ASCENT TRAJECTORY AND GUIDANCE DESIGN WITHOUT BLACK ZONE

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1 ASCENT TRAJECTORY AND GUIDANCE DESIGN WITHOUT BLACK ZONE Shii Mohan Aociat Profor, Dpt. of Elctrical Enginring, Mualiar Collg of Enginring, Chirayinkzh, Krala,(India) ABSTRACT Thi papr prnt an optial acnt guidanc algorith for a launch vhicl uch that th acnt trajctory i fr of black zon. An acnt trajctory with no black zon will nur that, in ca of a failur, th iion can b abortd afly without activ guidanc of th crw cap yt. Th guidanc probl i to axiiz th payload into orbit ubjct to th quation of otion of a rockt ovr a phrical non-rotating Earth. Th guidanc law dvlopd i bad on th ayptotic thod whrin a all xpanion paratr i ud to parat th dynaic of th probl into priary and prturbation dynaic. Th zroth ordr probl i obtaind by quating thi paratr to zro. Onc th olution to th zroth ordr probl i obtaind, th highr ordr corrction tr can b includd to copnat th ffct of prturbation forc which ar nglctd in th zroth ordr probl. Kyword: Ayptotic Mthod, Black Zon, Rgular Prturbation, Zro-Ordr Launch Probl I. INTRODUCTION Th objctiv of thi work i to dvlop an acnt trajctory for huan pac progra uch that black zon ar not prnt in th trajctory. Black zon i a portion of a annd rockt launch trajctory whr th pratur hutdown of any or all running bootr ngin will lad to lo of th vhicl and crw ubquntly du to th ovr tpratur or tructural load incurrd fro th rulting trajctory. Thu black zon i a ti priod during launch whn th crw would b unabl to afly cap or abort in th vnt of a failur of th launch vhicl. Thu an acnt trajctory with no black zon will nur that in ca of a failur, th iion can b abortd afly without activ guidanc of th crw cap yt. By optiizing th rockt trajctory and flight path, th black zon can b avoidd. Th acnt trajctory ut alo b hapd to avoid ubjcting th crw to fatal acclration during an abort. Rliabl and affordabl acc to pac with nhancd afty ha now bco a critical rquirnt for all pac iion. Nxt gnration yt ut b capabl of handling failur and abort rquirnt ffctivly. It i iportant to advanc th guidanc algorith dign to uch a point that it i poibl to uccfully tr and control th vhicl aftr any failur. Prior to th STS-51 Challngr accidnt, Spac Shuttl iion planning ainly focud on th noinal iion and th intact abort od of acnt and ntry flight. Th probl ot oftn aociatd with intact abort i that of a ingl Spac Shuttl Main Engin (SSME) failur which can b handld with vral availabl option lik abort to orbit and abort onc around. 84 P a g

