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Spacecraft Formation Flying Near Sun-Earth L2 Lagrange Point: Trajectory Generation and Adaptive Output Feedback Control

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Spacecraft Formation Flying Near Sun-Earth L2 Lagrange Point: Trajectory Generation and Adaptive Output Feedback Control 2005 American Control Conference ThB04.5 June 8-10, 2005. Portland, OR, USA Spacecraft Formation Flying near Sun-Earth L2 Lagrange Point: Trajectory Generation and Adaptive Output Feedback Control Hong Wong and Vikram Kapila Mechanical, Aerospace, and Manufacturing Engineering, Polytechnic University, Brooklyn, NY 11201 Abstract— In this paper, we present a trajectory gen- Spacecraft trajectory designs for single spacecraft eration and an adaptive, output feedback control design missions near the Sun-Earth Lagrange points include methodology to facilitate spacecraft formation flying near Lyapunov and Halo orbits [16], [18], [19]. These periodic the Sun-Earth L2 Lagrange point. Specifically, we create trajectories have the characteristic that spacecraft do a spacecraft formation by placing a leader spacecraft on a not require fuel to stay on these orbits. Thus, these desired Halo orbit and a follower spacecraft on a desired periodic trajectories are well suited as locations for a quasi-periodic orbit surrounding the Halo orbit. We develop leader spacecraft and a formation of follower spacecraft the nonlinear dynamics of the follower spacecraft relative can be placed in the vicinity of these trajectories. to the leader spacecraft, wherein the leader spacecraft is Current literature for formation design near the Sun- assumedtobeonadesiredHaloorbittrajectory.Inaddition, Earth L2 Lagrange point is scarce, with the exception of we design a formation maintenance controller such that the [9], [13], [17]. In [13], reference trajectories for follower follower spacecraft tracks a desired trajectory. Specifically, spacecraft are computed using classical orbital elements, we design an adaptive, output feedback position tracking resulting in bounded orbits around the leader spacecraft controller, which provides a filtered velocity measurement on a periodic orbit. In [17], feedback control is utilized to and an adaptive compensation for the unknown mass of the produce reference trajectories for follower spacecraft. In follower spacecraft. The proposed control law is simulated addition, [9] provides a method of generating reference for the case of the leader and follower spacecraft pair and is trajectories for follower spacecraft using a numerical shown to yield semi-global, asymptotic convergence of the method, where the resulting trajectories are quasi- relative position tracking errors. periodic. Current approaches for spacecraft control near the I. Introduction L2 Lagrange point require position and velocity sensors for feedback control purposes [7], [8], [11], [12], [14]. Equilibrium positions in the restricted three body However, exploiting the nonlinear, adaptive, output problem (RTBP) of the Sun-Earth system, known as the feedback control design methodologies of [3], [6], [15] to Lagrange points, have been exploited as key locations control spacecraft near the L2 Lagrange point eliminates for space-based astronomical observation stations [1], the need for velocity sensors, thus reducing the cost and [10]. As seen in Figure 1(a), Lagrange points L1,L2, mass of the spacecraft. and L3 are collinear with the Sun and Earth while L4 In this paper, we develop a leader-follower spacecraft and L5 each combined with the Sun and Earth yields formation, where the leader spacecraft is on a periodic, an equilateral triangle. A primary benefit of operating Halo orbit around the L2 Lagrange point in the Sun- observation stations in the vicinity of the Lagrange Earth system and the follower spacecraft is to track a points is that spacecraft near these points obtain nearly desired relative trajectory. Specifically, in Section II, we an unobstructed view of the galaxy, unhindered by the develop the dynamics of the follower spacecraft relative atmospheric and geomagnetic forces. to the leader spacecraft. Next, in Section III, we design a Spacecraft formation flying (SFF) has the potential to desired quasi-periodic relative trajectory for the follower enhance space-based imaging/interferometry missions spacecraft in the spirit of [9]. In contrast to [9], our by distributing mission tasks (usually conducted by trajectory design exploits the analytical properties of a monolithic spacecraft) to many small spacecraft. the quasi-periodic relative trajectories to characterize Incorporating this technology into future space mis- spacecraft formations using a parameter set. In Section sions near the Sun-Earth Lagrange points can enlarge IV, we formulate a trajectory tracking control problem. the sensing aperture and increase versatility of future In Section V, we develop an adaptive, output feedback observation platforms. However, effective utilization of control algorithm to enable the follower spacecraft to this new technology requires proper design of spacecraft track this desired quasi-periodic relative trajectory. formations and for each