A Generalzed Apprach On Desgn And Cntrl Methds Synthess Of Delta Rbt Trnh Duc Cung +84-90.89.28 cungtdc@htmal.cm Tung Phuc Th +84-909.160.264 tungphucth@gmal.cm Nguyen Trung Thnh +84-90.675.67 thnhnt@hcmute.edu.vn ABSTRACT Ths paper wll descrbe the knematcs and dynamcs f parallel rbt named Delta wth degree f freedm (d..f). The use f dynamcs cupled wth knematcs fr the cntrl f parallel rbt has been ganng ncreasng ppularty n recent years. Relatnshp between generalzed and artcular velctes s establshed, hence jacban and nverse jacban analyses are determnes. The nverse frmulas are generally shwn smply and the drect frmulas are als descrbed. Besdes, ths paper deal wth the drect and nverse dynamcs t determne the relatns between the generalzed acceleratns, velctes, crdnates f the end-effectr and the artcular frces based n smulatn and cntrl. Parallel rbts have becme the mprtant machnes t manufacturng. They are used fr varus purpses n ndustry and lfe. The dynamc mdel f parallel rbt wth df s presented, and an adaptve cntrl strategy fr ths rbt s descrbed. The rbustness f the cntrl system wth respect t the nnlnear dynamc behavr and parameter uncertantes s nvestgated by cmputer smulatn. Experments were mplemented t evaluate the respndng f cntrllng system based n dynamcs and knematcs cntrllng methd fr trackng desred trajectres. The results shw that the use f the sutable cntrl system based n dynamcs mdel can prvde the hgh perfrmance f the rbt. Categres and Subject Descrptrs B.1.2 [Cntrl Structures And Mcrprgrammng] Cntrl Structure Perfrmance Analyss and Desgn Ads -Autmatc synthess, Frmal mdels, Smulatn. General Terms Perfrmance, Desgn, Expermentatn, Verfcatn. Keywrds Delta platfrm, Desgn, Dynamcs, Delta Rbt, Parallel rbt,.. 1. INTRODUCTION Parallel rbts are clsed-lp mechansms presentng very gd perfrmances n terms f accuracy, regdty and abalty t manpulate large lads. Many applcatns n the feld f prductn autmatn, such as assembly and materal handlng, requre machnes capable f very hgh speeds and acceleratns. The parallel rbts are able t wrk n sme tasks wth a much better perfrmance. Hwever, there are stll several unanswered questns and few papers publshed studyng rbts wth parallel archtectures. Ths paper ntrduces a three d..f parallel manpulatr dedcated t pck-and-place: Delta Parallel Rbt. Frst a knematcs mdel f a Delta parallel rbt s btaned usng a generc gemetrcal frmulatn then the mdel s used fr a wrkspace analyss. Delta rbt has many advantages lke peratng requred accurary, rgdty and manpulatn f large lads. A Parallel Rbt s a mechansm that has lnks that frm clsed knematcs chans. Because f ths, Parallel mechansms have many advantages cmpared t seral mechansms, such as speed and accuracy. Generally, a parallel rbt s made up f a mble platfrm (end-effectr) wth n d..f, and a fxed base, lnked tgether by at least tw ndependent knematcs chans. Nrmally, each knematcs chan has a seres f lnks cnnected by jnts. Manpulatrs wth degrees prve extremely nterestng fr pck-and-place peratns. Several prttypes have been suggested. The mst famus rbt wth d..f s Delta. All the knematc chans f ths rbt are rtary actuatrs allwng t btan df n translatn. Ths paper ntrduces a -df parallel manpulatr archtecture Delta dedcated wth knematcs and dynamcs analyses t pck-and-place and develped t perfrm hgh speed and acceleratn. In ths artcle we have dscussed the nverse and drect knematcs slutn as well as dynamcs fr the Delta parallel rbt. Wth ths manpulatr t