CALCULUS OF JOINT FORCES IN DYNAMICS OF A 3-PRR PLANAR PARALLEL ROBOT

Transcription

1 LULUS OF JOINT FORES IN DYNMIS OF 3-PRR PLNR PRLLEL ROOT ŞTEFN STIU bstrct. Recurse mtrix reltions for the joint forces in dynmics of plnr 3-PRR prllel robot re estblished in this pper. Knowing the kinemtics of the pltform, we develop the inverse dynmic problem, using n pproch bsed on the principle of virtul work. Some grphs of simultion for the internl joint forces re obtined. Key words: plnr prllel robot, dynmics, joint forces, principle of virtul work, recurse mtrix reltions. 1. INTRODUTION Equipped with revolute or prismtic ctutors, prllel mnipultors hve robust construction nd cn move bodies of lrge dimensions with high velocities nd ccelertions. Tht is reson why the devices, which produce trnsltion or sphericl motion to pltform, technologiclly re bsed on the concept of prllel mnipultors [1]. Prllel mnipultors hve receed more nd more ttention from reserches nd industries. mong these, the clss of mnipultors known s Stewrt-Gough pltform focused gret ttention (Stewrt [2]; Merlet [3]). The prototype of Delt prllel robot (lvel [4]; Tsi nd Stmper [5]) nd the Str prllel mnipultor (Hervé nd Sprcino [6]) re both equipped with three motors, which trin on the mobile pltform in generl trnsltion motion. mechnism is sid to be plnr robot if ll the moving links in the mechnism perform the plnr motions tht re prllel to one nother. In plnr linkge, the xes of ll revolute joints must be norml to the plne of motion, while the direction of trnsltion of prismtic joint must be prllel to the plne of motion. onev, Zltnov nd Gosselin [7] describe severl types of singulr configurtions by studying the direct kinemtics model of 3-RPR plnr prllel robot with ctuted bse joints. Pennock nd Kssner [8] present kinemticl study of plnr prllel robot, where moving pltform is connected to fixed Politehnic Unersity of uchrest, Deprtment of Mechnics, Romni Rom. J. Techn. Sci. ppl. Mechnics, Vol. 57, N os 2 3 P , uchrest, 2012

2 2 lculus of joint forces in dynmics of 3-PRR plnr prllel robot 197 bse by three links, ech leg consisting of two binry links nd three prllel revolute joints. 2. KINEMTIS NLYSIS recurse method is introduced in the present pper, to reduce significntly the number of equtions nd computtion opertions by using set of mtrices for kinemtics nd inverse dynmics of the 3-PRR plnr prllel robot. The plnr prllel robot of three degrees of freedom is symmetricl mechnism composed of three plnr kinemticl chins 1 2 3, nd 1 2 3, hving vrible length nd identicl topology, ll connecting the fixed bse to the moving pltform The prllel mechnism with seven links consists of three prismtic joints nd six revolute joints (Fig. 1). Fig. 1 Plnr 3-PRR prllel robot.

