MPC for Coupled Station Keeping, Attitude Control, and Momentum Management of Low-Thrust Geostationary Satellites

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1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES h:// MPC for Couled Saon Keeng, Aude Conrol, and Momenum Managemen of Low-Thrus Geosaonary Saelles Walsh, A.; D Carano, S.; Wess, A. TR26-47 July 26 Absrac Ths aer develos a model redcve conrol (MPC) olcy for smulaneous saon keeng, aude conrol, and momenum managemen of a nadr-onng geosaonary saelle equed wh hree reacon wheels and four gmbaled elecrc hrusers ha are locaed on he an-nadr face of he saelle. The MPC olcy works n combnaon wh an nnerloo SO3q-based aude conroller ha ensures he saelle manans a nadr-onng aude. The MPC olcy s able o manan he saelle s oson whn a rescrbed laude and longude wndow, whle mnmzng he v requred by he hrusers. The MPC olcy also enforces hruser onng consrans and manages he saelle s sored momenum. Wh reference o smulaon resuls, we exlan how he MPC s uned for saon keeng, he need for an nner-loo aude conroller, and how hese searae sysems work ogeher o acheve all he conroller s objecves. 26 Amercan Conrol Conference (ACC) Ths work may no be coed or reroduced n whole or n ar for any commercal urose. Permsson o coy n whole or n ar whou aymen of fee s graned for nonrof educaonal and research uroses rovded ha all such whole or aral coes nclude he followng: a noce ha such coyng s by ermsson of Msubsh Elecrc Research Laboraores, Inc.; an acknowledgmen of he auhors and ndvdual conrbuons o he work; and all alcable orons of he coyrgh noce. Coyng, reroducon, or reublshng for any oher urose shall requre a lcense wh aymen of fee o Msubsh Elecrc Research Laboraores, Inc. All rghs reserved. Coyrgh c Msubsh Elecrc Research Laboraores, Inc., 26 2 Broadway, Cambrdge, Massachuses 239

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3 MPC for Couled Saon Keeng, Aude Conrol, and Momenum Managemen of Low-Thrus Geosaonary Saelles Alex Walsh, Sefano D Carano : and Avsha Wess ; Absrac Ths aer develos a model redcve conrol (MPC) olcy for smulaneous saon keeng, aude conrol, and momenum managemen of a nadr-onng geosaonary saelle equed wh hree reacon wheels and four gmbaled elecrc hrusers ha are locaed on he an-nadr face of he saelle. The MPC olcy works n combnaon wh an nnerloo SO3q-based aude conroller ha ensures he saelle manans a nadr-onng aude. The MPC olcy s able o manan he saelle s oson whn a rescrbed laude and longude wndow, whle mnmzng he v requred by he hrusers. The MPC olcy also enforces hruser onng consrans and manages he saelle s sored momenum. Wh reference o smulaon resuls, we exlan how he MPC s uned for saon keeng, he need for an nner-loo aude conroller, and how hese searae sysems work ogeher o acheve all he conroller s objecves. I. INTRODUCTION Recenly, here has been sgnfcan neres n he ulzaon of low-hrus, hgh-secfc-mulse elecrc engnes for saelle conrol boh for roellan-effcen nerlaneary rajecory desgn [], [2], and for roune orbal saon keeng [3] [8]. Elecrc hrusers have consderably hgher secfc mulse han ha of chemcal hrusers, allowng for a reducon n roellan mass, hereby lowerng launch coss and enablng larger ayloads [9]. Convenonal chemcal hrusers roduce hundreds of newons for orb conrol [], and n geosaonary Earh orb (GEO), may be fred oenloo once every wo weeks o comensae for saelle drf []. In conras, he maxmum hrus currenly generaed by sae-of-he-ar elecrc roulson sysems s n he hundreds of mllnewons [2], whch for GEO, s on he order of he erurbaon forces [3]. Thus, he ulzaon of elecrc hrusers oses new conrol challenges, requrng near connuous oeraon n order o rovde he necessary mared mulse, and sugges he use of closed-loo feedback conrol [3], [6], [7]. The roblem of mananng a