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Nonlinear Control of an Aerobatic RC Airplane

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Nonlinear Control of an Aerobatic RC Airplane Nonlinear Control of an Aerobatic RC Airplane MASSA CHUSETTS INSTITUTE by 0F TECHNOLOGY Joshua John Bialkowski J UN 2 3 2010 B.S., Georgia Institute of Technology (2007) LIBRARIES 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 ARCHIVES at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2010 @ Massachusetts Institute of Technology 2010. All rights reserved. Author .......... .~. .....-.--...--.-.- D ar o % autics and Astronautics May 21, 2010 Certified by..... Prof Emilio Frazzoli Associate Professor of Aeronautics and Astronautics Thesis Supervisor / / IA Accepted by............. / Eytan H. Modiano Associate Professor of Aeronautics and Astronautics Chair, Committee on Graduate Students Nonlinear Control of an Aerobatic RC Airplane by Joshua John Bialkowski Submitted to the Department of Aeronautics and Astronautics on May 21, 2010, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract An automatic flight controller based on the ideas of backstepping is applied to an aerobatic RC airplane. The controller asymptotically tracks a time-parameterized position reference, and depends on an orientation look-up rule to detemine the vehicle orientation from a desired acceleration. A coordinated-flight look-up rule compatible with the controller provides a nominal level of capability for traditional flight trajec- tories. A generalized coordinated look-up rule compatible with the controller provides more advanced capability, including stability for high angle of attack and hovering maneuvers, at the expense of an additional requirement from the reference trajectory. Basic simulation results are used to verify the controller, and a simulation software framework is described which will enable more extensive simulation and provide a platform for the final controller implementation. Thesis Supervisor: Prof. Emilio Frazzoli Title: Associate Professor of Aeronautics and Astronautics 4 Acknowledgments Foremost, I thank my family for supporting me in my academic pursuits and always taking an interest in my work, even if I couldn't always provide a simple explanation of what that was. I also thank my advisor, Emilio Frazzoli for the many excellent ideas and the advice he has given me, as well as the other professors here at MIT: Jon How, John Deyst, and Jean-Jaques Slotine, who have provided me with the foundations of everything I've learned in this field. Thank you also to my professors from Georgia Tech's Aerospace Engineering de- partment, especially J.P. Clarke, Wassim Haddad, Panagiotis Tsiotras, Eric Johnson, and Jerry Seitzman, for the excellent education that has prepared me well for my graduate studies. Lastly, I would like to thank the contributors, maintainers, and communities of the many open-source software packages that made it possible to produce this document, and that I've used in my research. In particular, thank you Irrlicht, wxWidgets, and FFMPEG communities for these excellent libraries I have used in my simulations, the GIT and SVN communities for the invaluable version control software that I've used. Thank you LaTeX, Tikz/PGF, and Inkscape maintainers and communities for the fantastic software that made creating this document possible. 6 Contents I Introduction 1.1 M otivation . .. .. .. .. ... .. .. .. 1.2 Problem Description .............. 1.3 Previous Work .. .. .. .. .. .. .. 1.4 Overview of Chapters . .. .. .. .. .. 2 Aircraft Model 17 2.1 The Aircraft .. .. .. .. .. .. .. .. 17 2.2 Aerodynamic Model . .. ... .. .. .. 20 2.3 Propeller Model . .. .. .. .. .. .. 24 2.4 Complete Force Model .. .. .. .. .. 25 2.5 Finite Section Model .. .. .. .. .. .. 25 3 State Parametrization 27 3.1 State ......................... 27 4 Design of a Non-linear Controller Inspired by Backstepping 31 4.1 Overview of Backstepping .. .. .. .. ... ...... .. 31 4.2 Backstepping Applied to Flight Control .. .... ..... .. 33 4.3 Translational Controller .. .. ... .. .. ...... ... 35 4.4 Orientation Controller ... .. .. ... .... ...... 36 4.5 Backstepping the Acceleration . .. .. .. ...... .... 38 7 5 Coordinated Flight 43 5.1 Orientation Look-up . .. ... .. .. .. ... .. .. ... .. .. 43 5.2 Lower-bound Saturation on Acceleration .. ... .. ... .. ... 47 5.3 Simulation Results and Limitations . .. ... .. ... .. ... .. 48 6 3-Coordinated Flight 51 6.1 Definition of 3-Coordinated Flight .. ... ... .. ... ... ... 52 6.2 #-Coordinated Orientation Solution . ... .. ... .. ... .. ... 53 6.3 Simulation Results and Limitations . .. ... .. ... .. ... .. 56 7 Simulation Architecture 59 7.1 A rchitecture .. ... ... .. ... ... ... .. ... ... ... .. 