Optimum Design of a New 4-DOF Parallel Mechanism. Jae Heon Chung*, Byung-Ju Yi*, and Whee Kuk Kim**

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1 CCAS Jne -, KNEX, yeongg-do, Kore Optmm Desgn of New -DO Pre Mechnsm Je Heon Chng*, Byng-J Y*, nd Whee Kk Km** * Schoo of Eectrc Engneerng nd Compter Scence, Hnyng Unversty, Ansn, yeongg, Kore (e : ; E-m: bj@hnyng.c.kr) **Deprtment of Contro nd nstrmentton, Kore Unversty, Chochwon, Kore Abstrct: Recenty, ots of pre mechnsms for spt -DO nd -DO were nvestgted. However, reserch on -DO nd -DO pre mechnsms hs been very few. n ths pper, we propose -DO pre mechnsm tht conssts of -rotton nd 1-trnston motons. he knemtc chrcterstcs of ths mechnsm re nyzed n terms of n sotropc ndex nd mxmm force trnsmsson rto, nd ts knemtc optmzton s beng condcted to ensre enhnced knemtc performnces Keywords: Pre mechnsm, Knemtc nyss 1. NRODUCON Recenty, ots of pre mechnsms for spt -DO nd -DO were nvestgted. However, when n operton spce s reqested ess thn -DO or more thn -DO, the cnddte w be to hve -DO nd -DO pre mechnsm. ng nd s [] proposed -DO pre mechnsm wth for-nk mbs nd three-nk mbs. Wng nd ossen [] nyzed pnr -DO knemtcy redndnt pre mechnsm nd spt spherc -DO knemtcy redndnt strctre. Zntnov nd ossen [] proposed pre rchtectre wth for degrees of freedom tht conssts of for R egs. Hng nd L [] proposed type synthess of -DO pre mechnsms tht hve three ctegores. t conssts of three rottons + one trnston, three trnstons + one rotton, nd two rottons + two trnstons. L nd Hng [, 9] nyzed -R pre mechnsm fmy whose mb conssts of R nd R pre sbchns or presented fmy of symmetrc ower-mobty pre mechnsms wth spherc nd pre sb chns, whch consst of two -DO, one -DO nd fve -DO pre mechnsms. Compny et. [1] presented fmy of fy pre robots prodcng motons of the Schoenfes dspcements sb-grop for hgh-speed hndng nd mchnng. Cho et. [11] de wth the desgn nd dynmc contro smton of -DOs pre mechnsm provdng trnstons nd 1 rotton for hgh speed hndng nd mchnng. D. chbt nd P. Wenger [1] proposed modr pre mechnsm wth -trnston nd -rotton motons. he performnce of mechnsm cn be mproved wth secton of optmzed nk prmeters fond by vros optmzton methods. Lee et. [1] pped Powe s method to nthropomorphc robot mode wth redndnt cttors. Lee et.[] so pped Powe s method by sng exteror penty fncton to fve br fnger mechnsm wth redndnt cttor nd pped genetc gorthm to -DO pre Hptc devce []. n ths pper, frsty, we nyze the mobty of the -DO system sng rüber s mobty form nd descrbe the new fetre of ths pre mechnsm. Secondy, we nyze the reverse poston of n ntern nd extern chn nd obtn the embedded Jcobn by sng extern Jcobn nd ntern Jcobn. hrdy, we descrbe knemtc desgn ndces (the sotropc ndex, mx force trnsmsson rto, nd workspce). orthy, we fnd optmzed nk prmeters by genetc gorthm to mprove the knemtc performnce ndces. ny, we show tht the performnce of the mechnsm wth the optmzed nk prmeter s speror to the non-optmzed mechnsm wth rbtrry prmeters.. SRUCURE.1 Mobty Mobty s defned s the nmber of ndependent vrbes tht re reqred to specfy the motons of the system retve to nother. he mobty of the mechnsm cod be fond from we-known rüber's mobty form s beow: j M = D( L 1) ( D ), (1) = 1 where D represents the mxmm moton spce ( for spt moton, for pnr moton), nd L,, nd J represent the nmber of nks ncdng the grond, the nmber of degree of freedom of the th jont, nd the nmber of jonts, respectvey. rom Eq. (1), when D=, L=11 nd J=1, the mobty of the mechnsm shown n g. 1 s. ( ) r R z b y x b b g. 1 he descrpton of mechnsm. Descrpton of mechnsm hs mechnsm conssts of for extern egs wth UPS (nvers, prsmtc, nd spherc) strctre, n ntern eg wth PS (prsmtc nd spherc) strctre, top pte, nd bse pte. he dvntge of the system ncdes rge workspce nd hgh stffness becse of the pre strctre. Specy, ths system s constrned by the ntern eg not to move n the x- nd y- drectons. here s no pror -DO mechnsm tht constrns the x nd y moton by ths wy. z t P yx t t