3 198 Ştefn Sticu 3 For the purpose of nlysis, we ttch rtesin frme Ox0y0z0( T 0) to the fixed bse with its origin locted t tringle centre O, the z 0 xis perpendiculr to the bse nd the x 0 xis pointing long the direction. 0 0 mobile reference frme GxGyGz G is ttched to the moving pltform, with the origin locted just t the centre G of the tringle. One of three cte legs (for exmple leg ) consists of prismtic joint, which is s well s piston of mss m 1 linked t the x 1 1y1z 1 frme, hving rectiliner trnsltion of displcement λ 10. Second element is rigid rod linked t the x 2 2 y2 z 2 frme, hving relte rottion bout z 2 xis with the ngle ϕ 21. It hs the length l 2, mss m 2 nd tensor of inerti Ĵ 2 with respect to T 2 frme. Finlly, revolute joint is introduced t plnr moving pltform, which is schemtised s n equilterl tringle with edge l = r 3, mss m 3 nd inerti tensor Ĵ 3 with respect to 3, which rottes with the ngle ϕ 32 bout z 3. t the centrl configurtion, we consider tht ll legs re symmetriclly extended with the ngles α = π /3, α = 3 α, α = α of orienttion of three edges of fixed pltform. We cll the mtrix ϕ kk, 1, for exmple, the orthogonl trnsformtion 3 3 mtrix of relte rottion with the ngle ϕkk, 1 of link T k round z k xis. Strting from the reference origin O nd pursuing three independent legs O0 12 3, O01 2 3, O01 2 3, we obtin the following trnsformtion mtrices where i ϕ ϕ 10 α q = θ, q = q θ, q = q θ ( q=, b, c) ( i=,, ), (1) ϕ i i qkk, 1 = rot( z, ϕkk, 1), θα = rot( z, αi ), θ = rot( z, π ). (2) 6 It cn be considered tht the position of the mechnism, in the inverse G G geometricl problem, is completely gen through the coordintes x0, y 0 of the mss centre G of the moving pltform nd the orienttion ngle φ of the mobile centrl frme GxGyGz G. The orthogonl known rottion mtrix of the pltform from Ox0y0z 0 to GxGyGz G reference system is R = rot( z, φ). We suppose tht the position vector of G centre r G 0 = [ x G G 0 y 0 0] T nd the orienttion ngle φ, which re expressed by following nlyticl functions

4 4 lculus of joint forces in dynmics of 3-PRR plnr prllel robot 199 G 0 G y0 G* 0 G* y0 * φ x x φ π = = = 1 cos t, (3) 3 cn describe the generl bsolute motion of the moving pltform in its verticl plne. From the conditions concerning the orienttion of the pltform T q q = R, q = q q q, q = θθθ α, ( q=, b, c) we obtin the following reltions between ngles ϕ21 + ϕ32 = φ ( i =,, ). Pursuing the kinemticl modeling developed in [9], six independent vribles λ10, ϕ 21, λ10, ϕ 21, λ10, ϕ 21 will be determined by the nlyticl equtions i i π G i π ( l1+ λ10)sinαi + l2sin( ϕ21+ + αi) = y0 y00 rcos( φ+ + αi) 6 3 i i π G i π ( l1+ λ10)cosαi + l2cos( ϕ21+ + αi) = x0 x00 + rsin( φ+ + αi) 6 3 ( i =,, ). Now, we compute in terms of the ngulr velocity of the pltform nd velocity of mss centre G the relte velocities v10, ω21, ω 32, strting from following mtrix conditions of connectity T T T T T G T T G T G v10u j 10u 1 + ω21u j 20u3 { r r 3 } + ω 32u j 30u3 r 3 = u j r 0 ( j = 1,2), ω + ω = (6) φ oncerning the first leg, the chrcteristic virtul velocities re expressed s functions of the pose of the mechnism by the generl kinemticl equtions (6), where we dd the contributions of successe virtul trnsltions or rottions during some fictitious displcements of the prismtic joint 1 nd of the revolute joints 2 nd 3, s follows: v T T yv T T v T T T T G v10 uj10u1 + v10 uj10u2 + ω10 uj10u 321{ r r3 } + xv T T yv T T v T T T G + v21 uj10u1 + v21 uj10u2 + ω21 uj20u 3{ r r3 } + xvt T yvt T vt T G T Gv + v u u + v u u + ω u ur 33 = uj v0 ( j= 1,2), (7) 32 j j j 30 v v v v ω + ω + ω = ω. i i i (4) (5)

