Weak Stability Boundary Research Articles

Leveraging the Interplanetary Superhighway for Propellant–Optimal Orbit Insertion into Saturn–Titan System

This paper presents an innovative approach using Dynamical Systems Theory (DST) for interplanetary orbit insertion into Saturn−Titan three−body orbits. By leveraging DST, this study identifies invariant manifolds guiding a spacecraft into Titan−centered Distant Retrograde Orbits (DROs), strategically selected for their scientific significance. Subsequently, Particle Swarm Optimization (PSO) is employed to fine−tune the insertion parameters, thereby minimizing ΔV. The results demonstrate that the proposed method allows for a reduction in ΔV of over 70% compared to conventional approaches like patched conics−based flybys (2.68 km/s vs. 9.23 km/s), albeit with an extended time of flight, which remains notably faster than weak stability boundary transfers. This paper serves as an interplanetary mission planning methodology to optimize spacecraft trajectories for the exploration of the Saturn−Titan system.

Extended perilune rendezvous method for low Delta-V transition to NRHO for cargo transportation mission to gateway

The construction of Gateway in lunar orbit is planned as a post-ISS mission by NASA, and JAXA is considering a cargo supply mission to provide service. The Near-Rectilinear Halo Orbit (NRHO) is a candidate for a Gateway orbit, and several methods for the transfer trajectory are being researched. The In-Direct Transfer (IDT) has a short transition period of seven days, but the ΔV application is inefficient. The Weak Stability Boundary (WSB) transfer maintains a small ΔV, but the transition takes over 100 days. However, a trajectory intermediate between IDT and WSB has not yet been actively studied. This paper proposes the Extended Perilune Rendezvous Method (EPRM) of intermediate transfer to NRHO. This method reduces ΔV by orbiting in an elliptical lunar orbit for over two weeks. EPRM has several transition methods corresponding to ΔV and transition periods. This paper systematically reveals the transition mechanisms. Moreover, an effective TCM, considering several error types, is presented for an actual mission. Finally, the rendezvous accuracy with Gateway is discussed.

Comparison of transfer trajectory to NRHO and operation plan for logistics resupply mission to gateway

NASA and other space agencies have been planning the construction of a space station in lunar orbit called the Gateway. They have also begun studies for its resupply missions using an HTV-X. There are several ways to transition to NRHO, a candidate orbit. In-Direct Transfer (IDT) has a short transition period of seven days, but the required ΔV is inefficient at more than 400 m/s. The Weak Stability Boundary (WSB) transfer requires a small ΔV, less than 100 m/s, but the transition period is more than 100 days. This paper proposes the Perilune Rendezvous Method (PRM), recently discovered in 2022. Its transition period is two weeks longer than IDT, but the period is much shorter than WSB. The required ΔV is 40 m/s lower than IDT but still larger than WSB. The PRM has an intermediate ΔV and a transition period between IDT and WSB. Since fuel consumption and transition duration requirements depend on the mission, the PRM expands the scope of future cargo transportation. To adopt the PRM in a mission, an efficient trajectory correction plan was found based on a Monte Carlo analysis that considered navigation and control errors. It confirmed that the PRM could rendezvous with Gateway with higher accuracy than IDT.

Low-Energy Transfers to Lunar Distant Retrograde Orbits from Geostationary Transfer Orbits

This study focuses on the low-energy transfers to lunar distant retrograde orbits (DROs) from geostationary transfer orbits (GTOs) in the bicircular-restricted sun–Earth–moon four-body problem. The low-energy transfer is essential for low-cost small satellites reaching out to the Moon, and the departure from GTO allows more rideshare opportunities. We first created several large-scale databases of trajectory segments, such as GTO to apogee in the weak-stability area, apogee to perilune, and DRO to perilune. Then, millions of GTO–DRO transfer trajectories with double powered lunar flybys (PLFs) and weak stability boundary (WSB) ballistic transfer were constructed through trajectory patching. The key flight information, such as the ΔV–time-of-flight Pareto fronts, launch windows, and the flight mode via WSB ballistic transfer, is obtained from feasible solutions. Results show that low-energy GTO–DRO transfers can be achieved by exploiting PLFs and WSB ballistic arcs, which suggests potential applications in the cislunar space.

