NONLINEAR TRAJECTORY NAVIGATION

by

Sang H. Park

A dissertation submitted in partial fulfillmentof the requirements for the degree of

Doctor of Philosophy(Aerospace Engineering)

in The University of Michigan2007

Doctoral Committee:

Associate Professor Daniel J. Scheeres, ChairProfessor Alfred O. Hero IIIProfessor Pierre T. KabambaProfessor N. Harris McClamrochResearch Scientist Paul W. Chodas, Jet Propulsion Laboratory

c© Sang H. ParkAll Rights Reserved

2007

To my parents.

ii

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 Adminis-

tration. 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 al-

most 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 parame-

iii

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, Navi-

gation 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 Can-

gahuala, 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 deep-

space 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 in-

tern, 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 astrodynam-

ics 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 Michi-

gan 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

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 Technol-

ogy,

• 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 Michi-

gan,

and the PhD thesis was defended on November 27, 2006. 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

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, Novem-

ber 2006.

• [69] R.S. Park and D.J. Scheeres, Nonlinear Mapping of Gaussian State Uncertain-

ties: Theory and Applications to Spacecraft Control and Navigation, Journal of

Guidance, Control, and Dynamics, Vol. 29, No. 6, 2006.

• [90] D.J. Scheeres, F.-Y. Hsiao, R.S. Park, B.F. Villac, and J.M. Maruskin, Fun-

damental Limits on Spacecraft Orbit Uncertainty and Distribution Propagation, ac-

cepted for publication, Journal of the Astronautical Sciences, 2005.

• [77] R.S. Park, D.J. Scheeres, G. Giampieri, J.M. Longuski, and E. Fischbach, Es-

timating Parameterized Post-Newtonian Parameters from Spacecraft Radiometric

Tracking Data, Journal of Spacecraft and Rockets, Vol. 42, No. 3, 2005.

• [53] J. Longuski, E. Fischbach, D.J. Scheeres, G. Giampieri, and R.S. Park, Deflec-

tion of Spacecraft Trajectories as a New Test of General Relativity: Determining the

PPN Parameters β and γ, Physical Review D, Vol. 69, No. 042001, 2004.

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, Key-

stone, Colorado, August 21-24, 2006, AIAA-2006-6399.

• [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

AAS/AIAA Astrodynamics Specialist Conference, Keystone, Colorado, August 21-

24, 2006, AIAA-2006-6398.

• [73] R.S. Park, D.J. Scheeres, Nonlinear Semi-Analytic Methods for Spacecraft Tra-

jectory Design, Control, and Navigation, paper presented at New Trends in Astrody-

namics and Applications Conference, Princeton, New Jersey, August 16-18, 2006.

• [67] R.S. Park, Expected Navigation Performance of the Radiation Belt Storm Probe

Mission, APL Internal Report, SEG-06-022, August 2006.

• [70] R.S. Park and D.J. Scheeres, Nonlinear Mapping of Gaussian State Uncertain-

ties: Theory and Applications to Spacecraft Control and Navigation, paper presented

at AAS/AIAA Astrodynamics Specialist Conference, Lake Tahoe, California, August

7-11, 2005, AAS-05-404.

• [75] R.S. Park and D.J. Scheeres, Nonlinear Mapping of Gaussian State Uncertain-

ties, paper presented at 15th Workshop on JAXA Astrodynamics and Flight Mechan-

ics, Kanagawa, Japan, July 25-26, 2005.

• [91] D.J. Scheeres, F.-Y. Hsiao, R.S. Park, B.F. Villac, and J.M. Maruskin, Funda-

mental Limits on Spacecraft Orbit Uncertainty and Distribution Propagation, invited

paper presented at the Shuster Symposium, Grand Island, New York, June 2005,

AAS-05-471.

• [74] R.S. Park and D.J. Scheeres, Nonlinear Mapping of Gaussian State Covari-

ance and Orbit Uncertainties, paper presented at AAS/AIAA Space Flight Mechanics

Meeting, Copper Mountain, Colorado, January 23-27. 2005, AAS-05-170.

• [78] R.S. Park, D.J. Scheeres, G. Giampieri, J.M. Longuski, and E. Fischbach, Or-

bit Design for General Relativity Experiments: Heliocentric and Mercury-centric

vii

Cases, paper presented at AAS/AIAA Astrodynamics Specialist Conference, Provi-

dence, Rhode Island, August 16-19, 2004, AIAA-2004-5394.

• [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 Ra-

diometric Tracking Data, Nuclear Physics B - Proceedings Supplement, Proceed-

ings of the Second International Conference on Particle and Fundamental Physics

in Space, 134(2004), 2004.

• [79] R.S. Park, D.J. Scheeres, G. Giampieri, J.M. Longuski, and E. Fischbach, Es-

timating General Relativity Parameters from Radiometric Tracking of Heliocentric

Trajectories, paper presented at AAS/AIAA Space Flight Mechanics Meeting, Ponce,

Puerto Rico, February 2003, AAS-03-205.

• [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 2003.

viii

TABLE OF CONTENTS

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

CHAPTERS

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Spacecraft Navigation and Uncertainty Propagation . . . . . . . 11.2 Specific Applications of Uncertainty Propagation to Spacecraft

Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Scope of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . 11

II. RELATIVE MOTION OF GENERAL NONLINEAR DYNAMICALSYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 General Trajectory Dynamics and Solution Flows . . . . . . . . 132.2 Higher Order Taylor Series Approximations and Solutions . . . . 16

2.2.1 Complexity of the Higher Order Solutions . . . . . . . 252.3 Dynamics and Properties of a Hamiltonian System . . . . . . . . 272.4 Symplecticity of the Higher Order Solutions of a Hamiltonian

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5 Convergence of the Higher Order Solutions . . . . . . . . . . . . 43

III. EVOLUTION OF PROBABILITY DENSITY FUNCTIONS IN NON-LINEAR DYNAMICAL SYSTEMS . . . . . . . . . . . . . . . . . . . 46

ix

3.1 Review of Probability Theory and Random Processes . . . . . . 463.2 The Gaussian Probability Distribution . . . . . . . . . . . . . . 493.3 Dynamics of the Mean and Covariance Matrix . . . . . . . . . . 523.4 The Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . 533.5 Integral Invariance of Probability . . . . . . . . . . . . . . . . . 563.6 Solution of the Fokker-Planck Equation for a Hamiltonian System 603.7 On the Relation of Phase Volume and Probability . . . . . . . . 633.8 Time Invariance of the Probability Density Function of the Higher

Order Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . 653.9 Nonlinear Mapping of the Gaussian Distribution . . . . . . . . . 673.10 Monte-Carlo Simulations . . . . . . . . . . . . . . . . . . . . . 693.11 Unscented Transformation . . . . . . . . . . . . . . . . . . . . . 71

IV. NONLINEAR TRAJECTORY NAVIGATION . . . . . . . . . . . . . 73

4.1 The Concept of Statistically Correct Trajectory . . . . . . . . . . 754.2 Nonlinear Statistical Targeting . . . . . . . . . . . . . . . . . . 79

4.2.1 On the Theoretic and Practical Aspects of NonlinearStatistical Targeting . . . . . . . . . . . . . . . . . . . 84

4.3 Higher Order Bayesian Filter with Gaussian Boundary Conditions 884.4 Implementation of a Nonlinear Filter . . . . . . . . . . . . . . . 98

4.4.1 Extended Kalman Filter . . . . . . . . . . . . . . . . 1004.4.2 Higher-Order Numerical Extended Kalman Filter . . . 1014.4.3 Higher-Order Analytic Extended Kalman Filter . . . . 1064.4.4 Unscented Kalman Filter . . . . . . . . . . . . . . . . 109

V. NONLINEAR SPACE MISSION ANALYSIS . . . . . . . . . . . . . 111

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2 Nonlinear Propagation of Phase Volume . . . . . . . . . . . . . 1125.3 Nonlinear Orbit Uncertainty Propagation . . . . . . . . . . . . . 120

5.3.1 Two-Body Problem: Earth-to-Moon Hohmann Transfer 1205.3.2 Hill Three-Body Problem: about Europa . . . . . . . . 123

5.4 Nonlinear Statistical Targeting . . . . . . . . . . . . . . . . . . 1255.4.1 Two-Body Problem: Earth-to-Moon Hohmann Transfer 1255.4.2 Hill Three-Body Problem: about Europa . . . . . . . . 128

5.5 Nonlinear Trajectory Navigation . . . . . . . . . . . . . . . . . 1315.5.1 Halo Orbit: Sun-Earth System . . . . . . . . . . . . . 1315.5.2 Halo Orbit: Earth-Moon System . . . . . . . . . . . . 1415.5.3 Potential Applications and Challenges . . . . . . . . . 152

VI. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS . . . . 156

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6.1 Concluding Remarks and Key Contributions . . . . . . . . . . . 1566.2 Future Research and Recommendations . . . . . . . . . . . . . . 159

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

xi

LIST OF FIGURES

Figure

3.1 Normalized two-dimensional Gaussian probability density function: p(x,y) =(1/2π) exp

−12(x2 + y2)

. . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Integral invariance of a phase volume. . . . . . . . . . . . . . . . . . . 57

4.1 Illustration of the statistically correction trajectory. . . . . . . . . . . . 75

4.2 Illustration of the nonlinear statistical targeting. . . . . . . . . . . . . . 79

4.3 Propagated mean and 1-σ error ellipsoid projected onto the spacecraftposition plane: comparison of the STT-approach and Monte-Carlo sim-ulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.1 Hill three-body trajectory plot at Europa for∼ 6.775 days: circled pointsare computed at t ∈ 0, 0.881, 2.26, 4.42, 5.38, 5.74 days. . . . . . . . 113

5.2 Trajectory norms and higher order solution magnitudes: circled pointsare computed at t ∈ 0, 0.881, 2.26, 4.42, 5.38, 5.74 days. . . . . . . . 115

5.3 Phase volume projections: ‘solid’ line represents integrated, ‘dotted’ linerepresents the 1st order, ‘dash-dot’ line represents the 2nd order, and‘dashed’ line represents the 3rd order solutions. . . . . . . . . . . . . . 117

5.4 Phase volume projections: ‘solid’ line represents integrated, ‘dotted’ linerepresents the 1st order, ‘dash-dot’ line represents the 2nd order, and‘dashed’ line represents the 3rd order solutions. . . . . . . . . . . . . . 118

5.5 Phase volume projections. . . . . . . . . . . . . . . . . . . . . . . . . 119

5.6 Two-body problem: Hohmann transfer trajectory. . . . . . . . . . . . . 121

5.7 Two-body problem: comparison of the computed mean and covarianceat apoapsis using STT-approach and Monte-Carlo simulations. . . . . . 121

xii

5.8 Hill three-body problem: a safe trajectory at Europa. . . . . . . . . . . 124

5.9 Hill three-body problem: comparison of the computed mean and covari-ance at periapsis using STT-approach and Monte-Carlo simulations. . . 124

5.10 Two-body problem: computed ∆Vk using the linear and nonlinear methods.126

5.11 Two-body problem: deviated position and velocity means at the target. . 127

5.12 Two-body problem: Monte-Carlo simulation using the linear and non-linear methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.13 Hill three-body problem: computed ∆Vk using the linear and nonlinearmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.14 Hill three-body problem: deviated position and velocity means at thetarget. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.15 Hill three-body problem: Monte-Carlo simulation using the linear andnonlinear methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.16 Nominal halo orbit about the Sun-Earth L1 point. . . . . . . . . . . . . 132

5.17 Nominal halo orbit about the Sun-Earth L1 point in x-y plane. . . . . . . 132

5.18 Sun-Earth halo orbit: covariance matrix computed after one orbital period.134

5.19 Sun-Earth halo orbit: comparison of the uncertainties computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurementsare taken every 20 days. . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.20 Sun-Earth halo orbit: comparison of the absolute errors computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurementsare taken every 20 days. . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.21 Sun-Earth halo orbit: comparison of the uncertainties computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurementsare taken every 5 days. . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.22 Sun-Earth halo orbit: comparison of the absolute errors computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurementsare taken every 5 days. . . . . . . . . . . . . . . . . . . . . . . . . . . 137

xiii

5.23 Sun-Earth halo orbit: comparison of the uncertainties computed usingthe HNEKFs for the cases m = 1, 2, 3. Measurements are taken every20 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.24 Sun-Earth halo orbit: comparison of the absolute errors computed usingthe HNEKFs for the cases m = 1, 2, 3. Measurements are taken every20 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.25 Sun-Earth halo orbit: comparison of the uncertainties computed usingthe HAEKFs for the cases m = 1, 2, 3. Measurements are taken every20 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.26 Sun-Earth halo orbit: comparison of the absolute errors computed usingthe HAEKFs for the cases m = 1, 2, 3. Measurements are taken every20 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.27 Sun-Earth halo orbit: comparison of the uncertainties computed usingthe EKF, UKF, and HAEKFs for the cases m = 1, 3. Measurementsare taken every 20 days based on the halo orbit Case 2. . . . . . . . . . 140

5.28 Sun-Earth halo orbit: comparison of the absolute errors computed usingthe EKF, UKF, and HAEKFs for the cases m = 1, 3. Measurementsare taken every 20 days based on the halo orbit Case 2. . . . . . . . . . 140

5.29 Nominal halo orbit about the Earth-Moon L1 point. . . . . . . . . . . . 143

5.30 Nominal halo orbit about the Earth-Moon L1 point in x-y plane. . . . . 143

5.31 Earth-Moon halo orbit: comparison of the uncertainties computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurementsare taken every 2 days. . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.32 Earth-Moon halo orbit: comparison of the absolute errors computed us-ing the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measure-ments are taken every 2 days. . . . . . . . . . . . . . . . . . . . . . . . 144

5.33 Earth-Moon halo orbit: comparison of the uncertainties computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurementsare taken every 2 days assuming zero initial mean. . . . . . . . . . . . . 147

5.34 Earth-Moon halo orbit: comparison of the absolute errors computed us-ing the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measure-ments are taken every 2 days assuming zero initial mean. . . . . . . . . 147

xiv

5.35 Earth-Moon halo orbit: comparison of the uncertainties computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurementsare taken every 2 days assuming zero initial mean and small initial co-variance matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.36 Earth-Moon halo orbit: comparison of the absolute errors computed us-ing the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measure-ments are taken every 2 days assuming zero initial mean and small initialcovariance matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.37 Earth-Moon halo orbit: comparison of the uncertainties computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurementsare taken every 6 hours. . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.38 Earth-Moon halo orbit: comparison of the absolute errors computed us-ing the EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measure-ments are taken every 6 hours. . . . . . . . . . . . . . . . . . . . . . . 149

5.39 Earth-Moon halo orbit: comparison of the uncertainties computed usingthe HNEKFs for the cases m = 1, 2, 3. Measurements are taken every2 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.40 Earth-Moon halo orbit: comparison of the absolute errors computed us-ing the HNEKFs for the cases m = 1, 2, 3. Measurements are takenevery 2 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.41 Earth-Moon halo orbit: comparison of the uncertainties computed usingthe HAEKFs for the cases m = 1, 2, 3. Measurements are taken every2 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.42 Earth-Moon halo orbit: comparison of the absolute errors computed us-ing the HAEKFs for the cases m = 1, 2, 3. Measurements are takenevery 2 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.43 Earth-Moon halo orbit: comparison of the uncertainties computed usingthe EKF, UKF, and HAEKFs for the cases m = 1, 3. Measurementsare taken every 2 days based on the halo orbit Case 2. . . . . . . . . . . 153

5.44 Earth-Moon halo orbit: comparison of the absolute errors computed us-ing the EKF, UKF, and HAEKFs for the cases m = 1, 3. Measure-ments are taken every 2 days based on the halo orbit Case 2. . . . . . . 153

A.1 Families of halo orbits about the Sun-Earth L1 point in non-dimensionalframe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

xv

LIST OF TABLES

Table

5.1 Local nonlinearity index. . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2 Halo orbit maximum amplitudes with respect to the Sun-Earth L1 point. 131

5.3 Halo orbit maximum amplitudes with respect to the Earth-Moon L1 point. 142

A.1 Properties of planets and satellites. . . . . . . . . . . . . . . . . . . . . 164

A.2 Properties of three-body systems. . . . . . . . . . . . . . . . . . . . . . 166

xvi

LIST OF APPENDICES

Appendix

A. EQUATIONS OF MOTION OF ASTRODYNAMICS PROBLEMS . . . . 163

A.1 The Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . 163A.2 The Three-Body Problem . . . . . . . . . . . . . . . . . . . . . 164

A.2.1 The Circular Restricted Three-Body Problem . . . . . 165A.2.2 The Hill Three-Body Problem . . . . . . . . . . . . . 166A.2.3 Halo Orbit . . . . . . . . . . . . . . . . . . . . . . . 167

B. PROPERTIES OF PROBABILITY DENSITY FUNCTIONS . . . . . . . 169

B.1 Integral Invariance of the PDF of a Linear Hamiltonian Dynami-cal System with Gaussian Boundary Conditions . . . . . . . . . 169

C. THE LINEAR KALMAN FILTER . . . . . . . . . . . . . . . . . . . . . 171

C.1 Kalman Filter Essentials . . . . . . . . . . . . . . . . . . . . . . 171C.2 Kalman Filter Derivation . . . . . . . . . . . . . . . . . . . . . 173

D. VECTORIZATION OF HIGHER ORDER TENSORS . . . . . . . . . . . 180

D.1 Specifications for MATLAB . . . . . . . . . . . . . . . . . . . . 180D.2 Specifications for C or C++ . . . . . . . . . . . . . . . . . . . . 181

xvii

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 xi or the vector φ is composed ofelements φi. Components of a vector are denoted with a boldface superscript, e.g.,given x = [rT , vT ]T , xr = r and xv = v, where xr and xv themselves are vectors.When a vector contains conflicting superscripts, parentheses and brackets are usedto 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 Aij , which indicates the ith-row andjth-column entry of A. When a matrix contains conflicting superscripts, parenthesesand brackets are used to clarify a component of a matrix, e.g., (P−)ij is an ith-rowand jth column entry of P−. A component of a matrix operation is denoted usingsuperscripts with parentheses or brackets, i.e., AijBjk = (AB)ik, or Aij(·)Bjk(·) =[A(·)B(·)]ik.

Subscripts:

• A plain subscript of a scalar, a vector, or a matrix denotes the time which the variableis computed, e.g., xi

0 = xi(t0), x0 = x(t0), or Hk = H(tk).

• A boldface subscript of a scalar denotes the row-wise partial derivative, e.g., Hx =∂H(x)/∂x or Hxx = ∂2H(x)/∂x∂x.

Superscripts:

• T : denotes the transpose of a vector or a matrix, e.g., xT or ΦT .

• −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

• i,γ1,···γp : denotes the pth order partial derivative of a scalar or a vector, e.g., yi,γ1···γp(x)= ∂pyi(x)/∂xγ1 · · · ∂xγp , where xγj indicates the γjth 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 probabilitydistribution.

Exceptions:

• ∆Vk ∈ R3 is a vector denoting an impulsive correction maneuver applied to a space-craft trajectory.

• Partial derivatives of a scalar are denoted with subscripts, e.g., Hi = ∂H(x)/∂xi orHijk = ∂3H(x)/∂xi∂xj∂xk.

Dimensions:

• A Lagrangian system has a dimension n, e.g., a Lagrangian system with position∈ R3 and velocity ∈ R3 has a dimension n = 3.

• A Hamiltonian system has a dimension 2n, e.g., a Hamiltonian system with gener-alized coordinate ∈ R3 and generalized momenta ∈ R3 has a dimension 2n = 6.

• When a Lagrangian system is transformed into a full-state system, it has a dimensionN = 2n, e.g., when a Lagrangian system with position ∈ R3 and velocity ∈ R3 istransformed 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

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 char-

acterized 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 ini-

tial 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 pre-

dict 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 as-

sumptions, 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

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 prob-

lems 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

3

uncertainties are small we often only need a linear characterization. For large uncertain-

ties, 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 in-

terplanetary 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 un-

stable, 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 cor-

rection 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-

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 space-

craft applications. One way to simplify the implementation would be to propagate each

random sample analytically based on the linearized model using the state transition ma-

trix. 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 consid-

ered. 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 nonlin-

early integrating a set of deterministically chosen sample points [45, 44].

1.2 Specific Applications of Uncertainty Propagation to SpacecraftNavigation

In the following we list a number of applications of orbit uncertainty propagation for

spacecraft trajectory problems. These applications apply to both traditional linear map-

pings and to nonlinear mapping methods:

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 ac-

curacy 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 celes-

tial 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 measure-

ments 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.

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 trajec-

tory. The uncertainty propagation then verifies that the applied correction maneuver

delivers the spacecraft to a desired target with tolerable error bounds. Using a sta-

tistical 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 tra-

jectory 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 informa-

tion 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-

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 in-

corporates 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 Ham-

merstein 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 ensem-

ble 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 navi-

gation 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-

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 space-

craft 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 condi-

tions [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

9

and can be easily adapted to many spacecraft applications that are based on linear the-

ory. 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 determin-

istic (i.e., diffusion-less) Hamiltonian systems and establish the time invariance property

of the probability density function. Also, assuming the local nonlinear solutions are com-

puted 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. Assum-

ing 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 sim-

ulations 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 proba-

bility theory.

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 nonlin-

earity 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 distribu-

tion.

11

• The concept of a statistically correct trajectory is introduced by incorporating navi-

gation 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 ef-

fects 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.

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 two-

body, 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 re-

sults. 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.

CHAPTER II

RELATIVE MOTION OF GENERAL NONLINEARDYNAMICAL 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 dis-

cuss the computation of higher order Taylor series terms that describe the local nonlinear

motion. Also presented are the properties of a Hamiltonian system concerning astrody-

namics 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), r(t)], (2.1)

where r(t) ∈ R3 represents the position vector and r(t) = v(t) ∈ R3 represents the

velocity vector.1 Considering Eqn. (2.1) and v = r, it is apparent that the system dynamics

can be transformed into first order differential equations. To show this, let x = [rT , vT ]T

1Examples 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 aregiven in Appendix A.

13

14

represent the spacecraft state vector. The governing differential equations for x can be

written as:

x(t) =

r(t)

v(t)

=

v(t)

f[t, r(t), v(t)]

. (2.2)

By letting g(t) = [vT (t) , fT (t)]T , the general dynamics of a spacecraft can be stated as:

x(t) = g[t, x(t)], (2.3)

with dimension N = 6 and an initial state x0 = x(t0).

Definition 2.1.1 (Solution Flow). A solution flow, φ(t; x0, t0), is a map of the initial state

x0 at time t0 to a state x at time t and is defined as:

x(t) = φ(t; x0, t0), (2.4)

where t0 and x0 are free variables and the solution flow is governed by:

dφ(t; x0, t0)

dt= g[t,φ(t; x0, t0)], (2.5)

φ(t0; x0, t0) = x0. (2.6)

Definition 2.1.2 (Phase Volume). A phase volume is a subset of Euclidean space RN 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 B0 = B(t0). Using the solution flow

notation, the evolution of B0 can be defined as the mapping of every point in an initial set

as:

B(t) = x | x = φ(t; x0, t0) ∀ x0 ∈ B0. (2.7)2A 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 Lagrangiansystem (in a position-velocity coordinate frame).

15

If B0 is defined locally with respect to a nominal initial condition ξ0, Eqn. (2.7) represents

all possible trajectories in a compact and closed neighborhood of the nominal trajectory

φ(tk; ξ0, t0).

Definition 2.1.3 (Inverse Solution Flow). Suppose we are given the solution flow of a

nominal initial state x0 for some time interval [t0, tk], i.e., xk = φ(tk; x0, t0). The inverse

solution flow ψ(t, x; t0) is defined as a map of x at a time t (t0 ≤ t ≤ tk) to a fixed state

x0 at time t0:

x0 = ψ(t, x; t0). (2.8)

The inverse flow can be defined using the solution flow as ψ(t, x; t0) = φ(t0; x, t). Also,

ψ(t, x; t0) satisfies a partial differential equation:

dx0

dt=

∂ψ

∂t+

∂ψ

∂x· g(t, x) = 0. (2.9)

Remark 2.1.4. Combining the definitions of the forward and inverse flows, an obvious, but

important, identity exists:

x0 = ψ[t, φ(t; x0, t0); t0]. (2.10)

Definition 2.1.5 (Relative Motion and Dynamics). Given a nominal initial state x0, the

relative motion δx(t) with respect to the reference (nominal) trajectory is defined as:

δx(t) = φ(t; x0 + δx0, t0)− φ(t; x0, t0), (2.11)

where δx0 represents a deviation in x0. The relative motion satisfies the equations of

motion:

δx(t) = g[t, φ(t; x0 + δx0, t0)]− g[t, φ(t; x0, t0)]. (2.12)

16

Remark 2.1.6. Using tensor notation, Eqns. (2.11) and (2.12) can be stated as:

δxi(t) = φi(t; x0 + δx0, t0)− φi(t; x0, t0), (2.13)

δxi(t) = gi[t, φ(t; x0 + δx0, t0)]− gi[t, φ(t; x0, t0)]. (2.14)

2.2 Higher Order Taylor Series Approximations and Solutions

Definition 2.2.1 (Taylor Series Expansion). Given an infinitely differentiable, real func-

tion s(x) with s, x ∈ RN , the Taylor series expansion about a point x = a is defined as

[8, 10, 62]:

si(x1, · · · , xN) =∞∑

j=0

1

j!

[N∑

k=1

(xk − ak)∂

∂ξk

]j

si(ξ1, · · · , ξN)

∣∣∣∣∣∣ξl=al

, (2.15)

where ξ represents a dummy variable. In vector form, the Taylor series expansion can be

stated as:

si(x) =∞∑

j=0

1

j![(x− a) · ∇ξ]

j si(ξ)

∣∣∣∣ξ=a

, (2.16)

where ∇ξ represents a gradient operator:

∇ξ =

[∂

∂ξ1· · · ∂

∂ξN

]. (2.17)

Definition 2.2.2 (Radius of Convergence). Consider the Taylor series expansion given in

Eqn. (2.16). A Taylor series is essentially a power series, and thus, the solution may con-

verge or diverge depending on the size of ‖x− a‖. To ensure that the series is convergent,

we define the radius of convergence Rc as:

• the series si(x) converges absolutely ∀ x such that 0 ≤ ‖x− a‖ ≤ Rc and

• the series si(x) diverges ∀ x such that ‖x− a‖ > Rc.

17

Remark 2.2.3. We can re-order the j = 0 term in Eqn. (2.16) and rewrite the Taylor series

expansion as:

si(x)− si(a) =∞∑

j=1

1

j![(x− a) · ∇ξ]

j si(ξ)

∣∣∣∣ξ=a

. (2.18)

In this form, it is apparent that the relative motion and dynamics defined in Eqns. (2.11)

and (2.12), respectively, can be approximated along a nominal trajectory solution.

Remark 2.2.4. An M th order Taylor series is defined as:

si(x)− si(a) =M∑

j=1

1

j![(x− a) · ∇ξ]

j si(ξ)

∣∣∣∣ξ=a

. (2.19)

Note that this is an approximation of Eqn. (2.18) by truncating the solution up to order M .

Definition 2.2.5 (Relative Trajectory Dynamics). Suppose we are given a system with the

governing differential equations g[t, x(t)] which is at least M times differentiable. By

applying the M th order Taylor series expansion about the reference trajectory φ(t; x0, t0),

where x0 is the initial condition, the relative trajectory dynamics can be stated using the

Einstein summation convention as:

δxi(t) =M∑

p=1

1

p!gi,γ1···γpδxγ1 · · · δxγp , (2.20)

where γj ∈ 1, · · · , N, superscripts γj denote the γjth component of the state vector, and

gi,γ1···γp(t, x) =∂pgi[t, ξ(t)]

∂ξγ1 · · · ∂ξγp

∣∣∣∣ξj=φj(t;x0,t0)

. (2.21)

Then, for a given initial deviation δx0 with respect to x0, the relative trajectory mo-

tion δx(t; δx0, t0) can be computed by integrating Eqn. (2.20) along the nominal flow

φ(t; x0, t0). Note that this formulation is an extension of the conventional linear dynamics

theory.

18

Definition 2.2.6 (State Transition Tensors). Suppose we are given a reference solution flow

φ(t; x0, t0) computed according to the governing differential equations g[t, x(t)], where x0

represents the nominal initial state. Moreover, assume g[t, x(t)] is at least m times differ-

entiable. By applying the mth order Taylor series expansion about the initial condition,

the relative trajectory motion can be stated as:

δxi(t) =m∑

p=1

1

p!φ

i,γ1···γp

(t,t0) δxγ1

0 · · · δxγp

0 , (2.22)

where

φi,γ1···γp

(t,t0) (t; x0, t0) =∂pφi(t; ξ0, t0)

∂ξγ1

0 · · · ∂ξγp

0

∣∣∣∣ξj0=xj

0

. (2.23)

The higher order partial derivatives of the solution flow, i.e., Eqn. (2.23), relate deviations

in the initial state at time t0 to deviations in the state at time t (i.e., δx0 7→ δx(t)): denoted

by the subscript (t, t0). We call these partial derivatives the state transition tensors (STTs).

Note that assuming the STTs are available, the relative solution flow is analytic in δx0.

Throughout this thesis, we note that an M th order solution refers to the integrated

relative motion according to Eqn. (2.20) and an mth order solution refers to the analyt-

ically mapped solution according to Eqn. (2.22). Moreover, unless stated otherwise we

assume m = M . In conventional practice, the higher order series are usually truncated at

M = m = 1, which is a first order or linear analysis. Then, Eqns. (2.21) and (2.23) are

the usual “linear dynamics matrix” and the “state transition matrix” (STM), respectively.

It is important to note that the relative trajectory motion δx0 7→ δx(t) computed ac-

cording to Eqns. (2.20) and (2.22) are fundamentally different. This is not only because

the relative motion is numerically integrated in Eqn. (2.20) and is analytically mapped

(assuming the STTs are available) in Eqn. (2.22), but also due to the difference in their

accuracies. To clarify these differences, consider an M th order relative motion. This inte-

grated solution is the M th order solution of a particular initial deviation δx0 and is accurate

19

up to order M since the M th order partial derivatives of the dynamics gi,γ1···γM are directly

incorporated in the integration.3 On the other hand, an mth order solution assumes that the

STTs are available and the relative motion is mapped analytically, and thus, the solution

accuracy depends on the accuracy of the STTs. If we consider the M = m = 1 case,

both approaches yield identical results. However, in general, numerically integrated and

analytically mapped solutions at the same order do not share the same level of accuracy.

Specifically, the analytic solution generally requires the STTs up to infinite order to capture

the full nonlinear effects captured by an M th order solution, which is discussed in Remark

2.2.11 in more detail. If we consider a sufficiently large m, however, the analytic solu-

tion should yield the same level of accuracy as the numerical solution, and ultimately give

an accurate approximation of the true nonlinear relative motion defined in Eqn. (2.11).

Considering this fact, in this thesis, we extend the analytic method to develop analytic

techniques for trajectory navigation problems.

Example 2.2.7. The summation convention applied to a second order expansion of the

solution is written as:

δxi(t) =2∑

p=1

1

p!φ

i,γ1···γp

(t,t0) δxγ1

0 · · · δxγp

0 ,

=N∑

γ1=1

φi,γ1

(t,t0)δxγ1

0 +N∑

γ1=1

N∑γ2=1

1

2φi,γ1γ2

(t,t0) δxγ1

0 δxγ2

0 , (2.24)

for γ1, γ2 ∈ 1, · · · , N.

Remark 2.2.8. The partial derivative of the mth order solution, Eqn. (2.22), can be stated

as:

∂(δxi)

∂(δxj0)

=N∑

p=1

1

(p− 1)!φi,γ1···γp−1jδxγ1

0 · · · δxγp−1

0 . (2.25)

3Note that this is not analytic in δx0 as the integration must be carried for different initial conditions.

20

Remark 2.2.9. The time derivative of the local trajectory solution, Eqn. (2.20), can be

obtained by directly differentiating Eqn. (2.22):

δxi(t) =m∑

p=1

1

p!φi,γ1···γpδxγ1

0 · · · δxγp

0 . (2.26)

To analyze the deviation δx as an analytic function of the initial deviations δx0, we

must solve for the STTs along the reference trajectory. To obtain differential equations for

the STTs, first substitute Eqn. (2.22) into Eqn. (2.20), which gives the equation of δxi as

a function of the STTs and the initial conditions. By equating this with Eqn. (2.26) and

balancing terms of the same order in δx0, the differential equations for the STTs (φi,γ1···γp)

can be obtained.4 Using the Einstein summation convention, the ODEs up to fourth order

can be stated as:

φi,a = gi,αφα,a, (2.27)

φi,ab = gi,αφα,ab + gi,αβφα,aφβ,b, (2.28)

φi,abc = gi,αφα,abc + gi,αβ(φα,aφβ,bc + φα,abφβ,c + φα,acφβ,b

)

+ gi,αβγφα,aφβ,bφγ,c, (2.29)

φi,abcd = gi,αφα,abcd + gi,αβ(φα,abcφβ,d + φα,abdφβ,c + φα,acdφβ,b + φα,abφβ,cd

+ φα,acφβ,bd + φα,adφβ,bc + φα,aφβ,bcd)

+ gi,αβγ(φα,abφβ,cφγ,d

+ φα,acφβ,bφγ,d + φα,adφβ,bφγ,c + φα,aφβ,bcφγ,d + φα,aφβ,bdφγ,c

+ φα,aφβ,bφγ,cd ) + gi,αβγδφα,aφβ,bφγ,cφδ,d, (2.30)

where all indices are over 1, · · · , N. At t = t0, the initial conditions of STTs are

φi,a(t0,t0) = 1 if i = a and all other initial STTs are zero.

4Another way of obtaining the differential equations of the STTs is through direct differentiations of Eqn.(2.23) according to the chain rule. An example of this approach is given in Example 2.2.10.

21

Example 2.2.10. The second order differential equations for the STTs can be obtained as

follows:

d

dtφi,ab

(tk,t0) =d

dt

∂2xik

∂xa0∂xb

0

=∂2gi

∂xa0∂xb

0

=∂

∂xb0

(∂gi

∂xa0

),

=∂

∂xb0

(gi,αφα,a

)= gi,α ∂(φα,a)

∂xb0

+∂(gi,α)

∂xb0

φα,a,

= gi,αφα,ab + gi,αβφα,aφβ,b.

