Gaussian state uncertainty, nonlinear mapping, state transition tensor, STT, Fokker-Planck equation, probability density function, two body problem, three body problem, Hohmann transfer orbit, mathematical analysis, Gaussian distribution, probability, Monte Carlo method
University of Michigan Department of Aerospace Engineering
University of Michigan Department of Aerospace Engineering
著者所属（英）
University of Michigan Department of Aerospace Engineering
University of Michigan Department of Aerospace Engineering
出版者
宇宙航空研究開発機構宇宙科学研究本部
出版者（英）
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency (JAXA/ISAS)
雑誌名
第15回アストロダイナミクスシンポジウム講演後刷り集 2005
雑誌名(英)
Proceedings of 15th Workshop on JAXA Astrodynamics and Flight Mechanics
ページ
364 - 377
発行年
2006-03
抄録（英）
This paper presents the nonlinear propagation of orbit uncertainties via solutions of the Fokker-Planck equation. We first derive an analytic expression of a nonlinear trajectory solution by incorporating the higher order Taylor series that describes the localized nonlinear motion, and by solving for the higher order state solution as functions of initial conditions. A systematic way to define the sufficient order of higher order solutions that defines the localized nonlinear motion is also presented. We then solve the Fokker-Planck equation for a deterministic system with a Gaussian boundary condition and discuss how the propagated phase volume and Gaussian statistics characterize the spacecraft orbit uncertainties. The phase volume approach presents the integral invariance of the probability density function, and thus, the probability of the initial confidence region remains the same as it is mapped over time. The statistical method shows that the propagated trajectory uncertainties remain no longer Gaussian in general; however, we can still approximate the first two moments (mean and covariance matrix) to define the true confidence region of the nonlinear system. We utilize the higher order Taylor series solutions to approximate the true nonlinear trajectory statistics and compare with the conventional linear theory and Monte-Carlo simulations to explain its significance. The result shows that the nonlinear solution provides a superior result than the linear solution when the system is under a strong nonlinearity or mapped over a long time span. Moreover, the higher order Taylor series approach becomes essentially the same as the Monte-Carlo result when sufficient order of Taylor series is considered. The two-body and Hill three-body problems are chosen as examples and realistic initial uncertainty models are considered.
Nonlinear mapping of Gaussian state uncertainties
