Sesrs & Trsducers, Vl. 7, Issue 8, August,. 8-89 Sesrs & Trsducers b IFSA Publshg, S. L. htt://www.sesrsrtl.cm Kemtcs Alss d Smult Trsfer Rbt wth S Degrees f Freedm Y Lu, D Lu Jgsu Jhu Isttute, Xuhu, Jgsu,, Ch Xuhu Cl Mg Gru,, Ch Tel.: 838898, f: 838898 E-ml: jjd@.cm Receved: M /Acceted: 3 Jul /Publshed: 3 August Abstrct: Stud fcuses trsfer rbt wth S Degrees f Freedm, estblshg kemtc equt b D-H methd, lg frwrd kemtcs d btg verse kemtcs b usg methd f verse trsfrm. Bsed vectr rduct, t devels velct Jcb mtr f rbt. The gemetrc mdel f rbt vrtul rtte s estblshed b SldWrks sftwre d geertes rmeters such s mss d mmet. Kemtc smult fr rbt s erfrmed b Mthemtc sftwre d devels curve grh f dslcemet, velct d ccelerted seed, d drect ed eecutr ceter f rbt wth mesuremet, lss d ssessmet, whch rvdes fudt fr further kemtcs lss d structure tmt s well s mt ctrl f rbt. Crght IFSA Publshg, S. L. Kewrds: Trsfer rbt, Kemtcs, Smult, Otmt, SldWrks, Mthemtc.. Prefce I recet ers, wth dee develmet f techcl reserch rbt d wder stud felds, kemtc smult lss f rbt ls mre mrtt rle stl lg, trjectr ctrl, mt ctrl, ff-le rgrmmg, tmt desg d ther sects. Rbt kemtc studes kemtc relt betwee ech jt d rgd bd, st f rts vlved d relt f velct d ccelertg seed wth tme. Rbt kemtcs cludes: e s drect kemtcs where jt vrble s gve d clculte hd erce, the ther s verse kemtcs where hd erce s gve d clculte jt vrble. Trsfer rbt s dvced utmtc devce whch culd crese rductvt d mrve wrkg cdt.. Rbt Kemtcs Mdel Artculted rbt wth -DOF hs ver cmlcted mechcl structure, csstg f bse, wst jts, uer rm, frerm, wrst jt d hds, ever mt ut f whch s cmsed f smll mechcl sstem, such s shft, berg, bushg, ke, ger, mtr s well s reducer. As shw Fg.. Mt rm f -DOF cssts f rms d jts. Ech rm s descrbed wth rmeters f, α, d, where d α mes feture f rm -; d d θ mes relt f rm - d rm. Fr rtr jt, jt gle s jt vrble d ther rmeters re cstt; fr rsmtc jt, lrt s jt vrble d ther rmeters re cstt. Ths methd t descrbe htt://www.sesrsrtl.cm/html/digest/p_3.htm 8
Sesrs & Trsducers, Vl. 7, Issue 8, August,. 8-89 mechsm kemtcs ws brught u b Devt d Hrteberg 9, clled D-H methd [, ]. Tble. D-H rmeters. α d Jt θ 9-9 θ -9 3 θ 3 +9 d θ -9 θ +9 d θ θ.. Drect Kemtcs Refer t lk rmeter t mke sure rm trsfrmt mtr eress I ccrdce wth the "left t rght" rcle: Fg.. Trsfer rbt mdel s degrees f freedm. Estblsh lk crdte sstem b D-H methd, s shw Fg.. Kemtcs d structure rmeters f relted rms re shw Tble. cθ sθcα T sθsα sθ cθ cα cθ sα sα cα dsα dcα, () Put rmeters Fg. t trsmss mtr (), t clculte ech rm hmgeeus trsfrmt mtr T, T, 3 T, 3 T, T, T. Multl bve mtres t get rbt ed cmet hmgeeus trsfrmt mtr: 3 ( θ ) T( θ ) T( θ ) T( θ ) T( θ ) T ( ) T 3 3 θ T, () Put rmeters Fg. t equt () d () t get st mtr T f rbt ed reltve t bsc crdte sstem: 3 T T T 3T T T T, (3) Equt (3) mes st reltsh f ed rm crdte sstem {} reltve t bsc crdte sstem {}, whch s the fudt f ech jt kemtcs lss f rbts. Itl st: θ 9 ; θ 9 ; θ 3 ; θ ; θ ; θ. Put tl vlue f θ t equt (3) t get: Where Fg.. Jt rm crdte sstem. mes dstce frm t + lg ; ; α mes gle frm t + rud mes dstce frm t lg ; gle frm t rud [3]. d θ mes T It ccrds wth the st Fg., whch rves bve clcult rcess s crrect. 8
Sesrs & Trsducers, Vl. 7, Issue 8, August,. 8-89 B bve lss, estblsh kemtcs equt f relt betwee trsfer rbt ed ctutr st d ther jt vrble rectgulr crdte sstem, whch gets frwrd slut f rbt kemtcs equt []... Reverse Kemtcs Reverse kemtcs s t get jt vrble b hd st, whch s used t mke sure mtr rmeters t drve jt d l trjectr []. Cmm reverse kemtcs methds clude cuter trsfrmt methd, gemetrc methd d Peer methd d s. The er les fr cuter trsfrmt methd t slve rbt reverse kemtcs, tht s, t multl reverse mtr the left f jt trsfrmt mtres d clculte t fd the vrble elemet the rght, d the mke these vrbles equl wth thse the left t get trgmetrc fuct. Accrdg t st f vectrl cmbed wth rbt rm rmeter d t get jt vrbles θ,. Iverse