INVERSE DYNAMICS OF THE CINCINNATI-MILACRON WRIST ROBOT

Transcription

1 INVERSE DYNMIS OF THE ININNTI-MILRON WRIST ROOT ŞTEFN STIU * Recusve matx elatons n dynamcs of the ncnnat-mlacon wst obot ae establshed n ths pape. The pototype of ths mechansm s a thee-degees-offeedom sphecal system wth sx movng lns and thee bevel gea pas. ontolled by electc motos thee actve elements of the obot have thee ndependent otatons. Supposng that the poston and the otaton moton of the end-effecto ae nown the nvese dynamc poblem s solved usng an appoach based on the pncple of vtual wo but the esults have been vefed n the famewo of the Lagange equatons of second nd. Fnally some ecusve matx elatons and some gaphs fo the toques and the powes of the actuatos ae obtaned.. INTRODUTION The oentng mechansms ae ncopoated n the stuctue of ndustal obots and have two o thee output otatons. Geneally these mechancal systems have concal and cylndcal toothed elements n the stuctue whle the nput axes ae paallel and the output axes ae othogonal. The thee oentaton motons ae usually pefomed aound the axes of a atesan othogonal fame havng ts axes lned to the last am of the obot s postonng mechansm. The ndustal obots wth oentng gea tans can pefom seveal opeatons such as weldng flame cuttng spay pantng mllng o assemblng. eng compaatvely smple and compact n sze the bevel-gea wst mechansms can be sealed n a metallc box that eeps the devce of contamnaton. Futhemoe usng bevel gea tans fo powe tansmsson the actuatos can be mounted emotely on the foeam theeby educng the weght and neta of a obot manpulato. Planetay gea tans wth thee degees of feedom ae adopted as the desgn concept fo obotc wst (Hseh and Sheu []; Paul and Stevenson []; Wlls []; Ma and Gupta [4]; Whte [5]; Stachouse [6]). Geneally the numbe of actuatos s typcally equal to the numbe of degees of feedom such that even nematcal chan can be contolled at o nea the fxed base. *Depatment of Mechancs Unvesty Poltehnca of uchaest Romana Rev. Roum. Sc. Techn. Méc. ppl. Tome 55 No. P ucaest

2 74 Ştefan Stacu. GEOMETRI MODELLING Recusve matx elatons fo nvese nematcs of a ncnnat-mlacon planetay bevel-gea mechansm whch has a non-symmetcal nematcal schema ae fst developed n the pape. The end-effectos must otate aound thee non-othogonal axes so that the oentng wst mechansm has thee degees of feedom. matx methodology fo the nematcs analyss based on the concept of fundamental ccut of an open-loop chan s pesented. Ths method nvolves the dentfcaton of all open-loop chans and the devaton of the geometc elatonshps between the oentaton of the end-effecto and the jont angles of compoundng lns ncludng the nput actuato dsplacements [7 8 9]. Let Ox yz be a fxed atesan fame about whch the oblque wst mechansm moves (Fg. ). The mechansm topology conssts of sx movng lns sx evolute jonts and thee bevel gea pas. Theefoe the wst s a -DOF sphecal mechansm whch has non-lmted otatonal anges about the jont axes. Thee ae two actve geas b c of ad = = 5 masses m m and neta tensos J ˆ Jˆ and thee nematcal chans a a a 4 a b b and c c ae dentfed statng fom the foeam and endng to a common element as end-effecto e = 4a. sgnfcant featue of ths wst devce s that thee s no physcal ntefeence between all the lns. l 4 a b z O b 4a e c a x l l a c l Fg. ncnnat-mlacon wst obot. In the ncnnat-mlacon wst the ln a as a box of chaactestc length l mass m and tenso of neta ˆ one of the thee dvng pats of the sphecal J

