1Black-box position and attitude tracking forunderwater vehicles by second-order

sliding-mode techniqueGiorgio BARTOLINI Alessandro PISANO

Abstract: In this work we address the trackingcontrol problems for autonomous underwater ve-hicles (AUVs). The proposed solution is based onthe Variable Structure Systems (VSS) theory, and,in particular, on the second-order sliding-mode (2-SM) methodology. The tuning of the controlleris carried out via black-box approach, dispensingwith the knowledge of the actual AUV parameters,by simply progressively increasing a single gainparameter. The presented stability analysis includesexplicitly the unmodelled actuator dynamics andthe presence of external uncertain disturbances.The good performance of the proposed scheme isverified by means of simulations on a 6-DOF AUV.Keywords: AUV guidance, AUV control, Second-order sliding-modes, Quaternion-feedback.

I. INTRODUCTION

In many operating situations of practical relevance(e.g., surveillance or remote sampling), autonomousunderwater vehicles (AUVs) are expected to feature ac-curate hovering capabilities and/or to follow preciselysome reference trajectories. To achieve this goal, thefollowing systems must be designed and implementedon board: i) navigation, to provide estimates of linearand angular positions and velocities of the vehicle,ii) guidance, to process navigation/inertial referencetrajectory data and output set-points for the vehicles(body) velocity and attitude, and iii) control, to gener-ate the actuation forces and moments required to drivethe actual velocity and attitude of the vehicle to thevalues provided by the guidance system.In this paper we disregard the navigation problemby assuming that precise absolute measurements areprovided by an ideal strapdown system.We concentrate on the control problem which is chal-lenging due to the strongly coupled nonlinear mul-tivariable AUV dynamics and, furthermore, since inpractical scenarios AUVs are affected by (possiblyheavy) parametric and non-parametric uncertainties, aswell as environmental disturbances. A good controller

Type of contribution: Regular paper. The authors are withthe University of Cagliari (Italy), Dept. of Electrical and ElectronicEngineering. E-mails: {giob,pisano}@diee.unica.it. Correspondingauthors is G. Bartolini.

should provide fast and accurate position and attitudetracking despite of the occurrence of unknown distur-bances and without relying on the precise knowledgeof the system parameters. For these reasons, we referto the sliding-mode control (SMC) theory as the can-didate approach to achieve the above specifications.The sliding mode control (SMC) approach was devel-oped in the late fifties [31], and by the end of theseventies it was recognized as one of most promisingrobust control techniques [30].However, the very first implementations showed thatthe real sliding-mode exhibits chattering due to theinteraction between the discontinuous control laws andthe nonidealities of the real control devices such dead-zones, backlashes, or parasitic dynamics of the overallsystem including sensors and actuators [33], [36].The most common method to attenuate the chatteringeffect is to use continuous approximations of the signfunction [26]. This approach allows for the controlcontinuity but cannot restrict the system dynamics ontothe switching surface. It only ensures the convergenceto a boundary layer of the sliding manifold, whose sizeis defined by the slope of the saturation characteristics,and causes oscillations of the control variable that mayexcite the unmodelled system resonances.Second-order sliding mode control (2-SMC) [3] isa more recent class of switching control algorithmswhich can lead to continuous control forces and mo-ments directly, without involving any smoothing func-tion [2]. This approach allows for finite-time conver-gence to zero of not only the so-called sliding variable,but also of its first time derivative. It was activelydeveloped over the last two decades, mainly for SISOsystems, as a major tool for chattering avoidance inSMC [2], [14], [18], [29]. Several successful applica-tions were recently demonstrated [7], [17], [19], [28],[32]. A multi-input version of 2-SMC was proposed in[5]. Here we propose a simpler solution, related to theone proposed in [21], which, unlike in [5], does notmake use of any observer.A basic reference in the field of AUVs guidance andcontrol is the paper [13], by T. Fossen and co-author,which consider a model-based feedback-linearizationapproach and also suggest its parameter-adaptive gen-eralization. Book [12] and the more recent surveypaper [38] provide valuable overviews of many control

2techniques. In [1], an interesting extensive comparisonamong several known controllers was made.Adaptive control (AC) techniques were widely incor-porated into AUVs control systems. Among the vastliterature concerning the application of AC, and itscombination with other robust control approaches, thefollowing works are cited [1], [10], [11], [27]. More re-cent fields of research entail the study of underactuatedAUVs [23] or account for possible actuator failures[25]. In this paper we are interested in the basiccontrol problem, i.e. the AUV is fully actuated and noactuator failures are allowed to occur.The use of sliding-mode control (SMC) techniques wasfound appropriate due to the well known robustnessproperties endowing such an approach. The seminalwork dealing with the application of SMC to under-water vehicles dates back to 1985 [35]. Subsequently,some improvements and further studies were presentedin [15], [11], [16]. In [24] and [9] a single-input2-SMC approach [24] and a multi-input combined1-SMC/2-SMC approach were proposed and testedexperimentally for controlling two jet-actuated marinevehicles prototypes.We consider a rather unconventional unit-quaternionbased implementation for the guidance system, andpropose a position/attitude tracking controller forAUVs based on the second-order sliding-mode ap-proach. Some features are summarized as follows:1. The proposed 2-SMC tracking controller providesthe exponential convergence towards the desired tra-jectory, as well as the complete rejection (more thanattenuation) of smooth disturbances, by means of con-tinuous control forces and moments.2. The parameters of the proposed controller can betuned following a black-box approach guaranteeinginsensitivity to large modeling uncertainties including,parameter uncertainty, environmental disturbances andunmodelled hydrodynamic phenomena.3. The presence of unmodelled actuator dynamics isaccounted for explicitly in the stability analysis

