J Intell Robot SystDOI 10.1007/s10846-015-0186-2

Modeling of a Complex-Shaped Underwater Vehiclefor Robust Control Scheme

Rui Yang · Benoit Clement · Ali Mansour ·Ming Li · Nailong Wu

Received: 7 July 2014 / Accepted: 9 January 2015© Springer Science+Business Media Dordrecht 2015

Abstract The two critical issues of robust controlare stable controller synthesis and control perfor-mance guarantee in the presence of model uncertain-ties. Inside all robust stable solutions, small modelingparametric uncertainties lead to better performancecontrollers. However, the cost to develop an accuratehydrodynamic model, which shrinks the uncertaintyintervals, is usually high. Meanwhile, when the robot

This work was supported by the China Scholarship Council.

R. Yang (�)Joint PhD candidate between ENSTA Bretagneand Ocean University of China, Brest, Francee-mail: yang.rui@ensta-bretagne.fr

B. Clement · A. MansourLab-STICC, UMR CNRS 6285, ENSTA Bretagne,29806 Brest Cedex 9, France

B. Clemente-mail: benoit.clement@ensta-bretagne.fr

A. Mansoure-mail: mansour@ieee.org

M. LiCollege of Engineering, Ocean University of China,Qingdao 266100, Chinae-mail: limingneu@ouc.edu.cn

N. L. WuState Key Laboratory of Ocean Engineering,Shanghai Jiao Tong University, Shanghai 200030, Chinae-mail: tale@sjtu.edu.cn

geometry is complex, it becomes very difficult toidentify its dynamic and hydrodynamic parameters.In this paper, the main objective is to find an effi-cient modeling approach to tune acceptable controldesign models. A control-oriented modeling approachis proposed for a low-speed semi-AUV (AutonomousUnderwater Vehicle) CISCREA, which has complex-shaped structures. The proposed solution uses costefficient CFD (computational fluid dynamic) softwareto predict the two hydrodynamic key parameters: Theadded mass matrix and the damping matrix. FourDOF (degree of freedom) model is built for CISCREAfrom CFD and verified through experimental results.Numerical and experimental results are compared. Inaddition, rotational damping CFD solutions are stud-ied using STAR-CCM+TM. A nonlinear compensatoris demonstrated to tune linear yaw model for robustcontrol scheme.

Keywords Underwater vehicle · Added mass ·Damping · CFD · Robust

1 Introduction

AUVs are increasingly playing significant roles inunderwater activities. For some critical applications:undersea pipeline survey, infrastructure inspectionsand large vehicle wet maintenance tasks, a small sizecubic AUV is generally preferred. Typically, cubicAUVs can be deployed to explore areas where HOVs

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Fig. 1 CISCREA robot, aview in water b framenotions of underwatervehicles

(Human Occupied Vehicles) and ROVs (RemoteOperating Vehicles) are limited to operate. Unliketorpedo-shaped AUVs, the cubic AUVs show moredegrees of freedom in motion. Especially, they canhover and enter complex underwater spaces.

Achieve necessary maneuverability depends on twokey factors: an accurate hydrodynamic model andan advanced control system. Generally, combinationsof various modeling methods and control algorithmshave been proposed, which led to countless spectrumof maneuvering solutions. In this case, our scope ismore focused on the combination of numerical mod-eling approaches and robust control schemes. In [25],Yamamoto pointed out that a model-based control sys-tem is more effective if the vehicle’s dynamics aremodeled to some extent. Meanwhile, in [10], Ferreiraet al. showed that an empirical linear model oftenfails to represent the dynamics of the AUV over awide operating region. Therefore, obtaining accept-able hydrodynamic models of the complex-shapedcubic AUV is the first key point for better maneuver-ability.

Many ways exist to model ocean vehicles, includ-ing scaled experiments, full-scale experiments,empirical formula approximations and computa-tional dynamic approaches. Scaled and full-scaleexperiments are capable to provide accurate hydrody-namic models. However, they usually need expensivedevices, such as towing tanks. Besides, note that,most of the time experimental modeling results arenot control-oriented. Experimental methods withouttowing tanks also exist as presented in [2] and [20];free decay approach is presented by Ross in [22].For regular geometry modeling, empirical formula

approximation is well proved on torpedo-shapedAUVs, usually slender bodies, as mentioned in [20]and [9]. The last method requires deeper knowledgeand experiences to simplify the AUV into elemen-tary components. Especially, for a complex-shapedAUV, the simplification is greatly complicated. Turnto computational approaches, potential theory andfinite element theory based CFD software such asWAMITTM, MCC (Marine Craft Characteristics, thisfreeware is created by ENSTA Bretagne which canbe downloaded from [16]), SHIPMOTM, ANSYS-CFXTM, ANSYS-FLUNTTM, ANSYS-AQWATM,STAR-CCM+TM and SeaFEMTM are capable to pre-dict hydrodynamic parameters for a complex-shapedAUV with very low cost. In [6], it is shown theefficiency of WAMITTM to predict the added massmatrix. ANSYS-CFXTM was employed in [7] forAUV damping analysis.

