Indirect Radau pseudospectral method for the receding horizon control problem

To solve the receding horizon control (RHC) problem in an online manner, a novel numerical method called the indirect Radau pseudospectral method (IRPM) is proposed in this paper. Based on calculus of variations and the first-order necessary optimality condition, the RHC problem for linear time-varying (LTV) system is transformed into the two-point boundary value problem (TPBVP). The Radau pseudospectral approximation is employed to discretize the TPBVP into well-posed linear algebraic equations. The resulting linear algebraic equations are solved via a matrix partitioning approach afterwards to obtain the optimal feedback control law. For the nonlinear system, the linearization method or the quasi linearization method is employed to approximate the RHC problem with successive linear approximations. Subsequently, each linear problem is solved via the similar method which is used to solve the RHC problem for LTV system. Simulation results of three examples show that the IRPM is of high accuracy and of high compu-tation efficiency to solve the RHC problem and the stability of closed-loop systems is guaranteed.     

利用温度测量结果反演三维热传导问题中热源项的算法研究

利用表面温度测量来反演热传导问题中的热源项是一类典型的热传导逆问题，在采用有限体积法对三维稳态热传导问题进行数值求解的基础上，将该热传导逆问题转化为优化问题，基于灵敏度分析建立了反演算法。采用该算法对一典型算例的计算结果表明：建立的算法是有效的，具有较好的抗噪性能。此外，对反演算法中计算收敛准则的选取进行了较深入的分析，结果表明，由于热传导逆问题的不适定性，优化过程中目标函数值越小并不意味着反演结果与真值更为接近，可以通过设定合适的收敛准则来模拟正则化项的作用，克服不适定性的影响。     

C—半群和对抽象Cauchy问题的应用

本文研究了Ｃ－半群的无穷小生成元定义中强弱极限的等价性及生成元的有关性质，讨论了Ｃ－半群在齐次与非齐次Ｃａｕｃｈｙ问题中的应用。     

Lagrangian coherent structures in various map representations for application to multi-body gravitational regimes

The Finite-Time Lyapunov Exponent (FTLE) has been demonstrated as an effective metric for revealing distinct, bounded regions within a flow. The dynamical differential equations derived in multi-body gravitational environments model a flow that governs the motion of a spacecraft. Specific features emerge in an FTLE map, denoted Lagrangian Coherent Structures (LCS), that define the extent of regions that bound qualitatively different types of behavior. Consequently, LCS supply effective barriers to transport in a generic system, similar to the notion of invariant manifolds in autonomous systems. Unlike traditional invariant manifolds associated with solutions in an autonomous system, LCS evolve with the flow in time-dependent systems while continuing to bound distinct regions of behavior. Moreover, in general, FTLE values supply information describing the relative sensitivity in the neighborhood of a trajectory. Here, different models and variable representations are used to generate maps of FTLE, and the resulting structures are applied to design and analysis within an astrodynamical context. Application of FTLE and LCS to transfers from LEO to the L₁ region in the Earth–Moon system are presented and discussed. In an additional example, an FTLE analysis is offered of a few stationkeeping maneuvers from the Earth–Moon mission ARTEMIS (Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon's Interaction with the Sun).     

Hill区域内基于推广的Poincaré映射的低能逃逸/捕捉轨道

给出了Hill区域内基于推广的Poincare映射的低能逃逸轨道的计算.文中的推广的Poincare映射需要在相空间中选取两个(而非一个)与流相截的面,即将包含某个天平动点的截面通过轨道来映射到近拱点的通道面上的,在数值上通过总变差减小的Runge-Kutta方法来实现两个相面之间的映射.捕捉轨道则可由Hill三体模型的对称性直接导出.最后,关于Rhea的二维、三维的逃逸和捕捉轨道的仿真结果验证了该方法的精度和鲁棒性.     

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.      

地月系统循环轨道初步设计与特性分析

基于平面圆型限制性三体问题(CR3BP)模型,根据轨道弧以及顺行和逆行特征采用圆锥曲线拼接法设计了5类地月系统周期轨道。以地月系统循环轨道的工程约束为出发点,从轨道周期、近地点高度、近月点高度、交会对接速度、轨道稳定性等方面分析了这5类周期轨道的特性,从中选择了适合地月系统循环轨道任务方案的周期轨道,并对周期轨道进行了优化。该研究为我国未来载人登月工程提供了一种新的思路与理论技术支持。     

大范围收敛的摄动Lambert问题新型解法:拟线性化-局部变分迭代法

研究实时、高效、稳定性强的高性能空间轨道计算方法对于中国未来航天工程具有重大应用价值。针对强非线性系统的多维两点边值问题,提出了一种拟线性化-局部变分迭代法（QL-LVIM）,通过拟线性化（QL）思想,将非线性两点边值问题转化为一系列具有一定迭代格式,并且成对出现的初值问题,进而通过局部变分迭代法（LVIM）对其进行求解。利用拟线性化的大范围收敛特性和局部变分迭代法的快收敛、高精度特性,该方法能够在较大的时间和空间尺度下快速精确获得摄动Lambert问题的初速度和转移轨道,其收敛域远大于传统的牛顿打靶法,为航天器轨道转移提供了一种简便高效、稳定性强的新型计算方法。在不同轨道情形下,与几类参考方法对比,结果表明本方法能够在计算效率方面实现大幅提升,并且能够在大范围内实现快速收敛。方法的有效性在地-月系三体问题中得到了进一步验证。     

Broad-search algorithms for the spacecraft trajectory design of Callisto–Ganymede–Io triple flyby sequences from 2024 to 2040, Part I: Heuristic pruning of the search space

Triple flybys of the Galilean moons of Jupiter can capture a spacecraft into orbit about Jupiter or quickly adjust the Jupiter-centered orbit of an already captured spacecraft. Because Callisto does not participate in the Laplace resonance among Ganymede, Europa, and Io, triple flyby sequences involving gravity-assists of Callisto, Ganymede, and Io occur only aperiodically for limited time windows. An exhaustive search of triple-flyby trajectories over a 16-year period from 2024 to 2040 using “blind” searching would require 8,415,358 Lambert function calls to find only 127,289 possible triple flyby trajectories. Because most of these Lambert function calls would not converge to feasible solutions, it is much more efficient to prune the solution space using a heuristic algorithm and then direct a much smaller number of Lambert function calls to find feasible triple flyby solutions. The novel “Phase Angle Pruning Heuristic” is derived and used to reduce the search space by 99%.     

Design of transfer trajectories between resonant orbits in the Earth–Moon restricted problem

The application of dynamical systems techniques to mission design has demonstrated that employing invariant manifolds and resonant flybys enables previously unknown trajectory options and potentially reduces the ΔV$\mathrm{\Delta }V$ requirements. In this investigation, planar and three-dimensional resonant orbits are analyzed and cataloged in the Earth–Moon system and the associated invariant manifold structures are computed and visualized with the aid of higher-dimensional Poincaré maps. The relationship between the manifold trajectories associated with multiple resonant orbits is explored through the maps with the objective of constructing resonant transfer arcs. As a result, planar and three-dimensional homoclinic- and heteroclinic-type trajectories between unstable periodic resonant orbits are identified in the Earth–Moon system. To further illustrate the applicability of 2D and 3D resonant orbits in preliminary trajectory design, planar transfers to the vicinity of L₅ and an out-of-plane transfer to a 3D periodic orbit, one that tours the entire Earth–Moon system, are constructed. The design process exploits the invariant manifolds associated with orbits in resonance with the Moon as transfer mechanisms.