Fundamentals of Kalman filtering: a practical approach

Numerical basics

Method of least squares

Recursive least-quares filtering

Polynomial Kalman filters

Kalman filters in a nonpolynomial world

Continuous polynomial Kalman filter

Extended Kalman filtering

Drag and falling object

Cannon-launched projectile tracking problem

Tracking a sine wave

Satellite navigation

Biases

Linearized Kalman filtering

Miscellaneous topics

Fading-memory filter

Assorted techniques for improving Kalman-filter performance

Fundamentals of Kalman-filtering software

Key formula and concept summary.

Chapter 1. Numerical Basics 1 --

Simple Vector Operations 1 --

Simple Matrix Operations 3 --

Numerical Integration of Differential Equations 13 --

Noise and Random Variables 19 --

Gaussian Noise Example 23 --

Calculating Standard Deviation 26 --

White Noise 28 --

Simulating White Noise 30 --

State-Space Notation 33 --

Fundamental Matrix 34 --

Chapter 2. Method of Least Squares 41 --

Zeroth-Order or One-State Filter 42 --

First-Order or Two-State Filter 46 --

Second-Order or Three-State Least-Squares Filter 50 --

Third-Order System 56 --

Experiments with Zeroth-Order or One-State Filter 59 --

Experiments with First-Order or Two-State Filter 64 --

Experiments with Second-Order or Three-State Filter 71 --

Comparison of Filters 78 --

Accelerometer Testing Example 80 --

Chapter 3. Recursive Least-Squares Filtering 91 --

Making Zeroth-Order Least-Squares Filter Recursive 91 --

Properties of Zeroth-Order or One-State Filter 93 --

Properties of First-Order or Two-State Filter 103 --

Properties of Second-Order or Three-State Filter 112 --

Chapter 4. Polynomial Kalman Filters 129 --

General Equations 129 --

Derivation of Scalar Riccati Equations 131 --

Polynomial Kalman Filter (Zero Process Noise) 134 --

Comparing Zeroth-Order Recursive Least-Squares and Kalman Filters 136 --

Comparing First-Order Recursive Least-Squares and Kalman Filters 139 --

Comparing Second-Order Recursive Least-Squares and Kalman Filters 142 --

Comparing Different-Order Filters 148 --

Initial Covariance Matrix 151 --

Riccati Equations with Process Noise 155 --

Example of Kalman Filter Tracking a Falling Object 159 --

Revisiting Accelerometer Testing Example 171 --

Chapter 5. Kalman Filters in a Nonpolynomial World 183 --

Polynomial Kalman Filter and Sinusoidal Measurement 183 --

Sinusoidal Kalman Filter and Sinusoidal Measurement 194 --

Suspension System Example 203 --

Kalman Filter for Suspension System 207 --

Chapter 6. Continuous Polynomial Kalman Filter 219 --

Theoretical Equations 219 --

Zeroth-Order or One-State Continuous Polynomial Kalman Filter 221 --

First-Order or Two-State Continuous Polynomial Kalman Filter 227 --

Second-Order or Three-State Continuous Polynomial Kalman Filter 232 --

Transfer Function for Zeroth-Order Filter 238 --

Transfer Function for First-Order Filter 243 --

Transfer Function for Second-Order Filter 248 --

Filter Comparison 251 --

Chapter 7. Extended Kalman Filtering 257 --

Theoretical Equations 257 --

Drag Acting on Falling Object 259 --

First Attempt at Extended Kalman Filter 261 --

Second Attempt at Extended Kalman Filter 274 --

Third Attempt at Extended Kalman Filter 284 --

Chapter 8. Drag and Falling Object 293 --

Problem Setup 293 --

Changing Filter States 309 --

Why Process Noise Is Required 311 --

Linear Polynomial Kalman Filter 320 --

Chapter 9. Cannon-Launched Projectile Tracking Problem 331 --

Problem Statement 331 --

Extended Cartesian Kalman Filter 334 --

Polar Coordinate System 349 --

Extended Polar Kalman Filter 354 --

Using Linear Decoupled Polynomial Kalman Filters 367 --

Using Linear Coupled Polynomial Kalman Filters 376 --

Robustness Comparison of Extended and Linear Coupled Kalman Filters 385 --

Chapter 10. Tracking a Sine Wave 395 --

Extended Kalman Filter 395 --

Two-State Extended Kalman Filter with a Priori Information 408 --

Alternate Extended Kalman Filter for Sinusoidal Signal 417 --

Another Extended Kalman Filter for Sinusoidal Model 431 --

Chapter 11. Satellite Navigation 443 --

Problem with Perfect Range Measurements 443 --

Estimation Without Filtering 447 --

Linear Filtering of Range 453 --

Using Extended Kalman Filtering 455 --

Using Extended Kalman Filtering with One Satellite 465 --

Using Extended Kalman Filtering with Constant Velocity Receiver 474 --

Single Satellite with Constant Velocity Receiver 479 --

Using Extended Kalman Filtering with Variable Velocity Receiver 493 --

Variable Velocity Receiver and Single Satellite 505 --

Chapter 12. Biases 515 --

Influence of Bias 515 --

Estimating Satellite Bias with Known Receiver Location 519 --

Estimating Receiver Bias with Unknown Receiver Location and Two Satellites 525 --

Estimating Receiver Bias with Unknown Receiver Location and Three Satellites 533 --

Chapter 13. Linearized Kalman Filtering 549 --

Theoretical Equations 549 --

Falling Object Revisited 552 --

Developing a Linearized Kalman Filter 556 --

Cannon-Launched Projectile Revisited 569 --

Linearized Cartesian Kalman Filter 570 --

Chapter 14. Miscellaneous Topics 587 --

Sinusoidal Kalman Filter and Signal-to-Noise Ratio 587 --

When Only a Few Measurements Are Available 595 --

Detecting Filter Divergence in the Real World 606 --

Observability Example 618 --

Aiding 629 --

Appendix Fundamentals of Kalman-Filtering Software 647 --

Software Details 647 --

MATLAB 648 --

True BASIC 654.