COMPLIANCE ANALYSIS, OPTIMISATION AND COMPARISON OF A NEW 3PUS-PU MECHANISM. B. Wei

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Intrnatonal Journal of Automotv and Mchancal Engnrng (IJAME) ISSN: 9-869 (Prnt); ISSN: 8-66 (Onln); Volum 7, pp. 9-99, Januar-Jun Unvrst Malasa Pahang DOI: http://d.do.org/.58/jam.7...9-99 COMPLIANCE ANALYSIS, OPTIMISATION AND COMPARISON OF A NEW PUS-PU MECHANISM B.

1 Intrnatonal Journal of Automotv and Mchancal Engnrng (IJAME) ISSN: (Prnt); ISSN: 8-66 (Onln); Volum 7, pp. 9-99, Januar-Jun Unvrst Malasa Pahang DOI: COMPLIANCE ANALYSIS, OPTIMISATION AND COMPARISON OF A NEW PUS-PU MECHANISM B. W Dpartmnt of Equpmnt Manufacturng, H Unvrst of Engnrng, 568, Handan, Chna E-mal: rav.987@6.com Phon: ABSTRACT Ths papr nvstgats th complanc of a nw PUS-PU hrd mchansm wth thr dgrs of frdom, ncludng translaton along th Z as and rotatons aout th X and Y as. Frstl, th knmatc analss of th mchansm s analsd and th complanc modl of th mchansm drvd. Scondl, th ffcts th gomtrc paramtrs and poston and orntaton paramtrs on th complanc of th mchansm n ach drcton ar nvstgatd, and th gntc algorthm s usd to optms th gloal complanc smultanousl adjustng dsgn varals. Fnall, th complanc of two smlar knds of PUS-PU mchansm n ach drcton s rvwd. Kwords: Paralll mchansm; complanc; gloal complanc complanc comparson. optmsaton; INTRODUCTION Ovr th past dcad, paralll mchansms hav n wdl usd n varous applcatons (Patl and Gorg, ), such as n machn tools (Hu t al., 6; Tsa and Josh, ; X t al., ; Zhang t al., 5), snsor dvcs (Zhang t al., 9), automotv ndustrs (Yu, ), mdcal dvcs (L and Xu, 8; Pan, ), mnng ndustrs (Wu t al., ), tc. Ths s largl du to th fact that paralll mchansms possss svral advantags compard to thr sral countrparts, such as a hgh stffnss, hgh prcson, hgh paload capact, good acclraton, tc. Ths attruts all contrut to th structur of paralll mchansms, n whch th movng platform s connctd to th as svral paralll lms. Stffnss or complanc s a vtal factor n valuatng th prformanc of paralll manpulators, snc a hgh stffnss can lad to hgh prcson (Zhang, 9). Hr a nw -DOF hrd mchansm s proposd, whr th movng platform s connctd to th as thr actv lgs and a cntral passv lg. Th complanc matr of th nw mchansm s drvd asd on th kntostatc modl and prncpl of vrtual work. Th ffcts th gomtrc paramtrs and poston and orntaton paramtrs on th complanc of th mchansm n ach drcton ar analsd. Th ladng dagonal lmnt of th complanc matr s th manpulator s pur complanc n ach drcton, and th sum of th ladng dagonal lmnts of th complanc matr s dfnd as th gloal complanc. Gloal complanc s rlatd to sstm rgdt; susquntl w optms th gloal complanc adjustng th dsgn varals. Fnall w compard th nw mchansm wth a mchansm n whch th gud-was ar ntrsctng ach othr, and w found that undr vr small structural paramtrs th complanc of th mchansm wth ntrsctng gud-was n ach 9

