An Initial Guess Generator for Launch and Reentry Vehicle Trajectory Optimization Von der Fakultät Luft- und Raumfahrttechnik der Universität Stuttgart zur Erlangung der Würde eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung vorgelegt von Albert Walter Markl aus München Hauptberichter: Prof. K.-H. Well, Ph. D. Mitberichter: Prof. Dr.-Ing. G. Sachs Tag der mündlichen Prüfung 11. Juni 2001 Institut für Flugmechanik und Flugregelung der Universität Stuttgart 2001 2001 Albert Markl Institut für Flugmechanik und Flugregelung Universität Stuttgart Pfaffenwaldring 7a 70550 Stuttgart Deutschland This document is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owners must be obtained. Printed in Germany Acknowledgements Tell my wife, I love her very much. She knows. David Bowie – Space Oddity Foremost I would like to express my thankfulness to my advisor, Professor Well, for giving me the opportunity to perform the studies which lead to this thesis. His many insightful inputs were invaluable to my work. My deeply felt appreciation extends to Prof. Dr. Sachs, for taking the task of co-referent and for teaching me the basics of flight mechanics and optimization, long time ago. Dr. Mehlem, Director of the Mathematical Studies Office of the European Space Agency, deserves my special thanks, since his unwavering support of the ASTOS program was and is one of the foundations of this project. To all my present and former colleagues, notably to Dr. Grimm and to Mr. Jänsch, I would like to express my gratitude for their stimulations and for the productive atmosphere, in which this work could grow. Finally I would like to thank my wife Darlina for all her love and for her understanding when I spent my time working on this thesis. I appreciate her support very much. An Initial Guess Generator for Launch and Reentry Vehicle Trajectory Optimization iii Acknowledgements iv An Initial Guess Generator for Launch and Reentry Vehicle Trajectory Optimization Table of Contents Acknowledgements iii Table of Contents v List of Figures xi List of Tables xv Notation xix Zusammenfassung xxv Abstract xxxiii 1 Introduction 41 1.1 The Role of an Initial Guess Generator (IGG) ............................................... 41 1.2 Objective of Advanced Guidance Laws ......................................................... 42 1.3 Structure of the Thesis .................................................................................... 43 An Initial Guess Generator for Launch and Reentry Vehicle Trajectory Optimization v 2 Classification of Vehicles 45 2.1 Conventional Launcher .................................................................................. 45 2.2 Advanced Launcher ........................................................................................ 48 2.3 Reentry Vehicles ............................................................................................. 50 3 Formulation of the Optimization Problem 53 3.1 Differential Equations of Motion ................................................................... 54 3.1.1 Inertial Velocity System ................................................................... 55 3.1.2 Flight-Path System ........................................................................... 56 3.2 Attitude Controls ............................................................................................ 57 3.2.1 Transformations ............................................................................... 57 3.2.2 Aerodynamic Angles ........................................................................ 59 3.2.3 Reduced Aerodynamic Angles ......................................................... 61 3.2.4 Euler Angles ..................................................................................... 62 3.2.5 Reduced Euler Angles ...................................................................... 63 3.2.6 Load Factor Controls ........................................................................ 64 3.3 Path Constraints .............................................................................................. 65 3.3.1 Dynamic Pressure ............................................................................. 65 3.3.2 Heat Rate .......................................................................................... 66 3.3.3 Normal Load Factor ......................................................................... 66 3.3.4 Loft Ceiling ...................................................................................... 67 3.3.5 Axial Acceleration ........................................................................... 68 3.4 Cost Functions ................................................................................................ 68 3.4.1 Maximum Payload ........................................................................... 68 3.4.2 Maximum Final Mass ...................................................................... 69 3.4.3 Minimum Fuel .................................................................................. 69 3.4.4 Minimal Heat Load .......................................................................... 69 vi An Initial Guess Generator for Launch and Reentry Vehicle Trajectory Optimization 3.4.5 Cross Range ..................................................................................... 70 3.4.6 Trajectory Smoothing ....................................................................... 71 4 Guidance Laws 73 4.1 Conventional Launcher ................................................................................... 74 4.1.1 Previous Work .................................................................................. 74 4.1.2 Pitch Push-Over ................................................................................ 75 4.1.3 Gravity Turn ..................................................................................... 76 4.1.4 Required Velocity Concept ............................................................... 76 4.1.5 Bi-Linear Tangent Law .................................................................... 79 4.1.6 Horizontal Guidance ........................................................................ 81 4.2 Advanced Launcher ........................................................................................ 82 4.2.1 Vertical Guidance ............................................................................. 82 4.2.1.1 Theory of Singular Perturbations ..................................... 82 4.2.1.2 Energy State Approximation ........................................... 84 4.2.1.3 Energy Method ................................................................ 86 4.2.1.4 Other Vertical Guidance Concepts ................................... 87 4.2.2 Horizontal Guidance ........................................................................ 87 4.3 Reentry Vehicles ............................................................................................. 89 4.3.1 Shuttle Guidance Law ...................................................................... 92 4.3.1.1 Angle of Attack Profile .................................................... 92 4.3.1.2 Side Slip Angle ................................................................ 92 4.3.1.3 Velocity Bank Angle ........................................................ 92 4.3.1.4 Drag Profile ..................................................................... 93 4.3.1.5 Range Control .................................................................. 94 4.3.1.6 Drag Controller ................................................................ 95 4.3.1.7 Lateral Steering ................................................................ 96 An Initial Guess Generator for Launch and Reentry Vehicle Trajectory Optimization vii 4.3.1.8 Literature on the Shuttle Guidance .................................. 96 4.3.1.9 Conclusions ..................................................................... 97 4.3.2 Adaptation of the Shuttle Guidance for the IGG ............................. 98 4.3.2.1 Cross and Down Range Prediction .................................. 98 4.3.2.2 Dynamic Pressure Control ............................................... 99 4.3.2.3 Cubic Spline .................................................................... 99 4.4 Reduced Order Equations ............................................................................... 100 4.5 Dynamic Pressure Controller ......................................................................... 103 4.6 Dynamic Pressure Controller Based on Load Factor Controls ...................... 105 4.7 Summary of Guidance Laws .......................................................................... 108 4.7.1 Conventional Launcher .................................................................... 108 4.7.2 Advanced Launcher .......................................................................... 109 4.7.3 Reentry Vehicles ............................................................................... 110 5 Comparison of Optimal and Guidance Law Solutions 111 5.1 Conventional Launcher .................................................................................. 111 5.1.1 Escape Mission ................................................................................