Port-Hamiltonian Based Control of the Sun-Earth 3D Circular Restricted Three-Body Problem: Stabilization of the <i>L</i>₁Lagrange Point

In this paper, we use Port-Hamiltonian framework to stabilize the Lagrange points in the Sun-Earth three-dimensional Circular Restricted Three-Body Problem (CRTBP). Through rewriting the CRTBP into Port-Hamiltonian framework, we are allowed to design the feedback controller through energy-shaping and dissipation injection. The closed-loop Hamiltonian is a candidate of the Lyapunov function to establish nonlinear stability of the designed equilibrium, which enlarges the application region of feedback controller compared with that based on linearized dynamics. Results show that the Port-Hamiltonian approach allows us to successfully stabilize the Lagrange points, where the Linear Quadratic Regulator (LQR) may fail. The feedback system based on Port-Hamiltonian approach is also robust against white noise in the inputs.

Restricted Three-Body Problem in Different Coordinate Systems

In this paper of the series, the equations of motion for the spatial circular restricted three-body problem in sidereal spherical coordinates system were established. Initial value procedure that can be used to compute both the spherical and Cartesian sidereal coordinates and velocities was also developed. The application of the procedure was illustrated by numerical example and graphical representations of the variations of the two sidereal coordinate systems.

Restricted Three-Body Problem in Cylindrical Coordinates System

In this paper, the equations of motion for spatial restricted circular three body problem will be established using the cylindrical coordinates. Initial value procedure that can be used to compute both the cylindrical and Cartesian coordinates and velocities is also developed.

Halo Orbits in the Photo-Gravitational Restricted Three-Body Problem

We study of halo orbits in the circular restricted three-body problem (CRTBP) with both the primaries as sources of radiation. The positioning of the triangular equilibrium points is discussed in a rotating coordinate system.

Oblateness Effect of Saturn on Halo Orbits of L1 and L2 in Saturn-Satellites Restricted Three-Body Problem

The Circular Restricted Three-Body Problem (CRTBP) with more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion of the primaries is considered to generate the halo orbits around L1 and L2 for the seven satellites (Mimas, Enceladus, Tethys, Dione, Rhea, Titan and Iapetus) of Saturn in the frame work of CRTBP. It is found that the oblateness effect of Saturn on the halo orbits of the satellites closer to Saturn has significant effect compared to the satellites away from it. The halo orbits L1 and L2 are found to move towards Saturn with oblateness.

Universal method for designing periodic orbits by homotopy classes in the elliptic restricted three-body problem

The current methods for designing periodic orbits in the elliptic restricted three-body problem(ERTBP)have the disadvantages of targeting limited orbits and ergodic searches and considering only symmetric orbits.A universal method for designing periodic orbits is proposed in this paper.First,the homotopy classes of orbits are structured based on their topological structures.Second,a dynamic model based on homotopy classes,ranging from the circular restricted three-body problem(CRTBP)to the ERTBP,can be built using the homotopy method.Third,a multi-and a single-period orbit were selected based on the resonance ratios.Finally,the corresponding orbit in the ERTBP was computed by modifying the initial condition of the orbit in the CRTBP.This method,without an ergodic search,can extend to any orbit,including an asymmetric orbit in the CRTBP,to the ERTBP model,and the two orbits are of the same homotopy class.Examples of the Earth–Moon ERTBP are presented to verify the efficiency of this method.

A novel method of periodic orbit computation in circular restricted three-body problem

Periodic orbits are fundamental keys to understand the dynamical system of circular restricted three-body problem, and they play important roles in practical deep-space exploration. Current methods of periodic orbit computation need a high-order analytical approximate solution to start the iteration process, thus making the computation complicated and limiting the types of periodic orbits that can be obtained. By utilizing the symmetry of the restricted three-body problem, a special kind of flow function is constructed, so as to map a state on the plane of symmetry to another state that also lies in this plane. Based on this flow function, a new method of periodic orbit computation is derived. This method needs neither a starting analytic approximation nor the state transition matrix to be computed, so it can be conveniently implemented on a computer. Besides, this method is unaffected by the nonlinearity of the dynamical system, allowing a large set of periodic orbits which have x-z plane symmetry to be computed numerically. As examples, some planar periodic orbits (e.g. Lyapunov orbit) and spatial periodic orbits (e.g. Halo orbit) are computed. By further combining with a differential correction process, the method introduced here can be used to design resonant orbits that can jump between different resonant frequencies. One such resonant orbit is given in this paper, verifying the efficiency of this method.

