Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport

Fiori Simone

doi:10.3934/mcrf.2020031

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2021, Volume 11, Issue 1: 143-167. Doi: 10.3934/mcrf.2020031

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Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport

Fiori Simone

Università Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy

Received: July 31, 2019

Revised: February 29, 2020

Early access: June 2020

Published: February 2021

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Abstract

The objective of the paper is to contribute to the theory of error-based control systems on Riemannian manifolds. The present study focuses on system where the control field influences the covariant derivative of a control path. In order to define error terms in such systems, it is necessary to compare tangent vectors at different points using parallel transport and to understand how the covariant derivative of a vector field along a path changes after such field gets parallely transported to a different curve. It turns out that such analysis relies on a specific map, termed principal pushforward map. The present paper aims at contributing to the algebraic theory of the principal pushforward map and of its relationship with the curvature endomorphism of a state manifold.

Keywords:

Error-based non-linear dynamical systems,
Riemannian state manifolds,
Parallel transport,
Covariant differentiation,
Principal pushforward map,
Curvature endomorphism.

Mathematics Subject Classification: Primary:93C10, 93C25, 58C25.

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Figure 1. A mass-spring-damper system

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Figure 2. Depiction of the parallel transport of a tangent vector field from a smooth curve to another

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Figure 3. Exemplification of two curves on a curved manifold $ {{{\mathbb{M}}}} $ that meet at a point $ p $

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Figure 4. Depiction of the homotopic net used in the proof of the Lemma 5.1.

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