PASSIVE ESTIMATION OF UNDERWATER MANEUVERING TARGETS

by

Pankaj M. Godiwala

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

Electrical Engineering

APPROVED:

R. L. Moose,Chairman

g_ t. vai'Y.tandingham C. E. Nunnal~y

November, 1982 Blacksburg, Virginia

PASSIVE ESTIMATION OE' UNDERWATER MANEUVERING TARGETS

by

Pankaj M. Godiwala

(ABSTRACT)

The initial portion of this thesis examines the problem

of tracking a maneuvering target in the 2-dimensional (X,Z)

plane, vertical to the ocean floor, using passive time-delay

measurements. The target is_ free to maneuver in velocity and

make depth changes at times unknown to the observer. In the

past, tracking systems have used Extended Kalman Filters to

process the nonlinear measurements,,. but these have inherent

divergence problems. To overcome this, a nonlinear prefilter

is added to linearize the measurements and thus allow the

use of a conventional Kalman E'ilter which makes the tracking

system more 'robust' and also decouples the depth estimator

from the polar range estimator. The depth estimator is dis-

cussed in detail here.

The latter part of this thesis introduces tracking in

the 2-dimensional horizontal (X, Y) plane, parallel to the

ocean floor, to observe polar range and target bearing an-

gle. The approach of using a nonlinear prefilter and a stan-

dard Kalman E'il ter is similar to the one described above.

Subsequently, the analysis is extended to a Kalman Filter

which is not 'matched', i.e. it does not possess any know-

ledge of the deterministic inputs which cause target motion.

This necessitates the use of a bank of Kalman Filters and an

adaptive weighting scheme. Test results are included to show

that all source maneuvers can be tracked with a relatively

high degree of accuracy.

ACI<N0WLEDGEMENTS

While working on this research, I was supported as a

Graduate Research Assistant on a contract from the Office of

Naval Research (ONR), Washington, D.C. I am deeply grateful

to my advisor and committee chairman, Dr. R. L. Moose, for

granting me this oportunity to work under his guidance. His

continuous encouragement and "discrete" criticism were of

immense help throughout the research work discussed here.

I must place on record the assistance of my graduate

committee members, Dr. H. F. Vanlandingham and Dr. C. E.

Nunnally, in reviewing the thesis and making valuable sug-

gestions for its improvement. Thanks are also due to Dr. I.

M. Besieris and Dr. R. Lumia for their discussions whenever

they were needed and to the various faculty members under

whom I have had the privilege of studying.

I feel that I speak for other foreign graduate students

as well as myself in acknowledging the constant help of all

the secretaries in the E. E. Dept. office during our course

of study. A special thanks goes out to Leslie Cobb whose

typing abilities have brought this thesis to its present

form.

Finally, I wish to extend my gratitude to all my family

and friends whose moral support has been invaluable in all

of my pursuits.

iv

TABLE OF CONTENTS

ABSTRACT ii

ACKNOWLEDGEMENTS . iv

Chapter

I.

II.

INTRODUCTION AND BASIC TARGET MODELING

Preliminary Remarks . . . . . . . Introduction to Target Tracking Singer Process and Target Dynamics Polar Target Model . . . . . . .

SIMPLE DEPTH ESTIMATION USING SONAR TIME DELAYS

Introduction . . . . . . . . . Sonar Time Delay Measurements . . . . .

1

1 2 4 6

11

11 . . 11

. 15 18 27

. 40

. 49

Incorporation of the Nonlinear Prefilter . Statistical Analysis of Depth Measurement Error Performance Analysis of the Depth Tracker Results for Depth Estimation . . . . . Conclusion . . . . . . . . . . . . . . . . .

III. DEVELOPMENT OF A RANGE/BEARING TRACKER IN THE

IV.

v.

HORIZONTAL PLANE . 51

Introduction . . . . . 51 Geometry and Time Delay Measurements . . 52 Discussion of the Nonlinear Prefilter . SS Statistics of Range and Bearing Measurement

Errors . . . . . . . . . . . . . 62 Performance Analysis of the Estimator . . . 73

THE POLAR RANGE ADAPTIVE STATE TRACKING SYSTEM

Introduction to the Adaptive Scheme Adaptive Filter Setup Simulation Results Conclusion .

CONCLUSION

v

. 94

94 97

101 112

120

Appendix

A. KALMAN FILTER EQUATIONS . . . 123

B. MODIFIED STATISTICAL ANALYSIS OF DEPTH MEASUREMENT ERROR . . . . . . . . . . . . . . . . . . . 125

C. MODIFIED STATISTICAL ANALYSIS OF RANGE MEASUREMENT ERROR . . . . . . . . . . . . . . . . . . . 131

BIBLIOGRAPHY ; . . . 135

VITA

vi

LIST OF FIGURES

Figure

1.

2.

3.

Geometry of Observer-Source Scenario in Polar Coordinates . . . . . . . . . . .

Geometry of Time Delay Measurements

Nonlinear Prefilter for Target Depth Measurement .

8

12

17

4. Block Diagram of vd Mean and Variance Calculations . 20

5. Mean Value of Depth Measurement Error vs Range 21

6. Standard Deviation of Depth Error vs Range . . 22

7. Normalized Density of Depth Measurement Error 24

8. Block Diagram of Measurement Compensation 26

9. Modified/Conventional Mean Results Comparison: High SNR . . . . . . . . . . . . . . . . . . . . 2 8

10. Modified/Conventional Variance Results Comparison: High SNR . . . . . . . . . . . . 29

11. Modified/Conventional Mean Results Comparison: Low SNR . . . . . . . . . . . . . . . 3 0

12. Modified/Conventional Variance Results Comparison: Low SNR . . . . . . . . . . 31

13. Data Generation for Depth Tracking 33

14. Basic Estimation Structure . 34

15. Noisy Depth Measurement Data 36

16. Large Measurement Error Effect on Depth Data 37

17. Depth Estimation vs Time for Low Noise Case 38

18. Depth Estimation for Longer Range, Higher Noise Case 39

19. Depth Estimate for Target Closing (80K to 20K) Range 41

vii

20. Depth Estimate for Target Closing (SOK to 20K), High Noise Case . . . . . . . . . . . . . . . . . 42

21. Short Variable Range, Variable Depth Target and Estimator Output . . . . . . . . . . . 44

22. Variable Range SOK to 20K and Variable Depth Target 46

23. Fixed Range (30K), Variable Depth Target ...... 47

24. Fixed Range, Fixed Depth Target: Mismatched Sta ti sties . . . . . . . . . . . . . . 48

25. Geometry for Puffs or Wavefront Curvature Analysis S4

26. Relationship between Range Bias and Bearing Angle . 61

27. Layout for Range/Bearing Measurement Error Statistics Calculation . . . . . . . 64

28. Range Statistics for Low-Noise (SOusec) Case 67

29. Range Statistics for High-Noise (200usec) Case . . 68

30. Range Meas. Error Probability Density Function (Low Noise) . . . . . . . . . . . . . . . . . . . . . 69

31. Range Meas. Error Probability Density Function (High Noise) . . . . . . . . . . . . . . . . . . 70

32. Bearing Meas. Error Probability Density Function (Low Noise) ................... 71

33. Bearing Meas. Error Probability Density Function (High Noise) . . . . . . . . . . . . . . . . 72

34. Noisy Time Delay Data Generation for Range/Bearing Tracking . . . . . . . . . . . . . . . . . . . 74

35. Basic Estimator Structure for Range/Bearing Tracking 76

36. Projected Geometry of the Observer-Source Scenario 78

37. Range Estimation vs Time (Fixed Range(2SK), Low Noise Case) ................... 80

38. Range Estimation vs Time (Fixed Range(2SK), High Noise Case) . . . . . . . . . . 81

39. Scenarios To Test Tracking Algorithm 82

viii

40. Range Estimation vs Time, Scenario # 1, High SNR Case . . . . . . . . . . . . . . 84

41. Range Estimation vs Time, Scenario # 1, Low SNR Case 85

42. Range Estimation vs Time, Scenario # 2, High SNR Case . . . . . . . . . . . . . . . . . 86

43. Range Estimation vs Time, Scenario # 2, Low SNR Case 87

44. Range Estimation vs Time, Scenario # 3, High SNR Case . . . . . . . . . . . . . . . 88

45. Range Estimation vs Time, Scenario # 3, Low SNR Case 89

46. Velocity Estimation for Scenario # 2, Low SNR Case . 90

47. Range Estimation, On-board Sensors, Scenario 1 ( lOusec) . . . . . . . . . . . . . . . . . . . 92

48. Range Estimation, On-board Sensors, Scenario 2 ( lOusec) . . . . . . . . . . . . . . . . . . 93

49. N Partially Overlapping Gaussian Curves, Displaced Mean Values . . . . . . . . . . . . . 96

50. Block Diagram of the Adaptive Range Tracker 102

51. Adaptive Range Estimation, Fixed Scenario, High SNR Case . . . . . . . . . . . . . . . 103

52. Adaptive Range Estimation, Scenario 1, High SNR Case . . . . . . . . . . . . . . . 104

53. Adaptive Range Estimation, Scenario 2, High SNR Case . . . . . . . . . . . . . . . . . . . 105

54. Adaptive Range Estimation, Scenario 3, High SNR Case . . . . . . . . . . . . . 106

55. Behavior of Weights for Special Scenario 108

56. Non-averaged Distribution of Measurement Residuals 3 and 5 . . . . . . . . . . . . . . 110

57. Distribution of Residuals 3 and 5 with Rolling Aver.age . 111

ix

58. Adaptive Range Estimation, Fixed Scenario, Low SNR Case . . . . . . . . . . . . . . 113

59. Adaptive Range Estimation, Scenario 1, Low SNR Case 114

60. Adaptive Range Estimation, Scenario 2, Low SNR Case 115

61. Adaptive Range Estimation, Scenario 3 I Low SNR Case 116

62. Tracking Results for Scenario 1, On-board Sensors, lOusec Case . . . . . . . . . . . . . . . 117

63. Tracking Results for Scenario 2, On-board Sensors, lOusec Case . . . . . . . . . . . . . . . . 118

x

LIST OF TABLES

Table

1. Accuracy of Range/Bearing Expressions .

2. Statistics of Bearing Measurement Error

xi

~

59

. 65

Chapter I

INTRODUCTION AND BASIC TARGET MODELING

1.1 PRELIMINARY REMARKS

In an era where a lot of time and money is devoted to

national defense, target tracking holds an important posi-

tion. Through the years, a lot of tracking systems for mane-

uvering targets have been developed, but the research for a

'best' tracking solution still goes on; 'best' in terms of

accuracy, speed and simplicity of implementation. This the-

sis addresses the problem of underwater passive tracking of

maneuvering targets in two distinct planes. The Observer

submarine tra~ks the target or Source submarine using sonar

signals emanating from the source. Because the Observer does

not use an active sonar but listens to sound waves coming

out of the Source, this method is denoted as passive track-

ing and explains the ti tl.es of the two submarines.

Underwater tracking, unlike airborne tracking systems,

has a lot of uncertainty and complexity due to the inherent

nature of the ocean environment; acoustical wave propagation

inefficiency is caused by the medium and the irregularities

of the ocean floor. Thus new and improved tracking systems

have to be constantly developed or current strategies have

1

2

to be approached with a different perspective. Here, we exa-

mine one current tracking system developed by Moose and

McCabe [l],[9]; several new approaches are tried out to va-

lidate their method and provide better results.

1.2 INTRODUCTION TO TARGET TRACKING

A survey of passive tracking literature shows that most

of the energy, in the recent past, has been expended on the

development of target models and digital filtering algor-

ithms for tracking maneuvering targets. A standard approach

has been to model the target dynamics in a rectangular coor-

dinate system; this produces a linear state model but the

measurements become nonlinear functions of the state varia-

bles. These nonlinear measurements necessitate the use of

an Extended Kalman Filter (EKF) to estimate current state

variables and linearize the next measurement vector. But,

EKF's work only moderately well under normal conditions and

in circumstances like abrupt target maneuvers, they can lead

to large bias errors or even complete filter divergence. To

overcome this problem, Moose and Gholson [1] have used polar

coordinates to develop a model of target and observer mo-

tion.

To linearize the measurements, a nonlinear prefilter is

added to the tracking system which leads to two major bene-

3

fits: the first is that an adaptive Kalman Filter can be

used to give the tracking system a larger degree of 'robust-

ness' and secondly, the range and depth estimators can be

uncoupled to reduce the complexity and computational level

of the adaptive tracker. The depth estimator forms the ini-

tial part of this thesis and is discussed in chapter 2.

Another area of interest is the method of modeling tar-

get maneuvers which will allow filter convergence. Earlier

work by Jazwinski [6] includes limited memory filtering whe-

re the filter gains are prevented from decaying to zero by

artificially maintaining them at a level to allow detection

of a maneuver. Another technique, by Thorp [7], proposes a

switching between two Kalman Filters in response to a de-

tected maneuver. A third approach, by Singer [3], uses a mo-

del which considers a typical target trajectory to be the

response to a time-correlated random acceleration. Large

scale trajectory changes are modeled by a semi-Markov pro-

cess [4]. The price one pays is that additional state varia-

bles are used to generate the correlated forcing functions

which increases the dimension of the Kalman filtering algor-

ithm. Thus, the filter is provided with statistical informa-

tion regarding target maneuvers based on an assumed range of

possible accelerations. Subsequently, this approach has

been used in this text and will be explained further in the

next section.

4

It should be noted at this point that most modern

submarines have a limited depth capability and thus are una-

ble to make large scale depth maneuvers. So, with respect to

the depth estimator, we do not need the modeling of deliber-

ate target maneuvers as a semi-Markov process and the use of

an adaptive scheme of a weighted bank of Kalman Filters.

