Earth-Moon Collinear Libration Point Research Articles

Quasi-periodic orbits of small solar sails with time-varying attitude around Earth–Moon libration points

This paper proposes new quasi-periodic orbits around Earth–Moon collinear libration points using solar sails. By including the time-varying sail orientation in the linearized equations of motion for the circular restricted three-body problem (CR3BP), four types of quasi-periodic orbits (two types around L1 and two types around L2) were formulated. Among them, one type of orbit around L2 realizes a considerably small geometry variation while ensuring visibility from the Earth if (and only if) the sail acceleration due to solar radiation pressure is approximately of a certain magnitude, which is much smaller than that assumed in several previous studies. This means that only small solar sails can remain in the vicinity of L2 for a long time without propellant consumption. The orbits designed in the linearized CR3BP can be translated into nonlinear CR3BP and high-fidelity ephemeris models without losing geometrical characteristics. In this study, new quasi-periodic orbits are formulated, and their characteristics are discussed. Furthermore, their extendibility to higher-fidelity dynamic models was verified using numerical examples.

Two-maneuver transfers from the collinear L2 point to the triangular L4 point in the planar Earth-Moon system

China’s Chang’E-4 (CE-4) probe has started its exploration on the far side of the Moon thanks to the relay satellite---Queqiao---which is around the Earth-Moon collinear libration point L2 and provides continuous communication links between the lander and ground stations. The triangular libration points of the Earth-Moon system, L4 and L5, have long been considered as potential locations for future space applications. As a possible extension of Queqiao or other future missions around L2, the work presented here contributes to constructing transfers from the collinear libration point (L2) to the triangular libration points. Taking the L4 point as an example, two types of two-maneuver transfers from a planar Lyapunov orbit around L2 to the periodic orbit around L4 are investigated in the planar Earth-Moon system and compared with each other, including the type-I transfer utilizing invariant manifolds along with powered lunar gravity assist and the type-II transfer using the Jacobi integral. Our studies indicate that type-I transfer saves more Δv cost than type-II transfer. For the cases studied by us, the minimum Δv cost is about 0.1636 km/s for type-I transfer, with a corresponding transfer time of 169.84 days. For type-II transfer and the cases studied by us, the corresponding values are 0.4745 km/s and 137.19 days. At the end of the paper, the type-I transfer is generalized (with some modifications) to the ephemeris model and some example transfer trajectories are given. Although only the transfer from L2 to L4 is studied in this paper, the same orbit design strategy can be generalized to the transfer to the triangular libration point L5, and the transfers from the collinear libration point L1 to triangular libration points.

The Characteristics and Related Problems of the Orbits Around the Earth-Moon Libration Points

Due to the specific dynamics, the probes located at the halo orbits or Lissajous orbits around the Earth-Moon collinear libration point L1 or L2 are always studied in the synodic system to understand their trajectories. In fact, they are also orbiting the Earth in a distant Keplerian ellipse. Because of their intrinsic orbital instability, in the orbit prediction the initial errors propagate more prominently than those of the normal orbiting satellites, this requires special attention in the orbit design, maneuver, and control. Despite of all this, they are similar to the normal orbiting satellites in orbit determination and hardly require other special attentions. In this paper, the quantitative results of error propagation under the unstable dynamics, together with the theoretical analysis are presented. The results of precise orbit determination and short-arc orbit predictions are also shown, and compared with the results from the Beijing Aerospace Control Center.

Scheme design of the CHANG’E-5T1 extended mission

Flight schemes for the CHANG’E-5T1 extended mission are investigated in this paper. In the flight scheme and trajectory design, the remaining propellant of the CHANG’E-5T1 mission is utilized. The CHANG’E-5T1 mission is firstly introduced with feasible flight goals derived based on the terminal trajectory and satellite status. The flight schemes are designed to include a lunar return and the libration points in the Sun-Earth/Moon and Earth-Moon systems, with an emphasis on the Earth-Moon triangle libration point thus far unexplored. Secondly, three schemes are proposed for the CHANG’E-5T1 extended mission with different flight goals. The direct libration point orbit transfer and injection method is adopted to solve the issue in the transfer trajectory design. Furthermore, an innovative concept is proposed to transfer from the Earth-Moon collinear libration point to the triangle point using the Sun-Earth/Moon libration point. Finally, the merits and drawbacks of the three schemes are discussed in terms of flight time, control energy and frequency, flight distance, and goal value. As a result, the scheme including a lunar return and the Earth-Moon L2 libration point is selected for the CHANG’E-5T1 extended mission. A flight to the Earth-Moon libration point is achieved, replicating the achievement of the ARTEMIS mission.

