design of an adaptive nonlinear controller for depth control of an autonomous underwater vehicle

Design of an adaptive nonlinear controller for depth

control of an autonomous underwater vehicle

Ji-Hong Li*, Pan-Mook Lee

Ocean Development System Division, Korea Research Institute of Ships & Ocean Engineering,

KORDI, 171 Jangdong, Yuseong-gu, Daejeon 305-343, South Korea

Received 7 September 2004; accepted 16 February 2005

Available online 13 June 2005

Abstract

This paper presents an adaptive nonlinear controller for diving control of an autonomous

underwater vehicle (AUV). So far, diving dynamics of an AUV has often been derived under various

assumptions on the motion of the vehicle. Typically, the pitch angle of AUV has been assumed to be

small in the diving plane. However, these kinds of assumptions may induce large modeling errors

and further may cause severe problems in many practical applications. In this paper, through a

certain simple modification, we break the above restricting condition on the vehicle’s pitch angle in

diving motion so that the vehicle could take free pitch motion. Proposed adaptive nonlinear

controller is designed by using a traditional backstepping method. Finally, certain numerical studies

are presented to illustrate the effectiveness of proposed control scheme, and some practical features

of the control law are also discussed.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Adaptive nonlinear control; Backstepping method; Diving dynamics; URVs; AUV

1. Introduction

In recent years, underwater robotic vehicles (URVs) have become an intense area of

oceanic research because of their emerging applications, such as deep sea inspections,

long range survey, oceanographic mapping, underwater pipelines tracking, and so on.

However, URVs’ dynamics are highly nonlinear and the hydrodynamic coefficients of

Ocean Engineering 32 (2005) 2165–2181

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0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2005.02.012

* Corresponding author. Tel.: C82 42 868 7507; fax: C82 42 868 7503.

E-mail address: [email protected] (J.-H. Li).

http://www.elsevier.com/locate/oceaneng

J.-H. Li, P.-M. Lee / Ocean Engineering 32 (2005) 2165–21812166

vehicles are difficult to be accurately estimated a priori, because of the variations of these

coefficients with different operating conditions. Therefore, conventional controllers such

as PID controller may not be able to handle these difficulties promptly and may cause poor

performance. For this reason, control systems of URVs need to have the capacities of

learning and adapting to the variations of the dynamics and hydrodynamic coefficients of

vehicles in order to provide desired performance (Choi and Yuh, 1996).

So far, AUV’s diving behavior has often been reduced to a certain multivariable linear

system such as in Healey and Lienard (1993), Fossen (1994) and Prestero (2001), where

two main assumptions were made on the AUV’s dynamics. One assumption is that the

pitch angle of the vehicle is small in diving behavior, and the other one is that the pitch

motion dynamics could be expressed as a certain linear equation. These two assumptions

are somewhat strong restricting conditions and may cause severe results in many practical

applications because of large modeling inaccuracy. In this paper, the AUV’s diving

dynamics is taken as a certain SISO system with the stern plane angle taken as control

input and the depth of the vehicle as output, and furthermore, above two restricting

conditions are broken so that the vehicle could take any pitch angle and the nonlinear

dynamics in pitch motion is assumed to be unknown. If the pitch angle is assumed to be

small, then the vehicle’s depth motion could be approximated by certain linear form of

pitch angle and the resulting diving equation could be expressed in a strict-feedback form

which could be resolved by using adaptive backstepping method (Krstic et al., 1995).

Adaptive backstepping, which was firstly developed by Kanellakopoulos et al. (1991), has

been a powerful design tool of adaptive controller for a class of nonlinear systems such as

strict-feedback systems, pure-feedback systems, and so on (Polycarpou, 1996; Yao and

Tomizuka, 2001). Furthermore, it influenced further developments in adaptive nonlinear

control (Marino, 1997; Marino and Tomei, 1877). Despite of these advances in adaptive

backstepping methodology, there are some difficulties still remain to directly apply this

kind of design method to many practical applications. For the depth control problem of an

AUV, if the pitch angle does not satisfy the small value assumption, then the AUV’s

diving equation could not be written properly in neither the strict-feedback form nor any

other form in Krstic et al. (1995), therefore, any design method introduced in Krstic et al.

