考虑地球扁率影响的任意椭圆轨道小推力最优交会控制

Optimum Low-Thrust Power Rendezvous between Neighboring Elliptic Orbits Considering the Oblateness of Earth

研究在地球扁率影响和航天器在小推力推进作用下,任意椭圆轨道最优转移的一种解析解法。将小推力和地球扁率影响视为摄动力,建立航天器转移的状态方程,应用Pontryagin极大值原理,得到航天器最优推力加速度控制方程。在假设简化时,将无量纲的推力加速度γ和地球扁率的二阶带谐系数J20视为同一阶量级,并只考虑它们的一次项,忽略其他的高次项,从状态方程中分解出小推力和地球扁率对轨道根数的影响,并分别进行了积分,得出最优转移轨道的一阶解析解。

An analytic solution is determined for optimum low-thrust power transfer between neighboring elliptic orbits considering the earth oblateness. The optimization problem is formulated as a Lagrange problem of optimal control with six elements of an elliptic orbit as state variables. For simplifying the problem, the coefficients for the second zonal harmonic J2 and the nondimensional thrust acceleration are supposed to be of the same order of magnitude. After applying optimal control theory and the Pontryagin maximum principle, the integrate equation are divided into three parts: the undisturbed part, the part due to thrust acceleration and the part due to the second zonal harmonic. The previous two parts are integrated with eccentric anomaly as independent variables; the first-order analytical solution of the last part can be obtained by mean element method. The solution equations are linear in both the changes on the six orbital elements and the corresponding adjoint variables. Finally a series of typical examples are given to show that the solution in this paper is accurate enough for magnitude of A between 10-3-10-5.