EARTH TO HALO ORBIT TRANSFER TRAJECTORIES
A Thesis
Submitted to the Faculty
of
Purdue University
by
Raoul R. Rausch
In Partial Fulfillment of the
Requirements for the Degree
of
Master of Science
August 2005
ii
ACKNOWLEDGMENTS
This research topic has been challenging and frustrating at times, rewarding and
fulfilling at others. I want to thank my advisor Professor Howell for her continuous
support and guidance and the seemingly infinite patience. The advice and recommendations she provided on reviews of this thesis, certainly surpassed anything that
could have been expected.
I also wish to thank the other members of my graduate committee, Professors
James M. Longuski and Martin Corless for their advice and reviews of this thesis.
I am grateful to all the past and present members of my research group. They have
provided much support and guidance and without their contributions, this work would
have been difficult to complete. They have helped me enhancing my understanding of
the three-body problem and inspired me to think about the technical issues at hand
more globally.
Additionally, I would like to thank my parents and my wife, Nicole, for their continuous support. Their combined energy has given me at times the extra motivation,
strength and confidence necessary to complete this work.
Finally, I wish to thank those who have provided the funding for my graduate
studies. For the last three years, I have been funded by the German section of the
Purdue School of Foreign Languages and Literatures. Teaching German has been an
enlightening and educational experience for me.
iii
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Previous Contributions . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1
Historical Overview . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.2
Transfer Trajectories . . . . . . . . . . . . . . . . . . . . . . .
8
Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2 BACKGROUND: MATHEMATICAL MODELS . . . . . . . . . . . . . . .
12
1.3
2.1
Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.1.1
Inertial Frame . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.1.2
P1 − P2 Rotating Frame . . . . . . . . . . . . . . . . . . . . .
13
2.1.3
Earth Centered “Fixed” Frame . . . . . . . . . . . . . . . . .
13
2.2
Transformations Between Different Frames . . . . . . . . . . . . . . .
15
2.3
Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.4
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.4.1
Singularities in the Equations of Motion . . . . . . . . . . . .
20
2.5
State Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.6
Differential Corrections . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.7
Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.8
Invariant Manifold Theory . . . . . . . . . . . . . . . . . . . . . . . .
28
2.8.1
Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.8.2
Periodic Orbits and Dynamical Systems Theory . . . . . . . .
33
iv
Page
2.9
Computing Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.10 Transition of the Solution to the Ephemeris Model . . . . . . . . . . .
42
3 TRANSFERS FROM EARTH PARKING ORBITS TO LUNAR L1 HALO
ORBITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.1
Stable and Unstable Flow that is Associated with the Libration Point
L1 in the Vicinity of the Earth . . . . . . . . . . . . . . . . . . . . . .
45
Design Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.2.1
Shooting Technique . . . . . . . . . . . . . . . . . . . . . . . .
49
3.2.2
Investigated Transfer Types . . . . . . . . . . . . . . . . . . .
50
Two-Level Differential Corrector . . . . . . . . . . . . . . . . . . . . .
51
3.3.1
First Step - Ensuring Position Continuity . . . . . . . . . . . .
52
3.3.2
Second Step - Enforcing Velocity Continuity . . . . . . . . . .
53
Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.4.1
Position and Epoch Constraints . . . . . . . . . . . . . . . . .
57
3.4.2
Parking Orbit Constraints . . . . . . . . . . . . . . . . . . . .
58
3.4.3
| ∆v̄ | Constraints . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.5
Direct Transfer Trajectories from Earth to Lunar to L1 Halo Orbits .
60
3.6
Transfer Trajectories with a Manifold Insertion
. . . . . . . . . . . .
64
3.7
Effects of a Cost Reduction Procedure . . . . . . . . . . . . . . . . .
66
3.8
Free Return Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.9
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
71
3.2
3.3
3.4
3.9.1
Numerical versus Dynamical Issues in the Computation of Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4 TRANSFERS FROM EARTH PARKING ORBITS TO
SUN-EARTH LIBRATION POINT ORBITS . . . . . . . . . . . . . . . . .
80
3.9.2
4.1
Stable Flow from the Libration Points in the Direction of the Earth .
80
4.2
Selection of Halo Orbit Sizes . . . . . . . . . . . . . . . . . . . . . . .
81
4.3
Transfer Trajectories From Earth to L1 Halo Orbits . . . . . . . . . .
83
v
Page
4.4
Transfer Trajectories From Earth to L2 Halo Orbits . . . . . . . . . .
84
4.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5 LAUNCH TRAJECTORIES . . . . . . . . . . . . . . . . . . . . . . . . . .
91
5.1
Equation of Motions with Constant Thrust Term . . . . . . . . . . .
91
5.2
State Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . .
94
5.3
Patch Points and Initial Trajectory . . . . . . . . . . . . . . . . . . .
95
5.3.1
96
5.4
Two-Level Differential Corrector with Thrust
. . . . . . . . . . . . .
97
Two-Level Differential Corrector with Thrust . . . . . . . . .
97
Trajectory from Launch Site into Parking Orbit . . . . . . . . . . . .
98
5.4.1
5.5
5.5.1
5.6
Determination of the Launch Site . . . . . . . . . . . . . . . .
Challenges with the Launch Formulation . . . . . . . . . . . . 100
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6 SUMMARY AND RECOMMENDATIONS . . . . . . . . . . . . . . . . . . 105
6.0.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.0.2
Recommendations and Future Work . . . . . . . . . . . . . . . 106
6.0.3
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 108
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
vi
LIST OF TABLES
Table
3.1
3.2
3.3
4.1
4.2
5.1
Page
Transfer Costs for Two Differently Sized Halo Orbits; TTI Maneuver
Constrained to the x − z Plane Crossing . . . . . . . . . . . . . . . .
60
Transfer Costs for Two Differently Sized Halo Orbits; Location of the
TTI Maneuver Determined by the Differential Corrections Scheme. .
65
Transfer Costs for Transfers with a Manifold Insertion for Two Differently Sized Halo Orbits. . . . . . . . . . . . . . . . . . . . . . . . . .
65
Transfer Costs for Transfers from a 200 km Altitude Earth Parking
Orbit to Two Differently Sized Sun-Earth L1 Halo Orbits . . . . . . .
86
Transfer Costs for Transfers from a 200 km Altitude Earth Parking
Orbit to Two Differently Sized Sun-Earth L2 Halo Orbits . . . . . . .
86
Changes in Thrust Parameters Throughout Trajectory Arc. . . . . . . 100
vii
LIST OF FIGURES
Figure
Page
2.1
Geometry of the Three-Body Problem. . . . . . . . . . . . . . . . . .
14
2.2
Zero Velocity Curve for C = 3.161. . . . . . . . . . . . . . . . . . . .
21
2.3
A Stylized Representation of 1 Step Differential Corrector. . . . . . .
24
2.4
Location of the Libration Points in the Earth-Moon System
Relative to a Synodic Frame. . . . . . . . . . . . . . . . . . . . . . . .
26
2.5
Northern Earth-Moon L1 Halo Orbit in the CR3BP. . . . . . . . . . .
28
2.6
Lissajous Trajectory at L1 in the Sun-Earth CR3BP. . . . . . . . . .
29
2.7
Stable and Unstable Eigenvectors and the Globalized Manifold for the
Earth-Moon L1 Point. . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Stable Eigenvectors of the Monodromy Matrix for an
Earth-Moon L1 Halo Orbit. . . . . . . . . . . . . . . . . . . . . . . .
38
Position and Velocity Components of the Stable Eigenvectors
of the Monodromy Matrix for an Earth-Moon L1 halo. . . . . . . . .
39
2.10 Various Manifolds Asymptotically Approaching the Orbit. . . . . . .
40
2.11 Stable (blue) and Unstable (red) Manifolds for a Sun-Earth Halo Orbit
near L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.12 Stable (blue) and Unstable (red) Manifold Tube Approaching the Earth
in the Sun-Earth System. . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.13 A ”Halo-like” Lissajous Trajectory in a Ephemeris model
with an Az Amplitude of approximately 15,000 km. . . . . . . . . . .
44
3.1
Stable and Unstable Manifold Tubes in the Vicinity of the Earth. . .
46
3.2
Minimum Earth Passing Altitudes for Trajectories on the Manifold
Tubes Associated with Various Earth-Moon L1 Halo Orbits. . . . . .
47
A Stylized Representation of Level II Differential Corrector
(from Wilson [1]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Transfer from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 15, 000 km) in the CR3BP (Location of HOI maneuver is constrained). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
2.8
2.9
3.3
3.4
viii
Figure
3.5
Page
Transfer from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 43, 800 km) in the CR3BP (Location of HOI maneuver is constrained). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az =
15, 000 km) in an Ephemeris Model (Location of HOI Manuever Constrained). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az =
43,800 km) in an Ephemeris Model (Location of HOI Manuever Constrained). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Transfer from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 15,000 km) in the CR3BP (Location of the TTI Maneuver
Determined by the Differential Corrections Scheme.) . . . . . . . . . .
66
Transfer from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 43,800 km) in the CR3BP (Location of the TTI Maneuver
Determined by the Differential Corrections Scheme.) . . . . . . . . . .
67
3.10 Transfer from an Earth Parking Orbit to L1 Halo Orbits
(Az = 15,000 km) in an Ephemeris Model (Location of the TTI Maneuver
Determined by the Differential Corrections Scheme.) . . . . . . . . . .
68
3.11 Transfer from an Earth Parking Orbit to L1 Halo Orbits
(Az = 43,800 km) in an Ephemeris Model (Location of the TTI Maneuver
Determined by the Differential Corrections Scheme.) . . . . . . . . . .
69
3.12 Transfer from an Earth Parking Orbit with a Manifold Insertion
into an L1 Halo Orbit (Az = 15,000 km) in the CR3BP. . . . . . . . .
70
3.13 Transfer from an Earth Parking Orbit with a Manifold Insertion
into an L1 Halo Orbit (Az = 43,800 km) in the CR3BP. . . . . . . . .
71
3.14 Transfer from an Earth Parking Orbit with a Manifold Insertion
into an L1 Halo Orbit (Az = 15,000 km) in an Ephemeris Model. . . .
72
3.15 Transfer from an Earth Parking Orbit with a Manifold Insertion
into an L1 Halo Orbit (Az = 43,800 km) in an Ephemeris Model. . . .
73
3.6
3.7
3.8
3.9
3.16 Effects of a Cost Reduction Procedure on the Transfer
Arcs Initially Using the Invariant Manifold on the Near Earth Side for
a
Halo Orbit with an Az Amplitude of 15,000 km in an Ephemeris Model. 74
ix
Figure
Page
3.17 x − y Projection of a Free Return Trajectory to a Halo Orbit
with an Az Amplitude of 15,000 km in an Ephemeris Model. . . . . .
75
3.18 x − z Projection of a Free Return Trajectory to a Halo Orbit
with an Az Amplitude of 15,000 km in an Ephemeris Model. . . . . .
76
3.19 y − z Projection of a Free Return Trajectory to a Halo Orbit
with an Az Amplitude of 15,000 km in an Ephemeris Model. . . . . .
77
4.1
Closest Approach Altitudes for L1 Sun-Earth Manifolds
Relative to the Earth. . . . . . . . . . . . . . . . . . . . . . . . . . .
82
Transfer from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 120, 000 km) in the CR3BP. . . . . . . . . . . . . . . . . . . .
84
Transfers from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 440, 000 km) in the CR3BP. . . . . . . . . . . . . . . . . . . .
85
Transfers from an Earth Parking Orbit to a L1 Halo Orbit
in an Ephemeris Model. . . . . . . . . . . . . . . . . . . . . . . . . .
87
Transfer from an Earth Parking Orbit to a L2 Halo Orbit
(Az = 120, 000 km) in the CR3BP. . . . . . . . . . . . . . . . . . . .
88
Transfers from an Earth Parking Orbit to a L2 Halo Orbit
(Az = 440, 000 km) in the CR3BP. . . . . . . . . . . . . . . . . . . .
89
Transfers from an Earth Parking Orbit to a L2 Halo Orbit
in an Ephemeris Model. . . . . . . . . . . . . . . . . . . . . . . . . .
90
Spherical Coordinates to Define the
Direction of the Thrust Vector. . . . . . . . . . . . . . . . . . . . . .
93
5.2
Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit.
101
5.3
Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit.
102
5.4
Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit.
103
4.2
4.3
4.4
4.5
4.6
4.7
5.1
x
ABSTRACT
Rausch, Raoul R. M.S., Purdue University, August, 2005. Earth to Halo Orbit
Transfer Trajectories. Major Professor: Kathleen C. Howell.
Interest in libration point orbits has increased considerably over the last few
decades. The Lunar L1 and L2 libration points have been suggested as gateways
to Sun-Earth libration points and to interplanetary space. The dynamics in the
vicinity of the Earth in the Earth-Moon system, where the Earth is the major primary in a three-body model, has only been of limited interest until recently. The new
lunar initiative is the origin of a wide range of studies to support the infrastructure
for a sustained lunar presence. A systematic and efficient approach is desirable to
ease the determination of viable transfer trajectories satisfying mission constraints.
An automated process is, in fact, a critical component for trade-off studies. This
work presents an initial approach to develop such a methodology to compute transfers from a launch site or parking orbit near the Earth to libration point orbits in
the Earth-Moon system. Initially, the natural dynamics in three-body systems near
both the smaller and larger primaries are investigated to gain insight. A technique
using a linear differential corrections scheme is then developed. Initial attempts to
incorporate the invariant manifolds structure in designing these transfers are presented. Simple transfers in the Earth-Moon system are computed and transitioned to
an ephemeris model. The methodology is also successfully applied in the Sun-earth
system. Challenges are discussed. The second task involves the determination of
launch trajectories.
A two-level differential corrections technique incorporating a constant thrust term
is developed and a sample launch scenario is computed. Limitations and extensions
are considered.
1
1. INTRODUCTION
Today, humankind continues to explore a new frontier as civilization expands into
space. In the near term, plans include a potential return to the Moon with both
robotic vehicles as well as human crews; an extended stay on the lunar surface or in
the vicinity of the Moon is possible. In addition to the exploration of the Moon (and
Mars), many other missions and observatories have been proposed that will make use
of libration point orbits, such as the James Webb Space Telescope [2] (formerly known
as the Next Generation Telescope), the Terrestrial Planet Finder [3] and the Europa
Orbiter Mission [4]. From these various scenarios, some of the new observatories will
take advantage of the prime location offered by the Sun-Earth libration points at L1
and L2 and of the efficient low-energy trajectories that are available throughout the
solar system. These low energy pathways are defined via the manifolds associated with
libration point orbits in all Sun-Planet and Planet-Moon three-body systems. These
pathways can be exploited due to only minor energy differences [5] between libration
point manifolds in different systems. The entire system of “tunnels” is made possible
by the chaotic environment resulting from multiple gravity fields. As stated by Lo et
al. [5] “...the tunnels generate deterministic chaos and for very little energy, one can
radically change trajectories that are initially close by.” This statement effectively
summarizes the physical basis and practicality of libration point trajectories. Future
space observatories can take advantage of the proximity of the Earth-Moon libration
points L1 and L2 and use them as inexpensive gateways [6] to the Sun-Earth libration
points L1 and L2 and to interplanetary space. The relatively short distance between
the Earth and the Earth-Moon lunar libration point L1 makes human servicing of
observatories possible [7]. A spacecraft can be delivered to this location within a
week from the Earth and within hours from the Moon’s surface [5]. In addition to
2
easy accessibility, spacecraft in an orbit near the Earth-Moon L1 libration point can
be continuously monitored from Earth.
The first spacecraft to make use of a libration point orbit was the International
Sun-Earth Explorer (ISEE-3) launched on November 20, 1978 [8]. The mission was
very successful at monitoring the solar winds. Subsequently, the vehicle was rerouted
to explore the Earth’s geomagnetic tail region before shifting to a new trajectory
arc that ultimately encountered the comet Giacobini-Zinner. The spacecraft was
renamed the International Cometary Explorer (ICE) [9] and, after a few close encounters with both the Earth and the Moon, ICE reached the comet on schedule.
Additional spacecraft have also been launched as part of successful libration point
missions, including WIND [10], SOHO [11], ACE [12], Genesis [13], and MAP. The
WIND, SOHO, and ACE missions were all part of the International Solar-Terrestial
Physics project. Genesis was the first mission to exploit dynamical systems theory
in the design and planning phases. This new analysis concept allowed the Genesis
spacecraft to take advantage of the complex dynamics in a multi-body regime, that is,
the Sun-Earth-Moon gravitational fields, and to fulfill the mission requirements with
a completely ballistic baseline trajectory [5] that required no maneuvers. Genesis also
returned the first samples of solar wind particles to the Earth from beyond the local
environment.
1.1
Problem Definition
The libration points L1 and L2 have been proposed as inexpensive gateways [5–7]
for performing transfers between different three-body systems. Many different types of
libration point trajectories are of current interest with the potential for inexpensive access to interplanetary space. The concept of system-to-system transfers was suggested
as early as 1968 [14]. New techniques, for example, the use of the invariant manifold
structure associated with three-body systems, that simplify the design of system-tosystem transfer are currently in development by various researchers [6, 15, 16]. How-
3
ever, servicing and repair missions to observatories in libration point orbits (LPO)
must meet very specific requirements such as time of arrival and precise target orbit
requirements [7]. Libration point missions involving humans must also have feasible
Earth return options in case of emergency. Meeting such requirements can quickly become a nearly impossible task. Therefore, future libration point missions will require
new and innovative design strategies to fulfill the mission requirements at low fuel
costs while satisfying an increasingly complex set of constraints. Two types of libration point orbits commonly investigated for applications are the three-dimensional
precisely periodic halo orbits and the quasi-periodic Lissajous trajectories. Much
progress has been made in the last decade by developing design strategies based on
these orbits and their associated manifolds.
One problem that influences every spacecraft is the launch. For libration point
missions, in particular, designing the launch leg, from a site on the Earth’s surface to a
transfer trajectory that delivers the vehicle into a libration point orbit, is challenging.
Traditionally, spacecraft enroute to libration point orbits have been first launched
into low Earth parking orbits, then depart along their transfer path to insert into a
baseline halo orbit or Lissajous trajectory. For preliminary analysis, this approach
allows the use of traditional two-body analysis tools to determine the trajectory arc
from the launch site into the parking orbit. Both the Earth-Moon and Sun-Earth
systems present unique challenges.
Regardless of the particular three-body system, determining any solution in an
ephemeris model without good baseline approximations is a tremendously complex
task. Simplified models are usually employed to obtain baseline solutions. However,
using a two-body model limits the solution-space considerably when the seemingly
chaotic motion present in a multi-body system is lost. To expose potential non-conic
solutions, a three-body model is therefore necessary. But, there are no known closedform solutions to the three-body problem (3BP). Some simplifying assumptions do
result in a model that retains the significant features of three-body motions; slight
modifications then offer a very useful formulation. The first assumption is one con-
4
cerning the masses; that is, the bodies are all assumed spherically symmetric, and
thus point masses. In fact, assuming one particle to be infinitesimal is key for another approximation. The Restricted Three-Body Problem (R3BP) consists of two
massive particles moving in undisturbed two-body orbits about their common center
of mass, the barycenter. It is assumed that the infinitesimal particle does not affect
the orbits of the primaries. If the relative two-body orbit of the primaries is assumed
elliptic, the resulting model is called the Elliptic Restricted Three-Body Problem
(ER3BP). If the two-body orbit of the primaries is assumed circular, the Circular
Restricted Three-Body Problem (CR3BP) yields even more insight into the motion.
Although the model defined as the CR3BP is the key to the solution of interest,
the design process remains challenging. Efficient and effective tools for the computation of transfer trajectories from the Earth to libration point orbits in both the
Sun-Earth and the Earth-Moon systems are of increasing interest. In the three-body
Sun-Earth-spacecraft system, the Earth is the smaller of the two primaries whereas
in the three-body Earth-Moon-spacecraft system, the Earth is the larger of the primaries. This difference is significant in the fundamental dynamical structure of the
solution space. The role that a planet, here the Earth, plays in a simplified model
as the larger or the smaller primary can have a large influence on the behavior and
characteristics of the solutions. First, the natural flow approaching and departing
libration point orbits in both cases requires investigation. A deterministic maneuver
will always be required to leave the Earth parking orbit. Additional maneuvers to
insert into the halo orbit, or into the transfer trajectory, may be necessary to satisfy
design requirements. Finally, tools to compute thrusting arcs from the launch site to
the transfer path require development.
5
1.2
Previous Contributions
1.2.1
Historical Overview
Investigations in the 3BP from a dynamical perspective began with Newton. His
development of the laws of motion and the concept of a gravity force, first published
in the Principia [17], enabled the formulation of a mathematical model. At the time,
predicting the motion of the Moon was of great interest with applications in naval
navigation [18]. Newton, of course, solved the two-body problem after reformulating
it as a problem in the relative motion of two bodies. But, to accurately predict the
Moon’s orbit, perturbations had to be incorporated. This led to the formulation
of the 3BP. A complete solution to the gravitational 3BP requires 18 integrals of
motion. Since only a total of ten integrals of motion exist, as later determined by
Euler [19], a closed-form solution was not straightforward. Six of these scalar integrals
result from the conservation of linear momentum, three from the conservation of total
angular momentum, and one integral results from the conservation of energy. In his
investigations of the 3BP formed by the Sun, the Earth, and the Moon, Newton
nevertheless managed to compute the motion of the lunar perigee to within eight
percent of the observed value in 1687. Continuing Newton’s work, Leonard Euler
proposed the highly special problem of three bodies known as the “problem of two
fixed force-centers,” solvable by elliptic functions, in 1760 [20]. Euler was also the
first to formulate the restricted three-body problem (R3BP) in a rotating frame. The
formulation was very significant. This step allowed him to predict the existence of
the three collinear equilibrium points L1 , L2 , and L3 [20]. Lagrange confirmed Euler’s
prediction in his memoir “Essai sur le problème des T rois Corps” published in 1772.
Lagrange deduces the existence of the collinear points and solves for two additional
equilibrium points. These additional two equilibrium points each form an equilateral
triangle with the primaries and are generally labelled the equilateral libration points
L4 and L5 . The five points are commonly denoted Lagrange or libration points. For
their work, Euler and Lagrange shared the P rix de l′ Academie de P aris in 1772 [18].
6
In the pursuit of additional integrals, Jacobi considered the concept of the R3BP
relative to a rotating frame. In 1836, he determined a constant of the motion by combining conservation properties of energy and angular momentum [21]. The constant
he discovered now carries his name [18]. In other research efforts, Hill’s Lunar Theory,
published in 1878, represents the result of an investigation of a satellite’s orbit around
a larger planet under the influence of solar and eccentric perturbations. Specifically,
Hill was interested in modelling the lunar orbit with the simplifying assumptions that
the solar eccentricity and parallax, as well as the Moon’s orbital inclination, are all
zero [21]. Hill’s work was revolutionary; he used the CR3BP as the base model and
was the first person to abandon a two-body analysis [22]. In his theory, Hill demonstrated that for a specified energy level, regions of space exist where motion is not
physically permitted [18].
Toward the end of the 19th century, Poincaré studied the 3BP seeking additional
integrals of the motion. Poincaré predicted an infinite number of periodic solutions
if two of the masses are small compared to the third [23]. Poincaré believed that
the primary problem in celestial mechanics was the behavior of orbits as time goes
to infinity [24] and focused on qualitative aspects of the motion. The planar R3BP
was of particular interest to him because it could be formulated as a Hamiltonian-like
system. Determined to investigate the problem further, he invented an analytical
technique called the ‘surface of section’ [24] that allowed him, in 1892, to describe the
phase space of a non-integrable system. Through his contributions to dynamics and
the invention of the method of the ‘surface of section,’ Poincaré is widely regarded
as the father of Dynamical Systems Theory (DST). In 1899, he proved that Jacobi’s
Constant is the only integral of motion that exists in the R3BP. Any other integral
would not be an analytical function of the systems coordinates, momenta, and the
time [24]. Poincaré’s insight into the 3BP, his contributions to mathematics, and
the eventual advent of high-speed computing makes most of today’s work possible.
At the time, his findings also resulted in a shift in the focus of research in the 3BP
toward determining specific trajectories rather than the general behavior.
7
At the beginning of the 20th century, Darwin, Plummer, and Moulton were all
seeking periodic orbits in the vicinity of the libration points [23, 25–28]. In 1899,
Darwin computed several approximate planar periodic orbits in the CR3BP using a
quadrature method. Plummer also accomplished the same feat in 1902 using an approximate, second-order analytical solution to the equations of motion in the CR3BP.
Between 1900 and 1917, Moulton developed several approximate analytical solutions
to the linearized equations of motion relative to the collinear points. Moulton’s solutions result in planar as well as three-dimensional periodic orbits [21]. Although
a series solution to the general three-body problem was produced by Sundman in
1912 [19, 29], it is useless for any practical purposes. Further exploration into Libration Point Orbits (LPO) were hindered by the computational requirements. Only
with the introduction of high-speed computers in the 1960’s did significant progress
occur.
In the late 1960’s, interest in the three-body problem (3BP) increased significantly. In 1967, Szebehely published a book summarizing all information to date
concerning the 3BP [21]. His compilation contained numerically integrated, periodic orbits in the planar CR3BP and the planar ER3BP as well as a few threedimensional notes. Motivated by new mission possibilities at NASA, Farquhar developed analytical approximations for three-dimensional periodic orbits in the translunar
Earth-Moon region [14] in the late 1960’s. Farquhar coined the term “halo” orbits
to describe periodic three-dimensional orbits in the vicinity of the collinear libration points because, when viewed from the Earth, they appear as a halo around the
Moon. Whereas halo orbits are precisely periodic in the CR3BP, Lissajous trajectories are quasi-periodic. Lissajous figures, in general, are named after the French
physicist Jules A. Lissajous (1822-1880) because planar projections of these curves
look similar to those studied by Lissajous in 1857 [9]. In 1972, Farquhar and Kamel
developed a third-order approximation for quasi-periodic motion near the translunar
libration point using a Lindstedt-Poincaré method [30]. Heppenheimer developed
a third-order theory for nonlinear out-of-plane motion in the ER3BP in 1973 [31].
8
Two years later, Richardson and Cary [32] obtained a third/fourth order approximation for three-dimensional motion in the elliptic restricted three-body problem in
the Sun-Earth/Moon barycenter system. An analytical approximation for halo-type
periodic motion about the collinear points in the Sun-Earth CR3BP was published by
Richardson in 1980. Breakwell and Brown numerically extended the work of Farquhar
and Kamel to yield a family of numerically integrated periodic halo orbits [33]. The
discovery of stable halo orbits in that family motivated future research in the halo
families near all three collinear libration points by Howell [34] in collaboration with
Breakwell. In 1990, Marchal published a book summarizing the more recent progress
on the CR3BP [35].
1.2.2
Transfer Trajectories
Much progress has occurred over the last decade in the development of analysis
strategies to determine transfer trajectories from the Earth to halo and Lissajous orbits near Sun-Earth L1 and L2 points. With few analytical tools, transfer design was
initially dependent on numerical techniques not available until the 1960’s. The advent of high-speed computers made the computation of transfers possible and, hence,
allowed the possibility of libration point missions. The first study published on transfer trajectories between a parking orbit and a libration point was by D’Amario in
1973 [36, 37]. D’Amario combined analytical and numerical techniques with primer
vector theory to develop a fairly accurate method for the quick calculation of transfer
trajectories from both the Earth and the Moon to the Earth-Moon libration point
L2 . With his multiconic approach, D’Amario determined families of locally optimal
two-impulse and three-impulse transfers [36]. Subsequently, ISEE-3 was planned, of
course, and successfully reached a Sun-Earth L1 halo orbit. The transfer trajectory
used for the ISEE-3 mission inserted the spacecraft into the halo orbit at the ecliptic plane crossing on the near-Earth side. The transfer was categorized as “slow”
because the transfer Time of Flight (TOF) was approximately 102 days versus the
9
approximately 35 days for expensive “fast” transfers. This type of transfer trajectory was selected because numerical studies had indicated that such a path is less
costly in terms of |∆V | than a “fast” transfer trajectory to the halo orbit [8, 38]. In
1980, Farquhar completed a post-flight mission analysis of the flight data from ISEE3 [39]. Simó and his collaborators were the first to publish details of a methodology
to use invariant manifold theory to aid in the design of transfer trajectories in 1991.
A manifold approaches a periodic orbit asymptotically and so eliminates, in theory,
any insertion maneuver cost. A year later, Hiday expanded primer vector theory to
the 3BP and studied impulsive transfers between parking orbits and Libration Point
Trajectories (LPT) near L1 in the Sun-Earth system. An extensive numerical study
using differential techniques was performed by Mains in 1993 [40]. Mains was interested in the development of approximations useful for future automated transfer
trajectory determination procedures. Mains studied transfers from a variety of different parking orbits with different times of flight (TOF) including a transfer similar
to that of the ISEE-3 spacecraft. Barden [28,41] later extended Mains’ investigations
through a combination of numerical techniques and dynamical systems theory. In the
mid-1990’s, Wilson, Barden, and Howell developed design methodologies used in the
determination of the Genesis trajectory [42–44]. In 2001, Anderson, Guzmán, and
Howell implemented an efficient procedure to investigate transfers from the Earth to
Lissajous trajectories by exploiting lunar flybys in an ephemeris model [45, 46]. Thus
far, most of the work has been focused on determining transfer trajectories from the
Earth to the Sun-Earth or Sun-Earth/Moon libration point orbits.
1.3
Present Work
The focus of this investigation is the continued development of techniques and
strategies to determine transfer trajectories from the Earth to LPOs in both the
Sun-Earth and the Earth-Moon systems. Most of the emphasis has been placed on
the development of the methodology used in the transfer design process. A useful
10
dynamical element that is significant in the determination of transfer trajectories is
the set of invariant manifolds approaching and departing periodic halo orbits. In the
Earth-Moon system, manifolds do not pass close by the Earth (larger primary) and
additional strategies are investigated. In the Sun-Earth system (where the Earth is
the smaller primary) manifolds that pass in the immediate vicinity of the Earth are
frequently available.
All of the analysis in this work is conducted numerically and the Circular Restricted Three-Body Problem (CR3BP) is used as the fundamental dynamical model
for baseline designs. The CR3BP is well-suited for qualitative analysis and the solution can easily be transferred into a more complex model making use of planetary
ephemerides. A few sample transfer trajectories are presented to underline the validity
of the CR3BP as the dynamical model. The first objective focuses on transfer trajectories from Earth parking orbits to halo and Lissajous trajectories located near L1
in the Earth-Moon system. This will improve knowledge of the dynamical structure
for trajectory arcs that depart from the major (first) primary. The goal is accomplished by considering both transfer arcs to manifolds as well as direct transfers into
the halo orbits. A second objective is the development of a procedure to compute
launch trajectories from a specific launch site on the surface of the Earth to an invariant manifold. Some initial results based on differential corrections algorithms are
presented that can be used to determine the transfer arcs and launch trajectories of
interest. The results are of a preliminary nature and the beginning of a more in-depth
investigation.
This work is arranged as follows:
Chapter 2:
This chapter summarizes the background material that underlies the foundations
of this study. The different reference frames are introduced and the mathematical
model used to represent the Earth-Moon and Sun-Earth dynamical environments,
that is, the circular restricted three-body problem, is developed. Assumptions employed in this model as well as special properties are discussed, followed by a derivation
11
of the linear variational equations. A method to solve for periodic solutions is introduced and particular solutions of general interest are presented. Invariant manifold
theory is introduced and the computation of manifolds is discussed.
Chapter 3:
The natural motion to and from halo orbits at L1 in the Earth-Moon system is
investigated. The numerical algorithm that forms the basis of this study is then presented and additional constraints are introduced. The numerical procedure is applied
to a number of different problems and sample transfer trajectories from an Earth
parking orbit to lunar L1 LPOs are presented. Preliminary results in an ephemeris
model are presented to demonstrate the validity of the obtained transfer arcs in the
CR3BP.
Chapter 4:
The natural motion to and from halo orbits at L1 in the Sun-Earth system is studied and a series of transfers from the Earth halo orbits at L1 and L2 are summarized.
Chapter 5:
A constant thrust term is added to the equations of motion. A modified version of
the differential corrector exploits the thrust parameters to determine launch trajectories with discretely varying thrust angles. A sample launch trajectory to an Earth
parking orbit is presented.
Chapter 6:
The work presented here is of a preliminary nature and more detailed investigations are required. A strategy for future work is presented and conclusions and
recommendations for further investigations are discussed.
12
2. BACKGROUND: MATHEMATICAL MODELS
The circular restricted three-body problem is the simplest model that can capture
the seemingly chaotic motion present in a multi-body system. Since this dynamical
structure is the focus in this study, the mathematical model is derived. Different
reference frames are introduced and the linear variational equations are developed.
The concept of differential corrections is introduced and particular solutions, such
as equilibrium points, as well as periodic and quasi-periodic orbits are discussed.
Some aspects of invariant manifold theory are presented and applied to illustrate
the computation of invariant manifolds. The transition of solutions to an ephemeris
model is also summarized.
2.1
Reference Frames
A variety of different reference frames are used in this investigation and are useful
for computation as well as visualization purposes. Their definitions and the associated
notation is detailed here for clarity.
2.1.1
Inertial Frame
Newton’s laws, as stated in their most original form, are valid relative to an
inertial reference frame. In this study, one inertial frame is defined to be centered
at the Earth (geocentric). The Earth Centered Inertial (ECI) frame is assumed to
be inertially fixed in space but, actually, is slowly moving over time. Since a truly
inertial system is impossible to realize, the standard J2000 system [47] has been
adopted as the best representation of an ideal, inertial frame at a fixed epoch. The
shift of this frame is so slow relative to the motion of interest, it can be neglected.
13
b − Yb plane. The unit vector X
b
The fundamental plane is the plane defined as the X
is directed toward the vernal equinox, the unit vector Yb is rotated 90 degrees to the
east in the ecliptic plane. The unit vector Zb is, then, normal to the plane defined by
b and Yb axes such that Zb = X
b × Yb . (Note that a caret indicates a vector of
the X
unit magnitude.) A formulation of the problem relative to the inertial frame is not
very convenient for investigations in the 3BP because the primaries are continuously
moving and no constant or fixed equilibrium solutions exist.
2.1.2
P1 − P2 Rotating Frame
The P1 − P2 rotating frame is the most convenient frame for visualization of the
motion of the infinitesimal body, P3 , moving near the libration points. The barycenter
of the two primaries that define the three-body system is typically used as the origin
(See figure 2.1). The unit vector x
b is defined such that it is always parallel to a line
between the primaries and is directed from the larger toward the smaller primary.
The unit vector yb is 90 degrees from x
b in the plane of motion of the primaries; it is
positive in the general direction of the motion of the second primary relative to the
first. The unit vector zb is defined to complete a right handed coordinate system and
is normal to the plane of motion spanned by x
b and yb. Only when the equations of
motion in the CR3BP are formulated relative to the rotating frame of the primaries
are the libration points the equilibrium solutions to the differential equations. This
will be apparent later. Libration point orbits only exhibit their periodicity relative
to this frame.
2.1.3
Earth Centered “Fixed” Frame
The Earth Centered Fixed Frame (ECF) is fixed in the Earth as it rotates on its
own axis. Thus, the ECF frame is useful when a specific launch site on the Earth
surface is defined and launch trajectories are computed. The ECF origin is at the
Earth center, and, thus, this geocentric coordinate system rotates with the Earth
14
Figure 2.1. Geometry of the Three-Body Problem.
relative to the inertial frame. The primary axis of ECF is always aligned with a
particular meridian, and the Greenwich meridian is very often selected. Since the
coordinate system is rotating relative to the inertial frame, it is necessary to specify
an epoch. In this model, the ECF frame is assumed to be aligned with the ECI
frame at time t = 0. This simplification is equivalent to assuming that the Greenwich
meridian is parallel to a line between the primaries at time t = 0. The rotation of
the Earth is approximated as constant. The approximation assumes that the Earth
completes precisely one revolution in a 24 hour period. By determining the angular
velocity of the Earth, the alignment between the ECF and the inertial frame can be
computed for any later time.
15
2.2
Transformations Between Different Frames
Transformations between different rotating and inertial frames are critically im-
portant to correctly model this problem. The alignment between the inertial frame
and the P1 − P2 frame is illustrated in figure 2.1. Transforming the position state
from an inertial frame to a rotating frame can be accomplished using the following
rotation matrix,


