Automatic Control in Aerospace 1994 (Aerospace Control '94). A Postprint Volume from the IFAC Symposium, Palo Alto, California, USA, 12–16 September 1994

432 Pages·1995·42.57 MB·English

by  D. Schaechter and K. R. Lorell (Auth.)

A U T O M A T IC C O N T R OL IN
A E R O S P A CE
(Aerospace Control '94)
A Postprint volume from the IF AC Symposium,
Palo Alto, California, USA, 12 -16 September 1994
Edited by
D.B. SCHAECHTER and K.R. LORELL
Lockheed Palo Alto Research Laboratory,
Palo Alto, California, USA
Published for the
INTERNATIONAL FEDERATION OF AUTOMATIC CONTROL
by
PERGAMON
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IFAC SYMPOSIUM ON AUTOMATIC CONTROL IN
AEROSPACE 1994
Sponsored by
International Federation of Automatic Control (IFAC) Technical Committee on
- Aerospace
Co-sponsored by
LEEE, Control Systems Society
Organized by
International Federation of Automatic Control (IFAC) Technical Committee on -
- Aerospace, on behalf of the American Automatic Control Council
Supported by
Lockheed Missiles & Space Company, Inc.
International Programme Committee (IPC)
D.B. DeBra (USA) (Chairman) M.J. Pelegrin (F)
G.D. Ianculescu (USA) (Co-Chairman) J.J.M. Prins (NL)
D.B. Schaechter (USA) (Co-Chairman) K. Reinel (D)
M. Arnoud (F) S. Rock (USA)
L.V. Basanez (E) G. Rodriquez (USA)
G. Bertonio (I) S.M. Seltzer (USA)
U. Butter (D) R.E. Skelton (USA)
S.J. Chand (USA) D. Soo (NL)
J.R. aothier(USA) J. Sunkel (USA)
E. Gottzein (D) T. Tanabe (J)
R. Graham (USA) H. Tolle (D)
S.C Gupta (IND) B.K. Walker (USA)
CJ. Harris (UK) K.H. Well (D)
J.W. Hursh (USA) B. Wie (USA)
R.J. Kline-Schoder (USA) P. Willems (B)
CF. Lin (USA) W. Wimmer (D)
K.R. Lorell (USA) J.C Yang (PRC)
S. Manabe (J) M.B. Zaremba (CDN)
S.C. Mohleji (USA) G.R. Zoyu (PRC)
J.L. Nevins (USA)
National Organizing Committee (NOC)
K.R. Lorell (Chairman)
G.D. Ianculescu (Co-Chairman)
D. Schaechter (Co-Chairman)
M.J. Wright
J. Gallup
C Efendaki
P.A. Berardo
J. Witt
Copyright © IFAC Automatic Control in Aerospace,
Palo Alto, California, USA, 1994
OPTIMAL MULTI-CRITERIA AEROASSISTED ORBITAL
TRANSFER TRAJECTORIES
C. JAENSCH and K.H. WELL
University of Stuttgart, Institute of Flight Mechanics and Flight Control, Stuttgart, Germany
Abstract Single and multiple atmospheric pass aeroassisted orbital transfers for three representative
AOTV configurations on a geostationary to space station orbit transfer mission are investigated using
multi-criteria trajectory optimization. The fuel optimal results indicate that independent of the vehicle
type, the maximum heat flux is reduced by 20% and 35%, respectively, using two and three atmospheric
passes in comparison to a single-pass. The multi-criteria results show that the ability to trade-off fuel
consumption for a lower peak heating rate improves for all transfer modes (single- and multiple-pass)
with increasing lift-to-drag ratio.
Key Words. Aerospace trajectories; Space vehicles; Optimal control; Multiobjective optimization;
Nonlinear programming
1. INTRODUCTION First of all, the thermal stress reduction potential and
the associated trade-offs in rocket fuel consumption
With the near realization of the space station Free- and mission duration of multiple versus single atmo-
dom, their may be a renewed international interest in spheric passes is examined for three representative
the commercial and scientific use of space. In order AOTV configurations on a geostationary to space sta-
to exploit the full potential of the space station, effi- tion orbit transfer mission. The benefits of multiple
cient and economically affordable transport systems atmospheric passes have been examined previously.
