A LOW-THRUST TRANSFER STRATEGY TO EARTH-MOON COLLINEAR LIBRATION POINT ORBITS
A Thesis
Submitted to the Faculty
of
Purdue University
by
Martin T. Ozimek
In Partial Fulfillment of the
Requirements for the Degree
of
Master of Science
December 2006
Purdue University
West Lafayette, IN
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For Mom, Dad, Sarah, and Bops
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ACKNOWLEDGMENTS
I would like to thank my parents for their seemingly never ending support and confidence that I would persevere in my personal quest for an advanced degree. The decision to commit to a higher degree has been an adventure that I hope to continue along, and I can’t begin to explain the importance of that priceless feeling of simply knowing that someone is there when needed. Professor Kathleen Howell, my advisor, is also owed a great deal of gratitude, not only for posing the fateful words “low-thrust” one day in her office during a discussion about NASA’s potential Jupiter Icy Moons missions, but also for her personal standard for excellence that she instills in each of her many successful graduate students. I’ve always felt that Purdue University reached out to me and offered me that “extra” indefinable something from the moment I began seriously considering a graduate institution. In no other person is this ambiguous something “extra” exemplified than in Professor Howell, who has sought to ensure that my research efforts are guided and ultimately shared with others in the best possible way. This notion has also been exemplified by Professor James Longuski, whose door has always been open to me from day one, and whom I must also credit in heavily influencing my decision to attend Purdue University. On more than one occasion, Professor William Crossley has also had his door open to help along my path of solving what turned out to be a difficult optimization problem. I also owe many thanks to Daniel Grebow. Dan has been a close friend, colleague, and even roommate throughout my stay at Purdue, and this work is the direct continuation of a mission analysis that we worked on together. Often, many of the new ideas I have for current and future research are a result of simple dialogs that we frequently engage in.
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The idea to study mission applications toward lunar south pole coverage would never have originated had I not fortuitously been privileged to work at the NASA Goddard Spaceflight Center during the summers of 2005 (as a member of the NASA Academy by way of the Indiana Space Grant Consortium) and 2006. There, I benefited from the knowledge of some of the greatest libration point mission experts in the world, and owe particular thanks to my mentor David Folta. Support from NASA under contract numbers NNG05GM76G and NNX06AC22G is greatly appreciated. Finally, I would like to thank Purdue University for financial support, including the Andrews Fellowship, for the entirety of my M.S. program.
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TABLE OF CONTENTS
Page LIST OF TABLES...... vii LIST OF FIGURES ...... viii ABSTRACT...... x 1 INTRODUCTION ...... 1 1.1 Historical Overview of the Three-Body Problem...... 3 1.2 Developments in Low-Thrust Transfer Trajectories ...... 5 1.2.1 Optimal Control ...... 5 1.2.2 Application to Orbit Problems ...... 6 1.3 Focus of this Work ...... 7 2 BACKGROUND ...... 10 2.1 The Circular Restricted Three-Body Problem...... 10 2.1.1 Assumptions...... 11 2.1.2 Geometry...... 11 2.1.3 Equations of Motion...... 12 2.1.4 Libration Points...... 15
2.1.5 Formulation Relative to P2 in the CR3BP...... 17 2.2 Natural Periodic Orbits in the CR3BP...... 18 2.2.1 First-Order Variational Equations Relative to the Collinear Points...... 19 2.2.2 The State Transition Matrix ...... 23 2.2.3 The Fundamental Targeting Relationships ...... 25 2.2.4 Periodic Orbits ...... 28 2.3 Invariant Manifolds ...... 31 2.3.1 Stable and Unstable Manifolds Associated with the Collinear Points...... 32
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Page 2.3.2 Invariant Manifolds Relative to a Fixed Point ...... 35 2.3.3 Computation of Manifolds Corresponding to Fixed Points Along an Orbit..... 37 2.4 Optimal Control Theory ...... 39 2.4.1 Summary of the First Necessary Conditions for Optimal Control...... 40 2.4.2 Tests for a Local Minimum Value of the Performance Index ...... 43 3 LOW-THRUST TRANSFER ALGORITHM ...... 45 3.1 Engine Model ...... 46 3.2 Control Law Derivation...... 48 3.3 Adjoint Control Transformation...... 55 3.4 Numerical Solution via Direct Shooting: A Local Approach...... 59 3.5 Shotgun Method for Initial Conditions: A Global Approach...... 63 4 MISSION APPLICATIONS...... 66 4.1 Orbits for Line-of-Sight Lunar South Pole Coverage (CR3BP) ...... 67 4.1.1 Three-Dimensional Periodic Orbits in the CR3BP ...... 67 4.1.2 Families of Orbits for Lunar South Pole Coverage ...... 68 4.1.3 Mission Orbit Selection Criteria ...... 73 4.2 Optimal Transfers to the Earth-Moon Stable Manifold...... 75
4.2.1 Transfers to a 12-Day L1 Halo Orbit ...... 78
4.2.2 Transfer to a 14-Day L1 Vertical Orbit ...... 87
4.2.3 Transfer to a 14-Day L2 Butterfly Orbit...... 91 5 SUMMARY AND RECOMMENDATIONS...... 96 5.1 Summary...... 96 5.2 Recommendations for Future Work ...... 98 LIST OF REFERENCES...... 99
