Quaternion-Based Tracking Control law Design for Tracking Mode

Transcription

1 Quatenion-ased aking Contol law Design fo aking Mode A. M. Elbeltagy Aiaft Mehanis Dep. MC, Caio, Egypt; Y. Z. Elhalwagy Guidane and Navigation Dep. MC, Caio, Egypt; A. M. ayoumy Aiaft Mehanis Dep. MC, Caio, Egypt; +767 A. M. Youssef Aiaft Eleti system Dep. MC, Caio, Egypt; SSC6-V-8 ASRAC Vaious ontol design tehniues ae model dependent. hey typially euie knowledge of the inetia mati. hee ae majo hallenges fo eah poposed ontolle to ope with spaeaft mission objetive in tems of pointing and jitte euiements. hese hallenges inlude sensitivity to noise effets ando modeling eos, while othes ae sensitive to etenal toue distubanes, suh as toues indued by sola adiation pessue. Robust ontolles have been developed to mitigate these sensitivities. n this pape, a obust nonlinea taking ontol algoithm intodued peviously in the open liteatue is modified and toleated to be utilized with ehange momentum atuatos, e.g. eation wheels. he ontol law is using the ommanded attitude ate, ommanded attitude aeleation, attitude eo uatenion and gyosopi tems. aking eo dynamis euivalent to ellite losed-loop time-vaying nonlinea dynami system is used altenatively to onfim that a globally stable taking ontolle always eists. he poposed ontolle is applied to meet euiements of a taking omple mode. Geneation of the needed taget attitude and attitude ate ae deived in details. he motion and kinematis of the gound taget elative to the in-obit ellite is analyzed and desibed in obitefeened oodinates. he ellite dynamis ae deived fom fist piniples and efomulated also in obit efeened oodinates.. A taking sheme fo the pointing ais along the body z-ais of ellite is highlighted. Consideing attitude and obit ontol system (AOCS with ideal attitude and obit detemination sensos with symmeti ellite inetia, the validity of poposed ontolle and taget data geneato is demonstated unde MALASMULNK envionment. MALA optimization tool is used fo optimal gains seletion. Robustness of the globally stable modified ontol law to spaeaft inetia mati unetainty is also disussed. Simulation esults show that the poposed ontol law an be used suessfully onboad fo fast taking and is obust enough to keep the pointing auay within aeptable limits with onsideable inetia unetainty. Elbeltagy 3 th Annual AAAUSU Confeene on Small Satellites

