# Low-Thrust Nonlinear Orbit Control for Very Low Lunar Orbits
## 1 Introduction
employed Lyapunov stability theory in order to derive a feedback controller [14], while [PERSON] et al. revisited the latter work to avoid crashing the planet surface during large-angle maneuvers and including the shadow effect [15]. However, these previous works take into consideration Keplerian trajectories and do not address the influence of perturbations. Recently, machine learning has been proposed as an effective strategy to develop lightweight and computationally efficient guidance algorithms, suitable for onboard software. In this context, neural networks have been applied to autonomous spacecraft guidance problems for proximity operations and low-thrust orbit transfers [16; 17; 18]. Although the use of neural networks enhances real-time guidance capabilities, there are still some issues. First of all, this technology relies on an extensive training, which is computationally expensive and can be only achieved through ground-based resources. This may limit the autonomy when a spacecraft must face new challenging scenarios, especially for missions that require limited ground support. On the other hand, nonlinear control provides a feedback law with guaranteed analytic stability within its convergence domain. Therefore, it is capable of providing online feasible trajectories without any reference path nor offline information except for gains, which is often more important than optimality [9]. Furthermore, the appropriate tuning of the parameters allows obtaining solutions that are very close to those found with optimization techniques [19; 20]. The aim of this study is to employ a saturated feedback control law to acquire a spacecraft in a very-low-altitude lunar orbit and to drive it toward a desired operational orbit, not necessarily designed as a frozen path. Similar very-low-altitude lunar orbits may represent interesting options for the deployment of constellations, provided that coordinated control of the orbit elements is feasible. In this study, orbital motion is describedin terms of modified equinoctial elements, which are nonsingular in most operational scenarios [21]. All relevant perturbations are included in the dynamical model, i.e., several harmonics of the selenopotential and third body gravitational attraction due to the Sun and the Earth, whose orbital motion is described in an ephemeris model. The spacecraft is assumed to be equipped with a low-thrust engine, which allows considerable propellant saving, with respect to a high-thrust propulsion system, at the price of a longer maneuver time. In particular, two different target operational orbits are considered: (i) a circular orbit with assigned radius; (ii) an elliptic orbit with specified minimum radius at periselenium and maximum radius at aposelenium. A similar feedback control law is employed also in previous works [22; 23; 24], and is here extended to focus on the following aspects:  
* The boundary conditions are being formulated in terms of modified equinoctial elements, with several positive analytical consequences;
* The stability analysis is being extended to target orbits with time-varying orbit elements. In the end, this research has the ultimate intent of extending the range of applicability of nonlinear orbit control, thus paving the way for its use in challenging, highly perturbed mission scenarios, such as low-thrust correction maneuvers in very low lunar orbits.