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Ocean Engineering 30 (2003) 453470www.elsevier.com/locate/oceaneng

Dynamics and control of a towed underwatervehicle system, part I: model development

B. Buckham a, M. Nahon c,, M. Seto b, X. Zhao a, C. Lambert a

a Mechanical Engineering, University of Victoria, P.O. Box 3055, Victoria, BC, Canada, V8N 6K8b Defence Research Establishment Atlantic, P.O. Box 1012, Dartmouth, Nova Scotia, Canada, B2Y 3Z

c Mechanical Engineering, McGill University, Montreal, Quebec, Canada, H3A 2K6

Abstract

Autonomous vehicles are being developed to replace the conventional, manned surfacevehicles that tow mine hunting towed platforms. While a wide body of work exists thatdescribes numerical models of towed systems, they usually include relatively simple models

of the towed bodies and neglect the dynamics of the towing vehicle. For systems in whichthe mass of the towing vehicle is comparable to that of the towed vehicle, it becomes importantto consider the dynamics of both vehicles. In this work, we describe the development of anumerical model that accurately captures the dynamics of these new mine hunting systems.We use a lumped mass approximation for the towcable and couple this model to non-linearnumerical models of an autonomous surface vehicle and an actively controlled towfish. Withinthe dynamics models of the two vehicles, we include non-linear controllers to allow accuratemaneuvering of the towed system. 2002 Elsevier Science Ltd. All rights reserved.

Keywords:Lumped mass; Underwater vehicle; Cable dynamics; Numerical simulation

1. Introduction

The use of towed cable systems in acoustic surveying, mine hunting, and underseacable laying applications has precipitated extensive research into the behaviour ofmarine cables. Because of non-linear cable geometry and the dominance of the non-linear hydrodynamic forces, the solution of the cable dynamics equations in the time

Corresponding author.

E-mail address: [email protected] (M. Nahon).

0029-8018/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 9 - 8 0 1 8 ( 0 2 ) 0 0 0 2 9 - X

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454 B. Buckham et al. / Ocean Engineering 30 (2003) 453470

domain requires that numerical models be used to approximate the governing non-

linear differential motion equations. Over the last thirty years, a significant body ofresearch has led to the development of numerical cable models which can assist in

the accurate positioning and navigation of long (kilometer scale), submergedtowed systems.

In general, numerical models of towed cables are obtained by spatially discretizing

the cable into finite linear segments. Walton and Polacheck (1960) described alumped parameter, or lumped mass, approach in which the cable was considered as

a concatenation of inextensible linear elements. The mass of the cable was concen-trated, or lumped, at frictionless spherical joints connecting these linear elements.

Applying an heuristic procedure, the hydrodynamic, buoyancy, and gravitational

forces acting over the linear elements were also lumped at the node points. This

lumping of the mass and environmental forces allowed the motion of the cable node

points to be represented by a series of ordinary differential equations. The main

advantages of such a scheme are the relative ease of dealing with the strong non-

linearities associated with the hydrodynamic loads, and the ability to assemble a

compact numerical model from the linear elements. As reported by Kamman and

Huston (1999), the transfer of inertia that occurs when defining the discrete cableelements in this way results in a computationally efficient and accurate discrete rep-resentation of a towcable. For these reasons the lumped mass approach has formed

the basis for a large body of towed systems research.

In earlier works, the lumped mass cable model was used to simulate the motion

of towed systems during relatively steady towship manoeuvers. Paul and Soler (1972)presented a two dimensional formulation in which the inertial forces were considered

insignificant. This quasi-static implementation was used to solve for the motion ofa towed system during straight tows at a steady speed. Both Chapman (1984) and

Sanders (1982) presented three-dimensional quasi-static lumped parameter models.

Chapman calculated the steady state profile of a towcable during straight tows andturns of varying diameters. This work showed that a towed system undergoes largetransient motions as it enters and exits the turns. This emphasized the need for cable

models to include inertial effects in order to accurately capture these transient

motions. Such lumped mass implementations have been presented by Huang (1994);

Vaz and Patel, (1995); Vaz et al. (1997), and Driscoll et al. (2000). Both Driscoll andHuang accounted for the elasticity of the towcable by using linear spring elements. In

both cases, the spring stiffnesses were calculated from the material properties of

the towcable.

