ON THE KINEMATICS OF THE SCORBOT ER-VII ROBOT

Transcription

1 U... Sci. ull. Series D Vol. 77 Iss. 05 ISSN 5-58 N THE KINEMATICS F THE SCRT ER-VII RT Lurenţiu REDESCU Ion STRE Due to the relly fst dynmics of the development nd diversifiction of industril robots nd to the ever wider res of their use structurl optimiztion is desired. Using computers with high processing power of the dt led to the possibility of modelling lrge number of vrints of structure to ccomplish similr functions. y modelling the corresponding direct nd inverse problems it is possible to determine positions in terms of the end-effector s timing nd tht it my impose different trvel pths within set time so tht the finl effect should chieve precise tsks. The prcticl spect cn be inferred by nlyzing the kinemtics of the seril robot SCRT ER-VII t the sme time determining velocity nd ccelertion chrts for chrcteristic point on the end-effector. Keywords: robot kinemtics model pth plnning. Introduction The tsk of robot mnipultor is to follow pth leding the end-effector towrds stble position. In order to cuse the robot to move ccording to pth between two or more well-defined trget positions being the required conditions to chieve these positions in given time t certin speed ccelertion nd so on it determines the movement of different components of the robot []. The movement it mkes cn be chieved by severl configurtions of elements in which cse the robot motion must be studied within the workspce [] []. Kinemtics nlysis is the study of the movement of robots tht mke up these mechnisms. Everything relted to kinemtic nlysis i.e. position velocity nd ccelertion of ll elements will be clculted in reltion to fixed given reference system. The kinemtics nlyses do not ccount for the forces nd momentums cting on the robot elements. Lecturer Deprtment of Food Engineering Vlhi University of Târgovişte Romni e-mil: prelur@yhoo.com rofessor Deprtment of Mechnics University LITEHNICA of uchrest Romni e-mil: ion.stroe@gmil.com

2 6 Lurenţiu redescu Ion Stroe To browse functionl trjectories the kinemtic model needs progrm to generte these trjectories which progrm will be ble to ccomplish the tsk either off-line or on-line. Most robot mnipultors re designed to ccomplish tsks in D workspce. There re two different pproches on the robot rm movement: specifying the loction of the end-effector in D coordintes; the individul movement of ech joint seprtely. The component the mnipulted tool or the end-effector must follow plnned trjectory. The kinemtic model provides reltions between the end-effector s position nd orienttion nd sptil positions of the other elements the joints. Kinemtics modeling is divided into two problems: direct kinemtics nd inverse kinemtics. The problem of direct kinemtics is to determine the position nd orienttion of the end-effector giving vlues for the vribles in the robot joints. Inverse kinemtics focuses on determining vlues for the vribles in the joints required to move the robot s end-effector in desired position nd orienttion.. The inverse geometric model The geometric model for geometric reverse order (inverse geometric model) consists in determining the vector of generlized coordintes (the robot Θ Θ q q bsed on the vector of the opertionl coordintes) ( ) q k 0 0 coordintes X ( p p p α β γ ) X x y z (the coordintes of the chrcteristic point nd the orienttion ngles for the end-effector in reltion to the system {0}) [][5][6][7][8]. The usul commnd lgorithms re mde up of reltions which express the movements of the motors in terms of the positioning prmeters of the directed body. As direct geometric modelling is defined by the vectoril expression: ( Θ) 0 X f (.) the inverse geometric modelling (the commnd geometric modelling) will hve the vector expression: 0 Θ f ( X ) (.) It is sid tht robot cn be solved for ctegory of tsks involving vector type 0 X of the opertionl coordintes if knowing the direct geometric 0 X ƒ Θ we cn mthemticlly obtin unique solution for the system model ( ) ( 0 X ) Θ ƒ.

