134 J. GUIDANCE VOL1 . . 8NO ,

Minimum-Fuel Aeroassisted Coplanar Transfer Using Lift-Modulation

Kenneth D. Mease* Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California and Nguyen X. Vinht University of Michigan, Arbor,Ann Michigan

Minimum-fuel trajectories and lift controls are computed for aeroassisted coplanar transfers from high orbit to low Orbit. The optimal aeroassisted transfer requires less fuel than the all-propulsive Hohmann transfer for a wide range of high orbit to low orbit transfers. The optimal control program for the atmospheric portion of the t maximua transfey fl o t ms ri positive lift-to-drag ratio (LID) initiall recoveo yt r fro downware mth d plunge, and then to fly at negative LID to level off the flight in such a way that the vehicle skips out of the atmosphere wit flighha t path angle near zero degrees avoio T . d excessive heating rate vehicle sth e flies initiall t higya h angle of attack in order to slow down higher in the atmosphere. This allows the subsequent recovery from the downward plunge, using the maximum positive LID, to take place at a higher altitude, where the atmosphere is less dense.

Nomenclature AF, A= deorbiFo t t froO mHE b A V2 circularizo t A= V O LE t a e x =c /ct B = PoSHeCl/(2m) L C = drag coefficient A =E*p,/(vpv) D — gravitational constant multiplied by mass of = valu Ceof D when C0 L= = lift coefficient (398601. 3 kmVs2) p = atmospheric density Q = valu f Ct eo (L/D)a L max E* P0 t //=4a valu = p f m 0eo k F T g acceleratio- f gravito n y GEO = geostationary Earth orbit Subscripts h valu= t atmospheriea e c entry H = altitude / = value at atmospheric exit HR = convective heating rat espher m fo 1 r e valu= t hypotheticaea p l perige transfef eo r orbit from HEO = high Earth orbit HEO to atmosphere K = coefficient in parabolic drag polar LID = lift-to-drag ratio LEO = Introduction m = vehicle mass orbitaN HE l transfe s requirei - r at d n thera an ds i e OTV = Orbital Transfer Vehicle mosphere-bearing celestial body in the vicinity, it may = adjoint variable associated with statej W Pj be advantageous to utilize aerodynamic force in effecting the r radia- l distance from Earth's center transfer n thiI . s paper e presenw , n investigatioa t f o n =O radiuHE f o s = radius of LEO aeroassisted coplanar transfer from a of radius r to a concentric circular obit of radius r where r is greater R = radiu f sphericao s l atmosphere } 2> 1 than r (Fig. 1). We will consider the to be about the S = effective vehicle surface area norma velocito t l y 2 vector _ Earth; however much of the analysis is more generally ap- v plicable r assumptionOu . s followsa e vehicle ar sa Th .s eha V inertial speed lifting configuration modulatee b n life ca tth varyiny d db an ; g the angle of attack. Lift modulation is the sole means of «/ rf/R inertial flight path angle controllin e flighth g e tatmosphere th pat n i h , propulsion 7 being used only outside the atmosphere. The vehicle has a 6 high-thrust propulsion system so that applications of the At;,- thrust can be considered to produce impulsive velocity = impulsive change in V AK changes (AFs) and the fuel consumption for an orbital transfe s thui r s indicate e characteristith y b d c velocitye th , Presente s Papea d r 83-209e AIAth t Aa 4 Atmospheric Flight AFe th su sf m o neede effeco dt transfere th t e transfeTh . r Mechanics Conference, Gatlinburg, Tenn., Aug. 15-17, 1983; received must involve only a single atmospheric pass. And finally, the Sept , 198314 . ; revision received Feb , 198424 . . Copyrigh© t atmospheric properties vehicle'e th , s aerodynamic properties, American Institut f Aeronauticeo Astronauticsd an s , Inc., 1983l .Al rights reserved. the equations of motion, and the initial position and velocity * Member Technical Staff, Navigation Systems Section. Member of the vehicle are all known precisely. AIAA. Under these assumptions, we determine the minimum-fuel fProfessor, Aerospace Engineering. aeroassisted transfer, the fuel requirements of which are then JAN.-FEB. 1985 MINIMUM-FUEL AEROASSISTED COPLANAR ORBIT TRANSFER 135

