Statistical Orbit Determination

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ISBN: 9780126836301 | 0126836302
Cover: Hardcover
Copyright: 5/27/2004

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This book presents fundmentals of orbit determination--from weighted least squares approaches (Gauss) to today's high-speed computer algorithms that provide accuracy within a few centimeters. Numerous examples and problems are provided to enhance readers' understanding of the material. *Covers such topics as coordinate and time systems, square root filters, process noise techniques, and the use of fictitious parameters for absorbing un-modeled and incorrectly modeled forces acting on a satellite. *Examples and exercises serve to illustrate the principles throughout each chapter. *Detailed solutions to end-of-chapter exercises available to instructors.

Preface

1 Orbit Determination Concepts

(16)

1.1 Introduction

(2)

1.2 Uniform Gravity Field Model

(10)

1.2.1 Formulation of the Problem

(1)

1.2.2 The Equation of the Orbit

(1)

1.2.3 The Role of the Observation

(4)

1.2.4 Linearization Procedure

(2)

1.2.5 State Transition Matrix

(2)

1.3 Background and Overview

(1)

1.4 Summary

(1)

1.5 References

(1)

1.6 Exercises

(1)

2 The Orbit Problem

(76)

2.1 Historical Background

(2)

2.2 Problem of Two Bodies: General Properties

(25)

2.2.1 Motion in the Plane

(7)

2.2.2 Motion in Space

(1)

2.2.3 Basic Coordinate Systems

(3)

2.2.4 Orbit Elements and Position/Velocity

(3)

2.2.5 Position/Velocity Prediction

(5)

2.2.6 State Transition Matrix and Error Propagation

(4)

2.3 Perturbed Motion

(27)

2.3.1 Classical Example: Lunar Problem

(3)

2.3.2 Variation of Parameters

(2)

2.3.3 Gravitational Perturbations: Mass Distribution

(6)

2.3.4 Gravitational Perturbations: Oblateness and Other Effects

(5)

2.3.5 Gravitational Perturbations: Third-Body Effects

(3)

2.3.6 Gravitational Perturbations: Temporal Changes in Gravity

(1)

2.3.7 Gravitational Perturbations: General Relativity

(1)

2.3.8 Nongravitational Perturbations: Atmospheric Resistance

(2)

2.3.9 Nongravitational Perturbations: Radiation Pressure

(1)

2.3.10 Nongravitational Perturbations: Other

(1)

2.3.11 Perturbed Equations of Motion: Summary

(2)

2.4 Coordinate Systems and Time: Introduction

(11)

2.4.1 Precession and Nutation

(3)

2.4.2 Earth Rotation and Time

(3)

2.4.3 Earth-Fixed and Topocentric Systems

(5)

2.4.4 Transformation between ECF and ECI

(1)

2.5 Orbit Accuracy

(2)

2.6 References

(3)

2.7 Exercises

(6)

3 Observations

(66)

3.1 Introduction

(1)

3.2 Observations

(5)

3.2.1 Ideal Range

(1)

3.2.2 Ideal Range-Rate

(1)

3.2.3 Ideal Azimuth and Elevation Angles

(1)

3.2.4 Examples: Ideal Observations

(4)

3.3 Conceptual Measurement Systems

(11)

3.3.1 Range

(8)

3.3.2 Range-Rate

107

(3)

3.4 Realization of Measurements

110

(6)

3.4.1 Considerations

110

(1)

3.4.2 Atmospheric Effects

110

(5)

3.4.3 General Relativity

115

(1)

3.5 Measurement Systems

116

(23)

3.5.1 One-Way Range

116

(9)

3.5.2 Two-Way Range

125

(7)

3.5.3 Doppler Systems

132

(2)

3.5.4 Examples

134

(5)

3.6 Differenced Measurements

139

(8)

3.6.1 Differenced GPS Measurements

140

(5)

3.6.2 Differenced Altimeter Data

145

(2)

3.7 Satellite Positions

147

(1)

3.8 Angles

148

(1)

3.9 References

149

(2)

3.10 Exercises

151

(8)

4 Fundamentals of Orbit Determination

159

(126)

4.1 Introduction

159

(1)

4.2 Linearization of the Orbit Determination Process

160

(13)

4.2.1 The State Transition Matrix

164

(3)

4.2.2 Solution for the State Transition Matrix

167

(5)

4.2.3 Relating the Observations to an Epoch State

172

(1)

4.3 The Least Squares Solution

173

(10)

