ORBIT uncertainty propagation plays an important role in

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1 JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 3, No. 6, Novembe Decembe 27 Nonlnea Sem-Analytc Metods fo Tajectoy Estmaton Ryan S. Pa and Danel J. Sceees Unvesty of Mcgan, Ann Abo, Mcgan 489 DOI:.254/.296 Nonlnea sem-analytc flteng metods to sequentally estmate sacecaft states and te assocated uncetantes ae esented. We fst dscuss te state tanston tensos tat caacteze te localzed nonlnea beavo of te tajectoy statstcs and llustate te motance of ge-ode effects on obt uncetanty oagaton. We ten esent a sem-analytc flteng metod by mlementng te state tanston tensos to sequentally udate te flte nfomaton wt contbutons fom eac measuement, wc eques no ntegaton once te tensos ae comuted. A sun Eat alo obt about te L ont s consdeed as an examle wt ealstc obt uncetantes, and te esults ae comaed wt te extended Kalman flte and unscented Kalman flte. Nomenclatue g, g = system dynamcs vecto and ts t comonent, = measuement vecto and ts t comonent K = Kalman gan comuted at tme t m, m = mean vecto and ts t comonent m, m = edcted mean vecto at tme t and ts t comonent m, m = udated mean vecto at tme t and ts t comonent N = system dmenson n, n = edcted measuement functon at tme t and ts t comonent P, P j = covaance matx and ts ; j enty P, P j = edcted covaance matx at tme t and ts ; j enty P, P j = udated covaance matx at tme t and ts ; j enty = obablty densty functon Q, Q j = dffuson (ocess nose) matx at tme t and ts ; j enty R, R j = measuement nose matx at tme t and ts ; j enty t = tme U = otental functon fo ccula estcted teebody oblem u, v, w = sacecaft velocty comonents v, v = measuement nose vecto and ts t comonent w, w = ocess nose vecto and ts t comonent x, x = state vecto and ts t comonent x, y, z = sacecaft oston comonents x, x = ntal state vecto and ts t comonent x, x = elatve state vecto and ts t comonent x, x = ntal elatve state vecto and ts t comonent j = Dac delta functon S = sola gavtatonal constant, : m 3 =s 2 = Eat gavtatonal constant, 398; 6:44 m 3 =s 2 = state tanston matx Pesented as Pae 6399 at te AAS/AIAA 26 Astodynamcs Secalsts Confeence, Keystone, CO, 2 24 August 26; eceved 3 Novembe 26; acceted fo ublcaton 24 Mac 27. Coygt 27 by te autos. Publsed by te Amecan Insttute of Aeonautcs and Astonautcs, Inc., wt emsson. Coes of ts ae may be made fo esonal o ntenal use, on condton tat te coe ay te $. e-coy fee to te Coygt Cleaance Cente, Inc., 222 Rosewood Dve, Danves, MA 923; nclude te code 73-59/7 $. n coesondence wt te CCC. P.D. Student, Deatment of Aeosace Engneeng; cuently Membe, Tecncal Staff, Oute Planet Navgaton, Jet Poulson Laboatoy, Pasadena, CA 99; Ryan.S.Pa@jl.nasa.gov. Membe AIAA. Assocate Pofesso, Deatment of Aeosace Engneeng; sceees@ umc.edu. Assocate Fellow AIAA. 668, = soluton flow and ts t comonent ; = t ode state tanston tenso, = nvese soluton flow and ts t comonent ; = t ode nvese state tanston tenso! E = mean moton of te Eat about te sun, AU 3 = S :99 7 s I. Intoducton ORBIT uncetanty oagaton lays an motant ole n vaous sace-elated alcatons, suc as obt detemnaton, aamete estmaton, coecton maneuve desgn, small-body collson/encounte analyss, etc. In actce, t s usually assumed tat te tue moton (n a statstcal sense) of a sacecaft wt esect to a nomnal tajectoy s wtn a bounday wee te lnea assumton suffcently aoxmates te elatve dynamcs and te covaance matx s maed usng te Rccat equatons. In some cases, oweve, te lnea assumton fals to ovde an accuate ealzaton of te local tajectoy moton and n suc cases a dffeent metod wc accounts fo te system nonlneaty must be mlemented. Te best nown tecnque fo nonlnea obt uncetanty oagaton s a Monte Calo (MC) smulaton, wc aoxmates te obablty dstbuton by aveagng ove a lage set of andom samles []. A Monte Calo smulaton can ovde tue statstcs n te lmt, but s comutatonally ntensve and only solves fo te statstcs of a secfc eoc and ts assocated uncetantes. Hence, fo msson oeatons, tese dffcultes mae Monte Calo