包含引力辅助变轨的三体Lambert问题求解算法

对包含引力辅助变轨的三体Lambert问题提出了一种数值求解算法,分为转移轨道初始设计和终值搜索两部分.采用伪状态理论,通过简单迭代求解高精度的转移轨道初始设计结果,在此基础之上,通过数值积分在更复杂的摄动环境中,计算精确的转移轨道和一二阶状态转移矩阵,并利用二阶微分修正算法搜索最终解.经过数值算例检验,这种方法具有较高的效率和鲁棒性,可以有效解决三体系统中引力辅助转移轨道的高敏感性问题.     

UKF参数估计在三体Lambert问题中的应用

为了快速精确地求解三体Lambert问题,提出了一种新的基于无损卡尔曼滤波(UKF)参数估计的数值求解算法,该算法由初值猜测和精确解求解两部分组成.首先,基于地月系统二体模型,通过简单迭代求解三体Lambert问题的初值.然后,将三体Lambert问题对应的两点边值问题转化为参数估计问题,通过UKF滤波算法求解,可得到收敛的精确解.该算法是基于概率估计理论的,不仅避免了传统数值方法推导相关梯度矩阵的复杂性,而且降低了三体Lambert问题对初值精确度的要求,从而显著降低了三体Lambert问题求解的难度.数值仿真表明,该方法求解效率较高,具有良好的鲁棒性,与微分修正算法、二阶微分修正算法对比具有更大的收敛域.      

基于引导型人工免疫算法的最优Lambert变轨

主要研究了燃料最省的Lambert双脉冲变轨问题.首先对普适变量法进行改进以避免奇异,并将其用于Lambert双脉冲变轨问题的求解.然后针对只给定初始时刻追踪航天器和目标航天器的轨道要素及总时间约束的交会问题,引入调相时间的概念,并将其和转移时间作为Lambert变轨的优化变量.最后采用引导型人工免疫算法GAIA(Guiding Artificial Im-mune Algorithm)对该优化问题进行寻优.仿真算例表明,与自适应遗传算法AGA(Adaptive Ge-netic Algorithm)相比,GAIA具有更强的寻优能力和更快的寻优速度,从而验证了GAIA用于最优Lambert变轨的有效性.     

Lambert转移中途修正的全局概率最优策略

应用Monte-Carlo法和遗传算法的联合仿真求解Lambert转移中途修正的全局概率最优策略.首先推广限制性三体问题中求解周期性特解的微分修正算法构造出考虑J₂项摄动下的Lambert转移轨道并以此作为参考轨迹,则中途修正策略仅需针对导航误差、初始偏差修正的控制偏差等进行补偿.应用微分修正算法导出的单值矩阵,设计出3类线性和非线性中途修正策略,以适应不同的精度需要.随后应用Monte-Carlo和遗传算法的联合仿真,可以得到实现代价函数(落点误差最小)在概率意义下的最优解.与直接利用优化算法寻优需要已知各种误差量不同,得到的最优修正策略更具有普适性.     

精确动力学模型中的行星引力辅助轨道设计

轨道设计分为初步设计和精确模型迭代两步,初步设计基于等效脉冲模型,用圆锥曲线拼接法确定时间窗口和引力辅助产生的速度脉冲。精确模型中引力辅助看作是一个连续的过程,将简化模型得到的引力辅助双曲线轨道化为行星心目标B平面参数,以地心逃逸速度作为设计变量,通过微分修正的方法进行求解。通过算例对比分析了简化模型和精确模型设计结果之间的关系,结果表明,引力辅助脉冲等效模型精度较好。     

一种无奇异的求解Lambert变轨的普适变量法

传统的普适变量法求解Lambert转移轨道的变轨点速度时,在两位置矢量夹角为180°处出现奇异,且用牛顿迭代法求解普适变量会丢失部分重要的解.通过公式变换消除奇异项而推导出一种无奇异的速度求解公式;且根据转移轨道的圈数对普适变量的取值范围进行分段,在每个区间用人工免疫算法AIA(Artificial Immune Algorithm)搜索普适变量正确解,解出所有Lambert转移轨道并从中选择最优转移轨道.仿真算例表明:这种改进的普适变量法能有效避免传统普适变量法的奇异性,而且能求得所有的Lambert转移轨道并得到燃料消耗最优的转移轨道.     