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LIST OF TABLES
Table Page Table 4.1 Dynamical and Propulsion Constants...... 77
Table 4.2 12-Day L1 Halo Orbit Transfer Data Summary ...... 82
Table 4.3 12-Day L1 Halo Orbit Long Transfer Data Summary...... 86
Table 4.4 14-Day L1 Vertical Orbit Transfer Data Summary...... 91
Table 4.5 14-Day L2 Butterfly Orbit Transfer Data Summary ...... 95
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LIST OF FIGURES
Figure Page Figure 2.1 Geometry in the Restricted Three-Body Problem...... 12 Figure 2.2 Equilibrium Point Locations for the CR3BP...... 16
Figure 2.3 Geometry of P2-Centered Rotating Frame ...... 18
Figure 2.4 Linearized L1 Periodic Orbit...... 23 Figure 2.5 Basic Diagram for a Free Final Time Targeting Scheme...... 27 Figure 2.6 Targeting a Perpendicular X-axis Crossing in the CR3BP ...... 30
Figure 2.7 Several L1 Lyapunov Orbits Obtained Via Continuation...... 31 Figure 2.8 Stable and Unstable Manifold at X eq ...... 34
Figure 2.9 Global Manifolds for an Earth-Moon L1 Lyapunov Orbit, Ay = 23,700 km.....38 Figure 3.1 CSI Engine Example – Smart-1 Ion Engine...... 46 Figure 3.2 VSI Engine Example - VaSIMR Rocket...... 47
Figure 3.3 Behavior of θΜ and τM Along the Stable Manifold Tube...... 51 Figure 3.4 Velocity Reference Frame...... 58 Figure 3.5 Numerical Algorithm – Direct Shooting Method via SQP ...... 62 Figure 3.6 Example Population Parameters for a 12-Day Halo Orbit ...... 65
Figure 4.1 Southern Halo Orbit Families: Earth-Moon L1 (Orange) and L2 (Blue); Moon Centered, Rotating Reference Frame ...... 70
Figure 4.2 Vertical Orbit Family of Interest: Earth-Moon L1 (Magenta) and L2 (Cyan); Moon Centered, Rotating Reference Frame...... 71
Figure 4.3 Southern L2 Butterfly Orbit Family; Moon Centered, Rotating Reference Frame...... 72 Figure 4.4 Period versus Maximum x-Distance from the Moon (Left); Definition of Maximum x-Distance (Right) ...... 74
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Figure Page Figure 4.5 Stability Index versus Maximum x-Distance from the Moon ...... 75 Figure 4.6 Optimal Orbit Raising from LEO...... 77
Figure 4.7 Stable Manifold Tube for 12-Day L1 Halo Orbit (Green) and Target Reference Trajectory Along the Manifold (Blue) ...... 79
Figure 4.8 Low-Thrust Short Transfer to a 12-Day L1 Halo Orbit ...... 80 Figure 4.9 Position and Velocity Costate Time Histories for the 12-Day L1 Halo Orbit Transfer ...... 81
Figure 4.10 Time History of Propulsion Related Parameters for the 12-Day L1 Halo Orbit Transfer ...... 82
Figure 4.11 Stable Manifold Tube for 12-Day L1 Halo (Green) and Initial Target Reference Trajectory Along the Manifold (Blue) for Long Transfer ...... 83
Figure 4.12 Low-Thrust Long Transfer to a 12-Day L1 Halo Orbit ...... 84
Figure 4.13 Position and Velocity Costate Time Histories for the 12-Day L1 Halo Orbit Transfer ...... 85
Figure 4.14 Time History of Propulsion Related Parameters for the 12-Day L1 Halo Orbit Long Transfer...... 86
Figure 4.15 Stable Manifold Tube for 14-Day L1 Vertical Orbit (Green) and Initial Target Reference Trajectory Along the Manifold (Blue)...... 88
Figure 4.16 Low-Thrust Transfer to a 14-Day L1 Vertical Orbit...... 89
Figure 4.17 Position and Velocity Costate Time Histories for the 14-Day L1 Vertical Orbit Transfer...... 90 Figure 4.18 Time History of Propulsion Related Parameters for the ...... 90
Figure 4.19 Stable Manifold Tube for 14-Day L2 Butterfly Orbit (Green) and Initial Target Reference Trajectory Along the Manifold (Blue) ...... 92
Figure 4.20 Low-Thrust Transfer to a 14-Day L2 Butterfly Orbit ...... 93
Figure 4.21 Position and Velocity Costate Time Histories for the 14-Day L2 Butterfly Orbit Transfer...... 94
Figure 4.22 Time History of Propulsion Related Parameters for the 14-Day L2 Butterfly Orbit Transfer...... 94
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ABSTRACT
Ozimek, Martin T. M.S.A.A., Purdue University, December, 2006. A Low-Thrust Transfer Strategy to Earth-Moon Collinear Libration Point Orbits. Major Professor: Kathleen Howell.
A strategy to compute low-thrust transfer trajectories in the Earth-moon circular restricted three-body problem is developed. The dynamical model is formulated assuming variable specific impulse engines, an advanced finite-thrust propulsion model. Originating in an Earth parking orbit, the spacecraft is delivered to a location along the stable manifold; the engines power off and the spacecraft asymptotically approaches the periodic libration point orbit of interest. Elements of optimal control theory are used to derive a primer vector control law as well as a set of additional dependent variables that characterize the solution to the corresponding two-point boundary-value problem (TPBVP). A hybrid direct/indirect method results in transfer trajectories that are associated with locally minimal propellant consumption. The generation of useful initial conditions is aided by an adjoint control transformation and a global “shotgun” method. The solution strategy is demonstrated in a detailed development of transfers to a 12-day
L1 halo orbit, a 14-day L1 vertical orbit, and a 14-day L2 “butterfly” orbit. These target orbits are all selected from the families that meet line-of-sight coverage requirements in support of lunar south pole mission architecture.
