Controllability for the Constrained Rolling Motion of Symplectic Groups

Marques, André; Leite, F. Silva

doi:10.1007/978-3-319-10380-8_1

Cited by 4 publications

(4 citation statements)

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“…Example 5 (Quadratic Lie groups). Rolling motions of quadratic Lie groups have been studied in [13] and [38] and we refer to these papers for more details. Here, we look at these motions in the context of the theory developed for symmetric spaces.…”

Section: Examplesmentioning

confidence: 99%

“…More recently, generalisations of rolling manifolds embedded in a general Riemannian manifold were presented in [23]. Extension of rolling notions to certain pseudo-Euclidean manifolds have also appeared in [29] for the Lorentzian sphere, and in [13] and [38] for pseudo-orthogonal groups and symplectic groups. In recent years, an intrinsic viewpoint has been successfully applied to study rolling motions.…”

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confidence: 99%

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A unifying approach for rolling symmetric spaces

Krakowski¹,

Machado²,

Leite³

2021

JGM

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The main goal of this paper is to present a unifying theory to describe the pure rolling motions of Riemannian symmetric spaces, which are submanifolds of Euclidean or pseudo-Euclidean spaces. Rolling motions provide interesting examples of nonholonomic systems and symmetric spaces appear associated to important applications. We make a connection between the structure of the kinematic equations of rolling and the natural decomposition of the Lie algebra associated to the symmetric space. This emphasises the relevance of Lie theory in the geometry of rolling manifolds and explains why many particular examples scattered through the existing literature always show a common pattern.

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Section: Examplesmentioning

confidence: 99%

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confidence: 99%

A unifying approach for rolling symmetric spaces

Krakowski¹,

Machado²,

Leite³

2021

JGM

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“…The spreading of the agents across the large search space is made possible through modeling and implementation of repulsive forces among the agents and crosswind biased motion. Marques et al (Marques et al, 2002) proposed a method for odor source localization by multiple robots. This method is based on genetic algorithms.…”

Section: Related Workmentioning

confidence: 99%

A Hybridization of Gravitational Search Algorithm and Particle Swarm Optimization for Odor Source Localization

Jain

Godfrey

Tiwari

2017

International Journal of Robotics Applications and Technologies

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This paper concerns with the problem of odor source localization by a team of mobile robots. The authors propose two methods for odor source localization which are largely inspired from gravitational search algorithm and particle swarm optimization. The intensity of odor across the plume area is assumed to follow the Gaussian distribution. As robots enter in the vicinity of plume area they form groups using K-nearest neighbor algorithm. The problem of local optima is handled through the use of search counter concept. The proposed approaches are tested and validated through simulation.

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“…In the case of pseudo-Riemannian manifolds, the work presented in 2008 by Jurdjevic and Zimmerman [16] was, as far as we know, the first attempt to extend results from Riemannian to pseudo-Riemannian manifolds. More recent results in this regard exist in [17] for the Lorentzian sphere, in [20] for pseudo-hyperbolic spaces, in [7] for pseudo-orthogonal groups, and in [21] for symplectic groups. The intrinsic approach has also been extended to pseudo-Riemannian manifolds in [19] and the case when the rolling manifolds have different dimensions was first tackled in [22].…”

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confidence: 99%

Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

Marques

Leite

2022

JGM

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<p style='text-indent:20px;'>This paper is devoted to rolling motions of one manifold over another of equal dimension, subject to the nonholonomic constraints of no-slip and no-twist, assuming that these motions occur inside a pseudo-Euclidean space. We first introduce a definition of rolling map adjusted to this situation, which generalizes the classical definition of Sharpe [<xref ref-type="bibr" rid="b26">26</xref>] for submanifolds of an Euclidean space. We also prove some important properties of these rolling maps. After presenting the general framework, we analyse the particular rolling of hyperquadrics embedded in pseudo-Euclidean spaces. The central topic is the rolling of a pseudo-hyperbolic space over the affine space associated with its tangent space at a point. We derive the kinematic equations, as well as the corresponding explicit solutions for two specific cases, and prove the existence of a rolling map along any curve in that rolling space. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. Finally, for the central theme, we write the kinematic equations as a control system evolving on a certain Lie group and prove its controllability. The choice of the controls corresponds to the choice of a rolling curve.</p>

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Controllability for the Constrained Rolling Motion of Symplectic Groups

Cited by 4 publications

References 7 publications

A unifying approach for rolling symmetric spaces

A unifying approach for rolling symmetric spaces

A Hybridization of Gravitational Search Algorithm and Particle Swarm Optimization for Odor Source Localization

Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

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