Micro-Satellite Attitude Determination with Only A Single Horizon Sensor

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1 MAEC Wb of Confrncs 5, 3 8 ( 6 DOI:.5/ matcconf/ C Ownd by th authors, publishd by EDP Scincs, 6 Micro-Satllit Attitud Dtrmination with Only A Singl Horizon Snsor Gaoxiang Ouyang a, Xin Dong, Xin Li Acadmy of Opto-Elctronics, Chins Acadmy of Scincs, Bijing, China Abstract. hrough using masurmnt from only a singl horizon snsor, this papr prsntd a quatrnion-basd 3-axis attitud dtrmination mthod, which can b implmntd on board micro-satllits and applid ovr a whol orbital priod. Firstly, a dscription of attitud rprsntation on th quatrnion is givn. Scondly, a dtaild modling formulation with nadir vctor and masurmnt quations on attitud stimation systm is dmonstratd. Aftrwards, a corrction is mad to liminat th stimation rror rsultd from Earth s oblatnss, and abl to furthr improv th accuracy of th attitud dtrmination algorithm. Finally, a six dgrof-frdom closd-loop simulation is usd to validat th accuracy of th attitud dtrmination mthod givn in this papr. Introduction At prsnt, micro-satllits ar incrasingly applid to diffrnt spac missions, at th sam tim also nd to fac th drastic rstrictions imposd by th Cubsat standard. aing into account powr consumption, wight and othr constraints on micro-satllits, th adoption of only a singl attitud snsor has bcom a practical rquirmnt of th micro-satllits such as Cubsat. If th final targt is to maintain th ovrall attitud accuracy, thn it will b accompanid by hug challngs. In fact, th attitud dtrmination basd on singl-snsor ssntially blongs to a class of th undrmasurmnt problm, th biggst difficulty lis in th lac of information dimnsion masurmnt, which mayb rsult in wa obsrvability of th dynamic systm 3. In attitud information acquisition procss, th attitud masurmnt is oftn achivd by dvics such as sun snsors, star snsors and magntomtrs. Howvr, ths snsors hav clar limitations: sun snsors will los thir functionalitis in priods of clips in orbit, magntomtrs cannot acquir high accuracy attitud masurmnts du to th constantly changing Earth magntic fild, star snsors ar too xpnsiv for micro-satllits. Compard with othr snsors, Earth horizon snsors (EHS hav mrgd as fficint and rlativly inxpnsiv mans of mor prcis attitud dtrmination, capabl of satisfying attitud nowldg rquirmnts of micro-satllits in low Earth orbit (LEO, spcially for missions with Earth-spcific scinc objctivs. Accordingly, this papr proposs a 3- axis attitud dtrmination mthod basd on only a singl scanning horizon snsor. o rduc th ffcts of nois on th masurmnt, it is mor practical for micro-satllits to introduc dynamic attitud stimation and filtring tchniqus, rlying on body masurmnts of nown vctors in th rfrnc fram, and us th dynamical quations of th systm undr a Corrsponding author : ouyanggx@ao.ac.cn 4

2 MAEC Wb of Confrncs considration. In this papr, a quatrnion-basd xtnd Kalman filtr (EKF is dvlopd to stimat th attitud history of LEO satllits using EHS and gyros, manwhil th 3-axis attitud pointing accuracy always rmains within th rror rang of 4 dg. marably, th pointing accuracy of th roll-pitch orintation, which mainly impacts on Earth obsrvation missions, stays within th dg of margin. Hr ar som coordinat systm dfinitions, th Local Vrtical Local Horizontal (LVLH fram dscribs th currnt orbit fram of th satllit, and has its origin at th cntr of mass of th satllit. h coordinat axis z L points towards th cntr of th arth (dirction of th nadir, y L points opposit to th satllit s angular momntum, and x L complts th right-handd triad. h instantanous LVLH fram is usd as rfrnc to masur th local attitud of th satllit. Body fram is an orthogonal coordinat systm fixd to th satllit body with origin at its cntr of mass. Masurmnt fram is nown in advanc, and all masurmnts of th snsor ar xprssd in th masurmnt fram. h symbols ˆx and x dnot th stimat and th cross-product matrix associatd with x, rspctivly. h papr is organizd as follows. Sction provids th modling on systm stat quation and masurmnt quation, and a full xprssion of th nadir vctor in th satllit s body fram. A simpl analytical mthod for Earth s oblatnss corrction is givn in Sction 3. Sction 4 prsnts a six dgr-of-frdom closd-loop simulation, which is usd to obsrv and compar th prformanc of th attitud dtrmination algorithm dvlopd by diffrnt typ of masurmnt rrors and stimators. Finally, Sction 5 summarizs this papr. Quatrnion-Basd EKF Estimator. Systm Modl A svn-lmnt stat vctor, x, consisting of th quatrnion q and th gyro-bias b, is dfind as x q b ( For simply, th angular rat ω m,, as th gyro outputs, is dirctly usd as an altrnativ to attitud dynamics, so th systm of diffrntial quations govrning th stat is dfind as q ( ωm b q ( whr th stat quation Ω is consistnt with dfinition givn in 4, th gyro bias b is assumd to b Gaussian whit nois, and othr nois is omittd. For implmntation of th discrt tim Kalman filtr (KF quations, w nd to discrtiz th abov stat propagation modl ( as follows qˆ ˆ Θ q (3 whr th matrix Θ is givn as ωω ˆ ˆ Θc ωi ˆ ˆ ˆ 33 s ω (c ω (4 whr th common practic of writing s for sin( and c for cos( is adoptd. h xprssion ω ˆ is rprsntd as ( ωm b t, and t is th sampling priod. h symbol I 33 is idntity matrix with thr dimnsions. Bcaus of singularity causd by th unit norm constrain of th quatrnion 4, a six-stat rror modl is usd for propagation of covarianc. Morovr, a multiplicativ rror modl is mployd, thn th discrt procss modl for th rror stat is obtaind as q Θ Ψ q x (5 b 33 I33 b whr th matrix Θ is th sam as (4, and th matrix Ψ is dfind as 38-p.

