Dexterous Manipulation of an Object by Means of Multi-DOF Robotic Fingers with Hemi-Spherical Rigid Tips

Transcription

1 Dtros Manplaton o an Objct by Mans o Mlt-DOF Robotc Fngrs wth Hm-Sphrcal Rg ps P..A. Ngyn +, S. Armoto ++, C.H. Ngyn +, M. Yosha ++ +Dpartmnt o Instral Atomaton, Hano Unvrsty o chnology ++ Dpartmnt o Robotcs, Rtsmkan Unvrsty Abstract hs artcl ams to sgn a ynamc mol an propos a nw control ramwork or moton o al mlt-dof robotc ngrs wth hm-sphrcal rg tps holng an objct wth non-paralll lat sracs n a horzontal workspac. Frstly, by sng th Lagrang mtho an Hamlton s prncpl, a mathmatcal mol o ynamcs o th gnral objct-ngrs stp has bn scrb as a systm o algbrac rntal qatons compos o ornary rntal qatons govrnng th ynamcs o th ngrs an th objct an a st o algbrac constrants govrnng tght rollng contacts btwn two ngr-tps an objct s sracs. Sconly, two nonlnar back schms or stabl graspng o th objct an controllng ts orntaton has bn propos an asymptotcal convrgnc o th clos-loop ynamcs o th gnral ngrs-objct has bn analyz. h notworthy pont n th propos control schm s that th sr orc s not a constant, bt a ncton o th actal rotatonal angl o th objct. Fnally, nmrcal smlaton rslts hav rconrm th ctvnss o th propos control law. Introcton In rcnt cas a goo nmbr o robotc hans whch can prorm tros opratons lk hman han hav bn sgn. ypcal sgns ar th thr-ngr Stanor/JPL han, or-ngr Utah/MI han, an - ngr MEL han []. h s o hman-lk hans nhancs th applcablty o manplator systms n varos tros opratons wth objcts or srronng nvronmnt to thr possblty o ast an n moton. Frthrmor, han s capabl o both compltng gross motons an n moton ajstmnts to a grasp objct. h scon capablty prsss trty o robotc ngr hans n varos manplaton wth objcts as compar wth only pck an plac opratons o almost nstral robots nowaays. In gnral, manplaton o an objct arsss th control problms as ollows: ( to manplat th grasp objct an ( to mantan graspng stablty rng manplaton. In orr to complt sccsslly sch goals, rstly a mathmatcal mol or moton o mlt-ngr hans manplatng a grasp objct shol b rv an thn a control law wol b sgn bas on t. Howvr, thr s a arth o paprs that trat th graspng stablty togthr wth anothr hghr lvl problms o ralzaton o tros moton or th objct rng stabl graspng n a ynamc sns. Accorng to laborat rvws o Shmoga [] an Bcch t al []., most pblsh works on mlt-ngr hans ocs manly on plannng or grasp an sng opn-loop control laws or objct hanlng. Among ths works, sch o-ln grasp synthss mthos can b class nto two man approachs as (gomtrc orc closr grasp synthss, an (optmzaton-bas grasp synthss. Both approachs am to trmn th bst grasp congraton an ngr motons bas on th knowlg o th sr moton o th objct. In tals, knowng th nomnal moton o th objct, an nvrs knmatc problm has mst b solv or th corrsponng ngr motons an contact orcs an thn postonng o ngr-tps shol b ct by an opn-loop control or by a spcal prpos robotc arm. hs las to a consrabl problm: th nvrs knmatc problm rqrs bg calclaton tm an provs act rslts only whn paramtrs o th systm ar prctly known. Howvr, t s mpossbl to trmn actly paramtrs o th ngr hans whos mchancal strctr s sgn or tros manplaton. hs mans that o-ln plannng an opn-loop controls or graspng an manplatng o an objct may not b val n ral task to ncrtanty. It s known that ctv snsory-back control algorthms or th ynamc systm to lll th rqr tasks shol b sgn bas on th ts ynamc mol. Howvr, only w sts al wth th problm o stabl grasp o an objct rng manplaton by mans o mlt-ngr robotc hans n a ynamc sns. Only n th rcnt yars, a ynamc mol o th lmp-paramtrzng systm wth snsory-back control sgnals hav bn analyz, whch nabl a stp o al mlt-dof ngrs to grasp an manplat a rg objct n -mnsonal workspac (Armoto t al [4,5,6]. h athors st nmros stps o al mlt-dof ngrs an pont ot that th trty o ths ngrs rng objct manplaton pns on th nmbr o DOF o ach stp. In th rst stag, th athors attmpt to