Regardless of modeling methods, the value of ahydrodynamic model depends on how robust yourcontrol scheme can adopt it. AUVs are generallydesigned to operate in the ocean environments. There-fore, numerous uncertainties are encountered, includ-ing parameter variations, non-linear hydrodynamiceffects, sensor measurement delays and ocean currentdisturbances. Owing to these unpredictable problems,traditional control methods, such as PID (Propor-tional integral derivative) and LQG (Linear quadraticgaussian), are not considered as efficient to guar-antee both stability and high performance, see [5].Consequently, advanced control algorithms shouldbe involved, such as the adaptive control schemein [15], robust control scheme in [3] and intervalapproach in [14]. Note that robust control schemes

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Table 1 CISCREA Characteristics

Size 0.525m (L) 0.406m (W) 0.395m (H)

Weight in air 15.56kg (without payload and floats)

Degrees of Freedom Surge, Sway, Heave and Yaw

Propulsion 2 vertical and 4 horizontal propellers

Speed 2 knots (Surge) & 1 knot (Sway, Heave)

Depth Rating 50m

On-board Battery 2-4 hours

were shown successfully in [21] and [8] for torpedo-shaped AUVs.

In this work, we appointed the semi-AUV CIS-CREA, as shown in Fig. 1a, and characterized in Table1. This fully functional AUV can turn into a ROVwhile connected to a shore based computer.

This paper is organized in six sections in order togive a complete method for modeling and control;main notions for the underwater vehicle are presentedin Section 2; the computational modeling approachand numerical results are given in Section 3; exper-iments to verify the numerical hydrodynamic modelis described in Section 4; Section 5 demonstrates anonlinear compensators; and finally conclusions aredrawn in Section 6.

2 AUV Modeling Framework

For marine systems, usually two coordinate sys-tems, NED-frame (North East Down) and B-frame(Body fixed reference) are introduced for conve-nience as presented by Fossen in [13] and shown inFig 1b.

In this section, CISCREA dynamics are repre-sented by the marine vehicle formulation proposed byFossen in [13] and [12], and the SNAME [1950](Society of Naval Architects and Marine Engineers)notions in [24]. Positions, angles, linear and angularvelocities, force and moment definitions are reflectedin Table 2. The position vector η, velocity vector ν andforce vector τ are defined as:

η = [x, y, z, φ, θ, ψ]T ; ν = [u, v, w, p, q, r]T ; τ = [X, Y, Z, K, M, N ]T

It is known that kinematic relation of velocity vector ν

and position vector η is expressed as Eq. 1 ([13]):

ν = J (�)η (1)

where, J (�) ∈ R6×6, stands for a transformation

matrix between B-frame and NED-frame, � =[φ, θ ,ψ]T , c(·) = cos (·), s (·) = sin (·) and t(·) = tan (·).

J (�) =[

R(�) 03×3

03×3 T (�)

], T (�) =

⎡⎣ 1 s(ψ)t (θ) c(φ)t (θ)

0 c(φ) s(φ)

0 s(φ)c(θ)

c(φ)c(θ)

⎤⎦

R(�) =⎡⎣ c(ψc(θ) −s(ψ)c(φ) + c(ψ)s(θ)s(φ) s(ψ)s(φ) + c(ψ)c(φ)s(θ)

s(ψc(θ) c(ψ)c(φ) + s(φ)s(θ)s(ψ) −c(ψ)s(φ) + s(θ)s(ψ)c(φ)

−s(θ) c(θ)s(φ) c(θ)c(φ)

⎤⎦

Depending on [13], rigid-body hydrodynamicforces and moments can be linearly superimposed.Therefore, the overall non-linear underwater modelcan be characterized by two parts, the rigid-bodydynamic (2) and hydrodynamic formulations (3)

(hydrostatics included). Table 3 shows parameter def-initions for this model.

