EARTH TO HALO ORBIT TRANSFER TRAJECTORIES

A Thesis

Submitted to the Faculty

of

Purdue University

by

Raoul R. Rausch

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science

August 2005 ii

ACKNOWLEDGMENTS

This research topic has been challenging and frustrating at times, rewarding and fulfilling at others. I want to thank my advisor Professor Howell for her continuous support and guidance and the seemingly infinite patience. The advice and recom- mendations she provided on reviews of this thesis, certainly surpassed anything that could have been expected. I also wish to thank the other members of my graduate committee, Professors James M. Longuski and Martin Corless for their advice and reviews of this thesis. I am grateful to all the past and present members of my research group. They have provided much support and guidance and without their contributions, this work would have been difficult to complete. They have helped me enhancing my understanding of the three-body problem and inspired me to think about the technical issues at hand more globally. Additionally, I would like to thank my parents and my wife, Nicole, for their con- tinuous support. Their combined energy has given me at times the extra motivation, strength and confidence necessary to complete this work. Finally, I wish to thank those who have provided the funding for my graduate studies. For the last three years, I have been funded by the German section of the Purdue School of Foreign Languages and Literatures. Teaching German has been an enlightening and educational experience for me. iii

TABLE OF CONTENTS

Page LIST OF TABLES ...... vi LIST OF FIGURES ...... vii ABSTRACT ...... x 1 INTRODUCTION ...... 1 1.1 Problem Definition ...... 2 1.2 Previous Contributions ...... 5 1.2.1 Historical Overview ...... 5 1.2.2 Transfer Trajectories ...... 8 1.3 Present Work ...... 9 2 BACKGROUND: MATHEMATICAL MODELS ...... 12 2.1 Reference Frames ...... 12 2.1.1 Inertial Frame ...... 12

2.1.2 P1 − P2 Rotating Frame ...... 13 2.1.3 Earth Centered “Fixed” Frame ...... 13 2.2 Transformations Between Different Frames ...... 15 2.3 Nondimensionalization ...... 16 2.4 Equations of Motion ...... 17 2.4.1 Singularities in the Equations of Motion ...... 20 2.5 State Transition Matrix ...... 22 2.6 Differential Corrections ...... 23 2.7 Particular Solutions ...... 25 2.8 Invariant Manifold Theory ...... 28 2.8.1 Brief Overview ...... 29 2.8.2 Periodic Orbits and Dynamical Systems Theory ...... 33 iv

Page 2.9 Computing Manifolds ...... 36 2.10 Transition of the Solution to the Ephemeris Model ...... 42

3 TRANSFERS FROM EARTH PARKING ORBITS TO LUNAR L1 HALO ORBITS ...... 45 3.1 Stable and Unstable Flow that is Associated with the Libration Point L1 in the Vicinity of the Earth ...... 45 3.2 Design Strategy ...... 48 3.2.1 Shooting Technique ...... 49 3.2.2 Investigated Transfer Types ...... 50 3.3 Two-Level Differential Corrector ...... 51 3.3.1 First Step - Ensuring Position Continuity ...... 52 3.3.2 Second Step - Enforcing Velocity Continuity ...... 53 3.4 Constraints ...... 56 3.4.1 Position and Epoch Constraints ...... 57 3.4.2 Parking Orbit Constraints ...... 58 3.4.3 | ∆¯v | Constraints ...... 59

3.5 Direct Transfer Trajectories from Earth to Lunar to L1 Halo Orbits . 60 3.6 Transfer Trajectories with a Manifold Insertion ...... 64 3.7 Effects of a Cost Reduction Procedure ...... 66 3.8 Free Return Trajectory ...... 69 3.9 Summary and Conclusions ...... 71 3.9.1 Numerical versus Dynamical Issues in the Computation of Trans- fers ...... 71 3.9.2 Discussion ...... 78 4 TRANSFERS FROM EARTH PARKING ORBITS TO SUN-EARTH LIBRATION POINT ORBITS ...... 80 4.1 Stable Flow from the Libration Points in the Direction of the Earth . 80 4.2 Selection of Halo Orbit Sizes ...... 81

4.3 Transfer Trajectories From Earth to L1 Halo Orbits ...... 83 v

Page

4.4 Transfer Trajectories From Earth to L2 Halo Orbits ...... 84 4.5 Conclusions ...... 85 5 LAUNCH TRAJECTORIES ...... 91 5.1 Equation of Motions with Constant Thrust Term ...... 91 5.2 State Transition Matrix ...... 94 5.3 Patch Points and Initial Trajectory ...... 95 5.3.1 Determination of the Launch Site ...... 96 5.4 Two-Level Differential Corrector with Thrust ...... 97 5.4.1 Two-Level Differential Corrector with Thrust ...... 97 5.5 Trajectory from Launch Site into Parking Orbit ...... 98 5.5.1 Challenges with the Launch Formulation ...... 100 5.6 Conclusion ...... 104 6 SUMMARY AND RECOMMENDATIONS ...... 105 6.0.1 Summary ...... 105 6.0.2 Recommendations and Future Work ...... 106 6.0.3 Concluding Remarks ...... 108 LIST OF REFERENCES ...... 109 vi

LIST OF TABLES

Table Page 3.1 Transfer Costs for Two Differently Sized Halo Orbits; TTI Maneuver Constrained to the x − z Plane Crossing ...... 60 3.2 Transfer Costs for Two Differently Sized Halo Orbits; Location of the TTI Maneuver Determined by the Differential Corrections Scheme. . 65 3.3 Transfer Costs for Transfers with a Manifold Insertion for Two Differ- ently Sized Halo Orbits...... 65 4.1 Transfer Costs for Transfers from a 200 km Altitude Earth Parking Orbit to Two Differently Sized Sun-Earth L1 Halo Orbits ...... 86 4.2 Transfer Costs for Transfers from a 200 km Altitude Earth Parking Orbit to Two Differently Sized Sun-Earth L2 Halo Orbits ...... 86 5.1 Changes in Thrust Parameters Throughout Trajectory Arc...... 100 vii

LIST OF FIGURES

Figure Page 2.1 Geometry of the Three-Body Problem...... 14 2.2 Zero Velocity Curve for C = 3.161...... 21 2.3 A Stylized Representation of 1 Step Differential Corrector...... 24 2.4 Location of the Libration Points in the Earth-Moon System Relative to a Synodic Frame...... 26

2.5 Northern Earth-Moon L1 Halo Orbit in the CR3BP...... 28

2.6 Lissajous Trajectory at L1 in the Sun-Earth CR3BP...... 29 2.7 Stable and Unstable Eigenvectors and the Globalized Manifold for the Earth-Moon L1 Point...... 34 2.8 Stable Eigenvectors of the Monodromy Matrix for an Earth-Moon L1 Halo Orbit...... 38 2.9 Position and Velocity Components of the Stable Eigenvectors of the Monodromy Matrix for an Earth-Moon L1 halo...... 39 2.10 Various Manifolds Asymptotically Approaching the Orbit...... 40 2.11 Stable (blue) and Unstable (red) Manifolds for a Sun-Earth Halo Orbit near L1...... 41 2.12 Stable (blue) and Unstable (red) Manifold Tube Approaching the Earth in the Sun-Earth System...... 42 2.13 A ”Halo-like” Lissajous Trajectory in a Ephemeris model with an Az Amplitude of approximately 15,000 km...... 44 3.1 Stable and Unstable Manifold Tubes in the Vicinity of the Earth. . . 46 3.2 Minimum Earth Passing Altitudes for Trajectories on the Manifold Tubes Associated with Various Earth-Moon L1 Halo Orbits...... 47 3.3 A Stylized Representation of Level II Differential Corrector (from Wilson [1])...... 52

3.4 Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 15, 000 km) in the CR3BP (Location of HOI maneuver is con- strained)...... 61 viii

Figure Page

3.5 Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 43, 800 km) in the CR3BP (Location of HOI maneuver is con- strained)...... 62

3.6 Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 15, 000 km) in an Ephemeris Model (Location of HOI Manuever Con- strained)...... 63

3.7 Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 43,800 km) in an Ephemeris Model (Location of HOI Manuever Con- strained)...... 64

3.8 Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 15,000 km) in the CR3BP (Location of the TTI Maneuver Determined by the Differential Corrections Scheme.) ...... 66

3.9 Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 43,800 km) in the CR3BP (Location of the TTI Maneuver Determined by the Differential Corrections Scheme.) ...... 67

3.10 Transfer from an Earth Parking Orbit to L1 Halo Orbits (Az = 15,000 km) in an Ephemeris Model (Location of the TTI Ma- neuver Determined by the Differential Corrections Scheme.) ...... 68

3.11 Transfer from an Earth Parking Orbit to L1 Halo Orbits (Az = 43,800 km) in an Ephemeris Model (Location of the TTI Ma- neuver Determined by the Differential Corrections Scheme.) ...... 69 3.12 Transfer from an Earth Parking Orbit with a Manifold Insertion into an L1 Halo Orbit (Az = 15,000 km) in the CR3BP...... 70 3.13 Transfer from an Earth Parking Orbit with a Manifold Insertion into an L1 Halo Orbit (Az = 43,800 km) in the CR3BP...... 71 3.14 Transfer from an Earth Parking Orbit with a Manifold Insertion into an L1 Halo Orbit (Az = 15,000 km) in an Ephemeris Model. . . . 72 3.15 Transfer from an Earth Parking Orbit with a Manifold Insertion into an L1 Halo Orbit (Az = 43,800 km) in an Ephemeris Model. . . . 73 3.16 Effects of a Cost Reduction Procedure on the Transfer Arcs Initially Using the Invariant Manifold on the Near Earth Side for a Halo Orbit with an Az Amplitude of 15,000 km in an Ephemeris Model. 74 ix

Figure Page 3.17 x − y Projection of a Free Return Trajectory to a Halo Orbit with an Az Amplitude of 15,000 km in an Ephemeris Model...... 75 3.18 x − z Projection of a Free Return Trajectory to a Halo Orbit with an Az Amplitude of 15,000 km in an Ephemeris Model...... 76 3.19 y − z Projection of a Free Return Trajectory to a Halo Orbit with an Az Amplitude of 15,000 km in an Ephemeris Model...... 77

4.1 Closest Approach Altitudes for L1 Sun-Earth Manifolds Relative to the Earth...... 82

4.2 Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 120, 000 km) in the CR3BP...... 84

4.3 Transfers from an Earth Parking Orbit to a L1 Halo Orbit (Az = 440, 000 km) in the CR3BP...... 85

4.4 Transfers from an Earth Parking Orbit to a L1 Halo Orbit in an Ephemeris Model...... 87

4.5 Transfer from an Earth Parking Orbit to a L2 Halo Orbit (Az = 120, 000 km) in the CR3BP...... 88

4.6 Transfers from an Earth Parking Orbit to a L2 Halo Orbit (Az = 440, 000 km) in the CR3BP...... 89

4.7 Transfers from an Earth Parking Orbit to a L2 Halo Orbit in an Ephemeris Model...... 90 5.1 Spherical Coordinates to Define the Direction of the Thrust Vector...... 93 5.2 Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit. 101 5.3 Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit. 102 5.4 Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit. 103 x

ABSTRACT

Rausch, Raoul R. M.S., Purdue University, August, 2005. Earth to Halo Orbit Transfer Trajectories. Major Professor: Kathleen C. Howell.

Interest in libration point orbits has increased considerably over the last few decades. The Lunar L1 and L2 libration points have been suggested as gateways to Sun-Earth libration points and to interplanetary space. The dynamics in the vicinity of the Earth in the Earth-Moon system, where the Earth is the major pri- mary in a three-body model, has only been of limited interest until recently. The new lunar initiative is the origin of a wide range of studies to support the infrastructure for a sustained lunar presence. A systematic and efficient approach is desirable to ease the determination of viable transfer trajectories satisfying mission constraints. An automated process is, in fact, a critical component for trade-off studies. This work presents an initial approach to develop such a methodology to compute trans- fers from a launch site or parking orbit near the Earth to libration point orbits in the Earth-Moon system. Initially, the natural dynamics in three-body systems near both the smaller and larger primaries are investigated to gain insight. A technique using a linear differential corrections scheme is then developed. Initial attempts to incorporate the invariant manifolds structure in designing these transfers are pre- sented. Simple transfers in the Earth-Moon system are computed and transitioned to an ephemeris model. The methodology is also successfully applied in the Sun-earth system. Challenges are discussed. The second task involves the determination of launch trajectories. A two-level differential corrections technique incorporating a constant thrust term is developed and a sample launch scenario is computed. Limitations and extensions are considered. 1

1. INTRODUCTION

Today, humankind continues to explore a new frontier as civilization expands into space. In the near term, plans include a potential return to the Moon with both robotic vehicles as well as human crews; an extended stay on the lunar surface or in the vicinity of the Moon is possible. In addition to the exploration of the Moon (and Mars), many other missions and observatories have been proposed that will make use of libration point orbits, such as the James Webb Space Telescope [2] (formerly known as the Next Generation Telescope), the Terrestrial Planet Finder [3] and the Europa Orbiter Mission [4]. From these various scenarios, some of the new observatories will take advantage of the prime location offered by the Sun-Earth libration points at L1 and L2 and of the efficient low-energy trajectories that are available throughout the solar system. These low energy pathways are defined via the manifolds associated with libration point orbits in all Sun-Planet and Planet-Moon three-body systems. These pathways can be exploited due to only minor energy differences [5] between libration point manifolds in different systems. The entire system of “tunnels” is made possible by the chaotic environment resulting from multiple gravity fields. As stated by Lo et al. [5] “...the tunnels generate deterministic chaos and for very little energy, one can radically change trajectories that are initially close by.” This statement effectively summarizes the physical basis and practicality of libration point trajectories. Future space observatories can take advantage of the proximity of the Earth-Moon libration points L1 and L2 and use them as inexpensive gateways [6] to the Sun-Earth libration points L1 and L2 and to interplanetary space. The relatively short distance between the Earth and the Earth-Moon lunar libration point L1 makes human servicing of observatories possible [7]. A spacecraft can be delivered to this location within a week from the Earth and within hours from the Moon’s surface [5]. In addition to 2

easy accessibility, spacecraft in an orbit near the Earth-Moon L1 libration point can be continuously monitored from Earth.

The first spacecraft to make use of a libration point orbit was the International Sun-Earth Explorer (ISEE-3) launched on November 20, 1978 [8]. The mission was very successful at monitoring the solar winds. Subsequently, the vehicle was rerouted to explore the Earth’s geomagnetic tail region before shifting to a new trajectory arc that ultimately encountered the comet Giacobini-Zinner. The spacecraft was renamed the International Cometary Explorer (ICE) [9] and, after a few close en- counters with both the Earth and the Moon, ICE reached the comet on schedule. Additional spacecraft have also been launched as part of successful libration point missions, including WIND [10], SOHO [11], ACE [12], Genesis [13], and MAP. The WIND, SOHO, and ACE missions were all part of the International Solar-Terrestial Physics project. Genesis was the first mission to exploit dynamical systems theory in the design and planning phases. This new analysis concept allowed the Genesis spacecraft to take advantage of the complex dynamics in a multi-body regime, that is, the Sun-Earth-Moon gravitational fields, and to fulfill the mission requirements with a completely ballistic baseline trajectory [5] that required no maneuvers. Genesis also returned the first samples of solar wind particles to the Earth from beyond the local environment.

1.1 Problem Definition

The libration points L1 and L2 have been proposed as inexpensive gateways [5–7] for performing transfers between different three-body systems. Many different types of libration point trajectories are of current interest with the potential for inexpensive ac- cess to interplanetary space. The concept of system-to-system transfers was suggested as early as 1968 [14]. New techniques, for example, the use of the invariant manifold structure associated with three-body systems, that simplify the design of system-to- system transfer are currently in development by various researchers [6, 15, 16]. How- 3

ever, servicing and repair missions to observatories in libration point orbits (LPO) must meet very specific requirements such as time of arrival and precise target orbit requirements [7]. Libration point missions involving humans must also have feasible Earth return options in case of emergency. Meeting such requirements can quickly be- come a nearly impossible task. Therefore, future libration point missions will require new and innovative design strategies to fulfill the mission requirements at low fuel costs while satisfying an increasingly complex set of constraints. Two types of libra- tion point orbits commonly investigated for applications are the three-dimensional precisely periodic halo orbits and the quasi-periodic Lissajous trajectories. Much progress has been made in the last decade by developing design strategies based on these orbits and their associated manifolds.

One problem that influences every spacecraft is the launch. For libration point missions, in particular, designing the launch leg, from a site on the Earth’s surface to a transfer trajectory that delivers the vehicle into a libration point orbit, is challenging. Traditionally, spacecraft enroute to libration point orbits have been first launched into low Earth parking orbits, then depart along their transfer path to insert into a baseline halo orbit or Lissajous trajectory. For preliminary analysis, this approach allows the use of traditional two-body analysis tools to determine the trajectory arc from the launch site into the parking orbit. Both the Earth-Moon and Sun-Earth systems present unique challenges.