compared to those of the minimum-fuel, all-propulsive generate enough drag to slow the vehicle in a reasonable transfer e characteristicTh . minimum-fuee th f o s l trajectory amount of time, and 2) that the vehicle has sufficient lift to during the atmospheric portion of the aeroassisted transfer maintain flight along the atmospheric boundary. e examinear detailn di additionn I . e effeca vehicl th ,f o t e characteristie Th c velocit f thiyo s idealized transfew no s i r heating constrain e atmospherith n o t c trajector s deteri y - compared with that of any realistic aeroassisted transfer. A mined. realistic transfer would require a larger AK7 to ensure suf- e motivatioTh r thinfo s study stems fro e currenmth - in t ficient penetration into the atmosphere such that the required terest in an orbital transfer, vehicle (OTV).1-2 This vehicle velocity is depleted before skipping back out, given the limited would transfe spacecrafa r t fro mspaca e shuttl highea o et r lifting capabilit vehiclee th f yo .idealize e Thu th AF e r sth fo 7 d and/o differena r t inclination orbit vehicle Th . e would then transfe lowea s i r r boun aeroassister dfo d transfers. Secondly, return, after delivering its cargo, to rendezvous with either a for the one-impulse transfer from atmospheric exit to LEO, shuttl a spac r o ee operation maneuverV s centerOT e Th .s exit wit hfligha t path angl f zereo o degrees (7/ = deg0 ) into which could benefit potentially from aeroassis orbitae th e lar t an elliptical transfer orbit t apogeea , tangenO LE , leado t o st plane change and the transfer from high Earth orbit (HEO) to the minimum circularizing AF2. The corresponding exit speed low Earth orbit (LEO) presene Th . t study concerns onle yth is Vf = ^2fjLr2/ [R(r2 +R) ] . Any other exit pair ( Vf, yf) will latter. lead to a higher A V2 . Consequently , the characteristic basie Th c sequenc aeroassistee f eventeo th r o t fo s O dHE velocity, AF AF7+ thir 2fo ,s idealized aeroassisted transfes i r coplanaO LE r orbit transfe followss a s i r . Referrin Figo gt . 1, a lower bound for the characteristic velocity of any realistic the transfer begins with an in-plane tangential retroburn aeroassited transfer. (AF7) at HEO which injects the vehicle into an elliptical An analytic expression for this lower bound can be derived. transfer orbit wit ha hypothetica l target perigee inside th e Let atmosphere. At point E, the vehicle enters the atmosphere. As vehicle th e flies throug atmospheree hth , s kinetisomit f eo c and energy is converted to heat, and, consequently upon skipping atmosphere th f o t t poinou e(a t apogeF) e orbie th , th s f i t eo ellipticale Th grazing trajectory require impulsn sa e decreased to the distance r2. Finally, at the new apogee, a ?7- V2/ [a1 (aj + 7) ] (1) second in-plane tangential burn (AF2) is executed to cir- culariz therebd ean y achiev desiree eth d LEO minimume Th . - fuel aeroassisted transfe s thai r e tminimu th whic s ha h m secone Th d impulse use circularizo dt orbie e th t rs 2i ta characteristic velocity, AF +AF . The flight path for the 7 2 (2) minimum-fuel transfe s effectei e ~AKr th 7 y b magnituded , which control atmospherie sth clife entrytth coefficiend an , t as a function of time during the atmospheric flight, which Thu e totath s l characteristic e idealizevelocitth r fo yd aeroassisted transfer is control exihence d sth tan e determine requiree sth d AK2. Analytic Solution for an Idealized Optimal Transfer (3) Consider an aeroassisted HEO to LEO transfer which Compare this to the characteristic velocity for the minimum- proceeds as follows. Referring to Fig. 1, a tangential fuel, all propulsive transfer, the Hohmann transfer, which is retrobur injectO n vehicle HE AFth st a 7 e int elliptican oa l transfer orbit with perigee at the distance R. When the vehicle t perigeeisa liftins it , g capabilit thin y(i s case, negative lifts i ) employe o effect d t flight alon- e boundarat th ge th f o y mosphere (i.e., along a circular orbit of radius R). Flight a2(K1+a2)] - (4) along the boundary is continued until sufficient velocity has

curve Th e plotte Fign di represent.2 s pairs (otwhicr ,afo ) h 2 been depleted (by atmospheric drag) such that upon reducing 1 the lift to zero, the vehicle ascends on an elliptical orbit to an AvA =Av r pairHFo . s belo curvee wth , AvA

mode is Av=^2/a — v^/a,. With respect to the charac- ]