地月历表误差对共线平动点周期轨道的误差影响分析

基于圆形限制性三体问题下地月平动点的3阶轨道解析解,分析不同初值条件下的轨道演化特性,并基于误差传播理论研究地月历表误差对地月共线平动点周期轨道在不同初值及初值误差时的影响。结果表明:1)轨道误差与地月历表误差在同一个量级,当地月历表误差均值不为0时,其峰值大约为地月历表误差的2.5倍,轨道误差随时间变化的无量纲周期约为轨道无量纲周期的0.5倍;2)在同一条轨道的不同位置,地月历表误差影响存在较大差异,平面Lyapunov轨道、Halo轨道和垂直Lyapunov轨道误差的峰值点分别位于或接近X向最大值与最小值处、Z向最大值与最小值处和z=0处。

不稳定地月循环轨道的混沌控制方法

针对现有的地月循环轨道稳定控制方法仅对少数轨道有效的缺点,本文提出了一种圆型限制性三体问题下不稳定地月循环轨道的混沌控制方法。首先在循环轨道上选择候选不动点及其对应的庞加莱截面图;然后通过混沌控制方法将轨道在庞加莱截面图上的投影点移动到不动点的稳定方向上;再计算轨道被控制后的目标点和轨道投影点到不动点之间偏差距离的比例,依次对比例最小的候选不动点进行筛选;最后根据所选择的不动点确定庞加莱截面图,计算控制脉冲量,对循环轨道进行稳定控制。仿真结果表明,本文所提出的方法适用于各种类型的地月循环轨道,相比现有方法至少节省了66%的能耗,能够使循环轨道长时间趋于稳定。

共线平动点中心流形上的轨道转移问题

圆形限制性三体问题共线平动点附近的平动点轨道由于其独特的动力学特性,在深空探测任务中有着重要价值,这些轨道间的轨道转移问题值得进行系统性研究.针对平动点轨道的计算与延拓,提出了一种基于数值的系统性计算平动点轨道的方法以及状态伴随法的轨道稳定维持策略.在此基础上,通过对大量平动点轨道不变流形以及平动点相空间中心流形的研究,设计了一套通过脉冲机动实现平动点轨道间轨道转移的系统性解决方案.该方法充分利用平动点动力学特性,在仿真验证中证实了方案的有效性,为平动点轨道转移研究提供了新的思路.

解析计算在月球中继卫星Halo轨道设计中的应用

面向月球中继卫星工程轨道设计需求,研究解析计算方法在地月系L2点halo轨道设计中的应用问题。在讨论圆型限制性三体问题三阶解析近似计算方法的基础上,分析了解析计算与数值计算的差异,给出了解析近似计算在工程约束下的适用范围,进而提出了基于解析计算的轨道设计和特征筛选方法。分别采用解析初值和数值初值进行halo轨道外推,比对验证采用解析计算设计轨道的可行性。研究结果表明,解析计算方法适用于月球中继卫星轨道的初步设计、特征分析和构型筛选。

基于圆型限制性三体模型的借力飞行机理研究

针对星际探测中的借力飞行技术,基于圆型限制性三体模型,深入研究了全相位角情况下借力飞行自由参数与借力飞行轨道变化之间的关系。在圆型限制性三体问题的假设下,通过正向与逆向积分相结合的方法,给出了圆型限制性三体系统中,通过借力飞行实现逃逸与俘获以及顺行轨道与逆行轨道转变的条件,并以地月系统为例,给出了通过月球借力飞行实现地月系统俘获与逃逸的借力飞行参数选择区域。同时,也给出了通过月球借力飞行实现轨道类型转变的参数变化范围。最后,讨论分析了借力飞行高度、速度和借力相位角对借力飞行轨道能量变化量的影响,进一步揭示借力飞行自由参数与借力飞行轨道变化之间的关系。

基于序优化理论的晕轨道转移轨道设计

利用晕轨道的稳定流形可以设计从地球到晕轨道的转移轨道。但由于小幅度晕轨道的稳定流形与地球停泊轨道无法相交,因此需采用两脉冲转移。微分修正法是求解两脉冲转移常用的优化方法,虽然收敛速度快,但很难获取全局最优解,而且收敛半径小,如果初始猜想与最优解相差很远,该方法可能会不收敛。将序优化理论与微分修正法相结合,利用序优化思想缩小搜索空间,得到足够好的初始猜想,然后利用微分修正法快速收敛到满足终端精度要求的解。仿真结果表明该方法有很好的收敛性,且计算量小。

