An Adaptive Control Scheme for 6-DOF Control of

3rd International Conference on Intelligent Computational Systems (ICICS'2013) April 29-30, 2013 Singapore

An Adaptive Control Scheme for 6-DOF Control of an AUV Using Saturation Functions Fahimeh Rezazadegan, Khoshnam Shojaei

with matched as well as unmatched uncertainties. It provides flexibility in the choice of Lyapunov functions and stabilizing virtual control signals at each step of the design process for shaping the closed- loop responses. The trajectory planning and tracking control of AUVs in the horizontal plane without parameter uncertainties using backstepping procedure has been attempted [8]. An adaptive backstepping control law has been developed for dive plane control using a single control surface (stern plane) [5]. In [9] a backstepping nonlinear controller with certain parameters without any disturbances was proposed. However, there has been very few works in the literature considering 6 DOF trajectory tracking control of AUVs. In this paper, we consider six DOF model to design a controller for an AUV [9]. In addition, we have assumed that the system parameters are unknown. The desired trajectory does not need to be of a particular type; it can be any kind of sufficiently smooth bounded curve parameterized by time. Infact, in this paper, an adaptive back-stepping controller is proposed using saturation function and parameter estimation done by projection algorithm. Then, suggesting a novel Lyapunov function, stability analysis will be carried out using Lyapunov theory and Barbalat’s lemma. The results verify our control design robustness in the presence of parameter uncertainty while tracking remains satisfactory a high extent.

Abstract— In this paper, the trajectory tracking control of autonomous underwater vehicles (AUVs) in six-degrees-of-freedom (6-DOFs) is analyzed. It is assumed that the system parameters are unknown and vehicle is underactuated. The desired trajectory can be any sufficiently smooth bounded curve parameterized by time even consist of straight line. To guarantee robustness against parameter uncertainty, an adaptive controller based on the Lyapunov’s direct method and the back-stepping technique, is proposed while Control signals are bounded using saturation functions. The nonlinear adaptive control scheme yields asymptotic convergence of the vehicle to the reference trajectory. The stability of the presented control law is proved in the sense of Lyapunov theory and Barbalat’s lemma. Numerical simulations are presented to illustrate the controller’ behavior.

Keywords— Adaptive control, Autonomous underwater vehicle, Back-stepping technique, Lyapunov theory, Saturaion function. I. INTRODUCTION

T

HE ocean covers about 70% of the earth surface and has great effect on the future existence of all human beings, beside the land and aerospace. Underwater robotics is no doubt an important scientific area due to its great applications that vary from scientific research of ocean, inspection of undersea facilities to military operations. Over the last few years, there has been considerable interest in the development of powerful methods for trajectory tracking control of underwater vehicles. Because of the highly nonlinear dynamics and the unpredictable operating environments of AUVs, conventional control schemes such as the PID controller may not be able to provide satisfactory outcomes in relation to the control problems experienced by underwater vehicles. Besides this, it is essential to deal with parametric uncertainties acting on the underwater vehicles in order to obtain a robust autopilot In the past, variety of design techniques based on optimal control, Lyapunov stability theory, feedback linearization, adaptive control, and sliding mode control have been attempted [1], [2], [3], [4], [5], [6]. For the control of nonlinear deterministic and uncertain MIMO systems, often a back-stepping design technique [7] is used. This is a sequential design process applicable to systems

II. VEHICLES MODEL AND CONTROL PROBLEM We adopt the standard notation for the motion equations of an AUV, see [6], [9]. Linear velocity v= [u v w]T consists of surge, sway and heave, angular velocity ω=[p q r]T consists of roll, pitch and yaw rate, and attitude η =[φ θ ψ]T consists of roll, pitch and heading angle. Furthermore, we assume that the center of gravity and the center of buoyancy are located vertically on the O b Z b -axis, in that there are no couplings (offdiagonal terms) in the matrices M, D, and D n (v). The mathematical model of an AUV in 6 DOF can be described as:

η 1 = J 1 (η 2 ) v1 , η 2 = J 2 (η 2 ) v 2 M v = −C (v) v − D (v) v − g (η ) − τ

(1)

η = [η 1 η 2 ] T is the position and orientation vector in the earth-fixed frame, where η 1 = [ x y z ] T and η 2 = [ϕ θ ψ ] T . v = [v1 v 2 ] T is the velocity and angular rate vector in the body-

