地月低能转移的发生条件及轨迹构造

应用平动点理论研究了地月低能转移的发生条件和轨迹构造. 由于空间双圆模型存在周期性扰动,传统的平动点概念不再适用;根据Hill瞬时边界的连接情况定义了等效平动点LL_{1}和LL_{2}. 通过构造合适的Poincaré截面以获得所有可能的月球捕获轨道的近月距和偏心率分布,从而获得了完全不同于Hill和圆型限制性三体问题的月面捕获能量. 在空间双圆模型下,平动点和Halo轨道不变流形的渐近结构遭到破坏:到达或离开平动点的时间,由无穷转变为有限值;运动方向由双向变为单向. 经由LL_{1}点穿越获得了最小能量的低能转移,借助LL_{1}-Halo轨道穿越得到了(M,N)-圈穿越轨道,经由LL_{2}点穿越获得了最小能量的弱稳定边界转移,借助LL_{2}-Halo轨道穿越得到了弱稳定边界逃逸和捕获窗口. 最后,以地月转移和大幅值逆行轨道的切入为例,给出低能转移的小推力、脉冲和弱稳定边界等转移的实现方式.      

基于多项式约束的三角平动点平面周期轨道设计方法

平动点是圆型限制性三体问题中的五个平衡解.其中,三角平动点在平面问题中具有"中心×中心"的动力学特性,其附近存在着大量的周期轨道,研究这些周期轨道的构建方法在深空探测中具有理论及工程意义.本文从振动角度分析周期轨道,通过多项式展开法构建出主坐标下周期轨道两个运动方向之间的渐近关系.从新的角度分析了系统的动力学特性和平面周期运动两个方向内在关联以及物理规律.这种多项式形式的关系式,可以作为约束条件用于数值微分修正算法中,通过迭代的方式寻找周期轨道.数值仿真算例验证了方法的正确性及精确性.文章从振动的角度对周期轨道进行分析,改进了微分修正算法.提出的方法可以被拓展至圆型/椭圆型限制性三体问题的三维周期轨道构建中.     

李级数算法和显式辛算法的相位分析

以线性可分Hamilton动力学系统为例,研究了李级数算法和显式辛算法的相位精度,研究了李级数算法的保辛精度及其保辛精度的提高方法;指出了显式辛算法相位精度与算法阶次的不协调性,印辛算法的阶次高并不意味着其相位精度也高,李级数算法不存在这种问题,指出了一个算法的相位可能超前也可能滞后.分析结果表明三阶显式辛算法具有比较高的相位精度.     

基于多项式约束的三角平动点平面周期轨道设计方法1）

平动点是圆型限制性三体问题中的五个平衡解.其中,三角平动点在平面问题中具有“中心×中心”的动力学特性,其附近存在着大量的周期轨道,研究这些周期轨道的构建方法在深空探测中具有理论及工程意义.本文从振动角度分析周期轨道,通过多项式展开法构建出主坐标下周期轨道两个运动方向之间的渐近关系.从新的角度分析了系统的动力学特性和平面周期运动两个方向内在关联以及物理规律.这种多项式形式的关系式,可以作为约束条件用于数值微分修正算法中,通过迭代的方式寻找周期轨道.数值仿真算例验证了方法的正确性及精确性.文章从振动的角度对周期轨道进行分析,改进了微分修正算法.提出的方法可以被拓展至圆型/椭圆型限制性三体问题的三维周期轨道构建中.     

带脉冲的三维引力辅助变轨研究

在引力辅助过程中施加脉冲可以有效地改善变轨效果.目前只能对施加小脉冲的情况进行近似计算,当脉冲大于近拱点速度的1%时无法进行分析.针对这一问题,提出了一种解析分析方法,可以计算施加任意大小和方向脉冲的三维引力辅助变轨.基于二体问题,建立了带任意脉冲的三维引力辅助模型,采用8个相互独立的参数对模型进行描述,其中5个参数表征三维引力辅助、一个参数表征脉冲的大小、两个参数表征脉冲的方向;建立了一组坐标系,可以方便地对轨道进行描述;以施加脉冲为界,将轨道划分为前后两段,分别进行公式推导;应用双曲线轨道动力学与坐标变换等技术方法,可以将飞行器的位置矢量和速度矢量表示为上述8个参数的解析公式,进而可以求出变轨导致的速度、能量和轨道倾角的变化量.通过与基于圆型限制性三体问题的数值仿真结果进行对比,验证公式的有效性.应用导出的解析公式分析了施加脉冲的大小和方向对飞行器能量和轨道倾角的影响,并给出了相应规律.结果表明:以最大能量改变为优化目标,施加脉冲的最优方向往往并不是该点速度方向;轨道倾角受到脉冲方向的影响显著.      

