Depth Control of an Over-Actuated, Hover-Capable Autonomous

Underwater Vehicle with Experimental Verification

Kantapon Tanakitkorna,c,1, Philip A. Wilsona,2, Stephen R. Turnocka,3, Alexander B. Phillipsb,4

aFluid Structure Interactions Research Group, University of Southampton, UKbMarine Autonomous Robotic Systems, National Oceanography Centre, Southampton, UK

cThis project is sponsored by the Faculty of International Maritime Studies, Kasetsart university, SriRacha Campus.

Abstract

A PI-D based control system is developed for over-actuated, hover-capable AUVs which enables asmooth transition from hover-style to flight-style operation. A system stability and convergence isproven using a Lyapunov-based approach. The performance of the controller is demonstrated bysimulation but crucially is proven to provide satisfactory performance experimentally. The approach isable to operate over a range of vehicle ballasting configurations, and to imposed external disturbances.The proposed system is computationally inexpensive and does not require a detailed hydrodynamicmodel to implement. By monitoring the energy consumption on board, the cost of maintaining depthat a range of forward speeds with different buoyancy conditions can be quantified and their impact oncost of transport is highlighted for future optimisation of energy consumption.

Keywords: Delphin2; AUV; depth control; over-actuated; hover-capable; energy consumption.

1. Introduction

Autonomous underwater vehicles (AUVs) arenormally designed with an emphasis on minimis-ing total resistance to suit long-range flight-styleoperation — they are of a torpedo-shaped bodythat is equipped with a main propeller and controlsurfaces [1, 2]. These conventional AUVs, how-ever, are not able to carry out detailed inspectiontasks at zero or slow forward speeds since the con-trol surfaces become ineffective in this regime [3,p.169].

Various concepts for next generation of hover-capable AUVs have been developed to overcomesuch a limitation [4, 5, 6, 7]. Typically, thrustersare added to the design to provide additional con-trol forces in low-speed operation, as illustrated

1K.Tanakitkorn@soton.ac.uk2Philip.Wilson@soton.ac.uk3S.R.Turnock@soton.ac.uk4abp@noc.co.uk

in figure 1. Since this class of the AUVs can beconsidered to have a redundant set of actuatorsfor controlling a given degree of freedom, they arereferred to as over-actuated AUVs.

Different control strategies are required for con-trolling an over-actuated AUV that is operatingin different styles. For flight-style operation, sternplanes are used to adjust the pitch angle of thevehicle which in turn results in depth changes.A controller for flight-style operation may be de-signed based on a cascaded depth and pitch con-trol structure [8, 9, 10, 11, 12]. In contrast, thecontrol problem for hover-style operation may beconsidered as two parts [13, 14, 15]. First, a con-trol law that determines the generalised forcesrequired. Second, a control allocation that dis-tributes the control forces onto vertical thrustersand stern planes. Such an approach has been im-plemented by [16, 17].

Alternatively, the vehicle depth, in hover-styleoperation, may also be controlled by adjusting

Preprint submitted to Mechatronics: The Science of Intelligent Machines September 15, 2016

its displacement via a buoyancy engine [18]. Al-though this approach consumes less energy com-pared to the thruster operations, the mecha-nism is typically slow [19] and additional tun-nel thrusters are still required for fast vehicle re-sponse.

Simulation studies on a concept of a unifiedcontrol system for over-actuated AUVs are pre-sented in [20, 21]. Weighting functions are usedto provide a seamless transition between hover-style and flight-style strategies.

Model predictive control (MPC) has also beenapplied to depth control of over-actuated AUVs[22]. This has been shown to provide acceptableresults over a range of speeds, but is computation-ally expensive due to the need to solve an opti-misation problem before each control allocation.Alternatively, sliding mode control (SMC) tech-nique can also be used [23, 24]. A downside ofthis approach is that it is susceptible to chatter; arange of approaches has been developed to over-come this [25, 26].

This paper focuses on design, simulationand experimental testing of proportional-integral-derivative (PID) based diving control system forover-actuated AUVs that can be used over a widerange of operating speeds with a seamless transi-tion between hover-style and flight-style control.

The paper is organised in a following order. TheDelphin2 AUV is presented in section 2, followedwith a mathematical model in section 3. Theproposed control system is explained in section4. Section 5 provides a stability and convergenceanalysis. Simulation and experimental setup areexplained in section 6 and 7 respectively. Resultsare discussed in section 8. This follows with aconclusion in section 9.

2. Delphin2 AUV

Delphin2 is a torpedo-shaped AUV with alength of 1.96m and mid-body diameter of 0.26m[27]. The AUV is equipped with a main rear pro-peller, four rear control surfaces in a cruciformarrangement. It is also fitted with four through-body tunnel thrusters: two horizontal and twovertical. (See figure 1.) Hence, it can effectively

perform a wide range of missions, ranging froma zero-speed hovering to longer range flight-stylesurvey missions at a maximum forward speed ofapproximately 1 m/s.