Fahimeh. Rezazadegan is with the Islamic Azad University of Najaf abad, Esfahan, Iran (phone: 00989131267857; fax: 00983117932066; e-mail: [email protected]). Khoshnam. Shojaei is with the Islamic Azad University of Najaf abad, Esfahan, Iran (e-mail: [email protected]). 67


m11 − m 22 d 66 1 fixed frame, where v1 = [u v w] T and v 2 = [ p q r ] T . The (3) = rd ud vd − qd + τ rd m66 m66 m66 positive definite inertia matrix M = M RB +M A includes the inertia M RB of the vehicle as a rigid body and the added inertia Then using definition (4), the problem of forcing the M A due to the acceleration of the wave. The skew symmetrical matrix C(v) is the matrix of Coriolis and centripetal. The underactuated underwater vehicle, given in (2), to track the hydrodynamic damping term D(v) takes into account the virtual vehicle (3), have been converted to stabilizing problem dissipation of energy due to the friction exerted by the fluid of the system (5).

surrounding AUV. g(η) is the vector of the gravitational forces xe cos(ψ ) cos(θ )( x − xd ) + sin(ψ ) cos(θ )( y − yd ) − sin(θ )( z − zd ) (η 2) and moments. τ is the input torque vector, and J1(η 2), J2= ye = − sin (ψ ) ( x − xd ) + cos (ψ ) ( y − yd ) are the transformation matrices which are related to functions = ze sin(θ ) cos(ψ )( x − xd ) + sin(θ )sin(ψ )( y − yd ) + cos(θ )( z − zd ) of Euler angles. The general mathematical model of an AUV in surge, sway, θ e= θ − θ d , ψ =e ψ −ψ d heave and heading motion ignoring roll motion is: (4) ue = u − ud , ve = v − vd , we = w − wd , qe = q − qd , re = r − rd

x cos (ψ ) cos (θ ) u − sin (ψ ) v + sin (θ ) cos (ψ ) w = = y sin (ψ ) cos (θ ) u + cos (ψ ) v + sin (θ ) sin (ψ ) w − sin (θ ) u + cos (θ ) w z = θ = q , ψ = r / cos(θ )

Error system equations : xe = ue − (cos (θ e) − 1 + cos (θ ) cos (θ d ) (cos (ψ e) − 1)) ud − cos (θ ) sin (ψ e) vd + (sin (θ e) − cos (θ ) sin (θ d ) × (cos (ψ e) − 1)) wd + (rd + re) ye + (qd + qe) ze

1 m 22 m33 d 11 u = vr − wq − u+ τu m11 m11 m11 m11 m11 d 22 v = − ur− v m 22 m 22 m11 d 33 = w uq− w m33 m33

m33 − m11 d 55 ρ g ∆GML sin (θ ) 1 uw− q− + τq m55 m55 m55 m55 m11 − m 22 d 66 1 = r uv− q+ τr m66 m66 m66

y e =+ ve cos (θ d ) sin (ψ e) − 1)) ud − (cos (ψ e) − 1)) vd − sin (θ d ) sin (ψ e) wd − ( xe + tan (θ ) ze) (rd + re) w e = we − (sin (θ e) + sin (θ ) cos (θ d ) (cos (ψ e) − 1)) ud − sin (θ d ) sin (ψ e) vd − (cos (θ e) − 1 + sin (θ ) × sin (θ d ) (cos (ψ e) − 1)) wd + tan (θ ) (rd + re) ye + (qd + qe) xe  θ e = qe ,

= q

ψ e = (re / cos(θ )) + (rd / cos(θ ) cos(θ d )) + (cos(θ d ) (1 − cos(θ e)) (5) + sin (θ d ) sin (θ e))

(2)

By replacing the nonlinear coordinate transformations, which is defined below (obtained from [9]):

The positive constant terms d ii and m ii (i=1, 2, 3, 4, 5) denote the hydrodynamic damping and AUV inertia including added mass in surge, sway, heave, pitch and heading, respectively. The available controls are the surge force τ u , pitch moment τ q and the yaw moment τ r . Since AUVs do not have independent actuators in the sway and heave axes, the vehicle represented by the mathematical model (1) is underactuated. Our objective is to design a controller (τ u , τ q , τ r ) to track the reference trajectory generated by the virtual vehicle model (3).

z1 =ψ e + arcsin (δ 1 ye / xe 2 + ye 2 + ze 2 )

z 2 =θ e − arcsin (δ 2 ze / xe 2 + ye 2 + ze 2 )

(6)

In the system equations (5), Error system equations (7) is achieved: xe = ue − (ϖ 2 − ϖ + cos (θ ) cos (θ d ) (ϖ 1 − ϖ ))ϖ −1ud + + cos (θ ) δ 1vd ϖ −1 ye + (δ 2 ze − cos (θ ) sin (θ d ) (ϖ 1 − ϖ ))ϖ −1wd

