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NASA/TP–2018–219822 Navigation Filter Best Practices Edited by J. Russell Carpenter Goddard Space Flight Center, Greenbelt, Maryland Christopher N. D’Souza Johnson Space Center, Houston, Texas April 2018 NASA STI Program. in Profile Since its founding, NASA has been dedicated CONFERENCE PUBLICATION. to the advancement of aeronautics and space • Collected papers from scientific and science. The NASA scientific and technical technical conferences, symposia, seminars, information (STI) program plays a key part or other meetings sponsored or in helping NASA maintain this important co-sponsored by NASA. role. SPECIAL PUBLICATION. Scientific, • The NASA STI Program operates under the technical, or historical information from auspices of the Agency Chief Information NASA programs, projects, and missions, Officer. It collects, organizes, provides for often concerned with subjects having archiving, and disseminates NASA’s STI. substantial public interest. The NASA STI Program provides access to TECHNICAL TRANSLATION. English- the NASA Aeronautics and Space Database • and its public interface, the NASA Technical language translations of foreign scientific Report Server, thus providing one of the and technical material pertinent to largest collection of aeronautical and space NASA’s mission. science STI in the world. Results are Specialized services also include organizing published in both non-NASA channels and and publishing research results, distributing by NASA in the NASA STI Report Series, specialized research announcements and which includes the following report types: feeds, providing information desk and TECHNICAL PUBLICATION. Reports of personal search support, and enabling data • completed research or a major significant exchange services. phase of research that present the results For more information about the NASA STI of NASA programs and include extensive Program, see the following: data or theoretical analysis. Includes compilations of significant scientific and Access the NASA STI program home page technical data and information deemed to • at http://www.sti.nasa.gov be of continuing reference value. NASA counterpart of peer-reviewed formal E-mail your question to professional papers, but having less • [email protected] stringent limitations on manuscript length and extent of graphic presentations. Phone the NASA STI Information Desk at • 757-864-9658 TECHNICAL MEMORANDUM. • Scientific and technical findings that are Write to: • preliminary or of specialized interest, e.g., NASA STI Information Desk quick release reports, working papers, and Mail Stop 148 bibliographies that contain minimal NASA Langley Research Center annotation. Does not contain extensive Hampton, VA 23681-2199 analysis. CONTRACTOR REPORT. Scientific and • technical findings by NASA-sponsored contractors and grantees. NASA/TP–2018–219822 Navigation Filter Best Practices Edited by J. Russell Carpenter Goddard Space Flight Center, Greenbelt, Maryland Christopher N. D’Souza Johnson Space Center, Houston, Texas National Aeronautics and Space Administration NASA Engineering and Safety Center Hampton, Virginia 23681 April 2018 The use of trademarks or names of manufacturers in this report is for accurate reporting and does not constitute an official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics and Space Administration. Available from: NASA STI Program / Mail Stop 148 NASA Langley Research Center Hampton, VA 23681-2199 Fax: 757-864-6500 NASA Engineering and Safety Center Navigation Filter Best Practices First Edition J. Russell Carpenter and Christopher N. D’Souza, Editors with contributions from J. Russell Carpenter, Christopher N. D’Souza, F. Landis Markley, Renato Zanetti April, 2018 This material is declared a work of the US Government and is not subject to copyright protection in the United States, but may be subject to US Government copyright in other countries. If there be two subsequent events, the probability of the second b/N and the probability of both together P/N, and it being first discovered that the second event has also happened, from hence I guess that the first event has also happened, the probability I am right is P/b. Thomas Bayes, c. 1760 The explicit calculation of the optimal estimate as a function of the observed variables is, in general, impossible. Rudolph Kalman, 1960. The use of Kalman Filtering techniques in the on-board navigation systems for the Apollo Command Module and the Apollo Lunar Excursion Module was an important factor in the overwhelming success of the Lunar Landing Program. Peter Kachmar, 2002. Dedicated to the memory of Gene Muller, Emil Schiesser, and Bill Lear. Contents Foreword iii Editor’s Preface v Notational Conventions ix Chapter 1. The Extended Kalman Filter 1 1.1. The Additive