APPLICATION OF KANE’S METHOD TO

INCORPORATE ATTITUDE DYNAMICS INTO THE

CIRCULAR RESTRICTED THREE-BODY PROBLEM

A Dissertation

Submitted to the Faculty

of

Purdue University

by

Amanda J. Knutson

In Partial Fulfillment of the

Requirements for the Degree

of

Doctor of Philosophy

December 2012

Purdue University

West Lafayette, Indiana

ii

For Mom, Dad, Ally, and Tim.

iii

ACKNOWLEDGMENTS

I would like to thank and acknowledge a number of individuals who have been

influential in the completion of this endeavor. First, I must recognize my family. My

parents, Lois and Timothy Knutson, have been a constant source of encouragement

and strength. Anything that I am able to accomplish in my lifetime is a direct result

of the time, effort, and energy they invested in raising me and, for that, I am forever

grateful. Without them, this work would not have been possible. I would like to thank

Timothy Phillips, as his support during my Ph.D. studies has been instrumental. His

positive attitude and outlook is inspirational. Thank you for keeping me grounded,

encouraging me to believe in myself, and for providing me with the confidence needed

to keep moving forward. My sister, Allysa Knutson, also deserves recognition for her

endless support, for being my best friend, and for always being able to make me laugh.

My wonderful extended family must also be thanked for all their encouragement and

support. I am so incredibly lucky to have such an amazing and loving family.

I owe much gratitude to my advisor, Professor Kathleen Howell, for her guidance

and contributions to this research. Professor Howell is a meticulous teacher, and I am

grateful for all of the tools, knowledge, and skills she has imparted on me during my

time at Purdue. I know everything she has taught me, both directly and indirectly,

will serve me well in the future. Thank you, additionally, for her careful review of this

dissertation. I would also like to express thanks to my committee members, Professor

Martin Corless, Professor James Longuski, and Professor Dengfeng Sun, for their

support of this work.

I truly appreciate the friendship and support of all my research group members. I

am especially grateful to Lucia Capdevila and Aurelie Heritier, for their close friend-

ship during my studies at Purdue. I would also like to acknowledge, Davide Guzzetti,

for the discussions, collaborations, and encouragement in regards to this research

iv

topic. Thank you to Geoff Wawrzyniak, for providing insight on solar sails, Tom

Pavlak, for his generous assistance with all things Linux related, and to Wayne Schlei

and Cody Short, for sharing their expertise in data visualization. Thanks also to

Natasha Bosanac, Loic Chappaz, Ashwati Das, Chris Geisel, Amanda Haapala, Galen

Harden, Masaki Kakoi, Bonnie Prado Pino, Raoul Rausch, Chris Spreen, Jeff Stu-

art, Mar Vaquero, and Ted Wahl. Additionally, I must acknowledge all my friends

back home from Thunder Bay and Queen’s University for their continual support,

especially Matt Fallen, Ellen Fitzsimons, Erin Hall, Tara Hassan, Kate Huntly, Jessie

Mundell-George, and Zoe Zimmerman.

Finally, I would like to recognize the following organizations for providing financial

support over the course of my Ph.D. studies: the Natural Science and Engineering

Research Council of Canada (NSERC), Zonta International, and the Canadian Space

Agency (CSA). I would specifically like to thank Dr. Alfred Ng from the CSA for his

support and guidance. My research program was also funded by the Purdue College

of Engineering through the Magoon Award, a Teaching Assistantship in School of

Aeronautics and Astronautics, and the Future Faculty Fellow (FFF) program in the

School of Engineering Education. I would especially like to thank Eric Holloway for

his continual support in the FFF program.

v

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Summary of Previous Contributions . . . . . . . . . . . . . . . . . . 11.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Development of a Model Framework . . . . . . . . . . . . . 41.2.2 Understanding the System Dynamics . . . . . . . . . . . . . 41.2.3 Application to Solar Sail Model . . . . . . . . . . . . . . . . 5

1.3 Contributions of This Work . . . . . . . . . . . . . . . . . . . . . . 51.4 Scope of the Current Investigation . . . . . . . . . . . . . . . . . . . 5

2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Kane’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Configuration Coordinates and Generalized Speeds . . . . . 102.1.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Momentum Principles . . . . . . . . . . . . . . . . . . . . . 112.1.4 Partial Velocity and Partial Angular Velocity Matrices . . . 122.1.5 Construction of Kane’s Equations . . . . . . . . . . . . . . . 13

2.2 Circular Restricted Three-Body Problem . . . . . . . . . . . . . . . 142.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . 142.2.2 Libration Points . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Linear Variational Equations . . . . . . . . . . . . . . . . . . 202.2.4 The State Transition Matrix . . . . . . . . . . . . . . . . . . 232.2.5 General Targeting Technique: Constraints and Free Variables 262.2.6 Targeting Periodic Orbits in the CR3BP . . . . . . . . . . . 26

2.3 Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.1 Rotational Kinematics . . . . . . . . . . . . . . . . . . . . . 312.3.2 Kinematic Differential Equations . . . . . . . . . . . . . . . 342.3.3 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 MODEL FRAMEWORK . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Spacecraft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Reference Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 41

vi

Page3.3.1 Generalized Speeds and Configuration Coordinates . . . . . 413.3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.3 Momentum Principles . . . . . . . . . . . . . . . . . . . . . 443.3.4 Partial Velocities and Partial Angular Velocities . . . . . . . 463.3.5 Constructing Kane’s Equations . . . . . . . . . . . . . . . . 48

3.4 Solution Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 MOTION ALONG PLANAR PERIODIC REFERENCES . . . . . . . . 504.1 Unperturbed L1 Asymmetric Rigid Body . . . . . . . . . . . . . . . 504.2 Unperturbed L3 Planar Lyapunov Orbits . . . . . . . . . . . . . . . 544.3 Changing Spacecraft Properties . . . . . . . . . . . . . . . . . . . . 554.4 Spin Stabilization of Axisymmetric Spacecraft . . . . . . . . . . . . 574.5 The Impact of a Perturbed Axisymmetric Spacecraft on the Orbit . 594.6 Perturbed Asymmetric Rigid Body . . . . . . . . . . . . . . . . . . 614.7 Summary of Results for L1 Lyapunov Family . . . . . . . . . . . . . 65

5 MOTION ALONG SPATIAL PERIODIC REFERENCES . . . . . . . . 715.1 Perturbed Asymmetric Rigid Body . . . . . . . . . . . . . . . . . . 715.2 Perturbed Axisymmetric Rigid Body . . . . . . . . . . . . . . . . . 755.3 Spin Stabilization of Axisymmetric Spacecraft . . . . . . . . . . . . 775.4 The Impact of a Perturbed Axisymmetric Spacecraft on the Orbit . 805.5 Summary of Results for L1 Northern Halo Family . . . . . . . . . . 82

6 APPLICATION TO SOLAR SAILS . . . . . . . . . . . . . . . . . . . . . 876.1 Reference Trajectories that Incorporate a Sail Force . . . . . . . . . 876.2 Modifications to Include Attitude and Develop a Coupled Model . . 906.3 Turn-and-Hold Model . . . . . . . . . . . . . . . . . . . . . . . . . . 906.4 Application of Sail Orientation Corrections to the Coupled Model . 966.5 Summary and Recommendations . . . . . . . . . . . . . . . . . . . 101

7 SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . 1027.1 Development of a Model Framework . . . . . . . . . . . . . . . . . . 1027.2 Understanding the System Dynamics . . . . . . . . . . . . . . . . . 1037.3 Application to Solar Sail Model . . . . . . . . . . . . . . . . . . . . 1037.4 Recommendations for Future Considerations . . . . . . . . . . . . . 104

7.4.1 Effects of Spacecraft Components . . . . . . . . . . . . . . . 1057.4.2 Orbit Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.4.3 Further Analysis of Attitude Maps . . . . . . . . . . . . . . 1057.4.4 Solar Sails and Other Applications . . . . . . . . . . . . . . 106

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

A Lyapunov Orbits: Unstable Eigenvalue/Time Constant . . . . . . . . . . 111

B Attitude Maps: L2 Lyapunov Family . . . . . . . . . . . . . . . . . . . . 112

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Page

C Attitude Maps: L2 Northern Halo Family . . . . . . . . . . . . . . . . . . 115

D Solar Sail: Turn-and-Hold Model Corrections . . . . . . . . . . . . . . . . 118

E Solar Sail: Coupled Model Position Offsets . . . . . . . . . . . . . . . . . 119

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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LIST OF TABLES

Table Page

D.1 Turn-and-Hold Model: Orientation Reconfigurations . . . . . . . . . . 118

E.1 Coupled Spinning Model: Without Orientation Reconfigurations . . . . 119

E.2 Coupled Spinning Model: With Orientation Reconfigurations . . . . . . 119

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LIST OF FIGURES

Figure Page

2.1 Circular Restricted Three-Body Problem . . . . . . . . . . . . . . . . . 15

2.2 Libration Point Locations in CR3BP . . . . . . . . . . . . . . . . . . . 19

2.3 Partial Family of L1 Planar Lyapunov Orbits in E-M System . . . . . . 29

2.4 L1 Planar Lyapunov Orbit that Bifurcates to L1 Halo Family in E-MSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Family of Northern Halo Orbits in the Vicinity of L1 in the E-M System 31

2.6 Gravity Force on an Infinitesimal Mass Element . . . . . . . . . . . . . 37

2.7 Gravitational Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1 Spacecraft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1 L1 Planar Lyapunov Reference Orbits in Earth-Moon System . . . . . 51

4.2 Orientation Response in the Planar L1 Lyapunov Orbits . . . . . . . . 52

4.3 Two Sets of Planar L1 Lyapunov Orbits . . . . . . . . . . . . . . . . . 52

4.4 Response in θ for Refined L1 Planar Lyapunov Orbit Sets . . . . . . . 53

4.5 L3 Planar Lyapunov Reference Orbits and Orientation Response in E-MSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.6 Orientation Response to Varied Spacecraft Inertia Properties . . . . . 56

4.7 Orientation Response to Spin Stabilization . . . . . . . . . . . . . . . 57

4.8 Motion of G∗ Relative to O, for ζ = 5 deg, over One Rev of Large E-MLyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.9 Motion of G∗ Relative to O, for ζ = 5 deg, over Two Revs of Large E-MLyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.10 Motion of G∗ Relative to O, for ζ = 5 deg, E-M L1 Lyapunov (Ay = 25469km) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.11 L2 Planar Lyapunov Orbit in the Sun-Earth (left) and Earth-Moon (right)Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.12 Orientation Response of Asymmetric Rigid Body . . . . . . . . . . . . 63

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Figure Page

4.13 Orientation Response for Ay = 125 km (left) and Ay = 13400 km (right)for Nonlinear E-M Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . 65

4.14 Orientation response in θ for L1 Lyapunov Family (E-M System) . . . 66

4.15 Set of L1 Planar Lyapunov Orbits (E-M System) in Blue Band Region 68

4.16 Response in θ in Blue Band Region (k3=0.5), 1 rev (left) 2.5 revs (right)) 68

4.17 Orientation Response to Small Perturbation Sets for L1 Planar LyapunovFamily (E-M System) . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.18 Orientation Response to Medium Perturbation Sets for L1 Planar Lya-punov Family (E-M System) . . . . . . . . . . . . . . . . . . . . . . . 70

5.1 Small Set of L1 Northern Halo Reference Orbits in Earth-Moon System 72

5.2 Asymmetric Orientation Response for Small L1 Northern Halo ReferenceOrbits in the E-M System . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 Set of Larger L1 Northern Halo Reference Orbits in Earth-Moon System 74

5.4 Orientation Response for an Asymmetric Vehicle in Large L1 NorthernHalo Reference Orbits in the E-M System . . . . . . . . . . . . . . . . 75

5.5 Four E-M Halo Reference Trajectories . . . . . . . . . . . . . . . . . . 76

5.6 Orientation Response to Perturbation ζ = 5 degrees in E-M Halos . . . 77

5.7 Orientation Response to Spin Stabilization in the E-M System . . . . . 78

5.8 Orbital Motion Response to Spin Stabilization in the E-M System . . . 79

5.9 Orbital Response to perturbation of ζ = 5 degrees in E-M Halos . . . . 81

5.10 Motion of G∗ Relative to O, for ζ = 5 deg, over Two Revs of E-M Halo 81

5.11 Orientation Response in L1 Northern Halo Family (E-M System) . . . 83

5.12 Orientation Response to Small Perturbation Sets for L1 Northern HaloFamily (E-M System) . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.13 Orientation Response to Medium Perturbation Sets for L1 Northern HaloFamily (E-M System) . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.1 Family of Solar Sail Reference Orbits in the Vicinity of L2 in the E-MSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Initial Condition for Multiple Shooting Algorithm . . . . . . . . . . . . 92

6.3 Corrections Determined Using Turn-and-Hold Model . . . . . . . . . . 92

6.4 Corrections Applied to Coupled Orbit and Attitude Model . . . . . . . 93

xi

Figure Page

6.5 Coupled Orbit and Attitude Response for Solar Sail in the Vicinity of L2

in the E-M System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.6 Multiple Shooting Initial Conditions . . . . . . . . . . . . . . . . . . . 99

6.7 Corrected Trajectory Using Turn-and-Hold Strategy; Decoupled Model 100

6.8 Orientation Corrections Applied to the Coupled Orbit and Attitude Model 101

A.1 Unstable Eigenvalue for L1 Planar Lyapunov Family (E-M System) . . 111

A.2 Time Constant for L1 Planar Lyapunov Family (E-M System) . . . . . 111

B.1 Orientation Response in L2 Planar Lyapunov Family (E-M System) . . 112

B.2 Orientation Response to Small Perturbation Sets for L2 Planar LyapunovFamily (E-M System) . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B.3 Orientation Response to Medium Perturbation Sets for L2 Planar Lya-punov Family (E-M System) . . . . . . . . . . . . . . . . . . . . . . . 114

C.1 Orientation Response in L2 Northern Halo Family (E-M System) . . . 115

C.2 Orientation Response to Small Perturbation Sets for L2 Northern HaloFamily (E-M System) . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

C.3 Orientation Response to Medium Perturbation Sets for L2 Northern HaloFamily (E-M System) . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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ABSTRACT

Knutson, Amanda J. Ph.D., Purdue University, December 2012. Application ofKane’s Method to Incorporate Attitude Dynamics into the Circular RestrictedThree-Body Problem. Major Professor: Kathleen C. Howell.

The three-dimensional, nonlinear, coupled orbit and attitude motion of a space-

craft in both planar and spatial reference orbits is explored in this investigation, within

the context of the Circular Restricted Three-Body Problem (CR3BP). The motion of

a multibody spacecraft, comprised of two rigid bodies connected by a single degree of

freedom joint, under the gravitational influence of the Earth and the Moon is exam-

ined. Kane’s dynamical formulation is employed and a model framework to develop

the fully coupled orbit and attitude differential equations that govern spacecraft mo-

tion in the CR3BP is realized. The nonlinear variational form of the equations of

motion is employed to mitigate numerical effects arising from large discrepancies in

the length scales. Within this framework, a process is summarized with steps for

implementation, resulting in a series of coupled nonlinear equations of motion for the

complex spacecraft model.

Several nonlinear, periodic reference orbits in the vicinity of the collinear libra-

tion points are selected for further investigation and the effects of the orbit on the

orientation, as well as the orientation on the orbit, are examined. Attitude maps are

employed to summarize the results; the inertia properties of the spacecraft are varied

across a periodic family of reference orbits in the vicinity of the collinear libration

points. These maps indicate regions, in terms of the orbit size and inertia properties,

where the spacecraft maintains the initial orientation with respect to the circular re-

stricted three-body problem rotating frame. Reference orbits for the system include

both planar Lyapunov orbits and three-dimensional halo trajectories. Additionally,

xiii

the model is extended to include solar radiation pressure and the motion of a solar

sail type spacecraft is examined.

1

1. INTRODUCTION

The Circular Restricted Three-Body Problem (CR3BP) has been examined exten-

sively in recent years; however, considerably fewer studies incorporate the effects of

spacecraft attitude in this regime. Several missions in the vicinity of the collinear

libration points have been completed, are ongoing, or are planned for the near fu-

ture. [1–3] Specifically, one example is the James Webb Space Telescope (JWST),

successor to the Hubble Telescope, that is to be deployed in a Sun-Earth L2 Lissajous

orbit within the next decade. [4] Due to the sensitive nature of the dynamics at the

libration points, it is desired to have a better understanding of the attitude motion of

spacecraft in orbit in the vicinity of these points. The ability to predict the attitude

motion of spacecraft as orbital parameters and spacecraft characteristics are varied

can be very useful. Once the behavior is investigated, the possibility of exploiting

the natural dynamics for passive attitude control in these regimes is also an option.

Furthermore, it is desired to examine the coupling effects between the orbital motion

and the orientation in such orbits to gain a better understanding of the system dy-

namics. Coupling effects are important when dealing with large spacecraft and could

prove useful in the application of solar sails. These issues combine to motivate the

development of a novel method to incorporate the effects of attitude dynamics into

the CR3BP.

1.1 Summary of Previous Contributions

While the study of attitude dynamics is well established for near Earth applica-

tions in the two-body problem, comparatively little work has been completed in the

CR3BP. Kane and Marsh first consider the attitude stability of a symmetric satellite,

located exactly at the equilibrium points. [5] In their analysis, the satellite is artifi-

2

cially maintained at the equilibrium points and only the attitude motion is considered.

Robinson continues this work with an examination of the attitude motion of both a

dumbbell satellite [6] and an asymmetric rigid body [7] artificially held at the equi-

librium points. Abad et al. introduce the use of Euler parameters in the study of a

single rigid body located at L4. [8] More recently, Brucker et al. explore the dynamics

and stability of a single rigid body spacecraft in the vicinity of the collinear points in

the Sun-Earth system, using Poincare maps, where the effects of the gravity gradient

torque are investigated, but the spacecraft is still artificially fixed to the equilibrium

point. [9] Wong et al. examine the motion of a single rigid body near the vicinity of

the Sun-Earth libration points in detail, using linearized Lyapunov and halo orbit ap-

proximations. [10] The work by Wong et al. details the response of the spacecraft in a

meaningful manner, offering orientation results for a number of numerical simulations

in terms of a body 3-2-1 (θ, φ, ψ) Euler angle rotation sequence. However, since the

Lyapunov and halo orbits are represented as linear approximations, the simulations

are only valid for relatively small orbits close to the equilibrium points. The behavior

of multibody spacecraft in the nonlinear spatial CR3BP has not yet been explored.

Additionally, none of the previous investigations consider the orbit and the attitude

coupling effects in the spatial CR3BP.

The motivation for the current effort is a more extensive model to explore the

orbit-attitude coupling as well as the dynamical influence on the attitude motion in a

nonlinear regime. Guzzetti examines the coupled motion of a spacecraft in nonlinear

Lyapunov reference orbits, considering the out-of-plane attitude motion. [11] Guzzetti

formulates the equations of motion using a Lagrangian approach and also incorporates

the effects of solar radiation pressure and flexible bodies; the out-of-plane attitude

response is compared across several test cases to those generated from the model

proposed by Knutson and Howell. [12] The results indicate good correlation between

the attitude response using the two models.

Another goal in the current investigation is the assessment of the departure from

the vicinity of the reference orbit that is predicted by the coupled orbit and atti-

3

tude model and the potential to employ this predicted deviation to determine the

required attitude configuration to control the orbital motion of the sail such that it

remains close to the reference. Due to continuously available solar radiation pressure

(SRP), solar sail type spacecraft are capable of a variety of periodic reference orbits,

that cannot be indefinitely maintained by traditional propulsion methods. [13] Nu-

merous authors have investigated periodic orbits near the libration points. [14–17]

First, a family of solar sail orbits in the vicinity of the L2 libration point in the

Earth-Moon system is computed using a single shooting algorithm as described by

Wawrzyniak. [18–21] The method is an extension of the vertically shifted linear ap-

proximation for elliptical orbits near the libration points originally introduced by

Simo and McInnes. [22–24] Each member in the family of orbits maintains a fixed

sail orientation with respect to the Sun frame in the Earth-Moon system. A multiple

shooting scheme is then employed by Wawrzyniak to determine the required correc-

tions to the attitude motion based on an offset in various orbital quantities from

the desired reference at some instant in time. A turn-and-hold strategy, similar to

one proposed by Wawrzyniak [21, 25], is used to determine attitude corrections. In

the turn-and-hold strategy, the spacecraft is assumed to maintain a fixed orientation

over each multiple shooting segment. Finally, the attitude corrections are applied to

the coupled orbit and attitude model, with a goal for the configuration to remain in

the vicinity of the reference orbit. [26] The corrections are computed iteratively, as

required, based on the departure of the state from the reference orbit.

One further application to motivate the development of a coupled orbit and at-

titude model framework is the area of space tethers. Many investigations have been

performed on this topic. [27–30] There is continued interest in the development of

models that incorporate space tether dynamics into the circular restricted three-body

problem. [31–33] The model framework developed in this investigation may be mod-

ified to include a space tether application in future work. The methods employed

are adaptable to include the motions of flexible bodies, an important aspect of space

tether research.

