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Transfers to Earth-Moon triangular libration points by Sun-perturbed dynamics

ADVANCES IN SPACE RESEARCH 2025年第3期75卷 2837-2855页

作者： Peng, Lei Liang, Yuying He, Xingji Beihang Univ Sch Astronaut Beijing 100191 Peoples R China Yunnan Police Coll Kunming 650223 Peoples R China

This paper is devoted to the construction of transfers to Earth- Moon triangular libration points based on their Sun-perturbed dynamics in the Sun-Earth-Moon system. In this bi-circular problem (B Moon triangular libration points with three remaining periodic substitutions. As the stable region, the invariant tori full of quasi the destination of triangular libration point transfers for a low st ation-keeping cost. However, the instability of the unstable periodic orbit is so mild that its stable and unstable manifolds are folded ar CP), the solar gravity perturbation breaks the stability of the Earth- orbits (two stable and one unstable) around them as their dynamical-periodic orbits around the two stable periodic orbits can be used as the authors approximate the stable and unstable manifolds of th from the Sun-perturbed Earth-Moon triangular libration points vi around triangular libration points. First, the transfers connecting t anew connection mechanism near L 4 and L 5 based on the N : M m skill to time-dependent situations. The quickest one is achieved by a 1:1 L 4-L5 heteroclinic-based connection with a TOF of less than 26.9 days at a fuel cost of 117.88 m/s. Note that such Sun-perturbed transfers cannot be produced by continuing any trajectory in the Earth-Moon circular restricted three-body problem but the continuation. ound the invariant tori surrounding the other two stable orbits. Thus, e unstable orbit through the parameterization method, which emerge cinity. They act as a bridge between outer space and the stable region he Earth's vicinity are constructed using the manifolds. Subsequently, ulti-loop transfers is developed by extending the conventional patching y can generate two-impulsive transfers in the ephemeris model by (c) 2024 COSPAR. Published by Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： triangular libration points Sun-Earth-Moon system Bi-circular problem Parameterization method

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Naturally bounded relative motion for formation flying near triangular libration points

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ADVANCES IN SPACE RESEARCH 2023年第12期71卷 5038-5049页

作者： He, Xingji Xu, Ming Sun, Xiucong Peng, Na Wang, Liang Liu, Lei Beihang Univ Sch Astronaut Beijing 102206 Peoples R China Beihang Univ ShenYuan Honors Coll Beijing 100191 Peoples R China Shanghai Aerosp Elect Technol Inst Shanghai 201100 Peoples R China Lanzhou Inst Phys Lanzhou 730000 Peoples R China Beijing Aerosp Control Ctr Beijing 100094 Peoples R China

This paper proposes a numerical searching and ergodic mappings method to seek the naturally bounded relative orbits near the triangular libration points in the Earth-Moon system. An orbital energy saved numerical continuation method along one axis is proposed and the Poincare ' map near the triangular libration points is obtained, based on which two characteristic parameters are defined to generate the naturally bounded relative motions, i.e., the crossing interval period and the reference angle. Then, a numerical searching method is employed to generate an ergodic mapping and a neural network is utilized to realize the transformation from the ergodic mappings to orbital states. By pairing the ergodic mappings of different initial orbital states, the naturally bounded relative motion realized can remain for a long time and does not diverge.(c) 2023 COSPAR. Published by Elsevier B.V. All rights reserved.

关键词： Natural formations triangular libration points Naturally bounded motions Satellite formation

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High-order solutions of motion near triangular libration points for arbitrary value of

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NONLINEAR DYNAMICS 2018年第2期93卷 909-932页

作者： Liang, Yuying Xu, Ming Xu, Shijie Beihang Univ Sch Astronaut Beijing 100191 Peoples R China

In this paper, a new methodology is proposed to derive the high-order approximations of motions near them in three cases, i.e., the mass ratio is greater than, smaller than and equal to Gascheau and Routh critical value. A preliminary analysis on the phase space structure of triangular libration points in the first case is accomplished from the point of view of the dynamical system theory, demonstrating that they have two-dimensional stable/unstable manifolds and zero-dimensional center manifolds, referred to as type. Further investigations show the topological type of triangular libration points evolves from center-center type to type as mass ratio increases above Gascheau and Routh critical value. The high-order approximations of motion near triangular libration points are constructed using a modified method of variation of parameters. The methodology developed in this paper can deal with triangular libration points of all three topological types, which remains unsolved by the traditional Lindstedt-Poincar, method. The simulation results demonstrate that the expansions up to 11th order are a good replacement of numerical computation with a tolerate error. The validity regions of initial position deviation are presented, and a modification of the algorithm is performed to guarantee the continuity near Gascheau and Routh critical value. Furthermore, the high-order solutions of two-dimensional stable/unstable manifolds are employed to search for homoclinic connections of triangular libration points of 2577 Litva.

