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Special Radon Semester on Computational Mechanics
Linz, October-December 2005
Abstract - Markus Schoeberl and Kurt Schlacher

Authors: Markus Schoeberl and Kurt Schlacher

Title of the contribution: Nonlinear Control of Mechanical Systems based on their Geometric Description

Abstract:
Mechanical control systems provide a challenging research area. Especially the introduction of geometric tools has greatly advanced the theory. Furthermore, studying classical mechanics illustrates the deep relationship between physics and geometry. The key idea of the geometric approach is to get rid of effects caused by the choice of a special coordinate system. This is achieved by describing the systems in an intrinsic manner as objects on certain manifolds and bundles. The first part of this talk is devoted to the description of differential geometric objects like connections and jet bundles, which are needed in this contribution. After the appropriation of the mathematical tools we will investigate the geometry of mechanical systems and take a close look at the Lagrangian and the Hamiltonian picture of lumped parameter systems. In this context also some remarks on time-variant systems will be given. The second part of this talk deals extensively with the control of nonlinear mechanical systems. In general one distinguishes fully actuated and underactuated systems. In this context underactuation means, that there are less control inputs available, than degrees of freedom. The control of fully actuated mechanical systems is much easier to handle because they admit an arbitrary shaping of the potential energy, on the other hand in the case of underactuation a severe restriction on the possibility of shaping the potential energy appears. Therefore the underactuated case is much more challenging. Two famous control strategies, namely "The method of the controlled Lagrangians" and "Interconnection and Damping Assignment - Passivity Based Control (IDA-PBC)" will be discussed. The applicability of the presented design concepts will be demonstrated on concrete examples. These two control concepts are based on the Lagrangian and the Hamiltonian description, respectively and fit in the geometric characterization, which was given in the first part of the talk.

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