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reaches a desired static reference or follows a desired dynamic reference. Although
IBVS approach is robust to modeling errors, several drawbacks can be mentioned
when the visual features are not correctly chosen. Besides the classical problem of
local minima and singularities in the interaction matrix [4], the constraint handling
is a tricky problem in IBVS. For instance, the 2D constraint, also named visibility
constraint, has to guarantee that the image measurements stay into the camera field
of view. Of course, if the visibility of the target is no longer ensured then the control
algorithm is interrupted. The 3D constraints such as workspace limits have to make
sure that the robot achieves admissible motions in its workspace all along the task.
Among the numerous works which have investigated this critical issue, three
points of view exist. The first one consists in designing adequate visual features.
In [17] for instance, the authors have shown that the system behavior explicitly de-
pends on the kind of features. Consequently, lines, spheres, circles, cylinders but
also moments may be used and combined to obtain good decoupling and linearizing
properties, implicitly ensuring the constraint handling. The control law is generally
a decoupled exponential decreasing law. Another way to deal with constraint han-
dling is to combine path-planning and trajectory tracking [7, 16, 19, 24]. When it is
successful, this solution allows ensuring both an optimal trajectory of the camera in
the Cartesian space and the visibility of the features. Path-planning via linear matrix
inequality (LMI) optimization has recently been proposed in [7] to fulfill 2D and
3D constraints. In the third approach, the effort is done on the control law design.
The visual features considered are generally basic, namely point-like features. Ad-
vanced control laws such as optimal control [14, 22], adaptive control [21], LMIs
[9, 10] and predictive control [2, 3, 12, 13, 23] have been reported in the literature.
In [12, 13], a predictive controller is used for motion compensation in target track-
ing applications. The prediction of the target motion is used to reject perturbation
in order to cancel tracking errors. In [23], the predictive controller is used from
ultrasound images for a medical application.
The strategy proposed in this chapter exploits nonlinear model predictive con-
trol for visual servoing tasks. The IBVS objective is formulated as solving on-line
a nonlinear optimization problem expressed in the image plane [2, 3]. This strategy,
named visual predictive control (VPC), offers two advantages. First, 2D and 3D con-
straints such as visibility constraints, mechanical constraints and workspace limits
can be easily taken into account in the optimization problem. Secondly, the image
prediction over a finite horizon plays a crucial role for difficult configurations. The
image prediction is based on the knowledge of a model. It can be a nonlinear global
model combining the robot model and the camera one. The image prediction can
also be obtained thanks to a linear model using the interaction matrix. The choice
of the model is addressed and discussed in the sequel. The interest of the image pre-
diction is pointed out through many simulations describing difficult configurations
for a free-flying perspective camera.
The chapter is organized as follows. In Section 20.2, the context of the study is
stated and the principle of VPC is presented. The control structure and the mathe-
matical formulation are addressed. Then, in Section 20.3, the choice of the image
prediction model is discussed. In Section 20.4, numerous simulations on a 6 DOF
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