Kinematics of A 3-PRP planar parallel robot

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Knematcs of -PRP planar parallel robot Damen hablat, Stefan Stacu To cte ths verson: Damen hablat, Stefan Stacu. Knematcs of -PRP planar parallel robot.

fr/hal-764 Submtted on 1 pr 9 HL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not.

1 Knematcs of -PRP planar parallel robot Damen hablat, Stefan Stacu To cte ths verson: Damen hablat, Stefan Stacu. Knematcs of -PRP planar parallel robot. UPB Scentfc Bulletn, Seres D: Mechancal Engneerng, 9, 71 (1), pp.-16. HL Id: hal Submtted on 1 pr 9 HL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 U.P.B. Sc. Bull., Seres D, Vol. 71, No., 9 KINEMTIS OF -PRP PLNR PRLLEL ROBOT Damen HBLT *, Stefan STIU ** rtcolul prezntă o modelare recurentă pentru cnematca unu robot paralel plan -PRP. Tre lanţur cnematce plane ce conectează platforma moblă a manpulatorulu sunt stuate în plan vertcal. unoscând mşcarea platforme, se dezvoltă cnematca nversă ş se determnă pozţle, vtezele ş acceleraţle robotulu. Unele ecuaţ matrceale oferă expres teratve ş grafce pentru deplasărle, vtezele ş acceleraţle celor tre acţonor de translaţe. Recursve modellng for the nematcs of a -PRP planar parallel robot s presented n ths paper. Three planar chans connectng to the movng platform of the manpulator are located n a vertcal plane. Knowng the moton of the platform, we develop the nverse nematcs and determne the postons, veloctes and acceleratons of the robot. Several matrx equatons offer teratve expressons and graphs for the dsplacements, veloctes and acceleratons of three prsmatc actuators. Keywords: nematcs, planar parallel robot, and sngularty 1 Introducton ompared wth seral manpulators, potental advantages of the parallel archtectures are hgher nematcal precson, lghter weght, better stffness, greater load bearng, stable capacty and sutable postonal actuator arrangements. However, they present lmted worspace and complcated sngulartes [1]. onsderable efforts have been devoted to the nematcs and dynamc analyss of fully parallel manpulators. The class of manpulators nown as Stewart-ough platform focused great attenton (Stewart []; Merlet []; Parent-astell and D regoro [4]) and s used n flght smulators and more recently for Parallel Knematcs Machnes. The Delta parallel robot (lavel [5]; Tsa and Stamper [6]; Stacu and arp-ocarda [7]) as well as the Star parallel manpulator (Hervé and Sparacno [8]) are equpped wth three motors, whch tran on the moble platform n a three-degree-of-freedom general translaton moton. ngeles, osseln, agné and Wang [9,, 11] analysed the nematcs, dynamcs and sngularty loc of a sphercal robot wth three actuators. * Dr., Insttut de Recherche en ommuncatons et ybernétque de Nantes, UMR NRS 6597, FRNE ** Prof., Department of Mechancs, Unversty Poltehnca of Bucharest, ROMNI

3 4 Damen hablat, Stefan Stacu mechansm s consdered a planar robot f all the movng lns n the mechansm perform the planar motons; the loc of all ponts n all lns can be drawn convenently on a plane and the axes of the revolute jonts must be normal to the plane of moton, whle the drecton of translaton of a prsmatc jont must be parallel to the plane of moton. B y y O B x x l l Fg. 1 The -PRP planar parallel robot In ther paper, radyfo and Qao [1] examne the nverse nematcs soluton for the three dfferent -DOF planar parallel robots. osseln and ngeles [1] and Pennoc and Kassner [14] each present a nematcal study of a planar parallel robot, where a movng platform s connected to a fxed base by three lns, each leg consstng of two bnary lns and three parallel revolute jonts. Sefrou and osseln [15] gve a numercal soluton n the nverse and drect nematcs of ths nd of robot. Mohammad-Danal et al. [16] present a study of velocty relatonshps and sngular condtons for general planar parallel robots. Merlet [17] solved the forward pose nematcs problem for a broad class of planar parallel manpulators. Wllams and Renholtz [18] analysed the dynamcs and the control of a planar three-degree-of-freedom parallel manpulator at Oho Unversty, whle Yang et al. [19] concentrate on the sngularty analyss of a class of -RRR planar parallel robots developed n ther laboratory. Bonev, Zlatanov and osseln [] descrbe several types of sngular confguratons by studyng the drect nematcs model of a -RPR planar parallel robot wth actuated base jonts. Mohammad-Danal et al. [1] analysed the nematcs of a planar -DOF parallel manpulator usng the three PRP legs, where the three revolute jont axes are perpendcular to the plane of moton whle the prsmatc jont axes le on the same plane.

