NONLINEAR TRAJECTORY NAVIGATION

Transcription

1 NONLINEAR TRAJECTORY NAVIGATION by Sang H. Park A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Aerospace Engineering) in The University of Michigan 2007 Doctoral Committee: Associate Professor Daniel J. Scheeres, Chair Professor Alfred O. Hero III Professor Pierre T. Kabamba Professor N. Harris McClamroch Research Scientist Paul W. Chodas, Jet Propulsion Laboratory

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3 c Sang H. Park All Rights Reserved 2007

4 To my parents. ii

5 ACKNOWLEDGEMENTS During the past five years at Michigan so many things have happened and there are so many people to thank. First and foremost, it s my parents who have encouraged me to pursue PhD studies. I thank them for their encouragements and supports throughout my academic career. To Prof. Daniel Scheeres, who has been my PhD advisor and a life-long mentor: it is his guidance and help that made this dissertation exist. I want to thank him, but no matter how much say here, I would not feel I have said enough. He has taught me the concept of how much one can owe someone so much. Hence, instead of thanking him, I promise that I will do the same as I have learned from him. Thank you for teaching me this valuable lesson! To Sophia Lim, who has patiently encouraged my studies and gave me the motivation for completing this dissertation: I thank you. Also, I am very grateful to my doctoral committee members, Dr. Paul Chodas, Prof. Alfred Hero III, Prof. Pierre Kabamba, and Prof. Harris McClamroch, for their helpful advice and critical comments. A part of research described in this dissertation was sponsored by the Jupiter Icy Moon Orbiter project through a grant from the Jet Propulsion Laboratory, California Institute of Technology which is under contract with the National Aeronautics and Space Administration. I thank Dr. John Aiello, Dr. Lou D Amario, Mr. Try Lam, Dr. Chris Potts, Dr. Ryan Russell, Dr. Jon Sims, and Dr. Mau Wong from the Jet Propulsion Laboratory for their helpful comments and suggestions. During my graduate studies, I have spent almost a year at the Jet Propulsion Laboratory as an intern/on-call employee. I was exposed to many interesting space mission projects: Pioneer anomaly, estimation of the parameiii

6 terized post-newtonian parameters, Cassini s synthetic aperture radar and altimetry data types, and Magellan orbit determination. I thank all the members of the Guidance, Navigation and Control Group and the Radio Science Systems Group for this great opportunity. Many thanks to Dr. John Anderson, Dr. Sami Asmar, Dr. Shyam Bhaskaran, Dr. Al Cangahuala, Dr. Paul Chodas, Dr. Bob Gaskell, Dr. Moriba Jah, Ms. Eunice Lau, Dr. Michael Lisano, Mr. Ian Roundhill, and Dr. Slava Turyshev. I have learned so much about deepspace spacecraft navigation and implementation of the radiometric measurements. I have also spent a few months at the Applied Physics Laboratory as a NASA/APL summer intern, where I have worked on the pre-flight navigation of the NASA s Radiation Belt Storm Probe. I thank Ms. Linda Butler, Ms. Julie Cutrufelli, Dr. Wayne Dellinger, Dr. David Dunham, Dr. Robert Farquhar, Dr. Yanping Guo, Dr. Jose Guzman, Mr. Gene Heyler, Mr. Daniel O Shaughnessy, Mr. Gabe Rogers, Dr. Tom Strikwerda, and Dr. Robin Vaughn. It was a great opportunity to learn about Earth-orbiting missions. Thank you all! I am also grateful to the collaborators on the General Relativity study during my early stage of the PhD program. I thank Prof. Ephraim Fischbach and Prof. James Longuski from Purdue University and Dr. Giacomo Giampieri from Imperial College. Special thanks to my former advisors, Prof. Robert Melton and Prof. David Spencer, from the Pennsylvania State University who have given me many reasons to study the astrodynamics and have encouraged me to continue graduate studies at the University of Michigan. Special thanks to Prof. Carlos Cesnik from the University of Michigan for being a great mentor and a teacher. Last but not least, I thank all my friends at the University of Michigan who I had technical discussions and have given me the motivation and encouragement for my PhD research: Steve Broschart, Nalin Chaturvedi, Vincent Guibout, Ji Won Mok, Rafael Palacios-Nieto, Leo Rios-Reyes, Yoshifumi Suzuki, Benjamin Villac, and many others. THANK YOU ALL! iv

7 PREFACE This dissertation was submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Aerospace Engineering at the University of Michigan. The doctoral committee members were Dr. Paul W. Chodas, The Jet Propulsion Laboratory, California Institute of Technology, Prof. Alfred O. Hero III, Electrical Engineering, The University of Michigan, Prof. Pierre T. Kabamba, Aerospace Engineering, The University of Michigan, Prof. N. Harris McClamroch, Aerospace Engineering, The University of Michigan, Prof. Daniel J. Scheeres (Chair), Aerospace Engineering, The University of Michigan, and the PhD thesis was defended on November 27, The following list of papers are either published (or submitted) full journal articles or proceedings presented at technical conferences. Note that some of these papers are based on the studies during my early stage of the PhD program and the contents are not discussed in this dissertation. However, these studies gave me the theoretical insights and technical background needed for this study and have led to the baseline for the topics of this thesis. v

