1 axis. F Input force for i th joint. f Force exerted on linki by linki 1at the coordinate frame ( i 1

Transcription

1 Internatonal Journal of Robotcs and Automaton (IJRA) Vol. 3, No., December, pp. ~33 ISSN: Dynamc Behavor of a SCARA Robot by usng N-E Method for a Straght Lne and Smulaton of Moton by usng Soldworks and Verfcaton by Matlab/Smulnk Brahm Fernn Depatement of Mechancal, Blda Unversty, Algera Desgn, path plannng and dynamc modelng for seral robots by usng Soldworks (3) and Matlab/Smulnk () Artcle Info Artcle hstory: Receved Jun 7, 3 Revsed Oct 6, 3 Accepted Apr, Keyword: Robotcs SCARA Robot Dynamc Behavor path plannng Smulaton Sold Works Matlab/Smulnk ABSRAC SCARA (Selectve Complant Assembly Robot Arm) robot of seral archtecture s wdely used n assembly operatons and operatons "pckplace", t has been shown that use of robots mproves the accuracy of assembly, and saves assembly tme and cost as well. he most mportant condton for the choce of ths knd of robot s the dynamc behavor for a gven path, no closed soluton for the dynamcs of ths mportant robot has been reported. hs paper presents the study of the knematcs (forward and nverse) by usng D-H notaton and the dynamcs of SCARA robot by usng N-E methods. A computer code s developed for trajectory generaton by usng nverse knematcs, and calculates the varatons of the torques of the lnks for a straght lne (path rest to rest) between two postons for operaton "pck-place". SCARA robot s constructed to acheve pck-place» operaton usng Sold Works software. And verfcaton by Matlab/Smulnk. he results of smulatons were dscussed. An agreement between the two softwares s certanly obtaned heren. Copyrght Insttute of Advanced Engneerng and Scence. All rghts reserved. Correspondng Author: Brahm Fernn, Departement of Mechancal, Blda Unversty, ALGERIA BP 7 route de soumaa, Blda, ALGERIA e-mal: fernn.brahm@gmal.com Nomenclature: A D-H transformaton matrx for adjacent frames, and C Cosne Cjk Cosne jk = cosne {( + j ) + k } d Dstance from the orgn of the ( ) the coordnate frame to the ntersecton of the X axs along Z axs e s the poston vector of the COM of lnk wth respect to frame F Input force for th jont f Force exerted on lnk by lnk at the coordnate frame ( the lnks above t Z axs wth the X, Y, Z ) to support lnk and Journal homepage:

2 ISSN: Input torque for th jont I Inerta matrx of lnk about ts center of mass wth reference to the coordnate system ( X, Y, Z ) J Inerta matrx of lnk about ts center of mass referred to ts own lnk coordnate system ( X, Y, Z ) l he shortest dstance between Z and Z axes meff Effectve mass. m Mass of the th lnk n Moment exerted on lnk- by lnk at the coordnate frame ( X, Y, Z ) p s the dsplacement from the orgn of frame to frame wth respect to frame he jont angle from X axs to the X axs about the Z axs (usng the rght hand rule) R A 3 3 rotaton matrx whch transforms any vector wth reference to coordnate frame ( X, Y, Z ) to the coordnate system ( X, Y, Z ) S Sne S jk Sne jk = sne {( + j ) + k } V Lnear velocty of the coordnate system ( X, Y, Z ) wth respect to base coordnate system ( X, Y, Z ) Angular velocty of the coordnate system ( X, Y, Z ) wth respect to base coordnate system ( X, Y, Z ). INRODUCION Pck And Place cycle s the tme, n seconds, to execute the followng moton sequence: Move down one nch, grasp a rated payload; move up one nch; move across twelve nches; move down one nch; ungrasp; move up one nch; and return to start locaton. he SCARA Selectve Complant Assembly Robot Arm or Selectve Complant Artculated Robot Arm s wdely used for operatons pck-place. he robot was called Selectve Complance Assembly Robot Arm, SCARA. Its arm was rgd n the Z-axs and plable n the XY-axes, whch allowed t to adapt to holes n the XY-axes. By vrtue of the SCARA's parallel-axs jont layout, the arm s slghtly complant n the X-Y drecton but rgd n the Z drecton, hence the term: Selectve Complant. hs s advantageous for many types of assembly operatons: pck-place, nsertng a round pn n a round hole wthout bndng. he second attrbute of the SCARA s the jonted two-lnk arm layout smlar to our human arms, hence the often-used term, Artculated. hs feature allows the arm to extend nto confned areas and then retract or fold up out of the way. hs s advantageous for transferrng parts from one cell to another or for loadng/ unloadng process statons that are enclosed. he SCARA robots are generally faster and cleaner than comparable Cartesan systems. her sngle pedestal mount requres a small footprnt and provdes an easy, unhndered form of mountng. On the other hand, SCARA's can be more expensve than comparable Cartesan systems and the controllng software requres nverse knematcs for lnear nterpolated moves. hs software typcally comes wth the SCARA though and s usually transparent to the end-user. In ths work, axes«r-r-p-r» robot systems for operaton pck and place wll be desgned and developed usng Soldworks program as shown n fgure, and modeled by Matlab/Smulnk as shown n fgure.smulaton by usng MALAB/Smulnk software wll be carred out. he Results of both sofwares IJRA Vol. 3, No., December : 33