Towed systems continue to play an increasingly important role in many marinemeasurement and salvage operations. This is especially true in the case of mine

countermeasures, a task for which semi autonomous towed vehicle systems are being

developed. These systems, such as International Submarine Engineerings DOL-PHIN-AURORA configuration, remove the human operator from a dangerous work-space, and this presents obvious advantages over conventional methods (Seto et al.,

1999). The DOLPHIN-AURORA system consists of a semi-autonomous, high speed(up to 15 knots) DOLPHIN semi-submersible towing an actively controlled AUR-

ORA towfish outfitted with sidescan sonar (Houle and Seto, 1999). In order to design

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455B. Buckham et al. / Ocean Engineering 30 (2003) 453470

the system, and devise control methodologies that provide reliable service, knowl-

edge of the systems dynamic response is essential. Such an evaluation must considerthe response of the surface vehicle to control inputs and surface effects, the dynamics

of the elastic tow cable, the response of the towfish to control surface deflections,and the interactions between all three components of the system. As it is much less

expensive than prototype testing, a computer model that accurately captures all of

these considerations is a preferred design tool.

The developments of previous works fall short of meeting these requirements.

Existing cable models have not considered the interaction of the towcable with boththe towing vehicle and the towed body. In a majority of earlier works the towed

body considered was a passive object: a spherical or cylindrical body with constant

drag coefficients. In this case, the towed body dynamics can easily be incorporatedinto the cable dynamics model at a node point as demonstrated by Paul and Soler

(1972); Sanders (1982); Huang (1994); Palo et al. (1983), and Delmer et al. (1983).

Recently, more realistic representations of the towfish have been coupled to the bot-tom end of numerical cable models. In these cases the full six degree of freedom

motion equations for the towfish are formed independent of the cable model and thetension of the towcable is applied as a thrust force within the towed vehicle model.

Such formulations have been given by Kamman and Huston (1999); Makarenko et

al. (1997); Banerjee and Do (1994); Sun and Leonard (1998), and Wu and Chwang

(2000 & 2001). However, in all of the surveyed literature it was assumed that the

towcable forces had an insignificant effect on the dynamics of the towing vehicle.

As such, the towing vehicle dynamics were replaced with a set of kinematic boundaryconditions. With the advent of smaller unmanned semi-autonomous towing vehicles

this assumption is no longer valid.

In this paper, Part I of a two part work, a lumped mass model of a towed, armoured

cable is developed and coupled to models of a semi-submersible DOLPHIN and a

neutrally buoyant AURORA towfish. The series of differential equations definingthe motion of the lumped mass nodes and the two vehicles are solved using anexplicit integration scheme in which all time differentials are expressed in terms of

values from the previous time step. We describe how the numerical models of two

streamlined vehicles are assembled to include the effects of the hull geometry and

the control surfaces. We also present a non-linear control methodology for the systemthat determines the control inputs to the towing vehicle and the towfish based onfeedback of the vehicle state vectors. In Part II, the mathematical model will be

validated and then used to optimize turning maneuvers.

2. Cable model

To aid in the modelling of the cable, we define an inertial reference frame, anda sequence of moving frames attached at points along the cable. Referring to Fig.

1 which illustrates the towed system, the inertial reference frame (X, Y, Z) is fixedat the surface. A horizontal direction, X, is defined positive to the right, the verticaldirection, Z, is defined positive downward, and the Ydirection completes this right

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Fig. 1. An illustration of the towed system taken from Houle and Seto (1999). In this planar view the

p2 axis is into the page for each body fixed frame.

handed coordinate system. The armoured cable is discretized into a series ofn elastic

elements with the element mass lumped at the n+1 node points. The lumped massapproximation allows for the motion of each node to be calculated independently in

the three degrees of freedom, but the elastic internal force of the element constrains

this motion of the nodes. The calculation of the elastic tension, damping, and hydro-

dynamic drag forces is performed in a body-fixed reference frame. Each cableelement has a body frame p1, p2, and q attached to it where p1 is normal, p2 is bi-

normal, and q is tangent to the cable element.

