IDENTIFICATION OF AN ELECTRICALLY DRIVEN MANIPULATOR USING THE DIFFERENTIAL FILTERS INPUT ERROR METHOD

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1 aa hania auoaia, o. no. 0 IDNTIFICATION OF AN CTICAY DIVN MANIPATO SING TH DIFFNTIA FITS INPT O MTHOD s CDO * * Ki nirsi of Thnoo, Fau of Mharonis and Mahinr Dsin, A. Tsiąia PP 7, 5- Ki, Poand [email protected] Absra: Th papr prsns an ap of soin h parar idnifiaion prob in as of robo ih hr drs of frdo has bn prsnd. Th idnifiaion has bn prford ih h us of aborad diffrnia firs. Th appid idnifiaion hod dos no rquir diffrnia quaions soin bu on drinin h appropria driais. Idnifiaion hod and is nraiaions usin h ob inrs od rquir inforaion on i driais of h inpu and oupu sinas. Th rquird driai ordr dpnds on h ordr of diffrnia quaions dsribin h ob. K ords: Diffrnia Firs, Idnifiaion, Inpu rror Mhod. INTODCTION Th rapid dopns in opur hardar and sofar and, onsqun, h oon us of opurs o onro prosss ha arousd id inrs in ahaia odin, onro prosss and, aordin, onro ss idnifiaion. Th hod of idnifiaion appid in h anasis inos fin-unin of h inrs od. Th hod an b usd on for suh aus of h inpu sinas ha ar drind fro h asurn daa. Idnifin a dnai ss b ans of h inpu rror hod Fi. rquirs ooin for a od ha nras h sa inpu as h ob. On in h as of od rrsibii is suh a produr possib. This rrsibii is ru for inar iniu-phas ods and a rain ass of non-inar ods hr h inpu is drind basin on h oupu daa Cdro and ani 009; Cdro and ani 0. us assu, for insan, ha h ob is dsribd b ans of a diffrnia quaion: Fi.. Shai diara of h idnifiaion pross. DIFFNTIA FITS us assu ha h diffrnia fir of h -h ordr is a sris onnion of a o-pass fir ih boundar frqun Ω and a diffrn quoin of h -h ordr Fi.. n n f,,...,, θ τ hr is a rain non funion, unnon parars and inpu sina. Thus, h idnifiaion rror is dfind as: τ ˆ τ, ˆ n n τ f,,...,, ˆ θ hr sia inpu sina and sia unnon parars. A draba of his hod is ha driai sias nd o b drind. An adana, on h ohr hand, is ha i is no nssar o so h diffrnia quaions dsribin h od a ah sp of iraion. Th fundana prob rad o h ipnaion of h inpu rror hod and is nraiaion is h nssi o drin h sias of sina driais. This is ahid b appin diffrnia firs ani and Cdro 007. Fi.. Bo diara of h diffrnia fir Th o-pass fir i b rsponsib firs for rduin h sina spru and sond for orrin h hararisis of h diffrn quoin in h ran of o frqunis. Thus, h fir i b ad a o-pass orrion fir. Th dsird ransfr funion of h o-pass fir is: H Ω H Ω for Ω Ω H or Ω 0 for Ω>Ω. As a rsu, h ransfr funion of h sris onnion of h diffrn quoin and h o-pass fir in h ran of o frqunis i b qua o h ransfr funion of an ida diffrnia fir. Th ransfr funion of h o-pass fir for Ω Ω is qua

