Depth and Heading Control for Autonomous Underwater Vehicle Using Estimated Hydrodynamic...

7/18/2019 Depth and Heading Control for Autonomous Underwater Vehicle Using Estimated Hydrodynamic Coefficients

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Depth an d Heading Control for Autonomous Underwater Vehicle

Using Estimated H ydrodynamic Coefficients

Joonyoung Kim, Kihun Kim, Hang S . Choi Woojae Seong,

and

Kyu-Yeul

Lee

Departmen t of Naval Architecture Ocean Engineering, Seoul National University, Seoul 15 1-742, Korea

Absfract

-

Depth and heading control of an

AUV

are

considered for the predetermined depth and heading angle.

The proposed control algorithm is based

on

a sliding mode

control using estimated hydrodyn amic coefficients. The

hydrodynamic coefficients are estimated with the help

of

conventional nonlinear observer techniques such as sliding

mode observer and extended Kalman filter.

By

using the

estimated coefficients, a sliding mode controller is constructed

for the combined diving and steering maneuver. The

simulation results

of

the proposed control system are compared

with those of control system with true coefficients. It is

demonstrated that the proposed control system makes the

system stable and maintains the desired depth and heading

angle with sufficient accuracy.

I.

INTRODUCTION

In recent years, intensive efforts are being concerted

towards the development of Autonomous Underwater

Vehicles (AUVs). In order to design an A W , it is usually

necessary to analyze its maneuverability and controllability

based on a mathem atical model. The mathem atical model

for

most 6

DOF contains hydrodynamic forces and m oments

expressed in terms of a set of hydrodynamic coefficients.

Therefore, it is important to know the true values

of

these

coefficients

so

as to simulate the performance of the AUV

correctly.

The hydrodynamic coefficients may be classified into 3

types;

linear damping coefficients, linear inertial force

coefficients, and nonlinear dampin g coefficients. The linear

damping coefficient is known to affect the maneuverability of

an AUV strongly. Sen

[l]

examined the influence of

various hydrodynamic coefficients on the predicted

maneuverability quality of submerged bodies and found that

the coefficients of significant effects on the trajectories

are

the linear dampin g coefficients. These coefficients are

normally obtained by exp erimental test, numerical analysis o r

empirical formula. Although Planar Motion Mechanism

(PMM) test is the most po pular among exp erimental tests, the

measured values are not completely reliable because of

experimental difficulties and errors.

Another approach is the observer method that estimates

the hydrody namic coefficients with the help

of

a model based

estimation algorithm. A representative method among

observer methods is the Kalm an filter, which has been w idely

used in the estimation of the h ydrodyn amic coeflficients and

This work

was supported

by the

NRL

program from the Ministry of Science

Technology

of

Korea.

state variables. Hwan g [2] estimated the maneuv ering

coefficients of a ship and identified the dynamic system of a

maneuvering ship using extended Kalman filtering technique.

These estimated coefficients are used not only for a

mathematical model to analyze AUV’s maneuvering

performance but also for a controller model to design AUV’s

autopilot. Antonelli

et

al. [3

J

estimated vehicle-manipulator

system’s velocity using observer and applied it in tracking

control law. Fossen and Blanke [4] designed propeller shaft

speed controller by using feedback from the axial water

velocity in the propeller disc. Farrell and Clauberg [ 5 ]

reported successful control of the Sea Squirt vehicle which

used an extended Kalm an filter as a parameter estimator with

pole placement to design the controller. Yuh [6] has

describes the functional form of vehicle dynamic equations

of

motion, the nature

of

the loadings, and the use of adaptive

control via online parameter iden tification.

Recently, advanced control techniques have been

developed for A W , aimed at improving the capability of

tracking desired position and attitude trajectories.

Especially, sliding mode control has been successfully

applied to AUV because of good robustness for modeling

uncertainty, variation from operating condition, and

disturbance. Yoerger and Slotine

[7]

proposed a series o f

SISO

continuous-time controllers by using the sliding mode

technique on an underwater vehicle and demonstrated the

robustness of their control system by computer simulation in

the presence of parameter uncertainties. Cristi

et

al.

