“Near Rectilinear Halo Orbit in the vicinity of L1 Point in the Earth-Moon System”

CRANFIELD UNIVERSITY
GEORGIOS GALANOS
Near Rectilinear Halo Orbit in the vicinity of L1 Point in the Earth-Moon System
SCHOOL OF AEROSPACE, TRANSPORT AND
MANUFACTURING
MSc in Astronautics and Space Engineering
MSc
Academic Year: 2018 - 2019
Supervisor: Marta Ceccaroni
August 2019

CRANFIELD UNIVERSITY
SCHOOL OF AEROSPACE, TRANSPORT AND
MANUFACTURING
MSc in Astronautics and Space Engineering
MSc
Academic Year 2018 - 2019
GEORGIOS GALANOS
Near Rectilinear Halo Orbit in the vicinity of L1 Point in the Earth-Moon System
Supervisor: Marta Ceccaroni
August 2019
This report is submitted in partial (30%) fulfilment of the requirements for the
degree of MSc In Astronautics And Space Engineering
© Cranfield University 2019. All rights reserved. No part of this publication may
be reproduced without the written permission of the copyright owner.

Georgios Galanos
iv
ABSTRACT
In the last few years, space agencies had their focus on the Moon for many
scientific reasons. Pointing to the L1 and L2 Lagrange points, the equilibrium of
the Earth-Moon system in the vicinity of the Moon, first however can provide
numerous of advantages. Indeed the dynamical properties of these points can be
used as possible gateways for missions to the Moon, but could also be exploited
as a trampoline for interplanetary mission. Many orbits around the L1 and L2 points
have been studied and Near Rectilinear Halo Orbits seems to be the ideal for this
purpose, for their marginal stability properties. The Lunar Orbital Platform-
Gateway is planned to be placed in such an orbit.
The main goal of this thesis is to obtain the manifolds associated with the Near
Rectilinear Halo orbits, in order to qualitatively study their stability properties. To
this aim initial conditions for generating approximated Halo orbits in the vicinity of
the L1 point around the Moon are derived via an analytical method. Different
techniques are thus used to correct these conditions and evaluate a periodic Halo
orbit to finally generate its whole family using a so called continuation method.
Finally, this thesis studies the stability of the NRHO and computes its manifolds.
Keywords:
Halo, NRHO, Manifolds, Family orbit generation, Approximation method.

Georgios Galanos
v
ACKNOWLEDGEMENTS
I would like to thank my advisor, Marta Ceccaroni for her patience and help during
this long trip. She was always there to pass me all her knowledge in the best
possible way.
I would not have been here if it wasn’t for my family, and so I am ever grateful to
them. Their sacrifices through all this year of my education create a better future
for me.
I would like to thank my beloved Marianna who was always by my side and was
giving me strength to achieve all my goals no matter the distance between us.
Last but not least, the family we choose in our life is the one, which hold our hand
when it is needed and it was always there to grab it. Thank you for being in my
life.
“Μια αστραπή η ζωή μας... μα προλαβαίνουμε”

Georgios Galanos
vi
TABLE OF CONTENTS
ABSTRACT............................................................................................................ iv
ACKNOWLEDGEMENTS ......................................................................................v
LIST OF FIGURES...............................................................................................viii
LIST OF TABLES.................................................................................................. ix
LIST OF ABBREVIATIONS ...................................................................................x
1 Introduction........................................................................................................11
1.1 Thesis Structure .........................................................................................13
2 Background .......................................................................................................14
2.1 The n-Body Problem ..................................................................................14
2.2 Adimensionalisation of the units of measure.............................................17
2.3 Equations of motion....................................................................................19
2.4 Integrals of motion......................................................................................20
2.5 Lagrange Points – Equilibrium Solutions...................................................22
2.6 Forbidden Regions and Zero Velocity Surfaces........................................25
3 Analytical Approximation of Halo Orbit.............................................................30
3.1 Reference Frames......................................................................................30
3.1.1 Inertial Barycentre System of Reference............................................30
3.1.2 Barycentric, non-inertial, rotating system of reference.......................33
3.1.3 Non Inertial, Rotating System of Reference, centred in L1................37
3.2 Rearranging the Hamiltonian .....................................................................39
3.2.1 Legendre Polynomial...........................................................................40
3.2.2 The truncated Hamiltonian ..................................................................42
3.2.3 The Linearized system ........................................................................43
3.3 Reduction to the central manifold ..............................................................44
3.3.1 Complexification ..................................................................................45
3.3.2 Eliminating the hyperbolic component................................................46
3.3.3 Eliminating the hyperbolic component................................................47
3.4 Action-Angle Variables...............................................................................50
3.5 Resonant Perturbation Theory...................................................................51
3.6 Finding Halo Orbits.....................................................................................57
3.7 Back to the Real World...............................................................................58
4 Periodic Orbits via Numerical Methods ............................................................61
4.1 State Transition Matrix (STM) ....................................................................62
4.2 Differential Corrections...............................................................................65
4.2.1 Single shooting method.......................................................................68
4.3 HALO and Near Rectilinear Halo Orbit Algorithm .....................................70
4.4 Single-Parameter Continuation Method ....................................................74
4.5 Pseudo-arc length Continuation method ...................................................74

Georgios Galanos
vii
5 Stability of Periodic Orbits.................................................................................75
5.1 Invariant Manifolds .....................................................................................77
6 Results and Conclusion ....................................................................................80
6.1 Near Rectilinear Halo Orbit Manifolds .......................................................81
6.2 Recommendations for Further Work..........................................................84
7 REFERENCES..................................................................................................86
8 BIBLIOGRAPHY................................................................................................89
9 APPENDICES ...................................................................................................90
Appendix A Diagonalising the Linearized System...........................................90
Appendix B Analytical process to define the Hamiltonian formula .................97

Georgios Galanos
viii
LIST OF FIGURES
Figure 1.1 Southern and Northern Halo and NRHO families. ............................12
Figure 2.1 Definitions of the n-Body Problem .....................................................15
Figure 2.2 Position of the Lagrange points. ........................................................23
Figure 2.3 Jacobi constant for C=C1....................................................................26
Figure 2.4 Jacobi constant for CL2 < C <CL1. ......................................................27
Figure 2.8 Jacobi constant for CL4,5 < C <CL3. ....................................................29
Figure 2.9 Jacobi constant for CL4,5 > C .............................................................29
Figure 3.1 Inertial (i) and rotating (r) reference frames in the CR3BP. ..............34
Figure 3.2 A Family of approximated Halo orbits around L1...............................60
Figure 4.1 Halo orbit seen from Earth. ................................................................70
Figure 4.2 Quasi Periodic Orbit. ..........................................................................71
Figure 4.3 Halo Periodic Orbit. ............................................................................73
Figure 5.1 Invariant manifolds associated to Halo orbit......................................79
Figure 5.2 Invariant manifolds – tube shape.......................................................79
Figure 6.1 3D representation of the Halo family in the vicinity of the L1 point. ..80
Figure 6.2 Halo and NRHO families (x-y axis). ...................................................81
Figure 6.3 Stable manifolds of the NRHO reference orbit towards the Moon....83
Figure 6.4 Stable manifolds of the NRHO reference orbit towards the Earth....84

Georgios Galanos
ix
LIST OF TABLES
Table 2.1 Parameters of the Earth-Moon system. ..............................................19
Table 2.2 Lagrange points in the Earth-Moon system........................................25
Table 2.3 Jacobi constant of the Langrange points. ...........................................26

Georgios Galanos
x
LIST OF ABBREVIATIONS
CU
CR3BP
3BP
Cranfield University
Circular Restricted Three Body Problem
Three Body Problem
ISSE-3
MAP
NASA
DART
ARM
NRHO
BKE
STM
Sun-Earth Explorer
Microwave Anisotropy Probe
National Aeronautics and Space Administration
Double Asteroid Redirected Test
Asteroid Redirect Mission
Near Rectilinear Halo Orbit
Basic Kinematic Equation
State Transition Matrix

Introduction Georgios Galanos
11
1 Introduction
From the very beginning of the existence of human being on planet Earth, the
Moon was always a source of fascination. Back then, our cosmic partner could
only be seen with the naked eyes, but as technology evolved, telescopes came
to our life and the Moon was closer than ever. In the 20th century the biggest
achievement became a reality as humans visited the Moon for the first time.
Prior to this, a decade before, some lunar exploration vehicles gave us the first
scientific insight of the Moon. Back in January 1969, the Soviets launched Luna
1 with the aim to land on the Moon [15]. Despite never landing on the Moon, it
was provided with a scientific equipment which for the first time revealed that the
Moon had no magnetic field. A number of missions followed to explore the Moon
in the next years.
To answer the need to improve communications with vehicles exploring the Moon
some other missions, focusing on orbits in the vicinity of the Lagrange points
started to be examined. These types of missions were focused on orbits in the
vicinity of the Lagrange points, the so called Halo orbits. Robert W. Farquhar was
the first to introduce these types of orbits back in 1968 in his PH.D. thesis [28].
Sun-Earth Explorer (ISEE-3) was the first spacecraft that made use of the
Lagrange points [24]. Several missions followed, such as WIND [16], GENESIS
[18] and MAP [17]. Orbits around these points possess properties, which make
them suitable for communication, scientific observation and many other scopes.
Moreover, the dynamic properties of these points can be used as possible
gateways for missions to the Moon. For example NASA is trying to take
advantage of these orbits to explore asteroids, thus planning to place a Lunar
Orbital Platform-Gateway in a periodic orbit around them.
Among all the periodic orbits around the equilibrium, the so called Near
Rectilinear Halo Orbits (NRHO) seems to be ideal especially due to their marginal
stability properties [29].
Moreover, the Asteroid Redirect Mission (ARM) designed by NASA aims to

12
capture a sample of an asteroid and place it in a Near Rectilinear Halo Orbit
(NRHO) in the Earth-Moon system. This project focuses on the study of the
NRHO in the Earth-Moon system. NRHO are periodic orbits around the
equilibrium points which arise from the continuation of the above cited Halo orbits,
as they get closer to the Moon.
Figure 1.1 Southern and Northern Halo and NRHO families. Blue and green orbits
represent the Halo orbits of the Northern and Southern respectively. Red and yellow
orbits represent the NRHO of the Southern and Northern respectively.
These two Halo families are symmetric with respect to the plane of motion. At the
borders of the L1 and L2 families of Halo orbits, the so called Near Rectilinear Halo
Orbits (NRHO) can be found. The dynamic behaviour of this type of orbits persists
in a higher-fidelity model and, hence, its advantage is the possibility of a long-
term manned mission around the Moon [19]. The fact that NRHO orbits are
characterised by the close approach over the polar regions of the Moon, a
spacecraft located in such an orbit provides access to the Polar Regions during
more than 90% of the total time [11]. On the other hand, Halo families can cover
approximately only the 50% of the total time. Moreover, NRHO and Halo orbits
can achieve a continuous coverage with Earth due to their motion around the
Equilibrium points. One of the most important advantages of the NRHO is
regarding the maintenance cost. NRHO families are characterised as marginally
stable orbits and in comparison with the Halo orbits, the cost of station-keeping
is much lower.

13
The main goal of this thesis is to obtain the manifolds associated with the Near
Rectilinear Halo Orbits, in order to qualitatively study their stability properties. To
this aim initial conditions for generating approximated Halo orbits in the vicinity of
the L1 point around the Moon are derived via an analytical method. Different
techniques are thus used to correct these conditions and evaluate a periodic Halo
orbit to finally generate its whole family using a so called continuation method.
Finally, this thesis studies the stability of the NRHO and computes its manifolds.
1.1 Thesis Structure
This thesis aims at determining and simulating NRHO in the vicinity of the L1 point
in the Earth-Moon system. At first stage, the initial Halo orbit is computed via an
approximation method and then the whole family is evaluated via different
techniques, which are discussed later on. The NRHO family of orbits that
generated is analysed and its manifolds are computed and presented.
The work of the thesis is organised as follows:
CHAPTER 2
Chapter 2 summarises the necessary background needed to design orbits
trajectories in the vicinity of the Equilibrium points. The general n-body problem
is discussed prior to the Circular Restricted Three Body Problem. The
adimensionalisation of the units of measure is derived as it is a very useful aid to
the numerical convergence of the problem. Different reference frames are
analysed and discussed, in order to understand the process of the approximation
method in Chapter 3. Finally, the equations and the integrals of motions are also
included in this Chapter, as is the computation of the equilibrium points in the
Earth-Moon system and their energy.
CHAPTER 3
In this Chapter the approximation method to finally provide the initial guess of a
approximated Halo periodic orbit is discussed. Information regarding different
reference frames discussed in Chapter 2 are used to change the reference

Background Georgios Galanos
14
frames from one to another until the orbit is found. Reduction of the central
manifold is included as well. Finally, the resonant perturbation theorem and the
process of finding the initial conditions for a Halo orbit is been discussed.
CHAPTER 4
This Chapter discusses the Numerical Methods that are employed to generate
Halo and NRHO families in the vicinity of the L1 point in the Earth-Moon system.
Different techniques are analysed first, to generate a perfect Halo periodic orbit
from the initial conditions guess provided in Chapter 3, and then the process to
generate the whole family is been discussed.
CHAPTER 5
The stability of the periodic orbits is analysed in this Chapter. Finally, techniques
to generate the manifolds for the Halo and NRHO periodic orbits are been
discussed.
CHAPTER 6
This Chapter provides the results of the thesis. The Halo Family and NRHO orbits
are been discussed and so the manifolds of the NRHO. Finally, some
recommendations for further work are been provided.
2 Background
2.1 The n-Body Problem
A key point of the success of any space mission, is the accuracy of its trajectory
design phase. In order to accomplish it, a deep insight in the dynamical
environment in which the spacecraft will move is necessary.
The general n-Body Problem is the problem of determining the motion of n
particles Pn where, n=1, 2, i, j and q (of mass mn respectively), moving under their
sole mutual gravitational attraction.
Setting an inertial reference frame X̂, Ŷ and Ẑ (foot note: where the X̂ and Ŷ axis
indicates some specific orthogonal directions of some specific reference plane to

