-
cb
a
.
l
nKeywords:Underwater vehicleTowing tank testsHydrodynamic
coefficientsManoeuvring simulation
was first calibrated using the sea-trial data, and then was used
to study the turning manoeuvres and tocompare the simulation
results with those from theoretical formulae based on the
linearized equations ofmotion. The simulation results show
non-linear changes in the hydrodynamic coefficients as the
turningmanoeuvre becomes tighter.
2010 Elsevier Ltd. All rights reserved.
1. Introduction
Underwater vehicles are being used increasingly in a variety
ofapplications, such as surveys, exploration,
inspection,maintenanceand construction, search and rescue,
environmental and biologicalmonitoring, military, undersea mining,
and recreation. Clearly,there are many parallels with the study of
the aerodynamics ofaircraft. However, for underwater vehicles, the
vehicle weight isbalanced by the buoyant force that is provided by
the surroundingfluid, so in that sense underwater vehicles are more
like airshipsthan traditional winged aircraft. Also, the
contribution of thehydrodynamic moment on the hull of an underwater
vehicleis much greater than the contribution of the fuselage on
awinged aircraft, so the traditional methods of computing
theaerodynamic coefficients for aircraft do not immediately
transferto the computation of hydrodynamic coefficients for
underwatervehicles [1,2].
Using both numerical simulations with a combination ofthe ANSYS
and LS-DYNA finite element codes, and physicalexperiments with the
Marine Dynamic Test Facility (MDTF), at theInstitute for Ocean
Technology, National Research Council, Canada(NRC-IOT), Curtis [3]
presented direct comparisons betweennumerical and experimental
results in the study of underwater
Corresponding author.E-mail addresses: [email protected]
(F. Azarsina),
[email protected] (C.D. Williams).
vehicle hydrodynamics. The bare hull of the DREA
(DefenseResearch Establishment Atlantic) Standard Submarine was
usedfor this purpose. The report by Jones et al. [2] provides
adiscussion and evaluation of three methods for the calculationof
hydrodynamic coefficients of simple and complex submergedbodies as
a function of their shape. Two of these methods werebased on the
techniques developed in the aeronautical industry:(i) the US Air
Force DATCOM method, which was applied byPeterson [4] to underwater
vehicles, and (ii) the Roskam method,as modified by Brayshaw [5]
for underwater vehicles. The thirdmethod was based on methods
applicable to the calculation ofthe coefficients of single-screw
submarines, and it was developedat University College, London. Many
semi-empirical relations tocalculate the hydrodynamic coefficients
are presented in the reportby Jones et al. [2], butmost of them are
only applicable over a smallrange of incidence angles, and the
effect of rate of change of angleis completely absent. One of the
few studies on large non-linearangles of attack has been done by
Finck [6], which provides someadditional techniques to use the
DATCOM method in a non-linearrange of angles of attack (AOAs).
For high-amplitude, high-rate manoeuvres, first-order
Taylorseries expansion is insufficient to capture the higher-order
non-linear dependence of the loads on the flow angle and the
vehicleturning rate. For example, Mackay et al. [7] show that
thetransverse force has a non-linear variation with the AOA;
abovean AOA of 10 the stability-derivative-based prediction
(slopethrough the data near the origin) underestimates the actual
loadby 50% or more. Also, a study of the available analytical
andApplied Ocean Resear
Contents lists availa
Applied Oce
journal homepage: www
Manoeuvring simulation of theMUN Exphydrodynamics of
axi-symmetric bare huFarhood Azarsina a,, Christopher D. Williams
ba Department of Marine Science and Technology, Science &
Research Branch, Islamic Azadb National Research Council Canada,
Institute for Ocean Technology, Box 12093, Station A
a r t i c l e i n f o
Article history:Received 17 March 2010Received in revised form16
September 2010Accepted 25 September 2010Available online 15 October
2010
a b s t r a c t
Straight-ahead resistance tesconfigurations of the
PhoenInstitute for Ocean Technolodrag force, lift force and
turforms. The empirical formulaplanar manoeuvres of the
M0141-1187/$ see front matter 2010 Elsevier Ltd. All rights
reserved.doi:10.1016/j.apor.2010.09.003h 32 (2010) 443453
le at ScienceDirect
n Research
elsevier.com/locate/apor
orer AUV based on the empiricallls
University, Tehran, Postal code: 1477893855, Iran, St. Johns,
NL, Canada, A1B 3T5
ts and static-yaw runs up to 20 yaw angle for the axi-symmetric
bare hullix underwater vehicle that were performed in the 90 m
towing tank at thegy, National Research Council, Canada, provided
empirical formulae for theing moment that are exerted on such
axi-symmetric torpedo-shaped hulle were then embedded in a
numerical code to simulate the constant-depthUN Explorer autonomous
underwater vehicle (AUV). The simulation model
-
O444 F. Azarsina, C.D. Williams / Applied
semi-empiricalmethods to estimate the hydrodynamic
derivativesfor an axi-symmetric autonomous underwater vehicle (AUV)
waspresented by Barros et al. [8].
In a previous project that was reported by Azarsina et al.
[9],manoeuvring of an underwater vehicle was studied under
theaction of its dynamic control systems. Sea-trial data for
severalmanoeuvres with the MUN Explorer have been reported by
Issacet al. [1012]. As a part of the underwater research at
NRC-IOT,the hydrodynamic coefficients for the AUV are directly
obtainedfrom these sea-trial data and are then substituted into a
simulationcode that was previously developed at the Memorial
University ofNewfoundland.
This paper presents original research in the following
sequence.First, the hydrodynamic axial and lateral forces and the
yaw
turning moment that are exerted on the bare hull of an
axi-symmetric underwater vehicle were modeled using the towingtank
test results which show clearly the effect of the hulllength to
diameter ratio. Next, the lift and drag forces thatare produced by
the four aft control planes of the AUV weremodeled. Instead of the
simplifying assumptions that may affectthe resulting hydrodynamic
model for the lift and drag forces, themodel effectively accounts
for the X-configuration of the sternplanes. Third, the thrust force
that is produced by the two-bladedsingle propeller of the AUV was
modeled using the curve of thevehicle speed versus the propeller
rpm based on measurementsfrom straight-ahead sea trials: this is a
simple and effectiveapproach which can be used in situations where
the actual thrustforce curves are not available or are commercially
confidential.Fourth, the known dry mass and flooded mass were used
toestimate the added masses and added moments of inertia for
thevehicle. Finally, manoeuvring simulations were performed and
thesimulation code was calibrated in order to minimize the errorin
turning diameter relative to the diameters measured duringsea
trials of the AUV. Also, the simulation results for the
turningmanoeuvres were compared to those from theoretical
formulaebased on the linearized equations of motion.
