Advanced Design Problems In Aerospace Engineering

Advanced Design Problems
in Aerospace Engineering
Volume 1: Advanced Aerospace Systems

MATHEMATICAL CONCEPTS AND METHODS
IN SCIENCE AND ENGINEERING

Series Editor: Angelo Miele
George R. Brown School of Engineering
Rice University

Recent volumes in this series:

31 NUMERICAL DERIVATIVES AND NONLINEAR ANALYSIS
Harriet Kagiwada, Robert Kalaba, Nima Rasakhoo, and Karl Spingarn
32 PRINCIPLES OF ENGINEERING MECHANICS
Volume 1: Kinematics— The Geometry of Motion M. F. Beatty, Jr.
33 PRINCIPLES OF ENGINEERING MECHANICS
Volume 2: Dynamics—The Analysis of Motion Millard F. Beatty, Jr.
34 STRUCTURAL OPTIMIZATION
Volume 1: Optimality Criteria Edited by M. Save and W. Prager
35 OPTIMAL CONTROL APPLICATIONS IN ELECTRIC POWER SYSTEMS
G. S. Christensen, M. E. El-Hawary, and S. A. Soliman
36 GENERALIZED CONCAVITY
Mordecai Avriel, Walter W. Diewert, Siegfried Schaible, and Israel Zang
37 MULTICRITERIA OPTIMIZATION IN ENGINEERING AND IN THE SCIENCES
Edited by Wolfram Stadler
38 OPTIMAL LONG-TERM OPERATION OF ELECTRIC POWER SYSTEMS
G. S. Christensen and S. A. Soliman
39 INTRODUCTION TO CONTINUUM MECHANICS FOR ENGINEERS
Ray M. Bowen
40 STRUCTURAL OPTIMIZATION
Volume 2: Mathematical Programming Edited by M. Save and W. Prager
41 OPTIMAL CONTROL OF DISTRIBUTED NUCLEAR REACTORS
G. S. Christensen, S. A. Soliman, and R. Nieva
42 NUMERICAL SOLUTIONS OF INTEGRAL EQUATIONS
Edited by Michael A. Golberg
43 APPLIED OPTIMAL CONTROL THEORY OF DISTRIBUTED SYSTEMS
K. A. Lurie
44 APPLIED MATHEMATICS IN AEROSPACE SCIENCE AND ENGINEERING
Edited by Angelo Miele and Attilio Salvetti
45 NONLINEAR EFFECTS IN FLUIDS AND SOLIDS
Edited by Michael M. Carroll and Michael A. Hayes
46 THEORY AND APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS
Piero Bassanini and Alan R. Elcrat
47 UNIFIED PLASTICITY FOR ENGINEERING APPLICATIONS
Sol R. Bodner
48 ADVANCED DESIGN PROBLEMS IN AEROSPACE ENGINEERING
Volume 1: Advanced Aerospace Systems Edited by Angelo Miele and Aldo Frediani

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Advanced Design Problems
in Aerospace Engineering
Volume 1: Advanced Aerospace Systems

Edited by

Angelo Miele
Rice University
Houston, Texas

and

Aldo Frediani
University of Pisa
Pisa, Italy

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Contributors

P. Alli, Agusta Corporation, 21017 Cascina di Samarate, Varese, Italy.

G. Bernardini, Department of Mechanical and Industrial Engineering,
University of Rome-3, 00146 Rome, Italy.

A. Beukers, Faculty of Aerospace Engineering, Delft University of
Technology, 2629 HS Delft, Netherlands.

V. Caramaschi, Agusta Corporation, 21017 Cascina di Samarate, Varese,
Italy.

M. Chiarelli, Department of Aerospace Engineering, University of Pisa,
56100 Pisa, Italy.

T. De Jong, Faculty of Aerospace Engineering, Delft University of

I. P. Fielding, Aerospace Design Group, Cranfield College of
Aeronautics, Cranfield University, Cranfield, Bedforshire MK43 OAL,
England.

A. Frediani, Department of Aerospace Engineering, University of Pisa,
56100 Pisa, Italy

M. Hanel, Institute of Flight Mechanics and Flight Control, University of
Stuttgart, 70550 Stuttgart, Germany.

J. Hinrichsen, Airbus Industries, 1 Round Point Maurice Bellonte, 31707
Blagnac, France.

v

vi Contributors

L. A. Krakers, Faculty of Aerospace Engineering, Delft University of
A. Longhi, Department of Aerospace Engineering, University of Pisa,
56100 Pisa, Italy.
S. Mancuso, ESA-ESTEC Laboratory, 2201 AZ Nordwijk, Netherlands.
A. Miele, Aero-Astronautics Group, Rice University, Houston, Texas
77005-1892, USA.
G. Montanari, Department of Aerospace Engineering, University of Pisa,
56100 Pisa, Italy.
L. Morino, Department of Mechanical and Industrial Engineering,
University of Rome-3, 00146 Rome, Italy.
F. Nannoni, Agusta Corporation, 21017 Cascina di Samarate, Varese,
Italy.
M. Raggi, Agusta Corporation, 21017 Cascina di Samarate, Varese, Italy.
J. Roskam, DAR Corporation, 120 East 9th Street, Lawrence, Kansas
66044, USA.
G. Sachs, Institute of Flight Mechanics and Flight Control, Technical
University of Munich, 85747 Garching, Germany.
H. Smith, Aerospace Design Group, Cranfield College of Aeronautics,
Cranfield University, Cranfield, Bedforshire MK43 OAL, England.
E. Troiani, Department of Aerospace Engineering, University of Pisa,
56100 Pisa, Italy.
M.J.L. Van Tooren, Faculty of Aerospace Engineering, Delft University
of Technology, 2629 HS Delft, Netherlands.
T. Wang, Aero-Astronautics Group, Rice University, Houston, Texas
77005-1892, USA.

K.H. Well, Institute of Flight Mechanics and Flight Control, University of

Preface
The meeting on “Advanced Design Problems in Aerospace Engineering”
was held in Erice, Sicily, Italy from July 11 to July 18, 1999. The occasion
of the meeting was the 28th Workshop of the School of Mathematics
“Guido Stampacchia”, directed by Professor Franco Giannessi of the
University of Pisa. The School is affiliated with the International Center
for Scientific Culture “Ettore Majorana”, which is directed by Professor
Antonino Zichichi of the University of Bologna.
The intent of the Workshop was the presentation of a series of lectures
on the use of mathematics in the conceptual design of various types of
aircraft and spacecraft. Both atmospheric flight vehicles and space flight
vehicles were discussed. There were 16 contributions, six dealing with
Advanced Aerospace Systems and ten dealing with Unconventional and
Advanced Aircraft Design. Accordingly, the proceedings are split into two
volumes.
The first volume (this volume) covers topics in the areas of flight
mechanics and astrodynamics pertaining to the design of Advanced
Aerospace Systems. The second volume covers topics in the areas of
aerodynamics and structures pertaining to Unconventional and Advanced
Aircraft Design. An outline is given below.

Advanced Aerospace Systems

Chapter 1, by A. Miele and S. Mancuso (Rice University and
ESA/ESTEC), deals with the design of rocket-powered orbital spacecraft.
Single-stage configurations are compared with double-stage configurations
using the sequential gradient-restoration algorithm in optimal control
format.
Chapter 2, by A. Miele and S. Mancuso (Rice University and
ESA/ESTEC), deals with the design of Moon missions. Optimal outgoing
and return trajectories are determined using the sequential gradient-
restoration algorithm in mathematical programming format. The analyses
are made within the frame of the restricted three-body problem and the
results are interpreted in light of the theorem of image trajectories in
Earth-Moon space.

vii

viii Preface

Chapter 3, by A. Miele and T. Wang (Rice University), deals with the
design of Mars missions. Optimal outgoing and return trajectories are
determined using the sequential gradient-restoration algorithm in
mathematical programming format. The analyses are made within the
frame of the restricted four-body problem and the results are interpreted
in light of the relations between outgoing and return trajectories.
Chapter 4, by G. Sachs (Technical University of Munich), deals with
the design and test of an experimental guidance system with perspective
flight path display. It considers the design issues of a guidance system
displaying visual information to the pilot in a three-dimensional format
intended to improve manual flight path control.
Chapter 5, by K.H. Well (University of Stuttgart), deals with the
neighboring vehicle design for a two-stage launch vehicle. It is concerned
with the optimization of the ascent trajectory of a two-stage launch vehicle
simultaneously with the optimization of some significant design parameters.
Chapter 6, by M. Hanel and K.H. Well (University of Stuttgart), deals
with the controller design for a flexible aircraft. It presents an overview of
the models governing the dynamic behavior of a large four-engine flexible
aircraft. It considers several alternative options for control system design.

Unconventional Aircraft Design

Chapter 7, by J.P. Fielding and H. Smith (Cranfield College of
Aeronautics), deals with conceptual and preliminary methods for use on
conventional and blended wing-body airliners. Traditional design methods
have concentrated largely on aerodynamic techniques, with some
allowance made for structures and systems. New multidisciplinary design
tools are developed and examples are shown of ways and means useful for
tradeoff studies during the early design stages.
Chapter 8, by A. Frediani and G. Montanari (University of Pisa), deals
with the Prandtl best-wing system. It analyzes the induced drag of a lifting
multiwing system. This is followed by a box-wing system and then by the
Prandtl best-wing system.
Chapter 9, by A. Frediani, A. Longhi, M. Chiarelli, and E. Troiani
(University of Pisa), deals with new large aircraft with nonconventional
configuration. This design is called the Prandtl plane and is a biplane with
twin horizontal and twin vertical swept wings. Its induced drag is smaller
than that of any aircraft with the same dimensions. Its structural,
aerodynamic, and aeroelastic properties are discussed.
Chapter 10, by L. Morino and G. Bernardini (University of Rome-3),
deals with the modeling of innovative configurations using

Preface ix

multidisciplinary optimization (MDO) in combination with recent
aerodynamic developments. It presents an overview of the techniques for
modeling the structural, aerodynamic, and aeroelastic properties of
aircraft, to be used in the preliminary design of innovative configurations
via multidisciplinary optimization.

Advanced Aircraft Design

Chapter 11, by P. Alli, M. Raggi, F. Nannoni, and V. Caramaschi
(Agusta Corporation), deals with the design problems for new helicopters.
These problems are treated in light of the following aspects: man-machine
interface, fly-by-wire, rotor aerodynamics, rotor dynamics, aeroelasticity,
and noise reduction.
Chapter 12, by A. Beukers, M.J.L Van Tooren, and T. De Jong (Delft
University of Technology), deals with a multidisciplinary design
philosophy for aircraft fuselages. It treats the combined development of
new materials, structural concepts, and manufacturing technologies
leading to the fulfillment of appropriate mechanical requirements and ease
of production.
Chapter 13, by A. Beukers, M.J.L. Van Tooren, T. De Jong, and L.A.
Krakers (Delft University of Technology), continues Chapter 12 and deals
with examples illustrating the multidisciplinary concept. It discusses the
following problems: (a) tension-loaded plate with stress concentrations, (b)
buckling of a composite plate, and (c) integration of acoustics and
aerodynamics in a stiffened shell fuselage.
Chapter 14, by J. Hinrichsen (Airbus Industries), deals with the design
features and structural technologies for the family of Airbus A3XX
aircraft. It reviews the problems arising in the development of the A3XX
aircraft family with respect to configuration design, structural design, and
application of new materials and manufacturing technologies.
Chapter 15, by J. Roskam (DAR Corporation), deals with user-friendly
general aviation airplanes via a revolutionary but affordable approach. It
discusses the development of personal transportation airplanes as
worldwide standard business tools. The areas covered include system
design and integration as well as manufacturing at an acceptable cost level.
Chapter 16, by J. Roskam (DAR Corporation), deals with the design of
a 10-20 passenger jet-powered regional transport and resulting economic
challenges. It discusses the introduction of new small passenger jet aircraft
designed for short-to-medium ranges. It proposes the development of a
family of two airplanes: a single-fuselage 10-passenger airplane and a
twin-fuselage 20-passenger airplane.

x Preface

In closing, the Workshop Directors express their thanks to Professors
Franco Giannessi and Antonino Zichichi for their contributions.

A. Miele A. Frediani
Rice University University of Pisa
Houston, Texas, USA Pisa, Italy

Contents

1. Design of Rocket-Powered Orbital Spacecraft 1
A. Miele and S. Mancuso

2. Design of Moon Missions 31
A. Miele and S. Mancuso

3. Design of Mars Missions 65
A. Miele and T. Wang

4. Design and Test of an Experimental Guidance System with a
Perspective Flight Path Display 105
G. Sachs

5. Neighboring Vehicle Design for a Two-Stage Launch Vehicle 131
K. H. Well

6. Controller Design for a Flexible Aircraft 155
M. Hanel and K. H. Well

Index 181

xi

1

Design of Rocket-Powered Orbital
Spacecraft1
A. MIELE2 AND S. MANCUSO3

Abstract. In this paper, the feasibility of single-stage-suborbital
(SSSO), single-stage-to-orbit (SSTO), and two-stage-to-orbit
(TSTO) rocket-powered spacecraft is investigated using optimal
control theory. Ascent trajectories are optimized for different
combinations of spacecraft structural factor and engine specific
impulse, the optimization criterion being the maximum payload
weight. Normalized payload weights are computed and used to
assess feasibility.
The results show that SSSO feasibility does not necessarily
imply SSTO feasibility: while SSSO feasibility is guaranteed for all
the parameter combinations considered, SSTO feasibility is
guaranteed for only certain parameter combinations, which might be
beyond the present state of the art. On the other hand, not only
TSTO feasibility is guaranteed for all the parameter combinations
considered, but a TSTO spacecraft is considerably superior to a
SSTO spacecraft in terms of payload weight.
Three areas of potential improvements are discussed: (i) use of
lighter materials (lower structural factor) has a significant effect on
payload weight and feasibility; (ii) use of engines with higher ratio
of thrust to propellant weight flow (higher specific impulse) has also

1
This paper is based on Refs. 1-4.
2
Research Professor and Foyt Professor Emeritus of Engineering, Aerospace Sciences,
and Mathematical Sciences, Aero-Astronautics Group, Rice University, Houston, Texas
77005-1892, USA.
3
Guidance, Navigation, and Control Engineer, European Space Technology and
Research Center, 2201 AZ, Nordwijk, Netherlands.

1

2 A. Miele and S. Mancuso

a significant effect on payload weight and feasibility; (iii) on the
other hand, aerodynamic improvements via drag reduction have a
relatively minor effect on payload weight and feasibility.
In light of (i) to (iii), with reference to the specific
impulse/structural factor domain, nearly-universal zero-payload
lines can be constructed separating the feasibility region (positive
payload) from the unfeasibility region (negative payload). The zero-
payload lines are of considerable help to the designer in assessing
the feasibility of a given spacecraft.

Key Words. Flight mechanics, rocket-powered spacecraft,
suborbital spacecraft, orbital spacecraft, optimal trajectories, ascent
trajectories.

1. Introduction

After more than thirty years of development of multi-stage-to-orbit
(MSTO) spacecraft, exemplified by the Space Shuttle and Ariane three-
stage spacecraft, the natural continuation for a modern space program is
the development of two-stage-to-orbit (TSTO) and then single-stage-to-
orbit (SSTO) spacecraft (Refs. 1-7). The first step toward the latter goal is
the development of a single-stage-suborbital (SSSO) rocket-powered
spacecraft which must take-off vertically, reach given suborbital altitude
and speed, and then land horizontally.
Within the above frame, this paper investigates via optimal control
theory the feasibility of three different configurations: a SSSO
configuration, exemplified by the X-33 spacecraft; a SSTO configuration,
exemplified by the Venture Star spacecraft; and a TSTO configuration.
Ascent trajectories are optimized for different combinations of spacecraft
structural factor and engine specific impulse, the optimization criterion
being the maximum payload weight. Realistic constraints are imposed on
tangential acceleration, dynamic pressure, and heating rate.
The optimization is done employing the sequential gradient-restoration
algorithm for optimal control problems (SGRA, Refs. 8-10), developed
and perfected by the Aero-Astronautics Group of Rice University over the
years. SGRA has the major property of being a robust algorithm, and it
has been employed with success to solve a wide variety of aerospace
problems (Refs. 11-16) including interplanetary trajectories (Ref. 11),

Design of Rocket-Powered Orbital Spacecraft 3

flight in windshear (Refs. 12-13), aerospace plane trajectories (Ref. 14),
and aeroassisted orbital transfer (Refs. 15-16).
In Section 2, we present the system description. In Section 3, we
formulate the optimization problem and give results for the SSSO
configuration. In Section 4, we formulate the optimization problem and
give results for the SSTO configuration. In Sections 5, we formulate the
optimization problem and give results for the TSTO configuration. Section
6 contains design considerations pointing out the areas of potential
improvements. Finally, Section 7 contains the conclusions.

2. System Description

For all the configurations being studied, the following assumptions are
employed: (A1) the flight takes place in a vertical plane over a spherical
Earth; (A2) the Earth rotation is neglected; (A3) the gravitational field is
central and obeys the inverse square law; (A4) the thrust is directed along
the spacecraft reference line; hence, the thrust angle of attack is the same
as the aerodynamic angle of attack; (A5) the spacecraft is controlled via
the angle of attack and power setting.

2.1. Mathematical Model. With the above assumptions, the motion
of the spacecraft is described by the following differential system for the
altitude h, velocity V, flight path angle and reference weight W (Ref.
17):

in which the dot denotes derivative with respect to the time t. Here,


where is the final time. The quantities on the right-hand side
of (1) are the thrust T, drag D, lift L, reference weight W, radial distance r,
local acceleration of gravity g, sea-level acceleration of gravity angle
of attack and engine specific impulse
In addition, the following relations hold:

where is the Earth radius, the Earth gravitational constant,
the exit velocity of the gases, and m the instantaneous mass. Note that, by
definition, the reference weight is proportional to the instantaneous mass.
The aerodynamic forces are given by

where is the drag coefficient, the lift coefficient, S a reference
surface area, and the air density (Ref. 18). Disregarding the dependence
on the Reynolds number, the aerodynamic coefficients can be represented
in terms of the angle of attack and the Mach number where
a is the speed of sound. The functions and used in this
paper are described in Refs. 1-4.
For the rocket powerplant under consideration, the following
expressions are assumed for the thrust and specific impulse:

where is the power setting, a reference thrust (thrust for and


a reference specific impulse. The fact that and are assumed to be
constant means that the weak dependence of T and on altitude and
Mach number, relevant to a precision study, is disregarded within the
present feasibility study.
The atmospheric model used is the 1976 US Standard Atmosphere
(Ref. 18). In this model, the values of the density are tabulated at discrete
altitudes. For intermediate altitudes, the density is computed by assuming
an exponential fit for the function This is equivalent to assuming that
the atmosphere behaves isothermally between any two contiguous
altitudes tabulated in Ref. 18.

2.2. Inequality Constraints. Inspection of the system (1) in light of
(2)-(4) shows that the time history of the state h(t), V(t), W(t) can be
computed by forward integration for given initial conditions, given
controls and and given final time In turn, the controls are
subject to the two-sided inequality constraints

which must be satisfied everywhere along the interval of integration. In
addition, some path constraints are imposed on tangential acceleration
dynamic pressure q, and heating rate Q per unit time and unit surface area,
specifically,

Note that (6a) involves directly both the state and the control; on the other
hand, (6b) and (6c) involve directly the state and indirectly the control.
Concerning (6c), is a reference altitude, is a reference velocity, and C
is a dimensional constant; for details, see Refs. 1-4.


In solving the optimization problems, the control constraints (5) are
accounted for via trigonometric transformations. On the other hand, the
path constraints (6) are taken into account via penalty functionals.

2.3. Supplementary Data. The following data have been used in the
numerical experiments:

3. Single-Stage Suborbital Spacecraft

The following data were considered for SSSO configurations designed
to achieve Mach number M= 15 in level flight at h = 76.2 km:

The values (8) are representative of the X-33 spacecraft.

3.1. Boundary Conditions. The initial conditions (t = 0, subscript i)
and final conditions subscript f) are


In Eqs. (9d), the reference weight is the same as the takeoff weight.

3.2. Weight Distribution. The propellant weight structural weight
and payload weight can be expressed in terms of the initial weight
final weight and structural factor via the following relations (Ref. 17):

with

3.3. Optimization Problem. For the SSSO configuration, the
maximum payload problem can be formulated as follows [see (10c)]:

The unknowns include the state variables h, V, W, control variables
and parameter

3.4. Computer Runs. First, the maximum payload weight problem
(11) was solved via the sequential gradient-restoration algorithm (SGRA)
for the following combinations of engine specific impulse and spacecraft
structural factor:


The results for the normalized final weight propellant weight
structural weight and payload weight associated
with various parameter combinations can be found in Refs. 1 and 4. In Fig.
1a, the maximum value of the normalized payload weight is plotted versus
the specific impulse for the values (12b) of the structural factor. The main
comments are that:
(i) The normalized payload weight increases as the engine specific
impulse increases and as the spacecraft structural factor
decreases.
(ii) The design of the SSSO configuration is feasible for each of the
parameter combinations (12).

Zero-Payload Line. Next assume that, for a given specific impulse in
the range (12a), the structural factor is increased beyond the range (12b).


Each increase of causes a corresponding decrease in payload weight,
until a limiting value is found such that By repeating this
procedure for each specific impulse in the range (12a), it is possible to
construct a zero-payload line separating the feasibility region (below)
from the unfeasibility region (above); this is shown in Fig. 1b with
reference to the specific impulse/structural factor domain. The main
comments are that:
(iii) Not only the zero-payload line supplies the upper bound
ensuring feasibility for each given but simultaneously supplies
the lower bound ensuring feasibility for each given
(iv) For a spacecraft of the X-33 type, with the limiting
value of the structural factor is Should the SSSO
design be such that it would become impossible for the
X-33 spacecraft to reach the desired final Mach number
in level flight at the given final altitude Instead, the
spacecraft would reach a lower final Mach number, implying a
subsequent decrease in range.


