Longitudinal Control of PirajubaAutonomous Underwater Vehicle, UsingTechniques of Robust Control LQG/LTR

Joao L. D. Dantas ∗ Ettore A. de Barros ∗∗ Jose J. da Cruz ∗∗∗

∗Mechatronics Engineering Department, Polytechnic School ofUniversity of Sao Paulo, Sao Paulo, Brazil (e-mail:

dantas.joao@gmail.com).∗∗Mechatronics Engineering Department, Polytechnic School of

University of Sao Paulo, Sao Paulo, Brazil (e-mail: eabarros@usp.br).∗∗∗ Electrical Engineering Department, Polytechnic School of

University of Sao Paulo, Sao Paulo, Brazil (e-mail: jaime@lac.usp.br)

Abstract: This work presents a study on a robust controller design for the motion on thevertical plane of an AUV at low depths. The LQG/LTR control strategy is proposed for rejectingdisturbances originated from sea waves on the depth, attitude and axial velocity of an AUV.The vehicle is an adapted version of the Pirajuba AUV, developed at the Unmanned VehiclesLaboratory, in the University of Sao Paulo. The plant modeling was based on previous studieson manoeuvrability predictions for the AUV, and a simplified model for wave disturbances.Modeling errors for manoeuvring coefficients were considered in the controller design. Thevalidation is based on simulation studies using a set of non-linear equations of motion.

Keywords: Robust control in marine systems; Autonomous Underwater Vehicles; Simulationsin marine systems.

1. INTRODUCTION

Currently, public institutions, industry and academy areincreasing the use of Autonomous Underwater Vehicles(AUVs) to perform a number of activities related tosea bottom survey, environment monitoring, inspection ofoffshore pipelines, and security related missions.

Usually, during those missions, the AUV navigates at sucha depth where the influence of the sea surface waves isnegligible. Therefore, researchers and engineers consideronly the calm water manoeuvring models for autopilotdesign and vehicle motion simulations.

However, some oceanographic and military related mis-sions may impose that the vehicle navigates close to the seasurface. In that case, it is important to consider the effectsof waves and currents into vehicle’s motion and controllerresponses to such disturbances.

In a number of papers, the analytical and semi-empiricalmodels considered in autopilot designs for submersiblevehicles included the wave forces and moments that are de-pendent of the body size and wave parameters (Dumlu andIstefanopulos, 1995; Liceaga-Castro and van der Molen,1995; Mandzuka, 1998; Moreira and Guedes Soares, 2008).They can be mathematically divided into first-order andsecond order components (Faltinsen, 1990). First orderefforts are proportional to each one of the wave componentamplitudes that can represent the sea spectrum. They are

? This work has been funded by FAPESP from Sao Paulo state underthe research project numbers 09/06726-8 and 09/10205-3.

responsible for imposing oscillations to a body close to thesea surface. The second order (drift) forces are propor-tional to the square of the wave components amplitudeand would be responsible for dragging the submersiblevehicle to the surface. However, the general formula pro-posed for modeling those forces and moments includedhydrodynamic coefficients, without any reference abouttheir prediction Moreover, since those models were derivedexperimentally for submarines, no consideration on thedifferences between the wave effects on those vehicles andAUVs was observed (Moreira and Guedes Soares, 2008).

This work analyses the performance of a LQG/LTR con-troller for compensating the wave disturbances on theAUV motion in the vertical plane. As long as a modelwhich includes the prediction of the hydrodynamic coef-ficients is not completed, it is proposed an approximatedmethod, where the wave effect is taken into account mainlyas a disturbance on the angle of attack at the AUVhydroplanes relative to the water flow. This approach issimilar to the one adopted by Song et al. (2003) whendesigning a real-time simulator for an AUV.

The vehicle considered for the simulations is a modifiedversion of the AUV Pirajuba, developed by the UnmannedVehicles Laboratory at the University of Sao Paulo. Com-pared to the current configuration, this model has theadditional effects of bow planes, that are included formanoeuvres close to the sea surface.