teristic velocitp y fpr the one-impulse mode, Eq. (1), the / ATMOSPHERE parabolic mod s optimai e l whenever a/>2(v 2 + 1)== 4.828. Hence, optimall t shoulyi case use e f th returdb eo n di n from GEO. However, in this case, the saving in the characteristic Fig 1 . Aeroassisted coplanar orbit transfer. velocity is only 21 m/s. Since the saving is small, while the 136 K.D. MEAS N.XD E.AN VINH J. GUIDANCE time required to complete the deorbit is greatly increased, the malized lift control parabolic mod efurthee shalb t lno r considered. The two-impulse mode of deorbit can be optimal for a (8) nonzero entry angle; but it can be shown that when it is, the savin n characteristii g c velocit s comparea y e oneth -o t d where C*L is the lift coefficient corresponding to the maximum l impulse mode is of the order of (ye) . With regard to the one- lift-to-drag ratio E*. In terns of CD and K, we have impuls insertioO LE e n following atmospherice th exit f i , apoge post-exie th f eo t orbi exactls i tdistance th t ya e r2, then insertion is optimally made by a tangential impulse at this E* = (9) pointapogee distanca th t f a I . s ei e slightly less tha greater no r than r2, the a ntwo-impuls e insertio s requiredi ne th t bu ; Usin followine gth g dimensionless variable parameterd san s incremen characteristin i t c velocit e orde s th onlyi f f o ryo (jf).2 Therefore s reasonabli t i , o considet e r only one- impulse deorbi d one-impulsan t insertioO e th LE e r fo n aeroassited coplanar transfer. proceedw Weno consideo t lesa r s idealized aeroassisted P0SHeCl = p/p ; = R/HB= ; orbit transfer in which the vehicle flies a skip trajectory 0 e 2m through the atmosphere. Determining the minimum-fuel trajectory and control in this case requires the formulation the equations of motion can be rewritten as and numerical solution of an optimization problem. dh Equations of Motion (lOa) ~dr The equations of motion for planar atmospheric flight are4 dv Bd • smy (lOb) (5a)

dy COST dV pSC V2 — =Bd\v+-——r—— (lOc) D -gsm-y (5b) dr (b-l + h) I (b-l h)v\+ 2m Besides being preferable for numerical computation, the - (g- — JCOS7 (5c) dimensionless equations of motion, Eq. (10), focus attention At 2m e criticath n o l aerodynamic parameters which affect flight, namely life tth , loading maximucoefficiene th d an mB t lift- assumin gnonrotatina g atmosphere assumes i t I . d that when to-drag ratio E*. modulatee Agaith s i nX d lift control, scaled r>R the flight is Keplerian. Hence, we shall consider a such tha correspondt1 X= maximue fligho st th t a t m lift-to- Newtonian gravitational attraction, that is drag ratio. (6) The Optimization Problem e optimizatioTh n proble o fine t e value th th s dmi f o s Furthermore, it is assumed that the drag polar is parabolic, tangential At>7, a scalar parameter, and the lift control X, a namely function of time, which minimize the total characteristic 2 velocity D = CDn+KCL (7)

AvjAv ^Jl/oij-+ 2= vecosye \ll/

With this relation, usin a gcontro s Ca L l corresponds physically to using pitch modulation to shape the trajectory. Equivalently, we can maximize the function CL is allowed to assume both positive and negative values. A /= vecosye/a v+ f}cosy f/a2 (12) negative CL interprete e valub n eca resultins da g either froma negative pitch angl r froeo mpositiva e pitch angle wite hth e atmospheriTh c entr d exian yt variables must satisfe th y vehicle flying upside down. It is convenient to use a nor : relations

2 2 (2-v )oi 1-2aJ (13) 2.0 and

2 HOHMANN (2-v})ot -2a (14) TRANSFER 2 2

which are derived from the conservation of energy and angular momentum equations for the HEO-to-entry and exit- AEROASSISTED to-LEO transfer orbits, respectively. At entry we have TRANSFER r=0d 1 / L= an (15) GEO t exia td an

fred hean f —1 (16) al = rl/R Fig. 2 Comparison of characteristic velocities for Hohmann and proceedw no e o derivt W e necessary conditione th r fo s idealized aeroassisted transfers. optimal solution. Introducing adjoint variables ph, pv, and JAN.-FEB. 1985 MINIMUM-FUEL AEROASSISTED COPLANAR ORBIT TRANSFER 137

py, we form the Hamiltonian and 2bsiny 2b dr ~ (b-l + h) E*(b-l + h)2 (b-l + h)v.

Bv2 FB (17) _ 'E* ~E*

With respec life tth controo t J, maximizeCs i lX d when AFcOS'y (23) E* (b-lh) + (b-l + h)v v=££ (18) vpv where However, realistically, the range of values which X can assum boundeds ei ; namely life th t, control must satisfe th y dh dH inequality constraint Writing the modified Hamiltonian in terms of A and F, we IXI

where X positiva ma s i x e constant whose valu dictates ei e th y db aerodynamic characteristic vehiclee th f o s . Accordine th o gt Maximum Principle fine w ,d thaoptimae th t l lift contros i l determine rule th e y db ABdv\ + ————+_ (24) E* E*(b-lh) + I (b-l h)v\+ when X>Xmax

when IX I

nr> f\_ /1O where A=E*py/ (vp determininvn )I . g this rule have w , e used at = 0 (25) e facth t thae Hamiltoniath t a quadrati s ni c functioX f o n whose second derivative with respecnegatives i X o t . Now, rather than the original six differential equations, we The adjoint variables satisfy the necessary conditions have five, namely, Eqs. (10) for the three states, and Eqs. (22) . IntegatioF d an A f thes(23 d nr o an fo ) e equations will yield extremal trajectorie numbea r fo s f problemo r s which differ (21) dr dh ' dr dv ' dr entre onlexid th yn an yti conditions which mus satisfiede b t . Besides having reduce dimensioe dth n o fivet fro x ,m si thi s However, note that to compute the optimal control only the formulation has the distinct advantage that four of the five value of A is required. This suggests replacing the three dependent variables are physical variables. (Actually, A has differential equations, Eqs. (21), wit whico htw h involve only physical meaning only if IAI