Maintenance of Relay Orbit About the Earth-Moon Collinear Libration Points

Relay orbits about the Earth-Moon collinear libration point shave significant valueon the exploration of the lunar farside, but have complex kinetic characteristics in the nature, thus the orbit maintenance has always been focused in the deep space navigation and control field. This paper explores orbit maintenance technology of the relay orbit about the collinear Earth-Moon libration points under the real dynamical conditions. First, based on the restricted three-body problem, the mathematic model of relay orbit station-keeping with the real dynamical model is analyzed. The continue-circling method is presented for the relay orbit maintenance with the two control styles, i.e., the Halo style and the Lissajous style. Second, with the third-body gravitation and the solar radiation pressure perturbations considered, the method is tested and analyzed by using the numerical simulations to achieve the control frequency and the corresponding velocity increment required by the relay orbits with different amplitudes. According to the simulations, the Lissajous style is suitable to the orbit maintenance with a control interval of 7.4 days and a velocity increment less than 20 m/s/a. Furthermore, the method has been successfully applied in Chang'e-2 and Chang'e-5T1 extended missions and can provide a beneficial reference for the future Chang'e-4 mission.

Long-Term Station Keeping of Space Station in Lunar Halo Orbits

Optimization methods are presented for long-term station keeping of a low-thrust space station in lunar halo orbits for unstable collinear libration points. The methods use discretization of each half-revolution into segments and a set of pseudoimpulses for each segment. A matrix inequality of the sum of the characteristic velocities for the pseudoimpulses is used to transform the problem into a large-scale linear programming form. Terminal constraints are presented as a linear matrix equation using numerical partial derivatives along a reference orbit. Possible station-keeping strategies for quasi-periodic orbits in the vicinity of the Earth–moon collinear libration point are considered. An iterative shooting algorithm, based on the full ephemeris model and differential correction, is used. Preliminary simulation and comparative results of long-term station-keeping strategies for one year are presented. As application examples, two quasi-periodic orbits are considered. The first is similar to a planar Lyapunov orbit. The second is a halo orbit, with continuous visibility of the space station from the Earth. An algorithm for very low-thrust station keeping with long-duration burns is also presented. The algorithm can be used to estimate the required thrust-to-weight ratio of the space station.

Optimal station-keeping near Earth–Moon collinear libration points using continuous and impulsive maneuvers

In this study the gravitational perturbations of the Sun and other planets are modeled on the dynamics near the Earth–Moon Lagrange points and optimal continuous and discrete station-keeping maneuvers are found to maintain spacecraft about these points. The most critical perturbation effect near the L1 and L2 Lagrange points of the Earth–Moon is the ellipticity of the Moon’s orbit and the Sun’s gravity, respectively. These perturbations deviate the spacecraft from its nominal orbit and have been modeled through a restricted five-body problem (R5BP) formulation compatible with circular restricted three-body problem (CR3BP). The continuous control or impulsive maneuvers can compensate the deviation and keep the spacecraft on the closed orbit about the Lagrange point. The continuous control has been computed using linear quadratic regulator (LQR) and is compared with nonlinear programming (NP). The multiple shooting (MS) has been used for the computation of impulsive maneuvers to keep the trajectory closed and subsequently an optimized MS (OMS) method and multiple impulses optimization (MIO) method have been introduced, which minimize the summation of multiple impulses. In these two methods the spacecraft is allowed to deviate from the nominal orbit; however, the spacecraft trajectory should close itself. In this manner, some closed or nearly closed trajectories around the Earth–Moon Lagrange points are found that need almost zero station-keeping maneuver.

Autonomous optical navigation for orbits around Earth–Moon collinear libration points

The analysis of optical navigation in an Earth–Moon libration point orbit is examined. Missions to libration points have been winning momentum during the last decades. Its unique characteristics make it suitable for a number of operational and scientific goals. Literature aimed to study dynamics, guidance and control of unstable orbits around collinear libration points is vast. In particular, several papers deal with the optimisation of the Δv budget associated to the station-keeping of these orbits. One of the results obtained in literature establishes the critical character of the Moon–Earth system in this aspect. The reason for this behaviour is twofold: high Δv cost and short optimal manoeuvre spacing. Optical autonomous navigation can address the issue of allowing a more flexible manoeuvre design. This technology has been selected to overcome similar difficulties in other critical scenarios. This paper analyses in detail this solution. A whole GNC system is defined to meet the requirements imposed by the unstable dynamic environment. Finally, a real simulation of a spacecraft following a halo orbit of the L2 Moon–Earth system is carried out to assess the actual capabilities of the optical navigation in this scenario.

Disturbance compensating control of orbit radially aligned two-craft Coulomb formation

This paper investigates the orbit radial stabilization of a two-craft virtual Coulomb structure about circular orbits and at Earth–Moon libration points. A generic Lyapunov feedback controller is designed for asymptotically stabilizing an orbit radial configuration about circular orbits and collinear libration points. The new feedback controller at the libration points is provided as a generic control law in which circular Earth orbit control form a special case. This control law can withstand differential solar perturbation effects on the two-craft formation. Electrostatic Coulomb forces acting in the longitudinal direction control the relative distance between the two satellites and inertial electric propulsion thrusting acting in the transverse directions control the in-plane and out-of-plane attitude motions. The electrostatic virtual tether between the two craft is capable of both tensile and compressive forces. Using the Lyapunov’s second method the feedback control law guarantees closed loop stability. Numerical simulations using the non-linear control law are presented for circular orbits and at an Earth–Moon collinear libration point.