Guidance of Underwater Vehicles With Cable Tug Perturbations Under Fixed and Adaptive Control...

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 33, NO. 4, OCTOBER 2008 579

Guidance of Underwater Vehicles With Cable TugPerturbations Under Fixed and Adaptive

Control SystemsMario Alberto Jordán and Jorge Luis Bustamante

Abstract—This paper is concerned with the study of guidance ofunderwater vehicles subject to cable perturbations due to actionof currents and harmonic waves. The general case involves a vari-able cable length that is usually deployed during vehicle tactics.Control situations are considered, namely, trajectory tracking andposition regulation when the reference trajectory for the vehicle isgiven beforehand. To this end, a control system is designed withadaptive features. A great effort is made in this paper for mod-eling the cable dynamics in a quasi-stationary state. This modelcan serve the operator to compensate for the perturbation auto-matically during teleoperation. Moreover, in the adaptive systems,the perturbation is filtered automatically without having to embedthe cable model in the controller design. Effects of the cable pertur-bation on the control system in adaptive and fixed systems are an-alyzed comparatively. Proofs of convergence to the tracking errorto a residual set are given rigorously in the framework of total sta-bility. The features of the controlled behavior are illustrated by nu-merical simulations in a case study in six degrees of freedom.

Index Terms—Adaptive control, cable dynamics, cable pertur-bation, guidance systems, path tracking, underwater vehicles.

I. INTRODUCTION

N ONLINEAR oscillations are widely considered as asymptom of unpredictability in the behavior of period-

ically perturbed both technical and biological systems. In atechnical sense, this is also broadly taken as a sign of instability.Particularly, in guidance of vehicles or positioning of structuressubject to wave and current perturbation, this means a lack ofachievable precision in reference tracking or regulation abouta fixed point.

For instance, the appearance of nonlinear oscillations inmoored floating structures has been pointed out by manyauthors (see, for instance, [1]–[3]). Therein, the role played bythe catenary stiffness of cables to account for these behaviors

Manuscript received November 23, 2007; revised June 09, 2008; acceptedAugust 26, 2008. Current version published February 06, 2009. This work wassupported by the National Council of Scientific and Technological Research atthe Argentinean Institute of Oceanography (IADO-CONICET) and the NationalUniversity of the South at the Department of Electrical Engineering and Com-puters (DIEC-UNS).

Associate Editor: H. Maeda.The authors are with the Argentinean Institute of Oceanography

(IADO-CONICET) and the Departamento de Ingeniería Eléctrica y deComputadoras, Universidad Nacional del Sur (DIEC-UNS), B8000FWB BahíaBlanca, Argentina (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JOE.2008.2005595

was highlighted. Similarly, in the case of submersed vehicles,such as tow fishes and remotely operated vehicles (ROVs),the occurrence of nonlinear oscillations in forces and positionmay indicate high cable tensions and a lack of maneuverability[4]–[6]. Particularly, in inspection tasks with camera-basedteleoperations, the appearance of low-frequency oscillationsmakes the manipulation of vehicle gaffs difficult [7]–[10].Moreover, such perturbed subaquatic scenarios are particularlycritical in estuarial applications in the presence of waves andstrong tide [11].

Usually, periodic waves accompanied with strong currentsmay excite the motion of the vehicle during teleoperation in thesame way as in a mass-spring system, but with nonlinear partic-ularities due to the hydrodynamic drag force on the cable and itstension-shape characteristic. A dynamic model of the cable-ve-hicle system subject to stationary flow is quite complex becauseof the combination of transverse and axial stiffness of the cable[12]. In [3], it was shown that when the ratio of the elastic to thecatenary stiffness is small and hydrodynamic drag loads are oflow frequencies, the mooring line preserves basically its cate-nary in-plane shape. In this case, a quasi-static cable behaviortakes place. This quasi-stationary model provides a straightfor-ward starting point for bifurcation analysis at regulation pointsof the vehicle, involving free parameters such as cable stiffness,length, wave amplitude and frequency, and current intensity,among others. Also, it could be useful to compensate pertur-bations directly during teleoperation or indirectly by includingthem as part of the control system.

One outstanding feature of a harmonic excited nonlinearmass-cable system is the appearance of tugs in the form ofperiodic cable pulls with a larger period than the excitation[2]. They may induce the advent of the so-called ”taut-slack”phenomenon in the cable tension giving rise to the appearanceof a wide behavior diversity covering high-period and chaoticmotions in both tracking and dynamic positioning issues ofthe vehicle [13], [14]. However, a behavior diversity canalso appear irrespectively of this phenomenon, i.e., under theprevalence of catenary stiffness over axial stiffness. For towedunderwater vehicles, this diversity will depend primarily onthe velocity of the towing ship [6]. Similarly, for teleoperatedunderwater vehicles, the strength of the diversity relies on theflow rate and cable catenary stiffness. In both cases, the cableshape determines the stiffness of the cable-body-environmentsystem. The identification of such behaviors is very importantin the design of vehicle autopilots so that they are able to tacklequalitative changes in the controlled behavior. Particularly,

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580 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 33, NO. 4, OCTOBER 2008

Fig. 1. Succession of ROV positions by sea bottom scanning with servo–con-trolled constant attitude.

for emerging limit-cycling modes of motion arriving in thevehicle teleoperation, there are required controllers that cancreate strong damping for nonlinear oscillations without losingtracking and regulation abilities [15].

In this paper, we will focus our attention on the study of theperturbed behavior in servo–controlled ROVs with full-actu-ation in six degrees of freedom, which can be operated witha variable cable length for achieving complex maneuvers,such as, for instance, scanning procedures over irregular oceanbottoms at both constant attitude or constant depth. Ambitioussetups in the tracking problem, such as, for instance, a rapidresponse of the vehicle with high-performance control underdynamic uncertainties in perturbed scenarios, are necessaryabove all in sampling missions, when the vehicle is requiredto move fast, nearly reaching down to the seafloor with rapidreactions, swerving possible collisions.

A key component of our intended analysis is the dynamicinteraction among waves and currents with a cable and a ve-hicle during operations that involve a deployment of the umbil-ical cable (see Fig. 1). Moreover, we will stress the benefits ofautomatization in path tracking for complex tactics in the pres-ence of perturbations, while human control could be delegatedfor teleoperation of a vehicle gaff or supervision of control tasksmainly.

