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OPTIMAL CONTROL of NONHOLONOMIC MECHANICAL SYSTEMS By OPTIMAL CONTROL OF NONHOLONOMIC MECHANICAL SYSTEMS by Stuart Marcus Rogers A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics Department of Mathematical and Statistical Sciences University of Alberta ⃝c Stuart Marcus Rogers, 2017 Abstract This thesis investigates the optimal control of two nonholonomic mechanical systems, Suslov's problem and the rolling ball. Suslov's problem is a nonholonomic variation of the classical rotating free rigid body problem, in which the body angular velocity Ω(t) must always be orthogonal to a prescribed, time-varying body frame vector ξ(t), i.e. hΩ(t); ξ(t)i = 0. The motion of the rigid body in Suslov's problem is actuated via ξ(t), while the motion of the rolling ball is actuated via internal point masses that move along rails fixed within the ball. First, by applying Lagrange-d'Alembert's principle with Euler-Poincar´e'smethod, the uncontrolled equations of motion are derived. Then, by applying Pontryagin's minimum principle, the controlled equations of motion are derived, a solution of which obeys the uncontrolled equations of motion, satisfies prescribed initial and final conditions, and minimizes a prescribed performance index. Finally, the controlled equations of motion are solved numerically by a continuation method, starting from an initial solution obtained analytically (in the case of Suslov's problem) or via a direct method (in the case of the rolling ball). ii Preface This thesis contains material that has appeared in a pair of papers, one on Suslov's problem and the other on rolling ball robots, co-authored with my supervisor, Vakhtang Putkaradze. Chapter4 contains much of the material published in the paper [1] on Suslov's problem: Vakhtang Putkaradze and Stuart Rogers, \Constraint Control of Nonholonomic Mechanical Systems," Journal of Nonlinear Science, DOI: 10.1007/s00332-017-9406-1, 2017. Chapter5 contains much of the material in the paper [2] on rolling ball robots that has been submitted for publication in Optimal Control, Applications and Methods and posted online on arXiv: Vakhtang Putkaradze and Stuart Rogers, \Optimal Control of a Rolling Ball Robot Actuated by Internal Point Masses," arXiv preprint arXiv:1708.03829, 2017. iii Rome wasn't built in a day, but they were laying bricks every hour. iv Acknowledgements I wish to thank my supervisor, Vakhtang Putkaradze, for bringing me to the University of Alberta, suggesting the topics researched in this thesis, and showing me how to apply the calculus of variations to investigate mechanical systems. There were fruitful discussions with Profs. A.M. Bloch, D.V. Zenkov, M. Leok, and A. Lewis concerning geometric mechanics and optimal control. In particular, Prof D.V. Zenkov suggested the new variation of Suslov's problem investigated in this thesis, in which ξ may vary with time. Prof. H. Oberle suggested that I try to use the ODE TPBVP solver bvp4c for numerically solving the controlled equations of motion. Prof. L.F. Shampine provided advice on using the ODE TPBVP solvers bvp4c and bvp5c. Prof. F. Mazzia provided information on using the ODE TPBVP solvers TOM and bvptwp. J. Willkomm provided help on using the automatic differentiation software ADiMat and suggested that I investigate the alternative automatic differentiation software ADiGator. M. Weinsten provided copious advice on using ADiGator and fixed numerous bugs in ADiGator that were revealed in the course of this research. Prof. A. Rao fixed a critical bug in the direct method optimal control solver GPOPS-II that was revealed in the course of this research. In return, Prof. A. Rao provided me with a free license to use GPOPS-II for this research. I would like to thank the following funding sources: the University of Alberta Doctoral Recruitment Scholar- ship, the NSERC Discovery Grant, the University of Alberta Centennial Fund, the FGSR Graduate Travel Award, the IGR Travel Award, the GSA Academic Travel Award, the AMS Fall Sectional Graduate Stu- dent Travel Grant, and the Alberta Innovates Technology Funding (AITF) which came through the Alberta Centre for Earth Observation Sciences (CEOS). Finally, I thank all my teachers, both past and present. v Table of Contents 1 Introduction 1 1.A Motivation and Methodology for this Research..........................1 1.B Background on Suslov's Problem..................................3 1.C Background on the Rolling Ball...................................3 1.D Contributions and Thesis Summary.................................5 2 Mechanics 8 2.A Hamilton's Principle, Symmetry Reduction, and Euler-Poincar´e'sMethod...........8 2.B Nonholonomic Constraints and Lagrange-d'Alembert's Principle................. 15 2.C A Simple Nonholonomic Particle.................................. 