s ften dffcult t determne the knematcs and dynamcs analyses. Thus, ths paper ncludes fve seperated sectns. The man prpertes f parallel rbt s descrbed n sectn II as well as fcusng n knematcs and dynamcs analyses, respectvely. Experment and dscussns s establshed n Sectn IV. Fnally, n Sectn V s shwn the cnclusn.. Research Ntes n Infrmatn Scence (RNIS) Vlume1, May 201 d:10.4156/rns.vl1.6 2. KINEMATIC AND JACOBIAN ANALYSES In ths sectn, the descrptn and knematcs f the parallel rbt df are shwn n Fg.1. Generally, parallel rbt s a 179
clsed lp manpulatr s mre dffcult t calculate the knematcs. The mvng plate always stays parallel t the base platfrm and ts rentatn arund the axs perpendcular t the base plate s cnstantly zer. Thus, the parallelgram type jnts (frearm) can be substtuted by smple rds wthut changng the rbt knematc behavur. The revlute jnts (between the base plate and the upper arms and between the frearms and the travellng plate) are dentcally placed n a crcle. Thus, the travellng plate can be replaced by a pnt P whch the three frearms are cnnected t. The mdellng f Delta rbt has the assumptns lke as: 1, 2, are the rtate angle f lnk, d A s the dstance frm the center f the base (rgn) t the spn axs f the transmssn, F 1; F 2; F are the center f the spndle attached t the transmssn, r A s the dstance frm the center stand n cmpared t the prjectn axs f the arm t stand n. And L 1, L 2 are the length f 2 lnk as descrbe n Fg. 2. Because, the nverse knematcs f Delta parallel rbt s mre easer than Drect Knematcs (Frward Knematcs), s frstly the nverse knematcs s shwn. The nverse knematcs f a parallel manpulatr determnes the angle f each actuated revlute jnt gven the (x,y,z) pstn f the travel plate n base-frame. z j1 (1) 1 arctan yf y 1 J1 Such algebrac smplcty fllws frm gd chce f reference frame: jnt F 1J 1 mvng n YZ plane nly, s we can cmpletely mt X crdnate. T take ths advantage fr the remanng angles 2 and, we shuld use the symmetry f delta rbt. Frst, lets rtate crdnate system n XY plane arund Z-axs thrugh angle f 120 cunterclckwse. Weve gt a new reference frame XYZ, and t ths frame we can fnd angle 2, usng the same algrthm that we used t fnd 1. x0 x.cs 120 y.sn 120 y0 x.sn 120 y.cs 120 z0 z0 Nw the three jnt angles 1, 2 and are gven, and we need t fnd the crdnates (x 0, y 0, z 0) f end effectr pnt E 0. (2) Fg.2. Shws mdel smplfcatn f the Delta parallel rbt. Use f the vectr translatn f y-axs dsplacement, we have: OJ1 OF1 F1 J1 J1J1 () Wth a length f the vectr, the dstance frm the rgnal quadrant t the swvel pnt f the transmssn are: OF OF OF r d 2 2 1 2 A A Dstance frm center f the three spheres ntersect at the center base J J J J J J r B Radus f the sphere s L 2, s: F J L cs 1 1 2 1 F J L cs F J L cs We have: 2 r r d r (9) 2 2 A A B OJ OF F J J J 1 1 1 1 1 1 And (x, y, z) s the crdnates f sphere centers J 1, J 2, J. S the crdnate f J 1 s: T 0 r L cs L sn d x y z (11) 2 1 2 1 A 1 1 1 Smlarly we have the crdnates f J 2 and J as fllws: J x ; y ; z (( r L cs ) cs0 ;( r L cs ) sn 0 ; L sn d ) 0 0 2 2 J x ; y ; z ( ( r L cs )cs0 ;( r L cs )sn 0 ; L sn d ) 0 0 2 2 2 S the ntersectn f sphere here: A A (4) (5) (6) (7) (8) (10) (12) (1) Fg.1. Mdellng f Delta parallel rbt. 180
