Low-energy Transfer Trajectories Research Articles

Trajectory optimization of near-Earth asteroids exploration by using reusable probes from cislunar space

Implementing the flyby to Near-Earth Asteroids (NEAs) with the potential impact risks to the Earth allows for obtaining detailed physical parameters, thereby supporting the high-precision orbit prediction and planetary defense strategy. Different from those conducted asteroid flyby missions, in the 12th China Trajectory Optimization Competition (CTOC-12), a NEAs flyby trajectory design problem using reusable probes that depart from a Lunar Distant Retrograde Orbit (DRO) station in the cislunar space was released. The objective was flyby to as many NEAs as possible using up to 20 probes within a total of 10 years. The ∑ team proposed a solution that can explore 47 NEAs using 11 probes, ranking the first in the competition. In this paper, the methods and results from the winning team are introduced, including mission analysis and preliminary design, and low-energy transfer trajectory optimization. In particular, a round-trip trajectory is divided into three phases: deep space transfer, indirect transfer between the Earth to DRO, and DRO phasing and rendezvous. With the combination of global optimization and local optimization algorithms, the required velocity increments to change the orbital planes are effectively reduced, thus increasing the number of the explored NEAs. The final solution of our team is presented and the results are compared with those of the top three teams. The competition demonstrates that the regularization of flyby missions from the cislunar space to explore NEAs with the potential impact risks to the Earth is the feasible and promising.

Design of a Cislunar space navigation constellation based on special long-period orbits

This paper proposes a method for designing a navigation constellation in the Cislunar space. Firstly, the dynamic characteristics of the Cislunar space were analyzed. Northern Halo orbits and southern Halo orbits around L1 and L2 libration point are designed. Then, based on the invariant manifold theory, the homoclinic low-energy transfer trajectories of L1 Halo orbits and the low-energy heteroclinic transfer trajectories between L1 and L2 Halo orbits are provided respectively. Afterthat, two types of special long-period orbits are attained for the constellation design. Based on the geometric dilution of precision and the minimum number of visible satellites, design requirements for two parts are given. Next, two navigation constellation configurations are proposed. Lastly, simulations are conducted on the performance of two constellation configurations with different number of satellites. Compared to constellations in one orbit, constellations that configured in two orbits require fewer satellites to meet the design requirements. In different lunar transfer orbits, the performance of the dual orbit constellation was validated.

Five-impulse low-energy earth-moon transfer using manifolds

The libration points in the Earth-Moon and Sun-Earth systems are widely regarded as cost-effective gateways into interstellar space. The associated manifold tubes have been utilized by numerous researchers in the development of transfers between distinct systems. In this paper, a five-impulse transfer method, based on the manifold, has been proposed for designing the low-energy transfer trajectory for the Earth-Moon system. The Sun-Earth-Moon-Spacecraft four-body problem is decomposed into two circular restricted three-body problems: Sun-Earth-Spacecraft and Earth-Moon-Spacecraft. Moreover, a time parameter is employed to describe the positional relationship between the Sun-Earth and Earth-Moon systems, thereby facilitating their conversion. A phase angle is utilized to characterize the position of the Poincare section, and multiple orbital elements, such as orbital altitude and inclination, are considered in the constraints. To solve the design of multi-constrained transfer trajectories, sequential differential correction algorithms are proposed. The results demonstrate that this method can be used to create compliant low-energy transfer trajectories for various combinations of time parameters and phase angles. Additionally, excellent velocity increments and time-of-flight metrics can be found within a single Sun-Earth-Moon conjunction cycle. In parallel, the method is validated by reconstructing transfer trajectories in a bicircular restricted four-body problem model (BCR4BP) of the Sun-Earth/Moon system.

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.

Hamiltonian Structure-Based Formation Flight Control Along Low-Energy Transfer Trajectory

This paper studies the three-dimensional formation-flying problem in multibody regime using the Hamiltonian structure-preserving (HSP) control strategy. A low-energy lunar halo transfer orbit is chosen as the reference trajectory, which can represent the dramatically varying dynamics in the vast Earth–Moon region, and demonstrates much more complex environments than the vicinity of libration points. This trajectory exhibits time-varying topology of four different types, with two new ones: saddle saddle center and complex saddle complex saddle center (to the authors’ best knowledge, never studied in the literature). Stability transitions occur with a tremendous variation of the eigenvalues, bringing new challenges to effective formation controller design. To obtain bounded motion, topology-type-dependent HSP controller is proposed using the local stable/unstable manifolds. Difficulty arises in the transition regions, where close-to-zero real (complex) eigenvalues appear and require huge control gains, which may be infeasible for onboard computers. Accordingly, the controller is improved by considering additionally the local center manifold, and the gains required are always lower than 25. The controller is simple to implement because only position information is required, and its validity and robustness in the presence of initial deviations and navigation errors are verified by Monte Carlo simulations.

