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Robust Trajectory Planning for Unmanned Aerial Vehicles in Uncertain Environments Brandon ... PDF

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Robust Trajectory Planning for Unmanned Aerial Vehicles in Uncertain Environments by Brandon Luders B.S., Aerospace Engineering Georgia Institute of Technology (2006) Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2008 (cid:13)c Massachusetts Institute of Technology 2008. All rights reserved. Author .............................................................. Department of Aeronautics and Astronautics August 28, 2008 Certified by.......................................................... Jonathan How Professor Thesis Supervisor Accepted by......................................................... Prof. David L. Darmofal Associate Department Head Chair, Committee on Graduate Students 2 Robust Trajectory Planning for Unmanned Aerial Vehicles in Uncertain Environments by Brandon Luders Submitted to the Department of Aeronautics and Astronautics on August 28, 2008, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract As unmanned aerial vehicles (UAVs) take on more prominent roles in aerial missions, it becomes necessary to increase the level of autonomy available to them within the mission planner. In order to complete realistic mission scenarios, the UAV must be capable of operating within a complex environment, which may include obstacles and other no-fly zones. Additionally, the UAV must be able to overcome environmental uncertainties such as modeling errors, external disturbances, and an incomplete situa- tional awareness. By utilizing planners which can autonomously navigate within such environments, the cost-effectiveness of UAV missions can be dramatically improved. This thesis develops a UAV trajectory planner to efficiently identify and execute trajectories which are robust to a complex, uncertain environment. This planner, namedEfficientRSBK,integratespreviousmixed-integerlinearprogramming(MILP) pathplanningalgorithmswithseveralimplementationinnovationstoachieveprovably robuston-linetrajectoryoptimization. Usingtheproposedinnovations, theplanneris abletodesignintelligentlong-termplansusingaminimalnumberofdecisionvariables. The effectiveness of this planner is demonstrated with both simulation results and flight experiments on a quadrotor testbed. Two major components of the Efficient RSBK framework are the robust model predictive control (RMPC) scheme and the low-level planner. This thesis develops a generalized framework to investigate RMPC affine feedback policies on the distur- bance, identify relative strengths and weaknesses, and assess suitability for the UAV trajectory planning problem. A simple example demonstrates that even with a con- ventional problem setup, the closed-loop performance may not always improve with additional decision variables, despite the resulting increase in computational com- plexity. A compatible low-level troller is also introduced which significantly improves trajectory-following accuracy, as demonstrated by additional flight experiments. Thesis Supervisor: Jonathan How Title: Professor 3 4 Acknowledgments First, I would like to thank Professor How for being a constant source of guidance and wisdom in my research pursuits at MIT, leading to this thesis. Regardless of the problem I encounter in my work, he always several handfuls of ideas to try. Professor How has played a critical role in helping me identify my personal research interests and apply them to projects such as this thesis. This research has been funded by the Air Force Office of Scientific Research, USAF, under grant FA9550-08-1-0086. Muchoftheworkrelatedtorobustmodelpredictivecontrol(RMPC)inthisthesis was inspired by the prior work and discussions of Professor Arthur Richards and Dr. Yoshiaki Kuwata, both of whom have provided valuable insights. Through frequent report revisions and e-mail exchanges, they (along with Professor How) have been essential in refining the state of our current RMPC research. The members of the Aerospace Controls Laboratory have been extremely support- ive throughout my work on this thesis, whether through proofreading this document, helping me reason through a problem, or simply offering helpful advice. I would like to offer a special thanks to Brett Bethke and Josh Redding, who graciously offered to read this document and suggest revisions. I would also like to thank Spencer Ahrens, Luc Brunet, Cameron Fraser, and Frant Sobolic for the specific ways in which they have assisted me. Above all, I would like to thank Kathryn Fischer for her support and resourcefulness throughout my time here. Finally, my family has been a constant source of love and inspiration for me throughout my education. So, to my parents, Brian and Lori, and my brother Chris, thank you for always being there for me and supporting my aspirations. 5 6 Contents 1 Introduction 17 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2.1 Robust Model Predictive Control . . . . . . . . . . . . . . . . 21 1.2.2 Mixed-Integer Linear Programming . . . . . . . . . . . . . . . 24 1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.2 Success Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.3.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 Affine Feedback Policies in Robust Model Predictive Control 29 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.3 Nominal MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Disturbance RMPC Policies . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1 Constraint Tightening . . . . . . . . . . . . . . . . . . . . . . 35 2.3.2 Affine Feedback Parametrization . . . . . . . . . . . . . . . . 38 2.3.3 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4.1 Disturbance Free Cost . . . . . . . . . . . . . . . . . . . . . . 43 7 2.4.2 Expected Cost . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.3 Worst Case Cost . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 Analysis and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5.2 Terminal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.5.3 Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.4 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3 Robust Trajectory Planning using Mixed-Integer Linear Program- ming 67 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 CT-Based Trajectory Planning Formulation . . . . . . . . . . . . . . 70 3.4 Robust Safe but Knowledgeable (RSBK) Algorithm . . . . . . . . . . 73 3.4.1 Trajectory Safety . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.2 Cost-to-Go Map . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5 Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5.1 Selective CT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5.2 Linear Interpolation Points . . . . . . . . . . . . . . . . . . . . 82 3.5.3 Detection Radius . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.5.4 Variable-Density Constraint Selection . . . . . . . . . . . . . . 83 3.6 Model-Specific Refinements . . . . . . . . . . . . . . . . . . . . . . . 88 3.6.1 Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.6.2 Obstacle Expansion . . . . . . . . . . . . . . . . . . . . . . . . 91 3.6.3 Obstacle Reachable Horizon . . . . . . . . . . . . . . . . . . . 92 3.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.7.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 8 4 Experimental Results 111 4.1 The Quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.2 RAVEN Testbed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.2.1 User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.2 Vehicle Manager . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.3 Low-Level Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.3.1 Termination Criteria . . . . . . . . . . . . . . . . . . . . . . . 118 4.3.2 Trajectory Generator . . . . . . . . . . . . . . . . . . . . . . . 118 4.3.3 Reference Controller . . . . . . . . . . . . . . . . . . . . . . . 120 4.4 Flight Results: Low-Level Controller . . . . . . . . . . . . . . . . . . 124 4.4.1 Stationary Hover . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4.2 Single-Dimension Tests . . . . . . . . . . . . . . . . . . . . . . 126 4.4.3 Advanced Trajectories . . . . . . . . . . . . . . . . . . . . . . 130 4.5 Flight Results: Efficient RSBK . . . . . . . . . . . . . . . . . . . . . 134 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5 Conclusion 139 5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.1.1 Disturbance-Aware Cost-to-go . . . . . . . . . . . . . . . . . . 141 5.1.2 Expansion of RMPC Analysis . . . . . . . . . . . . . . . . . . 141 5.1.3 Intelligent Selection of Cost-to-Go Nodes . . . . . . . . . . . . 142 5.1.4 Aerobatic Quadrotor Control . . . . . . . . . . . . . . . . . . 142 9 10

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As unmanned aerial vehicles (UAVs) take on more prominent roles in aerial missions, In order to complete realistic mission scenarios, the UAV must be.
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