Regardless of the particular three-body system, determining any solution in an ephemeris model without good baseline approximations is a tremendously complex task. Simplified models are usually employed to obtain baseline solutions. However, using a two-body model limits the solution-space considerably when the seemingly chaotic motion present in a multi-body system is lost. To expose potential non-conic solutions, a three-body model is therefore necessary. But, there are no known closed- form solutions to the three-body problem (3BP). Some simplifying assumptions do result in a model that retains the significant features of three-body motions; slight modifications then offer a very useful formulation. The first assumption is one con- 4 cerning the masses; that is, the bodies are all assumed spherically symmetric, and thus point masses. In fact, assuming one particle to be infinitesimal is key for an- other approximation. The Restricted Three-Body Problem (R3BP) consists of two massive particles moving in undisturbed two-body orbits about their common center of mass, the barycenter. It is assumed that the infinitesimal particle does not affect the orbits of the primaries. If the relative two-body orbit of the primaries is assumed elliptic, the resulting model is called the Elliptic Restricted Three-Body Problem (ER3BP). If the two-body orbit of the primaries is assumed circular, the Circular Restricted Three-Body Problem (CR3BP) yields even more insight into the motion. Although the model defined as the CR3BP is the key to the solution of interest, the design process remains challenging. Efficient and effective tools for the compu- tation of transfer trajectories from the Earth to libration point orbits in both the Sun-Earth and the Earth-Moon systems are of increasing interest. In the three-body Sun-Earth-spacecraft system, the Earth is the smaller of the two primaries whereas in the three-body Earth-Moon-spacecraft system, the Earth is the larger of the pri- maries. This difference is significant in the fundamental dynamical structure of the solution space. The role that a planet, here the Earth, plays in a simplified model as the larger or the smaller primary can have a large influence on the behavior and characteristics of the solutions. First, the natural flow approaching and departing libration point orbits in both cases requires investigation. A deterministic maneuver will always be required to leave the Earth parking orbit. Additional maneuvers to insert into the halo orbit, or into the transfer trajectory, may be necessary to satisfy design requirements. Finally, tools to compute thrusting arcs from the launch site to the transfer path require development. 5

1.2 Previous Contributions

1.2.1 Historical Overview

Investigations in the 3BP from a dynamical perspective began with Newton. His development of the laws of motion and the concept of a gravity force, first published in the Principia [17], enabled the formulation of a mathematical model. At the time, predicting the motion of the Moon was of great interest with applications in naval navigation [18]. Newton, of course, solved the two-body problem after reformulating it as a problem in the relative motion of two bodies. But, to accurately predict the Moon’s orbit, perturbations had to be incorporated. This led to the formulation of the 3BP. A complete solution to the gravitational 3BP requires 18 integrals of motion. Since only a total of ten integrals of motion exist, as later determined by Euler [19], a closed-form solution was not straightforward. Six of these scalar integrals result from the conservation of linear momentum, three from the conservation of total angular momentum, and one integral results from the conservation of energy. In his investigations of the 3BP formed by the Sun, the Earth, and the Moon, Newton nevertheless managed to compute the motion of the lunar perigee to within eight percent of the observed value in 1687. Continuing Newton’s work, Leonard Euler proposed the highly special problem of three bodies known as the “problem of two fixed force-centers,” solvable by elliptic functions, in 1760 [20]. Euler was also the first to formulate the restricted three-body problem (R3BP) in a rotating frame. The formulation was very significant. This step allowed him to predict the existence of the three collinear equilibrium points L1, L2, and L3 [20]. Lagrange confirmed Euler’s prediction in his memoir “Essai sur le probleme` des Trois Corps” published in 1772. Lagrange deduces the existence of the collinear points and solves for two additional equilibrium points. These additional two equilibrium points each form an equilateral triangle with the primaries and are generally labelled the equilateral libration points

L4 and L5. The five points are commonly denoted Lagrange or libration points. For their work, Euler and Lagrange shared the Prix de l′Academie de P aris in 1772 [18]. 6

In the pursuit of additional integrals, Jacobi considered the concept of the R3BP relative to a rotating frame. In 1836, he determined a constant of the motion by com- bining conservation properties of energy and angular momentum [21]. The constant he discovered now carries his name [18]. In other research efforts, Hill’s Lunar Theory, published in 1878, represents the result of an investigation of a satellite’s orbit around a larger planet under the influence of solar and eccentric perturbations. Specifically, Hill was interested in modelling the lunar orbit with the simplifying assumptions that the solar eccentricity and parallax, as well as the Moon’s orbital inclination, are all zero [21]. Hill’s work was revolutionary; he used the CR3BP as the base model and was the first person to abandon a two-body analysis [22]. In his theory, Hill demon- strated that for a specified energy level, regions of space exist where motion is not physically permitted [18].

Toward the end of the 19th century, Poincar´estudied the 3BP seeking additional integrals of the motion. Poincar´epredicted an infinite number of periodic solutions if two of the masses are small compared to the third [23]. Poincar´ebelieved that the primary problem in celestial mechanics was the behavior of orbits as time goes to infinity [24] and focused on qualitative aspects of the motion. The planar R3BP was of particular interest to him because it could be formulated as a Hamiltonian-like system. Determined to investigate the problem further, he invented an analytical technique called the ‘surface of section’ [24] that allowed him, in 1892, to describe the phase space of a non-integrable system. Through his contributions to dynamics and the invention of the method of the ‘surface of section,’ Poincar´eis widely regarded as the father of Dynamical Systems Theory (DST). In 1899, he proved that Jacobi’s Constant is the only integral of motion that exists in the R3BP. Any other integral would not be an analytical function of the systems coordinates, momenta, and the time [24]. Poincar´e’s insight into the 3BP, his contributions to mathematics, and the eventual advent of high-speed computing makes most of today’s work possible. At the time, his findings also resulted in a shift in the focus of research in the 3BP toward determining specific trajectories rather than the general behavior. 7

At the beginning of the 20th century, Darwin, Plummer, and Moulton were all seeking periodic orbits in the vicinity of the libration points [23,25–28]. In 1899, Darwin computed several approximate planar periodic orbits in the CR3BP using a quadrature method. Plummer also accomplished the same feat in 1902 using an ap- proximate, second-order analytical solution to the equations of motion in the CR3BP. Between 1900 and 1917, Moulton developed several approximate analytical solutions to the linearized equations of motion relative to the collinear points. Moulton’s so- lutions result in planar as well as three-dimensional periodic orbits [21]. Although a series solution to the general three-body problem was produced by Sundman in 1912 [19,29], it is useless for any practical purposes. Further exploration into Libra- tion Point Orbits (LPO) were hindered by the computational requirements. Only with the introduction of high-speed computers in the 1960’s did significant progress occur.

In the late 1960’s, interest in the three-body problem (3BP) increased signifi- cantly. In 1967, Szebehely published a book summarizing all information to date concerning the 3BP [21]. His compilation contained numerically integrated, peri- odic orbits in the planar CR3BP and the planar ER3BP as well as a few three- dimensional notes. Motivated by new mission possibilities at NASA, Farquhar devel- oped analytical approximations for three-dimensional periodic orbits in the translunar Earth-Moon region [14] in the late 1960’s. Farquhar coined the term “halo” orbits to describe periodic three-dimensional orbits in the vicinity of the collinear libra- tion points because, when viewed from the Earth, they appear as a halo around the Moon. Whereas halo orbits are precisely periodic in the CR3BP, Lissajous trajec- tories are quasi-periodic. Lissajous figures, in general, are named after the French physicist Jules A. Lissajous (1822-1880) because planar projections of these curves look similar to those studied by Lissajous in 1857 [9]. In 1972, Farquhar and Kamel developed a third-order approximation for quasi-periodic motion near the translunar libration point using a Lindstedt-Poincar´emethod [30]. Heppenheimer developed a third-order theory for nonlinear out-of-plane motion in the ER3BP in 1973 [31]. 8

Two years later, Richardson and Cary [32] obtained a third/fourth order approxi- mation for three-dimensional motion in the elliptic restricted three-body problem in the Sun-Earth/Moon barycenter system. An analytical approximation for halo-type periodic motion about the collinear points in the Sun-Earth CR3BP was published by Richardson in 1980. Breakwell and Brown numerically extended the work of Farquhar and Kamel to yield a family of numerically integrated periodic halo orbits [33]. The discovery of stable halo orbits in that family motivated future research in the halo families near all three collinear libration points by Howell [34] in collaboration with Breakwell. In 1990, Marchal published a book summarizing the more recent progress on the CR3BP [35].

1.2.2 Transfer Trajectories

Much progress has occurred over the last decade in the development of analysis strategies to determine transfer trajectories from the Earth to halo and Lissajous or- bits near Sun-Earth L1 and L2 points. With few analytical tools, transfer design was initially dependent on numerical techniques not available until the 1960’s. The ad- vent of high-speed computers made the computation of transfers possible and, hence, allowed the possibility of libration point missions. The first study published on trans- fer trajectories between a parking orbit and a libration point was by D’Amario in 1973 [36, 37]. D’Amario combined analytical and numerical techniques with primer vector theory to develop a fairly accurate method for the quick calculation of transfer trajectories from both the Earth and the Moon to the Earth-Moon libration point

L2. With his multiconic approach, D’Amario determined families of locally optimal two-impulse and three-impulse transfers [36]. Subsequently, ISEE-3 was planned, of course, and successfully reached a Sun-Earth L1 halo orbit. The transfer trajectory used for the ISEE-3 mission inserted the spacecraft into the halo orbit at the eclip- tic plane crossing on the near-Earth side. The transfer was categorized as “slow” because the transfer Time of Flight (TOF) was approximately 102 days versus the 9

approximately 35 days for expensive “fast” transfers. This type of transfer trajec- tory was selected because numerical studies had indicated that such a path is less costly in terms of |∆V | than a “fast” transfer trajectory to the halo orbit [8,38]. In 1980, Farquhar completed a post-flight mission analysis of the flight data from ISEE- 3 [39]. Sim´oand his collaborators were the first to publish details of a methodology to use invariant manifold theory to aid in the design of transfer trajectories in 1991. A manifold approaches a periodic orbit asymptotically and so eliminates, in theory, any insertion maneuver cost. A year later, Hiday expanded primer vector theory to the 3BP and studied impulsive transfers between parking orbits and Libration Point

Trajectories (LPT) near L1 in the Sun-Earth system. An extensive numerical study using differential techniques was performed by Mains in 1993 [40]. Mains was in- terested in the development of approximations useful for future automated transfer trajectory determination procedures. Mains studied transfers from a variety of dif- ferent parking orbits with different times of flight (TOF) including a transfer similar to that of the ISEE-3 spacecraft. Barden [28,41] later extended Mains’ investigations through a combination of numerical techniques and dynamical systems theory. In the mid-1990’s, Wilson, Barden, and Howell developed design methodologies used in the determination of the Genesis trajectory [42–44]. In 2001, Anderson, Guzm´an, and Howell implemented an efficient procedure to investigate transfers from the Earth to Lissajous trajectories by exploiting lunar flybys in an ephemeris model [45,46]. Thus far, most of the work has been focused on determining transfer trajectories from the Earth to the Sun-Earth or Sun-Earth/Moon libration point orbits.

1.3 Present Work

The focus of this investigation is the continued development of techniques and strategies to determine transfer trajectories from the Earth to LPOs in both the Sun-Earth and the Earth-Moon systems. Most of the emphasis has been placed on the development of the methodology used in the transfer design process. A useful 10

dynamical element that is significant in the determination of transfer trajectories is the set of invariant manifolds approaching and departing periodic halo orbits. In the Earth-Moon system, manifolds do not pass close by the Earth (larger primary) and additional strategies are investigated. In the Sun-Earth system (where the Earth is the smaller primary) manifolds that pass in the immediate vicinity of the Earth are frequently available.

All of the analysis in this work is conducted numerically and the Circular Re- stricted Three-Body Problem (CR3BP) is used as the fundamental dynamical model for baseline designs. The CR3BP is well-suited for qualitative analysis and the so- lution can easily be transferred into a more complex model making use of planetary ephemerides. A few sample transfer trajectories are presented to underline the validity of the CR3BP as the dynamical model. The first objective focuses on transfer tra-

jectories from Earth parking orbits to halo and Lissajous trajectories located near L1 in the Earth-Moon system. This will improve knowledge of the dynamical structure for trajectory arcs that depart from the major (first) primary. The goal is accom- plished by considering both transfer arcs to manifolds as well as direct transfers into the halo orbits. A second objective is the development of a procedure to compute launch trajectories from a specific launch site on the surface of the Earth to an in- variant manifold. Some initial results based on differential corrections algorithms are presented that can be used to determine the transfer arcs and launch trajectories of interest. The results are of a preliminary nature and the beginning of a more in-depth investigation.

This work is arranged as follows:

Chapter 2:

This chapter summarizes the background material that underlies the foundations of this study. The different reference frames are introduced and the mathematical model used to represent the Earth-Moon and Sun-Earth dynamical environments, that is, the circular restricted three-body problem, is developed. Assumptions em- ployed in this model as well as special properties are discussed, followed by a derivation 11

of the linear variational equations. A method to solve for periodic solutions is intro- duced and particular solutions of general interest are presented. Invariant manifold theory is introduced and the computation of manifolds is discussed. Chapter 3:

The natural motion to and from halo orbits at L1 in the Earth-Moon system is investigated. The numerical algorithm that forms the basis of this study is then pre- sented and additional constraints are introduced. The numerical procedure is applied to a number of different problems and sample transfer trajectories from an Earth parking orbit to lunar L1 LPOs are presented. Preliminary results in an ephemeris model are presented to demonstrate the validity of the obtained transfer arcs in the CR3BP. Chapter 4:

The natural motion to and from halo orbits at L1 in the Sun-Earth system is stud- ied and a series of transfers from the Earth halo orbits at L1 and L2 are summarized. Chapter 5: A constant thrust term is added to the equations of motion. A modified version of the differential corrector exploits the thrust parameters to determine launch trajec- tories with discretely varying thrust angles. A sample launch trajectory to an Earth parking orbit is presented. Chapter 6: The work presented here is of a preliminary nature and more detailed investi- gations are required. A strategy for future work is presented and conclusions and recommendations for further investigations are discussed. 12

2. BACKGROUND: MATHEMATICAL MODELS

The circular restricted three-body problem is the simplest model that can capture the seemingly chaotic motion present in a multi-body system. Since this dynamical structure is the focus in this study, the mathematical model is derived. Different reference frames are introduced and the linear variational equations are developed. The concept of differential corrections is introduced and particular solutions, such as equilibrium points, as well as periodic and quasi-periodic orbits are discussed. Some aspects of invariant manifold theory are presented and applied to illustrate the computation of invariant manifolds. The transition of solutions to an ephemeris model is also summarized.

2.1 Reference Frames

A variety of different reference frames are used in this investigation and are useful for computation as well as visualization purposes. Their definitions and the associated notation is detailed here for clarity.

2.1.1 Inertial Frame

Newton’s laws, as stated in their most original form, are valid relative to an inertial reference frame. In this study, one inertial frame is defined to be centered at the Earth (geocentric). The Earth Centered Inertial (ECI) frame is assumed to be inertially fixed in space but, actually, is slowly moving over time. Since a truly inertial system is impossible to realize, the standard J2000 system [47] has been adopted as the best representation of an ideal, inertial frame at a fixed epoch. The shift of this frame is so slow relative to the motion of interest, it can be neglected. 13

The fundamental plane is the plane defined as the X − Y plane. The unit vector X is directed toward the vernal equinox, the unit vector Y is rotated 90 degrees to the b b b east in the ecliptic plane. The unit vector Z is, then, normal to the plane defined by b the X and Y axes such that Z = X × Y . (Note that a caret indicates a vector of b unit magnitude.) A formulation of the problem relative to the inertial frame is not b b b b b very convenient for investigations in the 3BP because the primaries are continuously moving and no constant or fixed equilibrium solutions exist.

2.1.2 P1 − P2 Rotating Frame

The P1 − P2 rotating frame is the most convenient frame for visualization of the motion of the infinitesimal body, P3, moving near the libration points. The barycenter of the two primaries that define the three-body system is typically used as the origin (See figure 2.1). The unit vector x is defined such that it is always parallel to a line between the primaries and is directed from the larger toward the smaller primary. b The unit vector y is 90 degrees from x in the plane of motion of the primaries; it is positive in the general direction of the motion of the second primary relative to the b b first. The unit vector z is defined to complete a right handed coordinate system and is normal to the plane of motion spanned by x and y. Only when the equations of b motion in the CR3BP are formulated relative to the rotating frame of the primaries b b are the libration points the equilibrium solutions to the differential equations. This will be apparent later. Libration point orbits only exhibit their periodicity relative to this frame.

2.1.3 Earth Centered “Fixed” Frame

The Earth Centered Fixed Frame (ECF) is fixed in the Earth as it rotates on its own axis. Thus, the ECF frame is useful when a specific launch site on the Earth surface is defined and launch trajectories are computed. The ECF origin is at the Earth center, and, thus, this geocentric coordinate system rotates with the Earth 14

Figure 2.1. Geometry of the Three-Body Problem.

relative to the inertial frame. The primary axis of ECF is always aligned with a particular meridian, and the Greenwich meridian is very often selected. Since the coordinate system is rotating relative to the inertial frame, it is necessary to specify an epoch. In this model, the ECF frame is assumed to be aligned with the ECI frame at time t = 0. This simplification is equivalent to assuming that the Greenwich meridian is parallel to a line between the primaries at time t = 0. The rotation of the Earth is approximated as constant. The approximation assumes that the Earth completes precisely one revolution in a 24 hour period. By determining the angular velocity of the Earth, the alignment between the ECF and the inertial frame can be computed for any later time. 15

2.2 Transformations Between Different Frames

Transformations between different rotating and inertial frames are critically im- portant to correctly model this problem. The alignment between the inertial frame and the P1 − P2 frame is illustrated in figure 2.1. Transforming the position state from an inertial frame to a rotating frame can be accomplished using the following rotation matrix,

cθ −sθ 0 inert rot   A = sθ cθ 0 , (2.1)    0 0 1    where θ is the angle between the rotating and the inertial frame and appears in

figure 2.1. The trigonometric symbols are defined as sθ = sin(θ) and cθ = cos(θ). Transforming the velocity state between a rotating and an inertial frame requires the use of the basic kinematic equation (BKE), dr¯ dr¯ = +¯ω × r,¯ (2.2) dt dt  inertial  rot whereω ¯ is the angular velocity vector andr ¯ is the position vector in inertial coordi- nates. (Note that overbars denote vectors.) The cross productω ¯ × r¯ yields

A(2, 1) A(2, 2) A(2, 3) inertA˙ rot = θ˙ ∗ r ∗  A(1, 1) A(1, 2) A(1, 3)  , (2.3)    0 0 0      where r is the magnitude of the position vector. Thus, the transformation of the entire state vector from the rotating frame of the primaries to the inertial frame is represented AT 0 rotT inert = . (2.4)  A˙ T AT  In an ephemeris model, the θ angle does not remain constant and must be computed instantaneously; the rate θ˙ is also evaluated instantaneously. The inertial Xˆ − Yˆ plane represents the ecliptic plane, i.e., the plane of motion of the Earth about the Sun. The inertial axis Zˆ is normal to the ecliptic plane. The 16

Earth’s axis of rotation is inclined relative to Zˆ. The angle that denotes the inclina- tion of the Earth is i and is assumed to be constant such that i = 23.5 degrees. Thus, the transformation from the inertial to the inclined equatorial frame is written

eclipCequat 0 T = , (2.5)  0 eclipCequat  with   1 0 0 eclip equat   C = 0 ci si . (2.6)    0 −si ci    This transformation matrix, i.e., equation (2.5), allows conversions of the state be- tween the ecliptic, and the inertial equatorial plane.

2.3 Nondimensionalization

To eventually generalize the derived equations, it is advantageous to non-dimension- alize and, thus, express fundamental quantities in terms of relevant system param- eters. The characteristic dimensional quantities identified in the system are length, time, and mass. As illustrated in figure 2.1, the dimensional length between the barycenter and the first primary P1 is labelled l1 and the dimensional length between the barycenter and the second primary P2 is labelled l2. For the CR3BP, the refer- ence characteristic length is defined as the distance between the two primaries, that ∗ is, l = l1 + l2, the semi-major axis of the conic orbit of the second primary relative ∗ to the first. The reference characteristic mass is defined as m = m1 + m2, where m1

is the mass of the first primary and m2 the mass of the second primary. This allows a definition of the characteristic time as 1 ∗ 3 /2 (l ) t∗ = , (2.7) "G (m1 + m2)# where G is the dimensional gravitational constant. In standard models for the restricted problem, this specific form is employed to select t∗ such that the non- 17

dimensional gravitational constant Gnd is equal to 1. All other quantities can now be evaluated in terms of these three characteristic values. The non-dimensional distance between the two primaries is 1. Kepler’s third law yields the expression for the mean motion, or mean angular velocity, as 1 nd / G (m1 + m2) 2 n = ∗ 3 . (2.8)  (l )  As can be easily verified, the mean motion possesses a non-dimensional value of 1. The non-dimensional mass of the second primary is represented by the symbol µ, i.e., m µ = 2 . (2.9) m1 + m2 As a consequence, the mass of the first primary is represented as m 1 − µ = 1 . (2.10) m1 + m2 The non-dimensional quantities corresponding to the remaining distance elements, as seen in figure 2.1, are defined as r¯ ρ¯ = B3 , (2.11) l∗ r¯ d¯ = 13 , (2.12) l∗ and r¯ r¯ = 23 . (2.13) l∗ Non-dimensional time is then defined t τ = . (2.14) t∗ Nondimensionalization allows a more convenient and general derivation of the equa- tions of motion.