3 Invese dynamcs of the ncnnat-mlacon wst obot 75 mechansm seves as cae fo the b b and c c bevel gea pas whle the ln a of chaactestc length mass m and tenso of neta Jˆ seves as cae fo the a 4 a bevel gea pa. The otaton of gea c s tansmtted to ln a va a specal gea pa whle the otaton of second actve gea b s tansmtted to gea 4 a though two coaxal geas. The two geas of ad total mass m and total neta tensojˆ ae planet geas gdly connected togethe by a sngle shaft of nown length that s housed n both caesa and a. The axes of otaton of thee motos ae suppoted by beangs housed n the foeam. Includng the end-effecto of length l at the last gea 4 a = e of adus 4 = 4 we obtan an assembly of total mass m 4 and total tenso of neta Jˆ 4 that s fee to abtay undego thee concuent otatons wth espect to the common cente O. oncenng the output moton of the end-effecto we ema that the wst oll moton s acheved by otatng ln a about z -axs whle the ptch moton s accomplshed by elatve otatng ln a wth espect to ln a about the z -axs. In the followngs we apply the method of successve dsplacements to geometc analyss of closed-loop chans and we note that a jont vaable s the dsplacement equed to move a ln fom the ntal locaton to the actual poston. If evey ln s connected to at least two othe lns the chan foms one o moe ndependent closed-loops. z φ - z - z + φ + - α - O δ x α + + x - x + Fg. Gea fundamental ccut.

4 76 Ştefan Stacu 4 The vaable angles of otaton about the jont axs z ae the paametes needed to bng the next ln fom a efeence confguaton to the next confguaton. We call the matx a fo example the othogonal tansfomaton matx of elatve otaton wth the angle of ln aound z axs. In the study of the nematcs on constaned systems we ae nteested n devng a matx equaton elatng the locaton of an abtayt body to the jont vaables. When the change of coodnates s successvely consdeed the coespondng matces ae multpled. In what follows we ntoduce a matx appoach that utlzes the theoy of fundamental ccuts aoused by Tsa [7]. Thee exsts a eal o fcttous cae fo evey gea pa n a planetay gea tan and a fundamental matx equaton fo each loop can be wtten as δ a = a θ a = n δ = α + α + + y cosδ snδ θ δ = y () snδ cosδ whee and + denote the elatve angles of otaton of the caet and the planet geat + espectvely whleα α + ae the angles that chaacteze the geomety of the connected geast andt + (Fg. ). The ato of a gea pa s defned as n + = + / = z+ / z () whee + and z z+ ae the adus and the numbe of teeth of the two geas espectvely (Fg. ). x z α z z z z x z z O 4 z x z Fg. Movng fames sequence. α z z 4 T

5 5 Invese dynamcs of the ncnnat-mlacon wst obot 77 The motons of the sx pats of the wst mechansm ae all concuent otatons aound the fxed ponto. To smplfy the gaphcal mage of the nematcal scheme of the sphecal mechansm n what follows we wll epesent the ntemedate efeence systems by only two axes so as s used n most of obotcs papes. The z axs s epesented of couse fo each componentt. It s noted that the elatve otaton wth angle of the body T must always be ponted about the decton of the z axs. onsequently fou appopate fames fo the fst ccut thee fames fo the second nematcal chan and thee fames fo the last ccut ae fxed n a same ogn O (Fg. ). Let us consde the otatons angles of the thee actuatos as vaables that gve the nstantaneous poston of the mechansm. Statng fom the efeence ogn O and pusung the ndependent seal ccuts : a a a 4a b b and c c we obtan the followng successve matces of tansfomaton [ ]: a = a θ a T a = a α θ a = aθ a = 4 a4aα b = b θ b = b θ b = b a () α c = c θ c = c θ c = c a α θ whee ae denoted cosα snα θ = = θ a α = snα cosα cos sn p = sn cos ( p = a b c) ( = ) (4) p = p s+ s ( = 4). s= In what follows we consde that the end-effecto s ntally located at a cental confguaton whee ths s not otated wth espect to the fxed base. In the nvese geometc poblem howeve a complete descpton of the oentaton of the end-effecto n the fxed fame s nown by ntemedate of thee Eule angles φ φ φ assocated wth thee successve otatons whch can be expessed by the analytcal functons π φ * l = φl[ cos( t)] ( l = ) (5) 6

