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首页> 外文期刊>Celestial Mechanics and Dynamical Astronomy >Approximate analysis for relative motion of satellite formation flying in elliptical orbits
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Approximate analysis for relative motion of satellite formation flying in elliptical orbits

机译：椭圆轨道上卫星编队飞行相对运动的近似分析

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This paper studies the relative motion of satellite formation flying in arbitrary elliptical orbits with no perturbation. The trajectories of the leader and follower satellites are projected onto the celestial sphere. These two projections and celestial equator intersect each other to form a spherical triangle, in which the vertex angles and arc-distances are used to describe the relative motion equations. This method is entitled the reference orbital element approach. Here the dimensionless distance is defined as the ratio of the maximal distance between the leader and follower satellites to the semi-major axis of the leader satellite. In close formations, this dimensionless distance, as well as some vertex angles and arc-distances of this spherical triangle, and the orbital element differences are small quantities. A series of order-of-magnitude analyses about these quantities are conducted. Consequently, the relative motion equations are approximated by expansions truncated to the second order, i.e. square of the dimensionless distance. In order to study the problem of periodicity of relative motion, the semi-major axis of the follower is expanded as Taylor series around that of the leader, by regarding relative position and velocity as small quantities. Using this expansion, it is proved that the periodicity condition derived from Lawden’s equations is equivalent to the condition that the Taylor series of order one is zero. The first-order relative motion equations, simplified from the second-order ones, possess the same forms as the periodic solutions of Lawden’s equations. It is presented that the latter are further first-order approximations to the former; and moreover, compared with the latter more suitable to research spacecraft rendezvous and docking, the former are more suitable to research relative orbit configurations. The first-order relative motion equations are expanded as trigonometric series with eccentric anomaly as the angle variable. Except the terms of order one, the trigonometric series’ amplitudes are geometric series, and corresponding phases are constant both in the radial and in-track directions. When the trajectory of the in-plane relative motion is similar to an ellipse, a method to seek this ellipse is presented. The advantage of this method is shown by an example.
机译：本文研究了在无扰动的任意椭圆轨道上飞行的卫星编队的相对运动。引导卫星和跟随卫星的轨迹被投影到天球上。这两个投影与天赤道相交形成一个球面三角形，其中顶点角和弧距用于描述相对运动方程。该方法称为参考轨道元素方法。此处，无量纲距离定义为引导者卫星和跟随者卫星之间的最大距离与引导者卫星的半长轴之比。在紧密的地层中，该无量纲的距离以及该球形三角形的一些顶点角和弧距以及轨道元素的差异很小。进行了有关这些量的一系列量级分析。因此，相对运动方程通过被截断为二阶即无量纲距离的平方的展开来近似。为了研究相对运动的周期性问题，通过将相对位置和速度视为少量，将跟随器的半长轴扩展为围绕引导者的泰勒级数。利用这种展开，证明了从劳登方程导出的周期性条件等于一阶泰勒级数为零的条件。一阶相对运动方程（从二阶相对运动方程简化）具有与Lawden方程的周期解相同的形式。结果表明，后者是前者的进一步一阶近似。而且，与后者更适合研究航天器的交会对接相比，前者更适合研究相对轨道构型。一阶相对运动方程扩展为三角序列，偏心距为角度变量。除了一阶条件外，三角序列的振幅是几何序列，并且相应的相位在径向和轨道内方向上都是恒定的。当平面内相对运动的轨迹类似于椭圆时，提出了寻找该椭圆的方法。实例显示了此方法的优点。

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