Hayabusa2\u2019s superior solar conjunction mission operations: planning and post-operation results

In late 2018, the asteroid Ryugu was in the Sun’s shadow during the superior solar conjunction phase. As the Sun-Earth-Ryugu angle decreased to below 3°, the Hayabusa2 spacecraft experienced 21 days of planned blackout in the Earth-probe communication link. This was the first time a spacecraft had experienced solar conjunction while hovering around a minor body. For the safety of the spacecraft, a low energy transfer trajectory named Ayu was designed in the Hill reference frame to increase its altitude from 20 to 110 km. The trajectory was planned with the newly developed optNEAR tool and validated with real time data. This article shows the results of the conjunction operation, from planning to flight data.

A hybrid method based on invariant manifold and chaos control for earth-moon low-energy transfer

In this paper, a new hybrid method based on invariant manifold and chaos control is presented to design Earth-Moon low-energy transfer trajectory in the planar circular restricted three-body problem. In the proposed method, one segment trajectory emanates from the period orbit around the Earth to Lyapunov orbit associated with the Lagrange points L1, which is designed by chaos control method to reduce the transfer time wandering in the chaotic region. Another segment starts from the Lyapunov orbit and reaches the Moon orbit, which is designed by the invariant manifold method. The two segments are patched with varying Jacobi constant by utilizing a Poincaré section, which is determined by the pericenter relative to the Earth. Several candidate low-energy transfer trajectories can be obtained by the proposed method, among which a suitable trajectory could be selected via the trade-off between fuel and flight time. Extensive numerical simulations are implemented to demonstrate the effectiveness of the proposed method.

Multi-Step Control of Chaos in the Application of Earth-Moon Orbit Transfer

A multi-step control of chaos method is proposed in this paper to reduce the flight time in the chaotic region of Earth-Moon low-energy orbit transfer. According to the motion regularity and the fact that the chaos orbit is highly sensitive to the initial conditions in the circular restricted three-body system, the optimal perturbation of chaos control at each step is calculated by the adaptive particle swarm optimization algorithm, and then the low-energy earth-moon orbit transfer trajectory is obtained. This proposed method can efficiently construct the Earth-Moon low-energy orbit transfer trajectory without the relying on the periodic orbits and the requirement of random search, and the corresponding flight time in chaotic motion can be significantly reduced. Finally, numerical simulations are provided to demonstrate the correctness and efficiency of this proposed method.

Enhanced Hybrid Differential Evolution for Earth-Moon Low-Energy Transfer Trajectory Optimization

It is known that the optimization of the Earth-Moon low-energy transfer trajectory is extremely sensitive with the initial condition chosen to search. In order to find the proper initial parameter values of Earth-Moon low-energy transfer trajectory faster and obtain more accurate solutions with high stability, in this paper, an efficient hybridized differential evolution (DE) algorithm with a mix reinitialization strategy (DEMR) is presented. The mix reinitialization strategy is implemented based on a set of archived superior solutions to ensure both the search efficiency and the reliability for the optimization problem. And by using DE as the global optimizer, DEMR can optimize the Earth-Moon low-energy transfer trajectory without knowing an exact initial condition. To further validate the performance of DEMR, experiments on benchmark functions have also been done. Compared with peer algorithms on both the Earth-Moon low-energy transfer problem and benchmark functions, DEMR can obtain relatively better results in terms of the quality of the final solutions, robustness, and convergence speed.

Design of communication relay mission for supporting lunar-farside soft landing

ChangE-IV will be the first soft-landing and rover mission on the lunar farside. The relay satellite, which is located near the Earth-Moon L2 point for relay communication, is the key to the landing mission. Based on an analysis of the characteristics of the task and the technical difficulties associated with the relay satellite system, the overall design scheme of the relay communication mission is proposed in terms of trajectory design and communication system design among other aspects. First, according to the complex dynamic environment, a mission orbit that serves as an uninterrupted communication link is presented. A short-duration and low-energy transfer trajectory with lunar flyby is discussed. Orbital correction and a low-cost control strategy for orbit maintenance in the Earth-Moon L2 point region are provided. Second, considering the existing technical constraints, the requirement of relay communication in different stages and the design schemes of frequency division and redundant relay communication system are introduced. Finally, based on the trajectory design index and the performance of the communication system, the overall design scheme of the relay communication mission is proposed. This mission will provide the technical support and reference required for the ChangE-IV mission.