2 ralz stabl graspng rng hanlng a paralll lat srac objct by mans o al ngrs whos tps ar covr by sot an ormabl matral sch as slcol gl or rbbr. hror contacts btwn th ngrtps an th sracs o th objct ar ara-contact. ght ara contact constrants ar trmn bas on th assmpton that th ngrtps roll prly on th corrsponng sracs. By sng th Lagrang mltplr mtho an Hamlton s prncpl, w rv th ynamc qatons whch scrb th rlatons o ngrs-objct moton an constrant orcs nr knmatc constrants. Bas on th rv ynamc mol, snsory-back control algorthms hav bn vlop or th sr objct moton an sr contact orcs btwn th sot ngrtps an objct. In th scon stag, th problm has bn tn to al ngr robotc hans wth hm-sphrcalrg tps graspng an manplatng a rg objct whch has non-paralll lat sracs by Armoto t al. [7,8,9]. h athors hav propos snsory-back control npts constrct on th knowlg o ngr paramtrs, masrmnt ata o jont angls an vlocts, an th rotatonal angl o th objct. Howvr, th athors assm that th angl btwn two lat sracs an th vrtcal plan ar qal, whcg mans th objct s symmtrc. In ths papr, w vlop a nw control ram work or stabl graspng an manplatng a nonsymmtrcal rg objct by mans o mlt-dof ngrs wth hm-sphrcal rg tps. Consqntly, rst w apply th Lagrang mtho an Hamlton s prncpl to rv a mathmatcal mol o ynamcs o th objctv systm. Sconly, a nonlnar back schm or stabl graspng o th objct an controllng ts orntaton has bn propos an asymptotcal convrgnc o th clos-loop ynamcs has bn analyz. Fnally, nmrcal smlaton rslts hav rconrm th ctvnss o th propos control law. Systm scrpton A par o al mlt-dof ngrs wth sphrcal rgtps or ralzng stabl graspng an tros manplaton o a rg objct has bn llstrat n Fgr. hs ngrs ar attach to a ram an assm to work as two thr-dof planar robots. Vctor q =(q,q,q nots th jont angls o th lt han s ngr an vctor q =(q,q,q or thos o th rght han s ngr. h ngrtps ar rg an hmsphrcal-shap wth ras r. hroghot ths papr, w assm that th objct to b grasp hav two non-paralll lat sracs. o smply th problm, w assm that th workspac o th ngrs-objct systm s rstrct to a horzontal plan an thn th ct o gravty s nglct. h rrnc ram {oy} locats at th bas poston o th lt ngr, th objct ram {OXY} locats at th mass cntr o th objct. Poston o th movng ram {OXY} s n n th ram {oy} by q q q O o ( 0,y 0 o o y θ 0 λ l L Y θ O l Y θ 0 Y X q q q O o ( 0,y 0 Fg. : A par o al -DOF ngrs holng a nonparalll srac objct vctor Oc.m.=(,y an th rotatonal angl btwn ths rams s consr as th rotatonal angl θ o th objct. hn w n vctor z=(,y,θ notng both poston an orntaton o th objct n th bas ram {oy}. Frthr, th trms l j, m j, an I j not lngth, mass, an nrta momnt o lnk j o ngr rspctvly, or,j=, an th trm M, I not mass an nrta momnt o th objct, rspctvly. h stanc btwn two orgns o ths rams s not by L. Nt, ϕ nots th angl btwn th last lnk o th ngr an th X-as o th objct ram {OXY}. Coornats o O o -th npont o th last lnk o ngr as n n th bas ram {oy} ar 0,y 0. Nt, th coornats o th contact ponts O n th bas ram {oy} ar,y an n th objct ram {OXY} ar l an Y. h lt srac o th objct nclns to Y-as o th objct ram by an angl θ 0 an th rght srac to Y-as by an angl θ 0. In th rcnt pblsh work o Armoto t al., θ 0 qals θ 0. In ths papr, w consr th gnral cas o θ 0 os not qal to θ 0. From Fg., t s possbl to obtan gomtrc constrants as ollows: Q = ( + ( y Q = ( 0 0 ( y 0 cos( θ θ ysn( θ θ cos( ysn( 0 0 r l r l ( (