spacecraft in the formation In Section VI, we provide illustrative simulations to to be precisely controlled to maintain a meaningful demonstrate the efficacy of the proposed trajectory gen- baseline. eration and control design schemes. Finally, in Section VII, we give some concluding remarks. Research supported in part by the National Aeronautics and Space Administration–Goddard Space Flight Center II. System Model under Grant NGT5-151 and the NASA/New York Space In this section, we develop a nonlinear model charac- Grant Consortium under Grant 39555-6519. terizing the position dynamics of the follower spacecraft 0-7803-9098-9/05/$25.00 ©2005 AACC 2411 relative to the leader spacecraft near the L2 Lagrange spacecraft dynamics of (1), with the control thrust u point in the Sun-Earth system. Referring to Figure 1, set to zero. First, the Poincar´e-Lindstedt method is we assume that the Earth and the Sun rotate in a used to find a high order analytic approximation to circular orbit around the Sun-Earth system barycenter a periodic trajectory in the neighborhood of the L2 (center of mass) with a constant angular speed ω. Lagrange point. Next, the initial conditions, based on In addition, we attach an inertial coordinate system the Poincar´e-Lindstedt method, are used as an initial {X, Y, Z} to the Sun-Earth system barycenter and a seed in a numerical algorithm to find a better set of {x ,y ,z } rotating, right-handed coordinate frame L2 L2 L2 initial conditions leading to a periodic trajectory. This x to the L2 Lagrange point with the L2 -axis pointing numerical algorithm applies a Taylor series expansion z along the direction from the Sun to the Earth, the L2 - to the spacecraft states with respect to the initial axis pointing along the orbital angular momentum of conditions and time and truncates higher order terms, y the Sun-Earth system, and the L2 -axis being mutually such that for Halo orbits the result is a set of 3 linear x z perpendicular to the L2 and L2 axes, and pointing in equations with 4 unknown variables. Families of orbits the direction that completes the right-handed coordi- can be characterized by fixing one of the unknown nate frame. variables so that the result gives an equal number of equations to unknowns. Solving the aforementioned A. Dynamics of a Spacecraft Relative to the L2 La- linear matrix equation and using the result to update grange Point the previous set of initial conditions, we obtain a new initial condition guess. In order to describe the dynamics of a spacecraft The spacecraft dynamics are then propagated using formation near the L2 Lagrange point, we must first the new updated set of initial conditions to verify describe the dynamics of a spacecraft relative to the L2 T trajectory periodicity. If the trajectory is sufficiently q t xyz ∈ R3 Lagrange point. To do so, let ( ) = [ ] close to being periodic, then the initial conditions denote the position vector from the spacecraft to the {x ,y ,z } can be used for further simulation, else the above L2 Lagrange point, expressed in the L2 L2 L2 numerical algorithm is used to solve for a new set of R t ∈ R3 coordinate frame. In addition, let S→s( ) and initial conditions. Since the collinear Lagrange points R t ∈ R3 E→s( ) denote the position vectors from the are inherently unstable [19], long-term propagation of Sun and Earth, respectively, to the spacecraft. Finally, spacecraft dynamics using the initial conditions ob- R R R let L2 , E,and S denote the distances between tained in the above manner is futile. However, by the Sun-Earth system barycenter and the L2 Lagrange exploiting the symmetry property of Halo orbits (see point, the Earth, and the Sun, respectively. Then, below), we can artificially obtain a periodic orbit by the mathematical model describing the position of a computing trajectory information during half of a pe- spacecraft relative to the L2 Lagrange point is given by riod and reusing this trajectory data throughout other [21] simulations. mq¨+ Cq˙ + N(q, s) = u, (1) Halo orbits are classified as periodic trajectories that {x ,z } × are symmetric with respect to the L2 L2 plane (i.e., where m is the mass of the spacecraft, C ∈ R3 3 is y 0 −10 L2 = 0), and are not confined to be in the orbital C mω 100 plane of the Sun and Earth. Halo orbits have the a Coriolis-like matrix defined as = 2 000, 3 distinguishing characteristic that their projections on N ∈ R is a nonlinear term consisting of gravitational {y ,z } the L2 L2 plane are curves that resemble a Halo. effects and inertial forces T In this paper, we let q (t)=[x y z ] ∈ R3 denote ⎡ µ (x+R +R ) µ (x+R −R ) ⎤ H H H H S L2 S E L2 E − ω2 x R R 3 + R 3 ( + L2 ) the position vector from a point on a Halo orbit to ⎢ S→s E→s ⎥ the L Lagrange point, expressed in the {x ,y ,z } ⎢ ⎥ 2 L2 L2 L2 µS y µE y 2 coordinate frame. An initial seed for the numerical N m⎢ 3 + 3 − ω y ⎥, = ⎣ RS→s RE→s ⎦ algorithm of [19] consists of a spacecraft starting on {x ,z } y µS z µE z the L2 L2 plane with a nonzero initial L2 and 3 + 3 T RS→s RE→s z q x z L2 velocity (i.e., H(0) = [ H(0) 0 H(0)] and T 3 q y z x and u(t) ∈ R is the thrust control input to the ˙H(0) = [0 ˙H(0) ˙d(0)] ).
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