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Stochastic observers on Lie groups: a tutorial

Abstract : In this tutorial paper, we discuss the design of geometric observers on Lie groups in the presence of noise. First we review Lie groups, and the mathematical definition of noises on Lie groups, both in discrete and continuous time. In particular, we discuss the Ito-Stratonovich dilemma. Then, we review the recently introduced notion of group affine systems on Lie groups. For those systems, we discuss how using the machinery of Harris chains, (almost) globally convergent deterministic observers might be shown to possess stochastic properties in the presence of noise. We also discuss the design of (invariant) extended Kalman filters (IEKF), and we recall the main result, i.e., the Riccati equation computed by the filter to tune its gains has the remarkable property that the Jacobians (A,C) with respect to the system's dynamics and output map are independent of the followed trajectory, whereas the noise covariance matrices (Q,R) that appear in the Riccati equation may depend on the followed trajectory. Owing to this partial independence, some local deterministic convergence properties of the IEKF for group-affine systems on Lie groups may be proved under standard observability conditions.
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https://hal-mines-paristech.archives-ouvertes.fr/hal-01826025
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Submitted on : Thursday, June 28, 2018 - 9:36:05 PM
Last modification on : Wednesday, November 17, 2021 - 12:33:04 PM
Long-term archiving on: : Thursday, September 27, 2018 - 7:46:55 AM

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  • HAL Id : hal-01826025, version 1

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Axel Barrau, Silvere Bonnabel. Stochastic observers on Lie groups: a tutorial. 2018. ⟨hal-01826025v1⟩

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