The next sections of this paper are composed as follows.Section 2 introduces the non-linear equations of motionof the AUV, and the considered simplifications in order

Table 1. Dimensions of the fins.

Parameter Stern Plane Bow Plane

Height 132 mm 132 mmRoot Chord 155 mm 155 mmTip Chord 101 mm 101 mmSweep Symmetrical SymmetricalProfile NACA0015 NACA0015Leading edge 1317 mm 217 mm

position**w. r. t. the body nose.

to obtain a linear model for the controller design. Insection 3, the modeling errors assumed in the controllerdesign are described. Section 4 summarizes the LQG/LTRdesign procedure. Results of simulation tests are discussedin section 5. Finally, section 6 presents conclusions andfollowing steps of this research.

2. THE AUV DYNAMIC MODEL

The control algorithm was tested in simulations based ona modified version of the AUV Pirajuba (de Barros et al.,2010). The original vehicle includes 4 control surfaces atthe stern, in the conventional cruciform configuration.All of them have the same geometrical dimensions, andare based on the NACA0012 section profile. For thesimulations considered in this work, the NACA0015 profileis chosen, which provides a greater stall angle. The spanand chord dimensions are changed also so that eachhydroplane generates the same force per deflection angleas in the original AUV. Moreover, in order to provideboth depth and attitude control when navigating near thesea surface, an additional pair of control surfaces in thehorizontal plane are considered, the so called bow planes.Particulars of the hydroplanes are presented in table 1.

2.1 Equations of Motion

The nonlinear dynamic model is restricted to the motionin the diving plane. It is a reduced version of the sixdegrees of freedom model for the equations of motion of anunderwater vehicle (SNAME, 1950). The hydrodynamicefforts are considered through expressions for the normalforce, pitch moment and axial force that take into accounttheir variation with the angle of attack up to 25 degrees(de Barros et al., 2008a). The dynamic efforts (relatedto the angular velocities) and the forces and momentsgenerated by the control surfaces are predicted by adaptedformula originated in the aircraft industry (de Barroset al., 2008b).

The equations of motion in diving plane can be expressedin the matrix form, as follows:

[Mij +Aij ] · [u, w, q]T = FI + FW−B + FT + FH (1)

VxVzθ

=

[cos(θ) sin(θ) 0−sin(θ) cos(θ) 0

0 0 1

]·

[uwq

](2)

Where:

[Mij + Aij ] = (3)[m+Xu 0 m · zG

0 m+ Zw −m · xG + Zqm · zG −m · xG +Mw Iy +Mq

]

FI =

−m · w · q + xG · q2m · u · q + zG · q2

−m · zG · w · q −m · xG · u · q

(4)

FW−B =

[00

−m · g · zG · sin(θ)

](5)

The thruster force is given as a function of the propellerdiameter, D, it’s rotation speed, n, and the open waterthrust coefficient, KT :

FT =

ρ ·D4 · n2 ·KT (u, n)00

(6)

The hydrodynamic efforts are composed by the axialforce, X, the normal force, Z, and the pitch moment, M .The axial force is estimated by the method presented byHoak and Finck (1978). The normal force and the pitchmoment are estimated according to the formula presentedin de Barros et al. (2008b) and the equivalent angle ofattack method (Hemsch and Nielsen, 1985; Blake andGillard, 1988). They are calculated as functions of localangle of attack at the hull, αbody, the angle of attack atthe stern planes, αstern, and the angle of attack at the bowplanes, αbow:

FH =

[X(αbody, αstern, αbow)Z(αbody, αstern, αbow)M(αbody, αstern, αbow)

](7)

The body related components are defined as:

Xbody = 1/2ρU2L2CD0cos(αbody)2 (8)

Zbody =1

2ρU2L2

∫ L

0

cN (x)dx (9)

Mbody =1

2ρU2L2

∫ L

0

cN (x)(xm − x)dx (10)

cN (x) = 2ηCdc sin2(αbody)r(x)+ (11)

