Satellite Positioning with Koopman Filtering for Aging Data
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Satellite positioning systems face challenges in maintaining accurate knowledge of phase center locations over extended periods due to the growth of errors in position and time, particularly in adverse environments where frequent updates are not possible, leading to the 'age of data' problem.
Innovation Solution
Employing a dimension-limited Koopman processing technique with dynamic mode decomposition to separate deterministic components from chaotic residues, followed by adaptive filtering to update the state vector, thereby preserving positioning accuracy.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If Extended Kalman Filter is used to process measurement data, then positioning accuracy is improved locally, but error grows quadratically over time leading to the age of data problem
Solution Approach 1:
The patent transforms the nonlinear satellite orbital dynamics into a higher-dimensional linear state space using Koopman operator theory. By lifting the state vector into an extended dimensionality that includes polynomial terms of the original state variables, the system achieves global linearization, allowing the use of linear Kalman filtering while maintaining accuracy over extended periods without the quadratic error growth characteristic of local linearization methods.
Solution Approach 2:
The patent changes the mathematical representation parameters by transitioning from the original nonlinear orbital elements to a transformed state space defined by the Koopman operator. This parameter transformation converts the inherently nonlinear satellite motion equations into a linear form, enabling the application of linear filtering techniques that maintain positioning accuracy over longer time durations.
2Adaptability or versatility
If ground assets are destroyed or disabled, then satellite positioning system must operate independently, but error propagation increases without frequent updates
Solution Approach 1:
The patent performs preliminary linearization of the nonlinear orbital dynamics before operation by constructing the Koopman operator and extended state space in advance. This preliminary transformation creates a globally valid linear model that can be used for extended periods without ground updates, allowing the satellite positioning system to maintain reliability during independent operation after ground assets are compromised.
Solution Approach 2:
By elevating the state representation to a higher-dimensional space that captures global orbital dynamics, the system gains the ability to operate reliably independently for extended periods. The extended state space includes terms that account for long-term orbital evolution, enabling accurate positioning even when frequent ground updates are unavailable.
3Measurement precision
If dimensionality of measurement data is increased for Koopman processing, then global performance and accuracy are improved, but computational complexity increases
Solution Approach 1:
The patent segments the computational task by separating the offline construction of the Koopman operator (which is performed once to establish the linearized state space) from the online filtering operation (which uses efficient linear Kalman filtering). This segmentation allows the system to achieve global accuracy through dimensionality increase while managing computational complexity by performing the heavy lifting offline and using lightweight operations during real-time positioning.
Solution Approach 2:
The computationally intensive dimensionality increase and linearization operations are performed preliminarily offline to construct the Koopman operator and extended state space. Once this preliminary action is complete, the system can maintain global positioning accuracy using computationally efficient linear filtering operations, thereby reducing the online computational burden while preserving the benefits of higher-dimensional processing.
Data Source
Figure 1
Figure 2
Figure 3
AI summary
Embodiments regard improved satellite position determination with age of measurement data. A method includes receiving measurement data from a satellite, in a first iteration, increasing a dimensionality of the measurement data to a specified number of dimensions resulting in N-dimensional input data, performing dynamic mode decomposition on the N-dimensional measurement data resulting in a Koopman operator and modes of the N-dimensional measurement data, adaptive filtering a time domain residue resulting in a filtered residue, and updating, based on the filtered residue and a time domain deterministic component of the N-dimensional measurement data, a state vector of an object associated with the satellite measurement data.