Station-keeping under conical constraint on the control force
Authors:   Université Côte d'Azur, CNRS, Inria, LJAD, Nice, France, Alesia Herasimenka Miss , Farrés Ariadna , Dell'Elce Lamberto 
Published: January 2024

Due to specific mission goals, many satellites are subject to cone constraints on the thrust direction. For example, James Webb Space Telescope, launched on December 25, 2021 toward a Halo orbit around the Sun-Earth L2 libration point, has a thermal shield that must prevent the telescope and other instruments from overheating [1]. Therefore, it is constrained to always keep its attitude such that the angle between the normal to the shield and the Sun direction is smaller than 53 deg. It results in conical constraints for the propulsion directions. Using chemical propulsion to perform small impulsive corrections of the trajectory or a low-thrust satellite with very specific constraints on the control does not always allow to do any desirable maneuver, as we showed in [2], where the controllability of non-ideal solar sails in orbit about a planet was investigated.
In [2], we considered elliptic Keplerian orbits, and we formulated a convex optimization problem aimed at assessing whether some functions of the integrals of motion could not be decreased after one orbital period. Existence of such functions implies that there is a half-space of the neighborhood orbit's coordinates (orbital elements) where motion is locally forbidden [3]. In that paper, we strongly relied on the super-integrability of the Kepler problem. Here, we extend the methodology to infer local controllability of station-keeping satellites for any periodic orbit, regardless the dynamical system at hand. Given the projection of the nominal orbit on a surface of section, the methodology aims at verifying if a half space of such projection exists where the motion is forbidden after one orbital period. Variation of parameters is used to achieve a convex optimization problem that investigates the existence of obstructions to variations of local integrals of motion. Conical constraints are enforced by leveraging on the formalism of positive polynomials postulated by Nesterov [4], so that a finite-dimensional formulation of the convex program is achieved. Halo orbit in the CRTBP is eventually considered in the case study, but we emphasize again that the methodology is developed for a generic locally-integrable system. We compare the aforedescribed methodology with the results achieved in [5], where the authors looked at the controllability and the impact of limitations of the thrust direction on the station-keeping from a dynamical point of view. They use the Floquet Mode reference frame to describe the motion of the satellite in a close proximity to the orbit, and study the cost of station-keeping by projecting the thrust direction on the saddle plane.
The minimum requirement that we propose can be used for a design of space missions around any periodic orbit for satellites that have specific constraints on the thrust directions. It can be applied to a low-thrust satellite or even those with chemical propulsion under condition of using small impulses, so that the linearization of the dynamics holds.
[1] J. Petersen, “L2 Station Keeping Maneuver Strategy For The James Webb Space Telescope,” AIAA/AAS
Astrodynamics Specialist Conference, American Institute of Aeronautics and Astronautics, 2019.
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Journal of Guidance, Control, and Dynamics, 2022. in press.
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with a Periodic Drift,” 2022. HAL preprint no. 03779482.
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Boston, MA: Springer US, 2000. Series Title: Applied Optimization, 10.1007/978-1-4757-3216-0 17.
[5] A. Farres, C. Gao, J. J. Masdemont, G. Gomez, D. C. Folta, and C. Webster, “Geometrical Analysis of
Station-Keeping Strategies About Libration Point Orbits,” Journal of Guidance, Control, and Dynamics,
Vol. 45, June 2022, https://arc.aiaa.org/doi/10.2514/1.G006014.