will have to be made available. Besides launchers and However, because of their complexity multiple-pass
reentry vehicles, enhanced orbital transfer stages are AOTV trajectories have until now only been exam-
needed to accommodate the major transfer route ined in detailed simulation studies (Rehder 1984) or
between the space station and geostationary or sun- optimized with many simplifying model assumptions
synchronous orbits. These transfer vehicles will have (Vinh et al. 1990). In this paper, the optimization of
to be reusable and require considerably less fuel than multiple-pass maneuvers will be done with fairly
conventional orbital transfer stages. The aeroassisted realistic models.
orbital transfer vehicle (AOTV) concept seems to be
a promising candidate for fulfilling these require- Secondly, multi-objective trajectory optimization
ments. Numerous studies in the U.S. (Walberg 1985; problems are formulated for the three AOTV configu-
Mease 1988) have shown that considerable payload rations on the same orbital transfer mission. In addi-
gains can be expected. tion to the previously considered fuel consumption,
the maximum stagnation point heat flux and inte-
With the recent advancements in trajectory analysis grated heat load are considered as additional optimi-
tools, such as the software package GESOP1 devel- zation criteria. This approach accounts for the fact
oped in part by the first author while at the German that a vehicle design is generally a trade-off between
Aerospace Research Establishment (DLR), investiga- several possibly conflicting criteria. Disregarding
tions were initiated as part of the German hypersonic one, such as the thermal load, at the trajectory opti-
research program to advance on the state of knowl- mization step may lead to fuel optimal maneuvers
edge of these studies. Various aspects were investi- where the added weight of the thermal protection
gated that had not previously been examined in system surpasses the overall fuel savings. Vector-val-
detail. Two aspects are described in this paper. ued cost functions have until now been applied very
rarely to trajectory optimization problems. They are
more commonly used in mathematical economics
1. GESOP is an acronym for Graphical Environment for (Hwang et al. 1979). However, engineering applica-
Simulation and Optimization
1
tions of multi-objective optimization seem to be The atmospheric model is based on the 1976 U.S.
increasing (Stadler 1984; Rao et al. 1990). In the Standard Atmosphere whose properties are function
context of aeroassisted orbital transfer, no paper in of altitude only.
the open literature dealing with multi-criteria trajec-
tory optimization was found. Equations of Motion. For computational efficiency,
the inner- and outer-atmospheric flight segments are
handled differently. All space flight segments are
2. MATHEMATICAL MODEL treated as Keplerian arcs, while the equations of
motion (EQM) for the atmospheric portion are inte-
In this section the mathematical models for the vehi- grated numerically. In the atmosphere the vehicle is
cle, Earth, atmosphere, equations of motion and aero- treated as a point-mass model, i.e. the rotational
thermodynamics are briefly discussed. degrees of freedom are assumed to be instanta-
neously controllable. The resulting EQM over a
Vehicle. Three AOTV configurations that have sub- rotating Earth with the atmosphere at rest are:
ject of previous NASA studies (Wilhite et al. 1984;
Powell et al. 1985; Talay et al. 1985) are considered. dr .
— = Vsuiny (3)
These vehicles have been chosen because they cover dt
the aerodynamic performance range in a representa-
tive way. One is an aerobrake configuration with a dk Vcosycosx </φ Vcosysinx
(4)
maximum permissible lift-to-drag (L/D) ratio of 0.2. dt rcos<|> Jt = r
The other two AOTVs are typical medium and high
L/D configurations with a maximum achievable L/D
dV
of 0.8 and 1.8, respectively. As starting mass in geo- + ^cosysinx + ^ siny-i-
r
dt m
stationary orbit (GEO) 12000 kg is assumed for all
configurations. This corresponds approximately to a rcu^cos<t> (sinycosc)) - cosysinxsin<))) (5)
starting mass of 30000 kg for a LEO-GEO round trip
mission and is thus comparable to the data provided dy /^ακμ £ . . g
φ r
in Talay et al. 1985. The aerodynamic performance - = _ _ - 7s i n Ys . n X +- c o sY +
data (C and C) are taken from Wilhite et al. 1984
L D
rco,
because in addition to the angle of attack dependency
the reduction of the lift-to-drag ratio in the viscous —— α^φ (cosycos<{> + sinysinxsin<|>) +
interaction and transitional flow regime (high alti-
V (6)
tudes) is accounted for in that paper. This reduction is
— cosy + 2co^cosxcos<t)
particularly important for the multiple-pass missions,
since they generally operate at higher altitudes than
^ ί ημ ror
the corresponding single-pass missions. cos φ cos χ sin φ
dt m Vcosy Vcosy
Earth and Atmosphere. The Earth is modeled as an cos χ
cosγcosχtanφ +
axis-symmetric oblate spheroid that is flattened at its Vcosy r 1 Λ ψ
poles. The mass is assumed to be distributed in 2ω (sinxo^tany- είηφ) (7)
homogenous concentric layers, so that the gravita-
tional acceleration in increasing radial and latitudinal
where is the Earth's angular velocity, λ is the lon-
directions can be expressed as,
gitude, φ the latitude and r the radial distance from
the center of Earth to the center of mass of the vehi-
cle. V is the Earth relative velocity, γ the flight path
8 r 2 1- £ (n+l)/n/>„(sin<|» A (1) angle, χ the heading angle measured counter clock-
π = 2 wise from the east, μ the velocity bank angle and m
is the mass of the vehicle. Finally, the forces F and
A
F are for the considered class of vehicles identical
£ = -~cos<t> £ JP\{sm^) (-) (2) N
Φ n to the aerodynamic forces -D and L, respectively.
