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Jianping Yuan · Yu Cheng ·
Jinglang Feng · Chong Sun

Low Energy Flight:

Orbital Dynamics
and Mission
Trajectory Design
Low Energy Flight: Orbital Dynamics and Mission
Trajectory Design
Jianping Yuan Yu Cheng
• •

Jinglang Feng Chong Sun

•

Low Energy Flight: Orbital

Dynamics and Mission
Trajectory Design

123
Jianping Yuan Yu Cheng
School of Astronautics School of Astronautics
Northwestern Polytechnical University Northwestern Polytechnical University
Xi’an, China Xi’an, China

Jinglang Feng Chong Sun

Nanjing University School of Astronautics
Nanjing, China Northwestern Polytechnical University
Xi’an, China

ISBN 978-981-13-6129-6 ISBN 978-981-13-6130-2 (eBook)

https://doi.org/10.1007/978-981-13-6130-2
Jointly published with Science Press, Beijing, China
The print edition is not for sale in China Mainland. Customers from China Mainland please order the
print book from: Science Press.

Library of Congress Control Number: 2019930644

© Science Press and Springer Nature Singapore Pte Ltd. 2019

This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part
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Singapore
Preface

In recent years, space missions with the destinations to planets and small solar
system bodies have become more and more popular. For missions to such kind of
celestial bodies, one challenge comes from the trajectory design and optimization of
mission transfer that requires low energy, e.g., utilizing the continuous low thrust
and invariant manifold. The other challenge is the strong perturbation on the
spacecraft’s motion from the highly irregular gravity field. This book provides an
overview of the major issues related to the development of low-energy flight and
includes the continuous low-thrust transfer between the near earth asteroids using
the solar sailing and solar electrical propulsion systems that are covered in Chap. 2,
the low-energy transfer between libration point orbits and lunar orbits using
invariant manifold in Chap. 3, the Lorentz force formation flying under artificial
magnetic field in Chap. 4, and the highly non-linear dynamical environment in the
vicinity of small solar system bodies that are covered in the Chaps. 5 and 6.
Therefore, this book is suitable for both graduate students and researchers.
The contents of this book are based on the recent research and Ph.D. studies
of the authors, i.e., Prof. J. Yuan of Northwestern Polytechnic University (NWPU),
Dr. C. Sun from 2013 to 2015 of NWPU that contributes Chap. 2, Dr. Y. Cheng
from 2014 to 2016 as a joint Ph.D. student at Universitat de Barcelona and NWPU
contributing to Chaps. 3 and 4, and Dr. J. Feng from 2011 to 2016 of Delft
University of Technology for Chaps. 5 and 6.
In addition, the authors got valuable comments and suggestions from colleagues
and friends, and they express their deep gratitude toward Prof. X. Hou from Nanjing
University; Prof. G. Gómez, Prof. J. Masdemont, and Prof. À. Haro from Universitat
de Barcelona; and Ir. Ron Noomen, Prof. B. Ambrosius, Prof. B. Vermeersen, and
Prof. P. Visser from Delft University of Technology.

Xi’an, China Jianping Yuan

March 2018

v
Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Low-Thrust Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Transfer in the Earth-Moon System . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Orbital Dynamics Around Irregular Bodies . . . . . . . . . . . . . . . . . 4
2 Continuous Low Thrust Trajectory Design and Optimization . . . . . 7
2.1 The Virtual Gravity Field Method . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 The Definition of Virtual Central Gravitational Field . . . . . 8
2.2 Trajectory Design Using the VCGF Method . . . . . . . . . . . . . . . . . 9
2.2.1 Rendezvous Trajectory Design Using VCGF Method . . . . 9
2.2.2 Orbit Interception Trajectory Design Using VCGF
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14
2.2.3 Mission Applications . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15
2.2.4 Conclusion Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
2.3 Orbital Rendezvous Between Close Near Earth Asteroids
Considering the Third Body Perturbation . . . . . . . . . . . . . . . . . .. 24
2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24
2.3.2 The Variation of Orbital Elements Caused by the Earth
Gravitational Perturbation . . . . . . . . . . . . . . . . . . . . . . . .. 25
2.3.3 Orbital Rendezvous Considering the Third Body
Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29
2.3.4 Minimal Fuel Consumption Optimization Using Hybrid
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.5 Simulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

vii
viii Contents

3 Transfer Between Libration Point Orbits and Lunar

Orbits in Earth-Moon System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 The Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Change of Coordinates Between the Synodic CR3BP
and the Moon-Centred Sidereal Frames . . . . . . . . . . . . . .. 44
3.3 Computation of a Transfer from LPO to a Circular
Lunar Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46
3.3.1 Computation of the Transfer Manoeuvre to a Keplerian
Ellipse with a Fixed Inclination i . . . . . . . . . . . . . . . . . .. 48
3.3.2 Computation of the Departure Manoeuvre: First
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50
3.3.3 Refinement of the Departure Manoeuvre and
Determination of P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.4 Computation of the Insertion Manoeuvre at P2 . . . . . . . . . 53
3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Departing Halo Orbits and Their Invariant Manifolds . . . . 55
3.4.2 Selection of the Inclination i1 . . . . . . . . . . . . . . . . . . . . . . 56
3.4.3 Role of the Angles Between the Arrival and Departure
Velocities at P1 and P2 . . . . . . . . . . . . . . . . . . . . . . . . .. 59
3.4.4 The Role of the Orbit on the Unstable Manifold
with Dv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61
3.4.5 Varying P1 Along the Manifold Leg . . . . . . . . . . . . . . . .. 62
3.4.6 Setting P1 at the First Apolune . . . . . . . . . . . . . . . . . . . .. 64
3.4.7 Changing the Sizes of the Departing and the Target
Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68
4 Lorentz Force Formation Flying in the Earth-Moon System . ..... 71
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 71
4.2 Analysis of the Relative Dynamics of a Charged Spacecraft
Moving Under the Influence of a Magnetic Field . . . . . . . . ..... 73
4.2.1 Modelling Equations and Symmetries . . . . . . . . . . . ..... 73
4.2.2 Equations of Motion in the Normal, Radial and
Tangential Cases . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 75
4.2.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 77
4.2.4 Equilibrium Points, Stability and Zero Velocity
Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.5 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.6 Stability of the Equilibrium Points . . . . . . . . . . . . . . . . . . 79
4.2.7 Zero Velocity Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Contents ix