trsfrmt T ( θ) left multl fuct () left d rght, 3 ( θ ) T T ( θ ) T ( θ ) T ( θ ) T ( θ ) T ( ) T 3 3 θ, () c s s c T Gve fuct s equl rght d left sdes t get θ 3, there re tw ssbltes fr ech vrble. Iverse trsfrmt 3T left multl fuct () left d rght, T 3 ( θ ) T T T T θ, () 3, 3 Gve fuct s equl rght d left sdes t get θ, there re tw ssbltes fr ech vrble. Iverse trsfrmt T left multl fuct () left d rght, ( θ ) T T T T 3, θ, () Gve fuct s equl rght d left sdes t get θ, whch s clsed frm. I the sme w t get clsed frm f θ. Abve lss shws verse f kemtcs fuct hs ucertt d hs m grus f slut whch re ssble dfferet sce st f kemtc jt..3. Clculte Jcb Mtr Jcb mtr f mt rm s ts ler trsfrmt f ert seed d jt seed, whch c be see s trsmss rt f mvemet frm jt sce t ert sce [], tht s: V X J( q) q, (7) where X mes ert velct vectr, q mes jt velct vectr. The umber f mvemet dmess f rbt ert sce s equl t tht f rws f Jcb mtr d jts re equl t clum. Abve lss shws Jcb mtr f rbt wth s degrees f freedm s mtr. The er les fr vectrl methd t clculte rbt Jcb mtr. The le f Jcb mtr s: Z P Z ( R P) J, (8) Z Z where P mes rbt ed rg f crdte bsc crdte sstem {} reltve t st vectrl crdte sstem {}, Z mes ut vectrl f s f crdte sstem {} bsc crdte sstem {}. R mes drect cse mtr. Frm bve trsfrmt mtr, get R,Z, P (,,,), d ut them t fuct (8) t get le J (,,,) f Jcb mtr, s s t mke rbt Jcb mtr.. Jcb mtr f rbt s shw s belw: ( ) ( ) ( ) 3( ) ( ) ( ) ( ) (9) J q J q J q J q J q J q J q T T T T T T T 3. Kemtcs Smult Put mdel t Mthemtc, whch cludes 7 rts d ech mes bse, vertcl shft, bse turtble, uer rm, frerm, ws d wrst. Bse s fg rt d rttg jts re set betwee ther rts rud Z s. Gve vrble t ech jt, smulte, le d cmre kemtcs. 3.. Alss Smult Result Ale t the mddle f ws, whch s mdt f w d ccdes wth rgl t f crdte sstem. Outut dt s shw s Fg. 3. Fg. 3 shws tht whe tme t, ert rm s tl st, dslcemet,, drect s -,, 3 (ut: mm), whch s the sme s clcult result. 87
Sesrs & Trsducers, Vl. 7, Issue 8, August,. 8-89 Fg. shws tht whe tme t, velct,, drect s -97,,, d mmum velct s 97,, (ut: mm/s) whch s lmst the sme s clcult result. Chge legth f ech rm d mvemet rmeter f ech jt t get dfferet dslcemet, velct d ccelerted seed f rm dfferet cdts. Fg.. Agulr velct curve f w mdt. Fg. 3. Dslcemet curve f w mdt s tme chges. Fg. 7. Agulr ccelerted seed curve f w mdt. Fg.. Velct curve f w mdt s tme chges. Fg. shws tht whe tme t, ccelerted seed,, drect s, -3, d mmum ccelerted seed s 9, 3, (ut: mm/s), whch s the sme s theretcl lss. Fg.. Accelerted seed curve f w mdt s tme chges. The sme verfct s gt frm Fg. d Fg. 7.. Ccluss ) Bsed rbt D-H mtr ther, estblsh rbt kemtcs mthemtc mdel, deduce hmgeeus trsfrmt mtr f trsfer rbt kemtcs fuct b cmbg wth rbt structure, b whch gulr f ever jt d w st c be clculted mutull d get drect d verse kemtcs. ) Vectr rduct methd s led t deduce seed Jcb mtr, whch culd clculte dslcemet d seed f ts w wth ech jt gulr d seed mutull. 3) Smulte b sftwre t verf gve structure rmeters resble. ) Mthemtc smult sftwre s led t smulte kemtcs f trsfer rbt wth -degrees f freedm, mesure velct, ccelerted seed, dslcemet curve f rbt ed, le dmc mvemet fetures f rbt. Smult result shws: rbt mves stedl wrkg cdt whch c meet wrkg requremets. Refereces []. Zg C, Rbtcs, Tsghu Uverst Press, Bejg,. 88
Sesrs & Trsducers, Vl. 7, Issue 8, August,. 8-89 []. A. K. Bsu, Dm Nvee Kumr, Nmshkv Krthk, Vks Pdwekr, Mdelg d lss f mcr rbt, Jurl f Mechtrcs, Vl., Issue, 3,. 8-3. [3]. Yu Lu Xg et. l, Rbtcs, Mechcl Idustr Publshg Huse, Bejg, 993. []. htt://bbs.lcrbt.cm/thred-38--.html []. K. G. Sh, N. D. McK, Mmum-tme ctrl f rbtc multrs wth gemetrcth cstrts, IEEE Trscts Autmtc Ctrl, Vl. 3, Issue, 98,. 3- []. J. E. Bbrw, S. Dubwsk, J. S. Gbs, Tmetml ctrl f rbtc multrs lg secfed ths, Jurl f Rbtc Reserch, Vl., Issue 3, 98,. -. Crght, Itertl Frequec Sesr Assct (IFSA) Publshg, S. L. All rghts reserved. (htt://www.sesrsrtl.cm) 89