Holdings: Statistical orbit determination

Statistical orbit determination:

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Bibliographic Details
Main Author:	Tapley, Byron D. (Author)
Format:	Book
Language:	English
Published:	Amsterdam [u.a.] Elsevier Academic Press 2004
Subjects:	Mécanique orbitale Satellites artificiels - Orbites - Modèles mathématiques Trajectoires spatiales Mathematisches Modell Orbital mechanics Artificial satellites > Orbits > Mathematical models Space trajectories Bahnelement Satellit Umlaufbahn Statistik
Online Access:	Table of contents Publisher description Inhaltsverzeichnis
Physical Description:	XV, 547 S. Ill. 24 cm
ISBN:	0126836302

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650		4	\|a Mécanique orbitale
650		4	\|a Satellites artificiels - Orbites - Modèles mathématiques
650		4	\|a Trajectoires spatiales
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650		4	\|a Orbital mechanics
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adam_text	STATISTICAL ORBIT DETERMINATION BYRON D. TAPLEY CENTER FOR SPACE RESEARCH THE UNIVERSITY OF TEXAS AT AUSTIN BOB E. SCHUTZ CENTER FOR SPACE RESEARCH THE UNIVERSITY OF TEXAS AT AUSTIN GEORGE H. BORN COLORADO CENTER FOR ASTRODYNAMICS RESEARCH UNIVERSITY OF COLORADO, BOULDER ELSEVIER ACADEMIC PRESS AMSTERDAM * BOSTON * HEIDELBERG * LONDON * NEW YORK * OXFORD PARIS * SAN DIEGO * SAN FRANCISCO * SINGAPORE * SYDNEY * TOKYO CONTENTS PREFACE XI 1 ORBIT DETERMINATION CONCEPTS 1 1.1 INTRODUCTION 1 1.2 UNIFORM GRAVITY FIELD MODEL 3 1.2.1 FORMULATION OF THE PROBLEM 4 1.2.2 THE EQUATION OF THE ORBIT 5 1.2.3 THE ROLE OF THE OBSERVATION 5 1.2.4 LINEARIZATION PROCEDURE 9 1.2.5 STATE TRANSITION MATRIX 11 1.3 BACKGROUND AND OVERVIEW 13 1.4 SUMMARY 14 1.5 REFERENCES 15 1.6 EXERCISES 16 2 THE ORBIT PROBLEM 17 2.1 HISTORICAL BACKGROUND 17 2.2 PROBLEM OF TWO BODIES: GENERAL PROPERTIES . 19 2.2.1 MOTION IN THE PLANE 21 2.2.2 MOTION IN SPACE 28 2.2.3 BASIC COORDINATE SYSTEMS 29 2.2.4 ORBIT ELEMENTS AND POSITION/VELOCITY 32 2.2.5 POSITION/VELOCITY PREDICTION 35 2.2.6 STATE TRANSITION MATRIX AND ERROR PROPAGATION 40 2.3 PERTURBED MOTION 44 2.3.1 CLASSICAL EXAMPLE: LUNAR PROBLEM 45 2.3.2 VARIATION OF PARAMETERS 48 2.3.3 GRAVITATIONAL PERTURBATIONS: MASS DISTRIBUTION 50 2.3.4 GRAVITATIONAL PERTURBATIONS: OBLATENESS AND OTHER EFFECTS 56 2.3.5 GRAVITATIONAL PERTURBATIONS: THIRD-BODY EFFECTS 61 IN IV CONTENTS 2.3.6 GRAVITATIONAL PERTURBATIONS: TEMPORAL CHANGES IN GRAVITY 64 2.3.7 GRAVITATIONAL PERTURBATIONS: GENERAL RELATIVITY 65 2.3.8 NONGRAVITATIONAL PERTURBATIONS: ATMOSPHERIC RESISTANCE . 66 2.3.9 NONGRAVITATIONAL PERTURBATIONS: RADIATION PRESSURE .... 