2 Intact abort od ar aid at rcovring both th orbitr and th crw afly. But whn th probl i or vr a in th ca of ultipl ain ngin failur, it ay not b poibl to rcovr th orbitr and th crw urvival i of utot iportanc. So it i rquird to hav an acnt trajctory uch that, in th vnt of a ajor probl on acnt, th crw urvival i alway poibl. An Atla V 41 vhicl i capabl of launching annd payload without any black zon [1]. Th trajctory of th vhicl i hapd to avoid ubjcting th crw to fatal acclration during an abort and thi tranlat to a dprd acnt with a flattr trajctory. Indian Spac Rarch Organization i working toward it aidn annd iion for carrying thr atronaut to a Low Earth orbit. Thi ncitat th nd for digning an acnt trajctory that i copltly fr of black zon. Bcau thi will nur that in th vnt of a ajor probl on acnt, th crw urvival i alway poibl. Svral acnt guidanc algorith hav bn dicud arlir [6], [7], [8], [9]. Th ot challnging part of acnt guidanc li in th ndoatophric portion of flight. Thi i bcau of th fact that th prnc of arodynaic forc and wind in th atophric rgion coplicat th acnt probl and a a rult it i not poibl to obtain a fat convrgnc of th olution. Bcau of th raon opn loop guidanc i ud in th atophric flight pha of th vhicl and clod loop guidanc i ud in th xoatophric portion of flight. Th opn loop guidanc ch conit of a tabl of attitud coand, which will b a function of ti, altitud or vlocity. Th coand ar updatd with th condition on day of launch and loadd into th yt prior to launch for u during acnt through th atophr. Th clod loop guidanc algorith tak into account th chang in xtrnal variabl uch a wind, thrut, tc. during flight and accordingly gnrat attitud coand that th vhicl ha to follow. Dlivring th crw to orbit or afly i th ot iportant conidration govrning futur pac iion. On of th ot iportant rquirnt govrning thi i th capability of th guidanc ch to handl contingnci and abort. In th ca of opn loop guidanc, contingncy or abort planning following a failur i th ot labor intniv apct of pr-iion trajctory dign. Contingncy trajctori ut b dignd during pr-iion analyi to account for abort or ngin-out ituation. In th ca of opn loop ch, th gnration of th trajctori rquir any hour of analyi for both th dign and th vrification of th opn-loop tring profil. But vn with xtniv off-lin planning, th opn loop guidanc ch lack th capability to handl contingnci and abort. On th othr hand, a clod-loop yt can autoatically adapt to abort ituation and dign accptabl trajctori onboard th vhicl. Thi could av uch ffort a copard to opn loop ch. Acnt and ntry guidanc ch to iprov th probability of urviving a firt tag contingncy abort i dcribd in [1]. Th ca conidrd i that of ultipl ain ngin failur in which an intact abort i not poibl. Th acnt ch guid th vhicl out of th atophr a arly a poibl in ordr to rduc th dynaic prur at olid rockt bootr burnout. An attitud rror intgration tchniqu i ud to bia an initial pitch coand dtrind whn a cond Spac Shuttl Main Engin ha faild. Contraint on Mach nubr and tructural loading ar ipod for dvloping th guidanc ch. In thi papr a ral-ti optial guidanc law i dvlopd which i bad on ayptotic thod [1], [], [3], [4]. In ayptotic thod an approxiat olution to a probl i contructd in tr of a all paratr which i trd a th xpanion paratr. Th advantag of uing ayptotic thod i that th nonlinariti can b tratd a prturbation which can b nglctd in th forulation of th zroth ordr probl. Thi 85 P a g

3 ignificantly iplifi th probl and ak it poibl to obtain a clod for olution for th acnt probl. Sction dcrib th ayptotic thod in dtail including th rgular prturbation thod and ingular prturbation thod. In Sction 3 th acnt guidanc probl which i olvd uing ayptotic thod i dcribd. Sction 4 prnt th zro-ordr forulation of th acnt probl. Th iulation rult for zro-ordr probl ar givn in Sction 5. II. ASYMPTOTIC METHOD Thr xit any function ariing fro vryday probl that ar difficult to b valuatd xactly. Thi ay b du to th prnc of nonlinariti, variabl cofficint, coplx boundary hap, and nonlinar boundary condition which ay b known or unknown. Thu, in ordr to obtain inforation about olution of quation, it ay bco ncary to u approxiation, nurical olution, or cobination of both. Forot aong th approxiation thod ar th ayptotic (prturbation) thod. By uing ayptotic thod it i poibl to contruct an analytical approxiation to th olution uing ayptotic xpanion. On of th ain u of ayptotic analyi i to provid approxiation to diffrntial quation that cannot aily b olvd xplicitly. Th variabl with rpct to which th ayptotic bhaviour of a probl i tudid i known a th ayptotic variabl. In prturbation tchniqu, th olution of a probl i rprntd by th firt fw tr of an ayptotic xpanion, uually not or than thr tr. Th xpanion ay b carrid out in tr of a paratr which appar naturally in th quation, or which ay b artificially introducd for convninc. Such xpanion ar calld paratr prturbation. Prturbation thod ay b plit into rgular and ingular for..1 Rgular Prturbation Vry oftn, a athatical probl cannot b olvd xactly or, if th xact olution i availabl, it xhibit uch an intricat dpndncy on th paratr that it i hard to u a uch. But it ay b poibl to idntify a paratr, uch that th olution i availabl and raonably ipl for. Thn, thi olution can b altrd for non-zro but vry all 1. Thi for th bai of rgular prturbation thory. In rgular prturbation probl, th olution of a probl i ought a an xpanion in tr of th ayptotic qunc 1,,,... a. Th baic principl of th rgular prturbation xpanion i: a. St = and olv th rulting yt. b. Prturb th yt by allowing to b nonzro (but all). c. Forulat th olution to th nw, prturbd yt a a ri f f f whr f, f 1, f ar to b dtrind fro th boundary condition d. Expand th govrning quation a a ri in, collcting tr with a powr of ; olv th in turn a far a th olution i rquird. In rgular prturbation thory or dirct prturbation thod, it i xpctd that all th inforation rgarding th olution of a probl i capturd by th firt fw tr, idally, th firt two tr, i.. th olution ford by 86 P a g