Since the true anomaly difference Δν between the
where denotes the Earth's gravitational constant,
starting and end position of a de-orbit maneuver or a
r the equatorial radius, J gravitational coefficients
e n
ballistic coast arc connecting two consecutive atmo-
and /^(sin*))) Legendre polynomials in sin<|>. For
spheric passes can generally be determined a-priori,
the purposes of this paper, only terms up to J are
2
it is not necessary to solver Kepler's equation.
considered because they generate gravitational accel-
erations that are of the same order of magnitude as Instead, the following two equations are applied to
the maximum transport acceleration in the atmo- compute in an Earth-centered inertial reference
spheric equations of motion. frame the position r and velocity vector V on a
Keplerian orbit as a function of the initial position r
0
2
and velocity vector VO and a prescribed true anomaly r& cos φ si η χ
e
difference Δν. χ = χ- asin (16)
Vcosy
>(>ο, ν'ο.Δν) = [l--(l-cos(Av))]^0 + Aerothermodynamics. The mass of the thermal pro-
tection system (TPS) and thus the achievable payload
sin (Δν) V (8) capacity of an AOTV is significantly influenced by
0
the aerothermal loads experienced during the atmo-
spheric flight. For trajectory analysis purposes these
thermal loads can only be computed at critical points
ν(>ο,νΌ,Δν) 1 - - (1 - cos(Av)) \ V + of the vehicle with approximate correlation formulae.
0
For this paper, the convective and air radiative heat
1 F< X
~>o-V flux at the stagnation point are considered to assess
0 (1 - cos (Δν)) · /A (9)
"7V17s , n ( A) v r° these loads quantitatively. Amongst the different con-
L orP Γο v Ρ
vective stagnation point heat flux formulae used
where ρ = a (1 - e ) with a being the semi-major throughout the literature, the following is chosen
axis and e the eccentricity of the orbit. Note that for because it tends to produce estimates that are mid-
range conservative (Jaensch 1994).
both, the de-orbit and ballistic coast arcs, the value r
in Eq (8) is known to be the atmospheric entry radius.
if1
q = 1.83x10" 0.5f/3 (17)
De-orbit is accomplished by applying an impulse c
_ho_ where ρ is the atmospheric density, R the effective
(10) n
nose radius and g the ratio of the specific enthalpy
Aol w
at the wall to the local total specific enthalpy. The lat-
where ho = r χ V denotes the angular momentum ter usually takes values in the range [0.0.1]. All
0 0
vector of the starting orbit. For the purposes of this quantities are assumed to be in Si-units, [Rn] = m
paper, it is necessary to regard both Δν, and AV2 as [V] = m A f [p] =kg/ml , [qc] = W/m2.
optimization parameters. In order to guarantee atmo-
The air radiative heat flux formula used is based on a
spheric entry at a specified radius r for all possible
x
relatively new correlation of a radiation model that
combinations of the two impulse components it is
covers all essential equilibrium flow characteristics
best to split Δ V into the following two components:
x
(Tauber et al. 1991). It is given by the expression:
AVX = AV^ + AVj (11)
4rod = 4.736x10V 22/C/(V0 (18)
where the first component is a given function of Δ V
2 with a = 1.072x10V1 8ρ8~^°325 subject to the con-
that guarantees tangential entry at r as long as
x
ditions a < 0.6 if 1 < R < 2; a < 0.5 if 2 < R < 3.
|Δν | < V and the second component is free to be n n
2 T
fiV) is defined by tabulated values that are functions
optimized and controls essentially the value of the
of the velocity and the atmospheric composition. If
entry flight path angle. The expressions for AVn,n
x
Si-units are used for R, ρ and V, then q is given
and V are: n rad
T in W/cm2.
