Оглавление
Mohinder S. Grewal. Global Navigation Satellite Systems, Inertial Navigation, and Integration
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
Global Navigation Satellite Systems, Inertial Navigation, and Integration
Copyright
Preface to the Fourth Edition
Acknowledgments
About the Authors
Acronyms
About the Companion Website
1 Introduction
1.1 Navigation
1.1.1 Navigation‐Related Technologies
1.1.2 Navigation Modes
1.2 GNSS Overview
1.2.1 GPS
1.2.1.1 GPS Orbits
1.2.1.2 Legacy GPS Signals
1.2.1.3 Modernization of GPS
1.2.2 Global Orbiting Navigation Satellite System (GLONASS)
1.2.2.1 GLONASS Orbits
1.2.2.2 GLONASS Signals
1.2.2.3 Modernized GLONASS
1.2.3 Galileo
1.2.3.1 Galileo Navigation Services
1.2.3.2 Galileo Signal Characteristics
E5a–E5b Band
E6 Band
L1/E1 Band
1.2.4 BeiDou
1.2.4.1 BeiDou Satellites
1.2.4.2 Frequency
1.2.5 Regional Satellite Systems
1.2.5.1 QZSS
1.2.5.2 NAVIC
1.3 Inertial Navigation Overview
1.3.1 History
1.3.1.1 Theoretical Foundations
1.3.1.2 Development Challenges: Then and Now
1.3.2 Development Results. 1.3.2.1 Inertial Sensors
1.3.2.2 Sensor Attitude Control
1.3.2.3 Initialization
1.3.2.4 Integrating Acceleration and Velocity
1.3.2.5 Accounting for Gravity
1.4 GNSS/INS Integration Overview. 1.4.1 The Role of Kalman Filtering
1.4.2 Implementation
Problems
References
Notes
2 Fundamentals of Satellite Navigation Systems. 2.1 Chapter Focus
2.2 Satellite Navigation Systems Considerations
2.2.1 Systems Other than GNSS
2.2.2 Comparison Criteria
2.3 Satellite Navigation
2.3.1 GNSS Orbits
2.3.2 Navigation Solution (Two‐Dimensional Example)
2.3.2.1 Symmetric Solution Using Two Transmitters on Land
2.3.2.2 Navigation Solution Procedure
2.3.3 User Solution and Dilution of Precision (DOP)
2.3.4 Example Calculation of DOPs. 2.3.4.1 Four Satellites
2.4 Time and GPS. 2.4.1 Coordinated Universal Time (UTC) Generation
2.4.2 GPS System Time
2.4.3 Receiver Computation of UTC
2.5 Example: User Position Calculations with No Errors. 2.5.1 User Position Calculations
2.5.1.1 Position Calculations
2.5.2 User Velocity Calculations
Problems
References
3 Fundamentals of Inertial Navigation
3.1 Chapter Focus
Scope
3.2 Terminology
3.3 Inertial Sensor Technologies. 3.3.1 Gyroscopes
3.3.1.1 Momentum Wheel Gyroscopes (MWGs)
Bearing Technologies
Whole‐angle Gyroscopes
Rate Gyroscopes
Axial Mass Unbalance Torques
3.3.1.2 Coriolis Vibratory Gyroscopes (CVGs) Tuning Fork Gyroscopes
MEMS Tuning Fork Gyroscope
Hemispherical Resonator Gyroscopes
3.3.1.3 Optical Gyroscopes (RLGs and FOGs)
3.3.2 Accelerometers
3.3.2.1 Mass‐spring Designs
3.3.2.2 Pendulous Integrating Gyroscopic Accelerometers (PIGA)
3.3.2.3 Electromagnetic
3.3.2.4 Electrostatic
3.3.3 Sensor Errors. 3.3.3.1 Additive Output Noise
3.3.3.2 Input–output Errors
3.3.3.3 Error Compensation
3.3.4 Inertial Sensor Assembly (ISA) Calibration
3.3.4.1 ISA Calibration Parameters
Compensation
3.3.4.2 Calibration Parameter Drift
Predicting Incipient System or Sensor Failures
3.3.5 Carouseling and Indexing
3.4 Inertial Navigation Models
3.4.1 Geoid Models
3.4.2 Terrestrial Navigation Coordinates
3.4.3 Earth Rotation Model
3.4.4 Gravity Models
3.4.4.1 Gravitational Potential
3.4.4.2 Gravitational Acceleration
3.4.4.3 Equipotential Surfaces
WGS84 Ellipsoid
Geoid Models
3.4.4.4 Longitude and Latitude Rates
Meridional Radius of Curvature
Geodetic Latitude Rate
Transverse Radius of Curvature
Longitude Rate
WGS84 Reference Surface Curvatures
3.4.5 Attitude Models
