Prprnts of th 8th IFAC World Congrss Mlano (Italy) August 8 - Sptmbr, Rconfgurabl UKF for In-Flght Magntomtr Calbraton and Atttud Paramtr Estmaton Hall E. Son*. Chngz Hajyv.** * h Graduat Unvrsty for Advancd Studs (SOKENDAI), Dpartmnt of Spac and Astronautcal Scnc 3--, Yoshnoda, Sagamhara, Kanagawa, JAPAN (l: +8 4 759 83; -mal:rsn_son@ac.jaa.jp). ** Istanbul chncal Unvrsty Aronautcs and Astronautcs Faculty, 34469 Masla, Istanbul, URKEY (-mal: cngz@tu.du.tr). Abstract: In ths study a rconfgurabl unscntd Kalman fltr (UKF) basd algorthm for th stmaton of magntomtr bass and scal factors s proposd as a part of th atttud stmaton schm of a pco satllt. Algorthm s composd of two stags; n frst stag UKF stmats magntomtr bass and scal factors as wll as s atttud paramtrs of th satllt. Dffrntly from th stng algorthms, scal factors ar not tratd togthr wth th othr paramtrs as a part of th stat vctor; thr scal factors ar stmatd va a nwly proposd tnson for th UKF. Aftr a convrgnc rul for th bass scond stag starts and th UKF rconfgurs tslf for th stmaton of only atttud paramtrs. At ths stag fltr rgards th bass and scal factors stmatd at th ntal stag. Proposd algorthm s smulatd for atttud stmaton of a pco satllt whch has thr magntomtrs and thr rat gyros as masurmnt snsors. Kywords: Atttud algorthms, Calbraton, Kalman fltrs, Estmaton paramtrs, Satllt applcatons. INRODUCION Bcaus of advantags such as provdng contnuously avalabl two-as atttud masurmnts; rlatv low cost and almost nsgnfcant powr dmand, opratng magntomtrs as prmary snsor n small satllt mssons s a common mthod for achvng atttud nformaton. Howvr, ths snsors ar not rror fr bcaus of th bass, scalng rrors and msalgnmnts (nonorthagonalty). hs trms nhbt th fltr ffcncy and so atttud data accuracy and vn thy may brng about th fltr dvrgnc n long trms. h atttud accuracy rqurmnts dmand compnsaton for th magntomtr rrors such as msalgnmnts and bass (Wrtz, 988). Estmatng magntomtr bass and scal factors as wll as th atttud of th satllt s a proposd tchnqu to solv such problms and ncras on-board accuracy. In ltratur thr ar svral mthods for stmatng th magntomtr bas whn th atttud nowldg s not avalabl (.g. at th njcton phas whr th spaccraft s spnnng rapdly). As a rlatvly nw mthod, twostp algorthm, whch was proposd n ordr to ovrrun quartc cost functon problm mt durng atttud fr bas stmaton procss, may b gvn as an ampl (Alonso and Shustr, a). In (Alonso and Shustr, 3) t s also shown that twostp algorthm can b also mplmntd n cas of ncomplt obsrvablty of th magntomtr bas vctor. Morovr (Alonso and Shustr, b) uss algorthm to stmat scal factors and nonorthogonalty corrctons as wll as magntomtr bass. On th othr hand, Km and Bang ntgrats twostp algorthm wth th gntc algorthm, from whch th ntal stmats of th magntomtr bas stmaton ar provdd (Km and Bang, 7). wostp algorthm l mthods can b usd for gussng an ntal stmat for Kalman fltr typ stochastc atttud dtrmnaton mthods. hos a pror stmats for th stmator may b thn corrctd on an pandd stat vctor ncludng atttud paramtrs and bass. As anothr ampl for atttud-fr bas stmaton procdur, n (Crassds t al., 5; Huang and Jng, 8) t s provd that a full magntomtr calbraton can b prformd on-orbt durng typcal spaccraft msson-mod opratons by th us of ral-tm algorthms basd on both th Etndd Kalman Fltr (EKF) and Unscntd Kalman Fltr (UKF). Although ths studs can stmat th magntomtr charactrstcs such as bass, scal factors and nonorthogonalty corrctons, thy all dsrgard th atttud dynamcs of th satllt and f ths nformaton s possbl at any nstant, thr accuracs can b cdd. Hnc thy may b consdrd as a part of spaccraft mods whr atttud data s absnt. In (Styn, 995) both th magntomtr bass and scal factors ar stmatd by rcursv last squars mthod. h vctor of unnown paramtrs s formd of magntomtr bass and scal factors. Mthod s not atttud fr snc th atttud nowldg s ncssary to form th atttud matr. Howvr as a drawbac, usd atttud paramtrs ar not stmatd by th mthod tslf and th spaccraft dynamcs ar not tan nto consdraton n ths mthod. On th othr hand, a fw studs l (DaForno t al., 4; Ma, 5), whch handl magntomtr calbraton procss togthr wth th stmaton of atttud paramtrs, also do st. In (DaForno t al., 4) EKF s usd as a part of th stmaton schm. Va an UKF basd stmaton algorthm, whch has advantags ovr EKF such as absnc of Jacoban calculatons tc., ts accuracy may b surmountd. Ma and Jang solv magntomtr calbraton and atttud stmaton problm va two UKFs worng synchronously. As a Copyrght by th Intrnatonal Fdraton of Automatc Control (IFAC) 74
Prprnts of th 8th IFAC World Congrss Mlano (Italy) August 8 - Sptmbr, dsadvantag, ths approach rqurs a hgh computatonal ffort bcaus of two dstnct UKFs and t may b not sutabl for pco satllts whr th procssng capacty of th atttud computr s lmtd. In ths study a rconfgurabl unscntd Kalman fltr (UKF) basd algorthm s proposd for magntomtr calbraton and atttud paramtr stmaton. Algorthm s composd of two stags; n frst stag UKF stmats magntomtr bass and scal factors as wll as s atttud paramtrs of th satllt. Dffrntly from th stng algorthms, scal factors ar not tratd togthr wth th othr paramtrs as a part of th stat vctor; thr scal factors ar stmatd va a nwly proposd tnson for th UKF. At th scond stag whch starts aftr a convrgnc rul for th bas stmatons UKF rconfgurs tslf and rgardng th stmatd bass and scal factors, t runs for stmaton of only atttud paramtrs. Proposd algorthm s smulatd for atttud stmaton of a pco satllt whch has thr magntomtrs and thr rat gyros as masurmnt snsors.. MAHEMAICAL MODEL OF HE PICO SAELLIE If th nmatcs of th pco satllt s drvd n th bas of Eulr angls, thn th mathmatcal modl can b prssd wth a 9 dmnsonal systm vctor whch s mad of atttud Eulr angls (ϕ s th roll angl about as; θ s th ptch angl about y as; ψ s th yaw angl about z as) vctor, th body angular rat vctor wth rspct to th nrtal as fram, and th vctor formd of bas trms of magntomtrs. Hnc, For th frst stag whr bas trms of thr magntomtr ar stmatd; = ϕ θ ψ ω ω ω, y z bm b m b y m () z For th scond stag only atttud paramtrs ar stmatd; = ϕ θ ψ ω ω ω, () y z Hr subscrpt m for bas trms rprsnts magntomtr. Also for consstncy wth th furthr planatons, th body angular rat vctor wth rspct to th nrtal as fram should b statd sparatly as; ω = ω ω ω, BI y z whr ω BI s th angular vlocty vctor of th body fram wth rspct to th nrtal fram. Bsds, dynamc quatons of th satllt can b drvd by th us of th angular momntum consrvaton law; ω BI J = Ngg ωbi ( JωBI ), (3) dt whr J s th nrta matr conssts of prncpal momnts of nrta as J = dag ( J, J y, J z) and N gg s th vctor of gravty gradnt torqu affctng th