68 2.3.10 NONGRAVITATIONAL PERTURBATIONS: OTHER 68 2.3.11 PERTURBED EQUATIONS OF MOTION: SUMMARY 69 2.4 COORDINATE SYSTEMS AND TIME: INTRODUCTION 71 2.4.1 PRECESSION AND NUTATION 71 2.4.2 EARTH ROTATION AND TIME 74 2.4.3 EARTH-FIXED AND TOPOCENTRIC SYSTEMS 77 2.4.4 TRANSFORMATION BETWEEN ECF AND ECI 82 2.5 ORBIT ACCURACY 82 2.6 REFERENCES 84 2.7 EXERCISES 87 3 OBSERVATIONS 93 3.1 INTRODUCTION 93 3.2 OBSERVATIONS 94 3.2.1 IDEAL RANGE 94 3.2.2 IDEAL RANGE-RATE 95 3.2.3 IDEAL AZIMUTH AND ELEVATION ANGLES 95 3.2.4 EXAMPLES: IDEAL OBSERVATIONS 95 3.3 CONCEPTUAL MEASUREMENT SYSTEMS 99 3.3.1 RANGE 99 3.3.2 RANGE-RATE 107 3.4 REALIZATION OF MEASUREMENTS 110 3.4.1 CONSIDERATIONS 110 3.4.2 ATMOSPHERIC EFFECTS 110 3.4.3 GENERAL RELATIVITY 115 3.5 MEASUREMENT SYSTEMS 116 3.5.1 ONE-WAY RANGE 116 3.5.2 TWO-WAY RANGE 125 3.5.3 DOPPLER SYSTEMS 132 3.5.4 EXAMPLES 134 3.6 DIFFERENCED MEASUREMENTS 139 3.6.1 DIFFERENCED GPS MEASUREMENTS 140 3.6.2 DIFFERENCED ALTIMETER DATA 145 3.7 SATELLITE POSITIONS 147 3.8 ANGLES 148 3.9 REFERENCES 149 3.10 EXERCISES 151 CONTENTS V 4 FUNDAMENTALS OF ORBIT DETERMINATION 159 4.1 INTRODUCTION 159 4.2 LINEARIZATION OF THE ORBIT DETERMINATION PROCESS 160 4.2.1 THE STATE TRANSITION MATRIX 164 4.2.2 SOLUTION FOR THE STATE TRANSITION MATRIX 167 4.2.3 RELATING THE OBSERVATIONS TO AN EPOCH STATE 172 4.3 THE LEAST SQUARES SOLUTION 173 4.3.1 THE MINIMUM NORM SOLUTION 174 4.3.2 SHORTCOMINGS OF THE LEAST SQUARES SOLUTION 176 4.3.3 WEIGHTED LEAST SQUARES SOLUTION 176 4.3.4 AN ALTERNATE LEAST SQUARES APPROACH 178 4.4 THE MINIMUM VARIANCE ESTIMATE 183 4.4.1 PROPAGATION OF THE ESTIMATE AND COVARIANCE MATRIX .... 188 4.4.2 MINIMUM VARIANCE ESTIMATE WITH A PRIORI INFORMATION . . 188 4.5 MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION 190 4.6 COMPUTATIONAL ALGORITHM FOR THE BATCH PROCESSOR 194 4.7 THE SEQUENTIAL ESTIMATION ALGORITHM 199 4.7.1 THE SEQUENTIAL COMPUTATIONAL ALGORITHM 201 4.7.2 THE EXTENDED SEQUENTIAL ESTIMATION ALGORITHM 209 4.7.3 THE EXTENDED SEQUENTIAL COMPUTATIONAL ALGORITHM .... 210 4.7.4 THE PREDICTION RESIDUAL 210 4.8 EXAMPLE PROBLEMS 211 4.8.1 LINEAR SYSTEM 211 4.8.2 SPRING-MASS PROBLEM 216 4.9 STATE NOISE COMPENSATION ALGORITHM 220 4.9.1 THE GAUSS-MARKOV PROCESS 230 4.10 INFORMATION FILTER 233 4.11 BATCH AND SEQUENTIAL ESTIMATION 237 4.12 OBSERVABILITY 237 4.13 ERROR SOURCES 241 4.14 ORBIT ACCURACY 244 4.15 SMOOTHING 246 4.15.1 COMPUTATIONAL ALGORITHM FOR SMOOTHER 250 4.16 THE PROBABILITY ELLIPSOID 251 4.16.1 TRANSFORMATION OF THE COVARIANCE MATRIX BETWEEN COORDINATE SYSTEMS 257 4.17 COMBINING ESTIMATES 258 4.18 REFERENCES 261 4.19 EXERCISES 264 VI CONTENTS 5 SQUARE ROOT SOLUTION METHODS 285 5.1 INTRODUCTION 285 5.2 CHOLESKY DECOMPOSITION 286 5.2.1 THE CHOLESKY ALGORITHM 287 5.2.2 THE SQUARE ROOT FREE CHOLESKY ALGORITHM 288 5.3 LEAST SQUARES SOLUTION VIA ORTHOGONAL TRANSFORMATION 290 5.4 GIVENS TRANSFORMATIONS 291 5.4.1 A PRIORI INFORMATION AND INITIALIZATION 294 5.4.2 GIVENS COMPUTATIONAL ALGORITHM 298 5.4.3 SQUARE ROOT FREE GIVENS TRANSFORMATION 301 5.4.4 SQUARE ROOT FREE GIVENS COMPUTATIONAL ALGORITHM . . . . 