数学建模及其应用

针对嫦娥三号软着陆轨道设计与控制策略问题,在合理假设的前提下,建立动力学模型,求解得到了嫦娥三号着陆准备轨道近月点和远月点的速度。针对软着陆过程的6个阶段,通过受力分析,建立了嫦娥三号运动的微分方程模型,以燃料消耗最小为优化目标,以每个阶段的起止状态为约束条件,将软着陆轨道的优化设计问题转化为主发动机推力的泛函极值问题,并将其控制函数转化为近似的多项式函数优化问题。运用四阶Runge-Kutta差分迭代方法进行求解计算,从而得到各个阶段的最优控制函数和控制策略。结果表明,嫦娥三号软着陆过程耗时695s,消耗燃料1 269.1kg。

Under reasonable assumptions,a dynamic model is established for the trajectory design and control strategies of Chang'e-3's soft-landing.The specific positions of the perilune and apolune of the elliptical orbit where Chang'e-3orbits around the moon and gets prepared for landing are determined,as well as the magnitude of the spacecraft's velocity at both points.By force analysis for each sequence during the soft-landing,differential equation models are established,describing Chang'e-3's movements.The problem of the optimal design of the trajectory is converted into a functional optimization model concerning the thrust of the main engine,which aims at minimizing fuel consumption,subject to the starting-ending status of the spacecraft during the 6sequences of the landing process.In order to simplify the problem,the control function in the model is deemed as a third order polynomial.The fourth order Runger-Kutta difference iterative method is applied,and the optimal control function and control strategies for each sequence are obtained.The results show that Chang'e-3's soft-landing sequence takes about 695 s,with fuel consumption about 1 269.1kg.