Low-energy transfers to cislunar periodic orbits visiting triangular libration points

This paper investigates the cislunar periodic orbits that pass through triangular libration points of the Earth–Moon system and studies the techniques on design low-energy transfer trajectories. In order to compute periodic orbits, families of impulsive transfers between triangular libration points are taken to generate the initial guesses of periodic orbits, and multiple shooting techniques are applied to solving the problem. Then, varieties of periodic orbits in cislunar space are obtained, and stability analysis shows that the majority of them are unstable. Among these periodic orbits, an unstable periodic orbit in near 3:2 resonance with the Moon is taken as the nominal orbit of an assumed mission. As the stable manifolds of the target orbit could approach the Moon, low-energy transfer trajectories can be designed by combining lunar gravity assist with the invariant manifold structure of the target orbit. In practice, both the natural and perturbed invariant manifolds are considered to obtain the low-energy transfers, which are further refined to the Sun-perturbed Earth–Moon system. Results indicate that (a) compared to the case of natural invariant manifolds, the optimal transfers using perturbed invariant manifolds could reduce flight time at least 50days, (b) compared to the cheapest direct transfer, the optimal low-energy transfer obtained by combining lunar gravity assist and invariant manifolds could save on-board fuel consumption more than 200m/s, and (c) by taking advantage of the gravitational perturbation of the Sun, the low-energy transfers could save more fuel consumption than the corresponding ones obtained in the Earth–Moon system.

Near Earth asteroid resource utilisation for large in-orbit reflectors

The resources offered by the family of near Earth asteroids could provide bulk materials to support future space science ventures, both crewed missions and space-based astronomy. Using low-energy transfer trajectories small near Earth asteroids could be captured directly, or their material resources returned to Earth orbit or the Lagrange points. With novel fabrication methods, such as additive layer manufacturing, large-scale space structures including optical and radio telescopes could in principle be assembled from such resources. Indeed, with bulk materials readily available, very large numbers of structures could be fabricated in-situ for interferometry applications.

Analysis of Effect of Oblateness of Smaller Primary on the Evolution of Periodic Orbits

Evolution of periodic orbits in Sun-Mars and Sun-Earth systems are analyzed using Poincare surface of section technique and the effects of oblateness of smaller primary on these orbits are considered. It is observed that oblateness of smaller primary has substantial effect on period, orbit’s shape, size and their position in the phase space. Since these orbits can be used for the design of low energy transfer trajectories, so perturbations due to planetary oblateness has to be understood and should be taken care of during trajectory design. In this paper, detailed stability analysis of periodic orbit having three loops is given for A2 = 0.0001.

Analysis of Effect of Solar Radiation Pressure of Bigger Primary on the Evolution of Periodic Orbits

Evolution of periodic orbits in Sun-Mars and Sun-Earth systems are analyzed using Poincare surface of section technique and the effects of solar radiation pressure of bigger primary and actual oblateness of smaller primary on these orbits areconsidered. It is observed that solar radiation pressure of bigger primary has substantial effect on period, orbit’s shape, size and their position in the phase space. Since these orbits can be used for the design of low energy transfer trajectories, so perturbations due to solar radiation pressure has to be understood and should be taken care of during trajectory design. It is also verified that stability of such orbits are negligible so they can be used as transfer orbit. For each pair of solar radiation pressure q and Jacobi constant C we get two separatrices where stability of island becomes zero. In this paper, detailed stability analysis of periodic orbit having two loops is given when q = 0.9845.

Design of low-energy transfer from lunar orbit to asteroid in the Sun-Earth-Moon system

Asteroid exploration trajectories which start from a lunar orbit are investigated in this work. It is assumed that the probe departs from lunar orbit and returns to the vicinity of Earth, then escapes from the Earth by performing a perigee maneuver. A low-energy transfer in Sun-Earth-Moon system is adopted. First, the feasible region of low-energy transfer from lunar orbit to perigee within 5 000km height above the Earth surface in Sun-Earth-Moon system is calculated and analyzed. Three transfer types are found, i.e., large maneuver and fast transfers, small maneuver and fast transfers, and disordered and slow transfers. Most of feasibility trajectories belong to the first two types. Then, the low-energy trajectory leg from lunar orbit to perigee and a heliocentric trajectory leg from perigee to asteroid are patched by a perigee maneuver. The optimal full-transfer trajectory is obtained by exploiting the differential evolution algorithm. Finally, taking 4179 Toutatis asteroid as the target, some low-energy transfer trajectories are obtained and analyzed.