305 5.4.5 A SIMPLIFIED SQUARE ROOT FREE GIVENS TRANSFORMATION . .307 5.4.6 IMPLEMENTATION CONSIDERATIONS 309 5.5 THE HOUSEHOLDER TRANSFORMATION 310 5.5.1 APPLICATION TO THE SOLUTION OF THE LEAST SQUARES PROBLEM . 315 5.5.2 HOUSEHOLDER COMPUTATIONAL ALGORITHM 316 5.6 NUMERICAL EXAMPLES 318 5.6.1 CHOLESKY DECOMPOSITION 318 5.6.2 GIVENS TRANSFORMATION 320 5.6.3 HOUSEHOLDER TRANSFORMATION 322 5.6.4 A MORE ILLUSTRATIVE EXAMPLE 324 5.6.5 CHOLESKY DECOMPOSITION 325 5.6.6 SQUARE ROOT FREE GIVENS TRANSFORMATION 326 5.6.7 THE HOUSEHOLDER TRANSFORMATION 329 5.7 SQUARE ROOT FILTER ALGORITHMS 330 5.7.1 THE SQUARE ROOT MEASUREMENT UPDATE ALGORITHMS .... 332 5.7.2 SQUARE ROOT FREE MEASUREMENT UPDATE ALGORITHMS .... 340 5.8 TIME UPDATE OF THE ESTIMATION ERROR COVARIANCE MATRIX 343 5.9 CONTINUOUS STATE ERROR COVARIANCE PROPAGATION 346 5.9.1 TRIANGULAR SQUARE ROOT ALGORITHM 349 5.10 THE SQUARE ROOT INFORMATION FILTER 351 5.10.1 THE SQUARE ROOT INFORMATION FILTER WITH TIME-DEPENDENT EFFECTS 353 5.10.2 THE DYNAMIC CASE WITH PROCESS NOISE 358 5.10.3 SRIF COMPUTATIONAL ALGORITHM 363 5.10.4 SMOOTHING WITH THE SRIF 364 5.11 PROCESS NOISE PARAMETER FILTERING/SMOOTHING USING A SRIF . . . 369 5.11.1 EXPONENTIALLY CORRELATED PROCESS NOISE SRIF 371 5.11.2 SMOOTHING WITH A SRIF 377 5.12 REFERENCES 380 5.13 EXERCISES 381 CONTENTS VII 6 CONSIDER COVARIANCE ANALYSIS 387 6.1 INTRODUCTION 387 6.2 BIAS IN LINEAR ESTIMATION PROBLEMS 388 6.3 FORMULATION OF THE CONSIDER COVARIANCE MATRIX 389 6.4 THE SENSITIVITY AND PERTURBATION MATRICES 397 6.4.1 EXAMPLE APPLICATION OF A SENSITIVITY AND PERTURBATION MATRIX 398 6.5 INCLUSION OF TIME-DEPENDENT EFFECTS 400 6.6 PROPAGATION OF THE ERROR COVARIANCE 405 6.7 SEQUENTIAL CONSIDER COVARIANCE ANALYSIS 407 6.8 EXAMPLE: FREELY FALLING POINT MASS 410 6.8.1 PROPAGATION WITH TIME 415 6.8.2 THE SEQUENTIAL CONSIDER ALGORITHM 416 6.8.3 PERTURBATION IN THE STATE ESTIMATE 419 6.9 EXAMPLE: SPRING-MASS PROBLEM 420 6.10 ERRORS IN THE OBSERVATION NOISE AND A PRIORI STATE COVARIANCES . . 425 6.11 ERRORS IN PROCESS NOISE, OBSERVATION NOISE, AND STATE COVARIANCE 427 6.12 COVARIANCE ANALYSIS AND ORTHOGONAL TRANSFORMATIONS 430 6.13 REFERENCES 434 6.14 EXERCISES 435 A PROBABILITY AND STATISTICS 439 A.I INTRODUCTION 439 A.2 AXIOMS OF PROBABILITY 440 A.3 CONDITIONAL PROBABILITY 443 A.4 PROBABILITY DENSITY AND DISTRIBUTION FUNCTIONS 443 A.5 EXPECTED VALUES 445 A.6 EXAMPLES AND DISCUSSION OF EXPECTATION 446 A.7 MOMENT GENERATING FUNCTIONS 448 A.8 SOME IMPORTANT CONTINUOUS DISTRIBUTIONS 449 A.8.1 UNIFORM OR RECTANGULAR DISTRIBUTION 449 A.8.2 THE GAUSSIAN OR NORMAL DISTRIBUTION 450 A.9 TWO RANDOM VARIABLES 452 A. 10 MARGINAL DISTRIBUTIONS 453 A. 11 INDEPENDENCE OF RANDOM VARIABLES 454 