A Lie-Group Formulation of Kinematics and Dynamics of Constrained MBS and Its Application to Analytical Mechanics

A Lie-group formulation for the kinematics and dynamics ofholonomic constrained mechanical systems (CMS) is presented. The kinematics ofrigid multibody systems (MBS) is described in terms of the screw system of theMBS. Using Lie-algebraic properties of screw algebra, isomorphicto se(3), allows a purely algebraic derivation of the Lagrangian motion equations. As such the Lie-group SE(3) ⊗... ⊗ SE(3) (n copies) is theambient space of a MBS consisting of n rigid bodies. Any parameterizationof the ambient space corresponds to a chart on the MBS configuration space ℝn. The key to combine differential geometric and Lie-algebraic approaches is the existence of kinematic basic functions whichare push forward maps from the tangent bundle Tℝn to the Lie-algebra of the ambient space.MBS with kinematic loops are CMS subject to holonomic constraints, holonomicCMS. Constraint equations are formulated on the ambient space based on achart on ℝn. Explicitly considering the Lie-algebraicstructure of se(3) as semidirect sum of the algebra oftranslations and rotations enables to reduce the number of holonomicconstraints for cut joints. It is shown that third order Lie-brackets are sufficient to obtain any subalgebra of se(3).Differential geometric aspects of the kinematics of MBS with open and closedkinematic chains are considered. The distribution in any configuration andthe subalgebra generated by the screw system of a mechanism is the key todetect singularities and to analyze the structure of the set of singularpoints.

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