II. SYSTEM MODEL

A. Kinematics

Let us write down the kinematic equations of mo-tion for a rigid body. Let the B-frame and I-framedenote the body-fixed and inertial reference frames,respectively. Let r = [x, y, z]T denote the I-frameposition of the B-frame origin. Let the body attitudebe represented by the unit quaternion

q = [, T ]T , = [1, 2, 3]T (1)

whose elements (the so-called Euler parameters)

satisfy the following relationship

2 + 21 + 22 +

23 = 1. (2)

The terms R and R3 are normally referredto as the scalar and vector parts of quaternion q,respectively. Let us define an operator which extractsthe vector part of a given quaternion:

= vec(q) (3)

The Euler parameters are related to the axis-angleattitude parameters n and as follows:

q = [, T ]T , = cos

(

2

), = n sin

(

2

).

(4)Let = [u, v, w]T and = [p, q, r]T denote thelinear and angular body velocities in the B-frame. Theglobally nonsingular transformation from the B-frameto the I-frame is governed by [12]

r = R(q) (5)q =

1

2U(q) (6)

R(q) = I33 + 2S() + 2S2() (7)

U(q) =

[ TI33 + S()

](8)

with the skew-symmetric matrix S(a) = ST (a), a =[a1, a2, a3] 3, defined as follows:

S(a) =

0 a3 a2a3 0 a1a2 a1 0

(9)

Matrix S(a) is such that, for any vectors a,b 3,the identity a b = S(a)b holds.

B. AUV and actuator Dynamics

The B-frame dynamic model of an AUV is given asfollows [13]:M+C()+D()+g(q) = (q, , , t)+ (10)

where = [T , T ]T is the B-frame velocity vector, = [FT , TT ]T is the vector of actuator forces andmoments.

In the term (q, , , t) = [TF , TT ]T we repre-

sent the collective effect of any phenomenon whichgives rise to additional force and torque disturbances,whichever the complexity of the associated mathemat-ical representation is. The only requirement about thevector (q, , , t) is the existence of an upperboundto its norm and to the norm of its time derivative.M is the inertia matrix modeling the rigid-body andadded mass inertial effects, matrices C() and D()model the Coriolis and Damping effects, and vectorg(q) collects the restoring forces and moments.

3The gravitational and buoyant forces are written W =mg and B = g where is the mass density of waterand is the volume of the displaced water. They actthrough the center of gravity rG = [xG, yG, zG]Tand the center of buoyancy rB = [xB, yB, zB]T ,respectively.Matrix M is the sum of two components, the rigid-body inertia matrix MRB and the added masses inertiamatrix MA:

M =

[M11 M12M21 M22

]=MRB +MA (11)

MRB =

[m0I33 033033 I0

], MA =

[A11 A12A21 A22

](12)

where m0 is the vehicle mass and I0, A11, . . . A22 are

symmetric positive-definite matrices. C(), D() and

g(q) take the following form [12]

C() =

2

6

4

033 S(M11 +M12)

S(M11 +M12) S(M21 +M22)

3

7

5

(13)

D() = DL() +DQ() (14)

DL() = diag {DL1, DL2, . . . , DL6} (15)

DQ() = diag {DQ1|u|,

DQ2|v|, DQ3|w|, DQ4|p|, DQ5|q|, DQ6|r|}

(16)

It is assumed that the force and moment input vector is generated through the unmodelled actuator dynamics

= H + (18)

where H is an Hurwitz matrix and is the actuator in-put. The inertia matrix fulfills the following propertiesfor some constants m and m

M = MT , m M m (19)

The numerical values of the AUV and actuator pa-rameters are fairly known in practice and are assumeduncertain in the present treatment.

III. GUIDANCE EQUATIONS

Let rd be the I-frame desired profile for the vehicleposition and nd and d the desired axis-angle attitudeparameters. By (4), the desired unit quaternion is then

qd = [d, Td ]T , d = cos

(d2

), d = nd sin

(d2

).

