日地系L2点与日火系L1点转移轨道设计

将太阳-地球-火星-飞行器组成的四体问题分解成由太阳-地球-飞行器和太阳-火星-飞行器两个共面圆形限制性三体问题,设计日地系L2点与日火系L1点Lyapunov轨道之间的转移轨道,该转移轨道可以作为探测火星时的低能中间转移轨道.采用Richardson三阶近似解作为初始值,运用微分修正方法分别得到两个不同三体系统下拉格朗日点的精确Lyapunov轨道.基于Lyapunov轨道不变流形以及微分修正方法,设计了日地系L2点与日火系L1点间的转移轨道.将所得结果与基于Halo轨道不变流形设计的转移轨道进行了对比.结论表明:利用Lyapunov轨道不变形设计探火中间转移轨道相较于利用Halo轨道不变流形设计探火中间转移轨道在能量消耗以及飞行时间上都存在优势.     

受摄三体问题研究

圆型限制性三体问题模型忽略了摄动因素的影响,在很多情况下不能足够准确地描述三体系统的动力学性质。本文研究了考虑摄动影响的三体问题的动力学性质及其轨道设计。首先分析了运行在平动点附近的卫星所受的主要摄动因素;然后从系统在惯性坐标系中的动力学方程出发,推导了会合坐标系中考虑偏心率、第四体引力以及太阳光压摄动影响的一般动力学方程;最后使用两层微分修正方法将圆型限制性三体问题模型下设计的轨道转换到受摄三体问题模型下。     

限制性三体问题共线平动点轨道近似解析解

随着深空探测成为航天领域的研究热点,与其密切相关的三体问题基础研究也日益重要,尤其是在深空探测任务设计中处于基础地位的共线平动点附近运动的研究,更是具有重要的工程应用价值。在圆型限制性三体问题下,对共线平动点附近运动近似解析解的研究已经较为全面,但在更接近真实情况、更具一般性的椭圆型限制性三体问题下,相应的研究却相对较少。针对此背景,参考借鉴圆型限制性三体问题的研究方法,首先根据平动点的特性计算出平动点的位置,然后将非线性三体动力学模型在共线平动点处线性化,最后结合线性系统理论,获得了椭圆型限制性三体问题下共线平动点附近运动的近似解析解,并将其与经典的圆型限制性三体问题下的近似解析解进行对比分析,仿真结果证明了方法的有效性,同时也表明所推导的椭圆型限制性三体问题解析解相比圆型限制性三体问题解析解具有更高的精度。     

三维引力辅助机理分析

针对星际探测中的引力辅助技术,通过引入两个参数,将平面椭圆型限制性三体模型拓展到三维椭圆型限制性三体模型.通过逆向与正向积分,得到引力辅助前后飞行器的能量和角动量,据此将引力辅助轨道划分为16种类型.深入讨论了这两个参数对轨道类型、轨道能量和轨道倾角的影响,总结出了相应的变化规律.以地月系统为例,针对不同任务要求,给出了通过月球引力辅助,实现地月系统俘获与逃逸的引力辅助参数的选择区域,并对引力辅助参数进行了优化.     

Halo轨道族延拓方法及特性研究

对Halo轨道周期和运动范围等特性的研究是平动点任务设计的首要前提。针对大幅值Halo轨道和完整Halo轨道族的应用需求及其数值计算问题,面向当前应用广泛的地月系和日-地月系共线平动点,基于延拓法研究了圆型限制性三体问题下的Halo轨道族数值计算和运动学特性,给出了Halo轨道族延拓计算方法。数值仿真了族参数选择对轨道族计算的影响,得到了地月系和日-地月系共线平动点的大范围南北Halo轨道族,同时给出了轨道族的轨道周期变化和空间位置变化特性。研究结果表明,固定延拓步长下,L1点Halo轨道族应选择会合坐标系x坐标作为族参数,L2点Halo轨道族应选择y方向速度或者周期T作为族参数。方法适用于任意三体系统平动点的周期轨道族计算,特别是对其中的状态转移矩阵简单修改后可用于完整力模型下的Halo轨道（族）的数值设计。     

地月空间的远距离逆行轨道族及其分岔研究

地月系统中存在着一类绕月逆行、高度稳定的轨道族,称为远距离逆行轨道族(DRO)。以圆型限制性三体问题(CR3BP)为动力学模型研究了DRO轨道族周边的动力系统结构。利用Broucke稳定性图寻找分叉点,判断分叉类型,基于数值延拓计算分岔后产生的一系列新轨道分支。分叉类型主要有切分叉与多倍周期分叉(从3倍周期开始),轨道维度包含平面轨道族与三维轨道族。计算新轨道族的特征,包括形状、周期、能量、稳定性、双曲流形结构等。探讨周期轨道的轨道周期与能量的关系,以几何化的方式展现分叉结构、多周期轨道的双曲流形结构等。该动力结构将为基于DRO轨道族的地月空间任务提供重要的理论支持。      

