Kinematics Modeling of a 4-DOF Robotic Arm

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1 5 Iteatoal Cofeece o Cotol, Automato ad Robotcs Kematcs Model of a -DOF Robotc Am Am A. Mohammed ad M. Sua Mechacal Eee Depatmet K Fahd Uvey of Petoleum & Meals Dhaha, Saud Aaba e-mal: mehmets@kfupm.edu.sa Abact Ths wok pesets the kematcs model of a RA (a DOF) obotc am. The dect kematc poblem s addessed us both the Deavt-Hatebe (DH) coveto ad the poduct of expoetal fomula, whch s based o the scew theoy. By compa the esults of both appoaches, t tus out that they povde detcal solutos fo the mapulato kematcs. Futhemoe, a alebac soluto of the vese kematcs poblem based o toometc fomulas s also povded. Fally, smulato esults fo the kematcs model us the Matlab poam based o the DH coveto ae peseted. Sce the two appoaches ae detcal, the poduct of expoetal fomula s supposed to poduce same smulato esults o the obotc am uded. Keywods-Robotcs; DH coveto; poduct of expoetals; kematcs; smulatos I. INTRODUCTION Nowadays, obotcs s a ch aea of eseach, tems of the kematcs, dyamcs ad cotol. Kematcs, patcula, plays a sfcat ole obotcs ad especally fo the udy of dual mapulatos' behavo. Theefoe, a decsve ep ay obotcs syem s the aalyss ad model of the mapulato kematcs []. It ca be dvded eeally to fowad ad vese kematcs. The fome efes to dect fd of the ed effecto (EE) poo ad poue fo the ve jot coodates, o the cotay, the latte s the detemato of jot vaables tems of the EE poo ad poue []. I othe wods, t s the pocess of obta a cofuato space that coespods to a ve wok space. It has seveal dscples of applcatos clud the compute aphcs []. Whle the fowad kematcs s ahtfowad, the vese kematcs s ot a easy task ad t usually has multple solutos. Numeous techques ad the combatos could be used fo solv ad aalyz the vese kematcs poblem. Howeve each method besde ts advataes possesses some hadcaps. Theefoe, t s advataeous to apply moe tha oe method ad to teate them wth addtoal pocedues []. Seveal appoaches have bee veated ad compaed by Ades et al []. Jasjt Kau et al [5] have aalyzed ad smulated a obotc am hav thee lks-mapulato, whee the vese kematcs esults two solutos, out of whch the be oe has bee chose wth the usae of the eetc alothm. They have cocluded that the obotc am movemet optmzato ca be acheved by the eetc alothms a pactcal ad effectve way. A eometc appoach to solve the uspecfed jot ales eeded fo the autoomous poo of a obot am s peseted []. Based o few assumptos coce the wok evomet of the mapulato ad the kd of mapulato equed, addto to the basc toomety, a ease soluto has bee poposed. The toduced methodoloy has bee teed us fve deees of feedom obotc am compouded to a -Robot. By cay out the mechacal des, all compoets have bee assembled toethe addto to two faed sesos to detect the object ad to fd ts poo. Fally, the electcal des was doe, ad the poposed method was teed ad jufed. Howeve, ths appoach lacks ts eealty due to the abudat assumptos cosdeed, ad what s moe s that the la lk s ected to ju two oetatos. I what we toduce hee s that the la lk could have ay abtay oetato, whch makes the aalyss moe eeal to dffeet cofuatos. Ge ad J [6] poposed a alothm of a edudat obot kematcs based o poduct of expoetals (PE), whee the fowad ad vese kematcs poblems wee addessed. The vese kematcs poblem was decomposed to seveal sub poblems ad by udy