(1995) could not be directly applied to.

Our solution to this problem is very simple. We expand the Maclaurin series of the

trigonometric term (sine of pitch angle) in the depth dynamics around a certain stabilizing

function (Krstic et al., 1995) instead of zero value. Using this simple modification, the

diving dynamics of vehicle could be written in the strict-feedback form without any

restricting assumption on the vehicle’s pitch angle, and a stable adaptive nonlinear

controller for depth control of an AUV is proposed by using Lyapunov-based

backstepping method. Under the various assumptions on the unstructured uncertainties

(Slotine and Li, 1991), asymptotic tracking performances are discussed and some practical

features of proposed control law are also presented.

The remainder of this paper is organized as following. AUV’s diving equation is

derived and some assumptions are made in Section 2. In Section 3, a stable adaptive

nonlinear controller for depth control of an AUV is presented and some practical features

of the control law are also discussed. In order to demonstrate the effectiveness of proposed

J.-H. Li, P.-M. Lee / Ocean Engineering 32 (2005) 2165–2181 2167

control scheme, certain numerical studies are presented in Section 4. Finally, in Section 5,

we make a brief conclusion on the paper.

2. Problem statements

Dynamical behavior of an AUV can be described in a common way through six degree-

of-freedom (DOF) nonlinear equations in the two coordinate frames as indicated in Fig. 1

(Fossen, 1994)

MðnÞ _n CCDðnÞn CgðhÞCd Z t; _h1 Z J1ðh2Þn1; _h2 Z J2ðh2Þn2; (1)

where hZ ½hT1 ;h

T2 �

T with h1Z[x,y,z]T and h2Z[f,q,j]T is the position and orientation

vector in earth-fixed frame, nZ vT1 ; v

T2

� �Twith v1Z[u,v,w]T and v2Z[p,q,r]T is the velocity

and angular rate vector in body-fixed frame, M(n)2R6!6 is the inertia matrix (including

added mass), CD(n)2R6!6 is the matrix of Coriolis, centripetal and damping term,

g(h)2R6 is the gravitational forces and moments vector, d denotes the exogenous

disturbance term vector, t is the input torque vector, and the transformation matrices

J1(h2) and J2(h2) are as following

J1ðh2Þ Z

cðjÞcðqÞ KsðjÞcðfÞCcðjÞsðqÞsðfÞ sðjÞsðfÞCcðjÞcðfÞsðqÞ

sðjÞcðqÞ cðjÞcðfÞCsðfÞsðqÞsðjÞ KcðjÞsðfÞCsðqÞsðjÞcðfÞ

KsðfÞ cðqÞsðfÞ cðqÞcðfÞ

2664

3775;

J2ðh2Þ Z

1 sðfÞtðqÞ cðfÞtðqÞ

0 cðfÞ KsðfÞ

0 sðfÞ=cðqÞ cðfÞ=cðqÞ

2664

3775; ð2Þ

where s($)Zsin($), c($)Zcos($) and t($)Ztan($).

In general, most small underwater vehicles are designed to have symmetric structures.

And it is usually reasonable to assume that the body fixed coordinate is located at the

center of gravity with neutral buoyancy. Furthermore, for AUVs, whose shape could be

depicted as in Fig. 1 that having one propeller and two stern planes and two rudders, the

sway and heave velocities could be neglected. And in the diving plane, we assume that the

roll and yaw angle velocities are also could be neglected. This can be acquired by properly

adjusting the RPM of propeller and the rudders’ angles. Under these assumptions, the

heave dynamics of AUVs could be expressed as

_z ZKu sin q Cv cos q sin f Cw cos q cos fzKu sin qzKu0 sin q; (3)

where u0O0 is a forward constant speed, and the pitch kinematics could be written as

_q Z q cos f Kr sin fzq cos f: (4)

Since pz0, roll angle f is nearly constant. Without any loss of generality, we assume that

fz0. Therefore, Eq. (4) could be rewritten as

_qzq: (5)

Fig. 1. Body-fixed frame and earth-fixed reference frame for AUV.