3 IJRA ISSN: wll be presented and dscussed. In the paper, the equatons of knematcs for «R-R-P-R» robot wth the robot dynamcs for each jont were developed wth D-H formulaton. he paper s organzed as follows: Frst, an ntroducton to SCARA robot, knematcs spresented n secton. In secton 3, the dynamc behavor.in secton,the applcaton. Sectons 5, 6 and 7, the dynamcs smulaton, dscusson and concluson respectvely and followed by the the references. Fgure. SCARA robot modeled by Sold Works Fgure. SCARA robot modeled by Matlab/Smulnk Prevous Work he prevous work [] [] studed the dynamc of ths robot by usng N-L method, but ths method s not commonly used for real tme control as ts need large amount of computaton tme and space, and the study of the dynamc behavor s done for path created by the jont space, ths last does not gve the desred trajectores lke (straght lne, crcle,..). Present Work he present analyss of ths robot s carred out to study the dynamcs behavor for a straght lne (rest to rest path) by usng N-E method. he sgnfcance of ths study les n the fact that t gves nsght nto the dynamc behavor of ths robot. he drect knematcs allows us to fnd the relatonshp between the angular dsplacement and the poston of the end-effector, the nverse knematcs allow us to connect between two postons by a straght lne (rest to rest path). SoldWorks and Matlab Smulnk softwares are used to model and check the robot moton smulaton.. ROBO KINEMAICS. Drect Knematcs he Denavt-Hartenberg (D-H) parameters for SCARA robot shown n Fgare defned n table: Lnk able. D-H parameters of SCARA Robot. a d l l 3 3 d d : jont varables he expresson for the end effector frame relatve to the base frame s gven by the arm matrx ( ) as: Dynamc Behavor of a SCARA Robot by usng N-E method for a Straght Lne (Brahm Fernn)

4 ISSN: , where: 3 3 c s lc c s lc c s s c ls s c ls 3 s c 3 d3 d,,, After multplcaton and use of addton matrces, one gets the homogeneous transformaton matrx; c s lc lc s c ls ls d3 d.. Inverse Knematcs.. Inverse Soluton for Postons Desred locaton of Robot: nx ox ax px R ny oy ay py H nz oz az pz he fnal equaton representng the robot s [3]: R H We get: px lc lc, py ls l s. By usng Kramer methods we fnd []; Equaton of elbow up: a tan s, pxls py( llc ) a tan c p ( l l c ) ( p l s ) x y Equaton of elbow down: a tan s, pxls py( llc ) a tan c px( llc) ( pyls) Inverse soluton for velocty: Pc X Ps y P X ( lc lc) P y ( ) l s l s ls ll s Inverse soluton for acceleraton: ( Ps X Pc ) y ( Pc X Ps y ) lc ( Ps y Pc x ) l( Ps y Pc x ) l ls lls ( Pc y Ps x ) l ( Ps X Pc y ) l llc ll s 3. ROBO DYNAMICS We fnd the dynamcs equatons of moton of robots by two methods: Newton-Euler and Lagrange. he Newton-Euler method s more fundamental and fnds the dynamc equatons to determne the requred actuators force and torque to move the robot, as well as the jont forces. Lagrange method provdes only the requred dfferental equatons that determne the actuators force and torque. [5] he N-E method s based on two recursons forward and backward recursve equatons. he forward recursve equaton s used for the knematcs nformaton such as veloctes and acceleratons at the center of mass of each lnk. he backward recursve equaton s used for the forces and moments exerted on each lnk from the end effector to the base of the robot. IJRA Vol. 3, No., December : 33