them sepaately, a soluto of the vese kematc poblem was poposed. Kematcs ad teactve smulato syem model fo obot mapulatos s ve [7], whee the fowad ad vese kematcs of a obotc am called Kataa5 ae uded by the expoetal poduct method, the a teactve smulato syem s bult based o thee-met vtual scee model techques poposed fo obot mapulatos. These techques ae OpeGL, VRML ad aothe oe based o Vsual C, Lab Vew ad Matlab platfom. A obot smulato, whch s toduced by Lodes [8] ad s based o Matlab, eables the model of ay obotc am povded that the coespod DH paametes ae kow. Lks vsualzato tems of shape ad sze was made afte some cosdeatos. The poam was show to adequately model seveal example obots. Ths pape focuses o model ad smulato of the RA- obotc am, a small specal obot wth evolute jots [9]. A excellet suvey of obotcs kts fo posecoday educato s ve []. The e of ths pape s oazed as follows: Secto II below pesets the kematcs aalyss based o the DH coveto ad expoetals fomulato. The vese kematcs soluto /5/$. 5 IEEE 87

2 based o the alebac method s ve secto III. I secto IV, the mapulato smulato pocedue ad esults ae peseted. Secto V cocludes ths wok. II. MANIPULATOR KINEMATICS The DH coveto ad poduct of expoetals ofte used kematcs aalyss of obot mapulatos ad othe mechacal uctues. The below secto utlzes both methods to dve the kematcs model of the obot. To smplfy the aalyss, the effect of the ppe s oed mak the obotc am a fou deee of feedom, a fou-jot spatal mapulato. The teatmet of ths mapulato s aaloue to that of PUMA 56 afte emov the w ad add oe moe evolute jot. A. DH Coveto. Deavt-Hatebe (DH) coveto s commoly used the kematcs aalyss of the obotc mapulato []. It s based o attach a coodate fame at each jot ad specfy fou paametes kow as DH paametes fo each lk, ad utlz these paametes to couct a DH table. Fally, a tasfomato matx betwee dffeet coodate fames s obtaed. The majo objectve s to cotol both the poo ad oetato of the EE o the ppe ts wok space. We wll f deve the elatoshp betwee the jot vaables ad the poo, ad oetato of the ppe, us the DH method []. The obotc am ude udy s show below F. (RA-A of Imaes SI Ic [9] ) ad DH paametes F.. Vaous otay ad lea motos of the obotc am ae show below F. Each moto s made possble by a sevomoto (o ju a sevo) at the coespod locato. Fue. Robotc am wth 5 depedet motos [9]. B. Fowad Kematcs As F. depcts a coodate fame attached to evey lk ode to fd ts cofuato the ehbo fames us the d moto fomula. To do so a DH table s eeded as follows: Fame( ) TABLE I. a DH PARAMETERS. a d 9 l l l l Fue. The obot am. By apply the Deavt-Hatebe (DH) otatos [] fo the jots coodates, the DH-table ca be coucted as led above Table II. The lk leths as show F. ae l =.5 cm, l = cm, ad l = l = 9 c m. C. Fowad Tasfomato Matces Oce the DH table s eady, the tasfomato matces ae easy to fd. Geeally, the matx of tasfomato fom the fame B to the fame B fo the adad DH method s ve by []: cos s cos s s a cos s cos cos cos s a s T = s cos d () Fue. DH paametes. The the dvdual tasfomato matces fo =,,, a d ca be easly obtaed. Theeafte, the complete tasfomato T s foud fom: T = T T T T () Upo the detemato of, T we ca fd the lobal coodates of the ed effecto. The tp pot of the am s at the o of fame B (F. ),.e. t s at [ ] T. So, ts poo the lobal fame becomes: 88

