Gantry-Tau A New Three Degrees of Freedom Parallel Kinematic Robot

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Gabriella Price
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1 Gant-au A w h Dg of Fom Paa Kinmatic Robot La Johannon, Vikto Bbuk, ogn Bogåh. Dpatmnt of Machin an Vhic Stm Cham Univit of chnoog, 4 9 Götbog, Swn -mai: a.johannon@m.cham.. Dpatmnt of Machin an Vhic Stm Cham Univit of chnoog, 4 9 Götbog, Swn -mai: vikto.bbuk@m.cham.. ABB Automation chnoog Pouct/Robotic 7 8 Vätå -mai: togn.bogah@.abb.com Abtact In th at ca, an incaing attntion ha bn pai to th tu of iffnt paa tuctu mchanim an thi appication, main tigg b Stwat that pnt an aicaft imuato tm. Paa tuctu fatu povi big avantag in potntia appication. Fo amp, paa obot ma giv high p an accation, high tatic an namic accuac an high tiffn than what i poib with th inutia obot u toa. A tpica imitation with man of th paa tuctu i that thi wokpac i ma compa to th ia tuctu. hi pap pnt a nw paa tuctu, th Gant-au, which povi g of fom (DOF) tanationa motion with a ag wokpac. h tuctu of th obot i patnt b ABB. h Gant-au obot i a i ink paa kinmatic tuctu with th ink configu accoing to --. h -- notation f to how man ink fom ach uting kinmatic cut of th obot. Ointationa DOF of th obot cou b povi b a coup tm. Fo a convntiona DOF ia gant obot two of th actuato contibut to th moving ma. h Gant-au can b contuct with cptiona ow moving ma inc th actuato a tationa an th tuctu ha inhnt high tiffn. h tuctu i thu ia fo man appication with man on high accation, fo intanc fo th pick an pac opation. h nomina inv an ict kinmatic of th tuctu a vop an optimiation i u to fin a contuction of Gant-au with maimum wokpac voum. Intouction Mot of th obot u in th inut a ia manipuato. A ia manipuato ha an opn kinmatic chain tuctu. hi tp of obot off high gnait an can b u fo vaiou appication. Howv th ia manipuato uff fom a ow atio btwn oa capacit an obot ma. h main aon fo thi a that th obot actuato contibut to th moving ma an that ach ink i ubjct to th wight of th foowing ink. hu th ink hav to b imnion with pct to ag fu tou, which man that th tuctu ha to b tiffn, an thu bcom havi. Accuac i imit b th fact that th ink magnif o thoughout th chain. Fo intanc a ma angua o in a vout joint a in th chain wi inuc a ag o fo th too cnt point (CP).

2 A paa manipuato i a co kinmatic chain mchanim. h it a vait of achitctu ign fo iffnt appication. h paa manipuato can b chaacti with compaion to it ia countpat a a tm with [4]: high atio btwn oa capacit an obot ma, high tiffn, high about accuac, imp inv kinmatic, mo ifficut ict kinmatic, ma wokpac. h high atio btwn oa capacit an obot ma i u to that th actuato oftn a ocat on a fi patfom an fo man of th tuctu th ink a on ubjct to aia foc an that th oa i itibut ov th chain. High tiffn i u to that th tna foc i itibut ov th chain. High about accuac i u to non cumuativ joint o an th high tiffn. h inv kinmatic pobm i oftn ov ai inc th chain can b tui paat an that iffnt configuation a gna fi in th ign poc. h oution of th ict kinmatic pobm i oftn ifficut inc in th gna ca th i no uniu oution. h contant ointation wokpac i ao oftn imit fo mot DOF fu paa manipuato. On appoach to gt btt wokpac popti i to vop manipuato wh th tanationa g of fom a paat fom th otationa g of fom o to ign manipuato that a not fu paa. hi pap pnt a nw paa tuctu, th Gant-au, which povi DOF tanationa motion with a ag wokpac. h tuctu of th obot i patnt b ABB [] an a ut inicat that th tuctu cou outpfom th ia gant tuctu fo man appication. au bong to th PRRS fami of paa manipuato with th HaGi a on of it cot ativ [4]. h PRRS notation cib th joint in th kinmatic chain fom actuation to th CP. hu ach chain i fom b a pimatic joint with actuation (P), a univa joint (RR), an fina a phica joint which connct to th moving pat. h chain fom th kinmatic cut wh th chain a ogani a a oub paaogam, a ing paaogam an a ing ink which a connct to th moving pat. h pimatic joint a th paa ina tack. Figu how a chmatic fo th Gant-au tuctu. B moving A, B an C aong th tack fom S A,B,C, to S A,B,C, th tanationa motion i conto fo th CP whi th ointation of th moving pat i maintain. Z S A, Y X A S B, A B S C, Fig:. Schmatic Gant-au. Goba cooinat tm i fin with th X-ai aong th iction fom S A, to S A,. h back ot pnt phica joint. h vcto i i- fin th ocation fo th univa joint PU i i- fom point A, B an C (figu ). h vcto n i i- fin th ativ ocation fo th CP with pct to th phica joint PS i i- (figu ). C C PU PU PU S A, PS 4 B PS PU PU PU S B, CP S C, Kinmatic ciption PS PS 5 n 4 PS n h Gant-au i a i ink paa kinmatic tuctu with th ink configu accoing to --. h -- notation f to how man ink fom ach uting kinmatic am. Gant- PS n n 5 n n CP CP Z Y X Fig:. Schmatic moving pat. CP