c −sθ 0

 θ


inert rot
A =  sθ cθ 0  ,


0
0 1
(2.1)
where θ is the angle between the rotating and the inertial frame and appears in
figure 2.1. The trigonometric symbols are defined as sθ = sin(θ) and cθ = cos(θ).
Transforming the velocity state between a rotating and an inertial frame requires the
use of the basic kinematic equation (BKE),
dr̄
dr̄
=
+ ω̄ × r̄,
dt inertial
dt rot
(2.2)
where ω̄ is the angular velocity vector and r̄ is the position vector in inertial coordinates. (Note that overbars denote vectors.) The cross product ω̄ × r̄ yields


A(2, 1) A(2, 2) A(2, 3)




inert rot
Ȧ = θ̇ ∗ r ∗  A(1, 1) A(1, 2) A(1, 3)  ,


0
0
0
(2.3)
where r is the magnitude of the position vector. Thus, the transformation of the
entire state vector from the rotating frame of the primaries to the inertial frame is
represented
rot

T inert = 
T
A
0
ȦT AT

.
(2.4)
In an ephemeris model, the θ angle does not remain constant and must be computed
instantaneously; the rate θ̇ is also evaluated instantaneously.
The inertial X̂ − Ŷ plane represents the ecliptic plane, i.e., the plane of motion
of the Earth about the Sun. The inertial axis Ẑ is normal to the ecliptic plane. The
16
Earth’s axis of rotation is inclined relative to Ẑ. The angle that denotes the inclination of the Earth is i and is assumed to be constant such that i = 23.5 degrees. Thus,
the transformation from the inertial to the inclined equatorial frame is written

with
eclip
T =
eclip
C
equat
0
eclip
0

1
C
0
equat
0

,





C equat =  0 ci si  .