6 78 Ştefan Stacu 6 * whee φ l epesents the maxmum of the angle of otaton φ l. Snce all otatons tae place successvely about the movng coodnate axes the geneal otaton matx R = dd d of the end-effecto fom Ox yz to Ox4 y4 z4 efeence system s obtaned by multplyng thee tansfomaton matces T d = aθ d = aaα d = aaα (6) whee cosφl snφl a = l snφl cosφl ( l = ). (7) onstant geometc condtons fo the otaton of the end-effecto ae gven by the denttes a 4 = R a = b a = c. (8) Fom these equatons we obtan the eal-tme evoluton of all chaactestc jont angles as follows: φ φ = φ = φ + = 4 = φ n n φ φ φ φ φ = φ + = + = φ + n nn n nn n φ φ = φ + = = φ (9) n n 5 n = n = n =. 4. KINEMTIS OF THE WRIST ROOT The analyss of the nematcs of bevel-gea wst mechansms of gyoscopc stuctue s vey complex due to the fact that the caes and planet geas may possess smultaneous angula veloctes about nonpaallel axes. In the desgn of powe tansmsson mechansms t s often necessay to analyze the velocty atos between the nput and output pats and angula veloctes o angula acceleatons of the ntemedate pats. The conventonal tabula o analytcal method whch concentates on plana epcyclcal gea tans s no longe applcable. To ovecome ths dffculty Feudensten Longman and hen [] appled the dual elatve velocty and dual

7 7 Invese dynamcs of the ncnnat-mlacon wst obot 79 matx of tansfomaton fo the analyss of epcyclcal bevel-gea tans. Tsa hen and Ln [] hang and Tsa [4] and Hedman [5] showed that the nematcal analyss of geaed obotc mechansms can be accomplshed by applyng the theoy of fundamental ccuts. a a b a e b c 4a c Fg. 4 ssocated gaph of the mechansm. Snce a nematcal chan s an assemblage of lns and jonts these can be symbolzed n a moe abstact fom nown as equvalent gaph epesentaton (Fg. 4). Fo the eason that wll be clea late we use the assocated gaph to epesent the topology of the mechansm. In the nematcal gaph epesentaton we denote the lns by vetces and the jonts by edges (Yan and Hseh [6 7]. Two small concentc ccles label the vetex denotng the fxed foeam. To dstngush the dffeence between the pa types n the gaph epesentaton of the ncnnat-mlacon wst the thee gea pas b b c c a 4a ae dawn n heavy edges and the evolute jonts a a a a a a 4 a b and c ae setched n thn edges. Thee ae thee sgnfcant ndependent loops and we dentfy thee seal nematcal chans. The nematcs of an element fo each ccut s chaactezed by sewsymmetc matces gven by the ecusve elatons [8]: ~ T = p ~ p ~ + ~ = & ~ u p = abc ; = () whee ~u s a sew-symmetc matx assocated wth the unt vecto T u [ ]. These matces ae assocated to the angula veloctes = p & + = u. () Knowng the geneal otaton moton of the end-effecto attached at the planet gea 4 a by the elatons (5) one develops the nvese nematcal poblem and detemnes the absolute angula velocty and acceleaton ln. ε of each movng

8 8 Ştefan Stacu 8 ased on the mpotant ema = n + + () the devatves wth espect to tme of the elatons (9) lead to the elatve angula veloctes of all lns as functon of the angula veloctes = & φ & = φ = & φ of the end-effecto & & φ & = φ = & φ + φ = 4 = & φ n n φ & & φ ( & φ ) n + & φ & = + = ( & φ + ) φ = & φ + () n n n n & φ & = & φ + φ = = & φ. n n Statng fom the elatonshp = J the expesson of the Jacoban matx s easly wtten n an nvaant fom J = / n /( nn ) J = n n (4) / n n + ( n n) nn nn T whee [ ] = denotes the nput vecto of thee angula veloctes T and = [ & φ & φ & φ] s the vecto of jont angula veloctes of the end-effecto. Ths squae nvetble matx s an essental element fo the analyss of the loc nto obot wospace. Let us assume now that the obot has successvely thee ndependent vtual motons. haactestc vtual veloctes expessed as functon of mechansm s poston ae gven by the elatons (). Fst we consde the followng nput v v v angula veloctes a = a = a = and we obtan a set of vtual veloctes: v v v a = n a = ( n + n) 4a = n ( n + n). (5) v v second vtual moton s defned by the nput veloctes b = b = v b = and the followng esults: v b = n v b = n v = 4b nn. (6) v v v Fnally fom the thd vtual moton c = c = c = we obtan v v v = c = n 4c = nn. (7) c

