Low-Thrust Nonlinear Orbit Control for Very Low Lunar Orbits

Abstract

In the next decades, both space agencies and private competitors are targeting the lunar environment as a scientific and technological resource for future space missions. In particular, the confirmed existence of water-ice deposits in the vicinity of the poles (predominantly the south pole) makes polar or near-polar low lunar orbits attractive for the purpose of designing space missions that could search for suitable Lunar base sites. However, traveling very-low-altitude orbits is very challenging, as they are strongly perturbed by the Moon’s gravity field as well as third- and fourth-body effects due to the Earth and the Sun. Several studies demonstrate that these orbits are expected to impact the lunar surface in a few months. Therefore, the definition and implementation of an effective station-keeping strategy represents a crucial issue in order to extend satellites’ lifetime. In this paper, a feedback nonlinear control law is employed in order to perform corrective maneuvers aimed at keeping the state of the satellite within acceptable margins. The satellite is assumed to be equipped with a steerable and throttleable low-thrust propulsion system. The control law is based on the Lyapunov stability theory and does not require any reference path to track, with a considerable decrease in the computational cost. The proposed real-time control law includes control saturation, related to the maximum available thrust magnitude, and is developed employing modified equinoctial elements, in order to avoid singularities and extend its range of application. Finally, the strategy at hand is tested in the presence of all the relevant perturbations (i.e., harmonics of the selenopotential, third- and fourth-body effects) in order to show its effectiveness and efficiency.

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 outpost 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 Singh 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 Liu 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, Conway 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, Kluever 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]. Gurfil 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]. Chang et al. employed Lyapunov stability theory in order to derive a feedback controller [14], while Leeghim 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 described in 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.

2. Orbit Dynamics

An accurate description of the orbital dynamics requires a proper definition of an inertial frame. As a preliminary step, the Moon Centered Inertial frame (MCI) is introduced, with unit vectors $({\widehat{c}}_1^M,{\widehat{c}}_2^M,{\widehat{c}}_3^M)$ defined as follows: (i) ${\widehat{c}}_1^M$ lies in the Moon equatorial plane and is coplanar with the line that connects the Earth and the Moon at a reference time $t_{ref}$, directed toward the far side of the Moon; (ii) ${\widehat{c}}_3^M$ is aligned with the Moon rotation axis at $t_{ref}$; (iii) ${\widehat{c}}_2^M$ completes the right-handed set. The Local Vertical Local Horizontal (LVLH) frame is instead aligned with unit vectors $(\widehat{r},\widehat{\theta },\widehat{h})$: (i) $\widehat{r}$ is directed from the Moon center to the instantaneous position of the vehicle; (ii) $\widehat{h}$ is aligned with the spacecraft angular momentum; (iii) $\widehat{\theta }$ completes the right-handed set. If ${\mathbf{R}}_i\left(\alpha \right)$ denotes an elementary rotation matrix by angle $\alpha$ about axis i, then the two reference frames can be linked through three successive rotations,

$$\left[\begin{array}{c}\widehat{r}\\ \widehat{\theta }\\ \widehat{h}\end{array}\right]={\mathbf{R}}_3(\omega +\theta ){\mathbf{R}}_1\left(i\right){\mathbf{R}}_3(\Omega )\left[\begin{array}{c}{\widehat{c}}_1^M\\ {\widehat{c}}_2^M\\ {\widehat{c}}_3^M\end{array}\right],$$

(1)

where $\Omega$ is the Right Ascension of the Ascending Node (RAAN), i the inclination of the orbital plane, $\omega$ the argument of periselenium, and $\theta$ the true anomaly. Furthermore, we introduce the Local Horizontal (LH) frame, associated with unit vectors $(\widehat{r},\widehat{E},\widehat{N})$, where $\widehat{E}$ and $\widehat{N}$ are aligned with the local (lunar) East and North directions, respectively. If $\zeta$ is the heading angle, then the following relation holds,

$$\left[\begin{array}{c}\widehat{r}\\ \widehat{\theta }\\ \widehat{h}\end{array}\right]={\mathbf{R}}_1\left(\zeta \right)\left[\begin{array}{c}\widehat{r}\\ \widehat{E}\\ \widehat{N}\end{array}\right].$$

(2)

Figure 1 shows the reference frames and related angles. The spacecraft is modeled as a point mass and its motion is mainly affected by the Moon gravitational field. The presence of perturbing effects is also addressed: (i) the gravitational pull of Earth and Sun; (ii) the relevant harmonics of the selenopotential (down to two orders of magnitude less than the largest harmonic term $J_2$), that is, those associated with coefficients $|J_{lm}|>{10}^{-6}$. Appendix A collects all planetary data employed in this research. The gravitational influence of the Earth and Sun can be modeled as third body perturbations following Battin–Giorgi approach ([25,26]):

$$\begin{array}{c}{\overrightarrow{\mathit{a}}}_{3i}=-{\displaystyle \frac{{\mu }_i}{\left|\right|{\overrightarrow{\mathit{r}}}_i{\left|\right|}^3{(q_i+1)}^{3/2}}}\left(\overrightarrow{\mathit{r}}+{\overrightarrow{\mathit{r}}}_i{q}_i{\displaystyle \frac{q_i^2+3q_i+3}{{(q_i+1)}^{3/2}+1}}\right),q_i:={\displaystyle \frac{\overrightarrow{\mathit{r}}·\overrightarrow{\mathit{r}}-2\overrightarrow{\mathit{r}}·{\overrightarrow{\mathit{r}}}_i}{{\overrightarrow{\mathit{r}}}_i·{\overrightarrow{\mathit{r}}}_i}},\end{array}$$

(3)

where ${\overrightarrow{\mathit{r}}}_i$ denotes the vector position of the i-body with respect to the Moon and $\overrightarrow{\mathit{r}}$ the instantaneous position vector of the spacecraft with respect to the central body.

