三体轨道动力学研究进展

三体系统轨道动力学问题是航天动力学领域中的经典问题, 具有丰富的理论与工程意义, 并将在人类由近地延伸到深空的航天活动过程中起到至关重要的作用. 本文回顾并总结了三体系统轨道动力学相关研究进展, 并结合未来的深空探测的发展趋势, 展望了三体系统轨道动力学研究中的热点与挑战. 首先阐述了三体问题的研究背景及意义, 简要回顾了三体系统动力学模型的发展历程. 其次, 系统概述了三体系统平衡点附近的局部运动特性, 介绍了平衡点附近周期轨道解析与数值求解方法, 给出了拟周期运动的最新进展. 同时总结了共振轨道、循环轨道、自由返回轨道等三类三体系统全局周期运动的动力学特性与研究进展. 再次, 从不变流形理论和弱稳定边界理论两个方面综述了三体系统中低能量转移与捕获轨道设计的研究进展. 最后, 综述了三体系统轨道动力学在编队飞行、导航星座设计两方面的应用, 并展望了全月面覆盖轨道设计、三体系统下的小推力轨道优化和三体系统的三角平衡点开发利用中值得关注的轨道动力学与控制问题.      

Fourth body gravitation effect on the resonance orbit characteristics of the restricted three-body problem

In this paper, the gravitational effect of a fourth body on the resonance orbit defined in the restricted three-body problem (RTBP) is considered. In this regard, Resonance Hamiltonian of the RTBP and the Hamiltonian associated with the fourth gravitational body that perturbs the resonance orbit are computed. The Melnikov approach is utilized as a mean for the detection of chaos in resonance orbit under the influence of the fourth gravitation body. In addition, the numerical simulation of RTBP and bicircular four-body model, time–frequency analysis (TFA), and fast Lyapunov indicator (FLI) are performed to verify the results of the Melnikov approach. The results indicate that for the (2:1) resonance orbit, the Melnikov integral computed over outer loop of separatrix does not cross the zero line, and consequently chaos is unexpected. On the other hand, the Melnikov integral computed over the inner sepratrix loop crosses the zero line indicating a potential for chaos. Similarly, it is shown that inclusion of the fourth body gravitation leads the (3:1) as well as the (4:1) resonance orbits to chaos. Additionally, simulation results indicate that for some initial conditions on the separatrix, the fourth body effect bounds the amplitude of the resonance orbits while diffusing its corresponding trajectory in the bounded phase space. TFA and the FLI verify similar results.     

State-of-the-art and prospects for orbital dynamics and control near small celestial bodies

小天体探测是未来深空探测的重点领域之一, 而小天体附近轨道动力学与控制问题是小天体探测任务迫切需要解决的关键问题. 该问题涉及形状不规则小天体附近的动力学环境建模与小天体附近轨道动力学机理. 本文从不规则形状小天体引力场的建模、小天体附近的自然轨道动力学、小天体附近的受控轨道动力学3 个方面综述了小天体附近轨道动力学与控制的研究现状与发展趋势, 并分析了小天体附近轨道动力学所面临的挑战与难题, 最后对我国未来小天体探测任务可能涉及的轨道动力学与控制问题的发展方向进行了展望.     

基于星历匹配法的载人小行星探测轨迹优化问题求解

对基于星历匹配法的载人小行星探测轨迹优化问题解法进行了研究. 提出星历匹配法的概念,利用其求出最优发射窗口初值与候选最优探测序列,然后利用遗传算法对特定探测序列的时间节点等参数进行优化,最后利用基于庞德里亚金极大值原理的同伦法进行小推力转化以求得最终探测轨道及其推力控制律. 结果显示星历匹配法可以快速准确地求出最优发射窗口初值与候选探测序列,这大大提高了载人小行星探测等多目标交会探测问题的求解效率.     

