This paper presents a novel approach for propagating uncertainties in dynamical systems building on high-order Taylor expansions of the flow and moment-generating functions (MGFs). Unlike prior methods that focus on Gaussian distributions, our approach leverages the relationship between MGFs and distribution moments to extend high-order uncertainty propagation techniques to non-Gaussian scenarios. This significantly broadens the applicability of these methods to a wider range of problems and uncertainty types. High-order moment computations are performed one-off and symbolically, reducing the computational burden of the technique to the calculation of Taylor series coefficients around a nominal trajectory, achieved by efficiently integrating the system's variational equations. Furthermore, the use of the proposed approach in combination with event transition tensors, allows for accurate propagation of uncertainties at specific events, such as the landing surface of a celestial body, the crossing of a predefined Poincar\'e section, or the trigger of an arbitrary event during the propagation. Via numerical simulations we demonstrate the effectiveness of our method in various astrodynamics applications, including the unperturbed and perturbed two-body problem, and the circular restricted three-body problem, showing that it accurately propagates non-Gaussian uncertainties both at future times and at event manifolds.

翻译：本文提出了一种基于流和矩生成函数（MGFs）的高阶泰勒展开，用于传播动力系统中不确定性的新方法。与先前专注于高斯分布的方法不同，我们的方法利用矩生成函数与分布矩之间的关系，将高阶不确定性传播技术扩展到非高斯情形。这显著拓宽了这些方法对更广泛问题和不确定性类型的适用性。高阶矩计算是一次性且符号化执行的，从而将该技术的计算负担简化为围绕标称轨迹的泰勒级数系数的计算，这是通过高效积分系统的变分方程实现的。此外，将所提出的方法与事件转移张量结合使用，可以在特定事件处准确传播不确定性，例如天体着陆表面、预定庞加莱截面的穿越，或传播过程中任意事件的触发。通过数值模拟，我们在多个天体动力学应用中证明了该方法的有效性，包括无摄动和有摄动二体问题以及圆形限制性三体问题，结果表明它能够在未来时刻和事件流形上准确传播非高斯不确定性。