[SI

proposed an adaptive sliding mode controller for AUVs

based on the dominant linear model and the bounds of the

nonlinear dynam ic perturbations. Healey and Lienard [9]

described a

6

DOF model for the maneuvering of an

underwater vehicle and designed a sliding mode autopilot for

the combined steering, diving, and speed control functions.

Lea et

al.

[101 compared the p erformance of root locus,

uzzy

logic, and sliding mode control, and tested using a

experimental vehicle. Lee et al.

[l

13 designed a discrete-

time quasi-sliding mode controller for an AUV in the

presence of parameter uncertainties and a long sampling

interval.

In this paper, depth and heading control of an AUV are

proposed in order to maintain the desired depth and heading

angle in a towing tank. The proposed control algorithm

represents a sliding mode control using the estimated

hydrody namic Coefficients. The hydrodyn amic coefficients

are estimated based on the nonlinear observer such as Sliding

Mode Observer SMO) and Extended Kalman Filter

EKF).

Because the system to be controlled is highly nonlinear,

a

MTS

0-933957-28-9 429



sliding mode control is constructed to compensate the effects

of modeling nonlinearity, parameter uncertainty, and

disturbance.

Section

I1

describes

the nonlinear observers for estimation of the hydrodynamic

coefficients. Section

111

presents a sliding mode control for

depth and heading control. Section IV shows simulation

results. Finally, section V presents the conclusions.

This paper organized as follows:

11.

ESTIMATION OF THE HYDRODYNAMIC

COEFFICIENTS

The coefficients of significant effects on the dynamic

performance of an AUV are found to be the linear damping

coefficients. Especially, ten coefficients among the linear

damping coefficients are considered as highly sensitive

parameters and represented in

[ I ]

as M,.

A4 ,

N, N . Ny,

Z

Y,,,

Y, and Y In this paper, in order to estimate the

sensitive coefficients, the estimate system based on a

nonlinear observer is constructed as illustrated in Fig.

1.

The nonlinear observer block is composed of SMO and

EKF,

which is designed based on the A U V s

6

DOF equations of

motion. Based on the measured signal

of

the AUV’s motion,

two

nonlinear observers are developed for estimating the

sensitive coefficients. The AUV block represents the real

plant and includes a

6

DOF model of NPS AUV

I

[9]. The

value of the sensitive coefficients from this block is used

as

true value and compared with th e estimated ones.

In order to design a nonlinear observer, AUV’s equ ations

of

motion are needed. The observer model for this paper

includes

6

DOF AUV’s equations of motion and the

augmen ted states for the sensitive coefficients. The observer

Fig. I . Configuration of the estimate system.

Y

I

Top

I

Roarvlow ,

~

l i d o vlow

=

I

I

Fig. 2. Coordinate system.

~

430

model describes surge, sway, heave, roll, pitch, and yaw

motions and its coordinate system is shown in Fig. 2. The

general

6

DOF observer model is as follows:

m[u

- v r

+

wq - x G ( q 2 + r 2 ) + y G ( p q ) + , ( p r

+ q ) ] = x

m[V+

ur

-

wp + x , ( p q

+ i

y,(p2

+ r 2 ) zG qr-b ) ] =

m[W - uq +

vp

+x ( p r - q ) +y , qr + P) - ZG ( p 2+

q’)]

=z

1, j7

+

I ,

-

,

)qr

+

I , ( p r

-

q )

-

,

(q

-

2 )

Iy4+U,

, P - , ( q r + P

+ I , ( p q

-

+

1 , ( p 2

- 2 ) m[x,(W - uq

+

vp)

-

z, u - r

+

wq)]=M

I,+ +U, - r ) P 9 - I , b 2 - q 2 )- yI ( P ‘ + 4

+

I , ( q r - b)+ m[x,(V +ur - wp)- yG u- vr + wq)J=N

-

, ( p q

+ i

m b G

W

uq

+

vp)-

zG i’ +

ur -

~ p ) ] =

(1)

In the above equation, X, Y, Z, K,M, and N represent the

resultant force and moment with respect to

x , y,

and z axis,

respectively, and their detailed expressions are described in

[9]. In order to estimate the sensitive coefficients, these

coefficients have to be modeled as extra state variables.

Consequently, equation (1) is transformed into augmented

state-space form.