3.4.5.1 Coordinate Transformation Matrices and Rotation Vectors
3.4.5.2 Attitude Dynamics
3.5 Initializing The Navigation Solution. 3.5.1 Initialization from an Earth‐fixed Stationary State. 3.5.1.1 Accelerometer Recalibration
3.5.1.2 Initializing Position and Velocity
3.5.1.3 Initializing ISA Attitude
3.5.1.4 Gyrocompass Alignment Accuracy
Accuracy
3.5.2 Initialization on the Move. 3.5.2.1 Transfer Alignment
3.5.2.2 Initializing Using GNSS
3.6 Propagating The Navigation Solution. 3.6.1 Attitude Propagation
3.6.1.1 Strapdown Attitude Propagation. Strapdown Attitude Problems
Coning Motion
Rotation Vector Implementation
Bortz Model for Attitude Dynamics
3.6.1.2 Quaternion Implementation
Converting Incremental Rotations to Incremental Quaternions
3.6.1.3 Direction Cosines Implementation
Quaternions to Direction Cosines Matrices
Strapdown with Whole‐angle Gyroscopes
3.6.1.4 MATLAB® Implementations
3.6.1.5 Gimbal Attitude Implementations
Vehicle Attitude Determination
ISA Attitude Control
3.6.2 Position and Velocity Propagation. 3.6.2.1 Vertical Channel Instability
3.6.2.2 Strapdown Navigation Propagation
3.6.2.3 Gimbaled Navigation Propagation
3.7 Testing and Evaluation
3.7.1 Laboratory Testing
3.7.2 Field Testing
3.7.3 Performance Qualification Testing
3.7.3.1 CEP and Nautical Miles
3.7.3.2 Free Inertial Performance
Free Inertial Error Heuristics
CEP Rates
INS Performance Categories
Comparable Sensor Performance Ranges
CEP versus RMS
3.8 Summary
3.8.1 Further Reading
Problems
References
Notes
4 GNSS Signal Structure, Characteristics, and Information Utilization
4.1 Legacy GPS Signal Components, Purposes, and Properties
4.1.1 Signal Models for the Legacy GPS Signals
4.1.2 Navigation Data Format
4.1.2.1 Z‐Count
4.1.2.2 GPS Week Number (WN)
Frame and Subframe Identification
4.1.2.3 Information by Subframe
4.1.3 GPS Satellite Position Calculations
4.1.3.1 Ephemeris Data Reference Time Step and Transit Time Correction
4.1.3.2 True, Eccentric, and Mean Anomaly
4.1.3.3 Kepler's Equation for the Eccentric Anomaly
4.1.3.4 Satellite Time Corrections
4.1.4 C/A‐Code and Its Properties
4.1.4.1 Temporal Structure
4.1.4.2 Autocorrelation Function
4.1.4.3 Power Spectrum
4.1.4.4 Despreading of the Signal Spectrum
4.1.4.5 Role of Despreading in Interference Suppression
4.1.4.6 Cross‐correlation Function
4.1.5 P(Y)‐Code and Its Properties
4.1.5.1 P‐Code Characteristics
4.1.5.2 Y‐Code
4.1.6 L1 and L2 Carriers
4.1.6.1 Dual‐Frequency Operation
4.1.7 Transmitted Power Levels
4.1.8 Free Space and Other Loss Factors
4.1.9 Received Signal Power
4.2 Modernization of GPS
4.2.1 Benefits from GPS Modernization
4.2.2 Elements of the Modernized GPS
4.2.3 L2 Civil Signal (L2C)
4.2.4 L5 Signal
4.2.5 M‐Code
4.2.6 L1C Signal
4.2.7 GPS Satellite Blocks
4.2.8 GPS Ground Control Segment
4.3 GLONASS Signal Structure and Characteristics
4.3.1 Frequency Division Multiple Access (FDMA) Signals
4.3.1.1 Carrier Components
4.3.1.2 Spreading Codes and Modulation
4.3.1.3 Navigation Data Format
4.3.1.4 Satellite Families
4.3.2 CDMA Modernization
4.4 Galileo
4.4.1 Constellation and Levels of Services
4.4.2 Navigation Data and Signals
4.5 BeiDou
4.6 QZSS
4.7 IRNSS/NAVIC
Problems
References
5 GNSS Antenna Design and Analysis. 5.1 Applications
5.2 GNSS Antenna Performance Characteristics
5.2.1 Size and Cost
5.2.2 Frequency and Bandwidth Coverage
5.2.3 Radiation Pattern Characteristics
5.2.4 Antenna Polarization and Axial Ratio
5.2.5 Directivity, Efficiency, and Gain of a GNSS Antenna