Instead, the selenopotential U can be expanded in terms of Legendre polynomials $P_{l,m}$,

$$U={\displaystyle \frac{{\mu }_m}{r}}-{\displaystyle \frac{{\mu }_m}{r}}\sum _{l=2}^{\infty }{\left({\displaystyle \frac{R_m}{l}}\right)}^l{J}_l{P}_{l0}\left(sin\varphi \right)+{\displaystyle \frac{{\mu }_m}{r}}\sum _{l=2}^{\infty }\sum _{m=1}^l{\left({\displaystyle \frac{R_m}{r}}\right)}^l{J}_{l,m}{P}_{l,m}(sin\varphi )cos\left[m({\lambda }_g-{\lambda }_{l,m})\right],$$

(4)

where $R_m$ is the lunar equatorial radius, $\varphi$ the geographical latitude and ${\lambda }_g$ the geographical longitude, and the coefficients ($J_{lm}$ and ${\lambda }_{lm}$) of zonal, tesseral, and sectoral harmonics are provided by the Lunar Prospector LP100K model for the Moon gravitational field [27]. Then, the gravitational acceleration can be computed in the LH-frame,

$$\overrightarrow{\mathit{g}}=\nabla U,\mathrm{where}\nabla =\widehat{r}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{\widehat{E}}{rcos\varphi }}{\displaystyle \frac{\partial }{\partial {\lambda }_g}}+{\displaystyle \frac{\widehat{N}}{r}}{\displaystyle \frac{\partial }{\partial \varphi }}.$$

(5)

Finally, by means of Equation (2), it is straightforward to obtain the components of the perturbing acceleration in the LVLH-frame. The strongest gravitational perturbation is associated with the coefficient $J_2$. It is useful to express the perturbing acceleration due to the zonal harmonic $J_2$ in the LVLH-frame,

$${\overrightarrow{\mathit{a}}}_{J_2}={\displaystyle \frac{3{\mu }_m}{r^4}}{R}_m^2{J}_2{\left[\begin{array}{c}{\displaystyle \frac{3{sin}^2(\omega +\theta ){sin}^{2}i-1}{2}}\\ -{sin}^{2}isin(\omega +\theta )cos(\omega +\theta )\\ -sinicosisin(\omega +\theta )\end{array}\right]}^T\left[\begin{array}{c}\widehat{r}\\ \widehat{\theta }\\ \widehat{h}\end{array}\right]=\left[\begin{array}{ccc}{a}_r^{J_2}& a_{\theta }^{J_2}& a_h^{J_2}\end{array}\right]\left[\begin{array}{c}\widehat{r}\\ \widehat{\theta }\\ \widehat{h}\end{array}\right],$$

(6)

where ${\mu }_m$ is the Moon gravitational parameter. The $J_2$ harmonic is responsible for the time variation of the orbital elements, as predicted by the well-known Gauss equations [28],

$$\begin{array}{c}\dot{p}=2\sqrt{{\displaystyle \frac{p}{{\mu }_m}}}r{a}_{\theta }^{J_2},\end{array}$$

(7)

$$\begin{array}{c}\dot{e}=\sqrt{{\displaystyle \frac{p}{{\mu }_m}}}{a}_r^{J_2}sin\theta +\sqrt{{\displaystyle \frac{p}{{\mu }_m}}}{a}_{\theta }^{J_2}{\displaystyle \frac{e+e{cos}^2\theta +2cos\theta }{1+ecos\theta }},\end{array}$$

(8)

$$\begin{array}{c}\dot{i}=r{a}_{\theta }^{J_2}{\displaystyle \frac{cos(\omega +\theta )}{h}},\end{array}$$

(9)

$$\begin{array}{c}\dot{\Omega }=r{a}_h^{J_2}{\displaystyle \frac{sin(\omega +\theta )}{hsini}},\end{array}$$

(10)

$$\begin{array}{c}\dot{\omega }=-r{a}_h^{J_2}{\displaystyle \frac{sin(\omega +\theta )cosi}{hsini}}-{\displaystyle \frac{a_r^{J_2}}{e}}cos\theta \sqrt{{\displaystyle \frac{p}{{\mu }_m}}}+a_{\theta }^{J_2}\sqrt{{\displaystyle \frac{p}{{\mu }_m}}}sin\theta {\displaystyle \frac{ecos\theta +2}{e(1+ecos\theta )}},\end{array}$$

(11)

$$\begin{array}{c}\dot{\theta }=\sqrt{{\displaystyle \frac{{\mu }_m}{p^3}}}{(1+ecos\theta )}^2+{\displaystyle \frac{a_r^{J_2}}{e}}cos\theta \sqrt{{\displaystyle \frac{p}{{\mu }_m}}}-a_{\theta }^{J_2}\sqrt{{\displaystyle \frac{p}{{\mu }_m}}}sin\theta {\displaystyle \frac{ecos\theta +2}{e(1+ecos\theta )}},\end{array}$$

(12)

where p is the semilatus rectum and e the eccentricity. However, it is possible to apply averaging techniques in order to filter short period oscillations. This procedure yields the average time derivatives of the orbital elements, which equal zero for the semimajor axis, the eccentricy, and the inclination. Instead, the RAAN, the argument of pericenter and the true anomaly exhibit a secular variation. Therefore, the proposed strategy is to employ the control effort to maintain a constant semimajor axis, eccentricity and inclination, while complying with the variation of the RAAN, argument of pericenter and true anomaly, in order to achieve propellant saving.

The orbit dynamics can be described by means of modified equinoctial elements $x_i$, $i=1,\dots ,6$,

$$\begin{array}{c}{x}_1=a(1-e^2),\end{array}$$

(13)

$$\begin{array}{c}{x}_2=ecos(\Omega +\omega ),\end{array}$$

(14)

$$\begin{array}{c}{x}_3=esin(\Omega +\omega ),\end{array}$$

(15)

$$\begin{array}{c}{x}_4=tan{\displaystyle \frac{i}{2}}cos\Omega ,\end{array}$$

(16)

$$\begin{array}{c}{x}_5=tan{\displaystyle \frac{i}{2}}sin\Omega ,\end{array}$$

(17)

$$\begin{array}{c}{x}_6=\Omega +\omega +\theta ,\end{array}$$

(18)

where a is the semi-major axis. This set of elements is well-suited for numerical integration, as it does not exhibit singularities for every keplerian trajectories, with the exception of equatorial retrograde orbits. Given the vector of non Keplerian accelerations $\mathit{a}={\left[\begin{array}{ccc}{a}_r& a_{\theta }& a_h\end{array}\right]}^T$ expressed in the LVLH frame and $\mathit{x}:={\left[\begin{array}{ccccc}{x}_1& x_2& x_3& x_4& x_5\end{array}\right]}^T$, the dynamical equations can be written as

$$\begin{array}{c}\dot{\mathit{x}}=\mathbf{G}(\mathit{x},x_6)\mathit{a},\end{array}$$

