KINEMATIC ANALYSIS OF A PLANAR TENSEGRITY MECHANISM WITH PRE-STRESSED SPRINGS By VISHESH VIKAS A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008 1 °c 2008 Vishesh Vikas 2 Vakratunda mahakaaya Koti soorya samaprabhaa Nirvighnam kurume deva Sarva karyeshu sarvadaa. 3 TABLE OF CONTENTS page LIST OF TABLES ..................................... 5 LIST OF FIGURES .................................... 6 ACKNOWLEDGMENTS ................................. 7 ABSTRACT ........................................ 8 CHAPTER 1 INTRODUCTION .................................. 9 2 PROBLEM STATEMENT AND APPROACH ................... 12 3 BOTH FREE LENGTHS ARE ZERO ....................... 17 3.1 Equilibrium Analysis .............................. 17 3.2 Numerical Example ............................... 19 4 ONE FREE LENGTH IS ZERO .......................... 21 4.1 Equilibrium Analysis .............................. 21 4.2 Numerical Example ............................... 24 5 BOTH FREE LENGTHS ARE NON-ZERO .................... 28 5.1 Equilibrium Analysis .............................. 28 5.2 Numerical Example ............................... 31 6 CONCLUSION .................................... 36 APPENDIX A SHORT INTRODUCTION TO THEORY OF SCREWS ............. 37 B SYLVESTER MATRIX ............................... 40 REFERENCES ....................................... 44 BIOGRAPHICAL SKETCH ................................ 45 4 LIST OF TABLES Table page 3-1 Six solutions for Case 1 ................................ 20 4-1 List of operations to obtain Sylvester Matrix for case 2 .............. 23 4-2 Twenty solutions for Case 2 ............................. 25 5-1 List of operations to obtain Sylvester Matrix for Case 3 .............. 34 5-2 Twenty-four solutions for Case 3 .......................... 35 5 LIST OF FIGURES Figure page 1-1 Biological model of the knee ............................. 11 1-2 Tensegrity based model of cross-section of knee .................. 11 2-1 Tensegrity mechanism ................................ 12 3-1 Four real solutions for Case 1 ............................ 20 4-1 Bifurcation diagram for solution of x1 and varying parameter L01 ........ 26 4-2 Eight real solutions for Case 2 ............................ 27 5-1 Twenty-four real solutions for Case 3 (cases 1 to 12) ................ 32 5-2 Twenty-four real solutions for Case 3 (cases 13 to 24) ............... 33 6 ACKNOWLEDGMENTS I would like to express profound gratitude to my advisor, Prof. Carl Crane, for his invaluable support, encouragement, supervision and useful suggestions throughout this research work. His moral support and continuous guidance enabled me to complete my work successfully. I would like to thank Prof. John Schueller and Prof. Warren Dixon for serving on my committee. I am also grateful to Prof. Jay Gopalakrishnan and Dr. Jahan Bayat for listening to my queries and answering my questions regarding the research work. I would like to acknowledge the support of the Department of Energy under grant number DE-FG04-86NE37967. I am especially indebted to my parents, Dr. Om Vikas and Mrs. Pramod Kumari Sharma, for their love and support ever since my childhood. I also wish to thank my brother, Pranav, for his constant support and encouragement. I thank my fellow students at the Center for Intelligent Machines and Robotics. From them, I learned a great deal and found great friendships. I would also like to thank Nicole, Piyush, Rakesh, Rashi and Sreenivas for their friendships. 7 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Ful¯llment of the Requirements for the Degree of Master of Science KINEMATIC ANALYSIS OF A PLANAR TENSEGRITY MECHANISM WITH PRE-STRESSED SPRINGS By Vishesh Vikas August 2008 Chair: Carl Crane Major: Mechanical Engineering This thesis presents the equilibrium analysis of a planar tensegrity mechanism. The device consists of a base and top platform that are connected in parallel by one connector leg (whose length can be controlled via a prismatic joint) and two spring elements whose linear spring constants and free lengths are known. The thesis presents three cases: 1) the spring free lengths are both zero, 2) one of the spring free lengths is zero and the other is nonzero, and 3) both free lengths are nonzero. The purpose of the thesis is to show the enormous increase in complexity that results from nonzero free lengths. It is shown that six equilibrium con¯gurations exist for Case 1, twenty equilibrium con¯gurations exist for Case 2, and no more than sixty two con¯gurations exist for Case 3. 