THE EVOLUTION CHARACTERIZATION OF LIBRATIN POINT ORBITS AND APPLICATION RESEARCH

三体问题中, 轨道的受力和运动规律非常复杂. 对于特定的任务, 如何选择轨道的初始解是一大难题.针对平面三体问题, 利 用近拱点庞加莱映射, 对平动点顺行轨道和逆行轨道的长期和短期演化性质进行分析.根据轨道的初始状态将其分为逃逸轨道和捕获轨道.对于逃逸轨道, 给出了同宿轨道和异宿轨道的设计方法, 并利用两级微分修正法消除了拼接点处的位置不连续问题.对于捕获轨道, 得到了几类典型的周期和准周期轨道.对逆行轨道的演化性质进行分析时发现, 逆行轨道通常为准周期轨道, 比顺行轨道更加稳定.利用近拱点庞加莱映射可以快速确定不同类型轨道对应的初始状态, 为特定任务需求下的轨道设计提供了一种快速而有效的选择方案.     

Mechanical problems in momentous projects of aerospace engineering

近年来, 我国航天重大工程蓬勃发展, 航天工程中新的力学问题不断涌现, 开展航天工程力学问题研究在航天技术的发展中起到举足轻重的作用.随着航天器朝着超高速、深空探测、多功能方向的发展, 其面临的发射和运行环境也更加恶劣, 发射过程中的多场耦合、非线性等问题更加突出. 大阵面、大挠性的航天器对在轨结构展开、模态辨识、刚柔耦合控制提出新的要求, 而高精度、高分辨率的观测需求, 为航天器在轨微振动、热致振动的研究带来了新的课题. 同时, 这一系列的问题也对航天器的地面试验和仿真分析等提出了更高的要求, 在这些领域, 各国学者也积累了一定的成果. 本文概括介绍了近年来航天重大工程中出现的新的力学问题, 从航天器的发射、在轨运行、地面仿真和试验等方面对航天工程中的力学问题进行了综述. 内容主要集中在耦合动力学、空气动力学、多体动力学、结构动力学以及试验力学等方面, 同时提出了工程中力学方面所面临的问题以及下一步的发展方向.     

A survey on orbital dynamics and navigation of asteroid missions

Asteroid exploration is currently one of the most concerned topics among international space agencies. Or- bital dynamics and navigation are obviously crucial for asteroid exploration. This paper aims to give a brief review on the dynamics, control and navigation of asteroid reconnaissance orbits, including the heliocentric transfer orbit and near as- teroid orbit. The developments in optimization techniques of the transfer segment are discussed in detail. We surveyed global researches in this field and made comments on several important progresses. The final section proposed a prospec- tive of future studies with emphasis on the key techniques of these issues in the asteroid exploration missions.     

纯引力轨道检验质量的相对测量技术

纯引力轨道是物体在太空仅受引力作用的运行轨道, 通过构造纯引 力轨道, 可实现超高精度的空间引力探测, 也可为科学实验提供超稳定卫星 平台. 作为纯引力轨道构造的核心, 检验质量的相对测量不仅提供了部分任 务科学数据, 还为航天器平台的跟踪控制提供输入. 首先, 描述了纯引力轨道 的概念内涵, 总结了它在卫星重力测量、引力波探测等方面的应用情况. 其 次, 综述了不同任务对相对测量的需求, 给出了电容式测量、磁感应测量和 光学测量的原理, 总结了各自的优缺点. 根据检验质量的姿态运动, 将检验质 量质心相对状态解算问题分为3 类, 给出了基于检验质量姿态动力学与表面 建模的典型解算模型和质心速度估计方法. 最后分析了非引力干扰的理论计 算、地面实验验证和在轨实验验证问题.      

COLLINEAR LIBRATION POINTS WITH CONTINUOUS LOW THRUST

研究了圆型限制性三体问题的平动点在连续小推力作用下的具体位置和动力学特征的变化. 研究表明,随着小推力方向在空间中的变化,平动点的具体位置也会发生相应的变化,文章详细阐述了这些变化的特征.针对航天任务中应用较多的共线平动点L1和L2,研究了其附近运动的稳定性状态,给出了线性条件稳定解,并在此基础上,构造了条件稳定解的高阶形式,将其结果与数值积分轨道进行了比对,两者符合得很好. 最后,进一步研究了共线平动点附近周期轨道族的演化状态,由于连续小推力引入的非对称性,周期轨道族会发生分岔现象.     