2.4 Equations of Motion

A Newtonian approach is used to derive the equations of motion and, thus, it is necessary to begin with Newton’s law of gravity. With total force acting on the 18

infinitesimal particle P3 modelled in vector form, the law of motion can be written as follows, ¯ I ¨ Gm3m1 Gm3m2 F = m3 r3 = − 2 r¯13 − 2 r¯23, (2.15) r13 r23 I where r¨3 is the acceleration of P3 relative to the barycenter with respect to an inertial frame and dots indicate differentiation with respect to dimensional time. Multiplying

∗ 2 ∗ equation (2.15) by (t ) /l m3 yields:

2 ∗ d (¯r3/l ) Gm1 r¯13 ∗2 Gm2 r¯23 ∗2 ∗ 2 = − 3 ∗ t − 3 ∗ t . (2.16) d(t/t ) |r13| l |r23| l

With equations (2.11) - (2.14), the law of motion in equation (2.15) can be rewritten in the form 2 d ρ¯ m1 d¯ m2 r¯ 2 = − ∗ ∗ 3 − ∗ ∗ 3 . (2.17) dτ m |r¯13/l | m |r¯23/l | Using the non-dimensional quantities in equations (2.9) and (2.13), equation (2.17) can subsequently be expressed in the form

d2ρ¯ (1 − µ) µ = − d¯− r,¯ (2.18) dτ 2 d3 r3 where d¯=(x + µ)ˆx + yyˆ + zz,ˆ (2.19)

r¯ =(x − (1 − µ))ˆx + yyˆ + zz.ˆ (2.20)

The unit vectors Xˆ , Yˆ , and Zˆ are parallel to inertial directions as seen in figure

2.1. Then, ρx , ρy, and ρz are the non-dimensional coordinates of P3 with respect to the inertial, or sidereal, system. Recall that unit vectorsx ˆ ,y ˆ, andz ˆ are parallel to directions fixed in the rotating, or synodic, system. Note that the unit vectors comprise an orthonormal triad. Then the corresponding position coordinates are x, y, and z. The coordinates in the rotating and inertial system are related by a simple rotation, i.e.,

ρx ct −st 0 x       ρy = st ct 0 y . (2.21)        ρz   0 0 1   z              19

Differentiating equation (2.21), to obtain kinematic expressions for position, velocity and acceleration,

ρ˙x ct −st 0 x˙ − y       ρ˙y = st ct 0 y˙ + x , (2.22)        ρ˙z   0 0 1   z˙        and      

ρ¨x ct −st 0 x¨ − 2y ˙ − x       ρ¨y = st ct 0 y¨ + 2x ˙ − y , (2.23)        ρ¨z   0 0 1   z¨        where −2y ˙ and +2x ˙correspond  to the Coriolis  terms. Then, x and y represent non-dimensional terms that result from the centripetal acceleration. From a combination of the kinematic expansion with equation (2.18), equations (2.18) and (2.23) yield the scalar, second order, nonlinear set of differential equations: (1 − µ)(x + µ) µ(x − (1 − µ)) x¨ − 2y ˙ − x = − − , (2.24) d3 r3 (1 − µ)y µy y¨ + 2x ˙ − y = − − , (2.25) d3 r3 (1 − µ)z µz z¨ = − − . (2.26) d3 r3 As illustrated in Meirovitch [29], the Lagrangian of the CR3BP does not depend on time explicitly, that results in a constant Hamiltonian. It follows that the sys- tem possesses a constant of integration known as the Jacobi Constant. Physically, the gravitational forces, must be balanced by the centrifugal forces. It follows that a modified potential energy function corresponding to the differential equations in (2.24)-(2.26) can be identified, that is, 1 (1 − µ) µ U ∗ = (x ˙ 2 +y ˙2)+ + . (2.27) 2 d r Note that in the above definition, the potential is positive and is a convention in the formulation of the CR3BP. The equations of motion can now be expressed in terms of the following partial derivatives, ∂U ∗ x¨ = + 2y, ˙ (2.28) ∂x 20

∂U ∗ y¨ = − 2x, ˙ (2.29) ∂y ∂U ∗ z¨ = . (2.30) ∂z Equations (2.28)-(2.30) represent the traditional, convenient form of the equations of motion relative to the rotating frame in non-dimensional coordinates. Jacobi identified the constant of integration associated with the differential equa- tions that takes the following form,

C =2U ∗ − (x ˙ 2 +y ˙2 +z ˙2). (2.31)

Jacobi’s Constant is sometimes called the integral of relative energy [48]. It is important to note that it is not an energy integral but rather an energy-like constant partly due to the formulation of the problem relative to a rotating system. It is also notable that, in the restricted problem, neither energy nor angular momentum is conserved. The Jacobi Constant can be used to produce zero-velocity plots that identify regions of exclusion for a specific energy level. Figure 2.2 illustrates an example of a zero-velocity curve for a Jacobi Constant value of C=3.161. A particle would require an imaginary velocity to be within the region enclosed by the closed green curve. Awareness of these forbidden regions can offer much insight into the dynamics of the problem. In addition to insight, the Jacobi Constant is very often used as a method to check the accuracy of the calculations, particularly the accuracy of the numerical integration of the differential equations.

2.4.1 Singularities in the Equations of Motion

The equations of motion in the CR3BP possess singularities at the centers of the two primaries. The singularities result from terms of the form 1/r3 and 1/d3, where

“r” is the non-dimensional distance from the mass m2 to the spacecraft and “d” is the non-dimensional distance from the mass m1 to the spacecraft. When a transfer trajectory originates in a low altitude Earth parking orbit or a launch trajectory is computed, d or r are very small. Thus, the state is very close to a singularity that 21

1.5

1

0.5

L L 1 2 0

Earth Moon Y−Axis [lunar units]

−0.5

−1

−1.5 −1.5 −1 −0.5 0 0.5 1 1.5 X−Axis [lunar units]

Figure 2.2. Zero Velocity Curve for C = 3.161.

degrades accuracy and limits the effectiveness of the differential corrections technique in determining transfer trajectories beyond the Earth’s sphere of influence. The sin- gularity can be avoided through regularization [40] at the cost of physical insight. Previous studies [28], originally with similar difficulties, have demonstrated that reg- ularization is generally not necessary due to the current computational capabilities. Backward integration of the transfer trajectories is therefore employed to limit the sensitivity to the initial conditions.

From the differential equations, it is apparaent that the transformation [40] τ = −t results in a change in the derivatives with respect to the independent variable, such that d/dτ = −d/dt and d2/d2τ = d2/d2t. With this transformation, the integration 22

can be initiated on the halo orbit or along a manifold trajectory at time tf and computed backwards to determine the state close to the Earth at time t0. Both transfer trajectories, as well as launch trajectories, are determined using backward integration.

2.5 State Transition Matrix

Equations (2.28)-(2.30) can be rewritten as six first-order differential equations where the state vector is defined asx ¯ = [x y z x˙ y˙ z˙]T . These six first-order differential equations can be linearized relative to a reference solutionx ¯ref = [xref yref zref x˙ ref T y˙ref z˙ref ] by use of a Taylor series expansion and ignoring the higher order terms. Note that the reference solution can be a constant equilibrium state or a time-varying solution to the nonlinear differential equations. Define the linearized state relative to T x¯ref as δx¯(t) = [δx(t) δy(t) δz(t) δx˙(t) δy˙(t) δz˙(t)] . Then, the linear state variational equation can be expressed in the form

δx˙ (t)= A(t)δx(t), (2.32) where A(t)isa6 × 6, generally time-varying, matrix. It can be written in term of the following four 3 × 3 submatrices,

0 I3 A(t)= . (2.33)  ∗  UXX 2Ω Each one of the elements of the 6 × 6 matrix A(t)isa3 × 3 matrix, where 0 represents the zero matrix, and I3 the identity matrix of rank 3. Then Ω is defined as constant, 0 10 Ω=  −1 0 0  . (2.34)    0 00    ∗   The matrix of second partials, UXX , is comprised of elements ∗ ∗ ∗ Uxx Uxy Uxz ∗  ∗ ∗ ∗  UXX = Uyx Uyy Uyz , (2.35)  ∗ ∗ ∗   Uzx Uzy Uzz      23

where ∂2U ∗ U ∗ = . (2.36) ab ∂a∂b Of course, the partials are evaluated on the reference solution and it can be assumed ∗ that, if at least the first and second order derivatives are continuous, UXX is sym- metric. The form of the solution to equation (2.32) is well known, assuming the state transition matrix φ(t, t0) is available, that is,

δx(t)= φ(t, t0)δx0, (2.37)

where φ(t, t0) is the 6 × 6 state transition matrix (STM) evaluated from time t0 to

time t. The state transition matrix φ(t, t0) is a linear map that reflects the sensitivity

of the state at time t to small perturbations in the initial state at time t0. Any differential corrections scheme exploits the STM to predict the initial perturbations that yield some desired change in the final state. Differentiating equation (2.37) and substituting equations (2.32) and (2.37) into the result produces

φ˙(t, t0)= A(t)φ(t, t0). (2.38)

This matrix differential equation represents 36 scalar equations. The initial conditions

for φ are determined by evaluating equation (2.38) at time t0. So, the initial conditions for equation (2.38) yield the identity matrix of rank 6, or

φ(t0, t0)= I6. (2.39)

Adding the 6 scalar differential equations for the state yields 42 coupled scalar dif- ferential equations to be numerically integrated. A general analytical solution is not available because A(t) is time-varying.

2.6 Differential Corrections

Differential corrections (DC) schemes use the STM for targeting purposes. One application is an iterative process to isolate a trajectory arc that connects two points 24

in solution space. Differential corrections (DC) techniques can be used to quickly obtain a solution with the desired parameters in a wide range of problems. A suf- ficiently accurate first guess for the the initial state is always required. When only the natural motion is considered in three-dimensional space, the total number of parameters available in the problem is 14 [1]. This number of parameters will be termed the “dimension” of the problem. As illustrated in figure 2.3, there are two

seven-dimensional states, one at each end of the trajectory defined by the epochs, t1 and t2, their positions, R¯1 and R¯2, and the velocity components, V¯1 and V¯2; thus, the problem is parameterized by 14 scalar elements. This number also character- izes the sum of the fixed constraints, that is, the targets, the controls, and the free parameters in the problem. The fixed constraints are the parameters that are not allowed to vary, for example, the initial position vector R¯1. The controls are the set of parameters that the DC scheme is allowed to modify to achieve the desired target states. Free parameters are additional variables not used in the DC scheme as either fixed quantities, controls, or targets. These values will likely change in a way that may not be predictable. Differential correction schemes are often used to obtain periodic solutions to the nonlinear differential equations in the CR3BP. A common assumption, making use of the symmetry in this problem, is that the desired solution is symmetric about the x − z plane. Initially, the known states are given in the form T x¯0 = [x0 0 z0 0y ˙0 0] and, from the symmetry properties, it is concluded that for a simply symmetric periodic orbit, at the next x − z plane crossing, the trajectory

Figure 2.3. A Stylized Representation of 1 Step Differential Corrector. 25

T will be consistent with the values, i.e.,x ¯f = [xf 0 zf 0y ˙f 0] . Since the initial guess

for the state vectorx ¯0 is not likely to yield the necessary form ofx ¯f , the differential corrections process then uses the STM to compute the required changes in two of the initial non-zero variables to drive the velocitiesx ˙ f andz ˙f to zero. There are an infi- nite number of periodic orbits that satisfy these conditions. One approach to isolate a specific trajectory is to fix one non-zero quantity associated with the initial state as a constant. The fixed quantity can be varied, depending on the goals [34]. The process is repeated until the desired final result is achieved, i.e., the orbit crossing the x − z plane perpendicularly. Convergence to a solution is usually obtained after about four iterations. Numerically integrating equation (2.28)-(2.30) in three dimensions, with the appropriate initial conditions for one period, yields a halo orbit.

2.7 Particular Solutions

In 1772, Joseph Lagrange identified five particular solutions to equations (2.28)- (2.30) as equilibrium points in the 3BP, for a formulation relative to a rotating frame. As equilibrium points, the gravitational and centrifugal forces are balanced at these locations but it is important to note that the points are still moving in a circular orbit about the barycenter relative to the inertial frame. All five points lie in the plane of motion of the primaries and the location of the points in the Earth-Moon system appear in figure 2.4. Linear stability analysis can be employed to determine that the collinear Lagrange points L1, L2, and L3 are inherently unstable and that the equilat- eral points L4 and L5 are linearly stable. Placing a probe at the triangular points will result in oscillations in the vicinity of the equilibrium point. The equilibrium points obtained the name, libration points, from the oscillatory motion at the equilateral locations [20]. One type of periodic and quasi-periodic solutions that are the focus of a number of recent missions are the periodic, planar Lyapunov and three-dimensional halo orbits as well as the three-dimensional quasi-periodic Lissajous trajectories. 26

Libration Points in the Earth−Moon System 1.5

1 L 4

0.5

L Earth L L 3 1 2 0 Moon

−0.5

y [non−dimensional lunar units] L 5 −1

−1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x [non−dimensional lunar units]

Figure 2.4. Location of the Libration Points in the Earth-Moon System Relative to a Synodic Frame.

Poincar´erealized the importance associated with periodic orbits. In his conjecture [49] in 1895, he stated that an infinite number of periodic orbits exist in the 3BP. For Poincar´e, periodic motion appeared to be significant in nature and he considered the study of periodic orbits a matter of greatest importance. His investigations into periodic orbits in the 3BP were limited to an analytical investigation. Although considerable progress was made in approximation techniques to represent periodic orbits over the following 20 years, detailed investigations were hindered by the amount of computations involved.

Lyapunov, halo, ‘nearly-vertical’ orbits, and Lissajous trajectories each occur in families with similar characteristics. Lyapunov orbits are planar orbits that lie in the 27 plane of motion of the primaries. A pitchfork bifurcation of the Lyapunov orbits, both above and below the x − y plane, results in two halo families that are mirror images across the x−y plane [50]. When the maximum out-of-plane amplitude (Az) is in the +z direction, the halo orbit is a member of the northern family (NASA Class I) and if the maximum excursion is in the −z direction, the halo orbit is a member of the southern family (NASA Class II). Each member of a family corresponds to a slightly different energy level (Jacobi Constant). Halo orbits were first computed in the CR3BP and are defined as precisely periodic, three-dimensional libration point orbits.

They are typically characterized by their maximum out-of-plane amplitude (Az). An example of a halo orbit in the CR3BP appears in figure 2.5. The three-dimensional orbit is presented in terms of orthographic projections. For Lissajous trajectories, the amplitude of the in-plane motion and that of the out-of-plane motion are arbitrary and the frequencies are not commensurate [9, 51]. Orthographic projections of an example of a Lissajous trajectory appears in figure 2.6. Precisely periodic halo orbits do not exist in a more general model that incorporates additional perturbations. In an ephemeris model, Lissajous trajectories can always be generated and careful selection of in-plane and out-of-plane amplitudes yields Lissajous trajectories that are very close to periodic for a limited time interval. These orbits are generally denoted as halo orbits. Although Lissajous trajectories are not periodic, they are nevertheless bounded and exist on an n-dimensional torus [50, 51]. Another type of periodic motion, is the family of the ‘nearly-vertical’ orbits first visually identified by Moulton [50].

As is true with any nonlinear dynamical system, a low order approximation does not immediately yield a continuous trajectory, however, a differential corrections pro- cedure can result in a natural periodic solution if the initial guess is sufficiently ac- curate. If a linear approximation is not sufficient to create a Lissajous trajectory with the desired characteristics, the third-order analytical approximation developed by Richardson and Cary [32] is commonly deployed. Alternatively, patch points, consisting of the full six-dimensional state plus the time, along a halo orbit can be 28

4 x 10 3 2 1 Earth−Moon System A = 331,521 km x 0 L A = 25,991 km

Y [km] 1 y −1 A = 15,393 km z −2 −3 3 3.2 3.4 3.6 5 X [km] x 10 4 4 x 10 x 10

2 2 1 1 L L 0 1 0 1 Z [km] −1 Z (km) −1 −2 −2

3 3.2 3.4 3.6 −2 0 2 5 4 X [km] x 10 Y (km) x 10

Figure 2.5. Northern Earth-Moon L1 Halo Orbit in the CR3BP.

determined and imported into an ephemeris model to obtain a halo-like Lissajous trajectory. The patch points are corrected through a two-level differential corrections process (2LDC) developed by Howell and Pernicka [52]; details of this process are offered later.

2.8 Invariant Manifold Theory

In the late 19th century, the French mathematician Henri Poincar´esearched for precise mathematical formulas that would allow an understanding of the dynamical stability of systems. These investigations resulted in the development of what is now called Dynamical Systems Theory (DST). Dynamical systems theory is based 29

5 x 10

2 1 Sun Earth 0 L y [km] −1 1 −2

1.478 1.48 1.482 1.484 8 x [km] x 10 5 5 x 10 x 10

2 2 1 1 L L 0 1 0 1 z [km] −1 z [km] −1 −2 −2 −3 −3 1.478 1.48 1.482 1.484 −4 −2 0 2 4 8 5 x [km] x 10 y [km] x 10

Figure 2.6. Lissajous Trajectory at L1 in the Sun-Earth CR3BP.

on a geometrical view for the set of all possible states of a system in the phase space [53]. Detailed background information is available in various mathematical sources including Perko [54], Wiggins [55], as well as Guckenheimer and Holmes [56]. An extensive summary appears in Marchand [51].

2.8.1 Brief Overview

For a continuous nonlinear vector field of the form

x˙ = f(x), (2.40) the local behavior of the flow in the vicinity of a reference solution to the nonlinear equations can be determined from linear stability analysis for most applications. Note 30

thatx ¯ is the entire state vector, and letx ¯ =x ¯ref + δx¯ such that δx¯ is the vector of variations. Assuming that the above vector field is continuous to the second degree, it can be expressed in terms of a Taylor series expansion relative to this reference solution resulting in a vector variational equation of the following form

δx˙ (t)= A(t)x(t), (2.41) where A(t) is an n × n matrix of the first partial derivatives. If the reference solution is an equilibrium solution, such as a libration point, then the A matrix is constant. In general, however, the A matrix cannot be assumed constant and is time-varying, A = A(t). However, for the moment, assume a time-invariant system and variations relative to an equilibrium point. The algebraic technique of diagonalization can be used to reduce the linear system to an uncoupled linear system. Perko [54] states the following theorem from linear algebra that allows a solution to a linear, time-invariant system with real and distinct eigenvalues:

Theorem 2.1 If the eigenvalues λ1, λ2, ..., λn of an n × n matrix are real and n distinct, then any set of the corresponding eigenvectors η1 η2 ...ηn forms a basis R , the matrix P = [η1 η2 ...ηn] is invertible and

−1 P AP = diag[λ1, ..., λn].