3 h ngth of th ink i mut b th am fo ink bonging to th am cut. h vcto i i-5 an n i i-5 a puiit to fufi th conition that th vcto btwn PS an PS mut b paa to th vcto btwn PU an PU, an that th vcto btwn PS an PS 4 mut b paa to th vcto btwn PU an PU 4, an that th vcto btwn PS 4 an PS 5 mut b paa to th vcto btwn PU 4 an PU 5. Anoth phap obviou puiit i that PS 5 mut b ocat outi th pan PS PU PS 4. Sphica joint, aowing th ink to pin aoun thi pincipa ai, can of cou pac th univa joint. Fo om appication it might b favouab to u on univa joint. hi can b achiv b aing a vout joint on ach ink that pvnt th tuctu fom bing ov contain. A 4 DOF n too ointation aangmnt can b achiv b aing a oub caan ai a hown in figu. hi coup aangmnt i u in th Dta obot ign []. Anoth vaiant of th Gant-au i hown in figu 4. hi aangmnt off 5 DOF imit too tit an cou b u fo wat jt cutting, pama cutting an a cutting. Fig:4. 5 DOF oo tit.. Inv kinmatic Fo th coni paa obot th inv kinmatic pobm i fomuat a foow. Cacuat th ocation of point A, B an C aong th ina tack fo a givn CP ocation. Lt ( a ) S A, ( b ) SB, ( c ) SC, A +, B +, C +, ( ) CP. H th paamt a, b an c a to b tmin an can b foun a th intction btwn ph with mipoint at CP n, CP n an CP n an th pctiv ina tack. h phica uation can b wittn a foow: ( S + a +, + n, ) + ( S +, + n, ) + ( S + + n ),, ( S B,, + b +, + n, ) + ( S B,, +, + n, ) + ( S + + n ) C,,,, Fig:. 4 DOF En too otation. ( SC,, + c +, + n, ) + ( SC,, +, + n, ) + ( S + + n ) C,,,, hn w can tmin th paamt

4 ± ± ± a b c, ( S + + n ) ( S + + n ), n, S,,,, ( S + + n ) ( S + + n ) B,,, n, S B,,,, B,,,, ( S + + n ) ( S + + n ) C,, n, S C,,,, C,,, h ign bfo th oot pion ci th configuation of th obot., Mathmatica mboic oftwa can ov th phica uation, but pouc a ath tniv oution. Poficint u of impification u i n in o to impif th oution. hi pobm i avoi b oving th uation in two tp. Fit fin th intction btwn two of th ph. h intction i ith a cic o a point. Igno th point ca fo now. h intction btwn th thi ph an on of th oth fom of cou ao a cic. Div th pan wh thi cic i ocat. Scon th intction of thi pan an th fit cic cib th poib ocation fo th CP. In th oution bow th intction cic btwn ph with mipoint at A an C i cacuat. A cacuation a thn on in a cooinat tm with th -ai pointing fom A to C.. Dict kinmatic Fo th coni paa obot th ict kinmatic pobm can b fomuat a foow. Cacuat th ocation of th CP fo givn A, B an C. h ph with aiu, an cib a poib ocation fo th CP fo fi A, B an C. h intction point btwn th ph cib th ocation of th CP. h mipoint of th ph an th phica uation a: A B C [ a a a] S A, + a [ ] + + n [ b b b ] SB, + b[ ] + + n [ ] S, + [ ] + n c c c C c + ( ) + ( ) + ( ) a a a ( ) + ( ) + ( ) b b b ( ) + ( ) + ( ) c c c A Fig:5. Intction btwn two ph. + A C, A C Mipoint fo th cic: AC D A +, A C A point on th pan: E B + B C B C + B C B C h noma vcto fo th pan: B C Diving th otation mati: a c θ tan, c a β D A C in c AC a C 4