2 NRODUCON Fo seveal spae missions, Eath-pointing ellites ae euied to point a payload suh as a high-gain antenna, amea, and telesope to tak a fied taget on the Eath fo a etain peiod of time in ode to povide impoved long-peiod up-down-link ommuniation, low-distotion imaging and auate obsevation. hese missions euie gound taget taking ontol. -3 A linea uatenion feedbak egulato with open loop deoupling ontol toue fo gyosopi foes to ensue inetially efeened eigenais otations is poposed in Ref. 3. A taking ontolle whih deals with muh moe ompliated ontol task onsides the taget desied ate duing ealizing the desied taget attitude. A taking ontolle is ealized though a feedbak ontol sheme ontaining an inne veloity loop that taks the desied ate ommand and oute attitude loop that taks the desied attitude ommand to ahieve taking via an instantaneous eigenais otation. Goeee and Shuke developed a taking ontolle with a feedbak elasti tem fo taking attitude, a feedbak visous tem fo taking a desied angula ate, and a feed-fowad model-based ompenion tem. he aim of thei algoithm is to enable the telesope mounted along the z-ais of ellite body to tak a gound station while passing ove and to eeive the lase ommuniation signals fom the gound station. A new non-linea taking ontol algoithm based on an attitude eo uatenion uses the ommanded attitude ate without tansfomation into the body fame is also intodued. 4 he diet use of the ommanded attitude ate simplifies the alulation of its deivative, whih is used in the ontol law. his algoithm is poposed to be used with etenal atuatos fo geneal taking task. Petubation dynamis with seven state vaiables is used to analyze the stability of the system and the taking pefomane. Simulation esults show that the spaeaft an tak the ommanded attitude and ate uikly fo a non-zeo aeleation ate ommand. his poposed taking ontolle was the motivation to popose a modified non-linea ontolle fo fied gound taget taking puposes. Most of the ontol algoithms intodued peviously in the open liteatue ae about pointing auay and ignoe the time elapsed to eah these stit euiements. Atually thee is an ineasing demand fom ustomes fo a fast gound-taget taking even on the epense of pointing auay itself. When the taget loations and ellite gound station ae both within the ellite s footpint, it may be muh moe impotant fo the gound station to stee the main image boe sight towads the immediate euied taget aeas befoe aptuing the imaging data. Quik ellite esponse allows fo the militay intelligene data gatheing and downloading in the same ommuniation session. n this pape, a non-linea taking ontol algoithm intodued in Ref. 4 is modified and toleated to be utilized with ehange momentum atuatos, e.g. eation wheels, to tak any desied taget on the Eath fo a nominally Eath- pointing ellite as fast as possible. he poposed taking ontolle uses attitude eo epesented in uatenion, ate eo without the need to be epesented in ellite body oodinate fame, taget aeleation, gyosopi tems. he pape is oganized as following: Setion define the oodinates fo attitude tansfomation though the pape. he ellite non-linea ombined model euipped with thee-ais eation wheels (RW is pesented in setion 3. he RW non-linea attitude taking ontolle fo fied gound taget is pesented in setion 4 followed by stability analysis using taking eo dynamis in setion 5. he kinemati of fied gound taget taking inluding the taget desied ate and aeleation is totally oveed in details in setion 6. A design eample with its simulation esults using MALA Simulink softwae is pesented in setion 7 and finally a onlusion is given in setion in setion 8. DEFNON OF COORDNAE FRAMES Seveal oodinate fames ae used oodinates fo attitude tansfomation though the pape as follows Obit Refeened Coodinate fame ( he z-ais points towads the ente of the Eath stating fom the ellite in obit. he y-ais points to the negative obit nomal. he -ais is hosen to fom a ight-handed othogonal efeene fame. heefoe, fo a iula obit, the -ais will be along the veloity veto of the ellite. Spaeaft ody Fied Coodinate fame (SC he SC oodinate fame oiginates fom the ente of mass of ellite. he -ais points to the ente of the haness side of the ellite. he z-ais is othogonal to the -ais. he y-ais is hosen to fom a ight-handed othogonal efeene system. When the ellite is pefet nadi pointing without any otation aound the z-ais, the SC and oodinate fames ae assumed to be aligned. SC oodinates oiginate fom the ente of mass of ellite. he -ais points to the ente of the haness side of the ellite. he z-ais is othogonal to the -ais. he y-ais is hosen to fom a ight-handed othogonal efeene system. When the ellite is pefet nadi pointing without any otation aound the z-ais, the SC and oodinates ae assumed to be aligned. Elbeltagy 3 th Annual AAAUSU Confeene on Small Satellites

3 Eath Fied Coodinate fame (EFC he EFC fame is Eath-enteed. he -ais will point towads the pime meidian passing though the ente of the Royal Geenwih Obsevatoy in London. he z ais points to the noth elestial pole. he y-ais is hosen to fom a ight-handed othogonal efeene fame. Eath Centeed netial Coodinates (EC he EC fame is Eath-enteed. he -ais points towads the mean euino. he z-ais points to the elestial pole. he y-ais is hosen to fom a ight handed othogonal efeene fame. SPACECRAF COMNED NONLNEAR MODEL Assuming a igid ellite euipped with thee ais eation wheels (RW as intenal toue atuatos and by ealling the Eule s moment euation, then the dynami model of an Eath-pointing ellite is given by 5 (. J ω ( = N et ω Jω + h h ( v v ωo = [ ] (8 ( 4 ( ω = J ( N Ω ( ω ( Jω + h h et Whee J is the spaeaft inetia mati, ω = ( ω, ω, ω3 is the body angula veloity veto with espet to the inetial efeene fame, N et is the etenal distubane toue, h = ( h, h, h3 is the RW angula momentum veto, h is RW applied toue and Ω ( ω is a skew-symmeti mati defined by ω ω Ω ( ω ω ω 3 = 3 ω ω (3 Futhe assuming the ellite is 3-ais stabilized, then the absolute angula veloity in the inetial spae esolved in the fame is given by 5 ω = ω + A ω (4 SC O Whee ω O = obital angula ate veto of the ellite motion in fame. Substitute E. (4 into E. (, then the dynami euation of the Eath-pointing ellite beomes ω = J ( N + N Ω ( ω ( Jω + h h ωo et With SC SC N ωo = JA ωo JA ωo SC SC = JΩ( ω A ω JA ω Whee ω and O O O Whee N eff = effetive RW toue input. he nonlinea attitude kinematis euations of motion of an Eath-pointing ellite an be epesented by using vaious attitude paametes. Repesentation though uatenion paamete has the popety of (. nonsingulaity and it is fee fom the tigonometi omponent. heefoe, this epesentation is widely used to study the attitude behaviou of spaeaft. he kinematis of the ellite model is the pat whih epesses the elation between the attitude and angula veloities of the body and an be desibed by 6 (5 (6 ω O ae obit angula ate and obit angula aeleation epesented in fame and deived as follows v ωo = [ ] (7 An euivalent but moe useful fom of E. (5 fo ontol puposes is given by ω = J ( N ωo + N et Ω ( ω ( Jω + N eff h = Ω( ω ( h N eff = Ω ( ω + 4ω 4 ( = ω (9 ( Elbeltagy 3 3 th Annual AAAUSU Confeene on Small Satellites

