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Continuous-Discrete Extended Kalman Filter on Matrix Lie Groups Using Concentrated Gaussian Distributions

Abstract : In this paper we generalize the Continuous-Discrete Extended Kalman Filter (CD-EKF) to the case where the state and the observations evolve on connected unimodular matrix Lie groups. We propose a new assumed density filter called Continuous-Discrete Extended Kalman Filter on Lie Groups (CD-LG-EKF). It is built upon a geometrically meaningful modeling of the concentrated Gaussian distribution on Lie Groups. Such a distribution is parametrized by a mean and a co-variance matrix defined on the Lie group and in its associated Lie algebra respectively. Our formalism yields tractable equations for both non-linear continuous time propagation and discrete update of the distribution parameters under the assumption that the posterior distribution of the state is a concentrated Gaussian. As a side effect, we contribute to the derivation of the first and second order differential of the matrix Lie group logarithm using left connection. We also show that the CD-LG-EKF reduces to the usual CD-EKF if the state and the observations evolve on Euclidean spaces. Our approach leads to a systematic methodology for the design of filters, which is illustrated by the application to a camera pose filtering problem with observations on Lie group. In this application, the CD-LG-EKF significantly outperforms two constrained non-linear filters (one based on a linearization technique and the other on the unscented transform) applied on the embedding space of the Lie group.
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Contributor : Guillaume Bourmaud Connect in order to contact the contributor
Submitted on : Tuesday, May 3, 2016 - 8:40:37 PM
Last modification on : Tuesday, October 19, 2021 - 11:05:49 PM
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Guillaume Bourmaud, Rémi Mégret, Marc Arnaudon, Audrey Giremus. Continuous-Discrete Extended Kalman Filter on Matrix Lie Groups Using Concentrated Gaussian Distributions. Journal of Mathematical Imaging and Vision, Springer Verlag, 2015, 51 (1), ⟨10.1007/s10851-014-0517-0⟩. ⟨hal-01311170⟩



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