Parallel manipulator robots design and simulation

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1 Proceedngs of the 5th WSEAS Int. Conf. on System Scence and Smulaton n Engneerng, Tenerfe, Canary Islands, Span, December 16-18, Parallel manpulator robots desgn and smulaton SAMIR LAHOUAR SAID ZEGHLOUL LOTFI ROMDHANE Laboratore de Mécanque des Soldes Laboratore de Mécanque des Soldes Laboratore de Géne Mécanque UMR: 661 CNRS UMR: 661 CNRS Ecole Natonale d Ingéneurs Bd Perre et Mare Cure BP 3179, Futuroscope Bd Perre et Mare Cure BP 3179, Futuroscope Monastr 519 chasseneul Cedx chasseneul Cedx FRANCE FRANCE TUNISIA Abstract: - Parallel manpulator robots have complex knematcs and present sngular postons wthn ther workspace. For these reasons, n most software smulatng parallel robots, each knematc model should be gven n advance by users or programmers. In ths paper we present a new tool used to desgn and to smulate parallel manpulator robots. Explct knematc equatons are generated automatcally dependng on the archtecture of the robot. These equatons are used to smulate and detect sngulartes whle plannng robot s movement. Key-Words: - Parallel manpulator robots- smulaton- sngularty detecton. 1 Introducton Snce the 6 s, seral robots have taken a great place n ndustry. They are used n many ndustral applcatons and n many domans replacng men n hard tasks and hostle envronments. However, n the last 2 years, researchers have been nterested n another type of robots: parallel robots. Dfferent structures of parallel manpulators have been studed (see, for example [1-7]. Currently, more than 1 dfferent archtectures have been proposed and probably not all of them have been dscovered. A revew of most known archtectures s gven by Merlet [8]. Due to ther hgh accuracy and ther ablty to carry hgh loads, parallel robots are used n many domans such as fne postonng devces, moton generators, ultra fast pck and place tasks. They are also fndng ther way n the feld of machne tools and medcal applcatons. However, the closed nature of knematc chans formng parallel manpulators makes ther workspace lmted and ther knematc model complex and archtecture dependent. In addton, sngulartes may appear wthn the workspace. Sngulartes are dangerous and thus should be detected and avoded whle plannng parallel manpulator s moton. Ths problem s very mportant and s wdely studed n lterature, see for nstance [9-11] among many others. Moreover much work s carred out to study the dynamcs of parallel mechansms [12-14]. Desgn and smulaton are mportant ssues n studyng mechansms. The desgn s based on the topology and the geometry of the mechansm. Topology s the way that the onts, lnks and actuators are arranged, whle geometry deals wth the dmensons of lnks and the locatons of the onts. Smulaton conssts of showng the behavor of a mechansm gven ts topology and ts geometry. There s a real lack of software permttng to model and smulate mechansms wth closed chans. Commercal packages lke ADAMS, CATIA, DADS, SoldWorks can be used, but the problem wth most of them s that they use absolute coordnates as they are easer to program, they can not deal wth over constraned mechansms and they are black boxes and no addng nor modfcaton are allowed. In ths paper, we present our work on parallel manpulator robots modelng and smulaton. Our goal s to develop a tool able to model any parallel manpulator wth any combnaton of revolute and lnear onts and able to smulate the moton of many robots evolvng together n ther envronment. Based on the desgn of each parallel manpulator robot, explct geometrcal and knematcal equatons are automatcally generated. These equatons are used to perform possble motons and to detect sngulartes. In the next secton, we present how to model parallel manpulator robots by presentng some methods used to model them. Secton 3 shows how geometrc and knematc equatons are generated and how they are used to smulate moton. Fnally, we present an example of a 3-SPS robot whch s a spatal parallel manpulator composed of three legs. Each leg has two sphercal onts and a prsmatc ont. We gve the correspondng generated equatons to the 3-SPS robot.

