REPRESENTATIONS OF INVARIANT MANIFOLDS FOR APPLICATIONS IN
SYSTEM-TO-SYSTEM TRANSFER DESIGN
A Thesis
Submitted to the Faculty
of
Purdue University
by
Christopher E. Patterson
In Partial Fulfillment of the
Requirements for the Degree
of
Master of Science
May 2005 ii
For Betty and Ruth. iii
ACKNOWLEDGMENTS
First, I thank my parents. They have been with me through some very difficult times and supported me in more ways than I can mention. I could not have succeeded in school or in life without their encouragement. Because research does not occur in a vacuum, I thank all the past and current members of my research group for their interest in my work and for their help. Their friendship is greatly appreciated and I hope I succeed in returning the favor. I thank my adviser, Professor K. C. Howell, for her guidence throughout my graduate career. Her motivation has been invaluable. Finally, I thank the School of Aeronautics and Astronautics at Purdue University for support of my research and my education. Portions of this work were supported by NASA and the Goddard Mission Services Evolution Center under contract numbers NCC5-358 and NAG5-11839. iv
TABLE OF CONTENTS
Page LIST OF FIGURES ...... vii ABSTRACT ...... x 1 Introduction ...... 1 1.1 Problem Definition ...... 2 1.2 Previous Contributions ...... 5 1.2.1 Three-Body Problem ...... 5 1.2.2 Approximations and Transfers ...... 8 1.3 Current Work ...... 10 2 BACKGROUND: MATHEMATICAL MODELS ...... 12 2.1 The Circular Restricted Three-Body Problem ...... 13 2.1.1 Assumptions ...... 13 2.1.2 Reference Frames ...... 14 2.1.3 Characteristic Quantities ...... 15 2.1.4 Equations of Motion ...... 15 2.1.5 Particular Solutions ...... 18 2.1.6 The State Transition Matrix ...... 19 2.1.7 Differential Corrections and Halo Orbits ...... 21 2.2 Invariant Manifold Theory ...... 25 2.2.1 Invariant Manifolds ...... 25 2.2.2 Periodic Orbits and Maps ...... 29 2.2.3 Orbital Stability ...... 30 2.2.4 Stable and Unstable Manifolds of Periodic Orbits ...... 33 2.2.5 Eigenstructure of the Monodromy Matrix ...... 34 v
Page 2.2.6 Computation of the Stable and Unstable Manifolds of Periodic Halo Orbits ...... 35 2.3 Transition to an Ephemeris Model ...... 39 3 Manifold Approximations Using Cells ...... 44 3.1 Cell Approximations ...... 44 3.2 Cell Creation ...... 55 3.2.1 Division of Manifold Data ...... 55 3.2.2 Cell Expansion ...... 58 3.3 Applications Using Cell Approximations ...... 60 3.3.1 Determination of a Destination Halo Orbit ...... 60 3.3.2 Saving Computation and Storage ...... 63 4 Manifold Approximations as the Basis for Transfers ...... 65 4.1 Determination of Transfers to Periodic Halo Orbits ...... 65 4.1.1 Approaching the Halo Orbit ...... 65 4.1.2 Computation of Halo Orbit Insertion ...... 70 4.1.3 An Alternative to Direct Halo Orbit Insertion ...... 72 4.2 Generating System-to-System Transfers in the CR3BP ...... 76 4.2.1 Transition of CR3BP Transfers to an Ephemeris Formulation . 79 4.3 Adaptation for Lunar Encounters ...... 83 4.3.1 Cost Reduction ...... 90
5 Transfers to Orbits of the Sun-Earth L2 Point ...... 95 5.1 Varying Size of Initial Earth-Moon Lissajous Orbit ...... 96
5.1.1 Comparison With Southern Earth-Moon L2 Orbits ...... 100 5.2 Varying Initial Time of Computation ...... 100
5.3 Transferring From an Earth-Moon L1 Orbit ...... 104 6 Conclusion ...... 109 6.1 Summation ...... 109 6.2 Suggestions for Future Work ...... 109 vi
Page LIST OF REFERENCES ...... 111 vii
LIST OF FIGURES
Figure Page 2.1 Reference Frames ...... 14 2.2 Location of the Libration Points ...... 19 2.3 A Periodic Halo Orbit in the Earth-Moon System ...... 24 2.4 Stable and Unstable Manifolds Tangent to Subspaces ...... 28 2.5 Manifold Tubes in Earth-Moon CR3BP Collapsing on Two Halo Orbits ...... 38 2.6 A “Halo-Like” Lissajous Trajectory ...... 41 3.1 Each Tube is Contained within a Successively Larger Tube ...... 45 3.2 Cells are Defined to Contain Small Regions of Manifold Trajectories . 46 3.3 Manifold Data is Transformed to a Cell Frame with Origin Q.....¯ 47 3.4 A Manifold Data Point Represented in Spherical Coordinates Relative to a Cell Frame ...... 48 3.5 The N˜ and B˜ Vectors Placed at the Center of a Section of Data Points Directed Toward Two Potential Cell Origins ...... 50
3.6 (a) Using Q¯b as the Origin of a Cell Frame (b) Using Q¯n as the Origin of a Cell Frame ...... 51 3.7 Manifold Surfaces and Velocity Flow Approximated Within a Cell . . 53
3.8 Sun-Earth L1 Manifold Position and Velocity Fit Errors ...... 54 3.9 Dividing a Tube into Cells ...... 56 3.10 Dividing a Band Into Sectors ...... 56 3.11 Five Tubes through a Set of Cells ...... 59 3.12 Locating the Intersection Point within a Cell ...... 61 4.1 Trust Region Around the Surface Approximations in a Cell ...... 67 4.2 Improvement of an Approach to a Halo Orbit Using Scale Factors ...... 69 4.3 Transfer Determined using Cells ...... 72 viii
Figure Page 4.4 Initial Approach Arc to Halo ...... 73 4.5 Attempt to Target a Candidate HOI Point ...... 74 4.6 Multi-Point Targeting To Stay Near the Halo Orbit ...... 76
4.7 Earth-Moon L1 Manifold Paths Intersect Sun-Earth L2 Manifold Cells ...... 77
4.8 Transfer from the Earth-Moon L1 Halo Orbit to a Sun-Earth L2 Halo Orbit ...... 80
4.9 Transition of Earth-Moon L1 Manifold Arc to Ephemeris Model . . . 81 4.10 Transition of the Earth-Moon to Sun-Earth Transfer to the Ephemeris Model ...... 82
4.11 Earth-Moon L2 Tube Intersecting Sun-Earth L2 Tube Both Tubes Computed in CR3BP Models ...... 84
4.12 Earth-Moon L2 Manifold Surface Intersecting Sun-Earth L2 Tube (Earth-Moon Surface Computed in Ephemeris Formulation) ..... 85 4.13 Earth-Moon Manifold in Ephemeris Model; Transfer Arc and Halo Orbit in a Sun-Earth Circular Restricted Model ...... 87 4.14 The Entire Transfer is Converged in a Full Ephemeris Formulation . . 88 4.15 A Transfer using Multi-Point Targeting Near the Halo Orbit is Converged in a Full Ephemeris Formulation ...... 89 4.16 Transfer Computed with No Cost from HOI Targeting Routine ...... 91 4.17 Transfer Computed with No Cost from Multi-Point Halo Targeting Routine ...... 92 4.18 Earth-Moon Lissajous Orbit Changes Shape as Transfer Cost is Reduced ...... 93 5.1 Case 3a. Ephemeris Earth-Moon Manifold and CR3BP Sun-Earth Transfer Arc to Halo Orbit ...... 98 5.2 Case 3b. Sun-Earth View: Transfer Arc to LPO in Ephemeris Model 99 5.3 Case 3c. Free Ephemeris Transfer Between Systems ...... 99 5.4 Case 8b & c. Cost Reduction Facilitated by Large Changes in in Libration Point Orbits ...... 102 ix
Figure Page
5.5 Case 9b & c. Less Variation in Libration Point Orbits is Necessary to Reduce a Lower-Cost Preliminary Transfer ...... 103
5.6 Case 10b & c. Departing the Vicinity of the Earth-Moon L1 Point in a Direction Toward the Earth ...... 105
5.7 Manifolds of LPO Near Earth-Moon L1 Point Directed Toward the Moon ...... 106 5.8 Case 16b. Ephemeris Model Transfer Departing Earth-Moon System Via Lunar Flyby From an LPO Near the Earth-Moon L1 Point .... 108 x
ABSTRACT
Patterson, Christopher E. M.S., Purdue University, May, 2005. Representations of Invariant Manifolds for Applications in System-to-System Transfer Design. Major Professor: Kathleen C. Howell.
The Lunar L1 and L2 libration points have been proposed as gateways granting inexpensive access to interplanetary space. To date, only individual solutions to the problem of traveling between libration point orbits in different three-body systems have been computed. The methodology to solve the problem for transfers between arbitrary three-body systems and even entire families of orbits is under investigation. For preliminary design, manifolds associated with periodic halo orbits in the circular restricted problem are well-suited as the foundation for such transfers. This work presents an initial approach to solving the general problem of system- to-system transfers. A method is detailed for approximating and storing invariant manifold data. The method is based on a unique cell structure whereby invariant man- ifolds associated with halo orbits in the same family and occupying large volumes of space are approximated within smaller, overlapping cells. The approximations within the cells are stored in a manner that allows the parameterization of halo orbits, a capability that is very useful in a transfer design procedure, freeing the designer to select any orbit that can be reached from any manifold within the volume. A method is presented for computing low-cost transfers to and from libration point orbits using these approximations. Some particular solutions are presented for transfers between libration point orbits in the Earth-Moon and Sun-Earth systems, though the em- phasis is on developing the methodology for solving the more general problem. The particular solutions are successfully transitioned to a more robust ephemeris model and a procedure is presented that employs these solutions to produce natural (free) paths between systems. 1
1. Introduction
Currently, many scientific and exploratory missions are being proposed that include spacecraft trajectory arcs that exist only in a force model consistent with the multi- body problem. In preliminary analysis for such scenarios, it has also been suggested that libration point orbits and their associated manifolds may serve as gateways between three-body systems [1–5], suggesting a new approach to low-cost space travel.
For example, orbits at or near the Earth-Moon L1 libration point have been proposed as staging areas for missions to the Moon [4] and to orbits in the vicinity of the Sun-
Earth L2 point [2, 3]. Similarly, orbits near both the Earth-Moon L1 and the Sun-
Earth L2 libration points have been proposed as nodes or staging areas for missions
to Mars and beyond [1]. Of course, the Sun-Earth L2 region has repeatedly been proposed as an ideal location for large astronomical observatories. Such complex observatories, however, may require servicing and repair missions that, in turn, might
eventually involve spacecraft departing and returning to an Earth-Moon L1 orbit [2]. Should such missions be implemented, mission designers will require efficient tools to compute transfers not only to and from libration point orbits but also between libration point orbits in different systems. The trajectory design process for two recent missions, GENESIS and MAP, was facilitated by techniques that incorporate periodic halo orbits and their associated stable and unstable manifolds. However, there are an infinite number of individual trajectories that comprise the surfaces that define the stable and unstable manifolds corresponding to a single periodic halo orbit. When projected onto configuration space, these manifolds appear to form tubes asymptotically approaching (stable) and departing (unstable) the periodic halo orbits. Since no useful analytical solution to the problem of three bodies is currently known, the computation of an individual trajectory along a manifold requires numerical integration. Then, the determination 2
of the “best” trajectory to accommodate mission requirements has, thus far, required the numerical integration of many trajectories on the tube near the halo orbit. This process is inefficient and, for some applications, may be impractical. The difficult nature of the problem is apparent, in particular, when the choice of the halo orbit might itself be used as a design variable. If any of a potentially large number of periodic halo orbits could serve the mission goals equally well, and each has associated with it an infinite number of trajectories that comprise the manifold, then a robust search of all design parameters in the usual manner of trajectory propagation and selection is truly intractable. It is also noted that this problem — in terms of geometry, sensitivity, and topology — is not amenable to traditional optimizers.
This work is focused on development of a framework for the solution of one type of transfer problem. To replace preliminary propagation and a large search of manifolds, a reasonably accurate analytical approximation for a limited volume of the manifold space is constructed and, once an initial guess is available, the solution space is bounded and a corrections process in a higher fidelity model can successfully yield a transfer between the Earth-Moon and Sun-Earth systems. Also, of equal significance, this approach allows the halo orbit itself as a design parameter, enabling potentially lower-cost solutions. These approximations can be useful in many design problems such as transfers, error analysis and recovery, as well as contingency studies. Examples are presented here that demonstrate the use of approximations in system-to-system transfer problems.
1.1 Problem Definition
Since the 17th century, when Newton mathematically formulated the concept of gravity as a force, tremendous study and effort has focused on understanding the motion of multiple bodies that are free to interact gravitationally. The relative motion of only two bodies has been well understood kinematically since Kepler, of course; geometrically, the relative motion is described in terms of conic sections. Newton 3 could also prove this solution dynamically via the application of his force law. With the addition of even just one body to the problem, however, the complexity increases tremendously such that no closed-form analytical solution exists. But, simplifying assumptions do allow for some general insights into the three-body problem.
In simplifying the general problem of the motion of three arbitrary bodies, an important assumption involves the relative masses. Specifically, it is assumed that the mass of the third body is infinitesimal in comparison with the other two bodies such that it is gravitationally attracted to both larger bodies, yet exerts no influence upon their motion. The first two bodies, termed “primaries”, thereby comprise a two- body problem and their relative motions can be modeled as unperturbed conic orbits. For convenience, they are ordered according to their masses, so that the first primary is the most massive. If the primaries are restricted to move in elliptical paths, the problem is termed the elliptic restricted three-body problem (ER3BP); if the primary motion is circular, the problem is termed the circular restricted three-body problem (CR3BP). It is the CR3BP that has been employed whenever a simplified model is required throughout this work.
To further simplify the analysis, the motion can be viewed in a frame that rotates with the primary bodies. Although this does not make the problem solvable, it does allow significant insight and particular solutions are immediately available. Euler and Lagrange exploit this formulation to significant advantage [6]. The particular solution that emerges is the familiar set of five equilibrium points, Li,i = 1, 2, ...5, fixed relative to the rotating coordinate frame. These points are also commonly denoted as libration points or Lagrange points. The first three points are collinear with the positions of the primaries. One point lies between them, the other two on opposite sides of the massive bodies. The fourth and fifth libration points lie in the orbital plane of the primaries, and form equilateral triangles with them. Though the system of differential equations is nonlinear, a linear stability analysis relative to Li in the CR3BP does conclude that the collinear points are unstable, while the equilateral points may be linearly stable under suitable conditions. 4
This work focuses on trajectories near the collinear points. Analysis in the vicin- ity of these points yields initial conditions that demonstrate the possible existence of both two-dimensional periodic orbits existing in the orbital plane of the primaries and a family of three-dimensional periodic orbits leaving the plane. These types of orbits have been computed in the nonlinear system using numerical integration and a differ- ential corrections process. In the vicinity of each collinear point, the two-dimensional Lyapunov family of periodic orbits lies in the orbital plane of the primaries. A pitch- fork bifurcation of the Lyapunov orbits both above and below the plane, respectively, yields the “northern” and “southern” families of halo orbits [7]. The out-of-plane halo orbits are three-dimensional and precisely periodic in the CR3BP with a pe- riod equal to approximately half of the period of the circular orbit that defines the primary motion. As periodic orbits, halo orbits exist in the center subspace. From the eigenstructure in the local phase space near the periodic orbits, it is apparent that invariant stable and unstable manifolds exist. To compute trajectories on the stable/unstable manifolds, initial conditions can be approximated from linear analy- sis near fixed points along the halo orbits using the eigenstructure and propagating numerically. The paths of all six-dimensional manifold trajectories can be projected onto three dimensional configuration space; as a complete set, they appear as tubes either approaching (stable) or departing (unstable) the periodic halo orbit. Though a single tube associated with a given halo orbit can be visualized as a closed surface in R3, it is also a member of a continuous family of tubes associated with the con- tinuous family of halo orbits — parameterized in terms of out-of-plane amplitude — that are associated with the same collinear libration point. All such tubes, collected together, then define a certain volume in six-dimensional space. The first problem in the construction of system-to-system transfers is to accurately approximate the flow along the manifolds through this volume toward (or away from) the corresponding halo orbits.
Position and velocity states that define trajectory arcs that lie along a single manifold can be transformed from one rotating three-body system to another in a 5
straightforward manner. Such a transformation reveals that an unstable manifold tube departing a halo orbit in one system can intersect a stable manifold tube flowing toward a halo orbit in another system. This implies that a type of gateway exists between systems, one that is associated with the manifold tubes. A transfer between intersecting manifold tubes can shift a spacecraft between halo orbits in different three-body systems [5,8]. The subsequent problem is to use manifold approximations to determine such transfers and identify the shift that results in low maneuver costs.
1.2 Previous Contributions
1.2.1 Three-Body Problem
Newton was among the first to examine the dynamically difficult three-body prob- lem. His model for a gravity force first appears in The Principia, published in 1687. In the same document, Newton geometrically solves the two-body problem as well. But, the two-body problem is only integrable when it is reformulated as a problem in the relative motion of two bodies, effectively reducing the number of first-order differential equations. The introduction of even one additional body increases the number of equations, of course, and the ten known integrals of the motion are in- sufficient to completely solve the differential equations for three or more bodies in any formulation. Because it is not integrable, a general closed-form solution to the n-body problem remains unknown. However, the problem of three bodies was of great interest in the 18th and 19th centuries since navigators required accurate knowledge of the motion of the Moon, and its orbit around the Earth could not be understood without considering the influence of the Sun. In studies of the Moon, Newton began with a two-body analysis to determine an elliptical Lunar orbit and applied pertur- bation theory to predict variations in orbital parameters due to the Solar influence. Though the variation of parameters method is useful, even today, in predicting the influence of multiple bodies on low Earth-orbiting spacecraft, Newton’s approach was unsuccessful in predicting the shift of the Lunar apsides [6]. 6
In 1772, Euler formulated the restricted problem in a rotating frame and deter- mined some particular solutions relative to that frame, i.e., three locations collinear with the primaries [6]. That same year, Lagrange published a memoir in which he deduced Euler’s collinear solutions along with a second type, the equilateral points, and reduced the problem from a system of order 18 to a system of order seven. His method for reducing the problem was to determine: (a) the lengths of the sides of the triangle formed by the three bodies; (b) the plane in space containing the trian- gle; and, (c) the orientation of the triangle within the plane. For their work, Euler and Lagrange shared the Prix de l’Academie de Paris in 1772 [6]. Euler’s use of a restricted problem and a rotating frame is still dominant today in studies of the three-body problem and the complete set of particular solutions are now traditionally denoted as the Lagrange points. Later, in 1843, Jacobi explicitly reduced the general problem one step further, i.e., to a 6th order system, through the elimination of the nodes [6].