This method is utilised in the latter part of the thesis in-

volving range estimation in the horizontal plane and will be

discussed more fully in chapter 4.

1.3 SINGER PROCESS AND TARGET DYNAMICS

Moose and Gholson [1] have proposed a linearized polar

model to incorporate the correlated process. Using this, the

motion of a target in rectangular coordinates is described

by the following equations.

x + a.x = u + w x x . w + a.w = W(t) x x where

a is a drag coefficient.

(1.3.1)

(1.3.2)

u is the deterministic input, in the x-direction, which x

controls the target velocity and maneuvers.

w~ is the Singer correlated acceleration process in the x x-direction.

a is the Singer correlation time constant.

W(t) is a Gaussian white noise process.

5

1.3.1 describes the acceleration of a target in the x-

direction of the rectangular coordinate system; this accel-

eration is forced by a Singer correlated process wx (de-

scribed by 1.3.2) with a time constant r =1/a. c . By defining x, x and w as state variables, equations x

1.3.1 and 1.3.2 can be put in state variable form and yield

the following discrete state model.

~(k+l) = cp~(k) + r~(k) + '¥w(k)

or

x 1 cpl2 cpl3 x rl [ux(k)J '¥1 wk .

x = 0 c/l22 c/l23 x + r2 + '¥2 (1.3.3)

w 0 0 -aT 0 '¥3 e w x k+l x k

where

T = sampling interval.

cp 1 2 - a. T = (1-e )/a.

cjl 13 = -aT -aT [l+(ae -ae )/(a-a)]/(aa)

cp 2 2 = e - et T

-aT -aT c/l23 = (e -e )/(a-a)

rl -aT 2 = (a.T-l+e )/a.

r2 = cp 1 2

'¥1 = [T+(acp 12 -a.'¥ 3)/(a.-a)]/(aa)

'¥ 2 = C'¥3-c/l12) I (a-a)

6

qr = (1-e-aT)/a 3

k is the discrete time parameter.

Equation 1.3.3 describes target motion in the x-direc-

tion. Similar state models can be derived for the y and z

directions of the rectangular coordinate system. This state

model will henceforth be referred to as the linear drag mo-

del.

1.4 POLAR TARGET MODEL

We can now combine the state variable expressions for

the rectangular coordinates into a polar form which models

depth dT, and depth rate, d . T

One important point which

needs clarification at this stage is that the polar range

model development and analysis is explained in (11] and the

work discussed here in chapters 1 and 2 was done sirnultane-

ously with that of chapters 1,2 and 3 in (11]. This combina-

tion forms the complete polar target model in the vertical

plane.

Consider Figure 1 which shows the geometry of the Ob-

server-Source scenario in polar coordinates. The plane de-

fined by the X-Y axes is parallel to the ocean bed and fixed

with respect thereto. The subscripts 's' and 'o' ref er to

the source and observer, respectively. Hence, the vertical

distance z is simply the difference. in their depths. If so

7

d 0 = depth of observer, and dT = depth of target (both with

respect to the ocean surface), then z = dT - d , or in a so 0

discrete time parameter,

z = z - z 50k+ 1 5 k+ 1 °k+ 1

(1.4.1)

We also have the relations

z = z + T.z (1.4.2) Ok+! Ok Ok

z = z - z (l.4.3) sok sk Ok . . z = z + z (1.4.4)

sk sok Ok Using the linear drag model of 1. 3. 3 in the z-coordi-

nate, equations 1.4.1 through 1.4.4 and some algebraic mani-

pulation, [9], gives us the z-channel or the depth channel

model as

where the matrix entries are the same as in 1.3.3. It is to

be noted that state model 1.4.5 is subject to the constraint

of a constant velocity observer. This does not mean that the

observer cannot execute any maneuvers. What is implied is

that during the maneuver, the state model is inaccurate and

will not yield good estimates. Upon completion of the obser-

ver maneuver, the model is once again valid and will yield

good estimates. (Poor estimates during maneuvers can also be

x

x

x 0

z

OBS.

/ I

I/

SOURCE

r

1 z = depth difference I

elevation 'l - I ----- p I

I

I I

I

----- I ------1 I I

/

I I

/

/ /

--------~--/

I -- I / - I//

-------------------~~-~~

Figure 1. Geometry of Observer-Source Scenario in Polar Coordinates

00

y

9

attributed to filter characteristics because of the intro-

duction of transients.)

Comparing 1. 3. 3 and 1. 4. 5 reveals the similarity bet-

ween the models in the rectangular and polar coordinates ex-

cept that w~

equivalents w;z

and u have been replaced by their respective x

and u in the radial direction. sz

At this stage, a few assumptions need to be made in

order to facilitate the remaining analysis. As stated in

section 1.2, due to the depth maneuvering limitations of mo-

dern submarines, we can let u sz = 0 i the time correlated

w.. is enough of an input to the system. We can also let sz

the observer be stationary with respect to depth motion and .

so z = 0 . This makes the matrix r unnecessary in the 0

state model and the depth channel state model used hence-

forth becomes

z 1 <1>12 <1>22 z qr 1 w .. so so s

zk z = 0 <1>22 <1>23 z + qr 2 (1.4.6) so so

w" 0 0 -aT w" qr3 e sz +1 sz k

The last important assumption is with regard to the

distance between the observer and the target. The exact dis-

tance separating an observer and a source is the spherical

10

radius r, which must be measured in three-dimensions (figure

1). However, z, the depth difference is typically much less

than p, the range. So we can consider r to approximately

equal p and the two-dimensional polar model developed for

range in [11] can be used. It is seen in later expressions

(time-delay measurements, chapter 2) that if r=4z, the error

caused by the above assumption is of magnitude 3% and r=lOz

yields only a 0.5% error.

Chapter II

SIMPLE DEPTH ESTIMATION USING SONAR TIME DELAYS

2.1 INTRODUCTION

Having introduced the problem of target modeling in the

previous chapter, we will now develop an estimator to track

target depth. This chapter will discuss the set of passive

sonar time delay measurements and the use of a nonlinear

prefi 1 ter to obtain linearized depth measurements so that

they can be processed by a conventional Kalman Filter in

lieu of the Extended Kalman Fi 1 ter. A method of obtaining

'off-line' statistical mean and variance data on the depth

expression is presented. Finally, simulation results to

prove the validity of the above analysis are discussed.

2.2 SONAR TIME DELAY MEASUREMENTS

One particular set of measurements which is commonly

used in conjunction with the de~th channel model is that of

sonar time delays; these can be passively obtained by lis-

tening to sound waves emanating from the target. Referring

to Figure 2, there are three paths by which signals can

reach the observer from the source; (i) direct path without

undergoing any reflection, (ii) surface reflection path, and

(iii) bottom reflection path. This gives us two values of

time delay measurements.

11

Ocean Surface

Source

Observer

Ocean Bottom

Figure 2. Geometry of Time Delay Measurements

d w I-' N

13

1. Tl, the difference in propagation times between the

direct and the surface reflection path.

2. T2 , the difference in propagation times between the di-

rect and the bottom reflection path.

The notations used in Figure 2 are as follows

d = depth of observer. 0

d = depth of ocean. w

dk = keel depth (observer) .

dT = depth of target.

r = spherical radius; actual observer-source separation.

p = polar target range.

d = polar target depth.

In his work at NUSC, Hassab [ 8] has shown how these

measurements Tl and T2 are related to target range p and

depth dT . This derivation is briefly explained here for

the sake of completeness.

From Hassab, we have

= [(p2+dz+4dz-~d d)l/2 _ (p2+d2)1/2]/C T 1 0 ""'-:.._.o

T2 = [(p2+d2+4d~+4dkd)l/2 - (p2+d2)1/2]/C

where c is the speed of sound in water. 2 2 2 Using r = P + d , 2.2.1 becomes

2 (r2+4d -4-!i d// 2

o ,_o r c c

I 4d (d -d)] 112 = .E. l1 + 0 0 c 2

r

r c

(2.2.1)

(2.2.2)

14

Since (1+2ef12 =l+e , this can be further reduced to 2d (d -d) .

0 0 1' = ----1 c

But p = r and d -d=d o T

1' = 1

2d0~ pC

Similarly, 2.2.2 can be reduced as

(r 2+4d~+4dkd) 112

But (~ +d) =

T2 = C

= f [ 1 + 4dk~;+d)] = .E. r 1 + 2dk (dk +d)]

c 2 L r

= 2dk (dk +d)

rC

(d -d ) and w T p= r

2dk (dw -dT) 1'2 = pC

r - -c 1/2

r c

r - -c

(2.2.3)

(2.2.4)

In 2.2.3 and 2.2.4, we have two nonlinear algebraic

equations in terms of p and dT. Most of the work in the past

has made use of a Taylor's series expansion of these equa-

tions to yield a linear measurement of p and dT. But this

involves the use of an EKF tracking system [S] and combined

range/depth estimation which leads to system complexity and

15

computational burden. Elimination of this problem is sought

here by developing the nonlinear prefilter discussed in the

following section.

2.3 INCORPORATION OF THE NONLINEAR PREFILTER

In order to linearize the time delay measurements, we

note that T 1 and T 2 are nonlinear functions of the system

state variables, polar range and target depth (P,dT). By di-

viding T1 by T2 and letting a0 = d 0 /dk , the ratio of obser-

ver depth to keel depth (observer), we have the following

expression,

Solving for dT , the actual target depth, we get

Tl (dw-dT) = a 0 d'i'TZ

.. d T = (2.3.1)

Substituting 2.3.1 into 2.2.3, we can determine the true po-

lar target range P , as

• p = 2d d /C 0 w

Tl+aoT2

Define b = 2d d /C 0 0 w

16

(2.3.2)

Equations 2.3.1 and 2.3.2 form the basic nonlinear pre-

filter and provide us with a means for separate depth and

range estimation. However, in this thesis we shall deal with

only the depth portion of the prefilter. The relatively sim-

ple method of linearizing dT by 2. 3. 1 does not remain so

straight-forward in reality because we do not

have T1 and T2 given to us but only the noisy set of mea-

surements

(2.3.3) 2T2 =. T 2 + V 2

where v1 and v2 are Gaussian random processes with zero-mean 2 and variance a This results in the noisy set of depth n

measurements

z~ = z ld T W

z 1+a z 2 T O 't

(2.3.4)

Figure 3 shows the simple structure of the nonlinear prefil-

ter for target depth measurement.

Noisy Time Delay

Measurements +

Figure 3. Nonlinear Prefilter for Target Depth Measurement

18

2.4 STATISTICAL ANALYSIS OF DEPTH MEASUREMENT ERROR

Equation 2.3.4, the measurement equation at the output

of the prefilter, can be viewed as

(2.4.1)

where vd is defined as the measurement error random process.

vd = zd T -dT

Substituting for ~T and d from equations 2.3.1, 2.3.3 and

2.3.4 yields

or

(2.4.2)

where

The target depth error term is thus the ratio of two

Gaussian random processes x and y. The terms x and y are

both sums and differences of the zero-mean Gaussian process-

es v1 and v 2 . They are both strongly correlated, and in the

case of the denominator y, non zero-mean, which becomes very

important in determining the structure of the density func-

tion of vd. A detailed, theoretical, statistical analysis of

the target depth measurement error has been made by Moose

[2] but it is not included here in order to limit the length

and conserve the simplicity of this thesis.

19

Initially, when this part of the research was conduct-

ed, closed form solutions to the integrals required for vd

mean and variance calculations could not be found: hence,

these statistics were obtained by extensive simulation. The

layout for this data generation is shown in Figure 4. This

data generation system was exercised at a series of discrete

target ranges Pi=SK,lOK, .... ,lOOK. Additive Gaussian noises

v 1 and v 2 were generated from a series of ten independent

random generators. The noisy measurements ZTl and ZT 2 were

then fed into the nonlinear prefilter. Taking the output ZdT

and subtracting dT gives the measurement error vd . By aver-

aging a sequence of 500 noisy measurements, a value of mean

and variance of v d was obtained, for each target range Pi .

This was repeated for each of the ten random sequences to

produce a good 'Monti-Carlo' set of mean values and varianc-

es. The results are shown in Figures 5 and 6. Note that the

curves also show the effect of increasing the variance of

the Gaussian measurement errors v1 and v 2 from 2msec to

Smsec.

It has been determined that both range and depth esti-

mation improve as the observer increases its depth. Now

with target depth unknown, and this being an exploratory

study, we decided to investigate two scenarios (which ac-

counts for the two sets of plots in Figures 5 and 6). The

p

+ + +

SNR Control White Noise

5 msec 3 msec

Generator

"'-------1 a

White Noise Generator

+ + + I-------'

Prefilter

z 1 d T W Z 1+a Z 2 T 0 T

Mean Aver ager

Mean E[Vd]

Limiter dT ± 3000

+~

Variance Aver ager

Variance 0 Vd

Figure 4. Block Diagram of Vd Mean and Variance Calculations

N 0

-300

-200

-100

20K 40K

21

dT d

0

d w

dT ------- d

(J = 5ms

1

I

J J

)

J 1

J )

60K

I )

3ms

BOK

I I

I

0

d w

2ms J

1 3ms

lOOK

= 600

= 1000

= 3000

= 1000

= 600

= 3000

Figure 5. Mean Value of Depth Measurement Error vs Range

p

700

600

500

400

300

200

100

20K

22

a = Sms

40K

I

3ms J

60K 80K

dT = 600 d = 1000

0 d = 3000 w

dT = 1000 d = 600

0 d = 3000 w

------

2ms

lOOK p

Figure 6. Standard Deviation of Depth Error vs Range

23

first has the observer at 1000 ft. and the target at 600

ft., the ocean depth shallow at 3000 feet. The second is a

reversal of magnitudes with target at 1000 and observer at

600. The values obtained then would allow the estimator to

perform anywhere within this range without going to a new

set of tabulated means and variances.