0 −si ci
(2.5)
(2.6)
This transformation matrix, i.e., equation (2.5), allows conversions of the state between the ecliptic, and the inertial equatorial plane.
2.3
Nondimensionalization
To eventually generalize the derived equations, it is advantageous to non-dimension-
alize and, thus, express fundamental quantities in terms of relevant system parameters. The characteristic dimensional quantities identified in the system are length,
time, and mass. As illustrated in figure 2.1, the dimensional length between the
barycenter and the first primary P1 is labelled l1 and the dimensional length between
the barycenter and the second primary P2 is labelled l2 . For the CR3BP, the reference characteristic length is defined as the distance between the two primaries, that
is, l∗ = l1 + l2 , the semi-major axis of the conic orbit of the second primary relative
to the first. The reference characteristic mass is defined as m∗ = m1 + m2 , where m1
is the mass of the first primary and m2 the mass of the second primary. This allows
a definition of the characteristic time as
"
(l∗ )3
∗
t =
G (m1 + m2 )
#1/
2
where G is the dimensional gravitational constant.
,
(2.7)
In standard models for the
restricted problem, this specific form is employed to select t∗ such that the non-
17
dimensional gravitational constant Gnd is equal to 1. All other quantities can now be
evaluated in terms of these three characteristic values. The non-dimensional distance
between the two primaries is 1. Kepler’s third law yields the expression for the mean
motion, or mean angular velocity, as
Gnd (m1 + m2 )
n=
(l∗ )3
1/2
.
(2.8)
As can be easily verified, the mean motion possesses a non-dimensional value of 1.
The non-dimensional mass of the second primary is represented by the symbol µ, i.e.,
µ=
m2
.
m1 + m2
(2.9)
As a consequence, the mass of the first primary is represented as
1−µ=
m1
.
m1 + m2
(2.10)
The non-dimensional quantities corresponding to the remaining distance elements, as
seen in figure 2.1, are defined as
r̄B3
,
l∗
r̄13
d¯ = ∗ ,
l
ρ̄ =
(2.11)
(2.12)
and
r̄ =
r̄23
.
l∗
(2.13)
τ=
t
.
t∗
(2.14)
Non-dimensional time is then defined
Nondimensionalization allows a more convenient and general derivation of the equations of motion.
2.4
Equations of Motion
A Newtonian approach is used to derive the equations of motion and, thus, it
is necessary to begin with Newton’s law of gravity. With total force acting on the
18
infinitesimal particle P3 modelled in vector form, the law of motion can be written as
follows,
F̄ = m3 I r¨3 = −
Gm3 m1
Gm3 m2
r̄13 −
r̄23 ,
2
2
r13
r23
(2.15)
where I r¨3 is the acceleration of P3 relative to the barycenter with respect to an inertial
frame and dots indicate differentiation with respect to dimensional time. Multiplying
equation (2.15) by (t∗ )2 /l∗ m3 yields:
Gm1 r̄13 ∗2 Gm2 r̄23 ∗2
d2 (r̄3 /l∗ )
=− 3
t − 3
t .
∗
2
d(t/t )
|r13 | l∗
|r23 | l∗
(2.16)
With equations (2.11) - (2.14), the law of motion in equation (2.15) can be rewritten
in the form
d2 ρ̄
m2
d¯
m1
r̄
− ∗
.
=
−
3
2
∗
dτ
m |r̄13 /l∗ |
m |r̄23 /l∗ |3
(2.17)
Using the non-dimensional quantities in equations (2.9) and (2.13), equation (2.17)
can subsequently be expressed in the form
(1 − µ) ¯ µ
d2 ρ̄
=−
d − 3 r̄,
2
dτ
d3
r
(2.18)
d¯ = (x + µ)x̂ + y ŷ + zẑ,
(2.19)
r̄ = (x − (1 − µ))x̂ + y ŷ + zẑ.
(2.20)
where
The unit vectors X̂ , Ŷ , and Ẑ are parallel to inertial directions as seen in figure
2.1. Then, ρx , ρy , and ρz are the non-dimensional coordinates of P3 with respect
to the inertial, or sidereal, system. Recall that unit vectors x̂ , ŷ, and ẑ are parallel
to directions fixed in the rotating, or synodic, system. Note that the unit vectors
comprise an orthonormal triad. Then the corresponding position coordinates are x,
y, and z. The coordinates in the rotating and inertial system are related by a simple
rotation, i.e.,





x
c −st 0
ρ
 
 x   t
 
 

 ρy  =  st ct 0   y  .
 
 

z
0 0 1
ρz
(2.21)
19
Differentiating equation (2.21), to obtain kinematic expressions for position, velocity
and acceleration,
and


ρ̇x


ct −st 0
 

 

 ρ̇y  =  st
 

0
ρ̇z
ρ̈x


ct
0
ct
0
ẋ − y







ẏ + x  ,
0


ż
1
ct −st 0
 

 

 ρ̈y  =  st
 

0
ρ̈z


ẍ − 2ẏ − x
(2.22)





0   ÿ + 2ẋ − y  ,


z̈
1
(2.23)
where −2ẏ and +2ẋ correspond to the Coriolis terms. Then, x and y represent
non-dimensional terms that result from the centripetal acceleration.
From a combination of the kinematic expansion with equation (2.18), equations
(2.18) and (2.23) yield the scalar, second order, nonlinear set of differential equations:
ẍ − 2ẏ − x = −
(1 − µ)(x + µ) µ(x − (1 − µ))
−
,
d3
r3
(1 − µ)y µy
− 3,
d3
r
(1 − µ)z µz
− 3.
z̈ = −
d3
r
ÿ + 2ẋ − y = −
(2.24)
(2.25)
(2.26)
As illustrated in Meirovitch [29], the Lagrangian of the CR3BP does not depend
on time explicitly, that results in a constant Hamiltonian. It follows that the system possesses a constant of integration known as the Jacobi Constant. Physically,
the gravitational forces, must be balanced by the centrifugal forces. It follows that
a modified potential energy function corresponding to the differential equations in
(2.24)-(2.26) can be identified, that is,
1
(1 − µ) µ
U ∗ = (ẋ2 + ẏ 2) +
+ .
2
d
r
(2.27)
Note that in the above definition, the potential is positive and is a convention in the
formulation of the CR3BP. The equations of motion can now be expressed in terms
of the following partial derivatives,
ẍ =
∂U ∗
+ 2ẏ,
∂x
(2.28)
20
∂U ∗
− 2ẋ,
∂y
∂U ∗
z̈ =
.
∂z
ÿ =
(2.29)
(2.30)
Equations (2.28)-(2.30) represent the traditional, convenient form of the equations of
motion relative to the rotating frame in non-dimensional coordinates.
Jacobi identified the constant of integration associated with the differential equations that takes the following form,
C = 2U ∗ − (ẋ2 + ẏ 2 + ż 2 ).
(2.31)
Jacobi’s Constant is sometimes called the integral of relative energy [48]. It is
important to note that it is not an energy integral but rather an energy-like constant
partly due to the formulation of the problem relative to a rotating system. It is
also notable that, in the restricted problem, neither energy nor angular momentum
is conserved. The Jacobi Constant can be used to produce zero-velocity plots that
identify regions of exclusion for a specific energy level. Figure 2.2 illustrates an
example of a zero-velocity curve for a Jacobi Constant value of C=3.161. A particle
would require an imaginary velocity to be within the region enclosed by the closed
green curve. Awareness of these forbidden regions can offer much insight into the
dynamics of the problem. In addition to insight, the Jacobi Constant is very often
used as a method to check the accuracy of the calculations, particularly the accuracy
of the numerical integration of the differential equations.
2.4.1
Singularities in the Equations of Motion
The equations of motion in the CR3BP possess singularities at the centers of the
two primaries. The singularities result from terms of the form 1/r 3 and 1/d3, where
“r” is the non-dimensional distance from the mass m2 to the spacecraft and “d” is
the non-dimensional distance from the mass m1 to the spacecraft. When a transfer
trajectory originates in a low altitude Earth parking orbit or a launch trajectory is
computed, d or r are very small. Thus, the state is very close to a singularity that
21
1.5
1
Y−Axis [lunar units]
0.5
L
L
1
2
0
Earth
Moon
−0.5
−1
−1.5
−1.5
−1
−0.5
0
0.5
X−Axis [lunar units]
1
1.5
Figure 2.2. Zero Velocity Curve for C = 3.161.
degrades accuracy and limits the effectiveness of the differential corrections technique
in determining transfer trajectories beyond the Earth’s sphere of influence. The singularity can be avoided through regularization [40] at the cost of physical insight.
Previous studies [28], originally with similar difficulties, have demonstrated that regularization is generally not necessary due to the current computational capabilities.
Backward integration of the transfer trajectories is therefore employed to limit the
sensitivity to the initial conditions.
From the differential equations, it is apparaent that the transformation [40] τ = −t
results in a change in the derivatives with respect to the independent variable, such
that d/dτ = −d/dt and d2 /d2 τ = d2 /d2 t. With this transformation, the integration
22
can be initiated on the halo orbit or along a manifold trajectory at time tf and
computed backwards to determine the state close to the Earth at time t0 . Both
transfer trajectories, as well as launch trajectories, are determined using backward
integration.
2.5
State Transition Matrix
Equations (2.28)-(2.30) can be rewritten as six first-order differential equations
where the state vector is defined as x̄ = [x y z ẋ ẏ ż]T . These six first-order differential
equations can be linearized relative to a reference solution x̄ref = [xref yref zref ẋref
ẏref żref ]T by use of a Taylor series expansion and ignoring the higher order terms.
Note that the reference solution can be a constant equilibrium state or a time-varying
solution to the nonlinear differential equations. Define the linearized state relative to
x̄ref as δx̄(t) = [δx(t) δy(t) δz(t) δ ẋ(t) δ ẏ(t) δ ż(t)]T . Then, the linear state variational
equation can be expressed in the form
δ ẋ(t) = A(t)δx(t),
(2.32)
where A(t) is a 6 × 6, generally time-varying, matrix. It can be written in term of
the following four 3 × 3 submatrices,

A(t) = 
0
I3
∗
UXX
2Ω

.
(2.33)
Each one of the elements of the 6 × 6 matrix A(t) is a 3 × 3 matrix, where 0 represents
the zero matrix, and I3 the identity matrix of rank 3. Then Ω is defined as constant,


0 1 0




(2.34)
Ω =  −1 0 0  .


