# Statistical Orbit Determination

**By**

- Bob Schutz, University of Texas at Austin, U.S.A.
- Byron Tapley
- George Born, University of Colorado, U.S.A.

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.

View full description### Audience

Graduate students and researchers in aerospace engineering, astrodynamics, geodesy, and oceanography.Researchers in the aerospace industry and related research laboratories interested in satellite navigation and control.

### Book information

- Published: May 2004
- Imprint: ACADEMIC PRESS
- ISBN: 978-0-12-683630-1

### Table of Contents

1 Orbit Determination Concepts * Introduction * UniformGravity Field Model * Background and Overview * Summary * References * Exercises 2 The Orbit Problem * Historical Background * Problem of Two Bodies: General Properties * Perturbed Motion * Coordinate Systems and Time: Introduction * Orbit Accuracy * References * Exercises 3 Observations * Introduction * Observations * Conceptual Measurement Systems* Realization of Measurements * Measurement Systems* Differenced Measurements * Satellite Positions * Angles* References * Exercises4 Fundamentals of Orbit Determination * Introduction * Linearization of the Orbit Determination Process* The Least Squares Solution * The Minimum Variance Estimate * Maximum Likelihood and Bayesian Estimation * Compuational Algorithm for the Batch Processor* The Sequential Estimation Algorithm * Example Problems * State Noise Compensation Algorithm * Information Filter * Batch and Sequential Estimation * Observability * Error Sources * Orbit Accuracy * Smoothing * The Probability Ellipsoid * Combining Estimates * References * Exercises 5 Square-Root Solution Methods * Introduction * Cholesky Decomposition * Least Squares Solution via Orthogonal Algorithm* Givens Transformations * The Householder Transformation * Numerical Examples * Square Root Filter Algorithms * Time Update of the Estimation Error Covariance Matrix* Continuous State Error Corvariance Propagation* The Square Root Information Filter * Process Noise Parameter Filtering/Smoothing Using a SRIF* References * Exercises 6 Consider Covariance Analysis * Introduction * Bias in Linear Estimation Problems * Formulation of the Consider Covariance Matrix * The Sensitivity and Perturbation Matrices * Inclusion of Time-Dependent Effects * Propagation of the Error Covariance * Sequential Consider Covariance Analysis * Example: Freely Falling Point Mass * Example: Spring-Mass Problem * Errors in the Observation Noise and A Priori State Covariances * Errors in Process Noise, Observation Noise, and State Covariance* Corvariance Analysis and Orthogonal Transformations * References * Exercises A Probability and Statistics * Introduction * Axioms of Probability * Conditional Probability * Probability Density and Distribution Functions * Expected Values * Examples and Discussion of Expectation * Moment Generating Functions * Some Important Continuous Distributions * Two Random Variables * Marginal Distributions * Independence of Random Variables * Conditional Probability * Expected Values of Bivariate Functions * The Variance-Covariance Matrix * Properties of the Correlation Coefficient * Properties of Covariance and Correlation * Bivariate Normal Distribution * Marginal Distributions * The Multivariate Normal Distribution * The Central Limit Theorem * Bayes Theorem * Stochastic Processes * References B Review of Matrix Concepts * Introduction* Rank * Quadratic Forms * Determinants * Matrix Trace * Eigenvalues and Eigenvectors* The Derivatives of Matrices and Vectors * Maxima and Minima * Useful Matrix Inversion Theorems * Reference C Equations of Motion * Lagrange Planetary Equations * Gaussian Forms* References D Constants * Physical Constants * Earth Constants * Lunar, Solar and Planetary Masses * References E Analytical Theory for Near-Circular Orbits * Description * Example * References F Example of State Noise and Dynamic Model Compensation * Introduction * State Noise Compensation * Dynamic Model Compensation * References G Solution of the Linearized Equations of Motion * Introduction * The State Transition Matrix H ECI and ECF Transformation * Introduction * Matrix P * Matrix N * Matrix, S' * Matrix W * ReferencesBibliography AbbreviationsBibliographyAuthor IndexIndex