Neural Network Based Inverse Kinematics Solution for 6-R Robot Using Levenberg-Marquardt Algorithm

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1 Neurl Netwrk Bsed Inverse Kinemtics Slutin fr 6-R Rbt Using Levenberg-Mrqurdt Algrithm Prshnt Bdni Mechnicl Engineering Deprtment, Grphic Er Universit, Dehrdun , Indi Abstrct: The trditinl pprches re insufficient t slve the cmple kinemtics prblems f the redundnt rbtic mnipultrs. T vercme such intricc, ANNs re used nwds. The perfrmnce f the neurl netwrk is ffected b the trining lgrithm nd netwrk tplg. There re numerus trining lgrithms which re used in the trining f neurl netwrks. In this pper, Levenberg- Mrqurdt is used in trining lgrithm nd its effect n the perfrmnce f the neurl netwrk n the inverse kinemtics mdel lerning f 6-R rbt is studied. Kewrds: Inverse Kinemtics Slutin, MATLAB Tlb, Neurl Netwrks, Rbt mnipultr, Trining Algrithm. 1. Intrductin Neurl netwrk is ne f the prminent rtificil intelligent techniques used in the rbtics t ccmplish mre intelligence in sstems with high degree f utnm. ANN incrprtes lerning bilit which prvides fleibilit t the rbtic sstems. Neurl netwrk cn be implemented using MATLAB sftwre. Prcedure t trin the neurl netwrk mdel is s fllws: Dt cllectin Netwrk cretin Netwrk cnfigurtin Initilitin f the weights nd bises Netwrk trining Netwrk vlidtin Use the netwrk The wrking principl f neurl netwrk is bsed n lerning frm the frmerl btined dt set knwn s trining set, nd then g thrugh the success f sstem using test dt. The lerning lgrithm ffects the emplment f the neurl netwrk gretl. In this stud, the effects f Levenberg-Mrqurdt lerning lgrithm hve been tested fr the inverse kinemtics slutin f si jint rbtic mnipultr. This pper is rgnied s fllws: Sectin II prvides the kinemtics nlsis f the 6-R rbt. Sectin III f the pper dels with the neurl netwrk bsed inverse kinemtics slutin. Sectin IV describes trining nd testing. Sectin V gives results nd discussin, nd finll Sectin VI cncludes the pper. 2. Kinemtic Anlsis f 6-R Rbt A Rbt mnipultr is cmpsed f grup f links (rigid bdies) cnnected tgether b revlute r prismtic jints which llw mtin fr the desired link. Rbt Kinemtics refers t the nlticl stud f the mtin f rbt mnipultr withut regrd t n fctr (like frce) which influence the rbt mvement. Rbt Kinemtics cn be split int frwrd nd inverse kinemtics. In the frwrd kinemtics prblem, the end effectr s lctin in the wrk spce, tht is psitin nd rienttin, is determined bsed n the jint vribles [1] [2] [3]. The frwrd kinemtics prblem m epress mthemticll s fllws: F (θ 1, θ 2, θ 3...θ n ) = [p, p, p, R] Where, θ 1, θ 2, θ 3...θ n re the input vribles, [p, p, p ] re desired psitin nd R is the desired rttin. The inverse kinemtics prblem refers t finding the vlues f the jint vribles tht llws the mnipultr t rech the given lctin. The inverse kinemtics prblem cn be epressed mthemticll s fllws: F [p, p, p, R] = (θ 1, θ 2, θ 3...θ n ) The jint vribles re the link etensin in the cse f prismtic jints, r the ngles between the links in cse f rttinl jints. Figure 1: D-H crdintes f the rbt Figure 1 depicts the structure nd crdintes f 6-DOF rbt mnipultr which is studied in during the wrk. The D H prmeters f the mnipultr re listed in Tble1. Tble 1: D-H prmeters f the mnipultr Jints i-1 α i-1 d i θ i /4 2 3 l 2 /4 l 1 3 Pper ID: IJSER f 83