2.1. Kinematics

The orientation of each discretized cable element is represented using a Z-Y-X

(yi,qi,f i) Euler angle set (Etkin, 1972). These successive rotations align the inertialframe with the ith body frame. The torsion of the cable is not considered in the

simulation, and, as a result, only two of the three Euler angles are required to specify

the orientation of each cable element. As such, the initialyi rotation about the inertialZ axis is constrained to zero, an approach that is consistent with a wide body of

works (Ablow and Schechter, 1983) (Burgess, 1992). The orthogonal rotation

matrix, RiIB, describes the mapping from the local, body-fixed frame of the ith cable

element to the inertial frame. Applying the specified Euler angle set, the rotationmatrix from the body fixed frame of the ith cable element to the inertial framebecomes:

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RiIB

cosqi sinqisinfi sinqisinfi

0 cosfi sinfi

sinqi cosqisinfi cosqicosfi

(1)

The Euler angles can be calculated at any instant in the simulation provided thatthe endpoints of the cable element are known. For example, consider the ith element

of the cable, shown in Fig. 2, that is bounded by nodes i-1 and i. When expressed

in terms of the body fixed frame, the only non-zero component of the vector li is inthe tangential direction. Therefore we have:

RiIB00

li

ri

X

ri

X

riYriY

riZriZ

(2)

where li, the length of the ith element, at any instant in time is given by,

li (riXri1X )2 (riYri1Y )2 (riZri1Z )2

and ri is the position vector, with components riX, riY, r

iZ, describing the location of

the ith node in the inertial frame. Substitution of (1) into (2) results in the following

set of non-linear equations:

li sinqi cosfi (riXri1X ) (3)

li sin fi (riYri1Y ) (4)

licosqicosfi (riZri1Z ) (5)

Combining Eqs. (3) and (5),

qi atan2(riXri1X ,r

iZr

i1Z ) (6)

Fig. 2. The ith element of the discretized cable is bounded by the i-1st and ith nodes.

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The solution for fi can now be found from Eq. (4) and either of (3) or (5). Wetherefore choose to use the combination that leads to the best numerical stability,

as follows:

ifcosqi sinqi

fi tan1(riYri1Y ),(riZri1Z )cosqi , (7)

ifcosqi sinqi

fi tan1(riYri1Y ),(riXri1X )sinqi

, (8)Thus Eqs. (6), (7), and (8) are our desired relations which allow us to find theEuler angles from the node positions of the discretized cable.

2.2. Internal forces

Internal forces are generated by the cables elastic behaviour. The methodologyfor calculating these forces in three dimensional cable modelling applications has

been presented by Huang (1994). With this approach, the tension within the cable

element, Ti, acts in the tangential direction q of the element, and is modeled bya linear function of the strain within and the axial stiffness of the discrete

cable elements:Ti EAei

ei liliu

liu

where liu is the unstretched length of the ith cable element, A is the cross section

area of the cable element, E is the effective Youngs modulus of the cable, and ei

is the strain experienced within the ith element. The friction between the braids of

the cable, along with the polymer coatings that protect the conductors contained in

the cable core, create a damping effect. This effect is assumed to be linear with thefollowing relationship between tangential strain rate and damping force. The axial

force in the ith element generated by damping is:

Pi Cv(viqv

i1q )

where viq is the component of the ith node velocity in the tangential direction q, and

Cv is an internal viscous damping coefficient for the ith element.

