Modeling and Control of Multi-Elastic-Link Robots under Gravity From Oscillation Damping and Position Control to Physical Interaction DISSERTATION submitted in partial fulfillment of the requirements for the degree Doktor Ingenieur (Doctor of Engineering) in the Faculty of Electrical Engineering and Information Technology at TU Dortmund University by Dipl.-Ing. Jörn Malzahn Essen, Germany Date of submission: 10th February 2014 First examiner: Univ.-Prof. Dr.-Ing. Prof. h.c. Dr. h.c. Torsten Bertram Second examiner: Univ.-Prof. Dr.-Ing. Dr. h.c. Burkhard Corves Date of approval: 27th October 2014 Preface Robotics is a fascinating scientific discipline. By developing robots we assemble metal or plastic parts, wires as well as integrated circuits. Through software, we try to equip this assembly of lifeless components with a certain degree of apparent autonomy, which makes it move and interact with our environment. We struggle to convert basic movements and interactions into useful skills enabling e.g. robust locomotion on uneven terrain or dexterous manipulation of arbitrary objects. The harder we struggle, the more exciting becomes the moment, when a robot finally accomplishes some desired task. The struggle also completely changes our everyday perspective on how impressively rapid, easy and reliable humans learn and adapt to the various complex situations in life. Just look at young children starting their first grasping experiments... When I started studying electrical engineering at TU Dortmund University, my plan was to dive into this fascinating world of robotics with all its facets as a robot developer. While writing these lines I can look back and say the plan seems to have worked out pretty well so far. It would not have worked out without the continuous unconditional support and patience of my parents Traudel and Hein-Peter Malzahn, for which I am deeply grateful. I also thank Prof. Dr.-Ing. Prof. h.c. Dr. h.c. Torsten Bertram for his feedback and forward-looking strategic debates as well as the freedom to develop my work in the direction I wanted and to the extent it finally has. I would like to thank Prof. Dr.-Ing. Dr. h.c. Burkhard Corves, who agreed to review my thesis as the second examiner. Furthermore I would like to thank Prof. Dr.-Ing. Peter Krummrich for his valuable comments about my work as well as for being the third examiner. Many thanks go to my colleague Anh Son Phung, not only for all the time we spent working on TUDOR and writing papers together, but also for introducing me into the Vietnamese cuisine. A person who deserves many thanks is Jan Braun, who is not only a good friend to me. He proved to have a lot of patience with my personal impatience in learning SolidWorks and thought me a lot about mechanical design. He is a reliable source of valuable feedback on my ideas. This also applies to Johannes Krettek, who, as a friend, has played a good devil’s advocate so many times. I am thankful for the friendliness and the comradeship of the remaining staff at the Institute of Control Theory and Systems Engineering (RST). In particular I would like to mention Martin Keller and Malte Oeljeklaus for our discussions and their III comments on my work, Frank Hoffmann for vibrant controversies without resent- ment, Jürgen Limhoff for the quick assistance with the laboratory hard- and software infrastructure, Mareike Leber and Gabriele Rebbe for their kind assistance with ad- ministrative issues. I am happy to remember sharing experiences in thesis writing with Christian Häger- ling during our weekly "‘writer’s coffee corner"’. I would like to thank Arne Nordmann for inspiring conversations during our stud- ies as well as the Robotics Round Table NRW, which we founded together. The Ro- botics Round Table NRW brought me into contact with many other fascinating ro- boticists. One of them is Felix Reinhart from Bielefeld University, with whom I really enjoyed working together. It was a pleasure to supervise many students and especially Ribin Balachandran, Fabian Bürger, Philipp Gorzcak, Alexander Sapadinski, who allowed me to drop some of my ideas on them. The last words of gratitude are dedicated to all my friends, who I have not men- tioned individually by name. They helped me to find distraction and relaxation, but also understanding when it was needed. Thank you! IV Contents Nomenclature VIII 1. Introduction 1 1.1. Motivation.................................... 1 1.2. Relatedwork................................... 5 1.3. Contribution and outline . 11 2. Experimental Setup 14 2.1. Jointsandlinks ................................. 14 2.2. Strainsensors .................................. 15 2.3. Eyeinhandcamera............................... 19 2.4. Referencesensors ............................... 21 2.5. Communication architecture . 