A.12 CONDITIONAL PROBABILITY 454 A. 13 EXPECTED VALUES OF BIVARIATE FUNCTIONS 455 A. 14 THE VARIANCE-COVARIANCE MATRIX 456 A.15 PROPERTIES OF THE CORRELATION COEFFICIENT 458 A. 16 PROPERTIES OF COVARIANCE AND CORRELATION 460 A. 17 BIVARIATE NORMAL DISTRIBUTION 460 A.18 MARGINAL DISTRIBUTIONS 461 VIII CONTENTS A. 19 THE MULTIVANATE NORMAL DISTRIBUTION 462 A. 19.1 THE CONDITIONAL DISTRIBUTION FOR MULTIVARIATE NORMAL VARIABLES 464 A.20 THE CENTRAL LIMIT THEOREM 465 A.21 BAYES THEOREM 465 A.22 STOCHASTIC PROCESSES 467 A.22.1 DEFINITIONS FOR STOCHASTIC PROCESSES 469 A.23 REFERENCES 471 B REVIEW OF MATRIX CONCEPTS 473 B.I INTRODUCTION 473 B.2 RANK 475 B.3 QUADRATIC FORMS 475 B.4 DETERMINANTS 476 B.5 MATRIX TRACE 477 B.6 EIGENVALUES AND EIGENVECTORS 478 B.7 THE DERIVATIVES OF MATRICES AND VECTORS 478 B.8 MAXIMA AND MINIMA 480 B.9 USEFUL MATRIX INVERSION THEOREMS 481 B.10 REFERENCE 483 C EQUATIONS OF MOTION 485 C.I LAGRANGE PLANETARY EQUATIONS 485 C.2 GAUSSIAN FORM 485 C.3 REFERENCES 486 D CONSTANTS 487 D.I PHYSICAL CONSTANTS 487 D.2 EARTH CONSTANTS 487 D.3 LUNAR, SOLAR, AND PLANETARY MASSES 488 D.4 REFERENCES 490 E ANALYTICAL THEORY FOR NEAR-CIRCULAR ORBITS 493 E.I DESCRIPTION 493 E.2 EXAMPLE 497 E.3 REFERENCES 497 F EXAMPLE OF STATE NOISE AND DYNAMIC MODEL COMPENSATION 499 F.I INTRODUCTION 499 F.2 STATE NOISE COMPENSATION 501 F.3 DYNAMIC MODEL COMPENSATION 505 F.4 REFERENCE 509 CONTENTS IX G SOLUTION OF THE LINEARIZED EQUATIONS OF MOTION 511 G.I INTRODUCTION 511 G.2 THE STATE TRANSITION MATRIX 513 H TRANSFORMATION BETWEEN ECI AND ECF COORDINATES 517 H.I INTRODUCTION 517 H.2 MATRIX P 518 H.3 MATRIX N 519 H.4 MATRIX S 519 H.5 MATRIX W 521 H.6 REFERENCES 521 BIBLIOGRAPHY ABBREVIATIONS 523 BIBLIOGRAPHY 525 AUTHOR INDEX 537 INDEX 541
adam_txt	STATISTICAL ORBIT DETERMINATION BYRON D. TAPLEY CENTER FOR SPACE RESEARCH THE UNIVERSITY OF TEXAS AT AUSTIN BOB E. SCHUTZ CENTER FOR SPACE RESEARCH THE UNIVERSITY OF TEXAS AT AUSTIN GEORGE H. BORN COLORADO CENTER FOR ASTRODYNAMICS RESEARCH UNIVERSITY OF COLORADO, BOULDER ELSEVIER ACADEMIC PRESS AMSTERDAM * BOSTON * HEIDELBERG * LONDON * NEW YORK * OXFORD PARIS * SAN DIEGO * SAN FRANCISCO * SINGAPORE * SYDNEY * TOKYO CONTENTS PREFACE XI 1 ORBIT DETERMINATION CONCEPTS 1 1.1 INTRODUCTION 1 1.2 UNIFORM GRAVITY FIELD MODEL 3 1.2.1 FORMULATION OF THE PROBLEM 4 1.2.2 THE EQUATION OF THE ORBIT 5 1.2.3 THE ROLE OF THE OBSERVATION 5 1.2.4 LINEARIZATION PROCEDURE 9 1.2.5 STATE TRANSITION MATRIX 11 1.3 BACKGROUND AND OVERVIEW 13 1.4 SUMMARY 14 1.5 REFERENCES 15 1.6 EXERCISES 16 2 THE ORBIT PROBLEM 17 2.1 HISTORICAL BACKGROUND 17 2.2 PROBLEM OF TWO BODIES: GENERAL PROPERTIES . 