2 CCAS Another mert of ths mechnsm hvng n ntern eg s tht t cn spport rge pyod pped to the horzont drecton. Denote P = [ x y z ] s the poston vector from t t t the orgn of the bse frme to the orgn of the otpt frme. So, the poston vector of the otpt cn be expressed s [ z] becse of the fxed x nd y poston. r nd R represent the poston vector from the orgn of the bse pte to the th nvers jont of the extern eg nd the poston vector from the orgn of the top pte to the th spherc jont of extern eg, respectvey. denotes the poston vector from the th nvers jont to the th spherc jont of the extern eg.. Poston nyss.1 Reverse poston Reverse poston nyss s to fnd the ctve npt vector when the otpt poston/orentton vector of the mechnsm s gven. Denote the orentton mtrx of the otpt frme s [ R t ] sng the x y z Eer nge set: tht s, b [ R t ] = [ Rot( x, α)][ Rot( y, β)][ Rot( z, γ)] b cc cs s β γ β γ β. () = s sc + cs sss + cc sc α β γ α γ α β γ α γ α β csc + ss css + sc cc α β γ α γ α β γ α γ α β he poston vector P for the th extern eg cn be expressed s P = R + r, for = 1,,, () where ( ) r [ t ] = Rb r. () Rewrtng Eq. () wth respect to yeds = P+ r R. () he ve of the npt vrbe cn be fond by tkng nner prodct of tsef s foows: ( ) ( ) = = P+ r R P+ r R () Reverse poston of the ntern chn cn be drecty obtned from the gven otpt poston.. rst-order knemtcs o obtn reton between the ctve npt jont vector nd n ndependent otpt vector n operton spce, we strt to obtn the frst-order knemtcs for ech of fve egs, nmey for extern egs nd n ntern eg. rsty, the frst order knemtcs of the extern egs cn be obtned n the foowng wy. Assme tht ctve jonts re octed n prsmtc jonts of the extern eg. herefore, the frst-order knemtcs of the extern egs cn be obtned by dfferenttng Eq.() s foows = P P+ r R + r P+ r R, () ( ) ( ) ( ) = P+ r ω. () Eq. () cn be expressed s the mtrx form 1 ( r1 1) 1 1 ( r ) v = ( r ) ω, (9) ( r ) where v nd ω denote ner veocty vector nd n Jne -, KNEX, yeongg-do, Kore ngr veocty vector n the operton spce, respectvey. he retonshp between the ctve jont nd the otpt veocty vector cn be expressed s φ =, (1) where φ nd denote n ctve jont veocty vector nd otpt veocty vector n operton spce, respectvey. cn be expressed s 1 1 ( r1 1) 1 ( r ) ( r ) ( r ) =. (11) Secondy, the frst-order knemtcs of the ntern eg cn be obtned s 1 = φ = 1 s φ φ, (1) c sc s cc where φ nd φ denote jont veocty vector of the ntern eg nd the reton between the otpt veocty vector n the operton spce nd φ, respectvey. he otpt veocty of Eq (1) cn be seprted s the ndependent nd dependent otpt vectors P p U φ = = φ φ = φ, (1) φ where [ ] p = x y nd p = z ωx ωy ω z denote pssve otpt nd ctve otpt veocty vector, respectvey. ny, we embed Eq. (1) nto the extern Jcobn mtrx Eq. (1). An Embedded frst-order knemtcs cn be obtned from reton between ndependent otpt vector nd the ctve jont veocty vector by sng the constrnt reton between the ntern eg nd the extern eg. rom Eq. (1), we cn derve the reton between the pssve otpt ( P ) nd ndependent otpt veocty vector ( ) s the foows 1 P P p = φ φ =. (1) So, the retons between the otpt veocty ( ) nd the ndependent otpt veocty vector cn be expressed s = P = = P, (1) where denotes n by dentty mtrx. Sbstttng Eq. (1) nto Eq. (1), we hve φ =, (1) where cn be expressed s =. (1) ny, we cn obtn the frst-order knemtc reton = φ. (1)

