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In-Depth Information
•
control constraints such as actuator limitations in amplitude or velocity:
≤
≤
.
U
min
U
(
k
)
U
max
(20.11)
These constraints are added to the problem (20.4) which becomes a nonlinear con-
strained optimization problem:
min
U
J
(
U
)
(20.12)
∈
K
where
K
is the constraint domain defined by
C
(
U
)
0
Ceq
(
U
)=0
≤
(20.13)
.
The constraints (20.9), (20.10) and (20.11) can be formulated by nonlinear functions
C
(
U
) and
Ceq
(
U
) [8]. Numerous constrained optimization routines are available in
software libraries to solve this kind of problem: projected gradient methods, penalty
methods,
etc
. In our case, a sequential quadratic program (SQP) is used and more
precisely, the function
fmincon
from Matlab optimization toolbox.
The setting parameters of the predictive approach are the prediction horizon (
N
p
),
the control horizon (
N
c
) and the weighted matrix (
Q
(
j
)):
•
the prediction horizon is chosen in order to satisfy a compromise between scheme
stability (long horizon) and numerical feasibility in term of computational time
requirement (short horizon);
•
the control input is usually kept constant over the prediction horizon, which cor-
responds to a control horizon equal to 1. A
N
c
>
1 can be useful for stabilization
task of nonholonomic mobile robot for instance [1]; and
•
the matrix
Q
(
j
) is often the identity matrix but it can also be a time-varying
matrix useful for stabilizing the system. If
Q
(
k
+ 1)=
I
and
Q
(
k
+
l
)=0
∈
[2;
N
p
], the cost function J is then similar to the standard criterion of IBVS. It is
also equivalent to have a prediction horizon equal to 1.
∀
l
20.3
Model of Image Prediction
Here we focus on the model used to predict the image evolution. We consider a 6
DOF free-flying perspective camera observing fixed point features. A 3D point with
coordinates
P
=(
X
Z
) in the camera frame is projected in the image plane as a
2D point with coordinates
s
=(
u
,
Y
,
,
v
). The sampling period is
T
e
and the control input
U
is the camera velocity noted
W
z
).
The role of the model is to predict, over the horizon
N
p
, the evolution of the visual
features in regard to the camera velocity. The principle of the image prediction is
depicted in Figure 20.2. To perform this image prediction, two kinds of model can
be considered: a nonlinear global model and a local model based on the interaction
matrix. The identification of the model, described above by (20.7), is discussed with
respect to both cases in the next section.
τ
=(
T
x
,
T
y
,
T
z
,
W
x
,
W
y
,
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