Extended Kalman Filter 1 1.2. The Multiplicative Extended Kalman Filter 7 Chapter 2. The Covariance Matrix 9 2.1. Metrics for Orbit State Covariances 9 2.2. Covariance Propagation 15 2.3. Covariance Measurement Update 25 2.4. Sigma-Point Methods 28 Chapter 3. Processing Measurements 29 3.1. Measurement Latency 29 3.2. Invariance to the Order of Measurement Processing 30 3.3. Processing Vector Measurements 31 3.4. Filter Cascades 32 3.5. Use of Data from Inertial Sensors 32 Chapter 4. Measurement Underweighting 33 4.1. Introduction 33 4.2. Nonlinear E↵ects and the Need for Underweighting 34 4.3. Underweighting Measurements 38 4.4. Pre-Flight Tuning Aids 39 Chapter 5. Bias Modeling 41 5.1. Zero-Input Bias State Models 41 5.2. Single-Input Bias State Models 43 5.3. Multi-Input Bias State Models 49 Chapter 6. State Representations 53 6.1. Selection of Solve-For State Variables for Estimation 53 6.2. Units and Precision 53 6.3. Coordinate and Time Systems 54 6.4. Orbit Parameterizations 54 6.5. Relative State Representations 55 6.6. Modeling Inertial Components 58 i ii CONTENTS Chapter 7. Factorization Methods 63 7.1. Why Use the UDU Factorization? 63 7.2. Preliminaries 64 7.3. The Time Update of the Covariance 65 7.4. The Measurement Update 72 7.5. Consider Covariance and Its Implementation in the UDU Filter 74 7.6. Conclusions 76 Chapter 8. Attitude Estimation 77 8.1. Attitude Matrix Representation 77 8.2. Euler Axis/Angle Representation 79 8.3. Quaternion Representation 80 8.4. Rodrigues Parameter Representation 83 8.5. Modified Rodrigues Parameters 84 8.6. Rotation Vector Representation 85 8.7. Euler Angles 86 8.8. Additive EKF (AEKF) 87 8.9. Multiplicative EKF (MEKF) 88 Chapter 9. Usability Considerations 93 9.1. Editing 93 9.2. Reinitialization, Restarts, and Backup Ephemeris 93 9.3. Ground System Considerations 94 Chapter 10. Smoothing 95 Chapter 11. Advanced Estimation Algorithms 99 The Sigma-Point Estimator 99 Appendix A. Models and Realizations of Random Variables 105 Appendix B. The Mathematics Behind the UDU Factorization 107 B.1. The Partitioning into Two Subproblems 107 B.2. The Mathematics Behind the Second Subproblem 107 B.3. The Agee-Turner Rank-One Update 109 B.4. Decorrelating Measurements 112 B.5. The Carlson Rank-One Update 112 Appendix C. An Analysis of Dual Inertial-Absolute and Inertial-Relative Navigation Filters 117 C.1. Introduction 117 C.2. The Filter Dynamics 117 C.3. Incorporation of Measurements 120 C.4. Analysis of the Merits of the Inertial-Absolute and Inertial-Relative Filters 123 C.5. Conclusions 125 Bibliography 127 Foreword It certainly should not come as a surprise to the reader that navigation systems are at the heart of almost all of NASA’s missions, either on our launch vehicles, on robotic science spacecraft, or on our crewed human exploration vehicles. Clearly navigation is absolutely fundamental to operating our space systems across the wide spectrum of mission regimes. Safe and reliably performing navigation systems are essential elements needed for routine low Earth orbiting science missions, for rendezvous and proximity operation missions or precision formation flying missions (where relative navigation is a necessity), for navigation through the outer planets of the solar system, and for accomplishing pinpoint landing on planets/small bodies, and many more mission types. I believe the reader will find that the navigation filter best practices the team has collected, documented, and shared in this first edition book will be of practical value in your work designing, developing, and operating modern navigation systems for NASA’s challenging future missions. I want to thank the entire team that has diligently worked to create this NASA Engineering and Safety Center (NESC) GN&C knowledge capture report. I especially want to acknowledge the dedication, care, and attention to detail as well as the energy that both Russell Carpenter and Chris D’Souza, the report editors, have invested in producing this significant product for the GN&C community of practice. It was Russell and Chris who had the inspiration to create this report and they have done a masterful job in not only directly technically contributing to the report but also coordinating its overall development. It should be mentioned that some high-level limited work was previously performed under NESC sponsorship to capture the lessons learned over the course of the several decades NASA has been navigating space vehicles. This report however fills a unique gap by providing extensive technical details and, perhaps more importantly, providing the underlying rationale for each of the navigation filter best practices presented here. Capturing these rationales has proven to be a greatly needed but very challenging
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