modeling and identification of autonomous underwater...

Abstract—In this paper, horizontal plane models of AUV are studied. The linear models of AUV are simplified as vertical plane, which is helpful to identification of AUV model. A general frame of system identification for predictive control, predictive error method (PEM), is proposed. Discretization model of horizontal plane for identification and equivalent model for verification are given. The closed loop identification experiment is designed. Based on identified MFD model, the predictive control will be applied to depth control of AUV to promote the performance of AUV maneuvering.

Index Terms—APC; AUV; Identification; Depth control

I. INTRODUCTION

UV operating at deep submergence can be considered to be in a wave-free environment. This is due to the fact

that the effect of wave disturbances decreases exponentially with depth. In general, the design of high submergence AUV depth-keeping controllers is a straight forward task. On the other hand, at shallow submergence, under rough sea conditions and low speed, the design of effective depth-keeping autopilots requires further modeling and identification of the model of AUV. The paper aims to modeling the depth-keeping model of AUV for advanced control to promote the performance of AUV maneuvering. A paper titled “Generalized Predictive Control with Constraints for AUV Depth Control” will come soon.

Zhao Guoliang et al in Harbin Engineering University researched the adaptive control of ship direction[1][2][3]. Ozimina[4] studied the submarine maneuvering adaptive control, in which the state space equation’s states and parameters are estimated simultaneously by using Kalman, and the controller gain was get by solving the ARE on line. Kim et al[5] used sliding mode controller and extended Kalman to identify the hydrodynamic coefficients, and designed the sliding mode controller of AUV. Alessandri et al[6] identified the AUV horizontal plane model parameters by using Kalman and LS. Rentschler et al[7] identified the transfer functions of the rudder and heading angular velocity, the stern rudder and longitudinal angular velocity, and designed PI controller so as to improve the control results.

Manuscript received March 9, 2012. Geng Tao is with the Academy of Physics and Electroics, Henan

University, Kaifeng, China, 475004 (E -mail: [email protected]). Zhao Jin is with the Academy of Control Science and Engineering,

Huazhong University of Science and Technology (HUST), Wuhan, China, 430074(E-mail: [email protected]).

II. PROBLEM DESCRIPTION

A. Nonlinear Model of AUV

A uniform inertial reference frame should be selected for the study of AUV space motion, recommended by International Towing Tank Conference (ITTC), is used internationally, which is adapted in this paper. Shown as Fig.1, ITTC coordinate system includes fixed coordinate system and inertial coordinate system. The fixed coordinate system is set as -E , and any point in the ground or

ocean on earth can be chosen as the origin of coordinates E. The horizontal plane can be freely chosen as the -E axis

(just north of the earth is usually selected). The -E axis is

perpendicular to the -E axis and the forward direction of

-E axis is confirmed by “right-hand-rule”. -E axis is

perpendicular to the E plane, and its forward direction

points to the center of the earth. The inertial coordinate system is set as -O xyz , and any point on the AUV can be

selected as origin O. The vertical axis is parallel to the transverse shaft of AUV and points to the AUV bow. The horizontal axis -O y is parallel to the vertical shaft and points

to the starboard. The normal axis -O z points to the AUV bottom. Generally the AUV center of gravity is often selected as the origin O for analysis convenience.

E

yz

xO

Fig.1. Inertial coordinate system and AUV fixed coordinate system

1) motion in the x-direction 2 2[ ( ) ( ) ( )]g g gm u vr wq x q r y pq r z pr q X

(1) 2) motion in the y-direction

2 2[ ( ) ( ) ( )]g g gm v wp ur y r p z qr p x qp r Y

(2) 3) motion in the z-direction

2 2[ ( ) ( ) ( )]g g gm w uq vp z p q x rp q y rq p Z

(3)

Modeling and Identification of Autonomous Underwater Vehicle for Advanced Control