low-hrus saelle n GEO s reaed n [6], where he auhors roose a model redcve conrol (MPC) olcy for smulaneous saon keeng and momenum managemen. The auhors develo her conrol for a saelle equed wh hree reacon wheels and sx dual-axs hrusers mouned on each face of he saelle, enablng ure hruss and orques aled n any drecon and Ph.D. Canddae, Dearmen of Aerosace Engneerng, Unversy of Mchgan, Ann Arbor, MI 489, USA. Emal: aexwalsh@umch.edu : Senor Prncal Member Research Saff, Msubsh Elecrc Research Laboraores, Cambrdge, MA 239, USA. Emal: dcarano@merl.com ; Member Research Saff, Msubsh Elecrc Research Laboraores, Cambrdge, MA 239, USA. Emal: wess@merl.com abou any axs. In hs aer, we exand on [6] o consder a realsc hruser confguraon wh gmbaled hrusers locaed on he saelle s an-nadr face [3], [4], [5]. Ths arrangemen leaves room on he saelle s oher faces for anennas and solar anels, whou he rsk of hruser lume mngemen. However, he confguraon also oses new challenges for conrol desgn, severely resrcng he conrol acon and makng more dffcul o rejec orbal erurbaons. Furhermore, o boh effcenly counerac he erurbaon forces and allow for momenum unloadng, he hrusers are ermed o gmbal. In dong so, hey may also generae undesrable orques on he saelle, sgnfcanly coulng he saelle orbal and aude dynamcs. Thus, he need o rea concurrenly boh he fas aude dynamcs and he slow orbal dynamcs resen comuaonal challenges n resource consraned hardware. Wh fas dynamcs, a small dscrezaon me-se s requred. However, a long MPC redcon horzon s necessary for saon keeng. A small dscrezaon me se accomaned by a long redcon horzon resuls n a large and comuaonally challengng omzaon roblem. The man conrbuon of hs aer s he desgn of an nner-ouer loo conrol archecure ha overcomes he aforemenoned challenges, where an ouer-loo MPC olcy oeraes n conjuncon wh an nner-loo SO3q-based aude conroller for oson, aude, and momenum conrol. The aer begns wh he descron of he saelle model n Secon II, followed by saon-keeng MPC, model lnearzaon, and consrans n Secon III. The nnerloo aude conroller s nroduced n Secon IV. Closng remarks follow he man conrbuon n Secon V. The followng noaon wll be used n all dervaons. A frame of reference F a s defned by a se of hree orhonormal dexral bass vecors, and can be wren as a T vecrx a F, where F r a a 2 a a 3 s. A hyscal vecor descrbng he oson of a on y 2 relave o on y s y wren as v 2y. A hyscal vecor can be resolved n any reference frame, such as, v a v a ` a 2 v a2 ` a 3 v a3 T rv a v a2 v a3 sf a F v a a, where he subscr a on he column marx of comonens v a ndcaes ha he hyscal T vecor v s resolved n F a. Snce v F v T a a F v b b, alyng a lef do-roduc wh b F yelds v b b F F T v a a C ba v a. The marx C ba P SO3q s a drecon cosne marx (DCM) defned by C T ba b F F, where a SO3q C P R 3ˆ3 C T C, de C u, and s he deny marx. Prncal roaons abou he a bass vecor

4 by an angle α are denoed by C ba C αq. The cross oeraor mas a hree-dmensonal vecor o a skew symmerc marx, ha s qˆ : R 3 Ñ so3q, where so3q U P R 3ˆ3 U U T u. The uncross ma does he oose where q v : so3q Ñ R 3, where vˆq v P R 3. The skew-symmerc rojecon oeraor s P a Uq 2 U UT q, for all U P R nˆn. II. SATELLITE MODEL Consder he saelle deced n Fg. ha has hree axsymmerc reacon wheels aached o a rgd bus n an orhogonal and mass balanced confguraon. Four elecrc hrusers are locaed on he an-nadr face of he saelle. The hrusers rovde hrus n a lne concden wh her oson and he cener of mass of he saelle, bu are able o gmbal away from hs nomnal drecon. The saelle s cener of mass s denoed by c and has a bus-fxed frame F. The frame F g s he Earh Cenred Ineral (ECI) frame, w s an unforced arcle, and s assumed ha w s collocaed wh he cener of he Earh. Each hruser has a frame F, locaed a on for hrusers,..., 