4 Knematcs of a -PRP planar parallel robot 5 recursve method s developed n the present paper for dervng the nverse nematcs of the -PRP planar parallel robot n a numercally effcent way. Knematcs modellng The planar -PRP parallel robot s a specal symmetrcal closed-loop mechansm composed of three planar nematcal chans wth dentcal topology, all connectng the fxed base to the movng platform. Three ponts, B, represent the summts of a fxed trangular base and other three ponts defne the geometry of the movng platform. Each leg conssts of two lns, wth one revolute and two prsmatc jonts. The parallel mechansm wth seven lns ( T, 1,,...,7) conssts of three revolute and sx prsmatc jonts (Fg.1). rübler moblty equaton predcts that the devce has certanly three degrees of freedom. In the actuaton scheme PRP each prsmatc jont s an actvely controlled prsmatc cylnder. Thus, all prsmatc actuators can be located on the fxed base. We attach a artesan frame x y z ( T ) to the fxed base wth ts orgn located at trangle centreo, the z axs perpendcular to the base and the x axs pontng along the B drecton. nother moble reference frame x y z s attached to the movng platform. The orgn of the central reference frame x y z s located at the centre of the movng trangle (Fg. ). In the followng we wll represent the ntermedate reference systems by only two axes so as s used n many robotcs papers [1,, 9] and we note that the relatve translaton 1 and the rotaton angle, 1 pont along or about the drecton of z axs. We consder that the movng platform s ntally located at a central confguraton, where the platform s not rotated wth respect to the fxed base and the mass centre s at the orgn O of fxed frame. One of three actve legs (for example leg ) conssts of a prsmatc jont, whch s as well as a lnear drve 1 lned at the x1 y1 z1 frame, havng a rectlnear moton of dsplacement, velocty v and acceleraton. Followng ln of the leg s a rgd body lned at the x y z frame, havng a relatve rotaton about z axs wth the angle 1, velocty 1 1 and acceleraton 1 1. prsmatc passve jont s ntroduced at a planar movng platform as an equlateral trangle wth the edge l l, whch relatvely translates at the dsplacement and the velocty v along z axs.

5 6 Damen hablat, Stefan Stacu lso, we consder that at the central confguraton all legs are symmetrcally extended and that the angles of orentaton of three edges of fxed platform are gven by, B,. (1) y y O x x y y 1 v y ω 1 v x z z α Fg. Knematcal scheme of frst leg of the mechansm In the followng, we apply the method of successve dsplacements to geometrc analyss of closed-loop chans and we note that a jont varable s the dsplacement requred to move a ln from the ntal locaton to the actual poston. If every ln s connected to least two other lns, the chan forms one or more ndependent closed-loops. The varable angle, 1 of rotaton about the jont axs z s the parameter needed to brng the next ln from a reference confguraton to the next confguraton. We call the matrx q, for example, the orthogonal, 1 transformaton matrx of relatve rotaton wth the angle of ln around z., 1 T

6 Knematcs of a -PRP planar parallel robot 7 In the study of the nematcs of robot manpulators, we are nterested n dervng a matrx equaton relatng the locaton of an arbtrary T body to the jont varables. When the change of coordnates s successvely consdered, the correspondng matrces are multpled. So, startng from the reference orgn O and pursung the three legs O 1, OB B1 BB, O 1, we obtan the followng transformaton matrces []: T q 1, q1 q11, q 1 () wth ( q a, b, c), (, B, ) where q cos sn sn cos, 1 cos sn sn cos , 1 () 1 s1, s ( 1,, ) s1 q q. B The dsplacements,, of the actve lns are the jont varables that B T gve the nput vector [ ] of the poston of the mechansm. But, n the nverse geometrc problem, we can consder that the poston of the mechansm s completely gven by the coordnates, y of the mass centre x of the movng platform and the orentaton angle of the movable frame x y z. The orthogonal rotaton matrx of the movng platform from x yz to x y z reference system s cos sn R sn cos. 1 (4) Further, we suppose that the poston vector r [ x y T ] of centre and the orentaton angle, whch are expressed by followng analytcal functons