Geng Tao, Zhao Jin

A

Proceedings of 2012 International Conference on Modelling, Identification and Control, Wuhan, China, June 24-26, 2012

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4) rotation about the x-axis ( ) [ ( ) ( )]xx xx yy g gI p I I qr m y w vp uq z v ur wp K

(4) 5) rotation about the y-axis

( ) [ ( ) ( )]yy xx zz g gI q I I rp m z u wq vr x w vp uq M

(5) 6) rotation about the z-axis

( ) [ ( ) ( )]zz yy xx g gI r I I pq m x v ur wp y u wq vr N

(6) 7) relation between inertial coordinate and fixed

coordinate

c c s c c s s s s c s c u

s c c c s s s c s s s c v

s c s c c w

(7)

1

0

0 / /

s t c t p

c s q

s c c c r

(8)

where, ( gx , gy , gz ) is the coordinates of center of gravity（m ）.

( cx , cy , cz ) is the coordinates of center of buoyancy（ m）.

,u v w, are the velocity of surge, sway and heave（m s ）.

p q r、、 are the angular velocity of pitch, trim and yaw

（ rad s ）.

X Y Z、、 are the force of Ox , Oy and Oz （ N ）.

K M N、、 are hydrodynamic derivatives（ N m ）；、、 are the angle of trim, pitch and yaw（ rad ）.

xxI 、 yyI 、 zzI moment of inertia with respect to the Ox ,

Oy and Oz （ 2N m ）.

r ， b ， s are the angle of rudder, bow and stern

（ rad ）. L is the length of vessel（ m）.

is the density of liquid around vessel（ 3/kg m ）.

B. Linear Model of AUV

Assuming that the water area is deep and broadly wide, and is ideal fluid (viscosity ignored, and incompressible). The following assumptions are made.

1) Assumption I: All the second-order terms of increment are too small to be negligible.

2) Assumption II: The longitudinal velocity is approximately constant, and the increment can be ignored.

3) Assumption III: The hull is bilaterally symmetrical and approximately upper and down symmetrical.

4) Assumption IV: The hull has small amplitude motion at the equilibrium position.

5) Assumption V: The pitch angle is small, and the trim angle 0 .

The following approximation can be get by hypothesis 5) sin 0,sin ,cos 1,cos 1

So the pitch equations and vertical motion in equation (7) and (8) can be simplified as follows

q

w u

(9)

The equations (3) and (4) are expanded by using Taylor series. The first-order terms of equation (3) and (5) are just chosen and there is

0

0

b s

b s

w q w q b s

w q w q b s

Z Z Z w Z q Z w Z q Z Z

M M M w M q M w M q M M

Assuming that the AUV is well balanced in equation(5), B mg

P mg

g ch z z

So there is

sing cz P z B = sinmgh mgh

h is the distance between the center of buoyancy and

gravity. mgh is pitch correcting moment.

It can be seen from Assumption 3) that there is no steady-state bias, 0 00, 0Z M . the relative equations

about vertical and pitch motion are chosen to get that .

1 12 2

12

0 0

0

b s

b s

w q

w q

b

s

Z Z muw wT T

M M mghq q

Z ZT

M M

2

w q

w yy q

m Z ZT

M I M

11 12 13 11 12

21 22 23 21 22

0 1 0 0 0

b

s

w a a a w b b

q a a a q b b

(10)

11 12 12

21 22

w q

w q

Z Z mua aT

M Ma a

113 2 12

123 2 22

a T mgh

a T mgh

11 12 12

21 22

b s

b s

Z Zb bT

b b M M

Based on equation (9) and (10), the simplified motion equation of AUV can be written as state equation of standard form:

11 12 13 11 12

21 22 23 21 22

0

0

0 1 0 0 0 0

1 0 0 0 0

b

s

w a a a w b b

q a a a q b b

u

(11)


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0 0 1 0

0 0 0 1

w

q

III. THE VERTICAL PLANE MODEL ORDER

The nature of system identification is to solve over-determined equation, and when the length of measured data is N , the parameter number is increased to N , it is turned into determined equation, and residual error is zero. But meanwhile the number of system parameter is so large that there will be some problems like over-high order, over fitting noise, poor generalization ability, et al. Thus when the system order is determined, the criterion function needs not only “punish residual error” but also “punish residual error of system”, the system order is determined basically by the following criterion.