Next, a typical scenario was considered with p =SOK,

d =3000, d =1000, C=SOOO, an =3msec, which gives us w 0

the values 'i =9.6msec and 'z =76.Smsec. This data was pro-

cessed in the system configuration of Figure 4 with only one

random sequence generator to give us 500 noisy measurements

vd at the prefilter output. A typical distribution curve

(density function) of this additive noise was examined and

is shown in Figure 7.

Figures 5 and 6 show that vd has a non-zero mean and a

variance which are nonlinear functions of range (and SNR).

For the conventional Kalman Filter to process these measure-

ments, vd has to be zero-mean. So the set of values obtained

(as in Figure 4) are stored in a table which is looked up

(at every iteration of the tracking) based on the estimate

of P sent in by the polar range tracker and the SNR. This

'mean' is then subtracted from ·the current noisy measurement

Z dT producing a zero-mean noise process. The table look-up

procedure also gives the appropriate noise variance required

by the Kalman Filter.

P(Vd)

al a = 2 3ms

dT 600

d = 1000 0

d 3000 w p = SOK

N

""'

24

16

8

--1--•----+---t-------&----lK -800 -600 -400 -200 0 200 400 600 800

Figure 7. Normalized Density of Depth Measurement Error

25

~ very important point that needs to be stressed here

is that the depth error mean and variance are functions of

the range p , and ~ the polar depth estimator has to work

'hand-in-hand' with the polar range estimator. The range

tracker sends range estimates at every iteration to the

depth tracker and these are then used for the mean/variance

table look-up. A block diagram of the measurement compensa-

tion scheme is shown in Figure 8.

Further on in the research, while analysing the statis~

tics of the range measurement error in the horizontal plane

(chapter 3), the conventional method explained above failed

to work and gave erroneous results due to the extreme noise

sensi ti vi ty of the measurement expression. This led to a

search for various other methods and some results given in

(11] led to the development of a modified method to analyse

the measurement error statistics. Some of the advantages of

this modified method are that (i) it gives closed form ex-

pressions for the measurement error mean and variance, (ii)

it eradicates the existence of bad data points normally

found in the conventional method, and (iii) it is easily im-

plemented on a digital computer, cutting down the amount of

computer time utilised in the simulation by a factor of

twenty.

-"

Limiter ~

A

2dT > d - A T zdT(k) A Kalman

2dT < dT + A [ -+ ~

Filter A 3000 - ~ =

vd •f\

2 0 vd

SNR Mean I Variance Table - - Table· r-

Look-up Look-up

I A

p(k) from Range Tracker

Figure 8. Block Diagram of Measurement Compensation

.

N 0\

27

The details of the depth measurement error statistics

(modified method) are given in Appendix B. Comparisons of

the results obtained using the modified and conventional

methods are depicted in Figures 9 through 12.

2.5 PERFORMANCE ANALYSIS OF THE DEPTH TRACKER - -- --·__....;...___,;

Upto this point, the various sections in this text have

developed or discussed the separate portions of the tracking

scheme. This section puts all these different pieces into a

global perspective and introduces the reader to the struc-

ture of the complete tracking system including the data gen-

eration configuration and the estimator set-up. This is fol-

lowed by a discussion of the various types of scenarios

tested and the associated simulation results.

Figure 13 presents a discrete-time model showing the

development of noisy time delay measurements Z , 1 (k) and

z, 2 (k). Note that this data generation process is common to

both the range and depth estimators. The upper two blocks in

the figure show the generation of the actual polar range and

slowly varying target depth by using the polar target model-

ing technique described in chapter 1. Once rk and dT are

generated, they are acted upon in a nonlinear manner to gen-

erate 'i (k) and -r 2 (k), which, when added with the Gaussian

random measurement errors v1 and v2 , produce the noisy time

0

I I -100 1

-200 l I

-300 t I

-400 I I I

t -500 .l.

I I

-600 r

-700 I I r

-800 ~ l .,

20K

28

40K 60K

conventional

r:J = 3ms n dT = 1000 d = 600

0

d = 3000 w

SOK lOOK p

modified

Figure 9. Modified/Conventional Mean Results Comparison: High SNR

crvd j.

i

1200 I 1000 l

I aoo T I r !

600 +

400 j i f

I 200 i

i

0

I + I

29

conventional

20K 40K 60K

cr = 3ms n dT ::;: 1000 d = 600

0

d = 3000 w

modified

80K lOOK p

Figure 10. Modified/Conventional Variance Results Comparison: High SNR

0

-400

-800

-1200

-1600

-2000

-2400 t E[Vd]

20K 40K

conventional

a = 5ms n dT = 1000 d = 600

0

d = 3000 w

30

60K BOK lOOK

modified

Figure 11. Modified/Conventional Mean Results Comparison: Low SNR

p

31

2000

conventional 1600

modified

1200

800 (J = 5ms n dT = 1000 d = 600

0

d = 3000 w 400

0 20K 40K 60K 80K lOOK p

Figure 12. Modified/Conventional Variance Results Comparison: Low SNR

32

delay measurements zT1 (k) and ZT 2 (k). The values of matrix

entries in cpd and '¥ d are calculated by using the following

parameter values in equations 1.3.1 and 1.3.2.

Ct. = 0.04

a = 1/40 (and 1/200)

The sampling interval T was chosen to be 10 seconds.

Figure 14 shows the basic estimator system structure

where the nonlinear time delay measurements ZTl (k) and

ZT 2 (k) are fed into the nonlinear prefilter. This unit de-

velops a linearized measurement of target range ZP ( k) and

depth ZdT(k). The errors in measuring these target parame-

ters are both non-Gaussian and non-stationary depending upon

the geometry of the tracking situation. As target range

closes or opens, the mean value and variance of these errors

change, and are taken care of by the table look-up procedure

of section 2.4.

A conventional Kalman Filter was developed for the

state equation~(k+l) =cf>~ (k) +'!'wk where cf> and'¥ are given

by equation 1.4.5. The filter of the A

form ~ (k+l) = c/J!_(k) + Kk+l [ZdT(k+l) Ht/>~ (k)] gives esti-

mates of the target depth d (k+l) which is the first or up-T

per component of the estimated state vector ~(k+l). The

equations for a generalised, conventional, adaptive Kalman

Filter, used throughout this text, are discussed in Appendix

A.

Deterministic Input

Uk -

~+l

wk

Random Input

wk -

Random Input

~ ~ + r uk + $ wk = p- p p

pk = [l 0 OJ~

-

rk+1 = 4>ark + $dwk

dT = [l O OJrk

r------Actual Range I

I ~

~ I

I I "

pk 2 d o dT -c (.)

33

Polar Range

pk -

Actual

- I 2 2 rk =Jpk + (do-dT)k -

dT Slowly Varying -I Actual Depth I I I I I I

--~ I

I I I vl I I + ~ I 1' 1 + I

z,1 z I I Noisy I

I Time Delay r--------_J Measurements I

'ii

pk 2 dk (dw-dT) .,. ·2

E - z,2 c (.) + + ~ -

Figure 13. Data Generation for Depth Tracking

Range

rk ___,.

34

Bias ,_

Removal ,....

(Stored) . -- ii 2dT

A

Tl Target Depth dT + r Nonlinear Kalman Filter

z

Linearized Measurements

z T2 Prefilter zP A

Adaptive p . ~ +· r ~ Range .

-·"' Estimator

Bias ,_ .....

Removal (Stored) 1.-

"

Figure 14. Basic Estimation Structure

35

A large number of computer runs were made using noisy,

data generated by the technique shown in Figure 13. We will

now present some of these results which are typical of those

that were observed over many trial tests. Figure 15 is a

plot of the linearized non-Gaussian raw data ZdT (k). The

target is at a fixed range of p=25K and at a mean depth of

600 feet, making slow random changes on the order of 30-40

feet. Measurement errors of a1 = a 2 =3msec were added

to •i and • 2 • In Figure 16, the target range has been in-

creased to 40K and an has been increased to 5msec to incor-

porate a high noise case. Notice that the data shows a bias

due to an excessive number of large negative readings. This

-explains the need of a limiter to 'window-out' bad data

points, such as ZdT<O or ZdT>dw, the ocean depth.

Figures 17 and 18 'show the convergence of the tracking

filter for the previous sets of data, Figures 15 and 16, re-

spectively. The depth estimator provides a good track as

the target makes 40 feet depth changes about the mean value

of 600 ft., which is unknown to the tracking filter. Again,

the target range is fixed at 25K,40K with the observer at a

mean depth of 1000, target at 600 and ocean depth = 3000

feet.

The results presented so far dealt only with targets at

fixed ranges, which is a very unrealistic situation. Now, as

g 0 ('J

0 0

8

08 -;@-IH*~~.~~-*~il-M1iffi-*ll ~~!P.Hl*-Hf+ill:ii!filtlll-Hm~Jl!H!tl-llfl!-~JJ.llH·!IH ~mH4-lH-~~µ~~~*

8 0 N

0 0 9)+_-0-0~~-2~0-.00~~~4~0-.o-o~~-G~o-.o-o 80.00 '1110.00

r IME llE 10.1 120.00

Figure 15, Noisy Depth Measurement Data

140.00 161).00

g 0 N

0 0

8 -

8 0 N

~ Wlllll llHllU

I .11111111

I ~ I ' IHllll. I

llllllllM ,

I

--- -. ,- - - ,---.., 00 21.0C 4 l.00 60.00 80 I_ J II O.OC I'. 0.00 140.00 160.00

l I t - ti( 10

Figure 16. Large Measurement Error Effect on Depth Data

8 ~ -0 0

8

0 0

~ dT

- 0 Oo 1 ...... lllg-

' .._. Do dT 0

o_ ..,

0 0

~

.~ 9:LOO 80.00 160.00 240.00 320.00 400.00

TIME •101 480.00 560.00

Figure 17. Depth Estimation vs Time for Low Noise Case

w 00

8 0 N

0 0 0 0

8

0 0

0. N

0 0

9)'_-o-o~~--,0.--0.0-0~~~16~0--.o-o~--,2~40--.oo~~-3~20~.o-0~~4.~oo--.oo~~-4~00~.oo~~-5-60~_00~

TIME • 101

Figure 18. Depth Estimation for Longer Range, Higher Noise Case

w "'

40

the target range varies, so does the bias introduced in the

nonlinear data operation of the prefilter; this is where the

tabulated statistical data of section 2.4 is utilised.

2.6 RESULTS FOR DEPTH ESTIMATION

The simulation results shown herein were part of the

integrated range and depth estimation study and the inter-

ested reader is referred to [11] for results pertaining to

the range tracking.

In Figure 19, the target is closing range from P =80K to

20K at a constant depth of 900 feet. An initial depth esti-

mate d.rC0)=200 was chosen with the observer at 600 feet and

ocean depth 3000 ft. A standard deviation of 3ms (low noise

case) was chosen for additive noises v1 and v 2 . In this, and

in subsequent figures, raw depth measurements 2ciT out of the

nonlinear prefilter are shown to give an idea of the magni-

tude of the noisy data. Note the average decay of the magni-

tude of the noisy measurements as the target closes range,

making the received signal-to-noise ratio (SNR) increase.

Figure 20 illustrates the case of higher noise pow-

er cr =Smsec and target closing from SOK to 20K. The initial n

depth estimate was again chosen to be 200 with target at an

unknown fixed depth of 900.

-·

0 0

D "T. l'J

0 D 0 ()_ N

0 w

On .--.0 >t:c;

N.

T 1-{L ltJO [JC~

()_ 0.•

fl ()

0 'f

f) n <t,. 00 .

~T

.. T. -· ·- 1-- ----· - - . -- . --·- -----1-- - -------,------,---·-·--·--i-------·-I <10.00 no.oo 120.00 160.00 200.no · ? 110.00 200.00 320.00 JGo.oo

Tlt1E*IO

Figure 19, Depth Estimate for Target Closing (BOK to 20K) Range

0 0

n "' N

u ()

0 0. (J

-

Cl l:• .

.. UC> __ ,(.)

>¥ ci N.

:i: 1--CL LIO LI'~~

U. w

u 0 () ,,

n n

:

'l1. ,)ii

:,

i /\ +--

J_T N

I 1' &, ,1

:I 11 I

I ~

. .T.. ·-. I --- .. --

110. no AO. LIO 120. 01) --- ··-----------, · ------ -r··--- ------ -.-----------,------------- ··1

160. oo 200. no 2 ,,u. oo ?no. oo 320. on Jf:i TI t1E* I 0

Figure 20. Depth Estimate for Target Closing (SOK to 20K), High Noise Case

43

In Figure 21, the target closes range from lOK to lK

and makes large scale random changes in depth. It must be

clarified that this change was not produced by applying any

deterministic input uz but by changing the value of 'a' (the

Singer correlation time constant) to 1/200 and calling upon

a particular random generator from the IMSL computer li-

brary. The filter was also given this new value so as to

have a 'matched plant' case. The initial estimate was again

chosen to be 200 and it is seen from the figure that the es-

timator appears to follow the depth changes very accurately.

Note the fact that the unfiltered measurements are not very

noisy. This is due to the close range and the high SNR pre-

sent, even though crn=3msec.

Figure 22 is the same scenario as the previous figure,

but with target range increased, closing from SOK to 20K and

depth still making major random variations. Tracking is

still quite good with the exception of the tracker lag that

develops, which introduces an offset of about 100 feet. The

ability of the tracker to latch-on to these major random

depth changes is primarily due to the addition of the Singer

correlated acceleration, bui 1 t into the fi 1 ter. 'Fine tun-

ing' of the filter parameters was carried out so as to ob-

tain an optimum between filter speed of response and smooth-

ness of filter output. It must also be noted that the

--u

n n n N.