satllt as (Shavat t al., 7); N ( J ) y Jz A3 A gg 33 μ Ngg 3 3 ( ) y = Jz J A3 A33. (4) r N gg ( J Jy) A3 A3 z Hr μ s th gravtatonal constant, r s th dstanc btwn th cntr of mass of th satllt and th Earth and A j rprsnts th corrspondng lmnt of th drcton cosn matr of; c( θ) c( ψ) c( θ) s( ψ) s( θ) ( ϕ) ( ψ) ( ϕ) ( θ) ( ψ) ( ϕ) ( ψ) ( ϕ) ( θ) ( ψ) ( ϕ) ( θ). (5) ( ϕ) ( ψ) + ( ϕ) ( θ) ( ψ) ( ϕ) ( ψ) + ( ϕ) ( θ) ( ψ) ( ϕ) ( θ) s ar th cosns and snus functons A= c s + s s c c c + s s s s c s s c s c s c c s s c c In matr A, ( ) c and ( ) succssvly. Knmatc quatons of moton of th pco satllt wth th Eulr angls can b gvn as ϕ s( ϕ) t( θ) c( ϕ) t( θ) p θ ( ϕ) ( ϕ = c s ) q. (6) ψ ( ϕ) / ( θ) ( ϕ) / ( θ) s c c c r Hr t ( ) stands for tangnt functon and p, q and r ar th componnts of ω BR vctor whch ndcats th angular vlocty of th body fram wth rspct to th rfrnc fram. ω BI and ω BR can b rlatd va, [ ] BR = BI + A. (7) ω ω ω whr ω dnots th angular vlocty of th orbt wth 3 rspct to th nrtal fram, found as = ( r ) / ω μ/. 3. MEASUREMEN SENSOR MODEL 3. h Magntomtr Modl As th satllt navgats along ts orbt, magntc fld vctor dffrs n a rlvant way wth th orbtal paramtrs. If thos paramtrs ar nown, thn, magntc fld tnsor vctor that affcts satllt can b shown as a functon of tm analytcally (Shavat t al., 7). Not that, ths trms ar obtand n th orbt rfrnc fram. M H() t = { cos( ωt) 3 cos( ε) sn( ) sn( ε) cos( ) cos( ωt) r sn( ωt) sn( ε) sn ( ωt )} (8) M H () t = 3 cos( ε) cos( ) + sn ( ε) sn ( ) cos ( ωt), (9) r M H3() t = { sn( ωt) 3 cos( ε) sn ( ) sn ( ε) cos( ) cos( ωt) r sn( ωt) sn( ε) sn( ωt )} () Hr M s th magntc dpol momnt of th Earth as 5 M = 7.943 Wbm., μ s th Earth Gravtatonal 4 3 constant as μ = 3.986 m / s, s th orbt nclnaton as = 97, ω s th spn rat of th Earth as 5 ω 7.9 rad / s =, ε s th magntc dpol tlt as ε =.7 and r s th dstanc btwn th cntr of mass of th satllt and th Earth as r = 6,98,4 m. hr onboard magntomtrs of pco satllt masurs th componnts of th magntc fld vctor n th body fram. hrfor for th masurmnt modl, whch charactrzs th masurmnts n th body fram, gand magntc fld 74
Prprnts of th 8th IFAC World Congrss Mlano (Italy) August 8 - Sptmbr, trms must b transformd by th us of drcton cosn matr, A. Ovrall masurmnt modl may b gvn as; H ( ϕθψ,,, ) ( ) t H t S Hy ( ϕ, θψ,, t) = A H() t + bm + η, () ( ϕθψ,,, ) 3 () Hz t H t whr, H () t, H ( t) and H3 ( t ) rprsnt th Earth magntc fld vctor componnts n th orbt fram as a ϕθψ,,, H ϕθψ,,, t and functon of tm, and H ( t ), y ( ) ( ϕθψ,,, ) H z t show th masurd Earth magntc fld vctor componnts n body fram as a functon of tm and varyng Eulr angls. Furthrmor, S s th dagonal scal matr as, S = dag( s s, s ), (), 33 b m s th magntomtr bas vctor as b = m bm bm y b m z and η s th zro man Gaussan wht nos wth th charactrstc of E ηη j = I 33σmδj. (3) I33s th dntty matr wth th dmnson of 3 3, σ m s th standard dvaton of ach magntomtr rror and δ j s th Kroncr symbol. 