Kalman_and_Bayesian_Filters_in_Python.pdf卡尔曼与贝叶斯 P

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详细说明：卡尔曼与贝叶斯 Python版本本书教你如何解决这些过滤噪声问题，设计滤波器。使用了许多不同的算法，但它们都基于贝叶斯概率。简单来说，贝叶斯概率根据过去的信息确定可能是真实的。Contents Preface 0.1 Kalman and Bayesian Filters 0.2 Motivation for this book 12 0.3 Reading Online 13 0. 4 PDF Version 13 0.5 Downloading and running the book .....13 0.6 Jupyter 14 0.7 SciPy, NumPy, and matplotlib 15 0.7.1 Exercise-Create arrays .,,,,,,,,,19 0.7.2 Solution ..,,,,.,.19 0.8 Companion Software 20 0. 9 Thoughts on Python and Coding Math 20 0.10 License 21 0.11 Contact 21 0.12 Resources 21 1 The g-h Filter 1.1 Building Intuition via Thought Experiments 1.2 The g-h Filter 35 1. 3 Notation 38 1.4 Exercise: Write Generic Algorithm 1.4.1 Solution and Discussion 着垂 1.5 Choice of g and h 1.6 Exercise: create measurement function 1.6.1 Solution 1.6.2 Discussion 41 1. 7 Exercise: Bad Initial Conditions 41 1.7.1 Solution and Discussion 1.8 Exercise: Extreme Noise 42 1.8.1 Solution and Discussion 42 1.9 Exercise: The Effect of Acceleration 42 1.9.1 Solution and Discussion 1.10 Exercise: Varying g .43 1.10.1 Solution and Discussion 44 1.11V 46 1.12 Interactive Example 46 1.13 Don't lie to the filter 48 1. 14 Tracking a Train 1.15 g-h Filters with FilterP 53 1.16 Summar 3 CONTENTS 2 Discrete Bayes Filter 57 2.1 Tracking a Dog 57 2.2 Extracting Information from Sensor Readings 59 2.3 Noisy Sensors 60 2.4 Incorporating Movement 2.5 Terminology 2.6 Adding Uncertainty to the Prediction 66 2.7 Generalizing with Convolution 2.8 Integrating Measurements and Movement Updates 2. 9 The Discrete Bayes Algorithm ,,,,,,,74 2.10 The Effect of Bad Sensor Data 77 2.11 Drawbacks and limitations 2.12 Tracking and control 2.12.1 Simulating the Train behavior 2. 13 Bayes Theorem and the Total Probability Theorem 2. 14 Summary 85 2.15 Rcfo rences 85 3 Probabilities, Gaussians, and Bayes'Theorem 87 3.1 Introduction 87 3.2 Mean, Variance, and Standard Deviations 3.2.1 Random Variables 7 3.3 Probability distribution 3.3.1 The mean, Median, and mode of a random variable 3. 4 Expected Value of a Random Variable 90 3.4.1 Exercise 3.4.2 Solution 91 3.4.3Eⅹ ercise.., ..92 3.4.4 Solution 92 3.4.5 Variance of a Random variable ···· 92 3.4.6 Why the Square of the Differences 96 3.5 Gaussians 3.6 Nomenclature 3.7 Gaussian Distributions 3. 8 The Variance and belief 102 3.9The68-95-997Rule 103 3.10 Interactive Gaussians 104 3.11 Computational Properties of Gaussians ...105 3.12 Putting it all Together .· 107 3. 12.1 Bayes Theorem 108 3.12.2 Total Probability Theorem 110 3.13 Computing Probabilities with scipystats 111 3. 14 Limitations of Using Gaussians to Model the World 112 3.15 Product of Gaussians(Optional) 115 3.16 Sum of Gaussians(Optional) 116 3. 17 Summary and Key points .· ,,,,,,.,,,117 3.18 References 117 3.19 Useful Wikipedia Links l18 CONTENTS 4 One Dimensional Kalman filters 119 4.1 Problem Description 119 4.2 Beliefs as gaussians 120 4.3 Tracking with Gaussian Probabilities 121 4.4 Predictions with gaussians 122 4.5 Updates with Gaussians 124 4.5.1 Understanding Gaussian Multiplication 125 4.5.2 Interactive Example 128 4.6 First Kalman filter 129 4.7 Code Walkthrough 132 4.7.1 Exercise: Modify Variance Values .134 4.7.2 KF Animation .134 4. 