4.3 Periodic and Quasi-periodic Orbits Emanating from Equilibria . .. 87

4.3.1 Computation of Periodic Orbits Around the Equilibrium
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87
4.3.2 Computation of 2D Invariant Tori . . . . . . . . . . . . . . . . .. 92
4.3.3 Numerical Results on Periodic and Quasi-periodic
Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3.4 The Normal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3.5 The Radial Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.6 The Tangential Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4 Formation Flying Configuration Design . . . . . . . . . . . . . . . . . . . . 119
4.4.1 Formation Flying Configuration Using Equilibrium
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4.2 Formation Flying Configuration Using Periodic Orbits . . . . 121
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5 1:1 Ground-Track Resonance in a 4th Degree and Order
Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2 Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.1 Hamiltonian of the System . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.2 1:1 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 Primary Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3.1 EPs and Resonance Width . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.4 Secondary Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4.1 The Location and Width of Hreson2 . . . . . . . . . . . . . . . . . . 135
5.4.2 1996 HW1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4.3 Vesta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4.4 Betulia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.5 The Maximal Lyapunov Characteristic Exponent of Chaotic
Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6 Orbital Dynamics in the Vicinity of Contact Binary Asteroid
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2 Numerical Analysis of Orbital Motion Around Contact Binary
Asteroid System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.2.2 Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2.3 Contact Binary System 1996 HW1 . . . . . . . . . . . . . . . . . . 159
6.2.4 Orbital Motion Around the System . . . . . . . . . . . . . . . . . . 169
6.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
x Contents

6.3 Orbital Motion in the Vicinity of the Non-collinear Equilibrium

Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.3.2 Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.3.3 Non-collinear EPs and Their Stability . . . . . . . . . . . . . . . . 180
6.3.4 Motion Around the Stable Non-collinear EPs . . . . . . . . . . 182
6.3.5 Motion Around the Unstable Non-collinear EPs . . . . . . . . 189
6.3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Appendix A: The Primary Zonal and Tesseral Terms Contributing
to the 1:1 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Appendix B: The Un-normalized Spherical Harmonic Coefficients
to Degree and Order 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Appendix C: The Location of EPs and Resonance Width . . . . . . . . . . . . 203
Appendix D: The Second Derivatives of the Potential at the EPs
Located at (x0, y0, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Appendix E: The First and Second Derivatives of the n
Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Chapter 1
Introduction

The Sputniks 1 satellite, launched in 1957, was a milestone mission and marked the
start of human’s journey to space. Driven by our stronger dreams to explore the deep
space and with the aid of the fast-developing technologies, human’s activities have
extended to the space far away from Earth, with destinations including the Moon, the
planets and small bodies in our solar system. Numerous missions have been launched.
The one that travels the furthest from Earth is the Voyager 1 spacecraft (NASA),
which obtained images of Saturn and Jupiter during their flybys. Currently, it has
arrived at the outer solar system, i.e. the edge of the solar influence and interstellar
medium.
For the future deep space missions, the first challenge comes from the limited
amount of available propellant, since the flight destination is far from Earth and the
flight time is resultantly longer than Earth missions. Traditionally, the high-impulsive
chemical propulsions are used to change the velocity of the spacecraft. For instance,
the Hohmann transfer, based on the two-body problem, has been used for transfer
trajectory design. Although these techniques have been applied successfully in many
space missions, the most energy-efficient means should be addressed to handle the
new rising challenge.
Secondly, the mission requirements and flight environment will become more
and more complicated. For spacecraft around Earth, the forces in addition to the
Earth’s central gravity are viewed as small perturbations, which have been handled
with mature techniques based on the well-developed perturbed two-body problem.
However, in deep space, the gravitational forces from other celestial bodies are large
and even comparable with that of Earth. One example is the spacecraft around the
collinear libration points of the circular restricted three-body problem (CRTBP) such
as the Sun-Earth system, where both primary bodies exert forces of approximately
the same magnitude.
Thirdly, the destinations will be even more diverse, i.e. not restricted to large
and spherical planetary bodies. Small size and non-spherical bodies, e.g. small solar
system bodies including asteroids and comets, have become popular targets. The
© Science Press and Springer Nature Singapore Pte Ltd. 2019 1
J. Yuan et al., Low Energy Flight: Orbital Dynamics and Mission
Trajectory Design, https://doi.org/10.1007/978-981-13-6130-2_1
2 1 Introduction

explorations of them have significant scientific values, as they are recognized as

the remnants of the early solar system and contain rare materials that are potential
energy sources. For the spacecraft orbiting around highly irregular small bodies,
perturbations from the large non-spherical terms of these bodies make the dynamics
and the design of the spacecraft trajectory completely different from those of the
two-body problem. Therefore, efforts should be made to design orbits with these
large perturbations.
In these scenarios, the classical theory of astrodynamics based on the two-body
problem is no longer valid to describe and solve the orbital dynamics. Therefore, the
new kinds of propulsions, more accurate dynamical descriptions, and new techniques
that fully utilize the natural forces and inherent dynamics, are required to achieve
the low-energy flight, which is the focus in this book. These requirements bring new
challenges to mission design, such as efficient analytical/numerical ways to study
the orbital dynamics with low thrust, new methods to deal with strong perturbations,
new ways of designing optimal trajectories for transfer, station-keeping and formation
flying, etc. Therefore, modern astrodynamics needs to be applied, which basically
includes but not limited to the three-body problem, the dynamical system theory, the
low-thrust trajectory design, and orbital dynamics around highly irregular bodies.
For each space mission, mission orbits design mainly consists of two parts: the
first is the design of the trajectory transferring from Earth to the mission orbit around
the target celestial body; the second part is designing the mission orbits for scien-
tific observations and experiments. Therefore, the analysis of mission orbit design
includes the following two major concerns:
• How does the spacecraft been transferred to the target orbit?
• How can the spacecraft be kept on the target orbit or be transferred to new mission
orbits for difference mission scenarios or for mission extensions?
This book aims at investigating the above problems in the concept of low-energy
flight and in the framework of modern astrodynamics, using techniques such as
optimization, perturbation methods and dynamical tools for instance periodic orbits,
invariant manifolds, etc. The following three topics are the focuses:
• Low-thrust transfer trajectory design and formation flying;
• Transfer in the Earth-Moon system using invariant manifolds;
• Orbital dynamics around highly irregular asteroids.