AIC (Akaike's Information Criterion)

21AIC N

NV V

N

MDL (Rissanen's Minimum Description Length Criterion)

2 log1MDL N

N NV V

N

In which, NV is

2 2

1 1

1 1ˆ( ) ( | ) ( | )

N N

Nk k

V y k y k e kN N

, and N

is the total number of parameter.

Fig.2. Pitch subsystem’s order determine

res

idua

l err

or (%)

Fig.3. Depth subsystem’s order determine

In pitch subsystem, when the system orders are 4na , 0 0nb , 1 1nk , the residual error has been

reduced to about 43 10 % ; the orders under AICV criterion

are 5na , , 8 8nb , 5 5nk , the residual error is

about 42 10 % . The orders under MDLV criterion

are 4na , 3 3nb , 1 1nk , the residual error is

about 42 10 % . In which nk is lag, nk is equal to 1 inherently for discrete system, the orders that determined by

AICV criterion should be dropped due to the order of lag in

AUV control system. In order to avoid the impact that over-parameterized has on the control system, the system’s order should be reduced. When the orders

4,na 0 0nb are compared with the orders under

MDLV criterion, there is little difference in residual error and no

substantial difference. Thus the actual orders are 4,na 1 1nb . Heuristically, the pitch subsystem is a

self-balance system. Thus it is a four-order system after reduction to a common denominator.

In depth subsystem, when the system orders are 3,na 2 2 ,nb 1 1nk , the residual error has

been reduced to about 63 10 % . The orders under

AICV criterion are 5na , 4 4nb , 5 5nk . Then

residual error is about 63 10 % . The orders under

MDLV criterion are 5,na 0 0 ,nb 1 1nk . Then

residual error is about 42 10 % . The orders that determined by AICV criterion should be dropped due to the order of lag in

AUV control system.. The system’s order should be reduced either. When the orders 3na , 2 2nb are compared

with the orders under MDLV criterion, there is little difference

in residual error and no substantial difference, thus the actual orders are 3, 1 1na nb . Heuristically, the depth

subsystem is a non-self-balance system. Thus it’s a three-order system after reduction to a common denominator.

IV. THE IDENTIFICATION AND VIRIFICATION MODEL OF

VERTICAL PLANE

The vertical plane model of AUV is described by MFD, it includes pitch subsystem and depth subsystem, in this section the orders of pitch and depth subsystem are determined by using experimental method. Shown as Fig.2 and Fig.3, the system order is gradually increased, and the residual error is gradually decreased.

It can be known from what has been discussed above that the orders’ determination of AUV vertical plane is not only one solution because of the different orders when using different criterions. Smaller orders should be determined when the output precision of fitting system has been guaranteed.


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In order to make sure that the measured data used to identify by depth channel and pitch channel is in the same order of magnitude, degree should be taken as the unit of angle rather than radian. The vertical plane model is expressed as diagonalizable MFD as following:

1 1 11 11 12

1 1 12 21 22

( ) 0 ( ) ( )

0 ( ) ( ) ( )

A z B z B z

A z B z B z

y u

1

2

yy

y

1

2

b

s

uu

u

Depth subsystem model 1 1 1

1 1 11 1 12 2( ) ( ) ( )A z B z B z y u u 1 1 2 3

1 11 12 13

1 1 211 110 111

1 1 212 120 121

( ) 1

( )

( )

A z a z a z a z

B z b z b z

B z b z b z

Pitch subsystem model 1 1 1

2 2 21 1 22 2( ) ( ) ( )A z B z B z y u u 1 1 2 3 4

2 21 22 23 24

1 1 221 210 211

1 1 222 220 221

( ) 1

( )

( )

A z a z a z a z a z

B z b z b z

B z b z b z

There are seven model parameters for identification in depth subsystem, and eight for pitch subsystem. There are fifteen parameters in vertical plane model, each of the above two subsystems is identified by using online least squares, in

depth subsystem 1 1 2

T y u u , and in pitch

subsystem 2 1 2

T y u u .