2 s Cdro Idnifiaion of an ria Drin Manipuaor sin h Diffrnia Firs Inpu rror Mhod o: Hor Ω H Ω/ H Ω Ω/sinΩ Ω / osω Ω / sinω sinω. Th fir ipus rspons is h inrs Fourir ransfor of is frqun hararisi, hus: h Ω or n Hor Ω Ω Ω n dω. 5 nforuna, inra 5 anno b prssd b ans of h anai funions. I nds o b drind usin so approiaion. B pandin funion Ω ino a Taor sris around h au Ω0, obain: Ω O Ω Ω Hor Ω O Ω Ω O Ω. Th four-r approiaion of h pansion appars o b fair suffiin. Th inrs Fourir ransfor of h funion obaind b rin h rs of h hihr ordrs is qua o: h or nω os Ω n n n Ω n sin nω nω os Ω n n n n n Ω sin Ω n nω os nω n n n Ω sin nω. Assu ha h ipus rspons of h o-pass diffrnia fir is: hd n hor n WHarris n, 8 χ Whr is Harris indo dsribd b h fooin quaion: W Harris n os n/ M 0. osn/ M 0.0osn/ M. Th parar shoud b sd in suh a a ha h sop of h hararisi of h fir bin dsind a poin Ω 0 b h sa as ha of h ida diffrnia quaion, hus: H d Ω H Ω Ω Ω! A MATHMATICA MOD OF A OBOT MANIPATO In h n sions, h fooin probs i b sod: firs, i dri h quaions for h DC oors, hn, i dfin h ini and ponia nr of h ss, and fina, i sboia dri h robo dnai quaions, usin h sond ordr aran quaions. Fi.. An ria-drin anipuaor [ ] dno h or of oin ariabs ain as nraid oordinas, h ass, h ar nh, h disan fro h nr of rai and h oor of h in. sin pia quian diaras of DC oors aaiab in h iraur,.. Koa, 00, and h sond Kirhoff a, an ri h fooin ria quaion of h DC oor:, for,, hr is h oa suppid o h roor. Sin an opn-oop ss a b diffiu o onro, i is ssnia ha h idnifiaion b prford for a osd-oop ss ih propr sd PD onrors. us assu ha h quaions of h onrors ha h fooin for: K K, p hr:, h parars of h onrors, h onro sinas, h ariabs dsribin h posiion of h anipuaor ars. Th oa drops aross h roor indin rsisan and induan ar: d i, and di, d hr is h quian roor indin rsisan, is h quian roor indin induan, and is h urrn foin hrouh h roor indins. Th rooi induan for is:

3 aa hania auoaia, o. no. 0 5, 5 hr is an rooi onsan. Subsiuin h subsqun oponns o q., obain: ] [ K K i d di d p, for,,. Th roor orqu is: i M s, 7 hr is a hania onsan. us dfin h anipuaor ini and ponia nr. Th fooin oria raions a pa: os os, os os os os, sin os, sin os sin os, 8 sin, sin sin. Th oi of h nr of rai of h sond ar of h anipuaor is:,. 9 Thus, h ini nr of h ss is:, 0,,,,, hr ar ons of inria of h robo ars assud for a unifor ba. Th ponia nr of h ss is:,, sin, sin sin hr is h araion of rai. sin h prssions for h ini and ponia nr, obain o sond-ordr aran quaions: s M d d, for,,. Afr subsiuion and sipifiaion of a h ariabs, ha a ss of hr quaions hr:,,, : os os os os sin os os sin sin sin os os sin sin sin os os os sin os os os os os sin os os sin sin sin os os os 8 os sin os sin os sin sin os sin os os sin sin sin os os os os os sin sin sin os sin sin sin sin os sin os os os os os os os sin os os os sin os