15
be defined, and Ẑ completes the orthogonal triplet following the right-hand rule),
centred at the barycentre O of the n points of the system, by Newton’s Second
Law, the equations of motion of each of the particles Pn can be expressed as [25],
mi
d2
ri
dt2
= −G ∑
mimj
rji
3
n
j=1
j≠i
rji
2.1
Where G is the the universal gravitational constant and mi and mj are the masses
of the particles Pi and Pj respectively. Moreover, the distance between the origin
and the particle Pi is represeted as |ri|, whereas the distance between the
particles Pi and Pj is |rji| = |ri − ri|. The bar above the letters is used to indicate
vector as shown in Figure 2.1.
O
Figure 2.1 Definitions of the n-Body Problem [20]
Equation 2.1 is a vectorial equation of second order, as
d2
dt2 indicates a double
derivation with respect to time. As such it can be decomposed in two first order
equations for each of the three orthogonal directions of the reference frame. The
motion of each of the particles Pi is thus described by six first-order, scalar
equations, which are coupled and nonlinear. The system is therefore said to be
a 6n degrees of freedom. To solve such a system, 6n integrals of motion (i.e. any
function of the coordinates which in the phase space would be constant along a
trajectory) are needed [5]. The position of the barycentre of the n masses, for
example, is an integral of motion, as in the inertial frame is fixed at O. Moreover,

16
the Conservation Law applies for linear momentum, energy of n-body system and
angular momentum summing up to ten known integral of motion [27].
The fact that more integrals are required to solve n-body problem, implies that a
different approach is required to simplify the motion of the particles. In particular
we will considered the motion of n-1 particles as known, and not influenced by
the motion of so called particle of interest Pi. Thus, the system is reduced to a
three second order equations of motion. The Circular Restricted Three Body
Problem.
In the previous section, the general n-body problem was discussed and the
complexity of the n bodies system was noticed, stressing the necessity for some
simplifications. The Three Body Problem, hereafter 3BP, is a simplified model of
the n-body problem, with n=3. It is the problem of determining the dynamics of an
interest particle P, assuming that the positions of the two primary bodies are
known, the three body problem in 2.1 can be reduced to the form,
ms
d2
rs
dt2
= −G
m3mE
rEs
3 rEs − G
msmM
rMs
3 rMs 2.2
In Equation 2.2, mE and mM are the masses of the Earth and the Moon
respectively, as this thesis is focussing in these two planets, and ms is the mass
of the spacecraft. The distance between the origin O and the spacecraft is
represented as rs, while rEs and rMs are the distances from the spacecraft to the
Earth and the Moon respectively. Despite that the equation of motions have been
massively-reduced, the 3BP still cannot be analytically solved without some
further approximations, as it was proved caothic by Poincare [22].
To achieve a further reduction on the complexity of the problem, some critical
assumptions must be made and applied to the general three body problem, in
order to formulate the Circular Redistricted Three Body Problem. The
assumptions are listed below.
 The first assumption is that the particle Ps (the body of interest) is of
negligible size and mass in comparison to the primaries bodies PE and PM.
It is obvious that this assumption becomes reasonable when the body of

17
interest is modelled as an asteroid or spacecraft and interacts with the two
primary bodies being planets, or the Moon, or the Sun. It is important to
mention that the body of interest does not affect the motion of the primaries
bodies as it does not act any gravitational force to them. To conclude the
first assumption we need to keep in mind the relationship ms<<mE, mM.
 According to the first assumption Ps does not affect the motion of the two
primary bodies and thus PE and PM constitute an isolated two body system
with existing solutions being conics such as an ellipse or a circle. This
system of PE and PM is called the Primary system, in which PE is labelled
as the largest primary body and PM the smallest. The Secondary system
is the body of interest Ps. Joining the two primary bodies with a line, the
barycentre of the system can be found and so the centre of mass.
 Finally, the particle Ps can move in all three dimensions unlike the primary
system, which moves in circular motion around its centre of mass and
further restricts their motion to be planar instead of conic.
2.2 Adimensionalisation of the units of measure
To further simplify the study of the CR3BP, the quantities appearing in the
problem can be defined in adimensional units. This method sets the equations of
motion into a form that is easier to manipulate, as it normalises the difference in
the order of magnitudes for masses, velocities and distances, and makes them
applicable for any system (such as Earth – Moon system). In further analysis of
the circular restricted three body problem, the adimensionilisation of the units will
provide great assistance to numerical integration. In addition, shifting the system
into the rotating reference frame can help to simulate and determine periodic
orbits and equilibrium solutions in the dynamical system.
 All distances will be scaled by l∗
, the characteristic length, defined as the
distance between the two primary bodies.
l∗
= ||rE| − |rM|| 2.3

18
 All masses will be scaled by m* , the characteristic mass, defined as the
sum of the masses of the two primary bodies.
m∗
= mE + mM 2.4
With this definition the adimensional masses of the two primary bodies are
1 − μ = μE =
mE
mE + mM
μ = μΜ =
mΜ
mE + mM
2.5
 All times will be scaled by t∗
/2π, where t∗
is the orbital period of the primary
bodies around their centre of mass, defined using Kepler’s Third Law as,
t∗
= 2π (
l∗3
Gm∗)
1
2
=
2π
ω∗
2.6
G̃ =
Gl∗3
m∗t∗2 = 1 2.7
N = (
G̃m∗
l∗3 )
1
2 2.8
It must be noted that, in the adimensionilised units the gravitational constant
G̃ =
Gm∗t∗2
l∗3 = 1
2.9
While in the two last definition G is the universal constant and N is the conic
definition of the dimensional mean motion of the two primary bodies, the
adimensionalised value of N and time, which is independent in the differential
equations, can be defined as
𝑛∗
= 𝑁𝑡∗
= 1
𝜏 =
𝑡
𝑡∗
2.10
The summary of the parameters and characteristics of the Earth-Moon system
can be visualised in the next table.

19
Parameter Symbol Value Units
Mass parameter μ 0.012150535156801 -
Angular speed n 1 -
Characteristic distance l∗
384,400 km
Characteristic time t∗
23.58178 · 105
s
Characteristic mass m∗
6.0471754795301· 1024
kg
Table 2.1 Parameters of the Earth-Moon system.
2.3 Equations of motion
In this analysis, the gravitational forces generated by the Earth and the Moon are
the only ones that are considered. Under these circumstances, the Equation 2.2
can be written in the form,
d2rs
dτ2
= −
1−μ
|r⃗ Es|3
rEs −
μ
|r⃗ Ms|3
rMs
2.11
Obtain the equations of motion for the spacecraft requires the use of the Basic
Kinematic Equation (BKE) twice (expressed in the rotating frame coordinates) to
lead us to the acceleration of the spacecraft. First the BKE is used to express the
velocity of the spacecraft with respect to the inertial frame,
rs
′
=Ι
drs
dτ
=R
drs
dτ
+I
ωME
R
× rs
2.12
And then it is used to express the acceleration with respect to the inertial frame
of the spacecraft as,
rs
′′
=Ι
d2
rs
dτ2
=R
d2
rs
dτ2
+2I
ωME
R
× rs+I
ωME
R
× (.I
ωME
R
+ rs)
2.13
With, I
ωME
R
= θẑ being the angular velocity of the rotating frame with respect to
the inertial frame. From Equation 2.13 it can be observed that the acceleration
of the spacecraft is a function of the position vector of it. Knowing the relationship
between the acceleration and the position vector, the value rs is then substituted

20
in Equation 2.13 and provides the acceleration of the body of interest expressed
in the rotational coordinates,
rs
′′
= (ẍ − 2θẏ − θ2
x)x̂r + (ÿ + 2θẋ − θ2
y)ŷr + z̈ẑr
2.14
The acceleration of the spacecraft in terms of the rotating frame and the inertia
reference frame, can be expressed in two different equations, Equation 2.11 and
Equation 2.13. To reduce the resulting equation of motion to a known scalar, such
as a second order differential equation, the kinematic expression of the
acceleration expressed by Equation 2.13, must be substitute into the left hand
side of the Equation 2.11, resulting in the second order differential equation in the
CR3BP.
ẍ − 2θẏ − θ2
x = −
(1−μ)(x+μ)
rEs
3 −
μ
rMs
3
(x − 1 + μ) 2.15
ÿ − 2θẋ − θ2
y = −
(1 − μ)
rEs
3
y −
μ
rMs
3
y
2.16
z̈ = −
(1 − μ)
rEs
3
−
μ
rMs
3
z
2.17
In Equations 2.15 to 2.17, the magnitudes of the distances rMs and rEs can be
computed as,
rMs = √(x + μ)2 + y2 + z2 2.18
rEs = √(x − 1 + μ)2 + y2 + z2 2.19
2.4 Integrals of motion
A pseudo – potential function can be defined so to further simplify the differential
equations of motion in Equations 2.15 -2.17 [1]. This pseudo – potential function,
which labelled as U*, is defined as,
U∗
=
1−μ
d
+
μ
r
+
1
2
θ2(x2
+ y2) 2.20

21
The differential equations of motion defined in the previous section can be
rewritten in terms of the pseudo – potential function as,
ẍ − 2θẏ =
∂U∗
∂x
2.21
ÿ − 2θẋ =
∂U∗
∂y
2.22
z̈ =
∂U∗
∂z
2.23
In the equation system 2.21 - 2.23, a constant energy or energy-like quantity is
possible to be found in the rotating frame. It can be examined if this constant truly
exists by summing up the dot product of the rotating velocity vector Equation 2.17
and Equations (2.21-2.23).
ẋẍ + ẏÿ + żz̈ =
∂U∗
∂x
ẋ +
∂U∗
∂y
ẏ +
∂U∗
∂z
ż
2.24
The right hand side of the Equation 2.24, including the factor U*, which is only
function of position and so equals to
∂U∗
∂τ
, can be rewritten as,
ẋẍ + ẏÿ + żz̈ =
∂U∗
∂τ
2.25
The next step to make the constant appear is to integrate Equation 2.24 with
respect to the adimensional time τ and hence define the following expression as,
1
2
(ẋ2
+ ẏ2
+ ż2) = U∗
−
C
2
2.26
The integration constant C that appears in Equation 2.26 is called the Jacobi
constant or the Jacobian Integral of Motion. Rearranging Equation 2.26, the
Jacobi constant takes the form,
C = 2U∗
− V2
, 2.27
where V is the velocity in the rotating frame.

22
Since the equations of motion require six integrals of motion for the CR3BP to be
solved, the Jacobi constant is still not enough for a close-form solution and so a
numerical analysis cannot be prevented.
2.5 Lagrange Points – Equilibrium Solutions
In orbital mechanics, Lagrange points as they are commonly known, are
equilibrium points around a two body system (such as Earth – Moon) [14]. These
points are such that, if a massless object is located at one of this points, with zero
acceleration and velocity it would remain fixed in the rotating frame infinitely. At
the Lagrange points the sum of gravitational forces of the two large bodies, i.e.
the centripetal force of the orbital motion and the Coriolis acceleration, would
cause the small object to be stable or near stable with respect to the primary large
body.
Considering that the acceleration and the velocity at this points are zero for the
smaller body, in the CR3BP the derivatives of the rotating position and velocity
state components must be zero all the time. The equilibrium points can be
computed by substitute these conditions into the equations of motion by setting,
∂U∗
∂x
=
∂U∗
∂y
=
∂U∗
∂z
= 0
2.28
In the CR3BP five equilibrium points exist, notated as L1 – L5. All points are laying
in the orbital plane of the two primary bodies (Earth – Moon system), i.e. the out
of plane component z of all the points must be equal to zero. The location of these
points are displayed in Figure 2.2.

23
Figure 2.2 Position of the Lagrange points.
L1 – L3 are forming a line passing form the center of the two primary bodies. On
the other hand, L4 and L5 are forming an equilateral triangle with the larger body.
L1 point is always located between the two primary bodies on the x-axis while, L2
is always beyond the smaller of the two primary bodies on the positive x-axis as
well. Unlike with the L2 point, L3 is lying beyond the larger primary body, in the
negative side of the x-axis. L4 and L5 which called equilateral points, have a
positive y component and a negative y component respectively. The locations of
the Lagrange points can be evaluated by solving,
∂U∗
∂x
=
∂U∗
∂y
=
∂U∗
∂z
= 0 ,
xe −
(1−μ)(xe+μ)
|rEs|3
−
μ(xe−1+μ)
|rMs|3
= 0 2.29
ye (1 −
1−μ
|rEs|3
−
μ
|rMs|3
) = 0 2.30

24
−ze (
1−μ
|rEs|3
−
μ
|rMs|3
) = 0 2.31
From Equations 2.31 it can be noticed that z=0 and this leads that all five
equilibrium points are lying into the plane of motion of the primary bodies.
Moreover, to compute the location of the three collinear points, ye and ze are set
to be zero. Studying the geometry it can be shown that:
For L1: (xe − (1 − μ)) < 0 and (xe + μ) > 0.
For L2:(xe − (1 − μ)) > 0 and (xe + μ) > 0.
For L3: (xe − (1 − μ)) < 0 and (xe + μ) < 0.
Therefore, for L1, L2 and L3 yield,
xe +
μ
(xe(1−μ))
2 −
1−μ
(xe+μ)2 , ye = 0, ze = 0
xe −
μ
(xe(1−μ))
2 −
1−μ
(xe+μ)2
, ye = 0, ze = 0
xe +
μ
(xe(1−μ))
2 +
1−μ
(xe+μ)2
, ye = 0, ze = 0
In case that in Equation 2.30 y ≠ 0, the terms inside the brackets must be equal
to zero and yield |rEs|3
= |rMs|3
= 1. In this case, the equilibrium points L4 and L5
can be found as,
xL4 =
1
2
− μ xL5 =
1
2
− μ
yL4 =
√3
2
yL5 = −
√3
2
Substitute numerically the position of the two primary bodies (Earth and Moon)
by using Table 2.1, the equilibrium points can to be evaluated.