5 200 Ştefn Sticu 5 Now, let us ssume tht the robot hs successely some virtul motions determined by following sets of velocities: yv yv yv v =, v 10 = 1, v 10 = 0, v 10 = 0, ω 10 = 0, v 21 = 0, v 21 = 0, v 32 = 0, v 32 = 0 ; v =, v 10 = 0, ω 10 = 1, ω 10 = 0, ω 10 = 0, v 21 = 0, v 21 = 0, v 32 = 0, v 32 = 0 ; 10 0 v v v v =, v 10 = 0, ω 10 = 0, v 21 = 1, v 21 = 0, v 21 = 0, v 21 = 0, v 32 = 0, v 32 = 0 ; 10 0 xv xv yv yv yv v =, v 10 = 0, ω 10 = 0, v 21 = 0, v 21 = 1, v 21 = 0, v 21 = 0, v 32 = 0, v 32 = 0 ; (8) 10 0 v =, v 10 = 0, ω 10 = 0, v 21 = 0, v 21 = 0, v 32 = 1, v 32 = 0, v 32 = 0, v 32 = 0 ; 10 0 yv yv yv v =, v 10 = 0, ω 10 = 0, v 21 = 0, v 21 = 0, v 32 = 0, v 32 = 1, v 32 = 0, v 32 = 0. These virtul velocities re required into the computtion of virtul power nd virtul work of ll forces pplied to the component elements of the mnipultor. s for the relte ccelertions γ10, ε21, ε 32 of the robot, new conditions of connectity re obtined by the derte of bove equtions (6): γ u T T u + ε u T T u { r + T r G } + u T T ε u r G = 10 j 10 3 xv xv xv 21 j j T G T = u j r T T G 0 ω21ω21uj 20u3u3{ r r3 } ω ω 2ω21 ω32 u T j 20 T u332 T u3r3 G xv u T j 30 T u 3u3r3 G, ε21 + ε32 = φ ( j = 1, 2). (9) 3. DYNMIS EQUTIONS The dynmics of prllel mechnisms is complicted by existence of multiple closed-loop chins. In the context of the rel-time control, neglecting the friction forces nd considering the grvittionl effects, n importnt objecte of the dynmics is first to determine the input torques or forces which must be exerted by the ctutors in order to produce gen trjectory of the end-effector, but lso to clculte ll internl joint forces or torques. Upon to now, severl methods hve been pplied to formulte the dynmics of prllel mechnisms, which could provide the sme results concerning these ctuting torques or forces. First method pplied to formulte the dynmics modelling is using the Newton-Euler procedure [10], the second one pplies the Lgrnge s equtions nd multipliers formlism [11] nd the third pproch is bsed on the principle of virtul work [12]. Knowing the position nd kinemtics stte of ech link s well s the externl forces cting on the plnr 3-PRR prllel mnipultor, in the present pper we

6 6 lculus of joint forces in dynmics of 3-PRR plnr prllel robot 201 pply the principle of virtul work for the inverse dynmic problem in order to estblish some definite recurse mtrix reltions for the clculus of internl forces in the joints. Three independent mechnicl systems cting long the plnr directions x 1 1, 1x 1 nd x 1 1 with the forces 10 f = f10u1, f 10 = f10u, f = f u cn 1 control the generl motion of the moving pltform. The force of inerti in fk0 m = k γk0 + ( ωk0 ωk0 + εk0) r k nd the resulting moment of inerti forces in ˆ ˆ mk0 = [ mk r k γk0 + J kεk0 + ωk0j kωk0] of n rbitrry rigid body T k, for exmple, re determined with respect to the centre of joint k. On the other hnd, the wrench of two vectors fk nd mk evlutes the influence of the ction of the weight mk g nd of other externl nd internl forces pplied to the sme element T k of the mnipultor. Two significnt recurse reltions generte the vectors T Fk = Fk0 + k+ 1, kfk+ 1, T T M = M + M + r F, (10) k k0 k+ 1, k k+ 1 k+ 1, k k+ 1, k k+ 1 in with the nottions Fk0 = fk0 fk in, M k0 = mk0 mk. s exmple, strting from (10), we develop set of six recurse mtrix reltions for the leg : F3 = F30, 2 F F20 32 T = + F3, 1 F F10 21 T = + F2, M 3 = M30, T M = M + M3 + r32 32 T F3, (11) T M = M + M + r T F The fundmentl principle of the virtul work sttes tht mechnism is under dynmic equilibrium if nd only if the virtul work developed by ll externl, internl nd inerti forces vnish during ny generl virtul displcement, which is comptible with the constrints imposed on the mechnism. ssuming tht frictionl forces t the joints re negligible, the virtul work produced by ll remining forces of constrint t the joints is zero. Totl virtul work contributed by the inerti forces nd moments of inerti forces, by the wrench of known externl forces nd by some internl joint forces, for exmple, cn be written in compct form, bsed on the relte virtul velocities. pplying the fundmentl equtions of the prllel robots dynmics [13, 14, 15] the following compct mtrix reltions results:

