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Strategy for Optimal, Long-Term Stationkeeping of Libration Point Orbits in the Earth-Moon System

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Strategy for Optimal, Long-Term Stationkeeping of Libration Point Orbits in the Earth-Moon System Strategy for Optimal, Long-Term Stationkeeping of Libration Point Orbits in the Earth-Moon System Thomas A. Pavlak∗ and Kathleen C. Howelly Purdue University, West Lafayette, IN, 47907-2045, USA In an effort to design low-cost maneuvers that reliably maintain unstable libration point orbits in the Earth-Moon system for long durations, an existing long-term stationkeeping strategy is augmented to compute locally optimal maneuvers that satisfy end-of-mission constraints downstream. This approach reduces stationkeeping costs for planar and three- dimensional orbits in dynamical systems of varying degrees of fidelity and demonstrates the correlation between optimal maneuver direction and the stable mode observed during ARTEMIS mission operations. An optimally-constrained multiple shooting strategy is also introduced that is capable of computing near optimal maintenance maneuvers without formal optimization software. I. Introduction Most orbits in the vicinity of collinear libration points are inherently unstable and, consequently, sta- tionkeeping strategies are a critical component of mission design and operations in these chaotic dynamical regions. Stationkeeping is particularly important for libration point missions in the Earth-Moon system since fast time scales require that orbit maintenance maneuvers be implemented approximately once per week. Assuming that acceptable orbit determination solutions require 3-4 days to obtain, stationkeeping ∆V planning activities must be quick, efficient, and effective. Furthermore, the duration of a libration point mission is often dictated by the remaining propellant so a key capability is maintenance maneuvers that are low-cost. Thus, to accommodate a likely increase in future operations in the vicinity of the Earth-Moon libration points, fast, reliable algorithms capable of rapidly computing low-cost stationkeeping maneuvers, with little or no human interaction, are critical. The libration point stationkeeping problem has been explored in both the Sun-Earth and Earth-Moon systems by numerous researchers. The majority of these orbit maintenance methods can be classified as \short-term" approaches given that the goal for each individual maneuver is maintenance of the spacecraft within the vicinity of the libration point for only the immediate future. Optimal short-term strategies, some incorporating Floquet theory to eliminate the unstable component of spacecraft error, have been explored extensively by Farquhar,1,2 Breakwell et al.,3 and others.4{6 Globally optimal maneuvers have been sought via a global search stationkeeping algorithm designed for the Sun-Earth L2 James Webb Space Telescope by Janes and Beckman.7 During operation of the first Earth-Moon libration point mission, ARTEMIS,8 { depicted in Figure1 { the L1 and L2 quasi-periodic orbits were maintained over 1-2 revolutions at a time using direct optimization.9 Post-processing of maneuver data revealed that the ∆V direction aligned closely with the associated stable mode for virtually all of the approximately 60 stationkeeping maneuvers that were successfully implemented.10 A number of \long-term" stationkeeping approaches, specifically concerned with ensuring that the space- craft meets specific end-of-mission constraints, have also been developed, but are generally non-optimal. A method employed by Grebow et al.12 and Folta et al.13 targets a rigidly maintained baseline trajectory. The long-term stationkeeping strategy presented previously by Pavlak and Howell14 is flexible and able to ∗Ph.D. Candidate, School of Aeronautics and Astronautics, Purdue University, Armstrong Hall of Engineering, 701 W. Stadium Ave, West Lafayette, Indiana 47907-2045, Student Member AIAA. yHsu Lo Professor of Aeronautical and Astronautical Engineering, School of Aeronautics and Astronautics, Purdue Univer- sity, Armstrong Hall of Engineering, 701 W. Stadium Ave, West Lafayette, Indiana 47907-2045; Fellow AAS; Associate Fellow AIAA. 1 of 16 American Institute of Aeronautics and Astronautics Figure 1. ARTEMIS P1 Earth-Moon Trajectory11 maintain libration point orbits with a variety of mission requirements using multiple shooting. The algorithm employs stationkeeping maneuvers and targets many revolutions ahead { in contrast to the more limited 1-2 revolutions that serve as the basis in some short-term strategies { and, thus, is capable of maintaining tra- jectories for an arbitrary number of revolutions and/or meeting end-of-mission constraints such as a specific set of lunar arrival conditions. The analysis in this investigation is motivated by a need for flexible libration point orbit stationkeeping methods that are capable of reliably delivering a spacecraft to a set of desired end-of-mission conditions, final targets that may occur many revolutions downstream, and accomplish the goal for minimum cost in terms of ∆V . Furthermore, the empirical results generated from analysis of the ARTEMIS trajectory maneuvers currently represent only a single data set, so the observed