ème Congrès Franças de Mécanque Lyon, 4 au 8 Août 5 Inverse knematc Modelng of RRR Parallel Robot Ouafae HAMDOUN, Fatma Zahra BAGHLI, Larb EL BAKKALI Modelng and Smulaton of Mechancal Systems Laboratory, Abdelmalek Essaad, Faculty of Scences, BP., M hannech, 9, Tetouan, Morocco. Wafae.hamdoun@gmal.com Abstract: Ths work presents a knematc study of RRR parallel robot. Indeed, Parallel robot s fundamentally a closed-loop knematc chan mechansm n whch the end-effector s connected to the base by several ndependent knematc chans. Due to the potental advantages of parallel robots such as hgh rgdty, hgh accuracy and great carryng payload capablty, they have fascnated sgnfcant attenton and nterest amongst the researchers n the past decade. After descrbng the mechansm, a knematc model s establshed. Based on the geometry of the manpulator, a workspace analyss s performed n order to defne a trajectory plannng of the manpulator. Résumé: Dans ce traval on a élaborer un modèle géométrque du robot paralléle RRR, En effet, Les robots parallèles sont des mécansmes à chaînes cnématques fermées dont l organe termnal est relé à la base par pluseurs chaînes cnématques ndépendantes, ls ont pluseurs avantages par rapport aux robots sérels tels que la rapdté, la précson et un bon rapport charge utle/masse. Dans notre traval nous avons traté la modélsaton géométrque du robot paralléle RRR, Ans que l analyse de l une des caractérstques de ce robot : l espace du traval au but de planfer et d optmser le trajectore du robot. Keywords Inverse knematc, Workspace, Parallel Robot. INTRODUCTION : In recent years, parallel robots have receved huge attenton for dfferent researches and applcatons. Parallel manpulators are closed-loop mechansms n whch the separate seral chans (lnks and jonts) are connected to both the fxed base and the movng platform. Ths new type of robots can acheve certan tasks that seral robots are not able to perform. Even though seral robots have advantages lke smple dynamc and knematc models, large workspace, but parallel robots have advantages more potental than seral robots as better stffness, hgh speed, hgher knematcal precson, large load carryng capacty and the most mportant s the possblty of fxng actuators on the base, whch can be used for fast and precse operatons. Parallel robots have also some drawbacks, the major of them are a lmted workspace, complcated sngulartes.
ème Congrès Franças de Mécanque Lyon, 4 au 8 Août 5 Parallel manpulators can be equpped wth revolute or prsmatc actuators. They have a strong constructon and they are able to move bodes of large dmensons wth hgh veloctes and acceleratons []. Parallel manpulators can be dvded nto two fundamental categores, namely spatal and planar manpulators, The frst category composes of the spatal parallel manpulators that can translate and rotate n the three dmensonal space, ths category recognzed as Stewart Gough platform, has receved great attenton [-]. They are used n flght smulators. The planar parallel manpulator whch composes of second category, translate along the x and y-axes, and rotate around the z-axs, they are useful for manpulatng an object on a plane. Gosseln and Angeles [4] whch present a knematcal study of a planar parallel robot, where a movng platform s connected to a fxed base by three lnks, each leg consstng of two bnary lnks and three parallel revolute jonts. Sefrou and Gosseln [5] gve an nterestng numercal soluton n the nverse and drect knematcs of ths knd of planar robot. Thus, n ths work we can formulate our problem as follow: determne the nverse knematc problem, gven specfc ponts nsde the boundary of a desred workspace, establsh a workspace analyss of RRR planar parallel robot where a smple numercal mplementaton n Matlab software s carred out n order to obtan a trajectory