For computation, the STTs are put into a vectorized form as first order differential

equations and are numerically integrated along the reference trajectory.5 Moreover, the

numerical integration cost can be significantly reduced by considering the symmetry of

the STTs, excluding the m = 1 case. For example, a system with a dimension N = 6

requires integration of 1554 number of equations for the 3rd order solution; however,

considering the symmetry of the STTs, the total number of integrated solutions reduce to

504. The numerical complexity of the higher order solution is further discussed in §2.2.1.

Remark 2.2.11. In general, an analytic mth order solution requires STTs up to infinite

order if we want to accurately approximate the numerically integrated M th order solu-

tion, where M = m. To illustrate this point, consider the second order analytic solution

discussed in Example 2.2.7:

δxi(t) = φi,a(t,t0)δxa

0 +1

2φi,ab

(t,t0)δxa0δxb

0,

which depends on the first and second order STTs, φi,a(t,t0) and φi,ab

(t,t0), respectively. These

STTs, however, do not capture all first and second order dynamics. For example, from

Eqns. (2.27-2.30), we observe that there are first order dynamics terms in the third and

fourth order STT differential equations, i.e., gi,αφα,abc and gi,αφα,abcd, that are not included

in the second order analytic solution; the same analogy applies to the second order dynam-

ics, e.g., gi,αβφα,aφβ,b from the third order STT differential equation. Hence, in general,5See Appendix D for more detail.

22

we need STTs up to infinite order to accurately approximate the numerical solution at a

given order. Note that this does not apply to the case m = 1 since the higher order STTs

(m ≥ 2) are initially zero, and hence, the contributions due to the first order dynamics are

zero, e.g., gi,αφα,abc = gi,αφα,abcd = 0.

Once the STTs are computed by integrating along a nominal trajectory, they serve

an identical role to the STM except that the higher order effects are now included, and

thus, the solution is nonlinear. The main significance of the STTs is that they allow the

local nonlinear motion of a spacecraft trajectory to be mapped analytically. Given a set

of STTs, the evolution of a state relative to the nominal trajectory is a simple algebraic

manipulation, and any neighboring trajectories within the radius of convergence can be

mapped analytically with respect to the reference solution. The solution to the original

dynamical system is then found by adding the deviations to the reference solution, or

φi(t; x0 + δx0, t0) = φi(t; x0, t0) + δxi(t).

Similar to the state transition tensors, we can also define a series mapping a deviation

backward in time.

Definition 2.2.12 (Inverse State Transition Tensor). Suppose we are given the analytic so-

lution flow of a nominal initial state x0 for some time interval [t0, tk], i.e., xk = φ(tk; x0, t0),

so that the inverse solution flow is defined as x0 = ψ(t, x; t0). The inverse series mapping

deviations in the state x at time t (t0 ≤ t ≤ tk) to deviations in the initial state x0 along the

reference trajectory are:

δxi0 =

m∑p=1

1

p!ψ

i,γ1···γp

(t0,t) δxγ1 · · · δxγp , (2.31)

where γj ∈ 1, · · · , N and

ψi,γ1···γp

(t0,t) (t, x; t0) =∂pψi(t, ξ; t0)

∂ξγ1 · · · ∂ξγp

∣∣∣∣ξj=xj

. (2.32)

We call these higher order partials the inverse state transition tensors (ISTTs).

23

The ISTTs serve a role identical to the inverse of the STM and can be computed by

using a similar integration approach as in the STT computation. For example, consider

the state transition matrix Φ = ∂φ/∂x0, or Φia = φi,a. As shown in Eqn. (2.27), the

differential equations for Φ satisfy:

Φ =∂g[t, x(t)]

∂xΦ, (2.33)

and the differential equations for Φ−1 can be stated as:

Φ−1 = −Φ−1∂g[t, x(t)]

∂x. (2.34)

Note that we assume the inverse Φ−1 always exists and is well defined. The first order

STTs are ψi,a = (Φ−1)ia, and hence, the differential equations for ψi,a yield:

ψi,a = −ψi,αgα,a. (2.35)

If the STTs have been found already, however, it is more convenient to compute the

ISTTs via series reversion [62]. The inverse series can be computed by substituting Eqn.

(2.22) into Eqn. (2.31) and collecting the terms of same order in δx0. Hence, the ISTTs,

ψi,γ1···γp

(t0,t) , are also analytic in the STTs, φi,γ1···γp

(t,t0) . After carrying out the series reversion, the

ISTTs mapping from t to t0 up to fourth order are:

ψi,a =[Φ−1(t, t0)

]ia, (2.36)

ψi,ab = −ψi,αφα,j1j2ψj1,aψj2,b, (2.37)

ψi,abc = − [ψi,αφα,j1j2j3 + ψi,αβ

(φα,j1φβ,j2j3 + φα,j1j2φβ,j3 + φα,j1j3φβ,j2

)]

× ψj1,aψj2,bψj3,c, (2.38)

24

ψi,abcd = − [ψi,αφα,j1j2j3j4 + ψi,αβ

(φα,j1j2j3φβ,j4 + φα,j1j2j4φβ,j3 + φα,j1j3j4φβ,j2

+ φα,j1j2φβ,j3j4 + φα,j1j3φβ,j2j4 + φα,j1j4φβ,j2j3 + φα,j1φβ,j2j3j4)

+ ψi,αβγ(φα,j1j2φβ,j3φγ,j4 + φα,j1j3φβ,j2φγ,j4 + φα,j1j4φβ,j2φγ,j3

+ φα,j1φβ,j2j3φγ,j4 + φα,j1φβ,j2j4φγ,j3 + φα,j1φβ,j2φγ,j3j4)]

× ψj1,aψj2,bψj3,cψj4,d, (2.39)

where all indices are 1, · · · , N, ψ = ψ(t0,t) and φ = φ(t,t0) are used for concise notations,

and Φ(t, t0) in Eqn. (2.36) represents the STM. Note that Eqns. (2.36-2.39) are analytic

in the STTs and require no integration. Hence, given a reference trajectory with STT

solutions for some time interval [t0, tf ], any arbitrary trajectories in the neighborhood of

the reference solution can be mapped nonlinearly forward from t0 and backward from tf

to any time t ∈ [t0, tf ].

Now, consider the first order case. Given Φ(t, t0) for the entire trajectory t ∈ [t0, tf ],

the linear map from any given time tr to ts, where tr, ts ∈ [t0, tf ], can be represented as:

Φ(ts, tr) = Φ(ts, t0)Φ−1(tr, t0). (2.40)

Analogously, applying the higher order forward and inverse state transition tensors, the

STTs φi,γ1···γp

(ts,tr) which nonlinearly map the deviations from arbitrary time tr to ts (tr, ts ∈

[t0, tf ] and tr ≤ ts), can be represented as:

φi,a(ts,tr) =

[Φ(ts, t0)Φ

−1(tr, t0)]ia

= φi,αs ψα,a

r , (2.41)

φi,ab(ts,tr) = φi,α

s ψα,abr + φi,αβ

s ψα,ar ψβ,b

r , (2.42)

φi,abc(ts,tr) = φi,α

s ψα,abcr + φi,αβ

s

(ψα,a

r ψβ,bcr + ψα,ab

r ψβ,cr + ψα,ac

r ψβ,br

)

+ φi,αβγs ψα,a

r ψβ,br ψγ,c

r , (2.43)

25

φi,abcd(ts,tr) = φi,α

s ψα,abcdr + φi,αβ

s

(ψα,abc

r ψβ,dr + ψα,abd

r ψβ,cr + ψα,acd

r ψβ,br + ψα,ab

r ψβ,cdr

+ ψα,acr ψβ,bd

r + ψα,adr ψβ,bc

r + ψα,ar ψβ,bcd

r

)+ φi,αβγ

s

(ψα,ab

r ψβ,cr ψγ,d

r

+ ψα,acr ψβ,b

r ψγ,dr + ψα,ad

r ψβ,br ψγ,c

r + ψα,ar ψβ,bc

r ψγ,dr + ψα,a

r ψβ,bdr ψγ,c

r

+ ψα,ar ψβ,b

r ψγ,cdr

)+ φi,αβγδ

s ψα,ar ψβ,b

r ψγ,cr ψδ,d

r , (2.44)

where all indices are 1, · · · , N and ψr = ψ(t0,tr) and φs = φ(ts,t0) for concise notation.

Note that the ISTTs mapping from ts to tr, ψi,γ1···γp

(tr,ts), can computed by applying Eqns.

(2.36-2.39).

We have shown that once the STTs are computed for the entire reference trajectory,

any map from an arbitrary point in the relative space to some future time, or vice versa,

becomes a simple algebraic manipulation. Note that φi,γ1···γp

(ts,tr) can also be computed by

integrating the differential equations given in Eqns. (2.27-2.30) for the time interval [tr, ts].

2.2.1 Complexity of the Higher Order Solutions

When computing the state transition tensors by integrating Eqns. (2.27-2.30), one con-

cern is numerical precision if the integration is over a long time period. This problem, how-

ever, can be remedied by segmenting the reference trajectory arbitrarily to meet the desired

numerical accuracy, since in most space missions, the reference trajectory is known with

high precision. For example, given a nominal solution flow φ(tf ; x0, t0) for some time in-

terval [t0, tf ]. Assuming this nominal trajectory is precisely known, we can choose points

x0, x(t1), x2(t2), · · · from φ(tf ; x0, t0), where t0 ≤ tj ≤ tf , and integrate the STTs for

time intervals [tj, tj+1]. In this way, numerical round-off errors in the higher order STTs

from an integration can be reduced significantly. When computing the ISTTs via series

reversion, it is apparent from Eqns. (2.36-2.39) that a small error in the forward STTs can

result in a significant error int he ISTTs. For example, a small error in Φ can result in an

inaccurate determination of Φ−1, which yields accumulated errors in the computation of

26

the higher order ISTTs ψi,γ1···γp

(t0,t) . In case of a Hamiltonian system, an integration method,

such as a symplectic or variational integrator, which preserves the Hamiltonian structure

can be implemented to compute Φ more precisely.6 Thus, numerical errors in the STTs

and ISTTs can be reduced.

A main concern in computing the STTs is the number of computations as we consider

higher order solutions, especially for a large m. Specifically, assuming a system with a di-

mension N = 6, the mth order analysis requires integration of α =∑m

q=0 6q+1 equations.

For example, when m = 3, a total of 1554 equations must be integrated simultaneously.

However, the higher order solutions can be computed off-line, and when an orbit is peri-

odic (e.g., an elliptical or halo orbit), these only need to be computed once. In general, if

we consider the symmetry of the higher order partials the number of integrated equations

for the mth order system reduce to:

α = N

m∑j=0

(N − 1 + j

j

). (2.45)

For example, when m = 3, we need to integrate a total of 504 equations.

An additional burden is the computation of the partials of the dynamics. We note that

there are symbolic manipulators available which provide automatic differentiations, and

also note that many of these partials vanish for spacecraft applications. For the problems

considered in this dissertation, MATLAB R© symbolic toolbox is used for automatic differ-

entiation of the dynamics, and once computed, the partials are stored as an m-file. Also,

consider the governing differential equations given in Eqn. (2.2). All the partial derivatives

of v vanish for m ≥ 2.

6For a Hamiltonian system, the symplectic or the variational integrator preserves that (detΦ) = 1 for alltimes.

27

2.3 Dynamics and Properties of a Hamiltonian System

For spacecraft applications, the general dynamics model Eqn. (2.3) is usually stated

in a Lagrangian frame, i.e., in a position and velocity coordinate frame. However, many

astrodynamics problems can be restated as a Hamiltonian system, which provides several

useful properties that simplify the problem [16, 18]. This section presents these important

properties of a Hamiltonian system and extends the linear Hamiltonian dynamical theory

to the higher order Hamiltonian dynamics.

Definition 2.3.1 (Lagrangian System). The holonomic form of Lagrange’s equations are

defined as [29, 30, 31]:

d

dt

(∂L

∂qi

)− ∂L

∂qi= 0, (2.46)

for i ∈ 1, · · ·n. The scalar function L = L(q, q, t) is called the Lagrangian function

and the qi are called the generalized coordinates. The Lagrangian function L is defined as

L = T + V , where T represents the kinetic energy and V represents the potential energy.

Example 2.3.2. Consider the normalized two-body problem (GM = 1), where the kinetic

energy is defined as T (r) = 12r · r and the potential energy is defined as V (r) = 1/‖r‖.

The Lagrangian function for the two-body problem yields:

L(r, r) =1

2r · r +

1

‖r‖ ,

and the Lagrangian equations of motion becomes:

r(t) = − 1

‖r‖3r,

which satisfies the general dynamical system Eqn. (2.1).

Definition 2.3.3 (Hamiltonian System). A system is called Hamiltonian if there exists

a smooth scalar function H(q, p, t) such that the governing equations of motion can be

28

stated as [4, 13, 54]:

qi =∂H

∂pi, (2.47)

pi = −∂H

∂qi. (2.48)

The scalar function H , which is time-varying in general, is called a Hamiltonian function,

and qi and pi = ∂L/∂qi are called the generalized coordinates and generalized momenta,

respectively.

Given a Lagrangian system Eqn. (2.46) a Hamiltonian function H(q, p, t) can be de-

fined by the Legendre transformation as:

H(q, p, t) =n∑

i=1

piqi − L(q, q, t). (2.49)

After transforming a Lagrangian system into Hamiltonian form, there exist many useful

properties that simplify the problem. The rest of this section presents these properties.7

Definition 2.3.4 (Symplecticity). A 2n× 2n matrix M is called symplectic if:

MT JM = J, (2.50)

where

J = J2n×2n =

0n×n In×n

−In×n 0n×n

. (2.51)

Here, J is called the symplectic unit matrix.

Property 2.3.5. Properties of the symplectic unit matrix:

JT J = −JJ = I, (2.52)

JαiJαj = −JiαJαj = δij, (2.53)

det(J) = 1. (2.54)7The details of the definitions, properties, and proofs presented in this section can be found in Refs.

[30, 81].

29

where δij represents the Kronecker delta function.

Property 2.3.6. If a 2n× 2n matrix M is symplectic, then:

M−1 = −JMT J. (2.55)

Hence the inverse of a symplectic matrix can be computed without a matrix inversion.

Property 2.3.7. The governing equations of motion of a Hamiltonian system can be stated

as:

x =

q

p

= JHT

x , (2.56)

where Hx represents the row-wise partial derivatives of H . In tensor notation:

xi = JiαHα, (2.57)

where α ∈ 1, . . . , 2n. Note that Hα is an αth component of a vector Hx, which is

composed of partial derivatives ∂H/∂xi.8

Property 2.3.8. Liouville’s theorem states that if a 2n × 2n matrix M is symplectic, then

det(M) = 1.

Property 2.3.9. If a 2n× 2n matrix M is symplectic, then M−1 and MT are symplectic as

well.

Property 2.3.10. If M =

An×n Bn×n

Cn×n Dn×n

then M is symplectic if and only if:

1. ATn×nCn×n and BT

n×nDn×n are symmetric and

2. DTn×nAn×n − BT

n×nCn×n = In×n.

8See the Notation section for details.

30

Definition 2.3.11 (Canonical Transformation). A transformation of phase space (q, p) 7→

(Q, P) that preserves the Hamiltonian structure of the dynamical system is called a canon-

ical transformation.

Property 2.3.12. Consider a Hamiltonian system H(q, p, t) and let Q = Q(q, p, t) and

P = P(q, p, t). Also let MQP =∂(Q, P)

∂(q, p)be symplectic, which yields det(MQP) = 1, and

thus, there exists a unique inverse. This leads to q = q(Q, P, t) and p = p(Q, P, t), where

Mqp =∂(q, p)

∂(Q, P)is symplectic. Then:

q

p

=

∂q∂Q

Q +∂q∂p

P

∂p∂Q

Q +∂p∂P

P

= Mqp

Q

P

. (2.58)

Applying M−1qp = −JMT

qpJ gives:

Q

P

= JMT

qp

∂H

∂q∂H

∂p

=

∂K

∂P−∂K

∂Q

, (2.59)

for a Hamiltonian function K(Q, P, t). Therefore, a transformation (q, p) 7→ (Q, P) is

canonical if and only if M =∂(Q, P)

∂(q, p)is a symplectic matrix.

Property 2.3.13. The Hamiltonian system with q(q0, p0, t) and p(q0, p0, t) is a canonical

transformation between (q0, p0) and (q, p) with t as an independent variable, as the state

transition matrix Φ(t, t0) =∂(q, p)

∂(q0, p0)is a symplectic matrix.

Property 2.3.14. Given a Hamiltonian system (q, p, H(q, p, t)), the linearized model be-

comes:

δq

δp

= Φ(t, t0)

δq0

δp0

. (2.60)

The state transition matrix is symplectic, and thus, the transformation is canonical, which

yields a Hamiltonian system. Therefore, the linearized dynamics of a Hamiltonian system

are Hamiltonian.

31

Property 2.3.15. Properties of the state transition matrix of a Hamiltonian system:

Φ−1(t, t0) = Φ(t0, t) = − JΦT J, (2.61)

det[Φ(t, t0)] = 1, (2.62)

d

dtΦ = JHxxΦ. (2.63)

Hence the inverse of the state transition matrix (the first order ISTTs discussed in §2.2)

can be computed without a matrix inversion. Also, the determinant of the STM (the first

order STTs) can be used to check the numerical accuracy of an integration of the higher

order STTs.

By applying properties of a Hamiltonian system, we can also compute the inverse of

a STM (first order ISTTs) of a Lagrangian system without a matrix inversion. Consider

a Lagrangian system, (q, q, L(q, q, t)), and suppose there exists a unique state transition

matrix such that:

δq

δq

= Ψ

δq0

δq0

. (2.64)

Also consider the linearized Hamiltonian system:

δq

δp

= Φ

δq0

δp0

. (2.65)

Recall the Legendre transformation H = pT q − L(q, q, t) and p = ∂L/∂q, and let T be

the Jacobian of the Legendre transformation, i.e.,

T =∂(q, p)

∂(q, q)=

I 0∂2L

∂q∂q∂2L

∂q∂q

, (2.66)

δq(t)

δp(t)

= T(t)

δq(t)

δq(t)

. (2.67)

32

By substituting Eqn. (2.67) into Eqn. (2.65), we get:

Ψ(t, t0) = T−1(t)Φ(t, t0)T(t0), (2.68)

Φ(t, t0) = T(t)Ψ(t, t0)T−1(t0), (2.69)

which gives:

Ψ−1(t, t0) = −T−1(t0)JT−T (t0)ΨT (t, t0)TT (t)JT(t). (2.70)

If∂2L

∂q∂qis symmetric and

∂2L

∂q∂q= I, according Property 2.3.10, T is symplectic. Apply-

ing the symplectic condition for T, Eqn. (2.70) simplifies to:

Ψ−1(t, t0) = −JΨT (t, t0)J, (2.71)

which indicates that Ψ is symplectic as well.

Therefore, we can take advantage of a Hamiltonian system by computing the inverse

of a STM (Ψ−1) without a matrix inversion. This property is especially useful for the

computation of the ISTTs of a Lagrangian system, Eqn. (2.36-2.39), since the errors from

a matrix inversion can be avoided.

2.4 Symplecticity of the Higher Order Solutions of a HamiltonianSystem

In §2.3, we have shown that the solution flows of a Hamiltonian and a linearized Hamil-

tonian system are canonical transformations (Properties 2.3.13 and 2.3.14); thus the trans-

formations are symplectic. In this section, we extend the symplecticity of a linearized

Hamiltonian system and show that the higher order relative solutions are also canonical

transformations.

Consider a Hamiltonian system xi = gi = JiαHα with an initial state x0, where x =

[qT , pT ]T and H = H(x, t). Given the initial deviation δx0, the M th order differential

33

equations for the relative dynamics of a Hamiltonian system can be stated as:

δxi(t) =M∑

p=1

1

p!JiαHαγ1···γpδxγ1 · · · δxγp , (2.72)

where we apply the M th order Taylor series expansion given in Eqn. (2.20) and the higher

order partial derivatives of Hamiltonian dynamics:

gi,γ1···γp(t, x) = JiαHαγ1···γp = Jiα ∂pH[t, ξ(t)]

∂ξα∂ξγ1 · · · ∂ξγp

∣∣∣∣ξj=φj(t;x0,t0)

. (2.73)

To show that an integrated relative solution δx(t, δx) computed according to Eqn.

(2.72) is symplectic, we can show that there exists a Hamiltonian function K(δx, t) such

that the relative equations of motion can be written as:

δxi = JiαKα, (2.74)

Kα =M∑

p=1

1

p!Hαγ1···γpδxγ1 · · · δxγp . (2.75)

For example, when M = 1 the new Hamiltonian can be defined as K(δx, t) = 12Habδxaδxb,

so that the first order relative dynamics can be written as δxi(t) = JiαHαγ1δxγ1 . For higher

order equations, the new Hamiltonian function can be stated explicitly as [33]:

K(δq, δp, t) =M∑

p=2

p∑i1,··· ,i2n=0

i1+···+i2n=p

1

i1! · · · i2n!

∂pH(q0, p0, t)

(∂q1)i1 · · · (∂qn)in(∂p1)in+1 · · · (∂pn)i2n(δx1)i1 · · · (δx2n)i2n .

(2.76)

Therefore, the relative solution δx(t, δx0) can be written in a Hamiltonian form, and thus,

the solution flow is symplectic. Note that δx(t, δx0) is defined with respect to a nominal

solution of the original Hamiltonian system (x(x0), H(x, t)). Hence the relative dynamics

δx(t) must be integrated together with x(t) so that the higher order partials of H in Eqn.

(2.76) are computed along the nominal solution.

34

We have shown that the solution flow computed according to the differential equations

Eqn. (2.72) is symplectic. This, however, does not necessarily mean that an analytic

relative solution flow with respect to the original Hamiltonian flow is symplectic. To

clarify this point, consider the analytic relative flow Eqn. (2.22):

δxi(t) =m∑

p=1

1

p!φ

i,γ1···γp

(t,t0) δxγ1

0 · · · δxγp

0 , (2.77)

where the STTs φi,γ1···γp

(t,t0) are computed along the nominal solution flow φ(t; x0, t0) of the

original Hamiltonian system (x, H(x, t)) and δx0 is an initial deviation in x0. As discussed

in §2.2, the computation of the relative solution flow Eqn. (2.77) is fundamentally dif-

ferent than the relative solution integrated according to Eqn. (2.72). In limit (m → ∞),

the analytic map δx0 7→ δx(t), Eqn. (2.77), is a canonical transformation since δx(t) is

essentially the solution flow of a Hamiltonian system (δx(δx0), K(δx, t)) with m →∞ in

Eqn. (2.76). Also, when m = 1, the first order STTs φi,γ1

(t,t0) are symplectic, and thus, the

relative solution flow is canonical. However, for an arbitrary m, the map δx0 7→ δx(t) is

not necessarily guaranteed to be a canonical transformation.

To show this, consider the second order relative solution (m = 2):

δxi(t) = φi,γ1

(t,t0)δxγ1

0 +1

2φi,γ1γ2

(t,t0) δxγ1

0 δxγ2

0 , (2.78)

and suppose that the map δx0 7→ δx(t) is symplectic. Then, the Jacobian of Eqn. (2.78)

must satisfy the symplectic condition Eqn. (2.50):

Jij =

[∂(δxα)

∂(δxi0)

]Jαβ

[∂(δxβ)

∂(δxj0)

]= Jαβϕα,iϕβ,j, (2.79)

where ϕi,j represents the Jacobian. For m = 2 case, ϕi,j is defined as:

ϕi,j = φi,j + φi,γ1jδxγ1

0 . (2.80)

35

Assuming the symplectic condition is valid, the total time derivative of Eqn. (2.79) must

vanish, so that:

0ij =d

dt

(∂(δxα)

∂(δxi0)

)Jαβ

(∂(δxβ)

∂(δxj0)

)+

(∂(δxα)

∂(δxi0)

)Jαβ d

dt

(∂(δxβ)

∂(δxj0)

),

= ϕα,iJαβϕβ,j + ϕα,iJαβϕβ,j. (2.81)

Considering a Hamiltonian system, i.e., xi = gi = JiαHα, the time derivatives of the first

and second order STTs are (Eqns. (2.27) and (2.28)):

φi,a = JiλHλαφα,a, (2.82)

φi,ab = JiλHλαφα,ab + JiλHλαβφα,aφβ,b, (2.83)

where the partials of Hamiltonian dynamics gi,γ1···γp = JiαHαγ1···γp are substituted:

gi,α = JiλHλα

∣∣φj(t;x0,t0)

,

gi,αβ = JiλHλαβ

∣∣φj(t;x0,t0)

.

Now substitute Eqn. (2.80) into Eqn. (2.81) and collect the terms of the same order in δx0.

The symplectic condition yields:

0ij = (φα,i + φα,iνδxν0)J

αβ(φβ,j + φβ,jµδxµ0) + (φα,i + φα,iνδxν

0)Jαβ(φβ,j + φβ,jµδxµ

0)

=

zeroth order terms︷ ︸︸ ︷φα,iJαβφβ,j + φα,iJαβφβ,j

+

first order terms︷ ︸︸ ︷φα,iνJαβφβ,jδxν

0 + φα,iJαβφβ,jµδxµ0 + φβ,jφα,iνJαβδxν

0 + φβ,jµφα,iJαβδxµ0

+

second order terms︷ ︸︸ ︷φα,iνJαβφβ,jµδxν

0δxµ0 + φβ,jµφα,iνJαβδxν

0δxµ0 , (2.84)

where superscripts α, β, ν, µ ∈ 1, · · · , 2n are dummy variables. We shall analyze

each order of terms in Eqn. (2.84) to verify the symplectic condition.

36

Zeroth order terms:

The zeroth order terms in Eqn. (2.84) are from the first order relative dynamics and vanish

to zero:

φα,iJαβφβ,j + φα,iJαβφβ,j = JαλHλkφk,iJαβφβ,j + φα,iJαβJβλHλkφ

k,j,

= δλβHλkφk,iφβ,j − δαλHλkφ

α,iφk,j,

= Hβkφk,iφβ,j −Hαkφ

α,iφk,j,

≡ 0ij,

where the first step substitutes Eqn. (2.82) for time derivatives of the STTs, second and

third steps use the symplectic property JαiJαj = −Ji,αJαj = δij , and the last step uses the

symmetry of Hαβ and the fact that the dummy variables α, β, k can be swapped.

First order terms:

Substituting Eqns. (2.82) and (2.83) for the time derivatives of the STTs and re-ordering

give:

φα,iνJαβφβ,jδxν0 + φα,iJαβφβ,jµδxµ

0 + φβ,jφα,iνJαβδxν0 + φβ,jµφα,iJαβδxµ

0

= JβλHλγ1φγ1,jφα,iνJαβδxν

0 +(JαλHλγ1φ

γ1,iν + JαλHλγ1γ2φγ1,iφγ2,ν

)Jαβφβ,jδxν

0

+ JαλHλγ1φγ1,iJαβφβ,jµδxµ

0 +(JβλHλγ1φ

γ1,jµ + JβλHλγ1γ2φγ1,jφγ2,µ

)φα,iJαβδxµ

0 ,

where the superscripts α, β, γ1, γ2, λ, ν, µ are dummy variables. Applying the symplec-

tic properties JαiJαj = −JiαJαj = δij simplifies to:

− δαλHλγ1φγ1,jφα,iνδxν

0 + δλβHλγ1φγ1,iνφβ,jδxν

0 + δλβHλγ1γ2φγ1,iφγ2,νφβ,jδxν

0

+ δλβHλγ1φγ1,iφβ,jµδxµ

0 − δαλHλγ1φγ1,jµφα,iδxµ

0 − δαλHλγ1γ2φγ1,jφγ2,µφα,iδxµ

0 ,

=(δλαHλγ1φ

α,jφγ1,iν + δλαHλγ1γ2φγ1,iφγ2,νφα,j + δλαHλγ1φ

γ1,iφα,jν

− δαλHλγ1φγ1,jφα,iν − δαλHλγ1φ

α,iφγ1,jν − δαλHλγ1γ2φγ1,jφγ2,νφα,i

)δxν

0,

37

where the dummy variables β and µ are swapped with α and ν, respectively. Apply δii = 1

to get:

(Hαγ1φ

α,jφγ1,iν + Hαγ1φγ1,iφα,jν + Hαγ1γ2φ

γ1,iφγ2,νφα,j

− Hαγ1φγ1,jφα,iν −Hαγ1φ

α,iφγ1,jν −Hαγ1γ2φγ1,jφγ2,νφα,i

)δxν

0 ≡ 0ij,

since the second-rank tensor Hαγ1 and the third-rank tensor Hαγ1γ2 are symmetric. There-

fore, the second order terms in Eqn. (2.83) vanish to zero.

Second order terms:

Substitute Eqn. (2.83) for the time derivatives of the STTs in the second order terms of

Eqn. (2.84) and multiply out the terms to get:

φα,iνJαβφβ,jµδxν0δxµ

0 + φβ,jµφα,iνJαβδxν0δxµ

0

=(JαλHλγ1φ

γ1,iν + JαλHλγ1γ2φγ1,iφγ2,ν

)Jαβφβ,jµδxν

0δxµ0

+(JβλHλγ1φ

γ1,jµ + JβλHλγ1γ2φγ1,jφγ2,µ

)φα,iνJαβδxν

0δxµ0 ,

= JαλJαβHλγ1φγ1,iνφβ,jµδxν

0δxµ0 + JαλJαβHλγ1γ2φ

γ1,iφγ2,νφβ,jµδxν0δxµ

0

+ JβλJαβHλγ1φγ1,jµφα,iνδxν

0δxµ0 + JβλJαβHλγ1γ2φ

γ1,jφγ2,µφα,iνδxν0δxµ

0 ,

where the superscripts α, β, γ1, γ2, λ, ν, µ are dummy variables. Applying the symplec-

tic property JαiJαj = −JiαJαj = δij gives:

δλβHλγ1φγ1,iνφβ,jµδxν

0δxµ0 + δλβHλγ1γ2φ

γ1,iφγ2,νφβ,jµδxν0δxµ

0

−δαλHλγ1φγ1,jµφα,iνδxν

0δxµ0 − δαλHλγ1γ2φ

γ1,jφγ2,µφα,iνδxν0δxµ

0

=(Hαγ1φ

γ1,iνφα,jµ + Hαγ1γ2φγ1,iφγ2,νφα,jµ

− Hγ1αφα,jµφγ1,iν −Hαγ1γ2φγ1,jφγ2,µφα,iν

)δxν

0δxµ0 , (2.85)

where we apply δii = 1 and swap the dummy variable β with α. Considering the symmetry

of Hαγ1 , Eqn. (2.85) simplify to:

(Hαγ1γ2φ

γ1,iφγ2,νφα,jµ −Hαγ1γ2φγ1,jφγ2,νφα,iµ

)δxν

0δxµ0 ,

38

which does not vanish since when i 6= j, despite the symmetry of Hαγ1γ2:

(Hαγ1γ2φ

γ1,iφγ2,νφα,jµ −Hαγ1γ2φγ1,jφγ2,νφα,iµ

) 6= 0ij.

We have shown that the second order analytic relative flow is not a canonical trans-

formation. This result is generally true for any higher order solutions m ≥ 2 that are

computed analytically with respect to a Hamiltonian solution flow. However, we have also

shown that the second order solution is symplectic up to first order, O(δx). Based on this

observation, we introduce the following conjecture.

Conjecture 2.4.1. An mth order analytic solution of the relative trajectory motion of a

Hamiltonian system is symplectic up to order (m− 1).

To show this is true for a general mth order case, we can apply the similar approach

used for the m = 2 case. Recall the sufficiency condition:

ϕα,iJαβϕβ,j + ϕα,iJαβϕβ,j = 0ij,

where

ϕi,j =m∑

p=1

1

(p− 1)!φi,jγ1···γp−1δxγ1

0 · · · δxγp−1

0 . (2.86)

Substituting Eqn. (2.86) into the symplectic sufficiency condition yields:[

m∑p=1

1

(p− 1)!φα,iγ1···γp−1δxγ1

0 · · · δxγp−1

0

]Jαβ

[m∑

q=1

1

(q − 1)!φβ,jζ1···ζq−1δxζ1

0 · · · δxζq−1

0

]

+

[m∑

p=1

1

(p− 1)!φα,iγ1···γp−1δxγ1

0 · · · δxγp−1

0

]Jαβ

[m∑

q=1

1

(q − 1)!φβ,jζ1···ζq−1δxζ1

0 · · · δxζq−1

0

].

Now multiply out the terms and re-order to find:

m∑p=1

m∑q=1

Jαβ

(p− 1)! (q − 1)!φα,iγ1···γp−1φβ,jζ1···ζq−1δxγ1

0 · · · δxγp−1

0 δxζ10 · · · δxζq−1

0

+m∑

p=1

m∑q=1

Jαβ

(p− 1)! (q − 1)!φα,iγ1···γp−1φβ,jζ1···ζq−1δxγ1

0 · · · δxγp−1

0 δxζ10 · · · δxζq−1

0 . (2.87)

39

To show that an mth order solution is symplectic up to order (m−1), it suffices to show that

O(δxm−10 ) terms in Eqn. (2.87) vanish. This, however, is intuitively true since including

O(δxm0 ) terms in Eqn. (2.87) contribute no additional O(δxm−1

0 ) terms. In other words,

if we consider an (m + 1)th, or higher, order solution, the contributions due to O(δxm−10 )

are the same as from an mth order solution. Moreover, in limit (m → ∞), we know that

the symplectic condition must be satisfied, which indicates that the contributions due to

O(δxm−10 ) terms must vanish at some order of solution. Therefore, the contributions due

to O(δxm−10 ) terms must vanish at least for an mth order solution, and thus the Conjecture

2.4.1 is true. From a different perspective, for an mth order solution to be symplectic at

order m, it requires (m + 1)th order effects. This is as expected, as the symplecticity

condition is nonlinear. Hence satisfying the condition at one order requires contributions

from the next highest order, at least.

We can also show that Conjecture 2.4.1 is true more explicitly. From Eqn. (2.87) we

find that there are 2m terms that are O(xm−1):

Jαβ

m terms︷ ︸︸ ︷[φα,iγ1···γm−1φβ,j

(m− 1)!0!+

φα,iγ2···γm−1φβ,jγ1

(m− 2)!1!+ · · ·+ φα,iφβ,jγ1···γm−1

0!(m− 1)!

]δxγ1

0 · · · δxγm−1

0

+ Jαβ

[φα,iγ1···γm−1 φβ,j

(m− 1)!0!+

φα,iγ2···γm−1 φβ,jγ1

(m− 2)!1!+ · · ·+ φα,iφβ,jγ1···γm−1

0!(m− 1)!

]

︸ ︷︷ ︸m terms

δxγ1

0 · · · δxγm−1

0

= Jαβ

m∑p=1

1

(m− p)!(p− 1)!φα,iγ1···γp−1

if p=m, then φβ,j

︷ ︸︸ ︷φβ,jγp···γm−1 δxγ1

0 · · · δxγm−1

0

+ Jαβ

m∑p=1

1

(m− p)!(p− 1)!φα,iγp···γm−1

︸ ︷︷ ︸if p=m, then φα,i

φβ,jγ1···γp−1δxγ1

0 · · · δxγm−1

0 . (2.88)

40

Recall the time derivative of the STTs from §2.2:

φα,iγ1···γp−1 = gα,ζ1$1 + gα,ζ1ζ2$2 + · · ·+ gα,ζ1···ζp$p,

= JαλHλζ1$1 + JαλHλζ1ζ2$2 + · · ·+ JαλHλζ1···ζp$p,

=

p∑q=1

JαλHλζ1···ζq$q, (2.89)

where $j are some functions of the STTs up to order p, i.e., φi,γ1···γp . For example:

$1 = $1(i, γ1, · · · , γp−1) = φζ1,iγ1···γp−1 ,

$p = $p(i, γ1, · · · , γp−1) = φζ1,iφζ2,γ1 · · ·φζp,γp−1 .