4. w f f 2 3 F g ϕ hrus 25 P r c 2 F c m θ hrus 4 b cenrod.25 m 5 m 7.5 m 2» r cb 2.25 f 2.5 fl m 3.75 Fg. : Saelle B wh laform P showng elecrc hrusers. A force f s shown when he hrus s erfecly n lne wh he cener of mass of he saelle. F 3 The equaons of moon of he saelle are gven by J Bc 9ωg g ` µ rcw g r cw 3 a g ` f hrus m B :r cw ` ω gˆ g g, `JBc ω g ` J s 9γ ` J s η τ ` τ hrus, 9C g ω gˆ C g, :γ η, () where r cw g s he oson of he saelle and γ s a column marx conanng he angle of roaon of each reacon wheel. The vecor ω g s he angular velocy of F relave o F g resolved n F. The marx J Bc s he momen of nera of he saelle B relave o s cener of mass, resolved n F. Each hruser ales a force f a each on, and s resolved n a frame dedcaed o each hruser F. The reacon wheel array has a momen of nera J s and he wheels are conrolled wh an acceleraon η. The erm a g reresens he exernal erurbaons on he saelle due o Earh s non shercal gravaonal feld, solar and lunar gravaonal aracon, solar radaon ressure (SRP), and are defned n [6, Eq. (2)]. The erm τ reresens he SRP erurbaon orque, whch assumes oal absoron, and s gven by [6,. 229]. The hrusers are laced on he rear sde of he saelle, and each hruser has a oson r b. The nomnal hrus drecon s along he vecor r c 2. The frame F has s un vecor defned such ha 2 r c { r c. The DCM C s defned usng he geomery of he saelle. Each force f s aled a on. The oal force due o he hrusers aled o he saelle resolved n F g s gven by f hrus g f g C T g C T f. (2) The orque roduced by he elecrc hrusers relave o he saelle s cener of mass, resolved n he bus fxed frame F s gven by τ hrus r cˆ C T f. (3) The gmbal angle of he hruser s a quany ha can be evaluaed once he conroller has found he conrol nu f. The gmbal angles comrse wo angles, α and α 3, whch are deced n Fg. 2. The angles are defned such ha f can be descrbed by wo rncal roaons, yeldng f C T 3 α 3 qc T α q 2 f. (4) The negave sgn s due o he convenon ha a zero gmbal angle, he hrus s n he 2 drecon. The conroller comues he hrus comonens resolved n F, raher han a magnude and wo angles, because he laer roduces a nonlnear roblem and makes he omzaon much more dffcul. F 2 3 α3 Fg. 2: A saelle hruser wh gmbal angle and lnear onng consran. In hs fgure, he force f does no sasfy he onng consran as s ousde he blue shaded yramd. α f III. MPC FOR STATION KEEPING, MOMENTUM MANAGEMENT, AND ATTITUDE CONTROL The man conrbuon of hs aer s he desgn of an archecure ha allows for smulaneous saon keeng, aude conrol, and momenum managemen for he saelle shown n Fg.. The conroller s objecves are o

5 ) manan he saelle n he saon keeng wndow, 2) mnmze fuel consumon of he saelle, 3) ensure hrus remans whn rescrbed consrans, 4) manan a nadr-onng aude, and 5) manage he reacon wheel seeds. Followng [6], an MPC olcy s deal for hs ask as s naurally able o handle hese objecves. The MPC conroller solves a recedng-horzon fne-me omal conrol roblem ha s based on a sysem model subjec o onwse-nme sae and conrol consrans and a user-defned cos funcon. By usng lnearzed equaons of moon, lnear equaly and nequaly consrans, and quadrac coss on he saes and conrol, he MPC can be formulaed as a quadrac rogrammng (QP) roblem. A QP can be solved quckly and effcenly, whch makes sued for saelles where on-board comung ower may be lmed. A. Lnearzaon for MPC The dynamcs () are resolved n Hll s frame F h and lnearzed abou a nomnal crcular orb wh mean moon n, angular velocy ω, n a nadr-onng confguraon defned by F d wh zero reacon wheel seed. The aude error s aramerzed by a 3 2 Euler angle sequence gven by C d C g C T dg C φqc 2 θqc 3 ψq, where he angles ψ, θ and φ corresond o he yaw, ch and roll of he saelle. The Euler angles are condensed o a column marx δθ rδφ δθ δψs T. Defnng Ω dag 3n 2,, n 2 q, he lnearzed