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TABLE OF CONTENTS

  Fundamentals of High Accuracy Inertial Navigation

Chapter 1. Introduction
I. Forces Producing Motion
            A. Gravitation
            B. Inertia
II. Inertial Equivalence of Earth-Centered Frame
III. Fundamental Equation of Inertial Navigation
IV. Description of an Inertial Navigation System
V. Inertial Measurements
VI. Four Phases of Inertial Navigation
VII. Role of Geodesy .
VIII. Reference Earth Model
Part I Inertial Navigation
Chapter 2. Notation, Coordinate Systems, and Units
I. Notation Conventions
II. Coordinate System Definitions
A. Software Implemented
B. Hardware Implemented
III. Coordinate Transformation Characteristics
A. Orthogonal
B. Nonorthogonal
IV. Commonly Used Coordinate Rotations
A. Earth-Centered Inertial to Earth-Centered Earth-Fixed
B. Earth-Centered Inertial to Local Geodetic Vertical
C. Earth-Centered Inertial to Local Geocentric Vertical
D. Earth Centered Earth-Fixed to Local Geodetic Vertical
E. Earth-Centered Earth-Fixed to Local Astronomic Vertical
F. Star Line-of-Sight to Platform
G. Star to Earth-Centered Inertial
V. Units. ..............................................
Chapter 3. Equations of Motion in a Central Force Gravity Field
I. Motion in Inertial Coordinates with Zero-Specific Force
     A. Zero-Specific Force
     B. Schuler Frequency
II. State-Space Form
               A. Laplace Transform Form
B. Frequency Response
III. Motion in Inertial Computation Coordinates
 A. Transfer Functions
       B. Propagation of Initial State
       C. Frequency Response Functions
IV. Motion in Earth-Fixed Computation Coordinates
A. Significance of Terms in Equation of Motion
B. Transfer Functions
C. Propagation of Initial State

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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D. Frequency Response Functions
V. Effect of Velocity Damping
A. Propagation of initial State
B. Frequency Response Functions
Chapter 4. Inertial Instrumentation
I. Gyroscope
A. Rotating Wheel
B. Optical
         C. Recently Developed Instruments
II. Accelerometer
     A. Pendulous Integrated Gyro
B. Proof Mass
C. Vibrating String
D. Fiber Optic
III. Gradiometer

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                                                 (Continued)

 

 

 