It is typical for an AUV to be slightly posi-tively buoyant [28, p.89], allowing the vehicle tonaturally return to water surface in an event ofsystem failure. Due to this, Delphin2 is typicallyballasted to be 6 N positively buoyant.

Figure 1: Delphin2 AUV [29].

To conduct this work significant upgrades havebeen made to the vehicle that is described in [27];these include:

• The CPU was replaced with an Intel AtomD525 dual-core processor 1.8 GHz, reducingthe hotel load from 50 W to 30 W and en-hancing endurance.

• The OceanServer Compass and rate-gyrowere replaced by Xsens 4th generation MTi-30 IMU, providing higher precision naviga-tion performance.

• The 6-channel TSL motor control board wasreplaced by four identical maxon motor con-trol 1-Q-EC Amplifier DECS 50/5. Thesereduced the tunnel thruster deadband from450 rpm to 150 rpm, enhancing hover perfor-mance.

• A power monitoring system has been de-veloped: it is composed of Arduino Nano,and voltage and current sensors. This al-lows power consumption of the actuators tobe continuously recorded.

3. Vehicle Dynamics Modelling

The kinematics and dynamics model are devel-oped based on Delphin2 dynamics according to

2

the SNAME [30] notation. The two coordinatesystems used in the model are illustrated in figure2. The first is a body-fixed frame (b-frame) thatis fixed to the AUV body and aligned with thelongitudinal axis. The second is an earth-fixedcoordinate system where x-, y- and z-axis pointnorthward, eastward and downward respectively;hence the name NED-frame.

Figure 2: Coordinate systems.

Rigid-body parameters of Delphin2 AUV areobtained partly from a direct measurement andpartly from 3D CAD model. Most of the hy-drodynamic derivatives used are estimated fromplanar motion mechanism (PMM) and rotatingarm experiments performed on a torpedo-shapedAUV with a similar shape [31]. Surge dampingcoefficient and control surfaces derivatives are in-ferred from wind tunnel tests on Delphin2 [32]. Aquadratic heave damping coefficient is estimatedfrom a free ascent test, see Appendix B.

These model parameters are detailed in Ap-pendix A.

3.1. 3DOF Equations of Motion

The vertical plane AUV model may be de-scribed in a matrix form [24]:

M · ν+C(ν) · ν+D(ν) · ν+ g(η) = τ + ∆τ, (1)

where ν = [u,w, q]T is the velocity vector, η =[x, z, θ]T is position and orientation vector andτ = [X,Z,M ]T is a generalised force vector;∆τ represents a factor that is considered in thiswork but is missing from [24] — this is explained

later in this section. The inertia matrix, coriolis-centripetal matrix, damping matrix and hydro-static force vector for decoupled surge dynamicsare:

M =

m−Xu 0 00 m− Zw 00 0 Iyy −Mq

, (2)

C(ν) =

0 0 c1,3

0 0 c2,3

c3,1 c3,2 0

, (3)

where c1,3 = −c3,1 = −m(xgq − w) − Zww andc2,3 = −c3,2 = −m(zgq + u) +Xuu,

D(ν)=−

Xu+X|u|u|u| 0 0

0 Zw+Z|w|w|w| Zq+Z|q|q|q|0 Mw+M|w|w|w| Mq+M|q|q|q|

(4)

g(η) =

(W −B)sin(θ)−(W −B)cos(θ)

(zgW − zbB)sin(θ) + (xgW − xbB)cos(θ)

.(5)

The origin of the b-frame is chosen to be coinci-dent with the centre of buoyancy rb = [xb, yb, zg].The centre of gravity rg = [xg, yg, zg] is just belowrb.

A transformation matrix J(η) is used to corre-late the velocity vector ν in b-frame to the velocityvector η in NED-frame under the assumption ofzero heel angle:

η = J(η) ν, (6)

J(η) =

cos(θ) sin(θ) 0−sin(θ) cos(θ) 0

0 0 1

. (7)

3.2. Drag from Tunnel Thruster Operation

When a through-body tunnel thruster is oper-ating, it sucks water into one side of the tunneland ejects a jet of water on the opposite side. Thiswater jet may be considered as an increasing inthe effective frontal area of the vehicle, increasingthe vehicle drag. This effect is most pronouncedwhen the jet is strong relative to the ambient flow.

Palmer presents an extensive study of a tunnelthrusters performance on a torpedo-shaped AUV

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with a similar shape and size which utilised thesame thruster units as Delphin2 [33]. The exper-imental result in figure 3 suggests that a changein volumetric surge damping coefficient may bemodelled as a function of the ratio between surgespeed (u) and jet speed (uj):

∆Cd,th = 0.5660 · e(−7.6089( uuj

))(8)

+ 0.0565 · e(−0.8968( uuj

)),

where thruster jet speed is given by uj =√4Fth/(ρπD2

th) [34], Dth is thruster diameter,and Fth is thruster force that is determined from(16).