This is the first case in which the hydrodynamic loads on
thehull are based on the authors own experimental work, and
whichhas been published in the open literature. The simulation
resultsthat are presented in this research contribute to the
knowledgeof the hydrodynamics of underwater vehicles mainly in the
sensethat the hydrodynamic forces that are exerted on the vehicle
arenon-linearly changing as the rudder deflection angle
increasesand the turning manoeuvre becomes more abrupt in terms of
theturning rate r and the radius of turn R. Therefore, one clearly
seesthat the conventional practice to assume a constant value for
thehydrodynamic coefficients based on small angles (linear theory)
isno longer valid for an abrupt turn.
2. Dynamics of an underwater vehicle
The dynamics model to be used in this simulation wasintroduced
by Abkowitz [13] and Fossen [14]. The coordinatesystem is as shown
in Fig. 1: there is a global coordinate [X, Y , Z]in which the path
and orientation of the vehicle is recorded, anda body-fixed
coordinate system in which the velocities and forcesare
expressed.
The motion of an underwater vehicle with six degrees offreedom
can be expressed by the vectors ,v and , as follows: = [ 1, 2] (1)v
= [v 1,v 2] (2) = [ 1, 2], (3)
where the linear and angular displacement, velocity and
forcevectors, respectively, are 1 = [x, y, z], 2 = [, , ],v 1 =cean
Research 32 (2010) 443453
Fig. 1. Global and body-fixed coordinate systems for an
underwater vehicle.
[u, v, w],v 2 = [p, q, r], 1 = [Fx, Fy, Fz] and 2 =[Mx, My, Mz].
The captive tests on the bare hull series wereperformed in the XY
plane and the simulationmodel in this studywas programmed for the
horizontal plane manoeuvres in whichthe force vector has three
elements: surge and sway forces alongx-axis and the y-axis and
yawing moment around the z-axis in thebody-fixed coordinate system
(see Fig. 1). In the planarmanoeuvrewith surge, sway and yaw
degrees of freedom, the kinematics ofmotion simplify to the
following three equations:
mu vr xGr2 yG r
= Fx (4)mv + ur yGr2 + xG r
= Fy (5)Iz r +m[xG(v + ur) yG(u vr)] = Mz . (6)In the above
equations, r is the yaw rate of turn, m (as will beexplained later)
is the flooded mass of the underwater vehicle andIz is the moment
of inertia of the vehicle in the flooded state.The vertical axis
around which the moment of inertia is calculatedindicates the
origin of the body-fixed coordinate system relativeto which the
centre of gravity (CG) may have non-zero offsets xGand yG. In this
simulation, the origin of the body-fixed coordinatesystem is
assumed to be at the mid-length of the vehicle.
The vehicle acceleration is obtained as the inverse of the
massmatrix times the vector of forces and moments:v (t) = M1 (t).
(7)In (7), the acceleration vector
v (t) and the mass matrix M can
be identified from the left-hand side in Eqs. (4)(6) and
theforce vector (t) from the right-hand side in those
equations.Integration of the acceleration gives the velocity and
integrationof the velocity gives the position. Finally, the
position vector istransferred to the global coordinate system via
the axis rotationdefined by the Euler angles [, , ]. In order to
use thisprocedure we must formulate (t). The method chosen was
tomeasure the loads experimentally for a similar underwater
vehicle.
3. Bare hull hydrodynamics
3.1. Test set-up
Manoeuvring experiments were performed with a series of
fiveslender axi-symmetric bare hulls in the 90 m long, 12 m
widetowing tank at the National Research Council Canada,
Institutefor Ocean Technology (NRC-IOT). The original bare hull of
theunderwater vehicle Phoenix, shown in Fig. 2, had an overall
lengthof 1.641 m and a diameter of 0.203 m, that is, the original
lengthto diameter ratio (LDR) was about 8.5:1. In anticipation
thatthere would be a requirement to lengthen the vehicle in order
toaccommodate an increased payload or increased battery
capacity,extension pieces were designed and fabricated that would
permittesting hulls of the same diameter, 203mm, but with LDR 9.5,
10.5,11.5 and 12.5. Thus, a set of experiments was proposed that
wouldinvestigate the manoeuvring characteristics of the hull forms
ofLDR 8.512.5. The hydrodynamic loads were measured with an
internal three-component balance to record the axial force,
lateralforce and yaw moment [15,16].
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OF. Azarsina, C.D. Williams / Applied
Fig. 2. Bare hull Phoenix model installed on the planar motion
mechanism usingthe two vertical struts.
3.2. Resistance runs
Straight-ahead resistance runswere performed for the five
barehulls at fixed forward speeds of 1, 2, 3 and 4m/s. All the
resistanceruns were performed for zero drift angle, that is, with
each modelaligned with the direction of towing. The axial force
recordedduring the resistance testswasmodeled as a function of the
towingspeed and the bare hull LDR. The quadratic multiplier k for
thecurve fits to the resistance test data was used to model the
axialforce in straight-ahead motion (see [15]):
Fx = kU2, where: k = 0.162 LDR+ 0.681, (8)which is valid in the
range 8.5 < LDR < 12.5. Although thisdimensional model
captures the test data, it cannot be used topredict the resistance
for the bare hull of another underwatervehicle of different size.
