  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.