8 Journal Papers [76] R.S. Park and D.J. Scheeres, Nonlinear Semi-Analytic Methods for Trajectory Estimation, submitted to the Journal of Guidance, Control, and Dynamics, November [69] R.S. Park and D.J. Scheeres, Nonlinear Mapping of Gaussian State Uncertainties: Theory and Applications to Spacecraft Control and Navigation, Journal of Guidance, Control, and Dynamics, Vol. 29, No. 6, [90] D.J. Scheeres, F.-Y. Hsiao, R.S. Park, B.F. Villac, and J.M. Maruskin, Fundamental Limits on Spacecraft Orbit Uncertainty and Distribution Propagation, accepted for publication, Journal of the Astronautical Sciences, [77] R.S. Park, D.J. Scheeres, G. Giampieri, J.M. Longuski, and E. Fischbach, Estimating Parameterized Post-Newtonian Parameters from Spacecraft Radiometric Tracking Data, Journal of Spacecraft and Rockets, Vol. 42, No. 3, [53] J. Longuski, E. Fischbach, D.J. Scheeres, G. Giampieri, and R.S. Park, Deflection of Spacecraft Trajectories as a New Test of General Relativity: Determining the PPN Parameters β and γ, Physical Review D, Vol. 69, No , Conference Papers and Industrial Reports [72] R.S. Park and D.J. Scheeres, Nonlinear Semi-Analytic Method for Spacecraft Navigation, paper presented at AAS/AIAA Astrodynamics Specialist Conference, Keystone, Colorado, August 21-24, 2006, AIAA [71] R.S. Park, L.A. Cangahuala, P.W. Chodas, and I.M. Roundhill, Covariance Analysis of Cassini Titan Flyby using SAR and Altimetry Data, paper presented at vi

9 AAS/AIAA Astrodynamics Specialist Conference, Keystone, Colorado, August 21-24, 2006, AIAA [73] R.S. Park, D.J. Scheeres, Nonlinear Semi-Analytic Methods for Spacecraft Trajectory Design, Control, and Navigation, paper presented at New Trends in Astrodynamics and Applications Conference, Princeton, New Jersey, August 16-18, [67] R.S. Park, Expected Navigation Performance of the Radiation Belt Storm Probe Mission, APL Internal Report, SEG , August [70] R.S. Park and D.J. Scheeres, Nonlinear Mapping of Gaussian State Uncertainties: Theory and Applications to Spacecraft Control and Navigation, paper presented at AAS/AIAA Astrodynamics Specialist Conference, Lake Tahoe, California, August 7-11, 2005, AAS [75] R.S. Park and D.J. Scheeres, Nonlinear Mapping of Gaussian State Uncertainties, paper presented at 15th Workshop on JAXA Astrodynamics and Flight Mechanics, Kanagawa, Japan, July 25-26, [91] D.J. Scheeres, F.-Y. Hsiao, R.S. Park, B.F. Villac, and J.M. Maruskin, Fundamental Limits on Spacecraft Orbit Uncertainty and Distribution Propagation, invited paper presented at the Shuster Symposium, Grand Island, New York, June 2005, AAS [74] R.S. Park and D.J. Scheeres, Nonlinear Mapping of Gaussian State Covariance and Orbit Uncertainties, paper presented at AAS/AIAA Space Flight Mechanics Meeting, Copper Mountain, Colorado, January , AAS [78] R.S. Park, D.J. Scheeres, G. Giampieri, J.M. Longuski, and E. Fischbach, Orbit Design for General Relativity Experiments: Heliocentric and Mercury-centric vii

10 Cases, paper presented at AAS/AIAA Astrodynamics Specialist Conference, Providence, Rhode Island, August 16-19, 2004, AIAA [68] R.S. Park, E. Fischbach G. Giampieri, J.M. Longuski, and D.J. Scheeres, A Test of General Relativity: Estimating PPN parameters γ and β from Spacecraft Radiometric Tracking Data, Nuclear Physics B - Proceedings Supplement, Proceedings of the Second International Conference on Particle and Fundamental Physics in Space, 134(2004), [79] R.S. Park, D.J. Scheeres, G. Giampieri, J.M. Longuski, and E. Fischbach, Estimating General Relativity Parameters from Radiometric Tracking of Heliocentric Trajectories, paper presented at AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico, February 2003, AAS [17] C.E.S. Cesnik, R.S. Park, and R. Palacios, Effective Cross-Section Distribution of Anisotropic Piezocomposite Actuators for Wing Twist, paper presented at the SPIE 10th International Symposium on Smart Structures and Materials, San Diego, CA, March viii

11 TABLE OF CONTENTS DEDICATION ii ACKNOWLEDGEMENTS iii PREFACE v LIST OF FIGURES xii LIST OF TABLES LIST OF APPENDICES xvi xvii NOTATION xviii CHAPTERS I. INTRODUCTION Spacecraft Navigation and Uncertainty Propagation Specific Applications of Uncertainty Propagation to Spacecraft Navigation Scope of this Thesis Thesis Organization II. RELATIVE MOTION OF GENERAL NONLINEAR DYNAMICAL SYSTEMS General Trajectory Dynamics and Solution Flows Higher Order Taylor Series Approximations and Solutions Complexity of the Higher Order Solutions Dynamics and Properties of a Hamiltonian System Symplecticity of the Higher Order Solutions of a Hamiltonian System Convergence of the Higher Order Solutions III. EVOLUTION OF PROBABILITY DENSITY FUNCTIONS IN NON- LINEAR DYNAMICAL SYSTEMS ix

