JOURNA O GUIDANCE CONTROL AN DYNAMIC Vol  No  JulyAugus  Interplanetar Spacecraf Controller Usin Thruster Har B Hablani The Boeing Company Downey California  Desig o attitud controller fo interplanet

JOURNA O GUIDANCE CONTROL AN DYNAMIC Vol No JulyAugus Interplanetar Spacecraf Controller Usin Thruster Har B Hablani The Boeing Company Downey California Desig o attitud controller fo interplanet - Description

wit permission Principa Engineerin Specialist Avionic an Softwar Group Ad vance Programs Spac System Division Associat Fello AIAA atio comman profil i develope an th parameter fo integra puls frequenc modulatio IPFM thruste controlle are deter mine ID: 25118 Download Pdf

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JOURNA O GUIDANCE CONTROL AN DYNAMIC Vol No JulyAugus Interplanetar Spacecraf Controller Usin Thruster Har B Hablani The Boeing Company Downey California Desig o attitud controller fo interplanet

wit permission Principa Engineerin Specialist Avionic an Softwar Group Ad vance Programs Spac System Division Associat Fello AIAA atio comman profil i develope an th parameter fo integra puls frequenc modulatio IPFM thruste controlle are deter mine

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JOURNA O GUIDANCE CONTROL AN DYNAMIC Vol No JulyAugus Interplanetar Spacecraf Controller Usin Thruster Har B Hablani The Boeing Company Downey California Desig o attitud controller fo interplanet




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JOURNA O GUIDANCE CONTROL AN DYNAMIC Vol 21 No 4 July-Augus 199 Interplanetar Spacecraf Controller Usin Thruster Har B Hablani The Boeing Company, Downey, California 90241 Desig o attitud controller fo interplanetar spacecraf i addressed Th firs controlle i fo Rhumb-lin precessio usin a almos periodi trai o thruste pulse a a constan phas wit th sun Earlie relationship ar expande an interlace coherentl an demonstrate o a Mercato plot Th secon controlle i fo nutatio damping Fo spacecraf wit product o inertia a contro schem i formulate tha involve fou phas angles de apar i a

nutatio cycle wher th positiv o negativ contro impulse abou th transvers axe ar injecte t dam th nutations Th thir controlle i fo attitud contro durin aerobrakin fo planetar orbi insertion Th controlle seek an maintain a zer aerodynami torqu orientation Becaus thi orientatio ma correspon t a constan bia angl o attac measure fro th instantaneou velocit vecto turnin a a nonunifor pitc rate a pitc angula acceleratio comman profil i develope fo parameterizin a thruste controller Th fourth an last controlle i fo sof landing I comprise a longitudina velocit controlle enhance gravit tur an a constan

velocit phas a lo altitude nea ground a latera velocit controlle producin command t pitc spacecraf a varyin rates an a reactio je controlle fo trackin th commande pitc angl an rat unti touchdown Satisfactor performanc o al controller i ampl demonstrated Introductio HI pape describe variou controller fo interplanetar spacecraf suc a Mar Environmen Surveyor Mar Globa Surveyo (MGS) Mar Surveyo Program an Mar Surfac Ex ploratio Pre-Project Th pape deal wit bot spin-stabilized an three-axis-stabilize spacecraft an i consist o fou mai sec tions Sectio I describe a techniqu fo designin a Rhumb-lin

precessio controlle usin an almos periodi trai of thruste pulse a constan phas wit th sun Th relationship fro differen source i th literatur ar expanded coherentl interwoven an illustrate wit a detailed realisti simulatio o large-angl pre cessio dynamic o a propose interplanetar spacecraf (MGS) Rhumb-lin precession alon wit it ungrowin nutatio loops demonstrate o a Mercato plot Becaus th precessio o a mo mentu vecto inevitabl i accompanie b nutations thes high frequenc oscillation a th en o precessio mus b dampe fo satisfactor pointin accurac of the spi axis contro schem i devise an illustrate

fo thi purpos i Sec III A spinnin spacecraf wit product o inerti exhibit con stan biase i it transvers angula rates T effec nutatio damp in o suc spacecraft a contro schem i take u fro th litera tur an honed Th schem involves inte alia bandpas filter t eliminat biase fro th transvers rates Fou phas angles 9 de apar i a nutatio cycle ar determine analytically wher posi tiv o negativ contro impulse abou a transvers axi ar fire fo nutatio damping I i show tha whe a spacecraf ha smal prod uct o inertia th fina steady-stat nutatio angl equal th angl betwee th geometri spi axi an th adjacen principa

axis Sectio I present a attitud controller usin thrusters fo th aerobrakin phas o Mar orbi insertion A a spacecraf approache it destinatio planet i i despu an it velocit i reduce t establis a planetar orbit Th apoapsi o thi highl ellipti orbi i brough close t th plane b aerobrakin instea usin propellan fo velocit reduction Fo a portio o th orbi nea periapsis wher aerobrakin i effective a attitud controlle designe t see an maintai a zer aerodynami torqu orien tatio o th spacecraft I th presen application thi orientatio correspond t a constant positiv bia angl o attac measure fro th instantaneou

velocit vector Because i a ellipti orbit thi vecto turn a a nonunifor pitc rate a pitc angula acceler Receive Jan 31 1997 revisio receive Jan 29 1998 accepte fo publicatio Jan 29 1998 Copyrigh © 199 b Har B Hablani Pub lishe b th America Institut o Aeronautic an Astronautics Inc. wit permission Principa Engineerin Specialist Avionic an Softwar Group Ad vance Programs Spac System Division Associat Fello AIAA atio comman profil i develope an th parameter fo integra puls frequenc modulatio (IPFM thruste controlle are deter mine fo trackin time-varyin orientations Sectio V present a soft-landin