In the process of reducing the linear system in equation (2.41) to an uncoupled system, the linear transformation − y = P 1δx, (2.42) is defined, where P is the invertible matrix defined in theorem 2.1. Then

δx = P y, (2.43) and Perko demonstrates that the solution can be written

y(t)= P E(t)P −1δx(0), (2.44) 31

where E(t) is the diagonal matrix

E(t)= diag[eλ1t, ..., eλnt]. (2.45)

Rewriting equation (2.44) allows the more insightful expression

n λj t y(t)= cje ηj, (2.46) j=1 X

where the cj’s are scalar coefficients.

Perko continues by stating “The subspaces spanned by the eigenvectors ηi of the matrix A determine the stable and unstable subspaces of the linear system, equation (2.41), according to the following definition,”

Definition 2.1 Suppose that the n × n matrix A has k negative λ1,..., λk and n-k

positive eigenvalues λk+1,..., λn and that these eigenvalues are distinct. Let η1,...,ηn be a corresponding set of eigenvectors. Then the stable and unstable subspaces of s u the linear system (2.41), E and E , are the linear subspaces spanned by η1,...,ηk

and ηk+1,...,ηn respectively; i.e.,

s E = Span{η1, ..., ηk},

u E = Span{ηk+1, ..., ηn}.

Perko completes the above definition by adding “If the matrix A has pure imaginary eigenvalues, then there is also a center subspace Ec.” Define the complex vector

wj = uj + iηj, as a generalized eigenvector of the matrix A corresponding to a complex eigenvalue λj = aj + ibj and then let

B = {η1, ..., ηk, ηk+1, ηv+1, ..., um, vm}, (2.47) be the basis of Rn. Then definition 2.2 below allows a distinction between the stable, unstable, and center subspaces. 32

Definition 2.2 Let λj = aj + ibj, w¯j =u ¯j + iη¯j and B be as described above. Then s E = Span { uj, ηj k aj < 0 }, c E = Span { uj, ηj k aj =0 }, and

u E = Span { uj, ηj k aj > 0 }, i.e., Es, Ec, and Eu are the subspaces of the real and imaginary parts of the general- ized eigenvectors wj corresponding to eigenvalues λj with negative, zero, and positive real parts respectively.

Decomposing the phase space into three separate regions is the ‘dynamic approach’ [53]. The sum of the three fundamental subspaces spans the complete space Rn.

Selecting initial conditions carefully to ensure that certain specified coefficients cj are equal to zero in equation (2.46), results in the desired behavior inside a subspace for all time [53]. Once in a subspace, motion remains there for all time. From a linear perspective, this can be seen as only exciting the desired mode while eliminating any perturbations of the undesirable modes. Thus, solutions that originate in Es asymptotically approach y = 0¯ as t → ∞ and solutions with initial conditions in Eu approach y = 0¯ as t → −∞. Solutions in the center subspace Ec neither grow nor decay over time. Guckenheimer and Holmes [56] relate the stable and unstable manifolds to the invariant subspaces for an equilibrium point through the Stable Manifold Theorem.

Theorem 2.2 (Stable Manifold Theorem for Flows) Suppose that x˙ = f(x) has a hyperbolic equilibrium point xeq. Then there exist local stable and unstable man- s u ifolds Wloc(xeq), Wloc(xeq), of the same dimensions ns, nu as those of the eigenspaces s u s u E and E of the linearized system (2.41), and tangent to E and E at xeq. The s u local manifolds Wloc(xeq), Wloc(xeq) are as smooth as the function f.

Let xeq be the non-hyperbolic equilibrium point, or the libration point, at L1. Then, figure 2.7 can be used to illustrate the stable manifold theorem. In the imme- diate vicinity of the libration point, the eigenvectors are directed along the individual 33 local stable and unstable subspaces. The manifolds associated with the stable and unstable subspaces are globalized through numerical integration. In the CR3BP, the collinear libration points L1, L2, and L3 possess a four-dimensional center subspace and one-dimensional stable and unstable subspaces. Figure 2.7 represents the stable and unstable eigenvectors, Es and Eu, and the globalized manifolds, W s and W u, associated with the Earth-Moon L1 point. In the case of the libration points, the position and velocity components of the eigenvectors are always parallel. For the un- stable eigenvector, both position and velocity components are directed the same, i.e., away from the libration point. For the stable eigenvector, the position and velocity components of the eigenvector are oriented in opposite directions. Hence, a small displacement from the libration point along the stable eigenvector, results in motion toward the libration point. A small displacement along the unstable eigenvector, re- sults in motion away from the libration point. Planar Lyapunov orbits and nearly vertical out-of-plane orbits are examples of periodic solutions that exist in the center subspace near Li [51,57]. As the amplitude of the Lyapunov orbits increases to a crit- ical amplitude, a bifurcation point identifies the intersection of the planar Lyapunov orbits and the three-dimensional halo family of periodic orbits [50,51]. Note that the Lyapunov familly continues beyond the critical amplitude but the stability properties of the orbits have changed. The critical amplitude can be identified by monitoring the characteristics of the eigenvalues of the monodromy matrix.

2.8.2 Periodic Orbits and Dynamical Systems Theory

For a number of applications, the state transition matrix at the end of a full revolution, also termed the monodromy matrix φ(T, 0), must be available. Consider the point along the periodic orbit that was selected as the starting and ending point. In dynamical systems, this point is denoted as a fixed point in a stroboscopic map. Then, the monodromy matrix serves as a discrete linear map near the fixed point located at the origin of the map. Such a map is also often called a Poincar´emap [9,53]. 34

4 x 10 2

1.5

Wu+ 1 s+ s W Eu E 0.5

0 L 1 u

Y−Axis [km] E −0.5 Es Wu− −1 Ws− −1.5

−2 −1 0 1 2 4 X−Axis [km] x 10

Figure 2.7. Stable and Unstable Eigenvectors and the Globalized Manifold for the Earth-Moon L1 Point.

The eigenvalues and eigenvectors of the monodromy matrix can be used to estimate the local geometry of the phase space in the vicinity of the fixed point. It is important to note that the monodromy matrix possesses different elements for every fixed point along the periodic orbit. Thus, the eigenvectors change directions and, thus, the directions of the stable and unstable subspaces vary along the orbit. The eigenvalues, on the other hand, are a property of the orbit and remain constant as is apparent from equation (2.51). Linearizing relative to periodic orbits results in linear, periodic, differential equa- tions. The A matrix in equation (2.41) is now time-varying but periodic, i.e.,

δx˙ (t)= A(t)δx(t). (2.48) 35

According to Floquet theory [24,57], the STM can be rewritten as

φ(t, 0) = F (t)eJtF −1(0), (2.49)

where J is a normal matrix that is diagonal or block diagonal and the diagonal elements are the characteristic multipliers or Floquet multipliers. Note that F (t) is a periodic matrix. Solving for the matrices J and F that correspond to a periodic system yields φ(T, 0) = F (0)eJtF −1(0), (2.50)

since F (T )= F (0) and eJt = F −1(0)φ(t, 0)F (0). (2.51)

From equation (2.51), it is clear that F (0) and J contain the eigenvectors and eigen- values of the monodromy matrix. From equation (2.51), then,

̟iT λi = e , (2.52)

1 ̟ = ln(λ ), (2.53) i T i where ̟i are the Poincar´eexponents. The Poincar´eexponents are interpreted in a manner similar to the eigenvalues in a constant coefficient system. In Hamiltonian-like systems, such as the CR3BP, they must also occur in positive/negative pairs by Lyapunov’s theorem. From the stability properties associated with the Poincar´eexponents, conclusions about the location of the characteristic multipliers on the complex plane and, thus the stability of the fixed point and the periodic orbit, are potentially available [57]. A system with no characteristic multipliers on the unit circle is called hyperbolic and the nature of a hyperbolic system can be summarized as

|λi| < 1 ⇒ stable y = 0 as t → ∞

|λi| > 1 ⇒ unstable y = 0 ast → −∞

If |λi| = 1, no stability information can be obtained from the characteristic multipliers and they correspond to the center subspace. 36

Periodic orbits possess a monodromy matrix that yields at least one eigenvalue with a modulus of one [54]. The following theorem adds further significance to the eigenvalues.

Theorem 2.3 (Lyapunov’s Theorem) “If λ is an eigenvalue of the monodromy matrix φ(T, 0) of a time-invariant system, then λ−1 is also an eigenvalue, with the same structure of elementary divisors.”

1 1 1 λ1 = ,λ3 = ,λ5 = . (2.54) λ2 λ4 λ6

Thus, according to Lyapunov’s theorem, the six eigenvalues associated with a peri- odic orbit (via the map and associated monodromy matrix) also appear in reciprocal, complex conjugate pairs. So, two of the six eigenvalues of the monodromy matrix will always be precisely one. In the CR3BP, the center subspace has a dimension of four and two of the eigenvalues are real and equal to one for precisely periodic orbits. Eigenvalues with real parts smaller than 1 are considered stable eigenvalues and eigenvalues with real parts larger than 1 are considered unstable. Eigenvectors corre- sponding to a stable eigenvalue lie in the stable subspace and yield stable manifolds asymptotically approaching the periodic orbit as t →∞. Eigenvectors corresponding to an unstable eigenvalue lie in the unstable subspace and yield unstable manifolds asymptotically approaching the orbit as t → −∞.

2.9 Computing Manifolds

Computation of the globalized manifolds relies on the availability of initial condi- tions obtained from the stable and unstable subspaces, Es and Eu. The eigenvectors of the monodromy matrix offer local approximations of the stable and unstable sub- spaces for fixed points along periodic orbits. Note that an eigenvector only indicates orientation in space. The eigenvector does not yield a specific directional sense, that is multiplying by negative one, yields a valid eigenvector in the opposite direction. This results in manifolds approaching (stable) and departing (unstable) the orbit in two 37

different directions. Generally, one side of the globalized manifold enters the region

around the smaller primary P2 whereas the other enters the inner region around the

major primary P1. The directions spanned by the position components for 30 points

along the halo orbit(Az = 15,000 km) are illustrated in figure 2.8. The velocity (red) components relative to the position (blue) components for the stable eigenvector ap- pear in figure 2.9. This set results in the globalized manifold approaching the halo orbit from the larger primary. The angle between the position and velocity com-

ponents of the eigenvector varies between 146 and 170 degrees for a lunar L1 halo

orbit with an Az amplitude of 15,000 km. The angular range is dependent upon the orbit investigated, of course. While studying this figure, it is critical to realize that the eigenvectors only offer a linear approximation of the subspaces close to the halo orbit and, hence, only offer a valid approximation of the direction of these subspaces very close to the periodic orbit. Thus in addition to investigating the direction of the eigenvectors, it is crucial to consider the globalized manifolds to obtain a complete picture of the different subspaces. Applying a small perturbation in the direction of the eigenvector results in a local estimate of the one-dimensional manifold associated with the fixed point. After a local estimate has been determined, the trajectory on the manifold associated with the point can be globalized through numerical integra- tion. Given the eigenvectors of the monodromy matrix, the local estimate of the

stable and unstable manifolds, X¯s and X¯u, can be computed as

Ws X¯s =x ¯(ti)+ d · V¯ (ti), (2.55)

Ws where V¯ (ti) is defined by ˆ Ws Ws Y (ti) V¯ (ti)= , (2.56) x2 + y2 + z2 Ws T and Yˆ (ti) = [xs ys zs x˙ s y˙s z˙s] is thep stable eigenvector. Then,

Wu X¯u =x ¯(ti)+ d · V¯ (ti), (2.57)

Wu where V¯ (ti) is defined by ˆ Wu Wu Y (ti) V¯ (ti)= , (2.58) x2 + y2 + z2 p 38

4 x 10 2.5

2

1.5

1

0.5

0 Earth Moon L y [km] 1 −0.5

−1

−1.5

−2

−2.5

2.9 3 3.1 3.2 3.3 3.4 3.5 5 x [km] x 10

Figure 2.8. Stable Eigenvectors of the Monodromy Matrix for an Earth-Moon L1 Halo Orbit.

Wu T and Yˆ (ti) = [xu yu zu x˙ u y˙u z˙u] is the unstable eigenvector.

In equation (2.55) and equation (2.57), d is the initial displacement (perturbation) from the periodic orbit. Larger d values ensure an initial state along a manifold that is further advanced in departing from the periodic orbit or libration point. At the same time, the initial displacement cannot be selected arbitrarily large since the linear approximation must remain within the range of validity. In the Earth-Moon system, d values commonly range between 30 km and 70 km, whereas in the Sun-Earth system commonly used d values range between 150 and 200 km. The initial displacement can also be employed as a design parameter; changing the magnitude of d only affects the particular trajectory selected along the approaching manifold, but not the orbit 39

4 x 10 2.5

2

1.5

1

0.5 L Earth 1 Moon 0 y [km] −0.5

−1

−1.5

−2

−2.5

2.9 3 3.1 3.2 3.3 3.4 3.5 5 x [km] x 10

Figure 2.9. Position and Velocity Components of the Stable Eigenvectors of the Monodromy Matrix for an Earth-Moon L1 halo.

itself. The effects of different d values are illustrated in figure 2.10. Normalizing the eigenvectors relative to the position components ensures that the displacement along the eigenvector is uniform. Once the monodromy matrix is obtained for any one fixed point along the orbit, the manifolds associated with any other point along the orbit can be calculated two different ways. First, shifting the focus to another fixed point, the calculations can be repeated and a new monodromy matrix determined. Then, new eigenvectors can be computed. Alternatively, it is more efficient to exploit the state transition matrix. An eigenvector can be directly shifted along the orbit via the STM as follows,

¯ Ws ¯ Ws Yi = Φ(ti, t0)Y0 , (2.59) 40

Figure 2.10. Various Manifolds Asymptotically Approaching the Orbit.

¯ Wu ¯ Wu Yi = Φ(ti, t0)Y0 . (2.60)

This direct shift is quick and accurate. In the interest of numerical accuracy, it is also beneficial to explore the structure associated with the CR3BP to limit excessive numerical integrations of the STM as well. In the CR3BP, the monodromy matrix φ(T, 0) can be generated from the half-cycle STM φ(T/2, 0),

0 −I3 −2Ω I3 φ(T, 0) = G φT (T /2, 0) G−1φ(T /2, 0), (2.61)     I3 −2Ω −I3 0 where     100000  0 −10 0 0 0     001000  G =   . (2.62)    0 0 0 −10 0       000010       00000 −1      41

6 x 10 3

2

1

Earth 0 y [km] Sun

−1

−2

−3 1.44 1.45 1.46 1.47 1.48 1.49 1.5 1.51 8 x [km] x 10

Figure 2.11. Stable (blue) and Unstable (red) Manifolds for a Sun-Earth Halo Orbit near L1.

The proof of equation (2.61) uses the fact that the STM is a simplectic matrix [33]. Two halo orbits in the Sun-Earth system appear in green in figure 2.12. The halo orbits, plotted in green, possess an Az amplitude of 120,000 km. The stable

(blue) and unstable (red) manifolds associated with these L1 and L2 halo orbits form three-dimensional tubes in the vicinity of the Earth. The Earth in figure 2.12 is not plotted to scale and only the physical location is represented by the blue sphere. For larger halo orbits, part of the manifold tubes extend below the surface of the Earth, i.e., pass less than 6378 km from Earth’s center. As is illustrated in figure 2.12, the manifolds are separatrices for a given energy level. The manifold tubes separate different regions of motion in space. Additionally, if motion starts on a tube, it will not leave that tube unless the energy level is altered [58]. 42

6 x 10

1

0.5

0 y [km]

−0.5

−1

1.48 1.485 1.49 1.495 1.5 1.505 1.51 8 x [km] x 10

Figure 2.12. Stable (blue) and Unstable (red) Manifold Tube Approaching the Earth in the Sun-Earth System.

2.10 Transition of the Solution to the Ephemeris Model

Solutions in the CR3BP offer much insight into the solution space and are quali- tatively similar to solutions existing in higher order multi-body models. To improve the accuracy of transfer trajectories and to illustrate their credibility and robustness, they are transferred to an ephemeris model.

The ephemeris model that is employed here is the Generator software package [59] developed by Howell et al. at Purdue University. The primary motion is specified from planetary state ephemerides available from the Jet Propulsion Laboratory DE405 43 ephemeris files. In the ephemeris model, the Earth is used as the origin and Newto- nian equations of motion are integrated with respect to the inertial frame. From a number of possible integrators, an 8/9 order Runge-Kutta integrator is selected for this application to perform the integration with respect to the Earth. In the deter- mination of transfer trajectories in the Earth-Moon system, the Sun is also included in the force model, although any number of bodies can be incorporated. Since no as- sumptions about the primary motion are desired, the transformation from inertial to rotating coordinates is based on the instantaneous position of the primaries obtained from the ephemeris files. Much of the symmetry of the CR3BP is lost when solution arcs are transferred to an ephemeris model. The libration points are no longer fixed points relative to the rotating frame of the primaries. Instead, they oscillate. Precisely periodic halo orbits no longer exist, however, it is often desirable to maintain characteristics for the quasi-periodic orbits in the ephemeris model that are similar to those of the periodic halo orbits in the CR3BP. Therefore, transfer arcs that correspond to halo orbits in the CR3BP are employed as initial estimates in the ephemeris model. A two-level differential corrections (2LDC) scheme [52] (described later) computes a Lissajous trajectory, figure 2.13, resembling the original halo orbit. The “halo-like” Lissajous in figure 2.13 corresponds to a translunar halo orbit with Az amplitude of 15,000 km. The curve in figure 2.13 is plotted in the Earth-Moon rotating frame centered at the Moon. 44

4 x 10 5

0 L Y [km] 1 Moon L 2

Earth

−5 −6 −4 −2 0 2 4 6 4 X [km] x 10

4 4 x 10 x 10 4 4

2 2 Moon Moon 0 0 Z [km] Z [km] L L 1 2

−2 −2 Earth

−4 −4 −6 −4 −2 0 2 4 6 −5 0 5 4 Y [km] 4 X [km] x 10 x 10

Figure 2.13. A ”Halo-like” Lissajous Trajectory in a Ephemeris model with an Az Amplitude of approximately 15,000 km. 45

3. TRANSFERS FROM EARTH PARKING ORBITS TO

LUNAR L1 HALO ORBITS

Efficient determination and computation of transfer trajectories from parking orbits around the larger primary to L1 halo orbits remains a challenge. Direct insertion onto a manifold from the Earth is not possible, since the natural flow does not pass sufficiently close to the primary. The natural motion for arbitrary Earth-Moon halo orbits is presented to illustrate the difficulties in the determination of such transfers. Results in this chapter represent the most recent efforts in the analysis of this problem. The highest priority is an understanding of the difficulties associated with the computation of these transfer trajectories (including starting values) and the development of the methodology for detailed investigation. Various transfer trajec- tories between a low altitude Earth parking orbit and two different halo orbits are presented. A more complete understanding of the dynamical structure in this multi- body environment is the foundation for the design of future trajectories originating and returning to an Earth parking orbit. A free return transfer trajectory with poten- tial applications in human spaceflight is also included. The trajectory arcs computed here can ultimately be optimized as well.