25
Equilibrium
Points
x-dimensionless y-dimensionless z-dimensionless
L1 0.836915 0 0
L2 1.155681 0 0
L3 -1.005062 0 0
L4 0.487849 0.866025 0
L5 0.487849 -0.866025 0
Table 2.2 Lagrange points in the Earth-Moon system.
2.6 Forbidden Regions and Zero Velocity Surfaces
Forbidden regions are the regions in which the motion of the secondary body (i.e.
the spacecraft) cannot be possible [2]. These regions can be computed using
Equation 2.27 and in the same procedure, the possible regions of motion of the
secondary body can be found. Examining Equation 2.27, it can be observed that
if C>2U, then V2 would be smaller than zero and the motion of the body cannot
be existed. Given the value of the Jacobi constant C, in each point of the plane
that V2 is found to be zero, the kinetic energy is negative and the secondary body
cannot cross these regions. In the regions that the body cannot cross and the
relative velocity V2 is equal to zero, Equation 2.26 can be derived by inserting the
pseudopotential function U2.
x2
+ y2
+
2
rEs
(1 − μ) +
2
rMs
μ = C 2.32
An infinity number of different combinations of the rotating x, y and z coordinates
can represent a surface in 3D space which satisfy the Equation 2.32. The Jacobi
constant can always be expressed as a positive term, as the left hand side of the
Equation 2.32 is always positive since x2
and y2
are always positive and for 0 <
μ < 1 the terms
2
rEs
(1 − μ) and
2
rMs
μ are greater than zero, because rEs and rMs
are distances and so positive numbers. In any other case that the relative velocity

26
is positive, the motion of the secondary body is physically possible. The values
of the Jacobi constant at each of the equilibrium points for the system Earth –
Moon are displayed in Table 2.3 Jacobi constant of the Lagrange points., where
CL1, CL2, CL3, CL4 and CL5 are the Jacobi constants for each equilibrium point L1,
L2, L3, L4 and L5 respectively.
CL1 CL2 CL3 CL4 CL5
3.188340 3.172160 3.012147 2.987997 2.987997
Table 2.3 Jacobi constant of the Lagrange points.
These zero Velocity surfaces zones was first introduced by Hill and were known
as Hill’s regions [2]. The Lagrange points are matched with the Zero Velocity
surface at Jacobi constant values equal to CL1, CL2, CL3, CL4 and CL5. As Jacobi
constant decreases, the points tend to lie outside the surface, and while the
Jacobi constant increases they tend to lie inside the surface.
 For C=C1, the particle possesses the level of energy at its minimum value
and cannot move between the two primary bodies.
Figure 2.3 Jacobi constant for C=C1.
 If the Jacob constant would further decreases such that CL2 < C <CL1, the
gate between the two primary bodies would open and the particle will be

27
able to move as seen in Figure 2.4 and Figure 2.5. It is possible for the
trajectories, which starts close to one of the two primaries to move towards
to the other primary body through the path that links the two bodies.
Figure 2.4 Jacobi constant for CL2 < C <CL1.
 CL3 < C <CL2 : When the Jacobi constant is further decreased a new
gateway opens from the right side of the second primary body in the x-axis
where L2 point is located, and the spacecraft is able to move in the region
beyond the outer most surface as Figure 2.6 and Figure 2.7 display.

28
 CL4,5 < C <CL3 : At this point, the Zero Velocity Curve separated entirely
from the x-axis, and only the L4 and L5 points are enclosing. In addition, a
new gateway is now open form the side of the large primary body at the L3
Lagrange point as shown in Figure 2.8.

29
Figure 2.8 Jacobi constant for CL4,5 < C <CL3.
 Finally, for values of Jacobi constant smaller than CL4,5, the Zero Velocity
Curve does not exists and the x-y plane is open for the spacecraft to move
freely as Figure 2.9 shows.
Figure 2.9 Jacobi constant for CL4,5 > C .

Analytical Approximation of Halo Orbit Georgios Galanos
30
3 Analytical Approximation of Halo Orbit
In this chapter an analytical method for deriving initial conditions for approximated
Halo orbits will be illustrated. As the method is analytical, several reference
frames and changes of coordinates will be introduced and used to finally explicitly
build the approximated Halo.
3.1 Reference Frames
In this chapter, different coordinate frames are presented and analysed, as they
are used throughout this thesis in different forms. Transformation matrices to shift
from one frame to another are discussed as well. Most of the time, different
variables visualised in different frames, such as position and velocity, can be
understood much more easily. A simple example of this simplification is a periodic
orbit in a rotating frame, which will result in a stationary (i.e. fixed) equilibrium
point. The orbital plane of the Earth-Moon system is set to be the
𝐪̂ 𝐱
𝐪̂ 𝐲
plane. Later
on in this investigation, many change in coordinates are being applied, therefore,
the next sections provide useful information. In any system of reference the
following definitions are been defined:
 Position of the primaries
 Equations of motion
 Potential energy
 Kinematic Energy
 Lagrangian Energy
 Momentum of the system
 Hamiltonian form
3.1.1 Inertial Barycentre System of Reference
The inertial reference frame is very important as in it Newton’s laws holds, origin,
is set at the barycentre of masses of the system (in this case is the Earth-Moon
barycentre, as the spacecraft is considered massless).The origin of the inertial
frame is assumed to be fixed in space. The axis of the frame xi , yi lie on the plane

31
of motion of the two primaries. At the initial instant t=0 the xi-axis is assumed to
lie on the line connecting the Earth and the Moon, pointing to the Moon. The zi –
axis is fixed in the direction of the angular momentum, positive as the planets
revolve anticlockwise around it. The yi completes the right – handed Cartesian
frame.
Position of the primaries
Based on the inertial barycentre system of reference 𝐪 = {qx, qy, qz}, the positions
of the two primary bodies can be defined as,
𝐪 𝐄 = [μ cos(t + π) , μ sin(t + π) , 0] 3.1
𝐪 𝐌 = [(1 − μ) cos(t) , (1 − μ) sin(t) , 0] 3.2
Equations of motion
As we are in the inertial barycenter frame of reference, the equations of motion
come directly from Newton's Laws and the principles of dynamics (where, F = ma,
F = G
m1m2
r3
r being a central force, and can be expressed as F = ∇U) and
therefore:
q̈ x = −
∂U(q)
∂qx
q̈ y = −
∂U(q)
∂qy
q̈ z = −
∂U(q)
∂qz
3.3
Potential Energy
The potential energy comes straight forward form Gravity F and formed as,
U(𝐪) = −
1−μ
√(q−qE).(q−qE)
−
μ
√(q−qM).(q−qM)
=
−
1−μ
√(qx−μ cos(t+π))2+(qy−μsin(t+π))
2
+qz
2
−
μ
√(qx−(1−μ) cost)2+(qy−(1−μ)sin t)
2
+qz
2
3.4

32
Lagrangian
The Lagrangian of the system is defined as L=T-U, where T is the kinetic energy
of the system and U is the potential energy as defined previously. In the inertial
system of reference the kinetic energy is given by:
T =
1
2
𝐪̇ . 𝐪̇ =
1
2
(q̇ x
2
+ q̇ y
2
+ q̇ z
2
)
3.5
Where, the symbol . is the scalar product of two vectors, defined as,
[a,b,c].[d,e,f]=ad+be+cf. Therefore the Lagrangian L=T-U can be now formulated
as,
L =
1
2
(q̇ x
2
+ q̇ y
2
+ q̇ z
2
) − U(𝐪)
3.6
Momentum
The momentum of the system can be defined by the Lagrangian as,
px =
∂L
∂q̇ x
= q̇ x
py =
∂L
∂q̇ y
= q̇ y
pz =
∂L
∂q̇ z
= q̇ z
3.7
Hamiltonian
The Hamiltonian derives from the Lagrangian and the momentum as,
H = 𝐩. 𝐪̇ − L = pxq̇ x + pyq̇ y + pzq̇ z − L 3.8
In equation 3.8 we can use the notation q̇ x for the momentum, and the system is
now updated to,
H =
1
2
(q̇ x
2
+ q̇ y
2
+ q̇ z
2
) + U(𝐪)
3.9
Or in the same process Equation 3.7 can be expressed in terms of the momentum
of the system as,

33
H =
1
2
(px
2
+ py
2
+ pz
2
) + U(𝐪)
3.10
Finally the equations of motion can now derive straightforward from the
Hamiltonian as,
q̇ x =
∂H
∂px
= px
q̇ y =
∂H
∂py
py
q̇ z =
∂H
∂pz
= pz
ṗx = −
∂H
∂qx
= −
μ(qx−(1−μ)cos(t))
(qz
2+(qx+(−1+μ)cos(t))2+(qy(−1+μ) sin(t))
2
)
3/2
−
(1−μ)(qx+μ cos(t))
(qz
2+(qx+μ cos(t))2+(qy+μ sin(t))
2
)
3/2
ṗy = −
∂H
∂qy
= −
μ(qy−(1−μ)cos(t))
(qz
2+(qx+(−1+μ)cos(t))2+(qy(−1+μ) sin(t))
2
)
3/2
−
(1−μ)(qy+μcos(t))
(qz
2
)
3/2
ṗz = −
∂H
∂qz
= −
μ(qz−(1−μ)cos(t))
(qz
2+(qx+(−1+μ)cos(t))2+(qy(−1+μ) sin(t))
2
)
3/2
−
(1−μ)(qz+μ cos(t))
(qz
2
)
3/2
3.11
3.1.2 Barycentric, non-inertial, rotating system of reference
In this section the system of reference is transferred into the rotation
anticlockwise in the
𝐪̂ 𝐱
𝐪̂ 𝐲
plane of an angle θ. As the two primaries revolve in the
inertial frame around the origin with constant angular velocity, it is possible to
introduce a rotating reference frame, in which the position of the primaries will
result in being fixed. The new axis of the rotational reference frame are set as xr
and yr being perpendicular to each other and zr completing the right-handed
Cartesian frame. The zr axis of the rotating reference frame is always parallel to
the zi axis of the inertial reference frame and the angular speed of the reference
frame is the same of the Moon around the Earth. Therefore, the primaries will be
constantly lying on its x-axis.

34
Figure 3.1 Inertial (i) and rotating (r) reference frames in the CR3BP.
The transformation matrix to shift the coordinates to the rotational frame can be
defined as,
R(θ) = (
cos(θ) sin(θ) 0
− sin(θ) cos(θ) 0
0 0 1
) 3.12
The change in coordinates is developed to provide the new coordinate system so
that,
[
ξ
η
ζ
] = [
cos(θ) sin(θ) 0
− sin(θ) cos(θ) 0
0 0 1
].[
qx
qy
qz
] 3.13
Where ξ,η and ζ are the new axis of the system.
The change of coordinates takes the following form,
qx = cos(θ) ξ − sin(θ) η
qy = sin(θ) ξ + cos(θ) η
qx = ζ
3.14
The first derivative of the Equation 3.14 gives,
q̇ x = − sin(θ)θ̇ξ + cos(θ)ξ̇ − cos(θ)θ̇η − sin(θ)η̇
q̇ y = cos(θ)θ̇ξ + sin(θ)ξ̇ − sin(θ)θ̇η + cos(θ)η̇
q̇ z = ζ̇
3.15
Continuing the derivation, the second derivation takes the form,

35
q̈ x = − cos(θ) θ̇2
ξ − sin(θ)θ̈ξ − 2 sin(θ)θ̇ξ̇ + cos(θ)ξ̈ +
sin(θ) θ2̇ η − cos(θ)θ̈η − 2 cos(θ)θ̇η̇ − sin(θ)η̈
q̈ y = − sin(θ) θ̇2
ξ + cos(θ)θ̈ξ + 2 cos(θ)θ̇ξ̇ + sin(θ)ξ̈ −
cos(θ) θ2̇ η − sin(θ)θ̈η − 2 sin(θ)θ̇η̇ + cos(θ)η̈
q̈ z = ζ̈
3.16
Position of the Primaries
In the rotational reference frame system, the positions of the Earth can be
described as,
ξΕ = [
μ cos(θ) cos(t + π) + μ sin(θ) sin(t + π)
−μ sin(θ) cos(t + π) + μ cos(θ) sin(t + π)
0
]
T
=
[
μ cos(θ − t − π)
−μ sin(θ − t − π)
0
]
T
3.17
In the same way, the position of the moon in the rotational reference frame can
be described as,
ξΜ = [
(1 − μ) cos(θ − t)
−(1 − μ) sin(θ − t)
0
]
T
3.18
Equations of motion
The equations of motion can be determined from the second derivatives and after
some calculations, are defined as,
ξ̈ = θ̇2
ξ + θ̈η + 2θ̇η̇ −
∂U1(ξ)
∂(ξ)
η̈ = θ̇2
η − θ̈ξ − 2θ̇ξ̇ −
∂U1(ξ)
∂(η)
ζ̈ = −
∂U1(ξ)
∂ζ
3.19
Potential energy
The potential energy after the change of coordinates can be rewritten as,

