周期流形的不变环面和次调和分支

本文通过精华Ｆｌｏｑｕｅｔ方法在周期流形的周期轨道邻域建立起适当的局部坐标，然后应用平均法和积分流形及Ｆｅｎｉｃｈｅｌ不变流形理论来证明不变环面和次调和轨道的存在性和法向双曲性．大多数传统的假设被放弃，而大多数已知的结果被推广．文中还给出了一个例子作为应用     

一类含有参数和有理奇性哈密顿系统的同宿与异宿轨道

研究一类含有两个参数和有理奇性平面哈密顿系统的同宿与异宿轨道,该问题来源于一个关于聚合物流体剪切流动特性的研究．借助常微定性理论和不变流形分析的方法,文中给出了系统存在同宿与异宿轨道的条件,并通过数值计算检验了所得理论结果。     

伴随奇点分支的退化异宿轨道的保存和横截性

本文应用指数三分性和不变流形的局部几何表示方法,给出异宿流形上的轨道当两个奇点经历超;临界分支和摄动时保存和横截的条件．     

卫星编队飞行轨道和姿态控制研究

卫星编队飞行是一种卫星应用的新概念,通过一系列造价更便宜的小卫星的分布式合作,代替大卫星实现复杂功能．在编队飞行一些应用中,要求受控卫星对目标卫星保持要求的相对位置和姿态以观察目标卫星的特定面,特别的,目标卫星可能是失效的．研究在近地轨道如何控制追踪星在失效的目标卫星附近飞行以追踪目标卫星特定面 的问题,给出了相对姿态和一阶近似的相对轨道动力学方程．基于线性反馈和Liapunov稳定性理论设计了控制策略．进一步的,考虑目标卫星转动惯量的不确定性,通过自适应控制的方法,获得正确的转动惯量比率．数值仿真算例验证了该控制方法的有效性．     

一种便于摄动分析的编队飞行卫星相对运动的描述

定义了一组参数来描述卫星编队飞行的相对运动,称为相对轨道要素．利用它可以方便地分析摄动对相对轨道构形的影响以及卫星编队队形的几何特点．首先,对相对轨道要素给予了详细的推导,指出当主星偏心率为小量时,在主星轨道坐标系中相对轨道是一椭圆柱和一平面相交所得的交线,用描述该椭圆柱和平面的参数即可确定相对轨道构形,进而提出了相对轨道要素．其次,利用相对轨道要素对相对轨道进行地球扁率摄动分析,指出相对轨道构形的变化由两部分组成:一是椭圆柱的漂移导致相对轨道中心的漂移,二是平面法线的章动和进动引起相对轨道平面转动,同时还给出了地球扁率摄动下相对轨道构形漂移率及转动率的解析公式．最后,针对J2摄动分析了卫星编队相对轨道构形的变化以及相对轨道构形的漂移量和转动量．     

光压摄动对空间太阳能电站轨道的影响研究

针对以重力梯度稳定方式设计的3种典型空间太阳能电站轨道动力学问题,提出了考虑地影和有效截面积变化的太阳光压模型.首先,采用能量方法,通过Legendre变换,引入广义动量,建立了Hamilton体系下轨道的正则方程;其次,采用辛Runge-Kutta方法求解相应的正则方程;最后通过数值试验分析,验证了模型的有效性以及数值求解方法的稳定性.同时,说明了地影和有效截面积变化对空间太阳能电站轨道有显著的影响;给出了空间太阳能电站对其半长轴、离心率以及轨道倾角的轨迹曲线,为空间太阳能电站的设计提供一种理论参考.     

