Halo Orbit Research Articles

Comparison Study of Earth Observation Characteristics between Moon-Based Platform and L1 Point of Earth-Moon System

The unceasing quest for a profound comprehension of the Earth system propels the continuous evolution of novel methods for Earth observation. Of these, the Lagrange points situated in the cislunar space proffer noteworthy prospects for space-based Earth observation. Although extant research predominantly centers on Moon-based Earth observation and the L1 point within the Sun-Earth system, the realm of cislunar space remains relatively unexplored. This paper scrutinizes the overarching characteristics of the L1 point within the Earth-Moon system concerning Earth observation. A pivotal enhancement is introduced through the incorporation of the halo orbit. This research comprehensively analyzes the relative motion between the halo orbiter and the Earth, achieved via orbit determination within a rotating coordinate system, followed by a transformation into the Earth coordinate system. Subsequently, numerical simulations employing ephemeris data unveil the observing geometry and Earth observation characteristics, encompassing the distribution of nadir points, viewing angles, and the spatiotemporal ground coverage. As a point of reference, we also present a case study involving a Moon-based platform. Our findings reveal that the motion of the halo orbit, perpendicular to the lunar orbital plane, results in a broader range of nadir point latitudes, which can extend beyond 42°N/S, contingent upon the orbit’s size. Additionally, it manifests a more intricate latitude variation, characterized by the bimodal peaks of the proposed temporal complexity curve. The viewing angles and the spatiotemporal ground coverage closely resemble those of Moon-based platforms, with a marginal enhancement in coverage frequency for polar regions. Consequently, it can be deduced that the Earth observation characteristics of the L1 point within the Earth-Moon system bear a close resemblance to those of Moon-based platforms. Nevertheless, considering the distinct advantages of Moon-based platforms, the lunar surface remains the paramount choice, boasting the highest potential for Earth observation within cislunar space. In summation, this study demonstrates the Earth observation characteristics of the L1 point within the Earth-Moon system, emphasizing the distinctions between this and Moon-based platforms.

Thermal design and flight validation of the chang’e-4 relay satellite

The Chang’e-4 relay satellite operates in the HALO orbit at the Earth-Moon Lagrange L2 point, which can provide relay communication to the lander and the rover operating on the lunar far side to maintain their contacts with the earth and is a key link of the Chang’e-4 exploration mission. Unlike most LEO (Low Earth Orbit) satellites, the outer space environment of the Chang’e-4 relay satellite is much harsher. The Chang’e-4 relay satellite must suffer from a torrid environment that faces direct sunlight or direct exposure to 4K deep cold space in a shadow zone which may last longer than 6 hours. The Chang’e-4 relay satellite must ensure the operation and temperature control of instruments and equipment during the shadow zone with a limited battery level. This proposes high demands on the TCD (Thermal Control Design). The entire satellite resources were reasonably used for integrated TCD with limited battery level and limited weight. According to the working mode of the Chang’e-4 relay satellite and its torrid environment, its TCD was completed successfully, and on-orbit test data conformed to the expected results and theoretical calculation. TCD of the Chang’e-4 relay satellite has worked normally since the day that the Chang’e-4 relay satellite was launched into space.

Utilizing the geometric mechanics framework to predict attitude in a full ephemeris model of the Cislunar region

Predicting the attitude of spacecraft traveling within the Cislunar region is crucial to ensure the success of future missions planned within that realm of space. To consider the coupled nature of translational and rotational motion, the circular restricted full three-body problem (CRF3BP), previously proposed by the authors, is utilized to obtain an initial guess for the orbit and attitude motion of the spacecraft. Then, that initial guess is used to transition the trajectory into a full ephemeris model, which considers the additional perturbations due to the Sun’s gravitational effect and the eccentricity of the Moon’s orbit. Orbit/attitude coupling is studied for multiple orbits obtained within the CRF3BP considering spacecraft of different sizes. The trajectories selected for analysis are an L2 Lyapunov orbit, a distant retrograde orbit, a northern halo orbit, and a Lunar transfer from low-Earth orbit. The efficacy of the CRF3BP is demonstrated by studying how spacecraft motion, notably its attitude, is predicted in the full ephemeris model using this formalism.