36
U1(ξ) = −
1−μ
√((ξ−(μ cos(θ−π−t))2+(η−(−μ sin(θ−π−t)))
2
+ζ2
−
μ
√((ξ−(1−μ) cos(θ−t))2+(η−(−(1−μ)sin(θ−t)))
2
+ζ2
3.20
Lagrangian
The kinetic energy stems from Equations 3.5 and 3.15 as,
T1 =
1
2
(ξ̇2
+ η̇2
+ ζ̇2
) +
1
2
θ̇2(ξ2
+ η2) + θ̇(η̇ξ − ξ̇η) 3.21
And the Lagrangian is defined as L1=T1+U1(ξ),
L1 =
1
2
(ξ̇2
+ η̇2
+ ζ̇2
) +
1
2
θ̇2(ξ2
+ η2) + θ̇(η̇ξ − ξ̇η) − U1(ξ) 3.22
Momentum
The momentum of the system is now being defined from the new Lagrangian as,
pξ =
∂L1
∂ξ̇ = −θ̇η + ξ̇
pξ =
∂L1
∂η̇
= θ̇ξ + η̇
pξ =
∂L1
∂ζ̇ = ζ̇
3.23
Hamiltonian
We know already that the Hamiltonian derives from the Lagrangian and the
momentum of the system as, H1 = 𝐩. 𝛏̇ − L1 = pξξ̇ + pηη̇ + pζζ̇ − L1, thus the
Hamiltonian becomes,
H1 =
1
2
(ξ̇2
+ η̇2
+ ζ̇2
− ġ 2(ξ2
+ η2)) + U1(ξ) =
1
2
(pξ
2
+ pη
2
+ pζ
2
) −
ġ(ξpη − ηpξ) + U1(ξ)
3.24
Finally the equations of motion can derive from the Hamiltonian and be formed
as,

37
ξ̇ =
∂Η1
∂pξ
= pξ + θ̇η
η̇ =
∂Η1
∂pη
= pη − θ̇ξ
ζ̇ =
∂Η1
∂pζ
= pζ
ṗξ = −
∂Η1
∂ξ
=
1
2
(−
2μ(ξ+(−1+μ)cos(θ−t))
(ζ2+(ξ+(−1+μ) cos(θ−t))2+(η−(−1+μ)sin(θ−t))2)
3
2
−
2(ξ+μ cos(θ−t))
(ζ2+(ξ+μ cos(θ−t))2+(η−μ sin(θ−t))2)
3
2
ṗη = −
∂Η1
∂η
=
1
2
(−
2μ(η+(−1+μ)cos(θ−t))
(ζ2+(ξ+(−1+μ)cos(θ−t))2+(η−(−1+μ)sin(θ−t))2)
3
2
−
2(η+μ cos(θ−t))
3
2
ṗζ = −
∂Η1
∂ζ
=
1
2
(−
2μ(ζ+(−1+μ) cos(θ−t))
(ζ2+(ξ+(−1+μ) cos(θ−t))2+(η−(−1+μ)sin(θ−t))2)
3
2
−
2(ζ+μ cos(θ−t))
3
2
3.25
3.1.3 Non Inertial, Rotating System of Reference, centred in L1
This change of coordinates considers the transfer of the system so that the center
of the system is the L1 equilibrium point. In that case, the coordinates of the
system will be xe=[xe,0,0] where, xe is the x-axis coordinate defined in section 2.5.
The new shifted system is considered to be in the barycenter, non-inertial or
rotating system of reference. The change of coordinates is considered to be as,
ξ = x + xe
η = y
ζ = z
3.26
The first derivatives of Equations 3.26 are defined as,
ξ̇ = ẋ
η̇ = ẏ
ζ̇ = ż
3.27
And for the second derivatives we obtain,

38
ξ̈ = ẍ
η̈ = ÿ
ζ̈ = z̈
3.28
Position of the primaries
In this system of coordinates the position of the Earth and Moon is described as,
𝐱 𝐄 = [−μ − xe, 0,0] 3.29
𝐱 𝐌 = [(1 − μ) − xe, 0,0] 3.30
Equations of motion
By using the Equations (2.15 and 2.17) and the pseudopotential function with the
current system of coordinates, the equations of motion can be formed as,
ẍ = x + xe + 2ẏ −
∂U2(x)
∂x
ÿ = −2ẋ + y −
∂U2(x)
∂y
z̈ = −
∂U2(x)
∂z
3.31
Potential Energy
With the change in coordinates the potential energy becomes,
U2(𝐱) = −
1−μ
√(x−(1−μ)+xe)2+y2+z2
−
μ
√(x+μ+xe)2+y2+z2 3.32
Lagrangian
The kinetic energy with the change in coordinates transforms to,
T2 =
1
2
(ẋ2
+ ẏ2
+ ż2) +
1
2
((x + xe)2
+ y2) + (ẏ(x + xe) − ẋy) 3.33
And therefore the Lagrangian turns into,
L2 = T2 − U2(𝐱) =
1
2
(ẋ2
+ ẏ2
+ ż2) +
1
2
((x + xe)2
+ y2) + (ẏ(x + xe) −
ẋy) − U2(𝐱)
3.34
Momentum

39
From the new Lagrangian the momentum of the system becomes,
X =
∂L2
∂x
= ẋ − y
Y =
∂L2
∂y
= ẏ + x + xe
Z =
∂L2
∂z
= ż
3.35
Hamiltonian
The new Hamiltonian can now be represented as,
H2 = 𝐗. 𝐱 = Xẋ + Yẏ + Zż − L2 =
1
2
(ẋ2
+ ẏ 2
+ ż2) −
1
2
((x + xe)2
+ y2) +
U2(𝐱) =
1
2
(X2
+ Y2
+ Z2) + Xy − Y(x + xe) + U2(𝐱)
3.36
Equations of motion
The equations of motion can derive directly form the Hamiltonian as,
ẋ =
∂H2
∂X
= X + y
ẏ =
∂H2
∂Y
= Y + (−x − xe)
ż =
∂H2
∂Z
= Z
Ẋ = −
∂H2
∂x
= Y +
1
2
(−
2μ(−1+x+μ+xe)
(y2+z2+(−1+x+μ+xe)2)
3
2
−
2μ(x+μ+xe)
(y2+z2+(x+μ+xe)2)
3
2
+
2μ(x+μ+xe)
(y2+z2+(x+μ+xe)2)3/2
Ẏ = −
∂H2
∂y
= −X +
1
2
(−
2μy
(y2+z2+(−1+x+μ+xe)2)
3
2
−
2y
(y2+z2+(x+μ+xe)2)
3
2
+
2μy
(y2+z2+(x+μ+xe)2)3/2
Ż = −
∂H2
∂z
=
1
2
(−
2μz
(y2+z2+(−1+x+μ+xe)2)
3
2
−
2z
(y2+z2+(x+μ+xe)2)
3
2
+
2μz
(y2+z2+(x+μ+xe)2)3/2
3.37
3.2 Rearranging the Hamiltonian
This chapter provides an approach to reduce the complexity of the system, which
stems from the difficulty to integrate the system to its initial form, by a number of
changes in the coordinates. The first step for this simplification, is to rewrite the

40
Hamiltonian as a series of polynomials, and to truncate this series. In many cases
the polynomial can expand to a high number of orders, but this investigation only
considers the polynomials up to 4th order.
3.2.1 Legendre Polynomial
There are plenty of different directions to form the potential U2(x), but in this
section only the Legendre polynomial Pn(x) method is being used. The first orders
for the Legendre polynomial are:
P0(x) = 1
P1(x) = x
P2(x) =
1
2
(3x2
− 1)
P3(x) =
1
2
(5x3
− 3x)
P4(x) =
1
8
(35x4
− 30x2
+ 3)
3.38
Where, the series expansion that is been used is the
1
√1−2xt+t2
,for t < 1. That is,
1
√1−2xt+t2
= ∑ Pn(x)tn∞
n=0 3.39
The first step to perform this method is to modify the potential energy, so to finally
be reformulated in a similar way like the Equation (3.39) as follows,

41
U2(x) = −
1−μ
√(x−(1−μ)+xe)2+y2+z2
−
μ
√(x+μ+xe)2+y2+z2
=
−
1−μ
√(−(1−μ)+xe)2+2x(−(1−μ)+xe)+x2+y2+z2
−
μ
√(μ+xe)2+2x(μ+xe)+x2+y2+z2
=
−
1−μ
√(−(1−μ)+xe)2
1
√1−2√x2+y2+z2(−
x
(−(1−μ)+xe)√x2+y2+z2
)+(
√x2+y2+z2
(−(1−μ)+xe)
)
2
−
μ
√(μ+xe)2
1
√1−2√x2+y2+z2 (−
x
(μ+xe)√x2+y2+z2
)+(
√x2+y2+z2
(μ+xe)
)
2
(
√x2+y2+z2
(−(1−μ)+xe)
)
2
=
−
1−μ
−(1−μ)+xe
∑ Pn (−
x
−(1−μ)+xe√x2+y2+z2
)∞
n=0 (
√x2+y2+z2
(−(1−μ)+xe)
)
n
+
μ
μ+xe
∑ Pn (−
x
(μ+xe)√x2+y2+z2
)∞
n=0 (
√x2+y2+z2
(μ+xe)
)
n
= (
μ
−1+μ+xe
−
1−μ
μ+xe
) +
(
μ
(−1+μ+xe)2 −
1−μ
(μ+xe)2) x + (
μ
(−1+μ+xe)3 −
1−μ
(μ+xe)3) (x2
−
y2
2
+
z2
2
) +
(
μ
(−1+μ+xe)4
−
1−μ
(μ+xe)4
) (−x3
+
3
2
xz2
+
3
2
xy2
) + (
μ
(−1+μ+xe)5
−
1−μ
(μ+xe)5
) (
3
8
y4
+
3
8z4
+ x4
− 3x2
y2
− 3x2
z2
+
3
4
y2
z2
) −
1−μ
−(1−μ)+xe
∑ Pn (−
x
(−(1−μ)+xe)√x2+y2+z2
)n≥5 (
√x2+y2+z2
(−(1−μ)+xe)
)
n
+
μ
μ+xe
∑ Pn (−
x
(μ+xe)√x2+y2+z2
)n≥5 (
√x2+y2+z2
(μ+xe)
)
n
3.40
Where in L1 we know that:
(xe − (1 − μ)) < 0 and (xe + μ) > 0
Furthermore, the distance between the spacecraft and the L1 points is assumed
to be smaller than the distance between the spacecraft and primary bodies, such
as,
√x2 + y2 + z2
−(1 − μ) + xe
< 1 and
√x2 + y2 + z2
μ + xe
< 1

42
3.2.2 The truncated Hamiltonian
To obtain the Hamiltonian as a polynomial series, the Equation3.36 is plugged
into the Equation 3.40, considering the orders up to n=4 and neglecting the higher
orders. Finally, the truncated Hamiltonian will take the form as,
H2
trunc
= H2
(0)
+ H2
(1)
+ H2
(2)
+ H2
(3)
+ H2
(4)
3.41
Where the terms of the right hand are considered to be,
H2
(0)
=
μ
−1+μ+xe
−
1−μ
μ+xe
3.42
H2
(1)
= −Yxe − (
μ
(−1+μ+xe)2
−
1−μ
(μ+xe)2
) x 3.43
H2
(2)
=
1
2
(X2
+ Y2
+ Z2) + yX − Yx + (
μ
(−1+μ+xe)3
−
1−μ
(μ+xe)3
) (x2
−
1
2
y2
−
1
2
z2
)
3.44
H2
(3)
= (
μ
(−1+μ+xe)4
−
1−μ
(μ+xe)4
) (−x3
+
3
2
xy2
+
3
2
xz2
) 3.45
H2
(4)
= (
μ
(−1+μ+xe)4
−
1−μ
(μ+xe)4
) (x4
+
3
8
y4
+
3
8
z4
+
3
4
y2
z2
− 3x2
y2
− 3x3
z2
) 3.46
Each of the Equations 3.42 - 3.46 is the sum of the homogenous polynomials of
degree n for each of the cases, and thus Ḣ 2
(n)
is made of a sum of terms similar
to const xk1
yk2
zk3
Xk4
Yk5
Zk6
Where const is a constant and the sum of the exponents of all the variables is n
(k1+k2+k3+k4+k5+k6=n).
Finally, the equations of motion derives from the Hamiltonian and is defined as,
H2
trunc
= H2
(0)
+ H2
(1)
+ H2
(2)
+ H2
(3)
+ H2
(4)
3.47
While, the derivation of the Hamiltonian is computed as,
Ĥ2
trunc
= H2
(2)
+ H2
(3)
+ H2
(4)
3.48
Therefore, the Hamiltonian is formed as,

43
H = H2
(2)
+ H2
(3)
+ H2
(4) 3.49
3.2.3 The Linearized system
The linearized system is obtained by considering as Hamiltonian the sole second
order H2
(2) . By using the Equation3.37 and expanding the derivatives of the
Hamiltonian the equations of motion are reformulated and defined as,
ẋ =
∂H2
(2)
∂X
= X + y
ẏ =
∂H2
(2)
∂Y
= Y − x
ż =
∂H2
(2)
∂Z
= Z
Ẋ = −
∂H2
(2)
∂x
= Y − 2Ωex
Ẏ = −
∂H2
(2)
∂y
= −X + Ωey
Ż = −
∂H2
(2)
∂z
= Ωez
3.50
Where Ωe =
μ
(−1+μ+xe)3 −
1−μ
(μ+xe)3.
The equations of motions in Equation 3.50 can be represented in a matrix form
as,
A =
[
0 1 0 1 0 0
−1 0 0 0 0 0
0 0 0 0 0 1
−2Ωe 0 0 0 1 0
0 Ωe 0 −1 0 0
0 0 Ωe 0 0 0]
3.51
And the system in Equation 3.37 can be rewritten in the form,

