Rigidity analysis of T3R1 parallel robot with uncoupled kinematics B. C. Bouzgarrou, J.C. Fauroux, G. Gogu, Y. Heerah

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1 Rgdt analss of T3R parallel robot wth uncoupled nematcs B. C. Bouzgarrou, J.C. Faurou, G. Gogu, Y. eerah Laboratore de Recherche et Applcatons en Mécanque avancée (LaRAMA) Insttut Franças de Mécanque Avancée (IFMA) et Unversté Blase Pascal Campus unverstare de Clermont-Ferrand/ Les Cézeau, BP65, 6375 Aubère France Tel : , E-mal : bouzgarr/faurou/gogu@fma.fr Abstract Complet for common parallel robots s manl due to nematc couplng between jont and operatonal space coordnates. Desgnng a parallel robot wth uncoupled nematcs, such a Cartesan robot, smplfes consderabl robot modelng and control. In ths paper we stud statc and dnamc rgdt of a new parallel platform n order to optmze mechancal desgn and control strateg. Ths parallel robot wth four degrees of moblt and pseudo-decoupled motons s developed at LaRAMA - French Insttute of Advanced Mechancs. Introducton Ths paper presents rgdt analss of a new parallel mechansm wth four degrees of moblt and decoupled motons. Ths parallel robot, called T3R, s ssued from a general structural snthess approach of parallel platform mechansms wth, 3, 4, 5 and 6 degrees of moblt and decoupled motons. Ths approach and the structural solutons are etensvel presented n []. All ndependent motons of these robots can be partall or entrel decoupled. In a parallel platform mechansm wth entrel decoupled motons, the Jacoban matr of the lnear transformaton (mappng) between the jont veloct space and the operatonal veloct space s a dagonal matr. A one-to-one correspondence ests between the endeffector output veloctes and the nput jont veloctes. Several tpes of TPM (3-leg 3-degrees of moblt translatonal parallel manpulators) wth decoupled motons are presented b ervé and Sparacno [], Carrcato and Parent-Castell [3], Km and Tsa[4], Kong and Gosseln [5]. T3R adds a supplementar rotaton and mplctl a supplementar leg and supplementar jonts to the structure of a spatal 3-PRRR parallel manpulator TPM-tpe. One of the commonl admtted advantages of parallel robots over seral robots s ther mproved rgdt [6]. Parallel robots wth decoupled motons such as T3R have a wde range of potental applcatons n manpulaton, assemblng and machnng. Ths paper focus on the statc rgdt of T3R n order to evaluate ts performance for real applcatons. Knematc structure T3R s composed of a moble platform (5) and a base (0) connected b four legs (A, B, C and D) n parallel (Fgs. and ). Under the acton of the total constrants of ts four legs, the moble platform can do three ndependent translatons (,, z ) and one rotaton ϕ a (a=,,z). The solutons presented n Fgs. and allow a rotaton on as (a=). Each leg n Fg. has three revolute jonts wth aes parallel to the drecton of groundconnected prsmatc jont. Moble platform s connected b two revolute jonts parallel to a as (a=). Furthermore, the three prsmatc jont aes on the legs A, B and C are parallel to 0, 0, and z 0 aes, respectvel. The ground-connected prsmatc jont as on the leg D s also parallel to z 0 as. The four ground-connected prsmatc jonts are actuated. The four ndependent jont varables are d 0A, d 0B, d and d 0D. The drect nematc model of the soluton presented n Fg., defned b the matr mappng between the jont veloct space ( d & 0A,d & 0B,d &,d& 0D ) and the operatonal space ( &, &, &,ϕ& ) s : & & d0 A & d& 0B = z& d& & ϕ 0 0 b43 b44 d& 0D wth b43 = b44 =. () rcosϕ From Eq. () we can see that ths parallel robot acheves a one-to-one correspondence between the jont veloctes d & 0 A,d& 0B,d& and the operatonal translaton veloctes &, &, & but the rotaton veloct ϕ& depends on the dfference d& DC = d& 0 D d& ddc ϕ& = (d& 0D d & ) = &. (3) rcosϕ rcosϕ Ths robot has pseudo-decoupled motons. We can consder that a one-to-one correspondence est between ϕ& and d & DC. To obtan decoupled motons between d & 0 A,d& 0 B,d&,d& DC and &, &, &,ϕ& we can use the soluton presented n Fg.. In ths case, the prsmatc jont as on the leg D s not ground-connected. Ths prsmatc jont connects the nematc elements C and D. ()

