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Geometric Control Theory and Its Application To GEOMETRIC CONTROL THEORY AND ITS APPLICATION TO UNDERWATER VEHICLES A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI‘I IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN OCEAN & RESOURCES ENGINEERING DECEMBER 2008 By Ryan N. Smith Dissertation Committee: Monique Chyba, Chairperson Song K. Choi R. Cengiz Ertekin Geno Pawlak George R. Wilkens We certify that we have read this dissertation and that, in our opinion, it is satisfactory in scope and quality as a dissertation for the degree of Doctor of Philosophy in Ocean & Resources Engineering. DISSERTATION COMMITTEE —————————————— Chairperson —————————————— —————————————— —————————————— —————————————— —————————————— ii c 2008, Ryan Neal Smith. All rights reserved. iii To Neal iv Acknowledgments Of this entire dissertation, this single section has proven to be the most painful and difficult to write. Of course, after committing two-hundred and some odd pages to the discussion of my research over the last five years, I have few words left for the one section that will probably be read the most of any other in this dissertation. Additionally, recounting the memories of all the love and support which I have received from so many people brings tears to my eyes every time I have tried to work on this section. My cup runneth over. I have written this acknowledgments section one hundred forty-three times; yes, I have been counting. I can not simply list every person who has participated in the success of this work, as the list would be longer than the dissertation itself. So, I must restrict myself to acknowledging only those whom have truly made this all possible, and at times, very worthwhile. I will undoubtedly omit some individuals who do deserve recognition. For most of these, including you would require the inclusion of a multitude of others, as your contri- butions are of equivalent importance. I trust that these individuals know who they are, and willingly accept this blanket acknowledgment as sufficient recognition. Thank you! I would first like to extend a huge thank you to the Mathematics Department for their continued support throughout my graduate career. Emotionally, academically and finan- cially they have provided a stable foundation upon which I have based my studies. In particular, I would like to acknowledge my Hawaiian aunties, Susan and Shirley, for their guidance, wisdom and love. v I would like to thank the Ocean & Resources Engineering Department for taking me in and allowing me the freedom to follow my academic dreams. My experiences in this department have taught me things that I never knew I needed to know, and for this I am grateful. To Volker, Danny, Jordon, Kumar and Rob, I thank you for all your assistance that first year in teaching a mathematician how to be an engineer; I would not have made it through without you. I owe a debt of gratitude to the past and present members of the Autonomous Systems Laboratory and employees of Mase, Inc. For your instruction and guidance in the operation and maintenance of ODIN, I am truly grateful. In particular, I would like to give a huge thank you to Chris McLeod for his patience, support and friendship over the years; all three of which were tested many times when troubleshooting a malfunctioning ODIN. I think that I also owe an homage to the robotic gods for for keeping ODIN operational during the last experimental stages of my research. I want to extend a separate thank you to all of those students who assisted me in the collection and processing of the data gathered at the pool. Thank you Shashi, Jeff, Daniel, Will, Scott, Ben, Tim and Gina. Your dedicated efforts during those early morning experi- ments was greatly appreciated. A key component to this research was the implementation of the control strategies onto ODIN at the Duke Kahanamoku Aquatic Complex. None of that would have been possible without the help of Bruce Kennard. I want to thank him for his flexibility and willingness to provide ample space and time at the pool to conduct the experiments presented in this dissertation. vi I would like to thank Thomas Haberkorn for his guidance, assistance and tutelage over the past five years. And, of course, his persistence in treating every time we ”took a coffee.” Your padawan has finished his training. I would also like to thank George Csordas for his mentorship over the entire course of my graduate career. He taught me that nothing is obvious, and anything that is implied as such, clearly needs to be checked. Dr. Csordas, for the many times you have asked ”How is your thesis coming along?”, I can finally and proudly reply, ”It is finished.” I would also like to acknowledge Jonathan Gradie for his mentorship, support and friendship. Thank you for the opportunities, experiences, camaraderie and most