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Local approximation by linear systems and Almost-Riemannian Structures on Lie groups and Continuation method in rolling problem with obstacles

Abstract : The aim of this thesis is to study two topics in sub-Riemannian geometry. On the one hand, the local approximation of an almost-Riemannian structure at singular points, and on the other hand, the kinematic system of a 2-dimensional manifold rolling (without twisting or slipping) on the Euclidean plane with forbidden regions. A n-dimensional almost-Riemannian structure can be defined locally by n vector fields satisfying the Lie algebra rank condition, playing the role of an orthonormal frame. The set of points where these vector fields are colinear is called the singular set (Z). At tangency points, i.e., points where the linear span of the vector fields is equal to the tangent space of Z, the nilpotent approximation can be replaced by the solvable one. In this thesis, under generic conditions, we state the order of approximation of the original distance by d ̃ (the distance induced by the solvable approximation), and we prove that d ̃ is closer than the distance induced by the nilpotent approximation to the original distance. Regarding the structure of the approximating system, the Lie algebra generated by this new family of vector fields is finite-dimensional and solvable (in the generic case). Moreover, the solvable approximation is equivalent to a linear ARS on a homogeneous space or a Lie group. On the other hand, nonholonomic systems have attracted the attention of many authors from different disciplines for their varied applications, mainly in robotics. The rolling-body problem (without slipping or spinning) of a 2-dimensional Riemannian manifold on another one can be written as a nonholonomic system. Many methods, algorithms, and techniques have been developed to solve it. A numerical implementation of the Continuation Method to solve the problem in which a convex surface rolls on the Euclidean plane with forbidden regions (or obstacles) without slipping or spinning is performed. Several examples are illustrated.
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Submitted on : Thursday, July 7, 2022 - 11:20:13 AM
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  • HAL Id : tel-03716186, version 1

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Ronald Manríquez Peñafiel Manríquez. Local approximation by linear systems and Almost-Riemannian Structures on Lie groups and Continuation method in rolling problem with obstacles. Differential Geometry [math.DG]. Université Paris-Saclay, 2022. English. ⟨NNT : 2022UPAST104⟩. ⟨tel-03716186⟩

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