If the non-dimensional axial force isdefined by dividing the axial
force by the frontal area of the AUVtimes the dynamic pressure of
the free-stream as follows:
Fx = Cx1/2U2
d24
, q = 1/2U2 and
Af = d2/4(9)
Cx = Fx/(qAf ), (10)with fresh-water density = 1000 (kg/m3),
then the axial forcecoefficient for the Phoenix bare hulls in
straight-aheadmotions alsohas a linear variation over the bare hull
LDR, as follows:
Cx = 0.0117 LDR+ 0.038. (11)Note that (11) was derived for tow
speeds of 14 m/s andLDR values 8.512.5; however, due to the
relatively simplehydrodynamics of straight-ahead towing, it may be
used for smallextrapolations outside the above ranges.
3.3. Static yaw runs
All the static yaw runs were performed using a fixed sequenceof
yaw (drift) angles from 2 to +20 in steps of two degrees.All runs
were performed at a fixed speed of 2 m/s. For the purposeof
curve-fitting and modeling the data, the axial force, lateral
forceand yawing moment data versus yaw angle of attack that
were
presented by Williams et al. [15] (Figs. 46), were transformed
toglobal coordinate lift, drag and yaw moment coefficients
definedcean Research 32 (2010) 443453 445
Fig. 3. Drag coefficient versus yaw angle.
Fig. 4. Lift coefficient versus yaw angle.
as follows:
CD = D/(qAf ) (12)CL = L/(qAf ) (13)CM = M/(qAf l). (14)The
resulting non-dimensional coefficients along with the curvefits are
shown in Figs. 35. Due to the length parameter in thedenominator of
(14), the yaw moment coefficient for all the barehull
configurations is about the same in Fig. 5.
The drag coefficient data in Fig. 3 were fitted by
quadraticpolynomials which have no linear term, that is, an even
second-order polynomial of the form
CD = k12 + k2. (15)The constant value for the drag coefficient
is close to the axialforce coefficient value that was modeled in
Eq. (11) based on theresistance test results for tow speeds of 14
m/s instead of a singletow speed of 2 m/s. Thus, it is beneficial
to preserve the previousmodel for the constant value at zero yaw
angle and add to that thequadratic term. Also, the quadratic term
for the drag coefficient canbe averaged over the bare hull
configurations. Therefore, the dragcoefficient for the Phoenix hull
can be modeled as
1000CD = 1.882 + 11.7 LDR+ 38. (16)
Note that the yaw angle in (16) is in degrees.
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O446 F. Azarsina, C.D. Williams / Applied
Fig. 5. Moment coefficient versus yaw angle.
Cubic (third-order) odd polynomials were fitted to the curvesof
the lift and moment coefficients in Figs. 4 and 5, as follows:
CL = k33 + k4 (17)CM = k53 + k6. (18)The polynomial coefficients
to model the lift coefficient in (17)vary with length, and can be
approximated to have a closelylinear increase for longer
configurations. Thus, both the third-orderparameter k3 and the
linear parameter k4 aremodeled by linear fitsover the LDR, as
follows:
1000CL = (0.007LDR+ 0.011)3 + (4.87LDR+ 8.85), (19)where the yaw
angle in (19) is in degrees. However, for thePhoenix yawmoment
coefficient in (18), the cubic and linear termsare almost the same;
hence on average over all the bare hullconfigurations it is modeled
as follows:
1000CM = 0.013 + 17.92. (20)The empirical formulae (16), (19)
and (20) are valid over rangesof the bare hull length to diameter
ratio (LDR), yaw angle andforward speed factors that are,
respectively, 8.512.5, 20 20and 14 m/s.
4. Dynamic control systems
4.1. Control surfaces
The MUN Explorer AUV is shown in Fig. 6; its overall length
isabout 4.5 m and it has a maximum diameter of about 0.7 m.
Acylindrical main body is blended with an elliptical nose at its
frontand a tapered tail section at its rear. Manoeuvring of the
vehicleis facilitated by four aft planes arranged in an
X-configuration andtwo foreplanes which assist with precise depth
and roll control.The vehicle yaw, pitch and roll motions can be
independentlycontrolled by the aft planes. With proper control of
the vehiclepitch, the vehicle depth can also be controlled using
only the aftplanes. The planes have the symmetrical cross-section
of NACA0024 (see [10]). The MUN Explorer s control planes are about
35by 35 cm in chord and span; that is, an aspect ratio of 1.1
Numbering of the planes is compatible with
themanufacturersmanual, in which the two bow planes are numbered 1
and 2 andthe stern planes are numbered 3 and 4 on the port side and
5 and6 on the starboard side. All planes have a positive deflection
angle1 Each control plane consists of a stationary root-base of
about 3 cm span whichfairs to the hull and a moving main part of 35
cm span.cean Research 32 (2010) 443453
Fig. 6. TheMUN Explorer AUV being towed inwater in preparation
for the sea trials(MERLIN [17]).
when the leading edge turns upward. Thus the lift force of
eachplane is positive upward. As shown in Fig. 6, the angle between
theaxis of rotation of each stern plane and the horizontal, to be
calledthe X-angle, was manufactured to be = 45.
The lift and drag coefficients for an NACA 0024 airfoil
sectionare about the same as for an NACA 0025 airfoil section, for
whichextensive experimental results were given in NACA report 708by
Bullivant [18] for aspect ratio 6 and AOA range 8 24. Themaximum
lift coefficient is about 1, and it occurs at about 20,which
corresponds to a drag coefficient of about 0.2. The pitchingmoment
coefficient that was measured at an average Reynolds3.2106 for NACA
0025 had a linear trend increasing from zero toabout 0.05 at an
angle of attack (AOA) of 14, and reducing back tozero at an AOA of
24. The NACA tests were performed for airfoilsof aspect ratio (AR)
6, while theMUN Explorer planes have an AR of1. In a study by
Whicker and Fehlner [19], for NACA 0015 profiles,the effect of
aspect ratio was reported to be significant with higherlift
coefficient for larger aspect ratio. The following formulae [20,pp.
148167] can be used to correct the lift and drag coefficients ofa
two-dimensional (2D) section to those for a three-dimensional(3D)
wing:
CL(3D) = CL(2D)(AR/(AR+ 2)) (21)CD(3D) = CD(2D) + CL(2D)2/( AR).