1.1 Low-Thrust Propulsion

There are several ways to generate low thrust, e.g. solar power, electric energy, etc.
The electric propulsion is the most popular one, which uses electric or electromag-
netic energy to produce continuous low thrust. It has been well developed and widely
used in space missions, such as the orbit raising and station-keeping of geostationary
satellites. Compared with the chemical propulsion, it is more fuel efficient, which
1.1 Low-Thrust Propulsion 3

generates higher speed through the weaker thrust but much longer execution time.
Russian satellites have used electric propulsions for decades. In 2001, with elec-
tric propulsion, NASA’s Deep Space 1 spacecraft performed interplanetary flight for
more than 1000 h. In 2002, Japan launched its first electric-propellant GEO satellite
Kodama. In 2006, SMART-1 was ESA’s first solar-electric propulsion spacecraft to
the Moon and ended with a Moon impact. The Dawn spacecraft, launched in 2007,
was the first dual asteroid mission actuated by ion thruster. By 2013, approximately
350 spacecraft have flown using electric propulsions.
Chapter 2 is devoted to the optimal transfer trajectory design using the low thrust. A
new efficient method called virtual gravity field is proposed to provide initial guesses
for the optimization process. This method is demonstrated by several interplanetary
transfer examples and compared with other optimization methods.
Apart from the transfer trajectory, low thrust can also be used for keeping mission
orbits, e.g. formation flying. As an alternative of the single spacecraft mission, for-
mation flying provides the advantages of more flexibility for maintenance, upgrade,
and replacement of part of the formation. There are already some missions used
this technique, for instance, NASA’s Magnetospheric Multiscale Mission (MMS),
launched in 2015, investigating the interaction of the magnetic fields of Sun and Earth.
ESA’s Laser Interferometer Space Antenna (LISA) mission is going to use a trian-
gular formation in the orbit trailing the Earth in the heliocentric frame. Recently, a
new-concept propellant-less formation flying utilizing the Lorentz force is proposed.
The leader provides the magnetic field, and the motion of the charged follower in
this field generates the Lorentz force. Although it is currently in the research stage,
this technique demonstrates a promising application on future space explorations.
In spite of some preliminary analysis on the dynamics of this system, Chap. 4 of
this book provides systematic studies from different aspects, including equilibrium
points, periodic orbits, 2-dimensional invariant tori of three specific cases.

1.2 Transfer in the Earth-Moon System

For missions around major celestial bodies such as Earth, the gravity of the central
body is the dominant force, and other forces can be treated as small perturbations.
The resulting trajectories are perturbed Keplerian motions that are close to the unper-
turbed ones. In deep space missions, the circular restricted three-body CR3BP is usu-
ally applied to model the orbital motion of the spacecraft, instead of the perturbed
two-body problem. For this dynamical system, there are five libration points, around
which the dynamical structures such as periodic orbits and invariant manifolds are
often used in designing low-energy transfer trajectories in these systems. ISEE-3,
launched in 1978, was the first mission utilizes the invariant manifolds to trans-
fer to the first Lagrangian point (L1) of the Sun-Earth system. Launched in 2001,
the Genesis mission took the heteroclinic connections between orbits around differ-
ent collinear libration points as the transfer orbit for the first time. In 2009, NASA’s
ARTEMIS mission send two spacecraft to the Moon through the lunar ballistic trans-
4 1 Introduction

fer trajectories, which utilized the invariant manifolds of both the Sun-Earth and the
Earth-Moon systems. During the first mission extension after departing from the
lunar orbit, China’s Chang’e-2 spacecraft arrived the second Lagrangian point (L2)
of the Sun-Earth system for the first time ever in 2011. The following GRAIL mis-
sion, launched in 2011, uses similar transfer trajectories. Launched in 2014, China’s
CE-5/T1 mission also visited the L2 point of Earth-Moon system from Earth via a
lunar swing-by and returned to the Moon through the invariant manifolds associated
with this L2 point.
Apart from the CRTBP, there are some improved models such as the bi-circular
model, the Hill model, the restricted four-body problem. Nevertheless, only the
mostly applied CR3BP will be discussed in this book. Focusing on the Earth-Moon
system and using the CR3BP, Chap. 3 is devoted to studying the transfer from a halo
orbit to a lunar polar orbit. By varying the amplitude of the halo orbit, the specific
unstable orbit in the unstable invariant manifold, the specific point on this specific
unstable orbit, and the height of the lunar polar orbit, this problem is systematically
studied.

1.3 Orbital Dynamics Around Irregular Bodies

As mentioned, in addition to planets, small solar system bodies with irregular gravity
field are also interesting targets for both scientific and technical objectives and have
become popular destinations for space missions. In 2000, NEAR (NASA) space-
craft arrived at asteroid 433 Eros and determined its gravity, mass, spin rate and
orientation, density and internal mass distribution. In 2005, Hayabusa (JAXA) char-
acterized Itokawa’s surface and shape thoroughly. Regolith samples were collected
and returned to Earth for the first time in 2010. After ten years’ journey, Rosetta
(ESA) had a rendezvous with its target comet 67P/Churyumov–Gerasimenko in
2014 and released the lander Philae for the first landing on a comet ever. The comet
was revealed to have a highly irregular shape of a contact binary body. For missions
to such kind of bodies, one of the biggest challenges comes from the perturbation on
the spacecraft’s motion from the highly irregular gravitational field. Moreover, due
to the weak gravity field of the small bodies, the motion of the spacecraft is more
sensitive to perturbations, for instance, solar radiation pressure (SRP) and outgassing
etc.
Focusing on the non-spherical perturbation of the asteroids, Chap. 5 is devoted
to the ground-track resonances, which is the main cause of the instability of orbital
motions close to the asteroid’s surface. This gives the readers an intuitive picture of
“haotic” orbital environment around these highly irregular objects. After that, Chap. 6
focuses on families of periodic orbits and their associated stability properties. This
study provides a powerful way to select stable/unstable mission orbits out of the
“haotic sea” around these bodies.
1.3 Orbital Dynamics Around Irregular Bodies 5

