The Optimum Kinematic Design of a Spatial Nine-Degree-of-Freedom Parallel Manipulator
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1 ptmm Knmt Dgn of Spt Nn-Dg-of-Fdom P Mnpto ANNIN GALF nd RSARI SINARA Dptmnto d Inggn Indt Mn Unvtà d Ctn V A. Do, 955 Ctn IALY Att: - pp ond t nmt optmton of p mnpto ttd n dndnt fom, ng nm ondtonng of t on mt. nmt mod popod Zngn nd Ang 99 ont of nn pmt n-p tto,.., t tto, d tn g, onnt t movng ptfom dt to t ptfom p jont, nd t tto, tmd t pp ntn nd t ow ntn g, opd n p t onnt p jont t t ntn pont. Ft, t nd ffto vot pom ovd, nd tn t dntfton of nt onfgton of mnpto dvopd. Fn, t ptfom gomt pmt of t mnpto fomtd, nd ovd n optmton pom. K-Wod: - P dndnt mnpto, nmt nd, optmtton dgn pom. Intodton optm dgn fo p tt n opn nd dfft qton. A Mt notd n [], t v mpotnt to dgn p tt n odn wt t t to w t gnd, n pt fo mnm w t pfomn tt dpndnt on t mn tt nd t dmnon. In t oton of t pom, t ft dfft tt mot pfomn t ondd fo pt ppton tmm to fnton tt onfgton-dpndnt tffn, fo n t tto, dtt, t.. ond dfft tt of ng n ppopt dgn mtodoog to otn n optmm tt of t mnm fo t gnd t. Nmo td v nvtgtd t pom, nd on tntv to t ppo tt pntd Mt; t o-d pmt p ppo []. In t pp, t optmm dgn pom ondd fo nw dndnt p mnpto wt nn dg of fdom popod Zngn nd Ang n [, ] nd own n Fg.. mod ont of nn pmt n-p tto, nd q fo,,, nd, nd ˆ fo,, t nt vto ong A B, B nd A, ptv. Moov, t t tto { }, d tn g, onnt t movng ptfom dnotd MP dt to t ptfom dnotd BP p jont t { A } nd { B}. tto { q} nd { }, tmd t pp ntn nd t ow ntn g ptv, opd n p t onnt p jont t pont. A B ^ q p M B q B Fg. : mod of dndnt p mnpto f dt of t nmt mod, ndng t dt nd nv poton n, w dvd n [], t vot nd ton ton w tdd n []. n t of t nmt n, t po to ontt n nt of pototp w dff n t gomt of t ptfom nd n t poton of t jont t w fndmnt f to t m tt. Fo mp, t two pototp own n Fg. nd Fg. dff n t poton of t tn nd ntn g, w n opd o non-opd, tng n t tng o gon ptfom gomt. MP A q B BP A
![oot S Cg []. ondton nm td n o optmm dgn pom fo t dndnt p mnpto, tn.9, v w ow t tt to ondd o to t otop tt ondton. Condtonng Ind Sngt o n -ondtond on mt v to vodd n ovng t twt vto t fom t ton, Fg.](/docs-images/94/122286279/images/2-0.jpg ": Ft pototp of t mnpto θ t Fg. : Sond pototp of t mnpto pp pot td of t nmt pfomn of t dndnt p mnpto, ondng t nm ondtonng of t on mt.")
