In GNSS-denied and poorly-known environments, reliable autonomous navigation is a major challenge, as conventional data fusion algorithms require an extensive knowledge of their surroundings to accurately estimate the vehicle state. To address this issue, we propose to use an adaptive Gaussian process regression to model an approximation of the environment solely based on scarce and noisy samples. This paper takes advantage of the flexibility of Gaussian processes to dynamically model the underlying terrain by adapting the process to the relevant data at each step. To this end, we propose to locally fit the Gaussian process and perform regression by using only a subset of data points selected according to a proximity criterion. The developed method employs a regularised particle filter to effectively estimate the system state using the output of the regression. By integrating Gaussian process-based terrain predictions, the particle filter can effectively compensate for the lack of precise terrain information, thus enhancing navigation performance in GNSS-denied scenarios. To evaluate the effectiveness of the proposed approach, simulations were performed in terrain-aided navigation of an unmanned aerial vehicle. Comparative analysis with existing navigation methods illustrates the superiority of the proposed approach in terms of accuracy and robustness.

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.