Singularity-Free Lie Group Integration of Multibody System Models Described in Absolute Coordinates

Abstract

Abstract A classical approach to the modeling of multibody systems (MBS) is to use absolute coordinates, i.e. a set of (redundant) coordinates that describe the absolute position and orientation of the individual bodies w.r.t. to an inertial frame (IFR). A well-known problem for the time integration of the equations of motion (EOM) is the lack of a singularity-free parameterization of spatial motions, which is usually tackled by using unit quaternions. Lie group integration methods were proposed as alternative approach to the singularity-free time integration. Lie group methods are inherently coordinate free and thus incompatible with any absolute coordinate description. In this paper, an integration scheme is proposed that allows describing MBS in terms of arbitrary absolute coordinates and at the same using Lie group integration schemes, which allows for singularity-free time integration. Moreover, the direct product group SO (3) × ℝ3 as well as the semidirect product group SE (3) can be use for representing rigid body motions, which is beneficial for constraint satisfaction. The crucial step of this method, which renders the underlying Lie group integration scheme applicable to EOM in absolute coordinates, is the update of the (global) absolute coordinates in terms of the (local) coordinates on the Lie group by means of a local-global transitions (LGT) transition map. This LGT map depends on the used absolute coordinates and the local coordinates on the Lie group, but also on the Lie group itself used to represent rigid body configurations (respectively the deformation field of flexible bodies), i.e. the geometry of spatial frame motions. The Lie group formulation is thus embedded, which allows interfacing with standard vector space integration methods.