3 h tght contact constrants can b ormlat n th ollowng orm = Y ( 0 n = n= = Y c r ( + θ θ0 qn = n= R R [ c r π + θ θ q ] 0 ( [ ] 0 ( 4 π whry (=, can b calclat n th orm Y = ( 0 Y = ( 0 sn( θ θ 0 + ( y ( y 0 y 0 (5 y (6 0 or ngr : or th objct: H ( q + t J J 0 o H& ( q cos( S ( q r λ =, ( Formlaton o ynamc systm h ynamc qatons o th ovrall ngrs-objct systm ar ormlat bas on th Lagrang mtho an Hamlton s prncpl. Frstly, w n th Lagrangan ncton o th systm L=K-P+Q+R (7 n whch K s th knmatc nrgy gvn by M && + λ sn( θ θ My && + sn( θ θ 0 + λ Iθ&& Y + Y 0 o + + λ o + λ + λ l λ l whr J o or =, ar Jacoban matrcs n by ( (4 (5 ( & ( & & θ & H q q + M + My + I = K = (8 Potntal nrgy P qals zro to th ovrall systm works n th horzontal spac. Q an R ar scalar nctons whch ar gnrat rom th constrants Q an R. By ntrocng Lagrang mltplrs λ an, or =,, Q an R can b scrb as Q = + (9 R = λ + (0 Q Q R λr Nt, applyng Hamlton s prncpl, w obtan th ynamc qatons o th ovrall systm as ollows: t 0 J o = q It s possbl to obtan y0 q 0, (6 ( + τ = K( t K(0 (0 (7 K hror th systm satss passvty. Now t s possbl or s to scss th problms o sgnng control algorthms an smlaton rslts. 4 Ralzng stabl graspng an sr orntaton rglatng or ngr : H( q + t H& ( q + S( q, sn( θ θ0 J0 r cos( 0 λ θ θ cos( 0 θ θ + Jo = sn( 0 θ θ ( h control problm that shol b solv hr ncls two mans contnts:( to mantan stabl grasp an ( to rotat th grasp objct to th sr orntaton. For th rst contnt o th control, w mst rglat th contact orcs btwn th ngrtps an th corrsponng sracs to achv an approprat val. Armoto t al. hav propos control npts or th problm bas on balanc o orcs an momnts appl to th objct n whch th contact orcs ar constant. In ths papr, w propos a nw control schm to mantan th contact orcs at approprat varyng vals whl rglatng th objct to th sr rotatonal angl.