In 1836, Jacobi published his discovery of a constant that exists in Euler’s for- mulation of the circular restricted three-body problem relative to a rotating frame. Jacobi’s Constant further reduces the order of the problem to five. Regions in the three-body system where motion is physically permitted can be defined by the energy- like Jacobi Constant, a fact that was first demonstrated by Hill in 1878 [6]. Hill used this knowledge to argue that the Moon could never escape the region near the Earth. Hill’s Lunar theory is still considered exceptional since it was the first to abandon any two-body analysis as a starting guess in describing the motion of the Moon; al- ternately, Hill used a circular restricted three-body solution as a first guess before adding perturbations. Later in the 19th century, Henri Poincar´edetermined that no further constants could exist and that the problem therefore could not be solved an- alytically using this approach. However, Poincar´edid demonstrate that when two of the masses are small in comparison with the third, then an unlimited number of periodic orbits exist in the system. Much insight concerning the motions within the system is available from knowledge of these orbits. 7
By the end of the 19th century and into the 20th, many researchers, including Dar- win, Plummer, and Moulton redirected their efforts to the development of methods for the determination of periodic solutions in planar and three-dimensional problems, with heavy dependence on approximation techniques [6,9–13]. In 1897, G. H. Darwin employed “a laborious method of tracing orbits by quadratures, and finding periodic orbits by trial” and was among the first to numerically produce periodic results [10]. Henry C. Plummer used second and third-order approximate solutions to produce planar orbits near the collinear libration points in 1903 [11, 12]. Forest R. Moulton worked to numerically integrate periodic orbits in the system, a process hindered by the tedious computations required [13]. However, his results included periodic orbits near the equilateral points and three-dimensional periodic solutions relative to the collinear points determined from the linearized equations of motion. The advent of computers allowed extensive numerical studies and more accurate results from high speed numerical integration, but also offered a tool useful in generating more accurate analytical approximations. In 1967, Victor G. Szebehely published Theory of Orbits: The Restricted Problem of Three Bodies [13], the state-of-the-art at that time in the analysis of the three-body problem and a survey of the different types of orbits that have been produced. In the 1960’s, Robert Farquhar began the study of a particu- lar type of precisely periodic three-dimensional orbit in the vicinity of the collinear points, now denoted as a “halo” orbit [14]. By this time, interest in the problem was escalating as practical applications to space mission design were being considered. With Ahmed Kamel in 1973, Farquhar produced accurate approximations for quasi- periodic solutions near the translunar libration point in the Earth-Moon system [15]. Today, this is still considered a viable location for data-relay satellites to facilitate communication with the far side of the Moon. In 1975, David L. Richardson and Noel D. Cary developed third-order approximations for the motion near the interior Sun- Earth libration point [16], and in 1980, Cary published a more general third-order approximation for halo orbits near collinear libration points in the CR3BP. 8
1.2.2 Approximations and Transfers
Once three-dimensional periodic solutions were initially computed in the CR3BP, subsequent analytical and numerical work continued at an increasing rate. Spacecraft missions to libration point orbits originated in 1978 and ignited this ongoing interest from an astrodynamics perspective. In 1979, John V. Breakwell and John V. Brown extended Farquhar’s work to numerically produce an entire family of halo orbits, some of which were linearly stable [17]. The study of families of halo orbits (including the search for stable members) near all the collinear points as the relative masses of the bodies in the system is varied was continued in the 1980’s by K. C. Howell, collaborating with Breakwell and exploiting the numerical technique of continuation [18,19]. With Henry J. Pernicka, Howell developed numerical techniques for efficiently computing quasi-periodic Lissajous trajectories and, later, devised station keeping strategies for spacecraft in these orbits [20].
Space missions beyond the Earth are becoming increasingly complex and mission designers require more sophisticated mathematics to better understand the structure of the solution space as a basis for a more efficient design process [9]. Over the last ten years, invariant manifolds have emerged as useful tools for these applications. However, applications that exploit invariant manifolds are not yet fully developed. To be most useful, it is required that invariant manifolds be computed efficiently, yet modern methods of manifold computation can be cumbersome and involve excessive amounts of numerical simulations. A significant result of past research efforts is a method for the computation of halo orbits and Lissajous trajectories based on analytical approximations as initial guesses [17,20]. For application, it is critical that techniques are also available for approximations of the transfer paths to and from these unstable orbits. The stable and unstable manifolds associated with libration point orbits serve this purpose and their approximation greatly simplifies the determination of transfer paths. 9
Thus far, much of the mathematical research in the approximation of manifolds has not been applied to spacecraft trajectory design. Hobson [21] develops a method of quickly approximating a single manifold arc to a desired accuracy, yet a single arc is insufficient to approximate the global behavior of a set of trajectories computed on a manifold. Guder et al. [22] introduces a cell mapping method as a way to predict long term behavior; Dellnitz and Hohman [23] use similar cell mapping techniques combined with a subdivision algorithm to approximate unstable manifolds. However, they are not interested in an approximation of the full global behavior, nor do they apply their techniques to the problem of trajectory design. Even when using continu- ation to approximate the types of manifolds that are of interest — an approach that might yield a more global view — the method can encounter difficulties isolating the small details necessary for trajectory design.
The first transfer to a libration point orbit occurred when the spacecraft ISEE-3 (International Sun Earth Explorer 3) was launched from Earth on August 12, 1978, and injected into a Sun-Earth L1 halo orbit on November 20, 1978. The trajectory design methodology involves a shooting strategy to depart Earth and insert into a Sun-Earth L1 libration point orbit. However, the preliminary trajectory design for the GENESIS mission, launched on August 8, 2001, incorporates manifold arcs. More recently, many researchers are interested in the problem of transferring between systems from a new perspective. In this new concept, the transfer is between one three-body system and another. Methods of determining the intersections of manifold tubes and using such intersections to determine transfers between coupled three-body systems are presented by Koon et al. [4, 5]. Folta et al. [2] offer more examples, producing numerous trajectories to demonstrate the possibility that humans might service national resources in a Sun-Earth L2 orbit from an Earth-Moon L1 staging area. Investigation of this problem by various researchers has, thus far, produced only point solutions that emerge after either a time-consuming shooting method or from inspection of the intersections of manifold tubes. In some cases, the tubes are only generated in two dimensions, while real solar system applications would clearly be 10
three-dimensional. In actuality, these tubes are generated by propagation of numerous manifold arcs; the manifolds are associated with three-dimensional halo orbits that must be selected a priori. This study is based on an alternate view of the problem. By approximating a volume of manifolds rather than determining a single tube, and by incorporating halo orbit parameterizations, the result is a methodology that searches through many of the possible transfers in the design space and determines a “best” first guess for preliminary design. This initial approximation is transitioned to a higher fidelity model incorporating ephemeris data for a final end-to-end transfer design.
1.3 Current Work
The general focus of this study is the approximation of the stable and unstable manifolds that approach and depart periodic halo orbits. In particular, analytical approximations are sought for the position and velocity states along a number of tubes contained inside small volumes. These are subsequently used to meet design objectives without expensive calculation, storage, and data base searches in a large design space. This study also examines the intersections of tubes computed in dif- ferent three-body systems and exploits the manifold approximations to determine intersections that yield small differences in velocity states and, thus, accurately de- termine acceptable maneuvers for the transfers. By using manifold approximations in volumes rather than the determination of only single tubular surfaces, and by incorporating halo orbit parameterizations into the design process, an algorithm is developed that quickly searches through the possible transfers in the design space and determines the “best” choice for an initial guess. Then, a reduction or correc- tions strategy yields system-to-system transfers. Solutions computed from CR3BP analysis are then successfully transitioned to an ephemeris model, demonstrating the application for mission analysis. The cost (total ∆v) in the high fidelity model is reduced using a differential corrections scheme. 11
This document is comprised of the development of the methodology as well as a number of significant examples. It is organized as follows: CHAPTER TWO: The Circular Restricted Three Body Problem is defined, along with the associated characteristic quantities. The equations of motion for the space- craft are formulated relative to a rotating frame and the particular solutions (libration points) are identified. A differential corrections process is discussed for numerical de- termination of halo orbits in the vicinity of the collinear libration points, and linear analysis is used to compute their associated manifolds. A four-body model using ephemeris data is introduced to incorporate perturbations. Transfers initially com- puted in the CR3BP will be shifted to this model. CHAPTER THREE: A method for dividing a large volume of manifold tubes into smaller cells is introduced. Also, functional approximations are developed using standard fitting algorithms to accurately represent both position and velocity data along manifold tubes within the cells. CHAPTER FOUR: A parameterization of the halo orbits associated with mani- folds inside a cell is discussed. This parameterization is used along with the manifold approximations to determine a first guess for a low-cost transfer from the vicinity of the Earth to a halo orbit. This first guess does not reach the specified halo orbit precisely, but a corrections procedure is derived to correct the guess and complete a transfer. This method is subsequently employed to compute system-to-system trans- fers in coupled restricted problems; these solutions are transitioned to the ephemeris model and the costs of the maneuvers are lowered as a result of a differential correc- tions process. CHAPTER FIVE: A number of different transfers are presented and summarized. General trends are discussed regarding the variation of costs as parameters in the transfers are adjusted. CHAPTER SIX: A summary is included and further work is suggested. 12
2. Background: Mathematical Models
The motion of a spacecraft through the solar system is influenced by many different forces acting simultaneously. Dominant among these are the gravitational attrac- tions of the Sun and the nearest planets and/or moons. Other forces acting on the spacecraft include solar radiation pressure (SRP) and the gravity forces of smaller or more distant celestial bodies. A dynamical model that incorporates all of these forces is very complex and, without a good initial estimate, designing a trajectory in this model is an intractable problem. To simplify the problem, the force model is usually reduced to just two or three of the most significant gravity forces. In this work, the ephemeris data that is used for the trajectory design process includes the gravity of the Sun, the Earth, and the Moon. Solar radiation pressure can be incorporated as a force but is not included here. Yet, even in this simplified four-body model, a good starting point is necessary. This initial guess is obtained numerically from arcs in the circular restricted three-body problem. In this chapter, the mathematical tools are developed to generate trajectories for application to mission design. The circular restricted three-body problem (CR3BP) is the simplest model used; it incorporates a specific choice of coordinate frames and assumptions that govern the motion of the celestial bodies. The CR3BP models the motion of one body of infinitesimal mass (a spacecraft) that is free to move in the vicinity of two bodies of finite mass. These two massive bodies represent a two-body problem and, thus, their respective paths are predetermined. This simplified problem is used to design trajectories and/or arcs with certain desired characteristics, such as periodic orbits, or transfer paths between orbits within the same three-body system or even transfers between systems. These solutions are then employed as initial guesses and recomputed in a four-body model that models the motions of celestial bodies on ephemeris data. This transition to the four-body model adjusts for the 13
full effects of multi-body dynamics, removing the major assumptions of the CR3BP, while preserving the essential characteristics of the trajectory.
2.1 The Circular Restricted Three-Body Problem
2.1.1 Assumptions
A full dynamical model of the gravitational interaction of three massive bodies, even assuming them to be point masses, consists of 18 first-order, coupled, nonlinear differential equations. Only ten analytical integrals of the motion exist, so these equa- tions do not possess a known solution of closed form. A simpler model is developed here using assumptions that lower the number of equations from 18 to 6. Though these equations still yield no general analytical solution, they do allow particular solutions including periodic orbits that exist within the center subspace.
A fundamental assumption is that all bodies in the system are point masses.
Let the three bodies be identified as P1, P2, and P3, and define their dimensional masses to be M1, M2, and M3, respectively. Secondly, let M3 ≪ M2 < M1 such that the orbit of P2 relative to P1 is unperturbed by P3. The two larger bodies are labeled “primaries”. These assumptions result in the second primary, P2, moving in a
Keplerian orbit relative to P1. This orbit is either elliptical or circular since, of course, hyperbolic orbits are not of any practical interest here; the primary orbit defines the problem as either the elliptic restricted (ER3BP) or circular restricted three-body problem (CR3BP).
With the motion of two of the bodies completely specified, there remains only one free body in the system. Thus, only six first-order differential equations are necessary to describe its motion; the complexity of the problem has been reduced significantly from the original 18 degrees of freedom. 14
2.1.2 Reference Frames
To adequately derive the equations of motion for P3, appropriate reference frames are necessary. Since the orbits of P1 and P2 are unperturbed by P3, their mutual center of mass, i.e., the barycenter, B, is selected as a convenient inertially fixed origin. An inertial frame, I, appears in Figure 2.1, with axes and unit vectors Xˆ— Yˆ —Zˆ as indicated. The Xˆ—Yˆ plane is defined to be the plane of motion of the primaries and the Zˆ axis completes the right-handed triad. The Zˆ unit vector is parallel to the orbital angular momentum vector associated with the motion of the primaries. Recall that M1 > M2, and, thus, P1 lies closer to the origin than P2.A rotating frame, R, is then defined with axes and unit vectorsx ˆ—ˆy—ˆz such that the xˆ axis is directed from P1 toward P2, thez ˆ axis is coincident with the inertial Zˆ axis and they ˆ axis again completes a right-handed triad. The angle θ is defined to orient the rotating frame R relative to the inertial frame I, as seen in Figure 2.1. Since the primary orbits are circular with constant angular velocity, θ˙ is positive and constant.
Figure 2.1. Reference Frames 15
2.1.3 Characteristic Quantities
It is convenient to define characteristic quantities for the system, indicated with a superscript ‘*’. First, the characteristic mass of the system is defined to be the ∗ total mass of the primaries: m = M1 + M2. The characteristic length is the average distance between the primaries, i.e., the semi-major axis of the primary orbit: l∗ = 1 3 ∗ l∗ 2 a. The characteristic time quantity is t = ∗ , a convenient quantity since it h Gm i guarantees that the nondimensional values of both the gravitational constant, G, and the mean motion of the primaries, n, equal unity. In addition to characteristic quantities, define a nondimensional mass parameter, i.e., M µ = 2 . (2.1) m∗ Then, the nondimensional masses of the primaries can be written in the form
m1 = 1 − µ, (2.2)
m2 = µ, (2.3) and the nondimensional distance of each primary relative to the barycenter B is expressed as follows:
|BP1| = µ, (2.4)
|BP2| = 1 − µ. (2.5)
These expressions lead to a more elegant and useful form of the equations of motion for the system.
2.1.4 Equations of Motion
Using the above nondimensionalization, Newton’s gravitational force law describ- ing the motion of P3 under the gravitational influence of the two primaries can be written as (1 − µ) µ I ρ¯¨ = − d¯− r,¯ (2.6) d3 r3 16
whereρ ¯, d¯, andr ¯ are nondimensional position vectors: P3 relative to the barycenter, relative to P1, and relative to P2, respectively (Figure 2.1). The dots indicate the second derivative with respect to nondimensional time and the superscript I denotes the derivative is relative to an inertial observer. The kinematic expansion of the left side of the equation can be accomplished with the following steps. First, the position vectorρ ¯ is expressed in terms of coordinates relative to the rotating frame as follows
ρ¯ = xxˆ + yyˆ + zz.ˆ (2.7)
Then, the inertial derivative is evaluated from
I ρ¯˙ =R ρ¯˙ +I ω¯R × ρ,¯ (2.8)
where the left superscript R indicates a derivative with respect to the rotating frame and I ω¯R = nzˆ = nZˆ is the angular velocity of the rotating frame relative to the inertial frame. It is again noted that the mean motion n, and thus, the angular velocity are both in nondimensional units. This results in the following expression
I ρ¯˙ = (x ˙ − y)x ˆ + (y ˙ + x)y ˆ +z ˙z.ˆ (2.9)
Continuing, the second derivative is evaluated from
I ρ¯¨ = (¨x − 2y ˙ − x)x ˆ + (¨y + 2x ˙ − y)y ˆ +z ¨z,ˆ (2.10)
where the angular velocity of the rotating frame has been incorporated with a mag- nitude of one. To expand the right side of Equation (2.6) in terms of coordinates fixed in the rotating frame, note that
d¯=(x + µ)x ˆ + yyˆ + zz,ˆ (2.11)
r¯ =(x − (1 − µ))x ˆ + yyˆ + zz.ˆ (2.12) 17
Substituting Equations (2.10)–(2.12) into Equation (2.6) and equating components yields three scalar, second order differential equations:
(1 − µ)(x + µ) µ (x − (1 − µ)) x¨ − 2y ˙ − x = − − , (2.13) d3 r3 (1 − µ) y µy y¨ + 2x ˙ − y = − − , (2.14) d3 r3 (1 − µ) z µz z¨ = − − . (2.15) d3 r3
This is the standard form for the equations of motion in the CR3BP. Though the system is not conservative, it is nevertheless convenient to define a pseudo-potential, (1 − µ) µ 1 U ∗ = + + x2 + y2 . (2.16) d r 2 ¡ ¢ Then, by examining the partial derivatives of U ∗, Equations (2.13)–(2.15) can be easily rewritten in the form
∂U ∗ x¨ = + 2y, ˙ (2.17) ∂x ∂U ∗ y¨ = − 2x, ˙ (2.18) ∂y ∂U ∗ z¨ = . (2.19) ∂z
This yields an alternate form of the differential equations of motion. The system now consists of three coupled, second-order, nonlinear differential equations, or equivalently, six coupled, first-order, nonlinear differential equations. No analytical solution is currently available, yet a constant of integration is known to exist, as determined by Jacobi in 1836 [6]. Multiply Equations (2.17)–(2.19) by 2x ˙, 2y ˙, and 2z ˙, respectively, and a summation of the resulting equations produces the following scalar expression:
dx d2x dy d2y dz d2z dx ∂U ∗ dy ∂U ∗ dz ∂U ∗ 2 + 2 + 2 = 2 + 2 + 2 . (2.20) dt dt2 dt dt2 dt dt2 dt ∂x dt ∂y dt ∂z
Integrating this scalar expression with respect to time yields
C = 2U ∗ − V 2, (2.21) 18
where C is the Jacobi Constant and V = (x ˙ 2 +y ˙2 +z ˙2)1/2, that is, the nondimensional magnitude of the velocity relative to the rotating frame. The energy-like Jacobi Constant allows significant insight into the problem. It is commonly used as a tool to evaluate the accuracy of the solutions by noting any variation in C along a numerically integrated path. A more important use of the Jacobi Constant involves the definition of regions of exclusion. Any solution of Equations (2.17)–(2.19) is associated with a value of C that remains constant for all time. For the velocity along such a path to be real, it must therefore satisfy the expression V 2 = 2U ∗ (x, y, z)−C ≥ 0 (where it is noted that the pseudo-potential is explicitly stated here as a function of position). Any real solution with a given value of the Jacobi Constant C must forever be excluded from the region(s) in position space where U ∗ (x, y, z) < C/2 . These regions of exclusion bound trajectories and mission designers use this knowledge.
2.1.5 Particular Solutions
Since the equations of motion in the CR3BP are non-integrable and no complete analytical solution is available, much research has focused on particular solutions. The first types of particular solutions to consider are the equilibrium points. There are five known equilibrium solutions relative to the rotating frame, determined by solving ∇¯ U ∗ = 0¯ .
These equilibrium solutions, denoted libration points or Lagrange points, were first published separately by Leonard Euler and Joseph Lagrange in 1772 [6]. They repre- sent positions in the rotating frame where gravitational forces balance the centrifugal force resulting from the rotation of the frame. The libration points are numbered L1 through L5. All five points lie in the plane of motion of the primaries. Figure 2.2 (not to scale) illustrates their relative locations. The first three points are aligned along thex ˆ axis and are labeled the collinear points. The interior point, L1 lies between the primaries, L2 lies exterior to the primaries on the far side of P2, and L3 lies on the far side of the larger mass P1. The points L4 and L5 are located at vertices of two 19
Figure 2.2. Location of the Libration Points
equilateral triangles with a common edge. The primaries are the second and third vertex of each triangle as seen in Figure 2.2.
2.1.6 The State Transition Matrix
In nonlinear dynamical systems, it is often impossible to analytically determine a solution arc that will exactly meet desired objectives. A good estimate, perhaps from a lower order analysis, may be available, however. Differential corrections processes are often used to adjust estimated initial conditions to improve a solution. These corrections depend on predictions of the sensitivity of the solution to its initial condi- tions. These predictions of sensitivity are typically quantified in the state transition matrix, derived from a linearization relative to a reference path. As stated previously, the equations of motion (Equations (2.17)–(2.19)) are writ- ten as six first-order differential equations governing the dependent variables; these 20 are defined as elements of the six-dimensional state vectorx ¯ = [x, y, z, x,˙ y,˙ z˙]T . Math- ematically, the vector differential equation is expressed in the form x¯˙ = f¯(¯x). Assum- ing that a reference arc or solution,x ¯n(t) , is available that satisfies the differential equations, i.e., x¯˙ n = f¯(¯xn), a nearby solutionx ¯(t) =x ¯n(t)+ δx¯(t) can be represented via an expansion in a truncated Taylor series such that
f1x f1y f1z f1x ˙ f1y ˙ f1z ˙ ¯ δx ¯ f f f f f f ¯ δy 2x 2y 2z 2x ˙ 2y ˙ 2z ˙ ¯ ¯ f f f f f f ¯ δz ¯ 3x 3y 3z 3x ˙ 3y ˙ 3z ˙ ¯ x¯˙ = x¯˙ n + δx¯˙ = f(¯xn)+ ¯ + ··· (2.22) ¯ f4x f4y f4z f4x ˙ f4y ˙ f4z ˙ ¯ δx˙ ¯ ¯ f5x f5y f5z f5x ˙ f5y ˙ f5z ˙ ¯ δx˙ ¯ ¯ f6x f6y f6z f6x ˙ f6y ˙ f6z ˙ ¯ δz˙ ¯x¯n ¯ where δx¯ is the perturbation relative to the referencex ¯n. When the higher order terms are ignored, a linear system of differential equations is produced for δx¯, i.e.,
δx¯˙(t)= A(t)δx¯(t). (2.23)
The 6 × 6 matrix A(t) is evaluated along the reference trajectory arc and is time- varying. It has the form 0 I A(t)= , (2.24) UXX Ω where I is the 3 × 3 identity matrix,
0 2 0 Ω= −2 0 0 , (2.25) 0 0 0 and ∗ ∗ ∗ Uxx Uxy Uxz U = ∗ ∗ ∗ , (2.26) XX Uyx Uyy Uyz ∗ ∗ ∗ Uzx Uzy Uzz where the subscripts indicate second partial derivatives of the pseudo potential. 21
The solution to the linearized Equation (2.23) is
δx¯(t) = Φ(t,t0)δx¯(t0), (2.27)
where Φ(t,t0) is the state transition matrix (STM). The expression in Equation (2.27)
relates variations in the trajectory at time t to the initial perturbation at time t0. The STM is also described as a sensitivity matrix since it offers a linear prediction of the sensitivity of the trajectory to initial variations. Differentiation of Equation (2.27) and substitution into Equation (2.23) for A(t) yields a differential equation for the STM in the form
Φ(˙ t,t0)= A(t)Φ(t,t0). (2.28)
Since A(t) is evaluated along the nominal path and is time-varying with no analytical solution in the multi-body problem, Equation (2.28) must be numerically integrated
along with the state vector to determine the matrix Φ(t,t0). The initial condition
Φ(t0,t0) is a 6×6 identity matrix, as is apparent from Equation (2.27). The numerical integration to determine the state of the system and the corresponding STM involves the integration of 6 scalar equations for the 6 state components and 36 scalar equations for the STM elements for a total of 42 differential equations.