19 2.2.1 MOTION IN THE PLANE 21 2.2.2 MOTION IN SPACE 28 2.2.3 BASIC COORDINATE SYSTEMS 29 2.2.4 ORBIT ELEMENTS AND POSITION/VELOCITY 32 2.2.5 POSITION/VELOCITY PREDICTION 35 2.2.6 STATE TRANSITION MATRIX AND ERROR PROPAGATION 40 2.3 PERTURBED MOTION 44 2.3.1 CLASSICAL EXAMPLE: LUNAR PROBLEM 45 2.3.2 VARIATION OF PARAMETERS 48 2.3.3 GRAVITATIONAL PERTURBATIONS: MASS DISTRIBUTION 50 2.3.4 GRAVITATIONAL PERTURBATIONS: OBLATENESS AND OTHER EFFECTS 56 2.3.5 GRAVITATIONAL PERTURBATIONS: THIRD-BODY EFFECTS 61 IN IV CONTENTS 2.3.6 GRAVITATIONAL PERTURBATIONS: TEMPORAL CHANGES IN GRAVITY 64 2.3.7 GRAVITATIONAL PERTURBATIONS: GENERAL RELATIVITY 65 2.3.8 NONGRAVITATIONAL PERTURBATIONS: ATMOSPHERIC RESISTANCE . 66 2.3.9 NONGRAVITATIONAL PERTURBATIONS: RADIATION PRESSURE . 68 2.3.10 NONGRAVITATIONAL PERTURBATIONS: OTHER 68 2.3.11 PERTURBED EQUATIONS OF MOTION: SUMMARY 69 2.4 COORDINATE SYSTEMS AND TIME: INTRODUCTION 71 2.4.1 PRECESSION AND NUTATION 71 2.4.2 EARTH ROTATION AND TIME 74 2.4.3 EARTH-FIXED AND TOPOCENTRIC SYSTEMS 77 2.4.4 TRANSFORMATION BETWEEN ECF AND ECI 82 2.5 ORBIT ACCURACY 82 2.6 REFERENCES 84 2.7 EXERCISES 87 3 OBSERVATIONS 93 3.1 INTRODUCTION 93 3.2 OBSERVATIONS 94 3.2.1 IDEAL RANGE 94 3.2.2 IDEAL RANGE-RATE 95 3.2.3 IDEAL AZIMUTH AND ELEVATION ANGLES 95 3.2.4 EXAMPLES: IDEAL OBSERVATIONS 95 3.3 CONCEPTUAL MEASUREMENT SYSTEMS 99 3.3.1 RANGE 99 3.3.2 RANGE-RATE 107 3.4 REALIZATION OF MEASUREMENTS 110 3.4.1 CONSIDERATIONS 110 3.4.2 ATMOSPHERIC EFFECTS 110 3.4.3 GENERAL RELATIVITY 115 3.5 MEASUREMENT SYSTEMS 116 3.5.1 ONE-WAY RANGE 116 3.5.2 TWO-WAY RANGE 125 3.5.3 DOPPLER SYSTEMS 132 3.5.4 EXAMPLES 134 3.6 DIFFERENCED MEASUREMENTS 139 3.6.1 DIFFERENCED GPS MEASUREMENTS 140 3.6.2 DIFFERENCED ALTIMETER DATA 145 3.7 SATELLITE POSITIONS 147 3.8 ANGLES 148 3.9 REFERENCES 149 3.10 EXERCISES 151 CONTENTS V 4 FUNDAMENTALS OF ORBIT DETERMINATION 159 4.1 INTRODUCTION 159 4.2 LINEARIZATION OF THE ORBIT DETERMINATION PROCESS 160 4.2.1 THE STATE TRANSITION MATRIX 164 4.2.2 SOLUTION FOR THE STATE TRANSITION MATRIX 167 4.2.3 RELATING THE OBSERVATIONS TO AN EPOCH STATE 172 4.3 THE LEAST SQUARES SOLUTION 173 4.3.1 THE MINIMUM NORM SOLUTION 174 4.3.2 SHORTCOMINGS OF THE LEAST SQUARES SOLUTION 176 4.3.3 WEIGHTED LEAST SQUARES SOLUTION 176 4.3.4 AN ALTERNATE LEAST SQUARES APPROACH 178 4.4 THE MINIMUM VARIANCE ESTIMATE 183 4.4.1 PROPAGATION OF THE ESTIMATE AND COVARIANCE MATRIX . 188 4.4.2 MINIMUM VARIANCE ESTIMATE WITH A PRIORI INFORMATION . . 188 4.5 MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION 190 4.6 COMPUTATIONAL ALGORITHM FOR THE BATCH PROCESSOR 194 4.7 THE SEQUENTIAL ESTIMATION ALGORITHM 199 4.7.1 THE SEQUENTIAL COMPUTATIONAL ALGORITHM 201 4.7.2 THE EXTENDED SEQUENTIAL ESTIMATION ALGORITHM 209 4.7.3 THE EXTENDED SEQUENTIAL COMPUTATIONAL ALGORITHM . 