Robust hybrid global asymptotic stabilization of rigid body dynamics using dual quaternions

A hybrid feedback control scheme is proposed for stabilization of rigid body dynamics (pose and velocities) using unit dual quaternions, in which the dual quaternions and veloc- ities are used for feedback. It is well-known that rigid body attitude control is subject to topological constraints which often result in discontinuous control to avoid the unwinding phenomenon. In contrast, the hybrid scheme allows the controlled system to be robust in the presence of uncertainties, which would otherwise cause chattering about the point of discontinuous control while also ensuring acceptable closed-loop response characteristics. The stability of the closed-loop system is guaranteed through a Lyapunov analysis and the use of invariance principles for hybrid systems. Simulation results for a rigid body model are presented to illustrate the performance of the proposed hybrid dual quaternion feedback control scheme

Design and Development of a High-Performance Quadrotor Control Architecture Based on Feedback Linearization

The purpose of this thesis is to outline the development of a high-performance quadrotor control system for an AscTec Hummingbird quadrotor using direct motor speed control within a Vicon motion capture system environment. A Ground Control Station (GCS) acts as a user interface for selecting flight patterns and displaying sensor values. An on-board Intel Edison embedded Linux computer acts as the quadrotor\u27s controller. The Vicon system measures the quadrotor\u27s position and orientation, while the Hummingbird\u27s stock AscTec Autopilot board provides inertial measurements and receives motor speed commands. Based on the flight pattern set by the GCS, smooth and di erentiable trajectories are generated. A control program was written for the Edison to obtain measurements, receive flight pattern commands, perform state estimation, calculate control laws, send motor speed commands to the Autopilot board, and log values. The program was written as a multithreaded C++ program for increased performance. A feedback linearization of the quadrotor\u27s dynamics was performed to account for its nonlinearities. A controller structure designed to ensure exponential Lyapunov stability was applied to the input-output linearized dynamics. The simplex method was used to aid the controller in pushing the Hummingbird\u27s actuators for aggressive maneuvers within set input limitations. The Edison\u27s Wi-Fi capabilities enable it to contact the Vicon server directly for position and orientation measurements. Accelerations and angular velocities are measured by the Autopilot\u27s inertial measurement unit (IMU). A quick state estimation process was implemented to filter the measured states, and state prediction was used to compensate for latency in the system. A custom circuit board and communication framework was designed and assembled for interfacing the Edison with the Autopilot. The custom communication framework allowed for a 16 times speed improvement over the default settings while bypassing the stock wireless communication\u27s inherently unreliable timing. The Hummingbird\u27s physical properties, such as propeller performance and rotational inertias, were characterized via static and step response experiments. The control system\u27s flight performance was evaluated through simulation and experimental tests

Development of a Research Spacecraft Test-Bed with Implementation of Control Laws to Compensate Undesired Dynamics

The development of research spacecraft systems has a significant impact on the preparation and simulation of future space missions. Hardware, software and operation procedures can be adequately tested, validated and verified before they are deployed for the actual mission. In this thesis, a spacecraft vehicle test-bed named Extreme Access System (EASY) was developed. EASY aims at supporting validation and verification of guidance, navigation and control algorithms. Description of EASY spacecraft systems, sub-systems and integration is presented in this thesis along with an analysis of results from numerical simulation and actual implementation of control laws. An attitude control architecture based on quaternion feedback linearization is also described, and performance analysis in the compensation of undesired dynamics is presented. The results show the capabilities and potential of EASY to simulate missions that require validation and verification stages

A feedback linearization approach to spacecraft control using momentum exchange devices

