Introduction to Modern Navigation Systems

Contents......Page 12
Preface......Page 8
Introduction......Page 16
1.1 Introduction......Page 22
1.2 Vector Inner Product......Page 24
1.3 Vector Cross Products and Skew Symmetric Matrix Algebra......Page 25
2.1 Introduction......Page 32
2.2 Direction Cosine Matrices......Page 33
2.3 The Direction Cosine Matrix is a Unitary Matrix......Page 35
2.4 The Direction Cosine Matrix is a Transformation Matrix......Page 36
2.5 DCM Fixed Axis......Page 39
2.6 The Rotation Matrix......Page 41
2.7 Inner and Outer Transformation Matrices......Page 44
2.8 The Quaternion......Page 47
3.1 Introduction......Page 50
3.2 Simple Frame Rotations......Page 51
3.3 Euler Angles......Page 52
3.4 Rotation Vector......Page 53
3.5 Quaternion......Page 54
3.6 Simple Quaternions......Page 58
3.7.2 Conversion between DCM and Quaternion......Page 60
3.8 Dynamics of the Transformation Matrix......Page 62
3.8.1 DCM Differential Equation......Page 63
3.8.2 Quaternion Differential Equation......Page 65
3.8.3 Rotation Vector Differential Equation......Page 67
3.8.4 Euler Angles Differential Equation......Page 70
4.1 Introduction......Page 73
4.2 Earth, Geoid and Ellipsoid......Page 74
4.3 Radii of Curvature......Page 78
4.4 Earth, Inertial and Navigation Frames......Page 80
4.6 The Craft Rate n en ω......Page 82
4.8 Gravitational and Gravity Fields......Page 85
5.1 Introduction......Page 90
5.2 Body Frame of Reference......Page 91
5.3.1 The Accelerometer......Page 92
5.4 The Attitude Equation......Page 93
5.5 The Navigation Equation......Page 95
5.6 Navigation Equations Computational Flow Diagram......Page 98
5.7 The Navigation Equation in Earth Frame......Page 99
6.1 Introduction......Page 101
6.2 The Rotation Vector Differential Equation......Page 102
6.3 The Attitude Equation......Page 107
6.4 The Craft Velocity Equation......Page 110
6.5 The Craft Position Equation......Page 114
6.6 The Vertical Channel......Page 116
7.1 Introduction......Page 119
7.2 US Standard Atmosphere 1976......Page 120
7.3 Pressure Altitude......Page 122
7.4 Vertical Channel Parameter Estimation Using Inertial and Air Data......Page 126
7.5 Density Altitude......Page 131
7.7 Air Speed......Page 132
7.8 Indicated Air Speed (IAS)......Page 134
8.1 Introduction......Page 136
8.2 The Wander Azimuth Navigation......Page 138
8.3 Prospective of the Wander Azimuth Approach......Page 141
8.4 Polar Circle Navigation Algorithm......Page 143
8.5 Alternative Polar Circle Navigation Frame......Page 147
9.1 Introduction......Page 151
9.2 IMU Alignment......Page 152
9.3 Alternative Algorithm for b n C......Page 159
9.4 Estimation of the Accelerometer and Gyro Biases......Page 164
9.5 Effects of Biases on Estimate of b n C......Page 165
10.2 Attitude Initialization......Page 167
10.3 Heading Initialization......Page 170
10.4 Gyro Drift Compensation......Page 174
10.5.1 X-Gyro Bias......Page 175
10.5.2 Y-Gyro Bias......Page 177
10.6 Alternative Approach for Gyro Drift Compensation......Page 178
10.7.1 Rate Gyro Threshold Selection......Page 180
11.1 Introduction......Page 182
11.2 Navigation Frame Error Equation......Page 183
11.2.1 Craft Rate Error n en ω δ......Page 184
11.2.2 Earth Rate Error n ie ω δ......Page 185
11.2.3 Position Errors......Page 186
11.2.4 Attitude Error......Page 188
11.2.5 Gravity Error......Page 191
11.2.6 Velocity Error......Page 192
11.2.8 Error Block Diagram......Page 194
11.3 Earth Frame Error Equations......Page 195
11.3.1 Attitude Error......Page 196
11.3.2 Velocity Error......Page 197
11.4 Inertial Sensors Error Models......Page 198
11.5 The Global Positioning System......Page 202
11.6 Mechanization of the INS/GPS Equations......Page 206
Appendix A. The Vector Dot and Cross Products......Page 209
Appendix B. Introduction to Quaternion Algebra......Page 212
Appendix C. Simulink® Models......Page 217
Appendix D. Ellipse Geometry......Page 221
Appendix E. Vector Dynamics......Page 228
Appendix F. Derivation of Air Speed Equations......Page 234
Appendix G. DCM Error Algebra......Page 237
Appendix H. Kalman Filtering......Page 241
Index......Page 252