equaons of moon are gven by δ:r Ωδr 2ωˆ δ9r ` a h ` C T dh m B δθ 9 ωˆ δθ ` δω (5b) δ 9ω J Bc ωˆ JBc J Bc ω qˆqδω J Bc ωˆ J sδ 9γ J Bc J s η ` C T f (5a) r cˆ C T f. (5c) The dscree-me lnear model wh me se s x k` Ax k ` B w,d w k ` Bu k, (6) where w k models he dsurbances a h. Equaon (6) s used along a redcon horzon k,..., N. Durng he redcon horzon, we can aroxmae he dsurbances w k, k,..., N as he dsurbances assocaed wh he saelle a s desred oson, snce he saelle wll always be ke very close o he desred oson durng correc oeraon. The conrol nu u k s u k rf T... f T 4 4 η T s T. B. Consrans The lnearzed dynamcs are resolved n F h, wh δr rr r 2 r 3 s T. As such, he saon keeng consrans are gven by δr 2 ď r anλ,max q, δr 3 ď r anλ 2,max q, where λ,max and λ 2,max are he maxmum olerable longude and laude errors, resecvely, and where r s he nomnal orbal radus. The hrus onng consran s enforced by consranng he hrus o reman n he neror of four lanes, and s descrbed by he elemen-wse consrans D f ě, (7) where each of he four rows of D conans a normal vecor descrbng a lane. The four lanes n (7) are vsualzed n Fg. 2. The hrus magnude consran s a norm consran f ď f max, whch s nonlnear. Ths can be aroxmaed as lnear by mosng he consran ˇˇf ˇ ď f max, where f max r s T f max, and f max s he maxmum allowable hrus. Noe ha whle hs aroxmaon works well n our case, several oher aroxmaons of he norm are ossble. The aude error of he saelle s also subjec o he elemen-wse consran δθ ď θ max, where θ max s he maxmum allowable aude error. C. MPC for Saon Keeng Wh Aude Conrol The MPC olcy solves he fne-horzon omal conrol roblem a me subjec o N mn x T N Px ÿ N ` x T k Qx k ` u T k Ru k, (8) U k x k` Ax k ` B w,d w k ` Bu k, k,..., N, ˇ x xq, ˇδr2k ˇˇ ˇ ˇ ď r anλ,max q, ˇδθk ˇˇ ď θ max, ˇδr3k ˇˇ ď r anλ 2,max q, ˇ D f k ě, ˇf k ˇˇˇ ď fmax,,..., 4, where N s he redcon horzon, U u,..., u N u, and U u,..., u N u s he omzer of (8). The marces Q ě and R ą are sae and conrol weghs, and P ą s he ermnal cos deermned from he soluon of he Dscree Algebrac Rcca Equaon (DARE) for he nfne horzon roblem. The conrol s seleced as uq u. Snce he ermnal cos P s a soluon of he DARE, local sably of he equlbrum s guaraneed. Near equlbra, f sae and conrol consrans are nacve, he soluon of (8) s equvalen o ha of an LQR. Feasbly of (8) s guaraneed by relaxng he sae consrans o sof consrans wh he nroducon of slack varables. D. Smulaon Resuls The urose of he frs smulaon s o roduce resuls smlar o [6], demonsrang ha MPC aled o he nonlnear dynamcs () s able sasfy conrol objecves -5. The saelle s mass s m B 4 kg. The mean surface area s S 2 m 2, surface reflecance c refl.6, and he solar facng area s S facng 37.5 m 2. Each reacon wheel has a mass m Wk 2 kg, a radus r Wk.75 m, and a hegh of l Wk.2 m. For smlcy, he hree reacon wheels are assumed o concde wh he saelle s cener of mass. Earh s gravaonal arameer s µ E 3986 km 3 /s 2, he Sun s s µ S km 3 /s 2, he Moon s s µ M km 3 /s 2. Earh s J 2 arameer s J ,

6 s radus s R E 6378 km, and he solar radaon ressure consan s C sr N/m 2. The smulaon me s se o 9 mnues wh 5 mnues. The saelle mus reman n a saon keeng wndow of. n laude and longude. The maxmum hrus s f max. N, wh a onng angle α α The maxmum allowable error on he Euler angles s.2. The weghng marces for he saes n he smulaon are Q r Q 9r 3, Q θ 4, Q ω 2, and Q 9γ 5. The nal condon s se on January, 2 wh a semmajor axs of km and all oher orbal elemens zero. The aude nal condons are C g C 3 8 q, ω g ω, and 9γq r5 5 5s T rad/s, whch corresonds o C d and ω d. The resuls are shown n Fg. 3. As n [6], he MPC olcy s able o drve he reacon wheel seed o zero (Fg. 3a). However, he omal conrol u k causes he onng consran o be volaed, and hs s due o several reasons. Frs, he dynamcs of he nonlnear