3 d x d y p = T p = T = = d z Obvously, ths s the la colum of the tasfomato matx T. Afte smplfcatos us toometc fomulas, the pevous equato becomes: ( ) ( ) () dx = c l c l c l c c ( ) ( ) (5) dy = s l c l c l s c ( ) ( ) (6) dz = l l s l s l s whee s ads fo s () ad c ads fo c o s. I addto, dx, dy ad dz ae the lobal ed effecto coodates. Moeove, the ed effecto oetato s: = (7) The DH paametes ae ak to the cofuato of the obot. Fo dffeet mapulato uctues, the kematcs equatos ae ot uque. Moeove, kematcs equatos of the mapulato based o the DH coveto povde some sulaty mak the equatos dffcult to solve o usolvable some cases. I addto, the DH coveto, the commo omal s ot defed popely whe axes of two jots ae paallel. I ths case, the DH method has a sulaty, whee a lttle chae the spatal coodates of the paallel jot axes ca ceate a hue mscofuato epesetato of the DH coodates of the elatve poo. I the follow secto, a alteatve to the DH coveto, the PE s peseted. D. Poduct of Expoetals Asde fom the DH coveto, aothe method s the socalled poduct of expoetals (PE). Us the ealty that the moto of each jot s eeated by a tw accompaed wth the jot axs, a moe eometc epesetato of the kematcs ca be acqued. Remembe that f ξ s a tw, accod to efeece [], the fowad kematcs s ve by: ˆ ˆ ˆ ( ) = e e... e ( ) (8) The above equato s called the poduct of expoetal fomula fo the obot fowad kematcs, whee ( ) s the fal cofuato of the obot ad expoetal ve by [7]: T ˆ e ( I e )( v ) e = ˆ e s a matx (9) Fo a psmatc jot the tw s ve by: v =, ad fo a evolute jot by: q, = whee R s a ut vecto the decto of the axs of the tw, q R s ay pot o the axs, ad v R s a ut vecto dect the taslato decto. I ths case, the tw ξ s fo dffeet lks of the obot ae ve by: = L = L L = L ( L L ) = Moeove, ( ) s the tal cofuato of the obot ve by s t ( ) l l l = l The fowad kematcs map of the mapulato has the fom: ˆ ( ) ( ) ( ) ( ) ˆ ˆ ˆ R p e e e e = = By expad tems the poduct of expoetals fomula, Eq. yelds R cos( ) cos s s ( ) cos ( ) = cos( ) s cos s ( ) s s ( ) cos( ) ( ( ) L ( )) ( ( ) L ( )) L s ( ) L s L s ( ) cos L cos L cos cos p( ) = s L cos L cos cos L It s clea that the above equato () s detcal to those equatos () thouh (6). Thus, the kematcs model us the DH coveto ad poduct of expoetals aees wth each othe. III. INVERSE KINEMATICS We ed up wth a set of fou olea equatos wth fou ukows. Solv these equatos alebacally, kow as the vese kematcs, eques that we eed to kow the jot vaables,,, a d fo a ve EE poo [ dx, dy, dz ] ad oetato. We et fom equatos () to (7), by dvd, squa, add ad us some toometc fomulas: () () () 89

4 d y = ta () d x ( c c ) ( a b ) = ± () ta, ta, [ / / ] ad [ / / / ] depcted Fs. 6 ad 7, espectvely., whch ae A B C l l = c o s l l (5) whee a = l b = l l c = dz l l ad = a b s, cos, s,. I addto A = ( dx l c c ), B = ( dy l s c ), adc = ( dz l l s ) Hav detemed,, a d, we ca the fd fom the EE oetato of as follows: Fue. Home poo. = (6) IV. SIMULATION OF THE ROBOTIC ARM Sce obotcs s a multdscplay subject, ts udy eques sklls fom dffeet felds of kowlede, ot easly avalable. Thus, smulato has bee ecozed as a sutable tool that combes all these featues toethe eabl the use of a dect vsualzato of dffeet kds of moto that a obot may pefom, mak the ole of smulato vey mpotat obotcs []. Us the obotcs toolbox toethe wth the Matlab softwae []-[5], the kematcs of a obotc am ca be smulated ad aalyzed based o the DH coveto descbed befoe. The