(K2 −K1) sin(2αbody)dS(x)

dx

(1− sin(α)1.3

)The body drag coefficient, CD0 , is defined for the angleof attack equal zero. The parameter Cdc is the cross-flow drag coefficient for a infinitely long circular cylinder,taken as 1.2, and the factor η corrects this parameterfor the finite length cylinder. The coefficients K1 and K2

are, respectively, the longitudinal and transverse apparentmass coefficients of the body (de Barros et al., 2008b).The parameters S(x) and r(x) are the cross sectional areaand the radius of the body at station x, respectively. Thecoordinated xm is the distance of rotation reference centreto the nose, taken usually as equal to centre of gravity

The angle of attack in each station of the bare hull is givenby:

αbody(x) = α+ (xm − x)q/u (12)

Where the angle of attack of the vehicle, α, is calculatedas function of it’s motion and the wave induced velocity,ζ and ξ:

α = tan−1(

(w − ζ)/(u− ξ)

(13)

Table 2. Properties of the waves generated bythe wind considered in this project

Wave Parameters Symbol Value

Significant wave height H1/3 0.1 - 0.5 m

Mean wave period T1 5 - 10 sFrequency ωj 0 - 0.4 HzWave number kj 0 - 0.64 m−1

Wave length λ 46 - 184 mBottom depth h 100 m

The wave induced velocity components, ζ and ξ, aredefined by equations 20 and 21.

The axial force component acting in upon a hydroplane isgiven by:

Xfin = 1/2ρU2L2CA(αf ) (14)

The normal force and moment is given by (Blake andGillard, 1988):

Zfin = 1/2ρU2L2(1 +KB(W )/KW (B))CN (αf ) (15)

Mfin = Zfin(xm − xf ) (16)

The coefficients CA and CN were estimated by CFD sim-ulations. The coordinated xf corresponds to the distanceof the hydroplane to the nose, and the angle of attack ineach hydroplane, αf , is calculated as follows:

tanαeqδ=0=KW (B) tan(α) + tan

q(xm − xf )

u+ tan ∆αv (17)

tanαf = kW (B) [tan (αeqδ=0+ δf )− tanαeqδ=0

]

+ tanαeqδ=0(18)

Where:

• αeqδ=0is the hydroplane equivalent angle of attack for

δf = 0.• δf is the hydroplane deflection.• KW (B) represents the ratio of the hydroplane lift

of the hull-hydroplane combination to that of thehydroplane alone. It’s defined for the case in whichthe angle of attack of the combination is varying butthe hydroplane deflection is zero (Pitts et al., 1957).• kW (B) represents the same ratio as KW (B) for the case

in which the deflection angle is varying but the angleof attack of the body is zero.• ∆αv is the average angle of attack induced by hull

vortices on hydroplane (for the bow fins ∆αv = 0).

The wave effect upon the AUV can be modeled as a changein the longitudinal and vertical velocities, which results invariations of the angle of attack. The sea wave model whichwas chosen is a representation of the Pierson-Moskowitzspectrum:

S(ω)

H21/3T1

=0.11

2π

(ωT12π

)−5exp

[−0.44

(ωT12π

)−4](19)

Where, S(ω) is phase-modulation spectrum in the longitu-dinal direction. The other parameters are defined in table2.

The numerical implementation of a time-series represent-ing this spectrum is based on of a finite summation over Nsinusoidal components, calculated over spatial and tempo-ral arguments, and a uniformly distributed random phase(ε). Each sinusoidal function is attenuated exponentiallywith the sea depth z (Faltinsen, 1990). The longitudinal

(ξ) and vertical (ζ) wave velocity components, in theinertial coordinate system, are given as

ξ(x, z, t) =

N∑j=1

ωej

√2S (ωj) ∆ω

·cosh k(z + h)

sinh khsin (ωjt− kjxw + εj) (20)

and

ζ(x, z, t) =

N∑j=1

ωej

√2S (ωj) ∆ω

· sinh k(z + h)

sinh khcos (ωjt− kjxw + εj) (21)

Where xw is the position of vehicle in the inertial coordi-nated system, and ωe is the wave frequency of encounter,defined as

ωe = ω + ku (22)

In order to calculate the angle of attack disturbance dueto waves these components are combined in the bodyfixed frame. The values presented in table 2 are based ona model of the sea spectrum in the Brazilian SoutheastCoast.