Analysis, Design, and Optimization of Robust Trajectories in Cislunar Environment for Limited-Capability Spacecraft

Nowadays, the space exploration is going in the direction of exploiting small platforms to get high scientific return at significantly lower costs. However, miniaturized spacecraft pose different challenges both from the technological and mission analysis point of view. While the former is in constant evolution due to the manufacturers, the latter is an open point, since it is still based on a traditional approach, not able to cope with the new platforms’ peculiarities. In this work, a revised preliminary mission analysis approach, merging the nominal trajectory optimization with a complete navigation assessment, is formulated in a general form and three main blocks composing it are identified. Then, the integrated approach is specialized for a cislunar test case scenario, represented by the transfer trajectory from a low lunar orbit to an halo orbit of the CubeSat LUMIO, and each block is modeled with mathematical means. Eventually, optimal solutions, minimizing the total costs, are sought, showing the benefits of an integrated approach.

Off-nominal transfer analysis for NASA’s solar cruiser mission

In this work, off-nominal trajectory design options for NASA’s sail-based Solar Cruiser mission are examined. A novel method for constructing bounded trajectories about a periodic reference orbit is used to parameterize insertion into the target sub-L1 halo orbit. The nominal mission plan of reaching the sub-L1 halo orbit under purely sail power is then examined for a range of projected launch vehicle dispersions. Results indicate that cold (underburn) injections less than −0.2 m/s are incapable of arriving at the nominal sub-L1 mission orbit. However, hot (overburn) injections are shown to be much more robust to the nominal mission plan than the low-energy trajectories. The resilience of the overburn cases compared to the underburn cases reveals that the mission would significantly benefit from additional launch vehicle hyperbolic excess energy to ensure mission requirements are satisfied.

A Method for Calculating the Trajectory of a Single-Impulse Flight to a Halo Orbit around the L2 Libration Point of the Earth–Moon System

A Method for Calculating the Trajectory of a Single-Impulse Flight to a Halo Orbit around the L2 Libration Point of the Earth–Moon System

Methodology of Optimal Control of Low-Thrust Spacecraft Transfer Between Halo Orbits Around the Earth-Moon L2 Libration Point

Methodology of Optimal Control of Low-Thrust Spacecraft Transfer Between Halo Orbits Around the Earth-Moon L2 Libration Point

A Method for Calculating the Trajectory of a Single-Impulse Flight to a Halo Orbit around the L2 Libration Point of the Earth–Moon

The problem of calculation of low-energy impulse trajectories to halo orbits in the vicinity of the L2 point of the Earth–Moon system is considered. A new method for calculating the trajectories of a single-impulse low-energy flight to a halo orbit is presented. The limited problem of four bodies is analyzed, within which the attraction of the Earth, Moon, and Sun is taken into account, and their position and speed are calculated using high-precision ephemeris support. Particular attention in the development of the method is paid to ensuring its computational stability for calculating trajectories with a long stay of a spacecraft (SC) in the zone of weak stability near the boundary of the Hill sphere of the Earth. The results of the calculation of single-impulse transfer trajectories from low Earth orbit to halo orbit around the L2 point of the Earth–Moon system are given. The analysis of the dependence of the main characteristics of single-impulse trajectories from the date of approach to the halo orbit is carried out.

Two trajectory configurations for the low-thrust transfer between northern and southern halo orbits in the Earth-Moon system

This paper presents a study of the low-thrust transfer problem between L2 northern and southern halo orbits using the Pontryagin minimum principle (PMP) in the Earth-Moon circular restricted three-body problem (CRTBP). The northern and southern halo families exist in the vicinity of the Moon, and have the same orbital period distribution, which can be used for space missions such as lunar navigation constellations, lunar relay communications and lunar space stations. Traditional low-thrust transfer trajectories are multi-revolution structures, which are modeled in this problem as multiple rotations around the libration point. The characteristics of the transfer are affected by the number of revolutions. Moreover, we propose a novel approach using invariant manifolds associated with periodic orbits in the CRTBP to construct L2 north–south halo orbit transfers. This approach results in superior transfer performance, which involves shorter time of flight, lower fuel consumption and reduced engine thrust requirements. A complete set of low-thrust time-, energy- and fuel-optimal transfer algorithms using moving point strategies and continuation methods is proposed. The computational procedures and analysis presented in this paper provide a reliable reference for the design of related orbital transfer missions.

