Inverse Kinematic Analysis of Robot Manipulators

Transcription

1 Inverse Knemtc Anlyss of Robot Mnpultors A THEI UBMITTED IN FULFILMENT OF THE REQUIREMENT FOR THE AWARD OF THE DEGREE OF Doctor of Phlosophy IN INDUTRIAL DEIGN BY Pnchnn jh (Roll. No. ID0) NATIONAL INTITUTE OF TEHNOLOGY ROURKELA, INDIA July-0

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3 NATIONAL INTITUTE OF TEHNOLOGY ROURKELA, INDIA Dr. Bbhut Bhusn Bswl Professor Deprtment of Inustrl Desgn NIT, Rourkel ERTIFIATE Ths s to certfy tht the thess enttle Inverse Knemtc Anlyss of Robot Mnpultors beng submtte by Pnchnn Jh for the wr of the egree of Doctor of Phlosophy (Inustrl Desgn) of NIT Rourkel s recor of bonfe reserch work crre out by hm uner my supervson n gunce. He hs worke for more thn three yers on the bove problem t the Deprtment of Inustrl Desgn, Ntonl Insttute of Technology, Rourkel n ths hs reche the stnr fulfllng the requrements n the regulton reltng to the egree. The contents of ths thess, n full or prt, hve not been submtte to ny other unversty or nsttuton for the wr of ny egree or plom. (Dr. B. B. Bswl)

4 AKNOWLEDGEMENT Ths ssertton s result of the reserch work tht hs been crre out t Ntonl Insttute of Technology, Rourkel. Durng ths pero, the uthor cme cross wth gret number of people whose contrbutons n vrous wys helpe n the fel of reserch n they eserve specl thnks. It s plesure to convey the grttue to ll of them. Frst of ll the uthor expresses hs hertest grttue to hs supervsor n gue Dr. B. B. Bswl, Professor n He, Deprtment of Inustrl Desgn, NIT, Rourkel for hs vluble gunce, support n encourgement n the course of the present work. The successful n tmely completon of the work s ue to hs constnt nsprton n constructve crtcsms. The uthor cnnot equtely express hs pprecton to hm. The uthor recors hs grtefulness to Mm Mrs. Meent Bswl for her constnt support n nsprton urng hs work n sty t NIT, Rourkel. The uthor tke ths opportunty to express hs eepest grttue to Prof. M.R. Khn, He of the Deprtment n Prof. D.. Bsht, of the Deprtment of Inustrl Desgn, NIT Rourkel for constnt vce, useful scussons, encourgement n support n pursung the reserch work. The uthor s grteful to Prof..K. rng, Drector, NIT, Rourkel, Prof. R.K. hoo, former He of Mechncl Engneerng Deprtment, NIT, Rourkel, for ther kn support n concern regrng hs cemc requrements. The uthor lso expresses hs thnkfulness to Mr. H. Mshr, Mr. V. Mshr, Mr. A Jh, Mr. A. Agrwl, Mr. A. Gnguly n Mr.. hnrkr, Deprtment of Mechncl Engneerng n Mr. M. V. A. Rju, Mr. O. P. hu, Mr. R. N. Mhptr, Mr. B. Blbntry, n Mr. P. Pr, reserchers n NIT Rourkel for unhesttng cooperton extene urng the tenure of the reserch progrmme. The completon of ths work cme t the expense of uthor s long hours of bsence from home. Wors fl to express hs nebteness to hs wfe Vbh, lovng son mbhv, lovng Nece Mull, n nephew Osho-Prnce-Drsh for ther unerstnng, ptence, ctve cooperton n fter ll gvng ther tmes throughout the course of the octorl ssertton. The uthor thnks them for beng supportve n crng. Hs prents n reltves eserve specl menton for ther nseprble support n pryers. Lst, but not the lest, the uthor thnk the one bove ll, the omnpresent Go, for gvng hm the strength urng the course of ths reserch work. Pnchnn Jh

5 Abstrct An mportnt prt of nustrl robot mnpultors s to cheve esre poston n orentton of en effector or tool so s to complete the pre-specfe tsk. To cheve the bove stte gol one shoul hve the soun knowlege of nverse knemtc problem. The problem of gettng nverse knemtc soluton hs been on the outlne of vrous reserchers n s elberte s thorough reserche n mture problem. There re mny fels of pplctons of robot mnpultors to execute the gven tsks such s mterl hnlng, pck-n-plce, plnetry n unerse explortons, spce mnpulton, n hzrous fel etc. Moreover, mecl fel robotcs ctches pplctons n rehbltton n surgery tht nvolve knemtc, ynmc n control opertons. Therefore, nustrl robot mnpultors re requre to hve proper knowlege of ts jont vrbles s well s unerstnng of knemtc prmeters. The moton of the en effector or mnpultor s controlle by ther jont ctutor n ths prouces the requre moton n ech jonts. Therefore, the controller shoul lwys supply n ccurte vlue of jont vrbles nlogous to the en effector poston. Even though nustrl robots re n the vnce stge, some of the bsc problems n knemtcs re stll unsolve n consttute n ctve focus for reserch. Among these unsolve problems, the rect knemtcs problem for prllel mechnsm n nverse knemtcs for serl chns consttute ecent shre of reserch omn. The forwr knemtcs of robot mnpultor s smpler problem n t hs unque or close form soluton. The forwr knemtcs cn be gven by the converson of jont spce to rtesn spce of the mnpultor. On the other hn nverse knemtcs cn be etermne by the converson of rtesn spce to jont spce. The nverse knemtc of the robot mnpultor oes not prove the close form soluton. Hence, nustrl mnpultor cn cheve esre tsk or en effector poston n more thn one confgurton. Therefore, to cheve exct soluton of the jont vrbles hs been the mn concern to the reserchers. A bref ntroucton of nustrl robot mnpultors, evoluton n clssfcton s presente. The bsc confgurtons of robot mnpultor re emonstrte n ther benefts n rwbcks re elberte long wth the pplctons. The ffcultes to solve forwr n nverse knemtcs of robot mnpultor re scusse n soluton of nverse knemtc s ntrouce through conventonl methos. In orer to ccomplsh the esre objectve of the work n ttn the soluton of nverse knemtc problem n effcent stuy of the exstng tools n technques hs been one. A revew of lterture survey n vrous tools use to solve nverse knemtc problem on fferent spects s scusse. The vrous pproches of nverse knemtc soluton

6 s ctegorze n four sectons nmely structurl nlyss of mechnsm, conventonl pproches, ntellgence or soft computng pproches n optmzton bse pproches. A porton of mportnt n more sgnfcnt ltertures re thoroughly scusse n bref nvestgton s me on conclusons n gps wth respect to the nverse knemtc soluton of nustrl robot mnpultors. Bse on the survey of tools n technques use for the knemtc nlyss the bro objectve of the present reserch work s presente s; to crry out the knemtc nlyses of fferent confgurtons of nustrl robot mnpultors. The mthemtcl moellng of selecte robot mnpultor usng exstng tools n technques hs to be me for the comprtve stuy of propose metho. On the other hn, evelopment of new lgorthm n ther mthemtcl moellng for the soluton of nverse knemtc problem hs to be me for the nlyss of qulty n effcency of the obtne solutons. Therefore, the stuy of pproprte tools n technques use for the soluton of nverse knemtc problems n comprson wth propose metho s consere. Moreover, recommenton of the pproprte metho for the soluton of nverse knemtc problem s presente n the work. Aprt from the forwr knemtc nlyss, the nverse knemtc nlyss s qute complex, ue to ts non-lner formultons n hvng multple solutons. There s no unque soluton for the nverse knemtcs thus necessttng pplcton of pproprte prectve moels from the soft computng omn. Artfcl neurl network (ANN) cn be gnfully use to yel the esre results. Therefore, n the present work severl moels of rtfcl neurl network (ANN) re use for the soluton of the nverse knemtc problem. Ths moel of ANN oes not rely on hgher mthemtcl formultons n re ept to solve NP-hr, non-lner n hgher egree of polynoml equtons. Although ntellgent pproches re not new n ths fel but some selecte moels of ANN n ther hybrzton hs been presente for the comprtve evluton of nverse knemtc. The hybrzton scheme of ANN n n nvestgton hs been me on ccurces of opte lgorthms. On the other hn, ny Optmzton lgorthms whch re cpble of solvng vrous multmol functons cn be mplemente to solve the nverse knemtc problem. To overcome the problem of conventonl tool n ntellgent bse metho the optmzton bse pproch cn be mplemente. In generl, the optmzton bse pproches re more stble n often converge to the globl soluton. The mjor problem of ANN bse pproches re ts slow convergence n often stuck n locl optmum pont. Therefore, n present work fferent optmzton bse pproches re consere. The formulton of the objectve functon n ssocte constrne re scusse thoroughly. The comprson of ll opte lgorthms on the bss of number v

7 of solutons, mthemtcl opertons n computtonl tme hs been presente. The thess conclues the summry wth contrbutons n scope of the future reserch work. v

8 Tble of ontents ertfcte... Acknowlegements... Abstrct... -v Tble of ontents... v-x Lst of Tbles... x-x Lst of Fgures... x-xv Lst of ymbols... xv-xv Abbrevtons... xv INTRODUTION.... Overvew.... Evoluton of robot mnultors.... tructurl of nustrl robots..... lssfcton by Mechnsm..... lssfcton by of n relte componenets... 8 ) Generl Mnpultor... 0 b) Reunnt/Hyper-reunnt Mnpultor... 0 c) Flexble Mnpultor... 0 ) Defcent Mnpultor..... lssfcton by Actuton..... lssfcton by Workspce... ) rtesn Robot... b) ylnrcl Robot... c) phercl Robot... ) ARA Robot... ) Artculte/Revolute Robot..... lssfcton bse on regonl structure..... lssfcton by Moton hrcterstc lssfcton by Applcton Bsc Kenmtcs Motvton Bro Objectve Methoology... v

9 .8 Orgnzton of the Thess....9 ummry... REVIEW OF LITERATURE.... Overvew.... urvey of tools use for lterture soluton..... tructurl Anlsys of Mechnsm onventonl Metho for Knemtcs Intellgent or oft-omputng Approch Optmzton Approch Revew Anlyss n Outcome Problem ttement cope of Work ummry MATERIAL AND METHOD Overvew Mterls Descrpton of plnr -of revolute mnpultor Descrpton of -of ARA mnpultor Descrpton of -of revolute Poneer mnpultor Descrpton of -of PUMA 0 mnpultor Descrpton of -of ABB IRb-00 mnpultor Descrpton of -of AEA IRb- mnpultor Descrpton of -of TAUBLI RX 0 L mnpultor Methos onventonl pproch Intellgent bse pproches Optmzton bse pproches ummry... 9 MATHEMATIAL MODELLING AND KINEMATI ANALYI Overvew Representton methos n knemtcs Kenmtc Vrbles n Prmeters DH-prmeters DH-Algorthm for frme ssgnment Mthemtcl Moellng of -of revolute mnpultor Mthemtcl Moellng of -of ARA mnpultor Mthemtcl Moellng of -of revolute mnpultor v

10 ..7 Mthemtcl Moellng of -of PUMA 0 mnpultor.... Quternon Algebr Knemtcs..... Mthemtcl Bckgroun... ) onjugte of Quternon... b) Mgntue of Quternon... 7 c) Norm ) Quternon Inverse Quternon rotton n trnslton Knemtc soluton of ARA Mnpultor Knemtc soluton of -of Revolute Mnpultor..... Knemtc soluton of PUMA Mnpultor Knemtc soluton of ABB IRB-00 Mnpultor Knemtc soluton of TAUBLI RX 0 L Mnpultor Knemtc soluton of AEA IRb- Mnpultor.... ummry... 7 INTELLIGENT TEHNIQUE FOR INVERE KINEMATI OLUTION 8. Overvew Applctons of ANN Moels Mult-lyere Perceptron Network (MLPNN) Polynoml Preprocessor Neurl Network (PPNN)..... P-gm Neurl Network.... Applcton of Aptve Neurl Fuzzy Inference ystem (ANFI) Lernng Algorthms Hybrzton of ANN wth Metheurstc Algorthms Prtcle wrm Optmzton Algorthm(PO)..... Techng lernng bse optmzton(tlbo)..... Objectve Functon for trnng MLP..... Objectve Functon..... Weght n Bs Optmzton cheme.... ummry... OPTIMIZATION HEME FOR INVERE KINEMATI OLUTION Overvew Metheurstc Algorthms..... Genetc Algorthm Reprentton... ) Intlzton... v

11 b) Recombnton... c) Mutton... 7 ) electon Prtcle wrm Optmzton Overvew Grey Wolf Optmzton Algorthm Developement of Novel Optmzton Algorthm rb Intellgence Bse Optmzton (IBO)Algorthm Methoology of IBO lgorthm Mthemtcl Moellng... 7 ) Hypotheses IBO Algorthm Implementton for olvng Inverse Knemtcs Mthemtcl Moellng of Objectve Functon Poston Bse Error Orentton Bse Error oluton cheme of Inverse Knemtc Problem ummry REULT AND ANALYI Overvew Inverse Knemtc oluton Usng ANN Results for -of plnr Revolute Mnpultor Results for -of ARA Mnpultor ANFI Results for -of ARAMnpultor ANN Results for -of Revolute Mnpultor ANFI Results for -of Revolute Mnpultor Hybr ANN Approch for Inverse Knemtc Problem Results for -of ARA Mnpultor Results for -of revolute Mnpultor Results for -of PUMA Mnpultor Results for -of ABB IRb-00 Mnpultor Results for -of AEA IRb-Mnpultor Results for -of TAUBLI RX 0 L Mnpultor Methurstc Approch for Inverse Knemtc Problem Results for -of ARA Mnpultor Results for -of revolute Mnpultor Results for -of PUMA Mnpultor Results for -of ABB IRb-00 Mnpultor... x

12 7.. Results for -of AEA IRb-Mnpultor Results for -of TAUBLI RX 0 L Mnpultor Dscussons ummry ONLUION AND FURTHER WORK Overvew onclusons ontrbutons Future Reserch Work REFERENE Publctons... urrculum Vte... x

13 Lst of Tbles Tble.: Dfferent types of jonts....9 Tble.: onfgurtons n workspce.... Tble.: onfgurtons of -of revolute mnpultors.... Tble.: lssfcton bse on regonl structure.... Tble.: Lst of some mportnt ltertures....7 Tble.: Dfferent mechnsms n moblty.... Tble.: onfgurtons of robot mnpultors Tble.: Mnpultor jont lmts n knemtc prmeters....8 Tble.: Mnpultor jont lmts n knemtc prmeters....8 Tble.: Prm mnpultor jont lmts n knemtc prmeters....8 Tble.: Mxmum lmt of jont vrbles....8 Tble.: Jont vrble n prmeters of PUMA 0 robot....8 Tble.7: ABB IRB-00 mnpultor jont lmts n knemtc prmeters Tble.8: AEA IRb- mnpultor jont lmts n knemtc prmeters Tble.9: TAUBLI RX0L mnpultor jont lmts n knemtcprmeters..89 Tble.0: Aopte mterls n methos....9 Tble.: DH prmeters...00 Tble.: DH-prmeters for -of revolute mnpultor...0 Tble.: The DH Prmeters for ARA mnpultor....0 Tble.: The DH prmeters -of revolute mnpultor Tble.: The DH prmeters for PUMA mnpultor.... Tble.: hell selecton....7 Tble 7.: Poston of en effector n jont vrbles Tble 7.: onfgurton of MLPNN....9 Tble 7.: omprson between nlytcl soluton n MLPNN soluton....9 Tble 7.: Regresson nlyss....9 Tble 7.: onfgurton of ANFI Tble 7.: omprson of results Tble 7.7: Desre jont vrbles etermne through nlytcl soluton Tble 7.8: onfgurton of MLPNN Tble 7.9: onfgurton of ANFI....0 Tble 7.0: omprson of results....0 Tble 7.: Men squre error for ll trnng smples of hybr MLPNN Tble 7.: Men squre error for ll trnng smples of hybr MLPNN....0 Tble 7.: onfgurton of MLPNN....7 Tble 7.: Desre jont vrbles etermne through quternon lgebr...7 x

14 Tble 7.: Men squre error for ll trnng smples of hybr MLPNN....8 Tble 7.: onfgurton of MLPNN....7 Tble 7.7: Desre jont vrbles etermne through quternon lgebr...8 Tble 7.8: Men squre error for ll trnng smples of hybr MLPNN....9 Tble 7.9: onfgurton of MLPNN.... Tble 7.0: Desre jont vrbles etermne through quternon lgebr.... Tble 7.: Men squre error for ll opte lgorthms.... Tble 7.: onfgurton of MLPNN.... Tble 7.: Desre jont vrbles etermne through quternon lgebr... Tble 7.: Men squre error for ll trnng smples of hybr MLPNN....7 Tble 7.: Fve fferent postons n jont vrbles.... Tble 7.: Fve fferent postons n jont vrbles through opte lgorthm.... Tble 7.7: Fve fferent postons n jont vrbles.... Tble 7.8: Fve fferent postons n jont vrbles through opte lgorthm Tble 7.9: omputtonl tme for nverse knemtc evlutons.... Tble 7.0: Fve fferent postons n jont vrbles through quternon....0 Tble 7.: omprtve results for jont vrble n functon vlue.... Tble 7.: omputtonl tme for nverse knemtc evlutons....0 Tble 7.: Fve fferent postons n jont vrbles through quternon.... Tble 7.: omprtve results for jont vrble n functon vlue.... Tble 7.: Fve fferent postons n jont vrbles....8 Tble 7.: omprtve results for jont vrble n functon vlue....9 Tble 7.7: Fve fferent postons n jont vrbles through quternon....7 Tble 7.8: omprtve results for jont vrble n functon vlue....7 Tble 7.9: omprtve nlyss of conventonl tools Tble 7.0: omprtve nlyss of ntellgent pproches Tble 7.: omprtve nlyss of optmzton lgorthms x

15 Lst of Fgures Fgure.: () erl [], (b) Prllel n [] (c) Hybr mechnsms... Fgure.: Humn rm structures....7 Fgure.: Jont rottons of 7-of robotc rm....7 Fgure.: ARA robot.... Fgure.: Revolute robot.... Fgure.: Regonl mechnsm of robot mnpultors... Fgure.: Moel of -of revolute mnpultor....8 Fgure.: tructure of Aept One ARA mnpultor....8 Fgure.: tructure of the Pnoneer rm....8 Fgure.: tructure of PUMA 0 robot mnpultor...8 Fgure.: onfgurtons of ABB IRB Fgure.: onfgurtons of AEA IRb Fgure.7: onfgurtons of TAUBLI RX 0 L...89 Fgure.: Poston n recton of cylnrcl jont n rtesn coornte frme Fgure.: knemtc pr n DH prmeters Fgure.: Denvt-Hrtenberg prmeters for successve trnslton n rotton of lnks....0 Fgure.: Plnr -of revolute mnpultor...0 Fgure.: oornte frmes of -of revolute mnpultor....0 Fgure.: DH frmes of the ARA robot....0 Fgure.7: tructure of ARA mnpultor through MATLB....0 Fgure.8: Moel n coornte frmes of the mnpultor Fgure.9: onfgurton of -of revolute mnpultor...08 Fgure.0: Moel n coornte frmes of mnpultor.... Fgure.: Representton of rotton....7 Fgure.: ARA mnpultor....9 Fgure.: PUMA 0 mnpultor moel.... Fgure.: Bse frme n moel of -of revolute mnpultor....7 Fgure.: onfgurton n moel of ABB IRB-0 robot mnpultor.... Fgure.: oornte frme n moel of TAUBLI RX0L robot mnpultor.... Fgure.7: oornte frme n moel of AEA IRb- robot mnpultor.... Fgure.: Neurl network moels () Fee forwr n (b) bck propgton strtegy...0 Fgure.: Mult-lyere perceptron neurl network structure.... x

16 Fgure.: Flow chrt for MLPBP.... Fgure.: Polynoml perceptron network.... Fgure.: P-gm neurl network.... Fgure.: Trnng of ANFI structure....8 Fgure.7: Archtecture of ANFI....9 Fgure.8: Flow chrt for PO.... Fgure.9: Flow chrt for MLPPO.... Fgure.0: MLP network wth structure Fgure.: Bnry representtons of genes.... Fgure.: Exmples for smple crossover wth two fferent cses.... Fgure.: Mutton n genetc lgorthm....7 Fgure.: Roulette wheel selectons....8 Fgure.: Flow chrt for grey wolf optmzer Fgure.: Representton of swrm n serchng behvour Fgure.7: Flow chrt for IBO lgorthm Fgure.8: Poston bse error....8 Fgure.9: Orentton ngle between two frmes...8 Fgure 7.: omprson of esre n precte vlue of jont ngles for --- confgurton usng MLP moel....9 Fgure 7.: omprson of esre n precte vlue of jont ngles for --- confgurton usng MLP moel....9 Fgure 7.: omprson of esre n precte vlue of jont ngles for --- confgurton usng MLP moel....9 Fgure 7.: Men squre error for jont ngles usng PPN moel....9 Fgure 7.: Men squre error for jont ngles usng P-sgm network moel....9 Fgure 7.: Men squre error for thet....9 Fgure 7.7: Men squre error for thet....9 Fgure 7.8: Men squre error for....9 Fgure 7.9: Men squre error for thet...9 Fgure 7.0: Grphcl vew of regresson...97 Fgure 7.: Men squre error for thet...98 Fgure 7.: Men squre error for thet...98 Fgure 7.: Men squre error for...99 Fgure 7.: Men squre error for thet...99 Fgure 7.: Men squre error for thet...0 Fgure 7.: Men squre error for thet...0 Fgure 7.7: Men squre error for thet...0 Fgure 7.8: Men squre error for thet...0 xv

17 Fgure 7.9: Men squre error for thet...0 Fgure 7.0: Fgure 7.7: Men squre error for thet...0 Fgure 7.: Men squre error for thet...0 Fgure 7.: Men squre error for thet...0 Fgure 7.: Men squre error for thet...0 Fgure 7.: Men squre error for thet...0 Fgure 7.: Men squre error for thet...0 Fgure 7.: (), (b), (c), () n (e) re men squre error curve for ll jont ngles usng MLPGA...09 Fgure 7.7: (), (b), (c), () n (e) re men squre error curve for ll jont ngles usng MLPGA... Fgure 7.8: (), (b), (c), () n (e) re men squre error curve for ll jont ngles usng MLPTLBO... Fgure 7.9: (), (b), (c), () n (e) re men squre error curve for ll jont ngles usng MLPTIBO... Fgure 7.0: (), (b), (c), (), (e) n (f) re men squre error curve of MLPBP for ll jont ngles...0 Fgure 7.: (), (b), (c), (), (e) n (f) re men squre error curve of MLPGA for ll jont ngles... Fgure 7.: (), (b), (c), (), (e) n (f) re men squre error curve of MLPTLBO for ll jont ngles... Fgure 7.: (), (b), (c), (), (e) n (f) re men squre error curve of MLPIBO for ll jont ngles... Fgure 7.: (), (b), (c), (), (e) n (f) re men squre error curve of MLPGA for ll jont ngles... Fgure 7.: (), (b), (c), () n (e) re men squre error curve of MLPGA for ll jont ngles... Fgure 7.: (), (b), (c), () n (e) re men squre error curve of MLPGA...0 Fgure 7.7: omprson of jont vrbles for poston... Fgure 7.8: omprson of jont vrbles for poston... Fgure 7.9: omprson of jont vrbles for poston... Fgure 7.0: omprson of jont vrbles for poston... Fgure 7.: omprson of jont vrbles for poston... Fgure 7.: Functon vlue n jont vrbles for P...7 Fgure 7.: Functon vlue n jont vrbles for P...7 Fgure 7.: Functon vlue n jont vrbles for P...8 Fgure 7.: Functon vlue n jont vrbles for P...8 Fgure 7.: Functon vlue n jont vrbles for P...8 xv

18 Fgure 7.7: Functon vlue n jont vrbles for P...8 Fgure 7.8: Functon vlue n jont vrbles for P...9 Fgure 7.9: Functon vlue n jont vrbles for P...9 Fgure 7.0: Functon vlue n jont vrbles for P...9 Fgure 7.: Functon vlue n jont vrbles for P...9 Fgure 7.: Functon vlue n jont vrbles for P... Fgure 7.: Functon vlue n jont vrbles for P... Fgure 7.: Functon vlue n jont vrbles for P... Fgure 7.: Functon vlue n jont vrbles for P... Fgure 7.: Functon vlue n jont vrbles for P... Fgure 7.7: Functon vlue n jont vrbles for P... Fgure 7.8: Functon vlue n jont vrbles for P... Fgure 7.9: Functon vlue n jont vrbles for P... Fgure 7.0: Functon vlue n jont vrbles for P... Fgure 7.: Functon vlue n jont vrbles for P... Fgure 7.: Functon vlue n jont vrbles for P... Fgure 7.: Functon vlue n jont vrbles for P... Fgure 7.: Functon vlue n jont vrbles for P... Fgure 7.: Functon vlue n jont vrbles for P...7 Fgure 7.: Functon vlue n jont vrbles for P...7 Fgure 7.7: Functon vlue n jont vrbles for P...7 Fgure 7.8: Functon vlue n jont vrbles for P...7 Fgure 7.9: Functon vlue n jont vrbles for P...8 Fgure 7.70: Functon vlue n jont vrbles for P...8 Fgure 7.7: Functon vlue n jont vrbles for P...8 Fgure 7.7: Functon vlue n jont vrbles for P...8 Fgure 7.7: Functon vlue n jont vrbles for P...9 Fgure 7.7: Functon vlue n jont vrbles for P...9 Fgure 7.7: Functon vlue n jont vrbles for P...9 Fgure 7.7: Functon vlue n jont vrbles for P...0 Fgure 7.77: Functon vlue n jont vrbles for P... Fgure 7.78: Functon vlue n jont vrbles for P... Fgure 7.79: Functon vlue n jont vrbles for P... Fgure 7.80: Functon vlue n jont vrbles for P... Fgure 7.8: Functon vlue n jont vrbles for P... Fgure 7.8: Functon vlue n jont vrbles for P... Fgure 7.8: Functon vlue n jont vrbles for P... Fgure 7.8: Functon vlue n jont vrbles for P...7 xv

19 Fgure 7.8: Functon vlue n jont vrbles for P...7 Fgure 7.8: Functon vlue n jont vrbles for P...8 Fgure 7.87: Functon vlue n jont vrbles for P...70 Fgure 7.88: Functon vlue n jont vrbles for P...7 Fgure 7.89: Functon vlue n jont vrbles for P...7 Fgure 7.90: Functon vlue n jont vrbles for P...7 Fgure 7.9: Functon vlue n jont vrbles for P...7 Fgure 7.9: Functon vlue n jont vrbles for P...7 Fgure 7.9: Functon vlue n jont vrbles for P...7 Fgure 7.9: Functon vlue n jont vrbles for P...7 Fgure 7.9: Functon vlue n jont vrbles for P...7 Fgure 7.9: Functon vlue n jont vrbles for P...77 xv

20 Lst of ymbols Lnk Length Twst ngle Jont Dstnce Jont ngle c cos, (,,...n) s sn, (,,...n), Angulr rnom number h w j Hen weght k Quternon multplcton Locl grent of the kth lyer Momentum prmeter be T be R, Trnsformton quternon of en effector T O O q. Trnsfer mtrx Orgn Of Globl coornte system Orgns of Locl coornte systems of thumb n fngers Jont ngles of the hn moel 0 A...n Trnsfer mtrces from locl coornte system to globl coornte system. n e Hen lyer neurons nonlner ctvton functons, Lgrnge multpler xv

21 Abbrevtons of DH MLP PO GWO TLBO PPN IBO GA QA ANFI L BP FI MF GW F HT R P H Degrees of freeom Denvt- Hrtenberg Mult lyere perceptron Prtcle swrm optmzton Grey wolf optmzer Techng lernng bse optmzton Polynoml pre-processor network rb ntellgence bse optmzton Genetc lgorthm Quternon lgebr Aoptve Neuro-Fuzzy Inference ystem Lest squres Bck propgton Fuzzy Interfce ystem Membershp functon Grsp wrench spce Force closure Homogeneous trnsformton Revolute Prsmtc Quternon vector xx

22 hpter INTRODUTION. Overvew Over the lst few eces, use of nustrl robots cn be seen worlwe n hs sgnfcntly ncrese wth fster ncresngly tren. Mostly these re beng use for mterl hnlng, welng, pntng, ssemblng of prts, pckgng, hnlng hzrous mterls, unerse opertons, etc. Robot mnpultor mplctes n electromechncl evce tht requres humn exterty to perform vrety of tsks. Although few mnpultors re nthropomorphc n humno, most of these robots cn be trete s electromechncl evces from ther structure pont of vew. On the other hn, there re utonomous n semutonomous robots tht hve bro rnge of pplctons such s plnetry spce explorton, surgcl robotcs, rehbltton, n househol pplctons. A common chrcterstc of such pplctons s tht the robot nees to operte n unstructure envronments rther thn structure nustrl work cells. Moton control n trjectory plnnng for robots n unstructure envronments pose mportnt chllenges ue to uncertntes n envronment moellng, sensng, n robot ctuton. At the present sttus, the bro re of robot pplctons el wth nustrl robot rms opertng n both structure n unstructure envronments. A frst ntroucton to the subject of robotcs ought to nclue rgorous tretment of the topcs n ths text. Robots re lso concerne wth n re slowly becomng prt of humn lfe by ssstng them n professonl n personl lfe s well s nsultng humns from stuton nvolvng hzrs, scomfort, repettons, etc. Wth the vncement of vrous technology, the scope of the tsks performe by robots s wene so tht t s esrble for mchnes to exten the cpbltes of men n to replce them by robots n crryng out t tresome s well s hzrous jobs. In orer to ccomplsh the tsks n

23 humn-lke wys n to relze proper n sfe co-operton between humns n robots, the robots of the future must be thought of hvng humn excellence n terms of ts structure,ntellgence, smrtness n rectons.therefore, robot opertng uner some egree of utonomy cn be extremely complex electromechncl systems whose nlytc escrpton requres vnce methos. Desgn n evelopment of such evces present mny chllengng n nterestng reserch problems. The most mportnt thng s reprogrmmng blty of robot. It s computer controlle tht gves the robot ts utlty n ptblty. The so-clle robotcs revoluton s, n fct, prt of the lrger computer revoluton. There re mny fels for robot mnpultors to perform vrety of tsks. ome of these re utomobles, househol's proucts, pck-n-plce, unerse n plnetry explortons, stellte retrevl n repr, efusng of explosves n roctve fel. In the mecl fel robotcs fn pplctons n rehbltton n surgery tht nvolve knemtc, ynmc n control opertons. Robot mnpultors move long pre-specfe trjectores whch re sequence of ponts were en effector poston, n orenttons re known. Trjectores my be jont spce or rtesn spces tht re functon of tme. The nustrl robots cn be explctly consere s open chn mechnsms tht re systems of rg boes connecte by vrous jonts. Jonts llow prtculr types of reltve motons between the connecte boes. For exmple, rottonl jont cts s hnge n llows only reltve rotton between the connecte boes bout the xs of the jont. A system of rg boes nterconnecte by jonts s clle knemtc chn. Invul rg boes wthn the knemtc chn re clle lnks. A knemtc chn cn be serl, prllel, or serl, n prllel combne,.e. the knemtc chn cn be open, close, or brnche. It s requre to compute ll the necessry ponts n rtesn coornte to perform the smooth operton. The converson of trjectory loctons from rtesn coorntes to jont coorntes s referre to s the nverse knemtcs problem. Even though nustrl robots re n the vnce stge, some of the bsc problems n knemtcs re stll unsolve n consttute n ctve focus for reserch. Among these unsolve problems, the rect knemtcs problem for prllel mechnsm n nverse knemtcs for serl chns consttute ecent shre of reserch omn. The present reserch work prmrly focuses on Knemtcs of vrous nustrl mnpultors fferent confgurtons. The nverse knemtcs problem s funmentl, not only n the esgn of mnpultor but lso n other pplctons nclung computer nmtons n moleculr moellng. Ths problem s ffcult ue to ts nherent computtonl complexty (.e. NP-hr Problem) n ue to mthemtcl complexty tht oes not gurntee close form soluton.

24 . Evoluton of robot mnpultors The concept of the robot ws evently recognze by the zech plywrght Krel pek urng the twenteth century n hs ply Rossum s Unversl Robots (R.U.R.). The term robot s erve from robot whch mens subornte lbour n lve lnguges. In 90, the ethcs of the ntercton between robots n humns ws envsone to be governe by the well-known three funmentl lws of Isc Asmov, the Russn scence-fcton wrter n hs novel Run-roun. The mle of the twenteth century brought the frst explortons of the connecton between humn ntellgence n mchnes, mrkng the begnnng of n er of fertle reserch n the fel of rtfcl ntellgence (AI). Aroun tht tme, the frst robots were relze. They benefte from vnces n the fferent technologes of mechncs, controls, computers n electroncs. As lwys, new esgns motvte new reserch n scoveres, whch, n turn, le to enhnce solutons n thus to novel concepts. Ths vrtuous crcle over tme prouce tht knowlege n unerstnng tht gve brth to the fel of robotcs, properly referre to s the scence n technology of robots. The erly robots bult n the 90s stemme from the confluence of two technologes: numercl control mchnes for precse mnufcturng, n tele-opertors for remote roctve mterl hnlng. These mster slve rms were esgne to uplcte oneto-one the mechncs of the humn rm n h rumentl control n lttle percepton bout the envronment. Then, urng the m-to-lte twenteth century, the evelopment of ntegrte crcuts, gtl computers n mnturze components enble computer-controlle robots to be esgne n progrmme. These robots, terme nustrl robots, becme essentl components n the utomton of flexble mnufcturng systems n the lte 970s. Further to ther we pplcton n the utomotve nustry, nustrl robots were successfully employe n generl nustry, such s the metl proucts, the chemcl, the electroncs n the foo nustres. More recently, robots hve foun new pplctons outse the fctores, n res such s clenng, serch n rescue, unerwter, spce, n mecl pplctons. In the 980s, robotcs ws efne s the scence tht stues the ntellgent connecton between percepton n cton. Wth reference to ths efnton, the cton of robotc system s entruste to locomoton pprtus to move n the envronment (wheels, crwlers, legs, propellers) n/or to mnpulton pprtus to operte on objects present n the envronment (rms, en effectors, rtfcl hns), where sutble ctutors nmte the mechncl components of the robot. The percepton s extrcte from the sensors provng nformton on stte of the robot (poston n spee) n ts surrounng envronment (force n tctle, rnge n vson). The ntellgent connecton s entruste to progrmmng; plnnng n control rchtecture tht reles

25 on the percepton n vlble moels of the robot n envronment n explots lernng n skll cquston. In the 990s reserch ws booste by the nee to resort to robots to ress humn sfety n hzrous envronments (fel robotcs), or to enhnce the humn opertor blty n reuce hs/her ftgue (humn ugmentton), or else by the esre to evelop proucts wth we potentl mrkets me t mprovng the qulty of lfe (servce robotcs). A common enomntor of such pplcton scenros ws the nee to operte n scrcely structure envronment tht ultmtely requres ncrese bltes n hgher egree of utonomy. By the wn of the new mllennum, robotcs hs unergone mjor trnsformton n scope n mensons. Ths expnson hs been brought bout by the mturty of the fel n the vnces n ts relte technologes. From lrgely omnnt nustrl focus, robotcs hs been rply expnng nto the chllenges of the humn worl (humn-cntere n lfe-lke robotcs). The new generton of robots s expecte to sfely n epenbly co-hbtt wth humns n homes, workplces, n communtes, provng support n servces, entertnment, eucton, helthcre, mnufcturng, n ssstnce. Beyon ts mpct on physcl robots, the boy of knowlege robotcs hs prouce s revelng much wer rnge of pplctons rechng cross verse reserch res n scentfc scplnes, such s: bomechncs, hptcs, neuroscences, n vrtul smulton, nmton, surgery, n sensor networks mong others. In return, the chllenges of the new emergng res re provng n bunnt source of stmulton n nsghts for the fel of robotcs. It s nee t the ntersecton of scplnes tht the most strkng vnces hppen. Prctcl mplementton of nustrl robots ws frst strte urng the 90s, long wth numercl controlle n AD/AM systems. Now y these mnpultors reche the mturty stges. ome of the lnmrk evelopments n nustrl robots re mentone wth ths []: 97 The frst servo controlle electrc tele-opertor lunche 98 Introucton of force feebck n tele-opertor 9 Frst progrmmble esgn from George Devol 9 Founton of Unmton compny by Josh Engelberger the Unmton ompny 9 Generl Motors mplementton of Unmte robot n New Jersey 9 Frst vson system evelope for robots 97 - tnfor Unversty evelope robot rm

26 97 Frst computer controlle mnpultor ntrouce the MlcronT 978 Development of PUMA xs robot 979 Frst ssembly lne ARA robot esgne by Jpnese 98 - Mellon Unversty evelope frst rect rve mnpultor 989- H-tech chess plyng robot 99 - oncept of Hon's P humno robot Mrs spce explorton robot sojourner rover 00 - nrm ws mplemente nto I 00- Introucton of humno robot AIMO 00 - ornell Unversty expose robot sklle of self-replcton 00- Development of wreless operte n computer controlle HUBO robot by KIT 00- trfsh -legge robot evelope by ornell Unversty 007- Jpnese compny ntrouce entertnment robot TOMY 0-to present- Kuk Robotcs LBR w, lghtweght robot Rob coster for entertnment Toy, new communtes of users n evelopers re formng, wth growng connectons to the core of robotcs reserch. A strtegc gol for the robotcs communty s one of outrech n scentfc cooperton wth these communtes. Future evelopments n expecte growth of the fel wll lrgely epen on the reserch communty s bltes to cheve ths objectve.. tructure of nustrl robots Ths secton s evote to the clssfcton of nustrl robots, wth ttenton to serl structures. Bsc crter for clssfcton hve been resse stepwse, n concern mthemtcs behn the mechnsm hs lso been propose. The mjor m s restrcte to robots tht re mnly ntcpte for mnpulton tsks n serl knemtc chns. Robots cn usully be clssfe s per ther number of egree of freeom (of) or xes n ther knemtc chrcterstc. Workng profcences of robot mnpultor cn be evlute from ts egree of freeom. ommon -of robot mnpultor cn only cheve generl tsk n -menson spce contnng rbtrrly poston n orentton for ny object. On the other hn, for specfc pplcton one nees to esgn robot mnpultor s per of s well s knemtcs chrcterstc. However, there

27 re numerous crter for the clssfcton of robot mnpultor but typclly one cn select of or number of xes. On the other hn, Robotcs Insttute of Amerc (RIA), Assocton Frncse e Robotque (AFR) n Jpnese Inustrl Robot Assocton broly clssfe n verse moules tht re s follows:. Mnul hnlng evces. Fxe sequence robot. Vrble sequence robot. Plybck robot. Numercl control robot. Intellgent robot Other thn these bove mentone moules of nustrl robot mnpultor t cn lso be clssfe s per ther mechnsm, of, ctuton, workspce, control, moton n pplcton... lssfcton by mechnsm Typclly robot mnpultor my be ether serl one hvng open loop or prllel one hvng close loop structure. In nustrl robot mnpultors the jont type my be ether prsmtc (P) or revolute (R) wheres the lnk type my be ether rg or flexble. Moreover, there cn be hybr structure tht conssts of both open n close loop mechncl chns.the serl mnpultor cn be ctegorze bse on the frst jont wll lwys strtng from the fxe bse n en of the lnk wll free to move n spce, see Fgure. (). There re mny combntons of these jonts n lnks tht cretes fferent confgurtons of robot mnpultor smply ue to the jonts R n P, xes of two jcent my be ether prllel or orthogonl. Orthogonl jonts ntersect by 90 egrees wth respect to ther common norml n t cn be prllel when one xs rottes 90 egrees, see Fgure.(b). () (b) (c) Fgure. () erl [], (b) Prllel [] n (c) Hybr mechnsms Exmples of serl mnpultors re PUMA, ARA, KUKA, DENO etc., Gough pltform, Delt robot, RPR plnr prllel robot etc., re prllel mnpultors n

28 Fnuc -9000W s n exmple of hybr mnpultor s shown n Fgure.(c).Fgure. shows further exmples of mechnsms tht result from open, close n hybr open/close knemtc chns. Robotcs n lvng orgnsms resemble the serl/prllel or hybr mechnsm. The most common n well known exmple s the humn hn tht resembles s serl, prllel n hybr mnpultor s shown n Fgure.. The humn rm frme conssts of fferent number of bones, s shown n Fgure. tht cretes serl/hybr mnpultor or the knemtc chn. The humn shouler s ttche to the stem hvng sphercl jont. In the lter chpter, humn rm hs been consere only 7-of serl mnpultor s shown n Fgure. (). The clvcle jont s connecte to stem epcte s n Fgure n lso wth scpul v cromoclvculr jont (A). The scpul jont lter connecte wth glenohumerl jont (G) to the upper rm. A summrze exemplry of rm mechnsm entcl to shouler s shown n Fgure. (b). P ( (b Fgure. Humn rm structures [] As per epcte fgure the rm mnpultor structure hve -of. The upper rm wth humerl bone cn be ssume s serl mechnsm n elbow jont wth humerl bone tht connects the uln cn be consere s prllel mnpultor. Elbow houler Wrst Fgure. Jont rottons of 7-of robotc rm 7

29 Most of the mnpultor or common nustrl mnpultors re bse on the bove scusse humn rm mechnsm. As per Fgure. s the elborte vew of the humn rm wth 7-of mnpultor tht nclues the shouler, elbow n wrst... lssfcton by egree of freeom n relte components The specfc moton of lnks relte to ny mechnsm or mchne cn be efne s the egree of freeom. To execute specfc tsk egree of freeom wll lwys ply the mn role. The totl number of of wll lwys equl the number of nepenent splcement of lnks. As we know tht -of robot mnpultor s the bss to execute the specfc tsk n -mentonl spce. On the other hn, mthemtcl efnton of egrees of freeom wll be mnmum number of nepenent jont prmeters of ny mechnsm tht exclusvely escrbe the sptl poston n orentton of system/boy. On the other hn, no. of of n ny mechnsm cn be obtne by summton of the vlble of of movng lnks tht woul be then λn. Ths s no. of of f there re no jonts n from ths we cn subtrct the constrnt. Now ths cn be expresse s follows Where, constrnt n of N (.) f tht s the fference between the potentl of ( ) n no. of of permtte by jont (f). uppose there re f nepenent jont vrbles ssocte wth jont. We woul propose tht the jont permts f egrees of freeom. of (N J ) 8 J F (.) Ths s known s Grübler s formul for the egree of freeom. Where N s number of lnks nclung fxe or bse lnk, J s no. of jonts F s of t the th Jont. for sptl mnpultor n mechnsms (.) for plnr mnpultor n mechnsms A rg boy movng freely n -Dmensonl spce contns -of n ts poston n the spce cn be seprte wth three postonl n three orenttonl coorntes,.e. λ = prmeters s gven n equton (.). But n cse f λ = of then there wll be two postonl n one orenttonl coornte wll be there to expln. In ths context we cn expln no. of of n cse of rottonl jont, for exmple n ths jont we know f= n λ = therefor c=-=, mens rottonl jont reuces of of reltve movement between two lnks. The no. of of's permtte by jont n ther chrcterstc cn be etermne by the esgn constrnts mpose on boy or lnk. There re mny fferent types of jonts s

30 shown n Tble.. Among these fferent types of jont the two common jonts tht permt f= of n c= constrnts n sptl moton or nother c= constrnts for plnr moton. From Tble. bsc nottons for jonts re gven for exmple revolute n prsmtc jonts cn be enote s R n P. These jonts cn be escrbe by unt vector, whch efnes ther xs of ether rotton or trnslton. For exmple, revolute n prsmtc jonts hvng one of whle cylnrcl n Hooke jonts contn two egrees of freeoms. The versty of jonts n mny mechnsms s lrger, but these jonts re commonly use n the fel of robotcs. Tble. Dfferent types of jonts Jonts/Pr ymbol of Representton Revolute R Prsmtc P crew H s ylnrcl Hooke jont T phercl Plnr E In cylnrcl n screw, jont trnslton tkes plce n recton n rotton s bout the concent xs wth n ngle θ. Wheren, jont trnslton n rottons θ n re nepenent prmeters. Hence c wll be n f=. Therefore, nepenence of screw jont cn be explne wth the relton between, where n re the vrtons of jont n s ptch of the screw. Although screw jont s hvng two 9

31 nepenent jont prmeters n one jont ether θ or, therefore n ths cse f wll be n c=. mlrly, other jonts cn be elborte s per the egree of freeom n mpose constrnts. These nottons n representtons hve been opte throughout the text. Therefore on the bss of of the robots cn be clssfe s follows: ) Generl mnpultor Generl robots cn normlly hve -of ue to the vst pplcton n vrous fels. There re mny robots whch possess -of for exmple Fnuc -900W, where lst three jont xes ntersect t the wrst centre. The knemtcs soluton for ths clss of mnpultor cn be seprtely solve conserng frst three lnks n then lst three lnks cn be solve nepenently. A therefor soluton of nverse knemtcs wll be much eser thn the other clss of mnpultors. b) Reunnt/hyper reunnt mnpultor Knemtc reunncy of ny mechnsms rses when t hs more of thn those rgorously necessry to perform esre tsk. Most of the nustrl pplcton cn be execute by -of but f t s 7-of robot mnpultor, t cn be consere s the stnctve exmple of nherent reunncy. It s not lwys necessry tht the robot wth more of wll be reunnt, but sometmes t occurs for less of for specfc tsks, such s smple mnpultor tool postonng wthout hvng constrnts for the orentton. Hyperreunnt mnpultors for ny mechnsm occur when t hs lrger number of jonts. Its jont confgurtons of re exceee to ts tsk spce of. Therefore, 7-of or 8-of sptl mnpultor usully not consere s hyperreunnt mnpultor. A typcl exmple of hypereunnt s snke robot. In fct, reunnt mnpultors re mnly use ue to ts ncrese exterty; t my tolerte sngulrtes, jont vrble lmts, n obstcle vonce, but lso for mnmzng torque/energy for gven tsk. c) Flexble mnpultor The stnr hypothess reltng robot knemtcs, esgn of mnpultor n ynmcs s tht robot mnpultor generlly comprses of rg lnks n trnsmsson components. However t cn be ssume s stnr conton for generl pplcton whch my be effectve for less pylos or less nterctng forces n slow motons. Prctclly spekng, flexble robot mnpultor cn be useful ue to the reuce weght of movng lnks n slener esgn of lnks s well s use of complnt trnsmsson elements. Ths concept of flexblty usully hvng mjor pplcton n the re of spce robot becuse of very long lnks of mnpultor further requres resoluton of tme wth respect to elstc eformtons n lso nferor lnk weght to pylo rto long wth the 0

32 enhnce energy effcency. On the other hn, n cse of mecl surgery or nucler hzr pplctons tele-operte mnpultors epcts smlr concept lke spce mnpultor. Therefor t cn be unerstn tht n cse of flexble robot whch s hvng less control nputs s compre to number of of whch explns the esgn control prmeters for flexble mnpultor s more ffcult thn rg lnk mnpultor. Moreover, the executon of whole system wll efntely requres more number of sensors. Among these lmttons the flexble robot mnpultor unlkely use n vrous nustrl pplctons ue to the benefts of nertl ecouplng of the jont ctutor n the lnk, reuce n knetc energy consumptons n unesre collsons offere by obstcles s well s humns. ) Defcent mnpultor A robot s clle efcent robot f t possess less thn sx egrees of freeom n t cnnot postone or orent freely n spce, Aept-one ARA mnpultor s n exmple of efcent robot... lssfcton by ctuton Actutors re bsclly trnsmttng power s moton to rve rg or flexble lnks ttche to ny mechnsm or mnpultor. Actutors cn be ctegorze mnly s electrcl, pneumtc n hyrulc. There re other types of ctuton cn be consere s shpe memory lloys (MA), pezoelectrc, mgnetostrcton n polymerc. Among ll consere ctutors the bsc n most preferre ctutors re electrc whch re powere by A or D motors becuse of ther clener, precse n queter opertons s compre to other ctutors. Electrc rves re more effcent n precse t hgh spee becuse of ger box use n lso n cse of stepper motor precse moton n hgh torque re possble. However, for hgh spee n hevy lo crryng cpcty electrc motors oes not support s compre to hyrulc or pneumtc ctutors. Hyrulc rves re resonble becuse of ther hgh spee n effcent torque or power rtos. Therefore, hyrulc ctutors focuse mnpultors re mnly use for lftng hevy los. Mjor rwbcks of hyrulc ctutors re nosness, lekness of flu use n hevy pumps. Beses hyrulc ctute mnpultor Pneumtcs ctutors re smlr but t oes not hvng precse moton n ffculty n control of en effector... lssfcton by workspce In generl, workspce of ny mnpultor cn be efne s the totl volume covere by the en effector s the mnpultor fnshes mxmum possble movements. Workspce

33 cn be etermne by the lmts of jont vrbles n geometry of the mnpultor. There re bsclly two types of work spces whch re rechble n extrous; rechble workspce cn be unerstn by the totl locus pont trce by en effector n subset of these trce pont of en effector whle gvng rbtrry orentton s known s extrous workspce. But prctclly extrous workspce s sutble only for elze geometres n generlly t oes not possess for nustrl mnpultors. The bove mentone confgurtons n ther corresponng workspces re gven n Tble.. Tble. onfgurtons n workspce Types of robot tructure Jont type hpe of the workspce rtesn P-P-P ylnrcl R-P-P phercl R-R-P Revolute R-R-R ) rtesn robot rtesn robots re lso known s gntry robots, hvng three orthogonl rrngements of prsmtc jonts s shown n Tble.. Poston of wrst centre pont of rtesn robot cn be pproprtely etermne by ssocte coornte wth the three prsmtc jonts. Workspce of rtesn mnpultor

34 wll be rectngulr or cube n shpe, so tht performe work wll lwys be wthn the spce of jont moton. The robot confgurton wll be PPP lnerly rrnge three mutul xes, n the moton wll be n X, Y n Z recton. b) ylnrcl robot ylnrcl robot wll possess t lest one revolute jont long wth two prsmtc jonts (RPP) tht completely cretes cylnrcl coorntes of en effector. The workspce of ths confgurton s lmte by two concentrc structure of cylner of fnte length s shown n Tble.. Ths robot comprses of one revolute jont n bse n other two jonts hvng lner moton long Z n Y rectons. Frst jont of rotton long Z recton gves vntge to move rply n effcent pck n plce operton n ssembly. c) phercl robot phercl robots hvng frst two jonts revolute wth ntersectng xes n lst jont wll be lner or prsmtc jont (RRP) tht resembles sphercl coorntes of ll three jonts. The workspce of ths robot s lmte by two concentrc spheres s shown n Tble.. Frst lnk rottes long the Z-recton wth the bse n secon jont rottes n Y-recton whle lst jont moves left n rght lnerly overll crtes sphere envelope. ) ARA robot ARA (electve omplnce Assembly Robot Arm) robot mnpultor s bsclly esgne for ssembly tsks s t proves vertcl xs rgty n complnce n the horzontl xs. It mnly contns three revolute (-of revolute) n one prsmtc (P) jonts ltogether known s RRRP mnpultor s sown n Fgure.. In ths type robot frst three jonts re prllel to ech other n hvng ownwr recton grvty. Ths mnpultor s use mnly n eroplne prts n electronc prts ssembles. Aept one n ARA AR0 s the exmple of ths robot confgurton. Fgure. ARA robot

35 The mjor vntge of ths mnpultor s smll nstllton re n works s hgher of mnpultor. The mnml cqure re of ths mnpultor esgn les to mnmzng cost n mntennce. However, hvng less of or jonts wll be lmttons for rel worl pplcton n ton wth sngulrty, obstcle vonce n lmte workspce. e) Artculte/revolute mnpultor A robot mnpultor hvng ll three jonts revolute (RRR) s s to be rtculte or nthropomorphc mnpultor. The nthropomorphc resembles the esgn of humn hn tht nclues shouler, wst n elbow jonts. The workspce of ths type of robot s qute complex, mnly crescent-shpe crosssecton. Ths cn swept the volume n spce boune by sphercl outer surfce n consstng scllops of nner surfce to the constrnt jonts. Very wellknown exmples of ths ctegory re PUMA, KUKA, DENO, IRB etc s shown n Fgure.. Ths type of mnpultor generlly hvng -of consstng frst three revolute jont n X, Y n Z xes n lst three jont wll be ptch, yw n roll. Fgure. Revolute robot Mny serl mnpultors re esgne n regonl n orenttonl structure so tht t cn overcome the complexty of knemtc nlyss. The jont vrbles n regonl structure wll help for mjor splcement or postonng of en effector but n cse of prsmtc jont ts oesn t support for orentton of en effector. Now the revolute mnpultors cn lso be clssfe s type A, A, B, B, n D. The structures of the types of robots re shown n the Fgure.. Exmples of fferent types of robot re gven n Tble..