5.2.6 Antenna Impedance, Standing Wave Ratio, and Return Loss
5.2.7 Antenna Bandwidth
5.2.8 Antenna Noise Figure
5.3 Computational Electromagnetic Models (CEMs) for GNSS Antenna Design
5.4 GNSS Antenna Technologies. 5.4.1 Dipole‐Based GNSS Antennas
5.4.2 GNSS Patch Antennas
5.4.2.1 Edge‐Fed, LP, Single‐Frequency GNSS Patch Antenna
5.4.2.2 Probe‐Fed, LP, Single‐Frequency GNSS Patch Antenna
5.4.2.3 Dual Probe‐Fed, RHCP, Single‐Frequency GNSS Patch Antenna
5.4.2.4 Single Probe‐Fed, RHCP, Single‐Frequency GNSS Patch Antenna
5.4.2.5 Dual Probe‐Fed, RHCP, Multifrequency GNSS Patch Antenna
5.4.3 Survey‐Grade/Reference GNSS Antennas
5.4.3.1 Choke Ring‐Based GNSS Antennas
5.4.3.2 Advanced Planner‐Based GNSS Antennas
5.5 Principles of Adaptable Phased‐Array Antennas
5.5.1 Digital Beamforming Adaptive Antenna Array Formulations
5.5.2 STAP
5.5.3 SFAP
5.5.4 Configurations of Adaptable Phased‐Array Antennas
5.5.5 Relative Merits of Adaptable Phased‐Array Antennas
5.6 Application Calibration/Compensation Considerations
Problems
References
6 GNSS Receiver Design and Analysis. 6.1 Receiver Design Choices. 6.1.1 Global Navigation Satellite System (GNSS) Application to Be Supported
6.1.2 Single or Multifrequency Support
6.1.2.1 Dual‐Frequency Ionosphere Correction
6.1.2.2 Improved Carrier Phase Ambiguity Resolution in High‐Accuracy Differential Positioning
6.1.3 Number of Channels
6.1.4 Code Selections
6.1.5 Differential Capability
6.1.5.1 Corrections Formats
6.1.6 Aiding Inputs
6.2 Receiver Architecture
6.2.1 Radio Frequency (RF) Front End
6.2.2 Frequency Down‐Conversion and IF Amplification
6.2.2.1 SNR
6.2.3 Analog‐to‐Digital Conversion and Automatic Gain Control
6.2.4 Baseband Signal Processing
6.3 Signal Acquisition and Tracking
6.3.1 Hypothesize About the User Location
6.3.2 Hypothesize About Which GNSS Satellites Are Visible
6.3.3 Signal Doppler Estimation
6.3.4 Search for Signal in Frequency and Code Phase
6.3.4.1 Sequential Searching in Code Delay
6.3.4.2 Sequential Searching in Frequency
6.3.4.3 Frequency Search Strategy
6.3.4.4 Parallel and Hybrid Search Methods
6.3.5 Signal Detection and Confirmation
6.3.5.1 Detection Confirmation
6.3.5.2 Coordination of Frequency Tuning and Code Chipping Rate
6.3.6 Code Tracking Loop
6.3.6.1 Code Loop Bandwidth Considerations
6.3.6.2 Coherent Versus Noncoherent Code Tracking
6.3.7 Carrier Phase Tracking Loops
6.3.7.1 PLL Capture Range
6.3.7.2 PLL Order
6.3.7.3 Use of Frequency‐Lock Loops (FLLs) for Carrier Capture
6.3.8 Bit Synchronization
6.3.9 Data Bit Demodulation
6.4 Extraction of Information for User Solution
6.4.1 Signal Transmission Time Information
6.4.2 Ephemeris Data for Satellite Position and Velocity
6.4.3 Pseudorange Measurements Formulation Using Code Phase
6.4.3.1 Pseudorange Positioning Equations
6.4.4 Measurements Using Carrier Phase
6.4.5 Carrier Doppler Measurement
6.4.6 Integrated Doppler Measurements
6.5 Theoretical Considerations in Pseudorange, Carrier Phase, and Frequency Estimations
6.5.1 Theoretical Error Bounds for Code Phase Measurement
6.5.2 Theoretical Error Bounds for Carrier Phase Measurements
6.5.3 Theoretical Error Bounds for Frequency Measurement
6.6 High‐Sensitivity A‐GPS Systems
6.6.1 How Assisting Data Improves Receiver Performance. 6.6.1.1 Reduction of Frequency Uncertainty
6.6.1.2 Determination of Accurate Time
6.6.1.3 Transmission of Satellite Ephemeris Data
6.6.1.4 Provision of Approximate Client Location
6.6.1.5 Transmission of the Demodulated Navigation Bit Stream