AVmin = v0-Jv*.-AVl (12)
3. MULTI-CRITERIA TRAJECTORY
V= ^[2/r-2/(r + r)] (13) OPTIMIZATION
T e 0 0 x
Numerical methods to compute optimal trajectories
Finally, it is shown how the Earth-relative velocity in
with complex models, path constraints and a scalar
the atmospheric EQM is transformed into an Earth-
cost functional have considerably improved in the
centered inertial reference frame so that inner- and
last decade. There are basically two types of methods
outer-atmospheric flight segments can be connected.
in use today. The so-called indirect methods where
Let y» Ύ» X denote the inertial velocity in polar coor-
the optimal control problem (OCP) is reduced to a
dinates. It can be computed from the Earth relative
multi-point boundary value problem by applying the
velocity V, γ, χ using the following equations.
maximum principle. They are very accurate but gen-
erally require considerable pre-analysis to derive at
V2 = V2 + rcocos<l>(ra)^cos<|> + 2VcosYcosx) (14)
r the multi-point boundary value problem. On the other
hand, there are the so-called direct methods where
γ = asin (Vs\ny/V) (15) the control functions and sometimes also the states
are parameterized to transcribe the original OCP into
3
a possibly large but finite dimensional nonlinear pro- where the first k components are so-called Mayer and
gramming problem. Although less accurate than their last / components are Chebyshev functionals. To be
indirect counterparts, direct methods are by and large able to compute PO points for such a vector func-
more useful for engineering type problems. tional, the problem is transformed as follows. Define
ρ = [ρΓ,ξ] and minimize ξ instead of J subject to
v
Since direct methods are utilized in this paper, the the boundary and path constraints:
concept of Pareto-optimality is best discussed in the
framework of nonlinear programming (NLP) prob- F ^ x ^X u{t\ p)-ff }
f
lems. The objective of an NLP with a vector-valued (21)
cost is to determine a parameter vector ζ s Rn such
that cost vector
Fj(tx«) u(t) p ) - !^
t t t
B(z) = (*>,(z),...A(z))r E*0 <ξ, Vre [ί 0,ί7] (22)
is minimized w.r.t. ζ subject to the constraints:
for / = 1, 2 ,k and j = 1, 2 , /. F i¥ ) and /*>
g(z) = 0, h(z) £ 0, and z<z<z. (19) are so-called goal attainment values that specify
L v
what are considered to be good and bad values,
In general, it cannot be expected that for the above respectively, for each cost functional. There are other
multi-criteria NLP problem a vector minimal point is type of transformations for vector-valued cost func-
attained that simultaneously minimizes each compo- tions, see f.i. Rao et al. 1990, each producing in gen-
nent of B(z). Such "utopia" points are almost never eral a different PO point. However, since not a single
obtained. Thus a new optimality concept is necessary PO point but an approximation of the entire PO-set is
to define a solution for a multi-objective optimization to be computed for the AOTV problems, the other
problem. The concept of a Pareto-optimal solution transformations offer no additional advantages. The
(Leitmann 1981) has been recognized to be suitable PO-set is computed by systematically varying for
in this context. each cost functional the values for F*+) and F<').
Note that by using the scalar minima of each cost as
Definition. A feasible (w.r.t. the constraints) solution F<+,) the formulation in Eqs (21)-(22) is very well
z* is Pareto-optimal if and only if for any feasible z, suited for engineers because it allows them to think
z*z\ b{z) = fc,(z*) for all is {1,2, ...,*} or in terms of "How much am I willing to trade-in" for
there is at least one is {1,2,...,/:} such that the specifications of F ( ).
b{z)>bjiz).
i
In addition to the vector-valued cost 7 , the consid-
V
This means that a solution z* is Pareto-optimal (PO) ered OCPs have differential constraints given by the
if there exists no other ζ that would decrease an EQM, additional boundary and path constraints as
objective function without causing a simultaneous well as phase-transition conditions for multi-phase
increase in at least one other objective function. In OCP problems. For a detailed exposition on the gen-
general, the set of PO solutions is infinite. Thus, solu- eral form of these constraints and the direct numeri-
tions of multi-criteria optimization problems are not cal methods used to solve the OCPs described in this
unique; two conflicting criteria produce principally a paper, see f.i. Jaensch et al. 1994.