2.3. External forces

The external forces acting on a cable element are those generated by the surround-ing environment. These include hydrodynamic drag, weight, and buoyancy. The drag

forces on the ith cable element can be calculated according to:

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Dip1 1

2rwCddcl

iufp|v

i|2vip1

(vip1)2 (vip2)2

Dip2 1

2rwCddcl

iufp|v

i|2vip2

(vip1)2 (vip2)2

Diq sgn(viq)

1

2rwCddcl

iufq|v

i|2

where Dip1, Dip2

and Diq are the components of the hydrodynamic drag when rep-

resented in the body-fixed frame; rw is the density of the water; dc is the cablediameter; Cdis the normal drag coefficient of the cable; and v

i is the velocity of the

geometric centre of the ith cable element with respect to the surrounding fluid, with

components vip1, vip2

, and viq. The drag coefficient is modified by loading functionsfp and fq, which are functions of, h, the relative angle between the ith cable elementand the incident fluidflow. The loading functions account for the non-linear breakupof drag between the normal and tangential directions as discussed by Driscoll and

Nahon (1996).

fp 0.50.1cos(h) 0.1sin(h)0.4cos(2h)0.11sin(2h)

fq 0.01(2.0080.3858h 1.9159h24.1615h3 3.5064h41.1873h5

where h is expressed in radians and 0h

p

2. The relative velocity of the flow overa particular cable element is found by using the equations of relative motion between

points on the elements and the end nodes to interpolate the velocity of the geometric

centre of the element. Once the drag for ith and and i+1st element are calculated,half of each value is applied to the ith node that joins the two elements.

The mass and buoyancy of the ith cable element are given by,

mic rcVic

Bi rwVic

where rc is density of the cable, g is the acceleration due to gravity, and Vic is the

volume of the ith cable element.

The effects of added mass are accounted for in the cable element mass matrix.

Assuming there are no added mass effects in the tangential direction, the mass matrix

for the ith cable element, expressed in terms of the ith body frame, becomes:

MiB

mic mia 0 0

0 mic mia 0

0 0 mi

c

where mia rwVic is the added mass of the cylindrical cable element. In the govern-

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ing equations of motion, we require a representation with respect to the inertial frame.

The mass matrix is therefore transformed using the rotation matrix RiIB. The mass

attributed to the ith node is formed from contributions of elements above and below

the node. Thus, the ith nodal mass matrix, expressed in terms of the inertialframe, becomes:

MiI1

2RiIBM

iBR

iTIB

1

2Ri+1IBM

i+1B R

i+1TIB

2.4. Assembly of forces

Applying Newtons Second Law to each node of the cable, a series of equationsgoverning the motion of each node in the inertial frame is created. To evaluate

this equation of motion, the nodal forces described above are resolved into inertialrepresentation for use in the equations of motion of the cable element.

MiIri (Ti+1 Pi+1)(Ti Pi)

1

2(Di Di+1 micg m

i+1c g)

1

2(Bi Bi+1) (9)

where Di is the hydrodynamic drag on the ith element expressed in the inertial frame,

Ti is the elastic tension force generated in the ith element, P i is the internal damping

force generated in the ith element, and Bi is the buoyancy force on the ith element.

Eq. (9) is applied over the interior nodes of the discretized towcable, 1in1.

At the boundaries of the cable, the terminations to the towing vehicle and the tow fish,the cable motion is defined by the dynamics of the two vehicles. Considering thevehicles to behave as rigid bodies, the motion of the cable terminations can be

derived from the motion of the two vehicles. Thus, we eliminate the top (i 0) and

bottom (i n) cable nodes from the analysis and introduce dynamics models forthe towing vehicle and towfish.

3. Vehicle modelling

Referring to Fig. 3, one can see that the DOLPHIN and AURORA vehicles areboth streamlined and possess a longitudinal plane of symmetry. The DOLPHIN semi-

submersible is a diesel-powered vehicle. The faired mast houses an air intake and

the keel contains a winch that controls the payout of the towcable. The AURORA

towfish is actively controlled by a large hydrodynamic depressor (depth control) andfour tailplanes (attitude control). The tailplanes can be configured in either a standardcross-configuration or in an X-configuration (shown). Fortunately, for these vehiclegeometries it is possible to construct an accurate non-linear dynamics model using

the approach outlined by Nahon (1996).

The numerical non-linear models are constructed by decomposing each vehicle

into its constituent elements. We consider the hydrodynamic forces to act througha reference point on each of the component surfaces, called the centre of pressure.