12 3.1 Review of Probability Theory and Random Processes The Gaussian Probability Distribution Dynamics of the Mean and Covariance Matrix The Fokker-Planck Equation Integral Invariance of Probability Solution of the Fokker-Planck Equation for a Hamiltonian System On the Relation of Phase Volume and Probability Time Invariance of the Probability Density Function of the Higher Order Hamiltonian Systems Nonlinear Mapping of the Gaussian Distribution Monte-Carlo Simulations Unscented Transformation IV. NONLINEAR TRAJECTORY NAVIGATION The Concept of Statistically Correct Trajectory Nonlinear Statistical Targeting On the Theoretic and Practical Aspects of Nonlinear Statistical Targeting Higher Order Bayesian Filter with Gaussian Boundary Conditions Implementation of a Nonlinear Filter Extended Kalman Filter Higher-Order Numerical Extended Kalman Filter Higher-Order Analytic Extended Kalman Filter Unscented Kalman Filter V. NONLINEAR SPACE MISSION ANALYSIS Motivation Nonlinear Propagation of Phase Volume Nonlinear Orbit Uncertainty Propagation Two-Body Problem: Earth-to-Moon Hohmann Transfer Hill Three-Body Problem: about Europa Nonlinear Statistical Targeting Two-Body Problem: Earth-to-Moon Hohmann Transfer Hill Three-Body Problem: about Europa Nonlinear Trajectory Navigation Halo Orbit: Sun-Earth System Halo Orbit: Earth-Moon System Potential Applications and Challenges VI. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS x

13 6.1 Concluding Remarks and Key Contributions Future Research and Recommendations APPENDICES BIBLIOGRAPHY xi

14 LIST OF FIGURES Figure 3.1 Normalized two-dimensional Gaussian probability density function: p(x,y) = (1/2π) exp { 1 2 (x2 + y 2 ) } Integral invariance of a phase volume Illustration of the statistically correction trajectory Illustration of the nonlinear statistical targeting Propagated mean and 1-σ error ellipsoid projected onto the spacecraft position plane: comparison of the STT-approach and Monte-Carlo simulations Hill three-body trajectory plot at Europa for days: circled points are computed at t {0, 0.881, 2.26, 4.42, 5.38, 5.74} days Trajectory norms and higher order solution magnitudes: circled points are computed at t {0, 0.881, 2.26, 4.42, 5.38, 5.74} days Phase volume projections: solid line represents integrated, dotted line represents the 1st order, dash-dot line represents the 2nd order, and dashed line represents the 3rd order solutions Phase volume projections: solid line represents integrated, dotted line represents the 1st order, dash-dot line represents the 2nd order, and dashed line represents the 3rd order solutions Phase volume projections Two-body problem: Hohmann transfer trajectory Two-body problem: comparison of the computed mean and covariance at apoapsis using STT-approach and Monte-Carlo simulations xii

15 5.8 Hill three-body problem: a safe trajectory at Europa Hill three-body problem: comparison of the computed mean and covariance at periapsis using STT-approach and Monte-Carlo simulations Two-body problem: computed V k using the linear and nonlinear methods Two-body problem: deviated position and velocity means at the target Two-body problem: Monte-Carlo simulation using the linear and nonlinear methods Hill three-body problem: computed V k using the linear and nonlinear methods Hill three-body problem: deviated position and velocity means at the target Hill three-body problem: Monte-Carlo simulation using the linear and nonlinear methods Nominal halo orbit about the Sun-Earth L 1 point Nominal halo orbit about the Sun-Earth L 1 point in x-y plane Sun-Earth halo orbit: covariance matrix computed after one orbital period Sun-Earth halo orbit: comparison of the uncertainties computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 20 days Sun-Earth halo orbit: comparison of the absolute errors computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 20 days Sun-Earth halo orbit: comparison of the uncertainties computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 5 days Sun-Earth halo orbit: comparison of the absolute errors computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 5 days xiii

16 5.23 Sun-Earth halo orbit: comparison of the uncertainties computed using the HNEKFs for the cases m = {1, 2, 3}. Measurements are taken every 20 days Sun-Earth halo orbit: comparison of the absolute errors computed using the HNEKFs for the cases m = {1, 2, 3}. Measurements are taken every 20 days Sun-Earth halo orbit: comparison of the uncertainties computed using the HAEKFs for the cases m = {1, 2, 3}. Measurements are taken every 20 days Sun-Earth halo orbit: comparison of the absolute errors computed using the HAEKFs for the cases m = {1, 2, 3}. Measurements are taken every 20 days Sun-Earth halo orbit: comparison of the uncertainties computed using the EKF, UKF, and HAEKFs for the cases m = {1, 3}. Measurements are taken every 20 days based on the halo orbit Case Sun-Earth halo orbit: comparison of the absolute errors computed using the EKF, UKF, and HAEKFs for the cases m = {1, 3}. Measurements are taken every 20 days based on the halo orbit Case Nominal halo orbit about the Earth-Moon L 1 point Nominal halo orbit about the Earth-Moon L 1 point in x-y plane Earth-Moon halo orbit: comparison of the uncertainties computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2 days Earth-Moon halo orbit: comparison of the absolute errors computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2 days Earth-Moon halo orbit: comparison of the uncertainties computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2 days assuming zero initial mean Earth-Moon halo orbit: comparison of the absolute errors computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2 days assuming zero initial mean xiv