2.1.7 Differential Corrections and Halo Orbits
The most common application of the state transition matrix for this work is in targeting schemes. Given some reference pathx ¯n(t) with initial statex ¯(t0) and final statex ¯(tf ), a variation in the initial state is sought to shift the final state to a desired target pointx ¯T (tf ). Note that the time of propagation (tf − t0) may or may not be constrained. This shift is accomplished in an iterative differential corrections process. At each step, the linearized variational equations relative to the nominal path are
used to estimate an appropriate variation δx¯(t0). The result is propagated to a final state and the process repeats until the target is reached to within some specified tolerance. 22
The linearized equations relating initial to final variations are, from the equation for first variations in the final state,
∂x¯(tf ) δx¯(tf )= ¯ δx¯(t0)+ x¯˙(¯x(tf ),tf )(tf − t0) , (2.29) ∂x¯(t0)¯ ¯x¯n ¯ where ∂x¯(tf ) ¯ = Φ(tf ,t0), (2.30) ∂x¯(t0)¯ ¯x¯n and this STM is evaluated along the reference¯ path. Of course, the desired change in the final state is simply δx¯(tf ) =x ¯T (tf ) − x¯(tf ). With this substitution, an estimate for the initial variation can be computed. The process must be repeated until the resulting final state is within some specified tolerance of the target. Differential cor- rectors are very effective in computing periodic halo orbits, given reasonably accurate estimates for initial conditions. The numerical algorithm is not unique; and the steps detailed here are similar to those in [9, 17, 19]. Halo orbits are a particular type of periodic solution known to exist in the vicinity of the collinear libration points rela- tive to a rotating coordinate frame [7]. Characteristic of all periodic halo orbits is a perpendicular crossing of thex ˆ − zˆ plane. At any of these crossings,
y =x ˙ =z ˙ = 0. (2.31)
A differential corrections process is initiated via a guess that satisfies Equation (2.31): T x¯(t0) = [x0, 0, z0, 0, y˙0, 0] . This state is propagated forward in time until y(tf ) = 0, that is, the next crossing of thex ˆ − zˆ plane. In general, this second crossing is not
perpendicular, i.e.,x ˙(tf ) =6 0 andz ˙(tf ) =6 0, so variations in the non-zero initial states are sought to achieve a second perpendicular crossing. The number of linear equations in (2.29) can be reduced from 6 to 2. First, to maintain the initial perpendicular plane crossing, three of the initial state components cannot be varied (δy = δx˙ = δz˙ = 0). This constraint removes three equations. Then,
given y(tf ) = 0 and the subsequent requirement that δy(tf ) = 0, the variation in time can be resolved from the second scalar equation in (2.29). Thus, three unknown
variations in the initial state δx0, δz0, and δy˙0 remain to achieve two target conditions, 23
δx˙(tf )= −x˙(tf ), δz˙(tf )= −z˙(tf ). From an infinite variety of solutions, one is selected by fixing an initial state component and the system is solved with two equations in two unknowns. The variations are added to the initial state and the process is repeated until the final state yields a perpendicular plane crossing within some specified tolerance. A periodic halo orbit computed with such a differential corrections scheme appears in Figure 2.3 relative to a rotating frame in the CR3BP. This is a notably large orbit near L2 with amplitudes Az = 36, 346 km, Ay = 43, 599 km, and Ax = 15, 282 km. The Ay and Az amplitudes are defined as the maximum excursions from the xˆ − zˆ andx ˆ − yˆ planes, respectively, and the Ax amplitude is approximately half the difference between the maximum and minimum x−values along the orbit. Periodic halo orbits have periods nearly commensurate with the orbits of the primary bodies in the system. Typically, those in the vicinity of L1 and L2 have periods roughly half that of the primaries. The orbit in the figure possesses a precise period of 14.4955 days. Because the maximum excursion from thex ˆ − yˆ plane occurs in the positive z direction rather than below thex ˆ − yˆ plane, the orbit is defined as northern or class II ; otherwise, it is defined as southern or class I.A southern orbit of exactly the same amplitudes would appear as a reflection across thex ˆ − yˆ plane in configuration space. The projection of the southern orbit in thex ˆ − yˆ plane would appear the same while the projections in the other planes would appear as mirror images across the line z = 0. The direction of motion along the halo is indicated by arrows in the figure. In a northern orbit, the direction is clockwise as seen in they ˆ−zˆ plane; in a southern orbit it is counter-clockwise. 24
x 104
6 4
2 L 1 Moon 0 L y (km) 2 −2 Earth −4 −6
−5 0 5 x (km) x 104
x 104 x 104
6 6 4 4 2 2 L 1 Moon Moon 0 0
z (km) L z (km) −2 Earth 2 −2 −4 −4 −6 −6
−5 0 5 −5 0 5 x (km) x 104 y (km) x 104
Figure 2.3. A Periodic Halo Orbit in the Earth-Moon System 25
2.2 Invariant Manifold Theory
As mentioned previously, this investigation involves approximations of invariant manifolds associated with periodic halo orbits in the CR3BP as well as the use of these approximations in calculating transfers between orbits in three-body systems. Thus, some general background concerning the phase space near a periodic orbit in the CR3BP is necessary for an understanding of periodic orbits and their associ- ated manifolds. In addition, a method for the numerical computation of stable and unstable manifolds is a critical capability and is discussed in this section.
2.2.1 Invariant Manifolds
Consider an autonomous system of nonlinear differential equations
q¯˙ = f¯(¯q), (2.32)
where the state vector isq ¯(t) ∈ Rn and q¯˙(t) is its derivative with respect to the independent variable, t, time. The function f :Λ → Rn is a smooth function (at least
n n twice differentiable) defined on Λ ⊆ R that is said to generate a flow, φt(Λ) : Λ → R . The flow is the set of all trajectories originating from different initial conditions in
Λ. The dependence of the flow on initial conditionsq ¯0 =q ¯(0) can be explicitly demonstrated by defining φt(¯q0) as the trajectory initiated atq ¯0 and propagated for a time t.
An understanding of the global phase portrait generally begins with an exami- nation of the local flow near any equilibrium solutionq ¯eq. Behavior in these regions is approximated by the linear system resulting from retaining the linear terms in a Taylor series expansion about the equilibrium point. The accuracy of these approx- imations generally depends on the degree of nonlinearity in the real system and the size of the region being considered near the equilibrium solution. Letq ¯ =q ¯eq +y ¯, 26 such thaty ¯(t) is the perturbation from equilibrium, and then the linear system is represented as y¯˙(t)= A y¯(t), (2.33) where A = Df¯(¯qeq) is the Jacobian. Sinceq ¯eq is an equilibrium solution of Equation (2.32), the matrix A is constant and the linearized system is solvable. In general, when linearizing near a reference solutionq ¯ref (t) that may be a trajectory parameterized by time, the matrix A may itself be time-varying, A = A(t). However, for constant A, the solution to Equation (2.33) is the superposition of the fundamental solutions determined from the eigenvalues and eigenvectors of A. Thus the solution has the form n − y¯(t)= c eηj (t t0)ν¯ , (2.34) X j j j=1 where t0 is the initial time and ηj andν ¯j are the eigenvalues and eigenvectors of A, respectively. The constant coefficients cj are determined by the initial state. From Equation (2.34), it is clear that the flow corresponding to the linear system is dominated by the structure of the eigenvalues and eigenvectors of the Jacobian evaluated at the equilibrium point. Suppose there are ns eigenvalues with a positive real part and associated with eigenvectors {ν1, . . . , νns }, nu eigenvalues with a negative real part and associated with eigenvectors {νns+1, . . . , νns+nu }, and nc eigenvalues with zero real part and corresponding to eigenvectors {νns+nu+1, . . . , νns+nu+nc }. Then, assuming no repetition of eigenvalues — all eigenvectors are independent — n = ns + n n nu +nc, and R is spanned by the eigenvectors. So, R can then be divided into three s fundamental, independent subspaces: the stable subspace E = span {ν1, . . . , νns }, u c the unstable subspace E = span {νns+1, . . . , νns+nu }, and the center subspace E = span {νns+nu+1, . . . , νns+nu+nc }. These subspaces are invariant, that is, given an initial state that lies entirely within one subspace and propagating forward or backward in time under the linear system of equations, the resulting trajectory will remain within that subspace indefinitely. Given the structure of the eigenvalues in the linear system (with constant A), any trajectory with initial conditions lying within Es will tend toy ¯ = 0¯ as t → ∞. 27
Similarly, any trajectory with an initial condition lying within Eu will tend toy ¯ = 0¯ as t → −∞. Lastly, trajectories in the center subspace will remain within the vicinity ofy ¯ = 0,¯ neither growing or decaying as t → ±∞. Invariant manifolds in the nonlinear system are analogous to the eigenspaces of
s the linear system. The local stable manifold Wloc(¯qeq), associated withq ¯eq is defined
as the set of all trajectories tending toq ¯eq as t → ∞. The local unstable manifold u Wloc(¯qeq), associated withq ¯eq is defined as the set of all trajectories tending toq ¯eq as t → −∞. Though these stable and unstable manifolds are defined to be local to the
s u equilibrium point, they do have global analogs, W (¯qeq) and W (¯qeq), calculated by u propagating the flow forward in time from some initial states in Wloc(¯xeq) or backward s in time from some initial states in Wloc(¯xeq). All three invariant manifolds are tangential to the appropriate eigenspaces of the Jacobian as stated in the Center Manifold Theorem for Flows from Guckenheimer and Holmes [24]:
Theorem 2.1 (The Center Manifold Theorem for Flows) Let f¯ be a Cr vec- tor field on Rn vanishing at the origin (f¯(0) = 0¯) and let A = Df¯(0)¯ . Divide the
spectrum of A into three parts, ns, nc, nu with
< 0 η ∈ ns, ℜ(η) = 0 η ∈ nc, (2.35) > 0 η ∈ n . u s c u Let the (generalized) eigenspaces of ns, nc, nu be E , E , and E , respectively. Then there exist Cr stable and unstable invariant manifolds W u and W s tangent to W u and W s at 0¯ and a Cr−1 center manifold W c tangent to EC at 0¯. The manifolds are all invariant for the flow of f¯. The stable and unstable manifolds are unique, but W c need not be.
s For the application in this work, the important result to note is that Wloc and u s u Wloc are tangent to the linear subspaces E and E , respectively. This allows the natural asymptotic direction of motion toward (away from) an equilibrium point to 28
Figure 2.4. Stable and Unstable Manifolds Tangent to Subspaces
be determined by calculating the orientation of the eigenvectors in the linear system. Such an example is demonstrated in terms of a two-dimensional system in Figure 2.4. Thus, a simple method to determine the natural flow approaching (departing) an equilibrium point is the selection of initial conditions that are near the equilibrium point and shifted along the stable (unstable) eigenvector. The step in the eigenvector direction must be sufficiently small such that the linear assumption is valid. In the figure, the stable and unstable subspaces are both one-dimensional (Es =ν ¯s, Eu = ν¯u). Thus, each stable and unstable eigenvector yields two branches of a manifold trajectory. The stable manifold, for example, is tangent to both +¯νs and to −ν¯s . For notational purposes, W s+ is propagated in the direction of +¯νs and W s− is in the direction of −ν¯s.
If all the eigenvalues of A have non-zero real parts, then the equilibrium point is defined as hyperbolic, and does not possess a center manifold. The existence of the center subspace allows for more diverse flow within the vicinity of the equilibrium point since any bounded motion (including periodic or quasi-periodic orbits) near the point would exist in the center subspace. For example, the collinear libration points 29
in the CR3BP are all non-hyperbolic, and possess four dimensional center manifolds. Nearly vertical out-of-plane orbits, planar Lyapunov orbits, three-dimensional quasi- periodic orbits and periodic halo orbits are all examples of orbits that exist in the center subspace associated with the collinear libration points [7].
2.2.2 Periodic Orbits and Maps
The region of the phase space discussed thus far includes the eigenspaces and manifolds associated with equilibrium points of nonautonomous systems. But the phase space in the neighborhood of periodic orbits themselves is of great interest. For example, the periodic halo orbits near the collinear libration points in the CR3BP also have associated manifolds that asymptotically converge upon the orbits in positive or negative time. These are in many ways analogous to the manifolds approaching and departing the equilibrium points, and they reveal a rich solution space in the multi- body problem that astrodynamicists are only now beginning to exploit for spacecraft mission design. To explore the nonlinear phase space in the vicinity of periodic orbits, it is useful to represent the continuous flow in the vicinity of a periodic orbit as a discrete time map.
Let Γ be a limit cycle that exists as a solution to the system in Equation (2.32). Thus, Γ is an attracting set in Rn to which trajectories converge. Trajectories on Γ are periodic, with a period equal to T ; letx ¯∗ be a point in Γ. Define a hyperplane Σ of dimension (n − 1) atx ¯∗ and orthogonal to Γ . Then,x ¯∗ is a fixed point in Σ and completely defines the periodic orbit. Since Γ is periodic, the flow maps the pointx ¯∗ ∗ ∗ to itself after one period, i.e.,x ¯ = φT (¯x ). This mapping is assumed to be smooth and continuous with respect tox ¯∗, so that trajectories initiated in regions nearx ¯∗ will, when propagated for approximately T time units, again intersect Σ. Consider a small region aroundx ¯∗, U ⊂ Σ, where U is assumed sufficiently small such that initial conditions within U will be guaranteed to yield trajectories with a second intersection in Σ. In particular, they will intersect another region, V ⊂ Σ. The flow, 30
along with Σ, maps U to V , and that mapping is termed a Poincar´emap, P , i.e., V = P (U ). Similarly, the hyperplane Σ is denoted as a Poincar´esection.
The continuous time system described in terms of the flow φT is now described ∗ by a discrete time mapping for points in Σ nearx ¯ :x ¯k+1 = P (¯xk). The map is nonlinear, but invertible and differentiable, i.e., it is a diffeomorphism. Since the map is derived by propagating the flow from one crossing of Σ to the next, the propagation time is roughly equal to the period of the orbit. Thus, the best linear estimate for the propagation of initial conditions in U to their intersections with Σ in V is determined by evaluating the STM after exactly one period of the orbit, Φ(T, 0). This matrix is labeled the monodromy matrix. The monodromy matrix is a stroboscopic (time-sampling) map that has direct application to analyses in this investigation and is used to produce the linear map for the propagation of a state nearx ¯∗ after n periods:
x¯(nT ) =x ¯∗ + Φ(nT, 0)δx¯(0), (2.36) =x ¯∗ + δx¯(nT ), where Φ(nT, 0) = Φ(T, 0)n. The variational equation
δx¯(nT ) = Φ(nT, 0)δx¯(0), (2.37) = Φ(T, 0)nδx¯(0), has the solution n δx¯(nT )= c λnT s¯ , (2.38) X j j j j=1 where λj are the eigenvalues of Φ(T, 0), assumed to be distinct,s ¯j are the eigen- vectors, and the complex coefficients cj are determined by the initial perturbation
δx¯(0) = [c1s¯1 ...cns¯n]. Clearly, the structure of the eigenvalues of the monodromy matrix will determine the stability of Γ .
2.2.3 Orbital Stability
Examination of the flow in the vicinity of an equilibrium point in the continuous time system leads to the local stable, unstable, and center subspaces and, ultimately, 31
their associated global manifolds in the nonlinear system. It is also possible to lin- earize about a reference solution that is not an equilibrium point. In this case, use of a periodic orbit as a reference solution produces an A matrix that is time-varying and periodic with a period T . The first-order linear variational equations are of the following form, y¯˙(t)= A(t)¯y(t), (2.39) wherey ¯(t) again represents the variation relative to the reference solution, in this case a periodic orbit. Recall that the STM is the fundamental matrix solution in the linear system, and that the eigenvalues of the STM after one revolution (the monodromy matrix) reveal stability information about the orbit. A useful theorem appears in Perko [25]:
Theorem 2.2 (Floquet’s Theorem) If A(t) is a continuous T -periodic matrix, then for all t ∈ R any fundamental matrix solution for Equation (2.39) can be written in the form Φ(t, 0) = Q(t, 0)eBt, (2.40) where Q(t, 0) is a nonsingular, differentiable, T -periodic matrix and B is a constant
matrix. Furthermore, if Φ(0, 0) = In then Q(0, 0) = In.
Corollary 2.1 Under the hypothesis of Floquet’s Theorem, the nonautonomous linear system of Equation (2.39), under the linear change of coordinates
ω¯(t)= Q−1(t, 0)¯y(t), (2.41)
reduces to the autonomous linear system
ω¯˙ (t)= Bω¯(t). (2.42)
Since the nonautonomous linear system of Equation (2.39) is equivalent to an autonomous system of the form in Equation (2.42), the constant matrix B yields all of the relevant stability information associated with the periodic orbit, at least in the 32
linear sense. Note that the matrix B need not be unique. From Equation (2.40), the monodromy matrix may be written as
Φ(T, 0) = Q(T, 0)eBT , (2.43) since Q(t, 0) is periodic with period T ,
Φ(T, 0) = Q(0, 0)eBT , (2.44) = eBT .
Hence, the eigenvalues of B, which may be complex, βj = aj + ibj, are the character- istic exponents. These ultimately determine orbital stability. The eigenvalues of the monodromy matrix, the characteristic multipliers of Equation (2.38), are related to the characteristic exponents, i.e.,
βj T λj = e . (2.45)
1 Thus, the characteristic exponents are also written as βj = T ln λj. The real parts
of the characteristic exponents, aj, determine the magnitudes of the characteristic
aj T multipliers, kλjk = e . Table 2.1 relates this information to the stability of the orbit.
Table 2.1. Relating Orbital Stability, Characteristic Exponents, and Characteristic Multipliers
Subspace Characteristic Exponent Characteristic Multiplier
βj T βj = aj + ibj λj = e
Unstable aj > 0 kλjk > 1
Center aj = 0 kλjk = 1
Stable aj < 0 kλjk < 1
Thus, the stability of the orbit, in the linear sense, is determined by the location of the eigenvalues of the monodromy matrix in the complex plane relative to the unit circle. 33
2.2.4 Stable and Unstable Manifolds of Periodic Orbits
The flow in the region of phase space near a periodic orbit can be decomposed into invariant subspaces in a manner similar to the decomposition of the flow in the vicinity of the equilibrium solutions. The invariant manifolds associated with the pe- riodic orbit are related to the invariant subspaces via the Poincar´e(or stroboscopic) map. The monodromy matrix forms the basis for the linearization near a fixed point that is necessary to approximate the map. Given the fixed point,x ¯∗, and any stable, unstable, and center eigenvalues computed from the monodromy matrix (the charac- teristic multipliers) the corresponding eigenvectors approximate the directions of the flow in terms of the directions associated with the invariant manifolds. The following theorem, as stated in Perko, defines the stable and unstable man- ifolds corresponding to a periodic orbit and relates them to the characteristic expo- nents [25].