4.3.1 The Minimum Norm Solution

174

(2)

4.3.2 Shortcomings of the Least Squares Solution

176

(1)

4.3.3 Weighted Least Squares Solution

176

(2)

4.3.4 An Alternate Least Squares Approach

178

(5)

4.4 The Minimum Variance Estimate

183

(7)

4.4.1 Propagation of the Estimate and Covariance Matrix

188

(1)

4.4.2 Minimum Variance Estimate with A Priori Information

188

(2)

4.5 Maximum Likelihood and Bayesian Estimation

190

(4)

4.6 Computational Algorithm for the Batch Processor

194

(5)

4.7 The Sequential Estimation Algorithm

199

(12)

4.7.1 The Sequential Computational Algorithm

201

(8)

4.7.2 The Extended Sequential Estimation Algorithm

209

(1)

4.7.3 The Extended Sequential Computational Algorithm

210

(1)

4.7.4 The Prediction Residual

210

(1)

4.8 Example Problems

211

(9)

4.8.1 Linear System

211

(5)

4.8.2 Spring-Mass Problem

216

(4)

4.9 State Noise Compensation Algorithm

220

(44)

4.9.1 The Gauss-Markov Process

230

(3)

4.10 Information Filter

233

(4)

4.11 Batch and Sequential Estimation

237

(1)

4.12 Observability

237

(4)

4.13 Error Sources

241

(3)

4.14 Orbit Accuracy

244

(2)

4.15 Smoothing

246

(5)

4.15.1 Computational Algorithm for Smoother

250

(1)

4.16 The Probability Ellipsoid

251

(7)

4.16.1 Transformation of the Covariance Matrix between Coordinate Systems

257

(1)

4.17 Combining Estimates

258

(3)

4.18 References

261

(3)

4.19 Exercises

264

(21)

5 Square Root Solution Methods

285

(102)

5.1 Introduction

285

(1)

5.2 Cholesky Decomposition

286

(4)

5.2.1 The Cholesky Algorithm

287

(1)

5.2.2 The Square Root Free Cholesky Algorithm

288

(2)

5.3 Least Squares Solution via Orthogonal Transformation

290

(1)

5.4 Givens Transformations

291

(19)

5.4.1 A Priori Information and Initialization

294

(4)

5.4.2 Givens Computational Algorithm

298

(3)

5.4.3 Square Root Free Givens Transformation

301

(4)

5.4.4 Square Root Free Givens Computational Algorithm

305

(2)

5.4.5 A Simplified Square Root Free Givens Transformation

307

(2)

5.4.6 Implementation Considerations

309

(1)

5.5 The Householder Transformation

310

(8)

5.5.1 Application to the Solution of the Least Squares Problem

315

(1)

5.5.2 Householder Computational Algorithm

316

(2)

5.6 Numerical Examples

318

(12)

5.6.1 Cholesky Decomposition

318

(2)

5.6.2 Givens Transformation

320

(2)

5.6.3 Householder Transformation

322

(2)

5.6.4 A More Illustrative Example

324

(1)

5.6.5 Cholesky Decomposition

325

(1)

5.6.6 Square Root Free Givens Transformation

326

(3)

5.6.7 The Householder Transformation

329

(1)

5.7 Square Root Filter Algorithms

330

(13)

5.7.1 The Square Root Measurement Update Algorithms

332

(8)

5.7.2 Square Root Free Measurement Update Algorithms

340

(3)

5.8 Time Update of the Estimation Error Covariance Matrix

343

(3)

5.9 Continuous State Error Covariance Propagation

346

(35)

5.9.1 Triangular Square Root Algorithm

349

(2)

5.10 The Square Root Information Filter

351

(18)

5.10.1 The Square Root Information Filter with Time-Dependent Effects

353

(5)

5.10.2 The Dynamic Case with Process Noise

358

(5)

5.10.3 SRIF Computational Algorithm

363

(1)

5.10.4 Smoothing with the SRIF

364

(5)

5.11 Process Noise Parameter Filtering/Smoothing Using a SRIF

369

(11)

5.11.1 Exponentially Correlated Process Noise SRIF

371

(6)

5.11.2 Smoothing with a SRIF

377

(3)

5.12 References

380

(1)

5.13 Exercises

381

(6)

6 Consider Covariance Analysis

387

(52)

6.1 Introduction

387

(1)

6.2 Bias in Linear Estimation Problems

388

(1)

6.3 Formulation of the Consider Covariance Matrix

389

(8)