Trajectory Design for the Transfer from the Lissajous Orbit of Sun-Earth System to Asteroids

This paper investigates the trajectory design issue for the transfer from the Lissajous orbit of CHANGE 2 probe to asteroids. First, an intersection search method, which is a general design method of low-energy transfer trajectories by searching the intersection of unstable and stable manifolds in the circular restricted three-body problem (CRTBP), is applied to produce the zero-cost flyby trajectories for the asteroid flyby missions of CHANGE 2 probe, and the simulation result shows that this method is invalid. Then, for this trajectories design issue, a perturbation method, which consists of a process of searching initial trajectories by applying velocity perturbations in the direction of unstable eigenvectors of the Lissajous orbit and a trajectory correction process with two-level differential correction, is proposed. Finally, the transfer opportunities between the Lissajous orbit of CHANGE 2 and asteroids Toutatis and 2010 JK1 are searched by the perturbation method. The results show that the method proposed in this paper can identify low-energy transfer trajectories for asteroid flyby missions of CHANGE 2. Moreover, this method provides a global understanding of the trend of impulsive maneuvers over the transfer date.

Comparison of low-energy lunar transfer trajectories to invariant manifolds

In this study, transfer trajectories from the Earth to the Moon that encounter the Moon at various flight path angles are examined, and lunar approach trajectories are compared to the invariant manifolds of selected unstable orbits in the circular restricted three-body problem. Previous work focused on lunar impact and landing trajectories encountering the Moon normal to the surface, and this research extends the problem with different flight path angles in three dimensions. The lunar landing geometry for a range of Jacobi constants is computed, and approaches to the Moon via invariant manifolds from unstable orbits are analyzed for different energy levels.

Interplanetary transfers employing invariant manifolds and gravity assist between periodic orbits

The interest in the periodic orbits of the restricted three-body problem continues to grow for their significant practical application. This paper focuses on the interplanetary transfers between periodic orbits of two different three-body systems, whose invariant manifolds have no intersection in phase space. A novel design method is proposed to obtain the optimal transfer employing the invariant manifolds and planetary gravity assist. The periapsis Poincare map is used to analyze the periapsides of invariant manifolds. On the basis of hyperbola approximation, the impulses performed on the periapsis of invariant manifolds are calculated with a simple iterative algorithm. The propellant-efficient escape and capture trajectories can be found by comparing the impulses magnitudes corresponding to different invariant manifolds, which can provide the appropriate initial guess for optimization. Further, the trajectory design is formulated as an unconstrained optimization problem under the perturbed restricted three-body model. An efficient algorithm combining simplex method and differential correction is adopted to obtain the optimal solution. The validity of the proposed approach is demonstrated through several interplanetary low energy transfer trajectories.

Transfer Trajectory Design for Mars Exploration

With regard to the human exploration of Mars, low energy transfer trajectory is designed for Mars exploration based on the combination of invariant manifolds, differential correction and aerobraking methods. The whole transfer trajectory is composed of four stages: 1) from the Earth parking orbit to the Lyapunov orbit around Lagrange point L2 in the Sun-Earth system; 2) from the Lyapunov orbit around L2 to the Lyapunov orbit around L1 in the Sun-Mars system; 3) from the Lyapunov orbit around L1 in the Sun-Mars system to the large elliptical orbit around Mars; and 4) from the large elliptical orbit around Mars to the near-Mars parking orbit. In the first three stages, the circular restricted three-body problem is considered, and the trajectory is designed by using invariant manifolds and the differential correction method. The simulation results show that the transfer trajectory designed by means of the invariant manifolds of the Lyapunov orbit costs lower energy and shorter time of flight than that designed by means of the invariant manifold of the Halo orbit. In the fourth stage, the two-body problem is considered, and the aerobraking method is applied. A comparative performance analysis of static and rotating atmospheric models is carried out by using the details of duration, aerodynamic loading of the Mars vehicle, and other orbital parameters. It is shown that, on the low periareon where the influence of the atmospheric density increases, the changes of orbit parameters between rotating and static atmospheric environments are in large difference, such as orbital semimajor axis, orbital eccentricity, and so on. The influence of Martian rotating atmospheric environment should be considered.

Analytic approach on geometric structure of invariant manifolds of the collinear Lagrange points

An analytical method is proposed to find geometric structures of stable, unstable and center manifolds of the collinear Lagrange points. In a transformed space, where the linearized equations are in Jordan canonical form, these invariant manifolds can be approximated arbitrarily closely as Taylor series around Lagrange points. These invariant manifolds are represented by algebraic equations containing the state variables only without the help of time. Thus the so-called geometric structure of these invariant manifolds is obtained. The stable, unstable and center manifolds are tangent to the stable, unstable and center eigenspaces, respectively. As an example of applicability, the invariant manifolds of L1 point of the Sun-Earth system are considered. The stable and unstable manifolds are symmetric about the line from the Sun to the Earth, and they both reach near the Earth, so that the low energy transfer trajectory can be found based on the stable and unstable manifolds. The periodic or quasi-periodic orbits, which are chosen as nominal arrival orbits, can be obtained based on the center manifold.