regstered sub-workspes n (6). Here, though the workspe volume s used s the evluton ndex, shpe of the re nd other mesures n be used. Opertons n the steps (4) nd (5) for plnr se re vsulzed n Fg.4. For sptl se, ths lgorthm n be ppled. V. DESIGN OF 3-5R RANSLAIONAL PARALLEL MECHANISM WIH CONSIDERAION OF UILIY WORKSPACE A. Enlrgement of Rehble Workspe of 3- R RRR R Mehnsm Frst, we nvestgted the workspe boundres shown n Fg. 3 n order to enlrge the rehble workspe reltve to

5 Fgure 5. RUU-type onnetng hn () the se A (b) the se B Fgure 6. Lnk length rto vs. volume of the utlty workspe of 3-RUU prllel mehnsm the mehnsm sze. In Fg.3, three boundres A, B nd C re shown. Sne the boundry C s used by the lmtton of totl lnk length, enlrgement of the boundry C results n expnson of the mehnsm sze. hen, we do not dsuss bout the boundry C. In wht follows, enlrgement of boundres A nd B s dsussed. We found from numerl lultons tht the boundry A depends on the length 34. In order to enlrge ths boundry, t s neessry to lengthen 34. On the other hnd, we found tht the boundry B s dependent on the dfferene between the lnk length 12 nd the dstne of jonts 2 nd 5. hrough smultons, we found tht mehnsm of 23 = 45 = 0 n hve lrge rehble workspe. hs orresponds to the 3RUU mehnsm s spel se of 3- R R RRR trnsltonl prllel mehnsm. An RUU hn s shown n Fg. 5. B. Knemt Desgn of 3-RUU Prllel Mehnsm Wth Consderton of the Utlty Workspe Fgure 6 shows the reltonshp between the lnk length rto 12 /( ) ( = 1 ) nd the volume of the utlty workspe of 3-RUU prllel mehnsm. It s know from ths fgure tht there re three pek ponts () the se C Fgure 7. Dstrbutons of of the sgns of det J nd det J of 3-RUU trnsltonl prllel mehnsm A( 12 /( ) = 0.17 ), B( 12 /( ) = ) nd C( 12 /( ) = 0.37 ). Dstrbutons of the ombnton of sgns of determnnts det J nd det J re shown n Fg. 7 for these three ses. he se A leds to the mxmzton of the re of the sub-workspe (). he se C leds to the mxmzton of the re of the sub-workspe (). In the se B, the sub-workspes () nd () hve the sme volume. It s known from Fg. 6 tht the knemt onstnts for these two ses A nd C re dfferent from tht ledng to the mxmzton of the rehble workspe. When the knemt onstnts re hosen s the se C, tht s the se wth the lrgest utlty workspe, t s known from Fg. 7() tht one of the four sub-workspes, (), oupes most of the rehble workspe, nd ths results n the mxmzton of the utlty

6 () R RRRR -type (b) R RRR R -type Fgure 8. wo onnetng hns pplble to 3-5R trnsltonl prllel mehnsm workspe. herefore, n the desgn of prtl prllel mehnsm hvng lrge utlty workspe, t s mportnt to evlute the volume of the utlty workspe. he volume of the utlty workspe for the se C s Its NVI (normlzed volume ndex) [18] s From ths vlue, t s known tht the 3-RUU trnsltonl prllel mehnsm hvng knemt onstnts orrespondng to the se C hs lrge utlty workspe reltve to the sze of the mehnsm. C. Utlty Workspe of 3-5R Prllel Mehnsm Hvng Other Connetng Chns Connetng hns pplble to 3-5R trnsltonl prllel mehnsm other thn the R RRR R hn re shown n Fg. 8. Frst, RRRRR type onnetng hn s nvestgted. As s desrbed n [4], the RRRRR type onnetng hn s better thn the R RRR R type when the volume of the rehble workspe s onsdered. However, we found tht 3-RRRRR trnsltonl prllel mehnsm, n whh the frst revolute jonts on the bse re used s tve jonts, beomes sngulr (tuton sngulrty) n the XY plne. herefore, the RRRRR type onnetng hn s not useful s fr s the tutors re loted on the bse lnk. Next, we nvestgte the R RRR R type onnetng hn. When ths type of onnetng hn s used, the ondton for onstrnt sngulrty s ndependent of the poston of the pltform. herefore, t s possble to elmnte the onstrnt sngulrty by n pproprte knemt desgn. hs n be done by determnng the ngle β wth onsderton of the vlue of det J. We nvestgted knemt onstnts nludng β of ths onnetng hn evlutng the volume of the utlty workspe. As n exmple, dstrbuton of the sgns of det J s shown n Fg. 9, when knemt onstnts shown n ble I re used. o determne these knemt onstnts, we frstly nvestgted enlrgement of rehble workspe. hs resulted n RRUR onnetng hn. hen, we optmzed ts knemt onstnts tkng the volume of the utlty workspe nto onsderton. hs result s summrzed n the tble. It s known tht onstrnt sngulrty doesn't exst n Fg.9. he utlty workspe orresponds to the sub-workspe (). Its Fgure 9. Dstrbuton of the sgns of det J nd det J of 3- R RU R trnsltonl prllel mehnsm ABLE I. KINEMAIC CONSANS OF 3- R RU R MECHANISM ( γ 1 = 0, γ 2 = 120, γ 3 = 240, r p = rb = 1, j jk d j /4 2/4 β = 45 ) NVI s lulted s From the vew pont of the utlty workspe, the RUU-type onnetng hn s onsdered better for trnsltonl prllel mehnsm. D. Prototype of 3-RUU rnsltonl Prllel Mehnsm Wth Lrge Utlty Workspe Bsed on the results obtned n ths seton, we desgned nd bult prototype shown n Fg. 10. We onduted experments to nvestgte the workspe of the prototype expermentlly. As result, the prototype heved stroke of 420mm n the X-dreton whle the mxmum stroke n ths dreton s theoretlly 490mm. It ws lso observed through experments tht orentton of the pltform rpdly hnged s t losed to the sngulr pont. hs wll be nvestgted by senstvty nlyss s desrbed n [25][26]. VI. CONCLUSIONS In the present pper, we dsussed bout knmt desgn of 3-5R trnsltonl prllel mehnsm wth lrge utlty workspe. We defned the utlty workspe s losed re, from ny pont to other ponts n whh the mehnsm n move wthout sufferng from sngulrty nd workspe boundry. A omputtonl lgorthm of the volume of the utlty workspe ws proposed. We obtned 3-RUU trnsltonl prllel mehnsm wth lrge utlty workspe.

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