controlle fo th Mar Lan der I comprise 1 a longitudina velocit controlle employin a velocity-altitud referenc trajector fo guidance a gravit tur belo 4 m an a constant-velocit phas belo 5- altitud unti sof impac i sense b th navigatio system 2 a latera velocit controlle tha require pitchin th spacecraf firs a a rat inversel proportiona t th curren slan rang an the accordin t gravit tur activate a a suitabl lo altitude an 3 a IPF reactio je controlle fo trackin th commande pitc angl an rat pro fil unti touchdown A rada altimete i use unti th spacecraf descend t a 45- altitude Afterward a

inertia referenc unit in heritin th error o th rada altimete a 4 m i use fo landing Th optimu gain fo bot longitudina an latera controller ar determined an first-orde digita fadin memor filter ar used th coefficient o whic ar optimize wit stochasti respons analysi t smoot nois measurements Ampl numerica results withi th spac allowed ar presente demonstrat effectivenes o al controllers Th relationshi o th presen wor t th previou wor i identifie i eac sectio below II Rhumb-Lin Precessio Spin-Stabilize Spacecraf Th spi axi o a spacecraf ca b precesse alon eithe a grea ar o a Rhum lin betwee th

initia orientatio an th fina orientatio o a spacecraf i a celestia sphere. I th forme cas th phas angl o th thruste firin varie significantl wit eac spi cycle leadin t operationa complexities wherea i th latte cas th phas angl i constant easil controlle b a su sensor Rhumb-lin precession ar therefor mor commo tha great-ar precessions Analysi Figur 1 depict th geometr o spin-axi precessio produce thrusters Le x zi, b th spacecraft-fixe frame wit z/ a th spi axi o th spacecraft Th precessio torqu F ( = force = momen arm i generate abou a axi formin a angl ft wit th transvers axi */, Th spi angl

o th spacecraf i de note 0 an i measure fro th ascendin nod lin x' a show Fig 1 Le t denot th shor pulsewidt (20-20 ms o th thrusters Th angula momentu imparte b eac puls i the Fit an th precessio angl dP abou a axi transvers t bot torqu axi an spi axi i (se Fig 1 d = Fit /(I^a) wher 54
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HABLAN 54 d si ( + p + /2 y z = body-fixe frame spinnin abou z -axi Fig 1 Geometr o precessio o spi axi produce b thrusters i th spacecraf momen o inerti abou th spi axi an co th spi rate Th phas angl o th cente o th thruste fir ing measure fro th axi x i 0 = (0 + ft + a) /2), wher (/)o i th

spi angl 0 a whic thruster tur o instantaneousl fo t second fo precessin th spacecraft Th component o d th fram T"^:*^ ( = z/,)are[-d sin dPcos 0]-^ th componen abou th spi axi bein zer b definition T re lat thes component wit th change i longitud an colatitud (declination o th spi axi Z i a inertia frame le ^ denot th spin-axi longitud abou th inertia axi Zi measure fro inertiall fixe axi X an 9 denot it colatitude abou th ascendin nod lin x' Smal change d i longitud an d i colatitud the wil hav thes component i th x' y' z' frame [d d\ls si 0 dty co 0]-^* wher th spin-axi componen i no

nonzero implyin that notwithstandin earlie inference th spi angl doe chang slightl wit eac firing varyin th spi perio a consequence I vie o this th requiremen tha th phas angl $p b constan fo al firing wil b satisfie onl b monitor in thi angle clearly firing wil no tak plac exactl a th sam spin-perio interval Focusin o th equalit o x' an y' compo nent o th tw vector (d = d si 0^ d si 0 = d co 0^) divid x' component b y' component an obtai d< = dils ta 0 sin p (1 Becaus th angl 0 i controlle s tha i i th sam fo al firing wit a su senso withi it measuremen accuracies th integratio ofEq (1 lead t

—log ta ^ = V ta 0 + cons (2 Wit th longitud -fy a absciss an (—log< ta ^0 a ordinate—th coordinate o a Mercato plot—an th angl 0 a slope Eq (2 a straight-lin equation confirmin tha thi i indee a Rhumb lin maneuver Invokin th initia an th fina value 0\, 0 o colatitud an i/^ V" o longitude Eq (2 yield th followin desire equation -ta 0 = -log ta ±0 ta ±0 (3 whic ca b use t determin th phas angl 0^ Fo a agreemen betwee Eq (3 an Eq (1 o Ref 1 observ tha th ascendin nod lin x i typicall du eas (th solar senso emittin maxi mu output an therefor th phas angl 0 i sometime calle headin angle Rhum angle

denote <£ i measure fro eithe loca nort o loca south an i called accordingly 4> o <£$ On the ha 0 = n/2 + <& o 0 = n/2 + $5 Substitutin thes equation i th lef sid o Eq (3) on arrive a Eq (1 o Ref 1 Eq (30 o Ref 2 Th tw value o 0 ar i lin wit Eq (32 Ref 2 Th tota chang i th colatitude angl 0 i obtaine b integrat in th equatio d = d sin0^ yielding (0 -0 = { Pcos$>s} (4 wher P i th tota precessio angl effecte b N firing arrive b summin u al d fo N firings = (5 Wherea colatitude 9 an 0 ar dictate b th missio an spacecraft-su geometry th phas angl 0 i calculate fro Eq (3 an <£ o ® fro <£ = 0 n/2,