(19)

where $\eta =1+x_{2}cos{x}_6+x_{3}sin{x}_6$ and

$$\mathbf{G}=\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}\left[\begin{array}{ccc}0& {\displaystyle \frac{2x_1}{\eta }}& 0\\ sin{x}_6& {\displaystyle \frac{(1+\eta )cos{x}_6+x_2}{\eta }}& -{\displaystyle \frac{x_{4}sin{x}_6-x_{5}cos{x}_6}{\eta }}{x}_3\\ -cos{x}_6& {\displaystyle \frac{(1+\eta )sin{x}_6+x_3}{\eta }}& {\displaystyle \frac{x_{4}sin{x}_6-x_{5}cos{x}_6}{\eta }}{x}_2\\ 0& 0& {\displaystyle \frac{1+x_4^2+x_5^2}{2\eta }}cos{x}_6\\ 0& 0& {\displaystyle \frac{1+x_4^2+x_5^2}{2\eta }}sin{x}_6\end{array}\right].$$

(20)

The dynamical equation for the orbital element $x_6$ is instead given by

$${\dot{x}}_6=\sqrt{{\displaystyle \frac{{\mu }_m}{x_1^3}}}{\eta }^2+\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}{\displaystyle \frac{x_{4}sin{x}_6-x_{5}cos{x}_6}{\eta }}{a}_h.$$

(21)

Finally, if T denotes the instantaneous thrust magnitude and $m_0$ the initial mass, the dynamical equation for the mass ratio $x_7$ can be obtained,

$${\dot{x}}_7=-{\displaystyle \frac{u_T}{c}},\mathrm{with}{u}_T={\displaystyle \frac{T}{m_0}},$$

(22)

where c is the effective exhaust velocity of the propulsion system. The thrust can be modulated up to its maximum value $T_{max}$, and thus $0\le u_T\le u_{Tmax}=T_{max}/m_0$.

3. Nonlinear Orbit Control

The aim of this section is to introduce a nonlinear feedback control law able to perform an initial orbit acquisition and keep the spacecraft along the desired orbit.

3.1. Formulation of the Problem

Nonlinear orbit control is proposed as an effective strategy to drive a spacecraft toward a specified orbit (with subsequent maintenance), starting from a different initial orbit. As described in Section 2, the main effect the $J_2$ harmonic is to yield a secular variation in the orbital elements $(\Omega ,\omega ,\theta )$. Therefore, the state is integrated numerically by assigning the time variation of the desired orbital elements $(a_d,e_d,i_d,{\Omega }_d\left(t\right),{\omega }_d\left(t\right),{\theta }_d\left(t\right))$,

$$\begin{array}{c}{\dot{a}}_d={\dot{e}}_d={\dot{i}}_d=0,\end{array}$$

(23)

while ${\dot{\Omega }}_d$, ${\dot{\omega }}_d$, and ${\dot{\theta }}_d$ are given according to Equations (10)–(12). Then, through analytical differentiation, it is straightforward to convert the time variation of the classical orbital elements into equinoctial elements,

$$\begin{array}{c}{\dot{x}}_{1d}={\dot{a}}_d(1-e_d^2)-2e_d{\dot{e}}_d{a}_d=0,\end{array}$$

(24)

$$\begin{array}{c}{\dot{x}}_{2d}={\dot{e}}_{d}cos({\Omega }_d+{\omega }_d)-({\dot{\Omega }}_d+{\dot{\omega }}_d)e_{d}sin({\Omega }_d+{\omega }_d)=0,\end{array}$$

(25)

$$\begin{array}{c}{\dot{x}}_{3d}={\dot{e}}_{d}sin({\Omega }_d+{\omega }_d)+({\dot{\Omega }}_d+{\dot{\omega }}_d)e_{d}cos({\Omega }_d+{\omega }_d)=0,\end{array}$$

(26)

$$\begin{array}{c}{\dot{x}}_{4d}=-tan\left({\displaystyle \frac{i_d}{2}}\right){\dot{\Omega }}_{d}sin{\Omega }_d+{\displaystyle \frac{1}{2{cos}^2{i}_d}}{\dot{i}}_{d}cos{\Omega }_d,\end{array}$$

(27)

$$\begin{array}{c}{\dot{x}}_{5d}=tan\left({\displaystyle \frac{i_d}{2}}\right){\dot{\Omega }}_{d}cos{\Omega }_d+{\displaystyle \frac{1}{2{cos}^2{i}_d}}{\dot{i}}_{d}sin{\Omega }_d,\end{array}$$

(28)

$$\begin{array}{c}{\dot{x}}_{6d}={\dot{\Omega }}_d+{\dot{\omega }}_d+{\dot{\theta }}_d,\end{array}$$

(29)

where subscript d refers to the desired value of the equinoctial element. Equations (24) through (29) can be integrated to yield the time history of the desired orbit.

3.2. Feedback Control Law

The operational orbit conditions can be enforced as a nonlinear constraint,

$$\begin{array}{c}\mathbf{\Psi }(\mathit{x},t)=\left[\begin{array}{c}{\Psi }_1\\ {\Psi }_2\\ {\Psi }_3\end{array}\right]=\mathbf{0}.\end{array}$$

(30)

Specifically, the components of the constraint can be expressed as

$$\begin{array}{c}{\Psi }_1={(x_1-x_{1d})}^2=0,\end{array}$$

(31)

$$\begin{array}{c}{\Psi }_2={(x_2-x_{2d})}^2+{(x_3-x_{3d})}^2=0,\end{array}$$

(32)

$$\begin{array}{c}{\Psi }_3={(x_4-x_{4d})}^2+{(x_5-x_{5d})}^2=0.\end{array}$$

(33)

The first component ${\Psi }_1$ corresponds to a constraint on the semilatus rectum,

$${\Psi }_1={(p-p_d)}^2.$$

(34)