sysem and he lnear sysem dverge wh nonzero reacon wheel seeds. For examle, when he reacon wheel seeds ncrease from o 5 rad/s, a lghly damed mode aears a.5 rad/s, and a hgher damed mode aears a 95.3 rad/s. Snce here s a large dfference beween he lnearzed and nonlnear sysems, he MPC olcy redcs ha he aude consrans are sasfed when he lnear dynamcs are roagaed forward n me, as shown by he do-dashed lne n Fg. 3b. However, when he conrol s aled o he nonlnear sysem, he consrans are n fac volaed. Second, n [6], he hrusers are able o creae ure orques ha are decouled from body forces. Wh he hruser confguraon n hs aer, a hruser orque smulaneously creaes a ne body force. Ths effec coules he orbal and aude dynamcs, whch comlcaes conrol. Ths smulaon denfes addonal key roblems wh an MPC conrol olcy ha handles boh aude conrol and saon keeng. For aude conrol n [6], mn, and here 5 mn are used o handle he relavely fas aude dynamcs. Saon keeng dynamcs are much slower han he aude dynamcs, and hus he dscrezaon of he orbal dynamcs do no requre small me ses. To ake advanage of he saon keeng dynamcs, a horzon of a leas half an orb s needed. A small me se ( 5 mn) couled wh a large redcon horzon (5 h) dramacally ncreases he sze of he omzaon roblem (N 8). Moreover, hs omzaon would need o be solved every 5 mnues nsead of every hour, whch would requre a consderable ncreased use of onboard resources. IV. MPC WITH INNER-LOOP ATTITUDE CONTROL To overcome he roblems dscussed n he revous secon, an nner-loo aude conroller s roosed, wh conrol archecure descrbed n Fg. 4. The nner-loo conrols he aude va he reacon wheels, whle he MPC olcy conrols he closed-loo saelle/aude conroller sysem. Therefore, he lnearzaon of he oen-loo sysem (5) s no longer adequae, and he MPC requres he closed-loo saelle/aude conroller o be lnearzed. Seed (rad/s) δφ (deg) δθ (deg) δψ (deg) Tme (mn) (a) Reacon wheel seeds decreasng from 5 rad/s γ γ 2 γ 3 Aude Consrans Predced Aude Tme (mn) (b) Aude error (n sold blue), aude consrans (n dashed red), and redced aude from lnearzaon (n do-dashed yellow). Fg. 3: Smulaon for MPC conroller whou nner-loo conrol and 5 mn. A non-adave form of he conroller roosed n [7] s used for he nner-loo aude conroller as rovdes almos global asymoc sably for aude rackng and s caable of harmonc dsurbance rejecon. For an exlanaon of almos global, see [7]. Wh resec o he aude conroller, he SRP orque and he orques due o he elecrc hrusers are consdered o be aude dsurbances. I s assumed ha he orque dsurbance s he ouu of an LTI sysem A d, C d. Defnng S P a C d q v, we oban S 9 P a ω dˆ dsurbance τ s esmaed by C d q v, and ω d ω g C d ω dg d. The 9ˆd A dˆd ` B d ω d ` K Sq, (9a) ˆτ ds C dˆd, (9b) where K K T ą s a gan and where B d s desgned such ha (9) s osve real. Nex defne ν ω gˆ J Bc ω g ` J s 9γq J Bc K S 9 ` ω dˆ ω g q, ν 2 ˆτ ds, () ν 3 K v ω d ` K Sq K S, where K v K T v ą and K K T ą. The aude conrol law s gven by η J s ν ` ν 2 ` ν 3 q. ()

7 Subsung he lnearzaon of () no (5c) yelds δ 9ω τ hrus ` K ωˆ ` r K ` ωˆ JBc ωˆ ωˆ ` J Bc K v ωˆ K vk K q K v sδω J Bc C dˆd ı δθ, (2) where he SRP orque dsurbance has been gnored. The erm C dˆd s he ouu of (), and so he lnearzaon of (9) s also requred, and s gven by 9ˆd A dˆd ` B d δω ` B d K ωˆ qδθ, ˆτ ds C dˆd. (3a) (3b) The lnearzed redcon model for MPC s based on (5a), (5b), (2), (3), wh saes x rδr T δ9r T δθ T δω T δ 9γ T ˆd T s T. The change o he MPC olcy (8) s ha whle he regulaon of reacon wheel seeds 9γ s sll an objecve, her acceleraon η s no longer a conrol nu deermned as ar of he omzaon, bu s now rescrbed by he nner-loo conrol (). Ths ensures ha he MPC oeraes by conrollng he hrusers n a way ha causes he aude conroller () o reduce he reacon wheel seeds, and s ossble because he MPC olcy uses a closed-loo