3 CCAS. KNEMAC DESN NDCES.1 Workspce One of the bsc spects n robot desgn s to detere the workspce. he opertng regon or workspce of mnptor cod be defned s rechbe nd dexteros workspce, ccordngy. he vome of rechbe workspce s sed n ths nyss nd defned s = d. (19). sotropc ndex he knemtc sotropc ndex s defned s σ ( ) σ =, () σ mx ( ) where σ nd σ mx denotes the mm nd the mxmm sngr ve of, respectvey. When σ becomes nty, the end-effector cn generte nform veocty n drectons. n ddton, the gob desgn ndex, whch represents the verge of the mnptor s sotropc ndex over the whoe workspce, s defned s σ d =. (1) d he greter s, the better sotropy the mechnsm hs over the workspce. herefore, shod be mxmzed.. orce trnsmsson rto he mxmm force trnsmsson rto mpes the mxmm mgntde of n cttor od reqred for the nt end-effector force, nd t s defned s σ = σ. () mx ( ) σ becomes smer, the cttor od w be redced. hs mens tht the mnptor cn ber more weght wth ess cttor. he gob desgn ndex for σ s defned s σ d =. () d he smer s, the smer cpcty cttor reqred. hs, shod be mzed.. OPMUM DESN n order to mxmze the performnce of the proposed mechnsm, mt-crter desgn methodoogy bsed on genetc optmzton gorthm s empoyed.1 Composte desgn ndex Sever methodooges hve been proposed to cope wth mt-crter bsed desgn. However, vros desgn ndces re sy ncommensrte concepts becse of the dfferences n nt nd physc menngs, so shod therefore not be combned ness they re trnsferred nto common do. hs process conssts of normzton, whch trnsfers vros ndces to the sme do, nd then combnes sever ndces nto one. or both ndces nd, the most fvored preference s gven the mxmm ve, nd the est fvored preference s gven the mm ve of the crteron. hen, the preference desgn ndces, Jne -, KNEX, yeongg-do, Kore ~ nd ~ re expressed s ~ =, mx () ~ KS ( KS ) =, () KS ( ) ( ) KS mx KS where ~ on ech desgn ndex mpes tht t s trnsferred nto the common preference desgn do. Conversey, for, the hghest preference s gven the mxmm ve of the crteron. hen the preference desgn ndces, s expressed s ~ ( ) mx =. () ( ) ( ) mx o de wth ths mt-crter bsed desgn, we empoy knemtc composte desgn ndex (KCD) tht combnes sever ndvd preference desgn ndces nto nqe desgn ndex by sng the mx- prncpe of fzzy theory. KCD s expressed s ~ ~ ~ { KS } KCD =,,. () KCD s defned s the mm ve mong the preferred desgn ndces ccted for set of knemtc prmeters. A set of desgn prmeters, whch hve the mxmm ve of the KCD, s chosen s the optm set of desgn prmeters. When ny of snge desgn ndces, whch re ncded n constrcton of the KCD, s weghted more thn the others, weghted KCD cn be sed. A weghted KCD cn be represented s ~ β ~ γ α KCD =, KS,, () where α, β, nd γ represent the weghtng for ech desgn ndex. n ths work, of the weghtng s set to 1. n order to eveny stsfy the desgn objectves for desgn ndces.. Optmzton rest he knemtc desgn prmeters for the proposed mechnsm re the rds of the top pte, nk ength of ech eg, nd the rds of the bse pte. Bonds of these desgn prmeters re descrbed n be 1. enetc gorthm s empoyed to sove the nonner knemtc optmzton. We set the popton sze to 1. n ech generton, we evte the KCD of ech chromosome, seect new popton wth respect to the probbty dstrbton bsed on ftness ves, nd ter the chromosomes n the new popton by mtton nd crossover opertor. After nmber of genertons, we obtn the best chromosome (.e., prmeter set) tht represents n optm soton. he optmzton rest s represented n be 1. be 1 optmzton rest rbes Mn Mx Optmzton ve Prsmtc ength [m] op rds [m].1..9 Bse rds [m]. 1..1

4 CCAS. SMULAON RESUL Jne -, KNEX, yeongg-do, Kore sotropy ndex sotropy ndex... sotropy Rto bet [deg] () t the z= sotropy Rto bet [deg] (b) t the z=. sotropy ndex sotropy ndex... sotropy Rto bet [deg] (b) t the z= sotropy Rto bet [deg] (c) t the z=. g. sotropc ndex fter optmzton sotropy ndex. orce rnsmsson Rto.1. sotropy Rto bet [deg] (c) t the z=. g. sotropc ndex before optmzton orce rnsmsson Rto bet [deg] () t the z=..... orce rnsmsson Rto sotropy ndex.. sotropy Rto bet [deg] () t the z= orce rnsmsson Rto bet [deg] (b) t the z=....

