CDS TECHNICAL MEMORANDUM NO. CIT-CDS 95-027 September, 1995 "The Mechanics and Control of Undulatory Robotic Lo~ornotion" Jaines Patrick Ostrowski Control and Dynamical Systems California Institute of Technology Pasadena, CA 91125 The Mechanics and Control of Undulatory Robotic Locomotion Thesis by James Patrick Ostrowski Technical Report for the Department of Mechanical Engineering Division of Engineering and Applied Science California Institute of Technology Pasadena, CA 91125 California Institute of Technology Pasadena, California (Thesis defended S+tember 19, 1995) @ 1996 James Patrick Ostrowski All rights Reserved I would like to begin by thanking my advisor, Joel Burdick, who has shown me the depth and breadth of being an advisor, a mentor, and a friend, and without whom none of this could have happened. I would also like to give a special thanks to the members of my thesis committee for their support: Drs. Richard Murray, Jerry Marsden, Greg Chirikjian, and Steve Wiggins. In particular, I must credit Richard and Jerry, along with my office-mate, Andrew Lewis, with exposing me to the beauty and history of geometry and mechanics. I have spent my time at Caltech standing on the shoulders of giants, and I thank them very much for the magnificent view. Next, I would like to thank the folks at Caltech who have made the past five years much more than just so many days in the office. To Richard Tsuyuki, who left Caltech long before all of us, but who I'm sure I will know for so much longer; to Andrew Lewis, for always humoring my idiosyncrasies and never denying me a bite of his dessert; to Howie Choset, for providing me with laughter, friendship, and an endless supply of spontaneous diversions; to Victor Burnley, a true compadre whom I could always count on to jump off bridges or drive to Vegas; to Scott Kelly, for always keeping the office full of toys and for being such a marble-head; and finally to Ted Hubbard, for showing me the trivia of the world and the virtues of being Canadian. While I have obviously missed a great many people, many of them are included in this special thanks to a gang of folks I ate lunch with every week for the past four years, the Thomas Lunch Group. This is a great tradition which I hope will continue to support lively discussions and seek out new places to eat in Pasadena for many years to come. On a slightly different note, I must add that my stay at Caltech was profoundly influenced and greatly improved by the time I spent as a Resident Associate in Fleming House. A great deal of credit must go to Kim West for putting together an amazing and supportive st& of RA's. But primarily, I would like to thank the students of Fleming House for allowing me to enter so deeply into their lives. Certainly I will always remember the dinners we shared together (napkins and water included) and the athletic accomplishments we achieved. Most importantly, though, I take with me the memory of the students themselves, and the three event-filled years we shared together. Of course, I would not nearly be complete without thanking my family who have been so close and supportive over these many, many years of education: Sarah and Frank, my favorite brother, Mike, and my very special grandmother, Sarah Valderrama. It is with great fondness and respect that I dedicate this thesis to my grandmother, Sarah, and to my other grandmother, Mary Ostrowski, who is no longer with us, but who gave me a childhood full of happy memories and a father who supported me in my journey through math and sciences. Finally, I must give a plentitude of credit and thanks (though this is not nearly enough) to my best friend and now wife, Liz Price. She is a beautiful woman who has enriched my life in ways and measures that I don't think she, nor I, will ever fully comprehend. She has been there with love and support through the good times and the bad, but most importantly, she has been my connection to the real world and has kept me sane over these past few years. The Mechanics and Control of Undulatory Robotic Locomotion James Patrick Ostrowski In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Abstract In this dissertation, we examine a formulation of problems of undulatory robotic lo- comotion within the context of mechanical systems with nonholonomic constraints and symmetries. Using tools from geometric mechanics, we study the underlying structure found in general problems of locomotion. In doing so, we decompose lo- comotion into two basic components: internal shape changes and net changes in position and orientation. This decomposition has a natural mathematical inter- pretation in which the relationship between shape changes and locomotion can be described using a connection on a trivial principal fiber bundle. We