From a physical point of view, the cable shape and the po-sition of its extremes play a decisive roll in the vehicle oscil-latory behavior and path following. From a mathematical pointof view, particular simple motions of the cable-vehicle systemare described by Mathieu and Duffin-type nonlinear differen-tial equations [1], [13], [14]. These can be parametrized in aset of physical coefficients such as cable length, cable deploy-ment rate, stiffness, wave amplitude and frequency, and currentintensity, which are significant bifurcation parameters [14]. Onthe other hand, we aim to design high-performance controllersthat incorporate some information of the cable force togetherwith the vehicle dynamics, to perform path following with anachievable high performance. The control action will be cal-culated on the basis of current navigation measurements anda cable model. Although adaptive features of the control algo-rithm could be acquired without using a cable model, then an

open key problem would be to establish the ability of adaptivecontrollers to adapt the system behavior under cable perturba-tions [16]. Finally, a comparison of fixed and adaptive controllerfrom a stability point of view is a goal to be covered in this paper.Numerical examples will close the paper and illustrate the anal-ysis of the control system behavior under wide range perturba-tions and compare fixed and adaptive controllers from the pointof cable tugs rejections and achievable control performance.

II. QUASI-STATIONARY MODEL OF THE CABLE

A. Planar Configuration

A possible significant scenario of the ship-cable-ROV systemis illustrated in Fig. 1, where a submersed ROV hangs froma winch on a mother ship, from which it is required to per-form a constant attitude mission automatically maintaining thepitch angle constant to obtain a good visual scanning of thesea bottom. Herein, a perturbation originated by ocean streamscauses the ROV to be dragged along with the flow in the direc-tion . The cable bends in favor of the stream and producesmoments on the vehicle that can destabilize the desired course.Also, a wave action can eventually induce cable force perturba-tions at the support point whose effects propagate to the attachedpoint on the vehicle.

A more detailed scheme of the perturbed body-cable systemis shown in Fig. 2. In a general configuration, it is assumed thatthe ship is anchored in the direction of the flow and the cableremains entirely confined in a plane. This cable plane is definedby the directrix containing the support point on the winchtackle and the point on the attachment with the ROV. Theplane can eventually hinge about when the ROV moves side-ways with respect to the central plane of the ship. The hingeangle is determined by the ROV coordinates taken with re-spect to frame .

Let us suppose that the ship oscillates harmonically by effectof a wave. So, the winch tackle describes a stationary orbit abouta fixed point termed . The earth-fixed frame with center onand with ship orientation is referred to as the frame . For thesake of further analysis, the cable is divided appropriately intoan aerial stretch and a submersed stretch , where isthe moving contact point of the cable on the water surface.

The cable is also assumed to have neutral buoyancy in water.Additionally, the flow is considered stationary and uniform withan intensity that eventually changes in depth.

B. Orbit of the Support Point

Consider the frame and the axis coincident with thedirectrix (cf., Fig. 2). Suppose now that due to the action ofwaves on the ship along the main direction, an elliptic motion ofthe winch tackle is induced. So, the point moves about onthe vertical plane – as illustrated in Fig. 2 with coordinates inthe earth frame equal to

(1)

JORDÁN AND BUSTAMANTE: GUIDANCE OF UNDERWATER VEHICLES WITH CABLE TUG PERTURBATIONS 581

Fig. 2. Operation of the cable-ROV system from a mother ship under tidal flow.

where is the ship frequency and and are the oscillationellipse radii depending on wave parameters and the proper shipdynamics.

Let us assume that the coordinates of with respect tocan be obtained directly from measurable variables of the

vehicle navigation system and referred to as the coordinates. Thus, in the frame , they satisfy

(2)

with

(3)

A cable point on the slanted plane with coordinateshas global coordinates in the frame equal to

(4)

Similarly, the cable contact point to water also lies on theslanted plane with global coordinates

(5)

where and are the observed horizontal and vertical distancesfrom to , respectively.

C. Cable Equations

Besides the planar configuration of the cable introducedabove, it is supposed that the cable does not offer any resistanceto flexion at all, while the axial deformation is elastic obeyingYoung’s law. So, the relations between forces and cable shapebasically satisfy [3] (see Fig. 3)

(6)

(7)

(8)

(9)

where is the so-called unstretched Lagrangian coordinate,and analogously, is the stretched Lagrangian coordinateof the cable profile with , is the un-stretched cable length corresponding to with weight ,

is the load distribution function of the dragforce in the direction , is the cable tension at , and

are moduli of the vertical and horizontal reactive forces atextreme , respectively, and is the cable stiffness.

Clearly, due to the cable load and the fact that ,there exists a lengthening of the cable in the axial direction sothat with and only at .


Fig. 3. Forces in the slanted cable plane � –� .

D. General Solution of Cable Profile

Because a general solution of (6)–(9) is very involved, wetackle better solutions by sections instead. So let us considerthe sections of length and of lengthseparately, i.e., one searches for relations between forces andcable shape in the form of a continuous piecewise function.

1) Cable in Air: To this end, let us first deal with the upperportion . Accordingly, with and from(6)–(9), one achieves the solution (cf., [3])

(10)

(11)

(12)

The weight corresponding to the cable of length isthe same as for the unstretched portion ; it is

(13)

where is the cable diameter, is the water density, andis the gravity constant. The force components and willdepend on the next section .

The coordinates of the crossing point are calculated from(11) and (12) as and , respectively.The length will be determined later and is variable with

both the displacement of the vehicle and the wave effect. So,from (8) and with (11) and (12), one attains

(14)

which is a relation valid for .2) Cable in Seawater: On the other hand, for the section

, the weight disappears while the drag force plays the roleof the distributed load instead. The flow rate is referred to as

and assumed from now on constant in depth. Thus, one canapply Morison’s law and attain (Fig. 3)

(15)

(16)

where is the drag coefficient for slender cylinders. Com-bining (6) and (7) for

(17)

Then, integrating (17) results in

(18)Unlike the cable shape in the air with a catenary form, the

shape of the cable submersed assumes a quadratic form for con-stant rate profiles. Certainly, one attains an expression for thecable shape explicitly

(19)with obtained from (11).