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1. INTRODUCTION
As understanding of the solar system and the dynamical structure of the space environment increases, ever more complex questions continue to emerge. In the past 60 years, access to space has opened to both human-crewed and robotic spacecraft. Both scientific interest and engineering capability have invariably played a vital role in this expansion of knowledge. New scientific demands often spur engineering advancements to accomplish a set of mission objectives; breakthroughs in engineering capability enlighten the scientific community and serve as a catalyst for original mission concepts. Perhaps, as a consequence, it is not surprising that libration point orbits have relatively recently risen as venues for robotic spaceflight. Beginning with NASA’s 1978 solar wind measuring satellite, the International Sun-Earth Explorer (ISEE-3) [1], libration point orbits are now considered viable options to meet a range of scientific goals. Although originally proposed for manned Apollo missions [1], such orbits were not exploited prior to ISEE-3. However, with the ever increasing speed of computers, trajectory design within the context of the n-body problem is now feasible. Although the n-body problem is unsolvable in closed form, certain simplifying assumptions expose equilibrium solutions, i.e., the libration points. Periodic orbits in their vicinity can be computed numerically. Not coincidentally, the success of the early missions like ISEE-3 and the continuing increase in computational capabilities, have led to more contemporary libration point missions such as WIND [2], SOHO [3], ACE [4], MAP [5], and Genesis [6]. Clearly, such trajectory designs fill a particular niche in mission applications where long-duration scientific observation is required. More recently, a geometrical approach in studies of the multi-body problem has also led to alternative strategies for transfer trajectory design, as well as stationkeeping maneuvers. This approach is based on a complete analysis of the phase space in the
2 neighborhood of the periodic libration point orbits. In-depth analysis of the phase space, as suggested by Poincaré [7] in 1892, has evolved into dynamical systems theory (DST). An important astrodynamics application from DST is the exploitation of invariant manifolds to design trajectory arcs that asymptotically arrive at, and depart from, the vicinity of the periodic libration point orbits. Propagation of these manifolds often yields very efficient transfers. In some cases, these manifolds even pass within the vicinity of a planet [6]. Typically, such transfer arcs are more applicable to robotic spaceflight, since an additional time-penalty is often incurred. This DST approach was a key component in the design of the Genesis low-energy trajectory. The Genesis trajectory design incorporated heteroclinic and homoclinic arcs to deliver a spacecraft to a Sun-Earth L1 libration point orbit with a subsequent return to Earth [6]. Such manifold transfer trajectories are also useful for applications in the Earth-moon system. Recent studies have identified libration point orbits as a potential component in the development of a communications relay between a manned facility at the lunar south pole and ground stations on the Earth. In the Earth-moon problem, however, many of the viable libration point orbits possess manifolds that pass no closer than 50,000 km to the Earth. These manifolds may still serve as transfers, but additional fuel is necessary to incorporate a leg from an Earth parking orbit to the manifold. One type of intermediate transfer from the parking orbit to the manifold involves the use of low-thrust propulsion. Low-thrust propulsion introduces a time penalty, but can yield lower fuel expenditure due to higher specific impulse engines. Incorporating low- thrust also adds dynamical complexity because a “steering” law for the thrust direction and magnitude must be determined and successfully implemented. The objective of the current work is a method to design low-thrust transfers that deliver a vehicle onto a manifold trajectory. The specific application of interest supports the establishment of lunar south pole communications relay infrastructure.
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1.1 Historical Overview of the Three-Body Problem
The general problem of three bodies was first investigated by Isaac Newton in his 1687 landmark work, the Principia [8]. His successor, Leonhard Euler, receives much of the credit, however, for formulation of the restricted problem of three bodies. In 1765, Euler identified the equilibrium solutions in the restricted problem, i.e., the collinear libration points L1, L2, and L3 [9]. Euler also introduced a synodic reference frame in connection with the motion of the moon in 1772 [9]. Later, in 1772, the same year that Euler formulated the restricted problem, Lagrange determined the locations of two additional equilibrium points, the triangular libration points, L4 and L5 [9]. Euler’s formulation allows only one integral of motion in the restricted problem as determined by Jacobi in 1836, by balancing energy and angular momentum [10]. In 1878, George William Hill published his Researches in Lunar Theory [11], effectively modeling the motion of the moon as a satellite, exposed to the gravitational field of the Earth and the perturbing force of the Sun. In 1899, Henri Poincaré published his three-volume work, Les Méthodes Nouvelles de la Mécanique Celeste [7], the result of his unparalleled response to a contest in 1887. Participants were challenged to produce a definitive solution to the n-body problem. Ironically, Poincaré eventually won the prize by proving that the n-body problem cannot be solved in closed form. His work is highly regarded for the qualitative emphasis on behaviors in the n-body problem. In particular, Poincaré focused on the behavior of trajectories as time goes to infinity. Of course, the only trajectory that can be defined at infinite time is a periodic orbit. A detailed analysis of the phase space of a non-integrable system led Poincaré to invent a technique known as the “surface of section”. This work is considered the foundation of dynamical systems theory. In addition, Poincaré also proved that Jacobi’s Constant is the only integral of the motion in the restricted problem. Poincaré’s work generated great interest in the following decades. Periodic orbits are a common topic noted in the early 20 th century work of Darwin [12], Plummer [13], and Moulton [14]. In the absence of extensive computational capabilities, these researchers exploited expansion procedures to construct analytical approximations. Darwin and
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Plummer are noted for approximating planar periodic orbits in the restricted problem. Beyond a general focus on planar periodic orbits, Moulton also studied in-plane and out- of-plane orbits in the vicinity of the collinear libration points. The renewed attention on the restricted problem in the latter half of the 20th century is attributed primarily to the emergence of high-speed computing and the beginning of the space age. Several developments are notable. Szebehely’s 1967 book The Theory of Orbits [10], a comprehensive text on the three-body problem, is still regarded as one of the most thorough sources of information on the problem of three bodies. By the 1960’s, in support of the Apollo missions to the moon, trajectories computed in the restricted three-body problem were in development. Farquhar coined the term “halo” orbits and developed analytical approximations of these three-dimensional periodic orbits in the vicinity of L2 in the Earth-moon system. In addition to proposing orbits for use in the Apollo missions and ultimately for the unmanned ISEE-3 [1] spacecraft in the Sun-Earth system, Farquhar and Kamel [15] also developed third-order approximations for quasi- periodic orbits. Richardson and Cary [16] expanded these approximations to fourth order. Breakwell and Brown [17] are noted for a numerical study that generates families of periodic halo orbits. Howell [18] extended the analysis to include all collinear points and the families over all three-body systems. Hénon [19] has also produced thorough analyses of periodic orbits, including vertical orbits. Approximations for nearly rectilinear halo orbits at the collinear points were produced by Howell and Breakwell [20]. Perhaps the most rigorous numerical generation of periodic orbits is the investigation by Dichmann, Doedel, and Paffenroth [21], that uses the software package AUTO to detail periodic orbits as well as their interrelated orbits via bifurcation.
