Generalized restricted circular three-body problem as a model for dynamics of binary asteroids

Exploration of the Solar System has recently revealed the existence of a large number of asteroids with satellites, which has stimulated interest in studying the dynamics of such systems. This paper is dedicated to the analysis of the relative motion of a binary asteroid. The conditions of existence of such a system (i.e., when its components do not run away) are derived in the Introduction. Then it is assumed that the satellite has no significant effect on the motion of the main asteroid, the latter being modeled as a dumbbell-like precessing solid body. The equations of motion of this system are a two-parameter generalization of the corresponding equations of the restricted circular three-body problem. It is demonstrated that in the system under consideration there exist steady-state motions in which the small asteroid is equidistant from attracting centers at the ends of the dumbbell (an analog to triangle libration points). The conditions of existence of such motions are derived, and the positions with respect to the dumbbell are analyzed in detail. Examination of the stability of the triangle libration points is reduced to investigation of a characteristic equation of the sixth degree. The stability conditions are derived in the case when the main asteroid executes near-planar motion.     

Generalized problem of two fixed centers

In the problem under consideration, the libration points are determined, and their stability is investigated. The surfaces of zero velocity are constructed, and the regions where motion is possible are found. The limiting case of the problem is considered.     

系统参数对电动力绳系动力学的影响

将电动力绳系(EDT)的主星质量、子星质量、绳系质量以及绳系中的电流视为系统参数,研究这些参数对系统的摆动动力学和轨道动力学的影响。哑铃模型下的电动力绳系摆动动力学方程存在不稳定的周期解,通过Floquet理论来衡量周期解的不稳定程度,从而研究各系统参数对摆动动力学的影响。建立了用春分点轨道元素的形式描述的电动力绳系轨道动力学方程,并以降轨时间来衡量电动力绳系的降轨效率,从而研究系统参数对轨道动力学的影响。运用算例对周期解迁移矩阵的特征值、降轨时间随各系统参数的变化关系进行了仿真,分别得出了各系统参数对系统摆动动力学和轨道动力学的影响。综合本文的仿真结果,并考虑实际发射及空间运行中的其它因素,对电动力绳系的设计和降轨策略提出了建议。     

圆型限制性三体问题相对运动解析研究

针对圆型限制性三体问题共线平动点附近周期/拟周期轨道下的相对运动问题,提出一种新的、通用的解析研究方法。在周期/拟周期轨道近似解析解的基础上,结合微分修正方法,获得了精确的周期/拟周期轨道。对周期/拟周期轨道的单值矩阵进行分析,同时借鉴Floquet理论核心思想,建立了六个相对运动模态,并将相对运动表示为六个相对运动模态的线性组合,获得了相对运动的近似解析解。最后在地-月系统圆型限制性三体问题下,以L1点作为研究对象,分别以Halo轨道、Lissajous轨道和Lyapunov轨道为参考轨道,对相对运动模态和相对运动进行仿真分析,说明了相对运动模态的正确性以及相对运动近似解析解的有效性。     

Earth–Moon libration point orbit stationkeeping: Theory,modeling, and operations

Collinear Earth–Moon libration points have emerged as locations with immediate applications. These libration point orbits are inherently unstable and must be maintained regularly which constrains operations and maneuver locations. Stationkeeping is challenging due to relatively short time scales for divergence, effects of large orbital eccentricity of the secondary body, and third-body perturbations. Using the Acceleration Reconnection and Turbulence and Electrodynamics of the Moon's Interaction with the Sun (ARTEMIS) mission orbit as a platform, the fundamental behavior of the trajectories is explored using Poincaré maps in the circular restricted three-body problem. Operational stationkeeping results obtained using the Optimal Continuation Strategy are presented and compared to orbit stability information generated from mode analysis based in dynamical systems theory.     

太阳帆航天器三维人工平动点变化特性研究

本文研究了太阳帆航天器在光压因子、锥角和钟角共同作用下的三维人工平动点的变化特性。在光压因子较小的时候,五个平动点会拓展成五个不相连的“人工”平动点面。随着光压因子增大,平动点面SL 3 ,SL 4 和SL5将逐渐扩大并互相融合,最终延展至SL 1 与之融合。而平动点面SL2则始终保持独立的球面,只随着光压因子的增大而扩大但不与其它平动点面发生融合。平动点位置的改变,意味着对应的周期轨道也随之改变,这为平动点周期轨道的转移等任务提供了有效参考。     