210 4.7.4 THE PREDICTION RESIDUAL 210 4.8 EXAMPLE PROBLEMS 211 4.8.1 LINEAR SYSTEM 211 4.8.2 SPRING-MASS PROBLEM 216 4.9 STATE NOISE COMPENSATION ALGORITHM 220 4.9.1 THE GAUSS-MARKOV PROCESS 230 4.10 INFORMATION FILTER 233 4.11 BATCH AND SEQUENTIAL ESTIMATION 237 4.12 OBSERVABILITY 237 4.13 ERROR SOURCES 241 4.14 ORBIT ACCURACY 244 4.15 SMOOTHING 246 4.15.1 COMPUTATIONAL ALGORITHM FOR SMOOTHER 250 4.16 THE PROBABILITY ELLIPSOID 251 4.16.1 TRANSFORMATION OF THE COVARIANCE MATRIX BETWEEN COORDINATE SYSTEMS 257 4.17 COMBINING ESTIMATES 258 4.18 REFERENCES 261 4.19 EXERCISES 264 VI CONTENTS 5 SQUARE ROOT SOLUTION METHODS 285 5.1 INTRODUCTION 285 5.2 CHOLESKY DECOMPOSITION 286 5.2.1 THE CHOLESKY ALGORITHM 287 5.2.2 THE SQUARE ROOT FREE CHOLESKY ALGORITHM 288 5.3 LEAST SQUARES SOLUTION VIA ORTHOGONAL TRANSFORMATION 290 5.4 GIVENS TRANSFORMATIONS 291 5.4.1 A PRIORI INFORMATION AND INITIALIZATION 294 5.4.2 GIVENS COMPUTATIONAL ALGORITHM 298 5.4.3 SQUARE ROOT FREE GIVENS TRANSFORMATION 301 5.4.4 SQUARE ROOT FREE GIVENS COMPUTATIONAL ALGORITHM . . . . 305 5.4.5 A SIMPLIFIED SQUARE ROOT FREE GIVENS TRANSFORMATION . .307 5.4.6 IMPLEMENTATION CONSIDERATIONS 309 5.5 THE HOUSEHOLDER TRANSFORMATION 310 5.5.1 APPLICATION TO THE SOLUTION OF THE LEAST SQUARES PROBLEM . 315 5.5.2 HOUSEHOLDER COMPUTATIONAL ALGORITHM 316 5.6 NUMERICAL EXAMPLES 318 5.6.1 CHOLESKY DECOMPOSITION 318 5.6.2 GIVENS TRANSFORMATION 320 5.6.3 HOUSEHOLDER TRANSFORMATION 322 5.6.4 A MORE ILLUSTRATIVE EXAMPLE 324 5.6.5 CHOLESKY DECOMPOSITION 325 5.6.6 SQUARE ROOT FREE GIVENS TRANSFORMATION 326 5.6.7 THE HOUSEHOLDER TRANSFORMATION 329 5.7 SQUARE ROOT FILTER ALGORITHMS 330 5.7.1 THE SQUARE ROOT MEASUREMENT UPDATE ALGORITHMS . 332 5.7.2 SQUARE ROOT FREE MEASUREMENT UPDATE ALGORITHMS . 340 5.8 TIME UPDATE OF THE ESTIMATION ERROR COVARIANCE MATRIX 343 5.9 CONTINUOUS STATE ERROR COVARIANCE PROPAGATION 346 5.9.1 TRIANGULAR SQUARE ROOT ALGORITHM 349 5.10 THE SQUARE ROOT INFORMATION FILTER 351 5.10.1 THE SQUARE ROOT INFORMATION FILTER WITH TIME-DEPENDENT EFFECTS 353 5.10.2 THE DYNAMIC CASE WITH PROCESS NOISE 358 5.10.3 SRIF COMPUTATIONAL ALGORITHM 363 5.10.4 SMOOTHING WITH THE SRIF 364 5.11 PROCESS NOISE PARAMETER FILTERING/SMOOTHING USING A SRIF . . . 369 5.11.1 EXPONENTIALLY CORRELATED PROCESS NOISE SRIF 371 5.11.2 SMOOTHING WITH A SRIF 377 5.12 REFERENCES 380 5.13 EXERCISES 381 CONTENTS VII 6 CONSIDER COVARIANCE ANALYSIS 387 6.1 INTRODUCTION 387 6.2 BIAS IN LINEAR ESTIMATION PROBLEMS 388 6.3 FORMULATION OF THE CONSIDER COVARIANCE MATRIX 389 6.4 THE SENSITIVITY AND PERTURBATION MATRICES 397 6.4.1 EXAMPLE APPLICATION OF A SENSITIVITY AND PERTURBATION MATRIX 398 6.5 INCLUSION OF TIME-DEPENDENT EFFECTS 400 6.6 PROPAGATION OF THE ERROR COVARIANCE 405 6.7 SEQUENTIAL CONSIDER COVARIANCE ANALYSIS 407 6.8 EXAMPLE: FREELY FALLING POINT MASS 410 6.8.1 PROPAGATION WITH TIME 415 6.8.2 THE SEQUENTIAL CONSIDER ALGORITHM 416 6.8.3 PERTURBATION IN THE STATE ESTIMATE 419 6.9 EXAMPLE: SPRING-MASS PROBLEM 420 6.10 ERRORS IN THE OBSERVATION NOISE AND A PRIORI STATE COVARIANCES . . 