9 9 Invese dynamcs of the ncnnat-mlacon wst obot 8 oncenng the elatve angula acceleatons of the compoundng elements of the mechansm these ae mmedately gven by devng the elatons of the veloctes (): ε =&. The angula acceleatons ε and the useful squae matces ~ ~ ~ + ε ae calculated wth the followng fomulae [9]: ~ T ε = p ε + ε u + p p u (8) ~ ~ ~ T + ε = p ( ~ ~ + ~ ε ) p + ~ ~ ~ ~ T + ~ u u + ε u + p p u. The velocty v and the acceleaton γ of mass cente of T gd body ae calculated fom two nown matx elatons v = ~ γ = { ~ ~ ~ } + ε (9) whee z = z =.5( l l ) z = / z 4 = l +. 8l z = l z T = l4 = [ z ] ( = ). () 4. DYNMIS EQUTIONS 4.. PRINIPLE OF VIRTUL WORK Thee toques of moment m = mu m = mu m = mu can contol by ntemedate of electc motos the moton of the wst mechansm. The devaton of a dynamc model has a vey mpotant effect n the detemnaton of the actuato toques (Tsa []; Mulle Mannhadt and Glove []; astllo []). In the nvese dynamc poblem n the pesent pape one apples the pncple of vtual wo n ode to establsh some ecusve matx elatons fo the toques and the powes of the actve systems. The paallel mechansm can atfcally be tansfomed n a set of thee open seal chans ( = ) subjected to the constants. Ths s possble by cuttng open at thee bevel gea pas a 4a a b anda c and tang the effects nto account by ntoducng the coespondng constant condtons. onsdeng that the end-effecto moton s gven the poston angula velocty angula acceleaton as well as the velocty and acceleaton of the cente of mass ae nown of each element. The foce of neta and the esultng moment of the foces of neta of an abtay gd bodyt fo example n f m ~ ~ ~ n = + ) Jˆ ε ~ Jˆ = () ( ε m

10 8 Ştefan Stacu ae detemned wth espect to the common cente of otaton O. On the othe hand the wench of two vectos f and m evaluates the nfluence of the acton of the extenal and ntenal foces appled to the same elementt o of ts weght m g fo example: * f = m ga u * ~ m m g a u = ( =...6). () Fnally two ecusve elatons geneate the vectos T F = F + a+ F+ M = M + a M + % a F T T () whee one denoted F = f f n n M = m m. (4) In the context of the eal-tme contol neglectng the fctonal foces and consdeng the gavtatonal effect and the acton of a esstant toque M though * E * E the vectos f = m = M u the elevant objectve of a dynamc model s to detemne the nput toques whch must be exeted by the actuatos n ode to poduce a gven tajectoy of the end-effecto. The fundamental pncple of vtual wo states that a mechansm s unde dynamc equlbum f and only f the vtual wo developed by all extenal ntenal and neta foces vansh dung any geneal vtual dsplacement whch s compatble wth the constants mposed on the mechansm. pplyng the fundamental equatons of paallel obots dynamcs [] followng compact matx elatons esults fo the toques of thee actuatos T v v v v m = u { am + am + am + 4aM 4 } T v v v v m = u { bm + a M + am + 4bM 4 } (5) T v v v v m = u M + M + M + M }. { c c c 4c 4 The elatons () and (5) epesent the nvese dynamc model of the ncnnat-mlacon wst mechansm. The pocedue leads to vey good estmates of the actuatos toques fo gven dsplacement of end-effecto povded that the netal popetes of the geas ae nown wth suffcent accuacy and that fcton s not sgnfcant. Ths new dynamc appoach developed hee can be extended to any gyoscopc bevel-gea tan wth evolute actuatos.