MRBν + CRB(ν)ν = τenv + τhydro + τpro (2)

τhydro = −MAν − CA(ν)ν − D(|ν|)ν − g(η) (3)

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Table 2 The notation of SNAME for marine vessels

Positions & Angles Linear & Angular Velocities Forces & Moments

Coordinate NED-frame B-frame B-frame

Surge x u X

Sway y v Y

Heave z w Z

Roll φ p K

Pitch θ q M

Yaw ψ r N

Rigid-body mass inertia matrix MRB is defined inEq. 4, where m is the mass and rG = [xG, yG, zG]T isthe vector from Ob (origin of B-frame) to CG (centerof gravity). If rG = 0, i.e., Ob ≡ CG, then the matrixMRB will be extremely reduced. Moreover, symmetricproperties of cubic AUV, in x = 0 and y = 0 planes,can be used to simplify the inertia components to arough diagonal form.

MRB=

⎡⎢⎢⎢⎢⎢⎣

m 0 0 0 mzG −myG

0 m 0 −mzG 0 mxG

0 0 m myG −mxG 00 −mzG myG Ix −Ixy −Ixz

mzG 0 −mxG −Iyx Iy −Iyz

−myG mxG 0 −Izx −Izy Iz

⎤⎥⎥⎥⎥⎥⎦

(4)

CRB and CA contribute to the centrifugal force. Apractical way to calculate these two matrices usingMRB, MA, ν and the matlab function “m2c.m” isintroduced in MSS (Marine System Simulator) in [17].In our case, these two matrices are neglected, becausethe vehicle speed is low enough to be considered,C(v) ≈ 0.

For an AUV with neutral buoyancy, the weight W isapproximately equal to the buoyancy force B . In Eq. 5,

g is the gravity acceleration, ρ is the fluid density, and∇ is the displaced fluid volume.

W = mg, B = ρg∇ (5)

As pointed out by [13], the restoring forces andmoments vector g(η) can be simplified as in (6),where BG = [BGx, BGy, BGz]T is the distancefrom the CG to CB (the buoyancy center).

g(η)=

⎡⎢⎢⎢⎢⎢⎣

000−BGyWcos(θ)cos(φ)+BGzWcos(θ)sin(φ)

−BGzWsin(θ)+BGxWcos(θ)sin(φ)

−BGxWcos(θ)sin(φ)−BGyWsin(θ)

⎤⎥⎥⎥⎥⎥⎦

(6)

For CISCREA, CB and CG can be located usingtrial and error method on adding and removing thepayload and floats.

The marine disturbances, such as the wind, wavesand current contribute to the environmental effectτenv. But for an underwater vehicle, only currentis considered since wind and waves have negligibleeffects on an AUV during underwater operations.

Table 3 Nomenclature of the notations

Parameter Description

MRB AUV rigid-body mass and inertia matrix

MA Added mass matrix

CRB Rigid-body induced coriolis-centripetal matrix

CA Added mass induced coriolis-centripetal matrix

D(|v|) Damping matrix

g(η) Restoring forces and moments vector

τenv Environmental disturbances (wind, waves and currents)

τhydro Vector of hydrodynamic forces and moments

τpro Propeller forces and moments vector

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Fig. 2 Measure andCalculate Mass InertiaMatrix inPRO/ENGINEERTM

In order to put forward the model in the same coor-dinate, a transformation is made according to Eqs. 1, 2and 3. Cubic AUV model under both NED-frame andB-frame is transformed into NED-frame as follows:

M∗η + D∗(|ν|)(η) + g∗(η) = τpro + τenv (7)

M∗ = J−T (�)(MRB + MA)J−1(�);D∗(|ν|) = J−T D(|ν|)J−1(�);g∗(η) = J−T g(η)

Two hydrodynamic parameters added mass, MA ∈R

6×6, and damping, D(|ν|) ∈ R6×6, should be care-

fully involved in the AUV model. Added mass isa virtual conception representing the hydrodynamicforces and moments. Any accelerating emerged-objectwould encounter this MA due to the inertia of thefluid. For a cubic AUV, added mass in some direc-tions are generally larger than the rigid-body mass[26]. Damping in the fluid consists of four parts, asdescribed in Eq. 8: Potential damping DP (|ν|), skinfriction DS(|ν|), wave drift damping DW (|ν|) andvortex shedding damping DM(|ν|).D(|ν|) = DP (|ν|)+DS(|ν|)+DW (|ν|)+DM(|ν|) (8)

Details of the two hydrodynamic parameters are dis-cussed in the following section.

3 Computational Solutions for Dynamicand Hydrodynamic Parameters

This section is dedicated to calculate numericallydynamic and hydrodynamic parameters: Mass iner-tia matrix MRB , added mass matrix MA and dampingmatrix D(|ν|). Due to the complex structure, tra-ditional empirical formula approximation is not asefficient as for slender bodies [20]. To solve this prob-lem, we propose to calculate hydrodynamic modelsusing CFD software as follows:

– Mass inertia matrix MRB is calculated usingPRO/ENGINEERTM.