Note that $j is symmetric in the superscripts of φα,iγ1···γp−1 excluding α. Hence, we define

$j = $j(i, γ1, · · · , γp−1) to denote that it’s symmetric in (i, γ1, · · · , γp−1). Using Eqn.

(2.91) we have:

φα,iγ1···γp−1 =

p∑q=1

JαλHλζ1···ζq$q(i, γ1, · · · , γp−1), (2.90)

φβ,jγ1···γp−1 =

p∑q=1

JβλHλζ1···ζq$q(j, γ1, · · · , γp−1). (2.91)

Now substitute Eqns. (2.90) and (2.91) into Eqn. (2.88) to find:

Jαβ

m∑p=1

φβ,jγp···γm−1

(m− p)!(p− 1)!

[p∑

q=1

JαλHλζ1···ζq$q(i, γ1, · · · , γp−1)

]δxγ1

0 · · · δxγm−1

0

+ Jαβ

m∑p=1

φα,iγp···γm−1

(m− p)!(p− 1)!

[p∑

q=1

JβλHλζ1···ζq$q(j, γ1, · · · , γp−1)

]δxγ1

0 · · · δxγm−1

0 .

Applying the symplectic identity property, JαiJαj = −JiαJαj = δij , gives:

m∑p=1

1

(m− p)!(p− 1)!φα,jγp···γm−1

[p∑

q=1

Hαζ1···ζq$q(i, γ1, · · · , γp−1)

]δxγ1

0 · · · δxγm−1

0

−m∑

p=1

1

(m− p)!(p− 1)!φα,iγp···γm−1

[p∑

q=1

Hαζ1···ζq$q(j, γ1, · · · , γp−1)

]δxγ1

0 · · · δxγm−1

0 .

(2.92)

41

Without further simplification, we hypothesize that Eqn. (2.92) must vanish and consider

the m = 3 case as an example to show this is satisfied.

If we let m = 3, Eqn. (2.92) can be stated as:

1

2!0!φα,jγ1γ2 [Hαζ1$1(i)] δxγ1

0 δxγ2

0

+1

1!1!φα,jγ2 [Hαζ1$1(i, γ1) + Hαζ1ζ2$2(i, γ1)] δxγ1

0 δxγ2

0

+1

0!2!φα,j [Hαζ1$1(i, γ1, γ2) + Hαζ1ζ2$2(i, γ1, γ2) + Hαζ1ζ2ζ3$3(i, γ1, γ2)] δxγ1

0 δxγ2

0

− 1

2!0!φα,iγ1γ2 [Hαζ1$1(j)] δxγ1

0 δxγ2

0

− 1

1!1!φα,iγ2 [Hαζ1$1(j, γ1) + Hαζ1ζ2$2(j, γ1)] δxγ1

0 δxγ2

0

− 1

2!0!φα,i [Hαζ1$1(j, γ1, γ2) + Hαζ1ζ2$2(j, γ1, γ2) + Hαζ1ζ2ζ3$3(j, γ1, γ2)] δxγ1

0 δxγ2

0 .

(2.93)

Substituting expressions for $j gives:

1

2»»»»»»»»»: aφα,jγ1γ2Hαζ1φ

ζ1,iδxγ1

0 δxγ2

0

+ φα,jγ2(Hαζ1φ

ζ1,iγ1 + Hαζ1ζ2φζ1,iφζ2,γ1

)δxγ1

0 δxγ2

0

+1

2φα,j

[»»»»»»»: bHαζ1φ

ζ1,iγ1γ2 + Hαζ1ζ2

(φζ1,iφζ2,γ1γ2 + φζ1,γ1φζ2,iγ2 + φζ1,γ2φζ2,iγ1

)]

δxγ1

0 δxγ2

0

+1

2»»»»»»»»»»»»»»»»: c

φα,j(Hαζ1ζ2ζ3φ

ζ1,iφζ2,γ1φζ3,γ2)δxγ1

0 δxγ2

0

− 1

2»»»»»»»»»: bφα,iγ1γ2Hαζ1φ

ζ1,jδxγ1

0 δxγ2

0

− φα,iγ2(Hαζ1φ

ζ1,jγ1 + Hαζ1ζ2φζ1,jφζ2,γ1

)δxγ1

0 δxγ2

0

− 1

2φα,i

[»»»»»»»: aHαζ1φ

ζ1,jγ1γ2 + Hαζ1ζ2

(φζ1,jφζ2,γ1γ2 + φζ1,γ1φζ2,jγ2 + φζ1,γ2φζ2,jγ1

)]δxγ1

0 δxγ2

0

− 1

2»»»»»»»»»»»»»»»»: c

φα,i(Hαζ1ζ2ζ3φ

ζ1,jφζ2,γ1φζ3,γ2)δxγ1

0 δxγ2

0 .

42

Now simplify to find:

·︷ ︸︸ ︷φα,jγ2

(XXXXXXHαζ1φζ1,iγ1 + Hαζ1ζ2φ

ζ1,iφζ2,γ1)δxγ1

0 δxγ2

0

+1

2φα,jHαζ1ζ2

(»»»»»»φζ1,iφζ2,γ1γ2 + φζ1,γ1φζ2,iγ2 + φζ1,γ2φζ2,iγ1

)δxγ1

0 δxγ2

0

− 1

2φα,iHαζ1ζ2

(»»»»»»φζ1,jφζ2,γ1γ2 + φζ1,γ1φζ2,jγ2 + φζ1,γ2φζ2,jγ1

)δxγ1

0 δxγ2

0

−φα,iγ1(XXXXXXHαζ1φ

ζ1,jγ2 + Hαζ1ζ2φζ1,jφζ2,γ2

)δxγ1

0 δxγ2

0︸ ︷︷ ︸·

.

Note that · terms were the non-vanishing terms from the m = 2 case. Simplifying once

again results in:

φα,jγ2(Hαζ1ζ2φ

ζ1,iφζ2,γ1)δxγ1

0 δxγ2

0 +1

2φα,jHαζ1ζ2

(φζ1,γ1φζ2,iγ2 + φζ1,γ2φζ2,iγ1

)δxγ1

0 δxγ2

0

− φα,iγ1(Hαζ1ζ2φ

ζ1,jφζ2,γ2)δxγ1

0 δxγ2

0 − 1

2φα,iHαζ1ζ2

(φζ1,γ1φζ2,jγ2 + φζ1,γ2φζ2,jγ1

)δxγ1

0 δxγ2

0

= Hαζ1ζ2

(φζ1,iφζ2,γ1φα,jγ2 − φα,iγ1φζ1,jφζ2,γ2

)δxγ1

0 δxγ2

0

+ Hαζ1ζ2

(φζ1,γ1φζ2,iγ2φα,j

)δxγ1

0 δxγ2

0 −Hαζ1ζ2

(φα,iφζ1,γ1φζ2,jγ2

)δxγ1

0 δxγ2

0 ,

= 0ij.

Hence, the m = 3 case is symplectic up to order 2, which satisfies our conjecture.

Although an mth order solution is only symplectic up toO(δxm−1), we can still achieve

an approximate symplectic condition. That is, given a sufficiently high order analytic map

δx(t) 7→ δx0 computed relative to a Hamiltonian system (x(x0), H(x, t)), an accurate

approximation of the true nonlinear relative motion can be made. Therefore, we can say

that the analytic flow of a local relative solution is “almost” symplectic. This indicates

that, when an initial deviation is within the radius of convergence and m is large enough,

symplectic properties presented in §2.3 can be applied.

43

2.5 Convergence of the Higher Order Solutions

Given the Taylor series expansion of a solution Eqn. (2.22), it is non-trivial to deter-

mine what order of solution suffices to represent the local nonlinear motion for a given

set of initial deviations. If the initial conditions lie outside the region of convergence,

the higher order solutions diverge. In this dissertation, we assume the solution is within

the radius of convergence. Moreover, the convergence rate of the higher order solutions

vary along the trajectory, which directly depends on the fact that the system’s nonlinearity

varies along the trajectory. This is an interesting fact since it indicates that different or-

ders of solution can provide different levels of approximation of the relative motion of the

neighboring trajectory. In this section, we discuss a systematic way to find the necessary

order of Taylor series that captures the local nonlinear dynamics.

Consider an initial phase volume B0 defined locally with respect to a nominal initial

state ξ0. We can propagate B0 by nonlinearly mapping all points x0 ∈ B0, but since

B0 is defined locally, we can also propagate each point relative to the nominal trajectory

φ(t; ξ0, t0). Suppose we want to propagate B0 linearly to some future time. One important

issue in linear propagation is the significance of a coordinate system. For example, in the

two-body problem, the orbit elements are constant (except for the mean anomaly) whereas

the Cartesian coordinates are not. Hence, depending on the choice of a coordinate system,

different linear propagations can give different levels of approximation of the nonlinear

motion. This problem has been studied by Junkins et al. [47, 48], where the level of

nonlinearity (or linearity) of a trajectory is quantified using the nonlinearity index.

Definition 2.5.1 (Nonlinearity Index). Given an initial phase volume B0 with respect to

a nominal initial state x0, the nonlinearity index measures the level of nonlinearity (or

44

linearity) of the propagated phase volume. The nonlinearity index is defined as:

ν(t, t0) , supκ=1,··· ,Γ

‖Φκ(t, t0)−Φ(t, t0)‖f

‖Φ(t, t0)‖f, (2.94)

where Γ is the number of initial sample points, ‖ · ‖f represents the Frobenius norm,

Φ(t, t0) is the state transition matrix computed along the reference trajectory φ(t; x0, t0),

and Φκ(t, t0) represents the state transition matrix computed along the κth sample trajec-

tory φ(t; xκ0 , t0), where xκ

0 represents the κth sample point chosen from the boundary of

B0.9

The nonlinearity index computes the level of maximum linear deviation from the refer-

ence trajectory. However, our focus is more on deciding the sufficient order of the higher

order solution. For this reason, we apply a slightly different approach where we propagate

a relative trajectory instead of a state transition matrix.

Definition 2.5.2 (Local Nonlinearity Index). Given an initial phase volume B0 with respect

to a nominal initial state x0, define the local nonlinearity index as:

ηm(t, t0) , supi=1,··· ,Nκ=1,··· ,Γ

|(δxi)m(t; δxκ0 , t0)− δxi(t; δxκ

0 , t0)||δxi(t; δxκ

0 , t0)|, (2.95)

where | · | represents an absolute value, (δxi)m represents the ith component of the mth or-

der analytic solution Eqn. (2.22) computed with respect to the nominal trajectory φ(t; x0, t0),

δxi represents the ith component of the nonlinearly integrated solution vector according

to the governing equations of motion Eqn. (2.12), and δxκ0 represents the κth sample state

vector chosen from the boundary of the initial phase volume. Note that the superscripts

i ∈ 1, · · · , N and κ ∈ 1, · · · ,Γ, where N is the system dimension and Γ is the

number of initial sample points that characterize B0.10

9For space mission analysis, the initial phase volume B0 can be considered as an error ellipsoid, so thatthe sample points can be chosen from the worst-case initial conditions (e.g., boundary points of the 3-σellipsoid).

10If we have a single deviated state δx0 with respect to x0, the local nonlinearity index can be computedby considering δx0 as the only sample point. In other words, δx0 can be considered as B0.

45

In other words, we find the state that deviates the most from its reference value over all

initial samples and each component of the state vector. Hence, this is in a sense a Monte-

Carlo simulation which compares the higher order solution and the integrated solution,

which is a one time operation for the given reference trajectory and initial set B0. The

computed value of ηm then tells how well the mth order solution can approximate the true

nonlinear motion. Using Eqn. (2.22), ηm can be restated as:

ηm(t, t0) = supi=1,··· ,Nk=1,··· ,Γ

∣∣∣(∑m

p=11p!φi,γ1···γpδxγ1

0 · · · δxγp

0

)− δxi(t; δxκ

0 , t0)∣∣∣

|δxi(t; δxκ0 , t0)|

. (2.96)

As we consider higher order Taylor series, and if the series converge, ηm will converge

to zero, and by increasing the order of solution we can compute the percent difference

between the true and STT solutions. Assuming the initial sample points are within the

radius of convergence, ηm → 0 as m → ∞. As a result, a higher order solution needs to

be considered if ηm > ε, where ε stipulates the precision of approximation desired.

CHAPTER III

EVOLUTION OF PROBABILITY DENSITYFUNCTIONS IN NONLINEAR DYNAMICAL

SYSTEMS

The Gaussian distribution is widely used for astrodynamics applications due to its sim-

plicity and its invariance under linear operations. When we consider mapping a Gaussian

random vector under nonlinear orbital dynamics, however, the Gaussian structure is no

longer preserved, which is an important issue, but usually not considered in conventional

trajectory navigation. In this chapter, we start from the basics of probability theory and

discuss the integral invariance of the probability function of a Hamiltonian dynamical sys-

tem via solutions of the Fokker-Planck equation. We then apply the higher order solutions

from Chapter II and present an exact analytic representation of a non-Gaussian proba-

bility density, and an approximation of this distribution using the first few moments of

the original Gaussian distribution. Implicit in our discussion is that a well-defined initial

probability density function for our dynamical state exists. This is a standard and usual

assumption for spacecraft navigation and is uncontroversial.

3.1 Review of Probability Theory and Random Processes

In this section, we give a few definitions from probability theory that are used through-

out this thesis. The formal definitions and discussion can be found in Refs. [32, 51, 82, 88].

46

47

Definition 3.1.1 (Probability Density Function). Given a continuous random vector x ∈

RN , the probability of x in some volume B can be computed by:

Pr(x ∈ B) =

∫

Bp(ξ)dξ =

∫

Bp(ξ)dξ1dξ2 · · · dξN , (3.1)

where a function p(x) is called a probability density function (PDF).1 Note that the integral

is over dξ = dξ1dξ2 · · · dξN .

Remark 3.1.2. Since Pr(x ∈ B) ≥ 0 for all B, the PDF p(x) must be nonnegative for all x.

Remark 3.1.3. Given a PDF of a random vector x ∈ RN , the PDF must satisfy:∫

∞p(ξ)dξ = 1. (3.2)

Remark 3.1.4. Given a constant ξ ∈ RN , Pr(x = ξ) = 0 for all x.

Definition 3.1.5 (Marginal Density Function). Given two random variables x ∈ R and

y ∈ R, the marginal density function of x is defined as:

p(x) =

∫

∞p(x, y)dy. (3.3)

Definition 3.1.6 (Law of Total Expectation). For a continuous random vector x ∈ RN

with density p(x), the expected value of an arbitrary function g(x) can be computed by:

E[g(x)] =

∫

∞g(ξ)p(ξ)dξ, (3.4)

where E[·] represents the expectation operator.

Definition 3.1.7. Given a random vector x ∈ RN with a PDF p(x), the mean and covari-

ance matrix are defined as:

m = E[x] =

∫

∞ξp(ξ)dξ, (3.5)

P = E[(x−m)(x−m)T ] =

∫

∞(ξ−m)(ξ−m)T p(ξ)dξ, (3.6)

1Formally, the PDF should be written as pX(x, t), indicating that pX is a probability density function of arandom process X, where x ∈ X. In this thesis, the subscript is dropped for concise notation and p(x, t) isused to represent pX(x, t).

48

or in tensor notation:

mi = E[xi] =

∫

∞ξip(ξ)dξ, (3.7)

Pij = E[(xi − mi)(xj − mj)] = E[xixj]− mimj,

=

∫

∞ξiξjp(ξ)dξ− mimj. (3.8)

Remark 3.1.8. In general, an mth order moment is defined as:

E[xγ1 · · · xγm ], (3.9)

and an mth order central moment is defined as:

E[(xγ1 − mγ1) · · · (xγm − mγm)]. (3.10)

Definition 3.1.9 (Characteristic Function). The joint characteristic function (JCF) of a

continuous random vector x ∈ RN is defined as:

χ(u) = E[ejuT x], (3.11)

where j =√−1. The higher moments can be computed by:

E[xγ1 ] = j−1 ∂χ(u)

∂uγ1

∣∣∣∣u=0

, (3.12)

E[xγ1xγ2 ] = j−2 ∂2χ(u)

∂uγ1∂uγ2

∣∣∣∣u=0

, (3.13)

E[xγ1xγ2xγ3 ] = j−3 ∂3χ(u)

∂uγ1∂uγ2∂uγ3

∣∣∣∣u=0

, (3.14)

...

E[xγ1xγ2 · · · xγm ] = j−m ∂mχ(u)

∂uγ1∂uγ2 · · · ∂uγm

∣∣∣∣u=0

. (3.15)

Remark 3.1.10. A joint characteristic function is related to a probability density function

by:

χ(u) =

∫

∞ejuT xp(x)dx, (3.16)

49

and the probability density function can be recovered using the Fourier transform:

p(x) =1

(2π)N

∫

∞e−juT xχ(u)du, (3.17)

where N is the dimension of x.

Definition 3.1.11 (Conditional Density Function). Given two random variables x ∈ R and

y ∈ R, the conditional density function of y given x is defined as:

p(y|x) =p(x, y)p(x)

, (3.18)

for x with p(x) > 0.

Definition 3.1.12 (Transition Density Function). Given a random process x(t) ∈ RN , the

transition density function is defined as:

p(xk, tk|x0, t0) = p(xk|x0) = p[x(tk)|x(t0) = x0], (3.19)

which represents a PDF of x at time tk given x(t0) = x0.

Definition 3.1.13 (Markov Process). A random process x(t) ∈ RN is a Markov process if

for all 0 ≤ · · · ≤ tk−1 ≤ tk:

p(xk|xk−1, · · · , x1, x0) = p(xk|xk−1). (3.20)

3.2 The Gaussian Probability Distribution

Definition 3.2.1 (Gaussian Probability Density Function). Let x be a Gaussian random

vector, x ∼ N (m, P), where m is the mean vector and P is the covariance matrix. The

Gaussian probability density function for x is defined as:

p(x) =1√

(2π)N det Pexp

−1

2(x−m)T P−1 (x−m)

, (3.21)

where N is the dimension of the state. The Gaussian probability density function for the

case N = 2 is shown in Figure 3.1 in non-dimensional units.

50

−3−2

−10

12

3

−3−2

−10

12

30

0.05

0.1

0.15

xy

p(x,

y)

Figure 3.1: Normalized two-dimensional Gaussian probability density function: p(x,y) =(1/2π) exp

−12(x2 + y2)

An important property of the Gaussian distribution is that the statistics of the Gaussian

random vector x can be completely described by the first two moments, i.e., m and P

[20, 57, 93]. In other words, the higher moments of x, such as E[xixjxk] and E[xixjxkxl],

can all be computed as functions of m and P. In this thesis, we implement the JCF defined

in Definition 3.1.9 for higher moment computation.

Definition 3.2.2 (Gaussian Joint Characteristic Function). For a Gaussian random vector

the joint characteristic function is defined as:

χ(u) = exp

juT m− 1

2uT Pu

. (3.22)

For a nonzero mean Gaussian random vector, x ∼ N (m, P), using Eqn. (3.15), the

51

first four moments can be computed by:

E[xi] = mi, (3.23)

E[xixj] = mimj + Pij, (3.24)

E[xixjxk] = mimjmk +(miPjk + mjPik + mkPij

), (3.25)

E[xixjxkxl] = mimjmkml

+(mimjPkl + mimkPjl + mjmkPil + mimlPjk + mjmlPik + mkmlPij

)

+ PijPkl + PikPjl + PilPjk. (3.26)

Remark 3.2.3. The mth order central moments of x ∼ N (m, P) become [96]:

1. If m is odd, E[(xγ1 − mγ1) · · · (xγm − mγm)] = 0,

2. If m is even, where m = 2k (k ≥ 1),

E[(xγ1 − mγ1) · · · (xγm − mγm)] =∑perm

Pγ1γ2 · · ·Pγ2k−1γ2k , (3.27)

where the sum is computed over all permutations of 1, · · · , 2k and yields

(2k − 1)!

2k−1(k − 1)!, (3.28)

terms in the sum.

In conventional trajectory navigation, a spacecraft state uncertainty is usually modeled

using a Gaussian distribution. Assuming a spacecraft state x ∼ N (m, P) ∈ R6, we can

construct a confidence region (or a phase volume) within which the spacecraft is located

with a certain probability. The usual model is:

EM = x | (x−m)T P−1(x−m) ≤ M2, (3.29)

where M ∈ R and EM represents a 6-dimensional ellipsoid, and depending on the value

of M , the confidence region EM is called an M -σ error ellipsoid. If we consider a scalar

52

state x ∈ R ∼ N (m, ρ2):2

Pr(x ∈ E1) = 0.6827, Pr(x ∈ E2) = 0.9545, Pr(x ∈ E3) = 0.9973.

It is important to note that the probability of a spacecraft in an M -σ error ellipsoid depends

on the dimension of a state. For example, if x ∈ R3 ∼ N (m, P):3

Pr(x ∈ E1) = 0.3182, Pr(x ∈ E2) = 0.8696, Pr(x ∈ E3) = 0.9919.

In general, as the dimension of x increases, Pr(x ∈ EM) decreases.

3.3 Dynamics of the Mean and Covariance Matrix

To study how an initial probability ellipsoid evolves dynamically we can solve for the

mean and covariance matrix as functions of time. Consider the local trajectory dynamics,

i.e., δx = φ(t; x0 + δx0, t0)− φ(t; x0, t0), and its mean, covariance, and higher moments.

Differentiate the relative mean and covariance matrix to find:

δmi = E[δxi], (3.30)

Pij

= E[δxiδxj + δxiδxj]− (δmiδmj + δmiδmj

). (3.31)

Substituting the M th order relative dynamics from Eqn. (2.20), we find:

δmi = E

[M∑

p=1

1

p!gi,γ1···γpδxγ1 · · · δxγp

]=

M∑p=1

1

p!gi,γ1···γpE [δxγ1 · · · δxγp ] , (3.32)

Pij

=M∑

p=1

1

p!E

[gi,γ1···γpδxγ1 · · · δxγpδxj

]+

M∑p=1

1

p!gj,γ1···γpE

[δxγ1 · · · δxγpδxi

]

− (δmiδmj + δmiδmj

). (3.33)

2Probabilities are computed by integrating the normalized scalar Gaussian density function, i.e.,∫ M

−M1√2π

e−x2/2dx.3Probabilities are computed by integrating the normalized 3-dimensional Gaussian density function, i.e.,∫ M

−M

∫ M

−M

∫ M

−M1

(2π)3/2 e−(x2+y2+z2)/2dxdydz.

53

We observe from Eqns. (3.32) and (3.33) that the mean and covariances of the relative

motion cannot be computed unless we apply certain assumptions, such as linear dynamics

and Gaussian distribution. This is because, at each step of the integration, higher order ex-

pectations in Eqns. (3.32) and (3.33) are required, which are not directly available. These

higher order expectations can be computed, if we integrate the higher order moments, or

the probability density function, together with Eqns. (3.32) and (3.33); however, this is

not considered in this thesis. Also, we can incorporate the second order terms to derive the

prediction equations as given in the truncated second order filter or the Gaussian second

order filter;[6, 55] however, we want to generalize the propagation method by including

the higher order effects in order to compute more accurate orbit uncertainties. In §3.9,

we present a nonlinear analytic method to propagate the mean and covariance matrix by

mapping the initial statistics using the state transition tensors discussed in §2.2.

3.4 The Fokker-Planck Equation

In this section, we give a formal definition of the Fokker-Planck equation (also known

as the forward Kolmogorov equation). The Fokker-Planck equation (FPE) is a partial dif-

ferential equation that satisfies the propagation of a transition probability density function,

or a probability density function. Hence, the solution of the FPE gives a complete statisti-

cal description of a trajectory.

Definition 3.4.1 (Itô Stochastic Equation). Most orbital dynamics problems can be written

using the Itô stochastic differential equation:

dx(t) = g[x(t), t]dt + G[x(t), t]dβ(t), (3.34)

which is an extension of the general dynamical system in Eqn. (2.3), where G is an N -

by-s matrix characterizing the diffusion, and β is an s-dimensional Brownian motion or

54

Wiener process with zero mean and diffusion matrix Q, i.e.,

E[dβ(t)dβT (t)] = Q(t)dt, (3.35)

E[β(t2)− β(t1)][β(t2)− β(t1)]

T

=

∫ t2

t1

Q(t)dt. (3.36)

For example, the diffusion vector β can be modeled as stochastic acceleration or process

noise in case of a spacecraft orbit determination process. Note that systems with determin-

istic inputs (e.g., controls) can be simply rewritten by including them in the state vector.

Remark 3.4.2. The solution to the Itô stochastic differential equation is:

x(t) = x0 +

∫ t

t0

g[x(τ), τ ]dτ +

∫ t

t0

G[x(τ), τ ]dβ(τ). (3.37)

Definition 3.4.3 (The Fokker-Planck Equation for the Transition Probability Density Func-

tion). Let p(x, t|x0, t0) be the transition probability density function of the stochastic pro-

cess x(t). Given a system satisfying the Itô stochastic differential equation, the time evolu-

tion of the transition probability density function must satisfy the Fokker-Planck equation

[26, 55, 87]:

∂p(x, t|x0, t0)

∂t= −

N∑i=1

∂

∂xi

[p(x, t|x0, t0)gi(x, t)

]

+1

2

N∑i=1

N∑j=1

∂2

∂xi∂xj

p(x, t|x0, t0)

[G(x, t)Q(t)GT (x, t)

]ij

,

(3.38)

where the initial condition is simply p(x, t0|x0, t0) = δ(x− x0), (Dirac delta function).

From the FPE for the transition density function, we can also derive the FPE for the

probability density function. To show this, consider x in Eqn. (3.38) as a dummy variable.

Then we can view p(x, t|x0, t0) as a random variable as a function of x0. Computing the

expectation of the transition probability density gives:

p(x, t) =

∫

∞p(x, t|x0, t0)p(x0, t0)dx0, (3.39)

55

where p(x0, t0) is now an initial condition. Multiplying Eqn. (3.38) by p(x0, t0) and inte-

grating over x0 gives:

The left-hand side of Eqn. (3.38):

∫

∞

∂p(x, t|x0, t0)

∂tp(x0, t0)dx0 =

∂

∂t

∫

∞p(x, t|x0, t0)p(x0, t0)dx0,

=∂p(x, t)

∂t.

The first term in the right-hand side of Eqn. (3.38):

∫

∞−

N∑i=1

∂

∂xi

[p(x, t|x0, t0)gi(x, t)

]p(x0, t0)dx0

= −N∑

i=1

∂

∂xi

(∫

∞[p(x, t|x0, t0)p(x0, t0)] dx0

)gi(x, t)

,

= −N∑

i=1

∂

∂xi

[p(x, t)gi(x, t)

].

The second term in the right-hand side of Eqn. (3.38):

∫

∞

1

2

N∑i=1

N∑j=1

∂2

∂xi∂xj

p(x, t|x0, t0)

[G(x, t)Q(t)GT (x, t)

]ij

p(x0, t0)dx0

=1

2

N∑i=1

N∑j=1

∂2

∂xi∂xj

(∫

∞p(x, t|x0, t0)p(x0, t0)dx0

) [G(x, t)Q(t)GT (x, t)

]ij

,

=1

2

N∑i=1

N∑j=1

∂2

∂xi∂xj

p(x, t)

[G(x, t)Q(t)GT (x, t)

]ij

.

Thus, the probability density function satisfies the Fokker-Planck equation.

Definition 3.4.4 (The Fokker-Planck Equation for the Probability Density Function). The

probability density function must satisfy the Fokker-Planck equation:

∂p(x, t)

∂t= −

N∑i=1

∂

∂xi

[p(x, t)gi(x, t)

]

+1

2

N∑i=1

N∑j=1

∂2

∂xi∂xj

p(x, t)

[G(x, t)Q(t)GT (x, t)

]ij

. (3.40)

56

Remark 3.4.5. The FPE for the systems with no stochastic terms can be simplified by

setting β(t) = 0:

∂p(x, t)

∂t= −

N∑i=1

∂

∂xi

[p(x, t)gi(x, t)

]. (3.41)

Note that solutions of the FPE give the true evolution of the probability density func-

tion. However, including these partial differential equations in the trajectory navigation

problem introduces additional difficulties and is usually avoided for practical reasons. In

this thesis, we consider systems with no process noise and the probability density function

satisfies the Fokker-Planck equation defined in Eqn. (3.41). This is a reasonable model for

astrodynamics problems with no thrusters and no dissipative forces acting on the space-

craft.4

3.5 Integral Invariance of Probability

Definition 3.5.1 (Integral Invariance). Consider a dynamical system with the governing

equations of motion x = g(x, t) and let I(x, t) be an integral of a vector field M(x, t) over

some volume B:

I(x, t) =

∫

BM(x, t)dx. (3.42)

The integral I(x, t) is called an integral invariant if it is constant for all time, i.e., dI/dt =

0. In general, the sufficiency condition for integral invariance can be explicitly stated as

[30]:

∂M(x, t)

∂t= −

N∑i=1

∂

∂xi

[M(x, t)gi(x, t)

], (3.43)

which is known as Liouville’s equation.

4For interplanetary spacecraft motion such as Sun-Earth halo orbits or cruise from Earth to Mars, effectsattributable to stochastic acceleration can be reduced to 10−12 km/s2 or less, negligibly small compared toconservative forces acting on a spacecraft.

57

Initial

Phase

Volume

Final

Phase

Volume

Hamiltonian

Flow

Figure 3.2: Integral invariance of a phase volume.

For example, the phase volume in a Hamiltonian system,

B =

∫

Bdx =

∫

Bdx1dx2 · · · dx2n, (3.44)

is an integral invariant according to Liouville’s theorem and is illustrated in Figure 3.2.

By comparing Eqn. (3.41) with Eqn. (3.43), we see that p(x, t) satisfies the sufficiency

condition for the probability to be an integral invariant. Hence, this implies that probability

of any dynamical system with no diffusion term is an integral invariant.

Definition 3.5.2 (Time Invariance). Consider a dynamical system with the governing

equations of motion x = g(x, t). A scalar function f(x, t) is called a time invariant if

its total time derivative is zero for all t:

df(x, t)

dt= 0. (3.45)

Remark 3.5.3. If an integral I(x, t) is an integral invariant, it satisfies the time invariance

condition as well.

58

Suppose we are given a nominal initial state x0 with a PDF p(x0, t0). From the funda-

mental theorem of calculus and integral invariance of the probability we have:

Pr(x ∈ B) =

∫

Bp(x, t)dx, (3.46)

=

∫

B0

p[φ(t; x0, t0), t]

∣∣∣∣det

(∂φ(t; x0, t0)

∂x0

)∣∣∣∣ dx0, (3.47)

=

∫

B0

p(x0, t0)dx0, (3.48)

where x(t) = φ(t; x0, t0) is the solution flow defined in §2.1. By equating Eqns. (3.47)

and (3.48) we find:

∫

B0

p[φ(t; x0, t0), t]

∣∣∣∣det

(∂x∂x0

)∣∣∣∣− p(x0, t0)

dx0 = 0. (3.49)

Hence, the probability density functions at t0 and at t are related by [19, 93]:

p[φ(t; x0, t0), t] = p(x0, t0)

∣∣∣∣det

(∂φ(t; x0, t0)

∂x0

)∣∣∣∣−1

, (3.50)

where the value of | det(∂x/∂x0)| depends on the system dynamics.

Theorem 3.5.4. Given a system x(t) = g[x(t), t] with x(t0) = x0 and an associated

probability density function p(x0, t0), the probability density function p(x, t) satisfies the

time invariant condition:

p(x, t) = p(x0, t0), (3.51)

if and only if:

exp

(−

∫ t

t0

∇x · g(x, τ)dτ

)= 1, (3.52)

where ∇x denotes the gradient vector:

∇x =

[∂

∂x1· · · ∂

∂xN

], (3.53)

assuming a system with dimension N .

59

Proof. Under standard theories of ordinary differential equations, assume the solution flow

x = φ(t; x0, t0) is continuous and has continuous partial derivatives with respect to x0,

and let x0 = ψ(t, x; t0) be the inverse solution flow. The current state PDF p(x, t) can be

related to the initial state PDF by applying Eqn. (3.50):

p(x, t) = p[ψ(t, x; t0), t0] |det T| , (3.54)

where

T =

(∂ψ(t, x; t0)

∂x

)=

(∂φ(t; x0, t0)

∂x0

)−1

. (3.55)

To show the sufficiency condition, suppose the PDF satisfies the time invariance so that

| det T| = 1 and consider the Fokker-Planck equation for the PDF of a diffusion-less

system:

∂p(x, t)

∂t= −

N∑i=1

∂

∂xi

[p(x, t)gi(x, t)

]. (3.56)

Re-ordering Eqn. (3.56) gives:

∂p(x, t)

∂t+ p(x, t)

N∑i=1

∂gi(x, t)

∂xi+

N∑i=1

gi(x, t)∂p(x, t)

∂xi= 0, (3.57)

∂p(x, t)

∂t+ p(x, t)∇x · g(x, t) +

N∑i=1

gi(x, t)∂p(x, t)

∂xi= 0. (3.58)

The Lagrange system of Eqn. (3.58) can be stated as [30, 93]:

dt

1= − dp(x, t)

p(x, t)∇x · g(x, t)=

dx1

g1=

dx2

g2= · · · =

dxN

gN, (3.59)

where we assume a dimension N . The first equality provides an obvious relation:

dp(x, t)

dt= −p(x, t)∇x · g(x, t). (3.60)

Note that this can also be derived directly by substituting the Fokker-Planck equation into

60

the total time derivative of a PDF:

dp(x, t)

dt=

∂p

∂x· ∂x

∂t+

∂p

∂t,

=∂p

∂x· g(x, t)−

(p(x, t)∇x · g(x, t) +

N∑i=1

gi(x, t)∂p(x, t)

∂xi

),

=∂p

∂x· g(x, t)− p(x, t)∇x · g(x, t)− ∂p

∂x· g(x, t),

= −p(x, t)∇x · g(x, t).

The solution to Eqn. (3.60) is:

p(x, t) = p(x0, t0) exp

(−

∫ t

t0

∇x · g(x, τ)dτ

)∣∣∣∣xi0=ψi(x,t;t0)

. (3.61)

By comparing Eqns. (3.54) and (3.61), we have:

|det T| = exp

(−

∫ t

t0

∇x · g(x, τ)dτ

), (3.62)

which gives:

exp

(−

∫ t

t0

∇x · g(x, τ)dτ

)= +1. (3.63)

Also, the necessity condition is trivially satisfied since, if Eqn. (3.63) is true, we get

|det T| = 1, and thus, the PDF is a time invariant. Note that the PDF of any dynamical

system is not necessarily a time invariant (constant), although the probability satisfies the

integral invariance condition.