4

1.2 Research Objectives

The main goal in this preliminary investigation is an improved understanding

of the motion of a spacecraft, one comprised of multiple rigid bodies, under the

simultaneous influence of two point mass gravitational fields. The structure of this

analysis is focused on three main objectives.

1.2.1 Development of a Model Framework

Since few previous investigations have explored this topic, the first objective is

the development of a model framework to fully couple the orbital mechanics and the

attitude dynamics within the circular restricted three-body regime. Selection of a

dynamical formulation is an important first step in this process. The equations of

motion governing the orbit as well as the spacecraft attitude are brought together in

a cohesive manner that minimizes computational difficulties and allows for the prop-

agation of meaningful results. One of the major computational issues in this process

is the management of the disparity in length scales in this problem. The framework

is fully developed and is presented in detail in this document. The procedure supplies

clear steps for implementation and results in a series of coupled nonlinear equations

of motion for the complex spacecraft model.

1.2.2 Understanding the System Dynamics

The second objective is the use of the model to increase understanding of the

underlying system dynamics. Topics of interest facilitate an ordered investigation.

These areas are explored through numerical simulation studies for a variety of test

cases. The results of this analysis yield a greater understanding of the design space.

The focus of the analysis includes: the effects of orbit type, size and location on

attitude, the effects of spacecraft properties on attitude, the effect of attitude on the

orbital motion, and spin stabilization.

5

1.2.3 Application to Solar Sail Model

The final objective is to apply the model framework that has been developed to a

solar sail type application, that is, to investigate the potential departure of the vehicle

from the vicinity of the reference orbit as predicted by the coupled orbit and attitude

model. The attitude configuration may be potentially useful in a control scheme to

maintain the orbital motion of the sail to remain close to the reference.

1.3 Contributions of This Work

There are several key contributions of this investigation. First, this analysis ex-

pands the understanding of attitude dynamics within the context of the CR3BP.

Previously, the only investigation considering the attitude dynamics of a spacecraft

as it moves along a periodic orbit about a libration point in the CR3BP is that of

Wong et al., which employs linear orbit approximations and, as such, is valid only

for relatively small periodic orbits. In contrast, large nonlinear periodic orbits are

investigated here. Secondly, the model framework that is developed in this investi-

gation yields the fully coupled equations of motion that govern the orbital behavior

as well as the spacecraft orientation. No results from the literature are known to be

available that offer fully coupled results, for a three-dimensional spacecraft system in

the spatial CR3BP.

1.4 Scope of the Current Investigation

The current investigation concerns the motion of a spacecraft comprised of two

rigid bodies, connected by a joint that offers a single rotational degree of freedom,

within the context of the CR3BP. This formulation possesses some limitations as a

consequence of the assumptions; however, it is selected because it also offers several

locations where the rotating frame dynamics and the gravitational forces from the

attracting bodies effectively cancel. These areas are of interest for this investigation

6

because coupling effects that exist between the orbit and attitude are most visible in

the vicinity of these equilibrium points. The motion of the spacecraft system with

respect to Lyapunov and halo reference orbits are examined. The effects of the orbit

on the attitude and the impact of the attitude on the orbit are more closely examined

and the results are presented in the form of representative test cases. Additionally,

a solar sail family of orbits is incorporated to explore the application to solar sails.

The spacecraft system selected is sufficiently complex to offer some of the challenges

associated with an actual spacecraft system, while still allowing the model framework

to be established. The investigation is outlined as follows,

Chapter 2

Kane’s dynamical approach and methodology is introduced to formulate the equa-

tions of motion for a system. Relevant background information related to the circular

restricted three-body problem and attitude dynamics is incorporated to facilitate the

later development of the nonlinear variational coupled orbit and attitude equations

of motion for the spacecraft system. Notation standards are also established in this

chapter and are used throughout the rest of the analysis.

Chapter 3

The model framework to incorporate attitude dynamics into the CR3BP is pre-

sented in detail. All variables required to formulate the nonlinear variational equa-

tions of motion are discussed. Issues surrounding the computational difficulties that

arise naturally in this problem are presented along with the solution that is incorpo-

rated to overcome the problem.

7

Chapter 4

Analysis of the spatial motion of the spacecraft system is accomplished in various

planar periodic Lyapunov reference orbits. The effects of the orbit on the attitude

and the attitude on the orbit, as well as spin stabilization, are explored for a variety

of vehicle parameters and these are presented in the form of representative test cases.

Results reflecting the maximum orientation response are presented for a number of

representative orbits and inertia properties, and are generalized to create a tool in

the form of a map. These maps relate the maximum Lyapunov orbit amplitude, Ay,

and the inertia parameters to the maximum response observed for each angle in a

body-fixed Euler angle rotation sequence relative to the rotating CR3BP frame over

one revolution of the reference orbit.

Chapter 5

In this chapter, the spatial motion of the spacecraft system in three-dimensional

periodic halo references orbits is examined. Test cases similar to those examined

in Chapter 4 are presented. Again, the maximum orientation response, represented

by Euler angles relative to the rotating CR3BP frame, are generalized for a number

of representative orbits and inertia properties, in the form of attitude maps. These

maps relate the maximum out of plane excursion, Az and inertia parameters to the

maximum response observed for each angle in a body fixed Euler angle rotation

sequence relative to the rotating CR3BP frame over one revolution of the reference

orbit.

Chapter 6

The model framework is extended to include solar radiation pressure and the

motion of a solar sail type spacecraft is examined. An assessment of the departure

from the vicinity of the reference orbit as predicted by the coupled orbit and attitude

8

model, is summarized. The potential to employ this deviation to determine the

required attitude configuration to control the orbital motion of the sail such that it

remains close to the reference is examined.

Chapter 7

Concluding remarks and considerations for future work are included here.

9

2. BACKGROUND

The formulation of Kane’s equations is initially explored to model the problem. Sev-

eral key features of Kane’s method, such as generalized speeds and partial velocities

are defined in detail. Then, to establish the background necessary to formulate the

coupled orbit and attitude equations of motion in the dynamical regime represented

by the Circular Restricted Three-Body Problem (CR3BP), background related to the

CR3BP is introduced, followed by specifics related to attitude dynamics as applicable

to this investigation.

2.1 Kane’s Method

Kane’s approach to derive the equations of motion for a system is employed for

several specific reasons. Kane’s method is well-suited for deriving equations of motion

for systems of multiple rigid bodies moving under the influence of several gravitational

fields, such as spacecraft, that are composed of many individual bodies connected by

joints in a chain or tree-like structure. [34–40] Similar to the Lagrangian method [41],

assuming workless constraints, the contact forces are automatically eliminated. The

elimination of contact forces offers a great advantage over the Newtonian formulation,

which assumes six degrees of freedom for each body present. No rearrangement of the

derived equations of motion to achieve a form suitable for computer implementation is

necessary. Lagrange’s formulation requires the computation of the partial derivatives

of the Lagrangian, a function of the total kinetic and potential energy, with respect

to each configuration coordinate. Implementation of the Lagrangian method can be

very cumbersome when considering systems with many degrees of freedom. Kane’s

method employs simple matrix multiplication methods to derive the equations of

motion. Once the kinematics of the first body in the system are computed, the

10

kinematics for the remaining bodies are easily extrapolated. Kane’s method is based

on the principles of linear and angular momentum. The method partitions the general

formulas for the time rate of change of linear and angular momentum into separate

terms in the directions of a set of specified generalized speeds. [42,43]

2.1.1 Configuration Coordinates and Generalized Speeds

As an alternative to basing the analysis on position variables (configuration coor-

dinates), Kane’s method uses velocity variables identified as generalized speeds. The

generalized speeds, denoted ui, are selected in any manner, but it is usually desirable

to select an independent set such that changes to one speed do not affect any other

speed. If minimal variables are not assigned, constraint equations are incorporated

to reduce the order of the system. If minimal variables are selected, one generalized

speed is required for each degree of freedom. The total number of generalized speeds is

denoted n∗. Configuration coordinates may be required to fully describe the system.

The derivatives of the configuration coordinates are formulated by some combination

of a subset of the generalized speeds.

2.1.2 Kinematics

The kinematics of the system must be formulated before Kane’s equations are ap-

plied. A basic summary of the kinematics follows to facilitate an ordered presentation

of Kane’s method and to clearly define the required notation. First, the position of

the center of mass of body B, denoted rB∗, is defined relative to a fixed point. The

frame n represents the coordinate system fixed in the inertial reference frame. The

absolute velocity of the center of mass of body B is then determined by differentiating

the position vector as follows,

vn B∗=

dn

dtrB

∗

vn B∗=db

dtrB

∗+ ωn b × rB∗

(2.1)

11

The absolute angular velocity of some coordinate system b is defined as ωn b, while

db rB∗/dt is the rate of change of magnitude of the position vector with respect to

time as observed from frame b. Coordinate system b is, in fact, fixed in body B. This

process is repeated to determine the absolute acceleration of the center of mass of

body B. Once the kinematics corresponding to the center of mass of the first body in

the chain is available, the kinematical expression for the next body can be represented

by considering the motion relative to the first body. The bodies in the system are

connected by some type of mechanical joint that restricts motion, usually to just one

or two degrees of freedom, which renders these relative velocities and accelerations

easy to formulate.

2.1.3 Momentum Principles

The development of Kane’s equations of motion relies on the principles of linear

and angular momentum. The following notation is used throughout this investigation.

The rate of change of linear momentum, for any body, B, in a system is written,

∑FB =

dn

dt

(mb v

n B∗)

The sum of all the external forces on body B is then equal to the resultant∑FB. The

mass for any body, B, is defined by the scalar quantity mb. A single overbar denotes

a vector. A similar expression yields the rate of change of angular momentum,

∑MB/B∗

=dn

dt

(IB/B∗

· ωn b

)

The sum of all external moments on body B about the center of mass, B∗, is rep-

resented by the resultant sum∑MB/B∗

. The double overbar represents a dyadic.

Then, IB/B∗

is the inertia dyadic relative to B∗ for any three-dimensional body. The

components of the inertia dyadic are frequently collected in a matrix as,

12

IB/B∗

=

Ib1b1 Ib1b2 Ib1b3

Ib2b1 Ib2b2 Ib2b3

Ib3b1 Ib3b2 Ib3b3

where the vector basis is identified prior to computation. The mass moments of inertia

of body B, about the center of mass, B∗, are described by Ib1b1 , Ib2b2 , and Ib3b3 , and

the remaining terms are products of inertia. The equations for the time rate of change

of linear and angular momentum are rearranged so that all terms appear on the left

side of the equation, that is,

∑FB − dn

dt

(mb v

n B∗)= 0 (2.2)

∑MB/B∗ − dn

dt

(IB/B∗

· ωn b

)= 0 (2.3)

Equations (2.2) and (2.3) are then used to formulate Kane’s equations of motion.

2.1.4 Partial Velocity and Partial Angular Velocity Matrices

The partial velocity matrix is formed for body B from the derivative of the velocity

vector vn B∗with respect to each generalized speed, ui, for i = 1 to n∗. [42] This

partial velocity matrix, represented by ∂ vn B∗/∂u, determines the portion of the vector

reflecting the time rate of change of linear momentum that is aligned with the ui

direction, in each of the three spatial dimensions of the b-frame. The matrix has n∗

rows (one for each ui), and three columns (one for each spatial dimension of the b-

frame). Similarly, the partial angular velocity matrix is computed from the derivative

of the angular velocity vector, ωn b, with respect to each generalized speed, ui, for i

= 1 to n∗. This quantity indicates the precise portion of the vector that is computed

as the time rate of change of angular momentum is to be aligned in the ui direction;

appropriate terms appear in the matrix ∂ ωn b/∂u. Similarly, this matrix also possesses

n∗ rows and three columns.

13

2.1.5 Construction of Kane’s Equations

The partial velocity vectors and the momentum equations are combined to form

Kane’s equations of motion. [42] The product of the partial velocity matrix and the

time rate of change of linear momentum in Equation (2.2) as well as the product of

the partial angular velocity matrix and the time rate of change of angular momentum

in Equation (2.3) are determined for each body in the system. A coordinate frame

fixed in body B is selected as the common vector basis, and the partial velocity and

partial angular velocity matrices and Equations (2.2) and (2.3) are all represented in

terms of this frame. For body B, the equations are summarized,

∂ vn B∗

∂u

∑FB − ∂ vn B∗

∂u

dn

dt

(mb v

n B∗)= 0

∂ ωn b

∂u

∑MB/B∗ − ∂ ωn b

∂u

dn

dt

(IB/B∗

· ωn b

)= 0

These two vector equations for body B are now summed to create one vector equation.

Kane splits this equation into two sections, generalized inertia forces and generalized

active forces. [42] Generalized inertia forces, denoted Q∗, combine the contribution

of the quantities that depend on the time rate of change of both linear and angular

momentum, and the generalized active forces, denoted Q, collect the contributions

from external loads. The generalized inertia force and the generalized active force for

body B are described as follows,

Q∗B = −∂ vn B∗

∂u

dn

dt

(mb v

n B∗)− ∂ ωn b

∂u

dn

dt

(IB/B∗

· ωn b

)(2.4)

QB =∂ vn B∗

∂u

∑FB +

∂ ωn b

∂u

∑MB/B∗

(2.5)

To apply Kane’s method to a multibody system, the contributions from each body

must be added together to obtain a single equation. [42] A coordinate frame fixed

in body B remains as the common vector basis, and all contributions are expressed

14

in terms of this frame. Kane’s formulation for a system composed of m bodies is

represented by the following expression,

m∑

B=1

Q∗B +

m∑

B=1

QB = 0 (2.6)

Now that Kane’s method for deriving the governing equations of motion has been

summarized, it can be applied to the spacecraft system in the CR3BP.

2.2 Circular Restricted Three-Body Problem

For the purpose of this investigation, the reference trajectory is a path that reflects

the motion of a point mass spacecraft propagated in the CR3BP, where the spacecraft

is moving under the gravitational influence of two primary bodies modeled as point

masses. This regime has been selected specifically because there are locations in the

rotating system where the gravitational forces and rotating frame dynamic accelera-

tions cancel or the contributions from the combination of these two terms is small,

allowing the observation of coupling forces that exist between the orbital mechanics

and attitude dynamics.

2.2.1 Equations of Motion

The CR3BP models the motion of a small third body in the presence of the

gravitational influence of two larger primary bodies. There are several underlying

assumptions that are significant in this model. [44] Since the particle is assumed to

be of infinitesimal mass, the primary orbits are modeled as Keplerian. It is assumed

that the spacecraft has sufficiently small mass that it does not affect the motion of

either of the primary bodies. The two primary bodies then move in circular orbits

about their common barycenter at a constant rate Ω. A rotating coordinate system

is assigned, denoted (a1, a2, a3), as depicted in Figure 2.1. The angular velocity of

the a-frame relative to an inertial frame, n, is denoted ωn a and is equal to Ωa3. In

15

n1

a1

a2

n2

P1

P2

P3

D R

P

D1

D2

Figure 2.1. Circular Restricted Three-Body Problem

dimensional form, the position vector from the barycenter to the spacecraft center of

mass is defined by P = Xa1 +Y a2 +Za3. The dimensional equations of motion with

respect to the system barycenter are described by,

m3dn 2

dt2P =

−Gm1m3

D3D − Gm2m3

R3R (2.7)

where G represents the universal gravitational constant, the position from the center

of mass of the first primary to the spacecraft is represented by D = P − D1, and the

position from the center of mass of the second primary to the spacecraft is represented

by R = P − D2, with m1 as the mass of primary P1, m2 as the mass of primary P2,

and m3 as the mass of the spacecraft.

For convenience, a set of characteristic quantities are defined to assist in nondi-

mensionalizating Equation (2.7). The characteristic length, l∗ is the sum of D1 and

D2. The characteristic mass, m∗, is defined as the sum of m1 and m2. The charac-

teristic time is represented by,

t∗ =

[(D1 +D2)3

G (m1 +m2)

]1/2

=

[l∗3

Gm∗

]1/2

(2.8)

16

The barycenter of the primary system is located by considering the mass ratio of the

two primaries and the total distance between P1 and P2. In dimensional form, the

locations of the primaries with respect to the barycenter are given by

D1 = − m2l∗

m1 +m2

a1 D2 =m1l

∗

m1 +m2

a1 (2.9)

The introduction of the characteristic quantities yields a new nondimensional form of

the position vector from the barycenter to the spacecraft center of mass,

p =P

l∗= xa1 + ya2 + za3 (2.10)

The position vectors from the barycenter to the centers of mass of P1 and P2 are also

written in nondimensional form,

d1 =D1

l∗= −µa1 d2 =

D2

l∗= (1− µ) a1 (2.11)

respectively. Similarly, the position vectors from the centers of mass of P1 and P2 to

the spacecraft center of mass as nondimensional quantities are,

d =D

l∗r =

R

l∗(2.12)

respectively. A nondimensional time quantity is noted to be

τ =t

t∗(2.13)

The mass parameter that identifies a specific system, µ, is defined as the ratio of

the mass of the smaller primary body to the total mass of the two primaries. Since

m1 = m∗ −m2, the nondimensional expressions for m1 and m2 are,

1− µ =m1

m∗ µ =m2

m∗ (2.14)

Replacing the actual quantities with the product of the nondimensional terms and

the characteristic values, Equation (2.7) is rewritten,

dn 2

d (t∗τ)2 l∗p =

−Gm∗(1− µ)

(l∗d)3

(l∗d)− Gm∗µ

(l∗r)3 (l∗r) (2.15)

17

Rearranging Equation (2.15) such that the characteristic quantities are grouped to-

gether yields,l∗

t∗2

dn 2

d (τ)2 p =

(Gm∗

l∗3

)l∗(−(1− µ)d

d3− µr

r3

)(2.16)

where it is noted that the first term on the right side is simply the inverse of the

square of the characteristic time term. Thus, the characteristic quantities on both

sides cancel and the nondimensional equation simplifies to,

¨p = −(1− µ)d

d3− µr

r3(2.17)

where,

d = (x− ((1− µ)) a1 + ya2 + za3

d =

√(x− (1− µ))1/2 + y1/2 + z1/2

r = (x− µ) a1 + ya2 + za3

r =

√(x− µ)1/2 + y1/2 + z1/2

(2.18)

The resulting second order scalar nondimensional equations of motion are,

x− 2y − x =− (1− µ)(x+ µ)

d3− µ(x− 1 + µ)

r3

y + 2x− y =− (1− µ)y

d3− µy

r3

z =− (1− µ)z

d3− µz

r3

(2.19)

These equations of motion can be numerically integrated to obtain a time history of

the motion of the spacecraft center of mass, under the gravitational influence of the

two larger primary bodies. Since the system is conservative, [45] the following pseudo

potential function is defined, and the equations of motion can be further simplified,

that is,

U∗ =1− µd

+µ

r+

1

2

(x2 + y2

)(2.20)

18

x− 2y =∂U∗

∂x

y + 2x =∂U∗

∂y

z =∂U∗

∂z

(2.21)

An integral of the motion is realized by integrating the sum of the dot product of the

velocity of the spacecraft, P3, with respect to the rotating frame and the equations

of motion described in Equation (2.21). The velocity of the spacecraft with respect

the the a-frame is defined,

va P3 = xa1 + ya2 + za3 (2.22)

The dot product between Equation (2.22) and the vector form of Equation (2.21)

yields, in component form,

xx− 2xy = x∂U∗

∂x

yy + 2yx = y∂U∗

∂y

zz = z∂U∗

∂z

(2.23)

The addition of these components produces the following expression,

xx+ yy + zz = x∂U∗

∂x+ y

∂U∗

∂y+ z

∂U∗

∂z(2.24)

where,

x∂U∗

∂x+ y

∂U∗

∂y+ z

∂U∗

∂z=∂x

∂τ

∂U∗

∂x+∂y

∂τ

∂U∗

∂y+∂z

∂τ

∂U∗

∂z=∂U∗

∂τ(2.25)

Integration of Equation (2.24) and multiplication of all terms by a factor of 2 yields

the expression for the Jacobi Constant, JC.

x2 + y2 + z2 = 2U∗ − JC (2.26)

This integral of the motion is employed in the numerical integration strategies to

ensure accuracy of the propagated solution.

19

2.2.2 Libration Points

The CR3BP allows five equilibrium solutions as points relative to the rotating sys-

tem where the gravitational forces and the rotating frame dynamics cancel, assuming

a vehicle modeled as a point mass. Figure 2.2 illustrates the location of these points.

The three collinear points L1, L2, and L3, located along the line connecting the two

primaries, are of particular interest in this investigation. Two additional solutions are

the triangular or equilateral points, L4 and L5. The exact equilibrium point positions

are determined by considering the locations where the velocity and acceleration of the

spacecraft relative to the rotating frame is equal to zero. If this condition is satisfied,

then Equation (2.21) collapses to,

a1

a2

P2

P1

L2L1

L5

L4

L3

Figure 2.2. Libration Point Locations in CR3BP

∂U∗

∂x=∂U∗

∂y=∂U∗

∂z= 0 (2.27)

By considering the partial derivative of the pseudo potential, U∗, with respect to x,

y, and z, and equating these expressions to zero, the locations of the libration points

are determined. As an example, for the libration point L1,

20

∂U∗

∂xL1

=− (1− µ)(xL1 + µ)

d3− −µ(xL1 − 1 + µ)

r3+ x = 0

∂U∗

∂yL1

=− (1− µ)yL1

d3− −µyL1

r3+ y = 0

∂U∗

∂zL1

=− (1− µ)zL1

d3− −µzL1

r3= 0

(2.28)

The L1 libration point has yL1 = zL1 = 0. If xL1 is assumed to be xL1 = 1− µ− γ1,

then Equation (2.28) simplifies to,

1− µ− γ1 = − (1− µ)

(1− γ1)2− −µ

(γ1)2(2.29)

This equation is solved iteratively by employing a Newton method. The location of

the collinear libration points, in the Earth-Moon system, in nondimensional form are

xL1 = 0.836915 (nd), xL2 = 1.155682 (nd), and xL3 = −1.005062 (nd). Both two

and three-dimensional periodic orbits exist in the vicinity of these points. Numerical

techniques to produce these orbits are relatively straightforward to implement.