smulatons neffcent fo actcal sacecaft alcatons. Recently, Pa and Sceees [2 5] ave develoed a sem-analytc metod fo obt uncetanty oagaton by solvng fo te geode Taylo sees tems tat descbe te localzed nonlnea moton and by analytcally mang te ntal uncetantes. By consdeng suffcently g-ode solutons, tey ave sown tat te semanalytc aoac fo obt uncetanty oagaton can elcate Monte Calo smulatons wt te beneft of added flexblty n ntal obt statstcs. In ts ae, we deve nonlnea fltes by assumng a Gaussan statstc and by mlementng te sem-analytc obt uncetanty oagaton tecnque develoed by Pa and Sceees. A flte s usually comosed of two ats, edcton and udate. Obt uncetanty oagaton elates to te edcton oblem wle te uncetan dstbuton of a measuement and state nfluences te udate at. In conventonal tajectoy navgaton, one s usually gven a efeence (nomnal) tajectoy wt ecse eemedes and te extended Kalman flte () s used fo tajectoy estmaton. Te objectve of tajectoy navgaton s ten to follow te efeence tajectoy wle mnmzng some edefned otmalty constants, suc as te numbe of tajectoy coecton maneuves, flgt tme, fuel, etc. [Note tat wen te tajectoy devates fom te efeence tajectoy ove some eo bounday a coecton maneuve s aled

2 PARK AND SCHEERES 669 to ecouse te sacecaft to te efeence tajectoy (o to an altenate tajectoy tat satsfes msson objectves).] Te basc undelyng concet of suc a ocess s to stay wtn te lnea egon by tang a suffcent numbe of measuements and lnealy ma te devaton and statstcs va te state tanston (fundamental) matx wt esect to te nomnal tajectoy. Howeve, wen te nonlneaty s sgnfcant o wen only a lmted numbe of measuements ae avalable, t may be necessay to consde a flte tat ncooates system nonlneaty. Te sun Eat alo obt about te L ont based on te ccula estcted tee-body oblem (CR3BP) s cosen as an examle because te oveall nonlneaty s small, but te tajectoy s unstable and wen te sacecaft s not suffcently obseved, te nonlnea effect can become sgnfcant. Te oosed sem-analytc fltes ae comaed wt te extended Kalman flte and unscented Kalman flte wt ealstc obt uncetantes. Te esult sows tat ou ge-ode fltes ovde faste convegence and a sueo soluton as comaed to lnea fltes. II. Hge-Ode Petubaton Analyss Te moton of a sacecaft can be modeled wt fst-ode odnay dffeental equatons: dxt gt; xt () dt wee gt; xt eesents te system dynamcs vecto wt a dmenson N and x fx j ;...;Ng eesents te sacecaft state vecto wt te ntal condton xt x. Te soluton flow, wc mas te ntal state at t to t, s ten defned as Te soluton flow s govened by dt; x ;t dt x tt; x ;t (2) gt; t; x ;t (3) t ; x ;t x (4) By consdeng a smla notaton, we defne te nvese soluton flow tat mas te state at t to te ntal state as x t; x; t (5) In ts famewo we defne te local tajectoy dynamcs x by alyng a Taylo sees exanson about te efeence (nomnal) tajectoy, tat s, xtt; x x ;t t; x ;t, fo some ntal devaton x. Te mt ode soluton can be stated usng te Ensten summaton conventon as x t Xm! ; t;t x x (6) wee j 2f;...;Ng, suescts j denote te j t comonent of te state vecto, and ; t;t t; x t; j xj We call te ge-ode atals of te soluton flow Eq. (7) te state tanston tensos (STTs), wc ma te ntal devatons to te cuent tme. Note tat te fst-ode case ( ) educes to te usual state tanston matx (STM). Te dffeental equatons u to fout-ode devaton ae gven n Eqs. (8 ). Fo moe detals on ow to obtan tese dffeental equatons, eades ae efeed to [4,5]. _ ;a g ; ;a (8) _ ;ab g ; ;ab g ; ;a ;b (9) _ ;abc g ; ;abc g ; ;a ;bc ;ab ;c ;ac ;b g ; ;a ;b ;c () _ ;abcd g ; ;abcd g ; ;abc ;d ;abd ;c ;acd ;b ;ab ;cd ;ac ;bd ;ad ;bc ;a ;bcd g ; ;ab ;c ;d ;ac ;b ;d ;ad ;b ;c ;a ;bc ;d ;a ;bd ;c ;a ;b ;cd g ; ;a ;b ;c ;d () Te ntal condtons of te STTs ae ;a t ;t f a and zeo otewse. Once tese STTs ae comuted, tey seve a ole dentcal to te STM excet tat ge-ode effects ae now ncluded, and tus te soluton s nonlnea. Teefoe, a sgnfcance of te STTs s tat te local nonlnea moton of a sacecaft tajectoy can be maed analytcally and eques no ntegaton. Te nvese sees also exsts and s defned as x Xm! ; t ;t x x (2) wee j 2f;...;Ng