U

r;

P

4

i

e

~

i

x x,

Y

+

Y,

z z,

K + K ,

= [MI - { /

M + M ,

N + N ,

p +qsin

+ r cos4 tan B

qcos4 -

s in4

(qsin

4

+

r cos4)sece

0

where M i s the inertia matrix and extra state,

4

denotes,M,,

M , N,, Nh

Nv,

Z,, Yh,

Y, and

Y,,

respectively.

X,,

Y,,

Z,,

K,, M

and

N,

denote the components of the inertial

force and mome nt. The detailed expressions are shown in

APPENDIX. Nonlinear observers are designed based on

the above 2).

A . Sliding M ode Observer

The Sliding Mode Observer (SMO), which is developed

on the basis of the sliding surface concept [12], can set the

gain value according to the uncertainty range

of

the plant

model. The SMO is known to be robust under parameter

uncertainty and disturbance. Besides, it can be easily

applied to the nonlinear system. To estimate the

10

sensitive coefficients, the SMO is designed based on the

observer model o f (2).



i, =

h i,t)

where state variable x represents U, v, w , p ,

q ,

r,

4,

0, nd

v,

nd additional states

4

represents the sensitive

coefficients. The output variables are chosen as

U, ,

w,

, q ,

r,

4,

0,

and

v,

espectively.

L

means the nonlinear gain

value, which is determined by satisfying the sliding condition.

Detailed d erivation

of

the SM O is described in [13].

B, Extended Kalman Filter

The Extended KaLman Filter (EKF) can estimate the

state variables optimally in nonlinear stochastic cases that

include the plant perturbation and sensor noise. In

particular, unknown inputs

or

parameters can be estimated by

converting them into extra state variables [14]. The EKF is

designed utilizing the observer model of (2).

.-.

i, = h(2, t )

where state and output variables are equal to those of the

SMO. The gain matrix,

K

is determined from the Riccati

equations [15]. The ten sensitive coefficients, which is

represent by 6

,

re estimated from (4).

In order to estimate the sensitive coefficients associated

with horizontal and vertical motions, simulation is conducted

for

combined diving and steering motion

of

the AUV. The

sensitive coefficients from the AUV block in Fig.1 is used

as

true value and compared with the estimated ones. The

estimation performance of the SMO and the EKF is

compared when the AUV undergoes combined diving and

steering. The motion scenario is as follows: the AUV has

the initial speed

of

1.832 m/sec and the ruddedelevator angle

is applied to 0.35 rad from the start. The rudder and

elevator works within -0.4 to

0.4

rad.

Figure 3 compares the estimation results

of

the

SMO

nd

the EKF

for

the ten sensitive coefficients. The steady-state

error is compared in TABLE I. In the figures, thick solid

line represents the true value adopted from

[ 9 ]

and

dashedholid line represent the SM OE KF result. In general,

the EKF shows a good estimation performance, but

Y,, Y,

and

Yv,

hich are associated with sway motion, have steady-

state error. It

is

well known that the

SMO is

a robust

observer under parameter uncertainty and disturbance, but it

has large steady-state error and fluctuation at the transient

period. Based on a series

of

simulation, it is concluded that

the EKF estimates the sensitive coefficients with sufficient

accuracy. Although the nonlinear observers have been used

off-line in order to analyze system identification, those can

be im plemented online for estimating the state variables and

control

of an

AUV.

TABLE I

STEADY-STATE ERROR Yo)

..................

.....

.........

9

4.068

0

10

20 30

4

50 60 70 80 Bo 100

Tlme

sec)

4.038

4 04

...............

KF

0

10

20 30 40

50 80 70 80 Bo 100

.044

Time sec)

4.005

\ - T w

1

.........

nme

sec)

-

............

...............................

4.015

4 . M

10 20

30

40 50

60

70 80 Bo

100

Tlme sec)

x

l oJ

1

,

- 5 1

43

1



4.12

I / ' r '

;-True

11 ) -

SMO

........... i KF

......... ......

............................

' I

4 . 15

' ' '

0 10 20 30 40 50 6

70 80

90 100

nme

sec)

0.06 I

......................................

P

' 0

1 0 2 0 3 0 4 0 0 5 0 7 0 8 0 9 0 1 0 0

Time sec)

.....

