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intelligent technologies, tracking control of robotic
manipulators has a great of attention. Tracking con-
trol is needed to make each joint track a desired tra-
jectory as close as possible. Many control algorithm
such as computer torque method (Craig, 1989), opti-
mal control (Ruderman, 2014), adaptive control (Slo-
tine and Li, 1987), variable structure control (VSC)
(Cao and Ren, 2012), neural networks (NNs) (Yan
and Wang, 2012) and fuzzy system (Cruz and Mor-
ris, 2006) have been proposed to deal with this robotic
control problem. However, robotic manipulators are
highly nonlinear, highly time-varying and highly cou-
pled. Moreover, there always exists uncertainty in the
system model such as external disturbance, parame-
ter uncertainty, sensor errors and so on, which cause
unstable performance of the robotic system (Guo and
Woo, 2003).
In this paper, a novel adaptive sliding mode con-
trol (ASMC) algorithm is derived to deal with seam
tracking control problem of welding robotic manipu-
lator, during the process of large-scale structure com-
ponent welding. Its well known that classic sliding
mode control (SMC) will cause chattering, which is
a crucial disadvantage to the stability of the system,
making the controller designing become extremely
troublesome. The controllers robustness is verified by
the Lyapunov stability theory, and the analytical re-
sults show that the proposed algorithm enables better
high-precision tracking performance with chattering-
free than classic SMC.
The layout of the paper is as follows. Section 2
presents the dynamic model of welding robotic ma-
nipulator, and some relevant properties are discussed.
In Section 3, a novel adaptive sliding mode controller
is developed and analyzed for the tracking control of
welding robotic manipulators. Simulation examples
are given to demonstrate the performance of the pro-
posed controller in Section 4. Finally, we offer brief
conclusions and suggestion for further research.
2 DYNAMIC MODEL OF
WELDING ROBOTIC
MANIPULATORS
In general, the dynamic model of the 3-link welding
robotic manipulator is given as follows
M(q) ¨q+C(q, ˙q) ˙q+ g(q) + d(t) = u, (1)
where M(q) = M
T
(q) ∈ R
3×3
is the symmetric pos-
itive definite inertia matrix; q ∈ R
3
denotes the joint
position vector; C(q, ˙q) ∈ R
3×3
is the Coriolis and
centrifugal torques; g(q) ∈ R
3
is the vector of gravita-
tional torques; d(t) ∈ R
3
denotes the bounded distur-
bance; and u ∈ R
3
represents the torque input vector.
Several fundamental properties of the robot model
(1) can be obtained as follows.
Property 1. The matrix
˙
M(q) − 2C(q, ˙q) is skew sym-
metric matrix, i.e.,
x
T
˙
M(q) − 2C(q, ˙q)
x = 0, ∀x ∈ R
3
.
Property 2. For arbitrary x, y ∈ R
3
, we get that
M(q)x+C(q, ˙q)y+ G(q) = Y(q, ˙q, x, y)θ,
where Y(q, ˙q, x, y) denotes the regression matrix, θ is
the constant unknown parameter vector.
Property 3. The unknown disturbance d(t) is as-
sumed to be unknown, but bounded, i.e., kd(t)k < η.
3 CONTROLLER DESIGN
3.1 Adaptive Sliding Mode Controller
The objective of designed controller is to drive the
joint position q to the desired trajectory position q
d
.
First we define the tracking error as following:
˜q = q − q
d
. (2)
Let the sliding surface
s =
˙
˜q+ β ˜q, (3)
where β = diag[β
1
, β
2
, β
3
] in which β
i
is a positive
constant.
The objective of controller can be achieved by
choosing the control input u, so that the sliding sur-
face satisfies the sufficient condition (Slotine and Li,
1989; Slotine and Li, 1991). Let the reference state
˙q
r
= ˙q− s = ˙q
d
− β ˜q, (4)
and
¨q
r
= ¨q− ˙s = ¨q
d
− β
˙
˜q. (5)
Then the control law u is designed as
u =
ˆ
M(q) ¨q
r
+
ˆ
C(q, ˙q) ˙q
r
+ ˆg(q) − K
r
sgn(s)
α
, (6)
where
ˆ
M(q),
ˆ
C(q, ˙q) and ˆg(q) are the estima-
tions of M(q), C(q, ˙q) and g(q) respectively; K
r
=
diag[K
r11
, K
r22
, K
r33
] is a diagonal positive definite
matrix; sgn(s)
α
is defined as
sgn(x)
α
=
|x
1
|
α
sign(x
1
), |x
2
|
α
sign(x
2
), |x
3
|
α
sign(x
3
)
T
,
(7)
and x ∈ R
3
, 0 < α < 1.
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