7 8 Damen hablat, Stefan Stacu x x y 1 cos t, (5) * * y * can descrbe the general absolute moton of the movng platform n ts vertcal * * * plane. The values x, y, denote the fnal poston of the movng platform. The condtons concernng the absolute orentaton of the movng platform are expressed by three denttes q T q R, ( q a, b, ) (6) c where the resultng matrx q s obtaned by multplyng three basc matrces q q q q, q q ( t (, B, ). (7) 1 ) 1 From these condtons one obtans the frst relatons between the angles of rotaton B (8) Sx ndependent varables,, B B,,, wll be determned by several vector-loop equatons as follows T T r q r 1, q r r, ( q a, b, c) (, B, ) (9) 1 where T T r r ( l / ) qu, r l[ 1 ] B 1 T 1 T r l[ 1 ], r l[ 1 ] 1 r, r q u, r l [ 1 ] T T 1 u 1, u 1, u, ~ 1 1 u 1 1. () ctually, these vector equatons mean that l cos cos l x x cos( ) l sn( )

8 Knematcs of a -PRP planar parallel robot 9 l sn sn l y y sn( ) l cos( ) (11) wth (, B, ). Developng the nverse nematcs problem, we determne the veloctes and acceleratons of the manpulator, supposng that the planar moton of the movng platform s nown. So, we compute the lnear and angular veloctes of each leg n terms of the angular velocty u and the centre s velocty v r of the movng platform. The rotaton motons of the elements of each leg (leg, for example) are characterzed by recursve relatons of followng sew-symmetrc matrces ~ T ~ a u~ ( 1,,), (1) a 1 1, 1 1, 1 1 whch are assocated to the absolute angular veloctes, a 1 1 u, a u. (1) Followng relatons gve the veloctes v of jonts v u, v 1, v u v ~ a v a r v u. (14), 1 1,, 1 1,, 1, 1 The geometrcal equatons of constrants (8) and (9) when dfferentated wth respect to tme lead to the followng matrx condtons of connectvty [] T T T T T v u a u a u u r T T ~ T T ~ T u a u a u a u a r } ( j 1, ) j u v j j 1 j { 1, (15) where ~u s a sew-symmetrc matrx assocated to unt vector u pontng n the postve drecton of z axs. From these equatons, we obtan the relatve veloctes v, 1, v as functons of angular velocty of the platform and velocty of mass centre and the complete Jacoban matrx of the manpulator. Ths matrx s a fundamental element for the analyss of the robot worspace and the partcular confguratons of sngulartes where the manpulator becomes uncontrollable. By rearrangng, the dervatves wth respect to tme of the sx constrant equatons (11) lead to the matrx equaton T J J x y ] (16) 1 [

9 Damen hablat, Stefan Stacu for the planar robot wth prsmatc actuators. The matrces J 1 and J are the nverse and forward Jacoban of the manpulator and can be expressed as J dag { } 1 B J 1 B 1 1 B B, (17) wth sn( ) (, B, ) 1 sn( ), cos( ) (18) ( x x )cos( ) ( y y )sn( ) l ( )cos( ). The sngular confguratons of the three closed-loop nematcal chans can easly be determned through the analyss of two Jacoban matrces J 1 and J [4, 5]. For the matrx J 1, the determnant vanshes f /, whch leads also to a sngular confguraton of J. oncernng the relatve acceleratons, 1, of the robot, new connectvty condtons are obtaned by the tme dervatve of equatons n (15), whch are [6] T T u a T T T a u u r j u u j j } T 1 1 u T ~ ~ T T ~ T j { auuau auuar T T ~ T T ~ T T T u a u a u a u a r } ~ T 1vu j auau ( j 1, ) 1 j { 1. (19)

10 Knematcs of a -PRP planar parallel robot 11 The formulatons n (15) and (19) are for only and they also apply to the legs B and, f the superscrpt s replaced by ether B or. Followng recursve relatons gve the angular acceleratons and the acceleratons of jonts : u, 1, u, 1 u, a 1 1, 1 u, 1 a 1 ~ a u 1, T, 1 a a u u u T, 1 1, 1, 1,, 1, 1, 1, 1 a a u T, 1, 1 1,, 1 () a, 1 v, 1 1, a, 1 a, ~ 1, ~ ( ~ 1 1, 1, T a u, 1 ~ 1, ) r, 1 u, ( 1,, ), 1 We can notce that for ths robot, the dsplacement of the leg s very smple. The dsplacement of body s only along a fxed lne, and ts velocty and acceleraton s equal to those of actuated jont assocated. For smulaton purposes let us consder a planar robot, whch has the followng characterstcs: * * x.5m, y.5 m,, t s 1 l O OB O.m, l l program, whch mplements the suggested algorthm, s developed n MTLB to solve the nverse nematcs of the planar PRP parallel robot. For llustraton, t s assumed that for a perod of three second the platform starts at rest from a central confguraton and rotates or moves along two orthogonal drectons. numercal study of the robot nematcs s carred out by B computaton of the dsplacements,, B the acceleratons,, of three prsmatc actuators. B, the veloctes v, v, v, and Followng examples are solved to llustrate the smulaton. For the frst example we consder the rotaton moton of the movng platform about z axs wth varable angular acceleraton whle all the other postonal parameters are held equal to zero.