and ; t ;t t; x; t; (3) j x j We call tese ge-ode atals te nvese state tanston tensos (ISTTs). Te ISTTs can be comuted by usng a smla ntegaton aoac as n te STT comutaton; oweve, t s moe convenent to comute tem va sees eveson because numecal ntegaton can be costly fo lage m. As functons of te STTs, te ISTTs mang fom t to t ae ;a t; t a (4) ;ab ; ;j j 2 j ;a j 2;b (5) ;abc ; ;j j 2 j 3 ; ;j ;j 2 j 3 ;j j 2 ;j 3 ;j j 3 ;j 2 j ;a j 2;b j 3;c (6) ;abcd ; ;j j 2 j 3 j 4 ; ;j j 2 j 3 ;j 4 ;j j 2 j 4 ;j 3 ;j j 3 j 4 ;j 2 ;j j 2 ;j 3 j 4 ;j j 3 ;j 2 j 4 ;j j 4 ;j 2 j 3 ;j ;j 2 j 3 j 4 ; ;j j 2 ;j 3 ;j 4 ;j j 3 ;j 2 ;j 4 ;j j 4 ;j 2 ;j 3 ;j ;j 2 j 3 ;j 4 ;j ;j 2 j 4 ;j 3 ;j ;j 2 ;j 3 j 4 j ;a j 2;b j 3;c j 4;d (7) wee all ndces ae ;...;N, t ;t and t;t ae used fo te concse notatons. Note tat Eqs. (4 7) ae analytc n te STTs and eque no ntegaton. By alyng te fowad and nvese state tanston tensos, te STTs mang fom tme t to t s, wee t, t s 2t ;t f fo some fnal tme t f and t t s, can be eesented as ;abc t s ;t ; s ;abc ;a t s ;t t s;t t ;t a ; s ;a (8) ;ab t s ;t ; s ;ab ; s ;a ;b (9) ; s ;a ;bc ;ab ;c ;ac ;b ; s ;a ;b ;c (2)

3 67 PARK AND SCHEERES ;abcd t s ;t ; s ;abcd ; s ;abc ;d ;abd ;c ;ab ;cd ;ac ;bd ;ad ;bc ;a ;bcd ; s ;a ;bc ;ab ;c ;d ;d ;a ;bd ;ac ;b ;c ;d ;acd ;b ;ad ;b ;c ;a ;b ;cd ; s ;a ;b ;c ;d (2) wee all ndces ae ;...;N, t ;t, s ts ;t, and te ISTTs ae comuted by alyng Eqs. (4 7). In ote wods, once te STTs ae comuted fo te ente efeence tajectoy t ;t f, te ma fom an abtay ont n sace to some futue tme becomes a smle algebac manulaton. Note tat ; t s ;t can also be comuted by ntegatng te dffeental equatons gven n Eqs. (8 ) fo eac tme nteval t s ;t. One concen s numecal consstency wen Eqs. (8 ) ae ntegated ove a long duaton of tme. To addess ts, we note tat te efeence tajectoy can be segmented abtaly to meet te desed numecal accuacy. Anote queston tat may ase s te comutatonal dffculty (o te long ntegaton tme) as we consde te ge-ode solutons. Secfcally, assumng a system wt N 6, te mt ode analyss eques ntegaton of P m q 6q equatons. Fo examle, wen m 3, one must ntegate 554 equatons smultaneously. Howeve, te ge-ode solutons can be comuted offlne, and esecally wen te obt s eodc (e.g., alo obt), tese only need to be comuted once. Last, te comutaton of te atals of te dynamcs may be of concen. Hee, note tat tee ae symbolc manulatos avalable wc ovde automatc dffeentatons, and also note tat many of tese atals vans to zeo fo systems of sacecaft navgaton nteest. Once te atals ae comuted, tese equatons can be cast nto a fst-ode dffeental equaton fom and can be ntegated fowad n tme. III. Hge-Ode Extended Kalman Fltes Suose we ae gven te contnuous tajectoy model defned n Eq. (3). Because a sacecaft tacng model s usually dscete, consde te followng dscete system model: x t ; x ;t w (22) z x ;t v (23) wee x s te tue sacecaft state, w s te ocess nose etubng te sacecaft state, z s te actual measuement, s te measuement functon, and v s te measuement nose caactezng te obsevaton eo. Te ocess nose and measuement nose ae assumed to be noncoelated, tat s, Ev w T j, wt te autocoelatons: E w w T j Q j (24) E v v T j R j (25) fo all dscete tme ndexes and j. Hee, Q and R ae also nown as te dffuson and measuement nose matces, esectvely. A. Kalman Flte Altoug te Kalman flte algotm can be deved fom Bayes ule of condtonal denstes, as onted out by Jule and Ulmann [6], te Kalman flte can also be deved fom estmatons of a few exectatons nvolvng a state and a measuement [7]. To sow ts, consde te system model equatons (22) and (23) and suose we ae gven a state x wt mean m Ex j z and covaance matx P Ex m x m T j z at tme t. Te geneal flteng algotm can be defned as follows: Pedcton equatons: m Et ; x ;t w j z (26) P Eft ; x ;t w t ; x ;t w T j z g m m T (27) n Ex ;t v j z (28) wee n E j z s te exectaton of te measuement comuted at t. Udate equatons: K P xz P zz (29) m m K z n (3) P P K P zz KT (3) wee K s nown as te Kalman gan matx, P xz s te cosscovaance matx of te state and te measuement, P zz s te covaance