3 n the kinemtics of the scorbot ER-VII robot 7 In order to chieve the geometric commnd the (m) number of the 0 prmeters X 0 j X should not be lrger thn the (n) number of the degrees of freedom: T [ px p y pz α β γ ] if n 6 T [ p ] x p y pz α β if n T T X [ X j ; j m] [ px p y pz α ] if n (.) T [ px p y pz ] if n T [ p p ] if n x y where the first sitution shows the generl movement of the end-effector nd the lst cse reltes to trnsltion movement within the horizontl plne. The connection between the column vectors 0 X nd Θ is thus chieved by the f opertor: 0 0 T X [ X j ; j m] [ f j ( qi ; i n) ; j m; m n] T (.) Concerning the inverse geometric model the (.) eqution system stnds for non liner trnscending system of equtions for which there is no generl clcultion lgorithm. Under certin conditions connected to the position nd the reltive orienttion of the neighbouring kinemtic xes k i ki the (.) system cn be solved through lgebric methods or through methods belonging to the plne geometry. Unlike the geometric pproch of the inverse geometric model which differs from problem to nother the lgebric methods re bsed on the reduction of the trnscending equtions to lgebric ones using one unknown term nd s consequence they cn be generlized. The (.) eqution cn be written s follows: T 0 [ q ; i n] [ f ( X ; j m) ; i n ] T (.5) i i j The (.5) equtions express certin configurtion of the robot which stisfies the known position nd orienttion of the finl effector. The gret hindrnce of the (.) nd (.) eqution systems is tht they re non-liner. As we know such systems re solved using numericl methods which present unvoidble errors. The solving methods re divided into two ctegories: closed lgebricl or geometricl methods (pplicble on prticulr cses); numericl methods.

4 8 Lurenţiu redescu Ion Stroe Any of the previous methods led to multiple solutions for the generlized coordinte q i. The choice of the unique solution depends on the geometry of the mechnicl structure of the robot nd its interction with the environment.. The kinemtic model of the SCRT-ER VII robot The SCRT-ER VII robot is verticlly rticulted robot hving 5 rottion joints [9]. Arm Forerm End-Effector ody/ Shoulder se Fig. Elements nd joints of the SCRT-ER VII robot In figure both the joints nd the elements/components of the robotic system cn be identified. The kinemtics chin in which the chosen reference systems re highlighted too is presented in figure. Considering the problem of determining set of vribles respectively it is necessry to stisfy the demnd for position of the end-effector. The demnd for orienttion of the end effector nd determining the 5 vrible re neglected.

5 n the kinemtics of the scorbot ER-VII robot 9 z z y y y y d 5 5 y 5 ' d x x z y z z x x 5 x z 5 Fig.. Choosing the reference systems in the kinemtic chin of the SCRT -ER VII As fr s ick nd lce opertion is concerned the vribles cn be determined geometriclly imposing some of the conditions illustrted in figure. ne condition could be tht the end-effector is prllel to the bse plne. The representtions in figure nd figure re used to level the elements within the robot s configurtion which re constnt: d; ; ; ; 5 d5. The 5 point hs the coordintes expressed in terms of the bse frme xyz. Given the coordintes of 5 ( x y z ) on element 5 nd the constnt distnces ( d nd d 5 ) vribles cn be determined. In order to determine the vrible the fct tht the projection of 5 in the plne xy is point C with ( x y 0) coordintes cn be used. It results in: tn ( y x ) (.) The tn ( y x) function clcultes the rc tngent of the two vribles x nd y. In terms of the stndrd rctn function whose rnge is ( π π ) it cn be expressed s follows:

6 0 Lurenţiu redescu Ion Stroe z 5 A α β D y C x Fig. Simplified representtion of the robot s elements within the xyz reference system y rctn x > 0 x y rctn π y 0 x < 0 x y rctn π y < 0 x < 0 tn ( y x) x π y > 0 x 0 π y < 0 x 0 undefined y 0 x 0

