Hybrid Neural Network Based Prediction of Inverse Kinematics of Robot Manipulator

Transcription

1 5 th Itratoal & 6 th All Ida aufacturg Tchology, Dsg ad Rsrch Cofrc (AITDR 01) Dcmbr 1 th 1 th, 01, IIT Guwahat, Assam, Ida Hybrd Nural Ntw Basd Prdcto of Ivrs Kmatcs of Robot apulat Pachaad Jha 1* ad B. B. Bswal 1*, Dpartmt of Idustral Dsg, NIT Rourla, ad PIN: , E-mal: jha_p007@hotmal.com; bbbswal@trl.ac.; Abstract Th fudamtal of th vrs matcs of robot mapulat s to dtrm th jot varabls f a gv Cartsa posto ad tato of a d ffct. Covtoal mthods to solv vrs matcs such as gomtrc, tratv ad algbrac ar complx f rdudat mapulats. Thr s o uqu soluto f th vrs matcs thus csstatg applcato of approprat prdctv modls from th soft computg doma. Although artfcal ural tw (ANN) ca b gafully usd to yld th dsrd rsults, but th gradt dsct lrg algthm dos ot hav ablty to srch f global optmum ad t gvs a slow covrgc rat. Ths papr proposs structurg ANN wth hybrdzato of Partcl Swarm Optmzato to solv th vrs matcs of 6R robot mapulat. A vstgato has b mad o accuracs of adoptd algthm. Th ANN modl usd s mult-layrd prcptro ural tw (LPNN) wth bac-propagato (BP) algthm whch s compard wth hybrd mult layrd prcptro partcl swarm optmzato (LPPSO). A attmpt has b mad to fd th bst ANN cofgurato f th problm. It has b obsrvd that LPPSO gvs a fastr covrgc rat ad mprovs th problm of trappg local mma. It s foud that LPPSO gvs bttr rsult ad mmum rr as compard to LPBP. Kywds:Ivrs matcs, D-H algthm, PSO, LP 1 Itroducto A dustral robot cossts of a st of rgd ls coctd togthr by a st of jots. To cotrol th ovrall moto of a mchasm f ch ls coctd by varous jots l rvolut prsmatc s prfmd by mots. Grally tool d ffct prfms tass th Cartsa codat systm whch s cotrolld by jot codat systm. F bttr posto ad tato of robot d ffct to prfm th statd tas, t s sstal to udrstad th matcs rlatoshp btw th jot codat systm ad th Cartsa codat systm. Grally thr ar two typs of matc aalyss, whch s fward matcs ad vrs matcs. Fward matcs s a covrso of jot spac varabls to d-ffct posto ad tato. Covrso of th posto ad tato of robot mapulat d-ffcts from Cartsa spac to jot spac s calld as a vrs matcs problm. Ths s of fudamtal mptac calculatg dsrd jot agls f robot mapulat dsg ad postog. Th crspodg jot valus must b computd at hgh spd by th vrs matcs trasfmato Xu t al. (005). F a mapulat wth o dgr of frdom, at ay stat of tm jot varabl s dotd by θ= θ (t), I = 1,,... ad posto varabls by x j = x (t), j = 1,,... m. Th rlatos btw th d-ffcts posto x (t) ad jot agl θ (t) ca b rprstd by th fward matc qto x( t) = f ( θ( t)) (1) whr, f s a olr cotuous ad dffrtabl fucto. O th othr had, wth th dsrd d ffcts posto, th problm of fdg th valus of th jot varabls s vrs matcs, whch ca b solvd by, ' θ ( t ) = f ( x( t)) () Ivrs matcs soluto s ot uqu du to olr, ucrta ad tm varyg atur of th govrg qtos Chddarwar ad Babu (010).Th dffrt tchqus usd f solvg vrs matcs ca b classfd as algbrac, gomtrc ad tratvalavadar ad Ngam (008). Th algbrac mthods do ot gat closd fm solutos. I cas of gomtrc mthods, closd fm solutos f th frst thr jots of th mapulat must xst gomtrcallyhusty t al. (007). Th tratv mthods covrg to oly a sgl soluto dpdg o th startg pot ad may ot w r sgularts. I cas of umrcal mthod th maj dffculty of vrs matcs s that, wh th Jacoba matrx s sgular ll-codtod, t dos ot fd a soluto. I addto, f th tal approxmato of th soluto vct (.. Th vct of jot varabls) s ot suffctly accurat, ths mthod may bcom ustabl Olaru ad Olaru (011). Bcaus of th abov mtod rsos, varous 105-1