3.1 Stable and Unstable Flow that is Associated with the Libration Point

L1 in the Vicinity of the Earth

Exploration of the natural dynamics associated with invariant manifolds to and

from L1 halo orbits in the Earth-Moon system is the first step in developing an efficient design tool. Therefore, the stable and unstable manifolds have been computed for a series of different halo orbits. Although this behavior is similar in any 3BP system, the analysis presented here is limited to the Earth-Moon dynamical environment. 46

Figure 3.1. Stable and Unstable Manifold Tubes in the Vicinity of the Earth.

The stable and unstable flow around the first (largest) primary does not allow for natural solutions, i.e., manifolds, to pass close by the primary. The trajectories corresponding to different fixed points along the halo form a tube-like surface. Each individual trajectory wraps around the manifold surface that forms the tube. These stable (blue) and unstable (red) manifold surfaces appear in figure 3.1 for an Earth-

Moon halo orbit with an Az amplitude of 15,000 km. (The size of a halo orbit is typically characterized by the out-of-plane amplitude denoted Az.) The trajectories that correspond to specific L1 halo orbits first pass the Earth at altitudes ranging between 82,200 km to 93,000 km. Although, after multiple passes, a trajectory might subsequently pass closer to the Earth, the investigation here is limited to first passes. The closest Earth passing altitudes along the trajectories that correspond to the stable manifolds for differently sized Earth-Moon L1 halo orbits appears in figure 47

4 x 10

11

10.5

10

9.5

9

8.5

8

7.5 Lowest Passing Altitude [km] 7

6.5

6 1 2 3 4 5 6 7 4 Halo Orbit Az [km] x 10

Figure 3.2. Minimum Earth Passing Altitudes for Trajectories on the Manifold Tubes Associated with Various Earth-Moon L1 Halo Orbits.

3.2. Qualitively, the behavior for the stable and unstable manifold tubes is identical although they approach the Earth from different regions of space as is illustrated in

figure 3.1. Large halo orbits with an Az amplitude of approximately 43,800 km yield specific trajectories that offer the lowest Earth passing altitudes. As is apparent in figure 3.1, trajectories extend backwards from the halo orbit in a tube shape and, therefore, only manifolds that approach/depart a small region in the specific LPO pass close by the first primary. Some trajectories, those for large Az amplitudes, first pass the Earth at altitudes well about 240,000 km. 48

3.2 Design Strategy

The lunar initiative to return to the Moon and its vicinity requires the ability to compute transfers between the Earth, the Moon, and libration point orbits quickly and efficiently. In addition to robotic and human missions to the Moon, the deploy- ment of humans for servicing and repair missions to observatories or a space station in halo orbits at the collinear lunar libration points requires the determination of trans- fers meeting stringent rendezvous requirements. Extensive investigations of transfer trajectories to and from the Earth and the Moon to halo orbits near L1 and L2 are currently ongoing. The development of fast and efficient design tools and various types of tradeoff studies are, hence, of great interest and critical to the success of future efforts to expand scientific missions and to create a human presence in the lunar vicinity.

Recall that the pass distance between the Earth and the stable and unstable man- ifolds associated with Earth-Moon halo orbit is large. Since this is typically tens of thousands of kilometers, a broad range of concepts must be explored for transfers. In 1973, D’Amario first conducted an extensive study on transfers from both the

Earth and the Moon to the libration point at L2 using a multiconic approach and optimizing his results by applying primer vector theory [37]. Although it was cer- tainly not trivial, the methodology allowed him to limit the amount of numerical computations by obtaining the state transition matrices analytically. With minimal numerical computations, D’Amario was able to determine optimal transfers fairly accurately. It is noted that the ∆v to insert into L2 in the Earth-Moon system is not small. D’Amario never studied a halo orbit, however. In the late 1970’s, the determination of transfers between Earth parking orbits and the Sun-Earth libration points shifted to numerical integration and shooting techniques with the advent of high-speed computing. Although shooting techniques, in combination with differen- tial corrections schemes, have been employed very successfully by many researchers in the computation of transfers [8, 38, 40], they are time consuming when a large range 49

of parameters must be investigated. Nevertheless, straightforward shooting can be extremely useful in the determination of baseline solutions when no other tools are available. Shooting techniques are therefore examined first, and the results form the basis for future improvements. The ultimate goal of this investigation is the development of more efficient tech- niques to compute these transfers. The initial reference transfers obtained through shooting techniques, may form the baseline solutions. Differential correction tech- niques to solve for continuous transfers from initial to final orbits with tangential parking orbit departures are employed. The resulting transfers are investigated and compared to the local stable manifold structure to reveal potential improvements.

3.2.1 Shooting Technique

An initial reference transfer must exhibit the general characteristics of the desired trajectory. Shooting techniques are employed that consist of manually varying the velocity components of an initial state on the parking orbit and numerically integrat- ing forward in the CR3BP for a fixed time of flight. This process is repeated until a trajectory arc, one displaying the desired behavior, is determined. Patch points equally spaced in distance along the transfer are then isolated. The patch points are corrected to yield a tangential parking orbit intersection at the Earth and a contin- uous entry into the halo orbit using the differential correction algorithm that will be presented in Section (3.3). Previous work [60] has indicated that adding the transfer arc patch points to halo orbit patch points at the minimum and maximum x and y amplitudes for one or two revolutions along the halo, can substantially lower the halo orbit insertion cost (HOI). The patch points along the transfer trajectory are thus added to patch points corresponding to the desired halo orbit. The differential corrections process via backwards integration, will yield the solution. 50

3.2.2 Investigated Transfer Types

The investigations, thus far, have focused on two different types of trajectories. First, transfer arcs that directly connect the parking orbit with the halo orbit are ex- amined. Then, the second approach is insertion onto the stable manifold approaching the orbit. Clearly, these options do not encompass all possible transfer trajectories. Nor are any of the transfers optimized; but they will offer insight into the dynamics and characteristics of the transfers in this region of space for differently sized halo orbits. Two different halo orbits are examined. First, consider a halo orbit that possesses an out-of-plane Az amplitude of 15,000 km. Halo orbits near L2 with very small

Az amplitudes are of little practical value as they lie in the direct line of sight of the Moon, which results in communications difficulties. It is therefore necessary to select halo orbits with an out-of-plane component with a minimum Az amplitude of around 3,100 km to establish an efficient communications link [61]. The out-of- plane amplitude of 15,000 km was selected for comparison with other studies [58]. The second halo orbit corresponds to the one with the closest Earth approach man- ifold. The corresponding Az amplitude is approximately 43,800 km, obviously much larger than the first. Besides offering the closest Earth approach, this halo orbit is a representative Earth-Moon halo orbit that frequently results in low transfer costs in system-to-system transfers [6,58]. A spacecraft in a circular, Earth parking orbit, can, in theory, orbit the Earth in two different directions. Due to the Earth rotation, however, only one of these directions is obviously affordable for launch opportunities. Therefore only parking orbits with an eastward rotation have been considered. No assumptions about the inclination of the parking orbit have been made. 51

3.3 Two-Level Differential Corrector

A two-level differential corrector (2LDC) is a numerical scheme that can be used to solve a two point boundary value problem. The particular strategy employed here was originally developed by Howell and Pernicka [52] for the numerical determination of Lissajous trajectories. It employs an iterative process based on two distinct pro- cedures that alter elements of the six-dimensional state vectors, plus time, that are associated with points along a potential solution arc. (These points are termed ‘patch points’.) The values are all shifted simultaneously to yield a continuous trajectory that satisfies a set of constraints. As an initial guess, the algorithm requires a set of patch points sufficiently close to an acceptable dynamical solution. A minimum of three patch points is typically required. The first and last patch points are com- monly denoted the initial and final patch points. All other remaining patch points are labelled internal. Each one of n patch points is related to the previous and fol- lowing patch points that comprise (n−1) trajectory segments. The various positions, velocities, and times associated with three patch points are illustrated in figure 3.3.

The position stater ¯0, the velocity statev ¯0, and the time t0 define the initial point;

the positionr ¯f , the velocityv ¯f , and the time tf define the final point. At the inter- nal patch points, it is necessary to distinguish between the incoming state,r ¯p− ,v ¯p− ,

and tp− , and the outgoing state,r ¯p+ ,v ¯p+ , and tp+ . As noted, this is accomplished with the symbols ‘-’ and ‘+’. The first step in the corrections scheme employs a linear differential corrections process to ensure position and time continuity between all the segments but introduces velocity discontinuities at the internal patch points. The second level uses linear corrections, applied simultaneously, to the position vec- tors and time states corresponding to all of the target points, to reduce the velocity discontinuities and enforce any additional constraints. 52

Figure 3.3. A Stylized Representation of Level II Differential Corrector (from Wilson [1]).

3.3.1 First Step - Ensuring Position Continuity

The first step of the 2LDC ensures position and time continuity between all the trajectory segments. This initial procedure is implemented using a straightforward linear targeting scheme that modifies the velocity state and TOF (four potential scalar control variables) at the beginning of each segment to meet the position state requirements (3 scalar targets) at the end of the segment. For each segment, a vector relationship of the form δx¯ δx¯(t)= φ(t, t0)δx¯0 + δ∆t, (3.1) δt ∗ p

δx¯ where δt p∗ is the linear variation of the state with respect to time, is evaluated at the end of the segment and δ∆t is the variation in the time of flight along the trajectory

segment, such that ∆t = tf − t0. Thus, δ∆t = δtf − δt0. In matrix format, the linear targeter is then formulated as Lk¯ = ¯b, (3.2) with

φ14 φ15 φ16 x˙   L = φ24 φ25 φ26 y˙ , (3.3)    φ34 φ35 φ36 z˙      ¯ T k = [δx˙ 0, δy˙0, δz˙0, δ∆t0] , (3.4) 53

T ¯b = [δxp,δyp, δzp] . (3.5)

Since the number of control variables in k¯ exceeds the number of target states in ¯b, an infinite number of solutions exists. To minimize the changes in the elements of the control vector, the solution with the minimum Euclidean norm is selected, i.e.,

k¯ = LT (LLT )−1¯b. (3.6)

With the trajectory continuous in position, the second step of the 2LDC can now be deployed to decrease the known velocity discontinuities.

3.3.2 Second Step - Enforcing Velocity Continuity

The ultimate goal of the corrections process is to produce a trajectory that is continuous in position, velocity, and time, such that the design requirements and the constraints are both satisfied. So, in the second level, the positions of all of the patch points are varied to decrease the internal velocity discontinuities, ∆¯vp, introduced by the first step and, thus, the total cost. At all of the internal patch points, the incoming velocity state vectorv ¯p− is compared to the outgoing velocity state vector v¯p+ to compute the velocity discontinuity at that patch point, i.e.,

∆¯vp =v ¯p+ − v¯p− . (3.7)

Decreasing the velocity discontinuities simultaneously requires the formation of the State Relationship Matrix (SRM). A trajectory consisting of only three patch points

is described for clarity. This arc is composed of an initial patch point PP0, one

interior point PPp, and a final patch point PPf . The relationship between variations in the scalar elements of the initial, interior, and final positions, denoted δr¯, velocities, denoted δv¯, as well as the corresponding times, denoted δt, can then be written as follows

δr¯p− Ap0 Bp0 δr¯0 v¯p− = + (δt − − δt ) , (3.8)         p 0 δv¯p− Cp0 Dp0 δv¯0 a¯p−         54

δr¯p+ Apf Bpf δr¯f v¯p+ = + (δt + − δt ) . (3.9)         p f δv¯p+ Cpf Dpf δv¯f a¯p+         wherea ¯p− is the acceleration of the incoming state anda ¯p+ is the acceleration of the outgoing state at the internal patch point. The change in the time of flight on the

first segment is given by (δtp− − δt0) and the change in the time of flight along the second segment is (δtp+ − δtf ).

Evaluating the first vector equation in equation (3.8) in terms of δv¯0 and substi- tuting into the the second equation yields

− − − − 1 1 − 1 − − − δv¯p = Cp0 − Dp0Bp0 Ap0 δr¯0 + Dp0Bp0 δr¯p − Dp0Bp0 v¯p − a¯p (δtp − δt0) .

(3.10) Along the second segment, the same procedure is applied to equation (3.9) by solving for δv¯f and then substituting. The result is the expression

−1 −1 −1 δv¯p+ = Cpf − Dpf Bpf Apf δr¯f + Dpf Bpf δr¯p+ − Dpf Bpf v¯p+ − a¯p+ (δtp+ − δtf ) .

(3.11) Subtracting equation (3.10) from equation (3.11) and imposing position and time continuityr ¯p+ =r ¯p− , tp+ = tp− produces

δr¯0   δt0    δr¯p  δ∆¯vp =   , (3.12) M0 Mt0 Mp Mtp Mf Mtf    δtp  h i      δr¯f       δtf      where the M matrices are defined as follows,

−1 M0 = Dp0Bp0 Ap0 − Cp0 , (3.13)

−
− 1 − Mt0 =¯ap − Dp0Bp0 v¯p , (3.14)

−1 −1 Mp = Dpf Bpf − Dp0Bp0 , (3.15) 55

− − 1 − − 1 Mtp = Dp0Bp0 v¯p − a¯p − Dpf Bpf v¯p+ − a¯p+ , (3.16)

−1
Mf = Cpf − Dpf Bpf Apf , (3.17)

−1 Mtf = Dpf Bpf v¯p+ − a¯p+ . (3.18)

Since it is desired to achieve a value of zero for the interior velocity discontinuity, define the desired variation δ∆¯vp = −∆¯vp. The variations δr¯ in the position states can now be computed by solving the linear system given for δr¯,

δr¯ = M −1 [δ∆¯v] . (3.19)

For trajectories with multiple internal patch points, n, the M matrices must be com- puted for every internal patch point and collected in one equation for a simultaneous solution for all position variations. The resulting matrix is called the State Relation- ship matrix (SRM). It appears in the following general form

M01 Mt01 Mp1 Mtp1 Mf1 Mtf1 0 0 ... 0 0   0 0 M02 Mt02 Mp2 Mtp2 Mf2 Mtf2 ... 0 0    .. . .  M =   ,    .. . .       .. . .       0 0 0 0 0 M0n−1 Mt0 − Mpn−1 Mtp0 − Mfn−1 Mtf −   n 1 n 1 n 1   (3.20)  with δr¯ and δ∆¯vp in equation (3.19) defined as

δr¯0   δt0    δr¯1      δr¯ =  δt1  , (3.21)      :       δr¯f       δtf      56

−∆¯v1   −∆¯v2 δ∆¯v = . (3.22)          −∆¯vn−1      Multiple iterations of both steps are required to yield a trajectory continuous in position, velocity, and time.

3.4 Constraints

The transfer trajectory is assumed to originate in a low-altitude circular parking orbit around the Earth. To ensure that the differentially corrected transfer trajectory originates on the desired parking orbit, it is necessary to add constraints to the patch point states that are defined on the parking orbit. Constraints can be defined for any patch points if they can be expressed as functions of position, velocity, and time corresponding to the patch point [1]. Thus, constraints are of the form

αi = α(¯ri, v¯i, ti), (3.23) where the subscript i identifies the patch point at which the constraint is defined. The derivation of the second stage of the two-level corrector can be expanded in terms of constraint variations such that the goal, that is, no velocity discontinuities at the internal patch points, remains the same. When incorporating constraints into the differential corrections scheme, it is necessary to determine the variations of the constraints with respect to the independent parameters. It is possible to define multiple constraints at the same patch point.

A given constraint relationship at patch point PPi is influenced by, at most, the patch points immediately before (PPi−1) and after (PPi+1), as well as the point at which the constraint is defined. Position constraints only depend upon the patch point at which they are defined. Velocity dependent constraints may be functions of states at the previous, the current, and the following patch point. These constraints 57

are added to the SRM in the second step of the 2LDC and represented in the following expression,

d∆¯vi d∆¯vi δ∆¯vi δr¯k = dr¯k dtk . (3.24)    dαi dαij    δαi δtk dr¯k dr¯k

   Mα    where Mα represents the augmented| State{z Relationship} Matrix and the subscript k identifies all n patch points. Depending on the specific purpose of a velocity dependent

constraint α, the velocity at PPi,v ¯i, can be represented three different ways. The

velocityv ¯i in equation (3.23) can either be expressed solely as a function ofv ¯p− orv ¯p+ or as a combination of the two. Since the transfer trajectory is computed through backwards integration, the parking orbit constraints are applied at the last patch

point. In that case,v ¯i is defined asv ¯i =v ¯p−. On the other hand, if a velocity

dependent constraint α is applied at the first patch point,v ¯i must be defined as

v¯i =v ¯p+. The average value,v ¯i = 0.5(¯vp− +¯vp+) has proven very effective when a constraint α is defined at an internal patch point.

3.4.1 Position and Epoch Constraints

Constraining the position and epoch of a patch point is simply achieved by using the following two constraints,

α¯i1 =r ¯i − r¯des, (3.25)

and

αi2 = ti − tdes. (3.26)

Then, the related variations with respect to the independent variables are easily deduced as ∂α¯i1 = I3, (3.27) ∂r¯i and ∂α i2 =1. (3.28) ∂ti 58

The variations can be used to compute the elements of the augmented SRM through application of the chain rule.

3.4.2 Parking Orbit Constraints

At least two constraints are necessary to force a patch point to lie on a circular parking orbit around the Earth. At a minimum, a specified altitude and an apse condition must be ensured. In the present study, an altitude of 200 km above the Earth’s surface was somewhat arbitrarily selected for the parking orbit. Patch points and constraints must be defined with respect to the same center. Then, the altitude constraint is a modification of equation (3.25), i.e.,

αi1 = |r¯i| − rdes, (3.29)

th where |r¯i| is the altitude of the i patch point relative to the Earth. The apse constraint,r ¯ · v¯ = 0, is expressed in functional form as

αi2 =r ¯i · v¯i, (3.30)

th wherer ¯i is the position vector andv ¯i the velocity vector corresponding to the i patch point and with respect to the Earth. Since the altitude constraint is only a function of position, the only non-zero partial is ∂α r¯T i1 = i . (3.31) ∂r¯i |r¯i| For the apse constraint, the partials due to position and time are obtained as follows

∂αi2 T =v ¯i , (3.32) ∂r¯i ∂α i2 =0. (3.33) ∂ti

The partial due to velocity first requires selection of thev ¯i expression. The velocity

∂αij partial, − , can then be obtained through application of the chain rule such that ∂v¯i

∂αi2 ∂αi2 ∂v¯i ∂αi2 ∂αi2 ∂v¯i − = − ; + = + . (3.34) ∂v¯i ∂v¯i ∂v¯i ∂v¯i ∂v¯i ∂v¯i 59

The velocity partials corresponding to the apse constraint can then be formulated as

∂αi2 T ∂v¯i ∂αi2 T ∂v¯i − =r ¯i − ; + =r ¯i + , (3.35) ∂v¯i ∂v¯i ∂v¯i ∂v¯i and used in the computation of the elements of the augmented SRM.

