ESTIMATION RELATIVE AIRCRAFT USING Nicholas Brownsberger

/

ESTIMATION OF SONOBUOY POSITIONRELATIVE TO AN AIRCRAFT

USING EXTENDED KALMAN FILTERS

Nicholas Mason Brownsberger

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESISEstimation of Sonobuoy

Relative to an AireUsing Extended Kalman

Position.raftFilters

by


September 1979

Thesis Advisor: T) ,T Pollins

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Estimation of Sonobuoy PositionRelative to an AircraftUsing Extended Kalman Filters

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SRS (Sonobuoy Reference System)Kalman FilterSonobuoy

20. ABSTRACT 'Continue on reveree aid* II neceeeery end Identity by black number)

In airborne anti-submarine warfare there is a need to moreaccurately determine the positions of sonobuoys on the surfaceof the water. This report develops two algorithms which employextended Kalman filters to determine estimated position. Thebearing from the aircraft to the sonobuoy is the primarymeasurement. Range information is not available. The firstalgorithm is a six-state filter which was reduced from the

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13-state system developed, by the Orincon Corporation. Itsstates include relative position, relative velocity, andinertial misalignments. The second algorithm includes twocascaded Kalman filters. The primary two-state filter esti-mates sonobuoy position. A secondary filter estimates driftfrom information obtained from the primary filter. Bothalgorithms successfully estimated sonobuoy position forsimulated aircraft data. The effect of aircraft- to-sonobuoyrange, the frequency of measurement, and changes in altitudeare also analyzed.

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Estimation of Sonobuoy PositionRelative tc an Aircraft

Using Extended rCalman Filters

by

Nicholas Mason BrownsbergerLieutenant, United States Navy

B.S., United States Naval Academy, 1972

Submitted in partial fulfillment cf therequirements for the degrees of

MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING-AND

AERONAUTICAL ENC-INEER

from the

NAVAL POSTGRADUATE SCHOOLSeptember 1979

ABSTRACT

In airborne anti-submarine warfare there is a need to

more accurately determine the positions of sonobuoys on the

surface of the vater. This report develops two algorithms

which employ extended Kalman filters to determine estimated

position. The bearing from the aircraft to the sonobuoy is

the primary measurement. Range information is not available.

The first algorithm is a six-state filter which was reduced

from the 13-state system developed by the Orincon Corporation

Its states include relative jjosition, relative velocity, and

inertial misalignments. The second algorithm includes two

cascaded Kalman filters. The primary two-state filter

estimates sonobuoy position. A secondary filter estimates

drift from information obtained from the primary filter.

Both algorithms successfully estimated sonobuoy position for

simulated aircraft data. The effect of a ircraf t-to-sono bucy

range, the frequency of measurement, and changes in altitude

are also analyzed.

TABLE OF CONTENTS

I. INTRODUCTION 10

II. FILTERS 13

III. MODELS 20

A. GENERAL APPROACHES TO SYSTEM MODELING 20

B. THE COORDINATE SYSTEM 22

C. THE SIX-STATE SYSTEM 23

D. THE TWO-STATE SYSTEM 37

IV. ANAYLSIS 43

A. THE SIMULATION 43

3. THE RESULTS 56

V. CONCULSION 87

VI. SUMMARY 59

APPENDIX A: THE COMPUTER PROGRAMS 92

APPENDIX 3: COMPUTER PROGRAM FOR DATA GENERATION 127

APPENDIX C: RMS POSITION ERRORS EOR RANGE,

FREQUENCY, AND ALTITUDE ANALYSIS 140

3I3LICGRAFHY 149

INITIAL DISTRIBUTION LIST 150

LIST OF FIGURES

1. Kalman filter notation 15

2. Divergence 18

3- Earth fixed coordinate system 22

4. Aircraft fixed coordinate system 22

5. Cascaded Kalman filters 38

6. Aircraft track for the 15 NM circular patternwith initial sonobuoy location and directionof drift indicated 46

7. Aircraft track for the 15 NM square patternwith initial sonobuoy location and directionof drift indicated 47

8. Aircraft's navigational output for the 15 NMsquare; navigational drift is 2.5 NM/Hr tothe north 48

9. Aircraft's navigational output for the 15 NMsquare; navigational drift is 2.5 NM/Hr northwith a Schuler cycle 49

10. The effect of drift in the aircraft navigationalplot 52

11. RMS statistics at time = k 55

12. Orincon's simulation flight path 57

13. Estimated position and relative errors for thesix-state system using Orincon's pattern 67

14. Estimated position and relative errors for thetwo-state system using Orincon's pattern 68

15- Estimated position and relative errors for thetwo-state system using Orincon's pattern 69

16. RMS position errors for the six-state system whileflying the circular pattern developed by OrinconCorp. 70

17 • RMS position errors for the two-state system whileflying the circular pattern developed by OrinconCorp. 71

18. RMS position errors for the two-state system whileflying the circular pattern developed by OrinconCorp. 72

19. Estimated position and relative errors for the six-state system using circular pattern at 15 NM 73

20. Estimated position and relative errors for thetwo-state system using circular pattern at 15 NM - 7^

21. Estimated position and relative errors for thetwo-state system using circular pattern at 15 NM - 75

22. RMS position errors for the six-state system whileflying a 15 NM circular pattern around the sonobuoy 76

23- RMS position errors for the two-state system whileflying a 15 NM circular pattern around thesonobuoy 77

2^. RMS position errors for the two-state system whileflying a 15 NM circular pattern around thesonobuoy 78

25- Estimated position and relative errors for thesix-state system using square pattern at 15 NM 79

26. Estimated position and relative errors for thetwo-state system using square pattern at 15 NM 80

27- Estimated position and relative errors for thetwo-state system using square pattern at 15 NM 81

28. RMS position errors for the six-state system whileflying a 15 NM square pattern around thesonobuoy 82

29- RMS position errors for the two-state system whileflying a 15 NM square pattern around thesonobuoy 83

30. RMS position errors for the two-state system whileflying a 15 NM square pattern around thesonobuoy 8^

31. RMS position error as a function of aircraft-to-sonobuoy range, ax, for the six-state and two-state systems 85

32. RMS position error as a function of measurementinterval, At, for the six-state and two-statesystems 86

33- Factors which influence the covariance 64

34. Flow chart: main program 92

35' Flow chart: subroutine FILTER six-state 93

36. Flow chart: subroutine FILTER two-state 94

37. Flow chart: data generation program 127

38. RMS errors for two-state system and six-statesystem using circular pattern at 5 NM 140

39. RMS errors for two-state system and six-statesystem using circular pattern at 30 NM 141

40. RMS errors for two-state system and six-statesystem using circular pattern at 45 NM 1^-2

41. RMS errors for two-state system and six-statesystem using square pattern with At = 4 sec. 143

42. RMS errors for two-state system and six-statesystem using square pattern with At = 10 sec. 144

43. RMS errors for two-state system and six-statesystem using square pattern with At = 30 sec . 145

44. RMS errors for two-state system and six-statesystem using circular pattern, Alt = 300' 146

45. RMS errors for two-state system and six-statesystem using circular pattern, Alt = 10,000' 147

46. RMS errors for two-state system and six-statesystem using circular pattern, Alt = 20,000' 148

ACKNOWLEDGEMENT

I would like to thank Dr. Birnbaum and Ms. PeggyPembroke of NADC for their responsiveness in answeringquestions and supplying computer information. And toDr. Collins I would like to express my gratitude forhis help and guidance without which this thesis couldnot have been completed. To my wife I would like tooffer my special thanks and love for her patience,understanding, and fortitude during a time whichtested both of us.

Nick Brownsberger

I. INTRODUCTION

The P3 Orion is the U.S. Navy's primary long range

anti-submarine warfare (ASW) aircraft. It is outfitted with

equipment which allows it to search for, locate, and track

submarines. The aircraft carries a Univac digital computer,

the CF 901, v,hich performs much of the navigational and

tactical plotting chores. The primary sensor used by this

and many other ASW aircraft is an airdropped listening

device known as a sonobuoy. The Orion generally deploys

several scnobuoys (4 to 20) in patterns which can cover a

thousand square miles of ocean while searching for the

submarine. Once contact has been made, these sonobuoys

provide information which locates the submarine. The target

can then be tracked (or attacked if required) until the

Orion's mission is complete.

The sonobuoy is dropped from the aircraft at the

geographical location designated by the aircrev.. Once in

the water the sonobuoy floats and deploys a hydrophone to

depths varying from 50 to 500 feet. The information picked

up is transmitted back to the aircraft where it is analyzed.

This information can consist of the relative intensity of

target noise, bearings and sometimes ranges to the target,

all of vhich are used to fix the current position of the

submarine. In order to maintain close, accurate tracking and

be able to launch an attack, the submarine's position must

10

te accurate to within several hundred yards. The

information received from the sonobuoys has some error

inherent in the nature of the measurements made ty the

hydrophone. Sonobuoy position error also contributes

significantly to the submarine tracking inaccuracies.

Historically, the positions of the sonobuoys were

determined and updated by "mark-on-top"s . This required the

aircraft to home on the transmitting sonobuoy until the buoy

was overflown. At that instant the aircraft's position vas

entered into the on-board computer which slewed the buoy to

this updated position. After many of these updates the

computer was able to develop a bias which was applied to the

sonobuoy positions in the computer effectively allowing them

to drift. The methoa had several disadvantages. Error in the

updated position \*as at least as great as the aircraft

altitude at the time of the "mark" which could vary from 300

to 20,000 feet. The accuracy also depended en the

consistency of the several pilots who might be making the

mark-on-tcps during the flight. The updating was not

continuous in that it was several minutes between

consecutive marks on one buoy at best, and more likely 30 to

60 minutes. Not all buoys were even updated. Furthermore,

this method required the aircraft to overfly the submarine

many times in order to make the mark-on-tops. This should

be avoided.

The purpose of this thesis was to investigate some

alternative methods for accurately fixing the position of

11

sonobuoys. They should allow the aircraft to stand-off from

the sonobucy field and still produce more accurate fixing

than the historical method provided. The Naval Air

Development Center (NADC) at Warminster, PA. had already

partially developed such a system. This thesis was

undertaken in support of their work but v*as conducted

independently. Their system, the Sonobuoy Reference System

(SRS), was already installed on the aircraft and had the

capability of measuring the relative bearing to any

transmitting sonobuoy. Additional information available for

use included aircraft heading, altitude, and airspeed as

well as doppler velocity and drift angle. Also, the

aircraft's Inertial Navigational System (INS) provided

geographical position although the Schuler cycle and

inertial drifts could make this position several nautical

miles in error. On the other hand, aircraf t-to-scnobuoy

range and sea surface drift information were assumed not tc

be available. An attempt was made to determine sonobuoy

position at least relative to the aircraft with a less

accurate geographical position as a secondary objective.