3. h Rat Gyro Modl hr rat gyros ar algnd through thr as, orthogonally to ach othr and thy supply drctly th angular rats of th body fram wth rspct to th nrtal fram. Hnc th modl for rat gyros can b gvn as; ωbi, mas = ωbi + η. (4) whr, ω BImas, s th masurd angular rats of th satllt, and η s th zro man Gaussan wht nos wth th charactrstc of E η η j = I 33σδ g j, (5) Hr, σ g s th standard dvaton of ach rat gyro random rror. 4. UKF BASED BIAS AND SCALE FACOR ESIMAION ALGORIHM 4. Unscntd Kalman Fltr In ordr to utlz Kalman fltr for nonlnar systms wthout any lnarzaton stp, th unscntd transform and so Unscntd Kalman Fltr s on of th tchnqus. UKF uss th unscntd transform, a dtrmnstc samplng tchnqu, to dtrmn a mnmal st of sampl ponts (or sgma ponts) from th a pror man and covaranc of th stat. hn, ths sgma ponts go through nonlnar transformaton. h postror man and th covaranc ar obtand from ths transformd sgma ponts (Julr t al., 995). As t s statd, UKF procdur bgns wth th dtrmnaton of n + sgma ponts wth a man of ˆ ( ) and a covaranc of P( ). For an n dmnsonal stat vctor, ths sgma ponts ar obtand by ( ) = ( ) (6) ˆ ( ) ( κ ) ( ) = ( ) + ( + ) ( ) + ( ) ˆ n κ P Q (7) ( ) ˆ + = ( ) ( + ) ( ) + ( ) n n P Q, (8) whr, ( ), ( ) and + n( ) ar sgma ponts, Q( ) s th procss nos covaranc matr, n s th stat numbr and κ s th scalng paramtr whch s usd for fn tunng and th hurstc s to chos that paramtr as n + κ = 3 (Julr t al., 995). Also, s gvn as = n. Nt stp of th UKF procss s transformng ach sgma pont by th us of systm dynamcs, ( + ) = ( ), f. (9) hn ths transformd valus ar utlzd for ganng th prdctd man and th covaranc (Crassds and Marly, 3; Son and Hajyv 9). ˆ ( ) + = κ ( + ) + n ( + ), () n + κ = { P( + ) = κ ( + ) ˆ( + ) ( + ) ˆ( + ) + κ n + n ( + ) ˆ( + ) ( + ) ˆ( + ). () = Hr, ˆ ( + ) s th prdctd man and P( + ) s th prdctd covaranc. Nonthlss, prdctd obsrvaton vctor s, ˆ ( ) + = κ ( + ) + n y y y ( + ), () n + κ = whr, y ( + ) = ( + ), ( ), h v. (3) Aftr that, obsrvaton covaranc matr s dtrmnd as, { P ( + ) = κ ( + ) ˆ( + ) ( + ) ˆ( + ) yy + κ y y y y n + n y ( ) ˆ( ) ( ) ˆ( ) y y y, (4) = whr nnovaton covaranc s P + = P + + R + (5) vv ( ) yy ( ) ( ) Hr v ( ) s th wht Gaussan masurmnt nos and t s non-corrlatd wth th procss nos. Bsds, R( +) s th masurmnt nos covaranc matr. h cross corrlaton matr can b obtand as, { P ( + ) = κ ( + ) ˆ( + ) ( + ) ˆ( + ) y + κ y y n + n ( ) ˆ( ) ( ) ˆ( ) y y. (6) = 743
Prprnts of th 8th IFAC World Congrss Mlano (Italy) August 8 - Sptmbr, Followng part s th updat phas of UKF algorthm. At that phas, frst by usng masurmnts, y( +), rsdual trm (or nnovaton squnc) s found as ( + ) = y ( + ) y ˆ ( + ), (7) and thn Kalman gan s computd va quaton of, ( ) y ( ) vv ( ) K + = P + P +. (8) At last, updatd stats and covaranc matr ar dtrmnd by, ˆ + + = ˆ + + K+ +, (9) ( ) ( ) ( ) ( ) ( + + ) = ( + ) ( + ) ( + ) vv ( + ) Hr, ˆ ( + + ) P( + + ) s th stmatd covaranc matr. P P K P K. (3) s th stmatd stat vctor and 4. Scal Factor Estmaton Whn th condton of th masurmnt systm dos not corrspond to th modl usd n th synthss of th fltr that mans th ncomng masurmnts ar scald and thy nd to b r-scald aftr stmaton of scal factors. If th systm oprats normally, th ral and th thortcal nnovaton covaranc matr valus match as n (3). ( + ) ( + ) ( + ) + ( + ), μ = Pyy R (3) + w j μw hr, μ w s th wdth of th movng wndow. Howvr, whn th ral masurmnts ar scald as n () th ral rror wll cd th thortcal on. In ths cas th matr bult of scal factors, S( ), s addd nto th algorthm as, ( + ) ( + ) = ( + ) + ( ) ( + ) ( ). μ = Pyy SR S (3) + w j μw hn, snc t s now that S( ) s dagonal (thr n no msalgnmnt/nonorthagonalty for magntomtrs) and also R( +) s dagonal