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Optimal Reconfiguration Maneuvers for Spacecraft 58th International Astronautical Congress, Hyderabad, 24-28 September 2007. Copyright 2007 by Lindsay Millard. Published by the IAF, with permission and Released to IAF or IAA to publish in all forms. IACIAC----07070707----C1.7.03C1.7.03 OOOPTPTPTIMALPT IMAL RRRECONFIGURATION MMMANEUVERS FFFORFOR SSSPACECRAFT IIIMAGING AAARRAYS IN MMMULTI ---B-BBBODY RRREGIMES Lindsay D. Millard Graduate Student School of Aeronautics and Astronautics Purdue University West Lafayette, Indiana [email protected] Kathleen C. Howell Hsu Lo Professor of Aeronautical and Astronautical Engineering School of Aeronautics and Astronautics Purdue University West Lafayette, Indiana [email protected] AAABSTRACT Upcoming National Aeronautics and Space Administration (NASA) mission concepts include satellite arrays to facilitate imaging and identification of distant planets. These mission scenarios are diverse, including designs such as the Terrestrial Planet Finder-Occulter (TPF-O), where a monolithic telescope is aided by a single occulter spacecraft, and the Micro-Arcsecond X-ray Imaging Mission (MAXIM), where as many as sixteen spacecraft move together to form a space interferometer. Each design, however, requires precise reconfiguration and star-tracking in potentially complex dynamic regimes. This paper explores control methods for satellite imaging array reconfiguration in multi-body systems. Specifically, optimal nonlinear control and geometric control methods are derived and compared to the more traditional linear quadratic regulators, as well as to input state feedback linearization. These control strategies are implemented and evaluated for the TPF-O mission concept. IIINTRODUCTION Over the next two decades, one focus of NASA‘s Upcoming National Aeronautics and Space Origins program is the development of sophisticated Administration (NASA) mission concepts include telescopes and technologies to begin a search for satellite formations to facilitate imaging and extraterrestrial life and to monitor the birth of identification of distant planets. These scenarios are neighboring galaxies. Part of the program‘s mission is diverse, including designs such as the Terrestrial to (1) locate and image Earth-like planets orbiting Planet Finder-Occulter (TPF-O),2 where a monolithic other stars; and (2) determine the probability that life telescope is aided by a single occulter spacecraft, and has existed or could develop on the surface. 1 the Micro-Arcsecond X-ray Interferometry Mission (MAXIM),3 where as many as sixteen spacecraft move Traditional Earth-based telescopes must continuously together to form a space interferometer. Each design, expand in size to resolve increasingly distant and faint however, requires precise reconfiguration and star- objects. Unfortunately, as the size of a telescope tracking in potentially complex dynamic regimes. increases, it becomes too heavy and too expensive to Numerous inter-spacecraft positioning and system launch, thus motivating alternative designs for deep- geometry constraints must be satisfied to assure space observers. One such example is a free-flying successful imaging. Spacecraft separated over large array or constellation of smaller satellites that work distances must constantly be re-oriented and pointed together to form a single optical platform. at different stars, while the distance between the vehicles is expanding and contracting. All must be 1 accomplished with minimum fuel expenditure, and The goal in the present investigation is a comparison for many missions, autonomously. of control methods for array re-configuration in multi-body systems. Specifically, optimal nonlinear Extensive progress in spacecraft formation control has control as well as geometric control methods are been achieved. Previously, the focus for much of the derived and compared to more traditional linear analysis was on satellites moving under the quadratic regulators and feedback linearization. The gravitational influence of the Earth, or simply in free- control strategies are implemented and evaluated for space. 4,5 Recently, however, potential also applications the TPF-O mission concept. involve satellite arrays near the collinear libration points and, consequently, generate considerable DDDYNAMIC MMMODEL ::: CR3BP interest in formation control strategies in multi-body regimes. 6-14 The motion of a spacecraft in the CR3BP is described in terms of rotating coordinates relative to the Howell and Marchand successfully apply both discrete barycenter of the system primaries. For this analysis, and continuous control methods to deploy, enforce, emphasis is placed on formations near the libration and reconfigure formations near the vicinity of points in the Sun-Earth/Moon system. Thus, the libration points in the Sun-Earth/Moon system. 