Salman filtering techniques were used based primarily on

measurements of bearing from the aircraft to the sonobucy.

12

II. FILTEItS

Kalman filtering is a recursive technique for estimating

the state of a system. It was developed in the 1962's by R.

E. Kalman and improved upon previous methods by Wiener and

others. The Wiener filter is based on frequency domain

designs which are statistically optimal but are only

applicable to stationary processes. The Kalman filter is

based on state-space, time domain formulations and is

especially suited to digital computers. From a simplistic,

one-dimensional point of view, the Kalman filter recursively

averages noisy measurements to provide a more precise

estimate of the actual value.

Assume that a system can be linearly modeled with state

equations in matrix form as

X. = |, X, + A. U + T Vk ~k-l k4 k-i k-i k-t k-1

(1)

where X represents the states of the svstem at the Kthk

interval. (Only the discrete case \*as be considered in this

study.) <5 is the transition matrix and is used to"k

"propagate" the system from X tc £+1. U represents thek

control input to the system and W represents white,k

guassian noise with mean and Q variance, written N(3,Q )

k k

Measurements are required to update the system and are

13

described by the linear matrix equation

Zk - \\ + \ (2)

where Z, is the measurement. H, describes the relationshipk k

between the states and the measurement and V, is measurementk

noise described by N(0,R ).k

The discrete Kalman filter equations are:

Propagate

K K (+

k-i k-i(3)

?k

(

")

X p (+) (jjT + Qk-i k-i k-i k-

U)

Update

S - pk <-> < g p ( _ ) ff

T + dnk k v ; "k n

kj

-1

(5j

Xk

( + ) = I (-) + Gk

Zk "W (6)

P. (+)k

I - G Hk k V-' (?)

vhere

AX. = estimate of the state X.

covanance

Salman gain

14

AThe estimate prior to the measurement is denoted by X

K(-)

and can be updated after the measurement to a new estimate

Adenoted X

K(+). This notation is shown in figure 1. The

covariance matrix provides a statistical measure of the

Auncertainty of X. Consider a 2 x 2 covariance matrix where

the error T in the estimate is defined as x - x :

P =

E(x* )

E(x. lz )

E(x, xI

-""2

E(x* )

a, a-

(£)

The diagonal elements represent the mean square errors of

the corresponding state variables x and x2

. The

off-diagonal elements are indicators of cross-correlation

between the states. The Kalman gain J is an optimal gain

choosen so as to minimize the sum of the diagonal terms of

the covariance matrix.

Hk-i,

Rk-i

hl a

*k-iH

p k-.'-i

?k-.<*>

Vi<

£ k«-l

k,*k

*k (*l

*k-i,°k-iO *k.°k

S>

<k-i

TIME

Eigure 1. Salman filter notation

15

If either the system mcdel or the measurement model is

non-linear, then non-linear filters must be used. Let the

non-linear system be described by

\ - ^K~r> + v, ;w,^n(0,q, ) (9.

zk

= h(v + vk I V

k-K(0,R

k ) (10)

where ©(X, ) represents the non-linear equations cf ther k-i

system. They can be linearized cy a Taylor series expansion

around the latest estimate of the state tc obtain (j> ( X ,) .

Likewise, h(X ) represents the non-linear measurementA

equations and by a Taylor series expanicns yields H(X. ).

(xk

) = j*uk

) + I(i k ) X - Xk k

11)

h(X, h(Xk

) + H(Xfc

) Xk

" Xk

(12)

where

fu ) = —3_dx J x-x

(13)

H(X, )

ka

3xh(X,

x--x

14)

Second order Taylor series terms are neglected. Now, the

extended Kalman filter can be implemented with the following

equat ions

:

16

Propagate

X, (-)k

= 4 {*k-i

(15)

V"* - iUvnt-)) ^.,( +) i(x, ,(-

k-l •k-i(16)

Update

G = P (-) H(Xk ( -)) H(X,.(")) P. (-) E(X. (-) )

T- 17!

v + v-> + ^ zk

" H ( xfc(~)

)

(18)

¥.(+ )

= 1 " *kH(I

k (-))Pk(- (19)

Higher order filters can be used if the linearization

errors are large. The second order Kalman filter employs

one more term in the Taylor series expansion by modifing the

update equations of the extended Kalman filter to account

for this tern. The iterated extended Kalman filter uses the

same equations as does the extended Kalman filter. However,

the calculations are repeated, each time linearizing about

the most recent estimate, until there is little further

improvement with each new iteration. The iterated extended

filter can greatly reduce the errors due to non-linearities,

mere so than the second order filter.

Kalman filters should be based en correctly modeled

systems and accurate noise statistics to ensure proper

17

performance. This is not always possible either due to

ignorance about the system or lack cf sufficient statistical

information. A filter which is not operating properly may

diverge. Apparent divergence describes that situation where

the true estimation errors are larger than the filter

predicted errors although they are bounded. True divergence

is characterized by errors which continue to grovi with time

and eventually become infinite. These divergence phenomenon

are depicted in figure 2.

There are several ways to overcome the divergence

problem when the modeling is not completely accurate such as

adding fictitious noise. This allows the filter a little

more freedom to adjust to whatever modeling inconsistencies

may exist, but makes the filter estimate appear more

erratic. Another method *hich helps overcome divergence is

finite memory filtering. Since Kalman gains tend to grow

smaller and smaller as time passes, they may reach a point

TRUE

THEORETICAL

TIME

(a) Apparent Divergence

TIME

Ob) True Divergence

Figure 2. Divergence

18

where new measurement information has no effect on the

estimates. Finite memory filtering effectively eliminates

old data which is no longer useful by keeping the gains

significant. This is sometimes called a "moving window .

Some simplifing techniques used in Kalman filters

include precomputed gains. Although forfeiting the optimal

Kalman gains, this has the advantage of reducing

computational burden. More importantly, the gains can be

controlled to overcome modeling weaknesses if necessary.

Another technique which can be used when more than one

measurement is provided to a filter is processing them one

at a time. This avoids taking the inverse of more than a

scalar when computing the updated covariance matrix P„(+).

This is possible if the simultaneous measurements are

considered to be taken sequentially over a zero time span.

19

III. MODELS

A. GENERAL APPROACHES TO SYSTEM MODELING

The aircraf t-sonobuoy system must be modeled in

state-space for use with the Kalman filter. There are at

least two approaches to the modeling of this problem

depending on the point of view. One intuitive approach is to

assume the sonobuoy drifts at a constant velocity and then

use aircraf t-to-sonobuoy bearing measurements to locate the

sonobuoy. The states become sonobuoy position and sonobuoy

velocity. Unfortunately, this problem is not observable.

The bearing measurements provide only information about

position? there is no rate of change information in the

bearings themselves. In addition, the aircraft must

maintain a track of its geographical position between

updates in order to determine the next expected measurement

for the Kalman filter. As mentioned before, this aircraft

position is subject to non-linear as well as linear

navigational drifts which are not taken into account in this

model. However, an observable system can be obtained by

reducing the number of states to sonobuoy position only and

introducing fictitious process noise to account for the

drift. This noise effectively allows sonobuoy position to

update so as to keep up with the drift. One approach

developed by this thesis is a variation of this concept.

20

Another approach considers only the relative position of

the sonotmoy with respect to the aircraft. Relative

velocities and relative accelerations must be taken into

account and these change radically as the aircraft flies in

the tactical situation. Sonobuoy jjosition is not obtained

directly. (When only the word "position" is used it will

indicate the position relative to an earth fixed coordinate

system, such as latitude and longitude. "Relative position"

will always mean the location with respect to an aircraft

fixed coordinate system.)

There are some other considerations which should be

addressed. The sonobuoy drift is generally slow (less than

5 NM/Hr most of the time) and constant. It is not

unreasonable to assume that the entire sonobuoy field drifts

at the same velocity. Another point is that aircraft

navigational drift can not be distinguished from sonobuoy

drift. In other words, the drift that is preceived by the

aircraft is the combination of sonobuoy drift and aircraft

navigational drift. If this navigational drift is linear

and not excessive it causes few problems. dowever,

non-linear navigational drifts, such as the Schuler cycle,

can cause large errors.

21

B. THE COORDINATE SYSTEM

A right-handed coordinate system was chosen as depicted

in figures 3 and 4. This is a departure from the work of

Orincon Corporation which is described in the next section.

This system allows all angles to be measured positive in the

direction they are normally defined, i.e., aircraft heading

measured clockwise from north. It also coincides with the

usual aerodynamic coordinate system.

North

East

H = Aircraft heading, + clockwise

3

Doi*n

Eigure 3. Earth fixed

Nose

Aircraft pitch, + up

Aircraft roll, + r. wing down

Doppler drift angle, + r . drift

22

C. THE SIX-STATE SYSTEM

Orincon Corporation of La Jolla, CA., completed a

technical report in December of 1978 concerning the Sonobuoy

Reference System. They were contracted by NADC to

investigate the effect of errors in the aircraft

navigational system on the performance of the sonobuoy

tracking procedure. As a result of their v/ork they

developed a 13-state mathematical model which successfully

determined sonobuoy positions in spite of these navigational

errors. The complete state vector was

XT = AX

1AX

2AV

1AV

2X

lXZ

Vl

V2 *1

Xh *3 % m2

(22)

where

Ax

AV

x

V

m

relative position of the sonobuoy from the aircraft;

relative velocity cf the sonobuoy from the aircraft;

position of the sonobuoy on the ocean surface;

velocity of the sonobuoy on the ocean surface;

inertial system's platform azimuth alignment errors

inertial system's gyro drift rates assumed constant.

(Subscripts 1, 2, and 3 represent North, East, and dovn

respectively.) The model was developed using state equations

which described sonobuoy motion relative to the aircraft and

relative tc the ocean's surface. These were derived by

23

Orincon frcm

AX = X - (X1

+ Ea

(21

and other mechanization equations to obtain

A*. = A?"1

AX 2

Av1

Av,

1

= AV2

2= -R

j_

+ co ( x1

~R2

+oo U

2

= v.

- ial ~ AV " ^3 A

2(Kg

- Xa2

- A^2

) ~*3 A!"'^6

Z2

= V2

7=0V„ =

(22)

where

X1

H =

g =

to =

A =

aircraft inertial navigation system position

aircraft inertial navigation system error

vector from the center of the earth to the aircraft

acceleration due to gravity (assumed constant)

Schuler frequency yg/R

'

aircraft accelerometer outputs

The sonotuoys vere assumed to drift at a constant velocity

and the aircraft was assumed to fly at a constant altitude,

The inertial alignment errors were described by

2k

i>3

= ij^n cosx

(23)

inhere

Q, = earth rate

X = latitude

There were several sources of measuremer. t information.

