Fractional Order PID Design for Nonlinear Motion Control Based on Adept 550 Robot

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1 ape 65 ENG 6 Faconal Oe I esgn fo Nonlnea oon Conol Base on Aep 55 obo Absac Yuquan Wan uue Unves awn@puue.eu Haan Zhang uue Unves hhzhang@puue.eu cha a Fench uue Unves fench@puue.eu ulln obo sses ae pcal nonlnea sses whch geneousl nvolve n uncean subances hgh-ensonal sse a an non-unque chaacesc funcons. he auhos pesen he eho of applng faconal oe I conolle o such a nonlnea sse an show he avanages of hs faconal conolle. In he pape he nac oel of he sse seves as he founaon o eve he conol law an objecve funcon fo he opzaon esgn of he subjece faconal oe conol sse. he fequenc oan close loop ansacon funcon of he sue faconal sse has been evelope an he pape pocees b he su of conollabl obsevabl an obusl saabl. he pape eonsae he algohs o esgn an opze he faconal oe I o he nonlnea oon conol sse. B conucng sees nuecal copuaon hs pape shows ha he faconal oe I conolle coul enlage he sable egon of he ulln obo sse an heefoe bngs supeo conol pefoance n es of ajeco acng. he esuls an poceues nouce n hs pape coul be paccall genealze o ohe sla sses. Inoucon ulln obos ae wel use n anufacung nus an he oon conol ssues of hese obo sses have becae popula eseach opcs fo ecaes snce he fs appea of he nus obos. Geneall speang ulln obo sses ae pcal nonlnea sses an alwas nvolve n uncean subances hgh-ensonal sse a an non-unque chaacesc funcons. he fne conol of nus obos usuall eques cople conol sses caeful calbaons an opzaons. In nus pacce 9% of hese ulln obos ae conolle b I conolles whch have he es nclung effecveness splc an feasbl. Alhough he ona I conolles can acheve oceengs of he IAC-ASEE Inenaonal Confeence ISBN

2 sasfaco esuls n os coon anufacung ssons sll lacs enough pecsons n he fel eques pecse nsuens. he sucue of ona I conolle s fal saghfowa. Is popoonal negal an evave conol pas can pove esong coecve an couneacve foce especvel. Fuheoe he coecve foce nouce b negal conol can ovecoe he sea-sae eo bough b popoonal pa an he couneacve foce le b evave conol coul elnae he oveshoo poble cause b he negao. heefoe n coon suaon he ona I conolles can alwas effecvel acheve he conol objecons whou obvous awbacs. Howeve n oe nus he ean fo he pecse conol s vng people o seach fo poveen o he ona I conolles. Faconal oe I FoI nouce n hs pape s a naual eenson o he ona I conolles base on he faconal calculus heo. Snce n faconal calculus he oes of negal an evave ae no le o nege oes anoe people can esgn a new pe of I conolle b eplacng he ona oe negaos an ffeenaos b faconal oe ones. he an avanages of he FoI conolles wll nclue enlage sable egon elavel feasble sucue an ase conol pecson. As enone above faconal calculus aes he oe of negals an evaves as an eal nube. I has a hso neal as long as he ona calculus whch conses onl nege oes []. ecenl he applcaons of hs echnolog have been successfull foun n an fels such as vscoelasc [] [] conol heo [] [5] an eleco-analcal ches [6] [7]. In conol heo he geneal concluson abou faconal conol sse s ha coul enlage he sable egon [8] an el a pefoance a leas as goo as s nege counepa. An anohe poan avanage s ha faconal negals o evaves ae heea funconal whle he ona ones ae pon funconal. I s nown ha he heea funconal have he long eo chaacesc [9] whch eans a an e woul pocess a oal eo of pas saes. hs unque chaacesc seves as one of he poan easons fo he bee pefoance. Fo FoI conolles hee ae also an scholas have ae eenous conbuon n he pas eas [] especall n he unng ules [] [ ] appoaon [] an sabl conons []. All hese eseach wos geneall fos he sol founaons fo he wo one n hs su. In hs pape he auhos appl he FoI conolles o a nonlnea ulln obo sse an ae uncean subances no conseaon. Fuheoe he faconal oes of he negaos an ffeenaos use hee ae consee as esgn vaables ahe han pefe paaees. he auhos sue he sabl conons an opzaon esgn eho fo he oveall copehensve pefoance of he FoI conolles on he bass of he aheacal oel of Aep 55 obo. As one of he os coonl use obos n he nusal poucon lnes he Aep 55 obo s a fou-as SCAA obo wh hee oaonal jons an one anslaonal jon. Snce feaues a sall oon envelope whle s spees an paloas ae elavel hgh Aep 55 obo can be foun n echancal assebl aeal hanlng pacagng achne enng scew vng an an ohe opeaons equng fas an pecse auoaon. he auhos coplee su of FoI conolles on he oel of Aep55 obo shows ha he faconal conolle coul acheve oceengs of he IAC-ASEE Inenaonal Confeence ISBN