= xd cos (ψ d ) cos (θ d ) ud − sin (ψ d ) vd + sin (θ d ) cos (ψ d ) wd

+ (rd + re) ye − (qd + qe) ze + s1

sin (ψ d ) cos (θ d ) ud + cos (ψ d ) vd + sin (θ d )sin (ψ d ) wd

ye =ve − cos (θ d )δ 1 ud ϖ −1 ye − (ϖ 1 − ϖ )ϖ −1vd

− sin (θ d ) ud + cos (θ d ) wd

+ sin (θ d ) δ 1 wd ϖ −1 ye − ( xe + tan (θ ) ze) (rd + re) + s 2

θd = qd , ψ d = rd / cos(θ d )

ze = we − (δ 2 ze + sin (θ ) cos (θ d ) (ϖ 1 − ϖ ))ϖ −1ud + (rd + re) ye

m 22 m33 d 11 1 vd rd − w d qd − ud + τ ud m11 m11 m11 m11 m11 d 22 vd = − ud r d − vd m 22 m 22 m11 d 33 = w d ud qd − wd m33 m33 ud=

+ sin (θ ) δ 1 vd ϖ −1 ye − (ϖ 2 − ϖ + sin (θ ) sin (θ d ) × (ϖ 1 − ϖ ))ϖ −1wd + tan (θ ) (rd + re) ye − (qd + qe) xe + s 3 z1 = (1 − δ 1ϖ 1−1 (cos (θ ) xe + sin (θ ) ze)) (cos (θ )) −1 re

− δ 1ϖ 1−1ϖ −2 xe ye ue + h1 + s 4 z 2 = (1 − δ 2ϖ 2 −1 xe) qe − δ 2 tan (θ )ϖ 2 −1 ye re

m33 − m11 d 55 ρ g ∆GML sin (θ d ) 1 = qd ud w d − qd − + τ qd m55 m55 m55 m55

+ δ 2ϖ 2 −1ϖ −2 xe ze ue + h 2 + s 5 68


ϖ −1 ye − (ϖ 1 − ϖ )ϖ −1vd + sin (θ d ) δ 1 wd ϖ −1 ye) + ze ( we − (δ 2 ze + sin(θ ) cos(θ d ) (ϖ 1 − ϖ )ϖ −1ud + sin(θ )δ 1 vd ϖ −1 ye (8) −(ϖ 2 − ϖ + sin (θ ) sin (θ d ) (ϖ 1 − ϖ ))ϖ −1wd ))))

m 22 m33 d 11 1 vr − wq − u+ τ ud − ud m11 m11 m11 m11 m11 d 22 ve = − ve (ue re + ue rd + ud re) − m 22 m 22 m11 d 33 = w e we (ue qe + ue qd + ud qe) − m33 m33 ue =

III. ADAPTIVE NONLINEAR CONTROL DESIGN 2B

First, the virtual velocity controls of u e , q e and r e are designed to asymptotically stabilize x e , y e , z e , z 1 , z 2 , w e and v e at the origin. Then based on the back-stepping technique, the controls τ u , τ q , τ r will be designed to make the errors between the virtual velocity controls and their actual values exponentially vanished. It is clear that u e enters the v e and w e dynamics, so to simplify the stability analysis, we will design a bounded virtual control of u e . The virtual controls of q e and r e are chosen to stabilize the z 1 and z 2 dynamics. R

m33 − m11 d 55 ρ g ∆GML sin (θ ) 1 = qe ud w d − qd − + τ qd − qd m55 m55 m55 m55 m11 − m 22 d 66 1 (7) re uv− r+ = τ rd − rd m66 m66 m66

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ϖ = 1 + xe 2 + ye 2 + ze 2 , ϖ 1 = 1 + xe 2 + (1 − δ 12 ) ye 2 + ze 2

ϖ2=

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Where :

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Assumption1: The reference signals u d , q d , r d , ud , qd and rd are bounded. There exists a strictly positive constant ud min that ud (t ) ≥ ud min , ∀t ≥ 0 . This condition is much less

1 + xe 2 + ye 2 + (1 − δ 2 2 ) ze 2

6B

s1 = −((cos ( z 2) − 1)ϖ 2 − sin ( z 2)δ 2 ze + cos (θ ) cos (θ d ) ×

((cos ( z1) − 1)ϖ 1 + sin ( z1) δ 1 ye))ϖ −1ud + (sin ( z 2)ϖ 2 + (cos ( z 2) − 1) δ 2 ze − cos (θ ) sin (θ d ) ((cos ( z1) − 1)ϖ 1 +

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restrictive than a persistently exciting condition on the yaw reference velocity. The reference sway and heave velocities satisfy vd (t ) < ud (t ) , wd (t ) < ud (t ) , ∀t ≥ 0 .

sin ( z1) δ 1 ye))ϖ −1wd − cos (θ ) (sin ( z1)ϖ 1

− (cos ( z1) − 1) δ 1 ye)ϖ −1vd

Assumption2: The reference pitch angle satisfies θ d (t ) ≤ 0.5π , ∀t ≥ 0 because of singularity avoidance.