A. Gravity Gradient Tensor
B. Output Equations
C. Output Equation Processing
IV. Gimbal Configurations
A. Mechanical Frame
B. Floating Sphere
V. Strapdown Configuration
Chapter 5. Calibration
I. Physical Reference Vectors . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .
A. Specific Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Angular Rate ....... ................ ............. ........................
II. Calibration Procedure ................ ................................................
A. Inertial Measurement Unit Configuration . . . . . . . . . . . . . . . . .
B. Platform Rotation Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . .
III. Accelerometer Calibration . .......................................................
A. Observation Equation .. ................................................
B. Application of the Observation Equation ................... .
IV. Gyro Calibration ...................... ................................................
A. Observation Equation-Magnitude Form .......................
B. Observation Equation-Vector Form .............................
G. Geodetic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H. Geoid Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I. Height Above Mean Sea Level . . . . . . . . . . . . . . . . . . . . . . . . .
II. World Geodetic System 1984 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. Spherical Harmonic Coefficients . ..............................
B. Equipotential Surfaces Associated with SHCs . . . . . . . . . . . . . . .
C. Physical Meaning of the Low Degree and Order SHCs . . . . . . . . . .
D. Regional Datum Transformations ..............................
III.
Gravity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. Spherical Harmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Point Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C. Two-Dimensional Fourier Series . . . . . . . . . . . . . . . . . . . . . . . .
D. Two-Dimensional Table . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E. Other Types of Models . . . . . . . . . . . . . . . . . . . . . . .
IV. Useful Incremental Terms of Geodesy ..................................
A. Defections of the Vertical ............................................
B. Azimuth Differences . ..._.............................................
V. Extending Gravity Surveys with Internal Measurements . . . . . . . . . .
Chapter 8. Equations of Motion with General Gravity Model
I. State-Space Form in Earth-Centered Inertial Coordinates .
II. State-Space Form in Earth- Centered Earth-Fixed Coordinates
III. State-Space Form in Earth-Centered Earth-Fixed Coordinates with
Point-Mass Gravity Model . . . . . . . . . . . . . . . . . . . . . . . . . .
IV. State-Space Form in Local Geodetic Vertical Coordinates . . . . . . . . . .
A. Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Pseudo-Velocity Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V. Platform Control Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. Earth Centered Inertial . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Earth-Centered Earth-Fixed . . . . . . . . . . . . . . . . . . . .
C. Local Geodetic Vertical-Torqued Azimuth . . . . . . . . . . .
D. Local Geodetic Vertical-Free Azimuth . . . . . . . . . . . . . . . . . . 1
E. Local Geodetic Vertical-Platform Carousel . . . . . . . . . . . . . . . .
F. Local Geodetic Vertical-Platform Tumble . . . . . . . . . . . . . . .
VI. Integration of the Equations of Motion . . . . . . . . . . . . . . . . . . . . .
VII. Summary of Equations for Computing the Transition Matrix
A. Earth-Centered Inertial Coordinates-Stabilized Platform . . . . . .
B. Earth-Centered Earth-Fixed Coordinates-Stabilized Platform . . . .
C. Local Geodetic Vertical Coordinates-Standard
Form-Stabilized Platform . . . . . . . . . . . . . . . . . . . . . .
D. Local Gender, Vertical Coordinates-Pseudo-Velocity
Form-Stabilized Platform . . . . . . . . . . . . . . . . . . . . . . . . .
E. Earth-Centered Inertial Coordinates-Strapdown . . . . . . . . . .
F. Earth-Centered Earth-Fixed Coordinates-Strapdown ....... .
G. Local Geodetic Vertical Coordinates-Standard Form-Strapdown .
H. Local Geodetic Vertical Coordinates--Pseudo-Velocity
Form-Strapdown ....................... .................................
Part II Inertial Navigation with Aids
Chapter 9. Inertial Navigation with External Measurements
I. Basis for Using External Measurements . . . . .
A. Equations of Relative Motion . . . . . . .
B. Application of the Equations of Relative Motion
II. Kalman Filter State Updates . . . . . . . . . . . . . .
A. Overview of Navigation Computations Extended Kalman Filter
B. Gain Evaluation and Covariance Update . . .
D. Summary of Navigation Equations Extended Kalman Filter
E. Summary of Navigation Equations-Linearized Kalman Filter
F. Examples of External Measurement Predictions ..
G. Examples of Partial Derivative Evaluations ...
H. Example of a Suboptimal Filter . . . . . . . . . .
I. Aliasing............
Chapter 10. Error Equations for the Kalman Filter
I. Attitude Errors ............................... ...........................................
A. Delimit ................... .........................................................
B. Angular Equivalent of the Position Error . . . .
C. Actual Coordinate Rotations In Terms of Errors .
D. Attitude Error Vector Differential Equations . .
II. System Dynamic and Error Distribution Matrices in Earth-Centered
Inertial Coordinates . . . . . . . . . .
A. Acceleration-Earth-Centered Inertial Coordinates .
B. Velocity -Earth-Centered Inertial Coordinates .
C. State-Space Form of Error Equations-Earth-Centered Inertial Coordinates
III. System Dynamic and Error Distribution Matrices in Earth-Centered
Earth-Fixed Coordinates. .
A. Acceleration-Earth-Centered Earth-Fixed Coordinates .
B. Velocity-Earth-Centered Earth Fixed Coordinates
C. State-Space Form of Error Equations-Earth-Centered
Earth-Fixed Coordinates..................................
IV. System Dynamic and Error Distribution Matrices in Local
Geodetic Vertical Coordinates . .
A. Semiposition Error Definition .
B. Semivelocity Error Definition
C. Acceleration -Local Geodetic Vertical Coordinates
D. Velocity-Local Geodetic Vertical Coordinates
E. State-Space Form of Error Equations-Local Geodetic
Vertical Coordinates . . . . .
Chapter 11. Stale Variable Error Models
I. Inertial and External Measurement Equipment Error Shaping Functions
A. Random Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
B. Random Walk..................................................
C. Random Ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D. Markov .. .... ... ....... .......... ................ . .....
II. Omission Gravity Model Error Shaping Functions . . . . . . .
A Gravity Database Format . . . . . . . . . . . . . . . . .
B. Gravity Model Error Equations of Motion . .
C. Autocorrelation Function Approximation Method .2
D. Influence of Vehicle Velocity on the Power Spectral Density
E. Autoregressive Moving Average Method .
Part III Accuracy Analysis
Chapter 12. Accuracy Criteria and Analysis Techniques
I. Central LimitTheorem ......................................................
II. Standard Error . ....................................................
A. Uncorrelated Standard Errors for Circular-Error-Probable Calculation
B. Uncorrelated Standard Errors for Spherical-Error-Probable Calculation
III Gaussian Distribution Function for Navigation Position Errors .
IV. Circular Error Probable and Spherical Error Probable
A. CEP for Equal Standard Errors and Zero Means .
B. SEP for Equal Standard Errors and Zero Means .
C. CEP and SEP for Unequal Standard Errors and Nonzero Means
D. Verification of the CEP and SEP Formulas . .
V. Accuracy Analysis Techniques .....................
A. Types of Error . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Error Analysis Using Sensitivity Coefficients . . . . . . . . . . . .
Chapter 13. Error Equations for Calibration, Alignment,
and Initialization
I. Inertial Instrument Calibration ................................ ................
A. Apparent Gravity Magnitude . . . . . . . . . . . . . . .
B. Reference Rotation Rate ...............................
C. Pole Location . . . . . . . . . . . . . . . . . . . . . . . . . .              
II. Analytical Alignment ............... . .... ........ ....... . .... .... . ...
A. Astronomic Coordinates .............................. ......
B. Geodetic Coordinates . . . . . . . . . . . . . . . . . . . . . . . .
C. Specific Force and Pole Position . . . . . . . . . . . . .
III. Initialization . . ........ ....... .. ... ...... . ... . .......... ...
A. Initial Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Initial Position . . . . . . . . . . . . . . . . . . . . . . . . . . .
C. Conversion to Earth-Centered Inertial and Local Geodetic
Vertical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . .
IV. Kalman Filter Covarience, Initialization . . . .
Chapter 14. Evaluation of Gravity Model Error Effects
I. Spherical Harmonic Gravity Model Errors . .
II.. Point-Mass Model Generation . . . . . . . .
III. Sources of Error for Point-Mass Model . . .
A. Representation .....................................................
B. Reduction ............... ............................. ...............
C. Omission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A. Matrix Inverse Formulas
Appendix R. Laplace Transforms
Appendix C. Quaternions
Appendix D. Associated Legendre Functions
Appendix E. Associated Legendre Function Derivatives
Appendix F. Procedure for Generating Gravity Disturbance
Realizations
.
Appendix G. Procedure for Generating Specific Force Profile
Index
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