36 Tble. onfgurtons of -of revolute mnpultors Regonl structure Orentton structure Type: A IRb Exmple AEA Type: A 00 Exmple ABB IRb- Type: B T Exmple ncnnt Mlcron Type: B L Exmple TÄUBLI RX0 Type: Exmple Unmton PUMA 0 Type: D Exmple Fnuc-M-0A-80p.. lssfcton bse on regonl structure In cse of regonl structure of mnpultor etermnes the mjor splcement of en effector hvng three egrees of freeom. Now let us observe the frst regonl prt of mnpultor. In ths regonl prt of mnpultor both rottonl n trnsltonl cn be

37 use to poston the en effector of mnpultor. The recton of xes cn be rbtrrly n spce. But goo prctce s to poston of mechnsm wll lwys be prllel jont xes long wth fxe jont coornte frme x,y n z. Therefore the trnslton n rotton cn be represente s Rx, Ry, Rz n Px, Py, Pz, (see Tble.). Ths representton cn mke possble combntons of these sx jonts. wheres t s not lwys mportnt tht t coul mke sptl mechnsm for exmple PxPxPx or PxPyRz combntons cnnot move n t lest n one recton of the x, y n z xs. o to hve sptl mechnsm t s requre to hve moton n ll three rectons. Therefore moton of sngle jont shoul lwys be nepenent of other two jont motons. Tble. lssfcton bse on regonl structure Dfferent confgurtons RxRxRy RxRxRx RxPxRy RxPyPx PxRxPz RxRxRz RxRyPx RxPxRz RxPzPx PxRyPy RxRyRz RxRyPy RxPyRy PxRxRx PxPyRx RxRyRy RxRyPz RxPyRz PxRxRy PxPyRy RxRyRz RxRzPx RxPzRy PxRxRx PxPyPz RxRzRx RxRzPy RxPzRz PxRyRx RxRzRy RxRzPz RxPxPy PxRyRz RxRzRz RxPxRx RxPxPz PxRxPy P R P P R R P P P () R P (b) R (c) R R R () (e) Fgure. Regonl mechnsm of robot mnpultors () PRP cylnrcl mnpultor, (b) PPP rtesn mnpultor, (c) PRR ARA mnpultor, () RRP sphercl mnpultor n (e) RRR revolute mnpultor. As per Tble., only few of these combntons re use to form nustrl mnpultor. In generl there re fve types of postonng mechnsm re foun n nustrl mnpultor s shown n Fgure.. Another prt of mnpultor s orentton prt tht s requre to hve t lest -of jonts to cheve esre tsk n combnton of three rottons vrbles cn yel 7 fferent confgurtons of wrst. However, confgurton wth consecutvely perpenculr xes cn be consere s

38 shown n Fgure.. In cse the frst rotton s bout x xs then next lgnment of xs shoul be n y or z xs recton. If the next rotton s bout y xs, then lst xs of rotton shoul be bout z or x xs. onserng these crter there cn be fferent confgurtons, only the fference wll be there on the bss of ttchment orentton wth movng lnk of the mnpultor... lssfcton by moton chrcterstcs Robot mnpultors cn lso be clssfe ccorng to ther nture of moton such s; Robot mnpultor cn possess three fferent chrcterstcs of moton nmely plnr, sphercl n sptl. A mnpultor wll be known s plnr f the jont ssocte wth the lnks rbtrry trnsltes n rottes n the plne. In plnr mnpultor ll movng lnks performs plnr moton tht s ll jont xs re prllel to ech other. Prsmtc n revolute jonts re only llowble lower prs for plnr mechnsm. The moton of plnr jonts re lmte to E () group, consere s D subgroup of E (). But n cse of sphercl mnpultor ll the lnks ccomplsh sphercl moton wth respect to common fxe pont n ll other moton of jonts cn be etermne by the rl projecton of unt sphere. Revolute jonts re lmte to the constructon of sphercl lnkge tht cn be use s pontng evce. On the other hn, mnpultor movng n -mentonl spce or belongs to E () group n possesses three coorntes s s to be sptl mnpultor. Bse on the observton of growng tren of nustrl mnpultor pplcton the commonly use mnpultor re sptl...7 lssfcton by pplcton Regrless of structure, of n workspce, mnpultor cn lso be ctegorze s per ther pplcton. It cn be clssfe s follows; ) Assembly robot mnpultor b) Unerwter c) pce ) Agrculture e) Mnng f) urgcl n rehbltton g) Domestc h) Euctonl In cse of ssembly robot mnpultor the mjor pplcton wll lwys pck n plce, long/unlong, welng, pntng, nspecton, smplng, mnufcturng etc. As per confgurton n esgn, mostly nustrl robots re nthropomorphc, whch nclues shouler, n elbow n wrst. Therefore, fnlly most of the mnpultor possesses - 7

39 of to cheve esre poston n orentton the bsc fference s the pplcton. The bsc objectve of mnpultor s to hve hgh resoluton, energy effcent n cn crry mxmum lo. Hence, ll stngushe mnpultor possesses fferent knemtc structures.. Bsc knemtcs Ths secton scusse some bscs of knemtcs of rg boy n further ntrouce fferent types of mechnsm n prmeters ssocte wth t. Knemtc hn my consst of rg/ flexble lnks whch re connecte wth jonts or knemtcs pr permttng reltve moton of the connecte boes. For exmple, rottonl jont cts s hnge n llows only reltve rotton between the connecte boes bout the xs of the jont. The reltve movements llowe by jont re referre to s the jont vrbles or the nternl coorntes. The rottonl jont hs only one jont vrble n tht s the reltve rotton between the connecte boes. As we know bout the fferent types of knemtc chns for exmple serl, prllel or hybr whch my be open, close or brnche. For the postonng of en effector or bse t s requre to hve unerstnng of knemtcs of rg boy systems. The esgn of the lnks n jonts of ny mechnsm eces the orentton or postonl propertes tht ffect the overll knemtc chn. There re bsclly two types of knemtcs of ny mechnsm nmely forwr knemtcs n nverse knemtcs. The forwr knemtcs problem s concerne wth the reltonshp between the nvul jonts of the robot mnpultor n the poston n orentton of the tool or eneffector. The forwr knemtcs of ny mnpultor or mechnsm cn be etermne wth gven jont vrbles tht yel the poston n orentton of en effector. The jont vrbles my be revolute or prsmtc epenng of types of jonts use. On the other hn the secon problem of knemtc s resoluton of nverse knemtcs. Inverse knemtcs cn be efne s resoluton of jont vrbles n terms of gven en effector poston n orentton. ystemtc n generlze wys of knemtc nlyss re vectors n mtrx lgebrs tht represents n escrbe the locton of en effector n jont vrbles wth respect to efne reference frme. As we know tht the jonts cn be rotte or trnsltes so there s bsc mtrx lgebr known s X rotton mtrx s use to resolve the poston n orentton of en effector or tool. Ths rotton mtrx further mofe wth X homogeneous trnsformton mtrx to evlute the trnslton of the lnks n -mentonl spce. Ths representton concept ws frst pple by Denvt- Hrtenberg. 8

40 The secon problem ssocte wth robot mnpultors s nverse knemtc soluton. In orer to clculte the exct poston n orentton of the en effector of robot mnpultor to rech ts tsk the nverse knemtcs soluton s mntory. The nverse knemtcs of mnpultor s essentl not only for esgn synthess but lso to rech esre poston. The mjor problem ssocte wth nverse knemtc formultons s computtonl n mthemtcl complexty ue to hgher egree of polynoml hlf tngent equtons whch oes not gurntee close form soluton. Ths problem s mn re of reserch now y n the fel of nmtons n moleculr mthemtcl moellng. Overll t cn be summrze tht there re two bsc problems of knemtcs whch re forwr n nverse knemtcs. Now s t sgnfcnt to escrbe fferent prmeters tht crete knemtc n mthemtcl moellng of vrous mnpultor of ny mechnsm. Ths secton pertn only bref ntroucton of egree of freeom (of) n vrous types of jonts tht ltogether conclue the mechnsm of system wthout conserng ny forces/torque. In the lter chpter fferent methoologes to obtn knemtcs solutons wll be scusse n etl.. Motvton As we know tht n the trck of knemtc nlyss, rottonl, trnsltonl, DHlgorthm n homogeneous mtrces hve shown ther mportnce n the pplcton of postonl nlyss of fferent mnpultors. From mny eces these metho hve been opte by vrous reserchers n mplemente n fferent number of mnpultors. However, ensurng the bsence of proper mthemtcl formultons wth less computtonl n mthemtcl cost whch les to ecresng n mny pplctons, where quck clcultons re requre. These technques fl to prove when the mnpultor hvng hgher number of of's. In generl when there re hgher of mnpultor the nverse knemtc formultons re much more ffcult ue to nonlner, tme vryng n trnscenentl functons. There re mny other tools n technques re vlble to solve nverse knemtc problem for exmple lgebrc, Jcobn, or geometrc, nlytcl, pseuo nverse Jcobn etc. These methos re conventonl n they o not prove exct soluton. On the other hn, lterntve of these technques for representng n solvng knemtcs problem re quternon lgebr, Le lgebr, exponentl lgebr, epslon lgebr n screw theory whch re beng use from mny yers ue to less mthemtcl opertons. o the quternon lgebr s much more powerful metho for resolvng knemtc problem of ny mnpultor. Quternon cn be use for rotton s well trnslton of rg boy n Eucln spce. 9

41 onserng the complextes nvolve n the process of moellng n consequently solvng the nverse knemtc problem for chevng precse, optmze n fster soluton for rel tme pplcton. The present reserch problem n esgn, the reserch ssues wll prmrly focus on selectng/evelopng n pproprte tool for chevng the objectves fter vltng them on vrous confgurtons of nustrl robots. On the other se of these conventonl technques, ntellgent or soft computng technques re wely use to fn out the nverse knemtc solutons. Ths ntellgent technque nclues rtfcl neurl network, hybr ANN, fuzzy logc, hybr fuzzy, metheurstc lgorthms n bologclly-nspre pproches. In pst eces, mny others hve opte these technque becuse of ther less computtonl n mthemtcl cost. These technques re useful when the mnpultor hvng hgher number of of's where generlly conventonl metho fls. o the ultmte gol of ths ssertton s to fn out nverse knemtc soluton usng these technques n to evelop novel metho for resolvng nverse knemtc problem for ny confgurton of robot mnpultor.. Bro objectve The mjor objectve of ths ssertton s to resolve nverse knemtc problem. As per survey n nlyss of vrous ltertures n ths fel of mnpultor knemtcs recommens tht there s obvous requrement of some novel technque for solvng hgher of mnpultor knemtcs. It s lso requres to prouce nverse knemtc soluton effcently n shoul be cpble of onlne control of mnpultor. Therefore, ths work s plnne wth followng mjor objectves: ) To crry out crtcl stuy of fferent tools n technques sutble for solvng nverse knemtc problems. ) To evelop the nverse knemtc moel of vrous robot mnpultors n to opt some exstng technques for soluton of nverse knemtcs of selecte robot mnpultor confgurtons. ) Development of new lgorthm n mthemtcl moel for resolvng n smultng nverse knemtcs. ) To nlyze the effcency of newly evelope metho n comprson wth the obtne soluton through other exstng technques. ) To recommen the pproprte technques for solvng nverse knemtcs problem for vrous pplcton. 0

42 .7 Methoology Knemtc nlyss n synthess of plnr or sptl mnpultors lwys nee to follow through the nonlner equtons whch cn be complex n tme consumng. The conventonl methos re less effcent whch cn be the greter objectve for ny resercher to evelop novel metho to overcome the stte problem. onserng the bove stte objectve, knemtc reltonshp n mthemtcl moellng s requre to evelop. Therefore, to ccomplsh the fores mjor objectves of ths reserch work n to resolve perfect soluton of nverse knemtc one shoul evelop effcent metho. The opte methos n steps for chevng objectve hve been plnne s follows: Revew of lterture: onserng vrous confgurtons of revolute n prsmtc jonts of mnpultors n ther clssfcton on the bss of ther structures, mechnsm, ctutons, workspces, moton propertes, pplctons etc. hve been stue. Anlyss of the lterture survey wth the prme mportnce of mthemtcl moellng n knemtcs nlyss of robot mnpultor long wth the problem ssocte wth the evelope technques hs been one. Lterture revew hs been one relte to fferent methos lke lgebrc, nlytcl, ntellgent technques n optmzton lgorthms etc. onfgurtons of mnpultor: On the bss of lterture survey t hs been elberte ther outcome n ssocte problem so s to select pproprte moel of mnpultor. trtng from -of mnpultor up to 7-of reunnt mnpultor hs been tken for the resoluton of nverse knemtc. Dfferent confgurtons for -of mnpultor n -of revolute mnpultor hve been propose n mong them few confgurton hs been selecte for knemtc nlyss. Mthemtcl moellng n knemtc nlyss: Dfferent confgurtons of robot mnpultor hve been consere n ther mthemtcl moellng s presente. All consere mnpultor belongs to the ctegory of serl mnpultor n fferent confgurtons for -of n -of mnpultor hs been nlyse. Denvt-Hrtenberg lgorthms hve been use for knemtcs formulton n smulton of fferent jont confgurtons of robot mnpultor n lter the obtne soluton of nverse knemtcs hs been compre wth quternon lgebr. The forwr n nverse knemtcs of opte confgurtons hs been one long wth ther workspce nlyss n etle ervton of knemtcs. Intellgent pproch: Artfcl neurl network (ANN), fuzzy logc n hybr technques from the soft computng omn hve been wely use n lst

43 eces. These ntellgent technques o not requre hgher mthemtcl formulton n re cpble of solvng NP-hr, nonlner n hgher egree of polynoml equtons. As per revew of lterture fferent moels of ANN, ptve neurl fuzzy nference system (ANFI) hs been opte for the resoluton of nverse knemtcs of robot mnpultor. Although these ntellgent technques re not new n ths fel but few selecte moels of ANN long wth ANFI n hybr ANN methos hs been opte for the comprson. There re fferent optmzton technques lke Prtcle swrm optmzton (PO), genetc lgorthm(ga), rtfcl bee colony (AB), bogeogrphy bse optmzton(bbo), techers lerners bse optmzton(tlbo) etc. hve been pple for trnng of mult-lyer perceptrons (MLP) neurl network for the precton of nvers knemtc soluton of robot mnpultor. Optmzton lgorthm: Dfferent optmzton lgorthms lke GA, BBO, PO, TLBO n AB hve been opte for the soluton of nverse knemtc of robot mnpultor n novel rb ntellgence bse optmzton lgorthm (IBO) hs been propose. These lgorthms re compre wth new evelope IBO lgorthm. These opte optmzton lgorthms oes not requres ny computton of Jcobn mtrx only t nees forwr knemtc equtons whch cn be esly evelope. Dscusson n recommentons: onverston bout the results obtne through the opte methos n propose metho for robot mnpultor knemtcs. multon results for knemtcs n workspce nlyss hve been resse. Future recommentons for the opte confgurton of mnpultor n scope of the future work conserng mprovements of the qulty n effcency re gven..8 Orgnzton of the thess urrent chpter s the Introucton prt of the ssertton tht proves bref escrpton of hstory of evoluton of robots, types of mnpultor, clssfctons n pplcton n vrous fels. Forthcomng chpters prt from ntroucton chpter re orgnze s follows: hpter elvers revew of lterture on the bss of vrous spects of the robot mnpultor lke mechnsm, ctuton, workspce nlyss, moton types, fferent components consere, pplcton, ntellgent controls n optmzton. ome of the sgnfcnt ltertures re summrze n tble n bref explntons of the outcome n efcts wth respect to mnpultor n confgurtons re scusse. Fnlly the

44 objectves of the reserch work re etermne n explne on the bss of lterture nlyss. hpter proves the bref escrpton of selecte confgurtons of nustrl mnpultors for nverse knemtc nlyss. In lter secton, fferent methoologes to solve nverse knemtc problem s scusse n bref. hpter elvers the knemtc nlyss n mthemtcl moellng of vrous confgurtons of robot mnpultor. A bref scuss of vrous conventonl technques lke lgebr, nlytcl metho, tertve metho, numercl metho, geometrc metho, homogeneous mtrx, DH lgorthm n quternon lgebr re presente. lssfcton of -of n -of serl mnpultors long wth DH prmeters n ther mthemtcl moellng hs been scusse. The nverse n forwr knemtcs of opte mnpultor s erve usng opte metho. hpter proposes vrous ntellgent technques lke ANN, ANFI n hybr ANN for the precton of nverse knemtc soluton of robot mnpultor. Dfferent types of ANN moels lke mult-lyere perceptron (MLP), polynoml perceptron network (PPN) n P network re explne n bref n ther pplcton towrs the soluton of nverse knemtcs hs been presente. MLP moel s hybrze wth mny optmzton technques lke GA, GWO, PO, TLBO n propose IBO lgorthm to ncrese the performnce of MLP network. The en effector poston s consere s nput for the trnng of ANN moels n ANFI trnng s lso complete smlr to ANN trnng. Applcton of these lgorthms n strteges to use ANN n ANFI s resse. hpter scusses bout the opte optmzton lgorthms for the soluton of nverse knemtcs of robot mnpultors. In ths chpter forwr knemtcs equtons re use to fn out the jont vrbles of robot mnpultor usng Euclen stnce norm. rb Intellgence Bse novel Optmzton lgorthm (IBO) hs been propose n etl for the soluton of nverse knemtcs of robot mnpultor. For the comprson of the evelope optmzton lgorthm vrous metheurstc lgorthms re brefly explne. In hpter 7 presents the knemtc results cheve through ll opte technques n comprson hs been me wth other exstng technques. Forwr n nverse knemtcs long wth the workspce nlyss n jont ngle behvour hs been resse n compre. Progrmme output n the form of tbles n grphs re epcte n ths chpter. hpter 8 presents the conclusons of the ssertton n future reserch gunce wth summry of contrbuton.

45 .9 ummry In the current chpter, the generl synopss of the fferent types of robot mnpultor, clssfctons, hstory of evelopments re presente. onfgurtons of -of revolute mnpultors n combnton of -of mnpultor re presente. The chronologcl progresses of some selecte mnpultors re presente n lso current sttus hs been brefe. Bsc pplctons of knemtcs n objectves re lso scusse n ths chpter.

46 hpter REVIEW OF LITERATURE. Overvew Wth vncement of robot technology n ever ncresng pplcton of robots n vrous wlks of lfe, robotc reserch hs gnng pprecble momentum over the yers. Newer confgurtons, smrt behvours, utonomous robotcs, hgh level ntellgence, uncertnty n envronments hve been the vrous res for resercher n robotcs. All these res re nturlly connecte wth the subject of robot knemtcs; both forwr n nverse. Inverse knemtcs soluton for serl mnpultor s ffcult tsk, becuse the soluton s not unque ue to nonlner, uncertn n tme vryng nture of the governng equtons. There re vrous softwre's n lgorthms to smplfy the nverse knemtcs of robot mnpultor. Durng 980s wrst orentton ws ecouple from the trnslton by the rm by usng wrst xes tht ntersect wth the rm xes. But the mjor problem sngulrtes wth the robot rm were no longer bck rven, t lmts wth the structure of the mnpultor. There re vrous technques for solvng nverse knemtc problem. nce mny methos hve been presente to solve the IK problem such s homogeneous trnsformton metho, geometrc metho, ul number pproch n contnuton metho. However, the problem nvolves the solvng of hghly non-lner equtons. Mny ppers hve presente lgorthms gvng nlytcl solutons for the Inverse Jcoben. The formulton of the Jcoben mtrx n ts nverse hs to be one wthn very short spn of tme for rel-tme mplementtons. ome mthemtcl methos (such s MAYMA, REDUE, MP n EGM) re well- known, effcent tools n terms of ther spee n ccurcy. In the re of robotcs, reserchers such s (Krcnsk, 98 n Vukobrtovc, 98), (Morrs 987), (Hussn n Nobe, 98), n (Ts n hou, 989), hve use these kns of tools for ervng the rect knemtcs, Jcobn, n reverse Jcobn close- form equtons. Anlytc solutons, however, re only use for smple robot mnpultors.

47 . urvey of tools use for nverse knemtc soluton Beses bove mentone pproches, reserchers re up to evelopng newer technques whch woul mke the process eser n fster. In the current reserch fferent methoologes use by reserchers hve been stue. They re s follows: Anlytcl soluton Itertve soluton Geometrc soluton Quternon lgebr Theory of screws Exponentl rottonl lgebr Le lgebr Artfcl neurl network (ANN) Hybr ANN Aptve neuro-fuzzy nference system (ANFI) Genetc lgorthm multe nnelng Prtcle swrm optmzton Bee lgorthm Fuzzy lernng lgorthm Neuro-fuzzy Fuzzy-neuro Bse on comprehensve survey of lterture, lst of some the mjor work one n the re s presente n Tble..

48 Tble. Lst of some mportnt ltertures l. Yer Author Ttle Type ontrbuton Propose representton metho bse on screw splcement n me ther comprson on the bss of computtonl cost. In ths work they hve etermne the 990 Fun Pul [] n A computtonl nlyss of screw trnsformtons n robotcs PUMA 0 rottonl n trnslton representton of lne for the pplcton of generl splcement of rg boy. They hve compre four fferent mthemtcl formulztons whch rect ffects the rotton n trnslton of rg boy. The propose methos re ul orthogonl mtrx, ul unt quternon, ul specl untry mtrx n ul Pul spn mtrx. Propose work s bse on the nverse knemtc soluton 990 Fun et l. [] On homogeneous trnsform, quternons, n computtonl effcency PUMA 0 of robot mnpultor usng quternon vector pr bse metho. In ths work the propose metho s pple to solve nverse knemtc of the PUMA robot mnpultor n comprson hs been me on the bss of computtonl cost. 99 Mts et l. [] Optmzton of robot lnk moton n nverse knemtc -of revolutesptl n Propose nverse knemtc soluton of -of reunnt robot mnpultor bse on conventonl optmzton 7

49 soluton conserng collson reunnt metho. In ths work penlty functon optmzton vonce n jont lmt. metho opte for the problem resoluton. Moreover forwr knemtc soluton s one usng stnr nlytcl metho whch s lter use to formulte the objectve functon for the propose optmzton lgorthm. Propose nverse knemtc soluton of reunnt mnpultor usng mofe genetc lgorthm. They hve mplemente some ssumptons; frst they consere tht olvng the nverse knemtcs the mnpultor my be reunnt n rtculte. Then problem of reunnt robots the secon ssumpton s tht the mnpultor s n 998 Nerchou [] opertng n complex Pum movng object of ts workspce, n lst ssumpton s envronments v mofe tht they re not conserng ynmcs of the genetc lgorthm mnpultor. Therefter, genetc lgorthm s use n two fferent mnners, frst jont splcement ( ) error mnmzton n the secon pproch s bse on postonl error of en effector. 999 Ozgoren [] Knemtc nlyss of mnpultor wth ts poston n velocty relte sngulr confgurtons -of revolute Propose nverse n forwr knemtcs of generl R mnpultor conserng both poston n velocty usng exponentl rotton mtrx metho. They hve lso nvestgte the sngulr confgurton relte to the nverse poston nlyss terme s (POs).e. poston 8

50 relte sngulr confgurtons n other wth the velocty s terme s (VEOs). Propose nverse knemtc soluton of -of robot An mprove pproch to the mnpultor usng structure rtfcl neurl network 000 Krlk n Ayn [7] soluton of nverse knemtcs problems for robot -of revolute bse metho. In ths work they hve use DH-lgorthm to formulte forwr knemtc equton so s to mnpultors complete the tset for trnng ANN moel n the opte ANN moel s MLP. Proposenverse knemtc soluton of n -of plnr 7 00 Her et l. [8] Approxmtng robot nverse knemtcs soluton usng fuzzy logc tune by genetc lgorthms -of n -of plnner robot mnpultor usng fuzzy logc together wth the genetc lgorthm. They hve use trngulr membershp functon for fuzzy logc n center of grvty s use for the efuzzfcton. These prmeters re lter tunes by genetc lgorthm for the surety of exct nverse knemtc soluton Rue [9] Mnpultor knemtc error moel n clbrton process through quternon-vector prs PUMA 0 Propose nverse knemtc soluton of PUMA 0 robot mnpultor usng quternon vector pr bse metho. In ths work they hve clculte the geometrc error for ech jont vrbles n lnk. They hve use fferentl lgorthm for the resoluton of nverse knemtc to 9

51 cheve they formulte the objectve functon s poston error n orentonl error. Propose nverse knemtc soluton of R robot 9 00 Koker et l. [0] tuy of neurl network bse nverse knemtcs soluton for three-jont robot -of revolute mnpultor usng rtfcl neurl network technque. In ths work forwr knemtc s resolve usng nlytcl soluton whch s lter use to generte nput tset for the ANN trnng. Propose nverse knemtc soluton of -of robot omprson of nverse mnpultor usng rtfcl neurl network technque. In 0 00 Bngul [] knemtcs solutons usng neurl network for r robot -of revolute ths work forwr knemtc s resolve usng nlytcl soluton whch s lter use to generte nput tset for mnpultor wth offset the ANN trnng. Bck propgton lgorthm s use to clculte the output error n ths work. Propose nverse knemtc soluton of -of robot 00 Xu et l. [] An nlyss of the nverse knemtcs for -of mnpultor PArm (PRRPP) -of mnpultor usng nlytcl metho. In ths work DHlgorthm s use to resolve the forwr n nverse knemtc of opte mnpultor lter they hve scusse on trjectory plnnng n sngulrty nlyss of the mnpultor. 0

52 Propose nverse knemtc soluton of -of robot 00 Koker [] Relblty-bse pproch to the nverse knemtcs soluton of robots usng elmn s networks -of revolute mnpultor usng relblty bse rtfcl neurl network technque. In ths work forwr knemtc s resolve usng nlytcl soluton whch s lter use to generte nput tset for the ANN trnng. Bck propgton lgorthm s use to clculte the output error n ths work. 00 Myorg n nongboon [] Inverse knemtcs n geometrclly boune sngulrtes preventon of reunnt mnpultors: n rtfcl neurl network pproch -of revolute plnr reunnt Propose rtfcl neurl network technque to solve nverse knemtcs of the -of revolute plnr mnpultor n lso clculte the effectve geometrclly boune sngulrtes preventon of reunnt mnpultors. Propose nverse knemtc soluton of -of nustrl 00 Ayn n Kucuk [] Quternon bse nverse knemtcs for nustrl robot mnpultors wth euler wrst -of revolute mnpultor wth Euler wrst usng quternon vector pr metho. They hve gven etl ervton of forwr n nverse knemtc of R, RN n N type robot mnpultors. 00 Hsn et l. [] An ptve-lernng lgorthm to solve the nverse knemtcs problem of.o.f serl FANU M70 Propose ptve lernng pln of ANN for the soluton of nverse knemtc of -of mnpultor. Moreover they hve tre to resolve sngulrty n uncertnty problem

53 robot mnpultor. of the opte confgurton of the mnpultor. In ths work ANN moel hve been trne usng nlytcl soluton of the opte mnpultor. Generte tsets usng knemtcs equtons re use to trne n test the opte moel of ANN. They hve conclue tht the propose moel of ANN oes not nee to hve prevous nformton of the knemtcs of the system tht lerns through the ANN moel pplcton. Propose nverse knemtc solutons of -of PUMA mnpultor for the mjor splcement propose. In ths 00 Tbneh et l. [7] A genetc lgorthm pproch to solve for multple solutons of nverse knemtcs usng ptve nchng n clusterng. -of revolute work they hve opte genetc lgorthm wth ptve nchng n clusterng. Genetc lgorthm's prmeters re set by ptve nchng metho whch s lter requre the forwr knemtc equtons for the soluton of nverse knemtc of opte mnpultor. Forwr knemtc s smply clculte by stnr nlytcl metho. Therefter for processng the output flterng n clusterng s lso e to the genetc lgorthm Xe et l. [8] Inverse knemtcs problem for -of spce mnpultor bse on the theory of screws -of revolute Presente nverse knemtc soluton of -of mechncl rm wth the pplcton of free flyng spce usng screw lgebr bse metho. In ths work they hve complete the smulton moel for opte mnpultor long wth

54 knemtc nlyss. Propose nverse knemtc soluton of -of revolute robot mnpultor usng new effcent lgorthm bse on Husty et l. [9] A new n effcent lgorthm for the nverse knemtcs of generl serl r mnpultor -of revolute nlytcl metho. In ths work they hve use elmnton technque to reuce the complexty of nverse knemtc formulton. They hve use generl -of revolute robot mnpultor geometry for the elmnton nformton Prk [0] omputtonl spects of the prouct-of-exponentls formul for robot knemtcs -of sptl revolute Propose forwr knemtc nlyss of -of revolute sptl robot mnpultor usng prouct of exponentl lgebr bse metho. In ths work they hve lso focuse on the clculton of Jcobn mtrx usng POE metho Phm [] Lernng the nverse knemtcs of robot mnpultor usng the bees lgorthm -of revolute Propose nverse knemtc soluton of -of robot mnpultor usng Bee lgorthm. In ths work they hve compre three fferent methos lke evolutonry lgorthm, neurl network bck propgton metho n bee lgorthm. Neurl network structure s optmze by usng bee lgorthm to prect jont vrbles of the robot

55 mnpultor. Propose nverse knemtc soluton of -of n -of 008 Alvnr n Ngm [] Neuro-fuzzy bse pproch for nverse knemtcs soluton of nustrl robot mnpultors -of revolute n -of revolute plnr mnpultor usng ptve neurl fuzzy nference system (ANFI). In ths work, they hve opte ugeno type fuzzy rchtecture n hybrze wth smple neurl network for the precton of nverse knemtc of plnr mnpultor. Propose nverse knemtc soluton of -of revolute robot mnpultor usng rel tme genetc lgorthm. In 008 Albert et l. [] Inverse knemtc soluton n hnlng r mnpultor v rel-tme genetc lgorthm -of revolute ths work en-effector splcement form ts ntl pont to esre pont hs been optmze usng genetc lgorthm. Genetc lgorthm crossover selecton s bse on new metho whch s known s ynmc mult-lyere chromosome (DM) to prouce two offsprng's. The GUI smulton hs been verfe wth GA n DM. 008 Dutr [] New technque for nverse knemtcs problem usng smulte nnelng -of revolute Propose nverse knemtc soluton of -lnk plnr mnpultor usng smulte nnelng metho. In ths work stnr nlytcl soluton of forwr knemtc s presente usng forwr knemtc equton the

56 splcement bse error mnmzton objectve functon s use for the smulte nnelng pproch. 009 rylz n Temelts [] oluton of nverse knemtc problem for serl robot usng quternons -of revolute Propose nverse knemtc soluton of -of revolute robot mnpultor bse on quternon n the frmework of screw theory. In ths work they use quternon wth the screw theory to reuce the computtonl cost for nverse knemtcs ervton. 009 Ayz n Kucuk [] The knemtcs of nustrl robot mnpultors bse on the exponentl rottonl mtrces -of revolute Propose nverse n forwr knemtcs of -of robot mnpultor bse on exponentl rotton mtrx metho. In ths work they hve use the exponentl bse metho for ervton of nverse knemtcs of N n R type robot mnpultor. Propose nverse knemtc lernng of -of plnr n ARA mnpultor usng neuro-controller. 009 Mrtın [7] A metho to lern the nverse knemtcs of mult-lnk robots by evolvng neuro-controllers ARA Furthermore, they hve presente the some ssues of neurl network lernng such s clsscl supervse lernng scheme whch generlly converse n locl optmum soluton. Therefore they hve pple neuroevoluton lgorthm for the globl optmum soluton of the nverse knemtcs of the selecte mnpultor. In ths work DH-lgorthm s use to generte the nput t set

57 for the neurl network lgorthm. They hve reuce the rwbck of the grent escent lernng of ANN moel wth the help of evolutonry lgorthm. Propose mthemtcl moellng of -of robot mnpultor usng conventonl metho. In ths work they 7 00 Wenjun et l. [8] Numercl stuy on nverse knemtc nlyss of R serl robot -of revolute hve focuse on the nverse knemtc n forwr knemtc soluton of robot mnpultor. Lter secton els wth the pplcton of genetc lgorthm for the optmzton of jont vrbles of the opte mnpultor. Propose MLP n RBF neurl network moel for the soluton of nverse knemtc of the -of serl mnpultor. In ths work, fuson pproch of these omprson of RBF n MLP ANN moels s use wth the forwr knemtcs of the hrwr neurl networks to solve mnpultor. Forwr knemtcs equtons re use to 8 00 n Bbu nverse knemtc problem for -of revolute generte the t for trnng opte moels of ANN. [9] R serl robot by fuson They hve propose the rtesn pth to be followe by pproch the mnpultor en effector usng the generte ANN nverse knemtc soluton. KUKA -of mnpultor s teste wth the obtne results wheren DH-lgorthm s use to generte the nput for the ANN moels.

58 Artfcl neurl network-bse Propose nverse knemtc soluton of -of revolute 9 00 Hsn et l. [0] knemtcs Jcobn soluton for serl mnpultor s pssng through sngulr -of revolute robot mnpultor usng rtfcl neurl network bse technque. In ths work they hve lso focuse on the sngulrty vonce usng Jcobn bse metho confgurtons together wth the ANN pproch. Propose vrtul moel of n grculturl robot for frut Knemtcs smulton of n hrvestng n ther knemtcs nlyss usng DH 0 00 u [] e frut-hrvestng -of lgorthm. In ths work, the nverse knemtc s obtne mnpultor bse on ADAM usng lgebrc metho n smultons re crre out usng ADAM. Assste reserch n Propose nverse knemtc soluton of -of revolute 0 Olru et l. [] optmzton of the proper neurl network solvng the -of revolute robot mnpultor usng neurl network technque. In ths work DH-lgorthm s use to clculte the forwr nverse knemtcs problem knemtc of the opte mnpultor. Propose nverse knemtc soluton of -of revolute 0 Rmírez n Rubno [] Optmzton of nverse knemtcs of r robotc mnpultor usng genetc lgorthms. -of revolute sptl mnpultor usng genetc lgorthm. Forwr knemtcs formulton hs been complete by usng DHlgorthms n homogeneous mtrx multplcton bse metho. Ftness functon for the nverse knemtc soluton s bse on the en effectors ntl n esre poston error whch s lso known s Euclen stnce 7

59 norm. Propose nverse knemtc soluton of the serl 0 Zhng [] A psgo-bse metho for nverse knemtcs nlyss of serl ngerous rtcles sposl mnpultor Moble robot wth -of revolute mnpultor ngerous rtcles sposl mnpultor wth multple egrees of freeom usng prtcle swrm gene optmzton lgorthm. In ths work poston n orentton error of en effector s use s n objectve functon for trtonl PO n mofe PGO metho. A genetc lgorthm pproch Propose hybr pproch whch s combnton of to neurl-network-bse neurl networks n evolutonry technques (genetc 0 Köker [] nverse knemtcs soluton of tnfor robot lgorthms) to obtn more precse solutons. Three robotc mnpultors bse on Elmn neurl networks were trne usng seprte error mnmzton trnng sets. Propose nteger nverse knemtc soluton of mult-jont 0 Morsht n Tojo [] Integer nverse knemtcs metho usng fuzzy logc -of revolute robot mnpultor usng fuzzy logc bse metho. They hve evlute the effcency of the opte technque n teste t for trjectory generton n control pplcton 00 Perez n Mcrthy [7] Dul quternon synthess of constrne robotc systems,n -of Propose ul quternon lgebr bse knemtc synthess of constrne robotc system. They hve propose ths metho for one or more serl chn mnpultor conserng both prsmtc n revolute 8

60 jonts. In ths reserch they hve use DH lgorthm n successve screw splcement for etermnng the jont vrbles for the resoluton of en effector poston. Then ul quternons re use to efne the trnsformton mtrces obtne through DH lgorthm to smplfy the esgn formultons of fferent types of mnpultors Benezu et l. [8] ymbolc computton of robot mnpultor knemtcs 7-of nthropomorphc ymbolc robot rm tool softwre s ntrouce to solve nverse knemtc problem mth n Lpkn [9] Anlyss of fourth orer mnpultor knemtcs usng conc sectons -of revolute Introuce new technque bse on fourth orer nverse knemtc soluton. In ths work soluton of nverse knemtcs problem s consere s pencl of concs. 9

61 tuy confrms tht the number of reserch publctons whch ppers n vrous journls, conference proceengs n techncl rtcles verfy vrous spects of nverse knemtc nlyss of robot mnpultor. Inverse knemtc soluton of robot mnpultor cn be clssfe on the bss of fferent methoology. A lot of lterture survey hs been one regrng ths re, some of whch re scusse s follows. ) tructurl nlyss of mechnsm ) onventonl metho for knemtcs ) Intellgent or soft computng pproch v) Optmzton pproch.. tructurl nlyss of mechnsm The mjor m of ths lterture survey s lmte to the mechnsm serl, prllel or hybr mnly expecte for moblty's of the knemtc chns. As we know tht of or moblty of ny mechnsm or mnpultor s the bsc pproch for clssfcton or knemtc nlyss. Therefore t s requre hvng unerstnng of fferent wy of of or moblty for vrous mechnsm n ther formultons. On the other hn, workng bltes of mnpultor or ny mechnsm cn be evlute from ts of/moblty. A generl rg boy n spce hvng -of tht s the mxmum know of of the system. However, there re numerous crter for the lterture survey but typclly one cn go for the of/moblty nlyss of fferent mechnsm or structure of the mnpultor. The hstory of the structurl nlyss relte to moblty n bout the no. of nepenent knemtc chns ws one by L. Euler. Then fterwr n 9 th century, the frst mechnsm nlyss n structurl formul ws generte by [0]-[] s epcte n Tble.. The bsc concept of of s the totl number of nepenent loops (l), of of the mechnsm (M), number of jonts (j), movng lnks (n) of of knemtc prs (f), jont constrnts (s), no. of pssve of (Jp), no. of over closng constrnts (q), loop moton vrbles ( ) etc. Therefore bref lterture survey hs been one relte to structurl formul n the prmeters re presente n Tble.. Lter n the 0 th century structurl formul n smple groups hve evelope by []-[] s gven below n Tble.. Furthermore severl novel concept h been generte for the problem of confgurton nlyss n esgn synthess of mechnsm n mnpultors such s screw prs (c), some confgurtons wth zero of (M=0), no of vrble length lnks (n v ), generl vrble constrnts ( k ) n the close loop constrnts ( ). k k Afterwrs n 0 th century generl mthemtcl moellng of the etermnton of of for ny mechnsm h been cheve by []-[7] s shown n Tble.. Therefore followng up these reserch severl new prmeter for clculton of of of mechnsm 0

62 h been ntrouce these new prmeters re screw system for close loop (r), no. of nepenent of's ( k ), reltve splcement of jonts (m), coeffcent mtrx rnk (r(j)), new formul for nepenent loops L jb B cb where j B totl no. of jonts, c b s totl number of fxe lnks n B totl no. of movng lnks. Now n strtng of st century, rstc evelopment of mechnsm n mnpulton n the fel of robotcs hs been shown. There re severl new prmeters n structurl formul relte to the rel worl pplctons n mplementton hs been shown [7]- [7]. They hve clculte of of fferent mechnsm n ntrouce new prmeters for knemtcs n structurl nlyss. New formul for number of nepenent loops were L=c-B, n smple structurl formul for mechnsm (c B), where, c cb ch cl, l c. The knemtc formultons of the mechnsm or mnpultor re evlute through number of tsk postons n ther knemtcs to fn out the esgn equtons. These formuls re hvng both structurl n jont prmeters s unknown. These esgn equton re mnly bse on the knemtc nlyss n fferent prmeters. In ths re of reserch urng 0 th century [] evelope bsc theory of open loop serl chn n then ths ws utlze for fferent clssfctons of the structure. Therefter, ths theory ws nlyse by [7] for structurl synthess n knemtc nlyss. Ths problem of structurl synthess hs been one for the close loop problem. Ths clssfcton ws me on the bss of number of movng or fxe lnks, close loops n number of jonts. Boen [7] hs gven sptl n plnr confgurton relte to truss n lter efne by number of close loops. Kolchn [0] hs presente the theory of new constrnt.e. pssve constrnt but tht ws not for entfcton of geometrc contons t ws only for the generl constrnt problem of mechnsm. The problem relte to generl constrnt ws frst complete by Vone et l. [] through the rnk of mtrx n unknowns of the ngulr veloctes. In 9 Ozol hs presente the topologcl propertes of the mechnsm. The technque of confgurton synthess ws bse on grph theory to obtn knemtcs chns n mechnsm []-[]. Now for the hgher of or complcte structures the knemtcs nlyss of sptl n plnr cse ws one by []. Therefter, resolvng the clssfctons of structure s complete usng the theory of vng jonts by []-[]. In 0 th century, [] n [7] presente computer e technque for the structurl nlyss of sptl mnpultors. Lter [7] ntrouce computer e metho for plnr mnpultor or mechnsm n then loop formton for cncellng the somorphsm test ws ntrouce by [8] n [7]. f

63 Tble. Dfferent mechnsms n moblty N Authors Equtons Remrks L represents totl number of Euler L j l nepenent loops, l represents totl number of lnks, j represents totl number of jonts Frst equton represents the plnr l j 0 hebyshev mechnsm wth sngle of, [0] 0 j j l l represents the totl movng jonts n l =n represents number of j l l n l movng lnks ylvester l j 0 Ths equton represents the plnr [] j n mechnsm wth sngle of. M o represents the of of mechnsms. of epens on the rnk of functonl etermnnt Grübler [] M l l l or o l j q 0 j 0 j q 0 H l M (l j 7 0 ) p Frst equton s bse on plnr mechnsm. econ Eq. represents the knemtc chn of revolute n prsmtc jont. Thr eqn. s for plnr mechnsm wth only prsmtc jont. Fourth eqn. Eq. s for revolute, cm n prsmtc jonts. Ffth eqn. represents the of of sptl mechnsm of helcl jont. omov [] l ( )(v ) l q M o (l v L, ) K l v 7,, K ( )(v ) u u f j L q j p Frst equton s bse on both plnr n sptl mechnsm. econ one s lso for plne n sptl mechnsm where M o Equton thr s omov's unversl formul for structure where s the generl prmeter for constrnt.

64 Tble. Dfferent mechnsms n moblty (ontnue) N Authors Equtons Remrks Frst eqn. s for both plnr n sptl mechnsm, where ( ) f s the Gokhmn [] l( ) f L ( j L) totl no. of jont constrnts econ one represents the moblty crteron 7 Koengs [] M n 8 Assur [] n j 0 9 Muller [7] 0 Mlushev [8] Kutzbch ( ) M o s l n ( ) ( ) 0 s s the number of screw prs M o (l ) s p q n p s the knemtc prs wth clss = number of jont constrnt M (l [9] M o (l ) ( o j) j j f v )f Lst eqn. s ultmte resoluton of Euler eqn. usng frst n secon eqn. Koengs lso presente eqn. for the sptl mechnsm lke Gokhmn's eqn. Assur presente eqn. for smple mechnsm Ths eqn. represents the screw pr of the knemtc chn. Ths eqn. s the combne pproch of omov n Mlushev for moblty wth n no. of lnks, where p represents the knemtc pr wth no. of constrnt Kutzbch hs lso gven equton for unversl confgurton

65 Tble. Dfferent mechnsms n moblty (ontnue) N Authors Equtons Remrks Kolchn [0] M o (l ) (P R K) p P s the number of prsmtc prs R s the number of revolute prs Ths eqn. represents for plnr mechnsm where R s revolute, P s prsmtc n K represents hgher pr wth pure slp n roll vrbles wheres, p gves only for slppng n rollng hgher prs. Artobolevsk [] M o j n j L K K q Ths eqn. s lso represents the unversl moblty for fferent structure. Dobrovolsk [] M o n,... ( )p q Another moblty formul for fferent structure. Moroshkn [] M M o o,..., n L j n p r p p,... Frst eqn. represents the structurl form of ntegrl jonts econ eqn. represents of for vrble constrnts. Ths eqn. represents of for complex Vone n Atnsu [] M o j f L r K K j p mechnsm where r K s rnk for screw j jonts n f represents the totl no. of of for revolute, helcl n prsmtc jonts. 7 Pul [] L j l 0 Euler's formul for cretng topologcl stuton for plnr knemtc chn j 8 Rössner [] M o f (j l ) mlr to Euler's eqn. for moblty

66 Tble. Dfferent mechnsms n moblty (ontnue) N Authors Equtons Remrks M j 9 Boen [7] f (j l ) (j l ) o of eqn. for plnr n sptl mechnsm 0 Ozol [8] Wlron [9] Mnolescu [0] M M M M o o o o M o M o j j (l f L q f L q ) j q j L q F r l ( )p ( ) L Frst to thr eqn. represents the moblty wth vrble n excessve constrnt Lst eqn. represents the moblty for cylnrcl mechnsm. Eqn. for moblty of close loop mechnsm. Eqn. for close loop mechnsm wth elementry prmeter M Bgc [] ( )f K q o (l ) L K j p Mofe form of Artobolevsk's eqn. for moblty wth new ntrouce prmeter j p Antonescu [] Freuensten n Alze [] M M M M M o o o o o ( E E,,,, j j )(l ) m f m L f L L L K K K ( K j ) p mlr to Dobrovolsk's eqn. for moblty wth vrous moton coeffcent Moblty eqn. for vrous contons n prmeter for sptl n plnr mechnsm. Hunt [] M o (l j) f Ths eqn. s the extene for of eqn..