Consequently, the diving equation of an AUV could be expressed as following, which is a

certain modified expression from Fossen (1994) and Prestero (2001)

_z ZKu0 sin q; _q Z q; mq _q Z fq CFsu20ds Cdq; (6)

where mq is the inertia term including added mass, fqð _v; v; hÞincludes the cross added mass

terms, Coriolis, centripetal and damping terms, and gravitational force and moment terms,

Fs is the fin moment coefficient and ds is the stern plane angle as depicted in Fig. 2(b), and

dq denotes the disturbance terms, all of which are in the pitch motion behavior,

respectively.

Remark 1. In Fossen (1994) and Prestero (2001), fqð _v; v;hÞ is expressed as a certain

linearization form of v. However, this kind of assumption may cause severe result in

practical applications because of large modeling inaccuracy. In this paper, we approximate

the nonlinear function fq directly by using certain on-line adaptation scheme.

In fact, from a physics point of view, nonlinear dynamics fq, which includes inertia

terms, Coriolis, centripetal and damping terms, gravitational force and moment terms, is

Fig. 2. Effective fin angle of attack. (a) Rudder angle, (b) stern plane angle.


difficult to be exactly expressed in mathematical equation form. However, if fq satisfies to

be a smooth function, then it can always be written in the following parametric form

(Girosi and Poggio, 1990)

fqð _v; v;hÞ ZXN

iZ1

q*i f*

i ð _v; v;hÞ; (7)

where f*i , iZ1,.,n are the ‘basis functions’ of fq, q�i ,iZ1,.,n are constants values.

How to construct the basis function vector for a given unknown smooth function has

become one of the main issues in the research area of functional approximation theory (or

system modeling), and still remains open. In other words, while approximate a certain

given unknown function in practice, we could not exactly know the basis functions of it a

priori, therefore, there always remains certain mismatching. On the other hands, some

basis functions (or system dynamical modes) may be neglected in the practical

applications for the purpose of computational convenience. Some others are even

neglected unwillingly because of certain practical restrictions (Johansson, 1993).

Consequently, Eq. (7) should be modified as following

fqð _v; v;hÞ ZXn

iZ1

qifið _v; v;hÞC3ð _v; v; hÞ; (8)

where the first term is the practical model of fq, and 3($) is corresponding modeling error.

For more details about how to construct the basis function vector of a given function,

please refer to Lewis et al. (1998) and Li et al. (2002).

According to Eq. (8), Eq. (6) could be rewritten as following

_z ZKu0 sin q; _q Z q; mq _q Z FTQ CFsu20ds C3 Cdq; (9)

where FZ[f1, f2,.,fn]T, QZ[q1,q2,.,qn]T.

It is obvious that Eq. (9) could not be directly resolved without any restricting

conditions on it. For this reason, we make the following assumptions.

Assumption 1. The inertia term mq is strictly positive and taking a known constant value.

Remark 2. In general, most of AUVs are designed to move slowly in deep-sea

environment. Furthermore, the vehicles are often desired to keep constant forward speed

in diving motion. In this case, added mass term in mq varies slowly and this makes the

Assumption 1 be reasonable.

Assumption 2. Unstructured uncertainty term in Eq. (9) satisfies j3ð _v; v;hÞCdqðtÞj%cq,

ct2RC with cqR0 known constant.

The main focus of this paper is taken on the direct solution to the nonlinear depth

dynamics without any restricting assumption on the AUV’s pitch angle during diving

process, not on the robustness of proposed controller to various unstructured uncertainties.