4 Frstly, w n som mportant trms an Y = Y Y = Y l = Y 0 0 l = Y sn( θ θ 0 + l sn( θ θ l 0 + l l It s asy to obtan th ollowng rlatons an l l Y Y = ( = ( y = ( y = ( y r y + r r r sn( θ θ hat las to th ollowng rlatons: Y Y r r l + l = ( y sn( θ θ = ( 0 0 y 0 0 r 0 0 r (8 (9 (0 ( ( ( (4 (5 (6 (7 Eqs. (8 an (0 play an mportant rol n nng a control sgnal or balanc o two rotatonal momnts. W propos two optons or rglatng th trm (Y - Y to zro. In th rst opton, by notng q. (0 w propos a control npts or stabl graspng as ollows J K = J K 0 v J 0 sn( θ θ sn( θ θ cos( θ θ r r 0 = J 0 v 0 cos( whr =(,, r r 0 sn( θ θ Y 0 0 Y 0 ( ( Clarly that or =, can b wrttn n th ollowng compact orm: By rntatng q. (6 w obtan Y& Y& = ( y& whr [ r 0 = J y& J + r 0 0 Rθ& 0 ] θ& (8 R = r + r cos( θ + ] (9 [ 0 θ0 Frthrmor, rom qs. (5,6,8,9 w obtan Y& Y& = r 0 r + θ& ( (0 = J 0 r 0 r = J sn( θ θ 0 r 0 r Y 0 W n th trms as ollows: 0 K v Y K ( v (4 = cos(θ 0 -θ (5 = cos(θ 0 +θ (6 λ =λ - sn(θ 0 -θ (7 λ =λ - sn(θ 0 +θ (8

5 Frstly, w choos = an sbsttt qs. (, an (5,6,7,8 nto th ynamc qatons o th ovrall systm (,,,4,5, th clos-ynamc qatons o th systm can b wrttn n th ollowng orm: or ngr : H( q q H& && + ( q + S( q, cos( 0 θ θ + J0 sn( 0 θ θ J 0 sn( θ θ cos( θ θ r 0 + or ngr : 0 cos( r λ Y H ( q & + H& ( q + S ( q cos( 0 J 0 sn( 0 J 0 r or objct: M && + λ sn( θ θ My && + 0 sn( θ θ 0 + λ Iθ&& Y + Y + λ l r λ 0 λ l + λ = K λ, Y + v = K Y 0 v (9 (40 (4 (4 (4 t E = = K v (44 whr ( & + ( & + & + θ & H q q M My I E = = Y + ( ( D to th act that ( 0-0 s always postv, th qantty E s non-ngatv nt an ts rvatv s nonpostv nt, bt E can not b a Lyapnov ncton as pont ot n Armoto s paprs. Howvr, smlarly to thortcal analyss o Armoto s paprs, t can b provn th convrgnc o th clos-loop ynamc systm as ollows:, 0 &, y&, θ& 0 as t (46, 0 Λλ, λ 0 Nt, w propos control npts θ (=, or rglatng th rotatonal angl θ o th objct to th spc val θ. 0 θ = J 0 r 0 = J 0 r θ 0 cos( 0 β θ + αθ& l + l β θ + αθ& l + l (47 (48 whr θ=θ-θ. θ s th sr rotatonal angl o th objct. α an β ar postv constants. W constrct th ovrall control npts or both stabl graspng an rglatng orntaton o th objct by applyng th sprposton prncpl n sch a mannr that = + θ =, (49 Sbstttng nto th ynamc qatons (9,40 o th ovrall systm an mplmnt som mportant transormatons, t s possbl to obtan th ollowng rlaton E & & θ = αθ& q K vq (50 t = whr akng nnr procts btwn rvatvs o q, q,, y, θ wth qs. (9, 40, 4, 4, 4 rspctvly, thn smmng th rsltng qatons an rr to q. (0, t s possbl to obtan th ollowng rlaton