3.6 Solution of the Fokker-Planck Equation for a Hamiltonian Sys-tem

Consider a PDF of a Hamiltonian system, H(q, p, t) with dimension N = 2n, and re-

call the dynamics equations x(t) = JHTx . Assuming no diffusion in the dynamics (e.g., no

stochastic accelerations), the Fokker-Planck equation for the PDF, Eqn.(3.41), simplifies

61

to:

∂p(x, t)

∂t= −

[p(x, t)

∂xx + p(x, t)Trace

(JHT

xx)]

. (3.64)

The second term in the right-hand side vanishes since:

Trace(JHT

xx)

= Trace

Hpq Hpp

−Hqq −Hqp

,

=

(n∑

i=1

Hpiqi

)−

(n∑

i=1

Hqipi

),

= 0. (3.65)

and reduces the FPE to:

dp(x, t)

dt=

∂p

∂t+

∂p

∂xx = 0. (3.66)

Hence the solution of the FPE is:

p(x, t) = p[φ(t; x0, t0), t] = p(x0, t0), (3.67)

where p(x0, t0) is assumed to be specified.

We can also show this directly from Eqn. (3.50). For a Hamiltonian system the map-

ping from x0 to x, φ(t; x0, t0), is a canonical transformation, and thus, det(∂x/∂x0) = +1

for all t according to Liouville’s theorem. Therefore, for a Hamiltonian system, the PDF

must satisfy:

p[φ(t; x0, t0), t] = p(x0, t0), (3.68)

or we can also state this in terms of the inverse flow x0 = ψ(t, x; t0) as:

p(x, t) = p[ψ(t, x; t0), t0]. (3.69)

In other words, the current state PDF can be completely characterized by the initial PDF,

or vice versa. This means that if the solution is known as a function of initial conditions

62

(i.e., is integrated) and the PDF is known at any one time, it can be found for all time. This

derivation is of particular interest since, as we will see later, it provides a direct way to

evolve the current state statistics as functions of the initial state and its statistics.

This is an important result since it shows that not only is the probability a time in-

variant, but the probability density function also satisfies the time invariance condition.

This result is not true for just any dynamical system, but it is generally true for astrody-

namics problems represented in Lagrangian form, such as two-body problem, Hill three-

body problem, and circular restricted three-body problem, as they can be transformed into

Hamiltonian systems. This time invariance of a PDF, in conjunction with Liouville’s the-

orem, provides another proof that the probability over some volume B in a Hamiltonian

system is indeed an integral invariant.

Now consider a special case of practical interest. Suppose our system is Hamiltonian

and the initial state is Gaussian with mean m0 = m(t0) and covariance matrix P0 = P(t0),

which are constants, so that:

p(x0, t0) =1√

(2π)2n|P0|exp

−1

2(x0 −m0)

T Λ0 (x0 −m0)

, (3.70)

where Λ0 = P(t0)−1. Using Eqn. (3.69), the current state PDF can be stated as:

p(x, t) =1√

(2π)2n|P0|exp

−1

2[ψ(t, x; t0)−m0]

T Λ0 [ψ(t, x; t0)−m0]

.

(3.71)

The PDF Eqn. (3.71) is a valid probability density function since it satisfies:

∫

∞p(x, t)dx =

∫

∞p[ψ(t, x; t0), t]dx,

=

∫

∞p(x0, t0)

∣∣∣∣det

(∂x∂x0

)∣∣∣∣ dx0,

= 1,

63

where we apply the symplectic property of a Hamiltonian system (| det(∂x/∂x0)| = 1).

Also, the PDF is time invariant as it satisfies:

dp(x, t)

dt=

dp[ψ(t, x; t0), t0]

dt=

∂p(ψ, t0)

∂ψ

dψ(t, x; t0)

dt= 0 , (3.72)

since ψ(t, x; t0) = x0 are integrals of motion of our system, and thus, their total time

derivative is zero. A formal proof of the time invariance of a PDF for a linear Hamiltonian

system is presented in Appendix B.

An interesting observation that can be made from Eqn. (3.71) is that the maximum

value of the PDF (i.e., the mode) is always located at the propagated initial mean according

to the deterministic map (i.e., x = φ(t; m0, t0)). However, the solution flow is nonlinear

in general, and thus, the current state mean vector may no longer be located at the mode.

It is apparent from Eqn. (3.71) that once the solution flow φ(t; x0, t0) is represented by

higher order terms and an analytic expression can be obtained as a function of the initial

state, statistical moments of the current state can be obtained that are, by definition, more

accurate than the propagated statistics from linear theory. Obtaining an analytic framework

for statistics propagation is discussed in §3.9.

3.7 On the Relation of Phase Volume and Probability

Suppose we are given an initial confidence region B0 for the initial spacecraft state

x0 ∼ N (m0, P0), and without loss of generality, consider B0 to be a 1-σ error ellipsoid E1:

B0 =

x0 | (x0 −m0)T P−1

0 (x0 −m0) ≤ 1

. (3.73)

As the probability of a deterministic system is an integral invariant:

Pr(x0 ∈ B0) = Pr[x(t) ∈ B(t)], (3.74)

where B(t) = x(t) | x(t) = φ(t; x0, t0) ∀ x0 ∈ B0. This indicates that the propagated

phase volume is no longer ellipsoidal since the system dynamics are nonlinear in general.

64

Thus, this implies that the confidence region at some future time can be defined by nonlin-

early mapping the boundary of the initial error ellipsoid, B(t0). This mapped confidence

region retains its probability value but loses its relation to the moments of the statistical

distribution.

If we wish to use the confidence region to characterize the future distribution, this ap-

proach is computationally intensive since we must integrate many samples chosen from

the boundary of B(t0) according to the governing equations of motion. For this reason,

in practice, one usually works with the simple linear model with its penalty for ignoring

the higher order statistics. The linear method works well as long as the linear model suffi-

ciently approximates the true dynamics as the propagated phase volume and the statistics

are both available and are equivalent. That is, the propagated N -σ ellipsoid of p(x0, t0) is

the same as the N -σ ellipsoid of p(x, t). However, this is no longer true when the system is

subject to a highly unstable environment or when mapped over a long duration of time, as

the linear solution will fail then to characterize the true dynamics. For this reason we im-

plement the STT approach to allow us to analytically characterize both phase volume and

statistics. In other words, once we have the time solution of the STTs, computing the phase

volume incorporating the higher order effects becomes a simple algebraic manipulation,

which provides a more accurate solution than the linear case.

Considering the uniqueness of solutions, the boundary of an initial phase volume must

map to the outer boundary points of the phase volume computed at a later time. Hence we

only need to analyze the behavior of the surface of this N -dimensional object, the surface

being an (N − 1)-dimensional object. After the boundary of B0 is integrated to some

time t, it can be projected onto the position and velocity planes to compute the confidence

region B(t), where Pr[x(t) ∈ B(t)] is the same as Pr(x0 ∈ B0).

65

3.8 Time Invariance of the Probability Density Function of the HigherOrder Hamiltonian Systems

In §3.6, we showed that a PDF of a Hamiltonian system is constant for all times.

Hence, a PDF of the higher order Hamiltonian system must satisfy the time invariance

condition as well. In this section, we give a detailed derivation of this property by consid-

ering the symplectic structure of the higher order Hamiltonian system.

Consider the relative dynamics of a Hamiltonian system:

δxi(t) =M∑

p=1

1

p!gi,γ1···γpδxγ1 · · · δxγp , (3.75)

=M∑

p=1

1

p!JiαHαγ1···γpδxγ1 · · · δxγp , (3.76)

= ci(δx, t). (3.77)

To show the integral invariance of a PDF p(δx, t) of the relative motion δx(t), it suffices

to check if Theorem 3.5.4 is satisfied:

exp

(−

∫ t

t0

∇δx · c(δx, τ)dτ

)= 1. (3.78)

Let’s consider the integrand of the Eqn.(3.78) and substitute Eqn. (3.76) to get:

∇δx · c(δx, t) = ∇δx ·

∑Mp=1

1

p!J1αHαγ1···γpδxγ1 · · · δxγp

...∑M

p=1

1

p!J2nαHαγ1···γpδxγ1 · · · δxγp

, (3.79)

where the indices are 1, 2, · · · , 2n. We need to show that for every order M Eqn. (3.79)

vanishes. Without loss of generality, consider the case where p = q and carry out the dot

66

product to get:

(1

q!

) [∂

(J1αHαγ1···γqδxγ1 · · · δxγq

)

∂ (δx1)+ · · ·+ ∂

(J2nαHαγ1···γqδxγ1 · · · δxγq

)

∂ (δx2n)

]

=

(1

q!

) [q(J1αH1αγ1···γq−1δxγ1 · · · δxγq−1

)+ · · ·

+ q(J2nαH2nαγ1···γq−1δxγ1 · · · δxγq−1

)],

=

(q

q!

) (2n∑i=1

JiαHiαγ1···γq−1δxγ1 · · · δxγq−1

). (3.80)

Note that Hiαγ1···γq−1 is symmetric and Jab = −Jba and Jaa = 0. Hence, Eqn. (3.80)

trivially vanishes, i.e.,

(q

q!

) (2n∑i=1

JiαHiαγ1···γq−1δxγ1 · · · δxγq−1

)= 0. (3.81)

This can be shown explicitly as:

2n∑i=1

JiαHiαγ1···γq−1δxγ1 · · · δxγq−1

= J11H11γ1···γq−1 + J12H12γ1···γq−1 + · · ·+ J12nH12nγ1···γq−1

+ J21H21γ1···γq−1 + J22H22γ1···γq−1 + · · ·+ J22nH22nγ1···γq−1

...

+ J2n1H2n1γ1···γq−1 + J2n2H2n2γ1···γq−1 + · · ·+ J2n2nH2n2nγ1···γq−1 ,

= 0,

since JabHabγ1···γq−1 + JbaHbaγ1···γq−1 = 0 and JaaHaaγ1···γq−1 = 0. Note that this fact is

equivalent to:

(JiαHiα

)= Trace

(JHT

xx)

= 0.

Hence, the condition Eqn. (3.78) is satisfied, and thus, the PDF p(δx, t) of the higher order

Hamiltonian system is a time invariant so that dp(δx, t)/dt = 0.

67

Remark 3.8.1. The PDF of a higher order Hamiltonian system is a time invariant, and

hence, from Eqn. (3.50), we find:

∣∣∣∣det

∂ [φ(t; δx0, t0)]

∂ (δx0)

∣∣∣∣ =

∣∣∣∣det

∂ (ψ(t, δx; t0))

∂ (δx)

∣∣∣∣ = 1, (3.82)

which is computed along the nominal relative trajectory φ(t; δx0, t0).

3.9 Nonlinear Mapping of the Gaussian Distribution

In §3.3 we discussed the difficulties of mapping the orbit uncertainties from a direct

integration. In this section, we assume the STTs are computed for a given reference tra-

jectory and present an analytic mapping of orbit uncertainties as functions of the initial

statistics.

Let’s first consider computation of the mean. Assuming the system is symplectic and

applying the results from Eqns. (3.68) and (3.69), we have the following four equivalent

expressions for the mean of the state x(t):

E[x(t)] =

∫

∞x(t) p(x, t)dx, (3.83)

=

∫

∞φ(t; x0, t0) p(x0, t0)dx0, (3.84)

=

∫

∞φ(t; x0, t0) p[φ(t; x0, t0), t]dx0, (3.85)

=

∫

∞x(t) p[ψ(t, x; t0), t0]dx. (3.86)

We observe that Eqn. (3.84) is suitable for computation of the state uncertainties using the

STT formulation because the solution flow can be expanded using the Taylor series and

higher moments can be computed using the JCF of the initial Gaussian distribution.

Consider the Gaussian boundary condition for the PDF, Eqn. (3.21). Assuming a

nonzero mean for the initial state, the PDF for the state δx0 can be obtained via a linear

transformation, x0 = δx0 + m0 − δm0, where m0 is the initial mean and δm0 is the initial

68

mean of the deviation. We note that these variables are constants. Applying the change of

variable to the PDF yields:

p(δx0, t0) =1√

(2π)N det P0

exp

−1

2(δx0 − δm0)

T Λ0 (δx0 − δm0)

. (3.87)

Since the expectation of the nominal trajectory does not change, by definition, it is easier

to instead analyze the statistics of the relative motion.

Using the state transition tensor notation and applying Eqn. (3.84), the current state

mean and covariance matrix can be stated as:

δmi(t) =m∑

p=1

1

p!φ

i,γ1···γp

(t,t0) E[δxγ1

0 · · · δxγp

0

], (3.88)

Pij(t) =

(m∑

p=1

m∑q=1

1

p!q!φ

i,γ1···γp

(t,t0) φj,ζ1···ζq

(t,t0) E[δxγ1

0 · · · δxγp

0 δxζ10 · · · δxζq

0 ]

)− δmi(t)δmj(t),

(3.89)

where γj, ζj ∈ 1, · · · , N and the higher order expectations are defined as:

E[δxγ1

0 · · · δxγp

0 ] =

∫

∞δξγ1

0 · · · δξγp

0 p(δξ0, t0)d(δξ0). (3.90)

This result is in general true for any dynamical system with any initial distribution; how-

ever, note that we arrived at this result from the very basics of probability and time invari-

ance of the PDF. When m = 1, this propagation gives the ordinary first-order covariance

propagation, that is:

δm(t) = Φ(t, t0)δm0, (3.91)

P(t) = Φ(t, t0)P0Φ(t, t0)T − δm(t)δm(t)T , (3.92)

where Φ(t, t0) is the usual state transition matrix which linearly maps the deviation from

t0 to t. For the cases where m > 1 it is apparent from Eqn. (3.89) that we need to compute

2mth-order Gaussian moments, which is, however, a one time operation for the entire

69

trajectory since we only need to compute the moments of the initial distribution. If we

consider a different initial distribution these higher order moments have to be computed

again.

Considering that the initial distribution is Gaussian, i.e., Eqn. (3.87), the mean and

covariance matrix can thus be obtained as functions of time once we have the time solution

of STTs and the initial first two moments. Thus, the computation of the mean and the

covariance matrix is an algebraic operation. If we consider a zero initial mean, all the odd

moments of the initial conditions vanish, which is a property of the Gaussian distribution,

and the above equations simplify a great deal. Also, it is clear from Eqn. (3.88) that the

mean of the future trajectory will not be zero, indicating that the mean trajectory deviates

from the reference trajectory, whereas in the linear analysis the mean and the reference

trajectory coincide.

Example 3.9.1. Consider the case where m = 2 with zero initial mean. The mean and

covariance matrix become functions of the initial covariance matrix P0, that is:

δmi(t) =1

2φi,ab

(t,t0)Pab0 ,

Pij(t) = φi,a(t,t0)φ

j,α(t,t0)P

aα0 +

1

4φi,ab

(t,t0)φj,αβ(t,t0)

[Pab

0 Pαβ0 + Paα

0 Pbβ0 + Paβ

0 Pbα0

]− δmi(t)δmj(t).

Considering this fact, the mean trajectory deviates from the deterministic trajectory as

δm(t) 6= 0.

3.10 Monte-Carlo Simulations

In conventional trajectory navigation, Monte-Carlo simulation is often implemented

to analyze the nonlinear behavior of trajectory statistics. Monte-Carlo simulation is an

empirical method and approximates the orbit statistics by averaging over a large set of

random samples.

70

Definition 3.10.1 (Monte-Carlo Simulation). Given an initial distribution p(x0, t0), the

Monte-Carlo simulation for the mean and covariance matrix are defined as [47, 51]:

mi(t) =1

Ω

Ω∑κ=1

φi(t; xκ0 , t0), (3.93)

Pij(t) =1

Ω − 1

Ω∑κ=1

[φi(t; xκ

0 , t0)− mi(t)] [

φj(t; xκ0 , t0)− mj(t)

], (3.94)

where Ω represents the number of random samples and xκ0 represents the κth random

sample that is chosen according to the initial PDF p(x0, t0). For example, if x0 is Gaussian,

each sample point can be drawn using the Gaussian random number generator.

Remark 3.10.2. The Monte-Carlo simulation for the relative motion can be stated as:

δmi(t) =1

Ω

Ω∑κ=1

δxi(t; δxκ0 , t0), (3.95)

Pij(t) =1

Ω − 1

Ω∑κ=1

[δxi(t; δxκ

0 , t0)− δmi(t)] [

δxj(t; δxκ0 , t0)− δmj(t)

], (3.96)

where the random samples δxκ0 are chosen according to the initial PDF p(δx0, t0).

Based on the law of large numbers and convergence of the statistics, Eqns. (3.93)

and (3.94) become the true mean and true covariance matrix as we consider more sample

trajectories, i.e., Ω → ∞. In general, a sufficiently large number of samples give the true

probability distribution by taking higher order moments.

Precision orbit prediction often relies on Monte-Carlo simulations to predict the future

state for nonlinear dynamical situations. However, there are three critical disadvantages

when using this approach: (a) the number of sample trajectories may grow quite large to

obtain convergence of the statistics, (b) the simulation needs to be repeated for different

initial distributions, and (c) it does not provide an analytic framework. These problems

make the Monte-Carlo simulation computationally intensive and statistics-specific. For

71

practical problems, the initial samples are mapped based on a linearized model, assum-

ing the initial error is small and is within the linear regime. One way to improve this

linear assumption is to apply the STT approach to capture the system nonlinearity in the

propagation model.

As discussed in §2.2.1, the computation of the mth order STTs for a system with a

dimension N generally requires N∑m

j=0

(N−1+j

j

)differential equations to be integrated.

Monte-Carlo simulation, on the other hand, requires one to integrate NΩ equations un-

til the solution converges and it is often difficult to approximate a sufficient number of

samples for solution convergence. The number of integrated equations for the STTs may

exceed that of the Monte-Carlo simulation; however, the importance of the STT approach

comes from the fact that the STTs need be integrated only once and then can be used for

varying-epoch statistics, whereas the Monte-Carlo analysis needs to be recomputed for

each set of initial statistics.

3.11 Unscented Transformation

Another method for nonlinear orbit uncertainty propagation is the unscented transfor-

mation (UT), which was proposed by Julier et al. [45]. The UT is based on the idea that,

for a given system, it may be is easier to approximate the probability distribution than

the nonlinear transformation [43, 44, 46]. That is, instead of performing a higher order

analysis, the probability distribution at a future time can be approximated by nonlinearly

integrating a few samples that are deterministically chosen from the initial distribution.

Moreover, the weight of each sample does not have to lie in the range [0, 1], which is

different from the Monte-Carlo simulation.

Given an initial distribution with a mean m0 and a covariance matrix P0, the UT is

72

initialized with (2N + 1) sample points:

X 00 = m0, (3.97)

X i0 = m0 +

[√(N + α)P0

]i

, (3.98)

X i+N0 = m0 −

[√(N + α)P0

]i

, (3.99)

with sample weights:

W00 = α/(N + α) (3.100)

W i0 = W i+N

k = 1/[2(N + α)], (3.101)

where α ∈ <,X j0 are the sample vectors with associated weightsWj

0 , and[√

(N + α)P0

]i

are the ith row of the matrix square root of [(N + α)P0]. With this initialization, the UT is

defined as (assuming no process noise):

X i(t) = φ(t;X i0, t0), (3.102)

m(t) =2N∑i=0

W i0X i(t), (3.103)

P(t) =2N∑i=0

W i0

[X i(t)−m(t)] [X i(t)−m(t)

]T. (3.104)

The UT requires N(2N + 1) differential equations to be integrated, which is on the order

of solving the Riccati equation, making it fast compared to the higher order STT approach

or the Monte-Carlo simulation. However, it is important to note that the UT is not analytic

with respect to initial statistics so that it must be carried out for different initial conditions,

and is limited to only a second order approximation of the dynamics.

CHAPTER IV

NONLINEAR TRAJECTORY NAVIGATION

Generally, spacecraft navigation is composed of trajectory design, control, and esti-

mation. To illustrate this point, consider an interplanetary trajectory to Mars from Earth.

Initially, a nominal trajectory is provided by the mission design which satisfies the required

mission objective. However, upon the launch of a spacecraft from Earth, which usually

introduces large errors, the spacecraft deviates from the previously defined nominal trajec-

tory. Also, the spacecraft state is only known up to some uncertainties since the spacecraft

state is determined through an estimation/filtering process based on measurements such as

radiometric range, Doppler, or optical, that also have associated errors.

For this reason, a series of correction maneuvers must be applied to either target back

to the original nominal trajectory or to an alternate trajectory that also satisfies the mis-

sion requirement, which must be redesigned by considering the current spacecraft state.

Usually, a single correction maneuver cannot achieve this goal since 1) a spacecraft state is

uncertain, 2) each correction maneuver has associated uncertainties, 3) measurements have

associated uncertainties, and 4) the system model governing the spacecraft motion is not

perfect. Hence, when a correction maneuver is applied to refine its trajectory, the space-

craft is not precisely targeted back to a desired orbit due to attributed errors. Therefore,

the trajectory navigation is the continuation of trajectory design, targeting, and estimation,

73

74

and our goal is to more effectively perform these processes.

In conventional trajectory navigation, the nominal trajectory and correction maneuvers

are usually computed assuming the spacecraft state, control input, and the system model

are perfectly known [9, 83, 99]; hence, these computations are deterministically carried

out and related uncertainties are not taken into account. However, if the trajectory uncer-

tainties are available and can be characterized, it may be is more robust to incorporate them

into trajectory and maneuver design processes. Also, conventional trajectory estimation

is built around the linear theory [23, 61, 100], which assumes the trajectory dynamics are

locally linear. In general, there are two types of estimation methods. One type is the batch

least-squares approximation, or linear regression, which reconstructs the entire spacecraft

trajectory by minimizing the measurement residual errors given a set of measurements. In

practice, the square-root information filter is often implemented as a batch estimator for

orbit reconstruction/determination problem which is numerically more stable and robust

than the least-squares filter. Another type is the sequential estimation, such as the extended

Kalman filter, and is usually divided into two parts: prediction and update.1 In this setting,

the spacecraft uncertainties are linearly propagated and measurements are updated using

linearized measurement models. Hence, system nonlinearities are not incorporated in the

filtering process.

In this chapter we discuss the trajectory navigation where our nonlinear uncertainty

propagation technique can be utilized and implemented. We first introduce the concept

of a statistically correct trajectory where we incorporate statistical information of an orbit

into the trajectory design. We then extend this idea and present a method of nonlinear

statistical targeting by computing the correction maneuver that gives a statistically more

1The batch estimation is also divided into prediction and update parts, but the prediction part is performedover the entire batch of measurements and the update incorporates the information contents of the entirebatch. This is in contrast to sequential filters which alternate prediction and update at each measurementtime.

75

accurate target solution at a desired time. Our uncertainty propagation method is also im-

plemented using nonlinear filtering algorithms. We first derive an analytic expression of

the posterior density function for an optimal nonlinear filtering problem by using Bayes’

rule and by incorporating higher order solutions of the relative spacecraft motion. We then

present two Kalman-type filters by directly applying the higher order state transition ten-

sors to the Kalman filter algorithm.

desired

target

reference

trajectory

statistically

correct

trajectory

δx(t )

x(t )x(t )f

0

0

m(t )=E[x(t )]

neighboring

deterministic

trajectory

f

f

x(t )+δx(t )f

f

E[x(t)+δx(t)]

x(t )+δx(t )0 0

δx(t )f

mean of

reference

trajectory

p(t )0

Figure 4.1: Illustration of the statistically correction trajectory.

4.1 The Concept of Statistically Correct Trajectory

Conventional mission design usually relies on the deterministic solution of a boundary

value problem; no statistical information is taken into account in the design process. The

idea of a statistically correct trajectory is to improve on this deterministic trajectory by

incorporating trajectory navigation information. Consider Figure 4.1, which illustrates the

76

concept of a statistically correct trajectory. Suppose the deterministic reference trajectory

satisfies the desired target state at tf so that:

xf = x(tf ) = φ(tf ; x0, t0). (4.1)

In practice, however, x0 = x(t0) is always associated with a non-zero uncertainty (e.g.,

x0 ∼ N (m0, P0)), and thus, it is inevitable that the mean, mf = m(tf ) = E[φ(tf ; x0, t0)],

deviates from the desired target, i.e., mf − xf 6= 0. More specifically,

E[φ(tf ; x0, t0)] 6= φ(tf ; E[x0], t0). (4.2)

Definition 4.1.1. Suppose we are given an initial state x0 with the probability density func-

tion p(x0, t0) and it deterministically reaches the desired target xf = x(tf ) = φ(tf ; x0, t0).

The statistically correct trajectory is the expectation of a neighboring trajectory φ(tk; x0 +

δx0, t0) which satisfies:

E[φ(tk; x0 + δx0, t0)]− xf = 0, (4.3)

where x0 + δx0 is assumed to have the same probability distribution as x0, but the mean of

x0 is shifted by δx0 to satisfy Eqn. (4.3). In other words, we are varying the mean of the

initial distribution p(x0, t0) so that the propagated p(x0, t0) has the mean located at xf . In

terms of relative motion, this can be viewed as x0 being a constant vector and δx0 is a new

random vector with the same probability distribution as x0 with a mean shifted by x0. The

goal of the statistically correct trajectory is then to find the mean of δx0, δm0 = E[δx0]

such that Eqn. (4.3) is satisfied. Note that time tk does not have to equal tf in general.

Remark 4.1.2. Considering the relative motion, the condition for the statistically correct

trajectory can be restated as:

E[φ(tk; x0 + δx0, t0)− φ(tk; x0, t0)]− [xf − φ(tk; x0, t0)]

= E[δx(tk; δx0, t0)]− δc = 0, (4.4)

77

where δc = xf − φ(tk; x0, t0) and does not depend on δx0. Applying the STT solutions,

E[δx(tk; δx0, t0)] can be computed as:

E[δxi(tk; δx0, t0)] =m∑

p=1

1

p!φ

i,γ1···γp

(tk,t0) E[δxγ1

0 · · · δxγp

0

], (4.5)

where δx0 is the random variable and has the same probability distribution as x0. The

solution of the statistically correct trajectory is then the mean of δx0 that satisfies Eqn.

(4.4).

Definition 4.1.3 (Differential Correction or Newton’s Method). For y ∈ <N , suppose we

want to find a solution of y = h(x), i.e., find x ∈ <N such that y = h(x) or h(x)−y = 0. To

achieve this goal, construct a convergent sequence (x)ι such that limι→∞ h[(x)ι]− y = 0.

Applying a Taylor series expansion gives:

h[(x)ι + (δx)ι]− y = 0,

h[(x)ι] +∂h∂x

∣∣∣∣x=(x)ι

· (δx)ι − y ≈ 0,

(δx)ι ≈ −[∂h∂x

]−1∣∣∣∣∣

x=(x)ι

· h[(x)ι]− y .

Newton’s method is defined as:

(x)ι+1 = (x)ι −[∂h∂x

]−1∣∣∣∣∣

x=(x)ι

· h[(x)ι]− y , (4.6)

and its convergence varies depending on the initial guess of the solution.

Remark 4.1.4. Assuming the Jacobian of Eqn. (4.3) exists and is invertible, the statistically

correct trajectory can be solved by applying Newton’s method:

(δx0)ι+1 = (δx0)

ι −[∂[e(tk; δx0, t0)]

∂(δx0)

]−1∣∣∣∣∣δx0=(δx0)ι

· e[tk; (δx0)ι, t0], (4.7)

where

e[tk; (δx0)ι, t0] = Eφ[tk; x0 + (δx0)

ι, t0] − xf ,

= Eδx[tk; (δx0)ι, t0] − δc. (4.8)

78

Note that the superscript ι indicates the solution from the ιth iteration and the iteration

is carried out until δm0 = E[(δx0)ι] is sufficiently small. The iteration can be initialized

with:

E[(δx0)1] = Φ−1(tk, t0)δm(tk), (4.9)

where δm(tk) = E[δx(tk; δx0, t0)] is the deviated mean at tk due to the uncertainties in the

initial state.

Remark 4.1.5. If we consider the case where tk = tf , Eqn. (4.3) simplifies to:

E[φ(tf ; x0 + δx0, t0)− xf ] = E[δx(tf ; δx0, t0)] = 0,

and Newton’s method becomes:

(δx0)ι+1 = (δx0)

ι −[∂(E[δx(tf ; δx0, t0)])

∂(δx0)

]−1∣∣∣∣∣δx0=(δx0)ι

· δm[tf ; (δx0)ι, t0], (4.10)

with the initial guess of

E[(δx0)1] = Φ−1(tf , t0)δm(tf ), (4.11)

where δm(tf ) = E[δx(tf ; δx0, t0)] is the deviated mean of the initial state.

It is important to note that if we fix the target time to be tk = tf the solution exists

for cases with sufficiently small initial uncertainties. However, by varying tk, there is

more freedom in solving for a δx0 that converges. Once δm0 = E[δx0] is computed, the

statistically correct initial state x0 + E[δx0] can be determined for the given initial proba-

bility distribution. If we consider the initial state to be Gaussian, the deviated mean, i.e.,

δmk = E[δx(tk; δx0, t0)], can be represented as a power series in the initial mean δm0

and covariance P0 using Eqn. (3.88). Hence, once the STTs are integrated for the entire

trajectory up to tk, the Jacobian of δmk can be computed analytically and the iteration

process becomes a trivial problem assuming convergence. In general, for a fixed xf , there

79

are varying δmk = E[δx(tk; δm0, P0)] that satisfies the statistically correct trajectory con-

dition. Note that as P0 of the initial Gaussian distribution approaches zero, we recover the

conventional deterministic solution.

p(x ,t )

r(t )

v(t )

reference

trajectory

mean of

reference

trajectorydeterministically

corrected trajectory

0

δr(t ) = δm (t )

k

k

∆V

k

r(t )

k

r(t )

f

δr(t )

f

δm (t )=0

0

target

r

f r

0

δv(t ) = δm (t )

k

k

v

0

deterministially

corrected

trajectory

mean of

Figure 4.2: Illustration of the nonlinear statistical targeting.

4.2 Nonlinear Statistical Targeting

As an extension of the statistically correct trajectory, we describe how to practically

design a nonlinear statistical correction maneuver that uses this concept. For this example,

we focus on position targeting (i.e., interception) based on a single impulsive maneuver;

however, the result can be generalized to target the full state with two or more maneuvers.

Figure 4.2 illustrates the concept of nonlinear statistical targeting. In particular, suppose

we are given a reference trajectory φ(tf ; x0, t0) and let rf = r(tf ) = φr(tf ; x0, t0) be the

fixed target position and vf = v(tf ) = φv(tf ; x0, t0) be the corresponding velocity vec-

tor, which can vary. Note that φr represents the solution flow of the position components

(i.e., r ∈ 1, · · · , N/2). Also, suppose the initial state has a probability density func-

tion p(x0, t0). Under this assumption, the mean spacecraft trajectory deviates from the

80

reference trajectory. At time tk, let the estimated mean position and velocity be δr(tk) =

δmr(tk) and δv(tk) = δmv(tk), respectively. The goal of nonlinear statistical targeting is

to design a maneuver ∆Vk which satisfies E[φr(tf ; rk + δmrk, vk + δmv

k + ∆Vk, tk)] = rf .

In other words, we solve for ∆Vk such that δmr(tf ) = E[φr(tf ; rk + δmrk, vk + δmv

k +

∆Vk, tk)] − rf = 0, which can be stated analytically similar to Eqn. (3.88). Note that

∆Vk ∈ R3, which is an exception of the notation used in this thesis.

In order to make a distinction between the conventional deterministic method and

our statistical method, we first discuss the correction maneuver design assuming perfect

knowledge of the initial state.

Definition 4.2.1 (Nonlinear Deterministic Targeting). Consider a reference trajectory

which hits the desired target rf = r(tf ) = φr(tf ; x0, t0) at tf , where r denotes posi-

tion components of x. Suppose the spacecraft happens to be offset from the reference

trajectory at some time tk (t0 ≤ tk ≤ tf ) with deviations δrk and δvk due to errors in the

initial conditions. In order to re-target the spacecraft to hit the desired target, the goal of

conventional nonlinear targeting is to find a correction maneuver ∆Vk such that:

φr(ts; rk + δrk, vk + δvk + ∆Vk, tk)− rf = 0, (4.12)

where ts ≥ tk. Note that ∆Vk must be solved iteratively since φr is nonlinear in general,

and note that the time of interception ts does not necessarily have to be tf . In practice, ts is

usually chosen based on some optimality constraints, such as one that minimizes ‖∆Vk‖.

Remark 4.2.2. At the linear level, the mean trajectory can be propagated as:

δr(tf )

δv(tf )

=

Φrr(tf , tk) Φrv(tf , tk)

Φvr(tf , tk) Φvv(tf , tk)

δr(tk)

δv(tk)

, (4.13)

where Φ is the usual state transition matrix mapping the deviations from tk to deviations

at tf .

81

Definition 4.2.3 (Linear Deterministic Targeting). Assuming the interception time is ts,

the linear deterministic targeting problem solves:

δr(ts) =

[Φrr(ts, tk) Φrv(ts, tk)

]

δr(tk)

δv(tk) + ∆Vk

= 0, (4.14)

which gives the the linear correction maneuver:

∆Vk = − (Φrv)−1 Φrrδr(tk)− δv(tk). (4.15)

With a deterministic correction maneuver, the deviated position is zero; however, we

know that the true trajectory (in a statistical sense) will likely miss the desired target,

depending on the associated uncertainties at tk, the transit period tf − tk, system non-

linearity, etc., which may be significant.2 For this reason, additional maneuvers may be

needed. Our proposed statistical approach is to instead satisfy E[δr(tf )] = δmr(tf ) = 0.

In other words, instead of deterministically mapping the deviation, we apply the concept

of the statistically correct trajectory.

Definition 4.2.4 (Nonlinear Statistical Targeting). Suppose we are given a reference so-

lution which hits the desired target rf = r(tf ) = φr(tf ; x0, t0) at time tf . However, due

to uncertainties associated with x0, suppose at time tk the spacecraft is deviated from the

reference trajectory by position deviation of δr(tk) and velocity deviation of δv(tk), and

let p(xk, tk) be the probability distribution of the state xk = [rTk , vT

k ]T at time tk. The goal

of nonlinear statistical targeting is to find the correction maneuver ∆Vk that satisfies:

E[φr(ts; rk + δrk, vk + δvk + ∆Vk, tk)]− rf = 0, (4.16)

where ts ≥ tk in general and ts does not necessarily equal tf .

2In practice, the validity of this deterministic correction maneuver is checked by running Monte-Carlosimulations and making sure that the deviated mean is within some specified error bound with respect to thetarget rf .