6.4 The Sensitivity and Perturbation Matrices

397

(3)

6.4.1 Example Application of a Sensitivity and Perturbation Matrix

398

(2)

6.5 Inclusion of Time-Dependent Effects

400

(5)

6.6 Propagation of the Error Covariance

405

(2)

6.7 Sequential Consider Covariance Analysis

407

(3)

6.8 Example: Freely Falling Point Mass

410

(10)

6.8.1 Propagation with Time

415

(1)

6.8.2 The Sequential Consider Algorithm

416

(3)

6.8.3 Perturbation in the State Estimate

419

(1)

6.9 Example: Spring-Mass Problem

420

(5)

6.10 Errors in the Observation Noise and A Priori State Covariances

425

(2)

6.11 Errors in Process Noise, Observation Noise, and State Covariance

427

(3)

6.12 Covariance Analysis and Orthogonal Transformations

430

(4)

6.13 References

434

(1)

6.14 Exercises

435

(4)

A Probability and Statistics

439

(34)

A.1 Introduction

439

(1)

A.2 Axioms of Probability

440

(3)

A.3 Conditional Probability

443

(1)

A.4 Probability Density and Distribution Functions

443

(2)

A.5 Expected Values

445

(1)

A.6 Examples and Discussion of Expectation

446

(2)

A.7 Moment Generating Functions

448

(1)

A.8 Some Important Continuous Distributions

449

(3)

A.8.1 Uniform or Rectangular Distribution

449

(1)

A.8.2 The Gaussian or Normal Distribution

450

(2)

A.9 Two Random Variables

452

(1)

A.10 Marginal Distributions

453

(1)

A.11 Independence of Random Variables

454

(1)

A.12 Conditional Probability

454

(1)

A.13 Expected Values of Bivariate Functions

455

(1)

A.14 The Variance-Covariance Matrix

456

(2)

A.15 Properties of the Correlation Coefficient

458

(2)

A.16 Properties of Covariance and Correlation

460

(1)

A.17 Bivariate Normal Distribution

460

(1)

A.18 Marginal Distributions

461

(1)

A.19 The Multivariate Normal Distribution

462

(3)

A.19.1 The Conditional Distribution for Multivariate Normal Variables

464

(1)

A.20 The Central Limit Theorem

465

(1)

A.21 Bayes Theorem

465

(2)

A.22 Stochastic Processes

467

(4)

A.22.1 Definitions for Stochastic Processes

469

(2)

A.23 References

471

(2)

B Review of Matrix Concepts

473

(12)

B.1 Introduction

473

(2)

B.2 Rank

475

(1)

B.3 Quadratic Forms

475

(1)

B.4 Determinants

476

(1)

B.5 Matrix Trace

477

(1)

B.6 Eigenvalues and Eigenvectors

478

(1)

B.7 The Derivatives of Matrices and Vectors

478

(2)

B.8 Maxima and Minima

480

(1)

B.9 Useful Matrix Inversion Theorems

481

(2)

B.10 Reference

483

(2)

C Equations of Motion

485

(2)

C.1 Lagrange Planetary Equations

485

(1)

C.2 Gaussian Form

485

(1)

C.3 References

486

(1)

D Constants

487

(6)

D.1 Physical Constants

487

(1)

D.2 Earth Constants

487

(1)

D.3 Lunar, Solar, and Planetary Masses

488

(2)

D.4 References

490

(3)

E Analytical Theory for Near-Circular Orbits

493

(6)

E.1 Description

493

(4)

E.2 Example

497

(1)

E.3 References

497

(2)

F Example of State Noise and Dynamic Model Compensation

499

(12)

F.1 Introduction

499

(2)

F.2 State Noise Compensation

501

(4)

F.3 Dynamic Model Compensation

505

(4)

F.4 Reference

509

(2)

G Solution of the Linearized Equations of Motion

511

(6)

G.1 Introduction

511

(2)

G.2 The State Transition Matrix

513

(4)

H Transformation between ECI and ECF Coordinates

517

(6)

H.1 Introduction

517

(1)

H.2 Matrix P

518

(1)

H.3 Matrix N

519

(1)

H.4 Matrix S'

519

(2)

H.5 Matrix W

521

(1)

H.6 References

521

(2)

Bibliography Abbreviations

523

(2)

Bibliography

525

(12)

Author Index

537

(4)

Index

541

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Statistical Orbit Determination

Summary

Table of Contents

Supplemental Materials

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