4 olution y taking th firt two tr of th ayptotic xpanion will giv a good approxiation to th xact olution and that th highr-ordr tr rprnt only all corrction. Conidr a two point boundary valu probl: y y y y in( t) (1a) y( ), y (3.5) 1 (1b) Th olution of thi probl i contructd a a ri in : y y y () 1... Th zroth ordr probl i obtaind by ubtituting in (1) and taking th a boundary condition a in th original probl. So th probl bco: y y in t (3) For obtaining th firt ordr olution, th boundary condition ar takn a: y() ; y (3.5) 1 (4) Figur1 how th zro ordr olution y, firt ordr olution y 1, and copoit olution y y 1, for =.1. Th zro ordr olution giv a good approxiation of th actual olution. For vn allr valu of, th olution obtaind by ayptotic thod xactly atch th original olution of th probl. 3 1 y y1 copoit olution xact olution ti(c) Fig 1 Exact olution and ayptotic approxiation for two-point boundary valu probl. Singular Prturbation A prturbation probl i aid to b ingular whn th olution obtaind fro th rgular xpanion fail to b valid ovr th coplt doain. In ca whr i th lading ordr tr' only cofficint th ordr of th quation i rducd in th unprturbd quation. Whn i a ultiplir of th hight drivativ or lading tr of a polynoial quation it i known a a boundary layr probl or a atching probl. Th yt in which th upprion of a all paratr i rponibl for th dgnration or rduction of dinion (or ordr) of th yt ar trd a ingularly prturbd yt. Singular prturbation thory find wid application in guidanc and control of aropac yt [11]. Conidr a yt dcribd by a linar, cond ordr, initial valu probl: x ( t, ) x ( t, ) x( t, ) (5a) 87 P a g

5 olution x,z x( t ) x(), x ( t ) x () (5b) whr th all paratr ultipli th hight drivativ tr. Th dgnrat (outr or rducd-ordr) probl i obtaind by uppring th paratr ε a: Th olution to (6) i obtaind a: () () () x ( t) x ( t), x ( t ) x() (6) x () ( t) () t x () x() t Sinc th dgnrat probl i only of firt ordr and cannot atify both th givn initial condition givn in (5), on of th initial condition ha bn acrificd. Thu th ordr of th probl bco lowr for than for. Th iportant fatur of ingular prturbation ar a follow: 1) Th dgnrat probl, alo calld th unprturbd probl, i of rducd ordr and cannot atify all of th boundary condition of th original probl. ) Thr xit a boundary layr (rgion of rapid tranition) whr th olution chang rapidly. It i blivd that th boundary condition that ar lot during th proc of dgnration ar burid inid th boundary layr. 3) To rcovr th lot initial condition, it i rquird to trtch th boundary layr uing a trtching tranforation uch a t /. 4) Th ingularly prturbd probl will b having widly paratd charactritic root which giv ri to low and fat coponnt in th olution. To illutrat th fatur of ingular prturbation probl, conidr th probl givn by (5) in tat variabl for a: (7) dx( t, ) z( t, ), x( t ) x() (8a) dt dz( t, ) x( t, ) z( t, ), z( t ) z() (8b) dt Th variou olution of thi probl for o pcific valu of =.1, x () = and z () =3 ar hown in Figur low olution x dgnrat or outr olution for x -1 fat olution z dgnrat or outr olution for z ti(c) Fig Baic concpt of ingular prturbation 88 P a g