82

Remark 4.2.5. The condition for nonlinear statistical targeting can be restated as, find δuk

such that:

E[φr(ts; rk + δrk, vk + δvk + ∆Vk, tk)− φr(ts; x0, t0)]− δcr

= E[φr(ts; rk + δrk, vk + δuk, tk)− φr(ts; x0, t0)]− δcr,

= E[δr(ts; δrk, δuk, tk)]− δcr = 0,

where δcr = rf −φr(ts; x0, t0) and δuk = δvk + ∆Vk so that δxTk = [δrT

k , δuTk ]. Note that

δcr is constant for a fixed ts. Since the position is not changed instantaneously (i.e., δrk is

constant), we only need to solve for δuk such that:(

m∑p=1

1

p!φ

r,γ1···γp

(ts,tk) E[δxγ1

k · · · δxγp

k

])− δcr = 0. (4.17)

This nonlinear statistical targeting problem can be solved numerically using Monte-

Carlo simulations in an iterative way; however, a Monte-Carlo simulation does not provide

an analytic framework and is difficult to implement. By applying the higher order STT

solutions, depending on the initial probability distribution, the deviated position mean can

be determined analytically. If we assume the distribution at time ts can be approximated

by a Gaussian distribution, the position mean in Eqn. (4.17) is simply a power series in

the mean δm(tk) and covariance matrix P(tk). Note that the order of the solution can

be chosen depending on the system’s nonlinearity, which can be specified by the local

nonlinearity index. Therefore, we can compute ∆Vk analytically while incorporating the

system nonlinearity, finding analytical results similar to a Monte-Carlo simulation.

In general, as long as the Jacobian of the higher order expectations in Eqn. (4.17)

exists and is invertible, the correction maneuver can be solved by iteration according to:

(δuk)ι+1 = (δuk)

ι −[∂[er(ts; δrk, δuk, tk)]

∂(δuk)

]−1∣∣∣∣∣δuk=(δuk)ι

· er[ts; δrk, (δuk)ι, tk],

(4.18)

83

where

er[ts; δrk, (δuk)ι, tk] = E[φr(ts; rk + δrk, vk + δuk, tk)]− rf ,

and the superscript ι indicates an ιth iteration. For an initialization of the iteration, we can

use the solution from the linear correction maneuver as given in Eqn. (4.15), i.e.,

(δuk)1 = − [Φrv(ts, tk)]

−1 Φrr(ts, tk)δrk. (4.19)

Note that if we consider the case m = 1 in Eqn. (4.18), the solution is identical to the

linear targeting maneuver.

Remark 4.2.6. Consider the case where ts = tf . The nonlinear statistical targeting condi-

tion is:

δmrf (tf ; δrk, δuk, tk) =

m∑p=1

1

p!φ

r,γ1···γp

(tf ,tk) E[δxγ1

k · · · δxγp

k

],

= 0,

and the iteration formula becomes:

(δuk)ι+1 = (δuk)

ι −[∂[δmf (tf ; δrk, δuk, tk)

∂(δuk)

]−1∣∣∣∣∣δuk=(δuk)ι

· δmrf [tf ; δrk, (δuk)

ι, tk],

(4.20)

with the initialization:

(δuk)1 = − [Φrv(tf , tk)]

−1 Φrr(tf , tk)δrk. (4.21)

Example 4.2.7. Assume δxk ∼ N (δmk, Pk). The 3rd order STT propagated mean is:

δmrf =

3∑p=1

1

p!φ

r,γ1···γp

(tf ,tk) E[δxγ1

k · · · δxγp

k

],

= φr,γ1δmγ1

k +1

2!φr,γ1γ2 (δmγ1

k δmγ2

k + Pγ1γ2

k )

+1

3!φr,γ1γ2γ3 (δmγ1

k δmγ2

k δmγ3

k + δmγ1

k Pγ2γ3

k + δmγ2

k Pγ1γ3

k + δmγ3

k Pγ1γ2

k ) ,

(4.22)

84

where γj ∈ 1, · · · , N. In this example, the solution of the nonlinear statistical targeting

is δmk, which satisfies δmrf = 0, where Pk is a constant covariance matrix at the time of

the maneuver (i.e., tk) and can include maneuver execution uncertainties. Note that the

partial derivatives of δmrf with respect to δmk can be computed as:

∂(δmrf )

∂(δmjk)

= φr,j(tf ,tk) + φr,γ1j

(tf ,tk)δmγ1

k +1

2φr,γ1γ2j

(tf ,tk) (δmγ1

k δmγ2

k + Pγ1γ2

k ) . (4.23)

Assuming that the deviated mean computed using an mth order STT solution is in

good agreement with a Monte-Carlo simulation result, only one correction maneuver is

required to hit the desired target on average according to probability theory. In terms of

∆Vk cost, the linear or nonlinear deterministic targeting method predicts that it is better

to perform correction maneuvers at an early stage of the trajectory, since the deviated

mean is smaller at the time of the correction maneuver. This yields a trade-off between

the final deviation and the ∆Vk cost (i.e., if a maneuver is performed earlier the trajectory

deviates more). Using the nonlinear statistical method, the final deviated position mean is

zero. Moreover, we will see later that the nonlinear statistical targeting gives an optimal

time (minimum ∆Vk) to perform the correction maneuver, which is not found using the

deterministic method.

This assumes that the error due to ∆Vk can be cast into the velocity covariance, which

is a simple transformation in case we assume δvk and ∆Vk are locally Gaussian.

4.2.1 On the Theoretic and Practical Aspects of Nonlinear Statistical Targeting

Although, in theory, the nonlinear statistical targeting provides a maneuver that is sta-

tistically more accurate than the linear statistical theory, there are practical, as well as

fundamental, problems that must be discussed. First we note that the nonlinear statistical

targeting depends on statistical knowledge about the initial state whereas the linear cor-

rection maneuver is independent of the statistics, and hence, there is only one maneuver

85

(assumed impulsive) in the deterministic targeting problem. In practical implementation

of these maneuvers, the statistical part of the linear maneuver is then checked by Monte-

Carlo runs to ensure that the spacecraft resides within the necessary error bound at the

target, which is often carried out based on larger-than-estimated initial uncertainties so

that a more conservative distribution can be treated at the target. However, we must keep

in mind that at the time of the maneuver, past navigation data is the only information

we have about the trajectory, and that the linear correction ignores a component of this

information in the design process, the level of uncertainties or covariance.

For a nonlinear maneuver, however, dependency on the navigation data makes non-

linear targeting more difficult for maneuver designers to implement since the calibrated

initial uncertainties will provide a different maneuver.3 This is precisely the place where

we can make an important, yet distinct, comparison between the two methods. Our study

shows that the nonlinearly propagated mean provides a more conservative solution than

the linear solution. We mean conservative in that the resulting projection of the covariance

matrix using the nonlinear correction gives more dispersed uncertainties at a future time

since the covariance matrix is positively affected (i.e., the covariance matrix is increased)

by the deviated mean. Hence, it may not be necessary to calibrate the initial uncertainties

to encompass possible additional errors.

Another concern one may have is, what if the navigation data is perceived to be too

accurate, so the covariance matrix is increased to compensate for potential solution errors,

or there are other uncertainty data from different navigation sources? If there is only one

initial distribution (i.e., only one filter solution from a navigation team), we are actually

increasing the region of event space to obtain a higher probability by increasing the initial

uncertainties. Hence, the covariance matrix should remain unchanged in the maneuver

3Calibrated initial uncertainties mean increased initial covariances, which can occur if the initial stateerrors are believed to be smaller than expected.

86

design process. When the uncertainties are increased to check the linear maneuver using a

Monte-Carlo simulation, it actually means that a larger initial error ellipsoid is considered

to increase the probability, not the covariance matrix. This indicates that both methods are,

in a way, utilizing the navigation data. However, it is important to note that the nonlinear

statistical targeting includes the navigation data in the maneuver design process, which is

usually not considered in conventional mission operations. If there are initial distributions

from different navigation sources (i.e., multiple filter solutions from different navigation

teams), it will be the maneuver designer’s decision to choose which navigation data to be

used. If the navigation data to be used is extremely accurate and if the system behaves

linearly, both correction maneuvers are essentially the same; however, there are usually

significant levels of uncertainties associated at an initial epoch, and correction maneuvers

computed from linear and nonlinear targeting methods will be different. The importance

of the nonlinear method is that the number of correction maneuvers can be reduced when

the statistical-based nonlinear correction is used because the spacecraft will more likely

lie within the necessary confidence region for a longer period.

In nonlinear statistical targeting, we instead aim for the target on average with a known

probability distribution at the time of the maneuver whereas the linear correction aims for

the target on the reference trajectory. From the integral invariance of the probability, it

is easy to show that the reference trajectory always depicts the highest value (i.e., mode)

of the probability density function for an initial Gaussian state, and hence, the linear cor-

rection maneuver is in a sense the most probable targeting correction for the initial state.

However, the target point is a single event in the probability space, and hence, has a zero

probability. On the other hand, the mean is, by definition, the expected or averaged value

of the deviated state. It is true that an infinitesimal volume around the reference will have

the highest probability, but this probability is negligible when compared to the initial phase

87

−0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

x−coordinate

y−co

ordi

nate

m=1m=2m=3m=4Monte−Carlo

NominalTrajectory

Figure 4.3: Propagated mean and 1-σ error ellipsoid projected onto the spacecraft positionplane: comparison of the STT-approach and Monte-Carlo simulations.

volume we consider. To make a clear comparison, we must consider the probability of the

entire confidence region. In other words, both the target and mean have zero probabilities

in the probability space, but statistically speaking, the mean characterizes final location

more accurately given the initial phase volume.

As an example, consider Figure 4.3, which shows the propagated mean and 1-σ error

ellipsoid projected onto the spacecraft position plane considering an initial Gaussian dis-

tribution with zero mean.4 Here, m = i case represents the ith order STT solution, and

hence, m = 1 represents the linearly propagated uncertainties. One direct observation

from Figure 4.3 is that the linear solution gives a poor characterization of the propagated

uncertainties whereas higher order STT solutions (m ≥ 2) give more accurate description

of the propagated uncertainties as compared to the Monte-Carlo solution. Also, since the

initial distribution is Gaussian, the mean computed using the linear solution is located at

4A detailed discussion of Figure 4.3 is given in §5.3.

88

the mode of the propagated PDF, or at the deterministic solution flow, whereas the higher

order solutions approximate the propagated mean. The difference between the linear and

nonlinear targeting methods is that the nonlinear statistical targeting applies a correction

maneuver so that the deviated mean intercepts the target. Hence, in this example, a correc-

tion maneuver designed using the nonlinear targeting method can be quite different from

the solution of the linear targeting method. Statistically, however, targeting the nonlinearly

propagated mean is more accurate than targeting the mode.

From a different perspective, we can propagate uncertainties to the final state, and

approximate the PDF as a Gaussian, using the first two moments. Then, in essence, the

maneuver design approach targets the mode of the statistical distribution mapped forward

in time. This takes advantage of the fact that the mean of a distribution does not follow the

dynamical equations of motion.

4.3 Higher Order Bayesian Filter with Gaussian Boundary Condi-tions

In this section, we derive a Bayesian filter which incorporates the system nonlinear-

ity by applying the integral invariance property of the probability density function and

the higher order relative trajectory solutions. Here, we assume the spacecraft dynamics

model is known with perfect knowledge and only the initial state and measurements have

random errors that are modeled as Gaussian. This can be considered as the case where a

spacecraft is relatively quiet with no thrusters turned on (i.e., no process noise) and pre-

cise ephemerides are given. Also, we consider a measurement function which is a linear

function in state since our goal is to show the importance of the system’s dynamical non-

linearity acting on the filtering process. Examples of nonlinear measurement models can

be found ins Refs. [53, 77, 89]. In this section, a higher order Bayesian filter is derived

89

based on a single measurement taken at some future time, tk, and expressions for poste-

rior conditional mean and covariance matrix of the state after the measurement update are

presented. The results can be generalized to multiple measurement updates.

Suppose we want to estimate a spacecraft state x(tk) ∈ RN given an initial state x0 with

a probability density function p(x0, t0) and a set of measurements z1:k = z1, z2, · · · , zk,

which denotes all the measurements taken over the interval [t0, tk] and zj represents the

measurement vector taken at time tj . The goal of an optimal filtering problem is to com-

pute the posterior conditional density at tk conditioned on a set of measurements, i.e.,

p(xk|z1:k). The general Bayesian filtering problem is a two-step process, prediction and

update, defined as follows [21, 37]:

Definition 4.3.1 (Bayesian Prediction). General Bayesian prediction is defined as:

p(xk|z1:k−1) =

∫

∞p(xk|xk−1, z1:k−1)p(xk−1|z1:k−1)dxk−1. (4.24)

Definition 4.3.2 (Bayesian Measurement Update). General Bayesian measurement update

is defined as:

p(xk|z1:k) =p(zk|xk, z1:k−1)p(xk|z1:k−1)

p(zk|z1:k−1). (4.25)

It is evident from Eqns. (4.24) and (4.25) that the Bayesian formulation is not a re-

cursive process since we need to store and re-process all the measurements in order to

compute the terms p(xk|xk−1, z1:k−1) and p(zk|xk, z1:k−1), which are conditioned on z1:k−1.

This difficulty can be remedied by applying the following two assumptions.

Assumption 4.3.3 (Markov Process). The spacecraft state is a Markovian process:

p(xk|x0, x1, · · · , xk−1) = p(xk|xk−1). (4.26)

Assumption 4.3.4. Each measurement depends only on the state at the time the measure-

ment is taken and is independent of previous measurements. Thus, the density conditioned

90

on xk and zk is independent of the previous states and measurements:

p(z1, · · · , zk|x1, · · · , xk) =k∏

i=1

p(zi|xi). (4.27)

Example 4.3.5. Consider two measurements conditioned on two states:

p(z1, z2|x1, x2) =p(z1, z2, x1, x2)

p(x1, x2),

=p(z1|z2, x1, x2)p(z2, x1, x2)

p(x1, x2),

=p(z1|z2, x1, x2)p(z2|x1, x2)p(x1, x2)

p(x1, x2),

= p(z1|x1)p(z2|x2).

Based on these two assumptions, we drop the conditioning on z1:k−1 and we write

p(xk|xk−1, z1:k−1) = p(xk|xk−1). This leads to the following recursive prediction and

update equations:

Definition 4.3.6 (Recursive Bayesian Prediction).

p(xk|z1:k−1) =

∫

∞p(xk|xk−1)p(xk−1|z1:k−1)dxk−1. (4.28)

Definition 4.3.7 (Recursive Bayesian Measurement Update).

p(xk|z1:k) =p(zk|xk)p(xk|z1:k−1)

p(zk|z1:k−1). (4.29)

Remark 4.3.8. Note that applying the law of total probability, the denominator in Eqn.

(4.29) can be stated as:

p(zk|z1:k−1) =

∫

∞p(zk|xk)p(xk|z1:k−1)dxk, (4.30)

which is simply the integral of the numerator in Eqn. (4.29) over xk.

Now, consider the following discrete system realization of the relative dynamics:

δxik+1 =

m∑p=1

1

p!φ

i,γ1···γp

(tk+1,tk)δxγ1

k · · · δxγp

k + wik, (4.31)

δzk = Hkδxk + vk, (4.32)

91

where δzk is a single measurement taken at time tk, Hk is a row vector characterizing a

measurement function at time tk, vk is a measurement noise with zero mean5 and E[v2k] =

σ2k, and wk is the process noise which we assume to be zero. To ease the notation, let

ξk = δx(tk; δx0, t0) and zk = δzk throughout this section unless noted otherwise. Our

system can be restated as:

ξik+1 =

m∑p=1

1

p!φ

i,γ1···γp

(tk+1,tk)ξγ1

k · · · ξγp

k , (4.33)

zk = Hkξk + vk. (4.34)

Suppose the initial state ξ0 = δx0 can be characterized as Gaussian with zero mean and

covariance matrix P0, i.e.,

p(ξ0, t0) = Cξ0exp

−1

2ξT

0 Λ0ξ0

, (4.35)

where Cξ0= 1/

√(2π)N det P0 and Λ0 = P−1

0 .

According to the recursive Bayesian prediction (4.28) we find:

p(ξk, tk) =

∫

∞p(ξk, tk|ξ0, t0)p(ξ0, t0)dξ0. (4.36)

This is where the solution of the diffusion-less Fokker-Planck equation becomes useful.

Based on results from §3.5, the state PDF at time tk is related to the initial state PDF by:

p[ξ(tk; ξ0, t0), tk] = p(ξ0, t0)

∣∣∣∣det

(∂ξk

∂ξ0

)∣∣∣∣−1

, (4.37)

or

p(ξk, tk) = p[ξ0(tk, ξ; t0), t0]

∣∣∣∣det

(∂ξk

∂ξ0

)∣∣∣∣−1

, (4.38)

and gives an analytic solution of the recursive Bayesian prediction equation. Considering

the initial Gaussian boundary conditions, the state PDF at time tk is:5The higher order Bayesian filter can be generalized for a case where the mean is non-zero.

92

p(ξk, tk) = Cξkexp

−1

2ξ0(tk,ξk; t0)

TΛ0ξ0(tk,ξk; t0)

, (4.39)

= Cξkexp

−1

2

N∑

α,β=1

Λαβ0 ξα

0ξβ0

, (4.40)

where ξ0 = ξ0(tk,ξk; t0) denotes a function of ξk and

Cξk= Cξ0

∣∣∣∣det

(∂ξk

∂ξ0

)∣∣∣∣−1

. (4.41)

Now suppose a single (scalar) measurement zk is taken at tk. In this case, the recursive

Bayesian equation, i.e., Eqn. (4.29), becomes:

p(ξk|zk) =p(zk|ξk)p(ξk)∫

∞ p(zk|ξk)p(ξk)dξk

, (4.42)

where we drop the conditioning on z1:k−1. The PDF of the measurement conditioned on

the state in Eqn. (4.42) can be stated as:

p(zk|ξk) = Cz exp

−(zk −Hξk)

2

2σ2k

, (4.43)

where Czk= 1/

√2πσk. If p(zk|ξk) is assumed to be Gaussian, the numerator of the

posterior conditional PDF then becomes:

pN(zk,ξk) = p(zk|ξk)p(ξk), (4.44)

= CξkCzk

exp

−1

2

[ξT

0 (tk, ξk; t0)Λ0ξ0(tk, ξk; t0) +(zk −Hξk)

2

σ2k

].

(4.45)

Now consider the denominator of the Bayesian measurement update equation, pD =

p(zk|z1:k−1) = p(zk), which is the PDF of the measurement. Note that this is certainly

not Gaussian since:

E[zk] = E[Hξk + vk] = H1αE

[m∑

p=1

1

p!φ

α,γ1···γp

(tk,t0) ξγ1

0 · · · ξγp

0

], (4.46)

93

and obviously shows non-Gaussian structure unless m = 1. Following Remark 4.3.8, we

have:

pD(zk) =

∫

∞p(zk|ξk)p(ξk)dξk,

=

∫

∞Cξk

Czkexp

−1

2

[ξT

0 (tk, ξk; t0)Λ0ξ0(tk, ξk; t0) +(zk −Hξk)

2

σ2k

]dξk,

(4.47)

which is constant for the given measurement. However, we can map ξk back to the initial

epoch state and utilize the Gaussian form of the initial distribution to find its value. The

map (ξ0 → ξk) is bijective, and thus, by mapping it to the initial epoch we can write:

pD(zk)

=

∫

∞C exp

−1

2

(ξT

0 Λ0ξ0 +[zk −Hξk(tk; ξ0, t0)]

2

σ2k

)dξ0,

=

∫

∞Ce−

ξT0 Λ0ξ0

2 exp

−z2

k − 2zkHξk + ξTk HT Hξk

2σ2z

dξ0,

=

∫

∞Ce−

ξT0 Λ0ξ0

2 exp

− 1

2σ2k

(z2k − 2zk

N∑α=1

Hα1ξαk +

N∑

α,β=1

Hα1Hβ1ξαk ξβ

k

)

︸ ︷︷ ︸·

dξ0,

(4.48)

where C = Cξ0Czk

and ξk = ξk(tk; ξ0, t0) is a function of ξ0. Now, substitute Eqn. (4.33)

for ξk and decompose the second exponential function into linear and higher order terms

as:

· = − 1

2σ2k

[z2k − 2zk

N∑α=1

Hα1

(m∑

p=1

1

p!φ

α,···γ1···γp

(tk,t0) ξγ1

0 · · · ξγp

0

)

+N∑

α,β=1

Hα1Hβ1

(m∑

p=1

1

p!φ

α,···γ1···γp

(tk,t0) ξγ1

0 · · · ξγp

0

)(m∑

q=1

1

q!φ

β,ζ1···ζq

(tk,t0) ξζ10 · · · ξζq

0

)],

= − 1

2σ2k

(zk −HΦξ0)2 + u(ξ0), (4.49)

94

where Φ = Φ(tk, t0) is the usual state transition matrix and

u(ξ0) = − 1

2σ2k

(−2zk

N∑α=1

Hα1

m∑p=2

1

p!φ

α,γ1···γp

(tk,t0) ξγ1

0 · · · ξγp

0

)

− 1

2σ2k

N∑

α,β=1

Hα1Hβ1

m∑p,q=1p=q 6=1

1

p!q!φ

α,···γ1···γp

(tk,t0) φβ,ζ1···ζq

(tk,t0) ξγ1

0 · · · ξγp

0 ξζ10 · · · ξζq

0

.

(4.50)

By substituting this result into Eqn. (4.48), the measurement PDF becomes:

pD(zk) =

∫

∞C exp

−ξT

0 Λ0ξ0

2− (zk −HΦξ0)

2

2σ2k

exp u(ξ0) dξ0. (4.51)

Remark 4.3.9 (Matrix Algebra). See Appendix C for details.

[(y−Mx)T R−1(y−Mx) + (x−m)T S−1(x−m)

]

= yT R−1y + mT S−1m− aTΘ−1a + (x−Θ−1a)TΘ(x−Θ−1a), (4.52)

where

Θ = MT R−1M + S−1, (4.53)

a = MT R−1y + S−1m, (4.54)

Moreover, the determinants are related by:

(detΘ−1)1/2 =(det S)1/2(det R)1/2

[det(MSMT + R)

]1/2. (4.55)

Applying the results from Remark 4.3.9 gives:

pD(zk) =

∫C exp

−1

2

[z2k

HΦP0ΦT HT + σ2+

(ξ0 −Θ−1a

)TΘ

(ξ0 −Θ−1a

)]

× exp u(ξ0) dξ0, (4.56)

where

Θ =1

σ2k

ΦT HT HΦ + P−10 , (4.57)

a =1

σ2k

ΦT HT zk. (4.58)

95

Finally, applying the determinant identity gives the following expression for the measure-

ment PDF:

pD(zk) =

ρ(zk)

∫

∞

(detΘ)1/2 exp u(ξ0)(2π)N/2

exp

−1

2

(ξ0 −Θ−1a

)TΘ

(ξ0 −Θ−1a

)dξ0,

(4.59)

where ρ(zk) is a constant for the given measurement and is defined as:

ρ(zk) =1

(2π)1/2[det(HΦP0ΦT HT + σ2)

]1/2exp

−1

2

z2k(

HΦP0ΦT HT + σ2)

.

(4.60)

Note that if we consider a linear system, Eqn. (4.59) simply reduces to pD(zk) = ρ(zk)

since u(ξ0) = 0, and hence, the integral in Eqn. (4.59) becomes unity. In order to compute

its actual value, we can transform Eqn. (4.59) into an expectation form:

pD(zk) = ρ(zk) E[exp u(β0)], (4.61)

where u(ξ0) is replaced with u(β0) to denote that ξ0 now has an updated Gaussian prob-

ability density function of ξ0 = β0 ∼ N (Θ−1a,Θ−1). The mean and covariance matrix

of β0 can be re-written as:

Θ−1a = P0ΦT HT

[HΦP0Φ

T HT + σ2k

]−1zk, (4.62)

Θ−1 = P0 − P0ΦT HT

[HΦP0Φ

T HT + σ2k

]−1 HΦP0. (4.63)

Note that Eqns. (4.62) and (4.63) are essentially the update equations for a linear model,

where Θ−1a is the updated mean and Θ−1 is the updated covariance matrix. For compu-

tational purposes consider a Taylor series expansion of an exponential function:

eu =∞∑i=0

ui

i!= 1 + u1 +

u2

2!+

u3

3!+

u4

4!+

u5

5!+

u6

6!+

u7

7!+ · · · . (4.64)

96

We can apply this to calculate Eqn. (4.61) by considering a sufficient order of the solution.

That is, we can substitute u(β), defined in Eqn. (4.50), into Eqn. (4.64) and truncate the

series by fixing the degree of the higher order moments. Using this result, the posteriori

conditional PDF of the state can be approximated by computing:

p(ξk|zk) =p(zk|ξk)p(ξk)

ρ E[exp u(β0)], (4.65)

and its complete form is:

p(ξk|zk) =(det P0)

−1/2

ρσk

√(2π)N+1E[exp u(β0)]

∣∣∣∣det

(∂ξk

∂ξ0

)∣∣∣∣−1

× exp

−1

2

[ξT

0 (tk,ξk; t0)Λ0ξ0(tk,ξk; t0) +(zk −Hξk)

2

σ2k

]. (4.66)

If we consider a Hamiltonian system,6 the determinant of the Jacobian becomes unity, i.e.,

| det(∂ξk/∂ξ0)| = 1, and hence the posteriori conditional PDF simplifies to:

p(ξk|zk) =

(det P0)−1/2

ρσk

√(2π)N+1E[exp u(β0)]

exp

−1

2

[ξT

0 (ξk)Λ0ξ0(ξk) +(zk −Hξk)

2

σ2k

],

(4.67)

where ξ0(ξk) = ξ0(tk, ξk; t0) is to denote a function of ξk and can be computed analyti-

cally with an STT theory.

We are interested in computing the first and second central moments of the posterior

conditional density, i.e., the updated mean and updated covariance matrix, which are de-

fined as:

ξ+k = E[ξk|zk], (4.68)

P+k = E[(ξk − ξ+

k )(ξk − ξ+k )T |zk], (4.69)

where plus signs indicate updated values.6Assuming sufficiently high order solution is considered, the relative flow is almost symplectic.

97

First consider the updated mean equation. By directly applying expectation and map-

ping it back to the initial state, we find:7

(ξ+k )i =

∫

∞ξikp(ξk|zk)dξk,

=ρ(zk)

pD(zk)

∫

∞

(detΘ)1/2

(2π)N/2

(m∑

p=1

1

p!φi,γ1···γpξγ1

0 · · · ξγp

0

)exp u(ξ0)

× exp

−1

2

(ξ0 −Θ−1a

)TΘ

(ξ0 −Θ−1a

)dξ0,

=E[ξi

k(β0) expu(β0)]E[expu(β0)]

. (4.70)

Applying a similar method, the updated covariance matrix equation can be stated as:

(P+k )ij =

(∫

∞ξikξ

jkp(ξk|zk)dξk

)− (ξ+

k )i(ξ+k )j,

=ρ

pD

∫

∞

(m∑

p=1

1

p!φi,γ1···γpξγ1

0 · · · ξγp

0

)(m∑

p=1

1

q!φj,ζ1···ζpξζ1

0 · · · ξζq

0

)exp u(ξ0)

× (detΘ)1/2

(2π)N/2exp

−1

2

(ξ0 −Θ−1a

)TΘ

(ξ0 −Θ−1a

)dξ0 − (ξ+

k )i(ξ+k )j,

=E[ξi

k(β0)ξjk(β0) expu(β0)]

E[expu(β0)]− (ξ+

k )i(ξ+k )j, (4.71)

β0 ∼ N (Θ−1a,Θ−1). An ith order moment can be stated as:

E[ξγ1

k · · · ξγi

k |zk] =E[ξγ1

k (β0) · · · ξγi

k (β0) expu(β0)]E[expu(β0)]

, (4.72)

which is analytic in β0 and is completely described by a Gaussian distribution. Since

β0 is Gaussian with the mean Θ−1a the covariance matrix Θ−1, E[expu(β0)] can be

computed analytically after truncating the exponential series Eqn. (4.64) depending on the

desired accuracy of the solution.

We note that if we consider linear system dynamics, Eqns. (4.70) and (4.71) simplify

to the conventional Kalman filter algorithm for the initial state estimation:

7Note that this process is similar to the computation of the measurement PDF, pD = p(zk).

98

ξ+k = Θ−1a = P0Φ

T HT[HΦP0Φ

T HT + σ2k

]−1zk, (4.73)

P+k = Θ−1 = P0 − P0Φ

T HT[HΦP0Φ

T HT + σ2k

]−1 HΦP0. (4.74)

4.4 Implementation of a Nonlinear Filter

In §4.3 we have shown that, in theory, the higher order solutions of the relative dynam-

ics can be used to approximate a posterior conditional density function by using Bayes’

rule and we have derived an optimal nonlinear filter. This formulation, however, can be dif-

ficult to implement due to the series expansion of the exponential function in Eqn. (4.72),

which can be quite complicated. Hence, we present two sub-optimal nonlinear filters by

directly incorporating the higher order solutions into the Kalman filtering algorithm, which

are simpler and easier to implement than the Bayesian formulation.

Although the Kalman filter algorithm can be derived from Bayes’ rule of conditional

densities, as pointed out by Julier et al. [44], Kalman’s original derivation did not come

from the Bayesian approach [49], but rather from estimations of a few expectations in-

volving a state and a measurement. To show this, consider the following general system

model in discrete form:

xk+1 = φ(tk+1; xk, tk) + wk, (4.75)

zk+1 = h(xk+1, tk+1) + vk+1, (4.76)

where xk is the true spacecraft state, φ is the solution flow, wk is the white process noise

perturbing the spacecraft dynamics, zk is the actual measurement, h is the measurement

function, and vk is white measurement noise characterizing the observation error. The

process noise and measurement noise are assumed to be non-correlated, i.e., E[viwTj = 0],

99

with the autocorrelations:

E[wiwTj ] = Qiδij, (4.77)

E[vivTj ] = Riδij, (4.78)

for all discrete time indexes i and j, where δij represents the Dirac delta function. Here,

Qi and Ri are also known as the diffusion and measurement noise matrices, respectively.

Definition 4.4.1 (Kalman Filter Algorithm). Given the system model Eqns. (4.75) and

(4.76), suppose we are given a state xk with mean m+k = E[xk] and covariance matrix

P+k = E[(xk −m+

k )(xk −m+k )T ] at time tk. The Kalman algorithm is defined as follows:

Kalman Filter Prediction Equations:

m−k+1 = E[φ(tk+1; xk, tk) + wk], (4.79)

P−k+1 = E[φ(tk+1; xk, tk) + wk][φ(tk+1; xk, tk) + wk]

T

− (m−k+1)(m

−k+1)

T , (4.80)

n−k+1 = E[h(xk+1, tk+1) + vk+1], (4.81)

where n−k = E[hk] is the expectation of the measurement computed at tk.

Kalman Filter Update Equations:

Kk+1 = Pxzk+1(P

zzk+1)

−1, (4.82)

m+k+1 = m−

k+1 + Kk+1

(zk+1 − n−k+1

), (4.83)

P+k+1 = P−k+1 −Kk+1Pzz

k+1KTk+1, (4.84)

where Kk is known as the Kalman gain matrix, Pxzk is the cross-covariance matrix of the

state and the measurement, Pzzk is the covariance matrix of the measurement, zk is the

observation, and the difference between the actual and predicted measurement (i.e., zk −

n−k ) is called the residual or innovation.

100

4.4.1 Extended Kalman Filter

For estimation problems, the linear Kalman filter (LKF) is probably the most well

known filtering technique. The LKF allows one to compute the minimum mean-square-

error (MMSE) solution; however, it can only be used for linear systems, and in general,

cannot be used for trajectory navigation. In conventional spacecraft trajectory naviga-

tion, the extended Kalman filter (EKF) is usually implemented.8 The EKF is based on

the Kalman filter algorithm given in Eqns. (4.79-4.84), but assumes the true trajectory is

within the boundary where the linear approximation about a reference trajectory can suf-

ficiently model the trajectory dynamics and its statistics. Under this assumption, the mean

trajectory is propagated according to the deterministic solution flow and the covariance

matrix is linearly mapped assuming Gaussian statistics [2, 12, 100].

Definition 4.4.2 (Extended Kalman Filter Algorithm).

EKF Prediction Equations:

m−k+1 = φ(tk+1; m+

k , tk), (4.85)

P−k+1 = Φ(tk+1, tk)P+k ΦT (tk+1, tk) + Qk, (4.86)

n−k+1 = h(m−k+1, tk+1). (4.87)

EKF Update Equations:

Kk+1 = Pxzk+1(P

zzk+1)

−1,

= P−k+1HTk+1(Hk+1P−k+1HT

k+1 + Rk+1)−1, (4.88)

m+k+1 = m−

k+1 + Kk+1(zk+1 − n−k+1), (4.89)

P+k+1 = P−k+1 −Kk+1Pzz

k+1KTk+1,

= P−k+1 −Kk+1Hk+1P−k+1, (4.90)

8In practice, the extended Kalman filter is implemented for trajectory navigation often in square-rootinformation filter (SRIF) or in U-D filter formulation for numerical precision.

101

where h(m−k+1, tk+1) is the measurement function evaluated at tk+1 a function of m−

k+1 and

Hk = ∂hk/∂xk is the measurement partial computed at tk.

Among the many important properties of the extended Kalman filter, we point out two

which will be discussed in Chapter V in more detail. Considering the gain Eqn. (4.88) and

the mean update Eqn. (4.89), we observe that as the a priori covariance matrix becomes

more accurate (i.e., P−k+1 → 0) the filter values the residual less (i.e., the actual mea-

surement is trusted less). On the other hand, as the measurement becomes more accurate

(i.e., Rk+1 → 0) the filter values the residual more (i.e., the actual measurement is trusted

more). Therefore, optimally weighting the residual is a critical component of maximizing

the filter performance.

4.4.2 Higher-Order Numerical Extended Kalman Filter

In deriving the higher-order numerical extended Kalman filter (HNEKF), we assume

that the reference trajectory and its higher order state transition tensors are integrated for

each time interval between the measurements according to Eqns. (2.27-2.30). Under this

assumption the local trajectory motion can be mapped analytically over this time interval

while incorporating nonlinear effects, and the same analogy applies when mapping the

trajectory statistics. We note that this process is numerically quite intensive considering

higher order solutions; however, this can yield a more accurate filter solution.

Once the higher order state transition tensors are available for some time interval

[tk, tk+1], the mean and covariance matrix of the relative dynamics at tk can be mapped

analytically to tk+1 as functions of the probability distribution at tk as discussed in §3.9.

From tk to tk+1, the propagated mean and covariance can be stated as:

δmik+1(δxk) = E

[δxi

k+1

],

=m∑

p=1

1

p!φ

i,γ1···γp

(tk+1,tk)E[δxγ1

k · · · δxγp

k

], (4.91)

102

Pijk+1(δxk) = E

[(δxi

k+1 − δmik+1)(δxj

k+1 − δmjk+1)

],

=

(m∑

p=1

m∑q=1

1

p!q!φ

i,γ1···γp

(tk+1,tk)φj,ζ1···ζq

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

− δmik+1δmj

k+1, (4.92)

where γj, ζj ∈ 1, · · · , N. Now, the only unknowns in Eqns. (4.91-4.92) are the

expectations (i.e., moments) of the deviations. Even if the state at time tk is Gaussian,

except for the case m = 1, it is obvious that the mapped trajectory distribution is no longer

Gaussian due to system nonlinearity, and hence exact computation requires computation

of the higher order moments.

