平动点周期轨道间小推力转移的Gauss伪谱法

针对三体问题共线平动点附近周期轨道间的小推力转移问题，构造了一种新的形状函数，在此基础上提出了一种基于Gauss伪谱法的优化设计方法。首先，建立小推力轨道转移动力学模型，参考初始轨道和目标轨道的类型，构造一种新的形状函数以近似小推力转移轨道。为满足不同的约束要求，提出了振幅和相位按多项式变化的假设，推导了小推力转移轨道的近似解析解；然后利用Gauss伪谱法将小推力轨道转移的最优控制问题转化为非线性规划问题，并对推导的近似解析解进行解算和处理，为Gauss伪谱法求解非线性规划问题提供较为有效的控制变量的初始猜测值；最后以地月系统L1点附近Halo轨道间的小推力转移问题为例进行了仿真分析。仿真结果表明，小推力转移轨道近似解析解具备有效性和普适性，使得Gauss伪谱法的迭代效率提高55%以上，同时也表明Gauss伪谱法可有效解决平动点周期轨道间的小推力转移轨道优化设计问题。

Low-thrust transfers between libration point periodic orbits based on Gauss pseudospectral method

In order to solve the problem of low-thrust orbit transfers between periodic orbits around libration points in the three-body system, a novel shape function was constructed and a method based on Gauss pseudospectral method (GPM) was proposed. Firstly, the dynamics model of low-thrust transfers was built, a novel shape function was constructed to approximate the low thrust transfer trajectory according to the type of initial orbit and target orbit. The approximate analytic solution for the transfer trajectory was deduced under the assumption that amplitude and phases change in the form of a polynomial to satisfy different constraints. Secondly, the optimal control problem of low thrust transfer was discretized into a nonlinear programming problem based on the GPM, and the effective initial value of control variables of the nonlinear programming problem was obtained by solving and processing the approximate analytic solution. At last, the numerical simulation of transfers between Halo orbits around the L1 libration point of the Earth-Moon three-body system was conducted. The simulation result shows that the approximate analytic solution for low-thrust orbit transfers based on the new shape function is valid and general. The efficiency of GPM can be improved by 55%. Besides, the GPM can effectively solve the problem of low thrust orbit transfers between periodic orbits around libration points in the three-body problem.