2 Hybrd Nural Ntw Basd Prdcto of Ivrs Kmatcs of Robot apulat auths adoptd ANN.Th smulato ad computato of vrs matcs usg multlayr fd prcptro tw s partcularly usful whr lss computato tms ar dd, such as rlt al. (01). If tm adaptv robot cotrol rjall th umbr of dgrs of frdom crss, tradtoal mthods wll bcom m complx ad qut dffcult to solv vrs matcs Zhag t al. (007). Although th us of ANN s ot w th fld of mult-objctv ad NP-hard problm to arrv at a vry rsoabl optmzd soluto, th LPPSO has ot b trd to solv vrs matcs problm f 6R PUA robot mapulat. Thrf, th ma am of ths w s focusd o mmzg th m sq rr of th ural tw-basd soluto of th vrs matcs problm usg PSO. Th trag data of ural tw hav b slctd vry prcsly. Espcally, ulrd data ch ural tw hav b chos, ad usd to obta th trag st of th last ural tw. athmatcal odllg of 6R PUA apulat Dvt-Hartbrg (DH) algthm s usd to calculat th dvdl homogous trasfmato matrcs whch th us to drv th fward ad vrs matcs of 6R PUA robot mapulat. DH paramtrs ad assocatd valus f PUA mapulat hav gv tabl 1 ad assgd codat frams ar show Fg. 1,, Fram θ Tabl 1 D-H Paramtrs d (m) a (m) (dgr) 0 θ θ 0 0 θ d =0.1 a 1 =0.1 θ d =0.18 a = θ θ Fgur 1 odl ad codat frams of mapulat Ivrs matcs of PUA mapulat s gv blow: ) α (dgr) θ1 = a ta ( ± Px + Py d, d ) a ta ( Px, Py) () θ = a ta ( Pz, ± a ta ( ± whr, b a + d b, b + a ) = p Px + Py d ) a a d a p = Px + Py + Pz θ ca also b xprssd othr fm: θ = a ta ( ± a ta ( ± a + d Px + Py b, b d, Pz,) θ = a ta ( ± a + d b, b ) a ta π (5) + a) π a ta ( g, g1), f σ < 0 θ = (7) a ta ( g, g1), f σ > 0 Whr, σ = ± g 1 + g 5 = ta (, g) θ a σ a ta ( g, g1), f σ < 0 θ 6 = (9) π a ta ( g, g1), f σ > 0 Whr p,p,p ) rprsts th posto ad ( x y z {,, ),(o,o,o ),(a,a,a )} ( x y z x y z x y z d d a, () ( d, ) (6) (8) th tato of th d-ffct. It s obvous from th qtos () through (9) that thr xst multpl solutos to th vrs matcs problm. By comparg th rrs btw ths four gratd postos ad tatos ad th gv posto ad tato, o st of jot agls, whch producs th mmum rr, s chos as th crct soluto. 105-