3.4.3 | ∆¯v | Constraints

A transfer trajectory, from an Earth parking orbit to a halo orbit, that is com- pletely continuous in both position and velocity is likely to be impossible. This is true even when a number of revolutions of the halo orbit are incorporated in the differential corrections scheme to add additional freedom. Under these conditions, it is desirable in an initial step to allow deterministic maneuvers and constrain the magnitude of the maneuver. This is accomplished via a constraint on the magnitude of the velocity discontinuity that exists at an internal patch point. The constraint replaces the original formulation, that is, forcing the ∆¯v to zero, at the specified patch point. The constraint is defined as

+ − αi = ||v¯i − v¯i | − ∆vdes|, (3.36)

where ∆vdes is the desired scalar magnitude. Then the constraint partials evaluated as + − T ∂αi −(¯vi − v¯i ) − = + − , (3.37) ∂v¯i |v¯i − v¯i |

+ − T ∂αi (¯vi − v¯i ) + = + − . (3.38) ∂v¯i |v¯i − v¯i | The constraint is applied such that a deterministic maneuver of equal or smaller magnitude than that specified is allowed. No effort is made to constrain the direction of the maneuver. 60

3.5 Direct Transfer Trajectories from Earth to Lunar to L1 Halo Orbits

Direct transfers arcs that connect Earth parking orbits with the halo orbit gener- ally require at least two deterministic maneuvers, one at each end of the transfer arc. The maneuver to exit the Earth parking orbit and to escape the near-Earth envi- ronment is labelled the Transfer Trajectory Insertion (TTI) maneuver; the maneuver required to enter the halo orbit is labelled Halo Orbit Insertion (HOI) maneuver. The Time Of Flight (TOF) along the transfer is computed as the time interval between the TTI and the HOI maneuver. Observe that TOF is not defined consistently in the literature so comparison is sometimes difficult.

In a preliminary examination with a shooting technique, the transfer is computed using backwards integration. The HOI maneuver is initially constrained to occur at the x−z plane crossing on the far side of the L1 halo orbit. Earlier studies suggest that using the plane crossing as the HOI point, yields successful transfers [7, 37]. Sample transfer trajectories appear in figure 3.4 and figure 3.5. They are computed in the CR3BP for two differently sized halo orbits. The same transfers appear in figures 3.6 and 3.7, where the CR3BP results are transitioned to the ephemeris model. The corresponding |∆¯v| costs are listed in Table 3.1. The total cost for the transfer arc is

denoted by |∆¯vT otal|. It is the sum of the TTI and HOI maneuvers. Perhaps lifting the constraint on the location of the HOI maneuver would be beneficial.

Table 3.1. Transfer Costs for Two Differently Sized Halo Orbits; TTI Maneuver Constrained to the x − z Plane Crossing

Az Model |∆¯vT otal| TTI HOI TOF Figure (km) (m/s) (m/s) (m/s) (days) 15,000 CR3BP 3716.3 3115.4 600.9 3.98 3.4 43,800 CR3BP 3659.5 3118.0 541.5 4.67 3.5 15,000 Ephemeris 3733.2 3111.8 621.45 3.87 3.6 43,800 Ephemeris 3744.4 3129.6 565.8 4.58 3.7 61

4 x 10

15 Moon

10 Earth

5 y [km]

L 0 1

−5

0 0.5 1 1.5 2 2.5 3 5 x [km] x 10

Figure 3.4. Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 15, 000 km) in the CR3BP (Location of HOI maneuver is constrained).

An alternative strategy to design a transfer is to completely lift the constraint on the location of the HOI maneuver. Then, the patch point floats and the 2LDC scheme results in a new location for the patch point yielding a continuous transfer arc satisfying the applied constraints. The HOI maneuver costs in Table 3.1 are used as the maximal allowed cost for the HOI maneuver constraint. The resulting transfers in the CR3BP appear in figures 3.8 and 3.9. Note that the libration point orbit is no longer precisely periodic and has evolved into a Lissajous trajectory. Clearly, patch points along the Lissajous trajectory are included in the “transfer” arc. Thus, the shift in the location of the maneuver is the result of allowing the patch points of the orbit to move in the process of computing a continuous path. Details corresponding to the 62

5 x 10

1.5 Moon

1

0.5 y [km] Earth L 0 1

−0.5

−1 0 0.5 1 1.5 2 2.5 3 3.5 5 x [km] x 10

Figure 3.5. Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 43, 800 km) in the CR3BP (Location of HOI maneuver is constrained).

transfers in figures 3.8 and 3.9 appear in Table 3.2. Note that the ∆¯v’s are nearly equal to the results in Table 3.1 (CR3BP), when the location is completely constrained. But as suggested in figures 3.8 and 3.9, the new transfers are accomplished with a maneuver that involves more than just an energy change. Partially constraining the HOI location, based on the underlying dynamical structure, will likely yield results more consistent with the natural flow. Such an approach is beyond the scope of the current study since it requires more complete knowledge of the flow. Such work is continuing.

Transition of any trajectory arc from the CR3BP model to the ephemeris model is always nontrivial. Experience indicates that arcs including maneuvers with large 63

4 x 10

15

10

5 y [km]

0

−5

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 5 x [km] x 10

Figure 3.6. Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 15, 000 km) in an Ephemeris Model (Location of HOI Manuever Constrained).

directional changes are not good starting solutions. Thus, the transfers in figures 3.8 and 3.9 are not well-suited for transition to the ephemeris model, at least in their current form. Nevertheless, the transfers in figures 3.8 and 3.9 are used as starters for a DC scheme in the ephemeris model to explore the problem. Perhaps not unexpectedly, the differential corrections process does not yield a transfer that exhibits the same characteristics. These resulting transfer arcs appear in figures 3.10 and 3.11 and appear strikingly different. In fact, it is interesting to note, that these two transfers actually use the stable manifold structure approaching from the smaller primary. Such a result suggests a new direction for the study of transfer options. 64

4 x 10

15

10

5 y [km]

0

−5

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 5 x [km] x 10

Figure 3.7. Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 43,800 km) in an Ephemeris Model (Location of HOI Manuever Constrained).

3.6 Transfer Trajectories with a Manifold Insertion

Rather than inserting directly into the halo orbit, inserting onto a stable manifold associated with the specified halo orbit offers additional freedom. Many points along the manifold offer potential insertion points. The focus of the investigation, thus far, is limited to insertions along a relatively small region of the stable manifold near the approach to the halo orbit. This region is known to be successful in transfers from the smaller primary in the Sun-Earth system [44]. Sample transfers in the CR3BP appear in figures 3.12 and 3.13. The same transfers appear in figures 3.14 and 3.15, where the CR3BP results are transitioned to the ephemeris model. The cost and TOF for these transfers is summarized in Table 3.3. The maneuver to insert into the manifold occurs at the Manifold Insertion Point and, hence, is labelled MIP. No HOI maneuver is necessary as the manifold asymptotically approaches the halo orbit. The 65

Table 3.2. Transfer Costs for Two Differently Sized Halo Orbits; Location of the TTI Maneuver Determined by the Differential Corrections Scheme.

Az Model |∆¯vT otal| TTI HOI TOF Figure (km) (m/s) (m/s) (m/s) (days) 15,000 CR3BP 3700 3122. 579.0 4.19 3.8 43,800 CR3BP 3682.4 3122.6 539.2 4.95 3.9 15,000 Ephemeris 3808.4 3134.2 674.2 6.5 3.10 43,800 Ephemeris 3759.4 3142.5 616.9 9.2 3.11

time of flight for the transfer trajectory is measured from TTI to the first x − z plane crossing on the far side of the halo orbit relative to the Earth. The TOF is, therefore, considerably longer in comparison to previous cases, since the transfer path includes the slow loop along the manifold before approaching the halo orbit. The actual TOF on the transfer arc before the insertion onto a manifold trajectory is only 2.8 days for both cases. Note from Table 3.3, that the MIP maneuver is slightly higher than previous HOI ∆¯v’s. However, the approach is much smoother and may serve as a more efficient baseline solution prior to cost reduction or optimization.

Table 3.3. Transfer Costs for Transfers with a Manifold Insertion for Two Differently Sized Halo Orbits.

Az Model |∆¯vT otal| TTI MIP TOF Figure (km) (m/s) (m/s) (m/s) (days) 15,000 CR3BP 3802.46 3121 698.11 12.18 3.12 43,800 CR3BP 3859.10 3130.94 728.15 15.96 3.13 15,000 Ephemeris 3854.08 3126.86 727.22 15.0 3.14 43,800 Ephemeris 3886.40 3139.18 747.22 15.6 3.15 66

5 x 10

1.5 Moon

1

0.5

y [km] Earth L 0 1

HOI −0.5

−1

0 0.5 1 1.5 2 2.5 3 5 x [km] x 10

Figure 3.8. Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 15,000 km) in the CR3BP (Location of the TTI Maneuver Determined by the Differential Corrections Scheme.)

3.7 Effects of a Cost Reduction Procedure

A cost reduction procedure is a numerical continuation technique that can be employed to decrease maneuver costs. By slightly decreasing the magnitude of an allowed maneuver, it is often possible to determine trajectories near the initial base- line solution that exhibit similar behavior at a slightly lower cost. The process of decreasing the maneuver cost can then be repeated until either an acceptable maneu- ver cost is available or the differential corrections process fails to determine a solution satisfying all the constraints. Cost reduction procedures can often be successfully 67

5 x 10

1.5 Moon

1

0.5

Earth y [km] L 0 1

HOI −0.5

−1

0 0.5 1 1.5 2 2.5 3 3.5 5 x [km] x 10

Figure 3.9. Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 43,800 km) in the CR3BP (Location of the TTI Maneuver Determined by the Differential Corrections Scheme.)

applied to trajectories in an ephemeris model that originate from baseline solutions in the CR3BP. A preliminary investigation into the effects of a cost reduction procedure on the transfers in the ephemeris model, is promising. The results of a cost reduction pro- cedure applied to transfer trajectories that insert into a manifold on the near side

of the orbit is illustrated in figure 3.16 for transfers to the smaller halo orbit (Az = 15,000 km). The baseline trajectory that initiates the cost reduction procedure appears in figure 3.14. Note that the trajectory in black in figure 3.16 possesses a higher manifold insertion cost than the transfers in green. The MIP cost decreases from that associated with the transfer in black to the lower-cost transfer in green; the 68

5 x 10 3

2

1 y [km]

0

−4 −3 −2 −1 0 5 x [km] x 10 5 5 x 10 x 10

1 1

0 0 z [km] z (km)

−1 −1

−4 −3 −2 −1 0 0 1 2 3 5 x [km] 5 x 10 y (km) x 10

Figure 3.10. Transfer from an Earth Parking Orbit to L1 Halo Orbits (Az = 15,000 km) in an Ephemeris Model (Location of the TTI Maneuver Determined by the Differential Corrections Scheme.)

MIP cost corresponding to the black trajectory is 737.2 m/s compared to 692.2 m/s for the transfer in green. The lower cost transfer apparently exploits the invariant manifold departing toward the smaller primary. Since similar behavior is observed for both halo orbits, it suggests that cheaper transfers exist that approach the halo orbit from the far side. The larger halo orbit also changes its shape quite significantly and eventually possesses few similarities with the original orbit. 69

5 x 10 4

3

2

y [km] 1

0

−4 −2 0 5 x [km] x 10 5 5 x 10 x 10 2 2

1 1

0 0 z (km) z [km] −1 −1

−2 −2 −4 −2 0 0 2 4 5 y (km) 5 x [km] x 10 x 10

Figure 3.11. Transfer from an Earth Parking Orbit to L1 Halo Orbits (Az = 43,800 km) in an Ephemeris Model (Location of the TTI Maneuver Determined by the Differential Corrections Scheme.)

3.8 Free Return Trajectory

Transfer trajectories to halo orbits can be computed that naturally return to the vicinity of the Earth. One such type of transfer can be determined when a perpendicular x − z plane crossing on the far side of the halo orbit is targeted. The resulting transfer trajectory could be employed for a spacecraft in an Earth parking orbit to transfer to a lunar L1 halo orbit and return to an Earth parking orbit with no additional maneuver. Similar trajectories could find applications in bringing supplies to a space station in a lunar L1 orbit or even in human missions to L1 halo orbits requiring safe return options. The HOI cost to insert into a halo orbit from such a 70

5 x 10

Moon 1

0.5

Earth L 0 1 y [km]

−0.5

−1

0 0.5 1 1.5 2 2.5 3 5 x [km] x 10

Figure 3.12. Transfer from an Earth Parking Orbit with a Manifold Insertion into an L1 Halo Orbit (Az = 15,000 km) in the CR3BP.

trajectory is approximately 647 m/s in an ephemeris model, higher than alternate transfers previously discussed. The x − y projection of the trajectory arc appears in figure 3.17 and the x − z and y − z projections are plotted in figures 3.18 and 3.19. All three projections have their origin at the Moon. The Earth movement in the eight days it takes to complete the entire transfer from TTI to reentering the Earth parking orbit is displayed by a black line in figures 3.17 and 3.18. The 200 km altitude parking orbit at the beginning of the journey appears in black and the same parking orbit eight days later appears in red in figure 3.17. It is noted that the resulting trajectory is not a periodic cycler trajectory. Two deterministic maneuvers are required to enter and exit the Earth parking orbit. 71

5 x 10

1 Moon

0.5

Earth L 0 1 y [km]

−0.5

−1

0 0.5 1 1.5 2 2.5 3 3.5 5 x [km] x 10

Figure 3.13. Transfer from an Earth Parking Orbit with a Manifold Insertion into an L1 Halo Orbit (Az = 43,800 km) in the CR3BP.

3.9 Summary and Conclusions

The results here are of a preliminary nature and include the most recent efforts in an ongoing investigation. Before comparing the cost of the different transfers, some difficulties encountered in the differential corrections are discussed.

3.9.1 Numerical versus Dynamical Issues in the Computation of Transfers

In the determination of transfer trajectories using a linear differential correc- tions process, a number of difficulties are encountered in both the CR3BP and the ephemeris model. In the CR3BP model, the baseline transfers are obtained using 72

5 x 10

1

0.5

0 y [km] MIP

−0.5

−1

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 5 x [km] x 10

Figure 3.14. Transfer from an Earth Parking Orbit with a Manifold Insertion into an L1 Halo Orbit (Az = 15,000 km) in an Ephemeris Model.

shooting techniques and can be straightforwardly corrected when the HOI maneuver is completely constrained, where the original baseline solution intersects the orbit or the manifold. This is not surprising since only minor changes are required to yield a continuous transfer that tangentially departs the Earth parking orbit. In the case of the smaller halo orbit, it is also possible to decrease the HOI cost, when the location of the HOI maneuver is not constrained. For the larger halo orbit, the shape of the first revolution along the halo orbit is altered significantly in an attempt to reduce the cost. The freedom available in the process, formulated with patch points along the revolutions of the orbit, is designed to exploit this option. However, such a large maneuver in a relatively large orbit offers many search directions and the process 73

5 x 10

1

0.5

0

y [km] MIP

−0.5

−1

−1.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 5 x [km] x 10

Figure 3.15. Transfer from an Earth Parking Orbit with a Manifold Insertion into an L1 Halo Orbit (Az = 43,800 km) in an Ephemeris Model.

requires more guidance. Similar behavior is also observed, not surprisingly, in the ephemeris model, where it also requires more steps to correct transfers to the larger halo orbit compared to the smaller. The behavior of the numerical algorithm can be improved via an expanded understanding of the dynamical nature of the problem. More detailed investigations are required.

Reproducing or transitioning transfers to an ephemeris model is not a trivial task. One issue, in particular, gains new significance in the Earth-Moon system. First, the lunar mass is significant, resulting in a larger eccentricity of the relative primary orbit than in many other three-body systems. Secondly, the Earth is the larger primary. The impact of the first issue is apparent because computing trajectories in 74

5 x 10

1.5

1

0.5

y [km] 0

−0.5

−1

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 5 x [km] x 10

Figure 3.16. Effects of a Cost Reduction Procedure on the Transfer Arcs Initially Using the Invariant Manifold on the Near Earth Side for a Halo Orbit with an Az Amplitude of 15,000 km in an Ephemeris Model.

the ephemeris model, requires dimensionalizing the patch points obtained from the CR3BP. For a Sun-Earth trajectory, or an Earth-Moon trajectory not passing close to one of the primaries, using the mean value corresponding to the distance between the primaries, the characteristic distance, in the dimensionalization process is usually sufficient. But, in order to obtain a baseline solution close to the converged trajectory from the CR3BP model, the actual distance between the Earth and the Moon at the appropriate epoch is typically used for each patch point. This is necessary since the distance between the Earth and the Moon changes by about 13 percent over the course of one year. Using the mean value is, therefore, not sufficiently precise for trajectories very close to one of the primaries. Using the actual Earth-Moon distance 75

5 x 10

Moon 1

0.5 HOI

0 y [km] Direction −0.5 Earth is moving

−1

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 5 x [km] x 10

Figure 3.17. x − y Projection of a Free Return Trajectory to a Halo Orbit with an Az Amplitude of 15,000 km in an Ephemeris Model.

for the specific Julian Date associated with the patch points aids in assuring the validity of the baseline solution. This modification is reasonable and useful for a valid baseline, but one that is further from the desired result. Thus, without additional input to the differential corrections scheme, it is likely to converge to a solution with characteristics very different from those observed in the result from the CR3BP. A continuation scheme, therefore, is employed in the computation of some transfers. The continuation scheme initially computes a continuous trajectory and then enforces one constraint at a time. For this application, only the apse or the altitude constraint is imposed in the first step. The accuracy on the apse constraint is increased and the targeted altitude of the parking orbit is decreased over the course of multiple 76

5 x 10

1

Moon

0.5

0 Earth z [km] HOI

−0.5

−1

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 5 x [km] x 10

Figure 3.18. x − z Projection of a Free Return Trajectory to a Halo Orbit with an Az Amplitude of 15,000 km in an Ephemeris Model.

iterations in some of the cases. A nonlinear process in conjunction with a known dynamical structure is to be explored.