2 Proceedngs of the 5th WSEAS Int. Conf. on System Scence and Smulaton n Engneerng, Tenerfe, Canary Islands, Span, December 16-18, J3 5 J6 J3 1 2 J2 2 J4 4 J2 5 J5 1 J7 3 J5 3 4 J1 J6 J4 Fg. 1. A parallel robot and ts correspondng graph 2 Modelng parallel manpulator robots Parallel manpulator robots as any multbody mechansm are made of lnks and onts. The structure of the robot can be presented by a graph as shown n fg Lnks Each lnk has a poston and an orentaton n the 3D space. In order to determne the poston and the orentaton of the lnk we can use ether absolute coordnates or ont coordnates. Where ont coordnates correspond to onts n the mechansm and absolute coordnates correspond to poston and orentaton wth respect to reference frame. We propose three knds of lnks (see fg. 2: base lnk, seral lnk and moble lnk Base lnk We defne the base lnk as the frst lnk of a robot. Its poston and orentaton are gven wth respect to the envronment usng absolute coordnates. The base lnk s unque for each robot and s used to place the robot Seral lnk A seral lnk s connected to another lnk (called father lnk accordng to the ont between both lnks. The poston and orentaton of a seral lnk depends on the ont coordnates and on the poston and orentaton of the father lnk. Fg. 3. Concdence constrant between two ponts Supported onts are revolute onts, lnear onts, sphercal onts and unversal onts Moble lnk A moble lnk s postoned usng absolute coordnates wth respect to the base lnk. 2.2 Constrants Constrants are used to restrct moton between two dfferent lnks. In desgn, constrants are ether used to defne a ont between two lnks or to weld two lnks. We show n the next secton how constrants are used to replace onts and how wth a udcous use we can generate explct geometrcal model of the multbody mechansm. Supported constrants are: concdence between two ponts, parallelsm between two vectors and concdence between two lnks Concdence between two ponts The concdence between two ponts (see fg. 3 s gven by the equaton drven from a pont P belongng smultaneously to two lnks and. Ths equaton s wrtten n ths manner P T Where T = P (1 T the homogeneous matrx of the frame s attached to lnk and P s the homogeneous pont vector of coordnates of pont P n the frame. R R A O T = (2 1 Fg. 2. Lnk types

3 Proceedngs of the 5th WSEAS Int. Conf. on System Scence and Smulaton n Engneerng, Tenerfe, Canary Islands, Span, December 16-18, Fg. 4. Parallelsm constrant between two lnes Where A respect to frame. Whle s the orentaton matrx of frame wth O are the coordnates of the orgn of frame R wth respect to frame R. T ( x y z 1 P = (3 Where x, y and z are coordnates of P n frame R. O Parallelsm between two vectors Parallelsm constrant can be used to make two lnes or two planes of two dfferent lnks parallel (see fg. 4 and fg. 5. The equaton correspondng to ths constrant s wrtten as v T T = v (4 v s the homogeneous vector gven by: v ( v v v = (5 v x x v y y z Where, and are the coordnates of vector v n R the frame. v z Fg. 6. Concdence constrant between two lnks Concdence between two lnks Two frames R and R are attached to the same body (see fg. 6 f c T = T (6 Where T c s a constant homogeneous matrx. T = T ( T T T = T (7 3 Generatng closure equatons We propose two methods to desgn parallel robots. Then we show a possble use of our method n order to desgn seral manpulators and to generate automatcally ther knematcs. 3.1 Frst method As shown n fg. 7, a parallel manpulator s made of a base platform, a moble platform and seral legs. The dea s to replace a non actuated ont n each leg by the correspondng constrants. These onts are called closure constrants. Fg. 5. Parallelsm constrant between two planes Fg. 7. Frst modelng method