<&s = 0/ + JT/2 Eq (4 the yield th require precessio angl P an Eq (5 th require numbe N o firing - 0 Ftt (6 wher 3 = $> o <$ . Whe th pulsewidt t i no smal (tha is th conditio co /2 <$ 1 i no satisfied) a mor accurat versio Eq (6) presente i th unabridge versio o thi paper, i used Equatio (6 mus b integerize t it neares intege t mini miz deviatio o th actual fina longitud an colatitud fro th desire value ^ an 0 Moreover t asses th fue efficienc o a Rhumb-lin precession on ma compar numericall th preces sio angl P Eq (5) wit th grea ar lengt F betwee th initia an th fina latitude an longitude =

cos" [co 0 co 0 + si 0 si 0 co (1^ (7 state earlier th phas angl 0 i require t b th sam fo eac thruste firing Becaus th phas angl or equivalently 0 a t = 0) i measure fro th ascendin nod lin x' whos orientatio angl ty varie slowl wit eac thruste firing th spi angl 0 doe no cros th valu 0 a a unifor spin-perio interval Consequently a su senso mus b use t ensur tha th spi angl 0 equal 0 fo eac thruste firing Finally a commen abou th nutation o th spacecraf whil precessing Typically th spi perio o a spacecraf i a irrationa multipl o th nutatio perio an therefor th nutatio amplitud doe no gro

monotonicall wit thruste firings instead i waxe an wane an i self-limited Illustratio Si first-orde equation governin dynamic an inertia orien tatio o a spinnin spacecraf ar simulated Tabl 1 i Ref 3 fur nishe th parameter use fo simulatin a Rhumb-lin precessio illustrate conceptuall i Figs 19. an 19. o Wertz. Th contro torqu i produce b tw z thruster locate diametricall oppo sit on the rim of the spacecraft firin oppositely Figur 2 show dashe Rhum lin (no clearl visible joinin th initia an th fina specifie location o th spi axi o a Mercato plot Becaus th Rhumb-lin lengt P i 75. de wherea th

correspondin grea circl ar F i 5 deg thi Rhumb-lin precessio i inefficien compare wit th great-circl precession Th advantage nonethe less o th Rhumb-lin precessio i th constanc o th phas angl 0o equa t 31.7 de i thi example withi ±0.25-de measuremen accurac o th su sensor Figur 2 illustrates i addition th undulation o th nutatio loop accompanyin th precession thei amplitude d no gro unremittingl becaus th nutatio period 27.3 s i no a intege multipl o th spi period s Th correspondin nutatio angl o th spacecraft denote y an define a [(/ion) (8 varie discontinuousl wit eac firing remainin constan

i th interim Numerica result sho tha th angl y i withi 0. de an doe no increase Th longitud ^ an th colatitud 0 var wit time startin fro thei initia value an reachin thei fina desire value a th en o precessio i 563 s remainin essentiall constan afterward—essentiall becaus th spacecraf continue t
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54 HABLAN | ( - 0. 15. 30. 45. 60. 75. 90. 105. 120. 135. 150. 165. 180. Longitud (deg Fig 2 Rhumb-lin precessio o a spin-stabilize spacecraft latitud v longitud (Mercato projection) nutat wit som residua amplitud afte precession indicatin th nee fo activ o passiv nutatio damping III

Activ Nutatio Dampin Thi sectio i concerne wit spacecraf wit zer an y prod uct o inerti an i base o Ref 5 Althoug Ref 5 assume tha th spacecraf spin abou th mino axis th followin analysi i fo th majo spi axis Hence accordin t th definitio o th body-fixe fram xi,yi,zi, i Fig 1 7 > 7 > I[. Contro Strateg Conside th equation o motio governin th transvers rate OL>\ an 0)2, influence b th unspecifie contro torque N\ an A fo nutatio damping Followin Deve e al., defin quasinutatio frequencie (/ ~ /2 , , a>n = j (9a OJn2 = (/ V^ (9b (9c Assumin tha th transvers rate \o)j\ <£ co ( = 1 2) th spi spee &>

remain essentiall constant &> & o) Th fre response therefore o th tw equation o motio governin transvers an gula rate ca b writte a wher j 1 & = */(o) i/o) n2 ), oc = co (t to), an p (} an 0 ar initia amplitud and phas of the motio a)\ (t) and &> (0 at t = t () = (O)\Q + ^ <^7o) ( a (lib Further Ref 3 show that fo nutatio dampin wit th contro torqu N\ abou th x axis th jc-angula impulse imparte b th thruster mus b (12. (12b wher A = t - t () i selecte suc tha Eqs (12 ar satisfie fo smal k k = 0.05 fo example Fo y thrusters th angula impulse fo nutatio dampin ar foun t b sin(a/2 - ~ ~ ' * * L

Fo detaile derivation se Ref 3 sin(a/2 (13a (13b + jko) = A ex j (# (10 Spacecraf wit Product-of-Inerti /2 Here w assum tha onl 7 = - / y d produc o inerti i nonzero Becaus th product o inerti ar typicall muc smalle
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HABLAN 54 tha th moment o inertia i i beneficial fo analytica ease t trea eac produc o inerti separately Th followin analysi there for ca b retrace fo 7 an 7 products o inerti individually thoug w d no attemp tha here W als poin ou tha th abov definitio o 7 include th negativ sign unlik th conventio use b Deve e al. Proceedin wit th vecto equatio o motio o a rigi

body wit 7 i th inerti matrix thre scala equation ar derive fo zer externa torqu actin o th spacecraft W the invok typica an presentl legitimat assumption tha \a)j \ < | fo 7 = 1,2 hi I <3 (7i 7 7 an &> & co wher a) = a constan spi speed Th transvers equatio governin a>\ possesse a forcin ter du t th product-of-inerti 7 23 Denot thi b IJL an introduc nondimensiona product-of-inerti paramete / a follows /^ (14a = /*/(/2/3 (14b Th produc o inerti alter th nutatio frequenc a) Eq (9c) t I- (15 Th transvers rates obtaine b solvin th transvers equation exhibi th followin force response 0)1 = (ii/a)'