The current method to compute these transfers is based on a two-level linear dif- ferential corrections process, of course. A differential corrections scheme determines a solution satisfying the imposed constraints but makes no attempt to determine the optimal solution. Since it is based on a linear approximation, it requires a good base- line solution as an initial guess and requires that the underlying dynamics can locally be approximated by a linear approximation. If both of these conditions are satisfied, it returns a continuous solution close to the initial baseline solution. The natural flow in the neighborhood of the larger primary is very different from the dynamics near the 77

5 x 10

1

0.5

0 z [km] HOI

−0.5

−1

−1.5 −1 −0.5 0 0.5 1 1.5 5 y [km] x 10

Figure 3.19. y − z Projection of a Free Return Trajectory to a Halo Orbit with an Az Amplitude of 15,000 km in an Ephemeris Model.

smaller primary. No naturally occurring transfers exist that connect the environment close to the major primary to halo orbits at L1 or L2. Large deterministic maneuvers are therefore necessary to insert into the halo orbit. A linear differential corrections scheme is not very efficient in predicting the changes in the path point states to yield such a trajectory unless the initial baseline solution is in the immediate vicinity of the desired transfer. The underlying linear variations are not sufficient to approximate the dynamics close to the larger primary. It is important to note that most previous investigations into the determination of transfers in the 3BP have focused on transfers to the smaller primary. The natural dynamics and the computation of transfers from libration point orbits to the smaller primary is well understood and the developed 78 methodology will be discussed and used to compute a few sample transfers in chapter four.

3.9.2 Discussion

The TTI maneuver is consistently larger for transfers to the larger halo orbit. This is the result of a slightly higher energy level being required to reach and insert into the larger halo orbit, since it is slightly further away from the Earth due to the larger out-of-plane component. Overall, the changes in the TTI maneuver are fairly small, in agreement with previous investigations [5,7,62]. The HOI, respectively MIP, cost is lower for the larger halo. This is likely the result of the natural flow of the larger halo passing closer by the Earth than the flow corresponding to of the smaller halo. In this limited study, inserting into an invariant manifold on the near side of the orbit is more expensive than inserting into the halo orbit directly. Selecting chosen the closest Earth approach manifold for the specific halo, but arbitrarily identification of an initial manifold insertion point contributes to the increased cost. Investigating a larger range of manifolds and initial insertion points is certainly warranted and will likely result in significantly lower MIP costs. Another interesting observation is that the cost reduction procedure moves the manifold insertion point from the near Earth side to the far side of the halo orbit and uses the stable manifold approaching from the smaller primary. This suggests that transfers using the stable manifold structure approaching from the smaller primary exist for a lower cost. Transfers in the ephemeris model are always more expensive than the baseline solutions in the CR3BP. This is true for both the TTI as well as the HOI, as well as MIP, costs. Although, systematically applying a cost reduction procedure to the computed transfers is likely to decrease the cost, this is not surprising. The baseline solutions in the CR3BP do not incorporate any perturbations. Hence, the transfer determined in the ephemeris model does not necessarily correspond to the same transfer in the CR3BP. Incorporating the Sun into the model will yield more representative baseline solutions for correction in an 79 ephemeris model. Overall, the magnitude of the transfers is similar to the transfer costs in previous investigation [5,7,36,37,62]. 80

4. TRANSFERS FROM EARTH PARKING ORBITS TO SUN-EARTH LIBRATION POINT ORBITS

Transfers between Earth parking orbits and Sun-Earth halo orbits at L1 and L2 are computed. This is significant since the Earth is the smaller primary in the Sun-Earth system. An investigation into minimal passing altitudes for Sun-Earth halo orbits

at L1 is included and the passing distances and paths of manifolds near the smaller primary are clarified. Sample transfers between two differently sized halo orbits at both L1 and L2 in the Sun-Earth system are presented.

4.1 Stable Flow from the Libration Points in the Direction of the Earth

In the Sun-Earth system, the Earth is the smaller primary. The natural flow to and from the halo orbits is significantly different in the vicinity of the smaller primary compared to the larger primary. Manifolds corresponding to halo orbits near the collinear libration points, representing the natural flow, can be computed. These pass in the immediate vicinity of the Earth or even at a distance smaller than the radius of the Earth. In 1994, Barden completed an extensive study of the manifolds associated with specific halo orbits [28]. At the time, analysis of the manifolds associated with a large range of halo orbits was a challenging task due to the computations required.

Barden determined that for a halo orbit with an Az amplitude of 440,000 km, a range of manifolds passes the Earth at acceptable altitudes that could be used for transfers to such a halo orbit with minimal or no halo orbit insertion cost. Using modern high-speed computers, it is possible to compute the closest approach manifolds for a larger range of halo orbits.

The closest approach manifolds associated with halo orbits with Az amplitudes ranging between 0 km and 920,000 km have been computed. The relationship between 81

halo orbit amplitude and the closest approach altitude is plotted in figure 4.1. The

minimum occurs for a halo orbit possessing an Az value of approximately 691,800 km and the closest approach manifold passes only a few kilometers above the center of the Earth. A stable manifold intersecting the surface of the Earth could be used for direct launches onto a stable manifold and an asymptotic approach to the halo orbit. Any orbit (within the computed range) with an Az amplitude larger than 334,351 km has manifolds crossing an Earth parking orbit at a 200 km altitude. Two of these manifolds possess a periapsis distance at the parking orbit altitude. Thus, the velocity vector in the parking orbit and in the manifold are parallel and the TTI maneuver is a minimum. Manifolds that intersect the surface of the Earth (assuming

a perfect sphere) can be computed for halo orbits with Az amplitudes larger than

341,083 km. The relationship between the Az amplitude of the halo orbit and the altitude of the closest unstable manifolds with Earth approach is also represented by figure 4.1 due to symmetry considerations [51]. This is consistent with the findings

of other studies [28]. Halo orbits with relatively small Az amplitudes do not possess manifolds passing close by the Earth. Transfers from Earth parking orbits to small amplitude halo orbits can nevertheless be computed using the algorithm presented in section (3.3).

4.2 Selection of Halo Orbit Sizes

Spacecraft are not typically located precisely at the Sun-Earth libration point L1 because of the resulting Sun-Earth-satellite alignment. When viewed from the Earth,

a satellite at L1 is aligned with the Sun and downlink telemetry is overwhelmed by the intense solar noise background [8]. At the same time, large amplitude halo orbits also encounter communications issues. For large halo orbits, the Earth-Spacecraft-Sun angle can cause difficulty since it changes by a large amount. Therefore intermediate- sized halo orbits (or Lissajous orbits) are generally preferred. 82

8000

6000

4000

2000

0

−2000

Lowest Passing Altitude [km] −4000

−6000

−8000 0 2 4 6 8 10 5 Halo Orbit Az [km] x 10

Figure 4.1. Closest Approach Altitudes for L1 Sun-Earth Manifolds Relative to the Earth.

The halo orbits examined here possess Az amplitudes of 120,000 km and 440,000 km. The Az value of 120,000 km is selected because the analysis for the ISEE-3 mis- sion determined that such an orbit satisfies the communication constraints [8,38]. In addition to the ISEE-3 mission, other libration point missions have also used similar sized halo orbits [10, 63]. A halo orbit with an Az amplitude around 120,000 km is ideally suited to avoid interference with the Sun and allows the usage of standard com-

munication equipment. For a halo orbit with an Az value of approximately 440,000 km a large number of manifolds can be computed that pass the Earth at acceptable altitudes [28]. 83

4.3 Transfer Trajectories From Earth to L1 Halo Orbits

Transfer trajectories from an Earth parking orbit to two differently sized L1 halo orbits are presented. For the smaller halo orbit with an Az amplitude of 120,000 km, the manifold tube does not intersect the Earth at a radius of 6378 km or even at a 200 km altitude parking orbit. Hence, a transfer trajectory is computed using the 2LDC process. Before the 2LDC algorithm (with or without constraints) can be used to compute a transfer, it is necessary to obtain a baseline solution and generate patch points along the arc. The closest approach manifold can be selected as the baseline solution, since it is the closest, natural occurring solution compared to the desired transfer path. Patch points along the manifold are then generated and differentially corrected to yield a smooth, continuous trajectory between the parking orbit and the halo orbit. For the larger Az amplitude halo orbits, manifolds with periapsis at an altitude of approximately 200 km are selected as the baseline solutions. The ultimate transfer path, constitutes a differentially corrected version of the manifold that precisely connects the initial and final orbits with no insertion cost at the halo orbit. The maneuver cost associated with these transfers are summarized in Table

4.1. For the smaller halo orbit with an Az amplitude of 120,000 km, the HOI insertion cost is 20.5 m/s and the transfer appears in figure 4.2. This value is very close to the minimal halo orbit insertion cost from analysis by Barden for the same sized halo orbit. In a much more comprehensive investigation for the same sized orbit [28],

Barden computed a transfer trajectory from Earth parking orbit to an L1 halo orbit with a HOI insertion cost of only 20.3 m/s. Barden used a different differential corrections process incorporating a change of variables to compute the transfers. For the larger halo orbit (Az = 440,000 km), two transfer trajectories are computed with no HOI maneuver necessary. The two transfers appear in figure 4.3 in green and red. All three transfers are reproduced in an ephemeris model and appear in figure (4.3). 84

5 x 10

6

4

2

L 0 1 y [km] Earth

−2

−4

−6

1.48 1.482 1.484 1.486 1.488 1.49 1.492 1.494 1.496 8 x [km] x 10

Figure 4.2. Transfer from an Earth Parking Orbit to a L1 Halo Orbit (Az = 120, 000 km) in the CR3BP.

4.4 Transfer Trajectories From Earth to L2 Halo Orbits

Given the transfers to L1 halo orits in the Sun-Earth system, similar transfers are computed for L2 halo orbits with similar maneuver costs. The detailed transfer costs are summarized in Table 4.2. For a halo orbit with an Az amplitude of 120,000 km, this investigation resulted a transfer with an HOI cost of 20.60 m/s. This transfer appears in figure 4.5, with a HOI cost very close to the value computed by Barden, that is, 20.34 m/s. For the larger halo orbit with an Az amplitude of 440,000 km, two transfer trajectories are computed and appear in figure 4.6. No halo orbit insertion cost are necessary for either transfer, since they are trajectories on the manifold tube. 85

5 x 10 8

6

4

2 L 1 0

y [km] Earth

−2

−4

−6 Sun

−8 1.478 1.48 1.482 1.484 1.486 1.488 1.49 1.492 1.494 1.496 1.498 8 x [km] x 10

Figure 4.3. Transfers from an Earth Parking Orbit to a L1 Halo Orbit (Az = 440, 000 km) in the CR3BP.

All three transfers are reproduced in an ephemeris model and appear in figure (4.7).

4.5 Conclusions

The transfer trajectory insertion cost from an Earth parking orbit onto a transfer trajectory to a Sun-Earth halo orbit seem to be nearly constant for all cases consid- ered. The small differences in the TTI cost in an ephemeris model are caused by the varying location of the Moon. This is in agreement with previous previous findings published in the literature [5,7,62]. In the Sun-Earth system, the Earth is the smaller primary and trajectories between halo orbits and the vicinity of the Earth naturally 86

Table 4.1. Transfer Costs for Transfers from a 200 km Altitude Earth Parking Orbit to Two Differently Sized Sun-Earth L1 Halo Orbits

Az Model |∆¯vT otal| TTI HOI TOF Figure (km) (m/s) (m/s) (m/s) (days) 120,000 CR3BP 3270.7 3192.5 20.5 205.1 4.2 (red) 440,000 CR3BP 3193.5 3193.5 0 210.78 4.3 (red) 440,000 CR3BP 3193.5 3193.5 0 210.78 4.3 (green) 120,000 Ephemeris 3206.8 3196.26 14.74 208.91 4.4 (blue) 440,000 Ephemeris 3193.7 3193.7 0 210.78 4.4 (red) 440,000 Ephemeris 3196.4 3196.4 0 213.11 4.4 (green)

Table 4.2. Transfer Costs for Transfers from a 200 km Altitude Earth Parking Orbit to Two Differently Sized Sun-Earth L2 Halo Orbits

Az Model |∆¯vT otal| TTI HOI TOF Figure (km) (m/s) (m/s) (m/s) (days) 120,000 CR3BP 3213.13 3192.5 20.66 209.3 4.5 (red) 440,000 CR3BP 3193.4 3193.4 0 215.5 4.6 (red) 440,000 CR3BP 3193.4 3193.4 0 213.8 4.6 (green) 120,000 Ephemeris 3214.8 3192.4 18.66 209.25 4.7 (blue) 440,000 Ephemeris 3194.4 3194.4 0 215.5 4.7 (red) 440,000 Ephemeris 3194.2 3194.2 0 213.8 4.7 (green) 87

5 x 10 8

6

4

2

0 L y [km] 1 Earth −2

−4

−6 Sun

−15 −10 −5 0 5 x [km] x 10

Figure 4.4. Transfers from an Earth Parking Orbit to a L1 Halo Orbit in an Ephemeris Model.

occur. The halo orbit insertion cost is dependent on the size of the halo orbit. For larger halo orbits, it is possible to compute transfer trajectories with no halo orbit insertion cost, as naturally occurring manifolds exist. The HOI costs are comparable for both L1 and L2 halo orbits of identical size, again in agreement with previous findings [28]. Although, a slightly different methodology is used in this investigation compared to previous studies, the magnitude of the HOI maneuvers is consistent [28]. 88

5 x 10 8

6

4

2 Earth 0

y [km] L 2 −2

−4

−6

−8 1.494 1.496 1.498 1.5 1.502 1.504 1.506 1.508 1.51 1.512 1.514 8 x [km] x 10

Figure 4.5. Transfer from an Earth Parking Orbit to a L2 Halo Orbit (Az = 120, 000 km) in the CR3BP. 89

5 x 10 8

6

4

2 Earth 0

y [km] L 2 −2

−4

−6

−8 1.494 1.496 1.498 1.5 1.502 1.504 1.506 1.508 1.51 1.512 1.514 8 x [km] x 10

Figure 4.6. Transfers from an Earth Parking Orbit to a L2 Halo Orbit (Az = 440, 000 km) in the CR3BP. 90

5 x 10 8

6

4

2 L Earth 2 0 y [km]

−2

−4

−6 Sun

−8 0 5 10 15 5 x [km] x 10

Figure 4.7. Transfers from an Earth Parking Orbit to a L2 Halo Orbit in an Ephemeris Model. 91

5. LAUNCH TRAJECTORIES

As it is apparent from the previous chapter, the design of transfer trajectories from the Earth, as the smaller primary, to halo orbits in the Sun-Earth system is well es- tablished. When Earth is the larger primary, as in the Earth-Moon system, transfers are not as straightforward. However, an additional analysis capability that might be useful to mission designers, in either case, is the flexibility to determine the launch tra- jectory in conjunction with the transfer. The work presented in this chapter includes an investigation of preliminary methods for computing launch trajectories. The use of differential correction schemes for this purpose is discussed and a sample launch trajectory to a 200 km altitude parking orbit is presented. This proof of concept suggests more detailed investigations to be pursued.

5.1 Equation of Motions with Constant Thrust Term

To develop a mathematical model for launch, the equations of motion must be modified to incorporate an extra force. Because the launch leg is added to the transfer, the problem is still formulated in terms of the CR3BP. A constant thrust force is added to the equations of motion at some arbitrary angle relative to the direction of the velocity vector. Let the constant thrust term element, T¯, be equivalent to a constant force divided by the mass of the spacecraft. It is approximated as constant and, thus, the term possesses units of acceleration. The thrust term appears in the application of Newton’s law of motion, equation (2.15), i.e., Gm m Gm m ¯ I ¨ 3 1 3 2 F = m3 r¯3 = − 3 r¯13 − 3 r¯23 + m3T κ.ˆ (5.1) r13 r23 “Thrust” can also be nondimensionalized such that, nondimensional thrust isκ ¯ and, d2ρ¯ (1 − µ) µ = − d¯− r¯ +¯κ. (5.2) dτ 2 d3 r3 92

Then, the second order scalar differential equations appear in the following form (1 − µ)(x + µ) µ(x − (1 − µ)) x¨ − 2y ˙ − x = − − + κ , (5.3) d3 r3 x (1 − µ)y µy y¨ + 2x ˙ − y = − − + κ , (5.4) d3 r3 y (1 − µ)z µz z¨ = − − + κ . (5.5) d3 r3 z

where κ is the nondimensional thrust magnitude and κx, κy, and κz are thrust compo- nents in the various thrust directions as written in rotating coordinates and applied in three-dimensional space. Since it is desirable to apply the thrust in various di- rections relative to the velocity vector, spherical coordinates are introduced to define these angles. The spherical coordinates appear in figure 5.1 and relate the direction of the unit thrust to the velocity vector in the rotating frame of the primaries. The unit vectors

ξˆv, ξˆB and ξˆN , introduced by figure 5.1, are relative to the synodic frame of the primaries with ξˆv parallel to the velocity vector. Note thatr ¯κ is directed parallel to unit thrust in figure 5.1. Using the specified spherical coordinates, the unit thrust vector is defined by

κˆ = sin θ cos φξˆv + sin θ sin φξˆB + cos θξˆN . (5.6)

Thus the unit thrust vector relative to the rotating frame of the primaries becomes

κˆ = [sin θ · cos φ · C11 + sin θ · sin φ · C12 + cos θ · C13]ˆx

+ [sin θ · cos φ · C21 + sin θ · sin φ · C22 + cos θ · C23]ˆy (5.7)

+ [sin θ · cos φ · C31 + sin θ · sin φ · C32 + cos θ · C33]ˆz

where the Cij are scalar elements of the direction cosine matrix between the synodic frame of the primaries and the plane spanned by the position vector relative to the

Earth,r ¯E, and the velocity vector in the final orbit (at the parking orbit or manifold insertion point) in the rotating frame. To evaluate the elements of the direction cosine matrix, the position and velocity vector are crossed to obtain the unit angular momentum vector, normal to the orbital plane, such that rˆ × vˆ hˆ = E . (5.8) |rˆE × vˆ| 93

Figure 5.1. Spherical Coordinates to Define the Direction of the Thrust Vector.

The unit angular momentum vector, hˆ, is equal to the unit vector ξˆB that appears in figure 5.1. Now, crossing the unit position vector and the unit angular momentum vectors yields

ξˆN = hˆ × ξˆv, (5.9)

the third vector to form the dextral orthonormal triad ξˆv, ξˆB, and ξˆN . Thus, the

transformation matrix from triad ξˆv, ξˆB, and ξˆN to the synodic frame of the primaries in three-dimensional space can be defined as follows,

xˆ C11 C12 C13 ξˆv       yˆ = C21 C22 C23 ξˆN . (5.10)      ˆ   zˆ   C31 C32 C33   ξB              Equations (5.3)-(5.5) are used to compute the thrusted trajectory arcs presented in this investigation. 94

5.2 State Transition Matrix

To facilitate the implementation of the thrust parameters κ, θ, and φ into a differential corrections scheme, an augmented state vector is defined,

x¯ ≡ [x, y, z, x,˙ y,˙ z,φ,θ,κ˙ ]T , (5.11) and the corresponding augmented variational state vector then becomes

δx¯ ≡ [δx,δyδz,δx,˙ δy,˙ δz,δφ,δθ,δκ˙ ]T . (5.12)

It is assumed that the thrust magnitudes and the two angles remain constant along a trajectory segment. Thus, the state space form of the augmented variational equations is

δx˙ (t)= A(t)δx(t), (5.13)

where A(t) is now an 9 × 9 matrix, and, when written in terms of nine 3 by 3 submatrices has the general form

0 I3 0 A(t)=  B(t) 2Ω D(t)  , (5.14)    0 0 0     

with 0 representing the zero matrix, and I3 representing the identity matrix of rank 3. The addition of a thrust term to the equations of motion alters the elements of the submatrix B(t), in the variational matrix from equation (2.33), such that

Uxx + κxx Uxy + κxy Uxz + κxz   B(t)= Uyx + κyx Uyy + κyy Uyz + κyz , (5.15)    Uzx + κzx Uzy + κzy Uzz + κzz      95

∂κj where κjk = ∂k (k = x, y, z) are the partial derivatives of the thrust expression, in equation (5.7), with respect to the Cartesian coordinates. The time varying submatrix D(t) is 3 × 3 with the form

∂x ∂x ∂x ∂θ0 ∂φ0 ∂τ0 D(t)=  ∂y ∂y ∂y  . (5.16) ∂θ0 ∂φ0 ∂τ0  ∂z ∂z ∂z   ∂θ ∂φ ∂τ   0 0 0    With the addition of the variations in the thrust parameters, the state transition matrix, φ(t, t0), enlarges to become a 9 × 9 matrix.