67 Tble. Dfferent mechnsms n moblty (ontnue) N Authors Equtons Remrks 7 Herve [] M (l ) ( f Gronowcz 8 [] o ) j L L Moblty eqn. for the lgebrc formul of the structure for the splcement set. M o k FKj Eqn. of moblty for mult loop K jk mechnsm. j 9 Dves [7] M o f r Eqn. for moblty smlr to Moroshkn's eqn. Agrwl n 0 Ro [8] M N N o (ñ (n L K k ñ n L Kj K jk L )F ñ )F F n Eqn. of generl mechnsm for the moblty wth generl constrnt. Dut n Dconescu [9] M M o o j L f e K K L K j e K (L comj )f e comj Moblty eqn. for complex or elementry loop mechnsm. Angeles n Gosseln [70] Alze [7] M o nullty(j) nullty(j) (v) r(j) L j B c M M j o o B E j m (j B c ) q j f (j B c ) q j f (j B c ) B b B B b b b p p Eqn. of the mult-loop or close knemtc chn usng Jcobn mtrx where J represents the Jcobn mtrx of rnk r(j) n v mensonl spce Eqn. for no. of fferent loops n moble pltform to clculte ts moblty's.

68 Tble. Dfferent mechnsms n moblty (ontnue) N Authors Equtons Remrks Mcrthy [7] M c l o ( f ) Eqn. represents the moblty of prllel mechnsm Hung n L [7] j M o ( )(l j) f q mlr to Mccrthy's moblty formul for prllel mechnsms. Alze n Byrm [7] M j l o B j f (c B) L c B, c c c, c j c f (c B) b l b Moblty eqn. for smple n complex structurl mechnsms. 7 M Gogu [7] o f j p j l j mlr to Mccrthy, Alze, n Byrm. 8 Alze Byrm n Gezgn M M cl [7] l b h o o (B c) ( ) c c c l c cl j (f ) q j ( D) p f q j p Frst eqn. represents for moblty of robot mnpultor n secon eqn. gves moblty for prllel rtesn mnpultor... onventonl methos It s well known tht the three mensonl homogeneous trnsformton mtrx broly use n the robotcs fel. Homogeneous trnsformton mtrx mostly els n the fel of moble robot, nustrl robot n computer grphcs for moton nlyss. On the other hn there re severl conventonl tools to fn out the knemtc solutons of the robot mnpultor. Knym n Krhn [77] propose new heterogeneous two-mensonl (-D) trnsformton group to solve moton nlyss/plnnng problems n robotcs. In the new metho they use X mtrx to represent trnsformton whch s s cpble s the homogeneous theory. Ths requres less memory spce n less computton tme s oppose to X mtrx n the homogeneous formulton n t oes not hve the rottonl mtrx nconsstency problem. Ths heterogeneous formulton hs been 7

69 successfully mplemente n the MML softwre system for the utonomous moble robot Ymbco-. Pul n Zhng [78] presente homogeneous trnsformtons bse knemtc nlyss of Mnpultors wth phercl Wrsts n escrbe ts poston n orentton. They use propose technque to obtn knemtc equtons rectly n form sutble for computer mplementton. The equtons re numerclly stble n re obtne lmost utomtclly. The resultng equtons nvolve the mnmum number of mthemtcl opertons. Asprgthos n Dmtros [79] presente three methos for the formulton of the knemtc equtons of robots wth rg lnks. The frst n most common metho n the robotcs communty s bse on homogeneous mtrx trnsformton, the secon one s bse on Le lgebr, n the thr one on screw theory expresse v ul quternon lgebr. They compre these three methos for ther use n the knemtc nlyss of robot rms. They presente three nlytc lgorthms for the soluton of the rect knemtc problem corresponng to ech metho. Fnlly, comprtve stuy on the computton n storge requrements for the three methos s worke out. However the pplcton hs not been one n hgher DOF mnpultors n t s pple to fve DOF robots only. De Xu [80] propose n nlytcl soluton for -DOF mnpultor to follow gven trjectory whle keepng the orentton of one xs n the en-effector frme. They use homogeneous trnsformton mtrx for forwr knemtcs n nverse knemtcs of -DOF mnpultor. The sngulr problem s scusse fter the forwr knemtcs s prove. For ny gven rechble poston n orentton of the en-effector, the erve nverse knemtcs wll prove n ccurte soluton. In other wors, there exsts no sngulr problem for the -DOF mnpultor, whch hs we pplcton res such s welng, spryng, n pntng. Experment results verfy the effectveness of the methos evelope n ths pper. Mnoch n nny [8] propose n effcent lgorthm for nverse knemtcs soluton of generl -of revolute mnpultor wth rbtrry geometry. When strte mthemtclly, the problem reuces to solvng system of multvrte equtons. They use propertes of lgebr n symbolc formulton for reucng the problem to solve unvrte polynoml. However, the polynoml s expresse s mtrx etermnnt n ts roots re compute by reucng to n Egen vlue problem. These lgorthms nvolve symbolc pre-processng, mtrx computtons n vrety of other numercl technques. 8

70 L n Menq [8] Propose two lgorthms, the egenerte xs n tertve methos for the moton control of mnpultors wth close-form solutons n the neghbourhoo of sngulrtes. These two methos theoretclly gurntee robot's poston ccurcy. The egenerte xs metho my not work well when robot's orentton n locton ncrements become fnte. If robot s movng wth slow spee or the nterpolton tme s n the orer of mcrosecon, the locton n orentton ncrements re smll. In ths cse, the egenerte xs metho s fvoure for t hs less computton thn tht of the tertve metho. Although t cnnot be prove tht the tertve scheme gves the requre poston ccurcy n mnmzes the orentton error, the results seem to show tht ths scheme converges to n cceptble soluton. It s beleve tht the tertve metho s the frst of ts kn to solve the sngulr moton control problem by usng robot's close-form nverse knemtcs. mple computton for the tertve scheme mkes t possble to be mplemente n mny nustrl robots. Pennock n Rghvn [8] propose numercl lgorthm to solve the nverse knemtcs of prllel robots bse on numercl ntegrton. Inverse knemtcs lgorthms bse on numercl ntegrton nvolve the rft phenomen of the soluton; s consequence, errors re generte when the en-effector locton ffers from tht esre. The propose lgorthm ssoctes novel metho to escrbe the fferentl knemtcs wth smple numercl ntegrton metho. The methoology s presente n ths pper n ts exponentl stblty s prove. A numercl exmple n rel pplcton re presente to outlne ts vntges. Kucuk n Bngul [8] escrbe forwr n nverse knemtcs trnsformtons for n open knemtcs chn bse on the homogenous trnsformton. Then, geometrc n lgebrc pproches scusse wth explntory exmples. Fnlly, the forwr n nverse knemtcs trnsformtons re erve bse on the quternon moellng conventon n re explne wth the llustrtve exmples. Wlker [8] propose the poston of mnpultor expresse s ether n jont coorntes or n rtesn coorntes. A new lgebr hs been efne for the use n solvng the forwr n nverse knemtcs problem of mnpultors. The propertes of the lgebr re nvestgte n functons of n epslon numbers re efne. The A lnguge ws use for llustrton becuse of the ese n mplementng the lgebr n t s beng use to solve the forwr n nverse knemtcs problems. However, the progrm ctully use epslon numbers n use the overlong feture of the A lnguge to mplement the epslon lgebr. By smply chngng the orer of the lgebr, the resultng progrm cn compute tme ervtve of the en-effector s poston when use-to solve the forwr knemtcs problem n ny tme ervtve of jont postons when use to solve the nverse knemtcs problem. 9

71 Blkn et l. [8] presente nverse knemtc solutons nlytclly by mnpultng the trgonometrc equtons rectly wthout convertng them necessrly nto polynoml equtons. Four fferent subgroups re selecte for the emonstrton of the nverse knemtc soluton metho. Two of these subgroups re exmples to closeform n sem-nlytc nverse knemtc solutons for the most frequently seen knemtc structures mong the nustrl robots. Lpkn [87] escrbe the Denvt-Hrtenberg conventons moel chns of boes connecte by jonts. Orgnlly they were pple to sngle-loop chns but re now lmost unverslly pple to open-loop serl chns such s robotc mnpultors. Unfortuntely there re severl populr vrtons of the notton: the orgnl, the stl vrnt, n the proxml vrnt. These three cses re compre for ther pplcton to serl robots. The proxml vrte s vnce s the most notton lly trnsprent for the mechncl nlyss of serl mnpultors. eccrell n Ottvno [88] escrbe knemtc esgn proceure to obtn close-form formulton n/or numercl lgorthms, whch cn be use not only for esgn purposes but even to nvestgte effects of esgn prmeters on esgn chrcterstcs n operton performnce of mnpultors. Usully, there s stncton between open-chn serl mnpultors n close-chn prllel mnpultors. Ths stncton s lso consere s constrnt for the knemtc esgn of mnpultors n n fct fferent proceures n formulton hve been propose to tke nto ccount the peculr fferences n ther knemtc esgn. Nevertheless, recently, ttempts hve been me to formulte unque vew for knemtc esgn both of serl n prllel mnpultors, mnly wth n pproch usng optmzton problems. Low n Dubey [89] propose two fferent pproches to the nverse- knemtcs problem for sx-egree-of-freeom robot mnpultor hvng three revolute jont xes ntersectng t the wrst. One metho uses three rottonl generlze coorntes to escrbe the orentton of the boy. The other metho uses equvlent Euler prmeters wth one constrnt equton. These two pproches hve been ncorporte nto two fferent computer lgorthms, n the results from ech re compre on the bss of computtonl complexty, tme smulton, sngulrty, etc. It ws foun tht Euler prmeters were less effcent thn three rottonl ngles for solvng the nverseknemtcs problem of the robot consere, n tht the physcl sngulrtes cuse by the robot mechnsm coul not be elmnte by usng ether of the two pproches. Perez [90] propose lgorthms for computng constrnts on the poston of n object ue to the presence of other objects. Ths problem rses n pplctons tht requre choosng how to rrnge or how to move objects wthout collsons. They escrbe the 0

72 pproch bse on chrcterzng the poston n orentton of n object s sngle pont n confgurton spce, n whch ech coornte represents egree of freeom n the poston or orentton of the object. The confgurtons forben to ths object, ue to the presence of other objects, cn then be chrcterze s regons n the confgurton spce, clle confgurton spce obstcles. The pper presents lgorthms for computng these confgurton spce obstcles when the objects re polygons or polyherl. ngh n lssens [9] propose the nverse knemtcs soluton for the 7 Degrees of Freeom Brrett Whole Arm Mnpultor wth lnk offsets. The presence of lnk offsets gves rse to the possblty of the n-elbow & out-elbow poses for gven eneffector pose n s scusse. A prmetrc soluton for ll possble geometrc poses s generte for esre en-effector pose (poston n orentton). The set of possble geometrc poses re completely efne by three crcles n the rtesn spce. A metho of computng the jont vrbles for ny geometrc pose s presente. An nlytcl metho of entfyng set of fesble poses for some jont ngle constrnts s lso resse. Nelsen n Roth [9] propose soluton technques of nverse knemtcs usng polynoml contnuton, Gröbner bses, n elmnton. They compre the results tht hve been obtne wth these technques n the soluton of two bsc problems, nmely, the nverse knemtcs for serl-chn mnpultors, n the rect knemtcs of n-prllel pltform evces. Xn et l. [9] propose smple effectve metho for nverse knemtcs problem of generl -of revolute serl robot or forwr knemtcs problem of generl 7-of revolute sngle-loop mechnsm bse on one-menson serchng lgorthm. All the rel solutons to nverse knemtcs problems of the generl -of revolute serl robot or forwr knemtcs problems of the generl 7-of revolute sngle-loop mechnsm cn be obtne. They propose followng fetures of pple metho: () usng onemenson serchng lgorthm, ll the rel nverse knemtc solutons re obtne n t hs hgher computng effcency; n () compre wth the lgebrc metho, t hs evently reuce the ffculty of eucng formuls. The prncple of the new metho cn be generlze to knemtc nlyss of prllel mechnsms. Mvros et l. [9] propose geometrc esgn problem of R-R sptl mnpultors wth new metho tht uses the DH prmeters. They efne three en-effector postons n orenttons usng three by homogenous trnsformton mtrces. The loop-closure geometrc equtons prove the requre number of esgn equtons. Polynoml Elmnton technques re use to solve these equtons n obtn the mnpultor DH prmeters nclung DH prmeters tht escrbe the locton of the

73 bse frme wth respect to n rbtrry reference frme n prmeters ssocte wth the en-effector. A sxth orer polynoml s obtne n one of the esgn prmeters. Novel metho s pple to emonstrte tht the two sptl R-R chns obtne s rel solutons to the numercl exmple cn form four-br Bennett mechnsm. Fnlly, two specl cses where the orenttons of ny two or ll three precson ponts re entcl re solve usng the DH formulton. hen et l. [9] propose formulton of generc numercl nverse knemtcs moel n utomtc generton of the moel for rbtrry robot geometry, nclung serl n tree-type geometres. Both revolute n prsmtc types of jonts re consere. The nverse knemtcs s obtne through the fferentl knemtcs equtons bse on the prouct-of-exponentl POE formuls. The Newton Rphson terton metho s employe for soluton. The utomte moel generton s ccomplshe by usng the knemtc grph representton of moulr robot ssembly confgurton n the relte ccessblty mtrx n pth mtrx. Exmples of the nverse knemtcs solutons for fferent types of moulr robots re gven to emonstrte the pplcblty n effectveness of the propose lgorthm. Rco et l. [9] propose the pplcton of Le Algebr to the moblty nlyss of knemtc chns. The nstntneous form of the moblty crteron presente here s bse on the theory of subspces n sub lgebrs of the Le Algebr of the Euclen group n ther possble ntersectons. It s shown usng ths theory tht certn results on moblty of over-constrnt lnkges erve prevously usng screw theory re not complete n ccurte. The theory presente proves for computtonl pproch tht woul llow effcent utomton of the new group theoretc moblty crteron. Perez et l. [97] presente the smplest of the over-constrne lnkges, the close sptl RPRP lnkge. They hve use result n orer to synthesze RPRP lnkges wth postve moblty n for gven shpe of the screw system of reltve splcements. In orer to o so, they hve stte the esgn equtons usng the lffor lgebr of ul quternons []. The ul quternon expresson cn be esly relte to the screw system n t s lso use to ssgn the mgntue to the screws n orer to obtn the corresponence between the screw system n the trjectory of the en-effector. The esgn yels sngle RPRP lnkge. Perez n Mcrthy [98] propose ul quternon lgebr bse knemtc synthess of constrne robotc system. They hve propose ths metho for one or more serl chn mnpultor conserng both prsmtc n revolute jonts. In ths reserch they hve use DH lgorthm n successve screw splcement for etermnng the jont vrbles for the resoluton of en effector poston. Then ul quternons re use to

74 efne the trnsformton mtrces obtne through DH lgorthm to smplfy the esgn formultons of fferent types of mnpultors. Rvell et l. [99] propose knemtc soluton of -of revolute mnpultor usng ul quternon n they me comprson between DH lgorthm n ul quternon pproch. In ths work they hve clculte poston of en effector usng homogeneous trnsformton mtrx tht s lter compre wth propose metho. They hve performe the numercl robustness of opte technque.e. ul quternon. err n Grc [00] propose new metho for the escrpton of postonl mensonl synthess of robot en effector. The propose methoology of ths work s bse on roote tree grph system wheren, the grph nlyss s pple to etermne exct poston of en effector. They hve presente mny exmples of tree topologes. Krov et l. [0] propose esgn nlyss n knemtcs of sngle of novel couple serl chn mnpultor. In ths work they hve presente mensonl synthess for plnr mnpultor tsks, conserng motons n torques of en effector. They hve etermne the knemtc n knetosttc synthess of plnr mnpultor. Lee et l. [0] propose geometrc esgn problem of -of revolute serl mnpultor usng ntervl nlyss metho. They hve pple DH lgorthm for obtnng x homogeneous mtrces whch woul lter use for esgn nlyss. In ths reserch, fve sptl postons n orenttons of en effectors hs been preefne to check for the ccurcy of opte technque. Perez n Mcrthy [0] propose lffor lgebr for the serl couple n-r -of mnpultor to obtne esgn equtons n synthess. They presente the reltve knemtcs of serl chn n the mtrx exponentl form. In ths work the formultons of esgn equton usng lffor lgebr re shown effcent for mnpulton tsks. They hve lso presente the nverse knemtc soluton of the propose mnpultor. Hegeüs et l. [0] propose fctorzton theory usng moton polynomls over quternon lgebr for the soluton of -of revolute mnpultor knemtcs. In ths work they propose strtegy for pckng best solutons of the problem. Zhng n Nelson [0] propose knemtc esgn n optmzton of serl sphercl mechnsm usng genetc lgorthm, In ths work globl mnpulblty n the unformty of the mechnsm n ther workspce for synthess hs been nlyse. Müller [0] propose generc propertes of knemtc mppng for serl mnpultor. Frstly they hve presente the stblty of the property for smll chnges n geometry of the consere mechnsm n secon one s concern wth sngulrty nlyss. In ths work cler mnfestton of moton spces of ech jont n clsses of knemtc mppng s presente.

75 Mvros n Roth [07] presente new metho for the etermnton of uncertn confgurtons of generl -of revolute robot mnpultor. In ths work the propose novel metho for etermnng the uncertnty or reunncy s bse on nlytcl formultons for the loop closure equtons. In ths formulton generl -of revolute mnpultor s trnsforme nto mr onfgurton, n new structurl prmeters re efne. Blkn et l. [08] presente generl metho for the clssfcton of -of nustrl mnpultors bse on the knemtc structure n ther etl nlyses of knemtc equtons on the bss of clssfcton re gven. They hve opte the exponentl rotton mtrx lgebr to fn out the close form soluton of nverse knemtcs of robot mnpultor. Özgören [09] propose exponentl rotton bse mtrx metho for the knemtc nlyss of screw n crnk mechnsm. They hve presente the usefulness of the nlytcl tool for effectve soluton of knemtcs for sptl mechnsm nvolvng splcement, sngulrty, velocty n ccelerton. Pennestr n Vlentn [0] propose ul lgebr for the representton of vrous mechncl n mthemtcl enttes such s screws, lne vectors n wrenches. They hve gven fferent lgorthms for the hnlng of these vector n mtrces of ul number for the nlyss knemtc of fferent mechnsms. They hve lso propose the pplcton of the erve lgebr for the rg boy moton nlyss. Lee n Mvros [] propose polynoml contnuton metho for the nlyss of geometrc esgn problem of -of revolute mnpultor. They hve evelope the elmnton metho for pont precson geometrc nlyss of the mnpultor. In ths work, ech precson pont of the en effector hs been consere sptl confgurton. DH lgorthm s use n ths work for the formulton of the esgn equtons. Lng et l. [] presente pose error nlyss of ARA mnpultor usng screw theory. They hve presente the error prouce by DH lgorthm n compre the sme wth the output of the screw bse nlyss of the mnpulton. Zhung et l. [] propose the lner soluton of PUMA robot for the computton of trnsformtons of coornte from worl coornte to bse coornte. In ths work, soluton for loctng the robot en effector wth respect to reference frme hs been presente. They hve lso pple the quternon lgebr long wth the homogeneous trnsformton mtrx metho. mer Yhy et l. [] propose novel metho for the soluton of nverse knemtc of hyper reunnt mnpultor usng geometrc lgebr. In ths work, the jont ngles re set to smlr whch mkes fcng of two or more jont xes mpossble;

76 therefore t cn vo sngulrtes. They hve lso presente workspce nlyss of the propose mnpultor. u et l. [] propose vrtul moel of n grculturl robot for frut hrvestng n ther knemtcs nlyss usng DH lgorthm. In ths work, the nverse knemtc s obtne usng lgebrc metho n smultons re crre out usng ADAM. Ahme n Pechev [] propose pseuo-nverse bse technque for the control of feebck nverse knemtcs of Mtsubsh RV-A sx egree of freeom robotc mnpultor. In ths work, knemtc nlyss of -of mnpultor hs been one on the bss of DH lgorthm n lter compre wth mpe lest squre nverse knemtcs. We et l. [7] propose sem-nlytc metho for solvng nverse knemtcs of n-r robot mnpultor tht reuces the numercl metho's mrgns relte to ccurcy. In ths work, conforml geometrc theory s use for the generton of generl knemtc equton. Fnlly they hve teste the propose metho n -of revolute mnpultor to prove the effcency n qulty of the soluton. Plcos [8] propose severl pproch for the soluton of nverse knemtc of -of robot mnpultors wthout conserng explct soluton for the chosen mnpultor. In ths work, fferent structure or confgurtons of the -of mnpultor hs been presente n ther clssfcton on the bss of the structure. A complementry exmple s lso presente for the nverse knemtc soluton of -of mnpultor. Muszynsk [9] propose norml form pproch for the soluton of nverse knemtc of the AEA IRB- robot mnpultor. In ths work two steps hve been presente for the soluton of nverse knemtcs, frstly they hve consere the hyperbolc norml form of the sngulr knemtcs of the mnpultor n then nverson lgorthms s presente. BHATTI et l. [0] propose the problem of mtchng forwr n nverse knemtc moton of -mentonl chn usng pseuo-nverse Jcobn mtrx metho. Ths metho s propose for the soluton of nverse knemtcs of -metonl rg chrcter for nmtons. Herrer et l. [] presente ul number representton for solvng knemtcs problem of rg boy, wheren robot mnpultor hs been consere for the knemtc nlyss usng ul number theory prtculrly serl mnpultor. In ths work, cylnrcl, prsmtc n rottonl jonts re use for the nlyss of knemtcs usng the evelope metho. Luo et l. [] propose hyper-chotc lest squre metho for nverse knemtc soluton of -of revolute generl mnpultor. In ths work ll rel soluton of obtne nonlner equtons hs been propose n nverse splcement nlyss of -of

77 revolute mnpultor s complete. These obtne nonlner equtons re bsclly formulte by usng DH lgorthm n they hve presente the numercl exmple for the constrne equtons. Krpnsk et l. [] propose pproxmton problem of Jcobn bse nverse knemtc soluton of 7-of reunnt mnpultor. In ths pper they hve focuse on Jcobn pseuo nverse usng extene Jcobn lgorthm specfclly they hve exmne two methos, frst metho s referre to fferentl geometrc n lterntve metho s bse on mnmzton of pproxmton error usng clculus of vrtons.. Brnstotter et l. [] propose n effcent generc metho for the soluton of nverse knemtc of -of serl mnpultor. In ths work they hve mnly focuse on DH lgorthm conserng seven geometrc prmeters. Kofns et l. [] presente complete forwr n nverse knemtc nlytcl soluton of Alebrn NAO humno robot n ther softwre mplementton for rel tme on-bor executon. In ths work they hve ecompose NAO robot nto nepenent structure of the robot such s two rms, he, n two legs, then DH lgorthm s use for the knemtc resolutons. zkony [] presente ll equtons of forwr n nverse knemtcs of IRB- mnpultor usng mtrx bse metho. In ths work DH lgorthms n homogeneous trnsformton mtrces re use to formulte nverse n forwr knemtcs of IRB- robot mnpulton. Wng et l. [7] presente the geometrc structure, prtculrly Le group propertes of the ul quternon n the exponentl form of the ul quternon s erve. They hve lso presente the usefulness n pplcton of the propose moel for knemtc nlyss of robots. Feng n Wn [8] presente blenng lgorthm for quternon to ul quternon representtons of rg boy trnsformtons. Ths work mnly focuse on the chrcter nmton n knemtc nlyss of the chrcter usng the ul quternon n propose metho hs been presente. Gu n Luh [9] propose ul number theory for representton of lne trnsformton n ther pplcton to solve knemtc problem of robot mnpultor. Ths work s mnly focuse on n lgorthm whch pcts wth the symbolc nlyss of rotton n trnslton of lnks. Wenz n Worn [0] propose close form soluton of forwr n nverse knemtcs of -of mnpultor. In ths work DH lgorthm s use for ervton of nonlner nverse knemtcs equtons n these knemtcs equtons re smplfy usng Groebner bss elmnton metho.

78 Neppll et l. [] propose novel nlytcl metho for nverse knemtc soluton of mult secton contnuum mnpultor. In ths work, the knemtc of the mechnsm s ecompose nto some sub problems lke soluton of nverse knemtc for sngle trunk on the bss of known en ponts of trunk n then pplyng sngle secton nverse knemtcs to ll secton of the trunk. Fnlly, ths pproch computes fnl secton knemtcs of the propose moel of trunk. Olunloyo et l. [] propose nverse n forwr knemtc nlyss of -of robot mnpultor to compre the ccurcy n repetblty of the obtne solutons. In ths work, DH lgorthm s use for the ervton of knemtc of lnk n lnk mnpultors usng ll lgebrc equtons erve from the knemtc trnsformtons of the lnk. Ylrm n Byrm [] presente the mthemtcl moellng n knemtc nlyss of nustrl mnpultor usng Mple robotcs toolbox. In ths work poston n orentton of the tool cn be obtne by usng DH lgorthm n lso for jont vrbles ths metho s cpble of solvng Jcobn n ngulr veloctes. Der et l. [] propose reuce eformton moel bse lgorthm to solve nverse knemtcs of nmte chrcter. A propose lgorthm proves ntutve n rect control of the reuce eformble moels smlr to conventonl nverse knemtc lgorthm for the jont structure. They hve presente the fully utomtc ppelne trnsformtons of controllble shpes wth only few mnpultons tht reuce the mthemtcl complexty of the nverse knemtc of the mechnsm. Zorc et l. [] propose quternon pproch for the moellng knemtc n ynmcs of the rg mult-boy mechnsm. In ths work, regulr Newton-Euler n Lgrnge technque s sorte n the covrnt form by pplyng Rorguez pproch n quternon lgebr tht cn be useful for clculton of knemtc n ynmcs of ny mechnsm. leron et l. [] propose trjectory plnnng n nlytcl nverse knemtc soluton of -of Prm robot mnpultor. Ths work s bse on the hybr lgorthm of nlytcl nverse knemtc n splcement error. Furthermore resolve moton rte control usng Jcobn s use for the smooth moton of en effector. In ths work they hve use splcement error or Euclen stnce bse nverse knemtc soluton of -of mnpultor. Ahmm et l. [7] propose nverse knemtc soluton of -of reunnt mnpultor n vlte wth expermentl results. In ths work, prtton of the -of mnpultor nto, -of vrtul sub-robot n then solve the nverse knemtc nlytcl for both sub-robots. 7

79 Feák et l. [8] propose knemtc n ynmc nlyss of -of robot mnpultor. In ths work D AD moel of robot mnpultor s evelope n lter mporte to the MATLAB mulnk envronment. They hve worke on the MATLAB sm-mechncs for the evluton of knemtcs n ynmcs of the esgne mnpultor. Gousm et l. [9] propose knemtc nlyss n trjectory plnnng for -of n AARA mnpultor. They hve use olworks softwre for the moellng the mnpultor lter mporte n MATLAB mulnk envronment for smultons n moton nlyss. The mn tsk performe n ths pper s comprson of two robot postons wth the smlr trjectory long wth sme tme n estblshng computer progrm for the knemtc n ynmc nlyss. Rehr [0] propose nverse knemtc soluton of Aept three mnpultor. Forwr knemtcs of the selecte mnpultor ws clculte by DH-lgorthm whle nverse knemtc resolutons were complete by prncple of cosnes. A grphcl smultons n clcultons of robot knemtcs hve been presente by usng LbVIEW. Dhr n Tn [] propose forwr n nverse knemtc soluton for KUKA robot mnpultor for the welng pplcton. They hve selecte severl welng spot to be performe by the mnpultor. To o so they hve use nlytcl metho for solvng nverse knemtcs usng DH-lgorthm. ores et l. [] propose rhno mnpultor knemtcs n control usng Robm softwre. In ths work they hve focuse on mge cpturng evce for the poston n orentton of the en effector. Ths metho s evelope n MATLAB presente smultons for the selecte mnpultor. In ths pltform bsc unt whch s clle prmtves s use to smulte robot structure. Veo cpturng evce s use for the vson gue mnpultor experments n mge n postons re use for servong. Wng et l. [] propose nverse knemtc soluton of generl -of revolute serl mnpultor usng Groebner bses metho. They hve reuce the complexty of the nverse knemtc polynoml equton usng Groebner bse metho. From ths, they hve gven mxmum solutons for the nverse knemtc n lso conclue tht ths metho cn be esly mplemente on nonlner equtons wth the help of symbolc representtons. Gn et l. [] propose nverse knemtcs of 7-of robot mnpultor usng ul quternon lgebr. The consere mnpultor confgurton s serl 7-lnks wth revolute jont n ths work. They hve use Dxon's resultnt for nput-output; expresse 8

80 n x etermnnt equte to zero, n lso etermne the ngulr splcement of the jont vrbles. helnokov [] propose nverse knemtc soluton of robot mnpultor usng bquternon metho. In ths metho screw system s consere for the coornte frme representton... Intellgent or soft computng pproch onventonl methos for knemtcs nlyss re more exhustve n complex n nture s per lterture survey; there re numerous conventonl technques s explne erler such s nlytcl, lgebrc, numercl, Jcobn mtrx bse, n geometrc lgebr. These methos generlly yel nonlner, tme vryng n uncertn equtons for nverse knemtc. More over these equtons oes not prove sngle soluton for the nverse knemtc problem wheres n cse of forwr knemtcs lwys unque n sngle soluton exsts. Becuse of the bove-mentone resons, vrous uthors opte ntellgent technques to solve nverse knemtc. These ntellgent technques re rtfcl neurl network, fuzzy logc, support vector mchne, grey neurl network, hybr neurl network etc. However, to fn out the nverse knemtc soluton of the gven problem usng bove stte ntellgent technques, t s requre to clculte forwr knemtcs of the mechnsm whch wll be use to generte nput for the ntellgent system. Artfcl neurl network (ANN) prtculrly MLP (mult-lyere perceptron) neurl network s generlly use to lern forwr s well s nverse knemtcs equton of vrous confgurton of the mnpultor. Ths metho s bse on lernng process of some stnr t whch rely on the workspce of the mnpultor or mechnsm. In cse of ANN there re mny wys of lernng t such s supervse lernng, unsupervse or combnton of both. ANN follows the functonl reltonshp between the nput vrbles (rtesn coorntes) n output vrbles (jont coorntes) bse on the locl revson of mppng between nput n output. Ths concept s lso bss for fuzzy logc n hybr ntellgent technques whch les to smple soluton of nverse knemtc roppng the conventonl complex mthemtcl formule. The smulton n computton of nverse knemtcs usng ntellgent technques re preomnntly useful were less computton cost s requre, efntely for controllng n rel tme envronment. If the confgurton of mnpultor s well s conserng number of of ncreses, then the conventonl nlytcl methos wll turn nto more complex n ffcult mthemtcs. There re numerous reserch hs been one n the fel of ANN, fuzzy logc n lso for hybr technques. 9

81 Roríguez et l. [] propose rtfcl neurl ptve nterference system (ANFI) n ANN bse pproch for the soluton of the nverse knemtcs of the -of nthropomorphc mnpultor whch resembles the humn upper lmb. In ths reserch they hve use mult-lyere perceptron (MLP) n ANFI metho for the nverse knemtc precton n neuro-rehbltton purpose uner the ssste system. They hve pple MLP n ANFI trnng wth rtesn coorntes of the humn upper lmb for wter servng n bottle pckng pplcton. Fnlly they evlute the effcency n qulty of the opte technques. hrwr n Bbu [7] propose MLP n RBF neurl network moel for the soluton of nverse knemtc of the -of serl mnpultor. In ths work, fuson pproch of these ANN moels s use wth the forwr knemtcs of the mnpultor. Forwr knemtcs equtons re use to generte the t for trnng opte moels of ANN. They hve propose the rtesn pth to be followe by the mnpultor en effector usng the generte ANN nverse knemtc soluton. KUKA -of mnpultor s teste wth the obtne results wheren DH-lgorthm s use to generte the nput for the ANN moels. Koker [8] propose nverse knemtc soluton of the tnfor mnpultor usng neurl network n genetc lgorthm. In ths work, Elmn's neurl network hs been use n compre wth genetc lgorthm. A bsc clculton for the nput of the network hs been crre out wth the DH-lgorthm. Three Elmn's neurl network moels re trne wth DH-lgorthm output of knemtcs. In cse of genetc lgorthm the ftness functon s set to en effector poston error bse formul for the soluton of jont ngles. Krlk n Ayn [9] propose structure ANN pproch for the nverse knemtc soluton for -of mnpultor. In ths work, they hve use bck-propgton lgorthm for the trnng of the ANN moel n nput tsets were generte by usng DHlgorthm. They hve tre to fn out the excellent ANN confgurton for nverse knemtc resoluton. Hsn et l. [0] propose ptve lernng pln of ANN for the soluton of nverse knemtc of -of mnpultor. Moreover they hve tre to resolve sngulrty n uncertnty problem of the opte confgurton of the mnpultor. In ths work ANN moel hve been trne usng nlytcl soluton of the opte mnpultor. Generte tsets usng knemtcs equtons re use to trne n test the opte moel of ANN. They hve conclue tht the propose moel of ANN oes not nee to hve prevous nformton of the knemtcs of the system tht lerns through the ANN moel pplcton. 0

82 Bocs et l. [] propose nverse knemtc soluton of 7-of Brrett WAM usng support vector mchne. They hve explne the lernng problem of reunnt mnpultor usng neurl network bse moels. The mjor problem wth the soluton of nverse knemtc s non-unque n nture n generton of lrge tsets for nput. Therefore they hve propose sutble lgorthm for lernng the knemtcs n pple to rel worl problem of 7-of mnpultor. Hsn et l. [] propose ANN bse soluton of -of mnpultor to vo sngulrty n uncertnty of the confgurton. They hve use nput t for the trnng the ANN moel from the experments of the opte moel of robot usng vrous sensors. They hve esgne the ANN network for one hen lyer n nputs were tken s coorntes of the en effector of robot mnpultor. After trnng of neurl network moel they hve teste t for rel tme pplcton of the opte mnpultor wth vong the sngulrty problem. Obtne results through ther experments shown ther effcency n qulty. Olru et l. [] propose nverse knemtc soluton of the ctcl rm usng neurl network. In ths work they hve use two hen lyer n sgmo trnsfer functon for the trnng of the neurl network. Mthemtcl moellng ws crete by usng neurl network n LbVIEW. They hve one the experments for selectng number of hen neurons for trnng n better lernng o the network to o so they hve pple fferent number of neurons for evluton for trjectory error generte by the en effector. All gne results were teste by bsc knemtc through LbVIEW. Fnlly they obtne optml sgmo functon wth tme ely n recurrent network. Myorg n nongboon [] propose neurl network pproch for nverse knemtc soluton of the plnr reunnt mnpultor n effectve geometrc sngulrty vonce of the selecte mnpultor. Moreover they hve presente some geometrcl concept for the sngulrty vonce n obstcle vonce of the reunnt mnpultor. Fnlly they hve presente the performnce of the trne neurl network for the stte problem. Klr n Prksh [] propose neuro-genetc pproch for the resoluton of nverse knemtcs of plnr mnpultor. They hve use mssvely prllel rchtecture of ANN for the soluton of the stte problem. They hve selecte the MLP network n weghts of the ANN moel were optmze by rel coe genetc lgorthm so s to overcome the problem of bckpropgton lgorthm. Bhttchrjee n Bhttchrjee [] stue the problem of nverse knemtc soluton usng conventonl metho n therefor they pple ANN bse pproch for

83 the resoluton of nverse knemtc of the mnpultor. Frstly they hve obtne the jont ngles tset of the en effector so s to use s nput or trnng of ANN moel. They hve mnly focus on the obstcle vonce of the mnpultor usng ouble hen lyer ANN moel. Mrtn et l. [7] propose nverse knemtc lernng of -of plnr n ARA mnpultor usng neuro-controller. Furthermore, they hve presente the some ssues of neurl network lernng such s clsscl supervse lernng scheme whch generlly converse n locl optmum soluton. Therefore they hve pple neuroevoluton lgorthm for the globl optmum soluton of the nverse knemtcs of the selecte mnpultor. In ths work DH-lgorthm s use to generte the nput t set for the neurl network lgorthm. They hve reuce the rwbck of the grent escent lernng of ANN moel wth the help of evolutonry lgorthm. Feng et l. [8] propose novel neurl network bse pproch for the soluton of nverse knemtcs of the PUMA 0 robot mnpultor. In ths work they hve pple smple fee forwr neurl network to obtn the knemtc of the PUMA 0 mnpultor n compre wth the evelope ELM (extreme lernng mchne) bse neurl network. They hve use mchne lernng lgorthm to overcome the problem of trtonl grent escent lernng strtegy. Bngul et l. [9] propose nverse knemtc soluton of -of revolute robot mnpultor wth offset wrst usng ANN. Mnpultor wth offset wrst s consere becuse offset wrst bse structure generlly oes not gves the exct soluton usng some trtonl methos. Therefore they hve opte ANN moel for the nverse knemtc soluton. They hve use DH-lgorthm for the generton of nput tsets of MLP moel n lter precte soluton wll be use to compre wth the trtonl soluton. They hve presente the error occurre n the effcency of the opte technque. Hsn et l. [0] propose nverse knemtc soluton of -of robot mnpultor usng MLP neurl network wth fferent structures. In ths work they hve use fferent number of hen lyers for the precton of the soluton. In ther frst confgurton or rchtecture of the MLP moel they hve use three nputs (X, Y n Z coornte) n sx outputs of the jont ngles n n secon experment they hve use four nput.e. rtesn coorntes long wth velocty n clculte outputs re sx jont ngles n ther ngulr veloctes. Alsn n Gehlot [] propose moulr ANN bse nverse knemtc soluton for -of ARA mnpultor. They hve ssgne ech neurl moule n ech lnk n orer to fn out the nverse knemtcs. Ths pproch of neurl moules s connecte

84 n globl system for the uptng of the nverse knemtc soluton. In ths work three lyere neurl network s use wth sgmo ADLINE trnsfer functon. They hve consere -of n -of mnpultor for the smulton n verfcton of the solutons. Onozto n Me [] propose MLP neurl network bse soluton of -of ARA mnpultor. They hve use bsc nlytcl pproch for genertng the nput tset for lernng. A smultneous perturbton technque s pple for the lernng of network n clculte the nverse knemtc n ynmcs of the mnpultor. Al-Kheher n Alshmsn [] propose neurl network bse control of ARA mnpultor n compre wth the PD controller. In ths work they hve use DH lgorthm for the evluton of the nverse knemtc of the robot mnpultor. A serlprllel structure neurl network s use for poston control of ll jont vrbles. They hve use three lyere neurl network wth bck propgton supervse lernng. They hve lso optmze the number of hen lyer to obtne better result. Lter smultons re crre out n MATLAB mulnk. Myorg n nongboon [] propose neurl network bse pproch for nverse knemtc soluton n effectve sngulrty vonce of reunnt mnpultor. In ths pproch they hve estblshe some symbolzng mtrces, expressng some geometrcl es, so s to gn smple performnce nex for sngulrty vonce. These methos of mtrces re trne wth neurl network n fnlly compute the nverse knemtcs. Dch n Benllegue [] propose neurl network bse ptve controller for chevng en effector poston of reunnt mnpultor. They hve esgne the controller n rtesn spce so s to overcome the problem of pth n moton plnnng tht s well know problem of nverse knemtc. They hve -of reunnt plnr mnpultor. The unentfe moel of the scheme s pproche by ecompose structurl neurl network. Ths pproch s use to fn ptve stblty n the lgorthm s bse on Lypunov metho wth nherent propertes of robot mnpultors. Howr n Zlouchn [] propose fuzzy logc bse nverse knemtc soluton of -of robot mnpultor. In ths work herrchcl control bse metho s use for the controllng of robot mnpultor. The mppng of rtesn coornte wth the jont coornte s estblshe by fuzzy logc n orer to evlute ech jont vrble. The herrchcl control wth fuzzy logc mproves the robustness n lso ecreses the computtonl cost.

85 Kumr n Irsh [7] propose neurl network bse soluton for the nverse knemtc of -of serl mnpultor. They hve use MLP neurl network structure wth unsupervse lernng strtegy. They hve generte nput tsets usng forwr knemtc equton of the mnpultor. Bck propgton lgorthm s use for the trnng MLP neurl network. Oym et l. [8] propose novel moulr neurl network wth expert system for the precton of nverse knemtc of robot mnpultor. In ths metho ech expert estmtes the contnuous prt of the functon. The propose metho uses forwr knemtc for the selecton of experts. When the no. of consere expert ncreses the computton cost lso ncreses for the nverse knemtcs soluton, wthout usng ny prllel computng system. They hve use 7-of reunnt mnpultor for the nlyss of knemtcs. Tejomurtul n Kk [9] propose structures neurl network bse nverse knemtc soluton of plnr n sptl mnpultor. They hve use MLP neurl network wth two hen lyers for trnng of the network. In ths work bckpropgton lgorthm s use for -of plnr n sptl mnpultor nverse knemtc resoluton. Km n Lee [70] propose nverse knemtc soluton of reunnt mnpultor usng Jcobn mtrx n fuzzy logc methos. In ths work moton rte resolvng lgorthm s use whch s lter mprove by fuzzy logc. Furthermore, they hve obtne rough soluton of nverse knemtcs bse on grent metho whch s lter refne by fuzzy logc n extenson prncple. Alvnr n Ngm [7] propose nverse knemtc soluton of -of n -of plnr mnpultor usng ptve neurl fuzzy nference system (ANFI). In ths work, they hve opte ugeno type fuzzy rchtecture n hybrze wth smple neurl network for the precton of nverse knemtc of plnr mnpultor. Kozlzewcz et l. [7] propose nverse knemtc of -of mnpultor usng prttone neurl network whch s lso known s prllel neurl network. The selecte rchtecture s collecte of pre-processng lyer n prttone by moules contnng evote neurons. In ths work they hve use bck propgton lgorthm for the soluton of nverse knemtc. Kuroe et l. [7] propose nverse knemtc precton of -lnk robot mnpultor usng ANN. In ths work they pple supervse lernng theorem whch s bse on the Tellegen's theorem.

86 Jck et l. [7] propose nverse knemtc soluton of -of mnpultor usng fee forwr neurl network technque. In ths work they hve selecte three fferent confgurton of neurl network. Arstou n Lsenby [7] propose nverse knemtc soluton of vrous confgurton of revolute mnpultor usng novel evelope FABRIK (forwr n bckwr rechng nverse knemtcs) metho. FABRIK eves the necessty of conventonl rottonl ngle mtrces. Morten n Erleben [7] propose nverse knemtc soluton of nmte chrcter usng projecte-grent metho. Zhng et l. [77] propose ul neurl network bse knemtcs n moton plnnng of reunnt mnpultor. In ths work, lner vbrtonl nequltes (LVI) bse n smplfe LVI bse ul neurl network use for the problem resoluton. To ccomplsh ths rft-free conton s explote n qurtc form. Dugulen et l. [78] propose neurl network bse knemtc soluton of generl - of serl robot mnpultor. In ths work ul neurl network wth Q-lernng renforcement metho s use for the soluton of nverse knemtc n obstcle vonce. Dy et l. [79] propose nverse knemtc soluton of -of plnr mnpultor usng neurl network. In ths work neurl network rchtecture conssts of sx sub neurl network whch s bsclly extene form of MLP neurl network. Bck propgton lgorthm s pple for error mnmzton. Xuln et l. [80] propose nverse knemtc soluton of -of plnr mnpultor usng hybr neurl network. In ths work, fee forwr neurl network s frst optmze by prtcle swrm optmzton (PO) technque then use for nverse knemtc resoluton. Ths worke s compre wth the bckpropgton evluton of knemtcs wth PO bse ANN. Aghjrn n Kn [8] propose nverse knemtc soluton of PUMA 0 robot mnpultor usng ptve neurl fuzzy nference system (ANFI). In ths work MLP neurl network s hybrze wth fuzzy logc to obtn better result of nverse knemtc s compre to neurl network. hen et l. [8] propose nverse knemtc soluton of -lnk plnr mnpultor usng self- confgurton fuzzy logc. In ths work they hve pple fuzzy logc frst then self-confgurton pproch s ntrouce bse on nput-output prs. Knosht et l. [8] propose nverse knemtc soluton for -of plnr mnpultor usng MLP neurl network. In ths work forwr propgton lgorthm s use for the

87 estmton of output lyer error. The opte forwr propgton rule s bse on gol sgnl crre by Newton-lke metho, n then uptng of weght s complete by regresson coeffcent. Borbon [8] propose nverse knemtc soluton of smple ARA mnpultor usng fuzzy logc technque. In ths work they hve explne severl other lgorthms lke prllel chors lgorthm, Newton-Rphson n Rescon-Fgl lgorthms n compre wth the fuzzy logc solutons of nverse knemtc. Meshref n Vnlnnghm [8] propose forwr n nverse knemtc soluton of -of robot mnpultor usng mmune bse neurl network. In ths work forwr knemtc s complete by DH-lgorthm whch s lter use s nput for the propose mmune bse nverse knemtcs soluton. Al-Mshhny [8] propose nverse knemtc soluton for -of mnpultor usng loclly recurrent neurl network wth consere sphercl wrst. The opte metho LRNN (loclly recurrent neurl network) s progrmme n MATLAB n smulton hs been complete n mulnk. In ths work Levenberg-Mrqurt bse bck propgton lernng strtegy s pple for hgh computton n for soluton ccurcy of nverse knemtc. Asun et l. [87] propose nverse knemtc soluton of PUMA robot mnpultor usng self- orgnzng neurl network. In ths work Vso-motor coornton s use for lernng of neurl network. Ths metho s bse on bologcl nspre moel tht mttes humn brn power to crete reltonshp between motor n sensory t wth the help of lernng process. Ylrm n Esk [88] propose nverse knemtc soluton of PUMA 0 robot mnpultor usng neurl network metho. In ths work they hve pple fee forwr neurl network wth fferent lernng n weght uptng lgorthms. Frst they hve consere the Onlne bck propgton lgorthm n then elt br elt lgorthm n fnlly they hve pple quck propgton lgorthm for the nlyss of nvers knemtc of robot mnpultor. Zhng et l. [89] propose nverse knemtc soluton of MOTOMAN robot mnpultor usng neurl network. In ths work they hve use rl bss functon neurl network (RBFNN) for the evluton of nverse knemtcs. The neurl network system esgne s mult nput n sngle output (MIO) bse technque. Koker [90] propose nverse knemtc soluton of tnfor n PUMA 0 robot mnpultors usng neurl network technque. In ths work smulte nnelng (A) s pple long wth the neurl network to mnmze the error of the jont vrbles. Three Elmn's neurl network moel s use n trne wth the help of A lgorthm.