The premise of vertical plane identification is AUV’s good balance ability, that is when bow rudder and stern rudder are both zero, the depth is unchanged. There is a zero-pole in depth subsystem, and it will lead to accumulated error when having model verification, thus the zero-pole need to be cancelled.

1 1 222 0 1

1 1 22 1 2 2 1 2

1 1 1

Y zY s b z b zK

U s s s U z a z a z

(12)

1 13 23

113 1 2 3

21 1

s

Y z Y zY s K

U s s s s U z U zT

(13)

It can be known from equation (13)

1 1 13 2 2

1 2 31 11 2 1 23 2

1 1

s sY z Y z T Y z T

a z a a z a zU z U z

(14)

1 13 2

1 13 2

sY z Y z T

U z U z

(15)

After equation (14) is identified, according to the relationship between (12) and(14), the right of formula (15) can be get by just transforming the denominator. The model is verified by using depth difference (not derivation, because the denominator is only transformed, and there is a sT in

molecule, which is equal to the difference of equation left).

V. PEM IDENTIFICATION METHOD OF ARMAX MODEL 1

1 1 ( )( ) ( )

C zA z B z

y u e (16)

For ARMAX model above, its performance index into frequency domain is:

22

02

2

2

1

2

1

2

s

s

s

s

j j

T j jLS ujj

T

j

T jv

jT

A e B eV G e e d

A eC e

A ee d

C e

The 1( )C z is high pass filter, which can counteract the

high weighted function of 1( )A z and overcome the

shortcoming of ARX model. The PEM algorithm deduction of ARMAX is shown as

follows: The single step predicted value of ( )y k is

1 1 1ˆ 1y k A z y k B z u k C z e k (17)

Noted as 1 1( ) ( )A z I A z 1 1( ) ( )C z I C z

ˆ( 1) ( 1) ( 1)e k y k y k (18)

The equation (16) is expressed as

1 1

1 1

ˆ 1

ˆ( )

y k A z y k B z u k

C z y k C z y k

(19)

It can be known from equation (18) that numerical differential can be used for derivation

ˆ ˆˆ| |

ˆ ˆ

e k y k

Set Jaccobi matrix J[8], there is ˆˆ

J

y

Because of the iterative formula about y k in

equation(19), ˆˆ 1 |

ˆ

y k

should be solved when solving

ˆˆ |

ˆ

y k

, y k is related to all the historical data. The

partial derivative formula of , , i i ia b c is as follows:


627

1

ˆ ˆˆ ˆ| |, 1,...,

nC

jji i

y k y k jy k i c i nA

a a

(20)

1

ˆ ˆˆ ˆ| |1 , 0,...,

nC

jji i

y k y k ju k i c i nB

b b

(21)

1

ˆ ˆˆ ˆ| |ˆˆ |

, 1,...,

nC

jji i

y k y k jy k i y k i c

c c

i nC

(22)

Equations (20)~(22) can also be rewritten as

1

ˆˆ |, 1,...,

i

y k y k ii nA

a C z

1

ˆˆ | 1, 0,...,

i

y k u k ii nB

b C z

1

ˆˆ |, 1,...,

i

y k e k ii nC

c C z

ˆ ˆ ˆJ

A B C

y y y (23)

In which,

1 2

( ) ( 1) ( 1)

( 1) ( ) ( 2)