Instead, the second component ${\Psi }_2$ pursues the desired eccentricity of the orbit,

$$\begin{array}{c}{(x_2-x_{2d})}^2+{(x_3-x_{3d})}^2=\\ [ecos(\Omega +\omega )-e_{d}cos({\Omega }_d+{\omega }_d){]}^2+{[esin(\Omega +\omega )-e_{d}sin({\Omega }_d+{\omega }_d)]}^2=\\ e^2+e_d^2-2e{e}_{d}cos(\Omega -{\Omega }_d+\omega -{\omega }_d).\end{array}$$

(35)

Finally, we can prove that component ${\Psi }_3$ fixes the orientation of the orbital plane. Indeed, let us denote with $\overrightarrow{\mathit{h}}$ and ${\overrightarrow{\mathit{h}}}_d$ the actual and desired angular momentum unit vectors as expressed in the MCI-frame,

$$\begin{array}{c}\overrightarrow{\mathit{h}}={\left[\begin{array}{c}sin\Omega sini\\ cos\Omega sini\\ cosi\end{array}\right]}^T\left[\begin{array}{c}{\widehat{c}}_1^M\\ {\widehat{c}}_2^M\\ {\widehat{c}}_3^M\end{array}\right]={\left[\begin{array}{c}{\displaystyle \frac{2x_5}{1+x_4^2+x_5^2}}\\ {\displaystyle \frac{2x_4}{1+x_4^2+x_5^2}}\\ {\displaystyle \frac{1-x_4^2-x_5^2}{1+x_4^2+x_5^2}}\end{array}\right]}^T\left[\begin{array}{c}{\widehat{c}}_1^M\\ {\widehat{c}}_2^M\\ {\widehat{c}}_3^M\end{array}\right],\end{array}$$

(36)

$$\begin{array}{c}{\overrightarrow{\mathit{h}}}_d={\left[\begin{array}{c}sin{\Omega }_{d}sin{i}_d\\ cos{\Omega }_{d}sin{i}_d\\ cos{i}_d\end{array}\right]}^T\left[\begin{array}{c}{\widehat{c}}_1^M\\ {\widehat{c}}_2^M\\ {\widehat{c}}_3^M\end{array}\right]={\left[\begin{array}{c}{\displaystyle \frac{2x_{5d}}{1+x_{4d}^2+x_{5d}^2}}\\ {\displaystyle \frac{2x_{4d}}{1+x_{4d}^2+x_{5d}^2}}\\ {\displaystyle \frac{1-x_{4d}^2-x_{5d}^2}{1+x_{4d}^2+x_{5d}^2}}\end{array}\right]}^T\left[\begin{array}{c}{\widehat{c}}_1^M\\ {\widehat{c}}_2^M\\ {\widehat{c}}_3^M\end{array}\right].\end{array}$$

(37)

Then, the alignment condition between the actual and the desired orbital plane can be expressed as a scalar product,

$$\begin{array}{c}\overrightarrow{\mathit{h}}·{\overrightarrow{\mathit{h}}}_d=1.\end{array}$$

(38)

By expanding the scalar product, we obtain

$$\begin{array}{c}2x_4{x}_{4d}+2x_5{x}_{5d}+(1-x_4^2-x_5^2)(1-x_{4d}^2-x_{5d}^2)=(1+x_4^2+x_5^2)(1+x_{4d}^2+x_{5d}^2).\end{array}$$

(39)

Equation (39) can be easily simplified into

$$\begin{array}{c}-2x_4{x}_{4d}-2x_5{x}_{5d}+x_4^2+x_5^2+x_{4d}^2+x_{5d}^2={(x_4-x_{4d})}^2+{(x_5-x_{5d})}^2=0.\end{array}$$

(40)

Equations (31)–(33) define the target set. In order to achieve the desired final conditions, a nonlinear control law is identified through the definition of a meaningful Lyapunov function,

$$V={\displaystyle \frac{1}{2}}{\mathbf{\Psi }}^T\mathbf{K}\mathbf{\Psi },$$

(41)

where $\mathbf{K}$ is a symmetric and positive definite matrix. It is straightforward to recognize that $V>0$ unless $\mathbf{\Psi }=\mathbf{0}$. Let us introduce the auxiliary vectors

$$\begin{array}{c}\mathit{b}={\mathbf{G}}^T{\left({\displaystyle \frac{\partial \mathbf{\Psi }}{\partial \mathit{x}}}\right)}^T\mathbf{K}\mathbf{\Psi },\end{array}$$

(42)

$$\begin{array}{c}\mathit{d}={\mathit{a}}_P+{\left({\displaystyle \frac{\partial \mathbf{\Psi }}{\partial \mathit{x}}}\mathbf{G}\right)}^{-1}{\displaystyle \frac{\partial \mathbf{\Psi }}{\partial t}},\end{array}$$

(43)

where ${\mathit{a}}_P$ denotes the total perturbing acceleration. Then, as already proven in reference [23], the following nonlinear control law leads the dynamical system to converge to the target state, provided that $\mathbf{\Psi }$ and ${\displaystyle \frac{\partial \mathbf{\Psi }}{\partial \mathit{x}}}$ are continuous, ${\left[(\partial \mathbf{\Psi }/\partial \mathit{x}\mathbf{G})\right]}^{-1}(\partial \mathbf{\Psi }/\partial \mathit{t})$ is finite, $\left|\mathit{b}\right|>0$ unless $\mathbf{\Psi }=\mathbf{0}$, and $u_{Tmax}>x_7\left|\mathbf{d}\right|$:

$$\begin{array}{c}{\mathit{u}}_T=-u_T{\displaystyle \frac{\mathit{b}+\mathit{d}}{|\mathit{b}+\mathit{d}|}},\mathrm{with}\end{array}$$

(44)

$$\begin{array}{c}{u}_T=\left\{\begin{array}{c}{x}_7|\mathit{b}+\mathit{d}|,\mathrm{if}{x}_7|\mathit{b}+\mathit{d}|\le u_{Tmax}\hfill \\ \hfill 0,\mathrm{if}{x}_7|\mathit{b}+\mathit{d}|>u_{Tmax}\mathrm{and}{\mathit{b}}^T\mathit{b}+{\mathit{b}}^T\mathbf{d}<0\hfill \\ \hfill u_{Tmax},\mathrm{if}{x}_7|\mathit{b}+\mathit{d}|>u_{Tmax}\mathrm{and}{\mathit{b}}^T\mathit{b}+{\mathit{b}}^T\mathit{d}\ge 0.\hfill \end{array}\right.\end{array}$$