88 Her et l. [9] propose nverse knemtc soluton of n -of plnr robot mnpultor usng fuzzy logc together wth the genetc lgorthm. They hve use trngulr membershp functon for fuzzy logc n centre of grvty s use for the efuzzfcton. These prmeters re lter tunes by genetc lgorthm for the surety of exct nverse knemtc soluton. Hu et l. [9] propose nverse knemtc soluton of PUMA 0 robot mnpultor usng wvelet neurl network moel. Ths metho s workng on mult-nput mult output (MIMO) system. Neurl network trne s complete by Levenberg-Mrqurt lgorthm. Agrwl [9] propose nverse knemtc soluton of reunnt mnpultor usng fuzzy c-mens system. Novel evelope fuzzy clusterng metho s generlze bse on weghte sctter metrcs n cluster metrcs re evelope for mnpultor. Q n L [9] propose nverse knemtc soluton of -of robot mnpultor usng support vector mchne wth genetc lgorthm. upport vector coeffcent lke kernel functon, nsenstve coeffcent n penlty fctors re tunes by GA. Lu n Brown [9] propose extene pproch of fuzzy logc for the soluton of nverse knemtcs of robot mnpultor. In ths work they hve use PUMA 0 robot mnpultor for the mplementton of propose lgorthm. The propose lgorthm s bse on fuzzy trgonometry ervtves. Mrtn n Emm [9] propose rel tme neurl fuzzy trjectory generton of EPON robot mnpultor for the rehbltton purpose of ptents wth lmb ysfuncton. Netto et l. [97] propose nverse knemtc soluton of hexpo robot leg usng fuzzy system. In ths work hexpo robot's leg conssts of revolute jont smlr to nother leg. The knemtc nlyss s use to generte t for the blck box of fuzzy n neurl network, prtculrly forwr knemtc s use to generte the trnng t set for fuzzy n neurl network. ong n Jung [98] propose knemtc soluton of -of nthropomorphc robot mnpultor usng geometrc lgebr bse metho. In ths work trjectory hs been generte usng fuzzy controller. In ths work geometrc nverse knemtc soluton s use for the jont vrble control of mnpultor. The generte output from the opte technque s lter use s n nput of fuzzy logc controller. rengns et l. [99] propose mthemtcl moellng of 7-of humn rm lke mnpultor knemtcs. In ths work both forwr n nverse knemtc s presente n lter compre wth the (ANFI) fuzzy logc soluton of the knemtcs. They hve use MATLAB ANFI toolbox for knemtc resoluton. 7

89 Morsht n Tojo [00] propose nteger nverse knemtc soluton of mult-jont robot mnpultor usng fuzzy logc bse metho. They hve evlute the effcency of the opte technque n teste t for trjectory generton n control pplcton. Neumnn et l. [0] propose nverse knemtc precton bse on neurl network for humno robot AIMO, n whch they focuse on b-mnul tool. onsere humno robot hn s hghly reunnt n ths cse n recurrent reservor lernng strtegy hs been mplemente. Hshm et l. [0] propose mnpultor postonng nlyss usng rtfcl ntellgent technques. In ths work they hve opte three technques nmely fuzzy logc, genetc lgorthm n neurl network to solve nverse knemtcs of -of serl mnpultor. Forwr knemtc of serl mnpultor hs been tken s feeforwr control on the other hn ntellgence metho resolves the nverse knemtcs problem... Optmzton pproch Inverse knemtc close form solutons for severl confgurtons n smple structures re certn. Mthemtcl pproches re more complcte s per numercl, tertve or ntellgent bse methos n the obtne soluton usng these methos re not only confgurton epenent but lso mtters to mbguty of the mnufcturng errors. Therefore, to overcome mthemtcl complexty n mprove the effcency of the soluton, t s necessry to opt engneerng optmzton methos. Optmzton methos cn be pple to solve nverse knemtcs of mnpultors n or generl sptl mechnsm. Bsc numercl pproches lke Newton-Rphson metho cn solve nonlner knemtc formule or nother pproch s prector corrector type methos to ssmlte fferentl knemtcs formule. But the mjor ssues wth the numercl metho re tht, when Jcobn mtrx s ll contone or possess sngulrty then t oes not yel soluton. Moreover, when the ntl pproxmton s not ccurte then the metho becomes unblnce even though ntl pproxmton s goo enough mght not converge to optmum soluton. Therefore optmzton bse lgorthms re qute frutful to solve nverse knemtc problem. Generlly these pproches re more stble n often converge to globl optmum pont ue to mnmzton problem. The key fctor for optmzton lgorthms s to esgn objectve functon whch mght be complex n nture. On the other hn, metheurstc lgorthms generlly bse on the rect serch metho whch generlly o not nee ny grent bse nformton. In cse of heurstc bse lgorthms locl convergence rte s slow therefore some globl optmzton lgorthms lke GA, BBO, TLBO, AB, AO etc. cn be gnfully use. 8

90 Nerchou [0] propose nverse knemtc soluton of reunnt mnpultor usng mofe genetc lgorthm. They hve mplemente some ssumptons; frst they consere tht the mnpultor my be reunnt n rtculte. Then the secon ssumpton s tht the mnpultor s n movng object of ts workspce. An lst ssumpton s tht they re not conserng ynmcs of the mnpultor. Therefter, genetc lgorthm s use n two fferent mnners, frst jont splcement ( ) error mnmzton n the secon pproch s bse on postonl error of en effector. Wng n hen [0] propose nverse knemtc soluton of PUMA 0 robot usng optmzton metho. In ths work they hve consere postonl error n orentton error for robot mnpultor. The propose soluton s bse on cyclc coornte escent (D) n Broyen-Fletcher-hnno (BF) technque. Totl error s clculte bse on the en-effectors ntl n fnl splcement postons n reltve ngulr splcement error. Prker et l. [0] propose nverse knemtc soluton of -of PUMA mnpultor bse on genetc lgorthm. In ths work they hve consere two splcement mnmzton problems; frst problem of mnmzton s en-effector splcement from ntl poston to esre poston n the secon pproch s bse on the reltve jont rotton mnmzton. Both consere pproch s solvng together usng genetc lgorthm to fn out the globl soluton. Km n Km [0] propose trjectory plnnng of -of revolute mnpultor usng evolutonry lgorthm. In ths work they hve frst clculte optml nverse knemtc of -of reunnt mnpultor usng Jcobn mtrx metho. The optmzton objectve functon s selecte on the bss of jont n en-effector splcement from ntl rtesn coornte to esre locton, then evolutonry lgorthm s pple to fn out optml jont vrble. Pzzl nvsols et l. [07] propose nverse knemtcs soluton n trjectory plnnng for D-jont robot mnpultor bse on etermnstc globl optmzton bse metho. In ths work they clculte nvers knemtc to fn out the esre trjectory of -of mnpultor. They hve pple ntervl nlyss lgorthm for the globl optmzton of the pecewse moton of jont vrbles. Ahuctzn n Gupt [08] propose nverse knemtc soluton of reunnt mnpultor usng novel evelope globl optmzton lgorthm. In ths work they hve use the evelope lgorthm for pont to pont movement of en effector n then clculte the splcement error usng the propose INVIKIN lgorthm. The concept of the work s bse on the Arne s lew Algorthm (AA) whch s bsclly relte to moton plnnng. 9

91 hpelle n Bu [09] propose nverse knemtc soluton of PUMA robot mnpultor usng genetc progrmmng. In ths work, mthemtcl moellng s evolve usng genetc progrmmng through gven rect knemtc equtons. They hve represente the evolutonry symbolc regresson proceure for the nverse knemtcs of GMF Arc Mte n PUMA mnpultors. Khtm n ssn [0] propose knemtc sotropy for the performnce evluton of the -of mnpultor, where Globl sotropy Inex hs been use to mesure of the bove sotropy n epens on the entre workspce of the mnpultor. Genetc lgorthm s use to optmze the esgn prmeter of the mnpultor n the prmeter s lnk length. Lter, ths pproche s employe to optmze globlly throughout the mnpultor workspce. Klr et l. [] propose nverse knemtc soluton of -of rtculte robot mnpultor usng rel coe genetc lgorthm. In ths work they hve use Eucln stnce norm for the optmzton of jont vrble of robot mnpultor. Dsplcement error mnmzton objectve functon s subjecte to jont ngle constrnt n ths work, n bsc steps of rel coe genetc lgorthm re recombnton n mutton. Koren n Bler [] propose nverse knemtc soluton scheme of -of reunnt mnpultor bse on rech herrchy metho. In ths work they hve formulte nverse knemtc nlytcl n then usng Lgrngn multplers for mkng the problem wth equlty constrnt. Therefter they use numercl bse optmzton metho for mnmzng the propose objectve functon. Tbneh et l. [] propose nverse knemtc soluton of -of PUMA mnpultor for the mjor splcements propose. In ths work they hve opte genetc lgorthm wth ptve nchng n clusterng. Genetc lgorthm's prmeters re set by ptve nchng metho whch s lter requre the forwr knemtc equtons for the soluton of nverse knemtc of opte mnpultor. Forwr knemtc s smply clculte by stnr nlytcl metho. Therefter for processng the output flterng n clusterng s lso e to the genetc lgorthm. He et l. [] propose nverse knemtc soluton of -of MOTOMAN robot mnpultor for postonng of the en-effector. In ths work they hve opte ptve genetc lgorthm for optmum plcement of the en effector. There re severl prmeters lke en-effector splcement error crter, welng rechblty nex, moton stblty nex n exterty nex hve been consere for mkng of objectve functon. Rjpr et l. [] propose nverse knemtc n trjectory generton of humno rm mnpultor usng forwr recurson wth bckwr cycle computton metho. In 70

92 ths work DH-lgorthm s use to formulte the forwr knemtcs of humno rm mnpultor whch s lter use s n objectve functon for the optmzton process. En effector splcement n the orentton error re completely use s objectve functon for ths work. Lu n Zhu [] propose nverse knemtc soluton for -of revolute mnpultor usng rel tme optmzton lgorthm. DH-lgorthm s use to formulte the knemtcs equtons whch re lter reuce by the symbolc pre-processng. Lter Egen ecomposton s use to extrct roots from hgher egree polynoml knemtc equtons. Jryn [7] propose nverse knemtc soluton of -of robot mnpultor usng vrtul potentl fel metho. In ths work, set of ponts between ntl ponts to esre pont s obtne by en-effector wth fferent vrtul potentl fel metho. An optmum trjectory s crete by usng pttern serch metho whch explns the power of the potentl fle metho to optmze the vlue of generte objectve functon. In ths work cubc splnes re use to crete smooth trjectory jont spce obtne through nverse knemtc equton. Fnlly the effcency n effectveness of the opte metho s presente through smultons. Phm et l. [8] propose nverse knemtc soluton of -of robot mnpultor usng Bee lgorthm. In ths work they hve compre three fferent methos lke evolutonry lgorthm, neurl network bck propgton metho n bee lgorthm. Neurl network structure s optmze by usng bee lgorthm to prect jont vrbles of the robot mnpultor. Albert et l. [9] propose nverse knemtc soluton of -of revolute robot mnpultor usng rel tme genetc lgorthm. In ths work en-effector splcement form ts ntl pont to esre pont hs been optmze usng genetc lgorthm. Genetc lgorthm crossover selecton s bse on new metho whch s known s ynmc mult-lyere chromosome (DM) to prouce two offsprng's. The GUI smulton hs been verfe wth GA n DM. Hung et l. [0] propose nverse knemtc soluton of -of robot mnpultor usng mmune genetc lgorthm. In ths work forwr knemtc formulton s presente usng DH-lgorthms. En effector splcement error bse ftness functon s use for mplementton of propose lgorthm n results obtne through opte technque re compre wth neurl network bck propgton lgorthm. Blón et l. [] propose nverse knemtc soluton for the -of robot mnpultor usng genetc lgorthm. In ths work eght egree of polynoml equton s use to c pln trjectory for the robot mnpultor wth the help of DH-lgorthm. Genetc 7

93 lgorthm s use for the energy optmzton s well s trjectory optmzton of robot mnpultor. Prmn [] propose nverse knemtc nlyss of -of robot mnpultor compoun numercl optmzton metho. In ths work generl nlytcl soluton s fuse wth the numercl bse metho to solve nverse knemtc of -of mnpultor. The fuson pproch s gettng r of wth the problem of repetng vlue of numercl soluton n generlly tht gves slow convergence. Therefore n ths work combnton of nlytcl n numercl soluton for hgher orer polynoml functon s me. Le-png et l. [] propose nverse knemtc soluton of -of robot mnpultor usng genetc lgorthm. In ths work obstcle vonce s mjor crter, to vo the obstcle t s requre to clculte nverse knemtcs of the mnpultor. Then ths knemtc equton s moelle s n en effector splcement error bse ftness functon of the genetc lgorthm. MATLAB softwre s use to smulte the opte problem. Rubo et l. [] propose optmzton of pth plnnng of PUMA 0 robot mnpultor usng genetc lgorthm. In ths work severl fferent crter for ftness functon hve been tken to solve the pth plnnng of the robot. A frst crteron s splcement of en effector from ts ntl poston to fnl or esre poston, secon crter s bse on ts confgurton. The genetc lgorthm uses prllel popultons wth the mgrton for pth plnnng. Glck [] propose nverse knemtc soluton of moble mnpultor usng penlty functon bse optmzton metho. Ths work presents soluton on control feebck level whch s subject to stte equlty n nequlty constrnt for the opte mnpultor. In ths work Lypunove stblty constrnt s use for the control trjectory generton v nverse knemtc soluton. vr n Mln [] propose nverse knemtc soluton of -of PUMA mnpultor usng rtfcl bee colony lgorthm. In ths work DH-lgorthm s use to formulte ftness functon for the evluton of en effector trget poston bse on ntl gven poston. Llo et l. [7] propose nverse knemtc soluton of -of plnr mnpultor usng lner progrmmng metho. In ths work the mn e of genertng the objectve functon s bse on the mnmzng the jont vrbles to rech the esre locton. Lter secton els wth the sngulrty of opte mnpultor n they hve lso presente the formultons for the smooth trjectory generton for the -of plnr mnpultor. 7

94 Bernl et l. [8] propose metheurstc lgorthm pplcton n robotcs. In ths work, nt colony optmzton lgorthm n genetc lgorthm re use for pth plnnng of robot mnpultors en effector. The work s complete wth nturl selecton n evoluton, through two type of nts nmely job n explorer. The bsc prmeter of genetc lgorthm s hybrze wth nt colony optmzton to get globl soluton.. ubero [9] propose nverse knemtc soluton of serl mnpultor usng bln serch metho. In ths work stnr nlytcl soluton for forwr knemtcs s requre to prepre the ftness functon even ths cn be ccomplsh wth DH lgorthm. Konetschke n Hrznger [0] propose nverse knemtc soluton of hghly reunnt mnpultor wth combne optmzton lgorthm. In ths work close form soluton of nverse knemtcs of Justn robot s propose n lter t s combne wth the propose optmzton lgorthm for the globl; soluton. The propose optmzton lgorthm s bse on Levenberg-Mrqurt crter. Zhng et l. [] propose nverse knemtc soluton of -of robot mnpultor usng hybr genetc lgorthm metho. In ths work mechnsm n boy frmes re presente bse on the DH-lgorthm, whch s lter use to formulte the objectve functon. The objectve functon s bse on the en-effectors ntl poston to the esre poston splcement error mnmzton. Hung et l. [] propose nverse knemtc soluton of 7-of sptl mnpultor bse on the prtcle swrm optmzton (PO) technque. In ths work DH-lgorthm s use to formulte the forwr knemtc of the 7-of mnpultor whch s lter use to formulte the objectve functon for the prtcle swrm optmzton technque. En effectors; ntl poston esre poston bse splcement error long wth the orentton error s mnmze usng PO. Kumr et l. [] propose nverse knemtc soluton of reunnt mnpultor usng Lypunov metho. In ths work, optmzton pproch to solve nverse knemtc problem whch s converte nto nonlner problem solve by Lypunov metho. An mprove energy bse functon s etermne for the optmzton. Xu et l. [] propose nverse knemtc of the -of reunnt mnpultor usng two fferent optmzton crter. Frst optmzton crter s mnmzton of extr reunnt of n other crter s bse on totl potentl energy mnmzton of mnpultor lnks. They hve evelope numercl optmzton metho for clcultng the trjectory plnnng computton whch s bt more expensve. Therefore to overcome ths computton cost sequentl qurtc progrmmng n tertve Newton-Rphson metho s use. 7

95 Mzhr n Kumr [] propose knemtcs n ynmcs soluton of PUMA 0 robot mnpultor usng genetc lgorthm, smulte nnelng, n generlze pttern serch methos. They hve esgne controller for PUMA mnpultor usng bove opte lgorthms. Fne tunng s requrng for controller to cheve esre spee of smultons. MATLAB/mulnk softwre s use for smultons n ths work. Rmírez n Rubno [] propose nverse knemtc soluton of -of revolute sptl mnpultor usng genetc lgorthm. Forwr knemtcs formulton hs been complete by usng DH-lgorthms n homogeneous mtrx multplcton bse metho. Ftness functon for the nverse knemtc soluton s bse on the en effectors ntl n esre poston error whch s lso known s Euclen stnce norm. Feng et l. [7] propose nverse knemtc soluton of -of generl robot mnpultor usng Electromgnetsm-lke n mofe Dvson-Fletcher-Powell (DFP) metho. In ths work DH-lgorthm s use to formulte the forwr knemtcs of the -of generl robot mnpultor. The objectve functon s bse on the splcement error n orentton error of the en effector. The n totl combnton of the both error s use s ftness functon for the opte technque. DFP metho s hybrze usng EM lgorthm to get the best convergence rte. Henten et l. [8] propose nverse knemtc nlyss of 7-lnk robot mnpultor for the cucumber pckng operton usng nlytcl n numercl lgorthm bse methos. In ths work stnr DH-lgorthm s use for the nverse knemtc soluton of robot mnpultor n lter ths nlytcl metho s fuse wth the numercl nlyss bse lgorthm to get the optmum soluton of the robot mnpultor. Rokbn n Alm [9] propose nverse knemtc soluton of -of robot mnpultor usng prtcle swm optmzton lgorthm. In ths work ntl poston n esre poston error bse objectve functon s use whch s lso known s the Euclen stnce norm for en effector. In ths pproch norm nlytcl soluton of forwr knemtc s presente whch s lter use n objectve functon of PO lgorthm. Dutr [0] propose nverse knemtc soluton of -lnk plnr mnpultor usng smulte nnelng metho. In ths work stnr nlytcl soluton of forwr knemtc s presente usng forwr knemtc equton the splcement bse error mnmzton objectve functon s use for the smulte nnelng pproch. Zhng et l. [] propose nverse knemtc soluton of the serl ngerous rtcles sposl mnpultor wth multple egrees of freeom usng prtcle swrm gene optmzton lgorthm. In ths work poston n orentton error of en effector s use s objectve functon for trtonl PO n mofe PGO metho. 7

96 Rokbn n Alm [] propose nverse knemtc soluton of the bpe robot leg usng prtcle swrm optmzton (PO) metho. In ths work, two legs wth -of re consere for the optmzton purpose. Moreover both leg mnpultor's forwr knemtc s clculte nlytclly to obtn ftness functon for the PO. Luo n We [] propose knemtc nlyss of -of plnr reunnt mnpultor usng two fferent technques lke mmune bse n mmune genetc lgorthm methos. They hve clculte the forwr knemtc nlytcl for the pth plnnng of robot mnpultor. In lter secton mmune n mmune genetc lgorthm s use to evlute the effcency n performnce of the obtne soluton. Tylor et l. [] propose knemtcs n ynmcs of the -of plnr n sptl mnpultor bse on the complex optmzton metho. In ths work they hve use optmzton lgorthm for the evluton of the trjectory plnnng n nverse knemtc moellng of the -of mnpultor. For trjectory plnnng cubc splnes re use for the formultons. It hs been conclue tht the complex optmzton lgorthm s effectve n performng better for pth evlutons. Števo et l. [] propose nverse knemtc soluton of -of ABB IRB 00FHD robot mnpultor usng genetc lgorthm. In ths work forwr knemtc equton re generte by usng DH-lgorthm whch s lter use to obtn ftness functon for genetc lgorthm. In ths work splcement error of en-effector from pont to pont moton hs been clculte through opte metho. Objectve functon contnng three seprte prts whch re energy functon, operton tme n poston ccurcy to get combne ftness functon for genetc lgorthm.. Revew nlyss n outcomes Focusng the ttenton on the mnpultors confgurton n ther knemtcs, the revew epcts tht nverse knemtcs hs been trete s the gol mne for robot esgners. Numerous reserchers hve tre to evelop nverse knemtc soluton from lte 80's untl now wth vrous pproches n for vrous confgurtons of robot mnpultor. From the begnnng robot mnpultors re beng use for vrous nustrl pplctons lke pck n plce type work etc., so the mjor constrnt ws to fn jont vrbles of the mnpultor to rech the esre poston wth known object coornte ponts. Now y, robot mnpultors pplctons re wene long wth the nustrl pplctons to perform n vrous fle lke mecl rehbltton, uner wter pplctons, ssembly tsk, grculture, mnng, spce etc. long wth the humn nterctons. From the lterture revew t cn be summrze tht to cheve esre poston n orentton of the en-effector or tool long wth mnpulblty, exterty 7

97 n trjectory plnnng, the nee of nverse n forwr knemtcs rses. Almost ll revewe rtcles ncte tht the humn rm s the key pont of motvton n les to esgn of robot mnpultor. Now from esgn pont of vew t s requre to clculte knemtcs reltonshp of ech jont vrbles so tht the optmum esgn cn be obtn. There my be numerous confgurtons or structures of mechnsm or robot mnpultors but the mjor propertes of esgnng of ny mechnsm or mnpultor re knemtc nlyss, workspce nlyss, nthropomorphc vent, mnpulblty, trjectory generton n control. It s lso observe tht concernng wth the pplctons of the robot mnpultor the explct propertes re lwys gven mportnce. Workng spce n mnpulblty of robot mnpultor ncreses when number of jont vrbles ncreses whch generlly cuse more complex mthemtcl formultons for nverse knemtc resolutons n ffculty n control of the mnpultor. Numerous esgns n confgurtons of the robot mnpultors re beng use n mny humn envronment s well s nustrl pplctons. The mn m of the lterture survey s to explore fferent technques n methoologes vlble to solve nverse knemtcs of ny confgurton of robot mnpultor. But t s perceve from lterture revew tht DH-lgorthm, homogeneous trnsformton mtrx, nlytcl pproch, lgebrc pproch n geometrc pproches re mostly followe by vrous reserches. Among ll the evelope methoologes the most frequently use pproch s lgebrc soluton of nverse knemtc. Ths metho covers conventonl lgebr long wth quternon, ul quternon, quternon vector pr, screw lgebr, lffor lgebr n Le lgebr. DHlgorthm n ssocte prmeters re the best wy of representton rotton n trnslton of mnpultor lnks n jonts. The mjor rwbck n cse of conventonl lgebr s ts complextes n moellng n obtnng pproprte solutons when the robot confgurton s complex n hs lrger egree of freeoms. Ths problem of hgher mthemtcl formultons ws reuce by usng quternon n screw lgebr. Therefter few elmnton methos for reucng the complexty of nverse knemtcs formultons rse wth ther effectve performnce. In cse of geometrc lgebr the mjor problem s when the frst three jonts of ny mechnsm or mnpultor o not crete ny jont ngle n between them, n then t oes not gve exct soluton for the nverse knemtc problem. Moreover, the problem becomes unstble f the Jcobn mtrx s n ll conton or sufferng from sngulrty. Therefore, the conventonl metho re relble but there s lwys mthemtcl complexty problem rses wth the confgurtons n of's of mnpultor. To overcome these problem reserchers opte mny ntellgent technques for soft computng omn such s rtfcl neurl network technque, fuzzy logc, hybr ANN etc. These methos o not requre hgher 7

98 mthemtcl progrmmng n computton cost s lso less. Aprt from these, optmzton pproch lke heurstc, metheurstc, numercl bse pproch, etc. hve shown ther effcency n solvng nverse knemtc problem for ny confgurton of robot mnpultor. However, there remns scope to nvestgte further n work towrs fnng better solutons. Most of the optmzton lgorthms o not gve globl optmum pont becuse of trppng n locl optmum pont. Therefore, t s mportnt to work n to evelop n lgorthm so s to cheve globl optmum for the ftness functon.. Problem sttement The prme objectve of the present reserch work s to evelop n recommen n pproprte soluton technque for the nverse knemtc problem of nustrl robot mnpultor wth vew to obtn only fewer solutons tht coul be prctclly hnle n use. Further the obtne solutons shown to be optml n precse wth respect to orentton n poston. The evelope technque shoul yel fster results so s to mke t sutble for rel tme pplctons.. cope of work The evelopment n the fel of robots n s ever ncresng opton n nustres hs let to brng out mny esgn n opertonl chllenges. Reserchers re nvng lrge number of mcro s well s mcro problems to mke the robot system s user-frenly s possble. Every sngle component of the robot technology hs been, therefore, wene to prove reserch nterest n multple rectons. Wth lrge number of robot mnpultor confgurtons hvng ther own complexty/ smplcty, the evelopment of the opertonl coes hs been n nterestng n chllengng re of reserch. Focusng on nustrl robots n vogue, the present reserch work s envsge wth the followng scope of the work. The etl pln for the reserch work s gven s: Bse on the revew n nlyss of prevous lterture fferent confgurtons of nustrl robot mnpultor hve been chosen. In orer to nclue ll typcl confgurtons, the set of mnpultors conssts of rg s well s sem flexble confgurtons, the egree of freeom rngng from to. The knemtc moellng n nlyss woul use mthemtcl s well s ntellgent heurstc, sngle n hybr n orer to fn out ther sutblty n vew of ther moellng smplcty n soluton effcency. nce lrge numbers of such tools re vlble n the lterture, the present work ms t only lmte 77

99 to ol tools n new tools. The ol tools woul be pcke up on the bss of ther performnce n smlr stutons, wheres the new n hybr tools woul be chosen on the bss of ther fetures tht coul mtch the chrcter of the propose problem. The very purpose of ths work s lmte to only evelopng sutble methoology for solvng the nverse knemtc problem wth reltve ese n by checkng wth exstng tools n few recently evelope one nclung hybr ones.. ummry A bro stuy of lterture revews from ll ccessble sources n concerne sprghtly or nrectly wth the present prt of work hs been me. ome of the tonl sgnfcnt work hs been extrvgntly revewe so s to expn n recton of the reserch n ths re of work. Lterture from the pst tll present tme were explore n observe to fn out the exstence of present work scope for supportng the current work. A we-rngng preprton n presentton hs been covere throughout the complete work for the ssstnce of the reers. 78

100 hpter MATERIAL AND METHOD. Overvew In orer to nvestgte n compre the nverse knemtc soluton of the robot mnpultor, t s requre to select pproprte robot mnpultor confgurton. A robot mnpultor cn be consere s group of rg lnks or boes whch re connecte by specfc jonts. Jonts my be revolute, prsmtc, screw, unversl or cylnrcl etc. These jonts prove reltve movement n between the rg boes or lnks. Frst lnk s consere to be jone t the bse of robot mnpultor whle the lst lnk s free to move wthn the lmt of workspce. In ths work, some benchmrk mnpultors hve been consere n such wy tht the jont shoul possess -of n jonts re ether revolute or prsmtc. A revolute jont gves freeom to rotte bout ts xs, whle prsmtc jont offers jont to sle long the xs wthout ny rotton. The selecte benchmrk confgurtons of robot mnpultor hve been escrbe n the lter secton.. Mterls Robot mnpultors re generlly ctegorse ccorng to ther knemtc structure of open or close chns. In chpter, fferent types of robot mnpultors hve been escrbe. Despte of the types of robots, t s lso requre to mke the clssfcton on the bss of jont n lnks s explne erler. The mn focus of the reserch s prmrly on nustrl robot mnpultors of whch smplest confgurton re -of revolute robot. onserng the eployment of robots n nustres for vrous tsks, t s pprent tht robot mnpultor wth ARA confgurton n revolute robot wth - of re mostly use. Therefore, t hs been plnne to conser only these vrety of robot mnpultors, for bettng the propose reserch work n elbertng on the 79

101 vrous ssues roun the problem. Therefore, some selecte confgurtons of robot mnpultors hve been consere for the propose knemtc nlyss (see Tbles.). Tble. onfgurtons of robot mnpultors N. Knemtc tructure Degree of freeom Jont confgurton Knemtc moton erl RRR plnr erl RRPR ptl erl RRRRR ptl erl RRRRRR ptl In ths work both rg n flexble type robot mnpultors wth serl structure re consere for the knemtc nlyss. The serl robot mnpultors re extensvely use n nustrl pplcton ue to the fct tht they offer reltvely lrge work envelope s compre to prllel robots wth compct structure. Ths thess contns seven fferent types of robot mnpultor bse on the confgurtons from Tble.. The selecte mnpultors re -of revolute plnr, -of Aept One ARA, -of, Prm, -of AEA IRb-, -of PUMA 0, -of, ABB IRB-00 n -of TAUBLI RX 0 L. -of nustrl mnpultors re selecte from Tble. whch re type A, A, B n type of nustrl robots. The consere mnpultors re gven s: () -of revolute plnr mnpultor wth RRR confgurton (b) -of Aept One ARA mnpultor wth the jont confgurton of RRPR (c) -of Prm revolute mnpultor wth the jont confgurton of RRRRR () -of AEA IRb- robot mnpultor (e) -of PUMA 0 mnpultor wth revolute confgurton (f) -of ABB IRB-00 robot mnpultor (g) -of TAUBLI RX 0 L robot mnpultor The bove escrbe robot mnpultor confgurtons re the founton for the reserch work. From the lst mny eces resercher re workng on these ctegores of robot mnpultors s explne n prevous chpter. One of the most funmentl n mportnt problem for the postonng of the robot mnpultor s knemtc nlyss. Therefore, n ths work fferent confgurtons of robot mnpultors re selecte for the knemtc nlyss. For the knemtc nlyss of robot mnpultor one shoul strt wth the bsc -of revolute mnpultor. On the other hn, -of, - of n -of mnpultors re mostly preferre n nustrl pplctons ue to ts 80

102 hgh exterty n lrge workspce. The etl escrptons of the selecte mterls re presente n the subsecton... Descrpton of plnr -of revolute mnpultor A plnr robot mnpultor s cn be me of serl chns wth revolute or prsmtc jonts. Plnr -of revolute mnpultor s bsclly constructe by three revolute jonts. All the lnks or rg boes of serl chn re constrne to rotte n sme plne or prllel to ech other. A plnr mnpultor cn only hve revolute or prsmtc jonts. Inee the xes of ll revolute jonts shoul be perpenculr to the plnr chn whle the xes of prsmtc jont shoul lwys prllel to the plnr chn. Jont vrbles n prmeters of -of plnr mnpultor re gven n Tble.. Mn m of ths chpter s to prove etls stuy of selecte mnpultors for the knemtc nlyss n poston of the en effector t the esre pont. Ths secton els wth the fferent types of plnr mnpultor n selecton of pproprte plnr mnpultor for the knemtc nlyss. Fgure. Moel of -of revolute mnpultor Tble. Mnpultor jont lmts n knemtc prmeters l. (egree) (mm) (mm) (egree) = = = 0 0 The mthemtcl moellng of hgher of or sptl mnpultors s qute lengthy n tme consumng. Plnr mnpultors re smple to fgure knemtc reltonshp s well s for mthemtcl moellng. The plnr mnpultors exmples represent the founton for esgnng, knemtc nlyss n for controllng purpose wthout consumpton of tme n mthemtcl expressons. However, ths els wth the knemtc nlyss of plnr mnpultor but the sptl escrpton cn lso be 8

103 prolonge. We wll strt wth the exmple of the plnr -of revolute mnpultor s shown n Fgure.. There re mny nustrl mnpultors vlble whch resembles -of revolute plnr confgurton. For exmple, swvel of shouler, extenson of elbow n ptch of ncnnt Mlcron T mnpultor cn be trete s -of plnr mnpultor. mlrly, n cse of ARA mnpultor wthout conserng of prsmtc jont wll resemble the -of revolute mnpultor just to move en effector n up or own poston. Thus, t s useful to conser -of revolute plnr mnpultor for the nverse knemtc nlyss. The -of revolute plnr mnpultor cn be geometrclly specfe wth the lnk lengths, n. These lnks length re bsclly vrbles whch epen on the confgurton of robot mnpultor. The lnks lengths cn be efne n mny wys but the precse wy s the most stl lnk from stl jont xs to the en effector pont or tool pont. Other mportnt vrbles re coornte ponts of the en effector whch represents the poston n orentton of the en effector. The postons re efne s the coorntes (X n Y) whle orentton cn be efne s ngle. The overll vrbles ( X, Y n ) efnes the pose (poston n orentton) of the en effector. The proper efnton of these vrbles n prmeters cn be foun n the next chpter. The other possble confgurton of plnr mnpultor cn be R-P, P-P, n P- P-P. In ths thess -of revolute plnr mnpultor s consere for the further knemtc nlyss n the etl mthemtcl moellng of the mnpultor s presente n next chpter... Descrpton of -of ARA mnpultor The secon selecte confgurton for forwr n nverse knemtc nlyss s Aept One ARA mnpultor. The ARA (electve omplnt Assembly Robot Arm or electve omplnt Artculte Robot Arm) hs n RRPR structure. Ths mnpultor hvng jont xes consstng three revolute n one prsmtc jont whch s unlke from the sphercl robot mnpultor wth fferent pplctons. The jonts frst, secon n fourth re revolute n ther jont s prsmtc see Fgure. for overvew. The jont vrble n relte knemtc prmeters for nverse knemtc soluton re presente n Tble.. The jont motons of Aept One ARA mnpultor cn be escrbe s: () Jont moton Jont whch s lso known s shouler swvel gves the freeom for rotton of nner lnk n the column n the rnge of the rotton s

104 (b) Jont moton econ jont s lso referre s elbow jont whch s pvot pont n between nner lnk n outer lnk. The rnge of the moton s 9 0. Ths jont s responsble for the lefty n rghty confgurton of the mnpultor. (c) Jont moton The thr jont gves the vertcl trnslton of the qull t the en n outer lnk wth the stnr stroke of 9mm (optonl jont stroke my be vry up to 9mm). () Jont moton The lst jont s known s wrst jont whch proves the rotton of the qull wth the rnge lmte to 0. Ths jont moton s lke humn hn moton for unscrewng bottle cp or tghtenng bolt. Fgure. tructure of Aept One ARA mnpultor Tble. Mnpultor jont lmts n knemtc prmeters l. (egree) (mm) (mm) (egree) θ =±0 0 =0 0 θ =±0 0 = =0 0 0 θ =0 0 0 Ths mnpultor hvng one prllel shouler, one elbow n rottory wrst jonts long wth one lner vertcl xs for trnslton wrst. These confgurtons of robot 8

105 mnpultor re mostly use n lght uty pplctons ue to the hgh spee n precson of the mnpultor. ommon pplcton res re: electronc prt ssemblng, prntng of crcut bors, ssembly of tny prts of electromechncl evce, ssemblng of sk rvers. The ARA mnpultors re very compct n esgn n work spce s comprtvely lmte to less thn 000mm. But the pylos of the mnpultors re rnge to 0-00kg. Therefore, n orer to complete the knemtc nlyss n performnce, Aept One ARA mnpultor s selecte for reserch... Descrpton of -of revolute Poneer mnpultor The thr confgurton consere for the knemtc nlyss s Poneer mnpultor wth jont rottons. If the mnpultor s reunnt or hvng hgh of, thn conventonl soluton for nverse knemtc problem becomes more complcte. Therefore, conserng newly evelope Poneer mnpultor wth jont rottons for the knemtc nlyss s shown n Fgure.. Ths mnpultor s compct, low cost n lghtweght for the use n reserch s well s n cemc purpose. The ctuton of the jont s rven by open loop servo motors n grpper whch s ttche to the en of the lst lnk of mnpultor. Fgure. tructure of the Pnoneer rm The mjor pplcton of ths robot s for grspng n mnpulton of objects lke so cns up to the weght lmt of 0grms wthn the workspce. Jont of the Prm re; Jonts rottons: bse rotton shouler rotton 8

106 elbow rotton wrst rotton grpper mount grpper fngers All jonts re rven by servo motors except grpper fngers. The jont lmts n prmeters tken for the reserch hs presente n Tble.. Jonts 0 Tble. Prm mnpultor jont lmts n knemtc prmeters. (egree) (mm) From Tble. prmeters n jont vrbles re lste n the rnges re presente. These prmeters n jont vrbles re bss for the forwr n nverse knemtc nlyss of robot mnpultor. Lter usng MATLAB progrmmng the t sets for nverse knemtc soluton wll be use. 0.. Descrpton of -of PUMA 0 mnpultor The fourth mterl for the knemtc nlyss s PUMA 0 (Progrmmble Unversl Mchne for Assembly, or Progrmmble Unversl Mnpulton Arm) whch s n nustrl robot wth sx xs jonts see Fgure.. Tble. Mxmum lmt of jont vrbles. The en effector of the PUMA 0 robot s esgne to operte nomnl lo of.kg wth 0.mm postonl repetblty. The workspce of ths mnpultor s 0.9m from the centre xs to wrst centre n the mxmum en effector velocty reches m/s. All 8 (mm) 80 = 0 = = = = Jonts Lmts (egree) Wst 0 houler Elbow 8 Wrst ptch 00 Wrst roll 80 Wrst yw (egree)

107 sx jonts re ctute through the brushe D servo motors. The jonts lmts n prmeters re gven n Tble. n Tble.. Fgure. tructure of PUMA 0 robot mnpultor [0] Tble. Jont vrble n prmeters of PUMA 0 robot Jonts 0 (egree) 0 (m) (m) to to =0. = =0.8 = (egree) PUMA 0 robots re most use n hnlng of smll objects or prts n nustrl ue to ts compct esgn, hgh spee rto, repetblty n flexblty. Most complcte pplcton or ssembly of ntrcte prts cn be one by PUMA 0 robot. For exmple PUMA 0 robot cn be use for ssemblng of utomotve pnels, smll electrc motors, crcut bor prntngs, pplnces n so on. From Tble. jont vrbles n prmeters for DH-lgorthms wll be use to clculte the forwr n nverse knemtc of PUMA 0 mnpultor. Lter the generte t sets wll be nput for the ANN moels trnng n testng... Descrpton of -of ABB IRb-00 mnpultor The ABB IRB 00 s -of nustrl robot whch s speclly ese for the mnufcturng nustres. The confgurton of the mnpultor s -of revolute wth 8

108 rg structure see Tble.. Due to ts open structure t s esly opte for the flexble utomton use n lso flexble communcton wth externl systems. Ths type of robot mnpultor s known s nthropomorphc wth -of mechnsm. The 0 0 shouler jont wth roll n ptch motons moves the upper rm 70 n 70 ; the elbow jont wth ptch ctons rves the forerm 0 70 to 0 ; n the wrst roll n ptch rottons together wth the tool-plte roll move the hn (see Fgure.). The jont lmts n ssocte prmeters re lste n Tble.7 for ABB IRb-00 mnpultor. Jonts Fgure. onfgurtons of ABB IRB 00 Tble.7 ABB IRB-00 mnpultor jont lmts n knemtc prmeters (egree) (mm) to 0 (m) (egree) Ths type of robot mnpultor s mostly use for the rc welng process. The mjor vntges of ths type of mnpultor re ts rveblty, repetblty, ccurcy wth zero bcklsh n hgh resoluton wth nomnl pylos. Aprt from ts techncl vntges, t s commonly use n nustres n reserch work. Therefore, ths robot mnpultor s selecte for the forwr n nverse knemtc nlyss. 87

109 .. Descrpton of -of AEA IRb- mnpultor The AEA IRB s -of nustrl robot mnpultor whch llows movement n - xs wth mxmum lftng cpcty of kg. Ths type of mnpultor re commonly use n nustres n reserch work. The confgurton of the mnpultor s -of revolute wth rg structure see Tble.. The structure of the mnpultor s rg wth mxmum rech of mm. Due to ts hgh exterty t s ccepte n nustres for mterl hnlng, pckgng, pck-n-plce object, semblely etc. The bsc moel of ths mnpultor s presente n Fgure.. The shouler jont wth roll n ptch motons moves the upper rm to0 ; the elbow jont wth ptch ctons rves the forerm 0 to to0 n 0 0 to0 to ; n the wrst roll n ptch rottons together wth the tool-plte roll move the hn. The jont lmts n ssocte prmeters re lste n Tble.8 for AEA IRb- mnpultor. 0 Jonts Fgure. onfgurtons of AEA IRb- Tble.8 AEA IRb- mnpultor jont lmts n knemtc prmeters (egree) (m) to (m) to (egree) Ths mnpultor s bsclly esgne to work on utomte hnlng of grnng operton n lter t becme populr for mny other pplctons such s mterl hnlng, pckgng, ssemblng etc. The structure of ths mnpultor s compct n

110 rg. Ths type of mnpultor s lso ccepte for reserch work. Therefore, n ths reserch work, AEA IRb- robot mnpultor s selecte for the knemtc nlyss...7 Descrpton of -of TAUBLI RX0L mnpultor The täubl RX0L nustrl robot mnpultor s esgne to perform mny nustrl pplctons such s mterl hnlng, welng, spryng, n ssemblng n lso for reserch work. The structure of ths mnpultor s rg wth -xs of rottons. The mn feture of ths robot s hgh exterty n flexblty n nustrls pplctons. The täubl RX0L s resembles the humn hn exterty wth -of revolute jonts. Fgure.7 onfgurtons of TAUBLI RX0L The mjor splcement jont ngles re smlr to PUMA, IRB-00 but t llows more workspce s compre to other opte mnpultor. The mxmum pylo of ths mnpultor s 8 kg n nomnl lo s kg. Mxum rech of ths mnpultor n between xs to xs s 00mm whch llows more work envelope. The bsc moel of ths mnpultor s presente n Fgure.7 n jont vrble wth knemtc prmeters re presente n Tble.9. Jonts Tble.9 TAUBLI RX0L mnpultor jont lmts n knemtc prmeters. (egree) (m) (m) (egree)

111 . Methos The nverse knemtc robotcs problem hs been the focus of knemtc nlyss for robot mnpultors. In orer to etermne ll possble formtons to plce the en effector of robot mnpultor t prtculr pont n spce, one must compute the movements ssocte wth ech jont vrble. In ong so, over the spn of severl eces, uthors hve fce the followng ffcultes: The complexty of the nverse knemtc robotcs problem s etermne by the geometry of the robot mnpultor. Geometrc solutons epens on the frst three jonts shoul be geometrclly exst. ome clcultons to solvng the nverse knemtc problem cnnot be compute n rel-tme. It s not lwys possble to obtne close form or sngle solutons. It s lso ffcult to fn rels solutons for some confgurton of robot mnpultor. Algebrc solutons; knemtc equtons trnsform nto hgher orer polynoml n tngent of hlf ngle of jont ngels n then ll the roots of polynomls re numerclly etermne. ( -of mnpultor= th orer polynoml) Numercl solutons- when the Jcobn mtrx s sngulr (ll contone) or ntl pproxmtons s not ccurte then there wll be unstble soluton Mn m of ths thess s to fn out the jont vrbles or nverse knemtc solutons for the selecte benchmrk mnpultors. The mjor chllenge to clculte nverse knemtc problem s, t follows the non-lner trnscenentl equton wth complex mthemtcl formultons. The conventonl pproches re tme consumng s well s ffcult to unerstn s explne erler. Therefore, fter conserng ll the stte problems the mn objectve s to select the pproprte metho for the clculton of nverse knemtc problem. The opte methos n steps for chevng objectve hve been plnne s follows:.. onventonl pproch In the present reserch work DH-lgorthm, homogeneous trnsformton n quternon vector bse methos n ther sgnfcnce for the knemtc nlyss hve been consere. Mthemtcl moellng of the forwr n nverse knemtc problem of rg s well s sem-flexble robot wth to jont xs s one. To reuce the methemtcl complextes n computtonl tme quternon vector bse knemtc formultons hve been one for selecte confgurtons of the robot 90

112 mnpultor. The conventonl knemtc equtons of the open chn mnpultor re trnsforme nto consecutve quternon trnsformtons mtrces n then rtculte usng quternon. The Euler ngle representton contns three ngles whch s not enough for regulr representton n mostly trppe n sngulrty problem. Therefore, to overcome the problem of regul representton n for sngurty vonce qutrnon lgebr s much poerful. The number of mthemtcl oprtons cn be reuce by quternon vector bse metho. From the comprtve results of homogeneous trnsformton methos wth the quternon bse pproch, mthemtcl opertons re more n cse of homogeneous trnsformton metho. To mntn the ccurcy of the obtne soluton n reuce mthemtcl opertons, quternon bse pproch re much better. It cn be clerly unerstoo tht the quternon vector bse metho elvers very effectve n effcent tool s compre to other conventonl pproch. Further, ths pproch s cost effectve ue to ts less mthemtcl opertons. omprng wth the homogeneous trnsformton methos, t cn be observe tht quternon metho prouces sme results wth less tme consumpton. Therefore, ths metho cn be pple to ny confgurton of robot mnpultor. Ths cn be use s generl tool for the knemtc soluton of n-of robot mnpultor... Intellgence bse pproches The secon pproch s bse on the soft computng methos such s rtfcl neurl network, fuzzy logc, hybr fuzzy n hybr ANN. These vlble tools hve proven ther effcency to solve the non-lner n NP-hr problems. nce nverse knemtc soluton yels number of lternte solutons, n pproprte tertve or ntellgence bse technque cn be use. Forwr knemtc soluton of ny confgurton s proucng exct soluton. Therefore, usng forwr knemtc equtons the nput for the ANN moels cn be use to trn the opte network. Further trne network prects the nverse knemtc soluton of the selecte confgurton of the robot mnpultor. Although there re mny fferent neurl networks hve been teste on fferent confgurtons of the robot mnpultor but the most frequent moel s MLP neurl network. In the present work three fferent networks s consere for the nverse knemtc soluton. These opte network moels re not teste over the cnte mnpultor uner present nvestgton. The moels of neurl network whch s use for the nverse knemtc soluton re s follows: () Mult-lyere Perceptron Neurl Network (MLPNN) () Polynoml Pre-processor Neurl Network (PPN) 9

113 () P-gm Neurl Network ANN bse precton of nverse knemtc solutons re lter compre wth the ANFI n hybr ANNs. mlr to ANN moels, ANFI cn be trne from the generte tsets usng forwr knemtc equtons. The ptbuluty n lernng blty ncrese usng neurl network nto the fuzzy nference system. Ths metho hs lrey been pple n severl fferent confgurtons of robot mnpultors. mlr to the ANN moels, ANFI structure cn be engge to solve the nonlner functons, NP-hr problems n cn lso prect the chotc tme seres. The lernng cpblty of neurl networks s generlly use to tune the prmeters of the fuzzy logc. The lernng lgorthm proves the tunnng of the membershp functon of ugeno type FI (Fuzzy Inferencer ystem) usng the nput output trnng t. Therefore, ANFI s well s ANN moels wth lernng cpblty, ptblty, n hnlng of nonlner problems whch mkes t sutble to solve the nverse knemtc problem. For ll selecte confgurton of robot mnpultor FI (Fuzzy nference system) structure re obtne n pple for precton of the nvul jont ngles. Despte the vntges of the neurl network n ANFI pproch for nverse knemtc resoluton, chef concern tht often comes s bout the convergence n stblty of the soluton. These networks trnng generlly converge nto the locl optmum pont. Therefore, neurl network moels cn be hybrze wth populton bse optmzton lgorthm to upte the weght n bs of the network. The hybrzton scheme hs lrey been scusse n lter chpter. The MLP neurl network s most effcent n pple to mny nustrl mnpultors. Therfore, n present work MLP neurl network s hybrze wth severl optmzton lgorthms s well s comprson of grent escent lernng lgorthms n pproprte scheme. After the pplcton of the metheurstc lgorthms n trne neurl network, s pple to fn out the nverse knemtc soluton of the robot mnpultors. The opte hybr ANN moels re s follows: () MLPPO (Mult-lyere perceptron prtcle swrm optmzton) (b) MLPTLBO (Mult-lyere perceptron techer lerner bse optmzton) (c) MLPGA (Mult-lyere perceptron genetc lgorthm) () MLPGWO (Mult-lyere perceptron grey wolf optmzer) (e) MLPIBO (Mult-lyere perceptron crb ntellgence bse optmzton) Although there re mny vntges of ANN n hybr ANN tht cn be esly mplemente for the nverse knemtc soluton but mportnt concern s computtonl 9

114 cost n convergence spee of the lgorthm. ANN moels wth bck propgton lernng gves poor performnce for the hgher of robot mnpultors. The nonlner functonl reltonshp for hgher of problem become unstble n prouces uncceptble error t the en of lernng process... Optmzton lgorthm pproch Populton bse optmzton lgorthms cn be gnfully use to fn out the nverse knemtc soluton. The only requrement for the pplcton of optmzton lgorthms s to evelop the objectve functon for the concern mnpultor. In chpter, objectve functon formultons re scusse n etl whch cn be further pple wth mnor mofctons to ny confgurton of mnpultor. Moreover, the objectve functon prouces the cnte soluton of ech nvul jont vrble n tht cn be efne by the confgurton vector of mnpultor wth number of pont wthn the workspce lmt. Ths metho requres only the formulton of the forwr knemtc equtons of the robot mnpultor n ssocte constnt or prmeters. Ths metho proves flexblty to complete mny tsk relte to robot mnpultor lke esgn, knemtc nlyss, synthess of knemtc structures etc. On the other hn, for hgher of n complex tsk of robot mnpultor, populton bse optmzton lgorthms cn be use wth the generc formulton of objectve functon. The optmzton lgorthms shoul be ble to hnle the problem of nonlner, NP-hr n multmol serch problems. The fferent optmzton lgorthms re compre n use to clculte the nverse knemtc solutons re s follows: () Genetc Algorthm(GA) () Prtcle wrm Optmzton (PO) () Techer Lerner Bse Optmzton (TLBO) () Grey Wolf Optmzer (GWO) () rb ntellgence bse optmzton (IBO) These stte lgorthms re lter compre wth the novel evelope IBO lgorthms. Mny optmzton lgorthms requre the number of control prmeters settng n ths ncreses the complexty of the opte lgorthm. The prmeter ssocte wth the lgorthms cn mke the fferences n the results lke ccurcy, convergence spee, effcency, globl optmum pont n computtonl cost. Therefore, to vo mny prmeter settng, novel effectul nture-nspre metheurstc optmzton technque groune on crb behvour s propose (see chpter ). The propose rb Intellgence Bse Optmzton (IBO) technque s populton cntere tertve 9