Aircraft to sonobuoy bearing v,as available from the aircraft

interferometer system in the form of cosQ vhere 8 is the

angle between the direction to the scncbuoy and. the base

line of the antenna. Doppler velocity and. drift angle from

the aircraft doppler radar system provided information about

Av and Av„. The only other relevant source of measurement

information vould have come from JPS (Global Positioning

System) which would have provided error free aircraft

position.

Orincon performed a numerical observability analysis on

this system based on the observability matrix M, given by

\- Ii=l

£ H* R_1H $

k k k k24'

If this matriz is positive definite for H^2 then the system

is considered observable. In addition, an eigenvalue

analysis provided information as to the conditioning cf the

system. The result of their analysis indicated that the

25

system was not fully observable with measurements from the

interferometer and doppler systems alone. Only ten states

were observable; sonobuoy position most likely was not

observable and \p ~, m ,, m2

, v. and v were weakly observable

at test. They also noted that "observability is reduced

when the aircraft pursues a straight flight path". By

removing sonobuoy positions and gyro drifts from the system

they found the observability improved. Further, the system

became fully observable if sonobuoy velocity was also

removed from the state vector.

The Orincon report presented the results of a

Monte-Carlo simulation on the 13-state system. However,

they concluded that a reduced state filter made up of

TX - AX

1AX

2AV

1AV

2 *1 *2 (25)

would provide a workable solution to the problem although

they noted that it had a tendency toward divergence. It was

partly the intention of this report to further the

investigation of this reduced state filter.

26

The state equations for the reduced six-state filter can

be obtained directly from Equations (22) and (23).

Ax, = Av,

Ax_ = Av.

Av = -R. + (j {xl

Av2

= -I2

+ J(x2

ty.=

4>2

sinX

i>2

= -\pl

sinX

xal " A2

1}

+ VZa2 " Ax 2> " *1*

(25)

where ill , m , and m are set to zero. In matrix form3 1 2

X =

[

AVAx„

AvL =

AV2

*1

L *J

1

1

-co2 -g

012_

(JJ gu

--fisinX

f2sinX

Ax

Ax.

Av

Av.

n 2/ I-R + oil -I

, ,

1 i al

-K-+ 0)1 "I -,2 2 a2

which can be abreviated

X =AA

i k^X + B (27)

Expressed in discrete time the state equation is

\+l = »Xk

+*k (26)

27

The discrete formulation of the plant matrix can be obtained

from

i -ti Si - A•H At

= e (29)

Developing this

si - A

S 1 1

1

2

S 1 1

1

2 nOJ S '

1

g

. 2CO s

I

"_L

~s

0' s ft sin AI

' -ftsinA s

(30)

from which the determinant of the system is

si - A = (s2

+ u2

)

2(s

2+ n

2 sin 2\)

231)

The inverse of this matrix and its Laplace transform are

presented on the following pages as Equations (32) and (33)

respectively.

28

CM

00

<<CM-

e

+

3

+

cn

c:

00

(3

•Hcn

si

c:

+

3

+

-~c

C

oo

CM

+

3

+

ID

OO

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+

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+

to

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a.

+

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cn

c;

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+

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+

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c:

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+

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+

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+

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+

c-< •Hc CO

•H CMcn aa

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CMa

CMcHcn

CM

+CM

CD

r<CMc

<< •Hc 0)

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3

+

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+

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+

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+

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+

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+

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+

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29

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1

4-1

/'—

\

r< <<c1-1

c1

4J ' •H/-N CO CO CO

*< c: c: ac ••w

'

^—

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CN %-• CN 4-1 4-1

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CO 3 •7-1 3 •H o << -<

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1 l 1 ^< 1 •H T-l

CO u c << 4-1 r< CO CD

<< 3 r< •H CN 3 CN CS CI

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to CN •< CN 3 CJ *< a 1

o a C c: a a1°

1

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3 i-i

CD

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oo a.

31

c:00i

i i

0000

1

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1 u '

/—

s

! 4-1 1

^ f~*. a

t-I c•HCD

1 u '

CO •-N •H ac: r< CD •w CN•^ CN a CN a CN c 3 4-1 4-1

a 3 •H 3 n^ 3 •H ^"N >*"s

•H CO CD CD 1r< <.

CO 1 cs 1 O 1 r< c £1 CJ a -< T-l T-l

4J << CD << 1 << •H CN CD '/I

3 CN O CN 4-1 CM CD C C3 aC c o C 3 C! c: •H ••w' *—*

T-l H 1 T-t CO T-l i CD CD cCD CO 4-1 CO o CD 4-1 CN O H

CN 3 CN. °

1

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CD

1

31

CO

c: 00i cs

oo001

001

i

4-1 4-1

3 3c CD

o •HCD

-i| 3

o oCJ

o o

4-1 4-1

3 3C CO

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-<|3

4-1

3CO

4-1

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CD

31

o o

4J 4-1

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31

h©i

30

The discrete formulation of the continuous matrix S is

'k+l

Ak

= eAt

dX B where A = At - t (34)

Hence

{-h- + (x. - x1

. ) } (1 - cosojAtto 1 a 1

{-% + (x, - x1,) } (1 - coscoAt

to z a/

{-— + (i»(x. - x1

, ) } sincoAt1 al

{-

CO

io(x„ - x „ ) } sincoAt2 a2

3

3

(35)

The noise term is W ~ N(0,Q). The variances are estimated

to te .31 NM for relative position, 1.0 (NM/Hr) for relative

velocity, and .331 radians for inertial misalignments. As a

result, the following constant diagonal matrix is used for

Q:

Q =

.31

.31

1.0

1.3

.031

.301

36

The same measurement information is available as before

31

in the larger system, namely interferometer bearings and

doppler velocity and drift angle. These introduce non-

linearities and must he modeled as in Equation (10).

\ = h<V + vk

A Taylor series expansion as in Equation (11) about the

latest estimate of sonobuoy position provides a method of

linearization. As a result

K =3 Ax.

3 z

3 Ax.

3 z

3 Av.

*-*

3 z

3 Av,

3 z

3^3 z

3 i>,

(3?)

-i

To determine z and its partial derivatives, recall that

z = cosi = AX • R# (35

where AX is the unit vector representation of a line from

the aircraft to the sonobuoy and RB is the unit vector

describing the base line between the antenna pairs of the

in terf erome ter.

z . = cos 9

v--x

Ax.

AxRB

Ax,

AxRB

Ax.—- RB„Ax

Ax RB. + Ax„RB + h RB

(AxL

2+ Ax

2

2+ \})

h

where h is the aircraft altitude

(39)

32

inhere

" 9zl

3zi

La Ax. 3 Ax2

2 (40)

A-t a«x

a a.

3 Ax,•(z

A-K

X = X

(Ax.+Ax.+h )

1

(Ax.+Ax +h )

HB r

R32~

Ax/AXjRB, +A^IB2+HRBP

(AxJ + AxN

^ + h2

)

Axz(AxRB

;

+AxRBz+hRBp

(AXj + Ax2

+ h2

)

(41)

(42)

The doppler velocity, V,, is a measurement of aircraftd

speed relative to the ocean surface and doppler drift angle,

3, is a measure of the angle between aircraft track and

aircraft heading. These measurements can he related to the

state vector by assuming the sonobuoys drift at the same

velocity as the ocean surface. If this is true then

z2

- Vd

= (Av2

+ Av 2)

% (43)

Expanding in a Taylor series yields

z2

= (AvA2

+ AS 2)

1* (44

H -2

where

a z.

3 Av.

3 2,

3 Av.

x-x

(45

a z.

a Av.

a z.

a Av,

A-*

A* A

AAv

, . a 2 . a 2 , ^(Avj + Av2

)

Av

(AC2+ Av

2)

15

i^6;

(47:

33

Similarly,

z - B = Track - Heading3

-i/Av„\= tan - H

vAv

or since the aircraft heading, H , is deterministic

(46)

-i/Avz = H + g = tan f

3 e\Av

1'

2

a z.

3 Av.

3 z

3 Av.

x=* A^X

where

3z3

3 Av

3z3

3 Av2

Finally,

-Av,

jr-x

AVj + Av2

AAv.

Av + Avx=x

2

(49;

(50)

(51)

(52)

Z =

CCS0

V

H - 8e

H =

3 (cos 6) 3 (cos e)

x, x„

2

3(vrf) 3 (v.j )

3Av! 3Av29 0fe+8) 3(H tf +8)

3Av, 3Av„

(54)

3^

R = .0007 rad

R = 1.0 (NM/Hr)

R3

= .01 deg

The measurement noise, V,, is described by N(0,R). It

is assumed that the measurement noise R and the plant noise

Q are white, gaussian, and uncorrelated . Based on

information obtained from NACC, R was assigned the following

values :

( interferometer)

(doppler velocity) (55)

(floppier drift angle)

It should be pointed out that the doppler measurements

are not always available. The doppler system freezes the

last value of groundspeed and drift angle anytime the

doppler radar is not receiving good information. This

happens whenever the aircraft is above 50<30' altitude, or

can happen when flying above a cloud cover or ^hen the sea

surface is too smooth. As a result, the doppler could be

inaccurate during a significant portion of the flight.

The six-state system was programmed and tested for

several conditions. Flow charts for significant portions of

the program and the entire program listing are presented in

Appendix A. The main program has two purposes: first, to

control the simulation; and second, to read and prepare

measurement data for the subroutine FILTER. This subroutine

performs all the Salman filter computations as specified in

the above equations. The Carlson square root technique is

used to ensure a positive definite covariance matrix. Also,

in order to avoid inverting a matriz in the calculations for

the Kalman gain, the three measurements are processed one at

35

a time. The results of the simulation are presented in

section IV. 3.

36

D. THE TWO-STATE SYSTEM

An attempt was made earlier on in this research tc model

sonobuoy motion as

X, = X + U. Atk-H k k

U, = U,k+l k

(56

with a state vector made up of x , x , u ., and u . This

system proved to be unobservable although the simple

approach was appealing. Since sonobuoy drift rates are

generally slow, another attempt was made to model the system

without velocity. The equations reduced to

X, = X, + w,k+l k k

(57!

With only x and z as states, this system was observable.

3y describing large statistical values of system noise there

i*ould hopefully be enough freedom in the "update" tc

compensate for any sonobuoy drifts. Unfortunately, this

approach was not completely successful! hovever, it vas

observed that the general direction of sonobuoy drift kas

correct. As a result of these investigations a technique of

cascading Kalman filters was used to solve the tracking

problem. The intent was to use results from one filter as

measurement for a second filter. Then, the result cf the

37

second filter could be used as a deterministic control input

to better propagate the first filter.