6 β θ E θ = E + (5 It s possbl to show th convrgnc that q &,, &, y&, θ& 0 = cos( θ = cos( θ θ = θ θ 0 λ = λ sn( θ λ = λ sn( θ θ 0 + θ 0 θ 0 + θ 0 as t (5 Eθ = ( & + ( & + & H q q M My = ( + Y Y θ + β R W assm that π / < θ θ π / < h proo or th abov conclson s omtt n ths papr snc t s smlar to th thortcal analyss prsnt alray n th paprs o Armoto t al. [7,8]. 5 Smlaton In th scon opton, w propos th nw control npts by changng th last trm o qs. (,. Not that th nw trms ar sgn bas on th q. (8. = J 0 r sn( θ θ J 0 Y r cos( θ θ + r K v 0 = J 0 r J 0 Y + r cos( θ θ + r t K E v 0 It s possbl to pont ot that θ whr = = K v αθ& 0 0 (5 ( < π < π / / + Iθ& (56 (57 (58 thn th trm R n n q. (9 s postv an las to that E θ s also a non-ngatv nt ncton. hror, w hav smlar conclsons prsnt n (5. In orr to rconrm th thortcal nngs, w carry ot a smlaton work n whch al -DOF ngrs grasp an hanl an objct. Paramtrs o th ngrs an objct ar rport n abl. abl. Physcal Paramtrs Notaton Val m j lnk mass 0.05[kg] l j lnk lngth 0.05[m] I j lnk nrta momnt 5.08E- 6[kg.m ] M objct mass 0.05 I objct nrta momnt.0467e- 5[kgm ] θ 0 lt-ncln angl 0.7[Ra] θ 0 rght-ncln angl 0.[Ra] l stanc rom O c.m. to 0.05[m] lt-srac l stanc rom O c.m. to 0.055[m] rght-srac r =r ras o two ngrns 0.0[m] L stanc btwn two orgns o two ngrs 0.084[m] (55 In orr to smlat th ovrall ynamc systm, w s th constrant stablzaton mtho (CSM propos by Bamgart. CSM stats that th scon-orr rntal qatons wth gvn algbrac constrants ar stablz accorng to a lnar back control thory as ollows

7 Q&& + γ Q&& + γ R&& + γ R && + γ λ λ Q& + ω Q& + ω R& + ω λr R& + ω R λ Q Q (59 Paramtrs γ, γ λ, ω an ω λ n th qaton shol b chosn sch that th ampng pols o ths qatons ar plac ar n th lt-han s o th compl plan, n orr to mak nmrcal ampng sccssl. In ths cas, w choos ω = γ / 4 ω λ = γ λ / 4 (60 Fg. : Contact orc at ngr W consr two optons prsnt n scton Smlaton rslts o th rst opton W apply th control npt = + θ (or =, whr ar prsnt n qs. (,4 an θ (=, ar propos n qs.(47, 48. It s possbl to s that both th control sgnals an θ ar constrct bas on th masrmnt ata o jont angls an vlocts, an objct angls an vlocts. Not that (Y -Y an (l +l ar calclat by qs. (6, 7 rspctvly. Clarly th poston o mass cntr o th objct s not rqr to calclat thm. W nvstgat th contact orcs an normal to objct sracs, λ an λ tangnt to objct sracs, rotatonal angl o th objct, an or holonomc constrants Q, Q, R,R. h control paramtrs n ths nvstgaton ar chosn n abl. Fg. : Contact orc at ngr abl : Control paramtrs Symbol Notaton Val K vj D-gan α D-gan 0.0 β P-gan 0.77 sr orc 0.5[N] θ sr objct 0.0[Ra] angl γ λ =γ CSM paramtr 98 Fg. 4: Rotatonal angl o th objct