3 5 th Itratoal & 6 th All Ida aufacturg Tchology, Dsg ad Rsrch Cofrc (AITDR 01) Dcmbr 1 th 1 th, 01, IIT Guwahat, Assam, Ida Applcato of PSO f trag LP W propos th soluto usg a mult-layrd prcptro wth th bac-propagato algthms f trag. Th tw s th trad wth data f a umbr of d ffct postos xprssd Cartsa co-dats ad th crspodg jot agls. Th data cosst of th dffrt cofguratos avalabl f th arm. Th dffrt poss of th arm ar th usd to tra a thr-layr, fully coctd bac-propagato modl show Fg.,.Each of th sgals from th put uros s multpld by th valu of th wghts of th cocto wghts btw th rspctv put uros ad th hdd uro. Lrg rr E (ftss fucto) s calculatd from qto (10-11). E = E = m = 1 q = 1 ( o y ) (10) E q ( ) (11) whr q s th umbr of trag sampls, y s th dsrd output of th th put ut wh th th trag sampl s usd, ad o s th actl output of th th put ut wh th th trag sampl s usd. Coctos wght Bas Bas Start Iputs X Y Z Outputs θ, θ, θ, 1 5 θ, θ ad θ Slcto of tal radom vlocty ad posto of partcl I =1 Evalto of srchg locato of ch partcl (ftss) Chag th srchg locato of ch partcl Iput layr Hdd layr Output layr Fgur : A bloc dagram of th systm usg ANN Aalytcal soluto of vrs matcs problm s hghly o-lr ad mathmatcally complx atur. A ANN modl dos ot rqur hghr tal slcto of wght whch s vgous to yld local optma, covrgc spd ad trag tm f th tw. Grally wght s radomly slctd th rag of 0 to 1, aftr actvato fucto wght of ch uros adjustd f th xt trato. Th hurstc optmzato algthm optmzs th wghts of th ural tws. Wh crta trmato crtra ar mt, a maxmum umbr of tratos ar rchd, th tratos cs. From th prvous rsrch hybrd optmzato algthm startd volvg wth hgh ad rmarabl advacs thr prfmacs Kdy ad Ebrhart (1995). Ths tchqus producs bttr outflow from local optmum ad tstfd to b m opratv tha th stadard mthod. I ths papr w hav optmzd wght ad bas f ch uro usg PSO as show Fg.,. F th trag of tw t s mptat to hav all cocto wghts ad bass dr to mmz th m sq rr. To optmz LP ural tw t s mptat to hav ftss fucto PSO ad th t s rqurd to df th tal wght ad bas f th trag of LP ural twrjall (01). Th basc stps ad flow chart of LPPSO has gv Fg.,. I =I +1 No Fgur Flow chart f LPPSO Ftss fucto ca b calculatd from qto (1). Whr th umbr of put ods s ql to, th umbr of hdd ods s ql to h, ad th umbr of output ods s m. Thrf, th ftss fucto of th th trag sampl ca b dfd as follows: Ftss X ) = E( X ) (1) ( Ys Stop Updat th partcl posto ad vlocty usg qtos v t + 1 = wv + c rad ( p t rad ( g Updat p,bst ad g,bstwh codto s mt, p = p f f ( p ) > f ( p ) g, bst, bst 1, bst x = g I >ax I t x ) = x + v, bst t+1 t t +1 f f ( g ) > f ( g, bst, bst t x ) + c ) 105-

4 Hybrd Nural Ntw Basd Prdcto of Ivrs Kmatcs of Robot apulat Rsults ad Dscusso Th proposd w s prfmd o th atlab R01a. Bac-propagato algthm was usd f trag th tw ad f updatg th dsrd wghts. I ths w th trag data sts wr gratd by usg qto () through (15). A st of 1000 data sts wr frst gratd as pr th fmula f th put paramtr px, py ad pz codats mm. Ths data sts wr th bass f th trag, valto ad tstg th LP modl. Th followg paramtrs wr ta: lrg rat 0.6, momtum paramtr 0.1, umbr of poch 500, umbr of hdd layr, umbr of puts ad umbr of output 6. Th SE f LPBP algthm show Fg., th usd soluto mthod gvs th chac of slctg th output, whch has th lst rr th systm. So, th soluto ca b obtad wth lss rr. Tabl gvs th xprmtal rsults ad comparso btw th LPBP algthms wth rspct to hybrd LPPSO f two hdd layrs. Fg., (a), (b), (c), (d), () ad (f) shows th slctd bst m sq curv of LPBP f all jot varabls. Smlarly bst chos m sq curv of LPPSO from tabl dpctd Fg., (a), (b), (c), (d), () ad (f) f all jot varabls. Tabl sq rr f all jot agls S. sq rr of LPBP sq of LPPSO r ro a a f Thta Numbr of Epoch (b) f Thta Numbr of Epoch (c) f Thta Numbr of Epoch (d) r ro r f Thta 1 Numbr of Epoch (a) a 0. 5 f Thta 5 Numbr of Epoch () 105-