We apply kinematic relationships to determine the velocity of these reference points,

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Fig. 3. The DOLPHIN (a) and Aurora (b) vehicles (Seto et al., 1999). A vehicle frame of reference (x,

y, z) is defined for each vehicle.

and empirical relations to calculate the resulting hydrodynamic forces for the individ-ual components. Summing these component effects with the gravitational and buoy-

ancy forces attributed to each of the vehicles, we calculate the forces and moments

acting on the vehicles. Writing Newtons Second Law and Eulers equations in termsof the vehicle frames of reference, we solve for the resulting six degree of freedom

vehicle motion. Since this method does not linearize the hydrodynamic character-

istics, it captures the non-linear behaviour of the vehicle.

3.1. Vehicle kinematics

The orientation of the vehicle frames with respect to the inertial reference frameis defined by a Z-Y-X (V, V, V) Euler sequence of rotations. The superscript Vhas been introduced to differentiate between vehicles. As we are replacing the top

cable node with the towing vehicle dynamics, all quantities associated with the tow-

ing vehicle will be identified by V 0. Likewise, for the towfish, V n. Referringto Fig. 3, the vehicle reference frames are right handed systems attached to the hulls

at the mass centres. In each case, the surge direction, x, points to the nose of the

vehicle, the heave direction, z, points to the bottom of the vehicle, and the y axis

completes the coordinate system. Within these frames the vehicles translationalvelocities in the x, y, and z directions are given by u, v, and w respectively. The

angular rates about the x, y, and z axes are defined as p, q, and r respectively.The transformations between the vehicle frames and the inertial frames are defined

by RVIV, which is the rotation matrix generated from the Z-Y-X (V, V, V) Euler

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angle set. At any instant, the body rates p, q, and r can be used to obtain the time

rates of change of the Euler angles according to:

b V

V

V Qp

q

r,Q

1 sinVtanV cosVtanV

0 cosV sinV

0 sinVsecV cosVsecV

3.2. Formulation of the motion equationsApplying Newtons Second Law to the towing vehicle and towfish results in trans-

lational equations of motion of the form:

f(h)x f(p)x (m

VgbV)sinV mVx (u qwrv)

f(h)y f(p)y (m

VgbV)sinVcosV mVy (v rupw)

f(h)

z f(p)

z (mV

gbV

)cosV

cosV

mV

z (w pvqu)

(10)

where f(h)is the overall hydrodynamic force (with components f(h)x , f(h)y , and f

(h)z in the

vehicle frame), f(p)is the overall propulsive force (with components f(p)x , f(p)y , and

f(p)z ), and mVg and bV are the magnitudes of the gravitational and buoyancy forces,

respectively, which act in the Z direction. The plane of symmetry of each vehicle

reduce Eulers equations to:

m(h)x m(p)x (zbsincos)b

V IVxx p(IVyyI

Vzz)qrI

Vxz(rpq)

m(h)y m(p)y (xbcoscos zbsin)b

V IVyy q(IVzzI

Vxx) prI

Vxz(r

2p2)

m(h)

z m(p)

z (xbsincos)bV

IV

zz r(IV

xxIV

yy) pqIV

xz(pqr)(11)

where m(h) is the net moment of the hydrodynamic forces (with components m(h)x ,

m(h)y , and m(h)z in the vehicle frame) about the vehicles mass centre, m

(p) is the

moment produced by the propulsive device, and the buoyancy centre of the vehicle

is defined in terms of the vehicle frame by the vector, rb {xb 0 zb}T.

Added mass coefficients were calculated for the towing vehicle and towfish basedon the component surfaces of the vehicles. The added mass for each of the compo-

nent bodies in all three axes of the vehicle frame was calculated based on the

expressions presented by Newman (1977) for the added masses of cylinders andellipsoids that bounded each of the components. The added mass quantities were

included in the inertial terms of Eqs. (10) and (11). For example,

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mVx mV mAx

wheremAx is the added mass produced by acceleration of the vehicle in the x direction.