In fact, robustness has become one of the most important issues related to nonlinear

control problems, and considerable interests have been taken in to guarantee the stabilities

of the proposed nonlinear controllers under various assumptions on the unstructured


uncertainties (Jiang and Praly, 1998; Marino and Tomei, 1877; Arslan and Basar, 2001; Li

et al., 2004). In Li et al. (2004), a neural network adaptive controller was proposed for

autonomous diving control of an AUV with the depth dynamics reduced to _zZu0qCDZ ,

where the unstructured uncertainties were assumed to be unknown and unbounded, though

they still satisfy certain growth conditions characterized by ‘bounding functions’

composed of known function multiplied by unknown constants. In this paper, robustness

is out of the main interest and the focus is taken on the solution to the nonlinear depth

dynamics and the estimation performance of proposed adaptive scheme. For the purpose

of simplifications, in this paper, all the unstructured uncertainties are assumed to be

bounded by known constants.

3. Adaptive nonlinear diving control design

For a given desired trajectory zd, the control objective of this paper is to design a stable

adaptive nonlinear controller for Eq. (9) with asymptotic tracking performance by using

Lyapunov-based backstepping method. By defining new error variables x1ZzKzd, x2ZqKa1(x1,t), and x3ZqKa2(x2,t), where a1($) and a2($) are stabilizing functions (Krstic et

al., 1995), Eq. (9) can be rewritten as following

_x1 ZK_zd Ku0 sinðx2 Ca1Þ; _x2 ZK_a1 Ca2 Cx3;

mq _x3 ZKmq _a2 CFTQ CFsu20ds C3 Cdq:

(10)

Now, the tracking problem for Eq. (9) is transferred to the regulating problem for Eq. (10).

Step 1

Consider the first equation in Eq. (10)

_x1 ZK_zd Ku0 sinðx2 Ca1Þ: (11)

Here we use the following Maclaurin expansion of sine term in above equation

sinðx2 Ca1Þ Z sin a1 Ccos a1x2 Ksin a1

x22

2!Kcos a1

x32

3!C.: (12)

By neglecting the higher-order terms o(x2) in Eq. (12), Eq. (11) could be expressed as

following linearization form of x2

_x1 ZK_zd Ku0 sin a1 Ku0 cos a1x2: (13)

Assumption 3. The desired trajectory satisfies inequality j_zdðtÞj!u0, ct2RC.

Lemma 1. Consider the system expressed as Eq. (13) with Assumption 3. If x2Z0, then

the following control law

a1 Z arcsin satK_zd Ck1x1

u0

�� ; (14)

where k1R0 is a certain design parameter, and


satðxÞ Z

1; xO1

x; K1!x%1

K1; x%K1

8><>: (15)

could guarantee the tracking error x1 to asymptotically converge to zero.

Proof. Substituting Eq. (14) and x2Z0 into Eq. (13), we have

_x1 ZK_zd Ku0 satðbÞ; (16)

where bZ ðK_zd Ck1x1Þ=u0. Since u0O0, k1R0, and j_zdj!u0, saturation function in Eq.

(16) can be expressed as

satðbÞ Zsgnðx1Þ; jbjR1

b; jbj!1:

((17)

Consider the following Lyapunov function candidate

V1 Z1

2x2

1: (18)

Differentiating Eq. (18) and substituting (16) into it, we get

_V1 Z x1 _x1 ZK_zdx1 Ku0x1 satðbÞ: (19)

If bj jR1, according to Eq. (17) and Assumption 3, Eq. (19) can be expressed as

_V1 Z x1 _x1 ZK_zdx1 Ku0x1 sgnðx1Þ

ZK_zdx1 Ku0jx1j! j_zdjjx1jKu0jx1j!0: (20)

If bj j!1, then

_V1 Z x1 _x1 Z x1ðK_zd C _zd Kk1x1Þ ZKk1x21%0: (21)

Consequently, _V1%0, cx12R, therefore, the tracking error x1 always asymptotically

converges to zero.,

Using Eqs. (13) and (14), Eq. (19) can be rewritten as

_V1 Z x1 _x1 ZK_zdx1 Ku0x1 satðbÞKu0x1x2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 K ½satðbÞ�2

p: (22)

Step 2

Rewrite the second equation in Eq. (10)

_x2 ZK_a1 Ca2 Cx3: (23)

Here we choose the stabilizing function as following

a2 Z _a1 Cu0x1


pKk2x2; (24)

where k2R0 is a certain design parameter.