Figure 3: A change in volumetric drag coefficient due toan operation of front and aft thruster (adapted from [33]).

The drag for a thruster operation takes a form:

∆Xth = −1

2ρ∇2/3|u|u∆Cd,th, (9)

where ∇ denotes the volumetric displacement ofthe AUV.

This is incorporated into the model as a gen-eralised force along the longitudinal axis of thevehicle: ∆τ = [∆Xth,vf + ∆Xth,va, 0, 0]T , wheresubscription vf and va are used to refer tothe vertical-front and vertical-aft thruster respec-tively.

3.3. Actuator Modelling

Considering the system that has m degrees offreedom and involves n components of forces andmoments from actuators, the generalised forcevector (τ) due to actuators may be characterizedwith a form [24]:

τ = T f, (10)

where f = [f1, f2, f3, . . . , fn]T ∈ <n is a col-umn vector representing the total forces and mo-ments expressed in actuator local frame; T =[t1, t2, t3, . . . , tm] ∈ <m×n is a force configurationmatrix that transforms f into equivalent forcesand moments acting on the origin of b-frame.

3.3.1. Main Propeller Modelling:

Standard propeller theory [35] suggests that itis possible to characterise a thrust due to a pro-peller by advance ratio, J , and thrust coefficient,KT,prop. They are given by:

J =u

nprop Dprop

(1− wt), (11)

KT,prop = KT0,prop(1−[ J

cprop,1

]cprop,2), (12)

where wt denotes wake fraction estimated from[3]; Dprop is propeller diameter; the propellerspeed (nprop), measured in rev/s, in relation tothe propeller demand is found from the experi-ment to be

nprop = −0.0055 u2prop + 0.4136 uprop − 1.7895.

(13)The relation between KT,prop and J from (12) isillustrated in figure 4.

Figure 4: A KT,prop − J relation that is inferred fromexperiment data privided in [36].

To this end, the thrust due to the main pro-peller is expressed as:

Fprop = ρ n2propD

4propKT,prop(1− t), (14)

where t is thrust deduction estimated from [3].

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Components of the force configuration matrixand force vector for the main propeller are:

t1 =

100

, f1 = Fprop. (15)

3.3.2. Thruster Modelling:

A quasi-steady model is used for thruster mod-elling. Thrust deduction factors are included tomodel the thruster degradation due to forwardand lateral motions of the AUV [37]. The forcedue to a thruster is given by

Fth = ρ |nth|nthD4thKT,th︸ ︷︷ ︸

static thrust

(16)

· e(−cth,1(u)2) · [1− cth,2(v

nthDth

)]︸ ︷︷ ︸deduction factors due to motions

,

where nth denotes a thruster speed, Dth denotesa thruster diameter, KT,th is a thrust coefficient;constants for deduction factors were inferred from[33].

Components of the force configuration matrixand force vector for the thrusters are:

t2 =

0 01 1Lvf Lva

, f2 =

[Fth,vfFth,va

], (17)

where L denotes the thruster arm length relativeto a centre of rotation; subscriptions vf and varefer to the vertical-front and vertical-aft thrusterrespectively (see also figure 7).

3.3.3. Stern Plane Modelling:

The lift, drag and moment due to stern planesare modelled using a hydrodynamic derivative ac-cording to [30]. They are given by:

XS = X|u|uδSδS |u|uδ2S, (18)

ZS = Z|u|uδS |u|uδS, (19)

MS = M|u|uδS |u|uδS, (20)

where δS denotes stern plane deflection; thederivatives are deduced from results of wind tun-nel testing on Delphin2 AUV [32].

Although the AUV rarely reverses in nor-mal operations, it is possible that the for-ward speed becomes negative during accelera-tion/deceleration phases. Due to this reason, ab-solute signs are added to improve simulation sta-bility when the speed is nearly zero. However, it isimportant to emphasise that stern plane dynam-ics is not forward/backward symmetric and thisassumption is invalid for AUV operation at highreverse speeds.

Components of the force configuration matrixand force vector for the stern planes are:

t3 =

1 0 00 1 00 0 1

, f3 =

XS

ZSMS

. (21)

4. Control System Design

In this section, two control strategies are de-veloped for two operating conditions. The first isa hover-style control strategy that adjusts depthand pitch of the AUV by operating two verti-cal thrusters. The second is a flight-style con-trol strategy that relies on stern planes opera-tion for pitch angle control that, in turn, affectsvehicle depth. Actuator weighting functions areused to provide a seamless transition between thetwo control strategies when operating over a widerange of speeds.

A set of gain values are identified empiricallyusing a heuristic approach based on the Ziegler-Nichols method [38].

