Collinear Libration Points Research Articles

Dynamics of three-dimensional capture orbits from libration region analysis

Low-energy trajectories take advantage of the mutual action of multiple celestial bodies on the spacecraft, and can conclude with ballistic capture about the arrival body, thus allowing significant savings in terms of propellant consumption, if compared to more traditional transfers. Because of the chaotic nature of multibody environments, the design of low-energy trajectories with given constraints can be complex and it is often obtained after a long, iterative, and eventually computationally expensive process.This work is aimed at identifying a limited set of characteristic parameters related both to the time behavior of three-dimensional ballistic capture orbits and to some osculating orbit elements (i.e., inclination, semimajor axis, and eccentricity), relative either to the departure or to the arrival body. The analysis is performed using the linear expansion of the Hamiltonian equations of motion in the equilibrium region around the collinear libration point L1 (or L2), in the dynamical framework of the 3 dimensional circular restricted 3-body problem. A correlation among some Hamiltonian parameters in the equilibrium region and the trajectory osculating orbital elements at capture is established. This result is used to design missions with ballistic capture having required orbital parameters at the arrival planet and it provides a strategy to control the target orbital elements at capture by small thrust maneuvers at the equilibrium region. Because of the long flight time, the solar perturbation is considered in the analysis, and suitable launch dates for the ballistically captured missions are determined.

Structure of the centre manifold of the $$L_1,L_2$$ collinear libration points in the restricted three-body problem

We present a global analysis of the center manifold of the collinear points in the circular restricted three-body problem. The phase-space structure is provided by a family of resonant 2-DOF Hamiltonian normal forms. The near 1:1 commensurability leads to the construction of a detuned Birkhoff-Gustavson normal form. The bifurcation sequences of the main orbit families are investigated by a geometric theory based on the reduction of the symmetries of the normal form, invariant under spatial mirror symmetries and time reversion. This global picture applies to any values of the mass parameter.

A Modified Targeting Strategy for Station-Keeping of Libration Point Orbits in the Real Earth-Moon System

In this paper, a modified targeting strategy is developed for missions on libration point orbits (LPOs) in the real Earth-Moon system. In order to simulate a station-keeping procedure in a dynamic model as realistic as possible, LPOs generated in the circular restricted three-body problem (CRTBP) are discretized and reconverged in a geocentric inertial system for later simulations. After that, based on the dynamic property of the state transition matrix, a modified strategy of selecting target points for station-keeping is presented to reduce maneuver costs. By considering both the solar gravity and radiation pressure in a nominal LPO design, station-keeping simulations about fuel consumption for real LPOs around both collinear and triangular libration points are performed in a high-fidelity ephemeris model. Results show the effectivity of the modified strategy with total maneuver costs reduced by greater than 10% for maintaining triangular LPOs.

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.

The effect of small perturbations in the Coriolis and centrifugal forces in the axisymmetric restricted five-body problem

In the framework of the axisymmetric problem of restricted five bodies, the existence and stability of the libration points, the regions of possible motion are illustrated and analyzed numerically, under the effect of small perturbations in the Coriolis and centrifugal forces. It is explored how the parameters influence substantially the positions of the libration points and the possible regions of motion. In an attempt to understand how the parameters involved due to the small perturbations in the Coriolis and centrifugal forces affect the stability of the libration points, we perform a systematic investigation and reveal that some of the collinear and non-collinear libration points are stable under these perturbations, whereas none of these libration points are stable for any combination of the angle parameters when the effects of these forces are neglected.

Periodic and quasi-periodic orbit design based on the center manifold theory

This paper proposes a new numerical method for finding libration point orbits in the vicinity of collinear libration points in the circular restricted three-body problem. The main advantage of this method is that it requires neither an initial guess nor complex algebraic manipulations for finding both quasi-periodic and periodic orbits. The proposed method consists of two steps: center manifold design and differential correction. The first step provides a quasi-periodic orbit parametrized by a single parameter vector. In the second step, the parameter vector in the first step is used to obtain an exact periodic orbit. This method is applied to find periodic and quasi-periodic orbits in the Sun-Earth restricted three-body problem around the L1 and L2 libration points.

Mission Design Problems for the Spectrum-Roentgen-Gamma Project

The Spectrum-Roentgen-Gamma (SRG) project is intended for surveying the entire sky in the X‑ray band using two telescopes mounted onboard the spacecraft of the same name (SRG). In addition, some specially chosen areas of the sky and some radiation sources are planned to be investigated after the completion of the survey. The spacecraft was planned to be launched in 2018 into the vicinity of solar-terrestrial collinear libration point L2, using the Proton-M launch vehicle with a DM-03 block upper stage. The technology for completely surveying the celestial sphere consists in scanning by rotating the parallel axes of telescopes around the spacecraft axis, which roughly followed the direction to the Sun. The measurement data, stored onboard the spacecraft, would then be transmitted to the ground receiving stations using the mean gain antenna, the axis of which coincided with the spacecraft’s axis of rotation. This imposed some constraints on the mission design. These constraints included the permissible amplitude of the trajectory of spacecraft motion relative to the libration point in the direction orthogonal to the Sun–Earth line. In addition, the consumption of the onboard propellant was allowed only for orbit correction needed for keeping the spacecraft near the libration point. In this connection, a method was developed using the upper stage in order to decrease the mentioned amplitude down to acceptable values, in the conditions of the requirements imposed on the unit. The paper presents an assessment of its effectiveness.

