MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.merl.com Station-keeping and momentum-management on halo orbits around L2: Linear-quadratic feedback and model predictive control approaches Kalabi´c,U.; Weiss, A.; Di Cairano, S.; Kolmanovsky, I.V. TR2015-002 January 11, 2015 Abstract The control of station-keeping and momentum-management is considered while tracking a halo orbit centered at the second Earth-Moon Lagrangian point. Multiple schemes based on linear-quadratic feedback control and model predictive control (MPC) are considered and it is shown that the method based on periodic MPC performs best for position tracking. The scheme is then extended to incorporate attitude control requirements and numerical simulations are presented demonstrating that the scheme is able to achieve simultaneous tracking of a halo orbit and dumping of momentum while enforcing tight constraints on pointing error. 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Copyright c Mitsubishi Electric Research Laboratories, Inc., 2015 201 Broadway, Cambridge, Massachusetts 02139 AAS 15-307 STATION-KEEPING AND MOMENTUM-MANAGEMENT ON HALO ORBITS AROUND L2: LINEAR-QUADRATIC FEEDBACK AND MODEL PREDICTIVE CONTROL APPROACHES Urosˇ Kalabic´,∗ Avishai Weiss,y Ilya Kolmanovsky,z and Stefano Di Cairanox The control of station-keeping and momentum-management is considered while tracking a halo orbit centered at the second Earth-Moon Lagrangian point. Mul- tiple schemes based on linear-quadratic feedback control and model predictive control (MPC) are considered and it is shown that the method based on periodic MPC performs best for position tracking. The scheme is then extended to in- corporate attitude control requirements and numerical simulations are presented demonstrating that the scheme is able to achieve simultaneous tracking of a halo orbit and dumping of momentum while enforcing tight constraints on pointing error. INTRODUCTION The paper considers the control of a spacecraft near the second Lagrangian point L2 in the Earth- Moon orbital system. The objective is to determine an appropriate method for the stabilization of a spacecraft to a halo orbit while simultaneously stabilizing the attitude of the spacecraft to stay inertially fixed. Halo orbits are unstable limit cycles centered around the collinear Lagrangian points that are un- forced solutions to the restricted 3-body problem. The halo orbit about L2 is particularly interesting because L2 is behind the Moon and because it is the point with the lowest gravitational potential energy needed to escape the Earth-Moon system (see Figure1); a satellite or space station following a sufficiently large halo orbit trajectory could facilitate communication between Earth and the far side of the Moon and also serve as a launchpad for faraway space missions. The shape of a halo orbit is complex and a spacecraft would have complex dynamics when point- ing at an object in its vicinity. Because it is difficult and not necessary to point at a nearby object such as the Earth or Moon, we instead consider the stabilization of the attitude so that the spacecraft point at a distant star and thereby remain almost inertially fixed. A halo orbit trajectory coupled with an inertially fixed orientation are unforced solutions to the idealized and decoupled translational and attitude equations of motion governing the spacecraft. Using these, we are able to develop control schemes that stabilize these trajectories such that fuel is consumed only when correcting tracking and attitude errors. In addition to asymptotically stabi- lizing the error, the schemes under consideration are required to enforce system constraints; these ∗Graduate Student, Aerospace Engineering, University of Michigan, 1320 Beal Avenue, Ann Arbor, MI 48109. yMember Research Staff, Mechatronics, MERL, 201 Broadway, Cambridge, MA 02139. zProfessor, Aerospace Engineering, University of Michigan, 1320 Beal Avenue, Ann Arbor, MI 48109. xSenior Principal Member Research Staff, Mechatronics, MERL, 201 Broadway, Cambridge, MA 02139. 