Optimal Orbital Transfer Using a Legendre Pseudospectral Method by Stuart Andrew Stanton B.S. Astronautical Engineering United States Air Force Academy, 2001 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 at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUNE 2003 @2003 Stuart Andrew Stanton. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part. Signature of Author. Department of Aeronautics and Astronautics May 23, 2003 Certified by Christop er N. D'Souza, Ph.D. The Charles Stark Draper Laboratory, Inc. Thesis Supervisor Certified by Ronald J. Proulx, Ph.D. The Charles Stark Draper Laboratory, Inc. - ,Thesis Supervisor Certified by George T. Schmidt, Sc.D. Lecturer, Department of Aeronautics and Astronautics Director, Education The Charles Stark Draper Laboratory, Inc. Thesis Advisor Accepted by Edward M. Greitzer, Ph.D. H.N. Slater Professor of Aeronautics and Astronautics Chair, Committee on Graduate Students MASSAPiHuIIiiTUTn ()F TECHNM1.0GY AERO ep~IR E2S3 LIB1RARIES [Except for this sentence, this page intentionally left blank.] RPMEBW Optimal Orbital Transfer Using a Legendre Pseudospectral Method by Stuart A. Stanton Submitted to the Department of Aeronautics and Astronautics on May 23, 2003, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract Orbital transfer problems are solved successfully for several classes of transfers, in- cluding coplanar transfer, simple plane change, and complex orbital size/shape/plane changes. Both impulsive and finite-burn orbital transfers are constructed which mini- mize fuel costs. Using the direct Legendre pseudospectral technique software package (DIDO) developed by Ross and Fahroo of the Naval Postgraduate School, the effects of complex perturbations (like J2) are naturally included in the optimal solution. Written in Matlab, the DIDO package allows the user to incorporate complex cost functions, bounds, and constraints on the optimal transfer trajectory through a stan- dardized interface. Thesis Supervisor: Christopher N. D'Souza, Ph.D. Title: The Charles Stark Draper Laboratory, Inc. Thesis Supervisor: Ronald J. Proulx, Ph.D. Title: The Charles Stark Draper Laboratory, Inc. Thesis Advisor: George T. Schmidt, Sc.D. Title: Lecturer, Department of Aeronautics and Astronautics Director, Education The Charles Stark Draper Laboratory, Inc. 3 [Except for this sentence, this page intentionally left blank.] ACKNOWLEDGMENT May 23, 2003 This thesis could not have been written without the help of a number of different people and organizations. Thank you to everyone that has in any way contributed to making it possible for me to succeed. Thanks to the Charles Stark Draper Laboratory for the opportunity to work as a Draper Fellow for two years. My appreciation goes to all of the staff members that I have had the privilege of working with while I was here. Thanks to Tim Brand for providing direction for this research. A special thank you to Chris D'Souza and Ron Proulx for their wisdom and expertise. Chris, thank you for sticking with me from afar. Ron, thank you for putting up with me from anear. Both of you have provided me with so much technical support; I would have been lost without the benefit of your experience and knowledge. Thanks to my fellow Fellows, especially my officemates: Ted Dyckman, Stephen Long, David Woffinden, Kimberley Clarke, Christine Taylor. It was a trip. Ted, thanks for showing me the ropes and helping me get my feet wet. Steve, thanks for the comedy and thanks for being there. Dave, thanks for being the "grown-up" in our office. Thanks to Matt Obenchain for being a good friend and study partner. John Young, thanks for reminding me to lock my computer. Jen DiCarlo, thanks for the free food. Good luck to everyone in their future lives. Thanks to those in my life outside of the lab. To my parents, words cannot begin to express my gratitude. To my brother, Spencer, thanks for helping me get settled out here. Margaret O'Keefe, thank you for being my mentor. Thanks to the Air Force Institute of Technology for the opportunity to pursue a graduate degree. Thank you to Col Jack Anthony and LtCol Jerry Sellers for guiding my career as I prepare to finally join the Real Air Force. This thesis was prepared at The Charles Stark Draper Laboratory, Inc., under Draper Internal Research and Development Project 15269, GCDLF Support. Publication of this thesis does not constitute approval by Draper or the sponsor- ing agency of the findings or conclusions contained herein. It is published for the exchange and stimulation of ideas. Star&U. Stanton, 2Lt., USAF 5 [Except for this sentence, this page intentionally left blank.] Contents 1 Introduction 15 1.1 M otivation ... ..... ...... ..... ..... ..... 