Libration Points in the R3BP with a Triaxial Rigid Body as the Smaller Primary and a Variable Mass Infinitesimal Bod

The paper deals with the existence of the coplanar libration points in the restricted three-body problem when the smaller primary is a triaxial rigid body and the infinitesimal body is of variable mass. Following small parameter method, the coordinates of collinear libration points are established whereas the coordinates of triangular libration points are established by classical method. It is found that the mass reduction factor has small effect but triaxiality parameters of the smaller primary have great effects on the coordinates of the libration points.

Station-keeping of libration point orbits by means of projecting to the manifolds

There are many different types of (quasi)periodic orbits in the vicinity of collinear libration points. These libration point orbits (LPOs) are inherently unstable and must be controlled. This paper studies general station-keeping strategies that are suitable for various types of LPOs. Two complementary methods are proposed by exploiting the hyperbolic dynamics of the collinear libration points. One main advantage of these methods is that no information of the nominal orbit is needed in order to do the station-keeping. The first is a semi-analytical projection method which is based on the high order polynomial approximation of the center manifold. After the transformation from synodic coordinates to center manifold coordinates is derived, the error state can be projected to center manifold or its stable manifold by introducing a station-keeping Delta-V, effectively eliminating the unstable component. The second is a fully numerical method. An escape time algorithm is designed to find the intersection of stable and unstable manifolds, effectively projecting the error state to the center manifold. By applying these techniques to the collinear libration points of Earth–Moon system, it has been demonstrated that these proposed methods can achieve better maneuver cost performance and they can handle station-keeping problems of various types of LPOs in a unified manner.

Integrated orbit determination and maintenance in Earth-Moon unstable dynamics environment

The hyperbolic instability of collinear libration point orbits dominates the distribution of orbit uncertainty, which is closely related with the unstable manifold of the local orbit. Exploiting the distribution of orbit uncertainty in an unstable orbital environment, an integrated orbit determination and maintenance approach is proposed. X-ray pulsar navigation embedded in the Kalman filtering architecture is adopted to determine the orbit of a spacecraft flying along the Earth-Moon L2 halo orbit. To keep the spacecraft in the vicinity of the nominal trajectory, the newly proposed orbit maintenance approach based on the covariance matrix is proposed to design the orbit maintenance maneuver. The numerical simulations show that the proposed approach has a good performance in orbit determination and maintenance.

Lissajous motion near Lagrangian point $$ L_{2} $$ L 2 in radial solar sail

An attempt was made to study the dynamics close to the collinear libration point $$ L_{2} $$ of the radial solar sail circular-restricted three-body problem (RSCRTBP) in the Sun–Jupiter System, where the third massless body is a solar sail. We analyse the qausi-periodic (Lissajous solutions) orbits about the libration point $$ L_{2} $$ . The Lindstedt–Poincare approximation for the qausi-periodic orbits was used for numerical simulations. We utilized linear quadratic regulator (LQR) to stabilize the full nonlinear model, and linear state-feedback controller was designed to stabilize the trajectory.

Robust Station-Keeping Control of Sun-Earth/Moon Libration Point Orbits Using Electric Propulsion

AbstractA novel strategy combining an extended state observer (ESO) and a backstepping sliding mode (BSM) controller is proposed for station-keeping of unstable collinear libration point orbits (LP...

On the fractal basins of convergence of the libration points in the axisymmetric five-body problem: The convex configuration

In the present work, the Newton–Raphson basins of convergence, corresponding to the coplanar libration points (which act as numerical attractors), are unveiled in the axisymmetric five-body problem, where convex configuration is considered. In particular, the four primaries are set in axisymmetric central configuration, where the motion is governed only by mutual gravitational attractions. It is observed that the total number libration points are either eleven, thirteen or fifteen for different combinations of the angle parameters. Moreover, the stability analysis revealed that all the libration points are linearly stable for all the studied combination of angle parameters. The multivariate version of the Newton–Raphson iterative scheme is used to reveal the structures of the basins of convergence, associated with the libration points, on various types of two-dimensional configuration planes. In addition, we present how the basins of convergence are related with the corresponding number of required iterations. It is unveiled that in almost every case, the basins of convergence corresponding to the collinear libration point L2 have infinite extent. Moreover, for some combination of the angle parameters, the other collinear libration points L1,2 also have infinite extent. In addition, it can be observed that the domains of convergence, associated with the collinear libration point L1, look like exotic bugs with many legs and antennas whereas the domains of convergence, associated with L4,5 look like butterfly wings for some combinations of angle parameters. Particularly, our numerical investigation suggests that the evolution of the attracting domains in this dynamical system is very complicated, yet a worth studying problem.