5.3 Patch Points and Initial Trajectory

As discussed in Section (2.4.1), the region near the primaries is very sensitive. Backward integration from the manifold down to the surface of the Earth is used to limit the sensitivity to the initial conditions. To apply a 2-Level Differential Correc- tions scheme to a given problem, it is necessary that a discrete set of patch points be available to characterize the trajectory. An initial estimate for the launch trajec- tory is obtained through numerical integration of the differential equations (5.3)-(5.5) backwards to the surface of the Earth. The initial thrust angles can be determined either by trial-and-error or through a modified one-step targeting scheme. A set of patch points is created from the resulting, numerically determined, trajectory. It is desirable that the final launch trajectory tangentially approach (flight path angle 0o) the parking orbit to yield the lowest insertion cost. (Of course, if the “final” point is required to be the actual launch site on the rotating Earth surface, then backwards integration ceases at the tipoff location, where the flight path angle is between 0o and 90o. If the “final” point is the launch site, the angle is 90o.) These design require- ments are implemented in the 2LDC through an altitude and apse constraint at the parking orbit and a position constraint at the launch site. Additionally, the initial and final times are also fixed. 96

5.3.1 Determination of the Launch Site

The longitude and latitude of any launch site can be used in the algorithms. In the determination of the launch site, the oblate Earth is modelled as a perfect sphere ignoring the equatorial bulge. The resulting spherical angles are, thus, not precisely equal to longitude and latitude but qualitatively possess similar meanings. This allows the use of the locations of known launch sites, such as the European Launch site at Kourou, French Guyana, in the computation of transfer trajectories. Modifications to accommodate a non-spherical Earth can be computed later as required. The launch site on the surface of the Earth lies in the Earth-Centered-Fixed (ECF) frame. The ECF frame is a rotating frame centered at the Earth that is inclined with respect to the ecliptic plane. The inclination of the Earth is constant in the CR3BP. In an ephemeris model, the Earth’s inclination is determined instantaneously. Transformations between the ECF frame and the rotating frame of the primaries centered at the barycenter involve three steps. First, it is necessary to transform the state from the rotating frame to the inertial frame; then, the origin must be modified before the final transformation between the inertial and the desired rotating frame can commence. Two of these frame changes deal with a rotating frame, and therefore, it is necessary to follow the steps in Section (2.2). The launch site on the Earth’s surface is, hence, moving as a function of time with respect to the rotating frame of the primaries. Constraining the total time of flight for the launch trajectory results in a simplified model, where the location of the launch site remains constant. In an ephemeris model, the location of the launch site does not remain constant due to additional perturbations and is a function of the epoch. In the following launch scenario, the European launch site in Kourou, French Guyana (longitude - 52.8 degrees, latitude 5.2 degrees) is selected as the launch site. 97

5.4 Two-Level Differential Corrector with Thrust

The 2LDC from Section (3.3), developed by Howell and Pernicka [52], varies the position and velocity states of the patch points to achieve a continuous trajectory. The major difference between the current application and the differential corrections process as described in Section (3.3), is the set of parameters that define the thrust state. The augmented state vector and the augmented state transition matrix, includ- ing contributions from changes in the thrust terms, are incorporated. A differential corrections process is not limited to using the velocity and position states to yield a trajectory satisfying all constraints. Other variables present in the problem can be exploited. Variations in the parameters that define the thrust state take advantage of the acceleration introduced by the thrust term as it is added to the equations of motion. The process is most successful if each acceleration (dynamical plus thrust) is accommodated in a separate step. The thrust parameters are therefore varied in the first step of this modified process. Velocity discontinuities are introduced in the process of achieving position continuity. They are then reduced in the second step of the differential corrections scheme.

5.4.1 Two-Level Differential Corrector with Thrust

In this modified two-level differential corrections process, the first step involves variations of the thrust parameters to achieve a trajectory continuous in position. The thrust parameters, i.e., thrust magnitude and two angles, are maintained as con- stants along each trajectory arc but are allowed to vary between different segments. In matrix format, the linear targeter is formulated as

φ17 φ18 φ19 x˙   L = φ27 φ28 φ29 y˙ , (5.17)    φ37 φ38 φ39 z˙      98

consistent with equation (3.2) with the controls defined as

T k¯ = [δθ0,δφ0, δκ0, δ∆t] , (5.18)

where δ∆t = δtp − δt0 is the time of flight on each segment and the targets are

T ¯b = [δxp,δyp, δzp] . (5.19)

The state of the end of each segment is denoted by xp, yp, and zp, with the cor- responding time denoted tp. Since the number of control variables in k¯ exceeds the number of target states in ¯b, the Euclidean norm is computed to minimize the changes in the elements of the control vector, while achieving the desired target state. When a change in one particular control variable, such as the thrust magnitude or the time of flight, is not desirable but allowed to give the corrector more freedom, a weighted Euclidean norm, is employed, that is,

k¯ = W −1LT (LW −1LT )−1b. (5.20)

The weighting matrix, W , is a diagonal matrix with specific weights corresponding to each variable, on the diagonal. A larger specific weight in the weighting matrix, yields smaller changes in the corresponding control variable. Modification of the thrust parameters at the beginning of each segment results in velocity discontinuities between the individual segments along the trajectory. The second step, presented in section (3.3.2), varies the position and time components of the patch points to drive those internal velocity discontinuities to zero. The same process can be employed here to eliminate the velocity discontinuities introduced and produce a trajectory continuous in both position and velocity.

5.5 Trajectory from Launch Site into Parking Orbit

A sample launch scenario is computed using the modified differential correction scheme. The launch trajectory originates at the launch site on the surface of the Earth and inserts into a 200 km altitude parking orbit. Twelve patch points along 99

the trajectory, equally spaced in distance on the baseline solution, are used in the differential corrections process to yield the continuous trajectory arc satisfying the imposed constraints. The value of the initial thrust magnitude is 14.8245 m/s2. The x − y projection of the trajectory arc appears in figure 5.2 and the x − z and y − z projections are plotted in figures 5.3 and 5.4. The thrust parameters and times corresponding to the patch points of the converged trajectory are summarized in Table 5.1. Note that the trajectory arc is computed backwards from the parking orbit down to the launch site. Therefore the first patch point in Table 5.1 is on the parking orbit and the last patch, patch point number 12, is at the launch site. The time is also measured backwards from the parking orbit and, hence, is negative. The total time of flight on the trajectory is five minutes and one second and the cost measured in ∆v is 12.796 m/s. The time of flight along the launch trajectory for the Genesis mission to reach a 185 km parking orbit was 10 minutes and 34 seconds [64]. The ideal cost to bring a satellite into a low Earth orbit is usually on the order of 7.9 - 10 km/s [65]. The changes in the thrust magnitude introduced by the differential corrector are fairly small due to the weighting matrix in the computation of the Euclidean norm, equation o (5.20). The variations in the thrust angles from the initial angles (φ0 = 68 and θ0 =13o ) are small because the baseline solution is selected close to the desired final solution. This is necessary to ensure the validity of the linear approximations in the differential corrections process. 100

Table 5.1. Changes in Thrust Parameters Throughout Trajectory Arc.

PP Number Time κ φ θ (sec) (m/s2) (degrees) (degrees) 1 0 14.8245426 53.757684 5.38 2 -31.69 14.8245442 68.03 6.62 3 -59.18 14.8245387 39.32 4.05 4 -87.29 14.8245417 99.27 7.64 5 -115.01 14.8245397 77.04 7.80 6 -146.26 14.8245486 102.43 9.17 7 -174.54 14.8245454 64.84 18.16 8 -201.53 14.8245338 72.50 17.69 9 -231.04 14.8245342 67.55 12.51 10 -257.32 14.8245394 63.31 16.31 11 -284.04 14.8245502 66.54 9.32 12 -303.59 14.8245430 68.0 13.0

The computed launch segment can now be combined with the transfers presented in Chapter 4 to yield a complete transfer from the launch site on the Earth surface to a Sun-Earth halo orbit at L1 or L2. The deterministic maneuvers presented in tables 4.1 and 4.2 need to be added to the launch cost to estimate the cost of the whole transfer.

5.5.1 Challenges with the Launch Formulation

The launch trajectory design problem is formulated here in terms of the CR3BP to blend it easily with manifold trajectories for design applications in multi-body problems. However, some unique challenges are observed. These issues are discussed in terms of three separate categories. First, it is important to realize that the en- 101

6000

4000

Earth’ Surface 2000

0

y [km] Trajectory

−2000 Launch Site −4000 200 km Altitude Parking Orbit −6000 Parking Orbit Insertion Point

1.4959 1.4959 1.4959 1.496 1.496 1.496 1.496 1.496 1.4961 8 x [km] x 10

Figure 5.2. Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit.

vironment close to one of the primaries is very sensitive in the CR3BP due to the singularities at their center. Second, without modification, a differential corrections scheme is not developed to accommodate angles and, therefore, angles are not the best choice for dependent variables in a control strategy. Finally, parameterizing the thrust is a key step in the process.

The most severe numerical challenge is the sensitivity in the equations of motions near the Earth. Ill-conditioned STMs and SRM matrices are often the result of the sensitivities inherent in the near-Earth environment. These singular matrices often occur after a few iterations into a differential corrections process and, thus, the procedure fails to converge. Slight changes in the initial patch points or in the initial values of the thrust parameters may aid in the elimination of these problems. But 102

6000

Earth’ Surface 4000

Trajectory 2000 200 km Altitude Parking Orbit Launch Site

0 z [km]

Parking Orbit −2000 Insertion Point

−4000

−6000

1.4959 1.4959 1.4959 1.496 1.496 1.496 1.496 1.496 1.4961 8 x [km] x 10

Figure 5.3. Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit.

often in this regime, the initial baseline trajectory is simply too far from a continuous solution that meets all constraints. The linear approximations in the differential corrections process cannot be expected to adequately estimate the necessary changes in the patch points. Trajectories to higher altitude orbits or thrusted trajectories far away from the Earth are generally much easier to compute. Although, in any case, a good initial guess is necessary to ensure convergence onto an acceptable solution.

A differential corrections process treats angles like any other variable it encounters. But, of course, small changes in an angle can result in huge changes in other state variables downstream. Software logic, therefore, must be implemented to ensure that the angular changes are implemented in the intended direction and are not larger 103

6000

Earth’ Surface 4000 Launch Site

2000 Parking Orbit Insertion Point 200 km Altitude Parking Orbit

0 z [km]

Trajectory −2000

−4000

−6000 −8000 −6000 −4000 −2000 0 2000 4000 6000 8000 y [km]

Figure 5.4. Launch Trajectory from Kourou to a 200 km Altitude Parking Orbit.

than a few degrees per iteration. A better alternative is to represent the thrust in terms of different variables.

Depending on how the thrust parameters are incorporated into the differential corrections scheme, as well as the nature of the constraints, it may be impossible to force the internal ∆v’s to zero. Also, a staged approach is usually most successful. That is, both velocity/position are not modified in the same step of the differential corrections process as the thrust parameters. The internal ∆v’s are ultimately lower but not zero. The two accelerations in the current formulation, that is, the natural motion and the constant thrust term, are frequently in competition to eliminate the discontinuities. 104

Many of the difficulties challenges in this investigation are very similar to the problems highlighted by McInnes [66] in the computation of transfer trajectories using solar sails. In his efforts to compute transfers from Sun-Earth halo orbits to Earth parking orbits, he encountered difficulties with eliminating internal velocity discontinuities and was unable to compute transfers to low-Earth orbit. Limiting changes in thrust components and velocity/position components to separate stages did result in progress, however. Since the difficulties are most severe in the near-Earth environment in both cases, it suggests that the linear variations of the differential scheme are ineffective in approximating the changes required close to the primary.

5.6 Conclusion

This preliminary investigation is a proof of concept to illustrate that a differen- tial correction scheme can, in principle, be used to produce a baseline solution for a launch trajectory. The challenges here suggest that alternative methods be investi- gated to develop a more efficient and effective methodology for the computation of baseline launch trajectories using targetors and correctors. Using differential correc- tions techniques in the computation of launch trajectories is intended to be the first step in establishing a methodology that can be used efficiently. The use of optimal control methodology is therefore planned to be implemented in the future. But, per- haps of equal significance, the underlying dynamical structure can be more effectively exploited to create a more “intelligent,” adaptive procedure. 105

6. SUMMARY AND RECOMMENDATIONS

6.0.1 Summary

The primary goal of this study is an alternate methodology to determine transfers between the Earth and libration point orbits, such that the Earth is either the smaller or the larger primary. The work is decomposed into three separate problems. First,

transfers between the Earth and lunar halo orbits at L1 are considered. In this case, the Earth is the larger primary. Second, transfers are computed between the Earth and Sun-Earth libration point orbits, where the Earth is in the familiar role as the smaller primary. And finally, a simple scheme is introduced to yield launch trajectories from the launch site to Earth parking orbits. The goals are accomplished by combining invariant manifold theory and differential corrections techniques. In the Earth-Moon system, an initial guess featuring some of the characteristics of the desired trajectory is still required since no natural solution exists that connects the near-Earth region with lunar halo orbits.

The dynamical structure in the multi-body problem in the vicinity of the larger primary is very different than that around the smaller primary. Insight into the nat- ural dynamics is gained by computing the stable/unstable manifolds and identifying those with the closest Earth pass for variously sized halo orbits in both the Earth- Moon and the Sun-Earth systems. A range of sample transfers from the Earth to specific libration point orbits at L1 is then presented. Some of the transfers gen- erated in the CR3BP closely resemble transfers from the literature in appearance, cost, and time of flight. The transfer trajectory insertion cost from an Earth park- ing orbit is very similar for all the trajectories investigated. The transfers are then reproduced in an ephemeris model to demonstrate their validity. In the immediate vicinity of the halo orbits, it is noted that some of the transfers follow the invariant 106 manifold structure quite closely. Consistent with this observation, lower halo orbit insertion costs are achieved for larger halo orbits. The methodology developed for determination of transfers in the Earth-Moon system is very successfully used in the computation of transfers in the Sun-Earth system. In the Sun-Earth system, it is possible to use the manifold structure to aid substantially in the design of transfers.

Sample transfers to both L1 and L2 halo orbits can be computed without difficulty. The transfer trajectory insertion maneuver to depart an Earth parking orbit is nearly constant. Finally, a simple scheme is applied to determine thrust arcs and to offer a simple, but complete scenario from a launch site on the surface of the Earth to a halo orbit.

6.0.2 Recommendations and Future Work

Many challenges remain in the efficient and effective determination of transfer trajectories between the Earth and halo orbits in the Earth-Moon system. This is notable to ultimately develop an automated procedure. The work presented here is preliminary and serves as the basis for a more comprehensive investigation. Future work is likely a combination of developing more efficient techniques, as well as com- puting a larger range of transfers to gain more insight into the characteristics of the different types. The key to a more efficient computation of transfers, lies in a more complete understanding of the dynamics in the vicinity of the larger primary. Vari- ational methods based on linear approximations do not seem adequate in accurately predicting the motion close to the primaries. Other techniques such as nonlinear programming techniques might be more effective in determining transfers in these sensitive regions in the vicinity of the larger primary. Partial enforcement of con- straints or the application of constraints along a path rather than individual points could also offer improvement. Systematically incorporating the invariant manifold structure near the halo orbit would certainly be used. A methodology developed by Wilson [42] to project the end points of transfers onto the manifold tube could be one 107

way to achieve this. Alternatively, a multi-point targeting scheme or a differential corrections scheme able to vary the trajectory selected on the manifold tube could be helpful. To avoid using the current inefficient shooting techniques in that process, incorporating the energy-like Jacobi Constant might be helpful. Computing a larger range of transfers will offer insight into typical characteristics such as cost and TOF of the different types. Being aware of these characteristics will undoubtedly be helpful in identifying the optimal locations to perform maneuvers. Multiple maneuvers could yield transfers for lower total cost. Eventually, optimization of the transfer will be necessary. An optimization package in development at Purdue University specifically suited to the peculiarities of the three-body problem will be used for that purpose.

The design of transfers to/from the vicinity of the smaller primary to halo orbits is well understood. Use of this knowledge can be exploited in support of transfers from the larger primary, Earth, to lunar libration point orbits at L1 and L2 that include a lunar flyby. Transfers that pass the Moon on the far side could insert onto the stable manifold tube, and using the heteroclinic connections, offer lower cost transfer to lunar L1 halo orbits. Transfers to L2 halo orbits that include a lunar flyby could also be investigated.

The launch trajectory development is of a preliminary nature. Differential correc- tions techniques can be used to obtain rough approximations of baseline solutions. Though, more efficient and precise tools are desirable to extend the capabilities and the range of applications. Replacing the constant thrust term by a scheme using different launch vehicle stages with their varying masses and propellant limitations into account offers additional precision. Besides improving the model, a wider range of application could be investigated such as launches directly into stable manifolds or transfer trajectories. Eliminating the parking orbit could save valuable fuel. A scheme to compute such launches could use Wilson’s optimization technique based on projecting the launch trajectory onto the surface of the manifold tube. This would yield continuous transfers from the launch site to the libration point orbit of interest for a lower cost. Optimal control should eventually be investigated. 108

6.0.3 Concluding Remarks

Libration point trajectories and the associated low-energy pathways offer many low-cost, low-energy trajectory options for mission designers. However, no such path- ways naturally occur between the larger primary and the halo orbits. One technique, based on linear variational methodology, to compute such transfers is investigated in this work. The computation of launch trajectories using the same methodology is also considered. Although, the computation of transfers and launch trajectories using this technique is possible, more efficient methods are desirable. The immediate vicinity of either the smaller or the larger primary is highly nonlinear. Improvements in the effectiveness and efficiency of new design tools for these transfers are likely achieved through a more systematic use of the dynamics in these regions. 109

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