一种计算三体问题周期轨道的新方法

针对三体问题周期轨道计算方法存在计算量大、改变雅可比能量和局限于计算特定周期轨道等不足,本文提出了一种计算周期轨道的新方法。首先建立了一种初始点和投影点关系的改进型庞加莱截面图,能够更直观地反映随着初始点改变周期轨道的演变和分叉;其次基于改进的庞加莱截面图,通过初始点与投影点的对应关系筛选出可能存在周期轨道的候选区间;然后在该候选区间内利用状态转移矩阵给出距离周期轨道初始点真实解非常接近的初始猜想;最后采用打靶法求解能够快速得到周期轨道的数值解。本文方法不需要改变三体系统的雅可比能量,迭代次数少,能够快速计算得到大范围、具有x轴对称性的周期轨道。以地月圆形限制性三体问题为例进行仿真,验证了该方法的快速性和有效性。

圆型限制性三体问题下Halo-绕月轨道的星间测距定轨

在传统星座自主定轨中,SST(satellite to satellite tracking)可以同时提供轨道的大小、形状和星座相对方位信息,但不能确定星座的绝对定向。针对这一亏秩问题,联合圆型限制性三体模型CRTBP(circle restricted three bodyproblem)下的一种平动点周期轨道-Halo轨道飞行器,与二体问题轨道卫星组成扩展星座。利用两种力模型的特性差异,可以去除星座系统上的相关性,避免星座的整体旋转,从而确定星座的全部轨道状态参量。分析Halo轨道的力模型及性态特点,从系数矩阵的相关性角度讨论引进Halo轨道对定轨法矩阵正定性的改善作用,利用地月系L1平动点附近的Halo轨道与月球低轨卫星(LMO)的星间链路,在理想CRTBP框架下进行自主定轨仿真。初步验证了LMO-Halo星座定轨可行性,为开展附加平动点轨道的星座SST定轨提供了参考依据。

地月低能轨道转移的混沌控制方法

针对航天器在混沌区域的滑行时间过长,提出一种低能地月轨道转移的混沌控制方法。首先通过地月圆形限制性三体问题(CRTBP)的庞加莱截面图,将混沌区域进行分层并分析其规律,再利用混沌轨道对初始状态高度敏感的特性,在尽可能减少运送时间的前提下实现地月低能轨道转移。该方法的优点是利用混沌区域的固有规律,不需要依靠周期轨道特性。仿真结果表明,本文提出方法仅需施加2~4次脉冲,明显缩短混沌区域的滑行时间,从而实现地月低能转移。

地月循环轨道动力学建模与计算研究

根据地月循环轨道的概念,按照生成第二类周期轨道的弧段进行分类,并讨论了其共振性与对称性在轨道设计与应用中的作用。然后归纳了三种循环轨道的动力学建模与计算方法及其轨道延拓策略,最后总结了三种方法的利弊和应用轨道类型。对地月系统循环轨道的研究和分析,能够为我国未来载人登月工程提供一种新的思路与理论支持。

平动点双脉冲转移轨道的快速计算方法

针对限制性三体问题中的平动点双脉冲转移,提出一种高效的计算方法。通过利用基于二维插值的数值流形近似方法对流形进行近似计算,同时利用二体模型下的圆锥曲线近似流形拼接段,根据经典轨道要素推导得到完成拼接所需的速度增量,避免在优化过程中对流形的重复积分计算,以及在三体模型下对拼接段的迭代计算,从而显著提高计算效率。然后推导得到三体问题下的主矢量理论,可将其用于对优化所得的双脉冲转移轨道进行燃料最优性的验证。最后,以航天器从近地圆轨道到地月系L1点的halo轨道的双脉冲转移为例进行数值仿真,验证数值流形近似算法和二体模型近似脉冲的有效性,并表明该方法在优化过程中具有高效性。

混沌多步控制在地月轨道转移中的应用

为了减少低能地月轨道转移时航天器在混沌区域的滑行时间,提出一种混沌多步控制方法。利用地月圆形限制性三体问题下航天器轨迹的运行规律和混沌轨道对初始状态高度敏感的特性,通过自适应粒子群优化算法计算出混沌控制每一步的最优扰动,进而实施多步控制实现地月低能轨道转移。该方法不需要依靠周期轨道和随机大量搜索,能够有效地实现地月低能轨道转移,并在耗费能量很小的前提下显著地缩短航天器在混沌区域的滑行时间。最后,数值仿真结果验证了所提出算法的正确性和有效性。

基于不变流形的夸父卫星A轨道设计

介绍了夸父卫星A的轨道特点,分析了圆型限制性三体问题。采用基于不变流形的两脉冲转移方法为夸父卫星A设计出了转移轨道,并比较晕轨道近地点和远地点入轨的两种情况。仿真表明,晕轨道远地点入轨可以大大节省晕轨道入轨代价。