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1.2 Developments in Low-Thrust Transfer Trajectories
1.2.1 Optimal Control
The development of finite-burn trajectories (in particular, when the thrust level is low) is an application of optimal control theory and the calculus of variations. Optimization of curves and points can be traced to the 1600’s, when the calculus of variations was first introduced as an analysis tool for minimizing functions of functions, or “functionals”. It received particular attention with Johann Bernoulli’s proposal of the brachistochrone problem to the scientific community 1696 [22]. Generally, the focus is the set of conditions on the functions that drive the functional to a maximum or minimum. Such functions are termed “extremals”. In 1755, Joseph Lagrange wrote a letter to Leonhard Euler in connection with their mutual interest in an analytical solution to the tautochrone problem [23]. The analytical development resulted in the formulation of the Euler- Lagrange Equations (and the corresponding transversality condition). These equations have since served as the basis of a widely known technique for determining extremals, and a foundation of the calculus of variations, a term created by Euler in 1766. Classification of these extremals is accomplished via the Legendre-Clebsch necessary condition, the Weierstrass Condition, and in the most general of terms, Pontryagin’s Minimum Principle. Details of the generalized theory in support of the applications of this methodology appear in Bryson and Ho [24], Hull [25], and Kirk [26]. Low levels of thrust in the computation of finite-burn spacecraft trajectories is achieved by the seemingly parallel availability of high-speed computing and the continuing development of advanced propulsion systems. In his 1963 book, Optimal Spacecraft Trajectories [27], Lawden used primer vector theory to outline a general procedure for determining optimal low-thrust trajectories. Primer vector theory blends a control law and switching structure common in many “indirect” optimization methods. Indirect, low-thrust, trajectory optimization methods are typically characterized by the two-point boundary-value problem formulation from optimal control theory, with a continuous parameterization of the thrust direction via the tangent to the primer vector.
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Such approaches are termed “indirect” because once the two-point boundary-value problem is established, no minimization is required on the cost functional directly (although it can be included [28-30]). Alternatively, the solution involves a root-solving process on the kinematical as well as the natural boundary conditions as a result of the Euler-Lagrange equations, transversality condition, and a corresponding secondary test for a minimum. Conversely, “direct” methods involve an attempt to minimize the cost functional itself, and often use many variables to parameterize the thrust magnitude and direction. A common parameterization of the thrust direction is the development of a spline function [29,31]. While indirect methods typically require more precise initial conditions due to numerical sensitivities, the lower dimension on the search vector implies fewer computations. Marec [32] offers further mathematical analysis of the primer vector; examines high- and low-thrust propulsion systems and details the Contenson-Pontryagin Maximum Principle. Early applications to impulsive rendezvous problems are available in Jezewski [33] and to low-thrust rendezvous in Melbourne and Sauer [34]. Further increases in computing speed have allowed more sophisticated methods in numerical solutions to optimization problems. All trajectory optimization problems are typically solved with the following numerical methods: direct shooting, indirect shooting, multiple shooting, direct transcription, indirect transcription, dynamic programming, or genetic algorithms. For a detailed survey of all of these different methods, see Betts [35]. Thus, optimal control theory serves to set up the conditions and constraints that must be met to determine the existence of an optimal control, and the numerical optimization methods serve as the means to actually compute solutions that meet these exact conditions.
1.2.2 Application to Orbit Problems
A number of investigations are notable in examining optimal low-thrust trajectories. Many applications are formulated in the two-body problem [36-40]. In the three-body problem, there has been less attention. In one example, Herman and Conway [41]
7 compute optimal, low-thrust, Earth-moon transfers using equinoctial elements, with a direct collocation solution method. Kluever [29-30] also develops Earth-moon transfers, but utilizes a hybrid direct/indirect method. Golan and Breakwell [42] use matching of two trajectory segments to transfer into lunar orbit; transfers to L4 and L5 are also presented. In a Hill formulation of the three-body problem, Sukhanov and Eismont [43] establish a primer vector control law to develop a three-dimensional transfer into a Sun-
Earth L1 halo orbit that includes a constrained thrust direction. Although Seywald, Roithmeyer, and Troutman [44] do not employ the circular restricted three-body equations of motion, they are notable for attempting to provide approximate analytical solutions of circle-to-circle orbit transfers with a variable specific impulse engine, and, furthermore, compare the engine model with results using constant specific impulse engines. Primer vector theory is employed by Russell [45] to develop a global search and local optimization method and establish a Pareto front on his resulting fixed time solutions. Russell subsequently applies his method to produce transfers to planar Earth- moon and Sun-Earth distant retrograde orbits. Senent, Ocampo and Capella [28] also use primer vector theory for free final time transfers to Sun-Earth libration point orbits via the stable manifold.