(20)

d is evaluated by reversing (5) with r = rd, yielding:d = R

T (q)rd (21)

In order to derive d, let us first recall the definitionof the quaternion product between two generic unitquaternions q1 = [1, T1 ]T and q2 = [2, T2 ]T asfollows

q1 q2 =[1 1T1 1I33 + S(1)

] [22

](22)

The desired velocity profile d can be computedthrough the following relationship (see the appendix)

d = 2 vec(qd qd) (23)

where the vec operator extracts the vector part ofthe unit quaternion qdqd in accordance with (3). Theupper bar denotes the quaternion complex conjugate,obtained by changing the sign of the quaternion vectorpart:

qd = [d, Td ]

T = qd = [d,Td ]T (24)

In the next subsections, we derive reference velocityprofiles r and r guaranteeing the annihilation ofsuitably-defined attitude and position error variables.

A. Attitude control

Let qd represent the desired attitude profile. Consid-ering (7), and noticing that the relationship R(q) =R(q) holds, it follows that the unit quaternionsqd and qd both represent the same AUV attitudeconfiguration. This means that both conditions q = qdand q = qd are satisfactory in the steady state. Thisis a useful degree of freedom, since different transientbehaviours may occur while the actual attitude vectorq is driven either to qd or to qd, as investigated inthe simulation Section. An approach (see e.g. [13]) isbased on defining the attitude error as the differencebetween the actual and desired unit quaternion

Eq = q qd (25)

Considering (6), the corresponding attitude error dy-namics can be written as

Eq =1

2U(q) qd (26)

In order to steer to zero the attitude error Eq, thefollowing reference velocity profile can be used [13]r = 2U

T (q) [qd KpI44Eq] , Kp > 0 (27)

Using formula (27) one always obtains that q qd,thus the degree of freedom previously identified (i.e.,the alternative of driving q either to qd or to qd) isnot exploited. Note also that the attitude error Eq isno longer a unit quaternion.

4g(q) =

2(2 13)(W B)

2(1 23)(W B)

(2 + 21 + 32 23)(W B)

(2 + 21 + 32 23)(yGW yBB) + 2(1 + 23)(zGW zBB)

(2 + 21 + 32 23)(xGW xBB) + 2(2 13)(zGW zBB)

2(1 + 23)(xGW xBB) 2(2 13)(yGW yBB)

(17)

An alternative approach is to define the attitude errorvariable as follows

q = qd q (28)

Denote the real and vector part of q as and ,respectively, i.e.

q = [, T ]T , T = [1, 2, 3].(29)

The error variable (28) can be rewritten as:

q =

[d d

T

d dI33 S(d)]q (30)

Since the quaternion product is a closed operator inthe unit quaternion space, vector q is still a unitquaternion. This implies that the following conditionholds

2 + 21 + 22 +

23 = 1 (31)

The real part of q is:

= d + Td (32)

It is easy to note, considering (32) and (31), thatq = qd = 1 = 031. (33)q = qd = 1 = 031.(34)

The attitude tracking control problem can be thenreduced to the simple attainment of the scalar condition|| = 1, i.e., either = 1 or = 1. The timederivative of q, by elementary computation, is asfollows:

q = [, T ]T (35)

= 12T ( d) (36)

=1

2I33( d) + 1

2S(T )( + d)(37)

Consider the following scalar quantity

= + k sign()[|| 1], k > 0 (38)

with k being an arbitrary positive tuning coefficientand sign(0) set to 1 by convention. It will be shownthat the attainment of condition = 0 implies that|| 1 exponentially.Therefore, since = (), it makes sense to considerthe reference profile r such that (r) = 0. Substi-tuting (36) into (38) yields

() = 12T (d)+k sign()[||1] (39)

On () = 0,

T (r d) = 2k sign()[1 ||] (40)

System (40) has infinite solutions in the unknown r.The minimum-norm solution turns out to be

r = d [T ]12k sign()[1 ||] (41)

Since T = 21+ 22+ 23 = 1 2, relationship(41) can be simplified further as follows:

r = d 2k sign()[1||]12 == d 2k sign()[1||][1+||][1||] == d 2k sign()[1+||]

(42)

Eq. (42) represent a singularity-free reference profilefor the velocity vector guaranteeing the annihilationof quantity in (38).Proposition 1. Let = r = d 2k sign()[1+||] .Then, condition = sign((0)) is achieved expo-nentially, which corresponds to the attainment of thedesired attitude.Proof of Proposition 1. By construction, (r) = 0.This implies, by (38), that

= k sign()[|| 1], k > 0. (43)

Since || 1 we can consider the Lyapunov can-didate function V = 1 ||, whose time derivativeis

V = sign() = kV (44)

5Eq. (44) guarantees the global exponential convergenceof V to zero, i.e. of || to 1. Furthermore, || isalways increasing, which implies that will alwaystend to sign((0)). Conditions (33) and (34), in turns,imply the attainment of the desired attitude. The sign of at the the initial time determineswhether will tend to 1 or to 1. This means that theclosest steady state value is always achieved. Thisproperty is investigated in the simulation section.