by ts natur, t s possbl to rwrt (3), ( + ) ( + ) = ( + ) + ( ) ( ). μ + = Pyy S R (33) + w j μw Fnally, scal factor matr can b found as gvn blow, S( ) = ( + ) ( + ) Pyy ( + ) R ( + ). (34) μ w j= μw+ 4.3 Stoppng Rul for Bas Estmaton h followng stoppng rul may b ntroducd for th bas stmaton problm (Hajyv, 994; ): r = (bˆ -b ˆ ) D (bˆ - b ˆ ) = ε, (35) - - Δb - whr D Δb s th covaranc matr of th dscrpancy btwn two succssv bas stmats b ˆ and ˆb -, and ε s a prdtrmnd small numbr. Snc Kalman fltr stmats th paramtrs from a squnc of obsrvatons wth Gaussan tolrancs of th masurmnt rrors and systm nos, th fltr ylds an stmat wth an pctd valu qual to th stmatd quantty and a Gaussan dstrbuton functon. h dscrpancy bˆ -b ˆ - thn has a normal dstrbuton as wll, snc t s a lnar combnaton of two Gaussan random varabls. Wth ths consdratons n mnd t s nown that th statstc r has a χ dstrbuton wth n dgrs of frdom (n s th numbr of dmnsons of th bas vctor b ), and th thrshold valus of r can b found by dtrmnng th tabulatd valus of th χ dstrbuton for a gvn lvl of sgnfcanc. It s vdnt from rlaton (35) that th smallr th valu of r, th gratr wll b th consstncy of th stmats. Usually n th tstng of consstncy n such cass th lowr lmt of th confdnc ntrval must b qual to zro, and th uppr lmt s dtrmnd by th lvl of sgnfcanc α. o tst th consstncy of th stmats, th lvl of sgnfcanc α s adoptd, whch corrsponds to th confdnc coffcnt β = α. h thrshold χ β s spcfd n trms of ths probablty, usng th dstrbuton of th nvstgatd statstc r : { β } P χ < χ = β,< β <. (36) h stmaton procss s stoppd whn r < χ β, snc furthr obsrvatons yld nsgnfcant mprovmnt of th dntfd modl. If th quadratc form r s largr than or qual to th spcfd thrshold χ β, stmaton should b contnud. Nonthlss computaton of th covaranc matr of th dscrpancy btwn two succssv bas stmats b ˆ and ˆb - s nvstgatd n (Hajyv, ). h followng prsson for th covaranc matr D Δb s obtand n ths wor; D = P P. (37) Δb Dscrbd stoppng rul can b usd to ma a tmly dcson to stop th bas stmaton procss and t dos not rqur larg computatonal pndturs. 5. SIMULAIONS Smulatons ar ralzd for 5 sconds wth a samplng tm of Δ t =.sc. As an prmntal platform a cubsat modl s usd and th dagonal trms of th nrta matr s tan as J =. 3, J =. 3 y, J =.9 3 z, whr all non-dagonal trms ar zro snc th rotaton s about th 744
Prprnts of th 8th IFAC World Congrss Mlano (Italy) August 8 - Sptmbr, prncpal as of th satllt. Nonthlss th orbt of th satllt s a crcular orbt wth an alttud of ra = 55m. Othr orbt paramtrs ar sam as t s prsntd n th scton for th Magntomtr Modl (Scton 3.). Smulatons can b catgorzd n two: Estmaton of scal factors, bass and atttud paramtrs wthout stoppng rul for th fltr (wthout UKF rconfguraton) and stmaton ralzd by an UKF wth stoppng rul (wth UKF rconfguraton). Rsults gand va ths two dstnct smulaton scnaros ar compard n ordr to clarfy th ffctvnss of th proposd rconfgurabl UKF basd mthod. Frstly, va a sngl UKF algorthm t s possbl to calbrat th magntomtrs. In ths algorthm, stat vctor s formd of 9 stats; Eulr angls, angular rats and magntomtr bass. As wll scal factors ar stmatd va (34). Bas stmaton ampl s gvn n Fg.. As t s apparnt, bas trms ar stmatd accuratly. bz & bz a (tsla) rror(tsla) varanc (tsla ) 3.8 4-6 Magntomtr bas bz Estmaton Kalman Estmaton Actual Valu 3.6 5 5 