4. Single-Stage Orbital Spacecraft

The following data were considered for SSTO configurations designed
to achieve orbital speed at Space Station altitude, hence V = 7.633 km/s at
h = 463 km:

The values (13) are representative of the Venture Star spacecraft.

4.1. Boundary Conditions. The initial conditions (t = 0, subscript i)
and final conditions subscript f) are

In Eqs. (14d), the reference weight is the same as the takeoff weight.

4.2. Weight Distribution. Relations (10) governing the weight
distribution for the SSSO spacecraft are also valid for the SSTO
spacecraft, since both spacecraft are of the single-stage type.

4.3. Optimization Problem. For the SSTO configuration, in light of
Sections 3.2 and 4.2, the maximum payload problem can be formulated as
follows [see (10c)]:


The unknowns include the state variables h, V, W, control variables
and parameter

4.4. Computer Runs. First, the maximum payload weight problem
(15) was solved via SGRA for the following combinations of engine
specific impulse and spacecraft structural factor:



comments are that:
decreases.
(ii) The design of SSTO configurations might be comfortably
feasible, marginally feasible, or unfeasible, depending on the
parameter values assumed.

Zero-Payload Line. By proceeding along the lines of Section 3.4, a
zero-payload line can be constructed for the SSTO spacecraft.
With reference to the specific impulse/structural factor domain, the zero-
payload line is shown in Fig. 2b and separates the feasibility region
(below) from the unfeasibility region (above). The main comments are
that:
(iii) Not only the zero-payload line supplies the upper bound
ensuring feasibility for each given but simultaneously supplies
the lower bound ensuring feasibility for each given
(iv) For a spacecraft of the Venture Star type, with the
limiting value of the structural factor is Should the
SSTO design be such that it would become impossible
for the Venture Star spacecraft to reach orbital speed at Space
Station altitude. Instead, the spacecraft would reach a suborbital
speed at the same altitude.

5. Two-Stage Orbital Spacecraft

The following data were considered for TSTO configurations designed
to achieve orbital speed at Space Station altitude, hence V = 7.633 km/s at
h = 463 km:


The values (17) are representative of a hypothetical two-stage version of
the Venture Star spacecraft.
Let the subscript 1 denote Stage 1; let the subscript 2 denote Stage 2.
The maximum payload weight problem was studied first for the case of
uniform structural factor, and then for the case of nonuniform
structural factor,

5.1. Boundary Conditions. Equations (14), left column, must be
understood as initial conditions (t = 0, subscript i) for Stage 1; equations
(14), right column, must be understood as final conditions
subscript f) for Stage 2. Hence,


In Eqs. (18d), the reference weight is the same as the take-off weight.

Interface Conditions. At the interface between Stage 1 and Stage 2,
there is a weight discontinuity due to staging, more precisely [see (20)],

In turn, this induces a thrust discontinuity due to the requirement that the
tangential acceleration be kept unchanged,

where the tangential acceleration is given by (6a).

5.2. Weight Distribution. Relations (10), valid for SSSO and SSTO
configurations, are still valid for the TSTO configuration, providing they
are rewritten with the subscript 1 for Stage 1 and the subscript 2 for Stage 2.
For Stage 1, the propellant weight, structural weight, and payload
weight can be expressed in terms of the initial weight, final weight, and
structural factor via the following relations:

with

For Stage 2, the relations analogous to (20) are


with

For the TSTO configuration as a whole, the following relations hold:

with

5.3. Optimization Problem. For the TSTO configuration, the
maximum payload weight problem can be formulated as follows [see (21)
and (22)]:

The unknowns include the state variables and
the control variables and and the parameters and which


represent the time lengths of Stage 1 and Stage 2. The total time from
takeoff to orbit is

5.4. Computer Runs: Uniform Structural Factor. First, the
maximum payload weight problem (23) was solved via SGRA for the
following combinations of engine specific impulse and spacecraft
structural factor:

comments are that:
decreases.
(ii) The design of TSTO configurations is feasible for each of the
parameter combinations considered.
(iii) For those parameter combinations for which the SSTO
configuration is feasible, the TSTO configuration exhibits a much
larger payload. As an example, for s and the
payload of the TSTO configuration (Fig. 3a) is about eight times
that of the SSTO configuration (Fig. 2a).

Zero-Payload Line. By proceeding along the lines of Section 3.4, a
zero-payload line can be constructed for the TSTO spacecraft with
uniform structural factor. With reference to the specific impulse/ structural


factor domain, the zero-payload line is shown in Fig. 3b and separates the
feasibility region (below) from the unfeasibility region (above). The main
comments are that:
(iv) For the TSTO spacecraft, the size of the feasibility region is more
than twice that of the SSTO spacecraft.
(v) For a hypothetical two-stage version of the Venture Star
spacecraft, with s, the limiting value of the uniform
structural factor is This is more than twice the
limiting value of the single-stage version of the same
spacecraft.

5.5. Computer Runs: Nonuniform Structural Factor. The
maximum payload weight problem (23) was solved again via SGRA for
the following combinations of engine specific impulse and spacecraft


structural factor:

the specific impulse for the values (26c) of the Stage 1 structural factor
and k = 2. In Fig. 4b, the maximum value of the normalized payload


weight is plotted versus the specific impulse for and the values
(26d) of the parameter The main comments are that:
impulse increases, as the Stage 1 structural factor decreases, and
as the parameter k decreases, hence as the Stage 2 structural
factor decreases.
(ii) Even if the Stage 2 structural factor is twice the Stage 1 structural
factor (k = 2), the TSTO configuration is feasible; this is true for
every value of the specific impulse if or (Fig.
4a) and for if
(iii) For s and the maximum value of the parameter
k for which feasibility can be guaranteed is (Fig. 4b);
this corresponds to a Stage 2 structural factor

Zero-Payload Line. By proceeding along the lines of Section 3.4,
zero-payload lines can be constructed for the TSTO spacecraft with
nonuniform structural factor. With reference to the specific impulse/
structural factor domain, the zero-payload lines are shown in Fig. 4c for
the values (26d) of the parameter For each value of k, these lines
separate the feasibility region (below) from the unfeasibility region


(above). The main comments are that:
(iv) As the parameter k increases, the size of the feasibility region
decreases reducing, vis-à-vis the size for k = 1, to about 55
percent for k =2 and about 35 percent for k = 3.


(v) For the zero-payload line of the TSTO spacecraft
becomes nearly identical with the zero-payload line of the SSTO
spacecraft.
(vi) As a byproduct of (v), let us compare a TSTO configuration
with a SSTO configuration for the same payload
and the same specific impulse. For one can design a TSTO
configuration with considerably larger than implying
increased safety and reliability of the TSTO configuration vis-à-
vis the SSTO configuration. The fact that can be much larger
than suggests that an attractive TSTO design might be a first-
stage structure made of only tanks and a second-stage structure
made of engines, tanks, electronics, and so on.

6. Design Considerations

In Sections 3-5, the maximum payload weight problem was solved for
SSSO, SSTO, and TSTO configurations. The results obtained must be
taken “cum grano salis” in that they are nonconservative: they disregard
the need of propellant for space maneuvers, reentry maneuvers, and
reserve margin for emergency. This means that, with reference to the
specific impulse/structural factor domain, an actual design must lie wholly
inside the feasibility regions of Figs. 1b, 2b, 3b, 4c.

6.1. Structural Factor and Specific Impulse. With the above caveat,
the main concept emerging from Sections 3-5 is that the normalized
payload weight increases as the engine specific impulse increases and as
the spacecraft structural factor decreases. This implies that (i) the use of
engines with higher ratio of thrust to propellant weight flow and (ii) the
use of lighter materials have a significant effect on payload weight and
feasibility of SSSO, SSTO, and TSTO configurations.

6.2. SSSO versus SSTO Configurations. Another concept emerging
from Sections 3-4 is that feasibility of the SSSO configuration does not
necessarily imply feasibility of the SSTO configuration. The reason for
this statement is that the increase in total energy to be imparted to an
SSTO configuration is almost 4 times the increase in total energy of an


SSSO configuration performing the task outlined in Section 3. In short,
SSSO and SSTO configurations do not belong to the same ballpark; hence,
a comparison is not meaningful.

6.3. SSTO versus TSTO Configurations. These configurations do
belong to the same ballpark in that they require the same increase in total
energy per unit weight to be placed in orbit; hence, a comparison is
meaningful.
Figures 5a-5d compare SSTO and TSTO configurations for the case
where the latter configuration has uniform structural factor,
For the Venture Star spacecraft and s, Fig. 5a shows that, if
the TSTO payload is about 2.5 times the SSTO payload; Fig. 5b
shows that, if the TSTO payload is about 8 times the SSTO
payload; Fig. 5c shows that, if the TSTO spacecraft is feasible
with a normalized payload of about 0.05, while the SSTO spacecraft is
unfeasible. Figure 5d shows the zero-payload lines of SSTO and TSTO


configurations, making clear that the size of the TSTO feasibility region is
about 2.5 times the size of the SSTO feasibility region.
Figures 6a-6b compare SSTO and TSTO configurations for the case
where the latter configuration has nonuniform structural factor, and
with k = 1, 2, 3. Figure 6a refers to and shows that the
TSTO configuration with k = 2 (hence and ) has a
higher payload than the SSTO configuration. This implies that, vis-à-vis
the SSTO configuration, the TSTO configuration can combine the benefit
of higher payload with the benefit of increased safety and reliability.
Indeed, an attractive TSTO design might be a first-stage structure made of
only tanks and a second-stage structure made of engines, tanks,
electronics, and so on.

6.4. Drag Effects. To assess the influence of the aerodynamic
configuration on feasibility, a parametric study has been performed.
Optimal trajectories have been computed again varying the drag by ± 50%


while keeping the lift unchanged. Namely, the drag and lift of the
spacecraft have been embedded into a one-parameter family of the form

where is the drag factor. Clearly, yields the drag and lift of the
baseline configuration; reduces the drag by 50 %, while keeping
the lift unchanged; increases the drag by 50 %, while keeping the
lift unchanged.
The following parameter values have been considered:

with (28c) indicating that a uniform structural factor is being considered
for the TSTO configuration. The results are shown in Fig. 7, where the
normalized payload weight is plotted versus the drag factor for
the parameters choices (28).
The analysis shows that changing the drag by ± 50 % produces
relatively small changes in payload weight. One must conclude that the
payload weight is not very sensitive to the aerodynamic model of the
spacecraft, or equivalently that the aerodynamic forces do not have a large
influence on propellant consumed. Indeed, should an energy balance be
made, one would find that the largest part of the energy produced by the
rocket powerplant is spent in accelerating the spacecraft to the final
velocity; only a minor part is spent in overcoming aerodynamic and
gravitational effects.
For TSTO configurations, these results justify having neglected in the
analysis drag changes due to staging, and hence having assumed that the
drag function of Stage 2 is the same as the drag function of Stage 1.

7. Conclusions

In this paper, the feasibility of single-stage-suborbital (SSSO), single-


stage-to-orbit (SSTO), and two-stage-to-orbit (TSTO) rocket-powered
spacecraft has been investigated using optimal control theory. Ascent
trajectories have been optimized for different combinations of spacecraft
structural factor and engine specific impulse, the optimization criterion
being the maximum payload weight. Normalized payload weights have
been computed and used to assess feasibility. The main results are that:
(i) SSSO feasibility does not necessarily imply SSTO feasibility:
while SSSO feasibility is guaranteed for all the parameter
combinations considered, SSTO feasibility is guaranteed for only
certain parameter combinations, which might be beyond the
present state of the art.
(ii) For the case of uniform structural factor, not only TSTO
feasibility is guaranteed for all the parameter combinations
considered, but for the same structural factor a TSTO spacecraft
is considerably superior to a SSTO spacecraft in terms of payload
weight.
(iii) For the case of nonuniform structural factor, it is possible to
design a TSTO spacecraft combining the advantages of higher
payload and higher safety/reliability vis-à-vis a SSTO spacecraft.


Indeed, an attractive TSTO design might be a first-stage structure
made of only tanks and a second-stage structure made of engines,
tanks, electronics, and so on.
(iv) Investigation of areas of potential improvements has shown that:
(a) use of lighter materials (smaller spacecraft structural factor)
has a significant effect on payload weight and feasibility; (b) use
of engines with higher ratio of thrust to propellant weight flow
(higher engine specific impulse) has also a significant effect on
payload weight and feasibility; (c) on the other hand,
aerodynamic improvements via drag reduction have a relatively
minor effect on payload weight and feasibility.
(v) In light of (iv), nearly universal zero-payload lines can be
constructed separating the feasibility region (positive payload)
from the unfeasibility region (negative payload). The zero-
payload lines are of considerable help to the designer in assessing
the feasibility of a given spacecraft.
(vi) In conclusion, while the design of SSSO spacecraft appears to be
feasible, the design of SSTO spacecraft, although attractive from
a practical point of view (complete reusability of the spacecraft),
might be unfeasible depending on the parameter values consi-
dered. Indeed, prudence suggests that TSTO spacecraft be given
concurrent consideration, especially if it is not possible to achieve
in the near future major improvements in spacecraft structural
factor and engine specific impulse.

References

1. MIELE, A., and MANCUSO, S., Optimal Ascent Trajectories for a
Single-Stage Suborbital Spacecraft, Aero-Astronautics Report 275,
Rice University, 1997.
2. MIELE, A., and MANCUSO, S., Optimal Ascent Trajectories for
SSTO and TSTO Spacecraft, Aero-Astronautics Report 276, Rice
University, 1997.
TSTO Spacecraft: Extensions, Aero-Astronautics Report 277, Rice
University, 1997.


SSSO, SSTO, and TSTO Spacecraft: Extensions, Aero-Astronautics
Report 278, Rice University, 1997.
5. ANONYMOUS, N. N., Access to Space Study, Summary Report,
Office of Space Systems Development, NASA Headquarters, 1994.
6. FREEMAN, D. C, TALAY, T. A., STANLEY, D. O., LEPSCH,
R. A., and WIHITE, A. W., Design Options for Advanced Manned
Launch Systems, Journal of Spacecraft and Rockets, Vol.32, No.2,
pp.241-249, 1995.
7. GREGORY, I. M., CHOWDHRY, R. S., and McMIMM, J. D.,
Hypersonic Vehicle Model and Control Law Development Using
and Synthesis, Technical Memorandum 4562, NASA, 1994.
8. MIELE, A., WANG, T., and BASAPUR, V.K., Primal and Dual
Formulations of Sequential Gradient-Restoration Algorithms for
Trajectory Optimization Problems, Acta Astronautica, Vol. 13, No. 8,
pp. 491-505, 1986.
9. MIELE, A., and WANG, T., Primal-Dual Properties of Sequential
Gradient-Restoration Algorithms for Optimal Control Problems, Part
1: Basic Problem, Integral Methods in Science and Engineering,
Edited by F. R. Payne et al, Hemisphere Publishing Corporation,
Washington, DC, pp. 577-607, 1986.
2: General Problem, Journal of Mathematical Analysis and
Applications, Vol. 119, Nos. 1-2, pp. 21-54, 1986.
11. RISHIKOF, B. H., McCORMICK, B. R., PRITCHARD, R. E., and
SPONAUGLE, S. J., SEGRAM: A Practical and Versatile Tool for
Spacecraft Trajectory Optimization, Acta Astronautica, Vol. 26, Nos.
8-10, pp. 599-609, 1992.
12. MIELE, A., and WANG, T., Optimization and Acceleration
Guidance of Flight Trajectories in a Windshear, Journal of Guidance,
Control, and Dynamics, Vol. 10, No. 4, pp.368-377, 1987.
13. MIELE, A., and WANG, T., Acceleration, Gamma, and Theta


Guidance for Abort Landing in a Windshear, Journal of Guidance,
Control, and Dynamics, Vol. 12, No. 6, pp. 815-821, 1989.
14. MIELE A., LEE, W. Y., and WU, G. D., Ascent Performance
Feasibility of the National Aerospace Plane, Atti della Accademia
delle Scienze di Torino, Vol. 131, pp. 91-108, 1997.
15. MIELE, A., Recent Advances in the Optimization and Guidance of
Aeroassisted Orbital Transfers, The 1st John V. Breakwell Memorial
Lecture, Acta Astronautica, Vol. 38, No. 10, pp. 747-768, 1996.
16. MIELE, A., and WANG, T., Robust Predictor-Corrector Guidance
for Aeroassisted Orbital Transfer, Journal of Guidance, Control, and
Dynamics, Vol. 19, No. 5, pp. 1134-1141, 1996.
17. MIELE, A., Flight Mechanics, Vol. 1: Theory of Flight Paths,
Chapters 13 and 14, Addison-Wesley Publishing Company, Reading,
Massachusetts, 1962.
18. NOAA, NASA, and USAF, US Standard Atmosphere, 1976, US
Government Printing Office, Washigton, DC, 1976.

2

Design of Moon Missions
A. MIELE1 AND S. MANCUSO2

Abstract. In this paper, a systematic study of the optimization of
trajectories for Earth-Moon flight is presented. The optimization
criterion is the total characteristic velocity and the parameters to be
optimized are: the initial phase angle of the spacecraft with respect
to Earth, flight time, and velocity impulses at departure and arrival.
The problem is formulated using a simplified version of the
restricted three-body model and is solved using the sequential
gradient-restoration algorithm for mathematical programming
problems.
For given initial conditions, corresponding to a counterclockwise
circular low Earth orbit at Space Station altitude, the optimization
problem is solved for given final conditions, corresponding to either a
clockwise or counterclockwise circular low Moon orbit at different
altitudes. Then, the same problem is studied for the Moon-Earth
return flight with the same boundary conditions.
The results show that the flight time obtained for the optimal
trajectories (about 4.5 days) is larger than that of the Apollo
missions (2.5 to 3.2 days). In light of these results, a further
parametric study is performed. For given initial and final conditions,
the transfer problem is solved again for fixed flight time smaller or
larger than the optimal time.
The results show that, if the prescribed flight time is within one

1
77005-1892, USA.
2
Guidance, Navigation, and Control Engineer, European Space Technology and
Research Center, 2201 AZ, Nordwijk, Netherlands.

31


day of the optimal time, the penalty in characteristic velocity is
relatively small. For larger time deviations, the penalty in
characteristic velocity becomes more severe. In particular, if the
flight time is greater than the optimal time by more than two days,
no feasible trajectory exists for the given boundary conditions.
The most interesting finding is that the optimal Earth-Moon and
Moon-Earth trajectories are mirror images of one another with
respect to the Earth-Moon axis. This result extends to optimal
trajectories the theorem of image trajectories formulated by Miele
for feasible trajectories in 1960.

Key Words. Earth-Moon flight, Moon-Earth flight, Earth-Moon-
Earth flight, lunar trajectories, optimal trajectories, astrodyamics,
optimization.

1. Introduction

In 1960, the senior author developed the theorem of image trajectories
in Earth-Moon space within the frame of the restricted three-body problem
(Ref. 1). For both the 2D case and the 3D case, the theorem states that, if a
trajectory is feasible in Earth-Moon space, (i) its image with respect to the
Earth-Moon axis is also feasible, provided it is flown in the opposite
sense. For the 3D case, the theorem guarantees the feasibility of two
additional images: (ii) the image with respect to the Moon orbital plane,
flown in the same sense as the original trajectory; (iii) the image with
respect to the plane containing the Earth-Moon axis and orthogonal to the
Moon orbital plane, flown in the opposite sense.
Reference 1 establishes a relation between the outgoing/return
trajectories. It is natural to ask whether the feasibility property implies an
optimality property. Namely, within the frame of the restricted three-body
problem and the 2D case, we inquire whether the image of an optimal
Earth-Moon trajectory w.r.t. the Earth-Moon axis has the property of
being an optimal Moon-Earth trajectory.
To supply an answer to the above question, we present in this paper a
systematic study of optimal Earth-Moon and Moon-Earth trajectories
under the following scenario. The optimization criterion is the total
characteristic velocity; the class of two-impulse trajectories is considered;
the parameters being optimized are four: initial phase angle of spacecraft

Design of Moon Missions 33

with respect to either Earth or Moon, flight time, velocity impulse at
departure, velocity impulse at arrival.
We study the transfer from a low Earth orbit (LEO) to a low Moon
orbit (LMO) and back, with the understanding that the departure from
LEO is counterclockwise and the return to LEO is counterclockwise.
Concerning LMO, we look at two options: (a) clockwise arrival to LMO,
with subsequent clockwise departure from LMO; (b) counterclockwise
arrival to LMO, with subsequent counterclockwise departure from LMO.
We note that option (a) has characterized all the flights of the Apollo
program, and we inquire whether option (b) has any merit.
Finally, because the optimization study reveals that the optimal flight
times are considerably larger than the flight times of the Apollo missions,
we perform a parametric study by recomputing the LEO-to-LMO and
LMO-to-LEO transfers for fixed flight time smaller or larger than the
optimal time.
For previous studies related directly or indirectly to the subject under
consideration, see Refs. 1-9. References 10-11 are general interest papers.
References 12-15 investigate the partial or total use of electric propulsion
or nuclear propulsion for Earth-Moon flight. For the algorithms employed
to solve the problems formulated in this paper, see Refs. 16-17. For further
details on topics covered in this paper, see Ref. 18.


