Attitude Control of Rigid Bodies: A Survey of Representations, Topological Obstructions, and Stabilization Techniques

TL;DR

This survey comprehensively reviews various attitude representations for rigid bodies, their topological limitations, and stabilization techniques, highlighting the advantages and drawbacks of each method in control applications.

Contribution

It provides a detailed comparison of attitude representations, explores their topological issues, and analyzes different control laws and robustness techniques in a unified framework.

Findings

Rotation matrix offers global, unique attitude representation without singularities.

Euler angles and Rodrigues parameters have kinematic and geometric singularities.

Quaternion representation can cause unwinding despite computational efficiency.

Abstract

This paper reviews the attitude control problems for rigid-body systems, starting from the attitude representation for rigid body kinematics. Highly redundant rotation matrix defines the attitude orientation globally and uniquely by 9 parameters, which is the most fundamental one, without any singularities; minimum 3-parameter Euler angles or (modified) Rodrigues parameters define the attitude orientation neither globally nor uniquely, but the former exhibits kinematical singularity and Gimbal lock, while the latter two exhibit geometrical singularity; once-redundant axis-angle or unit quaternion globally define the attitude rotation but not uniquely using 4 parameters, but the former is not appropriate to define very small or very large rotations, while the latter shows unwinding phenomenon despite of the reduced computation burden. In addition, we explore the relationships among those attitude representations, including the connections among Gimbal lock, unwinding phenomenon and a nowhere dense set of zero Lebesgue measure. Based on attitude representations, we analyze different attitude control laws, almost global control and global attitude control, nominal and general robustness, as well as the technique tools.