2 Complanc analss, optmsaton and comparson of a nw PUS-PU mchansm drcton ar gratr than thos of th nw mchansm wth paralll gud-was, whl undr largr structural paramtrs th complanc of th mchansm wth ntrsctng gud-was n ach drcton ar smallr than thos of th nw mchansm wth paralll gud-was. KINEMATIC ANALYSIS OF THE PUS-PU MECHANISM Ths nw PUS-PU has thr dgrs of frdom, whch ar a translaton along th Z as and rotatons around th X and Y as. Thr ar thr dntcal actv lgs and on cntral passv lg. Th actv lg conncts th movng platform a sphrcal jont, and th othr nd conncts to th unvrsal jont that s actuatd n th gud-wa. Th gud-was ar paralll to ach othr. Th cntral passv lg s fd on th as, and conssts of a prsmatc jont, anothr movng lnk and a unvrsal jont attachd to th cntr of th movng platform. Fg. provds a schmatc rprsntaton of th nw PUS-PU mchansm. For th purpos of analss, th fd coordnat fram O (X,Y,Z) s attachd to th cntr of th as, and th movng coordnat fram O (,,) s attachd to th cntr of th movng platform. Th radus of th as s R, th radus of th movng platform s r, and l s th lngth of th actv lgs. Th poston vctor of B wth rspct to th fd fram can prssd as P ; th poston vctor of E wth rspct to th movng fram can prssd as O wth rspct to th fd fram can prssd as P ; th poston vctor of s th angl twn th X as and th attachmnt pont s th angl twn th Z as and th attachmnt pont Thn on can otan th followng: P R cos R sn, P Th poston vctor of r sn, r cos P, P. B on th as, and E on th movng platform. 8 6, 6 8 E wth rspct to th fd fram can prssd as: rcos sn sn rsn cos P Q P P r cos sn r sn sn sn r cos cos whr Q s th rotaton matr of th movng fram wth rspct to th fd fram. Equaton () can dffrntatd as follows: r sn sn sn r cos cos r cos cos sn rsn sn r sn cos sn r cos sn r sn sn cos () 95

3 W /Intrnatonal Journal of Automotv and Mchancal Engnrng 7() 9-99 E O E E l D Z B D Y u B X O B B E O r O Y E E R B B X Fgur. Schmatc rprsntaton of th nw PUS-PU mchansm. Snc th actv lgs hav a fd lngth, t can shown that Th aov quaton lds: O E O B B D D E (=,,) () k k k l () whr k Rcos, k Rsn, k u u (,,) s th jont moton, whch can prssd as follows: u P (,) ( l ( P (,) P (,)) ( P (,) P (,)) ) () / 96

4 Complanc analss, optmsaton and comparson of a nw PUS-PU mchansm Equaton () can dffrntatd as: k k k u k k k l k k k l l l (5) whr k k Th twst n th movng platform s prssd as t and T k k k J l l l l k k k l J p t J l l l l k k k J l l l (6) KINETOSTATIC MODEL Th knmatc chan of th passv lg can consdrd as a sral manpulator. Th knmatc structur of th passv lg s llustratd n Fgur. From Fgur on can otan th D-H paramtrs of th passv lg, as shown n Tal. Z Y O X Fgur. Knmatc structur of th passv l 97

5 W /Intrnatonal Journal of Automotv and Mchancal Engnrng 7() 9-99 Tal. D-H paramtrs for th cntral passv lg. a d 9 9 Th Cartsan coordnat fram s usd as fram, so that, on otans: Q, thn (7) whr Q s th rotaton matr from th fd rfrnc fram to th frst fram of th passv lg, and Q, Q cos sn sn cos, Q cos sn sn cos Th rotaton matr can wrttn as: Q QQ Q Q Qs th rotaton matr from th frst fram to th scond fram of th passv lg; Q s th rotaton matr from th scond fram to th thrd fram of th passv lg; and Q s th rotaton matr from th thrd fram to th fnal fram of th passv lg; B multplng quaton (7) n ach quaton followng: Q ; QQ ; QQ Q whr T and r, on otans th Th poston vctors ar prssd as: r Qa QQ a QQ Q a r QQ a QQ Q a ; r QQ Q a (8) whr a a a d cos sn 98

6 Complanc analss, optmsaton and comparson of a nw PUS-PU mchansm For th cntral passv lg, on has th followng quaton: J t (9) T whr Th Jacoan matr of th cntral passv lg can prssd as follows: J r r () Accordng to th prncpl of vrtual work, th Cartsan complanc matr of th PUS-PU mchansm can drvd as follows (Zhang, ): C J ( J J ) C( J J ) T J T () c p p whr C dag[ c, c, c], c s th complanc of th th c =.. actuator, hr w st EFFECTS OF POSITION AND ORIENTATION PARAMETERS ON COMPLIANCE W fd 9, and usd th dsgn knmatc paramtrs whch ar as follows: R =.m; r =.5m; l =.5m; [ 5,5 ], [.,.7] m Thn, complanc mappng s otand n ach drcton as shown n Fgur. From Fgur on can s that whn =,.. th movng platform s paralll to th as, a mnmum complanc sts n th drcton, whch mans a mamum stffnss sts n th drcton. Th valu of, whch s th hght of th movng platform, dos not mpos a fundamntal nflunc on th complanc n th drcton. As ncrass, th complanc n th drcton ncrass..8 Complanc n (m) Fgur. Complanc n th Complanc n drcton along wth and. 99

