Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

André Marques; Fátima Silva Leite

doi:10.3934/jgm.2021033

Article Contents

2022, Volume 14, Issue 1: 105-129. Doi: 10.3934/jgm.2021033

This issue Previous Article Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations Next Article

Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

André Marques^1, , and
Fátima Silva Leite^2,3,

1.
Scientific Area of Mathematics, School of Technology and Management of Viseu, Polytechnic Institute of Viseu, Campus Politécnico de Repeses, 3504-510 Viseu, Portugal
2.
Department of Mathematics, University of Coimbra, Largo D. Dinis, 3001-454 Coimbra, Portugal
3.
Institute of Systems and Robotics, University of Coimbra - Pólo II, Pinhal de Marrocos, 3030-290 Coimbra, Portugal

^* Corresponding author: André Marques
^* Corresponding author: André Marques

Received: June 2021

Revised: November 2021

Early access: January 11, 2022

Published online: January 11, 2022

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Abstract

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 [26] 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.

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Mathematics Subject Classification: Primary: 53A17, 53A35, 93C15; Secondary: 53C22, 53C50.

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Figure 1. Illustration of the no-twist condition (tangential part)

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Figure 2. Rolling $ H_0^2(r) $ over $ T_{p_{0}}^{\mathrm{aff}}H^{2}_0(r) $, with $ p_0 = \left[ r \,\,\, 0 \,\, \, 0 \right]^\top $

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Figure 3. Rolling of $ H_0^2(r) $, $ H_1^2(r) $ and $ H_2^2(r) $, in case II

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Figure 4. Rolling $ H^{n}_\kappa(r_1) $ over $ H^{n}_\kappa(r_2,\eta) $, with $ n = 1 $ and $ \kappa = 0 $

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Table 1. Equations of the curves which contain $ s(t) $, on plane $ x_2ox_3 $

hyperquadric	curve $t\mapsto s(t)$, when $r=1$, $c=\left[c_1 c_2\right]^\top$, $\overline{s}_0=0$
$H_0^2(r)$	$(x_2-c_1)^2+(x_3-c_2)^2=c_1^2+c_2^2$
$H_1^2(r)$	$x_3=\pm x_2+(c_2\mp c_1)$ if $c_1=c_2$ or $c_1=-c_2$ $(x_2-c_1)^2-(x_3-c_2)^2=c_1^2-c_2^2$ if $c_1\neq \pm c_2$
$H_2^2(r)$	$(x_2-c_1)^2+(x_3-c_2)^2=c_1^2+c_2^2$

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