3 IC4M 6 Ψ ti ˆ 33 ( c ω ( t s ( 3 h covarianc matrix Q of th nois in th discrt tim systm can b approximatd by twoordr aylor sris, according to rfrnc [5]. t Θ Ψ Θ Ψ Q ( Q Q 33 I33 33 I33 whr th Q is th spctral dnsity of th whit nois procss of systm dynamics. (6 (7. Roll/Yaw Masurmnts In th masurmnt configuration givn by this articl, th singl EHS fixd on body y axis dirctly provids pitch axis nowldg, bsids, basd on th gomtrical rlationship, th EHS can provid roll axis attitud information [6], which is calculatd as follows ( so si arcsin( / c s c.5so c.5 si (8 arctan(c / s c.5 si whr th scan angl width ω so and ω si ar dfind th arc lngth of th EHS s scanning axis which crosss from th Earth-in to Earth-out and from Earth-in to baslin rspctivly. Smi-scan angls γ is a nown hardwar paramtr. h Earth dis visibl from th satllit at an altitud h is of angular radius ρ is dfind by arcsin( (9 h whr th variabl R dnots th radius of th Earth. Although th EHS fixd on pitch axis is unabl to dirctly masur th yaw information, th attitud inmatics quation implis that th roll angl φ coupls with yaw attitud ψ. If th 3-axis angls ar small, th attitud coupld inmatics quation is givn by x y ( z whr orbital angl vlocity ω is always assumd to b constant. Whn ω x =ω z =, thn th trajctory of roll and yaw angl follow aftr solving diffrntial Equation( as ( t ( tcos ( t- t ( tsin ( t- t ( ( t ( tsin ( t- t ( tcos ( t- t As obviously sn from Equation(, vry /4 orbital priod, th roll and yaw angl valus xchang, which is also nown as th orbital gyrocompass in rfrnc 7-8. So it mas full us of th dynamical coupling ffct to stimat th yaw angl through roll axis masurmnts indirctly..3 Vctor Masurmnts W now will giv a dtaild drivation of th unit vctor in th coordinat systm of th horizon snsor, which points from th cntr of th pinhol camra to th cntr of th Earth. his vctor will b rfrrd to as th nadir vctor. h horizon snsor is always looing in th -z dirction, thus th pointing vctor E o of a nadir-pointing EHS should b [ -]. Sinc th EHS is not looing dirctly 38-p.3