6.6.1.6 Server‐Provided Location
6.6.2 Factors Affecting High‐Sensitivity Receivers
6.6.2.1 Antenna and Low‐Noise RF Design
6.6.2.2 Degradation due to Signal Phase Variations
6.6.2.3 Signal Processing Losses
6.6.2.4 Multipath Fading
6.6.2.5 Susceptibility to Interference and Strong Signals
6.6.2.6 The Problem of Time Synchronization
6.6.2.7 Difficulties in Reliable Sensitivity Assessment
6.7 Software‐Defined Radio (SDR) Approach
6.8 Pseudolite Considerations
Problems
References
7 GNSS Measurement Errors. 7.1 Source of GNSS Measurement Errors
7.2 Ionospheric Propagation Errors
7.2.1 Ionospheric Delay Model
7.2.2 GNSS SBAS Ionospheric Algorithms
7.2.2.1 L1L2 Receiver and Satellite Bias and Ionospheric Delay Estimations for GPS. System Model
Observation Model
UDUT Kalman filter (See Chapter 10)
SM‐to‐Earth‐centered, Earth‐fixed (ECEF) transformation
7.2.2.2 Kalman Filter
7.2.2.3 Selection of Q and R
7.2.2.4 Calculation of Ionospheric Delay Using Pseudoranges
7.3 Tropospheric Propagation Errors
7.4 The Multipath Problem
7.4.1 How Multipath Causes Ranging Errors
7.5 Methods of Multipath Mitigation
7.5.1 Spatial Processing Techniques. 7.5.1.1 Antenna Location Strategy
7.5.1.2 Ground Plane Antennas
7.5.1.3 Directive Antenna Arrays
7.5.1.4 Long‐Term Signal Observation
7.5.2 Time‐Domain Processing
7.5.2.1 Narrow‐Correlator Technology (1990–1993)
7.5.2.2 Leading‐Edge Techniques
7.5.2.3 Correlation Function Shape‐Based Methods
7.5.2.4 Modified Correlator Reference Waveforms
7.5.3 Multipath Mitigation Technology (MMT)
7.5.3.1 Description
7.5.3.2 Maximum‐Likelihood (ML) Multipath Estimation
7.5.3.3 The Two‐Path ML Estimator (MLE)
7.5.3.4 Asymptotic Properties of ML Estimators
7.5.3.5 The MMT Multipath Mitigation Algorithm
7.5.3.6 The MMT Baseband Signal Model
7.5.3.7 Baseband Signal Vectors
7.5.3.8 The Log‐Likelihood Function
7.5.3.9 Secondary‐Path Amplitude Constraint
7.5.3.10 Signal Compression
7.5.3.11 Properties of the Compressed Signal
7.5.3.12 The Compression Theorem
7.5.4 Performance of Time‐Domain Methods. 7.5.4.1 Ranging with the C/A‐Code
7.5.4.2 Carrier Phase Ranging
7.5.4.3 Testing Receiver Multipath Performance
7.6 Theoretical Limits for Multipath Mitigation. 7.6.1 Estimation‐Theoretic Methods
7.6.1.1 Optimality Criteria
7.6.2 Minimum Mean‐Squared Error (MMSE) Estimator
7.6.3 Multipath Modeling Errors
7.7 Ephemeris Data Errors
7.8 Onboard Clock Errors
7.9 Receiver Clock Errors
7.10 Error Budgets
Problems
References
8 Differential GNSS. 8.1 Introduction
8.2 Descriptions of Local‐Area Differential GNSS (LADGNSS), Wide‐Area Differential GNSS (WADGNSS), and Space‐Based Augmentation System (SBAS) 8.2.1 LADGNSS
8.2.2 WADGNSS
8.2.3 SBAS. 8.2.3.1 Wide‐Area Augmentation System (WAAS)
8.2.3.2 European Global Navigation Overlay System (EGNOS)
8.2.3.3 Other SBAS
8.3 GEO with L1L5 Signals
8.3.1 GEO Uplink Subsystem (GUS) Control Loop Overview
8.3.1.1 Ionospheric Kalman Filters
8.3.1.2 Range Kalman Filter
8.3.1.3 Code Control Function
8.3.1.4 Frequency Control Function
8.3.1.5 L1L5 Bias Estimation Function
8.3.1.6 Code‐Carrier Coherence
8.3.1.7 Carrier Frequency Stability
8.4 GUS Clock Steering Algorithm
8.4.1 Receiver Clock Error Determination
8.4.2 Clock Steering Control Law
8.5 GEO Orbit Determination (OD)
8.5.1 OD Covariance Analysis
8.6 Ground‐Based Augmentation System (GBAS) 8.6.1 Local‐Area Augmentation System (LAAS)
8.6.2 Joint Precision Approach and Landing System (ALS)
8.6.3 Enhanced Long‐Range Navigation (eLORAN)