7 n the kinemtics of the scorbot ER-VII robot Horizontl distnce: ( ) ( ) 5 d y x D D Verticl distnce: 5 5 d z DC C D (.) ( ) ( ) π (.) From (.) nd (.) results: (.) Replcing the horizontl nd verticl distnces in (.) nd previously determined this eqution is obtined: ( ) ( ) ( ) 5 d z d y x (.5) Using the previously determined vrible the α ngle in the A nd A tringles is determined. ( ) ( ) ( ) ( ) ( ) ( ) sin tn sin tn tn α A A A (.6) The β ngle will be determined within the tringle using the distnces nd ( ) ( ) ( ) 5 tn tn d y x d z β (.7) The vrible is determined through the difference between the two ngles α nd β : α β (.8) ( ) ( ) ( ) ( ) ( ) 5 sin tn tn d y x d z (.9) The condition of being horizontl of the end-effector is expressed by the reltion (.0) 0 (.0) From the (.0) condition nd by using (.9) nd (.5) the vlue of the ngle is determined: (.)

8 Lurenţiu redescu Ion Stroe. Trjectory plnning The purpose of trjectory plnning is to generte input dt for the motion control system so tht the mnipultor should execute specific tsk under imposed velocity nd ccelertion conditions [0]. In the cse of the previously studied mnipulted it is mndtory tht the end-effector trjectory should be segment defined by the strting point M by the (x M y M z M ) coordintes nd the finl point N by the (x N y N z N ) coordintes the bove mentioned trjectory being follow by the chrcteristic point 5 within the time intervl Δt (figure ). If we keep the imposed restriction ccording to which the end-effector cover the trjectory horizontlly nd we dd the restriction ccording to which the covered segment would be within xz plne the mnipultor movement will depend on the vrition of the ngle. z 0.75 [m] N M x [m] Fig. 5 MN trjectory in initil nd finl position of the robot mnipultor To set of constnt vlues representing the construction elements of the mnipultor [9] m 0. 00m 0. 50m d 0. 85m d5 0. m the M( ) N( ) trjectory nd for polynomil vrition of we cn obtin velocity nd ccelertion vritions of point 5. Using (.) (.5) (.9) (.) equtions for the coordintes of points M nd N we cn determine the initil nd finl nd ngles. y using these vlues (especil the ones for ) we cn determine the cubic vrition on Δt 0 seconds time intervl.

9 n the kinemtics of the scorbot ER-VII robot The lgebricl expression is grphiclly represented in figure 5 nd tkes the form of eqution (.). ( t) t 0. 6 t [ deg ] (.) According to the ngle we cn determine the segment which is prllel i to the trjectory s being the intersection between the segment f nd the i center circle nd the (respectively ) rdius (figure 6). We should observe tht is mobile ccording to the rc trjectory hving rdius (respectively ) with s its center [deg] time [s] Fig. 6 Cubic vrition in terms of time of the prmeter. We cn chieve the coordintes of the mobile point by solving the qudrtic eqution A X X C 0 using the following nottions: A m (.) [ m n m z ( t) x ( t) ] (.) [ x () t ] z () t [ ] n z () t n C (.) where: x () t ( ( t) ) nd z ( t) d ( ( t) ) sin z i z f z i z f m n z f x f x nd z the coordintes of the x i x f x i x f respective points. The coordintes thus determined re:

10 Lurenţiu redescu Ion Stroe z [m] 0.75 f N i M x [m] Fig. 7 Intersection between circle nd the segment to determine the current point A C x () t (.5) A z () t m x () t n (.6) Using equtions (.5) nd (.6) the coordintes of the point re given by: x () t x () t d 5 5 z ( t) z ( t) (.7) 5 We cn obtin velocity nd ccelertion of 5 which is ttched to the end-effector in figure 7. The velocity of point 5 [m/s] Vt () At () time [s] time t[s] t ) b) Fig. 8 Velocity nd ccelertion history of point 5 for the cubic vrition of ngle If we refer to sinusoid vrition of the ngle which is similr to the one in [] hs the structure (.8) nd the shpe in figure 8. The ccelertion of point 5 [m/s^]