It is stressed here that each chapter is self-contained, since they belong to relatively
different sub-topics of modern astrodynamics. The reason that we put Chap. 4 on low-
thrust after Chap. 3 is the system and the associated dynamical structures are similar
with those of the CR3BP studied in Chap. 3. The readers can better understand the
contents of this chapter after they read Chap. 3.
Chapter 2
Continuous Low Thrust Trajectory
Design and Optimization

2.1 The Virtual Gravity Field Method

2.1.1 Introduction

Continuous low thrust propulsion is an effective way for achieving space mission
trajectory design, which has attracted much attention in literatures. The fundamental
task of trajectory design is to find thrust profile which can change spacecraft from
one state to another state within a given flight of time. However, the trajectory design
is still challenging, because there are too many trajectory parameters which need
analyzing and those parameters associated search space is very large.
Analytical solutions have been proposed by several authors. Tsien [21], Boltz [6]
and Mengali [16] developed analytical solutions of orbit motion, under the assump-
tion of continuous thrust aligning along the radius direction. Following the same
formulation, Boltz [7] and Zee [25], studied the case of tangential thrust for continu-
ous low thrust trajectories. Furthermore, Gao [8] presented an averaging technique to
obtain analytical solution in case of tangential thrust. Although those methods can,
to a large extent, simplify the continuous low thrust problem, they are only suitable
for some special cases.
For general cases, there is no analytical solution in continuous thrust trajectory,
and the low thrust trajectory is designed typically using two optimization methods:
the indirect optimization method and the direct optimization method. The former is
based on the calculus of variation, and then the optimization problem is modeled
as a two-point boundary value problem. However, it is extremely sensitive to the
initial guess, so it is difficult to generate suitable solutions using the indirect method.
The latter parameterizes a trajectory using a few variables, and then the nonlinear
programming technique is used to optimize those variables to maximize objective
function. A variety of methods of this type have been examined [5]. A noticeable
direct method was proposed by Jon A. Sims et al. [11]. In his work, the trajectory was
divided into several segments at discrete points. The continuous low thrust trajectory
design problem was modeled into a nonlinear programming problem, and was solved
© Science Press and Springer Nature Singapore Pte Ltd. 2019 7
J. Yuan et al., Low Energy Flight: Orbital Dynamics and Mission
Trajectory Design, https://doi.org/10.1007/978-981-13-6130-2_2
8 2 Continuous Low Thrust Trajectory Design and Optimization

by the nonlinear programming software SNOPT. Another method that can provide
an initial guess for more accurate optimizers is the shape-based method developed
by [2]. In the shape-based method, the thrust is assumed to be aligned along the
velocity direction, and the radical vector of spacecraft is written as a function of
the transfer angle. Then the coefficients of the function are calculated to satisfy the
boundary constraints. Later, [1] extended the shape-based approximation method, in
order to reduce the required thrust to satisfy the thrust constraint. In their method,
the radius vector and polar angle are described as functions of flight time in form
of Fourier series. Those coefficients of Fourier series are optimized using Fmincon
tools in Matlab software to satisfy the boundary constraints and thrust constraints.
In this section, a virtual central gravitational field method (VCGF) is proposed
to determine continuous thrust trajectory. Instead of providing some special initial
guesses, the method can generate a large number of feasible initial guesses effi-
ciently and find the optimal one. There is no prior assumption about the direction
of thrust used in this method, and the solutions are analytically determined by the
virtual gravity. The basic idea of the method is that, without thrust and neglecting
perturbation, the spacecraft flies in a conic orbit in two-body gravitational field. Sim-
ilarly, if the thrust and the Earth gravity can form a virtual central gravitational field,
then the spacecraft can fly in a virtual conic orbit to accomplish trajectory maneu-
ver in that virtual gravity. In this way, feasible continuous thrust trajectories can be
parameterized, and expressed as a kind of displaced orbits named virtual conic orbits
analytically. Combined with the Particle Swarm Optimization (PSO) algorithm, the
proposed method provides a new way to obtain initial guess given an objective func-
tion. This method is not only intended for rendezvous case, but it is also used to solve
less constrained cases like orbit interception.

2.1.2 The Definition of Virtual Central Gravitational Field

In the two-body problem, spacecraft flies in a Keplerian orbit in the geocentric

gravitational field with no thrust. By analogy, if the continuous thrust and the Earth’s
gravity can form a virtual central gravitational field, the spacecraft can fly in a virtual
Keplerian orbit. A virtual central gravitational field can be defined by two parameters:
the magnitude of virtual gravity μvg , the displaced position parameter r0 , as shown
in Fig. 2.1. From the definition of the virtual central gravitational field, we can see
that the Earth’s gravity is a special case of a virtual gravity, whose gravity parameter
is μvg = 1 DU/T U 2 , r0 = 0 DU (here DU is the distance Unit, and the TU is the
time Unit).
As shown in Fig. 2.2, point A is an initial point and point B is a final point; ra , rb are
position vectors in point A and point B in geocentric coordinate system; rvga ,rvgb are
position vectors in point A and point B in the virtual central gravitational coordinates
system. Virtual Keplerian orbits can be classified into three types as shown in Fig. 2.2.
(1) μ = μvg , r0 = 0 as shown in Fig. 2.2a;
(2) μ = μvg , r0 = 0 as shown in Fig. 2.2b;
2.1 The Virtual Gravity Field Method 9

Fig. 2.1 The virtual central gravitational field

Fig. 2.2 Three types of virtual Keplerian orbits

The VCGF method requires that a spacecraft should fly in a virtual conic orbit and
in the virtual central gravitational field to satisfy the boundary constraints. There are
a few steps in the VCGF method. Firstly, feasible virtual gravity field determined by
parameter set (μvg , r0 ) are required, in which the spacecraft runs in the virtual conic
orbits to satisfy trajectory constraints. Secondly, the required thrust to form the virtual
gravity is computed, and the fuel consumption is calculated. Finally, considering the
objective function, the PSO algorithm is adopted to find the optimal initial guess for
more accurate optimizer.