(22)Therefore
CL(AR=6) = CL(2D) (6/8) and CL(AR=1) = CL(2D) (1/3) (23)CL(AR=1)
= CL(AR=6) (8/6) (1/3). (24)Note that the drag coefficient
resulting from (22) for the 3Dwing islarger than for a 2D section
and it occurs at a higher angle of attack(AOA) which is calculated
as follows [21]:
3D = 2D + CL(2D)/( AR) (rad), (25)where is the angle of attack
of the control plane relative to itsincident flow. The resulting
drag and lift coefficients for NACA0025 with AR = 1 were plotted
versus the plane AOA (see Fig. 7).According to (25), the curve of
the drag coefficient extends to largerAOAs, that is, the range of
3D is larger than that for 2D. Thepitching moment about an axis
through the quarter-chord pointwhich is the center of pressure of
the control plane, that is, at c/4distance from the leading edge,
is not influenced by the aspectratio because the lift and drag
forces are assumed to act at thatlocation. Thus, the NACA reported
values for the pitching momentcoefficient at c/4 for AR= 6 are used
for theMUN Explorer planes.
Fig. 8 is the view of stern planes looking from behind while
thevehicle has a surge velocity u, sway velocity v, and yaw rate
of
turn r . Also, the cut AA in Fig. 8 is a top view of plane 3
whileit is deflected by , during such a horizontal plane
manoeuvre,
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OF. Azarsina, C.D. Williams / Applied
Fig. 7. Lift, drag and pitching moment coefficients for the
control planes; NACA0025 airfoils corrected for AR= 1.
Fig. 8. View of the tail planes looking from behind.
Illustration of the flow velocityrelative to the stern planes
during a horizontal-plane manoeuvre.
as shown in Fig. 9(a). The resultant lateral velocity of the
planesrelative to flow which is corrected for the X-angle is as
follows2:
vplane = (v rxplane) sin(). (26)Then, the angle of incidence of
the flow relative to plane 3 asillustrated in Fig. 9(b) is
= tan1(vplane/u). (27)Then, the AOA for planes 3 and 6 is as
follows:
3,6 = 3, 6 + , (28)where is the controlled deflection angle of
the plane relative tothe hull, which is positive when the leading
edge of the plane turnsup and can reach a maximum of 25 for this
AUV due to physicallimits placed on the actuator mechanism. Note
that plane 6 sameas plane 3 in a positive constant-depth turn
(starboard turn) has itslower face facing the flow.
For plane 4, the angle of incidence of the flow relative tothe
plane is the same as in (27), but it is subtracted from
thedeflection angle of the plane, because in a positive starboard
turn,as illustrated in Fig. 8, the upper face of plane 4 faces the
flow.Therefore, the AOA of plane 4 is
4,5 = 4,5 . (29)2 All four stern planes were assumed at the same
longitudinal distance in thevehicle coordinate system.cean Research
32 (2010) 443453 447
a b
Fig. 9. Top view of plane 3 during a horizontal-plane manoeuvre:
(a) theperpendicular cut AA in Fig. 8, (b) the resultant inflow
velocity and drift angle.
Plane 5 is the same as plane 4, with the upper face facing
theflow during a positive turn. Drag, lift and moment
coefficientsare derived for the AOAs that are calculated with (28)
and (29).Note that the resultant lateral velocity was projected
along theplanes perpendicular in (26). If the planeswere in upright
position, = 90 for rudders and = 0 for horizontal planes, then,
in(26), for the rudders, sin() would reduce to unity, and for
thehorizontal planes it would diminish to zero. Also, note that
theprojected component of the resultant lateral velocity in the
planeof each stern plane which is directed along the planes span,
whichfor = 45 has equal magnitude as given by (26), may
introduceadditional complexity into the hydrodynamic performance of
thestern plane; however, that effect is neglected here.
Therefore, in summary, the lift and drag forces on each
sternplane are as follows:
L = 12U2ApCL, and D = 12U
2ApCD, (30)
where Ap is the planform area of each plane equal to the
chordlength, c , times the span, b. The lift and drag coefficients
in (30)are read from Fig. 7 at an angle of attack that is
calculated by either(28) or (29) for planes 36.
As shown in Fig. 9(b), the drag and lift forces should be
projectedalong the x-axis and the y-axis of the vehicle coordinate
system toobtain the net axial force and sway force that are
produced by thecontrol planes. Thus, the sway force that is
produced by plane 3,along its y3 axis, shown in Fig. 8, is
Fy,plane3 = Lplane3 cos( )+ Dplane3 sin( ). (31)Then, the net
sway force of the four stern planes is calculated bysumming up the
sway forces of each plane similar to (31) andcorrecting them for
the X-angle as follows:
Fy,planes = (Fy,plane3 Fy,plane4 Fy,plane5 + Fy,plane6) sin()=
[(Lplane3 Lplane4 Lplane5 + Lplane6) cos( )+ (Dplane3 Dplane4
Dplane5 + Dplane6) sin( )] sin()
= 12U2Ap[(CL,3 CL,4 CL,5 + CL,6) cos( )
+ (CD,3 CD,4 CD,5 + CD,6) sin( )] sin(). (32)During a simulation
run, for example, a turning manoeuvre, at thetime instant t ,
knowing the velocity vector of the vehicle, Eq. (32) isused to
calculate for the net sway force of the stern planes, which isthen
added up with other forces that act in the sway direction, andthe
resultant force produces the sway acceleration vector at thenext
time instant. The sway acceleration vector is then integrated
to produce the sway velocity vector fromwhere the loop
continues.For more details also see [16].
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O448 F. Azarsina, C.D. Williams / Applied
Fig. 10. TheMUN Explorer s forward speed versus propeller
rpm.
4.2. Propulsion
The AUV is propelled by a dp = 0.65mdiameter
high-efficiencytwo-bladed propeller, and it can achieve a maximum
speed of2.5 m/s. The propeller is blended into the tail cone to
maintainattached flow for better hydrodynamics (see [10]).
Straight-aheadtrials were performed with the vehicle to attain the
curve of thevehicle speed versus the propeller rpm (see. [11]),3 as
reproducedin Fig. 10. These data pointswere fittedwith the
following relation:
n = 109 U, (33)where U is the forward speed of the vehicle and n
is the propellerspeed of revolution in rpm. On the other hand, in a
constant-speed straight-ahead run, the propeller should produce a
thrustapproximately equal to the resistance force R plus the
thrustdeduction T ; that is, T = R + T . For the MUN Explorer,
thepropeller diameter to hull diameter ratio is about dp/d 1;also,
referring to the test results reported for C-SCOUT by Thomaset al.