11 1 Damen hablat, Stefan Stacu Fg. Dsplacements, B B, Fg. 4 Veloctes v, v, v Fg. 5 cceleratons, B B, Fg. 6 Dsplacements,, Fg. 7 Veloctes, B B v v, v Fg. 8 cceleratons,, s can be seen from Fgs., 4, 5, t s remared that durng the rotaton moton of

12 Knematcs of a -PRP planar parallel robot 1 the platform all dsplacements, veloctes and acceleratons of three actuators are dentcally dstrbuted. In a second example, the supposed moton of the platform s a translaton on horzontal x axs (Fgs. 6, 7, 8). Fg. 9 Dsplacements, B B, Fg. Veloctes v, v, oncernng the comparson n the case when the centre moves along a rectlnear trajectory along y axs wthout any rotaton of the platform, we remar that the dstrbuton of dsplacement, velocty and acceleraton calculated by the program and depcted n Fgs. 9,, 11 s the same, at any nstant, for two of three actuators. v B, Fg. 11 cceleratons, The smulaton through the MTLB program certfy that one of major advantages of the current matrx recursve approach t s the well structured way to formulate a nematcal model, whch leads to a computatonal effcency. The

13 14 Damen hablat, Stefan Stacu proposed method can be appled to varous types of complex robots, when the number of components of the mechansm s ncreased. onclusons Wthn the nverse nematcs analyss some exact relatons that gve n realtme the poston, velocty and acceleraton of each element of the parallel robot have been establshed n present paper. The method descrbed above s qute avalable n forward and nverse mechancs of all seral or planar parallel mechansms, the platform of whch behaves n translaton, rotaton evoluton or general -DOF moton. REFERENES [1]. Tsa, L-W., Robot analyss: the mechancs of seral and parallel manpulator, John Wley & Sons, Inc., 1999 []. Stewart, D., Platform wth Sx Degrees of Freedom, Proc. Inst. Mech. Eng., 1, 15, 18, pp , 1965 []. Merlet, J-P., Parallel robots, Kluwer cademc Publshers, [4]. Parent-astell, V., D regoro, R., new algorthm based on two extra-sensors for realtme computaton of the actual confguraton of generalzed Stewart-ough manpulator, Journal of Mechancal Desgn, 1, [5]. lavel, R., Delta: a fast robot wth parallel geometry, Proceedngs of 18 th Internatonal Symposum on Industral Robots, Lausanne, pp. 91-, 1988 [6]. Tsa, L-W., Stamper, R., parallel manpulator wth only translatonal degrees of freedom, SME Desgn Engneerng Techncal onferences, Irvne,, 1996 [7]. Stacu, S., arp-ocarda, D.., Dynamc analyss of lavel s Delta parallel robot, Proceedngs of the IEEE Internatonal onference on Robotcs & utomaton IR, Tape, Tawan, pp , [8]. Hervé, J-M., Sparacno, F., Star. New oncept n Robotcs, Proceedngs of the Thrd Internatonal Worshop on dvances n Robot Knematcs, Ferrara, pp , 199 [9]. ngeles, J., Fundamentals of Robotc Mechancal Systems: Theory, Methods and lgorthms, Sprnger-Verlag, []. osseln,., agné, M., Dynamc models for sphercal parallel manpulators, Proceedngs of the IEEE Internatonal onference on Robotcs & utomaton, Mlan, Italy, 1995 [11]. Wang, J., osseln,., new approach for the dynamc analyss of parallel manpulators, Multbody System Dynamcs,,, 17-4, 1998 [1].radyfo, D.D., Qao, D., Knematc Smulaton of Novel Robotc Mechansms Havng losed hans, SME Mechansms onference, Paper 85-DET-81, 1985 [1]. osseln,., ngeles, J., The optmum nematc desgn of a planar three-degree-of-freedom parallel manpulator, SME journal of Mechansms, Trans. and utomaton n Desgn., 1, 1, pp. 5-41, 1988 [14]. Pennoc.R., Kassner, D.J., Knematc nalyss of a Planar Eght-Bar Lnage: pplcaton to a Platfoem-type Robot, SME Mechansms onference, Paper DE - 5, pp. 7-4, 199 [15]. Sefrou, J., osseln,., On the quadratc nature of the sngularty curves of planar three-