)sma) OMrlX -COSO)' t) (16a (16b fro whic w conclud tha th transvers rat o> afte nutatio damping wil settl o it bia valu a)2b = M/ /»i A on woul anticipate a) i, i relate t th til angl A abou th geometri axi x/j, betwee th orthogona axe y/,z/ an th principa axe y particular i ca b show that fo smal til angle |A. <^ 1 th moments-of-inerti 7 an 7 ar essentiall th principa value an w (7 7 )A whic indicate that i th presen application becaus 7 > 7 th product-of-inerti 7 an th til angl A hav opposit signs Wit th ai o Eqs (9a an (14a) th bia rat o)2b i foun t b equa t Aa>, matchin wit Eq (39 o Ref 5

excep fo th negativ sig arisin fro th differenc tha Ref 5 deal wit spi abou the minimu momen of inerti and thi pape dealin wit spi abou th maximu moment-of-inertia Le b i , b bi b th uni vector alon th geometric spacecraft attache fram x zb> W the observ tha th steady-stat spi rat o th spacecraft u; equal &> b + o) b$ because i th prin cipa fram x (with/? a th uni vecto alon th principa spin-axi z ), i vie o th relationshi &> = A&; th angula velocit u equal (CD Xo> /,)p whic induce n angula motio abou th transvers principa axe x an y I th absenc o nu tation th steady-stat angula momentu h o th

spacecraf i th zh fram i = Substitutin coy, = A.o) 7 term h simplifie t w -M3 (17 (7 7 an ignorin th A + /3 (18 Th correspondin steady-stat nutatio angl y , define b Eq (8) thu y = \k\. Th steady-stat value o> 2/ = A&> an y = |X hel verif th numerica results implemen th nutatio dampin strategie (12 an (13) th phas angl 0 mus b determined Fo th zer product-of-inerti case thi i accomplishe usin th definitio (lib) wherei th unbiase transvers rate ar sense b gyros A nonzer produc o inertia however render thi procedur inadequat becaus gyro measur transvers rate containin biases Therefore th bia &>

determine above an &>i/, i i exist du t th product-of-inerti 7i no considere here mus b filtere ou fro a)\ an w sense th gyros Thi filterin i performe wit a bandpas filter on fo eac axis Bandpas Filter Th transfe functio o a bandpas filte i (19 centere a th nutatio frequenc o) here s i th Laplac variabl an f i a suitabl dampin factor Becaus thi filte i use a a par a sample contro system it followin discret for i derived () (20 wher z i th discret transfor an th polynomia coefficient ar define i Ref 3 i term o th semisampl angl 8 = a) /2, wit o) = (2/r tan(a) ts/2) an t equal th sampl period Le

a)2,k denot th £t sampl o th inpu angula rat &> a t = f* an a)2f,k denot th fct outpu sample Furthermore i th presen applicatio an typically th angl 8 i muc les tha unit an therefor 8~ ^ 1 . Usin thi assumption a simplifie time-domai versio o Eq (20 i then C0 f,k = (21 usin th discret time-domai version (21) w aver th nee fo th derivativ a) o th angula rat co i th continuou time domai versio o Eq (19) Thi i a significan benefi becaus th acceleratio 6) i no measured Numerica Result Fo result concernin zer produc o inertia se Ref 3 Figur 3 illustrate nutatio dampin whe th product-of-inerti

paramete i 0.01 Th correspondin til angl X betwee th geometri spin-axi Z an th principa axi z i —10. de an th bia rat (j)2b = A.a>. equal 3.2 deg/s Th filtere rat 0)2 / i obtaine usin th time-domai discret Eq (21 wit f equa t 0.2 Th correspondin tim dela fo th filte outpu co t ris t it steady-stat amplitud i nearl tw nutatio period (54. s) Moreover owin t th product-of-inerti 7 , th nutatio angl y , define by = (22 no constan now T dam th nutations th contro policy Eq (13) i invoke afte &> attain it stead state A dampin progresses th nutatio angl y (Fig 4 approache th magnitud th til angle |X

= 10. deg A th en o nutatio damping -2A -1. -1. -0. 0 0. 1. 1. 2 3 (deg/s Fig 3 Nutatio dampin i th plan uJi-U2 wit .y-torqu thrusters
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54 HABLAN 14.000- 11.875 9.750 7.625 5.50 6 12 18 21 30 36 42 48 54 60 Tim (s Fig 4 Nutatio angl 7 v time whil th filtere rat a> near zero 0) oscillate wit a smal amplitud (se Fig 3 ove th bia rat &> = 3.2 deg/s IV Thruste Attitud Contro Durin Aerobrakin Rea spacecraf ar seldo symmetri an streamline an there for atmospheri dra generate a torqu tha mus b nullifie t kee th spacecraf stable. Aerodynamic o th aerobrakin con figuratio o th spacecraf