Theorem 2.3 (The Stable Manifold Theorem for Periodic Orbits) Let f¯ ∈ C1(E) where E is an open subset of Rn containing a periodic orbit x¯ = Γ(t) of the ∗ system x¯˙ = f¯(¯x) of period T . Let φt be the flow of the system so that Γ(t)= φt (¯x ). If k of the characteristic exponents of Γ(t) have negative real parts where 0 ≤ k ≤ n − 1 and n − k − 1 of them have positive real parts then there exists a δ > 0 such that the stable manifold of Γ,
s W (Γ) = {x¯ ∈ Nδ (Γ) |d (φt(¯x), Γ) → 0 as t → ∞ (2.46) and φt(¯x) ∈ Nδ (Γ) for t ≥ 0} is a (k + 1)-dimensional, differentiable manifold which is positively invariant under the flow φt and the unstable manifold of Γ,
u W (Γ) = {x¯ ∈ Nδ (Γ) |d (φt(¯x), Γ) → 0 as t → −∞ (2.47) and φt(x) ∈ Nδ (Γ) for t ≥ 0} is an (n − k)-dimensional, differentiable manifold which is negatively invariant under the flow φt. Furthermore, the stable and unstable manifolds of Γ intersect transversally in Γ. 34
n In the theorem, Nδ(Γ) is a subset of R of size δ around Γ. The function d(φt, Γ) refers to the distance between its two arguments. As in the case of the manifolds for equilibrium points, these global manifolds have
s u local analogs, Wloc(Γ) and Wloc(Γ), that are unique, invariant, differentiable, and that possess the same dimensions as W s(Γ) and W u(Γ). The local stable and unstable manifolds are tangent to the stable and unstable eigenspaces of the monodromy matrix Φ(T, 0) at any given fixed point,x ¯∗, along the orbit Γ [25], that is,
s u E = span {s¯j| kλjk < 1} ,E = span {s¯j| kλjk > 1} , (2.48) where λj are the eigenvalues, or characteristic multipliers, ands ¯j are their associated eigenvectors. If there are k characteristic exponents with positive real parts, then the stable eigenspace has dimension k, yet the stable manifold is defined with dimension (k+1). The eigenspaces are defined local to a particular fixed point along the orbit and identified in the map on Σ; thus, the trajectories that comprise the local manifold are dependent on the choice ofx ¯∗ ∈ Γ. This free choice of a specific fixed point along the periodic orbit adds another dimension to the manifold as trajectories tangent to the k-dimensional eigenspaces at each fixed point along the orbit will combine to generate the (k + 1)-dimensional manifold over the entire orbit. Hence, the dimension of the manifold is always one greater than the dimension of the corresponding eigenspace.
2.2.5 Eigenstructure of the Monodromy Matrix
It is apparent that calculation of the manifolds of interest will depend greatly on the structure of the eigenvalues of the monodromy matrix. This structure is examined and determined to be limited to a particular form. As a consequence of the periodicity of the orbit, at least one eigenvalue of the monodromy matrix must be unity. In addition, Yakubovitch and Starzhinskii [26] demonstrate the time-invariance property of the STM and a significant theorem can be summarized:
Theorem 2.4 (Lyapunov’s Theorem) If λ is an eigenvalue of the monodromy matrix Φ(T, 0) of a t-invariant system, then λ−1 is also an eigenvalue, with the same 35 structure of elementary divisors. Consequently, the spectrum of the monodromy ma- trix of a t-invariant system is skew-symmetric with respect the unit circle (in the complex plane).
Since, for periodic orbits in the CR3BP, one eigenvalue is unity (say λ1 = 1) then a second eigenvalue is 1 as well. Thus,
−1 λj = λ , j = 1, 3, 5 , j+1 (2.49) λ1 = λ2 = 1.
Suppose (as is common for periodic halo orbits) that λ3 and λ4 are complex while λ5 −1 and λ6 are real, with λ5 = λ6 . Then, since the monodromy matrix must be real, ∗ any complex eigenvalue must be paired with its conjugate, implying λ3 = λ4 while −1 simultaneously λ3 = λ4 . This is only possible if λ3 and λ4 are a pair of complex conjugate eigenvalues on the unit circle in the complex plane. Most of the periodic halo orbits in the vicinity of the collinear libration points possess an eigenstructure of this type: two eigenvalues at unity, two on the unit circle, and the final two on the real axis. The eigenvectors associated with the two real eigenvalues yield the stable and unstable eigenspaces. Other configurations are possible and still consistent with Lyapunov’s Theorem including a set of six eigenvalues all on the unit circle. In fact, within the families of halo orbits there exist orbits that lie in a center subspace with no stable or unstable manifolds. These orbits are, thus, linearly stable. The change in stability character- istics along the family of orbits marks a bifurcation point. Other families of periodic orbits (some of different types) are known to emanate from the bifurcation points. Such bifurcations have only been pursued to a limited extent [27].
2.2.6 Computation of the Stable and Unstable Manifolds of Periodic Halo Orbits
To calculate the stable and unstable manifolds associated with periodic halo orbits, the eigenvalues and associated eigenvectors corresponding to the monodromy matrix 36
are used. They are obtained from the monodromy matrix as it is computed at fixed points along the orbit. The procedure is detailed here for the calculation of a stable manifold; the calculation of the unstable manifold is similar. At each fixed pointx ¯∗ ∈ Γ, the local stable eigenvector of the monodromy matrix, ∗ here denoted ass ¯s(¯x ), is computed. Then, the vector is normalized with respect to its position coordinates, that is,
s¯ Y¯ s = s . (2.50) s2 + s2 + s2 p sx sy sz The best guess for an initial state on the local stable manifold is then
∗ s x¯0 =x ¯ ± d Y¯ , (2.51) where a scalar distance d is selected to define the step in the direction of Y¯ s. The step must be small enough to justify the linear approximation. In the case of the Sun- Earth system, for computation of stable/unstable manifolds associated with periodic halo orbits near L1 or L2, d = 200 km is generally used. In the Earth-Moon system, d = 50 km is a satisfactory value. Note that the eigenvector defines a one-dimensional directional orientation for the local manifold; the positive or negative sense associated with the computed direction indicates that the stable manifold collapses onto Γ from two different directions. The initial state in Equation (2.51) is propagated backward in time to yield the trajectory on the stable manifold that is associated withx ¯∗ (if calculating an unstable manifold, the propagation would be forward in time). Numerous fixed points are identified along the orbit and the procedure is repeated. Note that the monodromy matrix need not be recomputed at each fixed point, since the stable eigenvector can ∗ ∗ be propagated from one fixed point to the next using the STM. Letx ¯1 andx ¯2 both be fixed points along the same orbit, corresponding to times t1 and t2, respectively. ∗ ∗ ∗ ∗ Then, ifs ¯s(¯x1) is known,s ¯s(¯x2) can be computed ass ¯s(¯x2) = Φ(t1,t2)¯ss(¯x1) [9]. All of the trajectories taken together comprise the manifold that is seen to resemble a surface in the shape of a tube approaching (departing) the periodic orbit. 37
Stable and unstable manifold tubes in the Earth-Moon system appear in Fig- ure 2.5. Half of the tubes are associated with a periodic orbit in the vicinity of L1. The stable (red) and unstable (blue) tubes are propagated in both the positive and negative directions. A similar set of tubes is associated with a periodic orbit near
L2. The two periodic halo orbits in the figure are selected to possess the same Jacobi Constant. This value of C is, then, also shared by all of the trajectories on the man- ifolds. The regions of exclusion (yellow) defined by the Jacobi Constant also appear in the figure. Some observations concerning the dynamical structure in Figure 2.5 are notable. The manifolds are separatrices in the system, thus, they divide the flow into regions. Trajectories initiated within the tubes (in six-dimensional space) will remain within the tubes and travel “through” the halo orbits. These are termed transit trajectories. Low-energy transfers (those that require little energy to change the Jacobi Constant) between regions inside and outside of the Earth-Moon system can be accomplished using the tubes and halo orbits as “gateways.” Thus, the arrows on the figure in- dicate the direction corresponding to the natural flow and immediately suggest the foundation for low-energy transfers [8]. 38
Figure 2.5. Manifold Tubes in Earth-Moon CR3BP Collapsing on Two Halo Orbits 39
2.3 Transition to an Ephemeris Model
Solutions determined in the restricted problem offer insight into the natural mo- tions that exist in more realistic multi-body systems. However, they lack the fidelity of solutions propagated directly in a multi-body simulation. Thus, trajectory arcs stemming from the coupling of the Earth-Moon and Sun-Earth circular restricted problems are routinely transitioned to a four-body model. This step guarantees the robustness necessary for actual spacecraft mission design.
The Generator suite of programs, developed at Purdue University, is used ex- tensively for work in the multi-body model. Within Generator, multi-body systems are simulated using Newtonian equations of motion. Ephemeris data specifies the motion of all gravitational bodies in the system, and solar radiation pressure can be included in the model [28] (though it is not incorporated in this work). Multiple op- tions are available for planetary and lunar ephemeris data. Integration of spacecraft trajectories is generally performed with a Runge-Kutta 8/9 integrator (though other integrators are also available). The program incorporates the definition of a restricted three-body system, however, the equations of motion for the spacecraft are integrated relative to “P2” in the standard inertial Mean of J2000 coordinate system. Although the integration is performed in inertial coordinates, transformations are available such
that the data may be output in the familiar P1-P2 rotating frame (as well as others). This transformation is defined at each epoch based on the instantaneous states of the primaries. The locations of the collinear libration points are also defined instanta-
neously as equilibrium states between P1 and P2 in a rotating frame. In the case of
a Sun-Earth/Moon system, the Sun is defined as P1 while “P2” is a body equal in mass to the sum of the mass of the Earth and Moon, and located at the barycenter of the Earth/Moon system. This model more correctly locates the libration points in the Sun/Earth/Moon gravitational environment [29].
Precisely periodic halo orbits do not exist in the four-body model. However, quasi- periodic Lissajous orbits exist in the vicinity of the collinear libration points and are 40
determined in Generator using a two-level differential corrections process described in Howell and Pernicka [20]. The process is initiated with a series of points that are seven-dimensional states (position plus velocity plus time) along some estimated ref- erence path, one that is generally determined in another model such as the CR3BP. These are termed “patch points.” If a path is propagated forward (or backward) in time from any individual patch point state, it will not reach the next (or previous) patch point. Thus, without any correction, the path determined by propagating from each patch point toward the next is discontinuous in both position and velocity. The first step in the corrections process uses a simple differential corrector to compute a path that is continuous in position; velocity continuities still exist. Then, a second corrections process is employed to determine shifts in patch point positions that will lower the velocity discontinuities. Both steps use state transition matrices to define linear relationships between the state variations. The process iterates until a tra- jectory is produced that is continuous in both position and velocity, to within some design tolerance. Such a trajectory is considered a natural motion in the system. It is possible to define a patch point as a maneuver location such that the velocity discontinuity will not be entirely eliminated, but constrained to a maximum magni- tude. Such a trajectory is not a natural motion, but consists of natural arcs linked by constrained maneuvers.
The initial set of patch points may be determined from any reference path. One option in Generator is to define the points based on an approximation to a Lissajous orbit near L1 or L2. Richardson and Cary [16] offer one possible third-order approx- imation for a Lissajous trajectory. Alternatively, states computed along a periodic halo orbit in the CR3BP are also successful starting points. Recall, a periodic halo orbit in Figure 2.3 is propagated in the Earth-Moon CR3BP in the vicinity of L2. The
amplitudes of the halo orbit are Az = 36, 346 km, Ay = 43, 599 km, and Ax = 15, 282 km. The trajectory in Figure 2.6 is a quasi-periodic Lissajous orbit generated in an ephemeris model from patch points obtained directly from the periodic halo orbit in Figure 2.3. The final orbit in Figure 2.6 is computed over 12 revolutions, corrected 41
x 104
5
L 1 Moon 0
y (km) L Earth 2
−5
−5 0 5 x (km) x 104
x 104 x 104
5 5
L 1 Moon Moon 0 0 z (km) L z (km) Earth 2
−5 −5
−5 0 5 −5 0 5 x (km) 4 y (km) 4 x 10 x 10
Figure 2.6. A “Halo-Like” Lissajous Trajectory 42 in the four-body model in Generator that includes the gravity of the Sun as well as the Earth and Moon. The orbit in Figure 2.6 is plotted relative to the Earth-Moon rotating frame, with the motion indicated by arrows, similar to Figure 2.3. The lo- cations of the libration points indicated in the figure are actually obtained from the CR3BP, though they suggest an accurate visual representation of the locations of the equilibrium points. Of course, in the rotating P1-P2 frame — based on ephemeris data — the locations of the libration points in the system are not fixed, but oscil- late slightly as the distance between the primaries contracts and expands with the primary motion. The amplitudes of the quasi-periodic Lissajous orbit remain only ∼ ∼ roughly equivalent to those of the periodic halo orbit: Az = 36, 000 km, Ay = 43, 000 ∼ km, and Ax = 15, 000 km. In fact, since the representation of a quasi-periodic orbit does include a mixture of frequencies, it is not surprising that the amplitudes vary with time. Recall that the period of a libration point orbit is approximately equal to half the period of the primaries. Thus, two revolutions in the libration point orbit are required to be commensurate with the motion in the primary system. This is most apparent in an elliptic restricted three-body model or an ephemeris model. Since a quasi-periodic Lissajous orbit appears to close after two revolutions, the “period” is now approximated over this time frame. For this Earth-Moon example, the time of flight over two revolutions is ∼ 29.05597 days. Note that the time corresponding to a revolution varies as well and, again, this is nearly commensurate with the Moon’s orbital period around the Earth. This example demonstrates a general method to transition solutions from the CR3BP directly into an ephemeris four-body model. The process is employed to determine quasi-periodic orbits as well as other, more general, solution arcs.
Although Lissajous orbits do not meet periodicity requirements, associated global stable and unstable manifolds can be approximated with the numerical procedure used for periodic orbits with small modifications. The eigenstructure of the STM after one two-revolution period is employed to yield approximations to the stable and 43 unstable manifold directions, just as the eigenstructure of the monodromy matrix is employed in the CR3BP. 44
3. Manifold Approximations Using Cells
Approximating the manifold tubes for application to mission design problems — such as transfers — is the first step in this investigation. The approximation process must be straightforward and offer accurate representations of the position and velocity states. It is also used to represent the volume that includes multiple manifold surfaces that aid in determining a specific halo orbit for a given scenario. Ultimately, the procedure is also applicable to error analysis, recovery, and design for contingencies. Recent studies suggest that these types of problems greatly benefit from another parameter such as halo orbit size; if the manifold is represented in an easily accessible form, analyses can be accomplished in a automated manner [30,31].
3.1 Cell Approximations
Manifolds are represented in a relatively straightforward cell structure such that each cell includes a polynomial model of one or more manifolds in a small volume. Thus, the potential set of solutions may incorporate manifolds from various halo orbits. The cell structure then includes volume elements that can be searched for intersections with other single trajectories or intersections with other cells. Consider a stable (unstable) manifold tube that is computed numerically from initial conditions near a periodic halo orbit. Given this initial surface, the manifold tubes that correspond to several additional, nearby halo orbits fill a volume in space as the individual manifold surfaces wrap around each other. Larger tubes are generally associated with larger halo orbits. The propagation of the tubes is limited to the region near the halo orbits such that the tubes are distributed from small to large, each successive tube contained within the next larger tube. Two such stable manifold tubes appear in Figure 3.1 in the vicinity of Sun-Earth L2 libration point. The larger 45
Figure 3.1. Each Tube is Contained within a Successively Larger Tube
tube is associated with a periodic halo orbit of amplitude Az = 280, 000 km and the
smaller corresponds to an out-of-plane amplitude of Az = 123, 000 km. This “volume” of stable manifold tubes is, then, associated with the distribution of unstable halo orbits in the specified region of this family. To model any particular tube, exploit the fact that over small regions the stable manifold associated with a particular halo orbit in the circular restricted three-body problem is nearly flat and can be represented in position and velocity by low order functions. A volume of space through which several manifold surfaces pass and that is sufficiently small to be approximated in this manner is denoted here as a cell. In Figure 3.2, cells are defined that encompass a set of three stable manifolds, shown in red, green, and blue,
associated with halo orbits near L1 in the Sun-Earth system. Within a cell, position data obtained from a single manifold tube are approximated using the fit function from Mathematica°c to produce an analytical function of a surface. This process is repeated, allowing the cell to enlarge to include other nearby manifolds, until surface approximation functions for several manifolds passing through the cell are available. With this same position data, velocity data are also used to generate a single fit of each velocity component within the cell as a function of position. Thus, if n halo orbit manifolds pass through a single cell, n fits to approximate the manifold surfaces are produced as well as three fits to approximate velocity components as a function of position. In visualizing this structure, it is important to recall that a family of halo orbits is continuous. Thus, the distribution of manifold surfaces throughout the cells is actually continuous and associated with the continuous distribution of 46
halo orbits. While surface functions are available to approximate only n manifolds, the existence of other manifolds within the cell is implied and velocity components on these additional surfaces can also be approximated. Knowledge of the subset of manifolds used to generate the cells, as well as their associated periodic halo orbits, can also be used to construct a model that maps points in cells to manifolds outside the subset and, thus, to other halo orbits in the family.
Figure 3.2. Cells are Defined to Contain Small Regions of Manifold Trajectories
A small section of M data points from a manifold tube do indicate some amount of curvature (Figure 3.3). The curvature and orientation of any such section of a surface is generally unpredictable. Nevertheless, a coordinate set must be defined for functional representations of each surface; the choice of coordinates must be designed to exploit the local characteristics of the surface. Therefore, the data is represented in terms of a spherical set of coordinates, r, θ, and φ, where r is measured from an origin point Q and the angles θ and φ are measured from reference axes fˆ1,2,3 in a “cell frame” defined at that origin. A functional approximation for the surface passing through the points is denoted asr ˜(θ, φ), where the symbol ‘˜’ indicates a functional representation rather than an actual coordinate. To encourage less variation in the functionr ˜ across the cell, the origin of the cell frame, Q¯, is placed in a location where the difference between the minimum and maximum value of r will be relatively small (Figure 3.3). 47
Figure 3.3. Manifold Data is Transformed to a Cell Frame with Origin Q¯
Within the cell frame, the axis fˆ2 is directed from Q¯ toward the mean center of
the data. Other axes, fˆ1 and fˆ3, are defined to complete a right-handed set in the steps that follow. Every point in the cell is defined in terms of its state,
T q¯j = [xj, yj, zj, x˙ j, y˙j, z˙j] , (3.1) T = [r ¯j, v¯j] ,
as expressed in terms of the rotating coordinate system. The angle between fˆ1 and
the vectorρ ¯j =q ¯j − Q¯ is defined as θj, and φj is the angle between fˆ2 and the
projection ofρ ¯j in the fˆ2 − fˆ3 plane (Figure 3.4). Thus, a cell, shaped as a spherical section, is entirely defined by three types of information: the point Q¯; the elements
R f of a direction cosine matrix C = [fˆ1, fˆ2, fˆ3] (orientation of the fˆ frame relative to the standard rotatingx ˆ − yˆ− zˆ frame of the CR3BP); and, all of the upper and lower
bounds on the data points as defined in terms of spherical coordinates: rmin, rmax,
φmin, φmax, θmin, and θmax. 48
Figure 3.4. A Manifold Data Point Represented in Spherical Coordinates Relative to a Cell Frame
The location of the origin Q¯ is determined from an analysis of the points in the T cell. For every point in the cell,q ¯j = [r ¯j, v¯j] , the curvature of the particular arc at that location is computed as kv¯j × a¯jk κj = 3 , (3.2) kv¯jk where dv¯ a¯ = j . (3.3) j dt
Thus, the radius of curvature is defined as Rj = 1/κj. A unit vector tangent to the
path atq ¯j is computed as v¯j Tˆj = . (3.4) kv¯jk A unit normal vector is determined from the expression R Nˆ = i a¯ − a¯ · Tˆ Tˆ , (3.5) j 2 ³ j ³ j ´ ´ kv¯jk and a unit vector in the binormal direction follows as
Bˆj = Tˆj × Nˆj. (3.6) 49
The center of curvature for the pointq ¯j is 1 c¯j =q ¯j + N.ˆ (3.7) κj Thus, an “average normal vector” for all the data points in the cell is then defined as
1 M N˜ = Nˆ , (3.8) M X j j=1
where M is the total number of points in the cell. An mean scalar radius of curvature is 1 M R = R , (3.9) M X j j=1 A mean center of the data points is computed, i.e.,
1 M q˜ = q¯ . (3.10) M X j j=1
Then an “average center of curvature” for the cell is defined as
Q¯n =q ˜ + RN˜ (3.11)
Another point,
Q¯b =q ˜ + RB,˜ (3.12) lies in the direction of an “average binormal vector”, B˜, defined similarly to N˜. Either of these points (Q¯n or Q¯b) serve as a potential origin for the cell frame. Consider Figure 3.5. The distance from Q¯ to data point j is r = q¯ − Q¯ . This distance is j ° j ° ° ° computed between each candidate origin and each point across the complete data set, and a variation, δr = rmax − rmin, is determined. Then δrb corresponds to the choice of origin Q¯b in the direction of B˜ and δrn results from the use of Q¯N in the direction of N˜. The decision between the two potential origins is based on a comparison of these variations across the data. A smaller variation will produce a smoother analytical representation within the cell. Compare the two cells that appear in Figure 3.6. Each cell could be produced from the same data that appears in Figure 3.5. In
Figure 3.6(a), the cell is created using an origin at Q¯b. The variation δrb is less than 50
Figure 3.5. The N˜ and B˜ Vectors Placed at the Center of a Section of Data Points Directed Toward Two Potential Cell Origins
δrn, corresponding to a cell with an origin in the direction at Q¯n (Figure 3.6(b)), i.e.,
δrb < δrn. Therefore, for a smoother functionr ˜(θ, φ), the origin Q¯ = Q¯b is selected for this set of data. This selection process is undertaken for every data set along the manifold.