4 he ellite attitude w..t fame as a efeene fame is detemined by its uatenion. oth and 4 of the uatenion ae defined as ϕ e sin( ϕ = esin( = 3 ϕ e 3 sin( ( ϕ 4 = os ( ( = 4 (3 Whee e, e and e 3 ae the omponents of the otation ais unit veto along the efeene fame; ϕ is the otation angle. he ombined dynami, E. (5 and kinemati, E. ( give the geneal nonlinea model fo the spaeaft angula motion with ten state vaiables d J ( N ωo + N et Ω ( ω ( Jω + N eff ω Ω ( ω + 4ω = 4 ( ω h ( ω ( h N Ω eff (4 RW NONLNEAR AUDE RACKNG CONROLLER ased on the non-linea taking ontol law intodued in Ref. 4 whih use attitude eo uatenion and desied ate without tansfomation in the body fame, a modified poposed RW nonlinea taking ontolle fo gound taget taking puposes is witten as follows h = N ω + N Ω ( ω ( J ω + h o et Ω ( ω J ω +Ω( ω J ω Ω( Jω ( ω ω J ω + D ( ω ω + K δ (5 Whee δ = attitude eo uatenion veto of δ attitude eo uatenion, whih is defined as follows δ δ (. δ 4 ( δ (7 Whee = the invese of ; (. = uatenion multipliation. Knowing thatδ is still a uatenion whih has a physial meaning as it epesents the attitude diffeene between and. n suh ase effetive RW toue input is N eff = N ωo N et +Ω( ω J ω + J ω D ( ω ω K δ +Ω( ω J ω +Ω( Jω ( ω ω Ω( ω J ω Substitute E. (5 into E. (5 gives ω ω = J ( D ( ω ω + K δ Ω( ω J ω +Ω( Jω ( ω ω (8 +Ω( ω J ω E. (9 and E. ( fom the losed-loop time-vaying nonlinea dynami system of the spaeaft attitude with the ontol law (5. SALY ANALYSS (9 Closed-loop solution t is lea that the following is a solution of euation (9 and euation ( O ω = ω = ( Elbeltagy 4 3 th Annual AAAUSU Confeene on Small Satellites

5 Solution ( means theoetially that it is possible fo the spaeaft to follow the ommanded attitude and attitude ate without estition although in eality thee ae always etain limits in using diffeent atuatos. he tems N ω, Ω ( ω ( J ω + h and o J ω ae essential to be inluded in the poposed RW ontolle dediated fo taking task of nominal eath pointing ellite. nluding these tems ensue that the solution ( always eist and hene an be applied in diffeent maneuve onditions (e.g. est to est maneuve, taking ommanded attitude with onstant ate. he stability analysis an daw a lea onlusion onening the ability to tak the ommanded attitude when solution ( eists. he ommanded ate and its deivative ω and ω espetively will estimated late fo fied gound taget with aid of GPS data. heefoe, the nonlinea ontolle is appliable fo spaeaft attitude ontol and ommand taking. aking eo dynamis Geneally asymptoti stability analysis is euied to ensue that the ellite an follow the ommanded attitude with ommanded attitude ate fom any ondition of both ω and. he stability analysis is stated by studying the loal asymptoti stability, at whih the initial values of both ω and ae nea the solution (. One this loal stability is onfimed, a global asymptoti stability, at whih any values of both ω and ae assumed, is futhe seahed. One of the objetives of this pape is to find the onstant gain maties D and K having a globally asymptoti stable solution ( and guaanteeing the needed solution onvege pefomane. hese gains ae used as initial design values to the Matlab optimization tools. aking eo may be used as paamete to be minimized in limited time fame.he taking eo dynamis intodued in Ref. 4 ae employed fo the indiet stability analysis altenatively to the nononvenient diet analysis of solution (. he taking eo dynamis defined in the following ae always valid whateve the diffeenes between the ommanded ates and the atual ates ae lage o small. We define the following state vaiable to epesent the taking eo dynamis as follows Whee ( ω 3 ω = ω ω = ω = ω 3 = = = = ( ( = ( ( 3 3 (3 46= With (4 Substituting (-4 into E. (9 and E. (, getting simila fomula dedued in Ref. 4 fo the taking eo dynamis with mino hanges as 46 3 = f3 = D3 K[ ] 7 46 = f 46 = Q Ω ( ω 46 + ω 7 X ( = = 3 + ω f (5 Elbeltagy 5 3 th Annual AAAUSU Confeene on Small Satellites