60 7.2 Object-Oriented Design. ... ... ... ... ... ... ... ... 62 7.3 Visualization .. ... ... ... ... ... .... ... ... ... 62 7.4 Math Library .. .... ... ... ... ... ... ... ... ... 64 7.5 User Interface .. ... ... .. ... ... ... .. ... ... ... 65 8 Conclusions 67 8.1 Conclusions ... .. ... ... ... .. ... ... ... .. ... .. 67 8.2 Future Work ... .. ... ... ... .. ... ... ... .. ... .. 67 A Aerodynamic Force Modeling 71 A.1 Two-Dimensional Thin Airfoil Theory ... ... ... ... .... .. 71 A.2 Prandtl's Lifting Line Method ... .... ... ... ... ... ... 76 B Proofs from Chapter 4 87 B.1 Proof of Geometric Relationship Between a Vector and Rotation . .. 87 B.2 Proof of Finite Limit .. ... .. ... .. ... .. ... .. ... .. 89 List of Figures 1-1 Recon Through a City .. .... ..... ..... ..... ..... 14 1-2 Task Assignment ........ ........ ........ ..... 14 2-1 Nexus Photograph ..... ....... ...... ...... 18 2-2 Nexus Solid Model ........ ......... ........ 18 2-3 Modeling Frame ... .... ..... ..... ..... ..... 19 2-4 Body Frame .... ........ ......... ........ 19 2-5 Velocity Components ... ........ ........ ........ 23 2-6 Velocity Components in Prop Wake ..... ...... ....... 23 2-7 Finite Element Model .... ....... ...... ...... .... 26 2-8 Quarter Chord ..... ...... ...... ....... ...... 26 4-1 General Non-linear System .. ..... ..... ..... ..... .. 32 4-2 Applied Inner Control .......... ............. ... 32 4-3 Backstepped Input ....... ........ ........ ..... 33 4-4 Backstepping Block Diagram ....... ............. ... 34 5-1 Coordinated Flight ........ ......... ........ ... 44 5-2 Coordinated Free Body Diagram ......... ........... 44 5-3 Feasible Accelerations .. ........ ........ ........ 48 5-4 Following a Helical Trajectory ...... ........ ........ 49 5-5 Hover-Hover Trajectory .. ........ ........ ........ 50 5-6 Angular Acceleration ...... ........ ........ ..... 50 6-1 #-Coordination ......... ............. ........ 52 6-2 Multi-point Hover ... ... ... ... ... ... ... .. ... ... 5 7 7-1 Scenario ........... .... .. .. .. .. 60 7-2 Nexus Data Flow ...... .... .. .. .. .. 62 7-3 Visualization .... .. .. .. .. .. 63 7-4 Thread Separation ......... .. ... .. .. 65 A-i Lifting Element as a Streamline .. .. 71 A-2 Vortex Sheet Model of Flat Plate .. ... 72 A-3 Downwash Due to Vortex Element ..... 72 A-4 Change of Variables ...... ... .. .. 73 A-5 Circulation Line Integral ...... ... 74 A-6 Geometry of Forces ......... .. .. 75 A-7 Wing as a Bound Vortex ...... .. .. 77 A-8 Finite Vortices .......... .. .. .. 77 A-9 Vortex Sheet ........... .. .. .. 77 A-10 Induced Angle of Attack ...... .. .. 78 A-11 Change of Variables ......... .. .. 80 A-12 Circulation Result ... ...... .. 85 A-13 Lift Coefficient Result ....... .. 85 B-1 Geometry of Vector Relationship . B-2 View From r ............. List of Tables 2.1 Component Mass Properties ... .. ... ... .. ... .. ... .. 19 2.2 Nexus Moments of Inertia . ..... ..... .... ..... .... 20 12 Chapter 1 Introduction 1.1 Motivation The history of Unmanned Aerial Vehicles (UAVs) is short, and their marketability is still being tested as their commercial use rises. T raditionally, their use has been most prominent in places where an autonomous flight vehicle can replace a human piloted one. In many cases, the mission parameters remain the same as for a human piloted mission, only the human pilot is taken out of the equation (to reduce the chance of injury or loss of life, for instance). A UAV's applicability to its mission, therefore, is often centered around its ability to perform like a traditional aircraft. Naturally, this leads their design and manufacture to somewhat mimic that of unmanned aircraft. However, as the public become more aware of their use and their potential, UAVs have begun to realize applicability in their own right. Carrying a human pilot often yields certain restrictions on how the craft is used, but without a human pilot as a factor, there is growing demand for vehicles that stretch the bounds of this perfor- mance envelope. There are many cases where small UAVs show significant potential. If we think about military applications, remote piloted drones are already used extensively for re- connaissance and munitions deployment. In reconnaissance missions in particular, the ability to deploy a large number of small vehicles to search a large area and identify potential threats, or potential target locations can greatly reduce the manpower re- quired for the task. An example mission might require a vehicle flying into a crowded city, maneuvering through various obstacles, observing certain locations, and then returning, as shown in figure 1-1. This kind of a mission may lead to some conflict- ing design requirements. The desire maneuver around buildings, and the potential
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