() D

0 0.

ll Cl o_ en

. ,<:) ::t(~

0 . ... ]_ I-fr. ldO Cle!

[J_ ~r

Cl (J

0. N

Cl D

s I IJMf-) EUUr~u_s JMSFC

: .. --·---·-·-----,---------·-r-------------,----------r----------,-----------,---·----.-'-lulO 40.00 00.00 120.0U IE0.00 200.00 240.00 2BO.OO

TIME:~ 10

,---------, 320 . 00 ]{j() . IJ()

Figure 21. Short Variable Range, Variable Depth Target and Estimator Output

+:-~

45

tracker continually uses new stored values of the mean and

variance of the non-Gaussian measurement error since the

target is constantly changing range.

A final set of runs was made to study some different

scenarios that might be of interest in real-life tracking.

The first, shown in Figure 23, is a tracking situation where

the observer is maintaining a fixed range of 30K and the

target is making rapid large scale random depth changes. The

filter was again initialised at a depth estimate of 200 feet

for the target. It is observed that the worst errors were on

the order of 100 feet lasting tor about 30 time samples, or

300 seconds of data.

Figure 24 shows the target at a fixed range of 30K pro-

ceeding at a constant depth of 250 feet. The observer is at

1000 feet maintaining the 30K fixed range, and using the

same set of means and variances tabulated for a fixed 600

feet depth target. A fixed bias is observed of about 60-70

feet, gradually decaying toward zero as a new set of means

and variances were automatically changed in the filter.

n u n ~

"'

{) 0

0 m_

On ~,Cl

*c~ N .

. .L I -f L (J 10 r.f~

[) 01

(] u Cl. ,,

0 (_)

SIOMA EDUALS JMSEC

\

'·

c·- -------··--·----,---------,-------,---· -·--·r------·-·-·1- ----·-·-·· ·-r·--··- ·---- - - .. ------.-------------, l.J . ()IJ ,II) . 00 00. 00 I 20 . 00 160 . ()() 2(10 . 00 2 ,10 . 00 2HO . on 320 . 00 ::160 . (Jfl

TI ME:.E ID

Figure 22. Variable Range SOK to 20K and Variable Depth Target

n n n 1'1.

n (_)

0 u

() 0

Cl .. (fl

L) ....• n )t, (~

n Ill

T 1--11. l1lfl ( l ( ~

u 'I

IJ I l

Cl. fJ

t I ()

n· ·· · ··· ·· --- · r-· · ·--- - ----1 --- ·-----. · · 1-·----·- -------- --r-- ----- --·-11.1111 -111.flll iU.1.1• I 1:-11.1'1·1 l[.f.1.(11_1 :·:;11J.i11.·1

I 11·11 · -. 111 I" -- - . . --·---·-·--r-----·-------.. - ·--·~-·-

~-ii.I J 11) . -~ .:(J .111 J :._-: ;-, . flf_I ~-:,

Figure 23. Fixed Range (30K), Variable Depth Target

..

n n u Cl

n 0

Ct (J)

() •.• t)

'.KC!

")

I. I . rt 11 JCl c J1 '.

0 '·I

'-.J CJ

C• f ,j

( J ( )

1 . I I I I

I/ !

I I' ~t Iii 11

fj I Oil() r 1111(11 S Jll(:lT

I ~.,IT

l ~

' "111 j 11 r~I :....- I+- I i

~ iii ... II

~ -n ,, I

( t. I llJ ·- .. ,. --·. f - ..

- -·--- -.---·-·--·-·---·----.---·------------71 UU.UU 11.~n. f11J 2n1J I 11 iL-..: I Ct

ru:i :.c-.111.un ::-f:U. :..o :i

Figure 24, Fixed Range, Fixed Depth Target: Mismatched Statistics

.p.. ():)

49

2.7 CONCLUSION

A state estimator has been developed and extensively

tested to track a target which is fixed in depth as well as

makes random depth variations. The target/observer scenario

is constrained to the vertical plane in the ocean environ-

ment so as not to compete with well established bearing

tracking programs. The estimator makes use of a nonlinear

prefilter to uncouple the state variables that model target

motion in both depth and range. An additional benefit is the

elimination of all EKF' s in the tracking system. This - re-

sults in a more robust tracker and significantly fewer com-

putations. The overall price one has to pay is that the li-

nearized measurements

non-Gaussian measurement

contain

errors and

non-stationary and

additional statistical

analysis has to be done to handle this problem.

System inputs to the tracking system consist of noisy

time difference measurements of bottom/direct and surface/

direct multipath time delays. The tracker prefil ters the

noisy multipath measurements in a nonlinear operation and

then transmits the new linearized depth and range measure-

ments into their respective filtering channels. The depth

channel gave excellent estimates as the target underwent

random depth changes. The overall tracking seems quite

good, especially in the high SNR cases.

so

This completes the study of a tracking strategy in the

vertical plane. The remaining portion of the thesis will be

devoted to the· study of a similar tracking system in the

horizontal plane, in the ocean environment, to track both

target range and bearing angle.

Chapter III

DEVELOPMENT OF A RANGE/BEARING TRACKER IN THE HORIZONTAL PLANE

3.1 INTRODUCTION

After going through the study and showing the validity

of an effective strategy to track target range and depth, we

now wish to extend our analysis to the horizontal underwater

plane, parallel to the ocean floor. Our aim is to measure

the bearing angle of the target so that, combined with the

previous analysis, we can track the target in all the three

dimensions. As shown in the following sections, an addition-

al advantage is that we can also obtain knowledge of the

target range in this plane. This can be used as a 'check' on

the.range estimate obtained in the vertical plane.

The approach used here is similar to that of chapter 2.

Sonar time delay measurements are obtained in a different

manner because of the change in geometry. A nonlinear pre-

filter is utilised to process these measurements and decou-

ple the range and bearing trackers. Initially, a 'matched'

Kalman Filter is used as an estimator; 'matched' because it

has prior knowledge of the deterministic inputs used to ma-

neuver the target. Later on, we advance to the case where

the Kalman Filter do·es not know the inputs governing target

51

52

motion. This requires the use of a bank of filters and an

adaptive weighting scheme, which will be discussed in the

next chapter.

3.2 GEOMETRY AND TIME DELAY MEASUREMENTS -- ---A brief look at past literature in the rangejbearing

field shows that the Puffs or Wavefront Curvature Analysis

provides a good way to obtain time delay measurements in the

horizontal plane. As opposed to a single listening device

used in the previous chapter, this method requires a set of

three listening sensors. Each of these sets may themselves

have an array of sensors but we shall treat each set as only

one sensor. Hassab [ 10] has considered the case where all

the three sets are on-board the observer, but the physical

dimensions of a submarine put a stringent limitation on the

distances separating the sensors. Statistical and test ana-

lysis prove that, as the distance between sensors increases,

the performance of the tracker improves in the sense of bet-

ter estimates at low ranges in a high-noise environment or

at large ranges in a low-noise environment. Thus, all the

work described here deals with the case of a 'towed' array

of sensors.

Figure 25 explains the geometry utilised in the Puffs

analysis. 'A' and 'C' are the two sets of on-board sensors

53

and 'F' is the towed array of sensors. Thus, S is the

length of towed array and is much larger than L2 , the sepa-

ration between the on-board sensors. RF , R and RA are the

three direct paths of the sonar signals from the source to

these sensors. This gives us the two values of time delay

measurements as

( i) ' 1 , the difference in propagation times between paths

RA and R,

( ii) 1° 2 I the difference in propagation times between paths R

and RF .

Target range and bearing are measured with respect to set

'C' and so, R is the actual range and e is the actual bear-

ing angle of the target from the observer.

We can now derive the relations between the sonar mea-

surements '1' '2 and target range Rand bearing e.

Using the law of cosines on triangle TFC in Figure 25, we

is

2 2 = R +L1 -2RL1 cos e = ( R 2 + r.i -2 R1:i_ cos e TI 2

defined as (~ -R)/C, where C=speed of

ter(approximately 2 2

5000 ft/sec)

-R+(R +L1-2RL1cos 6)1/2

'1 - c

sound in

(3.2.1)

A similar use of the law of cosines on triangle TCA gives 2 2 2 RA= R +L2 +2RL2 cosCrr-e)

wa-

11 = length of towed array

12 = distance between on-board sensors

F

r= 11

54

e

c

~JiE

• > A forward

.. , direction 12 of

observer

Figure 25. Geometry for Puffs or Wavefront Curvature Analysis

c 2 is defined as (R-RA)/C

R-(R2+L~+2RL2cos 8) 1 / 2 • T 2 = C

55

(3.2.2)

Note that the dilemma of positive/negative signs with the

square-root quantities can be solved by trying actual num-

bers in the scenario. Equations 3.2.1 and 3.2.2 are the two

nonlinear algebraic expressions for T 1 and • 2 ·1n terms of R

and 8 . We wi 11 now develop the nonlinear prefi 1 ter to get

separate expressions for the range and bearing angle.

3.3 DISCUSSION OF THE NONLINEAR PREFILTER

As shown in section 2.3, the nonlinear prefilter is ba-

sically an inversion of the expressions for the time delay

measurements i.e. it gives us linear equations for the tar-

get variables in terms of • 1 and • 2 .

From equation 3.2.1 we have

c • = - R + ( R 2 + L 2 - 2 RL cos 8) 112 1 1 1

= -R + R(l+Li- 2L1cos 8) 1/ 2

R.2 R For lxl < l, we have the series

2 3 ( l+x) 112 = (l~-~+~ ) 2 8 16 ..

Using only 3 terms in the series expansion yields

[ 1~2 211 ) 1 (Li 211 )21 C,. = -R+R 1 +- 1 --=-cos 9 -- - --=-cos 8 J ·1 • 2 z R 8 R2 R

(3.3.1)

56

Expanding, regrouping and neglecting terms higher than 2nd

order finally

Ci: = -L cos 1 1

yields 2

. 1 1 1 2 e + - - sin e 2 R

Also, equation 3.2.2 gives us

C i: 2 = R (R2+L 2 +2RL cose )1/ 2 22 2

= R - R(1+1 2 + 21 2 cos e) 1/ 2 . R2 R

(3.3.2)

A similar algebraic manipulation as above finally gives 2

1 1 2 2 ci: 2 = -12 cos e - z"'"R" sin e (3.3.3)

We can solve 3. 2. 2 and 3. 3. 3 simultaneously for R and e

Multiplying eqn. 3.3.2 by L 2 and eqn. 3.3.3 by L1 and sub-

tracting the second quantity from the first gives

(3.3.4)

2 2 Multiplying 3.3.2 by L 2 and 3.3.3 by L1 and adding the two

quantities gives

- -l [ C(L~T 1+LiT 2)] 6 - cos -L L (L +1 ) (3.3.5)

1 2 1 2 2 2

Manipulation of 3. 3. S using sin e =1-cos e and substituting

this into 3.3.4 gives us the final expression for R as

(3.3.6)

57

Equations 3. 3. 5 and 3. 3. 6 form the basic nonlinear

prefilter and provide us with a means for separate range and

bearing estimation. Of course, interaction between the two

estimators has to be maintained for proper processing of

statistical data in the Kalman Filters. An important differ-

ence between the prefilters discussed here and in the previ-

ous chapter is that the analysis here does not give us li-

near expressions for range and bearing. This makes further

analysis more challenging and interesting but does not

change the approach in any way.

Again, in real-time tracking, we do not

have T 1 and T 2 given to us but the set of noisy delay mea-

surements

Z Tl = Tl +Vl

Z T2 = T 2 +V2 2 where v 1, v 2 = N ( 0, a n)

Incorporation of 3.3.7 into 3.3.5 and 3.3.6 results in the

noisy range and bearing measurements.

An intermediate step is to check the validity and accu-

racy of equations 3.3.5 and 3.3.6. This is done by consider-

ing some typical scenarios like e =60° , 90° , 120° and

R=15K,30K,60K. Actual values of Tl and T2 are calculated

from the geometry; these are then substituted into 3.3.5 and

3. 3. 6 to give us the re spec ti ve bearing and target range.

58

The sensor lengths used are L1=2000, L2=250 and C=SOOO ft/

sec. The results of these mini-simulations are given in Ta-

ble 1.

It is clearly seen from Table 1 that the expression

for e gives almost precise results, but the range equation

yields values which are different from the actual target

range. This can be attributed to two factors: (i) the nonli-

near nature of the expressions involved, and (ii) the error

introduced in the development of the prefilter by using only

3 terms in the series expansion of 3.3.1.

As a first measure to.rectify this ambiguity, the pre-

filter derivation can be carried out again, using 4 terms in

the series expansion. Surprisingly, this gives the same ex-

pression for range R as equation 3.3.4; a new and better ex-

pression for e cannot be found because of the complexity of

algebra involved. It must be stressed again that throughout

this mini-simulation, no additive noise is used with Tl and

r 2 Realistically, we would have the noisy delay measure-

ments ZTl and ZT 2 which can lead to highly erroneous results

from the range estimator.

However, close examination of the set of calculated

ranges in Table 1 shows that the error (or bias, as it is

henceforth referred to) is dependent only on the bearing an-

gle and stays almost constant for a fixed e and changing

59

TABLE 1

Accuracy of Range/Bearing Expressions

Actual Actual e R

deg. feet

15K

60 30K

60K

lSK

0

90 30K

60K

15K

• 120 30K

60K

'r 1 msec.

-178.6528

-189.6644

-194.9164

26.549

13.3186

6.6648

218.6954

209.67

204.917

'r 2 msec.

-25.3099

-25.1556

-25.078

-0.417

-0.2083

-0.1042

24.685

24.843

24.9217

Cale. R

feet

Cale. e

deg.

