# Low-Thrust Nonlinear Orbit Control for Very Low Lunar Orbits
## 1 Introduction  
In recent years, space mission analysis and design in cislunar space has attracted an increasing interest among space agencies and private actors in the space economy. In particular, the ARTEMIS program led by NASA, in collaboration with the European Space Agency and other international partners, is foreseeing the construction of an orbital output along a Near-Rectilinear-Halo-Orbit (NRHO) to conduct scientific research in deep space and in a following phase to support the human exploration of the Moon [1]. Lunar satellite constellations are being proposed for a variety of purposes, including telecommunications, navigation, and remote sensing. However, lunar orbits are greatly perturbed by the gravitational harmonics of the selenopotential. To mitigate this issue, former research suggested the use of a variety of frozen orbits [2, 3]. However, if the spacecraft travels very-low-altitude orbits, then sectoral and tesseral harmonics play a significant role, and lead to differential perturbing actions on each satellite, depending on its longitude (and, ultimately, on its initial right ascension of the ascending node [4]). For these reasons, maintenance along very low lunar orbits is an open problem of practical interest. Station-keeping maneuvers along quasi-frozen, near-polar, and extremely low-altitude lunar orbits were investigated by [PERSON] et al. through impulsive maneuvers,though the authors suggested investigating the use of continuous low-thrust for correction station-keeping maneuvers [5]. A low-thrust propulsion system is instead employed by [PERSON] et al. to achieve lifetime extension of ultra low-altitude lunar spacecraft [6]. However, they identified the formulation of a complete mission scenario, including orbit transfers in cislunar space, as the next step in the research. From a technological point of view, the greater complexity of lunar missions and space vehicles will require the ability of the different subsystems to autonomously handle a larger amount of operations, especially for guidance, navigation, and control (GNC) tasks [7]. For example, the exploration vehicle Orion is equipped with a far larger autonomy and automation with respect to the Space Shuttle or the International Space Station, as it is required to autonomously perform rendezvous and docking operations or the deorbit burn [8]. Therefore, the design of future space vehicles must guarantee the autonomous execution of all GNC functions [9]. This involves the necessity to address two compelling requirements: first, develop a guidance algorithm able to manage a complex dynamical environment, which is nonlinear and highly perturbed, resulting in the inadequacy of Keplerian-based algorithms [7]; second, determine suitable techniques that can be implemented on an onboard software with limited computational resources. The present research focuses on nonlinear orbit control for orbit acquisition and maintenance by employing low thrust. Low-thrust orbit transfer can be achieved through optimal control, via either direct or indirect methods, which constitutes the bulk of most of previous research on this topic. With this regard, [PERSON] summarized several methods suitable for trajectory optimization problems [10]. However, optimal control is a time-demanding task that is usually accomplished offline, and thus, it cannot fulfill the necessity of a real-time low-thrust orbit maneuvering algorithm. Instead, feedback methods provide closed-loop control laws that can be employed for autonomous guidance, navigation, and control schemes [11]. For example, [PERSON] proposed a simple guidance scheme based on optimal control laws that maximize the time rate of the orbital elements according to the variational equations, suitable for an onboard implementation [12]. [PERSON] investigated the problem of orbital transfer under continuous low thrust and obtained a nonlinear feedback controller by showing the accessibility of Gauss variational Equations [13]. [PERSON] et al.