matx of te measuement, z s te obsevaton, and te dffeence between te actual and edcted measuement (.e., z n ) s called te esdual o nnovaton. Wen a lnea dynamcal system and Gaussan lnea measuement functon ae consdeed, Eqs. (26 3) smlfy to te conventonal lnea Kalman flte (LKF). B. Extended Kalman Flte Fo estmaton oblems, te LKF s obably te most wellnown flteng tecnque. Te LKF allows one to comute te mnmum mean-squae-eo (MMSE) soluton; oweve, t can only be used fo lnea systems, and n geneal, cannot be used fo tajectoy navgaton. In conventonal sacecaft tajectoy navgaton, te s usually mlemented. [In actce, te extended Kalman flte s mlemented fo tajectoy navgaton often n a squae-oot nfomaton flte (SRIF) o n U-D flte fomulaton fo numecal ecson.] Te s based on te flte algotm gven n Eqs. (26 3), but assumes te tue tajectoy s wtn te bounday wee te lnea aoxmaton can suffcently model te tajectoy dynamcs and ts statstcs. Unde ts assumton, te mean tajectoy s oagated accodng to te detemnstc soluton flow and te covaance matx s maed lnealy [8,9]. edcton equatons: m t ; m ;t (32) P t ;t P T t ;t Q (33) n m ;t (34) udate equatons: K P xz P zz P HT H P HT R m m K z n (35) (36) P P K P zz KT P K H P (37) wee m ;t s te measuement functon evaluated at t as a functon of m and =@x s te measuement atal comuted at t. Note tat te STM t ;t n Eq. (33), wc s te fst of te STTs, s comuted along te evously udated mean tajectoy, tat s, m.

4 PARK AND SCHEERES 67 Among te many motant oetes of te extended Kalman flte, we ont out two wc wll be dscussed n te examle secton n moe detal. Consdeng te gan Eq. (35) and te mean udate Eq. (36), we obseve tat as te a o covaance matx becomes moe accuate (.e., P! ) te flte values te esdual less (.e., te actual measuement s tusted less). On te ote and, as te measuement becomes moe accuate (.e., R! ) te flte values te esdual moe (.e., te actual measuement s tusted moe). Teefoe, effectve wegtng of te esdual s a ctcal comonent of maxmzng te flte efomance. C. Hge-Ode Numecal Extended Kalman Flte In devng te ge-ode numecal extended Kalman flte (HN), we assume tat te efeence tajectoy and ts geode state tanston tensos ae ntegated fo eac tme nteval between te measuements accodng to Eqs. (8 ). Unde ts assumton te local tajectoy moton can be maed analytcally ove ts tme nteval wle ncooatng nonlnea effects, and te same analogy ales wen mang te tajectoy statstcs. We note tat ts ocess s numecally qute ntensve consdeng ge-ode solutons; oweve, ts can yeld a moe accuate flte soluton. Once te ge-ode state tanston tensos ae avalable fo some tme nteval t ;t, te mean and covaance matx of te elatve dynamcs at t can be maed analytcally to t as functons of te obablty dstbuton at t. Fom t to t, te oagated mean and covaance can be stated as [4,5] m x E x Xm P j! ; t ;t E x x E x m X m X m t ;t j; q t ;t E x!q! ; q x j mj x (38) x x x q m mj (39) wee f j ; j g2f;...;ng. As te ode of te soluton nceases, tat s, m!, te ge-ode soluton yelds te tue mean and covaance matx comuted fom Monte Calo smulatons. Now, te only unnowns n Eqs. (38) and (39) ae te exectatons (.e., moments) of te devatons. Even f te state at tme t s Gaussan, excet fo te case m, t s obvous tat te maed tajectoy dstbuton s no longe Gaussan due to system nonlneaty, and ence exact comutaton eques comutaton of te ge-ode moments. In atcle-based fltes, ts oblem s emeded by usng an ensemble of samle onts to aoxmate te obablty dstbuton, weeas a moe fomal aoac s to use te Edgewot/Gam-Cale [] o Lalace aoxmatons to aoxmate te osteo densty functon. In tajectoy navgaton, oweve, te Gaussan assumton as sown to ovde a suffcently accuate statstcal aoxmaton. Hence, we assume tat te udated estmates ae Gaussan and we mlement te jont caactestc functon to comute te geode moments u to 2mt ode as aaent fom Eq. (39). By assumng te udated state can be aoxmated wt Gaussan statstcs, te ge-ode moments ae functons of te fst two moments. Moeove, f we consde a zeo ntal mean, all te odd moments of te ntal condtons vans and te equatons fo te oagated mean and