15 1.2 Background ..... ..... ...... ..... ..... .. 16 1.3 Thesis Overview ....... ....... ....... ..... 17 2 Optimal Control Theory 21 2.1 Function Minimization .... ...... ..... ..... .. 21 2.2 Functional Minimization . ..... ...... ..... .... 25 2.3 The Optimal Control Problem . ...... ...... ..... 27 2.3.1 Pontryagin's Minimum Principle ...... ...... 28 3 Numerical Methods for Solving Optimal Control Problems 31 3.1 Numerical Techniques .. ..... ...... ..... ..... ... 3 2 3.1.1 Lagrange Interpolation ......... 33 3.1.2 Orthogonal Functions .......... 34 3.1.3 Gaussian Quadrature .......... 36 3.2 Direct Optimization Methods .. ..... .. 37 3.2.1 Collocation Methods . ..... .... 37 3.2.2 Spectral Methods ... .... .... 40 3.2.3 The Legendre Pseudospectral Method . 42 4 General DIDO Problem Setup 45 4.1 Functions . ...... .... 45 4.1.1 Main Function . .. 45 4.1.2 Cost Function .. ... 48 4.1.3 Dynamics Function .. 48 4.1.4 Events Function . ... 48 4.1.5 Path Function .. ... 49 4.2 Scaling ..... ........ 49 5 Orbital Transfer Optimization 51 5.1 Problem Dynamics ...... ..... .. 51 5.1.1 State Definitions .. ..... 51 7 5.1.2 Perturbations .................... .. 54 5.1.3 Controls .......... ............. .. 57 5.2 The Performance Index ................... .. .. 58 5.3 Event Constraints ...................... .. 59 5.4 Design Choices for Specific Problems ........... .. 60 5.4.1 Pure Impulses .................. .. .. .. .. 60 5.4.2 High Thrust Impulsive Approximation . ..... .. .. ... 61 5.4.3 Finite-Burn ......... ......... ... .. ... .. 61 5.5 Other Tools ...... .......... ......... ... .. 62 6 Impulsive Burn Orbital Transfer 63 6.1 Capability Demonstration ................. 64 6.1.1 Hohmann Transfer ........... ...... 64 6.1.2 Plane Change .................... 70 6.1.3 Combined Plane Change ..... ......... 80 6.1.4 The J2 Perturbation. ................ 86. 6.2 Application ......................... 87 6.2.1 From LEO (ISS) to LEO (Sun-Synchronous) ... 87 6.2.2 From LEO (ISS) to Molniya Orbit .. ...... 92 6.3 Uncertainties in Solution Optimality ........... 99 6.3.1 Motivation ...................... 99 6.3.2 Understanding the Solution Space ... ...... 102 6.4 Providing a Good Guess for DIDO ..... ....... 109 6.4.1 Guess Consistency ........ ......... 109 6.4.2 Engineering Judgment ......... ...... 109 7 Finite Burn Orbital Transfer 113 7.1 Capability Demonstration ........... ...... .... .. 113 7.1.1 Orbit-Raising Transfer ............... ... .... 114 7.1.2 Inclination Change on a Circular Orbit ...... ... .. 122 7.2 Application: From LEO (ISS) to LEO (Sun-Synchronous) with J2 .. 134 7.2.1 Basic Problem .................... .. ... 134 7.2.2 One Burn at a Time ... ............. .. .. ... 136 7.3 Solution Feasibility ............... ...... .. .. .. 138 7.3.1 Orbit-Raising Transfer ............... .. .. 139 7.3.2 Inclination Change Transfer ............ .. ... .. 146 7.3.3 From LEO (ISS) to LEO (Sun-Synchronous) with J2 -- . 149 8 Conclusion 151 8.1 Sum m ary ..... ..... ...... ..... ...... .... 151 8.2 Future Work. ..... ...... ..... ...... ..... .. 152 8 List of Figures 1-1 Continuous Controls .. ...... ...... ...... ....... 17 1-2 "Bang-Off-Bang" Continuous Controls ......... 18 1-3 Transfer Series . ...... ...... ....... ...... .... 20 2-1 Constrained Minimization: The Method of Lagrange .... ..... 24 4-1 Standard Node Distribution for the Legendre Pseudospectral Method (100 nodes) ....... ..... ..... ........ ... .. ... 46 6-1 Circular-Circular Hohmann Transfer: Random Guess ......... 66 6-2 Circular-Circular Hohmann Transfer: Guess Trajectory = Initial Orbit 67 6-3 Elliptical-Elliptical Hohmann Transfer: Random Guess ........ 69 6-4 Inclination Change Transfer: Initial & Final Orbits .. ........ 71 6-5 Inclination Change Transfer: Single Impulse Guess (-z) ....... 72 6-6 Inclination Change Transfer: Two-Impulse Guess (+z, -z) ...... 73 6-7 Inclination Change Transfer: Two-Impulse Guess (-z, +z) ... ... 75 6-8 Inclination/Node Change Transfer: Random Guess .......... 77 6-9 Inclination Change Transfer (Elliptical): Random Guess ...... 79 6-10 Size and Inclination Change Transfer: Random Guess ...... ... 82 6-11 Size and Inclination Change Transfer (Elliptical): Random Guess .. 85 6-12 LEO-LEO Transfer: Continuous Solution ...... .......... 88 6-13 LEO-LEO Transfer: Continuous Solution with Transformed Parame- terized G uess) ............ ........... ........ 90 6-14 LEO-LEO Transfer: Parameterized Guess and Two Solutions ..... 91 6-15 LEO-Molniya Transfer: Continuous Solution ..........