2 4 l 3 / l 5 /4 l 4 6 Accrding the Denvit-Hrtenberg methd, the trnsfrmtin cn be frmulted in the chin prduct f si successive hmgeneus mtrices. is the hmgeneus trnsfrmtin mtri relting the i th crdinte frme t the (i-1) th crdinte frme [4]. =. (1) Equtin (1) cntins lrge set f trignmetric functins: sinθ i = S i nd csθ i = C i. cn be clculted b fllwing: In (2), n, n, n,,,,,, shw the rttinl elements f the trnsfrmtin mtri nd p, p, p refer t the elements f the psitin vectr. n = - C 1 S 2 C 3 C 45 S 6 + S 1 S 3 C 45 S 6 C 1 C 2 S 45 S 6 C 1 S 2 S 3 C 6 S 1 C 3 C 6 n = - S 1 S 2 C 3 C 45 S 6 - C 1 S 3 C 45 S 6 S 1 C 2 S 45 S 6 C 1 S 2 S 3 C 6 + C 1 C 3 C 6 n = C 2 C 3 C 45 S 6 - S 2 S 45 S 6 +C 2 S 3 C 6 = - C 1 S 2 C 3 C 45 S 6 + S 1 S 3 C 45 C 6 C 1 C 2 S 45 C 6 + C 1 S 2 S 3 S 6 + S 1 C 3 S 6 = - S 1 S 2 C 3 C 45 C 6 - C 1 S 3 C 45 C 6 S 1 C 2 S 45 C 6 + S 1 S 2 S 3 S 6 - C 1 C 3 S 6 = C 2 C 3 C 45 C 6 - S 2 S 45 C 6 - C 2 S 3 S 6 = - C 1 S 2 C 3 C 45 + S 1 S 3 C 45 + C 1 C 2 S 45 = - S 1 S 2 C 3 C 45 - C 1 S 3 C 45 + S 1 C 2 S 45 i n n T n p p p 1 (2) Then, θ 5 = rcsin [{p 2 + p 2 + (p - l 1 ) 2 - l S 4 2 l 4 2 (C 4 l 4 + l 23 ) 2 }/ A] - rctn [(C 4 l 4 + l 23 ) / S 4 l 4 ] - θ 4 (5) We culd ls btin the fllwing epressins: θ 2 = rcsin [(C 45 l 5 + C 4 l 4 + l 23 ) / ( p - l 1 ) 2 + p 2 ] - rctn [p / (p - l 1 )] θ 1 = rcsin [(S 2 C 3 S 4 - C 2 C 4 C 2 l 23 ) / ( p - l 5 ) 2 + (p l 5 ) 2 ] - rctn [(p - l 5 ) / (p - l 5 )] θ 3 = rcsin [(C 1 + p - l 5 ) / S 1 S 4 l 4 ( - S 2 C 45 )] - rctn [(p - l 5 ) / (p - l 5 )] The strteg t slve inverse kinemtics prblem tend t be time cnsuming, s there is usull lw interest in ppling this technique fr kinemtic clcultins. The trined neurl netwrk cn give the inverse kinemtics slutin quickl fr n given Crtesin crdinte in rbtic sstem. 3. Neurl Netwrk bsed Inverse Kinemtics Slutin Neurl netwrks re generll used in the mdeling f nnliner prcesses. ANN is prllel-distributed infrmtin prcessing sstem. T frm trinble nnliner sstem, it stres the smples with distributed cding. Trining f neurl netwrk cn be epressed s mpping between n given input nd utput dt set. Neurl netwrks hve sme dvntges, such s dptin, lerning nd generlitin. Implementtin f neurl-netwrk mdel requires us t decide the structure f the mdel, the tpe f ctivtin functin nd the lerning lgrithm. In Figure 3, the schemtic representtin f neurl netwrk bsed inverse kinemtics slutin is given. The slutin sstem is bsed n trining neurl netwrk t slve n inverse kinemtics prblem bsed n the prepred trining dt set using direct kinemtics equtins. In Figure 2, e refers t errr the neurl netwrk results will be n pprimtin, nd there will be n cceptble errr in the slutin. = C 2 C 3 S 45 + S 2 S 45 p = (- C 1 S 2 C 3 S 45 + S 1 S 3 S 45 + C 1 C 2 C 45 ) l 5 + ( C 1 S 2 C 3 S 4 + S 1 S 3 S 4 + C 1 C 2 C 4 ) l 4 + C 1 C 2 l 23 p = ( S 1 S 2 C 3 S 45 C 1 S 3 S 45 + S 1 C 2 C 45 ) l 5 + ( S 1 S 2 C 3 S 4 C 1 S 3 S 4 + S 1 C 2 C 4 ) l 4 + S 1 C 2 l 23 p = (C 2 C 3 S 45 + S 2 C 45 ) l 5 + (C 2 C 3 S 4 + S 2 C 4 ) l 4 + S 2 l 23 + l 1 The inverse kinemtics slutin fr the rbt is indicted s fllws: csθ 4 = - (l l 4 2 d 2 ) / 2 l 23 l 4 (3) The equtin btined frm (2) s: (S 45 l 5 + S 4 l 4 ) 2 + (C 45 l 5 + C 4 l 4 + l 23 ) 2 = p 2 + p 2 + (p - l 1 ) 2 (4) Pper ID: IJSER f 83