8 Fg. 5: Constrant orc at ngr Fg. 8: Constran R ( or =, It s possbl to concl that th smlaton rslts ar concnt to th thortcal nngs as pont ot n q. (46. h contact orcs an convrg to th sr nctons as shown n Fgs. (,. h rotatonal angl θ convrgs to th sr val θ as shown n th Fg. (4. h constrant orcs λ an λ convrg to th sr nctons as shown n Fgs. (5,6. Fnally th constrants Q an R convrg to zro as prsnt n Fgs. (7, 8 rspctvly. 5. Smlaton rslts o th scon opton Fg. 6: Constrant orc at ngr W apply th control npts = + θ whr ar propos n qs. (5, 54 an θ ar smlar to thos o th rst opton. Control paramtrs ar chosn n abl. abl : Control paramtrs Symbol Notaton Val K vj (=,; D-gan 0.0 j=, α D-gan 0.04 β D-gan 0.78 sr 0.5 orc γ CSM 90 paramtr γ λ CSM paramtr 90 Fg. 7: Constran Q ( or =, Smlarly to th rst cas, w nvstgat th transnt bhavors o th contact orcs,, λ, an λ, th rotatonal angl θ, an constrants Q an R as monstrat rom Fg. (9 to Fg. (5.

9 Fg. 9: Contact orc at ngr Fg. : Constrant orc at ngr Fg. 0: Contact orc at ngr Fg. : Constrant orc at ngr Fg. : Rotatonal angl o th objct Fg. 4: Constrant Q ( or =,

10 Rrncs [] A.M. Okamra, A. Smaby, an Mark R. Ctkosky, An ovrvw o tros manplaton, Proc. IEEE Int. Con. Robotcs an Atomaton, 000, pp [] K.B. Shmoga, Robot grasp synthss algorthms: A srvy, Int J Robotcs rsarch, 996,, pp [] A. Bcch, Hans or tros manplaton an robst graspng: A clt roa towar smplcty, IEEE rans. on Robotcs an Atomaton, 000, 6, pp Fg. 5: Constrant R ( or =, It s possbl to s that th prormancs o th ynamc systm n th opton ar approachng to thos o th opton to th constrants Q an R or =, whch convrg to zro wthn 0. scon. Clarly th smlaton rslts n ths cas also rconrm th ctvnss o th propos control algorthm. 6 Conclsons hs papr alt wth control problm o objct manplaton by mans o a par o robot ngrs: ( to manplat th grasp objct an ( to mantan graspng stablty rng manplaton. h rst problm was solv n a ynamc sns by controllng th rotatonal angl o th objct to th sr angl wthot knowlg o ynamc paramtrs o th objct as wll as ts mass cntr poston. h scon problm was solv also n a ynamc sns by controllng th contact orcs at an approprat stabl rang. It s notworthy that th sr contact orcs an λ n ths papr ar not constants, bt ar nctons o th rotatonal angl o th objct. hs mans controls or th corrsponng two problms shol b ct smltanosly. h mportant pont hr s th propos control algorthms or two problms ar smpl n a mathmatcal sns an applcabl n prmnts. h ctvnss o th propos control npts was monstrat by th smlaton rslts. [4] S. Armoto, P..A. Ngyn, H.-Y. Han, an Z. Dolgr, Dynamcs an control o a st o al ngrs wth sot tps, Robotca, 000, 8, Part, pp [5] P..A. Ngyn, S. Armoto, an H.-Y. Han, Comptr smlaton o controll moton o al ngrs wth sot-tps graspng an objct, Proc. Japan-USA Symposm on Flbl Atomaton, 000, pp [6] P..A. Ngyn an S.Armoto, Dtros manplaton o an objct by mans o mlt-dof robotc ngrs wth sot tps, J. o Robotc Systm, 7, No.7, 49-6, 00. [7] S. Armoto, K. ahara, J.-H. Ba, an M. Yosha, A stablty thory on a manol: concrrnt ralzaton o grasp an orntaton control o an objct by a par o robot ngrs, Robotca,, pp. 6-78, 00. [8] S. Armoto, Intllgnt control o mlt-ngr hans, to b pblsh n th Sprng 004 Iss o Annal Rvw n Control. [9] S. Armoto, M. Yosha, J.-H. Ba, an K. ahara, Dynamc orc/torq closr o D polygonal objcts by a par o rollng contacts an snsor-motor coornaton, J. o Robotc Systms, 0, pp