[22], the thrust deduction fraction T/T may be estimated ast 0.1.
Also, the resistance force exerted on the vehicle R equalsthe bare
hull drag as was modeled by Eq. (16), plus the drag forceon the
four stern planes and two bow planes, all at zero
deflection.Summing up, the thrust force is as follows:
T
11 t
[12U2(Af CD,hull + bcCD,planes)
],
CD, hull = 1.882 + 11.7 LDR+ 38, and CD,planes 0.01, (34)where b
and c are the span and chord length of the control planes.Then,
substituting the forward speed from (33) into (34) providesan
estimate of the propeller thrust versus its rpm, which is plottedin
Fig. 11. Although the sea trials were performed over a range109287
rpm,which corresponded to forward speeds of 12.5m/s,the curve in
Fig. 10 was extrapolated to the range 10287 rpmassuming that the
propeller has a similar performance.
5. Vehicle mass and the added mass of water
The dry mass of the MUN Explorer AUV was reported by Issacet al.
[10] as 630 kg. At the recovery stage of a sea trial, animmediate
reading of the weight scale indicated a total mass of
3 Those straight-ahead runs were performed in two phases:
accelerating anddecelerating. Thus, a total of eight data points
for the vehicle speed versus propeller
rpm were recorded. The average of the two phases is used as a
single set of data inFig. 10.cean Research 32 (2010) 443453
Fig. 11. Propeller thrust force estimated using test data for
the vehicle speed versuspropeller rpm.
about 1400 kg; a later calculation concluded a flooded mass
of1445 kg; that is, about 1445 630 = 815 kg of floodwater mass.The
moment of inertia of the AUV in yaw in the flooded state tobe used
in this simulation model is estimated as Iz = 2475 +844 (kg m2).
Also, the centre of gravity of the flooded vehicle wasestimated to
be about 2.33 m from the bow end and 0.02 m belowthe longitudinal
centerline; that is, xG = 2.25 2.33 = 0.08 maft of mid-length, with
yG = 0 and zG = 0.02 m below thelongitudinal centerline of the
hull.
Assuming potential flow about an ellipsoid with a length of land
maximum diameter d, Lamb [23] provides non-dimensionalfactors for
(i) the added mass in surge K1, (ii) the added massin sway and
heave K2, (iii) the added moment of inertia K forpitch and yaw, and
(iv) the added moment of inertia for roll beingzero. For the
forward acceleration state, the added mass accordingto Lambs [23]
curve for an ellipsoid, for the MUN Explorer AUVwith LDR 6.5, is
about 0.05. However, an additional amount ofadded mass is expected
since the vehicle, unlike an ellipsoid,has a blunt nose, a
constant-diameter mid-body, and a taperedtail section; also, it
includes appendages. Thus the axial addedmass was assumed to be
one-tenth of the vehicles flooded mass,i.e., one-tenth of 1445 kg.
The lateral (sway) added mass androtational (yaw) added moment of
inertia factors for the ellipsoidof LDR 6.5 are respectively about
K2 = 0.92 and K = 0.77(see [23]). Resulting values for an ellipsoid
equivalent to the barehull of theMUN Explorer are about 1057 kg for
the addedmass and1191 kg m2 for the added moment of inertia,
derived for a sea-water density of 1025 kg/m3.
To estimate the added mass effect for the control planes,
therelations for the added mass magnitude of a rectangular plate
ofspan b and chord length c accelerating normal to its face wasused
[24]. In a constant-depth manoeuvre, the total lateral addedmass
due to the four tail planes was predicted to be about 49 kg,and the
added moment of inertia due to these stern planes wasestimated as
90 kg m2 about the z-axis through the origin of thebody-fixed
coordinate system.
6. Simulation results
The simulation model was developed and its convergencewas
verified by performing straight-ahead manoeuvres. The MUNExplorer
AUV with an input propeller speed of 120 rpm startsto speed up
under a thrust force of about 71 N and in aboutthree minutes
attains a steady forward speed of about 1.03 m/s.
Changing the simulation time step slightly changes the
responsebut it converges close to the same forward speed.
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OF. Azarsina, C.D. Williams / Applied
Table 1Simulation results for the turningmanoeuvres at a
constant depthwith an approachspeed of 1 m/s compared to trial
results. T : tests, S: simulation.
Run Stern-plane deflection angles ()3 4 5 6
1 10.32 0.1 1.82 12.522 10.54 0.01 1.86 12.523 10.19 0.34 2.38
12.184 10.32 0.39 2.08 12.265 10.17 0.89 2.43 11.826 10.1 0.97 2.7
11.547 9.87 1.23 2.77 11.68 9.76 1.6 2.83 11.319 9.66 2.79 3.53
10.51
10 9.41 2.77 3.7 10.34Run R(T ) (m) R(S) (m) r(T ) (/s) r(S)
(/s) eR (%) er (%)
1 22.51 15.4 2.560 2.070 31.6 19.12 23.8 15.3 2.413 2.090 35.6
13.43 25.02 16.9 2.304 1.960 32.5 14.94 25.09 16.4 2.296 2.000 34.6
12.95 26.58 18.2 2.180 1.880 31.4 13.86 27.97 19.3 2.053 1.82 31.1
11.37 28.11 20.0 2.070 1.78 28.9 14.08 29.65 21.4 1.954 1.71 27.8
12.59 33.44 27.0 1.695 1.45 19.4 14.5
10 37.54 28.3 1.531 1.41 24.5 7.9
6.1. Turning manoeuvres: calibrating the simulation model with
thefree-running test results
In August 2006, at Holyrood Harbour, situated about 45 kmsouth
west of St. Johns, Newfoundland, a set of trials wasperformed with
the MUN Explorer AUV, some of which werereported by Issac et al.
[10,11] and Issac et al. [12]. Ten runs ofturning circle manoeuvres
with an approach speed of 1 m/s at aconstant depth of 3 m that were
reported by Issac et al. [11] asare reproduced in Table 1 were used
to evaluate and then calibratethe response of the simulation model.
