University of New Hampshire University of New Hampshire Scholars' Repository Master's Theses and Capstones Student Scholarship Spring 2007 A comparative study of the extended Kalman filter and sliding mode observer for orbital determination for formation flying about the L(2) Lagrange point Oliver Olson University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/thesis Recommended Citation Olson, Oliver, "A comparative study of the extended Kalman filter and sliding mode observer for orbital determination for formation flying about the L(2) Lagrange point" (2007). Master's Theses and Capstones. 275. https://scholars.unh.edu/thesis/275 This Thesis is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Master's Theses and Capstones by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. A COMPARATIVE STUDY OF THE EXTENDED KALMAN FILTER AND SLIDING MODE OBSERVER FOR ORBITAL DETERMINATION FOR FORMATION FLYING ABOUT THE L2 LAGRANGE POINT BY OLIVER OLSON B.S., University of New Hampshire, 2004 THESIS Submitted to the University of New Hampshire in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering May 2007 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 1443625 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 1443625 Copyright 2007 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This thesis has been examined and approved. ,/) // Thesis Director, Dr. May-Win L.Thein Associate Professor of Mechanical Engineering Dr. Baxrv Fussell Professor of Mechanical Engineering Dr. L. Gordon Kraft. Ill Professor of Electrical and Computer Engineering Date ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I would like to thank everyone who has helped me throughout this work. I am thankful to my thesis supervisor, Professor May-Win Them, for her guidance, help and encouragement. To Professors Barry Fussell and Gordon Kraft, for teaching me about controls and spending their time evaluating my thesis. To the New Hampshire Space Grant Consortium for funding this research. To Tracey Harvey, for her dedicated service to the students, faculty, and staff. To my family for their constant love and support since the day I was born. To all my friends for their encouragement along the way. To Rob S chum an. Thanks for all the help. To Jennie Mac. Thanks for all the love. And to my mother, Anne Olson, She did a great job. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iv TABLE OF CONTENTS ACKNOWLEDGEMENTS..................... ...................... iii LIST OF FIGURES......................................... ...................................................... ix ABSTRACT ........ ................................... x 1 INTRODUCTION 1 1.1 Distributed Spacecraft Systems and Formation Flying ................... 1 1.2 Orbit Determination ............................... 3 1.2.1 General Orbit D eterm in atio n ............... 3 1.2.2 Orbit Determination About, JL 2 ................................... 4 1.2.3 Orbit Determination for Formation Flying About L$ ............ 5 1.2.4 The Extended Kalman Filter ..................................... 6 1.2.5 The Sliding Mode Observer ...................................... 7 1.3 Thesis Outline . 8 2 FORMATION FLYING ABOUT L2 LIB RATION POINTS 11 2.1 The Restricted Three Body P ro b le m ...................................................... 11 2.2 Libration P o in ts ......................................................................... 15 2.3 Formation Flying in Orbit About an P o in t. ...................................... 18 2.4 Space Environment at the L 2 P o in t .................................................. 19 3 THE EXTENDED KALMAN FILTER 23 3.1 The Continuous-Time Extended Kalman Filter ................................... 23 4 SLIDING MODE OBSERVERS 30 4.1 Sliding 'Mode Observers ............................................................................ 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V 5 SIMULATION CHARACTERISTICS 36 5.1 Formation Flying Scenario ........................................................................... 36 5.2 VISNAV Measurement System .................................................................... 38 5.2.1 Measurement M o d e l ....................................................................... 40 5.2.2 Simulation C o n d itio n s ................................................................... 41 5.3 Control Law ..................................................................................................... 43 6 EXTENDED KALMAN FILTER FOR FORMATION FLYING 45 6.1 Extended Kalman Filter Simulations ...................................................... 45 6 . 2 Extended Kalman Filter Covariance Matrix Selection ......................... 49 6.3 Extended Kalman Filter Covariance Matrix Tuning ............................. 50 6.4 Extended Kalman Filter Results ................................................................ 51 6.4.1 Case One: Inaccurate Initial Conditions ..................................... 52 6.4.2 Case Two: Input Disturbances ........................................................ 54 6.4.3 Case Three: Parameter Uncertainty.............................................. 55 6.4.4 Case Four: Measurement N o is e ..................................................... 57 6.4.5 Case Five: Cumulative Inaccuracies and Disturbances ..... 58 7 SLIDING MODE OBSERVER FOR FORMATION FLYING 65 7.1 Sliding Mode Observer Simulations ........................................................ 65 7.2 Sliding Mode Observer Gain Selection ............................................. 65 7.3 Sliding Mode Observer Gain Tuning ........................................................ 67 7.4 Sliding Mode Observer R esults................................. 71 7.4.1 Case One: Inaccurate Initial Conditions ................................... 72 7.4.2 Case Two: Input Disturbances ............... 72 7.4.3 Case Three: Parameter U ncertainty ......................... 73 7.4.4 Case Four: Measurement N o is e ................................................... 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. v'i 7.4.5 Case Five: Cumulative Inaccuracies and Disturbances ............. 78 8 COMPARISON OF EKF AND SMO 83 8.1 Case Results ............................... 83 8.2 Sensitivity Comparisons .......................................................................... 92 8.3 Considerations ' ................................ 96 0 RESEARCH SUMMARY AND FUTURE WORK 99 9.1 Conclusions . 99 9.2 Future Work ............... 101 BIBLIOGRAPHY 103 APPENDICES . 107 APPENDIX A CONSTELLATION X MODELS 108 A.l Simulation Models .......................................................... 108 APPENDIX B EKF - DIAGRAMS & MATLAB FILES 111 B.l Extended Kalman Filter ......................... I l l APPENDIX C SMO - DIAGRAMS & MATLAB FILES 118 C.l Sliding Mode Observer ................... 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. v ii LIST OF FIGURES 2-1 Restricted Three Body Problem Geometry [1 ] ...................................... 13 2-2 Libration Point Locations [1] ...................................................................... 17 2-3 Two Spacecraft Orbiting in the Earth/Moon - Sun Rotating Frame [2] 20 4-1 Reaching And Sliding Phenomena of Sample State Trajectories, on the Phase Plane [3 ].............................................................................................. 31 5-1 VISNAV Measurement System ................................................................... 40 6-1 Case One - EKF Relative Position Error M agnitude ..................... 53 6-2 Case Two - EKF Relative Position Error Magnitude .................. 55 6-3 Case Three - EKF Relative Position Error Magnitude .......................... • 56 6-4 Case Four - EKF Relative Position Error Magnitude ......................... 57 6-5 Case Five - EKF Relative Position Estimates ■ ■ ■ 58 6 - 6 Case Five -EKF Relative Position Estimates (Magnified) ............ 59 6-7 Case Five - EKF Relative Position Estimate Error Magnitude .... 60 6 - 8 Case Five - EKF Relative Position Estimate Error Magnitude (Magnified) 60 6-9 Case Five - EKF Relative Velocity Estimates . ..............................