6 olution x, z It i found that th prdoinantly low olution i x( t, ) and prdoinantly fat olution i z ( t, ), which i aociatd (ultiplid) with. Th dgnrat olution of x ( t, ) and z( t, ) ar obtaind by olving th dgnrat probl with in quation (8). Thr xit a boundary layr nar th point t= and th dgnrat olution of z( t, ) i clo to it xact olution only outid th boundary layr. Alo on of th two condition, z (), i dtroyd in th proc of dgnration. Thi probl can b olvd uing th thod of atchd ayptotic xpanion. Th approxiat copoit olution i xprd in thr part, outr, innr, and coon olution. Th outr olution i valid in th rgion outid th boundary layr, whra th innr olution i valid inid th boundary layr. Bcau th two rgion ar bound to ovrlap, a atching proc i rquird to idntify th coon olution. A copoit olution, valid in th ntir rgion i contructd a th u of outr olution and innr olution fro which th coon olution i ubtractd. Thu atching i accoplihd by xtnding th outr olution into th innr rgion by tranforing th outr variabl to that of th innr variabl and taking th liit a. Thi i calld th innr liit of th outr olution or xpanion. Siilarly, th outr liit of th innr olution or xpanion i obtaind by xtnding th innr variabl into th outr rgion by tranforing th innr variabl to that of th outr variabl t and taking th liit a. By quating th innr liit of outr xpanion with th outr liit of innr xpanion, w can dtrin th coon olution. Th olution of (8) uing thi thod i hown in figur xact olution for x dgnrat or outr olution for x -1 xact olution for z zroth ordr olution for z dgnrat or outr olution for z ti(c) Fig 3 Solution of ingular prturbation tchniqu Thu, th copoit olution i copod of: x ) o i o i o i i o c x x ( x ) x x ( x (9) whr x o i th outr olution, x i i th innr olution, (x o ) i i th innr xpanion of th outr olution and (x i ) o i th outr xpanion of th innr olution. Alo, th innr xpanion of th outr olution i qual to th outr xpanion of th innr olution. Th ayptotic thod prov to b an fficint thod for olving th acnt guidanc probl. Thi i bcau th non-linariti prnt in th yt can b tratd a prturbation and thi iplifi th yt 89 P a g

7 dynaic. For thi purpo th quation of otion for th launch probl ar xprd in tr of an xpanion paratr, which parat th dynaic into priary and prturbation dynaic. III. ASCENT GUIDANCE PROBLEM Th quation of otion for a launch vhicl odld a a point a ovr a phrical, non-rotating Earth ar givn for flight in thr dinion a: h V in (1) T co D V g in (11) T in L V V r h g co V (1) V co (13) r h T vac (14) whr h i altitud, V i vlocity, i flight path angl, i longitud, i total a, g i gravity, T i thrut, T vac i vacuu thrut, i angl of attack, r i radiu of arth, i pcific ful conuption, and th arodynaic forc ar lift L and drag D, dfind a: L C qs, D C qs (15) L D whr C L and C D ar th lift and drag cofficint rpctivly, S i th rfrnc ara, and 1 q V i th dynaic prur. Th dnity i aud to b of th for: ( r h)/ h r r r h h h (16) whr h i th atophric cal hight and r i a rfrnc dnity. Th for of dnity i chon in ordr to parat th dynaic into priary and prturbation ffct. Th gnral quation of otion ar givn in tr of a all paratr, which i takn a th ratio of atophric cal hight to th radiu of arth: Th total thrut of th vhicl i odld a: h / r (17) T Tvac npa (18) whr T vac i th total valu of th thrut whn acting in vacuu, n i th nubr of ngin and A i th nozzl xit ara. 9 P a g