2.2.3 Linear Variational Equations

The linear variational equations are developed for various purposes including tar-

geting strategies for the computation of families of periodic orbits. [44–46] The vari-

ational equations are developed for L1, for example, by examining the expressions in

Equation (2.21) under perturbations,

x = xL1 + ξ

y = yL1 + ν

z = zL1 + χ

(2.30)

where xL1 , yL1 , zL1 locate the equilibrium solution, and ξ, ν, and χ are variations

relative to L1. The first and second derivatives of Equation (2.30) are evaluated with

respect to nondimensionalized time. The right side of Equation (2.21) is simplified

by using a Taylor series expansion about the equilibrium solution, neglecting all

21

higher-order nonlinear terms. The following linear variational equations with constant

coefficients are realized, i.e.,

ξ − 2ν =U∗xxξ + U∗

xyν + U∗xzχ

ν + 2ξ =U∗yxξ + U∗

yyν + U∗yzχ

χ =U∗zxξ + U∗

zyν + U∗zzχ

(2.31)

where U∗ij are the second-order partials of U∗ with respect to i, j, evaluated at L1.

Similar expressions can be formulated for the other libration points. All of the libra-

tion points are noted to be in the plane of the motion of the primary bodies, thus,

zLi= 0, and the expressions in Equation (2.31) simplify,

ξ − 2ν =U∗xxξ + U∗

xyν

ν + 2ξ =U∗yxξ + U∗

yyν

χ =U∗zzχ

(2.32)

A first-order form is realized by defining a six-element state vector,

δxL =

ξ

ν

χ

ξ

ν

χ

(2.33)

and Equation (2.32) is then written,

δ ˙xL = AδxL A =

0 I

U∗XX Ω

(2.34)

where,

22

U∗XX =

U∗xx U∗

xy U∗xz

U∗yx U∗

yy U∗yz

U∗zx U∗

zy U∗zz

Ω =

0 2 0

−2 0 0

0 0 0

The linear in-plane and out-of-plane motion are noted to decouple. Considering in-

plane motion, Equation (2.32) is expressed,

ξ − 2ν =U∗xxξ + U∗

xyν

ν + 2ξ =U∗yxξ + U∗

yyν(2.35)

which has a general solution of the form,

ξ =4∑

i=1

Aieλiτ

ν =4∑

i=1

Bieλiτ

(2.36)

where Ai and Bi are constants of integration, of which four of eight are independent.

A solution of the form,

ξ

ν

=

Ai

Bi

eλiτ (2.37)

is assumed for one particular i value. Differentiating Equation (2.37) twice with

respect to nondimensional time, and substituting into Equation (2.35) the resulting

expression is written,

λ2

i 0

0 λ2i

Ai

Bi

eλiτ+

0 −2λi

2λi 0

Ai

Bi

eλiτ

=

U∗

xx U∗xy

U∗yx U∗

yy

Ai

Bi

eλiτ

(2.38)

which is rearranged and simplifies to,

23

λ2

i − U∗xx −2λi − U∗

xy

2λi − U∗yx λ2

i − U∗yy

Ai

Bi

eλiτ = A∗

Ai

Bi

eλiτ =

0

0

(2.39)

It is clear from the form of Equation (2.39), that the λi represent the eigenvalues of

A∗ and Ai and Bi form the corresponding eigenvectors corresponding to each λi. For

any eigenvalue, the quantities Ai and Bi are scalar multiples, therefore Bi = αiAi.

Given the form of the equations and, due to the fact that the reference is a constant,

the eigenvalues of A∗ emerge in reciprocal pairs [45]. Evaluating these eigenvalues

reveals a real reciprocal pair of eigenvalues (λ1, λ2), and a reciprocal pair of imaginary

eigenvalues (λ3, λ4). By selecting initial conditions that suppress divergent modes (i.e.

positive real eigenvalues), the real roots are eliminated, and the solution in Equation

(2.36) is simplified to isolate the oscillatory mode,

ξ =A3eλ3τ + A4e

−λ3τ

ν =α3A3eλ3τ − A4α3e

−λ3τ(2.40)

The coefficients, A1, A2, correspond to the real eigenvalues and, therefore, are assigned

a value of zero. The set in Equation (2.40) offers a good initial guess for targeting

small in-plane periodic orbits about the libration points; the linear orbit can seed a

differential corrections continuation process to propagate a family of planar Lyapunov

orbits.

2.2.4 The State Transition Matrix

The equations of motion of the CR3BP cannot be solved analytically, thus nu-

merical tools are useful to gain insight into the behavior of the system. Numerical

integration is employed to determine the response of the system over a specific length

of time based upon an initial state. Given some six-dimensional initial state in posi-

tion and velocity, adjustments in this initial state affect the response of the system at

locations downstream. The State Transition Matrix (STM), defined as Φ(τ , τ0), pre-

dicts, through a linear estimate, the sensitivity of each element of some final state to

24

small perturbations in each element of the initial state and, thus, the state transition

matrix is frequently denoted a sensitivity matrix. The nonlinear equations of motion

for a particular system are represented by,

˙x = f(x, τ) (2.41)

where the vector,

x =

x

y

z

x

y

z

(2.42)

forms the full state vector in terms of the position and velocity of P3 within the con-

text of the CR3BP. Assume that a full state initial condition, x r0 , yields a general

reference trajectory xr (τ). The six-dimensional state along a nearby trajectory, de-

noted x (τ), is expressed as a combination of the state along the reference trajectory

and a variation, δx (τ), relative to the reference at each instant in time τ , i.e.,

x (τ) = xr (τ) + δx (τ) (2.43)

The derivative of Equation (2.43) with respect to time is substituted into the expres-

sions into Equation (2.41) and yields,

˙xr (τ) + δ ˙x (τ) = f (xr (τ) + δx (τ) , τ) (2.44)

To simplify the right side, a Taylor expansion is employed about the reference tra-

jectory, xr, where δx = 0, and assuming a small variation, all nonlinear higher-order

terms are truncated. Equation (2.44) is then written as a set of differential equations

of the form,

δ ˙x (τ) =∂f

∂xr

∣∣∣∣xrδx (τ) (2.45)

25

The term ∂f∂xr

∣∣xr

can also simply be written as A(τ) and with this final substitution,

the linear variational equations of motion relative to some reference trajectory, xr,

are described by,

δ ˙x (τ) = A (τ) δx (τ) (2.46)

The matrix A(τ) is of the form,

A =

03x3 I3x3

U∗XX Ω

(2.47)

where the matrix of second partial derivatives of the pseudo potential evaluated along

the reference trajectory (U∗XX) and the constant matrix (Ω) are evaluated as,

U∗XX =

U∗xx U∗

xy U∗xz

U∗yx U∗

yy U∗yz

U∗zx U∗

zy U∗zz

Ω =

0 2 0

−2 0 0

0 0 0

(2.48)

A general solution of Equation (2.46) is,

δx(t) = Φ (τ , τ0) δx0 (2.49)

where the STM, Φ (τ , τ0), is a 6x6 matrix, governed by the matrix differential equa-

tion,

Φ (τ , τ0) = A(τ)Φ (τ , τ0) (2.50)

Equation (2.50) is realized by differentiating Equation (2.49) and substituting the re-

sult into Equation (2.46). The additional 36 scalar equations represented in Equation

(2.50) are integrated along with the nonlinear equations of motion for the system, to

construct a representation for the state transition matrix at some final state along

the reference. Equation (2.49) must be true at the initial time τ0, thus, the initial

conditions for Equation (2.50), i.e., the STM at time τ = τ0, must be equal to the

6x6 identity matrix. The STM is employed in various iterative schemes to accom-

plish numerous objectives including design, control, stationkeeping, and navigation,

for example. It can also be useful for simple iterative targeting schemes. A general

targeting scheme is first considered.

26

2.2.5 General Targeting Technique: Constraints and Free Variables

There are many ways to implement a general targeting scheme. The generalized

method of free variables and constraints is one approach that has been successfully ap-

plied for astrodynamic applications. [47,48] To establish a general targeting method,

a vector of free design variables, X, is constructed. A constraint vector, F (X) is also

formed, containing all constraint equations; the ‘constraints’ incorporate the govern-

ing differential equations. The functional dependencies of the constraints with respect

to the free variables must be deduced. If the number of free variables is equal to the

number of constraints, the iterative Newton update equation is applied to determine

the solution, that is,

Xk+1 = Xk −DF (Xk)−1F (Xk) (2.51)

where DF (Xk) = ∂F /∂X is the Jacobian matrix that is evaluated on the reference

path. If there are more variables in the design vector than there are constraint

equations, a minimum norm solution is employed in the update step, i.e.,

Xk+1 = Xk −DF (Xk)T[DF (Xk) ·DF (Xk)T

]−1F (Xk) (2.52)

Simplifications that result due to symmetry properties and periodicity in the CR3BP

are now examined, where simple targeting schemes aid in the computation of two

types of families of nonlinear periodic reference orbits are explored.

2.2.6 Targeting Periodic Orbits in the CR3BP

Targeting planar and spatial periodic orbits in the vicinity of the collinear libra-

tion points is a well-known problem. [49–51] An infinite number of orbits exist in

these regions, many as members of various families. [44] Such periodic orbits can be

computed by a variety of targeting techniques. A simple, single arc targeting process

is employed here to compute the nonlinear, periodic orbits. There are many ways to

implement a single arc shooting scheme, but most are based on a Newton iteration

27

process. The equations of motion for P3 in the CR3BP are invariant under the trans-

formation to negative time and the Mirror Theorem, as noted Roy and Ovenden [52],

indicates that for every trajectory in forward time, there exist another trajectory in

reverse time that is reflected across a plane of symmetry (the a1 - a3 plane). Further-

more, for periodicity, the orbits must contain mirror configurations at two separate

points in time. [52] In the vicinity of the collinear libration points, the mirror theorem

allows periodic orbits to be determined by targeting perpendicular crossings. If the

initial state is located on the a1 - a3 plane and a perpendicular crossing condition is

enforced, i.e., there is no initial velocity component in either the a1 or a3 direction,

then another perpendicular crossing is sought at the next a1 - a3 plane crossing, by

employing the STM to iteratively correct the initial state. Once the half period tra-

jectory is determined, one that satisfies the requirements for initial and final crossings

of the a1 - a3 plane, the mirror configuration ensures that this trajectory is reflected

across the a1 - a3 plane, yielding a periodic orbit that returns to the initial state in

position and velocity, at least for the first revolution, if appropriate tolerances are

specified on the acceptable error. Since the vicinity of the collinear libration points

is numerically sensitive, the tolerance is set as low as computationally possible. The

general targeting approach for constraints and free variables is modified to exploit

the simplifications resulting from symmetry and periodicity.

Example 1: Planar L1 Lyapunov Family

The computation of the L1 Lyapunov family of periodic orbits is straightforward.

Given an initial guess for position and velocity, in the form noted in Equation (2.42)

for a spacecraft, the states are updated iteratively. To compute planar orbits in the

vicinity of L1, the following vector of design variables is assumed,

X =

y0

τ

(2.53)

28

where y0 represents the initial velocity in the a2 direction and τ is the final time,

corresponding to a half period. The constraint vector required at the final time to

enforce a perpendicular crossing of the a1 axis is,

F(X)

=

yf (y0, τ)

xf (y0, τ)

(2.54)

An initial guess is generated from a linear orbit in the vicinity of L1, as approximated

in Equation (2.40). The Jacobian is formed as the partial derivatives of the constraint

variables with respect to each free variable,

DF(X)

=

∂yf∂y0

∂yf∂τ

∂xf∂y0

∂xf∂τ

(2.55)

The partials in the Jacobian are replaced by elements of the STM as appropriate or

expressions for time derivatives, i.e,

DF(X)

=

φ25 yf

φ45 xf

(2.56)

Another simplification is realized for perpendicular crossing of the a1 axis. By defi-

nition, the a2 component of the final position vector must be zero. Combining this

observation and the fact that the Jacobian matrix is square, only the first of the vector

equations in Equation (2.51) is actually necessary. Thus, Equation (2.51) simplifies

to,

y0k+1 = y0

k − yfφ25xf − φ24yf

· xf (2.57)

Lyapunov orbits in the Earth-Moon system are employed in this analysis. By in-

creasing values of x0 in the a1 direction, while using the initial velocity, y0, from the

previous step as an initial guess, a set of orbits in the family of L1 planar Lyapunov

orbits in the Earth-Moon system, are constructed and appear in Figure 2.3.

29

2 3 4 5

x 105

−1.5

−1

−0.5

0

0.5

1

1.5x 10

5

a1 (km)

a 2 (km

)

Figure 2.3. Partial Family of L1 Planar Lyapunov Orbits in E-M System

Many different forms of the basic targeting algorithms are available; these steps can

easily be modified to obtain another formulation. Similar families of orbits have been

computed about the other collinear libration points.

Example 2: Spatial L1 Northern Halo Family

The three-dimensional halo family of orbits in the CR3BP has also been exten-

sively investigated. The halo family is numerically computed from a pitchfork bifur-

cation emerging from the Lyapunov family of orbits. [49–51,53] For the Earth-Moon

system, the Lyapunov orbit that bifurcates to the three-dimensional halo family is

highlighted in red in Figure 2.4.

30

3 3.2 3.4

x 105

−4

−3

−2

−1

0

1

2

3

4x 10

4

a1 (km)

a 2 (km

)

Figure 2.4. L1 Planar Lyapunov Orbit that Bifurcates to L1 HaloFamily in E-M System

In a method analogous to that presented for the L1 Lyapunov planar family, the

halo family of orbits are also targeted in the vicinity of the collinear libration points.

One three-dimensional periodic Northern halo orbit is computed by differentially

correcting initial conditions, seeded using the bifurcating Lyapunov orbit displayed

in red in Figure 2.4, in the vicinity of the L1 collinear libration point, and a small

positive Az component. Additional family members are determined by a differential

corrections and continuation process. Some trajectories in a family of northern Earth-

Moon (E-M) halo orbits in the vicinity of L1 appear in Figure 2.5. Additional families

are also computed about the other collinear libration points. Members of these orbit

families are used as reference paths for several test cases later in this investigation.

31

3.23.4

3.63.8

x 105

−5

0

5

x 104

−2

0

2

4

6

x 104

a1 (km)

a2 (km)

a 3 (km

)

Az

(km

)

385

76500

Figure 2.5. Family of Northern Halo Orbits in the Vicinity of L1 inthe E-M System

2.3 Attitude Dynamics

Attitude dynamics, as applied to the motion of space vehicles, concerns the ori-

entation of a spacecraft in three-dimensional space, with respect to some reference

frame. Attitude prediction is the focus of this investigation, although some aspects

of spacecraft control are briefly introduced. Equations of motion are formulated and

numerically integrated to gain understanding of the behavior of a specific body un-

der varying testing conditions. The basic concept and notation carried forward are

introduced in the following sections.

2.3.1 Rotational Kinematics

There are several ways to describe the orientation of a body in three-dimensional

space. It is usually most reasonable to relate some initial and final orientation to a

particular reference. At a minimum, three variables are required to determine the

orientation of a rigid body in three-dimensional space. [54] Incorporating additional

32

parameters assists in removing singularities; however, it also requires the implemen-

tation of appropriate constraint equations. Three different specific sets of variables

are employed in this investigation to describe orientation, and each set is described

in detail below.

Euler Parameters

Euler parameters, also denoted quaternions, are particularly useful for attitude

prediction investigations. By incorporating one redundant parameter, this set of four

quantities is free of possible singularities and trigonometric functions, ensuring that

the attitude prediction is valid for all possible rigid body configurations [43]. The

Euler vector, representing the rotation from some initial frame, denoted a, to a final

frame, denoted b, is defined as,

εa b = ε1b1 + ε2b2 + ε3b3 with the constraint ε12 + ε2

2 + ε32 + ε4

2 = 1 (2.58)

As is well known, a simpler rigid body rotation in three-dimensional space can be

described as a rotation through the angle θ about the unit direction λa b. [43] Then

the Euler parameters are correlated such that,

εa bi = λa b sin(θ/2) i = 1, 2, 3

εa b4 = cos(θ/2)

(2.59)

This set incorporates one redundant parameter and, therefore, one constraint equation

is required. This variable set does not offer a straightforward physical interpretation of

the individual parameters which creates challenges for visualization. To incorporate

the benefits of Euler parameters in this analysis, the results are transformed and

displayed in terms of meaningful variables such as Euler angles. Euler parameters are

selected as the working set, and are employed for all numerical integration purposes

in this investigation. The alternate types of variables are used solely for coordinate

frame transformations and visualization purposes.

33

Direction Cosine Elements

Direction cosine elements offer a simple means of transforming quantities from

one coordinate frame to another. [54] Direction cosine elements are a nine-parameter

set, that comprise the direction cosine matrix (DCM). [It is noted that a row vector

format is employed throughout this investigation.] Direction cosines are used only to

transform vectors and tensors from one frame to another. The transformation from

some initial frame, the a-frame, to some final frame, the b-frame, is written as Cab,

and the location of each direction cosine element in this matrix is identified by the

subscripts,

Cab =

C11 C12 C13

C21 C22 C23

C31 C32 C33

(2.60)

such that, [a1 a2 a3

]=[b1 b2 b3

]Cab (2.61)

Conversion from Euler parameters to direction cosine elements is completed using the

following relationships, [43]

C11 =1− 2ε22 − 2ε3

2 C12 = 2(ε1ε2 − ε3ε4) C13 = 2(ε3ε1 + ε2ε4)

C21 =2(ε1ε2 + ε3ε4) C22 = 1− 2ε12 − 2ε3

2 C23 = 2(ε2ε3 − ε1ε4)

C31 =2(ε3ε1 − ε2ε4) C32 = 2(ε2ε3 + ε1ε4) C33 = 1− 2ε12 − 2ε2

2

(2.62)

Given the DCM, the Euler parameters are computed from the following expressions,

[43]

ε4 =1

2(1 + C11 + C22 + C33)

12

ε1 =(C32 − C23)

4ε4ε2 =

(C13 − C31)

4ε4ε3 =

(C21 − C12)

4ε4

(2.63)

Equations (2.62) and (2.63) allow for straightforward transformations between vari-

able sets. The physical interpretation of both the DCM and Euler parameter sets are

not always clear, so an additional set is employed to aid in the visualization process.

34

Euler Angles

For visualization purposes, it is sometimes desirable to convert the orientation

in terms of the Euler parameters to an Euler angle rotation sequence. To later

compare with the results from Wong et al., a body 3-2-1 (θ, φ, ψ) sequence is defined

to represent the rotation from the a-frame to the b-frame, and is denoted C 3-2-1ab ,

that is,

C 3-2-1ab =

cos θ cosφ cos θ sinφ sinψ − sin θ cosψ cos θ sinφ cosψ + sin θ sinψ

sin θ cosφ sin θ sinφ sinψ + cos θ cosψ sin θ sinφ cosψ − cos θ sinψ

− sinφ cosφ sinψ cosφ cosψ

(2.64)

For an axisymmetric spacecraft, a body 3-2-3 (α, ζ, γ) sequence, denoted C 3-2-3ab , is

more intuitive to gain insight concerning the motion of the b-frame relative to the

a-frame. In this form, the angle α represents precession of the b-frame relative to the

a-frame, the angle ζ denotes nutation, and the angle γ reflects spin about the axis of

symmetry. This rotation matrix is defined,

C 3-2-3ab =

cosα cos ζ cos γ − sinα sin γ − cosα cos ζ sin γ − sinα cos γ cosα sin ζ

sinα cos ζ cos γ + cosα sin γ − sinα cos ζ sin γ + cosα cos γ sinα sin ζ

− sin ζ cos γ sin ζ sin γ cos ζ

(2.65)

Each of the three types of variable sets that are introduced for rotational kinemat-

ics possesses advantages and disadvantages. By combining these various sets, most

disadvantages are overcome. Generally, an Euler parameter set is employed for nu-

merical integration, the DCM is utilized for coordinate frame transformations and,

with several Euler Angle sequences, the process allows for seamless integration and

visualization of the resulting motion.