115 metheurstc lgorthm for D-mensonl n NP-hr problems. Beses usng Jcobn mtrx for the mppng of tsk spce to the jon vrble spce, forwr knemtc equtons re use. Knemtc sngulrty s voe usng these formultons s compre to other conventonl Jcobn mtrx bse methos. In generl, propose crb bse lgorthm gves generc soluton of the nverse knemtc problem for some selecte benchmrk mnpultors. But the propose IBO lgorthm hvng some lmttons lke, t cnnot pply for rel tme control n pplcton for hgher of mnpultor; t tkes tme to converge n sngle optmum pont, etc. A concse pln of pproch towrs soluton of the propose problem s presente n Tble.0. The tble proves uner nvestgton n the propose tool(s) to be use urng the reserch work. Tble.0 Aopte mterls n methos Methos Mterls Robots tructures Types onventonl pproches. HT. QA Methos Intellgence pproches. MLPBP. ANFI. MLPPO. MLPGWO. PMLTLBO. MLPGA 7. MLPIBO Methos -of revolute Rg(R-R-R) Plnr ARA(-of) Flexble(R-R-P-R) ARA Poneer rm(-of) Rg(R-R-R-R-R) ptl PUMA 0(-of) Rg (R-R-R-R-R-R) ptl Type- ABB IRb-00(-of) Rg(R-R-R-R-R-R) ptl Type-A AEA IRb (-of) Rg(R-R-R-R-R-R) ptl Type-A TÄUBLI RX0 L(-of) Rg(R-R-R-R-R-R) ptl Type-B Mterls Robots tructures Types -of revolute Rg(R-R-R) Plnr ARA(-of) Flexble(R-R-P-R) ARA Poneer rm(-of) Rg(R-R-R-R-R) ptl PUMA 0(-of) Rg(R-R-R-R-R-R) ptl Type- ABB IRb-00(-of) Rg(R-R-R-R-R-R) ptl Type-A AEA IRb (-of) Rg(R-R-R-R-R-R) ptl Type-A TÄUBLI RX0 L(-of) Rg(R-R-R-R-R-R) ptl Type-B Mterls Optmzton pproches. PO. GWO. TLBO. GA. IBO Robots tructures Types -of revolute Rg(R-R-R) Plnr ARA(-of) Flexble(R-R-P-R) ARA Poneer rm(-of) Rg(R-R-R-R-R) ptl PUMA 0(-of) Rg(R-R-R-R-R-R) ptl Type- ABB IRb-00(-of) Rg(R-R-R-R-R-R) ptl Type-A AEA IRb (-of) Rg(R-R-R-R-R-R) ptl Type-A TÄUBLI RX0 L(-of) Rg(R-R-R-R-R-R) ptl Type-B Where, HT- Homogeneous Trnsformton n QA- Quternon Algebr 9

116 . ummry Ths chpter presents the scusson of fferent mterls n methos opte for the knemtc nlyss. The mn purpose of ths chpter s to vl the etl escrpton of opte mterl for knemtc nlyss n fferent methos to cheve the objectve of the thess. The etle ervton of nverse knemtc soluton hs been gven n next chpter. In the result chpter nverse knemtc soluton for opte moels of mnpultor hs been tbulrse n comprson on the bss of mthemtcl complexty s me over other opte metho. 9

117 hpter MATHEMATIAL MODELLING AND KINEMATI ANALYI. Overvew The conventonl soluton pproch of knemtcs s mportnt n vrous fels of recent tren n moern technology, extenng through computer grphcs (e.g. nmton chrcter nlyss) to expnson of spce mnpulton n smultors. All these fels of pplctons re funmentlly requre to evlute both orentton n poston of the rtesn coorntes of en effector n jont vrbles of robot mnpultor. To evlute the poston n orentton of en effector n ts jont vrbles one cn opt homogeneous trnsformton mtrx metho. Ths metho s the conventonl tool to escrbe the knemtc reltonshp of jont n lnks. Moreover, ths metho of representton s use from mny eces for trcng the en effector poston of the robot mnpultor. On the other hn, t s extremely reunnt for the representton of -of of system. The reunncy generlly consumes more computtonl cost n more storge spce. Ths s lso relte to the problem of mthemtcl opertons whch generlly cretes more complexty. Therefore, mny lterntve methos for the representton of non-nertl coorntes n nertl coorntes hve been ntrouce. The propose metho shoul lwys be less complex n computtonlly effcent for the representton of mechnsm n trnsformton of the system. Keepng ll n mn, lterntve technques lke Epslon lgebr, quternon n ul quternon, Euler ngle, screw trnsformton, exponentl rotton mtrx n le lgebrs re requre to overcome the problem of nverse knemtc, for better unerstnng of representtons of sme n eucng the mthemtcl opertons n computtonl cost to ensure fst n responsve system n rel envronment. [89], 9

118 propose two fferent pproches to the nverse- knemtcs problem for sx-egreeof-freeom robot mnpultor hvng three revolute jont xes ntersectng t the wrst. One metho uses three rottonl generlze coorntes to escrbe the orentton of the boy. The other metho uses equvlent Euler prmeters wth one constrnt equton. These two pproches hve been ncorporte nto two fferent computer lgorthms, n the results from ech re compre on the bss of computtonl complexty, tme smulton, sngulrty, etc. It ws foun tht Euler prmeters were less effcent thn three rottonl ngles for solvng the nverse-knemtcs problem of the robot consere, n tht the physcl sngulrtes cuse by the robot mechnsm coul not be elmnte by usng ether of the two pproches. [8], propose the poston of mnpultor expresse s ether n jont coorntes or n rtesn coorntes. A new lgebr hs been efne for the use n solvng the forwr n nverse knemtcs problem of mnpultors. The propertes of the lgebr re nvestgte n functons of n epslon numbers re efne. The A lnguge ws use for llustrton becuse of the ese n mplementng the lgebr n t s beng use to solve the forwr n nverse knemtcs problems. However, the progrm ctully use epslon numbers n use the overlong feture of the A lnguge to mplement the epslon lgebr. By smply chngng the orer of the lgebr, the resultng progrm cn compute tme ervtve of the en-effector s poston when use-to solve the forwr knemtcs problem n ny tme ervtve of jont postons when use to solve the nverse knemtcs problem. [], propose polynoml contnuton metho for the nlyss of geometrc esgn problem of -of revolute mnpultor. They hve evelope the elmnton metho for pont precson geometrc nlyss of the mnpultor. In ths work, ech precson pont of the en effector hs been consere sptl confgurton. DH lgorthm s use n ths work for the formulton of the esgn equtons. [], propose soluton technques of nverse knemtcs usng polynoml contnuton, Gröbner bses, n elmnton. They compre the results tht hve been obtne wth these technques n the soluton of two bsc problems, nmely, the nverse knemtcs for serl-chn mnpultors, n the rect knemtcs of n-prllel pltform evces. [98], Propose ul quternon lgebr bse knemtc synthess of constrne robotc system. They hve propose ths metho for one or more serl chn mnpultor conserng both prsmtc n revolute jonts. In ths reserch they hve use DH lgorthm n successve screw splcement for etermnng the jont vrbles for the resoluton of en effector poston. Then ul quternons re use to efne the trnsformton mtrces obtne through DH lgorthm to smplfy the esgn formultons of fferent types of mnpultors. [08], presente generl metho for 97

119 the clssfcton of -of nustrl mnpultors bse on the knemtc structure n ther etl nlyses of knemtc equtons on the bss of clssfcton re gven. They hve opte the exponentl rotton mtrx lgebr to fn out the close form soluton of nverse knemtcs of robot mnpultor. [], presente pose error nlyss of ARA mnpultor usng screw theory. They hve presente the error prouce by DH lgorthm n compre the sme wth the output of the screw bse nlyss of the mnpulton. From the scusse lterture relte to fferent methos of representtons n knemtc nlyss t cn be unerstoo tht homogeneous mtrx wth DH-lgorthm metho s the well-known conventonl metho. Therefore the bove explne metho cn be the benchmrk metho for the comprson of other lterntve methos wth respect to the effcency n qulty of the soluton. Therefore, from bovementone technques n from the prevous lterture revew quternon, ul quternon, screw, exponentl rotton mtrx n Le lgebr re the methos whch expnsvely use for the knemtc nlyss of mnpultors. But stll etl escrpton n the eep theory behn the representton of these methos re not very much cler to most of the resercher. Therefore, further etl ervton of quternon lgebr n ts pplcton wthout mkng t hectc to the reers re prove n ths secton. On the other hn, bref escrptons of other methos re lso presente.. Representton methos n knemtcs Knemtcs cn be unerstoo wth the system of lnks or chn connecte wth jonts to crete reltve moton wthout nlysng the torque/forces or sources of the moton. Anlytcl stuy of the moton of robot lnk wth respect to one fxe coornte or bse coornte system wth functon of tme coul be unerstoo s robot knemtcs. The knemtcs of robot lnk lso proves the stuy of ts hgher ervtves lke velocty, jerk, ccelerton etc... Knemtc vrbles n prmeters A knemtc chn conssts of knemtc pr of lnks whch my be connecte by revolute or prsmtc jonts subjecte to rottonl or trnsltonl egree of freeom. As explne n the lterture there exst mny pproches for the mthemtcl representton of knemtc chn. The mjor fferences of these methos re the ttchment of coornte frmes. Therefore Denvt-Hrtenberg prmeters [], re commonly use. Homogeneous trnsformton mtrx bse methos re better for plcement of coornte's frmes to the lnks n jont vrbles. The metho conssts of four sclrs 98

120 whch re known s DH prmeters of knemtc chn. These sclrs re use to efne the geometry of lnk n reltve splcement of jont. Ths metho of representtons reuces the mthemtcl/rthmetcl opertons for the knemtc escrpton. In the Fgure. the poston n orentton of the xs of jont cn be etermne wth respecte to the bse coornte X,Y n Z wth mnmum four prmeters. To ccomplsh ths, common norml OP between xs of jont n Z xs of the bse frme hs been rwn. Therefore the mgntue of common norml s representng length, whch s locte from the offset stnce of Z xs from the orgn of bse frme to the pont O. θ s the ngle between OA whch s prllel to x-xs wth common norml OP. Ths ngle represents the rotton bout the z-xs whch s mesure n x-xs to the recton of common norml. Angle α represents the rotton of jont xs wth PQ whch s prllel to z-xs n mesure n recton of z-xs. These four sclr, θ, α n re the prmeters of Denvt Hrtenberg prmeters to represents the poston of the xs of ny jont n rtesn coornte system. In the lter secton etl scusson bout these four prmeters re gven. Z O θ P α Q A Y X Fgure. Poston n recton of cylnrcl jont n rtesn coornte frme.. DH-Prmeters Now let us observe ll chrcterstc propertes of sclr prmeters of DH metho for moellng of consere knemtc pr n Fgure.. tnr metho of representton hs been followe wthout lterng the concern propertes of knemtc pr. From Fgure. lnk - connecte by cylnrcl jont wth lnk, n + lnk s consecutve lnk wth sme jont. The ttche coornte frme wth lnk s 99

121 orentte n such wy tht the Z xs s lgne wth consecutve lnk + n X-xs s lgne wth common norml n between n +. Bse coornte frme s stute t the ntersecton of common norml wth + xs. An the lst coornte Y wll be plce s per rght hn rule whch s y z x. Therefore, from Tble., DH- prmeters cn be efne s wth consere geometry n orentton of ssocte lnks re s follows: Fgure. knemtc pr n DH prmeters Tble. DH prmeters : Jont rotton prmeter, whch cn be escrbe s the ngle of rotton of lnks n - whch s mesure from X to X bout the Z. : Lnk length prmeter, cn be represente s length of common norml of lnks + n, mesure n the recton of X,.e. xs to +. : Lnk offset prmeter; cn be escrbe s the stnce between common norml n the coornte X or stnce between strt ponts of n the recton of Z wth the orgn of coornte frme. : Twst ngle prmeter, cn be escrbe s the nclnton ngle between the xes of lnks mesure n X recton wth Z to Z. These prmeters escrbes the complete geometry of knemtc pr, f the jont s revolute then, wll be only vrbles n rest of the prmeters wll be constnt whle n cse of prsmtc jont wll be the vrble n smlrly other prmeters wll be constnt. From Fgure., the coornte frme X, Y, Z s over mpose wth frme of jont so tht the stnce or offset length cn be escrbe, n the 00

122 rotton ngle from Y wth jont cn be unerstn. The Z xs of coornte frme s prllel to the mpose consecutve jont +, whch gves the common norml n cn be escrbe s lnk length prmeter. Ths lnk length s perpenculr to xes + n n cretes the twst ngle bout xes n +. Therefore fter escrpton of DH prmeters, mthemtcl expresson of poston of coornte frmes n mpose frme cn be gves by homogeneous trnsformton mtrx A, B,, n, s,, whch s successve prouct of homogeneous trnsformton mtrces, escrbng ll DH prmeters. Therefore, DH mtrces cn be gves A, B, *, (.) B, cos sn 0 0 sn cos (.) Fgure. Denvt-Hrtenberg prmeters for successve trnslton n rotton of lnks, cos sn 0 (.) sn cos From equton A, cn be gven s, 0

123 A, cos sn 0 0 sn cos cos cos sn sn sn cos sn cos 0 0 cos sn (.) The bove A mtrc cn be use for ny knemtcs chn whch contns revolute, or prsmtc jont for the poston n orentton nlyss. But n cse f the jont xes n + re prllel then wll be zero n the mtrx wll be gven s follows: A, cos sn 0 0 sn cos cos sn T A A A A A A...A n nx n y nz 0 o o o x y z 0 x y z 0 X Y Z (.).. DH-lgorthm for frme ssgnment In the DH lgorthm, bse coornte frme X 0, Y0, Z0 s ttche to fxe bse of non-movng lnk n locl coornte wll be fxe t ech jont of movng lnks. The connecte lnks - to, where =,, n. Therefore, the bsc steps of DH-lgorthm for frme ssgnment re follows []: tep Bse frme X0, Y0, Z0 typclly ttche to the fxe boy t the orgn n such wy tht xs of rotton shoul concent wth the Z 0 xs, whle X 0 wll be plces rbtrrly recte towrs the perpenculr of the rotton xs or t cn be unerstn wth the forwr rechng recton of mnpultor. Usng rght hn coornte rule, the lst Y0 xs cn be plce.e. Y0 Z0 X0 tep Followng the secon step the subsequent secon jont, rotton xs wll be plce n the xs Z, whch goes to coorntes X, Y, Z. The secon coornte frme orgn wll be plce on the -th jont xs t the en of the common norml wy from the jont xs - to the jont xs. But n cse f the jont xes n - re prllel n jont type s revolute then the orgn of the frme wll be smply mpose to secon jont xs confrmng tht =o. otherwse n cse of prsmtc jont the orgn of frme cn be plces rbtrrly long the jont xs. Fnl conton of ntersecton of n -, the frme wll be postone t the pont of ntersecton. 0

124 tep For movng lnk -, xs X where =,,,..n, wll be recte towrs the common norml xes of jont n - from - to. If the jont xes n - ntersect, then xs X wll be perpenculr to the ntersectng plne n cn be recte towrs rbtrrly perpenculr xs. The rotton ngle, wll be chosen by norml recton of Z xs, whch s bsclly represente between the X n X through rotton xs Z. Therefore thr xs Y cn be evlute smlrly wth rght hn coornte rule Y Z X. tep Now the plcement of mnpultor en effector coornte frme X, Y, Z wll be on the reference pont of the grpper. Z e xs wll be recte nywhere n the orthogonl plne of X e, smlr to step three, X e wll be lgne wth common norml of Ze n Z e. But n cse of revolute jont xs of lst jont, Z e wll be consere s prllel to the prevous jont xs. The lst xs wll be gven s rght hn coornte rule X Z X e e e tep Fnlly fter ssgnment of ll coornte frmes for ll lnks =,,,, n, DH prmeters cn be evlute n cn be wrtten n tbulr form gven n the next secton n pctorl vew s presente n Fgure. n.... Mthemtcl moellng of -of revolute mnpultor The mthemtcl moelng of forwr n nverse knemtcs of robot mnpultor usng homogeneous trnsformton mtrx metho wth DH prmeters s presente. The purpose of ths pplcton s to ntrouce to robot knemtcs, n the concepts relte to both open n close knemtcs chns. The Inverse Knemtcs s the opposte problem s compre to the forwr knemtcs, forwr knemtcs gves the exct soluton but n cse of nverse knemtcs t gves multple solutons. The set of jont vrbles when e tht gve rse to prtculr en effectors or tool pece pose. Fgure. () shows the bsc jont confgurton of -of revolute plnr mnpultor n Fgure. (b) represents the moel of ncnnt Mlcron T n use s -of plnr mnpultor. Fgure. shows the smulton of -of revolute plnr mnpultor usng DH proceure. Poston n orentton of the en effectors cn be wrtten n terms of the jont coorntes n the followng wy, Tble. DH-prmeters for -of revolute mnpultor l. (egree) (mm) (mm) (egree) θ 0 0 θ 0 0 θ 0 0 e e e 0

125 () (b) Fgure. Plnr -of revolute mnpultor Fgure. oornte frmes of -of revolute mnpultor Trnsformton mtrx wll be gven by equton (.) A, cos sn 0 0 sn cos cos sn c s 0 c c c s c 0 s s s A, (.) where, c cos, s sn, c cos( ), s sn( ), cos( ) n sn( ) c s Therefore forwr knemtcs s gven by, X (.7) c c c 0

126 Y (.8) s s s (.9) ϕ represents orentton of the en effector. All the ngles hve been mesure counter clockwse n the lnk lengths re ssume to be postve gong from one jont xs to the mmetely stl jont xs. However, to fn the jont coorntes for gven set of en effectors coorntes (x, y, ); one nees to solve the nonlner equtons for,. n Inverse knemtcs, x y tn (s,c ) tn ( (c ), (.0) Where, k cos tn (y, x) tn (k, k ) (.) n k s (.).. Mthemtcl moellng of -of ARA mnpultor The Denvt-Hrtenberg (DH) notton n methoology re use n ths secton to erve the knemtcs of robot mnpultor. The coornte frme ssgnment n the DH prmeters re epcte n Fgure., n lste n Tble. respectvely, where to represents the locl coornte frmes t the fve jonts respectvely, represents the locl coornte frme t the en-effector, where θ represents rotton bout the Z-xs, α rotton bout the X-xs, trnston long the Z-xs, n trnston long the X-xs. Tble. The DH Prmeters l. (egree) (mm) (mm) (egree) θ =±0 0 =0 0 θ =±0 0 = =0 0 0 θ =0 0 0 The trnsformton mtrx A between two neghbourng frmes O n O s expresse n equton (.) s, 0

127 () (b) Fgure. DH frmes of the ARA robot Fgure.7 tructure of ARA mnpultor through MATLB By substtutng the DH prmeters n Tble. nto equton (.), the nvul trnsformton mtrces A to A cn be obtne n the generl trnsformton mtrx from the frst jont to the lst jont of the mnpultor cn be erve by multplyng ll the nvul trnsformton mtrces( 0 T ) n fnl confgurton of ARA s shown n Fgure.7. nx ox x X 0 n y oy y Y (.) T AA AA nz oz z Z Where ( X,Y, Z) represents the poston n( n x,n y,n z ),(ox,oy,oz ), n( x, y, z ) represents the orentton of the en-effector. The orentton n poston of the en- 0

128 effector cn be clculte n terms of jont ngles n the DH prmeters of the mnpultor re shown n followng mtrx s: c s 0 0 s c c c s s (.) s c tn tn (.) c p x p y c p y spx c px sp y s tn tn (.) c pz (.7) n xs n yc tn (.8) n xc n ys It s obvous from the representton gven n equtons (.) through (.8) tht there exst multple solutons to the nverse knemtcs problem. The bove ervtons wth vrous contons beng tken nto ccount prove complete nlytcl soluton to nverse knemtcs of rm. o to know whch soluton hols goo to stuy the nverse knemtcs, ll jonts vrbles re obtne n compre usng forwr knemtcs soluton. Ths process s been pple for,, n, to choose the correct soluton, ll the four sets of possble solutons (jont ngles) clculte... Mthemtcl moellng of -of revolute mnpultor mlrly Denvt-Hrtenberg (DH) lgorthm cn be use to fn out the en effector poston n orentton. DH prmeters n ssocte vlues for -of revolute mnpultor hve gven n Tble. n ssgne to coornte frmes re shown n Fgure.8 n.9. Tble. The DH prmeters Frme (egree) (mm) (mm) (egree) 0 θ = 0 = 0-90 θ 0 = θ θ = 0 90 θ =

129 () (b) Fgure.8 Moel n coornte frmes of the mnpultor Fgure.9 onfgurton of -of revolute mnpultor By substtutng the DH prmeters n Tble. nto equton (.), the nvul trnsformton mtrces A to A cn be obtne n the generl trnsformton mtrx from the frst jont to the lst jont of the mnpultor cn be erve by multplyng ll the nvul trnsformton mtrces gven n equton (.9) The orentton n poston of the en-effector cn be clculte n terms of jont ngles n the DH prmeters of the mnpultor re shown n followng mtrx s: 08

130 s s c s s c c c s s c c s c s s c c c s c s c s c c s s s c s c s s c c s s s c s c s s c c s s s s c s c s c c c s s c c c c c c c c s s s s c s c c c c s s s s c s c c s s s c s c c c s s c c s c (.9) From equton (.9), we cn get postonl equtons c c c c c c c c s s s s c s c X (.0) s c s c s c c s s s c s c s s Y (.) s s c s c s s c c Z (.) x s c c c s s c c c s n (.) y s c s c s c c c s s n (.) z s s c c c n (.) x c s s s c o (.) y c c s s s o (.7) z s o c (.8) x c c c c s s s c c s (.9) y c c s s s c s c s s (.0) z c s s c c (.) The poston n orentton of en effector cn be obtne from equtons (.9) through (.0). These equtons prove the forwr knemtc soluton of robot mnpultor. As we know the complexty of the bove equton cn le to more mthemtcl complexty for ervton of nverse knemtcs, ue to ts successve mthemtcl opertons. Therefore, t s requre to mke some technques to solve these equtons for nverse knemtc ervton of the mnpultor. Usng equtons (.9) n (.8),

131 X x mlrly by usng equtons (.) n (.), c ( c c ) (.) Y y s ( c c ) (.) It cn be unerstn tht the n jont ngles re totlly epenent on the poston of en effector so t cn be fxe s well s t generlly crets more effect on the entre system. In cse f ( c c ) 0 then X n x Y wll not be y equls to zero. If t s more thn zero then wll be gven by, Otherwse, tn(y,x ) (.) y x tn ( Y, X) (.) y x Now for the ervton of n, equtons (.) n (.) cn be mnpulte s, c c (X x)/c (.) c c (Y y ) / s Now usng equtons (.0) n (.), (.7) Z (.8) z s s Now conserng (.) n (.7), Let n r (X (.9) z x)/c r s s (.0) qurng n ng the equtons (.9) n (.0), (cc ss) r rz (.) olvng the terms cc ss n the bove equton (.), we get (c c s s ) cos cos( ) cos( ) cos( ) Therefore, gves mny possble solutons, Or, rz (.) = ± cos cos r z r (.) Rewrtng equton (.8) for the soluton of, s B s (.) 0

132 where, z pz B onserng the equtons (.) n (.7), equton (.) s erve s, Let B = ( x X) ( y Y), so equton (.) cn be rewrtten s, c + c = ( x X) ( y Y) c B c (.) (.) For soluton of B, B,rerrngng equton (.), (.) B (.7) = (c + ) s + (s) c B (.8) = (c + ) c - (s) s Dvng both se of (.7) n (.8), by B B, equton (.9) n (.0) s erve s, where, cos cos B *sn + sn *cos = (.9) B B B B *sn - sn *cos = (.0) B B ( c ) n cos = B sn = ( s ) B B The equton (.8) n (.9) re rewrtten s, An, sn( cos( Therefore, tn (B, B) n B + ) = (.) B B B + ) = (.) B B (c ) = ± cos B B It s cler tht coul be n 0, or,0. The rnge of wll epen on the rnge of. Therefore, f 0, thens 0 n sn( ) 0, thus 0. Then cn be erve s: (c ) = tn(b, B) - cos + (.) B B Otherwse, f 0, then s 0 nsn( ) 0, thus 0. Then the next possble soluton for s s: (c ) = tn(b, B) + cos + B B, (.)

133 Now tht, n re known, the solutons for n cn be foun by usng the remnng forwr knemtcs equtons. onserng equton (.8), the vlue of oz s =, when c 0 (.) c mlrly from equton (.) n (.7), the possble soluton for c s erve s: An gn c (o c s o / c ) x z (.) s c (o y ss o z / c ) (.7) c Usng equton (.) n (.7) for smll vlue of c, the soluton for s Otherwse for smll s, o tn c ) z (oz s oz / c, c (.8) s o (oy ss z tn, c c o / c ) z (.9) Now for soluton of, conserng equton (.), the vlue of c nz ss (.0) cc mlrly the vlue of s s erve by usng equton (.).e., s s c z (.) cc Usng equton (.7) n (.) n vce vers, the term c n s s rewrtten s: c n c c s An s z cc z s ( c c s z cc n z s Now usng ths bove ervton of c ns, s erve s follows: c c s n, n c c s (.) tn z z z z As per the nverse knemtc soluton of -of revolute mnpultor, t cn be unerstn smlr to ARA soluton, exst multple soluton whle n cse of forwr knemtcs t proves unque soluton. o to know whch soluton s gvng better results for ll jont vrbles re evlute usng MATLAB n compre the obtn soluton n the result chpter. )

134 ..7 Mthemtcl moellng of PUMA 0 robot mnpultor DH prmeters n ssocte vlues for PUMA 0 mnpultor hve gven n Tble. n ssgne coornte frmes re shown n Fgure.0, Tble. The DH prmeters Frme (egree) (m) (m) (egree) 0 θ θ θ =0. =0.8 0 θ =0.8 = θ θ () (b) Fgure.0 Moel n coornte frmes of mnpultor Forwr knemtcs of the PUMA 0 robot cn be gven from the trnsformton mtrx s: X c (.) (s c c) s Y s (.) (s c c) c Z ( c s s ) (.) x y z c (c c s s c ) s s s (.) s (c c s s c ) c c c (.7) s c s c c (.8) ox c [ c (c c s s c ) s s s ] s [ s c s c c ] (.9) oy s [ c (c c s s c ) s s s ] c [ s c s c c ] (.70) o z s (c c s s c ) c s s (.7)

135 n x c [c (c c c s s ) s s c ] s [s c c c s ] (.7) n y s [c (c c s s s ) s s c ] c [s c c c s ] (.7) n z s (c c c s s ) c s c (.7) Usng forwr knemtc equtons (.) through (.7), nverse knemtc of PUMA 0 mnpultor cn be erve s below, tn( X Y, ) tn(x,y) (.7) tn( Z, X Y ) tn( b, b ) (.7) where, form: b p, p Px Py Pz, cn lso be expresse n other tn ( X Y ), Z tn ( b, b ) (.77) tn( b, b ) tn(, ) (.78) We cn seprte the rm n wrst f the mnpultor hs sphercl wrst. Therefor rotton mtrx for rm cn be gven by: cc s cs R A sc c ss (.79) s 0 c Poston mtrx for rm cn be gven by: Px c(c s c ) s P A Py s(c s c ) c (.80) Pz s c s Now generl equton for sphercl wrst cn be evlute from mppng of z-y-z Euler ngle nto gven rotton mtrx: Z-Y-Z (,, ) =G (.8) Where, G R T A R Therefore we cn evlute elements of mtrx G from equton (.), g g (cc)r (sc)r sr (.8) g sr cr (.8) (cs )r (ss )r cr (.8)

136 Therefore, tn (g,g), f 0 (.8) tn (g,g), f 0 Where, g g tn (,g ) (.8) tn (g,g), f 0 (.87) tn (g,g), f 0 mlr to prevous ervton of forwr knemtc, equtons (.) through (.7) cn be mplemente for postonng of en effector wth known jont vrbles. Therefter nverse knemtcs soluton cn be foun usng equtons (.7), (.7), (.77), (.78), (.8), (.8) n (.87).. Quternon lgebr knemtcs There hve been tremenous work complete n the fel of knemtcs n recently fter evelopment of quternon lgebr some enttes re e to quternon for enhncng the effcency n qulty of results. lffor evelope ul number concept usng quternon lgebr n nme t ul quternon lgebr whch s power full mthemtcl tool for esgn, synthess n for computer grphcs pplctons. Ths metho s wely use n the fel of robot knemtcs usng few more enttes lke screw splcement, exponentl rotton mtrx etc. whch s use to represent poston n orentton of mechnsm. The most mportnt vntge of quternon lgebr reuces the mthemtcl opertons for knemtcs nlyss s well s gves the sngulrty free nlyss. Therefore, t yels numerclly stble equtons for the knemtc n synthess of mechnsm. On the bss of pplcton quternon lgebr cn be trete s powerful nlytcl tool for clculton the trnsformtons of mechnsm n ther representton. However, quternons re not tht much populr n the fel of robot knemtcs n ynmcs ue to the ffculty of nterpretton n D spce. Therefore to overcome ths problem, the quternon tretments for rel numbers usng lner lgebr n mtrces s propose. In ths work two opertors relte to rel quternon, re etermne n formulte. These opertors re use to trnslte quternon nto the mtrx whch s eser to unerstn n for pplctons. Quternons re bsclly extensons of complex number hvng four frctls, wth one rel number wth followng some rule three mgnry vlues. Ths s lso known s - mentonl components.

137 .. Mthemtcl bckgroun In ths secton mthemtcl bckgroun of quternon lgebr s presente n ts pplcton for the ervton of forwr n nverse knemtcs s scusse. Quternon cn be use for both rotton n trnslton of pont, lne, etc. wth references to bse coornte system wthout use of homogeneous trnsformton mtrx. Interpolton of the sequence of rottons n trnsltons re qute esy n quternon s compre to Euler ngles. It generlly les n sotropc spce tht s generlzton of sphere surfce topology. A bref scusson bout the quternon mthemtcs s escrbe n ths segment for evluton of references n to gve mportnt bckgroun for mthemtcl ervton of nverse knemtc of robot mnpultor. Quternon lgebr mplemente by Hmlton, hs shown ther potentl n vrous fels lke fferentl geometry, esgn, nlyss n synthess of mnpultors n mechnsm, smultons etc. In quternon lgebr hvng four mensons n ech menson conssts of four fferent sclr numbers, n whch one s rel number n rest re mgnry mensons. Ths three mgnry components hvng vlue of n ll re mutully orthogonl to ech other, n cn be represente s,j n k. therefore quternon cn be represente s; h r x jy kz h h (r, x, y,z) (r, v) (.88) where r s the sclr component of h, n v={x,y,z} form the vector prt, n whch r R, x, y,z R n, j, k re mutully orthogonl mgnry unts, whose composton rule cn be stte concsely s follows, =(,0,0,0), =(0,,0,0), j=(0,0,,0), k=(0,0,0,) where multplcton of mgnry vlues cn be explne s: j k jk n j k, jk,k j j k,kj,k j ) onjugte of quternon In ths cse mgntue wll be sme but the sgn of mgnry prts wll be chnges therefore from equton (.89), conjugte s s follows; conj(h) = r - x - jy - kz (.89) conj(h) cn lso be represente s h'.

138 b) Mgntue of quternon Mgntue of quternon cn be explne s, h r x jy kz r x y z (.90) c) Norm Norm for quternon cn be explne s h h *h' r x y z (.9) ) Quternon nverse Quternon nverse cn be clculte s rto of conjugte quternon to ts mgntue, h h' (h * h') (.9).. Quternon rotton n trnslton As t s cler from the bove scusson tht the quternon els wth four mensonl spces so t s qute ffcult to expln t physclly t cn be unerstoo wth quntty tht represents rotton s show n Fgure.. Now the rotton of pont n spce cn be explne from equton (.9). h cos *sn j*sn k *sn (.9) (b) () Fgure. Representton of rotton In mensonl spce t s qute ffcult to mgne th xs therefore n Fgure. (), pont roun the rotton xs (X, Y, Z), tht s unt stnce from the orgn n trcng plne of crcle. When the crcle s projecte to the rotton plne there s pont p rottng by ngle to pont p whch s pssng by m-pont p. Therefore p pont 7

139 s trnsformng to p followng by strght lne mkes cos / nsn / Fgure. (b) p s the pont vector representng ntl poston n. From p s the pont vector fnl conton to be trnsforme. Therefore, two quternons cn be represente on the bss of bove scusse concept. If there s subsequent rotton of two quternons h n h then the composte rottons h * h cn be gven by equton (.9) s, p (h (h h * h * h *(h ) * p ) * p * p *(h *(h Now pure trnsltons t r cn be one by quternon opertor tht s gven below, t Quternon trnsform cn be gven by, r * h ) * h * h * h ) ) (.9) h p (.9) p h *p *h (.9) Fnlly, n equvlent expresson for the nverse of quternon-vector prs cn be wrtten s, H [ h, h P h ] (.97) Where, h P h P [ s(v ( P)) v(v( P))] where t s mple tht the prouct of ny two terms n the bove expressons s nee quternon prouct, whch s efne n the most generl form for two quternons h = (r, v ), n h = (r, v ) s h h [r r v v r v r v v v ] (.98) where v v n v v enote the fmlr ot n cross proucts respectvely, between the three- mensonl vectors v n v. Obvously, quternon multplcton s not commuttve, snce the vector cross prouct s not. The set of elements {±, ±, ±j, ±k} form group (known s the quternon group) of orer 8 uner multplcton. mlrly the quternon multplcton for two pont vector trnsformton cn be clculte s H H (h,p ) (h,p ) h *h, h *P *h P Where, h *P *h P r (v P ) v (v P ).. Knemtc soluton of ARA mnpultor usng quternon (.99) ARA mnpultor moel n coornte frmes ttche to t s shown n Fgure.. Where, n represents the jont vrbles of revolute n prsmtc jonts n re lnks lengths. 8

140 9 Fgure. ARA mnpultor Quternon for ech jont vrble cn be clculte from eqn. (.99) ] j, k [ H (.0) ] j, k [ H (.0) ] k,,0 [ H (.0) ] k, k [ H (.0) Therefor nverse quternon for ech jont cn be clculte by usng equton (.97), ], k [ H (.0) ], k [ H (.0) ] k,,0 [ H (.0) ] k, k [ H (.07) Now clcultng quternon vector proucts usng equton (.99) n (.08) n H... H H Q (.08) Where n cse of ARA, n=. Therefore from equton (.08) nvul quternons cn be clculte s, ] k, k [ H Q (.09) ] )k (, k [ Q H Q (.0) From equton (.08), M H Q Where multplcton of ul quternon H H cn be clculte usng equton (.08) ] )k (, k [ ] j, k [ Q H Q Therefore,

141 Q Now clcultng Q from equton (.08), [ k, j ( )k ] (.) Q [ H Q [ k, j ] k, j ( ) k ] Therefore Q s expresse n equtons (.), Q [ k, ( ) ( ) j ( )k ] Now clcultng quternon vector prs usng equton (.) O (.) j H j O j (.) To solve the nverse knemtcs problem, the trnsformton quternon of en effector of robot mnpultor cn be efne s,t O [ w b j c k, X Y j Zk ] R be be Now usng equtons (.) n (.), O wll be gven by, O H O O [ k, ] [ w b j ck, X Y j Zk ] (.) O [ (w c ) ( b ) (b ) j (c w ) k, (X Y ) (Y X ) j Z k ] (.) O H O O 7 [ o s o j o k, o o j o k] (.) Where, o (w c ) c( ) w( ) o ( b ) (b ) o (b ) ( b ) o (c w) (w c ) o o o Z X Y Z Y 7 X Y Z Now, O H O O [ o o o j o k, o o j p k] (.7) z 0

142 Where, o o, o o, o o, o o, o o, o o Now ll the jont vrbles cn be clculte by equtng quternon vector proucts n quternon vector prs.e. Q, Q n Q to O, O n O respectvely. Therefore, p p x y (.8) (.9) tn (.0) px p y p p x p y ( ) x tn tn (.) p y pz (.) We know tht there s no trnslton n fourth jont of ARA robot t only gves orentton so we cn equte the sclr n vector prt of quternon vector prouct n quternon vector pr.e. Q, Q n Q to O, O n O respectvely. From equtons (.), (.) n (.7), w c c w c w w c w c cn be gven s, tn (.).. Knemtc soluton of -of revolute mnpultor knemtcs The confgurton n bse coornte frme ttchment of -of revolute mnpultor s gven n Fgure.. Where,,, n jont ngles for rtculte rm n, n re the lnk offset.,n represents lnk lengths.

143 Fgure. Bse frme n moel of -of revolute mnpultor Now quternon for successve trnsformton of ech jont cn be clculte from the equton (.99) s follows, ] k j, k [ H (.) ] k, j [ H (.) ] k, j [ H (.) ], [ H (.7) ] k, j [ H (.8) Inverse of ul quternon cn be clculte by equton (.08), ], k [ H (.9) ] k, j [ H (.0) ] k, j [ H (.) ], [ H (.) ] k, j [ H (.) n H... H H Q (.) Where n cse of -of revolute mnpultor rm n=. Now clcultng quternon vector proucts usng equton (.08) ] k, j [ H Q (.) ] k, j [ ], [ Q H Q (.) ] k j ) (, k j [ Q (.7)

144 ] k j ) (, k j [ ] k, j [ Q H Q (.8) ] )k ( j ( ) (, k j [ Q (.9) ] )k ( j ( ) (, k j [ ] k, j [ Q H Q (.0) Therefore, ] )k ( j ( ) (, )k ( )j ( ) ( ) ( [ Q (.) ] )k ( j ( ) (, ) k ( )j ( ) ( ) ( [ ] k j, k [ Q H Q (.) Therefore, ] )k ( j ) ( ) (, ) k ( )j ( ) ( ) ( [ Q (.) Now clcultng vector pr of quternon usng equton (.), to solve the nverse knemtcs problem, the trnsformton quternon of en effector of robot mnpultor cn be efne s

145 ,T O [ w b j ck, X Y j Zk ] R be be Now usng equtons (.08) n (.), O wll be gven by, (.) O H O O [ k, ] [ w b j c k, X Y j Z k ] O (o [ (o ) (o ) (o ) j o ) (o 7 k ] ) j (o ) k, (.) where, o w c o o o o o b b c w X Y Y X o 7 Z Now, O H O O Where, [ o o o j o k, o o j p k] (.) o o c( ) w( ) o o o o o o o o o o o o o o o o Z X Y Y X 7 Z X Y O 7 7 o o o o o o [ o o o j o k, o o j o k] (.7) o o z

146 X X Z Z Z Y X Z o X Y o 7 Y X Z Y X Z o Therefore, ll the jont vrbles cn be clculte by equtng quternon vector proucts n quternon vector prs.e. Q, Q n Q to O, O n O respectvely. X (.8) Form equton (.8), x u (.9) X u x x u X x u X ) ( ) ( u X x (.0) An Y (.) From equton (.) y u Y u y y u Y ) ( ) ( u Y y (.) From equtons (.0) n (.) ) ( u X ) ( u Y tn x y x y u X ) ( ) ( u Y tn

147 ) ( ) ( u X u Y tn x y (.) Now for thet Z (.) x v (.) Z v x (.) x v Z (.7) x v Z (.8) As we know tht ] tn [, n Therefore usng equtons (.)-(.8) x x v Z, v Z tn (.9) mlrly, Y X Z (.0) y v (.) y v Y X Z (.) y v Y X Z (.) y ) ( v Y X Z (.) y ) ( v Y X Z (.) Therefor thet usng equtons (.0)-(.),, ) ( v Y X Z, ) ( v Y X X tn y y (.) mlrly for thet n thet Y X Z Z Z Y X Z

148 7 Y X Z Z Z Y X Z Y X Z Z Z Y X Z (.7) Therefore from equton (.7) thet wll be Y X Z Z Z Y X Z, Y X Z Z Z Y X X tn (.8) Y X Z Y X Z Y X Z Y X Z (.9) Thet wll be gven by usng equton (.9),, Y X Z Y X Z, Y X Z Y X Z tn (.70).. Knemtc soluton of PUMA 0 mnpultor usng quternon PUMA 0 mnpultor moel n coornte frmes ttche to t s shown n Fgure.. Where n,,, represents the jont vrbles of revolute type jonts n, re lnks lengths n n,, re lnk offsets. Fgure. PUMA 0 mnpultor moel

149 8 Quternon of ech jont vrbles cn be clculte usng equton (.99), tht s smlr process lke ARA. ] j, k [ H (.7) ] k, j [ H (.7) ] k j, j [ H (.7) ], [ H (.7) ] 0,0,0, j [ H (.7) ] 0,0,0, [ H (.7) Now clcultng nverse of quternon usng equton (.08), ], k [ H (.77) ] k, j [ H (.78) ] j, j [ H (.79) ], [ H (.8) ] 0,0,0, j [ H (.8) ] 0,0,0, [ H (.8) Now clcultng quternon vector proucts usng equton (.99) n (.08) n H... H H Q ] k, j [ H Q (.8) ] k j [ Q H Q (.8) ], k j [ Q H Q (.8) ] ) k ( j ) (, )k ( j ) ( ) ( ) ( [ Q H Q (.87) ] k G j F E, k D j B A [ Q H Q (.88) Where,

150 9 ) ( A ) ( B j ) ( k ) ( D ) ( E j F k ) ( G Now, ] k G j F E, k D j B A [ Q H Q * * * * * * * (.89) Where, * D A A * B B * B * A D D ) ( E * j ) ( F * k ) ( K * Now clcultng vector pr of quternon usng equton (.90), to solve the nverse knemtcs problem, the trnsformton quternon of en effector of robot mnpultor cn be efne s ] Zk Y j X, ck j b w [ O,T R be be (.90) Now usng equtons (.08) n (.90), O wll be gven by, O H O ] Zk Y j X, ck j b w [ ], k [ O ] Z k j ) X (Y ) Y (X, ) k (o j ) (o ) (o ) (o [ O (.9) where, c w o

151 o o o b b c w Now, O H O Where, O [ o o o j o k, o o j p k] (.9) o o c( ) w( ) o o o o o o o o o o o X Y Y X Therefore, ll the jont vrbles cn be clculte by equtng quternon vector proucts n quternon vector prs.e. Q, Q n Q to O, O n O respectvely. z Where, X k X tn X Y Y 0 (.9) ( ) Z tn (.9) Z ( ) Y Z k tn (.9) k mlrly for, n, o o o O 7 7 [ o o o j o k, o o j o k] (.9) o o o o o o

152 o o o o o o o7 7 o7 o7 o o tn tn (.97) o o o o tn (.98) o o o o tn tn (.99) o o.. Knemtc soluton of ABB IRb-00 mnpultor usng quternon The bse coornte frmes n confgurton of the -of ABB IRB-00 robot mnpultor s presente n Fgure.. Where represents the jont vrbles of revolute type jonts n re lnks lengths n re lnk offsets. The bse frme s fxe rotton s fxe for ll jont rottons. Z Y X Fgure. onfgurton n moel of ABB IRB-0 robot mnpultor The quternon vector of ech jont cn be clculte by equton.98 n.99. H [ k, j k ] H [ j, k ] H [ j, k ] H [ k, k ] (.00) (.0) (.0) (.0)

153 H [ j, k ] (.0) H [ k, 0,0,0 ] Inverse of ul quternon cn be clculte by equton (.97), H (.0) [ k, k ] (.0) H [ j, k ] (.07) H [ j, k ] (.08) H [ k, 0,0, k ] (.09) H [ j, 0,0, k ] (.0) H [ k, 0,0,0 ] (.) mlr to prevous work, en effector poston cn be formulte s, X (.) c cs cs cc (( cs c ss )s ccc) Y (.) s ss ss sc (( ss c cs )s scc) Z c c s (c c s s c ) (.) mlr to the knemtc soluton of PUMA mnpultor, forwr knemtcs cn be clculte for TAUBLI RX 0L. Therefore, ll the jont vrbles cn be clculte by equtng quternon vector proucts n quternon vector prs.e. Q, Q n Q to O, O n O respectvely. f, then, where, ( s (c ((c c c c c s s s A tn(y,x) (.) Atn(Y,X) (.) tn (, ) (.7) s A )c )c c c s s s s )s )) )k (c (c c c c c s s s s )c )c c c s s s s )c )s k c ( (c ((c c c c c s s s s )c )c c s s )s )k css )) (c (c c c c c s s s s )c )c c c s s s s )c )s k A B c (ccc ss)c css ) c s (ccc ss)c css ) s ' ' tn (, ) A tn( / ) (.8) A

154 where, ' (c A B x c c s s )c c ' s s s ) A tn (ccs sc,(ccc ss)c css )) (.9) ( c A tn ( (( s s c c c s (c c s s c A tn( ((s s )c )s c c s c s c s s s c s c )c )c )c ) c s c (s s s s s s ) )s (c c c (s s )s c ) s s s c c c ) )c,, (.0) (.)..7 Knemtc soluton of TAUBLI RX0L mnpultor usng quternon The coornte frmes n confgurton of the -of TAUBLI RX0L robot mnpultor s presente n Fgure.. Where represents the jont vrbles of revolute type jonts n re lnks lengths n re lnk offsets. The bse frme s fxe rotton s fxe for ll jont rottons. Z Y X Fgure. oornte frme n moel of TAUBLI RX0L robot mnpultor The quternon vector of ech jont cn be clculte by equton.98 n.99. H [ k, j ] H [ j, k ] H [ j, k ] H [ k, 0,0,0 ] (.) (.) (.) (.)

155 ] 0,0,0, j [ H (.) ] 0,0,0, k [ H (.7) Inverse of ul quternon cn be clculte by equton (.98), ], k [ H (.8) ] k, j [ H (.9) ] k, j [ H (.0) ] 0,0,0, k [ H (.) ] 0,0,0, j [ H (.) ] 0,0,0, k [ H (.) mlr to the knemtc soluton of PUMA mnpultor, forwr knemtcs cn be clculte for TAUBLI RX 0L. Therefore, ll the jont vrbles cn be clculte by equtng quternon vector proucts n quternon vector prs.e. Q, Q n Q to O, O n O respectvely. Y Y X X tn (.) ) ( Z Z ) ( tn (.) k k tn (.) where, Z Y X k smlrly for, n, k] o j o o, k o j o o o [ O 7 7 (.7) o o o o o o o o o o o o

156 o o o o7 7 o7 o7 o o tn tn (.8) o o tn o o, o o (.9) o o tn tn (.0) o o..8 Knemtc soluton of AEA IRb- mnpultor usng quternon The bse coornte frmes n confgurton of the -of AEA IRb- robot mnpultor s presente n Fgure.7. Where represents the jont vrbles of revolute type jonts n re lnks lengths n re lnk offsets. The bse frme s fxe rotton s fxe for ll jont rottons. Fgure.7 oornte frme n moel of AEA IRb- robot mnpultor The quternon vector of ech jont cn be clculte by equton.98 n.99. H [ k, j k ] H [ j, k ] H [ j, k ] H [ j, 0,0,0 ] H [, 0,0,0 ] Inverse of ul quternon cn be clculte by equton (.08), (.) (.) (.) (.) (.)

157 ] k, k [ H (.) ] k, j [ H (.7) ] k, j [ H (.8) ] 0,0,0, k [ H (.9) ] 0,0,0, j [ H (.0) Quternon vector proucts cn be clculte by usng equton (.99) n (.08). Q, Q n Q cn be clculte usng equton bove, therefore forwr knemtc equton cn be gven s, ) ( X (.) ) ( Y (.) Z (.) mlr to prevous work nverse knemtcs cn be erve usng the equtons n equtng Q, Q n Q to O, O n O respectvely. Y X tn (.) tn (.) Where, ) ( ) A( B, ) ( ) B( A Y X A Z B tn (.) where, ) ( B A, ) ( Z Y X Z Y X tn (.7) ) ( ) ( ) ( ) ( tn (.8) where, n, re the orentton of the en effector.

