On the eigenstructure of rotations and poses: commonalities and peculiarities

摘要

Locating vehicles, targets and objects in three-dimensional space is key to many fields of science and engineering such as robotics, aerospace, computer vision and graphics. Rotations and poses (position plus orientation) of bodies can be expressed in a variety of ways. Rotation matrices constitute one of the classic matrix Lie groups, the special orthogonal group— <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SO</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> . Poses can likewise be represented by matrices. One such representation is embodied in a 4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo> </mml:mo> <mml:mo>×</mml:mo> <mml:mo> </mml:mo> </mml:math> 4 matrix establishing another famous matrix Lie group, the special Euclidean group— <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> . An alternative representation of pose uses 6 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo> </mml:mo> <mml:mo>×</mml:mo> <mml:mo> </mml:mo> </mml:math> 6 matrices and is referred to as the group of pose adjoints— <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>Ad</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . The eigenstructures of these representations reveal much about them from Euler’s theorem for rotations to the Mozzi–Chasles theorem for the general displacement of a rigid body. While the eigenstructure of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SO</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> has been extensively studied, those of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>Ad</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> have hardly received the same scrutiny yet their structure is much richer. Motivated by their importance in kinematics and dynamics, we provide here a complete characterization of rotations and poses in terms of the eigenstructure of their matrix Lie group representations. An eigendecomposition of pose matrices reveals that they can be cast into a form similar to that of rotations although the structure of the former can vary depending on the nature of the pose involved. In particular, the pose matrices of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>Ad</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mtext mathvariant="italic">SE</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="