17 5.35 Earth-Moon halo orbit: comparison of the uncertainties computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2 days assuming zero initial mean and small initial covariance matrix Earth-Moon halo orbit: comparison of the absolute errors computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2 days assuming zero initial mean and small initial covariance matrix Earth-Moon halo orbit: comparison of the uncertainties computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 6 hours Earth-Moon halo orbit: comparison of the absolute errors computed using the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 6 hours Earth-Moon halo orbit: comparison of the uncertainties computed using the HNEKFs for the cases m = {1, 2, 3}. Measurements are taken every 2 days Earth-Moon halo orbit: comparison of the absolute errors computed using the HNEKFs for the cases m = {1, 2, 3}. Measurements are taken every 2 days Earth-Moon halo orbit: comparison of the uncertainties computed using the HAEKFs for the cases m = {1, 2, 3}. Measurements are taken every 2 days Earth-Moon halo orbit: comparison of the absolute errors computed using the HAEKFs for the cases m = {1, 2, 3}. Measurements are taken every 2 days Earth-Moon halo orbit: comparison of the uncertainties computed using the EKF, UKF, and HAEKFs for the cases m = {1, 3}. Measurements are taken every 2 days based on the halo orbit Case Earth-Moon halo orbit: comparison of the absolute errors computed using the EKF, UKF, and HAEKFs for the cases m = {1, 3}. Measurements are taken every 2 days based on the halo orbit Case A.1 Families of halo orbits about the Sun-Earth L 1 point in non-dimensional frame xv

18 LIST OF TABLES Table 5.1 Local nonlinearity index Halo orbit maximum amplitudes with respect to the Sun-Earth L 1 point Halo orbit maximum amplitudes with respect to the Earth-Moon L 1 point. 142 A.1 Properties of planets and satellites A.2 Properties of three-body systems xvi

19 LIST OF APPENDICES Appendix A. EQUATIONS OF MOTION OF ASTRODYNAMICS PROBLEMS A.1 The Two-Body Problem A.2 The Three-Body Problem A.2.1 The Circular Restricted Three-Body Problem A.2.2 The Hill Three-Body Problem A.2.3 Halo Orbit B. PROPERTIES OF PROBABILITY DENSITY FUNCTIONS B.1 Integral Invariance of the PDF of a Linear Hamiltonian Dynamical System with Gaussian Boundary Conditions C. THE LINEAR KALMAN FILTER C.1 Kalman Filter Essentials C.2 Kalman Filter Derivation D. VECTORIZATION OF HIGHER ORDER TENSORS D.1 Specifications for MATLAB D.2 Specifications for C or C xvii

20 NOTATION Scalars, vectors, and matrices: Scalars are denoted by upper or lower case Roman or Greek letters in italic type, e.g., p, N, φ, or Γ. Vectors are denoted by lower case Roman or Greek letters in boldface type, e.g., x or φ. The vector x is composed of elements x i or the vector φ is composed of elements φ i. Components of a vector are denoted with a boldface superscript, e.g., given x = [r T, v T ] T, x r = r and x v = v, where x r and x v themselves are vectors. When a vector contains conflicting superscripts, parentheses and brackets are used to clarify a component of a vector, e.g., (x ) i is an ith component of x. Matrices are denoted by upper case Roman or Greek letters in boldface type, e.g., A or Φ. The matrix A is composed of elements A ij, which indicates the ith-row and jth-column entry of A. When a matrix contains conflicting superscripts, parentheses and brackets are used to clarify a component of a matrix, e.g., (P ) ij is an ith-row and jth column entry of P. A component of a matrix operation is denoted using superscripts with parentheses or brackets, i.e., A ij B jk = (AB) ik, or A ij ( )B jk ( ) = [A( )B( )] ik. Subscripts: A plain subscript of a scalar, a vector, or a matrix denotes the time which the variable is computed, e.g., x i 0 = x i (t 0 ), x 0 = x(t 0 ), or H k = H(t k ). A boldface subscript of a scalar denotes the row-wise partial derivative, e.g., H x = H(x)/ x or H xx = 2 H(x)/ x x. Superscripts: T : denotes the transpose of a vector or a matrix, e.g., x T or Φ T. 1 : denotes the inverse of a matrix with full-rank, e.g., Φ 1. : denotes predicted value of a scalar, a vector, or a matrix from a filter, e.g., x, m, or P. + : denotes updated value of a scalar, a vector, or a matrix from a filter, e.g., x +, m +, or P +. xviii