The present study is based on a simplified version of the restricted
three-body problem. More precisely, with reference to the motion of a
spacecraft in Earth-Moon space, the following assumptions are employed:
(A1) the Earth is fixed in space;
(A2) the eccentricity of the Moon orbit around Earth is neglected;
(A3) the flight of the spacecraft takes place in the Moon orbital plane;
(A4) the spacecraft is subject to only the gravitational fields of Earth
and Moon;
(A5) the gravitational fields of Earth and Moon are central and obey
the inverse square law;
(A6) the class of two-impulse trajectories, departing with an
accelerating velocity impulse tangential to the spacecraft velocity
relative to Earth [Moon] and arriving with a braking velocity
impulse tangential to the spacecraft velocity relative to Moon
[Earth], is considered.


2.1. Differential System. Let the subscripts E, M, P denote the Earth
center, Moon center, and spacecraft. Consider an inertial reference frame
Exy contained in the Moon orbital plane: its origin is the Earth center; the
x-axis points toward the Moon initial position; the y-axis is perpendicular
to the x-axis and is directed as the Moon initial inertial velocity. With this
understanding, the motion of the spacecraft is described by the following
differential system for the position coordinates and components
of the inertial velocity vector

with

Here are the Earth and Moon gravitational constants; are
the radial distances of the spacecraft from Earth and Moon; are the
Moon inertial coordinates; the dot superscript denotes derivative with
respect to the time t, with where 0 is the initial time and the
final time. The above quantities satisfy the following relations:


Here, is the radial distance of the Moon center from the Earth center,
is an angular coordinate associated with the Moon position, more
precisely the angle which the vector forms with the x-axis; is the
angular velocity of the Moon, assumed constant. Note that, by definition,

2.2. Basic Data. The following data are used in the numerical
experiments described in this paper:

2.3. LEO Data. For the low Earth orbit, the following departure data
(outgoing trip) and arrival data (return trip) are used in the numerical
computation:


corresponding to

The values (5a)-(5b) are the Space Station altitude and corresponding
radial distance; the value (5c) is the circular velocity at the Space Station
altitude.

2.4. LMO Data. For the low Mars orbit, the following arrival data
(outgoing trip) and departure data (return trip) are used in the numerical
computation:

corresponding to

The values (6a)-(6b) are the LMO altitudes and corresponding radial
distances; the values (6c) are the circular velocities at the chosen LMO
arrival/departure altitudes.

3. Earth-Moon Flight

We study the LEO-to-LMO transfer of the spacecraft under the
following conditions: (i) tangential, accelerating velocity impulse from
circular velocity at LEO; (ii) tangential, braking velocity impulse to
circular velocity at LMO.

3.1. Departure Conditions. Because of Assumption (A1), Earth fixed
in space, the relative-to-Earth coordinates are the same as
the inertial coordinates As a consequence, corresponding to
counterclockwise departure from LEO with tangential, accelerating


velocity impulse, the departure conditions (t = 0) can be written as
follows:

or alternatively,

where

Here, is the radius of the low Earth orbit and is the altitude of the
low Earth orbit over the Earth surface; is the spacecraft velocity in
the low Earth orbit (circular velocity) before application of the tangential
velocity impulse; is the accelerating velocity impulse; is the
spacecraft velocity after application of the tangential velocity impulse.
Note that Equation (8c) is an orthogonality condition for the vectors
and meaning that the accelerating velocity impulse is
tangential to LEO.

3.2. Arrival Conditions. Because Moon is moving with respect to
Earth, the relative-to-Moon coordinates are not the


same as the inertial coordinates As a consequence,
corresponding to clockwise or counterclockwise arrival to LMO with
tangential, braking velocity impulse, the arrival conditions can be
written as follows:

or alternatively,

where

Here, is the radius of the low Moon orbit and is the altitude of
the low Moon orbit over the Moon surface; is the spacecraft velocity


in the low Moon orbit (circular velocity) after application of the tangential
velocity impulse; is the braking velocity impulse; is the
spacecraft velocity before application of the tangential velocity impulse.
In Eqs. (10c)-(10d), the upper sign refers to clockwise arrival to LMO;
the lower sign refers to counterclockwise arrival to LMO. Equation (11c)
is an orthogonality condition for the vectors and meaning
that the braking velocity impulse is tangential to LMO.

3.3. Optimization Problem. For Earth-Moon flight, the optimization
problem can be formulated as follows: Given the basic data (4) and the
terminal data (5)-(6),

where is the total characteristic velocity. The unknowns include the
state variables and the parameters

While this problem can be treated as either a mathematical
programming problem or an optimal control problem, the former point of
view is employed here because of its simplicity. In the mathematical
programming formulation, the main function of the differential system (1)-
(2) is that of connecting the initial point with the final point and in
particular supplying the gradients of the final conditions with respect to
the initial conditions and/or problem parameters. In the particular case,
because the problem parameters determine completely the initial
conditions, the gradients are formed only with respect to the problem
parameters.
To sum up, we have a mathematical programming problem in which
the minimization of the performance index (13a) is sought with respect to
the values of which satisfy the radius condition
(11a)-(12a), circularization condition (11b)-(12b), and tangency condition
(10)-(11c). Since we have n = 4 parameters and q = 3 constraints, the
number of degrees of freedom is n – q = 1. Therefore, it is appropriate to
employ the sequential gradient-restoration algorithm (SGRA) for
mathematical programming problems (Ref. 16).


3.4. Results. Two groups of optimal trajectories have been computed.
The first group is formed by trajectories for which the arrival to LMO is
clockwise; the second group is formed by trajectories for which the arrival
to LMO is counterclockwise. For the results are shown in
Tables 1-2 and Figs. 1-2. The major parameters of the problem, the phase
angles at departure, and the phase angles at arrival are shown in Table 1
for clockwise LMO arrival and Table 2 for counterclockwise LMO arrival.


Also for the optimal trajectory in Earth-Moon space, near-
Earth space, and near-Moon space is shown in Fig. 1 for clockwise LMO
arrival and Fig. 2 for counterclockwise LMO arrival. Major comments are
as follows:
(i) the accelerating velocity impulse is nearly independent of
the orbital altitude over the Moon surface (see Ref. 18);
(ii) the braking velocity impulse decreases as the orbital
altitude over the Moon surface increases (see Ref. 18);
(iii) for the optimal trajectories, the flight time (4.50 days for
clockwise LMO arrival, 4.37 days for counterclockwise LMO
arrival) is considerably larger than that of the Apollo missions
(2.5 to 3.2 days, depending on the mission);
(iv) the optimal trajectories with counterclockwise arrival to LMO are
slightly superior to the optimal trajectories with clockwise arrival
to LMO in terms of characteristic velocity and flight time.


4. Moon-Earth Flight

We study the LMO-to-LEO transfer of the spacecraft under the
following conditions: (i) tangential, accelerating velocity impulse from
circular velocity at LMO; (ii) tangential, braking velocity impulse to
circular velocity at LEO.

4.1. Departure Conditions. Because Moon is moving with respect to
Earth, the relative-to-Moon coordinates are not the
same as the inertial coordinates As a consequence,
corresponding to clockwise or counterclockwise departure from LMO
with tangential, accelerating velocity impulse, the departure conditions (t
= 0) can be written as follows:

or alternatively,

where


Here, is the radius of the low Moon orbit and is the altitude of
the low Moon orbit over the Moon surface; is the spacecraft velocity
in the low Moon orbit (circular velocity) before application of the
tangential velocity impulse; is the accelerating velocity impulse;
is the spacecraft velocity after application of the tangential velocity
impulse.
In Eqs. (14c)-(14d), the upper sign refers to clockwise departure from
LMO; the lower sign refers to counterclockwise departure from LMO.
Equation (15c) is an orthogonality condition for the vectors and
meaning that the accelerating velocity impulse is tangential to
LMO.

4.2. Arrival Conditions. Because of Assumption (A1), Earth fixed in
space, the relative-to-Earth coordinates are the same as
the inertial coordinates As a consequence, corresponding to
counterclockwise arrival to LEO with tangential, braking velocity impulse,
the arrival conditions can be written as follows:

or alternatively,


where

Here, is the radius of the low Earth orbit and is the altitude of the
low Earth orbit over the Earth surface; is the spacecraft velocity in
the low Earth orbit (circular velocity) after application of the tangential
velocity impulse; is the braking velocity impulse; is the
spacecraft velocity before application of the tangential velocity impulse.
Note that Equation (18c) is an orthogonality condition for the vectors
and meaning that the braking velocity impulse is tangential
to LEO.

4.3. Optimization Problem. For Moon-Earth flight, the optimization
problem can be formulated as follows: Given the basic data (4) and the
terminal data (5)-(6),

where is the total characteristic velocity. The unknowns include the
state variables and the parameters

Similarly to what is stated in Section 3.3, we are in the presence of a
mathematical programming problem in which the minimization of the
performance index (20a) is sought with respect to the values of
which satisfy the radius condition (18a)-(19a),


circularization condition (18b)-(19b), and tangency condition (17)-(18c).
Once more, we have n = 4 parameters and q = 3 constraints, so that the
number of degrees of freedom is n – q = 1. Therefore, it is appropriate to
employ the sequential gradient-restoration algorithm (SGRA) for
mathematical programming problems (Ref. 16).

4.4. Results. Two groups of optimal trajectories have been computed.
The first group is formed by trajectories for which the departure from
LMO is clockwise; the second group is formed by trajectories for which
the departure from LMO is counterclockwise. The results are presented in
Tables 3-4 and Figs. 3-4. For the major parameters of the
problem, the phase angles at departure, and the phase angles at arrival are
shown in Table 3 for clockwise LMO departure and Table 4 for
counterclockwise LMO departure. Also for the optimal


trajectory in Moon-Earth space, near-Moon space, and near-Earth space is
shown in Fig. 3 for clockwise LMO departure and Fig. 4 for
counterclockwise LMO departure. Major comments are as follows:
(i) the accelerating velocity impulse decreases as the orbital
altitude over the Moon surface increases (see Ref. 18);
(ii) the braking velocity impulse is nearly independent of the
orbital altitude over the Moon surface (see Ref. 18);
(iii) for the optimal trajectories, the flight time (4.50 days for
clockwise LMO departure, 4.37 days for counterclockwise LMO
departure) is considerably larger than that of the Apollo missions
(2.5 to 3.2 days, depending on the mission);
(iv) the optimal trajectories with counterclockwise departure from
LMO are slightly superior to the optimal trajectories with
clockwise departure from LMO in terms of characteristic velocity
and flight time.


5. Earth-Moon-Earth Flight

A very interesting observation can be made by comparing the results
obtained in Sections 3 and 4, in particular Tables 1-2 and Tables 3-4. In
these tables, two kinds of phase angles are reported: for the phase angles
and the reference line is the initial direction of the Earth-Moon
axis; for the phase angles and the reference line is the
instantaneous direction of the Earth-Moon axis. The relations leading from
the angles to the angles are given below,

Thus, is the angle which the vector forms with the rotating
Earth-Moon axis, while is the angle which the vector forms
with the rotating Earth-Moon axis.
With the above definitions in mind, let the departure point of the
outgoing trip be paired with the arrival point of the return trip; conversely,
let the departure point of the return trip be paired with the arrival point of
the outgoing trip. For these paired points, the following relations hold (see
Tables 1-4):

showing that, for the optimal outgoing/return trajectories and in a rotating
coordinate system, corresponding phase angles are equal in modulus and
opposite in sign, consistently with the predictions of the theorem of the
image trajectories formulated by Miele for feasible trajectories in 1960
(Ref. 1).
To better visualize this result, the optimal trajectories of Sections 3 and
4, which were plotted in Figs. 1-4 in an inertial coordinate system Exy,
have been replotted in Figs. 5-6 in a rotating coordinate system here,
the origin is the Earth center, the coincides with the instantaneous
Earth-Moon axis and is directed from Earth to Moon; the is
perpendicular to the and is directed as the Moon inertial velocity.


For clockwise arrival to and departure from LMO, the optimal
outgoing and return trajectories are shown in Fig. 5 in Earth-Moon
space, near-Earth space, and near-Moon space. Analogously, for
counterclockwise arrival to and departure from LMO, the optimal
outgoing and return trajectories are shown in Fig. 6 in Earth-Moon
space, near-Earth space, and near-Moon space. These figures show that
the optimal return trajectory is the mirror image with respect to the
Earth-Moon axis of the optimal outgoing trajectory, and viceversa, once
more confirming the theorem of image trajectories formulated by Miele
for feasible trajectories in 1960 (Ref. 1).

6. Fixed-Time Trajectories

The results of Sections 3 and 4 show that the flight time of an optimal
trajectory (4.50 days for clockwise arrival to LMO, 4.37 days for
counterclockwise arrival to LMO) is considerably larger than that of the
Apollo missions (2.5 to 3.2 days depending on the mission). In light of
these results, the transfer problem has been solved again for a fixed flight
time smaller or larger than the optimal flight time.
If is fixed, the number of parameters to be optimized reduces to n =
3, namely, for an outgoing trajectory and
for a return trajectory. On the other hand, the number of
final conditions is still q = 3, namely: the radius condition, circularization
condition, and tangency condition. This being the case, we are no longer
in the presence of an optimization problem, but of a simple feasibility
problem, which can be solved for example with the modified
quasilinearization algorithm (MQA, Ref. 17). Alternatively, if SGRA is
employed (Ref. 16), the restoration phase of the algorithm alone yields the
solution.

6.1. Feasibility Problem. The feasibility problem is now solved for
the following LEO and LMO data:


and these flight times:

For LEO-to-LMO flight, the constraints are Eqs. (13b) and any of the
values (23c). For LMO-to-LEO flight, the constraints are Eqs. (22b) and
any of the values (23c). The unknowns include the state variables
and the parameters for LEO-to-
LMO flight or the parameters for LMO-to-LEO
flight.

6.2. Results. The results obtained for LEO-to-LMO flight and LMO-
to-LEO flight are presented in Tables 5-6. For LEO-to-LMO flight, Table
5 refers to clockwise LMO arrival; for LMO-to-LEO flight, Table 6 refers
to clockwise LMO departure. Major comments are as follows:
(i) if the prescribed flight time is within one day of the optimal time,
the penalty in characteristic velocity is relatively small;
(ii) if the prescribed flight time is greater than the optimal time by
more than one day, the penalty in characteristic velocity becomes
more severe;
(iii) if the prescribed flight time is greater than the optimal time by
more than two days, no feasible trajectory exists for the given
boundary conditions;
(iv) for given flight time, the outgoing and return trajectories are
mirror images of one another with respect to the Earth-Moon
axis, thus confirming again the theorem of image trajectories
(Ref. 1).

7. Conclusions

We present a systematic study of optimal trajectories for Earth-Moon
flight under the following scenario: A spacecraft initially in a
counterclockwise low Earth orbit (LEO) at Space Station altitude must be
transferred to either a clockwise or counterclockwise low Moon orbit
(LMO) at various altitudes over the Moon surface. We study a


complementary problem for Moon-Earth flight with counterclockwise
return to a low Earth orbit.
The assumed physical model is a simplified version of the restricted
three-body problem. The optimization criterion is the total characteristic
velocity and the parameters being optimized are four: initial phase angle
of the spacecraft with respect to either Earth (outgoing trip) or Moon
(return trip), flight time, velocity impulse at departure, velocity impulse on
arrival.
Major results for both the outgoing and return trips are as follows:


(i) the velocity impulse at LEO is nearly independent of the LMO
altitude (see Ref. 18);
(ii) the velocity impulse at LMO decreases as the LMO altitude
increases (see Ref. 18);
(iii) the flight time of an optimal trajectory is considerably larger than
that of an Apollo trajectory, regardless of whether the LMO
arrival/departure is clockwise or counterclockwise;
(iv) the optimal trajectories with counterclockwise LMO arrival/departure
are slightly superior to the optimal trajectories with clockwise


LMO arrival/departure in terms of both characteristic velocity
and flight time.
In light of (iii), a further parametric study has been performed for both
the outgoing and return trips. The transfer problem has been solved again
for a fixed flight time. Major results are as follows:
(v) if the prescribed flight time is within one day of the optimal flight
time, the penalty in characteristic velocity is relatively small;
(vi) for larger time deviations, the penalty in characteristic velocity
becomes more severe;
(vii) if the prescribed flight time is greater than the optimal time by
more than two days, no feasible trajectory exists for the given
boundary conditions.
While the present study has been made in inertial coordinates,
conversion of the results into rotating coordinates leads to one of the most
interesting findings of this paper, namely:
(viii) the optimal LEO-to-LMO trajectories and the optimal LMO-to-
LEO trajectories are mirror images of one another with respect to
the Earth-Moon axis;
(ix) the above result extends to optimal trajectories the theorem of
image trajectory formulated by Miele for feasible trajectories in
1960 (Ref. 1).


References

1. MIELE, A., Theorem of Image Trajectories in the Earth-Moon
Space, Astronautica Acta, Vol. 6, No. 5, pp. 225-232, 1960.
2. MICKELWAIT, A. B., and BOOTON, R. C., Analytical and
Numerical Studies of Three-Dimensional Trajectories to the Moon,
Journal of the Aerospace Sciences, Vol. 27, No. 8, pp. 561-573, 1960.
3. CLARKE, V. C., Design of Lunar and Interplanetary Ascent
Trajectories, AIAA Journal, Vol. 5, No. 7, pp. 1559-1567, 1963.
4. REICH, H., General Characteristics of the Launch Window for
Orbital Launch to the Moon, Celestial Mechanics and Astrodynamics,
Edited by V. G. Szebehely, Vol. 14, pp. 341-375, 1964.
5. DALLAS, C. S., Moon-to-Earth Trajectories, Celestial Mechanics
and Astrodynamics, Edited by V. G. Szebehely, Vol. 14, pp. 391-438,
1964.
6. BAZHINOV, I. K., Analysis of Flight Trajectories to Moon, Mars,
and Venus, Post-Apollo Space Exploration, Edited by F. Narin,
Advances in the Astronautical Sciences, Vol. 20, pp. 1173-1188,
1966.
7. SHAIKH, N. A., A New Perturbation Method for Computing Earth-
Moon Trajectories, Astronautica Acta, Vol. 12, No. 3, pp. 207-211,
1966.


8. ROSENBAUM, R., WILLWERTH, A. C., and CHUCK, W.,
Powered Flight Trajectory Optimization for Lunar and Interplanetary
Transfer, Astronautica Acta, Vol. 12, No. 2, pp. 159-168, 1966.
9. MINER, W. E., and ANDRUS, J. F., Necessary Conditions for
Optimal Lunar Trajectories with Discontinuous State Variables and
Intermediate Point Constraints, AIAA Journal, Vol. 6, No. 11, pp.
2154-2159, 1968.
10. D’AMARIO, L. A., and EDELBAUM, T. N., Minimum Impulse
Three-Body Trajectories, AIAA Journal, Vol. 12, No. 4, pp. 455-462,
1974.
11. PU, C. L., and EDELBAUM, T. N., Four-Body Trajectory
Optimization, AIAA Journal, Vol. 13, No. 3, pp. 333-336, 1975.
12. KLUEVER, C. A., and PIERSON, B. L., Optimal Low-Thrust
Earth-Moon Transfers with a Switching Function Structure, Journal
of the Astronautical Sciences, Vol. 42, No. 3, pp. 269-283, 1994.
13. R IVAS, M. L., and PIERSON, B. L., Dynamic Boundary
Evaluation Method for Approximate Optimal Lunar Trajectories,
Journal of Guidance, Control, and Dynamics, Vol. 19, No. 4, pp. 976-
978, 1996.
14. KLUEVER, C. A., and PIERSON, B. L., Optimal Earth-Moon
Trajectories Using Nuclear Electric Propulsion, Journal of Guidance,
15. KLUEVER, C. A., Optimal Earth-Moon Trajectories Using
Combined Chemical-Electric Propulsion, Journal of Guidance,
16. MIELE, A., HUANG, H. Y., and HEIDEMAN, J. C., Sequential
Gradient-Restoration Algorithm for the Minimization of Constrained
Functions: Ordinary and Conjugate Gradient Versions, Journal of
Optimization Theory and Applications, Vol. 4, No. 4, pp. 213-243,
1969.
17. M IELE, A., N AQVI, S., L EVY, A. V., and I YER, R. R.,
Numerical Solutions of Nonlinear Equations and Nonlinear Two-


Point Boundary-Value Problems, Advances in Control Systems,
Edited by C. T. Leondes, Academic Press, New York, New York,
Vol. 8, pp. 189-215, 1971.
18. MIELE, A. and MANCUSO, S., Optimal Trajectories for Earth-
Moon-Earth Flight, Aero-Astronautics Report 295, Rice University,
1998.

3

Design of Mars Missions
A. MIELE1 AND T. WANG2

Abstract. This paper deals with the optimal design of round-trip
Mars missions, starting from LEO (low Earth orbit), arriving to
LMO (low Mars orbit), and then returning to LEO after a waiting
time in LMO.
The assumed physical model is the restricted four-body model,
including Sun, Earth, Mars, and spacecraft. The optimization
problem is formulated as a mathematical programming problem: the
total characteristic velocity (the sum of the velocity impulses at LEO
and LMO) is minimized, subject to the system equations and
boundary conditions of the restricted four-body model. The
mathematical programming problem is solved via the sequential
gradient-restoration algorithm employed in conjunction with a
variable-stepsize integration technique to overcome the numerical
difficulties due to large changes in the gravity field near Earth and
near Mars.
The results lead to a baseline optimal trajectory computed under
the assumption that the Earth and Mars orbits around Sun are
circular and coplanar. The baseline optimal trajectory resembles a
Hohmann transfer trajectory, but is not a Hohmann transfer
trajectory, owing to the disturbing influence exerted by Earth/Mars
on the terminal branches of the trajectory. For the baseline optimal
trajectory, the total characteristic velocity of a round-trip Mars

1
77005-1892, USA.
2
Senior Research Scientist, Aero-Astronautics Group, Rice University, Houston, Texas
77005-1892, USA.