covaance matx smlfy a geat deal. Now, suose at tme t, te state estmate as mean m and covaance matx P. Also, let xt m x be te tue tajectoy we want to estmate. Followng te Kalman flte algotm, te HN algotm s gven as follows: HN edcton equatons: m E t ; m x ;t w t ; m ;t m x t ; m ;t Xm t ;t E x x (4) P! ; j E n t ; m x ;t w o t j ; m x ;t w j m m j (4) X m X m!q! ; t ;t j; q q t ;t E x x x x q m x m j x Q j (42) n E t ; m x ;t v t ; m ;t n x t ; m ;t Xm t ;t E x x! ; (43) wee te STTs [.e., t ; m ;t ] ae comuted along te soluton flow t ; m ;t and ; @x x t ;m ;t (44) Note tat t ; m ;t denotes tat te measuement functon s evaluated at t as a functon of te soluton flow t ; m ;t. [It s motant to note tat t ; m ;t m ;t n geneal because m t ; m ;t fo geneal nonlnea systems.] Te atal devatves ; t ;t u to fout ode ae defned as ;abc t ;t ; ;abc ;a t ;t ; ;a (45) ;ab t ;t ; ;ab ; ;a ;b (46) ; ;a ;bc ;ab ;c ;ac ;b ; ;a ;b ;c (47) ;abcd t ;t ; ;abcd ;acd ;b ;a ;bcd ;ab ;cd ;ad ;b ;c ;a ;b ;cd ; ; ;abc ;d ;abd ;c ;ac ;bd ;ad ;bc ;ab ;c ;d ;ac ;b ;d ;a ;bc ;d ;a ;bd ;c ; ;a ;b ;c ;d (48) wee t ;t s used fo a concse notaton and tat tese ae smla to te dffeental equatons of te STTs gven n Eqs. (8 ). Note tat ts edcton ste s a smle algebac oeaton once te STTs ae comuted fo te tme nteval t ;t.

5 672 PARK AND SCHEERES HN udate equatons: P zz j E z n z T j n E z z j n n j E n t ; m x ;t v o t j ; m x ;t v j R j Xm X m!q! ; t ;t j; q t ;t q E x x x x q j n n n n j (49) j P xz E x m z T j n E x z j m n j n t ; m x ;t t j ; m x ;t E v j X m o X m m n j!q! ; q t ;t j; q m w t ;t E x x x x q j n (5) K P xz P zz (5) m m K z n (52) P P K P zz KT (53) Note tat f we consde te measuement functon Eq. (23) to be lnea n x, Eq. (43) smlfes to n t ; m ;t n t ; m ;t X m ; t ;t E x x t ; m ;t ;! ; m m ;t (54) and gves n ; m. Alyng ts esult, Eqs. (49) and (5) smlfy to j X m X m P zz ; j; E x x x x q R j!q! ; t ;t ; q t ;t q ; j; E x x R j n j n n n H P HT R j (55) P xz j j; E x E X m X m!q! ; t ;t ; q t ;t q x x x q x x j; m m n n j j j P HT j (56) wc ndcates tat te measuement edcton and udate equatons ae dentcal to te algotm. Also note tat wen m, te HN becomes te algotm as sown n Eqs. (32 37). D. Hge-Ode Analytc Extended Kalman Flte Fom te devaton of te HN, t s obvous tat we can also deve a ge-ode analytc extended Kalman flte (HA) by assumng tat te efeence tajectoy and te ge-ode solutons (.e., STTs) ae comuted ove some tme san befoe flteng. Te flte algotm s smla to te HN excet tat te ont of sees exanson s now wt esect to te ntal efeence tajectoy, not te udated mean as n te HN algotm. Suose te STTs ae comuted fo te tme nteval of t ;t f and let x t ; x ;t eesent te efeence tajectoy fo t 2t ;t f, wee x as mean m and covaance matx P. Moeove, let xt x x be te tue tajectoy we want to estmate. Followng te Kalman flte algotm, te HA algotm s gven as follows: HA edcton equatons: m t ; x ;t Xm P j X m X m q E x x x x q! ;!q! ; t ;t j; q t ;t n x ;t Xm t ;t E x x (57) m x m j x Q j! ; t ;t E x x (58) (59) wee te STTs ae comuted along x t ; x ;t and ; @x (6) x x HA udate equatons: P zz j R j Xm X m q E x x x x q P xz j X m X m q!q! ; t ;t j; q t ;t n n j (6)!q! ; q t ;t j; t ;t E x x x x q m n j (62) K P xz P zz (63) m m K z n (64) P P K P zz KT (65) As n te HN case, te udate equatons fo te HA become te same as te wen we consde a measuement functon tat s lnea n x. Also, note tat wen m (.e., fst ode), te HA becomes te lnea Kalman flte, not te. (We call t te ge-ode analytc extended Kalman flte, not te ge-ode lnea Kalman flte, because te edcton equatons