toolbox takes a covetoal appoach to epeset the kematcs ad dyamcs of seal-lk obotc ams. Besdes, t povdes seveal fuctos ad outes, whch ae hady fo the smulato ad scuty of obotc mapulatos, lke kematcs, dyamcs ad tajectoy ceato. Based o the afoemetoed toolbox, may cofuatos of the obot mapulato ae easly vsualzed. I ode to smulate the obotc mapulato, f we eect a vecto of Lk objects ad set ou fou oups of DH paametes as Table I. The, these ae passed to the coucto SealLk, whch s the key ep to utlze the toolbox. A detaled pocedue ca be foud [5]. I ode to eeate the plot fou jot vaables ae eeded. I the f case all jot vaables ae eteed the fom of zeos (ow vecto), such as [ ] = [ ]. F. shows e o home poo of the obot, whee all jots vaables ae zeos. I the toolbox, each evolute jot s esembled by a small cylde. Sce we have fou evolute jots, fou cyldes ca be see Fs. -7. A secod tee cofuato s that whe all lks ae ad upht as F. 5 shows. The coespod put ths case s ve by: [ / ], whee all the ales ae defed ada by default. To te the fuctoalty of all the jots, othe two cofuatos ae selected as ve by the vectos of jot vaables Fue 5. Fue 6. Fue 7. Upht poo. Left-dow poo. All jots ae ve ales. V. CONCLUSION Kematcs model of a deee-of-feedom obotc am s peseted us both the DH method ad poduct of expoetals fomula. It s pove that both appoaches povde the same soluto fo the obot mapulato ude udy. I addto, the smulato of the obot mapulato s caed out us the Matlab softwae va the obotcs toolbox, thouh whch seveal poos of the mapulato ae ealzed based o the DH coveto. Althouh esults of the poduct of expoetals fomula ae ot ve, they ae expected to be same as those of the DH coveto. 9

5 ACKNOWLEDGMENT Authos atefully ackowlede the suppot povded by K Fahd Uvey of Petoleum & Meals thouh the NSTIP poject 9-ELE786-. REFERENCES [] K. E. Clothe ad Y. Sha, A Geometc Appoach fo Robotc Am Kematcs wth Hadwae Des, Electcal Des, ad Implemetato, J. Robot.,. [] R. N. Jaza, Theoy of Appled Robotcs. Boo, MA: Spe US,. [] Baka, Lukas, ad Roma Beka. Ivese kematcs-basc methods. Web.< cesc. o/cescg- /LBaka/pape. pdf (). [] A. Adou ad J. Laseby. "Ivese kematcs: a evew of ex techques ad toducto of a ew fa teatve solve," 9. [5] J. Kau ad V. K. Baa, "Smulato of Robotc Am hav thee lk Mapulato," Iteatoal Joual of Reseach Eee ad Techoloy (IJRET), vol., o., Mach,, ISSN: 77-78,. [6] X. F. Ge ad J. T. J, The alothm of edudat obotc kematcs based o expoetal poduct, Appl. Mech. Mate., vol. 58-6, pp. 9-97, Ju.. [7] X. Xao, Y. L, ad H. Ta, Depatmet of electomechacal eee, uvey of Macau, Macao SAR, Cha, IEEE Iteatoal Cofeece o Ifomato ad Automato (ICIA),, pp [8] A. Lodes, Robot Smulato MATLAB, upublshed. [9] [] M. Ruzzeete, M. Koo, K. Nelse, L. Gespa, ad P. Fo, A evew of obotcs kts fo tetay educato, Poceeds of Iteatoal Wokshop Teach Robotcs Teach wth Robotcs: Iteat Robotcs School Cuculum,, pp [] R. M. Muay, Z. L, S. S. Say, ad S. S. Say, A Mathematcal Itoducto to Robotc Mapulato, CRC pess, 99. [] L. Žlajpah, Smulato obotcs, Math. Comput. Smul., vol. 79, o., pp , Dec. 8. [] P. I. Coke ad othes, A compute tool fo smulato ad aalyss: the Robotcs Toolbox fo MATLAB, Poc. Natoal Cof. Auala Robot Assocato, 995, pp. 9-. [] P. I. Coke, Robotcs, Vso ad Cotol: Fudametal Alothms MATLAB. Bel: Spe,. [5] P. Coke, Robotcs, Vso ad Cotol, vol. 7. Bel, Hedelbe: Spe Bel Hedelbe,. 9