Note that, the PID control scheme in this workslightly differs from the convention: (1) a deriva-tive term is placed in a feedback path instead ofthe forward path, (2) an actual state of the sys-tem is fed to the derivative term instead of thetracking error. (See figure 5.) Thus, the deriva-tive terms are not affected by a sudden changein the required state, i.e. avoiding the setpointkick phenomenon [38]. By doing so, the deriva-tive gains must have a negative value; hence it iscommonly referred to as PI-D scheme.

4.1. Hover-Style Control Strategy

A control strategy for hover-style operationis used to continuously operate the two vertical

5

Figure 5: PI-D scheme.

thrusters to counter a net positive buoyancy, seefigure 6. It is also important that the forces fromthe two thrusters are balanced otherwise the AUVwill end up pitching and the thrust componentswill cause the AUV to move forward/backward.

Figure 6: Hover-style force diagram.

The AUV is considered as a rigid body thatneeds to be pushed down by a generalised forceZ, see figure 7. Based on a depth error signal (z),a generalised force control law is

Z = KP,Z z +KI,Z

∫zdt+KD,Z z. (22)

If a virtual centre of rotation (cr) is known,the total force Z can be allocated between thetwo thrusters so that the pitch error signal (θ) isdriven to zero. To this end, the cr shift controllaw is

∆cr = KP,cr θ +KI,cr

∫θdt+KD,cr θ. (23)

From figure 7, given an initial guess for a cen-tre of rotation (cr0), the virtual centre of rotationis computed: cr = cr0 + ∆cr. Arm lengths forthe front (Lvf ) and aft (Lva) vertical thrusters aredetermined accordingly and are used for a forceallocation: [

TvfTva

]=

[Lvf/LvLva/Lv

]Z. (24)

Figure 7: Parameters used for a generalised force alloca-tion.

Finally, forces are then converted to a thrusterdemand:

uth = sign(T )

√|T |

ρD4thKT,th

. (25)

Note that the thruster demand (uth) is equiva-lent to the thruster speed (nth), and (25) is in factan inverse of (16) where the deduction factors areneglected.

Figure 8: A hover-style control block diagram.

The hover-style control strategy is illustratedas a control block diagram in figure 8.

4.1.1. Anti-Windup Scheme:

It is found that an integrator for the generalisedthrust control law in (22) may become unneces-sary large when dealing with a big change in adepth demand. This phenomenon is known as in-tegral windup. This has a serious impact on thecontrol performance because the system needs toproduce an opposite sign of error signal, i.e. over-shoot, just to unwind the integrator.

A conditional integration technique is used toreduce the effect of the integral windup phe-nomenon. The integrator will be frozen if |z| >

6

zsat or |z| > zsat. That is, do not update the in-tegrator if the depth error is significantly large orthe depth is already changing quickly enough. Ineither case, a further effort from the integrator isnot yet required for steady-state error compensa-tion.

4.2. Flight-Style Control Strategy

The hover-style control strategy described inthe previous section cannot be used effectivelyat high-speeds. This is because thruster perfor-mance decreases when forward speed increases[37]; consequently, the power consumption be-comes unsustainable. Also, the AUV hull behaveslike a large low aspect ratio aerofoil, generatinga hydrodynamic force that dominates the forcesproduced by the thrusters.

In the high-speed regime, using stern planes is,therefore, a better strategy since this is more ef-fective and requires less energy. The AUV pitch isadjusted by using the stern planes in order to bal-ance the hydrodynamic force with the net buoy-ancy. This is illustrated in figure 9.

Figure 9: Flight-style force diagram.

To this end, a pitch demand is determined froma pitch bias control law:

θd = KP,θ z +KI,θ

∫zdt+KD,θ z. (26)

Then, the pitch error signal is used to determinea required stern plane deflection, δS, as follows:

δS = KP,S θ +KI,S

∫θdt+KD,S θ. (27)

These steps are illustrated as a control blockdiagram in figure 10.

Figure 10: A flight-style control block diagram.

4.3. Transition between two Control Strategies

Control strategies from section 4.1 and 4.2 areunified into one control system that works over anentire range of operating speeds.

This is done by applying actuator weightingfunctions as a gain to manage the contributionof each set of actuators according to the forwardspeed (figure 11). These functions are:

wth = 1− 1

2

(tanh(

u− u∗thσ∗th

) + 1), (28)

wS =1

2

(tanh(

u− u∗Sσ∗S

) + 1), (29)

where u∗th and u∗S are a mid-transition speed andσ∗th and σ∗S are a width of transition zone [20].

As speed increases, thruster weight, wth, de-creases from 1 to 0; therefore, the use of thrustersis totally removed at the high-speed regime. Bycontrast, the reliance on stern planes graduallyincreases with speed.

