The Libration Points in Photogravitational Restricted Three-Body Problem

The phologravilational restricled three-body problem in which the mass reduclionfactors of two primaries q_1 q_2( -∞, 1] are siudied and an analytic meihod toesli,;iale the number of libralion points ana io calculate lheir hoalion is given in thispaper. The results show lhal in phologravilalional reslricled three-body problem, thenumber of librafion poinis is .from one to seven for different q_1 and q_2. As application,the motion of dust grain like comet tail in the solar syslern is also discussed.

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. Semianalytical 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.

A new control law based on characteristic exponent assignment for libration point orbit maintenance

A new method is developed for stabilizing motion on collinear libration point orbits using the formalism of the circular restricted three body problem. Linearization about the collinear libration point orbits yields an unstable linear parameter-varying system with periodic coefficients. Given the variational equations, an innovative control law based on characteristic exponent assignment is introduced for libration point orbit maintenance. A numerical simulation choosing the Richardson＇s third order approximation for a halo orbit as a nominal orbit is conducted, and the results demonstrate the effectiveness of this control law.

Motion modeling and formation form analysis of spacecraft near collinear libration points

To meet the increasing research demand for deep space exploration,especially for the second libration point (L2) conditional periodic orbit (Halo orbit) in the Sun-Earth system,the methods to get analytical Halo orbit and differential-correction Halo orbit were described firstly,and the corresponding orbits accuracy was analyzed.Then,based on the results of third-order and differential-correction Halo orbits,the formation form was studied.Analysis was carried out to discuss the influence of system amplitude,initial phase,and phase difference on the formation form,as well as that of initial orbit values on form accuracy.Finally,some simulation results demonstrate the validity of the proposed methods.

A station-keeping strategy for collinear libration point orbits

Spacecrafts in periodic or quasi-periodic orbits near the collinear libration points are proved to be excellent platforms for scientific investigations of various phenomena.Since such periodic or quasi-periodic orbits are exponentially unstable,the station-keeping maneuver is needed. A station-keeping strategy which is found by an analytical method is presented to eradicate the dominant unstable component of the libration point trajectories.The inhibit force transforms the unstable component to a stable component.In this method,it is not necessary to determine a nominal orbit as a reference path.

Multi-modes control method for spacecraft formation establishment and reconfiguration near the libration points

This paper studies Multi-modes control method for libration points formation establishment and reconfiguration. Firstly, relations between optimal impulse control and Floquet modes are investigated. Method of generating modes is proposed. Characteristics of the mode coefficients stimulated at different time are also given. Studies show that coefficients of controlled modes can be classified into four types, and formation establishment and reeonfiguration can be achieved by multi-impulse control with the presented method of generating modes. Then, since libration points formation is generally unstable, mutli-modes keeping control method which can stabilize five Floquet modes simultaneously is proposed. Finally, simulation on formation establishment and reconfiguration are carried out by using method of generating modes and mutli-modes keeping control method. Results show that the proposed control method is effective and practical.

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.

Autonomous navigation to quasi-periodic orbits near translunar libration points

Libration-point missions have been very useful and successful. Due to the unstable natures of most of these orbits, the long-time stationkeeping demands frequent maneuvers and precise orbit determinations. Earth-based tracking will have to undertake much more responsibilities with the increasing number of libration missions. An autonomous navigation system could offer a better way to decrease the need for Earth-based tracking. Nevertheless, when an autonomous navigation system is applied, there are three important factors affecting autonomous navigation accuracy, i.e., the accuracy of initial conditions, the accuracy of measurements, and the accuracy of onboard dynamics for propagation. This paper focuses on analyzing the influence from the third factor and finding an appropriate navigation dynamics, which can satisfy the requirement of estimation accuracy but not cause too much burden for onboard computation. When considering the restricted three-body model and the bicircular restricted four-body model as navigation dynamics, the astrin- gency is not shown during the simulations. Meanwhile, when considering the influences of the Sun＇s direct and indirect perturbations and the eccentricity of the Moon＇s orbit, a new navigation dynamic model with the standard ephemerides is proposed. The simulation shows the feasibility of the proposed model.

Trajectory design for the Moon departure libration point mission in full ephemeris model

The lunar probe may still have some remaining fuel after completing its predefined Moon exploration mission and is able to carry out some additional scientific or technological tasks after escaping from the Moon orbit.The Moon departure mission for the lunar probe is the focus of this paper.The possibility of the spacecraft orbiting the Moon to escape the Moon's gravitational pull is analyzed.The trajectory design for the Earth-Moon system libration point mission is studied in a full ephemeris dynamical model,which considers the non-uniform motion of the Moon around the Earth,the gravity of the Sun and planets and the finite thrust of the onboard engine.By applying the Particle Swarm Optimization algorithm,the trajectory design for the transfer from the Moon-centered orbit to the L1 halo orbit,the station-keeping strategies for the Earth-Moon halo orbit and the construction of homoclinic and heteroclinic orbits are investigated.Taking the tracking conditions and engineering constraints into account,two feasible schemes for the Moon departure libration point mission for the lunar probe are presented.