Flight dynamics over irregular asteroids

小行星探测是当前深空探测的主要方向之一,具有重要的科学意义.绝大多数小行星引力场极不规则,探测器在小行星附近运动形态复杂多样.由于同时受到中心引力、快速自旋的不规则形状摄动力、以及光压摄动等作用,探测器容易与小行星发生碰撞或逃逸.概述小行星研究现状和不规则引力场建模方法.重点介绍不规则引力场内动力学特性,包括引力平衡点、局部流形、自然周期轨道和悬停探测轨道等,尝试提出新的研究方向.     

A method for classifying orbits near asteroids

A method for classifying orbits near asteroids under a polyhedral gravitational field is presented, and may serve as a valuable reference for spacecraft orbit design for asteroid exploration. The orbital dynamics near aster- oids are very complex. According to the variation in orbit characteristics after being affected by gravitational perturbation during the periapsis passage, orbits near an as- teroid can be classified into 9 categories： （1） surrounding- to-surrounding, （2） surrounding-to-surface, （3） surrounding- to-infinity, （4） infinity-to-infinity, （5） infinity-to-surface, （6） infinity-to-surrounding, （7） surface-to-surface, （8） surface- to-surrounding, and （9） surface-to- infinity. Assume that the orbital elements are constant near the periapsis, the gravitation potential is expanded into a harmonic series. Then, the influence of the gravitational perturbation on the orbit is studied analytically. The styles of orbits are dependent on the argument of periapsis, the periapsis radius, and the periapsis velocity. Given the argument of periapsis, the orbital energy before and after perturbation can be derived according to the periapsis radius and the periapsis velocity. Simulations have been performed for orbits in the gravitational field of 216 Kleopatra. The numerical results are well consistent with analytic predictions.     

The present status and prospects in the research of orbital dynamics and control near small celestial bodies

Small celestial body exploration is of great significance to deep space activities. The dynamics and control of orbits around small celestial bodies is of top priority in the exploration research. It includes the modeling of dynamics environment and the orbital dynamics mechanism. This paper introduced state-of-the-art researches, major challenges, and future trends in this field. Three topics are mainly discussed: the gravitational field modeling of irregular-shaped small celestial bodies, natural orbital dynamics and control, and controlled orbital dynamics. Finally, constructive suggestions are made for China's future space exploration missions.     

Symmetry-Breaking Homoclinic Bifurcations in Diffusively Coupled Equations

In this study we examine a symmetry-breaking bifurcation of homoclinic orbits in diffusively coupled ordinary differential equations. We prove that asymmetric homoclinic orbits can bifurcate from a symmetric homoclinic orbit when the equilibria to which the latter is homoclinic undergoes a pitchfork bifurcation. A condition which defines the direction of the bifurcation in a parameter space is given. All hypotheses of the main theorem are verified for a diffusively coupled logistic system and the twistedness of the bifurcating homoclinic orbits is computed for a range of coupling strengths.     

Motion of the moonlet in the binary system 243 Ida

The motion of the moonlet Dactyl in the binary system 243 Ida is investigated in this paper. First, periodic orbits in the vicinity of the primary are calculated, including the orbits around the equilibrium points and large-scale orbits. The Floquet multipliers' topological cases of periodic orbits are calculated to study the orbits' stabilities. During the continuation of the retrograde near-circular orbits near the equatorial plane, two period-doubling bifurcations and one Neimark–Sacker bifurcation occur one by one, leading to two stable regions and two unstable regions. Bifurcations occur at the boundaries of these regions. Periodic orbits in the stable regions are all stable, but in the unstable regions are all unstable. Moreover, many quasi-periodic orbits exist near the equatorial plane. Long-term integration indicates that a particle in a quasi-periodic orbit runs in a space like a tire. Quasi-periodic orbits in different regions have different styles of motion indicated by the Poincare sections. There is the possibility that moonlet Dactyl is in a quasi-periodic orbit near the stable region I, which is enlightening for the stability of the binary system.     

连续小推力航天器的深空探测轨道优化方法综述

连续小推力作用下航天器的深空探测轨道的优化设计是一个存在大量局部最优解的全局优化问题. 轨道设计流程总体上分为全局优化和局部优化. 全局优化为粗略设计, 通常在对航天器受连续推力作用下的轨道作近似处理的前提下大致确定探测序列和时间节点. 局部优化方法可分为直接法、间接法和混合法. 直接法是将连续的问题离散成一个参数优化问题. 间接法是求解由变分法和极大值原理推导的满足一阶最优必要条件的两点或多点边值问题. 混合法利用间接法推导的方程, 再离散后优化求解. 本文综述当前轨道优化设计领域最新和最常用的方法, 分析各种方法的优缺点.     

