More Effective Optimum Synthesis of Path Generating Four-Bar Mechanisms

Transcription

1 Junal f Multdscplnay Engneeng Scence and Technlgy (JMEST) ISSN: 59- Vl. Issue 5, May - 5 Me Effectve Optmum Synthess f Path Geneatng Fu-Ba Mechansms Wen-Y Ln Depatment f Mechancal Engneeng De Ln Insttute f Technlgy Tape, Tawan wyln@dlt.edu.tw Ku-M Hsa Depatment f Mechancal Engneeng Natnal Cha Tung Unvesty Hsnchu, Tawan kmhsa@cc.nctu.edu.tw Abstact The ne-phase synthess methd usng evlutnay heustc ptmzatn algthms can successfully slve the dmensnal synthess pblems f path geneatng fu-ba mechansms dscussed n the lteatue, whee the nput angles ae usually used as desgn vaables and the cnstant f the tatnal sequence f the nput angles s necessay. Hweve, thee s stll m f mpvement n slutn elablty and accuacy. T undestand the nfluence f the nput angles used as desgn vaables, a new ne-phase synthess methd wthut usng the nput angles as desgn vaables s pesented whee the squae devatn f the neaest dstance between the taget pnt and the cuple cuve s used as the e metc and the dffeental evlutn s used t slve the ptmum synthess pblems. A scale-tatn-tanslatn tansfmatn technque, whch can tansfm the cuple cuve f the ntal guess neae t the desed cuve n a smla scalng, entatn and lcatn, s ppsed. Fndngs shw that the new ne-phase synthess methd can mpve slutn accuacy makedly. Meve, the scale-tatn-tanslatn tansfmatn technque f the ntal guess s effectve t mpve slutn elablty, accuacy effcency. Keywds evlutnay algthm; dmensnal synthess; mechansm ptmzatn; path geneatn; scale-tatn-tanslatn tansfmatn I. INTRODUCTION The dmensnal synthess pblem f the path geneatn f plana mechansms (heen temed path synthess pblem) s t detemne the dmensns f a defnte mechansm whse cuple pnt can be used t tace a desed path taget pnts. The cntnuus path may be epesented by a sequence f dscete pnts. The methds f path synthess pblems dscussed n the lteatue may be dvded nt tw categes. The fst categy s the dect synthess methd wthut utlzng atlas database [-]; the secnd categy may be egaded as the ndect synthess methd usng (cmputezed) atlas database. The dect synthess methd may be futhe dvded nt tw subcateges. The fst subcategy s ne-phase synthess [-] and the secnd subcategy s tw-phase synthess [5-] f the dect synthess methd. The tw-phase synthess methd fst handles the shape synthess and then handles the scale-tatn-tanslatn synthess. In the tadtnal ne-phase synthess methd, the pblem may be cnsdeed a mechansm ptmzatn wth the gal f mnmzng an bjectve functn. The mst cmmn bjectve functn s the sum f the squae f the dstance between the desed pnt and the cespndng cuple pnt (heen temed the squae devatn f the dstance). The technque f the tadtnal ne-phase ptmum synthess whee the squae devatn f the dstance s used as the bjectve functn and the nput angles f the cank seve as desgn vaables s based n the pstn nfmatn f the desed pnts. In the wds, the technque f the ptmum synthess nt nly attempts t smultaneusly satsfy the shape, sze, lcatn and entatn nfmatn f the desed path but als t btan the nput angles f the cank,.e., the tme nfmatn egadng the desed pnts. The numbe f desgn vaables f such a technque nceases wth the ncement n the numbe f the desed pnts. F example, thee ae a ttal f desgn vaables f the path synthess f a fu-ba lnkage wth 5 desed pnts and wthut pescbed tmng. Hweve, thee ae 5 desgn vaables f the nput angles f the cank cespndng t the 5 desed pnts. The ppulatn-based heustc ptmzatn methds ae wdely used snce they ae smple, effectve