2.2 Trajectory Design Using the VCGF Method

2.2.1 Rendezvous Trajectory Design Using VCGF Method

2.2.1.1 Trajectory Parameterization

In the geocentric coordinate system, ra and va are the position vector and the velocity
vector at point A, rb and vb are the position vector and the velocity vector at terminal
10 2 Continuous Low Thrust Trajectory Design and Optimization

point B. (μvg , r0 ) are parameters of the virtual gravity. In the virtual central gravita-
tional coordinate system, rvga and vvga are the position vector and the velocity vector
at point A respectively; rvgb and vvgb are the position vector and the velocity vector at
point B respectively. Velocity vectors and position vectors of the spacecraft at initial
point A and terminal point B can be computed as,
⎧
⎪
⎪ rvga = ra + r0
⎨
vvga = va
(2.1)
⎪
⎪ rvgb = rb + r0
⎩
vvgb = vb

With the Keplerian orbit theory, one has that,

⎧
⎪
⎪ hvg = rvga × vvga = rvgb × vvgb
⎪
⎪ Δf
r ·rvgb
= arccos( rvga )
⎪
⎪ vga ·rvgb
⎨ hvg 2
rvga = μvg 1+evg cos fvga
1
(2.2)
⎪
⎪
⎪ vgb = fvga + Δf
⎪ f
⎪
⎪
⎩ r = hvg 2 1
vgb μvg 1+evg cos fvgb

where hvg , evg are the angular of momentum and the eccentricity of the virtual
Keplerian orbit, while fvga , fvgb are the true anomaly of point A and B in the virtual
central gravitational field respectively. The range of parameter fvga is [0, 2π ], while
r0 should satisfied,

hvg = rvga × vvga = rvgb × vvgb (2.3)

In the X-Y plane, we set

ra = [rax , ray ], rb = [rbx , rby ]

,
va = [vax , vay ], vb = [vbx , vby ]

and then Eq. 2.3 can be expressed as follow,

a1 · r0x + b1 r0y = c1 (2.4)

Where a1 , b1 , c1 are expressed as,

⎧
⎨ a1 = vay − vby
b1 = vbx − vax (2.5)
⎩
c1 = (rbx vby − rby vbx ) − (rax vay − ray vax )
2.2 Trajectory Design Using the VCGF Method 11

Fig. 2.3 Orbit rendezvous in

2-D space

Fig. 2.4 Orbit rendezvous in

3-D space

In the 2-dimensional plane, through Eq. 2.4, it can be obtained that vector r0 =
[r0x , r0y ] is located on a line, because the intersection angle between vvga and vvgb
is smaller than π , the range of r0 can be determined as r0x ∈ (xrl , xru ), as shown
in Fig. 2.3.
Once parameters (r0x , fvga ) are given, (r0 , μvg ) can be calculated using Eqs. 2.2
and 2.4. Each parameter set (r0 , μvg ) can determine a virtual gravity. Therefore, all
those feasible trajectories can be parameterized as virtual conic orbits (Fig. 2.3), and
analytically expressed as virtual conic orbits. The process of finding rendezvous
trajectory for 3-D space is similar as that for 2-D space, as shown in Fig. 2.4.
In the VCGF method, the transfer angle for one single virtual conic orbit is less
than 2π , because the spacecraft returns to the initial state after one revolution in the
virtual gravity. For multi-revolutions trajectory design problem, the orbit patching
technique discussed in [20] is used. The whole trajectory is divided into a few seg-
ments at discrete points, and those discrete points are target points or just the control
points. Each segment is a virtual conic orbit, and the continuous low thrust trajectory
design can be transformed into multi-segments patching problem. More detail can
be obtained in Ref. [20].

2.2.1.2 Trajectory Optimization

Through the Keplerian orbit theory, the flight time of a spacecraft can be expressed
as follows,
12 2 Continuous Low Thrust Trajectory Design and Optimization

 fvgb avg 3 (1 − evg 2 )3 1
ft = f (fvga , r0x ) = df (2.6)
fvga μvg (1 + evg cos f )2

Similarly, the energy consumption can be expressed as,

 fvgb
fe = f (fvga , r0x ) = (F2 − F1 )df (2.7)
fvga

here set r is the position vector from the spacecraft to the Earth,

μ μvg hvg 2 1
F1 = , F 2 = , r vg = (2.8)
r 2 rvg 2 μvg 1 + evg cos f vg

In minimum fuel consumption case, the objective function is,

 
fvgb
μvg μ
fΔV = fmin (fvga , r0x ) = − 2 df (2.9)
fvga rvg 2 r

In this optimization problem, r0x and μvg are independent variables. Feasible
variables r0x , μvg need to be optimized, so as to minimize the flight time or the fuel
consumption.

2.2.1.3 Thrust Acceleration

In the geocentric coordinate system, equation of motion of the spacecraft under

continuous thrust can be written as,

d 2r μ
+ 3 r = T ac (2.10)
d t2 r
Here we assume that the thrust can form a virtual gravity, and then in the virtual
central gravitational coordinate system, the equation of spacecraft motion under
continuous thrust can be written as,

d 2 rvg μvg
+ rvg = 0 (2.11)
d t2 rvg 3

where

rvg = r − r0 . (2.12)

Because r0 is a constant in one virtual gravity, we have (r0 ) = 0. The position

vector and the velocity vector in the virtual centric coordinate system can be computed
by Eq. 2.13,
2.2 Trajectory Design Using the VCGF Method 13

