Conference Paper

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

Robert Bosch Res. & Technol. Center, Palo Alto, CA, USA
Conference: American Control Conference (ACC), 2011
Source: IEEE Xplore

ABSTRACT 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.

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    • "when applied to actual attitude dynamics [11], and it may also exhibit unwinding behavior, where the controller rotates a rigid body through unnecessarily large angles [12], [13]. Attitude control systems also have been developed directly on the special orthogonal group, SO(3) to avoid the singularities associated with Euler-angles and the ambiguity of quaternions [14], [15], [16], [17]. "
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    ABSTRACT: The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Often, quaternion-based feedbacks require an additional mechanism that lifts a continuous attitude path to the unit quaternion space. When this mechanism is memoryless, it has a limited domain where it remains injective and leads to discontinuities when used globally. To remedy this limitation, we propose a hybrid-dynamic algorithm for lifting a continuous attitude path to the unit quaternion space. We show that this hybrid-dynamic mechanism allows us to directly translate quaternion-based controllers and their asymptotic stability properties (obtained in the unit-quaternion space) to the actual rigid-body-attitude space. We also show that when quaternion-based controllers are not designed to account for the double covering of the rigid-body-attitude space by a unit-quaternion parameterization, they can give rise to the unwinding phenomenon, which we characterize in terms of the projection of asymptotically stable sets.
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