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

https://doi.org/10.1016/j.cnsns.2017.05.031Get rights and content

Highlights

  • A set of cislunar periodic orbits visiting triangular libration points in configuration space are defined and computed, and their stability analysis is performed.

  • The lunar gravity assist is discussed in detail and applied to the strategy of trajectory design in combination with (natural and perturbed) invariant manifolds.

  • The gravitational perturbation of the Sun is taken into consideration, and the corresponding low-energy transfers are obtained.

Abstract

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 50 days, (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 200 m/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.

Introduction

Periodic orbits in the circular restricted three-body problem (CRTBP) have been widely investigated [1], [2], [3], [4], [5], and the critical role of periodic orbits in understanding the structure of a dynamical system has been recognized by poincaré [6]. In the CRTBP, there are varieties of basic families of simple periodic orbits, such as periodic orbits around libration points and around the primaries. Based on these basic families, plenty of bifurcated families of periodic orbits have been found [2], [4], [5]. These periodic orbits hold rich characteristics in terms of space configuration and dynamics [7], and they have played crucial roles in deep space exploration and attracted great interest of scientists and engineers.

The libration point orbits in the Sun–Earth and Earth–Moon systems have been applied to many scientific missions. For example, the periodic orbits around the Sun–Earth L1 point could provide suitable locations to observe the Sun and have been adopted by varieties of missions, such as ISEE-3 (1978), SOHO (1995), and GENESIS (2001), and the ones around the Sun–Earth L2 point possess ideal thermal and dynamical environment and have been taken to place astronomy telescopes, such as the missions MAP (2001), PLANK (2007) and GAIA (2012) [8]. For the periodic orbits around Li(i=3,4,5) points of the Sun–Earth system, their potential applications have been proposed by Hou et al. [9].

The periodic orbits around the equilibrium points of Earth–Moon system are very useful in lunar explorations, such as observing the opposite side of the Moon relative to the Earth [10], [11], providing gateway for space transportation system of the future [12], and placing communication or navigation satellites [13], [14]. The first mission that takes advantage of the Earth–Moon libration point orbits is the Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon's Interaction with the Sun (ARTEMIS) mission, which uses simultaneous measurements of particles and electric and magnetic fields from two different locations to provide three-dimensional information on how energetic particle acceleration occurs [15].

Periodic orbits around the primary bodies are becoming increasingly interesting and have been applied to practical missions due to their special configuration and long-term stability, especially for the periodic orbits in the Earth–Moon system. In 2008, the Interstellar Boundary Explorer (IBEX) vehicle [16] was launched into a high-altitude Earth orbit, and in the extension stage, it was inserted into a stable lunar resonant orbit, which is a member of families of periodic orbits around the Earth in the Earth–Moon CRTBP [17]. In 2013, NASA selected the Transiting Exoplanet Survey Satellite (TESS) for launch in 2017 as an astrophysics explorer mission, which aims to search for planets transiting bright and nearby stars [18]. The science orbit of the TESS mission is a long-term stable orbit in near 2:1 resonance with the Moon, which is also a periodic orbit around the Earth in the Earth–Moon CRTBP [19]. With the successful implementation of these missions, it is believed that there will be increasing number of practical missions that may take advantage of this kind of complex periodic orbits in the CRTBP to satisfy varieties of science and technology requirements.

