Kasdin N Jeremy, Gurfil Pini
Mechanical and Aerospace Engineering Department, Princeton University, Princeton, NJ 08544, USA. jkasdin,
Ann N Y Acad Sci. 2004 May;1017:138-57. doi: 10.1196/annals.1311.009.
This paper presents a Hamiltonian approach to modelling relative spacecraft motion based on derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations while allowing us to obtain closed-form solutions to the relative motion problem. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton-Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, they are called epicyclic elements. The influence of higher order terms and perturbations, such as the oblateness of the Earth, are incorporated into the analysis by a variation of parameters procedure. Closed-form solutions for J(2-) and J(4-)invariant orbits and for periodic high-order unperturbed relative motion, in terms of the relative motion elements only, are obtained.
本文提出了一种基于相对状态空间动力学的正则坐标推导来对航天器相对运动进行建模的哈密顿方法。哈密顿公式便于对高阶项和轨道摄动进行建模，同时使我们能够获得相对运动问题的闭式解。首先，将哈密顿量分为线性项和高阶项。通过分离求解线性部分的哈密顿 - 雅可比方程，得到相对运动的新常数，它们被称为本轮元素。通过参数变分法将高阶项和摄动（如地球扁率）的影响纳入分析。仅根据相对运动元素，获得了J(2-)和J(4-)不变轨道以及周期性高阶无摄相对运动的闭式解。