Once the origin, Q¯, is identified, the specific axes corresponding to the cell frame
N˜ fˆ2×B˜ are defined. If Q¯n is the origin, then fˆ2 is defined such that fˆ2 = − and fˆ3 = . |N˜| |fˆ2×B˜| B˜ fˆ2×N˜ Otherwise, fˆ2 = − and fˆ3 = . The remaining unit vector, fˆ1 = fˆ2 × fˆ3, |B˜| |fˆ2×N˜| completes a right-handed triad. Next, the data is transformed to a spherical set of coordinates relative to the cell frame, where the spherical coordinates θ and φ are defined to lie in the range −π < θ < π and −π < φ < π. The desired result is a cell in the shape of a spherical section in which surface approximations of manifold data can be expressed by a simple functionr ˜(θ, φ). The choice of axis directions, 51
Figure 3.6. (a) Using Q¯b as the Origin of a Cell Frame (b) Using Q¯n as the Origin of a Cell Frame 52
with fˆ2 directed from the origin Q¯ towards the data set, avoids discontinuities (such as angular boundaries, e.g., when θ changes from −π to π), that hinder attempts at automating the search process for functional approximations. Once position data within each cell have been transformed to spherical coordi- nates, a functional approximation of the surface is calculated. The routine fit in Mathematica°c [32] performs a least squares fit on the data to determine the coeffi- cients ai for the following surface function:
2 r˜(θ, φ) = a0 + a1 cos(φ)+ a2 cos (φ)+ a3 cos(θ)+ a4 cos(φ) cos(θ)+ 2 a5 cos (θ)+ a6 sin(φ)+ a7 cos(φ) sin(φ)+ a8 sin(φ) cos(θ)+ (3.13) 2 a9 sin (φ)+ a10 sin(θ)+ a11 cos(φ) sin(θ)+ a12 cos(θ) sin(θ)+ 2 a13 sin(φ) sin(θ)+ a14 sin (θ).
This function is essentially a truncated Fourier series in two variables. Of course, the velocity states that correspond to all of the position states are also available. The same standard Mathematica°c routine is incorporated to fit the veloc- ity data within the cells. The velocity components are fit as third-order polynomial functions of position in the standard rotating coordinates associated with the circular restricted three-body problem, that is,
2 3 2 v˜x,y,z(x, y, z) = a0 + a1x + a2x + a3x + a4y + a5xy + a6x y+ 2 2 3 2 a7y + a8xy + a9y + a10z + a11xz + a12x z+ 2 2 2 2 3 a13yz + a14xyz + a15y z + a16z + a17xz + a18yz + a19z . (3.14) This form is chosen for a good level of accuracy, though lower order functional forms have been investigated. In Figure 3.7, one cell, originally defined over a section of a single manifold, appears with three surfaces as a result expanding the boundaries of this cell to contain isolated sections of two neighboring manifolds. The curvature of the surfaces is apparent, as well as the spherical shape of the cell. The flow of the velocity field within the cell is indicated by arrows. The fits are reasonably accurate everywhere except very near the primary body where the curvature of the actual manifold is high. Along the manifolds, a high 53
Figure 3.7. Manifold Surfaces and Velocity Flow Approximated Within a Cell
curvature region requires the use of smaller cells within which the manifolds can be approximated as nearly flat. Also, the volume of space contained within a cell should not extend too far beyond the volume defined by the manifold. Consequently, cell sizes and shapes are tailored to the local shape of the manifold volume. As a demonstration, consider a stable manifold tube associated with a halo orbit with ∼ out-of-plane amplitude Az = 840, 000 km in the vicinity of Sun-Earth L1 libration point as it appears in Figure 3.8. Cell approximations are created in this example to represent a set of manifold tubes associated with periodic halo orbits in this region ∼ ∼ ranging in size from Az = 800, 000 km to Az = 880, 000 km (the cells themselves 54
Figure 3.8. Sun-Earth L1 Manifold Position and Velocity Fit Errors
appear in Figure 3.2). The maximum and median position and velocity errors of the fits are compared in the figure with numerically computed data along the actual manifold surface. Median errors are typically less than 500 km, and maximum errors typically less than 5000 km; the most notable exceptions are locations very near the Earth where the error actually extends beyond the range of the display. Also, the median and maximum percentage errors in the velocity functions are presented and the velocity errors as percentages are typically very small. This accuracy assessment is qualitative, but the results are sufficient for preliminary estimation of manifolds in a trajectory design process. A search for more quantitative metrics for the goodness of fit in these estimations is ongoing. 55
3.2 Cell Creation
3.2.1 Division of Manifold Data
Approximation of an entire volume of manifolds involves slicing the volume into many cells. In general, the accuracy of the approximation within a volume depends upon the smoothness of the data (i.e., that the points are on smooth surfaces). On a small enough scale, manifold surfaces will always appear smooth, thus, cell volumes are not uniform in size but vary to accommodate the curvature of the surface. Along the manifolds, a high curvature region requires the use of smaller cells so that the approximations remain accurate. The volume of space contained in a cell should not extend too far beyond the volume defined by the manifold data. Also, the cells are positioned and sized to overlap such that functional approximations of data within one cell transition smoothly with those of neighboring cells. This promotes continuity of the analytical functions over the combined volume, and, thus, the group of cells collectively represents a better global approximation of the volume of the data. A sample set of stable manifold trajectories associated with three large periodic halo
orbits (Az ranging from ∼ 800, 000 km to ∼ 880, 000 km) near L1 in the Sun-Earth system appears in Figure 3.2. The tubular shape of the overlapping surfaces formed by the manifold paths is evident. This set of cells has been designed to contain the three manifold surfaces in appropriately sized sections for analytical approximation.
The creation of the cells, such as those in Figure 3.2, begins with a single tube of manifold trajectories, each propagated to its first periapsis point near the second primary. Any other stopping criteria for the propagation would also be acceptable. To create the cells, a manifold tube is sliced into two types of sections (Figure 3.9). The first section (A) is formed by truncating the tube close to the periodic orbit. That is, the manifold trajectories are computed for a specified length of time from their initial points near the halo orbit. This length along the manifold is denoted as a “band.” In the first band, the time of propagation is close to the period of one revolution of the halo orbit about Li. The band is geometrically divided into a number of sectors 56
Figure 3.9. Dividing a Tube into Cells
Figure 3.10. Dividing a Band Into Sectors 57
(typically 64) as viewed in they ˆ − zˆ plane. Each sector is centered at the libration point or relative to a point further along the stable manifold that is associated with the libration point itself. Figure 3.10 displays the sectors that comprise the first band from the tube in Figure 3.9. The data within each sector are used to define a new cell and frame. This process is then repeated a predetermined number of times, adjusting the propagation times along the tube, to create two or three additional bands. Each band is defined with comparable width and each is similarly subdivided into cells. The second section, section (B) in Figure 3.9, farther from the halo orbit, is comprised of the rest of the tube. The method of cell creation is modified in this second type of section to more closely follow the evolution of the manifold trajectories. A limited number of neighboring trajectories on the manifold are collected and denoted here as a “ribbon.” They are computed from the last band to periapsis and sliced evenly in distance along their length. Each slice of data from the ribbon is used to define a new cell. The process is repeated using the last trajectory of the previous ribbon as the first trajectory of a new ribbon until the entire the tube is represented.
In Figure 3.9, a set of cells is generated from a stable manifold tube associated ∼ with a Sun-Earth L2 halo orbit of magnitude Az = 170, 000 km. The first band of data near the halo orbit is created from the data obtained by propagating the paths backward in time as they depart the halo orbit. The time interval is equal to 0.85 × P (the period of the periodic orbit). Two additional bands are created by collecting additional data points over two subsequent time intervals, each each equal to 0.08 × P. After three bands have been sliced off of the tube near the halo orbit, the rest of the tube is divided into ribbons, each comprised of five trajectories. Along the length of the ribbons, slicing of the data occurs in intervals of 150, 000 km, a parameter that is adjusted for different tubes.
Note that before this step is complete, some cells may require resizing to maintain accuracy. A maximum volume is selected and any cell that exceeds the maximum volume is sliced in half along the largest of either its θ or φ dimensions. It is also important to note that a minimum amount of data from a minimum number of 58
trajectories must always be present in a cell to ensure that the surface fits are accurate. One hundred points has been an acceptable minimum number, thus far, to adequately define a cell, and a minimum of five paths within a cell yields successful results. To meet these minimums, additional trajectories along the manifold tube can be computed with substantially more data points. The data can be added to any cell that does not possess the minimum number of points or trajectories. These minimums do exceed the saturation conditions for the fits and in the future a design of experiments approach may be investigated.
3.2.2 Cell Expansion
After initial cell creation, the cells contain the data corresponding to only one manifold tube. However, the ultimate goal is to approximate position and velocity states corresponding to any locations within the volume and correlate them with a particular halo orbit. To develop such approximations, data from a finite number of actual manifold tubes that pass through the volume are incorporated. Data from additional manifold tubes can be added to the cells; in this work, the cells always contain data from at least three complete tubes. This expansion of the cells to fully contain additional tubes can be performed with certain caveats. The cell shapes are originally tailored to the original tube shape and size, so the trajectories of any other manifold tube will not be fully contained within their volumes. Therefore, to contain as much data as possible from a nearby tube, the cells may be required to increase in size. However, the fitting process is premised on the idea that manifold data can be approximated only over small regions, so cell growth is limited to encompass only the new tube data.
A single cell originally contains all manifold data from one tube within a bounded volume. The data of a second tube is transformed to the original cell’s spherical coordinate frame, and the angular bounds of the cell are increased a small amount to include the new data. The growth is strictly monitored. Any part of the new 59 tube’s data that fits within the new angular bounds of the cell is considered as data for a second surface within the cell, regardless of the radial component — the radial bounds of the cell will grow as much as necessary to encompass new data. This process also ensures the overlap of cells, necessary for continuity in the global approximation. The amount of overlap increases with the growth in radial bounds. As each cell grows in size to include new manifolds, all previously included manifolds are re-evaluated for data points that fall within the resized cell boundaries to ensure that each cell contains as much data as possible for an accurate approximation. All the bounds of the cell are then reset to reflect the increase in the volume of the stored data. An example of multiple Sun-Earth L1 manifold tubes that have been placed within the ∼ same set of cells appears in Figure 3.11. They range in size from Az = 800, 000 km ∼ to Az = 976, 000 km. Originally, only the three tubes that appear in Figure 3.2 are placed in the cells. In Figure 3.11, two additional tubes have been accommodated, evidenced by the data of additional tubes in magenta and light blue.
Figure 3.11. Five Tubes through a Set of Cells 60
Creating and expanding the cells produces some cells with “vacant” areas. These areas are far removed from the data points and the velocity fits will be less accu- rate. This problem is addressed by continuing to add more manifold tubes without allowing further growth of the cells, thereby expanding the data within each cell into the vacant areas without expanding the cell itself. This is accomplished while main- taining the minimum number of points and trajectories. (Figure 3.11 includes five manifolds through the cells, though it was originally created for only three.) If, when determining the amount of data from a new manifold that lies within the bounds of a single cell, the minimum numbers of points and trajectories are not met, then the data for that surface does not get added to the cell. So some, generally larger, cells may contain more manifold tubes from a wider range of halo orbits than other, smaller, cells, while each still contains the minimum acceptable quantity of data. However, awareness of the vacant areas is generally sufficient to allow effective use of the cells. It is also possible to approximate the associated manifolds of a wider range of halo orbits by stacking cells, essentially implementing the cell creation process for the manifolds of one set of halo orbits, then creating entirely new cells for a larger or smaller set and repeating.
3.3 Applications Using Cell Approximations
3.3.1 Determination of a Destination Halo Orbit
The cells and approximations are created to determine transfers from one region of space to another. A particular application is a transfer originating from some point within a cell to some destination halo orbit, h¯(t), where the periodic halo orbit itself may be a design parameter. Determination of transfers departing halo orbits via unstable manifolds that are approximated with cells is a similar application. The fit functions corresponding to velocity data clearly yield the estimated components of velocity necessary at any point in the cell to approach a halo orbit. Since the functions are only approximations, it is still necessary to determine the destination 61
Figure 3.12. Locating the Intersection Point within a Cell
periodic orbit h¯(t) and use a corrections process to produce a complete transfer path. The information contained in the fit functions associated with the cell is used to determine an ideal destination halo orbit for the transfer.
Assume that some incoming spacecraft trajectory intersects a cell at a pointr ¯int identified via spherical coordinates (r, θ, φ) in the cell frame. This point may be inside, outside, or somewhere between the surfaces defined within the cell that corre- spond to actual, numerically computed manifold tubes as is apparent in Figure 3.12. To compare its location relative to the existing surface fits, the n surface functions r˜i(θ, φ),i = 1 ...n, are evaluated where θ and φ are the angular coordinates ofr ¯int. 62
Recall that each surface, and thus each value of the functions r˜i at the coordinates
(θ,φ), is associated with a specific halo orbit, h¯i(t). It is assumed thatr ¯int is likewise associated with another member of the halo orbit family, h¯(t). Since these halo orbits are periodic, each can be characterized by a single fixed point along the orbit. The six-dimensional state vector as the periodic orbit crosses the thex ˆ − zˆ plane is used ¯ i i i T th to identify the orbit. For example, hi(t0) = [x0, 0, z0, 0, y˙0, 0] corresponds to the i periodic halo orbit and can serve as a set of initial conditions to numerically compute the periodic orbit. Thus, each of the non-zero initial conditions of the n known halos,
i i i T [x0, z0, y˙0] , are associated with the functionsr ˜i(θ, φ) within the cell. In general, the halo orbit initial conditions are expressed as quadratic functions of the variable r at the location (θ, φ) within the cell, i.e.,
2 x0 = ax0 + ax1r + ax2r , 2 z0 = az0 + az1r + az2r , (3.15) 2 y˙0 = ay˙0 + ay˙1r + ay˙2r .
To estimate initial conditions corresponding to h¯(t), a specific halo orbit associated
withr ¯int, it is necessary to determine the coefficients of Equation (3.15). The relation- ships in Equation (3.15) are applied to each surface fit within the cell that includes
r¯int. The coefficients can be determined by solving the following three matrix equa- tions, 2 1 1r ˜1 r˜1 ax0 x0 : : : ax1 = : , (3.16) 1r ˜ r˜2 a xn n n x2 0 2 1 1r ˜1 r˜1 az0 z0 : : : az1 = : , (3.17) 1r ˜ r˜2 a zn n n z2 0 2 1 1r ˜1 r˜1 ay˙0 y˙0 : : : ay˙1 = : . (3.18) 1r ˜ r˜2 a y˙n n n y˙2 0 63
If the number of surfaces n ≥ 3, this is accomplished with a least squares fit. If n = 2, then a less accurate linear fit must suffice. However, an effort has been made to include at least three manifold surfaces in every cell.
The initial conditions for a halo orbit calculated by this process are only estimates and, thus, require correction before the final transfer can be attempted. However, this concept of parameterizing halo orbit sizes to the surfaces in the cells, by using their initial conditions, is essential to the process of generating transfers.
In the example in Figure 3.12, the pointr ¯int lies on a numerically computed stable manifold associated with a periodic halo orbit near the Sun-Earth L1 point. The actual barycentric initial conditions of the halo orbit in thex ˆ − zˆ plane are x0 = 148, 024, 188 km, z0 = 945, 481 km andy ˙0 = 380.66 m/s. Based on the location ofr ¯int relative to the manifold surface fits within the cell, the extrapolated initial conditions for a destination halo orbit that can be reached from the givenr ¯int are x0 = 148, 024, 182 km, z0 = 945, 413 km andy ˙0 = 380.71 m/s. These extrapolated initial conditions are sufficiently accurate to allow a differential corrections process to easily converge on a periodic halo orbit. The result is very close to the destination halo associated with the actual manifold passing throughr ¯int.
3.3.2 Saving Computation and Storage
The information that is necessary to completely describe a cell with sufficient detail to fully approximate a volume of manifolds based on the data of n computed manifolds now consists of the following: (a) 12 elements to describe the cell location and orientation of the cell frame (the 3 coordinates of the reference point Q¯ and the 9 elements of the direction cosine matrix RCf relating the frame to the standard rotating CR3BP frame); (b) 6 elements describing the maximum and minimum boundaries of the cell in spherical coordinates; (c) 1 set of 20 coefficients for the velocity fit; (d) n sets of 15 coefficients for the surface fits; and (e) n sets of 3 non-zero halo orbit initial
i i i T conditions, [x0, z0, y˙0] . In one example, approximating a volume of stable manifolds 64
associated with periodic halo orbits near the Sun-Earth L2 point, three manifolds are propagated to periapsis, with over 150 trajectories computed on each surface, and the data is divided into 283 cells for approximation. Direct storage of the numerically generated manifold data consistent with the requirements of mission designers to determine transfer arcs requires 183 megabytes of storage space. Of course, this is a small subset of the complete solution space. Storing the cell approximation data (fit coefficients and cell overhead) for the 283 cells requires only 2.5 megabytes. In addition, when the selection of the halo orbit is a design parameter in a transfer problem, using the cells allows a greater flexibility by allowing the designer to easily determine approximations of transfers to halo orbits, and search the transfer space, without requiring time-intensive recomputation of manifolds for every candidate halo considered in the search. Automation of the design process is the ultimate goal. 65
4. Manifold Approximations as the Basis for Transfers
Transfers from one multi-body system to another are the focus of a number of recent investigations [1–4,33]. An efficient method to compute such transfers is sought using the cell approximations of the manifold surfaces. The ultimate objective is to use the approximations as an initial step in producing the transfers in full, 3-D ephemeris models that may include other perturbations as well. A method is first detailed for using the cells to effect a transfer from an arbitrary point within a cell to some halo orbit; the transfer path will approximate a manifold. Then, transfers between halo orbits in different multi-body systems are considered. In particular, trajectory designs that depart halo orbits in the Earth-Moon system and arrive in L2 halo orbits in the Sun-Earth system are examined.
The problem of transitioning the transfers to a full, 3-D ephemeris model is solved in two steps. First, an approximate solution is obtained in the CR3BP using the cells. Then, the solution is transferred to a more complete model, using the software package Generator, developed at Purdue University [28]. The result is an end-to-end trajectory from an Earth-Moon libration point orbit to a Sun-Earth libration point orbit.