and easy t mplement f slvng cmplcated eal-wld ptmzatn pblems. In addtn, thee s n need f futhe demands petanng t the gadent f the bjectve functn. Futheme, f the bjectve functn f an ptmzatn task s nt expessed as an explct functn f the desgn vaables, t cmplcated t manpulate nn-dffeentable, the ppulatn-based heustc ptmzatn methds ae me sutable t slve such an ptmzatn task than the tadtnal detemnstc ptmzatn methds. The ppulatn-based heustc ptmzatn methds have tw mptant gups: evlutnay algthms and swam ntellgence based algthms. As dscussed n [6-,], the tadtnal ne-phase synthess methd usng evlutnay algthms swam ntellgence based algthms can successfully slve dffcult path synthess pblems. Hweve, the btaned values f the bjectve functn ae scatteed and the standad devatn s nt small f a cetan epeated uns, whch can be seen fm []. The mean value f the bjectve functn can be JMESTN569 95

2 much hghe than the best value f the bjectve functn f a cetan epeated uns [,]. Theefe, thee s stll m f mpvements n the mean value and the standad devatn (cnsdeed as elablty n ths study), the best value (cnsdeed as accuacy n ths study) the numbe f evaluatns f the bjectn functn (cnsdeed as effcency n ths study) f the ne-phase synthess methd usng evlutnay algthms swam ntellgence algthms. Besdes the easn f the emplyed ptmzatn algthm, the abve-mentned scatteed esults can be attbuted t the tw man easns descbed belw. One s the use f the nput angles as desgn vaables and the asscated cnstant f the tatn sequence f the nput angles. The the s the cnstant f Gashf s cndtn. T the auth s best knwledge, the nfluence f the nput angles used as desgn vaables and the asscated cnstant f the tatn sequence f the nput angles n the slutn qualty has nt been epted n the lteatue. In ths study, t nvestgate the nfluence f the nput angles n the slutn qualty, a new ne-phase synthess methd usng the sum f the squae f the neaest dstance between the taget pnt and the cuple cuve (heen temed the squae devatn f the neaest dstance) as the e metc s ppsed f the path synthess pblems wthut pescbed tmng. The use f the neaest dstance can avd the need f the nput angles as desgn vaables and als avd the cnstant f the tatn sequence f the nput angles f path synthess pblems wthut pescbed tmng. F the path synthess pblems wth pescbed tmng, the new ne-phase synthess methd and the tadtnal ne-phase synthess methd ae the same, snce the nput angles f the cank s specfed. Meve, n ths study, t mpve the slutn qualty f the ppsed ne-phase synthess methd f clsed paths, a scale-tatntanslatn tansfmatn technque, whch s gnally used n the secnd synthess f the tw-phase synthess methd, s executed f the ntal guess upn ntalzatn. The scale-tatn-tanslatn tansfmatn technque can tansfm the cuple cuves f the ntal guess neae t the desed cuve n a smla scalng, entatn and lcatn n de t execute the gal f mnmzng the squae devatn f the neatest dstance. T handle the cnstant f Gashf s cndtn, we adpt Deb s heustc cnstaned handlng methd [] whch uses a tunament selectn (suvv selectn) peat. Thee heustc ules ae shwn as fllws. Fst, f ne slutn s feasble and the the s nfeasble, then the feasble slutn s selected. Secndly, f bth the slutns ae feasble, then the slutn wth the bette value f the bjectve functn s selected. Lastly, f bth the slutns ae nfeasble, then the slutn wth the least cnstant vlatn s selected. The last ule can be neglected wth the help f the scheme [6,7] that the mechansm lengths ae eassgned untl the Gashf cndtn s satsfed dung ntalzatn. The mst