Uncertainty Propagation and Bayesian Fusion on Unimodular Lie Groups from a Parametric Perspective
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912263986741248 |
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| author | Ye, Jikai Chirikjian, Gregory S. |
| author_facet | Ye, Jikai Chirikjian, Gregory S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03425 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uncertainty Propagation and Bayesian Fusion on Unimodular Lie Groups from a Parametric Perspective Ye, Jikai Chirikjian, Gregory S. Systems and Control Statistics Theory 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. |
| title | Uncertainty Propagation and Bayesian Fusion on Unimodular Lie Groups from a Parametric Perspective |
| topic | Systems and Control Statistics Theory |
| url | https://arxiv.org/abs/2401.03425 |