On the Eigenstructure of Rotations and Poses: Commonalities and Peculiarities

TL;DR

This paper characterizes the eigenstructure of rotations and poses in matrix Lie groups, revealing their similarities and differences, which aids in understanding their properties and practical computations in various scientific fields.

Contribution

It provides a complete eigenstructure characterization of rotations and poses in SO(3), SE(3), and Ad(SE(3)), highlighting their similarities and differences.

Findings

Eigendecomposition reveals pose representations are similar to rotations

Structure varies depending on translation and rotation relationships

Facilitates understanding of minimal polynomials and Jacobians

Abstract

Rotations and poses are ubiquitous throughout many fields of science and engineering such as robotics, aerospace, computer vision and graphics. In this paper, we provide a complete characterization of rotations and poses in terms of the eigenstructure of their matrix Lie group representations, SO(3), SE(3) and Ad(SE(3)). An eigendecomposition of the pose representations reveals that they can be cast into a form very similar to that of rotations although the structure of the former can vary depending on the relative nature of the translation and rotation involved. Understanding the eigenstructure of these important quantities has merit in and of itself but it is also essential to appreciating such practical results as the minimal polynomial for rotations and poses and the calculation of Jacobians; moreover, we can speak of a principal-axis pose in much the same manner that we can of a principal-axis rotation.