Michigan Technological University Digital Commons @ Michigan Tech Dissertations, Master's Theses and Master's Dissertations, Master's Theses and Master's Reports - Open Reports 2011 Dynamic modeling, simulation and control design of a parafoil- payload system for ship launched aerial delivery system (SLADS) Anand S. Puranik Michigan Technological University Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mechanical Engineering Commons Copyright 2011 Anand S. Puranik Recommended Citation Puranik, Anand S., "Dynamic modeling, simulation and control design of a parafoil-payload system for ship launched aerial delivery system (SLADS)", Dissertation, Michigan Technological University, 2011. https://digitalcommons.mtu.edu/etds/401 Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mechanical Engineering Commons DYNAMIC MODELING, SIMULATION AND CONTROL DESIGN OF A PARAFOIL-PAYLOAD SYSTEM FOR SHIP LAUNCHED AERIAL DELIVERY SYSTEM (SLADS) By Anand S Puranik A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Mechanical Engineering – Engineering Mechanics) MICHIGAN TECHNOLOGICAL UNIVERSITY 2011 Copyright © 2011 Anand S Puranik This dissertation, “Dynamic Modeling, Simulation And Control Design Of A Parafoil- Payload System For Ship Launched Aerial Delivery System (SLADS),” is hereby approved in partial fulfillment for the requirements for the Degree of DOCTOR OF PHILOSOPHY IN Mechanical Engineering – Engineering Mechanics. Department of Mechanical Engineering – Engineering Mechanics Advisor: Dr. Gordon Parker Committee Member: Dr. Chris Paserrello Committee Member: Dr. Jason Blough Committee Member: Dr. Allan Struthers Department Chair: Professor William W. Predebon Date: Contents List of Figures :::::::::::::::::::::::::::::::::: x List of Tables ::::::::::::::::::::::::::::::::::: xii Acknowledgments :::::::::::::::::::::::::::::::: xiii Nomenclature :::::::::::::::::::::::::::::::::: xiv Abstract ::::::::::::::::::::::::::::::::::::: xxi 1. Introduction ::::::::::::::::::::::::::::::::::: 1 1.1 Background .................................. 2 1.2 Aerodynamic Decelerators .......................... 4 1.2.1 Parachute ............................... 4 1.2.2 Parasail ................................ 4 1.2.3 Paraglider ............................... 6 1.2.3.1 Parafoil Canopy ...................... 7 1.2.3.2 Leading Edge ....................... 7 1.2.3.3 Trailing Edge ....................... 7 1.2.3.4 Suspension Lines ..................... 7 iv 1.2.3.5 Control Lines ....................... 7 1.2.3.6 Risers ........................... 8 1.3 Ship Launched Aerial Delivery System (SLADS) .............. 8 1.4 Research Contributions ............................ 9 1.5 Dissertation Outline .............................. 10 2. Literature Review :::::::::::::::::::::::::::::::: 11 2.1 Parafoil Evolution ............................... 11 2.2 Hang gliders and Paragliders ......................... 13 2.3 Dynamic Models ............................... 13 2.4 Aerodynamic Theory ............................. 19 2.4.1 Angle Of Attack (α) ......................... 19 2.4.2 Side Slip Angle (β) .......................... 20 2.4.3 Aerodynamic Forces ......................... 20 2.4.3.1 Drag ............................ 20 2.4.3.2 Lift ............................. 21 2.4.3.3 Side Force ......................... 21 2.4.4 Aerodynamic Moments ........................ 22 2.4.4.1 Pitching Moment ..................... 22 2.4.4.2 Rolling Moment ...................... 22 2.4.4.3 Yawing Moment ...................... 23 2.5 Stability Analysis ............................... 23 2.6 Control Input ................................. 30 2.7 Apparent Mass Effects ............................ 31 v 3. Modeling Of Towed Parafoil-Payload System :::::::::::::::::: 32 3.1 Model Description And Modeling Assumptions ............... 32 3.2 Equations Of Motion ............................. 34 3.2.1 Rotation Matrices ........................... 36 3.2.2 Kinematics .............................. 37 3.2.3 Lagrange Equations ......................... 38 3.2.4 Generalized Forces .......................... 39 3.2.5 Dynamic Equations .......................... 41 3.3 Aerodynamic Model ............................. 41 3.3.1 Control Inputs ............................ 46 3.4 Geometric and Inertial Properties ....................... 50 3.4.1 Inertia Calculations .......................... 52 3.5 2D Simulation Results ............................ 55 3.6 Operator-In-The-Loop Simulation ...................... 60 4. Stability Analysis ::::::::::::::::::::::::::::::::: 62 4.1 Motivation ................................... 62 4.2 Linear Model ................................. 64 4.2.1 Lyapunov’s Linearization Method .................. 