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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
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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).
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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
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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
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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
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(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
0 I , - sKb
P
2
-m 4Yp
0
P
2
-
f L4Mw
- I v
0
- I , - ~ L ~ N ,
2
... ...
013x6
m G 0
0 - -L4 T :
-
f
4Zg 0
-1,
2
I)’ - - L M g
- 1 ,
-1,
2
2
2
- I , - - L 5 K , 0,,,3
I , - - L N ,5
s
... ..... ...
’ ‘13x13
435