2.2 Linear Dynamic Model

Considering its use in the controller design, a linear modelfor the AUV dynamics in the diving plane is derivedfrom the original nonlinear equations of motion. For thatpurpose, the linearization of the original model is donearound the cruising speed (u = 1m/s) and trimmingconditions (δstern = δbow = θ = 0). The linear modelobtained is represented as:

X =[M∗−1 ·A∗

]·X +

[M∗−1 ·B∗

]· Uc

Y = C ·X (23)

Where is considered the state variables u, w, q, θ, and thedepth z, and the input variables δstern, δbow and n. Thematrix M∗ is the inertia matrix, defined as follows,

M∗ = (24)m+Xu 0 m · zG 0 0

0 m+ Zw −m · xG + Zq 0 0m · zG −m · xG +Mw Iy +Mq 0 0

0 0 0 1 00 0 0 0 1

Where m is the vehicle mass, Iy is the vehicle inertia(about Y axis), xG and zG are the body fixed framecoordinates of the centre of gravity.

The matrix A∗ is defined by (25).

A∗ =

Xu 0 0 0 0Zu Zw Zq + u0m 0 0Mu Mw Mq + u0mxg Mθ 00 0 1 0 00 1 0 −u0 0

(25)

Where the coefficients X, Z and M represent the forcesand moment in the vertical plane differentiated withrespect to the motion variables (SNAME, 1950)

Similarly to the matrix A∗, the matrix B∗ is defined in(26).

B∗ =

0 0 Xn

Zδs Zδb ZnMδs Mδb Mn

0 0 00 0 0

(26)

The matrix C is defined in relation to the variable thatwill be controlled, in this case these variables are thelongitudinal velocity, angle of pitch and depth. So thematrix C can be write as in (27).

C =

[1 0 0 0 00 0 0 1 00 0 0 0 1

](27)

3. MODELING ERROR

In accordance to the LQG/LTR design strategy, plant andactuator related uncertainties around the nominal valueswere defined.

Uncertainties on the plant transfer functions are derivedfrom variations on the AUV mass distribution, hydro-static, propulsion and hydrodynamic related parameters.Added mass were considered separately from the other hy-drodynamic parameters. In the latter category, a uniformvalue named as “hydrodynamic efficiency” was adopted fordefining the error in the damping and control related pa-rameters. Conservative values were assumed for the uncer-tainties in each parameter (see table 3), and combinationsof all of them taking the minimum and maximum valueswere calculated for the considered frequency range.

The “actuators” represent the manoeuvring and propul-sion systems, responsible for generating the control vari-ables. They are modeled as second order systems, based onresults produced in laboratory and field tests. The actuatorerror model included variations on the natural frequencyand damping (see table 4).

Using the definition of multiplicative modeling error, givenby equation 28, in the controller design was considered theworst error considering the combining plant and actuatoruncertainties.

e(jω) = [GR(jω)−GN (jω)] · [GN (jω)]−1

(28)

Where GR is the real transfer function (considering theerrors of models), and GN is the nominal transfer function.

Table 3. Uncertainty of AUV parameters.

Parametros Erro [%]

Longitudinal velocity [m/s] ±20Hydrodynamic efficiency ±30Propulsion efficiency ±30Vehicle mass [kg] ±30Additional mass [kg - kg.m - kg.m2] ±50Inertia [kg ·m2] ±50Longitudinal variation of the CG [m] ±20Vertical variation of the CG [m] ±20Misalignment of propeller [◦] 8◦

Table 4. Uncertainty at actuators parameters.

Parametros Servo Motor

Minimum natural frequency [rad/s] 30 10Nominal natural frequency [rad/s] 40 13Maximun natural frequency [rad/s] 50 15Minimum damping coefficient 0.7 0.6Nominal damping coefficient 0.8 0.8Maximun damping coefficient 0.9 1.0

In Fig. 1 is seeing the variation of these erros as a functionof frequency.

4. LQG/LTR CONTROLLER DESIGN

The methodology used to define the LQG/LTR controllerparameters, and to analyze its performance was based onthe works of Doyle and Stein (1981).

Initially, the input and output variables of the plant weremade non-dimensional in order to produce the smallest dif-ference between minimum and maximum singular values.That was achieved through the function “goal attainment”(Gembicki, 1974) implemented in the “Matlab Optimiza-tion Tool Box”. This tool was applied to minimize thedifference between maximum and minimum singular val-ues in the low and high frequency ranges (as see in Fig.2). In Fig. 2 is possible verify the optimization results.

The design procedure continues with the adjustment ofthe open loop singular values of the target feedback looptransfer function to the stability and performance barriers.This was done in two steps. First, it’s found a value for thematrix L and the scalar µ that shape the singular valuesof the filter transfer function (Φ(jω)), defined by (29), torespect the boundaries of stability and performance. Thiswas done empirically (see Fig. 3).

Φ(jω) = 1/√µ[C(jωI −A)−1L

](29)

The next step is obtain the Σ matrix from the solutionof the Filter Algebric Riccati Equation (eq. (30)) for theKalman Filter gain matrix H (eq. (31)). From this matrix,the target feedback loop transfer function is calculated(GKF ) by the equation (32)). The singular values of thetarget system are then defined, and verified in the openloop and closed loop conditions according to stability andperformance criteria.

0 = AΣ + ΣA′ + LL′ − (1/√µ) ΣCC ′Σ (30)

H = (1/√µ) ΣC ′ (31)

Fig. 1. Modeling error considered in this project

Fig. 2. Difference between maximum and minimum singu-lar value of the nominal and dimensionless system.

GKF (jω) =[C(jωI −A)−1H

](32)

The feedback gain matrix G (eq. (33)) is calculeted bythe loop transfer recovery methodology, with also use theAlgebric Riccati equation solution (eq. (34)) for calculatethe value of the matrix K. The recovered open-loop systemkeep the singular values satisfactorily in the region definedby the stability and performance barriers.

G = (1/ρ)B′K (33)

0 = KA+A′K + C ′C − (1/ρ)KBB′K (34)

The final step is the verification of stability and robustnessspecifications in closed loop (Cn). As can be seen in Fig. 4,the stability barrier is not crossed by the singular values ofthe final closed loop system, ensuring the stability of thesystem for the errors specified in the section 3.

5. SIMULATION RESULTS

Simulation of the closed loop performance were carried outfor two sea states and wave directions.

The first case corresponds to the average wave height(H1/3 = 0.3m) of sea state 2 for the head sea condition.That is, defining β as the heading angle between theAUV and the wave propagation direction this conditionis represented by β = 0◦.

The second case corresponds to the average wave height(H1/3 = 0.5m) in the following sea condition, β = 180◦.

In the both cases the modal period of the sea spectrum is7.5s, and the reference depth is 1.0m. The controller must

Fig. 3. Singular values of the objective open-loop system,along with the barriers of stability and performance.

Fig. 4. Maximum and minimum singular values of thenominal system in closed-loop.

keep the cruising speed (u = 1.0m/s) and the trimmingcondition (θ = 0◦).

The controller can keep the depth with an error of 3%in the first case (Fig. 5). The standard deviations inthe trimming condition and cruising speed are 0.36◦ and0.03m/s, respectively.

In the second case, the control system shows a goodperformance as well (Fig. 6). In the depth it’s observeda maximum error of 5%. Standard deviations in thetrimming angle and cruising speed value are 0.63◦ and0.27m/s, respectively.

Figure 7 presents the control efforts of the worst case(following sea and H1/3=0.5m). The propeller speed showsa small variation around the nominal value at cruisingspeed (500 rpm). The bow plane and stern plane maximumdeflections are kept below the saturation level (30◦).

6. CONCLUSION

For the achievement of good closed loop performanceit was crucial to change the hydroplanes section profileand aspect ratio with relation to the original Pirajuba’sconfiguration (see table 1). The adopted hydroplanes could

Fig. 5. Closed loop response for a head sea, with H1/3 =0.3m and depth of 1m.

Fig. 6. Closed loop response for a following sea, withH1/3 = 0.5m and depth of 1m.

provide a linear behavior in the angle of attack rangewithout the occurrence of stall.

The control algorithm investigation should proceed withan improvement of the nonlinear dynamic model for thewave disturbance represented in the simulations. Firstorder wave disturbance should be further investigated, andthe inclusion of the second order wave forces should be alsoconsidered.

Additional sea states and directions of propagation shouldbe considered in order to extend the control system designto the motion in horizontal plane.

ACKNOWLEDGEMENTS

The first two authors would like to thank the Fundacaode Amparo a Pesquisa do Estado de Sao Paulo (FAPESP)

Fig. 7. Control effort for the case represented in Fig. 6.

for financial support. The third author would like to thankthe Conselho Nacional de Desenvolvimento Cientıfico eTecnologico (CNPq) for financial support.

REFERENCES

Blake, W. and Gillard, W. (1988). Prediction of vor-tex interference on a canard controlled missile. InAerospace and Electronics Conference, 1988. NAECON1988., Proceedings of the IEEE 1988 National, 444 –451vol.2.

de Barros, E., Dantas, J., Pascoal, A., and de Sa, E.(2008a). Investigation of normal force and momentcoefficients for an AUV at nonlinear angle of attack andsideslip range. Oceanic Engineering, IEEE Journal of,33(4), 538 –549.

de Barros, E.A., Freire, L.O., and Dantas, J.L.D. (2010).Development of the pirajuba AUV. In Proceedings ofCAMS 2010 - Control Applications in Marine Systems.

de Barros, E.A., Pascoal, A.M., and de Sa, E. (2008b). In-vestigation of a method for predicting AUV derivatives.Ocean Engineering, 35(16), 1627 – 1636.

Doyle, J.C. and Stein, G. (1981). Multivariable feedbackdesign concepts for a classical/modern synthesis. IEEETrans. Automatic Control, AC-26(1), 4–16.

Dumlu, D. and Istefanopulos, Y. (1995). H2 and H∞design of an adaptive controller for submersibles viamultimodel gain scheduling. Ocean Engineering, 22(6),593 – 614.

Faltinsen, O.M. (1990). Sea loads on ships and offshorestructures. Cambridge ocean technology series. Cam-bridge University Press, Cambridge ; New York :.

Gembicki, F. (1974). Vector Optimization for Control withPerformance and Parameter Sensitivity Indices. Ph.d.dissertation, Case Western Reserve Univ.

Hemsch, M.J. and Nielsen, J.N. (1985). Extension ofequivalent angle-of-attack method for nonlinear flow-fields. Journal of Spacecraft and Rockets, 22, 304.

Hoak, D.B. and Finck, R.D. (1978). USAF Stability andControl Datcom. Wright-Paterson Air Force Base.

Liceaga-Castro, E. and van der Molen, G. (1995). Sub-marine Hinfin; depth control under wave disturbances.Control Systems Technology, IEEE Transactions on,3(3), 338 –346.

Mandzuka, S. (1998). Mathematical model of a submarinedynamics at the periscope depth. Brodogradnja, 36, 5–6.

Moreira, L. and Guedes Soares, C. (2008). Designs fordiving and course control of an autonomous underwatervehicle in presence of waves. Oceanic Engineering, IEEEJournal of, 33(2), 69 –88.

Pitts, W.C., Nielsen, J.N., and Kaattari, G.E. (1957). Liftand center of pressure of wing-body-tail combinationsat subsonic, trans-sonic, and supersonic speeds. Nacatechnical report 1307, NACA Technical Report 1307.

SNAME (1950). Nomenclature for treating the motion ofa submerged body through a fluid. Technical ResearchBulletin N. 1-5, The Society of Naval Architects andMarine Engineers.

Song, F., An, P., and Folleco, A. (2003). Modeling andsimulation of autonomous underwater vehicles: designand implementation. Oceanic Engineering, IEEE Jour-nal of, 28(2), 283 – 296.