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Invariant extended Kalman filter
The invariant extended Kalman filter (IEKF)〔S. Bonnabel, Ph. Martin and E. Salaün, "Invariant Extended Kalman Filter: theory and application to a velocity-aided attitude estimation problem", 48th IEEE Conference on Decision and Control, pp. 1297–1304, 2009.〕 is a new version of the extended Kalman filter (EKF) for nonlinear systems possessing symmetries (or ''invariances''). It combines the advantages of both the EKF and the recently introduced symmetry-preserving filters. Indeed, instead of using a linear correction term based on a linear output error, it uses a geometrically adapted correction term based on an invariant output error; in the same way the gain matrix is not updated from a linear state error, but from an invariant state error. The main benefit is that the gain and covariance equations converge to constant values on a much bigger set of trajectories than equilibrium points than is the case for the EKF, which results in a better convergence of the estimation.
== Motivation ==
Most physical systems possess natural symmetries (or invariance), i.e. there exist transformations (e.g. rotations, translations, scalings) that leave the system unchanged. From mathematical and engineering viewpoint, it makes sense that a filter well-designed for the considered system should preserve the same invariance properties. The idea for the IEKF is a modification of the EKF equations to take advantage of the symmetries of the system.
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