Table 1 Characteristics of minimum-fuel trajectories LID capability Heating rate Low Moderate High Unconstrained Constrained

(L/D)m 6.845 1.5 2.9 1.5 1.5 Target perigee rpm ,k 6400.0 6400.0 6400.0 6415.0 6415.0 A,(Ap) 2.303247 2.701724 3.2586836 2.947660826 6.56(.79867315) Exit flight path angleg de , 0.3 0.4 0.3 0.45 0.49 LEO orbit radius, km 6558.8 6578.7 6557.6 6608.0 6625.0 AK2,m/s 26.1 31.0 25.2 40.5 45.1 Ideal AK2 for same LEO, m/s 18.3 24.0 18.0 32.6 37.7 Min altitudem k , 58.8 58.2 51.5 61.2 64.0 Max dynamic pressure, kN/m2 15.9 18.6 44.2 13.1 8.8 Max convective heating rate 193.1 222.8 361.4 190.8 150.0 metee foon ra sphere, W/cm2 Maxg's 3.6 2.7 1.8 1.9 3.7 138 K.D. MEAS N.XD E.AN VINH J. GUIDANCE

heatine Th g rate, HR9 alon atmospherie gth c trajectorys i , n AFo 7e idealize giveth y b n d transfer. Thus e characth , - computed according to the equation teristic velocity cannot get much smaller. Equally important, however, is whether or not the trajectory and/or control 4 /2 3 08 would change significantly if the characteristic velocity were R = (3.08xlO- )p' V - (26) reduce meterlase w d th fe tr second spe . Numerical experience 3 indicates tha t e exithe th o nott s yd fligh A . t path angls i e where p is the atmospheric density in kg/km and V is the reduced e atmospherith , c trajectory changes very little- In . spee km/sn di . Equation (26) give convective sth e heating rate deed value th , f Abeins ei o e g changed only slightly (partn si fo sphera r e wit hradiua m1 ,f undeso r condition laminaf so r exie th t t 10 angllessr ge o 6 o )et belotenthw degreea fe w f a so . flow. Since only relative change f concerno e ar s , this model This leve f chango l n Ai ee affect se trajectormosth f o t y will suffice. almost negligibly t extendbu , e trajectorth s n ordei y o t r achieve the lower exit angle. Furthermore, as mentioned Method of Numerical Solution abov zerr efo o exit flight path angle atmospheric trajectories, e shalW l only concern ourselves with minimizing AF2. ther one-to-ona s ei e mapping, base numerican do l experience, O AlthougHE e burnso th tw r e hfo largee , th AFth f s i o r 7 from values of rp to values of vf, and hence, to the LEO's for considered the difference between the value of A V , required 1 whic AFe minimuma hth s i 2 . Therefore f ri p, were increased to target to a perigee at the atmospheric boundary, R — 6498 in orde decreaso t r e AF correspondine 7th , g AF2 woule db kmthad an ,t require targeo d t perige a surface o t tth t ef a eo greater. Given the low sensitivity of AF7 to changes in the Earte th mera s m/shi 3 e 1 contrast n I . , AF vers i y sensitive 2 value of rp9 it is unlikely that the characteristic velocity could valueexie e th th t f o parametert so r example yd sFo f. van f , be reduced much f anyi , adjustiny b , g rconclusionn pI . a , the AF2 required for a given HEO to LEO transfer can in- trajectory and control solution computed in the manner crease by 100 m/s or more for each degree above zero in the described abov a realisti s i e c approximatio o thant t which exit flight path angle, T/. give e absolutth s e minimum characteristic velocityd an , Knowing tha skia t p trajectory wit e lead=/ g hth 7 0 de o st henceforth we shall refer to such a solution as a minimum- minimum AF2 at the LEO to which the ascending orbit is fuel solution. tangent employee w , e followinth d g approac o computt h e When a heat rate constraint is imposed, the solution minimum-fuel trajectorie controlsd an s targeA . t perigee, rp9 procedur somewhas ei t different folloe W . approacn wa h used is chosen and from this the entry parameters ve and ye are iheatine n RefTh . 6 g. skia rate r pfo ,trajectory , reaches sit determined accordin equatione th o gt s maximum value shortly after entry, in a monotonic fashion. It then decreases during the remainder of the flight, although 2 V =2[l-l/(a]+ap)] (27) some oscillation may occur. In order to satisfy a heating rate constraint, H < (H ) , we shall assume that it is sufficient 2 2 2 2 R R max cos ye=[2a]-(2-v e)a ]]/v e (28) to control the first peak of the heating rate function, such that