6 Whee D = J D (6 K = J K (7 [ ] Q 3 3 X 3 3 (8 (9 (3 he taking eo dynamis E. (5 is euivalent to E. (9 and E. (. E. (5 has an euilibium point =, whih oesponds to solution (. heefoe, instead of studying the deviation of ω and fom ω and fo E. (9 and E. (, we may simply study the stability of the taking eo E. (5 with espet to the euilibium point =. Global stability analysis A Lyapunov funtion based global stability analysis of the time-vaying non-linea taking eo dynamis E. (5 is employed. he states vaiable in this eo dynamis epesents the deviation fom obit efeened ellite body ate, uatenion and desied taget ate and uatenion espetively. Although the states vaiable of the nonlinea dynami system E.(9 and E. ( with the poposed ontol law (5 is diffeent fom those used in Ref. 5, similaity in the stutue of taking eo dynamis is enough to use theoem onditions n Ref. 4 to guaantee stability. t is poven that when K is symmeti and positive definite and that K D is positive definite, the euilibium point = of taking eo dynamis is globally stable. he andidate Lyapunov funtion V, whih is independent of time and is adially unbound, is defined by V( Whee P P = P ( = (3 P is a symmeti and positive definite mati. 4 4is an identity mati of ode 4. he total time deivative of V along the tajetoies of the taking eo dynamis is poved to be negative semidefinite when P is seleted P = K (33 hen (34 V( = PD 3 3 heefoe the euilibium point = of E. (5 is globally stable. Futhemoe using theoem 8.4 of Ref. appoahes zeo as t. 7, we onlude that 3 Assuming D and K ae diagonal maties epessed as D = diag ( d, d, d 3 K = diag ( k, k, k 3 (35 he dynami behaviou of the nonlinea system E.(5 an be ontolled aound the euilibium point in the dietion of eigenvalues with nonzeo eal pats of the Jaobian mati. 4 he haateistis euation of the Jaobian mati fo the speial ase when ω = an be epanded as 4 k k λ( λ + d λ + ( λ + dλ + k 3 ( λ + d 3λ + = (36 Elbeltagy 6 3 th Annual AAAUSU Confeene on Small Satellites