5 5 th Itratoal & 6 th All Ida aufacturg Tchology, Dsg ad Rsrch Cofrc (AITDR 01) Dcmbr 1 th 1 th, 01, IIT Guwahat, Assam, Ida a 0.1 f Thta 6 Numbr of Epoch (f) Fgur Fgur (a), (b), (c), (d), () ad (f) ar m sq rr curv of LPBP f all jot agls. r r. 5 f Thta 1 Itrato (a). f Thta Itrato (d) r r. 71 f Thta Itrato (b) 6. 1 f Thta 5 Itrato () r r f Thta Itrato (c) 7. 7 f Thta 6 Itrato (f) Fgur Fgur (a), (b), (c), (d), () ad (f) ar m sq rr curv of LPPSO f all jot agls

6 Hybrd Nural Ntw Basd Prdcto of Ivrs Kmatcs of Robot apulat 5 Coclusos I ths papr, w hav slctd two mthods whch ar LPBP ad LPPSO to obta th soluto of vrs matcs of 6R mapulat. I ths approach fward ad vrs matc modl of 6R mapulat s usd to grat th data st f trag th LP. Th dffrc dsrd ad prdctd data wth LPBP, gvs po rsults as compard to LPPSO. Also, th LPPSO accumulat small umbr of poch wth hybrd lrg algthm. Thrf, LPPSO ca b usd f accurat ad fast soluto of vrs matcs. Futur rsrch wll rvs th ruls, puts, umbr ad typ of mmbrshp fuctos, th poch umbrs usd, ad trag sampl to furthr rf th LPPSO modl. Rfrcs Alavadar S. ad Ngam. J., (008), Nuro- Fuzzy basd Approach f Ivrs Kmatcs Soluto of Idustral Robot apulats, It. J. of Computrs, Commucatos & Cotrol, vol.,pp. -. ChddarwarS. S ad BabuN. R. (010), Comparso of RBF ad LP ural tws to solv vrs matc problm f 6R sral robot by a fuso approach, Egrg Applcatos of Artfcal Itllgcvol., Husty. L., Pfurr. ad SchrocrH. P., (007), A w ad ffct algthm f th vrs matcs of a gral sral 6R mapulat, chasm ad ach Thy vol., Kdy J. ad Ebrhart R.C., (1995), Partcl swarm optmzato, : Procdgs of IEEE Itratoal Cofrc o Nural Ntws, vol., pp rjall S., Hashm S. Z. ad Sardroud,H.. (01), Trag fdfward ural tws usg hybrd partcl swarm optmzato ad gravtatoal srch algthm, Appld athmatcs ad Computato vol. 18, OlaruA. ad OlaruS. (011), Optmzato of th robots vrs matcs rsults by usg th Nural Ntw ad LabVw smulato, IPCSIT, vol.1. Sarafraz S., Nzamabad-pour H. ad Saryazd S. (011), Dsrupto: A w oprat gravtatoal srch algthm, SctaIraca, vol. 18 (), Wassrma P.D., (1989), Nural Computg, Va Nostrad Rhold, NwY, 1989, 0 pags. Xu D. t al., (005), A Aalyss of th Ivrs Kmatcs f a 5-DOF apulat, Itratoal Joural of Automato ad Computgvol., Zhag J. R. t al., (007), A hybrd partcl swarm optmzato bac-propagato algthm f fd fward ural tw trag, Appld athmatcs ad Computato vol