5 IJRA ISSN: he rotaton matrces are as follows: C S C S C S,, 3, R S C R S C R S C R3 C S C S C S, R R S C 3 R,, R S C R S C C S C S, R S C R S C 3 R R P lc, ls, P lc, ls, P,, d,,, 3 3 V, V g P,,,,, Forward recursve: R R Z R R R Z R R R R R R Z R R Z Z ,, R R p R RV R R R Z R Z R R R R R R Z R Z RV R R p R R R p R RV l l g RV R R p R l Sl Cl, l l S l C, g 3 3 R3RZ RV R Z RV R R p R R R p R R V R R p R R R p l Sl Cl, l Cl Sl, g Dynamc Behavor of a SCARA Robot by usng N-E method for a Straght Lne (Brahm Fernn)

6 6 ISSN: R R Rp 3 R3RV 3 RV R R p l Sl Cl C S, l Cl Sl S C, g he poston of center of mass: e lc /, ls /, /, /, 3,, d3 / RV RV l /, l /, g RV RV e l C l S e e,, R a R Re R R R e R a R Re R R Re /, /, RV l S l C l l C l S l g Ra3 R 3 Re 3 R 3 R 3 Re 3 RV l Sl Cl, l Cl Sl, g R R Re RV RV l Sl Cl C S Ra R Re, l C l S l S C, g Backward recursve: We have: f5n5 RfR Rf mramra l Sl Cl C S, m l C l S l S C, g IJRA Vol. 3, No., December : 33

7 IJRA ISSN: xl Sl C yl, R f R R f m R a l Sl Cl, m3m l Cl Sl, g 3 R f R R f m R a 3 3, x l C l S yl xg RfR Rf mra l xm yl C, l xm / xm m m / S, x yl S C 3 ym /mm 3 he moments exerted on the lnks: Rn R 5 Rn 5 Rp R f5 R p R e m R a J R RJR J RIR m g,, 3, Generally, the mass and the length of lnk () are very small n comparson to other lnks; the nerta of lnk () s evaluated to be zero. Rn 3 3 Rn 3 R Rn R p3 R f RpRe mra JR d3 m m R3 J3R3 ml 33/ l Cl Sl, l Sl Cl, Dynamc Behavor of a SCARA Robot by usng N-E method for a Straght Lne (Brahm Fernn)

8 8 ISSN: R p R e m R a J R R J R l C l S l l y l C l S l R n R R n R p R f, l S l C l l g y, d m m d m m m l m /3m m / 3 R p R e m R a J R R J R l l C l g y S l S R n R R n R p R f, l l S l C l g x y C m /, l x m /3 l ll C y l ll C y ll S y he jont torque of lnk (): l x m /3 R n R Z l ll C y l l l C y l l S y he jont torque of lnk (): he force exerted on the lnk (3): R n RZ ll yc l l ll ys F R f R Z m m g meff g. APPLICAION: Consder a rest-to-rest Cartesan path from pont (.5, ) to pont (.5,-) on straght lne x=.5 durng s wth l l.a cubc polynomal can satsfy the poston and velocty constrants at ntal and fnal ponts. y() y y() y y() y f y() y f he coeffcents of the polynomal are: a a a 6 a 3 ; he Cartesan path s : 3 y ( t) 6t t x.5 he trajectory smulaton: For the trajectory smulaton we use elbow down and elbow up. he fgure shows the smulaton block to smulate the trajectory by nverse knematcs of SCARA robot. IJRA Vol. 3, No., December : 33

9 IJRA ISSN: Fgure 3. he trajectory generaton of a SCARA robot wth Matlab/Smulnk by usng nverse knematcs rajectory smulaton: Elbow down Fgure. Matlab/Smulnk Fgure 5. Soldworks Elbow up Fgure 6. Matlab/Smulnk Fgure 7. Soldworks Dynamc Behavor of a SCARA Robot by usng N-E method for a Straght Lne (Brahm Fernn)

10 3 ISSN: he rajectory obtaned whether by usng Sold Works or by MALAB/Smulnk s exactly the same (a straght lne), so the poston constrant s verfed at ntal and fnal ponts. he jont velocty of the robot by Matlab/Smlnk:.5 elbow down elbow up.5 elbow down elbow up jont velocty (rad/sec) jont velocty(rad/sec) tme(sec) Fgure 8. he jont velocty() tme(sec) Fgure 9. he jont velocty() he angular velocty of the lnks of robot by Soldworks: Elbow down 3 3 angular velocty (deg/sec) angular velocty (deg/sec) tme (sec) tme (sec) Fgure. Lnk () Fgure. Lnk () Elbow up: Angular Velocty (deg/sec) me (sec) Angular Velocty5 (deg/sec) me (sec) Fgure. Lnk () Fgure 3. Lnk () IJRA Vol. 3, No., December : 33

