AUV

NN-Backstepping for Diving Control of an Underactuated AUV

Hong-jian Wang; Zi-yin Chen; He-ming Jia; Xing-hua Chen

Abstract—This paper develops a path following controller for an AUV in vertical plane. With the given tracking error expressed in Serret-Frenet frame, a backstepping method based on feedback gain is adopted rather than directly uncertainties[6]. Using backstepping and Lyapunov’s direct method, a controller is developed that guarantees global exponential stability of straight line and circle path following control[7-10]. Considering the coupled nonlinear nonlinear cancellation or feedback linearization. An adaptive neural network (NN) compensator is introduced due to the dynamics of AUV are highly nonlinear and the hydrodynamic coefficients are difficult to be accurately estimated at priori. The network weights adaptation law is derived from the Lyapunov stability analysis, and the resulting nonlinear feedback control scheme can guarantee that all the signals in the closed loop system are uniformly ultimately bounded (UUB). The simulation results demonstrate the effectiveness of proposed method with INFANTE AUV model.

Key words: autonomous underwater vehicle; path following; back stepping; adaptive control; neural network.

I. INTRODUCTION

In recent years AUV becomes one of the most efficient tools for ocean exploration and exploitation. Due to high intelligent and maneuver ability is required for different kind of ocean operations, the controller design itself is still an attractive topic and received considerable attention especially in nonlinear control domain[1]. The path following scheme is more suitable for practical implementation that numerous methods have been proposed for the path following control of marine vehicles. According to the underactuated characteristic for most surface ships, many path following controllers are based on a transformation of the ship kinematics to Serret-Frenet frame. Breivik proposed a path-following algorithm using the rudder angle with constant surge velocity, the tracking error is related to Serret-Frenet frame but with a simplified

orthogonal projection onto the path[2] that will be expanded

to a general forms to avoid the singularity in this paper. In [3-5] a nonlinear path following method is presented based on explicitly vehicle model using the back-stepping technique with the initial condition constraints are removed. Based on the decoupled dynamic of yaw motion, a

back-stepping technique is derived based on feedback

dominance and proven to be robust against model

This work was supported by the National Natural Science Foundation of China under Grant No.E091002/50979017; Ph.D. Programs Foundation of

Ministry of Education of China and Basic Technology (Grant

No.20092304110008); Research Operation Item Foundation of Central

University (Grant No.HEUCFU 1026).

Hong-jian Wang, Zi-yin Chen, He-Ming Jia and Xing-hua Chen are with the College of Automation, Harbin Engineering University, Harbin, Heilongjiang, CO 150001 China.

Corresponding author: Zi-yin Chen. tel: 86-0451-82518261; e-mail: chenziyin_heu@http://wendang.chazidian.com;

0-933957-39-8 ©2011 MTS

terms and hydrodynamic coefficients are difficult to be accurately obtained at priori, so the universal functional approximator is introduced to estimate the uncertainties of the model such as neural network and fuzzy logic. Li proposed a neural network based adaptive controller for diving control of an AUV, based on traditional backstepping method and Lyapunov theorem guarantee the system to be uniform ultimately bounded, however the existence of high order stabilizing function’s derivative leads to a complicate controller form hard for practical implementation[11-12]. That will be resolved in this paper.

Motivated by the results of the above mentioned papers, we proposed a backstepping method based on feedback domain for path following control of AUV in the vertical plane. A neural network is adopted to estimate the model uncertainties, and the adaptation law is derived through the Lyapunov stability analysis, so the UUB can be guaranteed for the closed loop system. The rest of this paper is organized as follows. Problem formulation for the path following is presented in Section II. Section III gives the methods for designing a dynamic controller for surge motion and path following respectively, by introducing the adaptive scheme in the design procedure to achieve the desired robustness against parameters uncertainty. Section IV contains the simulation results and discussions. Section V gives the conclusion.