3 Figure 2: ANN bsed inverse kinemtics slutin sstem [5] The designed neurl-netwrk tplg is given in Figure 3. A feed-frwrd multiler neurl-netwrk structure ws designed including 12 inputs nd 6 utputs. Onl ne hidden ler ws used during the stud. T trin the netwrk we must prvide the ANN with the dtset. ANN is trined with the dt which is generted b fifth-rder plnmil trjectr plnning lgrithm. The equtin fr fifth-rder plnmil trjectr plnning is given in fllwing equtin: θ i (t)= θ i + 1/t f 3 (θ if - θ i ) t /t f 4 (θ if - θ i ) t 4 + 6/t f 5 (θ if - θ i ) t 5 ; i = 1, 2, 3.. n (6) Where, θ i (t) = ngulr psitin t time t θ i = initil psitin f the i th jint θ if = finl psitin f the i th jint n = number f jints t f = rrivl time frm initil psitin t the trget Figure 3: Structure f neurl netwrk used in this stud Initil nd finl ngulr psitins re defined t prduce dt in the wrkspce f rbt. After gthering dt frm the whle netwrk is trined in the bck prpgtin mde nd ll the weighs re updted ccrding t the new trining dt. Fr the trining, 1 dt vlues crrespnding t the (θ 1, θ 2, θ 3...θ 6 ) jint ngles ccrding t the different (n,,, p, n,,, p, n,,, p ) Crtesin crdinte prmeters were generted b using (6) bsed n kinemtic equtins given in (2). A smple dt set prduced fr the trining f neurl netwrks is given in Tble 2 nd Trining Methd Tble 2: A smple input dt set fr the trining f neurl netwrks Inputs n n n p p p Pper ID: IJSER f 83

4 Tble 3: A smple trget dt set fr the trining f neurl netwrks Trgets Lerning / Trining Functin Trinlm frm MATLAB tlb is netwrk trining functin which updtes weight nd bis vlues ccrding t Levenberg-Mrqurdt ptimitin [6]. It is the fstest bck prpgtin lgrithm in the MATLAB tlb, nd is immensel suggested s first-chice supervised lgrithm. Frm the trining stte plt it is seen tht trining cntinued fr itertins befre the trining stpped. The perfrmnce plt shwn in Figure 6 des nt indicte n mjr prblems with the trining. The vlidtin nd test curves re ver similr. 5. Result nd Discussin In this stud, Neurl Netwrk Fitting Tl (using cmmnd: nftl) is used t crete nd trin the netwrk. The dtset is lded int selected dt windw [6]. The netwrk is trined using the input dt nd the perfrmnce plt, trining stte nd regressin plts re bserved. In this trining, Rndm (dividernd) rule divides the dt where 7% dt re ssigned t trining set, 15% t vlidtin nd 15% dt t test set. As shwn in Figure 4, this time the trining cntinued fr the mimum f 1 itertins. Figure 5: Trining Plt 6. Cnclusin Figure 6: Perfrmnce Plt Figure 4: Neurl Netwrk Trining Mstl mthemticl mdels fil t simulte the cmple nture f inverse kinemtics prblem. In cntrst, ANN is bsed n the dt input/utput dt pirs t determine the structure nd prmeters f the mdel. Als, ANN s cn lws be updted in rder t chieve better results b Pper ID: IJSER f 83

5 presenting new trining emples s new dt becme vilble. References [1] M.W. Spng, S. Hutchinsn nd M. Vidsgr, Rbt Mdeling nd Cntrl, 1 st Editin, Jn Wile & Sns, Inc, 25. [2] J. Angeles, Fundmentls f Rbtic Mechnicl Sstems: Ther, Methds, nd Algrithms, 2nd Editin, Springer, 23. [3] J. J. Crge, Intrductin t Rbtics Mechnics nd Cntrl, 3rd Editin, Prentice Hll, 25. [4] J. Denvit nd R. Hrtenberg, A Kinemtic Nttin fr Lwer Pir Mechnisms Bsed n Mtrices f Applied Mechnics, pp , [5] R. Köker, T. Çkr, Y. Sri, A neurl-netwrk cmmittee mchine pprch t the inverse kinemtics prblem slutin f rbtic mnipultrs. Engineering with Cmputers, Springer-Verlg Lndn, DOI 1.17/s , 213. [6] The Mthwrks Neurl Netwrk Tlb user guide. Avilble n line n: c/nnet/nnet.pdf Pper ID: IJSER f 83