Nonlinear Mapping of Gaussian Statistics: Theory and Applications to Spacecraft Trajectory Design
Nonlinear Mapping of Gaussian Statistics: Theory and Applications to Spacecraft Trajectory Design
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DOI：
10.2514/1.20177
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发表时间：
2006-11
影响因子：
2.6
通讯作者：
R. Park;D. Scheeres
中科院分区：
文献类型：
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作者：
R. Park;D. Scheeres
通过求解Fokker-Planck方程，讨论了航天器轨道不确定性的非线性传播问题。我们首先讨论了具有高斯边界条件的确定性系统的Fokker-Planck方程的解。其次，利用高阶泰勒级数方法得到了非线性轨道解的解析表达式，讨论了解的收敛区域，并将结果应用于航天器。这些应用包括均值和协方差矩阵的非线性传播、统计校正轨迹的设计和非线性统计目标。以二体问题和Hill三体问题为例，考虑了实际的初始不确定性模型。结果表明，弹道不确定性的非线性映射可以用解析形式来近似，并且存在一个线性方法找不到的最优修正机动位置。
This paper discusses the nonlinear propagation of spacecraft trajectory uncertainties via solutions of the Fokker– Planck equation. We first discuss the solutions of the Fokker–Planck equation for a deterministic system with a Gaussian boundary condition. Next, we derive an analytic expression of a nonlinear trajectory solution using a higher-order Taylor series approach, discuss the region of convergence for the solutions, and apply the result to spacecraft applications. Such applications consist of nonlinear propagation of the mean and covariance matrix, design of statistically correct trajectories, and nonlinear statistical targeting. The two-body and Hill three-body problems are chosen as examples and realistic initial uncertainty models are considered. The results show that the nonlinear map of the trajectory uncertainties can be approximated in an analytic form, and there exists an optimal place to perform a correction maneuver, which is not found using the linear method.