II. PROBLEM FORMULATION This section presents the simplified dynamic model of an AUV in the vertical plane and also develops a tracking error space based on Serret-Frenet frame that will be used for controller design. A. Vehicle modeling: kinematics and dynamics

The mathematical model of an underactuated AUV moving in surge, heave and pitch is obtained from decoupled six degrees of freedom by neglecting lateral motion, the body fixed frame coordinate origin is at the center of the ship, then them model is described as

muu =du+F m =muquq+d www (1) mqq

=dq+Γ+Δq

where

mu=m Xu

,mw=m Zw

mq=I y Mq ,muq=m Zuq d2X22u=Xuuu+www+Xqqq

dZ2

w=uwuw+(W B)cosθ+mzgq

dq

=Muwuw+Muquq mzgwq+(W B)zgsinθIn Eq. (1),mis the mass of the AUV; Iyis the AUV’s

inertia about the Ybaxis of the body fixed frame; zgis the

Zb coordinate of the AUV center of gravity (CG) in the body-fixed frame, the other symbols are referred as

hydrodynamic coefficients[13]. B. Path following: Error Coordinate

{U}Fig. 1 Coordinate frames

The solution to the problem of path following proposed

here is based on Serret-Frenet frame. Tracking error is established to express the distance between the vehicle’s center Q and the closest point P on the path and the angle between vehicle’s pitch axis and the tangent to the path at point P. The parameter s is a signed distance along the

path between P and an arbitrary fixed point on the path.

Consider Frenet frame is attached to the point P. The

AUV’s relative coordinate can be expressed either in the inertial frame as [x,0,z]T or in Frenet fame as [xT

e,0,ze]. The curvature of point ()

P

along the path is denoted by cs.

It is straightforward to compute for the velocity of point Q in the inertial frame as

ν 1

dPQ Q=νP+RF

dt +R 1

F(ωd×PQ) (2)

F

where ωd=c(s)s

, νP and νQ are velocities of point P and Q expressed in inertial frame. RF denotes

the rotation matrix from inertial frame to Serret-Frenet frame, which is given as cosθF0 sinθF R= 010 F (3)

sinθF0cosθF

Multiplying the above equation on both side by RF

gives

Rν dPQ

FQ=RFνP+ dt +(ωd×PQ) (4)

F

Then using the following coordinates

νQ=[x

0z ]T

，RFνP=[s 00]T

(dPQ

dt)F=[x

e0z

T

e] (ωT

d×PQ)= zec(s)s

0 xec(s)s

Resolving for Eq. (4), gives

x

e 0 =R

x

s zec(s)s F 0 z

e

0 0 (5) z 0

xec(s)s

Assume that the heave velocity is small compared with surge velocity that can be neglected, define θe=θ θF

we obtain

x

e=ucos(θe) (zec(s)+1)s z

e= usin(θe)+xec(s)s (6)

θ e=q θ F

III. CONTROLLER DESIGN This section proposes a solution for path following control of an underactuated AUV in vertical plane based on

backstepping method, with tracking error and AUV dynamic model are described as Eq. (1) and Eq. (6), given a sufficient

smooth reference path and desired forward speed profile, derive a feedback law for the evolution rate of virtual target,

propeller force and pitch moment to drive the tracking error

and surge speed asymptotically to zero.

A. Surge speed subsystem design

A proportional-derivative controller is designed to maintain the surge speed of the AUV at a desired speed profile

= kx2 cusinθez2 uzsinθez (14) V11e12ee

ee

ud>0. According to the Eq. (1), given the

To obtain the above equation we make use of the following control input

Fd=mu(u

d λ(u ud)) du (7)

whereλ>0, by defining u

=u udas speed error, replacing the Eq. (7) into the Eq. (1) we get

u

+λu =0 (8) Then the speed error u

can exponentially converge to zero.