65

66 A. Miele and T. Wang

mission is 11.30 km/s (5.65 km/s each way) and the total mission
time is 970 days (258 days each way plus 454 days waiting in
LMO).
An important property of the baseline optimal trajectory is the
asymptotic parallelism property: For optimal transfer, the spacecraft
inertial velocity must be parallel to the inertial velocity of the closest
planet (Earth or Mars) at the entrance to and exit from deep
interplanetary space. For both the outgoing and return trips,
asymptotic parallelism occurs at the end of the first day and at the
beginning of the last day. Another property of the baseline optimal
trajectory is the near-mirror property. The return trajectory can be
obtained from the outgoing trajectory via a sequential procedure of
rotation, reflection, and inversion.
Departure window trajectories are next-to-best trajectories. They
are suboptimal trajectories obtained by changing the departure date,
hence changing the Mars/Earth inertial phase angle difference at
departure. For the departure window trajectories, the asymptotic
parallelism property no longer holds in the departure branch, but still
holds in the arrival branch. On the other hand, the near-mirror
property no longer holds.

Key Words. Flight mechanics, astrodynamics, celestial mechanics,
Earth-to-Mars missions, round-trip Mars missions, mirror property,
asymptotic parallelism property, optimization, sequential gradient
restoration algorithm.

1. Introduction

Several years ago, a research program dealing with the optimization
and guidance of flight trajectories from Earth to Mars and back was
initiated at Rice University. The decision was based on the recognition
that the involvement of the USA with the Mars problem had been growing
in recent years and it can be expected to grow in the foreseeable future
(Refs. 1-15). Our feeling was that, in attacking the Mars problem, we
should start with simple models and then go to models of increasing
complexity. Accordingly, this paper deals with the preliminary results
obtained with a relatively simple model, yet sufficiently realistic to
capture some of the essential elements of the flight from Earth to Mars and
back (Refs. 16-19).

Design of Mars Missions 67

1.1. Mission Alternatives, Types, Objectives. There are two basic
alternatives for Mars missions: robotic missions and manned missions, the
latter being considerably more complex than the former. Within each
alternative, we can distinguish two types of missions: exploratory (survey)
missions and sample taking (sample return) missions.
Regardless of alternative and type, there is a basic maneuver which is
common to every Mars mission, namely, the transfer of a spacecraft from
a low Earth orbit (LEO) to a low Mars orbit (LMO) and back. For both
LEO-to-LMO transfer and LMO-to-LEO transfer, the first objective is to
contain the propellant assumption; the second objective is to contain the
flight time, if at all possible.

1.2. Characteristic Velocity. Under certain conditions, the propellant
consumption is monotonically related to the so-called characteristic
velocity, the sum of the velocity impulses applied to the spacecraft via
rocket engines. In turn, by definition, each velocity impulse is a positive
quantity, regardless of whether its action is accelerating or decelerating,
in-plane or out-of-plane.
In astrodynamics, it is customary to replace the consideration of
propellant consumption with the consideration of characteristic velocity,
with the following advantage: the characteristic velocity is independent of
the spacecraft structural factor and engine specific impulse, while this is
not the case with the propellant consumption. Indeed, the characteristic
velocity truly “characterizes” the mission itself.

1.3. Optimal Trajectories. This presentation is centered on the study
of the optimal trajectories, namely, trajectories minimizing the
characteristic velocity. This study is important in that it provides the basis
for the development of guidance schemes approximating the optimal
trajectories in real time. In turn, this requires the knowledge of some
fundamental, albeit easily implementable property of the optimal
trajectories. This is precisely the case with the asymptotic parallelism
condition at the entrance to and exit from deep interplanetary space: For
both the outgoing and return trips, minimization of the characteristic
velocity is achieved if the spacecraft inertial velocity is parallel to the
inertial velocity of the closest planet (Earth or Mars) at the entrance to and
exit from deep interplanetary space.


2. Four-Body Model

At every point of the trajectory, the spacecraft is subject to the
gravitational attractions of Earth, Mars, and Sun. Therefore, we are in the
presence of a four-body problem, the four bodies being the spacecraft,
Earth, Mars, and Sun (Fig. 1a). Assuming that the Sun is fixed in space,
the complete four-body model is described by 18 nonlinear ordinary
differential equations (ODEs) in the three-dimensional case and by 12
nonlinear ODEs in the two-dimensional case (planar case). Two possible
simplifications are described below.

2.1. Patched Conics Model. This model consists in subdividing an
Earth-to-Mars trajectory into three segments: a near-Earth segment in
which Earth gravity is dominant; a deep interplanetary space segment in
which Sun gravity is dominant; a near-Mars segment in which Mars
gravity is dominant. Under this scenario, the four-body problem is
replaced by a succession of two-body problems, each described in the
planar case by four ODEs, for which analytical solutions are available.


Then, the segmented solutions must be patched together in such a way that
some continuity conditions are satisfied at the interface between
contiguous segments.
Even though the method of patched conics has been widely used in the
literature, our experience with it has been rather disappointing for the
reason indicated below. Near the interface between contiguous segments,
there is a small region in which two of the three gravitational attractions
are of the same order. Neglecting one of them on each side of the interface
induces small local errors in the spacecraft acceleration, which in turn
induce large errors in velocity and position owing to long integration
times. In light of this statement, we discarded the patched conics model,
replacing it with the restricted four-body model.

2.2. Restricted Four-Body Model. This model consists in assuming
that the inertial motions of Earth and Mars are determined only by Sun,
while the inertial motion of the spacecraft is determined by Earth, Mars,
and Sun. In the planar case, this is equivalent to splitting the complete
system of order 12 into three subsystems, each of order four: the Earth,
Mars, and spacecraft subsystems. The first two subsystems can be
integrated independently of the third; the third subsystem can be integrated
once the solutions of the first two are known. This is the essential
simplification provided by the restricted four-body model, while avoiding
the pitfalls of the patched conics model.


Let LEO denote a low Earth orbit, and let LMO denote a low Mars
orbit. We study the LEO-to-LMO transfer [LMO-to-LEO transfer] of a
spacecraft under the following scenario (Fig. 1b). Initially, the spacecraft
moves in a circular orbit around Earth [Mars]; an accelerating velocity
impulse is applied tangentially to LEO [LMO], and its magnitude is such
that the spacecraft escapes from near-Earth [near-Mars] space into deep
interplanetary space. Then, the spacecraft takes a long journey along an
interplanetary orbit around the Sun, enters near-Mars [near-Earth] space,
and reaches tangentially the low Mars orbit [low Earth orbit]. Here, a
decelerating velocity impulse is applied tangentially to LMO [LEO] so as
to achieve circularization of the motion around Mars [Earth].


The following hypotheses are employed: (A1) the Sun is fixed in
space; (A2) Earth and Mars are subject only to the Sun gravity; (A3) the
eccentricity of the Earth and Mars orbits around the Sun is neglected,
implying circular planetary motions; (A4) the inclination of the Mars
orbital plane vis-à-vis the Earth orbital plane is neglected, implying planar
spacecraft motion; (A5) the spacecraft is subject to the gravitational
attractions of Earth, Mars, and Sun along the entire trajectory; (A6) for the
outgoing and return trips, the class of two-impulse trajectories is
considered, with the impulses being applied at the terminal points of the
trajectories; (A7) for the outgoing and return trips, circularization of
motion around the relevant planet is assumed both before departure and
after arrival.
Having adopted the restricted four-body model to achieve increased
precision with respect to the patched conics model, we are simultaneously
interested in five motions: the inertial motions of Earth, Mars, and
spacecraft with respect to the Sun; the relative motions of the spacecraft
with respect to Earth and Mars. To study these motions, we employ three
coordinate systems: Sun coordinate system (SCS), Earth coordinate
system (ECS), and Mars coordinate system (MCS).
SCS is an inertial coordinate system; its origin is the Sun center and its
axes x, y are fixed in space; in particular, the x-axis points to the initial
position of the Earth center and the y-axis is orthogonal to the x-axis. ECS
is a relative-to-Earth coordinate system; its origin is the Earth center and


its axes are parallel to the axes x, y of the Sun coordinate system.
MCS is a relative-to-Mars coordinate system; its origin is the Mars center
and its axes are parallel to the axes x, y of the Sun coordinate
system.
Clearly, ECS and MCS translate without rotation w.r.t. SCS. Their
origins E and M move around the Sun with constant angular velocities
and The angular velocity difference is also constant.
In this paper, the inertial motions of the spacecraft, Earth, and Mars
are described in Sun coordinates, while the spacecraft boundary conditions
are described in relative-to-planet coordinates. If polar coordinates are
used, a position vector is defined via the radial distance r and phase angle
while a velocity vector is defined via the velocity modulus V and local
path inclination If Cartesian coordinates are used, a position vector is
defined its via components x, y and a velocity vector via its components u,
w.
Let E, M, S denote the centers of Earth, Mars, and Sun; let
denote the gravitational constants of Earth, Mars, and Sun; let P denote the
spacecraft; let t denote the time, with 0 the initial time and the
final time. Below, we give the system equations for Earth, Mars, and
spacecraft in Sun coordinates; for details, see Refs. 16-19.

3.1. Earth. Subject to the Sun gravitational attraction and neglecting
the orbital eccentricity, we approximate the Earth (subscript E) trajectory
around the Sun with a circle. Hence, in polar coordinates, the position and
velocity of Earth are given by

(SCS)

In Cartesian coordinates, the position and velocity of Earth are described


by

(SCS)

with

(SCS)

Equation (3c) is an orthogonality condition between vec(SE) and
where vec stands for vector.

3.2. Mars. Subject to the Sun gravitational attraction and neglecting
the orbital eccentricity, we approximate the Mars (subscript M) trajectory
around the Sun with a circle. Hence, in polar coordinates, the position and
velocity of Mars are given by

(SCS)


In Cartesian coordinates, the position and velocity of Mars are described
by

(SCS)

with

(SCS)

Equation (6c) is an orthogonality condition between vec(SM) and
where vec stands for vector.

3.3. Spacecraft. Subject to the gravitational attractions of Sun, Earth,
and Mars along the entire trajectory, the motion of the spacecraft
(subscript P) around the Sun is described by the following differential
equations in the coordinates of the position vector and the
components of the velocity vector:

(SCS)


Here are the radial distances of the spacecraft from the Sun,
Earth, and Mars; these quantities can be computed via the relations

(SCS)

4. Boundary Conditions

4.1. Outgoing Trip, Departure. In polar coordinates, the spacecraft
conditions at the departure from LEO (time t = 0) are given by

(ECS)

Relative to Earth are the radial distance, phase angle,
velocity, and path inclination of the spacecraft; is the spacecraft
velocity in the low Earth orbit prior to application of the tangential,
accelerating velocity impulse; is the accelerating velocity impulse
at LEO; is the spacecraft velocity after application of the
accelerating velocity impulse.
The corresponding equations in Cartesian coordinates are


(ECS)

with

(ECS)

Equation (11c) is an orthogonality condition between vec(EP(0)) and
meaning that the accelerating velocity impulse is
tangential to LEO.

4.2. Outgoing Trip, Arrival. In polar coordinates, the spacecraft
conditions at the arrival to LMO are given by

(MCS)

Relative to Mars are the radial distance, phase angle,
velocity, and path inclination of the spacecraft; is the spacecraft


velocity in the low Mars orbit after application of the tangential,
decelerating velocity impulse; is the decelerating velocity impulse
at LMO; is the spacecraft velocity before application of the
decelerating velocity impulse.

(MCS)

with

(MCS)

Equation (14c) is an orthogonality condition between and
meaning that the decelerating velocity impulse is
tangential to LMO.

4.3. Return Trip, Departure. In polar coordinates, the spacecraft
conditions at the departure from LMO (time t = 0) are given by

(MCS)


Formally, Eqs. (15) can be obtained from Eqs. (12) by simply replacing
the time with the time t = 0. However, there is a difference of
interpretation: is now the spacecraft velocity in the low Mars orbit
before application of the tangential, accelerating velocity impulse;
is the accelerating velocity impulse at LMO; is the spacecraft
velocity after application of the accelerating velocity impulse.

(MCS)

with

(MCS)

Equation (17c) is an orthogonality condition between vec(MP(0)) and
meaning that the accelerating velocity impulse is
tangential to LMO.

4.4. Return Trip, Arrival. In polar coordinates, the spacecraft
conditions at the arrival to LEO are given by

(ECS)


Formally, Eqs. (18) can be obtained from Eqs. (9) by simply replacing the
time t = 0 with the time However, there is a difference of
interpretation: is now the spacecraft velocity in the low Earth orbit
after application of the tangential, decelerating velocity impulse; is
the decelerating velocity impulse at LEO; is the spacecraft velocity
before application of the decelerating velocity impulse.

(ECS)

with

(ECS)

Equation (20c) is an orthogonality condition between and
meaning that the decelerating velocity impulse is
tangential to LEO.


5. Coordinate Transformations

Due to the fact that the spacecraft equations of motion are given in
inertial coordinates (SCS), while the spacecraft boundary conditions are
given in relative-to-planet coordinates (ECS) or (MCS), coordinate
transformations are needed to pass from one system to another at the
terminal points of the outgoing and return trips. The transformations are
given below.
(i) ECS-to-SCS Transformation. For the outgoing trip, this
transformation is to be employed to convert spacecraft conditions at the
departure from LEO (time t = 0) from relative-to-Earth coordinates to
inertial coordinates. In Cartesian coordinates,

(ii) SCS-to-MCS Transformation. For the outgoing trip, this
arrival to LMO from inertial coordinates to relative-to-Mars
coordinates. In Cartesian coordinates,

(iii) MCS-to-SCS Transformation. For the return trip, this
departure from LMO (time t = 0) from relative-to-Mars coordinates to
inertial coordinates. In Cartesian coordinates,


(iv) SCS-to-ECS Transformation. For the return trip, this
arrival to LEO from inertial coordinates to relative-to-Earth
coordinates. In Cartesian coordinates,

6. Mathematical Programming Problems

In this section, we formulate the problem of the optimal round-trip
trajectory as a mathematical programming problem. The complete
problem can be decomposed into three separate problems to be solved in
sequence: (i) determination of the optimal trajectory for the outgoing trip;
(ii) determination of the optimal trajectory for the return trip; (iii)
determination of the waiting time in the low Mars orbit.

6.1. Outgoing Trip. The optimization of a LEO-to-LMO transfer can
be reduced to a mathematical programming problem involving the
following performance index, constraints, and parameters.
Performance Index. The most obvious performance is the characteristic
velocity,


which is the sum of the terminal velocity impulses: is the
accelerating velocity impulse at LEO (Earth coordinates) and is the
decelerating velocity impulse at LMO (Mars coordinates).
Constraints. The departure conditions include the radius condition
(11a), (9a), decircularization condition (11b), (9c), and tangency condition
(11c) [for brevity, constraints (11)]. Satisfaction of the departure
conditions is trivial for any choice of the parameters and By
the same token, the differential system (7) is never violated if a forward
integration is performed with SCS initial conditions consistent with (11)
and (21). The only constraints to be enforced are the final conditions,
which include the radius condition (14a), (12a), circularization condition
(14b), (12c), and tangency condition (14c) [for brevity, constraints (14)].
Parameters. Let a,b,c denote the following vector parameters:

The 7 × 1 vector a includes the major parameters governing a LEO-to-
LMO trajectory; the 2 × 1 vector b includes the components of x that are
fixed, namely, the radii of the terminal orbits and the 5 × 1
vector c includes the components of a that must be optimized, namely, the
terminal velocity impulses and transfer time
spacecraft/Earth relative phase angle at departure and Mars/Earth
inertial phase angle difference at departure Note
that if one sets by definition.
Problem P1. For the outgoing trip, given the vector parameter b [see
(26b)], minimize the performance index (25) w.r.t. the vector parameter c
[see (26c)], subject to the constraints (14).
Problem P1 is characterized by n = 5 variables and q = 3 constraints;
hence, the number of degrees of freedom is n – q = 2, implying that there
are only two independent parameters, for instance, and The
solution of Problem P1 is called the baseline optimal trajectory and yields


the smallest value of the characteristic velocity (25) compatible with a
given pair

6.2. Return Trip. The optimization of a LMO-to-LEO transfer can be
reduced to a mathematical programming problem involving the following
performance index, constraints, and parameters.
Performance Index. The most obvious performance is the
characteristic velocity,

which is the sum of the terminal velocity impulses: is the
accelerating velocity impulse at LMO (Mars coordinates) and is the
decelerating velocity impulse at LEO (Earth coordinates).
Constraints. The departure conditions include the radius condition
(17a), (15a), decircularization condition (17b), (15c), and tangency
condition (17c) [for brevity, constraints (17)]. Satisfaction of the departure
conditions is trivial for any choice of the parameters and
By the same token, the differential system (7) is never violated if a
forward integration is performed with SCS initial conditions consistent
with (17) and (23). The only constraints to be enforced are the final
conditions, which include the radius condition (20a), (18a), circularization
condition (20b), (18c), and tangency condition (20c) [for brevity,
constraints (20)].
Parameters. Let a, b, c denote the following vector parameters:

The 7 × 1 vector a includes the major parameters governing a LMO-to-
LEO trajectory; the 2 × 1 vector b includes the components of a that are
fixed, namely, the radii of the terminal orbits and the 5 × 1
terminal velocity impulses and transfer time


spacecraft/Mars relative phase angle at departure and Mars/Earth
inertial phase angle difference at departure Note
that if one sets by definition.
Problem P2. For the return trip, given the vector parameter b [see
(28b)], minimize the performance index (27) w.r.t. the vector parameter c
[see (28c)], subject to the constraints (20).
Problem P2 is characterized by n = 5 variables and q = 3 constraints;
hence, the number of degrees of freedom is n – q = 2, implying that there
are only two independent parameters, for instance, and The
solution of Problem P2 is called the baseline optimal trajectory and yields
the smallest value of the characteristic velocity (27) compatible with a
given pair

6.3. Waiting Time. For the outgoing trip, celestial mechanics requires
that Mars be ahead of Earth at departure from LEO, but behind Earth at
arrival to LMO; hence, for Problem P1, and For the
return trip, celestial mechanics requires also that Mars be ahead of Earth at
departure from LMO, but behind Earth at arrival to LEO; hence, for
Problem P2, and This implies that the spacecraft
cannot return immediately to Earth and is forced to wait a relatively long
time in LMO to allow the Mars/Earth inertial phase angle difference to
transition from the optimal arrival value of the outgoing trip to the optimal
departure value of the return trip.
For the optimal trajectory, the waiting time on LMO can be computed
with the relation

with angles measured in degrees and time in days.

6.4. Delay Time. Assume now that, due to technical difficulties, it is
not possible to fire the rocket engines at the appropriate departure day for
the return trip nor within the tolerance supplied by the departure window
(see Section 10). This implies that there is a further delay time in LMO,
which can be computed with the relation


with angles measured in degrees and time in days.

7. Computational Information

7.1. Algorithm. The sequential gradient-restoration algorithm (SGRA)
in mathematical programming format is used to solve the mathematical
programming problems of Section 6. SGRA is an iterative technique
which involves a sequence of two-phase cycles, each cycle including a
gradient phase and a restoration phase. In the gradient phase, the
augmented performance index (performance index augmented by the
constraints weighted via appropriate Lagrange multipliers) is decreased,
while avoiding excessive constraint violation. In the restoration phase, the
constraint error is decreased, while avoiding excessive change in the
problem variables. In a complete gradient-restoration cycle, the
performance index is decreased, while the constraints are satisfied to a
preselected accuracy. Thus, a succession of feasible suboptimal solutions
is generated, each new solution being an improvement over the previous
one from the point of view of the performance index (25) or (27).
Note that SGRA is available in both mathematical programming
format and optimal control format. For mathematical programming
problems, SGRA was developed by Miele at al in both ordinary-gradient
version and conjugate-gradient version (Ref. 20). Several variations of
SGRA were also developed, but the basic form proved to be the more
reliable, because of its robustness and stability properties (Ref. 21). For
optimal control problems, the development of SGRA by Miele at al has
been parallel to that for mathematical programming problems; see Refs.
22-24 for early versions and Refs. 25-27 for recent versions. Also for
optimal control problems, an industrial version of SGRA has been
developed by McDonnell-Douglas Technical Service Company under the
code name SEGRAM (Ref. 28) and is being used at NASA-Johnson Space
Center.

7.2. Integration Scheme. The achievement of constraint satisfaction
and optimality condition satisfaction requires multiple integrations of the
system equations of the restricted four-body model. The integration
process is computationally expensive and it is difficult to achieve the


desired accuracy, owing to the fact that the total gravitational acceleration
changes rapidly in near-Earth space and near-Mars space, but slowly in
deep interplanetary space. Indeed, orbital periods are of order one hour if
the Earth gravity or Mars gravity is dominant, but of order one year if the
Sun gravity is dominant. The above difficulties can be overcome by
properly designing a variable-stepsize integration scheme. Numerical
experiments show that good results can be obtained by linking the
integration stepsize to the total gravitational acceleration, with the stepsize
increasing whenever the total gravitational acceleration decreases, and
viceversa.

7.3. Remark. The computations reported here were done on a Unix
Sun Workstation using the C++ programming language. In particular, the
integrations were executed via a fifth-order Runge-Kutta-Fehlberg
scheme.

