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Ocean Engineering 26 (1999) 1183–1247www.elsevier.com/locate/oceaneng

A metamodel-based towed system simulation

D.E. Calkins*

Department of Mechanical Engineering, University of Washington, Box 352600, Seattle, WA, USA

Received 10 February 1998; accepted 9 April 1998

Abstract

The objective of this exercise was the development of a computer simulation of a US Navytowed system (towed body and line array) capable of near real-time predictive capabilities.The purpose of the computer simulation was to be able to determine the lateral motion of thetowed body in response to ship maneuvers. The theoretical approach was to develop a spatial-domain computer simulation of each of the towed system components treated as a system,and then develop a “metamodel” of the response by using simple algebraic expressions. At-sea trials were conducted aboard US Navy vessels to verify the simulation and fine-tune itsperformance. 1998 Elsevier Science Ltd. All rights reserved.

Keywords:Towed system simulation model; Towed body; Towed array; Metamodel

NomenclatureTowed body

L roll moment aboutX-axisM pitch moment aboutY-axisN yaw moment aboutZ-axisX force alongX-axisY force alongY-axisZ force alongZ-axisFxh

added mass force alongX-axisFyh

added mass force alongY-axisFzh

added mass force alongZ-axisP roll rate aboutX-axis

* Tel.: 1 1-206-543-9443; Fax:1 1-206-685-8047; E-mail: [email protected]

0029-8018/99/$ - see front matter 1998 Elsevier Science Ltd. All rights reserved.PII: S0029 -8018(98)00063-8

1184 D. Calkins /Ocean Engineering 26 (1999) 1183–1247

Q pitch rate aboutY-axisR yaw rate aboutZ-axisu velocity alongX-axisv velocity alongY-axisw velocity alongZ-axisu X component of linear accelerationv Y component of linear accelerationw Z component of linear accelerationP roll angular acceleration aboutX-axisQ pitch angular acceleration aboutY-axisR yaw angular acceleration aboutZ-axisi unit vector alongX-axisj unit vector alongY-axisk unit vector alongZ-axisf roll angleu pitch anglec yaw angleb sideslip angle (2 c)Ix moment of inertia aboutX-axisIy moment of inertia aboutY-axisIz moment of inertia aboutZ-axisJxz product of inertiaAw wing areaBw wing spanCw wing chorddr rudder deflectionde elevator deflectionW weight (air)B buoyancymb mass (W/g)Xw X-distance from towpoint to center of massZw Z-distance from towpoint to center of massXb X-distance from towpoint to center of buoyancyZb Z-distance from towpoint to center of buoyancyXh X-distance from towpoint to hydrodynamic centerZh Z-distance from towpoint to hydrodynamic centerXtp X-distance from towpoint to external forceZtp Z-distance from towpoint to external forceUs ship speedRs ship turn radiusUd towed body speedRd towed body turn radiusc turn rateTy0

9 externalY-forceTx0

9 externalX-force

1185D. Calkins /Ocean Engineering 26 (1999) 1183–1247

Tz09 externalZ-force

Tx9 towed bodyX-force on towlineTy9 towed bodyY-force on towlineTz9 towed bodyZ-force on towline

Faired tow cable

fw towline net weight force (weight minus buoyancy)ft towline tangential forcefs towline side forcefn normal forcefa added mass forcefi towline inertial (centrifugal) forcemt towline massmt9 towline added massi unit vector (X-axis)—towline natural coordinate systemj unit vector (Y-axis)—towline natural coordinate systemk unit vector (Z-axis)—towline natural coordinate systeme1 unit vector (Xbt-axis)—flow rectangular coordinate systeme2 unit vector (Ybt-axis)—flow rectangular coordinate systeme3 unit vector (Zbt-axis)—flow rectangular coordinate systeme− unit vector (X-axis)—flow cylindrical coordinate systemen unit vector (Y-axis)—flow cylindrical coordinate systeme3 unit vector (Z-axis)—flow cylindrical coordinate systemu towline kite angle (plan view)f towline cable angle (profile view)X position of material point “P”—rectangular systemY position of material point “P”—rectangular systemZ position of material point “P”—rectangular systemY position of material point “P”—cylindrical systemR position of material point “P”— cylindrical systemZ position of material point “P”—cylindrical systemS towline scope

1. Problem statement

The objective of this design exercise was to develop a computer simulation of aUS Navy towed system capable of near real-time predictive capabilities. A towedsystem typically comprises a towed underwater body and a towline which connectsthe underwater system to the towcraft at the surface, Fig. 1. The purpose of thecomputer simulation was to be able to determine the lateral motion of the towedbody in response to ship maneuvers. Specifically, the simulation was developed to


Fig. 1. AN/SQR-35//AN/SQR-18A(V)1 towed system configuration.

compute the difference between the heading angles of the ship and the towed bodyin response to changes in ship heading (ship maneuvers).

At the time of initiation of the project, the towed body heading angle was measuredby a heading gyro installed in the towed body. Because of the age of the gyrotechnology and its location in the towed body, the gyro suffered from reliabilityproblems. The goal of the development was to be able to predict the towed bodyheading using a computer simulation scheme. If successful, the goal was to replacethe gyro with the simulation technology for real-time analysis of the towed bodyheading. The design goals and constraints for the development of the computer modelwere for the simulation to:

1. have sufficient detail and accuracy in simulation to model the dynamic behaviorof the existing towed system;

2. predict the difference between the ship and towed body headings within6 5°under all operating conditions; and

3. operate at near real-time computational speed.

2. Towed array system description

2.1. Towed system

The towed system, shown in Fig. 2, comprised the Knox class frigate which servesas the tow vessel, 600 ft of faired tow cable, the AN/SQS-35 VDS depressor (towedbody), a 300 ft tether cable, the 311.4 ft AN/SQR-18A(VV)1 towed sonar hydro-


Fig. 2. AN/SQS-35//ANSQR-18A(V)1 towed system configuration.

phone array and the 51 ft rope drogue (Janes, 1990/91). The computer simulationsystem was termed the Fish Heading System (FHS) and was developed over thetime period between 1987 and 1991.

2.2. Knox class frigate

The tow vessels were of the series of 45 Knox class frigates, Fig. 3 (Janes,1990/91). The primary mission of this class of ships is antisubmarine warfare (ASW).The leading particulars of this class of ships include:

Fig. 3. Knox class frigate.


Displacement: 3011 tons (standard)4260 tons (full load)

Dimensions:Length: 439.6 ftBeam, extreme: 46.8 ftDraft, navigational: 24 ft 9 in

2.3. AN/SQS-35 IVDS towed system

The AN/SQS-35 IVDS towed sonar system comprises two components:

1. a faired tow-cable (600 ft), see Fig. 4; and2. a towed body/active sonar, see Fig. 5.

2.3.1. Tow cable/fairingThe segmented faired cable, manufactured by the EDO Corporation, is used to

reduce the drag of the structural cable member, Fig. 4. The fairing segments, whichare free swivelling around the steel cable to align with the flow and eliminate fairingside forces, are supported at intervals to remove the tangential hydrodynamic forceswhich build up along its length. The cable and fairing characteristics include:

Cable weight, air: 1.46 lb/ftFairing weight, air: 2.20 lb/ftTotal cable1 fairing weight, air: 3.66 lb/ftTotal cable1 fairing buoyancy, seawater: 1.78 lb/ftTotal cable1 fairing weight, seawater: 1.88 lb/ftFairing segment chord: 5.75 inFairing segment length: 4.89 inFairing segment thickness: 1.65 inFairing thickness/chord ratio: 0.20

2.3.2. Towed bodyThe towed body is depicted in Fig. 5. The mass properties of the towed body

were obtained from Walton and Brillhart (1966) and Patten and Olive (1967), andare listed in Table 1.