the peak valu s equaei o .(Ht l )Furthermore shale w , l max

Then, usin computee gth d value 7^a d f v,d an so e h1an ,e= assume that, once thiR s peak value is reached, flight does not

pair (A,F) as initial conditions, Eqs. (10), (22), and (23) are e e continue on the heating constraint boundary. These integrated fro 1= , m h usin . ro (20t = determingo 0 Eq )t e eth assumptions allow us to solve the constrained problem in two lift control paie Th r. (Ae,F determines i e) choosiny db g Ae stages, each requirin iteration ga onl n parametere no yon . usind . (24correspondinan e gsolvo Eq th ) t r efo g valu Ff eo e , s specifiea h d dan , abovewit7 , integratioe hv Th . pers ni - forme a variabl y b d e order, linear, multistep predictor- corrector routin Adams-Moultoe th f eo n type,5 wit locae hth l absolute error controlled to less than 1.0 x 10 ~8 for each of 10.0 five th e dependent variables l caseal n sI . studied s beeha nt i , 9.5 possibl fino et valua d f Aeo e suc t exia y hb g t thade 0 t y= f 9.0 iterative search correspondine Th . g valu f vefo determinee sth apogee of the transfer orbit following exit, and hence, the o whict vehicle O hth LE s optimalli e y transferrede th s A . 8.0 value of rp is lowered from R, 7/ = 0 deg continues to be reachable, but the exit speed decreases, resulting in lower 1 LEO transfers. There is a certain critical value of rp below o whic e liftinhth g capabilit e vehiclth f yo s insufficieni e o t t effect a skip trajectory. However, for the vehicle studied in the following, the range of reachable LEO's extended all the way down to altitudes of less than 200 km. trajectorA d controan y l compute n thii d s mannee ar r optima followine th n li g sense. First necessare th , y conditions, [Eqs. (10), (22), and (23)], the. entry and exit conditions = [Eqs. (15) and (16)], and Eqs. (13) and (14) are satisfied. Second, the lift control satisfies the constraint [Eqs. (19)]. Third, the near-zero degree flight path angle at exit ensures that the circularizing AF2 to achieve the LEO to which the E post-exit orbi s tangen i te absolut th s i t e minimum, when compared to those for all other aeroassisted transfers from £ same th eo t LEO same O th e.HE (The exit flight path angles achieved, as indicated in Table 1, are a few tenths of a degree. The iteration on Ae was stopped at this point because the associated value of AF2 was within 8 m/s of the lower bound idealizee th sey b t d transfer.) Fourth, although only AFs ha 2 10 15 20 25 been minimized e characteristith , c velocity, AF +AFs i , 7 2 TIME FROM ENTRY (min) very close to the absolute minimum. The value of AF7 for the

cases show Tabln i withi s i lowee e1 th f nro 10.bouns 4m/ d Fig. 3 Time histories of state variables, (Z,//)) = 1.5. x ma JAN.-FEB. 1985 MINIMUM-FUEL AEROASSISTED COPLANAR ORBIT TRANSFER 139