Although the constant rate profile is introduced here onlyfor mathematical simplification, more complex profiles canbe used, for instance, using acoustic Doppler current profiler(ADCP) technology and performing numerical integration of(15) to obtain .

E. Forces on the Extremes

The tension of the cable is obtained by combining (6)–(8).For (cable section in air)

(20)

and for the point , the forces are

(21)


from which one gets

(22)

F. Moments

The determination of forces and will be carried out later.However, to achieve this end, it is necessary to consider otherrelations such as those involving moments and lengths.

For the particular case (16), the moment of forces with respectto is

(23)

with being the coordinate of described in Fig. 3. Sofrom (11) with and using (13), one obtains

(24)

G. Cable Length

Let us consider the total stretched length

(25)

and the unstretched length

(26)

respectively. It is worth noticing that can be measured atthe revolving spool of the ship winch system exactly, while thestretched length has to be calculated from the model withpositions and forces on the cable ends and .

Accordingly, one can calculate from (12) and (13) withand under the hypothesis the

partial stretch

(27)On the other hand, the length of the section for the

submersed cable is achieved indirectly by first calculating thecable length . To this end, let us determine first

from (7) and (9) for and positionsand (see Fig. 3). So, after integration, it is valid that

(28)

The last integral has been solved by symbolic programming.Generically, this yields

(29)

where is a multivariable expression to be evaluated numer-ically in our next study.

III. DETERMINATION OF CABLE FORCES AND SHAPE

The determination of the cable forces at extremes is notstraightforward because the describing equations involve thesevariables implicitly, i.e., there exist no close expressions forsolving the problem of forces and cable profile analytically.So, we aim to an exact but numerical solution by which theimplicit function theorem guarantees solvability. Once forcesare determined, the cable shape can be identified from them.

Algorithm

Toward this goal, an algorithm is proposed in [11] for plasticcords, and redefined here for elastic cables. Particularly, a so-lution is developed for uniformly distributed drag loads. Thisconsists in the following steps.

1) Startup. Use starting values for , , , , , ,, , , and . Also, a constant flow rate is

provided together with known ROV absolute position.

2) Calculate according to (16).3) Solve the nonlinear (29) with (27) and (23) and (24) for

and recursively, e.g., by employing a gradient-basedsearching algorithm. For starting the iterations, use

and as initial conditions,


Fig. 4. Cable shape. Experimental determination.

TABLE ISETUP PARAMETERS

which are the adequate values coming from a simplifiedand symmetric cable shape.

4) Calculate the force components and on theROV by means of (21).

5) Calculate the cable form (11) and (12) for and (18)for

Our experience shows that the convergence of step 3) isreached after a few iterations [on average, less than 20 with theroutine ” fzero ” of MATLAB® 7.0 (R14)].

In this way, the cable perturbation is evaluated and can beemployed as a known force in the control system for guidingthe ROV automatically.

A. Experimental Results

The algorithm was tested in a laboratory flow canal with acord under conditions given in Table I.

The canal was set up to give approximately laminar flow andthe flow rate was measured in the midpoint of the depth bymeans of an axial turbine. The force values calculated from thealgorithm are summarized in Table II.

The cord curve is calculated in step 5) of the algorithmand thereafter superposed with a frame of the true cable formin the same scale (see Fig. 4). An error can be appreciated,

TABLE IIFORCE ESTIMATION

which is above all due to nonuniformity of the flow profile,which is slightly stronger close to the surface than near thebottom. Nevertheless, the degree of coincidence of the curvesis significant. The flow nonstationarity was not acute as verifiedfrom the observed small movements of the midpoint of thecord in videotapes. In other experiments with different setups,similar results were obtained. Moreover, one could concludethat vortex phenomena behind the cord did not appear duringthe experiment because of the small Reynolds number calcu-lated for large finite cylinders. The Reynolds number resultedin , which lies quite far away fromthe turbulence zone for the case of large finite cylinders.

IV. DESIGN OF A GUIDANCE SYSTEM

One of the main elements of the guidance system for an ROVis the controller for path following according to the prescribedreference trajectories or for regulation about desired fixed coor-dinates. Usually, highly elaborated controller designs are basedon a model of the vehicle. However, when a significant pertur-bation is involved as in the case being analyzed, a model of thecable may contribute to improvements in the navigation. As thiskind of perturbation can intensely influence the behavior in allmodes of motion the same way, a controller for six degrees offreedom of the ROV is needed. In this section, we will describea high-performance control system. Then, we will distinguish


Fig. 5. Remotely operated vehicle [model similar to a prototype at TechnischeUniversität Hamburg-Harburg (TUHH), Germany].

between the class of fixed and adaptive control systems for theperturbed guidance problem.

A. Vehicle Dynamics

Toward this end, let us consider a body-fixed frame with thecenter on some fixed point (see Fig. 5), which is not nec-essarily the mass center. Moreover, the position of the ROV ismeasured with respect to an earth-fixed frame with the centeron (see Fig. 3). Thus, the vehicle dynamics is given by [17]

(30)

(31)

with the matrices

(32)

(33)

(34)

(35)

where the generalized position in the earth-fixed frame is de-noted by , whileindicates the generalized relative velocity vector in its flight pathin a body-fixed frame, is the current flow rate in a vector formobserved from the inertia frame [ in (15) is the first compo-nent of ], is a well-known rotation matrix depending onthe Euler angles and , is the inertia matrix composed ofthe body inertia matrix and the added mass matrix thataccounts for the surrounded fluid mass, represents the cen-tripetal and Coriolis matrix, which can be decomposed into acombination of constant matrices and state-dependent ma-trices with the operation ” ” denoting the element-by-el-ement multiplication, is the damping matrix composed of aconstant matrix and a velocity-dependent matrix, is the netbuoyancy force decomposed into two terms with constant ma-trices and , is the cable reaction force, and is thegeneralized thrust [18]. All generalized forces are applied on

.

B. Cable Tug

The cable force on the ROV, in (22), lies in the slantedcable plane. First, we project it onto the main axis of the movingvehicle and it results in

(36)

where is the upper block matrix of and is the force on thepoint in the vehicle-fixed axis. Finally, the cable tug becomesa generalized force on (see Fig. 5) with the form

(37)

where and are the coordinates and , respectively, ofwith respect to , and are the components of .