5 CCAS orce rnsmsson Rto (c) t the z=. g. Mx force trnsmsson rto before optmzton orce rnsmsson Rto orce rnsmsson Rto orce rnsmsson Rto orce rnsmsson Rto orce rnsmsson Rto () t the z=. orce rnsmsson Rto (b) t the z=. orce rnsmsson Rto (c) t the z= bet [deg] bet [deg] bet [deg] bet [deg] g. Mx force trnsmsson rto fter optmzton g. shows the sotropc ndex before optmzton t z =.,., nd.. g. shows the sotropc ndex fter Jne -, KNEX, yeongg-do, Kore optmzton t z =.,., nd.. g. (c) shows tht sotropc performnce ws mproved mch compred to g. (c). g. shows the mx force trnsmsson rto before optmzton t z =.,., nd.. g. shows the mx force trnsmsson rto fter optmzton t z =.,., nd.. g. (c) shows tht force trnsmsson rto performnce ws mproved mch compred to g. (c). Concsvey, we cn concde tht the optmzed knemtc desgn prmeters mproved the knemtc performnces of the mechnsm sgnfcnty.. CONCLUSON We proposed new -DO pre mechnsm (-rotton + z-xs trnston). We derved Jcobn mtrx tht retes the ctve jont to the ndependent otpt. We nyzed the performnce of ths mechnsm n spect of three knemtc ndces. Moreover, we fnd optm knemtc desgn prmeters of the new mechnsm by sng genetc gorthm nd showed tht performnces of knemtc ndces were mproved. ACKNOWLEDMENS hs work ws spported by grnt of the Kore Heth 1 R&D Project, Mnstry of Heth & Wefre, Repbc of Kore. (-PJ-P-E-) REERENCES [1] S. H. Lee, B-J. Y nd Y.K. Kwk, Optm Knemtc Desgn of n Anthropomorphc Robot Mode wth Redndnt Acttors, Mechtroncs, o., No., pp. -, 199. [] J. H. Lee, B-J. Y, S-R. Oh nd. H. Sh, Optm Desgn nd Deveopment of ve-br nger wth Redndnt Actton, Mechtroncs, o. 11, No. 1, pp. -, 1. [] J. H. Lee, K. S. Eom, B-J. Y nd. H. Sh, Desgn of New -DO Pre Hptc Devces, Proc. nt. Conf. on Robotcs nd Atomton, pp.-91,. [] Y. ng nd L-W. s, Strctre Synthess of Css of -DO nd -DO Pre Mnptors wth dentc Lmb Strctres, Jorn of Robotcs Reserch, o. 1, No. 9, pp. 99-1,. [] J. Wng nd C. M. ossen, Knemtc Anyss nd Desgn of Knemtcy Redndnt Pre Mechnsms, Jorn of Mechnc Desgn, o. 1, No. 1, pp.19-11,. [] D. Zntnov nd C. M. ossen, A New Pre Archtectre wth or Degree of reedom, Proc. he nd workshop on Comptton knemtcs, pp.-, 1. [] Z. Hng nd Q. C. L, ype Synthess of Symmetrc Lower-mobty Pre Mechnsms Usng the Constrnt-synthess Method, Jorn of robotcs reserch, o., No. 1, pp. 9-9,. [] Q. C. L nd Z. Hng, Mobty Anyss of Nove -R Pre Mechnsm my, Jorn of Mechnc Desgn, o. 1, No. 1, pp. 9-,. [9] Q. C. L nd Z. Hng, A my of Symmetrc Lower-Mobty pre Mechnsms wth Spherc nd Pre Sbchns, Jorn of Robotc Systems o., No., pp-9-,. [1] O. Compny,. Mrqet, nd. Perrot, A New Hgh-Speed -DO Pre Robot Synthess nd Modeng sses, EEE rns. on Robotcs nd Atomton, o. 19, No., pp. 11-,.

6 CCAS [11] H. B. Cho, O. Compny,. Perrot, A. Konno,. Shbkw, nd M. Uchym, Desgn nd Contro of Nove -DOs Pre Robot H, Proc. nt. Conf. on Robotcs nd Atomton, pp ,. [1] D. Chbt nd P. Wenger, A New Concept of Modr Pre Mechnsm for Mchnng Appctons, Proc. nt. Conf. on Robotcs nd Atomton, pp.9-9,. Jne -, KNEX, yeongg-do, Kore