Simultaneous Position, Velocity, Attitude, Angular Rates, and Surface Parameter Estimation Using Astrometric and Photometric Observations

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1 Smultaneous Poston, Velocty, Atttude, Angula Rates, and Suface Paamete Estmaton Usng Astometc and Photometc Obsevatons Chales J. Wettee and C. Channng Chow Pacfc Defense Solutons, LLC Khe, Hawa, USA John L. Cassds and Rchad Lnaes Depatment of Mechancal & Aeospace Engneeng Unvesty at Buffalo Amhest, New Yo, USA Moba K. Jah A Foce Reseach Laboatoy, Space Vehcles Dectoate Ktland AFB, New Mexco, USA moba.jah@tland.af.ml Abstact Astometc and photometc data fuson fo the puposes of smultaneous poston, velocty, atttude, and angula ate estmaton has been demonstated n the past. Ths state estmaton s extended to nclude the vaous suface paametes assocated wth the bdectonal eflectance dstbuton functon (BRDF). Addtonally, a physcally consstent BRDF and adaton pessue model s utlzed thus enablng an accuate physcal ln between the obseved photometc bghtness and the atttudnal dynamcs and ultmately the obtal dynamcs. An example scenao s then pesented whee the model s an uncontolled Hgh Aea to Mass Rato (HAMR) object n geosynchonous Eath obt and the poston, velocty, atttude, angula ates, and suface paametes ae estmated smultaneously Keywods estmaton; data fuson; BRDF I. INTRODUCTION Wettee and Jah [] fst demonstated how bghtness (photometc flux ntensty) measuements can be used to estmate the atttude and angula ates of a space object (SO). Lnaes et al. [] ecently demonstated a mechansm that fuses both angles (lne-of-sght) and bghtness measuements fo the pupose of obt, atttude, and shape detemnaton. Wheeas angles measuements ae dect obsevatons of the SO s obtal poston, the bghtness measuement s dependent on obtal poston and the SO s shape, suface paametes and atttude, and thus povdes an ndect obsevaton of these othe attbutes. The physcal coelaton between the SO s shape/atttude and ts obtal poston ae caused by the vaous non-gavtatonal foces and toques, such as adaton pessues that poduce a lnea and angula acceleaton on the SO. These adaton pessues must be consstent wth the suface bdectonal eflectance dstbuton functons (BRDFs), as shown by Wettee et al. []. The wo pesented hee expands the paametes that ae ncluded n the state to nclude those assocated wth the space object s suface, namely, the vaous BRDF paametes. Fst, the paametes ncluded n the SO s augmented state ae pesented and the state functon detalng the dynamcs fo each of these paametes n how they ae popagated fowad n tme s defned. It s mpotant to note that though adaton pessue, the BRDF paametes and othe potental paametes n the augmented state nfluence the dynamcs that affect both the atttude and obt and thus nfluence the evoluton of the system fom pont to pont. As wth the classcal state paametes, these new paametes ae ntnsc and unque popetes of the system. Next, the unscented Kalman Flte (UKF) s befly evewed. Fnally, test cases usng an example scenao ae examned n detal to demonstate smultaneous estmaton, effect of model msmatch, and nfomaton dluton. II. BUILDING THE AUGMENTED STATE An Eath obtng SO s atttude, angula ates, poston, velocty, sze, shape, mass and suface chaactestcs ae all needed fo hgh fdelty obt popagaton and n calculatng assocated measuements that ae emotely obsevable, such as angles and bghtness. All of these paametes mae up the SO s augmented state. In ths pape, we wll nclude the SO s atttude, angula ates, poston, velocty, and suface chaactestcs n the augmented state. In ths pape, the quatenon, whch s based on the Eule angle/axs paametezaton and contans fou values, s used to specfy the SO s atttude. The quatenon s defned ξ ξ = q q q = eˆ sn and as q = [ ] wth [ ] ( ) q 4 υ