– Added mass matrix MA is calculated using radia-tion/diffraction program MCC and WAMITTM.

– Hydrodynamic programs ANSYS-CFXTM andSTAR-CCM+TM are studied to predict dampingbehavior D(|ν|).

Fig. 3 Geometry files ofMULTISURF (left) andGAMBIT(right)

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Table 4 Added mass of a sphere (Radius 1m, density 1kg/m3, depth 10m, 1024 surfaces)

Theoretical WAMIT MCC

Surge(kg) 2.0944 2.084236 2.106

Note that rotational damping effects are numeri-cally built using moving reference method in STAR-CCM+T M .

3.1 Rigid-Body Mass Inertia Matrix

Due to many different density components, the iner-tia parameters and CG of CISCREA are hard to becalculated according to [13] (Eq. (9)).

m =∫

V

ρmdV, I =∫

V

r2ρmdV (9)

Here ρm is the density of volume element dV , V is thevolume of the body, r is the distance between volumeelement dV and CG.

A practical way is to measure the principal com-ponents of the AUV and calculate MRB automaticallywith CAD software PRO/ENGINEERT M . This pro-cess is shown in Fig. 2.

The output of PRO/ENGINEERTM for CISCREAaround CG (Ob), MRB , is listed in Eq. 10 (kg andkg/m2). Floatings and payloads are not considered.

MRB =

⎡⎢⎢⎢⎢⎢⎣

15.643 0 0 0 0 00 15.643 0 0 0 00 0 15.643 0 0 00 0 0 0.2473 0 0.00290 0 0 0 0.3698 00 0 0 0.0029 0 0.3578

⎤⎥⎥⎥⎥⎥⎦

(10)

By neglecting small off-diagonal inertia elements,the above matrix can be simplified to a diagonalmatrix:

MRB =diag([

15.643 15.64315.643 0.2473 0.3698 0.3578])

(11)

3.2 Added Mass Matrix

As it is mentioned in [19], variations of underwatervehicle geometry play indeed a role on the added massof the AUV. Therefore, the hull approximation by ele-mentary shapes and the use of empirical formulas pre-dicting MA are inaccurate for complex-shaped AUV.To solve this issue, boundary element method basedWAMITTM and MCC are used. Both products offerthe function to compute the added mass matrix. Inour case, freeware MCC is used to guarantee commer-cial software WAMIT calculation are made with goodconfigurations.

WAMITTM requires close surface geometry file,potential control file and force control file asinput. To execute a WAMITTM calculation mightneed the basic configuration knowledge which isdescribed in software user manual [18]. Otherwise,an AUV WAMITTM application is briefly mentionedin [6]. MCC requires similar information [16]. Inour case, we only show the initial step, the loworder panel mesh results from MULTISURFTM (”.gdf” for WAMITTM) and GAMBITTM(”.neu” forMCC), shown in Fig. 3. It is worth mentioning that,elementary surfaces should be pointed to outsidedirection.

Knowing that the theoretical added mass of asphere is given by 2/3πρr3. This information wasused to verify the configuration of both prod-ucts. Results are compared in Table 4. Importsame input control files of the sphere calculationwith CISCREA geometry file to WAMITTM andMCC, added mass matrix results are calculated andcompared.

Table 5 Added mass result of CISCREA (Mass : kg, Inertia : kg · m2 )

Surge Sway Heave Roll Pitch Yaw

WAMIT 11.985 20.261 67.142 0.385 0.792 0.138

MCC 11.8 17.9 52.7 0.91 1.54 0.085

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Fig. 4 Mesh configurations

WAMITTM output is listed in MA1 (Eq. 12), andMCC in MA2 (Eq. 13) (Mass : kg, Inertia :kg · m2 ). As the vehicle speed is low enoughto neglect the small off-diagonal elements. InTable 5, only diagonal elements of MA areconsidered.

MA1 =

⎡⎢⎢⎢⎢⎢⎢⎣

11.985 −0.091 −0.105 0.039 0.308 0.0120.149 20.261 −0.147 0.085 −0.013 −0.7580.111 −0.129 67.141 −0.033 2.530 0.0640.122 0.319 −0.056 0.385 0.003 −0.0110.407 −0.001 2.543 −0.002 0.791 0.002−0.003 −0.758 0.064 −0.003 0.004 0.138

⎤⎥⎥⎥⎥⎥⎥⎦

(12)

MA2 =

⎡⎢⎢⎢⎢⎢⎢⎣

11.8 4.08 9.41 0.326 0.349 −0.2674.53 17.9 −10.3 0.492 −0.913 0.2338.6 −12.3 52.7 −2.88 −7.94 1.490.256 0.676 −2.74 0.91 0.573 0.0087−0.067 −0.628 −9.17 0.655 1.54 0.04−0.184 0.162 1.29 −0.0289 −0.0252 0.0854

⎤⎥⎥⎥⎥⎥⎥⎦

(13)

This work demonstrates besides empirical approxi-mation, the added mass of complex-shaped AUVs canbe numerically predicted and guaranteed using bothWAMITTM and MCC. For robust control scheme, thecalculation errors of our added mass results can beconsidered to be mass-inertia uncertainties.