.................................

o 06i _ i

. . . . ....... ' ' ' ' '

-

..

8

, , , , ' J

Time

(sec)

0 10 20 3 40 50 50 70 80

90

100

.................................................

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0

4.2

nme sec)

Fig. 3. Estimation

results

of

the

SMO

and EKF.

111.

CONTROLLER DESIGN

Although an AUV system is difficult to control due to

high

nonlinearity and motion co upling, sliding mode con trol

has been successfully appiied to underwater vehicles. In

this paper, sliding mode control

[9]

is adopted for an

AUV

with the uncertainties of system parameters. Especially,

when designing a sliding mode controller, the estimated

hydrodynamic coefficients

in

section

I1

are applied in the

controller model.

It is well known that sliding mode control provides

effective and robust ways of controlling uncertain plants by

means of a switching control law, which drives the plant's

state trajectory onto the sliding surface in the state space.

Any system is described

as

a single input, multi-state

equation.

i t )

=

A x ( t )

+

bu( t )

+

s f t )

/e\

x ( t ) E R ', A E

R ,

b

E R '

where 6 f ( t ) is a nonlinear function describing disturbances

and unmodelled coupling effects. The sliding surface is

defined as

s T z

~

432

where

sT

represents sliding surface coefficient and

2

the

state error,

i.e.

2

= x - d .

It is important that the sliding

surface is defined such that as the sliding surface tends to

zero, the state error also tends to zero. Sliding surface

reaches zero in a finite amount of time by the condition.

6.= --r)sgn(a) 7)

where q represents nonlinear switching gain.

and 7), we obtain

From

( 5 )

ST(AX

+

b U

+

sf

- i d )

-Sgn 0)

8)

and control input is determined as follows:

U

= - (sTb)- 'sTAX+ (sTb)- [-sTSf + S T

- l sgn(d1

9)

If the pair ( A , b ) is co ntrollable and

(s'b)

is nonzero,

then it may be shown that the sliding surface coeficients are

the elements of the left eigenvector of the closed-loop

dynamics matrix

( A - b k T )

corresponding to a pole at the

origin

S T [ A b k T ]

= 0

(10)

where the linear gain vector,

kT

is defined

( s T b ) - ' s T A

and can be evaluated from standard method such as pole

placement. It should be mentioned that one of the

eigenvalues of

(A -

k') must be specified to be zero.

The resulting sliding control law using a

'tunh'

function is

given as

U =

-kTX

- s Tb ) - * s TS f

( s T b ) - ' s T i

(J J )

-q(sTb)- '

tanh(a/cD)

where Q is the boundary layer thickness and it acts as a

low-pass filter to remove chattering and noise. The choice

of

the nonlinear switching gain, and the boundary layer

thickness,

Q

is selected to eliminate control chattering.

A . Depth C ontrol

linearized diving system dyn amics are developed as follows:

In

order to design a controller in the vertical plane, the

P P -

( I ~

-L~M,)~=(-L~uM~)~

2 - z G we

+ ( - L u3 2 - 6 ) S S

2

e = q

z = -ue

In

12),

the values

of

Gq and

Gh

re taken from the

estimated ones in section

11.

Then the dynamic model for



depth control yields the state equation as

The sliding

surface

is defined as

os

28.18?+14.378h-Z

(14)

when the poles are placed at [0 -0.25 -0.261. Finally, the

depth control law

is

determined as

S,

= 2.3531q

+

0.00628- 0.16982,

(15)

+

2.3

anh(a,

/4).

For implementation

of

the above depth control law an

AUV

has to be equipped with the pitch rate, pitch angle, and depth

sensors.

B.

Heading

Control

follows:

The linearized steering system dynamics are given as

P

P s

2 2

2

2

- - L 4 N v ) i f-

( I ,

-

N ,

)i

=

E

3 u i , ) v +

EL ir)r

@ = r

(16)

In (16), t he values of ~ , ~ , f & , $ v , $ , , a n d $&a re t aken

from the estimated ones. Then the dynamic model for

heading control yields the s tate equation as

[]=

1:;:; ] [ j + [ - o ; 5 2 / 5 r .

