椭圆相对运动方程的高阶分析解

相对动力学的研究在编队飞行应用中非常重要,比较精确的编队构型可以较大程度地减少队形保持所需的燃料消耗.传统的构型设计主要基于线性化方程即C-W方程(圆参考轨道)或Lawden方程(椭圆参考轨道)的周期解.可是,线性化解在编队尺度较大时不再适用.鉴于此,本文以椭圆参考轨道对应的非线性相对运动方程为基础,将伴星相对于主星的相对运动展开为参考轨道偏心率、平面内振幅以及垂直平面振幅的级数解形式,并以Lawden周期解为初始解,采用Lindstedt-Poincar′e方法构造任意高阶的分析解.特别地,本文构造的分析解在参考轨道偏心率为零时,可退化描述圆参考轨道对应的周期构型.最后,为了验证分析解的有效性,计算了分析解对应的收敛域.

High-order analytical solutions of the equations of relative motion with elliptic reference orbit

The study of the dynamics of relative motion plays an important role in the application of formation flying. For a more accurate configuration, much less fuel consumption is required for station-keeping. Traditionally, the configuration is designed based on the periodic solutions of the linear equation of relative motion called C–W equation for circular reference orbit or Lawden equation for elliptic reference orbit. However, the linear solutions are suitable for the configuration with small amplitudes. In this paper, based on the nonlinear equations of relative motion with elliptic reference orbit, the solution is expanded as formal series of the eccentricity of the reference orbit, in-plane amplitude and out-of-plane amplitude, then,taking the Lawden periodic solution as starting point, the high-order analytical solution is constructed by means of the Lindstedt–Poincar′e method. In particular, when the eccentricity is zero, the analytical solution constructed in this paper could be reduced to express the relative motion with circular reference orbit. At last, the practical convergence of the analytical solution is considered in order to check its validity and applicability.