Fig. 5 CISCREA set up intranslational and rotationalwater tanks

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Table 6 Configurations of CFX and STAR-CCM+

Parameter CFX Configuration STAR-CCM+ Configuration

Tank 10.5m (L) 4.5m (W) 4.5m (H) 9m (L)4m (W) 4m (H)

Cylinder - 8m(H) 8m(radius)

Fluid steady steady

Density ρ = 1023kg/m3 ρ = 1023kg/m3

Viscosity 1.56 × 10−6kg/(s · m) 1.56 × 10−6kg/(s · m)

Turbulence 1 % at inlet boundary, (k-ω) 1 % at inlet boundary, (k-ω)

Mesh 588221 elements for heave 1002637 cells, 3002121 faces for sway

(around for surge and sway) (around for surge and heave)

Convergence 10−4 100 steps (< 10−5)

Roughness PVC 0.0015 - 0.007 (mm) Wall

3.3 Damping Matrix

As mentioned before in Eq. 8, four elements con-tribute to D(|ν|), the damping matrix of marine vehi-cles. In this application, DP (|ν|) and DW(|ν|) canbe ignored. First, potential damping DP (|ν|) is neg-ligible comparing to other terms [13]. Second, wavesdrift damping DW(|ν|) is reduced since underwatervehicles operate under certain depth, and waves areassumed to act only on surface vehicles. Skin fric-tion damping DS(|ν|) and vortex shedding dampingDM(|ν|) are left to study.

As presented in [13], usually different dampingterms contribute to both linear and quadratic damp-ing, and it is difficult to separate these effects. So, theEq. 14 is introduced,

D(|ν|) = D + Dn(|ν|) (14)

where D is the linear damping matrix and Dn(|ν|) isa quadratic damping matrix. If vehicle velocities aresufficiently high the D can be neglected. Otherwise,Dn(|ν|) is negligible.

Before quantitative damping analysis of AUV,empirical formulas are used to roughly determinethe damping behaviors. The quadratic damping f (U)

force is represented as follows:

f (U) = −1

2ρdCD(Rn)AU |U | (15)

where U is the vehicle velocity, ρd is the fluid den-sity, A is the cross-sectional area projected on the fluidand CD(Rn) is the damping coefficient, which is afunction of Reynolds number Rn.

Rn = ρdUDCL

νis

(16)

Table 7 Damping forces and moments at different velocities

CFX 0.1m/s 0.2m/s 0.3m/s 0.4m/s 0.5m/s

Surge (N) 0.287 1.146 2.577 4.579 7.222

Sway (N) 0.537 2.14 4.815 8.561 13.382

Heave (N) 0.51 3.319 7.47 13.28 20.751

CCM+ 0.1m/s 0.2m/s 0.3m/s 0.4m/s 0.5m/s

Surge (N) 0.273 1.06 2.39 4.253 6.539

Sway (N) 0.5077 2.011 4.4531 8.0222 12.2759

Heave (N) 0.8393 3.298 7.43 12.974 20.745

CCM+ (rad/s) 0.5 1 1.5 2 2.5 3 3.5 4

Yaw (N · m) 0.038 0.149 0.338 0.593 0.932 1.33 1.792 2.381

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For Rn in Eq. 16, νis is the fluid viscosity and DCL isthe characteristic length. In our study, the fluid is cho-sen to be the seawater as described in [13]. CISCREAgenerally operates at a speed U range from 0 to 1m/s.Robot characteristic length is approximately 0.5m. Inthis case, the Rn of CISCREA is generally around 108

to 109. This implies that the damping of CISCREA isnot in the critical area, i.e., from 105 to 106 betweenthe laminar and the turbulent flow. Therefore, a con-verged constant damping coefficient is encountered,and nonlinear quadratic damping behavior is expected.Note that the variations of CD chart with respect to Rn

can be found in [4].Due to the complex-shaped geometry of CIS-

CREA, the empirical formulas are impractical to pre-dict the damping effects. Meanwhile, it is hard totell which part of the linear and quadratic dampingplay the major role for CISCREA. So, finite ele-ment theory based CFD software ANSYS-CFXTM andSTAR-CCM+TM are used to estimate the relationshipamong damping forces, damping moments and vehiclevelocities, angular velocities.

