Establishing long-term relative bounded motion between orbits in perturbed dynamics is a key challenge in astrodynamics to enable cluster flight with minimum propellant expenditure. In this work, we present an approach that allows for the design of long-term relative bounded motion considering a zonal gravitational model. Entire sets of orbits are obtained via high-order Taylor expansions of Poincarè return maps about reference fixed points. The high-order normal form algorithm is used to determine a change in expansion variables of the map into normal form space, in which the phase space behavior is circular and can be easily parameterized by action–angle coordinates. The action–angle representation of the normal form coordinates is then used to parameterize the original Poincarè return map and average it over a full phase space revolution by a path integral along the angle parameterization. As a result, the averaged nodal period and drift in the ascending node are obtained, for which the bounded motion conditions are straightforwardly imposed. Sets of highly accurate bounded orbits are obtained, extending over several thousand kilometers, and valid for decades.

中文翻译：

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基于微分代数的范式方法在地球纬向问题中的有界运动设计

在扰动动力学中建立轨道之间的长期相对有界运动是天体动力学中的一个关键挑战，以实现以最少的推进剂消耗实现集群飞行。在这项工作中，我们提出了一种方法，该方法允许设计考虑到纬向引力模型的长期相对有界运动。整个轨道集是通过关于参考不动点的庞加莱返回图的高阶泰勒展开获得的。高阶范式算法用于确定映射扩展变量到范式空间的变化，其中相空间行为是圆形的，可以很容易地通过动作-角度坐标进行参数化。然后使用正常形式坐标的动作角度表示来参数化原始庞加莱返回图，并通过沿角度参数化的路径积分在整个相空间旋转上对其进行平均。结果，获得了升交点的平均节点周期和漂移，为此直接施加了有界运动条件。获得了一组高度精确的有界轨道，延伸超过数千公里，并且有效数十年。