21 i,γ 1, γ p : denotes the pth order partial derivative of a scalar or a vector, e.g., y i,γ 1 γ p (x) = p y i (x)/ x γ1 x γp, where x γ j indicates the γ j th component of x. ι is reserved to denote an ιth solution of an iteration. κ is reserved to denote a κth sample chosen from a phase volume or a probability distribution. Exceptions: V k R 3 is a vector denoting an impulsive correction maneuver applied to a spacecraft trajectory. Partial derivatives of a scalar are denoted with subscripts, e.g., H i = H(x)/ x i or H ijk = 3 H(x)/ x i x j x k. Dimensions: A Lagrangian system has a dimension n, e.g., a Lagrangian system with position R 3 and velocity R 3 has a dimension n = 3. A Hamiltonian system has a dimension 2n, e.g., a Hamiltonian system with generalized coordinate R 3 and generalized momenta R 3 has a dimension 2n = 6. When a Lagrangian system is transformed into a full-state system, it has a dimension N = 2n, e.g., when a Lagrangian system with position R 3 and velocity R 3 is transformed into a full-state system, it has a dimension N = 6. Mathematical symbols: det : denotes a determinant. E( ) : denotes an expectation. E( ) : denotes a conditional expectation. exp : denotes an exponential function. lim : denotes a limit. sup : denotes a supremum. Trace( ): denotes the trace of a matrix. a b : denotes a dot product of a and b. : denotes an absolute value. : denotes a norm. : denotes an element of a vector or a set. : denotes a subset. xix

22 CHAPTER I INTRODUCTION "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." - Albert Einstein 1.1 Spacecraft Navigation and Uncertainty Propagation Given a dynamical system, the evolution of a particular state can be completely characterized by the system s governing equations of motion. In reality, however, the state is always associated with some errors that may be due to uncertain system models, inputs, or measurements. Hence, studying a deterministic trajectory may not provide sufficient information about the trajectory. For this reason, the problem of uncertainty propagation has received much attention in many engineering and scientific disciplines. Given an initial state and its associated uncertainties (usually described with a mean and a covariance matrix or a probability density function), the goal of uncertainty propagation is to predict the state and its statistical properties at some future time, or possibly along the entire trajectory, considering the statistical properties of the initial state. Except under certain assumptions, however, uncertainty propagation is an extremely difficult process if we want a complete statistical description. This is because it generally requires one to solve par- 1

23 2 tial differential equations such as the Fokker-Planck equation or to carry out particle-type studies such as Monte-Carlo simulations. Therefore, in practice, an approximation method is usually required. For spacecraft trajectory design and operations, uncertainty propagation usually refers to orbit uncertainty propagation, where the mean and covariance matrix of the spacecraft state are determined. Conventionally, the usual assumptions in spacecraft trajectory problems are: A linearized model sufficiently approximates the relative dynamics of neighboring trajectories with respect to a reference trajectory. The covariance matrix can be determined as the solution of a Riccati equation while assuming the reference trajectory is the mean trajectory. The orbit uncertainty can be completely characterized by a Gaussian probability distribution. These are usually good assumptions for spacecraft applications as we are usually given a reference (nominal) trajectory with a high degree of precision. The objective of trajectory navigation and spacecraft control is to follow a reference trajectory while minimizing some pre-defined optimality constraints, such as the number of trajectory correction maneuvers, flight time, fuel usage, etc. The basic underlying concept is to stay close enough to the reference trajectory so that the linear dynamics assumption applies. This can be achieved by taking a sufficient number of measurements along the trajectory so that the deviation and statistics can be accurately mapped linearly using the state transition (fundamental) matrix. The trajectory navigation problem is, at heart, a local problem, meaning that we wish to locate, control, and predict the spacecraft trajectory relative to a nominal path. When

24 3 uncertainties are small we often only need a linear characterization. For large uncertainties, however, the region of phase space where the spacecraft may be located is large and may require that we incorporate nonlinear local dynamics. For example, consider an interplanetary trajectory where the nominal trajectory leads to Mars. In reality, it is often the case that launch errors are large and lead to the actual trajectory deviating from the nominal. Also, orbit determination (OD) has a limited ability to locate the spacecraft, so even after tracking, the spacecraft s state is still only defined as a probability distribution. To correct the deviated trajectory, correction maneuvers must be made to target back to the Mars aim point. These target maneuvers have errors as well, due to the uncertainty of where the spacecraft lies; thus later corrections must be planned for and executed to provide additional corrections. Ideally, these sequences of correction maneuvers converge and a final, small maneuver days before arrival is all that is needed to hit the aim point with sufficient accuracy. However, when OD errors are large, the trajectory dynamics unstable, only a limited number of measurements available, or time spans are long, the use of linear dynamics models can result in additional errors and lead to more and larger correction maneuvers being required, inaccurate uncertainty predictions, and poorer filtering performance. It is these issues that this thesis deals with and solves. The linear dynamics assumption, when applied to spacecraft navigation, simplifies the problem a great deal, due mainly to the existence of a closed-form solution for the local dynamics. However, astrodynamics problems are nonlinear in general and the linear assumption can sometimes fail to characterize the true spacecraft dynamics and statistics when a system is subject to a highly unstable environment or when mapped over a long duration of time. Hence, in such cases, an alternate method which accounts for the system nonlinearity must be implemented. When such nonlinearities are important, the best known technique for propagating or-

