Gauss von Mises Distribution for Orbital Uncertainty
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Solution Overview
Problem
Current methods for characterizing uncertainty in space surveillance tracking, such as those used in space situational awareness, are inadequate due to their reliance on Gaussian assumptions and linear models, which fail to accurately represent the non-linear and non-Gaussian dynamics of Earth-orbiting space objects, particularly when dealing with angular coordinates on a cylindrical manifold.
Innovation Solution
The introduction of Gauss von Mises (GVM) distributions, which provide a more accurate and statistically rigorous treatment of uncertainty by defining probability density functions on a cylindrical manifold, enabling improved uncertainty propagation and data fusion algorithms that better handle non-linear and non-Gaussian phenomena.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If Gaussian assumptions and linear models are used for uncertainty characterization, then computational efficiency is improved, but accuracy in representing non-linear and non-Gaussian dynamics deteriorates
Solution Approach 1:
The patent changes the fundamental parameters of the probability distribution from Gaussian to Gauss von Mises distribution, which is defined on a cylindrical manifold. This parameter change allows the model to accurately represent non-Gaussian and non-linear dynamics of space objects while maintaining computational tractability through the structured form of the von Mises distribution.
Solution Approach 2:
The patent introduces a new dimensional aspect by moving from standard Euclidean space to a cylindrical manifold. The Gauss von Mises distribution adds an angular dimension (mean longitude) that wraps around periodically, properly capturing the periodic nature of orbital dynamics and resolving the dimensional mismatch in traditional Gaussian approaches.
2Ease of operation
If traditional filtering methods are used for space object tracking, then ease of operation is maintained, but reliability in handling non-Gaussian phenomena deteriorates
Solution Approach 1:
The patent creates a copy of the Gaussian framework adapted for cylindrical manifolds. The Gauss von Mises distribution mirrors the structure of the Gaussian distribution but is defined on a different manifold (cylinder instead of Euclidean space). This copying allows existing filtering algorithms to be adapted with minimal changes while gaining reliability for non-Gaussian space object dynamics.
3Device complexity
If angular coordinates are treated as unbounded Cartesian coordinates, then computational simplicity is improved, but measurement precision of orbital state deteriorates
Solution Approach 1:
The patent applies curvature to the angular coordinate representation by using a cylindrical manifold instead of flat Euclidean space. The Gauss von Mises distribution inherently accounts for the periodic boundary conditions of angular coordinates (mean longitude), properly representing the wrapped nature of orbital angles while maintaining computational feasibility through the exponential form of the distribution.
Data Source
AI summary
Embodiments of the present invention characterizing the uncertainty of the orbital state of an Earth-orbiting space object hereof using a Gauss von Mises probability density function defined on the n+1 dimensional cylindrical manifold n×. Additionally, embodiments of the present invention can include transforming a Gauss von Mises distribution under a diffeomorphism and approximating the output as a Gauss von Mises distribution. Embodiments of the present invention can also include fusing a prior state represented by a Gauss von Mises distribution with an update report, wherein the update can be either another Gauss von Mises distribution of the same dimension as the prior or an observation related to the prior by a stochastic measurement model. A Gauss von Mises distribution can be calculated from a plurality of reports, wherein the reports are either Gauss von Mises distributions or observations related to the state space by a stochastic measurement model.