7 W /Intrnatonal Journal of Automotv and Mchancal Engnrng 7() Complanc n (m) Complanc n Fgur. Complanc n th drcton along wth and n th From Fgur on can s that whn 7 drcton, whch mans a mamum stffnss sts n th, a mnmum complanc sts drcton. Th valu of, whch s th hght of th movng platform, dos not mpos a fundamntal nflunc on th complanc n th drcton. As ncrass, th complanc n th drcton ncrass. - - Complanc n Z (m) Complanc n Z Fgur 5. Complanc n Z drcton along wth and From Fgur 5 on can s that whn,.. th movng platform s paralll to th as, a mnmum complanc sts n th Z drcton, whch mans a mamum stffnss sts n th Z drcton. Th valu of dos not mpos a fundamntal nflunc on th complanc n th Z drcton. As ncrass, th complanc n th Z drcton ncrass. 9

8 Complanc analss, optmsaton and comparson of a nw PUS-PU mchansm EFFECTS OF GEOMETRIC PARAMETERS ON COMPLIANCE Th ffcts of R and r on complanc In addton w fd 9, and us th followng dsgn knmatc paramtrs: l =.5m;,.5m, R [.,.6] m, r [.5,.5] m Th complanc mappng along wth th chang n R and r can otand n Fgur 5. It can sn that whn R =. m, r =.5 m, a mnmum complanc sts n th drcton, whch mans a mamum stffnss sts n that drcton. From Fgur 7 on can s that th ffcts of R and r on th complanc n th drcton s th sam as n th drcton. From Fgur 8 on can s that whn R =. m, r =.5 m, th complanc n th Z drcton achvs th mnmum valu..6 Complanc n Complanc n r (m).5... R (m) R (m).6.. Complanc n r (m) Fgur 6. Complanc n th drcton along wth R and r. 9

9 W /Intrnatonal Journal of Automotv and Mchancal Engnrng 7() Complanc n.5..5 Complanc n r (m).5... R (m) R (m).6.. Complanc n Fgur 7. Complanc n th r (m) drcton along wth R and r Complanc n Z.5.5 Complanc n Z..8.5 r (m) R (m) R (m) Complanc n Z r (m) Fgur 8. Complanc n th Z drcton along wth R and r. 9

10 Complanc analss, optmsaton and comparson of a nw PUS-PU mchansm Th ffcts of l on complanc W fd 9, and usd th dsgn knmatc paramtrs whch ar as follows:,.5m, r.5m, R.m, l [.5,.65] m, thn w can otan th trnds n th complanc n ach drcton, along wth th chang n th lngth of th actv lgs as follows. Complanc n l (m) Complanc n l (m) Complanc n Z l (m) Fgur 9. Complanc n th, From Fgur 9 on can s that th complanc n th, Z drctons along wth l., and Z drctons rducs as l coms longr. Whn sttng 7,.5m, r.5m, R.m, l [.5,.65] m on has th followng trnds. From Fgur on can s that th complanc n th and Z drctons rducs as l coms longr, and th complanc n th drcton ncrass as l coms longr. Aftr man prmnts, w conclud that ths gomtrc paramtrs and th poston and orntaton paramtrs all affct th complanc of th mchansm n ach drcton. Th ladng dagonal lmnt of th complanc matr s th manpulator s pur complanc n ach drcton. Th sum of th ladng dagonal lmnts of th complanc matr s dfnd as th gloal complanc. Th gloal complanc s rlatd to sstm rgdt, 9