1 constraints can include limits on the available thrust as well as constraints on pointing error. Further- more, the constraints considered in this paper serve to couple the translational and attitude dynamics because the constrained thrusters need to simultaneously stabilize the tracking and attitude errors. In order to satisfy the above, in the following we propose discrete-time trajectory-tracking control schemes that are based on linear-quadratic (LQ) and model predictive control (MPC) methods. The schemes are based on the linearized discretization of the tracking error dynamics. These dy- namics are linear but time-varying and schemes are considered based on averaged, instantaneously fixed, and periodic dynamics. The schemes are tested in a nonlinear simulation of the Earth-Moon system and the scheme based on periodic MPC applied to dynamics discretized using one hour time- steps is shown to perform best while adhering to constraints on available computational power. The simulations show that MPC can be used to simultaneously guarantee trajectory-tracking while controlling the attitude and enforcing constraints. Furthermore, a more realistic simulation is provided for the case where the orbit between the primaries has non-zero eccentricity. The paper is organized as follows. The rest of the introduction describes the problem formulation. The next section presents three LQ control schemes and three MPC control schemes. The section after next presents numerical results and the final section is the conclusion. Station-Keeping on Halo Orbits about L2 Generally in a system of two bodies orbiting around a common barycenter, the Lagrangian (or libration) points are the points at which the gravitational and centrifugal forces cancel out in the orbital frame FO, which is scaled so that the distance between the two bodies remains constant. See Reference1 for a more detailed presentation than available here. There are five Lagrangian points, labeled L1 through L5; the first three points L1 through L3 are called collinear because they lie on the x-axis. The other two are called triangular. A contour plot of the rotated gravitational potential energy of the Earth-Moon system is shown in Figure1 with Lagrangian points labeled. It is well-known1,2,3 that the collinear points L1 and L2 are unstable and that there exist unstable orbits about them to which one can stabilize a spacecraft. These orbits are termed halo orbits.4 We are concerned with the development of control schemes for tracking halo orbits about Earth- Moon L2 and so we derive the equations of motion of a spacecraft of negligible mass in the frame FO. Let the coordinates of such a spacecraft be given by (x; y; z). The units are scaled so that both the distance between the Earth and the Moon and the angular momentum of the Earth-Moon system are fixed at 1. Under these assumptions, the equations of motion of the spacecraft are given by,5 00 0 (1 − ρ)(x + ρ) ρ(x − 1 + ρ) x − 2y = d(θ) x − 3 − 3 + ux; (1a) r1 r2 00 0 (1 − ρ)y ρy y + 2x = d(θ) y − 3 − 3 + uy; (1b) r1 r2 00 (1 − ρ)z ρz z + z = d(θ) z − 3 − 3 + uz; (1c) r1 r2 1 where d(θ) = 1+e cos θ , ρ is the mass ratio of the Earth-Moon system, e is the eccentricity of the orbit, the true anomaly θ is the independent variable, and a prime 0 denotes differentiation by θ, d i.e. dθ . The variables r1 and r2 denote, respectively, the spacecraft’s normalized distances from the 2 1 L4 0.5 L3 Earth L1 L2 y 0 Moon −0.5 L5 −1 −1 −0.5 0 0.5 1 x Figure 1. Plot of energy levels in the orbital frame of the Earth-Moon system where × indicate Lagrangian points 2 2 2 2 2 2 2 2 Earth and Moon and satisfy r1 = (x + ρ) + y + z and r2 = (x − 1 + ρ) + y + z . The variable ut = (ux; uy; uz) denotes the acceleration provided by the spacecraft thrusters. For convenience, we introduce a vector ξ = (x; y; z) to denote the position of the spacecraft in the orbital frame. A halo orbit is a solution to Eq. (1) and its computation for the case where e = 0 is described in detail in Reference3. The computation of halo orbits for the case where e > 0 is presented in Reference5 and references therein. Momentum-Management Denote the inertial reference frame by FI and the spacecraft body-fixed frame by FB. Let J be the moment of inertia of the spacecraft in FB. Along with thrust, the spacecraft can use its dual- axis thrusters to create a torque about the principal axes and control the attitude of the spacecraft; it can also use its three reaction wheels to dump the adverse momentum that this maneuver causes. Each wheel is placed on a unique principal axis, whose moments of inertia in the directions of the principal axes are α1; α2; α3 for the 1-, 2-, and 3-axes, respectively. Let ν be the 3-dimensional vector of rotational velocities of the three wheels so that the i-th com- ponent of ν corresponds to the i-th wheel. Assuming very fast controller dynamics, the rotational velocities are controlled by a vector of applied angular accelerations uα, so that the equations of motion governing ν are given by ν_ = uα.