where the estimated dynamics vector is defined as
^
f ¼
^
M
Z
ð
g
Þ
€
g
þ
^
hðq;
g
Þ (5)
with
^
h
Z
ðq;
g
Þ¼
^
C
Z
ðq;
g
Þ
_
g
þ
^
D
Z
ðq;
g
Þ
_
g
þ
^
g
Z
ð
g
Þ and the unknown
dynamics vector are defined as
~
f ¼
~
M
Z
ð
g
Þ
€
g
þ
~
h
Z
ðq;
g
Þþd (6)
with
~
h
Z
ð
_
q;
g
Þ¼
~
C
Z
ð
_
q;
g
Þ
_
g
þ
~
D
Z
ð
_
q;
g
Þ
_
g
þ
~
g
Z
ð
g
Þ,
~
M
Z
¼ M
Z
^
M
Z
;
~
C
Z
¼ C
Z
^
C
Z
;
~
D
Z
¼ D
Z
^
D
Z
, and
~
g
Z
¼ g
Z
^
g
Z
. Note that d 2
R
6
is
added as a disturbance force vector. The unknown dynamics
vector is also called lumped uncertainty vector (Lin and Wai,
2002).
Assumption 1. Nonlinear lumped uncertainty vector
~
f given
in (6) and its time derivative are bounded.
2.2. Chattering-free adaptive sliding-mode controller design
2.2.1. Sliding-mode controller with a switching term
A controller design based on the sliding-mode methodology
involves two main steps. The first step is to define the desired
dynamics in the form of a vector of sliding manifolds s 2
R
6
. The
second step is to find a control law
s
2
R
6
such that the system
trajectories move toward the sliding manifold, and once they hit
the manifold, remain on it in the presence of system uncertainties
and disturbances (Slotine and Li, 1991).
For a second-order system such as an underwater vehicle
system, reasonable desired dynamics would be a stable first-order
system. This first-order dynamics can be defined as (Slotine and
Li, 1991)
s ¼
d
dt
þ
K
2
Z
e dt
¼
_
e þ 2
K
e þ
K
2
Z
e dt (7)
where
K
2
R
66
is a constant, symmetric, positive definite and
diagonal matrix that defines the break frequency of the desired
error response. Each component of the sliding manifold repre-
sents a time-varying line in the state space that passes through
the desired state variables. When s ¼ 0, the system states are on
the surface meaning the system behaves consistently with the
desired dynamics. This implies that the value of s indicates the
extent of discrepancy between the desired state and the current
state. Any sliding-mode-based controller works to keep the value
of s at zero.
The tracking error between the measured state values and the
desired state values is given by e ¼
g
g
d
with the subscript d
denoting the desired position and attitude of the ROV produced by
a separate trajectory planner. The integral term ensures zero offset
error. For notational simplicity, Eq. (7) can also be written as
s ¼
_
g
_
g
r
(8)
where
_
g
r
¼
_
g
d
2
K
e
K
2
R
e dt, and the subscript r refers to
virtual reference trajectory (Slotine and Li, 1991).
The standard sliding-mode control law is in the form
s
¼
s
eq
þ
s
sw
(9)
where
s
corresponds to a generalized force acting at the centre of
mass of the ROV, and
s
eq
and
s
sw
symbolize the equivalent control
law and the switching control law, respectively.
The equivalent control law is continuous and model based. In
the absence of uncertainties in the system dynamics, this
equivalent control alone suffices to realize the desired dynamics.
However, due to model uncertainties, an auxiliary switching term
is needed that offsets the difference between the desired
dynamics and real dynamics. This switching term is defined as
s
sw
¼K sgn(s) in conventional sliding-mode control, where K 2
R
66
is the positive definite diagonal gain matrix that is defined
based on the upper bounds on the system parameter uncertain-
ties, and sgn( ) is the nonlinear signum function.
The switching term is a discontinuous feedback component
that is in charge of compensating for deviations from the desired
dynamics, and therefore is the source of the robustness of the
sliding-mode control law. The switching term acts on the system
in a bang–bang manner creating chatter in the actuators and
causing the system state to oscillate intensely across the sliding
manifold (Slotine and Li, 1991). The goal of this section is to
replace the discontinuous switching term
s
sw
with a continuous
adaptive term
s
ad
in an effort to eliminate the chattering problem.
2.2.2. Equivalent control law
The model-based equivalent control law component of the
sliding-mode control signal can be derived by assuming that the
motion is constrained to the sliding manifold. This implies that s
is a constant vector, and thus
_
s ¼ 0. The time derivative of s can be
defined based on Eq. (8), as
_
s ¼
€
g
€
g
r
(10)
where
€
g
r
¼
€
g
d
2
K
_
e
K
2
e. Multiplying both sides of Eq. (10) by
the inertia matrix
^
M
Z
, and substituting M
Z
€
g
¼ J
T
s
eq
ðC
Z
_
g
þ
D
Z
_
g
þ g
Z
Þ from Eq. (2) into the resulting equation yields
^
M
Z
_
s ¼ J
T
s
eq
ð
^
M
Z
€
g
r
þ
^
C
Z
_
g
þ
^
D
Z
_
g
þ
^
g
Z
Þ. (11)
Letting
_
s ¼ 0 and solving the resulting equation for
s
eq
yield the
equivalent control as
s
eq
¼ J
T
^
f
r
ð
g
;
_
g
;
€
g
r
Þ (12)
where
^
f
r
ð
g
;
_
g
;
€
g
r
Þ¼
^
M
Z
€
g
r
þ
^
C
Z
_
g
þ
^
D
Z
_
g
þ
^
g
Z
(13)
As mentioned before, in the absence of uncertainties in the
system dynamics, this equivalent control alone can keep the state
variables on the sliding surface.
2.2.3. Adaptive control law
With regard to the replacement of the discontinuous term, the
following continuous adaptive control law is proposed in place of
the switching term:
s
ad
¼ J
T
ð
~
f
est
ðK þ
^
C
Z
ÞsÞ (14)
where
~
f
est
is an adaptive term that estimates the lumped
uncertainty vector defined in Eq. (6), and K 2
R
66
is a diagonal
positive definite constant matrix that is related to the convergence
rate of the controller. The estimation of the lumped uncertainty
vector is proposed to follow:
_
~
f
est
¼
C
s (15)
where
C
2
R
66
is a positive definite diagonal constant design
matrix that determines the rate of adaptation. This adaptive term
relates the error metric s function to the dynamic uncertainties,
and acts on the controller in such a way that the estimated
dynamics reflect the unknown dynamics more closely to the
actual dynamics. The assumption of
~
f being bounded ensures that
Eq. (15) is bounded as well.
The total control input
s
is defined as the sum of the equivalent
control signal and the adaptive control signal, and is given by
s
¼
s
eq
þ
s
ad
¼ J
T
ð
^
f
r
þ
~
f
est
ðK þ
^
C
Z
ÞsÞ (16)
Assumption 2. The following inequality is assumed to hold:
s
T
KsXj
_
~
f
T
C
1
wj only when
_
~
f
T
C
1
wo0 (17)
ARTICLE IN P RESS
S. Soylu et al. / Ocean Engineering 35 (20 08) 1647–1659 1649