Ti*o Kalman filters are used in this approach. Figure 5

shows how they are related (or cascaded). The first filter

models soncbuoy position and is based on the discrete state

equation

X, = X + U, At + W,k k k k

(58)

The state vector consists of the sonobuoy positions, Xj and

X2 • U is a deterministic input accounting for soncbuoy

drift and W is system noise. The interferometerfcC

measurement, cosQ, is used to update this filter.

z cose

1

x k position

fil ter

^ .

/ i

XK-i

^delay

i' 1 ' X k

Xk ~ Xk-,

At

zu

u k drift u k

fil ter

Figure 5. Cascaded Kalman filters

38

The second filter models the sonobuoy drift with the state

equat ion

k+l k(59)

As the sonobuoy's position changes with each update of the

position filter, the velocity required to move from X to X

in the time interval At can be calculated. This value is

used as a measurement for the drift filter. All sonotuoys

contribute to this filter which produces one overall

estimate of drift for the sonobuoy field. This value is

then used as an input to the position filter when the next

interferometer measurement occurs.

The technique is understandably sensitive tc the output

of the drift filter. If the estimated drift is too much in

error, the estimated positions will be affected. Divergence

could result. 7or this reason, a simplified Salman filter

is used to estimate the drift. It is of the form

U, » U +\ { \ ' \ ed >

The gain is precomputed from l/(n+l) where n is the total

number of measurements and is limited to a value at least as

great as £t/(S4 min.). This limit is reached after the

first S4 minutes of tracKin- time. From that point en the

gain remains constant effectively averaging the drift over

one period of the Schuler cycle. It must be understood that

39

the drift this filter estimates consists net only of

sonobuoy drift but aircraft drift as well and is, in fact,

the vector sum of these. An estimate which attempts to

follow the changing velocities through the Schuler cycle

tends to cause the position filters to diverge. This might

be the case if larger values of gain are used, averaging

only the last few measurements of drift. Therefore, many

measurements of drift are averaged which provides an

estimate for linear drift only. The lower limit of gain

prevents new drift measurements from being ignored and also

provides for flexibility in estimating the drift.

Unlike the six-state system, the interferometer

measurement is the only measurement required by the position

filter. The measurement equation is essentially the same as

in the preceeding section where z is determined, as shown in

Equation (39). Since there is only one term in z, the H

matrix is a row vector consisting of

n = 3 z

3x3 z

3 x.(61)

where the terms are given in Equations (41) and (42). It is

necessary to determine the ai rcraf t-to-sonobuoy range for

these values from

AX = X - X (62,

A

where X is the estimated soncbuoy position. X1 represents

40

the aircraft position and is considered deterministic.

Undoubtedly, X is in error due to aircraft navigational

drifts and therefore causes errors in the estimated

geographical position of the sonobuoy. But, since the

measurement of bearing is based on relative positions, the

estimate is relatively correct.

An observability analysis on the 2-state position filter

was performed again using

N

lik=l

kEk *'\ ^k (63)

Subtitution into this formula with N=2 yields

M =

f 3Z

13 x

I dz

l3x,

/3 2

3x,

f 3 Z I Id z

\dx, /.\3x

Idz

\dx.

v 3x.t+i

'3z_

<3x,

/3z \ /a z

V3x

i"i^3x

21+)

i+A

3z,3x .

I+l

'3 2

13 x2A+I

(64)

Expanding

,

M ='32

\3xlA+l

/ 3z

Ux.

'3z (3z '3 2

^ xU+ iV9x

2A+iV3

v

xiA

' 3 2

i3x.

'3 2

\3x,

'd Z

\3x2/

32f3z

V3x

li+lV3x

2i + l

(65)

*H

This matrix is positive definite for all except the case

when

3x1/

3z3x and

14+1

3z3x.

3z3x,

21+1

This occurs whenever the relative bearing from the aircraft

to the sonobuoy is not changing (i.e. when flying directly

toward or away from the sonobuoy). This is intuitively

correct since two or more bearings must cross in order to

determine a position.

The two-state system was programmed and tested to

determine the usefulness of this simplified filter. The

majority of the program is the same as the one used for the

six-state system. The significant differences occur in

subroutine FILTER. The Carlson square root technique is

used again but only one measurement instead of three is

processed. The drift filter is programmed in this

subroutine along with the position filter. A flow chart and

program listing are ^resected in Appendix A.

^2

IV. ANAYLSIb

A. THE SIMULATION

Actual data from the Orion was not easily obtainable for

this research. Consequently, a computer program had to be

written to generate the information required by these

algorithms. A flow chart in Appendix 3 describes the

program. An aircraft track was created by alternating lines

and curves of various lengths and then a determination of

noise free measurements was made as the track was flown.

(Measurement noise was added later during the simulations.)

Sonotuoys i*ere allowed to drift at a constant velocity, and

the aircraft's navigational drift was modeled with a

constant velocity and Schuler cycle variations as

X1

= X + K At + A sinoota a x x

(66)

inhere

X = aircraft inertial position

X = aircraft true positiona r

ui = Schuler frequency -^/g/R

'

The constant drift rate, K , was found by NAOC to have d

mean and a standard deviation of 2.5 NM/HH. Likewise, the

amplitude of the Schuler cycle, A , had 2 mean and a

standard deviation of .5 NM. The effect of wind was also

available in the program but was never included for analysis

^3

It v.as obvious that both algorithms vere extremely

sensitive to the aircraft flight path. Since there is no

"typical" flight path for an Orion during its on-station

period, several uniform patterns were selected which would

provide meaningful information on eacn algorithm's

performance. These base parameters were chosen:

airspeed 180 kts

altitude 3000 ft

range 15 NM ai rcraf t-to-s onobuoy

frequency 20 sec between measurements on one sonobuoy

A circular pattern was flown around a sonobuoy at a range of

about 15 hiv

. Initial sonobuoy placement was at (1,1) with

the aircraft flying clockwise starting at (15,0). This is

shown in Figure 6 which depicts a portion of the aircraft's

true track. Sonobuoy drift was west at 1 N'M/HRj aircraft

navigational drift was south at 1 NM/HB and included a

Schuler cycle (amplitude .5 NM) when specified. (This drift

would be indicated in Figures 12 thru 30 by the following

notation: DR=(a/c 150-1 + Schuler, b 270-1) .) The sonobuoy

remained near the center of the pattern allowing the range

to remain relatively constant. This was desirable since

this basic flight path was used to test the response of the

algorithms to variations in altitude and rar.-e. Similarly,

a square pattern >as flown counter-clockwise around a

(x, ,Xj ) measured in nautical miles from an arbitraryorigin near the aircraft's starting point.

44

sonotuoy at a range of 15 MM. Ir. this case the aircraft

began on-top the initial position of the sonobuoy at (-3,-3)

and preceeded to fly the track shov.n in Figure 7. The

sonotuoy was allowed to drift to the east at 2 NM/HR and the

aircraft's navigational system to the north at 2.5 NM/HR.

This pattern was used eventually to analyze performance at

different measuring frequencies. Figure 8 shows the output

of the aircraft's navigational system when it has a

northerly drift of 2.5 NM/HR while the aircraft is flying

the square pattern. Figure 9 shows this same pattern when a

Schuler cycle with an amplitude of .5 NM is also present.

45

-

' '

i

. .1 ..,.j_,-i~ ! j i1 ..

: L•

i

. ! 4 - ..M . |._

!

| j|

ij i 1 1

i 1MJ .: :

: M1 I : 1 1 ! . L M !

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1 i —1

;

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|| j ! M M '

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1

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i ii

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: i _| I1 1 1

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III i i :i

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i i i i j| \

; i

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laxt^'x 5^»5> •: ; i ;

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1 '! j

! ! j?9" ~x ; i | ; j I i j

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.

„J_ | | j| | | j |

1 Ml , ,x*r

j i

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r M i M M 1

:

i X jf

!*'!, .

i; X

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1

' i ! -? i

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Xi

j !

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1 !1

X i\

rat

X? !

!

X

1'' X : !

;i ;

)

x1

t ;:

11

ii

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i,

1

'

|! f j i

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". , X

i

! ;

X .i Mi. . Jii

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X XKJ '

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xj \

| |I j i |

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i' ||

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: j j' 1

'1

; ji i 1

X JV- x -x-xf 8 < k y»x x

i!

|1 | i 1 |

| {i

! ; !

ill!1 1 1 ! | {

! 1 1|

! j |!

11 ; j j |1 j

ii

M 1 i 1 | S ! ! 1 ! M i

MM! ji j

| |i M M M ;

i

i i . •i

i I I l i it j I ! [Mi r ;

nmrii1 MM! I | ! i

Mil j

: ! i ; ;'

!i

; ;! M 1

i !

;:

Ij j

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-\ -:-1—

i j; ;

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MM | ! i > Mi i

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i i . - J 1 ; ! i '

-20. 10. 10. 20.

E-W POSITION (NM)

Figure 6. Aircraft track for the 15 NM circular patternwith initial sonobuoy location and direction of drift indicated

46

30.

20.

10.

COoQ_

eni

10.

-20.

-30.

!1

illi ! i

1

:

t„.L.

1

ii ! !

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; 1 1

:

- !| | i |

i H i ! ! !i 1

j1

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1 j j ! j'

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1[ ! s T ; ! i j 1 j '

j i j 1 1 ! |j ;

i i

1—;—i 1 1 1 11

i

— j . .1

I ..!. .. I. ..'. . i i

-20. 10. 10. 20.

E-W POSITION (NMJ

Figure 7. Aircraft track for the IS NM square patternwith initial sonobuoy location and direction of drift indicated.

ij-7

30.

20.

:o.

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-20.

!

i— j a j-

fflfO?

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!? & §9

g o g?oj'I g r§ § Sf'3-8

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-30.-20. -10. 0. 10. 20.

E-W POSITION (NM)

Figure 8. Aircraft's navigational output for the 15 NMsquare; navigational drift is 2.5 NM/Hr to the north.

^8

30.

20.

10.

! :

i

! !

d°b o d olb o d"o o e?a o e?'c

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-20.

d'Td o 'd'fc o d 135 a d 20

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j i_

-20. -10. 0. 10.

E-W POSITION (NM)

20.

Figure 9. Aircraft's navigational output for the 15 NM square;navigational drift is 2.5 NM/Hr north with a Schuler cycle.

^9

These patterns were used to generate the data vhich

included sonotuoy number, time of measurement and

measurement, aircraft latitude and longitude, altitude,

heading, pitch angle, roll angle, N-S acceleration, E-W

acceleration, and the antenna used for the measurement.

Also determined, tut used only for error analysis, were

sonobuoy position and true aircraft position. This

information was computed for the entire simulation period

and stored in a file to be used when needed.

Initial estimates of the state and the square root of

the covariance matrices were obtained as follows:

Six-state

X(0) =

x,(i3)-i (0)

zi(0)-2

ai(0) + .5

-1L cosK

-V, sinHS (0)D

.

