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Fast Online Control Based on Homotopies for Systems Subject To Damian Kontny Fast Online Control based on Homotopies for Systems subject to Time -Varying Constraints Damian Kontny kassel Fast Online Control based on Homotopies for Fast Online Control based on Homotopies for Systems subject to Time-Varying Constraints university press Damian Kontny Fast Online Control based on Homotopies for Systems subject to Time-Varying Constraints kassel university press This work has been accepted by Faculty of Electrical Engineering / Computer Science of the University of Kassel as a thesis for acquiring the academic degree of Doktor der Ingenieur- wissenschaften (Dr.-Ing.). Supervisor: Prof. Dr.-Ing Olaf Stursberg Co-Supervisor: Prof. Dr.-Ing Knut Graichen Defense day: 19th February 2020 This document – excluding quotations and otherwise identified parts – is licensed under the Creative Commons Attribution-Share Alike 4.0 International License (CC BY-SA 4.0: https://creativecommons.org/licenses/by-sa/4.0/) Bibliographic information published by Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.dnb.de. Zugl.: Kassel, Univ., Diss. 2020 ISBN 978-3-7376-0870-1 DOI: https://dx.doi.org/doi:10.17170/kobra-202008131567 © 2020, kassel university press, Kassel http://kup.uni-kassel.de Printed in Germany Contents Summary vii I. Introduction and Theoretical Background 1 1. Introduction 3 2. Literature Review 7 2.1.ConfigurationSpace.......................... 8 2.2.PlanningMethods............................ 9 2.2.1.Roadmaps............................ 9 2.2.2. Planning based on Artificial Potential Fields . 11 2.2.3. Incremental Sampling-Based Approaches . 12 2.2.4.Optimization-BasedApproaches................ 14 2.2.5. Homotopy Based Methods in Optimization . 15 2.3.ContributionofthisThesis....................... 16 3. Definitions and Preliminaries 19 3.1.SystemDynamics............................ 19 3.2.LinearMatrixInequalities....................... 20 3.3.Pseudoinverse.............................. 21 3.4.MatrixDecomposition......................... 22 3.5.LinearizationandTimeDiscretization................. 23 3.6.SimilarityTransformation....................... 24 3.7. Controllability . .................. 24 3.8.ModelPredictiveControl........................ 25 3.9.Mixed-IntegerProgramming...................... 26 II. Homotopic Control Algorithms for Linear Systems 29 4. Optimizing Control using Homotopy Properties 31 4.1.HomotopicFunctions.......................... 33 4.2.ProblemDescription.......................... 36 4.3. Transformation into the Homotopy Space . 39 4.4.OfflineControllerSynthesis...................... 43 iii Contents 4.5.OnlineControlwithStateConstraints................ 47 4.5.1. Mapping of Obstacle Passing Points into the Homotopy Space 48 4.5.2. Selection of a Homotopic Target Trajectory . 55 4.6.SimulationResults........................... 58 4.7. Discussion . .................... 64 5. Extension to Input Constraints 67 5.1.ProblemDescription.......................... 68 5.2.OfflineControllerSynthesis...................... 71 5.3.NumericalExample........................... 79 5.4. Discussion . .................... 83 6. Homotopic Control Algorithms for Predicted Constraints 85 6.1.ProblemDescription.......................... 86 6.2.TransitionBetweenHomotopicTrajectories.............. 87 6.3.OnlineControl............................. 92 6.3.1.OptimalHomotopicTrajectory................ 93 6.3.2. Transformation of the Obstacle into the Homotopy Space . 94 6.3.3.HomotopicControlAlgorithm................. 97 6.4.NumericalExample........................... 105 6.5. Discussion . .................... 108 III. Homotopic Control Algorithms for Nonlinear Systems 111 7. Homotopic Control for Nonlinear Systems 113 7.1.ProblemDescription.......................... 114 7.2. Online Determination and Adaptation of Base Trajectories . 115 7.3.OnlineControlforNonlinearDynamics................ 122 7.3.1. Optimal Homotopic Trajectory without Obstacles . 122 7.3.2. Obstacle Transformation into the Homotopy Space . 123 7.3.3. Online Homotopic Control Algorithm . 124 7.4.NumericalExample........................... 128 7.4.1.ModelofaRoboticManipulator................ 128 7.4.2.SimulationResults....................... 135 7.5. Discussion . .................... 142 8. Homotopic Control for Systems with Polytopic Space Occupancy 145 8.1.ProblemDefinition........................... 145 8.2. Consideration of Polytopes for Collision Avoidance . 146 8.3. Homotopic Control AlgorithmwithParticles............. 150 8.4. Simulation Results for Body Circumvention . 153 8.5. Discussions of the Particle Approach . 156 iv Contents 9. Cooperative and Distributed Homotopic Control 157 9.1.LiteratureReview............................ 