2.3.2 Kinematic Differential Equations

To discuss kinematic differential equations, the notation associated with the an-

gular velocity is first introduced. The angular velocity of the final frame, i.e., the

35

b-frame, with respect to the initial frame, the a-frame, is written as ωa b. The Euler

parameter set is used in the integration process; therefore, the time rate of change

equations for these variables are required. To formulate these equations, Poisson’s

kinematical equations for the change in the elements of the DCM are first introduced,

that is, [43]

Cab = Cab ωa b where ωa b =

0 − ωa b3 ωa b

2

ωa b3 0 − ωa b

1

− ωa b2 ωa b

1 0

(2.66)

This expression is rewritten in the form ωa b = CabT Cab. Relationships between the

angular velocity components and Equations (2.62) and (2.66) yield, [43]

ωa b1 = C13C12 + C23C22 + C33C32

ωa b2 = C11C13 + C21C23 + C31C33

ωa b3 = C12C11 + C22C21 + C32C31

(2.67)

Next, the time derivatives of the set described in Equation (2.62) are formulated and

the result is,

C11 =− 4ε2ε2 − 4ε3ε3

C12 =2(ε1ε2 + ε1ε2 − ε3ε4 − ε3ε4)

C12 =2(ε3ε1 + ε3ε1 + ε2ε4 − ε2ε4)

C12 =2(ε1ε2 + ε1ε2 + ε3ε4 + ε3ε4)

C11 =− 4ε1ε1 − 4ε3ε3

C12 =2(ε2ε3 + ε2ε3 − ε1ε4 − ε1ε4)

C12 =2(ε3ε1 + ε3ε1 − ε2ε4 − ε2ε4)

C12 =2(ε2ε3 + ε2ε3 + ε1ε4 + ε1ε4)

C33 =− 4ε1ε1 − 4ε2ε2

(2.68)

To obtain a relationship between the angular velocity components and the Euler

parameters and their associated rates, Equation (2.67) is augmented with the trans-

36

formations in Equation (2.62) and the time derivatives from Equation (2.68), yielding

the following expression,

ωa b1 =2ε4ε1 + 2ε3ε2 − 2ε2ε3 − 2ε1ε4

ωa b2 =2ε4ε2 + 2ε1ε3 − 2ε3ε1 − 2ε2ε4

ωa b3 =2ε4ε3 + 2ε2ε1 − 2ε1ε2 − 2ε3ε4

(2.69)

The expression in Equation (2.69) is represented in matrix form as follows, [43]

ωa b = 2 ˙εa bE (2.70)

where ωa b =[

ωa b1 ωa b

2 ωa b3 0

]and E =

ε4 −ε3 ε2 ε1

ε3 ε4 −ε1 ε2

−ε2 ε1 ε4 ε3

−ε1 −ε2 −ε3 ε4

(2.71)

Equation (2.70) is rearranged to obtain an expression for the time rate of change for

each Euler parameter, i.e.

˙εa b =1

2ωa bET (2.72)

This expression is integrated to obtain the orientation history in terms of Euler pa-

rameters.

2.3.3 Gravity

Formation of the equations of motion requires an understanding of the external

forces exerted on the system. In this investigation, the net force derived from the

gravitational potential due to both of the two primaries is assumed to be the only

external force applied to the system. To clarify the notation, the net force from

a single gravitational potential on a single rigid body is considered. The net force

expression is later extended to include both primaries, and all the connected bodies

37

that comprise the spacecraft system. An illustration of the simplified system appears

in Figure 2.6, where the primary body is denoted Pi and the single rigid body, S,

with center of mass S∗ are displayed.

Pi

S

dms

R s

ps=R sr

s rs

s1

s2

Figure 2.6. Gravity Force on an Infinitesimal Mass Element

The position vector from the center of mass of the gravitational primary to the

center of mass of rigid body S is defined as Rs, the position vector from the center of

mass of rigid body S to an infinitesimal mass element is written rs, and the sum of

these two quantities, denoted ps, yields the position vector from the primary center

of mass to an infinitesimal mass element.The force exerted on body S is then due to

the gravitational potential of this one attracting body, Pi, as represented by, [43]

F SPi

= −µPi

∫

S

psj(psj · psj

)−3/2ρdτj (2.73)

where µPiis the universal gravitational constant multiplied by the mass of the primary

and the infinitesimal mass element is denoted dmsj = ρdτ , with ρ representing the

density and dτ specifying the differential volume. Since the distance from the center

of the primary to the center of mass of the rigid body, Rs, is much greater than the

distance from the center of mass of the rigid body to the infinitesimal mass element, rs,

it is appropriate to expand the force expression in Equation (2.73) using the binomial

expansion. Simplification and truncation of all terms greater than second order leaves

38

the following expression for the force exerted on the rigid body and applied at the

center of gravity, [43]

F SPi

= −µPims

Rs2 s1−3µPi

ms

Rs4

[tr(I

S/S∗

)− 5s1 · IS/S∗· s1

]s1−

3µPims

Rs4 IS/S∗· s1 (2.74)

Note that ms is the mass of the rigid body, and the unit vector s1 is aligned with the

direction from the primary center of mass to the rigid body center of mass.

A complete model for the influence of the external forces includes the appropriate

moment. The gravity gradient moment about the center of mass of body S is delivered

by the two primary bodies. For each primary, this moment is derived in terms of the

distance vector from the attracting body to the center of mass of the rigid body, Rs,

as well as the force expansion in Equation (2.74) as displayed in Figure 2.7. The

gravitational moment, due to one primary body, Pi, is then written as, [43]

MS∗Pi

= −Rs × F SPi

(2.75)

The information supporting Equation (2.75) for the computation of the moment is

illustrated in Figure 2.7. The concepts introduced in this section are extended to

incorporate the effects of two primary bodies, for a spacecraft system comprised of

multiple rigid bodies, in Chapter 3.

Pi

S

RCG

rs1

s2

CGRS

FPi

rCG /S

Figure 2.7. Gravitational Moment

39

3. MODEL FRAMEWORK

The first research objective in this investigation is the development of a model to

formulate the fully coupled orbit and attitude equations in the CR3BP. Large dis-

crepancies in length scales exist in this problem. The distance from the barycenter

to the spacecraft is on the order of hundreds of thousands of km (depending on the

system), and the spacecraft dimensions are typically measured in terms of meters.

The presence of such large differences in the lengths presents a challenge for the com-

putational capabilities, as only finite precision is possible. Numerical challenges can

also prevent the propagation of meaningful results. Thus, a nonlinear variational form

of the equations of motion is employed to mitigate these numerical challenges. The

motion of the spacecraft is monitored relative to some known reference orbit (in any

desired form). The time rate of change of linear momentum, Equation (2.2), is used

to describe the absolute motion of the spacecraft and is later employed to describe

the motion of the reference system. The dynamical contributions from the reference

are subtracted from the absolute motion, eliminating the difficulties of very differ-

ent lengths coupled within the same differential equations, i.e., the position of the

spacecraft relative to the barycenter and the physical dimensions of the spacecraft.

The specific spacecraft configuration and reference orbits are initially introduced, the

framework is presented in detail; further complications that exist in the variational

force equations are discussed.

3.1 Spacecraft Model

The first spacecraft model is comprised of two asymmetric rigid bodies, that is,

body B and body R. Thus, two coordinate frames are assigned to the spacecraft.

The first coordinate frame is fixed in body B, denoted the b-frame. The second

40

set of unit vectors is fixed in body R and is denoted the r-frame. The coordinate

systems align with the principal directions of the individual bodies for each center of

mass. The mass of body B is mb and the center of mass location is denoted B∗. The

mass of body R is mr and the center of mass location is located at R∗. The inertia

matrices corresponding to the two bodies are expressed in terms of the appropriate

unit vectors, i.e., b-frame and r-frame coordinates, respectively,

IB/B∗

=

Ib1b1 0 0

0 Ib2b2 0

0 0 Ib3b3

IR/R

∗=

Ir1r1 0 0

0 Ir2r2 0

0 0 Ir3r3

The two bodies are connected by a revolute joint with a single degree of freedom

parallel to the b3 direction. Thus, the b3 and r3 directions are aligned for all time.

The location of the joint, denoted J , is determined by the following two vectors. In

the b-frame, the vector from B∗ to J is defined as rJ/B∗

= s1b1 + s2b2 + s3b3. In

the r-frame, the vector from J to R∗ is expressed by rR∗/J = j1r1 + j2r2 + j3r3. A

schematic representation of the spacecraft model appears in Figure 3.1.

3.2 Reference Trajectory

The reference trajectory for this analysis is a trajectory that reflects the motion of

a point mass spacecraft propagated in the CR3BP, where the spacecraft is moving

under the gravitational influence of two primary bodies modeled as point masses. The

rotating frame is defined as the a-frame and rotates with the line that connects the

two primary bodies as they move along circular paths about their common barycenter,

relative to the inertial n-frame, as previously discussed in Chapter 2. A feature that

demonstrates the flexibility of the model framework, is the fact that the reference

trajectory can be determined outside the model, and input as a set of discrete points

with a particular time history. As a consequence, the reference orbit can be of any

form, and when the path is precomputed, the reference points are interpolated using

41

b1

b2

b3

r1

r3

r2

B*

R*

J

Figure 3.1. Spacecraft Model

a cubic spline technique to determine the location along the reference orbit at a

specified instant in time, corresponding to the system integration time.

3.3 Equations of Motion

The approach introduced in the general discussion of Kane’s formulation is now

applied to the fully coupled orbit and attitude problem. The goal is to produce an

overall understanding of the dynamic model.

3.3.1 Generalized Speeds and Configuration Coordinates

To develop the equations of motion governing the behavior of the spacecraft,

generalized speeds are first assigned. The two-body spacecraft possesses seven degrees

of freedom; therefore seven independent generalized speeds, ui, are selected. The

angular velocity of the b-frame with respect to the a-frame is expressed,

42

ωa b = u1b1 + u2b2 + u3b3 (3.1)

Similarly, an expression for the angular velocity of the r-frame with respect to the

b-frame is formulated,

ωb r = u4b3 (3.2)

The remaining generalized speeds are used to describe the translational velocity of

B∗, i.e, the center of mass of body B, relative to the current location in the reference

orbit, denoted by O,

vn B∗/O = u5b1 + u6b2 + u7b3 (3.3)

Several other configuration coordinates are also required to fully describe the system

translation and orientation. Euler parameters, as elements in a four-vector, are se-

lected to describe the orientation of the b-frame with respect to the a-frame. The

Euler vector is defined as,

εa b = ε1b1 + ε2b2 + ε3b3 (3.4)

The angular velocity of the r-frame, fixed in the second body of the spacecraft, relative

to the b-frame is integrated directly to yield the angle between these frames. The

direction cosine matrix to convert from the b-frame to the r frame is evaluated as a

rotation about b3 as follows,

Cbr =

cos β − sin β 0

sin β cos β 0

0 0 1

(3.5)

The position of R∗ with respect to B∗ is eventually required and is written in b-frame

coordinates and evaluated as,

rR∗/B∗

= rJ/B∗

+ rR∗/JCT

br (3.6)

43

To determine the effects of the attitude motion on the orbit, it is assumed that

the spacecraft is slightly offset from the reference trajectory. The following scalar

variables expressed as coordinates in the rotating orbit frame, a, is used to track this

motion. Recall that O represents the current position along the reference orbit. This

position is determined by propagating the point mass system forward in time, or by

interpolating a known reference trajectory,

rG∗/O = g1a1 + g2a2 + g3a3 (3.7)

The motion of the actual center of mass of the spacecraft system, G∗, is tracked as

the simulation propagates. This quantity is required to determine the contributions

of the gravitational potential from the two primaries.

3.3.2 Kinematics

For an ongoing and complete assessment of the spacecraft location and orientation,

the computation of the absolute velocity and absolute angular velocity corresponding

to both body B and body R are required. These quantities are described using the

b-frame vector basis. Additionally, since variational force equations are employed,

the velocity associated with the point mass reference system must also be computed.

The absolute velocity is computed by applying Equation (2.1), i.e.,

vn B∗=

dn

dtrO + vn B∗/O (3.8)

vn R∗=

dn

dtrO + vn B∗/O +

dn

dtrR

∗/B∗(3.9)

vn O =dn

dtrO (3.10)

The absolute angular velocity of body B is represented as the angular velocity of the

b-frame and, similarly, the absolute angular velocity of body R is represented as the

angular velocity of the r-frame. Recall, from Figure 2.1, that the angular velocity of

the CR3BP rotating frame with respect to the inertial frame is ωn a.

44

ωn b = ωn a + ωa b ωn r = ωn a + ωa b + ωb r

The kinematic expressions are then employed to formulate the dynamic equations of

motion.

3.3.3 Momentum Principles

The momentum principles introduced in Equations (2.2) and (2.3) are now applied.

Equation (2.2) is first used to form an equation for the total motion of each body of

the multibody spacecraft and, then, to write the equation for the point mass reference

system. To remove the undesirable effects of the discrepancies in the length scales,

the point mass equation is then subtracted from the total motion of body B, yielding

the following variational form,

∑(FB − FO)− dn

dt

(mb v

n B∗/O)

= 0 (3.11)

Similar expressions are derived for the second body, R. Equation (2.3) is applied as

written since the point mass system includes no inertia properties about the spacecraft

center of mass and, therefore, no external moments are applied. External forces and

moments are discussed separately below.

External Forces

This model assumes that the net force derived from the gravitational potential of the

two primaries is the only external force applied to the system. The external field force

is assumed to be applied to the center of mass of the system, i.e, G∗. An equivalent

system is derived for the two individual bodies, B and R, by replacing the force

applied at G∗, with a force and moment pair applied at each of the centers of mass

of the individual bodies. The difference between the location of the center of mass of

the system, rG∗, and the point on the reference, rO is represented by δ. Both rG

∗and

45

rO have a base point fixed in the inertial frame at the barycenter of the Earth-Moon

system. The gravitational force exerted on the center of mass of the system G∗ is

represented by FG∗and the gravitational force exerted on the point mass reference

system at O, is represented by FO. The total mass of the spacecraft system of two

bodies is defined as mt and the difference between the two forces is denoted,

∆F = FG∗ − FO (3.12)

Since the gravitational force exerted on the center of mass of the spacecraft, G∗, is

nearly identical to the gravitational force exerted on the point mass reference system

at O, a special representation is employed to ensure that the appropriate precision

is maintained in the computation of this quantity. A general implementation of

Encke’s method is employed to achieve this task.[14] For the following derivation,

two attracting bodies are considered: the Earth and the Moon. The subscript e

denotes quantities related to the Earth, and the subscript m indicates lunar terms.

The distances from the barycenter to the Earth and from the barycenter to the Moon

are represented by D1 and D2. The distance from the Earth to the location on the

reference, O, is represented by Dref , and the distance from the Moon to the location

on the reference, O, is represented by Rref . The distances from the Earth and the

Moon to the center of mass of the spacecraft system, G∗, are denoted Dg and Rg

respectively. Equation (3.12) is recast using the point mass gravitational model as

follows,

∆F = mt

(− µeD3g

Dg −µmR3g

Rg

)−mt

(− µeD3ref

Dref −µmR3ref

Rref

)

where Dg = rG∗ − D1, Rg = rG

∗ − D2, Dref = rO− D1, and Rref = rO− D2.

Rearranging the above expressions and incorporating δ = rG∗ − rO yields,

∆F =

(− µeD3ref

δ − µeD3ref

(D3ref

D3g

− 1

)Dg −

µmR3ref

δ − µmR3ref

(R3ref

R3g

− 1

)Rg

)mt

46

Scalar functions, fe and fm, are used to represent the differences (Dref/Dg)3− 1 and

(Rref/Rg)3− 1, respectively. These differences are written in a more convenient form

by introducing scalar quantities qe and qm, [14]

qe =δ · δ − 2δ · Dg

‖Dg‖2and qm =

δ · δ − 2δ · Rg

‖Rg‖2(3.13)

The scalar quantities qe and qm are now employed to form new representations of fe

and fm as follows,

fe =

(qe

3 + 3qe + q2e

1 + (1 + qe)3/2

)fm =

(qm

3 + 3qm + q2m

1 + (1 + qm)3/2

)(3.14)

This approach produces results that are accurate as long as the actual spacecraft

remains close to the reference. The reference may require rectification for certain

simulations if the spacecraft center of mass is observed to shift sufficiently far from

the reference. The contributions applied to each body individually is determined by

partitioning the total mass, mt, into the mass of the two bodies, mb and mr. All force

terms are written in the b-frame vector basis.

External Moments

The gravity gradient moment in this problem is delivered by the two primary bodies.

For each primary, this moment is derived from the cross product of the distance from

the attracting body to the spacecraft and the force expansion (including second order

terms).[15] These moments are computed in terms of the b-frame vector basis.

3.3.4 Partial Velocities and Partial Angular Velocities

The partial velocity matrix is formed for each body from the partial derivative of the

center of mass velocity with respect to each generalized speed. The partial derivative

of the velocity vector is evaluated for each generalized speed and this vector quantity

is stored as a row in the partial velocity matrix. The matrices are defined below for

47

body B and body R, respectively. The partial velocity matrix and the partial angular

velocity matrix for both body B and body R are described in terms of the b-frame

vector basis. Each matrix has seven rows, one for each generalized speed, and three

columns, one for each spatial dimension of the b-frame.

∂ vn B∗

∂u=

0 0 0

0 0 0

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

(3.15)

∂ vn R∗

∂u=

0 −s3 − j3 s2 + j1 sinβ + j2 cosβ

s3 + j3 0 −s1 − j1 cosβ + j2 sinβ

−s2 − j1 sinβ − j2 cosβ s1 + j1 cosβ − j2 sinβ 0

−j1 sinβ − j2 cosβ j1 cosβ − j2 sinβ 0

1 0 0

0 1 0

0 0 1

(3.16)

The partial angular velocity matrix for each body also originates from the partial

derivative of the angular velocity vector with respect to each generalized speed. As

described previously, these matrices also have seven rows and three columns, and are

described in terms of the b-frame vector basis. The matrices for body B and body R,

respectively, are summarized as,

48

∂ ωn b

∂u=

1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

0 0 0

0 0 0

∂ ωn r

∂u=

1 0 0

0 1 0

0 0 1

0 0 1

0 0 0

0 0 0

0 0 0

(3.17)

These quantities are then employed in the formation of Kane’s equations.

3.3.5 Constructing Kane’s Equations

Using the quantities derived previously, the equations of motion are now constructed

using Kane’s formulation. The generalized inertia forces and generalized active forces,

described in Equations (2.4) and (2.5), are computed for each body using the partial

velocities in Equations (3.17), (3.15), and (3.16), and the appropriate parts of the

momentum equations, given in Equations (2.3) and (3.11). These two quantities are

combined for each body, as described in Equation (2.6). Finally the contributions

from the two separate bodies are added to obtain a single equation of motion for each

generalized speed. The governing equations are produced in the form Ax = b. This

system is combined with derivatives for the required configuration coordinates and

integrated to obtain the motion of the individual bodies in the system.

3.4 Solution Accuracy

In any investigation involving numerical integration of differential equations, an

awareness of the solution accuracy is an important issue. Since no Jacobi integral

is readily available in this system, other techniques are employed to ensure solution

accuracy. An Adams-Bashforth-Moulton, predictor-corrector type integrator of vari-

able order is employed for all numerical integrations. The relative and absolute error

49

tolerances are as low as numerically possible, that is, 10−12. All known integrals of

the motion are considered during propagation of numerical results. The reference

equations of motion are propagated in addition to the nonlinear variational coupled

orbit and attitude equations of motion. The Jacobi constant corresponding to the

path in the point mass reference system is monitored throughout the integrations to

ensure numerical error is not accumulating over the propagation. The coupled or-

bit and attitude equations of motion have been formulated for a system with a single

gravitational potential, with both a Newtonian and Kane’s dynamical approach, for a

spacecraft system comprised of two rigid bodies. These equations were first compared

term-by-term and then numerically integrated to test that the Kane formulation yields

the same results as the more familiar and intuitive Newtonian method. Within the

context of the planar circular restricted three-body problem, a comparison test was

completed with Guzzetti to test the behavior for a spacecraft system moving along a

number of planar Lyapunov orbits, [11] where Guzzetti formulates the nonlinear vari-

ational coupled orbit and attitude equations of motion using the Lagrangian method.

Although two completely different formulations are employed, the comparison results

yield excellent correlation between the two models. Additionally, test cases from the

literature are considered, especially those from Wong et al. [10] These test cases are

considered in detail in the following sections. Through all the test cases and system

checks, accuracy of the solution is confirmed.

50

4. MOTION ALONG PLANAR PERIODIC REFERENCES

Given the model framework that has been established, the effects of attitude dynamics

in the CR3BP are explored. Results are propagated in the Earth-Moon system using

planar Lyapunov orbits as reference orbits about the three collinear points. The

orbital initial conditions are differentially corrected to ensure periodic motion for

the necessary time intervals (the Lyapunov orbits are numerically continuous for a

minimum of 2.5 revolutions).

4.1 Unperturbed L1 Asymmetric Rigid Body

A set of planar Lyapunov orbits about L1 in the Earth-Moon system, as plotted

in Figure 4.1, are selected to explore the effects of orbital location and orbit size on

the vehicle orientation. The amplitude Ay is indicated for each orbit selected, and

Ay serves to denote each orbit. The smallest orbit is identified with an amplitude

Ay = 36 km and an approximate period of 11 days. In contrast, the largest orbit

possesses an amplitude, Ay = 144000 km, with an approximate period of 22 days. The

nonlinear nature of the larger Lyapunov orbits is clearly visible. The arrow indicates

the direction of motion along all the orbits.

