On Lambert’s problem and the elliptic time of flight equation: A simple semi-analytical inversion method

In the two-body model, time of flight between two positions can be expressed as a single-variable function and a variety of formulations exist. Lambert’s problem can be solved by inverting such a function. In this article, a method which inverts Lagrange’s flight time equation and supports the problematic 180°$180°$ transfer is proposed. This method relies on a Householder algorithm of variable order. However, unlike other iterative methods, it is semi-analytical in the sense that flight time functions are derived analytically to second order vs. first order finite differences. The author investigated the profile of Lagrange’s elliptic flight time equation and its derivatives with a special focus on their significance to the behaviour of the proposed method and the stated goal of guaranteed convergence. Possible numerical deficiencies were identified and dealt with. As a test, 28 scenarios of variable difficulty were designed to cover a wide variety of geometries. The context of this research being the orbit determination of artificial satellites and debris, the scenarios are representative of typical such objects in Low-Earth, Geostationary and Geostationary Transfer Orbits. An analysis of the computational impact of the quality of the initial guess vs. that of the order of the method was also done, providing clues for further research and optimisations (e.g. asteroids, long period comets, multi-revolution cases). The results indicate fast to very fast convergence in all test cases, they validate the numerical safeguards and also give a quantitative assessment of the importance of the initial guess.     

A robust semi-analytic method to solve the minimum ΔV two-impulse rendezvous problem

A novel semi-analytic approach is developed to determine the minimum $\mathrm{\Delta }V$ for a two-impulse rendezvous and validated both empirically and analytically. A previously published closed-form $\mathrm{\Delta }V$ estimate and the Lambert minimum energy transfer is used to establish upper and lower bounds of the minimum $\mathrm{\Delta }V$ transfer between two orbits. These bounds, in conjunction with the bisection method, operate on a nonlinear radical cost function to guarantee linear convergence. This approach has several real world applications including a low earth orbit (LEO) to highly elliptical orbit (HEO), and a HEO to retrograde geosynchronous orbit transfer. The minimum $\mathrm{\Delta }V$ estimates are better than those reported in the existing literature, while run times improved as much as two orders of magnitude over a fixed time Lambert solver. All singularity cases were addressed such that any orbital geometry, including Hohmann and radial elliptic transfers, converged to the global minimum $\mathrm{\Delta }V$. This approach will work for both coplanar and non-coplanar 3D geometries for any orbit type.     

变转速旋翼直升机单发失效低速回避区分析

利用最优控制方法研究变转速旋翼直升机在遭遇单发失效时,旋翼转速对自转着陆低速回避区的影响。首先,以UH-60A直升机为样机,建立三维刚体飞行动力学模型,并分析低速范围内旋翼转速对直升机需用功率的影响。然后,在模型中加入单发失效后自转着陆阶段发动机输出功率以及旋翼转速变化方程,并利用直接多重打靶法将直升机单发失效后的自转着陆过程转换为非线性最优控制问题进行数值求解。最后,基于最小化回避区面积的思想,得到并分析直升机在不同旋翼转速下单发失效后的自转着陆低速回避区,以及回避区高悬停点、拐点和低悬停点对应的最优着陆轨迹和操纵过程。结果表明：随着旋翼转速的降低,直升机单发失效后的低速回避区首先会逐渐缩小,然后迅速增大。最小回避区对应的旋翼转速略高于最小需用功率对应的旋翼转速。适当降低旋翼转速不仅能有效降低直升机的需用功率,还有利于提高直升机单发失效后的自转着陆性能。     

非光滑多体系统最大Lyapunov指数的计算方法

结合非光滑力学与非线性动力学的理论给出非光滑多体系统最大Lyapunov指数的数值计算方法.应用第一类Lagrange方法建立具有单边约束非光滑多体系统变拓扑结构动力学方程;应用线性互补理论与方法检测非光滑事件;应用混沌同步方法计算Lyapunov指数.在提高计算速度方面,提出了采用二分搜索法加快搜索速度以及采用调制参数方法加快系统达到同步状态的速度.通过算例说明本文算法的可行性与有效性.      

跟踪与数据中继卫星系统资源调度优化问题

现有跟踪与数据中继卫星资源调度研究常采取简化模型,它将天线的准备时间假定为固定值,由此成为工程应用中系统利用率提升的一个瓶颈。文章借鉴时间窗约束的车辆路径问题理论,提出天线准备时间变长的规划优化问题（LOPVAPT）模型,通过优化天线的扫描路径,实现天线准备时间的动态最优取值。采用STK模拟任务过程,基于Matlab实现蚁群算法,结果表明可将系统利用率从低于80%提高到90%以上。     

混合推进最省燃料轨道设计方法

使用混合推进方式设计地-月圆型限制性三体模型下的最省燃料转移轨道。将化学发动机以及电推进发动机的燃料消耗总和作为目标函数进行优化,推导一阶必要条件和雅可比矩阵。选择从近地圆轨道出发到达地-月L1附近Halo轨道的转移轨道为例测试上述方法。仿真结果表明,相比发射脉冲固定的情况,混合推进方式进一步降低了燃料消耗,而且给出了飞行时间和燃料消耗不同的组合方式,给予任务设计更大的灵活性。     

Multiobjective genetic optimization of Earth–Moon trajectories in the restricted four-body problem

In this paper, optimal trajectories of a spacecraft traveling from Earth to Moon using impulsive maneuvers (ΔV maneuvers) are investigated. The total flight time and the summation of impulsive maneuvers ΔV are the objective functions to be minimized. The main celestial bodies influencing the motion of the spacecraft in this journey are Sun, Earth and Moon. Therefore, a three-dimensional restricted four-body problem (R4BP) model is utilized to represent the motion of the spacecraft in the gravitational field of these celestial bodies. The total ΔV of the maneuvers is minimized by eliminating the ΔV required for capturing the spacecraft by Moon. In this regard, only a mid-course impulsive maneuver is utilized for Moon ballistic capture. To achieve such trajectories, the optimization problem is parameterized with respect to the orbital elements of the ballistic capture orbits around Moon, the arrival date and a mid-course maneuver time. The equations of motion are solved backward in time with three impulsive maneuvers up to a specified low Earth parking orbit. The results show high potential and capability of this type of parameterization in finding several Pareto-optimal trajectories. Using the non-dominated sorting genetic algorithm with crowding distance sorting (NSGA-II) for the resulting multiobjective optimization problem, several trajectories are discovered. The resulting trajectories of the presented scheme permit alternative trade-off studies by designers incorporating higher level information and mission priorities.     

次对称三对角矩阵特征值反问题

文章讨论了由两个特征对构造次对称三对角矩阵的特征值反问题。结合次对称矩阵中属于不同特征值的特征向量的次正交性,研究了解的存在性以及存在解的充要条件,并给出了相应的算法及数值例子。     

基于椭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,基于椭圆型限制性三体模型,深入研究了借力飞行 自由参数与借力飞行轨道变化之间的关系。在椭圆型限制性三体问题的假设下,通过正向和 逆向积分相结合的方法,重点研究了借力天体轨道偏心率和相位角对借力飞行逃逸和俘获区 域参数变化的影响,同时分析了借力天体轨道偏心率和相位角对借力飞行实现顺行轨道与逆 行轨道转变区域参数变化的影响,并以地月系统引力参数为例,分析了借力天体分别对应不 同轨道偏心率和相位角时,前向和后向飞越情况下,借力飞行对轨道能量和角动量的影响, 总结出了相应的变化规律。这些研究对于借力飞行轨道的设计与应用具有重要的参考意义。