三维引力辅助解析分析方法研究

针对星际探测中的引力辅助技术,基于二体问题,导出了三维引力辅助前后飞行器速度、能量、角动量和轨道倾角与引力辅助参数之间的解析公式. 以进行木星引力辅助为例,将解析公式计算的结果与基于圆型限制性三体问题的数值仿真结果进行了对比,各项绝对误差小于0.01,除极少数情况外,相对误差均小于0.7%. 分析结果表明导出的解析公式与实际情况对比具有较高的精度,可以满足引力辅助轨道初步设计的需要. 为了讨论和应用上的方便,对导出的公式进行了简化,据此进一步讨论了引力辅助参数对引力辅助效果的影响.     

MESHLESS METHOD FOR NUMERICAL SIMULATION OF SHOCK-INDUCED COMBUSTION

发展了基于无网格方法的激波诱导燃烧流场数值模拟算法. 该算法采用二维多组分Euler方程,在点云离散的基础上采用曲面逼近计算空间导数,引入多组分HLLC （Harten-Lax-van Leer-contact） 格式计算无黏通量,运用四阶Runge-Kutta 法进行时间显式推进,化学动力学采用有限速率反应模型. 对不同预混气体中的激波诱导燃烧流场进行了数值模拟,结果同相关文献吻合较好,验证了算法的正确性.     

多体动力学的几何积分方法研究进展

动力系统的几何积分研究是近20年来工程计算领域非常活跃的方向.多体动力学方程(微分方程, 微分代数方程)是一类典型的动力系统,将其从Lagrange体系向Hamilton系统过渡,目的在于从欧氏几何过渡到辛几何形态, 将对偶变量引入到力学研究中,然后利用辛几何的数学框架对多体系统动力学方程进行数值计算,可以预知多体动力学系统的一些定性信息,并在数值离散时能保持这些定性性质特征,尤其在表示关键的物理意义时需要强调保持这些几何性质.简要介绍多体系统(无约束多刚体系统、完整约束多刚体系统和柔性多体系统)的Hamilton正则方程的建立和几何积分方法的构造,着重介绍了在多体动力学计算中非常有应用前景的高阶辛算法(合成辛算法、分裂合成辛算法和辛精细积分法)、多辛算法,以及广义Hamilton 系统与Lie 群积分方法等计算几何力学方法, 并对Lie群积分的投影方法、流形局部坐标法等方法进行了阐述.      

基于新型解耦算法的激波诱导燃烧过程数值模拟

对一种模拟化学非平衡流动的时间和空间二阶精度新型解耦算法进行两方面改进,流动算子采用基于Runge-Kutta方法的时间格式以后, 可以推广到更多的空间差分格式,化学反应源项求解算子可以采用梯形公式、拟稳态逼近法和变系数常微分方程求解器. 对H化学非平衡流动; 解耦算法; 计算方法对一种模拟化学非平衡流动的时间和空间二阶精度新型解耦算法进行两方面改进,流动算子采用基于Runge-Kutta方法的时间格式以后,可以推广到更多的空间差分格式,化学反应源项求解算子可以采用梯形公式、拟稳态逼近法和变系数常微分方程求解器.对H_2/Air预混气体中激波诱导振荡燃烧的Lehr试验进行模拟,考察了化学动力学模型、网格尺寸和差分格式耗散大小对计算结果的影响,同时对不同的化学反应源项算子求解算法的计算效率进行了比较.     