2 oot S Cg []. ondton nm td n o optmm dgn pom fo t dndnt p mnpto, tn.9, v w ow t tt to ondd o to t otop tt ondton. Condtonng Ind Sngt o n -ondtond on mt v to vodd n ovng t twt vto t fom t ton, Fg. : Ft pototp of t mnpto θ t Fg. : Sond pototp of t mnpto pp pot td of t nmt pfomn of t dndnt p mnpto, ondng t nm ondtonng of t on mt. An ppopt dgn mtodoog d to dtmn n optmm tt n tm of nmt pfomn fo t gnd t. Att dgn nmt pfomn of mnpto o td to t nm tt of t on mt nd t to t nm ondton. Fo t ppo, v ndto of t nmt pfomn of mnpto nmt nd n tm of t nm ondton of on mt, v n ntodd: Yow, 985 ntodd mnpt, d on t dtmnnt of t podt of t mnpto on t tnpo []. P nd Stvnon 98 d t nmt pfomn of p wt t ot v of t on dtmnnt [5]. pfomn nd, onfgton-dpndnt,.. dpndng on jont v. Bt, t w nown tt moton pfomn o dpnd on t mnpto tt, nd t dpndn not gt n p tn n mnpto t m g dffn n t tt of t fom tn n tt of t tt. t o of tt nt fo good nmt pfomn. nott pfomn nd d n t pp t ondton nm of t on mt, ft ppd to w θ t vto vot of t ttd jont nd t [ ω, p ] wt ω nd p t ng vot nd t vto vot of t nd-ffto ptv. In ft, f ng o -ondtond, t, ptv, mpo to dtmn t o to dtmn t t. ngt o t -ondtonng of t on mt t of ot t onfgton nd t tt of t mnpto. Moov, t tt ngt nnot vodd tjto pnnng o onto, t n on mntd dgn. Sm, -ondtonng tng fom t tt o to mntd dgn. dg of t nfn of n tt on t nm ondtonng of tmd tt ondtonng nd t tt ontn v ndto to m tt ondtonng. n of t ndto ondton nm of t on mt. S θ Fg. : Ln tnfomton of t nt p nto n pod otd to on mt tnfom t nt p dfnd n jont vot p nto n pod, ottd o S θ mod, n t Ctn vot p S t ndtd n Fg.. If t on mt omogno mnt, t ondton nm n td t to twn t gt nd t mt of t ng v, σ σ M m S t
3 ng n m v twn on nd nfnt; t nt mnmm o wn t ng v t m, n t t tnfomton fom on p to not of dffnt d nd t mt d to otop o tt t mnpto dfnd otop. Conv, n ng mt t mt ng v q o nd t t ondton nm om nfnt. nd of nott pfomn n n pmt ndtng t dtoton of t p n jont vot p; t gt t dtoton,.. t gt t ondton nm, t wo t ondtonng of t on mt. It od notd tt t ondton nm n ndto of t mpfton of t omptton o n t oton of t n qton tm otd wt t mt tf, Fot nd Mo [7], Dqt nd Bjö [8]. ondton nm of t on mt, o t t po, twn o nd, tfo n nd of nmt pfomn fo mnpto w ow tt ompt ton. A notd Lpn nd Dff [9], t ondton nm of om dpvd of p gnfn wn t on mt dmnon togno: odng t ng v of dffnt dmnon fom t gt to t mt wod t n t ondton nm vng wt t nt of mmnt of t dmnon of t mnpto. nongt n ovd doptng dffnt nmt mod popod Gon nd Ang []. H t pom ovom dfnng tt n ngt w dvd t mnt of t on mt ngt. Dffnt tt ngt popod n t tt: nt ngt, mn ngt, nt ngt, ppopt fo pt ndvd ojtv. Ung tt n ngt, t ogn on mt omognd, dg,,,,, Now t ndto of nm ondtonng dfnd, In [], M gv mn mp of tt n w t ngt nd -ondtonng w n tn pom n mnpto pfomn. tt dgn wt t m ondton nm of t tt nmt pfomn. tt t t pfomn wn t ondton nm of t on mt q to nd tt d otop tt,.., σ 5 w σ potv nd t dntt mt. C, t mpo to v n otop tt t v onfgton t on mt o onfgton dpndnt. Hn n tt ondd otop ong t opondng on mt n om otop n on onfgton. ptmm Knmt dgn Att ondtonng md nott pfomn nd,.. t ondton nm of t mt, ntodd ov. In t pom ondd, t nd dpndng on t gomt pmt of t two ptfom mt td nd tn optmd mn of dgn v. B MP Fg. 5: Gomt nd pmt of mnpto ptfom gomt of t nd movng ptfom, own n Fg. 5,, fo mpt, gon wt t q ong d tntng wt t q ot d. tt of t ptfom mnpto tfo f dfnd t pmt,,,. pom of optmm dgn ondd fo pt ptfom onfgton, tmd t nt onfgton,.. t two m nt of t two ptfom nd t p jont t pont, gnd ong t. onfgton of t movng ptfom dfnd otton mt Q of fm M _,fd nto MP, pt to n fm B _, nd poton vto p of t nt m of t MP. ntn pont ndvdtd vto owd n Fg.. nt onfgton dfnd, p ; p ^ ^ ^ B BP ; Q w dntt mt nd p >. In t pt onfgton, t movng ptfom not ontd wt pt to t. Dpmnt ong t nd n p v, t onfgton fo gvn mnpto ^
4 not nq. pmt nd p o dgn v n t optmton pom. optmton dgn pom ont of fndng onfgton Q, p, nd n tt of t mnpto tt t nd mnmm. In mtmt fomton mn, p p θ θ θ nt onfgton t mot t o onfgton fo t ondton nm. onfgton of t mnpto n pd n tm of nn v,.. t t otton ng θ, θ, θ, ot t,, nd, ptv, of t nt fm of t BP, nd t Ctn oodnt of t pont p nd. fo, ndtng t gomt pmt of BP nd MP,,,, ptv, t po to vf tt t ondton nm dpndnt on nn v tt v t nt onfgton of t mnpto. Ung q., t Fg. ow t vton of fom nt onfgton wt pt to fo,.., of t nn v potd ov nton tnton ong [m] 8 - -,, nton tnton ong ong [m] 7 p > > p > 9 fo ontnt, q.9, n non-ngtv gomt dmnon fo t ptfom; t ft two ontnt of q.9 tng to t pont p nd, to vod t ng onfgton w ot pt ondnt. td ondton of q.9 vod t pont of onfn of t ntn tton ng on on of t. o dv t ojtv fnton, t on mt of t mnpto mt dvd n tm of t dgn v. gomt of ptfom own n Fg. 5. nmt qton of t p mnpto dvd n [] nd potd ow, p v p fo,, ˆ ˆ fo,, v q fo,, vto pd fng to t nt fm otd wt t ptfom. Ft, dffnttng q. wt pt to tm d to t ton, p v p ω fo,, 5 ˆ ˆ fo,, v ω q q fo,, 7,,9,8 -,, Rotton ot [d],5,95,9,85 -, -,, Rotton ot [d] nd pojtng ong t nt vto otd wt ttd g, t foowng qton otnd, p ω 8 ˆ ˆ ˆ ˆ ˆ 9 v ω q q Fg. : Condton nm v. dpmnt optmm dgn pom, wt dgn,,,, p,, n now mtmt fomtd : dtmn t mnmm of t ojtv fnton,,,,, p, mn 8 jt to t foowng ontnt, ; 9 Smpfng q.8- gv, p ω ˆ ˆ ˆ v ω q nd o ω p ˆ 5 q v ω