7 202 Ştefn Sticu 7 f u F u { M M M M } T T v v v v 10 y = ω ω ω ω21 2 for the externl joint force cting in the prismtic joint 1, T T v v v v m10 = u3 M 1 + u3 { ω21m 2 + ω32m 3 + ω21m 2 + ω21m 2 } for the joint torque cting in the prismtic joint 1, T T T v v v v f = u F + u { ω M + ω M + ω M + ω M } 21x for the first joint force nd T T T v v v v f = u F + u { ω M + ω M + ω M + ω M } 21y for the second joint force cting in the joint 2, T T T v v v v f = u F + u { ω M + ω M + ω M + ω M } 32x for the first joint force nd T T T v v v v f = u F + u { ω M + ω M + ω M + ω M } 32 y for the second joint force cting in the joint 3. In wht follows we cn pply the Newton-Euler procedure to estblish the set of nlyticl equtions for ech compounding rigid body of prototype robot in rel ppliction. These equtions ge ll connecting forces in the externl nd internl joints. Severl reltions from the generl system of equtions could eventully constitute verifiction for the input forces lredy obtined by the method bsed on the principle of virtul work. s ppliction let us consider sme plnr mechnism 3-PRR nlysed in [9], which hs the following geometricl nd rchitecturl chrcteristics: G* G* x0 = 0.025m, y0 = 0.025m, φ = π, m1 = 1kg, m2 = 1.5kg, m2 = 3kg 12 r = 0.1m, l = l = r 3, l = 0.3m, l = 0.2 m, t = 3 s Using the MTL softwre, computer progrm ws developed to solve the inverse dynmics of the plnr prllel mnipultor. To illustrte the lgorithm, it is ssumed tht for period of three second the pltform strts t rest from centrl configurtion nd rottes or moves long rectiliner directions. ssuming tht there re no externl forces nd moments cting on the moving pltform, dynmic simultion is bsed on the computtion of the joint forces 10 i i i i i f y, f21x, f21y, f32x, f 32 y nd of the externl joint torques m 10 i ( i=,, ) during the pltform s evolution. (12) (13) (14) (15) (16) (17)

8 8 lculus of joint forces in dynmics of 3-PRR plnr prllel robot 203 Fig. 2 Joint forces f 10 y, f 10 y, f 10 y. Fig. 3 Joint torques m 10, m 10, m 10. Fig. 4 Joint forces f 21x, f 21x, f 21x. Fig. 5 Joint forces f 21y, f 21y, f 21y. Fig. 6 Joint forces f 32x, f 32x, f 32x. Fig. 7 Joint forces f 32 y, f 32 y, f 32 y. If the pltform s centre G moves long rectiliner plnr trjectory without rottion of pltform, the intensities of internl joint forces or torques re clculted by the progrm nd plotted versus time s follows: Fig. 2 Fig. 7. For the second exmple we consider the rottion motion of the moving pltform bout z 0

