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

Jikai Ye, Gregory S. Chirikjian

TL;DR

This work tackles uncertainty propagation and Bayesian fusion for states evolving on unimodular Lie groups by linking non-parametric SDEs (McKean-Gangolli injection) to parametric SDEs in exponential coordinates. It introduces a mean-covariance fitting theorem for concentrated distributions on the exponential domain, enabling explicit propagation and fusion equations on the group, and it proposes a simple, computation-friendly modification to Kalman-filter updates that improves fusion accuracy. The approach is demonstrated through attitude estimation experiments on SO(3), showing improved posterior mean accuracy over standard Euclidean updates. The results provide a principled, SDE-based perspective on propagation and fusion on manifolds, with potential extensions to multimodal distributions and convergence analysis.

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.

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

TL;DR

This work tackles uncertainty propagation and Bayesian fusion for states evolving on unimodular Lie groups by linking non-parametric SDEs (McKean-Gangolli injection) to parametric SDEs in exponential coordinates. It introduces a mean-covariance fitting theorem for concentrated distributions on the exponential domain, enabling explicit propagation and fusion equations on the group, and it proposes a simple, computation-friendly modification to Kalman-filter updates that improves fusion accuracy. The approach is demonstrated through attitude estimation experiments on SO(3), showing improved posterior mean accuracy over standard Euclidean updates. The results provide a principled, SDE-based perspective on propagation and fusion on manifolds, with potential extensions to multimodal distributions and convergence analysis.

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.
Paper Structure (20 sections, 11 theorems, 52 equations, 4 figures)

This paper contains 20 sections, 11 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Stochastic Differential Equations on Lie Groups
  4. Propagation and Fusion on Lie Groups
  5. Mean and Covariance Fitting for Concentrated Distribution on Exponential Coorinates
  6. Propagation Equations
  7. Bayesian Fusion
  8. General Method
  9. Case 1: $G_2=\mathbb{R}^M$
  10. Case 2: $G_1=G_2$
  11. Numerical Experiments
  12. Observation Model Description
  13. 3D attitude estimation with gravitational and magnetic measurement
  14. 3D attitude estimation with state measurement
  15. Sampling and Evaluation Metric
  16. ...and 5 more sections

Key Result

Theorem 1

When the group is $SO(3)$ and the distance is chosen as $d(\mu,g)=|\log^{\vee}(\mu^{-1}g)|_2$, the Fréchet mean and the group-theoretic mean are the same.

Figures (4)

  • Figure 1: The plot of $c_1(\mu)$ for observation model 1 (on Euclidean space).
  • Figure 2: The plot of $c_2(\mu)$ for observation model 1 (on Euclidean space).
  • Figure 3: The plot of $c_1(\mu)$ for observation model 2 (on Lie group).
  • Figure 4: The plot of $c_2(\mu)$ for observation model 1 (on Lie group).

Theorems & Definitions (22)

  • Definition 1
  • Theorem 1
  • proof
  • Definition 2
  • Definition 3
  • Theorem 2
  • proof
  • Lemma 2.1
  • proof
  • Theorem 3
  • ...and 12 more