Uncertainty Propagation and Bayesian Fusion on Unimodular Lie Groups from a Parametric Perspective

2024/01/07 by Ye, Jikai, Chirikjian, Gregory S. · 1 citation
#FOS: Electrical engineering #FOS: Mathematics #Statistics Theory (math.ST) #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · doi:10.48550/arxiv.2401.03425

Abstract

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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• Means of Random Variables in Lie Groups

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