425 6.11 ERRORS IN PROCESS NOISE, OBSERVATION NOISE, AND STATE COVARIANCE 427 6.12 COVARIANCE ANALYSIS AND ORTHOGONAL TRANSFORMATIONS 430 6.13 REFERENCES 434 6.14 EXERCISES 435 A PROBABILITY AND STATISTICS 439 A.I INTRODUCTION 439 A.2 AXIOMS OF PROBABILITY 440 A.3 CONDITIONAL PROBABILITY 443 A.4 PROBABILITY DENSITY AND DISTRIBUTION FUNCTIONS 443 A.5 EXPECTED VALUES 445 A.6 EXAMPLES AND DISCUSSION OF EXPECTATION 446 A.7 MOMENT GENERATING FUNCTIONS 448 A.8 SOME IMPORTANT CONTINUOUS DISTRIBUTIONS 449 A.8.1 UNIFORM OR RECTANGULAR DISTRIBUTION 449 A.8.2 THE GAUSSIAN OR NORMAL DISTRIBUTION 450 A.9 TWO RANDOM VARIABLES 452 A. 10 MARGINAL DISTRIBUTIONS 453 A. 11 INDEPENDENCE OF RANDOM VARIABLES 454 A.12 CONDITIONAL PROBABILITY 454 A. 13 EXPECTED VALUES OF BIVARIATE FUNCTIONS 455 A. 14 THE VARIANCE-COVARIANCE MATRIX 456 A.15 PROPERTIES OF THE CORRELATION COEFFICIENT 458 A. 16 PROPERTIES OF COVARIANCE AND CORRELATION 460 A. 17 BIVARIATE NORMAL DISTRIBUTION 460 A.18 MARGINAL DISTRIBUTIONS 461 VIII CONTENTS A. 19 THE MULTIVANATE NORMAL DISTRIBUTION 462 A. 19.1 THE CONDITIONAL DISTRIBUTION FOR MULTIVARIATE NORMAL VARIABLES 464 A.20 THE CENTRAL LIMIT THEOREM 465 A.21 BAYES THEOREM 465 A.22 STOCHASTIC PROCESSES 467 A.22.1 DEFINITIONS FOR STOCHASTIC PROCESSES 469 A.23 REFERENCES 471 B REVIEW OF MATRIX CONCEPTS 473 B.I INTRODUCTION 473 B.2 RANK 475 B.3 QUADRATIC FORMS 475 B.4 DETERMINANTS 476 B.5 MATRIX TRACE 477 B.6 EIGENVALUES AND EIGENVECTORS 478 B.7 THE DERIVATIVES OF MATRICES AND VECTORS 478 B.8 MAXIMA AND MINIMA 480 B.9 USEFUL MATRIX INVERSION THEOREMS 481 B.10 REFERENCE 483 C EQUATIONS OF MOTION 485 C.I LAGRANGE PLANETARY EQUATIONS 485 C.2 GAUSSIAN FORM 485 C.3 REFERENCES 486 D CONSTANTS 487 D.I PHYSICAL CONSTANTS 487 D.2 EARTH CONSTANTS 487 D.3 LUNAR, SOLAR, AND PLANETARY MASSES 488 D.4 REFERENCES 490 E ANALYTICAL THEORY FOR NEAR-CIRCULAR ORBITS 493 E.I DESCRIPTION 493 E.2 EXAMPLE 497 E.3 REFERENCES 497 F EXAMPLE OF STATE NOISE AND DYNAMIC MODEL COMPENSATION 499 F.I INTRODUCTION 499 F.2 STATE NOISE COMPENSATION 501 F.3 DYNAMIC MODEL COMPENSATION 505 F.4 REFERENCE 509 CONTENTS IX G SOLUTION OF THE LINEARIZED EQUATIONS OF MOTION 511 G.I INTRODUCTION 511 G.2 THE STATE TRANSITION MATRIX 513 H TRANSFORMATION BETWEEN ECI AND ECF COORDINATES 517 H.I INTRODUCTION 517 H.2 MATRIX P 518 H.3 MATRIX N 519 H.4 MATRIX S' 519 H.5 MATRIX W 521 H.6 REFERENCES 521 BIBLIOGRAPHY ABBREVIATIONS 523 BIBLIOGRAPHY 525 AUTHOR INDEX 537 INDEX 541
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author	Tapley, Byron D.
