Physical Dissipation and the Method of Controlled Lagrangians

We describe the effect of physical dissipation on stability of equilibria which have been stabilized, in the absence of damping, using the method of controlled Lagrangians. This method applies to a class of underactuated mechanical systems including “balance” systems such as the pendulum on a cart. Since the method involves modifying a system’s kinetic energy metric through feedback, the effect of dissipation is obscured. In particular, it is not generally true that damping makes a feedback-stabilized equilibrium asymptotically stable. Damping in the unactuated directions does tend to enhance stability, however damping in the controlled directions must be “reversed” through feedback. In this paper, we suggest a choice of feedback dissipation to locally exponentially stabilize a class of controlled Lagrangian systems

Integral Control on Lie Groups

In this paper, we extend the popular integral control technique to systems evolving on Lie groups. More explicitly, we provide an alternative definition of "integral action" for proportional(-derivative)-controlled systems whose configuration evolves on a nonlinear space, where configuration errors cannot be simply added up to compute a definite integral. We then prove that the proposed integral control allows to cancel the drift induced by a constant bias in both first order (velocity) and second order (torque) control inputs for fully actuated systems evolving on abstract Lie groups. We illustrate the approach by 3-dimensional motion control applications.Comment: Resubmitted to Systems and Control Letters, February 201

Control of mechanical systems on Lie groups based on vertically transverse functions

"The transverse function approach to control provides a unified setting to deal with practical stabilization and tracking of arbitrary trajectories for controllable driftless systems. Controllers derived from that approach offer advantages over those based on more classical techniques for control of nonholonomic systems. Nevertheless, its extension to more general classes, such as critical underactuated mechanical systems, is not immediate. The present paper explores a possible extension by developing a framework that allows one to cast point stabilization problems for (left-invariant) second-order systems on Lie groups, including simple mechanical systems. The approach is based on "vertical transversality," a property exhibited by derivatives of transverse functions. In this paper, we lay out the theoretical foundations of our approach and present an example to illustrate some of its features.

Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry

This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages. This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottom-heavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove useful for the control of underwater vehicles

Advances in Underactuated Spacecraft Control

This dissertation addresses the control of a spacecraft which either becomes underactuated due to onboard failures or is made underactuated by design. Successfully controlling an underactuated spacecraft can extend spacecraft operational life in orbit and improve the robustness of space missions. The novel contributions of the dissertation include the following. Firstly, switching feedback controllers are developed for the attitude control of an underactuated spacecraft equipped with two pairs of thrusters, or two reaction wheels (RWs), or two control moment gyros (CMGs). The problem is challenging; e.g., even in the zero total angular momentum case, no smooth or even continuous time-invariant feedback law for stabilizing a desired orientation exists. The method exploits the separation of the system into inner-loop base variables and outer-loop fiber variables. The base variables track periodic reference trajectories, the amplitude of which is governed by parameters that are adjusted to induce an appropriate change in the fiber variables towards the desired pointing configuration. Secondly, nonlinear Model Predictive Control (MPC) is applied to the attitude dynamics of an underactuated spacecraft with two RWs and zero angular momentum. MPC has the remarkable ability to generate control laws that are discontinuous in the state. By utilizing nonlinear MPC, the obstruction to stabilizability is overcome and attitude maneuvers can be performed while enforcing constraints. Thirdly, an unconventional pathway is discussed for recovering the linear controllability of an underactuated spacecraft with two RWs by accounting for the effects of solar radiation pressure (SRP) in the spacecraft attitude model. Necessary and sufficient conditions for recovering linear controllability are given, and with linear controllability restored, conventional controllers can be designed for underactuated spacecraft. Lastly, two sets of coupled translational and rotational equations of motion for a spacecraft in a central gravity field are derived. The spacecraft is assumed to have only internal attitude actuators and the equations of motion are relative with respect to an equilibrium orbit. Under reasonable assumptions on the spacecraft configuration and equilibrium orbit, the coupled dynamics are small-time locally controllable (STLC), which opens a path to utilizing conventional control techniques to move translationally in space by employing attitude control only.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133430/1/cdpete_1.pd

Steering for a Class of Dynamic Nonholonomic Systems

In this paper we derive control algorithms for a class of dynamic nonholonomic steering problems, characterized as mechanical systems with nonholonomic constraints and symmetries. Recent research in geometric mechanics has led to a single, simplified framework that describes this class of systems, which includes examples such as wheeled mobile robots; undulatory robotic and biological locomotion systems, such as paramecia, inchworms, and snakes; and the reorientation of satellites and underwater vehicles. This geometric framework has also been applied to more unusual examples, such as the snakeboard robot, bicycles, the wobblestone, and the reorientation of a falling cat. We use this geometric framework as a basis for developing two types of control algorithms for such systems. The first is geared towards local controllability, using a perturbation approach to establish results similar to steering using sinusoids. The second method utilizes these results in applying more traditional steering algorithms for mobile robots, and is directed towards generating more non-local control methods of steering for this class of systems

ATTITUDE CONTROL ON SO(3) WITH PIECEWISE SINUSOIDS

This dissertation addresses rigid body attitude control with piecewise sinusoidal signals. We consider rigid-body attitude kinematics on SO(3) with a class of sinusoidal inputs. We present a new closed-form solution of the rotation matrix kinematics. The solution is analyzed and used to prove controllability. We then present kinematic-level orientation-feedback controllers for setpoint tracking and command following. Next, we extend the sinusoidal kinematic-level control to the dynamic level. As a representative dynamic system, we consider a CubeSat with vibrating momentum actuators that are driven by small $\epsilon$-amplitude piecewise sinusoidal internal torques. The CubeSat kinetics are derived using Newton-Euler\u27s equations of motion. We assume there is no external forcing and the system conserves zero angular momentum. A second-order approximation of the CubeSat rotational motion on SO(3) is derived and used to derive a setpoint tracking controller that yields order O(ε2) closed-loop error. Numerical simulations are presented to demonstrate the performance of the controls. We also examine the effect of the external damping on the CubeSat kinetics. In addition, we investigate the feasibility of the piecewise sinusoidal control techniques using an experimental CubeSat system. We present the design of the CubeSat mechanical system, the control system hardware, and the attitude control software. Then, we present and discuss the experiment results of yaw motion control. Furthermore, we experimentally validate the analysis of the external damping effect on the CubeSat kinetics