A symplectic moving horizon estimation algorithm with its application to the Earth-Moon L2 libration point navigation

Accurate state estimations are perquisites of autonomous navigation and orbit maintenance missions.The extended Kalman lter(EKF)and the unscented Kalman lter(UKF),are the most commonly used method.However,the EKF results in poor estimation performance for systems are with high nonlinearity.As for the UKF,irregular sampling instants are required.In addition,both the EKF and the UKF cannot treat constraints.In this paper,a symplectic moving horizon estimation algorithm,where constraints can be considered,for nonlinear systems are developed.The estimation problem to be solved at each sampling instant is seen as a nonlinear constrained optimal control problem.The original nonlinear problem is transferred into a series of linear-quadratic problems and solved iteratively.A symplectic method based on the variational principle is proposed to solve such linear-quadratic problems,where the solution domain is divided into sub-intervals,and state,costate,and parametric variables are locally interpolated with linear approximation.The optimality conditions result in a linear complementarity problem which can be solved by the Lemke's method easily.The developed symplectic moving horizon estimation method is applied to the Earth-Moon L2 libration point navigation.And numerical simulations demonstrate that though more time-consuming,the proposed method results in better estimation performance than the EKF and the UKF.

Control of orientation for spacecraft formations in the vicinity of the Sun-Earth L2 libration point

Formation flying in the vicinity of the libration point is an important concept for space exploration and demands reliable and accurate techniques for the control of a spacecraft.On the basis of previous works,this paper addresses the problem of relative orientation control of spacecraft formation flying utilizing the framework of the circular restricted three-body problem(CR3BP)with the Sun and Earth as the primary gravitational bodies.Two specific tasks are accomplished in this study.First,the tangent targeting method(TTM),an efficient two-level differential correction algorithm,is exploited to control the Chief/Deputy architecture to maintain a prespecified orientation.The time spent within the orientation error corridor between successive maneuvers is maximized while the relative separation between the vehicles is held constant at each target point.The second task is to further optimize the maneuver intervals by dropping the constraint imposed on the relative vehicle separation.Numerical investigation indicates that the number of maneuvers can be significantly reduced and the length of time between successive maneuvers can be greatly increased by utilizing the TTM.

Trajectory correction for lunar flyby transfers to libration point orbits using continuous thrust

Trajectory corrections for lunar flyby transfers to Sun–Earth/Moon libration point orbits(LPOs)with continuous thrusts are investigated using an ephemeris model.The lunar flyby transfer has special geometrical and dynamical structures;therefore,its trajectory correction strategy is considerably different from that of previous studies and should be specifically designed.In this paper,we first propose a control strategy based on the backstepping technique with a dead-band scheme using an ephemeris model.The initial error caused by the launch time error is considered.Since the perturbed transfers significantly diverge from the reference transfers after the spacecraft passes by the Moon,we adopt two sets of control parameters in two portions before and after the lunar flyby,respectively.Subsequently,practical constraints owing to the navigation and propellant systems are introduced in the dynamical model of the trajectory correction.Using a prograde type 2 orbit as an example,numerical simulations show that our control strategy can efficiently address trajectory corrections for lunar flyby transfers with different practical constraints.In addition,we analyze the effects of the navigation intervals and dead-band scheme on trajectory corrections.Finally,trajectory corrections for different lunar flyby transfers are depicted and compared.

Trajectory and Correction Maneuver During the Transfer from Earth to Halo Orbit

This article addresses the design of the trajectory transferring from Earth to Halo orbit, and proposes a timing closed-loop strategy of correction maneuver during the transfer in the frame of circular restricted three body problem （CR3BP）. The relation between the Floquet multipliers and the magnitudes of Halo orbit is established, so that the suitable magnitude for the aerospace mission is chosen in terms of the stability of Halo orbit. The stable manifold is investigated from the Poincar6 mapping defined which is different from the previous researches, and six types of single-impulse transfer trajectories are attained from the geometry of the invariant manifolds. Based on one of the trajectories of indirect transfer which are ignored in the most of literatures, the stochastic control theory for imperfect information of the discrete linear stochastic system is applied to design the trajectory correction maneuver. The statistical dispersion analysis is performed by Monte-Carlo simulation,

Spacecraft formation control strategy on Sun-Earth Lissajous orbit

To carry out the deep space exploration tasks near Sun-Earth Libration point L2, the CRTBP dynamic model was built up and the numerical conditional quasi-periodic orbit （Lissajons orbit） was computed near L2. Then, a formation controller was designed with linear matrix inequality to overcome the difficuhy of parameter tuning. To meet the demands of formation accuracy and present thruster＇s capability, a threshold scheme was adopted for formation control. Finally, some numerical simulations and analysis were completed to demonstrate the feasibility of the proposed control strategy.