= s 2 cos (θ d ) (sin ( z1)ϖ 1 − (cos ( z1) − 1) δ 1 ye)ϖ −1ud − ((cos ( z1) − 1)ϖ 1 + sin ( z1) δ 1 ye)ϖ −1vd − sin (θ d ) (sin ( z1)ϖ 1

7B

The virtual control errors are defined as: 8B

− (cos ( z1) − 1) δ 1 ye)ϖ −1wd

ue = ue − ued , qe = qe − qed , re = re − red

s3 = −(sin ( z 2)ϖ 2 + (cos ( z 2) − 1) δ 2 ze + sin (θ ) cos (θ d ) ((cos ( z1)

− 1)ϖ 1 + sin ( z1) δ 1 ye))ϖ ud − ((cos ( z 2) − 1)ϖ 2 − sin ( z 2) δ 2 ze + sin (θ ) sin (θ d ) ((cos ( z1) − 1)ϖ 1 + sin ( z1) δ 1 ye))ϖ −1wd − sin (θ ) (sin ( z1)ϖ 1 +

(9)

−1

where ued , qed and red are the virtual velocity controls of u e , q e and r e respectively. Since the standard application of backstepping leads to a complex controller, we have introduced virtual control laws without canceling some known terms using saturation functions: R

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+ (cos ( z1) − 1) δ 1 ye)ϖ −1vd = s 4 δ 1ϖ 1−1 ( s 2 − yeϖ −2 ( xe s1 + ye s 2 + ze s 3)) + (cos (θ )) −1 ×

ϖ −1 ((1 − cos ( z 2))ϖ 2 + sin ( z 2) δ 2 ze + tan (θ d ) (sin ( z 2) + (cos ( z 2) − 1) δ 2 ze))

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tanh(ued ) = −δ 0ϖ −1 xe + (ϖ 2 − ϖ + cos (θ ) cos (θ d ) × 1 − tanh(ue) tanh(ued )

−δ 2ϖ 2 −1 ( s 3 − zeϖ −2 ( xe s1 + ye s 2 + ze s 3)) s5 =

h1 (ϖ rd (1 + cos −1 (θ d ) cos (θ )) − (1 − cos ( z 2))ϖ 2 − sin ( z 2) δ 2 ze = −((cos ( z 2) − 1) + sin ( z 2)) tan (θ d ) δ 2 ze) (cos (θ )) −1ϖ −1 + δ 1ϖ −1 (ve − cos (θ d ) δ 1 ud ϖ −1 ye − (ϖ 1 − ϖ )ϖ −1 vd + sin (θ d ) δ 1 wd ϖ −1 ye − ( xe + tan (θ ) ze) rd − yeϖ −2 ( xe (−(ϖ 2 −

(ϖ 1 − ϖ ))ϖ −1ud − cos (θ ) δ 1 vd ϖ −1 ye − (δ 2 ze − cos (θ ) sin (θ d ) × (ϖ 1 − ϖ ))ϖ −1wd , qed = q1de + q2de , red = r1de + r2de , tanh(r1de ) (1 − δ 1ϖ 1−1 (cos (θ ) xe + sin (θ ) ze) )

(1 − tanh(r ) tanh(r ) )(1 − tanh(r

d 1e

) tanh(r2de ) )

=

 δ 1ϖ 1−1ϖ −2 xe ye tanh(ued )  − h1  cos (θ )  d 1 tanh( u ) tanh( u ) e − e  

(δ 2 ze − cos(θ ) sin(θ d ) (ϖ 1 − ϖ ))ϖ wd )) + ye (ve − cos(θ d ) × −1

tanh(r2de ) (1 − δ 1ϖ 1−1 (cos (θ ) xe + sin (θ ) ze) )

δ 1 ud ϖ −1 ye − (ϖ 1 − ϖ )ϖ −1vd + sin(θ d ) δ 1 wd ϖ −1 ye) + ze ( we −

= (−c1 z1 − s 4) cos (θ )

(1 − tanh(r ) tanh(r ) )(1 − tanh(r ) tanh(r ) ) tanh (q ) (1 − δ ϖ x ) = (1 − tanh (q ) tanh (q ) )(1 − tanh (q ) tanh (q ) )

(δ 2 ze + sin(θ ) cos(θ d ) (ϖ 1 − ϖ ))ϖ ud + sin(θ ) δ 1 vd ϖ ye − −1

d 1e

e

ϖ + cos (θ ) cos (θ d ) (ϖ 1 − ϖ ))ϖ −1ud + cos (θ ) δ 1 vd ϖ −1 ye +

−1

d 2e

e

(ϖ 2 − ϖ + sin (θ ) sin (θ d ) (ϖ 1 − ϖ ))ϖ −1wd ))))

d 1e

d 1e

h2 = −δ 2ϖ 2 ( we − (δ 2 ze + sin (θ ) cos (θ d ) (ϖ 1 − ϖ ))ϖ ud + −1

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−1

e

sin (θ ) δ 1 vd ϖ ye − (ϖ 2 − ϖ + sin (θ ) sin (θ d ) (ϖ 1 − ϖ )) × ϖ −1wd + xe qd + tan(θ ) ye rd − zeϖ −2 ( xe (−(ϖ 2 − ϖ + cos(θ ) × cos (θ d ) (ϖ 1 − ϖ ))ϖ −1ud + cos (θ ) δ 1vd ϖ −1 ye + (δ 2 ze − cos (θ ) sin (θ d ) (ϖ 1 − ϖ ))ϖ −1wd ) + ye (ve − cos (θ d ) δ 1 ud × −1