2 a) a) b) FIG. Parallel platform mechansm T3R wth pseudodecoupled motons: a) nematc structure, b) assocated graph (from Gogu, 00 []) b) FIG. Parallel platform mechansm T3R wth decoupled motons: a) nematc structure, b) assocated graph (from Gogu, 00 []) The drect nematc model becomes & d& 0 A & d& 0B = (4) z& d& & ϕ b44 d & 0D wth b44 =. (5) rcosϕ In the net sectons we present the stud of rgdt of a prototpe of T3R parallel manpulator wth respect to the soluton presented n Fg.. Ths prototpe, presented n Fgs. 3 and 4, s n constructon at LaRAMA - French Insttute of Advanced Mechancs. FIG 3. The CAD model of a T3R parallel mechansm [7] 3 Prelmnar desgn A frst prelmnar CAD model (Fg. 3) was constructed based on the nematc model of T3R robot Ths model demonstrates feasblt of the robot and absence of collson between parts [7].

Part Part F α FIG 5. A refned CAD model of one leg, made of two parts wth α angle and subject to force F. FIG 4.

The whte cube shown n Fg. 4 represents the mamum worspace correspondng to translaton jont lmts.

4 Stffness stud of one leg As T3R s made of four dentcal legs, we wll focus on the rgdt behavor of one leg (Fgs. 5 and 6).

Based on the CAD model of each leg, a fnte element model was constructed. Each leg part s made of two tps wth bearng cages connected b strengthenng rbs.

The rbs are meshed wth shell elements (4 nodes wth 6 DOF,, z, rot, rot, rotz on each) n order to tae bendng nto account.

The revolute jont between parts s created b four pars of sprng elements set on cardnal ponts of the jont wth quas nfnte stffness (Fg. 7).

3 Part Part F α FIG 5. A refned CAD model of one leg, made of two parts wth α angle and subject to force F. FIG 4. T3R robot n ts rgd frame [8] T3R robot s then ntegrated n a frame whch was desgned to be several orders of magntude more rgd than the robot (Fg. 4). The whte cube shown n Fg. 4 represents the mamum worspace correspondng to translaton jont lmts. It s now mportant to determne statc rgdt maps of the robot n order to understand ts wea ponts, sngular confguratons and later, use t for smart control. 4 Stffness stud of one leg As T3R s made of four dentcal legs, we wll focus on the rgdt behavor of one leg (Fgs. 5 and 6). It s made of two parts wth a foldng angle α varng from 5 to 65 n order to avod sngular confguraton. We note that sngular confguratons occur at α=π (=0; ). Based on the CAD model of each leg, a fnte element model was constructed. Each leg part s made of two tps wth bearng cages connected b strengthenng rbs. The bearng cages are meshed wth sold elements (8 nodes wth 3 DOF,, z on each). The rbs are meshed wth shell elements (4 nodes wth 6 DOF,, z, rot, rot, rotz on each) n order to tae bendng nto account. Both tpes of elements are connected b equalzng translaton DOF and cancelng rotaton DOF on the boundar. The revolute jont between parts s created b four pars of sprng elements set on cardnal ponts of the jont wth quas nfnte stffness (Fg. 7). In each par, one sprng s for blocng radal dsplacement, the other for aal dsplacement. The onl remanng DOF s rotaton around revolute as. The leg s consdered to be fed wth a gven α angle and submtted to a force F on one end along as. The FEM calculatons gves us the dsplacement of the applcaton pont of the force and equaton (6) gves rgdt along. = F /. (6) FIG 6. FEM model of one leg wth detals of the bearng cages on parts FIG 7. Revolute jont between parts s created va sprng element Matr 7 (onl one couple of sprngs s shown for smplct) Y X X Y 3

Rgdt (N/µm) 0.7 0.6 0.5 0.4 0.3 0. 0. 0 α angle 0 5 30 45 60 75 90 05 0 35 50 65 80 FIG 8.

