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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

* Corresponding author. Tel.: C82 42 868 7507; fax: C82 42 868 7503. E-mail address: [email protected] (J.-H. Li).

0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.02.012

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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

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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 degreeof-freedom (DOF) nonlinear equations in the two coordinate frames as indicated in Fig. 1 (Fossen, 1994) MðnÞn_ C CD ðnÞn C gðhÞ C d Z t; hZ ½hT1 ; hT2 T

h_ 1 Z J1 ðh2 Þn1 ;

T

h_ 2 Z J2 ðh2 Þn2 ;

(1)

T

where with h1Z[x,y,z] Tand h2Z[f,q,j] is the position and orientation vector in earth-fixed frame, nZ vT1 ; vT2 with 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 2 3 cðjÞcðqÞ KsðjÞcðfÞ C cðjÞsðqÞsðfÞ sðjÞsðfÞ C cðjÞcðfÞsðqÞ 6 7 KcðjÞsðfÞ C sðqÞsðjÞcðfÞ 7 J1 ðh2 Þ Z 6 4 sðjÞcðqÞ cðjÞcðfÞ C sðfÞsðqÞsðjÞ 5; KsðfÞ cðqÞsðfÞ 3 1 sðfÞtðqÞ cðfÞtðqÞ 6 7 KsðfÞ 7 J2 ðh2 Þ Z 6 4 0 cðfÞ 5; 2

0

cðqÞcðfÞ ð2Þ

sðfÞ=cðqÞ cðfÞ=cðqÞ

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_ Z Ku sin q C v cos q sin f C w cos q cos f zKu sin q zKu0 sin q;

(3)

where u0O0 is a forward constant speed, and the pitch kinematics could be written as q_ Z q cos f K r sin f zq 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 q_ zq:

(5)

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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_ Z Ku0 sin q;

q_ Z q;

mq q_ Z fq C Fs u20 ds C dq ;

(6)

_ v; hÞincludes the cross added mass where mq is the inertia term including added mass, fq ðv; 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. _ v; hÞ is expressed as a certain Remark 1. In Fossen (1994) and Prestero (2001), fq ðv; 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.

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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) _ v; hÞ Z fq ðv;

N X

_ v; hÞ; q*i f*i ðv;

(7)

iZ1

where f*i , iZ1,.,n are the ‘basis functions’ of fq, qi ,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 _ v; hÞ Z fq ðv;

n X

_ v; hÞ C 3ðv; _ v; hÞ; qi fi ðv;

(8)

iZ1

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_ Z Ku0 sin q;

q_ Z q;

mq q_ Z FT Q C Fs u20 ds C 3 C dq ;

(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. _ v; hÞC dq ðtÞj% cq , Assumption 2. Unstructured uncertainty term in Eq. (9) satisfies j3ðv; 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

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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 z_Z u0 qC DZ , 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, x2Z qKa1(x1,t), and x3ZqKa2(x2,t), where a1($) and a2($) are stabilizing functions (Krstic et al., 1995), Eq. (9) can be rewritten as following x_1 Z K_zd K u0 sinðx2 C a1 Þ;

x_2 Z K_ a1 C a2 C x3 ; (10)

mq x_3 Z Kmq a_ 2 C FT Q C Fs u20 ds C 3 C dq :

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) x_1 Z K_zd K u0 sinðx2 C a1 Þ:

(11)

Here we use the following Maclaurin expansion of sine term in above equation sinðx2 C a1 Þ Z sin a1 C cos a1 x2 K sin a1

x22 x3 K cos a1 2 C .: 2! 3!

(12)

By neglecting the higher-order terms o(x2) in Eq. (12), Eq. (11) could be expressed as following linearization form of x2 x_1 Z K_zd K u0 sin a1 K u0 cos a1 x2 :

(13)

Assumption 3. The desired trajectory satisfies inequality j_zd ðtÞj! u0 , ct2R . C

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

K_zd C k1 x1 a1 Z arcsin sat ; (14) u0 where k1R0 is a certain design parameter, and

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8 1; xO 1 > < satðxÞ Z x; K1! x% 1 > : K1; x%K1

2171

(15)

could guarantee the tracking error x1 to asymptotically converge to zero. Proof. Substituting Eq. (14) and x2Z0 into Eq. (13), we have x_1 Z K_zd K u0 satðbÞ;

(16)

where bZ ðK_zd C k1 x1 Þ=u0 . Since u0O0, k1R0, and j_zd j! u0 , saturation function in Eq. (16) can be expressed as ( sgnðx1 Þ; jbjR 1 satðbÞ Z : (17) b; jbj! 1 Consider the following Lyapunov function candidate 1 V1 Z x21 : 2

(18)

Differentiating Eq. (18) and substituting (16) into it, we get V_ 1 Z x1 x_1 Z K_zd x1 K u0 x1 satðbÞ:

(19)

If jbj R 1, according to Eq. (17) and Assumption 3, Eq. (19) can be expressed as V_ 1 Z x1 x_1 Z K_zd x1 K u0 x1 sgnðx1 Þ Z K_zd x1 K u0 jx1 j! j_zd jjx1 j K u0 jx1 j! 0:

(20)

If jbj ! 1, then V_ 1 Z x1 x_1 Z x1 ðK_zd C z_d K k1 x1 Þ Z Kk1 x21 % 0:

(21)

Consequently, V_ 1 % 0, cx12R, therefore, the tracking error x1 always asymptotically converges to zero. , Using Eqs. (13) and (14), Eq. (19) can be rewritten as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V_ 1 Z x1 x_1 Z K_zd x1 K u0 x1 satðbÞ K u0 x1 x2 1 K ½satðbÞ2 :

(22)

Step 2 Rewrite the second equation in Eq. (10) x_2 Z K_ a1 C a2 C x3 : Here we choose the stabilizing function as following pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2 Z a_ 1 C u0 x1 1 K ½satðbÞ2 K k2 x2 ; where k2R0 is a certain design parameter.

