Gauss von Mises Distribution for Orbital State Uncertainty
Find Innovative SolutionsGenerate Solutions
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 inaccurate tracking and collision predictions.
Innovation Solution
The implementation 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 support advanced space surveillance functions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If Gaussian assumptions and linear models are used for uncertainty characterization, then computational simplicity is maintained, but tracking accuracy and collision prediction reliability deteriorate due to inability to represent non-linear and non-Gaussian dynamics
Solution Approach 1:
The patent transforms the uncertainty representation by changing the parameter space from standard Gaussian covariance to a cylindrical manifold parameterization using von Mises distributions for angular coordinates. This allows the model to capture non-Gaussian characteristics while maintaining computational tractability through the specific mathematical structure of the cylindrical representation.
Solution Approach 2:
The patent introduces curvature by representing the state space as a cylindrical manifold rather than a flat Cartesian space. The cylindrical geometry naturally accommodates the periodic nature of orbital elements (right ascension, argument of perigee, mean anomaly) and allows uncertainty to be represented with appropriate angular periodicity, improving physical realism.
2Measurement precision
If cylindrical manifold representation with von Mises distributions is implemented, then uncertainty characterization accuracy improves, but computational complexity increases
Solution Approach 1:
The patent changes the distributional assumptions from Gaussian to von Mises for angular coordinates, which are specifically suited for circular data. This parameter change improves precision by respecting the periodic nature of orbital elements while the mathematical properties of von Mises distributions maintain computational feasibility through closed-form solutions for many operations.
Solution Approach 2:
The patent segments the six-dimensional orbital element space by treating the angular coordinates (right ascension, argument of perigee, mean anomaly) separately from the non-angular coordinates (semi-major axis, eccentricity, inclination). This segmentation allows application of von Mises distributions only where needed (angular dimensions) while using standard Gaussian representations for the remaining dimensions, reducing overall computational burden.
3Productivity
If standard Gaussian filtering is used, then computational efficiency is maintained, but the ability to handle non-Gaussian error terms and non-linear process models deteriorates
Solution Approach 1:
The patent modifies the filtering approach by changing the underlying distributional assumptions to accommodate non-Gaussian error terms. The von Mises distribution for angular coordinates and the cylindrical manifold representation enable the filter to naturally handle non-Gaussian characteristics while preserving the recursive efficiency of sequential filtering through the specific mathematical structure.
Solution Approach 2:
The patent enhances adaptability to non-linear dynamics by representing the state space with cylindrical geometry that naturally accommodates periodic orbital elements. This curved space representation allows the filter to properly propagate uncertainty through non-linear transformations without the approximations required in flat Cartesian spaces, improving both accuracy and physical consistency.
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.