7 Epessing E. (36 by the desied damping atio ξ and natual feuenyω, then the gains maties ae n elated dietly to the desied damping atio ξ and natual feuenyω as 4 n d= ξω n; k= ωn d= ξω n; k = ωn d3= ξω 3 n3; k3= ωn3 (37 Sine atuato onstaints ae onsideed in this pape, these gain maties figued using E. (37 an be used as initial values guide. A MatlabSimulink Optimization tool is used to adjust the dynami behaviou of the nonlinea system E. (5 aound the euilibium point. aking eo is the seleted paamete to be optimized within shot time. KNEMAC OF FXED GROUND ARGE RACKNG aget aking Mode will most likely be ativated fom Nominal Mode as soon as a pedefined gound taget loation (e.g. Gound station is in ange. Knowledge of the ellite position is theefoe euied by the on-boad ompute. he method that will be used to alulate the desied attitude and angula ates fo taking a taget was pesented by Chen et al. in. 8 he ellite will need to be able to detemine whethe the taget is in ange (i.e. in the ellite s field of view of the Eath. he distane D ma fom the ellite to the futhest point on Eath that the ellite an "see" an be alulated as D = ( R + h R (38 ma E E Whee R E = adius of the eath and altitude. is ellite An illustation of the above-mentioned geomety an be seen in Fig.. Compaing the uent distane to the taget with maimum taget taking distane allow the on-boad ontol algoithms to know if the taget is within the visibility zone o not and hene taking task an be stated. Figue : maimum taget taking distane Sine the nominal mode of eath obiting ellite usually maintains the ellite body fame aligned with obital ( fame, it is pefeable to desibe the kinematis of the gound taget with espet to the inobit ellite in fame. he veto fom the ellite to the gound taget epesented in will speify the taking dietion of the seleted pointing ais whih may be the positive mounting ais of a amea o antenna. Fistly the veto fom the ente of the Eath to the gound taget with espet to EFC fame fom the given geoenti latitude (assuming a spheial Eath and longitude will be deived as EFC = R E ( λ os( ϕ ( λ os( ϕ os sin sin( ϕ (39 Using the tansfomation mati fom EFC fame to EC fame,, the loation of the taget with espet to the EC fame is given by EC EC EFC AEFC = (4 Assuming that the Eath has a onstant angula ate aound its otation ais duing the taking maneuve, then the attitude tansfomation mati fom the EFC fame to EC fame an be epesented by os( ωet + α sin( ωet + α EC AEFC = sin( ωet α os( ωet α + + (4 Whee t is time sala and is the initial phase between the -aes of both EFC fame and EC fame. EC and (the ellite s position veto in EC fame, as obtained by onboad GPS eeive o Elbeltagy 7 3 th Annual AAAUSU Confeene on Small Satellites

8 podued by any obit popagato fo simulation puposes an then be used to alulate S, the veto fom the ellite to the taget epesented in obit fame, thus EC S = AEC ( (4 A EC As, is the tansfomation mati fom EC fame to fame and defined by A EC Whee wˆ [ uvw ˆ ˆ ˆ ] = (43 = (44 v ˆ = v v (45 uˆ = vˆ wˆ (46 f the magnitude of S is lage than ma, then the taget is not in ange and the ellite will ontinue opeating in nominal mode. Howeve, if < D, taget taking mode will be S ma ativated. atually the dietion of S is of use to the ellite s ADCS, hene the unit veto u S must be alulated by nomalised, thus u S S S S = (47 aget taking task euies that the mounting ais of the ommissioned payload being ontolled to point towads the dietion of u S. n most ases, the ommissioned payload, suh as a amea, u SC om antenna o telesope is usually mounted along the z- ais of the ellite body. his onfiguation theoetially allows the ADCS designes to ahieve the task with minimum esoues (i.e. task an be ahieved by only using the - and y- eation wheels. Nevetheless this onfiguation pemits speifially ontolling the angle aound the mounting ais. D SC Moeove, defining A _ d to epesent the desied attitude fom fame to SC fame, then A u u SC SC _ d S om = (48 Let the --3 seuene of otations is used fo the SC desiption of the desied attitude mati A _ d fo taking. Substituting suh attitude mati 9 in tems of φθ, and ψ espetively epesent the oll, pith and yaw angles in E. (48 leads to us = [sin θ osθsin φ osθosφ] (49 ased upon E. 49 the oll and pith angles φ and fo taking an be easily solved. Assuming a onstant yaw ψ is euied duing taking, the desied attitude SC mati A _ d an be omputed. heefoe, the efeened uatenion ommand an be deived. Aodingly, δ an be solved using E. (7. he ellite s angula ate fo taget taking will not be zeo, sine the taget is fied with espet to the Eath and will thus be otating elative to the obiting ellite. he ommanded angula ate ω an be alulated as epesented in 8 ω = u u (5 S S Due to the assumption of a onstant yaw angle duing taking, the desied angula ate along the z-ais will be zeo. ased upon E. (5, ω an be edued to 8 S _ y S _ us _ z us _ z ω = u u [ ] (5 he ommanded angula ate ω an be deived by taking the time deivative of E. (5 ω = u u + u u S S S S = us u S (5 he u S tem an be deived by fist taking the time deivative of E. (47 yields Elbeltagy 8 3 th Annual AAAUSU Confeene on Small Satellites