Low-Thrust Transfer Dynamics and Control Between Halo Orbits in the Earth–Moon System by Means of Invariant Manifold

The low-thrust transfer between halo orbits in the Earth-Moon system is studied in this paper. The optimization problem of low-thrust transfer is challenging in the circular restricted three-body problem (CRTBP). Making reasonable use of the three-body dynamics structure can reduce the thrust required for the transfer. The invariant manifolds associated with planar Lyapunov orbits are spliced at first to create homoclinic and heteroclinic connections that can be used as trial solutions for low-thrust transfer. After that, the transfer of spatial halo orbits is addressed using the continuation methods. The Pontryagin maximum principle is used to calculate the optimal control law for the transfer process. Trajectories with various numbers of rotations around the Moon are suitable for transfer between halo orbits with different orientations, depending on the category of the manifold splicing. Finally, a typical MPC controller is designed to track the transfer trajectory in the presence of perturbations. Numerical simulations show the efficiency and practicability of the optimal transfer trajectory design strategy by means of invariant manifold for future lunar exploration missions.

Solar Absorptivity Degradation of Spacecraft Materials Due to UV and Charged Particles in the Gateway Environment

Gateway will be an outpost in cislunar space designed to survive a 15-year mission to support human Moon landings and future deep space exploration. The Gateway Passive Thermal Control System (PTCS) Group has identified charged particle and UV degradation of potential Gateway materials as a knowledge gap for the Near Rectilinear Halo Orbit (NRHO) and transit environment, especially the slow spiral transit planned for the first two modules. Solar absorptivity degradation is important to quantify because the end-of-life absorptivity impacts thermal performance of the system. To address this, ground testing has been completed at NASA Goddard Space Flight Center in which potential Gateway materials, including radiators, multi-layer insulation (MLI), and structural materials are subjected to the expected transit plus 15-year on-orbit charged particle fluence (9.86x1015 protons/cm2 at 2.5 keV and 3.1x1016 electrons/cm2 at 10 keV) and 5093 Equivalent Solar Hours (ESH). Reflectivity of a single sample of each material was measured in atmosphere and in vacuum inside the chamber after exposure to incremental levels of ESH and charged particle fluence. Increase in absorptivity of most samples were seen throughout the test including large differences in absorptivity of materials of the same category. Future planned testing will validate these results and include additional materials of interest.

SRG/eROSITA X-ray shadowing study of giant molecular clouds

Context. SRG/eROSITA is situated in a halo orbit around L2 where the highly variable solar wind charge exchange (SWCX) emission from Earth’s magnetosheath is expected to be negligible. The soft X-ray foreground emissions from the local hot bubble (LHB) and the remaining heliospheric SWCX emissions could be studied in unprecedented detail with eROSITA All-Sky Survey (eRASS) data in a 6-month cadence and better spectral resolution than ROSAT. Aims. We aim to use eRASS data of the sight lines towards three giant molecular clouds away from the Galactic plane to isolate and study the soft X-ray diffuse foreground emission. These X-ray shadows will serve as calibration baselines for the future three-dimensional structural study of the LHB. Methods. We conducted spectral analysis on the diffuse X-ray spectra of these clouds from the first four eRASSs to estimate and separate the heliospheric SWCX contribution from the LHB emission. Results. We find the density of the LHB to be independent of the sight line with ne ~ 4 × 10−3 cm−3, but not the temperature. We report a lower temperature of kTLHB = 0.084 ± 0.004 keV towards Chamaeleon II & III (Cha II & III) than Ophiuchus (Oph) and Corona Australis (CrA), in which we measured 0.102 ± 0.006 and 0.112 ± 0.009 keV, respectively. We measured the emission measure of the LHB to be ~2 × 10−3 cm−6 pc at medium Galactic latitudes (|b| ~ 20°). A monotonic increase in the SWCX contribution has been observed since the start of 2020, coincidental with the beginning of solar cycle 25. For Oph, SWCX has dominated the LHB in the 0.3−0.7 keV band intensity since eRASS2. We observed lower SWCX contributions in Cha II & III and CrA, consistent with the expected decreasing solar wind ion density at high heliographic latitudes.

Rendezvous in cislunar halo orbits: Hardware-in-the-loop simulation with coupled orbit and attitude dynamics

Space missions to Near Rectilinear Halo Orbits (NRHOs) in the Earth–Moon system are upcoming. A rendezvous technique in cislunar space is proposed in this investigation, one that leverages coupled orbit and attitude dynamics in the Circular Restricted Three-body Problem (CR3BP). An autonomous Guidance, Navigation and Control (GNC) technique is demonstrated in which a chaser spacecraft approaches a target spacecraft in a sample southern 9:2 synodic-resonant L2 NRHO, one that currently serves as the baseline for NASA’s Gateway. A two-layer guidance and control approach is contemplated. First, a nonlinear optimal controller identifies an appropriate baseline rendezvous path for guidance, both in position and orientation. As the spacecraft progresses along the pre-computed baseline path, navigation is performed through optical sensors that measure the relative pose of the chaser relative to the target. A Kalman filter processes these observations and offers state estimates. A linear controller compensates for any deviations identified from the predetermined rendezvous path. The efficacy of the GNC technique is tested by considering a complex scenario in which the rendezvous operation is conducted with an uncontrolled tumbling target. Hardware-in-the-loop laboratory experiments are conducted as a proof-of-concept to validate the guidance algorithm, with observations supplemented by optical navigation techniques.