14173 60.028

29148 60.007

59137 60.002

15050 89.989

30030 89.999

60012 89.9999

15917 119.973

30894 119.993

60885 119.998

60

target range. This bias is on the order of approximately

-850 for e =60° , +30 for e =90° and +900 for e =120° . This. in-

teresting observation is further clarified by performing the

mini-simulation for the complete array of ranges and bearing

angles i.e. range R varying from SK to 80K in steps of SK

and e varying from 45 ° to 135 ° in steps of 5 ° . (Incidental-

ly, these are the limits of bearing angle that we w~ll work

with in all of the target tracking simulation scenarios).

The dependence of the abovementioned bias on angle e is por-

trayed in Figure 26. The bias is an 'almost' linear function

of the bearing and this reiationship can be approximated by

BIAS = -30(ANGLE-90)-80 (3.3.10)

This knowledge of the bias helps us to modify the 'calculat-

ed' range and obtain the 'actual' range. Equation 3.3.10 is

used as a bias-removal subroutine in all future simulations

to compensate for the errors caused by analytic assumptions.

In his work, Hassab [10) has used on-board sensors with

Li_ =L2=150. The above mini-simulation, carried out using

these lengths, produces 'no' bias in the calculated range

values. So, a notable point of interest is that the 'bias-

ing' problem seems inherent only to the towed array tracking

scheme.

Before starting the actual tracking algorithm, we need

to look into the statistics of the range/bearing expres-

BIAS (Ractual - R ) calc.

1200

1000

800

600

400

200

---~--------t---t-----f---t----+-~:-t------tr----;1--t---;---+-----t------.--- e (degrees) 50 60 70 80 100 110 120 130

-200

-400

-600

-800

-1000

-1200

Figure 26. Relationship Between Range Bias and Bearing Angle

62

sions; these will be utilised in the Kalman Filters of the

respective estimators.

3.4 STATISTICS OF RANGE AND BEARING MEASUREMENT ERRORS

The noisy measurement equations at the output of the

prefilter can be looked upon as

z = e + v e e ZR = R + VR

(3.4.1 a)

(3. 4.1.b)

where ve and vR are the bearing angle and range measurement

error random processes respectively.

v e = z6 - e

VR = ZR - R

(3.4.2 a)

(3.4.2 b)

We can obtain explicit expressions for v 6 and vR by substi-

tuting for ze I ZR, e and. R from equations developed in the

previous section.

Initially, closed form solutions for v 6 , v R mean and

variance calculations were not looked into. Instead, the

tried and tested method of extensive simulation (described

in section 2.4} was utilised to obtain these statistics. The

layout for this data generation is shown in Figure 27. This

system was exercised at a series of discrete target ranges

R=SK, lOK, .... , SOK and bearing angles e =45° , 50°, ... , 135 °. For

the time delay measurements, Hassab has used additive Gaus-

sian noise of a standard deviation of Susec. After some cur-

63

sory simulation runs, a 1 = a 2 =50, 200usec were chosen for the

low-noise and high-noise cases, respectively, for our towed

array scheme.

A 'Monti Carlo' set of means and variances for Ve shows

two interesting facts: ( i) a symmetry of values around

the e =90 ° case, and (ii) for a particular angle e i, the

mean and variance stay almost constant for changing range.

These condensed results are tabulated in Table 2. The magni-

tude of the mean/variance values signifies that the bearing

expression is quite insensitive to the levels of additive

noise used.

However, the data simulation did not fare very well

with the vR (range) expression and gave irrelevant, incon-

c lusi ve results. This was traced back to the term

(L2Z Tl - 11. z,2) I in the denominator of the VR equation,

which makes it ex"tremely sensitive to any additive noise.

So, an alternate method to obtain the vR mean/variance had

to be found. After looking at various approaches, some re-

sults given in [10] helped to develop a modified approxima-

tion method which gives closed form expressions for the mea-

surement error statistics. This modified method is explained

fully in Appendix C. One basic assumption was made for this

analysis; since the magnitudes of Ve statistics were so

small, e was not treated as a random process throughout the

R

6

Gen.

SNR Control

50 µsec 200 µsec

White Noise

Generator

White Noise

Generator

N.L.

Pre-

Filter

e

+ Mean/var.

Aver ager

Mean/var.

Aver ager

R

Figure 27. Layout for Range/Bearing Measurement Error Statistics Calculation

e deg.

45.0

50.0

55.0

60.0

65.0

70.0

75.0

80.0

85.0

90.0

65

TABLE 2

Statistics of Bearing Measurement Error

cr =SOusec n

E[v9 ] deg.

-0.00109934

-0.00100575

-0.00092728

-0.00087003

-0.00083082

-0.00079632

-0.00076791

-0.00075138

-0.00075192

-0.00073332

0.07133199

0.06585217

0.06158879

0.05826027

0.05567531

0.0537005

0.05224493

0.05124464

0.0506596

0.05049374

cr =200usec n

E [v6 ] deg.

-0.00490354

-0.00437856

-0.00398366

-0.00368854

-0.00346778

-0.00329475

0.28539883

0.26345789

0.24639064

0.2330665

0.22271952

0.21481663

-0. 00316681 ... 0. 20898873

-0.0030646 0.20498462

-0.00300807 0.20264276

-0.00295059 0.20189542

66

derivation. Typical results obtained for the low-noise and

high-noise cases with e =60° are shown in Figures 28 and 29.

(Note the vertical scale change in Figure 29)

As a final part of the statistical study, measurement

error probability density functions were obtained and ana-

lysed. For this, the specific scenarios considered were:

(a) L1=2000, ~=250, R=40K, 8=60°, crn =50usec,200usec for

the bearing error density function, and

(b) L1=2000, R=SOK, 6=45°, cr =50usec,200usec for n

the range error distribution curve.

The layout of Figure 27 was used with only one random se-

quence generator to give 500 noisy measurements z6 , ~ at

the prefilter output. Figures 30 through 33 show some typi-

cal density functions for v6 and vR. It is clearly seen that

the nonlinear operations make the density functions have

non-Gaussian structures with a finite (positive or negative)

mean value. This explains the need of a table look-up proce-

dure to subtract the mean error, so that the noisy target

measurements become zero-mean and can be processed by the

Kalman Filter.

67

4.8K

4.2K

3.6K

Ll = 2000

3K 12 = 2SO 8 = 60°

a n = SO µsec

2.4K

1.8K

l.2K

0.6K I

I 01 20K 40K 60K BOK Range

Figure 28. Range Statistics for Low Noise (SO µsec) Case

68

35K

11 = 2000 30K i

12 = 250 e = 60°

<J = n 200 µsec 25K t

I I

20K ! I I I I

I I

15K r E[vR]

lOK r I

SK j

SOK Range I

01 20K 40K 60K

Figure 29. Range Statistics for High Noise (200 µsec) Case

Ll 2000

L2 250 44

R 40K e 60°

a 50 µsec n

33

t-------+-- --+-------'-------t--·-----1---------------1----------_,_--5K -4K -3K -2K -lK 0 lK 2K 3K 4K

Figure 30. Range Meas. Error Probability Density Function (Low Noise)

24

12

-20K -16K -12K -8K -4K 0 -·------·t-----f

4K ·---f-----

8K 16K ·t----------t

20K vR 12K

Figure 31. Range Meas. Error Probability Density Function (High Noise)

P(v6)

55

11 2000

12 250

R SOK e = 45°

a 50 µsec n

t-------t -0.2 -0.16

-o:O_a _____ o ....... -0-4 --~-o--o-. ;----o ~~-8---o· . ..__1_2 ____ 0_ . ._1_6 ___ ___,o. 2 v e (deg) -0.12

Figure 32. Bearing Meas. Error Probability Density Function (Low Noise)

P(v8)

65 Ll 2000

L2 250

R SOK 8 45° 52

0 n 200 µsec

39

26

13

-0.8 -0.6 -0.4 -0.2 --------·--------- ----·---~

0 0.2 0.4 ~~~~-----r-·~~-

0. 6 0.8 -----t ve(deg)

Figure 33. Bearing Meas. Error Probability Density Function (High Noise)

73

3.5 PERFORMANCE ANALYSIS OF THE ESTIMATOR ---Prior to checking the performance of the estimator,

some typical scenarios have to be devised and the corres-

ponding time delay measurement data generated. Figure 34

shows this data generation set-up. The target motion is mo-

deled (similar to equation 1.3.3, the linear drag model) in

both the x and y directions as:

~(k+l) = <P x(k) +

x_(k+l) = <P y(k) +

ru (k) + '¥w (k) x x

ru (k) + '¥w (k) y y

(3.5.1)

with parameters T=S sec., drag coefficient a=0.04 and Sing-

er correlation time constant a=l/150. The range of inputs

u ,u is given by [-1,1] which gives the target a maximum x y

velocity of 25 ft/sec in both directions (max. radial vel-

ocity = 36 ft/sec). This velocity is calculated by consider-

ing the steady state condition of equation 1.3.1 which gives . x = ~/a = 25ux (as a =O. 04)

The observer motion is modeled in a simpler manner by

x 0 (k+l) = y (k+l) =

0

K .T + x (k) x 0

Ky. T + y0 (k) (3,5.2)

where K ,K govern the speed and direction of the observer, x y

relative to the coordinate system.

Thus, given the x and y coordinates of both the target

and observer at each time iteration, we can obtain the actu-

al target range and bearing angle by using

u XS

u ys

X (O) y (0) 0 0

OBSERVER

HOT ION

1'ARGET

MOTION

w XS

w xy

X}k) 1

RANGE/

BEARING

GEN.

X8 (k), Yx (k) DLOCK _A

TARGET HO'l'ION IS GIVEN BY 'l'llE GENERAL OIFF. EQN.

x+ax~u + w x x

R(k)

e <k>

BLOCK A I R(k) a /(X - X ) 2 + (Y - y ) 2

8 0 8 () . e (k) ~ tan- 1

Tl(k) - -R + (R2 + 1./ - 2Rl.l coee >'i c

2 2 ~ l 2 (k) • R - (R + 1,2 + 2RJ.2 coa 8 )

c

'1 (k)

'2 (k)

Gf.N.

I~] x - x 8 0

'1< k)

T 2 (k)

v1 m N(O,o 2) n

where L1 3 length of towed array

Figure 34. Noisy Time Delay Data Generation for Range/Bearing Tracking

Z1 l (k)

75

2 2 i12 I R ( k) = [ { xs - x o ) + ( Ys -yo ) ] k

= tan -l [ y 5 -y 0 ]· + o x -x

s 0 k

(3.5.3) 6(k)

where o is the observer course bearing relative to the fixed

origin of the x-y coordinates.

The outputs of equation 3. 5. 3 are used in 3. 2 .1 and

3.2.2 to give the time delays -r1 (k) and -r 2(k) respectively.

Additive zero-mean Gaussian noise of standard deviation 50

or 200usec is then used to obtain the low or high noise de-

lay measurements.

The set-up for processing this data to obtain range/

bearing estimates is shown in Figure 35. The only major

change here as opposed to Figure 14 is the addition of the

bi~s-removal block, discussed in section 3.3. As an initial

test case, the target was kept fixed at a range of 25K and

angle 50° while the observer was kept stationary at the ori-

gin. It was found that for both the low and high noise cas-

es, the noisy bearing measurements were very close to the

actual angle (maximum difference= 0.1°). Hence, the remain-

der of the text will deal only with the range estimator; the

bearing estimator, if needed, can be easily and similarly

implemented.

Before we discuss the actual filter performance, men-

tion must be made of the plant structure (target range mo-

Nonlinear

Prefilter +

BIAS Removal

for

Range Meas.

r-------1 I ~

E[vR]

Stored

0 vR

Range

Estimator

Bearing

Estimator

Value

t I

L _ ---------_ J_ Stored Value

~-1

I I I I I I

.._ ____ R(k)

I ~ - _j

8(k)

Figure 35. Basic Estimator Structure for Range/Bearing Tracking

77

del) in the filtering algorithm, as it is slightly different

from the one used in the depth tracker of the previous chap-

ter.

McCabe [9] has developed a polar target model of the form

p 1 ~12 p

p = 0 p +

(3.5.4) '¥ 3 w.> 0 0

Sp k+l 0

The entries in matrices ~, r and '¥ are the same as in chap-

ter 1, us is the radial deterministic input and v0 cos~0 is p

given by the relation (referring to Figure 36)

v cosS 0 so =

. = x cose + v sine

o 'o

where 8 = true bearing angle

x 0 = observer velocity in x-direction . Yo = observer velocity in y-direction

Thus, one more dimension has to be added to the r matrix,

and this is the target range model used in the estimator

here.

Figures 3 7 and 38 show the range estimation vs. time

for the fixed target/fixed observer, low noise (SOusec) and

high noise ( 200usec) cases respectively. (Target fixed at

y

SOURCE

OBS.

e

x x 0 s

Figure 36. Projected Geometry of the Observer-Source Scenario

x

...... 00

79

R=25K, 6 =50°, Observer fixed at origin). The noisy range

measurements are also superimposed to give an idea of their

magnitude. The filter has been initialised at lOK ft. The

range tracker provides a good estimate as the target

makes 200 feet changes about the mean value (of 25K) which

is unknown to the filter.

Moving on to more realistic situations, three scenarios

were devised to test the tracking algorithm comprehensively.

These are shown in Figure 39 a,b,c. The first one has the

observer fixed at the origin and the target moving across

from the 1st to the 2nd quadrant by deterministic inputs

u =-0. 4 and u =O. 0. This changes target range from 23K to x y

17.SK to 23K and bearing from 50° to 90° to 130°. The second

scenario also has the observer fixed at the origin but the

target makes two intermediate maneuvers. This gives R=l2K ->

26K -> 24.SK -> 46K and 6=55° -> 74° -> 116° -> 105°. The

final scenario incorporates observer motion while the target

goes through one 'elusive' maneuver, halfway through the

tracking process. Thus, relative to the observer, target

range R=40K -> 28K -> 61K and 6=135° -> 90° -> 54°.