Next, we will design fixed and adaptive controllers thatdetermine the generalized thrust by feeding back the states

and .According to (22), for given extreme coordinates points

and (see Fig. 2), a flow profile , and an areal cablelength , the tension on the extremes, i.e., , , , and

, will be bounded. So the cable tugs on and onare bounded.

C. Fixed Control System

At this point, we are interested in a design of a high-per-formance controller for the navigation problem with two givenuniformly continuous reference trajectories and

and for the respective measured vectors and.

Hence, we define the specifications of the servo–trackingproblem asymptotically in time as

for (38)

for (39)

for arbitrary bounded

(40)

The way to keep the spatial and kinematic vehicle trajectoriesclose to their respective references is achieved by manipulatingconveniently the rate on the vehicle thrusters.

To this end, let us define first a convenient expression thattakes into account the positioning and kinematic errors as (cf.,[19] and [20])

(41)

(42)

with . Clearly, from (41) and (42), if is zero,.

Accordingly, from (30) and (31) with (41) and (42), it is validthat

(43)


(44)

Now, we employ a speed-gradient strategy for the au-tonomous navigation problem with a fixed controller [21].Accordingly, let us define a cost function of the errors as

(45)

with the aim that for all initial mismatches and suchthat and , it is valid that

for (46)

The idea now is to employ the generalized thrust as the ma-nipulated variable so as to confer the function the property ofnonpositiveness. So we state

(47)

and choose a suitable generalized thrust as

(48)

with

(49)

where in (48) is a second design matrix nextto in (42). Later we will prove the convenience of (48) toestablish convergence of the tracking errors.

D. Adaptive Control System

To control the vehicle without requiring a priory knowledgeof it, the control action can alternatively be constructed bymeans of a dynamic state feedback of the tracking errors and ref-erence paths. To this end, a control strategy based on the adap-tive speed gradient is employed [21].

One defines first controller matrices referred to as ’s bymeans of an integral adaptive control law of type

for (50)

with positive–definite design gain matrices. Then,the generalized thrust is proposed as

(51)

for , the ’s accomplish for the centripetal andCoriolis matrix according to its decomposition into terms

as indicated in (33). For them those are valid laws

(52)

For the linear drag force component in (34), there corre-sponds the factor with the law

(53)

Analogously, for the quadratic drag force component in (34),there are associated ’s with and the laws

(54)

where is the th element of .For and related to the buoyancy vector components

of according to their decomposition form in (35), the fol-lowing laws are valid:

(55)

(56)

respectively.Finally, is connected with the inertia matrix by means

of the following law:

(57)

with being the auxiliary vector defined in (49).The integration of the adaptive laws with

, for , provides directly the adaptive con-troller matrices in (51). Asymptotic stability of theadaptive control system under the previously specified setup isproved in the next section.

Remark 1: It is noticed that the effects of the cable tug aresupposed to be canceled by embedding in the laws, i.e., in(48) for the fixed control and in (51) for the adaptive control,according to the cable model developed before with the pertur-bation given in (37). In the following, we will consider also thecase of the unknown perturbation, that is, when is not em-bedded in the laws, to investigate the influence of the cable as aperturbation on the stability of the vehicle.

E. Thruster Characteristic

By computing as in (48) or (51) according to the controlcase, with a suitable selection of the design matrices and(as shown in Remark 4; see the Appendix), it is expected that thevehicle path tracking can acquire a first-rate behavior in steady


Fig. 6. Thruster characteristic with hysteresis.

state after the fast adaptation transient. Even when thruster dy-namics can commonly be considered as parasitics in comparisonwith the often lazy vehicle dynamics, the static thruster char-acteristic is quite dominant, above all, when rapid kinematicsreference paths are aimed at high-performance controls. Thethruster characteristic reveals existing operating-rate-dependenthysteresis loops around the null shaft rate (see Fig. 6).

The generalized thrust acts on the center of the fixed-body frame (see Fig. 5) and it is implemented by the thrustvector composed of the individual forces acting on the shaftline in every actuator. It is expressed by

(58)

where is a constant matrix containing position coordinateswith respect to of every thruster.

On the other hand, the thrust vector fulfills

(59)

where and are constant matrices, is the axial ve-locity vector of the thruster, denotes an element-by-el-ement vector product, and is a vector with elements that arethe same for but in the absolute value. The family of curvesversus for different values of is depicted in Fig. 6. Clearly,as (59) is a quasi-convex vector function for , thenwill be finitely discontinuous every time some of its components

take the value , where is defined by the respec-tive component in the vector

(60)

or similarly, when some of its components take the value zerowhile the respective component of the thrust is .

Accordingly, when some thruster is required to regulate thepath with respect to zero, for instance, by zero-pitch angle, thecrosses of the shaft rate around zero will produce discontinuitiesin and a hysteresis loop, whose dimension is a function of

.Particularly, if there exists some hydrodynamic interaction

between thrusters due to their proximity, then and will benondiagonal. In this case, for a given vector at a time point ,

(59) describes the system of equations to be solved explicitly forall the components of . This has to be carried out numerically.

On the contrary, if thrusters are placed sufficiently far awayfrom each other, so that no hydrodynamic interaction can occuramong them, then the matrices and are diagonal and itis valid that

(61)where , , , , and are the elements of , , and

, and of the diagonals in and , respectively. Clearly, itis noticed from (61) that the jumps of are of magnitudesabout

with

(62)

with

(63)

where , is not a dense set ofdiscontinuous time points, where the jumps of occur, and

indicates a positive infinitesimal.Finally, as supposed before, in the absence of thruster dy-

namics, the needed reference rate vector to input the thrustersin the tracking problem is just conformed by the elements of(61). Moreover, the reference rate vector is piecewise contin-uous while in (58) is continuous for smooth path and kine-matic references and .