8.7 Measurement/Relative‐Based DGNSS
8.7.1 Code Differential Measurements
8.7.1.1 Single‐Difference Observations
8.7.1.2 Double‐Difference Observations
8.7.2 Carrier Phase Differential Measurements
8.7.2.1 Single‐Difference Observations
8.7.2.2 Double‐Difference Observations
8.7.2.3 Triple‐Difference Observations
8.7.2.4 Combinations of Code and Carrier Phase Observations
8.7.3 Positioning Using Double‐Difference Measurements. 8.7.3.1 Code‐Based Positioning
8.7.3.2 Carrier Phase‐Based Positioning
8.7.3.3 Real‐Time Processing Versus Postprocessing
8.8 GNSS Precise Point Positioning Services and Products
8.8.1 The International GNSS Service (IGS)
8.8.2 Continuously Operating Reference Stations (CORSs)
8.8.3 GPS Inferred Positioning System (GIPSY) and Orbit Analysis Simulation Software (OASIS)
8.8.4 Scripps Coordinate Update Tool (SCOUT)
8.8.5 The Online Positioning User Service (OPUS)
8.8.6 Australia's Online GPS Processing System (AUPOS)
8.8.7 National Resources Canada (NRCan)
Problems
References
9 GNSS and GEO Signal Integrity. 9.1 Introduction
9.1.1 Range Comparison Method
9.1.2 Least‐Squares Method
9.1.3 Parity Method
9.2 SBAS and GBAS Integrity Design
9.2.1 SBAS Error Sources and Integrity Threats
9.2.2 GNSS‐Associated Errors
9.2.2.1 GNSS Clock Error
9.2.2.2 GNSS Ephemeris Error
9.2.2.3 GNSS Code and Carrier Incoherence
9.2.2.4 GNSS Signal Distortion
9.2.2.5 GNSS L1L2 Bias
9.2.2.6 Environment Errors: Ionosphere
9.2.2.7 Environment Errors: Troposphere
9.2.3 GEO‐Associated Errors. 9.2.3.1 GEO Code and Carrier Incoherence
9.2.3.2 GEO‐Associated Environment Errors: Ionosphere
9.2.3.3 GEO‐Associated Environment Errors: Troposphere
9.2.4 Receiver and Measurement Processing Errors
9.2.4.1 Receiver Measurement Error
9.2.4.2 Intercard Bias
9.2.4.3 Multipath
9.2.4.4 L1L2 Bias
9.2.4.5 Receiver Clock Error
9.2.4.6 Measurement Processing Unpack/Pack Corruption
9.2.5 Estimation Errors
9.2.5.1 Reference Time Offset Estimation Error
9.2.5.2 Clock Estimation Error
9.2.5.3 Ephemeris Correction Error
9.2.5.4 L1L2 Wide‐Area Reference Equipment (WRE) and GPS Satellite Bias Estimation Error
9.2.6 Integrity‐Bound Associated Errors
9.2.6.1 Ionospheric Modeling Errors
9.2.6.2 Fringe Area Ephemeris Error
9.2.6.3 Small‐Sigma Errors
9.2.6.4 Missed Message: Old but Active Data (OBAD)
9.2.6.5 Time to Alarm (TTA) Exceeded
9.2.7 GEO Uplink Errors
9.2.7.1 GEO Uplink System Fails to Receive SBAS Message
9.2.8 Mitigation of Integrity Threats
9.2.8.1 Mitigation of GNSS Associated Errors. GNSS Clock Error
GPS Ephemeris Error
GNSS Code and Carrier
GNSS Signal Distortion
GNSS L1L2 Bias
Environment (Ionosphere) Errors
Environment (Troposphere) Errors
9.2.8.2 Mitigation of GEO‐Associated Errors. GEO Code and Carrier and Environment Errors
9.2.8.3 Mitigation of Receiver and Measurement Processing Errors. Receiver Measurement Error
Code Noise and Multipath (CNMP)
WRE L1L2 Bias
WRE Clock Error
9.2.8.4 Mitigation of Estimation Errors. Reference Time Offset Estimation Error
Clock Estimation Error, Ephemeris Correction Error, L1L2 WRE, and GNSS Satellite Bias Estimation Error
9.2.8.5 Mitigation of Integrity‐Bound‐Associated Errors. Ionospheric Modeling Error
Fringe Area Ephemeris Error
Small‐Sigma Errors
Missed Message: OBAD
TTA Exceeded
9.3 SBAS Example
9.4 Summary
9.5 Future: GIC
Problem
References
10 Kalman Filtering
10.1 Chapter Focus
10.2 Frequently Asked Questions
10.3 Notation
10.3.1 Real Vectors and Matrices. 10.3.1.1 Notation
10.3.1.2 Vector and Matrix Properties