(45)

The attracting set contains all the dynamical states that fulfill $\dot{V}=0$, that is, if all the components of $\mathit{b}$ are zero regardless of any choice of the gain coefficients $(k_1,k_2,k_3)$. Thus, the attracting set must be investigated in order to guarantee asymptotic stability via the Lyapunov theorem. Equations (46)–(48) show the analytical expression of the components of vector $\mathit{b}$, denoted with $(b_1,b_2,b_3)$,

$$\begin{array}{c}{b}_1=-2k_2{\Psi }_2\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}\left[(x_3-x_{3d})cos{x}_6-(x_2-x_{2d})sin{x}_6\right],\\ b_2=4k_1{\Psi }_1{\displaystyle \frac{x_1-x_{1d}}{\eta }}\sqrt{{\displaystyle \frac{x_1^3}{{\mu }_m}}}+\end{array}$$

(46)

$$\begin{array}{c}2k_2\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}{\displaystyle \frac{{\Psi }_2}{\eta }}\left\{(x_2-x_{2d})[x_2+(1+\eta )cos{x}_6]+(x_3-x_{3d})[x_3+(1+\eta )sin{x}_6]\right\},\\ b_3=2k_2\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}{\displaystyle \frac{{\Psi }_2}{\eta }}(x_{5}cos{x}_6-x_{4}sin{x}_6)\left[(x_2-x_{2d})x_3-(x_3-x_{3d})x_2\right]\end{array}$$

(47)

$$\begin{array}{c}+k_3{\Psi }_3\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}{\displaystyle \frac{1+x_4^2+x_5^2}{\eta }}\left[(x_4-x_{4d})cos{x}_6+(x_5-x_{5d})sin{x}_6\right].\end{array}$$

(48)

First of all, it is straightforward to recognize that if $x_1=0$ (rectilinear trajectories), $b_1=b_2=b_3=0$. If $x_1\ne 0$, then $b_1=0$ if either ${\Psi }_2=0$ or

$$(x_3-x_{3d})cos{x}_6-(x_2-x_{2d})sin{x}_6=0.$$

(49)

The latter relation is satisfied regardless of $x_6$ (which is time-varying also along the desired orbit) only if $x_2=x_{2d}$ and $x_3=x_{3d}$, which is equivalent to ${\Psi }_2=0$. This implies that also $b_2=0$ if and only if ${\Psi }_1=0$. Finally, $b_3=0$ regardless of $x_6$ if either ${\Psi }_3=0$ or $x_4=x_{4d}$ and $x_5=x_{5d}$, which is, however, equivalent to ${\Psi }_3=0$. To sum up, the attracting set includes the following subsets:

• $x_1=0$;
• ${\Psi }_1={\Psi }_2={\Psi }_3=0$ (target set).

However, as the condition $\mathit{b}=0$ implies $\dot{\mathit{b}}=0$, it is possible to rule out subset 1. Indeed, we have

$${\dot{b}}_i={\displaystyle \frac{\partial b_i}{\partial \mathit{x}}}\dot{\mathit{x}}+{\displaystyle \frac{\partial b_i}{\partial x_6}}{\dot{x}}_6+\sum _{j=1}^5{\displaystyle \frac{\partial b_i}{\partial x_{jd}}}{\dot{x}}_{jd}={\displaystyle \frac{\partial b_i}{\partial \mathit{x}}}\mathbf{G}\mathit{a}+{\displaystyle \frac{\partial b_i}{\partial x_6}}{\dot{x}}_6+\sum _{j=1}^5{\displaystyle \frac{\partial b_i}{\partial x_{jd}}}{\dot{x}}_{jd},$$

(50)

for $i=1,2,3$. If we evaluate this expression when $\mathit{a}=\mathbf{0}$, we obtain

$${\dot{b}}_i={\displaystyle \frac{\partial b_i}{\partial x_6}}\sqrt{{\displaystyle \frac{{\mu }_m}{x_1^3}}}{\eta }^2+\sum _{j=1}^5{\displaystyle \frac{\partial b_i}{\partial x_{jd}}}{\dot{x}}_{jd}.$$

(51)

For simplicity, let us rename

$$\begin{array}{c}{b}_1={\Psi }_2(x_2,x_{2d},x_3,x_{3d})\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}{f}_1(x_2,x_{2d},x_3,x_{3d},x_6),\\ b_2={\Psi }_1(x_1,x_{1d})\sqrt{{\displaystyle \frac{x_1^3}{{\mu }_m}}}{f}_2(x_1,x_{1d},x_2,x_3,x_6)\end{array}$$

(52)

$$\begin{array}{c}+{\Psi }_2(x_2,x_{2d},x_3,x_{3d})\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}{f}_3(x_2,x_{2d},x_3,x_{3d},x_6),\\ b_3={\Psi }_2(x_2,x_{2d},x_3,x_{3d})\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}{f}_4(x_2,x_{2d},x_3,x_{3d},x_4,x_5,x_6)\end{array}$$

(53)

$$\begin{array}{c}+{\Psi }_3(x_4,x_{4d},x_5,x_{5d})\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}{f}_5(x_2,x_3,x_4,x_{4d},x_5,x_{5d},x_6),\end{array}$$

(54)

where

$$\begin{array}{c}{f}_1=-2k_2\left[(x_3-x_{3d})cos{x}_6-(x_2-x_{2d})sin{x}_6\right],\end{array}$$

(55)

$$\begin{array}{c}{f}_2=4k_1{\displaystyle \frac{x_1-x_{1d}}{\eta }},\end{array}$$

(56)

$$\begin{array}{c}{f}_3=2k_2{\displaystyle \frac{(x_2-x_{2d})[x_2+(1+\eta )cos{x}_6]+(x_3-x_{3d})[x_3+(1+\eta )sin{x}_6]}{\eta }},\end{array}$$

(57)

$$\begin{array}{c}{f}_4=2k_2(x_{5}cos{x}_6-x_{4}sin{x}_6)\left[{\displaystyle \frac{(x_2-x_{2d})x_3-(x_3-x_{3d})x_2}{\eta }}\right],\end{array}$$

(58)

$$\begin{array}{c}{f}_5=k_3{\displaystyle \frac{1+x_4^2+x_5^2}{\eta }}\left[(x_4-x_{4d})cos{x}_6+(x_5-x_{5d})sin{x}_6\right].\end{array}$$