4.1 Determination of Transfers to Periodic Halo Orbits
4.1.1 Approaching the Halo Orbit
Recall, from Chapter 3, that n numerically integrated manifolds pass through any given cell. The n surfaces (i.e., sets of numerical data) are each approximated. The position states for each surface are related through the scalar-valued functions in Equation (3.13) and the functional form is noted here for convenience, 66
r˜i(θ, φ), i = 1,...,n. (4.1)
Then, the three approximations of the velocity components, i.e., those that define a vector field, are represented in the form,
˜ ˜ ˜ ˜ V (¯r)= ³Vx(x, y, z), Vy(x, y, z), Vz(x, y, z)´ . (4.2)
The velocity representation is expressed in more detail in Equation (3.14). Now, there are also n sets of three non-zero halo orbit initial conditions associated with the n surfaces; these are employed in the determination of destination halo orbits corresponding to desirable transfer paths. The vector field approximation is crucial to the determination of a maneuver that accomplishes a transfer from a point in a given cell to the halo orbit. However, not surprisingly, this approximation is most accurate near regions where data is sampled from the numerical integration. Thus, the velocity fits are only used within a “trust” region in the vicinity of the surface fits. The trust region is defined at any θ and φ coordinate pair within the cell by a parameter, w, the width between the surfaces of maximum and minimumr ˜i(θ, φ), w max min i.e., =r ˜i − r˜i . In Figure 4.1, three surface approximations are shown inside a cell, along with the upper and lower limits on the trust region (surfaces in black). With the approximations for the manifolds within the cell, the process of comput- ing a transfer from a given point within the cell is simplified. There is no restriction on the source of the incoming leg; any state that arrives within a cell can be delivered
to a halo orbit. Assume that an initial trajectory arc, ¯l1(t), originates from somewhere in or beyond the current system, and eventually intersects one or more cells. Note
that ¯l1(t) is assumed to be numerically produced and, hence, it is comprised of a set of
data points. All intersecting points (¯rint,j, v¯int,j), j = 1,...,J along ¯l1(t), and within the trust region, are tagged. For each intersection point, the velocity discontinuity between the point on the incoming arc and the manifold velocity approximation is
calculated, that is, ∆V¯cell,j = V˜ (¯rint,j) − v¯int,j. The location where ∆V¯cell,j is a min- ∗ ¯ ∗ imum is denoted asr ¯int. The computed minimum maneuver, ∆Vcell, is applied at 67
Figure 4.1. Trust Region Around the Surface Approximations in a Cell
that point and a second trajectory arc, ¯l2(t), is then propagated forward from the ∗ ∗ ¯ ∗ initial state at positionr ¯int and with initial velocityv ¯int + ∆Vcell. Using the method detailed in the previous chapter, initial conditions that correspond to a periodic halo orbit, h¯(t), are determined. This periodic orbit is, then, the intended destination of the transfer arc ¯l2(t).
Because of the accuracy of the velocity fits, the second arc, ¯l2(t), will usually approach the desired halo orbit. However, this approach is not asymptotic, and the vehicle inevitably leaves the vicinity of the libration point. From this initial guess, ¯ ∗ however, the maneuver ∆Vcell can be iteratively updated or corrected to reach the ¯ ∗ halo orbit successfully. One way to improve the arc is by scaling the vector ∆Vcell with a scalar attenuation factor K. An updated arc is propagated from an initial condition ∗ ∗ ¯ ∗ atr ¯int with velocityv ¯int +K∆Vcell and a good choice of K results in a closer approach 68
to the halo orbit. In evaluating the proximity of the updated approach arc to the halo orbit, all position and velocity states are significant. Neglecting velocity states can result in a large halo orbit insertion maneuver. Thus, a function D is defined such that the six-element state vectors for all points along the approach arc, beyond the cell intersection location, are subtracted from the six-element states corresponding to all points along the halo orbit. Then, the magnitudes of all these differences are computed to search for a minimum distance between the approach arc and the halo orbit in six- dimensional space. This is implicitly a function of K since the choice of K alters the shape of the arc approaching the halo. The states are dimensionalized such that positions are defined in terms of kilometers and velocities computed in centimeters per second. This choice of units yields relatively equal weighting in all 6 components. An ∗ ¯ ∗ optimal scaling factor K = K for ∆Vcell is determined by minimizing D(K) using the MATLABTM function fminbnd, a standard optimizer for functions of bounded scalar variables in the MATLABTM Optimization ToolboxTM. This optimizer uses a combination of a golden section search algorithm and parabolic interpolation to determine the optimal between an upper and lower bound. The method terminates when K changes by less than a given tolerance. The upper and lower bounds on K are 0.8 and 1.2 and the termination tolerance is 0.00005.
The transfer arc can be further improved by scaling each individual component ∗ ¯ ∗ of the maneuver, i.e., K ∆Vcell, by separate attenuation factors k1, k2, and k3, such ∗ ∗ ∗ ∗ that the total maneuver is written K k1∆V , k2∆V , k3∆V . The factors £ cell,x cell,y cell,z¤ k1, k2, and k3 are now variable parameters of the transfer arc while the parameter K ∗ remains fixed at K = K . Again, D(k1, k2, k3), now a function of the new attenuation ∗ ∗ ∗ factors, is minimized to determine optimal choices of k1 = k1, k2 = k2, and k3 = k3. In the MATLABTM Optimization ToolboxTM, the standard optimizer used to mini- mize multivariable functions is fmincon. For this problem, fmincon uses a sequential quadratic programming (SQP) method to determine the best choice of parameters to
minimize D(k1, k2, k3). The bounds on each parameter are 0.8 and 1.2 and again the tolerance for convergence is set at 0.00005. 69
x 106 1
0.8 Destination Halo Apply 0.6 Initial Trajectory ∆ V** cell 0.4
0.2 Sun Apply
0 K*∆ V* cell y [km] −0.2 Earth Approach −0.4 Arcs
−0.6 Apply Cell −0.8 ∆ V* Intersection cell
−1 −1 −0.5 0 0.5 1 1.5 2 2.5 x [km] x 106
Figure 4.2. Improvement of an Approach to a Halo Orbit Using Scale Factors
The maneuver to be performed within the cell can then be functionally represented as
∆V¯ ∗∗ = K∗ k∗∆V ∗ , k∗∆V ∗ , k∗∆V ∗ . (4.3) cell £ 1 cell,x 2 cell,y 3 cell,z¤
An approach trajectory to a Sun-Earth L2 halo orbit that is propagated from an arbi- trary point in a cell appears in Figure 4.2. In this case, the simple and straightforward ∗ ∆Vcell is first employed for the maneuver. This is compared with two improved arcs that employ scaling factors on the maneuver. The destination halo orbit is included and it is apparent that each iteration generates an arrival that is closer to completing a revolution around the halo orbit. 70
4.1.2 Computation of Halo Orbit Insertion
Once an approach to the halo orbit is determined, an insertion strategy is required
to complete the transfer. The halo orbit insertion point (HOI), [¯rHOI , v¯HOI ], is a significant parameter. Selection of HOI affects the cost of the transfer since targeting is necessary and a maneuver is likely required at an arbitrarily selected HOI point to insert into the periodic orbit. Therefore, it is desirable to search a set of candidate insertion points for the most efficient approach. Each such candidate HOI point is ∗ ¯ ∗∗ targeted fromr ¯int using ∆Vcell as an initial estimate for the targeting maneuver. In an automated process, the preliminary approach arc is propagated for the minimum length of time that is necessary to pass the libration point; this propagation time ensures that the arc is in the vicinity of the halo orbit and the closest approach to a candidate HOI can be straightforwardly computed. The initial velocity and time of propagation are then adjusted using a differential corrector to target the HOI point.
If the targeter is successful, the maneuver within the cell is adjusted by adding ∆V¯targ ¯ ∗∗ to ∆Vcell and the final point on the approach leg will lie on the halo orbit at HOI:
¯l2(tf ) = [¯rHOI , v¯f ]. The necessary insertion maneuver is simply ∆V¯HOI =v ¯HOI − v¯f .
To search a set of candidate insertion points along the destination halo orbit, each ∗ is targeted fromr ¯int in an automated process within an optimization routine that
determines the HOI point yielding the lowest insertion cost ∆V¯HOI . To accomplish this task, the points along the halo orbit are tagged with respect to time. The
parameter th indicates the time along the halo orbit since thex ˆ − zˆ plane crossing
at the maximum z location, and thus each point, h¯(th), can be referenced with this parameter. Candidate points for halo orbit insertion are typically represented by
values of th between 0 and roughly one third of the period of the halo orbit, P.
These bounds on the parameter th limit the search for insertion points to locations along the periodic halo orbit that are known, from experience, to generate manifolds that pass the second (smaller) primary at a reasonably close distance. Since the
candidate points are parameterized by a single variable, th, the standard univariate 71
function optimizer fminbnd in the MATLABTM Optimization ToolboxTM, is sufficient to perform the search within the given bounds and determine the value of th that is associated with the HOI point corresponding to the lowest insertion cost. Once an HOI point has been selected, the transfer is complete with an associated total cost of ∆V¯ = ∆V¯ ∗∗ + ∆V¯ + ∆V¯ . T ¯ cell targ¯ ¯ HOI ¯ ¯ ¯ ¯ ¯ Numerical values for a test case appear in Table 4.1 and the propagated trajectory appears in Figure 4.3 . Three halo orbits in the Sun-Earth system in the vicinity of
L1 are used to generate manifold volumes. These orbits are associated with the three ∼ out-of-plane amplitudes Az = 800, 000 km, 840, 000 km and 880, 000 km. A stable ∼ manifold corresponding to a fourth periodic halo orbit of amplitude Az = 945, 000 km — one that is not incorporated in any of the fittings — is used to test the transfer procedure. A point along this actual stable manifold is perturbed with a ∆V¯ of magnitude equal to 20 m/s at a point somewhere within a cell. At this precise intersection point, the approximations within the cells are employed to estimate the appropriate ∆V¯ and determine the specific halo orbit that will minimize the cost. The actual manifold should be the best solution with a cost of 20 m/s and no need for orbit insertion. The results of the approximations appear in the table. A very good estimate of the destination halo is achieved and the transfer is accomplished with a cost very close to the true 20 m/s perturbation.
Table 4.1. Targeting a Halo Orbit
Step in Process Destination Total ∆V at ∆VHOI Halo Orbit Intersection
Approximate Result Az = 945, 176 km 19.69 m/s 1.43 m/s at Same Intersection Point
Exact Manifold Az = 945, 481 km 20 m/s . . . Intersection 72
Figure 4.3. Transfer Determined using Cells
4.1.3 An Alternative to Direct Halo Orbit Insertion
In some examples, the HOI selection phase of the process fails to produce reason- able transfers. The optimization procedure successfully and accurately targets HOI
points, but the resulting ∆V¯targ is high despite a reasonably close and inexpensive initial guess for a halo orbit approach arc. Also, the resulting transfer arc bears little resemblance to the initial approach arc. This type of result, in this regime, is not unfamiliar to trajectory designers. Such an example appears in Figures 4.4–4.5 near
L2 in the Sun-Earth system. In this example, an incoming trajectory intersects a set of cells defined over a volume containing manifolds associated with halo orbits in the range 170, 000 km ≤ Az ≤ 220, 000 km. A reasonable initial guess yields an approach to the destination halo orbit that nearly wraps around the orbit and departs back toward the Earth (Figure 4.4). Attempts at targeting various HOI points consistently produce very expensive maneuvers within the cell and at the insertion points. Note that the HOI targeting is successful in the sense that the resulting trajectories do eventually reach the halo orbit, but, as evidenced in Figure 4.5, directional issues associated with the transfer arc yield the large cost. 73
x 106 1
0.8
0.6
0.4
0.2 Sun L 0 2
y [km] Earth ∆ V** −0.2 cell
−0.4
−0.6
−0.8
−1 −1 −0.5 0 0.5 1 1.5 2 x [km] x 106
Figure 4.4. Initial Approach Arc to Halo
Since direct targeting of HOI points is not successful in terms of cost, an alternate insertion scheme is required. Various strategies are options. However, an algorithm is implemented that is based on a multi-point targeting method developed by Howell and Gordon [34]. The procedure was originally developed for station-keeping to maintain a vehicle in the vicinity of a halo orbit; it requires that the approaching trajectory pass reasonably close to the halo orbit at some point. For example, the point indicated by the green symbol ‘*’ in Figure 4.4 passes very close to the maximum excursion along the halo orbit in the y-direction, indicated with the black symbol ‘*’. At this point, a single ∆¯v would alter the spacecraft path to track the destination orbit although it may not, in fact, insert directly into the periodic trajectory. The maneuver point (*) that is noted on the approach arc in Figure 4.4 can be represented in the form
x¯0 + δx¯0, wherex ¯0 corresponds to the fixed point of maximum y-excursion on the halo orbit (*). The maneuver point on the approach arc (*) is thus represented in terms of a perturbation relative to the halo orbit fixed point, that is,
p¯0 δx¯0 = , (4.4) e¯0 + ∆¯v 74
6 x 10 2.5
2
1.5
1 y [km] 0.5
Sun 0 Earth L ∆ v 2 1 −0.5 ∆ v 2
−1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x [km] 6 x 10
Figure 4.5. Attempt to Target a Candidate HOI Point
wherep ¯o is the position perturbation ande ¯0 is the velocity perturbation. The goal is to determine the maneuver ∆¯v in Equation (4.4) that is necessary to maintain proximity to the halo orbit. The state transition matrices that relate variations at three time increments further along the periodic orbit to the initial perturbation at
x¯0 are denoted in terms of the following submatrics:
Ak0 Bk0 Φ(tk,t0)= , k = 1, 2, 3. (4.5) Ck0 Dk0
Then, the linear representation for the relationship between variations at different times is written:
m¯ k δx¯k = = Φ(tk,t0)δx¯0 (4.6) v¯k 75
wherem ¯ k is the three dimensional vector representing the variation in position at a later time tk, andv ¯k is the corresponding variation in velocity. Define a cost function as
T T T T J(¯p0, e¯0, ∆¯v) = ∆¯v Q∆¯v +m ¯ 1 Rm¯ 1 +m ¯ 2 Sm¯ 2 +m ¯ 3 Tm¯ 3
T T T +v ¯1 Rvv¯1 +v ¯2 Svv¯2 +v ¯3 Tvv¯3, (4.7)
where Q, R, S, T, Rv, Sv, Tv are all 3 × 3 weighting matrices. The matrix Q is symmetric and positive definite. The others are positive semi-definite. This problem is solved for the ∆¯v that yields the minimal cost, that is,
∗ T ∆¯v = −[Q + B10RB10
T T T T T −1 + B20SB20 + B30TB30 + D10RvD10 + D20SvD20 + D30TvD30]
T T T T T T × [(B10RB10 + B20SB20 + B30TB30 + D10RvD10 + D20SvD20 + D30TvD30)¯e0+
T T T T T T (B10RA10 + B20SA20 + B30TA30 + D10RvC10 + D20SvC20 + D30TvC30)¯p0]. (4.8)
Observe that the ∆¯v∗ is computed based on the deviations of the current path relative to the reference at three future times. The time increment is a free parameter. In the examples here, a time increment of 60 days is incorporated. Thus, ∆¯v∗ is computed based on deviations from the halo orbit approximated at 60, 120, and 180 days beyond the maneuver point. The weighting matrices, from Howell and
Gordon [34] are specified as R = diag[100, 100, 100], Rv = diag[10, 10, 10], S = Sv = −14 −14 diag[1, 1, 1], T = Tv = diag[0.1, 0.1, 0.1] and Q = diag[0.01 × 10 , 0.1 × 10 , 1 × 10−14]. It is noted that the spacecraft is not delivered precisely into the destination orbit with this maneuver (Figure 4.6). Since the algorithm is developed for station- keeping, and based on a linearized model, this is not surprising. However, the resulting path remains close to the halo orbit. The use of ∆¯v∗ as an estimate for an insertion maneuver is sufficient to bring the solution to a higher-order model. 76
6 x 10 1
∆ v*
0.5
Sun 0 L
y [km] Earth 2 ∆ V** cell
−0.5
−1 −1 −0.5 0 0.5 1 1.5 2 x [km] 6 x 10
Figure 4.6. Multi-Point Targeting To Stay Near the Halo Orbit
4.2 Generating System-to-System Transfers in the CR3BP
More challenging than a transfer to a halo orbit from an arbitrary point is the determination of system-to-system transfers. In particular, the transfer from an L1 or L2 libration point orbit in the Earth-Moon system to an L2 halo orbit in the Sun-Earth system is considered. As noted previously, similar types of transfers are the focus of a number of researchers [1–4, 33]. In this work, an efficient method of computation is sought using approximations to the manifolds. The ultimate objective is to use the approximation to produce the transfers in full, 3-D ephemeris models that include other perturbations as well.
In the circular restricted Earth-Moon system, unstable manifolds for an L1 halo orbit of amplitude Az ∼= 39, 000 km are propagated toward the Moon. These man- ifolds form a tube, of course. In Figure 4.7, the Earth-Moon manifold trajectories are transformed to Sun-Earth rotating frame coordinates (red) and intersections are determined with cells that correspond to stable manifolds generated for Sun-Earth 77
x 106 2
Unstable Manifold Paths Cells Around Stable 1.5 Departing Earth−Moon Halo Orbit Manifolds L Halo Orbit Near Sun−Earth L 1 2
1
0.5 Earth y [km]
Sun 0
−0.5
−1 −1 −0.5 0 0.5 1 1.5 2 x [km] x 106
Figure 4.7. Earth-Moon L1 Manifold Paths Intersect Sun-Earth L2 Manifold Cells
L2 halo orbits (black). For this example, the cells are generated from the halo orbits ∼ ∼ ranging in size from Az = 170, 000 km to Az = 220, 000 km.
The Earth-Moon L1 arcs in Figure 4.7 are plotted after a transformation of the data from the Earth-Moon system to the Sun-Earth frame. The coordinate frames are assumed to be initially aligned and the origin of the Earth-Moon trajectories is translated from the Earth-Moon barycenter to the Earth. Denote the angular velocity of the Earth-Moon frame with respect to an inertial frame asω ¯EM and the angular velocity of the Sun-Earth frame with respect to an inertial frame isω ¯SE. It is assumed that both frames rotate about an inertially fixedz ˆ-axis. Then, the angular velocity 78
of the Earth-Moon frame relative to the Sun-Earth frame isω ¯ =ω ¯EM − ω¯SE. It is assumed that this rate remains constant. Thus, in the simplest approximation,
the time-varying angle between the frames is evaluated as Θ(t)= ωt + Θ(t0), where
ω = kω¯k and Θ(t0) represents an initial phasing angle between the two systems. It
is assumed here that Θ(t0) = 0 and that all trajectories on the Earth-Moon tube are initiated at the same time. A time-varying direction cosine matrix is defined as
cos(Θ(t)) sin(Θ(t)) 0 SE EM C = − sin(Θ(t)) cos(Θ(t)) 0 . (4.9) 0 0 1 This simple transformation is used to compute the positions along the Earth-Moon manifolds as viewed in the Sun-Earth frame. The direction cosine matrix in Equation (4.9) is also applied in a transformation of the velocity vectors corresponding to the position states along the Earth-Moon manifold trajectories. The velocities are then expressed with respect to the Sun-Earth frame. Thus, if EM r¯ is the position state along a manifold in the Earth-Moon frame, then SEr¯ = SECEMEM r¯ is the position
EM as it appears in the Sun-Earth frame. If v¯EM is defined as the velocity vector associated with a state relative to the Earth-Moon frame, as viewed in the Earth-
SE EM EM Moon frame, then v¯EM = v¯EM +ω ¯ × r¯ is the velocity with respect to the Sun-Earth frame, but expressed in Earth-Moon coordinates. Transforming to Sun-
SE SE EMSE Earth coordinates: v¯SE = C v¯EM . This transformation, as discussed previously, yields trajectories along the Earth- Moon manifold as viewed relative to the Sun-Earth frame. However, this derivation is based on two simplifying assumptions. First, the Earth-Moon plane of motion and the Sun-Earth plane of motion are identical. Thus, thex ˆ − yˆ plane of the Earth- Moon frame and thex ˆ − yˆ plane in the Sun-Earth system are coplanar. The second assumption is the initial orientation of the two frames, that is, initially the two systems are in phase. To change the initial phase, the resulting position and velocity vectors, SEr¯ and SEv¯, are transformed further with a simple direction cosine matrix of the same form as that in Equation (4.9). Also, the inclination of thex ˆ − yˆ plane corresponding 79
to the Earth-Moon system (i.e., the inclination of the Lunar orbit) within the Sun- Earth frame can be incorporated by including a second rotation about the Sun-Earth xˆ-axis. Then, the trajectories along the Earth-Moon manifolds, as viewed in the Sun- Earth frame, are available assuming some initial phase angle and a fixed inclination of the Earth-Moon plane of motion. Note that the line of nodes of the Lunar orbit is assumed to be in the Sun-Earthx ˆ-axis direction. The line of nodes can be shifted with another transformation if necessary.
