Sequential Markov Chain Monte Carlo methods on Matrix Lie Groups

Particle filters on Lie Groups represent a cutting-edge approach in nonlinear filtering and control. By generating randomly distributed particles from proposed densities, while accurately preserving rotation matrices, they offer a promising solution to nonlinear systems. However, particle filters grapple with numerous challenges such as high computational costs due to the large number of particles needed, vulnerability to particle degeneracy as well as the curse of dimensionality. To address these issues, Sequential Markov Chain Monte Carlo (SMCMC) methods aim to mitigate sensitivity to high-dimensional systems by iteratively sampling from the posterior density of the system state, gradually refining the estimation. In this paper we extend SMCMC techniques to matrix Lie groups, resulting in the Lie Groups Sequential Markov Chain Monte Carlo (LG-SMCMC) filter that circumvent the major drawbacks of particle filters. The proposed approach incorporates enhancements based on the Metropolis-Hastings algorithm, further improving the algorithms efficiency and robustness. To validate the effectiveness of these methods, the algorithms are tested on an Unmanned Aerial Vehicle (UAV) navigation scenario with challenging discrepancies in the noise tuning.