2

d 1e

2

−1

d 2e

e

d 1e

d 2e

δ 2 tan (θ )ϖ 2 −1 ye tanh( r1de ) δ 2ϖ 2 −1ϖ −2 xe ze tanh(ued ) 1 − tanh (re) tanh (r1de )

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−

1 − tanh (ue) tanh (u1de )

− h2


tanh (q2de ) (1 − δ 2ϖ 2 −1 xe )

(1 − tanh (q ) tanh (q ) )(1 − tanh (q d 2e

e

d 1e

δ 2 tan (θ )ϖ 2 −1 ye tanh( r1de ) 1 − tanh (re) tanh (r1de )

) tanh ( q2de ) )

V1 =

=

(10)

− c2 z 2 − s5

m11 2 m66 2 z1 + z 2 + m11 ln cosh (ue 2 ) + m55 ln cosh (qe 2 ) + 2 2 1 (13) m66 ln cosh (re 2 ) + ∑ θiT Γi −1θi 2 i =u , q , r

Where δ 0 , c 1 and c 2 are positive constants.

After substituting the mentioned terms and calculations, the following result will be achieved:

Remark1. qed and red are Lipschitz in (x e , y e , z e , v e , w e )

V 1 = − c1 z12 − c 2 z 2 2 − ( ρ 1 + d 11 m11−1 ) tanh 2 (ue) − ( ρ 2 + d 55 m55 −1 ) × tanh 2 qe 2 − ( ρ 3 + d 66 m66 −1 ) tanh 2 re 2 +

that play a crucial role in the stability analysis of the closed loop system. It could be shown that the virtual control ued is

(ϕ u T ue + ϕ qT qe + ϕ r T re) θi +

bounded as:

∑ θ Γ i

i



θîT

−1

(14)

i =u , q , r

ued ≤ δ 0 + (1 + δ12 + δ 22 ) ud + δ 1 vd + ( k 2 + δ12 ) wd : = ueb

Where θ=i θ i − θî . From (14), estimation rule is derived:

(11)

T

∑ θˆ

By differentiating (9) along the solutions of (10) and (5), the actual controls τ u , τ q , τ r by considering parameter uncertainties, without removing the useful damping terms, are chosen as following: R

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i =u , q , r

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i

= −Γi (ϕ u T ue + ϕ qT qe + ϕ r T re) ,

m112 d 22 m11 m112 d 33 m11 m66 ] , m 22 m 22 m33 m33 d 11 m55 m11 m55 d 22 m55 m11 m55 = θ Tq [ (m33− m11) d 55 m11 m 22 m 22 m33 d 33 m55 d 66 m55 m55 m66 ] , m33 m66 d 11 m66 m11 m66 d 22 m66 m11 m66 = θ Tr [ (m 22 − m11) m11 m 22 m 22 m33 d 33 m66 d 66 m66 m11 ] m33 V 1 = − c1 z12 − c 2 z 2 2 − ( ρ 1 + d 11 m11−1 ) tanh 2 (ue) − ( ρ 2 + d 55 m55 −1 ) × (15) tanh 2 qe 2 − ( ρ 3 + d 66 m66 −1 ) tanh 2 re 2 ≤ 0

θ Tu = [ m 22 m33 d 11

mˆ 22 mˆ 33 dˆ11 vr + wq + (u − tanh (ue)) + mˆ 11 mˆ 11 mˆ 11 mˆ 66 uˆd + uˆe d + δ 1ϖ 1−1ϖ −2 xe ye z1 − δ 2ϖ 2 −1ϖ −2 xe ze z 2 ) mˆ 11 mˆ 11 − mˆ 22 dˆ 66 uv + (r − tanh(re)) + rˆd + rê d τˆr= mˆ 66 ( − ρ 2 tanh(re) − mˆ 66 mˆ 66 mˆ 11 z1 (1 − δ 1ϖ 1−1 (cos(θ ) xe + sin(θ ) ze)) − + δ 2 tan(θ )ϖ 2 −1 ye z 2 ) mˆ 66 cos(θ ) mˆ 33 − mˆ 11 dˆ 55 uw+ (q − tanh(qe)) + τˆq =mˆ 55 ( − ρ 3 tanh(qe) − mˆ 55 mˆ 55 ρ g ∆GML sin (θ ) ˆ ˆ d mˆ 66 (12) + qd + qe − (1 − δ 2ϖ 2 −1 xe ) z 2 ) mˆ 55 mˆ 55

