Invariant Extended Kalman Filter for State Estimation
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Solution Overview
Problem
Conventional particle filtering methods, such as the Rao-Blackwellized particle filter, face high computational burdens and degeneration issues when estimating the state of non-linear dynamic systems with non-Gaussian measurement noises, particularly in systems like satellite navigation where position measurements are involved.
Innovation Solution
The method employs an invariant extended Kalman filter configured under specific invariance assumptions for orientation and position variables, and orientation and velocity variables, reducing the need for calculating multiple covariances and allowing for a common Ricatti equation and Cholesky decomposition across particles, thereby lowering the computational load.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional particle filtering (bootstrap) is applied to estimate the state of a non-linear dynamic system with non-Gaussian measurement noises, then the estimation can handle non-Gaussian noise, but the computational burden is high and requires a large number of particles
Solution Approach 1:
The state vector is segmented into two parts: parameters to be estimated (e.g., bias terms) and auxiliary variables (e.g., position, velocity, orientation). Particle filtering is applied only to the parameters while the auxiliary variables are processed via invariant EKF, reducing the number of particles needed and computational burden.
Solution Approach 2:
An invariant extended Kalman filter is introduced as an intermediary to process the auxiliary variables (position, velocity, orientation) that satisfy specific invariance properties. This mediator handles the computationally intensive parts efficiently, allowing particle filtering to focus only on the essential parameters.
2Productivity
If Rao-Blackwellized particle filter is applied to reduce the number of particles, then computational burden is reduced, but matrix inversion operations increase computation burden for each particle processed
Solution Approach 1:
The invariant EKF transforms the state representation and error variables to exploit invariance properties of orientation and velocity. This parameter transformation enables the use of a common Ricatti equation for all particles, eliminating the need for individual matrix inversions per particle and reducing computation complexity.
3Adaptability or versatility
If standard extended Kalman filter is used for non-linear systems, then the filtering can be applied to non-linear equations, but it proves inefficient when measurement noise is non-Gaussian causing divergence of state estimation
Solution Approach 1:
The state is segmented into variables with specific invariance properties (orientation, velocity) processed by invariant EKF and general parameters processed by particle filtering. This segmentation allows the system to handle non-Gaussian noise robustly while maintaining applicability to non-linear systems.
Data Source
AI summary
A method for estimating the state of a moving system, using a particle filter. The method implements on one reference particle an estimation of the state and includes a first set of variables and a second set of variables. A random mutation, by the particle filter, of the first set of variables is performed followed by parameterization of an extended Kalman filter using the mutated first set of variables. The Kalman filter produces a new particle using the second set of variables and data measured by at least one sensor. The second set of variables includes at least the variables: orientation, speed and position of the moving system. The Kalman filter is configured assuming that the pair of variables orientation and position has a property of invariance upon a rotation or translation and their orientation and speed has a property of invariance on application of a rotation or translation.