The lower portion of Table 1shows the reported results for the
radius of turn R and turning rater for ten turning manoeuvre
trials, indicated by T in parentheses(see [11]).4 Indicated by S in
parentheses are the respectivesimulation results.
Relative errors for the radius of turn R, if the test results
are usedas the reference values, are defined as follows:
eR = 100 (R(S) R(T ))/R(T ). (35)The relative errors between the
test and simulation results in theradius of turn R and the rate of
turn r for these ten runs are shownin Table 1 respectively by eR
and er , which vary between 10%and 35%.
At the time of the sea trials, the location of the CG was
notknown, so the longitudinal location of the CG was
approximated:xG = 0.08 m. For run numbers 1, 6 and 10 in Table 1,
simulationwas performed by changing the longitudinal location of
the CGfrom0.12m to+0.08mwith a step of 0.04m; that is, from 12
cmaft of mid-length to 8 cm forward of mid-length. Variation of
therelative error in the radius of turn as defined in (35) is
plotted inFig. 12 versus the longitudinal location of the CG for
run numbers1, 6 and 10. Thus the simulationmodel can be corrected
bymovingthe CG about 8 cm forward; then the longitudinal location
of theCG coincides with the origin of the body-fixed coordinate
system,xG = 0. Simulation results presented hereafter are based on
thecorrected CG location.4 The rate of turn in [11] was mistakenly
reported as being in (rad/s); the valueswere in (/s).cean Research
32 (2010) 443453 449
Fig. 12. Relative error in the radius of turns versus
longitudinal location of the CGof the vehicle.
Fig. 13. Predicted AOAof plane 3 during three turningmanoeuvres
at 290 rpmwithcommanded deflection angles of respectively7,10
and15.
6.2. Turning manoeuvres: radius of turn, turning rate, drift
angle andspeed reduction versus the stern-plane deflection angle
and theapproach speed
The simulation model is a useful tool to study the variationof
the indicators of turning manoeuvres such as radius and rateof
turn, drift angle and speed reduction versus the input
factors:stern-plane deflection angle and the approach speed. In
thefollowing simulations, the average plane angles were used for
allfour stern planes; i.e., planes 3 and 6 use and planes 4 and 5
use+ to perform a starboard turn. Fig. 13 shows the time history
ofthe predicted AOA of plane 3 during three turning manoeuvres
at290 rpm with commanded deflection of respectively 7, 10and 15.
After the vehicle obtains a steady forward speed, theplane starts
to deflect at a rate of 1/s, and the vehicles tail turns inthe
positive yawdirection, thus producing a negative sway velocityv and
a positive r xplane velocity. The predicted AOA of the MUNExplorer
s planes, at an approach speed of 2.5 m/s, will exceed 25for
average deflection angles larger than about 15,
whichproducesaminimum radius of turn about 4.7m, which is slightly
larger thanthe overall length of the vehicle, 4.5 m.
The drift angle is defined as the inverse tangent of the ratio
ofthe sway velocity to the surge velocity of the vehicle with a
minus
sign. For the starboard turns, the drift angle is in the
positive yawdirection, which means that the bow of the vehicle
points inside
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O450 F. Azarsina, C.D. Williams / Applied
Fig. 14. Drift angle of the AUV during turns at any approach
speed versus stern-plane deflection angle.
the circle. The drift angle increases for larger stern-plane
deflectionangles; however, it does not depend on the approach
speed. Duringa turning manoeuvre with an average of 4, the
magnitude ofthe drift angle for the AUV is about 5.3, which is
verified by thereported test results for the runs in Table 1 (see
[11] p. 7). The driftangle of the AUV versus the average deflection
angle of its sternplanes is plotted in Fig. 14. The simulation data
for drift angle showa linear increase as the deflection angle
increases, but at largerdeflection angles the data slightly move
off the linear trend, whichmay be due to the non-linear patterns in
the force and momentvectors, as will be explained later.
The radius of turn becomes smaller for larger deflection
angles,but it does not depend on the approach speed. While the
propellerrpm was maintained constant during the turns, the vehicle
surgevelocity notably decreased during the turn. The vehicles
totalspeed is the surge speed divided by the cosine of the drift
angle;that is,U = u/ cos(). The rate of turn is equal to the total
speed ofthe vehicle after it maintains a steady speed during the
turn, whichis tangent to the vehicle path, divided by the steady
radius of turn;that is, r = U/R. The data for the rate of turn at
different approachspeeds is plotted in Fig. 15.
Also, the non-dimensional turning rate (NDTR) is defined as
r = l/R, (36)where l is the overall length of the AUV and R is
the radius of turn.The NDTR does not depend on the approach speed,
and, based onsimulation data for the MUN Explorer, the results in
Fig. 15 can befitted with a simple linear relation versus the
average deflectionangle of the stern planes as follows:
r 0.04, (37)where is in degrees.
Also, the ratio of the steady speed of the vehicle during a
turnto the approach speed was calculated. It is observed that this
ratiohas the same variation versus the deflection of the stern
planesregardless of the magnitude of the approach speed.
Variationof the ratio steady-turning speed to approach speed versus
theratio turning diameter to vehicle length based on
empiricalrelationshipswas studied byDavidson [25] and Shiba [26]
PNA [27,p. 488]. Such a plot for the simulation data for the MUN
Explorerwas produced as shown in Fig. 16. The trend is the same as
ofthose empirical curves for ships; however, the simulation data
forthe MUN Explorer demonstrate a rather large drop in the
vehicle
speed compared to the surface ships. Note that no comparisons
areavailable for this trend from the sea-trial data since the
action ofcean Research 32 (2010) 443453
Fig. 15. Rate of turn versus stern-plane deflection angle for
theMUN Explorer AUVat the approach speeds 1, 1.5, 2 and 2.5
m/s.
Fig. 16. Speed reduction as a function of non-dimensional
turning diameter for theMUN Explorer AUV compared with surface
ships.
the vehicle controller is to increase the propeller rpm to keep
thespeed-over-ground constant throughout the turn.