9 u = [ u ( u ] (53 S 3 S S S S u So S an be deived by taking the time deivative of E. 53 yields d EC EC S = ( A EC ( AEFC d EC EC + ( A EC ( A EFC v (58 u d( S S S = S S S d u ( u S S S ( S (54 As d ( A EC EC EC ( A EFC = A ( A + A ( A EC EC EC EC EC EFC EC EFC A EC EFC EC (59 ut geneally fo any veto Α d ( Α = ( Α Α Α (55 Using E. (55 and substitute the oesponding elations in E. (54 and make some manipulation yields u S S = ( ( 3 S S S S + ( ( u S 3 S S S S ( u ( u ( u S S S S S + u ( u + u S S S S S S (56 Futhemoe fom E. (4, S an be detemined as = A ( A + A EC EC ( A v EC EC S EC EFC EC EFC (57 d ( A EC EC EC ( A EFC v = EC EC EC EC A EC ( AEFC v + AEC ( A EFC (6 a Whee A EC [ ˆ ˆ ˆ = uvw ] (3 ˆ w = ( ˆˆ 3 ww v (4 v ˆ = (5 uˆ = vˆ [( ˆˆ 3 ww v ] (6 sin( ωet + α os( ωet + α EC A EFC = ω E os( ωet α -sin( ωet α + + (7 [ ˆ ˆ ˆ A = uvw ] (8 EC Diffeentiating E and using E. 55 to get the olumn vetos of A EC as follows aking the deivative of E. (57 get Elbeltagy 9 3 th Annual AAAUSU Confeene on Small Satellites

10 ˆ a w = + ( v v ˆ ˆ ˆ ww wv ˆ + w a + v + ( ( ˆ ˆ v w v w (67 v fom a GPS eeive. Matlab Simulink Optimization tools is used to seah almost best gains to allow the taking poess meets the euied apid pointing to the taget with a little elaed auay. his yields the following gains v ˆ = (68 ˆ u = ˆ ( ˆ 3 v v v + v a + ( ˆ ˆˆ 3 v v ww v ˆ ( vˆ ( ww ˆ v + wˆˆ w a (69 Simulation is made fo one of the ovepass flight. Fig. 4-5 shows that the pointing eo of the poposed taking ontolle of E. (5 sueeded to maneuve ellite though a lage angle to a pedetemined attitude with the euied elaed auay (.8 within only (4 se whih is well below the euiements. Fig. 6-7 shows the uatenion eo and ate eo simulation esults. t is seen that both obit efeened ellite attitude and desied attitude ae almost oinide within only (3 se. Whee a is ellite aeleation veto as 5 µ asa t = = (7 3 So finally using E to get u S and substitute in E. 5 to obtain ω A MatlabSimulink softwae ode has been built to simulate the ellite fied gound taget taking task. he oveall system simulato inludes many subsystems o modules. DESGN EXAMPLE n this setion, a typial design eample has been used in simulation tests to veify the pefomane of the modified poposed taking ontolle pesented above. An imaginay ellite in a low-eath-eenti obit is used as an eample duing these simulations. n ode to investigate the attitude hange of the ellite, we define the oll, pith and yaw angles espetively to epesent the otations of the ellite body, y and z aes with espet to the oodinates. he simulation paametes ae given in able. Duing simulation, we assume pefet attitude knowledge and. We also assume pefet measuements of the vetos and Reuied aking auay aget position Unit veto of ommissioned ais in SC fame Moment of able : Simulation paametes (.3 within ( min of being in ellite FOV EFC [ ] = u = SC om [ ] inetia Reation wheel Obit paametes nitial phase α = kgm Ma toue = (. Nm, Ma momentum = (4 Nms, nitial Momentum = [ ] Nms peigee altitude = (65 km ; inlination (64.5 ; eentiity (.3, agument of peigee ( ; ight asension of asending node ( ; initial mean anomaly ( Elbeltagy 3 th Annual AAAUSU Confeene on Small Satellites

11 he obit efeened angula veloity ω of the ellite body does tak the desied angula ate ommand ω with aeptable limits duing the taking peiod at whih the taget is within the FOV of the ellite. he ativities of the thee-ais eation wheels epesented in wheel ealized toue and wheel momenta ae shown in Figs Duing the taking peiod, the toues of thee ais eation wheels is fa fom uation. he eation wheels will apply maimum toue (. Nm. to povide a bang-bang ontol manoeuve to meet the euied taget taking within speified limited time with elaed pointing auay. he wheel momentums ae kept well away below the pemissible limits (4 Nms. Although the gavity gadient distubane is onsideed in simulation, this safe patten an guaantee allowable gadually momentum build up aused by othe seula etenal distubane toues suh as aeodynami and sola pessue foes whih ae not onsideed in simulation in this pape. Figue 4: Quatenion Eo Figue 5: Rate Diffeene Eo Figue : Gound aget aking Eo Figue 6: Wheel oue Figue 3: Gound aget aking Eo (zoom in Elbeltagy 3 th Annual AAAUSU Confeene on Small Satellites

