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 引用本文: 何胜茂, 高扬, 张皓, 王扬鑫. 航天器“v∞转移轨道”的模型和解析法. 力学学报, 待出版.
He Shengmao, Gao Yang, Zhang Hao, Wang Yangxin. Model and analytic method of spacecraft“v transfer orbit”. Chinese Journal of Theoretical and Applied Mechanics, in press.
 Citation: He Shengmao, Gao Yang, Zhang Hao, Wang Yangxin. Model and analytic method of spacecraft“v∞ transfer orbit”. Chinese Journal of Theoretical and Applied Mechanics, in press.

## MODEL AND ANALYTIC METHOD OF SPACECRAFT“v∞ TRANSFER ORBIT”

• 摘要: 假定A和B是围绕一个引力中心按照开普勒轨道运行的卫星, 针对航天器从A出发转移至B的轨道确定问题, 本文提出了一个新的模型, 称为“v转移轨道”(v-Transfer-Orbit, 简称VTO). VTO选取航天器飞离A的时刻t0和逃逸速度大小v作为设计参数, 求解抵达B的开普勒轨道. 根据A和B运行轨道的空间位置关系, VTO分为A/B异面、A/B共面、A/B重合三种情况, 同时存在三类解: General-VTO、Backflip-VTO、Resonant-VTO. 本文建立了统一的VTO解析方法, 即将航天器抵达B的位置约束分解为轨道约束和时间约束, 根据轨道约束推导航天器轨道关于单个变量的解析式, 根据时间约束建立该变量的一元方程, 从而将原问题转化为一元方程寻根问题. 首先, 依次针对A/B异面、A/B共面、A/B重合情况构建了General-VTO的一元寻根方程, 详细介绍了一元方程的寻根区间, 并给出了一种基于三次样条插值的快速寻根算法; 然后, 构建了Backflip-VTO的一元寻根方程, 在分析一元方程函数单调性、极值点和拐点的基础上给出了一元方程寻根区间和寻根方法; 之后, 构建了Resonant-VTO的直接解析式. 最后, 给出算例并重点说明VTO多解性.

Abstract: Assuming that there exist the bodies A and B in Keplerian orbits around a single gravitational center and a spacecraft transfers from A to B, a new model called “v-Transfer-Orbit” (VTO-problem) is proposed for determining the spacecraft’s transfer orbit. In the VTO-problem, the escaping time t0 and the escaping velocity v departing from A are selected as the spacecraft’s orbital determination parameters. According to the spatial relative positions between A and B, the VTO-problem is divided into three cases: A/B is nonplanar, A/B is coplanar, and A/B is co-orbital, and there exist three types of solutions: General-VTO, Backflip-VTO and Resonant-VTO. In this paper, a uniform geometric analysis method for solving the VTO-problem is introduced, in which the position constraint of the spacecraft’s arrival at B is decomposed into orbital constraint and time constraint, the spacecraft’s orbital parameters are resolved by a single variable based on the orbital constraint, and an equation referring to this single variable is constructed based on the time constraint. According to the geometric analysis method, the VTO-problem is transformed into a one-variable equation-rooting problem. Firstly, the one-variable equation for General-VTO is derived in response to the cases of A/B nonplanar, A/B coplanar, and A/B co-orbital, and the intervals of the variable and an efficient equation-rooting algorithms based on the cubic spline interpolation are elaborated. Secondly, the different one-variable equation-rooting problem for Backflip-VTO is derived, and another set of equation-rooting algorithms are described on the basis of analyzing the equation function properties, such as monotonicity, extreme points and inflection points. Thirdly, the analytic solution is given directly for Resonant-VTO. Finally, examples are given to expound the solution multiplicity of the VTO-problem.

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