For the study on complex periodic orbits in the CRTBP, Folta et al. [20] provided a reference catalog for trajectory design in the Earth–Moon system to guide orbit design, including libration point orbits, resonant orbits and Moon-centered orbits. In [21], the planar and three-dimensional resonant orbits in the Earth–Moon system are analyzed and catalogued. They identified the planar and three-dimensional homoclinic- and heteroclinic-type trajectories between the unstable resonant orbits in the Earth–Moon system by taking advantage of their invariant manifold structures. In reference [22], Leiva and Briozzo presented a methodology to compute the set of unstable periodic orbits in the region of interest, and then developed a control algorithm allowing stabilizing the chosen unstable periodic orbit over long periods with very low fuel consumption. Parker et al. [23] developed numerical methods for constructing orbit transfers, complex chains and periodic orbits in the Earth–Moon system using the invariant manifolds of unstable periodic orbits. The complex orbit chains and periodic orbits obtained in reference [23] may have many applications in practical deep-space missions. In order to understand the dynamics of the IBEX extended mission orbit and its apparent long-term stability, Dichmann et al. [24] examined the dynamics of three types of near 3:1 resonant periodic orbits in the Earth–Moon system, namely, planar mirror, reflection and axial periodic orbits. The local stability of these resonant periodic orbits was analyzed by using Floquet analysis and the nonlinear stability was examined by using Poincaré maps. For the TESS mission orbit, Short et al. [25] applied the flow-based analysis strategies to characterizing the motion modes and exploring the long-term behavior of the orbit. By synchronizing the third body's orbital period to integer fractions of the Moon's sidereal period (3:1 or 2:1 resonance) and phasing apogee to stay away from the Moon, a new class of long-term stable lunar resonance orbits (IBEX-like and TESS-like orbits) have been obtained by McComas et al. [17]. These orbits provide cost-effective and nearly ideal locations for long-term space weather observations because (a) a spacecraft on these orbits can remotely image the Earth's magnetosphere from outside its boundaries, and (b) a spacecraft on most part of these orbits could provide external observations, such as solar wind, magnetosheath. To highlight the benefit of employing resonant periodic orbits for transfers, Vaquero and Howell [19] constructed planar and three-dimensional transfers from a low Earth orbit to the vicinities of the Earth–Moon libration points by taking advantage of the resonant arcs. Peng and Xu [26] utilized continuation methods together with the multiple-shooting differential correction method to generate two groups of multi-revolution elliptic halo orbits around the collinear libration points in the elliptic restricted three-body problem, and systematically investigated their stability evolution with respect to the eccentricity and the mass ratio of the system.

For a practical mission, after the science orbit is determined, how to transfer a spacecraft from the low Earth orbit (LEO) to the target orbit is of great significance. As the positions of starting and arriving points of a feasible transfer trajectory are constrained on two given orbits, the problem of transfer trajectory design can be considered as a two-point boundary value problem (TPBVP). In the two-body dynamical system, the TPBVP is also called the well-known Lambert problem, which can be easily solved by iterative method [27]. However, in the multi-body dynamical environment, the TPBVP becomes more complicated because of the following facts: (a) no analytical solutions, (b) complex dynamics, and (c) multiple solutions.

Usually, in the multi-body dynamical system the direct transfers from the LEO to the target orbit are fuel-consuming. For a long-term unmanned mission, it is essential to reduce the fuel cost in order to increase the scientific payload, thus low-energy transfers have attracted great interest of scientists and engineers. In literature, multiple fuel-saving techniques are incorporated into the natural dynamics of multi-body dynamical system in order to obtain fuel-saving transfer trajectories, such as low-thrust propulsion, gravity assist technique [28], and the like. Capdevila et al. [29], [30] systematically constructed a network of transfer trajectories connecting regions around the Earth, Moon, and the triangular libration points in the Earth–Moon system. Koon et al. [31] approximated the Sun–Earth–Moon–spacecraft four-body system as two restricted three-body problems, and they constructed low-energy transfer trajectories from the Earth to the Moon by matching the invariant manifold structures of periodic orbits around collinear libration points in a predefined section. The transfer trajectories they constructed are similar to the trajectory adopted by Hiten mission [32], which is designed based on the weak stability boundary theory (WSB) developed by Belbruno [33]. After the work of Koon et al. [31], the invariant manifold structures of Lagrange points have been widely used in the field of low-energy transfers. For lunar exploration mission, Mingotti et al. [34] combined the structures of invariant manifolds of two restricted three-body problems (Sun–Earth and Earth–Moon systems) with low-thrust propulsion to develop a systematic method, and the resulting low-energy, low-thrust transfers require less propellant than the standard low-energy transfers. The attainable sets proposed in [35], playing similar role to invariant manifolds in trajectory design, have been successfully applied to the design of lunar and interplanetary low-thrust, low-energy transfers [36], [37], [38]. In addition, the invariant manifold structures in the CRTBP have been successfully applied but not limited to the following problems: transfers to distant retrograde orbits of Earth–Moon system [39], transfers to the L3 halo orbits of Earth–Moon system [40], [41], indirect transfers to the Earth–Moon L1 point or halo orbit [42], [43], transfers between libration point orbits of Sun–Earth and Earth–Moon systems [44], [54], looking for the rescue trajectories that leave the lunar surface [45], and the like.

