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Technical Analysis
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iii
Contents
Translator's Preface .......................................................................................................... ix
Author's Preface ............................................................................................................... xi
Part I. Recursive Filter........................................................................................ 15
Chapter 1. Average filter ................................................................................................. 17
1.1 Recursive expression for average ........................................................................................ 17
1.2 Average filter function ........................................................................................................ 20
1.3 Example: Voltage measurement .......................................................................................... 21
1.4 Summary ............................................................................................................................ 24
Chapter 2. Moving average filter ..................................................................................... 25
2.1 Stock price and moving average .......................................................................................... 25
2.2 Recursive expression of moving average ............................................................................. 26
2.3 Moving average filter function ............................................................................................ 27
2.4 Example: Sonar ................................................................................................................... 30
2.5 Summary ............................................................................................................................ 33
Chapter 3. Low-pass filter ............................................................................................... 35
3.1 Limitation of moving average ............................................................................................. 35
3.2 1st order low-pass filter ........................................................................................................ 36
3.3 Low-pass filter function ...................................................................................................... 38
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3.4 Example: Sonar .................................................................................................................. 39
3.5 Summary ............................................................................................................................ 42
Chapter 4. Summary of Part I........................................................................................... 43
Part II. Theory of Kalman Filter ........................................................................ 45
Chapter 5. Introduction to Kalman filter .......................................................................... 47
5.1 Introduction ........................................................................................................................ 47
5.2 Kalman filter algorithm ...................................................................................................... 48
Chapter 6. Estimation process .......................................................................................... 53
6.1 Introduction ........................................................................................................................ 53
6.2 Computation of an estimate ................................................................................................ 53
6.3 Varying weight ................................................................................................................... 55
6.4 Error covariance ................................................................................................................. 56
6.5 Summary ............................................................................................................................ 58
Chapter 7. Prediction process ........................................................................................... 59
7.1 Computation of a prediction ............................................................................................... 59
7.2 Difference between prediction and estimation ..................................................................... 60
7.3 Reinterpretation of the expression for computing an estimate .............................................. 62
Chapter 8. System model ................................................................................................. 65
8.1 Introduction ........................................................................................................................ 65
8.2 System model ..................................................................................................................... 65
8.3 Covariance of the noise ...................................................................................................... 67
Chapter 9. Summary of Part II ......................................................................................... 71
Part III. Examples ............................................................................................... 73
Chapter 10. Extremely Simple Example ........................................................................... 75
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10.1 System model ................................................................................................................... 75
10.2 Kalman filter function ....................................................................................................... 77
10.3 Test program ..................................................................................................................... 79
10.4 Error covariance and Kalman gain ..................................................................................... 81
10.5 Summary .......................................................................................................................... 84
Chapter 11. Estimating velocity from position ................................................................. 85
11.1 System model ................................................................................................................... 86
11.2 Kalman filter function ....................................................................................................... 88
11.3 Result of the estimation ..................................................................................................... 90
11.4 Estimating position with velocity ...................................................................................... 93
11.5 Measuring velocity with sonar ........................................................................................... 98
11.6 Efficient Kalman filter function ....................................................................................... 101
11.7 Power of system model ................................................................................................... 105
Chapter 12. Tracking an object in an image ................................................................... 107
12.1 System model ................................................................................................................. 107
12.2 Kalman filter function ..................................................................................................... 109
12.3 Test program ................................................................................................................... 110
12.4 Test program 2 ................................................................................................................ 114
Chapter 13. Attitude reference system ........................................................................... 117
13.1 Introduction .................................................................................................................... 117
13.2 Attitude determination with gyros ................................................................................... 120
13.3 Attitude determination with accelerometers ..................................................................... 125
13.4 Attitude determination through sensor fusion................................................................... 131
13.4.1 System Model ........................................................................................................... 132
13.4.2 Kalman filter for sensor fusion .................................................................................. 136
Part IV. Nonlinear Kalman filter ..................................................................... 141
Chapter 14. Extended Kalman Filter ............................................................................. 143
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14.1 Introduction .................................................................................................................... 143
14.2 Linearized Kalman filter ................................................................................................. 144
14.3 Extended Kalman filter ................................................................................................... 144
14.3.1 Nonlinear system model ............................................................................................ 145
14.3.2 Extended Kalman filter algorithm ............................................................................. 145
14.4 Example 1: Radar tracking .............................................................................................. 149
14.4.1 System model ........................................................................................................... 150
14.4.2 Extended Kalman filter function ............................................................................... 151
14.4.3 Test program ............................................................................................................ 155
14.5 Example 2: Attitude reference system ............................................................................. 159
14.5.1 System model ........................................................................................................... 160
14.5.2 Extended Kalman filter function ............................................................................... 161
14.5.3 Test program ............................................................................................................ 165
14.6 Summary ........................................................................................................................ 167
Chapter 15. Unscented Kalman Filter ............................................................................ 169
15.1 Introduction .................................................................................................................... 169
15.2 Unscented transformation ............................................................................................... 170
15.2.1 Introduction .............................................................................................................. 170
15.2.2 Unscented transformation algorithm.......................................................................... 171
15.2.3 Unscented transformation function ............................................................................ 173
15.3 Unscented Kalman filter ................................................................................................. 175
15.3.1 Nonlinear system model ............................................................................................ 176
15.3.2 Comparison with an extended Kalman filter .............................................................. 176
15.3.3 Unscented Kalman filter algorithm ............................................................................ 178
15.4 Example 1: Radar tracking .............................................................................................. 180
15.4.1 System model ........................................................................................................... 180
15.4.2 Unscented Kalman filter function .............................................................................. 181
15.4.3 Test program ............................................................................................................ 184
15.5 Example 2: Attitude reference system ............................................................................. 187
15.5.1 System model ........................................................................................................... 187
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15.5.2 Unscented Kalman filter function .............................................................................. 188
15.5.3 Test program ............................................................................................................. 190
15.6 Summary ........................................................................................................................ 192
Part V. Frequency Analysis and Filter ............................................................. 193
Chapter 16. High-pass filter .......................................................................................... 195
16.1 Introduction .................................................................................................................... 195
16.2 Laplace transformation and filter ..................................................................................... 196
16.3 High-pass filter ............................................................................................................... 200
16.4 High-pass filter function .................................................................................................. 202
16.5 Example: Sonar ............................................................................................................... 203
16.6 Conclusion ...................................................................................................................... 205
Chapter 17. Complementary filter ................................................................................. 207
17.1 Introduction .................................................................................................................... 207
17.2 Concept of complementary filter ..................................................................................... 207
17.3 Example: Attitude reference system ................................................................................ 210
17.3.1 Complementary filter................................................................................................. 211
17.3.2 Complementary filter function ................................................................................... 215
17.3.3 Test program ............................................................................................................. 219
17.4 Another example of a complementary filter ..................................................................... 221
Index............................................................................................................................. 225