A large rigid asymmetric spacecraft modeled as a single body is tested in the

reference orbits described above. The single rigid body spacecraft is achieved by

locking the rotational degree of freedom between body B and body R, such that

the b-frame and r-frame are aligned for all time. The bodies have equal mass and

dimensions of 200 meters in b1, 100 meters in b2, and 50 meters in b3, such that the

total system possesses a mass of 2 ∗ 105 tons and dimensions of 200 meters in b1, 100

meters in b2, and 100 meters in b3. The simulation originates with the spacecraft

center of mass located along the a1 line connecting the two primary bodies, to the

51

1 2 3 4 5

x 105

−1.5

−1

−0.5

0

0.5

1

1.5x 10

5

a1 (km)

a 2 (km

)

Ay=36kmAy=360kmAy=3680kmAy=47600kmAy=100140kmAy=144000kmMoon

Figure 4.1. L1 Planar Lyapunov Reference Orbits in Earth-Moon System

left side of L1 (between the Earth and L1). No perturbations are applied to the

system. The response in angular velocity of the b-frame with respect to the a-frame

in the b3 direction is illustrated in Figure 4.2. A body fixed 3-2-1 (θ, φ, ψ) Euler

angle sequence is employed to visualize the orientation response. The sequence is

employed for asymmetric spacecraft systems for a later comparison with a previous

investigation. The response in θ is also displayed. Both φ and ψ remain very close to

zero throughout the simulation, for all the reference orbits.

The response in the large Lyapunov orbits is clearly dominated by the close ap-

proaches to the second primary, the Moon in this simulation. Both the angular

velocity and the orientation responses demonstrate sharp changes when the space-

craft is approaching the 0.5 and 1.5 revolutions along the orbit. Further investigation

is required to more extensively explore the behavior in the smaller Lyapunov orbits.

Two additional sets of orbits are displayed in Figure 4.3. The first additional set of

orbits, displayed on the left, aids in examining the behavior in an amplitude range

where the linear approximation of the orbit is still reasonable. Thus, the linear ap-

proximation appears reasonable in the figure for any Ay less than approximately 4000

km for L1 Lyapunov orbits in the Earth-Moon system. The Ay values are displayed

52

0 0.5 1 1.5 2−2

0

2

4

6

8x 10

−5

Rev

Aω

B (

rad/

s)

Ay=36kmAy=360kmAy=3680kmAy=47600kmAy=100140kmAy=144000km

0 0.5 1 1.5 2 2.5−2000

0

2000

4000

6000

8000

10000

12000

Rev

θ (d

eg)

Ay=36kmAy=360kmAy=3680kmAy=47600kmAy=100140kmAy=144000km

Figure 4.2. Orientation Response in the Planar L1 Lyapunov Orbits

for each orbit. Examining this plot, it is clear that the orbits are generally elliptical

in nature and not dominated by any nonlinear effects. The second set, on the right

in Figure 4.3, offers insight into the capabilities of the linear approximation. The

smaller orbits remain elliptical in nature, but as Ay is increased, the nonlinear effects

emerge.

3.16 3.18 3.2 3.22 3.24 3.26

x 105

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

a1 (km)

a 2 (km

)

Ay=360kmAy=720kmAy=1450kmAy=2180kmAy=2930kmAy=3680km

2.4 2.6 2.8 3 3.2 3.4 3.6

x 105

−5

0

5x 10

4

a1 (km)

a 2 (km

)

Ay=3680kmAy=7550kmAy=15960kmAy=25469kmAy=36130kmAy=47600km

Figure 4.3. Two Sets of Planar L1 Lyapunov Orbits

53

No perturbations are applied to the system for any spacecraft in the new sets

of reference trajectories. The response in θ is again displayed in Figure 4.4, for Ay

values less than 50000 km. Both φ and ψ remain very close to zero for all reference

orbits. The left plot demonstrates that the response in the smaller set of orbits is

cyclic in nature, with increasing amplitude in θ as the orbit size increases. In these

orbits, the spacecraft exhibits a slow oscillatory motion in the out-of-plane direction,

that appears to be bounded over the simulation time.

0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5x 10

−6

Rev

Aω

B (

rad/

s)

Ay=360kmAy=720kmAy=1450kmAy=2180kmAy=2930kmAy=3680km

0 0.5 1 1.5 2 2.5−500

0

500

1000

1500

Rev

θ (d

eg)

Ay=3680kmAy=7550kmAy=15960kmAy=25469kmAy=36130kmAy=47600km

Figure 4.4. Response in θ for Refined L1 Planar Lyapunov Orbit Sets

The rotational motion that is observed in the simulation is better understood by

considering the net moment, due to the gravity gradient torque, on the spacecraft

at four locations along the planar Lyapunov orbit. Temporarily assume that the

spacecraft frame (the b-frame) and the orbit frame (the a-frame) are aligned at all

locations along the orbit. The simulation originates with the spacecraft located along

the line connecting the two primary bodies, to the left of L1 (between the Earth and

L1). At this location the net moment is zero. The direction of motion in the orbit

is clockwise, and after 0.25 revolutions along the Lyapunov path, the spacecraft is

located at the maximum a2 amplitude along the orbit, at (L1,Ay). The net gravity

gradient moment at this location is negative. After 0.5 revolutions in the Lyapunov

54

orbit, the spacecraft is once again located along the line connecting the two primary

bodies and is now to the right of L1. The net moment at this location is once

again zero. The last location is at 0.75 revolutions in the Lyapunov orbit, where

the spacecraft is located again at the maximum a2 amplitude, i.e., at (L1,−Ay). At

this location, the net moment is positive. Thus, the angle θ becomes negative for

approximately the first half revolution of the Lyapunov and then reorients to become

positive for the second half of the orbit. For each orbit, the period of the oscillation

varies slightly over the simulation interval. This frequency response is unaffected by

increases in the orbit size. For all the relatively small orbits in this first set, the

period of oscillations appears to be slower than the period of the Lyapunov orbit for

the first cycle and slightly faster than the Lyapunov period over the second cycle.

The plot on the right in Figure 4.4 reflects the response in θ for the orbits with the

larger amplitudes. It is clear that the motion becomes unbounded if the Lyapunov

size is increased. This appears to coincide with the breakdown of the linear orbit

approximation (Ay ≈ 4000 km for L1 Lyapunov orbits in the Earth-Moon system).

4.2 Unperturbed L3 Planar Lyapunov Orbits

A set of planar Lyapunov orbits about L3 in the Earth-Moon system, depicted on

the left in Figure 4.5, are selected to explore the spacecraft behavior at a different

collinear point. The plot includes three periodic orbits, where the largest Lyapunov

orbit is clearly nonlinear in nature. Again, the arrow describes the direction of motion

in the planar L3 Lyapunov orbits. The same large asymmetric spacecraft as described

in the L1 simulation is tested with the L3 reference trajectories. The body fixed 3-

2-1 (θ, φ, ψ) Euler angle sequence again serves as a basis to visualize the orientation

response. The response in θ for all three orbits is plotted on the right in Figure

4.5. The response in φ and ψ is very close to zero for all reference orbits. This

response confirms that, for large Lyapunov orbits, the orientation response is again

dominated by the closest approach to the primary, in this case the Earth. The

55

simulation originates on the far side of L3; near 0.5 and 1.5 revolutions the spacecraft

is passing closest to the Earth. In the largest Lyapunov orbits, a distinct shift in the θ

response is clear near the closest approach to Earth. The asymmetric spacecraft in the

smaller Lyapunov trajectories demonstrate cyclic behavior similar to that displayed

in the vicinity of L1. The Ay values, where the motion remains bounded in the

L3 reference orbits, is much larger than in orbits near L1, due to the fact that the

attracting bodies are much further away from L3.

0 0.5 1 1.5 2 2.5−500

−400

−300

−200

−100

0

100

Rev

θ (d

eg)

Ay=2000kmAy=40000kmAy=528930km

Figure 4.5. L3 Planar Lyapunov Reference Orbits and OrientationResponse in E-M System

4.3 Changing Spacecraft Properties

The effects of changing the properties of the spacecraft are also examined. For

this investigation, the largest L1 planar Lyapunov orbits, as illustrated in Figure 4.1,

are again selected as a reference. Two effects are explored in detail. First, the ratio of

the in-plane moments of inertia is explored. A single asymmetric spacecraft is again

tested. Bodies B and R are locked together, each with a mass of 100 kg and nominal

dimensions of 10 meters in b1, denoted χ1, 10 meters in b2, denoted χ2, and 5 meters

in b3, denoted χ3. The dimension in the b1 direction is increased incrementally by 2

meters until the dimension in the b1 direction is 20 meters. The effects on the angular

56

velocity of the b-frame with respect to the a-frame are displayed in the left plot

in Figure 4.6. This plot demonstrates that, as the in-plane inertia ratio (I11/I22) is

increased from unity, the response in the angular velocity also increases (as expected).

The magnitude of the response in angular velocity for the largest in-plane inertia ratio

is observed to be on the order of ten times the orbital rate of the primaries, that is

2.665 ∗ 10−6 rad/s.

The effect of changing the total size of the spacecraft is also explored. In this

analysis, the inertia ratios remain constant, and all dimensions are modified accord-

ingly. The dimensions are of the form (χ1, 0.5χ1, 0.25χ1). Several values for χ1 are

selected and the other lengths are scaled accordingly. The results are displayed in the

right plot in Figure 4.6. No effect is noted on the angular velocities as the spacecraft

dimensions are increased uniformly. Again, this result is not unexpected and aids in

validating the algorithm. The effects of modifying the inertia ratios of an asymmetric

spacecraft on the out-of-plane orientation angle, without applying a perturbation, has

not been published in previously available literature for a vehicle moving in a planar

Lyapunov orbit.

0 0.5 1 1.5 2 2.5−2

0

2

4

6

8

10x 10

−5

Rev

Aω

B (

rad/

s)

χ

1=χ

2

χ1=1.2χ

2

χ1=1.4χ

2

χ1=1.6χ

2

χ1=1.8χ

2

χ1=2χ

2

0 0.5 1 1.5 2 2.5−2

0

2

4

6

8

10x 10

−5

Rev

Aω

B (

rad/

s)

χ

1=1m

χ1=40m

χ1=80m

χ1=120m

χ1=160m

χ1=200m

Figure 4.6. Orientation Response to Varied Spacecraft Inertia Properties

57

4.4 Spin Stabilization of Axisymmetric Spacecraft

The model is also used to predict the behavior of an axisymmetric body under an

angular velocity perturbation in the b1 direction. The spacecraft is now composed of

two rigid rods, each of length 10 meters and radius 0.1 meter. The rods are stacked,

configured such that the axial direction is aligned with b3. The top rod can spin about

b3 relative to the bottom rod. This spacecraft is placed in an L1 planar Lyapunov

orbit in the Earth-Moon system with an Ay value of approximately 3680 km. The

orientation response is displayed in Figure 4.7, in terms of the body fixed 3-2-3 (α, ζ, γ)

Euler angle sequence, for four test cases. Recall that the angle α represents precession

of the b-frame relative to the a frame, the angle ζ is the nutation angle, and the spin

about the axisymmetric axis is determined by the angle γ.

0 0.5 1 1.5 2 2.5−2000

0

2000

4000

6000

8000

Rev

α (d

eg)

unpertpertpert & spun (1rad/sec)pert & spun (100rad/s)

0 0.5 1 1.5 2 2.5−100

−50

0

50

100

150

Rev

ζ (d

eg)

unpertpertpert & spun (1rad/sec)pert & spun (100rad/s)

0 0.5 1 1.5 2 2.5−8000

−6000

−4000

−2000

0

2000

Rev

γ (d

eg)

unpertpertpert & spun (1rad/sec)pert & spun (100rad/s)

Figure 4.7. Orientation Response to Spin Stabilization

58

The unperturbed motion with the rods joined and no relative spin is displayed

in red. A perturbation of 0.00001 rad/sec is applied to body B in the b1 direction

while the rods are still fixed. This perturbation is noted to be 3.75 times the orbital

rate of the primaries and the response is plotted in blue. Once the perturbation is

applied, the rod is observed to fall into the plane of primary motion, as depicted

by the nutation angle, ζ, in Figure 4.7. This behavior is expected, as the rod is in

an unstable configuration at the beginning of the simulation. Spin stabilization is

attempted by spinning body R at a rate of 1 rad/sec in the b3 direction and this

motion is indicated in green. The effect of increasing the spin rate to 100 rad/sec

is also examined, and this response is plotted in magenta. It is apparent in Figure

4.7 that spin stabilization for this particular perturbation is possible. In response to

the perturbed motion, the spacecraft exhibits oscillatory motion in ζ and unbounded

motion in the other two angles which confirms that the spacecraft is not maintaining

the initial orientation. Using a spin rate of 1 rad/sec, the nutation angle is observed

to be better maintained; however, the spacecraft is precessing with respect to the

orbit frame at a higher rate than in the perturbed case. By increasing the spin rate

to 100 rad/sec, nearly all the effects of the initial perturbation on the nutation angle

are damped out, and the precession and spin angles also demonstrate improvement

relative to the perturbed case over the length of the simulation. When the spacecraft

is spinning, it appears that the vehicle is maintains a constant direction with respect

to the rotating frame, the precession angle and the spin angle effectively cancel, and

there is very little nutation of the body. This test case serves two purposes. First, the

results indicate that the spacecraft is displaying motion consistent with some results

observed in similar test cases in the two-body problem, which builds confidence in

the model formulation. Secondly, it demonstrates that spin stabilization is possible,

at least for some orientation parameters, in the CR3BP.

59

4.5 The Impact of a Perturbed Axisymmetric Spacecraft on the Orbit

With the fully coupled dynamical model, the effects of an attitude perturbation

on the orbit is also examined. The axisymmetric spacecraft is composed of two rigid

rods with a length-to-radius ratio of 100:1. The rods are stacked one on top the

other, and configured such that the axial direction is aligned with the out-of-plane

direction, b3. The effects of increasing the length, while maintaining a constant ratio

of length-to-radius, are examined for a L1 planar Lyapunov orbit in the Earth-Moon

system with an Ay amplitude of 144000 km. Two sets of perturbations are examined

to investigate the coupling effects and the length of the rod is increased from 10

meters to 100000 meters. Note that, since the Earth-Moon distance is approximately

384000 km, the largest length of is approximately 25 % of the Earth-Moon distance.

First, the generalized speed u1, the angular velocity of the body B in the b1 direction,

is perturbed from 0 rad/sec to 0.0001 rad/sec (37.5 times the orbital rate). For this

perturbation, the deviation of the system center of mass, G∗, relative to the reference

trajectory is very small. Secondly, a body fixed 3-2-3 Euler angle rotation sequence

(α, ζ, γ) is employed to produce an orientation offset of ζ = 5 deg. For this particular

test case, large motions of the center of mass of the spacecraft (G∗) with respect to

the location in the reference orbit (O) are observed. The motion of G∗ relative to O is

displayed in Figure 4.8 over the first revolution in the Lyapunov orbit. A significant

change in the response is observed as the spacecraft passes close to the Moon at 0.5

revolutions along the Lyapunov path. This effect is repeated in the second revolution

of the Lyapunov (see Figure 4.9) when the spacecraft again passes close to the Moon

at 1.5 revolutions. Prior to the encounter with the Moon, the spacecraft is observed

to be departing the reference orbit. As the spacecraft size increases, the motion of G∗

away from the reference location O is noted to also increase. Since the region is very

sensitive, fairly small perturbations in the initial conditions can result in a vehicle

that departs the L1 region significantly earlier than a spacecraft moving along the

60

reference. The unstable eigenvalues and time constants for departure, for members

of the L1 planar Lyapunov family of orbits, are included in Appendix A.

0 0.2 0.4 0.6 0.8 1−100

0

100

200

300

400

500

Rev

Mot

ion

of G

* in

a1 d

irect

ion

(m)

l=10ml=100ml=1000ml=10000ml=100000m

0 0.2 0.4 0.6 0.8 1−2000

−1500

−1000

−500

0

Rev

Mot

ion

of G

* in

a2 d

irect

ion

(m)

l=10ml=100ml=1000ml=10000ml=100000m

Figure 4.8. Motion of G∗ Relative to O, for ζ = 5 deg, over One Revof Large E-M Lyapunov

0 0.5 1 1.5 2−1

0

1

2

3

4

5

6

7x 10

4

Rev

Mot

ion

of G

* in

a1 d

irect

ion

(m)

l=10ml=100ml=1000ml=10000ml=100000m

0 0.5 1 1.5 2−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

5

Rev

Mot

ion

of G

* in

a2 d

irect

ion

(m)

l=10ml=100ml=1000ml=10000ml=100000m

Figure 4.9. Motion of G∗ Relative to O, for ζ = 5 deg, over Two Revsof Large E-M Lyapunov

The effect of decreasing the size of the L1 planar Lyapunov reference orbit in

the Earth-Moon system to an Ay amplitude of 25469 km is also examined. This

size is representative of Lyapunov orbits for actual mission applications. An initial

61

perturbation in ζ of 5 deg is applied and the length of the rod is increased from

10 meters to 100000 meters. The motion of G∗ relative to O is displayed on the

left in Figure 4.10 over two revolutions in the Lyapunov orbit, where the spacecraft

is observed to be departing the vicinity of the Lyapunov reference orbit. Further

examination confirms that the spacecraft are departing on the unstable manifold

associated with the particular reference path, as depicted on the right in Figure 4.10.

3.2 3.4 3.6 3.8 4 4.2 4.4

x 105

−5

0

5x 10

4

a1 (km)

a 2 (km

)

l=10ml=100ml=1000ml=10000ml=100000m

3.2 3.4 3.6 3.8 4 4.2 4.4

x 105

−5

0

5x 10

4

a1 (km)

a 2 (km

)

l=10ml=100ml=1000ml=10000ml=100000m

Figure 4.10. Motion of G∗ Relative to O, for ζ = 5 deg, E-M L1

Lyapunov (Ay = 25469 km)

4.6 Perturbed Asymmetric Rigid Body

One of a relatively few number of investigations into the problem, Wong et al. [10]

examine the motion of a single asymmetric rigid body near the vicinity of the Sun-

Earth libration points, using linear approximations for the planar Lyapunov orbits.

[10] The investigation is focused on the motion of a spacecraft in the vicinity of the L2

libration point and offers orientation results in terms of a body 3-2-1 (θ, φ, ψ) Euler

angle rotation sequence. The linear orbits are represented analytically, thus, the orbit

is not affected by the orientation. The results presented by Wong et al. [10] form a

series of test cases for the comparison with the coupled orbit and attitude model.

The current model framework to couple the translational and rotational motion is

62

employed to expand the investigation offered by Wong et al. The motion of an

asymmetric rigid body moving in the vicinity of L2 in the Earth-Moon system is

examined. Three different nonlinear planar Lyapunov reference orbits are considered,

with approximateAy values equal to 125 km, 6500 km, and 13400 km. These orbits are

selected for comparison to the results from Wong et al. in the Sun-Earth system. [10]

The left plot in Figure 4.11 represents the nonlinear orbits that correspond to the

linear approximations employed in the work by Wong et al. in the Sun-Earth system.

The similar nonlinear orbits in the Earth-Moon system are plotted on the right. The

orbits in the Earth-Moon system are scaled by the ratio of the distance between the

smaller primary and L2 in the two systems. The arrows indicate the direction of

motion in the orbits. It is noted that the mass ratio is larger in the Earth-Moon

system by about four orders of magnitude.

1.504 1.506 1.508 1.51 1.512 1.514

x 108

−4

−3

−2

−1

0

1

2

3

4x 10

5

a1 (km)

a 2 (km

)

Ay=3200kmAy=166000kmAy=350000km

4.2 4.25 4.3 4.35 4.4 4.45 4.5 4.55

x 105

−1.5

−1

−0.5

0

0.5

1

1.5x 10

4

a1 (km)

a 2 (km

)

Ay=125kmAy=6500kmAy=13400km

Figure 4.11. L2 Planar Lyapunov Orbit in the Sun-Earth (left) andEarth-Moon (right) Systems

The spacecraft is modeled as a single asymmetric rigid body, with a total mass

of 100 kg. The moment of inertia properties in b-frame coordinates for the system

are Ib1b1=7.083 kg·m2, Ib2b2=11.333 kg·m2, and Ib3b3=12.417 kg·m2, which are con-

sistent with the quantities employed by Wong et al. [10] All products of inertia are

63

equal to zero. A body fixed 3-2-1 (θ, φ, ψ) Euler angle sequence is employed to visu-

alize the orientation response. Propagating the system with no perturbations yields

zero response in φ or ψ; however, there is a response in the θ, which is amplified as

Ay increases. Two sets of perturbations in (θ, φ, ψ) are explored: (5, 5, 5) deg and

(−5,−5,−5) deg. The first perturbed set of initial angles, (5, 5, 5) deg, corresponds

to the results presented from Wong et al. [10] The impact on the Euler angles as

a result of the simulations initialized with both sets of initial perturbations are dis-

played in Figure 4.12. The positive perturbation yields a very similar response to the

results that appear in Wong et al., especially for the smaller orbits. As Ay increases,

differences are observed between this investigation and the work by Wong et al, likely

due to differences in the system and increases in the nonlinear effects of the orbit.