aude model ha caures he acon of he aude conroller. MPC Lnearzaon for MPC abou nomnal saes f Nonlnear Dynamcs Dsurbances Orbal Dynamcs Aude Dynamcs Aude Conrol Fg. 4: MPC wh nner-loo conroller. A. Smulaon Resuls The smulaon wh he nner-loo conroller has almos all he same arameers as n Secon III-D. The weghng marces of hese smulaons are Q r dag,, q, Q 9r and R 4 o ensure ha conrol s heavly enalzed. The sae wegh s laced on longude and laude devaons. I s ossble o use mnmal weghng on he saes because he saon keeng consrans are bul no he MPC olcy. The weghng for he aude relaed saes are Q θ Q ω 3, Q 9γ 2, and Qˆd. The marx B d s deermned by choosng Q d.5, solvng A T d P d `P d A d Q d for P d, and hen seng B d C TP d d. The horzon for MPC s chosen o be 5 hours o exlo he erodcy of he orbal dsurbances. In addon, he dynamcs are dscrezed usng h, as n [6], whch s now ossble because he fas aude dynamcs are managed by he nner-loo conroller, and hus N 5. The nal condons are he same as Secon III-D, exce wh 9γ. Cd! d 9 r 9r! 9 ˆd The smulaon s run for 425 orbs, bu only he las year (365 orbs) s used for analyss o solae he seady sae of he sysem (see Fg. 5). Fg. 5a shows ha he saelle remans whn he saon keeng wndow. Fg. 5c shows he aude error, whch remans whn.2. Fg. 5d shows ha he reacon wheel seed s conrolled and does no ncrease whou bound, as s he case n smulaons where momenum managemen s no erformed. Fg. 5b shows he cumulave v roduced by each hruser, and he v due o he ne force of he hrusers, resolved n F h. The v 3 s slghly less han 6 m/s, whch s execed due o he ou-of-lane erurbaons. The oal v.97 m/s s 22.3 m/s more han requred for saon keeng only. The exra hrus s needed o counerac he solar radaon ressure orque and o manage he momenum of he reacon wheels. Fg. 6 shows he las fve orbs are loed for enhanced deal. The reacon wheel seeds are cyclc over an orb as execed. The hrusers are held consan over a one hour erod, and he MPC soluon resuls n only wo hrusers beng acve a a me, as shown n Fg. 6d. Eher he wo Norh hrusers (forces a and 2 ), or he wo Souh hrusers (forces a 3 and 4 ) are acve smulaneously. Noe ha due o he naure of he MPC olcy, he hrus s held consan for one hour, and hus he gmbal angles are also consan. Gmbal angles do no change from `36 o 36 from one hour o he nex, bu do so n ncremens. The gmbal angles are ofen a her lms, whch ndcaes ha he hrusers are almos always creang some sor of orque, and mlyng ha he reacon wheels are always n use. V. CLOSING REMARKS Auonomous closed-loo feedback conrol for saon keeng oeraons wll ncrease he safey, robusness, and relably of saelles. In hs aer, we resened an MPC olcy wh an nner-loo SO3q-based aude conroller ha s able o sasfy saon keeng consrans, whle also rovdng momenum managemen. The SO3q-based aude conroller s also able o kee he saelle s aude error whn he onng consrans. All hs can be acheved for a saelle confguraon wh he hrusers locaed on he an-nadr face of he saelle, whch s convenen for saelle desgn bu challengng for conrol. REFERENCES [] J. A. Sarek and I. V. Kolmanovsky, Nonlnear model redcve conrol sraegy for low hrus sacecraf mssons, Omal Conrol Alcaons and Mehods, vol. 35, no.,. 2, 24. [2] O. Abdelkhalk and E. Taher, Aroxmae on-off low-hrus sace rajecores usng fourer seres, Journal of Sacecraf and Rockes, vol. 49, no. 5, , 22. [3] D. Losa, M. Lovera, J.-P. Marmora, T. Dargen, and J. Amalrc, Saon keeng of geosaonary saelles wh on-off elecrc hrusers, n Proc. 26 IEEE Inernaonal Conference on Conrol Alcaons, , 26. [4] A. Sukhanov and A. Prado, On one aroach o he omzaon of low-hrus saon keeng manoeuvres, Advances n Sace Research, vol. 5, no., , 22. [5] Y. Ulybyshev, Long-erm saon keeng of sace saon n lunar halo orbs, Journal of Gudance, Conrol, and Dynamcs, vol. 38, no. 6,. 63 7, 24.

