卫星编队飞行中C-W方程与轨道根数法的比较

目前卫星编队飞行动力学与控制的研究得到了广泛的重视,这些研究的理论主要是基于描述卫星相对运动的Clohessy-Wiltshire(C-W)方程。但根据特例及定性分析,表明C-W方程在初始条件的选取、解的周期性等方面与实际情况不符,从星的能量也不守恒。以卫星轨道根数为基础,提出了卫星编队飞行中的相对轨道根数法,克服了C-W方程的局限性,物理概念清楚,应用范围广,解的周期性成为自然结论。在主星为小偏心率的情况下,得到了简化的相对运动方程。最后对两种方法进行了比较,指出了C-W方程的局限性及其原因。     

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．     

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.     

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．     

基于全系数自适应预测校正算法的火星气动捕获制导研究北大核心CSCD

实现火星气动捕获制导对未来的返回式火星探测任务具有重大意义.针对火星大气环境密度不确定性大,大气接口初始条件不确定性大等问题,提出并分析比较了两类基于全系数自适应预测校正算法的火星气动捕获制导方法.其中第一类算法分为建立在以远拱点速度半径为控制输出和远拱点变轨速度增量为控制输出两种,另一类算法则是通过跟踪标称气动轨道的半长轴变化率实现制导目标.文章给出了以上三种算法建立的过程,并通过数学仿真验证比较了各个算法的特性,分析结果表明,基于全系数及适应预测校正算法的火星气动捕获方法在保证制导精度的前提下可以提高对环境及初始条件不确定性的鲁棒性.     

网络化Euler-Lagrange系统的分布式编队机动控制北大核心CSCD

研究了网络化Euler-Lagrange系统自适应编队机动控制问题.针对参数不确定的Euler-Lagrange系统,利用滑模控制方法提出了一种自适应编队机动控制算法.基于Lyapunov稳定性理论,证明了闭环系统的稳定性.该算法的显著特点是通过引入一种特殊的有向网络拓扑来描述智能体之间的通信交互行为,使得系统中跟随者在不需要知道或估计时变机动参数的情况下,能够实现编队的方向、平移、形状的连续改变.最后对提出的自适应编队机动控制算法进行数值模拟以验证该控制方案的有效性.     

基于全系数自适应预测校正算法的火星气动捕获制导研究

实现火星气动捕获制导对未来的返回式火星探测任务具有重大意义.针对火星大气环境密度不确定性大,大气接口初始条件不确定性大等问题,提出并分析比较了两类基于全系数自适应预测校正算法的火星气动捕获制导方法.其中第一类算法分为建立在以远拱点速度半径为控制输出和远拱点变轨速度增量为控制输出两种,另一类算法则是通过跟踪标称气动轨道的半长轴变化率实现制导目标.文章给出了以上三种算法建立的过程,并通过数学仿真验证比较了各个算法的特性,分析结果表明,基于全系数及适应预测校正算法的火星气动捕获方法在保证制导精度的前提下可以提高对环境及初始条件不确定性的鲁棒性.     

活用直线性质知识巧解题

本文举例说明直线的特殊性质在解题中 的应用． 一、两点确定一条直线 在解析几何中,若两点的坐标都满足一个 二元二次方程,则该方程就是过这两点的直线 的方程．     

一类时间分数阶偏微分方程的解

考虑一类时间分数阶偏微分方程,该方程包含几种特殊情况:时间分数阶扩散方程、时间分数阶反应-扩散方程、时间分数阶对流-扩散方程以及它们各自相对应的整数阶偏微分方程． 通过Laplace-Fourier变换及其逆变换,该方程在空间全平面和半平面内的基本解可以求出,但其表达式则是通过适当的变形来求．另外,对于有限域上的初边值问题,则可由Sine(Cosine)-Laplace变换导出该方程的一种级数形式的解,并通过两个数值例子来说明该方法的有效性．     

两种材料组成空间的弹性力学基本解

本文研究了由两种不同材料的半空间所组成弹性体的弹性力学基本解。应用三维弹性理论中的Papkovich-Neuber通解以及Kelvin特解,求解出了在空间内部作用有集中力时空间的弹性力学位移场。该位移场在两个半空间内部分别满足各自的位移平衡方程,在其交接面上满足位移及面力的连续条件。作为本文结果的几种特殊情况,半空间的Lorentz问题与Mindlin问题的解,以及Stokes流中类似问题的解均可从该解答中导出。     

Critical solution for a Hill's type problem

We study the problem of two satellites attracted by a center of force. Assuming that the motion of the center of mass of the two satellites is a Keplerian circular orbit around the center of force, we regularize the collision between them using the Levi-Civita procedure. The existence of a constant of motion in the extended phase space allows us to study the stability of the solution, where the two satellites are tied together in their circular motion around the center of force. We call this solution the critical solution. A theorem of M. Kummer is applied to prove, in specific conditions, the existence of two one-parametric families of almost periodic orbits for the satellites motion that bifurcates from the critical solution.     