τû =mˆ 11 ( − ρ 1 tanh (ue) −

z1, z 2, tanh (ue) , tanh(qe),

Considering (15), we observe

tanh(re) ∈ L∞ and θ ∈ L∞ ⇒ θˆ ∈ L∞ ⇒ V 1 ∈ L∞ , then by integrating

the last equation of (15), it could be shown that the Left side is bounded, so z1, z 2, tanh (ue) tanh(qe), tanh(re) ∈ L 2 , then using

Where ρ i , i=1, 2, 3 are positive constants and notation ( ∧ ) is used for estimated expressions. Substituting (12) and (10) into (7) the closed loop system equations are achieved. R

some

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previous observation: z1, z 2, ue, qe, re ∈ ( L 2  L∞ )

IV. STABILITY ANALYSIS

(16)

3B

In this section, we prove that the control signals defined above are well defined, bounded, and that the closed loop system is asymptotically stable at the origin. To simplify stability analysis of this closed loop system, we consider two subsystems (x e , y e , z e , v e , w e ) and ( z1 , z 2 , ue , qe , re) in an R

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Besides, considering five last equations of closed loop system, it is not hard to illustrate that: z1, z 2, ue, qe, re ∈ L∞

interconnected structure. Therefore, we first show the stability of the ( z1 , z 2 , ue , qe , re) -subsystem then move to (x e , y e , z e , v e , w e ) -subsystem. R

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(17)

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Now using (16), (17) and Barbalat Lemma, asymptotically convergence to zero is proved with an appropriate choice of the design constants δ 0 , δ 1 , and δ 2 .

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A. Stability Analysis of ( z1 , z 2 , ue , qe , re) - subsystem

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B. Stability Analysis of (x e , y e , z e , v e , w e )- subsystem

9B

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From the closed loop system equations, it is shown directly that this subsystem is exponentially stable at the origin by using the following Lyapunov function:

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From the closed loop system equations, it is direct to show that this subsystem is exponentially stable at the origin by using the following Lyapunov function:

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1 V2 = 1 + xe 2 + ye 2 + ze 2 − 1 + δ 3 (ve 2 + we 2 ) 2

(1 + tan 2 (θ ))) + δ 22 + δ12 ( ud + 3 vd + 2 wd ) + (δ12 + δ 22 ) ×

(18)

δ 1 ( ud + wd ) + δ 1 δ 2 ( ud + wd ) ) + δ13 ( ud + 2 vd +

δ 3 δ 22 m11 ueb tan (θ ) δ 3 δ 22 m11 ueb − m 22 (1 − δ 2 )8 γ 1 m 22 (1 − δ 2 )8 γ 1 δ 3 d 33 1 δ 3 m11 qd 2 − − λ w (t =) (δ 0 + (δ12 + δ 22 ) ud + δ 1 vd + m33 4γ 3 m33 4γ 1 δ 3 m11 ueb δ 2 wd + δ12 wd ) − ( δ 1 (ueb + 2.5 γ 1 + rd m33(1 − 2 δ 2 )8γ 1

Where δ3 is a positive constant to be specified later. The time derivative of (18) along with the solutions of the first five equations of closed loop system equations, after a lengthy but simple calculation using completed squares, satisfies:  2 ≤ −λ x (t )ϖ −2 xe 2 − λ y (t )ϖ −2 ye 2 − λ z (t )ϖ −2 ze 2 − λ v (t ) ve 2 V −λ w (t ) we 2 + (ζ 1(⋅) V 2 + ζ 2 (⋅)) e −σ 1 (t −t 0 )

wd ) + δ 2 ) −

(19)

×(1 + tan 2 (θ ))) + δ 22 + δ12 ( ud + 3 vd + 2 wd ) + (δ12 + δ 22 ) ×

δ 1 ( ud + wd ) + δ 1 δ 2 ( ud + wd ) ) + δ13 ( ud + vd +

Where ζ 1 (·) and ζ 2 (·) are some non-decreasing functions of ( z1(t 0), z 2 (t 0) , ue (t 0) , qe (t 0) , re (t 0)) , and: R

R

R

R

δ 3 δ 2 m11 ueb (δ 02 + (δ12 + δ 22 ) ud + δ 1 vd m33(1 − δ 2 ) 4 γ 1 δ 3 m11 ueb +( δ 2 + δ 22 ) wd ) − ( δ 2 (2.5 γ 1 + tan 2 (θ ) × m33 (1 − δ 2 ) 4γ 1 wd ) + δ 2 ) −