Fig. 2.5 Force analysis in

the 2-dimensional space

Fig. 2.6 Force analysis in

the 3-dimensional space

rvg = r − r0
(2.13)
vvg = v

where v, vvg are velocity vectors in geocentric coordinates system and virtual central
gravitational coordinate system. Through Eqs. 2.10, 2.11and 2.12, the required thrust
acceleration (TA) is,

d 2r μ d 2 rvg μvg
T ac = 2
+ 3
r − 2
− rvg (2.14)
dt r dt rvg 3

As shown in Fig. 2.5, in 2-dimensional space, given r0 x, μvg , the thrust accelera-
tion required in 2-D space can be obtained by Eq. 2.15,
⎧
⎪
⎪ rvg = r − r0
⎪
⎪ μ
⎨ Tar1 = |r|2
μ
⎪ Tar2 = vg 2 (2.15)
⎪
⎪ | rvg |
⎪
⎩ Tac = Tar · sin arccos Tar2
Tar1

where Tar1 , Tar2 , Tac are magnitude of gravity acceleration, virtual central gravita-
tional acceleration and required TA respectively.
In case of 3-dimensional trajectory design, as shown in Fig. 2.6, rM is the position
vector of spacecraft at point M, and rvgM is the position vector at point M in the
14 2 Continuous Low Thrust Trajectory Design and Optimization

virtual gravity. Assuming TA is T ac = [Tacn2 , Tacθ2 ], where Tacn2 is aligned along

the radical direction, and Tacθ2 is aligned along the circulation direction; Tar1 is the
geocentric gravitation force acceleration, and Tar2 is the virtual centric gravitational
force acceleration; α is the angle between Tacθ2 and Tar2 , and β is the angle between
rM 1 and h2 , then the required TA can be computed as,
⎧
⎪ |rM ·hvg |
⎪ β = arccos |rM |·|hvg |
⎪
⎪
⎪
⎪
⎪ Tar1 = rMμ2
⎪
⎪
⎨ μ
Tar2 = rvgMvg 2
(2.16)
⎪
⎪
⎪
⎪ α = arcsin Tar1Tar2 sin β
⎪
⎪
⎪
⎪ Tacθ2 = Tar2 sin β cos α
⎪
⎩
Tacn2 = Tar1 cos β

2.2.2 Orbit Interception Trajectory Design Using VCGF

Method

The VCGF method can also be applied to designing less constrained interception
trajectory. In Eq. 2.2, the unknown variables r0 , fvga or r0 , μvg and the virtual conic
orbit in Fig. 2.2a are used. In two dimension problem, we can set r0 = 0 to simplify
the problem, as shown in Fig. 2.7. Given parameters ra , rb , va , it needs to obtain
feasible parameter μvg to solve Eq. 2.2. While in the three dimension problem, both
r0 , μvg are required to optimize. A root finding function f-zeros in MATLAB can be
used to solve this problem.
In the PSO algorithm, the free parameters are the magnitude of virtual gravity
μvg and the projection of r0 in x axis: r0 . The objective function can be the fuel cost
of whole trajectory or the flight time. Here we set the maximal number of iteration
in PSO algorithm are N. Take the orbit rendezvous trajectory design as an example.
Given the range of free variables r0x ∈ (r0xl , r0xu ), the initial point (rA , vA ) and the
terminal point B (rB , vB ).
There is a large amount of parameter set (r0 , μvg ), and each one corresponds to
a trajectory, and the feasible trajectories are those can satisfy boundary constraints.

Fig. 2.7 Optimal trajectory

using PSO algorithm
2.2 Trajectory Design Using the VCGF Method 15

The PSO algorithm is adopted to obtain feasible parameter set (r0 , μvg ) from their
ranges to determine the virtual gravity and the corresponding virtual conic orbits.
There are a few steps in the algorithm. To begin with, the range of free parameters is
set, and a large number of possible trajectories, same as the number of parameter sets
(r0 , μvg ), are obtained. Furthermore, those feasible trajectories which can satisfy the
constraints are chosen. Finally, the value of the objective function is calculated, and
the optimal solution is obtained. The process of PSO algorithm is listed as follow:
Step1: Find the ranges of free variables μvg ∈ (μvgl , μvgu ), r0x ∈ (r0xl , roxu );
Step2: Initialize the particle position with a uniformly distributed random vector
(r0x , μvg ) in their ranges;
Step3: Calculate r0y through Eq. 2.4; then the parameter set (r0 , μvg ) is obtained;
Step4: Calculate the radius and velocity state of initial point A (rvga , vvga ), and
terminal point B (rvgb , vvgb )in the virtual gravity through Eq. 2.1;
Step5: Calculate transfer angle from point A and point B in virtual gravity field
through Eq. 2.2;
Step6: Calculate the orbital elements of point A through its radius and velocity in
virtual gravity field, coevgA =[avgA , evgA , ivgA , wvgA , ΩvgA , T AvgA ], here T AvgA are the
virtual conic orbital inclination, argument of the periapsis, longitude of the ascending
node, and true anomaly respectively.
Step7: Calculate the true anomaly at point B in the designed virtual gravity
(T Avgb =T Avga + Δθ );
Step8: Calculate the radius and velocity rvgb , vvgb through the orbital elements,
coevgB =[avgA , evgA , ivgA , wvgA , ΩvgA , T AvgB ];
Step9: Calculate the radius error Δεr = rvgB − rvgb and the velocity error
Δεv = vvgB − vvgb ; if Δεr ≤ 10−4 , ΔεV ≤ 10−4 ; the trajectory in the virtual grav-
ity can satisfy the boundary constraint. Then record the parameter set (r0 , fvga )i ,
and calculate the fuel consumption (or the flight time) as its fitness; else if Δεr >
10−4 , ΔεV > 10−4 ; then set the fitness of this particle as 1;
Step10: Initialize the particle’s best-known position to its initial position; then
update the swarms best known position and its fitness; If the number of iteration
i < N ; return to step4. If the number of iteration i > N , finished; It should be noted
that, there is a large number of feasible virtual Keplerian orbits that can satisfy
boundary constraints, and the optimal one is the subcategory of them.