44
[
ẋ
ẏ
ż
Ẋ
Ẏ
Ż]
= A.
[
x
y
z
X
Y
Z]
3.52
The diagonalization of the linearized system can be found on Appendix A.
3.3 Reduction to the central manifold
After analysing how H2
(2)
Hamiltonian, and therefore the linear system, changes
with the change of variables M, we need now to get back to the truncated
Hamiltonian in Equation 3.41 and examine the higher orders of the truncated
Hamiltonian with respect to the change of coordinates. In order to do so, we enter
Equation 9.23 into Equation 3.41 of the truncated Hamiltonian. As a result, the
transformed Hamiltonian occurs as,
H = H
(2)
+ H
(3)
+ H
(4)
3.53
As previously mentioned, the character of the collinear equilibrium points is a
saddle × centre × centre type and we need to proceed to the reduction to the
center manifold. The method is a straightforward extension of Jorba and
Masdemont [3]. Using as a reference Appendix A, the Hamiltonian H
(2)
can take
a simpler form as follows,
H
(2)
= λxX +
ω1
2
(y
2
+ Y
2
) +
ω2
2
(z
2
+ Z) 3.54
Where, Hamiltonian corresponds to the Equations of motion 9.22. As far as
the H
(3)
and H
(4)
is concerned, they will both be a sum of homogenous
polynomials in the variables x, y, z, X, Y and Z so that the sum of their exponents is
three and four respectively. So, considering only the third and fourth Hamiltonian
j=3,4 and,
H
j
= ∑ hk1,k2,k3,k4,k5,k6
(j)
x
k1
y
k2
z
k3
X
k4
Y
k5
Z
k6
k1+k2+k3+k4+k5+k6
3.55

45
Where, in the case of H
(3)
the exponents k1 + k2 + k3 + k4 + k5 + k6 must be
equal to 3 and in the case of H
(4)
the exponents k1 + k2 + k3 + k4 + k5 + k6 must
be equal to 4. hk1,k2,k3,k4,k5,k6
j
∈ ℝ are constants, which can be obtained by the use
of some algebra in Equation 3.41 after changing the system’s coordinates.
3.3.1 Complexification
A further (symplectic) change of variables now needs to be considered, following
[4] and [7]. Gomez applies the following complex transformation,
x = q1
y =
q2+ip2
√2
z =
q3+ip3
√2
X = x
Y =
iq2+p2
√2
Z =
iq3+p3
√2
3.56
Which, substituted in the Hamiltonian Equation 3.53 provides the new
Hamiltonian such as,
Ĥ = Ĥ2
+ Ĥ3
+ Ĥ4 3.57
For a better compliance of some of the following steps Equation 3.57 can be
modified as,
Ĥ = Ĥ2
+
Ĥ3
1!
+
Ĥ4
2!
3.58
With an abuse of notation, Ĥ4
= 2Ĥ4
.
Hence the quadratic part of the new Hamiltonian, takes the form,
Ĥ(2)
= λq1p 1 + iωq2p2 + iωq3p3
3.59
For Ĥ3
and Ĥ4
we have j=3,4 respectively and,
Ĥj
= ∑ ĥ
k1,k2,k3,k4,k5,k6
j
q̂1
k1
q̂2
k2
q̂3
k3
p̂1
k4
p̂2
k5
p̂3
k6
k1+k2+k3+k4+k5+k6
3.60

46
Where, ĥ
k1,k2,k3,k4,k5,k6
j
∈ ℝ are constants, which can be determined.
In addition to the quadratic part of Equation 3.59 the non-linear system must be
computed. It is a tedious process which is performed by Mathematica®. In this
way, the expressions of H3 and H4 can be obtained in complex variables.
Afterwards, we need to eliminate the hyperbolic component as explained by the
next section.
3.3.2 Eliminating the hyperbolic component
In the Hamiltonian equation the terms p1 and q1 are the conjugated variables
corresponding to the eigenvalue λ of the system, the only Real part of it. These
two components result in the hyperbolic motion of the system. The fact that this
investigation will focuses only in periodic orbits (i.e. Halo orbits), makes
necessary the elimination of these two components. The hyperbolic components
force the escape of the spacecraft-particle from the orbit and therefore not
providing a periodic orbit.
To obtain only periodic orbits, and hence halo orbits which is our scope, it is
important to define a new system of coordinates I = [Ix, Iy, Iz, θx, θy, θz], so that
the new plane Iy / Ix, contains the periodic orbits we are aiming to find. It is clear
that while this new plane is being defined, with the orbit starting on it, with Ix=0,
then the spacecraft- particle will remain in the same plane for the whole motion
Ix(t) = 0∀t and will not escape as the hyperbolic components have been
removed (i.e. İx =
∂H
∂θx
= 0 and H is not depending on θx).
The coordinate Ix of the new coordinate system, will be the link to the hyperbolic
component. The reduction to the central manifold is based on this kore procedure
[4].
The process of the reduction to the manifold starts with investigating the change
in coordinates, which have to be applied in order to secure the removal of he
hyperbolic components.

47
To find this new system of coordinates Q = [Q1, Q2, Q3, P1, P2, P3], the dependence
of the new Hamiltonian on the product of (P1, Q1)j
∀j ∈ N has to be taken into
account
Namely:
hk1,k2,k3,k4,k5,k6
j
= ĥ
k1,k2,k3,k4,k5,k6
j
if k1 = k4
hk1,k2,k3,k4,k5,k6
j
= 0 if k1 ≠ k4
3.61
H = H(2)
+
H(3)
1!
+
H(4)
2!
= ∑ ∑
1
(j−2)!
hk1,k2,k3,k4,k5,k6
(j)
Q1
k1
Q2
k2
Q3
k3
P1
k4
P2
k5
P3
k6
k1+k2+k3+k4+k5+k6=jj=2,3,4
3.62
Generally, in this paper the change in coordinates in any case, results in a
simplification of the Hamiltonian H(2), since the higher orders were neglected. In
this type of change in coordinates the main different is that the higher order terms
are included in the Hamiltonian. The theory introduced by Debrit [6] is based on
a form were the higher order terms are included.
This theory builds explicitly a change in variables through a generating function
G,
G = G(2)
+
G(3)
1!
+
G(4)
2!
=
∑ ∑
1
(j−2)!
gk1,k2,k3,k4,k5,k6
(j)
Q1
k1
Q2
k2
Q3
k3
P1
k4
P2
k5
P3
k6
k1+k2+k3+k4+k5+k6=jj=2,3,4
3.63
We notice that, Ĥ(2)
is already satisfying the Equation 3.61 (and so G(2)
= 0) and
for this reason Equation 3.63 can be updated and rewritten as,
G = G(3)
+ G(4)
=
∑ ∑
1
(j−2)!
gk1,k2,k3,k4,k5,k6
(j)
Q1
k1
Q2
k2
Q3
k3
P1
k4
P2
k5
P3
k6
k1+k2+k3+k4+k5+k6=jj=3,4
3.64
3.3.3 Eliminating the hyperbolic component
An operator called Poisson Brackets will be used in this section to
define H(3)
and H(4)
, and formed as,

48
{f, g} = ∑ (
∂f
∂Qj
∂g
Pj
−
∂f
∂Pj
∂g
∂Qj
)j=1,2,3, 3.65
With,
{f + h, g} = {f, g} + {h, g} 3.66
As Debrit describes, if f has a degree in the order of r (i.e. is made of
homogeneous polynomials so that the sum of the exponents is r) and g is in the
order of s, the {f,g} is of degree r+s-2. Moreover, the new Hamiltonian H will be
obtained by the use of the old Hamiltonian Ĥ generating a function G by solving
several Hamilton-Jacobi equations. Hence, we demonstrate the first three
equations accordingly,
H(2)
= Ĥ(2)
(𝐐, 𝐏)
H(3)
= Ĥ(3)
(𝐐, 𝐏) + {Ĥ(2)
(𝐐, 𝐏), G(3)
}
H(4)
= (Ĥ(4)
(𝐐, 𝐏) + {{Ĥ(2)
(𝐐, 𝐏), G(3)
}} , G(3)
}
+2{Ĥ(3)
(𝐐, 𝐏), G(3)
}) + {Ĥ(2)
(𝐐, 𝐏), G(4)
}
3.67
The analytical process to obtain the Hamiltonian formula can be found Appendix
B.
The Hamiltonian formula has now be obtained and is valid only for the L1 point
of the Earth-Moon system, which is the case we investigate,
H = H(2)
+
1
1!
H(3)
+
1
2!
H(4)
3.68
Where,
H(2)
= 2.93208P1Q1 + 2.3344iP2Q2 + 2.26885P3Q3
H(3)
= 1.22088P2
3
+ P1Q1(−3.11127P2 − 3.11127iQ2) − 0.980247iP2
2
Q2 +
1.00181iP3
2
Q2 − 1.22088iQ2
3
+ 2.00363P3Q2Q3 − 1.00181iQ2Q3
2
+
P2(1.00181P3
2
+ 0.980247Q2
2
− 2.00363iP3Q3 − 1.00181Q3
2
)
3.69

49
H(4)
= −9.76348P2
4
− 13.2937P2
2
P3
2
− 4.1593P3
4
+ 55.7423P1P2
2
Q1
+ 36.6035P1P3
2
Q1 − 10.9645P1
2
Q1
2
+ 10.9447iP2
3
Q2
+ 4.78258iP2P3
2
Q2 − 55.717iP1P2Q1Q2 + 7.92936P2
2
Q2
2
+ 11.8474P3
2
Q2
2
− 55.7423P1Q1Q2
2
− 10.9447iP2Q2
3
− 9.76348Q2
4
+ 15.0618iP2
2
P3Q3 + 9.10825iP3
3
Q3
− 49.8251iP1P3Q1Q3 − 5.16833P2P3Q2Q3
− 15.0618iP3Q2
2
Q3 + 11,8474P2
2
Q3
2
+ 9.8981P3
2
Q3
2
− 36.6035P1Q1Q3
2
− 4.78258iP2Q2Q3
2
− 13.2937Q2
2
Q3
2
− 9.10835iP3Q3
3
− 4.1593Q3
4
Following Deprit’s process, the transformation coordinates can be expressed
explicitly as,
q1 = Q1 +
∂G(3)
∂P1
+
G(4)
2! ∂P1
+
1
2!
{
∂G(3)
∂P1
, G(3)
}
q2 = Q2 +
∂G(3)
∂P2
+
G(4)
2! ∂P2
+
1
2!
{
∂G(3)
∂P2
, G(3)
}
q3 = Q3 +
∂G(3)
∂P3
+
G(4)
2! ∂P3
+
1
2!
{
∂G(3)
∂P3
, G(3)
}
p1 = P1 −
∂G(3)
∂Q1
−
G(4)
2! ∂Q1
+
1
2!
{−
∂G(3)
∂Q1
, G(3)
}
p2 = P2 −
∂G(3)
∂Q2
−
G(4)
2! ∂Q2
+
1
2!
{−
∂G(3)
∂Q2
, G(3)
}
p3 = P3 −
∂G(3)
∂Q3
−
G(4)
2! ∂Q3
+
1
2!
{−
∂G(3)
∂Q3
, G(3)
}
3.70
Defining the inverse coordinates of Equation 3.70 we get that,
Q1 = q1 −
∂G(3)
∂P1
(𝐪, 𝐩) −
G(4)
2! ∂P1
(𝐪, 𝐩) +
1
2!
{
∂G(3)
∂P1
, G(3)
} (𝐪, 𝐩)
Q2 = q2 −
∂G(3)
∂P2
(𝐪, 𝐩) −
G(4)
2! ∂P2
+
1
2!
{
∂G(3)
∂P2
(𝐪, 𝐩), G(3)
} (𝐪, 𝐩)
Q3 = q3 −
∂G(3)
∂P3
(𝐪, 𝐩) −
G(4)
2! ∂P3
+
1
2!
{
∂G(3)
∂P3
(𝐪, 𝐩), G(3)
} (𝐪, 𝐩)
P1 = p1 +
∂G(3)
∂Q1
(𝐪, 𝐩) +
G(4)
2! ∂Q1
+
1
2!
{−
∂G(3)
∂Q1
(𝐪, 𝐩), G(3)
} (𝐪, 𝐩)
P2 = p2 +
∂G(3)
∂Q2
(𝐪, 𝐩) +
G(4)
2! ∂Q2
+
1
2!
{−
∂G(3)
∂Q2
(𝐪, 𝐩), G(3)
} (𝐪, 𝐩)
P3 = p3 +
∂G(3)
∂Q3
(𝐪, 𝐩) +
G(4)
2! ∂Q3
+
1
2!
{−
∂G(3)
∂Q3
(𝐪, 𝐩), G(3)
} (𝐪, 𝐩)
3.71

50
3.4 Action-Angle Variables
Another change in variables needs to be implemented given by,
Q1 = −i√Ixeiθx
Q2 = −i√Iyeiθy
Q3 = −i√Izeiθz
P1 = √Ixe−iθx
P2 = √Iye−iθy
P3 = √Ize−iθz
3.72
Where θ is the coordinate’s position term and Is is the momentum term. So, the
Hamiltonian can be written as,
H = H
(2)
+
1
1!
H
(3)
+
1
2!
H
(4)
3.73
With H
(2)
= Ix + ωyIy + ωzIz and, H
(3)
and H
(4)
not depending on θx as Q1 and P1
continuously appear with the same exponent (i.e. the only two possibilities are
Q1P1 and Q1
2
P1
2
). Hence, these terms can be transformed to Ix and Ix
2
respectively.
Since, the Hamiltonian is independent on θx then its equation of motion becomes,
θ̇x =
∂H
∂Ix
θ̇y =
∂H
∂Iy
θ̇z =
∂H
∂Iz
İx = −
∂H
∂θx
= 0
İy = −
∂H
∂θy
İz = −
∂H
∂θz
3.74
As the Hamiltonian is independent from θx, the fourth Equation of 3.12 is in our
interest (İx = −
∂H
∂θx
= 0). It can be observed from İx = 0 that Ix is constant.
Setting the initial condition to be Ix(0) = 0 then Ix(t) = 0 ∀ t and the Hamiltonian
turns into,