From the viewpoint of practical application, the search, identification and computation of complex periodic orbits in the CRTBP as well as the techniques on how to design optimal transfer trajectories in terms of fuel consumption and time of flight are of great significance. In this paper, we concentrate on the following two aspects: (a) computation of a set of cislunar periodic orbits, and (b) design of low-energy transfer trajectories. Based on the families of impulsive transfers between triangular libration points computed in [46], a set of periodic orbits that pass through triangular libration points of Earth–Moon system are defined and computed, and their linear stabilities are analyzed. Among all the periodic orbits computed, an unstable periodic orbit in near 3:2 resonance with the Moon is taken as the science orbit of an assumed mission, and trajectory design techniques by combining lunar gravity assist with invariant manifolds are developed. In this study, the lunar gravity assist is studied in detail, and in the process of trajectory optimization, both the natural and perturbed invariant manifolds are taken into account. Finally, the low-energy transfers obtained in the Earth–Moon system are corrected to the Sun-perturbed Earth–Moon system.

The remainder of this paper is organized as follows. In Section. 2, the basic dynamical model adopted in this paper is briefly introduced, and then the periodic orbits passing through triangular libration points are computed in Section. 3. In Section. 4, discussions on how to design direct and low-energy transfer trajectories are performed. Finally, the conclusions are drawn in Section. 5.

Section snippets

Basic dynamical model

In the circular restricted three-body problem (CRTBP), the motions of an infinitesimal particle, such as a spacecraft, are governed by the gravitational field generated by two massive primaries, where the primary body P1 with mass m1 and the secondary body P2 with mass m2 move around their barycenter in circular orbits. Usually, the equations of motion are formulated in a synodic coordinate system, where the origin is fixed at the barycenter of primaries, the x-axis is directed from the massive

Computation of periodic orbits

According to our previous study [46], impulsive transfers between triangular libration points in the CRTBP are distributed on the characteristic plane (Period–Jacobi constant) in the form of families. Starting with the families of impulsive transfers from point L4 to L5 and from point L5 to L4, we aim to compute a type of periodic orbits that pass through triangular libration points in configuration space. The key concept is to match the states of termination points of the impulsive transfers

Low-energy transfer trajectories

As the periodic orbit A6 is suitable to place navigation satellites for communication in cislunar space, we will take it as the nominal orbit of an assumed mission and discuss how to construct low-energy transfers to it from low Earth orbit (LEO). In practical computation, the altitude of LEO is taken as 200 km.

Conclusions

In this paper, the set of periodic orbits passing through triangular libration points in cislunar space are computed, and taking an unstable periodic orbit in near 3:2 resonance with the Moon as the science orbit of an assumed mission, we investigate the techniques on how to design transfer trajectories.

Based on the families of impulsive transfers between triangular libration points, the periodic orbits that pass through triangular libration points are computed by means of multiple shooting

Acknowledgments

This work was carried out with financial support of the National Natural Science Foundation (No. 11603011), the Natural Science Foundation of Jiangsu province (No. BK20160612), the National Basic Research Program 973 of China (No. 2013CB834103 and 2015CB857100), the Satellite Communication and Navigation Collaborative Innovation Center (No. SatCN-201409) and National Defense Scientific Research Fund (No. 2016110C019).

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