begin by reviewing the processes of Lagrangian reduction and reconstruc- tion for unconstrained mechanical systems with Lie group symmetries, and present new formulations of this process which are easily adapted to accommodate exter- Z* nal constraints. Additionally, important physical quantities such as the mechanical connection and reduced mass-inertia matrix can be trivially determined using this formulation. The presence of symmetries then allows us to reduce the necessary calculations to simple matrix manipulations. The addition of constraints significantly complicates the reduction process; how- ever, we show that for invariant constraints, a meaningful connection can be syn- t hesized by defining a generalized momentum representing the momentum of the system in directions allowed by the constraints. We then prove that the generalized momentum and its governing equation possess certain invariances which allows for a reduction process similar to that found in the unconstrained case. The form of the reduced equations highlights the synthesized connection and the matrix quantities used to calculate these equations. The use of connections naturally leads to methods for testing controllability and aids in developing intuition regarding the generation of various locomotive gaits. We present accessibility and controllability tests based on taking derivatives of the connection, and relate these tests to taking Lie brackets of the input vector fields. The theory is illustrated using several examples, in particular the examples of the snakeboard and Hirose snake robot. We interpret each of these examples in light of the theory developed in this thesis, and examine the generation of locomotive gaits using sinusoidal inputs and their relationship to the controllability tests based on Lie brackets. Contents 1 Introduction 1 1.1 A Survey of Locomotion ......................... 3 1.2 Theoretical Statement of Purpose .................... 12 2 Background 19 2.1 Symbols and Notation .......................... 19 2.2 Lie Groups and Associated Structures ................. 21 2.2.1 Lie Groups ............................ 21 2.2.2 Algebras .............................. 23 2.2.3 The Lie Algebra of G ...................... 23 2.2.4 Principal Fiber Bundles ..................... 26 2.2.5 Symmetries and Invariances ................... 28 2.3 The Euler-Lagrange Equations ..................... 32 2.4 Noether's Theorem ............................ 34 2.5 Distributions and F'robenius' Theorem ................. 35 2.6 Equations of Motion with Nonholonomic Constraints ......... 36 3 Lagrangian Reduction in the Absence of Constraints 42 3.1 Connections on Principal Fiber Bundles ................ 42 3.2 The Momentum Map and Mechanical Connection One-Form .... 47 3.3 Reduction Using the Routhian ..................... 52 3.4 Reduction Using the Constrained Lagrangian ............. 56 3.4.1 The Reduced Lagrangian .................... 56 3.4.2 The Reduced Nonholonomic Variational Principle ...... 58 3.4.3 The Constrained Lagrangian Approach ............ 61 4 The Generalized Momentum 65 4.1 Assumptions ............................... 65 4.2 The Constrained Lie Algebra ...................... 69 4.3 An Alternative Derivation of the Generalized Momentum Equation . 73 4.4 Invariance of the Generalized Momentum ............... 76 4.5 The Nonholonomic Connection ..................... 78 4.6 Synthesizing the Connection ....................... 81 4.7 Invariance of the Generalized Momentum Equation .......... 84 4.8 Reduction ................................. 89 4.9 Reconstruction .............................. 95 5 Applications of the Theory 97 5.1 The Snakeboard ............................. 97 5.1.1 The Generalized Momentum Equation ............. 100 5.1.2 Reconstruction ..........................109 5.2 The Kinematic Snake of Hirose ..................... 110 6 Controllability and Gaits 116 6.1 Background and Formulation ...................... 116 6.1.1 Free Lie Algebras ......................... 118 6.1.2 Accessibility and Controllability ................ 121 6.1.3 The Lie Algebra Rank Condition ................ 123 6.1.4 The Principal Kinematic Case ................. 124 6.1.5 Unconstrained Systems with Symmetries ........... 129 6.2 Local Accessibility ............................ 130 6.3 Local Controllability ........................... 133 6.4 Locomotive Gaits ............................. 137 6.5 Examples .................................138 6.5.1 The Snakeboard ......................... 138 6.5.2 The Kinematic Snake ...................... 144 7 Conclusions and Future Work 152 7.1 Conclusions ................................ 152 7.2 Future Work ............................... 155 7.2.1 Averaging Theory for Lie Groups ...............