8 Th atophric prur i alo conidrd a an xponntial function: p p h/ h p (19) whr h p i th atophric prur cal hight and p i th a-lvl rfrnc prur. Th invr-quar-law gravity odl g g [ r / r h ] gh(r g g r h h r h h) gh(r h) r g can b xprd a: () whr th cond tr i all copard to g and i inrtd to forally introduc th approxiation. Evn though th paratr i artificially introducd, th tr ultiplying i of zro ordr. Thi approach i takn to includ all of th kinatic tr du to th phrical natur of th Earth bcau thy ar all copard to th priary forc. In tr of th all paratr, th quation of otion ar rwrittn a: h V in (1) T vac npa r SV CDr V co g in co h h gh r h h r r h in () Tvac g in V SVr C h L co npa r in V Vh V g r h h r V r h h r co h (3) V co h 1 (4) r h Thi paration of th dynaic into priary and prturbation tr allow a clod for olution of th zro -ordr probl. IV. ZERO-ORDER LAUNCH OPTIMIZATION PROBLEM Th zro-ordr launch probl i obtaind by putting in th quation of otion. Th probl i to axiiz th final a into orbit for th flight of a rockt in vacuu ovr a flat nonrotating Earth. Th quation of otion for th zro-ordr probl bco: h V in (5) 91 P a g

9 Tvac V co g in (6) Tvac g in V V co (7) V co r (8) T vac (9) Thu th zro-ordr optiization probl conit of iniizing J= - f, givn th initial condition, th dynaic and th trinal contraint on h, V and. Th Hailtonian for thi yt can thn b xprd a: whr, h, v and T H hv in v co g in T g in co Tvac V V ar th Lagrang ultiplir aociatd with th tat. (3) Th zroth ordr control law dtrind fro th optiality condition i: tan (31) V V Th optial control can b drivd in tr of th tat and Lagrang ultiplir but it i not poibl to obtain an analytic olution for th tat and Lagrang ultiplir. Thrfor, a coordinat tranforation into th Cartian rfrnc fra i prntd in th nxt ction. 4.1 Zro-ordr Coordinat tranforation Th analytic olution for th zroth-ordr probl can b found in th Cartian coordinat yt but th quation of otion of th full yt which includ th arodynaic forc ar writtn in th wind axi yt. Thrfor, to driv th zroth-ordr control and th firt-ordr corrction to th control th tranforation of coordinat and pcially th tranforation of th Lagrang ultiplir ut b known. Thi can b accoplihd by a canonical tranforation fro th (,, h) coordinat to th right-handd coordinat yt (X, Y, Z), whr X i poitiv in an atward dirction along th quator, Z i poitiv pointing toward th Earth, and Y i orthogonal to th X - Z plan. Th rlationhip btwn th two rfrnc fra i X= r, Y= r and Z = -h. In two-dinion, th corrponding vlocity coordinat (u, w) ar conidrd poitiv in th poitiv X and Z dirction, rpctivly. 9 P a g

10 Th tranforation: rquir that: u V co, w V in (3) V u / V u / / w / V u w w (33) Thi produc th tranforation of Lagrang ultiplir: co in (34) V u w V in co (35) u w 4. Solution to Launch Optiization Probl in Cartian coordinat Fra Th quation of otion in Cartian coordinat fra bco: h w (36) T w in p g (37) u T co p Th control variabl for thi probl bco th pitch attitud, p. Th optiization probl now to b olvd i to find th control qunc contraint on altitud, vrtical vlocity and horizontal vlocity. (38) p that axiiz th final a ubjct to th dynaic and trinal Th zroth ordr Hailtonian i : whr, h, w u and T H h w w in p g T u co p Tvac (39) ar th Lagrang ultiplir. Th optial control law i obtaind a: w tan p (4) u Thi control law i ubtitutd back into th quation of otion to obtain a clod for olution for th tat. V. SIMULATION AND RESULTS Th rult for th zro-ordr optiization probl ar prntd in thi ction. Th vhicl odl ud for iulation i th Advancd Launch Syt [5]. Th initial condition ar takn a V =1.16 /c, h =4 and =54 dgr. Th trinal contraint to b atifid ar V f = /c, h f =11458 and dgr. f = 93 P a g

