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free-flying camera illustrate the comparison of the different approaches: classical
IBVS, predictive control laws with local and global model. Difficult configurations
such as large displacements to achieve are tested under constraints. Finally, conclu-
sions are given in the last section.
20.2
Predictive Control for Constrained IBVS
The aim of visual servoing is to regulate to zero an error
e
(
t
) between the current
features
s
(
t
) and the reference features
s
. In IBVS, the features are expressed in the
image. The relationship between the camera velocity
(
t
) and the time variation of
the visual features
s
(
t
) is given by the interaction matrix noted
L
s
. Thus, specifying
a decoupled exponential decay law for the error
e
(
t
), we obtain the control input to
be applied to the camera:
τ
−
λ
L
s
+
e
(
t
) with
τ
(
t
)=
λ
>
0
,
(20.1)
where
L
s
+
is the approximate pseudo-inverse matrix of
L
s
. The classical IBVS is
very easy to implement but its weak points are the constraint handling and its possi-
ble bad behavior for large displacements to achieve as already mentioned in Section
20.1. The control objective of IBVS can also be formulated into an optimization
problem. The goal is to minimize an image error and to take into account con-
straints. When a model of the system is available, control predictive strategies are
well-adapted to deal with this kind of problem. The extension of predictive strategy
to visual servoing tasks is detailed below.
20.2.1
Visual Predictive Control
All predictive strategies are based on four common points: a reference trajectory, a
model of the dynamic process, a cost function and a solving optimization method.
The keystone of the predictive approach is the model used to predict the process
behavior over the future horizon. Its choice will impact on the tracking accuracy
and on the computational time. In VPC case, the process considered is generally
composed of the robotic system and the camera. For instance, the robotic system can
be a nonholonomic mobile robot [2], a drone or a robot arm. The camera system can
be a perspective or catadioptric camera [3] whatever its configuration with respect
to the robot, that is on board or remote. The model used is then a global model
describing the process. The model inputs are the control variables of the robotic
system. The model outputs are the visual features. The model is used to predict the
values of the features over a prediction horizon in regard to the control variables and
to satisfy the constraint handling. Before discussing the choice of the model, we first
introduce the control structure and then state the mathematical formulation of VPC.
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