2.2.3 Mission Applications

In order to verify the effectiveness of the proposed method, three application exam-
ples of the VCGF method are presented in this section. The first one is the Earth-Mars
orbital transfer trajectory design. The proposed method was used to generate initial
guesses of continuous thrust rendezvous trajectory. Those solutions were compared
with the shape-based (SB) method given in [24], in terms of the magnitude and
the direction of thrust, transfer angle and fuel consumption. The second example
is a fuel-optimal Earth-Mars-Ceres flight trajectory mission discussed in the same
16 2 Continuous Low Thrust Trajectory Design and Optimization

reference. The whole trajectory consists of an interception trajectory from the Earth
to the Mars, and a rendezvous trajectory from the Mars to the Ceres. The effective-
ness of the VCGF method is evaluated by providing an initial guess for the direct
optimizer. Solutions of the VCGF method are compared with the three-dimension,
shape-based method solution discussed in [24]. Here the direct optimizer GPOPS,
a Matlab software for solving a nonlinear optimal control problems, is selected to
generate an accurate solution based on the proposed method and the shape-based
method. The last case is a collision-speed-maximal interception trajectory design
problem in Ref. [12]. In this example, a spacecraft is transferred from the Earth to
intercept with a hazardous asteroid. The asteroid 99942 Apophis is the potentially
hazardous asteroid on an impact trajectory toward the Earth. The aim of optimiza-
tion is to maximize the relative speed between the spacecraft and the asteroid. For
well-documented reasons, heliocentric canonical units were used. In this paper, the
distance units (AU) and the time units (TU) are: 1 DU = 1 (AU) = 149596000 km, 1
TU = 1/2 π year = 58.17 days. All of the examples have been performed on an Intel
Core 2.6GHz with Windows 8. The computation time of optimization is calculated
by MATLAB tic-toc command.
Example A: The Earth-Mars orbit rendezvous
In this example, the two-dimension Earth-Mars orbit is performed. A spacecraft is
transferred from the Earth to rendezvous with the Mars. The boundary conditions
are listed in Table 2.1. Here, in order to analyze the relationship between the transfer
angle, the required thrust and the fuel cost, the VCGF method is applied to generate
initial guesses in case of three transfer angles (Δ θ ): a half revolution (Δ θ = π ),
one revolution (Δ θ = 2π )and two revolutions(Δ θ = 4π ). Using the VCGF method,
those feasible trajectories are expressed as virtual conic orbits and parameterized by
the parameter set (r0 , μ2 ). For one virtual conic trajectory, the transfer angle is less
than 2π (Δθ < 2π ). While in case of multi-revolutions condition (Δθ > 2π ), the
whole trajectory is divided into a few segments at discrete points, and each segment
corresponds to a virtual conic orbit. In order to testify the suitability of proposed
method, the results are compared with the five-degree inverse polynomial shaped
based method, in terms of magnitude and direction of required thrust, the flight time
and the fuel consumption. The boundary conditions of the Earth-Mars rendezvous
mission are listed in Table 2.1.
The transfer angles of those cases are assumed to be π, 2π, 4π , respectively. The
corresponding flight times are set as 3.75TU, 7.683TU and 16.69 TU respectively.
Trajectories generated by the VCGF method and the shape-based method are shown
in Fig. 2.8 (for the case of n = 2). The required thrust for all three cases is shown
in Figs. 2.9, 2.10 and 2.11. Table 2.2 shows the parameters of the virtual gravity

Table 2.1 Initial and final conditions for transfer trajectory

ri = 1AU θi = 0 ṙi = 0 θ̇i = 0.6564
rf = 1.52AU θf = nπ ṙf = 0 θ̇f = 0.5333
2.2 Trajectory Design Using the VCGF Method 17

Fig. 2.8 Rendezvous orbits

(n = 4)

Fig. 2.9 TA for rendezvous

orbits (n = 1)

Fig. 2.10 TA for rendezvous

orbits (n = 2)
18 2 Continuous Low Thrust Trajectory Design and Optimization

Fig. 2.11 TA for rendezvous

orbits (n = 3)

obtained by the VCGF method. The simulation results show that, in the shape-based
method, the maximum thrust accelerations are 0.3854AU/T U 2 , 0.1473AU/T U 2 ,
0.0372AU/T U 2 ; and the fuel costs are 0.638AU/T U , 0.559AU/T U and
0.3688AU/T U respectively; For the VCGF method, the maximum thrust
accelerations are 0.3561AU/T U 2 , 0.1852AU/T U 2 , 0.1096AU/T U 2 ; and corre-
sponding fuel consumptions are 0.6723 AU/TU, 0.6371 AU/TU, 0.6176 AU/TU
respectively. From the simulation results we know that there are two main differ-
ences between initial guesses generated by the VCGF method and the shape-based
method. Firstly, in the shape-based method, the magnitude of thrust and the fuel con-
sumption decrease greatly as increase of transfer angle. Secondly, compared with
the shape-based method, the VCGF needs larger thrust and more fuel consumption,
especially in the case of long flight time. The main reasons for the differences are
that, in the shape-based method, the tangential thrust is consist with large circumfer-
ential thrust and small radical thrust in the shape based method, as shown in Figs. 2.9,
2.10 and 2.11. The magnitude of thrust is determined by the transfer angle of trajec-
tory. The mathematical demonstration can be obtained from the dynamics model of
shape-based method in [24]. However, in the VCGF method, the key factor affecting
the virtual gravity field is the magnitude of virtual gravity and the position in the near
circular orbital rendezvous case, but those two parameters are not affected greatly
by the increase of the transfer angle. Thus, the required thrust acceleration does not
change greatly. Furthermore, the required thrust is consist with higher radical thrust
and lower circumferential thrust in the VCGF method. Conversely, the thrust in the
shape based method is consist with higher circumferential thrust and lower radical
thrust. While compared with orbital maneuver using radical thrust, it is more effi-
cient using circumferential thrust than that of radical thrust in terms of fuel cost, but
usually needs more flight time [6]. Thus, a conclusion that can be safely obtained
is, the shape-based method is more efficient than the VCGF method in terms of fuel
2.2 Trajectory Design Using the VCGF Method 19

Table 2.2 Parameters of the VCGF

Transfer angles r0 μvg
π, (n = 0.5) r0 = (−0.1286, 0) μvg = 0.8188
π, n = 1 r0 = (−0.05490) μvg = 0.9063
r0 = (−0.0748, 0) μvg = 0.8974
4π n = 2 r0 = (−0.03, 0) μvg = 0.9444
r0 = (−0.025, 0) μvg = 0.9573
r0 = (−0.0387, 0) μvg = 0.9415
r0 = (−0.0362, 0) μvg = 0.9504