0 0.5 1 1.5 2 2.5−60

−40

−20

0

20

40

60

Revs

θ (d

eg)

− unperturbed.... positive perturbation−− negative perturbation

Ay=125kmAy=6500kmAy=13400km

0 0.5 1 1.5 2 2.5−8

−6

−4

−2

0

2

4

6

8

Revs

φ (d

eg)

− unperturbed.... positive perturbation−− negative perturbation

Ay=125kmAy=6500kmAy=13400km

0 0.5 1 1.5 2 2.5−15

−10

−5

0

5

10

15

Revs

ψ (

deg)

− unperturbed.... positive perturbation−− negative perturbation

Ay=125kmAy=6500kmAy=13400km

Figure 4.12. Orientation Response of Asymmetric Rigid Body

64

By analyzing the linear attitude equations at the libration points, Wong et al.,

also identified a critical ratio of inertia values that excite large amplitude motion in

θ. For spacecraft in a planar Lyapunov orbit about L2, modeled as linear, as the

value of k3 = (Ib2b2Ib1b1)/Ib3b3 approaches a critical value of 0.36 in the Earth-Moon

system, the amplitude of θ grows to infinity. The critical value is determined by

applying the same process of linearizing the attitude equations of motion that Wong

uses in the Sun-Earth system. [10] Values of k3 further from the critical value yield a

more stable response, assuming that the inertias satisfy the additional condition that

Ib3b3 > Ib2b2 > Ib1b1 as noted in the linear analysis by Wong et al. [10] For relatively

small orbits, this observation holds over the propagation interval when a nonlinear

planar Lyapunov reference is incorporated. The inertia properties in terms of Ib1b1

are modified to supply various k3 values and the results for an Earth-Moon L2 orbit

(Ay = 125 km) are displayed on the left in Figure 4.13. The largest response in θ

occurs for the k3 value that is closest to the critical value of 0.36. The amplitude

response in θ diminishes for k3 values below and also above the critical value as

expected. However, when larger nonlinear orbits are selected as the reference paths,

large amplitude motion is apparent for a variety of k3 values. Given the same set of

k3 values, the response in θ for a planar Lyapunov L2 orbit (Ay = 13400 km) in the

Earth-Moon system is plotted on the right in Figure 4.13.

It is clear that the amplitude response in θ is no longer dominated by the critical

value of k3 and the nonlinear effects are significant in the attitude behavior over the

propagation interval. It should be noted that there is no motion of the spacecraft

system center of mass (G∗) relative to the location in the reference orbit (O) for any

k3 value investigated. Also, unlike the unperturbed rigid asymmetric spacecraft in

the large L1 Earth Moon Lyapunov orbits, the reference orbits are all sufficiently far

away from the Moon that the behavior is not dominated by a close lunar approach.

65

0 0.5 1 1.5 2 2.5−2

−1.5

−1

−0.5

0

0.5

1

Rev

θ (d

eg)

k3=0.671

k3=0.510

k3=0.342

k3=0.188

k3=0.027

0 0.5 1 1.5 2 2.5−100

−80

−60

−40

−20

0

20

40

60

Rev

θ (d

eg)

k3=0.671

k3=0.510

k3=0.342

k3=0.188

k3=0.027

Figure 4.13. Orientation Response for Ay = 125 km (left) and Ay =13400 km (right) for Nonlinear E-M Lyapunov

4.7 Summary of Results for L1 Lyapunov Family

Although the previous test cases offer insight into the attitude motion in the

circular restricted three-body problem, more global understanding of the behavior

in this regime is desired. To bring the results from simulations involving all planar

Lyapunov orbits in the family of reference orbits together, the concept of an “attitude

map” is now introduced. An attitude map is employed to visually display regions

where the orientation remains aligned with respect to the CRTBP rotating frame

and, in contrast, regions where the orientation is changing relative to the CR3BP

frame. The attitude motion of a general asymmetric rigid body is the focus. Three

parameters are defined: inertia ratios k1, k2, and k3.

k1 =I33 − I22

I11

(4.1)

k2 =I33 − I11

I22

(4.2)

k3 =I22 − I11

I33

(4.3)

66

For now, only the parameter k3 is explored. A body fixed 3-2-1 (θ, φ, ψ) Euler angle

sequence is employed to determine the maximum values of the attitude angles θ, φ,

and ψ, relative to the CR3BP rotating frame as observed for a spacecraft over one

revolution in an L1 Lyapunov orbit, as predicted by the coupled orbit and attitude

model. This process is repeated for many members of the Lyapunov family of reference

orbits, for inertia ratios in k3 varying from -1 to 1. The results are displayed in the

form of a map for each angle. Figure 4.14 displays the global response in θ, over

one revolution of each Lyapunov reference orbit. The body frame, the b-frame, is

initially aligned with the CR3BP rotating frame. The areas colored in blue are

regions where the body frame stays closely aligned with the CR3BP rotating frame,

for that particular orientation angle. Blue on the map corresponds to an orientation

angle of 0 degrees. The red areas are regions where the spacecraft orientation, in

terms of one of the orientation angles, is changing significantly relative to the CR3BP

rotating frame. Red on the map highlights angles greater than 90 degrees relative to

the initial orientation.

k3 (nd)

Ay

(km

)

−1 −0.5 0 0.5 1

194930

173190

150450

127830

105140

82460

59846

37184

16297

Am

plitu

de (

deg)

0

45

90

Figure 4.14. Orientation response in θ for L1 Lyapunov Family (E-M System)

67

The unperturbed maps for the other two orientation angles, φ and ψ, remained very

close to zero everywhere. It is observed from the map that there are specific areas

in terms of the inertia ratio k3 and the reference orbit Ay where the orientation is

not changing relative to the CR3BP rotating frame. The center line, where k3 = 0,

denotes the location where the spacecraft is a perfect cube; supporting the stability

observations in this region. A more interesting area of the map is the horizontal and

slightly diagonal band that crosses the map. In certain reference orbits, spacecraft

with specific inertia ratios falling on the blue line remain relatively stable in orien-

tation with respect to the CR3BP rotating frame. A set of planar Lyapunov orbits,

depicted in Figure 4.15, in the region of the blue horizontal band are selected for a k3

value of 0.6. The left plot in Figure 4.16 displays the orientation response for a vehicle

along these reference paths, over one revolution of the reference. The response on the

left of Figure 4.16 indicates that vehicles with a k3 value corresponding to 0.6 exhibit

relatively little orientation change relative to the CR3BP rotating frame between an

Ay amplitude of approximately 106000 km and 110000 km. It appears as though the

interaction with the Moon at the 0.5 revolution point alters the orientation motion

such that it remains relatively close to zero over the entire revolution. However, closer

examination of the motion over 2.5 revolutions, displayed on the right in Figure 4.16,

reveals that after the second pass near the Moon, the orientation in terms of the angle

θ is no longer remaining close to the a-frame. This interesting result is not necessarily

apparent when only considering specific test cases.

The maps can also be extended to include perturbations. The results for two

small perturbations and two larger perturbation sets (same perturbation applied to

all three angles simultaneously), appear in Figures 4.17 and 4.18. These maps are

useful to summarize the results observed over the planar Lyapunov orbits that are

employed as reference orbits. Additional maps for the L2 planar Lyapunov trajectories

are presented in Appendix B. The map could prove to be useful for mission design

applications.

68

1 2 3 4 5

x 105

−1.5

−1

−0.5

0

0.5

1

1.5x 10

5

a1 (km)

a 2 (km

)

Ay=101108kmAy=105619kmAy=110118kmAy=114621km

Figure 4.15. Set of L1 Planar Lyapunov Orbits (E-M System) in Blue Band Region

0 0.2 0.4 0.6 0.8 1

−80

−60

−40

−20

0

20

40

60

80

Rev

θ (d

eg)

Ay=101108kmAy=105619kmAy=110118kmAy=114621km

0 0.5 1 1.5 2 2.5−1000

0

1000

2000

3000

4000

Rev

θ (d

eg)

Ay=101108kmAy=105619kmAy=110118kmAy=114621km

Figure 4.16. Response in θ in Blue Band Region (k3=0.5), 1 rev (left)2.5 revs (right))

69

k3 (nd)

Ay

(km

) Response in θ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5 1

194940

173190

150450

127830

105130

82467

59837

37182

16298

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in θ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5 1

194940

173190

150450

127840

105140

82465

59838

37184

16298

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in φ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5 1

194940

173190

150450

127830

105130

82467

59837

37182

16298

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in φ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5 1

194940

173190

150450

127840

105140

82465

59838

37184

16298

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in ψ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5 1

194940

173190

150450

127830

105130

82467

59837

37182

16298

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in ψ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5 1

194940

173190

150450

127840

105140

82465

59838

37184

16298

Am

plitu

de (

deg)

0

45

90

Figure 4.17. Orientation Response to Small Perturbation Sets for L1

Planar Lyapunov Family (E-M System)

70

k3 (nd)

Ay

(km

) Response in θ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5 1

194930

173190

150460

127830

105140

82465

59843

37181

16298

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in θ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5 1

194940

173190

150460

127840

105140

82457

59836

37181

16297

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in φ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5 1

194930

173190

150460

127830

105140

82465

59843

37181

16298

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in φ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5 1

194940

173190

150460

127840

105140

82457

59836

37181

16297

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in ψ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5 1

194930

173190

150460

127830

105140

82465

59843

37181

16298

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in ψ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5 1

194940

173190

150460

127840

105140

82457

59836

37181

16297

Am

plitu

de (

deg)

0

45

90

Figure 4.18. Orientation Response to Medium Perturbation Sets forL1 Planar Lyapunov Family (E-M System)

71

5. MOTION ALONG SPATIAL PERIODIC REFERENCES

The CR3BP allows five equilibrium solutions as points relative to the rotating system

where the gravitational forces and rotating frame dynamical accelerations cancel,

assuming a vehicle modeled as a point mass. One of the collinear equilibrium points,

L1, located along the line connecting the two primaries, and located between them, is

of particular interest in this investigation. Both two and three-dimensional periodic

orbits exist in the vicinity of the libration points. Three-dimensional periodic orbits

have been computed by differentially correcting initial conditions in the vicinity of

the L1 collinear libration point. A family of northern Earth-Moon (E-M) halo orbits

in the vicinity of L1 is depicted in Figure 2.5. Members of this orbit family are used

as references for several test cases.

5.1 Perturbed Asymmetric Rigid Body

In Chapter 4, the motion of a perturbed asymmetric rigid body in a set of L2

Earth-Moon Lyapunov orbits is determined and compared to the results by Wong et

al. [10], for a similar rigid body moving in a similar set of L2 Sun-Earth Lyapunov

orbits. The results indicate good correlation between the two investigations, especially

for smaller orbit sizes. As an expansion of the previous contributions, the motion of

an asymmetric rigid body moving in the vicinity of L1 in the Earth-Moon system is

now examined. One set of nonlinear halo reference orbits is selected that corresponds

to orbits in the blue range of the L1 Northern halo family depicted in Figure 2.5. A

set of larger halo orbits are also employed as reference paths and these trajectories

are drawn from orbits in the red range of the family in Figure 2.5. The small set of

these halo orbits is displayed in Figure 5.1. The green orbit represents a halo orbit

corresponding to Az = 500 km, the red orbit is larger such that Az = 1000 km, and

72

the blue orbit includes more out-of-plane motion at Az = 5000 km. Note that Az

corresponds to the maximum out-of-plane excursion along the periodic orbit.

3.2 3.24 3.28−2−1

01

2−4000

−2000

0

2000

4000

a1 (105 km)

a2 (104 km)

a 3 (km

)

Az=500kmAz=1000kmAz=5000km

Figure 5.1. Small Set of L1 Northern Halo Reference Orbits in Earth-Moon System

The spacecraft is modeled as a single asymmetric rigid body. The inertia proper-

ties in b-frame coordinates for the rigid body include the moments Ib1b1=7.083 kg·m2,

Ib2b2=11.333 kg·m2, and Ib3b3=12.417 kg·m2, which are consistent with the quantities

employed by Wong et al [10]. All products of inertia are equal to zero. A body fixed

3-2-1 (θ, φ, ψ) Euler angle sequence is employed to visualize the orientation response.

A perturbation in (θ, φ, ψ) is explored: (5, 5, 5) deg. The impact on the Euler an-

gles as a result of simulations initialized with the set of perturbations is displayed in

Figures 5.2.

Assume that the vehicle is moving in a halo orbit from the larger set is displayed

below in Figure 5.3. The green orbit represents a halo orbit corresponding to Az =

10000 km, the red orbit possesses a maximum out-of-plane excursion equal to Az =

50000 km, and the path along the blue orbit reaches Az = 70000 km. Then again, a

perturbation in the attitude angles such that (θ, φ, ψ) equal to (5, 5, 5) deg is explored

for the large orbit as a reference path. The impact on the Euler angles as a result of

simulations that are initialized with the perturbations for the vehicles in larger halo

orbits is displayed in Figures 5.4.

73

0 0.5 1 1.5 2 2.5−60

−40

−20

0

20

40

60

Revs

θ (d

eg)

___ unperturbed− − perturbed

Az=500kmAz=1000kmAz=5000km

0 0.5 1 1.5 2 2.5−40

−30

−20

−10

0

10

20

30

40

Revs

φ (d

eg)

___ unperturbed− − perturbed

Az=500kmAz=1000kmAz=5000km

0 0.5 1 1.5 2 2.5−80

−60

−40

−20

0

20

Revs

ψ (

deg)

___ unperturbed− − perturbed

Az=500kmAz=1000kmAz=5000km

Figure 5.2. Asymmetric Orientation Response for Small L1 NorthernHalo Reference Orbits in the E-M System

In general, the orientation responses are observed to differ depending on the orbit

type. For the set of small reference orbits, the orientation response in terms of the

first Euler angle, θ, appears in Figure 5.2 to be oscillatory and relatively periodic

over the length of the simulation. Due to the perturbation, the response shifts and

the magnitude of the amplitude in θ varies noticeably for all three small reference

orbits. The response in θ for vehicles in one of the large halo orbits, as apparent in

Figure 5.4, do not exhibit similar behavior. As the orbit size increases, the motion

becomes unbounded and the divergence appears relatively quickly. The motion in

the second orientation angle, φ, for bodies in an orbit from the smaller orbit set is

displayed in Figure 5.2; the response appears to be oscillatory although the amplitude

74

3.23.4

3.6−4

−20

24

−2

0

2

4

6

a1 (105 km)

a2 (104 km)

a 3 (10

4 km

)

Az=10000kmAz=50000kmAz=70000km

Figure 5.3. Set of Larger L1 Northern Halo Reference Orbits in Earth-Moon System

is increasing with time. The magnitude of the attitude response for a vehicle in one

of the orbits from the larger set is plotted in Figure 5.4, diverges more quickly than

the response when the vehicle is in a smaller orbit, but overall, generally similar

characteristics appear. The exception is the orientation response for a vehicle the

largest orbit, with an Az amplitude of 70000 km, where the influence of the close

pass to the Moon at 0.5 rev and 1.5 rev locations is visible. In comparing Figures 5.2

and 5.4, the magnitude in the amplitude response of the third angle ψ, for bodies in

orbits between the small and large reference paths, is noted to increase much more

significantly. The test cases are explored to further expand on the analysis offered

in Wong et al. [10] by considering the orientation history resulting from selecting

nonlinear, L1, halo orbits as the reference path. While the results presented by Wong

et al. [10] display bounded, oscillatory motion for the first two Euler angles, θ and φ,

over the length of the simulation, recent results indicate that for larger orbit size, the

motion may become unbounded, less oscillatory, and less predictable in general.

75

0 0.5 1 1.5 2 2.5−1200

−1000

−800

−600

−400

−200

0

200

Revs

θ (d

eg)

___ unperturbed

− − perturbed

Az=10000kmAz=50000kmAz=70000km

0 0.5 1 1.5 2 2.5

−100

−50

0

50

100

Revs

φ (d

eg)

___ unperturbed

− − perturbed

Az=10000kmAz=50000kmAz=70000km

0 0.5 1 1.5 2 2.5−400

−200

0

200

400

600

800

Revs

ψ (

deg)

___ unperturbed

− − perturbed

Az=10000kmAz=50000kmAz=70000km

Figure 5.4. Orientation Response for an Asymmetric Vehicle in LargeL1 Northern Halo Reference Orbits in the E-M System

5.2 Perturbed Axisymmetric Rigid Body

The general model that has been developed is adaptable for exploring the behavior

of a range of vehicle masses and configurations. The behavior of an axisymmetric

body under a perturbation in the initial orientation is now explored. The spacecraft

is comprised of two rigid rods, each of length 10 meters and radius 0.1 meter. The

rods are stacked, one on top of the other, configured such that the axial direction

is aligned with b3. The top rod can spin about b3 relative to the bottom rod. For

visualization purposes, the orientation is represented in terms of a body fixed 3-2-3

(α, ζ, γ) Euler angle sequence, for a variety of test cases. The angle α represents

precession of the b-frame relative to the a frame, the angle ζ is nutation, and spin

76

about the axisymmetric axis is denoted by the angle γ. A 5 degree perturbation

in the nutation angle, ζ, is considered for an axisymmetric vehicle in four different

periodic L1 halo reference orbits in the Earth-Moon system with Az values of 500 km,

5000 km, 10000 km, and 50000 km. The reference trajectory corresponding to each

Az value is displayed in Figure 5.5.

Figure 5.5. Four E-M Halo Reference Trajectories

The attitude response, in terms of the body fixed 3-2-3 (α, ζ, γ) Euler angle se-

quence is considered. A perturbation in the nutation angle of ζ = 5 degrees, for the

four halo reference orbits is introduced. In Figure 5.6, the precession angle of the

b-frame with respect to the a-frame, α, is observed to be oscillatory and bounded

for motion in the orbit with the smallest Az value, irregular for intermediate values

of Az, and increasing at a fairly constant rate for the largest Az of 50000 km. The

nutation angle, ζ, for rods moving in halo orbits of all sizes is observed in Figure 5.6

to oscillate around ζ = 90 degrees, which is consistent with knowledge of attitude

motion in the two-body problem; in terms of the nutation angle, a long thin rod is

unstable in the vertical configuration and stable when the long axis is oriented in a

radial direction. The frequency of oscillations appears to increase with the size of

the halo reference trajectory as well. In Figure 5.6, the spacecraft spin angle, γ, is

77

noted to be increasing at a greater rate when moving in the relatively smaller halo

reference orbits in contrast to reference orbits possessing larger values of Az. With

the exception of the largest halo, orbit however, the spin angle, γ, appears to be fairly

irregular. Along the reference trajectory at Az = 50000 km, the overall response in

γ is observed to be oscillatory and increasing.

0 0.5 1 1.5 2 2.5−500

0

500

1000

1500

2000

Rev

α (d

eg)

Az=500kmAz=5000kmAz=10000kmAz=50000km

0 0.5 1 1.5 2 2.5−150

−100

−50

0

50

100

150

200

Rev

ζ (d

eg)

Az=500kmAz=5000kmAz=10000kmAz=50000km

0 0.5 1 1.5 2 2.5−200

−100

0

100

200

300

400

500

600

700

Rev

γ (d

eg)

Az=500kmAz=5000kmAz=10000kmAz=50000km

Figure 5.6. Orientation Response to Perturbation ζ = 5 degrees in E-M Halos

5.3 Spin Stabilization of Axisymmetric Spacecraft

Spin stabilization of the spacecraft in a halo reference orbit is now considered.

The spacecraft is comprised of two rigid rods, each of length 10 meters and radius

0.1 meter. The rods are stacked on top of each other, configured so that the axial

direction is aligned with b3. The top rod can spin about b3 relative to the bottom

78

rod. The orientation is expressed in terms of a body fixed 3-2-3 (α, ζ, γ) Euler angle

sequence. The spacecraft motion, under the perturbation ζ = 5 degrees, is explored

for one E-M halo reference orbit, i.e., Az = 5000 km and is attempted by spinning

body R with respect to body B at two rates: 0.1 rad/s (approximately 1 rpm) and

10 rad/s (approximately 95 rpm). In Figure 5.7, the perturbed response is displayed

in green, the red line indicates motion corresponding to a spin rate of 0.1 rad/s, and

the blue line reflects to a spin rate of 10 rad/s.

0 0.5 1 1.5 2 2.5−600

−500

−400

−300

−200

−100

0

100

200

Rev

α (d

eg)

perturbedw spin (0.1 rad/s)w spin (10 rad/s)

0 0.5 1 1.5 2 2.5−100

−50

0

50

100

150

200

Rev

ζ (d

eg)

perturbedw spin (0.1 rad/s)w spin (10 rad/s)

0 0.5 1 1.5 2 2.5−100

0

100

200

300

400

500

600

Rev

γ (d

eg)

perturbedw spin (0.1 rad/s)w spin (10 rad/s)

Figure 5.7. Orientation Response to Spin Stabilization in the E-M System

It is apparent in Figure 5.7 that spin stabilization to accommodate this partic-

ular perturbation depends on the spin rate employed, when focused specifically on

the nutation angle ζ. In the simulations of the perturbed motion, the spacecraft ex-

hibits oscillatory behavior in α and unbounded motion in the other two angles which

79

confirms that the spacecraft is not maintaining the initial orientation relative to the

rotating frame. Employing a spin rate of 0.1 rad/sec, the nutation angle response

is observed to be damped; however, the spacecraft is still oscillating about ζ = 90

degrees. This response is not unexpected; in addition, recall that this orbit possesses

an out-of-plane excursion of 5000 km The precession of the b-frame with respect to

the a-frame appears in Figure 5.7 to be bounded for this spin rate, and the spin angle

for body B of the spacecraft is growing less rapidly in comparison to the baseline

perturbed case, as apparent in Figure 5.7. When the spin rate of body R is increased

to 10 rad/s, nearly all effects of the initial perturbation on the nutation angle are

eliminated and the body is maintaining the initial orientation in ζ. Not surprisingly,

the spacecraft still precesses with respect to the orbit frame at a comparable rate

to the perturbed case, but the spin angle response is improved with respect to the

perturbed case.