alignment of the ∆V and stable mode directions warrants further exploration. The current investigation first seeks to augment the long-term stationkeeping scheme discussed by Pavlak and Howell14 with direct optimization to yield a flexible method for computing low-cost orbit maintenance maneuvers that is well-suited for automation. The optimal long-term strategy is also employed to further explore the empirical results extracted from the previous ARTEMIS stationkeeping analysis. Stationkeeping costs are reduced over results obtained via non-optimal long-term stationkeeping in the circular restricted three-body and higher-fidelity models. It is also demonstrated that knowledge obtained from ARTEMIS mission stationkeeping operations can be applied to develop algorithms to rapidly compute \near-optimal" stationkeeping maneuvers using only existing differential corrections strategies that do not require commercial optimization software. II. System Models Dynamical models of multiple levels of fidelity are utilized in this analysis. The circular restricted three-body problem (CR3BP) serves as the baseline model and is used for preliminary trajectory design and stationkeeping activities. A higher-fidelity ephemeris model incorporating lunar eccentricity and solar gravity { both of which can be significant when operating in the vicinity of the Earth-Moon libration point orbits { is included as well. II.A. Circular Restricted Three-Body Problem The circular restricted three-body problem represents a dynamical model that governs the motion of a \massless" spacecraft under the simultaneous influence of two gravitational bodies { in this case, the Earth and the Moon. The system is further simplified by assuming that the orbits of the Earth and Moon are both circular and coplanar; each body rotates about a common barycenter. It is often convenient to formulate the governing equations in the CR3BP with respect to a reference frame centered at the Earth-Moon barycenter; 2 of 16 American Institute of Aeronautics and Astronautics the rotational rate of the frame is equal to the orbital rate of the bodies. The rotating positive x-axis is aligned with the Earth-Moon line, the z-axis is defined to be orthogonal to the orbital plane of the gravitating bodies, and the y-axis completes the right-handed triad. The nondimensional mass parameter, µ, is defined as m µ = M (1) mE + mM and effectively represents the relative influence of the smaller body, i.e., the Moon, within the three-body system. The mass of the Earth and the Moon are denoted mE and mM , respectively. Spacecraft state information is described by a Cartesian six-dimesional state vector, h iT x = x; y; z; x;_ y;_ z_ (2) in which positions and velocities are expressed in the barycentered rotating frame. Note that bold symbols are used to represent vector quantities. Motion within the circular-restricted three-body problem is governed by the set of nondimensional second-order scalar differential equations, (1 µ)(x + µ) µ (x 1 + µ) x¨ 2y _ x = − 3 − 3 (3) − − − d1 − d2 (1 µ) y µy y¨ + 2x _ y = −3 3 (4) − − d1 − d2 (1 µ) z µz z¨ = −3 3 (5) − d1 − d2 with the scalar Earth-spacecraft and Moon-spacecraft distances { represented by d1 and d2, respectively { expressed as follows, q 2 2 2 d1 = (x + µ) + y + z (6) q d = (x 1 + µ)2 + y2 + z2 (7) 2 − Trajectories are computed in the CR3BP by numerically integrating the equations of motion written in first-order vector form, i.e., x_ = f (t; x) (8) where x is the six-dimensional state vector described previously. II.B. Libration Point Orbit Stability The equations of motion in the circular restricted three-body problem are autonomous when expressed in terms of a rotating frame and admit five well-known equilibrium solutions, i.e., the libration points. The three collinear libration points that lie along the Earth-Moon line are dynamically unstable as are many of the periodic orbits in their vicinity. The monodromy matrix, M, associated with a periodic libration point orbit is computed by integrating the state transition matrix (STM), Φ (t; ti), for one orbital period, P , i.e., M = φ (ti + P; ti) (9) Since the monodromy matrix corresponds to a discrete linear mapping, the orbital stability is assessed by examining the eigenvalues of M relative to a unit circle. The eigenvalues, λi, exist in reciprocal pairs and are interpreted as follows: λi 1 - Stable eigenvalue j j ≤ λi 1 - Unstable eigenvalue j j ≥ λi = 1 - Corresponds to orbit periodicity This classification is further illustrated in Figure2. Stable and unstable manifolds exist as higher dimensional surfaces governing flow into and out of un- stable periodic libration point orbits. Local manifold information is obtained from the eigenvectors, ν, of a 3 of 16 American Institute of Aeronautics and Astronautics Periodicity Neutrally Stable Stable Unstable Figure 2. Example Eigenvalue Diagram for Unstable Periodic Orbit monodromy matrix computed at a desired fixed point along the periodic orbit. The local stable and unstable manifolds { νs and νu, respectively { associated with a fixed point on an L2 Lyapunov orbit are projected onto configuration space and depicted in Figure 3(a). Note that the stable and unstable subspaces extend in both \+" and \ " directions.
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