In particle-based filters, this problem is remedied by using an ensemble of sample

points to approximate the probability distribution, whereas a more formal approach is

to use the Edgeworth/Gram-Chalier [55] or Laplace approximations to approximate the

posterior density function. In trajectory navigation, however, the Gaussian assumption

has proven to provide a sufficiently accurate statistical approximation. Hence, we assume

that the updated estimates are Gaussian and we implement the joint characteristic function9

to compute the higher order moments up to 2mth-order as apparent from Eqn. (4.92). As

the order of the solution increases, i.e., m → ∞, the higher order solution yields the true

Monte-Carlo mean and covariance matrix as discussed in §3.10.

Now, suppose at time tk, the state estimate has mean m+k and covariance matrix P+

k .

Also, let x(tk) = m+k + δxk be the true trajectory we want to estimate. Following the

Kalman filter algorithm, the HNEKF algorithm is given as follows:

9By assuming the updated state can be approximated with Gaussian statistics, the higher order momentsare functions of the first two moments. If we consider a zero initial mean, all the odd moments of theinitial conditions vanish, which is the unique property of the Gaussian distribution, and the equations for thepropagated mean and covariance matrix simplify a great deal.

103

HNEKF Prediction Equations:

(m−k+1)

i = E[φi(tk+1; m+k + δxk, tk) + wi

k],

= φi(tk+1; m+k , tk) + δmi

k+1(δxk),

= φi(tk+1; m+k , tk) +

m∑p=1

1

p!φ

i,γ1···γp

(tk+1,tk)E[δxγ1

k · · · δxγp

k

], (4.93)

(P−k+1)ij = E

[φi(tk+1; m+

k + δxk, tk) + wik][φ

j(tk+1; m+k + δxk, tk) + wj

k]

− (m−k+1)

i(m−k+1)

j, (4.94)

=

(m∑

p=1

m∑q=1

1

p!q!φ

i,γ1···γp

(tk+1,tk)φj,ζ1···ζq

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

− δmik+1(δxk)δmj

k+1(δxk) + Qijk , (4.95)

(n−k+1)i = E[hi(tk+1; m+

k + δxk, tk) + vk+1],

= hi(tk+1; m+k , tk) + δni

k+1(δxk),

= hi(tk+1; m+k , tk) +

m∑p=1

1

p!hi,γ1···γp

(tk+1,tk)E[δxγ1

k · · · δxγp

k

], (4.96)

where the STTs (i.e., φi(tk+1; m+k , tk)) are computed along the solution flow φ(tk+1; m+

k , tk)

and

hi,γ1···γp

(tk+1,tk) =∂phi

k+1

∂(δxγ1

k ) · · · ∂(δxγp

k )

∣∣∣∣xk+1=φ(tk+1;m+

k ,tk)

. (4.97)

Note that hi(tk+1; m+k , tk) denotes that the measurement function is evaluated at tk+1 as

a function of the solution flow φ(tk+1; m+k , tk).10 The partial derivatives hi,γ1···γp

(tk+1,tk) up to

fourth order are defined as:

hi,a(tk+1,tk) = hi,α

k+1φα,ak+1, (4.98)

hi,ab(tk+1,tk) = hi,α

k+1φα,abk+1 + hi,αβ

k+1φα,ak+1φ

β,bk+1, (4.99)

hi,abc(tk+1,tk) = hi,α

k+1φα,abck+1 + hi,αβ

k+1

(φα,a

k+1φβ,bck+1 + φα,ab

k+1φβ,ck+1 + φα,ac

k+1φβ,bk+1

)

+ hi,αβγk+1 φα,a

k+1φβ,bk+1φ

γ,ck+1, (4.100)

10It is important to note that h(tk+1; m+k , tk) 6= h(m−

k+1, tk+1) in general since m−k+1 6= φ(tk+1; m+

k , tk)for general nonlinear systems.

104

hi,abcd(tk+1,tk) = hi,α

k+1φα,abcdk+1 + hi,αβ

k+1

(φα,abc

k+1 φβ,dk+1 + φα,abd

k+1 φβ,ck+1 + φα,acd

k+1 φβ,bk+1 + φα,ab

k+1φβ,cdk+1

+ φα,ack+1φ

β,bdk+1 + φα,ad

k+1φβ,bck+1 + φα,a

k+1φβ,bcdk+1

)+ hi,αβγ

k+1

(φα,ab

k+1φβ,ck+1φ

γ,dk+1

+ φα,ack+1φ

β,bk+1φ

γ,dk+1 + φα,ad

k+1φβ,bk+1φ

γ,ck+1 + φα,a

k+1φβ,bck+1φ

γ,dk+1 + φα,a

k+1φβ,bdk+1φ

γ,ck+1

+ φα,ak+1φ

β,bk+1φ

γ,cdk+1 ) + hi,αβγδ

k+1 φα,ak+1φ

β,bk+1φ

γ,ck+1φ

δ,dk+1, (4.101)

where φk+1 = φ(tk+1,tk) is used for a concise notation and that these are similar to the

differential equations of the STTs given in Eqns. (2.27-2.30). Note that this prediction

step is a simple algebraic operation once the STTs are computed for the time interval

[tk, tk+1].

HNEKF Update Equations:

(Pzzk+1)

ij = E[(z−k+1 − n−k+1)(z

−k+1 − n−k+1)

T]ij

,

= E[(z−k+1)

i(z−k+1)j]− (n−k+1)

i(n−k+1)j,

= E[hi(tk+1; m+

k + δxk, tk) + vik+1][h

j(tk+1; m+k + δxk, tk) + vj

k+1]

− (n−k+1)i(n−k+1)

j,

=

(Rij

k+1 +m∑

p=1

m∑q=1

1

p!q!hi,γ1···γp

(tk+1,tk)hj,ζ1···ζq

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

− (δn−k+1)i(δn−k+1)

j, (4.102)

(Pxzk+1)

ij = E[(x−k+1 −m−

k+1)(z−k+1 − n−k+1)

T]ij

,

= E[(x−k+1)

i(z−k+1)j]− (m−

k+1)i(n−k+1)

j,

= E[φi(tk+1; m+

k + δxk, tk) + wik][h

j(tk+1; m+k + δxk, tk) + vj

k+1]

− (m−k+1)

i(n−k+1)j,

=

(m∑

p=1

m∑q=1

1

p!q!φ

i,ζ1···ζq

(tk+1,tk)hj,γ1···γp

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

− (δm−k+1)

i(δn−k+1)j, (4.103)

105

Kk+1 = Pxzk+1(P

zzk+1)

−1, (4.104)

m+k+1 = m−

k+1 + Kk+1(zk+1 − n−k+1), (4.105)

P+k+1 = P−k+1 −Kk+1Pzz

k+1KTk+1. (4.106)

Note that if we consider the measurement function Eqn. (4.76) to be linear in xk, Eqn.

(4.96) simplifies to:

(n−k+1)i = hi(tk+1; m+

k , tk) + (δn−k+1)i,

= hi(tk+1; m+k , tk) + hi,α

k+1

m∑p=1

1

p!φ

α,γ1···γp

(tk+1,tk)E[δxγ1

k · · · δxγp

k

],

= hi(tk+1; m+k , tk) + hi,α

k+1(δm−k+1)

α,

= hi(m−k+1, tk+1), (4.107)

and gives (δn−k+1)i = hi,α

k+1(δm−k+1)

α. Applying this result, Eqns. (4.102) and (4.103)

simplify to:

(Pzzk+1)

ij =

(hi,α

k+1hj,βk+1

m∑p=1

m∑q=1

1

p!q!φ

α,γ1···γp

(tk+1,tk)φβ,ζ1···ζq

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

+ Rijk+1 − (δn−k+1)

i(δn−k+1)j,

=(

Rijk+1 + hi,α

k+1hj,βk+1E[δxα

k+1δxβk+1]

)− (δn−k+1)

i(δn−k+1)j,

= (Hk+1P−k+1HTk+1 + Rk+1)

ij, (4.108)

(Pxzk+1)

ij =

(hj,α

k+1

m∑p=1

m∑q=1

1

p!q!φ

i,γ1···γp

(tk+1,tk)φα,ζ1···ζq

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

− (δm−k+1)

i(δn−k+1)j,

= E[δxik+1δxα

k+1]hj,αk+1 − (δm−

k+1)i(δn−k+1)

j,

= (P−k+1HTk+1)

ij, (4.109)

which indicates that the measurement prediction and update equations are identical to the

EKF algorithm.11 Moreover, for a linear measurement function, it is apparent that we11Note that when m = 1, the HNEKF becomes the EKF algorithm as shown in Eqns. (4.85-4.90).

106

can implement the Potter’s algorithm to develop a square-root filter for numerical stability

[12, 56]. This, however, is not obvious if we consider a nonlinear measurement function as

the Potter’s algorithm depends on there being a Cholesky decomposition, or alike, of the

covariance matrix, which may not be true for the higher order tensors. An extension to a

square-root filter, if possible, is not considered in this thesis since the focus our study is to

establish a general filter setup which incorporates the higher order dynamics and statistics.

4.4.3 Higher-Order Analytic Extended Kalman Filter

From the derivation of the HNEKF, it is obvious that we can also derive a higher-order

analytic extended Kalman filter (HAEKF) by assuming that the reference trajectory and

the higher order solutions (i.e., STTs) are computed over some time span prior to filtering.

The filter algorithm is similar to the HNEKF except that the point of series expansion is

now with respect to the initial reference trajectory, not the updated mean as in the HNEKF

algorithm.

Suppose the STTs are computed for the time interval of [t0, tf ] and let xk = φ(tk; x0, t0)

represent the reference trajectory for tk ∈ [t0, tf ], where x0 has mean m+0 and covariance

matrix P+0 . Moreover, let x(tk) = xk + δxk be the true trajectory we want to estimate.

Following the Kalman filter algorithm, the HAEKF algorithm is given as follows:

HAEKF Prediction Equations:

(m−k+1)

i = E[φi(tk+1; xk + δxk, tk) + wik],

= φi(tk+1; xk, tk) + δmik+1(δxk),

= φi(tk+1; xk, tk) +m∑

p=1

1

p!φ

i,γ1···γp

(tk+1,tk)E[δxγ1

k · · · δxγp

k

], (4.110)

107

(P−k+1)ij = E

[φi(tk+1; xk + δxk, tk) + wi

k][φj(tk+1; xk + δxk, tk) + wj

k]

− (m−k+1)

i(m−k+1)

j, (4.111)

=

(m∑

p=1

m∑q=1

1

p!q!φ

i,γ1···γp

(tk+1,tk)φj,ζ1···ζq

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

− δmik+1(δxk)δmj

k+1(δxk) + Qijk , (4.112)

(n−k+1)i = E[hi(tk+1; xk + δxk, tk) + vi

k+1],

= hi(tk+1; xk, tk) + δnik+1(δxk),

= hi(xk+1, tk+1) +m∑

p=1

1

p!hi,γ1···γp

(tk+1,tk)E[δxγ1

k · · · δxγp

k

], (4.113)

where the STTs are computed along xk+1 = φ(tk+1; xk, tk) and

hi,γ1···γp

(tk+1,tk) =∂phi

k+1

∂(δxγ1

k ) · · · ∂(δxγp

k )

∣∣∣∣xk+1=xk+1

. (4.114)

HAEKF Update Equations:

(Pzzk+1)

ij = E[(z−k+1 − n−k+1)(z

−k+1 − n−k+1)

T]ij

,

= E[(z−k+1)

i(z−k+1)j]− (n−k+1)

i(n−k+1)j,

= E[hi(tk+1; xk + δxk, tk) + vi

k+1][hj(tk+1; xk + δxk, tk) + vj

k+1]

− (n−k+1)i(n−k+1)

j,

=

(Rij

k+1 +m∑

p=1

m∑q=1

1

p!q!hi,γ1···γp

(tk+1,tk)hj,ζ1···ζq

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

− (δn−k+1)i(δn−k+1)

j, (4.115)

(Pxzk+1)

ij = E[(x−k+1 −m−

k+1)(z−k+1 − n−k+1)

T]ij

,

= E[(x−k+1)

i(z−k+1)j]− (m−

k+1)i(n−k+1)

j,

= E[φi(tk+1; xk + δxk, tk) + wi

k][hj(tk+1; xk + δxk, tk) + vj

k+1]

− (m−k+1)

i(n−k+1)j,

=

(m∑

p=1

m∑q=1

1

p!q!φ

i,ζ1···ζq

(tk+1,tk)hj,γ1···γp

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

− (δm−k+1)

i(δn−k+1)j, (4.116)

108

Kk+1 = Pxzk+1(P

zzk+1)

−1, (4.117)

m+k+1 = m−

k+1 + Kk+1(zk+1 − n−k+1), (4.118)

P+k+1 = P−k+1 −Kk+1Pzz

k+1KTk+1. (4.119)

As in the HNEKF case, the update equations for the HAEKF becomes the same as the

EKF when we consider a measurement function that is linear in xk:

(n−k+1)i = hi(xk+1, tk+1) + (δn−k+1)

i,

= hi(xk+1, tk+1) + hi,αk+1

m∑p=1

1

p!φ

α,γ1···γp

(tk+1,tk)E[δxγ1

k · · · δxγp

k

],

= hi(xk+1, tk+1) + hi,αk+1(δm−

k+1)α,

= hi(m−k+1, tk+1), (4.120)

(Pzzk+1)

ij =

(hi,α

k+1hj,βk+1

m∑p=1

m∑q=1

1

p!q!φ

α,γ1···γp

(tk+1,tk)φβ,ζ1···ζq

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

+ Rijk+1 − (δn−k+1)

i(δn−k+1)j,

=(

Rijk+1 + hi,α

k+1hj,βk+1E[δxα

k+1δxβk+1]

)− (δn−k+1)

i(δn−k+1)j,

= (Hk+1P−k+1HTk+1 + Rk+1)

ij, (4.121)

(Pxzk+1)

ij =

(hj,α

k+1

m∑p=1

m∑q=1

1

p!q!φ

i,γ1···γp

(tk+1,tk)φα,ζ1···ζq

(tk+1,tk)E[δxγ1

k · · · δxγp

k δxζ1k · · · δxζq

k ]

)

− (δm−k+1)

i(δn−k+1)j,

= E[δxik+1δxα

k+1]hj,αk+1 − (δm−

k+1)i(δn−k+1)

j,

= (P−k+1HTk+1)

ij. (4.122)

Also, note that when m = 1 (i.e., first order), the HAEKF becomes the linear Kalman

filter (LKF), not the EKF.12 The superiority of the EKF over the LKF is clearly demon-

strated in Maybeck [55]. However, when the true trajectory is within the convergence

radius of the reference trajectory, we shall see later that the HAEKF can provide a more12We call it the higher-order analytic extended Kalman filter, not higher-order linear Kalman filter, since

the prediction equations are nonlinear in general.

109

accurate solution and faster convergence than the EKF.

4.4.4 Unscented Kalman Filter

The unscented Kalman filter (UKF), first introduced by Julier and Uhlmann [44, 46,

98], has been implemented in diverse fields of engineering, science, and economics due to

its simplicity while providing faster convergence and better accuracy than the EKF. The

first implementation of the UKF to a trajectory navigation problem was discussed in Refs.

[39, 40], where Mars aerobraking spacecraft state is estimated using inertial measurement

unit data. The UKF is based on the unscented transformation discussed in §3.11, which

deterministically chooses the sample points to approximate the probability distribution

while keeping the computational cost at the order of the linear methods. The UKF provides

a good approximation of the true probability distribution and lower expected errors than

the EKF, and it does not require calculation of the Jacobian matrix. However, it depends

on how the initial sample points are chosen and parameterized, and generally captures only

the first three moments of an arbitrary distribution.13

Here, we do not go through the detailed derivation, as thorough discussions can be

found in Refs. [43, 44, 45, 46, 97]. We only present the UKF algorithm for additive

(linear) process and measurement noises with zero mean.

The UKF is initialized with the following deterministically chosen sample points:

X 0k = m+

k , (4.123)

W0k = α/(N + α), (4.124)

X ik = m+

k +[√

(N + α)Pk

]i

, (4.125)

W ik = 1/[2(N + α)], (4.126)

X i+Nk = m+

k −[√

(N + α)Pk

]i

, (4.127)

13In case of a Gaussian distribution, UKF captures the first four moments.

110

W i+Nk = 1/[2(N + α)], (4.128)

where α ∈ <,X jk are the sample points with associated weightsWj

k , and[√

(N + α)P(t0)]i

is the ith row of the matrix square root of [(N + α)P(t0)]. With this initialization, the UKF

algorithm is given as follows:

UKF Prediction Equations:

X ik+1 = φ(tk+1;X i

k, tk), (4.129)

m−k+1 =

2N∑i=0

W ikX i

k+1, (4.130)

P−k+1 =2N∑i=0

W ik

[X ik+1 −m−

k+1

] [X ik+1 −m−

k+1

]T+ Qk, (4.131)

Z ik+1 = h(X i

k+1), (4.132)

n−k+1 =2N∑i=0

W ikZ i

k+1. (4.133)

UKF Update Equations:

Pzzk+1 =

2N∑i=0

W ik

[Z ik+1 − n−k+1

] [Z ik+1 − n−k+1

]T+ Rk, (4.134)

Pxzk+1 =

2N∑i=0

W ik

[X ik+1 −m−

k+1

] [Z ik+1 − n−k+1

]T, (4.135)

Kk+1 = Pxzk+1(P

zzk+1)

−1, (4.136)

m+k+1 = m−

k+1 + Kk+1

(zk+1 − n−k+1

), (4.137)

P+k+1 = P−k+1 −Kk+1Pzz

k+1KTk+1. (4.138)

CHAPTER V

NONLINEAR SPACE MISSION ANALYSIS

5.1 Motivation

In this chapter, we apply the theoretical results derived previously and present exam-

ples and simulations of space mission problems. In these examples we implement our

nonlinear navigation results, the higher order solutions of relative dynamics to trajectory

and uncertainty propagations, statistical targeting, and higher order filtering.

The first example considered is an orbit about Europa in a Hill three-body formulation

(in the Jupiter-Europa system). Europa is one of the Jovian satellites, and currently, there is

a high interest in Europa by the science community as it is believed to have a vast reservoir

of water beneath its surface. This problem was originally motivated by the Jupiter Icy

Moon Orbiter (JIMO) mission (now the Europa Orbiter mission) which planned to study

the Jovian satellites Europa, Ganymede, and Callisto. We have specifically analyzed orbits

about Europa since its dynamical environment is more nonlinear than the other Jovian

satellites. The second problem is a Hohmann transfer orbit from the Earth to the Moon,

which is the simplest realization of an interplanetary trajectory. For example, considering

the Earth as the only gravitating body with uniform gravitational field, a cruise to the

Moon can be viewed as a Hohmann transfer orbit. The last problem is a halo orbit in the

Sun-Earth and Earth-Moon systems. A halo orbit is a periodic orbit found about Lagrange

111

112

points, and in the past, there have been a number of missions placed on halo orbits, e.g.,

the ISEE-3/ICE, SOHO, and Genesis missions. A halo orbit is a fundamentally fascinating

trajectory as it balances the perturbing forces of two massive bodies while preserving its

periodicity. Also, the halo orbit requires relatively little maintenance and has many useful

applications, such as solar activity monitoring station [27], a communication relay [24],

and a trajectory transit point [50, 52].

5.2 Nonlinear Propagation of Phase Volume

To show the effect of nonlinearity on a solution flow, this section presents a comparison

of linearly and nonlinearly propagated phase volumes.1 Consider the planar Hill three-

body problem applied to the Jupiter-Europa system in Lagrangian form (see Appendix A

for the governing equations of motion and required constants). Suppose a spacecraft is

initially located slightly below the L2 point with the initial conditions:

x(t0) = [ x(t0), y(t0), u(t0), v(t0) ]T ,

r(t0) = [ x(t0), y(t0) ]T = [ 13581.17, −1321.85 ]T km,

v(t0) = [ u(t0), v(t0) ]T = [ −44.56297, 12.84600 ]T m/s.

Figure 5.1 shows the position and velocity flows of this initial state plotted over tf = 12

units of time (∼ 6.775 days). This is a ‘safe’ trajectory since it does not collide with

Europa or escape from the system [80]. Such a trajectory is similar to a capture trajectory

at Europa. The boundary line of the shaded area in Figure 5.1(a) represents the zero-

velocity curve2 with Jacobi constant, J = −2.15. Figure 5.2(a) shows the distance (||r||)

and speed (||v||) of the spacecraft as functions of time. The circled points are computed at

t ∈ 0, 0.881, 2.26, 4.42, 5.38, 5.74 days and will be considered later when we compare

1Note that phase volumes are propagated deterministically assuming the initial state is perfectly known.2The zero-velocity curve is a boundary where a spacecraft with a given energy cannot cross.

113

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 104

−1.5

−1

−0.5

0

0.5

1

1.5

x 104

x−coordinate (km)

y−co

ordi

nate

(km

)

L2

(a) Position plane.

−2000 −1500 −1000 −500 0 500 1000 1500

−1500

−1000

−500

0

500

1000

1500

u−coordinate (m/s)

v−co

ordi

nate

(m

/s)

(b) Velocity plane.

Figure 5.1: Hill three-body trajectory plot at Europa for ∼ 6.775 days: circled points arecomputed at t ∈ 0, 0.881, 2.26, 4.42, 5.38, 5.74 days.

114

the propagated phase volumes. We consider this trajectory to be the reference trajectory

and analyze its neighboring phase volume with the STTs.

In order to show the effect of the higher order STTs, first consider the following initial

deviation in x0:

δx0 = [ 0 km, 10 km, 0 m/s, 0.1 m/s ]T .

The STTs, up to fourth order, are integrated with respect to the nominal flow φ(t; x0, t0)

over the time interval [t0, tf ]. Figure 5.2(b) shows the 1st-4th order STT effects as func-

tions of time. That is, the norm of each order of STT solutions:

1st order effect = φi,γ1δxγ1

0 ,

2nd order effect =1

2φi,γ1γ2δxγ1

0 δxγ2

0 ,

3rd order effect =1

6φi,γ1γ2γ3δxγ1

0 δxγ2

0 δxγ3

0 ,

4th order effect =1

24φi,γ1γ2γ3γ4δxγ1

0 δxγ2

0 δxγ3

0 δxγ4

0 ,

so that the sums represent the higher order contributions to the relative solution δx. It

shows that the higher order terms become larger than the lower order terms shortly after

6 days, which indicates that the higher order solutions diverge. This divergence of the

higher order solutions can be remedied by segmenting the orbit into piecewise trajectories;

however, that is not considered in this example.

Now consider an initial uncertainty distribution δx0 ∼ N (0, P0), where

P0 = diag[ (10 km)2, (10 km)2, (0.1 m/s)2, (0.1 m/s)2 ].

With this initial covariance matrix, we define the initial phase volume with respect to the

nominal initial state x0 as a 1-σ error ellipsoid:

B0 = δx0 | δxT0 P−1

0 δx0 ≤ 1,

115

0 1 2 3 4 5 62000

4000

6000

8000

10000

12000

||rre

f|| (k

m)

Time (days)

0 1 2 3 4 5 6

500

1000

1500

||vre

f|| (m

/s)

Time (days)

(a) Distance and speed of the spacecraft plotted as functions of time.

0 1 2 3 4 5 610

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

102

Time (days)

Hig

her

orde

r ef

fect

1st order effect2nd order effect3rd order effect4th order effect

(b) Effect of higher order solutions

Figure 5.2: Trajectory norms and higher order solution magnitudes: circled points arecomputed at t ∈ 0, 0.881, 2.26, 4.42, 5.38, 5.74 days.

116

or

δx20

(10 km)2+

δy20

(10 km)2+

δu20

(0.1 m/s)2+

δv20

(0.1 m/s)2≤ 1, (5.1)

which is a four dimensional hyper-ellipsoid. From several simulations based on integrated

nonlinear trajectories, we find that the position errors in P0 dominate the outer boundary

solutions of the future phase volume in this example. For this reason, we analyze the flow

of a string from the 4-dimensional object such that:

δx20

(10 km)2+

δy20

(10 km)2= 1,

which represents a boundary solution of B0 by simply changing the inequality sign in Eqn.

(5.1) with an equality sign and by assuming δv(t0) = 0, i.e., δu0 = δv0 = 0. In other

words, we study the evolution of a circular cross-section of the initial phase volume. The

projections onto the position and velocity planes are shown in Figures 5.3 and 5.4 for the

integrated3 and 1st-3rd order analytic solutions.4 We note that this is essentially the flow

of a string on the phase volume boundary. Several important facts can be observed from

these plots. First is that the convergence radius of the STTs varies with time. Specifically,

the convergence radius varies with the location of reference solution in phase space. As

an example, consider Figure 5.4(a). The linear phase curve solution computed at t = 4.42

days is closer to the true solution than the phase curve computed at t = 0.881 days (at

periapsis). One explanation for this is due to the difference in the higher order effects as

shown in Figure 5.2(b). Another observation is that the third order solutions are almost

identical to the true solution.

Another interesting observation is that the phase curve gets twisted along the trajectory.

Figures 5.3(b) and 5.4(b) show the twisting motion of the phase curve, which is shown by3Numerically integrated according to the nonlinear governing equations of motion.4Note that the 2nd and 3rd order solutions are sometimes overlapped with the integrated solution, which

indicates that they are good approximations of the true dynamics.

117

−10 −5 0 5 10−10

0

10

y−co

ordi

nate

(km

)

x−coordinate (km)

t =0 days

−200 0 200 400−1000

0

1000

y−co

ordi

nate

(km

)

x−coordinate (km)

t =0.881 days

−100 0 100−400−200

0200400

y−co

ordi

nate

(km

)

x−coordinate (km)

t =2.26 days

−200 0 200

−20

0

20

y−co

ordi

nate

(km

)

x−coordinate (km)

t =4.42 days

−1000 −500 0 500−1000

−500

0

500

y−co

ordi

nate

(km

)

x−coordinate (km)

t =5.38 days

−500 0 500 1000

−1000

100200

y−co

ordi

nate

(km

)

x−coordinate (km)

t =5.74 days

(a) Phase volume projected onto the position plane.

−200 0 200−50

0

50

y− c

oord

inat

e (k

m)

x− coordinate (km)

t =4.404 days

−200 0 200−40−20

02040

y− c

oord

inat

e (k

m)

x− coordinate (km)

t =4.409 days

−200 0 200

−200

2040

y− c

oord

inat

e (k

m)

x− coordinate (km)

t =4.415 days

−200 0 200

−20

0

20

y− c

oord

inat

e (k

m)

x− coordinate (km)

t =4.421days

−200 0 200

−20

0

20

y− c

oord

inat

e (k

m)

x− coordinate (km)

t =4.426 days

−200 0 200−40−20

02040

y− c

oord

inat

e (k

m)

x− coordinate (km)

t =4.432 days

(b) Phase volume projected onto the position plane with smaller time incre-ments.

Figure 5.3: Phase volume projections: ‘solid’ line represents integrated, ‘dotted’ line rep-resents the 1st order, ‘dash-dot’ line represents the 2nd order, and ‘dashed’ line representsthe 3rd order solutions.

118

−1 −0.5 0 0.5 1−1

0

1

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =0 days

−500 0 500

−1000

100200300

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =0.881 days

−20 0 20

−101

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =2.26 days

−5 0 5

−20

0

20

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =4.42 days

−100 0 100 200−80−60−40−20

02040

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =5.38 days

−10 −5 0 5

−50

0

50

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =5.74 days

(a) Phase volume projected onto the velocity plane

−2 0 2

−20

0

20

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =4.438 days

−1 0 1 2

−20

0

20

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =4.443 days

−0.5 0 0.5 1 1.5

−20

0

20

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =4.449 days

−2 −1 0 1 2

−20

0

20

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =4.455 days

−2 0 2 4

−20

0

20

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =4.46 days

−4 −2 0 2 4 6

−20

0

20

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =4.466 days

(b) Phase volume projected onto the velocity plane with smaller time incre-ments.

Figure 5.4: Phase volume projections: ‘solid’ line represents integrated, ‘dotted’ line rep-resents the 1st order, ‘dash-dot’ line represents the 2nd order, and ‘dashed’ line representsthe 3rd order solutions.

119

−10 −5 0 5 10−10

−5

0

5

10

y− c

oord

inat

e (k

m)

x− coordinate (km)

t =0 days

−100 0 100

−400

−200

0

200

400

y− c

oord

inat

e (k

m)

x− coordinate (km)

t =2.26 days

−200 0 200

−20

−10

0

10

20

30

y− c

oord

inat

e (k

m)

x− coordinate (km)

t =4.42 days

−500 0 500 1000

−100

0

100

200

y− c

oord

inat

e (k

m)

x− coordinate (km)

t =5.74 days

(a) Phase volume projected onto the position plane for different initial outerboundaries using the 3rd order solution.

−0.05 0 0.05

−0.05

0

0.05

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =0 days

−20 0 20

−1

0

1

v−co

ordi

nate

(m

/s)

t =2.258 days

u−coordinate (m/s)

−5 0 5−30

−20

−10

0

10

20

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =4.421 days

−10 −5 0 5

−50

0

50

v−co

ordi

nate

(m

/s)

u−coordinate (m/s)

t =5.742 days

(b) Phase volume projected onto the velocity plane for different initial outerboundaries using the 3rd order solution.

Figure 5.5: Phase volume projections.

120

the crossing motion of the phase curve. This is interesting since the linear solution only

predicts whether the phase curve gets stretched or contracted, but cannot lead to such a

trajectory. The points cross each other when projected to the position and velocity planes;

however, they are not intersecting in the actual 4-dimensional space. This is similar to a

twisted ball projected onto a plane.

Figures 5.5(a) and 5.5(b) show the projection of the phase volume onto the position and

velocity planes using the third order STT solutions while varying the initial conditions.

The initial conditions for the contour plots are computed by solving Eqn. (5.1) while

varying ‖δr0‖ =√

δx20 + δy2

0 from 0 km to 10 km with 0.2 km increment. The outer

boundary solutions are the same as in Figures 5.3(a) and 5.4(a) (except for the first plot

in Figure 5.5(b)), where ‖δr0‖ = 10 km resulting δu0 = δv0 = 0 m/s and the inner

most solutions correspond to ‖δr0‖ = 0 km resulting ‖δv0‖ =√

δu20 + δv2

0 = 0.1 m/s. In

addition to the twisting motion of the phase volume, the non-concentric distribution shown

in the contour plot is an interesting result.

5.3 Nonlinear Orbit Uncertainty Propagation

In this section, we present the simulations of the nonlinear orbit uncertainty propa-

gation discussed in §3.9, where we compare linear and nonlinear approaches. The first

example is based on the planar two-body problem where a spacecraft is on the Earth-

to-Moon Hohmann transfer. The second example is based on the planar Hill three-body

problem in Jupiter-Europa system.

5.3.1 Two-Body Problem: Earth-to-Moon Hohmann Transfer

Figure 5.6 shows a Hohmann transfer from near Earth (20, 000 km) to the Moon

(384, 400 km). The reference trajectory is propagated for the transfer period (∼ 5.24

days) and the initial statistics are assumed to be Gaussian with a zero mean and position

121

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

−1

−0.5

0

0.5

1

1.5

2

x 105

x−coordinate (km)

y−co

ordi

nate

(km

)

Earthθ

t0

tf

Figure 5.6: Two-body problem: Hohmann transfer trajectory.

3.5 3.6 3.7 3.8 3.9 4 4.1 4.2

x 105

−3

−2

−1

0

1

2

3

x 104

x−coordinate (km)

y−co

ordi

nate

(km

)

m=1m=2m=3m=4Monte−Carlo

Linearly Propagated

MeanNonlinearly Propagated

Means

~2400 km

Figure 5.7: Two-body problem: comparison of the computed mean and covariance atapoapsis using STT-approach and Monte-Carlo simulations.

122

Table 5.1: Local nonlinearity index.

ηm Two-Body Hill Three-Bodyη1 1.06 3.57η2 0.04 0.29η3 0.007 0.28η4 0.001 0.06

uncertainty of 100 km and velocity uncertainty of 0.1 m/s:

P0 = diag[ (100 km)2, (100 km)2, (0.1 m/s)2, (0.1 m/s)2 ].

With this initial distribution, the initial confidence region is defined with respect to the

initial state as:

B0 = δx0 | δxT0 P−1

0 δx0 ≤ 1.

To show the effect of nonlinearity on the relative motion, Table 5.1 shows the local non-

linearity index computed at the apoapsis. When computing ηm, eight sample points cor-

responding to the eigenvectors of the initial error ellipsoid are considered. As predicted

by the local nonlinearity index, for this Hohmann transfer example, the higher order series

are convergent and the second order solution (i.e., η2) provides results superior to the first

order solution (i.e., η1).

The initial uncertainties are then propagated to the apoapsis using the higher order

STT approach and the Monte-Carlo simulation. Figure 5.7 shows the propagated mean

and 1-σ error ellipsoid plotted with respect to the target state (i.e, apoapsis) where the

Monte-Carlo result is based on an ensemble of 106 sample points. The second and higher

order solutions are overlapped with the Monte-Carlo solution, which indicates that they

provide far more accurate estimates of the mean and the dispersion of the samples (i.e.,

projection of the covariance matrix) than the linear (m = 1) solution. Also, note that the

123

linear method predicts that the mean is located at the mode of the propagated distribution

(or the deterministic solution flow) whereas the true mean is deviated from the mode by ∼

2400 km. This shows that incorporating the system nonlinearity provides a more accurate

description of the propagated probability distribution.

5.3.2 Hill Three-Body Problem: about Europa

Consider the same planar Hill three-body problem discussed in §5.2. The reference

trajectory in non-dimensional coordinates is shown in Figure 5.8, where the final position

is located at a periapsis. The initial conditions used are:

r(t0) = [ 0.69010031015662 −0.06716709529872 ],

v(t0) = [ −0.11045639526249 0.03184084790390 ],

which are the same as in §5.2, but given in non-dimensional units. The initial state is

assumed to be Gaussian with zero mean and a diagonal covariance matrix with position

error of 10 km (5.1× 10−4 in normalized units) and velocity error of 0.1 m/s (2.5× 10−4

in normalized units):

P0 = diag[ (10 km)2, (10 km)2, (0.1 m/s)2, (0.1 m/s)2 ],

which defines the initial confidence region as B0 = δx0 | δxT0 P−1

0 δx0 ≤ 1. The local non-

linearity indices for this problem are computed at periapsis and are given in Table 5.1. The

result shows that the fourth order solution provides accuracy better than 10 percent error.

Note that the level of accuracy can be improved when a different final time is considered

since the strongest nonlinearity is at periapsis.

As in the two-body problem case, the initial uncertainties are propagated using the

higher order STT approach and a Monte-Carlo simulation based on an ensemble of 106

initial samples. Figure 5.9 shows the propagated mean and 1-σ covariance matrix plotted

124

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−0.2

−0.1

0

0.1

0.2

0.3

x−coordinate

y−co

ordi

nate

Europa

t0

tf

Figure 5.8: Hill three-body problem: a safe trajectory at Europa.

−0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

x−coordinate

y−co

ordi

nate

m=1m=2m=3m=4Monte−Carlo

LinearlyPropagated

Mean

NonlinearlyPropagated

Means

~0.0081

Figure 5.9: Hill three-body problem: comparison of the computed mean and covariance atperiapsis using STT-approach and Monte-Carlo simulations.