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spelling	Tapley, Byron D. Verfasser aut Statistical orbit determination Byron D. Tapley ; Bob E. Schutz ; George H. Born Amsterdam [u.a.] Elsevier Academic Press 2004 XV, 547 S. Ill. 24 cm txt rdacontent n rdamedia nc rdacarrier Mécanique orbitale Satellites artificiels - Orbites - Modèles mathématiques Trajectoires spatiales Mathematisches Modell Orbital mechanics Artificial satellites Orbits Mathematical models Space trajectories Bahnelement (DE-588)4326738-5 gnd rswk-swf Satellit (DE-588)4136498-3 gnd rswk-swf Umlaufbahn (DE-588)4238276-2 gnd rswk-swf (DE-588)4056995-0 Statistik gnd-content Satellit (DE-588)4136498-3 s Umlaufbahn (DE-588)4238276-2 s Bahnelement (DE-588)4326738-5 s DE-604 Schutz, Bob E. Sonstige oth Born, George H. Sonstige oth http://www.loc.gov/catdir/toc/els051/2004043275.html Table of contents http://www.loc.gov/catdir/description/els051/2004043275.html Publisher description GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014661574&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Tapley, Byron D. Statistical orbit determination Mécanique orbitale Satellites artificiels - Orbites - Modèles mathématiques Trajectoires spatiales Mathematisches Modell Orbital mechanics Artificial satellites Orbits Mathematical models Space trajectories Bahnelement (DE-588)4326738-5 gnd Satellit (DE-588)4136498-3 gnd Umlaufbahn (DE-588)4238276-2 gnd
subject_GND	(DE-588)4326738-5 (DE-588)4136498-3 (DE-588)4238276-2 (DE-588)4056995-0
title	Statistical orbit determination
title_auth	Statistical orbit determination
title_exact_search	Statistical orbit determination
title_exact_search_txtP	Statistical orbit determination
title_full	Statistical orbit determination Byron D. Tapley ; Bob E. Schutz ; George H. Born
title_fullStr	Statistical orbit determination Byron D. Tapley ; Bob E. Schutz ; George H. Born
title_full_unstemmed	Statistical orbit determination Byron D. Tapley ; Bob E. Schutz ; George H. Born
title_short	Statistical orbit determination
title_sort	statistical orbit determination
topic	Mécanique orbitale Satellites artificiels - Orbites - Modèles mathématiques Trajectoires spatiales Mathematisches Modell Orbital mechanics Artificial satellites Orbits Mathematical models Space trajectories Bahnelement (DE-588)4326738-5 gnd Satellit (DE-588)4136498-3 gnd Umlaufbahn (DE-588)4238276-2 gnd
topic_facet	Mécanique orbitale Satellites artificiels - Orbites - Modèles mathématiques Trajectoires spatiales Mathematisches Modell Orbital mechanics Artificial satellites Orbits Mathematical models Space trajectories Bahnelement Satellit Umlaufbahn Statistik
url	http://www.loc.gov/catdir/toc/els051/2004043275.html http://www.loc.gov/catdir/description/els051/2004043275.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014661574&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT tapleybyrond statisticalorbitdetermination AT schutzbobe statisticalorbitdetermination AT borngeorgeh statisticalorbitdetermination

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