B–plane assisted trajectory design with analytical mapping of swing-by parameters

The Artemis program has gathered further attention due to the transfer trajectories to the Moon and beyond. Trajectories such as weak stability boundary and ballistic lunar transfer are known as low-energy transfer toward the vicinity of the Moon. Among these categories, using the Moon swing-by contributes to the reduction of fuel consumption. In this study, we investigate trajectory design with the Moon swing-by as a constraint, and propose a methodology that projects swing-by parameters onto a conventional B–plane to facilitate the efficient analysis and design of post swing-by trajectories. This proposed approach not only enables a comprehensive understanding of all candidate solutions at each swing-by, but also provides flexibility in response to varied swing-by conditions. As illustrative examples, this paper demonstrates transfer problems to periodic orbits in the Earth–Moon system.

Low-energy Earth–Moon transfers via Theory of Functional Connections and homotopy

Numerous missions leverage the weak stability boundary in the Earth–Moon–Sun system to achieve a safe and cost-effective access to the lunar environment. These transfers are envisaged to play a significant role in upcoming missions. This paper proposes a novel method to design low-energy transfers by combining the recent Theory of Functional Connections with a homotopic continuation approach. Planar patched transfer legs within the Earth–Moon and Sun–Earth systems are continued into higher-fidelity models. Eventually, the full Earth–Moon transfer is adjusted to conform to the dynamics of the planar Earth–Moon Sun-perturbed, bi-circular restricted four-body problem. The novelty lies in the avoidance of any propagation during the continuation process and final convergence. This formulation is beneficial when an extensive grid search is performed, automatically generating over 2000 low-energy transfers. Subsequently, these are optimized through a standard direct transcription and multiple shooting algorithm. This work illustrates that two-impulse low-energy transfers modeled in chaotic dynamic environments can be effectively formulated in Theory of Functional Connections, hence simplifying their overall design process. Moreover, its synergy with a homotopic continuation approach is demonstrated.

Qualitative study of ballistic capture at Mars via Lagrangian descriptors

Lagrangian descriptors reveal the dynamical skeleton governing transport mechanisms of a generic flow. In doing so, they unveil geometrical structures in the phase space that separate regions with different qualitative behavior. This work investigates to what extent Lagrangian descriptors provide information about non-Keplerian motion in Mars proximity, which is modeled under the planar elliptic restricted three-body problem. We propose a novel technique to reveal ballistic capture orbits extracting separatrices of the phase space highlighted by Lagrangian descriptor scalar fields. The Roberts’ operator to approximate the gradient is used to detect the edges in the fields. Results demonstrate the chaos indicator ability to distinguish sets of initial conditions exhibiting different dynamics, including ballistic capture ones. Separatrices are validated against reference weak stability boundary derived on similar integration intervals. Compared to other techniques, Lagrangian descriptors provide dynamics insight bypassing the propagation of the variational equations.

Double Tisserand graphs for low-energy lunar transfer design

Tisserand graphs are a widespread tool for interplanetary trajectory and Moon tour design. They are based on the Jacobi constant being an integral of motion in the Circular Restricted Three Body Problem (CR3BP); as such, the classical Tisserand graph does not include the perturbation of bodies other than the flyby bodies. Low-energy transfers in the Earth-Moon system make use of the combined Earth, Moon, and Sun gravities, exploiting the third body perturbations to reach the Moon with a reduced transfer Δv. The paper, therefore, proposes a novel double Tisserand graph, where the level lines of both the Earth-Moon and the Sun-Earth CR3BPs are superimposed, portraying the complex 4-body dynamics into a single plot. Paths along such level lines correspond to trajectories utilizing the dynamical effect of the Moon or the Sun or a combination of both. We show how such a graph can be efficiently used for preliminary design of Weak Stability Boundary transfers, lunar resonance transfers, lunar flybys, weak lunar captures or any combination of them.