12 Figue 7: Wheel Momenta Due to the feedbak natue of poposed ontolle, this ontolle is obust against model unetainties. Simulations wee also done to investigate the obust behaviou of the taking ontolle against a ±% eo in the moment of inetia tenso of the ellite. hese simulations show in Fig. that the taking eo an still be maintained within euied limits duing the fast gound taget taking. Figue 9: Gound aget aking Eo Figue 8: Gound aget aking Eo with up to ±% inetia unetainty (Zoom in he gain seletion of the poposed RW non-linea ontolle an also be seleted to guaantee fo muh bette taking auay. Matlab Simulink Optimization tools is used again to seah fo almost best gains to allow the taking poess meets bette taking auay. his yields the following gain Figue : Gound aget aking Eo (Zoom in he simulation esults Figs - shows that the poposed RW non-linea ontolle is not limited to fast taking with elaed pointing auay. he pointing auay an eah (. and futhe enhanement an be aied out. he enhanement in taking auay will be on the epense of the time elapsed to tak the fied gound taget and atuato effot. CONCLUSONS n this pape a non-linea taking ontol algoithm intodued peviously in the open liteatue is modified and toleated to be utilized with ehange momentum atuatos, e.g. eation wheels, fo speifially fied gound taget taking task. he poposed ontolle aes about fast gound-taget taking. Quik ellite esponse allows fo the militay intelligene data gatheing and downloading in the same ommuniation session. he poposed RW non-linea taking ontolle is using the ommanded attitude ate, ommanded attitude aeleation, attitude eo uatenion and gyosopi tems. A systemati method fo detemine a omplete kinematis of fied gound taget elative to the obit fame has been pesented. he simulation esults show that RW non-linea Elbeltagy 3 th Annual AAAUSU Confeene on Small Satellites

13 ontolle is apable to ahieve a fast gound taget taking maneuve one the taget is being in the ellite FOV. he eation wheels will apply maimum toue to povide bang-bang ontol maneuves to meet the euied taget taking within speified limited time with elaed pointing auay. he wheel momentum is kept well away below the pemissible limits. he simulation esults also show well and onsideable obustness behavio of the poposed ontolle.he enhanement in taking auay is possible but it will be on the epense of the time elapsed to tak the fied gound taget and atuato effot. Refeenes. Weiss, H., "Quatenion-ased RateAttitude aking System with Appliation to Gimbal Attitude," Jounal of Guidane, Contol and Dynamis, Vol. 6, No.4, July-August 993, pp Goeee,., and Shuke,., "Geometi Attitude Contol of a Small Satellite fo Gound aking Maneuves," Poeedings of the AAAUSU Confeenes on Small Satellites, Utah State Univesity, Sept Wie,., and aba, P., "Quatenion Feedbak fo Spaeaft Lage Angle Maneuves," Jounal of Guidane, Contol, and Dynamis, Vol. 8, No. 3, May-June 985, pp Z. Zhou and R. COLGREN, "NONLNEAR AUDE CONROL FOR LARGE AND FAS MANEUVERS," AAA Guidane, Navigation, and Contol Confeene and Ehibit, 5-8 August 5 5. MARCEL J. SD, " Spaeaft Dynamis and Contol". 6. F. Landis Makley, Jhon L. Cassidis, "Fundamentals of Spaeaft Attitude Detemination and Contol", 4s 7. Hassan Khalil, "Nonlinea Systems," (3ed, Xiaojiang Chen, Willem H. Steyn, Yoshi Hashida"Gound-aget aking Contol Of Eath-Pointing Satellites," AAA Guidane, Navigation, and Contol Confeene and Ehibit,. 9. A. M. Elbeltagy, A. M. ayoumy Ali, A. M. Youssef, Y. Z. Elhalwagy, "Modeling And Simulation of Spaeaft Pointing Modes Using Quatenion-ased Nonlinea Contol Laws," AAASieh Confeene and Ehibit, 5. Elbeltagy 3 3 th Annual AAAUSU Confeene on Small Satellites