8. Planetary and Mission Data

The gravitational constants for the Sun, Earth, and Mars are given by

Earth and Mars travel around the Sun along orbits with average radii

The associated average translational velocities and angular velocities
(inertial coordinates) are given by

In particular, the angular velocity difference between Earth and Mars is


For the outgoing trip, the spacecraft is to be transferred from a low
Earth orbit to a low Mars orbit; for the return trip, the spacecraft is to be
transferred from a low Mars orbit to a low Earth orbit. The radii of the
terminal orbits are

corresponding to the altitudes

since the Earth and Mars surface radii are given by

The circular velocities at LEO and LMO (relative-to-planet coordinates)
are given by

and the corresponding escape velocities (relative-to-planet coordinates)
are given by

9. Baseline Optimal Trajectory Results

In this section, we present the results obtained by solving the
mathematical programming problems of Section 6 with the algorithm of
Section 7 in light of the planetary and mission data of Section 8.

9.1. Outgoing Trip. The optimal LEO-to-LMO trajectory is shown in
Figs. 2-3.


Figure 2a refers to deep interplanetary space (Sun coordinates). The
baseline optimal trajectory resembles a Hohmann transfer trajectory, but is
not a Hohmann transfer trajectory, due to the disturbing influence of the
gravitational fields of Earth and Mars on the terminal portions of the
trajectory.


Figure 3a refers to near-Earth space (relative-to-Earth coordinates, first
hour). The baseline optimal trajectory bends under the influence of the
Earth gravitational field, tending to become parallel to the Earth trajectory
at the end of near-Earth space. The asymptotic parallelism condition
(hinted by Fig. 3a, but not shown in Fig. 3a) is reached toward the end of
the first day (Earth gravitational attraction negligible w.r.t. Sun
gravitational attraction). See Ref. 18.


Figure 3b refers to near-Mars space (relative-to-Mars coordinates, last
hour). In reverse time, the baseline optimal trajectory bends under the
influence of the Mars gravitational field, tending to become parallel to the
Mars trajectory at the beginning of near-Mars space. The asymptotic
parallelism condition (hinted by Fig. 3b, but not shown in Fig. 3b) is
reached at the beginning of the last day (Mars gravitational attraction
negligible w.r.t. Sun gravitational attraction). See Ref. 18.
Major numerical results are given below:
(i) The terminal values of the Mars/Earth inertial phase angle
difference are

meaning that Mars is ahead of Earth by nearly 44 deg at departure and
behind Earth by nearly 75 deg at arrival.
(ii) The terminal values of the spacecraft/planet relative phase angle
are

meaning that the accelerating velocity impulse at departure must be
applied nearly 62 deg before the spacecraft becomes aligned with the
Sun/Earth direction, while the decelerating velocity impulse at arrival
must be applied nearly 141 deg before the spacecraft becomes aligned
with the Sun/Mars direction.
(iii) The characteristic velocity components are

implying that the total characteristic velocity is

(iv) The terminal values of the spacecraft inertial phase angle are


implying that the angular travel of the spacecraft is

which is within one degree of 180 deg, the value characterizing a
Hohmann transfer trajectory.
(v) The transfer time is

9.2. Return Trip. The optimal LMO-to-LEO trajectory is shown in
Figs. 2 and 4. Figure 2b refers to deep interplanetary space (Sun
coordinates). The baseline optimal trajectory resembles a Hohmann
transfer trajectory, but is not a Hohmann transfer trajectory, due to the
disturbing influence of the gravitational fields of Mars and Earth on the
terminal portions of the trajectory.


Figure 4a refers to near-Mars space (relative-to-Mars coordinates, first
hour). The baseline optimal trajectory bends under the influence of the
Mars gravitational field, tending to become parallel to the Mars trajectory
at the end of near-Mars space. The asymptotic parallelism condition
(hinted by Fig. 4a, but not shown in Fig. 4a) is reached toward the end of
the first day (Mars gravitational attraction negligible w.r.t. Sun
gravitational attraction). See Ref. 18.
Figure 4b refers to near-Earth space (relative-to-Earth coordinates, last
hour). In reverse time, the baseline optimal trajectory bends under the
influence of the Earth gravitational field, tending to become parallel to the
Earth trajectory at the beginning of near-Earth space. The asymptotic
parallelism condition (hinted by Fig. 4b, but not shown in Fig. 4b) is
reached at the beginning of the last day (Earth gravitational attraction
negligible w.r.t. Sun gravitational attraction). See Ref. 18.
Major numerical results are given below:
(i) The terminal values of the Mars/Earth inertial phase angle
difference are


meaning that Mars is ahead of Earth by nearly 75 deg at departure and
behind Earth by nearly 44 deg at arrival.
(ii) The terminal values of the spacecraft/planet relative phase angle
are

meaning that the accelerating velocity impulse at departure must be
applied nearly 141 deg after the spacecraft becomes aligned with the
Sun/Mars direction, while the decelerating velocity impulse at arrival must
be applied nearly 62 deg after the spacecraft becomes aligned with the
Sun/Earth direction.
(iii) The characteristic velocity components are

implying that the total characteristic velocity is

(iv) The terminal values of the spacecraft inertial phase angle are

implying that the angular travel of the spacecraft is

which is within one degree of 180 deg, the value characterizing a
Hohmann transfer trajectory.
(v) The transfer time is


9.3. Waiting Time. The waiting time in LMO is determined by the
need to allow the Mars/Earth inertial phase angle difference to transition
from the arrival value of the outgoing trip to the departure value of the
return trip. In light of the previous results, the waiting time on Mars is

Therefore, the total time for a round-trip LEO-to-LMO mission without
delay time becomes

and on account of the previous results,

9.4. Delay Time. If it is not possible to fire the rocket engines on the
appropriate departure day for the return trip nor within the tolerance
supplied by the so-called departure window (see Section 10), Eqs. (30)
and (34) yield the delay time

Therefore, the total time for a round-trip LEO-to-LMO mission with delay
time becomes

and on account of (40b) and (41),

9.5. Near-Mirror Property. In addition to the asymptotic parallelism
property noted in Sections 9.1 and 9.2, the optimal trajectories of the


outgoing and return trips have a near-mirror property, which emerges from
the comparison of Eqs. (37a), (37b),(37c) with Eqs. (38a), (38b),(38c).
These angular quantities can be grouped in pairs having nearly the same
modulus but opposite sign for the outgoing and return trips. Also, the
characteristic velocity components, total characteristic velocity, spacecraft
angular travel, and transfer time are the same or nearly the same for the
outgoing and return trips. The implication is that the optimal return
trajectory can be obtained from the optimal outgoing trajectory via a
sequential procedure of rotation, reflection, and inversion; see Ref. 19 for
details. The near-mirror property extends to the restricted four-body
problem the exact mirror property discovered by Miele for the restricted
three-body problem in connection with the flight of a spacecraft in Earth-
Moon space (Ref. 29).

10. Departure Windows for the Outgoing and Return Trips

In Section 6, we formulated the problems of minimizing the
characteristic velocity for the outgoing trip (Problem P1) and return trip
(Problem P2). In these problems, the vectors a,b,c appearing in Eqs. (25)
and (28) have dimensions 7, 2, 5 respectively. The vector a includes the
major parameters governing the transfer; the vector b includes the
components of a that are fixed, namely, the radii of the terminal orbits; the
terminal velocity impulses, transfer time, spacecraft/planet relative phase
angle at departure, and Mars/Earth inertial phase angle difference at
departure.
In this section, we modify the previous problems by assuming that the
departure date is fixed, hence by assuming that is given. In the
new problems, the vectors a, b, c have dimensions 7, 3, 4 respectively as
can be seen by transferring from (26c) to (26b) for the outgoing trip
and from (28c) to (28b) for the return trip. Thus, one can formulate the
following new problems:
Problem P3. For the outgoing trip, given the triplet
minimize the performance index (25) w.r.t the parameters
subject to the constraints (14).
Problem P4. For the return trip, given the triplet
minimize the performance index (27) w.r.t. the parameters
subject to the constraints (20).


While Problems P1 and P2 have two degrees of freedom, Problems P3
and P4 have one degree of freedom. For the outgoing trip, the true
independent variable is the spacecraft/Earth relative phase angle at
departure for the return trip, the true independent variable is the
spacecraft/Mars relative phase angle at departure
The following relations connect the departure dates and the Mars/Earth
phase angle differences:

Therefore, if one sets

and accounts for the baseline optimal trajectory results of Section 9, Eqs.
(43) become

with in days and in degrees. Equations (45) establish a one-to-one
correspondence between the departure date and the Mars/Earth inertial
angle difference at departure.
By varying the departure date, hence by varying the Mars/Earth
inertial phase angle difference at departure, one generates a one-parameter
family of mathematical programming problems, whose solutions form the
so-called departure windows for the outgoing and return trips. Note that, if
in Eq. (45a), the solution of Problem P3 reduces to that of Problem
P1; also note that, if in Eq. (45b), the solution of Problem P4
reduces to that of Problem P2.

10.1. Results. For the outgoing and return trips, Tables 1 and 2 list the
departure date, Mars/Earth inertial phase angle difference at departure,


spacecraft/planet relative phase angle at departure, spacecraft angular
travel, characteristic velocity components at departure and arrival, total
characteristic velocity, flight time, and Mars/Earth inertial phase angle
difference at arrival. In these tables, the central column refers to the
baseline optimal trajectory; the left column refers to the suboptimal
trajectory generated via anticipated departure by nearly 6 weeks; the right
column refers to the suboptimal trajectory generated via delayed departure
by nearly 3 weeks.
Major comments are as follows. For the suboptimal trajectories, the
Mars/Earth inertial phase angle difference at departure increases with
early departure and decreases with late departure; the angular travel and
flight time increase with early departure and decrease with late departure;
the characteristic velocity components and total characteristic velocity
increase with both early and late departures. The above statements hold for
both the outgoing and return trips. Finally, it must be noted that, for the
suboptimal trajectories of both the outgoing and return trips, the
asymptotic parallelism property no longer holds for the departure branch,
but still holds for the arrival branch. On the other hand, the near-mirror
property no longer holds.


11. Comments and Conclusions

From the previous analysis, the following comments and conclusions
emerge.
(i) The Mars mission is difficult because of the large distances
involved. In a round-trip LEO-LMO-LEO mission, the curvilinear
distance traveled along the trajectory exceeds one billion kilometers. At
some point of the trajectory, the spacecraft/Earth distance becomes larger
than the Earth/Sun distance.
(ii) The extremely long journey requires a long flight time, namely,
0.71 years for the outgoing trip, 0.71 years for the return trip, 1.24 years
waiting in LMO, plus a delay time of 2.13 years if the spacecraft is unable
to fire the rocket engines within the departure window tolerance for return.
The total round-trip time is 2.66 years without time delay and 4.79 years
with time delay.
(iii) If one converts the characteristic velocity results into mass ratios


using typical values of the spacecraft structural factor and engine specific
impulse, it can be seen that the required mass ratio for a round-trip LEO-
LMO-LEO mission is about 20. This means that, to return the mass of 1
kg to LEO, we need the mass of 20 kg at the departure from LEO.
If one includes the ascent from the Earth surface to LEO, the required
mass ratio becomes of order 300. If one further includes the ascent from
the Mars surface to LMO, the required mass ratio becomes of order 1000.
This means that, to return the mass of 1 kg to Earth, we need the mass of
1000 kg at the departure from Earth.
(iv) With reference to (iii), the required mass ratios can be decreased
via the use of aeroassisted orbital transfer maneuvers, also called
aerobraking maneuvers. See Refs. 30-33 for recent work on these special
maneuvers.
(v) The best trajectory is the baseline optimal trajectory. For the
outgoing trip, Mars must be ahead of Earth by nearly 44 deg at departure
and the accelerating velocity impulse must be applied 62 deg before the
spacecraft become aligned with the Sun/Earth direction. For the return
trip, Mars must be ahead of Earth by nearly 75 deg at departure and the
accelerating velocity impulse must be applied 141 deg after the spacecraft
becomes aligned with the Sun/Mars direction.
(vi) The baseline optimal trajectory resembles a Hohmann transfer
trajectory, but is not a Hohmann transfer trajectory, owing to the
disturbing influence exerted by the gravity fields of Earth and Mars on the
terminal branches of the trajectory.
(vii) An important property of the baseline optimal trajectory is the
asymptotic parallelism property: For optimal transfer, the spacecraft
inertial velocity must be parallel to the inertial velocity of the closest
planet (Earth or Mars) at the entrance to and exit from deep interplanetary
space. This asymptotic parallelism occurs at the end of the first day and at
the beginning of the last day for both the outgoing and return trips.
(viii) Another property of the baseline optimal trajectory is the near-
mirror property. The return trajectory can be obtained from the outgoing
trajectory via a sequential procedure of rotation, reflection, and inversion.
This property extends to the restricted four-body problem the exact mirror
property found for the restricted three-body problem in connection with
flight of a spacecraft in Earth/Moon space (Ref. 29).
(ix) Departure window trajectories are next-to-best trajectories. They
are suboptimal trajectories obtained by changing the departure date, hence
changing the Mars/Earth inertial phase angle difference at departure. For


the departure window trajectories, anticipated departure yields longer
flight time and wider angular travel; delayed departure yields shorter flight
time and narrower angular travel.
(x) For the departure window trajectories, the asymptotic parallelism
property no longer holds in the departure branch, but still holds in the
arrival branch. On the other hand, the near-mirror property no longer
holds.
(xi) While the present analysis is valid for both robotic and manned
missions, this author believes that, on account of the extremely long flight
times [see (ii)], robotic missions should be preferred for the time being.
Manned missions are extremely difficult and should not be attempted
unless one solves first all the problems that need to be solved to ensure the
survival of the astronauts in space and time.
(xii) It must be emphasized that the present study is preliminary.
Additional studies are under way to account for the ellipticity of the
motion of Earth and Mars around Sun. Further studies are under way to
account for the fact that the Earth and Mars orbital planes are not identical.
Also, aerobraking maneuvers are being considered as a means to reduce
propellant consumption through penetration of the Mars atmosphere in the
outgoing trip and penetration of the Earth atmosphere in the return trip
(Refs. 30-33).

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4

Design and Test of an Experimental Guidance
System with a Perspective Flight Path Display
G. SACHS1

Abstract. Design issues of a guidance system displaying visual
information in a 3-dimensional format to the pilot for improving
manual flight path control are considered. A basic concept of such
a synthetic vision system is described, yielding an integrated
presentation of the command flight path and the terrain,
supplemented by other guidance elements. The imagery is
generated by a computer in real time with an adequate update rate,
using attitude and position data from a precision navigation
system. This basic synthetic vision system was flight tested in an
experimental program consisting of several test series, with
demanding flight tasks aiming at different control aspects. The
flight test results show that the synthetic vision system enabled the
pilot to control precisely the aircraft and hold it on the command
trajectory. Furthermore, an extended 3-dimensional guidance
display concept is considered which employs a predictor indicating
the future position of the aircraft at a specified time ahead. Design
issues are described for achieving a predictor aircraft system
requiring minimum pilot compensation. Results from pilot-in-the-
loop simulation experiments are presented which provide a
verification of the design considerations.

Professor and Director, Institute of Flight Mechanics and Flight Control, Technische
Universität München, 85747 Garching, Germany.

105

106 G. Sachs

Key Words. Perspective flight path display, synthetic vision, flight
path predictor, manual flight path control, aircraft guidance.

Nomenclature

e = error,
g = acceleration of gravity,
K = gain,
= roll moment due to roll control input,
s = Laplace operator,
T = time constant,
Y(s) = transfer function,
y = lateral coordinate,
= perturbation of y,
= roll control,
= damping ratio,
= effective time delay,
= roll angle,
= azimuth angle,
= frequency.

1. Introduction

Innovative approaches for the cockpit instrumentation of aircraft are
displays which present guidance information in a 3-dimensional format to
the pilot. They show the future flight path in a perspective form and may
additionally depict a terrain imagery. Such displays, which are known as
tunnel or highway-in-the-sky displays, offer a fundamental enhancement
in the visual information of the pilot because they provide status and
command information not only of actual but also of future flight
situations. Furthermore, perspective flight path displays present the

Experimental Guidance System with Perspective Flight Path Display 107

information in a descriptive format and allow holistic perception. The
visual information can be perceived intuitively and directly by the pilot
and the scanning workload decreased. As a result, the mental effort for
reconstructing the spatial and temporal situation may be reduced
substantially when compared with current instrumentation.
Results from recent research including theoretical investigations as
well as simulation experiments and flight tests show that significant
improvements in aircraft guidance and control can be achieved with
displays presenting the flight path and other relevant information in a 3-
dimensional format (Refs. 1-18). The flight test verification includes the
worldwide first landing of an aircraft with a pictorial display presenting 3-
dimensional guidance information (synthetic vision) as the only visual
information for the pilot (Refs. 14, 15).
It is the purpose of this paper to describe design issues of perspective
flight path displays and to present experimental results from simulation
and flight tests.

2. Basic Concept of Three-Dimensional Guidance Display

The basic concept of the 3-dimensional guidance display comprises an
integrated presentation of the flight path and the terrain, supplemented by
other guidance elements. Such a display featuring synthetic vision
includes the following constituents (Fig. 1): 3-dimensional guidance
information; pictorial presentation of outside world; precision navigation.
Central element of the 3-dimensional guidance information is the
perspective flight path presentation in the form of a tunnel (Fig. 2). Further
guidance elements of primary significance are displayed in an integrated
manner. Indication of the command flight path provides the pilot with a
preview of the future trajectory. With command information and preview
available, the pilot can use this preview to structure a control feedforward.
This is illustrated in Fig. 3, which shows a simplified model for describing
general pathways of the human controller operating on visually sensed
inputs and exerting manual control outputs. Different control modes are
possible, one of which is compensatory control applied as a closed-loop
control for regulation tasks. The other control mode is pursuit/preview
control which is possible because of command information and preview.


The pictorial presentation of the outside world comprises an image of
the terrain, including all relevant information about its elevation and
features. This is illustrated in Fig. 4, which shows the integrated
presentation of the outside world image and the guidance tunnel. Two
groups of data are used for generating the outside world image: terrain
elevation and feature analysis data. The terrain elevation data are
referenced to a grid structure the elements of which have a size of 3" × 3"
or 1" × 1" (Fig. 5). A grid element represents an area of about 90 m × 60
m (or 30 m × 20 m) at the geographical latitude of the areas where the
flight test took place. Three data groups are applied for describing the
terrain features (Fig. 6): point features (buildings, bridges, power line
pylons, etc.); linear features (roads, railways, rivers, etc.); areal features
(cities, forests, lakes, etc.).
A special treatment of terrain elevation and features is applied for
areas where the aircraft operates close to the ground, like airports. It yields
a precise modeling as regards location, elevation, dimensions, objects, etc.
The precision navigation system provides the synthetic vision
computer with position and attitude data (Fig. 1). This is necessary for
generating an image according to the actual field of view of the pilot. In
the flight tests, a high-precision navigation system was applied using


differential global positioning and inertial sensor navigation data. The
navigation system was operated in local and wide area DGPS modes for
transmitting the differential correction data (Fig. 7). The local area DGPS
mode was used in flight tests for terminal flight operations (approach and
landing) using a customized ultra high frequency data link. The GPS
ground reference station was located close to the runway. The wide area
DGPS mode was applied in nonterminal flight tests (flights in river
valleys and mountainous areas) using a low-frequency transmitting
technique for providing the correction signal. Because of the low-
frequency transmitting technique, it was possible to receive the correction
data without having to cope with hiding effects due to terrain formations.
The technique was developed by the Institute of Applied Geodesy in
Potsdam, Germany (Ref. 19).

112 G. Sachs

3. Flight Test Results for Basic Three-Dimensional Guidance Display

A series of flight tests was performed aiming at a wide range of
guidance applications of the 3D-guidance display system, featuring the
above guidance information and terrain imagery. Demanding control tasks
were specified and investigated. The test program consisted of five flight
test series:
(i) precision approach and landing flight tests, Braunschweig Airport,
Germany, October 10-14, 1994;
(ii) low-level flight tests in a highly curved, narrow river valley,
Altmühl river, Germany, December 12-16, 1994;
(iii) curved and steep approaches in mountainous area, Lugano airport,
Switzerland, July 31 - August 4, 1995;
(iv) curved/steep/short approaches and low-level and terrain-following
flights in a mountainous area, Offenburg/Schwarzwald, Germany, March
18-22, 1996;
(v) curved/steep approaches and curved trajectory-following flights in
a mountainous area, Freiburg/Schwarzwald, Germany, July 7-10, 1997.


An overview of the flight test areas is provided by Fig. 8, which shows
the locations in Germany and Switzerland. The place of the ground
reference station for the wide area DGPS mode is also depicted.
The vehicle which is used in the flight test program is a twin engine
Dornier 128; it is operated by the Institute of Flight Guidance and Control
of the Technische Universität Braunschweig as a research aircraft (Fig. 9).
The aircraft is equipped with a high precision navigation system which
was developed by this Institute. The high navigation performance is
achieved by coupling differential global positioning and inertial sensor
systems to yield an integrated precision navigation system (Ref. 20). In
addition, computer and filter algorithms including error modeling are
applied. Thus, it is possible to achieve a high precision for static as well as
dynamic behavior.