空间交会中多圈Lambert问题的算法改进与实现

针对目前多圈Lambert问题多种算法的不足之处,如精确性不高和难以编程实现等,利用问题解集的分布特点改进算法,编写可直接调用的Matlab程序,为多圈Lambert转移的拓展提供必要条件.以空间交会两动点间的轨道转移为例,建立二层优化的数学模型,并提出结合智能算法的解题思路.通过应用算例验证了程序的正确性和有效性.     

行星三体引力摄动对卫星探测器大气制动的影响

考虑行星引力在其卫星探测器大气制动过程中的显著摄动,建立了基于Milankovitch参数的平均轨道动力学模型并对土卫六探测器进行仿真。首先,将轨道参数转换为无奇异的Milankovitch参数,考虑探测卫星的大气阻力、扁率摄动以及行星引力摄动,建立了半解析轨道方程。其次,以土卫六探测器为对象,选择不同的土星初始方位角进行有大气和无大气情况下的数值仿真,并进行比较分析。结果表明,土星初始方位角的选择会引起土卫六大气制动轨道偏心率和近拱点高度在不同范围内震荡,极大地影响大气制动效果。      

平面引力辅助中施加脉冲的最优位置

在引力辅助过程中施加脉冲可以改善变轨效果。虽然轨道设计中一般将施加脉冲的位置设置在轨道的近拱点,但是并没有研究表明近拱点是施加脉冲的最优位置。为了研究平面引力辅助中施加单次共面脉冲的最优位置,首先采用6个参数对变轨过程进行描述,其中3个参数表征平面引力辅助、2个参数表征脉冲的大小与方向、1个参数表征施加脉冲的位置。然后将航天器的状态向量表示成参数的函数,进而求出变轨之后航天器的能量变化。研究施加脉冲的位置对航天器能量的影响,结果表明最优位置一般并不是近拱点,在最优位置施加脉冲比在近拱点有时可高效20%以上。      

直接再入大气的月地转移轨道设计

针对从月球停泊轨道出发直接再入大气的月地转移轨道设计问题,提出了一种数值求解算法。该算法由初值设计和精确解求解两部分组成。首先,根据轨道设计的相应约束,采用伪状态理论,通过简单迭代求解高精度的初值。然后,考虑精确的动力学模型,通过数值积分计算真实轨道和状态转移矩阵,并利用微分修正方法搜索精确解。该算法通过设计高精度的初值,降低了月地转移轨道的设计难度。数值仿真表明:该算法求解效率高,具有良好的鲁棒性。      

On the Lagrange and Hill stability of motion in the three-body problem

In the three-body problem, we consider the Lagrange and Hill stability including the Lagrange stability for the manifold of symmetric motions that exists in the case where two of three bodies have equal masses. To analyze the stability, in addition to integrals of energy and angular momentum we use the Lagrange–Jacobi equality. We prove theorems on the Lagrange and Hill stability. The theorem on the Hill stability has effective application in the case where the mass of a body is much less than masses of two other bodies. In this case, as it is known, the model of the restricted three-body problem is usually applied.     

Resonance transitions associated to weak capture in the restricted three-body problem

An interesting dynamics is studied in the restricted three-body problem where a particle abruptly transitions between resonance states, called a resonance hop. It occurs in a region about the secondary mass point which supports weak capture. This region, called a weak stability boundary, was recently proven to give rise to chaotic dynamics. Although it was numerically known that the resonance hop was associated with this boundary, this process was not well understood. In addition, the dynamical structure of the weak stability boundary has not been well understood. In this paper, we give a way to reveal the global structure of the weak stability boundary associated to resonance motions. This structure is shown to be surprisingly rich in resonant periodic motions interconnected by invariant manifolds. In this case, nearly all the motions are approximately resonant in nature where resonance hops can occur. The correlation dimension of orbits undergoing resonant motions, associated to the weak stability boundary, is also examined. The dynamics analyzed in the present paper is related to that studied by J. Marsden et al. under the perspective of Lyapunov orbits and the associated invariant manifolds. Applications are discussed.     