2.3.3. AN/SQR-18A(V)1 towed array systemThe AN/SQR-18A(V)1 Tactical Towed Array System (TACTASS), Fig. 6, com-

prises three components: a coated tether cable which connects to the tail of theAN/SQS-35 IVDS towed body, the sonar hydrophone array with a VIM and a taildrogue. Their properties are listed in Table 2.


Fig

.4.

AN

/SQ

S-3

5to

wca

ble/

fairi

ng.


Fig

.5.

AN

/SQ

S-3

5IV

DS

tow

edfis

h.


Table 1Mass properties of the towed body

Item Wair (lb) Wwater (lb) Buoyancy,B Ref. Center of(lb) gravity, XW (ft)

Hull 2840 2404 436 Walton and —Brillhart (1966)

Transducer 960 596 364 Walton and —Brillhart (1966)

Body, total 3800 3000 800 Walton and 20.36Brillhart (1966)

Body, total 3825 2951 874 Patten and 21.09Olive (1967)

Fig. 6. AN/SQR-18A(V)1 towed array system.

3. Towed system simulation modeling scheme

3.1. Technical approach

The dynamic simulation was developed in a two-phase scheme designed to meetthe goals and constraints. The first step in the development of the system simulation


Table 2Properties of the towed array system

Component Length (ft) Diameter (in) Wa (lb/ft) Ww (lb/ft)

Tether cable 300.0 1.07 0.74 0.34Array 261.4 3.34 3.81 0.0

(995.0 lb) (0.0 lb)Drogue 51.0 0.5 — 0.0

was to develop a three-dimensional spatial-domain simulation model which wasbased on a complete representation of the geometry, mass properties and hydrodyn-amics of each of the system components. A limitation of the first simulation, knownas the MASTER program, was that the computational time for an individual casewas large enough that it was not possible to use it directly for the desired real-time computations. A second simulation, known as a metamodel and designatedQUICKTOW, was then developed based on the results from the MASTER program.The MASTER program was executed for a large matrix of ship operating conditionsand the towed body heading results were curve-fitted using a regression techniqueto develop simple algebraic expressions for an algorithm that could be solved inreal time.

3.2. Towed system simulation modeling

The development of a general mathematical model to describe the motions of thecombined towed array system will, in general, include consideration of the cableelastic, inertial and hydrodynamic forces, as well as the body tension, weight, andbuoyancy forces. To obtain the desired information on the towed body motions, andyet avoid undue complexity, two distinct mathematical models are generally required:

1. steady-state (spatial-domain) model to compute the tow-cable and array orien-tation in three-dimensional space; and

2. dynamic response (time-domain) model to predict towed system response to towvehicle maneuvers.

3.3. Three-dimensional spatial-domain simulation

The development of the towed system simulation begins with the MASTER pro-gram (Calkins, 1987, 1979). The MASTER program is based on a simulation of thetowed system. The purpose of the spatial-domain model is to provide a three-dimen-sional steady-state simulation of a towed body/towline/tow array system. The steady-state mathematical model computes the three-dimensional spatial configuration andtension distribution of the tow-cable either for a straight tow or a steady turn. Typi-cally, a steady-state model would include consideration of only the tension,weight/buoyancy and hydrodynamic forces. However, extension to the three-dimen-


sional steady turn case requires inclusion of the inertial centrifugal force. The devel-opment of the MASTER simulation was limited to the steady-state spatial-domainapproximation. Consideration of a time-domain simulation was eliminated becauseof the complexity and non real-time solutions typical of this type of simulation.

The MASTER simulation is an extension of existing three-dimensional cable pro-grams in that it is interfaced with a simulation of the towed body (towlinetermination), which is used to determine the towed body orientation and to providethe end-point inputs to the towline module. In addition, the program is able todescribe the system configuration in a steady-state three-dimensional turn. It also ispossible to include out-of-plane forces generated by the towline, the towed body orthe array. The tow-cable is modeled as a nonlinear, continuous line structure in three-dimensional space. The steady-state model predicts the three-dimensional configur-ation of the tow-cable in either a rectangular (straight ahead tow) or a cylindrical(steady turn) coordinate system. The tow-cable, which is modeled by six nonlinear,first-order, ordinary differential equations, is treated as continuous, inextensible,flexible, and nonuniform in density and geometry along its length.

The MASTER program is executed by inputting: (1) ship speed, (2) ship turn rate,(3) tow-cable scope and (4) water temperature. The outputs of the towed body mod-ule are its angular orientation, roll, pitch and yaw, and the force components imposedas the lower boundary conditions on the towline. The outputs of the towline andarray modules are the tension distribution and the towline configuration.

3.4. Metamodel simulation

The MASTER program is the transfer function that turns input parameters intooutput parameters, or performance measures. Since the computational time associatedwith the MASTER program exceeds the operational requirement, an alternative sol-ution is needed to develop the simulation. Needed is an approximate solution thatwill represent the results of a matrix of input parameter combinations when run inthe MASTER simulation. The solution is the so-called “response surface” (Law andKelton, 1991). A simple algebraic approximation of the response surface, called ametamodel, was developed to replace the MASTER simulation. The metamodel isthe second tier model which is an algebraic model of the results from the MASTERmodel. The MASTER model is executed for a matrix of cases which encompass thecomplete operational spectrum of the towed system, resulting in what are known assimulation response surfaces. The results, in terms of the difference between thetowed body and ship headings as a function of speed and turn rate, are fitted withan algebraic function that is then incorporated in the FHS production system.

4. Towed body equations of motion

A linearized six-degree-of-freedom simulation of the towed body provides thelower tension vector and towed body orientation, or lower boundary condition on


the faired towed cable. The differential equations, which model the system, aredeveloped in both Cartesian and cylindrical coordinate systems.

4.1. Coordinate systems—towed body

Two coordinate systems are used in the development of the towed body equationsof motion.

1. Towed body (body axis coordinate system)—Fig. 7: Cartesian coordinate systemaligned along the centerline of the towed body, origin at the towed body’s centerof gravity (CG).

X-axis: along longitudinal centerlineY-axis: to starboardZ-axis: downward

2. Towed body (flow axis coordinate system)—Fig. 7: Cartesian coordinate systemaligned with the velocity vector of the towed body, origin at the towpoint.

X-axis: parallel to towed body velocity vectorY-axis: to starboard and towards center of turnZ-axis: gravity vertical

Fig. 7. Towed fish embedded coordinate systems.


4.2. Equations of motion—towed body

The development of the equations of motion of the towed body is carried out inthe towed body aligned coordinate system. Since the motion is assumed to be steady-state, the turn rate of the towed body is equal to the towcraft turn rate. However,the turn radius of the towed body,Rd, is not equal to the turn radius of the ship,Rs,since the towed body may lie either to the inside or outside of the towcraft path. Acase is thus considered to be defined by specifying: (1) the towcraft speed and (2)the towcraft turn rate. The towcraft velocity (Us), towed body velocity (Ud) and turnradius (R) are related as:

c 5Us

Rs5

Ud

Rd5 towcraft turn rate (°/s) (1)

The towed body has six degrees of freedom and may assume a pitch angle,u, aroll angle,f, and a yaw angle,c. The towpoint is assumed to be a ball joint so thatthere is no moment transfer through it. A total of six unknowns therefore exist:

1. u—pitch angle;2. w—yaw angle;3. f—roll angle;4. Tx—towline tension component along thex-axis;5. Ty—towline tension component along they-axis; and6. Tz—towline tension component along thez-axis.

The equations of motion for the towed body may be derived from Newton’s secondlaw of motion, which states that the summation of all external forces acting on atowed body must equal the time rate of change of the momentum, and the summationof the external moments must equal the time rate of change of the moment ofmomentum (angular momentum). The time rates of change are all taken with respectto inertial space. These laws can be expressed by two vector equations:

ΣF 5ddT

(MU) 5 MFdUdt

1 (w 3 U)G (2)

and

ΣM 1 (an 3 Mrw) 5dHdt

(3)

where:

ΣF total external forcesΣM total external momentsM mass of towed bodyU linear velocity of towed bodyv angular velocity of towed body


H moment of momentum about towpointan absolute acceleration about towpointrw radius vector from towpoint to towed body center of mass

The second term in Eq. (3) arises from the fact that the moments are being takenabout the towpoint rather than the center of mass.

