On the non-robustness of inconsistent quaternion-based attitude control systems using memoryless path-lifting schemes

The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Quaternion-based feedback control laws that are inconsistent (i.e. do not have a unique value for a given attitude) require an additional mechanism that lifts a continuous attitude trajectory to the unit quaternion space. Lifting mechanisms that are memoryless, for example, selecting the quaternion having positive scalar component, have a limited domain where they remain injective and, when used globally, introduce discontinuities into the closed-loop system. We show that such discontinuities can be exploited by an arbitrarily small measurement disturbance to stabilize attitudes far from the desired attitude and destroy "global" attractivity properties.