0 0 0
∗
The matrix of second partials, UXX
, is comprised of elements


∗
∗
∗
U
Uxy Uxz

 xx


∗
∗
∗
∗
UXX =  Uyx Uyy Uyz  ,


∗
∗
∗
Uzx Uzy Uzz
(2.35)
23
where
∗
Uab
=
∂2U ∗
.
∂a∂b
(2.36)
Of course, the partials are evaluated on the reference solution and it can be assumed
∗
that, if at least the first and second order derivatives are continuous, UXX
is sym-
metric. The form of the solution to equation (2.32) is well known, assuming the state
transition matrix φ(t, t0 ) is available, that is,
δx(t) = φ(t, t0 )δx0 ,
(2.37)
where φ(t, t0 ) is the 6 × 6 state transition matrix (STM) evaluated from time t0 to
time t. The state transition matrix φ(t, t0 ) is a linear map that reflects the sensitivity
of the state at time t to small perturbations in the initial state at time t0 . Any
differential corrections scheme exploits the STM to predict the initial perturbations
that yield some desired change in the final state. Differentiating equation (2.37) and
substituting equations (2.32) and (2.37) into the result produces
φ̇(t, t0 ) = A(t)φ(t, t0 ).
(2.38)
This matrix differential equation represents 36 scalar equations. The initial conditions
for φ are determined by evaluating equation (2.38) at time t0 . So, the initial conditions
for equation (2.38) yield the identity matrix of rank 6, or
φ(t0 , t0 ) = I6 .
(2.39)
Adding the 6 scalar differential equations for the state yields 42 coupled scalar differential equations to be numerically integrated. A general analytical solution is not
available because A(t) is time-varying.
2.6
Differential Corrections
Differential corrections (DC) schemes use the STM for targeting purposes. One
application is an iterative process to isolate a trajectory arc that connects two points
24
in solution space. Differential corrections (DC) techniques can be used to quickly
obtain a solution with the desired parameters in a wide range of problems. A sufficiently accurate first guess for the the initial state is always required. When only
the natural motion is considered in three-dimensional space, the total number of
parameters available in the problem is 14 [1]. This number of parameters will be
termed the “dimension” of the problem. As illustrated in figure 2.3, there are two
seven-dimensional states, one at each end of the trajectory defined by the epochs, t1
and t2 , their positions, R̄1 and R̄2 , and the velocity components, V̄1 and V̄2 ; thus,
the problem is parameterized by 14 scalar elements. This number also characterizes the sum of the fixed constraints, that is, the targets, the controls, and the free
parameters in the problem. The fixed constraints are the parameters that are not
allowed to vary, for example, the initial position vector R̄1 . The controls are the
set of parameters that the DC scheme is allowed to modify to achieve the desired
target states. Free parameters are additional variables not used in the DC scheme as
either fixed quantities, controls, or targets. These values will likely change in a way
that may not be predictable. Differential correction schemes are often used to obtain
periodic solutions to the nonlinear differential equations in the CR3BP. A common
assumption, making use of the symmetry in this problem, is that the desired solution
is symmetric about the x − z plane. Initially, the known states are given in the form
x̄0 = [x0 0 z0 0 ẏ0 0]T and, from the symmetry properties, it is concluded that for
a simply symmetric periodic orbit, at the next x − z plane crossing, the trajectory
Figure 2.3. A Stylized Representation of 1 Step Differential Corrector.
25
will be consistent with the values, i.e., x̄f = [xf 0 zf 0 ẏf 0]T . Since the initial guess
for the state vector x̄0 is not likely to yield the necessary form of x̄f , the differential
corrections process then uses the STM to compute the required changes in two of the
initial non-zero variables to drive the velocities ẋf and żf to zero. There are an infinite number of periodic orbits that satisfy these conditions. One approach to isolate a
specific trajectory is to fix one non-zero quantity associated with the initial state as a
constant. The fixed quantity can be varied, depending on the goals [34]. The process
is repeated until the desired final result is achieved, i.e., the orbit crossing the x − z
plane perpendicularly. Convergence to a solution is usually obtained after about four
iterations. Numerically integrating equation (2.28)-(2.30) in three dimensions, with
the appropriate initial conditions for one period, yields a halo orbit.
2.7
Particular Solutions
In 1772, Joseph Lagrange identified five particular solutions to equations (2.28)(2.30) as equilibrium points in the 3BP, for a formulation relative to a rotating frame.
As equilibrium points, the gravitational and centrifugal forces are balanced at these
locations but it is important to note that the points are still moving in a circular orbit
about the barycenter relative to the inertial frame. All five points lie in the plane
of motion of the primaries and the location of the points in the Earth-Moon system
appear in figure 2.4. Linear stability analysis can be employed to determine that the
collinear Lagrange points L1 , L2 , and L3 are inherently unstable and that the equilateral points L4 and L5 are linearly stable. Placing a probe at the triangular points will
result in oscillations in the vicinity of the equilibrium point. The equilibrium points
obtained the name, libration points, from the oscillatory motion at the equilateral
locations [20]. One type of periodic and quasi-periodic solutions that are the focus of
a number of recent missions are the periodic, planar Lyapunov and three-dimensional
halo orbits as well as the three-dimensional quasi-periodic Lissajous trajectories.
26
Libration Points in the Earth−Moon System
1.5
y [non−dimensional lunar units]
1
L4
0.5
L3
Earth
L1
L2
0
Moon
−0.5
L5
−1
−1.5
−1.5
−1
−0.5
0
0.5
x [non−dimensional lunar units]
1
1.5
Figure 2.4. Location of the Libration Points in the Earth-Moon System
Relative to a Synodic Frame.
Poincaré realized the importance associated with periodic orbits. In his conjecture
[49] in 1895, he stated that an infinite number of periodic orbits exist in the 3BP.
For Poincaré, periodic motion appeared to be significant in nature and he considered
the study of periodic orbits a matter of greatest importance. His investigations into
periodic orbits in the 3BP were limited to an analytical investigation. Although
considerable progress was made in approximation techniques to represent periodic
orbits over the following 20 years, detailed investigations were hindered by the amount
of computations involved.
Lyapunov, halo, ‘nearly-vertical’ orbits, and Lissajous trajectories each occur in
families with similar characteristics. Lyapunov orbits are planar orbits that lie in the
27
plane of motion of the primaries. A pitchfork bifurcation of the Lyapunov orbits, both
above and below the x − y plane, results in two halo families that are mirror images
across the x − y plane [50]. When the maximum out-of-plane amplitude (Az ) is in the
+z direction, the halo orbit is a member of the northern family (NASA Class I) and
if the maximum excursion is in the −z direction, the halo orbit is a member of the
southern family (NASA Class II). Each member of a family corresponds to a slightly
different energy level (Jacobi Constant). Halo orbits were first computed in the
CR3BP and are defined as precisely periodic, three-dimensional libration point orbits.
They are typically characterized by their maximum out-of-plane amplitude (Az ). An
example of a halo orbit in the CR3BP appears in figure 2.5. The three-dimensional
orbit is presented in terms of orthographic projections. For Lissajous trajectories, the
amplitude of the in-plane motion and that of the out-of-plane motion are arbitrary
and the frequencies are not commensurate [9, 51]. Orthographic projections of an
example of a Lissajous trajectory appears in figure 2.6. Precisely periodic halo orbits
do not exist in a more general model that incorporates additional perturbations.
In an ephemeris model, Lissajous trajectories can always be generated and careful
selection of in-plane and out-of-plane amplitudes yields Lissajous trajectories that
are very close to periodic for a limited time interval. These orbits are generally
denoted as halo orbits. Although Lissajous trajectories are not periodic, they are
nevertheless bounded and exist on an n-dimensional torus [50, 51]. Another type of
periodic motion, is the family of the ‘nearly-vertical’ orbits first visually identified by
Moulton [50].
As is true with any nonlinear dynamical system, a low order approximation does
not immediately yield a continuous trajectory, however, a differential corrections procedure can result in a natural periodic solution if the initial guess is sufficiently accurate. If a linear approximation is not sufficient to create a Lissajous trajectory
with the desired characteristics, the third-order analytical approximation developed
by Richardson and Cary [32] is commonly deployed. Alternatively, patch points,
consisting of the full six-dimensional state plus the time, along a halo orbit can be
28
4
x 10
3
Y [km]
2
Earth−Moon System
A = 331,521 km
x
A = 25,991 km
y
Az = 15,393 km
1
0
L
1
−1
−2
−3
3
4
3.2
3.4
3.6
5
X [km]
x 10
4
x 10
x 10
2
1
L1
0
Z (km)
Z [km]
2
1
−1
−1
−2
−2
3
3.2
3.4
3.6
5
X [km]
x 10
L1
0
−2
0
Y (km)
2
4
x 10
Figure 2.5. Northern Earth-Moon L1 Halo Orbit in the CR3BP.
determined and imported into an ephemeris model to obtain a halo-like Lissajous
trajectory. The patch points are corrected through a two-level differential corrections
process (2LDC) developed by Howell and Pernicka [52]; details of this process are
offered later.
2.8
Invariant Manifold Theory
In the late 19th century, the French mathematician Henri Poincaré searched for
precise mathematical formulas that would allow an understanding of the dynamical
stability of systems. These investigations resulted in the development of what is
now called Dynamical Systems Theory (DST). Dynamical systems theory is based
29
5
x 10
y [km]
2
1
Earth
Sun
0
L1
−1
−2
1.478
5
1.48 1.482 1.484
8
x [km]
x 10
5
x 10
x 10
2
1
z [km]
z [km]
2
L
0
1
−1
−2
1
L1
0
−1
−2
−3
1.478
1.48 1.482 1.484
8
x [km]
x 10
−3
−4
−2
0
y [km]
2
4
5
x 10
Figure 2.6. Lissajous Trajectory at L1 in the Sun-Earth CR3BP.
on a geometrical view for the set of all possible states of a system in the phase
space [53]. Detailed background information is available in various mathematical
sources including Perko [54], Wiggins [55], as well as Guckenheimer and Holmes [56].
An extensive summary appears in Marchand [51].
2.8.1
Brief Overview
For a continuous nonlinear vector field of the form
ẋ = f(x),
(2.40)
the local behavior of the flow in the vicinity of a reference solution to the nonlinear
equations can be determined from linear stability analysis for most applications. Note
30
that x̄ is the entire state vector, and let x̄ = x̄ref + δx̄ such that δx̄ is the vector of
variations. Assuming that the above vector field is continuous to the second degree,
it can be expressed in terms of a Taylor series expansion relative to this reference
solution resulting in a vector variational equation of the following form
δ ẋ(t) = A(t)x(t),
(2.41)
where A(t) is an n × n matrix of the first partial derivatives. If the reference solution
is an equilibrium solution, such as a libration point, then the A matrix is constant.
In general, however, the A matrix cannot be assumed constant and is time-varying,
A = A(t). However, for the moment, assume a time-invariant system and variations
relative to an equilibrium point. The algebraic technique of diagonalization can be
used to reduce the linear system to an uncoupled linear system. Perko [54] states the
following theorem from linear algebra that allows a solution to a linear, time-invariant
system with real and distinct eigenvalues:
Theorem 2.1 If the eigenvalues λ1 , λ2 , ..., λn of an n × n matrix are real and
distinct, then any set of the corresponding eigenvectors η 1 η2 ...ηn forms a basis Rn ,
the matrix P = [η 1 η 2 ...η n ] is invertible and
P −1 AP = diag[λ1 , ..., λn ].
In the process of reducing the linear system in equation (2.41) to an uncoupled system,
the linear transformation
y = P −1 δx,
(2.42)
is defined, where P is the invertible matrix defined in theorem 2.1. Then
δx = P y,
(2.43)
and Perko demonstrates that the solution can be written
y(t) = P E(t)P −1δx(0),
(2.44)
31
where E(t) is the diagonal matrix
E(t) = diag[eλ1 t , ..., eλn t ].
(2.45)
Rewriting equation (2.44) allows the more insightful expression
y(t) =
n
X
cj eλj t η j ,
(2.46)
j=1
where the cj ’s are scalar coefficients.
Perko continues by stating “The subspaces spanned by the eigenvectors ηi of the
matrix A determine the stable and unstable subspaces of the linear system, equation
(2.41), according to the following definition,”
Definition 2.1 Suppose that the n × n matrix A has k negative λ1 ,..., λk and n-k
positive eigenvalues λk+1,..., λn and that these eigenvalues are distinct. Let η1 ,...,ηn
be a corresponding set of eigenvectors. Then the stable and unstable subspaces of
the linear system (2.41), E s and E u , are the linear subspaces spanned by η 1 ,...,ηk
and η k+1 ,...,ηn respectively; i.e.,
E s = Span{η 1 , ..., ηk },
E u = Span{ηk+1 , ..., η n }.
Perko completes the above definition by adding “If the matrix A has pure imaginary
eigenvalues, then there is also a center subspace E c .” Define the complex vector
wj = uj + iη j , as a generalized eigenvector of the matrix A corresponding to a
complex eigenvalue λj = aj + ibj and then let
B = {η 1 , ..., η k , η k+1 , ηv+1 , ..., um , vm },
(2.47)
be the basis of Rn . Then definition 2.2 below allows a distinction between the stable,
unstable, and center subspaces.
32
Definition 2.2 Let λj = aj + ibj , w̄j = ūj + iη̄j and B be as described above. Then
E s = Span { uj , η j k aj < 0 },
E c = Span { uj , η j k aj = 0 },
and
E u = Span { uj , η j k aj > 0 },
i.e., E s , E c , and E u are the subspaces of the real and imaginary parts of the generalized eigenvectors w j corresponding to eigenvalues λj with negative, zero, and positive
real parts respectively.
Decomposing the phase space into three separate regions is the ‘dynamic approach’
[53]. The sum of the three fundamental subspaces spans the complete space Rn .
Selecting initial conditions carefully to ensure that certain specified coefficients cj are
equal to zero in equation (2.46), results in the desired behavior inside a subspace for
all time [53]. Once in a subspace, motion remains there for all time. From a linear
perspective, this can be seen as only exciting the desired mode while eliminating
any perturbations of the undesirable modes. Thus, solutions that originate in E s
asymptotically approach y = 0̄ as t → ∞ and solutions with initial conditions in
E u approach y = 0̄ as t → −∞. Solutions in the center subspace E c neither grow
nor decay over time. Guckenheimer and Holmes [56] relate the stable and unstable
manifolds to the invariant subspaces for an equilibrium point through the Stable
Manifold Theorem.
Theorem 2.2 (Stable Manifold Theorem for Flows) Suppose that ẋ = f (x)
has a hyperbolic equilibrium point xeq . Then there exist local stable and unstable mans
u
ifolds Wloc
(xeq ), Wloc
(xeq ), of the same dimensions ns , nu as those of the eigenspaces
E s and E u of the linearized system (2.41), and tangent to E s and E u at xeq . The
s
u
local manifolds Wloc
(xeq ), Wloc
(xeq ) are as smooth as the function f .
Let xeq be the non-hyperbolic equilibrium point, or the libration point, at L1 .
Then, figure 2.7 can be used to illustrate the stable manifold theorem. In the immediate vicinity of the libration point, the eigenvectors are directed along the individual
33
local stable and unstable subspaces. The manifolds associated with the stable and
unstable subspaces are globalized through numerical integration. In the CR3BP, the
collinear libration points L1 , L2 , and L3 possess a four-dimensional center subspace
and one-dimensional stable and unstable subspaces. Figure 2.7 represents the stable
and unstable eigenvectors, E s and E u , and the globalized manifolds, W s and W u ,
associated with the Earth-Moon L1 point. In the case of the libration points, the
position and velocity components of the eigenvectors are always parallel. For the unstable eigenvector, both position and velocity components are directed the same, i.e.,
away from the libration point. For the stable eigenvector, the position and velocity
components of the eigenvector are oriented in opposite directions. Hence, a small
displacement from the libration point along the stable eigenvector, results in motion
toward the libration point. A small displacement along the unstable eigenvector, results in motion away from the libration point. Planar Lyapunov orbits and nearly
vertical out-of-plane orbits are examples of periodic solutions that exist in the center
subspace near Li [51,57]. As the amplitude of the Lyapunov orbits increases to a critical amplitude, a bifurcation point identifies the intersection of the planar Lyapunov
orbits and the three-dimensional halo family of periodic orbits [50,51]. Note that the
Lyapunov familly continues beyond the critical amplitude but the stability properties
of the orbits have changed. The critical amplitude can be identified by monitoring
the characteristics of the eigenvalues of the monodromy matrix.
2.8.2
Periodic Orbits and Dynamical Systems Theory
For a number of applications, the state transition matrix at the end of a full
revolution, also termed the monodromy matrix φ(T, 0), must be available. Consider
the point along the periodic orbit that was selected as the starting and ending point.
In dynamical systems, this point is denoted as a fixed point in a stroboscopic map.
Then, the monodromy matrix serves as a discrete linear map near the fixed point
located at the origin of the map. Such a map is also often called a Poincaré map [9,53].
34
4
x 10
2
1.5
Wu+
1
E
E
Y−Axis [km]
Ws+
s
u
0.5
0
L
Eu
1
−0.5
Es
Wu−
−1
Ws−
−1.5
−2
−1
0
X−Axis [km]
1
2
4
x 10
Figure 2.7. Stable and Unstable Eigenvectors and the Globalized Manifold for the
Earth-Moon L1 Point.
The eigenvalues and eigenvectors of the monodromy matrix can be used to estimate
the local geometry of the phase space in the vicinity of the fixed point. It is important
to note that the monodromy matrix possesses different elements for every fixed point
along the periodic orbit. Thus, the eigenvectors change directions and, thus, the
directions of the stable and unstable subspaces vary along the orbit. The eigenvalues,
on the other hand, are a property of the orbit and remain constant as is apparent
from equation (2.51).
Linearizing relative to periodic orbits results in linear, periodic, differential equations. The A matrix in equation (2.41) is now time-varying but periodic, i.e.,
δ ẋ(t) = A(t)δx(t).
(2.48)
35
According to Floquet theory [24, 57], the STM can be rewritten as
φ(t, 0) = F (t)eJt F −1 (0),
(2.49)
where J is a normal matrix that is diagonal or block diagonal and the diagonal
elements are the characteristic multipliers or Floquet multipliers. Note that F (t) is
a periodic matrix. Solving for the matrices J and F that correspond to a periodic
system yields
φ(T, 0) = F (0)eJt F −1 (0),
(2.50)
eJt = F −1 (0)φ(t, 0)F (0).
(2.51)
since F (T ) = F (0) and
From equation (2.51), it is clear that F (0) and J contain the eigenvectors and eigenvalues of the monodromy matrix. From equation (2.51), then,
λi = e̟i T ,
̟i =
1
ln(λi ),
T
(2.52)
(2.53)
where ̟i are the Poincaré exponents.
The Poincaré exponents are interpreted in a manner similar to the eigenvalues in
a constant coefficient system. In Hamiltonian-like systems, such as the CR3BP, they
must also occur in positive/negative pairs by Lyapunov’s theorem. From the stability
properties associated with the Poincaré exponents, conclusions about the location
of the characteristic multipliers on the complex plane and, thus the stability of the
fixed point and the periodic orbit, are potentially available [57]. A system with no
characteristic multipliers on the unit circle is called hyperbolic and the nature of a
hyperbolic system can be summarized as
|λi| < 1 ⇒ stable y = 0 as t → ∞
|λi| > 1 ⇒ unstable y = 0 as t → −∞
If |λi | = 1, no stability information can be obtained from the characteristic multipliers
and they correspond to the center subspace.
36
Periodic orbits possess a monodromy matrix that yields at least one eigenvalue
with a modulus of one [54]. The following theorem adds further significance to the
eigenvalues.
Theorem 2.3 (Lyapunov’s Theorem) “If λ is an eigenvalue of the monodromy
matrix φ(T, 0) of a time-invariant system, then λ−1 is also an eigenvalue, with the
same structure of elementary divisors.”
λ1 =
1
1
1
, λ3 = , λ5 = .
λ2
λ4
λ6
(2.54)
Thus, according to Lyapunov’s theorem, the six eigenvalues associated with a periodic orbit (via the map and associated monodromy matrix) also appear in reciprocal,
complex conjugate pairs. So, two of the six eigenvalues of the monodromy matrix will
always be precisely one. In the CR3BP, the center subspace has a dimension of four
and two of the eigenvalues are real and equal to one for precisely periodic orbits.
Eigenvalues with real parts smaller than 1 are considered stable eigenvalues and
eigenvalues with real parts larger than 1 are considered unstable. Eigenvectors corresponding to a stable eigenvalue lie in the stable subspace and yield stable manifolds
asymptotically approaching the periodic orbit as t → ∞. Eigenvectors corresponding
to an unstable eigenvalue lie in the unstable subspace and yield unstable manifolds
asymptotically approaching the orbit as t → −∞.
2.9
Computing Manifolds
Computation of the globalized manifolds relies on the availability of initial condi-
tions obtained from the stable and unstable subspaces, E s and E u . The eigenvectors
of the monodromy matrix offer local approximations of the stable and unstable subspaces for fixed points along periodic orbits. Note that an eigenvector only indicates
orientation in space. The eigenvector does not yield a specific directional sense, that is
multiplying by negative one, yields a valid eigenvector in the opposite direction. This
results in manifolds approaching (stable) and departing (unstable) the orbit in two
37
different directions. Generally, one side of the globalized manifold enters the region
around the smaller primary P2 whereas the other enters the inner region around the
major primary P1 . The directions spanned by the position components for 30 points
along the halo orbit(Az = 15,000 km) are illustrated in figure 2.8. The velocity (red)
components relative to the position (blue) components for the stable eigenvector appear in figure 2.9. This set results in the globalized manifold approaching the halo
orbit from the larger primary. The angle between the position and velocity components of the eigenvector varies between 146 and 170 degrees for a lunar L1 halo
orbit with an Az amplitude of 15,000 km. The angular range is dependent upon the
orbit investigated, of course. While studying this figure, it is critical to realize that
the eigenvectors only offer a linear approximation of the subspaces close to the halo
orbit and, hence, only offer a valid approximation of the direction of these subspaces
very close to the periodic orbit. Thus in addition to investigating the direction of the
eigenvectors, it is crucial to consider the globalized manifolds to obtain a complete
picture of the different subspaces. Applying a small perturbation in the direction of
the eigenvector results in a local estimate of the one-dimensional manifold associated
with the fixed point. After a local estimate has been determined, the trajectory on
the manifold associated with the point can be globalized through numerical integration.
Given the eigenvectors of the monodromy matrix, the local estimate of the
stable and unstable manifolds, X̄s and X̄u , can be computed as
X̄s = x̄(ti ) + d · V̄ Ws (ti ),
(2.55)
where V̄ W s (ti ) is defined by
V̄ W s (ti ) = p
Ŷ W s (ti )
x2 + y 2 + z 2
,
(2.56)
and Ŷ W s (ti ) = [xs ys zs ẋs ẏs żs ]T is the stable eigenvector. Then,
X̄u = x̄(ti ) + d · V̄ Wu (ti ),
(2.57)
where V̄ W u (ti ) is defined by
V̄ W u (ti ) = p
Ŷ W u (ti )
x2 + y 2 + z 2
,
(2.58)
38
4
x 10
2.5
2
1.5
1
y [km]
0.5
0
Earth
Moon
L1
−0.5
−1
−1.5
−2
−2.5
2.9
3
3.1
3.2
3.3
x [km]
3.4
3.5
5
x 10
Figure 2.8. Stable Eigenvectors of the Monodromy Matrix for an
Earth-Moon L1 Halo Orbit.
and Ŷ W u (ti ) = [xu yu zu ẋu ẏu żu ]T is the unstable eigenvector.
In equation (2.55) and equation (2.57), d is the initial displacement (perturbation)
from the periodic orbit. Larger d values ensure an initial state along a manifold that
is further advanced in departing from the periodic orbit or libration point. At the
same time, the initial displacement cannot be selected arbitrarily large since the linear
approximation must remain within the range of validity. In the Earth-Moon system,
d values commonly range between 30 km and 70 km, whereas in the Sun-Earth system
commonly used d values range between 150 and 200 km. The initial displacement
can also be employed as a design parameter; changing the magnitude of d only affects
the particular trajectory selected along the approaching manifold, but not the orbit
39
4
x 10
2.5
2
1.5
1
y [km]
0.5
L1
Earth
0
Moon
−0.5
−1
−1.5
−2
−2.5
2.9
3
3.1
3.2
3.3
x [km]
3.4
3.5
5
x 10
Figure 2.9. Position and Velocity Components of the Stable Eigenvectors
of the Monodromy Matrix for an Earth-Moon L1 halo.
itself. The effects of different d values are illustrated in figure 2.10. Normalizing the
eigenvectors relative to the position components ensures that the displacement along
the eigenvector is uniform.
Once the monodromy matrix is obtained for any one
fixed point along the orbit, the manifolds associated with any other point along the
orbit can be calculated two different ways. First, shifting the focus to another fixed
point, the calculations can be repeated and a new monodromy matrix determined.
Then, new eigenvectors can be computed. Alternatively, it is more efficient to exploit
the state transition matrix. An eigenvector can be directly shifted along the orbit via
the STM as follows,
ȲiWs = Φ(ti , t0 )Ȳ0Ws ,
(2.59)
40
Figure 2.10. Various Manifolds Asymptotically Approaching the Orbit.
ȲiWu = Φ(ti , t0 )Ȳ0Wu .
(2.60)
This direct shift is quick and accurate. In the interest of numerical accuracy, it is
also beneficial to explore the structure associated with the CR3BP to limit excessive
numerical integrations of the STM as well. In the CR3BP, the monodromy matrix
φ(T, 0) can be generated from the half-cycle STM φ(T /2, 0),




0 −I3
−2Ω
I
3
 φT (T /2, 0) 
 G−1 φ(T /2, 0),
φ(T, 0) = G 
I3 −2Ω
−I3 0
where

(2.61)