4.3. Towpoint forces

The force vector equation, Eq. (2), may be used to develop the towline forcesacting at the towpoint. The solution to the moment equation, Eq. (3), determines thethree equilibrium angles,u, b and f, of the towed body which satisfy the zeromoment condition about the towpoint. Using Eq. (2), and since

dUdt

5 0

then

ΣF 5 W 1 B 1 Fh 1 Fh 1 T 1 TEF 5 [MW 3 U] (4)

where

W weight forcesB buoyant forcesFh hydrodynamic forcesFh acceleration (added mass) forcesT towline tension5 Txi 1 Tyj 1 TzkTEF external force

4.4. Tension vector transformation

The tension vector was developed in the towed body axis coordinate system. Thisvector must be transformed into the flow axis coordinate system.

T 5 Txi 1 Tyj 1 Tzk (5)

where

Tx 5 Tx9 cosb cosu 1 Ty9 sin b cosu 1 Tz9 sin u

Ty 5 Tx9(cosb sin f sin u 2 sin b cosf) 1 Ty9(cosb cosf

1 sin b sin u sin f) 2 Tz9 cosu sin f

Tz 5 Tx9( 2 cosb sin f 2 sin b sin f) 1 Ty(cosb sin f 2 sin b sin u cosf)

1 Tz9 cosu cosf


Making the small angle approximation:

Tx 5 Tx9 1 Ty9b 1 Tz9u

Ty 5 Tx9b 1 Ty9 2 Tz9f (6)

Tz 5 2 Tx9u 1 Ty9f 1 Tz9

The projection of the towline onto the (yz) and (xz) planes is defined by the twoinitial towline angles:

f0 5 tan−1F 2 Tz9

Tx9G (7)

u0 5 tan−1F 2 Tz9

Tx9G (8)

The initial value of the towline tension is:

T0 5 √(TX9)2 1 (TY9)2 1 (TZ9)2 (9)

5. Tow cable/towed array equations of motion

The MASTER program contains a nonlinear continuous representation of each ofthe “line structures” in the towed cable and towed array systems. The mathematicalrepresentation provides the tension distribution and spatial configuration of eachcomponent. Both the faired tow cable and the towed array system components aremodeled in a similar manner. The following assumptions are made concerning allof the cable-like components in the towed system. They are:

1. continuous;2. inextensible;3. inelastic (EIx 5 EIy 5 GJ 5 0);4. uniform in density along the length; and5. uniform in geometry along the length.

5.1. Coordinate systems—tow cable/array

Since it is desired to analyze the towed system in both the straight tow and turncases, it is necessary to derive the towline equations of motion in two separate coordi-nate systems: a rectangular system for the straight tow case and a cylindrical systemfor the turn case. The derivations in both coordinate systems are based on the analysisof Calkins (1987).


5.1.1. Rectangular coordinate systemThe three coordinate systems to be used are defined. LetXe, Ye andZe, Fig. 8, be

Cartesian coordinates fixed in the ocean such that the vertical axis (Ze) is alignedwith gravity. These are the inertial coordinates. A second Cartesian coordinate (flow)system (Xbt, Ybt andZbt), Fig. 8, is fixed and moving with the ship such thatXbt, Ybt

and Zbt are mutually perpendicular toXe, Ye and Ze. The vectorse1, e2 and e3 areunit vectors in theXbt, Ybt, Zbt system. The position vector,r, to a point “P” on thetowline is

r 5 Xe1 1 Ye2 1 Ze3 (10)

Finally, a third Cartesian coordinate system, Fig. 8, (X, Y, Z), is fixed within thetowline such that thex-axis is the tangent vector to the towline along its length, ds.The vectorsi, j and k are unit vectors in this system. Thus

Fig. 8. Towed system coordinate systems.


i ; r/s (11)

for an inextensible towline. The velocity of the point “P” with respect to the inertialcoordinate system isU. The towline anglef is defined as the angle subtended bythe unit tangent vector,i, and the velocity vector,U. Thus

j 5 U 3 [ i/U sin f] (12)

k 5 j 3 i

The transformation from the flow system to the towline system is accomplishedby only one pair of angles and one order of rotation. These are a rotationu aboutthex-axis and a rotation off about they-axis, Fig. 9. Thus the directional derivativesin the rectangular coordinate system are given by:

dXds

5 cosf

dYds

5 sin u sin f (13)

dZds

5 cosu sin f

5.1.2. Cylindrical coordinate systemA fourth coordinate system is necessary for the turn case. This is a cylindrical

coordinate system (c, R, Z), Fig. 8, which rotates with the ship about thez-axis. Thevectors ec, er and e3 are unit vectors in this system. The orientation ofec and er

vary with S along the towline so that

Fig. 9. Towline transformation angles.


der

dc5 ec (14)

dec

dc5 2 er

The directional derivatives in the cylindrical coordinate system are:

dc

ds5

2 cosu

R

dRds

5 2 sin u sin f (15)

dzds

5 2 cosu sin f

5.2. Equations of motion—tow cable/array

5.2.1. Rectangular coordinate systemThe equations of motion in the rectangular coordinate system are:

mtS∂U∂t D 5 S∂T

∂sDi 1 (fw)e3 2 (ft)i 1 (fs)j 2 (fn)k (16)

where

∂Udt

5 0 (rectangular coordinate system) (17)

S∂T∂sDi 5 TSdi

dsD 1 SdTdsDi

Now

i vector:dTds

5 fw sin f cosu 1 ft

j vector:du

ds5

2 fw sin u 2 fsT sin f

(18)

k vector:df

ds5

fw cosf cosu 2 fnT

Eqs. (13) and (17) are the six, nonlinear, first-order ordinary differential equationsof motion of the towline in the rectangular coordinate system.


5.2.2. Cylindrical coordinate systemThe equations of motion in the cylindrical coordinate system are:

mtS∂U∂t D 5 S∂T

∂sDi 1 (fw)e3 2 (ft)i 1 (fs)j 2 (fn)k 1 (fa)er (19)

Now

i vector:dTds

5 fw sin f cosu 1 ft 1 mtRc2 sin f sin u

j vector:du

ds5

2 fw sin f 2 fs 1 [mtRc2(1 1 m22) 2TR

cos2 f cosu

T sin f(20)

k vector:df

ds5

fw cosf cosu 2 fn 1 FmtRc2(1 1 m33) 2TRG cosf sin u

T

Eqs. (15) and (20) are the towline equations of motion in the cylindrical coordi-nate system.

5.3. Solution procedure

A fourth-order Runge–Kutta numerical method is used to integrate the differentialequations. For the turn case, the turn radius of the towed body is not known, sinceit may follow a path either to the inside or the outside of the tow-ship’s path, Fig.10. In order to begin the integration, it is assumed that the towed body turn radiusis equal to the ship’s turn radius,Rs:

Rd15 Us/c 5 Rs (21)

After the first pass through the towline program, the towed body turn radius isadjusted using the turn radius to the last point on the towline,Rt1

:

Rd25 RsFRd1

Rt1

G (22)

The speed of the towed body is now

Ud25 Rd2

c (23)

which is used to re-compute the towed body forces, which are, in turn, used tocompute a new towline configuration. The corrective procedure

Rdn 1 15 RsFRdn

RtnG (24)


Fig. 10. Towed body turn radius correction.

Udn 1 15 Rdn 1 1

c

is repeated untilRdn 1 1andRdn

are approximately equal within a specified tolerance.

