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Chapter 7
A Variational Approach to Trajectory Planning
in Visual Servoing
Youcef Mezouar
Abstract. One deficiency of image-based servo is that the induced 3D trajectories
are not optimal and sometimes, especially when the displacements to realize is large,
these trajectories are not physically valid leading to the failure of the servoing pro-
cess. In this chapter, we address the problem of generating trajectories of some im-
age features that corresponds to optimal 3D trajectories in order to control efficiently
a robotic system using an image-based control strategy. First, a collineation path be-
tween given start and end points is obtained and then the trajectories of the image
features are derived. Path planning is formulate as a variational problem which al-
lows to consider simultaneously optimality and inequality constraints (visibility). A
numerical method is employed for solving the path-planning problem in the varia-
tional form.
7.1
Introduction
Image-based servoing is now a well-known local control framework [2, 3]. In this
approach, the reference image of the object corresponding to a desired position of
the robot is acquired first (during an off-line step) and some image features are
extracted. Features extracted from the initial image are matched with those obtained
from the desired one. These features are then tracked during the robot (and/or the
object) motion, using for example a correlation based method. An error is obtained
by comparing the image features in the current image and in the reference one. The
robot motion is then controlled in order to minimize the error (using for example
a gradient descent approach). Since the error is directly measured in the image,
image-based servo has some degrees of robustness with respect to modeling errors
and noise perturbations [24, 1]. However, sometimes, and especially when the initial
and desired configurations are distant, the trajectories induced by image-based servo
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