2 ( ) q 4 = cos υ, whee ê and υ ae the Eule axs of otaton and otaton angle, espectvely. In addton, each component s angula ate s denoted by ω = [ω x ω y ω z ] as defned n the SO s body-fxed fame. A sngle poston and velocty coespondng to the SO s cente of mass, denoted by I = [x y z] I and v I = [v x v y v z ] I espectvely, ae used, whee the supescpt I ndcates the netal fame. Ths s the classc 6DOF epesentaton of the SO s obt. Each mateal that maes up the SO could eflect lght dffeently. Thus, a SO mght have many paametes that specfy ts suface popetes. The functon that defnes how lght s eflected fom an opaque suface wth a gven suface nomal decton ( Nˆ ), llumnaton decton ( Lˆ wth angles θ and ϕ fom Nˆ ), and obseve decton ( Vˆ wth angles θ and ϕ fom Nˆ ) as shown n Fg. s called the bdectonal eflectance dstbuton functon (BRDF). each to the total (d and s espectvely whee d + s = ). These bdectonal eflectances ae calculated dffeently fo the vaous models. In ths pape we wll use the Ashhmn- Shley BRDF [4], also nown as the Ansotopc Phong BRDF, whee the dffuse and specula bdectonal eflectances ae calculated usng R s ( ) = ρ Nˆ Lˆ Nˆ ˆ R sf 0 V (4) d π = ( n + )( n + ) F n cos β sn β ( ) [ ] ( ) u + n u v v 8π Vˆ Hˆ mαx Nˆ Lˆ, Nˆ Vˆ cosα whee (4) s a non-lambetan dffuse BRDF, and the Fesnel eflectance (F) n (5) s gven by Schlc s appoxmaton [5] (5) ( Vˆ Hˆ ) 5 F = F0 + F0 (6) s Fg.. The geomety of eflecton The BRDF s gven by f ( θ f ; θ, f ; λ) dl ( θ, f ) ( θ, f ), = () de whee dl s the eflected adance n Wm - s - and de s the adance n Wm -. The bsecto vecto between the llumnaton souce and the obseve s In addton to d, ρ and F 0, the Ashhmn-Shley BRDF has two exponental factos (n u, n v ) that defne the ansotopc eflectance popetes of each suface. Wthout loss of functonalty, the dffuse eflectance and the specula eflectance at nomal ncdence can be set equal to each othe (ρ = F 0 ) and the dffeence between the dffuse and specula eflectances dsplayed n the dffuse facton paamete, d. Addtonally, fo the sae of smplcty, n ths pape the two exponental factos ae set equal to each othe as well (n u = n v = n). Thus, thee ae thee unque suface paametes pe suface (n, ρ, d). These suface paametes have constants (n > 0, 0 ρ, 0 d ). To account fo the constants wthn the flte, unconstaned poxy values ae used n the state vecto and estmaton flte and these poxy values ae conveted bac to the suface paamete value when needed. The conveson equatons to the poxy value and fom the poxy value fo each of the suface paametes ae ( n), n ( p ) p = ln = exp (7) ( Lˆ + Vˆ ) Lˆ Vˆ Hˆ = + () wth angles α and β fom Nˆ and s used n many analytc BRDF models. Thee ae many dffeent eflectance models that could be used, but all can be expessed n a common nomenclatue wth the geneal BRDF calculated by ( dr sr ) f = + () whch depends on the dffuse bdectonal eflectance (R d ) and the specula bdectonal eflectance (R s ) and the facton of d s p = ln ρ, ρ = ( tanh( p ) ) (8) ρ + = d p ln, d = ( tanh( ) + ) d p (9) The shape model wll be bult as the sum of facets whee each facet has a poston n the body fame and s specfed by a patcula aea and nomal vecto. The SO s bghtness s also calculated by summng the contbuton to the bghtness by each facet usng the BRDF n the equaton