结构非线性动力分析混合时间积分并行算法

综合隐式和显式时间积分技术,对结构非线性动力反应分析提出一种并行混合时间积分算法.该算法采用区域分解技术.将并发性引入到算法中,即利用显式时间积分技术进行界面节点积分而利用隐式算法求解局部子区域.为实现并行混合时间积分算法,设计了灵活的并行数据信息流.编写了该算法的程序,在工作站机群实现了数值算例,验证了算法的精度和性能.计算结果表明该算法具有良好的并行性能,优于隐式算法.     

Lunar landing trajectory design based on invariant manifold

The low-energy lunar landing trajectory design using the invariant manifolds of restricted three-body problem is studied.Considering angle between the ecliptic plane and lunar orbit plate the four-body problem of sun-earth-moon-spacecraft is divided into two three-body problems,the sun-earth-spacecraft in the ecliptic plane and the earth- moon-spacecraft in the lunar orbit plane.Using the orbit maneuver at the place where the two planes and the invariant manifolds intersect,a general method to design low energy lunar landing trajectory is given.It is found that this method can save the energy about 20% compared to the traditional Hohmann transfer trajectory,The mechanism that the method can save energy is investigated in the point of view of energy and the expression of the amount of energy saved is given.In addition,some rules of selecting parameters with respect to orbit design are provided.The method of energy analysis in the paper can be extended to energy analysis in deep space orbit design.     

Equilibrium configurations of the tethered three-body formation system and their nonlinear dynamics

This paper considers nonlinear dynamics of tethered three-body formation system with their centre of mass staying on a circular orbit around the Earth, and applies the theory of space manifold dynamics to deal with the nonlinear dynamical behaviors of the equilibrium configurations of the system. Compared with the classical circular restricted three body system, sixteen equilibrium configurations are obtained globally from the geometry of pseudo-potential energy surface, four of which were omitted in the previous research. The periodic Lyapunov orbits and their invariant manifolds near the hyperbolic equilibria are presented, and an iteration procedure for identifying Lyapunov orbit is proposed based on the differential correction algorithm. The non-transversal intersections between invariant manifolds are addressed to generate homoclinic and heteroclinic trajectories between the Lyapunov orbits. (3,3)and (2,1)-heteroclinic trajectories from the neighborhood of one collinear equilibrium to that of another one, and (3,6)and (2,1)-homoclinic trajectories from and to the neighborhood of the same equilibrium, are obtained based on the Poincaré mapping technique.     

空间刚性杆——弹簧组合结构轨道、姿态耦合动力学分析

以空间太阳帆塔在轨运行中遇到的强耦合动力学问题为研究背景,建立了空间刚性杆-- 弹簧组合结构轨道与姿态耦合 问题的动力学模型,采用辛 (几何) 算法研究了其轨道与姿态耦合的动力学行为,研究结果可以从系统的能量保持情况间接得到验 证. 首先,基于变分原理,通过引入对偶变量将描述空间刚性杆-- 弹簧组合结构动力学行为的拉格朗日方程导入哈 密尔顿体系,建立简化模型的正则控制方程;随后,采用辛龙格库塔方法模拟分析了地球非球摄动对轨道、姿态的影响及系统能 量的数值偏差问题. 数值模拟结果显示:随着初始姿态角速度增大,轨道半径的扰动 增大,轨道与姿态之间的耦合效应加剧; 带谐摄动对空间刚性杆-- 弹簧组合结构模型的轨道、姿态产生的影响比田谐摄动要高出至少两个数量级;同时辛龙格库塔方法能更好 地快速模拟地球非球摄动影响下空间刚性杆-- 弹簧组合结构的动力学行为,并能够长时间保持系统的总能量,有望为 超大空间结构实时反馈控制提供实时动力学响应结果.      

Solar sail dynamics in the three-body problem: Homoclinic paths of points and orbits

In this paper we consider the orbital dynamics of a solar sail in the Earth-Sun circular restricted three-body problem. The equations of motion of the sail are given by a set of non-linear autonomous ordinary differential equations, which are non-conservative due to the non-central nature of the force on the sail. We consider first the equilibria and linearisation of the system, then examine the non-linear system paying particular attention to its periodic solutions and invariant manifolds. Interestingly, we find there are equilibria admitting homoclinic paths where the stable and unstable invariant manifolds are identical. What is more, we find that periodic orbits about these equilibria also admit homoclinic paths; in fact the entire unstable invariant manifold winds off the periodic orbit, only to wind back onto it in the future. This unexpected result shows that periodic orbits may inherit the homoclinic nature of the point about which they are described.     