1 0 0 0 0 0




 0 −1 0 0 0 0 




 0 0 1 0 0 0 
.
G=


 0 0 0 −1 0 0 




 0 0 0 0 1 0 


0 0 0 0 0 −1
(2.62)
41
6
x 10
3
2
y [km]
1
Earth
0
Sun
−1
−2
−3
1.44
1.45
1.46
1.47
1.48
x [km]
1.49
1.5
1.51
8
x 10
Figure 2.11. Stable (blue) and Unstable (red) Manifolds for a Sun-Earth Halo Orbit
near L1 .
The proof of equation (2.61) uses the fact that the STM is a simplectic matrix [33].
Two halo orbits in the Sun-Earth system appear in green in figure 2.12. The
halo orbits, plotted in green, possess an Az amplitude of 120,000 km. The stable
(blue) and unstable (red) manifolds associated with these L1 and L2 halo orbits form
three-dimensional tubes in the vicinity of the Earth. The Earth in figure 2.12 is not
plotted to scale and only the physical location is represented by the blue sphere. For
larger halo orbits, part of the manifold tubes extend below the surface of the Earth,
i.e., pass less than 6378 km from Earth’s center. As is illustrated in figure 2.12,
the manifolds are separatrices for a given energy level. The manifold tubes separate
different regions of motion in space. Additionally, if motion starts on a tube, it will
not leave that tube unless the energy level is altered [58].
42
6
x 10
1
y [km]
0.5
0
−0.5
−1
1.48
1.485
1.49
1.495
x [km]
1.5
1.505
1.51
8
x 10
Figure 2.12. Stable (blue) and Unstable (red) Manifold Tube Approaching the
Earth in the Sun-Earth System.
2.10
Transition of the Solution to the Ephemeris Model
Solutions in the CR3BP offer much insight into the solution space and are qualitatively similar to solutions existing in higher order multi-body models. To improve
the accuracy of transfer trajectories and to illustrate their credibility and robustness,
they are transferred to an ephemeris model.
The ephemeris model that is employed here is the Generator software package [59]
developed by Howell et al. at Purdue University. The primary motion is specified from
planetary state ephemerides available from the Jet Propulsion Laboratory DE405
43
ephemeris files. In the ephemeris model, the Earth is used as the origin and Newtonian equations of motion are integrated with respect to the inertial frame. From a
number of possible integrators, an 8/9 order Runge-Kutta integrator is selected for
this application to perform the integration with respect to the Earth. In the determination of transfer trajectories in the Earth-Moon system, the Sun is also included
in the force model, although any number of bodies can be incorporated. Since no assumptions about the primary motion are desired, the transformation from inertial to
rotating coordinates is based on the instantaneous position of the primaries obtained
from the ephemeris files.
Much of the symmetry of the CR3BP is lost when solution arcs are transferred
to an ephemeris model. The libration points are no longer fixed points relative to
the rotating frame of the primaries. Instead, they oscillate. Precisely periodic halo
orbits no longer exist, however, it is often desirable to maintain characteristics for the
quasi-periodic orbits in the ephemeris model that are similar to those of the periodic
halo orbits in the CR3BP. Therefore, transfer arcs that correspond to halo orbits in
the CR3BP are employed as initial estimates in the ephemeris model. A two-level
differential corrections (2LDC) scheme [52] (described later) computes a Lissajous
trajectory, figure 2.13, resembling the original halo orbit. The “halo-like” Lissajous
in figure 2.13 corresponds to a translunar halo orbit with Az amplitude of 15,000 km.
The curve in figure 2.13 is plotted in the Earth-Moon rotating frame centered at the
Moon.
44
4
Y [km]
5
x 10
0
L
Moon
1
L
2
Earth
−5
−6
−4
−2
0
X [km]
2
4
6
4
x 10
4
4
x 10
4
Z [km]
2
x 10
2
Moon
0
Z [km]
4
L
L
2
1
Moon
0
−2
−2
Earth
−4
−6
−4
−2
0
X [km]
2
4
−4
−5
6
4
x 10
0
Y [km]
Figure 2.13. A ”Halo-like” Lissajous Trajectory in a Ephemeris model
with an Az Amplitude of approximately 15,000 km.
5
4
x 10
45
3. TRANSFERS FROM EARTH PARKING ORBITS TO
LUNAR L1 HALO ORBITS
Efficient determination and computation of transfer trajectories from parking orbits
around the larger primary to L1 halo orbits remains a challenge. Direct insertion
onto a manifold from the Earth is not possible, since the natural flow does not pass
sufficiently close to the primary. The natural motion for arbitrary Earth-Moon halo
orbits is presented to illustrate the difficulties in the determination of such transfers.
Results in this chapter represent the most recent efforts in the analysis of this
problem. The highest priority is an understanding of the difficulties associated with
the computation of these transfer trajectories (including starting values) and the
development of the methodology for detailed investigation. Various transfer trajectories between a low altitude Earth parking orbit and two different halo orbits are
presented. A more complete understanding of the dynamical structure in this multibody environment is the foundation for the design of future trajectories originating
and returning to an Earth parking orbit. A free return transfer trajectory with potential applications in human spaceflight is also included. The trajectory arcs computed
here can ultimately be optimized as well.
3.1
Stable and Unstable Flow that is Associated with the Libration Point
L1 in the Vicinity of the Earth
Exploration of the natural dynamics associated with invariant manifolds to and
from L1 halo orbits in the Earth-Moon system is the first step in developing an efficient
design tool. Therefore, the stable and unstable manifolds have been computed for a
series of different halo orbits. Although this behavior is similar in any 3BP system,
the analysis presented here is limited to the Earth-Moon dynamical environment.
46
Figure 3.1. Stable and Unstable Manifold Tubes in the Vicinity of the Earth.
The stable and unstable flow around the first (largest) primary does not allow
for natural solutions, i.e., manifolds, to pass close by the primary. The trajectories
corresponding to different fixed points along the halo form a tube-like surface. Each
individual trajectory wraps around the manifold surface that forms the tube. These
stable (blue) and unstable (red) manifold surfaces appear in figure 3.1 for an EarthMoon halo orbit with an Az amplitude of 15,000 km. (The size of a halo orbit is
typically characterized by the out-of-plane amplitude denoted Az .) The trajectories
that correspond to specific L1 halo orbits first pass the Earth at altitudes ranging
between 82,200 km to 93,000 km. Although, after multiple passes, a trajectory might
subsequently pass closer to the Earth, the investigation here is limited to first passes.
The closest Earth passing altitudes along the trajectories that correspond to the
stable manifolds for differently sized Earth-Moon L1 halo orbits appears in figure
47
4
x 10
11
Lowest Passing Altitude [km]
10.5
10
9.5
9
8.5
8
7.5
7
6.5
6
1
2
3
4
Halo Orbit Az [km]
5
6
7
4
x 10
Figure 3.2. Minimum Earth Passing Altitudes for Trajectories on the Manifold
Tubes Associated with Various Earth-Moon L1 Halo Orbits.
3.2. Qualitively, the behavior for the stable and unstable manifold tubes is identical
although they approach the Earth from different regions of space as is illustrated in
figure 3.1. Large halo orbits with an Az amplitude of approximately 43,800 km yield
specific trajectories that offer the lowest Earth passing altitudes. As is apparent in
figure 3.1, trajectories extend backwards from the halo orbit in a tube shape and,
therefore, only manifolds that approach/depart a small region in the specific LPO
pass close by the first primary. Some trajectories, those for large Az amplitudes, first
pass the Earth at altitudes well about 240,000 km.
48
3.2
Design Strategy
The lunar initiative to return to the Moon and its vicinity requires the ability to
compute transfers between the Earth, the Moon, and libration point orbits quickly
and efficiently. In addition to robotic and human missions to the Moon, the deployment of humans for servicing and repair missions to observatories or a space station in
halo orbits at the collinear lunar libration points requires the determination of transfers meeting stringent rendezvous requirements. Extensive investigations of transfer
trajectories to and from the Earth and the Moon to halo orbits near L1 and L2 are
currently ongoing. The development of fast and efficient design tools and various
types of tradeoff studies are, hence, of great interest and critical to the success of
future efforts to expand scientific missions and to create a human presence in the
lunar vicinity.
Recall that the pass distance between the Earth and the stable and unstable manifolds associated with Earth-Moon halo orbit is large. Since this is typically tens of
thousands of kilometers, a broad range of concepts must be explored for transfers.
In 1973, D’Amario first conducted an extensive study on transfers from both the
Earth and the Moon to the libration point at L2 using a multiconic approach and
optimizing his results by applying primer vector theory [37]. Although it was certainly not trivial, the methodology allowed him to limit the amount of numerical
computations by obtaining the state transition matrices analytically. With minimal
numerical computations, D’Amario was able to determine optimal transfers fairly
accurately. It is noted that the ∆v to insert into L2 in the Earth-Moon system is
not small. D’Amario never studied a halo orbit, however. In the late 1970’s, the
determination of transfers between Earth parking orbits and the Sun-Earth libration
points shifted to numerical integration and shooting techniques with the advent of
high-speed computing. Although shooting techniques, in combination with differential corrections schemes, have been employed very successfully by many researchers in
the computation of transfers [8, 38, 40], they are time consuming when a large range
49
of parameters must be investigated. Nevertheless, straightforward shooting can be
extremely useful in the determination of baseline solutions when no other tools are
available. Shooting techniques are therefore examined first, and the results form the
basis for future improvements.
The ultimate goal of this investigation is the development of more efficient techniques to compute these transfers. The initial reference transfers obtained through
shooting techniques, may form the baseline solutions. Differential correction techniques to solve for continuous transfers from initial to final orbits with tangential
parking orbit departures are employed. The resulting transfers are investigated and
compared to the local stable manifold structure to reveal potential improvements.
3.2.1
Shooting Technique
An initial reference transfer must exhibit the general characteristics of the desired
trajectory. Shooting techniques are employed that consist of manually varying the
velocity components of an initial state on the parking orbit and numerically integrating forward in the CR3BP for a fixed time of flight. This process is repeated until
a trajectory arc, one displaying the desired behavior, is determined. Patch points
equally spaced in distance along the transfer are then isolated. The patch points are
corrected to yield a tangential parking orbit intersection at the Earth and a continuous entry into the halo orbit using the differential correction algorithm that will be
presented in Section (3.3). Previous work [60] has indicated that adding the transfer
arc patch points to halo orbit patch points at the minimum and maximum x and
y amplitudes for one or two revolutions along the halo, can substantially lower the
halo orbit insertion cost (HOI). The patch points along the transfer trajectory are
thus added to patch points corresponding to the desired halo orbit. The differential
corrections process via backwards integration, will yield the solution.
50
3.2.2
Investigated Transfer Types
The investigations, thus far, have focused on two different types of trajectories.
First, transfer arcs that directly connect the parking orbit with the halo orbit are examined. Then, the second approach is insertion onto the stable manifold approaching
the orbit. Clearly, these options do not encompass all possible transfer trajectories.
Nor are any of the transfers optimized; but they will offer insight into the dynamics
and characteristics of the transfers in this region of space for differently sized halo
orbits.
Two different halo orbits are examined. First, consider a halo orbit that possesses
an out-of-plane Az amplitude of 15,000 km. Halo orbits near L2 with very small
Az amplitudes are of little practical value as they lie in the direct line of sight of
the Moon, which results in communications difficulties. It is therefore necessary to
select halo orbits with an out-of-plane component with a minimum Az amplitude
of around 3,100 km to establish an efficient communications link [61]. The out-ofplane amplitude of 15,000 km was selected for comparison with other studies [58].
The second halo orbit corresponds to the one with the closest Earth approach manifold. The corresponding Az amplitude is approximately 43,800 km, obviously much
larger than the first. Besides offering the closest Earth approach, this halo orbit is a
representative Earth-Moon halo orbit that frequently results in low transfer costs in
system-to-system transfers [6, 58].
A spacecraft in a circular, Earth parking orbit, can, in theory, orbit the Earth
in two different directions. Due to the Earth rotation, however, only one of these
directions is obviously affordable for launch opportunities. Therefore only parking
orbits with an eastward rotation have been considered. No assumptions about the
inclination of the parking orbit have been made.
51
3.3
Two-Level Differential Corrector
A two-level differential corrector (2LDC) is a numerical scheme that can be used
to solve a two point boundary value problem. The particular strategy employed here
was originally developed by Howell and Pernicka [52] for the numerical determination
of Lissajous trajectories. It employs an iterative process based on two distinct procedures that alter elements of the six-dimensional state vectors, plus time, that are
associated with points along a potential solution arc. (These points are termed ‘patch
points’.) The values are all shifted simultaneously to yield a continuous trajectory
that satisfies a set of constraints. As an initial guess, the algorithm requires a set
of patch points sufficiently close to an acceptable dynamical solution. A minimum
of three patch points is typically required. The first and last patch points are commonly denoted the initial and final patch points. All other remaining patch points
are labelled internal. Each one of n patch points is related to the previous and following patch points that comprise (n − 1) trajectory segments. The various positions,
velocities, and times associated with three patch points are illustrated in figure 3.3.
The position state r̄0 , the velocity state v̄0 , and the time t0 define the initial point;
the position r̄f , the velocity v̄f , and the time tf define the final point. At the internal patch points, it is necessary to distinguish between the incoming state, r̄p− , v̄p− ,
and tp− , and the outgoing state, r̄p+ , v̄p+ , and tp+ . As noted, this is accomplished
with the symbols ‘-’ and ‘+’. The first step in the corrections scheme employs a
linear differential corrections process to ensure position and time continuity between
all the segments but introduces velocity discontinuities at the internal patch points.
The second level uses linear corrections, applied simultaneously, to the position vectors and time states corresponding to all of the target points, to reduce the velocity
discontinuities and enforce any additional constraints.
52
Figure 3.3. A Stylized Representation of Level II Differential Corrector
(from Wilson [1]).
3.3.1
First Step - Ensuring Position Continuity
The first step of the 2LDC ensures position and time continuity between all the
trajectory segments. This initial procedure is implemented using a straightforward
linear targeting scheme that modifies the velocity state and TOF (four potential
scalar control variables) at the beginning of each segment to meet the position state
requirements (3 scalar targets) at the end of the segment. For each segment, a vector
relationship of the form
δx̄(t) = φ(t, t0 )δx̄0 +
δx̄
δ∆t,
δt p∗
(3.1)
where δδtx̄ p∗ is the linear variation of the state with respect to time, is evaluated at the
end of the segment and δ∆t is the variation in the time of flight along the trajectory
segment, such that ∆t = tf − t0 . Thus, δ∆t = δtf − δt0 . In matrix format, the linear
targeter is then formulated as
Lk̄ = b̄,
with

φ14 φ15 φ16


L =  φ24 φ25 φ26

φ34 φ35 φ36
(3.2)
ẋ



ẏ  ,

ż
k̄ = [δ ẋ0 , δ ẏ0 , δ ż0 , δ∆t0 ]T ,
(3.3)
(3.4)
53
b̄ = [δxp , δyp , δzp ]T .
(3.5)
Since the number of control variables in k̄ exceeds the number of target states in b̄,
an infinite number of solutions exists. To minimize the changes in the elements of
the control vector, the solution with the minimum Euclidean norm is selected, i.e.,
k̄ = LT (LLT )−1 b̄.
(3.6)
With the trajectory continuous in position, the second step of the 2LDC can now be
deployed to decrease the known velocity discontinuities.
3.3.2
Second Step - Enforcing Velocity Continuity
The ultimate goal of the corrections process is to produce a trajectory that is
continuous in position, velocity, and time, such that the design requirements and the
constraints are both satisfied. So, in the second level, the positions of all of the patch
points are varied to decrease the internal velocity discontinuities, ∆v̄p , introduced
by the first step and, thus, the total cost. At all of the internal patch points, the
incoming velocity state vector v̄p− is compared to the outgoing velocity state vector
v̄p+ to compute the velocity discontinuity at that patch point, i.e.,
∆v̄p = v̄p+ − v̄p− .
(3.7)
Decreasing the velocity discontinuities simultaneously requires the formation of the
State Relationship Matrix (SRM). A trajectory consisting of only three patch points
is described for clarity. This arc is composed of an initial patch point P P0 , one
interior point P Pp , and a final patch point P Pf . The relationship between variations
in the scalar elements of the initial, interior, and final positions, denoted δr̄, velocities,
denoted δv̄, as well as the corresponding times, denoted δt, can then be written as
follows


δr̄p−
δv̄p−


=
Ap0 Bp0
Cp0 Dp0


δr̄0
δv̄0


+
v̄p−
āp−

 (δtp− − δt0 ) ,
(3.8)
54


δr̄p+
δv̄p+


=
Apf Bpf
Cpf Dpf


δr̄f
δv̄f


+
v̄p+
āp+

 (δtp+ − δtf ) .
(3.9)
where āp− is the acceleration of the incoming state and āp+ is the acceleration of the
outgoing state at the internal patch point. The change in the time of flight on the
first segment is given by (δtp− − δt0 ) and the change in the time of flight along the
second segment is (δtp+ − δtf ).
Evaluating the first vector equation in equation (3.8) in terms of δv̄0 and substituting into the the second equation yields
−1
−1
−1
Ap0 δr̄0 + Dp0 Bp0
δv̄p− = Cp0 − Dp0 Bp0
v̄p− − āp− (δtp− − δt0 ) .
δr̄p− − Dp0 Bp0
(3.10)
Along the second segment, the same procedure is applied to equation (3.9) by solving
for δv̄f and then substituting. The result is the expression
−1
−1
−1
v̄p+ − āp+ (δtp+ − δtf ) .
Apf δr̄f + Dpf Bpf
δr̄p+ − Dpf Bpf
δv̄p+ = Cpf − Dpf Bpf
(3.11)
Subtracting equation (3.10) from equation (3.11) and imposing position and time
continuity r̄p+ = r̄p− , tp+ = tp− produces

δ∆v̄p =
h
M0 Mt0 Mp Mtp Mf Mtf
where the M matrices are defined as follows,
−1
M0 = Dp0 Bp0
Ap0 − Cp0 ,




i








δr̄0
δt0
δr̄p
δtp
δr̄f
δtf







,






(3.12)
(3.13)
−1
v̄p− ,
Mt0 = āp− − Dp0 Bp0
(3.14)
−1
−1
Mp = Dpf Bpf
,
− Dp0 Bp0
(3.15)
55
−1
−1
v̄p+ − āp+ ,
Mtp = Dp0 Bp0
v̄p− − āp− − Dpf Bpf
(3.16)
−1
Mf = Cpf − Dpf Bpf
Apf ,
(3.17)
−1
Mtf = Dpf Bpf
v̄p+ − āp+ .
(3.18)
Since it is desired to achieve a value of zero for the interior velocity discontinuity,
define the desired variation δ∆v̄p = −∆v̄p . The variations δr̄ in the position states
can now be computed by solving the linear system given for δr̄,
δr̄ = M −1 [δ∆v̄] .
(3.19)
For trajectories with multiple internal patch points, n, the M matrices must be computed for every internal patch point and collected in one equation for a simultaneous
solution for all position variations. The resulting matrix is called the State Relationship matrix (SRM). It appears in the following general form







M =






M01 Mt01 Mp1 Mtp1 Mf 1
Mtf 1
0
0
...
0
0
0
Mf 2
Mtf 2
...
0
0
0
M02 Mt02 Mp2
Mtp2
.
.
.
.
.
.
0
0
0
.
0
.
.
0
.
.
.
M0n−1 Mt0n−1 Mpn−1 Mtp0n−1 Mfn−1 Mtfn−1
(3.20)
with δr̄ and δ∆v̄p in equation (3.19) defined as

δr̄0





 δt0 




 δr̄1 




δr̄ =  δt1  ,




 : 




 δr̄f 


δtf
(3.21)







,






56


−∆v̄1




 −∆v̄2 
.
δ∆v̄ = 






−∆v̄n−1
(3.22)
Multiple iterations of both steps are required to yield a trajectory continuous in
position, velocity, and time.
3.4
Constraints
The transfer trajectory is assumed to originate in a low-altitude circular parking
orbit around the Earth. To ensure that the differentially corrected transfer trajectory
originates on the desired parking orbit, it is necessary to add constraints to the patch
point states that are defined on the parking orbit. Constraints can be defined for
any patch points if they can be expressed as functions of position, velocity, and time
corresponding to the patch point [1]. Thus, constraints are of the form
αi = α(r̄i , v̄i , ti ),
(3.23)
where the subscript i identifies the patch point at which the constraint is defined.
The derivation of the second stage of the two-level corrector can be expanded in
terms of constraint variations such that the goal, that is, no velocity discontinuities
at the internal patch points, remains the same. When incorporating constraints
into the differential corrections scheme, it is necessary to determine the variations of
the constraints with respect to the independent parameters. It is possible to define
multiple constraints at the same patch point.
A given constraint relationship at patch point P Pi is influenced by, at most, the
patch points immediately before (P Pi−1 ) and after (P Pi+1), as well as the point at
which the constraint is defined. Position constraints only depend upon the patch
point at which they are defined. Velocity dependent constraints may be functions of
states at the previous, the current, and the following patch point. These constraints
57
are added to the SRM in the second step of the 2LDC and represented in the following
expression,