4 Rgdt (N/µm) α angle FIG 8. Rgdt results for one leg B repeatng ths calculaton for varous values of angle α, a graph of rgdt can be drawn (Fg. 8). It appears that rgdt s far smaller for angles of 90. Ths can be eplaned b the combnaton of two phenomena: torson of part of the leg supermposed on bendng of the whole leg. Moreover, f we eamne orders of magntudes, t ma be notced that torson phenomenon (mamum at 90 ) s more dangerous than bendng (best vsble at 5 and 65 ). Ths s an mportant fact that wll be reproduced on each of the four legs of the robot. 5 Stffness stud of the whole robot After meshng the four legs, addng the connectng parts and defnng each revolute jont a global meshed FEM model of T3R can be obtaned (Fg. 9). It s full parametrzed b poston and angle of end effector. Several analss tpes are acheved from the fnte element model. Statc and modal performances reflect robot precson. Ther characterzaton overall the wor space s useful for mechancal desgn and control strateg optmzaton. FIG 9. FEM model of the full T3R robot translatons and 3 rotatons) of the end effector to generalzed eternal loads (3 forces and 3 torques) appled to t. Generalzed dsplacements are defned b the vector u = [ u u u u u u ] T where [ u u u ] T 3 are the three translatons and [ u u u ] T are the three rotatons. Generalzed eternal loads are defned b the F = F F F F F where vector [ F ] T [ F F F ] T 3 are the appled forces and [ F F F ] T the three appled torques at the end effector. We have F = Ru (7) In order to dentf the terms of the matr R, 6 lnearl ndependent load cases must be studed. B applng a force or a torque F, = L6, we obtan, usng the fnte element model a generalzed dsplacement vector u [ ] T = u u u3 u4 u5 u6. Complance terms s j, j = K 6 are defned b the followng relaton uj s j = (8) F Assumng that the structure s lnear, superposton prncple can be appled. Thereafter, for an arbtrar appled load F, terms of resultng dsplacement vector are gven b u j = n = F s j Whch can be wrtten n a matr form u = SF (0) where S s a complance matr havng s j as general term. Usng equatons (6) and (9), we obtan the rgdt matr R = S () In order to characterze robot statc rgdt over all the worspace, a desgn for eperment technque s adopted. It s based on the parametrzed fnte element model of the robot. The cost and the precson of such analss depend 3 on the number of calculaton ponts. A full factoral, 5, eperment desgn s used wth three varables and fve levels. We appl 6 load cases, F = F = F 3000 N and 3 = F 4 = F5 = F6 = 4000 N mm, for each measurement pont, = K5. Complance matr terms, s j,, are frstl determned at each pont and appromated b quadratc polnomal fttng accordng to Cartesan coordnates of the end effector. The adopted regresson functons have the followng form: (9) 5. Statc rgdt characterzaton Statc rgdt s defned as the 6 6 smmetrcal matr R that maps generalzed nfntesmal dsplacements (3 4

5 s j, = a 4, j 7, j 0, j, j 8, j 5, j, j z 9, j z 3, j z 6, j z () (,, z ) are the coordnates of th measurement pont. a l, j, l = K0, are the regresson functons coeffcents. The are dentfed n least square meanng. T Let = a,, a ] be the coeffcents vector and j [, j K 0, j 5 [ u ] T j, KK, u b j = j the vectors of dsplacements n measurement ponts. In optmal case, the coeffcents verf the relaton A = (3) j b j where A s a 5 0 matr gven b LL z M M A = LL z (4) M M 5 5 LL z5 A s the same for all complance terms and coeffcents can be determned b usng pseudo-nverse matr j T T ( A A) A bj = (5) Rgdt matrces at each pont are obtaned b the nverson of complance matrces. The evoluton of the robot rgdt can be represented b 3D maps for a constant values of the coordnate. In Fg. 0, terms of translaton rgdt are ftted for =50 mm. The hghest rgdt term s R zz along the Z drecton. Ths can be eplaned b the use of two legs n ths drecton. Rgdtes along X and Y drectons, R and R, are of same order. We notce the strong rgdt couplng, R z, between Z and Y drectons. On the other hand X and Y rgdtes are weal coupled especall for small values of. The global rgdt of robot along a drecton s greater then the rgdt of an solated leg due to rgdt couplngs. 5. Modal characterzaton A modal analss of the robot s performed. Natural frequences reflect the robot dnamc rgdt and precson. In order to characterze robot natural frequences over all the worspace, we adopt the same technque used for statc rgdt characterzaton. The three frst modes are consdered. The evoluton of natural frequences over the worspace s presented b 3D maps for a constant coordnate. Surfaces have sensbl the same shape for the 3 modes (Fg. ). The mnmum natural frequenc s about 40 z. R ( N/ µ m); = 50 mm R ( N/ µ m); = 50 mm R zz ( N/ µ m); = 50 mm R ( N/ µ m); = 50 mm R z ( N/ µ m); = 50 mm R z ( N/ µ m); = 50 mm FIG. 0. Statc rgdt maps 5