Figure 11: A unified diving control system: flight-styleand hover-style strategies are combined.

A critical speed [3, p.169], where stern planesbecome ineffective, is experimentally found to beapproximately 0.7 m/s. To ensure a sufficientcontrol authority, the weighting functions are de-signed to stop using the thrusters after this criti-cal speed and start using the stern planes before

7

this speed. This results in weighting functionsfor thruster and stern planes that are overlappingover the intermediate speed regime, see figure 12.

Figure 12: Actuator weighting functions: u∗th = 0.85,σ∗th = 0.03, u∗S = 0.5 and σ∗

S = 0.04.

5. Stability and Convergence Analysis

A Lyapunov-based approach is used to analysea system stability. Let’s simplify the dynamicsmodel, (1), by excluding the surge dynamics fromthe model. Also, it is assumed that the pitch an-gle, θ, is negligibly small; hence, J(η) becomes anidentity matrix. This gives a following form of aheave-pitch dynamics model:

M · ν + C(ν) · ν + D(ν) · ν = τ, (30)

η = ν, (31)

where ν = [w, q]T , η = [z, θ]T and τ = [Z,M ]T .An error tracking vector is η = ηd − η = [z, θ]T .The hydrostatic force vector, g(η), is assumed tobe cancelled out by the integral terms in the con-trol system; hence, they are not considered in thestability analysis.

The control reference in the stability analysis isassumed a constant, hence ηd = 0. This leads to:˙η = −η = −ν.

Note that, for the proposed hover-style controlstrategy, Z and M are proportional to (22) and(23) respectively. On the other hands, for theproposed flight-style control strategy, Z and Mare proportional to (26) and (27) respectively.

A Lyapunov function candidate is designedbased on kinetic and potential energy of the sys-tem [24, p.376]:

V =1

2νTMν +

1

2ηTKP η, (32)

where KP > 0 is a constant, proportional-gain,diagonal matrix. A time differentiation of theLyapunov function candidate is:

V = νTM ν + ηTKP˙η

= νT [M ν −KP η].(33)

By substituting (30) into (33), this yields:

V = νT [τ −C(ν)ν −Dν −KP η]. (34)

The generalised control force vector is chosen, ac-cording to the PI-D scheme, to be:

τ = KP η + KDη, (35)

where KD is a constant, derivative-gain, diagonalmatrix. (This complies with the proposed controlstructure for both hover and flight style.) Sub-stitute this generalised force vector into (34) andget:

V = νT [KP η+KDη−C(ν)ν−Dν−KP η]. (36)

Since νTC(ν)ν = 0 for all ν and η = ν, the aboveequation becomes:

V = −νT [−KD + D]ν. (37)

It is known that νTDν > 0. With KD < 0, theLyapunov stability criterion, V < 0, is satisfied,ensuring a global asymptotically stable (GAS).

Furthermore, at an equilibrium point (V = 0),(33) becomes:

ν = M−1KP η. (38)

This suggests, the system could only converge toan equilibrium point where η = [0, 0]T , and willnot get stuck at any other equilibrium points.

6. Simulation Parameters

All simulations were performed in MatlabSimulink with parameters provided in AppendixA. Initial conditions were ν = [0, 0, 0]T and η =[0, 0, 0]T , i.e., the AUV is at rest on the watersurface.

The solver was Runge-Kutta 4th order. Samplerates were fixed by the IMU at 20 Hz for the dy-namics modelling section and 5 Hz for the controlsection. These comply with actual sample ratesused on Delphin2 AUV.

8

(a) Alignment setup

(b) Delphin2 AUV is hovering at constant depth

Figure 13: Experiment performed in the Boldrewood tow-ing tank of dimensions 138 m long x 6 m wide x 3.5 mdeep.

7. Experiment Setup

The proposed control system was implementedon the Delphin2 AUV at 5 Hz so that it can syn-chronise to the sensor and actuator interfacingnodes.

All tests were performed in the freshwater Bol-drewood towing tank, see figure 13. Each testwas started when the AUV was at rest on the wa-ter surface, positioned in the centre of the tankwidth, and aligned with the tank length.

A heading controller was active to maintaina constant heading. A reference heading wasobtained from the Xsens 4th generation MTi-30IMU. Since the tank features a very strong mag-netic disturbance, the utilised heading is purelybased on gyroscope and accelerometer responses.

Propeller demands used in these tests wereuprop = {0, 10, 16, 22} — these numbers are set-

points that the motor control board requires whendriving the motor. They approximately corre-spond to forward speeds of u = {0, 0.26, 0.6, 1.0}m/s respectively, and are referred to in the resultsection as zero-, low-, mid- and high-speed cases.

When the AUV is at the surface the top sidesof the vertical thrusters are not sufficiently sub-merged for efficient operation. Due to this, the in-tegrator in the generalised thrust control law waspreloaded by 1.2 times of the integrator value ina steady-state. As a result, the AUV was able todive as soon as the mission starts. Previous exper-iments have shown that a depth of approximately0.1 m is sufficient for the thrusters to functioneffectively [39].

