18 research outputs found
Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus
We propose in this paper a constructive procedure that transforms locally,
even at singular configurations, the kinematics of a car towing trailers into
Kumpera-Ruiz normal form. This construction converts the nonholonomic motion
planning problem into an algebraic problem (the resolution of a system of
polynomial equations), which we illustrate by steering the two-trailer system
in a neighborhood of singular configurations. We show also that the n-trailer
system is a universal local model for all Goursat structures and that all
Goursat structures are locally nilpotentizable.Comment: LaTeX2e, 23 pages, 4 figures, submitted to International journal of
contro
Applications and extensions of Goursat normal form to control of nonlinear systems
The Goursat normal form theorem gives conditions under which a Pfaffian exterior differential system is equivalent to a certain normal form. This paper details how the Goursat normal form and its extensions provide a unified framework for understanding feedback linearization, chained form, and differential flatness
Contact systems and corank one involutive subdistributions
We give necessary and sufficient geometric conditions for a distribution (or
a Pfaffian system) to be locally equivalent to the canonical contact system on
Jn(R,Rm), the space of n-jets of maps from R into Rm. We study the geometry of
that class of systems, in particular, the existence of corank one involutive
subdistributions. We also distinguish regular points, at which the system is
equivalent to the canonical contact system, and singular points, at which we
propose a new normal form that generalizes the canonical contact system on
Jn(R,Rm) in a way analogous to that how Kumpera-Ruiz normal form generalizes
the canonical contact system on Jn(R,R), which is also called Goursat normal
form.Comment: LaTeX2e, 29 pages, submitted to Acta applicandae mathematica
Non-linear estimation is easy
Non-linear state estimation and some related topics, like parametric
estimation, fault diagnosis, and perturbation attenuation, are tackled here via
a new methodology in numerical differentiation. The corresponding basic system
theoretic definitions and properties are presented within the framework of
differential algebra, which permits to handle system variables and their
derivatives of any order. Several academic examples and their computer
simulations, with on-line estimations, are illustrating our viewpoint
Motion planning algorithms for stratified kinematic systems with application to the hexapod robot
The paper addresses the motion planning problem of legged robots. Kinematic models of these robots are stratified, i.e. the equations of motion differ on different strata. An improved version of the motion planning algorithm proposed in the literature is compared with two alternative solutions via the example of the six-legged (hexapod) robot. The first alternative solution uses explicit integration of the vector fields while the second one exploits the flatness of a restricted subsystem
Flat systems, equivalence and trajectory generation
Flat systems, an important subclass of nonlinear control systems introduced
via differential-algebraic methods, are defined in a differential
geometric framework. We utilize the infinite dimensional geometry developed
by Vinogradov and coworkers: a control system is a diffiety, or more
precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold
equipped with a privileged vector field. After recalling the definition of
a Lie-Backlund mapping, we say that two systems are equivalent if they
are related by a Lie-Backlund isomorphism. Flat systems are those systems
which are equivalent to a controllable linear one. The interest of
such an abstract setting relies mainly on the fact that the above system
equivalence is interpreted in terms of endogenous dynamic feedback. The
presentation is as elementary as possible and illustrated by the VTOL
aircraft