11 IJRA ISSN: he results obtaned by the two softwars Matlab/Smlulnk and Soldworks about the jont velocty and the angular velocty show us that the velocty constrant s verfed at ntal and fnal ponts by the two softwares, he smlarty of results of both softwars Soldworks and Matlab/Smulnk confrms the relablty of the knematc model. he SCARA robot acheved a straght lne moton between two postons wth respect the constrants poston and velocty Y(m) X(m) Fgure. Arbtrary change of the two lnks of SCARA robot (elbow up and elbow down) Orentatonof thehomogeneous transformaton matrx: -.6 Y Y Z Y(m) -. Y (m ) X X(m) Z X X(m) Fgure 5. Elbow up Fgure 6. Elbow down Dynamc Behavor of a SCARA Robot by usng N-E method for a Straght Lne (Brahm Fernn)

12 3 ISSN: DYNAMIC SIMULAION: 3 Lnk() elbow down Lnk() elbow down Lnk() elbow up lnk() elbow up rque varatons (N.m) tme (sec) Fgure 7. he torque varatons. 6. DISCUSSION: he dynamc equatons found by N-E method show that there s no couplng between the lnk and the lnk3 because the lnk3 has only moton n vertcal drecton and there s no torque actng for ths lnk there s only a force to acheve the vertcal moton. For these reasons the effectve mass can be added to the lnk and the lnk whle determnng the torques. hs fact s clear from the torque equatons. And t s found the torques are ndependents of angular postons and ths makes the robot very complant. Another fact that the jont torques are ndependent by the lengths of lnk3 and,they are dependent only by ther masses as shown the torque equatons. he torques tme analyss for SCARA robot s carred out takng the lnk masses of lnk, lnk, effectve mass are: 6.9Kg, 6.9Kg, Kg respectvely.he fg.7 llustrates the varatons of torques wth tme of elbow up and elbow down. It s found the magntude of the torque of lnk s hgher than the torque requrement for lnk. here s an ncreasng dfference n the two torques as tme ncreased for elbow up and elbow down. he followng table shows the torques value at =s and =s. able. he torque values at =s and =s for elbow up and elbow down Elbow up Elbow down me(sec) orque (N.m) Lnk orque (N.m) (Lnk) orque (N.m) (Lnk) orque (N.m) (Lnk) =s =s he torques of lnk and lnk of elbow down are symmetrc wth the torques of lnk and lnk of elbow down respectvely, ths fact s clear from the results of the table and the fgure 7. he results obtaned from the table and accordngly the fgure 7 show that energy consumpton s the same for elbow up and elbow down for ths operaton, n whch means elbow up and elbow down swept the same area as shown n fgure. 7. CONCLUSION he use of both softwares SoldWorks and Matlab/Smulnk permtted to us to qualtatvely develop and hghlght the relevance of the studed of the knematc model of SCARA robot. From the dynamc model by usng N-E, t s found that the effectve mass can be added to the lnk and the lnk whle determnng the torques, and there s no couplng between lnk 3 and lnk.another fact the torques are ndependents of angular postons and the masses of lnk3 and lnk and ths makes the robot very complant. IJRA Vol. 3, No., December : 33

13 IJRA ISSN: In our case, we can conclude dependng from the dynamc analyss by N-E the choce of elbow down or elbow up s the same for ths operaton pck-place wth straght lne whle respectng the constrants, because the fnal energy balance s the same. REFERENCES [] Mohamed Salah Khreddne and Abdelhalm Boutarfa, Reconfgurable Control for a SCARA Robot usng RBF Networks, Journal of Electrcal Engneerng, VOL. 6, NO.,, 6. [] Phlp Voglewede, Anton H.C. Smth, and Antonello Mont Dynamc Performance of a SCARA Robot Manpulator Wth Uncertanty Usng Polynomal Chaos heory, IEEE ransactons on Robotcs, Vol. 5, No., February 9. [3] Z. Robot modelng and control, JOHN WILEY & SONS, INC. New York, Chchester, Wenhem, Brsbane, Sngapore, oronto, page 97. [] Brahm Fernn, Mahmoud Gouasm, M hamed Meghatra, Knematc Modelng and Smulaton of a -R Robot by Usng Soldworks and Verfcaton by Matlab/Smulnk. Internatonal Journal of Advanced Robotc Systems. [5] Reza N Jazar, heory of appled robotcs, Knematcs, Dynamcs, and control, second edton sprnger, path plannng, ISBN Dynamc Behavor of a SCARA Robot by usng N-E method for a Straght Lne (Brahm Fernn)