8 . 3 v n frames F, oal v =.973m/s.5 v (m/s) τ w (N m) Seed (rad/s) Aude error (deg) Aude error (deg) - -2 Laude (deg) Longude (deg) (a) Laude and longude error. δφ δθ δψ Tme (orb) (c) Euler angle aude error. Seed (rad/s) v (m/s) v v2 v3 v n Hll s frame Fh Tme (orb) (b) v from each hruser and ne v resolved n F h. 4 2 γ γ 2 γ Tme (orb) (d) Reacon wheel seeds. Fg. 5: One year smulaon for MPC conroller wh nner-loo conrol Tme (orb) (a) Reacon wheel seeds and nu. γ γ 2 γ 3 J sη J sη 2 J sη 3-3 δφ δθ δψ Tme (orb) (c) Aude error. Maxmum error s.2. α (deg) α3 (deg) f (mn) 5 f f 2 f 3 f Tme (orb) (b) Gmbal angles for each hruser wh 36 consran Tme (orb) (d) Thrus magnude. Maxmum hrus s mn. Fg. 6: Close-u of 5 orbs dealng reacon wheels and hrusers [6] A. Wess, U. Kalabc, and S. D Carano, Model redcve conrol for smulaneous saon keeng and momenum managemen of low-hrus saelles, n Proc. 25 Amercan Conrol Conference, , 25. [7] U. Kalabc, A. Wess, S. D Carano, and I. Kolmanovsky, Saonkeeng and momenum-managemen on halo orbs around L2: Lnear-quadrac feedback and model redcve conrol aroaches, n 25h AAS/AIAA Sace Flgh Mechancs Meeng, AAS 5-37, 25. [8] A. Garull, A. Gannraan, M. Leomann, and F. Scorecc, Auonomous low-earh-orb saon-keeng wh elecrc roulson, Journal of Gudance, Conrol, and Dynamcs, vol. 34, no. 6, , 2. [9] M. Marnez-Sanchez and J. E. Pollard, Sacecraf elecrc roulson an overvew, Journal of Proulson and Power, vol. 4, no. 5, , 998. [] J. R. Werz and W. J. Larson, eds., Sacecraf Msson Analyss and Desgn. Mcrocosm Press, 3rd ed., 999. [] E. M. Soo, Handbook of Geosaonary Orbs. Srnger, 994. [2] D. Herman, G. C. Soulas, J. L. Van Noord, and M. J. Paerson, NASA s evoluonary xenon hruser (NEXT) long-duraon es resuls, Journal of Proulson and Power, vol. 28, , 25/9/ [3] R. R. Bae, D. D. Mueller, and J. E. Whe, Fundamenals of Asrodynamcs. Dover Publcaons Inc., 97. [4] J. Clmer, Boeng: World s frs all-elecrc roulson saelle begns oeraons, Inerne: h://boeng.medaroom.com/ndex.h?s=2295&em=2956, Se. 25, [22 Se. 25]. [5] B. M. Anzel, Mehod and aaraus for a saelle saon keeng. U.S. Paen , Augus 22, 995. [6] A. H. J. de Ruer, C. J. Damaren, and J. R. Forbes, Sacecraf Dynamcs and Conrol: An Inroducon. Chcheser, Wes Sussex, U.K.: John Wley & Sons, Ld, 23. [7] A. Wess, I. Kolmanovsky, D. S. Bernsen, and A. Sanyal, Inerafree sacecraf aude conrol usng reacon wheels, Journal of Gudance, Conrol, and Dynamcs, vol. 36, no. 5, , 23.