11 W /Intrnatonal Journal of Automotv and Mchancal Engnrng 7() 9-99 and w want th gloal complanc to as small as possl. Nt, w nd to dtrmn ths gomtrc paramtrs and th poston and orntaton paramtrs smultanousl to otan th mnmum summaton of th ladng dagonal lmnts of th complanc matr of th mchansm..6. Complanc n....8 Complanc n l l.9.85 Complanc n Z l Fgur. Complanc n th,, Z drctons along wth l OPTIMISATION Tradtonal optmsaton mthods us a local sarch a convrgnt stpws procdur, whch compars th valus of th followng ponts and thn movs to th optmal ponts. Gloal optma can onl found f th prolm has crtan convt proprts whch guarant an local optma s a gloal optmum It facs th dangr of fallng n local optma. Howvr, gntc algorthms ar asd on th populaton-to-populaton rul; whch can scap from local optma Gntc algorthms hav th advantags of good convrgnc and roustnss proprts: such as th followng, () Th rqur no knowldg or gradnt nformaton aout th optmsaton prolms; th can solv an knd of ojctv functons and an knd of constrants dfnd on dscrt, contnuous or md sarch spacs; () Dscontnuts prsnt on optmsaton prolms hav lttl ffct on th ovrall optmsaton prformanc; () It s ffctv at prformng gloal sarch nstad of local optma; (v) It prforms vr wll on larg-scal optmsaton prolms; (v) Th can mplod on a wd vart of optmsaton prolms. 9

Complanc analss, optmsaton and comparson of a nw PUS-PU mchansm Snc gloal complanc s rlatd to sstm rgdt, and th gomtrc paramtrs and poston and orntaton paramtrs all affct th complanc of th mchansm n

12 Complanc analss, optmsaton and comparson of a nw PUS-PU mchansm Snc gloal complanc s rlatd to sstm rgdt, and th gomtrc paramtrs and poston and orntaton paramtrs all affct th complanc of th mchansm n ach drcton as statd n sctons and 5, w wll us th gntc algorthm (GA) to optms th gloal complanc adjustng th gomtrc paramtrs and poston and orntaton paramtrs smultanousl. Th ojctv functon for optmsaton s as follows: C (,) C (,) C (,) C (,) C (5,5) C (6,6) () c c c c c c Dsgn varals ar R, r, l,, and Constrants ar st as follows: R [.,.6] m, r [.5,.5] m, l [.5,.65] m, [, ], [ 5,5 ], [.,.7] m Run th GA solvr, and th followng rsult s otand: Ftnss valu Bst:.875 Man:.878 Bst ftnss Man ftnss Currnt st ndvdual Gnraton Currnt Bst Indvdual Numr of varals (6) Fgur. Optmsaton rsults. 95

13 W /Intrnatonal Journal of Automotv and Mchancal Engnrng 7() 9-99 Aftr optmsaton, whn R.m, r.5m, l.59m, 89, 8.7,.57m, th complanc sum whch s th gloal complanc rachs th mnmum valu of.87. COMPLIANCE COMPARISON BETWEEN TWO PUS-PU MECHANISMS Th gud-was of th nw PUS-PU mchansm ar paralll to ach othr. Howvr, f th gud-was ar not paralll to ach othr, ut rathr ntrsctng ach othr as shown n Fgur (), hr w compar ths two mchansms complanc n ach drcton. Th angl twn th gud-was and th as s. E O E E l E O E E l l l D Z B D u O X B B Y u B D B D X Z u R O Y u B (a) Paralll gud-was () Intrsctng gud-was Fgur. Two knds of PUS-PU mchansm Whn 5, R.5m, r.5m, l.m, 9, [ 5,5 ], [.,.6] m, th complancs n ach drcton for th two mchansms ar shown n Fgur, th uppr plot fgur prsnts th complanc for th mchansm wth ntrsctng gud-was, and th lowr fgur prsnts th complanc for th nw mchansm wth paralll gud-was. On can clarl s that whn th structural paramtrs ar vr small, th complanc of th mchansm wth ntrsctng gud-was s largr than that of th nw mchansm wth paralll gud-was, whch mans that th stffnss of th nw mchansm s ttr than that of on wth ntrsctng gud-was. 96

14 Complanc analss, optmsaton and comparson of a nw PUS-PU mchansm Complanc n Z Complanc n Z (m) Complanc n Complanc n (m) Complanc n Complanc n (m) Fgur. Complanc comparson twn two mchansms whn th structural paramtrs ar small. Whn 5, R.7m, r.75m, [ 5,5 ], l.5m, 9, [.5,.6]m, th complanc n ach drcton of thos two mchansms ar prsntd n Fgur. On can also s that th complanc n ach drcton for th mchansm wth ntrsctng gud-was s smallr than thos of th nw mchansm wth paralll gud-was, whch mans th stffnss for th sstm wth ntrsctng gud-was s ttr than that of th nw mchansm wth paralll gud-was whn th structural paramtrs ar larg. 97