8 CHAPTER 1 INTRODUCTION The human musculoskeletal system is often described as combinations of levers and pulleys. However, at a number of places (particularly the spine) this lever-pulley-fulcrum model of the musculoskeletal system calculates such extreme amount of forces that will tear muscles o® the bones and shear bones into pieces. This, however, does not happen in real life and can be explained by the concept of `tensegrity'. Tensegrity (abbreviation of `tensional integrity [1], [2]), is synergy between tension and compression. Tensegrity structures consist of elements that can resist compression (e.g., struts, bones) and elements that can resist tension (e.g., ties, muscles). The entire con¯guration stands by itself and maintains its form (equilibrium) solely because of the internal arrangement of the struts and ties ([3],[4]). No pair of struts touch and the end of each strut is connected to three non-coplanar ties ([5]). More formal de¯nition of tensegrity is given by Roth and Whiteley([6]), introducing a third element, the bar, which can withstand both compression and tension. Tensegrity structures can be broadly classi¯ed into two categories, prestressed and geodesic, where continuous transmission of tensional forces is necessary for shape stability or single entity of these structures([7]). `Prestressed tensegrity structures', hold their joints in position as the result of a pre-existing tensile stress within the structural network. `Geodesic tensegrity structures', triangulate their structure members and orient them along geodesics (minimal paths) to geometrically constrain movement. Our bodies provide a familiar example of a prestressed tensegrity structure: our bones act like struts to resist the pull of tensile muscles, tendons and ligaments, and the shape stability (sti®ness) of our bodies varies depending on the tone (pre-stress) in our muscles. Examples of geodesic tensegrity structures include Fullers geodesic domes, carbon-based buckminsterfullerenes (buckyballs), and tetrahedral space frames, which are of great interest in astronautics because they maintain their stability in the absence of gravity and, hence, without continuous compression. Idea of combining several basic tensegrity 9 structures to form a more complex structure has been analyzed([8], [9]) and di®erent methods to do so have also been studied([10]). There has been a rapid development in static and dynamic analysis of tensegrity structures in last few decades ([11]). This is due to its bene¯ts over traditional approaches in several ¯elds such as architecture ([12]), civil engineering, art, geometry and even bi- ology. Bene¯ts of tensegrity structures are examined by Skelton et al.([13]). Tensegrity structures display energy e±ciency as its elements store energy in form of compression or tension; as a result of the energy stored in the structure, the overall energy required to activate these structures will be small([14]). Since compressive members in tensegrity structures are disjoint, large displacements are allowed and it is possible to create deploy- able structures that can be stored in small volumes. Deployable antennas and masts are notable space applications ([15],[16]). Kenner established the relation between the rotation of the top and bottom ties. Tobie ([3]), presented procedures for the generation of tensile structures by physical and graphical means. Yin ([5]) obtained Kenner's ([17]) results using energy considerations and found the equilibrium position for unloaded tensegrity prisms. Stern ([18]) developed generic design equations to ¯nd the lengths of the struts and elastic ties needed to create a desired geometry for a symmetric case. Knight ([19]) addressed the problem of stability of tensegrity structures for the design of deployable antennae. On macro level, tensegrity structures are used to model human musculoskele- tal system, deployable antennae, architecture structures, etc.; on cellular level, Donald Ingber ([7]) proposes Cellular Tensegrity Theory in which the whole cell is modelled as a prestressed tensegrity structure, although geodesic structures are also found in the cell at smaller size scales. Stephen Levin ([14]) proposed a truss-tensegrity model of the spine mechanics. Bene¯ts of tensegrity structures make them interesting for designing mobile robots. Aldrich ([20]) has built and controlled robots based on tensegrity structures and Paul et al. ([21]) have built triangular prism based mobile tensegrity robot.s 10 Most of the papers in the ¯eld of tensegrity assume zero free length of springs. In real life systems, esp. biological systems, such assumption does not hold true. Following research shows that this assumption is not trivial, complexities to ¯nd solutions increase tremendously and the number of static equilibrium con¯gurations also increase. Cross-