Christopher G. Mayhew

Quaternion-Based Hybrid Control for Robust Global Attitude Tracking

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.

Robust global asymptotic attitude stabilization of a rigid body by quaternion-based hybrid feedback

Global asymptotic stabilization of the attitude of a rigid body is hindered by major topological obstructions. In fact, this task is impossible to accomplish with continuous state feedback. Moreover, when the attitude is parametrized with unit quaternions, it becomes impossible to design a globally stabilizing state feedback (even discontinuous) that is robust to measurement noise. In this paper, we present a quaternion-based hysteretic hybrid feedback that robustly globally asymptotically stabilizes the attitude of a rigid body. The hybrid control laws are derived through Lyapunov analysis in kinematic and dynamic settings. In the dynamic setting, we provide two control laws: one derived from an energybased Lyapunov function and another based on backstepping. Analyzing the change in these Lyapunov functions due to switching of a logic variable yields a straightforward form for state-based hysteresis. A simulation study demonstrates how hysteresis provides robustness to measurement noise and highlights differences between the energy-based and backstepping control laws.

Robust Source-Seeking Hybrid Controllers for Autonomous Vehicles

We consider the problem of steering an autonomous vehicle to locate a radiation source utilizing measurements of the radiation intensity only. We propose a control algorithm that locates the source through a sequence of line minimizations of the radiation intensity. We implement in a hybrid controller, with sample-and-hold and logic variables, a discretized version of the algorithm suitable for steering a point-mass vehicle. The algorithm confers global convergence and practical stability properties to the closed-loop hybrid system. We discuss these properties and characterize the region of convergence for the vehicle. Convergence and stability results are supplemented with simulations.

On quaternion-based attitude control and the unwinding phenomenon

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.

Robust hybrid source-seeking algorithms based on directional derivatives and their approximations

A family of hybrid control algorithms is developed that steer a nonholonomic autonomous vehicle to the source of a scalar signal present in the environment. In an idealized setting, we develop a general hybrid control scheme that globally asymptotically stabilizes the vehicle position about the source. Pursuing a practical implementation, a series of perturbations to the family of controllers is introduced, resulting in a semi-global practical stability of the vehicle position about the source. An example of a recently developed conjugate direction-based controller fitting into this family is developed and demonstrated by simulation and experiment.

Quaternion-Based Hybrid Feedback for Robust Global Attitude Synchronization

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.

Robust source-seeking hybrid controllers for nonholonomic vehicles

We develop a hybrid controller that drives a nonholonomic vehicle to the source of a radiation-like signal. The control signal depends only on measurements of the signal, which may be corrupted by noise, and internal states of the controller. The control strategy is inspired by a line minimization-based algorithm for unconstrained optimization of nonlinear functions without gradient information. We state the main properties of the algorithm and then discuss its implementation in a hybrid controller that coordinates the optimization algorithm with vehicle steering. We present semi- global practical convergence and stability results for the closed-loop system. These results are illustrated by simulation and experiment.

On Path-Lifting Mechanisms and Unwinding in Quaternion-Based Attitude Control

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.

Synergistic Hybrid Feedback for Global Rigid-Body Attitude Tracking on

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.

\hbox{ SO }(3)^{\ast}

Abstract We show that when a compact set is globally asymptotically stable under the action of a differential inclusion satisfying certain regularity properties, there exists a smooth differential equation rendering the same compact set globally asymptotically stable. The regularity properties assumed in this work stem from the consideration of Krasovskii/Filippov solutions to discontinuous differential equations and the robustness of asymptotic stability under perturbation. In particular, the results in this work show that when a compact set cannot be globally asymptotically stabilized by continuous feedback due to topological obstructions, it cannot be robustly globally asymptotically stabilized by discontinuous feedback either. The results follow from converse Lyapunov theory and parallel what is known for the local stabilization problem.