cb

a

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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: Farhood.Azarsina@mun.ca (F. Azarsina),

Christopher.Williams@nrc-cnrc.gc.ca (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].

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.

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,

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].

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.

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

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,

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)

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

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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[20] Von Mises R. Theory of flight. Dover; 1959. pp. 148167.[21] Abbott IH, von Doenhoff AE. Theory of wing sections. New York: Dover

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hydrodynamic force and moment derivatives for the axi-symmetric bare hullof an underwater vehicle. In: Proceedings 29th ATTC. 2010.

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