158 . ummry Ths chpter elvers the bss of conventonl methos for moellng the fferent confgurtons of robot mnpultor n terms of knemtcs. The chef purpose of ths chpter s to prove bref scusson of DH-lgorthm n homogeneous mtrx metho for representton of rotton n trnslton of mnpultor lnk. In the lter secton quternon pplcton for forwr n nverse knemtc soluton hs been gven to show the effcency n esness of the metho. Therefore, nverse knemtc soluton of -of (Aept One ARA), -of (Prm), -of (AEA IRb-), -of (PUMA 0), -of (ABB IRB-00) n -of (TAUBLI RX 0 L) revolute mnpultors wthout Euler wrst re solve mthemtclly usng quternon vector bse metho. The opte metho s compct n effcent tool for representton of trnsformtons of en effector. The etle ervton of nverse knemtc soluton hs been prove to show the mthemtcl complexty of homogeneous mtrx bse soluton of robot mnpultor over quternon. In chpter 7 nverse knemtc solutons for opte mnpultor hs been tbulte n comprson on the bss of mthemtcl complexty s me over other conventonl bse metho. 7

159 hpter INTELLIGENT TEHNIQUE FOR INVERE KINEMATI OLUTION. Overvew ogntve process of lernng n usng t for ecson mkng n cse of hr to unerstn processes hs been well pprecte by the communty reserchers. Now ys, humn bengs re grspng the ntellgence from the nture n re tryng to mplement nto the mchne. The purpose s to retreve en effector poston of robot mnpultor, whch cn work n uncertn n cluttere envronment on the bss of knowlege or nformton so s to lern complex nonlner functons from outse nformton wthout the use of mthemtcl structures or ny geometry. The ntellgent methos mmc the cognton n conscousness n mny spects lke they cn lern from the experence or prevous trnng then t cn be unverslze to tht omn for testng, bsc concept s the mppng of nput output vrbles fster thn conventonl methos so s to reuce the computtonl cost. o the motvton s to reuce the computtonl cost n consequently ncrese the spee for robust control. On the other hn, nverse knemtc mppng for ny confgurton of robot mnpultor cn be nlytclly one but the process wll be long n slow for rel tme control. As explne n prevous chpter the nverse knemtc soluton of robot mnpultor s ffcult f followng the conventonl methos. The ffculty rses ue to fct tht nverse knemtc equtons re not true functon n gves multple solutons. In ton, nput-output mppng of nverse knemtcs problem s non-lner n tenency of the soluton s qulttvely ffers when en effector poston chnges wthn the workspce. On the other hn, conventonl methos yels effcent soluton of nverse knemtcs but suffer some rwbcks lke complex structure of mnpultor or hgher of cn be tme consumng n mthemtclly ffcult to obtn results, 8

160 sngulrtes occurs n some cses etc. Therefore, conserng overll complexty of nverse knemtc soluton n serch for effcent ntellgent technques lke rtfcl neurl network (ANN), fuzzy logc, ANFI n hybr neurl network wll be frutful. ANNs re extensvely opte technque to solve nverse knemtcs problem n generlly offers n lterntve pproch to hnle complex, NP-hr n ll-contone problems. ANN moels cn cqure prevous knowlege or nformton from exmples n re ble to tckle nosy n nequte t n to lern non-lner problems. Once the opte neurl network moels re trne then t cn perform precton of output wth hgher computtonl spee. These moels re pproprte n moellng n mplementton of system wth complex mppngs. A etl ntroucton of fferent opte moels of ANN hs been presente n ths chpter. However, ANN s qute ptve to the system n oes not requres hgher level of progrmng but prt from ths t hs some rwbck lke selecton of ANN rchtecture, numercl computton for weght uptng (.e. Grent escent lernng, Levenberg-Mrqurt bse bck propgton lernng etc.), etc. In contrst bove scusse nture of ANN moels, t s requre to set some rules for fuzzy logc to vl the vntges of nterpretblty n trnsprency of the metho. Fuzzy logc requres the pror knowlege of the problem n bse on the experence of expert ecson tht mkes use of lngustc nformton on the bss of ht n trl metho. Therefore, from lst eces, fuzzy logc becomes n lterntve metho over conventonl technques for nonlner nverse knemtc solutons. The mn e behn ths lgorthm s f-then logc whch s nherent to expert ecson. However, ths lgorthm s bse on trl n error logc therefore t cn be frutfully merge wth ANN moels. Fuzzy logc hs fferent membershp functon whch s fxe n mght be rbtrrly. An the shpe of the functon reles on few prmeters n ths cn be optmze usng ANN bck propgton rule. Ths metho s known s ptve neurl-fuzzy nference system (ANFI). Therefore, hybrzton of ANN wth fuzzy cn gve benefts of both metho. However, the mjor rwbck of ANFI s stuck n locl optmum pont. Therefore to overcome ths problem the wse ecson s to opt some metheurstc lgorthm for the optmzton of weght n bs of ANN moels. Therefore, n ths chpter hybr ANN moels re evelope to overcome the problem of ANN n ANFI wth the hybrzton strtegy. Detl scusson of ANN moels, ANFI n hybr ANN hs been presente n the lter secton.. Applcton of ANN moels ANN moels lke MLP, PPN, P-NN etc. generlly use to lern jont ngles of robot mnpultor n the t sets re generlly generte through some conventonl 9

161 methos lke DH-lgorthm, homogeneous trnsformton mtrx, lgebrc methos etc. Forwr knemtc equtons re mostly use to trn the neurl network moels wheres n ths chpter both forwr n nverse knemtcs equtons re use to trne the neurl network moels. The metho of lernng s bse on the stnr t whch generlly rely on the workspce of the mnpultor. The lernng cn be complete by supervse, unsupervse or both. ANN montors the nput-output reltonshp between rtesn coornte n jont vrbles bse on the mppng of t. Inverse knemtcs s trnsformton of worl coornte frme (X, Y, n Z) to lnk coornte frme (,,... n ). Ths trnsformton cn be performe on nput/output work tht uses n unknown trnsfer functon. A smple strtegy for nputoutput mppng s shown n Fgure. () n (b) whch re fee forwr n bck propgton for error mnmzton strtegy of ANN moels. Input Aopte Moel Output () Input Aopte Moel Actul Output Desre Output Error E - + (b) Fgure. () Fee forwrn (b) bck propgton strtegy In chpter mult-lyere neurl network, polynoml pre-processor neurl network n P-neurl network moels re presente. Bref scussons of these opte moels re gven n the next secton... Mult-lyere perceptron neurl network (MLP) It s well known tht neurl networks hve the better blty thn other technques to solve vrous complex problems. MLP neurl network's neuron s smple work element, n hs locl memory. A neuron tkes mult-mensonl nput, n then 0

162 elvers t to the other neurons ccorng to ther weghts. Ths gves sclr result t the output of neuron. The trnsfer functon of n MLP, ctng on the locl memory, uses lernng rule to prouce reltonshp between the nput n output. For the ctvton nput, tme functon s neee. We propose the soluton usng mult-lyere perceptron wth bck-propgton lgorthm for trnng. The network s then trne wth t for number of en effector postons expresse n rtesn co-orntes n the corresponng jont ngles. The t consst of the fferent confgurtons vlble for the rm. The fferent poses of the rm re then use to trn three-lyer, fully connecte bckpropgton moel (Fgure.). Ths result n two sets of weghts for ech mnpultor rm fter the trnng sesson ws over. A block grm of the propose work s shown n Fgure.. The sgnls, ojn, re presente to hen lyer neuron n the network v the nput neurons. Ech of the sgnls from the nput neurons s multple by the vlue of the weghts of the connecton, wj, between the respectve nput neurons n the hen neuron. onnecton weghts Bs Input X, Y n Z Output Input lyer Hen lyer Hen lyer Output Lyer Fgure.Mult-lyere perceptron neurl network structure A neurl network s mssvely prllel-strbute processor s shown n Fgure. tht hs nturl propensty for storng experentl knowlege n mkng t vlble for use. It resembles the humn brn n two respects; the knowlege s cqure by the network through lernng process, n nterneuron connecton strengths known s synptc weghts re use to store the knowlege [7].

163 Trnng s the process of mofyng the connecton weghts n some orerly fshon usng sutble lernng metho. The network uses lernng moe, n whch n nput s presente to the network long wth the esre output n the weghts re juste so tht the network ttempts to prouce the esre output. Weghts fter trnng contn menngful nformton wheres before trnng they re rnom n hve no menng [7]. Therefore flow chrt of MLP neurl network s presente n Fgure. n bsc steps re s follows: Uptng hen weghts trt Rnom weght vlues lculte the error grent for the neurons n the output lyer Input trnng t x lculte output of hen neurons lculte the error grent for the neurons n the hen lyer lculte output of output neurons No tsfctory performnce Yes En Lernng error Fgure. Flow chrt for MLPBP tep electon of hen lyers (L) n totl number of hen neurons ( n e, e=,,, L-) wth error tolernce 0. L j tep electon of weght vectors ( W ) on the bss of rnom number genertor, where =,,.. n e n L=. L tep Intlzton of weghts Wj wth rnom number n between 0- n gven s, W L j [0,]

164 tep lculton of output of neurons s hen nput, L n W *X B, j=,,.l. lculton of output of hen neurons s, h n output of output neurons cn be gven by, O o k exp( L W j * X ctvton functon * B L ) W kj * h 0 B tep Error estmton of output lyer neurons s, k, k=,, m. o k k k k (O D ) E. tep If the output of neuron s smlr to esre output then en else choose next step tep 7 Grent clculton of hen n output neurons cn be gven s L k ) k k k h k Ek *O *( O ) n h o *( h o ) * ( k * W tep 8 enstvty of hen n output lyers wll be gven by, for output lyer= L L L s E ( )(O D ) n s L E L k ( k L k )(W L,j ) T s L ( Oj )(O 0 j ) ( O 0 n j )(O n j w ) w tep 9 Uptng of weght, Wnew Wol (k) s (O ) n bnew bol (k) s where L=,,,,,l- tep 0 Evluton of termnton crter f error ter mn ton ( ) then go to step else step. tep Network s vlble for testng. The network uses lernng moe, n whch n nput s presente to the network long wth the esre output n the weghts re juste so tht the network ttempts to prouce the esre output. Weghts fter trnng contn menngful nformton L L wheres before trnng they re rnom n hve no menng. Net nput of hen neurons (for L nputs) = L n W *X B (.) o j The output, no of hen neuron s functon of ts net nput s escrbe n equton (.). The sgmo functon s: h o (.) exp( L W j *X B ) Once the outputs of the hen lyer neurons hve been clculte, the net nput to ech output lyer s clculte n smlr mnner s n equton (.). After clculton of T j j j s L L

165 output of output neurons comprson between esre vlue n network output s me on the bss of men squre error s gven n equton (.), m k k e (O D ) E (.) If the obtne men squre error s zero then lgorthm stops otherwse t goes to the error grent clculton of hen neuron usng the formul s show n equton (.), s L E L L k ( )(O D ) (.) Further error grent clculton of output lyer cn be gven s, s L E L k ( ( Oj )(O 0 j L k ) )(W L,j ( O 0 ) n j T s )(O L n j w ) w j j s L (.) The weght n bs uptng cn be performe ccorng to equton (.). W b new new W b ol ol (k) s (k) s L L (O L ) T (.) The mn m of ths overll trnng process of MLP network s to mnmze the men squre error of the prtculr opte network rchtecture. onvergence of network cn be tune wth the prmeters n. In ths work, two hen lyers re consere throughout the reserch wth three nputs X, Y n Z, whle output s epenng on the confgurton of the robot mnpultor... Polynoml pre-processor neurl network Polynoml pre-processor neurl network moel hvng stngushe property of summton of ll nputs s compre to MLP network t follows the Weerstrss pproxmton theorem tht sttes "Any functon whch s contnuous n close ntervl cn be unformly pproxmte wthn ny prescrbe tolernce over tht ntervl by some polynoml". Fgure. epcts PPN network where X, Y n Z re the nputs pttern gven by, ', x, x x m ] X [x (.7) For nstnt conserng D nput pttern X= [x x], to expln the Weerstrss pproxmton theorem wheren polynoml s orer of, therefore the functon of ecson cn be wrtten s ' X * D(X) W (.8) Where, ' W [w, x 0,x,...w m ] T * n X [,x,x,x,x, x,x ] T

166 Fgure. Polynoml perceptron network Now for m-mentonl pttern of nput cn be formulze usng generl qurtc cse wth conserng ll combnton of X elements, D(X) m m m M w jjx jj w jkx jxk J j k j j w x w j j 0 T * W X (.9) For the m-mensonl cse, the number of coeffcents n functon of r th egrees gven by N m r m r (m r)! r (.0) m!r! The nput pttern X to the PPN t tme n s the chnnel output vector X (n). Ths sthen converte nto X*(n) by pssng t nto polynoml pre-processor. Theweghte sum of the components of X*(n) s psse through nonlner functonsgmo n pure lner functon to prouce the output s shown n Fgure.... P-gm neurl network PNN (P-gm Neurl Network) s lso fee forwr or mult lyere neurl network consstng of one hen lyer. The mjor fferent of PNN s summng unts of hen lyer n prouct unt of output lyer s compre to MLPNN. The weghts of nput n hen lyer cn be obtne urng trnng process of network whle hen lyer to output lyer weghts re fxe to one. Ths network uses two fferent ctvton functons t hen lyer lner ctvton functon n t output lyer non-lner ctvton functon. Therefore p-sgm network evlutes the summng proucton of nput lyer n corresponng weghts n psses through nonlner ctvton functon. Ths concept of one hen lyer wth two ctvton functons rstclly mnmzes the totl trnng tme for the network. The p-sgm network structure s presente n Fgure..

167 Lner Trnsfer Functon Nonlner Trnsfer Functon Moreover, summng prouct lyer of p-sgm network proves hgher menson cpbltes through the expnson of nput menson nto hgher mensonl spce therefore t cn esly splt nonlner seprble clss to lner seprble clss. Fnlly ths network s cpble of provng the nonlner ecson wth better clssfcton of hgher menson t thn the norml network. Ajustble weghts Fxe weghts Input X, Y Z Output,,... n Input Lyer ummng Hen Lyer Output Lyer Fgure. Polynoml perceptron network Now conser p-sgm network wth n number of nputs, wth n h number of hen neurons n one output neuron. n h, efnes the orer of p-sgm network tht s n n h conserng ll summng unts re relte to n weghts. The output of the network wll be gven by the prouct of the output of n h hen unts whch psses through the nonlner ctvton functon; therefore t cn be gven s, n h O h k k (.) Where nonlner ctvton functons n h k s the output of kth hen lyer neurons whch s then clculte by summng the proucts of ll nputs (x, y, z) wth the corresponng weght ( W ) between th nput n kth hen unt. Therefore output of hen lyer wll be gven by: j n k (Wk X ) h (.)

168 . Applcton of ptve neurl-fuzzy nference system (ANFI) Aptve Neurl-Fuzzy Inference ystem (ANFI) evelope by Roger Jng [ ]. ANFI s hybrzton of neurl network n fuzzy logc methos. Ths s bsclly type of fee forwr neurl network whch nvolves fuzzy nference system through the structure of neurl network n ther neurons. It gves the lernng blty of neurl network to fuzzy nference system. The metho s mnly evelope for the evlutons of nonlner functons tht generlly entfes nonlner elements on lne for control system esgn n prects chotc tme seres. On the other hn, (FI) fuzzy nference system s most populr computng metho whch s bse on the fuzzy set theory wheren f-then rule n fuzzy resonng s mnly focuse. It s event from the lterture revew tht FI hvng lrge pplcton res such s control system, clssfcton of t, ecson nlyss, system of experts, precton of tme seres, robotcs, mge processng n recognton. The rchtecture of the FI conssts of three funmentl elements: rule element, tht covers the selecton of pproprte fuzzy rules: tbse, tht gves the reltonshp of membershp functon wth the estblshe fuzzy rules; then fnlly resonng components, whch gves the pproprte nference metho of opte rules n proves fcts to evelop resonble output or concluson. Ths cn tke ether fuzzy nput or crsp vlue, but prouce outputs re lmost fuzzy sets. But sometmes t s requre to hve crsp vlue, especlly where FI s use for controller. Therefore, efuzzyfcton s requre to ecoe the crsp vlue whchever best represent fuzzy set. Therefore FI wth neurl network s use to upte the prmeters of neurl network n cn perform mppng of nput to output t through pproprte lernng lgorthm. Ths process of tunng gves the optmze prmeter of neurl network. ANFI structure s conssts of fve fferent lyers such s fuzzy lyer, normlze lyer, prouct lyer, efuzzy lyer, n summton lyer. Bsc structure of the ANFI s gven n Fgure., n whch fxe noe s gven by crcle n justble noe s gven by squre. uppose f there s two nputs x n y wth one output z then ANFI cn be use s frst orer ugeno FI. There re mny fuzzy systems lke ugeno, Mmn etc., but most populr n wely use system s ugeno moel ue to ts hgh nterpretblty n computtonl effcency wth efult optml n ptve tools. Therefore frst orer ugeno fuzzy rule cn be expresse s, Frst rule: If x s A n y s B, then Z px qy r (.) econ rule: If x s A n y s B, then Z px qy r (.) 7

169 Where, A n B re fuzzy sets n p, q n r re prmeters whch s ssgne urng trnng process. From Fgure.7 ANFI structure conssts ll fve lyers. Now output noe wll be efne by, O (x),, (.) A O B (y),, where (x) n (y) cn hol ny membershp functon (MF). For exmple, n A B ths work wely use membershp functon.e. Gussn MF s use throughout the work. (Ac) B gussmf (A,B,) e (.) where, B re the prmeters whch chnges shpe of MF. econ lyer noes re represente by Π whch s fxe. O (x) (y),, (.7) A B Ech noe output represents the frng strength of rule. Fgure. Archtecture of ANFI Thr lyer fxe noes re represente by N. In ths lyer, the verge s clculte bse on weghts tken from fuzzy rules: 8

170 Fgure.7 Trnng of ANFI structure O,, (.8) Where re normlze frng strengths. Every th noe n the fourth lyer s n ptve noe gven by followng noe functon, O The prmeters (, p, q z (p x q y r ),, (.9) After rerrngement, the output cn be wrtten s lner combnton of the consequent prmeters: 9 n r ) of ths lyer re consequent prmeters. For the ffth lyer fxe noe s gven s Σ tht clcultes ll output s summton of ll nputs by, z z O (.0) z.. Lernng lgorthm In ANFI there s forwr lernng process whch s bse on lest squre metho n bckwr lernng s gven by grent escent lernng process. If the premse prmeters re fxe then the output of the ANFI cn be gven s, Replcng Eq. (.9) nto Eq. (.) gves, z z z (.) z z z (.) Replcng the fuzzy f-then rules nto Eq. (.), t becomes: z (p x q y r ) (p x q r ) (.)

171 z ( (.) x)p ( y)q ( )r ( x)p ( y)q ( ) r Lest squre metho s use to clculte the optml vlue of the consequent prmeters. When premse n consequent both prmeters re ptve, then t evelops hgher serch spce n ths les to solve convergence of trnng process. Therefore hybr lernng wth bck propgton s use to solve convergence problem. Ths hybr lernng reuces the serch spce menson. At the lernng stge, both premse n consequent prmeters re properly tune tll the esre output cheve by FI. Fgure. represents the trnng process of ANFI whch s one by usng MATLAB ToolBox of nfset commn. In ths work fferent ANFI structure wth frst orer ugeno fuzzy system s consere for vrous consere jont vrbles of robot mnpultor. Where nput s consere s the en effector postons (X, Y n Z) n t sets were generte by forwr n nverse knemtc equtons.. Hybrzton of ANN wth metheurstc lgorthms After ntroucton of smple neurl network wth the we pplcton of fee forwr neurl network wth bck propgton lgorthm s well s mult-lyere perceptron network yels mny troubles for trnng of n lgorthm. Bck propgton lgorthm s generlly rect serch metho wth weght uptng rule to ensure the mnmzton of the error. However, there re mny key ponts, whch ensure the lgorthm not efnte for the comprehensvely useful for mny pplctons. One of the mjor key pont of ths lgorthm s lernng rte prmeter whch s strctly requre to tune properly else t cretes fluctuton s well s more computtonl tme for trnng. On the other hn, weghts uptng les to long trnng tme for the specfc pplcton of the lgorthm. Furthermore, bck propgton lgorthm ultmtely gves slow convergence rte f the number of hen lyer ncrese ue to ts weght uptng rule. The most mportnt pont s the lernng lgorthms such s grent escent lernng of bck propgton lgorthm whch s generlly complex n lso contns vrous locl mnmum ponts. Therefore, ths lgorthm mostly gets stuck nto locl mnm, whch mke t utterly epenent on weght uptng n ntl settngs. Therefore, hybrzton of ANNs cn be one n mny wys to overcome ll stte problems. The ctegorzton of the hybrzton of ANN cn be explne s follows:. Archtecture optmzton. Weght n bs optmzton. Lernng rte n momentum prmeter optmzton In cse of optml rchtecture esgn, the number of hen lyers s the key fctor for esgnng the rchtecture. To fn out the best structure for specfc problem trnng 0

172 lgorthm prt from grent escent lernng n optmzton lgorthm s use. The rchtecture s epenent on the neurons connectons, no. of hen lyers n hen noes of neurl network. Mny reserches hve been one n ths fel for elementry solutons. The structure of the neurl network moel cn be gven by upper boun metho. But n cse of bounry t my only prove bsc e bout the structure but n cse of hgh nonlner functons n hghly ynmc nture cn cuse the network to go beyon the requrement. Therefore t cn be use s pproxmtons of the structure optmzton. There re few etermntons for the esgnng of systemtc rchtecture such s constructve n prunng lgorthms. onstructve metho ntlly ssume the neurl network wth mnmum noes n then strt ng noes n lnks untl to get optmum structure whle n cse of prunng metho t ssumes the lrge network whch procee wth prunng off the noes n lnks form the network to get best structure. These lgorthms re lso trppe n locl optmum structure becuse of the nonfferentble spce, complex n mult-moel structure. Therefore these lgorthms re lso fcng the smlr problem lke bck propgton lgorthms. Hence the secon cse.e. weght n bs optmzton s more promsng n stble metho to optmze the neurl network for better trnng thn optmzng of rchtecture. In cse of weght n bs optmzton lgorthms the rchtecture s constnt before the trnng of the neurl network moel. The trnng lgorthms cn be pplcton of ny metheurstc lgorthms whch mke sure the globl optmum pont for the specfc problem. Therefore the mn m of the trnng lgorthm s to fn n pproprte connecton weght n bs to reuce overll error. Therefore globl optmzton lgorthms lke, PO, WDO, GA, Evolutonry lgorthms, GWO, TLBO, BBO, AB, AO etc. re qute helthy to use for the trnng n fnng out the optmum weght n bs for the neurl network. The common fctor for ll globl optmzton lgorthms s populton bse stochstc metho n cn esly vo locl optmum ponts to get best soluton. Moreover, these lgorthms cn pple to ny moel of neurl network wth fferent number of ctvton functons. In ths work, PO, GA, GWO, IBO, TLBO etc. lgorthms re pple usng weght n bs bse optmzton crter n mult-lyere perceptron neurl network (MLP) s use throughout the reserch. The hybr ANN cn be clle s MLPPO (multlyere perceptron prtcle swm optmzton), MLPGA, MLPGWO etc. Therefore to esgn proper lgorthm objectve functon or ftness functon s most mportnt fctor for optmzton. In the lter secton men squre error bse objectve functon formulton s presente. Now hybrzton of metheurstc or populton bse lgorthms metho cn be strt wth the ntroucton of the opte lgorthms wth the

173 specfc moel of neurl network. Therefore the bsc of populton bse stochstc lgorthms re explne below... Prtcle swrm optmzton PO s populton-bse optmzton lgorthm mprnte from the smulton of socl behvour of br flockng. The populton comprses of the number of prtcles (cnte soluton) whch fles n serch spce to fn the out globl optmum pont. Intl pproxmton of prtcles for poston n velocty n serch spce s rnomly chosen s shown n Fgure.8. Ech nvul fles n the serch spce wth specfc velocty n crryng poston, smultneously ech prtcle upte ts own velocty n poston bse on the best experence of ts own n the socl populton [8]. The bsc steps wth mthemtcl moellng of Prtcle wrm Optmzton Algorthm re shown n flowchrt: trt electon of ntl rnom velocty n poston of prtcle I = Evluton of serchng locton of ech prtcle (ftness) I =I + hnge the serchng locton of ech prtcle Upte p,best n g,best when conton s met, I >Mx I Yes N top Upte the prtcle poston n velocty usng equtons Fgure.8 Flow chrt for PO

174 .. Techng lernng bse optmzton (TLBO) Techng-Lernng-Bse Optmzton (TLBO) s populton bse lgorthm works on the effect of mpct of techer on lerners. In ths lgorthm populton s consere s group of stuent where ech stuent ether lerns from techer clle s techer phse n they lso gn some knowlege from other clssmtes or stuents tht re lerners phse. Output wll be n terms of results or gres. In these lgorthms fferent subjects for lerner resembles the vrbles n lerner results s equvlent to the ftness functon for ny problem n fnlly the techer wll be consere s the best soluton cheve so fr. There re severl other populton bse methos hve been successfully mplemente n shown effcency. The etls bout ths lgorthm cn be foun on reference. [9]... Objectve functon for trnng MLP Anlytcl soluton of the nverse knemtcs problem s hghly non-lner n mthemtclly complex n nture. An ANN moel oes not requre hgher mthemtcl clcultons n complex computng progrm. ANN requres ntl selecton of weght, whch s vgorous to yel locl optm, convergence spee n trnng tme for the network. As we know tht the bs n weght for ech neuron rectly ffect the output vector of neurl network. Generlly, weght s rnomly selecte n the rnge of 0 to, fter ctvton functon weght of ech neuron juste for the next terton. The heurstc optmzton lgorthm optmzes the weghts of the neurl networks. When certn termnton crter re met, or mxmum number of tertons re reche, the tertons cese. From the prevous reserch hybr optmzton, lgorthm strte evolvng wth hgh n remrkble vnces n ther performnces, [0]-[]. These technques prouce better outflow from locl optmum n testfe to beng more opertve thn the stnr metho. All these pproches yel better results when neuron weght s juste. In ths work, optmze weght n bs for ech neuron usng vrous metheurstc lgorthms re use for the trnng of MLP network. For the trnng of network, t s mportnt to hve ll connecton weghts n bses n orer to mnmze the men squre error... Objectve functon From [], n ech epoch of lernng, the output of ech hen noe s clculte from equton (.). f (n k ) ( exp( ( n w j.x b j ))), j,,,...,h (.)

175 Where n k n w j.x b j,n s the number of the nput noes, w j s the connecton weght from the th noe n the nput lyer to the jth noe n the hen lyer, b j s the bs (threshol) of the jth hen noe, n x s the th nput. After clcultng outputs of the output noes from equton (.). o k h w kj.f (n k ) b k, k,,...,m (.) Where, w kj s the connecton weght from the jth hen noe to the kth output noe n b k s the bs (threshol) of the kth output noe. Fnlly, the lernng error E (ftness functon) s clculte from equton (.7-.8). m k k k (o y ) E (.7) q E E ( k ) (.8) q k Where, q s the number of trnng smples, when the kth trnng smple s use, n k k y s the esre output of the th nput unt o s the ctul output of the th nput unt when the kth trnng smple s use. Ftness functon cn be clculte from equton (.9). Where the number of nput noes s equl to n, the number of hen noes s equl to h, n the number of output noes s m. Therefore, the ftness functon of the th trnng smple cn be efne s follows:.. Weght n bs optmzton scheme Ftness(X ) E(X ) (.9) To represent weghts n bses t s requre to ncte the encong strtegy fter efnng the ftness functon for hybr ANN, []-[] From the ltertures, there re three encong strteges for representng the weghts n bses. Frst strtegy s vector metho n whch every gent s encoe s vector. For trnng MLP ech gent encoe s vector to represent ll weghts n bses for the MLP structure (see Fgure.0). The optmzton of weght n bs usng PO s shown n Fgure.0 cn be mplemente for ll other optmzton lgorthms. In mtrx encong, ech gent s encoe s mtrx. In cse of bnry encong, gents re encoe s strngs of bnry bts. From the ltertures [0]-[], n cse of vector encong strtegy, the encong s smple, but fter clculton of output of MLP, t s requre to ecoe ech prtcle nto weght mtrx, therefor econg process becomes complcte. Vector encong strtegy s generlly use n the functon optmzton fel. In cse of mtrx encong strtegy, the econg s smple for weght mtrx but the encong s ffcult

176 for neurl networks wth complex structures. Ths metho s very sutble for the trnng processes of neurl networks becuse the encong strtegy mkes t esy to execute econg for neurl networks. In the lst strtegy, ech prtcle shoul represent n the bnry form, so encong n econg becomes complcte for the complex network structure. An exmple of ths encong strtegy for the MLP hs gven n Fgure.9. trt electon of ntl rnom weght n bs for velocty n poston of prtcle I = Evluton of serchng locton of ech prtcle (ftness) hnge the serchng locton of ech prtcle Upte the prtcle poston n velocty usng equtons Upte p,best n g,best when conton s met, lculton of men squre error s ftness functon I =I+ N o Gen.>Mx Gen. Yes top Fgure.9 Flow chrt for MLPPO

177 X Y w w w w w b w w θ w w w b w b Z w Fgure.0 MLP network wth structure -- serch _ gents(:,:,) [W,B,W,B] w w w w b W w w w, W w, b B w w w w b B, b Where W s the hen lyer weght mtrx, B s the hen lyer bs mtrx, W s the output lyer weght mtrx, W s the trnspose of W, n B s the hen lyer bs mtrx. b. ummry Ths chpter elvers the bscs of rtfcl neurl network technque n ther hybrzton scheme wth metheurstc optmzton lgorthms. Furthermore, fferent types of mult-lyere perceptron network, ther lernng bltes re scusse. Moreover, t s lso covere the combnton of evolutonry lgorthms wth MLP neurl network s well s comprson of grent escent lernng lgorthms n pproprte scheme. In current scenro hybrzton of ANNs wth metheurstc lgorthms re populr n rechng to the vnce stge of the soft computng technques to hnle non-lner, NP-Hr problems, complex mthemtcs n nosy problems. Therefore n the lter secton, few fferent type of metheurstc lgorthms such s PO, GA, GWO, n IBO s scusse here whch s lter use to obtne the optmze weght n bs of the opte neurl network moel. After the pplcton of the metheurstc lgorthms n trne neurl network, s pple to fn out the nverse knemtc soluton of the robot mnpultors. Dfferent types of confgurton of the robot mnpultors hve been tken for the knemtc nlyss. The results obtne out of ll these moels re presente n chpter 7.

178 Therefore, nverse knemtc soluton of vrous confgurtons of the mnpultors wth n wthout Euler wrst re solve computtonlly usng trne hybr MLP neurl network. The opte metho s compct n effcent tool for knemtc nlyss. In the result chpter nverse knemtc soluton for opte mnpultor hs been tbulrse n comprson on the bss of mthemtcl complexty s me over other conventonl bse metho. 7

179 hpter OPTIMIZATION APPROAH FOR INVERE KINEMATI OLUTION. Overvew Optmzton s the metho whch yels best soluton of problem hvng number of vrbles n lterntves. From the efnton, t nclues the phenomenon or some bologcl concept n our ly lfe tht nspres to mnmze the energy, computtonl cost, mthemtcl opertons, tme, etc. n mxmses effcency, profts, power etc. wth the help of some rect n nrect prmeters. For exmple, computton of nverse knemtcs problem of robot mnpultor wth the rect relton of consere torque, energy n tme to be mnmze to get the esre poston. In ths exmple jont vrbles cn be clculte fter optmzton of the poston error, torque, energy etc. Therefore n bro sense, the mjor consttuents of the optmzton methos cn be recognze s ts objectve functon whch s generlly quntttve expresson of the system to be optmze n then the number of unknown prmeters or set of vrbles tht s requre proper settng to yel optmum vlue, fnlly the number of constrnts whch gves the complete objectve functon for the concern omn. These three consttuents s the bss to solve ny optmzton problem n ther objectve functon (ftness functon) formultons. On the other hn, the mjor objectves for optmzng of ny functon woul be the convergence of the soluton. Furthermore, optmzton lgorthms shoul lwys be flexble to mnge vrous problem such s nonlner, NPhr, screte, mult-objectves, mult-mols etc. Most mportnt property of ny optmzton lgorthms s to vo the locl optmum pont. onserng n equlty n nequlty constrnts problems, objectve functon cn be efne, 8

180 Mnmse F( ) ubject to: G ( ) G..G H. n l ( ) ( ) H ( ) b ( ) b l n.h ( ) b Further these constrnts G n ( ) n H l ( ) cn be hnle by Lgrngn formultons by equton (.) s, uppose tht n L (,, ) F( ) ( G ( )) j(b j H j( )) (.) * * * * n l j,,, s responsble for the mxmzton of the objectve functon () whch s subjecte to the constrnts G ( ) n H j( ) b j where =,,.n n j=,, l then the vectors ) n ) wll be lnerly G ( * n H ( * l nepenent to the problem. On the other hn, there my the Lgrngn vectors n * wll be gven by equton (.) s, * * * j ( j j n *) * * F( G ( ) j j * j * H ( j ) 0 * j H ( ) b ) 0 wll be complementrty n 0. (.) These bove mentone contons for the optmlty n complementrty s known s bsc concept of the Kuhn-Tucker. Therefore the constrnt optmzton problem cn esly hnle wth these concepts to mke the objectve functon unconstrnt. The optml soluton of the * cn be ether mnmum or mxmum epens on the consere problem n the soluton my be locl, globl optmum or ner optml. Further the optmzton problem cn lso be ctegorze s the consere objectve functon my be lner or non-lner followng lgebrc, polynomls or trnscenentl etc. formultons. It cn lso be bse on constrnt wth nteger or mxe nteger, n lso the problem my be the numerc or symbolc. Therefore the optmzton s epenng on the rel worl problem whch cn be formulte by bove consere cses. 9

181 In cse of tertve optmzton, ntl pproxmton of the soluton ccelertes the process by consequently uptng the current soluton wth the ol solutons untl t obtne the optml pont. The metho s bsclly mnfol but t requres the ttrbutes of objectve functon. On the other hn the conventonl methos grent bse serchng process s the key pont to obtn the locl optmum pont, whch mens the objectve functon s fferentble n the grent of the functon cn be evlute, then optml soluton yels wth escent recton serch wth ech grent pont. These methos re known s lne serch n some other conventonl methos re steepest-escent, qus-newton, Newton, Non-lner conjugte methos etc. The mjor vntges of the bove mentons methos re ts locl serch blty, convergence of the soluton for unconstrne problems, wheren ccurte soluton wth the help of grent bse metho s esy n computtonl cost s less. However, the ftness functon (objectve functon) shoul be unmol n cn be fferentble for two steps. The problem wth the metho s non-smoothness n nosy soluton of the objectve functon f t cnnot be expln by lgebrc or nlytcl formultons. On the other hn, zeroth orer metho oes not requre the hgher ervtves n grent bse pproxmton. The nterestng pont n ths metho s the efcency of the we ssumptons lke contnuty n fferentblty of the functon s not mportnt. Few exmples of the methos re rect serch, prtcle swrm bse optmzton, bcter forgng, evolutonry lgorthms etc. The rect serch methos re lso known s heurstc bse lgorthms whch contns the test n generton of the strtegy. Wheren, every nvul soluton for the functon s compre n evlute so s to fn the best soluton wth the constnt observton of the mprovement. There re two strteges for selecton or smplng nmely stochstc n etermnstc. tochstc serch cn be unerstn wth the rnom serch n the current menson whle n cse of etermnstc serch fxe or preefne coornte of serch for locl best soluton s known. Rnom wlk s the exmples for stochstc serch process n pttern serch, smplex metho re etermnstc methos for locl optmzton process. Due to ts rnom vrble epenences ts gves slow convergence whle ervtve bse metho performs fster. If the numbers of locl optmum ponts re more thn one then poor pproxmton whch s combne wth the greey serch coul be stuck t sub-optml pont. ubsequently, ntl pproxmton for the lgorthms s less mportnt f one consers the effectve selecton of the serch spce. Therefore, populton bse lgorthms gves soluton to the ntl pproxmton problem wth the help of selecton of the objectve functon tht cts s nrect locl optmzers. Furthermore, t s lso requre to fn the exct number of ntl 0

182 pproxmtons for the objectve functon so s to fn globl optmum. In ths regr, evolutonry lgorthms cn be frutful to opt wheren t helps not only for the populton generton of the cnte soluton even use prllel locl optmzers. On the other hn, explorton n explotton bltes re lso help to fn the globl ponts. The overvew of ths chpter s concerne wth the bref ntroucton of the conventonl n bologcl nspre lgorthms, wth the three mjor concepts of the optmzton lgorthms. The bove ntrouce clsscl methos on the bss of the conceptul frme wll be lter use to expln the evolutonry lgorthms, metheurstc lgorthms n swrm bse lgorthms etc. for the evluton of the nverse knemtc problem of robot mnpultors.. Metheurstc lgorthms In the lst few yers, metheurstc lgorthms hve been extensvely use for resolvng vrous complcte optmzton problem. Nture s plyng key role for evelopng mny optmzton lgorthms for exmple rtfcl bee colony lgorthm, frefly lgorthm, nt colony optmzton etc. We re lwys ttrctng by tny or lrge orgnsms lke mnutve nvertebrte, chrsmtc vertebrtes, brs, prmtes, bees, n nts etc. whch re often the source of nsprton for mny reserchers []. These orgnsms prove the most elcte systems for explorng nture n nswerng funmentl scentfc questons. omprtvely metheurstc lgorthms re more pproprte n omnnt thn the other nlytcl methos whch re bse on conventonl mthemtcs n ervtves. Metheurstc lgorthms commonly hve two elementry fetures lke ntensfcton n versfcton. Intensfcton normlly offers locl serch ner to exstng current best solutons wheres versfcton offers effcent explorton of serch spce, mostly bse on rnom numbers [], []. Metheurstc lgorthms re wely use becuse they prove globl soluton keepng the m of fster soluton, soluton of lengthy problems n obtnng robust technques. Metheurstc lgorthms cn fn proxmte optmum solutons t soun computtonl cost whch oesn t ssure fesblty or optmlty of the obtne soluton, on the other hn n most of the cses reserchers re keen to see the closeness to optml n fesblty of the soluton. []-[]. The nture s nfnte n there s no lmt for the source of nsprton for exmple prevously evelope lgorthms re nspre from nts, bees, frefles, bcter, musc, hbtts, frogs etc. There re mny nture-nspre optmzton lgorthms hve ppere for exmple the Genetc Algorthm (GA) [7], whch mmcs the genetc process of bologcl orgnsm. The concept of GA cme from the Drwn's prncple

183 ''survvl of the fttest'', whch escrbes the evoluton of populton on the bss of nturl selecton. In ths lgorthm ech nvul represente by gene n the combnton of gene cretes chromosome whch s ultmtely yels soluton. These chromosomes re recombnng usng crossover n mutton. Ths behvour les to globl soluton for the objectve functon [8]. In the process of evoluton of nturl thngs s mostly the source where selecton process of the concern orgnsms n populton keepng the best ftte to the envronment s lwys pte. Ths proves the most promnent optmzton lgorthms. As per Drwn theory, evoluton mnly concern wth the ntercton of the physcl mechnsm of selecton, reproucton, mutton, n competton wth other speces or orgnsms. In ths theory, ech nvul re compulsorly nee to compete the physcl process for the survvl, n ths wll le to fn the best or selecton of survvls wth better genetc chrcter for the concerne envronment so s to prouce offsprng or reproucton. Evolutonry lgorthms EAs re metheurstc lgorthm bse on the populton of the nvul soluton tht evolves by selecton, mutton n reproucton of best ft n the populton. The mjor vntge of these EAs lgorthms compre to conventonl metho, conventonl methos reles on the locl memory of one pont n ech step of terton whch les to locl optmzton process wheres populton bse metheurstc methos uses prllel serch mechnsm wth the mjor blty of explorton n explotton. The mjor pplcton fels of these lgorthms re mostly n reserch n nustres, mostly n robotcs, mchne egn, control pplcton, mge processng, moellng, sgnl processng etc. The bsc pseuo coe for evoluton lgorthms cn be gven s, Evolutonry lgorthm. Intlzton of populton. evluton of ech nvul. Whle termnton crter met o. selecton of prents. recombnton of prents. mutton yels offsprng 7. evluton of new nvuls 8. selecton for next generton 9. en whle In the bove pseuo coe conserng the frst exmple of mnmzton of F( ) functon optmzton where ntlzton cn be one wthn the fesble serch spce. Intl populton cn be selecte rnomly wthn the serch spce. Once ntlze, populton wll go through the terton n selecton of ech nvul best soluton untl t converges for e.g. functon threshol, no. of genertons, etc. Ths ech terton mechnsm gves nformton encoe n the current populton so s to cheve new

184 trl genertons, e.g. mutton n recombnton. electon process gves the solutons to replce the trl populton wthn the current populton so s to etermne the next generton. The whole process of optmzton of ech nvul soluton wthn the current n trl populton re ssgne wth the objectve functon vlue tht sgnfes the closeness to mnmum vlue. Moreover, ths objectve functon evluton helps the overll serch process to fn out whch nvul cn be use for reproucton n even survve for the more genertons. Therefore, convergence of the soluton woul be epenng on explorton n explotton blty of the serch process wthn certn regons. In evolutonry lgorthms, tertve process gves the blty to explore the new regons wthn the serch spce whle selecton s responsble for the explotton of nvul whch woul be crryng the nformton's to ensure the next generton to be complete. Fnlly, evolutonry lgorthms cn be ctegorze s, Genetc lgorthms (GAs), Genetc progrmmng (GP), Evolutonry Progrmmng (EP) n Evoluton strteges (Es). The bo-logclly nspre metheurstc lgorthms mmc the best feture of the nture whch coul turn nto better effcency s compre to other conventonl lgorthms. More often, these pproches re selectng the fttest vlue whch hs evolve by nturl selecton. Bo-nspre technques my be ctegorze nto: () Bcterl forgng lgorthms (b) Evolutonry lgorthms, (c) wrm ntellgence bse lgorthm [9]. Genetc lgorthm s the key fctor for the estblshment of evolutonlly lgorthms becuse t stsfes the prncple of "survvl of fttest" gven by Drwn. Ths clssfcton covers genetc progrmmng (GP), fferentl evoluton (DE), evolutonry strtegy (E) n bogeogrphy bse optmzton (BBO), but lso other. These re lso populton bse metheurstc lgorthms workng wth some form of the Drwn's prncple [9]. A swrm ntellgence bse lgorthm ntcptes specfc opertons, nterctons n shrng nformton wth other prtcles. These opertons cn be socl n cogntve, ue to ther socl behvour n knowlege shrng hbts turns nto ntellgence, whch cn be further known s swrm ntellgence. Ther cogntve n socl behvour yels globl results [8]. Another ctegory of populton bse metheurstc serch lgorthm s bcter forgng lgorthm. [0]-[]. The most well-known types of the bcterl forgng lgorthms re computng systems of mcrobl nterctons n communctons n rule-bse bcterl moellng. Bsc concept of these nture nspre populton bse lgorthms re, menson of the serch spce, number of nvuls, bsc relte prmeters, stoppng crter, number of evlutons etc. Ech nvul sgnfes

185 resoluton for the functon optmzton problem. In every generton set of new solutons re obtne n then best soluton re kept n memory to prouce new set soluton ths process en when t reches to certn termnton crter []. Mjor rwbck of these lgorthms s the number of nvuls whch shre nformton n ths my cuse hurle to yel best or globl soluton. On the bss of nccurte or nsuffcent nformton they my be converge n the locl optmum pont becuse the serchng spce or menson of the problem my not scovere equtely. Moreover, smlr nvuls on t yels fferent soluton tht cn lso be rwbck of the lgorthm when the functon hvng mny locl optmum ponts... Genetc lgorthms (GAs) representton Genetc lgorthm ws frst evelope by J. Holln bse on the rtfcl behvour of nturl system. GA's re encoe wth the bnry strngs of 0's n 's n t cn be represente s genes of bologcl or nturl systems. These genes re certn sequence of the chromosomes n etermne the behvourl n physcl chrcterstcs of n orgnsm n the envronment. In the sme wy ny evolutonry lgorthms cn be efne s two seprte serch spces n whch genes represents the vrbles to be optmze. Physcl or behvour prmeter of the system cn be represente by the soluton spce whle the encong wth the genes gves the representton spce. These physcl prmeters re known s phenotype n gene encong s genotype of the system. nce genotype nfluence the nvul solutons n the representton spce whle evluton s ccomplshe n the soluton spce, therefore t s requre to complete encong n econg of the vrbles from the soluton to representton spce. Moreover, the vrbles or prmeters cn lso be represente s -mensonl rrys, where ech nvul s ether bnry or rel vlue, tht s respectvely. [0,] or Fgure. Bnry representtons of genes

186 In genetc lgorthm bnry vlue s most often use for the representton of genes n the chromosomes, wheres n cse of mutton bse evolutonry progrmmng n evolutonry strteges re rel vlue representton. Fgure. represents chromosome wth n genes ll re coe s sx-bt bnry wors. Desgn vrbles n control prmeters for both lgorthms re encoe n sngle rry. In cse of evolutonry strteges ech nvul soluton cn be represente equton (.) s, (,, ) (.) n where, s the esgn vector, n, re the evoluton strtegy prmeters. Prmeter belongs to the vector spce of stnr evton whch mofes the mpltue urng mutton of n cn be gven s (n {,, n}) n. mlrly s set of rotton ngles tht gves the xes of orentton for the mutton n the serch spce topology n cn be represente s n (n {0, (n n )(n ) / }) [ ]. mlrly, evolutonry progrmmng ech cnte s represente s esgn vector n vector of vrnce n cn be gven equton (.) s s, (, ) (, n,, n ) (.) where, n v s rel vlue prmeter n n s rel postve vrnce. () Intlzton In most of the metheurstc lgorthms the ecson for the ntl pproxmton s ply crucl role to rech the optmum pont. nce the optmzton process s bsolutely bse on the ntl pproxmtons therefore t s requre to ensure the convenent proceure for rnom smplng of the ntl pproxmtons. Intlzton gves the hnt to bul the cnte solutons by smplng of the fesble serch spce. In most of the cses, rnom number generton s use to smple the ntl guessng so s to ensure the hgh versty n the ntl pont. Inste, f pror nformton bout the optmum pont s vlble, then ths nformton wll be frutful to use for the ntlzton process. In genetc lgorthm, ntlzton process s one wth the rnom smplng of the tmes the bnry vlue {0, }. In cse of evolutonry strteges, ntlzton process s me through the mutton upon whch strtng pont s selecte rnomly or efne by user, n smll stnr evton vlue s suggeste. Fnlly, evolutonry progrmmng uses the unform rnom strbuton for the ntlzton of esgn vector n vrnces.