114 G. Sachs

Results representative for the flight tests are presented in the
following. They are from the low-level flight tests in the Altmühl river
valley. The test course depicted in Fig. 10 shows that the Altmühl river
area represents a demanding test environment because of the highly
curved and narrow river valley with steep banks. The control task was to
follow precisely the command trajectory, which was referenced to the
course of the river, at an height of 100 m above the river as authorized by
the flight safety agency.
The flight test results presented in Figs. 11 and 12 show that the pilot
followed precisely the command trajectory, with only small deviations in
both the vertical and lateral directions. This holds generally for the whole
of the flight test course of about 70 km, and particularly for those sections
where the control tasks were very demanding in the vertical or lateral
direction. The command trajectory was indicated in the 3-dimensional
guidance display by a tunnel image, as shown in Fig. 4 for a flight
condition of the tests in the Altmühl river valley. From the results
presented in Figs. 11 and 12, it follows that the aircraft stayed well within
the tunnel.
For the motion in the vertical direction, there are three sections of
particular interest because evasive maneuvers were necessary (Fig. 11). In
two sections, electrical power lines intersect the river valley. This was
shown in the 3-dimensional guidance display, with a corresponding
change in the course of the tunnel. In a third section, there is a river bend


so tight that it could not be followed by the aircraft. For this part of the
trajectory, an evasive maneuver was specified according to which the pilot
left the river valley, flew over the bank at the riverside and entered again
the river valley afterward.

116 G. Sachs

4. Three-Dimensional Guidance Display with Predictor

An improvement in flight path control is possible by a predictor which
indicates the future position of the aircraft at a specified time ahead in the
3-dimensional guidance display (Fig. 13). This is because the pilot is
provided with precise information about the future aircraft position in
relation to the command flight path. As shown in Fig. 13, the deviation of
the predictor from the reference cross section of the command flight path
at the prediction time ahead yields an accurate error indication. The pilot
can act in response to this error for minimizing flight path deviations in
compensatory control mode.
In general, the overall predictive system consists of the 3-dimensional
guidance display with the tunnel and the predictor, the pilot and the
aircraft. The tunnel and the predictor present command and status
information about the present and the future. There are pilot-centered
requirements which result from the presence of the human operator in the
control loop, with the objective to achieve an overall predictive system
requiring minimum pilot compensation.
For achieving this objective, the predictive system should be
constructed to require no low-frequency lead equalization for the pilot and
to permit pilot-loop closure over a wide range of gains. This requirement
can be met when the equalizations and gains are selected so that the
effective transfer characteristic of the controlled element, the predictor-
aircraft system approximates a pure integration over an
adequately broad region centered around the pilot-predictor-aircraft
crossover (Refs. 21,22), i.e.,


This relation describes the desired dynamic characteristics of the
predictor-aircraft system as the controlled element. It can be used as key
requirement for designing the predictor to achieve appropriate dynamic
characteristics of the closed-loop pilot-predictor-aircraft system.
Besides this manual control-related predictor issue, there is another
point which is concerned with the role of the predictor as an indicator of
the future aircraft position. A realistic indication of the future aircraft
position can be considered a requirement for face validity according to
which the status information presented by the predictor in the 3-
dimensional guidance display should correspond to the actual situation.
Thus, geometric and kinematic relations come into consideration for
describing the continuation of the flight path to which the predicted
position can be referenced. This is illustrated in Fig. 14, which shows a
model for describing the continuation of the flight path in the lateral
direction, with particular reference to the situation at the prediction time
ahead.

118 G. Sachs

The pilot-centered requirement for best transfer characteristics of the
predictor-aircraft system, supplemented by the face validity
considerations, forms the basis for the predictor control law. With
reference to the block diagram in Fig. 15, the predictor law for lateral
flight path control can be constructed to yield

where is the prediction time related to the predictor position. Selecting
for the roll rate gain

and applying the aircraft dynamics model valid for the frequency region of
concern


the following relation for the predictor-aircraft transfer function is
obtained:

where

120 G. Sachs

From Eq. (5), it follows that there is a K/s frequency region above
By proper selection of the prediction time it is
possible to construct an adequately broad K/s frequency region centered
around the pilot-predictor-aircraft crossover. As a result, the objective of
an overall predictive system requiring minimum pilot compensation is
achieved. The described K/s properties are illustrated in Fig. 16, which
shows the frequency response characteristics of a predictor-aircraft
system. The data shown in Fig. 16 relate to an aircraft used in pilot-in-the-
loop simulation experiments; the relevant results are presented in a
subsequent section.
A further issue is closed-loop stability of the pilot-predictor-aircraft
system. In Fig. 17, the stability properties are evaluated with the root locus
technique yielding results of rather general nature. The following pilot
model valid for K/s characteristics is applied:


Basically, Fig. 17 shows that the system is stable for pilot gains above a
certain value Since the gain for pilot-system crossover is
significantly greater than it follows from the root locus result that
the pilot-predictor-aircraft system is stable. Furthermore, Fig. 17 shows
that there are basically two closed-loop modes, one primarily related to
path and the other to attitude motions.

5. Results of Simulation Experiments for Three-Dimensional
Guidance Display with Predictor

An experimental investigation of the described 3-dimensional
guidance display with predictor was the subject of pilot-in-the-loop
simulation tests. Five pilots with different professional background (airline
pilots, private pilot, student pilot) performed the simulation experiments
which were carried out at a fixed-base simulator. The layout of the 3-
dimensional guidance display developed for the experimental program
corresponds to the configuration shown in Fig. 13. The tasks of the pilot

122 G. Sachs

was to follow a curved trajectory (Fig. 18), indicated as command flight
path in the 3-dimensional guidance display. The sequence of the turns was
altered in order to avoid familiarization of the pilots with a fixed
trajectory. In the simulation experiments, a nonlinear six degree-of-
freedom aircraft model was used, which can be regarded as representative
of a small twin jet engine aircraft.
A primary purpose of the simulation experiments was to investigate
the effect of the prediction time because of its significance for the
K / s frequency region. Simulation results on predictor position control
are presented in Fig. 19 (box plot technique, 95 % confidence interval).
From Fig. 19, it follows as a basic result that the predictor position is
controlled effectively by the pilot, with rather small deviations from the
command flight path. Concerning the prediction time it turns out that
it has a substantial effect on the predictor position control, showing a
decrease of the predictor error as is decreased and vice versa. Control
activity results are depicted in Fig. 20 which shows the correcting aileron

124 G. Sachs

commands given by the pilot. The effect of the prediction time is
again significant, showing now an increase of the control activity as is
decreased.
The described effects of the prediction time on the predictor
position error and control activity can be attributed to pilot-loop closure
behavior. Reference is made to Fig. 21, which shows that a decrease of
yields a downward shift of the K / s frequency region. For loop closure, the
downward shift of the K / s frequency region requires an increase of the
pilot gain. As a consequence, the predictor position deviations are reduced
when is decreased. Furthermore, the pilot control activity is increased.
The predictor, which indicates the position at the prediction
time ahead, is basically related to a future state. But it is also an
efficient means for controlling the current position y(t). Using
the relation between the current position error
and the future position error (predictor error) can
be expressed as

Accounting for it follows that

This relation shows that the current position error is basically smaller
than the predictor error The reduction of relative to
increases significantly in the frequency region above Furthermore,
both errors approach zero in steady-state reference conditions. This is
because


These results were confirmed by pilot-in-the-loop simulation
experiments. This is illustrated in Fig. 22, which shows that the deviations
of the current position are smaller than those of the predictor position as
depicted in Fig. 19.

6. Conclusions

A guidance display is considered providing the pilot with status and
command information in a 3-dimensional format for current and future

126 G. Sachs

flight situations. The basic concept features a computer-generated imagery
of the command flight path, other guidance information, and the outside
world. The required attitude and position data for a correct adjustment of
the displayed imagery are transferred from a precision navigation system
using differential global positioning and inertial sensor data. A series of
flight tests aiming at a wide range of applications of the 3D-guidance
display were performed, with demanding control tasks for the pilots like
precision approach and landing, low-level flights in highly curved, narrow
river valleys, curved/steep/short approaches and low-level and terrain-
following flights in mountainous areas. The flight test results show that the
pilot controlled precisely the aircraft and held it on the command
trajectory.
An extended display concept for presenting guidance information in a
3-dimensional format features a predictor which indicates the future
position of the aircraft at a specified time ahead. For best results in terms


of performance and workload, the predictive system should be designed
such that the controlled predictor-aircraft element requires minimum pilot
compensation. A predictor control law is developed for achieving this
objective. Results from pilot-in-the-loop simulation experiments
concerning significant predictor law parameters were performed, yielding
verification of the design considerations.

References

1. THEUNISSEN, E., Integrated Design of a Man-Machine Interface
for 4D-Navigation, PhD Thesis, Delft University of Technology,
Delft, Netherlands, 1997.
2. THEUNISSEN, E., and MULDER, M., Availability and Use of
Information in Perspective FlightPath Displays, Proceedings of the
AIAA Flight Simulation Technologies Conference, pp. 137-147,
1995.
3. GRUNWALD, A.J., ROBERTSON, J.B., and HATFIELD, J.J.,
Experimental Evaluation of a Perspective Tunnel Display for Three-
Dimensional Helicopter Approaches, Journal of Guidance, Control,
and Dynamics, Vol. 4, No. 6, pp. 623-631, 1981.
4. GRUNWALD, A.J., Tunnel Display for Four-Dimensional Fixed-
Wing Aircraft Approaches, Journal of Guidance, Control, and
5. GRUNWALD, A.J., Predictor Laws for Pictorial Flight Displays,
Journal of Guidance, Control, and Dynamics, Vol. 8, No. 5, pp. 545-
552, 1985.
6. GRUNWALD, A.J., Improved Tunnel Display for Curved Trajectory
Following: Control Considerations, Journal of Guidance, Control,
7. GRUNWALD, A.J., Improved Tunnel Display for Curved Trajectory
Following: Experimental Evaluation, Journal of Guidance, Control,

128 G. Sachs

8. H ASKELL , I.D., and W ICKENS , C.D., Two- and Three-
Dimensional Displays for Aviation: A Theoretical and Empirical
Comparison, International Journal of Aviation Psychology, Vol. 3,
No. 2, pp. 87-109, 1993.
9. WICKENS, C.D., FADDEN, S., MERWIN, D., and VERVERS,
P.M., Cognitive Factors in Aviation Display Design, Proceedings of
the 17th AIAA/IEEE/SAE Digital Avionics Systems Conference,
Bellevue, Washington, 31 October – 6 November 1998, 0-7803-5086-
3/98, 1998.
10. HELMETAG, A., MAYER, U., and KAUFHOLD, R., Improvement
of Perception and Cognition in Spatial Synthetic Environment,
Proceedings of the 17th European Annual Conference on Human
Decision Making and Manual Control, Valenciennes, France, 14-16
December 1998, pp. 207-214, 1998.
11. LENHART, P.M., PURPUS, M., and VON VIEHBAHN, H.,
Flight Testing of Cockpit Displays with Sinthetic Vision, Yearbook
1998-I, German Society of Aeronautics and Astronautics, pp. 707-
713, 1998 (in German).
12. FUNABIKI, K., MURAOKA, K., TERUI, Y., HARIGAE, M., and
ONO, T., In-Flight Evaluation of Tunnel-in-the Sky Display and
Curved Approach Pattern, Proceedings of the AIAA Guidance,
Navigation, and Control Conference, pp. 108-114, 1999.
13. MULDER, M., Cybernetics of Tunnel-in-the-Sky Displays, Delft
University Press, Delft, Netherlands, 1999.
14. SACHS, G.; and MÖLLER, H., Synthetic Vision Flight Tests for
Precision Approach and Landing, Proceedings of the AIAA
Guidance, Navigation, and Control Conference, pp. 1459-1466, 1995.
15. SACHS, G., DOBLER, K., and HERMLE, P., Flight Testing
Synthetic Vision for Precise Guidance Close to the Ground,
Proceedings of the AIAA Guidance, Navigation, and Control
Conference, pp. 1210-1219, 1997.
16. SACHS, G., DOBLER, K., and THEUNISSEN, E., Pilot-Vehicle
System Control Issues for Predictive Flight Path Displays,


Proceedings of the AIAA Guidance, Navigation, and Control
Conference, pp. 574-582, 1999.
17. SACHS, G., Flight Path Predictor for Minimum Pilot Compensation,
Aerospace Science and Technology, Vol. 3, No. 4, pp. 247-257, 1999.
18. SACHS, G., Perspective Predictor/Flight Path Display with
Minimum Pilot Compensation, Journal of Guidance, Control, and
19. D ITTRICH , J., K ÜHMSTEDT , E., L ECHNER , W., et al,
Experiments with Real Time Differential GPS Using a Low
Frequency Transmitter in Mainflingen, Germany: Results and
Experiences, Paper Presented at EURNAV-94 Land Vehicle
Navigation, Dresden, Germany, 14-16 June, 1994.
20. VIEWEG, S., and SCHÄNZER, G., Precise Flight Navigation by
Integration of Satellite Navigation Systems with Inertial Sensors,
Yearbook 1992-I, German Society of Aeronautics and Astronautics,
pp. 171-177, 1992.
21. MCRUER, D.T., Pilot Modeling, AGARD Publication LS-157,
Chapter 2, pp. 1-30, 1988.
22. HESS, R. A., Feedback Control Models: Manual Control and
Tracking, Handbook of Human Factors and Ergonomics, 2nd Edition,
Edited by G. Salvendy, Wiley, New York, NY, pp. 1249-1294, 1997.

5

Neighboring Vehicle Design for a Two-Stage
LaunchVehicle1
K. H. WELL2

Abstract. The paper presents numerical results of a study concerned
with the simultaneous optimization of the ascent trajectory of a two-
stage launch vehicle and some significant vehicle design parameters.
Besides the trajectory design, models are given that relate (i) the
propulsion mass to a desirable increase in the mass flow for the
rocket engines and (ii) the structural mass of the fuel tanks to a
desirable increase in the propellant mass. Using these models, it is
shown how the example vehicle should be modified in order to carry
a higher payload into an Earth escape orbit. It is shown that an
overall increase of the vehicle liftoff mass of about 4% will result in
a payload increase of about 11%.

Key Words. Trajectory optimization, launch vehicles, concurrent
engineering.

1
The author gratefully acknowledges the financial support provided by the European
Space Technology Center (ESTEC) through its Contract Monitor Klaus Mehlem.
2
Professor and Director, Institute of Flight Mechanics and Control, University of

131

132 K. H. Well

1. Introduction

Reference 1 gives an overview of a trajectory optimization software
(ASTOS) which has been developed over the past ten years for the
European Space Agency. This software enables a user to specify a
particular launch or reentry vehicle and a particular mission solely by data.
It generates an initial estimate for the solution automatically, and it assists
the user in the solution process via a user interface. Among many other
features, it contains a particular capability which links the vehicle design
to trajectory optimization and allows the combined optimization of the
trajectory and the vehicle parameters. The purpose of this paper is to
demonstrate this capability taking as an example the ascent of a two-stage
launch vehicle into an Earth escape orbit, while simultaneously answering
the question of how the nominal vehicle should be modified in order to
increase the payload in that orbit.
Traditionally, vehicle design is mostly separated from atmospheric
trajectory optimization. At most, atmospheric trajectories are simulated
using particular guidance laws during the design process. However, in
recent years, attempts have been made to link the task of finding the best
ascent trajectory to the task of designing the vehicle size.
Reference 2 presents a design tool to this end. There, the optimization
is organized hierarchically: The design optimization is performed in an
outer loop; the trajectory optimization is performed in an inner loop. In the
outer loop, the trajectory is frozen; in the inner loop, the design is frozen.
The software has been applied successfully to reentry vehicle design as
well as to design modifications of a winged launcher with air breathing
propulsion.
In principle, the design process must take into consideration that, when
changing the geometry of the vehicle, not only the mass data change but in
particular the aerodynamic data do.
Therefore, once a particular change in geometry has occurred,
appropriate aerodynamic methods have to be used to recalculate the
aerodynamic coefficients. Depending on the required accuracy, more or
less sophisticated aerodynamic codes have to be used which may lead
easily to rather large amounts of computing times.
In this paper, it is assumed that the modifications from a reference
design are small enough such that a recalculation of the aerodynamic
coefficients is not needed. In addition, the diameter of the cylindrical
vehicle stages is kept constant. This leads to the assumption that the drag

Neighboring Vehicle Design for a Two-Stage Launch Vehicle 133

forces will not be affected by the modifications. The essential design
modifications are changes in the engine masses and tank sizes for the two
stages. By limiting the modifications to 20% from the nominal values, it is
conjectured that the results are realistic and can serve as guidelines for
eventual modifications.

2. Reference Vehicle

Figure 1 shows the reference vehicle. It consists of two main stages,

134 K. H. Well

the lower cryogenic stage with the H155 engine and the two boosters P230
and the upper stage with the L9 engine. Figure 1 shows the version of the
vehicle carrying two payloads; SPELTRA is the device which holds and
separates the two payloads once orbital target conditions have been
achieved by the upper stage. Table 1 contains the mass flow for a P230
booster and the drag coefficient of the vehicle (see also Fig. 2).


Table 2 contains the masses of the various vehicle stages. Included in the
total structural mass are 1401 kg for the vehicle equipment bay which is
attached to the L9 stage and 1935 kg for the payload fairing which is
ejected after burnout of the main stage. Table 3 gives the engine data, the
x designating that these data are for experimental engines. About 0.8% of
the fuel (unburned propellant) for both the H155x engines and the L9x
engines cannot be utilized in the combustion process and thus does not
contribute to the propulsion of these two engines.

136 K. H. Well

3. Mathematical Model of the Rocket Vehicle

3.1. Equations of Motion. The equations of motion of the center of
mass over an oblate, rotating Earth are taken from Ref. 3. The state
variables are: Inertial velocity components (see Fig. 3), position
variables and appropriate equations for the mass change of the
vehicle during the ascent


where is the Earth rotational speed and is the vectorial sum of all
the external forces. The subscript L indicates inertial variables in the local
horizontal coordinate system. The external forces are the thrust forces

the aerodynamic forces

138 K. H. Well

and the gravitational forces

The thrust force for the boosters is a function of time as given in Table 1;
is the gravitational acceleration at sea level; and are
the specific impulses, mass flows, and engine exit areas of the various
propulsion systems as given in Table 3; p is the ambient pressure as a
function of altitude. In this paper, a special pressure profile for the launch
site Kourou (French Guyana) is taken, but the model of the US Standard
atmosphere might be taken as well; q and are the dynamic pressure
and reference area for the aerodynamic forces. The two
components in Eq. (3) containing the partial derivatives of the side forces
and normal forces of the vehicle with respect to angle of attack and
sideslip angle are not available for the reference vehicle. Therefore, no
aerodynamic normal forces or side forces are computed in the model. The
symbols are the Earth gravitational constant and the oblateness
and triaxiality constant of the Earth gravitational potential, is the
equatorial radius of the Earth; their values are given, for example, in Ref.
3. The subscripts B in equations (2) and (3) describe the forces in body
axes. To transform them into the local horizontal axes, the transformation
relations


are to be applied where are the azimuth, pitch, and roll angles
describing the launchers attitude with respect to the local horizontal
system. The transformation matrices are

It is assumed that the vehicle will not roll during launch and, therefore,
is set. The same transformation applies to the aerodynamic forces in
(3). With these definitions, the controls for the ascent problem are the
pitch and yaw angles.

3.2. Mass Models for Engine and Tank Sizing. For the task at hand,
it is assumed that the boosters are given and are not to be modified.
Modifiable are the two rocket engines and the size of the tanks for the fuel
of the main and the upper stages. By scaling the engine up, the mass flow,
nozzle area, and thrust of a particular engine can be increased causing, of
course, an increase in the engine mass. Simple models describing the
interrelation of these data are taken from Ref. 4. Given a sizing parameter
the mass flow is modeled as

where is the reference mass flow and is the throttle
setting (an additional control). The nozzle exit area is

the maximum thrust is

the actual thrust itself is

With these definitions, the engine mass can be calculated as

140 K. H. Well

where a,b are correlation coefficients. Similarly for tank sizing, the
propellant mass and the structural mass are

Here, is the reference value for the propellant mass and is the
reference value for the structural mass; b is another correlation
coefficient. As mentioned in the introduction, the tank size is to be varied
assuming a constant diameter. Then, the change in tank volume is
computed from

with as mean density of the fuel. Assuming a cylindrical shape of the
tank, this results in a change in length of the tank,

Tables 4 and 5 present the data used in the subsequent calculations. Figure
4 shows how the dry engine masses change with increasing fuel flow and
how the tank masses change with increasing amounts of fuel. Altogether
there are four design parameters to be chosen in the
optimization.

142 K. H. Well

4. Multiphase Optimal Control Problem

The main constraints of the trajectory optimization problem are the
differential system described by equations (1)-(6) and the additional
differential equation for the change in mass,

where is the fuel flow in the ith phase, i = 1...4. Table 6 contains the
mass flow data for each phase. The first phase consists of the simultaneous
burn of the main engine and the two boosters, the second and third phase
are with the main engine only, and the fourth phase is the burning of the
upper stage engine. The booster burn time is fixed, so is the overall burn
time of the main engine. The times for the fairing jettisoning and for the
L9 engine cut-off are kept free in the optimization process.

4.1. Initial Conditions. The vehicle is supposed to be launched from
Kourou (French Guyana). The initial values for the position are altitude
geographical longitude and latitude of Kourou. The initial velocity
components are taken to be The liftoff mass is computed
according to

Here, the subscripts s,p,e designate structural, propellant, and engine
masses, the subscripts H155, P230, L9 identify the stage association.
VEB stands for vehicle equipment bay. By defining a sizing parameter


according to Section 3.2 for the amount of propellant to be used in each of
the main stage and the upper stage as well as in each of the engines, there
are altogether five parameters to be optimized, the fifth being the payload
mass In this way, the initial mass is a function of these five
parameters.