well-knwn heustc Junal f Multdscplnay Engneeng Scence and Technlgy (JMEST) ISSN: 59- Vl. Issue 5, May - 5 ptmzatn methd usng the suvv selectn s the dffeental evlutn (DE) [], and the DE algthm s used t slve the ptmum path synthess pblems n ths study. The effectveness f the ppsed nephase synthess methd wthut wth the scaletatn-tanslatn tansfmatn technque s demnstated usng the cmpasns wth the cuent best esults dscussed n the lteatue f seveal epesentatve path synthess pblems f fu-ba mechansms wth wthut pescbed tmng based n the best value, the mean value and the standad devatn f the bjectve functn ut f 5 epeated uns. II. PROBLEM FORMULATION Fgue depcts a knematc stck dagam and all the gemetc paametes f the fu-ba mechansm. The sum f the squae f the neaest dstance ( the dstance e) between the taget pnt and the cespndng cuple pnt s ppsed as the e metc ( f bj ) f path synthess pblems wthut ( wth) pescbed tmng. The path synthess pblem s cnsdeed a mechansm ptmzatn wth the gal f mnmzng the e metc. Mnmze N fbj [( X X d ) ( Y Yd ) ] () whee ( X d, Yd ) s the pescbed cdnates f the - th taget pnt, and ( X, Y ) s the cdnates f the cespndng -th cuple pnt f path synthess pblems wth pescbed tmng, the cdnates f the cespndng cuple pnt neaest t the -th taget pnt f path synthess pblems wthut pescbed tmng; N s the numbe f the taget pnts. Y y O Y x C Fg.. Fu-ba mechansm n the glbal cdnate system. The pstn equatn f the cuple pnt s expessed by X X JMESTN569 96

3 X cs sn sn cs Y x y cs sn cx cx cs sn cy cy sn cs Thee ae nne desgn vaables,,,, cx, cy, x, y and t be ptmzed f path synthess pblems wthut pescbed tmng, whch des nt need nput angles as desgn vaables. In addtn t the nne desgn vaables, the nput angles,, cespndng t the fst pstn may be cnsdeed as desgn vaables f path synthess pblems wth pescbed tmng n ths study. Thus, the desgn vaable vect can be expessed as: X x, x,..., x } (n s the numbe f the desgn p { n vaables). The cdnates f the cuple pnt f path synthess pblems wth pescbed tmng can be btaned fm Eq. (), whee angle can be slved by the Feudensten equatn []. In de t use Eq. () and the Feudensten equatn t detemne the cdnates f the cuple pnt neaest t the taget pnt f path synthess pblems wthut pescbed tmng, the cespndng nput angle must be slved fst. The nput angle cespndng t the extema f the vaatn f the dstance between a taget pnt and cuple pnts s detemned by the fllwng equatn, whch s deved by dffeentatng the dstance wth espect t. whee () ( X Xd ) X ( Y Yd ) Y () ' X [ cx sn( ) cy cs( )] sn( ) Y [ cx cs( ) cs( ) cy ' sn( )] () (5) ' d [ sn( ) sn ] (6) d [ sn( ) sn] Equatn () can be slved usng the bsectn methd and the maxma f dstances between a taget pnt and cuple pnts can be excluded when the fllwng nequalty s satsfed. X Y ( X X d ) X ( Y Y d ) Y (7) In ths study, the fllwng tw cnstants ae cnsdeed: ) the desgn vaables ae wthn the specfed anges. ) the Gashf cndtn s satsfed t allw f the ente tun f at least ne lnk. It may be expessed by [max(,,, ) mn(,,, )] ( ) (8) Junal f Multdscplnay Engneeng Scence and Technlgy (JMEST) ISSN: 59- Vl. Issue 5, May - 5 The value f a desgn vaable s andmly geneated usng a unfm dstbutn dung ntalzatn. If the value btaned by the mutatn peatn f DE s nt wthn the pescbed ange, we use a smple methd whch sets the vlatng value t be the mddle f the vlated bund and the cespndng value f the paent ndvdual []. Dung ntalzatn the vaables f mechansm lengths ae eassgned untl the Gashf cndtn s satsfed. The cnstant f Gashf s cndtn can be handled usng