1.3 Focus of this Work
Optimal, free final time low-thrust transfer trajectory profiles to several Earth-moon libration point orbits is the objective of this work. The target orbits are selected based on lunar south pole coverage applications, including halo orbits, vertical orbits, and “butterfly” orbits. These transfer trajectories represent an extension of the exhaustive coverage analysis by Grebow et al. [46].
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The work is organized as follows. Chapter 2: All background material is presented in Chapter 2. The equations of motion that govern the nondimensional, barycentric, cartesian, circular restricted three-body problem are developed. A moon-centered model is also introduced. The linear variational equations that result from using the collinear libration points as a reference are introduced, and, thus, form the basis for generating initial conditions in the nonlinear problem. Then, an alternate set of linear variational equations is developed, where the time-varying orbital trajectory is exploited as a reference. The second set of variational equations is useful for iterative orbit targeting. Invariant manifold theory is summarized for both libration points and for fixed points along a periodic libration point orbit. Finally, the necessary conditions for establishing a stationary value of a generalized performance index are introduced. These include both the Euler-Lagrange equations and Pontryagin’s Minimum Principle. Such conditions ultimately yield the full two-point boundary-value problem that may be solved via nonlinear programming methods. Chapter 3: Optimal control theory is applied to develop the well-known primer vector control law parameterization. Variable specific impulse engines (e.g., those in development for the VaSIMR rocket project) are modeled and produce notable improvement in numerical convergence. The problem is formulated as a free final time transfer from an Earth parking orbit to the stable manifold associated with the libration point orbit of interest. Using the stable manifold allows a stopping condition, and a secondary coast phase; the spacecraft asymptotically converges to the desired orbit. Once the control law and the elements of the two-point boundary-value problem are established, a solution method is presented. This scheme incorporates a “global” search method, and a subsequent local sequential quadratic programming (SQP) algorithm. Rather than solve the complete two- point boundary-value problem, the local approach is set up as a direct shooting method, and is termed a hybrid direct/indirect approach. The adjoint control transformation (ACT) is used to map the initial costates into more meaningful physical quantities that are useful in the global method and the first step of the local, direct shooting method.
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Chapter 4: Once the methodology is established, the criteria for target orbit selection is summarized and the orbit families of interest are introduced. A halo orbit, vertical orbit, and a “butterfly” orbit are selected and the solution procedure is applied to generate low- thrust transfer trajectories. The resulting thrust profiles, trajectory plots, and overall performance parameters are then discussed. Chapter 5: Conclusions concerning the solution methodology are presented. Then, recommendations on future work, including different solution methods, equations of motion, and higher fidelity models are detailed. Finally, potential future applications as a result of this work are presented.
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2. BACKGROUND
Some fundamental mathematical tools and concepts are necessary for the development of the trajectories and transfers in the current application. The cartesian, nondimensional, barycentric Circular Restricted Three-Body Problem (CR3BP) is initially formulated, and the corresponding dynamical equations of motion are derived. The five equilibrium points are the basis for the computation of special periodic orbits. The state transition matrix is introduced as a tool for predicting linear, and approximating nonlinear, motion. An alternate formulation of the equations of motion, centered at the smaller primary, is also presented. The fundamental structure underlying the dynamics in the restricted problem is analyzed with invariant manifold theory, yielding natural pathways to and from a periodic orbit. Basic numerical methods for a targeting procedure are detailed, forming the groundwork for determining periodic orbits. Finally, the basic concepts of optimal control theory are introduced, including the necessary conditions along a trajectory for the existence of a locally optimal transfer trajectory.
2.1 The Circular Restricted Three-Body Problem
The rapid growth of high-speed computing in the last three decades has helped spark the discovery of new and exciting trajectories. Many of these trajectories exist within the context of the Circular Restricted Three-Body Problem (CR3BP). An exact analytical solution to the CR3BP does not exist; thus, any solutions beyond the equilibrium points require numerical integration. Nevertheless, at the expense of additional numerical exploration, propagation of trajectories in this model result in non-Keplerian orbital motion, such as “figure-eight” orbits, “halo” orbits, and an infinite variety of other periodic orbits; quasi-periodic trajectories have also been identified.
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2.1.1 Assumptions
Given an arbitrary inertial reference point, dynamical analysis indicates that 18 first- order differential equations of motion are required to mathematically model the system comprised of the three bodies. This number, however, is reduced by considering the relative motion. An infinitesimally small point mass, P3 (of mass m3), moving with respect to two point masses, or primaries, P1 (of mass m1) and P2 (of mass m2), appears in > Figure 2.1. The masses are defined such that m1 m 2≫ m 3 , restricting the problem in
the sense that all gravitational influence exerted by m3 is neglected. With this assumption, the motion of P1 and P2 is entirely Keplerian, and reduced to the solution of the two-body problem. Additionally, this two-body motion is constrained by assuming that the primaries move in a circular orbit about their common center of mass, or barycenter, B. As a result, the problem only requires 6 first-order differential equations.
2.1.2 Geometry
An inertial reference frame, I, described in terms of unit vectors Xˆ− Y ˆ − Z ˆ , is centered at B such that the Xˆ− Y ˆ plane is defined to be coincident with the orbital plane of the primaries. Since the primary motion is Keplerian, it is constrained to the Xˆ− Y ˆ plane, however, the third body can move in any of the three spatial dimensions. A rotating frame, S, with coordinate axes xˆ−y ˆ − zˆ is initially aligned with I, then rotates through the angle θ , such that the xˆ -axis is always directed from P1 toward P2. Both the zˆ - direction and Zˆ -direction are parallel to the orbital angular velocity vector of the primaries, and, thus, the yˆ and Yˆ axes complete the respective right-handed systems.