8. Results and Discussion

There are two scenarios used for testing sys-tem stability: the first is zero-speed operation,aiming to demonstrate performance for the hover-style control strategy; the second is operation ata constant forward speed with initial and final op-eration at zero speed, aiming to demonstrate theperformance when the vehicle is accelerating fromat rest to a constant speed and vice versa.

Additional studies on power consumption andsensitivities to disturbances are also presented.

8.1. Hover Style Operation at Zero-speed

In each run the AUV started from the sur-face, and is pre-programmed with three consec-utive depth demand of 1, 2 and 1 m. Simulationand experimental results are presented in figure14.

The experimental and simulation results illus-trate the same qualitative behaviours. However,the experiments exhibit a reduced overshoot andthe oscillations are more heavily damped. Withthe conventional integrator, the vehicle takes lessthan 30 sec to settle and eventually converged tothe desired depth with no steady-state error (fig-ure 14a). As the depth demand changes, thrusterdemands are generated to accelerate the AUV to-ward the desired depth. These quickly convergeto the level that is needed to counter the positivenet buoyancy.

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(a) without anti-windup scheme

(b) with anti-windup scheme

Figure 14: Hover-style operation at zero-speed.

There is, however, a considerable overshoot indepth (approximately 0.5 m) which occurs whenthe depth demand changes. This is due to theconventional integrator continuously accumulat-ing the error signal. Hence, the integrator mo-mentarily becomes unnecessary huge, i.e. windup,when dealing with a big change in step demandor when a transient is long. The system needed

to overshoot just to produce an opposite sign oferror signal so that the integrator is unwound.

To improve the transient performance, the anti-windup scheme, described in section 4.1.1, wasimposed to freeze the integrator when a steady-state error compensation is not yet required. Asshown in figure 14b, there is a significant improve-ment in overshoot which reduces to about 0.2 m,

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and the settling time is reduced to less than 15sec. Also, the integrator with anti-windup schemeis more stable during the transient, see figure 15.

Figure 15: A comparison on the integrators from simula-tion results, tracking three consecutive depth demands of1, 2 and 1 m.

Figure 16: Simulation results for a comparison of depthtracking performance with different sizes of changes in stepdepth demands.

It is also important to note that, with the con-ventional integrator, the magnitude of the over-shoot grows in proportion to the size of a stepchange in the depth demand. The overshoot maybe as high as 1 m for 4 m depth change whichis considered unacceptably large, see figure 16.However, this issue can be alleviated when theanti-windup scheme is imposed.

Considering the pitch tracking error in figure14, the AUV is pitching towards the desired depthwhen initially adjusting the depth. This is due

to the presence of the stern planes at the far aftthat influence pitch dynamics when undergoingheave motions. The degree of pitch variation fromthe experiment is more significant, and may beas high as ±10 deg. This implies that the valueof M|q|q is underpredicted. However, the ratio offorces between front and rear thruster is adaptingto quickly dampen the pitch dynamics.

8.2. Operation at Depth with a Speed Transition

Each run starts when the AUV is at rest on thewater surface. The AUV is demanded to dive ver-tically to 1m depth at a zero propeller demand.It is then programmed to maintain depth whileexecuting a constant propeller demand for a fixedperiod of time. Following this, the propeller stopsand the AUV needs to maintain depth at a zeropropeller demand for 30 sec before the run is fin-ished, and the vehicle rises to the surface underits positive buoyancy.

The simulation and experimental results fromflight-style operation are presented in figure B.22.These clearly shows that the proposed control sys-tem yields a good depth tracking performanceover an entire speed range. Although there isa slight deviation in depth during the acceler-ation and deceleration phase, the fluctuation isbounded within ±0.2 m and quickly decays.

In the low-speed case, the AUV is able to main-tain depth with no steady-state depth error byusing only the thrusters. However, since ∆cris bounded, the pitch moment generated by thethrusters is also limited and this results in asteady-state error in pitch. The controller yieldsa good depth tracking performance.

As forward speed increases, the controller usesthe thrusters less and relies on the stern planesmore. When operating at mid-speed and high-speed the vehicle needs to pitch down by -5degapproximately to balance the net buoyancy force.

8.3. Influence of Positive Buoyancy

The actuator demands during steady operationare highly dependent on the magnitude of the netbuoyancy. The impact on controller performanceis considered in this subsection. The procedure for

11

Figure 17: Experimental results show control performance for the AUV with +1N net buoyancy.

testing with a speed transition, described in theprevious subsection, is repeated for Delphin2 bal-lasted to be almost neutrally buoyant (the resid-ual buoyancy is measured to be +1N); this buoy-ancy condition remains unchanged for every casein this scenario.