51
H = ωyIy + ωzIz + H
(3)
+
1
2!
H
(4)
3.75
Where, H
(3)
and H
(4)
are functions of the sole Iy, Iz, θy and θz. Finally, the
hyperbolic components have been removed successfully. It is important to notice
that the Hamiltonian has developed as a Fourier series in the two variables θy
and θz.
3.5 Resonant Perturbation Theory
After considering the obit to be in the plane of Iy/Iz, it is essential to consider the
fact that a Halo orbit needs the frequencies of the two angles θy and θzto be in
1:1 resonance. This statement leads us to consider θ̇y =
∂H
∂Iy
= ωy +
∂H
(3)
∂Iy
+
∂
1
2!
H
(4)
∂Iy
to be equal with the corresponding equation for z.
To change the coordinates to the resonant angle we need to set α = θy − θz.
The remaining coordinate angle β can be any other arbitrary combination of the
angles m1θy + m2θzwith m ∈ N, with the only constraint not to be resonant
(i.e. m1θy + m2θz ≠ 0) and so, according to [7], β is chosen to be equals to θz.
The new variable J needs to take a generic form, so to understand the
transformation of the I terms. This form is,
J1 = m1Iy + m2Iz
J2 = m3Iy + m4Iz
3.76
According to Equation 3.14, the transformation matrix is now described by the
following matrix F as,
F = (
1 −1 0 0
0 1 0 0
0 0 m1 m3
0 0 m2 m4
)
3.77
Applying the definition of the symplectic matrix we get that,

52
Ft
. J. F = (
0 0 m1 m2
0 1 −m1 + m3 −m2 + m4
−m1 m1 − m3 0 0
−m2 m2 − m4 0 0
) 3.78
Where, m1 = 1, m2 = 0, m3 = 1 and m4 = 1, so that,
α = θy − θz
β = θz
J1 = Iy
J2 = Iy + Iz
3.79
With the new coordinate system, the Hamiltonian is taking the following form,
H
(2)
= 0.0655545J1 + 2.26885J2
H
(3)
= −2.00363J1
3
2
cos(α − β) + 2.00363J1
1
2
J2 cos(α − β) + 2.04676J1
3
2
cos(α + β) −
4.00726J1
1
2
J2 cos(α + β) − 2.00363J1
3
2
cos(α + 3β) + 2.00363J1
1
2
J2 cos(α +
3β) + 2.44175J1
3
2
J2 cos(3α + 3β)
1
2!
H
(4)
= −45.9916J1
2
+ 49.9291K1 𝐽2 − 19.7962J2
2
+ (47.3897𝐽1
2
−
47.3897𝐽1 𝐽2) cos(2𝛼) + (17.3031𝐽1
2
− 53.7364𝐽1 𝐽2 + 36.4334𝐽2
2) cos(2𝛽) +
(−16.6372𝐽1
2
+ 33.2744𝐽1 𝐽2 − 16.6372𝐽2
2) cos(4𝛽) + (−16.4683𝐽1
2
+
60.2471𝐽1 𝐽2) cos(2𝛼 + 2𝛽) + (53.1747𝐽1
2
− 53.1747𝐽1 𝐽2) cos(2𝛼 + 4𝛽) −
39.0539𝐽1
2
cos(4𝛼 + 4𝛽)
3.80
Where,
H = H
(2)
+ H
(3)
+
1
2!
H
(4)
3.81
To transform the Hamiltonian into a Fourier series in sole the resonant angle α a
change of coordinates needs to be found to transform α, β, J1and J2 to α̃, β̃, J1
̃ and J2
̃
respectively.

53
The new Hamiltonian will now include only terms which depend on the new
actions J1
̃ and J2
̃. Moreover, the terms with the actions J1
̃ and J2
̃ and solely
cos(kα̃) or sin(kα̃) with k=1, 2, 3, 4 will appear.
Terms containing β will not only disappear, but in fact will be brought to the higher
orders (i.e. higher than the 4th order) so that they can be neglected.
The new process of changing the coordinates will be done by the use of Debrit’s
theorem as in previous sections. In Equation 3.81, the Hamiltonian developed in
a Fourier series of the two angles can be updated as,
H
(j)
= ∑ ∑ (hck1,k2,k3,k4
j
J1
k1
J2
k2
cos(k3α + k4β) +k3,k4≤jk1+k2=
j
1
hsk1,k2,k3,k4
j
J1
k1
J2
k2
sin(k3α + k4β))
3.82
Therefore, the change in coordinates that must be applied has the form of F =
F(3)
+ F(4)
, so that the Hamiltonian will be transferred to the form of,
K = K(2)
+ K(3)
+
1
2!
K(4)
3.83
Where in this new Hamiltonian,
K
(j)
= ∑ ∑ (kck1,k2,k3,0
j
J1
k1
J2
k2
cos(k3α) +k3≤jk1+k2=
j
1
ksk1,k2,k3,0
j
J1
k1
J2
k2
sin(k3α))
3.84
Which gives the same result as imposing,
kck1,k2,k3,k4
j
= 0
ksk1,k2,k3,k4
j
= 0
if k4 ≠ 0 3.85
Following Debrit’s process as done in previous sections, the following equations
are set to be solved,

54
K(2)
= H
(2)
(𝛂̃, 𝐉̃)
K(3)
= H
(3)
(𝛂̃, 𝐉̃) + {H
(2)
(𝛂̃, 𝐉̃), F(3)
}
K(4)
= (H
(4)
(𝛂̃, 𝐉̃) + {{H
(2)
(𝛂̃, 𝐉̃), F(3)
}} , F(3)
} + 2{H
(3)
(𝛂̃, 𝐉̃), F(3)
}) +
{H
(2)
(𝛂̃, 𝐉̃), F(4)
}
3.86
Note that in this process the Poisson Bracket method is defined by taking into
consideration the coordinates α̃, β̃, J1
̃ and J2
̃.
The first equation is been solved by substituting J̃ in the Hamiltonian,
K(2)
= (ωy − ωz)J̃1 + ωzJ̃2
3.87
As for the second equation the term {H
(2)
(𝛂̃, 𝐉̃), F(3)
} is being firstly analysed,
{H
(2)
(𝛂̃, 𝐉̃), F(3)
} = (ωy − ωz)J̃1 + ωziJ̃2, F(3)
=
= − (
∂
∂J̃1
((ωy − ωz)J̃1 + ωzJ̃2))
∂
∂α̃
F(3)
− (
∂
∂J̃2
((ωy − ωz)J̃1 + ωzJ̃2))
∂
∂β̃ F(3)
= −(ωy − ωz) ∑ k3(−fck1,k2,k3,k4
(3)
k1+k2=
3
2
k3,k4≤3
J̃1
k1
J̃2
k2
sin(k3α̃ + k4β̃) +
fsk1,k2,k3,k4
(3)
J̃1
k1
J̃2
k2
cos(k3α̃ + k4β̃) −
(ωz) ∑ k4(−fck1,k2,k3,k4
(3)
k1+k2=
3
2
k3,k4≤3
J̃1
k1
J̃2
k2
sin(k3α̃ + k4β̃) +
fsk1,k2,k3,k4
(3)
J̃1
k1
J̃2
k2
cos(k3α̃ + k4β̃)
3.88
− ∑ ((ωy − ωz)k3 + (ωz)k4) (fsk1,k2,k3,k4
(3)
J̃1
k1
J̃2
k2
cos(k3α̃ + k4β̃) −k1+k2=
3
2
k3,k4≤3
fck1,k2,k3,k4
(3)
J̃1
k1
J̃2
k2
sin(k3α̃ + k4β̃)
Inserting Equation 3.88 into the second Equation of 3.86 we obtain that,

55
kck1,k2,k3,k4
(3)
= hck1,k2,k3,k4
(3)
− ((ωy − ωz)k3 + (ωz)k4) fsk1,k2,k3,k4
(3)
ksk1,k2,k3,k4
(3)
= hsk1,k2,k3,k4
(3)
+ ((ωy − ωz)k3 + (ωz)k4) fck1,k2,k3,k4
(3)
3.89
For constraints in Equation 3.85 to be satisfied we need to,
fsk1,k2,k3,k4
(3)
= 0
fck1,k2,k3,k4
(3)
= 0
if k4 = 0 3.90
fsk1,k2,k3,k4
(3)
=
hck1,k2,k3,k4
(3)
((ωy−ωz)k3+(ωz)ik4)
fck1,k2,k3,k4
(3)
=
hsk1,k2,k3,k4
(3)
((ωy−ωz)k3+(ωz)ik4)
if k4 ≠ 0 3.91
For the last Equation of 3.86 the process is the same, and the first to be analyzed
is,
H(4)∗
= (H
(4)
(𝛂̃, 𝐉̃) + {{H
(2)
(𝛂̃, 𝐉̃), F(3)
}} , F(3)
} + 2{H
(3)
(𝛂̃, 𝐉̃), F(3)
}) 3.92
Which, can be written as,
∑ ∑ (hck1,k2,k3,k4
(j)∗
J̃1
k1
J̃2
k2
cos(k3α̃ + k4β̃) +k3,k4≤jk1+k2=
j
1
hsk1,k2,k3,k4
(j)∗
J̃1
k1
J̃2
k2
sin(k3α̃ + k4β̃))
3.93
Then, the last part of the third Equation of 3.86 can be described as,
{H
(2)
(𝛂̃, 𝐉̃), F(4)
} = − ∑ ((ωy − ωz)k3 +k1+k2=2
k3,k4≤4
(ωz)k4) (fsk1,k2,k3,k4
(4)
J̃1
k1
J̃2
k2
cos(k3α̃ + k4β̃) −
fck1,k2,k3,k4
(4)
J̃1
k1
J̃2
k2
sin(k3α̃ + k4β̃)
3.94
Again we substitute Equation 3.94 into 3.86 in order to get the following,

56
kck1,k2,k3,k4
(4)
= hck1,k2,k3,k4
(4)∗
− ((ωy − ωz)k3 + (ωz)k4) fsk1,k2,k3,k4
(4)
ksk1,k2,k3,k4
(4)
= hsk1,k2,k3,k4
(4)∗
+ ((ωy − ωz)k3 + (ωz)k4) fck1,k2,k3,k4
(4)
3.95
For constraints in Equation 3.85 to be satisfied we need to,
fsk1,k2,k3,k4
(4)
= 0
fck1,k2,k3,k4
(4)
= 0
if k4 = 0
fsk1,k2,k3,k4
(4)
=
hck1,k2,k3,k4
(4)∗
((ωy−ωz)k3+(ωz)k4)
fck1,k2,k3,k4
(4)
=
hsk1,k2,k3,k4
(4)∗
((ωy−ωz)k3+(ωz)k4)
if k4 ≠ 0
3.96
The new Hamiltonian becomes,
K(2)
= 0.0655545J̃1 + 2.26885J̃2
K(3)
= 0
K(4)
= 43.5696J̃1
2
− 44.0268J̃1J̃2 + 22.6167J̃2
2
−44.15747J̃1
2
cos(2α̃) + 44.15747J̃1J̃2 cos(2α̃)
3.97
With,
K = K(2)
+ K(3)
+
1
2!
K(4)
3.98
Following Deprit’s paper, the change of coordinates can be explicitly evaluated
by the following formulas,
α = α̃ +
∂F(3)
∂J̃1
+
F(4)
2! ∂J̃1
+
1
2!
{
∂F(3)
∂J̃1
, F(3)
}
β = β̃ +
∂F(3)
∂J̃2
+
F(4)
2! ∂J̃2
+
1
2!
{
∂F(3)
∂J̃2
, F(3)
}
J1 = J̃1 −
∂F(3)
∂α̃
−
F(4)
2! ∂α̃
+
1
2!
{
− ∂F(3)
∂α̃
, F(3)
}
J2 = J2 −
∂F(3)
∂β̃ −
F(4)
2! ∂β̃ +
1
2!
{
− ∂F(3)
∂β̃ , F(3)
}
3.99
Defining now the inverse coordinates of Equation 3.99 we get that,

57
α̃ = α −
∂F(3)
∂J1
(α, 𝐉) −
1
2!
F(4)
∂J1
(α, 𝐉) +
1
2!
{
∂F(3)
∂J1
, F(3)
} (α, 𝐉)
β̃ = β −
∂F(3)
∂J2
(α, 𝐉) −
1
2!
F(4)
∂J2
(α, 𝐉) +
1
2!
{
∂F(3)
∂J2
, F(3)
} (α, 𝐉)
P1 = J1 +
∂F(3)
∂α
(α, 𝐉) +
1
2!
F(4)
∂α
(α, 𝐉) +
1
2!
{−
∂F(3)
∂α
, F(3)
} (α, 𝐉)
P2 = J2 +
∂F(3)
∂β
(α, 𝐉) +
1
2!
F(4)
∂β
(α, 𝐉) +
1
2!
{−
∂F(3)
∂β
, F(3)
} (α, 𝐉)
3.100
3.6 Finding Halo Orbits
From Equation 3.83 and 3.97 and the change in coordinates that we obtained in
the previous section, the equations of motion can be defined as,
α̃̇ =
∂K
∂J̃1
= 0.0655545 + 43.5696J̃1 − 22.0134J̃2
+(−44.1547J̃1 + 22.0773J̃1)cos(2α̃)
β̃̇ =
∂K
∂J̃2
= 2.26885 − 22.0134J̃1 + 22.6167J̃2 + 22.0773J̃1cos(2ã)
J̃̇
1 = −
∂K
∂α̃
= J̃1(−44.1547J̃1 + 44.1547J̃2)sin(2ã)
J̃̇
2 = 0
3.101
Solving these equations of motion results in different periodic orbits around the
L1 point in the Earth-Moon system.
Setting J̃1 = J̃2 the so called horizontal Lyapunov is obtained.
Setting J̃1 = 0 the so called vertical Lyapunov is obtained.
Setting α̃ = ±
π
2
the so called Halo orbits are obtained.
Setting α̃ = 0 another periodic orbit is obtained.
So, to obtain the Halo orbit we need to set α̃ = ±
π
2
, and wanted to be constantly
π
2
abd therefore we impose also that α̃̇ = 0 and the following equations are been
formulated,