The block coefficient for the surface ships is defined as the
ratioof the submerged hull volume to the volume of a rectangular
prismwith dimensions overall length bymaximumbreadth by ship
draft.If the block coefficient CB for an underwater vehicle in a
similarwayis defined as the ratio of the enclosed hull volume to a
rectangularprismof volume, overall length timesmaximumdiameter
squared,then for theMUN Explorer we have CB = 0.66. Note that the
curvesgiven in Davidson [25] and Shiba [26] were for ships with
blockcoefficients CB of respectively 0.8 and 0.7, and it may be
concludedthat a finer body experiences a larger speed reduction
than a morefull body during a turn. The abscissa in Fig. 16 for
theMUN Explorerdata increases up to about 2 RLOA = 60 and reaches
an asymptotictrend at higher values; however, only a portion of the
data areshown so as to be in the same range as the data for the
surfaceships.
6.3. Vehicle path, velocity, hydrodynamic forces and moments
The XY path of the vehicle at a propeller speed of 290
rpmturning with the stern-plane average deflection angle
ofrespectively 3, 6 and 9 are shown by solid, dashed anddotted
curves in Fig. 17. All the turns shown in Fig. 17 are initiatedat
time t = 60 s, which corresponds to position (X, Y ) of (150 m,0).
Clearly, a larger average produces a smaller radius of turn. At
an average of 3, the solid curve, the turn is a circle which
isinitiated tangent to the X-axis. However, at6, the dashed
curve,
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OF. Azarsina, C.D. Williams / Applied
Fig. 17. Turning manoeuvres at 290 rpm with average deflections
of 3, 6 and 9for the X stern planes.
Fig. 18. Surge velocity of the AUV corresponding to turns in
Fig. 17.
the AUV turns around and crosses the X-axis. Then, at an average
of9, the dotted curve, the AUV first turns in a smaller circle
andthen maintains a larger steady radius.
Also, time histories of the surge and sway velocities of the
AUVare plotted in Figs. 18 and 19. The radius of curvature of the
AUVspath defined as the speed of the vehicle divided by its rate of
turnR = U/r is plotted versus time during t = 80200 s of the =
9manoeuvre in Fig. 20. Obviously, the radius of curvature is
changingduring the transient portion until the vehicle speed and
its turningrate approach their respective steady values, and thus
the radius ofcurvature reaches a steady value of about 10.5 m.
Note that the turn at = 9 and 290 rpm initiates at t = 73 s,and
the radius of curvature of the vehicles path is of course
infinitebefore it starts to turn. Also, during the same length of
time, with alarger stern-plane deflection angle , and the
sameapproach speed,the vehicle performs a larger number of
turns.
The time history of the net sway force that was produced by
thestern planes during these turns is shown in Fig. 21. The net
swayforce in the starting phase of the turnwith average of9
reachesa maximum of about 40 N directed to port. However, as
was
described before in Section 4.1, the AOA of each stern plane,
dueto the relative flow velocity vector, changes during the
manoeuvrecean Research 32 (2010) 443453 451
Fig. 19. Sway velocity of the AUV corresponding to turns in Fig.
17.
Fig. 20. Radius of curvature of the vehicles path at average
stern-plane deflectionangle = 9 and 290 rpm.
(see Fig. 13), and thus the net sway force generated by the
sternplanes, after the turn becomes steady, is to starboard in Fig.
21.The net yawingmoment of the stern planes has the same
variationbut in the opposite direction: for a starboard turn first
a positiveyaw moment is produced; however, the steady turning
momentbecomes negative due to the change in the incidence angle of
theflow with respect to each stern plane. Also, the axial force
that isexerted on the bare hull is shown in Fig. 22. Note that the
AUV inthese simulations is initially at rest while its propeller is
rotatingat speed n rpm; then the vehicle starts to move under the
effect ofthe almost constant thrust force.
7. Verifying the simulation results with the theoretical
formu-lae for turning manoeuvres
Solving the linearized equations of motion for a vessel
duringthe steady phase of a turning manoeuvre, the following
equationsfor the steady radius of turn and the steady drift angle
have beenpresented (PNA [27], Chapter VIII, p. 484):
R = L
Y vN r mxG
N v(Y r m)Y vN N vY
(38)
= N Yr m Y(Nr m xG)
Y vN r mxG
N v(Y r m) . (39)
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O452 F. Azarsina, C.D. Williams / Applied
Fig. 21. Net sway force that is produced by the stern planes
during turns tostarboard.
Fig. 22. Axial force that is exerted on the bare hull during
turns to starboard.
For the notation of the terms in (38) and (39) see PNA [27] (p.
484)or [16] (p. 173). The contributions by the bare hull to thesway
force derivative Yv and yaw moment derivative Nv wereestimated
using the static yaw test results for the Phoenix bare
hullconfigurations. For the Phoenix bare hull with LDR 8.5, it was
foundthat, approximately, Y v = 0.037 and N v = 0.011 (see [16],p.
66). A negative value for the moment derivative Nv meansthat the
effect of the bow dominates. Converting the above non-dimensional
derivatives to dimensional form for theMUN ExplorerAUV with overall
length l = 4.5 m (and LDR 6.5) at a forwardspeed U = 2.5 m/s gives
the prediction Yv 958 N/(m/s) andNv 1363 N m/(m/s) [28].
At a propeller speed of 290 rpm and approach speed 2.5 m/s,the
simulations were performed for turning manoeuvres at anaverage
stern-plane deflection angle of 19, and the steadyvalues of sway
force and yaw moment that are exerted on thebare hull were
extracted and recorded as are shown in Table 2.Variations of sway
force versus sway velocity and yaw momentversus yaw rate of turn at
approach speeds of 1, 1.5, 2 and 2.5 m/sare respectively plotted in
Figs. 23 and 24. Also, values for thenet steady sway force and yaw
moment that were produced by
the stern planes during the steady phase of the turns are
shownin Table 2. In the lower part of Table 2, the force and
momentcean Research 32 (2010) 443453
Table 2Simulation results for the steady values of sway force
and yaw moment that areexerted on the bare hull and produced by the
stern planes for theMUNExplorer AUVat 290 rpm: 2.5m/s approach
speed and the resulting non-dimensional derivatives.