157 9.2.ProblemDefinition........................... 159 9.3. Cooperative Homotopic Control Algorithm . 161 9.4. Simulation Results for MultipleCooperativeRobots......... 170 9.5. Discussion of the Cooperative Control Method . 175 IV. Conclusion 177 10.Conclusion and Future Research 179 10.1.Conclusion................................ 179 10.2.FutureResearch............................. 183 List of Symbols 185 References 197 v Summary The integration of intelligent, autonomously acting systems into modern society is a rapidly growing field. After robots are established for simple, recurring processes in industry and everyday life, more complex tasks are of interest which require the systems to consider the environment in their decision making process. In the future, intelligent systems will get access to fields like autonomous driving cars, unmanned aerial vehicle (UAV), manufacturing processes, household, or the assistance to peo- ple in need of care. The fast calculation of optimal circumventing trajectories is therefore an essential component to be able to integrate intelligent systems into our environment at all. The ambitious goal of real-time interaction between a human and an autonomous system is challenging. An autonomous system has to react timely on human motion such that a real interaction can be established. Thus, the autonomous system must continuously capture its environment and adapt its solution. If the system additionally determines a solution with respect to a certain optimization criterion, the computation times quickly rise, and real-time capability moves far away. To solve this problem, the thesis proposes an algorithmic control procedure which determines optimal collision-free trajectories fast. Therefore, a concept which uses homotopy properties in the control procedure is introduced. This allows to deter- mine near-optimal solutions much faster than by commonly used techniques. At the beginning, an algorithmic procedure is shown for linear systems. It selects an optimized, circumventing trajectory based on the current obstacle location, and adapts its trajectory when the obstacle moves. Since real physical systems always underlie actuator limitations like e.g. motor torques, the provided method also considers input constraints. The developed method is subsequently extended to consider predictions of a moving obstacle. Thus, the procedure can further reduce the costs of the executed trajectory and is able to detect and avoid a collision at an early stage. Since many real-world system are described by nonlinear dynamics, the intro- duced method is also extended to nonlinear systems. The effectiveness of the pro- posed approach is shown in several simulations. Motivated by the results, the developed approach is extended to more complex tasks, like the collision avoidance between geometric bodies, and to multi agent systems that cooperatively determine solution trajectories in real-time, by means of the homotopy properties. Simulation results of a collision avoidance scenario for a robotic manipulator show, that with the proposed technique good results can be obtained in real-time. vii Contents Acknowledgments First of all, I thank my supervisor Prof. Dr.-Ing. Olaf Stursberg for the continuous support of my research, for his motivation and ideas. I am grateful for the freedom that he gave me and his great guidance. I also would like to thank the complete team of the Institute of Control and System Theory, where the research for this thesis was conducted. The fruitful discussions regarding all questions in control were always helpful. Furthermore, I am grateful for the financial support by the EU through the H2020-project UnCoVerCPS. Finally, I would like to thank my family and friends for the encouragement and enduring support during the past years. viii Part I. Introduction and Theoretical Background 1 1. Introduction In this thesis, control algorithms for non-convex point-to-point optimization prob- lems of dynamical systems are developed. Point-to-point optimization problems appear in many situations, and thus represent a basic problem in control theory. The task of such an optimization problem is to bring the dynamical system from an initial state to a target state optimally regarding to a given performance func- tion. A practical example is the movement of a robotic manipulator as typically used in industrial processes. The task of the robot may be to grasp a workpiece and bring it to a new point, where, e.g. further processing steps take place. The dynamic behavior of the system is usually described by mathematical equations, or in particular, by ordinary differential equations (ODE). These equations describe the change of the system states
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