unconstrainefirse e Inth th t n stagei s begia e =r w 0 , t na d case, except that now the goal is to choose Ae such that HR = (HR)max at the time when the derivative of the heating rate with respec timo t equas ei zeroo lt . Once this valu Af eo e is foun minimum-fuee dth l trajectorpeae th ko t heatin p yu g rate is determined. The second stage is to find a value for A, A = A,, such that, when Eqs. (10), (22) (23 d integratee an ),ar d from the time of the peak heating rate to atmospheric exit, the value of the exit flight path angle is zero degrees. Indeed, in the cases studied, it has been possible to find such values of Ae and Ap. Thus, while the functions A and F are, in general, discontinuou peae time th kth f eo t heatin sa g rate statee th , s alwaye ar 7 sd continuousan h,, v . Minimum-Fuel Trajectories e caseth l s al reporte r Fo d below e transfeth , s froi r m goestationary Earth orbit (GEO), for which rl =42,241 km. radiue atmosphere th Th f so 649 s eAbov. i 8km e this distance, the density is identically zero. Over the altitudes of at- mospheric flight, 40-12 (wherm 0k radiuEarte e eth th f s hi so taken to be 6378 km), the density is approximated by a fifth- degree Chebyshev polynomial whose coefficients were determined by a least-squares fit to the U.S. Standard At- mosphere, 197 vehicle 6 (RefTh . e7) . mass-to-surface ares ai 300 kg/m2 for all cases. Concernin radiuatmospheree e gth th f so remarkw fe a ,e sar 10 15 20 25 in order. Since the atmospheric density variation with altitude TIME FROM ENTRY (min) is exponential in nature, according to the U.S. Standard Fig. 4 Time histories of heating rate, dynamic pressure, and g load, Atmosphere e interfacth , e betwee e atmosphernth e th d an e (L/D)mm=l.5. vacuu t welno lms i defined commos i t I . n practic chooso et e the entry (and exit) radius such that the aerodynamic force produces a slight but detectable acceleration to modify the CDo and K in the altitude range of effective aerodynamic Keplerian orbit. Thus criticae th , l paramete determininn i r g maneuvering. Since aeroassisteth e d coplanar transfer the radius value is the lift loading coefficient B. Accordingly, requires only energy depletion, it is reasonable to expect that a the value of 6498 km for the radius is consistent with the more precise treatment of the vehicle aerodynamics would not value useB r numericaf dfo o s l computation thin i s s study. drastically alter the characteristic behavior of the lift control With e optimizatioregarth o t d n results e characteristith , c resultine th r o g performance. velocit aeroassistee th r yfo d transfer shows little sensitivito yt For each of the three maximum LID cases, we have fixed the value of R. For the GEO to 350 km altitude circular LEO havd an 640 re= p m searche0k found dan Ae edth such that transfer eithen i m ,r k changindirectio0 1 y b valu e R gnth f eo / =deg0 7 thin I . s manner a minimum-fue, l trajectorr fo y alters the characteristic velocity by less than 3 m/s. each case was generated. The corresponding LEO orbits, to which the transfers are optimal, are not exactly the same, but Unconstrained are close enough to permit comparisons. The alternative We begin by presenting some minimum-fuel trajectories approac f specifyin horbio O priora LE t e gth i would require under condition unboundef so d unconstraine d lifan t d heating seachin parameterso tw n go Ad ,orde n eri ,an p determino rt e rate. Three vehicle configurations were considereds a , the minimum-fuel trajectory. distinguishe theiy db r respective maximu capabilitiesD mLI , Certain characteristics of the minimum-fuel unconstrained namely, 0.845, 1.5, and 2.9. Data from wind tunnel tests is trajectorie firsgivee e th sar t n threi e column Tablf so ee 1W . available for vehicles with these maximum LID capabilities see that the high LID vehicle penetrates farthest into the , respectively10 e e d valueth th an r d , (Refsfo s9 an ), 8 . atmospher experienced ean highese sth t dynamic pressurd ean parameters CDo and K which appear in the parabolic drag heatinvehiclD LI w eg lo experience ratee Th . highese th s g t polar were chos°e datae valuee besth o n t t Th .fi t e s th use r dfo load r comparisonFo . shuttle th , e design limit r dynamifo s c 2 11 pair (CDo, K) were (0.21, 1.67), (0.10, 1.11), and (0.017, pressure and g-load are 16 kN/m and 2.5, respectively. 1.76), respectively. Time historie state th ef o svariable e (L/D)th 5 r 1. fo smax = Note that although constant values for CD and K are used case are shwon in Fig. 3; those for the heating rate, dynamic numericae th her r efo l computations precedine th , g theors yi pressure, and g load are shown in Fig. 4. The behaviors more generally applicable. For precise computation of an illustrate thesn di figureo e tw qualitativel e ar s y representative optimal trajectory, it is necessary to model the coefficients casee th l so al fdiscusse thin di s paper. CDo and K as continuous functions of the altitude h and the Figur showe5 lift-to-drae th s g ratia functio e s oa th f o n speed v since at low altitude and low hypersonic speed these time from atmospheri cthree entrth r e yfo cases similaA . r coefficients are functions of the Mach number and the pattern is followed in each case. The maximum positive LID Reynolds number Eqsn I . .s a (10) thee * E w , n d havan eB is used initially to recover from the downward plunge. As the Hamiltoniae Th . v functiond an nh integraf o s . (25)Eq l , with flight path angle becomes positive, the maximum negative . (24)3s giveCa Eq ,n ni remain e sth unchangedf o e us e Th . LID is used to level off the flight. These first two phases occur adjoin stils i ltF validratiod onle an Th s.yA chang thas ei n i t within the first four minutes of flight. Of course, although the Eqs. (22(23)d woule )an w , d have additional terms containing basic patter s similari n , quantitatively ther e definitar e e the nonvanishing partial derivatives Bh and E*h , with respect difference e flighth n ti s characteristic e threD th LI ef o s to h, and Bv and E*, with respect to v. The method of vehicles indicates a , Tabln di e 1. Afte firse th r t four minutes, numerical solution remains applicable and involves no ad- negativea useis maintaidto LID n fligh smala at t l positive ditional difficulty. Example f modelino s e aerodynamith g c flight path angle in order to achieve the desired shallow exit. coefficients as functions can be found in Ref. 4. For our The required negative L/D increases as the flight proceeds to purposes here, it suffices to use the best fit constant values for compensate for the decreasing atmospheric density. 140 K.D. MEAS N.XD E.AN VINH J. GUIDANCE