25 4 bit uncertainty is a Monte-Carlo (MC) simulation, which approximates the probability distribution by averaging over a large set of random samples. A Monte-Carlo simulation can provide true statistics in the limit, but is computationally intensive and only solves for the statistics of a specific epoch and its associated uncertainties. Hence, for mission operations, these difficulties make Monte-Carlo simulations inefficient for practical spacecraft applications. One way to simplify the implementation would be to propagate each random sample analytically based on the linearized model using the state transition matrix. It is, however, inapplicable for highly nonlinear trajectories or problems with large initial errors, since the true trajectory (in a statistical sense) may not lie entirely within the linear regime. Other approaches to orbit uncertainty propagation have also been considered. Junkins et al. [47, 48] analyzed the effect of the coordinate system on the propagated statistics, and found that using osculating orbit elements improved future prediction. Their propagation method was still based on a linear assumption and system nonlinearity was not incorporated in the mapping. Another approach to orbit uncertainty propagation is a reduced Monte-Carlo method, such as approximating a probability distribution by the line-of-variation (LOV), which is a line chosen along the most uncertain direction of the uncertainty distribution [60]. An additional example of a reduced Monte-Carlo method is the unscented transformation, which approximates the probability distribution by nonlinearly integrating a set of deterministically chosen sample points [45, 44]. 1.2 Specific Applications of Uncertainty Propagation to Spacecraft Navigation In the following we list a number of applications of orbit uncertainty propagation for spacecraft trajectory problems. These applications apply to both traditional linear mappings and to nonlinear mapping methods:

26 5 Pre-mission covariance analysis: Covariance analysis is a design tool which is often used in spacecraft missions to characterize the navigation performance considering the statistical properties of the system and measurements. Examples of covariance analysis are predictions of how well estimation of some parameters can be made [53, 58, 59, 77] or how well a spacecraft state can be estimated using radiometric measurements, such as range and Doppler data [71, 89]. Orbit uncertainty propagation in a covariance analysis refers to the computation of a covariance matrix which characterizes the level of accuracy to which a spacecraft orbit can be estimated from an OD process, and provide predictions of delivery accuracy. Therefore, improved orbit uncertainty propagation can provide a more accurate determination of a covariance matrix, and thus, a more realistic estimation and mission scenario can be simulated. Mission prediction of future uncertainties and confidence regions: Trajectory prediction is an important problem in mission design as well as in celestial mechanics. The goal is to propagate the estimated spacecraft state to a future time and construct a confidence region where a spacecraft, or a celestial body, should be located with some probability level. For example, consider a small-body, such as an asteroid or a comet. We want to map both the body s state and its probability distribution so that we can project its uncertainties onto a target plane to obtain a confidence region [60]. When the confidence region is large, additional measurements may be necessary to reduce the error bounds. Hence, it is not sufficient to only consider a deterministic trajectory, but the statistical properties of a trajectory are also important.

27 6 Design and planning of statistical correction maneuvers: In mission operations, a spacecraft often deviates from the nominal trajectory due to uncertainties in the state and measurements and unmodeled accelerations acting on the spacecraft. Therefore, a spacecraft performs series of correction maneuvers to converge to a target or back to the nominal trajectory. Conventionally, trajectory correction maneuvers are computed based on the solution of a deterministic trajectory. The uncertainty propagation then verifies that the applied correction maneuver delivers the spacecraft to a desired target with tolerable error bounds. Using a statistical targeting method, however, it is also possible to incorporate the trajectory s statistical information to improve targeting performance. For example Ref. [84] discusses the computation of statistical correction maneuvers that optimizes the trajectory maintenance problem within an unstable dynamical environment. Computation of probability density functions using first few moments: In conventional navigation, a spacecraft state is usually modeled with Gaussian statistics, and hence, its probability distribution is completely characterized by the first two moments (mean and covariance matrix). Considering nonlinear trajectory dynamics, however, the propagated probability distribution no longer preserves the Gaussian structure. An approximation to the mapped probability distribution can be made using the first few moments of the system. This can provide useful information in case mission operations do not require a complete statistical description of a trajectory. Filtering and orbit determination: A filter is usually composed of two parts, prediction and update. Orbit uncertainty propagation relates to the prediction problem while the uncertain distribution influ-

28 7 ences the update part. In conventional trajectory navigation, a spacecraft is initially assumed to lie within a certain probability ellipsoid and the spacecraft trajectory is sequentially estimated until the solution converges. Mission operations usually implement an extended Kalman filter for trajectory navigation; however, when the trajectory is significantly nonlinear, it may be necessary to consider a filter that incorporates system nonlinearity. Examples of such filters include: Divided difference filter: approximates state using polynomial approximations from a multi-dimensional interpolation formula [63], Gaussian sum filter: approximates the conditional probability density using the sum of Gaussian distributions [38, 95], Hammerstein filter: incorporates the system nonlinearity using a linear Hammerstein system [11, 28, 41, 42], Higher order filter: applies higher order Taylor series expansion to estimate the mean and covariance matrix [6, 14, 55], Particle filter: approximates the conditional probability density using ensemble of random sample points [5, 7, 15, 21, 34, 35], and Volterra filter: applies Volterra series to estimate the higher order moments for system identification and estimation [1, 3, 64, 65, 66]. These filters, however, have not been implemented to real spacecraft trajectory navigation problems, except for a few special cases. This is mainly because the extended Kalman filter has, so far, provided sufficient accuracy for mission operations and the development of software tools for a nonlinear filter can be quite costly, and thus, may not be cost-effective. However, once a nonlinear filtering capability is devel-

