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In this paper, we propose two hybrid feedbacks based on “synergistic” potential functions to achieve robust global asymptotic tracking of rigid-body attitude, a task that is impossible by classical state feedback—be it smooth, nonsmooth, or periodic—due to the topological structure of the special orthogonal group. Both hybrid feedbacks are based upon a proportional-derivative structure where the proportional term is generated from a finite family of control-induced artificial potential energies and the derivative term is due to a damping injection. In a patient-yet-greedy fashion, the hybrid feedback hysteretically switches to the minimum control-induced artificial potential energy in a finite family. The synergy property—which requires that for each undesirable critical point of each potential energy, there exists a lower potential energy in the family—guarantees the globality of robust asymptotic tracking. The first hybrid feedback is the natural extension of existing PD controllers and results in discontinuous jumps in the control signal due to a direct switch the potential energy. Using a backstepping procedure, we design a second hybrid feedback that smooths the jumps in the control torque by dynamically interpolating between the switching potential-energy terms. We show that while the class of “modified trace functions” is not wide enough to generate a “centrally synergistic” potential function, relaxing the centrality assumption allows one to construct a synergistic family and we provide explicit guidelines for doing so. [ABSTRACT FROM PUBLISHER]

The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Because the space of unit quaternions constitutes a double cover of the rigid-body-attitude space, quaternion-based control laws are often—by design—inconsistent, i.e., they do not have a unique value for each rigid-body attitude. Inconsistent quaternion-based control laws require an additional mechanism that uniquely converts an attitude estimate into its quaternion representation; however, conversion mechanisms that are memoryless—e.g., 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—through an explicit construction and Lyapunov analysis—that such discontinuities can be hijacked by arbitrarily small measurement disturbances to stabilize attitudes far from the desired attitude. To remedy this limitation, we propose a hybrid-dynamic algorithm for smoothly lifting an 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. Finally, we employ the main results to show that certain hybrid feedbacks can globally asymptotically stabilize the attitude of a rigid body. [ABSTRACT FROM AUTHOR]

We apply recent results on robust global asymptotic stabilization of the attitude of a single rigid body to the problem of globally synchronizing the attitude of a network of rigid bodies using a decentralized strategy. The proposed hybrid feedback scheme relies on the communication of a binary logic variable between each pair of neighboring rigid bodies that determines the orientation of a torque component acting to reduce their relative error. Through a hysteretic switch of this logic variable, the hybrid feedback achieves global synchronization under the assumption that the network is connected and acyclic. The hysteresis eliminates chattering while preventing the “unwinding phenomenon” apparent in some quaternion-based attitude control schemes. The results are exercised in a numerical example. [ABSTRACT FROM AUTHOR]

It is well known that controlling the attitude of a rigid body is subject to topological constraints. We illustrate, with examples, the problems that arise when using continuous and (memoryless) discontinuous quaternion-based state-feedback control laws for global attitude stabilization. We propose a quaternion-based hybrid feedback scheme that solves the global attitude tracking problem in three scenarios: full state measurements, only measurements of attitude, and measurements of attitude with angular velocity measurements corrupted by a constant bias. In each case, the hybrid feedback is dynamic and incorporates hysteresis-based switching using a single binary logic variable for each quaternion error state. When only attitude measurements are available or the angular rate is corrupted by a constant bias, the proposed controller is observer-based and incorporates an additional quaternion filter and bias observer. The hysteresis mechanism enables the proposed scheme to simultaneously avoid the “unwinding phenomenon” and sensitivity to arbitrarily small measurement noise that is present in discontinuous feedbacks. These properties are shown using a general framework for hybrid systems, and the results are demonstrated by simulation. [ABSTRACT FROM AUTHOR]