11 Invese dynamcs of the ncnnat-mlacon wst obot EQUTIONS OF LGRNGE soluton of the dynamcs poblem of the ncnnat-mlacon mechansm can be developed based on the Lagange equatons of second nd. The genealzed coodnates of the obot ae epesented by the otaton angles of the thee actuatos: q = q = q =. The Lagange s equatons wll be expessed by thee dffeental elatons that contan followng genealzed foces Q d L L { } = Qj ( j = ) (6) dt q& q j j = m + M n ( n + ) Q = m M nn Q = m M nn. (7) n The components of the geneal expesson 4 L = L + L + L = of the Lagange functon ae expessed as analytcal functons of the genealzed coodnates and the fst devatves wth espect to tme: L T T T = J ˆ m gu a T L Jˆ T = L Jˆ = (8) whee the angula veloctes have the expessons: = q& u = n ( q& q& ) u = ( n + n)( q& q& ) u 4 = n[( n + n) q& nq& nq& ] u = q& u = q& u. (9) The absolute angula veloctes and the fst devatves of othogonal matces p ae expessed as follows: = a + ~ = a + T T T p& = u p p = & p & = 4 a4 + 4 & ~u ()

12 84 Ştefan Stacu p ~ T = u p p T p T = ~ u ( p = a b c). In the nvese dynamcs poblem a long calculus of the devatves wth espect to tme d L ( j ) dt = q of above functons leads fnally to the same & j expessons (5) fo the nput toques m m m equed by actuatos. Fo smulaton puposes let us consde a mechansm whch has the followng chaactestcs l =.75 m l =. 45m l =. 5 m l =. 5 m l4 =. 55 m =.65 m =. 5 m =. 5 m =. m 4 =. m 5 =. m m =.75g m =. 6 g =. m 6 g =. m4 5 g m =. 5 g m =. 4 g * π * π * π φ = φ = π φ = π α = M =. 5 Nm t = 6 s. ased on the algothm deved fom the above elatons () (5) a compute pogam solves the nvese dynamcs modellng of the mechansm usng the MTL softwae. ssumng that a esstant toque of constant moment M =.5Nm appled at the end-effecto and the weghts m g of compoundng gd bodes consttute the extenal foces actng on the mechansm dung ts evoluton a numecal computaton n the dynamcs s developed based on the detemnaton of the thee nput toques m m m (Fg. 5) and the actve powes P m = b P m = and P m = (Fg. 6). 5. ONLUSIONS Wthn the nvese nematcs analyss some exact matx elatons gvng the poston velocty and acceleaton of each ln fo the ncnnat-mlacon -DOF wst mechansm have been establshed. Most of dynamcal models based on the Lagange fomalsm neglect the weght of ntemedate bodes and tae nto consdeaton the actve foces o moments only and the wench of appled foces on the end-effecto. The numbe of elatons gven by ths appoach s equal to the total numbe of the poston

13 Invese dynamcs of the ncnnat-mlacon wst obot 85 vaables and Lagange multples nclusve. lso the analytcal calculatons nvolved n these equatons pesent a s of eos. The commonly nown Newton- Eule method whch taes nto account the fee-body-dagams of the mechansm leads to a lage numbe of equatons wth unnowns ncludng also the connectng foces n the jonts. Fnally the actuatng toques could be obtaned. Fg. 5 Input toques m m m of the thee actuatos. The dynamcs model developed n the pesent pape taes nto consdeaton the mass the tenso of neta and the acton of weght and neta foce ntoduced by all compoundng elements of the mechansm. ased on the pncple of vtual wo ths appoach s vey effcent can elmnate all foces of ntenal jonts and establshes a dect detemnaton of the tme-hstoy evoluton of powes equed by the actuatos. The method descbed above s qut avalable n fowad and nvese mechancs of all seal o paallel mechansms the platfom of whch behaves n tanslaton otaton evoluton o geneal sx-degees-of-feedom moton. The matx elatons gven by ths dynamc smulaton can be tansfomed n a model fo automatc contol of the sphecal mechansm.

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