δ 3 m11 rd γ 1 δ 3 m11 ueb γ 1  δ 1 rd λ x (t ) = δ0− − + δ 13 ud  m 22 m 22 (1 − 2 δ 1 )  (1 − δ 12 ) δ 3 m11 qd γ 1 δ 3 m11 ueb γ 1 + δ12 vd + (δ12 + δ 22 ) δ 1wd ) − − × m33 m33 (1 − δ 2 )

qd ) + δ12 δ 2 (3 ud + 2 vd + 2 wd ) +δ 22 ( ud + 2 wd ) +

δ 23 ( ud + 2 wd ) + 2 δ 1 δ 2 ( ud + vd + wd ) )

 δ 2 qd  δ 3 m11 ueb δ 2 tan (θ ) γ 1 + δ13 ud + δ 12 δ 2 vd  − ×  2 m33 (1 − δ 2 )  (1 − δ 1 ) 

(δ r

1 d

) + δ13 ud + δ 12 vd + (δ12 + δ 22 ) δ 1 wd + ueb

λ y (t ) δ 1 rd cos (θ d ) − δ 1 wd − γ 3 − δ12 ( ud + vd + wd ) − = × (δ 1 ud + wd + δ 1 vd ) −

δ 3 m11 qd γ 1 m33

(δ

2 1

Where γ i , i=1, 2, 3 are positive constants, and:

)

R

δ 3 m11 ueb γ 1 m 22 (1 − δ 2 )

( ud + wd ) + δ 1 vd ) −

R

θ (t ) = θ d (t ) + z 2 (t ) + arcsin(δ 2 ze (t ) / 1 + x (t ) + y (t ) + z (t )) (21)

δ 3 m11 ueb γ 1 × ( ueb δ 1 + δ12 (2 ud m 22 (1 − 2 δ 1 )

+ vd + 2 wd ) δ13 wd + 2 δ13 vd + (δ12 + δ 22 ) δ 1ud ) −

R

δ 2 rd  − 1 − δ 22 

+ δ13 (2 vd + wd ) + (δ12 + δ 22 ) δ 1 ud )

m 22

(δ

2 2

ud + δ 2 w d ) −

δ 1 δ 2 ( ud + vd + 2 wd ) ) −

δ 3 m11 qd γ 1 m33

(δ

2

R

R

R

R

for first 150 seconds, and τ rd = −(m11 − m 22) ud vd + 5sin (0.003t ) for the rest of simulation time. −0.5sin (0.005t )

δ 3 m11 ueb γ 1 2 δ 23 wd + (δ 22 + δ12 δ 2 ) ( ud + vd + wd ) + m33 (1 − δ 2 )

(

2

e

4B

δ 3 m11 ueb γ 1 2 ( δ1 δ 2 × m33 (1 − δ 2 )

δ 3 m11 rd γ 1

2

e

In this section, to illustrate the performance of the tracking control algorithm, simulations have been carried out assuming that the vehicle is directly actuated in force in the x B direction and in torque about the y B and z B axes. Results are shown in Fig. 1-4, verified that desired trajectory does not need to be of a particular type. In fact proposed adaptive controller can track any sufficiently smooth curve parameterized by time. Reference trajectory generated by (4) is considered as: τ ud = −(m 22 vd rd − m33 wd qd ) + 5 d 11 , τ rd = −(m11 − m 22) ud vd τ qd = −(m33 − m11) ud wd − d 55 qd − 69.42sin (θ d ) + m55 (−θ d + 0.2 − θd )

δ 3 m11 ueb δ 1 tan (θ ) γ 1 ( ueb δ 1 + δ12 (2 ud + vd + 2 wd ) m33 (1 − δ 2 )

λ z (t=) δ 2 ud − δ 1 vd − γ 3 − δ 22 wd −

2

e

V. SIMULATION RESULTS

δ 1 vd − 4γ 2

R

( ud + wd ) + δ 1 δ 2 ( ud + vd ) +

(20)

This choice consists of straight line and helix with constant curvature and torsion, in the reference trajectory. The initial conditions are picked as:

wd + δ 22 ud )

η d (t 0) [0,0, -20,0,0,30,0,0,0,0], η (t 0) [-50,80,0,0,0,0,0,0,0,0] =  δ 3 m11 ueb δ 2 tan (θ ) γ 1 δ 3 m11 ueb γ 1 =  2 −  +  (δ 2 + m33 (1 − δ 2 ) m 22 (1 − 2 δ 1 )   and the initial parameter estimation values are picked as 30% of real parameter values of the system. The parameters δ 1 qd + δ 2 + δ 1 δ 2 ( ud + w d ) ) used for producing reference trajectory are taken from [9]. 1 − δ12

λ v (t=)