14 Knematcs of a -PRP planar parallel robot 15 degrees-of-freedom parallel manpulators, Mechansm and Machne Theory,, 4, pp , 1995 [16]. Mohammad-Danal, H., Zsombor-Murray, H.P., ngeles, J., Sngularty nalyss of Planar Parallel Manpulators, Mechansm and Machne Theory,, 5, pp , 1995 [17]. Merlet, J-P., Drect nematcs of planar parallel manpulators, Proceedngs of the IEEE Internatonal onference on Robotcs & utomaton, Mnneapols, Mnnesota, pp , 1996 [18]. Wllams II, R.L., Renholtz,.F., losed-form Worspace Determnaton and Optmzaton for Parallel Mechansms, The th Bennal SME, Mechansms onference, Kssmmee, Florda, DE, Vol. 5-, pp , 1988 [19]. Yang,., hen, W., hen, I-M., eometrcal Method for the Sngularty nalyss of - RRR Planar Parallel Robots wth Dfferent ctuaton Schemes, Proceedngs of the IEEE/RSJ Internatonal onference on Intellgent Robots and Systems, Lausanne, Swtzerland, pp. 55-6, []. Bonev, I., Zlatanov, D., osseln,., Sngularty analyss of -DOF planar parallel mechansms va screw theory, Journal of Mechancal, 5,, pp , [1] Mohammad-Danal,H.,P. Zsombor-Murray, P.., J. ngeles, J., The nematcs of -DOF planar and sphercal double-trangular parallel manpulators, omputatonal Knematcs, edted by J. ngeles et al., Kluwer cademc Publshers, Dordrecht, Netherlands, pp , 199 [] Stacu, S., Lu, X-J., Wang, J., Inverse dynamcs of the HLF parallel manpulator wth revolute actuators, Nonlnear Dynamcs, Sprnger Press, 5, 1-, pp. 1-1, 7 []. Stacu, S., Zhang, D., Rugescu, R., Dynamc modellng of a -DOF parallel manpulator usng recursve matrx relatons, Robotca, ambrdge Unversty Press, 4, 1, 6 [4]. Pashevch,., hablat, D., Wenger, P., Knematcs and worspace analysys of a three-axs parallel manpulator: the Orthoglde, Robotca, ambrdge Unversty Press, 4, 1, 6 [5]. Lu, X-J., Tang, X., Wang, J., Knd of Three Translatonal-DOF Parallel ube- Manpulator, Proceedngs of the 11 th World ongress n Mechansm and Machne Scence, Tanjn, hna, 4 [6] Stacu, S., Inverse dynamcs of a planetary gear tran for robotcs, Mechansm and Machne Theory, 4, 7, pp , 8 [7]. L, Y-W., Wang, J., Wang, L-P., Lu, X-J., Inverse dynamcs and smulaton of a -DOF spatal parallel manpulator, Proceedngs of the IEEE Internatonal onference on Robotcs & utomaton IR, Tape, Tawan, pp , [8]. Dasgupta, B., Mruthyunjaya, T.S., Newton-Euler formulaton for the nverse dynamcs of the Stewart platform manpulator, Mechansm and Machne Theory, 4, 1998 [9]. uegan, S., Khall, W., hablat, D., Wenger, P., Modélsaton dynamque d un robot parallèle à -DDL: l Orthoglde, onférence Internatonale Francophone d utomatque, Nantes, France, 8- Jullet []. eng, Z., Haynes, L.S., Lee, J.D., arroll, R.L., On the dynamc model and nematc analyss of a class of Stewart platforms, Robotcs and utonomous Systems, 9, 199 [1]. Mller, K., lavel, R., The Lagrange-Based Model of Delta-4 Robot Dynamcs, Robotersysteme, 8, pp , 199 []. Zhang, -D., Song, S-M., n Effcent Method for Inverse Dynamcs of Manpulators Based on Vrtual Wor Prncple, Journal of Robotc Systems,, 5, pp , 199 [] Stacu, S., Zhang, D., novel dynamc modellng approach for parallel mechansms analyss Robotcs and omputer-integrated Manufacturng, 4, 1, pp , 8 [4] Stacu, S., Relatons matrcelles de récurrence en dynamque des mécansmes, Revue Roumane des Scences Technques - Sére de Mécanque pplquée, 5, 1-, pp. 15-8, 5

Inverse kinematic Modeling of 3RRR Parallel Robot

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