3 hgh pecse conol an bng feasble appoaches o opze he esgn of FoI n ohe applcaons. nacs oel of Aep 55 obo We have nclue he sple sucue of Aep 55 obo as n fgue. When applng enav-haenbeg -H coonaes one woul noce ha Aep 55 has he specal case of paallel aes whch s connece wh he g nne an oue lns. he ajeco of hs obo s eene b he oon of hese wo lns an a he ws he oaonal jon oaes abou he z as o ajus he gppe angle bu no o change he ajeco. Snce we anl focus on he pefoance of ajeco acng n hs su In oe o focus on he ajeco su whou losng geneal we woul fal assue he ws s oa angle s zeo. If oo he noaons as labele n fgue one coul ge he -H paaees as showe n he able. Fgue : splfe sucue of Aep 55 obo able. -H paaees n Inne Oue able lss he -H paaees of he nne an oue lns. Fo he nne ln an oue ln he coonae ansfoaon wh oaon an anslaon coponens s escbe wh he a: oceengs of he IAC-ASEE Inenaonal Confeence ISBN

4 oceengs of he IAC-ASEE Inenaonal Confeence ISBN sn sn sn sn sn sn sn z z A α α α α α α α he splfe ansfoaon a of Aep 55 fo he base o he gppe s as follows: sn sn sn sn A A Nocng he elaonshps he angula poson of he oos an he angle abou pevous z fo ol o new : he gppe s hozonal poson can be epesse as sn sn Fo Equaon he oo angula posons can be eve: ± ± an an Whee: 5 he fowa veloc v can be foun fo Equaon : a v 6

5 oceengs of he IAC-ASEE Inenaonal Confeence ISBN Whee a s acoban a sn sn a 7 An he bacwa veloc can be eve as follows: 8 Also he elaonshps of he fowa acceleaon can be eene: v a 9 Whee: sn sn v he elaonshps of he bacwa acceleaon ae as follows: v a a Applng he agange eho one coul ge he nacs of Aep 55 obo as escbe below [5]: apng G H Whee: 7 7.a

6 oceengs of he IAC-ASEE Inenaonal Confeence ISBN sn sn H.b s s g g G.c. C C C apng apng apng.e An a C s copose of he apng coeffcens. oel of faconal oe I conolles Base on equaon we can assue ha he oos vng he nne an oue lns ae n he sae pe. nacs of he wo lns s escbe as: b n j j j n j j j V B C g h Fo : Snce whee s he gea ao he wo nac equaons of obo ln an s vng oo epesse n Equaon can be cobne no a sngle one: C V B eff eff 5 Now fo a faconal oe I conolle µ I one has he fve esgn paaees as suaze n able :