Due to the circular primary motion, the angular rate, θɺ , is constant and equal to the mean motion, n. The position of each body, Pi, with respect to the barycenter is defined by Ri , and the relative position of P3 with respect to P1 and P2 is defined by R13 and R23 , respectively. Note that the overbars ( ) indicate vectors.
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Yˆ yˆ ( ) P3 m 3 R23 xˆ R3 ( ) P2 m 2 R13 θ R2 ( ) Xˆ P1 m 1 B R1
Figure 2.1 Geometry in the Restricted Three-Body Problem
2.1.3 Equations of Motion
One goal of this analysis into the restricted problem of three bodies is a description of the motion of the infinitesimal mass, P3, subject to the gravitational influence of the primaries. From Newton’s Second Law, the vector differential equation for motion of P3 is written I 2 =dR =− Gmm13 − Gmm 23 ∑ Fm3 2 3 R13 3 R 23 , (2.1) dt R13 R 23 where the superscript I represents differentiation in the inertial frame. A standard nondimensionalization used in the CR3BP is employed here. Since the mass of the third body is negligible, the characteristic mass, m*, is the sum of the two primary masses, i.e., ∗ = + m m1 m 2 . (2.2) The characteristic length, l*, is then the constant distance between the primaries, i.e., the scalar distance, ∗ = + l R1 R 2 . (2.3)
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Finally, the characteristic time, τ*, is defined such that the nondimensonal gravitational constant, G, is unity, i.e., G = 1. This property is accomplished through the use of Kepler’s third law, i.e.,
*3 ∗ l τ = , (2.4) Gmɶ * where Gɶ represents the dimensional value of the gravitational constant for clarity. These newly defined natural units lead to the following nondimensional quantities, R R m τ r=i , r =ij , µ =2 , t = , (2.5) il* ij l * m *τ * where µ is denoted as the mass ratio. The motion of the third mass is now expressed in
terms of these quantities by dividing equation (2.1) by m3 and the appropriate characteristic units in equations (2.2)-(2.4), I 2 − µ µ d r 3 = −1 − 2 3r13 3 r 23 . (2.6) dt r13 r 23 The kinematical expansion of the (inertial) first and second derivatives on the left side of the expression in equation (2.6) exploits the well-known operator relationship, Idr S dr Irɺ =3 = 3 + Iω S × r , (2.7) 3 dt dt Idr2 S dr 2 S dr Iɺɺr =3 = 3 +2 ISω ×+×× 3 ISIS ω ω r , (2.8) 3 dt2 dt 2 dt 3 where Iω S is the angular velocity of the rotating frame, S, with respect to the inertial Sd2 r frame. The second derivative, 3 , in equation (2.8) can also be expanded dt 2 kinematically in terms of the nondimensional, cartesian rotating frame, S, r= xxˆ + yy ˆ + zzˆ , (2.9) Sdr 3 =xxɺˆ +yy ɺ ˆ + zzɺˆ , (2.10) dt Sd S dr 3 =ɺɺxxˆ + ɺɺyy ˆ + ɺɺ zzˆ , (2.11) dt dt
14 where dots indicate derivatives with respect to nondimensional time. In this case, Iω S =nzˆ = z ˆ , since the nondimensional mean motion is equal to one. The inertial acceleration in the rotating frame is expressed by substituting equation (2.11) into equation (2.8), resulting in the kinematical expansion, I ɺɺ =−−( ) ++−( ) + r3 ɺɺ x2yxx ɺˆ ɺɺy 2 xyyzz ɺ ˆ ɺɺ ˆ . (2.12) The radius vectors of relative position can also be expanded in terms of the rotating coordinate frame, i.e., =−( µ) + + r13 x xyyzzˆ ˆ ˆ , (2.13) =−−( µ) ++ r23 ( x1 ) xyyzzˆ ˆ ˆ . (2.14) Finally, the equations of motion in the rotating frame are derived by combining the kinematics (equations (2.12)-(2.14)) and the kinetics (equation (2.6)) associated with m3 to yield the following scalar, second-order differential equations, (1−µ)(x − µµ) ( x +− 1 µ ) ɺɺ−=− ɺ − x2 y x 3 3 , (2.15) r13 r 23
(1− µ) y µ y ɺɺ+=− ɺ − y2 x y 3 3 , (2.16) r13 r 23 ( − µ) µ = −1 z − z ɺɺz 3 3 . (2.17) r13 r 23 Equations (2.15)-(2.17) are also written more compactly by introducing the pseudo- potential function, U, (1− µ) µ 1 U= ++() x2 + y 2 , (2.18) r13 r 23 2 reducing equations (2.15)-(2.17), i.e., − = ɺɺx2 y ɺ U x , (2.19) + = ɺɺy2 x ɺ U y , (2.20) = ɺɺz U z . (2.21)
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∂U where U = . Equations (2.15)-(2.17) are particularly useful in numerical methods j ∂ x j due to the inherent nondimensional scaling.