δ∆v̄i
δαi


=
|
d∆v̄i
dr̄k
dαi
dr̄k
{z
d∆v̄i
dtk
dαij
dr̄k
Mα


}
δr̄k
δtk

.
(3.24)
where Mα represents the augmented State Relationship Matrix and the subscript k
identifies all n patch points. Depending on the specific purpose of a velocity dependent
constraint α, the velocity at P Pi, v̄i , can be represented three different ways. The
velocity v̄i in equation (3.23) can either be expressed solely as a function of v̄p− or v̄p+
or as a combination of the two. Since the transfer trajectory is computed through
backwards integration, the parking orbit constraints are applied at the last patch
point. In that case, v̄i is defined as v̄i = v̄p− . On the other hand, if a velocity
dependent constraint α is applied at the first patch point, v̄i must be defined as
v̄i = v̄p+ . The average value, v̄i = 0.5(v̄p− + v̄p+ ) has proven very effective when a
constraint α is defined at an internal patch point.
3.4.1
Position and Epoch Constraints
Constraining the position and epoch of a patch point is simply achieved by using
the following two constraints,
ᾱi1 = r¯i − r̄des ,
(3.25)
αi2 = ti − tdes .
(3.26)
and
Then, the related variations with respect to the independent variables are easily
deduced as
∂ ᾱi1
= I3 ,
∂r̄i
(3.27)
∂αi2
= 1.
∂ti
(3.28)
and
58
The variations can be used to compute the elements of the augmented SRM through
application of the chain rule.
3.4.2
Parking Orbit Constraints
At least two constraints are necessary to force a patch point to lie on a circular
parking orbit around the Earth. At a minimum, a specified altitude and an apse
condition must be ensured. In the present study, an altitude of 200 km above the
Earth’s surface was somewhat arbitrarily selected for the parking orbit. Patch points
and constraints must be defined with respect to the same center. Then, the altitude
constraint is a modification of equation (3.25), i.e.,
αi1 = |r̄i | − rdes ,
(3.29)
where |r̄i | is the altitude of the ith patch point relative to the Earth. The apse
constraint, r̄ · v̄ = 0, is expressed in functional form as
αi2 = r̄i · v̄i ,
(3.30)
where r̄i is the position vector and v̄i the velocity vector corresponding to the ith
patch point and with respect to the Earth. Since the altitude constraint is only a
function of position, the only non-zero partial is
∂αi1
r̄iT
=
.
∂r̄i
|r̄i |
(3.31)
For the apse constraint, the partials due to position and time are obtained as follows
∂αi2
= v̄iT ,
∂r̄i
(3.32)
∂αi2
= 0.
∂ti
(3.33)
The partial due to velocity first requires selection of the v̄i expression. The velocity
∂α
partial, ∂ v̄−ij , can then be obtained through application of the chain rule such that
i
∂αi2 ∂v̄i
∂αi2
;
− =
∂v̄i ∂v̄i−
∂v̄i
∂αi2
∂αi2 ∂v̄i
.
+ =
∂v̄i ∂v̄i+
∂v̄i
(3.34)
59
The velocity partials corresponding to the apse constraint can then be formulated as
∂αi2
T ∂v̄i
;
− = r̄i
∂v̄i
∂v̄i−
∂αi2
T ∂v̄i
,
+ = r̄i
∂v̄i
∂v̄i+
(3.35)
and used in the computation of the elements of the augmented SRM.
3.4.3
| ∆v̄ | Constraints
A transfer trajectory, from an Earth parking orbit to a halo orbit, that is completely continuous in both position and velocity is likely to be impossible. This is
true even when a number of revolutions of the halo orbit are incorporated in the
differential corrections scheme to add additional freedom. Under these conditions,
it is desirable in an initial step to allow deterministic maneuvers and constrain the
magnitude of the maneuver. This is accomplished via a constraint on the magnitude
of the velocity discontinuity that exists at an internal patch point. The constraint
replaces the original formulation, that is, forcing the ∆v̄ to zero, at the specified patch
point. The constraint is defined as
αi = ||v̄i+ − v̄i− | − ∆vdes |,
(3.36)
where ∆vdes is the desired scalar magnitude. Then the constraint partials evaluated
as
∂αi
−(v̄i+ − v̄i− )T
,
=
∂v̄i−
|v̄i+ − v̄i− |
(3.37)
(v̄i+ − v̄i− )T
∂αi
.
=
∂v̄i+
|v̄i+ − v̄i− |
(3.38)
The constraint is applied such that a deterministic maneuver of equal or smaller
magnitude than that specified is allowed. No effort is made to constrain the direction
of the maneuver.
60
3.5
Direct Transfer Trajectories from Earth to Lunar to L1 Halo Orbits
Direct transfers arcs that connect Earth parking orbits with the halo orbit gener-
ally require at least two deterministic maneuvers, one at each end of the transfer arc.
The maneuver to exit the Earth parking orbit and to escape the near-Earth environment is labelled the Transfer Trajectory Insertion (TTI) maneuver; the maneuver
required to enter the halo orbit is labelled Halo Orbit Insertion (HOI) maneuver. The
Time Of Flight (TOF) along the transfer is computed as the time interval between
the TTI and the HOI maneuver. Observe that TOF is not defined consistently in the
literature so comparison is sometimes difficult.
In a preliminary examination with a shooting technique, the transfer is computed
using backwards integration. The HOI maneuver is initially constrained to occur at
the x−z plane crossing on the far side of the L1 halo orbit. Earlier studies suggest that
using the plane crossing as the HOI point, yields successful transfers [7, 37]. Sample
transfer trajectories appear in figure 3.4 and figure 3.5. They are computed in the
CR3BP for two differently sized halo orbits. The same transfers appear in figures
3.6 and 3.7, where the CR3BP results are transitioned to the ephemeris model. The
corresponding |∆v̄| costs are listed in Table 3.1. The total cost for the transfer arc is
denoted by |∆v̄T otal |. It is the sum of the TTI and HOI maneuvers. Perhaps lifting
the constraint on the location of the HOI maneuver would be beneficial.
Table 3.1. Transfer Costs for Two Differently Sized Halo Orbits; TTI Maneuver
Constrained to the x − z Plane Crossing
Az
Model
(km)
|∆v̄T otal |
TTI
HOI
TOF
(m/s)
(m/s)
(m/s)
(days)
Figure
15,000
CR3BP
3716.3
3115.4
600.9
3.98
3.4
43,800
CR3BP
3659.5
3118.0
541.5
4.67
3.5
15,000 Ephemeris
3733.2
3111.8 621.45
3.87
3.6
43,800 Ephemeris
3744.4
3129.6
4.58
3.7
565.8
61
4
x 10
15
y [km]
10
Moon
Earth
5
L1
0
−5
0
0.5
1
1.5
x [km]
2
2.5
3
5
x 10
Figure 3.4. Transfer from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 15, 000 km) in the CR3BP (Location of HOI maneuver is constrained).
An alternative strategy to design a transfer is to completely lift the constraint
on the location of the HOI maneuver. Then, the patch point floats and the 2LDC
scheme results in a new location for the patch point yielding a continuous transfer arc
satisfying the applied constraints. The HOI maneuver costs in Table 3.1 are used as
the maximal allowed cost for the HOI maneuver constraint. The resulting transfers
in the CR3BP appear in figures 3.8 and 3.9. Note that the libration point orbit is no
longer precisely periodic and has evolved into a Lissajous trajectory. Clearly, patch
points along the Lissajous trajectory are included in the “transfer” arc. Thus, the shift
in the location of the maneuver is the result of allowing the patch points of the orbit
to move in the process of computing a continuous path. Details corresponding to the
62
5
x 10
Moon
1.5
y [km]
1
0.5
Earth
L1
0
−0.5
−1
0
0.5
1
1.5
2
x [km]
2.5
3
3.5
5
x 10
Figure 3.5. Transfer from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 43, 800 km) in the CR3BP (Location of HOI maneuver is constrained).
transfers in figures 3.8 and 3.9 appear in Table 3.2. Note that the ∆v̄’s are nearly equal
to the results in Table 3.1 (CR3BP), when the location is completely constrained.
But as suggested in figures 3.8 and 3.9, the new transfers are accomplished with a
maneuver that involves more than just an energy change. Partially constraining the
HOI location, based on the underlying dynamical structure, will likely yield results
more consistent with the natural flow. Such an approach is beyond the scope of the
current study since it requires more complete knowledge of the flow. Such work is
continuing.
Transition of any trajectory arc from the CR3BP model to the ephemeris model
is always nontrivial. Experience indicates that arcs including maneuvers with large
63
4
x 10
15
y [km]
10
5
0
−5
−3.5
−3
−2.5
−2
x [km]
−1.5
−1
−0.5
5
x 10
Figure 3.6. Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az =
15, 000 km) in an Ephemeris Model (Location of HOI Manuever Constrained).
directional changes are not good starting solutions. Thus, the transfers in figures
3.8 and 3.9 are not well-suited for transition to the ephemeris model, at least in
their current form. Nevertheless, the transfers in figures 3.8 and 3.9 are used as
starters for a DC scheme in the ephemeris model to explore the problem. Perhaps
not unexpectedly, the differential corrections process does not yield a transfer that
exhibits the same characteristics. These resulting transfer arcs appear in figures 3.10
and 3.11 and appear strikingly different. In fact, it is interesting to note, that these
two transfers actually use the stable manifold structure approaching from the smaller
primary. Such a result suggests a new direction for the study of transfer options.
64
4
x 10
15
y [km]
10
5
0
−5
−3.5
−3
−2.5
−2
x [km]
−1.5
−1
−0.5
5
x 10
Figure 3.7. Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az =
43,800 km) in an Ephemeris Model (Location of HOI Manuever Constrained).
3.6
Transfer Trajectories with a Manifold Insertion
Rather than inserting directly into the halo orbit, inserting onto a stable manifold
associated with the specified halo orbit offers additional freedom. Many points along
the manifold offer potential insertion points. The focus of the investigation, thus far,
is limited to insertions along a relatively small region of the stable manifold near the
approach to the halo orbit. This region is known to be successful in transfers from
the smaller primary in the Sun-Earth system [44]. Sample transfers in the CR3BP
appear in figures 3.12 and 3.13. The same transfers appear in figures 3.14 and 3.15,
where the CR3BP results are transitioned to the ephemeris model. The cost and
TOF for these transfers is summarized in Table 3.3. The maneuver to insert into the
manifold occurs at the Manifold Insertion Point and, hence, is labelled MIP. No HOI
maneuver is necessary as the manifold asymptotically approaches the halo orbit. The
65
Table 3.2. Transfer Costs for Two Differently Sized Halo Orbits; Location of the
TTI Maneuver Determined by the Differential Corrections Scheme.
Az
Model
(km)
|∆v̄T otal |
TTI
HOI
TOF
(m/s)
(m/s)
(m/s)
(days)
Figure
15,000
CR3BP
3700
3122.
579.0
4.19
3.8
43,800
CR3BP
3682.4
3122.6
539.2
4.95
3.9
15,000 Ephemeris
3808.4
3134.2
674.2
6.5
3.10
43,800 Ephemeris
3759.4
3142.5
616.9
9.2
3.11
time of flight for the transfer trajectory is measured from TTI to the first x − z plane
crossing on the far side of the halo orbit relative to the Earth. The TOF is, therefore,
considerably longer in comparison to previous cases, since the transfer path includes
the slow loop along the manifold before approaching the halo orbit. The actual TOF
on the transfer arc before the insertion onto a manifold trajectory is only 2.8 days
for both cases. Note from Table 3.3, that the MIP maneuver is slightly higher than
previous HOI ∆v̄’s. However, the approach is much smoother and may serve as a
more efficient baseline solution prior to cost reduction or optimization.
Table 3.3. Transfer Costs for Transfers with a Manifold Insertion for Two
Differently Sized Halo Orbits.
Az
Model
(km)
|∆v̄T otal |
TTI
MIP
TOF
(m/s)
(m/s)
(m/s)
(days)
Figure
15,000
CR3BP
3802.46
3121
698.11
12.18
3.12
43,800
CR3BP
3859.10
3130.94 728.15
15.96
3.13
15,000 Ephemeris
3854.08
3126.86 727.22
15.0
3.14
43,800 Ephemeris
3886.40
3139.18 747.22
15.6
3.15
66
5
x 10
1.5
Moon
y [km]
1
0.5
Earth
L
1
0
HOI
−0.5
−1
0
0.5
1
1.5
x [km]
2
2.5
3
5
x 10
Figure 3.8. Transfer from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 15,000 km) in the CR3BP (Location of the TTI Maneuver
Determined by the Differential Corrections Scheme.)
3.7
Effects of a Cost Reduction Procedure
A cost reduction procedure is a numerical continuation technique that can be
employed to decrease maneuver costs. By slightly decreasing the magnitude of an
allowed maneuver, it is often possible to determine trajectories near the initial baseline solution that exhibit similar behavior at a slightly lower cost. The process of
decreasing the maneuver cost can then be repeated until either an acceptable maneuver cost is available or the differential corrections process fails to determine a solution
satisfying all the constraints. Cost reduction procedures can often be successfully
67
5
x 10
1.5
Moon
y [km]
1
0.5
Earth
L1
0
HOI
−0.5
−1
0
0.5
1
1.5
2
x [km]
2.5
3
3.5
5
x 10
Figure 3.9. Transfer from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 43,800 km) in the CR3BP (Location of the TTI Maneuver
Determined by the Differential Corrections Scheme.)
applied to trajectories in an ephemeris model that originate from baseline solutions
in the CR3BP.
A preliminary investigation into the effects of a cost reduction procedure on the
transfers in the ephemeris model, is promising. The results of a cost reduction procedure applied to transfer trajectories that insert into a manifold on the near side
of the orbit is illustrated in figure 3.16 for transfers to the smaller halo orbit (Az
= 15,000 km). The baseline trajectory that initiates the cost reduction procedure
appears in figure 3.14. Note that the trajectory in black in figure 3.16 possesses a
higher manifold insertion cost than the transfers in green. The MIP cost decreases
from that associated with the transfer in black to the lower-cost transfer in green; the
68
5
3
x 10
y [km]
2
1
0
−4
−3
−2
−1
x [km]
0
5
x 10
5
x 10
5
x 10
1
z (km)
z [km]
1
0
−1
0
−1
−4
−3
−2
−1
x [km]
0
5
x 10
0
1
2
y (km)
3
5
x 10
Figure 3.10. Transfer from an Earth Parking Orbit to L1 Halo Orbits
(Az = 15,000 km) in an Ephemeris Model (Location of the TTI Maneuver
Determined by the Differential Corrections Scheme.)
MIP cost corresponding to the black trajectory is 737.2 m/s compared to 692.2 m/s
for the transfer in green. The lower cost transfer apparently exploits the invariant
manifold departing toward the smaller primary. Since similar behavior is observed
for both halo orbits, it suggests that cheaper transfers exist that approach the halo
orbit from the far side. The larger halo orbit also changes its shape quite significantly
and eventually possesses few similarities with the original orbit.
69
5
4
x 10
y [km]
3
2
1
0
−4
−2
x [km]
5
0
5
x 10
5
x 10
2
2
1
1
z (km)
z [km]
x 10
0
0
−1
−1
−2
−2
−4
−2
x [km]
0
0
5
x 10
2
y (km)
4
5
x 10
Figure 3.11. Transfer from an Earth Parking Orbit to L1 Halo Orbits
(Az = 43,800 km) in an Ephemeris Model (Location of the TTI Maneuver
Determined by the Differential Corrections Scheme.)
3.8
Free Return Trajectory
Transfer trajectories to halo orbits can be computed that naturally return to
the vicinity of the Earth. One such type of transfer can be determined when a
perpendicular x − z plane crossing on the far side of the halo orbit is targeted. The
resulting transfer trajectory could be employed for a spacecraft in an Earth parking
orbit to transfer to a lunar L1 halo orbit and return to an Earth parking orbit with no
additional maneuver. Similar trajectories could find applications in bringing supplies
to a space station in a lunar L1 orbit or even in human missions to L1 halo orbits
requiring safe return options. The HOI cost to insert into a halo orbit from such a
70
5
x 10
Moon
1
y [km]
0.5
Earth
L1
0
−0.5
−1
0
0.5
1
1.5
x [km]
2
2.5
3
5
x 10
Figure 3.12. Transfer from an Earth Parking Orbit with a Manifold Insertion
into an L1 Halo Orbit (Az = 15,000 km) in the CR3BP.
trajectory is approximately 647 m/s in an ephemeris model, higher than alternate
transfers previously discussed. The x − y projection of the trajectory arc appears
in figure 3.17 and the x − z and y − z projections are plotted in figures 3.18 and
3.19. All three projections have their origin at the Moon. The Earth movement in
the eight days it takes to complete the entire transfer from TTI to reentering the
Earth parking orbit is displayed by a black line in figures 3.17 and 3.18. The 200
km altitude parking orbit at the beginning of the journey appears in black and the
same parking orbit eight days later appears in red in figure 3.17. It is noted that the
resulting trajectory is not a periodic cycler trajectory. Two deterministic maneuvers
are required to enter and exit the Earth parking orbit.
71
5
x 10
1
Moon
y [km]
0.5
Earth
L1
0
−0.5
−1
0
0.5
1
1.5
2
2.5
3
x [km]
3.5
5
x 10
Figure 3.13. Transfer from an Earth Parking Orbit with a Manifold Insertion
into an L1 Halo Orbit (Az = 43,800 km) in the CR3BP.
3.9
Summary and Conclusions
The results here are of a preliminary nature and include the most recent efforts in
an ongoing investigation. Before comparing the cost of the different transfers, some
difficulties encountered in the differential corrections are discussed.
3.9.1
Numerical versus Dynamical Issues in the Computation of Transfers
In the determination of transfer trajectories using a linear differential corrections process, a number of difficulties are encountered in both the CR3BP and the
ephemeris model. In the CR3BP model, the baseline transfers are obtained using
72
5
x 10
1
y [km]
0.5
0
MIP
−0.5
−1
−4
−3.5
−3
−2.5
−2
x [km]
−1.5
−1
−0.5
5
x 10
Figure 3.14. Transfer from an Earth Parking Orbit with a Manifold Insertion
into an L1 Halo Orbit (Az = 15,000 km) in an Ephemeris Model.
shooting techniques and can be straightforwardly corrected when the HOI maneuver
is completely constrained, where the original baseline solution intersects the orbit or
the manifold. This is not surprising since only minor changes are required to yield a
continuous transfer that tangentially departs the Earth parking orbit. In the case of
the smaller halo orbit, it is also possible to decrease the HOI cost, when the location
of the HOI maneuver is not constrained. For the larger halo orbit, the shape of the
first revolution along the halo orbit is altered significantly in an attempt to reduce
the cost. The freedom available in the process, formulated with patch points along
the revolutions of the orbit, is designed to exploit this option. However, such a large
maneuver in a relatively large orbit offers many search directions and the process
73
5
x 10
1
y [km]
0.5
0
MIP
−0.5
−1
−1.5
−4
−3.5
−3
−2.5
−2
x [km]
−1.5
−1
−0.5
5
x 10
Figure 3.15. Transfer from an Earth Parking Orbit with a Manifold Insertion
into an L1 Halo Orbit (Az = 43,800 km) in an Ephemeris Model.
requires more guidance. Similar behavior is also observed, not surprisingly, in the
ephemeris model, where it also requires more steps to correct transfers to the larger
halo orbit compared to the smaller. The behavior of the numerical algorithm can be
improved via an expanded understanding of the dynamical nature of the problem.
More detailed investigations are required.
Reproducing or transitioning transfers to an ephemeris model is not a trivial task.
One issue, in particular, gains new significance in the Earth-Moon system. First,
the lunar mass is significant, resulting in a larger eccentricity of the relative primary
orbit than in many other three-body systems. Secondly, the Earth is the larger
primary. The impact of the first issue is apparent because computing trajectories in
74
5
x 10
1.5
1
y [km]
0.5
0
−0.5
−1
−4
−3.5
−3
−2.5
−2
x [km]
−1.5
−1
−0.5
5
x 10
Figure 3.16. Effects of a Cost Reduction Procedure on the Transfer
Arcs Initially Using the Invariant Manifold on the Near Earth Side for a
Halo Orbit with an Az Amplitude of 15,000 km in an Ephemeris Model.
the ephemeris model, requires dimensionalizing the patch points obtained from the
CR3BP. For a Sun-Earth trajectory, or an Earth-Moon trajectory not passing close
to one of the primaries, using the mean value corresponding to the distance between
the primaries, the characteristic distance, in the dimensionalization process is usually
sufficient. But, in order to obtain a baseline solution close to the converged trajectory
from the CR3BP model, the actual distance between the Earth and the Moon at the
appropriate epoch is typically used for each patch point. This is necessary since the
distance between the Earth and the Moon changes by about 13 percent over the
course of one year. Using the mean value is, therefore, not sufficiently precise for
trajectories very close to one of the primaries. Using the actual Earth-Moon distance
75
5
x 10
Moon
1
HOI
y [km]
0.5
0
−0.5
Direction
Earth is
moving
−1
−4
−3.5
−3
−2.5
−2
x [km]
−1.5