4 MAEC Wb of Confrncs at th Earth cntr in th prsnc of attitud angls, a squnc of two rotations can b dfind to dscrib th alignmnt of th EHS coordinat systm with that of a nadir-pointing EHS. hs consist of (i rotating th EHS again about itslf z-axis by an angl φ; (ii rotating a nadir-pointing EHS about itslf x-axis by an angl θ, thn. hs angls hav alrady bn masurd with EHS: thy ar th pitch and ngativ roll angls givn in Equation(8 rspctivly. o comput th nadir vctor in th coordinat systm of th rotatd EHS, w simply apply th rotation matrics to th nadir vctor E o as follows: sin( Es Rx( Rz( E o sin( cos( ( cos( cos( Aftrwards, combining with th nown of EHS stup matrix, th abov vctor E s is transfrrd and xprssd in th body fram as follows, sin( sin( cos( Eb sin( cos( sin( (3 cos( cos( cos( cos( Now, th nadir vctor E b is usd as th masurmnt in body fram to yild filtring rsidual..4 Masurmnt Modl As shown in th prvious sction, th EHS masurmnts hav bn usd to construct th nadir vctor in th body fram. Hnc, th masurmnt quation can b writtn as, z E b v (4 Which rsults in th masurmnt rsidual, z ˆ ( ˆ z Eb C q E o v (5 h masurmnt nois covarianc matrix is givn as R Evv (6 Using an approximation to attitud dirction matrix C( qˆ ( I θ C( q as shown in 4, so th 33 rquird snsitivity matrix for our masurmnt modl can b drivd as z ( ˆ Eb θ rx z 3 H (7 E ( y ˆ b θ x r z 3 whr th vctor r x and r y ar th first and scond row of th transformation matrix from th body to th snsor fram, rspctivly. In ordr to rduc th amount of computations in th filtring procss, w draw lssons from rfrnc 9 to transfr th abov masurmnt modl (4-(7 to quivalnt form as blow z ˆ Eb( C( q Eo v (8 R, H I 4 h advantag of th masurmnt modl (8 is that both th snsitivity matrix H and th discrt procss modl (5 ar constant, so a stady-stat KF is applid by solving th corrsponding Riccati quation. h rsulting filtr calculation cost will b gratly rducd compard with th situation using th masurmnt modl (4-(7, manwhil th attitud stimation prcision obtaind by th stady-stat KF is not dcrasd, as shown in following simulation..5 Corrction Equations 38-p.4

5 IC4M 6 In th cas, Murrll s approach to squntially procss th vctor masurmnts is implmntd 4. h rcursiv calculation form of KF gain is givn as follows K P H HP H R (9 For covarianc propagation, Josph s form is usd as follows P ( I66 KH P ( I 66 KH KRK ( hn, th stat vctor x is updatd as follows ˆ q qˆ ˆ ˆ ˆ q q ˆ q q ( ˆ ˆ b b bˆ 3 Earth S Oblatnss Corrctions Actually, th Earth s oblatnss contributs th most of rror sourc of attitud dtrmination basd on horizon snsor. Instad of trating th total masurmnt rrors as whit nois in this papr, an ffort is mad to dvlop a ralistic systmic rror modl. Assuming th spaccraft s position nown, th rror duo to th Earth s oblatnss can b liminatd by considring variation of Earth s radius with latitud. h corrction basd on arth modl for th Earth s oblatnss is givn as cos cos f F(, ( whr η is th nadir angl ignoring th ffct of flattning, th f is Earth llipsoid, F(φ,σ is th function of rror modification for Earth s oblatnss, as follows, F(, 4 R ( r R r R (3 s c s css r ( r R whr r rprsnts th distanc btwn th satllit and th Earth cntr, φ is th latitud of th subsatllit point, σ is th azimuth of horizon point with rspct to th local astward dirction. hn th corrction for Earth s oblatnss is givn by f c c s ( (, (, ( (, (, F F F F s s c (4 f c s c ( F(, F(, ( F(, F(, s s c whr th variabls σ, σ and χ ar calculatd rspctivly as follows, arcsin arcsin r c r c R r c arcctg(tg( icos( u, s r s Finally, th corrct calculation of th roll and pitch attitud angls ar shown as follows, ˆ ˆ, ˆ ˆ (5 4 Simulation sults h initial conditions considrd for our simulation wr chosn as shown in abl : 38-p.5

6 MAEC Wb of Confrncs abl Paramtr stting for simulation Paramtr Valu Unit a a m b i c dg q [-.7,.33,.8,.] d ω angl [,-.4,.] rad/sc γ 55 dg b.5 dg/hr t.5 sc I s f diag{57,5,5} g m σ φ - σ θ - a h variabl a is orbital smi-major. b h variabl is orbital ccntricity. c h variabl i is orbital inclination. d h ω angl is initial angl rat. h b is initial gyro drift bias. f h I s is th momnt of inrtia matrix,which is diagonal matrix. W assum that no Sun obtrusions ar prsnt (although this is not truly ralistic. At ach tim instant all availabl LVLH nadir vctor and body masurmnts ar usd to form th KF gain matrix in Equation(9. hn, th quatrnion stimat is found using Equation(. It is clarly shown in th Figur that th yaw rrors ar much largr than othr axis. his is sinc th nowldg of yaw axis is not dirctly acquird by th EHS. Furthrmor, as xpctd, th accuracy changs as th orbital priod changs. In contrast, th roll and pitch attitud stimation ar all pt within a small margin of rror. Estimation Error:Attitud(LVLH (dg (dg (dg tim (sc Figur. Estimation Errors of rcursiv EKF without Corrction hrough corrcting th rror causd by th Earth's oblatnss, th roll axis pointing accuracy has bn furthr improvd, but for improvmnt of pitch axis was not obvious, as shown in Figur. h stimat accuracy of th roll-pitch axis mt th pointing accuracy rquirmnt of dg, in spit of th rlativly largr stimation rror in yaw dirction. 38-p.6