Dynamics in the Vicinity of the Stable Halo Orbits

This work presents a study of the dynamics in the vicinity of the stable L2 halo orbits in the Earth-Moon system of the circular restricted three-body problem. These solutions include partially elliptic, partially hyperbolic, and elliptic quasi-halo orbits. The first two types of orbits are 2-dimensional quasi-periodic tori, whereas the elliptic orbits are 3-dimensional quasi-periodic tori. Motivated by the Lunar Gateway, this work computes these orbits to explore the 3-parameter family of solutions lying in the vicinity of the stable halo orbits. An algorithm is presented to quantify the size of the invariant surfaces which gives perspective on the size of the orbits. A stability bifurcation is detected where the partially elliptic tori become partially hyperbolic. A nonlinear behavior of the Jacobi constant is observed which differs from the behavior of the quasi-halo orbits emanating from the unstable halo orbits which makeup the majority of the quasi-halo family. Uses of the orbits in the vicinity of the stable L2 halo orbits are identified, and the results highlight characteristics and structure of the family to broaden the understanding of the dynamical structure of the circular restricted three-body problem.

Multi-orbit lunar GNSS constellation design with distant retrograde orbit and Halo orbit combination

The Moon is the closest natural satellite to mankind, with valuable resources on it, and is an important base station for mankind to enter deep space. How to establish a reasonable lunar Global Navigation Satellite System (GNSS) to provide real-time positioning, navigation, and timing (PNT) services for Moon exploration and development has become a hot topic for many international scholars. Based on the special spatial configuration characteristics of Libration point orbits (LPOs), the coverage capability of Halo orbits and Distant Retrograde Orbit (DRO) in LPOs is discussed and analyzed in detail. It is concluded that the Halo orbit with a period of 8 days has a better coverage effect on the lunar polar regions and the DRO has a more stable coverage effect on the lunar equatorial regions, and the multi-orbital lunar GNSS constellation with the optimized combination of DRO and Halo orbits is proposed by combining the advantages of both. This multi-orbital constellation can make up for the fact that a single type of orbit requires a larger number of satellites to fully cover the Moon, using a smaller number of satellites for the purpose of providing PNT services to the entire lunar surface. We designed simulation experiments to test whether the multi-orbital constellations meet the full lunar surface positioning requirements, and compare the coverage, positioning, and occultation effects of the four constellation designs that pass the test, and finally obtain a set of well-performing lunar GNSS constellations. The results indicate that the multi-orbital lunar GNSS constellation combining DRO and Halo orbits can cover 100% of the Moon surface, provides there are more than 4 visible satellites at any time on the Moon surface, which meets the navigation and positioning requirements, and the Position Dilution of Precision (PDOP) value is stable within 2.0, which can meet the demand for higher precision Moon surface navigation and positioning.

Particle Swarm Optimization-based co-state initialization for low-thrust minimum-fuel trajectory optimization

In this paper, Particle Swarm Optimization with energy-to-fuel continuation is proposed for initializing the co-state variables for low-thrust minimum-fuel trajectory optimization problems in the circular restricted three-body problem. Particle Swarm Optimization performs a search of the solution space by minimizing the weighted sum of squares of the two-point boundary-value problem final boundary condition residuals for the minimum-energy problem. Next, an energy-to-fuel homotopy is employed to transition the minimum-energy trajectory to a minimum-fuel trajectory, starting from the generated guess. The proposed methodology is applied to two low-thrust transfer problems in the Earth–Moon system: a transfer from a geostationary transfer orbit to an L1 halo orbit, and a transfer from an L2 halo orbit to an L1 halo orbit. The resulting minimum-fuel trajectories are validated with the literature. It is demonstrated that the methodology can successfully generate guesses for the initial co-state variables which converge to a solution for both scenarios. A strategically chosen particle swarm size is shown to improve the efficiency of the methodology. The proposed approach is of simple implementation, can be easily extended to other trajectory optimization problems, facilitates the discovery of multiple candidate trajectories, and does not require a user-provided guess, all of which are advantageous features for the preliminary phase of mission design.

