Elastic Systems for Static Balancing of Robot Arms

Transcription

1 . th World ongress n Mechans and Machne Scence, Guanajuato, Méco, 9- June, 0 _ lastc Sstes for Statc alancng of Robot rs I.Sonescu L. uptu Lucana Ionta I.Ion M. ne Poltehnca Unverst Poltehnca Unverst Poltehnca Unverst Poltehnca Unverst Unverst of Quebec ucharest, Roana ucharest, Roana ucharest, Roana ucharest, Roana Quebec, anada bstract The purpose of the statc balancng of the robot ars s to decrease the drvng forces or oents. The weght forces of the ars represent prncpal charge of the drvng sste of an ndustral robot. The paper shown soe consderatons on the statc balancng of the ndustral robot ars and new constructonal solutons for the elastc sstes wth nonlnear characterstcs used for these. nall, the result of solvng a nuercal eaple s presented Kewords: elastc sstes, robot ar, statc balancng I Introducton The echanss of the ndustral robots consttute a partcular categor of echancal sstes whch are buldng b bg ass coponents that ove n vertcal planes wth relatvel low veloctes and acceleratons. The ost energ consupton s used to overcoe the echancal work of weght forces. The decreasng of the energ consupton can be done balancng the weght forces of the robot ars,.e. b statc balancng. Ths balancng can be perfored b gravtatonal forces of the counterweghts or b elastc forces of the sprngs. The used of the counterweghts ncrease the overall sze and the nerta of the sste wth unfavorable effect over the dnac workng of robot. The splest constructve soluton of balancng the weght force of an ar whch rotates around a horzontal fed as used a helcal etended sprng joned between a pont of the ar and another f pont []. In ths sple case, soe dffcultes arse n the constructon of ths sste, where ust be use the zero-free length helcal sprng. In order to avod the constructve coplcatons whch results b the account the real helcal sprngs, followng optons can be consdered:. the aes of the clndrcal pars between the helcal etended sprng and the frae or the robot ar are not fed wth respect to these,. the usng of the helcal etended sprng wth a nonlnear elastc characterstc, sonescu@ahoo.co lvuc@ahoo.co lucana_onta@ahoo.co on@ahoo.co arene@uqat.ca. the helcal etended sprng wth lnear characterstc s changed wth an elastc sste buld b helcal sprngs and lnks. The elastc sstes for statc balancng of the robot ars have been eplored b a bg nuber of researches. The dfferent structures of elastc sstes have been advanced, for dfferent applcatons: parallel robots [], [], planar lnkages wth revolute jonts [] etc. Soe constructve solutons realzed onl an approate balancng, or have the ver lted feld of applcatons [], []. II lastc Sstes wth Nonlnear haracterstcs In [] and [7] are shown soe solutons whch correspond to the frst opto The second opton s ver dffcult to buld fro technologcal pont of vew, because the elastc eleent ust not be a clndrcal helcal sprng. The thrd opton can be soetes a ver good choce. ut, the kneatcs densons nuber of the coponent eleents s fnte, and as result, the postons nuber of the robot ar n whch the statc balancng are accuratel acheved s relatvel sall. These elastc sstes ust have nonlnear elastc characterstcs, but the elastc eleent s a helcal sprng wth lnear characterstc. The snthess equatons of these balancng echanss consst of: - kneatcs equatons, whch epressed the projectons of the vectoral equatons of the ndependent loops on the fed coordnate aes - knetostatcs equatons, whch defned the forces and oents equlbru whch act on the coponent eleents. sple elastc sste wth nonlnear characterstc s dsplaed n g.. Ths sste s ade up b two helcal sprngs ( and (, connected n seres. rst sprng s joned to the robot ar (, and second to the frae. The connecton pont s joned to the lnk (, whch rotates around a horzontal fed as. The unknown kneatcs densons of the coponents, nael:,,,,, β, and the characterstcs of the sprngs: unstressed lengths l 0, l 0 and stffness coeffcents k, k, are calculated as solutons of the followng equatons [9]: cos( cos cos 0 sn( sn sn 0

2 th World ongress n Mechans and Machne Scence, Guanajuato, Méco, 9- June, 0 _ cos( cos cos 0 sn( sn sn 0 S cos S cos R cos 0 S sn S sn R sn ( g 0 S ( sn cos g( ( G cos, cos( 0,, n, β arctan ( ( cos( cos sn( sn cos( β sn( β. G g g.. lastc sste wth two sprngs and one lnk The asses of the eleents and the postons of the ass centers are assued as know These asses are consdered statc concentrate to the centers of adjacent jonts. In fact, the proble ust be solved n an teratve anner, because at the begnnng of the desgn, the asses of the helcal sprngs and of the lnks are unknowns. To the unknowns above entoned are added the poston angles φ, φ, φ the lengths, and the agntude of the reacton force R. ro copatblt condton of snthess equatons s obtaned n. The statc equlbru of the ar s accuratel realzed n these postons defned b the angles,, onl. ue to contnut accounts, the unbalanced values are ver sall between these postons. The au agntudes of the unbalanced oents, between calculaton postons, are nverse proportonal to the nuber of the unknowns of the elastc sste []. In g. s shown another elastc sste, fored b two helcal sprngs and two stff eleents. The snthess equatons are: α G g cos( cos sn( g. lastc sste wth two sprngs and two lnks sn cos( cos sn( sn cos 0 sn 0 0 cos sn 0 cos( sn( cos sn γ γ S cos S cos R cos 0 sn sn R sn S ( S (( S g(( (( S ( S cos ( cos g 0 ( sn ( ( cos( 0,, n, cosα β arctan cos sn cos sn sn ( 0 cos g ( ( G cos