plannng of the robot.. Descrpton of RRR Parallel Robot In ths paper we present a geometrc descrpton of RRR parallel robots whch composed by a moble platform (MP) and three RRR seral chans that jon t to a fxed base (Fg. ). Each RRR chan s a seral chan composed by three rotatonal R jonts. Allow P(x, y) be the end-effector poston n the plane and ts orentaton. Let O be the orgn of the fxed reference frame and let A, B andc wth,, defne the rotatonal artculatons of each leg. A, A Ponts are actuated, so that the actuators are fxed to the base. l B s the length of Lnk B. l A s the length of Lnk A B M l d l d l d P M B l B M l A y B A x A Fg.: Geometrc Descrpton of RRR parallel robot
ème Congrès Franças de Mécanque Lyon, 4 au 8 Août 5. Knematc nverse model of the robot : The dervaton of the nverse knematcs gven from the loop closure equaton for the -RRR parallel robot s as follow: OP OA A B B M M P (Eq.) Wrtng the equatons n component form: Where: P l cos( ) l cos( ) l cos( ) x x A B d A P l sn( ) l sn( ) l sn( ) y y A B d A s the angle between the lne ld from the pont P to the jont at M and t s consdered from the lne between M andm on the movng platform. (Eq.) To compute the nverse knematcs, we can rewrte the equaton as follows: Where : K snf cos E (Eq.) E P P l l l x y P l cos( ) P x x y A B d A A x d x A l x cos( ) P l sn( ) P y l y sn( ) d A y d y A d A F l l l cos( ) l x A A d A A K P l l l sn( ) l y y A A d A A The nverse knematcs soluton for equaton () s: A tan ( K, F ) A tan ( ( K F E ), E ) Eq. (4) Where:,, 4. Numercal Implementaton of RRR Parrallel Robots The equaton obtaned can smply be solved for any number of legs by smply changng the subscrpt n equatons. For the partcular robot there are two possble solutons gven by the solutons to equatons. When the roots are magnary (.e. there s no real soluton) for some gven pose, t means that the poston/orentaton of that confguraton s outsde the reachable workspace
Y Y Y ème Congrès Franças de Mécanque Lyon, 4 au 8 Août 5 5 nverse knematc of RRR Parallel robot platform 5 Base -5 Base - Base -5-5 - -5 5 5 Fg.: A confguraton of RRR parallel robot 5. The workspace of each lnk of the parallel robot: The workspace s the characterstcs for a class of -DOF. Usng numercal smulatons, the complete workspace s dentfed for each lk of the lnk The workspace for the frst lnk The workspace of the second lnk.5.5.5 -.5 - - - - - - -.5 - -.5 - -.5.5.5 The workspace of the thrd lnk - - - -.5 - -.5 - -.5.5.5 Fg.: the workspace of each lnk of the manpulator 4
Y ème Congrès Franças de Mécanque Lyon, 4 au 8 Août 5 The total workspace obtaned by the ntersecton of three regons of the legs of the robot.4 workspace of RRR parallel robot..8.6.4. -. -.4 -.6 -.4 -...4.6.8..4 Fg : Intersecton between the three regons of moblty of the robot 6. Concluson In ths work we present an approach for determnng the nverse knematcs appled on -RRR planar parallel, Whch have mportant potental for dfferent applcatons. Thus the analyss of the workspace of the robot lead us to a great performance and a hgher prscon n postonng of the platform n order to do a trajectory plannng of the manpulator. References: [] J.-P. Merlet, Parallel Robots, Kluwer Academc Publshers,. [] V. Parent Castell, R. D Gregoro, A new algorthm based on two extra sensors for real-tme computaton of the actual confguraton of generalzed Stewart Gough manpulator, J. Mech. Des. (). [] D. Stewart, A platform wth sx degrees of freedom, Proc. Inst. Mech. Eng. 8 (5) (965) 7 78. [4] Gosseln CM, Angeles J. The optmal knematc desgn of a planar three-degree-of-freedom parallel manpulator. JMech, Trans Automat Des 988;():5 4 [5] Sefrou J, Gosseln CM. On the quadratc nature of the sngularty curves of planar three-degreeof-freedom parallel manpulators. Mech Mach Theory 995;(4):5 5. 5