Remark 3. From (14), it is obvious that a1 is a piecewise smooth function. In order to

calculate _a1 at its jump points jbjZ1� �

, we make following definition,

_a1jjbjZ1 Z _a1jjbjO1Z0.

Substituting above equation into Eq. (23), we have

_x2 Z u0x1


pKk2x2 Cx3: (25)

Consider the following Lyapunov function candidate

V2 Z V1 C1

2x2

2: (26)

Differentiating Eq. (26) and substituting Eqs. (22) and (25) into it, we have

_V2 Z _V1 Cx2 _x2 ZK_zdx1 Ku0x1 satðbÞKu0x1x2


pCu0x1x2


pKk2x2

2 Cx2x3

ZK_zdx1 Ku0x1 satðbÞKk2x22 Cx2x3: (27)

Step 3

This is the final step and the actual control input would be derived. Consider the final

equation in Eq. (10)

mq _x3 ZKmq _a2 CFTQ CFsu

20ds C3 Cdq: (28)

Lemma 2. Consider the depth equation of an AUV expressed as Eq. (6) with Assumptions

1–3. Choose the control law as

ds Z ðFsu20Þ

K1½mq _a2 Kx2 Kk3x3 KFTQ̂ Kcq sgnðx3Þ�; (29)

where k3R0 is a certain design parameter, the stabilizing functions a2 and a1 are taken as

Eqs. (24) and (14), and Q̂ is the estimation of the unknown constant vector Q expressed in

Eq. (9). Furthermore, the adaptation law is chosen as

_̂Q Z x3F: (30)

Then, the tracking errors x1, x2 and x3 all asymptotically converge to zeroes.

Proof. Substituting Eq. (29) into Eq. (28), we get

mq _x3 ZKx2 Kk3x3 CFT ~Q C3 Cdq Kcq sgnðx3Þ; (31)

where ~QZQKQ̂.

Here we consider the following Lyapunov function candidate

V3 Z V2 C1

2mqx2

3 C1

2~Q

T ~Q: (32)

Differentiating Eq. (32) and substituting Eqs. (27) and (31) into it, we have


_V3 Z _V2 Cx3mq _x3 C ~QT _~Q

ZK_zdx1 Ku0x1 satðbÞKk2x22 Cx2x3 Kx2x3 Kk3x2

3 Cx3FT ~Q C ~QT _~Q

Cx3ð3 CdqÞKcqx3 sgnðx3Þ%K_zdx1 Ku0x1 satðbÞKk2x22 Kk3x2

3

Cx3FT ~Q K ~QTx3F C jx3jj3 CdqjKcqjx3j%K_zdx1 Ku0x1 satðbÞ

Kk2x22 Kk3x2

3: (33)

According to Lemma 1, the following inequality holds for cx1, x2, x32RC and ct2R

C

_V3%0: (34)

Consequently, the tracking errors x1, x2 and x3 all asymptotically converge to zeroes.,

Remark 4. From Eqs. (14) and (24), we can see that the stabilizing functions a1 and a2 are

all smooth known functions, and therefore, the time derivatives of them could be

computed directly without using any differentiator. This is also a key feature of

backstepping introduced in Krstic et al. (1995). However, in many practical applications,

system dynamics always include certain unstructured uncertainties such as exogenous

disturbances and measurement noises. In this case, the time derivatives of stabilizing

functions could not be directly computed. How to deal with this kind of problem is in the

outside of this work and introduced in Li et al. (2004).

Remark 5. It is well known that if the known function vector F satisfies the persistency

excitation condition, then the adaptation law expressed as Eq. (30) could result in the exact

estimation of Q. However, the persistency excitation conditions are difficult to satisfy in

many practical applications. In this case, the adaptation law Eq. (30) should be made a

certain modification to avoid the divergence of estimation of Q (Li et al., 2004).