3 th World ongress n Mechans and Machne Scence, Guanajuato, Méco, 9- June, 0 _ sn cos( cos sn( sn cos( β sn( β snα γ π arcsn(,, The elastc forces of the sprngs ( and ( respectvel are: S ( l0 k S ( l0 k, ( ( ( (,, The unknowns of the sste of snthess equatons are:,,,,,,, α,, β, l 0, l 0, k, k,,, R, φ, φ, φ, φ,, ro copatblt condton, the nuber of coputaton postons s: n. G _ g g. lastc sste wth one sprng and three lnks or the desgn of elastc sste dsplaed n g., the snthess equatons are wrtten usng the vrtual work prncple [8], [9]: gg cos ( g ( g ( g ( g ( g ( 0 ( l0 k 0,,, ( ( ( ( ( ( cos( α sn( sn α cos cos sn cos cos α α sn( cos( cos( β sn( sn sn( cos( cos β sn cos β β sn cos sn( sn cos sn( sn cos. sn( These equatons are solved wth respect to the followng unknowns:,,,,,,, α,, β,,, 0 and k. The statc balancng s realzed theoretcal eact for dstnct values of the poston angels. The length l 0 of sprng ( s posed fro constructve condtons. ertanl, can be posed 0 0 and s calculated the unstressed length l 0. The poston angles,, and are calculated for each poston as solutons of the followng equatons: cos cos 0 sn sn 0 sn cos sn(

4 th World ongress n Mechans and Machne Scence, Guanajuato, Méco, 9- June, 0 _ cos cos 0 sn sn 0, nael: U U V W VW V U V W U W U U V W VW V U V W U W U U V W VW V U V W U W U U V W VW, V U V W UW U ( V ( W ( ( U ( V ( W ( ( U ( V ( W ( ( U ( V ( W ( ( In g. a few ore elastc sstes wth nonlnear characterstcs for statc balancng of the robot ars are show The nuber n of postons of the ar where the statc balancng s realzed theoretc eact depends on the coplet of elastc sste,.e. on the nuber of unknowns of the snthess equatons. Thus, ths nuber on unknowns whch ust be calculated for desgn the elastc sste dsplaed n g., a s 0, n g, c s, and ncrease to for elastc sstes fro gs., d and e, to for g., f and to fro the last fgure. The equatons whch epressed kneatcs operatng condtons and the equatons whch defned the forces and oents equlbru condtons ust be resolved together, wth ecepton of frst eaple (g., c. Ths condton has as effect the ncrease the nuber of nonlnear equatons of the snthess sste. G g a G g b c G g G g d

5 th World ongress n Mechans and Machne Scence, Guanajuato, Méco, 9- June, 0 _ e G g G f g G g G g H G g h g. lastc sstes wth nonlnear characterstcs for statc balancng of robot ars III aple The statc balancng of the ar of ass kg s taken as eaple to llustrate the usefulness of the elastc sste shown n g.. The calculated densons of the sprngs and lnks are: -.09,.90, -.08, 0.0, 8.7,.09,.98, α.07,.0, β , l 0.77, l 0 0.0, K 0.7, K The asses of the eleents are:.,.,.,.. IV onclusons The choce of structure of the elastc sste for statc balancng of robot ars s not a sple proble. Where the oveent of the ar has the large range of speeds and acceleratons, the structure ust be as sple as possble. In each case, ust calculate the effcacous coeffcent [0], whch lts the usage doan of a specfc structure. References [] Sonescu I., uptu L.The statc alancng of Industral Robot rs, Part I: screte alancng, Mechans and Machne Theor, (9, pp , 000. [] Lessard S., gras P., onev I.., rot S., rakelan V., ptu statc alancng of the Parallel Robot for Medcal Ultrasound, th IToMM World ongress, esançon, June 8-, 007 [] aradat., rakelan V., rot S., Guegan S., esgn and Prototpng of a New alancng Mechans for Spatal Parallel Manpulators, Journal of Mechancal esgn, Jul 008, Vol. 0 [] Po-ang Ln, Wn-n Shch, ar-zen hen, esgn of Perfectl Statc alanced ne- Planar Lnkage Wth Revolute Jont nl, Proceedngs of the SM 008 IT/I, ugust -, rookln, New ork, US [] Herder J.L., Gravt alancers wth onl Sprngs onnectons to Ground, th IToMM World ongress, esancon, rance, June 8-, 007 [] Stret.., Glore.J., Perfect Sprng qulbrators for Rotatable odes, Journal og Mechanss, Transssons and utoaton n esgn no., (989, pp. 8. [7] Sonescu I., uptu L.The statc alancng of Industral Robot rs, Part II: ontnuous alancng, Mechans and Machne Theor, (9, pp. 99, 000. [8] ppell P., Trate de Mechanque Ratonnelle, Gauther Vllars, Pars, 90 [9] Sonescu I., Mose V., Mecanse, dtura Tehnca, ucurest, 999 (n Roanan. [0] Sonescu I., Ion I., uptu L., Mecansele robotlor ndustral, dtura GIR, ucurest, 008 (n Roanan