of all, the stories. You are an inspiration. It has been an amazing experience to work under and collaborate with Monique Chyba for the better part of the last six years. What started as a simple conversation after a collo- quium has blossomed into the work presented in this dissertation. It has been exciting to be a part of her programs and visions from the very beginning. I can not thank Monique enough for her patience, guidance, support, enthusiasm, coaching and friendship. Most of all, I would like to thank my family for providing a solid foundation of uncon- ditional love and support. I thank my uncle Tom for the inspiration to follow my dreams. Thank you Mom and Dad for letting me lead during this journey; following, supporting and encouraging every decision along the way. Thank you Troy and York for never letting me forget who I am and where I came from. To my wife Ellen, a woman who fell in love with, and married, a mathematician turned ocean engineer during the middle of his Ph.D. I will do my very best for the rest of my life vii to return the love and support you have given me over the past four years. Without you, this would never have been possible. viii Abstract This dissertation is based on theoretical study and experiments which extend geometric control theory to practical applications within the field of ocean engineering. We present a method for path planning and control design for underwater vehicles by use of the archi- tecture of differential geometry. In addition to the theoretical design of the trajectory and control strategy, we demonstrate the effectiveness of the method via the implementation onto a test-bed autonomous underwater vehicle. Bridging the gap between theory and application is the ultimate goal of control the- ory. Major developments have occurred recently in the field of geometric control which narrow this gap and which promote research linking theory and application. In particular, Riemannian and affine differential geometry have proven to be a very effective approach to the modeling of mechanical systems such as underwater vehicles. In this framework, the application of a kinematic reduction allows us to calculate control strategies for fully and under-actuated vehicles via kinematic decoupled motion planning. However, this method has not yet been extended to account for external forces such as dissipative viscous drag and buoyancy induced potentials acting on a submerged vehicle. To fully bridge the gap between theory and application, this dissertation addresses the extension of this geometric control design method to include such forces. We incorporate the hydrodynamic drag experienced by the vehicle by modifying the Levi-Civita affine connection and demonstrate a method for the compensation of potential forces experienced during a prescribed motion. We present the design method for multiple ix different missions and include experimental results which validate both the extension of the theory and the ability to implement control strategies designed through the use of geometric techniques. By use of the extension presented in this dissertation, the underwater vehicle application successfully demonstrates the applicability of geometric methods to design implementable motion planning solutions for complex mechanical systems having equal or fewer input forces than available degrees of freedom. Thus, we provide another tool with which to further increase the autonomy of underwater vehicles. x Contents Acknowledgments . v Abstract . ix List of Tables . xvii List of Figures . xix List of Symbols . xxii Chapter 1: Introduction . 1 Chapter 2: Equations of Motion . 13 2.1 Rigid Body Kinematics . 14 2.2 Rigid Body Dynamics . 21 2.3 Submerged Rigid Body Dynamics . 29 2.3.1 Added Mass . 29 2.3.2 Viscous Damping . 31 2.3.3 Restoring Forces and Moments . 33 2.4 Geometric Equations of Rigid-Body Dynamics . 34 2.4.1 Ideal Fluid, Including Added Mass . 36 2.4.2 Geometric Forces . 48 2.4.3 Viscous Drag Geometrically . 51 2.4.4 Restoring Forces Geometrically . 54 xi 2.4.5 Geometric Equations of Motion in a Viscous Fluid . 56 2.5 Closing Remarks . 63 Chapter 3: Control of Underwater Vehicles . 64 3.1 Motion Planning for the Rigid Body . 64 3.2 Motion Planning for the Submerged Rigid Body . 67 3.3 Control of Autonomous Underwater Vehicles . 68 3.3.1 PID Control . 69 3.3.2 Adaptive Control . 70 3.3.3 Sliding Mode Control . 71 3.3.4 Neural Network Control . 72 3.3.5 Fuzzy Logic . 73 3.3.6 General Control . 74 Chapter 4: Geometric Control of Underwater Vehicles . 76 4.1 Literature Review . 76 4.2 Research Motivation and Direction . 78 4.3 Kinematic Reduction . 80 4.3.1 Decoupling Vector Fields . 85 4.3.2 Under-Actuation . 86 4.4 Control Strategy Design Using Decoupling Vector Fields . 87 Chapter 5: Application to the Omni Directional intelligent Navigator .
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