Uncertainty Propagation and Bayesian Fusion on Unimodular Lie Groups from a Parametric Perspective
CoRR(2024)
摘要
We address the problem of uncertainty propagation and Bayesian fusion on
unimodular Lie groups. Starting from a stochastic differential equation (SDE)
defined on Lie groups via Mckean-Gangolli injection, we first convert it to a
parametric SDE in exponential coordinates. The coefficient transform method for
the conversion is stated for both Ito's and Stratonovich's interpretation of
the SDE. Then we derive a mean and covariance fitting formula for probability
distributions on Lie groups defined by a concentrated distribution on the
exponential coordinate. It is used to derive the mean and covariance
propagation equations for the SDE defined by injection, which coincides with
the result derived from a Fokker-Planck equation in previous work. We also
propose a simple modification to the update step of Kalman filters using the
fitting formula, which improves the fusion accuracy with moderate computation
time.
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