Abstract:

The dynamic equations of spacecraft formation flying are weakly nonlinear equations since the distance between spacecrafts is quite small compared with the orbital radius of the spacecrafts. To solve weakly nonlinear equations effectively, a precise exponential integrator (PEI) was proposed. Precise integration method (PIM) was applied to calculate exponential function in the formulas of exponential integrators (EI). Firstly, PEI was validated by solving a weakly nonlinear equation compared with Runge-Kutta method. Secondly, the dynamic equations of spacecraft formation flying were obtained through Lagrange equations, and then the equations were tansfered into semi-linear form. Ultimately, PEI and Runge-Kutta method were comparatively used to solve these equations. Through numerical analysis, PEI gave higher precision of the dynamic equations of spacecraft formation flying, indicating that PEI can be applied to other weakly nonlinear problems as well.

Key words: exponential integrator, precise integration method, spacecraft formation flying, Runge-Kutta method

摘要：

编队飞行卫星间的距离远小于卫星的轨道半径, 其动力学方程表现为弱非线性。针对弱非线性方程的求解, 提出精细指数积分方法, 用精细积分法求解指数积分方法中的指数矩阵。用精细指数积分法和Runge-Kutta方法, 在不同条件下求解弱非线性方程的算例, 验证了精细指数积分法的有效性。通过Lagrange方程, 建立卫星编队飞行动力学方程的半线性形式, 用精细指数积分方法与Runge-Kutta方法求解方程。数值计算结果表明, 与同阶的Runge-Kutta求解弱非线性微分方程相比, 精细指数积分法具有更高的精度, 为卫星编队飞行动力学仿真提供了一种有效的数值算法。

关键词: 指数积分方法, 精细积分法, 卫星编队飞行, Runge-Kutta方法