B. Position control

Let rd represent the desired attitude profile. The posi-tion error is defined as follows

er = r rd (45)

Proposition 2. Let = r, with the reference profiler given as follows

r = d RT (q)er , > 0 (46)Then the position error er tend to zero exponentially.Proof of Proposition 2Consider the time derivative of er, er = r rd, whichcan be rewritten as follows by (5) and (21):

er = R(q)[ d] (47)

Substituting for r in (47) one achieves the expo-nentially stable position error dynamics er = er.

IV. THE 2-SM TRACKING CONTROLLER

The six-dimensional sliding vector is defined as fol-lows

=

[

]= r (48)

with r = [rT , rT ]T containing the referencevelocity profiles given in the Propositions 1 and 2, and

= r = [u, v, w]T = r = [p, q, r]T (49)

Differentiating (48), and considering (10), it yields = r = M1C() M1D()

M1g(q) +M1 r +M1 ==M1

[1(, r,q,) +

](50)

with the vector field

1(, r,) = C() D() + g(q)+

+(q, , , t) Mr

(51)

considered uncertain in the present treatment.Differentiating further (50)-(51), and considering (10),one obtains after some manipulation the second-ordersliding variable dynamics where the actuator vectorinput explicitly appears

= r =M1[2(, , , r, ,q,) +

](52)

2(, , , r, ,q,) =d

dt1(, , r,,q, t)

(53)In any bounded domain of the system variables, aslarge as one desires, the uncertain vector field 2 isnorm-bounded by a sufficiently large constant

2(, , , r, ,q,)1 (54)

For the stabilization of system (52)-(53) satisfying (54)the following algorithm is proposed

(t) = a sign() b sign() (55)

where the bold sign function, with a vector as ar-gument, is understood component-wise and a, b arepositive constants satisfying the following inequalities

b > a > b+ (56)

Clearly, controller tuning is simply accomplished bytaking a sufficiently large value of into the aboveinequalities.The implementation of the controller (55)-(56) steersvectors and to zero in finite time along withtheir first time derivatives. According to Propositions1 and 2, the enforced conditions = r and = rguarantee that the position and attitude errors tend tozero exponentially. The above result is summarized inthe following Theorem 1. The proof is basically anapplication of the extended invariance principle [22],and, in particular, the presented result is a generaliza-tion of Theorem 4.2 in [21], where the multivariabledecoupled problem was dealt with in detail (with refer-ence to the present application, this would correspondto studying the special case when M is diagonal).Theorem 1 Consider the AUV dynamics (10), (18),with uncertain parameters. Let rd and qd representthe desired position and attitude parameters. Computethe reference velocities r and r as follows

r = d RT (q)er , > 0 (57)r = d 2k sign()

[1 + ||] , k > 0(58)[

, T]T

= qd q (59)

Define the sliding vector according to (48)-(49), letcondition (54) hold and apply the controller (55)-(56).

6Then, the sliding vector and its derivative are steeredto zero in finite time which implies that the desiredposition and attitude are achieved exponentially.Proof of Theorem 1.Let us define some quantities that shall be used withinthe proof. Let x R6 be a generic vector. Define

x1 =6i=1

|xi|, x2 =[

6i=1

x2i

]1/2(60)

It is known that

x2 x1 6x2 (61)

With reference to the symmetric positive definite in-ertia matrix M = {mij}, i, j = 1, 2, . . . , 6, denoteas i (i = 1, 2, . . . , 6) its eigenvalues. By definition,i R+, then it makes sense to define its maximaland minimal eigenvalues

M = supi{i}, M = inf

i{i} (62)

and the maximal entry of M as follows

m = supi,j=1,...,6

{mij} (63)

The proof, which is an application of the extendedinvariance principle [22], is broken in two steps.1. Equiuniform stabilityLet us introduce the Lyapunov function

V =1

2TM + a1 (64)

the time derivative of V along the trajectories ofthe uncertain system (52)-(55) is negative semidefiniteaccording to the following sequence of relationships:

V = T [2() + ] + aT sign() == T [2() a sign() b sign()]+

+a T sign() =

= T2() b1 (b)1

(65)It follows that the uncertain system (52)-(55) is equiu-niform stable, i.e., due to (65), the positive definitefunction (64) does not increase along the solutionsof (52)-(55). Thus, once initialized in an arbitrary -vicinity

D ={(, ) : V (, )

}(66)

of the origin, the uncertain system cannot leave suchvicinity.2. Semiglobal Lyapunov stabilityWe now construct a parameterized family of localLyapunov functions VR(, ), R > 0, such that

each VR(, ) is well defined on the correspondingcompact set

DR ={(, ) : V (, ) R

}(67)

More clearly, we require that VR is positive definiteon DR\{(0, 0)}, whereas its time derivative, computedalong the trajectories of the uncertain system (52)-(55)with initial conditions within DR, is to be negativedefinite for all (, ) DR\{(0, 0)}.A parameterized family of Lyapunov functions withthe properties above can constructed as follows,by augmenting V with the sign-indefinite termU(, ) = kR