Deb s heustc cnstaned handlng methd, as ntduced n Sectn. III. DE ALGORITHM DE s a well-knwn evlutnay algthm f eal paamete ptmzatn. The ntal desgn vaable vect s andmly geneated usng a unfm dstbutn. Afte ntalzatn, a lp f evlutnay peatns cntanng mutatn, cssve and selectn s mplemented. A. Mutatn A dn vect V { v, v,..., vn} cespndng t each taget (paent) vect X p f the next geneatn s btaned accdng t the mutatn peatn. The thee mst fequently used mutatn stateges ae lsted n the fllwng: ) DE/and/ V X F( X X ) (9) ) DE/and/ V Xbest F( X X ) () ) DE/cuent-t-best/ V X F X X ) F( X X ) () p ( best p whee the DE/x/y/z wth x: bject t be dstubed, y: numbe f dffeental vects, z: cssve way (nt shwn); the ndces, and ae dstnct nteges unfmly chsen fm [, N p ] ( N p s the ppulatn sze); and a scalng fact F s a use suppled cnstant n the ange (,+). B. Cssve Afte mutatn, a bnmal cssve peatn fms the tal (new) vect X p, new { u, u,..., un} v u x f thewse and(,) C () whee and(,) s a unfm andm numbe n the nteval [,]; s a andmly chsen ntege fm [, n], whch ensues that the tal vect gets at least ne cmpnent fm the dn vect; C s the cssve at. JMESTN569 97

4 C. Selectn The supe X p, new X P s selected as the ffspng vect and then used as the paent vect n the next geneatn. IV. SCALE-ROTATION-TRANSLATION TRANSFORMATION FOR A CLOSED PAT In ths study, a scale-tatn-tanslatn tansfmatn technque usng the ppety f a plygnal cuve f a clsed path s ntduced t tansfm the cuple cuve f the ntal ppulatn neae t the desed cuve n a smla scalng, entatn and lcatn n de t execute the gal f mnmzng the squae devatn f the neatest dstance. The ppsed e metc cmbned wth the gemetc tansfmatn technque f a clsed path may lead t an mpvement n slutn qualty. Thee ae tw gemetc-based appaches ppsed by Smal and Dab [7] and Buśkewcz et al. [8] ndvdually f the secnd synthess f the tw-phase synthess methd f a clsed path. Buśkewcz et al. [8] uses a gemetc-based pcedue (tanslatn fst, fllwng by tatn and fnally scalng) afte shape ptmzatn t cmplete the secnd synthess, whch uses the ppetes f the centd, the dectn f the maj pncpal axs and the pemete f a plygnal cuve t tanslate, tate and scale the mechansm t the desed cnfguatn, espectvely. If tanslatn s pefmed befe tatn, then a secnd tanslatn pcess shuld be pefmed afte tatn because a pue tatn f the mechansm abut the pvt causes tatn and tanslatn f the cuple cuve. In the wds, tatn synthess shuld be pefmed befe tanslatn synthess s as t avd a secnd tanslatn pcess [7,]. Fst, a scale tansfmatn s pefmed, ne may have the scale f a clsed path and Sd s () S X = [ s, s, s, s, whee s cx, s cy, x, y,, ] () S d s the pemete f plygnal cuve C d cnnectng the taget pnts, and S s the pemete f plygnal cuve C cnnectng cuple pnts. In ths study, a plygnal cuve C cnnectng 8 cuple pnts fm t wth an ncement f / 8 ae used t epesent the cuple cuve. Secndly, a tatn tansfmatn s pefmed and the vaable s detemned by and p p f ( I y I x )( I y I x ) (5) p ( p / ) f ( I y I x )( I y I x ) Junal f Multdscplnay Engneeng Scence and Technlgy (JMEST) ISSN: 59- Vl. Issue 5, May - 5 p p whee tan tan p and p I xy ( ) I y I x I xy ( ) I I axs f plygnal cuves y tansfmatn, espectvely; and I x I xy x (6) ae the angles defnng a pncpal C d and C afte scale ( I x ), I y ( I y ) and ( I xy) ae the secnd mment wth espect t the centdal axes f plygnal cuve scale tansfmatn. C d ( C ) afte Lastly, a tanslatn tansfmatn s pefmed and the vaables x and y ae detemned by x x c xc y y c yc (7) V. IMPLEMENTATION OF THE PROPOSED NEW