4. Numerical studies

In this section, we validate the proposed control laws expressed as Eqs. (29), (24) and

(14) by simulating them on a six degree-of-freedom nonlinear dynamical model of ASUM

AUV, which is under development in KRISO (Lee et al., 2003). The diving behavior of

ASUM AUV can be expressed as

_x1 ZK_zd Ku0 sinðx2 Ca1Þ; _x2 ZK_a1 Ca2 Cx3;

3:495 _x3 ZK3:495 _a2 CFTQ C6:541u20ds C3 Cdq;

(35)

where


F Z ½ _u; _w; uq; vp; vr;w;wjwj;wq; q; qjqj; rp; sin q; cos q cos f�T;

Q Z ½q1; q2;.; q13�T:

(36)

In Section 2, while derive the diving dynamics of an AUV, we use the assumptions that the

vehicle’s forward speed is constant and the roll and yaw angle velocities are close to

zeroes. To realize these assumptions, forward speed and steering control should be

considered firstly.

4.1. Decoupled forward speed and steering control

Forward speed dynamics of ASUM AUV could be expressed as

_u Z fuð _v; v;hÞC0:048tu; (37)

where fuð _v; v; hÞ could be expressed as a similar form as Eq. (36), and estimated by certain

adaptation scheme. However, for the purpose of simplification, in this paper we assume

that fu($) is exactly known a priori. Using new error variable euZuKu0 with u0 forward

constant speed, rewrite the Eq. (37) as following

_eu Z fuð _v; v;hÞC0:048tu: (38)

Then, the forward speed control law is selected as

tu Z 20:665½Kfuð _v; v;hÞKkueu�; (39)

where kuR0 is a certain design parameter. It is obvious that the above control law can

guarantee the tracking error eu to asymptotically converge to zero.

Similar to above discussion, for yaw dynamics of ASUM AUV described by

_r Z frð _v; v;hÞC0:987u20dr; (40)

where fr($) is exactly known priori, we choose the steering control law as

dr Z ð0:987u20Þ

K1½Kfrð _v; v;hÞKkrr�; (41)

where krR is a certain design parameter. Also, Eq. (41) can guarantee the yaw angle

velocity to asymptotically converge to zero.

4.2. Proposed adaptive nonlinear diving control design

In this simulation, we assume that the unstructured uncertainty in Eq. (35) is 3CdqZ3K4 rand ($), where rand ($)2[0, 1] is a random noise. Here we select the constant, which

is defined in Assumption 2, as cqZ5. According to Lemma 2, the control law of proposed

adaptive nonlinear diving control is chosen as

ds Z ð6:541u20Þ

K1½3:495 _a2 Kx2 Kk3x3 KFTQ̂ K3 sgnðx3Þ�; (42)

where a2 and a1 are defined in Eqs. (24) and (14). Furthermore, we take the following

design parameters k1Z2, k2Z8, k3Z120 in this simulation.


Consequently, the final control input torque t defined in Eq. (1) is taken as

t Z ½tu; Yuuru2dr; Zuusu

2ds; 0;Muusu2ds;Nuuru

2dr�; (43)

where the fin lift coefficients are taken as YuurZ13.917, ZuusZK13.917, MuusZ6.651,

NuurZK6.541, and the propeller torque in the roll motion is neglected.

4.3. Simulation results

The main focus of this paper is taken on the direct solution to the nonlinear depth

dynamics without any restricting assumption on the AUV’s pitch angle during diving

process. In order to illustrate the advantage of proposed control scheme, we compare the

control performance with a case where the depth dynamics is linearized as _zZKu0q. In

this case, the diving equation cab be expressed as

Fig. 3. Tracking error x1ZzKzd in the case zZKu0q. (a) For desired trajectory 1, (b) for desired trajectory 2, (c)

for desired trajectory 3.


_z ZKu0q; _q Z q; mq _q Z fq CFsu20ds Cdq; (44)

which is in the strict-feedback form and the backstepping method introduced in Krstic

et al. (1995) could be directly applied. Corresponding control law is chosen as Eq. (42),

where the stabilizing functions a2 and a1 are selected as

a1 Z uK10 ðK_zd Ck1x1Þ; (45)

a2 Z _a1 Cu0x1 Kk2x2; (46)

and all design parameters are chosen as Section 4.1–4.2.