11 n the kinemtics of the scorbot ER-VII robot 5 π ω ( 0 0 π ω0 i t) i A t sin( ω t) [ ] π deg (.8) For this vrition prmeter [ rd] A. 975 [ rd ] [ ] ω 0 s re considered. 5 0 nd 5 00 [deg] [deg] time [s] Fig. 9 Sinusoid vrition in terms of time of the prmeter The velocity of point 5 [m/s] Vt () time t [s] ) Fig. 0 Velocity nd ccelertion history of point Velocity nd ccelertion history of point 5 re represented in figure 9 ) m / s. nd b) nd re expressed in [ s] The ccelertion of point 5 [m/s^] m / nd [ ] ) time [s] t b) 5 for the sinusoid vrition of the ngle.

12 6 Lurenţiu redescu Ion Stroe 5. Conclusions This pper wishes to illustrte the stges of plnning nd the motion control of robot with rotry couplers. In order to chieve complex trjectories we cn pln nd control the movement interpolting trjectories belonging to the stright line on plne. The method presented here llows the clcultion of the geometricl prmeters of the decoupled Scorbot-ERVII robot nlysing seprtely the position equtions nd the orienttion ones. Even if this procedure simplifies nd fcilittes the clcultion effort the nlyticl pproch of the kinemtic control solutions still remins complex mtter. The bse concept of the proposed pproch is constituted by the fct tht determining vribles involves geometricl modelling of the robotic structure which leds to multiple solutions mening tht for certin positioning severl configurtions re obtined in which cse it is necessry to intervene in the choice of the vrible sets to generte the tsk. The comprtive study of the grphic representtion genertes the possibility of n optiml pproch to the rel work version in terms of the tsk imposed to the end-effector nd its lod. R E F E R E N C E S []. Sicilino L. Scivicco L.Villni G. riolo Robotics Modelling lnning nd Control Springer 009 [] J.A. Snymn n non-ssembly in the optiml synthesis of seril mnipultors performing prescribed tsks J. Lenrcic nd. Roth (eds.) Advnces in Robot Kinemtics Springer 006 pp [] R.N. Jzr Theory of Applied Robotics nd ed. Springer Scienceusiness Medi 00 [].opescu I.Negren I.Vuşcn N.Hiduc R.opescu Mecnic mnipultorelor şi roboţilor (Mechnics of Mnipultors nd Robots) Vol. şi Editur Didctică şi edgogică R.A. ucureşti 99 [5] S.Sticu Modèle dynmique en robotique U Scientific ulletin Series D: Mechnicl Engineering 6 - pp [6] V.Filip Modelre mnipultorelor robot. Clcul simbolic. Geometri directă şi inversă. Cinemtic directă (Modelling of robot mnipultors. Symbolic clculus.. Direct nd inverse geometry. Direct Kinemtics) Editur RINTECH ucureşti 999 [7] L-W. Tsi Robot nlysis: the mechnics of seril nd prllel mnipultors Wiley 999 [8] S.Sticu Méthodes mtricielles en cinémtique des mécnismes U Scientific ulletin Series D: Mechnicl Engineering 6 pp [9] *** User s Mnul SCRT-ER VII nd ed. Eshed Robotec 998 [0] S.R. Wng Z.Z. Qio nd.c. Tung Appliction of the force control on the working pth trcking Journl of Mrine Science nd Technology vol. 0 no. 00 pp [] I. Stroe S. Sticu A. Criflenu Internl Forces clculus of Compss Robotic Arm Using Lgrnge Equtions th Symposium on Advnced Spce Tehnologies for Robotics nd Automtion ASTRA 0 ESTEC Noordwijk The Nederlnds April - 0.