5 3 35 4 45 5-7 - 5 5 5 3 35 4 45 5 5-4 5 5 5 3 35 4 45 5 tm(sc) Fg.. Estmaton of th bas for magntomtr algnd through z as. Bsds, n smulatons, scal factors ar mplmntd to th magntomtr masurmnts n ordr to smulat stmaton procss ( S ). In rturn, scal factors gvn at th rght-hand sd ( Ŝ ) ar stmatd va th proposd mthod (gvn rsults ar th man of scal factor stmaton sampls gand tll th bas stmaton stoppng nstant):..4 S =.3 Sˆ.356 =.5.55 As sn, stmaton rror for th scal factors s not hgh. Dffrnc btwn th actual valu and th Kalman stmaton s crtanly causd by th random procss wht nos of th masurmnts. On th othr hand, t s possbl to stmat also all othr paramtrs accuratly va ths algorthm. As sn n Fg. atttud paramtr stmaton rrors ar n accptabl lmts for a pco satllt. Not that smlar rsults hav bn obtand for all othr paramtrs. thta(dg) rror(dg) Ptch Angl Estmaton Kalman Estmaton Actual Valu 5 5 5 3 35 4 45 5.5 -.5 5 5 5 3 35 4 45 5 varanc (dg ) 4-5 5 5 5 3 35 4 45 5 tm(sc) Fg.. Ptch angl stmaton va UKF wthout rconfguraton. It s obvous that wth an UKF algorthm t s possbl to stmat all 9 paramtrs of th stat vctor () and also th scal factors prcsly. Howvr, a 9 dmnsonal stat vctor and th addtonal part to UKF for scal factor stmaton dmand a hgh computatonal ffort. In gnral, Kalman fltr s computatonal burdn ncrass n a drct rlaton wth th cub of th numbr of stats. Hnc, consttutng a stat vctor wth 9 stats mans mor computatonal burdn than a stat vctor wth 6 stats. hat s an mportant rdundancy for consdrably lmtd atttud computr of a pco satllt. As a rsult t s bttr to rconfgur fltr rgardng th scal factors and bass of magntomtrs and to procd th stmaton procdur by only stmatng Eulr angls and angular rats. As gvn n scton 4.3, a stoppng rul s proposd n ordr to rconfgur UKF aftr convrgnc of th bas stmatons to th actual valus. In that scond stag, prvously stmatd magntomtr bas trms and scal factors ar tan nto account by UKF algorthm and numbr of stats to b stmatd rducs to 6 as shown wth (). Ptch angl stmaton rsult s gvn n Fg.3. thta(dg) rror(dg) Ptch Angl Estmaton Kalman Estmaton Actual Valu 5 5 5 3 35 4 45 5.5 -.5 5 5 5 3 35 4 45 5 varanc (dg ) -5 Bas Estmaton Stoppng Instant 5 5 5 3 35 4 45 5 tm(sc) Fg. 3. Ptch angl stmaton va UKF wth rconfguraton. 745
Prprnts of th 8th IFAC World Congrss Mlano (Italy) August 8 - Sptmbr, Hr n Fg. 3 rconfguraton occurs at about 8 th scond and t s mard on fgur wth dashd ln. As t s apparnt from th stmaton varanc, rconfguraton brngs about an nhancmnt to fltr accuracy. So as to undrstand th ffct of rconfguraton mor clarly, absolut valus of rror ar tabulatd for two fltrs; UKF wth and wthout rconfguraton (abl ). abl. Comparson of Absolut Valus of Error for UKFs wth and wthout rconfguraton. Abs. Valus of Err. for UKF wth Rconfguraton Abs. Valus of Err. for UKF wthout Rconfguraton Par. s. 4s. s. 4s. ϕ(dg).48.673.44.45 θ(dg).5.3.44.93 ψ (dg).5.85.43.58 ω (dg/s).4..3. ωy (dg/s).3..7. ωz (dg/s).9...7 As sn from tabl rducng th sz of th stat vctor wthout any chang at masurmnt vctor brngs about an ncrmnt n th accuracy of stmaton of th stats. Nonthlss, a smallr stat vctor mans lowr computatonal burdn and rs for th dvrgnc of th fltr. 6. CONCLUSIONS In ths study a rconfgurabl unscntd Kalman fltr (UKF) basd algorthm s proposd for magntomtr calbraton and atttud paramtr