Discrete Mechanics and Optimal Control Der Fakult¨at f¨urElektrotechnik, Informatik und Mathematik der Universit¨at Paderborn zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – vorgelegte Dissertation von Sina Ober-Bl¨obaum Paderborn, 2008 THE FUNDAMENTAL VARIATIONAL PRINCIPLE Namely, because the shape of the whole universe is the most perfect and, in fact, designed by the wisest creator, nothing in all the world will occur in which no maximum or minimum rule is somehow shining forth... Leonhard Euler (1744) iii Acknowledgements To begin, I would like to thank my advisor Prof. Dr. Michael Dellnitz for his guidance, support, and motivation, and, in particular, for the great freedom he has given me during my PhD research. I would also like to thank Prof. Dr. Oliver Junge from the Technische Univer- sit¨atM¨unchen for his extensive supervision over the last years. Professor Jerrold E. Marsden from the California Institute of Technology has provided valuable guidance during my studies. He introduced me to the area of discrete variational mechanics and provided interesting ideas for application. Mutual visits and discussions contributed to the progress of my research as well. Special thanks goes to my co-workers Dr. Sigrid Leyendecker, Dr. Kathrin Padberg and Mirko Hessel-von Molo for many interesting and enlightening dis- cussions, exciting joint work, and their constant support. I gratefully acknowledge support under the DFG-sponsored research project SFB 376 on Massively Parallel Computation. My colleages in Paderborn participated in lengthy discusssions on scientific and unscientific topics and I appreciate their technical and administrative sup- port, including Alessandro Dell’Aere, Olaf Bonorden, Tanja B¨urger,Sebastian Hage-Packh¨auser,Marianne Kalle, Stefan Klus, Anna-Lena Meyer, Marcus Post, Dr. Robert Preis, Marcel Schwalb, Stefan Sertl, Bianca Thiere, Julia Timmer- mann, and Katrin Witting. I am very grateful to all my friends, primarly Oliver Krimmer and Farina Schneider for proof-reading the manuscript. Special thanks goes to Kay Klobe- danz, who has been a great support and motivator over the last years. For support during the last months I would especially like to thank Helene Waßmann. Most of all, I thank my family. In particular, the constant support and en- couragement from my parents Ingetraud and Hartmut and my brother Mark has accompanied me through many years of studies. Finally, I thank Andreas Kohlos - for a number of things. iv Abstract The optimal control of physical processes is of crucial importance in all modern technological sciences. In general, one is interested in prescribing the motion of a dynamical system in such a way that a certain optimality criterion is achieved. Typical challenges are the determination of a time-minimal path in vehicle dy- namics, an energy-efficient trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In order to solve optimal control problems for mechanical systems, this thesis links the theory of optimal control with concepts from variational mechanics. The application of discrete variational principles allows for the construction of an optimization algorithm that enables the discrete solution to inherit characteristic structural properties from the continuous problem. The numerical performance of the developed method and its relationship to other existing optimal control methods are investigated. This is done by means of theoretical considerations as well as with the help of numerical examples arising in problems from trajectory planning and space mission design. The development of efficient approaches for exploiting the mechanical system’s structures reduce for example the computational effort. In addition, the optimal control framework is extended to mechanical systems with constraints in multi- body dynamics and applied to robotical and biomechanical problems. v Zusammenfassung Die optimale Steuerung physikalischer Prozesse ist in allen modernen techno- logischen Wissenschaften von wichtiger Bedeutung. Das Ziel ist es, die Bewe- gung eines dynamischen Systems so vorzuschreiben, dass ein bestimmtes Op- timlit¨atskriterium erreicht wird. Typische Anwendungen sind die Bestimmung zeitoptimaler Wege in der Fahrzeugdynamik, energieeffizienter Trajektorien von Raumfahrtmissionen oder optimaler Bewegungsabl¨aufein der Robotik und der Biomechanik. Diese Arbeit vereint die Theorie der optimalen Steuerung mit den Konzepten der Variationsmechanik, um Steuerungsprobleme mechanischer Systeme zu l¨osen. Die Anwendung diskreter Variationsprinzipien erm¨oglicht es, einen Optimierungs- algorithmus zu konstruieren, dessen L¨osungcharakteristische strukturelle Eigen- schaften des kontinuierlichen Problems erbt. Die numerische Effizienz der entwickelten Methode, sowie Vergleiche und Re- lationen zu existierenden optimalen Steuerungsmethoden, werden sowohl anhand theoretischer Betrachtungen als auch anhand numerischer Beispiele untersucht. Die Entwicklung effizienter Ans¨atze zur Ausnutzung der speziellen Struktur des mechanischen Systems reduziert beispielsweise den rechnerischen Aufwand. Abschließend wird die vorgestellte Methode dahingehend erweitert, dass sie sich auf mechanische Systeme mit Zwangsbedingungen in der Mehrk¨orperdynamik anwenden l¨asst. Dabei werden Probleme aus der Robotik und der Biomechanik behandelt. vii Contents 1 Introduction 1 2 Optimal control 13 2.1 Optimal control problem ....................... 