TM, where kR is an appropriateconstant. Thus define

VR = V +U(, ) =1

2TM+a1+kRTM

(68)with the weight parameter kR small enough accordingto

kR < min

{M6m

,a2

3mR,

M (b )M

2R

}(69)

By Lemma 1, (68) can be further manipulated asVR 12 TM + a1 3kRm

[Ra 1+

+ 1M

TM]=

=(

12 3kRm

M

)TM +

(a 3kRmRa

) 1(70)

The restrictions (69) on kR clearly guarantee that thefamily of functions VR thus constructed is positivedefinite in the domain DR\{(0, 0)}.Now let us compute the derivative VR = V + U alongthe trajectories of the uncertain system (52)-(55).U(, ) = kR

TM + kR

T [2()a sign() b sign()]

(71)Define vector

z = 2() a sign() b sign() = col(zi) (72)

Since, according to (56), a + b > , then by con-sidering (54) it is straightforward to express the i-th entry of z as zi = i(t)sign(i), where i(t)(i = 1, 2, .., 6) are strictly positive functions such thati(t) a b. It is therefore immediate to derivethat

T z [a b ]1 (73)

Now considering (65), (71) and (73) it can be writtenthat

VR (b)1 + kRTM [a b]1(74)

7By considering Lemma 2 it is possible to furthermanipulate (74) as

VR [b kRM

2R

M

]1[ab]1

(75)Thus taking into account once more the restrictions(69) on kR it is concluded that VR is negative definitefor all (, ) DR\{(0, 0)}.Because of constant R can be taken arbitrarily largeand function VR is radially unbounded it follows thatthe uncertain system (52)-(54) is semiglobally stableat the origin.

REMARK 1. Controller tuning. The controllerdepends on two constant parameters, a and b, thatmust satisfy the conditions (56). In order to find apair of admissible controller parameters one might tryto make a worst-case analysis of the system uncertaindynamics devoted to overestimate the constant . Thisprocedure, thought feasible, would give rise to a con-servative overestimation, and, in turns, to unnecessarilylarge values of the control parameters. This woulddecrease the actual control system performance. Asusually done in the actual implementation of VSCwith sliding modes [37], it is more effective andappropriate to exploit a black-box approach, i.e. to tunethe controller by means of the formula (56) startingfrom a reasonably small value of and progressivelyincreasing it until satisfactory system behaviour isobserved.

REMARK 2. Evaluation of sign() The controller(55) can be effectively implemented without requiringthe knowledge of . Indeed the componentwise signof can be well approximate by the sign of thedifference between the current and past sample of (first-difference approximation)

sign (k) sign (k k1) , > 0 0(76)

where xk denotes the value of the vector signal x atthe current sampling instant t = kTs. This operationdoes not involve any division by small numbers, henceit is more reliable than, e.g., standard approximatedifferentiation via the backward difference method,which is actually not necessary.

V. SIMULATIONS

The proposed controller has been simulated for theunderwater vehicle in 6 DOF (10)-(16). The model

matrices are taken as follows

M =

215 80 100 30 25 2080 265 70 20 10 30100 70 265 15 10 2030 20 15 40 30 3525 10 10 30 80 4020 30 20 35 40 80

(77)

DL() = diag {70, 100, 100, 30, 50, 50} (78)

DQ() = diag {100|u|, 200|v|, 200|w|, 50|p|, 100|q|, 100|r|}

(79)

rG = [0.2, 0.2, 0.2], rB = [0.15, 0.15, 0.15] (80)

W = 1850N B = 1800N (81)

The considered actuator dynamics is a decoupled seriesof first-order filters with time constants = 2s, i.e.

H = 2I66 (82)

The vehicle is initially at rest in the zero posi-tion/attitude configuration, i.e., (0) = [0, 0, 0]T ,r(0) = [0, 0, 0]T , q(0) = [1, 0, 0, 0]T . Simulationshave been done in the Matlab-Simulink environment.The fixed-step ODE5 integration method has beenused, with step Ts = 0.2ms. The controller parametersare b = 1000, a = 1200, = 2, k = 2.

The three components of the force disturbance vectorF are taken coincident, and the same has been donefor the components of the torque disturbance vectorT . The first entry of vectors F and T are reportedin the left and right Figure 1.

0 5 10

0

20

40

60

Time [sec]

The force disturbance F

0 5 100

50

100

Time [sec]

The torque disturbance T [N m]

Fig. 1. First entries of the force and torque disturbance vectors.