V. CONTROL PERFORMANCE UNDER CABLE TUGS

The study of influence of the cable tugs on the control dy-namics can be posed as a problem of stability. On one side, thecable model developed in Section II can be employed to pre-dict the perturbation and then to embed it in the force (30) as

to completely cancel its effect on the controlled dy-namics. Moreover, the asymptotic convergence of the trackingerrors should be ensured with more certainty with this procedure(cf., Remark 1). On the other side, the presence of perturbationcan be tolerated and partially rejected by the (fixed/adaptive)control law itself. In this case, the controllers should have theability to ensure bounded tracking errors for bounded tugs andthese errors should be maintained by them proportional to theorder of magnitude of the perturbation.

In the following, we present the results that illustrate the con-trol performance of the guided vehicle under cable tugs in amore precise description. The results are enunciated as theoremsand their proofs are presented in the Appendix.

The first theorem describes the possibility of the controllersto track paths with asymptotic null errors when the cable modelis taken into account in the calculation of the control action.

The second theorem is the counterpart of the first one andit quantifies the bound of the tracking errors according to themagnitude of the perturbation.

The third theorem attempts to compare the control perfor-mance between the fixed and adaptive control systems presented


in this paper. Far from the advantages of the adaptation features,these results will give information of the damping features insteady state in both cases comparatively.

Theorem 1 (Tracking error convergence with existing but can-celled cable perturbation): Consider the vehicle dynamics (30)and (31) with actuator characteristic (59), the fixed control ac-tion (48) on one side, and the adaptive control action (51) withthe adaptive laws (52)–(57) on the other side. Moreover, letus assume that the cable-vehicle dynamics obeys the model ofSection II that prescribes faithfully a cable tug calculated ac-cording to (37) and it is fully compensated for directly throughthe control action. Then, both control systems (fixed and adap-tive) can ensure asymptotic tracking with a vanishing null errorfor every continuous reference and . Besides, allvariables in the respective control loop are bounded.

Proof: See the Appendix.Let us consider now the cable tug as a pure perturbation in

both control systems. Before drawing some new results, let usintroduce a type of stability that will be beneficial later for thenext convergence proof and further results.

Definition 1 [23]: The equilibrium point described byis said to be totally stable, if for each , there exist

two positive real values and such that the solutionsand of (43) and (44) fulfill

for (64)

for (65)

provided that

and (66)

and, in the state–space region delimited by (64) and (65), that

(67)

Theorem 2 (Tracking error convergence with existing andnoncompensated cable perturbation): Consider the same infor-mation as in Theorem 1 for the vehicle dynamics, the fixed andadaptive controls. Similarly, let us assume that the cable-vehicledynamics obeys the model of Section II, however, the cable tug

in the laws (48) and (51) is not considered in the designs, i.e.,it acts as an unknown force perturbation.

Then, the asymptotic equilibrium point of the perfecttracking, i.e., , is totally stable in both controlsystems.

Proof: See the Appendix.Theorem 3 (Convergence comparison between fixed and

adaptive cases with cable perturbation): Consider the devel-oped speed-gradient-based controls in the fixed and adaptiveforms comparatively under the same perturbation, initial condi-tions, and any continuous bounded references and .Suppose that the cable tug acts as perturbation in the vehicledynamics and it is not considered in the controller designs.Then:

i) the error trajectories enter asymptotically the invariant

(68)

ii) the adaptive control will always give a stronger (at worst,the same) asymptotic attenuation of the perturbation ef-fects on the vehicle tracking behavior than its fixed coun-terpart.

Proof: See the Appendix.The previous result of performance in the steady state is very

useful in cases when the cable perturbation can be considered asgenerated by the proper vehicle motion in a regulation positionwith actuation of the periodic waves.

VI. FORCED NONLINEAR OSCILLATIONS

The quasi-stationary model for the cable developed beforeprovides a mean for analysis of a qualitative behavior of thecontrolled dynamic system in the stationary state. When the con-trolled vehicle is excited periodically by the wave actions, spe-cific patterns in the cable shape will indicate the route to the ap-pearance of nonlinear oscillations. Moreover, the analysis of therelations between forces and catenary stiffness when the cable isdeployed and the vehicle moves under excitation of waves andcurrent is significant to both the adaptive and fixed approachesdeveloped in this paper.

A. Critical Vehicle Velocity

The navigation may be subject to sudden cable tugs if theROV velocity is relatively large in comparison with cable de-ployment. Particularly, in a guidance at constant depth along thecurrent direction, the condition for tug avoidance is such that theROV forward speed must accomplish with (6)–(8)

(69)

(70)

where is the time point of the starting position, which cor-responds to a loose cable of length . Not only the rate forcable deployment to avoid the so-called taut-slack phenomenonbut also the stiffness provided by the cable during the vehiclemotion are important. Moreover, cable shapes have a significa-tive influence on the appearance of nonlinear oscillations andintensity of the cable force.

B. Catenary Stiffness

The order of magnitude of cable force components acting onthe system depends strongly on the cable shape. As these areinvolved in the perturbation , it is significant to analyze theconnection between them with vehicle position, cable length,and environmental variables.

To this end, let us consider Fig. 7 that illustrates the cableshape sequence for different lengths and constantand (i.e., navigation at constant depth). Without loss of gen-erality, let us assume , (i.e., cable completelysubmersed), [by (13)], and no excitation of waves otherthan the uniform flow rate .


Fig. 7. Sequence of a cable shape for different cable lengths and vehicle positions.

TABLE IIIFORCE RELATIONS IN FIG. 7

Using (6) and (7) one attains the slopes for the cable extremesand as

(71)

(72)

So, we can now define a useful qualitative feature for furtheranalysis related to the cable spire by means of the angle

(73)

When , the spire is null and the cable shape is rectilinear.Otherwise, when increases, the length is large and the cableshape takes the form of a catenary. At the limit when ,the length is infinite with a spire infinite as well.

On the other hand, from (36), it is valid that

(74)

Clearly, the norm is independent of. If the flow is stationary, then

(75)

(76)

Moreover, from (22) and (23), with the conditions stated previ-ously, it is valid that

(77)

(78)

Now if one compares in Fig. 7 the cases (a) and (e) or (b) and(d) with the same pair, one obtains the results presentedin Table III.

Clearly, if decreases, then and diminish andand increase, so the catenary stiffness decreases and

vice versa. So, when the spire tends to zero, the cable taut takesplace and and are limited only by the axial stiffness,which is generally much larger than the catenary stiffness.Moreover, it is seen from Table III that the navigation alongwith the flow produces a smaller cable force perturbation thanin the other direction [this is deduced from (76) and (77) with

]. Finally, one sees from (75) and (77) that the rise of theflow intensity causes the increase of both and as well.