5 co( θ ) in( θ ) Rot in( θ ) co( θ ) Rot co( β π / ) in( β π / ) in( β π / ) co( β π / ) Rot Rot Rot P + P P P S,,, P, +, Q, R P + P h noma vcto fo th pan an point D an E a tanfom into a cooinat tm with th -ai pointing fom A to C. ( ) Rot ( ) Rot D ( ) Rot E h phica uation can now b wittn in th nw cooinat tm a th intction btwn a cic an a ph. ( ) + ( ) ( ) + ( ) + ( ), wh ( ) Rot CP + P P S S CP Rot ( + P )( P ) P R + Q R h configuation of th obot ci which oution i vai. Wokpac optimiation In o to chaacti th wokpac of th manipuato th foowing optimiation pobm i fomuat. Fin th itanc btwn th tack that giv th agt co-ction wokpac fo a manipuato with ink of ua ngth. On mmtica pacmnt of th tack a coni. h wokpac i futh tict in th iction with th uimnt that th wokpac mut b a pat of th opn ctangua aa fom b th ina tack. wo tp of joint a coni both hown in figu. h caan joint put no tiction on th co-ction wokpac whi th ba an ockt joint imit th wokpac aong on iction. Caan joint Ba an ockt joint + P P + S o imit + P P + S Fig:. Joint -β β 5

6 h pobm i ov inpnnt fom th vcto n i an i b impoing th paamtiation on th mipoint of th th ph that intct at th CP. h optimiation paamt a an a hown in figu 7. Whn th ba an ockt joint i u, th optima ointation of th joint mut ao b coni β.8 A C B h ation btwn optima aa an ink ngth i a foow: Aa.95545* 4 Concuion Fig:7. Optimiation paamt. h ah in fom th opn ctangua aa which tict th wokpac. h optimiation pobm i ov b uing a non ina pogamming outin. h obtain optima co ction aa i hown in figu 8. In th pap th oution of th inv kinmatic an ict kinmatic pobm fo th nov paa tuctu obot hav bn obtain. h initia tu of th Gant-au tuctu ha montat goo wokpac popti of th obot. Futh tui a n in o to amin how comptitiv th coni tuctu i. Rfnc Z C A B [] Bogåh,., PKM ach impotant iu, a n fom a pouct vopmnt ppctiv at ABB Robotic, in Poc. Of th Wokhop on Funamnta Iu an Futu Rach Diction fo Paa Mchanim an Manipuato, (E. Cémnt M., Goin an Imm Ebt-Uphoff), Octob -4,, Qubc Cit, Qubc, Canaa, 8-8. Y Fig:8. Optima co ction wokpac. h taight in how th imit fo ba an ockt joint with β.8 o. h cic how th maimum achabiit fo ach cut without imitation impo b th joint. A ong a β in( ) an th ba an ockt joint a ointat a in figu 8 th optima ation btwn ink ngth an th optimiation paamt a th am fo both joint tp, nam: [] Bogåh,., Inutia Robot, Intnationa Pubication umb WO /448 A. [] Cav, R. 988, DELA, a fat obot with paa gomt, in Poc. Of th 8 th Intnationa Smpoium on Inutia Robot, (Eito H. van Bu), 9-. [4] Mt, J.-P., Paa Robot, Kuw Acamic Pubih, Docht, h than. [5] Stwat, D. 95, A patfom with i g of fom, Pocing of th Intitut of Mchanica Engin, Lonon, 8, 7-8.

Differential Kinematics

Differential Kinematics Lctu Diffntia Kinmatic Acknowgmnt : Pof. Ouama Khatib, Robotic Laboato, tanfo Univit, UA Pof. Ha Aaa, AI Laboato, MIT, UA Guiing Qution In obotic appication, not on th poition an ointation, but th vocit