134 J. GUIDANCE VOL1 . 8NO , Minimum-Fuel Aeroassisted Coplanar Orbit 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 Earth (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 = low Earth orbit 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 circular orbit 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 orbits 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 <Ay//, i.e.e th , apogee at r . Finally, at r , a tangential circularizing burn, idealized aeroassisted transfer requires less fuelr exampleFo . , 2 2 idealized aeroassisted transfer from geostationary orbit to AF execute2s i , achievo dt desiree eth d LEO idealizatione Th . s here are 1) that the atmospheric density at R is sufficient to LEO requires less fuel than the Hohmann transfer, if the LEO radius, rs les2i , s than abou case th t e n 12,00I wher . 0ekm r2 =6728 km (a typical shuttle orbit radius), the characteristic velocitye transfeth r fo ,r iO fros 1.5mGE 5e th km/ r fo s aeroassisted mode, as compared to 3.87 km/s for the all- propulsive mode.