C. Nonlinear Oscillations

It is well known that the appearance of taut-slack phenomenaunder wave perturbations is related to nonlinear oscillationssuch as low period or chaotic behaviors (see, for instance,[2] and [14]). Such oscillations are undesirable for high-per-formance tracking targets. According to the previous section,it is not surprising that the route to chaos is connected withcritical situations where is small, i.e., when the cable shapeis nearly rectilinear. Next, we will investigate the ability ofthe controllers in the adaptive and fixed cases to attenuatesteady-state oscillations in tracking and regulation cases.

As proved before in the theory, the adaptive control providesa better all-around performance than the fixed-control counter-part. Generally, it can ensure the vehicle regulation and guid-ance more accurately in steady state. On the other hand, forsmall cable spires, the adaptive controller will attempt to dampinduced oscillations on the vehicle caused by waves more ef-fectively than the fixed controller without frequently falling intosaturation. So, one can expect that the nonlinear oscillations are


TABLE IVSETUP PARAMETERS FOR EXPERIMENTS

Fig. 8. Evolution of the cable shape during rectilinear path tracking in adaptive and fixed control. At endpoints: taut-slack phenomenon.

generally associated with the fixed case and not the adaptivecase, and that they would be larger in intensity than in the adap-tive case with a higher energetic cost involved in the controlaction.

The main bifurcation parameters in our study were focusedon the cable length, spire, wave, and flow coefficients. The cablelength and spire can be combined together to define the catenarystiffness.

VII. CASE STUDY

To illustrate the features of the proposed control systems of anROV subject to cable perturbations, numerical simulations werecarried out according to the control algorithms and models de-veloped previously. Herein, we will distinguish between, on oneside, the path-tracking problem along a reference trajectory, and

on the other side, the regulation problem around a fixed point.In the first case, both controls are applied to a vehicle moving inboth directions with a variable cable length subject to a regularflow and wave action. We will also point out the differences inperformance when adaptive features of the controller are incor-porated with respect to the behavior of fixed control systems.From them, conclusions will finally be drawn.

To this end, let us consider a setting according to Fig. 1, whichwas taken as case study. The setup is given in Table IV.

The setup describes the scenario for a ship-cable-ROV systemperturbed with a wave with a frequency and a current . Thedepth from the surface to the bottom is . These conditions pro-duce the motion of according to an elliptic path of amplitudes

and frequency . The ROV is guided with a variablevelocity according to a reference rate while the cable isbeing deployed with a variable rate . Both guidance targets,


Fig. 9. Evolution of the angle � during path tracking in adaptive and fixed control.

Fig. 10. Evolution of the cable force during path tracking in adaptive and fixed control.

path following and regulation, are pursued for constant vehicledepth as a reference.

The design matrices and are the same for the adaptiveand fixed control as well. The adaptive gain matrices togetherwith and are given in the following:

(79)

Here and were selected as diagonal matrices. To im-prove the convergence of both and (roll and pitch errors)for a rapid advance with cross current, the respective elements

in are magnified, while in , the emphasis is put after allin the rate mode errors and (heave and pitch rates), to re-duce the effects of cable tugs. Additionally, coefficients cor-responding to the inertia and buoyancy coefficients in the di-agonal matrices , , and , were ponderated stronger asthe ones in the damping and Coriolis coefficients in the morecomplex matrices and , respectively. This enables a bettertransient performance.

A. Path Tracking

Let us consider a rectilinear motion for the ROV at constantdepth along the -axis in two opposite directions. Fig. 8 illus-trates the cable shape during the motions. Both experiments startjust below the ship ( ): one with and the other against the


Fig. 11. Evolution of the tracking error �� in adaptive and fixed control.


tidal flow with the same flow intensity . In both mis-sions, the cable is deployed slowly, beginning from 20 mup to 218 m at a rate of 0.09 m/s. It is noticed that atthe start phase the cable spire is small. Then, as the vehicle be-gins to accelerate from rest until the cruise reference rate0.1 m/s is achieved, the cable increases its spire. On the otherhand, since , the cable spire begins to diminish gradu-ally until condition (69)–(70) is finally violated coming to thetaut-slack phenomenon. During this process, the cable continuesdeploying and the controllers attempt to maintain the course at

.It is observed that the cable shape evolves similarly in the

two control modes. Judging by the straight forms of the cable atthe endpoints, the taut-slack phenomenon seems to actuate with

comparable severity on the vehicle dynamics in both controlmodes, though the effect of the perturbation is more acute inthe column for the adaptive control. However, to fairly judgethe control performances, we will go into more details in Fig. 9.

Fig. 9 shows the spire evolution through the angle . Ini-tially, the angle is small and it increases during the deploymentof the cable. The tracking errors tend practically to null in bothcontrol systems during the elastic catenary form . Afterthe advent of the null spire, the vehicle behavior is rather chaoticindependently of how the control system is working and of thepath direction.

One can conclude that when the cable has a small spire andis slanted, the wave oscillations will have more influence onin comparison with the cable with the same spire and extremes



located approximately in vertical position. This effect is morepronounced when the taut-slack phenomenon takes place.

In Fig. 10, one sees comparatively that the adaptive controlsystem behavior generates a more energetic cable perturbation

than in the case of the fixed control, however a better perfor-mance of the former can be appreciated mainly during the startphase when smaller path errors , , and are produced. Onthe other hand, during the tugs, saturation in both control casesoccur. Here, the fixed controller reacts pushing the vehicle in alarge pitch angle while the adaptive controller pushes it upward(see Figs. 11–13). Another property of the path-following con-trol is that the navigation against the flow causes a higher cableforce as pointed out in Table III and Fig. 7 in both control cases.

One can conclude that outside the period of appearance ofthe taut-slack phenomenon, both control systems can performvery well, generally when , or more precisely whenconditions (69) and (70) are satisfied.