5.4. Ship coordinate system

When full-scale towing trials are conducted, all measurements of the towed systemare made relative to the ship. For that reason, the output of the towline program istransformed into a Cartesian coordinate system, Fig. 11, aligned with ship velocityvector with its origin at the towpoint. The coordinates ofPn, the termination of thetowline, are (Rx, Ry, Rz) in the Xbt, Ybt, Zbt system. The coordinate transformation isaccomplished as follows:

j 5 sin−1SRx

RsD

g 5 tan−1SRy

RxD


Fig. 11. Transformation to ship coordinate system.

Rp 5 √R2x 1 R2

y

and where

Xp 5 2 Rp cos(j 2 g) (25)

Yp 5 Rp sin(j 2 g)

are coordinates ofPn in the ship system. The coordinates in the ship system of anypoint, Pn−1(x, y, z), along the towline are

R 5 √x2 1 y2

g 5 tan−1SyxD

Xpn 2 15 Xp 1 R cos(j 2 g) (26)

Ypn 2 15 Yp 1 R sin(j 2 g)

Zpn 2 15 Zp 2 Z


5.5. Tow ship orientation correction

It was determined that an additional computation module needed to be added tothe basic simulation to account for the orientation of the ship relative to the turncircle (Calkins, 1987). It was found necessary to apply a correction to the trail andsidetrail data. This was due to the following effect. In the case where the length ofthe ship is not small compared with the turn radius, the ship orientation during theturn must be taken into account and the trail and cross trail output of the simulationmust be corrected for this geometry. This correction accounts for what is known asthe ship drift angle.

Fig. 12, taken fromPrinciples of Naval Architecture(SNAME, 1989), shows theexaggerated orientation of a ship during a turn. In order to turn, the ship must havea sideslip angle,b, relative to the inflow velocity. It is assumed that the ship willrotate about its center of mass (CG), which is assumed to be at mid-length. It isseen that the bow of the ship therefore will lie to the inside of the steady turn circle,while the stern of the ship will lie to the outside of the steady turn circle. Accordingto PNA (SNAME, 1989), the ship drift angle lies in range fromb1 to b2 where:

b1 5 (22.5Ls/Rs) 1 1.45 (°)

b2 5 18Ls/Rs (°)

Fig. 12. Steady-state turn maneuver.


Taking the average

b 5 (b1 1 b2)/2 (27)

where:

Ls ship length5 414.5 ftRs ship turn radius5 96.7156(Ukn)/cUkn ship speed (kn)c ship turn rate (°/s)

5.6. Ship/towed body heading difference

The corrections that must be made to the simulation output data involve takinginto account the angular difference between the centerline of the ship and the velocityvector measured at the stern of the ship. There are four angular components thatconstitute the ship/towed body heading difference angle,DH, Fig. 13:

DH 5 cb 1 h 1 a 1 b (28)

where

1. cb 5 towed body yaw angle;2. h 5 towed body/ship towpoint velocity vector angular difference;3. a 5 ship towpoint/ship CG velocity vector angular difference; and4. b 5 ship drift angle.

Fig. 13. Steady turn ship geometry.


Now

1. cb is determined from the towed body simulation program;2. h 5 tan−1(Yb/Xb) (29)

whereXb is towed bodyX-coordinate in the ship towpoint coordinate system andYb

is the towed bodyY-coordinate in ship towpoint coordinate system;3. a 5 tan−1(Xtp/Y9) (30)

whereXtp 5 (Ls/2) cosb, Y9 5 Rs 1 Ytp andYtp 5 (Ls/2) sinb; and4. b is determined from Eq. (27).

6. QUICKTOW metamodel development

6.1. Version A metamodel basis

The execution time for an individual case using the MASTER program (onespeed/turn rate combination) is on the order of 15 to 30 s depending on the hostmachine. This execution time precludes its use as the production software whichrequires near real-time execution. The solution to this problem is the developmentof the QUICKTOW program. The development scheme is as follows. The MASTERprogram was executed for a large matrix of speed, turn rate, cable scope and watertemperatures values. These data were then used to develop the QUICKTOW algor-ithm through a sequential curve fitting process. The final simplified algebraic equa-tions which resulted allow near real-time execution.

The first step in the development of the program was to delineate the value rangefor each variable. This was based on the actual ranges from the full-scale systemtrials. The maximum ship speed is listed as 27 kn in Janes (1990/91), while the flankspeed is listed as 25 kn on the ship. The maximum tow speed (system deployed) is20 kn based on conversations with the ships’ crews. The selected ranges are shownin Table 3.

6.2. Sign convention

The sign convention follows the right-hand orthogonal coordinate system shownin Fig. 14. A positive turn (1 turn rate,c) is defined as a turn to starboard, and isaccomplished by a positive or “right” rudder deflection (trailing edge starboard). The

Table 3Value ranges for variables in the Version A metamodel

Variable Range Increment

Speed 7.5 to 30 kn 5 knTurn rate 0.1 to 1.0°/s 0.25°/sCable payed out (CPO) 100 to 600 ft 100 ftWater temperature 32 to 86°F Std. temp. (59°F)


Fig. 14. QUICKTOW sign convention.


ship/towed body heading difference is defined as positive for a positive turn as shownin Fig. 14. Thus the ship/towed body heading difference (°) is given by:

DH 5 Hs 2 Hf (31)

where

Hs ship heading (°)Hf towed body heading (°)

Positive turn5 ship heading increasing5 right rudder (1 dr)

5 positive turn rate (1 c) 5 positive ship/towed body heading difference

( 1 DH)

Negative turn5 ship heading decreasing5 left rudder (2 dr)

5 negative turn rate (2 c) 5 negative ship/towed body heading difference

( 2 DH)

6.3. Ship input

The inputs to the QUICKTOW simulation algorithm include the ship speed andturn rate which are computed as follows. Ship turn rate is not measured directly,and therefore must be computed from the ship heading signal, Fig. 15. The signalis assumed to be recorded over a time incrementDt, and the turn rate is then:

c 5 (Hs22 Hs1

)/Dt 5 turn rate (°/s) (32)

whereHs1andHs2

are the ship heading at timest1 and t2.Since the turn rate is based on ship heading at two discrete times, the value of

turn rate computed must be applied at a time,t 5 t1 1 Dt/2, midway between thetwo values. The ship speed is measured directly, but must be averaged to obtain avalue for timet 5 tavg:

U0 5 (U011 U02

)/2 (33)

The cable scope, or immersed portion of the cable, is obtained from the recordedCPO, or cable payed out. It must be corrected for the approximately 10 ft length ofcable which runs from the storage drum tangency point to the water surface:

S 5 CPO2 10.05 scope (ft) (34)

6.4. QUICKTOW basic

Ship/towed body heading difference as a function of speed from 7.5 to 30 knotsfor turn rates of 0.1. 0.25. 0.5, 0.75 and 1.0°/s are shown plotted as the data points


Fig. 15. QUICKTOW input parameter computation.

in Fig. 16. The curves are shown for a scope of 600 ft. After determining that theeffect of water temperature was negligible, all computations were performed at astandard ocean water temperature of 59°F. These data were then curve-fitted overthe variable ranges so that the ship/towed body heading difference was a functionof speed and turn rate. The form of the curve-fit equation used was:

DH 5 A 1 B 3 (log10(U0)) (35)

where:

DH ship/towed body heading difference (°)

U0 average ship speed over time interval (kn)

A, B curve fit constants

The values of the constantsA andB are listed at the top of each individual graph


Fig. 16. MASTER/QUICKTOW comparison—scope5 600 ft.


for a single scope/turn rate combination. The constantsA and B were then curve-fitted as a function of turn rate,c, for each value of scope, Fig. 17, using second-order polynomials:

A 5 A0 1 A1 3 (c) 1 A2 3 (c2) (36)

and

B 5 B0 1 B1 3 (c) 1 B2 3 (c2) (37)

wherec is the turn rate (°/s).Finally, the coefficientsA0, A1, A2, B0, B1 and B2 were determined as a function

of the scope,S. The individual graphs with the equations at the top are shown inFigs. 18 and 19 as a function of scope,S. These coefficients were curve-fitted usingboth second- and fourth-order polynomials.