During the first simulation tests, a problem of attituderegulation is considered. The desired position and unit-quaternion attitude are

rd = [0, 0, 0]T qd = [

2/2, 0, 0,

2/2]T . (83)

The desired attitude qd corresponds to the axis/angleparameters d = (3/2) and nd = [0, 0, 1]T . There-fore, the desired attitude correspond to the vehiclebeing rotated of an angle (3/2) around the z axis (the

8desired yaw angle is (3/2), the desired roll and pitchangles are both zero). The final position can be reachedby rotating either clockwise or counterclockwise . Theclockwise angular path is shorter, and requires lesscontrol effort.The angular velocity reference profiles (27) (withKP = 2) and the proposed one (58) were comparedin the TEST 1 and TEST 2. During TEST 1 thewrong (i.e. longest) rotation is selected as shownin the Figs. 2. This is a consequence of the fact thatby using (27) one always steers q to qd . In TEST 2,whose results are depicted in the Figs. 3, we observethat the proposed profile (58) selects automatically theshortest transient path (q is actually driven to qd).The comparison between the Figures 1 and 2 showsthat by using formula (27) the body velocities aresmaller in magnitude.

0 5 10 15 20-100

0

100

200

300

Time [sec]

Angl

e [d

eg]

0 5 10 15 20-500

0

500

1000

1500

Time [sec]

Velo

city

[d

eg / s

ec]

0 5 10 15 20-1.5

-1

-0.5

0

0.5

1

1.5

Time [sec] 0 5 10 15 20-1

-0.5

0

0.5

1

1.5

2

Time [sec]

Yaw, Pitch and Roll angles [deg] B-frame velocities p, q, r [deg/sec]

Actual unit quaternion q=[ 1 2 3 ] Quaternion error q-qd

Yaw angle

r

3

Fig. 2. TEST 1. Attitude regulation using the reference profile (27)

0 5 10 15 20-1

-0.5

0

0.5

1

Time [sec]

0 5 10 15 20-800

-600

-400

-200

0

200

Time [sec]

Velo

city

[ra

d / s

ec]

0 5 10 15 20-100

-80

-60

-40

-20

0

20

Time [sec]

An

gle

[d

eg]

Yaw, Pitch and Roll angles [deg] B-frame velocities p, q, r [deg/sec]

Actual unit quaternion q=[ 1 2 3 ] Quaternion error q

Yaw angle

r

3

0 5 10 15 20-1.5

-1

-0.5

0

0.5

1

Time [sec]

1 2 3

3

Fig. 3. TEST 2. Attitude regulation using the reference profile (58)

Only the proposed velocity profile (58) will be im-plemented from this point on. In TEST 3 a po-sition/attitude tracking problem is considered. Thedesired attitude profile has a constant axis nd =[1/3, 1/

3, 1/3]T while the desired angle d

and position vector rd are time varying and reported in

the next Figure 4 together with the desired quaternionqd.

0 5 100

50

100

150 The desired angle d [deg]

Time [sec] 0 5 100.5

0

0.5

1

Time [sec]

The desired unit quaternion qdd

1d,3d

4d

0 2 4 6 8 100

1

2

3

4

Time [sec]

The desired position vector rd [m]

zd

yd

xd

Fig. 4. The desired attitude and position profiles in TEST 3

The results are reported in the Figures 5. An almost-instantaneous convergence to the desired profile isachieved. Note that the control torques and forcesare continuous functions of time. The results confirmthat the proposed sliding-mode controller is able tocompletely reject smooth disturbances.

0 5 10 15 200.999

0.9995

1

1.0005

1.001

Time [sec]

Quaternion error component

Position error r-rd

0 5 10 15 20-0.06

-0.04

-0.02

0

0.02

Time [sec]

Posi

tion

er

ror

[m]

Control forces [N]

0 5 10 15 20-1000

-500

0

500

1000

Time [sec]

Con

tro

l fo

rces

[N

]

Control torques [Nm]

0 5 10 15 20-800

-600

-400

-200

0

200

400

Time [sec]

Con

tro

l mo

men

ts [N

m]

Fig. 5. TEST3: position and attitude errors; control forces and

moments

In the final TEST 4, all AUV parameters have beenrandomly changed up to 30% of their nominal values(this modification was made by multiplying each modelparameters for a proper coefficient randomly chosenin the interval [0.7, 1.3]). The position and attitudereference profiles are the same as in the previous TEST3. The position and attitude errors, and the controlforces and moments are depicted in the followingFigure 6, showing that satisfactory performance ispreserved.

9

0 5 10 15 200.999

0.9995

1

1.0005

1.001

Time [sec]

Quaternion error component

Position error r-rd

0 5 10 15 20-0.08

-0.06

-0.04

-0.02

0

0.02

Time [sec]

Pos

itio

n e

rro

r [m

]

10 12 14 16 18 20-5

0

5x 10-4

Zoom

Control forces [N] Control torques [Nm]

0 5 10 15 20-1000

-500

0

500

1000

Time [sec]

Con

tro

l fo

rce

s [N

]

0 5 10 15 20-800

-600

-400

-200

0

200

400

Time [sec]

Con

tro

l mo

me

nts

[N

m]

Fig. 6. TEST4: position and attitude errors; control forces and

moments.