.2025

.0025

(6?

Two-state

X(0) =*l(0)

x2

( ) + . 5

5D(0) = r Alt

Alt(68)

where S (0) indicates the diagonal elements of 5(0D

50

For the circular pattern

xl(d) = 1.

x2(0) = 1.

x (0) = 15.a

l

X (0) =0.a2

and for the square pattern

xL(0) = -3.

x2(0) = -3.

x (0) = -3,a

l

x (0) = -3.a 2

The initial values of the covariance were chosen so as to

describe the errors in the state associated with dropping a

sonobuoy from an aircraft. The two-state system operated

tetter vith a lower initial variance than did the six-state

system .

Simulations were run for a nominal period of two hours.

Each unit of data was sequentially read into the simulation

program from the storage file. Measurement noise was added

with the following normal distributions:

ccsQ N(0,.0007) in radians

doppler velocity N( 0,1.0) in NM/EH

doppler drift angle N(0,.01) in degrees

These values agree with the ones chosen for H as described

previously. A random number generator using an initial seed

was used to create the noise from the proper distributions.

After the estimate was made, errors were measured ard

manipulated in subroutine RESULT. This cycle was repeated

51

until the simulation was complete. (Refer again to Appendix

A for a flow diagram of the simulation process.)

Information vas gathered and is presented primarily in

two ways. First, for both the square and circular patterns

the results of the first run of the simulation are plotted.

(Figure 13 is an example.) The top portion of the results

shows the estimated positions of the sonobuoy on the

aircraft's navigational plot. It should be noted that these

are not true positions since inertial errors may be causing

this plot to drift. The bottom portion shows the

North-South and East-West errors in tne estimated positions.

These errors are the differences between the true and the

estimated relative positions of the sonobuoy. They are

depicted in Figure 10 and were computed from

AX, = AX, - AX,k k k

AX, = (X - X1

),k a k(X - X ),

a k

(69)

(70)

where AX, is the relative error at time k. The true aircraftk

/

N

[

N

I

/ AX

AX^

/ /

/ / A

// AX

y aX

//

\/

^ **

yX1

a

a^ Aircraft navigat ional 'clot

EE arth fixed coordinate system

Figure 10. The effect of the aircraft navigational plot

52

position is denoted by X and the true geographical position

of the sonobuoy is denoted by X. These values are known

from the simulated data. X is the filter's estimate of

position and X is a deterministic input from the aircraft'scL

navigational system. The errors indicated in these plots

are different from those perceived in the upper plot

whenever the aircraft's navigational picture is drifting.

The positive and negative values of the square root of the

covariance are also shown on these plots as solid lines.

Specifically,

N-S error = ±-/P'

E-W error = ± -/?Y 22

(Only jjosition errors are analyzed in this report.)

The second way in which the information was gathered

consisted of a shortened Monte-Carlo simulation. 7or each

scenerio under study, 20 two-hour simulations were run, each

with new values of measurement noise provided by the random

number generator from their respective distributions. The

objective was to compare the RMS errors predicted by the

covariance matrix in the filters to the actual RMS errors

observed in the simulations. Three RMS statistics v.ere

collected and plotted versus time as follows:

1. The square root of the covariance computed by the

Kalman filter is represented by a solid line on the

clots. It was commuted from

-LVp +lYp , n=20,n^ 11 n <—' 22

k = 1,2, (71

53

This standard deviation is the filter's estimate of

its accuracy.

2. The mean and standard deviation of the error in the

estimated sonobuoy position relative to the aircraft

was determined by

y = -X A^ki'n = 20, k = 1,2,3... (72)

ni=l

a2

= -V^AX 2 - p.2

, n = 20, k = 1,2,3. ..x. nZ_i ki kk 1=1

This standard deviation is represented by an 'X' on

the plot and is a measure of the variability in the

filter's estimated .position.

3. The RMS value of the actual relative error was also

also determined. It was computed as

Li

n -^—

i

i=lki

n = 20, k = 1,2,3. (74

and is represented by an '0' on the plot. This value

is a measure of the error which occurred between the

actual relative position of the sonobuoy and the

estimated one. (See Figure 11.)

These values are plotted over the two hour simulation period

(k = 1,2,3...) and provide a measure of the accuracy

obtained by each of the algorithms.

5k

-S 36-

Gx—*~

XXX XXXXXX 3-3—X—8 5 X-

-ax

x - X (X - x-

Fiture 11

a i a 1

RMS statistics at time = k.

55

B. THE RESULTS

Orincon Corporation performed much of their analysis

using the small circular pattern shovn in Figure 12. The

aircraft sped around this track at 36£ knots taking a

measurement every 20 seconds. Altitude was constant at 3000

feet. The initial position cf the sonobuoy was at (2,2) and

it drifted on a heading of 345 degrees at 5.5 N'M/HH while

the aircraft's navigational system drifted in the opposite

direction at 5.2 NM/HR. A Schuler cycle was alsc

superimposed on this drift.

The initial estimates of sonotuoy position and variances

used by the algorithms were different for this particular

pattern ir. order to coincide with the Orincon simulation as

much as possible. They were

Two-s tate

M) = sD(0) =

2.](74.

Six-state

-5.4 2

.

2 .

X<3) --Va cosH

-Va sinHSD (0)

-

5.

. 0025

.0025

(75)

56

> E

CIRCLEa = ~H m/s =-0.4 £

YA

= 200 m/s

Vfi

= 2.8 M/S

Figure 12. Orincon's simulation flight path.

57

Initially, both the six-state and the two-state

algorithms were tested using this flight path and

comparisons >ere made with the results cf Orincon's 13-state

system. It was discovered that the six-state system's

results did not change as a consequence of vhether or not a

Schuler cycle existed in the navigational system.

Therefore, the six-state system was run only with the

Schuler cycle active. The two-state system was run with and

without a Schuler cycle.

The results of the first run of the simulation are shown

in Figures 13, 14, and 15. The sawtooth shape of the

covariance (solid lines) is the result of the circular

pattern and is caused in two ways. Primarily, since the

soncbuoy remains closer to one side of the circle, there is

a minimum value of the variance each time the aircraft

passes on this side. A second reason for a change in

variance is the aircraft's location in the pattern.

However, these modulations are not as apparent since the

range variation dominates. RMS errors were obtained by

repeating these runs 20 times and are shown in Figures 16

thru 18. The fluctuations generated by the flight path are

apparent, more so in the six-state system than in the

two-state system. Also, steady state values cf the error

gradually increase. This is true because the average range

to the so no buoy increases as the buoy moves farther away

58

from the center of the pattern. (It will be demonstrated

later that the errors are range dependent.) The covariance

of the six-state system increases from 400 yards at 25

minutes to 800 yards at the end of the simulation. The

two-state system increases slowly from 250 yards until it is

hammered by the Schuler cycle. It is worthy of note that

vithout the navigational errors stemming from the Schuler

cycle the RMS errors of the two-state system are

considerably lover than those of the six-state system.

Table 1 compares the RMS errors of these two systems to

the results obtained by Orincon for their 13-state system.

(The values for the two-state and six-state systems are

taken from the covariance at a point 35 minutes into the

simulation.) The first line considers measurements of

bearing, doppler velocity, and drift angle. Navigational

errors are present including a Schuler cycle. As expected

there is an increase in relative position error because of

the reduction from 13 to 6 states. The next line considers

the effect of the Jlooal Positioning System (3-PS), with very

accurate aircraft positioning information, on the relative

13-state 6-sta te 2-s tate

Schuler

SRS-dv*DA394 453

No Schuler

SHS+DV+DA+GPS450 200

Table 1. Comparison cf RMS results in yards.

59

position errors of the sonotuoy. In the 13-state system

Orincon observed that the errors in the states pertaining to

the aircraft inertials improved markedly; relative position

errors most likely remained, the same. This is exactly vhat

was found in the analysis of the six-state system: that the

relative position errors were unaffected by navigational

errors. However, the two-state system is very much affected

by navigational errors. Relative position errors were

significantly less for the two-state system when accurate

aircraft positions were known.

To obtain a more realistic analysis of their performance,

each algorithm was tested using the circular and square

flight patterns of Figures 6 and 7. The results of the

first simulation run for the circular path at a range cf 15

NM are shown in Figures 19 thru 21. In this case, the v.avy

nature of the covariance is due to the location of the

aircraft in the pattern since the range to the sonotuoy

remains relatively constant. For instance, as the aircraft

passes directly north or south of the sonobuoy the ability

to correct E-W errors is greatest. Therefore, the E-W

variance reaches a minimum value at this time. The N-S

variance is affected in a similar manner. The two-state

filter is not as sensitive to this as the six-state filter.

Also, the effect of the Schuler cycle on the two-state

filter can be seen in Figure 20.

The RMS errors are shown in Figures 22 thru 24. The

steady state RMS value of error predicted by the filter is

60

425 yards for the two-state and 750 yards for the six-state

system. The actual RMS error in the relative position of

the sonobuoy is 250 to 1200 yards for the two-state and 400

yards for the six-state system. Without Schuier cycle error

the two-state system drops to a steady 250 yards, better

than the six-state system. It is interesting to note that

the measured errors are significantly less than those

predicted ty the filter. Both measured values of RMS error,

a and a , are in close agreement; that is, the deviation ofx a

the filter's estimated position about its mean is generally

the same as the deviation of the estimated position about

the actual relative position of the sonobuoy. The closer

these values are to one another, the more confidence can te

placed in the analysis. The only exception to this is

Figure 23 which shows the reaction of the two-state filter

to the Schuier cycle. One complete cycle with a period of

84 minutes is obvious. The Kalman filter does not recognize

this cycle since the modeling equations do not account for

it. The dip in the center of the two peaks is once again

the result of flight path geometry. The aircraft is in such

a position relative to the sonobuoy that the measurements

provide enough information to correct for Schuier cycle

errors. However, it is not in this position long enough to

influence the errors anymore than it does.

The same analysis was performed using the square

pattern of Figure 7. This scenerio has mere drift than the

other and in a different direction providing the algorithms

61

with another motion to track. The outcome was generally the

same as can be seen in Figures 25 thru 30. There is no

measurable difference in the results from the tvo different

flight paths. Flying a straight path does not adversely

affect the results as was concluded by the Oricon

Corporation for their 13-state system. The wavy nature of

the covariance is again a result of the flight path. It is

not as smooth as before because a square pattern was flown

instead of a circular one. Another look at errors caused in

the two-state system by the Schuler cycle can be seen in

Figure 30. The algorithm does make some corrections to

these errors but they are not as effective as before. In

this case the time the aircraft was in a position to ma^ce

the corrections aid not coincide v,ith the time the peak

errors occurred. There appears to be no way to predict the

optimum time and place for the aircaft to te without knov.ing

when and hew the Schuler cycle is occurring.

