Multiple-Revolution Solutions of the Transverse-Eccentricity-Based Lambert Problem
Multiple-Revolution Solutions of the Transverse-Eccentricity-Based Lambert Problem
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DOI：
10.2514/1.45041
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发表时间：
2010
影响因子：
2.6
通讯作者：
Quan He;Jun Yu Li;Chao Han
中科院分区：
文献类型：
--
作者：
Quan He;Jun Yu Li;Chao Han
T HIS工程笔记遵循以前的论文，其中Avanzini [1]提出了一种基于横向偏心率矢量的算法来解决经典的Lambert问题：即确定具有指定飞行时间并连接两个位置矢量的转移轨道[2]。在Avanzini的文献[1]中，转移轨道的偏心率矢量可以分解为轨道平面上平行于两点连线弦的常分量和垂直于它的变横向分量。给定两个固定位置矢量，转移时间可以表示为横向偏心率eT的函数。与优雅的Battin方法相比，这种简单的Lambert算法的推导从数学角度看似乎要求更低，并且在物理上更直观[1]。然而，在文献[1]中，由于只考虑了直接转移弧，既没有给出转移时间对横向偏心率的导数的显式表达式，也没有给出基于新方法的多转解。自20世纪80年代以来，多革命Lambert问题一直在讨论[3]。20世纪90年代以后，许多专家主要在Ochoa的工作基础上[4]，利用经典的椭圆转移轨道的拉格朗日公式研究了多次旋转，其中转移时间表示为半长轴的函数。Prussing [5]基于经典的拉格朗日公式，提出了一种椭圆轨道的最小燃料多转算法，以确定所有可能的转移轨道。在此基础上，Shen和Tsiotras [6]提出了两个共面圆轨道之间的最小V、固定时间、双脉冲转移算法。在Shen和Tsiotras的算法中，只需要计算和比较两个多圈解，就可以获得最小成本的转移轨道。除了拉格朗日公式化方法外，还可以使用通用变量来解决多转问题[7]。Han和Xie [8]也提出了一个相应的算法来获得使用这些变量的多圈解。在文献[8]中，转移时间被写为普适变量z的函数，其值被限制在边界4N2 < z <4N122
T HIS Engineering Note follows a previous paper in which Avanzini [1] presented a transverse-eccentricity-vector-based algorithm to solve the classical Lambert problem: that is, the determination of a transfer orbit having a specified flight time and connecting two position vectors [2]. In Avanzini’s [1] paper, the eccentricity vector of the transfer orbit can be decomposed into a constant component parallel to the chord connecting the two points and a variable transverse component in the direction perpendicular to it on the orbit plane. Given the two fixed position vectors, the transfer time can be expressed as a function of the transverse eccentricity eT. Compared with the elegant Battin’s method, the derivation of this simple Lambert algorithm seems to be considerably less demanding from the mathematical standpoint and physically more intuitive [1]. However, with the only consideration of direct-transfer arcs, neither the explicit expression of the derivative of the transfer time with respect to the transverse eccentricity nor the multiple-revolution solutions based on the novel method were given in [1]. The multiple-revolution Lambert’s problem has been being discussed since the 1980s [3]. After the 1990s, mainly based on Ochoa’s works [4], many experts studied the multiple-revolution using the classical Lagrange formulation for an elliptic transfer orbit, where the transfer time was expressed as a function of semimajor axis. Prussing [5] developed a minimum-fuel multiple-revolution algorithm for an elliptic orbit to determine all the possible transfer trajectories, based on the classical Lagrange formulation. Also based on this formulation, Shen andTsiotras [6] presented aminimumV, fixed-time, two-impulse transfer algorithm between two coplanar circular orbits. In Shen and Tsiotras’s algorithm, only two of the multiple-revolution solutions need to be calculated and compared to obtain the minimum-cost transfer orbit. In addition to the Lagrange formulation method, it is also possible to use universal variables to solve the multiple-revolution problem [7]. Han and Xie [8] also presented a homologous algorithm to obtain the multiple-revolution solutions using these variables. In [8], the transfer timewaswritten as a function of universal variable z, for which the value is limited to the boundary 4N 2 < z < 4 N 1 2 2