The Pennsylvania State University The Graduate School College of Engineering AUTOMATIC CONTROLLER SYNTHESIS FOR A RECONFIGURABLE UAV A Thesis in Aerospace Engineering by Joseph D. Harkins © 2017 Joseph D. Harkins Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2017 The thesis of Joseph D. Harkins was reviewed and approved∗ by the following: Jacob W. Langelaan Associate Professor of Aerospace Engineering Thesis Advisor Michael A. Yukish Assistant Professor of Aerospace Engineering George A. Lesieutre Professor of Aerospace Engineering Acting Head of the Department of Aerospace Engineering ∗Signatures are on file in the Graduate School. ii Abstract This thesis describes the development of an adaptable control architecture that can be used to automatically synthesize inner-loop controllers for new configurations of modular aircraft. This will allow for new configurations to be flown without the need for extensive flight testing to tune the controller gains for favorable flying characteristics. This is achieved by breaking the aircraft geometry into finite strips and calculating the forces and moments on each strip using 2D airfoil approximations. The contributions of each finite strip are then summed to obtain the approximate equations of motion for the aircraft. These equations are then linearized about a cruise setpoint. This linearized dynamics model for the aircraft is then used to synthesize inner-loop controllers for roll, pitch, and yaw. These controllers can be implemented within an on-board autopilot system, therefore enabling autonomous flight ’out of the box’. The performance of these controllers was simulated and compared to controllers generated using an established vortex-lattice code to determine the accuracy of the finite strip method. iii Table of Contents List of Figures vii List of Tables ix List of Symbols x Acknowledgments xiv Chapter 1 Introduction 1 1.1 Research Motivation . 1 1.2 Control Law Synthesis for Modular Aircraft . 4 1.3 Pixhawk Autopilot . 6 1.4 Related Work . 7 1.4.1 Aerodynamic Modeling of Flight Vehicles . 7 1.4.2 Controller Synthesis . 10 1.5 Theoretical Background . 12 1.5.1 Prandtl Lifting-Line Theory . 12 1.5.2 Small Disturbance Theory . 13 1.6 Potential Applications . 14 1.7 Reader’s Guide . 15 Chapter 2 Problem Definition 16 2.1 Coordinate Frames and Equations of Motion . 17 2.2 Forces and Moments on ith Section . 19 2.3 Transformation of Forces to Global Coordinate Frame . 22 iv Chapter 3 Linearized Aerodynamic Model 25 3.1 Aerodynamic Coefficients . 26 3.1.1 Total Coefficient Equations . 26 3.1.2 Linearized Airfoil Data . 28 3.1.3 Corrections for 3D Aerodynamic Effects . 34 3.2 Linearization of Section EoMs . 35 3.3 Stability and Control Derivatives . 38 3.3.1 Assembly of Section Equations of Motion . 38 3.3.2 Linear Equations of Motion and Nondimensional Derivatives 39 Chapter 4 Control Law Derivation 43 4.1 Controller Description . 43 4.2 Controller Gain Calculation . 44 Chapter 5 Verification via Vortex Lattice Methods 49 5.1 AVL Aircraft Model Setup . 49 5.2 Model Validation Tests . 50 5.2.1 3 Wing Segments, 2 Tails . 51 5.2.2 5 Wing Segments, 2 Tails . 52 5.2.3 5 Wing Segments, V-Tail . 54 5.3 Comparative Simulation of Controller Response . 56 5.3.1 Test Case 1 . 60 5.3.2 Test Case 2 . 63 5.3.3 Test Case 3 . 66 Chapter 6 Conclusions 70 6.1 Results . 71 6.2 Sources of Error . 72 6.3 Future Work . 73 Appendix A B and D Matrix Components 74 A.1 Section with "All-Flying" Control Surface . 74 A.1.1 D Matrix Components . 75 A.1.2 U Matrix Components . 77 A.2 Section with Traditional Control Surface . 77 v A.2.1 D Matrix Components . 78 A.2.2 U Matrix Components . 80 References 81 vi List of Figures 1.1 Example of modular aircraft configuration . 2 1.2 CAD model of example UAV configuration . 4 1.3 Block diagram of system with controller . 5 1.4 Pixhawk Hardware Module[15] . 6 1.5 PX4 Autopilot Pitch Control . 7 1.6 Blade Element Theory Discretization[5] . 8 1.7 Blade Element Theory Elemental Forces[5] . 8 1.8 Panel Method Vortex Distribution[3, p. 285] . 9 1.9 Vortex Lattice Vortex Distribution[3, p. 458] . 10 1.10 Finite Wing Modeled as a Horseshoe Vortex[3, p. 424] . 12 1.11 Superposition of an Infinite Number of Horseshoe Vortices[3, p. 427] 13 2.1 Aircraft Body Frame . 17 2.2 Section Coordinate Frame . 17 3.1 All-Flying Control Surface . 27 3.2 Traditional Control Surface . 28 3.3 PSU-94-097 Airfoil . 29 3.4 NACA-63A010 Airfoil . 31 3.5 NACA-63A010 with various flap deflection angles . 33 3.6 NACA-63A010 with Control Surface Deflections . 34 4.1 Controller Block Diagram . 