The cells in Figure 4.7 contain position and velocity fits for the data representing only three stable Sun-Earth manifold tubes. The Earth-Moon L1 manifold arcs are transformed to the Sun-Earth system assuming that, initially, the phase angle between the systems is 45o. It is also noted that the Earth-Moon frame is inclined relative to the Sun-Earth frame by a rotation of −5o about the Sun-Earthx ˆ-axis to more accurately represent the inclination of the Moon’s orbit. After the transformation to the Sun-Earth system, the Earth-Moon manifold trajectories are searched to locate all intersections with the cells, and the transfer approximation process proceeds. In
Figure 4.7, all Earth-Moon manifold trajectories are overlaid with the Sun-Earth L2 cells. The estimated total cost for the “best” transfer is 24.52 m/s, as determined in the CR3BP. The maneuver within the cell possesses a magnitude of 21.15 m/s, and the insertion at HOI costs 3.37 m/s. The transfer appears in Figure 4.8 in both the circular restricted Earth-Moon view and the circular restricted Sun-Earth view.
In the Earth-Moon view, it is observed that the manifold from the vicinity of L1
does encircle L2 as it departs the system. This characteristic is actually familiar and
resembles a transit orbit flowing past L2.
4.2.1 Transition of CR3BP Transfers to an Ephemeris Formulation
Transfers are, thus far, computed in two separate CR3BP systems. First, manifold arcs generated in the Earth-Moon system are transformed to the Sun-Earth rotating coordinate frame. Then cell approximations and a corrections scheme are used to 80
Figure 4.8. Transfer from the Earth-Moon L1 Halo Orbit to a Sun-Earth L2 Halo Orbit
develop a transfer from the intersection point in the cell to the halo orbit. However, this preliminary transfer arc is computed in the Sun-Earth circular restricted problem with no regard for the dynamical effects of the Moon. To include these effects, the solution is transferred into a four-body model with complete ephemeris formulation.
Points along the Earth-Moon arc, including three revolutions of the Earth-Moon L1 halo orbit, are used to define a set of patch points that are corrected in position and velocity in the four-body ephemeris model in the Generator 3.0.2 software (Fig- ure 4.9). For this application, the ephemeris model includes the gravity of the Sun, Earth, and Moon as well as the modeling for their ephemeris locations using Solar Lunar Planetary (SLP) ephemeris files, but no solar radiation pressure force is added. Of course, time is a significant factor when the model involves actual ephemeris data. The time to initiate the transfer is not arbitrary. The initial Julian date correspond- ing to the first patch point along the path is selected such that the departure along the manifold occurs at a time when the Moon’s orbit, as viewed in the Sun-Earth frame, closely reflects the conditions that defined the transfer in the CR3BP. 81
x 105 1
0.5 Moon
0 Earth −0.5
−1
−1.5 y [km]
−2
−2.5
−3 ∆ v 1 −3.5
−4 −4 −3 −2 −1 0 1 2 x [km] x 105
Figure 4.9. Transition of Earth-Moon L1 Manifold Arc to Ephemeris Model
A second set of patch points is generated from the halo approach arc in the Sun-
Earth rotating frame and appended to 3 revolutions along the Sun-Earth L2 halo. The patch points from the Earth-Moon arc are transformed to the ephemeris rotating Sun- Earth frame, and the two sets of points are joined. Two maneuvers are defined at two of the patch points; these locations are previously identified along the transfer arc computed in the simpler CR3BP model. The first maneuver is initially assumed at the breakpoint between the Earth-Moon and Sun-Earth frames. The initial estimate ¯ ∗∗ ¯ ¯ is the computed value (∆Vcell + ∆Vtarg). A second ∆V is assumed at the halo orbit insertion point, estimated as ∆V¯HOI . The full set of patch points is then converged in the Sun-Earth frame using the differential corrector program in the Generator software. The Sun-Earth view of the complete transfer is plotted in Figure 4.10. In the figure, the Moon’s orbit is plotted in blue and the trajectory of the spacecraft is shown in red. Some differences relative to the solution in the circular problem now appear. The patch points along the periodic halo orbits naturally generate quasi- 82
x 106 1
0.8
0.6
0.4 Moon 0.2 Sun 0
y [km] L Earth 2 −0.2
−0.4 ∆ v −0.6 1 ∆ v 2 −0.8
−1 −1 −0.5 0 0.5 1 1.5 2 x [km] x 106
Figure 4.10. Transition of the Earth-Moon to Sun-Earth Transfer to the Ephemeris Model
periodic Lissajous orbits in the nonperiodic ephemeris model. The Earth-Moon L1 ∼ Lissajous orbit has a magnitude of Az = 39, 000 km, consistent with the Earth- Moon halo orbit in the circular restricted problem. The magnitude of the Sun-Earth Lissajous trajectory varies with each revolution but the range of variation is relatively ∼ ∼ small, from Az = 115, 000 to Az = 117, 000. This contrasts with the original Sun-
Earth L2 halo orbit corresponding to the transfer computed in the circular restricted ∼ problem, an original magnitude of Az = 123, 000 km. The cost at the breakpoint is 25.36 m/s in the ephemeris model, compared to 21.15 m/s in the circular restricted model. The cost of Lissajous Orbit Insertion (LOI) is 4.00 m/s in the ephemeris model, compared to 3.37 m/s for halo orbit insertion in the circular restricted model. This level of agreement in the transfer characteristics between the different models is common using the manifold approximation in the circular problem. It is noted, however, that the proximity of the Moon is the most significant perturbing body in the problem. 83
4.3 Adaptation for Lunar Encounters
Transformation of an arc from the CR3BP to an ephemeris formulation that in- cludes four-body dynamics may pose an additional challenge. In particular, the vari- ability of the Moon’s orbit in the ephemeris model plays a significant role in perturbing arcs computed in the Earth-Moon CR3BP, most noticeably when those arcs pass near the Moon. Maneuvers that are incorporated along Earth-Moon manifolds may occur close to the Moon. It is also possible that the path departing the halo orbit may include several flybys of the Moon prior to departure from the Earth-Moon system. Thus, inclusion of ephemeris data from four-body dynamics in the computation of the Earth-Moon manifold is beneficial.
The Generator software version 4.0.1 is used to generate Earth-Moon Lissajous orbits and their associated manifolds. They can be transformed to the ephemeris Sun-Earth rotating coordinate system. Since the ephemeris formulation is time- dependent, the states corresponding to the manifold trajectories computed in the Earth-Moon frame are time-varying. In particular, the initial states along each man- ifold arc are associated with states along the Lissajous orbit at different times. Also, since the transformation of each arc to the ephemeris Sun-Earth rotating frame is time-dependent, each arc undergoes a different transformation. This contrasts with the manifolds in the circular restricted problem where each arc is assumed to be ini- tiated at the same time since the model is autonomous. When a manifold tube of the circular restricted Earth-Moon problem is transformed to the circular restricted rotating Sun-Earth frame the result resembles a tube in the Sun-Earth frame with all arcs initiated in the vicinity of the Earth-Moon libration point at the same initial time. This fact is evident in Figure 4.11 where an Earth-Moon unstable manifold tube (red), generated in a circular restricted model, retains a tube-like shape even when transformed to the Sun-Earth frame. It intersects a Sun-Earth manifold tube (blue) and a transfer trajectory between the tubes is plotted in black. This is the type of transfer that is determined when coupling systems in the circular restricted 84
Figure 4.11. Earth-Moon L2 Tube Intersecting Sun-Earth L2 Tube Both Tubes Computed in CR3BP Models
model. However, manifolds computed in the Earth-Moon ephemeris system, then transformed to the Sun-Earth ephemeris rotating frame, no longer resemble a tube. Each arc undergoes a unique transformation defined by the state of the Earth-Moon system and, what appears to be a tube in the Earth-Moon frame, stretches out to define a different type of surface in the Sun-Earth frame. It might be visualized as a sheet that is “unrolling” and spiraling outward. This difference can be viewed when examining an intersection of one such surface, generated in an Earth-Moon ephemeris system, with a tube generated in a Sun-Earth CR3BP model, as seen in
figure 4.12. The red unstable manifold surface departs an Earth-Moon L2 orbit and, transformed to the Sun-Earth frame, intersects a blue Sun-Earth L2 stable manifold tube computed in a circular restricted model. A transfer trajectory is determined at the intersection of the surfaces (with the aid of cell approximations). It is plot- ted in red and includes several revolutions of both the initial Earth-Moon and final Sun-Earth Lissajous orbits. The ephemeris Lunar orbit is plotted in blue and the locations of Earth and Sun-Earth L2 are indicated.
For the example in Figure 4.12, twelve revolutions of a Lissajous orbit in the vicin-
ity of the Earth-Moon L2 point are computed. Since the timing of both the orbit and the departure along the manifold affects the transformation to the Sun-Earth frame, 85
Figure 4.12. Earth-Moon L2 Manifold Surface Intersecting Sun-Earth L2 Tube (Earth-Moon Surface Computed in Ephemeris Formulation)
time becomes a significant parameter in the search for ephemeris transformations. Thus, it is noted in this example that the Lissajous orbit is initiated at the Julian date 2455201.665077 (Calendar Date 01/04/2010). The period for two revolutions ∼ of the obit is ∼ 29.4 days. The magnitude of the orbit is Az = 15, 000 km (recall that this is a quasi-periodic orbit and hence the magnitude changes from revolution to revolution). Departures from the Lissajous orbit along unstable manifold arcs ex- panding away from the Earth-Moon system are computed beginning at Julian date 2455295.70083 (Calendar Date 04/09/2010). There are fifty-seven manifold arcs gen- erated over a full period (two revolutions) of the Lissajous orbit. Recall that the time span corresponding to two revolutions of the Lissajous is roughly the time needed for one revolution of the Moon around the Earth. The Generator software produces a set of patch points defining both the manifold arcs and the Lissajous orbit in the Earth- Moon frame. Once joined to patch points along a transfer into a Sun-Earth libration point orbit, correction of these states to converge upon a continuous transfer generally requires flexibility in the locations of states along the Earth-Moon Lissajous orbit. Such flexibility is attained by including more patch points. Hence several revolutions of the Lissajous orbit are included before manifold departure is initiated. 86
Once transformed to the Sun-Earth rotating frame, the Earth-Moon manifold arcs are available in the ephemeris model for the determination of transfers. In contrast to previous analysis, solely in the circular restricted model, the arcs now result from exploiting the ephemeris model to incorporate the Lunar perturbation. A transfer is subsequently designed by searching for the lowest cost intersections between all of the manifold arcs and all of the cells, in exactly the same manner as accomplished previ- ously for CR3BP system-to-system transfers. Since the cells are defined on manifold volumes computed in the circular restricted model, the approximations are used only in that model. A transfer arc and destination halo orbit, along with the breakpoint ¯ ∗∗ ¯ ¯ maneuver (∆Vcell + ∆Vtarg) and insertion maneuver ∆VHOI are computed for a trans- fer in the CR3BP, though the incoming Earth-Moon unstable manifold is computed in an ephemeris formulation. The result appears in Figure 4.13 where the initial arc of the transfer, up to the breakpoint, is determined in an ephemeris model and the transfer arc and halo orbit are determined in the circular restricted model. In this example, the initial maneuver magnitudes are ∆V¯ ∗∗ + ∆V¯ = 40.11 m/s ° cell targ° ° ° and ∆V¯ = 1.60 m/s. The destination periodic halo orbit is determined with ° HOI ° ° ° ∼ a magnitude of Az = 125, 000 km in the circular restricted model. The trajecto- ries produced in the circular restricted model are intended to be transitioned to the Generator software to produce a complete solution.
Within the full ephemeris model in Generator, the solution in Figure 4.13 is not initially continuous. Along the Sun-Earth leg of the transfer, patch points are selected that include some points along the transfer path and additional points along four rev- olutions of the destination libration point orbit. Of course, a maneuver exists at the insertion point, initially estimated as ∆V¯HOI . The patch points from the ephemeris Earth-Moon manifold, transformed to the Sun-Earth frame, are concatenated with the points along the Sun-Earth transfer arc from the circular restricted model and a ¯ ∗∗ ¯ maneuver is defined at the breakpoint that is estimated to be (∆Vcell + ∆Vtarg). The final set of patch points is then corrected in the four-body problem using the differen- tial corrector in Generator, and the result appears in Figure 4.14. After converging, 87
x 106 1
0.8 Transfer Arc and Halo in Restricted Model 0.6
0.4 Moon 0.2 Sun
0
y [km] L Earth 2 −0.2
−0.4 ∆ v −0.6 Earth−Moon Manifold 1 Arc in Ephemeris Model ∆ v 2 −0.8
−1 −1 −0.5 0 0.5 1 1.5 2 x [km] x 106
Figure 4.13. Earth-Moon Manifold in Ephemeris Model; Transfer Arc and Halo Orbit in a Sun-Earth Circular Restricted Model
the maneuver magnitudes are 34.82 m/s at the system breakpoint and 2.47 m/s at Lissajous Orbit Insertion (LOI). The revolutions along the Sun-Earth Lissajous orbit ∼ ∼ vary in size from Az = 114, 000 km to Az = 123, 000 km, slightly smaller than the halo orbit determined in the circular restricted model. Most significantly, the transfer trajectory is continuous in the ephemeris model. Recall that an alternate method of determining transfers uses multi-point target- ing near the libration point orbit as opposed to direct orbit insertion. The multi-point targeting method is also employed to this same transfer to determine its effectiveness upon transitioning to the ephemeris model. The results of the two methods are com- ¯ ∗∗ pared. The process of determining ∆Vcell, a maneuver estimate to best approach the halo orbit from the breakpoint of the systems, remains unchanged. The point at which the multi-point targeting maneuver is implemented occurs near the location of maximum y-excursion along the periodic halo orbit, though this may not be the closest approach to the orbit. At this point, a maneuver ∆¯v∗ is computed that results 88
x 106 1
0.8
0.6
0.4 Moon 0.2 Sun 0
y [km] L Earth 2 −0.2
−0.4 ∆ v −0.6 1 ∆ v 2 −0.8
−1 −1 −0.5 0 0.5 1 1.5 2 x [km] x 106
Figure 4.14. The Entire Transfer is Converged in a Full Ephemeris Formulation
in an arc to track the halo orbit without direct insertion. However, the computed ∆¯v∗ is employed as an estimate for an insertion maneuver transitioned to the ephemeris model. Subsequent points along several revolutions of the libration point orbit are included. Once all patch points and maneuvers are corrected in Generator, a trans- fer into a Lissajous orbit in the ephemeris four-body model is achieved. The result is plotted in figure 4.15. The multi-point targeting maneuver (originally defined in the CR3BP) is not an optimal insertion maneuver, and, thus, the corresponding cost is slightly higher than that for the previously computed insertion maneuver. Com- pensation for this increase in cost near the libration point orbit is the fact that no targeting is required at the breakpoint to exactly reach an insertion point. Thus, the breakpoint maneuver may decrease. In fact, the maneuvers have magnitudes of 37.90m/s at the system breakpoint and 4.07 m/s near the halo orbit as computed in the CR3BP. After converging in the ephemeris model, the magnitudes have been altered to 27.39 m/s at the breakpoint and 5.00 m/s at LOI, a value that exactly 89
Figure 4.15. A Transfer using Multi-Point Targeting Near the Halo Orbit is Converged in a Full Ephemeris Formulation
meets a constraint placed on the corrections process. Contrast this with the costs of 34.82 m/s at the breakpoint and 2.47 near the destination orbit that were attained using the HOI searching method (Table 4.2). As noted, the multi-point targeting procedure avoids the requirement of searching for, and targeting, a good orbit inser- tion point. In this example, a lower maneuver cost at the breakpoint also results and the search of HOI candidates is eliminated as well. Ultimately, for an insertion esti- mate to be transitioned to the full ephemeris model, the multi-point targeting method proves more efficient here than HOI searching. In the final ephemeris transfer, the ∼ ∼ Sun-Earth Lissajous orbit varies in size from Az = 119, 000 km to Az = 122, 000 km, similar to the final Lissajous orbit determined from the HOI searching method. 90
4.3.1 Cost Reduction
A system-to-system transfer is represented by a set of patch points that are con- verged using a differential corrections scheme in an ephemeris model. As noted previ- ously, the corrector minimizes the velocity discontinuities at all points, unless a point is designated as a maneuver point, in which case the corrector only constrains the velocity discontinuity below some specified maximum |∆¯v|. This formulation allows the user to specify any upper limit to the ∆¯v magnitude at a maneuver point, as- suming that it is not unreasonable that such a linearized model can approximate a solution. Thus, a procedure to incrimentally lower the cost of a transfer is devised.
Given a set of patch points defining a transfer that is a solution in the ephemeris model, the constraints on the magnitudes of the maneuvers are lowered a small amount. The transfer is converged to meet the constraints, and the process is re- peated. In the above examples, using both the HOI targeting and multi-point halo targeting procedures, the cost is reduced to zero at all maneuvers, generating essen- tially free, natural, transfers between systems. The free transfer that originated with the HOI targeting method appears in Figure 4.16 in the Sun-Earth frame. The loca- tions of patch points are indicated along the transfer path. Figure 4.17 represents the natural transfer that originated from the multi-point halo targeting procedure. The two appear almost identical.
At each iteration, the corrector requires only small changes to states at each of the patch point positions to accommodate the lower maneuver costs. It is noted, however, that after many iterations in the cost reduction procedure, the orbits and transfer arcs undergo significant changes in size and shape as the overall reductions in cost are imposed. As the Earth-Moon Lissajous orbit is modified to reduce the cost, ∼ flexibility is demonstrated in Figure 4.18. From an initial magnitude of Az = 15, 000 km (blue) the shape of the orbit changes significantly and the magnitude increases to ∼ ∼ the range of Az = 21, 000 km to Az = 29, 000 km (red). 91
x 106 1
0.8
0.6
0.4 Moon
0.2 Sun 0 y [km] Earth L −0.2 2
−0.4 Patch Point −0.6 Locations
−0.8
−1 −1 −0.5 0 0.5 1 1.5 2 x [km] x 106
Figure 4.16. Transfer Computed with No Cost from HOI Targeting Routine
Table 4.2 summarizes the comparison of the two cases demonstrated here. The first originating from HOI targeting, the second from multi-point halo targeting. There is little difference between the two. 92
x 106 1
0.8
0.6
0.4 Moon 0.2 Sun 0 y [km] L Earth 2 −0.2
−0.4
−0.6 Patch Point Locations −0.8
−1 −1 −0.5 0 0.5 1 1.5 2 x [km] x 106
Figure 4.17. Transfer Computed with No Cost from Multi-Point Halo Targeting Routine 93
x 105 1
0.5
0 Moon z [km]
−0.5
−1 −1 −0.5 0 0.5 1 y [km] x 105
Figure 4.18. Earth-Moon Lissajous Orbit Changes Shape as Transfer Cost is Reduced 94
Table 4.2. Comparing Transfer Computation Methods After Transition to Ephemeris
Using HOI Targeting Using Multi-Point Halo Targeting
Original E-M L2 Lissajous ∼ 15, 000 ∼ 15, 000
Az [km] (Ephemeris)
Destination S-E L2 Lissajous 125, 000 125, 000
Az [km] (CR3BP)
Converged S-E L2 Lissajous 114, 000 119, 000
Az [km] (Ephemeris) —123, 000 —122, 000 Maneuvers [m/s] At Intersection 34.82 27.39 Near Lissajous 2.47 5.00 After Cost E-M Lissajous 21, 000 21, 000
Reduction Az [km] — 29, 000 — 28, 000 S-E Lissajous 126, 000 123, 000
Az [km] — 132, 000 — 129, 000 95
5. Transfers to Orbits of the Sun-Earth L2 Point
A number of sample transfers have been computed that originate at quasi-periodic Lissajous orbits near libration points in the Earth-Moon system and arrive at libra- tion point orbits (LPO’s) in the vicinity of the Sun-Earth L2 libration point. Some examples are summarized here. Several parameters are varied in the search for cases of lowest cost. For all of the examples presented in this chapter, computation of trajectory arcs in the Earth-Moon system is accomplished using the full ephemeris model including the gravitational forces of Sun, Earth, and Moon, but not the SRP force. Initial estimates for transfers into the Sun-Earth system employ the cell representations of manifolds computed in the Sun-Earth CR3BP. The cells used for these estimates are generated from stable manifolds associated with periodic halo orbits in the vicinity ∼ of the Sun-Earth L2 point with magnitudes ranging in size from Az = 170, 000 km ∼ to Az = 220, 000 km. In all cases considered here, the original estimate for the transfer arc from the intersection within a cell to the destination libration point orbit is computed in the Sun-Earth CR3BP, a model that necessarily neglects the Moon’s gravitational influence. The destination orbit is always a periodic halo orbit. The transfers are always accomplished with two maneuvers; the magnitudes listed in the tables as |∆¯v1| and |∆¯v2|. The first maneuver is at the breakpoint between systems, the second is at LPO insertion. When necessary, the multi-point targeting scheme is employed as an alternative to direct HOI targeting, as indicated in the tables with a symbol ‘ * ’. Such an indicator implies no direct insertion occurs in the halo orbit as computed in the Sun-Earth system. In most cases, the resulting transfer (comprised of an Earth-Moon arc generated in an ephemeris model and Sun-Earth arcs generated in a CR3BP model) is transitioned to a full ephemeris model employing patch points as discussed previously in Chapter 96
4. In this process, the destination orbit transitions from a periodic halo orbit to a quasi-periodic Lissajous orbit, a trajectory that is bounded and possesses time- varying amplitudes. Thus, any amplitudes listed for LPO’s that are generated in the ephemeris model are actually mean values over several revolutions. In some of these cases, the cost reduction procedure is employed to seek free system-to-system transfers in the ephemeris model. It is worth noting that the amount of variation in Lissajous orbit amplitudes tends to increase as the overall cost of a transfer is reduced.