Bounded Lift Constrained Heating Rate For the vehicle, with (L/D)max = 1.5, wind tunnel data Usin . (26) heatine gEq th , g rate alon minimum-fuee gth l show that the lift coefficient does not exceed 0.9 in absolute trajectory can be calculated. Referring to this as the un- value. Thus impose w , constraine eth t constraine questionde heatinth k as gn : Wharatee ca e th w s ,i t minimum-fuel trajectory if the maximum heating rate is \CL\<0.9 constraineo greaten e b r o t thad n some fractioe th f o n maximum unconstrained heating rate? which corresponds to setting Xmax = 3.0 in Eq. (20). The effece heatina I th nf o e torde se g o ratt r e constrainte w , resulting lift contro dashe e gives li th y nb d curvr Fign Fo ei . 6 . again consider the configuration with (L/D)mSLX = 1.5, as comparison, the corresponding curve with CL unbounded is described earlier, except that the target perigee is taken to be given by the solid curve. We see that flight is along the 6415 km. The minimum AF2 trajectory is computed first with constraint boundary for much of the flight. The important the heating rate unconstrained. The maximum heating rate is point, however, is that a near-zero exit flight path angle is still foun190.e b o 8dt W/cm referenca r fo 2sphere m 1 e . Next, reachable by proper choice of Ae. minimum-fuee th l trajector computes yi d wit l conditionhal s identical, except that the heating rate is constrained not to exceed 150.0 W/cm botn 2I . h caseslife th t coefficiena , s i t 3.0 bounded describes a , d earlier. 2.5 Certain characteristic e unconstraineth f o s d conan d- strained case e give ar r scompariso fo n o e lasth tw t n i n 2.0 column Tablf so e bot1n I . h cases near-zera , o exit flight path 1.5 angl reachabls ei AFe withith s i 2 d e e an m/n 8 th f tha sr o fo t idealized transfer. With the heating rate constrained, the 1.0 vehicle does not penetrate the atmosphere as deeply; the 0.5 maximum dynamic pressure is reduced; but the maximum g 0 loa increaseds di . optimae Th l lift control r eacfo , h case plottes i , tims dv n ei -0.5 tha e vehicle se th t e FigW e . flie7 . s initiall t (Cya e Lth ) maxn i -1.0 constrained case; wherea e unconstraineth n i s d case, Cs i L decreasing steadily during the same period. By flying at -1.5 (CL)max initially correspondingld an , highea t ya r Ce Dth , -2.0 vehicle slows down e atmospherehigheth n i r , allowing -2.5 recovery from the downward plunge, which occurs sub- sequently at the maximum positive LAD, to take place at a 0 2 5 1 10 25 30 lower atmospheric density r equivalentlyo , a highe t a , r TIME FROM ENTRY (min) altitude thin I . s manner, higher heating rate avoidede sar . Fig 5 . Time historie f lift-to-draso g ratio. As a final note, the minimum entry flight path angle from which the vehicle can recover and achieve the prescribed exit state conditions s raisei , d whe na heatin g rate constrains i t 1.0 imposed (tha , raiseis t d with respece unconstraineth o t t d case). The reason is that if the entry is too steep, excessive 0.0 heating rates cannot be avoided, even by flying at the 1 0 maximum positive CL. In the particular case investigated •g - - here optiman a , l solutio founs nentrr wa dfo s ya anglew lo s sa o -6.5 °e unconstraine th ( ^n i = ) 640km 0 de th case n I . t -2.0 constrained case, the lowest entry angle that could be tolerate -6.0s dwa ° (^=6415 km). -3.0 5 2 0 2 5 1 0 1 30 35 Summary and Conclusions TIME FROM ENTRY (min) Unde e giveth r n assumption d restrictionsan s , minimum- Fig 6 .Tim e historie f optimao s l lift contro r unboundefo l d an d fuel aeroassisted coplanar transfer from highw orbilo o t t bounded cases. orbit using lift modulatio bees nha n considered idealizen A . d version of the transfer lent itself to analytic treatment and allowe loweda r characteristiboune th n do c velocity an r yfo aeroassisteO LE o t giveO d ntransfeHE determinede b o t r n I . orde examino t r e minimum-fuel transfer under more realistic conditions n optimizatioa , n proble s formulatemwa d an d solved numerically. It was found that for each given HEO to transfeO LE r considered, even with bounded lift and/oa r heating rate constraint a characteristi, c velocity withi- 10 n 20 m/s of the lower bound is achievable. The optimal aeroassisted mode of transfer requires less fuel than the optimal all-propulsive modwida r efo rang f higeo h orbio t low orbit transfers transfee th circula a r o t Fo r. froO r mGE 350 km altitude orbit, the characteristic velocity is 2.32 km/s less for the aeroassisted mode. The characteristic lift program for the atmospheric portion of the minimum-fuel transfer is to fly at the maximum positive LID initially to recover from the downward plunge, 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 flighe t negativ a leve th o t y f thed tfl of lsucD an o nt e LI h that TIME FROM ENTRY (min) e vehicl th e atmospherth ef o skip t ou se wit a hfligh t path Fig. 7 Time histories of optimal lift control for unconstrained and angle near zero degrees. This program is modified at the constrained heating rate cases (only firsminuteo tw t s shown). beginning if high heating rates are to be avoided. Flight, JAN.-FEB. 1985 MINIMUM-FUEL AEROASSISTED COPLANAR ORBIT TRANSFER 141