125

with respect to the periapsis. The result shows that the linear (first order) solution captures

the semi-major axis of the true covariance projection (i.e., Monte-Carlo solution) and the

semi-minor axis depends on the deviated mean. Also, the linearly propagated mean (i.e.,

mode) is offset from the mean of the true probability distribution by 0.0081 ≈ 160 km

and lies inside the true covariance projection. As a result, it is clear that the higher order

solution provides a superior result as compared to the linear case. We can see this from

the location of the deviated mean and the covariance projection of the Monte-Carlo result.

This is important since a more conservative and statistically more accurate error bound

can be computed. Note that, assuming the STT series are convergent, we can analytically

propagate different values of the initial mean and covariance matrix without any additional

numerical integrations.

5.4 Nonlinear Statistical Targeting

In this section, we present simulations of the nonlinear statistical targeting discussed in

§4.2 where we consider the two-body Hohmann transfer and the Hill three-body problem

discussed in the previous section, §5.3.

5.4.1 Two-Body Problem: Earth-to-Moon Hohmann Transfer

In the two-body case, we assume that the 3rd order solution is considered to be the

truth model (i.e., solution based on the 3rd order STT series) as the local nonlinearity

index predicts that it captures most nonlinear effects. Thus, the ∆Vk computed using the

STT-based nonlinear statistical targeting method satisfies the statistically correct trajectory

concept. To simulate nonlinear statistical targeting, we first solve for the 3rd order solution

of the entire Hohmann transfer (t0 to tf ) and compute the deviated mean for the entire

trajectory. Note that the deviated mean is the difference between the mean computed

using the 3rd order solution and the reference trajectory. We then re-solve for the 3rd order

126

0 20 40 60 80 100 120 140 160 18010

−3

10−2

10−1

100

101

102

103

θ (degrees)

||∆V

k|| (

m/s

)

LinearNonlinear

Figure 5.10: Two-body problem: computed ∆Vk using the linear and nonlinear methods.

solutions from the deviated mean computed at tk to tf (tk ≥ t0) and find the correction ∆Vk

using Newton’s method as discussed in §4.2. We can avoid integrating the STT solutions

for each tk by applying the series reversion discussed in §2.2; however, this is not used in

this particular example. At every instance of a ∆Vk execution, the state is assumed to be

Gaussian with a zero mean and position and velocity uncertainties of 100 km and 0.1 m/s,

respectively, which is the same as the initial errors.

Figure 5.10 shows the magnitude of ∆Vk corrections computed using the linear and

nonlinear methods as functions of the true anomaly value which corresponds to the time tk

the impulse is applied. Both solutions become essentially the same after θ ≈ 90 degrees;

indicating small nonlinearity for the later part of the transfer. As expected, the linear

solution ∆Vk grows as the correction maneuver is made at a later time due to a larger

deviation in the mean trajectory. We note that the nonlinear statistical targeting method

gives an optimal time (i.e., minimum ‖∆Vk‖) to perform a correction maneuver that targets

127

0 20 40 60 80 100 120 140 160 18010

−2

10−1

100

101

102

103

104

θ (degrees)

||δm

r (tf )

|| (

km)

0 20 40 60 80 100 120 140 160 18010

−2

10−1

100

101

102

θ (degrees)

||δm

v (tf )

|| (

m/s

)

Linear Solutions

LinearNonlinear

Figure 5.11: Two-body problem: deviated position and velocity means at the target.

−2000 −1500 −1000 −500 0 500 1000 1500

−800

−600

−400

−200

0

200

400

600

x−coordinate (km)

y−co

ordi

nate

(km

)

TargetMC mean based on linear correctionMC mean based on nonlinear correction

MC 1−σ covariance projection based on nonlinear correction

MC 1−σ covariance projection based on

linear correction

~166 km

Figure 5.12: Two-body problem: Monte-Carlo simulation using the linear and nonlinearmethods.

128

the position mean. Figure 5.11 shows the deviated position and velocity means at the

target. The nonlinear deviated position mean is not shown since it is zero by definition.

When the correction maneuver is made at an early stage of the trajectory using the linear

theory, the position mean may deviate quite a bit at the final target. For the deviation in

the velocity mean, the difference between the linear and nonlinear solutions is very small.

Lastly, for a verification purpose, Figure 5.12 shows Monte-Carlo simulations of the

linearly and nonlinearly corrected trajectories, where the correction maneuvers are per-

formed at θ = 90 degrees. Both cases are based on ensembles of 106 sample points. We

observe that, on average, the nonlinear correction maneuver computed using the 3rd order

STT approach intercepts the target whereas the linear correction results in a deviation of

∼ 166 km. Hence, this confirms that our nonlinear analytic targeting approach gives a

correction maneuver that satisfies the statistically correct trajectory.

5.4.2 Hill Three-Body Problem: about Europa

Similar to the two-body example, in the Hill three-body problem example, we assume

the uncertainties computed using the 4th order STT solution is the truth model. Once the

mean trajectory is computed for the time interval [t0, tf ], at every tk ∈ [t0, tf ], we compute

∆Vk using the nonlinear and linear methods. At every time of maneuver, we assume the

spacecraft state has the same Gaussian statistics as the initial state, i.e., zero mean and

a covariance matrix with 10 km uncertainties and 0.1 m/s uncertainties for position and

velocity components, respectively.

Figure 5.13 shows the magnitude of ∆Vk applied at tk ∈ [t0, tf ] that are solved using

both the linear and nonlinear methods. The correction maneuvers are plotted as functions

of time and we observe little nonlinearity from tk ≈ 15 hours onwards. In this case,

however, ∆Vk fluctuates around tk ≈ 4 hours in both cases, unlike the two-body case. This

129

0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

100

101

102

Time (hours)

||∆V

k|| (

m/s

)

LinearNonlinear

Figure 5.13: Hill three-body problem: computed ∆Vk using the linear and nonlinear meth-ods.

is due to a sudden change in the velocity direction, and hence, the system nonlinearity is

varied. There is also an optimal ∆Vk (i.e., minimum ‖∆Vk‖), which occurs around tk ≈ 15

hours. Figure 5.14 shows the deviated position and velocity means. As in the two-body

case, when the correction maneuver is made at an early stage of the trajectory using the

linear theory, the position mean may deviate noticeably. The overall difference between

the linear and nonlinearly computed velocity mean is very small. A high fluctuation around

tk ≈ 4 in the nonlinearly solved δmv(tf ) is due to the higher correction maneuvers in that

time frame.

As in the two-body case, Figure 5.15 shows the Monte-Carlo simulation results for

the correction maneuvers computed, at time tk = 3 hours, using the linear and nonlinear

methods. The result shows that linearly corrected mean results in a deviation of ∼ 60 km

whereas the nonlinear correction maneuver intercepts the target on average.

130

0 5 10 15 2010

−1

100

101

102

103

Time (hours)

||δm

r (tf )

|| (

km)

0 5 10 15 2010

−1

100

101

102

103

104

Time (hours)

||δm

v (tf )

|| (

m/s

) LinearNonlinear

Linear Solutions

Figure 5.14: Hill three-body problem: deviated position and velocity means at the target.

−200 −150 −100 −50 0 50 100 150 200 250−800

−600

−400

−200

0

200

400

600

800

x−coordinate (km)

y−co

ordi

nate

(km

)

TargetMC using linear correctionMC using nonlinear correction

MC 1−σ covarianceprojection based on

linear correction

MC 1−σ covarianceprojection based on nonlinear correction

~60 km

Figure 5.15: Hill three-body problem: Monte-Carlo simulation using the linear and non-linear methods.

131

Table 5.2: Halo orbit maximum amplitudes with respect to the Sun-Earth L1 point.

Cases Ax (km) Ay (km) Az (km)1 245924 668228 1379082 246069 668416 139015

5.5 Nonlinear Trajectory Navigation

In this section, we present simulations of the nonlinear Kalman filters discussed in

§4.4. We give several examples based on halo orbits of the Sun-Earth and Earth-Moon

systems [22, 25, 36, 52]. A halo orbit is a Lissajous-type periodic orbit where the in-

plane and out-of-plane frequencies are the same, which we compute based on the circular

restricted three-body problem (see Appendix A for details).

5.5.1 Halo Orbit: Sun-Earth System

Consider a halo orbit about the Sun-Earth L1 point in a non-dimensionalized frame,

which can be dimensionalized by applying the length scale of ` = 1 AU = 1.49597870691×

108 km, where AU stands for “astronomical unit”, and time scale of τ = 1/ωE , where ωE

is the mean motion of the Earth about the Sun (i.e.,√

µS/AU3 = 1.991× 10−7 s−1). Fig-

ures 5.16 and 5.17 show the reference (nominal) trajectory for one orbital period (∼177.86

days) in 3-dimensions and in the x-y plane, respectively, which corresponds to the Case 1

given in Table 5.2. The initial conditions for these orbits are (in non-dimensional units):

rcase1(t0) = [ 0.988884102845168, 0.0, 0.000921858528329094 ]T ,

vcase1(t0) = [ 0.0, 0.00893471471659142, 0.0 ]T ,

rcase2(t0) = [ 0.98888423093423, 0.0, 0.000929261736280955 ]T ,

vcase2(t0) = [ 0.0, 0.00893688204973967, 0.0 ]T .

132

1.481.481

1.4821.483

x 108

−6

−4

−2

0

2

4

6

x 105

−1

0

1

x 105

x−coordinate (km)

y−coordinate (km)

z−co

ordi

nate

(km

)

L1

Figure 5.16: Nominal halo orbit about the Sun-Earth L1 point.

1.472 1.474 1.476 1.478 1.48 1.482 1.484 1.486 1.488 1.49

x 108

−8

−6

−4

−2

0

2

4

6

8x 10

5

x−coordinate (km)

y−co

ordi

nate

(km

)

L1

Figure 5.17: Nominal halo orbit about the Sun-Earth L1 point in x-y plane.

133

For the measurement model, we assume a simple linear model where only the y-

coordinate is observed, i.e.,

zk = yk + vk, (5.2)

where yk represents the vertical position component of the state vector and vk represents

the measurement error. This measurement model can be viewed as a range measurement

obtained by optical imaging of the Earth relative to distant stars or a Very Long Baseline

Interferometry (VLBI) measurement. The measurement noise is assumed to be 0.1 m for

each range measurement. This linear assumption simplifies the problem a great deal since

the measurement sensitivity does not require the computation of the higher order partials.

This way, it is easier to understand the effect of the nonlinear orbit uncertainty propagation

on filter performance.

Initially, the spacecraft state is assumed to be a zero mean Gaussian with position

uncertainties of 100 km and velocity uncertainties of 0.1 m/s.5 The initial mean and co-

variance matrix are mapped using the STT approach for m = 1, 3, unscented trans-

formation, and Monte-Carlo simulations based on 106 sample points. Figure 5.18 shows

the mean and the projection of the 1-σ covariance matrix onto the x-y plane after one or-

bital period. Assuming the MC simulation is the true solution, the result shows that the

3rd order solution is the most accurate approximation, whereas the linear solution fails to

characterize the orbit uncertainty distribution.

We now consider the same initial uncertainties, but assume the initial guess (mean) is

off by 100 km for the position components and 0.1 m/s for the velocity components so

that they lie on the boundary of the initial 1-σ ellipsoid. A set of pseudo-measurements

are computed based on the reference trajectory with a 20-day increment. Using the same

5The initial covariance matrix is a diagonal matrix with (100 km)2 and (0.1 m/s)2 for position componentsand velocity components, respectively.

134

1.477 1.4775 1.478 1.4785 1.479 1.4795 1.48 1.4805 1.481 1.4815

x 108

−8

−6

−4

−2

0

2

4

6

8x 10

4

y−co

ordi

nate

(km

)

x−coordinate (km)

m=1m=3UTMC

(a) Projected onto the x-y plane.

1.479 1.4791 1.4792 1.4793 1.4794 1.4795 1.4796 1.4797

x 108

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x 104

y−co

ordi

nate

(km

)

x−coordinate (km)

m=1m=3UTMC

(b) Larger plot of (a).

Figure 5.18: Sun-Earth halo orbit: covariance matrix computed after one orbital period.

135

0 100 200 300 40010

−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

0 200 40010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)

σ V (

m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.19: Sun-Earth halo orbit: comparison of the uncertainties computed using theEKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 20days.

0 100 200 300 40010

−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

Time (days)

||Pos

ition

Err

or||

(km

)

0 100 200 300 40010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)

||Vel

ocity

Err

or||

(m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.20: Sun-Earth halo orbit: comparison of the absolute errors computed using theEKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 20days.

136

measurements, the initial mean and covariance matrix are mapped and solved using the

EKF, UKF, 3rd order HNEKF, and 3rd order HAEKF. For the HAEKF, since the trajec-

tory is periodic, the STTs are computed and stored for only one orbital period, which is

divided into two segments for numerical consistency, and reversion of series is applied to

map states analytically. Figure 5.19 shows the a priori and a posteriori position and ve-

locity root-sum-square errors, where σR =√

σxx + σyy + σzz and σV =√

σuu + σvv + σww,

and σii represents (i, i) component of the covariance matrix. A sudden drop in the uncer-

tainties right after 100 days is due to the fact that the initial covariance matrix is quite large

and requires at least six independent measurements to obtain a well-defined (i.e., reduced

to the measurement noise level in all directions) a posteriori covariance matrix. The result

shows that the EKF overestimates the solution accuracy (i.e., the uncertainties are smaller

than they are in actuality) while the UKF, HNEKF, and HAEKF provide conservative un-

certainty estimates. Figure 5.20 shows the magnitude of the absolute position and velocity

errors, i.e., the magnitude of the difference between the updated mean and the true state.

The result shows that the EKF does not perform well as compared to the higher order fil-

ters. This clearly explains the importance of nonlinear orbit uncertainty propagation. The

covariance matrix computed by using the first order method (i.e., EKF) overestimates the

solution accuracy, and hence, the residual is trusted less. On the other hand, the UKF and

the higher order filters predict more conservative uncertainties and more effectively bal-

ance the a priori uncertainties and the actual measurements (i.e., measurements are valued

more than the a priori information in this case). Figures 5.21 and 5.22 are based on the

same filter setup except that the measurements are updated every 5 days. It shows that

there is not much difference in the propagated uncertainties, but the absolute errors are

computed more accurately in UKF and higher order filter runs.

Figures 5.23 and 5.24 show the HNEKF results for cases m ∈ 1, 2, 3. As mentioned

137

0 100 200 300 40010

−4

10−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

0 200 40010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)

σ V (

m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.21: Sun-Earth halo orbit: comparison of the uncertainties computed using theEKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 5days.

0 100 200 300 40010

−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

Time (days)

||Pos

ition

Err

or||

(km

)

0 100 200 300 40010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)

||Vel

ocity

Err

or||

(m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.22: Sun-Earth halo orbit: comparison of the absolute errors computed using theEKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 5days.

138

0 100 200 300 40010

−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

0 200 40010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)σ V

(m

/s)

HNEKF (m=1)HNEKF (m=2)HNEKF (m=3)

HNEKF (m=1)HNEKF (m=2)HNEKF (m=3)

Figure 5.23: Sun-Earth halo orbit: comparison of the uncertainties computed using theHNEKFs for the cases m = 1, 2, 3. Measurements are taken every 20 days.

0 100 200 300 40010

−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

Time (days)

||Pos

ition

Err

or||

(km

)

0 100 200 300 40010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)

||Vel

ocity

Err

or||

(m/s

)

HNEKF (m=1)HNEKF (m=2)HNEKF (m=3)

HNEKF (m=1)HNEKF (m=2)HNEKF (m=3)

Figure 5.24: Sun-Earth halo orbit: comparison of the absolute errors computed using theHNEKFs for the cases m = 1, 2, 3. Measurements are taken every 20 days.

139

0 100 200 300 40010

−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

0 200 40010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)σ V

(m

/s)

HAEKF (m=1)HAEKF (m=2)HAEKF (m=3)

HAEKF (m=1)HAEKF (m=2)HAEKF (m=3)

Figure 5.25: Sun-Earth halo orbit: comparison of the uncertainties computed using theHAEKFs for the cases m = 1, 2, 3. Measurements are taken every 20 days.

0 100 200 300 40010

−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

Time (days)

||Pos

ition

Err

or||

(km

)

0 100 200 300 40010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)

||Vel

ocity

Err

or||

(m/s

)

HAEKF (m=1)HAEKF (m=2)HAEKF (m=3)

HAEKF (m=1)HAEKF (m=2)HAEKF (m=3)

Figure 5.26: Sun-Earth halo orbit: comparison of the absolute errors computed using theHAEKFs for the cases m = 1, 2, 3. Measurements are taken every 20 days.

140

0 100 200 300 40010

−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

0 200 40010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)

σ V (

m/s

)

EKFUKFHAEKF (m=1)HAEKF (m=3)

EKFUKFHAEKF (m=1)HAEKF (m=3)

Figure 5.27: Sun-Earth halo orbit: comparison of the uncertainties computed using theEKF, UKF, and HAEKFs for the cases m = 1, 3. Measurements are taken every 20days based on the halo orbit Case 2.

0 100 200 300 40010

−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

Time (days)

||Pos

ition

Err

or||

(km

)

0 100 200 300 40010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)

||Vel

ocity

Err

or||

(m/s

)

EKFUKFHAEKF (m=1)HAEKF (m=3)

EKFUKFHAEKF (m=1)HAEKF (m=3)

Figure 5.28: Sun-Earth halo orbit: comparison of the absolute errors computed using theEKF, UKF, and HAEKFs for the cases m = 1, 3. Measurements are taken every 20 daysbased on the halo orbit Case 2.

141

earlier, note that the case m = 1 is identical to the EKF formulation. The result shows that

the higher order filters, m ∈ 2, 3, provide superior filter performance over the first order

case and it is observed that the second order effect contains most of the system nonlinearity,

indicating that the second order filter is sufficient for an accurate nonlinear filter in our

example. Figure 5.25 and 5.26 show the HAEKF uncertainties and absolute error plots,

respectively, for m ∈ 1, 2, 3. The uncertainties for m = 1 are similar to the EKF solution

and for m = 2 are similar to the case m = 3 as shown in Figure 5.19. The absolute error

plot shows that all three filters provide good estimation performance even for the case

m = 1. This is expected since the pseudo-measurements are computed based on the

reference trajectory which the STTs are computed based on. In other words, the reference

trajectory can be thought of as a regression solution for the simulated measurements.

In order to analyze the higher order effect, the pseudo-measurements are now gen-

erated from the Case 2 halo orbit given in Table 5.2. Figures 5.27 and 5.28 show the

simulated filter solutions. The results show that the higher order solutions are superior

over the linear filters, i.e., EKF and HAEKF for m = 1. As expected, this indicates that

the linear Kalman filter is only feasible when the reference trajectory is sufficiently close

to the true trajectory. The HAEKFs for m > 1, however, have more flexibility in the ref-

erence trajectory. The overall filter convergence is slightly slower than the previous cases

since the initial mean is assumed to be the same as in the previous cases, and thus, it is

farther away from the true trajectory (i.e., the trajectory which the pseudo-measurements

are generated).

5.5.2 Halo Orbit: Earth-Moon System

As another example of nonlinear filtering, we present a similar simulation as in §5.5.1

based on a halo orbit about the Earth-Moon L1 point. This is a much more nonlinear

142

Table 5.3: Halo orbit maximum amplitudes with respect to the Earth-Moon L1 point.

Cases Ax (km) Ay (km) Az (km)1 6934 21612 21322 6968 21667 2665

system since the mass ratio constant is much larger, i.e., µEM = 0.01215 > µSE =

3.003 × 10−6. For this system, the length scale is ` = 384400 km and the time scale

is τ = 4.3691 days. Figures 5.29 and 5.30 show the reference trajectory for one orbital

period (∼11.99 days), which is the Case 1 given in Table 5.3. Note that this orbit has

a much smaller orbit size as well as orbital period than the Sun-Earth halo orbit. The

corresponding initial conditions are (in non-dimensional units):

rcase1(t0) = [ 0.823386040115578, 0.0, 0.00554618294369076 ]T ,

vcase1(t0) = [ 0.0, 0.126839229387039, 0.0 ]T ,

rcase2(t0) = [ 0.823384852005846, 0.0, 0.00693385322129738 ]T ,

vcase2(t0) = [ 0.0, 0.127125596330242, 0 ]T .

Also, we assume the same measurement model considered in §5.5.1, i.e., zk = yk + vk,

with 0.1 m accuracy.

At epoch, the initial state is assumed to be Gaussian with:

δm+0 = [ 100 km, 100 km, 100 km, 0.1 m/s, 0.1 m/s, 0.1 m/s ]T ,

P+0 = diag[ (100 km)2, (100 km)2, (100 km)2

(0.1 m/s)2, (0.1 m/s)2, (0.1 m/s)2 ],

and we assume the measurements are taken every 2 days, which we consider to be a

baseline case. Note that the mean vector δm+0 lies on the boundary of the 1-σ ellipsoid,

which simply means that the initial guess is offset from the initial state of the true tra-

jectory. This navigation scenario is simulated using the EKF, UKF, 3rd order HNEKF,

143

3.23.25

3.3

x 105

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x 104

−20000

2000

x−coordinate (km)

y−coordinate (km)

z−co

ordi

nate

(km

)

L1

Figure 5.29: Nominal halo orbit about the Earth-Moon L1 point.

3 3.1 3.2 3.3 3.4 3.5

x 105

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

4

x−coordinate (km)

y−co

ordi

nate

(km

)

L1

Figure 5.30: Nominal halo orbit about the Earth-Moon L1 point in x-y plane.

144

0 10 20 3010

−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

EKFUKFHNEKF (m=3)HAEKF (m=3)

0 10 20 3010

−5

10−4

10−3

10−2

10−1

100

101

Time (days)

σ V (

m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.31: Earth-Moon halo orbit: comparison of the uncertainties computed using theEKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2days.

0 10 20 3010

−5

10−4

10−3

10−2

10−1

100

101

102

103

Time (days)

||Pos

ition

Err

or||

(km

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

0 10 20 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Time (days)

||Vel

ocity

Err

or||

(m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.32: Earth-Moon halo orbit: comparison of the absolute errors computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2days.

145

and 3rd order HAEKF,6 and Figure 5.31 shows the a priori and a posteriori position and

velocity root-sum-squares errors (i.e., σR and σV ) and we observe that the UKF, HNEKF,

and HAEKF provide conservative uncertainty estimates whereas EKF overestimates the

solution accuracy. Figure 5.32 shows the absolute errors of the position and velocity of

the trajectory, i.e., ‖m+k − xk‖, where xk represents our reference halo orbit which the

pseudo-measurements are generated about. The result shows that the EKF provides a slow

convergence as compared to the higher order filters. Again, this is because the EKF over-

estimates the solution accuracy whereas the higher order filters compute the Kalman gain

more effectively by balancing the a priori state error and the measurement residual.

Figures 5.33 and 5.34 are based on the exact same filter setup except that the initial

mean is assumed to be on the true trajectory, i.e., δm+0 = 0. In this case, the result shows

that the EKF provides a faster solution convergence than the higher order filters. This is

expected since the propagated initial mean of the EKF is the true trajectory and can be

considered as an already converged solution, and thus, the EKF only improves the covari-

ance matrix.7 The higher order filters, on the other hand, propagate the mean nonlinearly,

which deviates from the reference trajectory because of the large initial uncertainties. The

higher order filters then try to simultaneously reduce the errors in the state and the co-

variance matrix, which requires additional measurements to obtain a converged solution.

Therefore, the EKF solution is, in a sense, a false result which is obtained by disregarding

the system nonlinearity. To make this case more realistic, the initial uncertainties (i.e.,

6For HAEKF runs, as in the Sun-Earth case, the STTs are computed and stored for only one orbitalperiod since the orbit is periodic; the orbit is divided into two segments and integrated independently fornumerical consistency.

7If started out with small initial uncertainties and large initial deviations from the true state, a linear filtersolution usually does not converge since the filter trusts the a priori covariance strictly and does not includethe measurement (residual) contribution in the update process. If started out with large initial uncertaintiesand also large initial deviations from the true state, but within the initial error bounds, a linear filter solutionusually converges since the filter has enough flexibility to adjust so that the true solution can be found fromsequential updates.

146

P+0 ) should be reduced to a level where the linear assumption gives a good approximation

of the dynamics. This is because we want to show that when a linear approximation is

sufficient, both nonlinear filters and EKF should result in similar filter solutions. This case

is shown in Figures 5.35 and 5.36, where the initial state is assumed to have zero mean

with a covariance matrix:

P+0 = diag[ (1 km)2 (1 km)2, (1 km)2 (0.01 m/s)2 (0.01 m/s)2 (0.01 m/s)2 ].

We observe that both linear and nonlinear filters provide the same level of filter perfor-

mance.

Now let’s consider the baseline case with different measurement scheduling. Figures

5.37 and 5.38 show the uncertainties and absolute errors where the measurements are now

updated every 6 hours (n.b., 8 times more frequently than the baseline case). Overall, the

result shows that both linear and nonlinear filters provide essentially the same filter output

since measurements are taken frequently enough to maintain the phase volume within a

linear boundary.

Figures 5.39 and 5.40 show uncertainties and absolute errors based on the HNEKF

result for cases m ∈ 1, 2, 3. As in the Sun-Earth halo orbit case, the higher order

filters, m ∈ 2, 3, provide superior filter performance over the EKF, and we observe

that the second order solution (i.e., m = 2) captures most of the trajectory dynamics in

this case. Figures 5.41 and 5.42 show the HAEKF uncertainties and absolute error plots,

respectively, for m ∈ 1, 2, 3. The uncertainties for m = 1 are similar to the EKF

solution and for m = 2 are similar to the case m = 3 as shown in Figure 5.31.

The absolute error plot shows that all three filters provide good filter solutions and

convergence even for the first order case, despite that fact the initial state, δm+0 , is not

within the linear regime. This is expected since the m = 1 case is, in a sense, similar to

147

0 10 20 3010

−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

EKFUKFHNEKF (m=3)HAEKF (m=3)

0 10 20 3010

−5

10−4

10−3

10−2

10−1

100

101

Time (days)

σ V (

m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.33: Earth-Moon halo orbit: comparison of the uncertainties computed using theEKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2days assuming zero initial mean.

0 10 20 3010

−5

10−4

10−3

10−2

10−1

100

101

Time (days)

||Pos

ition

Err

or||

(km

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

0 10 20 3010

−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Time (days)

||Vel

ocity

Err

or||

(m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.34: Earth-Moon halo orbit: comparison of the absolute errors computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2days assuming zero initial mean.

148

0 10 20 3010

−3

10−2

10−1

100

101

102

Time (days)

σ R (

km)

EKFUKFHNEKF (m=3)HAEKF (m=3)

0 10 20 3010

−5

10−4

10−3

10−2

10−1

Time (days)

σ V (

m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.35: Earth-Moon halo orbit: comparison of the uncertainties computed using theEKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2days assuming zero initial mean and small initial covariance matrix.

0 10 20 30

10−4

10−3

10−2

10−1

100

101

Time (days)

||Pos

ition

Err

or||

(km

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

0 10 20 30

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Time (days)

||Vel

ocity

Err

or||

(m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.36: Earth-Moon halo orbit: comparison of the absolute errors computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 2days assuming zero initial mean and small initial covariance matrix.

149

0 10 20 3010

−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

EKFUKFHNEKF (m=3)HAEKF (m=3)

0 10 20 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)

σ V (

m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.37: Earth-Moon halo orbit: comparison of the uncertainties computed using theEKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 6hours.

0 10 20 30

10−4

10−3

10−2

10−1

100

101

102

103

Time (days)

||Pos

ition

Err

or||

(km

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

0 10 20 3010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Time (days)

||Vel

ocity

Err

or||

(m/s

)

EKFUKFHNEKF (m=3)HAEKF (m=3)

Figure 5.38: Earth-Moon halo orbit: comparison of the absolute errors computed usingthe EKF, UKF, HNEKF (m = 3), and HAEKF (m = 3). Measurements are taken every 6hours.

150

0 10 20 3010

−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

HNEKF (m=1)HNEKF (m=2)HNEKF (m=3)

0 10 20 3010

−5

10−4

10−3

10−2

10−1

100

101

Time (days)σ V

(m

/s)

HNEKF (m=1)HNEKF (m=2)HNEKF (m=3)

Figure 5.39: Earth-Moon halo orbit: comparison of the uncertainties computed using theHNEKFs for the cases m = 1, 2, 3. Measurements are taken every 2 days.

0 10 20 3010

−5

10−4

10−3

10−2

10−1

100

101

102

103

Time (days)

||Pos

ition

Err

or||

(km

)

HNEKF (m=1)HNEKF (m=2)HNEKF (m=3)

0 10 20 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Time (days)

||Vel

ocity

Err

or||

(m/s

)

HNEKF (m=1)HNEKF (m=2)HNEKF (m=3)

Figure 5.40: Earth-Moon halo orbit: comparison of the absolute errors computed usingthe HNEKFs for the cases m = 1, 2, 3. Measurements are taken every 2 days.

151

0 10 20 3010

−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

HAEKF (m=1)HAEKF (m=2)HAEKF (m=3)

0 10 20 3010

−5

10−4

10−3

10−2

10−1

100

101

Time (days)σ V

(m

/s)

HAEKF (m=1)HAEKF (m=2)HAEKF (m=3)

Figure 5.41: Earth-Moon halo orbit: comparison of the uncertainties computed using theHAEKFs for the cases m = 1, 2, 3. Measurements are taken every 2 days.

0 10 20 3010

−5

10−4

10−3

10−2

10−1

100

101

102

103

Time (days)

||Pos

ition

Err

or||

(km

)

HAEKF (m=1)HAEKF (m=2)HAEKF (m=3)

0 10 20 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Time (days)

||Vel

ocity

Err

or||

(m/s

)

HAEKF (m=1)HAEKF (m=2)HAEKF (m=3)

Figure 5.42: Earth-Moon halo orbit: comparison of the absolute errors computed usingthe HAEKFs for the cases m = 1, 2, 3. Measurements are taken every 2 days.

152

the EKF run of the previous case where we assumed the initial mean is zero, and thus, the

solution is false since the filter completely ignores the system nonlinearity. To be more

specific, because the reference trajectory is the true trajectory,8 the solution converges no

matter how offset the initial state is from the reference trajectory. We shall see that this

is no longer true if the pseudo-measurements are generated based on a trajectory that is

different than the reference trajectory.

In order to show this, the pseudo-measurements are generated from the Case 2 halo

orbit given in Table 5.2 while keeping the Case 1 halo orbit as the reference trajectory.

Figures 5.43 and 5.44 show that the higher order solutions are superior over the linear

filters, i.e., EKF and HAEKF for m = 1. As in the Sun-Earth simulation, we observe

that the linear Kalman filter (i.e., first order HAEKF) solution does not converge, which

implies that the first order HAEKF is only feasible when the reference trajectory is suf-

ficiently close to the true trajectory. We note that the HAEKFs for m > 1 incorporates

the system nonlinearity and provide solution essentially equivalent to the numerical higher

order filters.

5.5.3 Potential Applications and Challenges

In this study, the EKF required integration of N + N2 = 42 equations and the UKF

required integration of (2N + 1)N = 78 equations between each measurement update,

and in the actual filter runs, the EKF was slightly faster than the UKF. The HNEKFs

for m > 1 provided superior results over the linear filters; however, we note that the

computational load increases significantly as m increases. For example, the third order

HNEKF requires integration of 1554 equations for STT computation. On the other hand,

the HAEKF does not require any integration in the the actual filtering process. The most

8In other words, the reference trajectory is the reconstructed (i.e., a regression solution) solution of thesimulated data.

153

0 10 20 3010

−3

10−2

10−1

100

101

102

103

Time (days)

σ R (

km)

EKFUKFHAEKF (m=1)HAEKF (m=3)

0 10 20 3010

−5

10−4

10−3

10−2

10−1

100

101

Time (days)

σ V (

m/s

)

EKFUKFHAEKF (m=1)HAEKF (m=3)

Figure 5.43: Earth-Moon halo orbit: comparison of the uncertainties computed using theEKF, UKF, and HAEKFs for the cases m = 1, 3. Measurements are taken every 2 daysbased on the halo orbit Case 2.

0 10 20 3010

−5

10−4

10−3

10−2

10−1

100

101

102

103

Time (days)

||Pos

ition

Err

or||

(km

)

EKFUKFHAEKF (m=1)HAEKF (m=3)

0 10 20 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Time (days)

||Vel

ocity

Err

or||

(m/s

)

EKFUKFHAEKF (m=1)HAEKF (m=3)

Figure 5.44: Earth-Moon halo orbit: comparison of the absolute errors computed usingthe EKF, UKF, and HAEKFs for the cases m = 1, 3. Measurements are taken every 2days based on the halo orbit Case 2.

154

expensive numerical operation in the HAEKF is the higher order moment computation;

however, we note that there exist various techniques for efficient computation of moments.

The examples presented are filter initialization problems, where an initial state is as-

sumed to have a large covariance matrix and has a poor initial guess that may be far from

the true trajectory. The goal is then to sequentially find a converged solution with accu-

rate covariances. Our examples show that a nonlinear filter can have more flexibility in

the initial guess and obtain a converged solution whereas a linear filter diverges when the

initial guess is far from the true trajectory. Also, it is apparent that a nonlinear filter can

provide a more accurate filter solution than a linear filter even when the initial state is

precisely known. For example, when a trajectory is propagated over a long time period

or when there exists strong system nonlinearity, the number of measurements to maintain

a certain error limit can be reduced since a nonlinear filter provides a faster convergence

than a linear filter. Also, the integration between the dynamics and filtering can be more

effectively modeled.

For a given reference trajectory, we have shown that the higher-order analytic EKF is

essentially equivalent to higher order numerical filters (i.e., HNEKF and UKF) if we con-

sider a sufficient order of solutions and assume the series is within the radius of conver-

gence. Hence, for missions with pre-determined reference trajectories, the HAEKF may

be suitable for the trajectory navigation while obtaining faster convergence and a computa-

tionally faster filter9 than the EKF. Such applications of the HAEKF consist of interplan-

etary cruises, trajectories about complex dynamical environments, spacecraft launches,

orbit insertions, atmospheric re-entries, etc.

We have shown examples where the the trajectory dynamics are assumed to be known

with perfect knowledge by setting the process noise to be zero. This is a valid assumption if

9The HAEKF process is faster than the EKF or other numerical filters since the higher order solutionscan be stored onboard before the launch or up-linked to a spacecraft during the mission.

155

a spacecraft is quiet and thrusters are turned off. However, this not true for some problems,

such as spacecraft launch or atmospheric re-entry since the un-modeled accelerations (e.g.,

rocket thrusters or atmospheric drag) can be significant. In practice, a covariance matrix

is usually integrated including a process noise matrix according to the Riccati equation.10

This problem is not considered in this dissertation since the focus of our study is to show

the importance of the system nonlinearity on a filtering problem. However, we note that, in

discrete form, the process noise matrix can be estimated by integrating it over some time

interval [55, 56], which is simple if we assume an additive white noise. This, however, can

be a difficult process if we consider a nonlinear stochastic acceleration and apply no linear

assumption.