ONE- PHASE SYNTHESIS METHOD The pcedue f the ppsed new ne-phase synthess methd f the clsed path s descbed belw. ) Cmpute the pemete, the entatn f the pncpal axs and the centdal pstn f the plygnal cuve cnnectng the desed pnts. ) Intalzatn: Randmly geneate X p { x, x,..., xn}, whee the desgn vaables x n each X P s gven by x mn( x ) and(,) (max( x ) mn( x )). F any X P, the vaables f mechansm lengths ae eassgned untl Gashf s cndtn s satsfed. ) Pefm the scale-tatn-tanslatn tansfmatn f the ntal ppulatn. ) Calculate the ftness value f X P and dentfy thus fa, f necessay. Xbest 5) Cmpute the dn vect V { v, v,..., vn} usng a mutatn stategy. 6) Check the bunday cnstants f each v. Let the vlatng value t be the mddle f the vlated bund and the cespndng value f the paent ndvdual. 7) F any X P, cmpute the tal (new) vect X p, new { u, u,..., un} usng a bnmal cssve peatn. If dnated t X p, new s supe t X P, X p, new s X P. Othewse, XP emans unchanged. 8) Calculate the ftness value f X P and dentfy thus fa. Xbest JMESTN569 98

5 9) If the temnatn ctea ae satsfed, the ptmal slutn s btaned. Othewse, etun t step 5. VI. RESULTS AND DISCUSSION The effectveness f the new ne-phase synthess methd the scale-tatn-tanslatn tansfmatn technque f a clsed path n slutn accuacy (the best value f the bjectve functn) elablty (the mean value and the standad devatn f the bjectve functn) at a cetan numbe f evaluatns f the bjectve functn s demnstated usng 5 epeated uns n fu epesentatve path synthess pblems wth and wthut pescbed tmng. The use-suppled paametes f the DE algthm ae as fllws: ndvduals (ppulatn sze) f pblems and and ndvduals f pblems, and 5, geneatns, scalng fact F =., cssve at C =.9. In ths study, the esults btaned usng DE/and/ s manly shwn because DE/and/ can usually btan me elable slutns, except that the DE/best/ DE/cuent-t-best/ can btan bette accuacy. Because the synthess pblem slved des nt belng t a eal-tme cmputng pblem, the executn tme f btanng ptmum esults s nt shwn. A. Pblem The synthess pblem wth sx taget pnts aanged n an vetcal staght lne and wthut pescbed tmng has been studed by Achayya and Mandal [6], Ln [7], Cabea et al. [8] and Otz et al. []. Taget pnts: C d =[(,),(,5),(,),(,5),(,),(,5)]. Lmts f the vaables:,,, x, y [-6,6]., [, ]. [5,6]; cx, cy, The numbe ( N ) f evaluatns f the bjectve functn, the synthess desgn vaables and the best value (Best), the mean value (Mean) and the standad devatn (SDev) f the bjectve functn btaned usng the ppsed e metc ae shwn n Table, tgethe wth the synthess slutns btaned by Cabea et al. [8] (usng mdfed DE) and Otz et al. [] (usng new DE wth aut-adaptve cntl paametes). F thee mutatn stateges, DE/and/ btans the best elablty and the wst accuacy; cnvesely, DE/best/ btans the best accuacy and the wst elablty. It can be seen that the best value f the bjectve functn btaned usng the ppsed e metc (the squae devatn f the neaest dstance) wth 9 desgn vaables and the DE/best/ 7 algthm s f the de f, n cntast wth the de f btaned usng the tadtnal e metc (the squae devatn f the dstance) wth 5 desgn vaables dscussed n the lteatue. Theefe, the ppsed new ne-phase synthess methd can Junal f Multdscplnay Engneeng Scence and Technlgy (JMEST) ISSN: 59- Vl. Issue 5, May - 5 mpve the slutn accuacy makedly. The nput angles (,,6) f the cank f the pesent slutns cespndng t the cuple pnt clsest t the taget pnt ae shwn n Tables and. TABLE I. SYNTHESIS RESULTS FOR PROBLEM Pesent wthut DE/and/ DE/best/ Cabea et al. [8] N p Otz et al. [] N,,,, cx cy x y Best.6E-5.5E-7.57E-.7E- Mean SDev TABLE II. INPUT ANGLES OF THE CRANK OF THE PRESENT SOLUTION USING DE/RAND/ FOR PROBLEM TABLE III. INPUT ANGLES OF THE CRANK OF THE PRESENT SOLUTION USING DE/BEST/ FOR PROBLEM JMESTN569 99