Adaptive closed-loop maneuver planning for low-thrust spacecraft using reinforcement learning

Autonomy is an increasingly essential component of future space missions, and new technologies are necessary to accommodate off-nominal occurrences onboard that may pose risks to mission success. Identifying a suitable maneuver plan for onboard, low-thrust mission applications in cislunar space remains challenging, particularly in the case of unanticipated events. This investigation addresses this challenge by demonstrating artificial neural networks as promising tools for accurately estimating startup solutions for a conventional targeting-based iterative guidance and control algorithm. This blended approach produces a robust ‘hybrid’ architecture that simultaneously benefits from the computational simplicity of neural networks and the robustness of differential corrections to satisfy mission requirements. In this paradigm, a multiple shooting scheme is incorporated directly into a reinforcement learning process, tasking the resulting neural network controller with convergence basin identification for trajectory recovery. Rapid low-thrust maneuver planning is demonstrated in a ‘runaway’ spacecraft scenario, where deviation over time from a planned near rectilinear halo orbit path renders stationkeeping ineffective and demands an alternative approach to determine an effective recovery plan. The proposed Neural Network-Initialized Targeting (NNIT) algorithm significantly extends the recovery window compared to a crossing-control orbit maintenance approach and exhibits promising results in identifying both the timing and control components for low-thrust maneuver plans.

Uncertainty Propagation and Filtering via the Koopman Operator in Astrodynamics

The Koopman Operator (KO) theory is applied to generate an analytical solution of dynamical systems. The approach proposed in this work exploits a novel derivation of the KO with orthogonal polynomials to represent and propagate uncertainties, where the polynomials are modified to work with stochastic variables. Thus, a new uncertainty quantification technique is proposed in which the KO solution is expanded to include the prediction of central moments, up to an arbitrary order. The propagation of uncertainties is then expanded to develop a new filtering algorithm, for which the measurements are considered as additional observables in the KO mathematics. The uncertainty propagation technique is tested by predicting the state probability density function of a spacecraft in a halo orbit. The performance of the technique is assessed with a Monte Carlo analysis and proven to obtain accurate estimates for the state covariance, skewness, and kurtosis. The novel filtering methodology is then applied to an orbit determination application regarding a Lyapunov orbit, where an analysis on the filter accuracy and consistency shows that the new KO filter outperforms other common estimators.

Optimal time-fixed impulsive non-Keplerian orbit to orbit transfer algorithm based on primer vector theory

The work applies a systematic method to compute an optimal transfer trajectory between two non-Keplerian halo orbits in the Circular Restricted Three-body formulation of the Earth–Moon dynamics. The paper exploits the knowledge of the natural non-linear dynamics and the primer vector theory applied to the Circular Restricted Three Body Problem to design optimal multi-impulsive Halo-to-Halo transfers with fixed and limited Time of flight, from every point of the departure orbit to every point of the arrival orbit with an optimal number of impulses. The used method is constituted by three systematic steps with the goal of facilitating the mission design process and the on-board autonomous guidance implementation of future missions in highly non-linear dynamics.

Transfer from Lunar Gateway to Sun-Earth Halo Orbits Using Solar Sails

To extend the usability of solar sails in the sun–Earth–moon system, we analyze the transfer trajectories from the 9:2 Earth–moon near-rectilinear halo orbit (NRHO) to halo orbits around the sun–Earth L1 and L2 points under the assumption of a future mission for a solar sail spacecraft equipped with a solar electric propulsion (SEP) system deployed from the Lunar Orbital Platform-Gateway. The dynamics are modeled using the bicircular restricted four-body problem, where the gravitational forces from the sun, Earth, and moon as well as solar radiation pressure (SRP) are considered. We propose a trajectory design method that utilizes both SRP and SEP. The method consists of initial guess generation and optimization steps. The initial guess generation comprises the forward propagation of the escape trajectory from the NRHO, the backward propagation of the stable manifold of the target halo orbits, and their apoapsis patching process. Optimization is conducted to minimize propellant consumption by effectively controlling SRP. We perform optimizations with various parameters, namely, the sail area-to-mass ratio (), specifications of SEP, target sun–Earth halo orbit, and departure direction. The results validate the proposed trajectory design method and verify that solar sail acceleration can reduce the necessary amount of propellant, which indicates that such missions can be realized by small CubeSats.