Figures 40, 42 and 44 illustrate the filter performance

for the above scenarios for the high SNR case whereas Fig-

ures 41, 43 and 45 show the tracking results for the same

scenarios with a low SNR. In the latter cases, it is seen

8 D fr!-

8 D ~-

Id el Zo ([O n:: .

0 w_

0 0

~-

Q ____ _

"b.on 40.00

SIGMR EQ 50 USEC

1--80.00

R

R

----.------.-----.-------r-----.----. 120.00 160.00 200.00 240.00 280.00 320.00 360.00

TIMEiidO

Figure 37. Range Estimation vs Time (Fixed Range (25K), Low Noise Case)

00 0

N

D w lJ)

g 0

~

0 0

~

Do ~•O *o ~

l-\:_J CJ Zo ([O Ct:ci ~ N

0 0

Po

8

-

-

-

-l I/

-

- ~-----.--fit.LIO 40.00

~

SIGMR EQ 200 USEC

z V"R

~

R

j ~l~~~f

J /l ~1 ~

,------r-----,-----,-80.00 120.00 160.00 200.00

TI ME~dO

' f I l r

I 240.00

~ R

I '-280.00 ------.

320.00 360.00

Figure 38. Range Estimation vs Time (Fixed Range (25K),' High Noise Case)

Scenario 1

0

Initial Co-od (15K,17.5K)

u -0.4 x

u 0.0 y

Observer fixed at origin

(a)

Scenario 2

T

0

Initial Co-od (7K, lOK)

(i) u 0, u = 0.6 x y (ii) u 0.8, u = 0.0 x y

(iii) u 0, u 0.8 x y

Observer fixed at origin

(b)

Scenario 3

0

Initial Co-od (OK,40K)

(i) u x 0.4, u -0.4 y (ii) u = 0.4, u 0.4 x y

OBS. Motion Initial Co-od (15K,-15K) vel. x-dir. -10 ft/sec vel. y-dir. = 10 ft/sec

(c)

Figure 39. Scenarios to Test Tracking Algorithm

co N

83

that the filter estimates are not exact but vary from the

actual target range by about 700-1000 feet (higher for rang-

es> 40K). This is caused by the very large magnitude of the

noisy measurements at long ranges and the high degree of

nonlinearity in the entire processing. On the other hand,

the filter gives excellent. 'on-target' results for the high

SNR cases.

It is also interesting to note that for all the cases

shown here, the initial range estimate for the KF was cho-

sen, arbitrarily, to be lOK. In all later runs, a new and

simple method was used to obtain an initial estimate; the

first 10 noisy range measurement values were averaged to

give the initial estimate and test results prove that this

method seems to work equally well without affecting the fil-

tering algorithm. Moreover, in some cases, the filter con-

verges marginally faster to the true target range.

Another test of the tracking algorithm was to observe

its performance when required to estimate the target veloci-

ty profile even though this was not a design objective. Fig-

ure 46 shows the excellent results for the high-noise case

in scenario number 2. This can be treated as an added advan-

tage of this estimator.

As a final check, it was decided to check the estimator

performance with on-board sensors as in [10]. For this, the

N

0 0

0

~

D 0 0 ~ N

g 0 o_ N

Do ~-•O *o ltJ (~

(0.

Zo a:o lYci

N.

0 0

0 D

o_ .. -··---,---~CJ. OU 40.00

SIGMR EQ SO USEC

--, 80.00

·r-------r··-----,----,------.---- --1 120.00 160.00 200.00 240.00 280.00 320.00 360.00

TIMElll:lO

Figure 40. Range Estimation vs Time, Scenario #1, High SNR Case

0 D

8

D N ..

(J D

SIDM~ EfJ 200 USLC.

R

R

C>_ -----·--------------~-·---.-------.--------.--·----.------.--------. °lJ.OO 40.00 80.00 120.00 160.00 200.00 240.00 200.00 320.0U 360.00

TI MEitdO

Figure 41. Range Estimation vs Time, Scenario #1, Low SNR Case

00 V1

8 .

8

g

8 CJ (()_ ...

8 fib:oo __ · 4b.oo

SIGMn EQ 50 USEC

-----.-------,-----,----- -1---, BO.OD 120. 00 160.00 200.00 240.00 280.00 320.00 360.00

TI ME~;tO

Figure 42. Range Estimation vs Time, Scenario #2, High SNR Case

00 0\

8 g

0 0

0 0

@-

g ~·

8 9------, 0.00 40.00

SIOMn EQ 200 USEC

I BO.OD

R

,-----.-----,-----.-----~----.--·-----.

120.00 160.00 200.00 240.00 280.00 320.00 360.00 1 I MEildO

Figure 43. Range Estimation vs Time, Scenario #2, Low SNR Case

00 .......

u n n. ['

0 0

0 .. w

0 0 D_ 10

0 0

~-

n 0

R

R

~! - -- -·--- -- ,---·-----r---- ---·---r-···· - ··-·-·T_. ______ ·--· 1··------·--·--·--r-·· ·--------,---------- r· --·-·- ---· -- , lJ.00 40.00 80.00 120.IJ(J 160.UO 2ll0.0U 240.0U 200.00 ]20.00 360.00

·1 I ME* l 0

Figure 44. Range Estimation vs Time, Scenario #3, High SNR Case

00 00

Ill Cl

n 0 C> N_

Cl 0 0 o_

D D

@-

.-10

*~ ' Po-l\J CJ z erg n::: .

0 -q-

0 0 (_l_ N

SIDMn EU 20CI Uf>[C

R

R

Figure 45. Range Estimation vs.Time, Scenario #3, Low SNR Case

00

'°

D 0

0 0

£-

8 -le) u.1...,. >

0 0

0

a 0 0 'I. -I

0 0 0

SIOMR EQ 200 USEC

. R

/ ,-' ' ~ R

ro ·-----·--·-·----.-------r ---·-,--·--·----.-----r---------,------. 10.00 20.00 40.00 60.00 BO.OU 109.00 120.00 140.00 160.00 180.00

TIME*2 *10

Figure 46. Velocity Estimation for Scenario #2, Low SNR Case

I.() 0

91

lengths L1 and ~2 were changed to 150 ft. and the bias remo-

val block was bypassed. (As discussed in section 3.3, the

on-board sensors case does not have the 'biasing' problem.)

The additive noise levels were chosen as crn=S,10 usec and

the filter gave very good results for all the above scenar-

ios. Typical performance results for cr =lOusec are shown in n

Figures 47 and 48 for scenarios 1 and 2 respectively.

In summary, the KF here provides excellent tracking in

giving good range estimates, particularly at close ranges

and for the high SNR cases. However, for test purposes, the

tracker has a knowledge of u ,u which govern the target mo-x y

tion; in a practical situation, this never exists and the

final tracking system must be able to track the target with-

out apriori knowledge of these parameters. This leads us to

the next chapter where this problem is overcome by develop-

ing an 'adaptive' range tracker which uses a bank of Kalman

Filters instead of the single KF used till now.

N

0 0 0 ({) m

0 0 Cl ~-

8 0 00. N

Oo ~•O

it: c) ~­N

Lil (_') Zo <IO C~o

2-

0 0

0 1.11_

n D 0 N. ~o. no

SIGMR EQ 10 USEC

R

R

--- --,--··--- ---i-------·---·--·-------, ·-·-·-------··-----.------·--··-··-··-------··-------,-·-··-----------1 OU.DO lfi0.00 2·10.00 321).0fJ 400.CJO <!BO.OU 560.00 b40.0ll 7!0.0fl

TIME*

Figure 47. Range Estimation, On-board Sensors, Scenario 1 (10 µsec)

"'

n D [_) (()_ l/l

0 ()

u (J.)_

"' 0 0 Cl n •t

Clo . ,(.)

>t:ci

It.I (.')

l'I (ll

7'.u rro ft~ ci .,,.

(',,.

fl 0

n en.

u n Cl . . . u:r,. 111 I

--, ---·- ·-. .. I fJll . 01 I 11 )t I . I Ill

. . .,. !·111.llfJ

1···· - .. ·- I··-·

:-1: ''I. I JU '1DI J .t.111 Tl MF:.:

R

I .

•li:Jl.111) - .. -·

!;t,11 .no

R

I - . b1I0.111·1

Figure 48. Range Estimation, On-board Sensors, Scenario 2 (10 µsec)

I 'I .. , , . ( 111

Chapter IV

THE POLAR RANGE ADAPTIVE STATE TRACKING SYSTEM

4.1 INTRODUCTION TO THE ADAPTIVE SCHEME ---The technique of adaptive state estimation allows us to

tackle the problem of tracking target range of a maneuvering

target without prior knowledge of up which governs target

motion. This involves the formulation of the total estimate

from a weighted sum of state estimates conditioned on N pos-

sible discrete input levels. u(i) This section qualitatively p •

explains the operation of a general adaptive filter; the

specific filter structure used is discussed in the next sec-

tion.

In the state model of 3.5.4, the polar target range can

be represented as being derived from a time correlated Gaus-

sian density having a mean value u . This input corresponds p

to a particular target velocity and the Gaussian distribu-

tion accounts for any random fluctuations. A 'matched'

tracking filter, which knows the value of up, can easily

follow these fluctuations around u and provide good esti-P

mates of target range. If the filter_is given a value u~)

which is unequal to u , it will assume a displaced distribu-P

tion and the random fluctuations are assumed to be around a (i) mean value of up rather than up. This will result in the

94

95

mismatched filter's output having a bias proportional to the

difference between u (i) and u . p p

Now consider a series of N such partially overlapping

Gaussian curves with displaced mean values u~), i=l,2, ... ,N

as shown in Figure 49. By using a bank of N Kalman Filters,

each conditioned on a different u(i) and corresponding Gaus-P

sian curve, a series of N estimates is obtained.

Case 1: If actual u equals, p say, (2) u I p

give the true range estimates.

Case 2: If u is located between u(3) p p

then KE' # 2 will

and u( 4) then the p I

true range estimate will lie between the estimates

of KE''s 3 and 4.

Thus, an adequate 'weighting' scheme is necessary so

that the final unconditioned estimate of the adaptive filter

can be formed by the 'weighted' sum of the N filter outputs.

The jth weighting factor is related to how close the target

velocity distribution is located to that of the jth filter.

The calculation of the weighting factors is based on

the fact that the distribution of the i th filter can be

viewed as the distribution for the probability that a mea-(i)

surement resulted from a target whose up equals up . Thus,

N new probabilities are calculated at every measurement it-

eration, one for each filter, and the weights are propor-

tional to these probabilities. At every iteration, the sum

(1) u

. (2) u

(3) u

(4) u

(N) u

Figure 49. N Partially Overlapping Gaussian Curves, Displaced Mean Values

97

of these weights must equal unity. So for case 1 described

above, weighting factor 2 (w2) would equal one while all

other weights would be zero. In case 2, w3 and w4 would have

much larger magnitudes as compared to the other (N-2)

weights.

For a particular scenario, one would expect the weights

to remain constant until u changes, because they are based p (i) I on the relationship between up and the fixed up s. Howev-

er, the noisy nature of the measurements and minor random

fluctuations of target range make these weights fluctuate,

even for a constant up. This results in a non-smooth final

range estimate and necessitates the use of a smoothing fil-

ter on the weights in order to stabilise them. As test re-

sults prove, this approach works well° for the final estimate

and cuts down 'spikes' in the adaptive filter output.

4.2 ADAPTIVE FILTER SETUP

The mathematical development of the basic adaptive fil-

tering technique is quite involved and does not explain the

operation of the adaptive filter. Hence, only the final ex-

pressions used and the setup of the adaptive filter in the

tracking algorithm are given here. Simulation results in the

following pages will clarify the working of the weighting

scheme and the performance of the tracker. The interested

98

reader is referred to [l],[2] for a complete discussion of

the mathematics involved.

The specific adaptive filter used with the range track-

er consists of six Kalman Filters. The u~i) for each of

these estimators is chosen so that the filter bank spans the

entire velocity range of typical targets to insure proper

tracking at any velocity. The weighting factors for the six

KF's can be mathematically expressed by

_2!(k+l) = c(k+l)*P (k)*qiT *w(k) w w - (4.2.1)

where

~(k+l) is the 6xl vector containing the new weighting fac-

tors.

c(k+l) is the normalising constant computed at each itera-

tion to ·make the sum· of the weighting factors equal

unity.

~ (k) is a 6x6 diagonal matrix. The (i,i) element is the

probability that a measurement resulted from a tar-(i)

get whose up =u P

¢w is a 6x6 matrix which models the semi-Markov nature

of changing input associated with target maneuvers.

The (i,i) element is the probability that the target

will remain in the i th velocity state at time tk+l

given that it was in the ith state at time tk. The

( i, j) element is the probability that the target

99

will change from the i th (at time ~) to the jth

velocity state (at time tk+l ) . The sum of the ele-

ments in each row/column equals unity.

~(k) is the 6xl vector containing the weighting factors

at the previous iteration.

The elements of the P matrix are evaluated by using the w

Gaussian probability distribution of the form

... 2 -1/2 zi P (i,i) = e ~ • constant w v

where Zi = ZR (k+l) - H(<j>~ (i) (k) + ru~i)]

v = [ H. M ( k+ 1) . HT + RR]

(4.2.2)

These values of zi and v are calculated and used from each

of the KF's in the filter bank. Henceforth, the zi 's are re-

ferred to as 'measurement residuals'.

Evasive target maneuvers are considered to have semi-

Markov characteristics and the matrix <I> defines the prob-w

abilities of such maneuvers. The values of probability used

in this matrix are typically, 0.95 for maintaining the same

velocity profile and 0.01 for any change in velocity; these

numbers stay_ constant for the entire scenario and are not

recalculated at every iteration.