7 able. esgn paaees fo he conolle Coeffcen fo he popoonal e Coeffcen fo he evave e Coeffcen fo he negal e I µ Faconal oe fo he evave e Faconal oe fo he negal e he close loop conol aga s showe n fgue an equaon 6 escbes he ansfe funcon of hs close loop sse. he faconal evave use n hs su s efene a Capuo s faconal evave [6]. Fgue close loop aga of faconal oe conolle obo a S 6 n n I n n n eff ns Beff n Cn S p ns I n In hs eseach we conse he FoI conolles n he wo as have he sae faconal oe an µ bu ffeen coeffcens. Beses boh he faconal oe of he negao an he ffeenao ae boune n he ange of n hs su. In equaon 6 he non-lnea es n s as equaon 7 { sn } g { sn g 7 Appl he Capuo s faconal oe evave o 7 an snce we coul have he e oan sse funcon n a epesenaon as n 8. oceengs of he IAC-ASEE Inenaonal Confeence ISBN

8 oceengs of he IAC-ASEE Inenaonal Confeence ISBN I I I I eff eff eff eff c B c B 6 5 µ µ µ µ 8 Whee n 8: 9.a sn Γ Γ 9.b π 9.c sn π 9. 5 π g 9.e 6 π g 9.f In equaon 8 he ffeenal oe of he ae µ an. Snce hese oes ae no equall space s no eas o ecl e-we 8 n a lnea a foaon. Inspe b he wo of Galows Bachele an ue [7] we assue an µ ae aonal nubes whch coul be epesse b b a an c n he elavel pe foas especvel. B nong ] ; [ equaon 8 coul be wen as n : 5 U b a bc ac ac ac In an U ae coeffcen a conanng he coesponng es n 8. One oe hng o enon hee s no all of hese coeffcens ae consan snce he uncean subance. We wll show s acuall a e vaan sse lae. B nseng zeo aes s equvalen o we as n :

9 ac ac N N a ac N ac bc a N ac N 5 j b a U ac Whee N N N N N.Base on one has an equall space j faconal oe sse on eve e an heefoe he sae space coul be efne as: a a a An he sse s: X AX BU Whee: [a ] I[a a ] A 5.a [a ] B.b In.a an.b [a ] s a zeo a whose enson s [ a ] an I s he en a has he enson of [ a a ]. [ a a ] Equaon s he sae space epensaon of ou sse funcon. he sse a A has he eson of [ a a ] an B has he enon of [ a ]. he sabl su an he esgn of he faconal oe I conolle wll focus on he a A. Alhough A coul have a ve hgh enson wh he ffeen faconal oe he fac ha a A s a spase a aes he as ease n os cases. Conollable obsevable an obus sable of he sse: Snce a A s n he conollable canoncal fo an consequenl one sae coul be ansfee o anohe he sse s conollable an obsevable. he esgn focuses on he obus sable of hs sse. Fo a faconal oe sse he sse woul be guaanee sable f all he sse a s egenvalues sasf he followng cea [8]. oceengs of he IAC-ASEE Inenaonal Confeence ISBN

10 π ag > 5 heefoe n hs su he ao of sable egon of FoI o he nege I s One sees coul sng b an o ge a lage sable egon. Howeve sng he wll cause a lage enson of a A an nvolves oe egenvalues snce he oal nube of egenvalues s a. oe egenvalues woul ae hae o ganee all of he ae sele n he sable egon. oeove snce A s a boune spase a wh neval unceanes hee shoul be nfne nubes of egenvalues o chec o sasf he sable egon f one ecl use he eho n 5. In hs case we woul le o chec he bounaes of each egenvalue [8] [9] an connue o analss he sables of he sse base on he behavos of all egenvalues bounaes []. heefoe we nee o chec he bounaes of hs sse a A. Base on 8 an 9 he followng nequal hols: > [ 6 eff eff ] hus he eenan of a sasfes: eff eff 7 he fac ha conon 7 alwas hols ples he a s alwas nonsngula an consequenl he a A wll neve be sngula f. An n hs esgn we wll eep hs conon. hus we have: e [ ] 8.5 eff eff eff eff An n hs obo conol su he an ae also boune because of eal. heefoe one coul fn he a A s boune. lug n he paaees we use n hs su we ge he followng bouna funcons fo each vaan es n A hough nuecal copuaon he bounaes ae funcons of esgn paaees. I I µ I 8.76 I I I I I I I 9.a oceengs of he IAC-ASEE Inenaonal Confeence ISBN