Fig. 4. Tracking error x1ZzKzd in the case zZKu0 sin q. (a) For desired trajectory 1, (b) for desired trajectory 2,

(c) for desired trajectory 3.


Effectiveness comparisons of above two cases are performed for three different desired

trajectories as following

Desired trajectory 1 : zd Z 10; Desired trajectory 2 : zd

Z 10 C5 sin 0:1t; Desired trajectory 3 : zd Z 10 C5 sin 0:3t: (47)

Simulation results are depicted in Figs. 3–7. Fig. 3 shows the tracking errors for above

three different desired trajectories with _zZKu0q, while Fig. 4 is for the cases with

_zZKu0 sin q. In the simulations for the first two desired trajectory 1 and 2 in Eq. (47), the

vehicle’s pitch angle is small enough (%G208 shown in Fig. 5) so that there is not notable

difference in tracking performance between above two cases _zZKu0q and _zZKu0 sin q.

However, for the desired trajectory 3 in Eq. (47), the pitch angle of vehicle is near to G908

Fig. 5. Vehicle’s pitch angle in the case zZKu0 sin q. (a) For desired trajectory 1, (b) for desired trajectory 2, (c)

for desired trajectory 3.

Fig. 6. Vehicle’s motion in the diving plane. (a) For the case zZKu0q, (b) for the case zZKu0 sin q.


in some sections (Fig. 5). In this case, the linearization of sin qzq induces large modeling

error and therefore may cause severe problems such as the divergence of the control

system (Fig. 6(a)), while the proposed scheme can guarantee the stability of the same

dynamical system (Fig. 6(b)). These observations coincide with the objective of proposed

control scheme discussed in previous sections.

Fig. 7. Control inputs (stern plane angle) in the case zZKu0 sin q. (a) For desired trajectory 1, (b) for desired

trajectory 2, (c) for desired trajectory 3.


According to the physical property of an AUV, the fin angle could not be taken as any

free value. In other words, the fin angle is bounded in a section [Kp/2,Cp/2]. In the

simulation studies, we saturate the stern plane angle asKp/3%ds%p/3, and

corresponding control inputs (stern plane angle) are shown in Fig. 7.

As discussed in Remark 4, if the known function vector F satisfies the persistency

excitation condition, then the adaptation law expressed as Eq. (30) could result in an exact

estimation of unknown constant vector Q. However, this persistency excitation condition

is difficult to be satisfied in many practical applications, typically for the systems such as

underwater vehicles whose dynamics could be expressed as a six DOF nonlinear equation,

it is hard to plan a desired trajectory such that the function vector F to be persistently

exciting. In order to avoid the divergence of estimation of Q, we modify the adaptation law

Eq. (30) as_̂

QZx3FCaðQ̂KQ0Þ with aR0 and Q0 certain design parameters (Li et al.,


2004), which are taken as aZ3, Q0Z0.5Qtrue with Qtrue known vector directly calculated

by coefficients of ASUM AUV.

5. Conclusions

This paper presents an adaptive nonlinear control scheme for diving dynamics of an

AUV without any restricting condition on the vehicle’s pitch angle in the diving plane.

Different from the existing method where the Maclaurin series of the trigonometric term in

the depth dynamics of an AUV has been expanded around zero value, we expand the same

series around a certain stabilizing function, which can get any value. Using this kind of

modification, we can break the small value assumption on the vehicle’s pitch angle made

in the existing method, further the diving dynamics of vehicle could be written in the strict-

feedback form so that a traditional backstepping method could be directly applied to.

Some numerical studies are also presented. From these simulations, we can see the

proposed scheme is more effective than the existing method, typically in the case where

the vehicle’s pitch angle takes large value.

For the future development of proposed scheme, the persistency excitation conditions

for the vehicle model will have to be analyzed and reflected in the design of controller.

Acknowledgements

This work was supported by the Ministry of Maritime Affairs and Fisheries in Korea.

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