行星悬浮轨道附近的编队

研究了不同类型的地球和火星悬浮轨道附近的相对运动．首先,推导了悬浮轨道附近的相对运动方程并将相对运动方程在悬浮轨道附近线性化．利用两种半自然编队控制率进行编队控制,其中一种为被动控制,对工程应用有很大的价值．在两种控制率下,讨论了每种悬浮轨道的稳定区域．由于两种控制率都不能满足特殊的火星悬浮轨道任务要求,于是,根据该任务的要求设计了一种特殊的半自然控制率．该控制率既能满足编队任务的要求也能使相对运动稳定．     

基于伴随同化方法的空间飞行器轨道估计

基于空间几何原理提出了一个双星无源定位方案,方案中包括两个观测坐标系和一个基础坐标系.每个观测坐标系中分别存在一条由观测卫星发出指向目标飞行器的射线,考虑到实际观测中存在各种误差,任意两颗观测卫星对目标飞行器的定位方向一般不在同一个平面内(即为异面直线),假定两条异面直线的公垂线段的中点(相交的情况下为交点)是目标飞行器所在的位置,而公垂线段的长度表征两颗卫星的观测误差.利用空间飞行器的观测数据计算了飞行器在基础坐标系中的位置坐标并进行了误差估计.利用一个基于伴随同化方法的数值模型估计了空间飞行器主动段的轨道.正向模型用来模拟飞行器的运动过程,伴随模型用来优化参数.通过同化空间飞行器在基础坐标系下的观测数据得到了空间飞行器的轨道.设计了数值实验以检验该模型的合理性,实验结果表明了双星无源定位系统的正确性,通过同化卫星对飞行器的观测数据,该伴随同化模型可以对空间飞行器的轨道进行成功估计.     

利用太阳光压力进行Halo轨道编队控制

利用太阳光压力可以实现地日限制性三体问题中L2点附近编队控制.该编队需要的控制力量级小,常规的推进方式难以实现.太阳帆能产生微小的连续光压力,可以用于Halo轨道附近的编队控制.由于太阳光压力的方向受到限制,只有部分编队构型可以利用太阳光压力实现.该文主要讨论了两种常见的编队构型--直线编队和圆编队,给出了太阳帆能实现的编队构型需要满足的条件.最后,对每种构型的编队进行了数值仿真,仿真结果表明太阳帆能有效的进行编队控制.     

纯引力轨道验证质量辐射计效应分析

纯引力轨道飞行器在精密导航、重力场测量以及基础科学研究等方面具有重要意义,辐射计效应是纯引力轨道验证质量的重要干扰力之一．针对内编队重力场测量系统,利用解析和数值计算相结合的方法,分析了内卫星辐射计效应与内编队系统参数的关系,并给出了适宜于工程计算的内卫星辐射计效应近似函数及其修正因子．分析可知,内卫星辐射计效应与腔体平均压力成正比,与腔体平均温度成反比;随腔体温差的增加而增加,随外卫星腔体半径的增加存在极小值,并且取极小值时外卫星腔体半径和内卫星半径比为常数1.189 4,这一常数是由内外卫星的球形腔体构型决定的,与腔体内温度和压力无关．当内外卫星半径比大于10时,可认为外卫星腔体充分大,此时内卫星辐射计效应与内卫星半径的平方近似成正比,随外卫星腔体半径的变化可忽略．     

BepiColombo gravitational capture and the elliptic restricted three-body problem

This paper shows how the BepiColombo trajectory near Mercury follows the invariant manifolds to its final capture. The BepiColombo gravitational capture provides several recovery opportunities at nominal conditions and was designed by exploring the solution space entirely, without knowledge of the invariant manifolds. In this work we reproduced the trajectory in the model of the elliptic restricted three body- problem (due to the high eccentricity of Mercury's orbit) and showed that it does follow the manifolds. Consequently we envision that manifolds should be used to give insight on the solution space, speeding up the design and optimization process and there by saving cost. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics

We consider periodic perturbations of conservative systems. The unperturbed systems are assumed to have two nonhyperbolic equilibria connected by a heteroclinic orbit on each level set of conservative quantities. These equilibria construct two normally hyperbolic invariant manifolds in the unperturbed phase space, and by invariant manifold theory there exist two normally hyperbolic, locally invariant manifolds in the perturbed phase space. We extend Melnikov’s method to give a condition under which the stable and unstable manifolds of these locally invariant manifolds intersect transversely. Moreover, when the locally invariant manifolds consist of nonhyperbolic periodic orbits, we show that there can exist heteroclinic orbits connecting periodic orbits near the unperturbed equilibria on distinct level sets. This behavior can occur even when the two unperturbed equilibria on each level set coincide and have a homoclinic orbit. In addition, it yields transition motions between neighborhoods of very distant periodic orbits, which are similar to Arnold diffusion for three or more degree of freedom Hamiltonian systems possessing a sequence of heteroclinic orbits to invariant tori, if there exists a sequence of heteroclinic orbits connecting periodic orbits successively.We illustrate our theory for rotational motions of a periodically forced rigid body. Numerical computations to support the theoretical results are also given.     