10.3.2 Probability Essentials
10.3.2.1 Basic Concepts
10.3.2.2 Linearity of the Expectancy Operator
10.3.2.3 Means and Covariances of Linearly Transformed Variates
10.3.3 Discrete Time Notation. 10.3.3.1 Subscripting
10.3.3.2 A Priori and A Posteriori Values
10.3.3.3 Allowing for Testing and Rejecting Measurements
10.4 Kalman Filter Genesis
10.4.1 Measurement Update (Corrector)
10.4.1.1 Linear Least Mean Squares Estimation: Gauss to Kalman. Least Squares
Gauss–Markov Theorem and Homoscedasticity
Generalization by Aitken
Means and Mean Squared Estimation Error
Covariance of Estimation Error
Recursive Linear Least Mean Squares Estimation
Duncan–Guttman Formula
Kalman Gain Matrix
10.4.1.2 Kalman Measurement Update Equations
10.4.2 Time Update (Predictor)
10.4.2.1 Continuous‐Time Dynamics
Homogeneous and Nonhomogeneus Differential Equations
Example 10.1 ( Order Nonhomogeneous Ordinary Linear Differential Equation)
Homogeneous Equation Solutions
Matrix Exponentials
State Transition Matrices
Nonhomogeneous Solutions
Linear Stochastic Differential Equations
Stochastic Integrals and Markov Processes
10.4.2.2 Discrete‐Time Dynamics
10.4.3 Basic Kalman Filter Equations
10.4.4 The Time‐Invariant Case
10.4.5 Observability and Stability Issues
10.5 Alternative Implementations
10.5.1 Implementation Issues
10.5.2 Conventional Implementation Improvements
10.5.2.1 Measurement Decorrelation by Diagonalization
10.5.2.2 Exploiting Symmetry
10.5.2.3 Information Filter
10.5.2.4 Sigma Rho Filtering
10.5.3 James E. Potter (1937–2005) and Square Root Filtering
10.5.4 Square Root Matrix Manipulation Methods
10.5.4.1 Cholesky Decomposition
10.5.4.2 Modified Cholesky Decomposition
10.5.4.3 Nonuniqueness of Matrix Square Roots
10.5.4.4 Triangularization by QR Decomposition
10.5.4.5 Householder Triangularization
10.5.5 Alternative Square Root Filter Implementations. 10.5.5.1 Potter Implementation
Measurement Update
Time Update
10.5.5.2 Carlson “Fast Triangular” Square Root Filter
10.5.5.3 Bierman–Thornton UD Filter
10.5.5.4 Unscented Square Root Filter
10.5.5.5 Square Root Information Filter (SRIF)
10.6 Nonlinear Approximations
10.6.1 Linear Approximation Errors
Example 10.2 (Nonlinearity of satellite pseudorange measurements)
10.6.2 Adaptive Kalman Filtering
10.6.3 Taylor–Maclauren Series Approximations
10.6.3.1 First‐Order: Extended Kalman Filter
Iterated Extended Kalman Filtering
10.6.3.2 Second‐Order: Bass–Norum–Schwartz Filter
10.6.4 Trajectory Perturbation Modeling
10.6.5 Structured Sampling Methods
10.6.5.1 Sigma‐Point Filters
Sigma‐Point Transform
10.6.5.2 Particle Filters
10.6.5.3 The Unscented Kalman Filter
10.7 Diagnostics and Monitoring. 10.7.1 Covariance Matrix Diagnostics
10.7.1.1 Symmetry Control
10.7.1.2 Eigenanalysis
10.7.1.3 Conditioning
10.7.2 Innovations Monitoring. 10.7.2.1 Kalman Filter Innovations
10.7.2.2 Information‐Weighted Innovations Monitoring
Example 10.3 (Simulated Innovations Monitoring)
10.8 GNSS‐Only Navigation
10.8.1 GNSS Dynamic Models. 10.8.1.1 Receiver Clock Bias Dynamics
10.8.1.2 Discrete Time Models
10.8.1.3 Exponentially Correlated Random Processes
10.8.1.4 Host Vehicle Dynamics for Standalone GNSS Navigation
10.8.1.5 Point Mass Dynamic Models
Velocity Random Walk Models
Exponentially Correlated Velocity Models
Exponentially Correlated Acceleration Models
Models for Bounded RMS Velocity and Acceleration
10.8.2 GNSS Measurement Models. 10.8.2.1 Measurement Event Timing
10.8.2.2 Pseudoranges
10.8.2.3 Time and Distance Correlation