4 Proceedngs of the 5th WSEAS Int. Conf. on System Scence and Smulaton n Engneerng, Tenerfe, Canary Islands, Span, December 16-18, Fg. 8. Second modelng method The base platform s modeled usng a base lnk whle the moble platform s modeled by a moble lnk. Legs are modeled usng seral lnks. By replacng only non actuated onts by constrants, we keep the actuated ont coordnates n the generated equatons. Moble lnk has 6 absolute coordnates among whch 3 are used for postonng and 3 used for orentaton. These coordnates are attached to the moble platform whch s the end effector of the manpulator, so they are called the output varables. Constrants between moble lnks and some seral lnks gve equatons nvolvng operatonal varables and ont coordnates. 3.2 Second method To model parallel manpulators we can consder that the moble platform s a part of each leg as shown n fg. 8. Then by applyng constrants of concdence between the lnks representng the moble platform we get closure constrants that need to be satsfed to generate a vald model of the manpulator. Ths method s used n order to generate equatons wth all ont coordnates. If n addton operatonal varables are needed n the generated equatons, a moble lnk concdent wth one of the lnks representng the moble platform s added. The same method s used n the next paragraph n order to generate geometrcal model of seral manpulators. 3.3 Geometrc model of seral manpulators Geometrcal model s a set of equatons between output varables and ont coordnates. The dea s to model the manpulator wth seral onts and to use a moble lnk attached to the last lnk wth a concdence constrant as shown n fg. 9. Fg. 9. Modelng seral manpulators 4 Equaton analyss and nterpretaton The number of equatons depends on the constrants used to model the mechansm. A constrant of concdence between two ponts gves 3 equatons, whch s equvalent to a constrant of parallelsm between two vectors. Whle a constrant of concdence between two lnks gves 6 equatons. The varables used to descrbe the moton of the mechansm depend on the lnks used to buld the model. A moble lnk has sx varables. The number of varables gven by a seral lnk depends on the nature of ts correspondng ont. For a revolute and a lnear ont there s only one varable whle for sphercal ont there are three varables and for unversal ont there are two varables. We call X the vector of varables. Ths vector ncludes all the varables of the system. It contans actve and passve onts as well as output varables. The equatons can be wrtten as F ( X = (8 Where F s a vector functon. 4.1 Extractng the knematcal model In order to obtan the knematc model we only have to derve the functon F wth respect to tme, whch yelds: [ E ] X& = k (9 X & s the dervatve of vector X wth respect to tme. The moblty of the system s gven by ([ E ] m = N rank (1 k Where N s the dmenson of the vector X.

5 Proceedngs of the 5th WSEAS Int. Conf. on System Scence and Smulaton n Engneerng, Tenerfe, Canary Islands, Span, December 16-18, ( x, x1, x2 J1 J5 J3 ( x8, x9, x1 1 ( x4, x5, x6 3 5 ( x 3 J2 ( x 7 J4 J6 ( x 11 Fg SPS Manpulator graph ( x12, x13, x14, x15, x16, x17 Fg. 1. Modelng the 3SPS Manpulator We generate automatcally an explct expresson of the matrx [ E k ] dependng on the geometry of the robot. The varables are dvded nto actve, passve and operatonal varables. By rearrangng equaton (9 we can wrte [ M ] q [ N] q& + [ K] X& = & (11 a + p op Where q& s the vector of actve onts, X & the vector of a output varables whle q& p s the vector of the remanng varables, we call them passve varables. [ M ] and [ K] should be full rank matrces otherwse an actuated varable could be changed wthout movng the moble platform or an output varable could be changed whle actuators are locked, whch means that the model does not completely descrbe the behavor of the robot. [ M ], [ N ] and [ K] are evaluated at each step whle movng the parallel manpulator. 4.2 Movng the parallel manpulator robot There are two ways to control the robot. In one hand the user controls the actve varables and n the other hand the user controls the output varables. In order to do ths, t s necessary to elmnate the matrx N from equaton (11. Ths operaton conssts n elmnatng passve varables n the model. Consequently we obtan a relatonshp between actve and output varables. To do so, a matrx O s computed. The columns of O] [ ] the matrx [ are generated n the null space of [ ] matrx N usng Gram-Schmdt orthonormalzaton. We have: T [ O ] [ N] = [ ] T op (12 By applyng the matrx O to equaton (11 we obtan the followng form of the knetc model. [ B ] q [ A] X& = & (13 a + op T T Where [ A] = [ O] [ K] and [ B] = [ O] [ M] Once q& and X & are known, q& are computed by a op solvng equaton (11. In order to move the platform, the explct form of matrx [ O ] s not compulsory snce t has usually a complcated form, t s ust evaluated at each step. It s A and [. the same for matrces [ ] 4.3 Sngularty detecton Whle movng the robot, sngulartes could be detected. They correspond to a loss of rank of one of the matrces [ N ], [ A ] or [ B ]. The rank of these matrces s contnuously checked whle movng the parallel manpulator. 5 Example We gve a smple example of a parallel manpulator robot and we show the generated equatons. The stude robot s a 3SPS manpulator shown n fg. 1. Ths robot s made of three legs wth a sphercal ont, a lnear ont and a second sphercal ont, the correspondng graph s gven n fg. 11. Jont J1 s a sphercal ont t has three revolute varables ( x, x1, x2. It s the same for J3 and J5. Whle J2, J4 and J6 are lnear onts, each one has an actve varable whch are x3, x7 and x11 respectvely. Lnk 7 s a moble lnk, t s orented usng three varables x, x x and t s postoned usng ( 12 13, 14 varables ( x, x x , 17 Ths robot s modeled wth a base lnk, 6 seral lnks and a moble lnk. The geometrcal dmensons are gven n fg. 12. B] p

