求解最优月球软着陆轨道的隐式打靶法

基于间接法思想推导出一种隐式打靶法对月球软着陆轨道优化问题进行了研究。建立月球软着陆轨道归一化系统模型，利用庞特亚金极大值原理将月球软着陆轨道优化问题转化为满足最优必要条件的两点边值问题（TPBVP），采用一种新的时间变量使两点边值问题的终端时刻固定，同时将终端时刻看作状态变量并引入终端时刻的哈密尔顿函数值作为隐式终端条件，提出一种隐式打靶法对含有隐式终端条件的两点边值问题进行迭代求解，从而得出燃料消耗最优的月球软着陆轨道。仿真结果表明，与直接法和混合法相比，隐式打靶法优化精度高，收敛速度快，实现了月球软着陆过程燃料消耗最优。同时应用隐式打靶法求解不同发动机推力值的最优月球软着陆问题，得到燃料消耗最小的最优推重比，可为月面着陆器下降级发动机选型提供参考。

Implicit shooting method to solve optimal Lunar soft landing trajectory

Based on the idea of indirect method, an implicit shooting method is derived to solve the optimal Lunar soft landing trajectory. Firstly, a normalization system model of Lunar soft landing trajectory is established. Secondly, based on the Pontryagin's maximum principle, the optimization of Lunar soft landing trajectory is converted into a Two-Point Boundary Value Problem (TPBVP) of satisfying the necessary conditions for optimality. A new time variable is adopted, and then the terminal time-free TPBVP is changed into terminal time-fixed TPBVP. Moreover, the terminal time is considered as state variable and the value of the Hamiltonian function in terminal time is introduced as an implicit terminal condition. Finally, an implicit shooting method is proposed to solve the TPBVP with implicit terminal condition. The simulation results show that the proposed method has better convergence speed and higher optimization precision than the direct method and the hybrid method, achieving the goal of optimal fuel consumption in Lunar soft landing process. Meanwhile, the optimal Lunar soft landing trajectory with different engine thrusts is studied by this method, obtaining the optimal thrust-weight ratio with lowest fuel consumption. This can provide references for the engine selection of Lunar lander descent stage.