4.2. Target and Intermediate Conditions, Cost Function. In order to
define the final boundary conditions, a few auxiliary variables need to be
defined (see Ref. 3). The inertial path inclination, inertial azimuth, and
velocity are computed as

Furthermore, with the parameter f is defined
as

and the semimajor axis and eccentricity are defined as

These parameters take on different values for different kind of conic
sections,

From orbital mechanics, it is known that the true anomaly can be
computed from

144 K. H. Well

The orbital elements can be calculated by applying the laws of
spherical trigonometry to the triangle with the sides in
Fig. 5. Here, is the inertial longitude of the vehicle at a particular time.
It can be defined as

where is the geocentric longitude at the time of launch. From this
figure, one obtains


Since the true anomaly is known from (23), these three equations can be
used to determine the unknown orbital elements.
A hyperbolic target orbit can be defined by its excess velocity

its true anomaly for

and the declination of its asymptote for

By specifying the excess velocity and the declination of the asymptote, the
semimajor axis and the inclination are defined via (28) and (25) for a
given velocity vector, the parameter f and the eccentricity are calculated
from (19) for a given value of R, and are computed from (29),
(26), and (27). As intermediate conditions, the perigee altitude

and the heat flux

are needed.
Finally, the cost function for the optimal control problem is to
maximize Table 7 summarizes the trajectory optimization
problem.
The design parameters are with the
subscripted notation as described above. Of course, the integrals over the
mass flow for each vehicle stage must satisfy the conditions

146 K. H. Well

5. Solving the Trajectory Optimization Problem

The ASTOS software has been used to solve the above described
problem. Inside the software, two methods are implemented; one is a
direct multiple shooting method, first suggested in Ref. 5; the other
method is based on direct collocation; see Ref. 6. Both methods transcribe
the continuous optimal control problem into high parametric nonlinear
programming problems which are solved by standard software. Inside
ASTOS, two nonlinear programming solvers are implemented: A
sequential linear least squares quadratic program solver (SLLSQP, Ref. 7)
and a sparse nonlinear optimization solver (SNOPT, Ref. 8).

5.1. Initial Guess. In order to generate the initial time histories for the
controls and the states, a guidance law based on the required velocity


concept (Ref. 9) is used. Although in principle only applicable for
elliptical target orbits, it can be used for the above problem as an
approximation by choosing a sufficiently large apogee altitude for the
makeshift target orbit.
For given orbital parameters of such a highly eccentric target
orbit, one computes the reference velocity, that is, that particular velocity
which the vehicle should have in the desired orbit at that radius vector.
The components in a local horizontal system are

where and

with

are the inertial elevation and the azimuth angles of the required
velocity vector with respect to the local horizontal system; see Fig. 3. The
difference between the required velocity vector and the actual velocity
vector is

148 K. H. Well

This velocity difference must ultimately be zero. One can show (see e.g.
Ref. 10) that, by accelerating the vehicle in the direction of this goal
can be achieved.
According to Fig. 6, the vehicle acceleration is The direction and
magnitude of are obtained from

where is the effective gravitational acceleration, with
From the figure, one gets

and after some manipulations, as the solution of the quadratic equation

with


where the appropriate sign has to be chosen. The yaw direction of the
thrust vector is simply

With this guidance law, a nominal trajectory can be obtained by
integrating the equations of motion from the initial state. The complete
initial guess is obtained in the following three major steps,
Step 1: Vertical ascent followed by a constant pitch rate until a
prescribed pitch attitude is obtained. Flight with this attitude
until the angle of attack is zero.
Step 2: Gravity turn, that is, flight with zero angle of attack until some
user specified event or time, usually until the burnout of the
major stage.
Step 3: Guidance steering according to the above procedure until a
specified time or until the desired orbit has been reached.

5.2. Optimal Solutions. Figure 7 shows the state time histories
and as well as the control time histories and for both the
initial guess and the nominal solution, that is, the solution with fixed
values of the design parameters. These nominal values are given in Table
8 together with the optimal values. Figure 8 shows the altitude, ground
track, inertial speed, and osculating perigee altitude of the H155x stage.
These as well as other state and control time histories of the nominal case
do not differ much from those of the optimal case. Both altitude and speed
are somewhat smaller for a given time, due to the fact that the vehicle is
heavier initially. Both trajectories have to satisfy the intermediate


boundary condition on the osculating perigee altitude at 580 sec. The
overall flight time is approximately the same. The main difference
between the nominal and the optimal case is shown in Table 8. The upper
part of the table contains the structural, propulsive, and engine masses for
both the nominal case and the optimal case. The VEB mass and fairing
mass are included in the overall structural mass. The five design parameter
values are given and can be compared to the nominal values. Due to the
increased engine and fuel mass, the main stage needs to be extended by
approximately 5m, while the geometric modifications of the upper stage
are small. The lower part of the table gives the resulting changes in
percent compared to the nominal design. The order of magnitude of the
changes is between 15 to 20% for each stage; the changes are rather small
for the vehicle altogether, since the booster mass contributes significantly
to the overall mass of the vehicle. The overall increase in engine mass is
17% with respect to the engine masses without boosters. The increase in
liftoff mass is about 4%; the increase in payload is about 11%.

152 K. H. Well

6. Conclusions

The paper addresses the simultaneous optimization of both the ascent
trajectory and some typical vehicle design parameters of a two-stage
launch vehicle. It is shown that the modified vehicle does not influence the
ascent trajectory to a great extent, which is not surprising, since for rocket
propelled conventional launch vehicles the performance of the propulsion
system depends weakly on the atmospheric conditions through the back
pressure. This is due partly to the modeling assumptions that the diameter
of the vehicle geometry has been held constant and that the dynamic lift of
the vehicle has not been taken into account. By removing these
restrictions, a greater interdependence between design and trajectory is
conjectured to be observed. The approach presented here is applicable to
launch vehicles with airbreathing propulsion as well where the interaction
between design and trajectory is much more predominant.

References

1. W ELL, K. H., M ARKL, A., and M EHLEM. K., ASTOS: A
Trajectory Analysis and Optimization Software for Launch and
Reentry Vehicles, Paper IAF-97-V4.04, 48th International
Astronautical Congress, Turin, Italy, 1997.
2. R AHN, M., S CHOETTLE, U. M., and M ESSERSCHMID, E.,
Multidisciplinary Design Tool for System and Mission Optimization
of Launch Vehicles, 6th AIAA/NASA/ISSMO Symposium on
Multidisciplinary Analysis and Optimization, Bellevue, Washington,
USA, 1996.
3. BUHL, W., EBERT, K., and WOLFF, H., Technical Report 2,
Modelling: Advanced Launcher Trajectory Optimization Software
Technical Documentation, European Space Technology and Research
Center, Nordwijk, Netherlands, Contract 8046-88-NL-MAC, 1992.
4. S CHÖTTLE , U., and R AHN , U., Fahrzeugmodelle für
Sensitivitätsstudien konventioneller Trägerraketen (Vehicle Modelling
of Conventional Launch Vehicles for Sensitivity Analysis), Institute
for Space Systems, University of Stuttgart, Stuttgart, Germany,
Report IRS 95-IB-11, 1995 (in German).


5. BOCK, H. G., and PLITT, K. J., A Multiple Shooting Algorithm for
Direct Solution of Optimal Control Problems, Proceedings of the 9th
IFAC World Congress, Budapest, Hungary, pp. 243-247, 1984.
6. HARGRAVES, C. R., and PARIS, S. W., Direct Trajectory
Optimization Using Nonlinear Programming and Collocation,
Journal of Guidance, Control, and Dynamics, Vol. 10, pp. 338-342,
1987.
7. KRAFT, D., TOMP - FORTRAN Modules for Optimal Control
Calculations, VDI Fortschrittsberichte, Volume 8, No. 254, 1991.
8. GILL, P. E., MURRAY, W., and SAUNDERS, M. E., Users Guide
for SNOPT 5.3: A Fortran Package for Large-Scale Nonlinear
Programming, Department of Mathematics, University of California,
San Diego, Report NA 97-5-4, 1997.
9. BATTIN, R., Astronautical Guidance, McGraw-Hill, New York,
NY, 1964.
10. GRIMM, W., and WELL, K.H., Guidance, Lecture Notes, Institute
for Flight Mechanics and Control, University of Stuttgart, 1994 (in
German).

6

Controller Design for a Flexible Aircraft
M. HANEL1 and K. H. WELL2

Abstract. The paper presents an overview of modeling the dynamic
behavior of a large four-engine flexible aircraft and considers some
of the options for control system design. The first part describes how
to build an integral model, which can be used for simulating the
rigid motion as well as the flexible motion of the aircraft. The result
is a system of nonlinear equations of motion. The second part
analyzes the dynamic properties of a sample aircraft by considering
the linearized equations of motion for flight in a vertical plane at
several operating points in the flight envelope. Here, it is shown how
the eigenfrequencies of the rigid body and the elastic motion change
with the load and flight conditions. In the third part, three options
for control system design are discussed: (i) a conventional SAS
controller, which does not influence actively the elastic behavior;
(ii) an output feedback controller; and (iii) a robust controller. It is
concluded that, using an integral controller, certain flying quality
criteria can be met and damping of all the elastic modes can be
improved.

Key Words. Flight control, aeroservoelasticity, flexible aircraft.

Research Scientist, Institute of Flight Mechanics and Control, University of Stuttgart,
70550 Stuttgart, Germany.
Professor and Director, Institute of Flight Mechanics and Control, University of

155

156 M. Hanel and K. H. Well

1. Introduction

The evolution of large transport aircraft is characterized by fuselages
getting longer and wing spans getting wider, while efforts to reduce the
structural weight reduce the structural stiffness. Both effects lead to more
flexible aircraft structures with significant aeroelastic coupling between
flight mechanics and structural dynamics, especially at high speed, high
altitude cruise. This means that flight maneuvers and gusts may incite
strong elastic reactions, which influence also the rigid-body flight
mechanics.
Ride comfort and structural loads, especially for flight in a turbulent
atmosphere, are influenced strongly by the vibrations of the aircraft
structure. Since these vibrations cannot be controlled by conventional
stability augmentation systems (SAS), some modern aircraft are equipped
with additional control loops to improve the ride comfort (Ref. 1).
Stability augmentation and aeroelastic control loops are separated by
dynamic filters. As rigid body dynamics and low frequency elastic modes
get closer with increasing structural flexibility, the separate design of
stability augmentation system and aeroelastic control loops becomes more
difficult. Therefore, several recent studies (Refs. 2-5) have investigated the
integration of flight mechanics and aeroelastic control design.
As the aircraft rigid-body motion and the elastic degrees of freedom
are highly coupled, with mode shapes and frequencies changing with the
flight conditions and loading, a realistic aircraft model has to be generated.
Here, linearized integrated flight mechanics and aeroelastics models are
generated as outlined in Ref. 4. In addition, a simulation model with
nonlinear rigid-body dynamics is used for flight maneuver verification.
Model reduction techniques (Ref. 6) are employed to generate separate
control design models for the longitudinal motion.
In addition to the sensor information obtained from an inertial
platform, accelerometers placed along the aircraft structure are used.
Control is based on conventionally available control surfaces for primary
flight control, i.e., elevator, rudder and inner and outer ailerons. While in
Ref. 2 symmetrically deflected inner ailerons are available as means of
direct lift control, here symmetric inner and outer aileron activity is
restricted to low authority aeroelastic control purposes.
The flight control system for the longitudinal motion is divided into an
outer-loop flight path and attitude control and an inner-loop stability
augmentation and aeroelastic control. Emphasis in this paper is put on the

Controller Design for a Flexible Aircraft 157

inner-loop control, which is assumed to be embedded in an outer-loop
structure. This outer-loop controller may be based on the concept of total
energy control (TECS, Ref. 7) or may be even more elaborate with various
autopilot functions. For this paper, it is assumed that the outer loop
produces essentially a reference command for the desired C* command,
where C* is a combination of the vertical acceleration at the pilot position
and pitch rate.
Three alternatives are discussed for the inner-loop control system
design. First, a conventional cascaded single-input-single-output (SISO)
design is presented, which improves the flying qualities of the aircraft
without any active aeroelastic control. The second approach is based on
output feedback and does influence the rigid body as well as the
aeroelastic dynamic behavior of the aircraft. As a third approach,
optimization is used to design the controller for the inner loop. This gives
a robust design with respect to the different operating points of the
aircraft.

2. Modeling the Dynamic Aircraft Behavior

In general, the rigid-body dynamics of an aircraft is described by the
equations of motion consisting of 12 nonlinear scalar differential
equations with 3 states x,y,z for the position of the aircraft center of mass,
3 states u,v,w for the velocity components in a body-fixed reference
coordinate system, 3 states that is, azimuth, pitch, and roll angles
to describe the attitude with respect to an Earth-fixed reference coordinate
system, 3 states p,q,r, that is roll, pitch, and yaw rates around the body-
fixed axes. The controls are the elevator, rudder, and aileron angles and
the power setting. A detailed description of these equations is given for
instance in Ref. 8.

2.1. Structural Dynamics. The structural dynamics for the static and
dynamic deformations of the aircraft is described by linear differential
equations, which are generated using the finite-element method (FEM, see
e.g. Ref. 9). To arrive at such a model, the structure is assumed to consist
of many geometrically simple parts, the finite elements. In every element,
a space-dependent displacement function is approximated by a fixed
number of interpolation functions, describing the displacement behavior of


the element as a function of the displacement z of the discrete nodes.
Integrating over the element volume (and the known interpolation
functions), the work done by the inertial, elastic stiffness, and external
forces is expressed as a function of the nodal displacements and the
external nodal point forces. Assembling the results for the individual
elements, a system of second-order differential equations for the nodal
displacements is obtained,

In equation (1), is the vector of nodal displacements and is the vector
of external point forces. denotes the mass matrix, the stiffness
matrix and the load matrix of the aircraft. For the determination of the
static deformation of the structure the algebraic equation
has to be solved. For the dynamic deformation, the solution
of the homogeneous differential equation is determined by setting
thereby separating the time-dependent and space-dependent
components of the solution and solving the resulting eigenvalue problem,

The eigenvectors describe the mode shapes (normal modes) of the
undamped structure. For a free-flying aircraft structure, 6 zero
eigenvalues, representing the rigid body motion are obtained. The
corresponding eigenvectors can be chosen to represent the unit
displacements in the direction of the axes of the center-of-mass-based,
body-fixed reference frame and the unit rotations about these axes. The
eigenvectors (orthogonal to the rigid body motion) associated with the
negative eigenvalues describe the elastic deformations of the structure at
the fixed center of mass. They are normalized with respect to the mass
matrix,

The corresponding eigenmotions of the undamped structure are
harmonic oscillations with eigenfrequency Now, any small arbitrary


motion and deformation of the aircraft can be represented (within the
resolution of the discretization) by the superposition of the free undamped
normal modes,

with Here is the matrix of the eigenvectors for the rigid
body motion, is the matrix of the eigenvectors for the elastic motion,
and q is a vector of generalized coordinates.
A good approximation can be achieved by retaining only a small
number of modes at the low-frequency end of the set. For the flight-
mechanical and aeroelastic analyses addressed in this paper, the aircraft
motion can be described with sufficient precision using modes (6 rigid
body modes plus up to 60 low-frequency elastic modes for a full aircraft
model) up to about 20Hz,

Inserting the approximation of equation (5) into equation (1) and left
multiplying by a compact representation of the aircraft motion and
deformation can be achieved using a relatively small number of
generalized coordinates in the vector

Additional vectors, the control modes describing the unit deflections
of the control surfaces, are added to the eigenvector matrix
The control surface motion is appended with given spring
constants and mass and stiffness matrices in generalized coordinates. The
inertial coupling of the control surface motion and elastic deformation is
neglected. In a later step, a transfer function representing the actuator
dynamics is added.

2.2. Aerodynamic Forces and Moments. The air flow around a
flexible aircraft is modeled as an inviscid compressible flow. For the
purpose of aeroelastic calculations, the doublet-lattice-method (DLM) for


the approximate numeric calculation of the unsteady pressure distribution
on harmonically oscillating surfaces in three-dimensional subsonic flow
was developed by Albano and Rodden (Ref. 10). In this approach, the
surface of the aircraft structure is discretized by means of trapezoidal
boxes arranged in columns parallel to the free stream, the so-called panels.
The 1/4-chord line of each box is taken to contain a distribution of
acceleration potential doublets, expressed as local pressure differences, of
uniform but unknown strength. Then, an integral equation for the induced
downwash can be solved approximately for individual reduced frequencies
with as the undamped rigid body or structural frequency, c
the wingspan or the mean chord, and the free stream velocity.
The resulting forces (normal to the plane of the box) and moments
(about the 1/4-line of the box) are obtained by multiplying the pressure
difference over each box with the box area. Using the above technique, it
is possible to calculate a matrix of influence coefficients
that relates the changes in the lifting force at box i to the changes in the
induced downwash at box j. This influence coefficient matrix has to be
calculated for different Mach numbers and a number of frequencies in the
range of interest.
The DLM is well suited to account for the influence of wings and tail
planes. The power plants are modeled as annular wings. The influence of
the fuselage can be treated approximately. To extend the use of the DLM
to the transonic flight regime, the calculated pressure distributions can be
calibrated using a nonlinear Euler solution for steady flows.
The result of the aerodynamic force and moment calculations using
this method is

Here, is the dynamic pressure, G is a matrix that provides an
interpolation between the structural noding and the boxes used for the
aerodynamic calculations. Introducing this into equation (1) yields the
relation

where is a transformation matrix from aerodynamic to body-fixed
axes. Transforming equation (8) to the frequency domain using the


reduced Laplace variable and assuming a structural damping
matrix gives

with the reduced frequency and the complex matrix
Equation (9) is called the flutter equation. The complex
eigenvalues and eigenvectors of the nonlinear eigenvalue problem
have to be determined iteratively. Flutter occurs if for any
eigenvalue . The corresponding eigenvector determines the flutter
shape. Flutter calculations are described in Ref. 9.
With the aerodynamic forces available, a steady-state trim solution of
the flexible aircraft can be computed. To this end, the differential system

must be solved for the deformation vector with given acceleration due
to gravity and a transformation matrix from a geodetic system to a
body-fixed coordinate system.
Having obtained the deformation vector a coordinate transformation
to the stability axes at a particular Mach number is performed. With the
definition

the generalized aerodynamic forces in the new coordinate system can be
expressed as

Here, the term represents the steady-state aerodynamic force
corresponding to the trim angle of attack at a particular Mach number.

2.3. State Space Description. Aerodynamic forces based on the
generalized DLM aerodynamic force coefficient can be evaluated only for


harmonic oscillations at given discrete reduced frequencies k. To realize a
time domain simulation model, the unsteady aerodynamic forces have
to be represented in the time domain. This can be achieved first by
approximating the tabulated force coefficients by rational functions of the
Laplace variable s and then by transforming the resulting transfer
functions to the time domain.
The major difficulty associated with rational function approximation is
the matching of phase responses dominated by phase lags due to dead
times. Dead times occur frequently in unsteady aerodynamic responses,
representing for example the transit time from the wing to the tailplane.
However, rational transfer functions and continuous-time state-space
representations allow no exact representation of dead times and delays.
Instead, a large number of additional lag states is required to provide the
necessary phase lag. While the lag states are a common feature of all
rational function approximations, the number of lag states required for
different methods varies considerably. As a large number of lag states
means higher model complexity and increased calculation effort, methods
requiring a lower number of lag states are preferable for industrial-size
problems.
The minimum-state method of Ref. 11 formulates a general rational
transfer function matrix,

in the reduced Laplace variable to match the tabulated coefficient
matrix on the imaginary axis, that is,

where The diagonal matrix R in equation (13) is used to define
the aerodynamic lag states. Usually, roots with absolute values spread
within the range of the tabulated reduced frequencies are chosen. The
elements of the matrices D, E are determined from a
nonlinear weighted least-square solution minimizing, under some
constraints, the total weighted least-square approximation error.
Up to 3 constraints for every element of can be introduced to
enforce perfect data fit at specific frequencies (for example at k=0).
Weights are used to improve data fitting for selected elements at specific
frequencies, or simply for normalizing the tabulated data.


From the transfer function matrix in equation (13), a time domain
state-space representation can be derived by changing from to s (the
unscaled Laplace variable) and then applying the inverse Laplace
transform. It should be remembered though that both the aerodynamic
coefficient matrix and the set of approximation matrices resulting
form the minimum-state method are valid for only a single Mach number
and trim condition. Therefore, the simulation of a flight trajectory, when
the Mach number changes, requires an interpolation between different sets
of matrices. A time-domain representation of the aerodynamic forces, with
coefficient matrices D, E (R is assumed constant) scheduled
with Mach number is given in equations (15)-(16), where the vector
representing the aerodynamic lag states is introduced,

For the scheduling, an interpolation scheme based on third-order Hermite
polynomials and the (evidently false) assumption of zero tangent at the
Mach grids is used.
With an approximation of induced drag in place, it is possible to
complete the translational equation of motion in the body-fixed x-direction
by adding thrust and ram drag. In a flexible aircraft, the thrust vector
moves with the powerplant during vibrations, while the ram drag depends
on the local flow condition. After summing up the forces of the
powerplants and generalizing, the thrust forces can be described by
where denotes the thrust forces at the trim
condition, and denote the linearized thrust forces depending on
aircraft motion and deformation, and denotes the thrust forces due to
changes in the throttle position (throttle position vector
In stability axes, the coupled flight mechanics and aeroelastic equation
can finally be described as


Here, M is the generalized mass matrix, H is a generalized matrix that
contains the Coriolis terms due to the moving coordinate system, is the
generalized stiffness matrix, B is the input matrix obtained from a
partitioning of the doublet lattice matrix, is the control input consisting
of the inner ailerons and outer ailerons, rudder, and elevator, is a
transformation matrix from geodetic coordinates to stability axes, is the
gravitational acceleration, and has been described above. For a detailed
derivation of these equations, see e.g. Ref. 4.