FIG. Natural frequences maps (z) = 50 mm 6 Concluson The paper presents statc rgdt and natural frequences of T3R, a new parallel platform mechansm wth four degrees of moblt and pseudo-decoupled

6 FIG. Natural frequences maps (z) = 50 mm 6 Concluson The paper presents statc rgdt and natural frequences of T3R, a new parallel platform mechansm wth four degrees of moblt and pseudo-decoupled motons developed at LaRAMA - French Insttute of Advanced Mechancs. T3R s composed of a moble platform and a base connected n parallel b four legs. Under the acton of the total constrants of ts four legs, the moble platform can do three ndependent translatons (,, z ) and one rotaton ϕ a (a=,,z). The soluton analzed n ths paper allows a rotaton on as (a=). Each leg has three revolute jont aes parallel to the ground-connected prsmatc jont as and the moble platform s connected b two revolute jonts parallel to a as (a=). The four ground-connected prsmatc jonts are actuated. Rgdt stud s acheved b usng FEM. The nfluence of the foldng angle on the rgdt of each leg has shown that rgdt s far smaller for angles of about 90. Ths can be eplaned b the combnaton of two phenomena: torson of part of the leg supermposed on bendng of the whole leg. Moreover, f we eamne orders of magntudes, t ma be notced that torson phenomenon (mamum at 90 ) s more dangerous than bendng (mamum at 5 and 65 ). Based on the parametrzed FEM model, a desgn for eperment technque was adopted n order to characterze robot statc rgdt and natural frequences over all the worspace. A full factoral, 3 5, eperment desgn was used wth three varables and fve levels. Rgdt stud of the whole robot have ndcated a strong rgdt couplng, R z, between Z and Y drectons. On the other hand X and Z rgdtes are weal coupled. The global rgdt of robot along a drecton s greater then the rgdt of an solated leg. Ths s due to rgdt couplngs. The modal analss of the robot frst three modes have ndcated a mnmum natural frequenc of about 40 z. These prelmnar results allow us to conclude that parallel robots wth decoupled motons such as T3R have a wde range of potental applcatons n manpulaton, assemblng and machnng. Acnowledgment Ths stud was jontl funded b the MAX project wthn the CNRS-ROBEA program and b CPER Auvergne program. References [] G. Gogu, Structural snthess of parallel robotc manpulators wth decoupled motons. Internal Report ROBEA MAX - CNRS, 00. [] J.M. ervé, F. Sparacno, Structural snthess of parallel robots generatng spatal translaton. Proceedngs of the 5 th Internatonal Conference on Advanced Robotcs, Psa, Ital, vol., [3] M. Carrcato, V. Parent-Castell, Sngulart-free full-sotropc translatonal parallel mechansm, Internatonal Journal of Robotcs Research () (00) [4]. S. Km, L-W. Tsa, Evaluaton of a Cartesan parallel manpulator, Advances n Robot Knematcs, edted b J. Lenarcc and F. Thomas, Kluwer Academc Publshers, 00, pp.-8. [5] X. Kong, C. M. Gosseln, Tpe snthess of lnear translatonal parallel manpulators, Advances n Robot Knematcs, edted b J. Lenarcc and F. Thomas, Kluwer Academc Publshers, 00, pp [6] J-P. Merlet, Les Robots Parallèles, ermes, Pars, 997. [7] F. Morellec, Concepton du robot parallèle T3R, Projet CNAM, 003, LaRAMA-IFMA. [8] L. Lecerf, Concepton du bât du robot parallèle T3R, Projet de Fn d Etudes 003, LaRAMA-IFMA. [9] Y. eerah, Etude de la rgdté statque et dnamquedu robot parallèle T3R, Master Thess, LaRAMA-IFMA and Unverst of Bath. 6

Inverse kinematic Modeling of 3RRR Parallel Robot

Inverse kinematic Modeling of 3RRR Parallel Robot 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,