Fuzzy Attitude Control of a Magnetically Actuated CubeSat Alex R. Walker B.S. University of Cincinnati A Thesis submitted to the University of Cincinnati School of Aerospace Systems in partial fulfillment of the requirements for the degree of Master of Science in Aerospace Engineering October 2013 Committee Chair: Kelly Cohen, Ph.D. Abstract The problem of magnetic attitude control of a CubeSat is analyzed. Three controller types are examined: a Constant-Gain Simple PD controller, a Linear Constant-Gain Optimal PD controller (i.e. an LQR), and a Fuzzy Gain-Scheduled PD controller. Each subsequent controller type utilizes a more-complex design algorithm. The Simple PD controller is tuned by hand iteration, the LQR is tuned using rule-of-thumb algorithms, and the Fuzzy Gain-Scheduled PD controller is designed using a Genetic Algorithm operating on two Fuzzy Inference Systems. Though the basic structures of these three controllers are identical, the differing design processes lead to different controller performance. The use of a Genetic-Fuzzy System is of particular interest, because this demonstrates the use of an intelligent algorithm to automate the controller design process. The techniques presented herein are directly applicable to any magnetically actuated satellite that can be modeled as a rigid body, although the mass distribution, geometry, and orbit of the satellite will determine controller-specific constants. ii iii Contents Abstract ...................................................................................................................................... ii Contents .................................................................................................................................... iv List of Tables............................................................................................................................. ix List of Figures ............................................................................................................................ x Nomenclature .......................................................................................................................... xvi Introduction ................................................................................................................................ 1 1.1 Impetus ......................................................................................................................... 1 1.2 Literature Review ......................................................................................................... 3 1.2.1 CubeSat Attitude Control ...................................................................................... 3 1.2.2 Controllability and Control Algorithms for Magnetically Actuated Satellites ..... 5 1.2.3 Observability and Satellite Attitude Estimation .................................................... 8 1.3 Design Approach and Assumptions ........................................................................... 10 1.4 Thesis Outline ............................................................................................................. 13 Satellite Model ......................................................................................................................... 15 2.1 Reference Frames ....................................................................................................... 15 2.1.1 Earth Centered Inertial ........................................................................................ 15 2.1.2 Spacecraft-Centered Inertial ................................................................................ 17 2.1.3 Spacecraft-Centered Body-Fixed ........................................................................ 18 2.1.4 Earth-Centered Earth-Fixed ................................................................................ 19 iv 2.2 Attitude Parameterization and Kinematics ................................................................. 20 2.2.1 Parameterization Evaluation ............................................................................... 20 2.2.2 Euler-Rodrigues Symmetric Parameters (Quaternion) ....................................... 22 2.3 Attitude Dynamics ...................................................................................................... 25 2.3.1 Momentum Equations ......................................................................................... 25 2.3.2 Magnetic Control Torque .................................................................................... 27 2.3.3 Aerodynamic Drag Torque.................................................................................. 28 2.3.4 Solar Pressure Torque ......................................................................................... 30 2.3.5 Gravity Gradient Torque ..................................................................................... 30 2.4 Linearized Attitude Model .......................................................................................... 31 2.4.1 Aerodynamic Drag Torque.................................................................................. 31 2.4.2 Gravity Gradient Torque ..................................................................................... 33 2.4.3 Complete Linearized System Dynamics ............................................................. 35 2.5 Magnetic Field Modeling ........................................................................................... 36 2.5.1 Earth’s Magnetic Field ........................................................................................ 36 2.5.2 Orbit Propagation ................................................................................................ 37 PD Controller and the LQR ..................................................................................................... 41 3.1 General Controller Structures ..................................................................................... 41 3.2 PD Design Process...................................................................................................... 46 3.3 LQR Design Process ................................................................................................... 46 v 3.3.1 Optimal Solution of Linear Constant-Gain Feedback Control............................ 46 3.3.2 Defining the Optimal Control Problem ............................................................... 48 3.4 Summary ..................................................................................................................... 50 Genetic Algorithm-Tuned Fuzzy Logic System ...................................................................... 51 4.1 Fuzzy Logic Overview ............................................................................................... 51 4.1.1 Fuzzy Sets and Membership Functions ............................................................... 52 4.1.2 Fuzzy Inference System ...................................................................................... 54 4.1.3 Fuzzification ........................................................................................................ 55 4.1.4 Inference Method ................................................................................................ 55 4.1.5 Defuzzification .................................................................................................... 57 4.2 Genetic Algorithm Overview ..................................................................................... 57 4.2.1 Strings – Problem Parameterization .................................................................... 58 4.2.2 Population Size .................................................................................................... 59 4.2.3 Selection – Roulette Wheel ................................................................................. 59 4.2.4 Crossover ............................................................................................................. 60 4.2.5 Mutation .............................................................................................................. 61 4.2.6 Elitist Selection ................................................................................................... 61 4.3 Genetic-Fuzzy System Implementation ...................................................................... 62 4.3.1 Fuzzy Inference Systems ..................................................................................... 62 4.3.2 Genetic Algorithm ............................................................................................... 67 vi 4.4 Summary ..................................................................................................................... 75 Design Results .......................................................................................................................... 76 5.1 Simple PD Design Results .......................................................................................... 80 5.1.1 First Simple PD Design Snapshot ....................................................................... 80 5.1.2 Second Simple PD Design Snapshot ................................................................... 83 5.1.3 Third Simple PD Design Snapshot ..................................................................... 84 5.1.4 Fourth Simple PD Design Snapshot .................................................................... 86 5.1.5 Fifth Simple PD Design Snapshot ......................................................................