一种新的共线平动点拟周期轨道稳定保持策略

限制性三体问题下共线平动点附近的拟周期轨道在深空探测中具有重要的实际应用价值,得到了各航天大国的广泛重视。通过将动力学中心流形结构引入轨道控制方法的设计之中,得到了基于投影到中心流形的共线平动点拟周期轨道稳定保持策略。首先推导了会合坐标到中心流形坐标的正则变换方法,在此基础上设法通过引入轨道机动,将偏差状态点投影到中心流形上,从而达到消除不稳定分量的目的。该方法充分整合了平动点的动力学特性,并且也适用于周期轨道的稳定保持。通过对Lissajous轨道和晕轨道的数值仿真表明,该方法较以往方法具有更强的稳定性,能在显著降低轨控燃料消耗的基础上达到较好的稳定保持效果。     

嫦娥五号平动点拓展任务轨道方案研究

中国嫦娥五号探测器成功实现月球采样返回任务,为最大限度利用任务资源,研究了利用嫦娥五号轨道器的平动点拓展任务轨道方案,设计了平动点轨道及其转移轨道.首先,给出了任务轨道设计的轨道动力学模型,包括圆型限制性三体问题模型和精确力模型.其次,基于嫦娥二号和嫦娥5T1平动点拓展任务设计经验,介绍了平动点轨道直接转移与入轨等轨道...     

Solving Optimization Problems for the Flight Trajectories of a Spacecraft with a High-Thrust Jet Engine in Pulse Formulation for an Arbitrary Gravitational Field in a Vacuum

A mathematically well-posed technique is suggested to obtain first-order necessary conditions of local optimality for the problems of optimization to be solved in a pulse formulation for flight trajectories of a spacecraft with a high-thrust jet engine (HTJE) in an arbitrary gravitational field in vacuum. The technique is based on the Lagrange principle of derestriction for conditional extremum problems in a function space. It allows one to formalize an algorithm of change from the problems of optimization to a boundary-value problem for a system of ordinary differential equations in the case of any optimization problem for which the pulse formulation makes sense. In this work, such a change is made for the case of optimizing the flight trajectories of a spacecraft with a HTJE when terminal and intermediate conditions (like equalities, inequalities, and the terminal functional of minimization) are taken in a general form. As an example of the application of the suggested technique, we consider in this work, within the framework of a bounded circular three-point problem in pulse formulation, the problem of constructing the flight trajectories of a spacecraft with a HTJE through one or several libration points (including the case of going through all libration points) of the Earth–Moon system. The spacecraft is launched from a circular orbit of an Earth's artificial satellite and, upon passing through a point (or points) of libration, returns to the initial orbit. The expenditure of mass (characteristic velocity) is minimized at a restricted time of transfer.     

Periodic orbits in the restricted four-body problem

One solution to the restricted three-body problem is that of three-dimensional, periodic halo orbits. These orbits emanate from the three collinear libration points and exist at all possible mass ratios of the primaries, with infinite possible trajectories. Their existence motivated this study of the effect of an additional gravitational influence on the motion. The approach is to seek a solution in the restricted four-body problem using halo solutions as an initial approximation. The method first solves for periodic, coplanar motion of the three primaries under their mutual gravitational attractions and represents them as trigonometric series. Then, under the modified gravity force model, a three-dimensional solution is obtained in the problem of four bodies, periodic with respect to the synodic system of the three primaries. The additional primary remains relatively far removed and acts as a perturbing influence on the original motion. Some shape and stability characteristics are presented for three such solutions.     

Calculation and Study of Limited Orbits around the <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> Libration Point of the Sun–Earth System

A procedure has been proposed for calculating limited orbits around the L₂ libration points of the Sun–Earth system. The motion of a spacecraft in the vicinity of the libration point has been considered a superposition of three components, i.e., decreasing (stable), increasing (unstable), and limited. The proposed procedure makes it possible to correct the state vector of the spacecraft so as to neutralize the unstable component of the motion. Using this procedure, the calculation of orbits around various types of libration points has been carried out and the dependence on the orbit type on the initial conditions has been studied.     

地月平动点轨道应用与研究进展

面向未来月球和深空探测任务的需求,调研了地月平动点应用与研究的国内外现状与进展,着重分析了近年来的研究方向、研究内容、技术方法与特点,提出了面向未来月球和深空探测任务的地月平动点应用构想,梳理总结了需解决的相应关键技术,可为未来平动点相关研究与应用提供有益借鉴,以及为我国后续月球和深空探测任务的规划与论证提供参考。     

Planar oscillations of a vibrating dumbbell-like body in a central field of forces

The problem of planar motions of a dumbbell-like body of variable length in a central field of Newtonian attraction is considered both in the exact formulation and in satellite approximation. In the satellite approximation the true anomaly of the center of mass is used as an independent variable, which has allowed us to represent the equation of planar oscillations of the dumbbell in the form similar to the Beletskii equation. Some exact solutions to the inverse problem are given both in the complete formulation and in satellite approximation. Under an assumption of small orbit eccentricity and amplitude of the dumbbell vibrations the conditions of existence are found for families of almost periodic motions and splitting separatrices. The phenomena of alternation of regular and chaotic motions are established numerically, as well as chaos suppression with increasing frequency of vibrations. Using the method of averaging the stabilization of tangent equilibria is proved to be impossible.     