a han reveal tha th pitc torqu coefficient—th onl coefficien require i thi study—i zer fo positiv angle-of-attac a atkt equa t 3.1 de and fo aerody nami stabilit o equivalentl fo positiv pitc stiffnes a angle of-attac a at equa t zero th slop o thi coefficien nea th equilibriu angl a atk ,/ i negative, a desired T nul th aerody nami torqu actin o th spacecraft a at mus b commande t equa t th equilibriu angl a atk> /, Analysi Figur 5 depict th geometri relationshi betwee th angle of-attac a at pitc angl a an th angl £ betwee th velocit vecto v an th loca horizonta O\ i th orbi plane Th loca

vertica loca horizonta orbi tria O\, O O an th spacecraf attache triad b\ b # show i Fig 5 ar define a usual Becaus th nos o th spacecraf i alon b\ th angl o attac is b definition or at = a £ Th angl £ varie wit th spacecraf tru anomal 0 measure fro th periapsi o wit r\ measure fro th lin o ascendin nod (Fig 5) an th objectiv o th pitc controlle i t ensur tha a at = or atk) /, tha is t kee th pitc angl equa t th commande pitc angl a 2c wher a 2c = a tk,/ - £ an £ varyin wit th tru anomal 9 thu esinf + e co 6 (23 wher e i th eccentricit o th orbit desig a reactio je controlle fo trackin a

varyin a 2c correspondin inertia pitc acceleratio comman i determined fo i govern th minimu contro torqu required Th inertia pitc rat comman co i term o th pitc rat comman 6t 2c an th tru anomal rat 0 i co 2c = d 2c - 0 = £ - 9. Differentiatio Eq (23 the lead t Q) 2c = - 0( + e + 2 co 9 (24 furthe differentiatio o whic yield th sough inertia pitc acceleratio comman a) 2c 0) 2c = si 9 ( + 3e + 4 co 9 (25 Wit th pitc comman a 2c an th inertia pitch-rat comman (i> 2c th angl o attac i commande t b equa t th constan bia angl a atk) /, Th actua angl o attack o course wil depen th performanc o th

IPF reactio je controlle use here. Numerica Result th initia phas o aerobraking th altitud o periapsi h an o apoapsi h ar h = 10 k an h = 33,26 km wherea th fina phas th apoapsi altitud decrease t h = 76 km I th initia phase th atmospheri torqu (Fig 6 cause b th rol an ya dra force i significan fo nearl 1 mi centere aroun th periapsis I th fina phase thi duratio expand t 2 min Destinatio Plane Lin o Ascendin Nod Fig 5 Attitud geometr o spacecraf fo aerobrakin i a ellipti orbit 6 12 18 240 30 36 42 48 54 60 Tim (s Fig 6 Pitc atmospheri torqu (#2,air nea periapsi o th initia aerobrakin orbit

0.06500 0.04675 0.0000 -1 1.500- 6 12 18 24 30 36 42 48 54 60 Tim (s Fig 7 Trackin a bia angl o attack angl o attack pitch an pitc rat v time Figur 1 o Ref 3 displayin co 2c pertainin t th initia aero brakin altitudes indicate tha th maximu commande torqu wil be pitc inerti * &>2c,ma = 1 -43e N m Th actua thruste torque however i fou order o magnitud larger 4.5 N m T mitigat th impac o thi disparity thruster ar turne o fo onl 2 ms impartin a pitc angula momentu o 0.0 N m o angula rat chang o 0.012 deg/ o th spacecraft Figur 7 depict th performanc o th IPF reactio je controlle designe kee a at

equa t a tk Th contro parameter correspon t a limit-cycl attitud 9 equa t 1 de an th dampin coefficien o a equivalen linea controlle equa t 0.707 Figur 7 show tha th angle-of-attac a at startin fro zero i brough clos t <*atk,/> equa t 3.1 deg I furthe show tha th pitc angl a
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HABLAN 54 become approximatel equa t a at k,/ + f an tha th pitc rat ai track £ I vie o th constraint o give thruste parameters th performanc o th controlle i thu ver satisfactory Referenc 3 ma b reviewe fo furthe details Sof Martia Landin Controlle Whe a spin-stabilize interplanetar spacecraf approache

Mars i i despu and a Martia atmospher become palpable th tucked-i parachut i deployed reducin th approac velocit t a constan termina velocity sa 8 m/s wit possibl a sizabl loca horizonta component Th parachut i jettisone a a altitud a lo a th thruster onboar an landin dynamic woul permit an bot horizonta an vertica velocit component ar reduce almos zer b a activ landin controlle i th remainin alti tud befor touchdown Th controller develope her accomplis thi purpos an ar significantl enhance version o th classica longitudina an latera controllers fo luna landing Longitudina Controlle Th

objectiv o thi controlle i t reduc th vertica velocit th tim o parachut jettisonin (a' Fig 8 t a specified low constan velocit v ( m/s b a certai altitud h ( m poin b Fig 8 abov th destinatio plane surface A additiona objectiv of the controlle is to lowe the spacecraf at a constan velocit fro th poin b t touchdown I Refs 9 an 10 th controlle i turne of a a sligh altitud (poin c Fig 8) e.g. 1 m t avoi raisin dus wit thruste plum an thereb contaminatin th planetar surface However thi require a accurat altimete t detec th poin c escalatin th spacecraf cos an weight Th alternativ adopte her i t le