7. Pre-production FHS towing trials

The validity of the metamodel FHS simulation system was then verified througha set of validation “at-sea” trails aboard naval vessels. These trials required that theexisting measurement system, the towed body mounted vertical gyro, was func-tioning so that actual body heading measurements could be made. Data recordingtechniques varied from recording by hand, to recording on a data measurement sys-tem to recording with the pre-production FHS. The trials spanned a time period oftwo years over which the software was validated and modified and the hardwareproduction was initiated. The towing trials schedule is shown in Table 4.

The conditions listed in Table 5 applied during the trials.

7.1. USS Bearysea trials data analysis

During the period from 11 through 19 August 1988, the first towing trials of the(AN/SQS-35)/(AN/SQR-18A(V)1) towed VDS system were conducted aboard theUSS Donald B. Beary. The purpose of the trials was to establish a comparative database for correlation studies with the computer simulation program. Each data eventconsisted of a ship speed/rudder deflection combination. The ship course during anevent included straight runs, and port and starboard turns. Each turn began at aspecific heading and then was followed through a complete circle (360°) until theship was on the same heading, at which time the rudder was returned to zero. Thedata analysis process was conducted in two phases. Turns were conducted in 360°circles both to port and to starboard, to determine the degree of symmetry of thetowed system. It was determined that an asymmetry existed and that the towed sys-tem configurations were different from port to starboard turns.

The turn rate was determined by examining the data over a complete manoeuvre,e.g., a 360° turn. Shown as a function of time are the ship and towed body headingsin Fig. 20 for all six rudder angle settings. The average ship and towed body turnrates are shown in Fig. 21, which is derived from the heading data. The turn rate


Fig. 17. (a) Curve fit “A” coefficients for turn rate; (b) curve fit “B” coefficients for turn rate.


Fig. 17. Continued.


Fig. 18. Curve fit “A” coefficients for cable scope.


Fig. 19. Curve fit “A” coefficients for cable scope.


Table 4FHS development program chronology

Date Event

14–19 Aug 1988 USS Donald B. BearyBoston, MA to Norfolk, VAHand-recorded data

20–23 Feb 1989 USS Jesse L. BrownCharleston, SC to Savannah, GAData measurement system

9–11 March 1990 USS MillerNewport, RIFirst trials of production FHS

16–19 April 1990 USS MoinesterPuerto RicoSecond set of trials of production FHS

Table 5Conditions during pre-production FHS towing trials

Item USS Beary USS Brown USS Miller

Date 14–19 Aug 1988 20–23 Feb 1989 6, 10–11 Mar 1990Latitude 36°309N 31°309N 38°359NLongitude 74°309W 78°309W 72°009WWater temperature (°F) 62.8 75 naCPO (ft) 540 520 520Array tether cable length (ft) 300 300 1000Data time interval (s) 30 2 0.2Data sets 362 11 700 na

was computed by dividing the difference between two successive headings by thedata time interval of 30 s. The ship turn rate/rudder angle relationship is shown inFig. 22. The ship/towed body heading difference is shown in Fig. 23. This representsthe primary goal for conducting the trials. Immediately obvious is the fact that theheading difference is not constant during the steady portions of the turn, and thatthe entire data set is skewed.

The first version of the QUICKTOW metamodel (symmetrical version) was usedduring these trials. The design goal was to have the difference between the trialsheading difference and the simulation results be within6 5°. The ship/towed bodyheading difference from the simulation, using turn rate and speed as an input, isshown in Fig. 24, with towed body depth and roll shown in Figs. 25 and 26. It isimportant to note the contrast between the test values and simulation values for theship/towed body heading difference. In addition, the character of the towed bodydepth trace is important as a strong cyclic behavior is noted. Next, the ship/towedbody heading difference and towed body depth variation as a function of towed body


Fig. 20. Ship/fish heading variation with time (USS Beary).

heading are shown in Figs. 27 and 28. The cyclic behavior is emphasized, with theminimum values occurring at a heading of 140° and the maximum occurring at aheading of 320°, which are 180° apart.

The difference between the test and simulation heading difference, or headingdifference error, is shown as a function of towed body heading in Fig. 29. It is seento be cyclic with a maximum and minimum, as would be expected from the previousdiscussion, and extend beyond the6 5° tolerance band that was desired. At thispoint, the first two problem areas in the development of the QUICKTOW algorithmwere identified:

1. cyclic variation with heading during a steady turn; and2. offset value so that results are not symmetrical.

At this point, an investigation was undertaken to identify the cause of and quantifythe magnitudes of these problems. It was felt that the cyclic variation was associatedwith the ocean currents, since the cycle took place over 360° of heading change.The existence of the ocean current profile, with current magnitude less at the towedbody depth than at the surface, was determined to be the cause. Based on the trialsdata, it was determined that the ocean current was on a heading of 50°, with a currentvelocity differential magnitude of about 1.95 kn. The static offset was believed tobe due to lateral towed body asymmetries, or to towed body gyro errors.


Fig. 21. Turn rate variation with time (USS Beary).

7.2. USS Brownsea trials

At-sea tests of the AN/SQS-35/18 towed sonar system were conducted aboard theUSS Jesse L. Brownduring the period from 20 through 23 February 1989. Thepurpose of the tests was to provide an updated database for continuing studies ofthe system simulation. These tests were conducted off the east coast sailing fromSavannah, Georgia. In contrast to the first at-sea tests conducted aboard theUSSDonald B. Bearyin August 1988, data were recorded automatically on tape using asystem developed by the EDO Corporation. Data were recorded at 2 s intervals, incontrast to the 30 s intervals by visual observation.

A master test maneuver was developed to ensure that sufficient data were obtainedto isolate the effect of ocean currents. The maneuver involved running on four pointsof the compass: North, East, South and West. On each leg, 360° turns were madeboth to port and to starboard at various ship rudder commands. The tests were con-ducted at a nominal speed of 15 kn on the straight run portions, with rudder com-mands of 15, 10 and 5°. A total of 6.5 h of data were recorded at 2 s intervals fora total of 11 700 data sets. The data sets are divided into three master files, one foreach rudder command.

A serious limitation was discovered during the process of analyzing the data. Theresolution accuracy of the data during the recording process was not high enoughto allow the data to be processed in the manner that was desired. The data accuracylimits were as shown in Table 6.


Fig. 22. USS Bearyturn rate/rudder characteristics.

The accuracy limitations on speed and heading are especially critical, since theyform the basis of the computations necessary to provide the input to the computersimulation. The problem manifests itself in the computation of the ship turn ratefrom the heading change. As an example, the data file from the 15° turn (file #2)indicated that the heading was changing between 1 and 2° over the 2 s time intervals.This, in turn, results in turn rates alternating between2 0.5 and2 1.0°/s over thetime intervals. When combined with the measured speed, which is alternatingbetween 13, 14 and 15 kn, the ship/towed body heading difference computed fromthe program QUICKTOW produces results which also oscillate.

7.2.1. Trials data analysisAs noted, the accuracy limits that were used in the on-board data system were

not high enough to evaluate the computer simulation in a real-time manner. Thisrequired that the data be processed in a time-averaged manner. Time samples of theorder of 5 min were chosen for the straight runs, while the time for a complete 360°turn chosen for the turns. These time samples were also of the order of 5 min. Theaveraged values of speed and ship turn rate were used as the input to the QUICK-TOW computer simulation.


Fig. 23. Ship/fish heading difference (trials data) variation with time (USS Beary).