8 Kalman gain ....135 4.9 Full Description of the algorithm .137 4.10 Comparison with g-h and discrete Bayes Filters .138 4.11 Introduction to Designing a Filter 140 4.11.1 Animation 143 4.12 Example: Extreme Amounts of noise .143 4.13 Example: Incorrect Process variance 144 4.14 Example: Bad Initial Estimate .....145 4.15 Example: Large noise and Bad Initial Estimate 146 4.16 Exercise: Interactive Plots 垂垂 ,,,,..148 4.161 Solution 4.17 Exercise- Nonlincar Systems 150 4.17.1 Solution 150 4.17.2 Discussion 151 4.18 Fixed Gain Filters 151 4.19 FilterPy's Implementation 152 4.20 Summary 152 5 Multivariate gaussians 155 5. 1 Introduction 155 5.2 Multivariate Normal distributions ..155 5.3 Correlation and Covariance 156 5.4 Multivariate Normal Distribution Equation 161 5.4.1 Pearson's Correlation Coefficient 171 5.5 Using Correlations to Improve Estimates 5.6 Multiplying Multidimensional Gaussians ...173 5.7 Hidden variables .· 5.8 Higher Dimensions 183 5.9 Summary 183 5.10 References ,,,,,184 6 Multivariate Kalman Filters 185 6.1 Introduction 185 6.2 Newtons equations of Motion .185 6.3 Kalman Filter Algorithm 186 6.4 Tracking a Dog 188 6.5 Predict Step 188 6.5.1 Design State Variable 188 6.5.2 Design State Covariance 190 6.5.3 Design the Process Model 191 6.5.4 Design Process Noise ...194 6.5.5 Design the Control Function 195 CONTENTS 6.6 Update Step ...196 6.6.1 Design the Measurement Function 196 6.7 Implementing the Kalman Filter 198 6. 8 The Saver Class .202 6.9 The Kalman Filter equations .204 6.9.1 Prediction Equations 205 6.9.2 Update Equations .208 6.9.3 An Example not using FilterPy .210 6.9.4 Summary 211 6.10 Exercise: Show Effect of hidden variables ,,,,,,.213 6.10.15olu 213 6.10.2 Discussion .214 6. 11 How Velocity is Calculated 215 6. 12 Adjusting the Filter 217 6.13 A Detailed Examination of the Covariance matrix .220 6. 14 Question: Explain Ellipse Differences 225 6. 14.1 Solution 225 6.15 Filter Initialization 226 6.16 Batch Processing 6.17 Smoothing the Results ,,,,229 6.18 Exercise: Compare Velocities 230 6.18.1 Solution .,,231 6.19 Discussion and Summary 231 6.20 References 232 7 Kalman filter math 233 7.1 Modeling a Dynamic System 233 7.2 State-Space Representation of Dynamic Systems 235 7.2.1 Forming First Order Equations from Higher Order Equations ,,,236 7.2.2 First Order Differential Equations In State-Space Form 236 7.2.3 Finding the Fundamental Matrix for Time Invariant Systems 7.2.4 The Matrix Exponentia .237 7. 2.5 Time Invariance 239 7.2.6 Linear Time Invariant Theory 240 7.2.7 Numerical Solutions 240 7.3 Design of the Process Noise Matrix 241 7. 3.1 Continuous white noise model ,,,,,,,241 7.3.2 Piecewise White Noise model 243 7.3.3 USing FilterPy to Compute Q .· ..245 7.3.4 Simplification of Q 245 7.4 Stable Compution of the Posterior Covariance 24 7.5 Deriving the Kalman gain equation 248 7.6 Numeric Integration of Differential Equations 249 7.6.1 Eulers method 249 7.6.2 Runge Kutta Methods 7.7 Bayesian Filtering .