In previous work [26], only translational directionsare calculated using ANSYS-CFXTM. For rotationalmotions, the CFD calculation were considered inap-propriate. The remesh solution in [11] shows even for2D rotational simulation, the calculation is too expen-sive. In this paper, we introduce the move referencetechnique of STAR-CCM+TM. The idea is estimatingrotational damping with less expensive calculation.

In STAR-CCM+, three ways exist to analyzerotational motion: moving reference method, DFBI(Dynamic Fluid Body Interaction) or similarly rota-tion motion method, and overset mesh method ([23]).Moving reference approach convert an unsteadymotion problem into a steady-state problem by impos-ing a moving frame on a static mesh. In this case,steady fluid are available to make the rotationalproblems fast to be solved. Rotation motion orDFBI involve actual displacement of regions. Mov-ing regions and static regions exchange physics amongcontacting interfaces. Overset mesh method appliesreal time displacement between background mesh andoverlap mesh. In every calculation iteration, over-set mesh interact with background mesh on oversetboundaries. The last two methods bind to implicitunsteady fluid, and calculation load is generally heav-ier than moving reference. Therefore, we choose mov-ing reference method for rotational simulation. To

achieve better CFD results, overset mesh solution isalso suggested. A streamline plot of moving referencemethod and static overset mesh results are shown inFig. 9a and b.

To execute a water tunnel simulation, five stepsare required: Generate model geometry, set up mesh,configure physics conditions, run calculation and pro-cess results. In our case, AUV is fixed in rectangular(surge, sway, heave) or cylindrical (yaw) water tanks,as shown in Fig. 5. Fluid moves in the tank withthe speed variations from 0 to 0.5m/s for transla-tional motion and 0 to 5rad/s for rotational motion.Respectively, speed growth interval of 0.1m/s and0.5rad/s are selected. Note that mesh quality isusually important for CFD analysis, but our objec-tive is to capture major damping effects. Therefore,we chose the mesh configurations in Table 6 inorder to make good compromise between the calcu-lation time and accuarcy. Mesh results are shown inFig. 4.

The configurations used in CFX and STAR-CCM+are listed in Table 6, and the damping force (moment)results are shown in Table 7. Velocity streamlineviews of CFX and STAR-CCM+ results in differentdirections are shown in Figs. 6, 7, 8 and 9. Secondorder polynomial lines are implemented to approxi-mate the relationship between damping and velocities,see Fig. 10 for translational direction, Fig. 14d forrotational direction and Table 8 for parameters. Ourresults show that quadratic damping dominates thedamping effects.

4 Experimental Results

In order to verify the efficiency of the numericalmodel obtained in Section 3, real world experimentshave been conducted on the open-loop CISCREAto verify the translational and rotational dampingparameters.

Bollard thrusts of propellers are measured, seeFig. 11. During experiment, CISCREA is driving inthe surge and sway directions inside a pool (4m ×4m × 3.5m) from one end to another, and dive inthe heave direction from top to bottom. The pro-cess of yaw rotation drives CISCREA spin in themiddle of the pool until it reaches a constant angu-lar velocity. Meanwhile, all these moving processeswere captured by a 15 fps CCD camera on top and

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Fig. 6 CISCREA at 0.5m/s

Fig. 7 CISCREA at 0.5m/s

Fig. 8 CISCREA at 0.5m/s

Fig. 9 CISCREA at 4rad/s(STAR-CCM+)

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Fig. 10 Damping force andvelocity (CFX: solid line,STAR-CCM+: dash line)

0 0.1 0.2 0.3 0.4 0.502468

101214161820

Velocity (m/s)

Dam

ping

For

ce (

N)

Surge (A)

Sway (B)

Heave (C)

C

B

A

another camera with 25 fps underwater, as shown inFig. 12.

The propelling force, the pool size and the time hit-ting the wall or single rotational lap are known, wecan build damping and velocity relationship based onthis information. First of all, we can verify that giv-ing any propelling input, the final velocity of AUVshould converge to a steady constant speed. We sim-ply assume the surge dynamic is a linear equation(Eq. 17), where DL is the unknown linear dampingcoefficient, x is the surge position of CISCREA. Asa result, the convergence in Fig. 13 indicates that theaverage speed can be measured after a specific time.Position and camera frame information provide thisaverage speed for surge, sway and yaw motions.

