Intuitive Understanding of Kalman Filtering with MATLAB

Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Acknowledgments......Page 12
Authors......Page 14
Introduction......Page 16
Part I Background......Page 20
  1.1 DETERMINISTIC AND RANDOM MODELS AND VARIABLES......Page 22
  1.2 HISTOGRAMS AND PROBABILITY FUNCTIONS......Page 25
  1.3 THE GAUSSIAN (NORMAL) DISTRIBUTION......Page 31
  1.4 MODIFICATION OF A SIGNAL WITH GAUSSIAN DISTRIBUTION THROUGH A FUNCTION REPRESENTED BY A STRAIGHT LINE......Page 33
  1.5 EFFECTS OF MULTIPLYING TWO GAUSSIAN DISTRIBUTIONS......Page 40
  2.1 JOINT DISTRIBUTIONS—BIVARIATE CASE......Page 44
  2.2 BIVARIATE GAUSSIAN DISTRIBUTION—COVARIANCE AND CORRELATION......Page 51
  2.3 COVARIANCE MATRIX......Page 57
  2.4 PROCESSING A MULTIDIMENSIONAL GAUSSIAN DISTRIBUTION THROUGH A LINEAR TRANSFORMATION......Page 58
  2.5 MULTIPLYING TWO MULTIVARIATE GAUSSIAN DISTRIBUTIONS......Page 59
  3.1 CONDITIONAL PROBABILITY AND THE BAYES’ RULE......Page 64
  3.2 BAYES’ RULE FOR DISTRIBUTIONS......Page 69
Part II Where Does Kalman Filtering Apply and What Does It Intend to Do?......Page 72
  4.1 A SIMPLE MODELING SCENARIO: DC MOTOR CONNECTED TO A CAR BATTERY......Page 74
  4.2 POSSIBILITY TO ESTIMATE THE STATE VARIABLE BY PREDICTION FROM THE MODEL......Page 76
   4.2.2 External Uncertainty Impacting the System......Page 77
   4.3.1 Uncertainty in the Values Read of the Measured Variable......Page 78
 Chapter 5 General Scenario Addressed by Kalman Filtering and Specific Cases......Page 80
  5.1 ANALYTICAL REPRESENTATION OF A GENERIC KALMAN FILTERING SITUATION......Page 81
  5.2 UNIVARIATE ELECTRICAL CIRCUIT EXAMPLE IN THE GENERIC FRAMEWORK......Page 86
  5.3 AN INTUITIVE, MULTIVARIATE SCENARIO WITH ACTUAL DYNAMICS: THE FALLING WAD OF PAPER......Page 89
  6.1 GOALS AND ENVIRONMENT FOR EACH ITERATION OF THE KALMAN FILTERING ALGORITHM......Page 94
  6.2 THE PREDICTION PHASE......Page 95
  6.3 MEASUREMENTS PROVIDE A SECOND SOURCE OF KNOWLEDGE FOR STATE ESTIMATION......Page 97
  6.4 ENRICHING THE ESTIMATE THROUGH BAYESIAN ESTIMATION IN THE “CORRECTION PHASE”......Page 98
  7.1 SO, WHAT IS THE KALMAN FILTER EXPECTED TO ACHIEVE?......Page 106
  7.2 EACH ITERATION OF THE KALMAN FILTER SPANS “TWO TIMES” AND “TWO SPACES”......Page 107
  7.3 YET, IN PRACTICE ALL THE COMPUTATIONS ARE PERFORMED IN A SINGLE, “CURRENT” ITERATION—CLARIFICATION......Page 109
  7.4 MODEL OR MEASUREMENT? KG DECIDES WHO WE SHOULD TRUST......Page 110
Part III Examples in MATLAB®......Page 120
  8.1 DATA AND COMPUTATIONS NEEDED FOR THE IMPLEMENTATION OF ONE ITERATION OF THE KALMAN FILTER......Page 122
  8.2 A BLOCK DIAGRAM AND A MATLAB® FUNCTION FOR IMPLEMENTATION OF ONE KALMAN FILTER ITERATION......Page 125