7.2.2. Ship/towed body heading differenceThe averaged values of the ship/towed body heading difference for the seven

rudder deflection sets are shown in Fig. 30. The total number of data points that areplotted is equal to the total number of individual data files, which is 57. The nominal0° rudder deflection points represent the straight run tests. The remaining data pointsrepresent the turns for nominal rudder deflections of6 5, 10 and 15°. A linear curve-fit through the data shows an offset at the 0° rudder deflection. The linear fit indicatesan offset of 5.89°, while the numerical average of the 0° rudder deflection datais 4.60°.

The average speed and ship turn rate were computed for these same events. Usingthese values, the ship/towed body heading difference was computed using the com-puter simulation QUICKTOW. These data are shown in Fig. 31, also with a linearcurve-fit. The simulation at this point contains no correction for a yaw bias at the0° rudder case. Comparing Fig. 31 with Fig. 30 shows that the slope of curve is thesame; however, it is offset by a constant amount. The average of the straight towship/towed body heading difference offsets, 4.32°, is then added to the computedship/towed body heading difference in Fig. 32. This now brings the computed valuesby the simulation into excellent agreement with test values shown in Fig. 30.

This comparison is exaggerated if the difference between the test and the computedvalues of the ship/towed body heading difference, heading error, is plotted as shownin Figs. 33 and 34. Fig. 33 shows the values before the constant offset of 4.32° is


Fig. 24. Ship/fish heading difference (simulation data) variation with time (USS Beary).

added, and Fig. 34 shows the values after it is added. Also shown are the6 5°tolerance bands that have been the goal for the computer simulation. It is seen that,on the whole, the computer results are within this band with the exception of threepoints at the6 15° rudder deflections.

7.2.3. Ocean-current-induced behaviorNext, the magnitude of the ocean-current-induced changes in the ship/towed body

heading difference was evaluated. Cyclic behavior of the heading difference wasalso observed in theUSS Browndata similar to the observed behavior aboard theUSS Beary. During the tests conducted aboard theUSS Brown, complete 360° turnswere conducted which would allow a complete cyclic change to be observed. Twoocean current parameters may be extracted from the test data. These are:

1. difference in speed between the ship and towed body due to the depth-dependentocean current velocity profile; and

2. ocean current direction or heading.

Two assumptions are made when applying the ocean current correction. First, thecurrent does not affect the heading of the ship, but does affect the heading of thetowed body relative to the ship. This is reasonable since the tow cable is flexibleenough to provide a yaw degree of freedom, so that the towed body will try to alignitself with the current.

Second, it is the difference in speed between the ship and towed body that is


Fig. 25. Fish depth variation with time (USS Beary).

important. If the ocean current profile were constant with depth, no speed changein the towed body would be observed. This speed differential may be observed, inthe absence of a speed measurement at the towed body, by observing the change intowed body depth during the turn. The ocean current velocity typically decreaseswith depth. This means that the towed body speed will increase relative to the shipspeed when running with the current, and decrease when running into the current.This indicates that the towed body depth will decrease when running with the current,and increase when running into the current. These two conditions will thus determinethe direction of the current.

The data from theUSS Browntests were examined to determine the maximumship/towed body speed differential,DUc, and the current direction,uc. The averageship/towed body speed differential and current direction were:

Ship DUc (kn) uc (°)USS Beary 1.95 50USS Brown 1.13 346

The ocean current ship/towed body speed differential of 1.13 kn will result in acyclic heading difference between the ship and towed body headings which dependson the ship speed. TheUSS Browntests were conducted at a nominal speed of 15 knduring the straight runs. The ship speed decreased as the ship went into the steady


Fig. 26. Fish roll variation with time (USS Beary).

turns, down to approximately 13 kn for the6 15° rudder deflections. This equatesto approximately a6 5° cyclic variation during a 360° turn.

Two turn cases,6 15° rudder, were selected to demonstrate the effect of thecorrections for both the straight tow offset of 5.89° and the cyclic variation due tothe ocean current. Figs. 35 and 36 show the ship/towed body heading difference forthese turns. The two solid straight lines represent the computed values based on theaverage speed and turn rate for the event. The lower line is the value uncorrectedfor the straight tow offset, and the upper line is the value after the correction hasbeen applied. As can be seen, it is very close to the mean of the cyclic data. Thecorrection for the ocean current based on the mean values determined from the datawere then used to compute the cyclic variation, which is also shown as a solid line.Note that the spikes at about 270° are the data from the transient phases of the turnand can be ignored.

The difference between the heading difference test data and the computed meanwith the straight tow correction is shown in Figs. 37 and 38. Again the spikes at270° should be ignored since they represent the transient phase. The6 5° tolerancelines are shown to indicate that even without making the correction for the oceancurrent, the data are within the required tolerance. The difference between the testand computed data for these two turns are shown in Fig. 39 after applying the oceancurrent correction. The differences are seen to be reduced even further within the6 5° tolerance.


Fig. 27. Ship/fish heading difference variation with fish heading (USS Beary).

7.3. USS Bearyand USS Browndata comparison

The data that are presented in this section represent the mean, or averaged, datafor complete 360° turns. That is, the cyclic effect of the ocean current has beenfiltered out, but the static offset is still included. These data are presented in termsof the towed body/ship heading difference and towed body roll as a function of cablescope, speed and turn rate, i.e.,

DH, f 5 f(U0, c, S) (38)

where:

DH heading difference (towed body/ship) (°)f roll (towed body) (°)U0 speed (kn)c turn rate (°/s)S tow cable scope (immersed) (ft)

The data presented are from the tests aboard theUSS Bearyand aboard theUSSBrown. Data are presented for test conditions which were determined by each shipcrew to represent actual operation conditions. Table 7 outlines the conditions.

The data are presented in terms of towed body/ship heading difference,DH, and


Fig. 28. Fish depth variation with fish heading (USS Beary).

towed body roll,f, as a function of turn rate,c, for a fixed value of speed,U0, andscope. Thus:

(DH, f) 5 f(c) (39)

where (U0, S) 5 constant.Figs. 40 and 41 show the towed body/ship heading difference as a function of

turn rate for the fixed values of speed and scope. Both data sets have been fittedwith a linear curve and show that an offset at a zero turn rate of approximately thesame amount exists for both data sets. In fact, the offset and slope are close for bothdata sets, indicating that the system behavior for each ship is identical.

Both data sets are plotted together and fit with a single curve in Fig. 42. Theoffset is 4.82°, which agrees with the average for theUSS Brownstraight run dataof 4.69°. The average of these two values, 4.75°, is the value that was used as theoffset in the QUICKTOW program. The data from the MASTER simulation program,with the offset constant of 4.75° added, are also shown in Figs. 40 and 41. Thesedata are seen to lie almost exactly on the linear fit to the experimental data.


Fig. 29. Heading difference error variation with fish heading (USS Beary).

Table 6Data accuracy limits

Measurand Trials resolution/accuracy Required resolution

Speed 6 1 kn 6 0.1 knHeading (ship/towed body) 6 1° 6 0.1°Depth 6 1 ft 6 0.1 ftPitch 6 1° 6 0.1°Roll 6 1° 6 0.1°Water temperature 6 1° 6 0.1°CPO 6 1 ft 6 0.1 ftRudder 6 1° 6 0.1°

8. QUICKTOW metamodel modification

8.1. Version B metamodel

Version B of QUICKTOW differs from Version A by accounting for the offsetvalue of the ship/towed body heading difference when the system is towed on astraight course. The value of this offset was determined to be 4.75°, towed bodybow port relative to the ship heading. This constant is added to the symmetrical


Fig. 30. Fish yaw relative to ship (USS Brown).

ship/towed body heading difference computed from Version A. Thus the ship/towedbody heading difference, corrected for offset (°) is given by:

DH9 5 DH 1 4.75 (°) (40)

where DH is the computed symmetrical ship/towed body heading difference (°),Eq. (31).

