PD control and the unwinding effect in quaternion parameterized attitude control

The unit quaternion is popular in the representation and control of rigid body attitude systems. However, it is known that two equilibrium points in the unit quaternion attitude formulation represent a single equilibrium point in a rotation matrix in SO(3). This often leads to a non-efficient control effort as the quaternion transients can settle to either of the equilibrium point regardless of the path taken or, even undergo a homoclinic orbit between the two equilibrium points. This behaviour is referred to as the unwinding effect. This paper presents solutions to cure the unwinding effect in unit quaternion parameterized attitude control systems through the use of a switching parameter or a saturation function.