7 IC4M 6 Estimation Error:Attitud(LVLH (dg (dg (dg tim (sc Figur. Estimation Errors of rcursiv EKF with Corrction As mntiond arlir, th rcursiv KF gain filtr can b modifid to its stady-stat implmntation, to do away with its long transints and rduc computing cost. h constant gain K is [.8697, ,., -.495,.869, -.497], basd on th masurmnt quation (8. h prformanc with th rcursiv gain filtr for a sampl run is illustratd in Figur 3. (dg Estimation Error:Attitud(LVLH (dg (dg tim (sc Figur 3. Attitud Errors of Stady-stat EKF with Corrction As sn from Figur 3, th stimats of 3-axis attitud angls quicly tnd to convrgnc aftr th transition procss. his indicats that th stady-stat KF can rciv th sam stimation accuracy as th KF with rcursiv gain, but contrast it will gratly rduc th amount of calculations. h xistnc of th attitud lads to th projctd componnt of orbital spd in thr axial dirctions. So th gyro output includs th disturbanc rsultd from orbital spd, which causs th gyro drift stimats dgradd. It is clarly illustratd in Figur 4 that th stimation rrors of gyro drift in th roll-yaw dirction ar rlativly largr. In addition, th coupling ffct of clos loop control will also incras th complxity of th drift stimation procss. 38-p.7

8 MAEC Wb of Confrncs b x ( \ hr Estimation Error:Drift Rat Bias b y ( \ hr b z ( \ hr Figur 4. Gyro Drift Estimation Error tim (sc 5 Conclusion In this papr, a quatrnion-basd EKF attitud stimation algorithm with only a singl horizon snsor has bn studid, analyzd and simulatd. Although th EHS can't provid dirctly yaw axis information, but basd on th rat-gyro and satllit attitud inmatics quation, th orbital gyrocompass can giv th yaw angl stimation and gradually tracs th ral yaw attitud. On th othr hand, bcaus of no dirct accss to th yaw masurmnt, th accuracy of yaw attitud stimation is rlativly poor. In spit of this, for arth obsrvation satllit, its normal mission will b not affctd, as long as roll and pitch attitud can maintain a crtain lvl of angl stimation accuracy. o furthr nhanc th prformanc of th stimation algorithm, an ffort is mad to considr dtaild modls of snsor masurmnts and to mnd th algorithms in prsnc of ralistic snsor rrors, which ar rsultd from Earth s oblatnss. h simulation rvals that th stimation prcision of th roll axis obtaind furthr improvmnt. It is anothr important supriority that th attitud dtrmination basd on a singl EHS can b suitabl for a whol orbital priod. hrfor, th attitud stimation mthod givn by this papr will b an conomic and rliabl masurmnt configuration, spcially for a class of arth obsrvation micro-satllits. frncs. Sandau, R., Rosr, H. P., and Valnzula, A. 8. Small Satllits for Earth Obsrvation. Brlin: Springr.. CubSat Dsign Spcification v.. California Stat Polytchnic Univrsity. trivd August 5 from: 3. rntlman, H. L., Stoorvogl, A. A., and Hautus, M.. Control hory for Linar Systms. Lond: Springr. 4. Marly, F. L. and Crassidis, J. L. 4. Fundamntals of Spaccraft Attitud Dtrmination and Control. Nw Yor: Springr. 5. Farrl, J. A. 8. Aidd Navigation GPS with High Rat Snsors. Nw Yor: McGraw-Hill Companis. 6. Wrtz, J. R.,tc Spaccraft Attitud Dtrmination and Control. Nthrlands: D.idl Publishing Company. 38-p.8

9 IC4M 6 7. Bowrs, A. E., Roddn, J. J., and Scott. E. D., t al. Orbital Gyrocompassing Hading frnc Journal of Spaccraft and Rocts. 5: Bryson, A. E. and Kortum, W. 97. Estimation of th Local Attitud Orbiting Spaccraft. Automatica. 7(: Shustr, M.D. 99. Kalman filtring of spaccraft attitud and th QUES modl. Journal of th Astronautical Scincs. 38(3: Crassidis, J. L., and Junins, J. L.. Optimal Estimation of Dynamic Systms. Boca Raton: CRC Prss. awy, J. A., Wang, P., and Gray, C. W Scanning Horizon Snsor Attitud Corrction for Earth Oblatnss. Journal of Guidanc, Control, and Dynamics. 9(3: p.9