Collaborative Research: Control of Underactuated Mechanical Systems through Lyapunov Direct Methods

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AbstractCollaborative Research: Control of Underactuated Mechanical Systems through Lyapunov Direct MethodsThe proposed project involves developing Lyapunov direct methods for designing stabilizing controllers for nonlinear underactuated mechanical systems that render the system into closed loop Lyapunov form as opposed to closed loop Hamiltonian form. A goal is to show that the controller associated with the closed loop Lyapunov form can be implemented through either an analytical or a numerical solution (done in the feedback) of certain linear (partial) differential equations constituting asymptotic stability. The advantage of the proposed methodology is that the class of systems in Hamiltonian form is a strict subset of the class of systems in Lyapunov form thus yielding a larger class of stabilizing controllers that could offer improvements in system performance. It is further expected that the proposed research will provide a basis on which to treat problems that are intractable using other methods, while also leading to a unifying framework for methods utilizing controlled Lagrangians and controlled Hamiltonians. Another significant result is that the obtained controller is linear with respect to the velocities, whereas existing energy based nonlinear controllers are quadratic in the velocities. Comparisons between control laws designed through the proposed method and those designed through controlled Lagrangian, lambda, and IDA-PBC methods will be made.The primary thrust of the proposed activity is to analyze and further develop a new strategy, Lyapunov direct methods, for designing stabilizing controllers for underactuated mechanical systems, systems characterized by having fewer drives than degrees of freedom such as a robot where one or more joints are not powered. Examples of such systems are spacecraft, underwater vehicles, satellites, hovercraft and cargo cranes for container ships. The proposed method offers an improvement over the current art in that better performance is achieved while not suffering some common problems of existing methods. It is expected that there are systems that can be treated by the proposed techniques that cannot be treated by current techniques. The analysis will include comparisons of control laws produced by Lyapunov direct methods and current published methods. Undergraduate students in both mathematics and engineering will be encouraged to participate in the project through employment and course credit. Some students will help with development of an REU program at KSU. To facilitate participation of underrepresented groups, the KSU Minority Engineering Program and Women in Engineering and Science Program offices have committed to help in student recruitment, an activity that dovetails with their NSF funded STEP program.