64 4.2.2 Linearization of Dynamic System .................. 65 4.3 Steady State Flight .............................. 69 4.4 Linear and Nonlinear Model Comparison .................. 71 4.5 Stability Boundaries .............................. 73 4.5.1 Elevation Angle η .......................... 75 4.5.2 Symmetric Brake Deflection δs ................... 81 vi 4.5.3 Effect of Tow Point Location ..................... 82 4.6 Summary ................................... 85 5. Control System Design :::::::::::::::::::::::::::::: 86 5.1 Control Objectives .............................. 86 5.2 Controllability And Observability ...................... 87 5.2.1 Definition ............................... 89 5.2.2 Case I ................................. 90 5.2.3 Case II: ................................ 93 5.2.4 Case III: ................................ 95 5.3 Tow Up Control Strategy ........................... 97 5.3.1 Controller for Ascent Rate ...................... 99 5.4 Lateral Stability Control ........................... 104 5.4.1 Lateral State Feedback Controller .................. 104 5.4.2 Full State Feedback Controller .................... 108 5.5 Conclusions .................................. 113 6. Conclusions And Future Work :::::::::::::::::::::::::: 115 References :::::::::::::::::::::::::::::::::::: 118 A. Nonlinear Dynamic Equation Coefficients :::::::::::::::::::: 122 B. Derivation of Equations of Motion: Maple Script :::::::::::::::: 149 C. Steady State Values ::::::::::::::::::::::::::::::: 158 D. Copyrights :::::::::::::::::::::::::::::::::::: 166 List of Figures 1.1 An American paratrooper using a MC1-1C series parachute (1). (See Ap- pendix D for the documentation that this material is in public domain.) ... 5 1.2 Parasail is a towed parachute (2). (See Appendix D for the documentation that this material is in public domain.) .................... 5 1.3 Significant parts of a paraglider or ram-air parachute (3). (See Appendix D for the documentation of the permission to republish this image). ...... 6 1.4 Stages of operation of SLADS. ........................ 9 2.1 Francis Rogallo with his parawing in the wind tunnel. (Photo credit: NASA/courtesy of nasaimages.org. See Appendix D for the documentation that this image is in the public domain.) ........................... 12 2.2 NASA X-38 re-entry vehicle (4). Photo credit: NASA/courtesy of nasaim- ages.org. (See Appendix D for the documentation that this image is in the public domain.) ................................ 16 2.3 Nine DOF dynamic model of parafoil-payload system used by Slegers and Costello (5). (See Appendix D for the documentation of permssion to re- publish this figure from (5).) ......................... 17 viii 2.4 Panel discretization used by Slegers and Costello (5). (See Appendix D for the documentation of permssion to republish this figure from (5).) ..... 18 2.5 Angle of attack α ............................... 19 2.6 Side slip angle β ............................... 20 2.7 Longitudinal static analysis (6). (See Appendix D for the documentation of permssion to reprint this figure from (6).) .................. 25 3.1 Significant points of the model. ........................ 33 3.2 Schematic diagram of the states and reference frames of the model. ..... 35 3.3 Aerodynamic forces .............................. 42 3.4 Modeling control input or parafoil brake deflection ............. 48 3.5 Steady state roll angle ϕ20 for various control inputs ............. 48 3.6 Steady state yaw angle 20 for various control inputs ............ 49 3.7 Steady state off plane tow angle 10 for various control inputs ....... 49 3.8 Parafoil used for testing. (courtesy: Flight Concepts Int’l, Inc. See Ap- pendix D for the documentation of the permission to republish this image) . 50 3.9 Payload dimensions .............................. 51 3.10 Side view of a parafoil payload system for net CG calculation ....... 54 3.11 Steady state values of in-plane tow angle θ1 and pitch angle θ2 ....... 56 3.12 Steady state values of in-plane tow angle θ1 for various symmetric brake inputs ..................................... 57 3.13 Steady state values of pitch angle θ2 for various symmetric brake inputs .. 58 3.14 Inertial X component of the relative vector between point of attachment at ship and the parafoil-payload system ..................... 59 ix 3.15 Inertial Z component of the relative vector between point of attachment at ship and the parafoil-payload system ..................... 59 3.16 Operator-In-Loop simulation setup ...................... 60 4.1 Steady state flight condition for symmetric brake input δs0 = 0:2 and asym- metric brake input δa0 = 0:1 ......................... 70