( 1) ( 2) ( )

ˆ ˆ ˆ

ˆ

f f f

f f f

f f f

nA

n n n nA

n n n nA

N N N nA

a a a

y y y

y y y

Ay y y

y y y

y

0 1

( ) ( ) ( 1)

( 2) ( 1) ( 2)

( ) ( 1) ( )

ˆ ˆ ˆ

ˆ

f f f

f f f

f f f

nB

n n n nB

n n n nB

N N N nB

b b b

u u u

u u u

Bu u u

y y y

y

1 2

( ) ( ) ( 1)

( 2) ( 1) ( 2)

( ) ( 1) ( )

ˆ ˆ ˆ

ˆ

f f f

f f f

f f f

nC

n n n nC

n n n nC

N N N nC

c c c

e e e

e e e

Ce e e

y y y

y

1 1

1

( ) ( ) ( ) ( )( )

( )

A z y k B z u ke k

C z

(24)

, ,n MAX nA nB nC , f means wave filtering by 1( )C z .

When 1( )C z is variable, all the measured data in each

iteration should be recalculated to get gradient J . All the measured data involved in identification should be saved, and

equivalent online algorithm is not existed.

Algorithm – ARMAX model identified by LM method N is the length of sample data, and 0 is the initial value

of which is got from the extended least square method.

As setting decay factor as 0 , threshold value of TJ e

as 1 , threshold value of 2lmh as 2 , threshold value of step

size 2

e as 3 , maximum iterative times as maxit . 0J ，

0e can be calculated a according to (23)，(24).

while (not found) and (k< maxit )

1

1 1 1 1 ;T T

lm k k k kh J J I J e

if 2 1 2lm kh or 32e

found:=true; else 1 ;new k lmh

1 1

1

2

T T

k k new new

TTlm lm new new

e e e e

h h J e

if 0

;k new

found:= 1 ;T

k kJ e

31max ,1 2 1 ; : 2;

3v

else : ; 2 ;v v v

end if end if : 1; k k

end while return k

After the gradient is solved, identification problem is

turned into nonlinear least square fitting problem, Ljung[9]used Gauss-Newton method to search optimization solution for above identification problem. In order to obtain better convergence rate, Levenberg-Marquardt method as well as Levmar software package is used to solve the above optimization problem in this paper, the algorithm description is as above, the initial value 0 of prediction error method is

obtained according to the iteration final value of extended least squares.

VI. IDENTIFICATION OF AUV MODEL

Based on the theory talked above, simulation experiments are done to make identification of AUV vertical plane model. Simulation parameters are set as that in TABLE 1. We will consider a AUV with data given in the Matlab GNC toolbox


628

version2.3[11]. The vessel model can be found in it. TABLE 1 describes the autopilot parameters setting.

TABLE 1 NUMERICAL SIMULATION PARAMETERS SETTING

Parameters setting Quantity Symbol Value

sample time sT 4s

Data accuracy 0.1°

Levenberg-Marquardt

decay factor 0 510 maximum iterative times maxit 100

TJ e

1 1010

2lmh 2 1010

2e 3 1010

Fig 4. Closed loop identification experiment

0 200 400 600-40

-20

0

20

40

time/s

b

Fig.5. Bow rudder excitation of vertical plane model identification

Closed loop identification experiment for identification is as Fig 4 shown. The PRBS is designed as[10]. The identification data of vertical plane model are shown in Fig.5, Fig.6, Fig.7, Fig.8 After the excitation signal PRBS is applied, the depth is fluctuated between ±10m, and the pitch is fluctuated between ±5°, the depth and pitch are fully incented to make sure the safety of AUV. PEM is used to identify the AUV ARMAX model.