11 b 9.c AX AX AX AX AX 9. Now we coul su he obus sabl of hs FoI conolle sse a ffeen esgn I I µ. An hs feaue acuall poves a ceon o opze he esgn of he conolles. Ne we woul le o show how he esgn paaees whch ae he coeffcens an he faconal oe of he wo FoI conolles affec he obus sabl. he fgue shows hs effec. ang he uppe lef fae n fgue as an eaple he ecangles ew b blue sol lnes show he bounaes of each egenvalues. Snce hee ae unceanes nvolve n hs sse he egenvalues ae acuall locae n a ange ahe han sngle spos. An ecangles pove suffcen bounaes fo hs egenvalues [9]. o ensue he sse s obusl sable he egenvalues bounaes ae no allowe o coss he sable bouna whch essenall epesen he angle epesens he angle ± π / n hs eseach. Fo a bee eonsaon we plo hose non-volae sable bounaes b can sol lnes an hose volae sable bounaes b e sol lnes. oceengs of he IAC-ASEE Inenaonal Confeence ISBN

12 5 p Inegal/ evave/ Sable-Unsable Bouna ange of Each Egenvalues 6 p 5 Inegal/ evave/ Sable-Unsable Bouna ange of Each Egenvalues Iage As - Iage As eal As eal As 7 p5 5 Inegal/ evave/ Sable-Unsable Bouna ange of Each Egenvalues 8 p5 Inegal/ evave/ Sable-Unsable Bouna ange of Each Egenvalues Iage As - Iage As eal As eal As Fgue : he effec of changng esgn vaables o he oveall sabl Fgue cleal shows ha changng he cobnaon of he esgn vaables coul bng changes on he oveall sabl o he whole sse. ung he esgn of he whole paaees se hee coul be unle peuaons fo he choces of esgn vaable se I I µ. he auhos woul le o appl soe opzaon algoh o acheve he copehensve opze esgn. Snce as of opzaon esgn nvolves he peuaon of each paaee he genec algoh s a naual choce fo hs sson. Opzaon esgn Fo hs esgn he sse conans unceanes an one coul onl oban he anges fo each egenvalues. e showe n fgue he anges ae he ecangles boune b eve fou cone egenvalues. If we aw own hs cone egenvalues n a cople plan an noe he aguens of he b n; j one coul hen easue he ffeence j of hese aguens o he sable bouna. In hs wa an cobne wh he facs ha all egenvalues ae secal o eal as n he cople plan a naual opzaon objecve s o nze he ffeence of sable aguens π / o he absolue value of each. heefoe he opzaon funcon use n hs eseach s epesse as followng j equaon : Opalesgn π n I I µ ag n ψ j j j In ψ seves as he coeffcen of penalzaon. hee coul be an ehos o assgn j he values ofψ. An also one coul sepaae he cople plan no ffeen segens j oceengs of he IAC-ASEE Inenaonal Confeence ISBN

13 accong o vaous cea. In hs pape we sue he wo-zone penalzaon eho an hee-zone penalzaon ehos an he ae all sepwse penalzaon ehos. An able n he followng suaze hese wo ffeen ehos. wo-zone eho j ψ j able : value of penalzaon coeffcen hee-zone eho j ψ j [ π / ] e5 [ π / ] e π / π ] π /.5π π / ] e N/A.5π π / π ] Befoe eplong he ajeco acng pefoance we woul le o nouce he ajeco plannng use n hs su. We ae gong o le he obo a ove n boh - econ an -econ. We se he ognal pon a 5 an allow secon fo he obo a o ove o poson 6. Fgue eonsaes he ajeco plan. We also suaze he opzaon esuls n he able. oson Spee /s Acceleaon /s ajeco lan of X lanne oson of X lanne Spee of X lanne Acceleaon of X e S oson Spee /s Acceleaon /s Fgue : jeco plan ajeco lan of Y lanne oson of Y lanne Spee of Y lanne Acceleaon of Y e S able. opzaon esuls of esgn paaees wo-zone eho hee-zone eho oceengs of he IAC-ASEE Inenaonal Confeence ISBN