3. Analysis of the Aircraft Dynamics

In this paper, a heavy four-engine transport aircraft is chosen as an
example. The cruise condition is set at a speed of Mach 0.86 and an
altitude of 30000ft. Three additional flight and load conditions are chosen
for detailed analysis, see Table 1. Although they represent only a small
part of the flight envelope, they allow us to develop an understanding of
the basic phenomena related to changes in the flight condition (speed,
altitude, and correspondingly Mach number and dynamic pressure) and the
load condition (tank loading, changing mass, moments of inertia, and e.g.
position). Flight condition 1 represents the cruise condition and is chosen


as the control design flight condition. It is the most challenging from an
aeroelastic point of view. Flight conditions 2 and 3 are encountered during
climb and flight condition 4 is encountered during descent.
In order to analyze the dynamic response of the flexible structure as a
function of various input signals, the frequency responses are investigated.
The analysis is based on individual single-input-single-output (SISO)
transfer functions. For the example aircraft, measurements at the cockpit,
at the center and aft fuselage positions, on the wing, and at the engines are
chosen. The main results are discussed using only a limited number of
transfer functions from the longitudinal motion.
Figure 1 shows the transfer functions from elevator deflection on
cockpit vertical acceleration; Fig. 2 shows the transfer function from
symmetric inner aileron deflection on midwing vertical acceleration. The
rigid-body modes (phugoid and short period mode) can be identified easily
in the low-frequency domain. In the 1-10 Hz frequency range however,
the aircraft response is dominated by weakly damped elastic modes,
especially wing bending, engine and fuselage modes. Higher-frequency


modes also have considerable influence, but are more difficult to identify
on the basis of these transfer functions. Comparing the two transfer
functions of the longitudinal motion, it can be noticed that the first
(symmetric) wing bending mode is not perceived in the cockpit but
dominates the response at the wing, while the outer engine vertical
vibration mode (coupled with the wing torsion) interacts with the first
fuselage bending mode and can be measured over all of the aircraft. For
the given configuration, the outer engine vertical vibration mode (together
with the fuselage bending) is the most critical with respect to flutter.
From the gain amplitudes in the aeroelastic frequency range, it can be
concluded that the control bandwidth for a flight mechanics stability
augmentation system that does not affect aeroelastics must not exceed
1Hz. This restriction limits severely the achievable handling qualities. It is
less severe for an integrated flight and aeroelastic control law. But even
then, a steep descent to cut off those elastic modes that are to remain
unaffected by the control law is required.
The load distribution (fuel and payload) influences strongly the
dynamic behavior of the elastic structure and consequently the aeroelastic


coupling. Figure 3 shows the frequency response for three different load
conditions (IE0-70, IET-70 and 000-70) at constant flight condition. While
changes in the elastic mode shapes and frequencies are not unexpected
(wing bending frequency should increase with fuel consumption), strong
changes in the damping (maximum amplitudes) and phase response are
also observed. As expected, the low-fuel configuration (000-70) turns out
to be the least critical with respect to aeroelastics. With mass and moments
of inertia reduced, comparatively more control power is available.
Therefore, the analysis described in this paper is concentrated on the high-
load cases.
Although frequency responses are the preferred means of analysis,
time responses are of interest for an assessment of the aircraft handling
qualities and for developing a physical understanding of the accelerations
and the level of vibration experienced by the pilot and the passengers. The
input signals used for the simulations shown subsequently have been
designed not to contain frequencies beyond about 5 Hz and could be
reproduced by a pilot. Therefore, the curves represent the basic low-
frequency response felt by the pilot and the passengers and targeted by the
flight and aeroelastic control effort.
Figure 4 shows the time response to the same elevator pulse for
different flight conditions. It can be seen that the amplitude varies
differently with the flight condition for the pitch rate and cockpit vertical


acceleration. This is due to the fact that, at lower speed, a larger angle of
attack is required to generate the same amount of lift and consequently
vertical acceleration. This relationship should be kept in mind for control
design. The acceleration response at different positions of the structure is
dominated by the fuselage bending and outer engine mode vibrations. As
said before, coupling between these two modes is strong and intensifies
with increasing speed and dynamic pressure.
It has been argued above that aeroelastic coupling would increase as
rigid-body motion and aeroelastic modes get closer in frequency. As a
consequence, integrated models for flight mechanics and aeroelastics were
deemed necessary; further, an integrated flight and aeroelastic control law
is envisaged in this paper. In this context, it is interesting to investigate the
influence of the frequency neighborhood between rigid-body motion and
aeroelastic modes on aeroelastic coupling. To that end, two state-space
models for the example aircraft in cruise flight have been generated with
stiffness changed to 50% and 200% of the nominal value. Figure 5
compares the frequency responses of these models to the response of the
nominal model.


It can be seen that aeroelastic coupling indeed increases with
decreasing stiffness and vice versa. The rigid-body mode frequencies are
also strongly affected by the elastic stiffness, with phugoid frequency
decreasing and short-period frequency increasing with the stiffness. As
could be expected, the frequency response of the stiffer aircraft tends
toward the response of the models with fewer or no elastic modes. For the
aircraft model with reduced stiffness, damping of the fuselage bending and
outer engine vertical vibration modes is lower than for the nominal model
and coupling between short-period motion and wing bending is significant
(see the change in the phase response).
Table 2 contains the modes of a reference aircraft for One can
see easily that the phugoid is unstable. Due to the long period of
approximately five minutes, however, this would be controllable easily by
a pilot. The short period mode is rather well damped, its frequency is
about one fourth of the frequency of the lowest elastic mode, which is the
first wing bending mode. All elastic modes are close to the imaginary axis,
that is, they have low damping. Due to the lag states, which are used to
approximate the aeroelastic phase lags, the pole positions loose some
significance, as the frequency and damping of the aeroelastic modes are
not determined uniquely by the dominant (2nd order) poles.


4. Controller Design

As mentioned above, the controller design of the outer loop is not
considered. This controller may be a flight path angle controller, or a
speed hold controller, or an altitude hold controller. Here, a C* command
is the reference command for the inner loop with


where is position of the pilot with respect to the center of gravity; the
other variables are explained below. This function is commonly used to
specify the flying qualities of an aircraft. The time history of C* is
supposed to be between the lower and upper time history bounds. For the
control system design, the function is computed in the feedback loop in
Fig. 6 and is compared to a commanded value
The design goals are: (i) to stabilize the phugoid and to increase the
damping of the short period mode; (ii) to reduce structural vibrations as
well as to increase passenger comfort; (iii) to increase the damping of the
aeroelastic modes up to about four Hz. In addition, it is required that the
closed-loop system is robust with respect to various operating points, if
possible without scheduling the controller.
Figure 6 shows a possible architecture for the inner-loop control. At
the core of the model is the state space system describing the linearized
equations of motion at a particular operating point, here operating point 1,
see Table 1. The state vector is defined as

Here, are associated with the rigid-body motion,
with the structural motion, and are the lag


states. The aircraft is controlled by

where the components are the elevator and the inner and outer symmetric
ailerons, which can be used with low authority for longitudinal control. It
is assumed that the following measurements are available:

where the first two components are attitude, attitude rate at the center of
mass, the third component is the acceleration at the center of mass. In the
order of appearance, the following components are the vertical
accelerations at the forward fuselage, at the rear fuselage, the midwing
acceleration at the wings, the acceleration at the winglets, and the lateral
accelerations of the inner and outer engines. These measurements can be
used for the control system design.
The matrices A, B, C, D in Fig. 6 are the results of the linearization
process. Between the controller, there is a low-pass filter which filters out
any higher-frequency signals which the controller might produce; in front
of the controller, there is an additional low-pass filter which filters out any
high frequency commands in Below the plant dynamics box, there is
a measurement box which selects those output signals to be fedback to the
controller. On top of the figure, the actuator signals are recorded in the
simulation of the closed-loop system (CLS); on the right side of the figure,
the output signals are recorded. The box entitled “test signals” generates
perturbations while simulating the CLS.

4.1. Stability Augmentation. If one disregards the aeroelastic
behavior in the control system design, like in conventional SAS
controllers, C* is fedback. With this signal, the design goal (i) can be
achieved. The low-pass filter 1 avoids the excitation of the elastic modes,
but there is no artificial damping. Figure 7 shows the time responses due
to a reference input of the commanded C* satisfying certain flying quality
criteria and due to an impulsive perturbation of the inner ailerons of 3.5
degrees magnitude after four seconds. Considerable vibrations with low


damping in the cockpit and at the outer engines are observed. In practice,
this would not be tolerable and a separate aeroelastic controller could be
designed which improves the damping of the elastic modes. This simple
SAS controller is used in the sequel as a reference in order to quantify the
improvements which advanced control design methods may offer.

4.2. Integral Controller Using Output Feedback. In addition to the
elevator, symmetric inner and outer ailerons are used as actuators.
Furthermore, all or some of the available output signals are fedback. Then,
the control design problems is formulated as a quadratic output feedback
control problem in which the cost functional


is minimized with respect to the elements of the gain matrix K. The choice
of the constant weighting matrices Q,R determines the quality of the
resulting feedback law,

In Fig. 8, this integral controller shows an improved time response for the
flexible motion of the aircraft (solid lines) in comparison to the reference
controller (dashed lines). In Fig. 9, it can be seen that the damping
increases for all poles. The open-loop modes presented in Table 2
represent the dynamics of a typical four-engine aircraft. Thus, the multi-
variable control system design achieves goals (i) and (ii). Concerning the
third goal, it is stated without additional results that robustness with
respect to varying operating points cannot be achieved without some
scheduling for the gain matrix K.

4.3. Integral, Robust Controller Using the Control Design
Method. The advantages of the previous design methods are a clear
structure with a unique assignment of dynamic elements (filters,


integrators, sensors) to particular tasks. This makes it possible to define
structured redundancy concepts for those cases where for instance sensors
or actuators degrade. The disadvantage of these methods is their lack of
robustness.
To compensate the deficiency observed in the output feedback design,
the design method considers modifications of the nominal plant in the
design process; that is, error models for various dynamic components of
the system are defined and considered in the design process. In addition,
nonmeasured states are estimated through an observer. The design goals
are defined in terms of the norm of particular transfer functions of the
closed-loop system. This norm is a metric of all gains as a function of the
frequency. For the transfer function from to C *, for instance,
one could demand that the closed-loop system should perform like a
second-order system with the transfer function Then, the
requirement in terms of the controller is formulated as


with respect to the controller contained in I is 1 in the scalar case
and an appropriately dimensioned identity matrix in a multivariable case.
The transfer matrix can be viewed as a frequency dependent

weighting function. Alternatively, should the influence of a gust with gust
velocity on the pitch rate q be minimized, then the criterion should be

with a specified weight. In a similar way, the modeling errors can be
formulated. The approach can be extended to multivariable problems; see
e.g. Ref. 12. If all the design goals are formulated in this way, then the
design task consists of finding a controller K(s), s being the Laplace
variable, which minimizes the infinity norm of a transfer matrix
describing the influence of the external inputs on the external output z.
This approach has been used here and details about the design procedure
are given in Ref. 4. Figure 10 shows the time histories for the same


variables as presented in Fig. 8. It can be observed that the robust
controller shows actuator activity at the inner and outer ailerons, which the
reference controller did not have. Figure 11 shows the pole migration. It
can be observed that damping is increased for all modes. It is rather
difficult to increase the damping of the engine modes, like in the output
feedback controller design. Figure 12 (in two parts) demonstrates that the
controller is robust indeed. Here, the same controller is used for simulating
unit step responses in C* with a perturbation after 4 sec at the inner
ailerons. The variables shown are defined in equation (20). is the flight
path inclination. The aircraft response is quite similar for all flight
conditions shown and the elastic mode damping is satisfactory.


5. Conclusions

Based on an integral model describing the dynamic behavior of the
rigid motion as well as the elastic motion of a flexible aircraft, it has been
shown that an integral controller can achieve desired flying qualities as
well as dampen the elastic vibrations considerably. This is achieved by
feeding back to the control system not only pitch attitude, pitch rate, and
vertical acceleration at the center of gravity, but in addition, various
accelerations measured at certain positions of the aircraft. The least
damped eigenmode is a symmetric vibration of both the outer engines, in
the y-direction of the lateral aircraft axis, which can only be improved
marginally through the control system.

References

1. SEYFFARTH, K., et al., Comfort in Turbulence for a Large Civil
Transport Aircraft, Proceedings of the International Forum on
Aeroelasticity and Structural Dynamics, Strasbourg, France, 1993.
2. SCHULER, J., Flugregelung und aktive Schwingungsdämpfung für
flexible Großraumflugzeuge, Dissertation, Universität Stuttgart,
Stuttgart, Germany, 1997.
3. KUBICA, F., and LIVET, T., Flight Control Law Synthesis for a
Flexible Aircraft, Proceedings of the AIAA Guidance, Navigation and
Control Conference, Scottsdale, Arizona, Paper AIAA 94 - 3630, pp.
775-783, 1994.
4. HANEL, M., Robust Flight and Aeroelastic Control System Design
for a Large Transport Aircraft, Dissertation, University of Stuttgart,
Germany, 2000.
5. TEUFEL, P., HANEL, M., and WELL, K. H., Integrated Flight
Mechanics and Aeroelastic Modelling and Control of a Flexible
Aircraft Considering Multidimensional Gust Input, NATO Research
and Technology Organization (RTO), Specialist Meeting on
Structural Aspects of Flexible Aircraft Control, Ottawa, Canada,
1999.


6. MOORE, B., Principal Component Analysis in Linear Systems:
Controllability, Observability and Model Reduction, IEEE
Transactions on Automatic Control, Vol. 26. No. 1, pp. 17-32, 1981.
7. LAMBREGTS, A., Vertical Flight Path and Speed Control Autopilot
Design Using Total Energy Principles, Paper AIAA 83-2239, 1983.
8. STEVENS, B. L., and LEWIS, F. L., Aircraft Control and
Simulation, John Wiley and Sons, New York, NY, 1992.
9. DOWELL, E. H., et al., A Modern Course in Aeroelasticity, Kluwer
Academic Publishers, Dordrecht, Holland, 1995.
10. ALBANO, E., and RODDEN W., A Doublet-Lattice Method for
Calculating Lift Distributions on Oscillating Surfaces in Subsonic
Flows, AIAA Journal, Vol. 7, No. 2, pp. 279 – 285, 1969.
11. K ARPEL, M., and S TRUL, E., Minimum-State Unsteady
Aerodynamic Approximations with Flexible Constraints, Journal of
Aircraft, Vol, 33, No. 6, pp. 1190-1196, 1996.
12. D OYLE , J. C., G LOVER , K., K HARGONEKAR , P., and
FRANCIS, B., State-Space Solutions to Standard and Control
Problems, IEEE Transactions on Automatic Control, Vol. 34, No. 8,
pp. 831-847, 1989.

General Index

Aeroservoelasticity, 155 Manual flight path control, 105
Aircraft guidance, 105 Mirror property, 65
Ascent trajectories, 1 Moon-Earth flight, 31
Astrodynamics, 31, 65
Asymptotic parallelism property, 65 Optimal trajectories, 1, 31, 65
Optimization, 1, 31, 65
Celestial mechanics, 31, 65 Orbital spacecraft, 1
Concurrent engineering, 131
Perspective flight path display, 105
Earth-to-Mars missions, 65
Earth-Moon flight, 31 Round-trip Mars missions, 65
Earth-Moon-Earth flight, 31 Rocket-powered spacecraft, 1

Flexible aircraft, 155 Sequential gradient-restoration algorithm, 1, 31,
Flight control, 155 65
Flight mechanics, 1, 31, 66 Suborbital spacecraft, 1
Flight path predictor, 105 Synthetic vision, 105

Launch vehicles, 1, 131 Trajectory optimization, 1, 31, 65, 131
Lunar trajectories, 31

181

Subject Index

Aeroservoelasticity, 156, 164–169; see also Hohmann transfer trajectory, 90, 92
Flexible aircraft, modeling
Aircraft, flexible: see Flexible aircraft Launch vehicles, 132–136, 152
Aircraft guidance: see Guidance display mathematical model of rocket vehicle
Ascent trajectories, 2 equations of motion, 136–139
ASTOS software, 132, 146 mass models for engine and tank sizing,
Astrodynamics, 31, 65, 67 139–141
multiphase optimal control problem, 142
Design cost function, 143–146
controller for a flexible aircraft, 15 initial conditions, 142–143
experimental guidance system, 105 target and intermediate conditions, 143–146
Mars mission, 65 reference vehicle, 133–136
Moon mission, 31 trajectory optimization problem, 146
perspective flight path display, 105 initial guess, 146–149
rocket-powered orbital spacecraft, 1 optimal solutions, 149–151
two-stage launch vehicle, 131 Low Earth orbit (LEO), 33, 45, 69, 82, 83, 86–88,
Differential global positioning system (DGPS), 90, 92, 98
111–113 Low Earth orbit (LEO) data, 35–36, 45–47, 51,
Drag, 24, 26 56–58
Low Mars orbit (LMO), 69, 82, 83, 86–88, 90, 93
Earth coordinate system (ECS), 70–71, 74–80 Low Mars orbit (LMO) data, 85–86
Earth-Moon-Earth flight, 53–62 Low Moon orbit (LMO), 33, 45–46
Earth-Moon flight, 36, 40–44 Low Moon orbit (LMO) data, 36, 40–44, 48–52,
arrival conditions, 37–39 54–62
departure conditions, 36–37 Lunar trajectories, 32
optimization problem, 39
Earth-to-Mars missions: see Mars missions Manual control, 116–118; see also Guidance
Exploratory Mars missions, 67 display
Mars coordinate system (MCS), 70–71, 75–77
Flexible aircraft, 156–157, 179 Mars missions, 66, 97–99
aerodynamic forces and moments, 159–161 baseline optimal trajectory results, 86
analysis of aircraft dynamics, 164–170 delay time, 93
controller design, 170–172 near-mirror property, 93–94
integral controller using control, 174–178 outgoing trip, 86–90, 94–96
integral controller using output feedback, return trip, 90–92, 94–97
173–174 waiting time, 93
modeling aircraft dynamic behavior, 157 boundary conditions
stability augmentation, 156, 172–173 outgoing trip, arrival, 75–76
state space description, 161–164 outgoing trip, departure, 74–75
structural dynamics, 157–159 return trip, arrival, 77–78
Flight path control, 105; see also Flexible aircraft, return trip, departure, 76–77
controller design characteristic velocity, 67
Flight path predictor, 116–127; see also Guidance computational information
display algorithm, 84
Global positioning system, 111–113 integration scheme, 84–85
Guidance display, three-dimensional, 106–107, coordinate transformation, 79–80
125–127 delay time, 83–84
basic concept, 107–112 four-body model, 68
flight test results, 112–116 mathematical programming problems, 80
with predictor, 116–121 mirror property, 94
results of simulation experiments, 121–125 mission alternatives, types, and objectives, 67

183

184 Subject Index

Mars missions (continued) Rocket-powered orbital spacecraft (continued)
optimal trajectories, 67 specific impulse, 21
outgoing trip, 80–82 structural factor, 21
patched conics model, 68–69 system description, 3
planetary and mission data, 85–86
restricted four-body model, 69 Sample taking (sample return) Mars missions, 67
return trip, 82–83 Sequential gradient-restoration algorithm
system description, 69–71 (SGRA), 2, 7, 11, 39, 48, 84
Earth, 71–72 Single-stage orbital spacecraft: see SSTO
Mars, 72–73 spacecraft
spacecraft, 73–74 Single-stage-suborbital (SSSO) spacecraft, 2, 6,
waiting time, 83 26–28
Moon-Earth flight, 45 boundary conditions, 6–7
arrival conditions, 46–47 computer runs, 7–8
departure conditions, 45–46 optimization problem, 7
optimization problem, 47–49 weight distribution, 7
trajectories, 48–52 zero-payload line, 8–9
Moon missions, 32–33, 58–62; see also Earth- Single-stage-to-orbit (SSTO) spacecraft, 10, 16,
Moon flight 27, 28
differential system, 34–35 boundary conditions, 10
feasibility problem, 57–58 computer runs, 11–12
fixed-time trajectories, 57–58 optimization problem, 10–11
system description, 33–34 weight distribution, 10
Multi-stage-to-orbit (MSTO) spacecraft, 2 zero-payload line, 12, 13, 21
Stability augmentation systems (SAS), 156,
Optimal trajectories, 2, 32–33, 53 172–173
Optimization, 2, 32–33, 53 Suborbital spacecraft: see SSSO spacecraft
Sun coordinate system (SCS), 70–74, 79–80
Patched conics model, 68–69 Survey missions to Mars, 67
Perspective flight path display, 106–107, 116, Synthetic vision, 107; see also Guidance display
117; see also Guidance display
Pilot-predictor-aircraft crossover, 116–117 Terrain elevation modeling, 109–111
Planned Mars missions, 67 Trajectory optimization, 2, 32–33, 53
Predictor-aircraft transfer function, 119–120 Two-stage orbital spacecraft: see TSTO spacecraft
Two-stage-to-orbit (TSTO) spacecraft, 2, 12–13,
Robotic Mars missions, 67 27–28
Rocket-powered orbital spacecraft, 2–3, 26–28 boundary conditions, 13–14
design considerations, 21 computer runs, 16–21
drag effects, 24, 26, 27 interface conditions, 14
inequality constraints, 5–6 optimization problem, 15–16
mathematical model, 3–5 weight distribution, 14–15
SSSO vs. SSTO configurations, 21–22 zero-payload line, 16–17, 19–21
SSTO vs. TSTO configurations, 22–26

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