Relative-kinematic formulation of geometrically exact beam dynamics based on Lie group variational integrators

Geometrically exact beam models can accurately describe the statics and dynamics of elastic, slender beams undergoing large deformations. This work introduces a variational integrator for such models based on a relative-kinematic formulation, in which positions and velocities of nodes in the discretized beam are expressed relative to the respective preceding nodes in the kinematic chain. The resulting model is especially suited for applications in robotics and control, since it has a minimum number of states and possesses the same structure as rigid mechanical systems, simplifying the combined modeling of rigid–flexible systems. Moreover, this approach allows to specifically select the modeled deformation modes, avoiding numerical issues associated with stiff, high-frequency modes and greatly increasing numerical efficiency. The proposed model is derived fully variationally in the discrete mechanics framework and under consistent consideration of the underlying Lie group structure, which translates into beneficial numerical properties. We provide a detailed treatment of the inclusion of dissipation and propose two solver strategies for time integration, including a linear-time solver based on recursive rigid-body dynamics algorithms. The model is validated and analyzed in detailed simulation studies underlining its efficiency for the simulation of stiff and slender beams, where the straightforward, but exact reduction to a Kirchhoff beam in the presented frame leads to a substantial speed-up of the simulations.     «

Geometrically exact beam models can accurately describe the statics and dynamics of elastic, slender beams undergoing large deformations. This work introduces a variational integrator for such models based on a relative-kinematic formulation, in which positions and velocities of nodes in the discretized beam are expressed relative to the respective preceding nodes in the kinematic chain. The resulting model is especially suited for applications in robotics and control, since it has a minimum num...     »