(59)

Therefore,

$$\begin{array}{c}{\dot{b}}_1={\Psi }_2{\displaystyle \frac{{\eta }^2}{x_1}}{\displaystyle \frac{\partial f_1}{\partial x_6}}+\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}\left[\left({\displaystyle \frac{\partial {\Psi }_2}{\partial x_{2d}}}{f}_1+{\Psi }_2{\displaystyle \frac{\partial f_1}{\partial x_{2d}}}\right){\dot{x}}_{2d}+\left({\displaystyle \frac{\partial {\Psi }_2}{\partial x_{3d}}}{f}_1+{\Psi }_2{\displaystyle \frac{\partial f_1}{\partial x_{3d}}}\right){\dot{x}}_{3d}\right],\\ {\dot{b}}_2={\Psi }_1{\eta }^2{\displaystyle \frac{\partial f_2}{\partial x_6}}+{\Psi }_2{\displaystyle \frac{{\eta }^2}{x_1}}{\displaystyle \frac{\partial f_3}{\partial x_6}}+\sqrt{{\displaystyle \frac{x_1^3}{{\mu }_m}}}\left(f_2{\displaystyle \frac{\partial {\Psi }_1}{\partial x_{1d}}}+{\Psi }_1{\displaystyle \frac{\partial f_2}{\partial x_{1d}}}\right){\dot{x}}_{1d}\end{array}$$

(60)

$$\begin{array}{c}+\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}\left[\left({\displaystyle \frac{\partial {\Psi }_2}{\partial x_{2d}}}{f}_3+{\Psi }_2{\displaystyle \frac{\partial f_3}{\partial x_{2d}}}\right){\dot{x}}_{2d}+\left({\displaystyle \frac{\partial {\Psi }_2}{\partial x_{3d}}}{f}_3+{\Psi }_2{\displaystyle \frac{\partial f_3}{\partial x_{3d}}}\right){\dot{x}}_{3d}\right],\\ {\dot{b}}_3=\left({\Psi }_2{\displaystyle \frac{\partial f_4}{\partial x_6}}+{\Psi }_3{\displaystyle \frac{\partial f_5}{\partial x_6}}\right){\displaystyle \frac{{\eta }^2}{x_1}}\\ +\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}\left[\left({\displaystyle \frac{\partial {\Psi }_2}{\partial x_{2d}}}{f}_4+{\displaystyle \frac{\partial f_4}{\partial x_{2d}}}{\Psi }_2\right){\dot{x}}_{2d}+\left({\displaystyle \frac{\partial {\Psi }_2}{\partial x_{3d}}}{f}_4+{\displaystyle \frac{\partial f_4}{\partial x_{3d}}}{\Psi }_2\right){\dot{x}}_{3d}\right]\end{array}$$

(61)

$$\begin{array}{c}+\sqrt{{\displaystyle \frac{x_1}{{\mu }_m}}}\left[\left({\displaystyle \frac{\partial {\Psi }_3}{\partial x_{4d}}}{f}_5+{\displaystyle \frac{\partial f_5}{\partial x_{4d}}}{\Psi }_3\right){\dot{x}}_{4d}+\left({\displaystyle \frac{\partial {\Psi }_3}{\partial x_{5d}}}{f}_5+{\displaystyle \frac{\partial f_5}{\partial x_{5d}}}{\Psi }_3\right){\dot{x}}_{5d}\right].\end{array}$$

(62)

It is evident that, as investigated in [23], the subset associated with the rectilinear trajectories can be ruled out as the time derivative $\dot{\mathit{b}}$ diverges. Instead, it is straightforward to see that, when the target set is reached, the following relations hold:

$$\begin{array}{c}{f}_1=f_2=f_3=f_4=f_5=0,\end{array}$$

(63)

and therefore, $\dot{\mathit{b}}=\mathbf{0}$ as well. Since the invariant set does not contain other subsets than the target set, the asymptotic stability is quasi-global. Finally, the quantity ${\left[(\partial \mathbf{\Psi }/\partial \mathit{x})\mathbf{G}\right]}^{-1}(\partial \mathbf{\Psi }/\partial t)$ must be investigated to check any singularity issue. Let us denote with $({\overline{d}}_1,{\overline{d}}_2,{\overline{d}}_3)$ the components of vector $\mathit{d}-{\mathit{a}}_P$. First of all, component ${\overline{d}}_2$ is inspected. Using Mathematica [29], one can prove that

$$\begin{array}{c}\underset{\mathbf{x}\to {\mathbf{x}}_d}{lim}{\overline{d}}_2=-\sqrt{{\displaystyle \frac{{\mu }_m}{x_{1d}}}}{\dot{x}}_{1d}{\displaystyle \frac{(1+x_{2d}cos{x}_6+x_{3d}sin{x}_6)}{2x_{1d}}},\end{array}$$

(64)

which is finite if and only if $x_{1d}\ne 0$. Instead, the analytical expression for component ${\overline{d}}_1$ is

$${\overline{d}}_1=\sqrt{{\displaystyle \frac{{\mu }_m}{x_1}}}{\displaystyle \frac{(x_2-x_{2d}){\dot{x}}_{2d}+(x_3-x_{3d}){\dot{x}}_{3d}}{(x_3-x_{3d})cos{x}_6-(x_2-x_{2d})sin{x}_6}}.$$

(65)

The relation for ${\overline{d}}_1$ can be written as

$$\begin{array}{c}{\overline{d}}_1=\sqrt{{\displaystyle \frac{{\mu }_m}{x_1}}}\left\{{\displaystyle \frac{{\dot{x}}_{2d}}{{\displaystyle \frac{x_3-x_{3d}}{x_2-x_{2d}}}cos{x}_6-sin{x}_6}}+{\displaystyle \frac{{\dot{x}}_{3d}}{cos{x}_6-{\displaystyle \frac{x_2-x_{2d}}{x_3-x_{3d}}}sin{x}_6}}\right\}.\end{array}$$

(66)

The following limits turn out to be indeterminate:

$$\begin{array}{c}\underset{x_2\to x_{2d},x_3\to x_{3d}}{lim}{\displaystyle \frac{x_2-x_{2d}}{x_3-x_{3d}}},\end{array}$$