6 PARK AND SCHEERES 673 ae nonlnea n geneal.) Te sueoty of te ove te LKF s clealy demonstated by Maybec []. Howeve, wen te tue tajectoy s wtn te convegence adus of te efeence tajectoy, we sall see late tat te HA can ovde a moe accuate soluton and faste convegence tan te. An advantage of te HA s tat t can be ecomuted and used onlne wt no numecal ntegatons. E. Unscented Kalman Flte Te unscented Kalman flte (), fst ntoduced by Jule and Ulmann, s beng mlemented n a dvese feld of engneeng, scence, and economcs due to ts smlcty wle ovdng faste convegence and bette accuacy tan te extended Kalman flte. Te s ntalzed wt a set of edetemned sgma onts and ncooates te second-ode tajectoy nfomaton n te flte model. Detaled devaton and algotm can be found n [6, 4]. IV. Examles In ts secton, we esent seveal smulatons of a alo obt, wc s a eodc obt wee te n-lane and out-of-lane fequences ae te same, comuted based on te CR3BP. Te govenng equatons of moton fo CR3BP, n nondmensonal fom, ae gven as [5] x wee U @z x2 y (67) (68) (69) x 2 y 2 z 2 =2 (7) 2 x 2 y 2 z 2 =2 (7) Hee, U s te CR3BP otental and = S. Consde a alo obt about te sun Eat L ont n a nondmensonalzed fame, wc can be dmensonalzed by alyng te lengt scale of AU : m, wee AU stands fo astonomcal unt, and te tme scale of =! E. Fgue sows te efeence (nomnal) tajectoy fo one obtal eod (77:86 days), wc coesonds to case gven n Table. Te ntal condtons fo tese obts ae (n nondmensonal unts) case t : ; :; : T v case t :; : ; : T case 2 t : ; :; : T v case 2 t :; : ; : T Fo te measuement model, we assume a smle lnea model wee only te y coodnate s obseved, tat s, z y v (72) y coodnate (m) z coodnate (m) x x coodnate (m) x 8 x 5 z coodnate (m) 5 x 5 y coodnate (m) x 8 x coodnate (m) x x coodnate (m) y coodnate (m) x 8 x 5 Fg. Nomnal alo obt about te sun Eat L ont. wee y eesents te vetcal oston comonent of te state vecto and v eesents te measuement eo. Ts measuement model can be vewed as a ange measuement obtaned by otcal magng of te Eat elatve to dstant stas o a vey long baselne ntefeomety (VLBI) measuement. Te measuement nose s assumed to be. m fo eac ange measuement. Ts lnea assumton smlfes te oblem a geat deal because te measuement senstvty does not eque te comutaton of te ge-ode atals. Ts way, t s ease to undestand te effect of te nonlnea obt uncetanty oagaton on flte efomance. Intally, te sacecaft state s assumed to be a zeo mean Gaussan wt oston uncetantes of m and velocty uncetantes of : m=s. [Te ntal covaance matx s a dagonal matx wt m 2 and : m=s 2 fo oston comonents and velocty comonents, esectvely.] Te ntal mean and covaance matx ae maed usng te STT aoac fo m f; 3g, unscented tansfomaton, and Monte Calo smulatons based on 6 samle onts. Fgue 2 sows te mean and te ojecton of te - covaance matx onto te x y lane afte one obtal eod. Assumng te Monte Calo smulaton s te tue soluton, te esult sows tat te td-ode soluton s te most accuate aoxmaton, weeas te lnea soluton fals to caacteze te obt uncetanty dstbuton. We now consde te same ntal uncetantes, but assume te ntal guess (mean) s off by m fo te oston comonents and : m=s fo te velocty comonents so tat tey le on te bounday of te ntal - ellsod. A set of seudomeasuements ae comuted based on te efeence tajectoy wt a 2-day ncement. Usng te same measuements, te ntal mean and covaance matx ae maed and solved usng te, te, te td-ode HN, and te td-ode HA. Fo te HA, because te tajectoy s eodc, te STTs ae comuted and stoed fo only one obtal eod, wc s dvded nto two segments fo numecal consstency, and eveson of te sees s aled to ma states analytcally. [Note tat all flte smulatons ae based on sngle uns wee te same set of seudomeasuements ae consdeed (.e., andom noses added to efect measuements). We ave also smulated many dffeent set of seudomeasuements on te sde and ave obtaned neglgble dffeence n te flte efomance gven te state and measuement uncetantes consdeed n ts ae.] z coodnate (m) x 5 Table Halo obt maxmum amltudes wt esect to te sun Eat L ont Cases A x,m A y,m A z,m 245, ,228 37, ,69 668,46 39,5 5 5 L