Recent developments in the area of nonlinear control theory have shown how coordiante changes in the state and input spaces can be used with nonlinear feedback to transform certain nonlinear ordinary differential equations into equivalent linear equations. These feedback linearization techniques are applied to resolve two problems arising in the control of spacecraft equipped with control moment gyroscopes (CMGs). The first application involves the computation of rate commands for the gimbals that rotate the individual gyroscopes to produce commanded torques on the spacecraft. The second application is to the long-term management of stored momentum in the system of control moment gyroscopes using environmental torques acting on the vehicle. An approach to distributing control effort among a group of redundant actuators is described that uses feedback linearization techniques to parameterize sets of controls which influence a specified subsystem in a desired way. The approach is adapted for use in spacecraft control with double-gimballed gyroscopes to produce an algorithm that avoids problematic gimbal configurations by approximating sets of gimbal rates that drive CMG rotors into desirable configurations. The momentum management problem is stated as a trajectory optimization problem with a nonlinear dynamical constraint. Feedback linearization and collocation are used to transform this problem into an unconstrainted nonlinear program. The approach to trajectory optimization is fast and robust. A number of examples are presented showing applications to the proposed NASA space station

Nonlinear Control for Dual Quaternion Systems

The motion of rigid bodies includes three degrees of freedom (DOF) for rotation, generally referred to as roll, pitch and yaw, and 3 DOF for translation, generally described as motion along the x, y and z axis, for a total of 6 DOF. Many complex mechanical systems exhibit this type of motion, with constraints, such as complex humanoid robotic systems, multiple ground vehicles, unmanned aerial vehicles (UAVs), multiple spacecraft vehicles, and even quantum mechanical systems. These motions historically have been analyzed independently, with separate control algorithms being developed for rotation and translation. The goal of this research is to study the full 6 DOF of rigid body motion together, developing control algorithms that will affect both rotation and translation simultaneously. This will prove especially beneficial in complex systems in the aerospace and robotics area where translational motion and rotational motion are highly coupled, such as when spacecraft have body fixed thrusters. A novel mathematical system known as dual quaternions provide an efficient method for mathematically modeling rigid body transformations, expressing both rotation and translation. Dual quaternions can be viewed as a representation of the special Euclidean group SE (3). An eight dimensional representation of screw theory (combining dual numbers with traditional quaternions), dual quaternions allow for the development of control techniques for 6 DOF motion simultaneously. In this work variable structure nonlinear control methods are developed for dual quaternion systems. These techniques include use of sliding mode control. In particular, sliding mode methods are developed for use in dual quaternion systems with unknown control direction. This method, referred to as self-reconfigurable control, is based on the creation of multiple equilibrium surfaces for the system in the extended state space. Also in this work, the control problem for a class of driftless nonlinear systems is addressed via coordinate transformation. It is shown that driftless nonlinear systems that do not meet Brockett\u27s conditions for coordinate transformation can be augmented such that they can be transformed into the Brockett\u27s canonical form, which is nonholonomic. It is also shown that the kinematics for quaternion systems can be represented by a nonholonomic integrator. Then, a discontinuous controller designed for nonholonomic systems is applied. Examples of various applications for dual quaternion systems are given including spacecraft attitude and position control and robotics

Koopman Operator Based Modeling and Control of Rigid Body Motion Represented by Dual Quaternions

In this paper, we systematically derive a finite set of Koopman based observables to construct a lifted linear state space model that describes the rigid body dynamics based on the dual quaternion representation. In general, the Koopman operator is a linear infinite dimensional operator, which means that the derived linear state space model of the rigid body dynamics will be infinite-dimensional, which is not suitable for modeling and control design purposes. Recently, finite approximations of the operator computed by means of methods like the Extended Dynamic Mode Decomposition (EDMD) have shown promising results for different classes of problems. However, without using an appropriate set of observables in the EDMD approach, there can be no guarantees that the computed approximation of the nonlinear dynamics is sufficiently accurate. The major challenge in using the Koopman operator for constructing a linear state space model is the choice of observables. State-of-the-art methods in the field compute the approximations of the observables by using neural networks, standard radial basis functions (RBFs), polynomials or heuristic approximations of these functions. However, these observables might not providea sufficiently accurate approximation or representation of the dynamics. In contrast, we first show the pointwise convergence of the derived observable functions to zero, thereby allowing us to choose a finite set of these observables. Next, we use the derived observables in EDMD to compute the lifted linear state and input matrices for the rigid body dynamics. Finally, we show that an LQR type (linear) controller, which is designed based on the truncated linear state space model, can steer the rigid body to a desired state while its performance is commensurate with that of a nonlinear controller. The efficacy of our approach is demonstrated through numerical simulations