6 Proceedngs of the 5th WSEAS Int. Conf. on System Scence and Smulaton n Engneerng, Tenerfe, Canary Islands, Span, December 16-18, Fg. 12. Dmensons of the 3SPS manpulator There are three constrants of concdence between two ponts defned between each leg and the moble lnk. The generated equatons are: F x sn x x 3 ( X = x sn x cos x + 5 x = 1 15 x3 sn x cos x1 x16 x3 cos x cos x1 x 17 x7 sn x5 x x7 cos x4 cos x5 x17 x11 sn x x15 x11 sn x8 cos x x16 x11 cos x8 cos x9 x17 wth X = ( L x T s x 17 (14 Thereafter, we use the followng notaton: sn and c = cos x. = x Dervng F wth respect to tme gves the matrx [ E k ] [ M ] N ] [ K ]. that we dvde nto three matrces, [ and We choose q & = &, X & = (& T and q & = p a ( T x (& T x op x Concluson We presented a new methodology for modelng parallel manpulator robots and we showed how to generate closure equatons automatcally whch gves the knematc model. We also proposed the way to elmnate the passve parameters n the model. We also proposed the way to elmnate the passve parameters n the model. Ths work s the frst step of a tool that we are stll developng n order to model parallel manpulators n ther envronments and to smulate and plan ther moton avodng sngulartes and wthout collson wth obstacles. References: [1] V.E. Gough, Contrbuton to dscusson of papers on research n automoble stablty, control and tyre performance, Proc. Auto Dv. Inst. Mech. Eng [2] D. Stewart, A platform wth 6 degrees of freedom. Proc. of the Insttuton of mechancal engneers, 18(Part 1, 15: , [3] R. Clavel, DELTA, a fast robot wth parallel geometry, In 18th Int. Symp. on Industral Robot, pages 91-1, Lausanne [4] A. Goudal, J-P. Lallemand, S. Zeghloul, Espace de traval de la nouvelle structure 2-Delta. Revue d'automatque et de Productque Applquée, 8(2-3:25-21, [5] J. Hesselbach, N. Pltea, M. Frndt, A. Kusek, A new parallel mechansm to use for cuttng convex glass panels. ARK, pages , [6] O. Company, F. Perrot, A new 3T-1R parallel robot. ICAR, Tokyo, [7] L. Romdhane, Z. Aff, M. Fayet, Desgn and sngularty analyss of a 3 translatonal-dof nparallel manpulator. ASME Journal of Mechancal Desgn 124, pp , 22. [8] J.-P. Merlet, Parallel robots. Kluwer Academc Publshers, 2. [9] C. Gosseln, J. Angeles, Sngularty analyss of closed-loop knematc chans, IEEE Transactons on Robotcs and Automaton, vol. 6, pp , 199. [1]Guanfeng Lu, Yunang Lou, Zexang L, Sngulartes of parallel manpulators: a geometrc treatment, IEEE Transactons on Robotcs and Automaton, vol.19, No.4, pp , 23. [11]Sandpan Bandyopadhyay, Ashtava Ghosal, Analyss of confguraton space sngulartes of closed-loop mechansms and parallel manpulators, Mechansm and Machne Theory 39, pp , 24. [12]X.Sh, R.G. Fenton, Soluton to the forward nstantaneous knematcs for a general 6 d.o.f. Stewart platform. Mechansm and Machne Theory, 27(3: , Ma [13]V. ParentCastell, R. D Gregoro, Determnaton of the actual confguraton of the general Stewart platform usng only one addtonal sensor, ASME Journal of Mechancal desgn 121, pp , [14]J. Gallardo, J.M. Rco, A. Frsol, D. Checcacc, M. Bergamasco, Dynamcs of parallel manpulators by means of screw theory, Mechansm and Machne Theory 38, pp , 23.

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