PROGRESS OF THREE-BODY ORBITAL DYNAMICS STUDY

The orbital dynamics in the three-body system is a classical problem in the field of astrodynamics. It has rich theoretical and engineering significance, and have played an important role in the process of space activities extending from near-earth space to deep space. This paper reviews and summarizes the progress of the study of orbital dynamics in the three-body system. Combined with the development trend of deep space exploration in the future, the hotspots and challenges in the research of three-body orbital dynamics are prospected. First, the research background and significance of the three-body problem are surveyed, and the development of the dynamics model of the three-body system is briefly reviewed. Secondly, characteristics of the local motion near the equilibrium point in the three-body problem are systematically summarized. The analytical and numerical methods for periodic orbits are introduced. The latest development of quasi-periodic motion is discussed. Meanwhile, the characteristics and research progress of global periodic motions in the three-body system including resonance orbits, cycler trajectories, and free return orbits are summarized. Next, the research progress of the low-energy transfer and capture trajectory design in the three-body system is analyzed from two aspects of invariant manifold theory and weak stability boundary theory. Finally, the applications of orbital dynamics in the three-body system in formation flight and navigation constellation design are summarized. Several orbital dynamics and control problems in the design of landing trajectories for full lunar-surface coverage, the low thrust orbit optimization of the three-body system, and the utilization of non-linear equilibrium points in the three-body system are addressed for future study.

BepiColombo Ground Segment and Mission Operations

The ESA/JAXA BepiColombo mission to Mercury was launched in October 2018. It includes two scientific Mercury orbiters, flying as a single stack during the seven years interplanetary transfer, and as individual spacecraft operated by their respective space Agencies ESA and JAXA once deployed in their respective scientific orbit around Mercury. Three ground segments are closely collaborating on this mission. The European Space Operations Center of the European Space Agency, in Darmstadt, Germany, responsible for mission operations of the composite stack during the interplanetary transfer and of the Mercury Planetary Orbiter (MPO) once at Mercury, the JAXA Sagamihara Space Operation Center (SSOC) in Sagamihara, Japan, responsible for Mercury Magnetospheric Orbiter (MMO) operations throughout the mission, and the European Space Astronomy Center of the European Space Agency, in Villanueva de la Canada, Spain, responsible for MPO scientific archiving and Mercury science operations. The mission benefits from infrastructure and concepts developed on previous interplanetary missions of ESA and JAXA. However, it also includes specific operational challenges, such as electric propulsion, and a complex Mercury Orbit Insertion sequence, exploiting weak stability boundaries, and interleaving three module separations with 15 chemical propulsion manoeuvers to deliver MPO and MMO in their respective science orbits, requiring specific developments and operational approaches. In view of the nominal mission duration of 8.5 years, the mission also requires the adoption of specific measures for knowledge preservation and long-term maintenance.

Low-energy weak stability boundary transfers to Pluto's moons: Preliminary trajectory design via triangular libration point

By patching the invariant manifolds of collinear libration points Lyapunov orbits of two restricted three body problem (CR3BP), Koon (2001) successfully constructed low-energy Earth-Moon transfers in the Sun-Earth-Moon system. It reveals the essential mechanism of low-energy cislunar transfers, i.e., weak stability boundary (WSB), from the viewpoint of dynamical system. This paper is devoted to the construction of low-energy WSB transfers using the invariant manifolds of triangular libration points. Koon's method is applied to two of Pluto's moons, Charon and Hydra. Specifically, Hydra is treated as a contact binary system due to its irregular shape. Numerical Results show a remarkably small fuel consumption, i.e., ∼39 m/s for Pluto-Charon transfers and ∼159 m/s for Pluto-Hydra transfers. Finally, combining with the heteroclinic connections around Hydra, a sample trajectory for end-of-life mission of Pluto probe is built up achieving Pluto-Hydra low-energy transfer and encircling Hydra body at the cost of 7 m/s fuel consumption. Therefore, Charon and Hydra can act as potential targets of Pluto end-of-life mission from the engineering viewpoint. The WSB trajectories obtained in this paper can work as initial guess of differential correction scheme to achieve real trajectories in more complicated gravity fields.