To evaluate Eqs. (10) and (11) for both the towing vehicle and towfish, it remainsto define the hydrodynamic and propulsive forces and moments.

3.3. Towing vehicle

The DOLPHIN is decomposed into ten basic components: the streamlined hull,

four horizontal planes (two forward and two aft), the upper and lower vertical tail-

planes, the mast, the keel, and the streamlined bulb attached to the bottom of thekeel. As described by Nahon (1996), the hydrodynamic forces acting over each

component are calculated as follows:

1. Determine the angle of attack the horizontal planes and the angle of sideslip for

the rudder, keel and the mast, based on the velocity of the centre of pressure of

each surface. In terms of the vehicle frame, the kth component surface has a centre

of pressure located at rk {rkx rk

y rk

z}T. The velocity of the centre of pressure is

given by:

Vk

u rkzqr

kyr

v rkx

rrkzp

w rkyprk

xq

The angle of attack, ak, for the four horizontal planes, and the angle of sideslip,bk, for the keel, mast and rudder, are given by:

ak tan-1vkzvkx

b

ktan

-1

vky

vkx2. Superimpose the control deflections of each active surface, dk, with the angle of

attack of the four horizontal planes and with the angle of sideslip of the rudder

to produce effective angles of attack, ak, and sideslip, bk, for these active sur-faces. ak ak dk , bk bkdk

3. Calculate the three dimensional lift and drag coefficients of the surfaces, basedon the effective angles of attack and sideslip, for the horizontally and vertically

oriented surfaces respectively.

4. Using empirical relationships, calculate the lift and drag coefficients for the hull

and the keel bulb.5. Dimensionalize the lift and drag forces for each component using the velocity of

the centre of pressure of each surface.

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6. Evaluate f(h) by transforming the resulting lift and drag forces through the angles

of attack and sideslip into a vehicle frame representation.

7. Sum the moments of the lift and drag forces about the vehicle centre of mass to

solve for m(h).

The modelling of the mast involves some additional intricacies. The vehicle oper-

ates at a depth of 2.5 to 3 hull diameters to its hull centreline, at which point the

mast is approximately 70% submerged. The mast consists of a vertical pipe faired

by free-swiveling drag-reducing sections that pivot about the pipes centreline. Sincethey are free to swivel, it is assumed that the chord line of the mast will remain

parallel to the flow. This implies that side forces will not be generated and only thedrag need be considered. The wave drag acting on the mast is included in the model

as a discrete, velocity-dependent drag force acting on the mast at the free surface.

The magnitude of the wave drag is taken to be proportional to the speed squared

and is determined from measured data on a scaled model of DOLPHIN (Butt et

al., 1998).

The DOLPHIN vehicle is propelled by an engine-propeller combination which is

represented by including a thrust, tp, directly in the vehicle motion equations. As

this thruster is in line with the surge axis of the vehicle, it produces no moment

about the mass centre of the DOLPHIN. The effect of the towcable on the towing

vehicle is included in the propulsion term. The termination of the towcable on DOL-

PHIN is defined by a constant vector, rtow, expressed in the vehicle frame of refer-

ence. The termination point is coincident with thefirst node, node 0, of the discretizedtowcable, and the internal and external forces associated with the top cable node areapplied at this point on the towing vehicle. Referring to Eq. (9), the DOLPHIN

propulsive force is thus defined by:

f(p)x

f(p)y

f(p)z tp

0

0 RTIV(T1 P1) 12(D1 m1cgB1) (12)3.4. Towfish

The AURORA towfish is decomposed into six components: the hull, the hydrody-namic depressor, and the four tailplanes. For the hydrodynamic depressor and hull,

the calculation of the hydrodynamic forces and moments follows the process outlined

above. However, theX configuration of the tailplanes requires an additional refer-ence frame to be introduced. As shown in Fig. 4, this new frame is aligned with the

tailplanes and facilitates a calculation of the tailplane forces that is consistent with

the procedure outlined for the towing vehicle. The tailplane forces are transformed

back to a vehicle frame representation prior to the assembly of the vehicle hydrodyn-amic forces. The tailplane frame is related to the vehicle frame by a rotation about

the towfish x-axis of45.