10In general the differential equation for the covariance matrix can be written as: P(t) = A(t)P(t) +P(t)AT (t) + Q(t), where A(t) represents the linear dynamics matrix.

CHAPTER VI

CONCLUSIONS AND FUTURE RESEARCHDIRECTIONS

This last chapter summarizes the theory developed in this thesis and gives an overview

of the key contributions made for general space applications. Also, we present several

future research directions/projects where our study can be used as a baseline and can be

extended to practical problems.

6.1 Concluding Remarks and Key Contributions

The main contribution of this thesis is the development of a nonlinear analytic theory of

spacecraft navigation. To achieve this goal, we have developed higher order state transition

tensors, nonlinear orbit uncertainty propagation, the concept of the statistically correct

trajectory, nonlinear statistical targeting, and nonlinear Bayesian and Kalman filters.

Higher-Order Relative Dynamics:

In this dissertation, we have developed an analytic expression for the solution of

relative dynamics by solving for state transition tensors that describe the localized

nonlinear motion. These tensors are computed by applying a Taylor series expan-

sion and by numerically integrating them along a nominal solution flow. Once the

state transition tensors are available for some time interval, the inverse series can be

156

157

computed via a series reversion, which requires no numerical integration. Then, as-

suming an initial condition is within the radius of convergence and a sufficient order

of solution is considered, the nonlinear relative motion can be completely charac-

terized and the deviations can be mapped analytically from an arbitrary point in the

relative space to some future time, or vice versa. In this way, we can also propagate

a phase volume analytically by mapping the boundary of the phase volume using the

higher state transition tensors. A convergence criteria for these higher order series

is discussed by introducing the local nonlinearity index. Also presented is the sym-

plecticity of the higher solutions of a Hamiltonian system. As many astrodynamics

problems can be transformed into a Hamiltonian form, we can take the advantage of

useful properties that are available from a Hamiltonian dynamical system.

Time Invariance of Probability Density Function:

From the Fokker-Planck equation for the probability density function, we have shown

that the probability function of any dynamical system (without diffusion) satisfies

the integral invariance condition. Then, by applying the integral invariance of the

probability and conservation of the phase volume, we presented the relation between

the probability function and the phase volume, since they possess the same statistical

information, but are represented in different ways. We have also solved the Fokker-

Planck equation for a deterministic system and derived a sufficiency condition for

the time invariance of a probability density function. This result was then applied to

show that the probability density function of a higher order Hamiltonian system is a

time invariant.

Nonlinear Orbit Uncertainty Propagation:

Applying the higher order state transition tensors and the time invariance of the prob-

158

ability density function, we have derived an analytic representation of the nonlinear

uncertainty propagation as a function of an initial distribution. When a sufficient

order of the higher order solutions are considered, we have shown that our analytic

approach can replace Monte-Carlo simulations while providing the same level of ac-

curacy. This serves as the baseline of this thesis and was applied to several trajectory

navigation problems.

The Statistically Correct Trajectory and Nonlinear Statistical Targeting:

The analytical uncertainty propagation method enabled us to introduce the concept

of the statistically correct trajectory, where we solve for an initial state that satisfies

the target condition on average. As a practical application of the statistically correct

trajectory, we have developed the nonlinear statistical targeting method by utilizing

the statistical property of the trajectory in the maneuver design process, and thus

providing a statistically more accurate solution. When the initial uncertainties are

small, the nonlinear method essentially becomes the linear solution; however, when

there are sufficiently large initial uncertainties the solution gives superior statistical

performance. The results from the two-body and Hill three-body examples show

that there appear to exist an optimal time to perform a correction maneuver, a result

that is not possible using the linear method.

Nonlinear Filtering:

As an extension of the nonlinear orbit uncertainty propagation, we have derived

an analytic expression for the posterior conditional density function by solving the

Bayes’ rule for conditional probability distributions. This showed that the optimal

nonlinear filtering problem can be solved by applying the higher order relative dy-

namics solutions. For practical purposes, we then derived two Kalman-type filters,

159

called higher-order numerical extended Kalman filter and higher-order analytic ex-

tended Kalman filter, by directly applying the higher order solutions to the Kalman

filter algorithm. These higher order filters were compared with the conventional

extended Kalman filter and the unscented Kalman filter based on halo orbits com-

puted in restricted three-body problem frame about the Sun-Earth and Earth-Moon

L1 points. The filter simulations were carried out assuming the dynamics of the

system are perfectly known, but there are errors in the initial state and in the mea-

surements. The results showed that a higher order filter provides faster convergence,

a superior filter solution, and more flexility in the initial guess over linear filters.

Also, the Gaussian assumption of the a posteriori state yielded a sufficient approx-

imation even for nonlinear filters. For the cases where the reference trajectory was

relatively close to the true trajectory, the higher order analytic filter provided solu-

tions essentially equivalent to both the UKF and HNEKF, and yielded a much faster

filter process.

6.2 Future Research and Recommendations

There are a number of open questions and future research directions for the research

presented in this thesis. In the following we discuss several potential applications where

our analytic trajectory navigation techniques can be implemented and utilized.

Low-Thrust Trajectory:

The problem of low-thrust trajectory is one area where both the statistically correct

trajectory and the higher-order analytic extended Kalman filter can be applied. The

challenge in the low-thrust trajectory problem is that the nature of the small, con-

tinuous thrust is stochastic in general, and may be modeled as non-additive white

noise. Thus, a critical research for this problem would be to incorporate the nonlin-

160

ear noise effect into our analytic trajectory navigation framework, or to effectively

estimate the process noise (i.e., stochastic acceleration). Additional applications of

this type are spacecraft launch, atmospheric re-entry/aero-braking, and near solar

trajectories.

Autonomous Trajectory Navigation:

In recent proposals for mission to the Moon and Mars, precision entry, descent, and

landing problems are often discussed. This can be considered as a problem where

we have a fairly accurate initial solution, which is subject to a highly nonlinear tra-

jectory environment. However, we are still given a desired landing site or a reference

trajectory. The nature of these problems must consider autonomous navigation tech-

niques since it is not possible to perform open-loop control from a ground station

during the critical periods of the mission. This is another main research area where

our method can be directly implemented. Once a reference trajectory is computed,

an autonomous navigation processor that incorporates the trajectory nonlinearity in

a filtering process can be developed and a faster convergence may be feasible in

practice. Note that the processor would not require any numerical integration if

it uses the higher-order analytic extended Kalman filter (HAEKF) technique, and

thus, it is also possible to implement semi-analytic Monte-Carlo/particle type filter-

ing techniques by mapping each random sample analytically using the STT approach

instead of numerically integrating the samples. The analytic method would require

less computational power; however, one must study the trade-off between large data

storage and numerical integration onboard a spacecraft.

Small-Body Collision/Encounter Analysis and Space Surveillance:

In small-body collision/encounter studies, the uncertainties associated with an initial

161

state are usually the dominating error source and process noise is almost negligible

(depending on the size of a small-body). However, we are still given an estimated

orbit from trajectory reconstruction, which we can consider as the reference tra-

jectory. After integrating this reference orbit to some arbitrary time in the future,

which can be determined according to the local nonlinearity index, we can establish

a nonlinear filtering algorithm which includes the higher order statistics whenever

a measurement (e.g., a radar/optical measurement) becomes available. One chal-

lenge with this approach is that there exist a large number of small-bodies in our

solar system, and thus, computation of the higher order solutions can be computa-

tionally quite intensive. However, it is also important to note that this only needs

to be carried out once over a long time period. For the space surveillance problem

(e.g., low Earth space debris), the same analogy applies as in the small-body study.

One critical difference is that the process noise due to the atmospheric drag must be

estimated as a part of a filtering process.

Robust Computation of the State Transition Tensors:

Assuming the trajectory dynamics can be modeled as a Hamiltonian system, the

numerical errors (e.g., truncation error) associated with computing the higher order

state transition tensors can be significant reduced by implementing the variational

or symplectic integrator. This is possible since these integrators preserve the Hamil-

tonian structure in each step of integration, and thus, the nominal trajectory solution

and the state transition matrix can be computed with high accuracy, which yields a

more robust computation of the higher order solutions and can be integrated for a

long period of time.

APPENDICES

162

163

APPENDIX A

EQUATIONS OF MOTION OF ASTRODYNAMICSPROBLEMS

A.1 The Two-Body Problem

The two-body problem describes the relative motion of two bodies, represented as

point particles, under their mutual gravitational attractions. The simplest generalization

of the two-body problem is when one of the bodies has negligibly small mass and the

central body has a uniform gravitational force, such as a spacecraft orbiting about the Earth

(assuming Earth is spherical with uniform gravitational force). This approximation is what

we usually referred to as the two-body problem. The governing equations of motion for

the two-body problem are defined as:

In Hamiltonian Form:

H =1

2(p2

x + p2y + p2

z )−GM

r, (A.1)

where the Hamiltonian H is simply the specific energy of an orbit, r = ‖r‖ is the radius,

G is the universal gravitational constant, and M is the mass of the central body. The state

vector x = [ qT , pT ]T , where q = [ x , y , z ]T is the position (generalized coordinate)

vector and p = [ px , py , pz ]T is the velocity (generalized momentum) vector.

164

Table A.1: Properties of planets and satellites.

Planets GM (km3/s2) Radius (km)Sun 1.327×1011 695990

Earth 398600 6378Moon 4903 1738Europa 3201 1565Jupiter 1.267×108 71492Titan 9028 2575

In Lagrangian Form:

r = −GM

r3r, (A.2)

where r = [ x , y , z ]T is the position vector and v = [ x , y , z ]T is the velocity vector. Note

that for two-body problem, both the Lagrangian and Hamiltonian equations of motion

are identical to each other, i.e., the generalized momentum are the same as the velocity

components. The properties of planets and satellites considered in this thesis are given in

Table A.1.

A.2 The Three-Body Problem

The three-body problem is an extension of the full two-body problem, where a third

body is added to the mutual gravitational potential, and hence, it describes the dynamics

of three point mass particles in space. The three-body problem is particularly useful for

trajectory design and analysis since it can approximate the majority of the dynamical en-

vironments that are present in our solar system. The two most widely used models for the

three-body problem are the circular restricted three-body problem (CR3BP) and the Hill

three-body problem.

165

A.2.1 The Circular Restricted Three-Body Problem

The CR3BP assumes the third body has negligible mass compared to the other two

bodies and both bodies are in a mutually circular orbit, such as the Sun-Earth-spacecraft

system. Assuming the coordinate system is centered at the center of mass in a rotating

frame the governing equations of motion in non-dimensional units are defined as follows:

In Hamiltonian Form:

H =1

2(p2

x + p2y + p2

z )− (xpy − ypx) +1

2(x2 + y2)− V (x, y, z), (A.3)

V =1

2(x2 + y2) +

1− µ

r1

+µ

r2

, (A.4)

r1 =√

(x + µ)2 + y2 + z2, (A.5)

r2 =√

(x− 1 + µ)2 + y2 + z2, (A.6)

where µ = M2/(M1 + M2) are shown in Table A.2 and M2 ≤ M1 are the body masses.

Note that q = [ x , y , z ]T is the generalized coordinate vector and p = [ px , py , pz ]T is

the generalized momentum vector.

In Lagrangian Form:

x− 2y =∂V

∂x, (A.7)

y + 2x =∂V

∂y, (A.8)

z =∂V

∂z, (A.9)

where v = [ x , y , z ]T is the velocity vector. The generalized momentum is related to the

velocity components as:

p =

x− y

y + x

z

. (A.10)

166

Table A.2: Properties of three-body systems.

Systems µ Distance (km) ω (s−1)Sun-Earth 3.003×10−6 149597871 1.991×10−7

Earth-Moon 0.01215 384400 2.649×10−6

Jupiter-Europa 2.526×10−5 670900 2.048×10−5

Note that, according to Eqn. 2.70 in §2.3, the inverse of the state transition matrix in a

Lagrangian system can be computed as:

Φ−1 = −T−10 JT−T

0 ΦT TT JT, (A.11)

where

T =∂(q, p)

∂(q, q)=

I 0∂2L

∂q∂q∂2L

∂q2

=

I 0

ω I

, (A.12)

ω =

0 −1 0

1 0 0

0 0 0

. (A.13)

The units can be dimensionalized by applying the length scale of ` = distance between

the two massive bodies, and the time scale of τ = 1/ω, where ω is the mean motion of the

secondary body about the primary body. Moreover, there exists a Jacobi constant which is

preserved for all time:

J =1

2(x2 + y2 + z2)− V (x, y, z). (A.14)

A.2.2 The Hill Three-Body Problem

The Hill three-body problem is an approximation of the CR3BP where the mass of the

primary is assumed to be much larger than the secondary, i.e., M2 ¿ M1 or µ1/3 ¿ 1.

167

After shifting the coordinate center to the secondary point mass, the governing equations

of motion can be defined as follows:

In Hamiltonian Form:

H =1

2(p2

x + p2y + p2

z )− (xpy − ypx) +1

2(x2 + y2)− VH(x, y, z), (A.15)

VH =1√

x2 + y2 + z2+

1

2(3x2 − z2), (A.16)

In Lagrangian Form:

x− 2y =∂VH

∂x, (A.17)

y + 2x =∂VH

∂y, (A.18)

z =∂VH

∂z, (A.19)

and a Jacobi constant exists:

J =1

2(x2 + y2 + z2)− VH(x, y, z). (A.20)

The units can be dimensionalized by applying the length scale of ` = (GM2/ω2)1/3 and

the time scale of τ = 1/ω, where M2 is the mass of the secondary body and ω is the mean

motion of the secondary body about the primary body.

A.2.3 Halo Orbit

In the circular restricted three-body problem, there exists a special type of orbit called

a ‘halo orbit’, which is of particular interest for space missions. A halo orbit is a Lissajous-

type1 periodic orbit where the in-plane and out-of-plane frequencies are the same [22, 25,

36, 52]. Figure A.1 shows families of halo orbits about the Sun-Earth L1 point in a non-

dimensional frame.2 As shown, there exist many families of halo orbits; however, the1A Lissajous orbit is an orbit where the in-plane and out-of-plane frequencies are not necessarily the

same. There is also an orbit called a Lyapunov orbit, which is periodic in the rotating frame of the primarybodies, i.e., periodic in the x-y plane only.

2In the CR3BP in a rotating frame, there exist five equilibrium points called the Lagrangian points. TheL1 point is a collinear Lagrange point located between the primary and secondary bodies.

168

0.99 0.995

−50

5x 10

−3

−8

−6

−4

−2

0

2

4

6

8

x 10−3

y−coordinate

x−coordinate

z−co

ordi

nate

Figure A.1: Families of halo orbits about the Sun-Earth L1 point in non-dimensionalframe.

computation of a halo orbit requires a numerical technique since the problem is not inte-

grable in closed form. The examples considered in this dissertation implemented a third

order analytic solution as the initial guess and applied differential corrections to obtain

convergence to the true halo orbit solution [85, 86].

169

APPENDIX B

PROPERTIES OF PROBABILITY DENSITYFUNCTIONS

B.1 Integral Invariance of the PDF of a Linear Hamiltonian Dynam-ical System with Gaussian Boundary Conditions

Here, we show the integral invariance of the Gaussian probability density function

subject to a linear Hamiltonian dynamical system. Consider a linearized Hamiltonian

system:

δx =

δq

δp

= JHxxδx = Aδx, (B.1)

which gives the solution flow:

δx(t; δx0, t0) = Φ(t, t0)δx0, (B.2)

Φ = JHxxΦ = AΦ. (B.3)

Now, without loss of generality, suppose δx = (δq, δp) ∼ N (0, P) so that:

p(δx, t) =(detΛ)1/2

(2π)N/2exp

(−1

2δxTΛδx

), (B.4)

with initial conditions:

p(δx0, t0) =(detΛ0)

1/2

(2π)N/2exp

(−1

2δxT

0 Λ0δx0

), (B.5)

170

where Λ = P−1, which is usually called the information matrix. In order to show the

integral invariance, it suffices to show:

dp(δx, t)

dt=

(detΛ)1/2e−12δxT Λδx

2(2π)N/2

·

︷ ︸︸ ︷˙(detΛ)

(detΛ)−

(δxTΛδx + δxT Λδx + δxTΛδx

),

= 0, (B.6)

and thus we only have to show · = 0.

Recall from the basic probability theory that P = E[δxδxT ]. Taking the time derivative

gives:

P(t) = E[δxδxT + δxδxT ],

= E[AδxδxT + δxδxT AT ],

= A(t)P(t) + P(t)AT (t), (B.7)

which is the Riccati equation for the covariance matrix. Now consider the identity, ΛP =

I. Taking the total time derivative of ΛP and substituting Eqn. (B.7) give:

Λ = −ATΛ−ΛA. (B.8)

The time derivative of the information matrix can be stated as [92]:

d(detΛ)

dt= −2(detΛ)Trace(A), (B.9)

Trace(A) =N∑

i=1

(∂2H

∂qi∂pi− ∂2H

∂pi∂qi

)= 0. (B.10)

Then Eqn. (B.7) simplifies to:

· = 0− [δxT ATΛδx + δxT

(−ATΛ−ΛA)δx + δxTΛAδx

],

= δxT ATΛδx− δxT ATΛδx− δxTΛAδx + δxTΛAδx,

= 0, (B.11)

which satisfies the necessary condition for the integral invariance.

171

APPENDIX C

THE LINEAR KALMAN FILTER

In this appendix, following a similar derivation as in Maybeck [56], we give a formal

derivation of the current state linear Kalman filter based on the Bayes’ rule for the posterior

conditional density function.

C.1 Kalman Filter Essentials

Here, we first present two useful matrix identities that will be used to derive the Kalman

Filter in the next section.

Lemma C.1.1 (Matrix Inversion Lemma). Given matrices PN×N , Hm×N , and Rm×m, the

following matrix inversion identities exist:

(P−1 + HT R−1H

)−1= P− PHT

(HPHT + R

)−1 HP, (C.1)

(P−1 + HT R−1H

)−1 HT R−1 = PHT(HPHT + R

)−1, (C.2)

R−1H(P−1 + HT R−1H

)−1=

(HPHT + R

)−1 HP, (C.3)

H(P−1 + HT R−1H

)−1 HT = R− R(HPHT + R

)−1 R, (C.4)

where we assume PN×N and Rm×m are semi-positive definite matrices.

172

Proof. Define:

A =

P−1N×N HT

m×N

Hm×N −Rm×m

, (C.5)

A−1 =

DN×N FN×m

GTN×m Em×m

. (C.6)

Setting AA−1 = I(N+m)×(N+m), and solving for A−1, we get:

D =(P−1 + HT R−1H

)−1,

F = PHT(HPHT + R

)−1,

GT = R−1H(P−1 + HT R−1H

)−1,

E = − (HPHT + R

)−1.

Substituting these matrices into A−1 and evaluating A−1A = I(N+m)×(N+m), the matrix

inversion identities, Eqns. (C.1-C.4), can be attained.

Remark C.1.2 (Determinant Identity). Suppose we are given:

Θ = HT R−1H + P−1. (C.7)

The following identity holds:

(detΘ−1

)1/2=

(det P)1/2 (det R)1/2

[det(HPHT + R)

]1/2(C.8)

Proof. Recall from basic linear algebra, given AN×N and BN×N , the following determi-

nant identities hold:

det(AB) = det(A) det(B), (C.9)

det(A) = det(AT ). (C.10)

173

Also, given CN×m and Dm×m:

det

A C

0 D

= det(A) det(D). (C.11)

Now consider:

Γ =

P PHT

HP HPHT + R

,

=

(P−1 + HT R−1H)−1 PHT

0 HPHT + R

I 0

(HPHT + R)−1HP I

,

which gives det Γ = det[(HT R−1H + P−1)−1] det(HPHT + R). The following matrix

partition holds for a positive-definite symmetric matrix:

X11 X12

XT12 X22

−1

=

I −X−111 X12

0 I

X−111 0

0 (X22 − XT12X−1

11 X12)−1

I 0

−XT12X−1

11 I

, (C.12)

which gives det Γ = (det P)(det R). Combining det Γ from two approaches, we get:

det[(HT R−1H + P−1)−1] =(det P)(det R)

det(HPHT + R), (C.13)

(detΘ−1

)1/2=

(det P)1/2 (det R)1/2

[det(HPHT + R)

]1/2. (C.14)

C.2 Kalman Filter Derivation

Consider the following system dynamics:

dx(t) = F(t)x(t)dt + G(t)dβ(t), (C.15)

174

where x is the state vector with a dimension N , F is an N × N matrix characterizing the

system dynamics, G is an N × s noise input matrix characterizing system diffusion, and

β is an s-dimensional Brownian motion with:

E[β(t)] = 0, (C.16)

E[dβ(t)dβT (t)] = Q(t)dt, (C.17)

E[β(t2)− β(t1)][β(t2)− β(t1)]

T

=

∫ t2

t1

Q(τ)dτ. (C.18)

Since measurements are usually obtained in discrete form, we will consider the following

linear dynamics and linear measurements in discrete form:

xk = Φ(tk, tk−1)xk−1 + wk−1, (C.19)

zk = Hkxk + vk, (C.20)

where wk−1 and vk are white noises with E[wk] = E[vk] = 0, E[wkwTk ] = Qk, and

E[vkvTk ] = Rk. Note that wk−1 can be understood as a linear stochastic integral:

wk−1 =

∫ tk

tk−1

Φ(tk, τ)G(τ)dβ(τ), (C.21)

and E[wkwTk ] can be computed as:

E[wkwTk ] = Qk =

∫ tk

tk−1

Φ(tk, τ)G(τ)Q(τ)GT (τ)ΦT (tk, τ)dτ. (C.22)

Using Bayes’ rule, the posterior conditional density (i.e., measurement update proba-

bility density function) can be stated as:

p(xk|z1:k) =p(xk|z1:k−1)p(zk|xk, z1:k−1)

p(zk|z1:k−1), (C.23)

=p(xk|z1:k−1)p(zk|xk, z1:k−1)∫

∞ p(xk|z1:k−1)p(zk|xk, z1:k−1)dxk

, (C.24)

where z1:k = z1, z2, · · · , zk. Now suppose a state at tk−1 can be characterized as Gaus-

sian, i.e., xk−1 ∼ N (m+k−1, P+

k−1). The system equations, Eqn. (C.19) and (C.20) are both

175

linear and process (wk) and measurement (vk) noises are Gaussian. Thus, it is apparent that

the probability density functions in Eqn. (C.23) must be Gaussian as well. First consider

p(xk|z1:k−1) in the numerator, which describes the evolution of the probability density of

xk. The mean and covariance matrix can be mapped to tk by computing:

m−k = E [xk|z1:k−1] = Φ(tk, tk−1)m+

k−1, (C.25)

P−k = E[(

xk −m−k

) (xk −m−

k

)T |z1:k−1

],

= Φ(tk, tk−1)P+k−1Φ

T (tk, tk−1) +

∫ tk

tk−1

Φ(tk, τ)G(τ)Q(τ)GT (τ)ΦT (tk, τ)dτ,

(C.26)

which gives:

p(xk|z1:k−1) =(detΛ−

k )1/2

(2π)N/2exp

−1

2

(xk −m−

k

)TΛ−

k

(xk −m−

k

), (C.27)

where Λ−k = (P−k )−1. Now, consider the second numerator in the Bayes’ rule equation.

The mean and covariance matrix of the measurement at tk yields:

E [zk|xk, z1:k−1] = Hkxk, (C.28)

E[(zk −Hkxk)(zk −Hkxk)

T |xk, z1:k−1

]= Rk, (C.29)

and the probability density function can be stated as:

p(zk|xk, z1:k−1) =(det Rk)

−1/2

(2π)m/2exp

−1

2(zk −Hkxk)

T R−1k (zk −Hkxk)

. (C.30)

Remark C.2.1. The measurement probability density function, p(zk|z1:k−1) can be directly

computed as:

E [zk|z1:k−1] = E [Hkxk + vk|z1:k−1] = Hkm−k , (C.31)

E [Pzk|z1:k−1] = E

[(zk −Hkm−

k )(zk −Hkm−k )T |z1:k−1

],

= E[(Hkxk −Hkm−

k + vk)(Hkxk −Hkm−k + vk)

T |z1:k−1

],

= HkP−k HTk + Rk, (C.32)

176

and the resulting probability density function is:

p(zk|z1:k−1) =e−

12(zk−Hkm−k )

T(HkP−k HT

k +Rk)−1

(zk−Hkm−k )

(2π)m/2[det(HkP−k HTk + Rk)]1/2

.

Note that the measurement probability density function, p(zk|z1:k−1), can be computed

directly by applying the linearity of the measurement function and the Gaussian assump-

tion. However, we carry out the integration given in Eqn. (C.33) to show that these two

approaches yield the same solution.

Now, consider the denominator of the posterior conditional density function:

p(zk|z1:k−1) =

∫

∞p(zk|xk, z1:k−1)p(xk|z1:k−1)dxk

=

∫

∞

exp−12

·

︷ ︸︸ ︷[(zk −Hkxk)

T R−1k (zk −Hkxk) + (xk −m−

k )TΛ−k (xk −m−

k )]

(2π)N+m

2 (det Rk)1/2(detΛ−k )1/2

dxk.

(C.33)

· =

zTk R−1

k zk − xTk HT

k R−1k zk − zT

k R−1k Hkxk + xT

k HTk R−1

k Hkxk

+ xTΛ−k xk −mT

k Λ−k xk − xT

k Λ−k mk + mT

k Λ−k mk

,

=

xTk (HT

k R−1k Hk + Λ−

k )xk − 2xTk (HT

k R−1k zk + Λ−

k mk) + zTk R−1

k zk + mTk Λ−

k mk

,

(C.34)

where we let Pk = P−k and mk = m−k to ease the notation. Now define:

Θk = HTk R−1

k Hk + P−1k , (C.35)

ak = HTk R−1

k zk + P−1k mk. (C.36)

177

Then Eqn. (C.34) can be re-written as:

· =

xTk Θkxk − 2xT

k ak + zTk R−1

k zk + mTk P−1

k mk

,

=

xTk Θkxk − 2xT

k ΘkΘ−1k ak + aT

k Θ−1k ΘkΘ

−1k ak

− aTk Θ−1

k ak + zTk R−1

k zk + mTk P−1

k mk

,

=

zTk R−1

k zk + mTk P−1

k mk − aTk Θ−1

k ak + (xk −Θ−1k ak)

TΘk(xk −Θ−1k ak)

.

The measurement density function then becomes:

p(zk|z1:k−1)

=e−

12zT

k R−1k zk+mT

k P−1k mk−aT

k Θ−1k ak ∫

e−12(xk−Θ−1

k ak)T Θk(xk−Θ−1k ak)

(2π)N+m

2 (det Rk)1/2(det Pk)1/2dxk,

=e−

12zT

k R−1k zk+mT

k P−1k mk−aT

k Θ−1k ak(2π)N/2 |Θ−1|1/2

(2π)N+m

2 (det Rk)1/2(det Pk)1/2,

=1

(2π)m/2∣∣HkPkHT

k + Rk

∣∣1/2e−

12

·

︷ ︸︸ ︷(zT

k R−1k zk + mT

k P−1k mk − aT

k Θ−1k ak

), (C.37)

where the third equality applies the result from Remark C.1.2. Using the matrix inversion

lemma, we can factor out the exponential function in Eqn. (C.37) as:

· = zTk R−1

k zk + mTk P−1

k mk − aTk Θ−1

k ak,

= zTk R−1

k zk + mTk P−1

k mk − zTk R−1

k

[Rk − Rk

(HkPkHT

k + Rk

)−1 Rk

]R−1

k zk

− 2mTk P−1

k PkHTk

(HkPkHT

k + Rk

)−1 zk

− mTk P−1

k

[Pk − PkHT

k

(HkPkHT

k + Rk

)−1 HkPk

]P−1

k mk,

= zTk R−1

k zk + mTk P−1

k mk − zTk

[R−1

k − (HkPkHT

k + Rk

)−1]

zk

− 2mTk HT

k

(HkPkHT

k + Rk

)−1 zk −mTk

[P−1

k −HTk

(HkPkHT

k + Rk

)−1 Hk

]mk,

= zTk

(HkPkHT

k + Rk

)−1 zk − 2mTk HT

k

(HkPkHT

k + Rk

)−1 zk

+ mTk HT

k

(HkPkHT

k + Rk

)−1 Hkmk.

178

Applying this result, the measurement probability density function becomes:

p(zk|z1:k−1) =1

(2π)m/2 det(HkP−k HTk + Rk)]1/2

e−12(zk−Hkmk)T (HkPkHT

k +Rk)−1

(zk−Hkmk),

(C.38)

which is identical to Eqn. (C.33) as expected.

At this point, simplifying the numerator of the posterior conditional density function

becomes straightforward as the process is similar computing p(zk|z1:k−1). The complete

representation of the updated probability density function can be stated as:

p(xk|z1:k) =p(xk|z1:k−1)p(zk|xk, z1:k−1)∫

∞ p(xk|z1:k−1)p(zk|xk, z1:k−1)dxk

,

=1

(2π)N/2(detΘ−1k )1/2

exp

−1

2(xk −Θ−1

k ak)TΘk(xk −Θ−1

k ak)

,

(C.39)

which gives:

m+k = E [xk|z1:k] = Θ−1

k ak,

= m−k + P−k HT

k

[HkP−k HT

k + Rk

]−1(zk −Hkm−

k ), (C.40)

P−k = E[(

xk −m+k

) (xk −m+

k

)T |z1:k

]= Θ−1

k ,

= P−k − P−k HTk

[HkP−k HT

k + Rk

]−1 HkP−k , (C.41)

where the matrix inversion lemma is applied again.

The conventional Kalman Filter algorithm is then stated as follows:

Definition C.2.2. Prediction:

m−k = Φ(tk, tk−1)m+

k−1, (C.42)

P−k = Φ(tk, tk−1)P+k−1Φ

T (tk, tk−1) +

∫ tk

tk−1

Φ(tk, τ)G(τ)Q(τ)GT (τ)ΦT (tk, τ)dτ.

(C.43)

179

Definition C.2.3. Update:

m+k = m−

k + Kk

(zk −Hkm−

k

), (C.44)

P+k = P−k −KkHkP−k , (C.45)

Kk = P−k HTk

(HkP−k HT

k + Rk

)−1, (C.46)

where Kk = Pxzk (Pzz

k )−1 is called the Kalman gain.

180

APPENDIX D

VECTORIZATION OF HIGHER ORDER TENSORS

When integrating the higher order state transition tensors, it is more convenient to put

them into a vectorized form.

D.1 Specifications for MATLAB

Consider a system with dimension N . Let S(i) be the ith element of a tensor φγ1,γ2···γp ,

where the element number increases from the last index γp to the first index γ1. For

example, a system with dimension 2, the second order tensor φγ1,γ2 can be stated as:

S(1) = φ1,1, S(2) = φ1,2, S(3) = φ2,1, S(4) = φ2,2. Now, let U(i) be the ith element

number of the tensor in a vector form. The previous example can be stated as: U(1) = φ1,

U(2) = φ2, U(3) = φ1,1, U(4) = φ1,2, U(5) = φ2,1, U(6) = φ2,2. Note that U(1) = φ1

and U(2) = φ2 include zeroth-order tensors, i.e., the state components.

Example D.1.1. An element of a second order tensor can be stated as:

φi,j = S[N(i− 1) + j],

= U [N + N(i− 1) + j],

= U(iN + j).

181

Example D.1.2. An element of a third order tensor can be stated as:

φi,jk = S[N2(i− 1) + N(j − 1) + k],

= U [N + N2 + N2(i− 1) + N(j − 1) + k],

= U(iN2 + jN + k).

Example D.1.3. An element of a fourth order tensor can be stated as:

φi,jkl = S[N3(i− 1) + N2(j − 1) + N(k − 1) + l],

= U(iN3 + jN2 + kN + l).

In general, an element of a tensor can be stated as:

φγ1,γ2···γp = S

[1 +

p∑i=1

Np−i(γi − 1)

], (D.1)

= U

(p∑

i=1

Np−iγi

), (D.2)

where γj ∈ 1, · · · , N. Note that MATLAB initializes an array with an integer 1, and

thus, this generalization cannot be applied to C++ or alike.

D.2 Specifications for C or C++

In C or C++ programming language, an array is initialized with an integer 0. In general,

an element of a tensor can be stated as:

φγ1,γ2···γp = S

(p∑

i=1

Np−iγi

), (D.3)

= U

[(p∑

i=1

Np−iγi

)+

(p∑

i=1

Np−i

)− 1

], (D.4)

= U

[p∑

i=1

Np−i (γi + 1)

]− 1

, (D.5)

where γj ∈ 0, · · · , N − 1.

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182

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ABSTRACT

NONLINEAR TRAJECTORY NAVIGATION

by

Sang H. Park

Chair: Daniel J. Scheeres

Trajectory navigation entails the solution of many different problems that arise due

to uncertain knowledge of the spacecraft state, including orbit prediction, correction ma-

neuver design, and trajectory estimation. In practice, these problems are usually solved

based on an assumption that linear dynamical models sufficiently approximate the local

trajectory dynamics and their associated statistics. However, astrodynamics problems are

nonlinear in general and linear spacecraft dynamics models can fail to characterize the true

trajectory dynamics when the system is subject to a highly unstable environment or when

mapped over a long time period. This limits the performance of traditional navigation

techniques and can make it difficult to perform precision analysis or robust navigation.

This dissertation presents an alternate method for spacecraft trajectory navigation based

on a nonlinear local trajectory model and their statistics in an analytic framework. For a

given reference trajectory, we first solve for the higher order Taylor series terms that de-

scribe the localized nonlinear motion and develop an analytic expression for the relative

solution flow. We then discuss the nonlinear dynamical mapping of a spacecraft’s proba-

bility density function by solving the Fokker-Planck equation for a deterministic system.

From this result we derive an analytic method for orbit uncertainty propagation which

can replicate Monte-Carlo simulations with the benefit of added flexibility in initial orbit

statistics.

Using this approach, we introduce the concept of the statistically correct trajectory

where we directly incorporate statistical information about an orbit state into the trajectory

design process. As an extension of this concept, we define a nonlinear statistical target-

ing method where we solve for a correction maneuver which intercepts the desired target

on average. Then we apply our results to a Bayesian filtering problem to obtain a gen-

eral filtering algorithm for optimal estimation of the posterior conditional density function

incorporating nonlinearity into the filtering process. Finally, we derive practical Kalman-

type filters by applying our nonlinear relative solutions into the standard filters and show

that these filters provide superior performance over linear filtering methods based on re-

alistic trajectory and uncertainty models. The examples we consider are a conventional

Hohmann transfer from the Earth to Moon using a simple two-body model, a strongly un-

stable transfer trajectory in the Hill three-body problem from the vicinity of L2 through

several orbits, and to the navigation of a spacecraft in a halo orbit in the restricted three-

body problem. For each of these examples we show the benefits of using our nonlinear

trajectory navigation techniques as compared to traditional linear navigation techniques.