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arXiv · 1910.00537

Excessive Transverse Coordinates for Orbital Stabilization of (Underactuated) Mechanical Systems

Christian Fredrik Sætre·Anton Shiriaev·Stepan Pchelkin·Ahmed Chemori
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Abstract

Transverse linearization-based approaches have become among the most prominent methods for orbitally stabilizing feedback design in regards to (periodic) motions of underactuated mechanical systems. Yet, in an $n$-dimensional state-space, this requires knowledge of a set of $(n-1)$ independent transverse coordinates, which can be nontrivial to find and whose definitions might vary for different motions (trajectories). In this paper, we consider instead a generic set of $excessive$ transverse coordinates which are defined in terms of a particular parameterization of the motion and a projection operator recovering the "position" along the orbit. We present a constructive procedure for obtaining the corresponding transverse linearization, as well as state a sufficient condition for the existence of a feedback controller rendering the desired trajectory (locally) asymptotically orbitally stable. The presented approach is applied to stabilizing oscillations of the underactuated cart-pendulum system about its unstable upright position, in which a novel motion planning approach based on virtual constraints is utilized for trajectory generation.

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Christian Fredrik Sætre, Anton Shiriaev, Stepan Pchelkin, Ahmed Chemori. 2019-10-01. Excessive Transverse Coordinates for Orbital Stabilization of (Underactuated) Mechanical Systems. https://arxiv.org/abs/1910.00537

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