th controlle brin th spacecraf dow a a constan velocit unti touchdown whereupo th inertia referenc uni (IRU indicate zer vertica velocit an th controlle the wil tur itsel off Th nomina free-fal phas c an th actua free-fal phas c'd henc d no exis i th presen controller Figur 9 (adapte fro Fig 13.12 p 583 Ref 10 depict th geometr o descent Le a b th nomina spacecraf deceleratio Altitud Termina ^- Velocity Parachut Descen ^Nomina Trajector "* Parachut jettisons controlle Actua start Trajector Nomina Constan Velocit Phas -Actua Constan Velocit Phas Nomina Fre Fal -Actua Fre Fal d d Downwar Velocit

z o v Fig 8 Differen phase o longitudina motio i altitud an vertica velocit plane Thrus Axi Win (Horizonta Velocity (dow rang direction = pitc (abou y-axis (slan range Martia Surfac Mar Cente Fig 9 Latera geometr fo activ Martia Lande (adapte fro Ref 10) th body-fixe —zi, directio produce b selecte thrusters Also denot th Mar gravitationa acceleratio alon th vertica z axi g Becaus th pitc angl 0 i small |0 <£ 1 rad th ne deceleratio o th spacecraf alon th z axi i essentiall (a gM). Becaus th spacecraft' referenc trajector passe throug th poin b(h ct v i th h - v plane Fig 8 th commande vertica

velocit z a a altitud h mus b = v +2(a -g )(h-h (26 Thi provide a descen contou show i th contro bloc dia gram Fig 10 Th actua velocit z however measure b a rada a th instantaneou spacecraf altitud o estimate b a IR system is usuall differen fro z A nois velocit erro the is obtaine show an filtere t arriv a a velocit erro s T eliminat thi error th referenc acceleratio a^ i augmente b a incrementa acceleratio K an th thruster ar turne o appropriately A ne velocit i measure o estimate agai an th abov procedur repeat itself T determin a suitabl valu o th contro gai K not tha th referenc tim T

require t brin th initia vertica velocit zo t v i T = (z v )/(a g ). Referenc 9 show tha th erro 8h betwee th actua altitud an th curren referenc altitud decrease exponentiall an K > 5 i a suitabl gai becaus exp(—5 = 0.0067 Th nomina deceleratio a i deter mine fro Eq (26) knowin th initia velocit z an th nomina initia altitud /z/, a whic th landin controlle i turne on •^v^/.no ^c gM (27 Thi i th minimu deceleratio tha th thruster mus supply I the delive more thei widt i modulate s a t trac a wit littl error Th abov controlle i use als fo th constant-velocit phas (Fig 8 execute nea ground Thi

phas i introduce because otherwise th latera controller discusse below cause larg pitc rat an pitc acceleratio command nea th landin site T effec constant-velocit descent th nomina acceleratio a i se equa to g in the descen contou (26 whe the measure altitud h (wit measuremen errors i les tha o equa t th specifie altitud h (e.g. 5 m) Also th nomina duratio r = h /v o th constant velocit phas i smaller tha th tim duratio T an therefor th acceleratio gai K fo thi phas i se t b muc large tha wha K cv = 5 woul allow Latera Controlle Th objectiv o thi controlle i t reduc t zer th initia horizonta

velocit o th spacecraf usin th sam thruster tha concurrentl decelerat th lande i th vertica direction Conse quently th lande mus b pitche fo latera contro b a angl 9 (Fig 9) no necessaril th sam a th angl ft the angl o th velocit vecto wit th vertical Whe th pitc angl 9 i equa t ft, tha result i gravit turn latera contro schem wit certai advantage elucidate an utilize subsequently Th bloc diagra th latera controlle i show i Fig 11 I i a amende versio th controlle i Fig 13.1 o Ref 1 wherei th commande pitc rat 9 yc i proportiona t v /R, wher V i th latera ve locit an R i th slan rang (Fig 9)

Thoug O yc s calculate effectiv initially th schem slowl degrade inasmuc a i cause th spacecraf t swin whil approachin th ground wit yc exhibitin divergen oscillations Becaus thi i unsatisfactory yc i calculate instea fro a suitabl altitud downwar (4 m presently accordin t gravit turn a portraye i Fig 11 Th ad vantag o doin s i tha th divergen oscillation i th pitch-rat comman the ar replace b stable smoot commands Th pitc comman fo gravit turn i th linea regime i 6 yc « x/z. Bu eve th gravit tur ha it limitation because a th spacecraf reache ver nea th ground bot latera an longitudina veloc

it component approac zer an th rat an acceleratio com mand (O yc an O yc becom intolerabl large Th latera controlle therefor i turne of altogethe whe |jc < 0. m/s acceptin th residua smal pitc angl an rate an residua latera velocit (1-1. m/s an longitudina velocit (0.5- m/s t b withstoo th spacecraf structur an th residen electroni components
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54 HABLAN RADA NOIS I Z-VELOCIT VERTICA + ? NOIS COMMANDE VERTICA VELOCIT DESCEN CONTOU a h h 2 2(a N" )(h +v v i h < h c (const vel descent c = 5. m c ACCN v QA| ACCN INCREMEN VELOCIT K K £ i >* a NOMINA DECELERATIO THRUSTE 49 N /