VI. CONCLUSIONS

A multi-input second-order sliding mode controlscheme for uncertain systems has been developed andapplied to solve the position and attitude trackingproblems for uncertain AUVs.The robustness to unmeasurable matching disturbanceshas been proved theoretically and demonstrated bysimulations.Important features of the proposed method are thehigh tracking accuracy and simple calibration andimplementation procedures.

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[2] G. Bartolini, A. Ferrara, and E. Usai, Chattering avoidanceby second order sliding mode control, IEEE Trans. AutomaticControl,vol. 43, n. 2, pp.241-246, 1998.

[3] G. Bartolini, A. Ferrara, A. Levant, E. Usai, On Second OrderSliding Mode Controllers, in Variable Structure Systems, Slid-ing Mode and Nonlinear Control, K.D. Young and U. Ozgunereds., Lecture Notes in Control and Information Sciences, vol.247, pp. 329350, Springer-Verlag, 1999.

[4] G. Bartolini, A. Pisano, E. Usai Variable Structure Controlof Nonlinear Sampled Data Systems by Second Order SlidingModes, in Variable Structure Systems, Sliding Mode andNonlinear Control, K.D. Young and U. Ozguner eds., LectureNotes in Control and Information Sciences, Springer-Verlag, vol.247, pp. 43-68, 1999.

[5] G. Bartolini, A. Ferrara, E. Usai, V.I. Utkin On multi-inputchattering-free second order sliding mode control, IEEE Trans.Aut. Contr., vol. 45, no. 9, pag. 1711-1717, (2000).

[6] G. Bartolini, A. Pisano, E. Usai Global Stabilization for Non-linear Uncertain Systems With Unmodeled Actuator Dynamics,IEEE Trans. Aut. Contr., 46, 11, 1826-1832 (2001).

[7] Bartolini G., Pisano A., Punta A., Usai E. A survey of ap-plications of second-order sliding mode control to mechanicalsystems, Int. J. of Control, 76, n. 9/10, pp. 875-892, 2003.

[8] I. Boiko, L. Fridman, A. Pisano and E. Usai PerformanceAnalysis of Second-Order Sliding-Mode Control Systems withFast Actuators, IEEE Trans. on Aut. Contr., to appear in themarch 2007 issue.

[9] G. Bartolini, N. Orani, A. Pisano, E. Punta and E. Usai A com-bined first-/second-order sliding-mode technique in the controlof a jet-propelled vehicle Int. J. Rob. Nonl. Contr., XXXXXX,2007.

[10] G. Conte and A. Serrani Robust nonlinear motion control forAUVs, IEEE Robotics & Automation Magazine, 1, pp. 3338,1999.

[11] M. L. Corradini and G. Orlando, A discrete adaptive variable-structure controller for MIMO systems, and its application to anunderwater ROV, IEEE Trans. Contr. Syst. Technol., vol. 5, pp.349359, 1997.

[12] T.I. Fossen, Guidance and Control of Ocean Vehicles, Wiley,UK; 1995.

[13] O.E. Fjellstad and T.I. Fossen Position and attitude trackingof AUVs: a quaternion feedback approach, IEEE Journal ofOceanic Engineering, vol. 19, n. 9, pp. 512518, 1994.

[14] L. Fridman and A. Levant Higher OrderSliding Modes as aNatural Phenomenon in Control Theory, in Robust Control viaVariable Structure & Lyapunov Techniques (Lecture Notes inControl and Information Science), F. Garofalo, L. Glielmo eds.,Springer-Verlag, London, vol. 217, pp. 107-133, (1996).

[15] A. J. Healey and D. Lienard, Multivariable sliding mode con-trol for autonomous diving and steering of unmanned underwatervehicles, IEEE J. Ocean. Eng., vol. 18, pp. 327339, 1993.

[16] P.M. Lee, S.W. Hong, Y.K. Lim, C.M. Lee, B.H. Jeon, J.W. ParkDiscrete-Time Quasi-Sliding Mode Control of an AutonomousUnderwater Vehicle, IEEE J. Oceanic Eng, vol. 24, n. 3, pp.388-395, 1999.

[17] A. Levant, A. Pridor, R. Gitizadeh, I. Yaesh, J.-Z. Ben-AsherAircraft Pitch Control via Second Order Sliding Technique IAAJournal of Guidance, Control and Dynamics, vol. 23, n. 4, pp.586-594, 2000.

[18] A. Levant, Sliding order and sliding accuracy in sliding modecontrol, Int. J. Control,vol. 58, pp.1247-1263, 1993.

[19] Y. Orlov, L. Aguilar, J. C. Cadiou Switched chattering controlvs. backlash/friction phenomena in electrical servo-motors, Int.J. Control, vol.76 ,no. 9/10, pp.959-967, 2003.

[20] Y. Orlov, Finite Time Stability of Homogeneous Switched Sys-tems, Proc. 2003 CDC, paper ThP05-2, Maui, US, December2003.