Design and analysis of Weak Stability Boundary trajectories to Moon

An algorithm is developed to find Weak Stability Boundary transfer trajectories to Moon in high fidelity force model using forward propagation. The trajectory starts from an Earth Parking Orbit (circular or elliptical). The algorithm varies the control parameters at Earth Parking Orbit and on the way to Moon to arrive at a ballistic capture trajectory at Moon. Forward propagation helps to satisfy launch vehicle’s maximum payload constraints. Using this algorithm, a number of test cases are evaluated and detailed analysis of capture orbits is presented.

The MEOW lunar project for education and science based on concurrent engineering approach

The use of concurrent engineering in the design of space missions allows to take into account in an interrelated methodology the high level of coupling and iteration of mission subsystems in the preliminary conceptual phase. This work presents the result of applying concurrent engineering in a short time lapse to design the main elements of the preliminary design for a lunar exploration mission, developed within ESA Academy Concurrent Engineering Challenge 2017. During this program, students of the Master in Space Systems at Technical University of Madrid designed a low cost satellite to find water on the Moon south pole as prospect of a future human lunar base. The resulting mission, The Moon Explorer And Observer of Water/Ice (MEOW) compromises a 262 kg spacecraft to be launched into a Geostationary Transfer Orbit as a secondary payload in the 2023/2025 time frame. A three months Weak Stability Boundary transfer via the Sun-Earth L1 Lagrange point allows for a high launch timeframe flexibility. The different aspects of the mission (orbit analysis, spacecraft design and payload) and possibilities of concurrent engineering are described.

Cislunar Navigation Constellation by Displaced Solar Sails

Increasing lunar exploration activities are giving rise to higher demands for a navigation constellation system in cislunar space. A novel constellation of solar sails around the Sun-Earth Artificial Lagrangian Points (ALPs) is proposed for cislunar navigation in this paper, which benefits from the numberless and out-of-plane advantages of ALPs compared with the classical Lagrangian points. To relieve the technical pressure on sail equipment, a two-layer optimisation strategy including the navigation constellation architecture and trajectory design is developed to reduce the desired lightness number of the sail's motion. The constellation architecture is constructed in the shape of a regular tetrahedron, whose size and orientation are derived from the realisable lightness number at the ALPs. The powerful Hamiltonian structure-preserving controller and differential evolution algorithm are adopted to propagate the bounded quasi-periodic trajectory with minimum lightness number variation. With the premise of the sail's high feasibility in the mechanism, the numerical navigation simulations for a typical trans-lunar weak stability boundary trajectory indicate that the proposed navigation constellation has a low geometric dilution of precision factor and a good navigation performance.

Design of two-impulse Earth–Moon transfers using differential correction approach

In this paper, a novel optimized algorithm of two-impulse Earth–Moon transfers in the circular restricted three-body problem (CR3BP) is proposed based on differential correction approach. In this innovative approach, the initial and final states of transfer orbit are estimated reasonably by fully considering the periapsis map (PM) condition. Then the Newton–Raphson method is utilized iteratively to deduce the differential equations of transfer orbit and further to yield the accurate initial state. Thus, the planar two-impulse Earth–Moon transfer can be accomplished. In addition, a simple design procedure is developed to solve the problem that arises from more unknown parameters in the spatial CR3BP. Therefore, the proposed method can be applied not only for the planar CR3BP but also the spatial CR3BP. Numerical results show the two-impulse Earth–Moon transfers need less energy than the Hohmann transfers and shorter flight time than the Weak Stability Boundary (WSB) transfers. Moreover, the proposed method can effectively enable a large set of two-impulse Earth–Moon transfers to be computed numerically.