6 Junal f Multdscplnay Engneeng Scence and Technlgy (JMEST) ISSN: 59- Vl. Issue 5, May B. Pblem The synthess pblem wth taget pnts aanged n an ellpse and wthut pescbed tmng has been studed by Achayya and Mandal [6], Ln [7], Cabea et al.[8], Otz et al. []. Taget pnts: [(,),(7.66,5.),(.76,7.878),(5,6.98),(. 67,.76),(.67,7.68),(5,.78),(.76,. 5),(7.66,.8577),(,)]. Lmts f the vaables:,,, x, y [-8,8]., [, ]. [5,8]; cx, cy, The numbe f evaluatns f the bjectve functn, the synthess desgn vaables and the best value, the mean value and the standad devatn f the bjectve functn btaned usng the ppsed ne-phase synthess methd ae shwn n Table, tgethe wth the synthess slutns btaned by Cabea et al. [8] and Otz et al. []. The dffeences n accuacy and elablty amng the thee DE mutatn stateges ae small. It can be seen fm Table that the best value f the bjectve functn btaned usng the ppsed e metc wth 9 desgn vaables and the DE/and/ algthm s f the de f, n cntast wth the de f t btaned usng the tadtnal e metc wth 9 desgn vaables dscussed n the lteatue. Theefe, the ppsed new ne-phase synthess methd can mpve the slutn accuacy makedly agan. In cntast t the slutn accuacy, the slutn elablty f the ppsed new ne-phase synthess methd wthut s p. Hweve, wth the help f the ppsed the scale-tatn-tanslatn tansfmatn technque, the values f Mean and SDev f the ppsed new ne-phase synthess methd can be educed t be the de f fm the de f. It can be als seen that n slutn accuacy and elablty the ppsed new ne-phase synthess methd wth s much supe t Otz et al. []. The nput angles f the cank f the pesent slutn cespndng t the cuple pnt clsest t the taget pnt ae shwn n Table 5. TABLE IV. SYNTHESIS RESULTS FOR PROBLEM DE/and/ wthut Pesent DE/and/ wth Cabea et al. [8] Otz et al. [] N p N,,,, cx cy x y Best.787E-.555E-.7.9 Mean E-.7 SDev E-.66 TABLE V. INPUT ANGLES OF THE CRANK OF THE PRESENT SOLUTION WITH FOR PROBLEM C. Pblem Ths s a vey epesentatve path synthess pblem wth 8 taget pnts and pescbed tmng. Taget pnts: [(.5,.),(.,.),(.,.),(.,.),(.,.9),(.5,.7 5),(.,.6),(.,.5),(.,.),(.,.),(.,.5),(.5,.),(.,.),(.,.),(.,.5),(.5,.7),(.6,.9),(.6,.)]. Pescbed tmng: ( ), = t 8. 9 Lmts f the vaables:,,, x, y [-,]., [, ]. [,]; cx, cy, The pblem has been studed by many eseaches and the cuent best esults ae btaned by Penunu et al. [9]. The dffculty f the pblem JMESTN569 9

7 cnssts n the qute uneven allcatn f the taget pnts. In the wds, the cuve cnnectng the taget pnts s nt smth. The dffculty can als be seen fm the study that Penunu et al. [9] used DE/and/ algthm wth ndvdual and,87 geneatns t btan the best value f the bjectve functn,.e., Best = 9.88E-. T futhe mpve the accuacy, they use ndvduals and, t btan Best = 9.6E-. Meve, by makng a thd efnement f the seach space (the fst tw Best values ae btaned by educng the seach space tw tmes), they btaned Best= 9.E-, as epted n [9]. Nte that f the path synthess pblems wth pescbed tmng, the new ne-phase synthess methd and the tadtnal nephase synthess methd ae the same and that the tw-phase synthess methd cannt be appled t the path synthess pblems wth pescbed tmng. The numbe f evaluatns f the bjectve functn, the synthess desgn vaables and the best value, the mean value and the standad devatn f the bjectve functn btaned usng the ppsed ne-phase synthess methd