16 3 Optimal Control 22 3.A Calculus of Variations........................................ 22 3.B Pontryagin's Minimum Principle.................................. 23 3.C Implementation Details for Solving the Optimal Control ODE TPBVP............. 26 4 Suslov's Problem 33 4.A Suslov's Optimal Control Problem for an Arbitrary Group.................... 33 4.A.1 Derivation of Suslov's Uncontrolled Equations of Motion................. 33 4.A.2 Derivation of Suslov's Controlled Equations of Motion.................. 35 4.B Suslov's Optimal Control Problem for Rigid Body Motion.................... 38 vi 4.B.1 Derivation of Suslov's Uncontrolled Equations of Motion................. 38 4.B.2 Controllability and Accessibility of Suslov's Uncontrolled Equations of Motion.... 42 4.B.3 Suslov's Optimal Control Problem............................. 46 4.B.4 Derivation of Suslov's Controlled Equations of Motion.................. 47 4.C Numerical Solution of Suslov's Optimal Control Problem..................... 57 4.C.1 Analytical Solution of a Singular Version of Suslov's Optimal Control Problem.... 57 4.C.2 Numerical Solution of Suslov's Optimal Control Problem via Continuation....... 58 4.C.3 Numerical Solution of Suslov's Optimal Control Problem via the Indirect Method and Continuation......................................... 59 5 The Rolling Ball 64 5.A Mechanical System and Motivation................................. 64 5.A.1 Coordinate Systems and Notation............................. 65 5.B Uncontrolled Equations of Motion................................. 67 5.B.1 Kinetic Energy, Potential Energy, and Lagrangian.................... 67 5.B.2 Rolling Constraint and Lagrange-d'Alembert's Principle................. 69 5.B.3 Uncontrolled Equations of Motion for the Rolling Ball.................. 73 5.B.4 Uncontrolled Equations of Motion for the Rolling Disk.................. 77 5.C Controlled Equations of Motion................................... 84 5.C.1 Controlled Equations of Motion for the Rolling Disk................... 84 5.C.2 Controlled Equations of Motion for the Rolling Ball................... 89 5.D Numerical Solutions of the Controlled Equations of Motion................... 98 5.D.1 Simulations of the Rolling Disk............................... 99 5.D.2 Simulations of the Rolling Ball............................... 102 6 Conclusions 111 6.A Summary and Conclusions...................................... 111 6.B Future Work............................................. 113 vii Bibliography 116 Appendices 125 A Survey of Numerical Methods for Solving Optimal Control Problems: Dynamic Pro- gramming, the Direct Method, and the Indirect Method 125 B Calculation Connecting Classical and Reduced Costates for Suslov's Optimal Control Problem 129 C Predictor-Corrector Continuation Method for Solving an ODE TPBVP 132 C.1 Introduction.............................................. 132 C.2 A Hilbert Space............................................ 133 C.3 The Fr´echet Derivative and Newton's Method........................... 133 C.4 The Davidenko ODE IVP...................................... 134 C.5 Construct the Tangent........................................ 135 C.6 Normalize the Tangent........................................ 138 C.7 Construct the Tangent Predictor.................................. 138 C.8 Construct the Corrector....................................... 139 C.9 Polish the Corrector......................................... 142 C.10 Pseudocode for Predictor-Corrector Continuation......................... 143 D Sweep Predictor-Corrector Continuation Method for Solving an ODE TPBVP 146 D.1 Introduction.............................................. 146 D.2 Construct the Tangent........................................ 146 D.3 Determine the Tangent Direction.................................. 148 D.4 Sweep along the Tangent....................................... 149 D.5 Pseudocode for Sweep Predictor-Corrector Continuation..................... 151 E Quaternions 153 viii List of Figures 1.1 Examples of real rolling ball robots.................................1 1.2 Suslov's problem studies the motion of a rotating rigid body subject to hΩ(t); ξ(t)i = 0. In Suslov's original formulation, ξ(t) ≡ ξ is fixed (i.e. does not vary with respect to time) inthe body frame...............................................3 1.3 Idealized realization of Suslov's problem using a pair of diametrically opposed ice skates wedged inside a spherical ice shell. The grey shaded blob represents the rigid body rotat- ing about the fixed point O. At this time instant, the fixed vector ξ lies in the plane of the page and is orthogonal to the rotation axis, which points out of or into the page and which is parallel
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