29 8 oped and made feasible, it can provide more accurate science return and potentially reduced mission cost. A posteriori reconstruction: Given a set of measurements of a spacecraft, the OD process reconstructs the spacecraft trajectory to some level of accuracy. This is usually carried out using the batch least-squares approximation, which minimizes the sum of squared measurement residuals. From the trajectory reconstruction, an a posteriori spacecraft trajectory can be estimated and its statistics can be characterized. This is often useful for interpretation of scientific measurements. Among the many applications of orbit uncertainty propagation, this thesis mainly focuses on the trajectory navigation problem where we consider the problems of nonlinear orbit prediction, nonlinear statistical maneuver design, and nonlinear trajectory filtering and orbit determination. 1.3 Scope of this Thesis The goal of this thesis is to develop an analytical framework for nonlinear trajectory navigation. Throughout this dissertation, we assume a trajectory can be modeled as a Hamiltonian dynamical system with no diffusion (i.e., no process noise). Although this is not a completely realistic assumption, it is a standard and quite useful first approach to this problem. For a given reference trajectory, the nonlinear relative motion can be approximated by applying a Taylor series expansion of the solution function in terms of the initial conditions [62, 94]. The nonlinear local trajectory dynamics is then characterized by the higher order Taylor series terms that are extensions of the state transition matrix (STM) to higher orders. The theory and implementation of this approach are reasonably straightforward

30 9 and can be easily adapted to many spacecraft applications that are based on linear theory. However, there have been almost no studies on the use of higher order analysis for trajectory navigation problems. The significance of such a higher order approach is that it provides an analytic expression of the local nonlinear trajectory as a function of initial conditions. Thus, trajectory propagation only requires a simple algebraic manipulation. Using this nonlinear local solution we solve the Fokker-Planck equation for deterministic (i.e., diffusion-less) Hamiltonian systems and establish the time invariance property of the probability density function. Also, assuming the local nonlinear solutions are computed and the initial probability distribution is precisely known, we derive an analytic expression for propagation of the orbit uncertainties. Note that this analytic formulation determines the probability distribution from the solution of a nonlinear function, rather than from an empirical sampling technique such as a Monte-Carlo simulation. Assuming a Gaussian initial state with this approach, the statistics (mean and covariance matrix) computed using the higher order approach provide good agreement with Monte-Carlo simulations over a reasonable time period in the presence of strong nonlinearities. Using this result, we first consider the design of a statistically correct trajectory. Given a trajectory with an initial probability distribution, the mean trajectory will, depending on the system s nonlinearity, deviate from the reference (nominal) trajectory. To compensate for this, we introduce the concept of the statistically correct trajectory by solving for the initial state where the trajectory will satisfy a desired target state condition on average (e.g., the final state of a boundary value problem), not from a deterministic solution. This is a seemingly counter-intuitive approach since this implies that the initial state must be different from the solution of the deterministic boundary value problem. However, this should yield a better, more realistic (and practical), trajectory design according to probability theory.

31 10 As an extension of the statistically correct trajectory, we define a nonlinear statistical targeting method where we solve for a correction maneuver based on the mean trajectory. The usual linear targeting method solves for a maneuver for a deterministic trajectory. Due to orbit uncertainties, however, the final target will be offset from the desired target and additional correction maneuvers may be needed. If we solve for a correction maneuver using our nonlinear approach the number of correction maneuvers may be reduced, as it delivers the mean trajectory to the target. Our approach can also be applied to Bayesian filtering. We present a general filtering algorithm for optimal estimation of the conditional density function incorporating nonlinearity in the filtering process. We then derive practical Kalman-type filters based on our formulation. The first type, the higher-order numerical Extended Kalman filter, is based on an extension of the extended Kalman filter, where we integrate the nonlinear flow and the higher order solutions between each measurement update. This is in a sense similar to the second order filter by Athans et al. [6], but can be generalized to higher orders. The second type is called the higher-order analytic extended Kalman filter and is an extension of the linear Kalman filter, which assumes that the nonlinear flow and the higher order solutions are available prior to filtering, and hence, requires no on-line integration. For a nonlinear spacecraft trajectory we show that these filters have superior performance as compared with the conventional EKF. The following is a list of the key contributions of this thesis: A general theory for nonlinear relative motion is developed. A semi-analytic method for orbit uncertainty propagation is derived by applying the solutions of higher order relative dynamics with a known initial probability distribution.

32 11 The concept of a statistically correct trajectory is introduced by incorporating navigation information in the trajectory design process. A nonlinear statistical targeting method is developed by analytically computing a correction maneuver that hits the target on average. An optimal solution of the posterior conditional density function is presented by solving Bayes rule for state and measurement probability density functions. Practical Kalman-type filters are derived by incorporating nonlinear dynamical effects in the uncertainty propagation. 1.4 Thesis Organization This thesis starts from the basics of a general dynamical system and probability theory, and poses trajectory problems in an analytic framework. In Chapter II we develop an analytic trajectory propagation method. We first discuss the dynamical aspects of general astrodynamics problems and define the nonlinear relative dynamics with respect to a reference trajectory. We then present how the relative motion of a spacecraft can be completely characterized by computing the forward and inverse state transition tensors. A review of Hamiltonian dynamical systems is also given and their unique properties and facts are discussed. In Chapter III, we present a review of probability theory and give a discussion of the Fokker-Planck equation, which governs the evolution of the probability density function. The solution of the Fokker-Planck equation is then analyzed for a deterministic system and the time invariance of the probability density function for a Hamiltonian dynamical system is derived. This is then combined with results from Chapter II and applied to orbit uncertainty propagation as a function of the initial state and associated uncertainties.