· 254 7.8 Converting Kalman Filter to a g-h Filter 255 7.9 References 257 CONTENTS 8 Designing Kalman Filters 259 8.1 Introduction 259 82 Tracking a Robot· .259 8.2.1 Choose the State Variables 260 8.2.2 Design State Transition Function 261 8.2.3 Design the Process Noise matrix 262 8.2.4 Design the Control Function 262 8.2.5 Design the Measurement Function .262 8.2.6 Design the Measurement Noise Matrix 263 8.2.7 Initial Conditions .263 8.2.8 Implement the filter 263 8.3 Filter Order 266 8.3.1 Zero Order Kalman filter .268 8.3.2 First Order Kalman filter 269 8.3.3 Second Order Kalman Filter 270 8.4 Evaluating Filter Order 271 8.5 Excrcisc: State Variable Design 284 8.5.1 Solution .285 8.6 Detecting and Rejecting Bad Measurement 287 8.6.1 Gating and data Association Strategies .....291 8.7 Evaluating Filter Performance 293 8.7.1 Normalized Estimated Error Squared(NEES) 8. 7.2 Likelihood Function 8.8 Control Inputs 8.9 Sensor Fusion 297 8.9.1 Exercise: Can you Filter gPS outputs? 302 8.9.2 Exercise: Prove that the position sensor improves the filter 306 8.10 Nonstationary Processes 307 8.10.1 Sensor fusion Different data rates 308 8.11 Tracking a Ball 垂 ..310 8.11.1 Choose the state Variables 8.11.2 Design State Transition Function ..313 8.11.3 Design the Control Input Function 313 8.11. 4 Design the Measurement Function 314 8. 11.5 Design the Measurement Noise Matrix 314 8.11.6 Design the process Noise matrix 315 8.11.7 Design the Initial Conditions 315 8.12 Tracking a Ball in Air .316 8.12. 1 Implementing Air Drag 317 8.13 References 325 9 Nonlinear Filtering 327 9.1 Introducti 327 9.2 The Problem with non linearity 328 9.3 An Intuitive Look at the problem 328 9. 4 The Effect of nonlinear Functions on gaussians .· 9.5 A 2D Example 334 9.6 The A t orithms 337 9.7 Summar .338 9. 8 Refe 338 CONTENTS 10 The Unscented Kalman filter 339 10.1 Sigma Points -Sampling from a Distribution 341 10. 2 A Quick Example .342 10.3 Choosing Sigma Points 344 10.4 The Unscented Transform 10.4.1 Accuracy of the Unscented Transform 347 10.5 The Unscented Kalman Filter .349 10.5.1 Predict Step 349 10.5.2 Update Step 349 10.6 Van der Merwe's Scaled Sigma Point Algorithm .351 10.6.1 Sigma Point Computation 351 10.6.2 Weight Computation ...352 10.6.3 Reasonable Choices for the parameters ....352 10.7 Using the UKF .352 10.8 Tracking an airplane 356 10.8.1 Tracking Maneuvering Aircraft 360 10.8.2 Sensor Fusion 10.8.3 Multiple position Sensors 365 10.9 Effects of sensor Error and Geometr 368 10.10Exercise: Explain Filter Performance .....369 10.10.1 Solution .369 10.11Implementation of the uKF 垂垂 ....370 10.11.1 Weights 10.11. 2 Sigma Points 371 10. 11.3 Predict Step .373 10.11. 4 Update Step 373 10.11.5 FilterPy's Implementation 374 10.12Batch Processing 374 10.13Smoothing the Results 377 10.14 Choosing the Sigma Parameters 378 10. 15Robot Localization- A Fully Worked Example 380 10.15.1 Robot motion model 10.15.2 Design the State Variables 10. 15.3 Design the System Model 382 10. 