−1
−0.5
5
x 10
Figure 3.17. x − y Projection of a Free Return Trajectory to a Halo Orbit
with an Az Amplitude of 15,000 km in an Ephemeris Model.
for the specific Julian Date associated with the patch points aids in assuring the
validity of the baseline solution. This modification is reasonable and useful for a valid
baseline, but one that is further from the desired result. Thus, without additional
input to the differential corrections scheme, it is likely to converge to a solution with
characteristics very different from those observed in the result from the CR3BP. A
continuation scheme, therefore, is employed in the computation of some transfers. The
continuation scheme initially computes a continuous trajectory and then enforces one
constraint at a time. For this application, only the apse or the altitude constraint
is imposed in the first step. The accuracy on the apse constraint is increased and
the targeted altitude of the parking orbit is decreased over the course of multiple
76
5
x 10
1
Moon
z [km]
0.5
0
Earth
HOI
−0.5
−1
−3.5
−3
−2.5
−2
x [km]
−1.5
−1
−0.5
5
x 10
Figure 3.18. x − z Projection of a Free Return Trajectory to a Halo Orbit
with an Az Amplitude of 15,000 km in an Ephemeris Model.
iterations in some of the cases. A nonlinear process in conjunction with a known
dynamical structure is to be explored.
The current method to compute these transfers is based on a two-level linear differential corrections process, of course. A differential corrections scheme determines
a solution satisfying the imposed constraints but makes no attempt to determine the
optimal solution. Since it is based on a linear approximation, it requires a good baseline solution as an initial guess and requires that the underlying dynamics can locally
be approximated by a linear approximation. If both of these conditions are satisfied,
it returns a continuous solution close to the initial baseline solution. The natural flow
in the neighborhood of the larger primary is very different from the dynamics near the
77
5
x 10
1
z [km]
0.5
0
HOI
−0.5
−1
−1.5
−1
−0.5
0
y [km]
0.5
1
1.5
5
x 10
Figure 3.19. y − z Projection of a Free Return Trajectory to a Halo Orbit
with an Az Amplitude of 15,000 km in an Ephemeris Model.
smaller primary. No naturally occurring transfers exist that connect the environment
close to the major primary to halo orbits at L1 or L2 . Large deterministic maneuvers
are therefore necessary to insert into the halo orbit. A linear differential corrections
scheme is not very efficient in predicting the changes in the path point states to yield
such a trajectory unless the initial baseline solution is in the immediate vicinity of the
desired transfer. The underlying linear variations are not sufficient to approximate
the dynamics close to the larger primary. It is important to note that most previous
investigations into the determination of transfers in the 3BP have focused on transfers
to the smaller primary. The natural dynamics and the computation of transfers from
libration point orbits to the smaller primary is well understood and the developed
78
methodology will be discussed and used to compute a few sample transfers in chapter
four.
3.9.2
Discussion
The TTI maneuver is consistently larger for transfers to the larger halo orbit. This
is the result of a slightly higher energy level being required to reach and insert into
the larger halo orbit, since it is slightly further away from the Earth due to the larger
out-of-plane component. Overall, the changes in the TTI maneuver are fairly small,
in agreement with previous investigations [5, 7, 62]. The HOI, respectively MIP, cost
is lower for the larger halo. This is likely the result of the natural flow of the larger
halo passing closer by the Earth than the flow corresponding to of the smaller halo. In
this limited study, inserting into an invariant manifold on the near side of the orbit is
more expensive than inserting into the halo orbit directly. Selecting chosen the closest
Earth approach manifold for the specific halo, but arbitrarily identification of an initial
manifold insertion point contributes to the increased cost. Investigating a larger range
of manifolds and initial insertion points is certainly warranted and will likely result
in significantly lower MIP costs. Another interesting observation is that the cost
reduction procedure moves the manifold insertion point from the near Earth side to
the far side of the halo orbit and uses the stable manifold approaching from the smaller
primary. This suggests that transfers using the stable manifold structure approaching
from the smaller primary exist for a lower cost. Transfers in the ephemeris model
are always more expensive than the baseline solutions in the CR3BP. This is true for
both the TTI as well as the HOI, as well as MIP, costs. Although, systematically
applying a cost reduction procedure to the computed transfers is likely to decrease the
cost, this is not surprising. The baseline solutions in the CR3BP do not incorporate
any perturbations. Hence, the transfer determined in the ephemeris model does not
necessarily correspond to the same transfer in the CR3BP. Incorporating the Sun
into the model will yield more representative baseline solutions for correction in an
79
ephemeris model. Overall, the magnitude of the transfers is similar to the transfer
costs in previous investigation [5, 7, 36, 37, 62].
80
4. TRANSFERS FROM EARTH PARKING ORBITS TO
SUN-EARTH LIBRATION POINT ORBITS
Transfers between Earth parking orbits and Sun-Earth halo orbits at L1 and L2 are
computed. This is significant since the Earth is the smaller primary in the Sun-Earth
system. An investigation into minimal passing altitudes for Sun-Earth halo orbits
at L1 is included and the passing distances and paths of manifolds near the smaller
primary are clarified. Sample transfers between two differently sized halo orbits at
both L1 and L2 in the Sun-Earth system are presented.
4.1
Stable Flow from the Libration Points in the Direction of the Earth
In the Sun-Earth system, the Earth is the smaller primary. The natural flow to and
from the halo orbits is significantly different in the vicinity of the smaller primary
compared to the larger primary. Manifolds corresponding to halo orbits near the
collinear libration points, representing the natural flow, can be computed. These pass
in the immediate vicinity of the Earth or even at a distance smaller than the radius of
the Earth. In 1994, Barden completed an extensive study of the manifolds associated
with specific halo orbits [28]. At the time, analysis of the manifolds associated with
a large range of halo orbits was a challenging task due to the computations required.
Barden determined that for a halo orbit with an Az amplitude of 440,000 km, a range
of manifolds passes the Earth at acceptable altitudes that could be used for transfers
to such a halo orbit with minimal or no halo orbit insertion cost. Using modern
high-speed computers, it is possible to compute the closest approach manifolds for a
larger range of halo orbits.
The closest approach manifolds associated with halo orbits with Az amplitudes
ranging between 0 km and 920,000 km have been computed. The relationship between
81
halo orbit amplitude and the closest approach altitude is plotted in figure 4.1. The
minimum occurs for a halo orbit possessing an Az value of approximately 691,800
km and the closest approach manifold passes only a few kilometers above the center
of the Earth. A stable manifold intersecting the surface of the Earth could be used
for direct launches onto a stable manifold and an asymptotic approach to the halo
orbit. Any orbit (within the computed range) with an Az amplitude larger than
334,351 km has manifolds crossing an Earth parking orbit at a 200 km altitude. Two
of these manifolds possess a periapsis distance at the parking orbit altitude. Thus,
the velocity vector in the parking orbit and in the manifold are parallel and the TTI
maneuver is a minimum. Manifolds that intersect the surface of the Earth (assuming
a perfect sphere) can be computed for halo orbits with Az amplitudes larger than
341,083 km. The relationship between the Az amplitude of the halo orbit and the
altitude of the closest unstable manifolds with Earth approach is also represented by
figure 4.1 due to symmetry considerations [51]. This is consistent with the findings
of other studies [28]. Halo orbits with relatively small Az amplitudes do not possess
manifolds passing close by the Earth. Transfers from Earth parking orbits to small
amplitude halo orbits can nevertheless be computed using the algorithm presented in
section (3.3).
4.2
Selection of Halo Orbit Sizes
Spacecraft are not typically located precisely at the Sun-Earth libration point L1
because of the resulting Sun-Earth-satellite alignment. When viewed from the Earth,
a satellite at L1 is aligned with the Sun and downlink telemetry is overwhelmed by the
intense solar noise background [8]. At the same time, large amplitude halo orbits also
encounter communications issues. For large halo orbits, the Earth-Spacecraft-Sun
angle can cause difficulty since it changes by a large amount. Therefore intermediatesized halo orbits (or Lissajous orbits) are generally preferred.
82
8000
Lowest Passing Altitude [km]
6000
4000
2000
0
−2000
−4000
−6000
−8000
0
2
4
6
Halo Orbit Az [km]
8
10
5
x 10
Figure 4.1. Closest Approach Altitudes for L1 Sun-Earth Manifolds
Relative to the Earth.
The halo orbits examined here possess Az amplitudes of 120,000 km and 440,000
km. The Az value of 120,000 km is selected because the analysis for the ISEE-3 mission determined that such an orbit satisfies the communication constraints [8, 38]. In
addition to the ISEE-3 mission, other libration point missions have also used similar
sized halo orbits [10, 63]. A halo orbit with an Az amplitude around 120,000 km is
ideally suited to avoid interference with the Sun and allows the usage of standard communication equipment. For a halo orbit with an Az value of approximately 440,000
km a large number of manifolds can be computed that pass the Earth at acceptable
altitudes [28].
83
4.3
Transfer Trajectories From Earth to L1 Halo Orbits
Transfer trajectories from an Earth parking orbit to two differently sized L1 halo
orbits are presented. For the smaller halo orbit with an Az amplitude of 120,000 km,
the manifold tube does not intersect the Earth at a radius of 6378 km or even at a
200 km altitude parking orbit. Hence, a transfer trajectory is computed using the
2LDC process. Before the 2LDC algorithm (with or without constraints) can be used
to compute a transfer, it is necessary to obtain a baseline solution and generate patch
points along the arc. The closest approach manifold can be selected as the baseline
solution, since it is the closest, natural occurring solution compared to the desired
transfer path. Patch points along the manifold are then generated and differentially
corrected to yield a smooth, continuous trajectory between the parking orbit and
the halo orbit. For the larger Az amplitude halo orbits, manifolds with periapsis
at an altitude of approximately 200 km are selected as the baseline solutions. The
ultimate transfer path, constitutes a differentially corrected version of the manifold
that precisely connects the initial and final orbits with no insertion cost at the halo
orbit. The maneuver cost associated with these transfers are summarized in Table
4.1. For the smaller halo orbit with an Az amplitude of 120,000 km, the HOI insertion
cost is 20.5 m/s and the transfer appears in figure 4.2. This value is very close to
the minimal halo orbit insertion cost from analysis by Barden for the same sized
halo orbit. In a much more comprehensive investigation for the same sized orbit [28],
Barden computed a transfer trajectory from Earth parking orbit to an L1 halo orbit
with a HOI insertion cost of only 20.3 m/s. Barden used a different differential
corrections process incorporating a change of variables to compute the transfers. For
the larger halo orbit (Az = 440,000 km), two transfer trajectories are computed with
no HOI maneuver necessary. The two transfers appear in figure 4.3 in green and red.
All three transfers are reproduced in an ephemeris model and appear in figure (4.3).
84
5
x 10
6
4
y [km]
2
0
L1
Earth
−2
−4
−6
1.48
1.482 1.484 1.486 1.488
x [km]
1.49
1.492 1.494 1.496
8
x 10
Figure 4.2. Transfer from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 120, 000 km) in the CR3BP.
4.4
Transfer Trajectories From Earth to L2 Halo Orbits
Given the transfers to L1 halo orits in the Sun-Earth system, similar transfers are
computed for L2 halo orbits with similar maneuver costs. The detailed transfer costs
are summarized in Table 4.2. For a halo orbit with an Az amplitude of 120,000 km,
this investigation resulted a transfer with an HOI cost of 20.60 m/s. This transfer
appears in figure 4.5, with a HOI cost very close to the value computed by Barden,
that is, 20.34 m/s. For the larger halo orbit with an Az amplitude of 440,000 km, two
transfer trajectories are computed and appear in figure 4.6. No halo orbit insertion
cost are necessary for either transfer, since they are trajectories on the manifold tube.
85
5
8
x 10
6
4
y [km]
2
L1
0
Earth
−2
−4
−6
−8
1.478
Sun
1.48
1.482 1.484 1.486 1.488 1.49
x [km]
1.492 1.494 1.496 1.498
8
x 10
Figure 4.3. Transfers from an Earth Parking Orbit to a L1 Halo Orbit
(Az = 440, 000 km) in the CR3BP.
All three transfers are reproduced in an ephemeris model and appear in figure (4.7).
4.5
Conclusions
The transfer trajectory insertion cost from an Earth parking orbit onto a transfer
trajectory to a Sun-Earth halo orbit seem to be nearly constant for all cases considered. The small differences in the TTI cost in an ephemeris model are caused by the
varying location of the Moon. This is in agreement with previous previous findings
published in the literature [5,7,62]. In the Sun-Earth system, the Earth is the smaller
primary and trajectories between halo orbits and the vicinity of the Earth naturally
86
Table 4.1. Transfer Costs for Transfers from a 200 km Altitude Earth Parking
Orbit to Two Differently Sized Sun-Earth L1 Halo Orbits
Az
Model
(km)
|∆v̄T otal |
TTI
HOI
TOF
(m/s)
(m/s)
(m/s)
(days)
Figure
120,000
CR3BP
3270.7
3192.5
20.5
205.1
4.2 (red)
440,000
CR3BP
3193.5
3193.5
0
210.78
4.3 (red)
440,000
CR3BP
3193.5
3193.5
0
210.78 4.3 (green)
120,000 Ephemeris
3206.8
3196.26
14.74
208.91
4.4 (blue)
440,000 Ephemeris
3193.7
3193.7
0
210.78
4.4 (red)
440,000 Ephemeris
3196.4
3196.4
0
213.11 4.4 (green)
Table 4.2. Transfer Costs for Transfers from a 200 km Altitude Earth Parking
Orbit to Two Differently Sized Sun-Earth L2 Halo Orbits
Az
Model
(km)
|∆v̄T otal |
TTI
HOI
TOF
(m/s)
(m/s)
(m/s)
(days)
Figure
120,000
CR3BP
3213.13
3192.5
20.66
209.3
4.5 (red)
440,000
CR3BP
3193.4
3193.4
0
215.5
4.6 (red)
440,000
CR3BP
3193.4
3193.4
0
213.8
4.6 (green)
120,000 Ephemeris
3214.8
3192.4
18.66
209.25
4.7 (blue)
440,000 Ephemeris
3194.4
3194.4
0
215.5
4.7 (red)
440,000 Ephemeris
3194.2
3194.2
0
213.8
4.7 (green)
87
5
x 10
8
6
4
y [km]
2
0
L1
Earth
−2
−4
−6
Sun
−15
−10
−5
x [km]
0
5
x 10
Figure 4.4. Transfers from an Earth Parking Orbit to a L1 Halo Orbit
in an Ephemeris Model.
occur. The halo orbit insertion cost is dependent on the size of the halo orbit. For
larger halo orbits, it is possible to compute transfer trajectories with no halo orbit
insertion cost, as naturally occurring manifolds exist. The HOI costs are comparable
for both L1 and L2 halo orbits of identical size, again in agreement with previous
findings [28]. Although, a slightly different methodology is used in this investigation
compared to previous studies, the magnitude of the HOI maneuvers is consistent [28].
88
5
8
x 10
6
4
y [km]
2
Earth
0
L2
−2
−4
−6
−8
1.494 1.496 1.498
1.5
1.502 1.504 1.506 1.508
x [km]
1.51
1.512 1.514
8
x 10
Figure 4.5. Transfer from an Earth Parking Orbit to a L2 Halo Orbit
(Az = 120, 000 km) in the CR3BP.
89
5
8
x 10
6
4
y [km]
2
Earth
0
L2
−2
−4
−6
−8
1.494 1.496 1.498
1.5
1.502 1.504 1.506 1.508
x [km]
1.51
1.512 1.514
8
x 10
Figure 4.6. Transfers from an Earth Parking Orbit to a L2 Halo Orbit
(Az = 440, 000 km) in the CR3BP.
90
5
x 10
8
6
4
y [km]
2
L2
Earth
0
−2
−4
−6
Sun
−8
0
5
10
x [km]
15
5
x 10
Figure 4.7. Transfers from an Earth Parking Orbit to a L2 Halo Orbit
in an Ephemeris Model.
91
5. LAUNCH TRAJECTORIES
As it is apparent from the previous chapter, the design of transfer trajectories from
the Earth, as the smaller primary, to halo orbits in the Sun-Earth system is well established. When Earth is the larger primary, as in the Earth-Moon system, transfers
are not as straightforward. However, an additional analysis capability that might be
useful to mission designers, in either case, is the flexibility to determine the launch trajectory in conjunction with the transfer. The work presented in this chapter includes
an investigation of preliminary methods for computing launch trajectories. The use
of differential correction schemes for this purpose is discussed and a sample launch
trajectory to a 200 km altitude parking orbit is presented. This proof of concept
suggests more detailed investigations to be pursued.
5.1
Equation of Motions with Constant Thrust Term
To develop a mathematical model for launch, the equations of motion must be
modified to incorporate an extra force. Because the launch leg is added to the transfer,
the problem is still formulated in terms of the CR3BP. A constant thrust force is
added to the equations of motion at some arbitrary angle relative to the direction
of the velocity vector. Let the constant thrust term element, T̄ , be equivalent to a
constant force divided by the mass of the spacecraft. It is approximated as constant
and, thus, the term possesses units of acceleration. The thrust term appears in the
application of Newton’s law of motion, equation (2.15), i.e.,
F̄ = m3 I r̄¨3 = −
Gm3 m1
Gm3 m2
r̄13 −
r̄23 + m3 T κ̂.
3
3
r13
r23
(5.1)
“Thrust” can also be nondimensionalized such that, nondimensional thrust is κ̄ and,
(1 − µ) ¯ µ
d2 ρ̄
=−
d − 3 r̄ + κ̄.
2
dτ
d3
r
(5.2)
92
Then, the second order scalar differential equations appear in the following form
(1 − µ)(x + µ) µ(x − (1 − µ))
−
+ κx ,
(5.3)
d3
r3
(1 − µ)y µy
− 3 + κy ,
(5.4)
ÿ + 2ẋ − y = −
d3
r
(1 − µ)z µz
z̈ = −
− 3 + κz .
(5.5)
d3
r
where κ is the nondimensional thrust magnitude and κx , κy , and κz are thrust compoẍ − 2ẏ − x = −
nents in the various thrust directions as written in rotating coordinates and applied
in three-dimensional space. Since it is desirable to apply the thrust in various directions relative to the velocity vector, spherical coordinates are introduced to define
these angles.
The spherical coordinates appear in figure 5.1 and relate the direction of the unit
thrust to the velocity vector in the rotating frame of the primaries. The unit vectors
ξˆv , ξˆB and ξˆN , introduced by figure 5.1, are relative to the synodic frame of the
primaries with ξˆv parallel to the velocity vector. Note that r̄κ is directed parallel to
unit thrust in figure 5.1. Using the specified spherical coordinates, the unit thrust
vector is defined by
κ̂ = sin θ cos φ ξˆv + sin θ sin φ ξˆB + cos θ ξˆN .
(5.6)
Thus the unit thrust vector relative to the rotating frame of the primaries becomes
κ̂ = [sin θ · cos φ · C11 + sin θ · sin φ · C12 + cos θ · C13 ] x̂
+ [sin θ · cos φ · C21 + sin θ · sin φ · C22 + cos θ · C23 ] ŷ
(5.7)
+ [sin θ · cos φ · C31 + sin θ · sin φ · C32 + cos θ · C33 ] ẑ
where the Cij are scalar elements of the direction cosine matrix between the synodic
frame of the primaries and the plane spanned by the position vector relative to the
Earth, r̄E , and the velocity vector in the final orbit (at the parking orbit or manifold
insertion point) in the rotating frame. To evaluate the elements of the direction
cosine matrix, the position and velocity vector are crossed to obtain the unit angular
momentum vector, normal to the orbital plane, such that
ĥ =
r̂E × v̂
.
|r̂E × v̂|
(5.8)
93
Figure 5.1. Spherical Coordinates to Define the
Direction of the Thrust Vector.
The unit angular momentum vector, ĥ, is equal to the unit vector ξˆB that appears
in figure 5.1. Now, crossing the unit position vector and the unit angular momentum
vectors yields
ξˆN = ĥ × ξˆv ,
(5.9)
the third vector to form the dextral orthonormal triad ξˆv , ξˆB , and ξˆN . Thus, the
transformation matrix from triad ξˆv , ξˆB , and ξˆN to the synodic frame of the primaries
in three-dimensional space can be defined as follows,


