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s
*
(k)
s
d
(k)
s(k)
U(k)
+
+
_
Process
Optimization
_
Model
+
ѓ(k)
s
m
(k)
s
m
(k)
Controller
Model
_
Fig. 20.1
IMC Structure
20.2.2
Internal Model Control Structure
The control structure considered is the well-known internal model control (IMC)
structure [20] (see Figure 20.1). The process block contains the robotic system and
the camera. The input
U
is the robotic control variable and the output
s
is the cur-
rent value of the visual features. For IBVS, the reference
s
∗
is expressed in the image
plane, as the visual features, and can be static or dynamic. The error signal
rep-
resents all modeling errors and disturbances between the current features and the
values predicted from the model of the system:
ε
ε
(
k
)=
s
(
k
)
−
s
m
(
k
)
.
(20.2)
The usual controller is replaced, in the predictive approach, by an optimization al-
gorithm. The latter minimizes the difference between a desired trajectory
s
d
and the
predicted model output
s
m
. Indeed, according to Figure 20.1, we can write (
k
is the
current iteration)
s
d
(
k
)=
s
∗
(
k
)
−
ε
(
k
)
,
s
d
(
k
)=
s
∗
(
k
)
−
(
s
(
k
)
−
s
m
(
k
))
,
(20.3)
s
m
(
k
)=
s
∗
(
k
)
s
d
(
k
)
−
−
s
(
k
)
.
Consequently, the tracking of the reference features
s
∗
by the process output
s
is
thus equivalent to the tracking of the desired features
s
d
by the model output
s
m
.
The model predicts the behavior of the features over a finite prediction horizon
N
p
.
The difference
s
d
(
k
)
−
s
m
(
k
) between the desired features and the predicted model
features is used to define the cost function
J
to be minimized with respect to a
control sequence
U
. Only the first component
U
(
k
) of the optimal control sequence
is really applied to the process. At the next sampling time, due to disturbances and
model mismatches, the measurements are updated, the finite horizon moves one step
forward and the procedure starts again.
20.2.3
Mathematical Formulation
The cost function
J
is defined as a quadratic function of the error to be minimized.
Due to the IMC structure, the mathematical formulation of VPC strategy can be
written in discrete-time as
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