Rotary Positional Embeddings (RoPE) have demonstrated exceptional performance as a positional encoding method, consistently outperforming their baselines. While recent work has sought to extend RoPE to higher-dimensional inputs, many such extensions are non-commutative, thereby forfeiting RoPE's shift-equivariance property. Spherical RoPE is one such non-commutative variant, motivated by the idea of rotating embedding vectors on spheres rather than circles. However, spherical rotations are inherently non-commutative, making the choice of rotation sequence ambiguous. In this work, we explore a quaternion-based approach -- Quaternion Rotary Embeddings (QuatRo) -- in place of Euler angles, leveraging quaternions' ability to represent 3D rotations to parameterize the axes of rotation. We show Mixed RoPE and Spherical RoPE to be special cases of QuatRo. Further, we propose a generalization of QuatRo to Clifford Algebraic Rotary Embeddings (CARE) using geometric algebra. Viewing quaternions as the even subalgebra of Cl(3,0,0), we extend the notion of rotary embeddings from quaternions to Clifford rotors acting on multivectors. This formulation enables two key generalizations: (1) extending rotary embeddings to arbitrary dimensions, and (2) encoding positional information in multivectors of multiple grades, not just vectors. We present preliminary experiments comparing spherical, quaternion, and Clifford-based rotary embeddings.

Global registration is a fundamental task that estimates the relative pose between two viewpoints of 3D point clouds. However, there are two issues that degrade the performance of global registration in LiDAR SLAM: one is the sparsity issue and the other is degeneracy. The sparsity issue is caused by the sparse characteristics of the 3D point cloud measurements in a mechanically spinning LiDAR sensor. The degeneracy issue sometimes occurs because the outlier-rejection methods reject too many correspondences, leaving less than three inliers. These two issues have become more severe as the pose discrepancy between the two viewpoints of 3D point clouds becomes greater. To tackle these problems, we propose a robust global registration framework, called \textit{Quatro++}. Extending our previous work that solely focused on the global registration itself, we address the robust global registration in terms of the loop closing in LiDAR SLAM. To this end, ground segmentation is exploited to achieve robust global registration. Through the experiments, we demonstrate that our proposed method shows a higher success rate than the state-of-the-art global registration methods, overcoming the sparsity and degeneracy issues. In addition, we show that ground segmentation significantly helps to increase the success rate for the ground vehicles. Finally, we apply our proposed method to the loop closing module in LiDAR SLAM and confirm that the quality of the loop constraints is improved, showing more precise mapping results. Therefore, the experimental evidence corroborated the suitability of our method as an initial alignment in the loop closing. Our code is available at https://quatro-plusplus.github.io.