Dynamics of weak stability boundary transfer trajectories to Moon

In order to design a WSB trajectory to Moon, highly elliptical geocentric orbit is propagated forward in time so that its perigee increases to Earth-Moon distance. Another highly elliptical selenocentric orbit is propagated backward in time till it starts moving towards Earth. The two trajectories are patched using Fixed Time of Arrival targeting method to obtain a WSB transfer trajectory. In this paper, various geocentric and selenocentric orbits are considered and eligible candidates for WSB transfer are represented on the phase space with color code on transfer time. Genetic Algorithm is used to find patching points to reduce the sum of \(\Delta V\) required at the patching points. Lunar fly-by on the way to apogee is also studied.

Earth–Moon transfer with near-optimal lunar capture in the restricted four-body problem

In this paper, the low energy transfer (LET) from low Earth orbit to a permanent lunar orbit is investigated under the Sun–Earth–Moon-spacecraft restricted four-body problem (RFBP). Based on the near-optimal lunar capture, we propose an optimization method to construct the three-impulse Earth–Moon transfer in the RFBP. In the design process, a hybrid model is used to reduce the difficulty of the optimization problem. Compared with the Weak Stability Boundary transfer, the transfer orbit obtained has more chances to insert the spacecraft into low lunar orbit. Besides, compared with the patched-conic transfers, the LETs obtained can efficiently reduce the propellant costs, but the transfer durations apparently increase. Finally, the effect of solar gravitation on the transfer is analyzed.

Trajectory optimization for a lunar orbiter using a pattern search method

The Republic of Korea plans to launch a lunar orbiter and lander by 2020. There are several ways to enter lunar orbit: direct transfer, phasing loop transfer, weak stability boundary transfer, and spiral transfer trajectory. In this study, trajectory optimization is investigated for a lunar orbiter using a pattern search method that minimizes the required delta-V for direct lunar transfer. This method generates neighborhood points near the initial condition and then determines whether there is a new point that can reduce the value of the objective function. Classical methods require the gradient and acceleration of the objective function, but pattern search does not. Six poll methods and nine search methods are chosen; thus, 54 combinations of poll and search methods are available. The pattern search method can reduce the required delta-V on average by a few meters per second for a time of flight of five days and more than 10 m/s for a time of flight of four or six days, regardless of whether translunar injection is performed at the ascending or descending node.

A survey of weak stability boundaries in the Sun-Mars system**Supported by the National Natural Science Foundation of China.

We investigate the weak stability boundary (WSB) for a new primary, Mars, in the framework of the planar circular restricted 3-body problem, and also in the planar bicircular restricted 4-body problem by including a perturbation due to Jupiter. For the sake of a simple stability/instability criterion, our computations have been done using the equations of motion in polar coordinates. It is found that the relative size of the weakly stable sets around Mars is much larger than that of the Earth-Moon and the Sun-Jupiter systems, as the mass ratio of the Sun-Mars system is significantly smaller. We propose that this difference could be scaled by the Hill radius. In an enlarged view of the domain close to Mars, the geometry of the WSB has been presented for various parameters and compared to previous works. Our results also show that Jupiter's gravitational force would strongly affect the Martian stable regions and should be taken into account to design a ballistic capture trajectory.

Contingency and recovery options for the European Student Moon Orbiter

This paper presents an overview of the analysis performed on the lunar orbit and some of the possible contingencies for the European Student Moon Orbiter (ESMO). Originally scheduled for launch in 2014 –2015 as a piggyback payload, it was the only ESA planned mission to the Moon. By way of a weak stability boundary transfer, ESMO is inserted into an orbit around the Moon. Propellant use is at a premium, so the operational orbit is selected to be highly eccentric. In addition, an optimization is presented to achieve an orbit that is stable for 6 months without requiring orbit maintenance. A parameter study is undertaken to study the sensitivity of the lunar orbit insertion. A database of transfer solutions across 2014 and 2015 is used to study the relation between the robustness of weak capture and the planetary geometry at lunar arrival. A number of example recovery scenarios, where the orbit insertion maneuver partially or completely fails, are also considered.