.145

(17)

The values to place the poles of the steering system at

[O

-0.41 -0,421become

(18)

T,

=

0.15V

+

1.657+@

and the heading control law is as follows:

6,

=

0.5 260 ~ 0.1621r+4 .3465~ ,

19)

+

1.5

- ar

/ 0.05)

In order to implement the above heading control law, it is

necessary to measure the signals of v, Y , and v

.

IV.

SIMULATION RESULTS

Numerical simulations have been performed in order to

show the effectiveness of the proposed control system. The

simulation program, which is developed using

MATLAB

6.0

with

SIMULINK

4.0 environment, is shown in Fig. 4. The

controller block is composed

of

a sliding mode controller for

the depth and heading control. The

AUV

data of input and

output as the true plant are taken from NPS

AUV

I1 [9].

The depth and heading controls are simulated with the full

nonlinear equation and the sliding mode controller developed

in (15)

and

19).

The responses

of

control law using the

estimated hydrodynamic coefficients

are

compared with

those o f control law with true coefficients.

Figure

5

shows the desired depth, tracking trajectory, and

other controlled variables for the depth control simulation.

The desired depth is given 1 m down from the initial depth

during the first 50 secs, and then returning back to the initial

depth after 50 sec. From these figures, the performance of

the control law using estimated coefficients is similar to that

of the control law with true coefficients. Especially, the

proposed control system has accurate tracking performance

and almost no drift in sway direction

as

can be seen from Fig.

5

(d). Also, sliding mode control shows the robustness on

the presence

of

parameter uncertainty.

Figure 6 shows the response of the heading control.

The heading control simulations are performed together with

depth control in order

to

prevent the vertical motion

occurring from the coupling. In order to follow the desired

path, we define the line of sight

[9]

in terms of a desired

heading angle. The proposed heading control follows the

desired heading angle and is compared with the control law

with true coefficients. The desired path is chosen 1 m

towards y-direction during the first

50

secs, and then

returning to the initial position. The proposed control law

follows the desired path accurately similar to the depth

control law.

433



(c)

Pltch angle

E.

4 5

-

esired Depth

stimate

1

-4

0

10 20

30

40 50

60

7 80

90

100

0

10 20 3

40

50

60

70 80

90

100

@)

Elmt or ang le

d) Horizontal

dedation

P

ii -20 0.1

o i o 20

30

40

50 60 70

80

90

100 -O20

1 0 2 0 3 0 4 0 5 6 6 0 7 0 8 0 9 0 1 0 0

Time (sac) nm e sec)

Fig. 5. Simulation

results

of depth control.

(a) Hodzontal trajactoty

Desired Path

stimate

(a) Hodzontal trajactoty

DeSiredPath 11

I

0

10

20 30 40 50 60 70 80 90

100

(b) Rudderangle

40

(c) Yaw

angle

-5

0 10

20

30 4

50

60

70

80

90

100

d)

Vertical

de\latlm

0.01

0 m

O

Fig.

6

Simulation resultsof heading control.

V. CONCLUSIONS

A sliding mode control using the estimated

hydrodynamic coefficients is proposed in this paper to

maintain the desired depth and heading angle. The

hydrodynamic coefficients are estimated based on the

nonlinear observer such

as

SMO and

EKF.

Especially, the

EKF

has a good estimation performance and estimates the

coefficients with sufficient accuracy. Using the estimated

coefficients, a sliding mode controller is designed for the

diving and steering maneuver. The control system using

estimated hydrodynamic coefficients is compared with the

contro l system with true coefficient. It is demon strated that

the proposed control system is table and follows the desired

depth and path accurately. It means that the sliding mode

control shows the robustness under parameter uncertainties.

The proposed estimation method is believed to reduce the

PMM test for m easuring the hydrodynamic coefficients. In

addition, the proposed control system makes the AUV stable

and controllable in the presence

of

parameter uncertainties

and ex ternal disturbances.

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APPENDIX

M =

0

- - L X ,

3

2

0

~ - - L ~ Y ,

2

0 0

0 -m zG - - L ~ K +

2

m G 0

2

0

-

4N,

... ...

0 0

0

m - ~ Z ,

2

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Depth and Heading Control for Autonomous Underwater Vehicle Using Estimated Hydrodynamic...

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Transcript of Depth and Heading Control for Autonomous Underwater Vehicle Using Estimated Hydrodynamic...