B. Fixed-Point Regulation

Figs. 14 and 15 describe a vehicle behavior concerning theregulation of the ROV about 45.65, 0, 20 m in the twocontrol systems, where the cable length is extended from 50to 60 m at a rate 0.05 m/s. Previously, the systems were insteady state in forced oscillation due to wave action. Clearly, thecable spire is smaller at the beginning. In both cases of control,the oscillations at the start phase are much larger than the ones atthe end phase where is larger due to the cable length enlarge-ment. This points out that a small cable spire near tautly is moredifficult to regulate due to the rigid catenary stiffness in compar-ison with the opposite end situation with a more elastic catenarystiffness. By the adaptive controller, the nonlinear stationary os-cillations are always of period-1, while by the fixed controller,a period-3 oscillation takes place initially, which vanishes withtime and turns finally into a period-1 oscillation. This shows thatthe route to chaos in the cable length parameter is connected tosmall spires, i.e., to a high catenary stiffness. Comparatively,

Fig. 14. Oscillations in the adaptive control system during the regulation withvariable cable length ��.

the oscillations in the adaptive case are much smaller than inthe fixed case, allowing a more accurate regulation.

Fig. 16 displays the force cable generated during the regula-tion in both cases. Clearly, small cable spires affect more thefixed control system, in which case a twice as large cable per-turbation is created than in the adaptive control system.

Figs. 17 and 18 present the evolution of the thrusts during theregulation period. It is seen that the adaptive control can reducethe oscillation employing a much lower energy in the controlaction, while the fixed control cannot avoid saturation mainlywhen the cable is more rigid. This saturation in the case of fixedcontrol arrived in the vertical thrusters only.

VIII. CONCLUSION

This paper has dealt with the guidance of ROVs under per-turbations that are present during path tracking and regulationtargets in oceanic applications. We have placed special emphasis


Fig. 15. Oscillations in the fixed control system during the regulation with vari-able cable length ��.

Fig. 16. Cable force generated during the regulation in adaptive and fixed sys-tems.

on cable perturbation as being induced on the vehicle by actionof harmonic waves and currents. Particularly, in the accomplish-ment of sampling missions with variable cable length, such asa path at constant depth or attitude, systematic sampling on seabottom or in the water column, a considerable high precision isdemanded in maneuvers involving low energetic cost and min-imal execution times. These maneuvers cannot generally be cov-ered appropriately by the vision-based teleoperation subject tostrong perturbations and eventual nonlinear oscillations. So, itis shown that automatic control, especially adaptive control, canbring numerous advantages comparatively.

A model for quasi-stationary dynamics of a ship-cable-ROVsystem was developed. The model is based on locations of thecable extremes and flow intensity, and it gives estimations offorces on extremes and the cable shape with sufficient accu-racy for normal motions of the vehicle. The estimation of theforce at the attached point on the vehicle can be employed di-rectly to compensate for the perturbation and give more capa-bility of maneuvering even in the case of teleoperation from theship. Moreover, it can be included in the structure of high-per-formance controllers to attain path following and regulation au-tomatically without the intervention of an operator.

Fig. 17. Thrust evolution in the regulation with adaptive controller.

Fig. 18. Thrust evolution in the regulation with fixed controller.

In this paper, high-performance control systems based onspeed gradient are developed in a general structure that can


embrace an adaptive and fixed controller as well. Their per-formance features in steady state were compared and stabilityproperties proved rigorous using concepts of total stabilitywhen cable force acts as perturbation without its compensationby the cable model. The stability results are valid also whenadaptive control is turned off.

Theoretical and simulation results show that in the case nocable model is embedded in the control, adaptive controllerscan damp induced nonlinear oscillation more effectively thanthe fixed controller in stationary state, and this occurs generallyat a lower cost of control energy without causing thrust satura-tion. This feature is quite important in autonomous vehicles. Ad-ditionally, employing the adaptive control systems, no knowl-edge of the hydrodynamics and vehicle parameters is required,achieving a high-performance quickly. It is seen that nonlinearoscillations of low periods can appear when the catenary stiff-ness is high (i.e., cable spire is small). The design parametersto tune in both the adaptive and fixed controllers are almost thesame with additional gain matrices for the adaptive laws thatregulate the time of transients during adaptation. The featuresin both control cases were illustrated by numerical simulationsin a case study in six degrees of freedom.

APPENDIX

Proof of Theorem 1: Letand . Additionally, let and be boundedand possibly piecewise continuous. With existing and bounded

and in (31) being bounded for and for anattraction domain and

, one sees that the functions on the right-hand side ofin (30) and in (31) are uniformly bounded

on . Then, by the theorem of Picard and Lindelöf(cf., [22]), one concludes that there exist unique solutions in

and that are Lipschitz continuous for.

For the fixed controller, we choose a candidate Lyapunovfunction . Thus, with (48) in (47), with fully com-pensated through , it is valid that

(80)

Taking into account that , , and are positive–definitematrices, one gets

(81)

where are eigenvalues of the matrix indicated in paren-theses and is the identity matrix. Accordingly, as is nonin-creasing, one achieves for a particular error path

(82)

with being a positive constant given by the rational value in(81).

One infers from (81) that is also a decreasing functionfor every .

Clearly, as is well defined for , thenthe limit of (82) does exist for . Moreover, is uni-formly continuous and bounded on , so one can invoke thelemma of Barbalat (see, for instance, [21]) to show thattends to zero for . Consequently, it yields that the errortrajectories and converge to zero asymptotically. Fi-nally, using (42)

as well. This result proves asymptotic convergence of the errortrajectories and to zero.

Now, for the adaptive controller, we choose the candidateLyapunov function

(83)and the column vector of , and analogously, thecolumn vector of every matrix defined as

with (84)

(85)

with (86)

(87)

(88)

(89)

according to the constant matrices given in (32)–(35).In the following, consider with (51) in (47). Thus

(90)

As is chosen globally convex in any compact convexset in the space of all elements of the ’s, it is valid that for anypairs of controller matrices

(91)for any , , , and

.Hence, for the particular choice of pairs in (91) with

in (84)–(89), from (52)–(57)and the ’s inserted in (48) with fully compensated forthrough , one achieves from (90) and (91)

(92)

Thus, using in (51) and (49), one attains in the error space

(93)


and so together with (45), (47) can be further bounded, similarlyto (81), so that it is valid for every error path that

(94)With the same arguments as used previously, the error trajec-

tories and vanish asymptotically in time.To demonstrate the boundness of the variables in the

closed loops, first, we state from the previous results that, and with (45) and the boundness of

and , one infers that the thrust in (48) is also bounded.For the adaptive control, the boundness of ensures

and . Thus, from (52)–(57),it follows:

for (95)

Hence, it is concluded also in the adaptive case that the termsare also bounded.