10.8.2.4 Measurement Sensitivity Matrix
10.8.2.5 Noise Model
Example 10.4 (State Vector Example for Standalone GNSS Navigation)
10.9 Summary
Problems
References
Notes
11 Inertial Navigation Error Analysis
11.1 Chapter Focus
11.2 Errors in the Navigation Solution
11.2.1 Navigation Error Variables
11.2.2 Coordinates Used for INS Error Analysis
11.2.3 Model Variables and Parameters
11.2.3.1 INS Orientation Variables and Errors
Misalignments and Tilts
Effect of INS Misalignments
Small‐Angle Rotation Rate Approximation
Effects of Position Errors
11.2.4 Dynamic Coupling Mechanisms. Dynamic coupling
11.3 Navigation Error Dynamics
11.3.1 Error Dynamics Due to Velocity Integration
11.3.2 Error Dynamics Due to Gravity Miscalculations. 11.3.2.1 INS Gravity Modeling
11.3.2.2 Navigation Error Model for Gravity Calculations
11.3.3 Error Dynamics Due to Coriolis Acceleration
11.3.4 Error Dynamics Due to Centrifugal Acceleration
11.3.5 Error Dynamics Due to Earthrate Leveling
11.3.6 Error Dynamics Due to Velocity Leveling
11.3.7 Error Dynamics Due to Acceleration and IMU Alignment Errors
11.3.8 Composite Model from All Effects
11.3.9 Vertical Navigation Instability
11.3.9.1 Altimeter Aiding
Example 11.1 (Three‐state model for altimeter aiding)
Example 11.2 (Ten‐state INS error model for altimeter aiding)
11.3.9.2 Using GNSS for Vertical Channel Stabilization
11.3.10 Schuler Oscillations
11.3.10.1 Schuler Oscillations with Coriolis Coupling
11.3.11 Core Model Validation and Tuning
11.3.11.1 Horizontal Inertial Navigation Model
11.4 Inertial Sensor Noise Propagation
11.4.1 Noise
11.4.2 White Noise
11.4.3 Horizontal CEP Rate Versus Sensor Noise
11.5 Sensor Compensation Errors. Sensor compensation parameters
Parameter drift
11.5.1 Sensor Compensation Error Models
11.5.1.1 Exponentially Correlated Parameter Drift Models
11.5.1.2 Dynamic Coupling into Navigation Errors
11.5.1.3 Augmented Dynamic Coefficient Matrix
11.5.2 Carouseling and Indexing
11.6 Chapter Summary
11.6.1 Further Reading
Problems
References
Notes
12 GNSS/INS Integration
12.1 Chapter Focus
12.2 New Application Opportunities
12.2.1 Integration Advantages. 12.2.1.1 Exploiting Complementary Error Characteristics
12.2.1.2 Cancelling Vulnerabilities
12.2.2 Enabling New Capabilities. 12.2.2.1 Real‐Time Inertial Sensor Error Compensation
12.2.2.2 INS Initialization on the Move
INS Alignment Using GNSS
12.2.2.3 Antenna Switching
12.2.2.4 Antenna‐INS Offsets
12.2.3 Economic Factors. 12.2.3.1 Economies of Scale
Market Size
Manufacturing Technology
12.2.3.2 Implementation Tradeoffs
The Curse of Dimensionality
Sensor Output Rates
Early Implementation Alternatives
Loosely Coupled Implementations
More Tightly Coupled Implementations
Ultra‐tightly Coupled Integration
Limitations
12.3 Integrated Navigation Models. 12.3.1 Common Navigation Models
12.3.2 GNSS Error Models. 12.3.2.1 GNSS Time Synchronization. UTC and GNSS time
12.3.2.2 Receiver Clock Error Model
Model Parameter Values
Cold‐start Initialization
12.3.2.3 Propagation Delay. Dual‐frequency GNSS
Steady‐state Delay Covariance
12.3.2.4 Pseudorange Measurement Noise
12.3.3 INS Error Models. 12.3.3.1 Navigation Error Model
12.3.3.2 Sensor Compensation Errors
12.3.4 GNSS/INS Error Model. 12.3.4.1 State Variables
12.3.4.2 Numbers of State Variables
12.3.4.3 Dynamic Coefficient Matrix
12.3.4.4 Process Noise Covariance
12.3.4.5 Measurement Sensitivities
12.4 Performance Analysis. 12.4.1 The Influence of Trajectories