B. Path tracking subsystem design

In this work the backstepping method together with

Lyapunov stability theorem is adopted to derive the feedback law for path tracking control of an underactuated AUV in diving plane, based on recursive design procedure,

first we define new error variables as z1=,

z2=θe α1, z3=q α2, where α1and α2are

stabilizing functions, the design procedure is delineated as follows: Step 1:

Define the first Lyapunov function as

V11:=

2

z2

1 (9) Differentiating V1along the Eq. (6) yields

V 1=z1z 1=xex e+zez e

= xes

+xeucosθe zeusinθe (10) If we select the evolution rate of curvilinear abscissa s

as an additional control input, so we can relax the assumption of underactuated control system, so by designing

s

=ucosθe+k1xe,k1>0 (11) Replacing the Eq. (10) with Eq. (11) yields

V 1= k1x2e

zeusinθe (12) Furthermore by selecting the stabilizing function α1 as below

α1=c1ze,c1>0 (13)

The Eq. (12) can be rewritten as

property as

0<

sinθe

<1, θe∈( π,π).

e

By differentiating z2 with respect to time, we obtain

z 2=θ e

α 1=q c(s)s c1z e =q+csinθe

1uθ(z2+α1) c(s)s

(c1xe+1)(15) e

Step 2:

Augment the first Lyapunov function into the second

Lyapunov function as

VV12

2:=1+2p1z2

(16) where the control parameterp1>0and will be designed in the subsequent analysis. Differentiating V2

along the solution of Eq. (14) and Eq. (15) yields

V 2= k1x2

e csinθez2sinθe1

uθe uzez2+p1z2z

2eθe

Vθ

2= k1x2e c1usineθz2e+p1z2

(q c(s)s (c1xe+1)e

(17)

+cθe

1u

sinz 21 sinθe

2+ ce

1 p uze 1 e

Since both c1and p1are controller parameter to be

designed, so we can choose p1

=c21 to meet the

condition that the uncertain signed term in equation can be removed to guarantee the stability. Then we get

V = k1x2e c1z2e

usinθe2θ+e

(18)

p sinθe

1z2

q c(s)s (c1xe+1)+c1uz2θ

e By selecting the stabilizing function α2as below

α2= c2z2+c(s)s

(c1xe+1) (19) wherec2

>0, then the Eq. (18) can be rewritten into

the following forms

= kx2 cusinθez2Ve21e1

θe

the control law and the neural network weights adaptation

law are selected as

(20)

c1usinθe 2 p1c2z2 1 +p1z2z3c e2

According to the above equation, by selecting the

p TΦ 2 c3z3 1z2 WΓd=mq α (25)

p2

W (26) W=Γ p2z3Φ λW0

()

parameterc2 to satisfy the conditionc2>c1u, so inequality

meets the condition 0<c1u

c2<1. Due to hydrodynamic coefficients can be varying a lot according to the ocean environment changes, so there exists the parameter uncertainty and modeling errors which can lead the controller to be poor performance, to avoid these an online neural network controller is proposed to compensate the modeling error, the artificial neural network with three layer is known to be a universal approximator of continuous nonlinearities with squashing activation functions, a linearly parameterized RBF neural networks can be described as

fq(u,w,q)=W*TΦ(x)+ε(x) (21)

where fq=(dq+Δq)

mq,Φ(x)is vector of

activation function, ε

(x)is the functional reconstruction

error. In general, given a constant positive ε*

>0, we

assume that for all x∈ n

holds with

ε<ε*. By

defining the optimal neural network weight vector as

W*=argminW∈R

n

(supW TΦ fq(u,w,q))

(22) so the Eq. (1) can be replaced with

q =Γ

m+W*TΦ+ε (23)

q

Due to lack of prior knowledge about the nonlinear

function, so we use the variable W

be the estimation of optimal weight vector W*

that will be used in final

controller design. Defining weights error as W

=W W*

By differentiating z3 with respect to time, we obtain

z

3=Γ

m+W*TΦ+ε α

2 (24) q

C. Neural network compensator design

Theorem. Considering the diving dynamics of an

underactuated AUV expressed as Eq. (1). If we choose the stabilizing control functions as Eq. (13) and Eq. (19), while

where

c3>0, λ>0 and Γis a strictly positive

definite diagonal matrix are adaptive gains and W0is the neural network initial estimates of unknown optimal weight vector W*

that can be obtain through off-line identification

or other methods, and also provide robustness[14], then all the signals in the closed loop system are guaranteed to be uniformly ultimately bounded.