7 674 PARK AND SCHEERES Fgue 3 sows te a o (edcted) and a osteo (udated) oston and velocty oot sum squae eos, wee R xx yy zz and V uu vv ww, and eesents te ; comonent of te covaance matx. A sudden do n te uncetantes gt afte days s due to te fact tat te ntal covaance matx s qute lage and eques at least sx ndeendent measuements to obtan a well-defned (.e., educed to te measuement nose level n all dectons) a osteo covaance matx. Te esult sows tat te oveestmates te uncetantes (.e., assumes tey ae smalle tan tey ae n actualty) wle te, HN, and HA ovde consevatve uncetanty estmates. Fgue 4 sows te magntude of te absolute oston and velocty eos, tat s, te magntude of te dffeence between te udated mean and te tue state. Te esult sows tat te does not efom well as comaed to te ge-ode fltes. Ts clealy exlans te motance of nonlnea obt uncetanty oagaton. Te covaance matx comuted by usng te fst-ode metod (.e., ) oveestmates te soluton, and ence, te esdual s tusted less. On te ote and, te and te ge-ode fltes edct moe consevatve uncetantes and moe effectvely balance te a o uncetantes and te actual measuements (.e., measuements ae valued moe tan te a o nfomaton n ts case). Fgues 5 and 6 ae based on te same flte setu excet tat te measuements ae udated evey 5 days. It sows tat tee s not muc dffeence n te oagated uncetantes, but te absolute eos ae comuted moe accuately n and ge-ode flte uns. y coodnate (m) x 4 m= m=3 UT MC x coodnate (m) x 8 a) Pojected onto te x-y lane y coodnate (m) x m= m=3 UT MC x coodnate (m) x 8 b) Lage lot of (a) Fg. 2 Sun Eat alo obt: mean and - eo ellsod ojected onto te oston lane afte beng oagated fo one obtal eod. Fgues 7 and 8 sow te HN esults fo cases m 2f; 2; 3g. As mentoned eale, note tat te case m s dentcal to te fomulaton. Te esult sows tat te ge-ode fltes, m 2f2; 3g, ovde sueo flte efomance ove te fst-ode case and t s obseved tat te second-ode effect contans most of te system nonlneaty, ndcatng tat te second-ode flte s suffcent fo an accuate nonlnea flte n ou examle. Fgues 9 and sow te HA uncetantes and absolute eo lots, esectvely, fo m 2f; 2; 3g. Te uncetantes fo m ae smla to te soluton and fo m 2 ae smla to te case m 3 as sown n Fg. 3. Te absolute eo lot sows tat all tee fltes ovde good estmaton efomance even fo te case m. Ts s exected because te seudomeasuements ae comuted based on te efeence tajectoy wc te STTs ae comuted based on. In ote wods, te efeence tajectoy can be tougt of as a egesson soluton fo te smulated measuements. To analyze te obustness of te ge-ode flteng tecnques, te seudomeasuements ae now geneated fom te case 2 alo obt gven n Table. Fgues and 2 sow te smulated flte solutons. Te esults sow tat te ge-ode solutons ae sueo ove te lnea fltes, tat s, and HA fo m. As exected, ts ndcates tat te lnea Kalman flte s only feasble wen te efeence tajectoy s suffcently close to te tue tajectoy. Te HAs fo m>, oweve, ave moe flexblty n (m) 3 2 Fg. 3 ncement measuement udate. Poston Eo (m) Comason of te oot sum squae eos wt a 2-day Fg. 4 measuement udate. Velocty Eo 7 8 Comason of te absolute eos wt a 2-day ncement

8 PARK AND SCHEERES HN (m=) HN (m=2) HN (m=) HN (m=2) (m) (m) Fg. 5 ncement measuement udate Comason of te oot sum squae eos wt a 5-day Fg. 7 Comason of te oot sum squae eos wt a 2-day ncement measuement udate HN (m=) HN (m=2) HN (m=) HN (m=2) Poston Eo (m) Velocty Eo Poston Eo (m) Velocty Eo 7 7 Fg. 6 measuement udate. 8 Comason of te absolute eos wt a 5-day ncement Fg. 8 measuement udate. 8 Comason of te absolute eos wt a 2-day ncement te efeence tajectoy. Te oveall flte convegence s slgtly slowe tan te evous cases because te ntal mean s assumed to be te same as n te evous cases, and tus, t s fate away fom te tue tajectoy (.e., te tajectoy n wc te seudomeasuements ae geneated). In ts study, te equed ntegaton of N N 2 42 equatons [o N NN =2 27 equatons f te covaance matx s dectly ntegated] and te equed ntegaton of 2N N 78 equatons between eac measuement udate, and n te actual flte uns, te was slgtly faste tan te. Te HNs fo m> ovde sueo esults ove te lnea fltes (even wen m>2); oweve, te comutatonal load nceases sgnfcantly as m nceases. Fo examle, te td-ode HN eques ntegaton of 554 equatons. On te ote and, te HA does not eque any ntegaton n te actual flteng ocess. Te most exensve numecal oeaton n te HA s te ge-ode moment comutaton; oweve, tee exst vaous tecnques fo effcent comutaton of moments. Hence, fo mssons wt edetemned efeence tajectoes, te ge-ode analytc flte may be sutable fo te tajectoy navgaton wle obtanng faste convegence and a moe accuate flte soluton tan te. To be moe secfc, avng te ge-ode sem-analytc metod onboad a sacecaft o a launc vecle ovdes a comlete soluton sace fo te gven efeence tajectoy deendng on te ode of soluton. Hence, fo oblems wee te nonlneaty s sgnfcant and eques a ad convegence, suc as sacecaft launc, lanetay/small-body obt nseton, o autonomous ecson landng, alyng te HA can esult n a moe ad, accuate, and obust state estmaton tan te. (m) 3 2 HA (m=) HA (m=2) Fg. 9 ncement measuement udate. HA (m=) HA (m=2) Comason of te oot sum squae eos wt a 2-day