12.4.2 Performance Metrics. 12.4.2.1 Application‐Dependent Performance Metrics
12.4.2.2 General‐Purpose Metrics
12.4.2.3 Mean Squared Error Metrics
12.4.2.4 Probabilistic Metrics. CEP
Medians of PDFs
12.4.3 Dynamic Simulation Model. 12.4.3.1 State Transition Matrices
12.4.3.2 Dynamic Simulation
12.4.4 Sample Results. 12.4.4.1 Stand‐Alone GNSS Performance. Simulating Satellite Selection
GNSS‐Only MATLAB® Implementation
12.4.4.2 INS‐Only Performance
12.4.4.3 Integrated GNSS/INS Performance
12.5 Summary
Problems
References
Notes
Appendix A Software
A.1 Software Sources
A.2 Software for Chapter 2
A.3 Software for Chapter 3
A.4 Software for Chapter 4
A.5 Software for Chapter 7
A.6 Software for Chapter 10
A.7 Software for Chapter 11
A.8 Software for Chapter 12
A.9 Software for Appendix B
A.10 Software for Appendix C
A.11 GPS Almanac/Ephemeris Data Sources
Appendix B Coordinate Systems and Transformations
B.1 Coordinate Transformation Matrices. B.1.1 Notation
B.1.2 Definitions
B.1.3 Unit Coordinate Vectors
B.1.4 Direction Cosines
B.1.5 Composition of Coordinate Transformations
B.2 Inertial Reference Directions
B.2.1 Earth's Polar Axis and the Equatorial Plane
B.2.2 The Ecliptic and the Vernal Equinox
B.2.3 Earth‐Centered Inertial (ECI) Coordinates
B.3 Application‐dependent Coordinate Systems
B.3.1 Cartesian and Polar Coordinates
B.3.2 Celestial Coordinates
B.3.3 Satellite Orbit Coordinates
B.3.4 Earth‐Centered Inertial (ECI) Coordinates
B.3.5 Earth‐Centered, Earth‐Fixed (ECEF) Coordinates
B.3.5.1 Longitude in ECEF Coordinates
B.3.5.2 Latitudes in ECEF Coordinates
B.3.5.3 Latitude on an Ellipsoidal Earth
B.3.5.4 Parametric Latitude
B.3.5.5 Geodetic Latitude
B.3.5.6 WGS84 Reference Geoid Parameters
B.3.5.7 Geocentric Latitude
B.3.5.8 Geocentric Radius
B.3.6 Ellipsoidal Radius of Curvature
B.3.7 Local Tangent Plane (LTP) Coordinates
B.3.7.1 Alpha Wander Coordinates
B.3.7.2 ENU/NED Coordinates
B.3.7.3 ENU/ECEF Coordinates
B.3.7.4 NED/ECEF Coordinates
B.3.8 Roll–Pitch–Yaw (RPY) Coordinates
B.3.9 Vehicle Attitude Euler Angles
B.3.9.1 RPY/ENU Coordinates
B.3.10 GPS Coordinates
B.4 Coordinate Transformation Models
B.4.1 Euler Angles
B.4.2 Rotation Vectors
B.4.2.1 Rotation Vector to Matrix
B.4.2.2 Matrix to Rotation Vector
B.4.2.3 Special Cases for
B.4.2.4 MATLAB® Implementations
B.4.2.5 Time Derivatives of Rotation Vectors
B.4.2.6 Time Derivatives of Matrix Expressions
General Formulas
Time Derivative of
Time Derivative of
B.4.2.7 Partial Derivatives with Respect to Rotation Vectors
Derivatives of Scalars
Derivatives of Vectors
General Formula
B.4.3 Direction Cosines Matrix
B.4.3.1 Rotating Coordinates
B.4.4 Quaternions
B.4.4.1 Quaternion Matrices
B.4.4.2 Addition and Multiplication
B.4.4.3 Conjugation
B.4.4.4 Representing Rotations
B.5 Newtonian Mechanics in Rotating Coordinates
B.5.1 Rotating Coordinates
B.5.2 Time Derivatives of Matrix Products
B.5.3 Solving for Centrifugal and Coriolis Accelerations
Notes
Appendix C PDF Ambiguity Errors in Nonlinear Kalman Filtering
C.1 Objective
C.2 Methodology
C.2.1 Computing Expected Values
C.2.2 Representative Sample of PDFs
Example C.1 (Verifying PDF‐independence of Eq. (10.20))
C.2.3 Parametric Class of Nonlinear Transformations Used
C.2.4 Ambiguity Errors in Nonlinearly Transformed Means and Variances
C.3 Results. C.3.1 Nonlinearly Transformed Means
C.3.2 Nonlinearly Transformed Variances
C.4 Mitigating Application‐specific Ambiguity Errors
References
Index
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