Table 2.3 Orbit elements of Mars and Ceres (MJD2000:4748.5)

Planet a/AU e i/deg Ω/deg ω/deg f/deg
Mars 1.524 0.0934 1.8506 348.7422 114.2112 41.990
Ceres 2.7658 0.078 10.607 80.329 72.522 57.3248

cost, especially in case of long flight time rendezvous. Indeed, from the perspective
of trajectory shape, the VCGF method can be regarded as a special case of the shape
based method, because each arc generated by the VCGF method is a conic, and its
shape is determined by variables. But the difference lies in, the required thrust in the
VCGF method is not necessary to be aligned along the direction of the spacecraft
velocity, and it needs large radical thrust and small circumferential thrust to form the
virtual gravity.
Example B: The Earth-Mars Flyby-Ceres flight trajectory
In this example, the proposed method is applied in a more complicated problem given
in [12]: the Earth-Mars-Ceres rendezvous mission. The flight trajectory includes an
interception trajectory form the Earth to the Mars, and a rendezvous trajectory from
the Mars to the Ceres. The spacecraft is given the same launch window: year 1990–
2049 and the same launch velocity ranges. The total time of flight constraints: less
than 1133 days. The classical orbital elements of the Mars and the Ceres are listed
in Table 2.3. The planetary orbital elements corresponding to the date on January 01,
2013 (MJD 2000: 4748.5) are interpreted from the HORIZONS Web Interface of
JPL, and the free parameters boundaries are listed in Table 2.4.
The aim of preliminary design using proposed method is to provide a reason-
able initial guess to approximately determine the optimal launch time, the time of
flight and revolutions to more accurate optimizer. In order to verify the suitability of
the VCGF method, the three-dimensional shape-based method proposed in [24] is
applied in this mission. After obtaining the initial guess, a direct method GPOPS, is
applied to find the optimal result. A total number of 30 nodes are used for the Earth-
Mars-Ceres trajectory. The thrust acceleration is assumed constant, 0.2694DU/T U 2 .
In the optimization process, there are seven state parameters (r, θ , z, vr , vθ , m), and
three controls parameters (Tn , Tc , Tr ).
20 2 Continuous Low Thrust Trajectory Design and Optimization

Table 2.4 Orbit elements of Mars and Ceres (MJD2000:4748.5)

Optimization parameter Bounds
Launch date 01/01/1990-12/31/2049
Magnitude of V∝ at Earth departure [0.75,200] km/s2
In plane angle of V∝ at Earth departure [−π ,π ] rad
Out plane angle of V∝ at Earth departure [−π ,π ] rad
Time of flight from Earth to Mars [0,3] years
Magnitude of V∝ at Mars arrival [0,3] km/s2
In-plane angle of V∝ at Mars arrival [−π ,π ] rad
Out-plane angle of V∝ at Mars arrival [−π ,π ] rad
Out-of-plane angle of Mars flyby plane [−π ,π ] rad
Time of flight from Mars to Ceres [0,3] years

Table 2.5 The comparison of preliminary design and accurate optimization

Flight parameters Shape-based Optimal result VCGF Optimal result
Launch date 2016/05/29 2016/07/26 2016/07/23 2016/09/17
(yy/mm/dd)
Arrive date 2019/07/06 2019/06/30 2019/02/04 2019/09/25
(yy/mm/dd)
Time-of-flight 1133 1130 927 1103
(day)
Final mass/initial 0.290 0.252 0.4270 0.3210
mass
CPU time (s) 686.9 1560 653.3 1023.3

The optimal variables in the preliminary stage and optimization stage are shown
in Table 2.5. The final optimal Earth-Mars-Ceres trajectory using the VCGF initial
guess is shown in Fig. 2.12, and the direction of optimal thrust is shown in Fig. 2.13.
From the simulation results, two conclusions can be obtained. Firstly, the optimal
result based on the VCGF method is larger than that of the shape-based method
in terms of fuel consumption. It is important to point out that, in order to form
the virtual gravity field, the required thrust is consist with large radical thrust and
small circumferential thrust; hence in terms of fuel cost, it is less efficient than the
shape-based method which uses tangential thrust. This result is consistent with the
conclusion obtained in Example A. However, it should be noticed that the VCGF
method is more efficient than the shape based method in terms of computation time,
just as shown in Table 2.5. The main reason lies in the fact that the thrust direction
is assumed to be along the tangent direction in the shape based preliminary design.
However, the optimal results using the shape-based method are inconsistent with
the assumption. Conversely, in the VCGF method, thrust directions are determined
by the parameters of virtual gravity field, which is similar as that of optimal result.
2.2 Trajectory Design Using the VCGF Method 21

Fig. 2.12 The Earth-Mars-Ceres trajectory

Fig. 2.13 The thrust

direction in the
Earth-Mars-Ceres trajectory
mission

Secondly, the VCGF method can parameterize feasible trajectories using only two
variables, so it is simpler to get the initial guess for continuous thrust trajectory.
Example C: Collision-speed, maximal interception trajectory
In order to verify the efficiency of the proposed method, it is applied in the near earth
asteroid collision-speed maximal interception trajectory design and optimization
problem given in [9]. The relative velocity between the spacecraft and the hazardous
asteroid is considered as the performance index, and the aim of optimizing is to max-
imize it. Here Asteroid 99942 Apophis is assumed to be the target of the potentially
hazardous asteroid on an impact trajectory toward the Earth. The launch window is
supposed to be located between January 1, 2016 and January 1, 2018. The departure
energy is assumed to be no more than 1. The time-of-light is limited to be no more
than 800 days, and the search step is set as 5 days. Here the solar-electric propulsion
model is utilized, and the performance parameters of solar-electric propulsion engine
are given in [9]. The classical orbital elements of the Earth and the Apophis are listed
in Table 2.6. The planetary orbital elements corresponding to the date on January 01,
2013 (MJD2000: 4748.5) are interpreted from the HORIZONS Web Interface of
JPL.
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