HFiRMTATt- 3 ENGINE + =500 + } a cosf3__* z 1 VERTICA ^ ^ AEROD INITIA VERT VEL 57. m/ ^ s 9 (MAR |T M GRAVITY DECEL DU T IPF ATTITUD CONTROLLER NE DECELERATIO INST VERT VELOCIT Fig 1 Closed-loo longitudina velocit controller RADA NOIS I HORIZ VEL VE ALON THRUS AX ALTiTUDE SLAN RANG H I UNBALANCE THRUS FORC T LONGITUDINA CHANNE 1 + LJ_ CN DU OLLE THRUS ACCN FRO LONGI CHANNE l^,5m^l.K LAT ^X/" PITC RAT COMMAN YE GRAVIT TUR PITC RAT COMMAN ~~1 PITC IPF CONTROLLE y PITC RAT COMMAN % Ll_ PITC RAT PITC ACCN DU T LATERA WIN LONGI CHANNE , r\ = Latera Geometr Angles v define i Latera Geometr (Fig 10

Fig 1 Latera velocit controlle fo activ Martia Lander Also thes touchdow condition and the cente of gravit of the lande ar stipulate t b suc tha th lande doe no overturn Regardin th gai K\ &i i Fig 11 th pitc rat comman 0 yc linea regim (tha is smal latera velocit compare t th lon gitudina velocit an smal pitc angle) accordin t Pfeffer, i yc KI^VI/R, wher th angl r\ an th longitudina velocit ar define i Fig 9 Th gai K\ at i optima whe i render th tw root o th quadrati equatio (13.52 o Ref 1 equal leadin use belo a certai altitude e.g. 4 m (Fig 11) replacin th earlie problemati pitch-rat

command Definin a quantit ctj = b /2 ( fo jets) th linea pitc com man i foun t be (29 (28 wher i an zt ar th initia velocit component a th instan whe th gravit tur i initiated Th pitc rat an acceleratio command ar derive b differentiatin 9 yc an substitutin z = -a + g arrivin a wher th positiv constan b i define a b =t 2a /(a —g > 2 Becaus th plu sig yield large AT la tha th minu sig does adop th plu sig an th large gain a gravit turn th angl rj (Fig 9 i zer b definitio an th commande pitc angl 0 yc varie continuousl a th horizonta an vertica velocit component i an z decreas an th spacecraf

descend t th Martia surface Knowin x an z wit th ai o a rada o IRU th idea angl 9 yc rat 9 yc an acceleratio 6 yc ca calculate t contro th pitc attitude Thi schem therefor VC = 3 -a )x (30 (31 Comparin I yc ( = pitc momen o inertia wit th availabl pitch-contro torque th gravit tur ca b terminate whe th for me exceed th latter an th constan velocit descen the begins
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HABLAN 54 45.0 35.0 25.0 15.0 5.0 0. -5. Actua (6 8 Commande (6^ 8 J 1. 3. 4. .6. 8 9. 11. 12. 14. 1 Time(s 20. - 13.0 6, - -1,0 -8.0 -15. Pitc Rate Limi Cycl Rate 1. 3. 4. 6. 8 9. 11. 12. 14. 1 Fig 1 Pitc an

pitch-rat control actua an commanded Numerica Result Th performance o increasingl mor complet landin con troller wer examine an extensiv simulatio studie wer con ducte but t conserv space onl limite result ar show here Referenc 3 illustrate a unsatisfactory oscillator pitch-contro performanc o th origina landin controlle o Refs 9 an 1 withou gravit tur fo pitc contro a lo altitudes Figur 12 contrast illustrate performanc o th pitc controlle whe th gravit tur i incorporate belo 45- altitude provide th hori zonta velocit magnitud \v \ i greate tha 0. m/s Th unaccept abl divergen oscillation ar

absen i Fig 1 becaus 1 whe h < m an | v \ > 0. m/s th smoot pitc an pitc rat command ar obtaine fro Eqs (29 an (30) 2 whe th horizonta veloc it \v < 0. m/s regardles o th altitude th commande pitc angl an th rat co yc ar bot zeroed an 3 th pitc angl an rat command ar bot zeroe i th constant-velocit phas belo 5 m Th performanc o th landin controlle which i additio t gravit turn use a rada altimete wit certai measuremen nois characteristic an digita fadin memor filter t smoot nois altitud an velocit measurement i illustrate i Figs 1 an 14 Becaus o excessiv nois i th measurement belo 45- alti

tude th rada altimete i cu of a 4 m an IR navigatio o th lande i initialize wit th las measuremen o th velocit an altitud b th rada assembly Althoug th altitud measurement hav rando errors the ar no modele here onl thei bia erro modeled A 45- altitude th IR inherit a bia altitud erro —2.1 m (2.58o valu t includ 95 erro probability o th altimeter A positive-bia velocit erro an a negative-bia altitud erro ar chose fo illustratio becaus th actua velocit the i les an th actua altitud i mor tha th quantitie measured Consequently th lande take longe t touc th groun tha th referenc constant-velocit

duration r cv yieldin a estimat o th require wors propellan consumption Moreover l valu o th rando velocit nois i no constan (nois i nonstationary) i instea proportiona t velocity. Figure 1 an 1 illustrat a sampl performanc o th multi axi landin controlle amids velocit an altitud noise filtere a first-orde filter th coefficient o whic ar selecte accord in t it optimu stochasti respons t nois an it ris time Th initia altitud i thes figure i se at 348.8 m base o a referenc altitud o 35 m an a 1.66 l bia erro i altitud measurements Th actua an referenc vertica velocit profile (Fig 13 confir th

earlie conclusio that becaus o a negativ altitud bia erro o —2.1 m an a positiv vertica velocit bia erro of 0.4 m/s the actua touchdow take longe tha the ref erenc touchdow (nearl 1 s v 15.8 s) Th vertica touchdow velocit i abou 1. m/s—th constan velocit o th fina descent Also becaus th horizonta velocit bia erro i positiv (0. m/s) 21.0 15.0 _ 9.0 3 '° 0. -3. 60.0 48.0 36.0 24.0 12.0 1 H 1 1 2 2 4 6 8 1 1 H 1 1 2 Tim (s Fig 1 Performanc o a sof landin controlle wit a nois rada altimete abov 4 m a noiseles IR belo 4 m an a digita nois filter horizonta an vertica velocit components 1 2 3 3 4 4