3 m object = m Sun.5log 0 N facets = A ( f ) ( Nˆ Lˆ )( Nˆ Vˆ ) (0) whee A s the aea of the th facet, s the dstance between the SO and obseve, and m Sun s the appaent magntude of the llumnaton souce (n ths case the Sun). So, n summay, the augmented state s I I [ q ω v p] T xˆ = () whee xˆ s a 6 vecto of eal numbes and p = [p p p ] s a vecto contanng the suface paamete poxy values. III. PARAMETER PROPOGATION The Newtonan two-body gavtatonal equatons of moton wth adaton pessue acceleaton n Eath-centeed netal coodnates (ECI) ae gven by I µ I I I = + a J + a () total I whee the tems μ epesents the gavtatonal paamete of the I Eath, a s the gavtatonal petubaton due to nonsymmetc dstbuton of mass along the lnes of lattude, J and N facets I I a = a epesents the acceleaton petubaton due the total = vaous adaton pessues and summed ove all the sufaces. Detals egadng the calculaton of ths last tem can be found n []. The atttude matx fo each component of the SO can be wtten as a functon of the component s quatenon by whee and Ξ Ψ T ( ) ( ) A = Ξ q Ψ q () ( q) ( q) q = q = [ ξ ] 4 I + (4) T ξ [ ξ ] 4 I (5) T ξ [ a ] 0 = a a a 0 a a a 0 (6) s the sew-symmetc matx epesentaton of the coss poduct fo any geneal vecto a. The quatenon nematcs equaton s gven by q = Ξ( q)ω (7) whee ω s the component s angula velocty. The angula velocty dynamc equaton can be wtten as ( M [ ω ] Jω) ω J (8) = whee J the neta tenso fo the SO and M ae any extenal appled toques. The adaton pessue moments can be calculated by consdeng that the foces act though the cente of each facet N facets M = m = B [ ] ( ) I A q a (9) B whee s the locaton of the geometc cente of each facet wth espect to the cente of mass of the SO n body coodnates and A(q) s the atttude matx calculated by the quatenon q. The suface popetes ae assumed to be constants, and so the dynamcs equaton can be wtten as p = 0 (0) IV. UNSCENTED KALMAN FILTER In ths pape, the quatenon-based unscented Kalman flte (UKF) of Cassds and Maley [6,7] s employed wth the atttude state eos epesented as eo Genealzed Rodgues Paametes (GRPs) [8]. The augmented state s the SO s oentaton and otaton ate, and the SO s poston, velocty, and suface paametes gven by (). The paamete popagaton equatons n (), (7), (8), and (0) can be wtten n the geneal state functon whch gves the detemnstc pat of the stochastc model + ( xˆ w ) ˆ f whee w s the pocess nose vecto x =, () In ths appoach, the obsevaton vecto ncludes the appaent magntude as computed wth (0), as well as

4 astometc measuements of the ght ascenson and declnaton of the SO: [ m RA Dec] T y ˆ = () object whee ŷ s a vecto of eal numbes. The geneal measuement functon used n the estmaton flte s: ( xˆ v ) y ˆ = h, () In the fst two tests, the physcally-consstent BRDF/Sola Radaton Pessue (SRP)/TRP model was used to geneate the tuth. In the fst test, ths same model was used n the state functon of the UKF estmaton whle n the second test, a smplfed SRP model (whee the BRDF has a Lambetan dffuse component and a mo-le specula component) and no TRP model was used n the state functon. Fgs. -4 plot the measuements ove the obsevaton peod. Fgs. 5-7 plot selected components of the estmated state and covaance as a functon of tme fo Test # whle Fgs. 8-0 plot the same fo Test #. whee v s the measuement