LAUSR

In this letter, we propose a novel method to merge a Gaussian mixture on matrix Lie groups and present its application for a simultaneous localization and mapping problem with symmetric objects. The key idea is to predetermine the weighted mean called a midway point and merge Gaussian mixture components at the associated tangent space. Through this rule, the covariance matrix captures the original density more accurately, and the need for the back-projection is spared when compared to the conventional merge. We highlight the midway-merge by numerically evaluating dissimilarity metrics of density functions before and after the merge on the rotational group. Furthermore, we experimentally discover that the rotational error of symmetric objects follows heavy-tailed behavior and formulate the Gaussian sum filter to model it by a Gaussian mixture noise. The effectiveness of our approach is validated through virtual and real-world datasets.