0 200 400 600-40

-20

0

20

40

time/s

s

Fig.6. Stern rudder excitation of vertical plane model identification

trim/o

Fig.7. Pitch output of vertical plane model identification

depth/m

Fig.8. Depth output of vertical plane model identification

TABLE 2 PARAMETER OF 8KN AUV’S VERTICAL PLANE MODEL

Depth subsystem value Symbol

pitch subsystem

11a -2.950 21a -3.520

12a 2.903 22a 4.638

13a -0.953 23a -2.712

24a 0.594

110b -0.00166 210b 0.000894

120b -0.000438 220b -0.00149

111b 0.00153 211b -0.000890

TABLE 2 shows the results of identification model parameter by PEM. In order to avoid the absolute error of model verification caused by integral element, the model is


629

transformed by using formula (15) to get the verification model, and it is verified by using depth difference data, another group of bow rudder, stern rudder shown as figure Fig.9 and Fig.10 is get, in which the bow rudder and stern rudder is taken as input of the verification model, and the output is shown as Fig.11. The identification model is well fitted into the AUV depth subsystem model. The fitting results of pitch subsystem is shown as Fig.12. It can be seen from the figure that PEM identification model has good fitting precision, especially the pitch channel.

0 200 400 600-40

-20

0

20

40

time/s

Fig.9. Bow rudder excitation of vertical plane model verification data

0 200 400 600-40

-20

0

20

40

time/s

Fig.10. Stern rudder excitation of vertical plane model verification data

Fig.11. Pitch output of vertical plane model verification data

0 100 200 300 400 500 600 700 800-3

-2

-1

0

1

2

3

4

time/s

PEM methodverification data

Fig.12. Depth difference output of vertical plane model verification

VII. CONCLUSION

AUV vertical plane model is a multiple input and multiple output control system. This paper figured out the order number of vertical plane model, designed experiment of closed loop identification of AUV vertical plane model . And with describing AUV depth model with MFD, this paper successfully identified vertical plane model by using PEM method of ARMAX model. It also proved the identified precision of the model is well. The identifying experiment has important significance to the advanced control for AUV.

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[2] Zhao Guoliang, Jiang Renfeng and Cong Wang. Self-Tuning Course-Keeping Control for Ships. Journal of Harbin Engineering University, 1989, 10(1): 13~20(in Chinese);

[3] Meng Hao and Zhao Guoliang. Intelligent adaptive ship course control system based on neural network reference model. Journal of Harbin Engineering University, 2003, 24(4): 395~399;

[4] C. D. Ozimina and G. J. Bierman. Navigation, control, and parameter identification of an unmanned submersible. Oceans, 1980, 12: 172~180;

[5] Joonyoung Kim;Kihum Kim;Hang S. Choi;Woojae Seong and Kyu-yeul Lee Depth and heading control for autonomous underwater vehicle using estimated hydrodynamic coefficients. Annual conference of the marine-technology-society, 2001, 429~435.

[6] A. Alessandri;M. Caccia;G. Indiveri and G. Veruggio Application of LS and EKF Techniques to the Identification of Underwater Vehicles. Proceedings of the 1998 IEEE International Conference on Control Applications, 1998, 1084~1088.

[7] Mark E. Rentschler;Franz S. Hover and Chrssostomos Chryssostomidis. System Identification of Open-Loop Maneuvers Leads to Improved AUV Flight Performance. IEEE Journal of Oceanic Engineering, 2006, 31(1): 200~208;

[8] K. Madsen;H. B. Nielsen and O. Tingleff. Methods for non-linear least squares problems. 2nd EDITION. Technical University of Denmark, 2004;

[9] L. Ljung. System Identification. 2nd EDITION. Englewood Clifs, New Jersey: Prentice Hall, 1999;

[10] D. E. Rivera;S. V. Gaikwad and X. Chen. A demonstration prototype for control-relevant identification. American Control Conference, Baltimore, 1994;

[11] T. I. Fossen. Guidance and Control of Ocean Marine Vehicles. New York: John Wiley and Sons Ltd., 1994;


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