An overview of the mission and technical characteristics of Change'4 Lunar Probe

Change＇4 Lunar Probe will softly land on the farside of the Moon for the first time of all mankind and carry out in-situ and rovering exploration. In this paper, the scientific significance and engineering difficulties of Change＇4 are introduced and the probe＇s general design, including the aspects of landing site selection, relay communication, trajectory design of relay satellite is explained. Besides, four key technologies, namely safe landing strategy on complex terrain, orbit design and control of libration point 2, relay communication on L2, radioisotope thermoelectric generator （RTG） and electric-thermal utilization, as well as how to realize them are also discussed. Finally the prospect of the prominent technological breakthrough of Change＇4 is described.

Research on the transfers to Halo orbits from the view of invariant manifolds

This paper discusses the evolutions of invariant manifolds of Halo orbits by low-thrust and lunar gravity. The possibility of applying all these manifolds in designing low-thrust transfer, and the presence of single-impulse trajectories under lunar gravity are also explained. The relationship between invafiant manifolds and the altitude of the perigee is investigated using a Poincare map. Six types of single-impulse transfer trajectories are then attained from the geometry of the invariant manifolds. The evolutions of controlled manifolds are surveyed by the gradient law of Jacobi energy, and the following conclusions are drawn. First, the low thrust （acceleration or deceleration） near the libration point is very inefficient that the spacecraft free-flies along the invariant manifolds. The purpose is to increase its velocity and avoid stagnation near the libration point. Second, all con- trolled manifolds are captured because they lie inside the boundary of Eatlh＇s gravity trap in the configuration space. The evo- lutions of invariant manifolds under lunar gravity are indicated from the relationship between the lunar phasic angle and the altitude of the perigee. Third and last, most of the manifolds have preserved their topologies in the circular restricted three-body problem. However, the altitudes of the perigee of few manifolds are quite non-continuous, which can be used to generate single-impulse flyby trajectories.

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.

On the dynamics and control of the Sun-Earth L_(2) tetrahedral formation

The dynamics and control of a tetrahedral spacecraft formation flying in the Sun-Earth L2 region is initiatively studied,based on the circular restricted three-body problem(CR3BP).Driven by the science goal of identifying extra-solar terrestrial planets and the requirement of imaging optics,a conceptional four-spacecraft triangular pyramid configuration has been proposed for the Multiple-spacecraft Exoplanet Aperture sYnthetic INterferometer(MEAYIN)project,China’s first mid-infrared interferometric imaging mission.Although it looked promising from an optical perspective,the configuration has not been verified dynamically.The formation is required to be virtually“rigid”,because its mutual distances and inertial pointing direction must be maintained with very high accuracy during each observation.In this study,the spatial geometrical relationship between the four spacecraft was established by introducing the parameters of lengths,angles,and a reference vector.The first contribution is that a compact set of normalized factors and critical time indices are defined,which can provide a complete description of the drift of the shape and pointing direction of the configuration,caused by the unstable dynamical environment.Five design variables are isolated and analyzed,and their individual impacts on the uncontrolled evolution of the formation are studied.The main results obtained reveal that the dimensions of the rigid configuration allow a free drift for a time period on the order of tens of hours,while the inertial pointing direction will be lost within merely tens of seconds.Therefore,to form a rigid configuration,the control challenge lies in the fact that control efforts are frequently required for each spacecraft in the fleet,owing to the diverging dynamics.As a second contribution,a simple and feasible control algorithm is proposed to maintain the rigidity of the formation configuration.The results indicate that the associated energy cost is merely 0.05 m/s per observation on average.

A novel method of periodic orbit computation in circular restricted three-body problem

Periodic orbits are fundamental keys to understand the dynamical system of circular restricted three-body problem, and they play important roles in practical deep-space exploration. Current methods of periodic orbit computation need a high-order analytical approximate solution to start the iteration process, thus making the computation complicated and limiting the types of periodic orbits that can be obtained. By utilizing the symmetry of the restricted three-body problem, a special kind of flow function is constructed, so as to map a state on the plane of symmetry to another state that also lies in this plane. Based on this flow function, a new method of periodic orbit computation is derived. This method needs neither a starting analytic approximation nor the state transition matrix to be computed, so it can be conveniently implemented on a computer. Besides, this method is unaffected by the nonlinearity of the dynamical system, allowing a large set of periodic orbits which have x-z plane symmetry to be computed numerically. As examples, some planar periodic orbits (e.g. Lyapunov orbit) and spatial periodic orbits (e.g. Halo orbit) are computed. By further combining with a differential correction process, the method introduced here can be used to design resonant orbits that can jump between different resonant frequencies. One such resonant orbit is given in this paper, verifying the efficiency of this method.

A note on the full two-body problem and related restricted full three-body problem

Truncating at the second order of the mutual potential between two rigid bodies,time-explicit rst order solutions to the rotations and the orbital motion of the two bodies in the planar full two-body problem(F2BP)are constructed.Based on this analytical solution,equations of motion(EOMs)for the related restricted full three-body problem are given.In the case of the synchronous or double synchronous states for the full two-body problem,EOMs for the related restricted full three-body problem(RF3BP)are also given.At last,one example-the"collinear libration point"in the binary asteroid system-is given.