关键词： triangular libration points Invariant manifolds Phase space structure Homoclinic/heteroclinic connections Circular restricted three-body problem

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Analytical and numerical construction of vertical periodic orbits about triangular libration points based on polynomial expansion relations among directions

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ASTROPHYSICS AND SPACE SCIENCE 2017年第8期362卷 136-136页

作者： Qian, Ying-Jing Yang, Xiao-Dong Zhai, Guan-Qiao Zhang, Wei Beijing Univ Technol Coll Mech Engn Beijing Key Lab Nonlinear Vibrat & Strength Mech Beijing 100124 Peoples R China

Innovated by the nonlinear modes concept in the vibrational dynamics, the vertical periodic orbits around the triangular libration points are revisited for the Circular Restricted Three-body Problem. The. zeta-component motion is treated as the dominant motion and the xi and eta-component motions are treated as the slave motions. The slave motions are in nature related to the dominant motion through the approximate nonlinear polynomial expansions with respect to the zeta-position and zeta-velocity during the one of the periodic orbital motions. By employing the relations among the three directions, the three-dimensional system can be transferred into one-dimensional problem. Then the approximate three-dimensional vertical periodic solution can be analytically obtained by solving the dominant motion only on zeta-direction. To demonstrate the effectiveness of the proposed method, an accuracy study was carried out to validate the polynomial expansion (PE) method. As one of the applications, the invariant nonlinear relations in polynomial expansion form are used as constraints to obtain numerical solutions by differential correction. The nonlinear relations among the directions provide an alternative point of view to explore the overall dynamics of periodic orbits around libration points with general rules.

关键词： triangular libration points Vertical periodic orbits Polynomial expansion method

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High-order solutions around triangular libration points in the elliptic restricted three-body problem and applications to low energy transfers

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2014年第9期19卷 3374-3398页

作者： Lei, Hanlun Xu, Bo Nanjing Univ Sch Astron & Space Sci Nanjing 210093 Jiangsu Peoples R China

High-order series expansions around triangular libration points in the elliptic restricted three-body problem (ERTBP) are constructed first, and then with the aid of the series solutions, two-impulse and low-thrust low energy transfers to the triangular point orbits of the Earth-Moon system are designed in this paper. The equations of motion of ERTBP in the pulsating synodic reference frame have the same symmetries as the ones in circular restricted three-body problem (CRTBP), and also have five equilibrium points. Considering the stable dynamics of triangular libration points, the analytical solutions of the motion around them in ERTBP are expressed as formal series of four amplitudes: the orbital eccentricity of the primary, the long, short and vertical periodic amplitudes. The series expansions truncated at arbitrary order are constructed by means of Lindstedt-Poincare method, and then the quasi-periodic orbits around triangular libration points in ERTBP can all be parameterized. In particular, when the eccentricity of the primary is zero, the series expansions constructed can be reduced to describe the motion around triangular libration points in CRTBP. In order to check the validity of the series expansions constructed, the domain of convergence corresponding to different orders is studied by using numerical integration. After obtaining the analytical solutions of the bounded orbits around triangular points, the target orbits in practical missions can be expressed by several related parameters. Thanks to the series expansions constructed, two missions are planned to transfer a spacecraft from the Earth to the short periodic orbits around triangular libration points of Earth-Moon system. To complete the missions with less fuel cost, low energy transfers (two-impulse and low-thrust) are investigated by means of numerical optimization methods (both global and local optimization techniques). Simulation results indicate that (a) the low-thrust, low energy transfers

关键词： Elliptic restricted three-body problem triangular libration points Series expansions Lindstedt-Poincare method Two-impulse low energy transfer Low-thrust low energy transfer

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On Resonance Values of Parameters in the Problem of Stability of Lagrange Solutions in a Restricted Three-Body Problem Close to a Circle

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MECHANICS OF SOLIDS 2023年第7期58卷 2531-2543页

作者： Markeev, A. P. Moscow Aviat Inst Res Univ Moscow Russia

The restricted problem of three bodies (material points) moving under the action of gravitational attraction according to Newton's law is considered. The orbits of the main attracting bodies are considered to be ellipses with a small eccentricity, while a passively gravitating body can perform arbitrary spatial motion near the triangular libration point. For the Hamiltonian function corresponding to such a motion, the structure of the normal form is found in the case of third-order resonances. In the planar restricted three-body problem, the equations are derived up to the second degree of eccentricity for resonance curves for all resonances up to the sixth order inclusive.