Experimental results in figure 17 show thatthe almost neutrally buoyant AUV could steadilymaintain a constant depth for the entire rangeof propeller setpoints as well as during the accel-eration/deceleration phases. Thruster setpointsmostly fluctuate around the zero value instead ofholding at a constant demand like they do forthe typical buoyancy (+6N) case. In addition,it is described in section 7 that the integral termis preloaded so that the AUV with the typicalbuoyancy can quickly dive. However, this pre-loading technique only delivers the best perfor-mance for a certain buoyancy condition. For the

neutrally buoyant cases, the pre-loading causes0.3 m overshoot in depth before the integral termunwinds and the AUV gets back to the desireddepth within 35 sec.

8.4. Sensitivity to Large Disturbance

An effect of a sudden change in the net buoy-ancy on the control performance is examined bysimulation for both zero- and high-speed cases:the AUV is demanded to maintain constant depthat constant propeller demand, then an extraweight is added and, later on, removed.

A 3N variation in buoyancy is considered. Sucha change could be observed in operation in areasof varying salinity. An experiment for the samescenario was carried out on the AUV for only zerospeed because the ballast needed to be manuallyremoved during the experiment.

Figure 18 shows that when a 3N change in thenet buoyancy suddenly occurs, the AUV depth

12

Figure 18: Results show control performance when the system is subjected to a variation in the net buoyancy.

tracking performance momentarily degrades andthe AUV depth deviates from the depth setpointby 0.3 m. It then recovers within 20sec by wind-ing/unwinding the integral terms in the controllaws. In the zero-speed case, the system adaptsby adjusting the thruster setpoints so they matchto the new net buoyancy. On the other hand,in the high-speed case, the system adapts by ad-justing the AUV pitch through the use of sternplanes.

8.5. Energy Consumption

The power consumption, Ptot, of a neutrallybuoyant flight style AUV can be assumed to besplit between hotel load, PH , which is invariant toforward speed, and propulsion power, PP , whichis related to propulsion speed raised to the power3 [40, 41]. For a positively buoyant over-actuatedvehicle, additional power is required to maintaindepth, PB:

Ptot = PH + PP + PB. (39)

Power consumption profiles, for the Delphin2AUV operating at constant depth with a typical

(+6N) and almost neutral (+1N) buoyancy, arepresented in figure B.23, and table 1 and 2; theyare compared to predicted power consumption foran ideal neutrally buoyant AUV with the samepropulsion power requirements and hotel load.

The results presented here indicate that, forthe typical buoyancy cases, PB dominates atslow speeds due to the comparative inefficiencyof thrusters in generating stationary thrust. Asudden decrease in total power consumption oc-curs at high speeds when the thruster is switchedoff and only the stern planes are used to main-tain depth, see figure 19. This decrease in powerconsumption is also directly linked to the selectedweighting function. The mid-transition speed inthe thruster weighting function could be modi-fied to lower the energy spent on thrusters earlier,e.g. starts from mid-speed onwards. However, agreater pitch down angle, up to -20deg, is requiredas a trade-off [29], and, below a critical speed, theAUV will be unable to maintain control authorityusing the stern planes alone [12].

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Table 1: Average power required when operating at constant depth at different speeds (+6N buoyant).

upropu

[m/s]power [W] COT

[kJ/kg/km]hotel th-frt th-aft fins prop total0 0.00 30 10.73 11.11 0.21 0.00 52.05 –10 0.26 30 12.62 13.17 0.16 3.12 59.06 2.8616 0.60 30 11.98 13.05 1.00 11.77 67.80 1.4222 1.00 30 1.42 5.40 0.89 22.19 59.90 0.75

Table 2: Average power required when operating at constant depth at different speeds (+1N buoyant)

.

upropu

[m/s]power [W] COT

[kJ/kg/km]hotel th-frt th-aft fins prop total0 0.00 30 2.22 3.59 0.58 0.00 36.39 –10 0.32 30 3.31 2.79 0.58 2.74 39.42 1.8616 0.68 30 2.24 2.12 1.25 11.79 47.40 1.0422 1.01 30 0.12 0.55 1.11 21.29 53.07 0.77

Figure 19: Power consumption at different speeds.

Figure 20: Cost of transport (COT) at different speeds.

The total power requirement may be analysedas a cost of transport (COT), which is a nor-malised measure of the energy required to trans-port the mass of the vehicle over a unit distanceat a speed u. The general formulation of cost of

transport for an individual is given by

COT =PH + PP + PB

mu. (40)

Figure 20 concludes that it is more energy effi-cient for the AUV to perform flight-style oper-ation at high speeds, with an optimal forwardspeed of approximately 1.1 m/s. As the AUVbecomes more neutrally buoyant, COT tends toconverge to the predicted COT for the ideal AUV.