The distance the aircraft is from the sonotuoy is

directly related to the accuracy the algorithms can achieve.

Simulations were performed at ranges of 5, 15, 33, and 45

nautical miles using a circular flight path with the

sonobuoy in the center. (The RMS plots can be fcund in

Appendix G.) The steady state errors were observed and are

plotted in Figure 31. Solid lines represent the RMS values

of the covariance and dashed lines represent the actual

errors. In all cases the errors increase with distance

>.hich is intuitively satisfying. Mathematically, the

62

covariance increases because the H matrix becomes smaller

with an increase in range. Consequently, the updated value

of the covariance, P(+), is larger from Equation (19). For

the six-state system there is an increase in actual error

from 240 yards at 5 NM to 750 yards at 45 NM . And for the

two-state system, actual errors increase from 13*3 yards at 5

NM to 500 yards at 45 NM. (Note that for the Uo-state

system, Schuler cycle errors, indicated by circles, peak

approximately 1300 yards above these plotted values! the

lower values, indicated by x's, were used since they were

more reliable for comparison between ranges.) There is a

slight decrease in the slopes of all the error curves as

distance increases. Hovever, this decrease is small and the

curves might well have been interpeted as linear within the

limits of the analysis.

The frequency at which measurements are made also

affects the accuracy of the estimate. Each algorithm \>as

tested with measurement intervals of 4, 10, 20, and 30

seconds on one sonobuoy. The outcomes of the steady state

RMS errors are plotted in Figure 32. (Once again the

two-state system's errors do not show the effect of the

Schuler cycle.) The actual errors in the six-state system

increase steadily as the measurement interval increases.

However, the covariance decreases rapidly at first to a

minimum value somewhere around 13 seconds and then increases

with increasing interval. An abbreviated run with a two

second measurement interval confirmed these results.

63

Propagation effect

Observability effect

Figure 33. Factors which influence the covariance.

Figure 33 helps to explain this outcome. There are tv.o

conditions which affect the covariance. The most obvious is

the increase in covariance due to an increase in the

propagation interval. This occurs because the plant matrix,

vhich is a function of time, affects the covariance in

Equation (16). The second condition is a consequence of the

model's observability. If the relative bearing betv-een the

aircraft and the scnobuoy does not change then the state is

not observable. In other words, two successive bearings

must intersect to determine an estimated position. By

allowing the interval between measurements to become too

small, the aircraft is unable to make a significant change

relative to the sonobuoy. The covariance increases as the

conditions approach those which maKe the problem

unobservable . There appears to be an optimum frequency with

which to make measurements en one buoy, namely 10 seconds

for this range of 15 NM and speed of 180 knots.

6^+

The two-state system exhibited similar characteristics

in Figure 32. In this case the covariance never does

increase in the range of intervals studied. The actual

errors remain constant. Since the plant matrix fcr this

system is equal to the identity matrix and the Q matrix is

prescribed to be constant, the propagation interval, &t,

does not effect the covariance. However, the effects of the

observability conditions do cause the covariance to increase

vhen At becomes too small. It is believed that as At

becomes smaller the actual errors would begin to increase

also .

The statistical plots for the range tests found in

Appendix B show that the Schuler cycle causes larger peaks

in the actual error as the range from the sonobuoy

increases. However, the peaks are decreasing with time.

Further simulation revealed that the peak errors decrease to

a steady value about 10J0 yards greater than non-Schuler

cycle errors. This coincides with the amplitude of the

Schuler cycle as it was programmed for this particular

flight path. Therefore, in the steady state a Schuler cycle

may cause additional estimation errors approximately equal

to its amplitude.

Altitude was also tested for any effect it might have en

these algorithms. Using the circular pattern at a range of

15 MM, altitudes of 300, 3000, 10,000, and 20,000 feet were

tested. Neither the six-state nor the two-state system

shoved any effect for these changes in altitude. It is

65

believed that at very small ranges (less than two nautical

miles) altitude might hamper good position estimation.

However, for the majority of the time altitude is of no

concern .

66

+

** V

* dfflt

-J. 0.

E-W POSITION iNM)

U V wi

w #-*J.-v .- <* V *

TIME (MINUTES)

TIME (MINUTESI

Figure 13. Est. position and relative errors for the six-state system usingOrincon's pattern. DR=(a/c 225-5.2 + Schuler, b 045-5.5), At=20s, Alt=3000'.

67

^

-2. 0.

E-W POSITION tNMI

TIME (MINUTES!

TIME (MINUTES!

Figure 14. Est. position and relative errors for the two-state system usingOrincon's pattern. DR=(a/c 225-5.2 + Schuler, b 045-5.5), At=20s, Alt=3000'.

68

w<1

.

'

E-W POSITION INM!

^y^^^^^S^Zrs^'

T IME (MINUTES)

TIME :M!NU T ES)

Figure 15. Est. position and relative errors for the two-state system usingOrincon's pattern. DR=(a/c 225-5.2 + No Sen., b 045-5.5), At=20s, Alt=3000'.

69

u -

CO mi—

1

c3 CM

II

u 4J•H <t

~

01 ^SX! u"i

u •

LOM I

CJ m•H <r>%o

r-l

1+H j2

o> „

l~4 M•H 0)~* -—

<

3 3

£ U01 CO4J

CO +>igo ~l

u un4-1 i

CO uO4J CMen CN

X u•H ^^0) CO

*—

'

01 II

J3 UtJJ a1-4

•

IW a1-4

en•-j uM 31-4

HI U3

3 i-l

U•H o4J

1-1 >191 -a

a. T3i)

rn a.^pd —

i

i>

>• 01 .

vO T3 -i

—

o3 C

0) M Ou 0) m3 u

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M 4-1 4-1

H CO r-4

In a <

CSQJL) yoya3 NQinsod sny

70

enjj

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1-1 ••

cfl 'Si

H o3 OJU II

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•H CU

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0) mu 1

tfl LTl

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oU-| Q.

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C/3 Q.S oPi H

tu

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i^~ T3 -<—

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u l-i o1-4 0) CO3 4-1 II

04 4-1 4-1

•rt m .-n

Pn a. <;

tsaji) youy3 NOinsod swy

71

2:o

co

1_o

Q_+

Q_

CC —

'

LU COJ— O_J Q_

QCoCO

ex

txTD O"51 CC

CCLU

"O O"CO COcr cc

i i

a

o-on

X

$>

^6

T2^

a o

ucO

H •»

3 en

U ou CN(

•H II

U 4-1

<j0)

J3 *u00 •

c m1-1 i

>* lOH ~j14-1 o0) jo^H•H «

x: •

3u

e woi

u o'Si z>>00 CN

on V lO

UJ a 1

U CN^•J] CNz1

1—'J

s: 3 —

^

U CO

LU M II

z: fi Dei

Q

•

M-l ft

WU uou eMcu

ca —1

M—i Oi-l

—

1

5*>

[fl X3

O.T3OJ

-VI ft

§3H• OJ •

oo TD -FH O

c oCD l-i ou cu en3 U II

0C -J 4-1

•H CO iHfc. CL <

;qju yosdU3 Nouisod swy

72

+

!

* 6

% «&

I1

I

E-W POSITION (NM)

TIME IMINUTES)

[ME (MINUTES)

Figure 19. Est. position and relative errors for the six-state system usingcircular pattern at 15NM. DR=(a/c 180-1 + Schuler, b 270-1), At=20s, Alt=3000'

73

,__i*

1""""*

w

E-W POSITION (NM)

TIME (MINUTE:

TIME (MINUTES!

Figure 20. Est., position and relative errors for the two-state system usingcircular pattern at 15NM. DR=(a/c 180-1 + Schuler, b 270-1), At=20s, Alt=3000'

7^

1

r

<

i

E-W POSITIGN INM)

\ '1

1

|

_i^_-X ' /

^

*X

/^^—***** ^—-^^ s?^-—<^ ^j-—-^_ " ^s-— ^ -y

- / \ *?

TIME (I1INUTE5I

TIME (MINUTES!

Figure 21. Est. position and relative errors for the two-state system usingcircular pattern at 15NM. DR=(a/c 180-1 + No Sen., b 270-1), At=20s, Alt=3000'

75

CL+

azC3CO

ccLU

Q

O

CO zo o

CC —

<

LU COI— o_J Q_

CL

cr

cr

ZD

cr cr

X

J en v

a; «_e

qr„ ox

^ "*

mr^3

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E-W POSITION INMJ

TIME (MINUTES!

TIME 'MINUTE5I

Figure 25. Est. position and relative errors for the six-state system usingsquare pattern at 15NM. DR=(a/c 000-2.5 + Schuler, b 090-2), At=20s, Alt=3000'

79

t*

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80

/J

(i

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Figure 27. Est. position and relative errors for the two-state system using

square pattern at 15NM. DR=(a/c 000-2.5 + No Sch. , b 090-2), At=20s, Alt=3000

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81

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86

V. CONCULSION

The following are the conculsions reached at the end of

this research:

1. The six-state algorithm developed in this report

produced estimates of relative sonobuoy position with a

standard deviation in the error of 500 yards at a range

of 20 NM. Aircraft drift, which included the Schuler

cycle, had no effect on this system since the states

were defined relative to the aircraft.

2. The two-state algorithm developed in this report

produced estimates of relative sonobuoy position with a

standard deviation of 300 yards plus an amount equal to

the amplitude of the Schuler cycle error at a range of

20 NM . When nc Schuler cycle navigational errors were

present, the two-state system cut-performed the

six-state system.

3. Both algorithms shoved an increase in steady state RMS

error with an increase in range.

4. The six-state system showed an increase in actual

steady state RMS error for an increase in the time

interval between measurements? however, the covariance

of this system indicated minimum error occurred with a

10 second interval on one sonobuoy at a range of 15 NM

and an aircraft speed of 180 lets. (This translates to a

lh degree change in relative bearing.) The two-state

87

system shewed no change in the actual steady state RMS

error for intervals between 4 and 33 seconds and only a

slight increase in these errors as a 60 second interval

was approached. The covariance of the system actually

decreased with increasing interval.

5. Neither system showed any dependence on altitudes

within the operational limits at a range of 15 MM.

6. No adverse effect was observed while flying straight

tracks as opposed to curved ores.

7. Steady state RMS error was affected by two conditions

as measurement intervals changed. The longer the

interval the more the error tended to increase,

especially for the six-state algorithm, as a result of

system propagation. The shorter the interval the more

the error tended to increase as the systems encroached

upon their non-observability condition. An optimum

measurement interval existed for each system which was a

function of range.