one-dimensional set of optimal solutions. But this is
exactly the advantage, since f.i. a vehicle designer
has a whole set of optimal trajectories to choose from 4. THE AEROASSISTED ORBITAL TRANSFER
when design trade-off decisions have to be made. PROBLEM
Returning to trajectory optimization, the type of vec- In this section, the aeroassisted orbital transfer prob-
tor-valued cost functionals considered in this paper lem from a geostationary orbit (GEO) to a candidate
are now described. Let x(t) denote the state vector, US space station orbit (LEO) with an inclination of
u{t) the control vector and ρ the vector of model 28.5° is formulated as an optimal control problem.
design parameters, then the most general type of vec- Note that the maneuver is optimized in it's entirety,
tor-valued cost functional considered is: since it is known from previous studies (Mease 1988)
that considering only the atmospheric segment may
f Fx(tpx{tf\u(tf\p) lead to wrong conclusions. First, the single and mul-
tiple atmospheric pass variants are described with
F(t x(t), u(t), p)
k f f f fuel consumption as cost. Then the multi-criteria cost
max f^U x{t),u(t),p) I (20) functionals are considered.
ίν,]
tS
4.1 Single-Pass OCP Formulation
^ mjix^FK/, x(/) «(/), p)J
t
Figure 2 shows an overview of a typical GEO-LEO
4
single-pass ^AOTV maneuver. At (1) a de-orbit | 5( f) | £ 5 B (34)
impulse AV is applied causing the vehicle to
d
descend to the atmospheric boundary (2) defined at where the indices 1 to 4 in Eqs (24)-(31) refer to the
h = \20km. In the atmosphere the vehicle performs points along the maneuver as shown in Fig. 1 and
an aerodynamic turn while dissipating all excess subscript Τ denotes the transfer orbit defined by the
energy. At (3) the vehicle leaves the atmosphere and coast arc from point 3 to 4. Finally, the GEO starting
coasts until (4) where the LEO circularization position r\ and velocity are defined as:
impulse AV is applied.
C
R ~ GEOR
r\ = λ = 0 γ=0 • (35)
_ φ = 0 _ χπΟ
4.2 Multiple-Pass OCP Formulation
Figure 2 shows an overview of a typical GEO-LEO
multiple-pass AOTV maneuver. In contrast to the sin-
gle-pass maneuver, their are two or more atmospheric
passes to perform the orbital transfer. Up to three
Fig. 1. Single-pass aeroassisted GEO-LEO transfer
atmospheric passes are considered in this paper,
for a high L/D vehicle.
because the simulation studies of Rehder 1984 sug-
The corresponding minimum-fuel optimal control gest that only small additional gains can be expected
for more than three atmospheric passes.
problem is formulated as follows:
jffin J = ||ΔΡ, ,|| = ||AVJ+||AV || (23)
0 C
subject to the initial boundary constraints,
HKYu*v(AV))-r = 0 (24)
l2 d 2
V(.re vi,Av (AVrf))-V = 0 (25)
u 12 2
/π,βχρ(-||Δ^||/(/ ))-ιη = 0 (26)
ν ί ο 2 Fig. 2. Multiple-pass aeroassisted GEO-LEO trans-
fer for a medium L/D vehicle.
AtCr\, PI Av(AV)) - 1 = 0 (27)
l2 u d 2
The OCP formulation of multiple-pass transfers
final boundary constraints, extends the specification given by Eqs (23)-(35) with
additional phase-transition equations for the ballistic
Λ(/) = 120km , γ(/ )>0 (28) flight segments using Eqs (8M9). Due to lack of
3 3
space these are not presented. A detailed exposition
/|_£Q — acos (COS<|(JCOSJ(J) =0 (29) is given in Jaensch 1994.
4.3 Vector-valued Cost Functions
(30)
In addition to the 5ca/jr-valued cost functional given
1 by Eq (23), different vector-valued formulations are
) = 0 (31)
considered. They replace the heat flux constraint Eq
\Ί LrEO V LEO
(33) with a comparable cost functional and consider
in addition the integrated heat load as cost functional.
and path constraints on the dynamic pressure, total
stagnation point heat flux and acceleration load,
(32)
min Jv = mρi , 9(0 (36)
/ e
(33)
5
Note that the third cost functional in Eq (36) also fits gle-pass case, shown in Fig. 3.
into the class of vector-valued cost functional con-
sidered in Eq (20), because it is easily converted into
FFuueell--ooppttiimmaall ssoolluuttiioonn
a Mayer functional by augmenting the state vector by
one state which holds the value resulting from the • He at flux-op lima] sol.