11 altitud() vrtical coponnt of vlocity(/c) horizontal coponnt of vlocity(/c) ti(c) Fig 4 Horizontal vlocity V ti ti(c) Fig 5 Vrtical vlocity V ti 1 x ti(c) Fig 6 Altitud V ti 94 P a g

12 pitch angl(dgr) flight path angl(dgr) ti(c) Fig 7 Flight path anglv ti ti(c) Fig 8 Pitch angl V ti Th final ti for th zroth ordr probl i obtaind a c. Th final a i obtaind a kg. Fro figur 4 it i found that th horizontal coponnt of vlocity at th final ti i /c. Th trinal contraint on altitud i t at th final ti of c. Thi i hown in figur 6. Th flight path angl at th final ti i zro dgr a can b n fro figur 7. Th variation of pitch angl with ti i hown in figur 8. Th acnt trajctory gnratd by th zro-ordr probl can b furthr optiizd by iplnting firt ordr corrction. Thi can ignificantly rduc th final ti and thu axiiz th final a injctd into th orbit. VI. CONCLUSION Th tchniqu for dvloping an approxiat optial guidanc law for a launch vhicl i prntd in thi papr. Th quation of otion for th acnt guidanc probl ar forulatd in tr of a all xpanion paratr which i th ratio of atophric cal hight to th radiu of arth. Th forc acting on th vhicl ar paratd into doinant forc and prturbation forc. A a firt tp in olving th guidanc probl, th zroth ordr probl i forulatd by quating th xpanion paratr to zro. Onc th olution to th zroth ordr optiization probl i obtaind, th highr ordr corrction tr can b includd to copnat th ffct of prturbation forc which ar nglctd in th zro ordr probl. Futur work includ iplnting 95 P a g

13 firt ordr corrction to th zro-ordr launch trajctory in ordr to optiiz th trajctory for axiizing th final a and digning th trajctory uch that it i fr of black zon. Thi i for nuring af rcovry of th crw in ca of a ajor failur during acnt. REFERENCES [1] T. S. Fly and J. L. Spyr, Tchniqu for dvloping Approxiat Optial Advancd Launch Syt Guidanc, Journal of Guidanc, Control, and Dynaic, Volu 17, No. 5, pp ,Sptbr- Octobr [] T. S. Fly and J. L. Spyr, Ral- Ti Approxiat Optial Guidanc Law for Flight in a Plan, Procding of th Arican Control Confrnc, San Digo, 199. [3] T. S. Fly and J. L. Spyr, Approxiat Optial Guidanc for th Advancd Launch Syt, NASA Contractor Rport 4568, [4] J. L. Spyr and E. Z. Cru, "Approxiat Optial Atophric Guidanc Law for Aroaitcd Plan- Chang Manuvr", Journal of Guidanc, Control, and Dynaic, Volu 13, No. 5, Sptbr-Octobr 199. [5] A. J. Cali, and Martin S. K. Lung, Optial Guidanc Law Dvlopnt for an Advancd Launch Syt, NASA-CR-19189, Jun [6] Mc Hnry, R. L., T. J. Brand, A. D. Long, B. F. Cockrll, and J. R. Thibodau, Spac Shuttl Acnt Guidanc, Navigation and Control, Journal of th Atronautical Scinc, Volu 7, No.1, pp.1-38, January-March, [7] John M. Hanon, M. Wad Shradr, and Craig A. Cruzn, Acnt guidanc coparion, Th Journal of th Atronautical Scinc, Volu 43, No. 3, pp , July-Sptbr [8] A. J. Cali, Mlad, N., and L, Sungja, Dign and Evaluation of a Thr- Dinional Optial Acnt Guidanc Algorith, Journal of Guidanc, Control, and Dynaic, Volu 1, No. 6, pp , Novbr-Dcbr [9] Ping Lu, and Honghng Sun, Clod-Loop Endoatophric Acnt Guidanc, Journal of Guidanc, Control and Dynaic, Volu 6, No., March-April 3. [1] Sponagl, Stvn J., Frnand, Stanly T., Spac Shuttl Guidanc for Multipl Main Engin Failur during Firt Stag, Journal of Guidanc, Control and Dynaic, Volu 1, No. 6, pp , P a g

[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then

[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd