Literature review

Ocean Engineering 72 (2013) 441–447

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Ocean Engineering

0029-80 http://d

n Corr E-m

jgpelaez

journal homepage: www.elsevier.com/locate/oceaneng

On a submarine hovering system based on blowing and venting of ballast tanks

Roberto Font a,n, Javier García-Peláez b

a Departamento de Matemática Aplicada y Estadística, ETSI Industriales, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain b Direction of Engineering, DICA, Navantia S. A., Cartagena 30205, Spain

a r t i c l e i n f o

Article history: Received 27 July 2012 Accepted 27 July 2013 Available online 19 August 2013

Keywords: Manned submarines Underwater hovering Variable buoyancy Sliding control

18/$ - see front matter & 2013 Elsevier Ltd. A x.doi.org/10.1016/j.oceaneng.2013.07.021

esponding author. Tel.: +34 968 338 947; fax: ail addresses: [email protected], robertojav @navantia.es (J. García-Peláez).

a b s t r a c t

A submarine hovering system based on the blowing and venting of a set of dedicated tanks is investigated. We review the mathematical models involved and propose a sliding mode controller for the input–output linearized system. Numerical simulation results support the idea that this could be a promising hovering strategy for manned submarines, autonomous underwater vehicles or other plat- forms.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Underwater hovering, the ability to statically keep a desired depth, is at the same time a challenge, due to many uncertainties associated with the underwater environment, and a very impor- tant feature for both small size autonomous underwater vehicles (AUVs) and large manned submarines. In the last years several hovering AUVs have been developed (see Vasilescu et al., 2010 for a survey on the subject). The hovering facility expands the capabilities of AUVs allowing them to perform more complex missions that previously could only be carried out through Remotely Operated Vehicles (ROVs). For manned submarines, accurate hovering can be an invaluable tool, for example, for safe swimmer delivery, cover supply replacement or the deployment and recovery of AUVs, a subject that has recently raised an extraordinary interest (see for example Hardy and Barlow, 2008; Martínez-Conesa and Oakley, 2011).

From the technological point of view, AUVs usually hover by using thrusters (Choi et al., 2003; Li et al., 2011). Due to the large energy requirements of this approach, however, buoyancy control by pumping seawater in or out of ballast tanks can be used to save energy (Vasilescu et al., 2010) or in larger designs (Tangirala and Dzielski, 2007). In manned submarines, hovering is traditionally performed using hydraulic pumps (Yang and Hao, 2010; Ying and Jian, 2010), although not very much information about hovering

ll rights reserved.

+34 968 338 916. [email protected] (R. Font),

systems is available in the literature due to the military nature of these vehicles.

The aim of this work is to investigate the feasibility of a hovering system in which dedicated tanks are blown and vented similarly to the way the main ballast tanks are traditionally operated in manned submarines.

In these vehicles, a variable number of main ballast tanks are distributed along the hull. In case of emergency the main ballast tanks can be emptied by blowing into them air from high pressure bottles. This way the water is expelled from the tanks, the vehicle gains buoyancy and can rise more quickly. To fill the tanks with water, air is vented out of the ballast tanks. In the previous works (Font et al., to appear, 2013) we proposed mathematical models for the blowing and venting of ballast tanks and showed that the implementation of a control system for these processes, usually performed manually, can improve in a significant way the perfor- mance and stability in emergency rising manoeuvres. Our objec- tive is to extend the approach used with the main ballast tanks to a set of dedicated hovering tanks (see Section 2 for details) in order to test the feasibility of a hovering system based on blowing and venting of tanks. Although throughout this paper we will use a manned submarine as test platform, it is worth noting that the use of blowing and venting of tanks as hovering control is not limited to these vehicles nor is our intention to carry out the discussion of conceptual designs for the compromise between efficiency and stealth in a hovering system ready for military applications. Indeed, this technology could be applicable to manned submar- ines, AUVs, ROVs or any offshore platform requiring variable buoyancy control.

The rest of the paper is organized as follows. In Section 2 we formulate the problem and describe the mathematical models

Table 1 Summary of symbols introduced in Section 2.1.

Av Vent pipe cross-section (m 2)

hwc(t) Height of water column in the tank (m) mB(t) Mass of air in ballast tank (kg) mF(t) Mass of air in pressure bottle (kg) mF0 Initial mass of air in pressure bottle (kg) _mFðtÞ Mass flow rate from pressure bottle (kg/s) _m ðtÞ Mass flow rate through venting valve (kg/s)

R. Font, J. García-Peláez / Ocean Engineering 72 (2013) 441–447442

for blowing and venting processes, vehicle motion and external disturbances. In Section 3 we propose a feedback control scheme for the hovering submarine consisting of a sliding mode controller acting on a previously input–output exactly linearized system. Section 4 is devoted to the results of numerical simulations testing the performance of the proposed hovering system. Finally, in Section 5 we discuss the obtained results and present some conclusions.

v

pB(t) Pressure in ballast tank (Pa) pext(t) Pressure outside the venting valve (Pa) pF(t) Pressure in bottle (Pa) pF0 Initial pressure in bottle (Pa) pSEA(t) Pressure outside the outlet hole (Pa) qB(t) Water flow through outlet hole (m

3/s) Rg Gas constant for air (J/kg K) TB Water temperature (K) VB0 Initial air volume in ballast tank (m

3) VB(t) Volume of air in ballast tank (m

3) VF Pressure bottle volume (m

3) γ Isentropic constant ρ Density of water (kg/m3)

2. Problem formulation. Mathematical models

As we said above, we will consider a manned submarine, particularly the Navantia P-650 design, as the test platform for our hovering system. Details about the hydrodynamic character- istic and the blowing/venting system can be found in García et al. (2011) and Font et al. (to appear) respectively.

Our objective is to maintain a desired depth with no propulsion (and thus without any help from the control surfaces) in the face of external disturbances like changes in water density or forces induced by the sea state. In the next sections we review the mathematical models for the blowing/venting system, vehicle motion and external disturbances.

2.1. Blowing/venting system

The blowing and venting system is composed of the tank, the pressure bottle, the blowing and venting valves and the outlet/ inlet hole located at the bottom of the tank. When the blowing valve is opened, air flows into the tank from the bottle increasing the pressure and forcing the water to flow out through the outlet hole. When the venting valve is opened, air can flow out from the tank letting the water flow back into the tank. Fig. 1 shows a schematic view of these processes. The subindex F denotes conditions in the bottle, the subindex B denotes conditions in the tank, _mF and _mv are respectively the mass flow rates through blowing and venting valves, qB is the water flow through the tank hole, hwc is the height of the water column in the tank and pSEA, pext are respectively the hydrostatic pressures outside the flood port and venting outlet (they differ in the depth at which each one is evaluated). We will use 3 variables for each tank to completely describe its state: mass of air in the bottle, mF, mass of air in the tank, mB, and pressure in the tank, pB. We refer the reader to Font et al. (to appear) and Font and García (2011) for a more detailed description of the model presented below. The symbols introduced in this section are summarized in Table 1.

Due to the high pressure difference between bottle and tank, the flow from the bottle will usually be supersonic. As the bottle empties, however, this difference decreases and the flow can become subsonic if the pressure ratio is below the critical pressure

Fig. 1. Blowing and venting processes.

ratio Pc ¼ ððγ þ 1Þ=2Þγ=ðγ�1Þ, with γ the isentropic constant. Let s denote the variable aperture of the blowing valve, the equation for the mass of air in the bottle in both the supersonic and the subsonic cases is

_mF tð Þ ¼ s tð ÞA mFðtÞγþ1pF0

mγF0VF

!1=2 μ pB tð Þ; mF tð Þ � �

ð1Þ

where A ¼ _mFmaxðð2=ðγ þ 1ÞÞ�ðγþ1Þ=ðγ�1ÞVF=γpF0mF0Þ1=2, with _mFmax the maximum mass flow rate from the bottle, experimentally measured, VF is the bottle volume, pF0, mF0 are respectively the initial pressure and mass of air in the bottle and

μ pB; mF � �

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ

2 γ þ 1

� �ðγþ1Þ=ðγ�1Þs ; Pc r

pF pBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2γ γ�1

pB

pF0 mF mF0

� �γ 0 BB@

1 CCA

2=γ

� pB pF0

mF mF0

� �γ 0 BB@

1 CCA

ðγþ1Þ=γ0 BB@

1 CCA

vuuuuut ; 1opFpB oPc

0; pF pB

r1:

8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>:

The mass flow through the venting valve is obtained similarly. The variation in the mass of air in the tank is the difference between the mass flow rate from the bottle and the mass flow rate through the venting valve. Let s denote the aperture of the venting valve. Then, the equation for the mass of air in the tank is

_mB tð Þ þ _mF tð Þ ¼ �μ pextðtÞ pBðtÞ

� � sðtÞAvpBðtÞffiffiffiffiffiffiffiffiffiffi

RgTB p ; ð2Þ

where Av is the venting pipe system cross-section, Rg is the gas constant for air, TB is the temperature in the tank and μðpextðtÞ=pBðtÞÞ is a function of the tank and outside pressures obtained by curve fitting from experimental measures.

Finally, the variation in the tank pressure is obtained from the perfect gas equation as

mBðtÞ pBðtÞ

_pB tð Þ� _mB tð Þ ¼ � pBðtÞqBðtÞ

RgTB ; ð3Þ

where qB(t) is the water flow through the flood port. We will consider two hovering tanks, bow and aft, with their

respective bottles and blowing and venting valves. The geometric characteristics of the blowing and venting system have been adapted from the characteristics of the main ballast tanks blowing and venting system which can be found in Font et al. (to appear).

Table 2 Hovering system characteristics.

Parameter Stern tank Bow tank

Flood port area, Ah ðm2Þ 0.1 0.1 Venting pipe system cross-section, Av ðm2Þ 9.62 � 10�4 9.62 � 10�4 Tank height, Htk (m) 1.5874 1.5874 Maximum mass flow rate, _mF;max ðkg=sÞ �2 �2 Initial bottle pressure, pF0 (Pa) 2.5 � 107 2.5 � 107 Initial bottle temperature, TF0 (K) 4 4 Tank volume, VBB ðm3Þ 293 293 Bottle volume, VF ðm3Þ 0.8 0.8 Tank location (m)

xb �28.6 23.7 yb 0 0 zb 0.595 0.595

R. Font, J. García-Peláez / Ocean Engineering 72 (2013) 441–447 443

However, since the size of the main ballast tanks is not adequate for the hovering system requirements, the size of the hovering tanks has been considerably reduced with respect to the main ballast tanks. Precisely, each hovering tank has a volume of 4 m3

and is initially filled up to half of its capacity. In the same way, the maximum mass flow rate from the bottle has been reduced to 2 kg/s. The main characteristics of the hovering system are summarized in Table 2. We note that these are tentative values in the context of this preliminary study. Of course, the implemen- tation in a real ship would require the careful selection of the most appropriate values.

2.2. Vehicle motion

Vehicle motion is given, as usual, by Feldman's (1979) 6 degree of freedom equations of motion. The final form of these equations adapted to the particular vehicle we are considering can be found in García et al. (2011).

These equations assume the mass of the submarine to be constant. In our case, however, water flowing in or out of the hovering tanks will cause mass variations at several points of the vehicle. We will therefore need to write vehicle mass, weight, moments and products of inertia and location of center of gravity as a function of the amount of water in the tanks.

Let m0 be the initial mass of the vehicle and Δmstern, Δmbow the mass variations in the stern and bow tanks respectively (positive when mass is added). This mass variation can be obtained by multiplying the increase in the volume occupied by water with respect to the initial condition by the density of water, ρ. It is easy to see that this increase coincides with the decrease in the volume of air, i.e. the difference between the initial and momentary volume of air in the tank. This way

Δmfstern;bowg tð Þ ¼ ρ VB0�VBfstern;bowg tð Þ � �

¼ ρ VB0� mBfstern;bowg ðtÞRgTB

pBfstern;bowg

! ;

and the momentary mass of the vehicle is given by

mðtÞ ¼ m0 þ ΔmsternðtÞ þ ΔmbowðtÞ: The rest of parameters can be obtained analogously (Font et al., to appear).

It is worth noting that coefficient based models can lead to inaccurate results in certain situations. In movements at high angles of incidence, like hovering, vortices generated by crossflow separation of the hull boundary layer and the trailing wakes from appendages can produce significant loads (Watson et al., 1993). We do think, however, that this standard model is enough for the purposes of this preliminary study. Up to our best knowl- edge, it is not easy to account for these phenomena without using

computationally intensive techniques that are not suitable for control system design (see for instance Bettle et al., 2009). Our objective (see Section 3) is to design a robust controller that can maintain consistent performance in the presence of large external disturbances and/or modelling uncertainties. This approach has been successfully used in the design of a variety of hovering AUVs. We refer the reader to, for example, Riedel et al. (2005) where a sliding mode controller is used to address a similar problem.

2.3. Complete model in affine form

Let the state vector be

x ¼ ½x; y; z; ϕ; θ; ψ; u; v; w; p; q; r; …

mFstern ; mBstern ; pBstern ; mFbow ; mBbow ; pBbow � ð4Þ

composed of the state variables for the stern and bow tanks, the vehicle position and orientation, x, y, z and ϕ (roll), θ (pitch) and ψ (yaw) respectively, and the linear and angular velocities, u, v, w and p, q, r. Let u ¼ ½sstern; sstern; sbow; sbow� be the control vector. The state law is composed of the equations of motion (see Section 2.2) plus a set of Eqs. (1)–(3) for each of the two hovering tanks. This state law can be expressed in compact form as

AðxÞ _x ¼ Φðx; uÞ ð5Þ

where A (x) is the mass matrix of the system. As we will see in Section 3, it is convenient, from the point of

view of control system design, to express (5) in the affine form

_x ¼ fðxÞ þ ∑ m

i giðxÞui: ð6Þ

This expression can be obtained from (5) as follows. It is easy to see that the state law (5) can be expressed as

AEMðxÞ 012�6 06�12 ABV ðxÞ

" # _x ¼

ΦEMðxÞ ΦBV ðx; uÞ

" #

where AEM (x) and ΦEMðxÞ are respectively the mass matrix and the right hand side of the equations of motion, depending only on the state variable, ΦBV ðx; uÞ is the right hand side of Eqs. (1)–(3), and

ABV xð Þ ¼

1 0 0 0 0 0 1 1 0 0 0 0

0 �1 mBstern pBstern

0 0 0

0 0 0 1 0 0 0 0 0 1 1 0

0 0 0 0 �1 mBbow pBbow

2 6666666666664

3 7777777777775

Since both AEM (x) and ABV (x) take non-singular values for any x (Font et al., to appear), the state law can be expressed in explicit form as

_x ¼ A�1EMΦEMðxÞ A�1BV ΦBV ðx; uÞ

" #

It is immediate to separate the terms where the control input appears to obtain the desired form (6) with

f xð Þ ¼ A�1EMΦEM; 0; 0; pBstern qBstern mBstern RgTB

; 0; 0; pBbow qBbow mBbow RgTB

T

R. Font, J. García-Peláez / Ocean Engineering 72 (2013) 441–447444

and gi; 1rmr4, the columns of the matrix

012�4 αstern 0 0 0 �αstern βstern 0 0

�pBstern mBstern

αstern pBstern mBstern

βstern 0 0

0 0 αbow 0 0 0 �αbow βbow 0 0 �

pBbow mBbow

αbow pBbow mBbow

βbow

2 6666666666666664

3 7777777777777775

with

αfstern;bowg ¼ μfstern;bowgA mγþ1Ffstern;bowg pF0

mγF0VF

0 @

1 A

1=2

;

βfstern;bowg ¼ �μfstern;bowg AvpBfstern;bowgffiffiffiffiffiffiffiffiffiffi

RgTB p :

2.4. External disturbances

The main disturbances that a submerged submarine must face are the wave induced forces and the changes in its buoyancy caused by seawater density changes and hull compressibility.

2.4.1. Wave induced forces The surface elevation of a long-crested irregular sea can be

modelled as a sum of N regular wave components. The amplitude of each regular wave is given by a wave energy spectrum, in this case the Pierson–Moskowitz spectrum. A detailed description of this approach can be found in Fossen (1994).

To model the effect of an irregular sea over the vehicle it is usual to consider 1st and 2nd order wave disturbances. Using the superposition principle the 1st order forces (in this work we will consider a heave force and a pitching moment) can be modelled as the sum of the forces caused by each individual regular compo- nent, i.e. a wave with the same frequency and different amplitudes and phase angles.

For this work, the amplitude and phase angle of the vertical force and pitching moment, Fi; Mi; ΦF;i; ΦM;i, have been obtained experimentally using captive tests on a scale model. The ampli- tudes and phase angles under regular waves of different wave- lengths were measured for several heading directions. These data were then adjusted using a least-squares fit. Fig. 2 shows the results for the vertical force and pitching moment at 1801 wave

Fig. 2. Non-dimensional force (left) and moment (right) vs non-dimensional wave leng solid line.

heading. In this case the fitted curves are

F

ρgHL2 ¼

0:01161 λ

L λ

L

� �2 �3:898λ

L þ 5:046

M

ρgHL3 ¼

0:001932 λ

L λ

L

� �2 �2:398λ

L þ 1:917

where L is the vehicle length, λ the wave length, H the wave height and g the acceleration due to gravity. A similar methodology was used in Byström (1988).

The 2nd order force, or suction force, is an upward force, usually small in comparison with 1st order forces. Under an irregular sea, this is a slowly varying force (Booth, 1983). For the purpose of this work, however, it suffices to consider a mean constant value. The series of captive tests mentioned above provided a mean value for the suction force under Sea State 5 of 0.8 ton.

Both the 1st and 2nd order forces are assumed to decay exponentially with depth in the form e�kiðzðtÞ�z0Þ where ki is the wave number of each component. For the suction force, k was taken corresponding to the peak of the spectrum.

2.4.2. Compressibility and water density Vehicle buoyancy can be expressed as B ¼ ρg∇fd, where ρ is the

water density and ∇fd is the form displacement, i.e. the total displaced volume in submerged condition. In trim condition the vehicle buoyancy is equal to its weight. If the submarine moves down from the equilibrium position, however, the outside pres- sure will compress the hull and other materials reducing the vehicle buoyancy, which will result in a downward acceleration. In the same way, moving the submarine up from its equilibrium position will result in an upward acceleration (see, for instance, Booth, 1983). The displacement variation caused by pressure hull compressibility can be considered linear with depth in the form ΔVcðzÞ ¼ αðz�z0Þ, where α is a compression coefficient that can be easily determined for each particular vehicle.

Similarly, changes in seawater density will modify the vehicle buoyancy also causing a destabilizing effect. The seawater density as a function of temperature and salinity can be obtained using the algorithms described in Fofonoff and Millard (1983). Let us assume that, for a certain temperature/salinity vertical profile, the water density can be expressed as a function of depth. Let ρðzÞ denote this variable density (we will continue to denote the nominal value by ρ). The vehicle buoyancy is given by

BðzÞ ¼ ρðzÞgð∇fd�ΔVcðzÞÞ: ð7Þ

th for wave heading 1801. Measured values are shown as circles and fitted curve as

R. Font, J. García-Peláez / Ocean Engineering 72 (2013) 441–447 445

3. Sliding mode controller

Sliding mode control (Slotine and Li, 1991) has been success- fully applied in several AUV designs and has been identified as one of the most promising control strategies to account for large disturbances or model uncertainties (Lea et al., 1999). Although sliding mode control is often applied over a linearized model (Demirci and Kerestecioğlu, 2004; Riedel et al., 2005), our previous experience with the dynamic model of the submarine leads us to think that in this case, due to high coupling and severe nonlinea- rities, this could be inaccurate. Instead, we propose an exact input–output feedback linearization (Isidori, 1995). The sliding control of a nonlinear system using input–output linearization is discussed more generally in Fossen and Foss (1991).

For the hovering control we will consider a single output, the depth z, and 4 inputs, the apertures of the blowing and venting valves. This way, the system can be expressed (see Section 2.3) as

_x ¼ fðxÞ þ ∑ m

i giðxÞui

y ¼ hðxÞ ¼ z: ð8Þ

Let Lfh, Lgi h denote the Lie derivatives of h with respect to f and gi respectively. The repeated Lie derivatives are recursively defined as Ljf h ¼ Lf ðL

j�1 f hÞ. In the same way, Lgi L

j f h ¼ Lgi ðL

j f hÞ. The computa-

tion of Lgi L j f h; j ¼ 1…r, shows that the system has a well defined

relative degree r¼3. We refer the reader to Isidori (1995) for more detail on this subject.

Define FðxÞ ¼ Lrf h and GðxÞ ¼ ½Lg1 L r�1 f h ⋯ Lgm L

r�1 f h�. This way (8)

can be expressed as

yðrÞ ¼ FðxÞ þ GðxÞ u1 ⋮ um

2 64

3 75 ð9Þ

that can be reduced to

yð3Þ ¼ v ð10Þ

Fig. 3. Evolution of depth for Scenario 1.

Fig. 4. Flooded volume in stern (left) an

using the control law

u ¼ GnðxÞðv�FðxÞÞ ð11Þ

where GnðxÞ is a vector such that GðxÞGnðxÞ ¼ 1. Once the input–output relation in system (8) has been reduced

to the linear form (10) it is easy to use linear control techniques to stabilize (10). The exact cancellation of the nonlinear terms, however, relies on the perfect knowledge of fðxÞ and giðxÞ, some- thing that can hardly be achieved due to modelling errors or unknown external disturbances. We propose the use of sliding mode control to account for this issue.

Let y ¼ ½y _y ⋯ yðr�1Þ� be the output vector, yd ¼ ½yd _yd ⋯ yðr�1Þd � a desired output vector and ~y ¼ y�yd ¼ ½~y _~y ⋯ ~yðr�1Þ� the tracking error vector. For r¼3 the sliding surface can be defined as

s ¼ €~y þ 2λ _~y þ λ2 ~y:

Taking the derivative with respect to time in the above equation and taking into account that in our case yð3Þd ¼ 0 and in the absence of uncertainties yð3Þ ¼ v, the equivalent control is veq ¼ �2λ €~y�λ2 _~y .

Let F̂ðxÞ; ĜðxÞ be the estimates of FðxÞ and GðxÞ, respectively. If the signum function is linearly smoothed inside a boundary layer of thickness Φ (Slotine and Li, 1991), then the proposed control law is

u ¼ ĜnðxÞð�2λ €~y�λ2 _~y�k satðs=ΦÞ�F̂ðxÞÞ ð12Þ

with satð�Þ the saturation function. If we consider no errors in the modelling of the hovering

system, then ĜðxÞ ¼ GðxÞ. Let ΔZjFðxÞ�F̂ðxÞj be an upper bound of the error in the estimation of FðxÞ, then it suffices to choose k4Δ.

Considering extreme values for seawater density, the value of Δ for changes in water density can be obtained. In the case of wave induced forces the value of Δ cannot be explicitly obtained due to the random nature of these forces. However, for a given sea state, upper bounds can be estimated. The choice of k¼0.001 is enough to account for both the density changes and the wave induced forces in Sea State 5. For the rest of parameters we have taken

d bow (right) tanks for Scenario 1.

Fig. 5. Depth evolution for Scenario 2.

Fig. 6. Blowing valve aperture (top) and venting valve aperture (bottom) for Scenario 2.

R. Font, J. García-Peláez / Ocean Engineering 72 (2013) 441–447446

λ ¼ 0:05, Φ ¼ 0:001. If modelling uncertainties, like those discussed in Section 2.2, were considered, an upper bound of the associated error should also be estimated and taken into account in the choice of k.

4. Numerical simulations

The aim of this section is to test the performance of the proposed hovering system against the disturbances treated in Section 2.4. Two different scenarios are considered. In both cases the submarine starts at straight and level flight at 4 kn and at t¼0 the propeller is stopped and the vehicle progressively loses forward speed (it is around 0.4 kn at the end of simulation time).

In the first scenario (Figs. 3 and 4) the submarine is at 50 m depth under the action of sea wave disturbances (Sea State 5, with H1/3¼3.25 m, Tz¼6.2 s). Additionally, the hovering system has to compensate for sudden changes in water temperature. For 0rtr1250 s temperature varies with depth as TðzÞ ¼ 20�0:05z 1C. At t¼1250 s, the temperature gradient is changed to TðzÞ ¼ 5 þ 0:05z 1C. The vehicle buoyancy is affected by these gradients according to (7).

Flooded volume in stern and bow tanks is plotted in Fig. 4. We can see how the buoyancy variation is rapidly compensated by emptying the tanks. Depth evolution is plotted in Fig. 3. Although depth keeping is satisfactory for both temperature gradients, we can see how, after the instability caused by the density jump, the second gradient, where the density increases with depth, is more favourable. Indeed, density increasing with depth tends to com- pensate the effect of hull compressibility while decreasing density adds up to this destabilizing effect (see Section 2.4).

For the second scenario we consider a more unfavourable situation, with a depth of 35 m and Sea State 5. Fig. 5 shows the evolution of vehicle depth. We can see how, again, the system shows excellent performance with maximum error around 1.5 m. The control inputs over a 250 s interval are shown in Fig. 6. In all cases there is no appreciable chattering and the oscillations are smooth and compatible with the physical restrictions considered.

The autonomy of the air bottles is around 30 min in this operating condition and above 90 min at the same depth and Sea State 4.

5. Conclusions

The performance of a hovering system based on blowing and venting of ballast tanks has been investigated using a manned submarine as test platform. Simulation results show that this system has the ability to maintain a desired depth in the presence of significant external disturbances.

The blowing and venting of tanks, alone or in conjunction with another control mechanism, like pumping, seems to be a promis- ing hovering strategy to be used in manned submarines, AUVs or other submarine platforms.

Accuracy and fast response are the potential advantages of this approach. On the other hand, the dependence on air bottles with their associated payload and limited autonomy are the main disadvantages for small platforms like AUVs. In the case of manned submarines, the main limitation is the noise generation associated with the blowing process. However, since the needed flow rates are considerably lower than the flow rates used when blowing the main ballast tanks, we do believe that this problem could be mitigated with an appropriate design of the blowing system. This will be the subject of our future work.

Acknowledgements

This work was supported by projects 2989/10MAE from Navantia S.A. and 08720/PI/08 from Fundación Séneca, Agencia de Ciencia y Tecnología de la Región de Murcia (Programa de Generación de Conocimiento Científico de Excelencia, IIPCTRM 2007-10).

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