15.4 Design the Measurement Model 382 10.15.5 Design Measurement noise 383 10.15.6 Implementation .383 10. 15.7 Steering the Robot .387 10.16 Discussion 10.17References 11 The Extended Kalman Filter 391 11. 1 Linearizing the Kalman Filter 391 11.2 Example: Tracking a Airplane 393 11.2.1 Design the State Variables 394 11.2.2 Design the Process Model .· 11.2.3 Design the Measurement Model .395 11.2.4 Design Process and Measurement noise 397 11.2.5 Implementation 397 11.3 Using SymPy to compute Jacobians 400 11.4 Robot localization 400 11.4.1 Robot motion model 401 11.4.2 Design the State variables 401 11.4.3 Design the System Model 402 CONTENTS 11.4.4 Design the measurement model 404 11.4.5 Design Measurement Noise 405 11.4.6 Implementation 405 11.4.7 Discussion 11.5 UKF VS EKF 22 12 Particle filters 415 12.1 Motivation 415 12.2 Monte Carlo Sampling 416 12.3 Generic Particle Filter Algorithm 417 12.4 Probability distributions via Monte Carlo b翻 418 12.5 The Particle Filter 421 垂垂 12.5.1 Predict Step 422 12.5.2 Update Step 422 12.5.3 Computing the State Estimate 423 12.5.4 Particle resampling ....423 12.6 SIR Filter-A Complete example 424 12.6.1 Effect of Sensor Errors on the filter 428 12.6.2 Filter Degeneracy From Inadequate Samples 429 12.7 Importance Sampling 430 12. 8 Resampling Method ..,,,,,,,,.432 12.8. 1 Multinomial Resampling ..,,,,,,.432 12.8. 2 Residual Resampling ...433 12.8.3 Stratified Resampling 434 12.8.4 Systematic Resampling .435 12.8.5 Choosing a Resampling Algorithm 435 12.9 Summary 436 12.10References ..437 13 Smoothing 439 13.1 Introduction 439 13.2 An Overview of How smoothers work 441 13. 3 Types of Smoothers 441 134Ch f filters 442 13.5 Fixed-Interval Smoothing 442 13.6 Fixed-Lag Smoothing 13.7 Referen 448 4 Adaptive Filtering 449 14.1 Introduction 449 14.2 Maneuvering Targets 449 14.3 Detecting a Maneuver 456 14.4 Adjustable Process noise 458 14.4.1 Continuous Adjustment .458 14.4.2 Continuous Adjustment-Standard Deviation Version 460 14.5 Fading Memory Filter 14.6 Multiple model Estimation 14.6.1 A Two Filter Adaptive Filter 470 147 MMAE 472 147.1 Limitations of the mmae filter 474 14.8 Interacting Multiple Models(IMM) .475 14.8.1 Mode Probabilitic 476 14.8. 2 Mode Transitions 476 14.83 Computing the Mode Probabilities 478 CONTENTS 14.9 Mixing Probabilities 149.1 IMM Estimate 14.9.2 Tracking Maneuvering Target with the IMM 480 480 14.9.3 Limitations of the IMM 482 14.10Summary ,,484 14.11 References· 484 A Installation 487 A1 Installing the SciPy Stack 487 A 2 Installing FilterPy 488 A3 Downloading and Running the Book b翻 488 A4 Companion Software 489 A.5 Using Jupyter Notebook 489 A 6 SymP 489 A7 Various links B Symbols and notations 493 B.1 Labbe 493 B2 Wikipedia B 3 Brookner 493 B 4 Gelb 494 B5 Brown 494 B6 Zarchan 494 C INfinity filter 495 D Ensemble Kalman Filters 497 D. 1 The Algorithm 498 D.1.1 Initialize Step 499 D 1.2 Predict Ste 499 D 1.3 Update Ste 499 D2 Implementation and example 501 D 3 Outstanding Questions 503 D 4 References 504

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