25 3. Joint Dynamics and Control 27 3.1. Jointdynamics.................................. 27 3.2. Jointangularcontrol .............................. 29 3.3. Controllerevaluation. 30 3.4. Generalcontrolarchitecture . 32 4. Elastic Link Dynamics Analysis 33 4.1. Preliminaryassumptions. 33 4.2. Theequationofmotion ............................ 33 4.3. Thegeneralsolution .............................. 35 4.4. Special solutions to the boundary value problem . 36 4.5. Natural frequencies under varying boundary conditions . 38 4.6. Mode shapes under varying load mass and inertia . 40 4.7. Frequencymeasurements . 41 4.8. Impactofbacklashundergravity. 42 4.9. Conclusions for elastic link robots . 44 5. Proportional Oscillation Feedback 46 5.1. Linktransferfunctionmodel . 46 5.2. Controllersynthesis. 50 5.3. Controllerevaluation. 53 V Contents 6. Lumped Parameter Wave Echo Control 55 6.1. Wave properties in a lumped mass model . 55 6.2. Waveabsorption................................. 57 6.3. Wave component separation . 58 6.4. Lumpedwaveimpedance ........................... 58 6.5. Controllerreduction .............................. 59 6.6. Controllerevaluation. 61 7. Spatially Continuous Wave Echo Control 62 7.1. Continuouswavevariables . 62 7.2. Reflection and transmission at junctions and boundaries . ..... 63 7.3. Nearfieldcontribution............................. 64 7.4. Reflectionmatrixshaping . 65 7.5. Approximation of the half-integrator . 69 7.6. Controllerevaluation. 70 8. Experimental Damping Comparison 71 8.1. Experimentdesign ............................... 71 8.2. Whole workspace step responses . 72 8.3. Varyingpayloads ................................ 74 8.4. Disturbancerejection ... .... .... .... .... .... .... ... 75 8.5. Dampingwithasingleactuator. 78 8.6. Discussion .................................... 79 9. End Effector Control 81 9.1. Visualservoing ................................. 81 9.2. Databasedkinematics ............................. 85 9.3. Ballcatching................................... 88 10. Damped Dynamics Modelling 92 10.1.Motorcurrentmodel .............................. 92 10.2.Linkstrainmodel................................ 93 10.3. Data-driven reference model . 94 10.4.Identification................................... 95 10.5.Validation .................................... 97 10.6.Discussion .................................... 100 11. Collision Detection and Reaction 102 11.1.Collisiondetectionandisolation . 102 11.2.Collisionreaction . 105 11.3.Experimentalresults . 106 11.4.Discussion .................................... 112 12. Conclusion and Outlook 113 A. Hardware Parameters 117 A.1.Elasticlinks ................................... 117 VI Contents A.2.Computersystems ............................... 117 A.3.Actuators..................................... 118 A.4.Sensors...................................... 119 B. Mathematical Definitions and Derivations 121 B.1. Equivalences of trigonometric, hyperbolic and exponential functions . 121 B.2. Derivatives of general solutions to the boundary value problem . 122 B.3. Characteristic equation of the boundary value problem . 122 B.4. Performancemetrics .............................. 127 B.5. Stereocameraaccuracy. 128 B.6. Jointaccelerationprofile . 130 C. Supplemental Collision Experiments 131 C.1. Bluntimpactswithacompliantobject . 131 C.2. Sharpimpactswithafragileobject . 131 C.3. Sharpimpactsonahumanarm. 134 D. Steps to Deploy the Techniques 135 Bibliography 137 VII Nomenclature The following list explains all abbreviations and symbols used throughout this work. In general scalar symbols are represented by normal font letters. Vectors are expressed as bold lower case letters, matrices are indicated by bold upper case letters. If not ob- vious, coordinate frames of reference are given by leading superscripts to the symbol. For coordinate transformations the leading superscript indicates the original frame, while the leading subscript denotes the target frame. Where required, integration variables are written in Gothic print. L general beam length aˆ1...4 coefficients of the hyperbolic solution to the beam deflection ODE aˆ5,6 coefficients of the solution to the beam temporal ODE + + aˆ, −aˆ, aˆn, −aˆn amplitudes of propagating wave and near field components + a, −a vectors of rightwards and leftwards directed wave components + + a, −a, an, −an wave variables for the propagating wave and near field com- ponents ag vector of gravitational acceleration C joint referred robot matrix of Coriolis and centrifugal torques Cε strain referred robot matrix of Coriolis and centrifugal torques vc camera velocity cu modal stiffness Du modal attenuation factor du modal damping E Youngs modulus E unit matrix aE end effector acceleration
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