4 s Cdro Idnifiaion of an ria Drin Manipuaor sin h Diffrnia Firs Inpu rror Mhod. SIMATION This sion disusss h rsus of a siuaion of osdoop quaions inudin a robo od ih PD onrors Fi.. Th od daa i hn b usd in h idnifiaion aorih. Firs, h pr-drind sina as dfind: [ ]. Th sina as assud o b a propr dad sp funion ah ar ih a diffrn da passin hrouh an addiiona o-pass fir ih a boundar frqun Ω 0.05 [rad/s]. Th firin as rsponsib for iiin h sina spru. Fi.. sponss of,, Th rsponss ar no saisfaor fro h poin of i of ruaion. Th ai of h sud as o nra sinas o b usd in h idnifiaion pross. I is adisab ha h prdrind sinas and h onror parars b arfu sd so ha h sinas proid suffiin inforaion abou h ob dnais. 5. IDNTIFICATION WITHOT MASMNT NOIS us ra ha h robo ass and ar nh ar h unnon parars dnod as [,,,,,,,, ]. Th hod usd for h parar idnifiaion is rprsnd raphia in Fi.. I is assud ha h asurn daa onrnin h raoris of h nraid ariabs and h nssar inpu sinas ar aaiab. Th sia of h inpu sinas,, is drind basin on h urrn sias of h ob parars [,,,,,,,, ]. Ths quaions ha h sa sruur as q. ;, h unnon parars, ar rpad b h sias, h nraid ariabs ar rpad b ariabs fird hrouh a o-pass fir, and hir driais hih ar no asurd ar rpad b hir sias obaind b usin ran diffrniain firs. us assu ha h boundar frqun of h diffrniain firs is: Ω 0. [rad/s]. Th idnifiaion rquirs drinin h sias of h parars rsponsib for h quai faor iniiaion. T f f ˆ θ ˆ τ τ d, T 0 hr is an inpu sina fird ih a o-pass fir. Th idnifiaion produr is ond for h fooin iniia aus: [99.8, 5.7, 9.9, 0.9,.0, 0.75,.95,.95,.05]. Th fina aus of h parars ar drind afr 5 iraions of h iniiaion aorih. Th sias [7., 50., 9.95, ,.000, ,.00,.999,.000] sih dpar fro h ra aus of h parars, [00, 50, 50, 0.5,, 0.7,,, ].. IDNTIFICATION AND MASMNT NOIS In his poin i ain ho far h aborad firs iina h asurn and quaniaion nois Moa and ohr 007; abinr and God 975. W i aso ain h infun of h asurn and quaniaion nois on h rsu of idnifiaion pross ih h us of fini ns diffrniaion hod and aborad firs. Th sina prossin hor opriss aiiis aid on sion of subsania inforaion on h aind phnona and iinaion of rdundan inforaion. I is oon non ha h asurd sinas onain oponns rsuin fro h disurbans. In our as h quaniaion nois au is onnd dir ih h nubr of bis of h n-bi A/D onrr ons 999; Pinon and Shouns 990. sin h sa idnifiaion hod and aborad firs fooin parars ha bn obaind for h nois sina n [9.705, 9.78, , 0.50, 0.99, ,.005,.99,.000], n θˆ [.87, 5.0, 8.5, -.05,.007, ,.99,.9787,.008]. sin h fini ns hod fooin parars ha bn obaind for h nois sina n [-.7*0^7, -.9*0^, , , , , 5.75,.77,.599]. Coparin h obaind rsus an sa ha h diffrnia firs iina h asurn nois in a aor dr and h parars drind in h idnifiaion pross ar os o h aua ons. Tradiiona diffrniaion dos no nsur nois iinaion and h idnifid parars diffr sinifian fro h aua ons. sin h aborad firs in idnifiaion hods obain drind parars in as of quaniaion on h of -bi ards. 7. CONCSIONS In onras o h onniona oupu rror hod, hih inos oparin and siain inpu sinas, h inpu rror hod is onsidrab fasr. Th idnifiaion produr dos no rquir soin a sris of diffrnia quaions in ah iraion of h aorih iniiin h quai faor. I shoud b nod ha h spru of h pr-drind sinas is iid. In spi of h fa ha h robo ss is a non-inar ss, h fooin raionship is obaind for h fird sinas: if. As h sih diffrns ar du o h ss non-inari and quaniaion rrors, h quaion an b sod approia. aborad diffrnia firs ha o-pass harar. This faur nabs roin of hih-frqun oponns of h

5 aa hania auoaia, o. no. 0 sina, for ap h nois. Diffrnia firs nsur drinin of appropria driais of sina ih rrors far or ss han sip diffrniaion hods, ha pas pariuar iporan ro in h idnifiaion pross. In arious auaions hih ha bn prford, propr opraion of h hod for or opiad hania sss and for sss of rar nubr of idnifid parars has bn sad. FNCS. Cdro., ani D. 009, Diffrnia firs and h idnifiaion of a anipuaor usin Mahaia sofar, XXXIV. Sinar AS '009 Insruns and Conro, Osraa, ISBN Cdro., ani D. 0, Mod parar indnifiaion ih noninar parariaion appid o a anipuaor od, Monoraphi sris of pubiaions-copur sin in h a of XXI nur, ISBN , ISBN ani D., Cdro. 007, Diffrnia Firs Wih Appiaion To Ss Idnifiaion, 7h uropan Confrn of Youn sarh and Sin Worrs in Transpor and Touniaions TANSCOM 007, śiina, Soaia, p. 5.. Koa. 00, Fundanas of onro ninrin. Vo. II, WND, Kraó. 5. ons. G.999, An inroduion o diia sina prossin in Poish, WKiŁ, Warsa.. Moa., ania I. I, ia M., Busin D. 007, Th s of Fraiona Diffrniaion or Inraion for Sina Ipron, Masurn Sin i, Vou 7, Sion, No Pinon., Shouns. 990, a-ti Inraion and Diffrniaion of Anao Sinas b Mans of Diia Firin, I Transaions on Insrunaion and Masurn. Vo. 9. No.. 8. abinr.. and God B. 975, Thor and Appiaion of Diia Sina Prossin, Prni-Ha, In.: nood Ciffs, N. 7