stmaton. Smulaton rsults show that, t s possbl to stmat both, magntomtr bass and scal factors as wll as th atttud paramtrs va th proposd algorthm. Nvrthlss, proposd rconfgurabl UKF algorthm s advantagous bcaus smulaton rsults provs that rducng th sz of th stat vctor wthout any chang at masurmnt vctor brngs about an ncrmnt n th accuracy of stmaton of th stats. Also a smallr stat vctor mans lowr computatonal burdn and rs for th dvrgnc of th fltr. Snc magntomtr utlzaton has sgnfcanc spcally for pco satllt mssons, proposd algorthm may affct th msson prformanc and rlablty n a consdrabl dgr. ACKNOWLEDGMEN hs wor was supportd n part by UBIAK (h Scntfc and chnologcal Rsarch Councl of ury) undr Grant 8M53. REFERENCES Alonso, R. and M.D. Shustr (a). wo Stp: A Fast Robust Algorthm for Atttud- Indpndnt Magntomtr-Bas Dtrmnaton. h Journal of th Astronautcal Scncs, Vol.5, No.4, 433-45. Alonso, R. and M.D. Shustr (b). Complt lnar atttud-ndpndnt magntomtr calbraton. h Journal of th Astronautcal Scncs, Vol.5, No.4, 477-49. Alonso, R. and M.D. Shustr (3). Cntrng and Obsrvablty n Atttud-Indpndnt Magntomtr- Bas Dtrmnaton. h Journal of th Astronautcal Scncs, Vol.5, No., 33-4. Crassds, J.L. and F.L. Marly (3). Unscntd Fltrng for Spaccraft Atttud Estmaton. Journal of Gudanc, Control and Dynamcs, Vol.6, No.4, 536-54. Crassds, J.L., K. La, and R.R. Harman (5). Ral-tm atttud-ndpndnt thr-as magntomtr calbraton. Journal of Gudanc, Control, and Dynamcs, Vol.8, No., 5-. DaForno R. and t al. (4). Autonomous navgaton of mgsat : Atttud, snsor bas and scal factor stmaton by EKF and magntomtr-only masurmnt. Proc. nd AIAA Intrnatonal Communcatons Satllt Systms Confrnc and Ehbt, Calforna, USA. Hajyv, Ch.M.(994). Som Problms of Dynamc Systms Paramtrcal Idntfcaton. In Dgst of Frst Asan Control Confrnc, July 7-3 (994), Japan,. Vol.: 9-94. Hajyv, Ch.M.(). Stoppng ruls formaton and faults dtcton n paramtrc dntfcaton problms. IMchE Procdngs, Part I, J.Systms & Control Engnrng, Vol.5 (I4): 357-364. Huang, L. and W. Jng (8). Atttud-ndpndnt gomagntc navgaton usng onboard complt thras magntomtr calbraton. Proc. 8 IEEE Arospac Confrnc, Montana, USA, -7. Julr, S.J., J.K. Uhlmann and H.F. Durrant-Whyt (995). A Nw Approach for Fltrng Nonlnar Systms. Procdngs of Amrcan Control Confrnc, Vol.3, 68-63. Km, E. and H. Bang (7). Bas stmaton of magntomtr usng gntc algorthm. Proc. Intrnatonal Confrnc of Control Automaton and Systm 7, Soul, Kora, 95-98. Ma, G.F. and X.Y. Jang (5). Unscntd Kalman Fltr for Spaccraft Atttud Estmaton and Calbraton Usng Magntomtr Masurmnts, Proc. 4 th Intrnatonal Confrnc on Machn Larnng and Cybrntcs, Guangzhou, Chna, 56-5. Shavat P., Q. Gong and I.M. Ross (7). NPSA I Paramtr Estmaton Usng Unscntd Kalman Fltr. Procdngs of 7 Amrcan Control Confrnc, 4445-445, IEEE, Nw Yor, USA. Styn, W.H. (995). A Mult-mod Atttud Dtarmnaton and Control Systm for Small Satllts. Dssrtaton Prsntd for h Dgr of Doctor of Phlosophy, Unvrsty of Stllnbosch, South Afrca. Son, H.E. and Ch. Hajyv (9). UKF for th Idntfcaton of th Pco Satllt Atttud Dynamcs Paramtrs and th Etrnal orqus on IMU and Magntomtr Masurmnts. Proc. 4th Intrnatonal Confrnc on Rcnt Advancs n Spac chnologs, Istanbul, ury, 547-55. Wrtz, J.R. (988). Spaccraft Atttud Dtrmnaton and Control, Kluwr Acadmc Publshrs, Dordrcht, Holland. 746