2.1.4 Libration Points
Since the equations of motion in the restricted problem do not possess time explicitly due to the formulation in a rotating frame, the possibility exists for time invariant equilibrium locations. Such solutions are characterized by stationary position and velocity in the synodic frame corresponding to the nonlinear system of differential equations. These particular solutions are determined by nulling the velocity and acceleration terms in equations (2.15)-(2.17), resulting in the scalar equations, ( −µ) − µµ +− µ 1(xeq) ( x eq 1 ) −=−x − , (2.22) eq r3 r 3 13eq 23 eq
(1− µ) y µ y −=−eq − eq yeq , (2.23) r3 r 3 13eq 23 eq
(1− µ) z µ z = −eq − eq 0 3 3 . (2.24) r13 r 23 = Equation (2.24) is readily solvable, that is, zeq 0 . Substitution of this result into equations (2.22) and (2.23) produces a coupled system of two equations and two = = unknowns, xeq and yeq . As discovered by Lagrange [9], if r13 r 23 1, then equations (2.22) and (2.23) reduce to identity, implying that two of the equilibrium points are located at vertices of two unique equilateral triangles. Thus, in cartesian coordinates, the primaries comprise two of the common vertices of both triangles, with the remaining
1 3 vertex defined by x = − µ and y = ± . eq 2 eq 2
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Three other equilibrium points also exist along the x-axis. Discovered first by Euler = = [5], they are denoted the collinear points and can be computed by forcing yeq z eq 0 . Substitution into equation (2.22) yields,
(1−µ)(x − µµ) ( x +− 1 µ ) −eq − eq = xeq 0. (2.25) −µ3 + − µ 3 xeq x eq 1
Equation (2.25) is a quintic equation in xeq . These solutions require numerical root-
solving methods that ultimately yield three real solutions, labeled L1, L2, and L3. The L1 and L2 points are defined such that L1 is between the primaries, L2 is on the far side of the smaller mass, and L3 is nearly a unit distance from the larger primary. All five libration points appear in Figure 2.2.
yˆ
L4
30
B L3 60 L1 L2 60 xˆ
30
L5
Figure 2.2 Equilibrium Point Locations for the CR3BP
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2.1.5 Formulation Relative to P2 in the CR3BP
Using the same rotating frame and characteristic quantities as in the original development of the equations of motion, an alternate formulation in the CR3BP is employed when low-thrust terms are included; this alternative formulation can reduce numerical sensitivity when an additional force of very low magnitude is added to the model. The origin is shifted from the barycenter to the smaller primary, P2 (as defined in Figure 2.3), and the equations of motion are rewritten as a function of position, r , and velocity, v , relative to the rotating frame, ɺɺr= gr( ) + hv( ) , (2.26) where (x+−1µκ) +( x ++ 1 ) µρ = + κ g() r y ()1 , (2.27) κ r3
2 yɺ () = − h v2 xɺ . (2.28) 0 Note that the kinematical terms in h( v ) have been shifted to the right side of the equation. The intermediate term, κ , simply allows equation (2.27) to be written in a more compact form; it is defined, (1− µ) µ κ = − − . (2.29) ρ 3r 3
The unit vectors associated with the P2-centered frame are defined parallel to xˆ−y ˆ − zˆ , − − and are defined as rˆ1 r ˆ 2 r ˆ 3 . The radius vectors of relative position are expanded as follows, = + + r rr11ˆ rr 22 ˆ rr 33 ˆ , (2.30) ρ =( −) + + r11 rˆ 1 rr 22 ˆ rr 33 ˆ . (2.31) This formulation is preferred when attempting to accurately propagate finite-thrust transfers to libration point orbits at L1 and L2.
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rˆ2
( ) P3 m 3 ρ r l* rˆ ( ) 1 P ( m ) P2 m 2 1 1
Figure 2.3 Geometry of P2-Centered Rotating Frame
2.2 Natural Periodic Orbits in the CR3BP
In his 1892 study of the n-body problem, “ Méthodes Nouvelles de la Méchanique Celeste ” [7], Poincaré focused on the behavior of the nonintegrable solutions as t → ∞ . There is no practical method of numerical integration to evaluate absolute behavior in this problem in terms of these conditions. Thus, Poincaré’s investigation focused on the behavior of periodic orbits – the only viable subset of solutions in the problem of three bodies for which motion can be predicted as t → ∞ . For nonintegrable dynamical systems, complete information on an orbit requires either an asymptotic, periodic, or almost periodic structure. In the CR3BP, the Hamiltonian consistent with a formulation relative to the rotating frame is time invariant, and an infinite number of periodic solutions are available. Initially, insight concerning the different types of orbital motion in the restricted problem is gained by linearizing relative to the libration points. A first- order linearization process yields approximate gradient information for use in evaluating the stability of the equilibrium points, as well as applications to some nonlinear targeting algorithms. Also, the linear solutions are eventually extended into the actual nonlinear problem by numerically solving a two-point boundary-value problem that exploits symmetry across the x-axis.