(MRB + MA)x + DLx = τm (17)

We should mention that, it is hard to identify dis-tance for calculating average speed on our underwater

camera. There is no clear marks on the wall. There-fore, we use the final speed of the linear model(Eq. 17) for the heave motion.

Finally, experimental damping and velocity resultsand polynomial lines are compared to CFD resultsin Fig. 14 and Table 9. The gap in Fig. 14a andb are mainly caused by the drag of cables andropes, which are playing opposite efforts, as shownin Fig. 14c. In additions, the propulsion decrease,in case of moving fluid, contributes to experimenterror [13]. Rotational Damping of STAR-CCM+ andexperiment are compared in Fig. 14d. Notice thatrotational CFD results can be improved by furthersimulations. In addition, as propulsion decrease andrope drag indicate a smaller hydrodynamic damp-ing, therefore, we assume ideal nominal modelsby taking average of CFD and experiment resultsfor every direction (Table 9). Nominals are usedto demonstrate the nonlinear compensator in nextsection.

Fig. 11 Forces and torquesmeasurements

30 40 50 60 70 80 90 100 110 1200123456789

101112

Input signal

Pro

pelle

r F

orce

(N

) &

Tor

que(

N*m

) Surge force from 2 horizontal propellerSway force from 4 horizontal propellerHeave force from 2 vertical propellerTorque from 4 horizontal propeller (YAW)

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Fig. 12 CISCREA movingin surge, sway, heave andyaw directions

5 Nonlinear Compensator

A control-oriented underwater vehicle model bringsmany benefits to the control designs. For example, toestimate unmeasurable or noisy states: speed, posi-tion or angles, it is possible to build state spaceobservers like Kalman filters [1]. Otherwise, it is pos-sible to build smith compensators [28] (require accu-rate model) to handle system or sensor delays. Moreimportantly, we propose here a method to deal withsystem nonlinearities under a robust control scheme.

As found in above sections, damping is a majornonlinearity in underwater vehicle models. However,

most of the control theories including robust con-trol are developed on linear system, and nonlinearproblems are largely related to fit a linear theory onnonlinear models [1]. Generally, the first option isto linearize the model on equilibrium points. Sec-ondly, nonlinear components can be assigned to uncer-tainties using robust control scheme. Otherwise, wecompensate nonlinear behaviors by creating a linearbehavior, the compensation error is assigned to beuncertainty.

Without loss of generality, we only demonstrate theidea in yaw direction. The rotational model is sim-plified as Eq. 18 (neglecting buoyancy and gravity).

Fig. 13 Convergence ofvelocities after 10s for 10constant force points

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Vel

ocity

of t

he L

inea

r M

odel

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/s)

time (s)

J Intell Robot Syst

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7

8

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Dam

ping

For

ce (

N)

Experiment(A)

ANSYS−CFX(B)

STAR−CCM+(C)

Assumed Nominal(D)

AB

C

D

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14

Velocity (M/s)

Dam

ping

For

ce (

N)

Experiment(A)

ANSYS−CFX(B)

STAR+CCM+(C)

Assumed Nominal(D)

AB

C

D

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

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14

Velocity (m/s)

Dam

ping

For

ce (

N)

Experiment(A)

ANSYS−CFX(B)

STAR+CCM+(C)

Assumed Nominal(D)A

D

C

B

0 1 2 3 40

1

2

3

4

5

6

Tor

que

N*m

Velocity (rad/s)

STAR−CCM+(A)Rotation(left,B)Rotation(right,C)Assumed Nominal(D)

A

B

C

D

Fig. 14 Comparison of ANSYS and experiment damping and velocities

Definitions in this section are listed in Table 10.

(IYRB + IYA)xr + DYN |x|x + DYLxr = τi (18)

For the linearization process, Eq. (18) is linearizedinto (19) according to equilibrium point xr0.

(IYRB + IYA)xr + (DYL + 2DYN |xr0|)xr − DYN |xr0|xr0 = τi (19)

However, the critical issue is that xr0 is usuallyequal to zero, but xr move in a wide range. Systemdoes not stay around a equilibrium point as supposed.The second method is more straight forward, as shownin Eq. 20.

