Dynamical equations of multibody systems on Lie groups

The Euler–Poinaré principle is a reduced Hamilton’s principle under Lie group framework. In this article, it is applied to derive a hybrid set of dynamical equations of rigid multibody systems, which include four parts: the classical Euler–Lagrange equations of rigid bodies in their translational coordinates of mass center; Euler–Poinaré equations via orientation matrices and their related angular velocities; the constraint equations due to different joints in Cartesian coordinates and Lie groups; and the reconstruction equations between special orthogonal groups and their Lie algebras. The generalized mass matrices of dynamical equations are constant, which is computationally efficient. All the equations can be constructed systematically and can be solved easily. The construction equations can be used to design Lie group integrators of multibody system dynamics. The procedure presented in this article can be extended easily to flexible multibody systems, systems with non-holonomic constraints, and so on