Homoclinic Twist Bifurcation in a System of Two Coupled Oscillators

We analyse the dynamics of two identical Josephson junctions coupled through a purely capacitive load in the neighborhood of a degenerate symmetric homoclinic orbit. A bifurcation function is obtained applying Lin's version of the Lyapunov–Schmidt reduction. We locate in parameter space the region of existence of n-periodic orbits, and we prove the existence of n-homoclinic orbits and bounded nonperiodic orbits. A singular limit of the bifurcation function yields a one-dimensional mapping which is analyzed. Numerical computations of nonsymmetric homoclinic orbits have been performed, and we show the relevance of these computations by comparing the results with the analysis.     

THRUST-AMPLITUDE CONTINUATION DESIGN APPROACH FOR SOLVING SPACECRAFT OPTIMAL CONTROLLED FLY-AROUND TRAJECTORY

针对航天器燃料最优可变周期绕飞轨迹的求解问题, 提出了一种以推力幅值为延拓参数的延拓方法. 问题求解从最为简单的双脉冲绕飞轨迹出发, 首先利用有限推力替代脉冲推力, 设定推力序列为“开— 关— 开”,然后逐步减小推力幅值, 最终得到最小推力绕飞轨迹; 此后, 再逐步增加推力幅值, 结合主矢量曲线判断最优推力开关序列, 将最小推力解延拓至有限推力以及脉冲推力燃料最优解. 该方法通过对推力幅值的延拓, 实现了有限推力bang-bang 控制与脉冲推力燃料最优绕飞轨迹优化问题的一并求解, 同时避免了最优控制问题中协态变量的随机猜测. 慢速与快速绕飞算例的优化结果验证了所提出方法的有效性.     

火星引力捕获动力学与动态误差分析

火星探测的制动捕获机会唯一,是影响任务成败的关键. 从限制性三体问题出发,推导了火星引力球、作用球与希尔球半径的计算公式,比较了三者的特点与适用范围,并结合作用球的定义与物理意义,给出了一种火星探测制动捕获段的工程定义. 在作用球范围内建立了火星制动捕获段动力学模型,给出了对捕获轨道精度产生影响的各项误差源. 通过蒙特卡洛仿真,定量分析了导航初始误差、发动机推力误差、制动点火时间误差等对捕获轨道近火点与远火点高度的影响,并对不同误差源可能导致的超差概率进行了分析,指出了影响捕获精度的主导误差源,可为我国未来火星探测制动捕获段的任务实施提供参考.      

基于虚拟中心引力场方法的航天器转移轨道设计

空间机动技术是实现空间操作任务的基础,具有重要的研究价值. 研究了连续推力作用下航天器转移轨道设计问题,提出了一种基于虚拟中心引力场的轨道设计方法. 该方法有两大特点:(1) 能够将机动轨道设计问题转化为虚拟中心引力场参数的优化问题,简化了设计过程;(2) 对轨道形状或推力方向、大小不做任何假定,能够应用于一般情况下的机动轨道设计. 将该方法应用于航天器二维和三维的转移轨道设计,并和形状方法进行了对比分析. 仿真结果分析表明,采用该方法简化了轨道设计过程,为航天器快速轨道设计提供了新思路.      

The geometry of the orbits of artificial satellites

A novel approach to the study of the orbits of artificial satellites is presented. Emphasis is placed upon the basic geometry and other aspects of satellite motion which are of first importance to satellite engineering. The motion of the orbital plane as a rigid body is introduced and a non-elliptical orbit motion in this plane is defined. The plane orbit so defined possesses the very desirable feature of representing a succession of satellite positions and hence reveals the true motion of the satellite. An analytical treatment yields a completely general second order theory of earth satellite motion which is suitable for engineering purposes. In the latter development, particular attention is paid to the apsidal motion of the orbit and the concomitant resonance effects at the critical orbit inclination. The basic nonlinear features of the apsidal motion, which have not been recognized in earlier theories, are incorporated in the analytical development so as to produce a theory valid at all angles of inclination of the orbit.