地月L2点周期轨道的月球背面覆盖分析

地月L2点附近轨道具备独特的动力学和运动学特性,是月球背面探测任务的中继卫星首选布设位置。面向未来月球背面探测任务的中继通信需求,分析并研究了地月L2点周期轨道(halo轨道)对月球背面的覆盖。在圆型限制性三体问题模型下,研究并给出了halo轨道族延拓计算方法,基于延拓法设计了地月系大范围南北halo轨道族;给出了中继卫星的月球背面覆盖计算模型,定义了相应的时间覆盖因子;数值仿真了地月系南北halo轨道族的月球背面覆盖情况。研究结果表明:地月L2点周期轨道幅值和类型决定其对月面的覆盖性,幅值较小的轨道的月面整体覆盖性较好,幅值较大的轨道对月球南北极覆盖较好,南北族轨道分别有利于月球南北半球的覆盖。文章研究可为我国"嫦娥4号"月球背面探测任务的中继星轨道设计提供有益参考和借鉴。     

双小行星系统引力场建模方法研究

针对弱引力双小行星系统的引力场建模问题,本文采用复杂度和精度依次递增的球体–球体模型、椭球体–球体模型和改进的限制性椭球体–椭球体模型来进行引力场建模,并分别采用椭圆积分以及无积分环节、计算效率高的二阶二次球谐函数来表征引力势,从而比较精确地刻画双小行星系统和探测器构成的限制性全三体问题的动力学模型;针对双小行星系统1999KW4,对其不同的引力场模型进行了仿真研究,分别给出了不同模型下的等效势能函数曲面及零速度曲线,比较了不同模型下的平动点位置坐标偏差。结果表明,二阶二次球谐函数计算引力势的椭球体-椭球体模型计算精度高,复杂程度低,计算量更少,计算速度更快,能够较精确的对双小行星系统进行引力场建模。     

日地平动点卫星两脉冲转移轨道设计

研究基于最小二乘微分修正方法的平动点卫星两脉冲转移轨道设计,推导了考虑高度和航迹角约束的微分修正公式,讨论了该方法的收敛性.以日地L₁点附近的Halo轨道为目标轨道,在圆型限制性三体问题模型下设计了其转移轨道,系统地研究了HOI(Halo Orbit Insertion)点和Halo轨道幅值对转移轨道的影响,给出了HOI点的选择策略,并讨论了应急情况下快速转移轨道设计.数值仿真验证了方法的有效性,选择Halo轨道靠近地球侧的点作HOI点可以获得飞行时间适中的转移轨道.     

高精度模型下Halo轨道设计研究

针对未来地月L2点Halo轨道空间站长期停泊任务,研究了高精度模型下Halo轨道设计方法。首先,详细推导了圆型限制性三体问题(Circular Restricted Three Body Problem,CR3BP)质心会合坐标系与高精度模型地心J2000坐标系之间的转换关系,并在此基础上,将CR3BP下的闭合Halo轨道转换到地心J2000坐标系得到了高精度模型下Halo轨道迭代初值。其次,采用序列二次规划(Sequence Quadratic Program,SQP)构造多层迭代格式,在高精度模型下对初值进行逐层修正。最后,通过仿真测试,验证了该方法的可行性与有效性。该研究结果可为未来平动点任务标称轨道设计方案的制定提供参考。     

Earth–Moon low energy trajectory optimization in the real system

The problem of the Earth–Moon low energy trajectory optimization in the real system (the model defined by the JPL ephemeris DE405) is considered in this paper. First, this problem is investigated in the model of circular restricted three-body problem, since the fuel consumption is closely related to the Jacobi integral of the transfer trajectory, a method based on Jacobi integral is proposed and eight optimal trajectories are obtained. These optimal trajectories provide initial information (the flight time and the braking velocity impulse) to search the optimal low energy trajectories in the real system through optimization techniques. Considering the merit and drawback of particle swarm optimization and differential evolution algorithm in solving the space trajectory problem, an improved cooperative evolutionary algorithm is put forward. Result shows that the low energy trajectories in the real system are more fuel-efficient than the corresponding ones under the circular restricted three-body problem.     