一类超混沌离散系统的控制

研究一类超混沌离散系统的控制问题.基于局部线性化建立了时变线性反馈控制律.采用Liapunov直接法估计了控制律可以有效作用的邻域范围.分别给出了应用该控制律解决不稳定周期轨道的镇定问题和任意给定周期轨道的追踪问题的算例.     

Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips

Homoclinic bifurcations in four-dimensional vector fields are investigated by setting up a local coordinate near a homoclinic orbit. This homoclinic orbit is principal but its stable and unstable foliations take inclination flip. The existence, nonexistence, and uniqueness of the 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold 1 -periodic orbit and three-fold 1 -periodic orbit are also obtained. It is indicated that the number of periodic orbits bifurcated from this kind of homoclinic orbits depends heavily on the strength of the inclination flip.     

High-order solutions around triangular libration points in the elliptic restricted three-body problem and applications to low energy transfers

High-order series expansions around triangular libration points in the elliptic restricted three-body problem (ERTBP) are constructed first, and then with the aid of the series solutions, two-impulse and low-thrust low energy transfers to the triangular point orbits of the Earth–Moon system are designed in this paper. The equations of motion of ERTBP in the pulsating synodic reference frame have the same symmetries as the ones in circular restricted three-body problem (CRTBP), and also have five equilibrium points. Considering the stable dynamics of triangular libration points, the analytical solutions of the motion around them in ERTBP are expressed as formal series of four amplitudes: the orbital eccentricity of the primary, the long, short and vertical periodic amplitudes. The series expansions truncated at arbitrary order are constructed by means of Lindstedt–Poincaré method, and then the quasi-periodic orbits around triangular libration points in ERTBP can all be parameterized. In particular, when the eccentricity of the primary is zero, the series expansions constructed can be reduced to describe the motion around triangular libration points in CRTBP. In order to check the validity of the series expansions constructed, the domain of convergence corresponding to different orders is studied by using numerical integration. After obtaining the analytical solutions of the bounded orbits around triangular points, the target orbits in practical missions can be expressed by several related parameters. Thanks to the series expansions constructed, two missions are planned to transfer a spacecraft from the Earth to the short periodic orbits around triangular libration points of Earth–Moon system. To complete the missions with less fuel cost, low energy transfers (two-impulse and low-thrust) are investigated by means of numerical optimization methods (both global and local optimization techniques). Simulation results indicate that (a) the low-thrust, low energy transfers outperform the corresponding two-impulse, low energy transfers in terms of propellant fraction, and (b) compared with the traditional Hohmann-like transfers, both the two-impulse, low energy and low-thrust, low energy transfers perform very efficiently, at the cost of flight time.     

Phase-locking for dynamical systems on the torus and perturbation theory for mathieu-type problems

Summary When several oscillators are coupled together and the parameters of their coupling are varied, the oscillators pass through so-called phase-locked regimes. In physical terms this means that the oscillators tend to synchronize their motion. To describe this phenomenon, we frame the concepts ofpartial phase andphase-locking. A partial phase of a toral flow puts emphasis on how orbits of the flow drift around the torus in some fixed direction. The partial phase is locked if it grows in time along some orbit slower than any linear function. When a toral flow is given by a trigonometric polynomial, its phase-locked regions are quite narrow. With the coupling amplitude increasing, each region grows in width as some power of the amplitude. That power can be calculated in terms of both the partial phase and degree of the trigonometric polynomial.     

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.     

A new method to find homoclinic and heteroclinic orbits

A new series method is provided for continuous-time autonomous dynamical systems, which can find exact orbits as opposed to approximate ones. The method can reduce the connecting orbit problem as a boundary value problem in an infinite time domain to the initial value problem. It consists of transforming time to the logarithmic scale, substituting a power series around each fixed point of interest for each of the unknown functions into the system, and equating the corresponding coefficients. When solving for the power series coefficients, additional parameters are used in order to find the intersections of the unstable manifold and the stable manifold of the equilibria. This paper demonstrates how the new method allows to obtain heteroclinic and homoclinic orbits in some well-known cases, such as Nagumo system, stretch-twist-fold flow or mathematical pendulum.     

Dynamics of perturbed equilateral and collinear configurations of three point vortices

Using the technique of asymptotic expansions, we calculate trajectories of three point vortices in the vicinity of stable equilateral or collinear configurations. We show that in an appropriate rotating coordinate system each vortex moves in an elliptic orbit. The orbits of the vortices have equal eccentricities. The angle and ratio between the major axes of any two orbits have a simple analytic representation.      

Replicate periodic windows in the parameter space of driven oscillators

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have the same periodicity and pattern of stable and unstable periodic orbits already existent for the unperturbed oscillator. Moreover, these unstable periodic orbits are embedded in chaotic attractors in phase space regions where the new stable orbits are identified. Thus, the observed periodic window replication is an effective oscillator control process, once chaotic orbits are replaced by regular ones.     

Quaternion solution for the rock’n’roller: Box orbits, loop orbits and recession

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock’n’roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.