Proof. Considering the following Lyapunov candidate

V3:=V2+

12pz2123+2

W TΓ 1W (27) where

p2>0, differentiating V3with respect to time

along the solution of Eq. (20) we obtain

V 3= k1x2e c1usinθez2c1usinθe e p1c2z2 2

1 e

c2e (28) c2

p 3p2z3+z3

q α pz T 1 2+12 +WΓW

2and consequently

V 3= k1x2e cc1usinθe 1usinθez2e p1c2z2 2

1 e

c2e (29) c3

p2z23

+p2z3

( W

TΦ+ε)+W TΓ 1W Define the following notations

dθ

1=c,d cusinθ 1u

sin2=c2 1

1c

2 By substitute Eq. (26) into and consequently

V ≤ kx2 dz2 pdz2 pcz231e1e122233

+p2z3ε*

λ T W (30)

then

V ≤ k1x2e d1z2e p1d2z22 p2c3z233

(31)

+p21*2 2 T γz3+4ε

λ W0

and consequently

V 3≤ k1x2e d1z2e p1d2z22 p2(c3 γ)z23

λ 1 2+1

W21*2 (32) 220 2 W0

+1

p*242εwhere

γ

is a positive constant such that c3 γis

strictly positive then

V 3≤ k1x2e d1z2e p1d2z22 p2(c3 γ)z23

λ 21*21*

2(33) 2W+4p2ε+2

λW W0Comparing with the Eq. (27) we obtain

V 3< cV3

+μ (34) where

c:=min 2kdλ 1,2d1,22,2 (c3 γ),λ 1minΓ (35)

μ:=121

24p2ε*+2

λ* W0

selecting ρ:=

μc>0

0≤V(t)≤ρ+(V(0) ρ)e ct (36) From Eq. (34), it is seen that V3 exponentially converges to a ball centered the origin with the radius

RV=ρ，so all the signals in the closed loop are UUB.

Substituting the expression for z2, z3 and α

2 in terms of original state variables, Eq. (25) becomes

Γd= 1q 2θe+3ze mqW TΦ+fn

(37) where

1=mq(c2+c3)m 1

2=q c

2c3+c2

1p2 1

3=mq c

1c2c3+c

1p2 fn= c2mq(c1usin(θe) c(s)s

(1+c1xe)+m()(38) ( c(s) s)s 2

q

+c(s)s

)(c1xe

+1)+c1

mq

c(s)s (ucosθe

(c(s)ze

+1)s )+c3mqc(s)s

(c1xe+1)IV. SIMULATION RESULTS

This section illustrates the effectiveness of our proposed controller for path tracking of an AUV in vertical plane. The non-dimensional parameters of the INFANTE AUV are given in [15]. A radial basis function (RBF) NN with three layers is used in the adaptive scheme. The functional form for each RBF neuron in hidden layer is defined by

ΦT

η cii(η)=e

(η ci)()2

(39)

where η is each hidden layer neuron input vector, the centers ci, i=1,2,"12 are randomly selected over the range

[

0.5,0.5], the width is selected as σ

=, and

the neural network gains are, Γ

12×3

=diag{10,",10},

λ=1.5, the initial neural network weights are set to be

W0=0.

In the simulation, the initial position and attitude

variables are given as x

(0)=0, z(0)=45,θ(0)=0

linear and angular velocities are given asu(0)=0, w(0)=0, q(0)=0. The desired forward speed profile is

set to ud=1(m/s).

The controller design parameters are given as follows that satisfies the condition mentioned in design procedure.

p1=30,p2=103, k1=0.5 c1=0.18, c2=0.5, c3=1

The reference parameterization path to be tracked is

given as

xp(μ)=μ

2π

zp(μ)=15cos

100μ

(40) +30To illustrate the validation of the neural network

controller’s performance against the external disturbance together with parameters uncertainty, the term in Eq. (1) is set to as follows

Δq=50sin 2π 100t

+0.5w+0.5q (41)

The reference and actual AUV tracking paths are shown in Fig. 2, compared to the PID controller, clearly demonstrating with the on-line neural network compensator, proposed controller can still track the reference path accurately, while the PID controller results almost unstable