(67)

$$\begin{array}{c}\underset{x_2\to x_{2d},x_3\to x_{3d}}{lim}{\displaystyle \frac{x_3-x_{3d}}{x_2-x_{2d}}},\end{array}$$

(68)

However, the following parameterization can be used as $x_2\to x_{2d}$ and $x_3\to x_{3d}$:

$$\begin{array}{c}{x}_2=x_{2d}+z_1\tau ,\end{array}$$

(69)

$$\begin{array}{c}{x}_3=x_{3d}+z_2\tau ,\end{array}$$

(70)

where $\tau$ is a parameter that tends to zero, whereas $z_1$ and $z_2$ are two constant values that identify the direction traveled as $x_2\to x_{2d}$ and $x_3\to x_{3d}$. This means that the following limits depend on the ratio $z_1/z_2$:

$$\begin{array}{c}\underset{x_2\to x_{2d},x_3\to x_{3d}}{lim}{\displaystyle \frac{x_2-x_{2d}}{x_3-x_{3d}}}={\displaystyle \frac{z_1}{z_2}},\end{array}$$

(71)

$$\begin{array}{c}\underset{x_2\to x_{2d},x_3\to x_{3d}}{lim}{\displaystyle \frac{x_3-x_{3d}}{x_2-x_{2d}}}={\displaystyle \frac{z_2}{z_1}}.\end{array}$$

(72)

Therefore, the limit of Equation (66) as $x_2\to x_{2d}$ and $x_3\to x_{3d}$ depends on the direction (i.e., the ratio $z_1/z_2$) and is

$${\overline{d}}_1=\sqrt{{\displaystyle \frac{{\mu }_m}{x_{1d}}}}\left\{{\displaystyle \frac{{\dot{x}}_{2d}}{{\displaystyle \frac{z_2}{z_1}}cos{x}_6-sin{x}_6}}+{\displaystyle \frac{{\dot{x}}_{3d}}{cos{x}_6-{\displaystyle \frac{z_1}{z_2}}sin{x}_6}}\right\}.$$

(73)

As $x_6$ is time varying, the component ${\overline{d}}_1$ still remains finite almost everywhere, provided $x_{1d}\ne 0$. Finally, component ${\overline{d}}_3$ is written as

$${\overline{d}}_3=-2\sqrt{{\displaystyle \frac{{\mu }_m}{x_1}}}{\displaystyle \frac{\eta }{1+x_4^2+x_5^2}}\left\{{\displaystyle \frac{(x_4-x_{4d}){\dot{x}}_{4d}+(x_5-x_{5d}){\dot{x}}_{5d}}{(x_4-x_{4d})cos{x}_6+(x_5-x_{5d})sin{x}_6}}\right\}.$$

(74)

The same considerations developed for Equation (66) still hold and ensure that the feedback law at hand is feasible.

4. Numerical Simulations

The spacecraft is assumed to be equipped with a low-thrust propulsion system, characterized by the effective exhaust velocity c and the maximum thrust acceleration $u_{Tmax}$,

$$c=30\mathrm{km}/\mathrm{s},u_{Tmax}=5·{10}^{-5}{g}_0,$$

(75)

where $g_0=9.81{\mathrm{m}/\mathrm{s}}^2$ is the gravity acceleration at sea level. Canonical units are employed for the numerical simulations; the distance unit (DU) is equal to the Moon radius, while the time unit (TU) is selected in order to obtain a unit gravitational parameter,

$$\begin{array}{c}DU=1738.1\mathrm{km},\end{array}$$

(76)

$$\begin{array}{c}TU=\sqrt{{\displaystyle \frac{1}{{\mu }_m}}D{U}^3}.\end{array}$$

(77)

The following weighting coefficients are employed:

$$k_1={10}^5,k_2={10}^5,k_3={10}^7.$$

(78)

Two target orbits are selected, with a different inclination of the orbital plane.

Table 1. Target orbits.

Orbital Element	${\mathit{i}}_{\mathit{d}}$ [deg]	${\mathbf{\Omega }}_{\mathit{d},0}$ [deg]
Target orbit 1	60	300
Target orbit 2	90	300

collects the initial conditions for these two orbits. The initial argument of latitude is set to 10°.

Instead, the initial state of the spacecraft is specified by assigning a random value of the initial radii at periselenium ($r_P^0$) and aposelenium ($r_A^0$), selected using a stochastic uniform distribution over the following intervals:

$$\begin{array}{c}{r}_P^0\in [50,100]\mathrm{km},\end{array}$$

(79)

$$\begin{array}{c}{r}_A^0\in [100,500]\mathrm{km}.\end{array}$$

(80)

Furthermore, the initial inclination and RAAN are chosen according to a normal distribution with mean value, respectively, $i_d$ and ${\Omega }_d^0$, and standard deviation is set to 5 deg.

For each target orbit, two different operational scenarios are analyzed: (i) final circular orbit at an altitude of 100 km; (ii) final elliptic orbit with assigned minimum periselenium radius ($r_{Pmin}=R_m+50$ km) and maximum aposelenium radius ($r_{Amax}=R_m+150$ km). The latter operational scenario can be specified through the following relations:

$$\begin{array}{c}{r}_P={\displaystyle \frac{p}{1+e}}\ge r_{Pmin}\to p\ge r_{Pmin}(1+e),\end{array}$$

(81)

$$\begin{array}{c}{r}_A={\displaystyle \frac{p}{1-e}}\le r_{Amax}\to p\le r_{Amax}(1-e).\end{array}$$

(82)

These two inequalities identify a triangular region in the $(p,e)$ plane, where the motion of the spacecraft is constrained. In particular, after orbit injection, the propulsive effort is employed in order to release the vehicle inside this specific region and the thrust is turned off when the inequalities (81) and (82) are satisfied, together with

$$|{\Psi }_3|<{10}^{-6}.$$

(83)

For each Monte Carlo campaign, 100 simulations are run. The initial reference epoch is set to 1 January 2025 and the total simulation time is about 1 year (370 days, 2 h, 51 min). For each simulation, the mean values of the semimajor axis ($\overline{a}$), eccentricity ($\overline{e}$), inclination ($\overline{i}$), and RAAN deviation ($\Delta \overline{\Omega }$) are computed as

$$\begin{array}{c}\overline{\mathit{a}}={\int }_{t_s}^{t_f}adt,\end{array}$$

(84)

$$\begin{array}{c}\overline{\mathit{e}}={\int }_{t_s}^{t_f}edt,\end{array}$$

(85)

$$\begin{array}{c}\overline{\mathit{i}}={\int }_{t_s}^{t_f}idt,\end{array}$$

(86)

$$\begin{array}{c}\Delta \overline{\Omega }={\int }_{t_s}^{t_f}(\Omega -{\Omega }_d)dt,\end{array}$$

(87)

where $t_f$ is the final time, while $t_s$ is conventionally set to 1 month, when the acquisition phase is completed. For each campaign, the mean values and standard deviations of each mean variable are provided. The run time associated with a single 1-year simulation is about 215 s on an Apple M2 Pro processor, 32 GB RAM, 10 cores (4 high-efficiency + 6 high-performance).