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I

We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo–Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper.A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues–Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given.Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo–Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass.The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.     

Hamilton体系下介电弹性体圆形薄膜的动力学建模与辛求解

采用辛算法研究了Hamilton体系下介电弹性体圆形薄膜的动力学响应。首先，将该问题引入Hamilton对偶变量体系，借助Legendre变换，给出系统的广义动量和Hamilton函数，通过对Hamilton函数作用量的变分，得到Hamilton体系下的正则方程。其次，对于得到的正则方程给出了辛Runge-Kutta的计算格式。最后，采用二级四阶辛Runge-Kutta算法对动力学系统进行了数值求解，和四级四阶经典Runge-Kutta算法进行对比，结果表明，二级四阶辛Runge-Kutta算法具有保能量以及长时间数值稳定的优势，同时说明四级四阶经典Runge-Kutta算法对于步长依赖的局限性。     

An improved nonlinear control strategy for deep space formation flying spacecraft

This paper aims to provide further study on the nonlinear modeling and controller design of formation flying spacecraft in deep space missions. First, in the Sun-Earth system, the nonlinear formation dynamics for the circular restricted three-body problem （CRTBP） and elliptic restricted three-body problem （ERTBP） are presented. Then, with the Floquet mode method, an impulsive controller is developed to keep the Chief on the desired Halo orbit. Finally, a nonlinear adaptive control scheme based on Nonzero set- point LQR and neural network is proposed to achieve high precision formation maneuver and keeping. The simulation results indicate that the proposed nonlinear control strategy is reasonable as it considers not only the orbit keeping of the Chief, but also the formation modeling inaccuracy. Moreover, the nonlinear adaptive control scheme is effective to improve the control accuracy of the formation keeping.     

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.      

Apollonius coordinates,theN-body problem,and continuation of periodic solutions

This paper treats theN-body problem and its relation to various restricted problems. For each solution of the Kepler problem a generalization of the pulsating coordinates used to express the Hamiltonian of the elliptic restricted three-body problem is given. These coordinates are called Apollonius coordinates. The method of symplectic scaling is used to give a precise derivation of the elliptic restricted problem showing the precise asymptotic relationship between the restricted problem and the full three-body problem. This derivation obviates the proof of the fact that a nondegenerate periodic solution of the elliptic restricted three-body problem can be continued into the full three-body problem under mild nonresonance assumptions. Also, the method of symplectic scaling is used to give a precise derivation of the elliptic Hill lunar equation showing the precise relationship between the elliptic Hill lunar equation and the full three-body problem. A similar continuation theorem is established.     

Multi-objective transfer to libration-point orbits via the mixed low-thrust and invariant-manifold approach

The multi-objective optimization of transfer trajectories from an orbit near Earth to a periodic libration-point orbit in the Sun–Earth system using the mixed low-thrust and invariant-manifold approach is investigated in this paper. A two-objective optimization model is proposed based on the mixed low-thrust and invariant-manifold approach. The circular restricted three-body model (CRTBP) is utilized to represent the motion of a spacecraft in the gravitational field of the Sun and Earth. The transfer trajectory is broken down into several segments; both low-thrust propulsion and stable manifolds are utilized based on the CRTBP in different segments. The fuel cost, which is generated only by the low-thrust trajectory for transferring the spacecraft from an orbit near Earth to a stable manifold, is minimized. The total flight time, which includes the time during which the spacecraft is controlled by the low-thrust trajectory and the time during which the spacecraft is moving on the stable manifold, is also minimized. Using the nondominated sorting genetic algorithm for the resulting multi-objective optimization problem, highly promising Pareto-optimal solutions for the transfer of the spacecraft are found. Via numerical simulations, it is shown that tradeoffs between time of flight and fuel cost can be quickly evaluated using this approach. Furthermore, for the same time of flight, transfer trajectories based on the mixed-transfer method can save a larger amount of fuel than the low-thrust method alone.