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We address the problem of uncertainty propagation and Bayesian fusion onunimodular Lie groups. Starting from a stochastic differential equation (SDE)defined on Lie groups via Mckean-Gangolli injection, we first convert it to aparametric SDE in exponential coordinates. The coefficient transform method forthe conversion is stated for both Ito's and Stratonovich's interpretation ofthe SDE. Then we derive a mean and covariance fitting formula for probabilitydistributions on Lie groups defined by a concentrated distribution on theexponential coordinate. It is used to derive the mean and covariancepropagation equations for the SDE defined by injection, which coincides withthe result derived from a Fokker-Planck equation in previous work. We alsopropose a simple modification to the update step of Kalman filters using thefitting formula, which improves the fusion accuracy with moderate computationtime.

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