58
0 = 0.0655545 + 87.7243J̃1 − 44.0907J̃2
β̃̇ = 2.26885 − 44.0907J̃1 + 22.6167J̃2
J̃̇
1 = 0
J̃̇
2 = 0
3.102
Solving the first Equation of 3.102 and substituting it into the second we obtain
that,
2.30179 + 0.456447J̃2 3.103
Hence, integrating the differential equations,
This is the closed form analytic 4th order approximation of a Halo Orbit in the
vicinity of the L1 point around the Moon. The term β̃(0) is the initial phase between
the angle θ, while the term J̃1(0) is the amplitude of the orbit [23]. The different
choices of the term β̃(0) and θ, lead to the generation of a whole family of
approximated Halos. This orbit has to be traced back through all the changes of
coordinates until the x, X coordinates.
3.7 Back to the Real World
We now need to get the initial conditions back to the inertial, Cartesian
coordinates, getting through all the coordinates used so far.
By 9.23 we know that,
α̃(t) = ±
π
2
J̃2(t) = J̃2(0)
β̃(t) = (2.30179i + 0.456447J̃2(0)) t + β̃(0)
J̃1(t) = −0.000747279i + 0.502606J̃1(0)
3.104

59
𝑥 = −
2𝑌√ 𝜔
√−3+𝜔4+𝜔2(2−4𝛺 𝑒)+4𝛺 𝑒+4𝛺 𝑒
2
+
√2(𝑥+𝑋)𝜆
√𝜆(−3+𝜆4+4𝛺 𝑒+4𝛺 𝑒+𝜆2(−2+4𝛺 𝑒))
𝑥 = −
𝑦
(𝜔√−3+𝜔4+𝜔2(2−4𝛺 𝑒)+4𝛺 𝑒+4𝛺 𝑒
2
+
(𝑥−𝑋)(−1+𝜆2+2𝛺 𝑒)
√2𝜆(−3+𝜆4+4𝛺 𝑒+4𝛺 𝑒+𝜆2(−2+4𝛺 𝑒))
𝑧 =
𝑧
−𝛺 𝑒
1
4
3.105
Then by 3.56
𝑥 = −
√2(𝑝2+𝑖𝑞2)√ 𝜔
√−3+𝜔4+𝜔2(2−4𝛺 𝑒)+4𝛺 𝑒+4𝛺 𝑒
2
+
√2(𝑝1+𝑞1)𝜆
√𝜆(−3+𝜆4+4𝛺 𝑒+4𝛺 𝑒+𝜆2(−2+4𝛺 𝑒))
𝑥 = −
(𝑖𝑝2+𝑞2)(1+𝜔2−2𝛺 𝑒)
(2𝜔√−3+𝜔4+𝜔2(2−4𝛺 𝑒)+4𝛺 𝑒+4𝛺 𝑒
2
+
(𝑞1−𝑝1)(−1+𝜆2+2𝛺 𝑒)
√2𝜆(−3+𝜆4+4𝛺 𝑒+4𝛺 𝑒+𝜆2(−2+4𝛺 𝑒))
𝑧 =
𝑖𝑝3+𝑞3
√2(−𝛺 𝑒
1
4)
3.106
We keep going using Equations 3.70, 3.72, 3.79, 3.99 and finally 3.104 yielding,
𝑥 =
1
𝐽̃1(0)
(−396152 × 10−10
− 0.00420527𝐽̃1(0) + 2.66644𝐽̃1(0)2
−
27.2742𝐽̃1(0)3
+ (2.73943 × 10−10
+ 0.00168008𝐽̃1(0) − 0.0399274𝐽̃1(0)2
−
2.71122𝐽̃1(0)3) cos (4.6688𝑡 + 2𝛽̃(0)) + (3.52765 × 10−10
+
0.0000240644𝐽̃1(0) + 0,00439217𝐽̃1(0)2
− 2.696𝐽̃1(0)3) cos (9,3376𝑡 + 4𝛽̃(0)) +
√ 𝐽̃1(0)(−5.25038 × 10−7
+ 0.272656𝐽̃1(0) + 5.14961𝐽̃1(0)2) sin (2.3344𝑡 +
𝛽̃(0)) + √ 𝐽̃1(0)(−5.54177 × 10−7
+ 0.00203499𝐽̃1(0) +
0.354513𝐽̃1(0)2) sin (7.0032𝑡 + 3𝛽̃(0)))
𝑦 =
(−7.82169×10−6+1.02329𝐽̃1(0)−1.22625𝐽̃1(0)2 cos(2.3344𝑡+𝛽̃(0))
√𝐽̃1(0)
+
(4.68393×10−7−0.00191351𝐽̃1(0)−1.16773𝐽̃1(0)2) cos(7.0032𝑡+3𝛽̃(0))
√𝐽̃1(0)
+ (−0.000964576 −
1.85911×10−9
𝐽̃1(0)
+ 0.578258𝐽̃1(0) − 14.977𝐽̃1(0)2
) sin (4.6688𝑡 + 2𝛽̃(0)) +

60
(0.0000230158 +
(3.88588×10−10)
𝐽̃1(0)
− 0.0216355𝐽̃1(0) − 2.4114𝐽̃1(0)2
) sin (9.3376𝑡 +
4𝛽̃(0))
𝑧 =
1
.00331736+1.07649𝐽̃1(0)
(0.0058932√ 𝐽̃1(0) − 1.86188𝐽̃1(0)
3
2 + 14.6727𝐽̃1(0)
5
2 +
√ 𝐽̃1(0)(0.00174145 − 0.58425 𝐽̃1(0) + 6.21255𝐽̃1(0)2) cos (4.6688𝑡 + 2𝛽̃(0)) +
√ 𝐽̃1(0)(−0.0000300729 + 0.00465797𝐽̃1(0) + 1.65521𝐽̃1(0)2) cos (9.3376𝑡 +
4𝛽̃(0)) − 0.00309337 sin (2.3344𝑡 + 𝛽̃(0)) + 1.00454𝐽̃1(0) sin (2.3344𝑡 +
𝛽̃(0)) − 0.238511𝐽̃1(0)2
sin (2.3344𝑡 + 𝛽̃(0)) + 0.0000114286 sin (7.0032𝑡 +
3𝛽̃(0)) − 0.000970249𝐽̃1(0) sin (7.0032𝑡 + 3𝛽̃(0)) −
0.888602𝐽̃1(0)2
sin (7.0032𝑡 + 3𝛽̃(0))
3.107
Setting, for example, 𝛽̃(0) = 0 and varying the amplitude 𝐽̃1(0), a family of
approximated Halo orbits is generated,
Figure 3.2 A Family of approximated Halo orbits around L1.

Periodic Orbits via Numerical Methods Georgios Galanos
61
As we need one initial Halo orbit to be used as initial guess for our continuation
method, we set 𝛽̃(0) = 0 and 𝐽̃1(0) = 0.0048, obtaining,
𝑥(0) = 0.8462799363093021
𝑦(0) = 0.06986089490865859
𝑧(0) = −0.006266339315212618
𝑢 𝑥(0) = 0.049834157010533287
𝑢 𝑦(0) = 0.005569784497144666
𝑢 𝑧(0) = 0.0912838257451067
3.108
Since, the order higher than four was previously neglected, the differential
corrector will be needed to find the refined initial conditions.
Moreover, the bigger the amplitude of the Halo orbit the more this guess will be
wrong (it is based in the assumption that the orbit is close to the equilibrium point
L1), the more the differential correction will be needed. At a certain point, these
might not converged ant the continuation method will probably be needed.
4 Periodic Orbits via Numerical Methods
In previous sections, a closed-form solution of an approximated CR3BP was
explicitly built. A solution of the full CR3BP, however, cannot be available.
Periodic and quasi–periodic orbits require numerical methods for simulate the
orbits, and this section introduces some fundamental tools to build these orbits.
Such tools will be applied later on, in this thesis. Tools such as the State
Transition Matrix (STM), the Mirror Theorem, Differential Corrections and its
techniques, and the Continuation Method are included in this introduction section.
STM can be the base of all the tools as it is the foundation for many linear
differential correction algorithms. To simplify the process of finding the desired
orbit, Mirror Theorem helps to generate symmetric periodic orbits in the CR3BP
after the differential correction method has introduced, which divided into two
different process depending on the complexity of the general problem. Single
shooting method and Multiple shooting method are the two techniques used for
the differential corrections in the CR3BP. Multiple shooting method will not be

62
discussed in this work as it will not be used. After a desire orbit has been obtained,
the Continuation method is being applied to generate the whole orbit family.
4.1 State Transition Matrix (STM)
Two different approaches of finding a desired orbit can be applied in any case.
One tedious way is to integrate the equations of motion with several initial
conditions until the desired orbit be found. On the other hand, State Transition
Matrix is a visualized approach of the sensitivity of any point of the orbit, which
combined with the linearization of the equations of motion, to a reference solution,
helps to determine many of the periodic orbits. Following this method, information
about all the nearby orbits can be collected at the same time. In the vicinity of the
CR3BP, the non-differential equations and the dynamical behavior of the system
can provide many different types of solutions. In this thesis only periodic orbits
near the collinear libration points are investigated.
To produce a particular trajectory (a reference solution Xn(τ)) the set of initial
conditions X0, must be integrated for a non–dimensional time 𝜏 considering the
first – order form of the nonlinear equation of motion. The nearby trajectory of the
reference solution of this integration can be defined as,
X(τ) = Xn(τ) + δX(τ) 4.1
Where, X(τ) is the nearby trajectory and δX(τ) is the variation with respect to the
reference trajectory. By substituting the Equation 4.1 into the nonlinear equation
of motion of the particle in interest, results the variation equations such as,
Χ̇ = f(Χ, τ) 4.2
An expansion in Taylor series relative to the reference trajectory, is able to create
the desired solution close to the reference one. By keeping only the linear part of
Taylors series expansion, the first order variations relative to the reference can
be obtained and represented by the linear equation expressed in terms of the
variation state δΧ̇ as follows,

63
δΧ̇ = Α(τ)δX 4.3
Where A(τ) is the Jacobian Matrix and δΧ̇(τ) = [δx δy δz δẋ δẏ δż ]T
. Equation 4.3
can now be rewritten in a matrix form as,
[
δẋ
δẏ
δż
δẍ
δÿ
δz̈]
=
[
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Uxx
∗
Uxy
∗
Uxz
∗
0 2 0
Uyx
∗
Uyy
∗
Uyz
∗
−2 0 0
Uzx
∗
Uzy
∗
Uzz
∗
0 0 0] [
δx
δy
δz
δẋ
δẏ
δż]
4.4
The second partial derivatives of the pseudo-function U* of the lower left block of
the Equation 4.4 can be individually computed as,
Uxx
∗
=
∂2
U∗
∂x2
= 1 −
1 − μ
rEs
3
+
3(1 − μ)(x + μ)2
rEs
5
−
μ
rMs
3
+
3μ(x − 1 + μ)2
rMs
5
4.5
Uyy
∗
=
∂2
U∗
∂y2
= 1 −
1 − μ
rEs
3
+
3(1 − μ)y2
rEs
5
−
μ
rMs
3
+
3μy2
rMs
5
4.6
Uzz
∗
=
∂2
U∗
∂z2
= −
1 − μ
rEs
3
+
3(1 − μ)z2
rEs
5
−
μ
rMs
3
+
3μz2
rMs
5
4.7
Uxy
∗
=
∂2
U∗
∂x ∂y
= Uyx
∗
=
3(1 − μ)(x + μ)y
rEs
5
+
3μ(x − 1 + μ)y
rMs
5
4.8
Uxz
∗
=
∂2
U∗
∂x ∂z
= Uyx
∗
=
3(1 − μ)(x + μ)z
rEs
5
+
3μ(x − 1 + μ)z
rMs
5
4.9
Uyz
∗
=
∂2
U∗
∂y ∂z
= Uyx
∗
=
3(1 − μ)zy
rEs
5
+
3μzy
rMs
5
4.10
The Jacobian Matrix is a visualized method for the sensitivity, as the information
of the final state at any time τ with respect to the variation of the initial state at
initial time τ0 contained into the Jacobian Matrix and most of the times the matrix
can be referred as the Sensitivity Matrix. This form of the matrix is an alternative
display of the derivative of the states at non-dimensional time τ with respect to

64
the initial condition at τ0 and it is a fundamental tool in any targeting scheme.
STM can be defined as,
[
δxf
δyf
δzf
δxḟ
δyḟ
δzḟ ]
=
[
∂x
∂x0
∂x
∂y0
∂x
∂z0
∂x
∂ẋ0
∂x
∂y0̇
∂x
∂ż0
∂y
∂x0
∂y
∂y0
∂y
∂z0
∂y
∂ẋ0
∂y
∂ẏ0
∂y
∂ż0
∂z
∂x0
∂z
∂y0
∂z
∂z0
∂z
∂ẋ0
∂z
∂ẏ0
∂z
∂ż0
∂ẋ
∂x0
∂ẋ
∂y0
∂ẋ
∂z0
∂ẋ
∂ẋ0
∂ẋ
∂ẏ0
∂ẋ
∂ż0
∂ẏ
∂x0
∂ẏ
∂y0
∂ẏ
∂z0
∂ẏ
∂ẋ0
∂ẏ
∂ẏ0
∂ẏ
∂ż0
∂ż
∂x0
∂ż
∂y0
∂ż
∂z0
∂ż
∂ẋ0
∂ż
∂ẏ0
∂ż
∂ż0 ]
[
δx0
δy0
δz0
δẋ0
δẏ0
δż0 ]
4.11
Examining the partial derivatives of the matrix A(τ), it can be said that A(τ) is not
a constant matrix but in fact is a function of the sate on the reference trajectory
with a resulting general solution of the Equation 4.3 taking the form,
δX(τ) = Φ(τ, τ0)δX0
4.12
where, Φ(τ, τ0) is the first order 6x6 STM and δX0 is the six elements initial
variation form the given initial conditions which are connected to the reference
trajectory. Matrices A(τ) and Φ(τ, τ0) are not constant, except in the case of
variations to a constant reference solution i.e. Lagrangian points. Moreover, as
both matrices are time dependent, they can be propagated by the differential
equation, as below,
Φ̇ (τ, τ0) = Α(τ). Φ(τ, τ0) 4.13
Since Equation 4.13 represents the dot product between the two matrices, matrix
Φ̇ (τ, τ0) provides a 36 scalar differential equations where, in time τ=0 matrix
Φ(τ, τ0) becomes a 6x6 identity matrix.

