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# Aspects of Geometric Mechanics and Control of Mechanical Systems

Lewis, Andrew D. (1995) Aspects of Geometric Mechanics and Control of Mechanical Systems. California Institute of Technology , Pasadena, CA. (Unpublished) https://resolver.caltech.edu/CaltechCDSTR:1995.017

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## Abstract

Many interesting control systems are mechanical control systems. In spite of this, there has not been much effort to develop methods which use the special structure of mechanical systems to obtain analysis tools which are suitable for these systems. In this dissertation we take the first steps towards a methodical treatment of mechanical control systems. First we develop a framework for analysis of certain classes of mechanical control systems. In the Lagrangian formulation we study "simple mechanical control systems" whose Lagrangian is "kinetic energy minus potential energy." We propose a new and useful definition of controllability for these systems and obtain a computable set of conditions for this new version of controllability. We also obtain decompositions of simple mechanical systems in the case when they are not controllable. In the Hamiltonian formulation we study systems whose control vector fields are Hamiltonian. We obtain decompositions which describe the controllable and uncontrollable dynamics. In each case, the dynamics are shown to be Hamiltonian in a suitably general sense. Next we develop intrinsic descriptions of Lagrangian and Hamiltonian mechanics in the presence of external inputs. This development is a first step towards a control theory for general Lagrangian and Hamiltonian control systems. Systems with constraints are also studied. We first give a thorough overview of variational methods including a comparison of the "nonholonomic" and "vakonomic" methods. We also give a generalised definition for a constraint and, with this more general definition, we are able to give some preliminary controllability results for constrained systems.

Item Type: Report or Paper (Technical Report) This thesis would not have been possible without the cooperation of my advisor, Richard Murray. His insightful thoughts and our lengthy discussions added a great deal to the quality of this work. I would also like to extend thanks to the other members of my thesis committee, Drs. Burdick, Caughey, and Marsden. During my none too brief stay at Caltech, my mates in Thomas lab have always provided a healthy environment for work and play. The Thomas Lunch Group, first led by Jim Ostrowski, then by Michael Scott, is a wonderful institution which I hope will have a long and distinguished history. Special mention must go to Bob M'Closkey as the only guy who has been around here as long as I have, Kevin Otto for providing me with an opportunity to diversify my research portfolio, Howie Choset for always providing me with entertaining and enlightening company, Victor Burnley for being degenerate with me, Jim Ostrowski for showing keen interest in my work and for being in the unfortunate position of being my office mate, and Ted Hubbard just for being Canadian. On a personal note, I would like to extend my most sincere love and appreciation to my family in Prince Edward Island: Elizabeth, Gordon, Cathryn, Marc, Monic, and Eric. It is to my mother, Elizabeth, that I dedicate this thesis. The last part of my time at Caltech has been made very special by a lovely lady, Andria Costello. She has been very patient and loving. I look forward to the future if it is with her. A final word of thanks I extend to the subject of mechanics. Anything in this thesis which is remotely interesting may be attributed to the subject and to the researchers who have laid so beautiful a foundation for its future. Control and Dynamical Systems Technical Reports CaltechCDSTR:1995.017 https://resolver.caltech.edu/CaltechCDSTR:1995.017 You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format. 28094 CaltechCDSTR Imported from CaltechCDSTR 26 Sep 2006 03 Oct 2019 03:29

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