() u (m/s) v (m/s) r (rad/s) Fy,hull (N) Mz,hull(N m)
Fy,planes (N) Mz,planes(N m)
0 2.50 0 0 0 0 0 01 2.47 0.057 0.019 65.6 123.7 8.3 10.82 2.36
0.108 0.036 121.2 225.6 15.49 20.163 2.2 0.152 0.052 161.6 294.2
20.98 27.294 2.02 0.187 0.066 186.7 329.8 24.75 32.215 1.83 0.213
0.078 199.5 339.3 27.16 35.346 1.65 0.232 0.089 204 331.1 28.6
37.227 1.48 0.246 0.099 203.1 312.3 29.41 38.298 1.34 0.254 0.108
199.2 288.1 29.83 38.859 1.2 0.26 0.117 193.7 262 30.02 39.12 () Y
v 103 N r 103 Y 103 N 1030 1 44.4 12.4 7.33 2.122 42.0 11.1 6.35
1.843 35.3 8.27 4.85 1.44 27.6 4.95 3.33 0.975 19.1 1.49 2.13 0.616
9.03 1.46 1.27 0.377 2.34 3.61 0.72 0.218 18.9 4.91 0.37 0.119 35.0
5.52 0.17 0.05
Fig. 23. Steady sway force exerted on the bare hull of the MUN
Explorer duringturning manoeuvres at various propeller rpm and
approach speeds.
derivatives were calculated using the following formulae:
Yv = Fy,hull/v, Nr = Mz,hull/r,Y = Fy,planes/, N = Mz,planes/,
(40)
where is in radians. In (40), is 1 = /180 rad betweensuccessive
rows in Table 2, and all other parameters vary as in partone of
Table 2 between two successive rows.
According to Figs. 23 and 24, the assumption of constant
slopes(and hence values of the hydrodynamic derivatives) is valid
forthis vehicle only for the range of rudder deflection angle 3
< < +3. Therefore, in the vicinity of zero stern-plane
deflectionangle , where the variation of forces and moments as
shown inFigs. 23 and 24 are linear, if the first three values for
the non-dimensional derivatives in the second part of Table 2,
i.e., at of 1,2 and 3, are averaged it indicates that Y v = 40.6
103,N r =3 3 310.6 10 , Y = 6.18 10 ,N = 1.79 10 . Only theforce
derivative Y r remains unknown, which using the available
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OF. Azarsina, C.D. Williams / Applied
Fig. 24. Steady yaw moment exerted on the bare hull of the MUN
Explorer duringturning manoeuvres at various propeller rpm and
approach speeds.
data for ships, e.g. in [27] (pp. 526540) can be estimated as
one-tenth of Y v; that is, Y r = 4.1 103. If all the
non-dimensionalderivatives are substituted into (38) and (39), the
resulting curvescompared to the simulation results for the radius
of turn and driftangle were observed to be in good agreement.
8. Conclusions
Regression models for the hydrodynamic coefficients of thebare
hull of a torpedo-shaped underwater vehicle, using fixed-attitude
test results, were usefully embedded within a simulationmodel to
predict the manoeuvring behaviour of the full-scalevehicle, the MUN
Explorer AUV. The stern planes, which are inan X-configuration,
were modeled to produce the required swayforce and yawing moment
for constant-depth manoeuvres, andthe propeller thrust force was
modeled using the test results fromstraight-line sea trials. To
predict the radius of turn within 5%relative error compared to the
test results for ten turning circlesea trials, the initial estimate
for the longitudinal location of theCG was corrected. The
calibrated simulation model was then usedto simulate turning
manoeuvres for various approach speeds andvarious deflection angles
of the stern planes. It was observed that(i) the radius of turn,
drift angle and the speed reduction ratio areindependent of the
approach speed, (ii) the radius of turn has aninverse relation to
the stern-plane deflection angle, (iii) the rateof turn is faster
at higher approach speeds and higher stern-planedeflection angles,
(iv) the drift angle during a starboard turn ispositive, which
means that the vehicles bow is pointing insidethe circle during the
steady-state turn; the drift angle increaseslinearly with at larger
deflection angles, (v) the speed reductionratio increases
asymptotically to unity at higher radius of turns,i.e. for smaller
stern-plane deflection angles, and (vi) the speedreduction during a
turn is larger for bodies with higher finenessratio; that is, for
bodies of smaller block coefficient.
The time histories of the path, velocity, hydrodynamic forcesand
moments that are experienced by the MUN Explorer duringturning
manoeuvres were also plotted. At larger deflection anglesof the
stern planes, non-linear patterns in those signals are
clearlyobservable, as were shown in Figs. 21 and 22. Using the
steadyvalues for the sway force and yaw moment that were
recordedfor the bare hull and the stern planes during the turns,
non-dimensional force and moment derivatives were calculated, andit
was observed that in the vicinity of zero sway velocity v,
turningrate r and stern-plane deflection angle , the simulations
produced
similar results for the radius of turn and drift angle as did
thetheoretical expressions (38) and (39).cean Research 32 (2010)
443453 453
Acknowledgements
The authors gratefully acknowledge the Memorial Universityof
Newfoundland, the Institute for Ocean Technology, NationalResearch
Council Canada (NRC-IOT) and the National Scienceand Engineering
Research Council (NSERC) for their technical,intellectual and
financial support of this research. Also, we thankthe reviewers for
their comments, which helped us to improve thispaper.
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Manoeuvring simulation of the MUN Explorer AUV based on the
empirical hydrodynamics of axi-symmetric bare
hullsIntroductionDynamics of an underwater vehicleBare hull
hydrodynamicsTest set-upResistance runsStatic yaw runs
Dynamic control systemsControl surfacesPropulsion
Vehicle mass and the added mass of waterSimulation
resultsTurning manoeuvres: calibrating the simulation model with
the free-running test resultsTurning manoeuvres: radius of turn,
turning rate, drift angle and speed reduction versus the
stern-plane deflection angle and the approach speedVehicle path,
velocity, hydrodynamic forces and moments
Verifying the simulation results with the theoretical formulae
for turning manoeuvresConclusionsAcknowledgementsReferences