(IYRB + IYA)xr + (DYN |xr | + DYL)xr = τi (20)

Table 8 CFD results curve fitting

CFX & RMSE STAR-CCM+ & RMSE

Surge y = 28.6x2 + 0.0089x 0.01773 y = 25.75x2 + 0.2406x 0.02294

Sway y = 53.52x2 0.00188 y = 48.39x2 + 0.4512x 0.0595

Heave y = 83.42x2 0.103 y = 82.44x2 0.1144

Yaw - - y = 0.1479x2 + 0.001328x 0.009881

J Intell Robot Syst

Table 9 Experimental results curve fitting (Benchmark: STAR-CCM+)

Experiment RMSE STAR-CCM+ RMSE

Surge y = 21.4x2 + 10.75x 0.214 y = 25.75x2 + 0.2406x 0.02294

Sway y = 61.39x2 + 9.775x 0.3817 y = 48.39x2 + 0.4512x 0.0595

Heave(dive) y = 83.42x2 0.5924 y = 82.44x2 0.1144

Yaw(left) y = 0.3513x2 + 0.0321x 0.119 y = 0.1479x2 + 0.001328x 0.009881

Yaw(right) y = 0.3338x2 + 0.1081x 0.117 y = 0.1479x2 + 0.001328x 0.009881

Assumed nominal model

Surge y = 25x2 + 5.379x

Sway y = 57.48x2 + 4.88x

Heave(dive) y = 80.37x2

Yaw(left) y = 0.2496x2 + 0.021x

Fig. 15 Nonlinearcompensator

Nonlinear Yaw Model

Nonlinear Compensator

Linear system with uncertainties

AngleTorque

Angular Velocity

Nonlinear Compensation

Propellers

Linear Command

Table 10 Rotational model notions of yaw direction

Parameter Description Value

IYRB Rigid-body inertia 0.3578kg · m2

IYA Added mass inertia 0.138kg · m2

DYN Nominal quadratic factors Ideal 0.2496

DYL Nominal linear factors Ideal 0.021

xr Angular Velocity 0 to 4rad/s

τi Torque input 0 to 6N · mτcom Compensation Torque 0 to 6N · mxr0 Equilibrium velocity 0 to 4rad/s

DYND CFD quadratic factors 0.1479

DYLD CFD linear factors 0.0013

DYLA Artificial linear factors <Moto limit (select 1.2)

J Intell Robot Syst

Table 11 Linear damping uncertainty margin

Methods Nominal Linear Factor Uncertainty Margin

Compensate DYLA:1.8 (for example) DYLA : [1.3735, 2.2265], 23.7 %

DYN |xr | is supposed to be an uncertainty add onDYL. This method lead to a linear nominal model,of which DYL uncertainty is large. To the last,the proposed approach feedback the CFD nonlinearbehavior to real world propellers and nonlinear yawmodel, as shown in Fig. 15. The idea is to cancelthe original nonlinear behavior, and create a roughartificial linear damping behavior for robust con-trol nominal. A nonlinear compensation is given inEq. 21.

τcom = (DYLA − DYLD − DYND|xr |)xr (21)

DYLA is the artificial linear factor given in Table 10.DYND and DYLD are CFD damping estimations. Thelinear model result of compensation is described inEq. 22.

(IYRB + IYA)xr + (DYLA + (DYN |xr | − DYND |xr |+DYL − DYLD))xr = τi (22)

The term δ = DYN |xr | − DYND|xr | + DYL −DYLD is calculated as an uncertainty added on DYLA.Generally, this δ remains small with DYLA.

If we calculate δ using that,xr ∈ [−4, 4] rad/s; DYLA=1.8; DYN=0.2496;DYL=0.021; DYND=0.1479; DYLD=0.0013;

we can then consider that DYLA has a dynamicuncertainty of 23.7 %, listed in Tab le11. At the end,the proposed model, Eq. 23 is a first order linearsystem.

(IYRB+ IYA)xr + (DYLA + δ)xr = τi;δ ∈ [−0.4265, 0.4265] (23)

To verify this approach, Matlab simulation and realworld experiments are implemented. Results are vali-dated and presented on MOQESM conference [27].

6 Conclusion

We proposed an AUV modeling approach to avoidthe deployment of expensive devices. The quantita-tive model is built for CISCREA, and validated byrealistic experiments. We estimated numerically twoimportant hydrodynamic parameters: the added mass(Predicting by WAMITTM and MCC) and the damp-ing effects (Predicting by ANSYS-CFXTM, STAR-CCM+TM and experiments). Our experiment resultsshowed that the quadratic damping is the domi-nant component of all damping. With our numericalmodel, we propose nonlinear compensator to shrinkthe uncertainty margin for linear-based robust con-trol designs. Note that with even our unimprovedrotational CFD results, we can guarantee an uncer-tainty margin less than 23.7 % (linear damping fac-tor). By assuming the nominal values of hydrody-namic parameters, this model can shrink the 6 DOFsparameter uncertainty margins for robust controldesign.

Acknowledgments The authors would like to express theirgreat appreciation to the fellows of ENSTA Bretagne: Prof. J.M. Laurens, I. Probst, F. Le Bars and J.S. Zhang, for theirsupports to complete this work successfully.

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