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Algèbre de Lie et cinématique des mécanismes en boucles fermées

Abstract : The aim of this work is the study of the kinematic behavior of the closed mechanisms composed of rigid bodies. The mathematical model of such a mechanism is the closure equation f (q1,..., qm) = e where q1,...,qm are the articulate coordinates and f is an analytic function valued in the Lie displacement group. The study of the kinematic property lies to the one of the set of admissible configurations f-1 ( e ) which is a submanifold in the regular case where f is a subimmersion; however, the research is much more difficult when f has some singularities. We use as a fundamental tool the formalism of the differential geometry of the Lie groups for the displacement group and the Δ - modulo structure of its Lie algebra, that permits a simple and condensed writing of the kinematic equations and makes easier their symbolic treatment. We have demonstrated that the analyze of the closure equation up to the second order is sufficient for the 6R paradoxical mechanisms. An algorithm to estimate the rank of a set of skew-symmetric fields is developed and the generations of Lie sub algebra are studied using this algorithm. We have proposed also some methods for inverse kinematic for the spatial mechanisms permitting to solve the closure equation independently of the choice of coordinates and to obtain the necessary and sufficient conditions for the solutions. Especially, the method for the spatial 6R mechanisms simplify considerably the procedure of resolution.
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Kuangrong Hao. Algèbre de Lie et cinématique des mécanismes en boucles fermées. Géométrie algébrique [math.AG]. Ecole Nationale des Ponts et Chaussées, 1995. Français. ⟨pastel-00569136⟩

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