65
4.2 Differential Corrections
The computation of periodic or quasi periodic orbits, requires an analytical
solution of the dynamical system, which as it has been mentioned, cannot be
found. Different set of tools can be used to obtain numerical results for the
CR3BP with a set of initial and final conditions. The use of such a tool contains
the use of Newton-Raphson differential corrections process usually called ‘’single
shooting method’’ and ‘’multi shooting method’’. This method, provided an initial
conditions guess, combined with the associated STM, targets to predict the
appropriate adjustments that must be applied to the initial conditions so the
desired orbit can be achieved. In this section, only the single shooting method will
be discussed, to generate periodic orbits and manifolds in next sections. In any
case differential corrections are applied, two main matrices must be defined, the
so called free variable matrix and constraint matrix. Free variable matrix (X
0
)
consists of variables that all are allowed to vary, so that a set of constrains could
be satisfied. Usually, velocity, position and integration time are the main variables
that are used to form the free variable matrix. The constraint variables are
equations ( F (X
0
)) that must be satisfied by the propagated trajectory. Most of
times, especially in finding periodic orbits these constraints consists of the
position, velocity and time of flight.
Differential corrections, aim to find a desired set of free variables (X) close to the
initial guess, provided at the beginning of the process, so that the constraint
vector is fulfilled. Taylor series expansion is being used for this process about the
initial guess of the free variables and the following expression can be defined,
F(X) = F (X
0
) +
∂F (X
0
)
∂X
0 (X − X
0
) + ⋯ 4.14
Where the term
𝛛𝐅(𝐗
𝟎
)
𝛛𝐗
𝟎 which, appears in the right hand side of the Equation 4.14
, is the Jacobian matrix of the constraints vectors with respect to the initial free
variable set and can be defined as,

66
DF(X) =
∂F (X
0
)
∂X
0
(
∂F1
∂X1
⋯
∂F1
∂Xn
⋮ ⋱ ⋮
∂Fm
∂X1
⋯
∂Fm
∂Xn )
4.15
Where, n and m are the size of the free variable vector and the constraint vector
respectively. Equation 4.14 can be rearranged to,
F(X) = F (X
0
) + DF (X
0
) (X − X
0
) 4.16
While the constraint equations are not satisfied and F(X) ≠ 0, Equation 4.16 can
be updated to,
X
j+1
= X
j
− DF (X
j
)
−1
F (X
j
)
4.17
Where X
j
, is the current initial conditions of the process and X
j+1
represents the
initial conditions of the next iteration in case the constraint vector is not satisfied.
On the right hand side of the Equation 4.17, X
j
shows the free variables of the
current iteration, F (X
j
) is the current constraint vector with respect to the current
free variables vector X
j
. For each of the iterations, the Jacobian matrix is modified
with respect to the corresponding free variable vector and the constraint vector
of the specific iteration. The correction stops when an acceptable tolerance, that
has been set to the algorithm, is larger than the Euclidean norm of the constraint
vector, such that for example,
ej = ||F (X
j+1
)|| < 10−12
4.18
Where ej is the error of the current constraint vector. In every next iteration the
error must be smaller such that ej+1<ej.
This approach and Equation 4.16 can be used only in the case that that the free
variable vector and the constraint vector are consist of the same number of

67
elements (n=m). In case that n>m, the system referred as an under determined
system and leads to infinity solutions. Hence, Equation 4.16 must be modified as
follows,
X
j+1
= X
j
− DF (X
j
)
T
(DF (X
j
) DF (X
j
)
T
)
−1
F (X
j
) 4.19
On the other hand, if the elements of the free variables vector are fewer than the
elements of the constraint vector i.e. n<m, Equation 4.16 must take the form as,
X
j+1
= X
j
− (DF (X
j
)
T
DF (X
j
))
−1
DF (X
j
)
T
F (X
j
)
4.20
To conclude the general process of the correction method the following steps
must be taken:
1. Ensure the problem is completely understood and the free variable
vector X, can be obtained at the best approach.
2. Secondly, an initial guess of the free variable vector X0 , can be
defined.
3. Constraint vector F(X), is being specified to satisfy the problem.
4. Create the Jacobian Matrix DF(X), with respect to the free variable
vector and the constraint vector.
5. The appropriate equation between Equations (4.17, 4.19 and 4.20)
must be selected and solved.
6. Finally, check if the error occurred in the constraint vector is
acceptable with respect to the one was set. If the error is larger than
the tolerance, then repeat the process from step 5. Otherwise the
method is converged.
This process of the differential corrections forms the basic step for any kind of
targeting algorithm. In the next section single shooting method is discussed.

68
4.2.1 Single shooting method
A single shooting method or otherwise a simple targeting problem, is a
fundamental correction process. By giving an initial guess of the initial conditions,
single shooting method aims to find a nearby solution to the reference trajectory
such that the final conditions (the constraint vector) would be satisfied. In any
kind of correction process, the iteration scheme developed in the previous section
is been used. The process of a single shooting method is a simple method with
a number of standards steps. The initial conditions that should be set at the very
beginning of the process are being propagated for a given time T, by the use of
the nonlinear equations of motion which has been discussed in previous section.
As the objective of any correction method is to satisfy a number of constraints, a
single shooting method aims to reach the desire final conditions. In this case, by
propagating the trajectory for a specific time T, the final conditions are being
obtained, without necessarily being the desired. If we let a target final conditions
exists, close enough to the final conditions that evaluated by propagating the
initial conditions, then the single shooting method is able to find a closer approach
to the target trajectory by modifying the initial conditions and/or the propagation
time. In many cases, different problems allow different variables to be free. In the
following process the initial positions are considering to be fixed and not subject
to change. Following the scheme of the differential correction, the free variables
vector is defined firstly and can be written as,
x(t0) =
[
x(t0)
y(t0)
z(t0)
ẋ(t0)
ẏ(t0)
ż(t0)
T ]
4.21
Or considering the fact that the initial position is not free to vary, Equation 4.21
can be rewritten as,

69
x(t0) = [
ẋ(t0)
ẏ(t0)
ż(t0)
T
]
4.22
Following the steps from the differential correction scheme, the constraint vector
must be obtained, considering that the trajectory must targets the desire final
conditions xd(t0 + T). Finally, including all the scalar constraint equations that
must be satisfied, the constraint vector can be formed as,
G(xd(t0 + T)) =
[
xd(t0 + T) − xd
yd(t0 + T) − yd
zd(t0 + T) − zd
ẋ 𝑑(t0 + T) − ẋd
ẏ 𝑑(t0 + T) − ẏd
ż 𝑑(t0 + T) − żd]
= 0
4.23
The propagated final conditions are represented as xd(t0 + T) (notice that these
final conditions are still not the desired). In the option that the final conditions to
be targeted are only the final positions, Equation 4.23 can be rewritten as,
G(xd(t0 + T)) = [
xd(t0 + T) − xd
yd(t0 + T) − yd
zd(t0 + T) − zd
] = 0
4.24
After obtaining the free variable vector and the constraint vector of the system,
the Jacobian Matrix is following with respect to these two matrices in equations
4.22 and 4.24. Jacobian matrix can be now be created and formed as,
DG(x(t0 + T)) =
[
∂x(t0+T)
∂ẋ(t0)
∂x(t0+T)
∂ẏ (t0)
∂x(t0+T)
∂ż(t0)
∂x(t0+T)
∂T
∂y(t0+T)
∂ẋ(t0)
∂y(t0+T)
∂ẏ (t0)
∂y(t0+T)
∂ż(t0)
∂y(t0+T)
∂T
∂z(t0+T)
∂ẋ(t0)
∂ ∂z(t0+T)
∂ẏ (t0)
∂z(t0+T)
∂ż(t0)
∂z(t0+T)
∂T ]
4.25
The Jacobian matrix has been already defined as a matrix that indicates the
sensitivity of the system. This definition leads to obtain that the three first columns
of the matrix in Equation 4.25 quantify the sensitivity of the final states with
respect to the initial velocities and the elements of the last column indicate the

70
sensitivity of propagation time on the final states and thus the derivatives of these
elements are elements of the STM.
4.3 Halo and Near Rectilinear Halo Orbit Algorithm
In the previous section, the technique to generate a periodic orbit was discussed.
In this section, the generation of Halo orbit and their specific properties will be
analysed. Halo orbit took its name from the characteristic shape of the trajectory
as seen from the Earth with respect to the Moon as in Figure 4.1. Robert Farquhar
first introduced the term Halo orbit for the need of communication for the Apollo
program. He proposed to use the Halo orbit near the L2 point in the Earth-Moon
system to provide continuously communication to the Apollo program [8].
Figure 4.1 Halo orbit seen from Earth.
Section 3 provides via the approximation method a quasi-periodic solution in the
CR3BP displayed in Figure 4.2. Hence, to generate a periodic solution such as a
halo periodic orbit, the differential corrector must be applied. Considering the
initial conditions provided, x0 = [x0 y0 z0 ux0 uy0 uz0]
T
, it can be observed

71
that the fact that the components y0, ux0 and uz0 are not zero, the algorithm is not
converged. To generate a periodic solution in the CR3BP, the initial conditions
must be in the form of x0 = [x0 0 z0 0 uy0 0]
T
. Hence, it is needed to
propagate the initial conditions of the quasi-periodic orbit until the point where the
conditions will be suitable to apply the differential corrector. So, the initial state
vector would be,
x0 = [x0 0 z0 0 uy0 0]
T 4.26
Figure 4.2 Quasi Periodic Orbit.
The process of the differential corrector requires firstly to define the free variable
vector and the constraints vector. In this design, three components must be
satisfied and be equal to zero at the beginning and the end of on period of the
orbit (y0 = ux0 = uz0 = 0). The differential corrector that is applied to this orbit
determination is the single shooting method. Usually, the single shooting method
is been applied with fixed integration time. Unfortunately, this method is working
only for a certain number of halo orbits and in some occasions the method

72
diverges. Therefore, the problem can be solved by considering the integration
time as a free variable. The free variable vector contains four components and
the system would provide infinite solutions. Therefore, one of the free variables
should be fixed to find a specific orbit. So, considering the x0 component to be
fixed, the free variable vector can be formed as,
X = [z0 uy0 T]
T
→ fixed x0
4.27
The constraint vector must include the final y position despite the fact that this
constraint will be fulfilled thanks to the stopping conditions that would be set.
However, it is necessary to be included so the method converges. Finally, the
constraint vector can be formed as,
F(X) = [y uxf uzf]T 4.28
The Jacobian matrix based on the free variable vector with respect to the
constraint vector must now account the partial derivatives with respect to
integration time T such that,
DG(X) =
[
∂y
∂x0
∂y
∂ẏ 0
∂y
∂T
∂ẋ
∂x0
∂ẋ
∂ẏ 0
∂ẋ
∂T
∂ż
∂x0
∂ż
∂ẏ 0
∂ż
∂T]
4.29
For the single shooting method the following steps must be followed,
1. The initial guess provided from the approximation method must be used
to generate the initial conditions for the process.
2. The free variable and constraint vector must be defined.
3. The Jacobian matrix must be determined considering the free variables
and constraint vector.
4. Generate the next initial conditions by using the appropriate equation
between Equations (4.17, 4.19 and 4.20).
5. The dynamic equations of the system must be propagated until the half of
the period of the orbit is completed.

73
6. After step 5, the error for the constraint vector must be defined. The error
is not standard and can be vary depending on the accuracy of the method.
If the error after step 5 is acceptable, then the method has converged and
a periodic orbit has been generated. If the error is not acceptable the
process starts again from step 4.
7. When the periodic orbit is finally be obtained, the mirror theorem can be
applied to generate the whole orbit.
The mirror theorem is a simple process to generate the other half or the periodic
orbit. Once, the differential corrector has been obtained the half of the orbit, due
to the stopping conditions that has been set, the following conditions must be
applied to determine the other half of the orbit,
x0 = [x0 − y0 z0 − ux0 uy0 − uz0]
T
Integration time T = −T
4.30
Figure 4.3 Halo Periodic Orbit.

“Near Rectilinear Halo Orbit in the vicinity of L1 Point in the Earth-Moon System”

“Near Rectilinear Halo Orbit in the vicinity of L1 Point in the Earth-Moon System”

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