187 (b) Recombnton Recombnton process cn be unerstn wth the mechnsm of nvolvng of two or more prents whch my be sexul, sexul or pnmctc to prouce new offsprng s. It cn be represente s p q R : n ths cn be unerstn wth the sexul or pomctc gene opertor where p. The bove mechnsm of recombnton mttes the bologcl process to generte new nvul solutons or offsprng by shrng genetc nformton tht re mprnte n ll nvuls of the prents. In genetc lgorthm, ths recombnton process s generlly bse on the selecton of chromosome n ech nvul rnomly where p represents the no. of chromosome for selecton n t cn generte by the crossover probblty p c [0, ]. Ths probblty vlue s compre n mesure wth the smple rnom number r= [0, ]. If the r pc then rnom crossover of chromosome n the bt strng cn be selecte for the next generton of offsprng. After crossover of the selecte bt other t wll be swppe to crete chlren chromosome wth the replcement of the rnom crossover pont. If r p c then the prent chromosomes cn be uplcte s shown n Fgure.. () Frst cse r pc (b) econ cse r pc Fgure. Exmples for smple crossover wth two fferent cses. On the other hn, recombnton n Es cn be sexul or pnmctc for the generton of new offsprng wth consere rnom prents. onsequently, sexul recombnton wll be on the pr bss where p=, for ech new offsprng's. In cse of pnmctc recombnton one prent wll be constnt n nother wll be rnomly selecte from the prent populton ( p ) for ech nvul offsprng. Recombnton cn be ntermete or screte, screte recombnton s rnom selecton of the ech

188 component of the offsprng n ntermete clcultes the rthmetc men of the ech component of the offsprng. (c) Mutton Mutton cn unerstn wth the mechnsm smlr to the recombnton process beses sexul or pnmctc opertors t works on sexul opertor n cn be represente s M :. Ths gves the smll rnom chnges nto the gene cong for ech nvul. Mutton opertor bsclly works on the populton multplcty wth the ton of smll perturbtons on the nvuls wth further blty of explorton of new regons wthn the serch spce. It lso helps to overcome the problem of trppng n locl mnm. Genetc mutton s smlr to recombnton process prt from nvertng the vlue of rnom bts of chromosomes. orresponngly, one pont crossover, mutton s generte by some ctvton of mutton probblty p m [0, ]. Ths mutton probblty wll then be compre wth the unformly rnomly generte number r [0, ] such tht f r pm then bt wll be nverte otherwse t wll be unchnge (Fgure.). () electon Fgure. Mutton n genetc lgorthm As we know tht recombnton n mutton gves the blty of explorton, wheres selecton s mnly responsble for the explotng the cnte soluton wth the vncement of the next genertons. nce the selecton explots the fvourble ponts n the serch spce, the ftness of ech nvul must be mesure n the populton. To ccomplsh the most promsng re n the serch spce, t s requre to efne the objectve or ftness functon whch confrms the closeness of the soluton towrs the optml vlue. Let us ssume the ftness functon f to elborte the selecton proceure n genetc lgorthm. Therefore the probblty of selecton cn be gven for ech chromosome s,,, n the populton equton (.) s, 7

189 f (s ) p (.) s k f (s k ) Where, represents the populton sze. The most common type of selecton s roulette wheel selecton proceure whch s prttone nto tmes n the sze of the ech prttone s proportonl to selecton probblty of ech nvul. New populton cn be generte by spnnng the roulette wheel tmes n n every spn rnom selecton of the chromosome s one from the current generton (Fgure.). Therefore hgher selecton probblty s les towrs the generton of new nvuls n the populton. Fgure. Roulette wheel selectons. The selecton probblty for ll four chromosomes s 0., 0.0, 0.0 n Prtcle swrm optmzton Kenney et l. [8], propose n effcent evolutonry lgorthm whch s bse on swrm ntellgent whch s known s Prtcle swrm optmzton (PO). It s populton-bse optmzton lgorthm mprnte from the smulton of socl behvour of br flockng. Here n ths lgorthm populton comprses of the number of prtcles (cnte soluton) whch fles n serch spce to fn the out globl optmum pont. Ech nvul fles n the serch spce wth specfc velocty n crryng poston, smultneously ech nvul upte ts own velocty n poston bse on the best experence of ts own n the socl populton [8].The bsc steps n mthemtcl moellng of Prtcle wrm Optmzton Algorthm hs been scusse n prevous chpter. In ths chpter optmzton lgorthm wll be use to evlute the jont vrbles of vrous confgurton of mnpultor. 8

190 .. Grey wolf optmzton lgorthm In ths lgorthm leershp of grey wolves re rrnge herrchl nmely lph, bet, elt n omeg wth the mn purpose of huntng, encrclng of vctm, lookng for vctm, ttck on vctm. These strteges gve ntellgent n socl behvour of the grey wolf for the rrngement of ther foo source wth the mnmum lbour. The strtng steps of ths lgorthm re () fnng of foo source (.e. vctm), (b) chsng the vctm n (c) pprochng the vctm. Therefter confrmton of vctm or foo source, grey wolves encrcles n hrsses the vctm so s to not lose the foo source whch s lter fnshe wth the kllng. erchng for the vctm represents the explorton blty of the wolves for the evelopment of the lgorthm n explotton cn be unerstoo wth the huntng of vctm. Therefore the mn theme of the lgorthm s to uptng the serchng gents of ther postons n clculton of ftness for ll gents []. Dfferent prmeters of GWO re ntlzton of lph, bet n elt, mx tertons, serchng gents, neghbourhoo ste selecton n termnton crter. After ntlzton GWO follows certn steps such s, () Trckng, chsng n pprochng the prey () pursung the prey then enclosng n hrssng the prey tll t qute the movements () kllng the prey Mthemtcl moellng of wolves behvour cn be gven s the best ftness vlue wll be consere s lph, then secon n lst best cn be nme s bet n elt. Other nvul solutons cn be consere s omeg. Now encrclng of the grey wolves cn be clculte s equton (.)-(.7), D.X (t) X(t) (.) p X(t ) X (t) A.D (.7) Where, t represents ol terton n t+ s new terton, p X p s the poston vector of vctm, A, re the coeffcent vector, X s the poston vector of the grey wolf. Now the corresponng vectors cn be clculte equton (.8)-(.9) s, *r (.8) A r (.9) where r, r re rnom number vector of [0, ] n ecreses lnerly from to 0 through the complete terton. 9

191 Now huntng behvour of the grey wolves cn be mthemtclly escrbes s, lph wolf s the best nvul together wth the bet n elt whch serch for the potentl locton of the prey or ths cn be unerstn wth the optmum locton. Therefore keepng the postons of these wolves cn be consere s best locton n cn keep n memory so s to upte the ol poston wth the comprson of memory. Therefore the concerne mthemtcl formuls cn be gven equtons (.0)-(.) s, D *X X, D *X X, D *X X (.0) (X A )* D (X A )* D (X A )* D (.) X, X, X X X X X(t ) (.) trt Intlzton of wolves' populton,, A n. I = lculton of ftness for ech nvul Upte the poston of ech nvul usng Upte, A n lculton of ftness for ll nvul n lculton of overll error of en effector poston n orentton s ftness functon I =I+ No Gen.>Mx Gen. Yes top Fgure. Flow chrt for grey wolf optmzer 70

192 Therefore, the postons of lph, bet n elt cn be clculte from the bove formul n the fnl poston wll be gven by rnom plce wthn the rus of serch meter. Alph, bet n elt behves lke leer to get the poston of the vctm n ccorng to ths other wolves upte ther poston rnomly roun the vctm. Flow chrt of GWO s presente n Fgure. for the corresponng ftness functon of overll error mnmzton of the en effector poston n orentton. Another effcent n fmous metheurstc lgorthm bse on populton of bees s rtfcl bee colony (AB) lgorthm. AB s lso populton bse optmzton technque lke PO whch resembles the ntellectul performnce of honey bee swrm. The honey bee socety comprses of three groups nmely employe, onlookers n scouts. Onlookers bees gves the hnt for the foo source whch lter scover through employe bees n then scout bees serch for new sources. In ths system, the locton of foo source sgnfes potentl soluton of the concern problem n nectr mount of foo source resembles to the ftness of the relte soluton []. Techng-Lernng-Bse Optmzton (TLBO) s populton bse lgorthm works on the effect of mpct of techer on lerners. In ths lgorthm populton s consere s group of stuent where ech stuent ether lerns from techer clle s techer phse n they lso gn some knowlege from other clssmtes or stuents tht re lerners phse. Output wll be n terms of results or gres. In these lgorthms fferent subjects for lerner resembles the vrbles n lerner results s equvlent to the ftness functon for ny problem n fnlly the techer wll be consere s the best soluton cheve so fr. There re severl other populton bse methos hve been successfully mplemente n shown effcency [9]. However, t s not lwys necessry tht every lgorthm cn solve complex problem n proves best soluton n fct t ws mthemtclly prove by Wolpert et l. [].. Development of novel metheurstc optmzton lgorthm Ths secton ntrouces fferent nture nspre lgorthm, clle rb Intellgence bse optmzton (IBO), for optmzng vrous unmol, multmol, seprble, non- seprble problems n for nverse knemtcs soluton of robot mnpultors. The IBO lgorthm s bse on the swrm, crossng n shell selecton behvor of the crbs. Ech crb represents the nvul or cnte soluton of the problem n ftness evluton cn be one by shell selecton behvor of the crb. When ll crb occupes the shell then t cn be unerstn wth the convergence of the soluton whch s evlute by poston vector of ech crb. The best poston wll be kept n memory 7

193 n shorte to best ft vlue for the next serch. Ths process of serchng stops when t reches to mxmum tertons... rb ntellgence bse optmzton lgorthm Novel effectul nture-nspre metheurstc optmzton technque groune on crb behvor s propose n ths pper. The propose rb Intellgence Bse Optmzton (IBO) technque s populton centere tertve metheurstc lgorthm for D- mensonl n NP-hr problems. The populton of swrm represents the group of crbs whch hve socl behvor s well s nterct wth ther reltves n neghbors. Populton of smll group's moves over D-mensonl serch spce collectvely behves lke swrm. In ths work postonl vector of ech nvul whch permts mutul movements of other nvuls wthn the swrm s ntrouce. Ths lgorthm consers three prts of crb behvor nlyss: the frst prt s swrm behvor of crb, secon prt s relte to crossng behvor n the thr prt s shell selecton or recognton behvor of crbs. The mthemtcl moelng of the lgorthm n the source of nsprton of the IBO lgorthm re explne n etl. In ths work, the effcency of the suggeste lgorthm wth verse nvultes hs been teste n then compre ts performnce wth well-known populton bse metheurstc optmzton lgorthms. In the lter secton ths lgorthm hs been pple for nverse knemtcs soluton for R robot mnpultor. Most of the bo-nspre processes cn be nferre n terms of computtonl cost. ocl behvor nctes ntellgence on crb whch cn be founton for nsprton. rbs hve Intellgence for survvng the pretors, lookng for the rght pth for grbbng foo n fnlly shelter for ther sfety. Vrous stues of crb's lfe whch coul be perfectly sutble to evelop n optmzton lgorthm hve been one. The fferent behvorl stues of crbs re: () wrm behvor, (b) Forgng behvor, (c) Pretor Protecton, () hell selecton (Recognton behvor) n (e) rossng behvor. Among bove mentone behvors only three of them nmely swrm behvor, shell selecton n crossng behvor of crbs hve consere n ths pper. Pretor protecton n forgng behvor hs not been consere n ths reserch whch coul be prt of future work. Few speces of crbs (e.g. Mctyrs gunote) populte on flt lgoons n form mssve groups of severl hunres n sometmes hunres of thousns of crbs. It hs been observe tht crbs show serchng n swrm behvor s per bologcl experments []. A front group of ther swrm s rven by nherent turbulence tht cuses ech nvul lwys chngng ther poston n entre serch re. Ths nherent turbulence helps to fn out locl serch ponts. wrm behvor gves potentl to cross wter pools n 7

194 vonce re wheren sngle nvul or group of nvuls never tres to cross vonce re; however, huge swrm enters the wter n crosses lgoon wthout reluctnce. In the swrm crossng preventon or vonce re conssts of forwr fcng n submssve tl. Bckwr or submssve tl smply keeps n eye on forwr group. It hs been ssume n here tht there re two types of neghborhoos frst one s optmstc nterctve n nother for observng n succeeng flock-mtes. It hs been observe tht the swrm or group of swrm cn mngle wth ther reltves or even wth non- reltves ue to ther ffuson mechnsm. Mostly crbs spen ther whole mture lves on ln, but to reprouce they choose se n nto t they schrge ther evelopng lrve. The mn reson of swrm behvor s collectvely efen gnst pretors, or to come together to et stmpe foo resources. The bologcl orgnsms ntercte ue to shrng or extrctng relevnt nformton from the envronment. Envronment prouces vrous physcl or chemcl sgnls whch coul be extrcte by orgnsm though ther evolve sensory mechnsms []. Terrestrl n qutc orgnsms hve chemcl, vson n tctle sensors n mong these sensors chemcl sensors ply crucl role to extrct ecologcl nformton. These chemcl sensors prouce sgnls for presence of pretors, convenence of foo resources, n sttus of compnons n vlblty of shell [7]-[9]. From the prevous experments t hs been shown tht chemcl recogntons re the metor for the behvorl stuy of crbs n other crustcens. Most of the speces gves ttentons to ptve behvor when expose to oors to recognze the vlblty of shell [7]-[9]. It hs lso been observe tht P. longcrpus spens more tme nvestgton n empty shell. Mny speces of crbs re epenent on shell prouce by gstropos for ther protecton. They generlly oes not nterfere on lvng gstropos shell, rther they compete wth ech other for gstropos shells tht e by other orgnsm or other mens. The most mportnt behvor of crbs s they contnully serch for new shells ue to ther boy growth n for gettng hgher qulty of shell thn ther current shell. When they leve the current shell other crbs occupes vcte shell. For better unerstnng of shell selecton behvor we hve followe some prevous reserch of vcncy chn [70] - [7]. ynchronous n synchronous, these re two stngush ctegory of shell selecton whch ffer n ther behvorl n ecologcl cost n benefts s shown n Tble., ths stuy gves socl n lone serch n strght vergence to shell reltons comprsng ech nvul for sngle shell selecton. ynchronous vcncy chns rse when mny crb stns n queue n front of shell. When bgger sze crb occupes the vcnt shell, others wll wt s per ther escenent orer of ther sze. On the other hn, synchronous vcncy chn wll be occupe by 7

195 nvul wthout mkng queue or ny socl ntercton wth others. In both cses, f shell qulty s too low or mge ll nvuls wll scr the shell. Bse on these evences, our objectve ws lernng from the mechnsm tht unerle for ech nvuls s chemcl recognton behvor for the selecton of pproprte shell. Lter we observe tht whether the crbs re ble to clssfy two fferent shells or trget on the bses of ther sze, rnk n shell qulty. Ths behvor yels better clue for evelopng the lgorthm. Tble. hell selecton Asynchronous Low potentl for fnng n optml shell Esly reversble shell swtchng No rsk of njury for pretors Decrese vulnerblty to pretors ynchronous Greter potentl for fnng n optml shell Greter potentl to get strne n suboptml shell Pretor competton requres tme n energy; cretes rsk of njury Lrge crb ggregtons Recently H. Murkm et l [7], conucte n experment wth Mctyrs gunote for the swrm behvor n nvng vonce re. Through numerous experments n fel stuy of these crbs, they exmne followng observtons of swrmng behvor: ) Movng swrm n the teln hs nherent turbulence n fferent veloctes n ech nvul. ) If the swrm fces wter pool or vonce re they o not enter nto the pool untl n unless they hve ense populton. ) Ech nvul follows ther preecessor. o these bove stte observtons lke crossng behvor of the swrm hs been opte for the evelopment of the lgorthm... Methoology of IBO lgorthm ocl behvor specfes ntellgence on crb whch cn be orgn for nsprton. rb ntellgence for vong ngerous wter pool n fnng ther shelter for the protecton re the mn m of the evelopment of the lgorthm. As we re now well fmlr wth the swrm behvor of the crb but there re mny other behvor lke 7

196 forgng, pretor protecton, olfctory behvor etc. Beses ther behvorl nlyss some ssumptons hs been estblshe whch re s follows: ) wrm my be movng on shore or nse wter confrmng tht the sze n ntl stnce between them stsfy the mnmum stnce crter. ) It s mportnt for swrm to cross wter pool or vonce re to get the shell. ) Every shore or teln contns unknown number of shell. ) hell cquston my be synchronous or synchronous epens on swrm. ) hell esgn prmeters lke volume, weght n geometry etc. hve not consere... Mthemtcl moelng Ths secton escrbes the mthemtcl moellng of swrm n shell selecton behvor of crbs. In ths moel swrm hvng N-nvuls movng n D-mensonl spce. Where N {,,,,...N mx}. Bounry conton belongs to serch spce D. The locton of -th crb t the p-th step s gven by equton (.), Where, x D n I {,,...M} x L (x, x ) p (.) Poston vector for ech -th nvul t p-th step P v (,p,v) wth v V {0,,...V } n P v (,p,v). If v=0, the poston vector (, p,0), wll be present poston vector n cn be exemplfe by equton (.), P v P (, p,0) z{(r cos v,p ),(R sn, p )} (.) Where z s nteger n R s the length of current poston vector from orgn. If v 0 the vector wll be efne by rnom number α [0, ] n ngulr rnom vlue δ [-π, π] by equton (.) s, P (,p, v) z{(r cos( v,p )),(R sn(, p ))} (.) Poston vector of shell cn be obtne by equton (.), pos L,p P v (, p, v) (.) where [0,], re postonl constnt. As we know tht ech crb nterctng n, shrng nformton for crossng n shell selecton. Here we cn efne the sutblty of shell on the bss of ther sze n stnce.utblty nex for p-th poston vector ( x, x ), ( x, x ) D woul be gven by equton (.7) s, 7

197 f (x, x, p) Lp (x, x ) where I (,,...M) n vv (0,,...V ) (.7) Now settng up memory for uptng the poston vector ( x, x ) t the p-th poston by equton (.8) s, m (x, x, p) 0 (.8) Uptng the poston of nvuls my be synchronous n synchronous wll be bse conton gven below: onton : ze of nvuls of the -th nvul s bse on rnom number γ [0, ]., sze, sze onton : Mnmum number of nvul n sngle swrm wll lwys be more thn., mn ze of the -th shell wll be, sze onton : Mnmum stnce between crbs s lso bse on rnom vlue µ [0, ].,mn_ st,mn_st onton : If number of -th nvul wll be more thn the number of shell present on shore then the shell selecton wll be on the bss of musculr power of crb Mp. M p [0,] The next poston of shell for the -th nvul wll be gven by equton (.9), L,p pos, j (.9) Where j stsfes the conton for vv (0,,...V ) by equton (.0) s, L,p f (x,x,p) (.0) 7

198 Now the upte poston wll be gven by equton (.), P new {(x, x ) D, L,p (x, x ) new} (.) Now we cn set upte memory by equton (.), m(x, x, p) Pnew {(x, x ) D, L,p (x, x ) new } (.) Now we cn mplement the en crter for globl pont by equton (.), Where (.) s number of nvul n s number of shell present, = (,.M). A. Hypotheses Ths secton frst gves some bsc n mportnt efntons for vltng the IBO lgorthms then n lter we ntrouce the bsc steps of lgorthm. Hypothess : A swrm z,mn z,mn_st result n epenent upon rnom vlue genertor. Hypothess : ze of ech nvul s vector of z ntegers tht sgnfes the fesblty of, sze n sze of trget or shell, sze prmeters n represente by rnom vlue genertor. Hypothess : A poston vector Where R s set of rel number. Hypothess : re two epenent pos R of swrm s mesure of the ftness of the soluton. Proportonlty of, sze n, sze represents the globl serchng blty. Hypothess : Musculr power of ll nvul cn be efne by the rnom number genertor M p [0,]. 77

199 .. IBO lgorthm Ths secton elbortes the bsc phses of the propose lgorthm. As shown n Fg. there s N-number of swrm contnng n-number of nvuls tht represents the serch meter n mnmum stnce n between them. Therefore, by genertng huge serch spce, the swrm coul cross the vonce re n cn rech to the sutble trget. The shell s entfe when swrm of crbs foun locl optmum. Ientfcton of locl optmum s bse on mnmum fference of pst postons of crb. In nture crbs re fghtng for the better shell n the best shell wll be occupe by stronger one. Once they reche to the better shell they test t for sutblty n f t not then strts serchng for next pont (swtchng from locl serch to globl serch). In ths pproch ech crb s ntlze wth vlue of hs sze (best foun soluton shell); when crb fns shell they test t for better sze. If ftness vlue of locl optmum sze of shell s better thn sze of crb t wll occupy. The testng of shell n levng t for smller crbs s cusng some ely where some crbs cn escpe the swrm. Becuse crbs cnnot swtch to globl serch lone they nee mnmum sze of swrm. If enough crbs leve the shell n others re lrey testng for the sze they cn mke globl serch, respectvely mgrte to other prt of spce n leve others behn. It hs been observe through vrous reserches for fferent orgnsms tht they spontneously nve from vonce re wth the nsprton of rch foo sources []-[7]. Ths behvor emonstrtes the power of ther neurl processng. Another mportnt ssumpton hs been me tht s swrm nees to cross the wter pool or vonce re n fter crossng, there wll be chnces of gettng ther shells. Ths mentone ssumpton s helpng to fn out globl optmum pont on the entre serch spce. Globl Mxmum Mn stnce Locl erch Dmeter Locl Mnmum Globl Mnmum Fgure. Representton of swrm n serchng behvour 78

200 These crbs hve greter blty of recognton of better qulty, sze n emptness of shell becuse of ther chnces of survvl n protecton from preton s explne erler. In Fg..7 represente the bsc steps of the lgorthm. IBO lgorthm cn be escrbe brefly s: ) Intlzton of IBO prmeters lke swrm conserng N-nvuls movng n D-mensonl spce, bounry conton belongs to serch spce D, the ntl locton of ech nvul, meter of locl serch, ntl stnce between crbs n ntl swrm sze. ) Intlzton of mnmum number of nvuls s per conton, compre wth rnom vlue genertons n rnom vlue for mnmum stnce n sze of ech nvul s escrbe n efnton f not then go to step. ) Recognton of empty shell ner the serch spce f not go to. ) Ftness evluton of ech nvul for ther current poston vector s explne n efnton n evluton of sutblty of shell s per efnton. ) lcultons of sze of ll nvul. ) hell selecton: comprng the sze of ll nvul wth the sutblty of shell f nvul sze s not less thn shell sze then go to step. 7) Tkng the shell f ll nvul stsfes the conton for selecton of shell then the ecson wll be bse on musculr power of ech nvul generte rnomly s per conton. 8) Uptng the vlue of poston vector of nvul. 9) If ll nvul occupes the shell then stop else go to step. 79

201 Intlzton erchng spce, nvuls, meter for locl serch, swrm sze, ntl stnce, ntl swrm number Mnmum number of nvuls = In mn n Mn crb stnce =mn_st No Look/wt for others (locl serch, lone serch behvour) Yes Wether empty shell present nerby No Yes Ftness evluton shell lculte sze of ll nvuls ze of crb No elect next trget for shell (erch for globl) Where no. of shell s equl to no. crbs Mnmum stnce to be trvelle Yes Occupy the shell If more thn one crb stsfy the conton then occupy by crb who hs hghest Musculr power (Mp= [0,]) Upte the vlue of Poston vector of occupe crb All crbs occupe the shell No Yes top Fgure.7 Flow chrt for IBO lgorthm 80

202 . Implementton for solvng nverse knemtcs Inverse knemtcs of ny robot mnpultor cn generlly be efne s fnng out the jont ngles for specfe rtesn poston s well s orentton of n en effector n opposte of ths, etermnng poston n orentton of n en effector for gven jont vrbles s known s forwr knemtcs. Forwr knemtc hvng unque soluton but n cse of nverse knemtcs t oes not prove ny close form soluton thus t s requre to hve some sutble technque to solve nverse knemtcs of robot mnpultor. In ths secton opte lgorthmsre pple to fn out the nverse knemtcs of fferent confgurtons of robot mnpultor. However, there re mny optmzton lgorthms tht cn be frutfully use to prouce the esre results, but most of the populton bse lgorthm oes not hve n blty to serch for globl optmum, n t gves slow convergence rte. On the other hn, evelope optmzton lgorthm wll be use to overcome the problem of Jcobn n other numercl bse methos for nverse knemtc soluton. Further mthemtcl moellng of objectve functon s scusse n etl to vl the nverse knemtc soluton... Mthemtcl moellng of objectve functon Any Optmzton lgorthms whch re cpble of solvng vrous multmol functons cn be mplemente to fn out the nverse knemtc solutons. In ths secton generl moel of objectve functon s ntrouce for further mplementton of optmzton lgorthm. In chpter, vrous confgurton of robot mnpultor hs been consere for nverse knemtc soluton. Now let's ssume N-of serl robot mnpultor hvng jont confgurton. Therefore, poston n orentton of the en effector cn be gves by equton (.) s, R P 0 P e or n x o x x X n y o y y Y P e (.) n z oz z Z Where R represents the orentton of the en effector poston mnpultor. n n R O n en effector P these re reltve to the fxe reference frme t the bse of Now the current poston P c of the en effector cn be clculte from the equton (), n for the gven esre poston P, conserng ths problem to fn out mnmum one fesble jont vrble whch gves the poston of en effector t the trget poston coornte. On the other hn, t s qute ffcult to fn the close form soluton lke 8

203 conventonl nverse knemtc ervton therefore wthout loss of the generlty optmzton pproch s necessry. Now optmzton pproch to solve nverse knemtc problem cn be solve conserng the error E between the current poston n esre poston of the en effector. Ths error s known s Euclen stnce norm n conton of optmzton of objectve functon s when P P. Therefore, for the specfc pose P wth corresponng jont vrbles c * consere soluton of nverse knemtc problem. wth the mjor m to mnmze error E cn be Above scusse error E efnes two fferent serch res for the optmzton process; () pose coorntes of the en effector n () fesble jont vrbles n the confgurton spce. Therefore the frst pont cn be known s representton spce or explorton n secon pont efnes the soluton spce for the metheurstc lgorthms. Now these spces cn be mppe together to form functon n such wy tht fesble jont vrbles cn be trnsform onto pose coorntes. Fnlly the trnsformton between the forwr knemtcs n nverse knemtcs coul be the functon for the evluton of nverse knemtc problem smlr to conventonl metho. Forwr knemtc of the mnpultor wll be useful for the generton of objectve functon n terms of current poston wth unknown jont vrbles. Further, prevously efne error E wll mesure the fference between the current en effector pose wth respect to the gven gol. But to rech the esre poston, wll not only helpful for obtnng the jont vrbles. Hence, the en effector coornte wth reltng to poston wll be known s poston error P E n on the secon se esre orentton of the en effector coul be helpful to rech to the exct pont s known s orentton error O E. Therefore, totl error E wll be the functon of poston error s well s orentton error s gven by equton (.), E E P E O (.) where s constnt weghtng fctor n cn be efne s rnom number n between P 0 to, the nvul errors re E n E. The weghtng fctor cn lso be set to zero tht wll yel the prtculr tsk requrements. Fnlly objectve functon formulton for the nverse knemtc soluton cn be gven by equton (.) s, O mn E q sub. to FK( ) P c (.) 8

204 q { g( ) 0 h( ) 0} Where g n h represents the lmt of the jont vrbles of robot mnpultor. The constrnt g n h gves the mxmum n mnmum vlue of the jont vrbles so s to obtn specfc workspce. Hence these constrnts cn be formulte s equton (.7), g( ) h( ) lower hgher where lower n hgher represents the lower n hgher lmts of the jont vrbles. n [ [ [ mn mn n mn,,, mx ] mx n mx ] ] (.7) The bove generl formultons of objectve functon for the soluton of nverse knemtc problem proves rect soluton wth respect to jont vrbles. In ths concept mnpultor sngulrtes re lso voe s compre to conventonl methos n the objectve functon cn esly be mofe s per the tsk requrements... Poston bse error To optmze the jont ngle of rotton of robot mnpultor, one escrbes ftness functon tht s compose of the fference between current poston of n effector to the esre poston tht s known s postonl functon n secon pproch s to clculte jont ngle error or orentton error. Mnpultor ccurcy cn be mesure by ts blty to rech to the esre poston wthn the workspce. Therefore, the stnce between the current poston n esre poston shoul be zero so s to cheve hgher ccurcy. The fference between the esre postons to trget poston E s kwon s poston bse error ( P ) s shown n Fgure.8. To resolve the problem of nverse knemtc the poston bse error wll be efne by the stnce norm (Euclen stnce) s gven n equton (.8), P E P where represents the Euclen stnce norm functon E P P c n. (.8) The norm functon efnes the specfc sclr metrc of vector spce elements, n cn be efne n mny wys. ommonly use norm on n s bove efne Euclen stnce norm or l -norm n ths norm cn be gven by euqton (.9), 8

205 P E n P E (.9) The secon mportnt norm s Mnhttn norm or l-norm, n cn be efne s gven n equton (.0), P E n P E (.0) Fgure.8 Poston bse error ( P ) The l-norm s wely use for the robot pose estmton uner vrous levels of nose contgon n the sensory output n proves to be more robust n ccurte s compre to other norm. mlrly, l -norm s lso use for the estmton of robot pose but t proves better convergence spee n mostly opte for the nverse knemtc optmzton. Therefore, n ths work l -norm s opte throughout the ssertton... Orentton bse error Orentton of the en effector of robot mnpultor cn be represente by the most common Euler ngles metho. Orentton between two fferent orthogonl rtesn coornte system xyz n uvw s generlly escrbe by the rotton mtrx R, whch s prmeterze by Euler ngles α, β n γ. Nonetheless, orentton ngles cn be obtne from the Euler ngles representton but they unergo the problem of sngulrtes. Therefore, to vo the problem of sngulrty t hs to be replce wth the Quternon vector metho. Quternon vector metho hs lrey been scusse n chpter whch gves stble n compct representton of the knemtcs representton. Hence usng the formultons of quternon vector metho, orentton error coul be formulze. E 8

206 Fgure.9 Orentton ngle between two frmes Now the orentton error cn be resolve usng the current frme fference wth the esre frme. Frme XYZ n UVW represents two fferent frmes tht re fxe t the sme orgn s shown n Fgure.9. Therefore, the current frme XYZ shoul rotte to lgn wth frme UVW so s to formulte for the orentton error R E. Now the orentton error cn be formulze on the bss of current frme rotton to esre frme. Orentton error cn be gven by equton (.) s, R E R R c R R T c (.) Where frme. R rotton mtrx of esre frme n R c s the rotton mtrx of current E Rotton mtrx R efnes the requre rotton to obtn the en effect poston to the esre coornte wthn the workspce. It s requre to fn out the sclr functon whch cn express the orentton of en effector error. As per Euler ngles representton theorem, ny orentton O(n) wll be equvlent to rotton of fxe xs n k through n ngle [ 0, ]. Therefore, to fn out the equvlent xs representton quternon vector metho cn be use. Now the quternon expresson cn be gven by equton (.) s, e e e e e H R rotton (, k ) cos, sn k where, s cos quternon metho. e s sclr component n v e e represents the ngle of error n represents the bsolute error ngle between the mtrx (.) e e sn k s vector prt of the R n e k s rotton xs. e R c. Therefore, Euler rotton E R cn convert nto the quternon representton s gven n equton (.), s n e x o e y e z (.) 8

207 e e e E where n x, o y n z re the elements of rotton mtrx R. Therefore from equton (.) n (.) orentton error cn be gven s equton (.), O e e e E cos n x o y z (.). oluton scheme of nverse knemtcs problem In ths chpter, numercl optmzton contexts bse on metheurstc lgorthms re propose to resolve nverse knemtc problem. Dfferent confgurtons of the robot mnpultor wth severl metheurstc lgorthms bse soluton s propose n the next chpter. The mjor vntge of the mplementton of optmzton lgorthms re compct n ccurte soluton of the nverse knemtc problem. ompre to the conventonl methos lke mpe lest squre or Jcobn bse methos t oes not suffer sngulrty. Metheurstc lgorthms re generlly populton bse methos whch re cpble of solvng vrous multmol functons. The ftness or objectve functon cn be gven by the totl error of the mnpultor pose. Ech nvul represents the jont vrble wth n the populton. The optmum set of jont vrbles cn be obtne by usng optmzton pproches. In cse of nverse knemtcs problem, multple solutons exst for the sngle poston of the en effector so t s requre to fn out the best set of jont ngle n orer to mnmze whole movement of mnpultor. As we know tht objectve functon (pose error) s efne on the tsk spce whle the jont vrbles re the subset of confgurton spce of the mnpultor. Therefore, objectve functon wll be evlute on the bss of forwr knemtc mppng n totl error obtne by the opte metho.. ummry Inverse knemtcs of ny robot mnpultor cn generlly be efne s fnng out the jont ngles for specfe rtesn poston s well s orentton of n en effector n opposte of ths, etermnng poston n orentton of n en effector for gven jont vrbles s known s forwr knemtcs. Forwr knemtc hvng unque soluton but n cse of nverse knemtcs t oes not prove ny close form or unque soluton thus t s requre to hve some sutble technque to resolve the problem for ny confgurton of robot mnpultor. Therefore optmzton bse lgorthms re qute frutful to solve nverse knemtc problem. Generlly these pproches re more stble n often converge to globl optmum pont ue to mnmzton problem. Moreover, selecton of pproprte optmzton technque les to globl optmum results. 8

208 Ths chpter proves the bscs of metheurstc optmzton lgorthms n fferent types of metheurstc optmzton lgorthms re escrbe. Novel effectul nturenspre metheurstc optmzton technque groune on crb behvour s propose n ths chpter. The propose rb Intellgence Bse Optmzton (IBO) technque s populton cntere tertve metheurstc lgorthm for D-mensonl n NPhr problems. A generl formulton of objectve functon for the nverse knemtc soluton s erve n lter s use for vrous confgurtons of the robot mnpultor. In the erve objectve functon, optmzton lgorthms re use to mnmze the en effector pose error for the gven tsk. Beses usng Jcobn mtrx for the mppng of tsk spce to the jon vrble spce, forwr knemtc equtons re use. Knemtc sngulrty s voe usng these formultons s compre to other conventonl Jcobn mtrx bse methos. Dfferent types of confgurton of the robot mnpultors hve been tken for the knemtc nlyss. Therefore, nverse knemtc soluton of vrous confgurtons of the mnpultors wth Euler wrst re solve computtonlly usng severl popultons bse metheurstc lgorthms. The opte metho s compct n effcent tool for knemtc nlyss. In the result chpter nverse knemtc soluton for opte mnpultor hs been tbulrse n comprson on the bss of mthemtcl complexty s me over other conventonl bse metho. 87

209 hpter 7 REULT AND DIUION 7. Overvew The prevous chpters hve been essentlly evote to unerstnng the bckgroun of the reserch topc, vrous confgurtons of nustrl robot mnpultors for knemtc nlyss, moellng of the nverse knemtc problem n soluton technques for selecte robot mnpultors. Although, the topc of nverse knemtc of robot mnpultors s n ntensvely reserch topc, vrous new technques re ttempte by vrous reserchers n orer to mke the soluton metho eser n/or fster epenng upon the requrements. The metho coul be sngle tool bse or hybr one. The requrements coul be to smplfy the soluton metho, obtn precse result, or to mke t sutble for the rel tme pplctons. Moellng of the nverse knemtc problem usng erve mthemtcl tools hs been one n chpter. These mthemtcl moellng of knemtcs s use to generte nput t set for the trnng of ANN moels n hybr ANN s. The trnng schemes of the opte ANN moels hve been scusse n chpter. In chpter, fferent optmzton lgorthms n ther pplcton to solve nverse knemtc problem s scusse n etls. The formulton of objectve functon for the soluton of nverse knemtc problem bse on postonl s well s orentton bse error hs been scusse n etl. Therefore, current chpter s summrze uner the followng objectves.. Inverse knemtc soluton of the selecte mnpultors usng conventonl mthemtcl tools such s homogeneous trnsformton mtrx n quternon lgebr s presente.. Inverse knemtc soluton usng ANN moels n obtne results re presente. 88

210 . Results of some selecte mnpultors usng ANFI moels re presente n comprsons hve been me wth hybr ANN moels.. Results of nverse knemtc soluton through opte optmzton lgorthms n ther comprtve nlyss s presente. To hve n ncorporte perspectve of the reserch work one n o legtmte exmnton, ll the outcomes hve been gthere n re ntrouce n ths chpter. 7. Inverse knemtcs soluton usng ANN Typclly, nustrl mnpultors re esgne to perform vrous tsks n sptl s well s n plnr spce. The en effector/tool of mnpultor s progrmme to follow specfc trjectory to execute the esre tsk wthn the workspce. The control over ech jonts n lnks of the mnpultor s requre to rech to the esre poston long wth the control of en effector for explct orentton n poston wthn prescrbe lmt. Therefore, knemtc reltonshp of the jonts n lnks plys crucl role to obtne esre poston n orentton of the en effector. In cse of forwr knemtc nlyss, jont vrbles re known whch helps en-effector to rech t esre locton whle nverse knemtc requres orentton s well s poston of the object wthn the workspce. In chpter, DH-lgorthms n homogeneous mtrx bse methos re use for the knemtc ervton for vrous confgurtons of mnpultor. Inverse knemtc soluton of -of ARA, -of PUMA n -of mnpultors re escrbe wthout use of Euler wrst usng quternon lgebr. In ths secton ANN moels lke MLP, PPN, n P-gm NN re use to lern jont ngles of robot mnpultor n the t sets re generte through conventonl methos lke DH-lgorthm, homogeneous trnsformton mtrx, quternon lgebr metho. Forwr knemtc equtons from chpter re use to trn the neurl network moels wheres n ths chpter both forwr n nverse knemtcs equtons re use to trne the neurl network moels. ANN montors the nput-output reltonshp between rtesn coornte n jont vrbles bse on the mppng of t. Inverse knemtcs s trnsformton of worl coornte frme (X, Y, n Z) to lnk coornte frme (, ). Ths trnsformton cn be performe on,... n nput/output work tht uses n unknown trnsfer functon. In ths secton results re prouce for the mnpultors strtng from -of plnr to 7- of nthropomorphc mnpultor usng conventonl forwr knemtc equtons s well s ANN bse moels. In ths chpter nverse knemtc soluton for opte mnpultor hs been tbulrze n comprson on the bss of mthemtcl complexty s me over other conventonl bse metho. MATLAB softwre s use 89

211 for the clculton of nverse knemtc soluton of selecte mnpultors s presente n chpter. Further results of ANN presente n the followng sectons. 7.. Result for -of plnr revolute mnpultor usng ANN The propose robot mnpultor moel s consere s -of plnr mnpultor (see Fgure.). The length of the ech lnks re 0, 7 n. Let s the ngle between bse n frst lnk smlrly n mkes ngle wth secon n thr rm. onserng ll jont ngles lmts 0 to 80 egrees. Now usng forwr knemtc equton from chpter, trnng t for MLP, PPN n P-gm network hs been obtne through MATLAB progrm. The generte smple t sets for of trnng opte neurl network moels re gven n Tble 7.. Tble 7. Poston of en effector n jont vrbles N. X Y Three fferent moels hve been tken for the vlton of the results n the moels re: MLP (Mult-lyer perceptron), PPN (Polynoml perceptron network) n P-sgm network whch re consere for the nlyss of nverse knemtcs problem, smulton stues re crre out by usng MATLAB. A set of 000 t sets were frst generte s per the formul for ths the nput prmeter X n Y coorntes. These t sets were bss for the trnng n evluton or testng the ANN moels. Out of the sets of 000 t ponts, 900 were use s trnng t n 00 were use for testng for ANN moels. Bck-propgton lgorthm ws use for trnng the network n for uptng the esre weghts. In ths work epoch bse trnng metho ws pple. The comprsons of esre n precte vlue of jont ngles of MLP moel for 00 epochs hve been represente n Fgure 7. through 7.. Where, Fgure 7. represent the --- confgurton of network n whch (), (b) n (c) epcts the precte ngles respectvely. Another two fferent confgurtons hve been tken for the testng of network shown n Fgure 7. n Fgure

212 Fgure 7. omprson of esre n precte vlue of jont ngles for --- confgurton usng MLP moel Fgure 7. omprson of esre n precte vlue of jont ngles for --- confgurton usng MLP moel Fgure 7. omprson of esre n precte vlue of jont ngles for --- confgurton usng MLP moel 9

213 To test the stblty of the MLP two other moels.e. PPN n P-sgm network moels hve been stue. Fgure 7. (), (b) n (c) shows the men squre error of jont ngles usng PPN network whch gves reltve poor result s compre to MLP moel n n cse of P-sgm network the obtne result s lso poor to compre wth MLP s shown n Fgure 7. (), (b) n (c). Fgure 7. Men squre error for jont ngles usng PPN moel Fgure 7. Men squre error for jont ngles usng P-sgm network moel 7.. Results of -of ARA mnpultor To vlte the opte MLP neurl network, the propose work s performe on the MATLAB Neurl Networks Toolbox. The trnng t sets were generte by usng forwr knemtc equtons from chpter. A set of 000 t ws frst generte s per the formul for the nput prmeter X, Y n Z coorntes. These t sets were the begnnng for the trnng, evluton n testng the opte MLP neurl network 9

214 moel. Out of the sets of 000 t, 900 were use s trnng t n 00 were use for testng for MLP s shown n Tble 7.. The followng prmeters were tken, Tble 7. onfgurton of MLPNN l. Prmeters Vlues tken Lernng rte 0.99 Momentum prmeter 0.0 Number of epochs 0000 Number of hen lyers Number of nputs Number of output 7 Trget tsets Testng tsets Trnng tsets 00 Tble 7. omprson between nlytcl soluton n MLPNN soluton.n. Jonts vrbles through nlytcl metho Jonts vrbles through MLPNN metho θ θ θ θ θ θ mlr to prevous work, bck-propgton lgorthm ws use for trnng the network n for uptng the esre weghts. The formulton of the MLPNN moel s generlze one n t cn be use for the soluton of forwr n nverse knemtcs 9

215 problem of mnpultor of ny confgurton. However, specfc confgurton hs been consere n the present work only to llustrte the pplcblty of the metho n the qulty of the soluton vs-à-vs other lterntves methos. Tble 7. gves the t for poston of jonts etermne through nlytcl soluton n tht obtne from MLPNN moel. l. Tble 7. Regresson nlyss Ths s consstent wth our ttle tht t s goo pproch to trn the ANN wth goo representtve set of fxe trgets postons nste of vrble trget postons for the lernng process tht wll ntrouce nose n the cost functon n my result n poor convergence. The men squre curves, shown n Fgure 7. through Fgure 7.9 exhbt the proper escrpton of the men squre error of trne network. As shown n result, the use soluton metho the chnce of selectng the output, whch hs the lest error n the system. Hence, the soluton cn be obtne wth less error s shown n Fgures 7. through 7.9 for the best vlton performnce of the obtne t wth the esre t. Regresson coeffcent (r) Men squre error Epoch number Resoluton through ept one robot wth smrt controller user s gue Resoluton through MLPNN mm mm

216 Fgure 7. Men squre error for Fgure 7.7 Men squre error for 9

217 Fgure 7.8 Men squre error for Fgure 7.9 Men squre error for 9

218 () (b) (c) () Fgure 7.0 Grphcl vew of regresson Generlzton tests were crre out wth rnom trget postons showng tht the lerne MLP generlze well over the whole spce. From Tble 7. t cn unerstn tht the men squre error for ll jont vrbles s qute closer to zero. The regresson coeffcent nlyss s per Tble 7. tht shows 99.9% mtchng for ll jont vrbles whch s cceptble for obtnng nverse knemtcs of the ARA mnpultor. Resolutons of the AeptOne ARA robot gven n Tble 7. (obtne from AeptOne robot wth smrt controller user s gue) re compre wth the resoluton obtne from the MLPNN moel. Fgure 7.0 represents the grphcl vew of regresson nlyss. 7.. ANFI results of -of ARA mnpultor The propose work s performe n MATLAB toolbox. The coorntes (X, Y n Z) n the ngles (,, n ) re use s trnng t to trn ANFI network wth Gussn membershp functon wth hybr lernng lgorthm. The trnng t for ANFI moel were tken from the Tble 7. smlr prevous work. Tble 7. shows confgurton of ANFI. Fgure 7. through Fgure 7. shows the vlton curve for the problem of lernng the nverse knemtcs of the -DOF ARA mnpultor. Tble 7. gves the verge errors of jont vrbles usng ANFI n MLPNN. These 97

219 errors re smll n the ANFI lgorthm s, therefore, cceptble for obtnng the nverse knemtcs soluton of the robotc mnpultor. l. Tble 7. onfgurton of ANFI Number of noes 7 Number of lner prmeters Number of nonlner prmeters Totl number of prmeters 0 Number of trnng t prs 700 Number of fuzzy rules Tble 7. omprson of results ME of MLPNN ME of ANFI Fgure 7. Men squre error for Fgure 7. Men squre error for 98

220 Fgure 7. Men squre error for Fgure 7. Men squre error for 7.. Results of -of revolute mnpultor mlr to ARA mnpultor propose work s performe on the MATLAB Neurl Networks Toolbox. Premnmx functon s use for preprocessng of nput n outpur t. Then, the functon newff s use to crete fee forwr network for nverse knemtcs. Further, the sme network s trne ccorng to tnsg n logsg trnsfer functon. The trnng functons employe re trnoss n trnlm, to vlte the performnce of MLPNN neurl network for nverse knemtcs problem. Then, the weghts n bses re clculte for the network. To smulte the t corresponng to the tsk consere here, the new nput t to the trne network re preprocesse wth the trmnmx functon. Then, the outputs smulte by the trne network re post processe bck usng the postmnmx functon. The generte smple t sets for trnng opte neurl network moels re gven n Tble

221 Tble 7.7 Desre jont vrbles etermne through nlytcl soluton Postons n jont vrbles etermne through quternon lgebr N θ θ θ θ θ X Y Z These t sets re use for trnng, evluton n vltng the neurl network moel for nverse knemtc soluton. Out of the sets of 000 t ponts, 900 were use s trnng t n 00 were use for testng for MLPNN s shown n Tble 7.8. The followng prmeters were tken: Tble 7.8 onfgurton of MLPNN l. Prmeters Vlues tken Lernng rte 0.9 Momentum prmeter 0.8 Number of epochs 0000 Number of hen lyers Number of nputs Number of output 7 Trget tsets Testng tsets Trnng tsets 00 Bck-propgton lgorthm ws use for trnng the network n for uptng the esre weghts. The men squre curve shown n Fgure 7. through Fgure 7.9 n result, the use soluton metho gves the chnce of selectng the output, whch hs the lest error n the system. o, the soluton cn be obtne wth less error. Fgure 7. through Fgure 7.9 shows the vlton curve for the problem of lernng the nverse knemtcs of the -DOF mnpultor. These errors re smll n the MLPNN lgorthm s, therefore, cceptble for obtnng the nverse knemtcs soluton of the robotc mnpultor. Fgure 7.0 shows the grphcl vew of regresson wth respect to number of epochs n t s lmost gves 99.99%. In the next secton, the comprson of MLPNN moel wth ANFI hs been presente. 00

222 Fgure 7. Men squre error for Fgure 7. Men squre error for Fgure 7.7 Men squre error for 0

223 Fgure 7.8 Men squre error for Fgure 7.9 Men squre error for () (b) (c) () (e) Fgure 7.0 Regresson coeffcent plot for jont vrbles 0

224 7.. ANFI results of -of revolute mnpultor In ths secton MLPNN result of -of mnpultor hs been compre wth the ANFI results of nverse knemtc problem. Generte t sets for MLPNN trnng s use to trnng ANFI network. mlr to prevous work, the coorntes (X, Y n Z) n the ngles (,,,, n ) re use s trnng t to trn ANFI network wth Gussn membershp functon wth hybr lernng lgorthm. A set of 000 t sets were frst generte s per the formul for the nput prmeter X, Y n Z coorntes. Out of the sets of 000 t ponts, 700 were use s trnng t n 00 were use for testng the performnce of ANFI.In the trnng phse, the membershp functons n the weghts wll be juste such tht the requre mnmum error s stsfe or f the number of epochs reche. At the en of trnng, the trne ANFI network woul hve lerne the nput/output mp n t s teste wth the euce nverse knemtcs. Fgure 7. through Fgure 7. shows the fference n jont vrbles nlytclly n the t precte wth ANFI. l. Averge testng Error of MLPNN Tble 7.9 onfgurton of ANFI Number of noes 7 Number of lner prmeters Number of nonlner prmeters Totl number of prmeters 0 Number of trnng t prs 700 Number of checkng t prs 0 Number of fuzzy rules Epoch Number MLPNN Tble 7.0 omprson of results Averge testng Error of ANFI Epoch Number ANFI Tble 7.9 shows confgurton of ANFI. Fgure 7. through Fgure 7. shows the vlton curve for the problem of lernng the nverse knemtcs of the -DOF mnpultor. Tble 7.0 shows comprson between the MLPNN wth respect to ANFI. Generlzton tests were crre out wth new rnom trget postons showng tht the lerne MLPNN gves evton of of the error gol urng the lernng process n ANFI gves verge errors whch s better thn the 0

225 men squre error of MLPNN. The obtne errors re smll n the ANFI lgorthm s, therefore, cceptble for obtnng the nverse knemtcs soluton of the robotc mnpultor. Fgure 7. Men squre error for Fgure 7. Men squre error for 0

226 Fgure 7. Men squre error for Fgure 7. Men squre error for 0

227 Fgure 7. Men squre error for 7. Hybr ANN pproch for nverse knemtcs soluton In ths secton metheurstc lgorthm lke IBO, PO, GA, GWO, TLBO etc. re pple usng weght n bs bse optmzton crter n MLP neurl network s use throughout ths secton. From the prevous secton neurl network moels re pproprte for solvng nverse knemtc problem but the opte moels proucng npproprte men squre error for the fferent confgurtons of the robot mnpultor. Therefore, hybrzton of metheurstc or populton bse lgorthms metho cn be strt wth the ntroucton of the opte lgorthms wth the specfc moel of neurl network. The etle scussons of nverse knemtc problem hve been presente n chpter. Hybr ANN moel re use for solvng the nverse knemtc problem n the present work. hpter gves the etl scusson of hybrzton scheme of MLP network wth metheurstc lgorthms. The results obtne by solvng the nverse knemtc problem for fferent confgurtons of -of mnpultor usng hybr ANN n MATLAB pltform s presente n followng sectons. 7.. Inverse knemtc soluton of -of ARA mnpultor In ths secton, -of ARA mnpultor s selecte for the nverse knemtc nlyses. ARA robot mnpultors re mostly use n lght uty pplctons ue to the hgh spee n precson of the mnpultor. The etl knemtc moellng of ths 0