1 I
integration of q(t). 1 low LTD
1 1
t 1 med. L/D
\ 1
--].-• big η u υ
\ !
5. RESULTS AND DISCUSSION
f— |
The results obtained by numerical optimization of the ΙΛΟ ΙΛ2 lAi \JH IM 1.1· 1.12 U4 1.1* l.lt 1.2·
total impulse / foci-optimal total Impulse
single- and multiple-pass AOTV optimal control
problems formulated above are discussed in this sec-
Fig. 3. Pare to-optimal single-pass solutions for the
tion. First, the fuel optimal results and then the vec-
low, medium and high L/D vehicles.
tor-valued results obtained by considering either all,
only the first two or the first and the third functional
This difference is best explained by the fact that the
in Eq (36) are presented.
medium and high L/D vehicles have more to trade-in.
Their portion of atmospheric plane change that is
reduced with increasing fuel consumption is larger in
5.1 Minimum Fuel Trajectories
the first place.
The fuel optimal results indicate that independent of
the vehicle type, the maximum heat flux is approxi-
mately reduced by 20% and 35%, respectively, using
two and three atmospheric passes in comparison to a
single-pass. The optimal fuel consumption for a par-
ticular vehicle is essentially independent of the num-
ber of atmospheric passes and the mission duration
increases on the average by a factor of 1.6 and 2.4 for
two and three atmospheric passes, varying slightly
with the type of vehicle. Although the optimal
amount of atmospheric plane change varies with
vehicle type (3 to 26 for the considered AOTVs),
the single- and multiple-pass solutions exhibit an
interesting invariance. For a particular vehicle type,
the fuel optimal amount of atmospheric plane change J
c
is primarily a function of the specific energy differ- -x
5.
ence of the initial and final orbit. This is independent
of the number of atmospheric passes as long as the
vehicle aerodynamic doesn't degrade significantly at
velocity (km/s)
higher altitudes. This observation corroborates earlier
results by Vinh et al. 1985 who had observed the
same characteristic in the context of fuel optimal sin-
gle-pass transfers. They also coined the term "free
atmospheric plane change" to this phenomena. A
comparison of fuel optimal single-, double- and tri-
ple-pass AOTV trajectories are shown in Fig. 4.
5.2 Minimum Fuel and Heat Trajectories
The vector-valued results indicate that the ability to
Fig. 4. Fuel optimal single- and multiple-pass tra-
trade-ofT additional fuel consumption for a lower
jectories for the medium L/D vehicle.
peak heating rate improves for all transfer modes
(single- and multiple-pass) with increasing aerody- The multiple-pass results exhibit similar trends with
namic lift-to-drag ratio. For example, the typical aer- an additional interesting feature. The initial reduction
obrake configuration (low L/D) can reduce the of the peak heating rate is achieved without any sig-
maximum heating rate by at most 20%, whereas the nificant increase in fuel consumption for all vehicle
high L/D configuration is able to reduce it by up to types. This is explained by the fact that the maximum
65% from the nominal fuel optimal value in the sin- heating rate is unevenly distributed amongst the dif-
6
ferent atmospheric passes for the fuel optimal solu- Acknowledgments
tions and redistributing it to an equally lower level
The invaluable support provided by all other mem-
doesn't require any significant amount of fuel. Some
bers of the GESOP developer team at DLR is grate-
typical multi-objective trajectories where this redis-
fully acknowledged. Finally, without the financial
tribution can be observed are shown in Fig. 4.
support of the Deutsche Forschungsgemeinschaft
(DFG) through SFB-255 Transatmospharische Flug-
systeme this work would not have been possible.
7. REFERENCES
Hwang, C.L. and A.S.M. Masud (1979). Multiple Objec-
2 tive Decision Making - Methods and Applications. Lec-
ture Notes in Economics and Mathematical Systems, No.
lAl 142 1*3 ΙΛ4 145 IM ΧΛΊ 164, Springer Verlag, Berlin.
total maneuver Impulse (km/t)
Jaensch, C. (1994). Untersuchungen zur Wechselwirkung
von Bahn- und Thermaldynamik bci der Auslegung opti-
maler Wiedereintritts- und AOTV Bahnen. Dr.-Ing. Dis-
sertation, University Stuttgart. Germany.
κ Jaensch. C, K. Schnepper and K.H. Well (1994). Multi-
9
C phase Trajectory Optimization Methods with Applica-
δ
X tions to Hypersonic Vehicles. In: Applied Mathematics in
c
"δ Aerospace Science and Engineering (Miele and Salvetti,
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