5.1 Varying Size of Initial Earth-Moon Lissajous Orbit
Unstable manifolds associated with several Lissajous orbits of differing ampli- tudes in the vicinity of the Earth-Moon L2 libration point are computed and low-cost transfers to Sun-Earth L2 libration point orbits are determined. Initially, only Earth- Moon Lissajous orbits of the northern type are considered, and all of their associated manifolds are propagated in the direction away from the Moon.
Table 5.1 includes various transfers from Earth-Moon Lissajous orbits of four different amplitudes to LPO’s near the Sun-Earth L2 libration point. The total cost in all cases remains in the range of 70 to 89 m/s for transfers originating in Lissajous ∼ ∼ orbits ranging in size from Az = 10, 000 km to Az = 40, 000 km. The cost reduction procedure successfully reduces three of these to free system-to-system transfers.
The transfer represented as Case 3 in Table 5.1, from an Earth-Moon LPO of ∼ amplitude Az = 15, 000 km, is examined in additional detail in Figures 5.1–5.3. In the figures, the LPO’s, manifolds, and transfer arcs appear in red; the periodic halo orbit near Sun-Earth L2 is plotted in black for clarity. The Moon’s orbit (in the Sun-Earth view) appears in blue and the location of the Sun-Earth libration point is indicated in black. Note the location of the libration point oscillates with time in the full ephemeris computations, but is fixed in the CR3BP computations. This example is notable for a relatively low cost after preliminary design. Also of significance, 97
Table 5.1. Comparing Transfers From Earth-Moon L2 Lissajous Orbits Of Various Amplitudes
Earth-Moon Final Sun-Earth |∆¯v1| |∆¯v2|
Case LPO Az (km) LPO Az (km) (m/s) (m/s) 1a Eph. & CR3BP 40, 000 127, 000 69.90 2.35 1b Full Ephemeris 120, 000 70.00 2.38 2a Eph. & CR3BP 23, 000 118, 000 118.02 4.33 2b Full Ephemeris 116, 000 88.92 5.50 2c Cost Reduction more variable 93, 000 0 0 3a Eph. & CR3BP 15, 000 123, 000 78.20 2.40 3b Full Ephemeris 125, 000 73.16 3.00 3c Cost Reduction more variable 100, 000 0 0 4a Eph. & CR3BP 10, 000 124, 000 76.38 1.63 4b Full Ephemeris 125, 000 78.60 1.97 4c Cost Reduction more variable 110, 000 0 0 98
5 x 10 Earth−Moon View 1 6 x 10 Sun−Earth View Moon 1 0 ∆ v Earth 1 −1 0.5 Moon L −2 Sun 2 0 Earth y [km] −3 y [km]
−4 −0.5 ∆ v 1 ∆ v −5 2 −1 −1 −0.5 0 0.5 1 1.5 2 −6 6 −4 −3 −2 −1 0 1 2 x [km] x 10 5 x [km] x 10
Figure 5.1. Case 3a. Ephemeris Earth-Moon Manifold and CR3BP Sun-Earth Transfer Arc to Halo Orbit
the cost is reduced entirely to zero with the cost reduction procedure, generating a free transfer between systems. This transfer is plotted in Figure 5.3. Note that the Earth-Moon LPO has undergone a noticeable change in shape. This is also true of the Sun-Earth LPO, though it less noticeable on the scale of the figure. 99
x 106 1
0.5
Moon L Sun 2 0
y [km] Earth
−0.5 ∆ v 1 ∆ v 2
−1 −1 −0.5 0 0.5 1 1.5 2 x [km] x 106
Figure 5.2. Case 3b. Sun-Earth View: Transfer Arc to LPO in Ephemeris Model
5 x 10 Earth−Moon View 1 6 x 10 Sun−Earth View Earth 1 0 Moon −1 0.5 Moon −2 Sun 0 Earth L y [km] −3 y [km] 2
−4 −0.5
−5 −1 −1 −0.5 0 0.5 1 1.5 2 −6 6 −4 −3 −2 −1 0 1 2 x [km] x 10 5 x [km] x 10
Figure 5.3. Case 3c. Free Ephemeris Transfer Between Systems 100
5.1.1 Comparison With Southern Earth-Moon L2 Orbits
Two alternative types of transfers appear in Table 5.2. From Table 5.1, Cases 2 and 3 are recomputed assuming that the departure orbit is a southern Earth-Moon Lissajous trajectory (rather than a norther type). The magnitudes of the orbits are identical to those in Table 5.1. The resulting transfers converged in the full ephemeris model are very similar, with small differences in cost.
Table 5.2. Comparing Transfers From Southern Earth-Moon L2 Lissajous Orbits Of Various Amplitudes
Earth-Moon Final Sun-Earth |∆¯v1| |∆¯v2|
Case LPO Az (km) LPO Az (km) (m/s) (m/s) 5a Eph. & CR3BP 23, 000 125, 000 90.79 3.80 5b Full Ephemeris 124, 000 90.00 4.47 6a Eph. & CR3BP 15, 000 125, 000 79.02 3.50 6b Full Ephemeris 125, 000 75.36 3.98
5.2 Varying Initial Time of Computation
In searching for low-cost transfers between systems, it is necessary to consider the relative location of the spacecraft in the initial orbit before the transfer is initiated, i.e., before departure along the manifold. This phasing is altered by varying the initial time for computation of the Earth-Moon Lissajous orbit. Consider the spacecraft of Case 3 in Table 5.1. Along the initial Earth-Moon Lissajous orbit, an infinite number of locations exist that might serve as suitable departure points from the Earth-Moon system along the manifold. Departure from a number of those paths have, in fact, been computed and a transfer path to a Sun-Earth libration point orbit constructed. Now consider another vehicle initially on a similar orbit in the Earth-Moon system, but lagging behind the first by a few days. This 101
vehicle eventually reaches similar manifold departure points along the Lissajous orbit, but reaches these points several days later than the first vehicle. At the times when the second vehicle may achieve departure, the dynamics of the system are changed and the manifold paths are altered. Thus, there is potential for an improvement in transfer cost. The transfer identified as Case 3 from Table 5.1 is initiated with a start date of 01/08/2010. This marks the initial date along the Lissajous orbit in the Earth-Moon system. Three other transfers are computed for spacecraft either lagging or leading the vehicle associated with the original Case 3. The resulting transfers appear in Table 5.3. Each is initiated from a Lissajous orbit in the vicinity of the Earth-Moon ∼ L2 libration point with an out-of-plane amplitude of Az = 15, 000 km, consistent with Case 3.
Table 5.3. Varying Start Dates for Transfers From Orbits in the Earth-Moon System; for Comparison with Case 3 of Table 5.1
Variation in Final Sun-Earth |∆¯v1| |∆¯v2|
Case Start Time (days) LPO Az (km) (m/s) (m/s) 7 Eph. & CR3BP + 6 127, 000 161.36 1.33 8a Eph. & CR3BP + 3 128, 000 187.36 12.36 8b Full Ephemeris 129, 500 172.29 14.99 8c Cost Reduction 92, 000-148, 000 0 0 9a Eph. & CR3BP - 3 125, 000 40.11 1.60 9b Full Ephemeris 118, 000 34.82 2.47 9c Cost Reduction 130, 000 0 0
The transfer presented as Case 8 in Table 5.3 is remarkable for the success of the cost reduction procedure to reduce a transfer cost from 187.28 m/s to zero, i.e., a free transfer. However, it can be noted in Figure 5.4 that the Earth-Moon and Sun-Earth libration point orbits undergo considerable variation to facilitate the process. In the figure, the change in the Earth-Moon Lissajous orbit is most apparent. 102
5 x 10 Earth−Moon View 1 6 x 10 Sun−Earth View Earth 1 ∆ v 0 1 Moon −1 0.5 Moon L −2 Sun 2 0 y [km] −3 Total Cost 187.28 m/s y [km] Earth
−4 −0.5 ∆ v ∆ v 2 1 −5 −1 −1 −0.5 0 0.5 1 1.5 2 −6 6 −4 −3 −2 −1 0 1 2 x [km] x 10 5 x [km] x 10
5 x 10 Earth−Moon View 1 6 x 10 Sun−Earth View Earth 1 0 Moon −1 0.5 Moon −2 Sun 0
y [km] y [km] Earth L −3 Free Transfer 2 −4 −0.5
−5 −1 −1 −0.5 0 0.5 1 1.5 2 −6 6 −4 −3 −2 −1 0 1 2 x [km] x 10 5 x [km] x 10
Figure 5.4. Case 8b & c. Cost Reduction Facilitated by Large Changes in in Libration Point Orbits 103
5 x 10 Earth−Moon View 1 6 x 10 Sun−Earth View Earth 1 0 Moon −1 0.5 Moon L −2 Sun 2 Total Cost 0 Earth y [km] −3 37. 29 m/s y [km]
−0.5 −4 ∆ v ∆ v 1 ∆ v 1 2 −5 −1 −1 −0.5 0 0.5 1 1.5 2 −6 6 −4 −3 −2 −1 0 1 2 x [km] x 10 5 x [km] x 10
5 x 10 Earth−Moon View 1 6 x 10 Sun−Earth View Earth 1 0 Moon −1 0.5 Moon −2 Sun 0
y [km] −3 y [km] Earth L Free Transfer 2
−4 −0.5
−5 −1 −1 −0.5 0 0.5 1 1.5 2 −6 6 −4 −3 −2 −1 0 1 2 x [km] x 10 5 x [km] x 10
Figure 5.5. Case 9b & c. Less Variation in Libration Point Orbits is Necessary to Reduce a Lower-Cost Preliminary Transfer
The preliminary cost computed for Case 9 is quite reasonable, before cost reduc- tion. This preliminary and final transfer arcs associated with this case are displayed in Figure 5.5, where it is apparent that the libration point orbits undergo less change than was necessary to reduce the considerably larger cost in Case 8. 104
5.3 Transferring From an Earth-Moon L1 Orbit
Thus far, the example transfers have all been initiated from Lissajous orbits in the vicinity of the Earth-Moon L2 libration point using unstable manifolds departing in a direction away from the Moon. This choice immediately inserts the spacecraft into a region of space exterior to the Earth-Moon system (outside the orbit of the Moon), but it is not the only choice. Unstable manifolds departing LPO’s near the
Earth-Moon L1 libration point may also depart the Earth-Moon system via a Lunar flyby. Such an encounter is necessary since, in the restricted approximation, an orbit near the L1 point generally possesses a Jacobi Constant such that the associated zero velocity curve will bound the interior of the system everywhere except in the vicinity of L1, the Moon, and L2. Thus, the only natural escape from the vicinity of L1 is via a Lunar flyby along an unstable manifold or transit orbit. Any other option for escape requires a maneuver of sufficient magnitude to alter the Jacobi Constant (and, thus, the zero-velocity curve bounding the motion) and escape the system. Table 5.4 contains data on two such transfers.
Table 5.4. Transfers From An Orbit in the Vicinity of the Earth-Moon L1 Point
Departure Final Sun-Earth |∆¯v1| |∆¯v2|
Case Direction LPO Az (km) (m/s) (m/s) 10a Eph. & CR3BP Toward Earth 147, 000 438.68 8.07 10b Full Ephemeris 148, 000 439.04 8.49 10c Cost Reduction 100, 000 320.00 8.50 11a Eph. & CR3BP Toward Moon 124, 000 131.85 1.76 11b Full Ephemeris 124, 000 126.99 3.00
First, the example in Case 10 represents a departure along an unstable manifold in a direction toward the Earth. The best transfer in this case does not include a Lunar flyby and, thus, a large maneuver is necessary to achieve escape. This cost can be reduced, as is apparent in the table and displayed in Figure 5.6, but the 105
6 Sun−Earth View 5 x 10 x 10 Earth−Moon View 1 3
2 0.5 Moon 1 Sun Earth 0 0 L y [km] Earth Moon y [km] 2 −1 Total Cost ∆ v −0.5 ∆ −2 1 447.53 m/s v 1 ∆ v −3 2 −6 −4 −2 0 2 −1 x [km] 5 −1 −0.5 0 0.5 1 1.5 2 x 10 6 x [km] x 10
6 Sun−Earth View 5 x 10 x 10 Earth−Moon View 1 3
2 0.5 Moon 1 Sun Earth 0 0 Earth L y [km] Moon y [km] 2 −1 ∆ v 1 Total Cost −0.5 −2 328.50 m/s ∆ v ∆ v 1 2 −3 −6 −4 −2 0 2 −1 x [km] 5 −1 −0.5 0 0.5 1 1.5 2 x 10 6 x [km] x 10
Figure 5.6. Case 10b & c. Departing the Vicinity of the Earth-Moon L1 Point in a Direction Toward the Earth
preliminary design could not be reduced to a free transfer (and maintain the same transfer characteristics). This result implies that some minimal amount of energy increase is always required to exit the system from an unstable manifold of this type.
Alternatively, the example in Case 11 involves departure of the Earth-Moon sys- tem Lissajous orbit in the direction of the Moon. The spacecraft does undergo a Lunar flyby, but immediately returns to the interior of the system (inside the orbit of the Moon). This motion is often observed for these types of manifolds. After the Lunar flyby, many trajectories along the manifold return to the interior of the system, some remain in the vicinity of the Moon, and some escape to the exterior of the system (Figure 5.7). 106
x 105 3
Manifolds Returning to 2 Interior of System
1 Moon
0
y [km] Earth
−1 Manifolds Escaping −2 System Via Lunar Flyby
−3 −7 −6 −5 −4 −3 −2 −1 0 1 2 x [km] x 105
Figure 5.7. Manifolds of LPO Near Earth-Moon L1 Point Directed Toward the Moon
Those few manifolds that depart the system naturally, via Lunar flyby, are not necessarily departing in the proper manner for intersections with Sun-Earth stable manifold surfaces. Once again, the start time corresponding to initiation of the Lis- sajous orbit in the Earth-Moon system is an important parameter since, by varying the timing along the Lissajous orbit and, thus, the timing of departure along the manifolds, it is possible to alter the departing manifolds until a more appropriate intersection with the Sun-Earth stable manifold surface is identified. The more suc- cessful intersections are generally along unstable manifolds that exploit a Lunar flyby to escape the system freely, then allowing a maneuver at an intersection with a Sun- Earth stable manifold surface.
The initial Earth-Moon LPO corresponding to Case 11 in Table 5.4 is recomputed and transfers are determined for vehicles leading the original by a number of days. Results appear in Table 5.5. Recall that preliminary design in the restricted model results in a reasonably good prediction of the costs in a (non-optimized) ephemeris model. Thus, from the results in Table 5.5, only Case 16 was pursued and transitioned 107 to a solution in the ephemeris model, though the cost reduction procedure was not performed in this case. Recall that multi-point targeting to simulate LPO insertion is indicated with ‘ * ’. The trajectory associated with the lowest cost result, Case 16, is plotted in Figure 5.8. A free Lunar flyby facilitates an escape of the Earth-Moon system before the maneuver at the breakpoint with the Sun-Earth system.
Table 5.5. Varying Start Dates for Transfers From Orbits Near the Earth-Moon L1 Libration Point; for Comparison with Case 11 of Table 5.4
Variation in Final Sun-Earth |∆¯v1| |∆¯v2|
Case Start Time (days) LPO Az (km) (m/s) (m/s) 12a Eph. & CR3BP * - 6 130, 000 529.37 15.01 12b Full Ephemeris 134, 500 529.31 16.00 13a Eph. & CR3BP - 9 104, 000 1634.39 147.12 14a Eph. & CR3BP * - 11.75 172, 000 131.89 0.34 15a Eph. & CR3BP * - 12.75 137, 000 107.65 0.15 16a Eph. & CR3BP * - 13.75 125, 000 101.48 2.48 16b Full Ephemeris 129, 000 88.50 4.99 108
5 x 10 Earth−Moon View 6 Sun−Earth View 1 x 10 1 Earth ∆ v 2 0 0.5 Moon −1 Moon Sun 0 Earth y [km] y [km] L −2 2 ∆ v 1 −0.5 −3 ∆ v 1
−1 −4 −1 −0.5 0 0.5 1 1.5 2 −4 −3 −2 −1 0 1 2 6 5 x [km] x 10 x [km] x 10
Figure 5.8. Case 16b. Ephemeris Model Transfer Departing Earth-Moon System Via Lunar Flyby From an LPO Near the Earth-Moon L1 Point 109
6. Conclusion
6.1 Summation
Invariant manifolds in the three-body problem have consistently proved an accu- rate, sometimes computationally expensive, tool in the design of trajectories in multi- body systems. One aim of this study is to develop useful manifold approximations that can be employed efficiently in trajectory design problems, replacing intensive computation and storage of large amounts of manifold data for analysis. The approx- imations that are developed accomplish this goal, providing accurate representations of manifold surfaces and velocities within localized cell volumes.
A second aim of this study is the development of an initial approach to the prob- lem of system-to-system transfers. This is accomplished, simultaneously offering a demonstration of the usefulness of the cell approximations, by employing the cells as first estimates in the transfer design. Trajectories intersecting cells that include representations of manifolds are analyzed and estimates for low-cost maneuvers are determined that ultimately can be corrected to yield complete transfers between li- bration point orbits in different systems. Transfers computed in coupled restricted models (or patched ephemeris-to-restricted models) are transitioned to a full 4-body ephemeris model as a demonstration of their applicability to realistic designs.
6.2 Suggestions for Future Work
The manifold representations presented here can be used for any type of problem requiring data along manifold trajectories. One particular application is the design of transfer arcs between orbits in the vicinity of libration points — whether these orbits are in different systems, the same system, or even the same family. An estimated 110
preliminary solution can be determined using cell approximations. However, for the method of computing transfers in this work, there is currently a limitation concerning the departure orbit — it must always be determined a priori. Alternatively, stable and unstable manifold volumes can be approximated as separate sets of cells, and those cells can be searched for low-cost intersections within their volumes to determine transfers in a manner where both the origin and destination orbits serve as variable design parameters. Such an approach would allow a more rapid search of a broader space of the transfer problem. There is undoubtedly a need for further optimization of the transfer routines in this work. The process of transitioning a solution from coupled restricted problems to a full ephemeris model via patch point corrections is well determined, but not yet automated. The design of free transfers between systems is encouraging, but no attempt has been made to constrain such paths to enforce mission requirements. Ultimately, it is desirable to build automated approximation and transfer routines, which might include mission constraints, into tools for use in mission design in an ephemeris model. LIST OF REFERENCES 111
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