As previously mentioned, a simple first order digital

filter (or averager) is used on the weights to obtain a

smoothing effect. This is given by

w (k+l) = a *w (k) + b *w(k+l) (4.2.3) --s w--s w-

100

From this point onwards, all mention of the weights is with

respect to the 'smoothed' weighting and the subscript 's'

will not be used.

Finally, the total unconditioned estimate of -the target

range is obtained from the filter bank as

R(k+l) = ~ R(i)(k+l) • wi(k+l) i=l

(4.2.4)

At this stage, it might appear from the above equation

that an entire Kalman Filter algorithm is being executed N

times (for each of the possible discrete input levels) at

each time iteration, but such is not the case since the pro-

cess and measurements covariances Q and RR are the same for

each filter. The target dynamics remain unchanged for each

filter and the entire covariance and gain analysis of the KF

algorithm becomes identical for each state in a given chan-

nel. Consequently, these computations need to be made just

once rather than ~ times per iteration.

Figure 50 shows the complete layout of the adaptive

filter to track target range. The input to the structure is

the noisy range measurement from the prefilter and the mea-

surement compensation block consists of the table look-up of

mean and variance. Some of the blocks in this diagram are

used for the purpose of 'engineering approximation and re-

finement'; these are necessary to improve the performance of

101

the tracking algorithm in the low SNR cases and will be dis-

cussed alongwith the results in the next section.

4.3 SIMULATION RESULTS

In the estimator of Figure 50, 6 levels of input u(i) p

were chosen to span the expected target velocity· range of

+30 (ft/sec) for an opening target and -20 (ft/sec) for a

closing target. If the velocity range was greater than these

numbers, N ( 6 in this case) could be increased or !J. u (the p

difference between adjacent input levels) could be slightly

expanded. The u 's were chosen with a difference !J. u of 0. 4 p p

to model -20, -10, 0, 10, 20, 30 (ft/sec).

The weights were all initialised, arbitrarily, at

0.1666 and the estimator set-up was exercised for the high

SNR case (cr =SOusec) in all the scenarios discussed in the n

previous chapter. The results are depicted in Figures 51 -

54; the noisy measurements are again left out of the plots

for the sake of clarity. The only refining process used here

is the rolling averager on the weights and it is observed

that the tracker performs extremely well, even at high range

values. The sampling time used is 5 sec. and the weights di-

gital filter coefficients aw' bw are 0.8, 0.2 respectively.

To provide an insight into the behavior of the weight-

ing schemes, a special scenario was devised. Here, the tar-

A(l) R

Estimate x Conditioned ..... (1) -(1)

(1) z w on u

ZR A(2) R Output

Estimate moo thing Conditioned -(2)

(2) z Filter

Measurement on u p

Compensation

R I-' 0 N

Kalman • -(6) Gain/Cov. A(6) w

R Computation Estimate

Conditioned ~(6) (6) z

on u Avg. p

Weighting eij Aver ager Factor

Cal cu la tion Semi-Markov Parameters

Figure 50. Block Diagram of the Adaptive Range Tracker

() ()

() (fl_ N

D 0 0 r,_ ('J

0 Cl u (() N

N Ori • _,(J

*ci ~-

Ld C1 Zn ffO rv • ,._U

v_ (\f

n ()

n (1), ('J

() ()

n

S l GMH El~ SD lJSEC

(\I. ------ - - - -·------r------ ---- .. -----,---·--.. -----------.-.. ------- -----------,---- ___ .. _ ---·-- --,----------------- .. -,--------------.. ·--·--r----------·--r· ------------------, <ti.no B0.00 160.00 :?•10.0U ~20.0IJ ¥10.00 480.0U bl)(l.00 G40.ll!l '/?11.IJil

T J Mf *f_)

Figure 51. Adaptive Range Estimation, Fixed Scenario, High SNR Case

I-' 0 w

r·•

n [)

(.) en_ N

D 0 n lO C\I

n 0 n '<J". N

c )Cl ·-·Cl *c)

Lil (.')

~-

Zu crn (Y(~ n_

(\J

(J D (_) en_

() D Cl w __ • CJ. uo

SI GMO ECJ !.-:iO U~>EC

/ ~-r ,, . ..... -

-~"Y~,o::::::.-

,------------ ·----.--------------- --. ·-··· ----- ---- ,--· ---· --· ·-· ---,------- -- ----··-· ---------------,----------.------- ---- -------. nu.Ou lGtl.00 :-'40.0l) ]:?0.!Jl) '100.IJl) 4llll.OO bGU.PO 640.011 7:!0.00

l l Ml:.iKh

Figure 52. Adaptive Range Estimation, Scenario 1, High SNR Case

N

0 0 0 (O_ If)

C> 0 . () ro_ ...

n 0 u o __ ..,

Oo ~--•C> W<c)

N. (I)

ltJ C> Zo ((0 O·' • ·o

"I" N-

0 0 0 lO.

n {.)

SIOMA LO 50 USEC

R

_,_~1~f R

(:)_ . ·---·-· ... ----.----·- - ···--·- -· • ··-·. ····------ --· , .... - -··· ·····-· .. ··1-··-··---·--·-- ..• ··--· ------·· ---- ,---· ...... ·-- ------,-----··-- - .. -·-· ··-·· 0b. nu l:iO. oo H .o. oo ?.·m. ou 370. on .t1on. un 4fJfJ. nu !:»1m. no £-Mn. oo '/%( 1. no

TI MElKG

Figure 53. Adaptive Range Estimation, Scenario 2, High SNR Case

8 n ;},-

n 0

0 <O .. tn

N Un ~•O *a (D_

"'" Lil Cl Zo cro we;

n "" CJ u (I N (1)

n Cl ( ) '-l -· -f'lt.t ii l

··r-··--·- -·- ·--,------- - -----,-- ·-·----..------------------,------- ------,----- -· -----.-------- -·-r-------- -·- 1 fllJ.UO lhO.tJO ~' 1lfl.llll :ci:-:'U.OU "IUll.tlll 400.UO f)ljO.flO C·lll.flO '/?0.0LI

l 1 MEidi

Figure 54. Adaptive Range Estimation, Scenario 3, High SNR Case

I-' 0 (J'\

107

get is initialy at coordinates (lOK,O) for the first 300 it-

erations corresponding to u =O and then moves away (at the p

same bearing angle of 90°) at the rate of 20 ft/sec corres-

ponding to up=0.8. These values of up are the exact values

of u (i) used by filters 3 and 5 in the filter bank. In the p

tracking process, we would expect the weight w3 to approach

unity, initially, while w5 approaches zero. After the maneu-

ver, these weights should switch values. Figure 55 ·shows the

precise behavior of these weights as expected. Obviously,

the exact weighting scheme performance results in a good

range tracking for the entire scenario.

Moving along to the low SNR cases (cr =200usec), prelim-n

inary simulation results showed the need for some modifica-

tions to the adaptive tracker in order to enable it to han-

dle the high level of noise involved. These refining

strategies are listed below:

(i) A rolling average on each of the 6 measurement residu-

als to Gaussianize their distribution. As shown in

equation 4.2.2, the elements of the P matrix are cal-w

culated assuming a Gaussian distribution. Figure 56

shows the non-averaged distribution of measurement re-

siduals z3 and Zs for the special scenario (low SNR)

described above. Comparing this to Figure 57, where

the same residuals have been subjected to a rolling

0 N

0 ()

Cl Ul

Cl

({) l··D ]W C.~c~· ...... I.ti ~

() '<;!

0

0 N (j

n Cl

cf1: r-11] ·-""'--f-rK-l .-c-_10----., G--L-l-. u-o---2.-4-0-. C-)-0--3~0. oo 400. oo 480.00 --,-----------,

5f:>O . 00 640 . 00 720 . 00 TIME*S

Figure 55. Behavior of Weights for Special Scenario

...... 0 ex:>

109

average, the difference is quite noticeable; only 3

previous values are averaged to avoid excessive time

lag.

(ii) Use of 0.5, 0.1 instead of 0.95, 0.01 in the semi-Mar-

kov matrix ~w • This tells the weighting scheme that

the probability of a change in target state ( i->j) is

high and enables it to detect a maneuver which may be

masked in noisy data. An unavoidable drawback is that

the estimator output becomes non-smooth; so this stra-

tegy has to be implemented with (iii).

(iii) A simple first order digital filter on the final un-

conditioned estimate of equation 4.2.4. h h ~

R (k+l) = a • R (k) + b • R(k+l) s r s r (4.3.1)

with a =0.7 and b =0.3 r r (i) and (ii) are needed to help the characteristics of the

weighting scheme in the high-noise environment while the fi-

nal range estimate is slightly smoothed by using (iii) and

the digital filter on the weights.

Figures 58 - 61 illustrate the results for the high ad-

ditive noise case with all the modifications built-in. The

estimates are still quite good though they tend to fluctuate

around the actual trajectory. There is a slight time lag in

the final output due to the compensation, averaging and

smoothing involved but this is considered an acceptable

--------·-+-

-24K -20K

"'(5) z

-16K -12K -BK -4K

l3B

115

92

69

-(3) z

------t--·--t-----0 4K BK 12K

Figure 56. Non-averaged Distribution of Measurement Residuals 3 and 5

., 16K

-(5) z

_____ , ______ o::__. ___ ,_. ___ ..._ _______ _

-20K -16K -12K -BK -4K

22B

190

152

l14

0 4K

-(3) z

--t-----BK 12K

Figure 57. DistribuUon of Residuals 3 and 5 with Rolling Average

112

trade-off for the more desirable quality of filter conver-

gence and smooth estimates. Figure 61 shows that the estima-

tor tends to lose exact track of the target beyond ranges of

45K in the presence of a high bearing rate. Further study

into this problem is being carried out at this time.

Various other scenarios were devised and simulated to

include rapidly closing range (target in attack mode) and

the tracker performance held-up to its standards in all the

cases. Also, Hassab (10] type (L1=L2 =150) simulations were

carried out with cr =5,lOusec for all the above scenarios and n

the overall algorithm performed adequately giving good range

tracking. Figures 62 and 63 show some typical results

(lOusec case) for scenarios 1 and 2 respectively.

4.4 CONCLUSION

A feasible solution to the problem of passive tracking

of maneuvering targets without any knowledge of the inputs

(up) involved has been presented here in the form of the

adaptive range estimator. The performance of this adaptive

tracker is good in all the test cases with the addition of

several compensation and modification techniques but there

is still room for improvement, especially in the low SNR

cases. The nonlinear nature of the data pre-processing makes

the measurement errors and residuals non-Gaussian which re-

ld (.')

8 0 0 ... 0 0

0 ~-

0 0

0 N_ en

Zo ([O O'.'.c)

"'" -N

0 0 0 o_ N

u 0 0

SIOMA EQ 200 USEC

R

R

~- --------T- ·---------r----·--·---.------------,---·----.----------.-----b. on an. oo t bo. oo 240. oo 320. oo 400. oo 4BO. oo

TIME*S .------1-------,

660. 00 640. 00 720. 00

Figure 58. Adaptive Range Estimation, Fixed Scenario, Low SNR Case

Cl 0 0 (D SI GMR [() 200 Uf.iEC m

D 0 0 N_ (tl

0 D

0 en. N

"' C10 r ;0

*c~ ""- R N ld C> Zo (.[O ft'. c:i

0. N

0 0 D co_

R

0 n (J ~- ----------.---·----.------r---------r·--·---,--·---,---------,------.-------1 tum 80.00 160.00 240.00 320.00 400.00 4U0.00 560.00 640.00 72U.OO

TIME*5

Figure 59. Adaptive Range Estimation, Scenario 1, Low SNR Case

N

0 0 0 m D 0 0 m ..,.

0 0 0 o_ .....

Oo .--.a lKa ~-

LIJ (_') Zo cro er .

0 ...... N

n 0 n (0.

D 0

fib.oo

SI GMi:l ECJ 200 USEC

80.00 nm.oo -.-----r------.------1-------2 40 . 00 320 . 00 400 . 00 480 . 00 bf){) . 00

---------- ·-·1 6,10 . 00 720 . 00

TIMEllE5

Figure 60. Adaptive Range Estimation, Scenario 2, Low SNR Case

Pl

() 0

g-

0 0

R-

Cl 0 0 .. lD

D ~--•O

*~ Cl UJ-

ld CJ z ao cr:o

n. "' n CJ

CL (l)

n D

n_ ---·-··----·--r---"l.J . no Oo . no

·-r--------r-----------, -···--· - · -·-·-r-------·----,---·-·--··-----·-·1-------,------· ---··---, 160.00 240.00 320.00 40U.OO 4BU.OIJ l~t:>0.00 6'10.00 72ll.UO

TIMElKS

Figure 61. Adaptive Range Estimation, Scenario 3, Low SNR Case

.,

D n () ((l _ (f)

n 0

CJ ('.J (l)

() 0

0 (()_ N

CJo ~-10

;ii:c)

ill (')

,, N-

Zu QO We)

D N

(_) 0

0 (O __

u Cl

u (\1

·i:J.Ot)

R

---r EltJ . on

~:; I nM(-) Ff J I U U~_il L

- I 11;n .un

I -:·"-'II). (II I

I -- I - - - I - -J'.,'.0. UI I ·lllU. fJlJ /flU I_ (111 TIML~G

I - - ---- -- - -- ·1 !)lilJ.(11) 1;.11J.(lll

Figure 62. Tracking Results for Scenario 1, On-board Sensors, 10 µsec Case

I '/?IJ .I 111

n 0 (J (0 lfJ

Cl 0

Cl Q) "t -

D 0

0 o __ ~r

N Ou _,o 11'.l)

N en Ld (') Zo a:o n--:c; ...,,._

('J

D D

u (fl_

0 ()

n m· --U.IJll

1----- -- -fU).IJll

El I UMf-) FU 1 n UhEC

- I ICO.CIU

T --- - -- - - · 1 -- - - --- - -- -- I -- - - - - I - - -- -- --- - -- - I --- - - - --- -T-- -- - -? •II). (II J ~J'.:U . 00 41111. ()IJ :Jun -I llJ f,r:;n JJ( J fi,11) . I JI I

TI ME*F>

Figure 63. Tracking Results for Scenario 2, On-board Sensors, 10 µsec Case

I "//ll .l_llJ

119

sults in the weighting scheme not behaving 'precisely'. An

effort is underway to study techniques to improve this facet

of the algorithm so that we can come up with a faster con-

verging and more reliable (long range) adaptive filter.