228 mnpultor usng conventonl metho s presente n chpter. Usng forwr knemtc equtons from secton.. s use to crete t sets for the trnng of MLPNN moel. The results usng bck propgton lgorthms wth hen noes for MLPNN s obtne n secton 7... The present secton gves the comprtve nlyss of nverse knemtc soluton wth the hybr MLPNN technque. From secton 7.. the trnng of MLPNN moel MATLAB neurl network toolbox s use n lter the obtne results re compre wth the hybr MLPNN metho. A set of 000 t sets re frst generte s per the formul for the nputs X, Y n Z coornte. The prmeters for trnng MLP network s presente n secton 7.. (see Tble 7.). Tble 7. Men squre error for ll trnng smples of hybr MLPNN Output Algorthms MLPBP MLPPO MLPGA Number of Hen Noes e-.e-.88e- 9.7e-.e-.9e-.e- 0.e-.e- 8.9e-.0e-.08e-.e-8 0.e-.8e- 7.90e-7.e-.70e-.e-9.e-.0e-.e- 9.09e-.8e-.e-.e e-9.e-8.8e-.09e- MLPTLBO.7e-07.00e-08.9e-.8e e-09.9e e-.e-0.e-.e- MLPIBO.e-0.9e-07.8e-09 9.e-.e-0 0.e e-.0e- 7.7e e- MLPBP MLPPO.e-0 0.e-0.9e-0.e-0.8e-0 0.7e-0.e-0.e-0.9e-0 8.e e-0.e-09.7e e- 0.e-07.0e-.e- 7.e- 8.e-.9e- MLPGA MLPTLBO e-9.e-.e-.8e-8.00e e-.0e- 0.89e-9 MLPIBO.e-.e- 7.7e- 8.8e- 0.9e-.8e-.e-.7e-7.87e- 9.9e- MLPBP MLPPO.87e-0.e-0.e-0 7.e-0.7e-0.e-0.77e-0.7e-0 0.e-0 8.0e-0 7.7e-0.e-07.e-0.7e- 0.e e- 7.0e-.7e- 8.e-.77e- MLPGA e 9 9.e 7.9e e 9.9e 9.7e 7.7e 9 MLPTLBO.8e 8.7e.9e 8.7e.e- 9.99e- 7.80e 9 7.8e 9.e 7.e MLPIBO.e-0.80e-09 7.e- 8.88e- 0.e-09.88e-.e-.e-.9e-.e- MLPBP MLPPO 8.9e-0.e-0.e-0.99e-0 7.e-0.7e-0 0.7e-0.07e-0.7e-0 8.e-0.7e-0.78e-07.e-08.7e e-.8e-.9e-.e-.e-9 9.7e- MLPGA.7e-.09e-9.77e-.00e MLPTLBO MLPIBO 8.7e-.7e-.e-.0e- 8.7e- 7.8e-9.9e-.e- 8.e- 0.0e- In ths secton, fferent numbers of hen noes re use n numbers of hybr lgorthms such s MLPPO, MLPGA, n MLPTLBO MLPIBO etc. hve been compre wth the MLPBP lgorthm. Hybr scheme hs been presente n secton., usng the scheme of hybrzton trnng of MLNN moel s presente. The weght n bs of MLPNN moel s optmze wth the selecte optmzton lgorthms. The prmeters of the optmzton lgorthms hve been chosen rnomly. After the optmze trnng of the MLPNN moel, the trne network s use to clculte the nverse knemtc of the ARA mnpultor. After clculton of the nverse knemtc soluton the men squre error s obtne wth the comprson of ctul soluton n esre solutons. Best men squre errors for ll jont vrbles re 07

229 presente n Fgure 7.. Men squre error for ll jont vrbles n comprson of fferent lgorthms re presente n Tble 7.. () Best men squre for (b) Best men squre for (c) Best men squre for 08

230 () Best men squre for Fgure 7. (), (b), (c), () n (e) re men squre error curve for ll jont ngles usng MLPGA. 7.. Inverse knemtc soluton of -of mnpultor In ths secton, -of revolute mnpultor s consere for the knemtc nverson. Ths mnpultor s extensvely use n nustres s well s n reserch work. The nverse knemtc soluton for the opte mnpultor s performe on the MATLAB R0. From the prevous reserch work the opte MLPNN moels perform poor. Therefore, performnce of hybr ANN moel n evlutons hve been me. For the trnng of MLP network, MATLAB Neurl Networks Toolbox s use (see secton 7..). Premnmx functon s use for preprocessng of nput n output t. Then, the functon newff s use to crete fee forwr network for nverse knemtcs. Further, the sme network s trne ccorng to tnsg n logsg trnsfer functon. The trnng functons employe re trnoss n trnlm, to vlte the performnce of MLP neurl network for nverse knemtcs problem. Then, the weghts n bses re clculte for the network. To smulte the t corresponng to the tsk consere here, the new nput t to the trne network re preprocesse wth the trmnmx functon. Then, the outputs smulte by the trne network re post processe bck usng the postmnmx functon. The trnng t sets were generte by usng forwr knemtc equton from chpter. A set of 000 t sets re frst generte s per the formul for the nputs X, Y n Z coornte. The generte t sets re use to trn the MLP network. The prmeters for trnng MLP network s gven n Tble 7.8. The bltes of severl fferent hybr lgorthms such s MLPPO, MLPGA, MLPTLBO n MLPIBO hve been compre. In the trnng phse, the weghts n bses wll be juste such tht the requre mnmum error s stsfe or f the 09

231 number of terton reche. At the en of trnng, the trne MLP network woul hve lerne the nput/output mp, n t s teste wth the euce nverse knemtcs. The results obtne through the MLPBP hve been escrbe n secton 7... There s no specfc tunng of the ssocte prmeters re consere. The prmeters for the PO lgorthm s scusse further n for the rest of the lgorthm prmeters re selecte rnomly wthout ny tunng. The ntl pproxmtons of every prtcle hve been chosen rnomly n the rnge of [0, ]. For MLPPO, mxmum n mnmum nert weghts re ecresng lnerly from 0.9 to 0.. n re set to ; r n r re two rnom numbers n the ntervl of [0, ] n the ntl veloctes of prtcles re rnomly selecte n the ntervl of [0, ]. Fnlly the populton szes for ech lgorthm re 00. From secton 7.., Tble 7.9 gves some of the esre t for the poston of jonts etermne through nlytcl soluton, whch wll further be use to clculte the ME of MLPBP, MLPPO, MLPGA, MLPTLBO n MLPIBO. Output Algorthms Tble 7. Men squre error for ll trnng smples of hybr MLPNN MLPBP Number of Hen Noes e-.e- 9.0e- 7.7e-.e-.9e-.e-.e- 7.e- 9.9e- MLPPO.0e-9.80e-9.e-9.e-8.8e e-.e-.07e-.07e-.e- MLPGA MLPTLBO MLPIBO.e-.9e e-9.e e-0 MLPBP.e-.e-.9e-.e- 7.8e- 7.7e-.e-.9e- 7.89e- 9.e- MLPPO.90e-.e-.7e-9.7e-0.e-8.0e-.e- 8.e- 9.09e-7.9e- MLPGA MLPTLBO 9.e-9.e-0.e-8.e-9.e-8.8e-7.9e-9 8.e-7 8.e-9.09e- MLPIBO.e-.e-7.e-9 7.7e e-7.7e-.e-.e-9.e-9.e- MLPBP.e-.e-.9e- 8.e-.e-.e-.e-.8e-.e- 9.0e- MLPPO 8.e-7.8e-.e-.7e-.e-9.8e-0.0e-.e-.e-.87e- MLPGA.e 9.7e 7.e.9e.e.9e.e 7.7e 9.7e.7e 9 MLPTLBO.8e 9.8e 7.e 9.7e e 0.8e 0.e 8.88e 7 MLPIBO.e-7.e- 8.e- 9.99e-9.e- 0.9e-9.0e- 8.88e-.9e-.e- MLPBP 9.9e-0.e-0.e-0.e-0 8.7e-0.78e-0.e-0.9e-0.8e-0.e-0 MLPPO.8e-07.e-08.8e-09.e-0.09e-08.7e-09.e-.7e-.e-0.7e- MLPGA.e e e-.7e e-9 9.8e-.e- MLPTLBO e-79.e-78.e MLPIBO e- 7.e-9 9.8e- 8.7e-.8e-.9e e MLPBP.8e-.e-.e-.e-.e- 7.e-.e-.e-.e- 8.87e- MLPPO.e-0.e-.8e-9.8e-.7e-9.8e-9.e-7.e- 9.0e-.e- MLPGA MLPTLBO MLPIBO.89e-.e-7.8e e-79.e

232 The ME for MLPBP lgorthm shown n Fgure 7. through Fgure 7.9 n result secton 7.., the use soluton metho proves the crter for selecton of the output f t prouces less error. o, the soluton cn be obtne wth less error. Tble 7. gves the expermentl results n comprson of ll opte lgorthms for fferent hen noes. Best results for jont vrbles specfe n bol letters n presente n Fgure 7.7 through Fgure 7.9. Fgure 7.7 (), (b), (c), () n (e) shows the selecte best men squre curve of MLPGA for ll jont vrbles. mlrly best chosen men squre curve of MLPTLBO from Tble 7. epcte n Fgure7.8 (), (b), (c), () n (e) for ll jont vrbles. From Tble 7., MLPBP oes not gve better results thn the other opte lgorthms, ue to trppng n locl mnm. Also t hs been observe tht MLPBP hs been slow serchng blty, n t consumes more PU tme. MLPGA gves the best results for ll jont vrbles of the robot mnpultor. MLPTLBO s more stble lgorthm s compre to MLPGA. Although results obtne through MLPTLBO s less s compre to MLPGA but ue to better stblty, MLPTLBO oes not consume much PU tme s compre to MLPGA. It hs been lso observe tht the MLPIBO hvng slow serchng process over MLPTLBO n MLPGA. However, IBO hs the strong explorton blty mong ll heurstc lgorthms. For trnng MLP network t hs been observe tht both opte heurstc lgorthm yel goo results of ll jont vrbles. In other wors, the proper utlzton of n evolutonry lgorthm wth MLP networks gves outstnng performnce for trnng the network. o these opte hybr technques cn be use for NP-hr problem, whch generlly suffers from trppng n locl mnm. Also these technques gurntee fster convergence rte. () Best men squre for

233 (b) Best men squre for (c) Best men squre for () Best men squre for

234 (e) Best men squre for Fgure 7.7 (), (b), (c), () n (e) re men squre error curve for ll jont ngles usng MLPGA. () Best men squre error for (b) Best men squre error for

235 (c) Best men squre error for () Best men squre error for (e) Best men squre error for Fgure 7.8 (), (b), (c), () n (e) re men squre error curve for ll jont ngles usng MLPTLBO.

236 () Best men squre error for (b) Best men squre error for (c) Best men squre error for

237 () Best men squre error for (e) Best men squre error for Fgure 7.9 (), (b), (c), () n (e) re men squre error curve for ll jont ngles usng MLPIBO. 7.. Inverse knemtc soluton of -of PUMA mnpultor In ths secton, PUMA 0 robot mnpultor s selecte for the nverse knemtc nverson. Ths mnpultor s one of the benchmrk nustrl mnpultor whch s wely use n nustres n reserch work. The etl escrptons bout ths mnpultor hve been presente n chpter. The forwr n nverse knemtc ervton usng quternon lgebr s lrey scusse n chpter. Usng the forwr knemtc equtons the t sets for the trnng of MLNN neurl network s generte. MATLAB progrm s use to generte the t sets of the en effector coorntes n jont vrbles. These t sets were the bss for the trnng, evluton n testng the MLP moel.

238 mlr to prevous work hybr ANN metho s use to resolve the problem of nverse knemtcs of -of PUMA mnpultor. The comprson hs been me wth the severl hybr moels lke MLPPO, MLPGA, MLPTLBO n MLPIBO. Intl pproxmtons for ll opte optmzton lgorthms re smlr to the cse of -of mnpultor. The esre jont vrbles re tken s nput for the trnng of hybr ANN moel. The confgurton n prmeters s gven n Tble 7.. Quternon vector metho s use to clculte the nverse knemtc soluton for the PUMA mnpultor n smple t sets re gven n Tble 7.. Tble 7. onfgurton of MLPNN l. Prmeters Lernng rte 0.8 Momentum prmeter 0. Number of hen lyers Number of nputs Number of output Trget tsets Testng tsets 00 8 Trnng tsets 700 Vlues tken Tble 7. Desre jont vrbles etermne through quternon lgebr Jonts vrbles n postons etermne through quternon lgebr N θ θ θ θ θ θ X Y Z mlr to the results of prevous reserch work MLPBP oes not prouce better results s compre to other hybr lgorthms. On the other hn, MLPGA gves best results mong ll other hybr lgorthms. But the convergence rte of MLPTLBO s more stble thn GA. It hs been observe tht MLPGA s gvng fst serchng blty n cse of -of PUMA mnpultor over other opte lgorthms. However, TLBO show strong explorton cpblty thn others. Hybrzton of metheurstc 7

239 lgorthms gves better performnce s compre to bck propgton lgorthm. In other wors, the proper utlzton of n evolutonry lgorthm wth MLP networks gves outstnng performnce for trnng the network. o these opte hybr technques cn be use for NP-hr problem, whch generlly suffers from trppng n locl mnm. Also these technques gurntee fster convergence rte. Tble 7. Men squre error for ll trnng smples of hybr MLPNN Output Algorthms Number of hen neurons MLPBP.e-.e-.8e-.e- 9.e- 7.7e- ANFI.e-7.9e e-.e-.98e-.e- MLPGA MLPPO.9e-0.8e- 8.8e-9.e e- 0.e-90 MLPTLBO MLPIBO 8.e-.89e- 7.8e- 8.8e-8.7e-.e-7 MLPBP.e- 8.e-.e-.89e-.9e- 8.88e- ANFI 0.88e-9 8.e-8.e-9.e- 8.08e-8.e- MLPGA MLPPO.e-08.e-9.88e-7.8e-07.e-9.e- MLPTLBO.e-0.e-8.e-8.e-8 9.7e-8.87e- MLPIBO.e-0.e-.9e- 8.78e-.97e- 8.00e- MLPBP.e- 9.e- 0.9e-.8e-.e- 7.e- ANFI.e-9.e- 8.78e e- 9.7e-.88e-9 MLPGA.e 0.00.e 89.87e 9.e 9.e MLPPO 9.e.e 09.7e 08 0.e e 09.e 08 MLPTLBO e 90 MLPIBO 8.e-.8e- 9.0e-.e- 8.89e-.e- MLPBP.99e-0 9.9e-0.9e-0.e-0 7.7e-0.8e-0 ANFI.8e-08.e-.8e-.e-0.e-.8e- MLPGA e-.8e MLPPO.e-8 7.e-7 7.e-.e- 7.7e-.e-9 MLPTLBO MLPIBO.9e-0.e-7.9e-8.e-8.e-0 8.9e-9 MLPBP.9e-.e-.7e-0.e-0.7e e-0 ANFI.e- 7.9e e-9.e- 0.09e-.e- MLPGA MLPPO 7.7e-9 7.e- 7.e-9.e e- 7.e- MLPTLBO MLPIBO e-79.8e-.e MLPBP.e-0 8.e-0.e-0.e-0.e-0.e-0 ANFI.78e-.e-78.e-9 8.9e MLPGA MLPPO.e-09.e-.e-9.e-7 8.9e-09.e- MLPTLBO MLPIBO.87e-09.8e-.e-09.0e e- 9.e- 8

240 Tble 7. gves the expermentl results n comprson of ll opte lgorthms for fferent hen noes. Best results for jont vrbles specfe n bol letters n presente n Fgure 7.0 through Fgure 7.. Fgure 7. (), (b), (c), () n (e) shows the selecte best men squre curve of MLPGA for ll jont vrbles. mlrly best chosen men squre curve of MLPTLBO from Tble 7. epcte n Fgure 7. (), (b), (c), () n (e) for ll jont vrbles. () Best men squre error for (b) Best men squre error for (c) Best men squre error for 9

241 () Best men squre error for (e) Best men squre error for (f) Best men squre error for Fgure 7.0 (), (b), (c), (), (e) n (f) re men squre error curve of MLPBP for ll jont ngles. 0

242 () Best men squre error for (b) Best men squre error for (c) Best men squre error for

243 () Best men squre error for (e) Best men squre error for (f) Best men squre error for Fgure 7. (), (b), (c), (), (e) n (f) re men squre error curve of MLPGA for ll jont ngles.

244 () Best men squre error for (b) Best men squre error for (c) Best men squre error for

245 () Best men squre error for (e) Best men squre error for (f) Best men squre error for Fgure 7. (), (b), (c), (), (e) n (f) re men squre error curve of MLPTLBO for ll jont ngles.

246 () Best men squre error for (b) Best men squre error for (c) Best men squre error for

247 () Best men squre error for (e) Best men squre error for (f) Best men squre error for Fgure 7. (), (b), (c), (), (e) n (f) re men squre error curve of MLPIBO for ll jont ngles.

248 7.. Inverse knemtc soluton of -of ABB IRB-00 mnpultor Ths secton pertns, ABB IRB-00 (Type A) robot mnpultor for the nverse knemtc soluton. The etl escrpton of the opte robot mnpultor moel n knemtc prmeters re presente n chpter. The quternon vector bse mthemtcl moellng of the opte robot mnpultor s gven n chpter. Usng knemtc equtons from secton.., severl jont vrbles n en effector coorntes re epcte n Tble 7.7. MATLAB cong s use to generte the jont vrbles n en effector coorntes. The generte t sets re use for trnng n testng of the opte ntellgence bse methos. Once the trnng s complete the ctul output s compre wth the esre output so s to get men squre error for ll jont vrbles. The men squre errors for ll jont vrbles re obtne usng ANN n hybr ANN moels. The confgurton of the MLPNN wth fferent number of hen neurons s presente n Tble 7.. Tble 7. onfgurton of MLPNN Followng the smlr proceure of PUMA 0 mnpultor, the nverse knemtc soluton s presente usng MLPNN n hybr MLPNN methos. From the prevous reserch work MLPNN prouces poor results for the precton of nverse knemtc solutons. Therefore, The ANN moel s hybrze wth the fuzzy logc n other optmzton bse lgorthms. The comprson hs been me wth the severl hybr moels lke MLPPO, MLPGA, MLPTLBO, ANFI n MLPIBO. Intl pproxmtons for ll opte optmzton lgorthms re smlr to the cse of -of n PUMA mnpultors. Quternon vector metho s use to clculte the nverse knemtc soluton for the PUMA mnpultor n smple t sets re gven n Tble 7.7. The esre jont vrbles re tken s nput for the trnng of hybr ANN moel. l. Prmeters Lernng rte 0.9 Momentum prmeter 0. Number of epochs 000 Number of nputs Number of output Trget tsets 00 7 Testng tsets 00 8 Trnng tsets 700 Vlues tken Tble 7.8 proves the expermentl results n comprson of ll opte lgorthms for fferent hen noes. Best performnce of the lgorthm s presente n bol letters wth men squre error plots. The obtne men squre error vlues re very 7

249 low s compre to MLPNN n ANFI moels. Therefore, fuson of MLPNN moel wth optmzton lgorthms proves strong convergence blty long wth better precton of nverse knemtc soluton. On the other hn, ANFI perform better thn MLPBP see Tble 7.8. Although, the results prouces through the ANFI s qute cceptble s compre to MLPBP but hybr ANN's re more effcent n ccurte. The convergence of MLPBP n ANFI re poor ue to ts locl serchng blty. Therefore, hybrs ANN re more effcent n yels globl serchng blty. Fgure 7. (), (b), (c), () n (e) shows the selecte best men squre curve of MLPGA for ll jont vrbles. Tble 7.7 Desre jont vrbles etermne through quternon lgebr N Jonts vrbles n postons etermne through quternon lgebr θ θ θ θ θ θ X Y Z

250 Tble 7.8 Men squre error for ll trnng smples of hybr MLPNN Output Algorthms Number of Hen Noes MLPBP 0.e-0.8e-0.e-0.8e-0.e-0.e-0 ANFI 9.0e-0.e-0 7.e e-0 8.7e-0.e-0 MLPGA MLPTLBO e-9.8e- 8.e-7 MLPIBO.e-.e-.9e-9.8e-7.99e- 7.e- MLPPO 9.e- 9.0e-7.e-7.0e-.87e-09.e-08 MLPBP 9.99e-0.8e-0 0.7e-0.7e-0.8e e-0 ANFI.0e-0.e-0 7.e-0.e-0.0e-0 7.e-0 MLPGA MLPTLBO.0e e-80 7.e-9.e-9.7e-90.9e-90 MLPIBO.e-98 7.e-89.9e-0 9.9e- 8.89e-0.8e- MLPPO.9e e-9.e-89.87e- 9.78e- 0.e-90 MLPBP.0e-0.e-0.8e-0.e-0 7.e-0 0.8e-0 ANFI.e-0.7e-0.0e-0 8.e-0.e-0.89e-0 MLPGA e-.e-.e MLPTLBO.e-0 7.e-0.89e-.e-.e-7.e- MLPIBO.9e-80.e-.8e- 9.0e-79.9e-.e- MLPPO.e-.9e- 9.99e-9.9e-.78e- 7.78e- MLPBP 0.07e-0.e e-0 0.e-0.8e-0.e-0 ANFI 8.e-0 0.0e-0.e-0 0.8e-0.e-0.9e-0 MLPGA.e-9.9e- 0.8e-.0e MLPTLBO.7e-.e- 8.e-.8e- 8.e-9.e-9 MLPIBO.e-08.09e-09.e-09.e-9.e-8.88e-7 MLPPO.87e-8 9.e-9 8.7e-8 8.e-.e-08 7.e-9 MLPBP 9.e-0.8e-0 0.0e-0.7e-0 8.e-0.9e-0 ANFI.08e-0.e-0.8e-0 0.7e-0.e-0.9e-0 MLPGA 0.09e-8 8.e-9.0e-.e- 7.9e-9.8e-8 MLPTLBO.8e-07 8.e-0.e-9.e- 7.7e-.7e- MLPIBO.9e-.e-.7e-.8e-7.00e e-08 MLPPO.8e-.e-.e-7.87e-.7e-.e-0 MLPBP 0.0e-0.7e-0 9.e-0.e-0.00e-0 0.e-0 ANFI.87e-0.8e-0.77e-0 8.e-0.7e-0 0.8e-0 MLPGA.9e-.8e- 0.89e-.8e-9.e-7.e- MLPTLBO 0.0e-9.e-.7e- 7.e e-.e-8 MLPIBO 9.e 8.7e.e.e 09 8.e 08.7e 08 MLPPO 0.e 07.9e e 09.e 08 8.e e-08 9

251 () Best men squre error for (b) Best men squre error for (c) Best men squre error for 0

252 () Best men squre error for (e) Best men squre error for (f) Best men squre error for Fgure 7. (), (b), (c), (), (e) n (f) re men squre error curve of MLPGA for ll jont ngles.

253 The optmzton of ANN prmeters such s weght n bs proves the better effcency n precton of results. MLPGA n MLPTLBO eqully perform better thn MLPPO n MLPIBO. On the other hn, performnces of MLPGA compre to ll opte lgorthms re cceptble. It cn lso be observe tht hybrzton wth GA shows the fst serchng blty n wth better explotton of the soluton. However, TLBO shows strong explorton blty thn other lgorthms. The results of the MLPTLBO re epcte n Tble 7.8. Therefore, the hybrzton of optmzton lgorthms wth MLP moels prouces better results s compre to trtonl bck propgton lgorthms. Also these technques gurntee fster convergence rte. 7.. Inverse knemtc soluton of -of AEA IRb- mnpultor In ths secton, AEA IRb- robot mnpultor s consere for the nverse knemtc soluton. Ths mnpultor s wely use n nustres s well s n reserch res. The etl escrpton of the opte robot mnpultor s presente n chpter. The mthemtcl moellng of the opte robot mnpultor s gven n chpter. Followng the knemtcs equtons n DH-prmeters of the opte robot, the jont vrbles n en effector coorntes re obtne usng MATLAB cong. The prepre t sets re use s n nput to trn the opte ANN moels for the precton of nverse knemtc of the mnpultor. After trnng of the opte confgurtons of the neurl network moels n hybr neurl networks, the nverse knemtc solutons re compre wth the conventonl metho bse soluton. Further the men squre errors for ll jont vrbles re clculte on the bss of ctul t n esre t sets. The confgurton of the MLPNN wth fferent number of hen neurons s presente n Tble 7.9. Tble 7.9 onfgurton of MLPNN l. Prmeters Lernng rte 0.9 Momentum prmeter 0. Number of epochs 000 Number of nputs Number of output Trget tsets 00 7 Testng tsets 00 8 Trnng tsets 700 Vlues tken

254 Tble 7.0 Desre jont vrbles etermne through quternon lgebr N Jonts vrbles n postons etermne through quternon lgebr θ θ θ θ θ X Y Z Tble 7. Men squre error for ll opte lgorthms Output Algorthms Number of Hen Noes MLPBP.e-0.e e-0.98e-0 7.8e-0.7e-0 ANFI 9.7e-0.e-0.00e-0.0e-0.e-0 9.e-0 MLPGA MLPTLBO.e-.e-.08e-.79e-8.e-.e- MLPIBO.e- 0.e-.e-08.8e-0.00e- 0.e- MLPPO.8e-0.70e-07.e-09.e-.09e-0.0e-9 MLPBP 7.e-0.0e-0 0.e-0.e-0.e-0 0.0e-0 ANFI.e-0.77e-0 0.e-0.e-0.e-0.e-0 MLPGA 0.00.e-.8e-8.7e MLPTLBO.e-8.e- 7.78e-8 7.e-7.e- 7.e- MLPIBO.e-.e- 7.7e-.e-9.7e-0.0e-0 MLPPO.0e-08.0e-.8e- 7.0e-.07e-0.e- MLPBP.e-0 0.e-0.8e-0.78e-0 7.e-0 9.8e-0 ANFI.e-0 0.e-0.8e-0.89e-0.e-0 7.9e-0 MLPGA.e 7.e.e 0.0e MLPTLBO 0.99e-.e-8.e-0.e-.7e-0.e-8 MLPIBO 7.7e-9.78e-8.00e-9 7.e- 0.8e- 7.e-9 MLPPO.e-.8e- 9.e- 0.9e- 9.e-7 0.e-8 MLPBP.88e e-0.e-0.00e-0 0.e-0 7.e-0 ANFI.8e-0 7.0e-0.7e-0.e-0 8.e-0.00e-0 MLPGA e-9.e-79 MLPTLBO.e-09.e e- 7.e-.e-8.e- MLPIBO.78e-9.e-8.78e-7.e-.8e- 7.e- MLPPO 9.87e-.8e-.8e-9 7.7e-09.e-09.e-9 MLPBP 7.e-0.0e-0 7.e-0.e-0.e-0 7.e-0 ANFI.e-0.7e-0.e-0 8.0e-0.8e e-0 MLPGA MLPTLBO.e-.e-.0e-.e-9.e-7.e- MLPIBO 8.9e-09.e- 8.e-.e-.07e-.8e- MLPPO 0.8e-9.e-.e

255 Followng the smlr proceure of PUMA 0 mnpultor, the nverse knemtc soluton s presente usng MLPNN n hybr MLPNN methos. The hybr MLP moels re s follows MLPPO, MLPGA, MLPTLBO, ANFI n MLPIBO. Intlzton of lgorthms n certn pproxmtons re selecte rnomly wthout ny specfc tunng smlr to the prevous reserch work. After ntlzton of the lgorthm the jont vrbles for the selecte mnpultor s obtne n compre wth the quternon vector metho bse soluton. The smple t set generte by the conventonl tool s presente n Tble 7.0. From the generte t 00 sets of jont vrbles re consere for the trnng of the network. The comprsons of ll opte lgorthms re presente n Tble 7. wth fferent number of hen noes. The precte nverse knemtc solutons from the opte moels re compre wth the ctul soluton n lter the men squre error s clculte. The best men squre error s presente n bol letters (see Tble 7.). Fgure 7. (), (b), (c), () n (e) shows the selecte best men squre curve of MLPGA for ll jont vrbles. mlrly best chosen men squre error of other opte lgorthm s presente n Tble 7.. () Best men squre error for (b) Best men squre error for

256 (c) Best men squre error for () Best men squre error for (e) Best men squre error for Fgure 7. (), (b), (c), () n (e) re men squre error curve of MLPGA for ll jont ngles 7.. Inverse knemtc soluton of -of TAUBLI RX0 L mnpultor In ths secton, -of TAUBLI RX0 L robot mnpultor s opte for the nverse knemtc soluton. The etl explnton of the opte robot mnpultor s presente n chpter. The mthemtcl moellng usng conventonl tool of the opte robot mnpultor s gven n chpter. Usng the knemtc equtons of the opte

257 mnpultor the trnng t sets re prepre for the precton of nvers knemtc soluton. The work s perform n the MATLAB 0. All jont vrbles n en effector postons re clculte usng the knemtc equtons. The prepre t sets re presente n Tble 7. whch s lter use s n nput for the trnng of the neurl network moels. Once the trnng s complete, the esre output s compre wth the ctul t sets. Hence the men squre error from the esre vlue n ctul vlue hs been clculte. The confgurton of the MLPNN wth fferent number of hen neurons s presente n Tble 7.. Tble 7. onfgurton of MLPNN l. Prmeters Lernng rte 0.0 Momentum prmeter 0. Number of nputs Number of output Trget tsets 00 Testng tsets 00 7 Trnng tsets 00 Vlues tken The confgurton of the neurl network moel from Tble 7. gves the fferent prmeters for the trnng the neurl network moels. Bse on the bove confgurtons the precton of nverse knemtc soluton s one. Lter the other prmeters such s weght n bs re upte usng the bck propgton lgorthm. The expermentl results n comprson wth other hybr moels re presente n Tble 7.. The opte lgorthms for the precton of nverse knemtc soluton re smlr to the prevous work. The comprson hs been me on the bss of men squre error of the soluton. There s no specfc tunng of the ssocte prmeters for the trnng of ANN moels. The esre jont vrbles n en effector postons re presente n Tble 7.. Tble 7. Desre jont vrbles etermne through quternon lgebr N θ Jonts vrbles n postons etermne through quternon lgebr θ θ θ θ θ X Y Z

258 Tble 7. Men squre error for ll trnng smples of hybr MLPNN Output Algorthms Number of Hen Noes MLPBP 0.0e-0.09e-0 7.e-0 7.e-0 0.e-0.e-0 ANFI.e-0.78e-0 0.8e-0.07e-0.99e-0 7.9e-0 MLPGA.e-78 8.e-7.8e-89.8e MLPTLBO 8.e- 8.70e-7.88e-7.0e-0.78e-80.e-7 MLPIBO.9e- 8.e-.e-.9e-.8e-8.99e-9 MLPPO.70e-.e- 7.e-8.e- 7.99e-.e- MLPBP 7.e-0.e e-0 7.e-0 0.e-0.70e-0 ANFI 9.e-0.99e-0.78e-0 7.e-0.00e-0.08e-0 MLPGA.e-99.9e MLPTLBO.0e-9.7e-8.e-80 8.e-90.7e-9.89e-80 MLPIBO.e-.7e-.e-0.09e MLPPO.e- 8.9e-0.9e-.e-9 7.e- 9.8e- MLPBP.e-0.77e-0.99e-0.7e-0 0.e-0 7.e-0 ANFI 8.e-0.e-0 7.9e-0 9.e-0.07e-0.e-0 MLPGA.e-9 9.8e- 8.e-8.e e- 7.89e-79 MLPTLBO.e-09.e- 8.e-07.09e-08.e-09.8e-09 MLPIBO.e-09.e-09.89e-08 9.e-09.e-0 8.7e- MLPPO.7e-.0e-7.e-.0e-0.e-0.e- MLPBP 0.0.e-0.00e-0.89e-0.7e-0 0.7e-0 ANFI 7.89e e-0.7e-0.e-0.e-0.0e-0 MLPGA.9e.88e e 7.e 0 MLPTLBO 7.97e-8.7e-.e-.7e-9.7e 8.e MLPIBO.0e e e-09 7.e-0.e-09.e-0 MLPPO.7e-.8e-7.e-.8e-.9e-.9e-0 MLPBP 7.e-0.89e-0.e-0.99e-0.00e-0 0.0e-0 ANFI 8.99e-0.0e-0 8.9e-0.88e-0 7.8e-0.e-0 MLPGA 7.e-.7e MLPTLBO.7e 9 9.0e 9.97e-.8e-9.0e-.e- MLPIBO.7e 0.7e 0.07e 0.7e 0.0e 0 8.9e 0 MLPPO.e 0.e 0.7e 0 9.e 0.e-7.e- MLPBP 7.e-0.0e-0.e-0.e-0.99e-0.0e-0 ANFI.7e e-0 7.9e-0.0e-0.0e-0 8.e-0 MLPGA.0e-.e- 7.7e- 7.0e MLPTLBO.e-7.e-8.e-.e-.e-.e-7 MLPIBO.7e-0.9e-0.0e-08.0e-0.98e-.9e-8 MLPPO.e-8.e-.0e-.e- 7.7e- 7.0e-9 The expermentl results n comprson of ll opte lgorthms wth severl fferent numbers of hen noes re presente n Tble 7.. The perform work s smlr to prevous sectons. The use of fuzzy sets wth the neurl network moels s proucng better results s compre to MLPNN moel. The results for MLPBP n ANFI re gven n Tble 7.. The uptng of the weght n bs usng bck propgton lgorthm s slow n stgnte t locl optmum pont. Moreover, bck 7

259 propgton lgorthm consumes more computtonl tme s compre to ANFI n other hybr lgorthms. Therefore, to ncrese the explorton n explotton blty, t s requre to fuse optmzton lgorthm wth MLPNN moel. The results of the hybr ANN moels re presente n Tble 7.. The best men squre error s specfe n bol letter for ll opte lgorthms. Fgure 7. gves the overll best performnce of the hybr moel usng genetc lgorthm. Fgure 7. (), (b), (c), () n (e) shows the selecte best men squre curve of MLPGA for ll jont vrbles. mlrly best chosen men squre error for ll other lgorthms cn be obtne from Tble 7.. () Best men squre error for (b) Best men squre error for 8

260 (c) Best men squre error for () Best men squre error for (e) Best men squre error for 9

261 (f) Best men squre error for Fgure 7. (), (b), (c), () n (e) re men squre error curve of MLPGA 7. Metheurstc pproch for nverse knemtcs soluton Inverse knemtcs soluton of robot mnpultors hs been consere n evelope fferent soluton scheme n lst recent yer becuse of ther multple, nonlner n uncertn solutons. Optmzton methos cn be pple to solve nverse knemtcs of mnpultors n or generl sptl mechnsm. Bsc numercl pproches lke Newton-Rphson metho cn solve nonlner knemtc formule or nother pproch s prector corrector type methos to ssmlte fferentl knemtcs formule. But the mjor ssues wth the numercl metho re tht, when Jcobn mtrx s ll contone or possess sngulrty then t oes not yel soluton. Moreover, when the ntl pproxmton s not ccurte then the metho becomes unblnce even though ntl pproxmton s goo enough mght not converge to optmum soluton. Therefore optmzton bse lgorthms re qute frutful to solve nverse knemtc problem. Generlly these pproches re more stble n often converge to globl optmum pont ue to mnmzton problem. The key fctor for optmzton lgorthms s to esgn objectve functon whch mght be complex n nture. On the other hn, metheurstc lgorthms generlly bse on the rect serch metho whch generlly o not nee ny grent bse nformton. In cse of heurstc bse lgorthms locl convergence rte s slow therefore some globl optmzton lgorthms lke GA, TLBO, PO etc. cn be gnfully use. Therefore, the key purpose of ths work s focuse on mnmzng the Eucln stnce of en effector poston bse resoluton of nverse knemtcs problem wth comprson of opte optmzton lgorthms obtne soluton for -of n -of revolute robot mnpultors. The objectve functon (ftness functon) mthemtcl 0

262 moellng s gven n prevous chpter. The result of ech lgorthm s weghe by usng nverse knemtcs equtons to obtn sttstcs bout ther error. In other wors, rtesn coorntes hve been use s n nput to clculte ech jont ngle. Fnlly th orer splne s use to generte trjectory n corresponng jont ngles of mnpultor usng optmzton lgorthms n quternon for -of mnpultor. The mthemtcl moellng of the opte confgurton of robot mnpultors n etl ervton of forwr n nverse knemtcs of -of mnpultor usng quternon lgebr s gven n chpter. The expermentl results s obtne from smultons re scusse elbortely n lter secton. 7.. Inverse knemtc soluton for -of ARA mnpultor In ths secton, nverse knemtc soluton of -of ARA robot mnpultor s presente. The mthemtcl formultons n bckgroun of the reserch topc hs been presente n chpter. The formulton of the objectve functon (ftness functon) s obtne usng poston n orentton bse error metho. The etl escrpton of the formulton of the objectve functon s presente n secton.. Usng the poston n orentton error subjecte to the jont vrbles constrne s solve usng severl optmzton lgorthms. Moreover, the obtne nverse knemtc solutons re compre wth the conventonl soluton of the nverse knemtc problem. multons n MATLAB progrms re use to check the performnce n effectveness of the opte optmzton lgorthms for the nverse knemtc solutons. Fve fferent postons of the en effector hve been consere for the comprtve evluton of the nvers knemtc solutons. Fve fferent postons of the en effector re presente n Tble 7. usng conventonl tool. Tble 7. Fve fferent postons n jont vrbles Jont ngles Postons P(0.8, 0., -.) P(-.9, 9., -0.00) P( -.0, -9., -.7) P(.7, -77.9, -8.) P(.9, 7.9, ) Tble 7. represents the comprtve results of the ll opte lgorthms for the evluton of the nverse knemtc problem of -of ARA mnpultor. The experment of the opte optmzton lgorthms oesn t follow ny speclze tunng for the ssocte prmeters. From Tble 7., t cn be observe tht the objectve functon vlue for genetc lgorthm s better thn ll other lgorthms. On the other hn, TLBO n GWO re performng eqully on the bss of functon

263 evluton. The optmum vlue for the Euclen stnce norm s 0; the obtne results for opte lgorthms re cceptble f t vres wthn the lmt of Therefore, t cn be observe tht the optmzton bse nverse knemtc solutons re cceptble. The comprsons on the bss of computtonl tme for ll opte lgorthms re scusse n chpter 8. Tble 7. Fve fferent postons n jont vrbles through opte lgorthm Postons PO Jont ngles Functon Vlue P P P P P Postons GWO Jont ngles Functon Vlue P P P P P Postons TLBO Jont ngles Functon Vlue P P P P P Postons GA Jont ngles Functon Vlue P P P P P Postons IBO Jont ngles Functon Vlue P P P P P The comprson of the obtne solutons usng optmzton lgorthms re presente n Fgure 7.7 through Fgure 7.. The soluton obtne through the optmzton methos re compre wth the quternon bse soluton. All jont vrbles for fve

264 postons of en effector re ncte n Fgure 7.7 through Fgure 7.. Fgure 7.7 gves comprtve results for the poston one (P) usng ll opte lgorthms. For poston P the jont vrbles usng PO n GWO re close to the soluton obtne through quternon metho. mlrly for postons P to P, the solutons obtne through the GA re better s compre to other optmzton lgorthms. Therefore, the functonl vlue n jont vrbles s cceptble usng GA lgorthm. Moreover, TLBO n GWO eqully perform for the nverse knemtc soluton. Fgure 7.7 omprson of jont vrbles for poston Fgure 7.8 omprson of jont vrbles for poston

265 Fgure 7.9 omprson of jont vrbles for poston Fgure 7.0 omprson of jont vrbles for poston

266 Fgure 7. omprson of jont vrbles for poston 7.. Inverse knemtc soluton for -of mnpultor multons hve been me to check the performnce n effectveness of opte optmzton lgorthms n comprson to solve nverse knemtc problem of -of mnpultor. Fve fferent postons of en effector hve been consere for nverse knemtcs evluton s presente n Tble 7.7. The propose work s performe on the MATLAB R0. In ths work, comprson t sets were generte by usng quternon vector bse metho from chpter. Tble 7.7 Fve fferent postons n jont vrbles Postons Jont ngles P( -7.09,., -.9 ) P( 89.9, 9., 90.87, ) P( -., 9.08, 97. ) P(9.0,.0, -.) P(-8., -., 8.7) Tble 7.7 gves some of the esre t for the poston of jonts etermne through nlytcl soluton (Quternon), whch wll further be use to clculte the fference between the opte lgorthms n nlytcl soluton of nverse knemtc. Tble 7.8 represents the comprtve results of optmzton lgorthms for the evluton of nverse knemtc ftness functon n jont vrbles. onucte experment oesn't follow ny specl tunng of ssocte prmeters of ll lgorthms. It s observe from Tble 7.8 TLBO performng well s compre to GA on the bss of ftness evlutons for poston. In cse of the Euclen stnce norm the mnmum vlue of the consere ftness functons s 0, the obtne result s ccepte f t vres from

267 the optmum vlue by less thn 0.0 n ll lgorthms re ner to the stte vlue. Hence t cn be unerstoo from the obtne results tht the propose soluton scheme performng qute well for metheurstc lgorthms. Tble 7.8 Fve fferent postons n jont vrbles through opte lgorthm PO Jont ngles Postons Functon Vlue P P P P P GWO Jont ngles Postons Functon Vlue P P P P P Postons TLBO Jont ngles Tble 7.9 omputtonl tme for nverse knemtc evlutons N Metho omputtonl tme TLBO.7s GA 7.9s GWO.s PO.88s IBO 9.s Functon Vlue P P P P P GA Jont ngles Postons Functon Vlue P P P P P IBO Jont ngles Postons Functon Vlue P P P P P

268 Fgure 7. through 7. represents the best functon vlue n corresponng jont vrbles for ll postons. These fgures gve the performnce of the opte lgorthms for the soluton of nverse knemtc problem of -of mnpultor. The convergence of the ftness functon goes to zero error for both opte lgorthm but n cse of TLBO t gves 0.0 errors for poston. Therefore t cn be sy tht the TLBO s performng less ccurte s compre to GA. Fgure 7.7 through 7. gves the performnce of the GA for the opte moel n hstogrms gves the vlue n rns whch s converte nto egree n presente n Tble 7.8. Usng GA MATLAB toolbox the progrm ws testes n the results converge to zero splcement error n corresponng jont vrble for sngle run s shown n Fgure 7. through 7.9. In ths fgure the opte lgorthm s proucng multple solutons for the sngle poston but s per gve termnton crter n mong those generte results mnmum vlue of jont ngle hs tken for the comprson. It hs been observe tht the convergence of the soluton for GA s tkng less computton tme s compre to lgorthms. omputtonl tmes for ll opte lgorthms re gven n Tble 7.9. Overll computton tme for the clculton of nverse knemtc soluton s.7secons for TLBO whch s more thn other lgorthms, whle the GA s tkng only 7.9 secon. Therefore t cn lso be compre on the bss of computtonl cost tht quternon lgebr s tkng slowest tme mong other opte metho. Fgure 7. Functon vlue n jont vrbles for P Fgure 7. Functon vlue n jont vrbles for P 7

269 Fgure 7. Functon vlue n jont vrbles for P Fgure 7. Functon vlue n jont vrbles for P Fgure 7. Functon vlue n jont vrbles for P Fgure 7.7 Functon vlue n jont vrbles for P 8

270 Fgure 7.8 Functon vlue n jont vrbles for P Fgure 7.9 Functon vlue n jont vrbles for P Fgure 7.0 Functon vlue n jont vrbles for P Fgure 7. Functon vlue n jont vrbles for P 9

271 7.. Inverse knemtc soluton of -of PUMA mnpultor In ths secton, type PUMA 0 mnpultor s use for the nverse knemtc nlyss. multons stues re crre out to check the performnce n effcency of propose GWO, PO, IBO, TLBO lgorthm n comprson wth the GA to solve nverse knemtc problem of PUMA mnpultor. Fve fferent postons of en effector hve been consere for nverse knemtcs evluton s ncte n Tble 7.0. The propose work s mplemente on the MATLAB R0. Tble 7. proves smple of the trget t for the poston of en effector etermne through nlytcl soluton (Quternon), whch s use to etermne the fference between the opte lgorthms n nlytcl soluton of nverse knemtc. Tble 7. enotes the comprtve results of GWO, PO, TLBO, IBO n GA lgorthms for the evluton of nverse knemtc ftness functon n jont vrbles. The prmeters of ll opte lgorthms re use wthout ny speclze tunngs smlr to prevous work. The objectve functon formultons re presente n chpter whch s bse on the poston n orentton bse error. The mnmum functonl vlue of the stnt bse norm s 0 whle n ths thess the mnmum functonl vlue of objectve functon s llowe to the lmt of 0.0 for ll lgorthms. It hs been observe from the comprson Tble 7.; ll opte lgorthms re performng precsely to cheve mnmum functonl vlue. The overll performnce to get mnmum functon vlue for the objectve functon usng GA s yelng better results s compre to other opte lgorthm. It s lso observe from Tble 7. GWO performng well s compre to other opte lgorthms on the bss of ftness evlutons for postons P, P, P n P whle TLBO s performng better n cse of P. Hence t cn be unerstoo from the obtne results tht the propose soluton scheme performng qute well for metheurstc lgorthms. Tble 7.0 Fve fferent postons n jont vrbles through quternon Postons (X, Y, Z) P(., 7.9, 0.7) P(-., -.9, 9.87) P-., 9.0, 8.00) P(-., 8.8,.) P(-0., 8.7, 0.7) Jont ngles

272 Tble 7. omprtve results for jont vrble n functon vlue Postons (X, Y, Z) Fgure 7. through 7.7 represents the best functon vlue n corresponng jont vrbles for ll consere postons. Jont ngles by PO Vsulzton of the ftness functon usng fferent omns of the vrbles cn be foun n the left se (frst column) of the Fgure 7. through 7.7. urf plot functon s use n n re of gven rnge of vrbles, where focus or mpresson s gven on the XY plne epctng the globl optmum rnge from [-0, 0]. For ll surf plot n the left se of the fgure represents the fferent propertes of the consere ftness functon. When lookng t the nner surfce re, the ftness functon looks fferent, wheren mny smll vlleys n peks re vsble. These peks n vlleys ncreses when the consere problem s hgher mensonl. Moreover zoomng of the surf plot cn yel the esre locton of the Functon Vlue P P P P P Postons Jont ngles by GWO Functon Vlue P P P P P Postons Jont ngles by TLBO Functon Vlue P P P P P Postons Jont ngles by GA Functon Vlue P P P P P Postons Jont ngles by IBO Functon Vlue P P P P P

273 optmum pont wthn the smll serch re. In orer to nlyses the convergence behvors of the opte lgorthms serch hstory n corresponng jont ngles of the mnpultor s presente n Fgures 7. through 7.7 n secon columns. In ths work three serch gents for GWO hs been consere to fn out the optmum vlue of ftness functon, smlrly three sets of lerner for TLBO, three prtcle for PO n three genes consere for GA. It hs been observe tht the ll consere serch gents or nvuls for opte lgorthm hvng blty of explorton n explotton of best ftness evlutons. From Fgure 7., TLBO s gvng mnmum of ftness functon for the poston P s compre to other opte lgorthms whle for rest of the consere postons GWO s performng better. The convergence of the ftness functon goes less thn zero error for ll opte lgorthm but n cse of PO t gves.9 errors for poston P. Therefore t cn be sy tht the PO s performng less ccurte s compre to other lgorthms. Fgure 7.7 through 7.7 gves the performnce of the GA for the opte moel n hstogrms gves the vlue n rns whch s converte nto egree n presente n Tble 7.. Usng GA MATLAB toolbox the progrm ws teste n the results converge to zero splcement error n corresponng jont vrble for sngle run s shown n Fgure 7.7 through 7.7. In ths fgure the opte lgorthm s proucng multple solutons for the sngle poston but s per gve termnton crter n mong those generte results mnmum vlue of jont ngle hs tken for the comprson. Fgure 7. PO Functon vlue n jont vrbles for P

274 Fgure 7. PO Functon vlue n jont vrbles for P Fgure 7. PO Functon vlue n jont vrbles for P Fgure 7. PO Functon vlue n jont vrbles for P

275 Fgure 7. PO Functon vlue n jont vrbles for P Fgure 7.7 GWO Functon vlue n jont vrbles for P Fgure 7.8 GWO Functon vlue n jont vrbles for P

276 Fgure 7.9 GWO Functon vlue n jont vrbles for P Fgure 7.0 GWO Functon vlue n jont vrbles for P Fgure 7. GWO Functon vlue n jont vrbles for P

277 () (b) Fgure 7. TLBO Functon vlue n jont vrbles for P () Fgure 7. TLBO Functon vlue n jont vrbles for P (b) () Fgure 7. TLBO Functon vlue n jont vrbles for P (b)

278 () Fgure 7. TLBO Functon vlue n jont vrbles for P (b) () (b) Fgure 7. TLBO Functon vlue n jont vrbles for P Fgure 7.7 GA Functon vlue n jont vrbles for P Fgure 7.8 GAFuncton vlue n jont vrbles for P 7

279 Fgure 7.9 GAFuncton vlue n jont vrbles for P Fgure 7.70 GAFuncton vlue n jont vrbles for P Fgure 7.7 GAFuncton vlue n jont vrbles for P Fgure 7.7 IBO Functon vlue n jont vrbles for P 8

280 Fgure 7.7 IBO Functon vlue n jont vrbles for P Fgure 7.7 IBO Functon vlue n jont vrbles for P Fgure 7.7 IBO Functon vlue n jont vrbles for P 9