Finally, the impact of spin stabilization on the orbit is considered. The drift in

the center of mass location relative to the reference orbit is displayed in Figure 5.8.

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Rev

Mag

nitu

de o

f M

otio

n of

G*

wrt

0 (

km)

perturbedw spin (0.1 rad/s)w spin (10 rad/s)

Figure 5.8. Orbital Motion Response to Spin Stabilization in the E-M System

The orbital motion for the perturbed spacecraft is plotted in green, and the spin

stabilized spacecraft, with spin rate equal to 0.1 rad/sec, in red; the center of mass,

80

G∗, divergence given a spin rate of 10 rad/sec is plotted in blue. This figure illustrates

that spinning the spacecraft results in only very marginal improvement in maintaining

the desired center of mass motion relative to the reference orbit. This test case serves

two purposes. First, the results confirm that the spacecraft is displaying motion

consistent with previous results and serves to verify the model formulation. Secondly,

it demonstrates an application of spin stabilization, at least for some orientation

parameters, in the coupled orbit and attitude model in the CR3BP.

5.4 The Impact of a Perturbed Axisymmetric Spacecraft on the Orbit

Employing the fully coupled dynamical model,the effects of an attitude pertur-

bation on the orbit are considered for an axisymmetric spacecraft, comprised of two

rigid rods with a length-to-radius ratio of 100:1. The rods are stacked one-on-top of

the other, and configured such that the axial direction is aligned with the out-of-plane

direction, b3. A 5 degree perturbation in the nutation angle, ζ, is considered for four

periodic L1 halo reference orbits in the Earth-Moon system with Az values of 5000

km, 20000 km, and 50000 km. In Figure 5.9, the departure of the spacecraft from

the reference is evident. It is observed that smaller values of Az actually correspond

to a larger motion of the spacecraft center of mass, G∗, away from the isochronous

location along the reference path, O. In any family of L1 halo orbits, the orbits with

smaller Az values are unstable. The perturbations in these smaller, more unstable

orbits appear to induce a more significant departure than in larger, less unstable or-

bits based on the magnitude of the unstable eigenvalue of the associated monodromy

matrix.

The effects of increasing the length, while maintaining a constant ratio of length-

to-radius, are examined for a L1 halo orbit in the Earth-Moon system with an Az

amplitude of 5000 km. A perturbation in the nutation angle is examined to investigate

the coupling effects and the length of the rod is increased from 10 meters to 10000

meters. A 5 degree perturbation in the nutation angle, ζ, is considered. The response,

81

Figure 5.9. Orbital Response to perturbation of ζ = 5 degrees in E-M Halos

in Figure 5.10, indicates that as the size of the spacecraft increases, the effects of the

attitude on the orbit also increase. This result is not unexpected, and helps to confirm

the functionality of the simulation.

0 0.5 1 1.5 2 2.50

2000

4000

6000

8000

10000

Rev

Mot

ion

of G

* in

a1 d

irect

ion

(km

)

l=10ml=100ml=1000ml=10000m

0 0.5 1 1.5 2 2.5−14000

−12000

−10000

−8000

−6000

−4000

−2000

0

Rev

Mot

ion

of G

* in

a2 d

irect

ion

(km

)

l=10ml=100ml=1000ml=10000m

Figure 5.10. Motion of G∗ Relative to O, for ζ = 5 deg, over TwoRevs of E-M Halo

82

5.5 Summary of Results for L1 Northern Halo Family

The attitude behavior for bodies moving in the CR3BP is further clarified if

spatial, i.e., 3-dimensional reference trajectories are employed. To bring the results

from all the reference trajectories in any given family together, the concept of an

“attitude map”, as was introduced in the Chapter 4 for planar reference orbits, offers

a quick to visual display of the relationship between the size of the reference orbit

and the shape (through inertia ratios, k) of the body. In particular, the maps reflect

the maximum orientation over one revolution relative to an initial alignment with

the rotating frame. Thus, each point in the map reflects an orbit and inertia ratio

combination. Observations of the map indicate regions where the orientation remains

aligned with respect to the CRTBP rotating frame and regions where the orientation is

changing relative to the CR3BP frame. The attitude motion of a general asymmetric

rigid body in the CRTBP is again considered but now the reference paths are 3-

dimensional. Three parameters representing the inertia ratios, k1, k2, and k3, as

previously defined in Equations (4.1) - (4.3) are again employed. For this discussion,

the focus is only on the parameter k1. A body fixed 3-2-1 (θ, φ, ψ) Euler angle

sequence is employed to determine the maximum values of θ, φ, and ψ, relative to

the CR3BP rotating frame observed over one revolution of a reference path that is

a member of the L1 Northern halo family, as predicted by the coupled orbit and

attitude model. This process is repeated for many periodic members of the family,

for inertia ratios in k1 varying from -1 to 1. The results are displayed in the form

of a map for each angle. The global response in θ, φ, and ψ, appears in Figure 5.11

displayed over one revolution of each periodic halo reference orbit. Recall that the

body frame, i.e., the b-frame, is initially aligned with the CR3BP rotating frame.

The areas colored in blue are regions where the body frame stays closely aligned

with the CR3BP rotating frame, for the specific orientation angle. Blue on the map

corresponds to an orientation angle of 0 degrees. The red areas are regions where

the spacecraft is changing orientation significantly relative to the CR3BP rotating

83

frame. A red point on the map correlates to angles with maximum absolute values

that exceed 180 degrees relative to the initial orientation.

k1 (nd)

Az

(km

)

Response in θ − unperturbed

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in φ − unperturbed

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in ψ − unperturbed

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

Figure 5.11. Orientation Response in L1 Northern Halo Family (E-M System)

It is observed from the map that there are clearly defined areas where the orien-

tation is not changing significantly relative to the CR3BP rotating frame. The center

line, where k3 equals to zero, denotes the inertia ratio that reflects the fact that the

spacecraft is a perfect cube; thus the stability observed in this region is not unex-

pected. Similarly to the L1 Lyapunov maps, a horizontal and slightly diagonal band

also exists for all orientation angles in the spatial halo family, but it does not extend

as far in the horizontal direction as in the Lyapunov family and it is pushed to high

values of Az in the range of 67500 km to 75000 km. In certain orbits, spacecraft with

specific inertia ratios falling on the blue line remain relatively stable in orientation

84

with respect to the CR3BP rotating frame. The maps also indicate that some trends

in attitude “stability” are visible in all three orientation angles, i.e., there are some

regions that are blue in all three maps in Figure 5.11 and other areas that are pre-

dominately red for the three orientation angles. Since three-dimensional motion of

the vehicle occurs along the members of the halo family of orbits, all three orientation

angles must be considered simultaneously to determine regions where the vehicle is

maintaining a similar orientation as the CR3BP rotating frame. The maps are again

extended to include perturbations. Two small perturbations sets and two larger per-

turbation sets (same perturbation applied to all three angles simultaneously) are also

considered. The results, displayed in Figures 5.12 and 5.13, indicate the evolution

of the “stability” for each orientation angle in response to the applied perturbation.

Considering the smallest perturbation set in Figure 5.12, it appears that potentially

some new areas of “stability” are realized in response to the applied perturbation, for

orbits with a small Az amplitude and negative values of k1. As larger perturbations

are applied, the blue and red regions continue to shift. The horizontal and slightly

diagonal band, observed for large values of Az in the unperturbed case, begins to

extend in the negative k1 direction; however, with the excepts of this band, all other

locations corresponding to negative k1 values shift to become more red, considering

the orientation angles θ and ψ. The attitude maps developed are useful to summarize

the all the results observed over for the L1 spatial Northern halo family. Additional

maps for the L2 Northern halo family are presented in Appendix C. The maps could

prove to be useful for mission design applications. Given vehicle properties for a

specific mission, the regions of “stability” for all potential reference orbits can be

considered. The maps could also be used to make predictions of how the orienta-

tion motion might change in the case where spacecraft inertia properties are altered

on-orbit, i.e, for cases where a telescoping arm is extended or a sail is deployed.

85

k1 (nd)

Az

(km

) Response in θ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in θ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in φ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in φ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in ψ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in ψ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

Figure 5.12. Orientation Response to Small Perturbation Sets for L1

Northern Halo Family (E-M System)

86

k1 (nd)

Az

(km

) Response in θ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in θ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in φ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in φ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in ψ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in ψ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

Figure 5.13. Orientation Response to Medium Perturbation Sets forL1 Northern Halo Family (E-M System)

87

6. APPLICATION TO SOLAR SAILS

One further objective in the development of the coupled orbit and attitude model

is the capability to directly influence the orbit with the orientation. A solar sail

offers a potentially significant coupling. Thus, the departure of the spacecraft from

the vicinity of the reference orbit is considered, as predicted by the coupled orbit

and attitude model. The potential for the attitude configuration to be employed to

control the orbital motion of the solar sail is explored.

6.1 Reference Trajectories that Incorporate a Sail Force

A solar sail immediately introduces the challenge of a more complex reference

trajectory. Thus the first step in developing tools for the analysis is the construction

of a family of periodic orbits, in the vicinity of the libration points, that incorporate a

solar sail. For this analysis, the L2 point in the Earth-Moon system is selected as the

candidate libration point. The computation is accomplished using a fixed-time single

shooting algorithm, where the period of each full reference trajectory, in dimensional

form, is equal to the system synodic period. [18,21,25] This period is required to ensure

that the Sun returns to the same location in the Earth-Moon rotating frame after

one revolution along the reference orbit. The scheme is formulated by originating the

search for a family of solar sail orbits with the vertically-shifted linear approximation

for elliptically-shaped orbits near the libration points that is explored in Simo and

McInnes [22–24] and, then, employing the numerical extension process in Wawrzyniak

and Howell. [18,21,25] In the investigation here, an ideal solar sail model is assumed,

where the nondimensional acceleration associated with the sail is,

as(τ) = ac

(l1(τ) · us

)2

us (6.1)

88

where ac is the characteristic sail acceleration, l1 is the sunlight direction, τ is the

nondimensional time and us is the direction of the sail normal. In the CR3B Earth-

Moon rotating frame, the sunlight direction frame, denoted the l-frame, is defined,

l1 = cos(Ωlτ)a1 − sin(Ωlτ)a2

l2 = sin(Ωlτ)a1 + cos(Ωlτ)a2

l3 =a3

(6.2)

where Ωl is the rate at which the sunlight direction is changing in the Earth-Moon

reference frame, described as the ratio of the sidereal to synodic periods of the Earth-

Moon system [25].

The orientation of the solar sail normal direction, us, with respect to the Earth-

Moon rotating frame, the a-frame, is described by a body 3-2-1 (θs − Ωlτ , αs, γs)

Euler angle rotation sequence. The third angle, γs, is set equal to zero since the sail

normal direction is fully described by solely two angles. The sequence consistently

conveys the sail orientation relative to the Earth-Moon rotating frame. The family of

orbits computed by Wawrzyniak [18, 21] is based on the assumption that the vehicle

maintains a fixed sail orientation with respect to the sunlight direction in the Earth-

Moon system. To generate this particular family, the value of θs is held constant at 0

degrees and the value of αs is fixed at −35.264 degrees. This value for αs corresponds

to the angle that generates the maximum out-of-plane force, based on the linear

equations of motion as described by Simo and McInnes [22]. Using Equations (6.1)

and (6.2) as well as the Euler angle rotation sequence, the contributions of the sail

are incorporated into the traditional circular restricted three-body problem equations

of motion. Note that the characteristic acceleration is permitted to vary across the

family and serves as one of the continuation parameters. The equations of motion,

augmented with the sail force, are written,

x = 2y + x− (1− µ)(x+ µ)

d3− µ(x− 1 + µ)

r3

+ ac cos(αs)3 cos(Ωlτ)

(6.3)

89

y = −2x+ y − (1− µ)y

d3− µy

r3− ac cos(αs)

3 sin(Ωlτ) (6.4)

z = −(1− µ)z

d3− µz

r3− ac cos(αs)

2 sin(αs) (6.5)

From these equations of motion, a family of periodic, three-dimensional, solar sail ref-

erence orbits is computed via a single shooting differential corrections process. Since

a fixed-time targeter is employed, the characteristic acceleration is included as one

of the free variables. The characteristic acceleration is a property of the sail, and as

such, it does not vary over one trajectory; however, it is permitted to change between

members of the family. The family of orbits is displayed in Figure 6.1. The color vari-

ation in Figure 6.1 indicates the range in values of the nondimensional characteristic

acceleration across the family of trajectories. The smallest orbit, displayed in blue

possesses a characteristic acceleration of 0.0001 (0.0003 mm/s2). The largest orbit,

displayed in red, corresponds to a value of 0.0894 (0.2450 mm/s2).

to Earth

Figure 6.1. Family of Solar Sail Reference Orbits in the Vicinity ofL2 in the E-M System

90

6.2 Modifications to Include Attitude and Develop a Coupled Model

The model framework that was previously developed to formulate the coupled or-

bit and attitude equations of motion is now modified slightly to incorporate the effects

of solar radiation pressure. A particular reference orbit is selected, and accordingly,

the characteristic acceleration corresponding to that reference is, thus, available. The

rotational degree of freedom between the two possible bodies in the spacecraft model

is assumed to be fixed, creating one rigid body. Realistic parameters are selected to

reflect the mass and inertia of an approximate solar sail type spacecraft. The pa-

rameters are described such that the sail normal, us, corresponds to the body-fixed

b1 direction. The motion of the sunlight direction, as described by l1 in Equation

(6.2) is monitored and the force contribution due to the solar sail is computed by

incorporating the mass of the solar sail and the acceleration as modeled in Equation

(6.1), dimensionalizing as appropriate. The initial conditions are consistent with the

selected solar sail reference orbit. As expected, due to the orbit and attitude coupling,

when the system equations of motion are propagated forward in time, the solar sail

does not remain in the vicinity of the reference solution.

6.3 Turn-and-Hold Model

The orbit and the sail orientation are both very sensitive to perturbations in this

regime and the coupling heightens such an impact. To explore the influence of the

orbit and attitude coupling, the drift of the center of mass is monitored and the sail

orientation is explored for orbit control. A multiple shooting scheme is employed in

an attempt to control the orbital motion of the spacecraft such that it remains in

the vicinity of the desired reference. The applied “control” modifies the sail angles

discretely and is not optimal. At some point in time, τi, observations indicate that

the orbital motion in the coupled orbit and attitude model deviates sufficiently far

from the reference orbit and, therefore, some correction is required. The position and

velocity state representing the coupled system at τi is denoted xi and reflects the state

91

of the system center of mass, G∗. A turn-and-hold strategy, similar to one proposed

by Wawrzyniak [21, 25], is then employed to determine the attitude corrections that

are required to guide the solar sail back to the vicinity of the reference orbit at some

specified final time, τf . Intermediate patch points are introduced at times, τi+1, τi+2,

... τi+np , where np represents the number of intermediate patch points. The position

and velocity states at each intermediate point are described by, xi+1, xi+2, ... xi+np

and the position and velocity at the final state, is denoted xf , and corresponds to a

state at τf on the reference trajectory. Initial conditions corresponding to the position

and velocity states are selected to be close to those associated with the reference

trajectory; however, the position and velocity state at each patch point is part of the

design vector and vary during the multiple shooting process. In the turn-and-hold

strategy, the spacecraft is assumed to maintain a fixed orientation with respect to

a frame associated with the sunlight direction over each multiple shooting segment.

This simplification to the model facilitates the computation of discrete corrections

to the spacecraft orientation at a number of intermediate patch points. Finally, if

the attitude corrections are successfully designed, their application to the coupled

orbit and attitude model produces a configuration that remains in the vicinity of the

reference orbit for a pre-determined total time interval. The corrections are actually

determined iteratively, as required, based on the departure from the reference path.

A simple example serves to illustrate this strategy. A schematic of the process is

displayed in Figures 6.2 to 6.4, using three internal patch points (depicted in green).

The starting location at τ1 is denoted by the red point and the desired final location

at τ5 is black. The orientation corrections are applied at τ1 through τ4, computed

via the turn-and-hold model, to the coupled orbit and attitude model. These cor-

rections should aid the spacecraft in maintaining an orbital response that remains

in the vicinity of the reference path. The general targeting scheme is implemented

via the technique of constraints and free variables that was explored in Chapter 2

and is combined with the multiple shooting strategy, established through the simple

example, for the more challenging solar sail application. Specifically, the formulation

92

t i

t 2t 3

t 4

t 5

Reference Trajectory

Coupled Orbit and Attitude Trajectory

Multiple Shooting with Turn-and-Hold Model

Figure 6.2. Initial Condition for Multiple Shooting Algorithm

t it 2

t 3

t 4

t 5

Reference Trajectory

Coupled Orbit and Attitude Trajectory

Multiple Shooting with Turn-and-Hold Model

Figure 6.3. Corrections Determined Using Turn-and-Hold Model

of the constraint equations that are required to govern the motion in the turn-and-

hold model are explored. The normal directions associated with the sail at the initial

point and at each intermediate patch point are included as parameters in the design

vector. Over each multiple shooting segment, the sail normal direction is fixed with

93

t i

t 2

t 3

t 4

t 5

Reference Trajectory

Coupled Orbit and Attitude Trajectory

Apply Corrections from Turn-and-Hold Model to Coupled Model

Figure 6.4. Corrections Applied to Coupled Orbit and Attitude Model

respect to a frame associated with the sunlight pointing direction, l1, as described in

Equation (6.2). At the initial time, τi, the sail normal direction is introduced in the

form,

usi = Us1i l1 + Us2i l2 + Us3i l3 (6.6)

Since the sail normal direction is a unit vector, the following equality constraint

equation is applied at the initial point,

Us1iUs1i + Us2iUs2i + Us3iUs3i − 1 = 0 (6.7)

Similar sail normal directions are described to correspond to each patch point, ui+1

through ui+n, and the associated constraint equations are applied, similar to Equation

(6.7). The description of the sail normal from Equation (6.6) is applied to Equation

(6.1) and the following updated equations of motion are derived that govern the

behavior under the turn-and-hold model,

94

x = 2y + x− (1− µ)(x+ µ)

d3− µ(x− 1 + µ)

r3

+ ac(U3s1 cos(Ωlτ) + U2

s1Us2 sin(Ωlτ)) (6.8)

y = −2x+ y − (1− µ)y

d3− µy

r3

+ ac(U2s1Us2 cos(Ωlτ)− U3

s1 sin(Ωlτ)) (6.9)

z = −(1− µ)z

d3− µz

r3− acU2

s1Us3 (6.10)

Note that, in this formulation, the characteristic acceleration is fixed and specified

by the selection of the reference orbit. Of course, the sail normal direction in these

equations can change discretely at each patch point. Consistent with the analysis

by Wawrzyniak, [21,25] continuity is enforced in position and velocity at each of the

intermediate patch points. The equations of motion in Equations (6.8) - (6.10) are

propagated forward using the initial conditions at times τi, τi+1 ... τi+np until reaching

the time associated with the next patch point or the final time in the case of the last

patch point, i.e. τi+1, τi+2 ... τf . The position and velocity states at the end of

each integration interval are denoted, xi+1(usi), xi+2(xi+1, usi+1), ... xf (xi+np, usi+np

).

Therefore, the position and velocity states at the end of the intermediate propagation

segments are functions of both the state and the sail normal vectors that are delivered

at the first point along the propagation; at the start of the first arc, where the final

states are merely functions of the sail normal vector, the initial position and velocity

state are fixed at time τi for each iteration. To force continuity in position and

velocity, the states rendered by the propagation of the equations of motion must be

equal to the states that are introduced to define the next patch point state vector.

Inequality constraints are also implemented at the initial point and each subsequent

patch point to ensure that the sail remains oriented with the appropriate side facing

the Sun. Slack variables are employed, denoted ηi, ηi+1 ... ηi+n, to formulate the

inequality constraints. [25] In particular, inequality constraints must be added for

this particular application, to allow for some variation in the position and velocity at

95

the final time, yet remain within some “box” defined to bound the final state along

the reference, i.e.,

xflb < xf(xi+np, usi+np

)< xfub (6.11)

The vector xflb defines the lower bound and xfub specifies the upper bound. Thus,

twelve additional scalar slack variables are required to enforce the inequality con-

straints in Equation (6.11), six associated with the lower bound constraint, denoted

slow and six associated with the upper bound constraint, denoted supp. All the con-

straint equations are required to fully describe the motion of the turn-and-hold model,

and these equation are now employed in the general method of constraints and free

variables to target a specific position and velocity state downstream at time τf .