nose vecto. V. SIMULATIONS The patcula example wll be that of smultaneously estmatng the poston, velocty, atttude, angula ates, and suface paametes of a HAMR object n geosynchonous Eath obt (GEO). Table I lsts the ntal tuth state, the ntal estmated state, and the ntal uncetanty. The shape s defned as a cube wth -m sdes and a mass of g. Each suface of the cube s coated wth the same BRDF suface paametes. The obt was set to geosynchonous (a = m, e = 0, = 0 o, M 0 = 9 o, ω = 0, Ω = 0). Obsevatons wee smulated statng at 00 Ma 5 at 4:00:00 UT, 800 obsevatons evey s (fo a total of hou), wth an obsevaton ste coespondng to the top of Haleaala on Mau (lattude = 0.7 deg, longtude = deg, alttude =.0586 m). The Themal Radaton Pessue (TRP) paametes of each suface wee C = 9000 J/K, K = 5.5 W/K, and T body = 4.5 K. The measuement nose s 0. mag fo the bghtness obsevaton and 0 ac-sec n the ght ascenson and declnaton obsevatons. Fg.. Bghtness n magntudes as functon of tme TABLE I. TESTS #/# AUGMENTED STATE SETUP Value n State Intal Tuth Intal Estmate Uncetanty q deg ω (ad/s) x I (m) v I (m/s) n Fg.. Rght Ascenson as functon of tme ρ d Fg. 4. Declnaton as functon of tme

5 Fg. 5. Atttude dffeence fom tuth and -σ eo bounds fo Test # Fg. 8. Atttude dffeence fom tuth and -σ eo bounds fo Test # Fg. 6. Angula ate dffeence fom tuth and -σ eo bounds fo Test # Fg. 9. Angula ate dffeence fom tuth and -σ eo bounds fo Test # Fg. 7. Suface paamete dffeence fom tuth and -σ eo bounds fo Test # Fg. 0. Suface paamete dffeence fom tuth and -σ eo bounds fo Test #

6 TABLE II. TESTS #/# RESULTS Value n State Eule angle (deg) Dffeence fom Tuth Test # Test # Uncetanty Dffeence fom Tuth Uncetanty ω (deg/h) I (m) v I (m/s) n ρ Fg.. Atttude dffeence fom tuth and -σ eo bounds fo Test # d Table II lsts the dffeence between the fnal estmated state and tuth, and the -σ uncetanty as calculated by the fnal covaance matx fo both tests. The atttude quatenon dffeence has been conveted to the equvalent oll/ptch/yaw Eule angle dffeences. Of patcula note n Test # s that all paametes n the state ae obsevable by the flte and convege to the coect values (.e. all paametes to wthn -σ of the quoted uncetanty). In Test #, howeve, due to the dynamcs model msmatch, some paametes n the state ae not convegng, and the dffeences fom tuth of the fnal state fo these paametes ae well outsde -σ of the quoted uncetanty (e.g. atttude Eule angle components > 0-σ fom tuth). Some of the paametes ae unaffected (e.g. poston and velocty) by the model msmatch ove the one hou tme scale sampled, but would undoubtedly be affected ove a longe tme span. In the next two tests, the same ntal condtons wth the physcally-consstent BRDF/SRP/TRP used to both geneate the tuth and n the estmaton flte as n Test # ae duplcated except that the uncetantes on the suface paametes ae nceased by a facto of fom those shown n Table I. Addtonally, n Test #4, the dffeence fom the tuth and the uncetantes of the obtal and atttudnal state paametes ae deceased by a facto of 0. Fgs. -5 dsplay how the nceased ntal covaance fo the suface paametes esults n flte dvegence by plottng all the state paamete dffeences fom tuth and the -σ uncetanty as a functon of tme fo Test #. In ths test, even the poston and velocty begn to dvege. In contast, when the obtal and atttudnal state paamete uncetantes ae deceased and the estmaton eun, as n Test #4, the flte s able to convege (although outsde the -σ bound), despte the nceased suface paamete uncetanty as shown n Fg. 6. Fg.. Angula ates dffeence fom tuth and -σ eo bounds fo Test # Fg.. Poston dffeence fom tuth and -σ eo bounds fo Test #