8.2. Version C metamodel

The effect of the ocean current is a cyclic variation in the ship/towed body headingdifference as the ship executes a 360° turn. The effects of the ocean current on thetowed system are assumed to be as follows:

1. the ocean current magnitude and direction affect only the towed body headingrelative to the ship; and

2. the difference between the ocean current magnitude at the surface and at thetowed body depth affects the towed body.

The required inputs to the simulation are the direction of the ocean current,uc,and the difference in the ocean current magnitude at the sea surface and at the depth


Fig. 31. Ship/fish heading difference (sim.) variation with rudder angle (USS Brown).

of the towed body,DUc. The vector sum of the towed body velocity and the oceancurrent effect is then used to compute the new towed body heading due to the oceancurrent, Fig. 43. It is assumed that the towed body will align itself with this newvelocity vector. The computation begins with the computation of the towed bodyheading without the ocean current effect from QUICKTOW:

Hf9 5 Hs 2 DH9 (41)

u 5 uc 2 Hf9 (42)

5 difference between towed body and ocean current heading

a 5 tan−1F DUc(sin u)U0 1 DUc(cosu)G (43)

Hf" 5 Hf9 1 a (44)

where

Hs ship heading (trials data) (°)DH9 computed ship/towed body heading difference, corrected for offset (°),

Eq. (40)uc ocean current heading (°)DUc ocean current magnitude difference, ship to towed body (kn)


Fig. 32. Corrected heading difference (sim.) variation with rudder angle (USS Brown).

U0 ship speed (kn)

For comparison with the test data, the new ship/towed body heading differenceis computed:

DH" 5 Hs 2 Hf" (°) 5 ship towed body heading difference, (45)

corrected for offset and ocean current

whereHf" is the towed body heading corrected for ocean current (°), Eq. (44).

8.3. USS Browntrials data example

The above analysis may also be used to determine the ocean current direction,uc,and magnitude differential,DUc, from the trials data for a 360° steady turn. If it isassumed that the difference between towed body and ocean current heading,u, is90°, that is the towed body is directly abeam of the ocean current, then:

a 5 tan−1[DUc/U0] (46)

and


Fig. 33. Heading difference error variation with rudder angle (USS Brown).

DUc 5 U0 tana (kn) (47)

where a is determined from the minimum and maximum excursions of theship/towed body heading difference during a 360° steady turn when shown as afunction of towed body heading. The ocean current direction,uc, is determined fromthe towed body depth trace over the turn as a function of towed body heading. Whenthe towed body is going with the current, the depth will be a minimum, and if thetowed body is going into the current, the towed body depth is a maximum.

8.4. Test data comparison

The Version C algorithm is now compared to the test data from theUSS Browntrials. Two runs were used for this comparison (Table 8), rudder deflections of215° (port turn) and1 15° (starboard turn) at a nominal speed of 15 kn. The followingassumptions are made:

1. the transient conditions into and out of the turns are not modeled (transient datanot considered); and

2. the speed and turn rate are constant during the turn.

The test data are compared sequentially in Figs. 44 and 45 with the cumulativeresults from Versions A, B and C of QUlCKTOW:

1. symmetrical value (Version A);


Fig. 34. Heading difference error variation with rudder angle (USS Brown).

2. static offset added (Version B);3. ocean current added (Version C); and4. cumulative (Versions A1 B 1 C).

The comparison shows that QUICKTOW: Version C effectively models thebehavior of the towed body during the turn under the assumptions made.

9. Production FHS towing trials

9.1. USS Miller trials data analysis

The analysis of theUSS Browndata involved examining the test data to determinethe required modifications to the QUICKTOW algorithm. These modifications werethen integrated into the final production version of the FHS and evaluated duringthe last set of trials aboard theUSS Miller which were conducted on 6, 10 and 11March of 1990.

The following procedure was used to analyze theUSS Miller test data. The com-puted towed body heading is composed of the following components:

Hf 5 Hs 2 (DH 1 DH0 1 DHoc) (48)


Fig. 35. Heading difference (2 15° rudder)/ocean current correction (USS Brown).

5 towed body Heading (computed) (°)

where

Hs ship heading, measured (°)DH ship/towed body heading difference (°) (computed from QUICKTOW)DH0 static offset due to lateral towed body (°)DHoc cyclic variation due to ocean-current-induced difference in velocity

between ship and towed body (°)

The performance goal established for the FHS was that the difference betweenthe measured and computed towed body heading be between6 5°, where the erroris defined as:

Error 5 DH (computed)2 DH (measured) (49)

9.2. QUICKTOW—validation

The pre-production version of the FHS was installed on theUSS Millerand wasfunctioning, thus providing real-time speed and turn rate signal input into theQUICKTOW program. In addition to the towed body static offset and the ocean


Fig. 36. Heading difference/ocean current correction (USS Brown).

current cyclic effect problems that were encountered during the trials aboard theUSSBeary and theUSS Brown, two new developmental problems were encountered:

1. turn rate and speed input signal filtering; and2. transient ship/towed body phase lag.

9.2.1. Static offset correctionThe static offset was evaluated from a series of test courses where the ship ran

on headings of 0, 90, 180 and 270°. The mean of the data from theUSS Millertrialsdoes not compare well with the data from the previous trials:

Ship Static offset—bow port,DH0 (°)USS Beary 4.83USS Brown 4.69Mean 4.75USS Miller 2.39

The static offset correction was particularly bothersome. The difference in thevalues from theUSS Bearyand USS Brownand theUSS Miller (4.75 and 2.39°)suggest that a single value of the correction does not exist. Further analysis of datafrom the USS Miller for values of CPO of 100 and 300 ft showed:


Fig. 37. Heading difference error (2 15° rudder) (USS Brown).

CPO (ft) DH0 (°)100 0.0300 4.0520 2.39

This variation suggested that the problem may be due to gyro alignment error,rather then lateral asymmetries. The value used in the production FHS remainedat 4.75°.

9.2.2. Ocean current cyclic correctionBy analyzing the test data, the ocean current direction,uc, and the difference in

current speed between the ship and towed body,DUc, may be determined. An oceancurrent database, (Meserve, 1974), were used to determine the two required variablesat the locations of all three trials, and then compared with the results from the trialsdata analysis, Table 9.

While the values ofDUc anduc from the ocean current database are not in exactagreement with the values determined from the trials data, they are sufficiently closeto be used in the operational FHS, if desired. It should be pointed out that thelocations of the trials for the three ships were in an area of high ocean currentactivity. The maximumDH deviation,a, due to the ocean current will depend onthe ship speed andDUc. For example, taking the minimum and maximum values ofthe ship towing speed, as well as the minimum and maximum values ofDUc, then:


Fig. 38. Heading difference error (1 15° rudder) (USS Brown).

Speed (kn) DUc (kn) a (°)8 0.5 3.588 2.0 14.010 0.5 2.8610 2.0 11.3

The effect of the ocean current will be thus be within the6 5° margin if thevalue of DUc is less then about 1 kn. The inclusion of the ocean current correctionrequires that a database of ocean current statistics be available. However, it wasdecided that since all of the tests conducted were in area of high ocean currentactivity, and since the performance of the pre-production FHS was within the6 5°tolerance, this correction would not be included.

9.2.3. Transient effects during turnThe data from Course 1 for the transition from a 180° to the 270° course is used

to demonstrate the towed system behavior during the transition into and out of aturn. The ship and towed body heading signals are shown in Fig. 46. The steadyturn rate,c, is 1.238°/s while turning. What is evident is the lag between the com-puted heading difference and the measured value during the transient phases intoand out of the turn. The difference between the ship and towed body heading isshown in Fig. 47.


Fig. 39. Heading difference error with ocean current correction (USS Brown).

Table 7Comparison of test conditions

Ship Speed (kn) CPO (ft) Scope (ft) Rudder (°)

USS Beary 15 540 530 6 5, 10, 15USS Brown 15 520 510 6 5, 10, 15

This lag may be examined as the ship/towed body heading difference error. Thegoal during the development of the FHS was to maintain the error to within6 5°.The error, shown in Fig. 48, is seen to be within the desired6 5° during the steadyportion of the turn, while rising to6 30° during the transient portions into and outof the turn.

