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Chapter 10
Multicriteria Analysis of Visual Servos through
Rational Systems, Biquadratic Lyapunov
Functions, and LMIs
Patrick Danes, Daniel F. Coutinho, and Sylvain Durola
Abstract.
This chapter outlines a generic method to the analysis of eye-in-hand
position-based or image-based visual servos. This analysis is said to be “multicri-
teria” as both the convergence and the fulfillment of important constraints can be
assessed, including the target visibility, the avoidance of actuators' saturations, and
the exclusion of 3D areas. The field of nonlinear “rational” systems is first shown to
constitute a sound and versatile framework to the problem. The fundamentals of a
solution based on Lyapunov theory are overviewed next, together with the notewor-
thy difficulties raised by robotics. Constructive results are finally presented, on the
basis of biquadratic or piecewise-biquadratic Lyapunov functions, leading to fea-
sibility/optimization programs subject to linear matrix inequalities (LMIs). A case
study illustrates the approach.
10.1
Introduction
Visual servoing has been a very active field of research for the last two decades [4].
Indeed, visual servos are generally more robust to modeling and measurement errors
than proprioceptive control, and participate to the perspective of enabling robots to
evolve in dynamically changing environments. Within the eye-in-hand approaches,
two recurrent strategies can be exhibited depending on the data transmitted to the
controller. On the one hand, position-based (or 3D) servos define the controlled vari-
able as the relative situation of the target with respect to the camera. As the quality
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