13 2.1.1 Problem formulation ..................... 13 2.1.2 Necessary conditions for optimality ............. 15 2.2 Solution methods for optimal control problems ........... 16 2.2.1 Indirect methods ....................... 17 2.2.2 Direct methods ........................ 18 2.3 Solution methods for nonlinear constrained optimization problems 20 2.3.1 Local optimality conditions ................. 20 2.3.2 Sequential Quadratic Programming (SQP) ......... 21 2.4 Discussion of direct methods ..................... 23 3 Variational mechanics 25 3.1 Lagrangian mechanics ........................ 26 3.1.1 Basic definitions and concepts ................ 26 3.1.2 Discrete Lagrangian mechanics ............... 29 3.2 Hamiltonian mechanics ........................ 32 3.2.1 Basic definitions and concepts ................ 32 3.2.2 Discrete Hamiltonian mechanics ............... 35 3.3 Forcing and control .......................... 37 3.3.1 Forced Lagrangian systems .................. 37 3.3.2 Forced Hamiltonian systems ................. 38 3.3.3 Legendre transform with forces ............... 39 3.3.4 Noether’s theorem with forcing ............... 40 3.3.5 Discrete variational mechanics with control forces ..... 40 3.3.6 Discrete Legendre transforms with forces .......... 42 3.3.7 Discrete Noether’s theorem with forcing .......... 43 ix 4 Discrete mechanics and optimal control (DMOC) 45 4.1 Optimal control of a mechanical system ............... 45 4.1.1 Lagrangian optimal control problem ............. 46 4.1.2 Hamiltonian optimal control problem ............ 47 4.1.3 Transformation to Mayer form ................ 48 4.2 Optimal control of a discrete mechanical system .......... 49 4.2.1 Discrete Optimal Control Problem ............. 51 4.2.2 Transformation to Mayer form ................ 52 4.2.3 Fixed boundary conditions .................. 53 4.3 Correspondence between discrete and continuous optimal control problem ................................ 55 4.3.1 Exact discrete Lagrangian and forcing ........... 55 4.3.2 Order of consistency ..................... 58 4.3.3 Discrete problem as direct solution method ......... 62 4.4 High-order discretization ....................... 64 4.4.1 Quadrature approximation .................. 64 4.4.2 High-order discrete optimal control problem ........ 67 4.4.3 Correspondence to Runge-Kutta discretizations ...... 69 4.5 Adjoint system ............................ 76 4.5.1 Continuous setting ...................... 76 4.5.2 Discrete setting ........................ 79 4.5.3 The transformed adjoint system ............... 82 4.6 Convergence .............................. 85 5 Implementation, applications and extension 89 5.1 Implementation ............................ 89 5.2 Comparison to existing methods ................... 91 5.2.1 Low thrust orbital transfer .................. 91 5.2.2 Two-link manipulator ..................... 94 5.3 Application: Trajectory planning .................. 99 5.3.1 A group of hovercraft ..................... 99 5.3.2 Perfect underwater glider ................... 102 5.4 Application: Optimal control of multi-body systems ........ 104 5.4.1 The falling cat ........................ 104 5.4.2 A gymnast (three-link mechanism) ............. 109 5.5 Reconfiguration of formation flying spacecraft – a decentralized approach ................................ 116 6 Optimal control of constrained mechanical systems in multi-body dynamics 127 6.1 Constrained dynamics and optimal control ............. 128 x 6.1.1 Optimization problem .................... 128 6.1.2 Constrained Lagrange-d’Alembert principle ......... 129 6.1.3 Null space method ...................... 129 6.1.4 Reparametrization ...................... 130 6.2 Constrained discrete dynamics and optimal control ........ 130 6.2.1 Discrete constrained Lagrange-d’Alembert principle .... 130 6.2.2 Discrete null space method .................. 131 6.2.3 Nodal reparametrization ................... 132 6.2.4 Boundary conditions ..................... 133 6.2.5 Discrete constrained optimization problem ......... 135 6.3