1 δ 3 m11 rd 2 − (δ 0 + (δ12 + δ 22 ) ud + δ 1 vd + m 22 4γ 3 m 22 4γ 1 δ 3 m11 ueb δ 2 wd + δ12 wd ) − ( δ 1 (ueb + 2.5 + rd × m 22 (1 − 2 δ 1 ) 4γ 1

δ 3 d 22

M=1089.8kg, L=5.56m, m 11 =1116kg, m 22 = m 33 =2133kg, m 44 =36.7 kgm2, m 55 = m 66 =4061 kgm2, d 11 =25.5 kgs-1, d 22 =138 kgs-1, d 33 =138 kgs-1, d 44 =10 kgm2s-1, d 55 = d 66 =490 kg m2s-1, du 2 = du 3 = dp 2 = dp 3 =0, dq 2 = dq 3 = dr 2 = dr 3 =0, dv 2 = dw 2 =920.1 kgm-2s, dv 3 = dw 3 =750 kgm-3s2.

−

R

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P

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71

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P

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P

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P

P

R

R

R

R

R

R

R

R

R

R

R

R

R

P

R

R

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unknown parameters. Saturation functions have been used to avoid actuator saturation, so the estimations remain bounded. It was revealed that in the closed-loop system including the adaptive law, the actual trajectory asymptotically converge on the reference trajectory. Simulation results verified the asymptotical convergence of position and orientation tracking errors even when the reference trajectory consists of straight line. However, in this paper, the system matrices were assumed diagonal and roll motion was neglected. Besides, we concentrated on the case of a vehicle workspace free of obstacles. In the future, more work will be done considering collision avoidance to improve the design.

Vehicle Trajectory

Reference Trajectory

REFERENCES Fig. 1. Reference and vehicle trajectory in three dimensions

[1]

[2]

[3]

[4]

[5]

[6]

Fig. 2. Reference and vehicle trajectory in two dimensions [7] [8]

[9]

O.E. Fjellstad, T.I. Fossen, “Position and attitude tracking of AUV̕s: Aquaternion feedback approach,” Journal of Oceanic Engineering, vol. 19, no. 4, October 1994, pp. 512-518. M. Narasimhan, S. N. Singh, “Adaptive optimal control of an autonomous underwater vehicle in the dive plane using dorsal fins,” Elsevier Journal of Ocean Engineering vol. 33, 2006, pp. 404–416. J. Guo, F. Chiu, C. Huang, “Design of a sliding mode fuzzy controller for the guidance and control of an autonomous underwater vehicle,” Ocean Engineering, vol. 30, no. 16, 2003, pp. 2137–2155. A. Healey, D. Lienard, “Multivariable sliding mode control for autonomous diving and steering of unmanned underwater vehicles,” IEEE Journal of Oceanic Engineering, vol. 18, no.3, July 1993, pp. 327338. J. Li, P. Lee, “Design of an adaptive nonlinear controller for depth control of an autonomous underwater vehicle,” Elsevier Journal of Ocean Engineering, vol. 32, 2005, pp. 2165-2181. T. Prestero, “Verification of a six-degrees of freedom simulation model for the REMUS autonomous underwater vehicles,” Master Thesis, Department of Ocean Engineering and Mechanical Engineering, MIT, USA, 2001. M. Krstic, I. Kanellakopoulos, P. V. Kokotovic, “Nonlinear and adaptive control design,” Willey, New York, 1995. F. Repoulias, E. Papadopoulos, “Planar trajectory planning and tracking design for underactuated AUV̕s,” Journal of Ocean Engineering, vol. 34, 2007, pp. 1650-1667. K. Do, J. Pan, “Control of ships and underwater vehicle,” Springer, Scotland, UK, 2009.

Fahimeh Rezazadegan was born in Esfahan, Iran on September 12, 1983. She received her B. Sc. in electronic engineering from the IUT (Isfahan University of Technology), Esfahan, Iran in 2006. Currently, she is a master student in control engineering at Islamic Azad University of Najaf abad, Esfahan, Iran. She has worked at the IURRC (Isfahan University Radar Research Center) for six years. Her fields of interests are in control, robotics, autonomous underwater vehicle and adaptive design.

Fig. 3. Tracking position and orientation errors (xe, ye, ze, θe, ψe)

Khoshnam Shojaei was born in Esfahan, Iran, on March 8, 1981. He received his B.S-degree, MS-degree, Ph.D.degree in electrical engineering from Iran University of Science and Technology in 2004, 2007 and 2011, respectively. Currently, he is an assistant professor at Islamic Azad University of Najaf abad, Esfahan, Iran. His research areas are adaptive control of nonlinear systems, robotics and mechatronic applications.

VI. CONCLUSION In this paper, a 6-DOF trajectory tracking control scheme was presented for an under actuated underwater vehicle in the presence of parameter uncertainties. Besides an adaptive controller was designed using the Lyapunov theory and backstepping technique. To avoid parameter drift instability, projection algorithm was used to update the estimation of the 72