9 204 Ştefn Sticu 9 horizontl xis with vrible ngulr ccelertion while ll the other positionl prmeters re held equl to zero (Fig. 8 Fig. 13). Fig. 8 Joint forces f 10 y, f 10 y, f 10 y. Fig. 9 Joint torques m 10, m 10, m 10. Fig. 10 Joint forces f 21x, f 21x, f 21x. Fig. 11 Joint forces f 21y, f 21y, f 21y. Fig. 12 Joint forces f 32x, f 32x, f 32x. Fig. 13 Joint forces f 32 y, f 32 y, f 32 y. The simultion through the MTL progrm certify tht one of the mjor dvntges of the current mtrix recurse formultion is reduced number of

10 10 lculus of joint forces in dynmics of 3-PRR plnr prllel robot 205 dditions or multiplictions nd consequently smller processing time of numericl computtion. 4. ONLUSIONS The present dynmics model tkes into considertion the mss, the tensor of inerti nd the ction of weight nd inerti force introduced by ll compounding elements of the prllel mechnism. sed on the principle of virtul work, this pproch estblishes direct determintion of the time-history evolution for the internl forces or torques in joints. hoosing pproprite seril kinemticl circuits connecting mny moving pltforms, the present method cn esily be pplied in forwrd nd inverse mechnics of vrious types of prllel mechnisms, complex mnipultors of higher degrees of freedom nd prticulrly hybrid structures, when the number of components of the mechnisms is incresed. REFERENES 1. TSI, L-W., Robot nlysis: the mechnics of seril nd prllel mnipultors, Wiley, STEWRT, D., Pltform with Six Degrees of Freedom, Proc. of the Institution of Mechnicl Engineers, Prt 1, 180, 15, pp , MERLET, J-P., Prllel robots, Kluwer cdemic Publishers, LVEL, R., Delt: fst robot with prllel geometry, Proceedings of 18 th Interntionl Symposium on Industril Robots, Lusnne, 1988, pp TSI, L-W., STMPER, R., prllel mnipultor with only trnsltionl degrees of freedom, SME Design Engineering Technicl onferences, Irvine,, HERVÉ, J-M., SPRINO, F., Str. New oncept in Robotics, Proceedings of the Third Interntionl Workshop on dvnces in Robot Kinemtics, Ferrr, pp , ONEV, I., ZLTNOV, D., GOSSELIN,., Singulrity nlysis of 3-DOF plnr prllel mechnisms vi screw theory, Journl of Mechnicl Design, 125, 3, pp , PENNOK, G.R., KSSNER, D.J., Kinemtic nlysis of Plnr Eight-r Linkge: ppliction to Pltform-type Robot, SME Mechnisms onference, Pper DE - 25, pp , STIU, S., Inverse dynmics of the 3-PRR plnr prllel robot, Robotics nd utonomous Systems, 57, 5, pp , DSGUPT,., MRUTHYUNJY, T.S., Newton-Euler formultion for the inverse dynmics of the Stewrt pltform mnipultor, Mechnism nd Mchine Theory, 33, 8, pp , GENG, Z., HYNES, L.S., LEE, J.D., ROLL, R.L., On the dynmic model nd kinemtic nlysis of clss of Stewrt pltforms, Robotics nd utonomous Systems, 9, 4, pp , ZHNG,.D., SONG, S.M., n Efficient Method for Inverse Dynmics of Mnipultors sed on Virtul Work Principle, Journl of Robotic Systems, 10, 5, pp , STIU, S., LIU, X-J., LI, J., Explicit dynmics equtions of the constrined robotic systems, Nonliner Dynmics, 58, 1-2, pp , STIU, S., Dynmics nlysis of the Str prllel mnipultor, Robotics nd utonomous Systems, 57, 11, pp , STIU, S., Dynmics of the 6-6 Stewrt prllel mnipultor, Robotics nd omputer- Integrted Mnufcturing, 27, 1, pp , 2011.