5 ton n wttn n t foowng w, At 7 B 8 Dω v C q 9 w A ] ˆ ˆ ˆ [ B ] [ C D Ung t tt ngt, t mt A nd D, ndtd n q. nd q. ptv wttn n omogno fom. Rwtng q.7-9 n ompt fom nd n omogno fom gv, G Ft w G 5 ; ; CB F E A 5 [ C]; D E q A dtmnd Zngn nd Ang [], nd doptng t notton ntodd n tt pp, t qton tng t twt of t MP to t tto vot, n t fom of t fowd vot pom,, G F t ' In t fom, ' t on mt otd wt t fowd vot pom, nd dmnon omognd tt n ngt. Conqnt t optmton pom n fomtd ng t on mt otnd t t nv o t fowd pom. t dgn pom o po to otop tt. A good on ondtonng w v n fft on t nm oton of t IKP, ndng t nmt mod of t mnpto t. Bd on t mpton tt t mnpto n nt onfgton, t jont poton vto wt pt to t fm own n Fg. 5, pd, 7 ˆ ˆ ˆ Emnng t ondton nm v own n Fg. fo gn tjto, t tt t mnmm ondton opond to t onfgton w ot ntn pont nd t ogn of t oodnt fm otd wt t movng ptfom, ot ong to t of t nt fm otd wt t ptfom.,5,5,5 5 5,5,5,5 5 / Fg. 7: Vton of v. t to of /
6 It o n td tt t ojtv fnton d on nng t to twn t dmnon of t d of t two ptfom /, nd / vng t ot pmt f to v own n Fg Nm optmton optmton pom fomtd n Ston w ovd ng oftw d on t tnq of Sqnt Qdt Pogmmng SQP. oftw ffnt n ovng optmm ontn pom. oton of dgn v :. m 5.8 m p.988 m nd t opondng v of t ojtv fnton, o otn pt dmnon fo t v fond n 5, mng.5, t v om, m.9 m t otnd ndt tt t t pototp, n tm of vtd nmt pfomn, tt n w t nv jont of t ntn g ntd wt t jont of t tn g, own n Fg.. In t w mnpto wt tng ptfom podd. It mpotnt to not tt p ptfom mnpto wt pmt jont, wd d fo fgt mton n, v gon t fd tw t of t mo,.., wt on d m ot tn t ot. fom o to t tng ptfom tt w optm fom t nmt pont of vw. Conon pp pnt t nmt optmton of dndnt p mnpto ng t ondton nm. t otnd ow tt t tt w t pont of ttmnt of t tn g ottd wt pt to t pp ntn g on t movng ptfom opond to t t nmt pfomn. dgn oton v mpotnt n pvntng nmt ngt. Moov, fom n of ondton nm v, t w ptn of otop onfgton n t tjto mnd. Fn, t nt onfgton opondng to t ondton nm mnmm w dtmnd fo gvn gomt mnpto pmt. Rfn: [] Mt,.P., mpotn of optm dgn fo p tt, Ft Eopn-Amn Fom on PKM. Ag- Spt, 998 Mno. [] K. E. Zngn nd. Ang, Mot nd Poton An of Nov Rdndnt P Mnpto, IEEE Int. Confn on Root nd Atomton, 99. [] K. E: Zngn nd. Ang, Intntno Knmt nd Dgn of Nov Rdndnt P Mnpto, IEEE Int. Confn on Root nd Atomton, 99. [] Yow,., Mnpt of oot mnm, Int.. f Root R Vo., 985, -9. [5] P, R. P. nd Stvnon, C. N. 98. Knmt of oot wt. Int.. Root R. :-. [] Sg,. K. nd Cg,.., Attd nd: Fo nd nmt, ntnton on of Root R, Vo. n., 98, pp.-7. [7] Fot, G.E., nd Mo, C. B., Compt Soton of Ln Ag Stm. Engwood Cff N: Pnt-H, 97. [8] Dqt, C. nd Bjo, A. 97, Nm Mtod, Pnt-H, In., Engwood Cff.97 [9] Lpn, H. nd Dff,.. Hd twt nd wn onto of oot mnpto. ASME. Mnm nmon Atomton D. Vo., 988, pp. 8-. [] Gon C., nd Ang.,, optmm nmt dgn of pn t-dg of-fdom p mnpto, n.asme,. M., nd Ato n dgn, Vo.. 988, pp.5-. []. M, Mn An of P Mnpto wt Smton, dgn nd Conto Appton P.D. MG Unvt, Mont, 99.
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