7 TABLE III. TESTS #/#4 RESULTS Fg. 4. Velocty dffeence fom tuth and -σ eo bounds fo Test # Value n State Eule angle (deg) Dffeence fom Tuth Test # Test #4 Uncetanty Dffeence fom Tuth Uncetanty ω (deg/h) I (m) v I (m/s) n ρ d Table III lsts the dffeence between the fnal estmated state and tuth, and the -σ uncetanty as calculated by the fnal covaance matx fo Test # and Test #4. The atttude quatenon dffeence has agan been conveted to the equvalent oll/ptch/yaw Eule angle dffeences. In Test #, the flte s suffeng fom nfomaton dluton. Thee ae smply too many paametes n the state whose uncetantes ae lage. When, howeve, as n Test #4 the uncetantes fo only a lmted numbe of state paametes ae lage, the flte s able to convege to close to the tuth. Fg. 5. Suface paamete dffeence fom tuth and -σ eo bounds fo Test # VI. CONCLUSIONS Physcally consstent BRDF, SRP, and TRP modelng enables astometc and photometc data fuson fo the puposes of smultaneously estmatng obtal, atttudnal, and suface paametes of a space object. Othe paametes assocated wth the space object, such as the mass and shape, could also be added to the augmented state vecto and estmated povded that the effect on the system s obsevable wth an appopate measuement model. Cauton s waanted, howeve, concenng model msmatches (both n the dynamcs and the BRDF model) and the possblty of nfomaton dluton when too many quanttes wth lage uncetantes ae estmated at once. Fg. 6. Suface paamete dffeence fom tuth and -σ eo bounds fo Test #4 ACKNOWLEDGMENT Ths wo s sponsoed by a Small Busness Innovatve Reseach (SBIR) contact FA945--C-054 by the A Foce Reseach Laboatoy (AFRL) at Ktland A Foce Base. Some of the code was also poduced unde the SBIR Phase I contact FA945--M-0, also sponsoed by AFRL. The authos also wsh to acnowledge useful dscussons and techncal contbutons made by othe membes of the SBIR team: Paul Cefola of Unvesty at Buffalo, and Kec Hll of Pacfc Defense Solutons, LLC.

8 REFERENCES [] Wettee, C. J. & Jah, M. K., Atttude estmaton fom lght cuves, Jounal of Gudance, Contol, and Dynamcs : , 009. [] Lnaes, R., Cassds, J. L., Jah, M. K., and Km, H., Astometc and Photometc Data Fuson fo Resdent Space Object Obt, Atttude, and Shape Detemnaton Va Multple-Model Adaptve Estmaton, AAS/AIAA Astodynamcs Specalsts Confeence, Vol. AIAA-84, 00. [] Wettee, C. J., Lnaes, R., Cassds, J. L., Kelecy, T. M., Zebat, M. K., Jah, M. K., and Cefola, P. J., Refnng Space Object Radaton Pessue Modelng wth Bdectonal Reflectance Dstbuton Functons, Poceedngs of the AAS/AIAA Spaceflght Mechancs Confeence, Kaua, HI, Febuay 0 [4] Ashhmn, M., and Shley, P., "An Ansotopc Phong BRDF Model," Jounal of Gaphcs Tools, Vol. 5, No., pp.5-, 000. [5] Schlc, C., An nexpensve BRDF model fo physcally-based endeng, Compute Gaphcs Foum, (), pp. -46, 994. [6] Jule, S. J., Uhlmann, J. K., and Duant-Whyte, H. F., A New Appoach fo Flteng Nonlnea Systems, Poceedngs of the Amecan Contol Confeence, Seattle, WA, 68 6, June 995. [7] Cassds, J. L. and Maley, F. L., Unscented Flteng fo Spacecaft Atttude Estmaton, Jounal of Gudance, Contol and Dynamcs 6, 56 54, 00. [8] Schaub, H. and Junns, J. L., Steeogaphc Oentaton Paametes fo Atttude Dynamcs: A Genealzaton of the Rodgues Paametes, Jounal of the Astonautcal Scences 44, 0, 996.