14 Sulaon esuls an conclusons We have ploe he sulaon esuls abou he ajeco acng n he fgue 5. hs fgue nclues he esuls fo he sse opze b boh he wo-zone eho an heezone eho. An o copae wh we have also nclue an ona I conolle esul [5]. As showe n fgue 5 he opze FoI conolles have ace he ajeco plan successfull. In es of acng eo he faconal sse acheve a hghe pecson when copang wh he ona I sse. Boh of he wo-zone eho an hee-zone eho pove sasfaco opzaon esuls an heefoe he opzaon eho sue n hs pape has been jusfe as effecve one. We also ecoe he acng eo a each saplng pon an copue he aveage squae acng eo as suaze n able 5. Fo able 5 one coul cleal see ha he FoI sses have ase he pecson of acng b one oe of agnue Inege I obo oson n X-As Acual oson of X lane oson of X Inege obo oson n Y-As oson..5. oson Acual oson of Y lane oson of Y e S e S oson wo-zones Wegh Faconal-I obo oson n X-As Acual oson of X lane oson of X oson wo-zones Wegh Faconal-I obo oson n Y-As Acual oson of Y lane oson of Y e S e S oson hee-zones Wegh Faconal-I obo oson n X-As Acual oson of X lane oson of X oson hee-zones Wegh Faconal-I obo oson n Y-As Acual oson of Y lane oson of Y e S e S Fgue : Sulaon copasons n es of ajeco acng oceengs of he IAC-ASEE Inenaonal Confeence ISBN

15 able 6. copasons n es of ean squae acng eos Ona I FoI wo-zones FoI hee-zones ean Squae Eo n X.8859e-5.79e e-6 ean Squae Eo n Y.879e e-6.768e-6 Evence b he sulaon esuls he FoI conolle Aep55 obo sse coul acheve a bee esul n es of ajeco acng. An he esgn ehos nouce n hs pape s effecve when fnng he opze esgn of he faconal conolles. hs eho coul be easl ansfee no ohe applcaons elae o faconal conol an consequenl bng valuable esuls o nus pacce. In he en we woul le o conclue he followng pons:. he faconal oe conol of ulln obo sse alwas nvolves subance o ohe unceanes heefoe sung he ls of each egenvalues s a feasble eho o analss he oveall sabl. Fuheoe he bouna a coul be helpful n fnng he opzaon esgn of he faconal oe conolles.. he sepwse penalze eho coul be use o opze he esgn of FoI sse an hs eho allows people o ove he sse egenvalues owa o he ese egons. he eho popose n hs pape coul be genealze o ohe applcaon n he esgn of faconal oe conolles.. he opze faconal sse wll ae he avanage of he enlage sable egon whle avong an negave effecs bough b he ncease nube of egenvalues. Sulaon esuls show ha he opze FoI conolle Aep55 sse coul ac he planne ajeco successfull an ase he pecson geal ung he acng pocess. hs chaacesc woul bng valuable esul o he anufacung nus. efeences [] oveo A. Faconal Calculus: Hso efnons an Applcaons fo he Engnee [] ana F. Applcaons of Faconal Calculus n echancs ansfo ehos an Specal Funcons Vana 96 SC ublshes Sngapoe 997. [] osshn Y.A Shova.V. Applcaons of Faconal Calculus o nac obles of nea an Nonlnea Heea echancs of Sols Appl. ech. ev [] Bagle.. an Calco.A. Faconal Oe Sae Equaons fo he Conol of Vscoelascall ape Sucues.Guance vol. no. 5 pp.- 99 oceengs of he IAC-ASEE Inenaonal Confeence ISBN

