# Statistical Orbit Determination

## 1st Edition

**Authors:**Bob Schutz Byron Tapley George Born

**eBook ISBN:**9780080541730

**Hardcover ISBN:**9780126836301

**Imprint:**Academic Press

**Published Date:**27th May 2004

**Page Count:**547

## Description

*Statistical Orbit Determination* 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.

## Key Features

- 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.

## Readership

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.

## 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
- Exercises

4 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
- References

Bibliography Abbreviations Bibliography Author Index Index

## Details

- No. of pages:
- 547

- Language:
- English

- Copyright:
- © Academic Press 2004

- Published:
- 27th May 2004

- Imprint:
- Academic Press

- eBook ISBN:
- 9780080541730

- Hardcover ISBN:
- 9780126836301

## About the Author

### Bob Schutz

### Affiliations and Expertise

University of Texas at Austin, U.S.A.

### Byron Tapley

### George Born

### Affiliations and Expertise

University of Colorado, U.S.A.

## Reviews

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