88

VI- SUMMARY

Both the six-state and the tv,o-state algorithms were

atle to successfully track the sonobuoy. However, the

six-state system was the best choice given the navigational

system in use by the Orion. It was capable of providing

estimated sonobuoy positions at a range of 22 NM such that

the errors had a standard deviation of 502 yards. The

computer time required to process one measurement was on the

order of 222 milliseconds. On the other hand, the much

simpler two-state system required only 2S milliseconds or

one seventh (y) the amount of computer time. However, its

estimate of sonobucy position was degraded by Schuler cycle

errors in the navigational system. Without these, this

system was subject to errors having a standard deviation of

322 yards at a range of 20 NM. The Schuler cycle

periodically increased these errors by an amount equal tc

its amplitude.

The algorithms i*ere very much affected by the aircraft's

flight path including the relative location of the sonobucy,

the range to the sonobuoy, and the amount of angular change

in the bearing to the sonobuoy as measurements v,ere taken.

Since the sonobuoy position was to be estimated in

tv» o-dimensions , the aircraft was required tc maneuver in

such a vay so as to provide information in both dimensions.

For example, as the aircraft crossed north or south cf the

89

sonobuoy the E-W errors decreased while the N-S errors

increased. The trend reversed when the aircraft flew east

or vest of the sonobuoy. It \»as also observed that steady

state RMS errors grew with range at a slightly decreasing

rate. They were twice as large at 45 NM as they were at 15

NM. Finally, the sonobuoy position was not observable if

the relative bearing to the sonobuoy did not change implying

that two lines of bearing must cross in order to obtain an

estimated position. If the amount of angular change between

each measurement was too small it began to affect the errors

because this condition was being approached. Anytime the

aircraft flew in such a way as to reduce the rate of bearing

change, or the frequency at which measurements were made >as

too high, this condition was prevalent. It was observed

that at least 1% degrees of change was required to avoid

this problem.

The sii-state system showed no tendency toward

divergence as long as initial conditions were reasonable.

The two-state system, on the other hand, did have a tendency

to diverge when the gains on the drift filter were not

adjusted correctly. For example, if the gain was to high

the estimated drift was heavily dependent on the last fev

measurements and was too quickly affected by changes in the

position filter. But, it was also necessary for the drift

filter to have non-zero gains in the steady state in order

to provide some insurance against an initially wrong

estimate of sonobuoy drift. With these modifications maae

90

the algorithm operated correctly during all tests and, in

fact, had less error in some cases than did the six-state

system.

It is the opinion of the author that either one of these

algorithms is an improvement on the nistorical method of

"mariting-on-top" the sonobuoy. further, it adds a great

deal mere uncertainty as to exactly ho* much accuracy could

be obtained in the old way. Non-linear navigational effects

were not even considered and the number of measurements made

to determine sonobuoy position was on the order of 10 or

less for any one sonobuoy pattern. Nevertheless, in an

operational sense the accuracies achieved here would need to

be improved. Without the non-linear navigational errors the

accuracies could have been much better. Eowever, the

results of this research do provide another perspective for

the problem at hand.

91

APPENDIX ATHE COMPUTER PROGRAMS

MAIN

Simulation setup

Do for the no.of simulations

Read measurement

i

cos & and antennadoppler velocity and drift angle -*

bouy, time, a/c lat . and long,altitude, heading, pitch, rollaccelerometer outputs*

YES

Initialize buoy'sstate vector andcovariance matrix

YES

Process results

Retrieve buoy's-state vectorcovariance matrixtime of measurement

(Stop)

Convert lat/long to NMDetermine antenna baseline vectorAt = present time -last timeZ3 = heading - drift angle*

CALLFILTER

Store new-state vectorcovariance matrixtime of measurement

Process simulation results

Figure Jk- . Flow chart:main program

* Required for six-statesystem only.

92

SUBROUTINE FILTERSIX -3 TATE

Calculateplant matrix ($.

constant matrix A,,.

Propagatestate vector

\ - h-A-i + Kcovariance matrix

3k

aK*ic-i<-i sk-AVVl ]

*

Do 1=1.3

YES

z. - cos a

Calculate

dz = z, - z.

H = Hj

R = R,

no

YES

'2 dCalculate

dz = z~ -

H = H2

R = R,

z3

B He

Calculate

dz = z, -

H = H..

R = R,

Updategain

-T Tf - vv

state vector

A - f ft< k

k kcovariance

\ sk

ci

TP = SS

+ G dz

T Jff ]*

MO

YES Xk " X

k

Sk "

_( Next)

©Figure 35 Flow chart

six-stateSubroutine FILTER

93

SUBROUTINE FILTERTWO -STATE

Save last position &velocity + .

Xk-f Uk-1

Propagatestate vector

Xk - X

k-1+

°k-lAt

covariance

sk

= K-i sk-i

+ Qk-i]i

Calculate

X " Xk

+ Xa

H(l,l) = HtlH(l ,2 )

= H12

Updategain

-T T T t

f = S^H^ C< = fXf + R G = ~ V

state vector

dz = a - z z = cose

x; = x- g dz

covariance

<= 3ktl-i ffT]

P = SST

Driftmeasurement

Zu

= (x£ - X^jJ/at , limit to 20 kts

gain

C = i limi 1- tn ^ rlinimumn + 1 34 min

inhibit first 5 dr ift calculations (C = 0)

n = 1+1

Drift

U* = U* + C (Z - V? ,k k-1 u k-1

Figure 36. Flow chart: SubroutineFILTER two-state

9/4.

GLOSSARY OF COMPUTER VARIABLES

SIMULATION PROGEAM

A Vector of aircraft accelerations

ALPHAI Result of -^—

^

or -fTf + R

ALT Altitude

ANGLE Direction of sonobuoy drift from North

ANT Antenna

AVEL Initial aircraft velocityT

AS Matrix result of I - ^~

B Matrix result of (eSS 1+ Q>

BOUY Sonobuoy RE channel or number

C Gain of drift filter

CA Result Of At/(84 min.)

C^ Constant matrix in six-state system's equations

CKL Rate of change of nautical miles for a change indegrees of longitude

D Array which holds all items requiring storage fromone measurement to the next

CELT Interval between measurements, At

DELX1 Relative position, axx

DELX2 Relative position, ax 2

DELX3 Relative position, ax3

DRIFT Magnitude of sonobuoy drift in yards

DSEED Seed for the random number venerator

DTIMS Array of initial drop times for the sonobuoy

DZ Res idual , dz

SDELX1 Error in the est. relative position of sonobuoy, &Lx l

95

ELELX2 Error in the est. relative position of sonobuoy, ax 2

EOSSO Matrix result of e $TSTS 23

EPSLN Age weighting factor,

ESST Matrix result of e S 5T

EX1 Error in est. geographical position of sonobuoy, a*!

EX2 Error in est. geographical position of sonobuoy, Ax2

F f vector from Carlson square root technique

TFFT Matrix result of ff

Tff

EFTA Matrix result of

FT Transpose of f

3- Gain matrix

GAMMA Aircraft pitch angle, y

GDZ Vector result of G dz

GR Gamma in radians

H Aircraft heading

H The H matrix when used in subroutine FILTER

HOUR Time in hours

HR Heading in radians

HT Transpose of the H matrix

I Index

I Identity matrix when used in subroutine FILTER

IEND Indicator of EOFT

ffIFFTA Matrix result of I - —^-

IN File from which data is read

INRES File from which true scnobuoy positions are read

IOUT Device on which print is made

IOUTF File in which results are written

96

IOUTZ File in which random vectors are inspected

IRUN Index for run number

IZ Index for measurement, z = 1,2,3

K Incremental measure of time

LAT Latitude

LATI The latitude of the origin of the local earth fixedcoordinate system

LONG Longitude

LONSI The longitude of the origin of the local earth fixedcoordinate system

MARK Indicates whether information is to be storedor retrieved

MEAM Mean estimated sonobuoy position, mi

MEAN2 Mean estimated sonobuoy position, u

2

MIN Time in minutes

N Total number of measurements made on all sonobuoys

N Index *hen used in subroutines IN ITAL and DATA

MRUNS Integer equivalent of RUNS

Plant matrix, $

OS Matrix result of $ S

OST Transpose of $ ST

OSSO Matrix result of $TSTS $

OX Matrix result of § X

? Covariance matrix

PHI Aircraft roll angle, f

PI TT

PR Phi in radians

Q Variance cf plant noise from N(0 f Q)

R Variance of measurement ncise from N(3,R)

97

RAD

RANGE

RANSE2

RBI

RB2

RB3

RBX

RNO

RUNS

S

SI

S2

SEC

SE

SIGMA

SIGMA?

SIGMAX

SKL

SNSXT

SST

ST

SUMP1

SUMP2

SUMSQ1

SUMSQ2

SUMX1

tt /180

Range to sonobuoy from aircraft

Squared value of range

Unit vector component of antenna baseline



Result of (RB1&x

x+ RB

2 ax 2+ RB

3ax

3)

Normally distributed random number from N

(

d , 1

)

Number of simulation runs to be made

Array in Carlson's square root technique whichrepresents square root of covariance matrix

Square root of variance, /Pn

Square root of variance, ^fiz

Time in seconds

Matrix result of S f

RMS error in the estimate about the actual location

RMS value from the Kalman filter, /Pj \ + F 2 2

RMS error in the estimate about the mean estimate

Rate of change of nautical miles for a change indegrees of latitude

Propagated matrix of S

Matrix result of S ST

Transpose of S matrix

Array for the sum of P(l,l

Array for the sum of P(2,2

over NRUNS

over NRUNS

Array for sum of ax over NRUNS

Array for sum of ax over NRUNS

Array for sum of a'x. over NRUNS

98

SUMX2 Array for sum of Ax 2 over NRUNS

TIME

Ul

U2

VAfil

VAR2

VARX1

VARX2

VELCT

VELCT2

x

XAI

XAT

XI

X2

X3

XB1

X32

XB1I

XB2I

XNEXT

Z

Last time in seconds the sonobuoy was processed Dy

the filter

Time from aircraft clock in seconds

Sonobuoy drift vector

Old value of U(l)

Old value of U(2)

Variance of the estimated sonobuoy position aboutthe actual location af

Variance of the estimated sonobuoy position aboutthe actual location o\

Variance of the estimated sonobuoy position aboutthe mean estimated oosition, a 2

,xl

Variance of the estimated sonobuoy position aboutthe mean estimated position, a 2

Relative velocity, aV

Squared value of relative veclocity

State vector

Aircraft inertial position, Xa

Aircraft true position, Xa

Old value of X(l)

Old value of X(2)

Vector including XB1, XB2, velocity of sonotuoy, andtwo zeros used in the six-state system

True geographical location of sonobuoy, x

True geographical location of sonobuoy, x

Array of initial sonobuoy positions

Array of initial sonobuoy positions

Propagated value of X

Measurement vector

1

99

ZHAT Estimate of z

ZN Normally distributed random number from N(z,R)

ZU Vector of velocity measurement for drift filter

100

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APPENDIX B

COMPUTER PROGRAM FOR DATA GENERATION

DATA GENERATION

Set initial conditions:

sonobouy position and velocityaircraft position and velocityaircraft headingdelt

LINE

Set:

PhiAccel -

Calculate:

aircraft position

CallFILE

CURVE

Calculate

:

PhiAccel

Calculate

:

aircraft position

FILE

CallTIME

Calcul ate

:

aircraft driftsonobouy numbersonobouy positionmeasu rements, I

Set:

gamnla «

Write

data intostorage file

CallFILE

©

Figure 37. Flow chart: data generation program.

