Controllability of Mechanical Systems with Potential and External Forces

University Of Hawaii, Honolulu

Investigators

Abstract

A very important class of control systems, even though they are non-generic, is the class of controlled mechanical systems. Examples of these systems include the planar rigid body with a single variable direction thruster, the snakeboard. Trajectory design problems for such systems are of particular interest in this project. The main application considered in this project is the control of a submerged rigid body. Clearly, this application is particularly well adapted to analysis, both due to the practical motivation coming from the recent trend to build autonomous underwater vehicles and for more mathematically oriented reasons. Indeed, an underwater vehicle can be modeled as a simple mechanical control system, with dissipative forces. A major goal is to establish a mathematical formulation of the switching time parametrization algorithm, developed in a previous work, based on the differential geometric properties of the system such as the notion of decoupling vector field. The key notions involved in this project are the ones of decoupling vector fields for invariant systems on a Lie group, and the notion of singular extremals coming from optimal control. Indeed, a recent observation concerns a possible relationship between singular extremals of order greater than 1 and decoupling vector fields. The goal is to develop a proper generalization for the notion of decoupling vector field for forced affine-connection control systems. In the exploration of our world's oceans, there comes a time when human controlled vehicles are not sufficient. Whether in the context of a deep ocean survey or a long term monitoring project, autonomous underwater vehicles are the natural choice to take the burden. Like any other machine, we need the ability to accurately control the underwater vehicle to perform even the simplest of tasks. We propose to combine our theoretical methods with experiments to show the efficiency of our algorithms versus the classic model-free controllers developed in the autonomous underwater vehicle field. The impact of the proposed research will reach several communities: mathematics, engineering and oceanography. Another major concern of the project is the involvement of graduate and undegraduate students at every level of the research as well as the development of outreach educational programs for younger generations.