Improved semi-analytical computation of center manifolds near collinear libration points

In the framework of the circular restricted three-body problem, the center manifolds associated with collinear libration points contain all the bounded orbits moving around these points. Semi-analytical computation of the center manifolds and the associated canonical transformation are valuable tools for exploring the design space of libration point missions. This paper deals with the refinement of reduction to the center manifold procedure. In order to reduce the amount of calculation needed and avoid repetitive computation of the Poisson bracket, a modified method is presented. By using a polynomial optimization technique, the coordinate transformation is conducted more efficiently. In addition, an alternative way to do the canonical coordinate transformation is discussed, which complements the classical approach. Numerical simulation confirms that more accurate and efficient numerical exploration of the center manifold is made possible by using the refined method.

Polynomial expansions of single-mode motions around equilibrium points in the circular restricted three-body problem

In this work, the single-mode motions around the collinear and triangular libration points in the circular restricted three-body problem are studied. To describe these motions, we adopt an invariant manifold approach, which states that a suitable pair of independent variables are taken as modal coordinates and the remaining state variables are expressed as polynomial series of them. Based on the invariant manifold approach, the general procedure on constructing polynomial expansions up to a certain order is outlined. Taking the Earth–Moon system as the example dynamical model, we construct the polynomial expansions up to the tenth order for the single-mode motions around collinear libration points, and up to order eight and six for the planar and vertical-periodic motions around triangular libration point, respectively. The application of the polynomial expansions constructed lies in that they can be used to determine the initial states for the single-mode motions around equilibrium points. To check the validity, the accuracy of initial states determined by the polynomial expansions is evaluated.

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.

Lunar Gravity-Assist Maneuver As a Way of Reducing the Orbit Amplitude in the Spectrum–Röntgen–Gamma Project

Spectrum–Rontgen–Gamma (SRG) is a space observatory designed to observe astrophysical objects in the X-ray range of the electromagnetic spectrum. SRG is planned to be launched in 2019 by a Proton-M launch vehicle with a DM3 upper stage. The spacecraft will be delivered to an orbit around the Sun–Earth collinear libration point L2 located at a distance of ~1.5 million km from the Earth. Although the SRG launch scheme has already been determined at present, in this paper we consider an alternative spacecraft transfer scenario using a lunar gravity-assist maneuver. The proposed scenario allows a oneimpulse transfer from a low Earth orbit to a small-amplitude orbit around the libration point to be performed while fulfilling the technical constraints and the scientific requirements of the mission.

An improved numerical method for constructing Halo/Lissajous orbits in a full solar system model

An improved numerical method that can construct Halo/Lissajous orbits in the vicinity of collinear libration points in a full solar system model is investigated. A full solar system gravitational model in the geocentric rotating coordinate system with a clear presentation of the angular velocity relative to the inertial coordinate system is proposed. An alternative way to determine patch points in the multiple shooting method is provided based on a dynamical analysis with Poincaré sections. By employing the new patch points and sequential quadratic programming, Halo orbits for L1, L2, and L3 points as well as Lissajous orbits for L1 and L2 points in the Earth-Moon system are generated with the proposed full solar system gravitational model to verify the effectiveness of the proposed method.

Periodic motion around libration points in the Elliptic Restricted Three-Body Problem

Periodicity of motion around the collinear libration point associated with the Elliptic Restricted Three-Body Problem is studied. A survey of periodic solutions in the Circular Restricted Three-Body Problem is presented considering both Sun–Earth and Earth–Moon systems. Halo, Lyapunov and Vertical families around L1, L2 and L3 points are investigated, and their orbital period ranges through the entire family are reported. Resonant motions within the orbit families in the circular problem are identified and selected as suitable initial guess to find periodic orbits in the elliptic problem, which are targeted using a differential correction algorithm. Periodic solutions found are cataloged depending on the number of revolutions around libration points. Geometry, dynamical behavior and stability properties of single-revolution orbits are shown, as well as double-, triple- and quadruple-revolution solutions.

Modeling low-thrust transfers between periodic orbits about five libration points: Manifolds and hierarchical design

The low-thrust version of the fuel-optimal transfers between periodic orbits with different energies in the vicinity of five libration points is exploited deeply in the Circular Restricted Three-Body Problem. Indirect optimization technique incorporated with constraint gradients is employed to further improve the computational efficiency and accuracy of the algorithm. The required optimal thrust magnitude and direction can be determined to create the bridging trajectory that connects the invariant manifolds. A hierarchical design strategy dividing the constraint set is proposed to seek the optimal solution when the problem cannot be solved directly. Meanwhile, the solution procedure and the value ranges of used variables are summarized. To highlight the effectivity of the transfer scheme and aim at different types of libration point orbits, transfer trajectories between some sample orbits, including Lyapunov orbits, planar orbits, halo orbits, axial orbits, vertical orbits and butterfly orbits for collinear and triangular libration points, are investigated with various time of flight. Numerical results show that the fuel consumption varies from a few kilograms to tens of kilograms, related to the locations and the types of mission orbits as well as the corresponding invariant manifold structures, and indicates that the low-thrust transfers may be a beneficial option for the extended science missions around different libration points.