15 W /Intrnatonal Journal of Automotv and Mchancal Engnrng 7() Complanc n Z 5 Complanc n Z (m) Complanc n 6 Complanc n (m) Complanc n 5 Complanc n (m) Fgur. Complanc comparson twn two mchansms whn th structural paramtrs ar larg. CONCLUSION Th artcl nvstgats th complanc of a nw PUS-PU mchansm, and compars th complanc of two smlar PUS-PU mchansms. Th ffcts th gomtrc paramtrs and poston and orntaton paramtrs on th complanc of th nw PUS-PU mchansm n ach drcton s analsd, and th gntc algorthm s usd to optms th gloal complanc adjustng th dsgn varals. It s found that th 98

16 Complanc analss, optmsaton and comparson of a nw PUS-PU mchansm gloal complanc rachs th mnmum valu whn R=. m, r =.5 m, l =.59 m, = 89 o, = -8.7 o, =.57 m. Scondl, th complanc of two smlar knds of PUS-PU mchansm n ach drcton ar compard, and t s shown that th stffnss of th mchansm wth ntrsctng gud-was n ach drcton s smallr than that of th nw mchansm wth paralll gud-was whn th structural paramtrs ar vr small. Howvr th stffnss of th mchansm wth ntrsctng gud-was n ach drcton s largr than that of th nw mchansm wth paralll gud-was whn th structural paramtrs ar rlatvl larg, whch provds ngnrs wth a gud to slctng th corrct mchansm tp accordng to thr dffrnt practcal rqurmnts and scnaros, such as whthr t s to usd n machn tools or mcromanpulaton. For futur work, a prototp should farcatd asd on th proposd modl to tst th thortcal rsults. REFERENCES Hu, Y., L, B., Hu, H. and Yng, W. 6. Knmatcs, workspac and dtrt analss for a PUS-RPU paralll knmatc platform. Procdngs of th 7th Intrnatonal Confrnc on Frontrs of Dsgn and Manufacturng Scncs (ICFDM), pp L, Y.M. and Xu, Q.S. 8. Dsgn, analss and applcatons of a class of nw -DOF translatonal paralll manpulators. In: Ru, J.-H. (d.) Paralll Manpulators, Nw Dvlopmnt, pp Pan, M.. Improvd dsgn of a thr-dgr of frdom hp osklton asd on ommtc paralll structur. Mastr thss, Unvrst of Ontaro Insttut of Tchnolog, Canada. Patl, Y.D. and Gorg, P.M.. Paralll manpulators applcatons - a surv. Modrn Mchancal Engnrng, : Tsa, L.W. and Josh, S.. Knmatc analss of -DOF poston mchansms for us n hrd knmatc machns. Journal of Mchancal Dsgn, : 5-5. Wu, C., Lu, X.J., Wang, L.P. and Wang, J.S.. Dmnson optmaton of an orntaton fn-tunng manpulator for sgmnt assml roots n shld tunnlng machns. Automaton n Constructon, (): X, F.F., Zhang, D., Mchfsk, C.M. and Lang, S.Y.T.. Gloal kntostatc modlng of trpod-asd paralll knmatc machn. Mchansm and Machn Thor, 9: Yu, H.J.. Rsarch on paralll root asd fll fturs for automotv sht mtal assml. PhD thss, Harn Insttut of Tchnolog, Chna. Zhang, D.. Kntostatc analss and optmaton of paralll and hrd archtctur for machn tools. PhD thss, Laval Unvrst, Canada. Zhang, D. 9. Paralll rootc machn tools. Sprngr. Zhang, D., Gao, Z., Song, B. and G, Y.J. 9. Confguraton dsgn and prformanc analss of a multdmnsonal acclraton snsor asd on RRPRR dcouplng paralll mchansm. Jont 8th IEEE Confrnc on Dcson and Control and 8th Chns Control Confrnc, pp Zhang, D., Wang, L.H. and Lang, S.Y.T. 5. Paralll knmatc machns: dsgn, analss and smulaton n an ntgratd vrtual nvronmnt. Journal of Mchancal Dsgn, 7:

Heisenberg Model. Sayed Mohammad Mahdi Sadrnezhaad. Supervisor: Prof. Abdollah Langari

Heisenberg Model. Sayed Mohammad Mahdi Sadrnezhaad. Supervisor: Prof. Abdollah Langari snbrg Modl Sad Mohammad Mahd Sadrnhaad Survsor: Prof. bdollah Langar bstract: n ths rsarch w tr to calculat analtcall gnvalus and gnvctors of fnt chan wth ½-sn artcls snbrg modl. W drov gnfuctons for closd