16 [5] aoglou A. lle.. an Sa S. Copuaonal esuls fo a Feebac Conol fo a oang Vscoelasc Bea of Guance Conol an nacs vol. 7 no. pp [6] Olha.B. A Sgnal Inepenen Eleco-analcal eho Anal. Che. vol.7 pp [7] Goo. an Ish. Se-ffeenal Eleco-analss. Eleco anal. Che. an Inefacal Elecochecal. vol.6 pp [8]. agnon Genealze Faconal ffeenal an ffeence Equaons: Sabl opees an oellng Issues. oc. of ah.heo of Newos an Sses Sposu aova Ial 998. [9] ehel. N.. Fo e al. 5. "Algohs fo he faconal calculus: A selecon of nuecal ehos." Copue ehos n Apple echancs an Engneeng 96-8: [] I. olubn Faconal-Oe Sses an Faconal-Oe Conolles Ins. Ep. hs. Slova Aca. Sc. Vol. No. 99 pp.8-. [] Yng uo; YangQuan Chen; "Faconal-oe [popoonal evave] conolle fo obus oon conol: unng poceue an valaon" Aecan Conol Confeence 9. ACC '9. vol. no. pp.-7 - une 9 [] ngu Xue; YangQuan Chen; "Faconal Oe Calculus an Is Applcaons n echaonc Sse Conols Oganzes" echaoncs an Auoaon oceengs of he 6 IEEE Inenaonal Confeence on vol. no. pp.nl-nl une 6 [] ngu Xue; Chunna Zhao; YangQuan Chen; "A ofe Appoaon eho of Faconal Oe Sse" echaoncs an Auoaon oceengs of he 6 IEEE Inenaonal Confeence on vol. no. pp une 6 [] oceln Sabae aheu oze Chsophe Fages I sabl conons fo faconal oe sses Copues aheacs wh Applcaons Volue 59 Issue 5 Faconal ffeenaon an Is Applcaons ach ages ISSN 898- [5] Hen Zhang I Conolle esgn fo A Nonlnea oon Conol Base on oelng he nacs of Aep 55 obo Inenaonal ounal of INusal Engneeng an anageen IIE Vol. No. [6] El-Sae A.. A. ulvalue Faconal ffeenal Equaons Apple. ah an Copue vol. 8 pp [7] Galows.; Bachele O.; ue A.; "Faconal olnoals an n Sses: A Connuous Case" ecson an Conol 6 5h IEEE Confeence on vol. no. pp ec. 6 [8] A.S. ef he neval egenvalue poble Z. Angew. ah. ech [9] YangQuan Chen; Ho-Sung Ahn; olubn I.; "obus sabl chec of faconal oe lnea e nvaan sses wh neval unceanes" echaoncs an Auoaon 5 IEEE Inenaonal Confeence vol. no. pp. - 5 Vol. 9 ul- Aug. 5 [] Z. Qu.C. ülle A. Foe An applaon eho fo he sana neval egenvalue poble of eal nonsec neval aces Co. Nue. ehos Eng oceengs of he IAC-ASEE Inenaonal Confeence ISBN

17 Bogaph YUEQUAN WAN s cuenl a h canae n uue Unves hs eseach focuses on apple faconal calculus an faconal oe conol. HAIYAN ZHANG s an asssan pofesso a uue Unves hs eseach coves conol apple faconal calculus haulcs achnng an ul-ssnplne esgn opzaon.. Zhang go hs h fo unves of chgan. ICHA A FENCH s an assocae pofesso a uue Unves. hs eseach focuses on apple faconal calculus usc acouscs aeoelascs an opzaon.. Fench go hs h fo unves of aon. oceengs of he IAC-ASEE Inenaonal Confeence ISBN