33 12 Chapter IV presents several applications where orbit uncertainty propagation can be utilized. We introduce the concept of the statistically correct trajectory, and extend the idea to a nonlinear statistical targeting problem. Also, we discuss how the higher order solutions can be implemented in a Bayesian filtering algorithm to compute the optimal posterior conditional density function. We then extend this idea and derive Kalman-type filters based on numerical and analytical propagation of the orbit statistics. Chapter V gives several examples and simulations of our methods based on the twobody, the Hill three-body, and the circular restricted three-body problems. The examples assume realistic initial conditions and errors, and we compare the nonlinear STT-approach to conventional linear methods. Finally, conclusions and future work are presented in Chapter VI. Our study shows that a nonlinear relative trajectory of sufficient order recovers Monte-Carlo simulation results. Also, nonlinear statistical targeting provides a statistically more accurate correction maneuver than the conventional linear method. For a nonlinear filtering problem, we show that our higher order filters provide faster convergence and a superior solution as compared to linear filters.

34 CHAPTER II RELATIVE MOTION OF GENERAL NONLINEAR DYNAMICAL SYSTEMS In this chapter, relative motion about a nominal spacecraft trajectory is presented. We first express the general dynamics of a spacecraft as first order differential equations and define the solution flow which represents the nominal path of a spacecraft. We then derive the relative motion by applying a Taylor series expansion to a nominal trajectory and discuss the computation of higher order Taylor series terms that describe the local nonlinear motion. Also presented are the properties of a Hamiltonian system concerning astrodynamics problems that can be modeled as a Hamiltonian system. 2.1 General Trajectory Dynamics and Solution Flows In this thesis, we consider astrodynamics problems that can be modeled as: r(t) = f[t, r(t), ṙ(t)], (2.1) where r(t) R 3 represents the position vector and ṙ(t) = v(t) R 3 represents the velocity vector. 1 Considering Eqn. (2.1) and v = ṙ, it is apparent that the system dynamics can be transformed into first order differential equations. To show this, let x = [r T, v T ] T 1 Examples of this model consist of problems such as the two-body problem, Hill three-body problem, and circular restricted three-body problem. The governing differential equations for these examples are given in Appendix A. 13

35 14 represent the spacecraft state vector. The governing differential equations for x can be written as: ẋ(t) = ṙ(t) v(t) = v(t) f[t, r(t), v(t)]. (2.2) By letting g(t) = [v T (t), f T (t)] T, the general dynamics of a spacecraft can be stated as: ẋ(t) = g[t, x(t)], (2.3) with dimension N = 6 and an initial state x 0 = x(t 0 ). Definition (Solution Flow). A solution flow, φ(t; x 0, t 0 ), is a map of the initial state x 0 at time t 0 to a state x at time t and is defined as: x(t) = φ(t; x 0, t 0 ), (2.4) where t 0 and x 0 are free variables and the solution flow is governed by: dφ(t; x 0, t 0 ) dt = g[t, φ(t; x 0, t 0 )], (2.5) φ(t 0 ; x 0, t 0 ) = x 0. (2.6) Definition (Phase Volume). A phase volume is a subset of Euclidean space R N that is compact (closed and bounded) 2 ; e.g., the closed unit interval [0, 1] is a phase volume in R. Suppose we are given an initial phase volume B 0 = B(t 0 ). Using the solution flow notation, the evolution of B 0 can be defined as the mapping of every point in an initial set as: B(t) = {x x = φ(t; x 0, t 0 ) x 0 B 0 }. (2.7) 2 A phase volume is usually defined for the phase space of a Hamiltonian system (in a generalized coordinate-momenta coordinate frame), but in this thesis, a phase volume is also defined for a Lagrangian system (in a position-velocity coordinate frame).

36 15 If B 0 is defined locally with respect to a nominal initial condition ξ 0, Eqn. (2.7) represents all possible trajectories in a compact and closed neighborhood of the nominal trajectory φ(t k ; ξ 0, t 0 ). Definition (Inverse Solution Flow). Suppose we are given the solution flow of a nominal initial state x 0 for some time interval [t 0, t k ], i.e., x k = φ(t k ; x 0, t 0 ). The inverse solution flow ψ(t, x; t 0 ) is defined as a map of x at a time t (t 0 t t k ) to a fixed state x 0 at time t 0 : x 0 = ψ(t, x; t 0 ). (2.8) The inverse flow can be defined using the solution flow as ψ(t, x; t 0 ) = φ(t 0 ; x, t). Also, ψ(t, x; t 0 ) satisfies a partial differential equation: dx 0 dt = ψ t + ψ g(t, x) = 0. (2.9) x Remark Combining the definitions of the forward and inverse flows, an obvious, but important, identity exists: x 0 = ψ[t, φ(t; x 0, t 0 ); t 0 ]. (2.10) Definition (Relative Motion and Dynamics). Given a nominal initial state x 0, the relative motion δx(t) with respect to the reference (nominal) trajectory is defined as: δx(t) = φ(t; x 0 + δx 0, t 0 ) φ(t; x 0, t 0 ), (2.11) where δx 0 represents a deviation in x 0. The relative motion satisfies the equations of motion: δẋ(t) = g[t, φ(t; x 0 + δx 0, t 0 )] g[t, φ(t; x 0, t 0 )]. (2.12)