initiall t maximuya m lift, and, correspondingl t higya h drag, 4Vinh, N.X., Optimal Trajectories in Atmospheric Flight, Elsevier lowers the vehicle's speed higher in the atmosphere, allowing Scientific Publishing Company Yorkw Ne , , 1981. recovery froe downwarth m d plunge (which occurs sub- 5Krogh, F.T., "VODQ/SVDQ/DVDQ—Variable Orde- In r sequently usin maximue gth m positive L/D) tako t e placa t ea tegrators for the Numerical Solution of Ordinary Differential lower atmospheric density r equivalentlyo , a highe t a , r Equations, L TechnicaJP " l Utilization Documen . CP-2308No t , altitude. NPO-11643 1969y Ma , . 6Chern, J.S., Yang, C.S., Vinh, N.X. Hwangd an , , G.R., "Op- Acknowledgment timal Three-Dimensional Reentry Trajectories Subjec Deceleratioo t n The research described in this paper was carried out by the Heatind an g Constraints," Pape . 82-309No r , 33rd Congresse th f o International AstronauticalFederation, Paris, France, Sept. 1982. t Je Propulsion Laboratory, California Institutf o e 7 Technology, under contract with the National Aeronautics "U.S. Standard Atmosphere, 1976," NOAA-S/T 76-1562, U.S. and Space Administration authore Th . s thank E.A. Rinderle Government Printing Office, Washington, D.C. for developing the software used for the numerical com- 8Penland, J.A., "A Study of the Stability and Location of the putations. Center of Pressure on Sharp, Right Circular Cones at Hypersonic Speeds," NAS D-2283N AT , 1964. 9Miller, C.G. and Gnoffo, P.A., "An Experimental Investigation References of Hypersonic Flow Over Biconic t Incidenca s Comparisod ean o nt ^ruz, M.I., French, J.R. Austind an , , R.E., "System Design Prediction," AIAA Paper 82-1382, Aug. 1982. Concept d Requirementan s r Aeroassistefo s d Orbital Transfer 10Books, C.W., Jr. and Cone, D., Jr., "Hypersonic Aerodynamic Vehicles," AIAA Paper 82-1379, Aug. 1982. Characteristics of Aircraft Configurations with Canard Controls," 2Walberg, G.D., "A Review of Aeroassisted Orbit Transfer," NAS D-3374N AT , 1966. AIAA Paper 82-1378, Aug. 1982. 3Buseman, A. and Vinh, N.X., "Optimum Constrained Disorbit HHarpold, J.C. and Graves, C.A., Jr., "Shuttle Entry Guidance," by Multiple Impulses," Journal f Optimizationo - TheoryAp d an Journal Astronauticale th f o Sciences, , Jul3 . y 1979No Vol . , pp 27 ., plications, , 19681 . 40-64. Volpp , No , ..2 239-268.

From the AIAA Progress in Astronautics and Aeronautics Series...

INTERIOR BALLISTICS OF GUNS—v. 66

Edited by Herman Krier, University of Illinois at Urbana-Champaign, and Martin Summerfield, YorkNew University

In planning this new volume of the Series, the volume editors were motivated by the realization that, although the science of interior ballistics has advanced markedly in the past three decades and especially in the decade since 1970, there exists no systematic textboo r monograpko h today importand thaan t w coverne e t developmentssth . This volume, composed entirely of chapters written specially to fill this gap by authors invited for their particular expert knowledge, was therefore planned in textbookpara s a t , with systematic coverag fieleditorse e sees th th d a f y eno b . Three new factors have entered ballistic theory during the past decade, each it so happened from a stream of science not directly related to interior ballistics. First and foremost was the detailed treatment of the combustion phase of the ballistic cycle, including the details of localized ignition and flame spreading, a method of analysis drawn largely from rocket propulsion theory e formulatioe seconth Th s . e dynamicadwa th f o n l fluid-flow equation two-phasn i s e flow form with appropriate relations for the interactions of the two phases. The third is what made it possible to incorporate the first two factors, namely, the use of advanced computers to solve the partial differential equations describing the nonsteady two- phase burning fluid-flow system. The book is not restricted to theoretical developments alone. Attention is given to many of today's practical questions, particularly as those questions are illuminated by the newly developed theoretical methods. It will be seen in several of the articles that many pathologie f interioso r ballistics, hitherto called practical problem relegated san empiricao dt l description treatmentd an yieldine ar , theoreticao gt l analysi meany newee sb th f sro method f interioso r ballistics. In this way booe th , k constitutes a combined treatment of theroy and practice. It is the belief of the editors that applied scientists in many fields will find materia f intereso l thin i t s volume.

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