ae shwn n Table 6, tgethe wth the synthess slutns btaned by Penunu et al. [9]. It can be seen that the best value f the bjectve functn btaned usng the ppsed ne-phase synthess methd wthut and wth geneatns s hghe abut.6% than that btaned by Penunu et al. [9]. T btan me an accuate slutn, we use geneatns and btan the almst same accuacy wth [9]. In cntast wth the numbe f evaluatns f the bjectve functn,.e.,, f ppsed ne-phase synthess methd wthut and,6, f [9], the slutn effcency f the ppsed ne-phase synthess methd wthut s supe t that f [9]. Ths wk and [9] use the same e metc and the DE/and/ algthm; hweve, the dffeence f the slutn effcency s cnsdeable. Ths mght be attbuted t the scalng fact, the cssve at and the handlng f the bunday cnstants. They used a andm numbe f the scalng fact and. f the cssve at. It als can be seen fm Table 6 that the slutn elablty (Mean and SDev) f the ppsed ne-phase synthess methd can be mpved usng the scaletatn-tanslatn tansfmatn technque. It s wth ntng that the slutn effcency ( N ) f the ppsed ne-phase synthess methd s als mpved usng the scale-tatn-tanslatn tansfmatn technque f the almst same accuacy (.9 and.99). In fact, the Best value s acheved at 5 geneatns and theefe the numbe f evaluatns f the bjectve functn s nly, f DE/and/ wth. By the way, the 8-pnt path synthess pblem wth pescbed tmng has been studed by Ln [7], but the sxth taget pnt n [7] s (.5,.75). The slutns btaned by Ln have been checked usng the SldWks D sketch [7]. Theefe, the cntvesy f dffeences n the values f the bjectve functn Junal f Multdscplnay Engneeng Scence and Technlgy (JMEST) ISSN: 59- Vl. Issue 5, May - 5 dscussed by Penunu et al. [9] ases fm the tw dffeent sxth taget pnts f (.5,.75) and (.5,.75), nt undng es. TABLE VI. SYNTHESIS RESULTS FOR PROBLEM DE/and/ wthut Pesent DE/and/ wthut DE/and/ wth Penunu et al. [9] N p N,,,,6, cx cy x y Best Mean SDev D. Pblem Pblem s the same wth pblem but wthut pescbed tmng. The pblem was studed by Penunu et al. [9]. The numbe f evaluatns f the bjectve functn, the synthess desgn vaables and the best value, the mean value and the standad devatn f the bjectve functn btaned usng the ppsed ne-phase synthess methd ae shwn n Table 7, tgethe wth the synthess slutns btaned by Penunu et al. [9]. It can be seen that the best value f the bjectve functn btaned usng the ppsed ne-phase synthess methd wthut and wth JMESTN569 9

8 geneatns s hghe abut.% than that btaned by Penunu et al. [9]. T btan me an accuate slutn, we use geneatns and btan Best =.7E-, abut.% smalle than that btaned n [9]. Meve, n cntast wth the numbe f evaluatns f the bjectve functn, the slutn effcency f the ppsed ne-phase synthess methd wthut s supe t that f [9]. It als can be seen fm Table 7 that the slutn accuacy, elablty and effcency f the ppsed ne-phase synthess methd f ths pblem can be mpved usng the scale-tatn-tanslatn tansfmatn technque. The nput angles f the cank f the pesent slutn cespndng t the cuple pnt clsest t the taget pnt ae shwn n Table 8. TABLE VII. SYNTHESIS RESULTS FOR PROBLEM DE/and/ wthut Pesent DE/and/ wthut DE/and/ wth Penunu et al. [9] N p N,,, cx cy x y Best Mean SDev TABLE VIII. INPUT ANGLES OF THE CRANK OF THE PRESENT SOLUTION WITH FOR PROBLEM 5 Junal f Multdscplnay Engneeng Scence and Technlgy (JMEST) ISSN: 59- Vl. Issue 5, May VII. CONCLUSIONS The ppsed new ne-phase synthess methd can mpve the slutn accuacy makedly f the synthess pblems f pen clsed paths. Wth the help f the scale-tatn-tanslatn tansfmatn technque, the