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Modelling and Control of Tensegrity Structures Modelling and Control of Tensegrity Structures Anders Sunde Wroldsen A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of PHILOSOPHIAE DOCTOR Department of Marine Technology Norwegian University of Science and Technology 2007 NTNU Norwegian University of Science and Technology Thesis for the degree of philosophiae doctor Faculty of Engineering Science & Technology Department of Marine Technology c Anders Sunde Wroldsen ISBN 978-82-471-4185-4 (printed ver.) ISBN 978-82-471-4199-1 (electronic ver.) ISSN 1503-8181 Doctoral Thesis at NTNU, 2007:190 Printed at Tapir Uttrykk Abstract This thesis contains new results with respect to several aspects within tensegrity research. Tensegrity structures are prestressable mechanical truss structures with simple dedicated elements, that is rods in compression and strings in tension. The study of tensegrity structures is a new field of research at the Norwegian University of Science and Technology (NTNU), and our alliance with the strong research community on tensegrity structures at the University of California at San Diego (UCSD) has been necessary to make the scientific progress presented in this thesis. Our motivation for starting tensegrity research was initially the need for new structural concepts within aquaculture having the potential of being wave com- pliant. Also the potential benefits from controlling geometry of large and/or interconnected structures with respect to environmental loading and fish welfare were foreseen. When initiating research on this relatively young discipline we discovered several aspects that deserved closer examination. In order to evaluate the potential of these structures in our engineering applications, we entered into modelling and control, and found several interesting and challenging topics for research. To date, most contributions with respect to control of tensegrity structures have used a minimal number of system coordinates and ordinary differential equa- tions (ODEs) of motion. These equations have some inherent drawbacks, such as singularities in the coordinate representation, and recent developments there- fore used a non-minimal number of system coordinates and differential-algebraic equations (DAEs) of motion to avoid this problem. This work presents two gen- eral formulations, both DAEs and ODEs, which have the ability to choose the position on rods the translational coordinates should point to, and further con- strain some, or all of these coordinates depending on how the structural system should appear. The DAE formulation deduced for tensegrity structures has been extended in order to simulate the dynamics of relatively long and heavy cables. A rigid bar cable (RBC) model is created by interconnecting a finite number of inextensible thin rods into a chain system. The main contribution lies in the mathemati- i Abstract cal model that has the potential to be computationally very efficient due to the small size of the matrix that needs to be inverted for each iteration of the numer- ical simulation. Also, the model has no singularities and only affine coordinate transformations due to the non-minimal coordinate representation. The proposed model is compared to both commercial finite element software and experimental results from enforced oscillations on a hanging cable and show good agreement. The relationship between enforced input frequency and the number of elements, indirectly the eigenfrequency of individual elements, to be chosen for well behaved simulations has been investigated and discussed. This thesis only considers tensegrity structures with control through strings alone, and all strings actuated. In the first control contribution ODEs of mo- tion are used. Due to the inherent prestressability condition of tensegrity struc- tures the allocation of constrained inputs (strings can only take tension) has traditionally been solved by using numerical optimization tools. The numerical optimization procedure has been investigated with an additional summation in- equality constraint giving enhanced control over the level of prestress throughout the structure, and possibly also smooth control signals, during a change in the structural configuration. The first contribution with respect to the control of three-dimensional tenseg- rity structures, described by DAEs of motion, uses Lyapunov theory directly on a non-minimal error signal. By using this control design it is able to solve for the optimization of control inputs explicitly for whatever norm of the control signal is wanted. Having these explicit analytical solutions this removes the potentially time-consuming iteration procedures of numerical optimization tools. The sec- ond contribution introduces a projection of the non-minimal error signal onto a subspace that is orthogonal to the algebraic constraints, namely the direction of each rod in the system. This gives a minimal error signal and is believed to have the potential of reducing the components of control inputs in the direction of the algebraic constraints which produce unnecessary internal forces in rods. Also this control design uses Lyapunov theory and solves for the optimization of control inputs explicitly. The concept of using tensegrity structures in the development of wave com- pliant marine structures, such as aquaculture installations, has been introduced through this work. In order to evaluate such a concept one should first have a clear understanding of the possible structural strength properties of tensegrities and load conditions experienced in marine environments. An approach to mod- elling tensegrity structures in marine environments has also been proposed. A tensegrity beam structure, believed to be suitable for the purpose, has been used to both study axial and lateral strength properties. ii Acknowledgements This thesis represents the work in my doctoral studies carried out in the pe- riod from August 2004 through September 2007. This work has been conducted both at the Norwegian University of Science and Technology (NTNU) under the guidance of my supervisor Professor Asgeir J. Sørensen and at the University of California at San Diego (UCSD) under the guidance of my co-supervisors Pro- fessor Robert E. Skelton and Dr. Maur´ıcio C. de Oliveira. My Ph.D. study has been funded through the IntelliSTRUCT program, a joint collaboration between the Centre for Ships and Ocean Structures (CeSOS) and SINTEF Fisheries and Aquaculture (SFH), sponsored by the Research Council of Norway (NFR). My research at UCSD was funded by the RCN Leiv Eiriksson mobility program, Det Norske Veritas (DNV) and Kjell Nordviks Fund. I would first like to thank Professor Asgeir Sørensen for giving me the op- portunity to enroll in the Ph.D. program. His guidance has been founded on his broad experience in research and life in general and has been highly appreciated. The motivational skills of this man have made me feel ready for any task when leaving his office. I am grateful to Professor Robert E. Skelton for giving me an unforgettable period at UCSD from January through August 2007. I would like to thank him for his generosity and hospitality and for including me among his family, friends and colleagues. I admire his creativity in research and passion for tensegrity structures. His many stories and our long discussions were enriching and have given me many good memories. I deeply appreciate the close cooperation and guidance of Dr. Maur´ıcio C. de Oliveira since the beginning of my research stay in San Diego. In addition to having benefitted from his exceptional skills with respect to control theory and hands-on experience, I have enjoyed his good company, sense of humor and numerous discussions over espressos by the bear statue, and over the Internet. Dr. Østen Jensen should be thanked for being an excellent research partner. His ABAQUS simulations have been the core content of the investigations on strength properties of tensegrity structures, and are also important with respect to validating the cable model proposed in this work. The ideas and first steps iii Acknowledgements towards understanding the proposed cable model were done in cooperation with Vegar Johansen, who generously shared his knowledge and provided experimental results needed for model verification. I have enjoyed many conversations with Dr. Jerome Jouffroy on the philosophy of research and analytical mechanics. His support and creative thinking have been very motivating in my work. Anne Marthine Rustad has been my sporty office mate and she has with her tremendous organizational skills helped me out with nearly anything. I give her my warm thanks for being a good friend and discussion partner throughout this period of both joy and frustration. Sondre K. Jacobsen, Erik Wold and Jone S. Rasmussen at NTNU Technology Transfer Office (TTO) should be thanked for good cooperation and their con- tributions with respect to patenting the ideas of using tensegrity structures in marine systems. I have enjoyed working with Per R. Fluge and Hans-Christian Blom in Fluge & Omdal Patent AS. They have contributed with creative input and expertise throughout the process of securing our intellectual property. I am forever grateful to the father of tensegrity structures, the world famous artist Kenneth Snelson, for letting me visit him in New York. We had a very interesting conversation in his Manhattan studio over a large cup of coffee one early morning in late November 2006. I would like to thank all my friends and research colleagues during the last three years. Thanks
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