最优双冲量交会问题的数学建模与数值求解

基于普适变量法研究了两个共面轨道的最优双冲量交会问题。具体地，基于求解Lambert问题的普适变量法，在将给定时间段划分初始飘移阶段、轨 道转移阶段与终端停泊阶段的前提下，对两圆轨道及两拱线相同的椭圆轨道的最优双冲量交 会问题分别进行了优化数学建模，并利用数学软件Lingo进行了数值求解。数值结果表明，划分给定时间段可以得到更优解。

Mathematical Modeling and Numerical Solving of the Optimal Two-Impulse Rendezvous Problem

Technology of space rendezvous is one of the key technologies of space operations. Based on the Universal Variable's method, in this paper, we study the optimal two-impulse rendezvous problem of two coplanar orbits. Specifically, we firstly review the Lambert two-impulse rendezvous problem, which can be solved by the classical Gauss method based on the Universal Variable's method, discuss the optimal Lambert two-impulse rendezvous problem with the assumption that the rendezvous time and the beginning and ending positions are all fixed, and then build a corresponding two-dimensional optimal mathematical model. Secondly, for further energy saving, we extend this optimal mathematical model by introducing two additional time variables to divide the given fixed time period into three distinct stages (i.e., initial coasting stage, transfer orbit period and finial coasting stage) for two specific problems: one is the optimal two-impulse rendezvous problem of coplanar circular orbits and the other is the optimal two-impulse rendezvous problem of coplanar elliptical orbits with the same arch line, and build general four-dimensional optimal mathematical models for these two problems, respectively. Because of the existence of singular points in the optimal mathematical models, by taking advantage of the continuity of the object functions within the neighborhood of singular points and by introducing a sufficiently small positive constant to approximate the zero, we add a new constraint condition to remove singular points and obtain approximate optimal mathematical models. Lastly, we solve these approximate optimal mathematical models using Lingo, which is a mathematical software tool for solving non-linear programming, and get the global optimal numerical solution to the two-impulse rendezvous problems. The numerical results of four examples show that our approximate optimal mathematical models are feasible and that we can get a better solution by dividing the given time period.