[21] Y. Orlov, Finite Time Stability and Robust Control Synthesisof Uncertain Switched Systems, SIAM J. Control Optim., Vol.43, No. 4, pp. 12531271, 2005.

[22] Y. Orlov, Extended Invariance Principle for NonautonomousSwitched Systems, IEEE Trans. Aut. Contr., vol. 48, No. 8, pp.14481452, 2003.

[23] K. Y. Pettersen and O. Egeland, Time-Varying ExponentialStabilization of the Position and Attitude of an UnderactuatedAutonomous Underwater Vehicle, IEEE Trans. Aut. Contr., vol.44, n. 1, pp. 112-115, 1999.

[24] A. Pisano and E.Usai, Output-feedback control of an underwa-ter vehicle prototype by higher-order sliding modes Automatica,Vol. 40, pp.1525-1531, 2004.

[25] N. Sarkar, T. K. Podder and G. Antonelli, Fault-Accommodating Thruster Force Allocation of an AUV Consider-ing Thruster Redundancy and Saturation, IEEE Trans. Robotics& Automation, vol. 18, n. 2, pp. 223-233, 2002.

[26] J.J. Slotine, W. Li, Applied Nonlinear Control, Prentice Hall,Englewood Cliffs, NJ.,1991.

[27] D. A. Smallwood and L. L. Whitcomb Model-Based DynamicPositioning of Underwater Robotic Vehicles: Theory and Exper-iment IEEE J. of Oceanic Engineering, vol. 29, n. 1, 2004.

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10

Control in Dynamic Sliding Manifolds , Int. J. Control, Vol. 76,n. 9/10, pp.1000-1017 2003.

[29] Y. B. Shtessel, I. A. Shkolnikov, M.D.J. Brown. An asymptoticsecond-order smooth sliding mode control, Asian J. of Con-trol,vol. 5 , no. 4, pp.498-5043, 2003.

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[32] V. Utkin, J.Guldner, and J Shi, Sliding Modes in Electrome-chanical Systems, Taylor and Francis, London,1999.

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[34] J. T.-Y. Wen and K. Kreutz-Delgado, The Attitude ControlProblem, IEEE Trans. Aut. Contr. VOL. 36, NO. 10, pp. 1148-1162, 1991.

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[36] K.D. Young, V.I. Utkin, and U. Ozzguner, A Control Engi-neers Guide to Sliding Mode Control, IEEE Trans. Contr. Syst.Technol., vol. 7, n. 3, pp. 328342, 1999.

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APPENDIX

Lemma 1 The following relationship is true when(, ) DR

TM 3m[R

a1 + 1

MTM

](84)

Proof of Lemma 1. By definition we have

TM =

i,j=1,...,6

mijij (85)

with i and j being the i-th and j-th entry of and respectively. A trivial inequality establishes thatgiven any two real numbers x, y it can be written thatxy 12 (x2 + y2). On the basis of such inequality,and taking into account that mij 0 it follows that

TM 12

i,j=1,...,6 mij(2i +

2j)

12m

i,j=1,...,6(2i +

2j ) =

= 3m [22 + 22](86)

Within DR one has that

V =1

2TM + a1 R a1 R (87)

The last term of (87) implies that 1 R/a. Nowtaking into account (61) we get

22 21 R

a1 (88)

For the last term in (86), from the known formulaM22 TM M22 (89)

it derives that

22 1

MTM (90)

Now considering (88) and (90) into (86) it followsdirectly the claimed inequality (84).Lemma 2 The following relationship is true when(, ) DR

TM M

2R

M1 (91)

Proof of Lemma 2.Within DR one has that

V =1

2TM + a1 R 1

2TM R

(92)The last term of (92), together with (89), implies that

22 2R/M 2

2R

M(93)

Again exploiting (89), and considering (93), it can bewritten the following chain of inequalities, the last ofwhich proves the Lemma.

TM M22 M

2RM2

M

2RM1

(94)

Derivation of formula (24)From (20), and taking into account that d = dnd,write down the derivative qd in explicit form.

qd = [d, Td ]T (95)

d = 1

2dsin

(d2

)(96)

d =1

2d cos

(d2

)nd + sin

(d2

)nd = (97)

=1

2cos

(d2

)d

In order to derive (97), the condition nd = dnd =dnd nd = 0, derived from Poissons differentiationrule, is exploited. Now let us consider (24) togetherwith (22), which yields

qd qd =[d Tdd dI33 + S(d)

] [dd

](98)

The vector part of the unit quaternion qdqd, saysz, is

z = vec(qd qd) = dd dd S(d)d (99)

11

Since vectors d and d are parallel, then their vec-tor product d d = S(d)d is the null vector:S(d)d = 0, which means that

z = dd dd (100)

Now considering (20), (96) and (97) into (100) it iseasy to derive that

z = vec(qd qd) =1

2d (101)

which substantiates formula (24).