15,16 rotating x-axis is directed from the Sun toward the Specifically, these methods include input-state and Earth/Moon barycenter. The z-axis is normal to the input-output feedback linearization, linear quadratic plane of motion of the primaries, and the y-axis regulator theory, and impulsive maneuvers derived completes the right-handed triad. The system is via a differential corrections scheme. In their represented in Figure 1. examples, spacecraft follow both natural and non- natural trajectories in the Circular Restricted 3-Body xÙ yÙ inertial Problem (CR3PB). This analysis is also transitioned to yÙ P2 an ephemeris model, including solar radiation L2 pressure. L1 r2 Marchand and Howell further indicate that the required discrete velocity changes for individual G satellites within a formation are often impossibly xÙ 17 P1 inertial small, considering current thruster capabilities. r1 Carlson, Pernicka, and Balakrishnan expand upon this result via the development of a method for identification of formation size and control tolerances for which impulsive maneuvers become a practical Figure 1: Circular Restricted Three-Body Problem. option. 18,19 This research focuses on the analysis of discrete maneuvering techniques using differential Let Χ = [ x y z x& y& z &] T , then the general non- corrections in the CR3BP. dimensional equations of motion of a spacecraft, relative to the system barycenter, are Very recently, control of interferometric satellite arrays has been pursued. Because of the complexity of 1 − x + x − x − x&& − 2y& − x = − ( µ)( µ) − µ ( ( µ)) [1] 3 3 the problem, much of the available research assumes r1 r2 satellites within the formation are moving through 1 − y y y&& + 2x& − y = − ()µ − µ 20-25 3 3 free space. However, Millard and Howell develop r1 r2 [2 ] two control strategies that are advantageous for 1 − z z z&& = − ()µ − µ 3 3 interferometric imaging arrays in multi-body regimes. r1 r2 [3 ] A continuous geometric control algorithm is based on the dynamical characteristics of the phase space near where periodic orbits. 26 By adjusting parameters in the 2/1 control algorithm appropriately, satellites in the r = x + 2 + y 2 + z 2 1 (( µ) ) formation follow trajectories that aid image r = x − 1− 2 + y 2 + z 2 2 ()()µ reconstruction. Secondly, a nonlinear, locally-optimal () control law is developed that minimizes fuel while optimizing image resolution in the CR3BP. 27 2 The quantity is a non-dimensional mass parameter For the nominal Lissajous orbit, A(t) is time-varying associated with the system. For the Sun-Earth/Moon and of the form system, ≈ 3.0404 þ 10 -6. 0 0 0 1 0 0 A more compact notation incorporates a pseudo- 0 0 0 0 1 0 potential function, U, such that 0 0 0 0 0 1 A()t = [9] U xx U xy U xz 0 2 0 1− 1 U = ( µ) + µ + x2 + y 2 [4] U U U − 2 0 0 r r 2 () yx yy yz 1 2 U U U 0 0 0 zx zy zz Then, the scalar equations of motion are rewritten in the form The partials Ukl are also functions of time, although the functional dependence on time, t, has been x&& = U + 2y& = f x, x&, y, y&, z, z& [5 ] dropped. x x ( ) y&& = U − 2x& = f x, x&, y, y&, z, z& [6 ] y y () The general form of the solution to the system in z&& = U = f x, x&, y, y&, z, z& [7 ] z z () equation [8] is where the symbol Uj denotes the partial derivative of δΧ t = Φ ,tt δΧ t [10] ( ) ( 0 ) ( 0 ) U with respect to j. Equations [1]-[3] or equations [5]- [7] comprise the dynamic model in the circular where (t,t 0) is the state transition matrix (STM). The restricted three-body problem. STM is a linear map from the initial state at the initial time, t0, to a state at some later time, t, and thus offers Given a solution to the nonlinear differential an approximation for the impact of the variations in equations, linear variational equations of motion in the initial state on the evolution of the trajectory. the CR3BP can be derived in matrix form as The STM must satisfy the matrix differential equation & Χ = Χ & δ t A t δ t [8] Φ ,tt = A t Φ ,tt ()() () ( 0 ) ( ) ( 0 ) [11] δΧ = δ δ δ δ& δ& δ& T where [ x y z x y z ] represents given the initial condition, variations with respect to a reference trajectory. Generally, the reference solution of interest is not Φ ,tt = I ( 0 ) 6 x6 [12] constant. In this analysis, the reference trajectory is a Lissajous orbit, or a quasi-periodic orbit, near L 2. This where I is the 6 þ 6 identity matrix. particular orbit was identified by NASA Goddard 6x6 Space Flight Center (GSFC) as a possibly advantageous The STM at time t can be numerically determined by orbit for the TPF-O mission, and appears in Figure 2. integrating equation [10] from the initial value. Since the STM is a 6 þ 6 matrix, propagation of equation [10] requires the integration of 36 first-order, scalar 5 x 10 differential equations. Because the elements of A(t) depend on the reference trajectory, equation [10] 1 must be integrated simultaneously with the nonlinear [km] z 0 equations of motion to generate reference states. This -1 requirement necessitates the integration of an additional 6 first-order equations, for a total of 42 5 differential equations.
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