This suggests that, for hover-style operation, agreat deal of energy could be saved by employinga buoyancy engine to actively adjust the netbuoyancy to as close to zero as possible; hence,fewer efforts are required for holding the AUV at

depth. The main concern is that this solutionrequires significant modifications to the presentAUV. It also needs to mention that the typicalbuoyancy engine response is slow and the verticalthrusters are still required for faster actions.

9. Conclusions

This paper presents a robust PI-D basedcontroller for operation of an over-actuated,hover-capable AUV. By utilising an appropri-ate weighting function, the appropriate alloca-tions of control forces are shared across actua-tors. A stability and convergence is proven us-ing a Lyapunov-based approach. The controlleris shown to provide good performance both insimulations and experiments. Typical depth over-shoots/variations of less than 0.2 m are observed

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when charging depth or speed. Such small devia-tions are acceptable operationally, and are consis-tent with or better than results using more com-putationally expensive MPC scheme for the samevehicle [22].

The impact of net positive buoyancy is con-sidered and its impact on control performance isexplained. In low speeds, as the positive buoy-ancy decrease, less effort from thrusters are re-quired to maintain a constant depth. On the otherhand, the impact is subtle for high speeds whenonly stern planes are utilised. Control perfor-mance slightly degrades when operating in neu-trally buoyant condition. More overshoot andlonger settling time are observed when changingdepth at zero speed. This is because the PI-D is alinear controller that yields the best performancefor a specific condition of that the controller istuned. Despite this, the system could toleratea fair level of net buoyancy variation, yieldingsatisfactory outcomes over the range of forwardspeeds.

The ability of the controller to respond to ex-ternal variations, such as typical changes in buoy-ancy which could be observed in operation, areconsidered. The controller responds to this ac-ceptably well.

By measuring the power utilised by the actua-tors, the impact of speed and net buoyancy havebeen recorded. The power consumption is signif-icantly dependent on the magnitude of positivebuoyancy. The vehicle hovering at zero speed con-sumes 52.05 W at 6 N net buoyancy whereas itconsumes only 36.39 W at 1 N. The buoyancycondition has less impact on this at high speeds.

Acknowledgement

This research is sponsored by the Faculty of In-ternational Maritime Studies, Kasetsart Univer-sity, SriRacha Campus, Thailand.

The gratitude is also extended to BertrandMalas and Dave J. Lynock for their great supportin using boldrewood tank, and Sophia Schillai forher assistance during the vehicle preparation andexperiment.

Appendix A. Simulation Parameters

Parameter Value Unit

m 79.40 kg∇ 0.08 m3

W −B -6 NLAUV 1.96 mIyy 35 kg·m2

rb [0, 0, 0] mrg [0, 0, 0.06] mXu -2.4 kgZw -65.5 kgMq -14.17 kg·m2/radZw 0.5ρL2

AUV u (-28.5e-3) kg/sZq 0.5ρL3

AUV u (-12.6e-3) kg·m/rad/sMw 0.5ρL3

AUV u (-4.5e-3) kg·m/sMq 0.5ρL4

AUV u (-5.3e-3) kg·m2/rad/sX|u|u -6.5 kg/m

Z|w|w -183 kg/m

Z|q|q 0 kg·m/rad2

M|w|w -59 kg

M|q|q -82 kg·m2/rad2

KT0,prop 0.0946 -Dprop 0.305 mcprop,1 0.6999 n/acprop,2 1.5205 n/aKT,th 1.2870e-4 -Dth 0.07 mwt 0.36 -t 0.11 -Lv 1.06 mLvf 0.52 mLva -0.54 mcth,1 0.35 n/acth,2 1.5 n/aXuuδSδS -0.0036 kg/m/deg2

ZuuδS 0.3241 kg/m/degMuuδS 0.3254 kg/deg

Appendix B. Free Ascent Test

The AUV was programmed to dive to 2mdepth, at a zero propeller demand, and stayedsteadily for 30sec. Then, the two verticalthrusters were suddenly turned off, allowing thevehicle to freely float back to the water surface.

15

(a) depth response

(b) pitch response

Figure B.21: Results from free ascent test.

The vehicle response is presented in figure B.21.Since a pitch angle is negligibly small, a heave ve-locity may be directly inferred from a depth re-sponse.

Assuming the total force along z-axis is dueto only the net buoyancy, i.e., Z = W − B. Aquadratic heaving damping coefficient is deter-mined as follows:

Z|w|w =Z

|w|w. (B.1)

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(a) low-speed case (uprop = 10)

(b) mid-speed case (uprop = 16)

(c) high-speed case (uprop = 22)

Figure B.22: Flight-style operation at speeds with a speed transition.

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(a) +6N buoyant

(b) +1N buoyant

Figure B.23: Experimental results show power consumption on actuators when travelling at different speeds and buoyancyconditions.

19