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

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.