关键词： restricted three-body problem triangular libration points resonance

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On the Metric Stability and the Nekhoroshev Estimate of the Velocity of Arnold Diffusion in a Special Case of the Three-body Problem

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REGULAR & CHAOTIC DYNAMICS 2021年第4期26卷 321-330页

作者： Markeev, Anatoly P. Ishlinsky Inst Problems Mech RAS Pr Vernadskogo 101-1 Moscow 119526 Russia Natl Res Univ Moscow Aviat Inst Volokolamskoe Sh 4 Moscow 125080 Russia

A study is made of the stability of triangular libration points in the nearly-circular restricted three-body problem in the spatial case. The problem of stability for most (in the sense of Lebesgue measure) initial conditions in the planar case has been investigated earlier. In the spatial case, an identical resonance takes place: for all values of the parameters of the problem the period of Keplerian motion of the two main attracting bodies is equal to the period of small linear oscillations of the third body of negligible mass along the axis perpendicular to the plane of the orbit of the main bodies. In this paper it is assumed that there are no resonances of the planar problem through order six. Using classical perturbation theory, KAM theory and algorithms of computer calculations, stability is proved for most initial conditions and the Nekhoroshev estimate of the time of stability is given for trajectories starting in an addition to the above-mentioned set of most initial conditions.

关键词： restricted three-body problem triangular libration points stability Arnold diffusion

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On the Stability of Lagrange Solutions in the Spatial Near-Circular Restricted Three-Body Problem

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MECHANICS OF SOLIDS 2021年第8期56卷 1541-1549页

作者： Markeev, A. P. Russian Acad Sci Ishlinsky Inst Problems Mech Moscow 119526 Russia Moscow Inst Aviat Technol Moscow 125993 Russia

The restricted problem of three bodies (material points) is considered. The orbits of the main attracting bodies are assumed to be ellipses of small eccentricity, and the passively gravitating body during its motion can leave the plane of the orbits of the main bodies (spatial problem). The stability of body motion corresponding to triangular Lagrangian libration points is investigated. A characteristic feature of the spatial problem under study is the presence of resonance due to the equality of the Keplerian motion period of the main bodies and the linear oscillation period of the passively gravitating body in the direction perpendicular to the plane of their orbits. Using the methods of classical perturbation theory, Kolmogorov-Arnold-Moser (KAM) theory and computer algebra algorithms, the nonlinear problem of stability for most (in the Lebesgue-measure sense) initial conditions and formal stability (stability in any arbitrarily high finite approximation with respect to the coordinates and impulses of perturbed motion) are investigated.

关键词： restricted three-body problem triangular libration points stability

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Low-thrust transfer to the Earth-Moon triangular libration point via horseshoe orbit

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ACTA ASTRONAUTICA 2020年 177卷 111-121页

作者： He, Xingji Liang, Yuying Xu, Ming Zheng, Yaru Beihang Univ Sch Astronaut Beijing 102206 Peoples R China York Univ Toronto ON M3J 1P3 Canada Beihang Univ Shenyuan Honors Coll Beijing 100191 Peoples R China 9th Designing Cas Wuhan 430040 Peoples R China

This paper proposes a novel method for the low-thrust transfer to the triangular libration points in the Earth-Moon system. Horseshoe orbit is explored as the coasting arc in this transfer scenario. By tangential thrust control strategy, the low-thrust transfer from the Earth to the short-period orbit around L-4 point is realized. The pseudospectral method is used to obtain another Earth-L-4 transfer trajectory by solving an energy-optimal low-thrust trajectory optimization problem. Interestingly, these two kinds of Earth-L-4 low-thrust transfer trajectories both share similar geometric configuration and velocity trend with a natural Horseshoe orbit, which evokes the main contributions of this paper that leveraging Horseshoe orbit as the coasting arc. The complete transfer trajectory is divided into three phases: Earth spiral phase, Horseshoe coasting phase and L-4 point insertion phase, where the first and the third are powered phases. The tangential thrust control strategy is used in the first phase to generate the transfer from the Earth to the Horseshoe orbit, and the pseudospectral method is utilized in the third phase to enable the transfer from the Horseshoe coasting arc to the L(4 )point by solving a time-optimal transfer problem. Then, by patching these two powered segments with the unpowered Horseshoe coasting arc, the Earth-L-4 low-thrust transfer is constructed.

关键词： Low-thrust transfer Horseshoe orbit triangular libration points Earth-moon system Pseudospectral method

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Non-linear Stability of triangular Equilibrium points in ER3BP Under Radiating Primaries in Presence of Resonance

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DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS 2018年第4期26卷 331-354页

作者： Narayan, A. Singh, Nutan Bhilai Inst Technol Durg India

The non linear stability of the triangular equilibrium points are studied considering the radiation pressure of both the primaries in the presence of third and fourth order resonance. Markeev (libration points in celestial mechanics and cosmodynamics, 1978) method is applied throughout our analysis to study the non linear stability of the system. In order to do so, the Birkhoff 's normal form in the vicinity of the triangular equilibrium points up to fourth order terms of the Hamiltonian is computed. The normal forms of the Hamiltonian contains both the resonance cases omega(1) = 2 omega(2)/ and omega(1)= 3 omega(2)). It is also observed that the triangular equilibrium points satisfying the conditions mu <= 1.4, mu(c) = 0.0385209;are unstable in the third order resonance but stable in the fourth order resonance for different values of radiation pressure taken.

关键词： ER3BP Hamiltonian functions triangular libration points Resonance Radiation pressure Stability

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