Chapter V

CONCLUSION

This thesis addresses passive underwater tracking of

maneuvering targets in two distinct planes with respect to

the ocean floor, viz. (a) the vertical plane which gives

target depth and range, and (b) the horizontal plane which

involves target range and bearing angle. Trackers in both

the planes process similar types of passive sonar time delay

measurements and are based on identical basic state equa-

tions to model target dynamics in the radial direction.

The depth tracker of chapter 2 and the range tracker of

chapter 3 utilise a nonlinear prefilter to linearize and in-

vert the time delay measurements. The advantages of this

prefilter are two-fold:

(i) It decouples the range and depth or the range and bear-

ing estimators and considerably reduces the order of

the computation.

(ii) It eliminates the use of all extended Kalman Filters

with their associated divergence problems. Instead,

standard Kalman Filters are used resulting in more ro-

bust trackers with a significant decrease in the com-

plexity of the total algorithm.

120

121

The cost that one pays is that the processed measure-

ments contain non-stationary and non-Gaussian measurement

errors. This necessitates an additional statistical analysis

of these errors but the convergence characteristics of the

estimator more than make up for this added burden. Chapter

4 introduces the reader to the adaptive fi 1 tering scheme

which produces reasonably accurate range estimates with only

a statistical apriori knowledge of the target deterministic

inputs.

The results shown throughout this text are good but

there is definitely room for improving and 'polishing' the

design strategies presented. It must be put in perspective

that the main purpose of this thesis is to introduce viable

algorithms for passive tracking and the performance plots

demonstrate the validity and potential of the schemes used.

Some suggestions for further study include:

(a) The use of real tracking data on these algorithms.

(b) The inclusion of automatic 'fine-tuning' processes which

can optimise the various parameter values used in the

models.

(c) The use of higher order digital filters used for smooth-

ing in the adaptive tracker.

(d) The test of the bearing angle estimator (KF) with addi-

tive noise levels higher than 200usec.

122

(e) The introduction of some randomness in the length L 1 of

the towed array for bearing/range estimation.

( f) An alternate approach to obtain velocity estimates in

the event of unavailability of time delay measurements

and the combination of such alternatives with the ap-

proach presented to give a more reliable real-life

tracker.

A final comment is appropriate at this time, concerning

the precision of the computer used to obtain the results

shown here. It must be stressed that most tracking algor-

ithms are implemented on tactical computers which have a re-

latively small word length. This implies round-off errors

which can significantly alter the performance of sensitive

tracking algorithms. All of ·the results in this thesis were

obtained on the IBM 370/158 digital computer in the VPI&SU

Computing Center, using single precision. At several points

in the research, the small word length of the IBM computer

presented a few problems but the easy solution of using dou-

ble precision was avoided; instead, the algebraic and sta-

tistical analysis were made more rigorously with stringent

limitations on the various assumptions used. Thus, the al-

gorithm was modified and improved to perform successfully

using only single precision.

Appendix A

KALMAN FILTER EQUATIONS

The equations for a generalized basic Kalman Filter are

given below.

Given a discrete time state variable model of the form

!_(k+l) = ~!,(k) + r.!:!,(k) + 'f'w(k)

~ where

x is the state variable vector.

~ is the state transition matrix.

u is the set of deterministic inputs to the plant.

r is the transition matrix associated with u

'¥ is the input transition matrix (random inputs).

w is a Gaussian zero-mean white noise process.

k is the discrete time parameter.

and the measurement equation

z(k+l) = ~(k) + v(k+l)

where

z is the measurement or plant output,

H is the vector relating the state variables to the

measurements,

v is an additive zero-mean Gaussian noise,

the Kalman equations are :

1. The estimate equation

123

124

A A

.?S_(k+l) = ~.?£(k) + r~(k) + Kk+l[z(k+l)-H(~x(k)+r~(k))]

2. The gain ·equations

M(k+l) = ~P(k)~T + ~Q~T

K(k+l) = M(k+l)HT (HM(k+l)H T + RR]-l

P(k+l) = [I-K(k+l)H]M(k+l) where

x is the estimate of the state variables.

K is the filter gain matrix. A A

P(k) = E[ (x-~) (~-~) ]k , the error covariance matrix.

Q = E[w.wT]

I is the Identity matrix.

RR = variance of the measurement error.

Note that the filter equations used in the depth track-

er of chapter 2 will·not have any r-related terms because

of the non-existence of that matrix in the depth channel mo-

del.

Also note that the range-tracking adaptive Kalman Fil-

ter in chapter 4 requires a third term in the equation for

M(k+l). This term is 2

+rt.u rT 12

h , th d, ff b .._ t t' (i) I • w ere ~u is e 1 erence e .... ween wo consecu ive u P s in

the filter bank. The reason for this term and its derivation

are discussed by Moose [2].

Appendix B

MODIFIED STATISTICAL ANALYSIS OF DEPTH MEASUREMENT ERROR

The derivation for the modified statistical analysis on depth measure-

ment error, referred to in Chapter 2, is discussed here.

We have

where

a = (obs. depth)/(obs. keel depth) 0

and

d· = depth of ocean w

2 -r2 = '2 + v2

where

Assuming depth measurement error Vd = zdT-dT it follows that

- dT z 1+a z 2 T 0 T

125

(Bl)

126

Substituting from (Bl)

Let

=

Using a series expansion for the terms in parentheses, we get

2 3 4 dT 2 3 4 Vd = dT[-e+e -e +e .... ] + , 1 (v1-v1e+v1e -v1e +v1e . . . ]

(B2)

Now, as a digression, let us find the expected values of each of the

terms separately.

vl+aov2 € =

'1+ao'2

E[e] = E (v 1)+a0 E (v 2)

'1+ao'2

2 €

2 = N(O,cr ) n

= 0

3 €

4 €

127

(B3)

=

3 E[c: ] = 0 , odd moments of zero-mean Gaussian r.v's.

th Using the 4 moment rule for zero mean Gaussian random variables

vlc: =

= 3cr 4 €

2 vl +aovlv2

•1+ao'2

2 (J

n T +a Tz 1 0

using

Using (Bl) E[v1 c:] =

(B3)

= a (B4)

128

odd moments.

3cr 4 (l+a 2 )d-] = n o · T using (Bl)

3 d 3 Tl W

3cr 2 d cr 2 = n T e: using (B3)

T d 1 w

3 3acr 2 using (B4) E[v1e: ] = e:

all odd moments.

We can use these sub-results now in (B2) to get E[Vd] by neglecting the

higher order terms in the series expansion.

234 dT 2 3 4 = d T E [ -e:+e: -e: +e: ] + - E [ v -v1e:+v e: -v1e: +v1e: ] Tl 1 1

129

Grouping terms

E [ vd] = d' ·[cr 2 -..£_] [1 +3cr 2] ·T e: 't' e: - 1

Now

Squaring (B5) and neglecting H.O.T.

2 [

2 2acr 2 E 2 [ v ] = d .2 ~ _ e: + 6a cr 2

d '-T 2 't' 2 e: 't'l 1 't'l

4 12acr e: + 9a 2 0 4]

't'l 2 e: 't'l

Squaring (B2) and neglecting H.O.T. gives

d ? T . 2 2 2 3 2 2 4 2 3) + -2 ( v 1 - v 1 e:+ v 1 e: - v 1 e:

't'l

(B5)

(B6)

Again, a digression is needed to obtain the expected values of the

intermediate terms.

The results become 2 E[v1 e:] = 0

2 2 E[v1 e: ] = 2 3 E[v1 e: ] = 0

-2 2 4 = dT [cre: +9cre: ]

2d 2 + _T_ [ -a-9acr 2]

't' 1 e: (B7)

130

Finally using (B6) and (B7) and some algebraic manipulation yields

the variance term

a2v = d .2[a.2 (8+3a 2)+a 2(1-6a.\+3a 4(~-3a\ d T 2 o e: 2 e: '!l 2 1 •1 •1

- 2a(l+8a 2) I an 2] (B8) •1 e: 2

'! 1

Equations (B5) and (B8) are the closed form expressions which give

the statistical data on the depth measurement error.

Appendix C

MODIFIED STATISTICAL ANALYSIS OF RANGE MEASUREMENT ERROR

The modified method to obtain first and second order statistics

on the range measurement error, referred to in Chapter 3, is explained

here.

Also

Using our geometry where

z = 'z+vz T2

where

L1 = length of towed array

Assuming range measurement error v = zR - R R

L1L2 (11+1 2) sin2e VR = 2C [Lzz Tl-11 z,21

- R

L1L2 CL1+L2)sin28 = - R 2C[(L2Tl-LlT2)+(L2vl-Llv2)]

131

(Cl)

132

Using Cl to substitute for R we have

=R [-1 - l] 1+£

where

is I < 1

2 3 4 VR = R[-s+s -£ +£ • . + H.O.T.] (C2)

Now a degression to find out expected values of each term separately.

2 £

(0 (J 2) Vl' V2 = N ' n

= 0

Using (Cl) to substitute for (1 2, 1-11, 2) gives

= cr £

2 (C3)

3 e:

4 e:

3 E[e: ] = 0

133

all odd moments.

th Using the 4 moment rule for zero mean Gausian random variables

Using (Cl) and (C3) simplifies this to

Now from C2 we have

2 2 E[VR] = Rcr [1+3cr ] e: e:

Also 2 2 2 3 4 2 V = R [-e:+e: -e: +e: ] R

neglecting H.O.T.

(C4)

134

Finally

Expanding, grouping and neglecting H.O.T. yields

2 cr v

R (CS)

Equations (C4) and (CS) thus give us closed form expressions to find

the mean and variance, respectively, of the range measurement error.

If v1 is N(O,cr12) and v 2 is N(O,cr 22) where cr 1 , cr 2 are different,

as may be the case in practice, then (C3) becomes

cr €

2

However, the final expressions (C4) and (CS) remain unchanged.

(C3-i)

BIBLIOGRAPHY

[1] Gholson N.H. and Moose R.L., 'Maneuvering target tracking using adaptive state estimation', IEEE Trans. Aerosp. Electron. Syst., May 1977.

[ 2] Moose R. L. , 'Adaptive target tracking of underwater maneuvering targets using passive measurements', 1981 Annual Report, ONR.

[3] Singer R.A., 'Estimating optimal tracking filter per-formance for manned maneuvering targets', IEEE Trans. Aerosp. Electron. Syst., July 1970.

[4] Moose R.L., 'Adaptive estimator for passive range and depth determination of a maneuvering target (U)', U.S. Naval J. Underwater Acoustics, July 1973.

[5] Howard R.A., 'System analysis of semi-Markov process-es', IEEE Trans. Mil. Electron., vol. MIL-8, April 1964.

[ 6]

[ 7]

Jazwinski A.H. , 'Limited memory optimal fi 1 tering' , IEEE Trans. Automat. Contr., vol. AC-13, Oct. 1968.

Thorp J. S., IEEE Trans. 1973.

'Optimal tracking of maneuvering targets', Aerosp. Electron. Syst., vol. AES-9, July

[8] Hassab J.C., 'Passive tracking of a moving source by a single observer in shallow water' , Journal of Sound and Vibration (1976), 44(1)

[9] McCabe D.H., 'Adaptive estimation techniques for tracking airborne and underwater maneuvering targets', Ph.D. dissertation, VPI&SU, May 1979.

[10] Hassab J.C., et al, 'Estimation of location and motion parameters of a moving source observed from a li·near array', Journal of the Acoustical Soc. of America, vol 70, No. 4, Oct. 1981.

[11] Dailey T.M., 'Examination of selected passive tracking schemes using adaptive Kalman filtering', M.S. thesis, VPI&SU, June 1982.

135

136

[12] Hassab J.C. and Boucher R.E., 'Passive Ranging Estima-tion from an array of sensors', J. Sounq Vib., 67(2), 289-292 I ( 1979) •

[13] Ludeman L., 'Bias and variance of a sound ranging es-timator', New Mexico St. Univ., Las Cruces, NM, May 1979.

[ 14] Carter G.C., 'Variance bounds for passively locating an acoustic source with a symmetric line array', J. Acoust. Soc. Am., 62, 922-926, (1977).

(15] Moose R.L., Vanlandingham H.F., McCabe D.H., 'Modeling and estimation for tracking maneuvering targets', IEEE Trans. Aerosp. Electron. Syst., 1979.

[16] Knapp C.H. and Carter G.C., 'Estimation of time delay in the presence of source or receiver motion' , J. Acoust. Soc. Am., 61, 1545- 1549, (1977).

[17] Hinich M.J. and Bloom M.C., 'Statistical approach to passive target tracking' , J. Ac oust. Soc. Am. , 69, 738-743, (1981).

[18] Schultheiss P.M. and Weinstein E., 'Passive localiza-tion of a moving source', Eascon 78 Con. Rec., 1978.

[19] Gong K.F. and Davis J.S., 'Evaluation of Target motion analysis in a multipath environment', NUSC Tech. Re-port 4814, March 1976.

[20] Suggested by Moose R.L., Department of Electrical En-gineering, Virginia Polytechnic Institute & State University, Blacksburg, Virginia.

The vita has been removed from the scanned document