NDLI: Rolling of manifolds without spinning

Content Provider	Springer Nature Link
Author	Kokkonen, P.
Copyright Year	2013
Abstract	The control model of rolling of a Riemannian manifold (M; g) onto another one $ \left( {\hat{M},\hat{g}} \right) $ consists of a state space Q of relative orientations (isometric linear maps) between their tangent spaces equipped with a so-called rolling distribution $ {\mathcal D} $ $_{R}$, which models the natural constraints of no-spinning and no-slipping of the rolling motion. It turns out that the distribution $ {\mathcal D} $ $_{R}$ can be built as a sub-distribution of a so-called no-spinning distribution $ {{\mathcal{D}}_{\overline{\nabla}}} $ on Q that models only the no-spinning constraint of the rolling motion. One is thus motivated to study the control problem associated to $ {{\mathcal{D}}_{\overline{\nabla}}} $ and, in particular, the geometry of $ {{\mathcal{D}}_{\overline{\nabla}}} $ -orbits. Moreover, the definition of $ {{\mathcal{D}}_{\overline{\nabla}}} $ (contrary to the definition of $ {\mathcal D} $ $_{R}$) makes sense in the general context of vector bundles equipped with linear connections.The purpose of this paper is to study the distribution $ {{\mathcal{D}}_{\overline{\nabla}}} $ determined by the product connection $ \nabla \times \hat{\nabla} $ on a tensor bundle $ {E^{}}\otimes \hat{E}\to M\times \hat{M} $ induced by linear connections ∇, $ \hat{\nabla} $ on vector bundles $ E\to M,\,\,\,\hat{E}\to \hat{M} $ . We describe completely the orbit structure of $ {{\mathcal{D}}_{\overline{\nabla}}} $ in terms of the holonomy groups of ∇, $ \hat{\nabla} $ and characterize the integral manifolds of it. Moreover, we describe the general formulas for the Lie brackets of vector elds in $ {E^{}}\otimes \hat{E} $ in terms of $ {{\mathcal{D}}_{\overline{\nabla}}} $ and the vertical tangent distribution of $ {E^{*}}\otimes \hat{E}\to M\times \hat{M} $ .In the particular case of tangent bundles $ TM\to M,\,\,\,T\hat{M}\to \hat{M} $ and Levi-Civita connections, we describe in more detail how $ {{\mathcal{D}}_{\overline{\nabla}}} $ is related to the above mentioned rolling model, where these Lie brackets formulas provide an important tool for the study of controllability of the related control system.
Ending Page	156
Page Count	34
Starting Page	123
File Format	PDF
ISSN	10792724
e-ISSN	15738698
Journal	Journal of Dynamical and Control Systems
Issue Number	1
Volume Number	19
Language	English
Publisher	Springer US
Publisher Date	2013-01-09
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometric methods rolling of manifolds linear connections riemannian geometry Systems Theory, Control geometry of vector bundles Connections, general theory Issues of holonomy Vibration, Dynamical Systems, Control Vector and tensor analysis Calculus of Variations and Optimal Control; Optimization Analysis Development Controllability Applications of Mathematics
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Control and Systems Engineering Algebra and Number Theory Numerical Analysis

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