Sommario ii

Contents

Abstract i

Sommario ii

Acknowledgement iii

Ringraziamenti iv

List of notations vi

Contents ix

List of figures xi

List of tables xii

1 Introduction 1

1.1 Background . . . . 1

1.2 System Overview . . . . 3

1.3 Thesis Outline . . . . 4

2 Acoustic Positioning Systems 5 2.1 Principles of Sound Propagation . . . . 5

2.2 Conventional Systems . . . . 8

2.2.1 Ultra Short or Super Short Baseline . . . . 8

2.2.2 Short Baseline . . . . 9

2.2.3 Positioning Modes for USLB/SBL . . . . 10

Transponder . . . . 10

Free Running Pinger . . . . 10

2.2.4 Long Baseline . . . . 10

2.2.5 Positioning Modes for LBL . . . . 10

Direct Ranging . . . . 11

Intelligent Acoustic Remote . . . . 11

Relay . . . . 11

Pseudo Ranging . . . . 12

vii

CONTENTS

2.3 Latest Architecture Developments . . . . 12

2.3.1 System Integration . . . . 12

2.3.2 Inverted architecture . . . . 12

2.4 G.I.B. System . . . . 13

3 Instantaneous Positioning Algorithms 16 3.1 Trilateration . . . . 16

3.1.1 3D Trilateration . . . . 16

3.1.2 2D Trilateration . . . . 18

Weighted Trilateration . . . . 19

3.2 Maximum Likelihood Estimation . . . . 21

3.2.1 MLE: Squared ranges . . . . 22

3.2.2 MLE: Pure ranges . . . . 23

3.2.3 Stepsize: Armijo rule . . . . 24

3.2.4 Simulation Results . . . . 24

3.3 Cramer-Rao Lower Bound . . . . 31

3.3.1 Buoy Geometry . . . . 32

3.3.2 Algorithm Comparison . . . . 32

4 Filtering Methods 36 4.1 Linear Estimation Problem . . . . 36

4.1.1 The Discrete Kalman Filter . . . . 37

4.1.2 Uncertainty Ellipsoid . . . . 37

4.1.3 Alternative formulation . . . . 38

4.1.4 Example . . . . 39

Linear system . . . . 39

4.2 Nonlinear Estimation Problem . . . . 40

4.2.1 Extended Kalman Filter . . . . 41

4.2.2 EKF Design . . . . 43

Nonlinear System . . . . 43

Jacobians . . . . 44

Simulation Results . . . . 44

5 Initialization and Outlier rejection algorithms 47 5.1 Initialization algorithm . . . . 47

5.2 Outlier Rejection . . . . 49

6 Graphical User Interface 51 7 Experimental Results 54 7.1 Experimental scenario . . . . 54

7.2 Results analysis . . . . 54

viii

CONTENTS

8 Conclusions and future work 64

8.1 Conclusions . . . . 64 8.2 Future work . . . . 65

Bibliography 66

A MLE: Gradient and Hessian derivation 69

A.1 MLE: Squared ranges . . . . 69 A.2 MLE: Pure ranges . . . . 71

B Chi-Square distribution 74

C Coordinate Transformations 76

C.1 Earth Model . . . . 76 C.2 ECEF coordinates from Longitude and Latitude . . . . 77 C.3 NED coordinates from ECEF coordinates . . . . 78

ix

List of Figures

1.1 Multipaths propagation. . . . . 2

1.2 The G.I.B. system. . . . . 3

2.1 Typical variable characteristics . . . . 6

2.2 Typical sound velocity profile . . . . 6

2.3 Sound propagation in isovelocity water . . . . 7

2.4 Sound propagation in positive gradient water . . . . 7

2.5 Sound propagation in negative gradient water . . . . 7

2.6 Layer Depth Phenomenon . . . . 8

2.7 Ultra Short Baseline . . . . 9

2.8 Short Baseline . . . . 9

2.9 Long Baseline and Direct Ranging . . . . 11

2.10 USBL/INS integration scheme, from [16] . . . . 13

2.11 Inverted USBL, from [32] . . . . 13

2.12 Inverted LBL, from [32] . . . . 14

2.13 GIB system . . . . 14

2.14 GIB pinger (up), GIB buoy (left) and GIB central unit (right) . . . . 15

2.15 Buoy deployment . . . . 15

3.1 Step size determination by Armijo Rule. . . . . 25

3.2 Gradient and pure ranges. . . . . 26

3.3 Newton and pure ranges. . . . . 27

3.4 Gradient and squared ranges. . . . . 28

3.5 Newton and squared ranges. . . . . 29

3.6 Local and global minimums for pure ranges log-likelihood function. . . . . . 30

3.7 Local and global minimums for squared ranges log-likelihood function. . . . . 30

3.8 Configuration of 4 buoys in square shape. . . . . 33

3.9 Configuration of 4 buoys in rhombus shape. . . . . 33

3.10 Configuration of 3 buoys in triangle shape. . . . . 34

3.11 Algorithm accuracy compared with CRLB. . . . . 35

4.1 Gaussian probability density for K = 2. . . . . 38

4.2 Kalman Filter simulation results. . . . . 40

4.3 Uncertainty ellipsoid graph. . . . . 41

x

LIST OF FIGURES

4.4 Ranges components. . . . . 43

4.5 EKF simulation results. . . . . 45

4.6 State error evolutions. . . . . 46

5.1 Integration of EKF, initialization and outlier rejection procedures. . . . . 50

6.1 GUI logical scheme. . . . . 52

6.2 Read Binary Data block. . . . . 53

7.1 Experimental trajectory. . . . . 56

7.2 Times of Arrival of acoustic pulses at each buoy. . . . . 57

7.3 Details of TOA at buoys. . . . . 58

7.4 State vector evolution. . . . . 59

7.5 Experimental trajectory of second set of data. . . . . 60

7.6 Detailed of TOA validation in the time range [300 − 340]. . . . . 61

7.7 Times of Arrival of acoustic pulses at each buoy (second set of data). . . . . 62

7.8 State vector evolution (second set of data) . . . . 63

B.1 χ

distribution. . . . . 74

C.1 Earth shape model. . . . . 76

C.2 ECEF coordinate frame with z-axis along Earth’s rotation axis. . . . . 78

xi

List of Tables

3.1 MLE: Simulation parameters . . . . 26

4.1 Kalman Filter Matrices . . . . 40

4.2 EKF: Simulation parameters . . . . 45

7.1 Experimental filter parameters . . . . 54

B.1 Table of critical values of χ

distribution. . . . . 75

C.1 WGS-84 ellipsoid constants. . . . . 77

xii