( x x ) ( y y ) ( z z ) L ( x x ) ( y y ) ( z z ) L ( x x ) ( y y ) ( z z ) L (14) 1 1 1 1 2 2 2 1 1 And, we have slutns lke as: x a z b d 1 0 1 0 (1) y a z b d 2 0 2 0 (14) b z0 (15) 2a Wth help frm cmputer ths equatn system can be slved. There wll be tw slutns that descrbe the tw ntersectn pnts f the three spheres. Then the slutn that s wthn the rbts wrkng area must be chsen. Wth the base frame {R} n ths case t wll lead t the slutn wth negatve z crdnate.. DYNAMIC ANALYSIS OF DELTA ROBOT One mprtant step n desgn prcess f a rbt s t understand the behavur f the devce as t mves arund ts wrkspace r dng a specfc task. Ths behavur s determned thrugh the study f the dynamcs f the mechansm, where the frces actng n the elements and trques requred by the actuatrs can be determned. Cnsequently, each cmpnent must be ptmzed n dmensns and materal t be used n the manufacturng prcesses. In sectn, the dynamcs f Delta parallel rbt s descrbed based n Lagrangan frmulatn, whch s based n calculus varatns, states that a dynamc system can be express n terms f ts knetc and ptental energy leadng n an easy way the slutn t the prblem. In addtn, t s cnsdered a gd ptn t be used fr real-tme cntrl fr parallel manpulatrs [4]. The Lagrange equatns can be derved. d L L g dt q q q k Q j j j 1 k (16) Where L s the Lagrange functn, where L = T - V, T s the ttal knetc energy f the bdy, V s the ttal ptental energy f the bdy, q s the k th generalzed crdnate, Q s a generalzed external frce, λ s the Lagrange multpler and g s the cnstran equatn. By emplyng the frmula abve t s pssble t determne the external frces f a bdy. Hwever, frctn frces are nt cnstrants even thugh they play an mprtant rle n the dynamcs analyss s they can be treated separately. The Lagrange multplers are derved as. 2 ( p h cs r cs a cs cs ) ( m m ) (17) p x 1 p b x 1 1 2 ( p h sn r sn a sn cs ) ( m m ) p y 1 p b y (18) pz a 1 m p m b pz mp mb g (19) c 1 2 ( sn ) ( ) ( ) When the Lagrangan multples are fund the actuatr trque can be determned as. 1 2 2 1 1 maa mba 11 ma mb gca cs11 2 2a 1 px cs 1 p y sn 1 h r sn 11 pz cs 11 1 2 2 1 2 maa mba 12 ma mb gca cs12 2 2a 2 px cs2 py sn2 h rsn12 pz cs 12 (20) (21) 1 2 2 1 ma a mba 1 ma mb gca cs1 2 (22) 2a px cs py sn h rsn1 pz cs 1 The analytcal nverse dynamcs slutns fr Delta parallel rbt can be btaned frm Eqs.(20-22) 4. EXPERIMENTS AND DISCUSSIONS T vald the analyses f knematcs and dynamcs n prevus sectn, an expermental setup was bult t perfrm the cntrl f Delta parallel rbt (Fg.). The specfcatns f Delta parallel rbt s shwn n Table 1. Table 1. Specfcatns f Delta parallel rbt Parameters Value Upper rbt arm m a [kg] 1.1 Parallelgram m b [kg] 0.9 Mvng platfrm m p [kg] 0.2 Radus f the fxed base a [mm] 150 Radus f the mvng platfrm b [mm] 100 Upper arm length l 1 [mm] 250 Parallelgram length l 2 [mm] 480 N. f AC Serv mtr Mtr pwer [W] 200 Encder reslutn [ppr] 1000 Maxmum lad capacty [kg] 5 Maxmum mvng platfrm velcty [m/s] 5.0 Pstn repeatblty [mm] 0.2 Dameter [mm] 500 Wrkspace Heght [mm] 200 Ths expermental mplementatn s bult n the PC and Delta rbt. The sftware fr Delta parallel rbt s mplemented n Matlab usng the knematcs and dynamcs analyses frm abve slutns t cntrl the mvng platfrm. The prpsed analyses are appled the Delta parallel rbt fr materal cuttng and drawng. The prgram s used t cntrl the mvng platfrm wth predefned trajectry. We wll apply the knematcs, Jacb and dynamcs t cntrl sutable trajectry f parallel rbt based n pstns, velctes. In the sectn, sme expermental results by knematcs - dynamcs cntrl are addressed. T demnstrate the capablty cntrller, several respnses were taken nt accunt wth several varus trajectres. In these experments, a pen attached t mvng platfrm f Delta parallel rbt s regulated fllwng the several predefned paths ncludng curves f crcle, butterfly, flwer, heart. 181