9 676 PARK AND SCHEERES 3 2 HA (m=) HA (m=2) HA (m=) HA (m=2) 3 2 HA (m=) HA (m=) Poston Eo (m) Velocty Eo Poston Eo (m) Velocty Eo 7 7 Fg. measuement udate. 8 Comason of te absolute eos wt a 2-day ncement Fg. 2 8 Comason of te absolute eos wt a 2-day ncement measuement udate based on te alo obt case 2. (m) 3 2 HA (m=) Fg V. Conclusons We deved two Kalman-tye fltes, called ge-ode numecal extended Kalman flte and ge-ode analytc extended Kalman flte, by dectly alyng te ge-ode solutons to te Kalman flte algotm. Tese ge-ode fltes wee comaed wt te conventonal extended Kalman flte and te unscented Kalman flte based on alo obts comuted n a estcted tee-body oblem fame about te sun Eat L ont. Te flte smulatons wee caed out assumng te dynamcs of te system ae efectly nown, but tee ae eos n te ntal state and n te measuements. Te esults sowed tat a ge-ode flte ovdes faste convegence, a sueo flte soluton, and moe flexblty n te ntal guess ove lnea fltes. Also, te Gaussan assumton of te a osteo state yelded a suffcent aoxmaton even fo nonlnea fltes. Fo te cases wee te efeence tajectoy was elatvely close to te tue tajectoy, te HA ovded solutons essentally equvalent to bot te and te HN, and yelded a muc faste flte ocess. Ts ndcates tat once tajectoy solutons ae stoed on a sacecaft, an autonavgaton ocesso tat ncooates tajectoy nonlneaty and allows fast convegence may be feasble n actce. HA (m=) Comason of te oot sum squae eos wt a 2-day ncement measuement udate based on te alo obt case 2. Acnowledgement Te eseac descbed n ts ae was sonsoed by a gant fom te Jet Poulson Laboatoy, Calfona Insttute of Tecnology wc s unde contact wt te Natonal Aeonautcs and Sace Admnstaton. Refeences [] Doucet, A., de Fetas, N., and Godon, N., Sequental Monte Calo Metods n Pactce, Snge Velag, Beln, 2. [2] Pa, R., and Sceees, D., Nonlnea Mang of Gaussan State Covaance and Obt Uncetantes, AAS Pae 5-7, Jan. 25. [3] Pa, R., and Sceees, D., Nonlnea Mang of Gaussan State Uncetantes: Teoy and Alcaton to Sacecaft Contol and Navgaton, AAS Pae 5-44, Aug. 25. [4] Pa, R., and Sceees, D., Nonlnea Mang of Gaussan Statstcs: Teoy and Alcatons to Sacecaft Tajectoy Desgn, Jounal of Gudance, Contol, and Dynamcs, Vol. 29, No. 6, 26, [5] Pa, S., Nonlnea Tajectoy Navgaton, P.D. Tess, Unvesty of Mcgan, Ann Abo, MI, 27. [6] Jule, S., and Ulmann, J., Unscented Flteng and Nonlnea Estmaton, Poceedngs of te IEEE, Vol. 92, No. 3, Mac 24, [7] Kalman, R., A New Aoac to Lnea Flteng and Pedcton Poblems, Tansactons of te ASME, Jounal of Basc Engneeng, Sees D, Vol. 82, Mac 96, [8] Beman, G., Factozaton Metods fo Dscete Sequental Estmaton, Vol. 28, Academc Pess, New Yo, 977. [9] Montenbuc, O., and Gll, E., Satellte Obts, 2nd ed., Snge, New Yo, 2. [] Maybec, P., Stocastc Models, Estmaton, and Contol, Vol. 2, Academc Pess, New Yo, 982. [] Jule, S., Ulmann, J., and Duant-Wyte, H., A New Aoac fo Flteng Nonlnea Systems, Poceedngs of te Amecan Contol Confeence, June 995, Vol. 3, [2] Jule, S., Ulmann, J., and Duant-Wyte, H., A New Metod fo te Nonlnea Tansfomaton of Means and Covaances n Fltes and Estmatos, IEEE Tansactons on Automatc Contol, Vol. 45, No. 3, Mac 2, [3] Jule, S., and Ulmann, J., Te Scaled Unscented Tansfomaton, Poceedngs of te Amecan Contol Confeence, 22, Vol. 6, [4] Wan, E., and van de Mewe, R., Kalman Flteng and Neual Netwos: Te Unscented Kalman Flte, Wley, New Yo, 2, ca. 7. [5] Pollad, H., Celestal Mecancs, Te Matematcal Assocaton of Ameca, 976.