三角平动点附近高阶解在轨道位置保持中的应用

首先给出三角平动点附近的高阶解析解,并计算了三种特殊的运动类型。以日–地+月系三角平动点附近无长周期运动分量的拟周期轨道作为目标轨道,探讨轨道保持问题。针对三角平动点任务的轨道保持问题,我们研究了两种轨道保持策略,分别为多点打靶轨道保持与重构目标轨道的策略。计算中,将轨道控制问题转化为非线性规划问题,并以优化方法求解。仿真表明优化方法在轨道保持问题求解方面非常有效。     

Satellite dynamics under the influence of gravitational and aerodynamic torques. A study of stability of equilibrium positions

The dynamics of the rotational motion of a satellite, moving in the central Newtonian force field under the influence of gravitational and aerodynamic torques, is investigated. The paper proposes a method for determining all equilibrium positions (equilibrium orientations) of a satellite in the orbital coordinate system for specified values of aerodynamic torque and the major central moments of inertia; the sufficient conditions for their existence are obtained. For each equilibrium orientation the sufficient stability conditions are obtained using the generalized energy integral as the Lyapunov function. The detailed numerical analysis of the regions where the stability conditions of the equilibrium positions are satisfied is carried out depending on four dimensionless parameters of the problem. It is shown that, in the general case, the number of satellite’s equilibrium positions, for which the sufficient stability conditions are satisfied, varies from 4 to 2 with an increase in the value of the aerodynamic torque magnitude.     

Equilibrium positions of a weight on a cable fixed to a dumbbell-shaped space station moving along a circular geocentric orbit

The motion of a space object in the gravitational field of the Earth is considered. The object consists of an extended space station and a weight, which is free to move along the cable fixed to the ends of the station. It is assumed that the station is composed of two masses coupled by a weightless rod, while the cable is weightless and non-stretched. The equations of motion of such a system are derived for the case when the motion proceeds in a single plane, while the center of mass of the system moves along a circular geocentric orbit. The conditions of the cable tension (conditions of being on tie) are derived. The phase portrait of the weight motion along the cable is constructed when the station is oriented to the attracting center or is perpendicular to this position. The possibility to leave the tie in this case is analyzed. Equilibrium configurations of the system are found, i.e., such motions of the object under consideration at which the weight does not change its position relative to the station. Lyapunov stability of such configurations is analyzed for two situations: when the station is composed of equal masses and when masses at the ends of the station are different. In particular, for the case of different masses it is established that there exist such positions of equilibrium at which the dumbbell is located at an angle to the direction to the attracting center. In some cases these positions can be stabilized (if the weight is fixed on the cable).     

Dynamics and control of an orbiting flexible mobile manipulator

This paper presents a Lagrangian formulation for studying the dynamics and control of the proposed Space Station based Mobile Servicing System (MSS) for a particular case of inplane libration and maneuvers. The simplified case is purposely considered to help focus on the effects of structural and joint flexibility parameters of the MSS on the complex interactions between the station and manipulator dynamics during slewing and translational maneuvers. The response results suggest that under critical combinations of parameters, the system can become unstable. During maneuvers, the deflection of the MSS can become excessive, leading to positioning error of the payload. At the same time the libration error can also be significant. A linear quadratic regulator is designed to control the deflection of the manipulator and maintain the station at its operating configuration.     

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.     

Stable spatial orbits around collinear libration points

A technique of generation of spatial periodic solutions to the restricted circular three-body problem from periodic orbits of the planar problem has been used for the families of orbits around collinear libration points L ₁ and L ₂. Developing the families obtained at the 1: 1 resonance, we have obtained stable solutions both in the Earth-Moon system and in the Sun-Earth system. Of course, the term “around the libration point” is rather conventional; the obtained orbits become more similar to the orbits around the smaller attracting body. The further development of the family of orbits “around” the libration point L ₂ in the Sun-Earth system made it possible to find the orbits satisfying the new, much more rigorous constraints on cooling the spacecraft of the Millimetron project.     

Periodic solutions in three-dimensional restricted three-body problem: 2. Loss of symmetry at the 1:1 resonance

Within the framework of the circular restricted three-body problem a family of inverse periodic orbits around the two attracting bodies (the Egorov’s family) and families generated by it at the 1:1, 2:1, and 3:1 resonances for three-dimensional orbits in the Sun-Earth and Earth-Moon systems are considered. Their relationship with families generated by orbits around the libration points L ₁, L ₂ and L ₃ is investigated. One of the families contains periodic solutions that seem promising as possible orbits for the space radio telescope of the Millimetron project.