Vertica Velocit v (meters/s 6 0. Fig 1 Altitud v vertica velocit v compariso o actua an ref erenc trajectories th actua touchdow horizonta velocit v i negative, —1.3 m/ (Fig 13) Becaus th IR navigatio syste operate essentiall noise-fre belo h = 4 m th velocitie ar see t b suc afte = 8 s wher th lande crosse 45- altitude Th h v guid anc trajector an actua trajector ar portraye i Fig 14 wher observ tha th longitudina controlle track th guidanc tra jector easil durin bot th constant-acceleratio phas an th constant-velocit phase th latte startin a a tru altitud o 5 + 2.1 = 7.1 m instea o 5 m du t

-2.1 m o /z error Figur 2 Ref 3 show picturesquel th poignan descen o th lande i th altitude-downrang plane Therei enthrallin i it sligh hori zonta backwar motio jus befor touchdown preenin wit pride VI Concludin Remark Fou attitud controller ar presente fo spinnin an nonspin nin phase of interplanetar spacecraft For Rhumb-lin precessio th spi axis i i crucia t maintai a constan phas o thruste pulse wit th sun T mee thi condition thes pulse mus b slightl aperiodic exactl periodic pulse a spi perio woul vee th spi axi t a unintende direction Thruster ca dam ver effectivel nutation o spacecraf

wit product o inerti i it transvers angula rate ar passe throug suitabl designe band pas filter tha eliminat constan rat biases Whe th spacecraf arrive a th destinatio planet th initia highl eccentri orbi
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55 HABLAN circularize wit aerobraking Attitud o th spacecraf ca b controlle durin aerobraking ami a order-of-magnitud uncer taint i th ai density wit IPF reactio je controller provide th referenc pitc an pitch-rat command ar inputte t th controller Finally th vertica an horizonta velocit controller availabl i th literatur fo sof landin ar inadequat becaus th latte controlle

cause divergen oscillation i pitc an pitch rat command a th spacecraf approache th ground Thi short comin i remove b switchin a a suitabl intermediat altitude e.g. 5 m t pitc an pitch-rat command associate wit gravit turn Whe ver clos t th ground e.g. 5- altitude a constant velocit referenc trajector i introduce i th vertica velocit controlle fo a sof touchdow o th spacecraft Acknowledgmen Wit grea pleasure I wis t acknowledg m numerou in sightfu technica discussions ove th years wit th Guidance Navigation an Contro Lea Enginee A Cormack Withou hi guidance i i inconceivabl ho thes effort woul

hav bee brough t fruition Reference ^reene R H. "Earl Bir Placemen i a Stationar Orbit Launc an Contro Syste Maneuvers, AIA Pape 66-262 Ma 1966 Furukawa M. "Precessio Maneuve Performe b Applyin a Unifor Trai o Thrus Pulses, Journal o Spacecraft an Rockets, Vol 1 N 1 1976 pp 600-604 ' ' Hablani H B. "Interplanetar Spacecraf Controller Usin Thrusters, AIAA Guidance, Navigation, an Control Conference, AIAA Reston VA 1997 pp 1590-161 (AIA Pape 97-3754) ' Wertz J R. Spacecraft Attitude Determination an Control, Reidel Boston MA 1984 pp 649-654 Devey W J. Field C F. an Flook L. "A Activ Nutatio

Contro Syste fo Spi Stabilize Satellites, Proceedings o 6t IFAC Triennial World Congress (Boston/Cambridge MA) 1975 pp 14.1/1-10 Carpenter A S. "Th Magella Aerobrakin Experiment Attitud Con tro Simulatio an Preliminar Fligh Results, AIAA Guidance, Naviga- tion, an Control Conference, AIAA Washington DC 1993 pp 1148-115 (AIA Pape 93-3830) Carpenter A S. an Dukes E. "Contro o th Magella Spacecraf Durin Atmospheric Drag, America Astronautica Society 17t Annua AA Guidanc an Contro Conf. Pape 94-064 Keystone CO Feb 1994 Hablani H. B. "Targe Acquisition Tracking Spacecraf Attitud Con trol an

Vibratio Suppressio wit IPF Reactio Je Controllers, Journal Guidance, Control, an Dynamics, Vol 17 No 4 1994 pp 831-839 Cheng R K. an Pfeffer I. "Termina Guidanc Syste fo Sof Luna Landing, Guidance an Control, edite b R E Roberso an J S Farrior Vol 8 Progres i Astronautic an Rocketry Academic Ne York 1962 pp 217-239 1() Pfeffer I. "Termina Guidanc fo Sof Luna Landing, Guidance an Control o Aerospace Vehicles, edite b C T Leondes McGraw-Hill Ne York 1963 pp 563-587 7t Annua AIAA/BMD Technolog Readines Conferenc an Exhibi Fo th lates technolog concernin hardwar an softwar i ballisti missil

defens i area o surveil lance weapons an testing mak sur yo atten thi nationa confer enc an exhibition Admissio t session require (Secret/U.S.-only clearance receiv a preliminar progra o register contac AIA a 800/639-242 o visi AlAA' We sit a http://www.aiaa.org America Institut Aeronautic an Astronautic