Nonlinear optimal feedback control for lunar module soft landing

In this paper, the task of achieving the soft landing of a lunar module such that the fuel consumption and the flight time are minimized is formulated as an optimal control problem. The motion of the lunar module is described in a three dimensional coordinate system. We obtain the form of the optimal closed loop control law, where a feedback gain matrix is involved. It is then shown that this feedback gain matrix satisfies a Riccati-like matrix differential equation. The optimal control problem is first solved as an open loop optimal control problem by using a time scaling transform and the control parameterization method. Then, by virtue of the relationship between the optimal open loop control and the optimal closed loop control along the optimal trajectory, we present a practical method to calculate an approximate optimal feedback gain matrix, without having to solve an optimal control problem involving the complex Riccati-like matrix differential equation coupled with the original system dynamics. Simulation results show that the proposed approach is highly effective.     

Numerical approximation of connecting orbits with asymptotic rate

Summary. In this paper we set up and analyze a numerical method for so called {\bf connecting orbits with asymptotic rate } in parameterized dynamical systems. A connecting orbit with asymptotic rate has its initial value in a given submanifold of the phase space (or its cross product with parameter space) and it converges with an exponential rate to a given orbit, e. g. a steady state or a periodic orbit. It is well known that orbits with asymptotic rate can be used to foliate stable or strong stable manifolds of invariant sets. We show that the problem of determining a connecting orbit with asymptotic rate is well-posed if a certain transversality condition is made and a specific relation between the number of stable dimensions and the number of parameters holds. For the proof we employ the implicit function theorem in spaces of exponentially decaying functions. Using asymptotic boundary conditions we truncate the original problem to a finite interval and show that the error decays exponentially. Typically the asymptotic boundary conditions by themselves are the result of a boundary value problem, e. g. if the limiting orbit is periodic. Thus it is expensive to calculate them in a parameter dependent way during the approximation procedure. To avoid this we develop a boundary corrector method which turns out to be nearly optimal after very few steps. Received April 28, 2000 / Revised version received December 18, 2000 / Published online May 30, 2001     

Isotropy Subgroups of Transformation Groups

In this paper we consider transitive actions of Lie groups on analytic manifolds. We study three cases of analytic manifolds and their corresponding transformation groups. Given a free action on the left, we define left orbit spaces and consider actions on the right by maximal compact subgroups. We show that these actions are transitive and find the corresponding isotropy subgroups. Further, we show that the left orbit spaces are reductive homogeneous spaces. This article thus forms the basis of a forthcoming paper on invariant differential operators on homogeneous manifolds. Partially supported by a Carver Research Initiative Grant.     

On manifolds of initial conditions leading to intersection of orbits of satellites with planet under weak gravitational perturbations

We consider the problem of initial conditions that lead to the intersection of a satellite orbit with planetocentric sphere of a radius R. The problem is considered in frame of the satellite version of the double-averaged restricted three body problem with taking into account gravitational perturbations caused by the polar oblateness of the planet. For some integrable cases we provide the boundaries of the manifolds of the initial orbital elements leading (or not leading) to the intersection of the satellite orbit with the planet surface.     

The limit set of trajectory in quasi-homogeneous system in ℝ3

In this article, we study the limit set of trajectory in three-dimensional quasi-homogeneous system. For the trajectory which is close to a singular curve, we show that either it approaches a fixed point or infinity, or it oscillates. Moreover, an oscillating example is given. The behaviour of trajectory which is near a closed orbit is also studied. At the end, we classify the integral manifolds of the system with isolated singularities.     

Symmetric Periodic Orbits and Uniruled Real Liouville Domains

A real Liouville domain is a Liouville domain with an exact anti-symplectic involution. The authors call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asymptotically, an invariant finite energy plane converges to a symmetric periodic orbit. In this note, they work out a criterion which guarantees uniruledness for real Liouville domains.