Properties of transit trajectory in the restricted three and four-body problem

In the restricted three-body problem if the Jacobi constant is just below the value corresponding to Lagrangian point only a little neck exists around the equilibrium point and capture trajectories are indicated as low-energy. Capture properties depend on the dynamics around these critical points and qualitative results can be obtained using linearized systems. In this paper, to study transit trajectory properties in the restricted three and four-body problem, the Earth–Moon–Sun–Satellite system is considered as example and studied using different models. In the circular restricted three-body problem (Earth–Moon–Satellite), transit, non transit and asymptotic trajectories, are easily identified by using the principal reference frame. Dynamics around Lagrangian point are then studied introducing the Moon eccentricity into the elliptical restricted three-body model. A preferential region for transit orbit is individuated and studied as a function of eigenvalue properties. To introduce the Sun effect, the bi-circular four-body model is considered and dynamics around Lagrangian points studied as a function of angular distance between Earth–Sun and Earth–Moon line. Finally, results obtained in the elliptical three-body model and bi-circular four-body model, are compared with numerical simulations using real Sun–Moon–Earth ephemeris.     

Optimal station-keeping near Earth–Moon collinear libration points using continuous and impulsive maneuvers

In this study the gravitational perturbations of the Sun and other planets are modeled on the dynamics near the Earth–Moon Lagrange points and optimal continuous and discrete station-keeping maneuvers are found to maintain spacecraft about these points. The most critical perturbation effect near the L₁ and L₂ Lagrange points of the Earth–Moon is the ellipticity of the Moon’s orbit and the Sun’s gravity, respectively. These perturbations deviate the spacecraft from its nominal orbit and have been modeled through a restricted five-body problem (R5BP) formulation compatible with circular restricted three-body problem (CR3BP). The continuous control or impulsive maneuvers can compensate the deviation and keep the spacecraft on the closed orbit about the Lagrange point. The continuous control has been computed using linear quadratic regulator (LQR) and is compared with nonlinear programming (NP). The multiple shooting (MS) has been used for the computation of impulsive maneuvers to keep the trajectory closed and subsequently an optimized MS (OMS) method and multiple impulses optimization (MIO) method have been introduced, which minimize the summation of multiple impulses. In these two methods the spacecraft is allowed to deviate from the nominal orbit; however, the spacecraft trajectory should close itself. In this manner, some closed or nearly closed trajectories around the Earth–Moon Lagrange points are found that need almost zero station-keeping maneuver.     

Design and analysis of an extended mission of CE-2: From lunar orbit to Sun–Earth L2 region

Chang’E-2 (CE-2) has firstly successfully achieved the exploring mission from lunar orbit to Sun–Earth L2 region. In this paper, we discuss the design problem of transfer trajectory and at the same time analyze the visible segment of Tracking, Telemetry & Control (TT&C) system for this mission. Firstly, the four-body problem of Sun–Earth–Moon and Spacecraft can be decoupled in two different three-body problems (Sun–Earth + Moon Restricted Three-Body Problems (RTBPs) and Earth–Moon ephemeris model). Then, the transfer trajectory segments in different model are computed, respectively, and patched by Poincaré sections. The full-flight trajectory including transfer trajectory from lunar orbit to Sun–Earth L2 region and target Lissajous orbit is obtained by the differential correction method. Finally, the visibility of TT&C system at the key time is analyzed. Actual execution of CE-2 extended mission shows that the trajectory design of CE-2 mission is feasible.     

Numerical techniques for generating and refining solar sail trajectories

Like all applications in trajectory design, the design of solar sail trajectories requires a transition from analytical models to numerically generated realizations of an orbit. In astrodynamics, three numerical strategies are often employed. Differential correctors (also known as shooting methods) are perhaps the most common techniques. Finite-difference methods and collocation schemes are also employed and are successful in generating trajectories with pseudo-continuous control histories. These three numerical techniques are employed here to generate periodic trajectories displaced below the Moon in a circular restricted three-body system. All these approaches reveal trajectory options within the design space for solar sail applications.     

Design and analysis of a direct transfer trajectory from a near rectilinear halo orbit to a low lunar orbit

A near rectilinear halo orbit (NRHO) is regarded as a potential orbit for a future deep space station that can effectively support sustainable crewed lunar exploration missions. In this paper, the L₂ southern NRHOs are selected as the research object, and a direct transfer trajectory from an NRHO to a low lunar orbit (LLO) is designed and analyzed. First, based on the circular restricted three-body problem, the characteristics of NRHOs are discussed including geometric behaviour, stability and synodic resonance. Second, optimal direct transfer trajectories are obtained by combining a local gradient optimization with a numerical continuation strategy. A reachable domain calculation model is established. Finally, simulations are carried out to verify the feasibility of the trajectory design method. The relationship between the velocity change and the reachable domain is further analysed through simulation calculations.