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2.2.1 First-Order Variational Equations Relative to the Collinear Points
Analyzing the linear equations in the neighborhood of a libration point offers information concerning potential bounded behavior. Let the variables, ξ, η, and ζ indicate the relative position of the third body, or spacecraft, with respect to the collinear libration point, i.e., ξ = − η = − ζ = − x x eq , y y eq , z z eq . (2.32) Linearizing the equations of motion in equations (2.15)-(2.17) relative to the libration point and using a first-order Taylor series expansion, results in an expression of the following form,
ξɺɺ ξ ξ ɺ η= η + η ɺɺB C ɺ , (2.33) ζɺɺ ζ ζ ɺ where the matrices B and C are defined as follows, Uxx U xy U xz = B Uyx U yy U yz , (2.34) U U U zx zy zz = X X eq
0 2 0 = − C 2 0 0 . (2.35) 0 0 0
The subscripts “ ij ”, on Uij denote evaluation of the second partial derivative of the
∂2U pseudopotential function, . Note that the B matrix is evaluated on the reference, in ∂j ∂ i = this case, the libration point, X X eq . Thus, the differential equation in equation (2.33) is linear with constant coefficients. The relationship is rewritten in state-space form, γɺ = A γ , (2.36) where
γ ɺ = ξηζξηζɺ ɺ T ɺ , (2.37)
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0 I 3 A = . (2.38) B C The A matrix is a 6x6 matrix composed of four 3x3 submatrices, the matrix 0 is simply a
matrix of zeros, and I3 is defined as the 3x3 identity matrix. Evaluating the C-matrix at the collinear libration points, where yeq = zeq = 0, yields
U = U = U = U = 0 , U < 0 , xz= yz = zx = zy = zz = (2.39) XXeq XX eq XX eq XX eq X X eq
U = U = 0 ,U > 0 ,U > 0 . xy= yx = xx = yy = (2.40) XXeq XX eq X X eq X X eq Equations (2.38)-(2.40) simplify the linearized equations of motion to the form, ξɺɺ ξ ξ ɺ U xx 0 0 ηɺɺ= 0U 0 η+ C η ɺ . yy (2.41) ζɺɺ0 0 U ζ ζ ɺ zz = X X eq From inspection of the third row of equation (2.41), it is clear that the out-of-plane component, ζ , is completely decoupled from the independent variables, ξ and η . The characteristic equation for this out-of-plane motion in equation (2.41) also possesses purely imaginary roots. Thus, the solution is oscillatory, ρ= ω + ω C1cos tC 2 sin t , (2.42) where the frequency is ω = U . The first two rows in equation (2.41) are coupled zz = X X eq
through the matrix C and represent the in-plane motion, with a general solution of the following form,
4 λ ξ = it ∑ Aei , (2.43) i=1
4 λ η = it ∑ Bi e , (2.44) i=1 λ where Ai and Bi are dependent. To determine the eigenvalues, i , Szebehely [10] uses a special form of the characteristic equation (a quadratic), i.e., Λ+2β Λ− β 2 = 21 2 0 , (2.45) where
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U + U xxX= X yy = β =2 − eq X X eq , (2.46) 1 2
2 β = −U U > 0 , 2 xx= yy = (2.47) XXeq XX eq
λ = ± Λ . (2.48) The quadratic roots are determined first, i.e.,
Λ=−+β β2 + β 2 > 1 1 1 2 0 , (2.49)
Λ=−−β β2 + β 2 < 2 1 1 2 0 . (2.50) Substituting equations (2.49)-(2.50) into equation (2.48) reveals the four characteristic roots, λ = ± Λ 1,2 1 (real), (2.51) λ = ± Λ 3,4 2 (imaginary). (2.52) Of course, the positive real root in equation (2.49) results in a positive real eigenvalue in equation (2.51) and unbounded behavior in the general solution in equations (2.43)-(2.44) as t → ∞ . The dependency between Ai and Bi is resolved by substituting equations (2.51)-(2.52) into the first two expressions in equations (2.41) and simultaneously solving the equations, thus,
λ −U i xx = β=X X eq A = α A . (2.53) iλ iii 2 i Equation (2.43)-(2.44) and their derivatives are evaluated at the initial time to determine the unknown initial conditions. Equation (2.53) is substituted into equation (2.44) so that the results are entirely in terms of the independent variables Ai, i.e.,
4 λ ξ = it0 0 ∑ Aei , (2.54) i=1
4 λ ξɺ = λ it0 0 ∑ iAe i , (2.55) i=1
4 λ η= α it0 0 ∑ iAe i , (2.56) i=1
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4 λ η= α λ it0 ɺ0 ∑ i iAe i . (2.57) i=1
Since the coefficients A1 and A2 are associated with the real eigenvalues in equations
(2.51)-(2.52), fixing the values of A1 and A2 to zero ensures that any exponential increase and decay is suppressed. Thus, a bounded planar solution emerges, η ξ= ξ cosstt() −+0 sin stt() − , 0 0β 0 (2.58) 3 ηη=( −−) βξ ( − ) 0cosstt 030 sin stt 0 , (2.59) where λ = 3 is , (2.60)
=β + β2 + β 2 1 2 s 1() 2 3 , (2.61)
s2+ U * β = xx , (2.62) 3 2s α= β 3i 3 . (2.63) ξ η ξɺ η Once the initial conditions 0 and 0 are selected, 0 and ɺ0 are predetermined to
enforce A1 = A2 = 0. The resulting orbit is an ellipse with a collinear libration point at the center . The semimajor axis is parallel to the unit vector yˆ and the semiminor axis is in the x-direction. An example of such an orbit about L1 in the Earth-moon system appears 2π in Figure 2.4. The period is IP = for the planar motion, and the root, s, may be s selected to achieve a specific value of IP . In this linear model, the out-of-plane frequency is not commensurate with the in-plane frequency. Nevertheless, these planar solutions form the basis for determining initial conditions consistent with planar nonlinear orbits; such planar orbits that are solutions in the nonlinear problem are then used in computing fully three-dimensional nonlinear orbits. Although the first-order linear set of initial conditions do not result in a closed periodic orbit in the nonlinear problem, initial conditions “sufficiently close” to the libration point still yield an initial guess that may be used in targeting algorithms.
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Earth-Moon System γ = [ ]T 0 0.1 0 0 0 -0.837227214342055 0 µ = 0.0121505649407351 , 1 unit = 385692.5 km
To Earth Moon
L1
Figure 2.4 Linearized L1 Periodic Orbit
2.2.2 The State Transition Matrix
Besides stationary equilibrium points, dynamical information is also sought relative to time-varying solutions in the nonlinear problem. Numerical computation of trajectory arcs, such as periodic orbits and orbital transfers to a specific target in the CR3BP, requires the use of differential corrections procedures. Before introducing these corrections procedures, an important result from linearization of the equations of motion is required. Define X* ( t ) as a time-varying reference solution such that Xt( ) = Xt* ( ) + δ Xt( ) . Let the perturbed state relative to the reference trajectory be
defined as,
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