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Special Orthogonal Group SO(3), Euler Angles, Angle-axis, Rodriguez Vector and Unit-Quaternion: Overview, Mapping and Challenges

Hashim A. Hashim

TL;DR

An overview of the rotation matrix, attitude kinematics and parameterization is given and the main weaknesses of attitude parameterization using Euler angles, angle-axis parameterization, Rodriguez vector, and unit-quaternion are illustrated.

Abstract

The attitude of a rigid-body in the three dimensional space has a unique and global definition on the Special Orthogonal Group SO (3). This paper gives an overview of the rotation matrix, attitude kinematics and parameterization. The four most frequently used methods of attitude representations are discussed with detailed derivations, namely Euler angles, angle-axis parameterization, Rodriguez vector, and unit-quaternion. The mapping from one representation to others including SO (3) is given. Also, important results which could be useful for the process of filter and/or control design are given. The main weaknesses of attitude parameterization using Euler angles, angle-axis parameterization, Rodriguez vector, and unit-quaternion are illustrated. Keywords: Special Orthogonal Group 3, Euler angles, Angle-axis, Rodriguez Vector, Unit-quaternion, SO(3), Mapping, Parameterization, Attitude, Control, Filter, Observer, Estimator, Rotation, Rotational matrix, Transformation matrix, Orientation, Transformation, Roll, Pitch, Yaw, Quad-rotor, Unmanned aerial vehicle, Robot, spacecraft, satellite, UAV, Underwater vehicle, autonomous, system, Pose, literature review, survey, overview, comparison, comparative study, body frame, identity, origin, dynamics, kinematics, Lie group, inertial frame, zero, filter, control, estimate, observation, measurement, 3D, three dimensional space, advantage, disadvantage.

Special Orthogonal Group SO(3), Euler Angles, Angle-axis, Rodriguez Vector and Unit-Quaternion: Overview, Mapping and Challenges

TL;DR

An overview of the rotation matrix, attitude kinematics and parameterization is given and the main weaknesses of attitude parameterization using Euler angles, angle-axis parameterization, Rodriguez vector, and unit-quaternion are illustrated.

Abstract

The attitude of a rigid-body in the three dimensional space has a unique and global definition on the Special Orthogonal Group SO (3). This paper gives an overview of the rotation matrix, attitude kinematics and parameterization. The four most frequently used methods of attitude representations are discussed with detailed derivations, namely Euler angles, angle-axis parameterization, Rodriguez vector, and unit-quaternion. The mapping from one representation to others including SO (3) is given. Also, important results which could be useful for the process of filter and/or control design are given. The main weaknesses of attitude parameterization using Euler angles, angle-axis parameterization, Rodriguez vector, and unit-quaternion are illustrated. Keywords: Special Orthogonal Group 3, Euler angles, Angle-axis, Rodriguez Vector, Unit-quaternion, SO(3), Mapping, Parameterization, Attitude, Control, Filter, Observer, Estimator, Rotation, Rotational matrix, Transformation matrix, Orientation, Transformation, Roll, Pitch, Yaw, Quad-rotor, Unmanned aerial vehicle, Robot, spacecraft, satellite, UAV, Underwater vehicle, autonomous, system, Pose, literature review, survey, overview, comparison, comparative study, body frame, identity, origin, dynamics, kinematics, Lie group, inertial frame, zero, filter, control, estimate, observation, measurement, 3D, three dimensional space, advantage, disadvantage.

Paper Structure

This paper contains 42 sections, 8 theorems, 216 equations, 12 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Math Notation
  3. $\mathbb{SO}\left(3\right)$ Preliminaries
  4. Angle-axis parameterization preliminaries
  5. Rodriguez vector preliminaries
  6. Unit-quaternion preliminaries
  7. Useful math identities
  8. Special Orthogonal Group $\mathbb{SO}\left(3\right)$
  9. Attitude dynamics
  10. Normalized Euclidean distance
  11. Normalized Euclidean distance dynamics
  12. Discrete Attitude dynamics
  13. Attitude error and attitude error dynamics
  14. Euler Parameterization
  15. Attitude parameterization through Euler angles
  16. ...and 27 more sections

Key Result

Lemma 1

Let $R\in\mathbb{SO}\left(3\right)$, $u\in\mathbb{S}^{2}$ denote a unit vector such that $\left\Vert u\right\Vert =1$, and $\alpha\in\mathbb{R}$ denote the angle of rotation about $u$. Then, the following holds: Proof. Seeeq:OVERVIEW_att_ang_VEX,eq:OVERVIEW_att_ang_RI, andeq:OVERVIEW_att_ang_Vex1.

Figures (12)

  • Figure 1: The orientation of a rigid-body in body-frame relative to inertial-frame in the 3D-space hashim2018SO3Stochastic.
  • Figure 2: Graphical representation of Euler angles with respect to the reference-axis of the body-frame.
  • Figure 3: The orientation of a rigid-body in body-frame relative to inertial-frame.
  • Figure 4: Angular velocity - Example 1
  • Figure 5: Euler Angles ($\dot{R}=R\left[\Omega\right]_{\times},\space R\rightarrow\xi$) vs ($\dot{\xi}=\mathcal{J}\Omega$) - Example 1
  • ...and 7 more figures

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8