Gauss von Mises Distribution for Orbital Uncertainty Characterization
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Solution Overview
Problem
Current methods for characterizing uncertainty in space surveillance, 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, leading to inaccuracies in tracking and collision prediction.
Innovation Solution
The introduction of Gauss von Mises (GVM) distributions, which provide a more accurate characterization of uncertainty on a cylindrical manifold, enabling improved algorithms for uncertainty propagation and data fusion, particularly in the Bayesian non-linear filter, to better model the orbital state of space objects.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Device complexity
If Gaussian assumptions and linear models are used for uncertainty characterization, then computational simplicity is maintained, but accuracy in representing non-linear and non-Gaussian dynamics of space objects deteriorates
Solution Approach 1:
The patent transforms the uncertainty representation from traditional Gaussian parameters (mean and covariance) to cylindrical manifold parameters including angular coordinates and radial components. This parameter transformation enables accurate representation of non-Gaussian orbital uncertainties while maintaining computational tractability through specialized mathematical operations on the cylindrical manifold.
Solution Approach 2:
The patent applies curvature by representing uncertainty on a cylindrical manifold rather than flat Euclidean space. This geometric transformation accounts for the periodic nature of angular orbital elements and the non-linear relationship between position and velocity uncertainties, thereby accurately capturing non-linear dynamics without requiring complex non-linear filters.
2Ease of operation
If traditional Gaussian filters are used, then ease of operation is maintained, but reliability of tracking and collision prediction deteriorates due to non-Gaussian phenomena
Solution Approach 1:
The patent changes the parameterization of the filter from Gaussian moments to cylindrical manifold coordinates, allowing the filter to naturally handle non-Gaussian orbital uncertainties. This parameter transformation maintains the recursive filtering structure for ease of operation while improving reliability through accurate representation of angular coordinates and their periodic boundary conditions.
Solution Approach 2:
The patent employs curved manifold geometry to represent orbital state space, where angular coordinates are treated with proper periodicity and radial components capture non-Gaussian spread. This geometric approach maintains computational simplicity similar to Gaussian filters while significantly improving tracking reliability in the presence of non-Gaussian phenomena.
3Device complexity
If angular coordinates are treated as unbounded Cartesian coordinates, then mathematical simplicity is maintained, but accuracy of location representation deteriorates due to periodicity
Solution Approach 1:
The patent represents angular coordinates on a cylindrical manifold where the angular dimension wraps periodically. This geometric representation naturally enforces the periodicity of orbital elements (e.g., right ascension of ascending node, argument of perigee) while maintaining mathematical tractability through specialized operations on the curved manifold.
Solution Approach 2:
The patent transforms angular coordinates from unbounded Cartesian representations to bounded periodic parameters on the cylindrical manifold. This parameter change introduces proper periodicity constraints while maintaining computational simplicity through adapted mathematical operations that respect the manifold's geometry.
4Productivity
If non-linear dynamics are approximated with linear models, then computational efficiency is maintained, but accuracy of collision prediction deteriorates
Solution Approach 1:
The patent uses the intrinsic curvature of the cylindrical manifold to naturally represent non-linear orbital dynamics. By performing uncertainty propagation on the curved manifold rather than linearizing around an operating point, the method captures non-linear effects accurately while maintaining computational efficiency through the manifold's geometric structure.
Solution Approach 2:
The patent changes the state representation to cylindrical manifold coordinates where non-linear dynamics are more naturally expressed. This parameter transformation allows for more accurate collision prediction by properly representing the non-linear relationship between orbital elements and spatial position, while maintaining computational efficiency through specialized filtering operations.
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.