  
ξˆv
C11 C12 C13
x̂


  
 ˆ 
  
=
 ŷ   C21 C22 C23   ξN  .


  
C31 C32 C33
ξˆB
ẑ
(5.10)
Equations (5.3)-(5.5) are used to compute the thrusted trajectory arcs presented in
this investigation.
94
5.2
State Transition Matrix
To facilitate the implementation of the thrust parameters κ, θ, and φ into a
differential corrections scheme, an augmented state vector is defined,
x̄ ≡ [x, y, z, ẋ, ẏ, ż, φ, θ, κ]T ,
(5.11)
and the corresponding augmented variational state vector then becomes
δx̄ ≡ [δx, δyδz, δ ẋ, δ ẏ, δ ż, δφ, δθ, δκ]T .
(5.12)
It is assumed that the thrust magnitudes and the two angles remain constant along a
trajectory segment. Thus, the state space form of the augmented variational equations
is
δ ẋ(t) = A(t)δx(t),
(5.13)
where A(t) is now an 9 × 9 matrix, and, when written in terms of nine 3 by 3
submatrices has the general form


0
I3
0




A(t) =  B(t) 2Ω D(t)  ,


0
0
0
(5.14)
with 0 representing the zero matrix, and I3 representing the identity matrix of rank
3. The addition of a thrust term to the equations of motion alters the elements of
the submatrix B(t), in the variational matrix from equation (2.33), such that


U + κxx Uxy + κxy Uxz + κxz

 xx


B(t) =  Uyx + κyx Uyy + κyy Uyz + κyz  ,


Uzx + κzx Uzy + κzy Uzz + κzz
(5.15)
95
∂κ
where κjk = ∂kj (k = x, y, z) are the partial derivatives of the thrust expression, in
equation (5.7), with respect to the Cartesian coordinates. The time varying submatrix
D(t) is 3 × 3 with the form

∂x
∂θ0

 ∂y
D(t) =  ∂θ
 0
∂z
∂θ0
∂x
∂φ0
∂x
∂τ0
∂y
∂φ0
∂y
∂τ0
∂z
∂φ0
∂z
∂τ0



.

(5.16)
With the addition of the variations in the thrust parameters, the state transition
matrix, φ(t, t0 ), enlarges to become a 9 × 9 matrix.
5.3
Patch Points and Initial Trajectory
As discussed in Section (2.4.1), the region near the primaries is very sensitive.
Backward integration from the manifold down to the surface of the Earth is used to
limit the sensitivity to the initial conditions. To apply a 2-Level Differential Corrections scheme to a given problem, it is necessary that a discrete set of patch points
be available to characterize the trajectory. An initial estimate for the launch trajectory is obtained through numerical integration of the differential equations (5.3)-(5.5)
backwards to the surface of the Earth. The initial thrust angles can be determined
either by trial-and-error or through a modified one-step targeting scheme. A set of
patch points is created from the resulting, numerically determined, trajectory. It is
desirable that the final launch trajectory tangentially approach (flight path angle 0o )
the parking orbit to yield the lowest insertion cost. (Of course, if the “final” point is
required to be the actual launch site on the rotating Earth surface, then backwards
integration ceases at the tipoff location, where the flight path angle is between 0o and
90o . If the “final” point is the launch site, the angle is 90o .) These design requirements are implemented in the 2LDC through an altitude and apse constraint at the
parking orbit and a position constraint at the launch site. Additionally, the initial
and final times are also fixed.
96
5.3.1
Determination of the Launch Site
The longitude and latitude of any launch site can be used in the algorithms. In
the determination of the launch site, the oblate Earth is modelled as a perfect sphere
ignoring the equatorial bulge. The resulting spherical angles are, thus, not precisely
equal to longitude and latitude but qualitatively possess similar meanings. This allows
the use of the locations of known launch sites, such as the European Launch site at
Kourou, French Guyana, in the computation of transfer trajectories. Modifications
to accommodate a non-spherical Earth can be computed later as required.
The launch site on the surface of the Earth lies in the Earth-Centered-Fixed (ECF)
frame. The ECF frame is a rotating frame centered at the Earth that is inclined
with respect to the ecliptic plane. The inclination of the Earth is constant in the
CR3BP. In an ephemeris model, the Earth’s inclination is determined instantaneously.
Transformations between the ECF frame and the rotating frame of the primaries
centered at the barycenter involve three steps. First, it is necessary to transform
the state from the rotating frame to the inertial frame; then, the origin must be
modified before the final transformation between the inertial and the desired rotating
frame can commence. Two of these frame changes deal with a rotating frame, and
therefore, it is necessary to follow the steps in Section (2.2). The launch site on the
Earth’s surface is, hence, moving as a function of time with respect to the rotating
frame of the primaries. Constraining the total time of flight for the launch trajectory
results in a simplified model, where the location of the launch site remains constant.
In an ephemeris model, the location of the launch site does not remain constant
due to additional perturbations and is a function of the epoch. In the following
launch scenario, the European launch site in Kourou, French Guyana (longitude 52.8 degrees, latitude 5.2 degrees) is selected as the launch site.
97
5.4
Two-Level Differential Corrector with Thrust
The 2LDC from Section (3.3), developed by Howell and Pernicka [52], varies the
position and velocity states of the patch points to achieve a continuous trajectory.
The major difference between the current application and the differential corrections
process as described in Section (3.3), is the set of parameters that define the thrust
state. The augmented state vector and the augmented state transition matrix, including contributions from changes in the thrust terms, are incorporated. A differential
corrections process is not limited to using the velocity and position states to yield a
trajectory satisfying all constraints. Other variables present in the problem can be
exploited. Variations in the parameters that define the thrust state take advantage
of the acceleration introduced by the thrust term as it is added to the equations of
motion. The process is most successful if each acceleration (dynamical plus thrust)
is accommodated in a separate step. The thrust parameters are therefore varied in
the first step of this modified process. Velocity discontinuities are introduced in the
process of achieving position continuity. They are then reduced in the second step of
the differential corrections scheme.
5.4.1
Two-Level Differential Corrector with Thrust
In this modified two-level differential corrections process, the first step involves
variations of the thrust parameters to achieve a trajectory continuous in position.
The thrust parameters, i.e., thrust magnitude and two angles, are maintained as constants along each trajectory arc but are allowed to vary between different segments.
In matrix format, the linear targeter is formulated as


φ
φ
φ
ẋ

 17 18 19


L =  φ27 φ28 φ29 ẏ  ,


φ37 φ38 φ39 ż
(5.17)
98
consistent with equation (3.2) with the controls defined as
k̄ = [δθ0 , δφ0, δκ0 , δ∆t]T ,
(5.18)
where δ∆t = δtp − δt0 is the time of flight on each segment and the targets are
b̄ = [δxp , δyp , δzp ]T .
(5.19)
The state of the end of each segment is denoted by xp , yp , and zp , with the corresponding time denoted tp . Since the number of control variables in k̄ exceeds the
number of target states in b̄, the Euclidean norm is computed to minimize the changes
in the elements of the control vector, while achieving the desired target state. When
a change in one particular control variable, such as the thrust magnitude or the time
of flight, is not desirable but allowed to give the corrector more freedom, a weighted
Euclidean norm, is employed, that is,
k̄ = W −1 LT (LW −1 LT )−1 b.
(5.20)
The weighting matrix, W , is a diagonal matrix with specific weights corresponding
to each variable, on the diagonal. A larger specific weight in the weighting matrix,
yields smaller changes in the corresponding control variable.
Modification of the thrust parameters at the beginning of each segment results
in velocity discontinuities between the individual segments along the trajectory. The
second step, presented in section (3.3.2), varies the position and time components of
the patch points to drive those internal velocity discontinuities to zero. The same
process can be employed here to eliminate the velocity discontinuities introduced and
produce a trajectory continuous in both position and velocity.
5.5
Trajectory from Launch Site into Parking Orbit
A sample launch scenario is computed using the modified differential correction
scheme. The launch trajectory originates at the launch site on the surface of the
Earth and inserts into a 200 km altitude parking orbit. Twelve patch points along
99
the trajectory, equally spaced in distance on the baseline solution, are used in the
differential corrections process to yield the continuous trajectory arc satisfying the
imposed constraints. The value of the initial thrust magnitude is 14.8245 m/s2 .
The x − y projection of the trajectory arc appears in figure 5.2 and the x − z and
y − z projections are plotted in figures 5.3 and 5.4. The thrust parameters and times
corresponding to the patch points of the converged trajectory are summarized in Table
5.1. Note that the trajectory arc is computed backwards from the parking orbit down
to the launch site. Therefore the first patch point in Table 5.1 is on the parking orbit
and the last patch, patch point number 12, is at the launch site. The time is also
measured backwards from the parking orbit and, hence, is negative. The total time
of flight on the trajectory is five minutes and one second and the cost measured in ∆v
is 12.796 m/s. The time of flight along the launch trajectory for the Genesis mission
to reach a 185 km parking orbit was 10 minutes and 34 seconds [64]. The ideal cost
to bring a satellite into a low Earth orbit is usually on the order of 7.9 - 10 km/s [65].
The changes in the thrust magnitude introduced by the differential corrector are fairly
small due to the weighting matrix in the computation of the Euclidean norm, equation
(5.20). The variations in the thrust angles from the initial angles (φ0 = 68o and θ0
=13o ) are small because the baseline solution is selected close to the desired final
solution. This is necessary to ensure the validity of the linear approximations in the
differential corrections process.
100
Table 5.1. Changes in Thrust Parameters Throughout Trajectory Arc.
PP Number
Time
κ
φ
θ
(sec)
(m/s2 )
(degrees)
(degrees)
1
0
14.8245426 53.757684
5.38
2
-31.69
14.8245442
68.03
6.62
3
-59.18
14.8245387
39.32
4.05
4
-87.29
14.8245417
99.27
7.64
5
-115.01 14.8245397
77.04
7.80
6
-146.26 14.8245486
102.43
9.17
7
-174.54 14.8245454
64.84
18.16
8
-201.53 14.8245338
72.50
17.69
9
-231.04 14.8245342
67.55
12.51
10
-257.32 14.8245394
63.31
16.31
11
-284.04 14.8245502
66.54
9.32
12
-303.59 14.8245430
68.0
13.0
The computed launch segment can now be combined with the transfers presented
in Chapter 4 to yield a complete transfer from the launch site on the Earth surface to
a Sun-Earth halo orbit at L1 or L2 . The deterministic maneuvers presented in tables
4.1 and 4.2 need to be added to the launch cost to estimate the cost of the whole
transfer.
5.5.1
Challenges with the Launch Formulation
The launch trajectory design problem is formulated here in terms of the CR3BP
to blend it easily with manifold trajectories for design applications in multi-body
problems. However, some unique challenges are observed. These issues are discussed
in terms of three separate categories. First, it is important to realize that the en-
101
6000
4000
Earth’ Surface
y [km]
2000
0
Trajectory
−2000
Launch Site
−4000
−6000
200 km
Altitude
Parking Orbit
1.4959 1.4959 1.4959 1.496
Parking Orbit
Insertion Point
1.496
x [km]
1.496
1.496
1.496 1.4961
8
x 10
Figure 5.2. Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit.
vironment close to one of the primaries is very sensitive in the CR3BP due to the
singularities at their center. Second, without modification, a differential corrections
scheme is not developed to accommodate angles and, therefore, angles are not the
best choice for dependent variables in a control strategy. Finally, parameterizing the
thrust is a key step in the process.
The most severe numerical challenge is the sensitivity in the equations of motions
near the Earth. Ill-conditioned STMs and SRM matrices are often the result of
the sensitivities inherent in the near-Earth environment. These singular matrices
often occur after a few iterations into a differential corrections process and, thus, the
procedure fails to converge. Slight changes in the initial patch points or in the initial
values of the thrust parameters may aid in the elimination of these problems. But
102
6000
Earth’ Surface
4000
Trajectory
z [km]
2000
200 km Altitude
Parking Orbit
Launch Site
0
Parking Orbit
Insertion Point
−2000
−4000
−6000
1.4959 1.4959 1.4959
1.496
1.496
x [km]
1.496
1.496
1.496
1.4961
8
x 10
Figure 5.3. Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit.
often in this regime, the initial baseline trajectory is simply too far from a continuous
solution that meets all constraints. The linear approximations in the differential
corrections process cannot be expected to adequately estimate the necessary changes
in the patch points. Trajectories to higher altitude orbits or thrusted trajectories far
away from the Earth are generally much easier to compute. Although, in any case, a
good initial guess is necessary to ensure convergence onto an acceptable solution.
A differential corrections process treats angles like any other variable it encounters.
But, of course, small changes in an angle can result in huge changes in other state
variables downstream. Software logic, therefore, must be implemented to ensure that
the angular changes are implemented in the intended direction and are not larger
103
6000
Earth’ Surface
4000
Launch Site
Parking Orbit
Insertion Point
z [km]
2000
200 km Altitude
Parking Orbit
0
Trajectory
−2000
−4000
−6000
−8000
−6000
−4000
−2000
0
y [km]
2000
4000
6000
8000
Figure 5.4. Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit.
than a few degrees per iteration. A better alternative is to represent the thrust in
terms of different variables.
Depending on how the thrust parameters are incorporated into the differential
corrections scheme, as well as the nature of the constraints, it may be impossible to
force the internal ∆v’s to zero. Also, a staged approach is usually most successful.
That is, both velocity/position are not modified in the same step of the differential
corrections process as the thrust parameters. The internal ∆v’s are ultimately lower
but not zero. The two accelerations in the current formulation, that is, the natural
motion and the constant thrust term, are frequently in competition to eliminate the
discontinuities.
104
Many of the difficulties challenges in this investigation are very similar to the
problems highlighted by McInnes [66] in the computation of transfer trajectories
using solar sails. In his efforts to compute transfers from Sun-Earth halo orbits to
Earth parking orbits, he encountered difficulties with eliminating internal velocity
discontinuities and was unable to compute transfers to low-Earth orbit. Limiting
changes in thrust components and velocity/position components to separate stages
did result in progress, however. Since the difficulties are most severe in the near-Earth
environment in both cases, it suggests that the linear variations of the differential
scheme are ineffective in approximating the changes required close to the primary.
5.6
Conclusion
This preliminary investigation is a proof of concept to illustrate that a differen-
tial correction scheme can, in principle, be used to produce a baseline solution for a
launch trajectory. The challenges here suggest that alternative methods be investigated to develop a more efficient and effective methodology for the computation of
baseline launch trajectories using targetors and correctors. Using differential corrections techniques in the computation of launch trajectories is intended to be the first
step in establishing a methodology that can be used efficiently. The use of optimal
control methodology is therefore planned to be implemented in the future. But, perhaps of equal significance, the underlying dynamical structure can be more effectively
exploited to create a more “intelligent,” adaptive procedure.
105
6. SUMMARY AND RECOMMENDATIONS
6.0.1
Summary
The primary goal of this study is an alternate methodology to determine transfers
between the Earth and libration point orbits, such that the Earth is either the smaller
or the larger primary. The work is decomposed into three separate problems. First,
transfers between the Earth and lunar halo orbits at L1 are considered. In this
case, the Earth is the larger primary. Second, transfers are computed between the
Earth and Sun-Earth libration point orbits, where the Earth is in the familiar role
as the smaller primary. And finally, a simple scheme is introduced to yield launch
trajectories from the launch site to Earth parking orbits. The goals are accomplished
by combining invariant manifold theory and differential corrections techniques. In
the Earth-Moon system, an initial guess featuring some of the characteristics of the
desired trajectory is still required since no natural solution exists that connects the
near-Earth region with lunar halo orbits.
The dynamical structure in the multi-body problem in the vicinity of the larger
primary is very different than that around the smaller primary. Insight into the natural dynamics is gained by computing the stable/unstable manifolds and identifying
those with the closest Earth pass for variously sized halo orbits in both the EarthMoon and the Sun-Earth systems. A range of sample transfers from the Earth to
specific libration point orbits at L1 is then presented. Some of the transfers generated in the CR3BP closely resemble transfers from the literature in appearance,
cost, and time of flight. The transfer trajectory insertion cost from an Earth parking orbit is very similar for all the trajectories investigated. The transfers are then
reproduced in an ephemeris model to demonstrate their validity. In the immediate
vicinity of the halo orbits, it is noted that some of the transfers follow the invariant
106
manifold structure quite closely. Consistent with this observation, lower halo orbit
insertion costs are achieved for larger halo orbits. The methodology developed for
determination of transfers in the Earth-Moon system is very successfully used in the
computation of transfers in the Sun-Earth system. In the Sun-Earth system, it is
possible to use the manifold structure to aid substantially in the design of transfers.
Sample transfers to both L1 and L2 halo orbits can be computed without difficulty.
The transfer trajectory insertion maneuver to depart an Earth parking orbit is nearly
constant. Finally, a simple scheme is applied to determine thrust arcs and to offer a
simple, but complete scenario from a launch site on the surface of the Earth to a halo
orbit.
6.0.2
Recommendations and Future Work
Many challenges remain in the efficient and effective determination of transfer
trajectories between the Earth and halo orbits in the Earth-Moon system. This is
notable to ultimately develop an automated procedure. The work presented here is
preliminary and serves as the basis for a more comprehensive investigation. Future
work is likely a combination of developing more efficient techniques, as well as computing a larger range of transfers to gain more insight into the characteristics of the
different types. The key to a more efficient computation of transfers, lies in a more
complete understanding of the dynamics in the vicinity of the larger primary. Variational methods based on linear approximations do not seem adequate in accurately
predicting the motion close to the primaries. Other techniques such as nonlinear
programming techniques might be more effective in determining transfers in these
sensitive regions in the vicinity of the larger primary. Partial enforcement of constraints or the application of constraints along a path rather than individual points
could also offer improvement. Systematically incorporating the invariant manifold
structure near the halo orbit would certainly be used. A methodology developed by
Wilson [42] to project the end points of transfers onto the manifold tube could be one
107
way to achieve this. Alternatively, a multi-point targeting scheme or a differential
corrections scheme able to vary the trajectory selected on the manifold tube could
be helpful. To avoid using the current inefficient shooting techniques in that process,
incorporating the energy-like Jacobi Constant might be helpful. Computing a larger
range of transfers will offer insight into typical characteristics such as cost and TOF
of the different types. Being aware of these characteristics will undoubtedly be helpful
in identifying the optimal locations to perform maneuvers. Multiple maneuvers could
yield transfers for lower total cost. Eventually, optimization of the transfer will be
necessary. An optimization package in development at Purdue University specifically
suited to the peculiarities of the three-body problem will be used for that purpose.
The design of transfers to/from the vicinity of the smaller primary to halo orbits is
well understood. Use of this knowledge can be exploited in support of transfers from
the larger primary, Earth, to lunar libration point orbits at L1 and L2 that include
a lunar flyby. Transfers that pass the Moon on the far side could insert onto the
stable manifold tube, and using the heteroclinic connections, offer lower cost transfer
to lunar L1 halo orbits. Transfers to L2 halo orbits that include a lunar flyby could
also be investigated.
The launch trajectory development is of a preliminary nature. Differential corrections techniques can be used to obtain rough approximations of baseline solutions.
Though, more efficient and precise tools are desirable to extend the capabilities and
the range of applications. Replacing the constant thrust term by a scheme using
different launch vehicle stages with their varying masses and propellant limitations
into account offers additional precision. Besides improving the model, a wider range
of application could be investigated such as launches directly into stable manifolds
or transfer trajectories. Eliminating the parking orbit could save valuable fuel. A
scheme to compute such launches could use Wilson’s optimization technique based
on projecting the launch trajectory onto the surface of the manifold tube. This would
yield continuous transfers from the launch site to the libration point orbit of interest
for a lower cost. Optimal control should eventually be investigated.
108
6.0.3
Concluding Remarks
Libration point trajectories and the associated low-energy pathways offer many
low-cost, low-energy trajectory options for mission designers. However, no such pathways naturally occur between the larger primary and the halo orbits. One technique,
based on linear variational methodology, to compute such transfers is investigated in
this work. The computation of launch trajectories using the same methodology is also
considered. Although, the computation of transfers and launch trajectories using this
technique is possible, more efficient methods are desirable. The immediate vicinity
of either the smaller or the larger primary is highly nonlinear. Improvements in the
effectiveness and efficiency of new design tools for these transfers are likely achieved
through a more systematic use of the dynamics in these regions.
109
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