4.1. Target Orbit 1

Table 2. Results from Monte Carlo campaign—circular orbit.

Parameter	Mean Value	Standard Deviation
$\overline{\mathit{a}}$ [km]	1837.96	4.83 · 10−2
$\overline{\mathit{e}}$	3.20 · 10−4	4.15 · 10−6
$\overline{\mathit{i}}$ [deg]	59.99	3.38 · 10−5
$\Delta$$\overline{\Omega }$ [deg]	4.87 · ${10}^{-4}$	9.15 · 10−5
Final Mass Ratio	0.625	5.29 · 10−3

and

Table 3. Results from Monte Carlo campaign—elliptic orbit with assigned $r_{Pmin}$/ $r_{Amax}$.

Parameter	Mean Value	Standard Deviation
$\overline{\mathit{a}}$ [km]	1838.83	3.38 · 10−2
$\overline{\mathit{e}}$	1.54 · 10−3	4.34 · 10−6
$\overline{\mathit{i}}$ [deg]	59.99	8.47 · 10−6
$\Delta$$\overline{\Omega }$ [deg]	8.63 · ${10}^{-2}$	8.63 · 10−4
Final Mass Ratio	0.890	2.94 · 10−3

collect the results for the first target orbit. Figure 2 and Figure 3 show the time histories of the orbital elements, respectively, under no tolerances and when tolerances are included.

4.2. Target Orbit 2

Table 4. Results from Monte Carlo campaign—circular orbit.

Parameter	Mean Value	Standard Deviation
$\overline{\mathit{a}}$ [km]	1838.33	3.37 · 10−3
$\overline{\mathit{e}}$	3.11 · 10−4	1.80 · 10−6
$\overline{\mathit{i}}$ [deg]	90.00	2.45 · 10−7
$\Delta \overline{\Omega }$ [deg]	1.81 · ${10}^{-4}$	3.68 · 10−7
Final Mass Ratio	0.661	1.50 · 10−3

and

Table 5. Results from Monte Carlo campaign—elliptic orbit with assigned $r_{Pmin}$/ $r_{Amax}$.

Parameter	Mean Value	Standard Deviation
$\overline{\mathit{a}}$ [km]	1837.34	1.61 · 10−1
$\overline{\mathit{e}}$	2.31 · 10−3	6.41 · 10−5
$\overline{\mathit{i}}$ [deg]	90.00	7.43 · 10−7
$\Delta \overline{\Omega }$ [deg]	4.42 · ${10}^{-4}$	2.12 · 10−6
Final Mass Ratio	0.911	3.56 · 10−3

collect the results for the second target orbit. Figure 4 and Figure 5 show the time histories of the orbital elements, respectively, under no tolerances and when tolerances are included.

5. Discussion of Results

From inspection of the results, it is evident that the proposed feedback control law is capable of adjusting the injection errors and keeping the spacecraft along the desired operational state, despite the active perturbations. In particular, in both operational scenarios the orbital elements are kept close to the desired value, with limited standard deviations. Unsurprisingly, when tolerances are introduced, the accuracy of the control law tends to decrease. For instance, the comparison between Figure 2a and Figure 3a or between Figure 4a and Figure 5a show that, when the constraint on the final orbit is relaxed, the oscillations in the semimajor axis are wider as expected, though very limited. However, despite a marginal increment in the error, the fuel saving is considerable, as the final mass ratio increases by 43% and 45%, respectively, for target orbit 1 and 2. The same considerations apply from inspection of Figure 2b, Figure 3b, Figure 4b and Figure 5b, by comparing the time histories of the eccentricity. Figure 2d and Figure 4d highlight the capability of the control strategy at hand to keep the RAAN close to the desired value either when the secular variation vanishes or when the gravitational perturbations yield a near-linear drift.

Finally, Figure 6 depicts the time histories of the mass ratio for the first target orbit under the two operational scenarios. In both cases, the thrust is always on and at maximum value during the acquisition phase, and consequently, all the curves overlap, resulting in a global linear trend with respect to time. During the maintenance phase, the perturbation accelerations are comparable with the thrust acceleration. Therefore, when no tolerances are set, the thrust acceleration is variable. Instead, when tolerances are included, orbit maintenance is achieved through the succession of either zero thrust or variable thrust intervals, resulting in a shallow profile that mimics a step function. This results in a globally lower fuel expenditure.

6. Concluding Remarks

This research addresses the application of a Lyapunov feedback controller to inject a spacecraft into a very low lunar orbit, with subsequent maintenance maneuvers. The space vehicle is assumed to be equipped with a steerable and throttleable low-thrust engine. The formulation of the nonlinear constraints is simplified with respect to previous works by employing modified equinoctial elements. The stability analysis is developed in a general framework that includes time-varying orbit elements and proves that the asymptotic stability is quasi-global. Monte Carlo campaigns are carried out for two target orbits with different inclination and two operational scenario, i.e., a circular target orbit or an elliptic target orbit with assigned minimum radius at periselenium and maximum radius at aposelenium. The numerical simulations demonstrate the capability of the strategy at hand to perform orbit acquisition and maintenance despite third body and gravitational perturbations. Very-low-altitude lunar orbits with tight control of the orbit elements appear very attractive for the deployment of satellite constellations. Even though this research investigates operational scenarios with the only precession of the right ascension, the proposed feedback control law can be applied to target non-Keplerian trajectories with time-varying orbital elements.