Proof of Theorem 2: Let us consider the Lyapunov functionin (83) with for the fixed controller first. Thus

(96)

Using (45), it is valid that

(97)Moreover, as and are uniformly continuous, and thesolutions and are Lipschitz continuous, then there existpositive constants and such that

and (98)

are valid for some domain , where isa positive real constant, and .

As is a decreasing function for the equilibrium pointin the unperturbed system (cf., Theorem 1), there exist

three functions of class (see, for instance, [22]), referred toas , , and , such thatand in for all . Considering (43) and(44), we have

(99)

where and are the time derivatives of evaluated alongthe solutions of the perturbed and unperturbed fixed control sys-tems, respectively. Given any , there exist a and a

such that , and with and in (21)bounded, one gets from (36)

(100)

for and . Let

(101)

For all solutions and with, entering the region ,

for some time , one has

(102)

Choosing , where ,for all time , so that

and . Hence,the equilibrium point described by is totally stablefor the fixed control.

Now, for the adaptive control, the procedure is somewhat dif-ferent. Accordingly, we consider the candidate Lyapunov func-tion in (83). The main difference is shown from (91)for the particular choice of pairs in (84)–(89) and thefeedback with the adaptive laws (52)–(57). So it is valid with(92) that

(103)

Since is bounded (as established in the first part ofthe theorem), there exist bounded partial derivatives and

for all and for some domain, with .

As is a decreasing function for the equilibriumpoint in the unperturbed system (cf., The-orem 1), there exist two functions of class , referredto as , and , such that

and in for all . Considering (43) and(44), we have

(104)

where and are the time derivatives of evaluated alongthe solutions of the perturbed and unperturbed fixed control sys-tems, respectively. Given any , there exist a and asuch that , and similarly as before, one gets

(105)

for and . Let

(106)


For all solutions and with, entering the region

and for some time , one has

(107)

Choosing , where ,for all time , so that

and . Hence, the equilibrium point de-scribed by is totally stable also for the adaptivecontrol.

Proof of Theorem 3: Let us consider the Lyapunov function(83) denoted by for the adaptive controller and for in (48)without compensating . Then, it is valid that

(108)Next, by the attainability condition (91), one gets (92) for theperturbed case in the form

(109)

(110)

On the other hand, let in (45) be the Lyapunovfunction for the fixed controller. Similarly to , one achieves

(111)

(112)

with given in (93).Finally, one can see that for an invariant set

for , , and and continuous bounded referencesand , in both the adaptive and fixed cases, there exists an

invariant set where it is valid that andoutside it, and so every bounded solution starting in

converges asymptotically to for . Clearly,this invariant is given in (68) and can be calculated from (110)and (111) separately.

For the proof of ii), let us compare (109) and (111) for anycontinuous bounded references and , and every initial con-ditions with common . So, one attains outside

(113)

for all . Thus, due to the influence of arbitrarily chosenin the adaptive control behavior, one could not ensure a

better convergence of the error paths in the adaptive control overthose in the fixed control during transient periods. However, ingeneral, the error trajectories in the adaptive case enter moredeeply in the residual set for for any arbitraryand common than in the fixed control case. This fact en-sures a better stationary performance for the adaptive controlsystem over the fixed control one.

Remark 2: It is clear from the comparison between (80)and (90) that if the integrals of the speed-gradient control laws

(52)–(57) are frozen at constant values , one concludes by(91) that the same asymptotic properties would be achieved forthis control system under identical path references and initialconditions as in the fixed control case.

Remark 3: Theorem 2 asserts that the solutions of the per-turbed fixed and adaptive control systems are small if the initialconditions and as well as the perturbation are suf-ficiently small. As the cable perturbation depends mainly on thecurrent and cable shape given in (19), consequently,the bounds for the tracking errors will increase with the strengthof the flow and catenary stiffness but they will remain bounded.The role played by the cable shape in the strength of the pertur-bation will be pointed out in Section VI.

Remark 4: It is worth noticing from (68) that if bothand are chosen relatively large, then the boundary of theresidual set will be correspondingly small, and consequently, fora given , the norms and will be advantageouslyreduced in this way. Moreover, the transient performance in theadaptive control can be improved by using known techniquesto ensure such as by initialization of thetrajectories [24].

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Mario Alberto Jordán received the Dipl.-Ing.degree in electro-mechanical engineering withorientation in control systems from the NationalUniversity of San Juan (UNSJ), San Juan, Argentina,in 1983, and the Dr.-Ing. degree in control systemsfrom the Institute of Control at the Technical Uni-versity of Darmstadt (TUHH), Darmstadt, Germany,in 1990.

From 1992 to 1994, he held a fellowship ofthe Foundation for Industrial Research, Cologne,Germany, for pursuing postdoctoral work. He was

an Assistant Professor at the Institute of Control Systems (INAUT-UNSJ),San Juan, Argentina, from 1992 to 1994, and an Associated Professor at theNational University of South (UNS), Bahía Blanca, Argentina, from 1995to 2008. He has been a Research Fellow of the Council for Scientific andTechnological Research (CONICET), Argentina, since 1995. He has authoredmore than 150 publications and participated in a textbook entitled AdaptiveControl Systems (New York: Prentice-Hall, 1992). His main interests areapplied nonlinear and adaptive control theory, and systems for autonomousguidance and vision-based control of underwater vehicles.

Jorge Luis Bustamante received the professionaldegree in electronic engineering from the Univer-sidad Nacional del Sur, Bahía Blanca, Argentina, in2002, where he has been working towards the Ph.D.degree in control systems, under the direction ofDr. M. A. Jordán. Currently, his Ph.D. dissertation“Adaptive control of autonomous and teleoperatedunderwater vehicles under perturbations” is underfinal examination. The research was developedat the Argentinean Institute of Oceanography(IADO-CONICET).

His research areas include adaptive control, underwater vehicles, system iden-tification, and optimal control.

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