  8.3 RECURSIVE EXECUTION OF THE KALMAN FILTER ALGORITHM......Page 127
  8.4 THE KALMAN FILTER ESTIMATOR AS A “FILTER”......Page 129
  9.1 IDENTIFICATION OF THE KALMAN FILTER VARIABLES AND PARAMETERS......Page 132
  9.2 STRUCTURE OF OUR MATLAB® SIMULATIONS......Page 133
  9.3 CREATION OF SIMULATED SIGNALS: CORRESPONDENCE OF PARAMETERS AND SIGNAL CHARACTERISTICS......Page 135
  9.4 THE TIMING LOOP......Page 138
  9.5 EXECUTING THE SIMULATION AND INTERPRETATION OF THE RESULTS......Page 140
  9.6 ISOLATING THE PERFORMANCE OF THE MODEL (BY NULLIFYING THE KALMAN GAIN)......Page 144
  10.1 OVERVIEW OF THE SCENARIO AND SETUP OF THE KALMAN FILTER......Page 150
  10.2 STRUCTURE OF THE MATLAB® SIMULATION FOR THIS CASE......Page 153
  10.3 TESTING THE SIMULATION......Page 159
  10.4 FURTHER ANALYSIS OF THE SIMULATION RESULTS......Page 163
  10.5 ISOLATING THE EFFECT OF THE MODEL......Page 167
Part IV Kalman Filtering Application to IMUs......Page 170
 Chapter 11 Kalman Filtering Applied to 2-Axis Attitude Estimation from Real IMU Signals......Page 172
  11.2 REVIEW OF ESSENTIAL ATTITUDE CONCEPTS: FRAMES OF REFERENCE, EULER ANGLES AND QUATERNIONS......Page 173
  11.3 CAN THE SIGNALS FROM A GYROSCOPE BE USED TO INDICATE THE CURRENT ATTITUDE OF THE IMU?......Page 178
  11.4 CAN WE OBTAIN “MEASUREMENTS” OF ATTITUDE WITH THE ACCELEROMETERS?......Page 179
  11.5 SUMMARY OF THE KALMAN FILTER IMPLEMENTATION FOR ATTITUDE ESTIMATION WITH AN IMU......Page 184
  11.6 STRUCTURE OF THE MATLAB® IMPLEMENTATION OF THIS KALMAN FILTER APPLICATION......Page 185
  11.7 TESTING THE IMPLEMENTATION OF KALMAN FILTER FROM PRE-RECORDED IMU SIGNALS......Page 189
 Chapter 12 Real-Time Kalman Filtering Application to Attitude Estimation from IMU Signals......Page 198
  12.1 PLATFORM AND ORGANIZATION OF THE REAL-TIME KALMAN FILTER IMPLEMENTATION FOR ATTITUDE ESTIMATION......Page 199
  12.2 SCOPE OF THE IMPLEMENTATION AND ASSUMPTIONS......Page 200
  12.3 INITIALIZATION AND ASSIGNMENT OF PARAMETERS FOR THE EXECUTION......Page 203
  12.4 BUILDING (COMPILING AND LINKING) THE EXECUTABLE PROGRAM RTATT2IMU.EXE—REQUIRED FILES......Page 204
  12.5 COMMENTS ON THE CUSTOM MATRIX AND VECTOR MANIPULATION FUNCTIONS......Page 205
  12.6 INPUTS AND OUTPUTS OF THE REAL-TIME IMPLEMENTATION......Page 208
  12.7 TRYING THE REAL-TIME IMPLEMENTATION OF THE KALMAN FILTER FOR ATTITUDE ESTIMATION......Page 210
  12.8 VISUALIZING THE RESULTS OF THE REAL-TIME PROGRAM......Page 211
APPENDIX A LISTINGS OF THE FILES FOR REAL-TIMEIMPLEMENTATION OF THE KALMANFILTER FOR ATTITUDE ESTIMATION WITH ROTATIONS IN 2 AXES......Page 216
REFERENCES......Page 236
INDEX......Page 240