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Fig. 4. The X configuration of the AURORA tailplanes requires an additional tailplane fixed frame

of reference (tp) to be used in the calculation of the tailplane hydrodynamic forces.

The towing vehicle propels the towfish vehicle via the tension in the towcable.Analogous to the case of the towing vehicle, the internal and external forces attri-

buted to the last cable node, node n, are applied at the cable termination point. Thus,

the propulsive force for the towfish becomes:

f(p)x

f(p)y

f(p)z RTIV12(Dn mncgBn)(Tn Pn) (13)The propulsive moment for the AURORA is found by taking the moment of the

force given in (13) about the mass centre of the AURORA.

4. Control methodology

The control surfaces of the towing vehicle and towfish are actively controlledusing both closed loop feedback of the vehicle state, which is provided by on board

instrumentation, and open loop control signals. To facilitate the study of turningmanoeuvers, the controller is designed to provide:

1. constant power propulsion.

2. a desired operating depth for the towing vehicle and the towfish.3. a level pitch and roll of both the towing vehicle and the tow fish.4. tracking of a turn profile, defined by waypoints.

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4.1. Towing vehicle controllers

The constant power is the product of the equilibrium thrust, TE, and steady speed,Usteady, during a straight steady tow. During manoeuvering, the thrust, T, for an

instantaneous speed U is calculated using:

T TEUsteady

U

Since the forces and moments transferred from the towcable are too large for the

horizontal tailplanes (tp) to equalize, active control of the front (fb) and rear (rb)ballast is also required in order to maintain a desired depth and level attitude. To

dedicate the foreplanes (fp) to depth correction, the tailplanes are used to maintaina level attitude and a zero roll angle by superimposing a differential aileron deflectionover the tailplane deflection produced for the correction of the vehicle pitch.

The desired path of the towing vehicle is defined by a series of waypoint locationsthat were calculated along the desiredflight path of the towing vehicle. To manoeuverthe towing vehicle through the waypoints, rudder (r) inputs were generated based

on proportional feedback of the difference between the current heading of the towing

vehicle and the direction to the next waypoint location. As shown in Fig. 5, V is

the current vehicle heading and P is the direction to the desired waypoint. Both

angles are measured relative to the X axis of the inertial frame of reference. When

the towing vehicle reaches within two meters of the desired waypoint, it is considered

to have reached its target and the next waypoint in the sequence becomes the new tar-

get.

To ensure a straight steady tow of the system at the nominal speed it is necessary

to also make provision for open loop control inputs for the horizontal tailplanes,

aileron deflection, front and rear ballast, foreplanes, and rudder, dtp0 , da0, d

fb0 , d

rb0, d

fp0 ,

and dr0 respectively. We combine the closed loop PID feedback and the open loopinputs into the following control scheme:

Fig. 5. Definition of waypoint navigation angles.

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dtp

da

dfb

drb

dfp

dr dtp0

da0

dfb0

drb0

dfp0

dr0 0 0 0 K

0

Ptp K

0

Dtp 0 0 0

0 0 0 0 0 K0

Pa K

0

Da 0

KZPfb KZIfb

KZDfb K0

Pfb K

0

Dfb 0 0 0

KZPrb KZIrb

KZDrb K0

Prb K

0

Drb 0 0 0

KZPfp KZIfp

KZDfp 0 0 0 0 0

0 0 0 0 0 0 0 KPr

r0Z

r0Zdt

r0Z

0

0

0

0

0

where dtp, da, dfb, drb, dfp, anddr are the closed loop horizontal tailplane deflections,aileron deflection, buoyancy changes of the front and rear ballast, foreplane deflec-

tions, and rudder deflections respectively; r0Z,r0Zdt, and r0Zare the DOLPHINdepth error, integral of depth error, and time derivative of depth error; 0

VP is the error in the heading of the DOLPHIN; 0 and 0 are the

DOLPHIN pitch angle error and its time derivative; 0 and 0 are the DOLPHINroll angle error and its time derivative; and dk0 is the open loop control input for thekth control surface.