ppsed new ne-phase synthess methd can mpve the slutn elablty, even accuacy effcency f the synthess pblems f clsed paths. The mpvement f the slutn elablty f the synthess pblems f pen paths shuld be a futhe wk. REFERENCES [] A.P. Mgan, C.W. Wample, Slvng a plana fuba desgn pblem usng cntnuatn, Junal f Mechancal Desgn (99) [] S. Kshnamuty, D.A. Tucc, Optmal synthess f mechansms usng nnlnea gal pgammng technques, Mechansm and Machne They 7(99) [] J. Maappan, S. Kshnamuty, A genealzed exact gadent methd f mechansm synthess, Mechansm and Machne They (996) -. [] E.C., Knzel, J.P. Schmedele, G.R. Pennck, Knematc synthess f fntely sepaated pstns usng gemetc cnstant pgammng, Junal f Mechancal Desgn 8 (6) [5] G.R. Pennck and A. Isa, Knematc analyss and synthess f an adjustable sx-ba lnkage, Mechansm and Machne They (9) 6-. [6] S. Achayya, M. Mandal, Pefmance f EAs f fu-ba lnkage synthess, Mechansm and Machne They (9) [7] W.-Y. Ln, A GA-DE hybd evlutnay algthm f path synthess f fu-ba lnkage, Mechansm and Machne They 5 () [8] J.A. Cabea, A. Otz, F. Nadal, J.J. Castll, An evlutnay algthm f path synthess f mechansms, Mechansm and Machne They 6 () 7. JMESTN569 9

9 [9] F. Peñuñu, R. Peón-Escalante, C. Vllanueva, D. Pech-Oy, Synthess f mechansms f sngle and hybd tasks usng dffeental evlutn, Mechansm and Machne They 6 () 5-9. [] W.Y. Ln, Optmum path synthess f a fulnk mechansm wth llng cntacts, Pceedngs f the Insttute f Mechancal Engnees, Pat C: Junal f Mechancal Engneeng Scence, 6 (), [] W.-Y. Ln, Optmum path synthess f a geaed fve-ba mechansm, Advances n Mechancal Engneeng, 5 (), [] A. Otz, J.A. Cabea, F. Nadal, A., Bnlla, Dmensnal synthess f mechansms usng Dffeental Evlutn wth aut-adaptve cntl paametes, Mechansm and Machne They 6 () -9. [] J.P. Schmedele, B.C. Clak, E.C. Knzel, G.R. Pennck, Knematc synthess f nfntesmally and multply sepaated pstns usng gemetc cnstant pgammng, Junal f Mechancal Desgn 6 () 5. [] S. Ebahm and P. Payvandy, Effcent cnstaned synthess f path geneatng fu-ba mechansms based n the heustc ptmzatn algthms, Mechansm and Machne They 85 (5) 89-. [5] I. Ullah, S. Kta, Optmal synthess f mechansms f path geneatn usng Fue descpt and glbal seach methds, Junal f Mechancal Desgn 9 (997) 5-5. [6] S. Dbaka, T.S. Muthyunjaya, Synthess f wkspaces f plana manpulats wth abtay tplgy usng shape epesentatn and smulated Junal f Multdscplnay Engneeng Scence and Technlgy (JMEST) ISSN: 59- Vl. Issue 5, May - 5 annealng, Mechansm and Machne They (999) 9-. [7] A. Smal, N. Dab, A new appach t shape ptmzatn f clsed path synthess f plana mechansms, Junal f Mechancal Desgn 9 (7) [8] J. Buskewcz, R. Stasta, T. Walczak, On the applcatn f the cuve cuvatue n path synthess, Mechansm and Machne They (9) -9. [9] J. Buśkewcz, Use f shape nvaants n ptmal synthess f geaed fve-ba lnkage, Mechansm and Machne They 5 () 7-9. [] W.-Y. Ln, Optmum synthess f plana mechansms f path geneatn based n a cmbned dscete Fue descpt, Junal f Mechansms and Rbtcs Tansactns f the ASME (5) Accepted, d:.5/.58. [] K. Deb, An effcent cnstant handlng methd f genetc algthm, Cmpute Methds n Appled Mechancs and Engneeng 86 () 8. [] K. Pce, R. Stn, J. Lampnen, Dffeental Evlutn-A Pactcal Appach t Glbal Optmzatn, Spnge-Velag, Beln Hedelbeg, 5. [] Fedensten, An analytcal appach t the desgn f fu-lnk mechansms, Tansactns f the ASME 76 (95) 8-9. [] W.-Y. Ln, Optmzatn f scale-tatntanslatn synthess afte shape synthess f path geneatn f plana mechansms, Junal f the Chnese Insttute f Engnees 7 () JMESTN569 9