A new class of intermediate thrust arcs in the restricted three-body problem

The Mayer’s variational problem of determining spacecraft optimal trajectories in the context of a classical restricted three-body problem is considered. Integrability of differential equations of a controllable motion in the non-central gravitational field represents well known and challenging task. It is shown that in the absence of sufficient number of first integrals, an explicit dependability of unknown integrals on certain variables can be used to explore the existence of previously unknown particular integrals and solutions of these equations utilizing Dokshevich’s method of analytical dynamics. A new class of extremal analytical solutions of the problem for intermediate thrust arcs is presented. An illustrative example of utilizing these solutions in minimizing the spacecraft characteristic velocity of a transfer from a specified initial position to some final position in the Earth-Moon system is discussed.     

Periodic attitude motions along planar orbits in the elliptic restricted three-body problem

A way to improve the accuracy of the three-body problem model is taking into account the eccentricity of primary attractors. Elliptic Restricted Three-Body Problem (ER3BP) is a model for studying spacecraft trajectory within the three-body problem such that the orbital eccentricity of primaries is reflected in it. As the principal cause of perturbation in the employed dynamical model, the primaries eccentricity changes the structure of orbits compared to the ideal Circular Restricted Three-Body Problem (CR3BP). It also changes the attitude behavior of a spacecraft revolving along periodic orbits in this regime. In this paper, the coupled orbit-attitude dynamics of a spacecraft in the ER3BP are exploited to find precise periodic solutions as the spacecraft is considered to be in planar orbits around Lagrangian points and Distant Retrograde Orbits (DRO). Periodic solutions are repetitious behaviors in which spacecraft whole dynamics are repeated periodically, these periodic behaviors are the main interest of this study because they are beneficial for future mission designs and allow delineation of the system’s governing dynamics. Previous studies laid the foundation for spacecraft stability analysis or studying pitch motion of spacecraft in the ER3BP regime. While in this paper, at first, initial guesses for correction algorithms were derived through verified search methods, then correction algorithms were used to refine calculated orbit-attitude periodic behaviors. Periodic orbits and full periodic solutions are portrayed and compared to previous studies and simpler models. Natural periodic solutions are valuable information eventuate in the longer functional lifetime of spacecraft. Since the problem assumption considered in this paper is much closer to real mission conditions, these results may be the means to use natural bounded motions in the actual operational environment.     

Families of halo orbits in the elliptic restricted three-body problem for a solar sail with reflectivity control devices

Solar sail halo orbits designed in the Sun-Earth circular restricted three-body problem (CR3BP) provide inefficient reference orbits for station-keeping since the disturbance due to the eccentricity of the Earth’s orbit has to be compensated for. This paper presents a strategy to compute families of halo orbits around the collinear artificial equilibrium points in the Sun-Earth elliptic restricted three-body problem (ER3BP) for a solar sail with reflectivity control devices (RCDs). In this non-autonomous model, periodic halo orbits only exist when their periods are equal to integer multiples of one year. Here multi-revolution halo orbits with periods equal to integer multiples of one year are constructed in the CR3BP and then used as seeds to numerically continue the halo orbits in the ER3BP. The linear stability of the orbits is analyzed which shows that the in-plane motion is unstable while the out-of-plane motion is neutrally stable and a bifurcation is identified. Finally, station-keeping is performed which shows that a reference orbit designed in the ER3BP is significantly more efficient than that designed in the CR3BP, while the addition of RCDs improve station-keeping performance and robustness to uncertainty in the sail lightness number.     

背包问题的量子算法分析

对可用于密码体制设计的NP完全问题——背包问题,进行了量子算法分析.从复杂度理论角 度出发,讨论了如何用量子搜索算法加速背包问题等NP完全问题的求解.并从群论的角度与S hor的大数分解算法做了比较,讨论了影响算法速度一些因素.对量子算法的特性和前景做了展望.      

多机多窗口优化调度问题的模型与算法

最小化完工费用且具有多个时间窗口的多机调度问题,一直是组合优化领域的一个研究难点.首先给出描述问题的2种模型:整数规划IP(Integer Programming)模型,约束规划CP(Constraint Programming)模型.通过对IP模型和CP模型各自缺点的讨论,引出一个新的模型--混合IP-CP模型,重点讨论了该混合模型的求解方法,给出一个模型求解的启发式算法,经测试表明新模型和算法能极大地提高问题求解效率,为解决此类大规模优化调度问题提供了方法.