4.2. AURORA controllers

The control scheme for the AURORA is similar to that of the DOLPHIN. Refer-

ring to Fig. 4 for the tailplane numbering, the controller is implemented in the form:

dtp1

dtp2

dtp3

dtp4

Kn

Ptp1K

n

Ptp10

Kn

Ptp2K

n

Ptp20

Kn

Ptp3K

n

Ptp30

Kn

Ptp4K

n

Ptp40

n

n

n

Kn

Dtp1K

n

Dtp1K

n

Dtp1

Kn

Dtp2K

n

Dtp2K

n

Dtp2

Kn

Dtp3K

n

Dtp3K

n

Dtp3

Kn

Dtp4K

n

Dtp4K

n

Dtp4

n

n

n

Since the tailplanes are arranged in an X-configuration, the task of maintaininglevel flight is distributed over the entire arrangement. The hydrodynamic depressor(w) is used to attain the desired depth. The separate PID scheme for the depressor

deflection is:

d

w d

w

0

[K

rnZ

Pw K

rnZ

Iw K

rnZ

Dw]

rnZ

r

n

Zdt

rnZ

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where dw0 is a set equilibrium wing deflection established during the approach tothe turn.

5. Implementation

Eqs. (10) and (11) are rewritten in augmented form for each vehicle

Mvq fv:

where

q {u v w p q r }T

is the vector of generalized body-frame velocities, MV is the vehicle mass matrixincluding added mass and inertia, and fV is the assembled, generalized force vector,

composed of the hydrodynamic, gravitational, buoyancy, and propulsive forces acting

on the vehicle. Numerical solution of the six coupled equations for each vehicle

produces the body frame accelerations of each vehicle. For the purposes of coupling

the vehicle models to the cable model, it is desirable to represent the accelerations

in the inertial frame of reference. This is accomplished by application of the global

transformation, T.

X v Tq Tq

T RIV 0

0 Qwhere xV {r

VX r

VY r

VZ

V V V}T defines the vehicle state in the inertial frame.The complete system, including a cable composed ofnvisco-elastic elements, the

towing vehicle, and towfish is modeled by 3(n 1) 6 second order differentialequations. For implementation of a fourth order RungeKutta integration scheme,these equations are rewritten as a system of 6(n 1) 12first order equations. Thecorresponding global state vector, XG, is:

XG

{0

,0

,0

,0

,0

,0

, r

0

X, r

0

X, r

0

Y, r

0

Y, r

0

Z, r

0

Z, r

1

X, r

1

X, , rnX, r

nX, r

nY, r

nY, r

nZ, r

nZ,

n, n, n, n, n, n}

where the vectors r0 and rn have been used to represent the positions of the towing

vehicle and towfish, respectively, rather than the top and bottom cable node positions.Assembly of the first order equations, and implementation of the RungeKutta inte-grator was done in C/C++ on a Pentium III 700Mhz desktop computer.

6. Conclusions

A mathematical model for a semi-submersible towing vehicle pulling a towfishhas been developed and implemented. The dominant component of the system, the

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armoured towcable, is represented using a lumped mass method that treats the cable

as a series of point masses and elastic elements. Taking advantage of the modularity

of the lumped mass discretization, the vehicle dynamics have been seamlessly

included as boundary conditions within the cable model. After decomposing thestreamlined vehicles into component surfaces, empirical relationships have been used

to define the hydrodynamics within each vehicle representation. As such, the systemmodel is non-linear and can be used in long time domain simulations during which

the system is traversing a wide range of operating conditions. A controller has been

implemented which governs the deflections of the towing vehicle and towfish controlsurfaces based on both open loop inputs and closed loop PID feedback of the vehicle

states. By incorporating realistic, active vehicles at the cable boundaries, while main-

taining the non-linear capabilities of the lumped mass cable model, this work captures

the complexity of this modern towed vehicle system and thus allows detailed study

and optimization of the systems design and operation.

Acknowledgements

The authors would like to thank International Submarine Engineering and Defence

Research Establishment Atlantic for permission to publish this work.

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