Fg.. Delta parallel rbt fr experments. The frst expermental results fr cntrllng the mvng platfrm wth cntur f flwer are llustrated n Fg.4. A curve has the shape f a petalled flwer and the plar equatn f the rse s fllws. r a sn n (2) The drawng n paper r cuttng n acrylc reveals that the analyss results are almst near the desred nes shwn n Fg.4(a). Cmpared desred cntur, we can see that the very small dfferences between the desred and expermental values may be attrbuted t the fllwng reasns: frst, there s errr f mechancal transmssn and calculatn f knematcs and dynamcs f Delta rbt. The mprvements wll brng better results fr generatng trajectres. And respndng f three AC serv mtrs wth tme s shwn n Fg.4(b). Next, ther respnses fr reference cmmands fr butterfly cntur are presented t evaluate the perfrmance f the cntrller based knematcs and dynamcs. The equatn f butterfly curve s fllws. sn 5 2 r e 2cs 4 sn 24 (24) Fg.5 shws utput f respndng trajectry and nput respnses fr cntur f butterfly. The cntrl results fr butterfly are gd enugh t track the perfect shape whle mvng path f pen has a lttle bt errr. Fg.4. Trajectry f mvng platfrm (a) and respndng f mtrs(b) wth flwer curve path. Fg.5. Trajectry f mvng platfrm (a) and respndng f mtrs (b) wth butterfly curve path. 182
5. CONCLUSION Ths paper s manly cncerned wth knematc and dynamc analyses as well as the applcatn f slutns f knematcs and dynamcs t mdelng and cntrl f parallel manpulatrs. A practcal mplementatn s cmpleted t evaluate the results f an desgned cntrller fr Delta manpulatr cntrl system. It can be sad that, excepted results has been acheved fr these cases. The nverse and frward knematcs and velcty equatns have been derved. The results presented n the paper wll be valuable fr bth the desgn and develpment f Delta parallel rbt fr varus applcatns. Wth the ad f cmputer, these equatns wth the desgn f ths rbt base n dynamc mdelng and dynamc cntrl n rder t mprve the behavr f the rbt whle reachng hgh acceleratn. By fttng grppers r ther tls t ths small platfrm the delta rbt can handle all srts f tems. Ther desgn enables them t mve bth rapdly and accurately, and they are deplyed fr tasks varyng frm hghspeed packagng t the assembly f mnature prducts. Fg.6. Trajectry f mvng platfrm (a) and respndng f mtrs(b) wth heart curve path. Besdes, we als generate the trajectry f heart curve wth ple equatn lke as: sn cs (25) r 2 2sn 7 sn 5 Fg.6(a) shws the actual tme respnse sgnals and the cmmand sgnals f parallel t a heart prfle, and the tme hstry f the cntrlled pstn utput. The mvement f mvng platfrm fllwed the cmmanded sgnals qute well fr lng tme. Present results shw that the analyses f knematcs and dynamcs can be successfully appled t the dynamc trackng f varus cntur prfles. 6. ACKNOWLEDGMENTS Ths study was fnancally supprted H Ch Mnh cty, Vet Nam (HCMUTE). 7. REFERENCES [1] André Olssn, Mdelng and cntrl f a Delta- rbt, 2009. [2] Jn Martínez García, Inverse-Frward Knematcs f a Delta Rbt, 2010. [] Manuel Naple and Cardna Guterrez, Knematcs Analyss f a Delta Parallel Rbt, 2011. [4] S.M.Ha, P.V.B. Ngc and H.S.Km, Dynamcs Analyss f a Delta-type Parallel Rbt, 2011 11 th Internatnal Cnference n Cntrl, Autmatn and System, 2012. [5] S.M.Ha, P.V.B.Ngc and H.S.Km, Dynamcs Analyss f a Delta-type Parallel Rbt, 2011 11 th Internatnal Cnference n Cntrl, Autmatn and Systems, pp.855-857, 2011. 18