44 4.2 Elevator deflection for a range of ωn . 47 4.3 Aileron deflection for a range of ωn . 47 4.4 Rudder deflection for a range of ωn . 48 5.1 3 Wing Segment, 2 Tail AVL Model . 51 5.2 5 Wing Segment, 2 Tail AVL Model . 53 5.3 5 Wing Segment, V Tail AVL Model . 55 5.4 Simulink Model of the Roll Rate Controller . 57 5.5 Simulink Model of the Pitch Rate Controller . 57 vii 5.6 Simulink Model of the Yaw Rate Controller . 58 5.7 Simulink Model of the Linear Aircraft Roll Dynamics . 58 5.8 Simulink Model of the Linear Aircraft Pitch Dynamics . 59 5.9 Simulink Model of the Linear Aircraft Yaw Dynamics . 59 5.10 Case 1 Roll Response . 61 5.11 Case 1 Pitch Response . 62 5.12 Yaw Rate Response . 62 5.13 Case 1 Control Surface Response . 63 5.14 Case 2 Roll Response . 64 5.15 Case 2 Pitch Response . 65 5.16 Case 2 Yaw Rate Response . 65 5.17 Case 2 Control Surface Response . 66 5.18 Case 3 Roll Response . 67 5.19 Case 3 Pitch Response . 68 5.20 Case 3 Yaw Rate Response . 68 5.21 Case 3 Control Surface Response . 69 viii List of Tables 1.1 Examples of commercially available 3D printers . 2 3.1 Aerodynamic Coefficients for PSU-94-097 Airfoil . 30 3.2 Aerodynamic Coefficients for NACA-63A010 Airfoil . 32 3.3 Longitudinal Dimensionless Derivatives . 41 3.4 Lateral Dimensionless Derivatives . 41 5.1 Key Stability and Control Derivative Comparison for 3 Segment, 2 Tail Configuration . 52 5.2 Key Stability and Control Derivative Comparison for 5 Segment, 2 Tail Configuration . 54 5.3 Key Stability and Control Derivative Comparison for 5 Segment, V Tail Configuration . 56 5.4 Test Case Geometry and Trim Conditions . 60 5.5 Test Case 1 PI Controller Gains . 61 5.6 Test Case 2 PI Controller Gains . 64 5.7 Test Case 3 PI Controller Gains . 67 ix List of Symbols a Lift curve slope of a finite wing a0 Lift curve slope of an infinite wing AR Aspect Ratio b Wing span c Wing chord CD Drag coefficient CD0 Profile drag coefficient CL Lift coefficient CL0 Zero angle of attack lift coefficient CLα Lift curve slope CLδ Change in CL with δ Clp Non-dimensional stability derivative of roll moment with roll rate C Non-dimensional stability derivative of roll moment with aileron deflection lδa CM Moment coefficient CM0 Average moment coefficient CMδ Change in CM with δ Cmq Non-dimensional stability derivative of pitch moment with pitch rate x C Non-dimensional stability derivative of pitch moment with elevator deflection mδe Cnr Non-dimensional stability derivative of yaw moment with yaw rate C Non-dimensional stability derivative of yaw moment with rudder deflection nδr e Span efficiency F Force g Gravitational acceleration Ix Moment of inertia about x-axis Iy Moment of inertia about y-axis Iz Moment of inertia about z-axis KI Integral controller gain KP Proportional controller gain L Aerodynamic moment about x-axis (roll moment) L0 Lift per unit span M Aerodynamic Moment about y-axis (pitch moment) m Aircraft mass N Aerodynamic moment about z-axis (yaw moment) P Roll rate Q Pitch rate R Yaw rate r Location vector S Wing area si Panel length u Velocity in x-direction V Velocity xi v Velocity in y-direction w Velocity in z-direction X Aerodynamic force in x-direction Xˆ X-direction Y Aerodynamic force in y-direction Yˆ Y-direction Z Aerodynamic force in z-direction Zˆ Z-direction α Angle of attack β Angle of sideslip Γ Total vortex strength γi Discretized vortex strength ∆ Disturbance value δ Control surface deflection angle δa Aileron deflection angle δe Elevator deflection angle δr Rudder deflection angle θ Pitch angle ξ Damping ratio ρ Air density P Summation φ Roll angle ψ Yaw angle ω Angular velocity vector xii ωn Natural frequency (∗)b In the body reference frame (∗)E In the Earth reference frame (∗)i In the section reference frame (∗)b Of the aircraft (∗)i Of the segment AM Additive Manufacturing AVL Athena Vortex Lattice CFD Computational Fluid Dynamics CG Center of Gravity EoM Equation of Motion FDM Fused Deposition Modeling SLM Selective Laser Melting SLS Selective Laser Sintering UAV Uninhabited Air Vehicle xiii Acknowledgments I would like to thank my advisor, Jack Langelaan, for his advice and assistance throughout the process of completing this thesis as well as during the extent of my graduate career at Penn State. I would like to thank my co-workers at the Penn State Applied Research Lab for their support and guidance throughout my time at Penn State. Lastly I would like to thank my family for continually providing support and advice, which has helped me reach this point in my career despite numerous roadblocks and detours.