Now that the problem is completely formulated, the variables and constraints

are collected in vectors. The design vector for the multiple shooting scheme, X, is

comprised of the sail normal directions, the slack variables, and the position and

velocity states of the patch points as follows,

X =

usi

ηi

xi+1

usi+1

ηi+1

...

xi+np

usi+np

ηi+np

slow

supp

(6.12)

The constraints are collected into the constraint vector, F (X), as follows,

96

F(X)

=

u2si− 1

−Us1i + η2i

xi+1(usi)− xi+1

u2si− 1

−Us1i+1+ η2

i+1

xi+2(xi+1, usi+1)− xi+2

...

−(xf(xi+np, usi+np

)− xflb

)+ s2

low

−(xfub − xf

(xi+np, usi+np

))+ s2

upp

(6.13)

The remainder of the corrections scheme is implemented in the same manner as

Wawrzyniak. [21, 25] Once the corrections are determined for the orientation via the

turn-and-hold model, the prescribed orientations are applied to the coupled orbit and

attitude model.

6.4 Application of Sail Orientation Corrections to the Coupled Model

The orientation corrections for the turn-and-hold model, determined at the initial

point and subsequent patch points, are now applied to the coupled-orbit and attitude

model. It is desired to investigate if these orientation reconfigurations, applied to the

coupled model, return the solar sail center of mass to the vicinity of the reference

trajectory at a specified point downstream. The objective is the implementation of

the corrections predicted by the turn-and-hold model, to control the sailcraft motion

in the coupled orbit and attitude model. A rigid body system representative of a

solar sail type spacecraft is created by fixing the degree of freedom between body B

and body R. Each body is assigned a sail thickness of 7.5 µm in the b1 direction,

and dimensions of 14 m in the b2 direction and 7 m in the b3 direction. This creates

a single rigid body with inertia properties on the same order of magnitude as the

IKAROS sailcraft [55]. A single reference orbit is selected in the blue region from

the solar sail family displayed in Figure 6.1. This reference orbit is constructed with

97

a nondimensional characteristic acceleration such that ac = 0.01 (0.0282 mm/s2).

Given the sail reference orbit, the initial state from this reference trajectory, with the

position depicted by the blue dot in Figure 6.5, is employed to seed the numerical

propagation of the coupled orbit and attitude solar sail model with the sail force

included. The response of the coupled orbit and attitude (with sail force) model, in

terms of the system center of mass behavior, appears in red in Figure 6.5; the path

of G∗ is propagated for 0.25 revolutions of the reference path and clearly departs

the vicinity of this orbit. Consistent with the IKAROS sailcraft, spin stabilization

is employed and the sail is given a spin rate of 0.1 rad/s [55] about the b1 axis

in anticipation that spinning the sail improves the orbital motion response. [56] In

Figure 6.5, the spinning sail response, as predicted by the coupled orbit and attitude

(with sail force) model, depicted in black, is observed to remain near the reference

for approximately 0.25 rev, after which it departs the vicinity. The multiple shooting

strategy is now employed, in conjunction with the spin stabilized results, to further

improve the sail orbital response.

The black path in Figure 6.5 indicates that after a time interval corresponding to

approximately 0.35 revolutions along the reference orbit, the response in the coupled,

spinning model departs from the vicinity of the reference trajectory. The location

of the spacecraft at this particular time is depicted by the red dot in Figure 6.6.

The turn-and-hold multiple shooting scheme is now applied. The red dot indicates

the “initial point” for the multiple shooting algorithm, that is, the location where the

sailcraft center of mass has sufficiently departed the vicinity of the reference orbit. The

“initial point”, corresponding to the red dot in Figure 6.6 should not to be confused

with the location of the initial conditions, displayed in blue in Figure 6.5, where the

propagation of the reference orbit and the spinning coupled orbit and attitude, with

sail force, model originates. Seven patch points, represented by the green dots in

Figure 6.6, are assigned. The paths corresponding to the initial propagations along

each of the intermediate multiple shooting arcs, as predicted by the decoupled turn-

and-hold model, are plotted in magenta, and the full reference orbit, predicted by the

98

Figure 6.5. Coupled Orbit and Attitude Response for Solar Sail inthe Vicinity of L2 in the E-M System

decoupled model, is displayed in blue. The target location at the final time, indicated

by the black dot, is selected to occur at a point located after 0.7 rev along the reference

orbit. The decoupled turn-and-hold model, originating from the location indicated

by the red dot in Figure 6.6, is now employed to determine the discrete orientation

corrections that are required to bring the sailcraft center of mass back to the vicinity

of the reference at the final time, i.e., the location depicted by the black dot in Figure

6.6.

Using the turn-and-hold strategy and applying corrections or updates to the orien-

tation of the spacecraft at the “initial point” and each subsequent patch point, a new

path, as predicted by the decoupled turn-and-hold model, is determined that allows

the spacecraft to return to the vicinity of the reference by the final time correspond-

ing to 0.7 rev along the reference trajectory. The resulting trajectory is displayed in

Figure 6.7 in magenta where, again the reference is represented in blue, the initial

position is indicated by the red dot, the final position is highlighted by the black dot;

99

Figure 6.6. Multiple Shooting Initial Conditions

then, the converged patch point locations are represented by the green dots along the

final corrected arc. The orientation reconfigurations are summarized in Appendix D.

The discrete corrections to the sail orientation angles that are determined for the

“initial” point and all the intermediate patch points, for motion predicted by the

decoupled turn-and-hold model, are then applied to the vehicle in the coupled, spin-

ning, orbit and attitude model that includes the sail force. Once the corrections are

implemented in the coupled model, the sail still departs the vicinity of the reference

as evidenced by the black trajectory in Figure 6.8 due to the impact of the coupling

on the center of mass, G∗, behavior. The converged turn-and-hold trajectory, as pre-

dicted by the turn-and-hold model, is again represented in magenta and follows near

the blue reference path. The green dots in Figure 6.8 indicate patch point locations,

the red dot highlights the “initial” point location for initiation of the corrections pro-

cess, and the black dot specifies the desired position at the final time. The control

strategy combines spin stabilization with the discrete corrections to the orientation

100

Figure 6.7. Corrected Trajectory Using Turn-and-Hold Strategy; Decoupled Model

angles as determined by the decoupled turn-and-hold multiple shooting technique.

For comparison purposes, the position offset of the coupled, spinning orbit and at-

titude model (with sail force) from the reference trajectory, both with and without

the orientation reconfigurations are displayed in Appendix E. The result of this ap-

proach demonstrates a significant improvement over simply allowing the sail natural

dynamics to dominate the response. However, at the final time, the location of the

center of mass G∗ in the coupled, spinning, orbit and attitude model does not reach

the desired target location, although the reference path is maintained over a longer

time interval than the response to spin stabilization alone, plotted in red. Further

iterations to update the corrections process do improve the orbital response, but min-

imally. Further improvements may be possible if the drift of the center of mass, G∗,

is predicted and incorporated into any predictions used in the control strategy,

101

4.44 4.44 4.44−2

−1

0

1

2400

600

x (105 km)

y (103 km)

z (k

m)

referenceturn−and−holdcoupled − no corcoupled − with cor

Figure 6.8. Orientation Corrections Applied to the Coupled Orbitand Attitude Model

6.5 Summary and Recommendations

In summary, observe that applying the orientation corrections determined by the

decoupled turn-and-hold model to the coupled orbit and attitude model (with sail

force) does not meet the final desired position constraints. The dynamics introduced

by considering a coupled orbit and attitude model do not permit the orientations

corrections, determined by the decoupled turn-and-hold model and applied at the

patch points, to effectively control the motion of the center of mass of the sail, when

considering the motion predicted by the coupled orbit and attitude model. Further

analysis is required to determine if it is possible to incorporate the coupled orbit and

attitude model into the multiple shooting corrections process. The key step is the

prediction of the nonlinear drift of the center of mass due to the coupling.

102

7. SUMMARY AND CONCLUSIONS

The motion of point mass spacecraft under the influence of two gravitational bod-

ies is currently the focus of numerous investigations, especially within the context

of the circular restricted three body problem; however, considerably fewer studies

incorporate the effects of spacecraft attitude in this regime. The overall goal in this

investigation is an improved understanding of the motion of such a spacecraft, that

is, one comprised of multiple rigid bodies, under the simultaneous influence of two

point mass gravitational fields. Three main research objectives are established: to de-

velop a model framework to incorporate attitude dynamics within the context of the

circular restricted three body problem, to employ the model framework to formulate

a better understanding of the system dynamics as it relates to attitude dynamics in

the CR3BP, and to investigate the application of this model framework to solar sails.

7.1 Development of a Model Framework

A framework has been developed to create a model governing coupled orbit and

attitude motion in the spatial circular restricted three-body problem. The three-

dimensional motion of a spacecraft, comprised of two rigid bodies, connected by a

single degree of freedom joint, is examined. Kane’s dynamical formulation is employed

as it is well-suited for systems containing multiple rigid bodies. Large discrepancies in

length scales exist in this problem. The distance from the barycenter to the spacecraft

is on the order of hundreds of thousands of km (depending on the system), and the

spacecraft dimensions are typically measured in terms of meters. The presence of such

large differences in the lengths presents a challenge for the computational capabilities,

as only finite precision is possible. Numerical challenges can prevent the propagation

of meaningful results. Thus, a nonlinear variational form of the equations of motion is

103

employed to mitigate these numerical challenges. The equations of motion governing

the orbit as well as the spacecraft attitude are integrated to minimize computational

difficulties and to allow for the propagation of meaningful results. Within this frame-

work, a process is summarized with steps for implementation resulting in a series of

coupled nonlinear equations of motion for the complex spacecraft model.

7.2 Understanding the System Dynamics

The second objective in this investigation is to increase the understanding and

knowledge of spacecraft attitude dynamics in the vicinity of the libration points within

the context of the circular restricted three-body problem. Several test cases are

introduced for both planar and spatial reference orbits to validate the procedure

and highlight results. For both the planar and spatial references, overall results are

presented in the form of an attitude map. These maps convey relationships between

specified inertia ratios and orbit amplitude (represented in terms of the Ay direction

for the planar Lyapunov reference family and the Az direction for the spatial halo

family of periodic orbits). From the maps, it is apparent that the spacecraft undergoes

relatively stable motion in specific regions relative to the rotating frame in the circular

restricted three body problem. Combinations are also identified for which unstable

attitude behavior is likely, in terms of a body fixed (3-2-1) Euler angle sequence for

asymmetric spacecraft. The maps can serve as design tools in an orbit selection

process. Attitude maps might also be exploited as a prediction tool, to determine the

impact on attitude stability with respect to the rotating frame as a result of changing

inertia properties, e.g., in the analysis of the spacecraft deploying a boom or sail.

7.3 Application to Solar Sail Model

The last objective is the application of the model framework to a solar sail type

application. The additional acceleration terms introduced due to the sail force are

included in the nonlinear, variational, coupled orbit and attitude equations of mo-

104

tion. As expected, the sail departs the reference trajectory when the coupled model

is propagated; therefore, spin stabilization is employed to ensure that the sail follows

the reference closely for, at least, the first quarter revolution of the reference orbit. At

some point downstream from the starting configuration, the departure of the sailcraft

center of mass with respect to the reference orbit increases and, eventually, is suffi-

ciently large to warrant corrections. A decoupled turn-and-hold model is employed

to compute corrections to the orientation angles at the departure location and at a

set of subsequent patch-points further downstream. The goal is a the sailcraft center

of mass that returns to the the vicinity of the reference trajectory by some desired

final time, within a control box of predetermined size. Corrections are computed

using the turn-and-hold model that ensures that the trajectory predicted by the de-

coupled spacecraft model returns to the vicinity of the reference. Initially, when the

corrections are applied to the coupled orbit and attitude model, the corrections are

successful in maintaining the location of the spacecraft center of mass nearby the ref-

erence solution; however, at the final time, the final desired position is not achieved.

This result indicates the potential importance of considering the effects of attitude

motion on the orbital mechanics.

7.4 Recommendations for Future Considerations

There are numerous recommendations for future work in the incorporation of the

effects of attitude dynamics into the circular restricted three body problem. Discus-

sion of future research initiatives focuses on four main concentrations: studying the

effects of various spacecraft components, exploring the effects of attitude dynamics

along additional reference orbit families within the context of the CR3BP, further

analysis of the attitude maps, and extending the investigation of solar sail missions

and other applications.

105

7.4.1 Effects of Spacecraft Components

Since the framework established can easily accommodate spacecraft comprised of

multiple rigid bodies, numerous modification can be implemented to create a more

realistic spacecraft model. Modeling complex joint types and increasing the number

of components included in the model is an important first step. Incorporating flex-

ible bodies may also be of interest, especially in the solar sail or space tether type

application. Additional considerations may include extending or collapsing bodies,

and bodies with changing inertia properties.

7.4.2 Orbit Type

Many families of orbits are available within the context of the circular restricted

three body problem, and only two types of periodic families in the vicinity of the

collinear points, the planar Lyapunov family and the Northern halo family, have been

considered in this work. Many other types of periodic orbit families remain to be

investigated near the collinear and equilateral libration points. Additionally, attitude

motion in quasi-periodic orbits is another area of interest for future work.

7.4.3 Further Analysis of Attitude Maps

The attitude maps developed may prove useful for mission design. To determine

the most appropriate frame to present the orientation response with respect to, con-

sideration of actual mission requirements is important. For particular applications,

it may be useful to formulate the maps with respect to the inertial frame. For a

solar sail type example, perhaps the frame fixed to the sunlight direction is more

appropriate. Additionally, the information presented on the map can be altered. The

maximum magnitude of each orientation angle over one revolution of the reference

may not contain sufficient information. It may be important to include the direc-

tion of rotation (positive or negative). It may also be insightful to generate many

106

instantaneous attitude maps, to determine how the orientation angles evolve over one

revolution of the reference path.

7.4.4 Solar Sails and Other Applications

As only an ideal sail is considered in this analysis, increasing the fidelity of the

solar sail model is an important feature to consider in future work. In the solar sail

application, the orientation corrections computed via the turn-and-hold model did

not successfully maintain the sailcraft center of mass in the vicinity of the reference

trajectory, when the corrections were applied to the coupled orbit and attitude model.

Additional development is necessary to determine if the coupled orbit and attitude

model can be integrated into a multiple shooting algorithm. The ultimate objective

is an orientation corrections strategy that results in the sail spacecraft returning to

the vicinity of the reference solution, given some initial perturbation.

LIST OF REFERENCES

107

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APPENDICES

111

A. Lyapunov Orbits: Unstable Eigenvalue/Time Constant

0 0.5 1 1.5 2

x 105

0

500

1000

1500

2000

2500

Ay (km)

Eig

enva

lue

Figure A.1. Unstable Eigenvalue for L1 Planar Lyapunov Family (E-M System)

0 0.5 1 1.5 2

x 105

0

1

2

3

4

5

6

7

8

9x 10

−3

Ay (km)

Tim

e C

onst

ant (

nd)

Figure A.2. Time Constant for L1 Planar Lyapunov Family (E-M System)

112

B. Attitude Maps: L2 Lyapunov Family

k3 (nd)

Ay

(km

)

Response in θ − unperturbed

−1 −0.5 0 0.5

143620

126640

108900

91195

73684

55938

38168

20519

4230.6A

mpl

itude

(de

g)0

45

90

k3 (nd)A

y (k

m)

Response in φ − unperturbed

−1 −0.5 0 0.5

143620

126640

108900

91195

73684

55938

38168

20519

4230.6

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in ψ − unperturbed

−1 −0.5 0 0.5

143620

126640

108900

91195

73684

55938

38168

20519

4230.6

Am

plitu

de (

deg)

0

45

90

Figure B.1. Orientation Response in L2 Planar Lyapunov Family (E-M System)

113

k3 (nd)

Ay

(km

) Response in θ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73683

55934

38171

20520

4230.9

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in θ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73684

55938

38167

20520

4231.3

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in φ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73683

55934

38171

20520

4230.9

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in φ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73684

55938

38167

20520

4231.3

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in ψ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73683

55934

38171

20520

4230.9

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in ψ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73684

55938

38167

20520

4231.3

Am

plitu

de (

deg)

0

45

90

Figure B.2. Orientation Response to Small Perturbation Sets for L2

Planar Lyapunov Family (E-M System)

114

k3 (nd)

Ay

(km

) Response in θ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73684

55934

38171

20520

4231.1

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in θ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73683

55938

38169

20519

4231.4

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in φ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73684

55934

38171

20520

4231.1

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in φ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73683

55938

38169

20519

4231.4

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in ψ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73684

55934

38171

20520

4231.1

Am

plitu

de (

deg)

0

45

90

k3 (nd)

Ay

(km

)

Response in ψ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5

143620

126640

108900

91195

73683

55938

38169

20519

4231.4

Am

plitu

de (

deg)

0

45

90

Figure B.3. Orientation Response to Medium Perturbation Sets forL2 Planar Lyapunov Family (E-M System)

115

C. Attitude Maps: L2 Northern Halo Family

k1 (nd)

Az

(km

)

Response in θ − unperturbed

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86A

mpl

itude

(de

g)0

90

180

k1 (nd)

Az

(km

)

Response in φ − unperturbed

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in ψ − unperturbed

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

Figure C.1. Orientation Response in L2 Northern Halo Family (E-M System)

116

k1 (nd)

Az

(km

) Response in θ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in θ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in φ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in φ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in ψ − perturbed (1 deg, 1 deg, 1 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in ψ − perturbed (5 deg, 5 deg, 5 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

Figure C.2. Orientation Response to Small Perturbation Sets for L2

Northern Halo Family (E-M System)

117

k1 (nd)

Az

(km

) Response in θ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in θ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in φ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in φ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in ψ − perturbed (10 deg, 10 deg, 10 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

k1 (nd)

Az

(km

)

Response in ψ − perturbed (20 deg, 20 deg, 20 deg)

−1 −0.5 0 0.5 1

76502

67276

57665

48054

38443

28832

19222

9610.8

768.86

Am

plitu

de (

deg)

0

90

180

Figure C.3. Orientation Response to Medium Perturbation Sets forL2 Northern Halo Family (E-M System)

118

D. Solar Sail: Turn-and-Hold Model Corrections

The orientation corrections, determined for the turn-and-hold model are displayed

in Table D.1. The corrections are converted to a body 3-2-1 (θs, αs, γs) Euler angle

sequence, representing a rotation from the sunlight frame (l-frame) to the body frame

(b-frame). The initial configuration, corresponding to the orientation of the reference,

requires αs to be equal to -35.264 degrees, and the other two angles are zero.

Table D.1 Turn-and-Hold Model: Orientation Reconfigurations

Orientation Reconfigurations θs (deg) αs (deg) γs (deg)

Start Point (0.35 rev) -14.1 -33.3 0Patch Point 1 (0.40 rev) 5.6 -33.3 0Patch Point 2 (0.45 rev) 3.2 -34.1 0Patch Point 3 (0.50 rev) 1.5 -33.9 0Patch Point 4 (0.55 rev) -10.6 -28.5 0Patch Point 5 (0.60 rev) 8.5 -38.5 0

119

E. Solar Sail: Coupled Model Position Offsets

The position offset from the reference, prior to applying the orientation reconfigura-

tions to the coupled, spinning orbit and attitude (with sail force) model, are sum-

marized in Table E.1. After applying the orientation reconfigurations, determined

by the turn-and-hold model, to the coupled spinning orbit and attitude (with sail

force) model, improvements are observed in terms of offset from the reference orbit.

The results in Table E.2 confirm that the addition of the orientation reconfigurations

assist to reduce the position offset from the reference trajectory.

Table E.1 Coupled Spinning Model: Without Orientation Reconfigurations

Distance From Reference a1 (km) a2 (km) a3 (km) Magnitude (km)

Start Point (0.35 rev) -8.7 20.0 -13.6 25.70.40 rev 30.0 16.5 -19.5 39.50.45 rev 81.2 12.4 -25.8 86.10.50 rev 187.9 -89.6 -31.7 210.60.55 rev 405.7 -254.2 -36.4 480.70.60 rev 850.3 -577.9 -39.4 1028.90.65 rev 1736.4 -1189.1 -39.6 2104.9

Table E.2 Coupled Spinning Model: With Orientation Reconfigurations

Distance From Reference a1 (km) a2 (km) a3 (km) Magnitude (km)

Start Point (0.35 rev) -8.7 20.0 -13.6 25.70.40 rev 7.9 16.3 -15.0 23.60.45 rev 18.4 9.5 -17.3 27.00.50 rev 40.0 1.9 -10.7 41.40.55 rev 76.8 -15.2 14.9 79.70.60 rev 129.6 -51.0 67.7 154.90.65 rev 214.2 -103.2 145.2 278.6

VITA

120

VITA

Amanda Knutson earned her B.S. Degree in Mechanical Engineering at Queen’s Uni-

versity, in Kingston, Ontario, Canada, in May, 2005. She also completed a M.Sc.

Degree in Mechanical Engineering at Queen’s, specializing in dynamic modeling in

May, 2008. She began her Ph.D. studies in Astrodynamics at Purdue in the fall of

2008 with Professor K.C. Howell. Her research interests include Kane’s dynamical

formulation, and attitude dynamics in the circular restricted three-body problem.