(23)

(24)

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Remark 3. From (14), it is obvious thata1 is a piecewise smooth function. In order to calculate a_ 1 at its jump points jbjZ 1 , we make following definition, a_ 1 jjbjZ1 Z a_ 1 jjbjO1 Z 0. Substituting above equation into Eq. (23), we have pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x_2 Z u0 x1 1 K ½satðbÞ2 K k2 x2 C x3 :

(25)

Consider the following Lyapunov function candidate 1 V2 Z V1 C x22 : 2

(26)

Differentiating Eq. (26) and substituting Eqs. (22) and (25) into it, we have pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V_ 2 Z V_ 1 C x2 x_2 Z K_zd x1 K u0 x1 satðbÞ K u0 x1 x2 1 K ½satðbÞ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C u0 x1 x2 1 K ½satðbÞ2 K k2 x22 C x2 x3 Z K_zd x1 K u0 x1 satðbÞ K k2 x22 C x2 x3 :

(27)

Step 3 This is the final step and the actual control input would be derived. Consider the final equation in Eq. (10) mq x_3 Z Kmq a_ 2 C FT Q C Fs u20 ds C 3 C dq :

(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 ðFs u20 ÞK1 ½mq a_ 2 K x2 K k3 x3 K FT Q^ K cq sgnðx3 Þ;

(29)

where k3R0 is a certain design parameter, the stabilizing functions a2 and a1 are taken as Eqs. (24) and (14), and Q^ is the estimation of the unknown constant vector Q expressed in Eq. (9). Furthermore, the adaptation law is chosen as Q_^ Z x3 F:

(30)

Then, the tracking errors x1, x2 and x3 all asymptotically converge to zeroes. Proof. Substituting Eq. (29) into Eq. (28), we get mq x_3 Z Kx2 K k3 x3 C FT Q~ C 3 C dq K cq sgnðx3 Þ;

(31)

~ QK Q. ^ where QZ Here we consider the following Lyapunov function candidate 1 1 T~ V3 Z V2 C mq x23 C Q~ Q: 2 2 Differentiating Eq. (32) and substituting Eqs. (27) and (31) into it, we have

(32)

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T V_ 3 Z V_ 2 C x3 mq x_3 C Q~ Q_~ T Z K_zd x1 K u0 x1 satðbÞ K k2 x22 C x2 x3 K x2 x3 K k3 x23 C x3 FT Q~ C Q~ Q_~

C x3 ð3 C dq Þ K cq x3 sgnðx3 Þ%K_zd x1 K u0 x1 satðbÞ K k2 x22 K k3 x23 T C x3 FT Q~ K Q~ x3 F C jx3 jj3 C dq j K cq jx3 j%K_zd x1 K u0 x1 satðbÞ

K k2 x22 K k3 x23 :

(33)

According to Lemma 1, the following inequality holds for cx1, x2, x32RC and ct2RC V_ 3 % 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 x_1 Z K_zd K u0 sinðx2 C a1 Þ;

x_2 Z K_ a1 C a2 C x3 ;

3:495x_3 Z K3:495a_ 2 C FT Q C 6:541u20 ds C 3 C dq ; where

(35)

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_ w; _ uq; vp; vr; w; wjwj; wq; q; qjqj; rp; sin q; cos q cos fT ; F Z ½u; 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 _ v; hÞ C 0:048tu ; u_ Z fu ðv;

(37)

_ v; hÞ could be expressed as a similar form as Eq. (36), and estimated by certain where fu ðv; 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 _ v; hÞ C 0:048tu : e_u Z fu ðv;

(38)

Then, the forward speed control law is selected as _ v; hÞ K ku eu ; tu Z 20:665½Kfu ðv;

(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 _ v; hÞ C 0:987u20 dr ; r_ Z fr ðv;

(40)

where fr($) is exactly known priori, we choose the steering control law as _ v; hÞ K kr r; dr Z ð0:987u20 ÞK1 ½Kfr ðv;

(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 3CdqZ 3K4 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 ð6:541u20 ÞK1 ½3:495a_ 2 K x2 K k3 x3 K FT Q^ K 3 sgnðx3 Þ;

(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.

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Consequently, the final control input torque t defined in Eq. (1) is taken as t Z ½tu ; Yuur u2 dr ; Zuus u2 ds ; 0; Muus u2 ds ; Nuur u2 dr ;

(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 z_ZKu0 q. 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.

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z_ Z Ku0 q;

q_ Z q;

mq q_ Z fq C Fs u20 ds C dq ;

(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 zd C k1 x1 Þ; a1 Z uK1 0 ðK_

(45)

a2 Z a_ 1 C u0 x1 K k2 x2 ;

(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.

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Effectiveness comparisons of above two cases are performed for three different desired trajectories as following Desired trajectory 1 : zd Z 10; Z 10 C 5 sin 0:1t;

Desired trajectory 2 : zd

Desired trajectory 3 : zd Z 10 C 5 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 z_ ZKu0 q, while Fig. 4 is for the cases with z_ZKu0 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 z_ZKu0 q and z_ZKu0 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.

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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.

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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 _^ x FC aðQK ^ Q0 Þ with aR0 and Q0 certain design parameters (Li et al., Eq. (30) as QZ 3

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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 strictfeedback 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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