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In this thesis, we develop a feedback-invariant theory of local controllability for affine distributions. We begin by developing an unexplored notion in control theory that we call proper small-time local controllability (PSTLC). The notion of PSTLC is developed for an abstraction of the well-known notion of a control-affine system, which we call an affine system. Associated to every affine system is an affine distribution, an adaptation of the notion of a distribution. Roughly speaking, an affine distribution is PSTLC if the local behaviour of every affine system that locally approximates the affine distribution is locally controllable in the standard sense. We prove that, under a regularity condition, the PSTLC property can be characterized by studying control-affine systems. The main object that we use to study PSTLC is a cone of high-order tangent vectors, or variations, and these are defined using the vector fields of the affine system. To better understand these variations, we study how they depend on the jets of the vector fields by studying the Taylor expansion of a composition of flows. Some connections are made between labeled rooted trees and the coefficients appearing in the Taylor expansion of a composition of flows. Also, a relation between variations and the formal Campbell-Baker-Hausdorff formula is established. After deriving some algebraic properties of variations, we define a variational cone for an affine system and relate it to the local controllability problem. We then study the notion of neutralizable variations and give a method for constructing subspaces of variations. Finally, using the tools developed to study variations, we consider two important classes of systems: driftless and homogeneous systems. For both classes, we are able to characterize the PSTLC property. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2010-01-11 20:11:45.466

Tato diplomová práce se zabývá popisem řiditelnosti specifického robotického hada, který se nazývá trident snake robot. Tento robot je řazen mezi neholonomní systémy. Model je převeden do jazyka diferenciální geometrie a řízen pomocí vektorových polí a operace na nich zavedené (Lieova závorka). Je také uvažována aproximace řídicí distribuce. Dále jsou formulovány pohyby hada ve směru vektorových polí a jejich kombinace, které zajišťují základní pohyby v prostoru (rotace a translace). Tyto pohyby jsou na závěr simulovány v prostředí V-REP.

This Master's thesis clarifies the significance of Frölicher-Nijenhuis bracket and its applications in problems of physics. The basic apparatus for these applications is differential geometry on manifolds, tensor calculus and differential forms, which are contained in the first part of the thesis. The second part summarizes the basic theory of calculus of variations on manifolds and its selected applications in the field of physics. The last part of the thesis is devoted to the applications of Frölicher-Nijenhuis bracket in the derivation of Maxwell's equations and to the description of the geometry of ordinary differential equations.

Tato diplomová práce se zabývá popisem kinematického modelu řízení neholonomního mechanismu, konkrétně robotického hada. Model je zkoumán prostředky diferenciální geometrie. Dále je odvozena jeho nilpotentní aproximace. Lokální říditelnost je zjištěna pomocí dimenze Lieovy algebry generované řídícími vektorovými poli a jejich Lieovými závorkami. V závěru jsou navrženy dva jednoduché řídící algoritmy, jeden pro globální a druhý pro lokální řízení, a poté následuje srovnání jednotlivých modelů.

This thesis deals with the theory of geometric control of the trident robot. The thesis describes the basic concepts of differential geometry and control theory, which are subsequently used for describing various mechanisms. Finally, the thesis proposes the management using inferred results.

In this thesis I describe construction of Lie group and Lie algebra and its following usage for physical problems. To be able to construct Lie groups and Lie algebras we need define basic terms such as topological manifold, tensor algebra and differential geometry. First part of my thesis is aimed on this topic. In second part I am dealing with construction of Lie groups and algebras. Furthermore, I am showing different properties of given structures. Next I am trying to show, that there exists some connection among Lie groups and Lie algebras. In last part of this thesis is used just for showing how this apparat can be used on physical problems. Best known usage is to find physical symmetries to establish conservation laws, all thanks to famous Noether theorem.