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In-Depth Information
min
U
∈
R
J
(
U
)
(20.4)
m
×
N
p
with
k
+
Np
∑
j
=
k
+1
s
m
(
j
)]
T
J
(
U
)=
[
s
d
(
j
)
−
Q
(
j
)[
s
d
(
j
)
−
s
m
(
j
)]
(20.5)
subject to
s
d
(
j
)=
s
∗
(
j
)
−
ε
(
j
)
,
(20.6)
x
(
j
)=
f
(
x
(
j
−
1)
,
U
(
j
−
1))
(20.7)
s
m
(
j
)=
h
(
x
(
j
))
.
p
are respectively the state, the input and
the output of the model. We will see, in the next section, that the state can be differ-
ently chosen in regard to the prediction model used and in regard to the constraints
to be handled. The first nonlinear equation of (20.7) describes the dynamics of the
system where
x
(
j
) represents the predicted state at time
j
,
n
,
U
m
The variables
x
∈
R
∈
R
and
s
m
∈
R
[
k
+ 1;
k
+
N
p
].For
j
=
k
+ 1, the predicted state
s
m
is initialized with the system state
s
at time
k
which
guarantees the feedback of the IMC structure. Moreover, in case of modeling errors
and disturbances, a second feedback is ensured by the error signal
∀
j
∈
ε(
j
) which modi-
fies the reference trajectory accordingly. The second equation of (20.7) is the output
equation. To compute
s
d
(
j
)
(
j
)
defined in (20.2). This error depends on
s
m
(
j
) that is available but also on
s
(
j
) that
is unknown over the prediction horizon. Consequently, the error
, ∀
j
∈
[
k
+ 1;
k
+
N
p
], we need to compute the error
ε
ε
(
j
) is assumed to
be constant over the prediction horizon:
ε
(
j
)=
ε
(
k
)=
s
(
k
)
−
s
m
(
k
)
, ∀
j
∈
[
k
+ 1;
k
+
N
p
]
.
(20.8)
Finally,
U
=
{
U
(
k
)
,
U
(
k
+ 1)
,...,
U
(
k
+
N
c
)
,...,
U
(
k
+
N
p
−
1)
}
is the optimal control
sequence. From
U
(
k
+
N
c
+ 1) to
U
(
k
+
N
p
−
1), the control input is constant and
equal to
U
(
k
+
N
c
) where
N
c
is the control horizon. The weighted matrix
Q
(
j
) is a
symmetric definite positive matrix.
One of the main advantages of VPC is the capability to explicitly handle con-
straints in the optimization problem. Three kinds of constraints are distinguished:
•
constraints on the state of the robotic system. It can typically be a mechanical
constraint such as workspace limit when the state represents the camera pose:
x
min
≤
≤
x
(
k
)
x
max
;
(20.9)
•
2D constraints also named visibility constraints to ensure that the visual features
stay in the image plane or to represent forbidden areas in the image. The latter
can be very useful to deal with obstacle avoidance or image occlusion:
s
min
≤
s
m
(
k
)
≤
s
max
;
(20.10)
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