These transient errors are due to the fact that the MASTER program which wasused to develop QUICKTOW is a steady-state program. To model the transient, ortime-dependent, behavior would require a time-domain program. This was recog-nized during the development of the FHS; however, until the mathematical basis forthe MASTER program had been verified through field tests, it was decided to expendno effort in this direction. A further reason was the fact that the MASTER programrequires from 30 to 60 s to execute. A time-domain program would require evenmore time, thus violating the requirement for real-time computation.


Fig. 40. Ship/fish heading difference characteristics (USS Beary).

9.3. QUICKTOW modifications

9.3.1. Turn rate signal filterAn alterative solution to the problem was developed by the EDO Corporation and

APS Corporation, the subcontractor responsible for the design and manufacture ofthe pre-production and final versions of the FHS system. As noted, the turn rate,c, signal required filtering before being input to the QUICKTOW algorithm. Inaddition, this filter was designed to address the lag and lead exhibited by the towedbody going into and out turns. The turn rate filter that was implemented in the FHSsoftware is a two-stage filter providing “slow attack” when entering turns, and “fastrelease” when exiting turns. The Stage 1 filter is a linear control filter. Asimplemented, this is an exponential IIR filter (infinite impulse response) the orderof which is a function of two variables:

1. ship speed (U0) (kn); and2. cable payed out (CPO) (ft).

The filter operates on the turn rate,c, signal input which is computed from twoconsecutive ship heading,Hs, readings. The magnitude of this value (filtered turnrate) is used to provide variable bandwidth control for the Stage 2 filter. The Stage2 filter is a nonlinear exponential IIR filter implemented in two parts. Part 1 utilizesthe present and previous values of the filtered turn rate signal and CPO to calculate


Fig. 41. Ship/fish heading difference characteristics (USS Brown).

a control parameter that is used in computing the filter order for Part 2, which isthe final exponential IIR filter. The difference in magnitude of the two values is anindication of whether the ship is entering a turn or leaving it. Entering a turn setsthe control parameter to high filter order and a correspondingly fast (lightly damped)filter response, providing the “slow-attack” performance. Leaving a turn results in arelatively low value of the control parameter and a correspondingly fast (lightlydamped) response from the IIR filter, providing the “fast-attack” performance. Thisvalue of the two-stage filtered turn rate signal is input to the QUICKTOW algorithmfor computing the ship/towed body heading difference.

10. Conclusions

The purpose of the towed body heading system (FHS) was to replace the towedbody mounted heading gyro which has been an unreliable piece of equipment. Thegoal of the development was to develop a computer simulation that could beimplemented in hardware (FHS) to replace the heading gyro. The purpose of theFHS would be to determine the towed body heading in response to ship maneuvers,so that it could be related to the ship heading.

A three-dimensional steady-state mathematical simulation of the towed systemwas developed that models each of the five system components including:


Fig. 42. Heading difference composite (USS Bearyand USS Brown).

Fig. 43. Ocean current vector diagram.

Table 8Data used for comparison of Version C algorithm withUSS Browndata

File # Rudder (°) Speed (kn) Turn rate (°/s) Scope (ft) DH DH9

21 215.0 12.825 20.81 510 225.10 220.3523 115.0 12.798 1.022 510 31.21 35.96


Fig. 44. QUICKTOW/test data comparison (2 15° rudder).


Fig. 44. Continued.


Fig. 45. QUICKTOW/test data comparison (1 15° rudder).


Fig. 45. Continued.


Table 9Comparison ofuc and DUc from trials analysis and database

Ship Latitude Longitude DUc (kn) uc (°)

Database Trials Database Trials

USS Beary 36°309N 74°309W 2.2 1.95 54 50USS Brown 31°309N 78°309W 0.8 1.13 38 346USS Miller 39°409N 72°009W 1.2 0.53 231 15

Fig. 46. Ship/towed body heading signals during turn (USS Miller).

1. the faired tow cable;2. the towed body;3. the array tow cable;4. the towed array; and5. the drogue.

The simulation was able to compute the difference between the ship and towedbody headings in response to ship turn maneuvers. Inputs to the simulation includeship speed and ship turn rate, as well as the geometry and mass properties of eachsystem component. The execution time precludes its use in the production version


Fig. 47. Turn rate signal comparison (USS Millerdata/QUICKTOW simulation).

of the Fish Heading System (FHS), since it requires essentially real-time compu-tation. In order to solve this problem, a metamodel was developed which is analgebraic-based simulation. The MASTER model was executed over a matrix ofvalues of the ship speed and turn rate to determine the ship/towed body headingdifference. The results were then curve-fitted with algebraic equations that wereimplemented in a second mathematical model known as the QUICKTOW model.

A series of at-sea towing trials was conducted over a period from December 1988through April 1990 aboard five different Knox class Frigates off of the east coastof the United States. The trials were conducted with AN/SQS-35 towed body thathad still functioning towed body heading gyros to provide evaluation data for themetamodel.

The data validated the basic QUICKTOW algorithm, but also indicated problemsthat required modifications: (1) a static towed body/ship heading bias exists whichis due to lateral asymmetries in the towed body geometry. Although all of the towedbody tested had values of the same sign, there were slight variations in the absolutevalues; (2) as the ship executed a full 360° turn maneuver, a cyclic variation in thetowed body/ship heading difference was found. This was determined to be due tothe existence of ocean currents in the test areas. Because of the ocean current velocitydistribution with depth, a difference in velocity existed between the ship and thetowed body. This velocity difference, in addition to the ocean current direction,resulted in the cyclic variation.


Fig. 48. Heading difference error during turn (USS Miller).

The MASTER simulation, which was a steady-state model, does not address thetransient or time-dependent behavior, which existed into and out of the turns, forexample. As an alternative to modifying the MASTER algorithm to the time domain,an additional filtering scheme was developed by EDO personnel and the subcontrac-tor that solved this problem. The MASTER and QUICKTOW algorithms were vali-dated for use in the production FHS version. Required modifications were identifiedand solutions were developed and implemented. The final algorithm version wassuccessful in meeting the goal of measuring the ship/towed body heading differencewithin 6 5°.

Acknowledgements

The research in this paper was sponsored by Mr Eman Marder, formerly ProgramManager of USN/FMS In-Service Sonar Systems for the EDO Corporation.

References

Calkins, D.E., 1979. Hydrodynamic analysis of a high-speed marine towed system. Journal of Hydronaut-ics 13 (1), 000–000.


Calkins, D.E., 1987. Phase I: A Three-Dimensional Steady State Towed Body/Towline Computer Pro-gram. University of Washington.

Janes, 1990/91. Janes Fighting Ships. Janes Publishing Company Limited, London, England.Law, A.M., Kelton, W.D., 1991. Simulation Modeling and Analysis. McGraw-Hill, Inc. New York, NY.Meserve, J.M., 1974. US Navy Marine Climatic Atlas of the World, Volume I—North Atlantic Ocean.

Naval Weather Service Command, NAVAIR 50-1C-528 U.S. Government Washington, DC.Patten, M.J., Olive, R.W., 1967. Underwater towed sonar vehicles, stability derivatives for two, the

AN/SQA-13 and the hydrospace towed bodies. Naval Undersea Warfare Center (NUWC)-TN-19, SanDiego, CA.

Society of Naval Architects and Marine Engineers, 1989. Principles of Naval Architecture, Volume III—Motion in Waves and Controllability, Jersey City, NJ.

Walton, C.O., Brillhart, R.E., 1966. The stability derivatives of the scheme a towed body used withthe AN/SQA-13 (XN-1) variable depth sonar system. David Taylor Model Basin Report 153-H-01,Carderock, MD.

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