127

GLOSSARY 01 COMPUTER VARIABLES

DATA GENERATION PROGRAM

A Vector of aircraft accelerations

AL Altitude in nautical miles

ALT Altitude

ANT Antenna

AVEL Initial aircraft velocity

AX1 Amplitude of Schuler cycle, A,.

AX2 Amplitude of Schuler cycle, A^

3 Intercept on N-S axis of straight track

BANG Sonotuoy drift direction

BOUY Sonobucy RE channel cr number

BVEL Sonobuoy drift direction

CKL Rate of change of nautical miles for a change indegrees of longitude

D Length of straight track

DELT Interval between measurements, At

DELV1 Relative velocity, Avj

DELV2 Relative velocity, Av 2

DELX1 Relative position, ^xl

DELX2 Relative position, ax 2

DH Total change in heading while on curved track

DSEEI) Seed for the random number generator

END Indicator of EOE

i Gravitational acceleration

GAMMA Aircraft pitch angle, y

128

GR Gamma in radians

H Aircraft heading

KNEW Final heading on curved track

HOUR Time in hours

HR Heading in radians

IOUT Device on \*hich print is made

IOUTP File in which results are written

IOUTR File in which true sonobuoy positions are written

IOUTZ File in fchich random vectors are inspected

ITIME Time from aircraft clock in seconds

KXl Rate of linear navigational drift, K

KX2 Rate of linear navigational drift, K

LAT Latitude

LATI The latitude of the origin of the local earth fixedcoordinate system

LONG Longitude

LONGI The longitude of the origin of the local earth fixedcoordinate system

LTIME Array of initial drop times for the sonobuoy

M Slope of straight track

MARK Indicates the first time a buoy is to be proccessed

MIN Time in minutes

N Index

NBOUY Total number of sonobuoys

NSTOP End of line or curve series

PHI Aircraft roll angle,ft

PI TT

FI2 2^

129

PR Phi in radians

R Radius of curved track

RBI Unit vector component of antenna baseline

RB2 Unit vector component of antenna baseline

RB3 Unit vector component of antenna baseline

S Distance aircraft travels in time At

SEC Time in seconds

SKL Rate of change of nautical miles for a change indegrees of latitude

TRACK Direction of aircraft travel over the ground

U Direction to center cf curved track

V Direction from center of curved track to aircraft

VA1 Velocity of aircraft

VA2 Velocity of aircraft

V31 Sonobuoy drift

VB2 Sonobucy drift

WANG Wind direction

WVEL Wind velocity

XAI Aircraft inertial position, Xa

XAT Aircraft true position, Xi

i nf <;oTinhiinv. x1

a

XB1 True geographical location of sonobuoy, x

XB2 True geographical location of sonobuoy, x

XC1 Location of the center of the curved track

XC2 Location of the center of the curved track

XXI Incremental change in X

XX2 Incremental change in X2

Z Measurement vector

130

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APPENDIX C

Range Analysis

C OLID - FILTER: SORT Pl+P^j

X

CD

- ABOUT MRAN FILTER POSITION

- ABOUT ACTUAL POSIT ION

!

•A 9. a a4o

On ° 3 e°% °3

QJk, L S a j% ,a$k

\

;' IIITIME IMINUTE5I

SOLID FILTER: SORT (P1+P3)

X - ABOUT M

CD!

- ABOUT A

AN FflLTER iPOSITI

TUAL POSITION

ION

TIME (MINUTES)

Figure 38. RMS errors for two-state system (top) and six-state system (bottom) using circular pattern at 5 NM.

140

-

SOLID - FILTER: SQRT P1+P2)

-

X

o

- ABOUT MEAN FILTER POSITION

- ABCJUT AQTUAL POSITION

L

C

o

D

S

oo

iHft.\ °

3/

\3.

°

3 \

L o %Sy^/Lr

°

*>

j» ; i ?^ 3

9 \ sa

JS

•

?Ai »X ^m ,'-

:^v<***%#*»^v

-

TIME (MINUTES!

:

'1 C OLID - FIL TER: SQRT

;

P1+P2)

- .1

«x<

X

o

- ABC

- =E

UT MEAN FILTER IpOSITIONJ J

UT ALlT ION

\

o9?

\x

^—,

—

J

i |

»

, 1

-

s?

* " n »,H *** j

-

- . 1i

TI ME (MINUTES)

Figure 39. RMS errors for two-state system (top) and six-state system (bottom) using circular pattern at 30 NM.

lto.

TIME IMINUTE31

o

:\OLID - FIL

I |

TER: isQRT JPI^PS!

i

( ^X - ABC

Q |

• RBC

!JT MFiflN FCLTER 'pOSIlilON1

!

UT PtfTUPL .position

- '.\\| |

|

i

1

- <**1

i

i' :

-

s 31i

-

f 11

J &3 5 «* A * "b°

-

F\*$?

* * It*

5* 9\ &

-

i

60.

TIME (MINUTES!

Figure ^0 . RMS errors for two-state system (top) and six-

state system (bottom) using circular pattern at k$ NM.

1^2

Frequency Analysis

'

I ME MINIJTES1

-; :

|

SOLID j-- FILJTER: |SQRT ( P i + P 2 J

.

X

o

- fiBC

- RBC

UT M^AN F IJL TEP °0S I TiIONi

'

UT ACTUAL POSITION

-

:

-

\^/"^

A V-^.

i !

7: * aaJto

KiiJkJM¥**$^SA*JiJ

'WWf ^ »? "¥Tnjnr" 1

TIME (MINUTES!

Figure 4-1. RMS errors for two-state system (top) and six-state system (bottom) using square pattern with t=4 sec.

143

'[ME (MINUTES)

Figure 42. RMS errors for two-state system (top) and six-state system (bottom) using square pattern with t=10 sec

144

-

c— OLID - FILjTER: SORT PI +P2)

-

X

o

- ABOUT M^HN FILTER POSITION

- flBCUT RGTUflL POSIT ION

ACL

-

c "6

o %

oj

if*

otO

O

1 \I

°0

\^jd1 O

oo

(SB\!

o ^_>-^^<^^—-f-—-v_

"l ^ z^~" i

° *i

1

1

TIME (MINUTES)

TIME (MINUTES)

Figure kj. RMS errors for two-state system (top) and. six-state system (bottom) using square pattern with t=30 sec

145

Altitude Analysis

SOLID - FILTER: SQRT PI +P^J

-

X

CD

- ABOUT MEAN FILTER POSlTjION

- fiBfJUT ACTUAL POSITION

-

-

<$?»3

CD

<#h>

/ °3

s •

'1

a

CD

a ° o

* Is•° * °l

c

a

.?&>

° urnfffli]

S.

*"$ °| &$ ^

- V^,

w~^

—

;

—"" ^—^< 1

—""A ly

-^

r* * '«Mti "f ,. o" T

,i

TIME (MJNUTE5I

SOLID FILTER: SORT PI t-P^Ax

o

ABOUT MEiRN FILTER iPOSlilON

ABOUT ACTUAL POSITION

==h-

T!ME (MINU7E3)

Figure ^3 • RMS errors for two-state system (top) and six-state system (bottom) using circular pattern, Alt=300'.

146

TIME (l-'INUTESi

SOLID TER: iSQRT (PI+P2

ABOUT M^RN [ |POSIlt[QN

o - RB(iuT rqturl position

;,?*•%»

',» » *

1 1 . ,' i l-l, *-m- — 1 *^&> . 1

TIME (MINUTES)

Figure 4-5. RMS errors for two-state system (top) and six-state system (bottom) using circular pattern, Alt=10,000'.

147

TIME (MINUTES)

50LID - FILTER: ,5QR' + P3)

X j- ABOUT M^flN FILTER |POS

O '- ReqUT fidTURL ;P0S I T* I ON

ON

%&>.&x -f tf

*

- ItA

1ft

»*#

ST ^»

TIME ;M[NUTE5I

Figure 46. RMS errors for two-state system (top) and six-state system (bottom) using circular pattern, Alt=20,000'

148

BIBLIOGRAPHY

Bryson, Jr., A. E. and Yu-Chi Ho, Applied Optimal Control ,

Hemisphere Publishing Corp. 1975-

Carlson, N. A., Fast Triangular Formulation of the Square RootFilter, Vol. II, No. 9, AIAA Journal, September 1973-

Demetry, J. S., Notes on the Theory and Applications of OptimalEstimation , Naval Postgraduate School, Monterey, California1970.

Gelb, A. and others, Applied Optimal Estimation , M.I.T. Press1974.

Leondes, C. T. Editor, Control and Dynamic Systems , Vol. 12,Academic Press, 1976.

Orincon Corporation Report 0C-R-78-A003-1 Sonobuoy ReferenceSystem , by Asher, R. M. , Judge, R., and Sorenson, H. W.,December 1978.

System Functional Description for Sonobuoy Reference System ,

FD-17 (I ^.3), Naval Air Development Center, June 1978.

System Functional Description for Navigation , FD-9 (I 3A-2)Naval Air Development Center, November 1977-

1^9

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center 2Cameron StationAlexandria, Virginia 22314

2. Library, Code 0142 2

Naval Postgraduate SchoolMonterey, California 93940

3. Department Chairman, Code 62 1

Department of AeronauticsNaval Postgraduate SchoolMonterey, California 93940

4. Prof. D. J. Collins, 67Co 3Department of AeronauticsNaval Postgraduate SchoolMonterey, California 93940

5. LT N. M. Brownsberger 2

3802 La Miranda PIPittsburg, California 94565

6. Dr. D. Birnbaum, Code 2031 1

Naval Air Development CenterWarminister, Pennsylvania 18974

150

Thesis1 8 4 3 i

B8243 Brownsbergerc.l Estimation of

sonobuoy positionrelative to an aircraftusing extended kalmanfilters.

K) m »] 5 7 1 IB

Thesis 184 310B8243 Brownsbergerc.l Estimation of

sonobuoy positionrelative to an aircraftusing extended kalmanfilters.

thesB8243

Estimation of sonobuoy position relative

3 2768 002 08054 1

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ESTIMATION RELATIVE AIRCRAFT USING Nicholas Brownsberger

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