Lie-group integration method for constrained multibody systems in state space

Abstract

Coordinate-free Lie-group integration method of arbitrary (and possibly higher) order of accuracy for constrained multibody systems (MBS) is proposed in the paper. Mathematical model of MBS dynamics is shaped as a DAE system of equations of index 1, whereas dynamics is evolving on the system state space modeled as a Lie-group. Since the formulated integration algorithm operates directly on the system manifold via MBS elements’ angular velocities and rotational matrices, no local rotational coordinates are necessary, and kinematical differential equations (that are prone to singularities in the case of three-parameter-based local description of the rotational kinematics) are completely avoided. Basis of the integration procedure is the Munthe–Kaas algorithm for ODE integration on Lie-groups, which is reformulated and expanded to be applicable for the integration of constrained MBS in the DAE-index-1 form. In order to eliminate numerical constraint violation for generalized positions and velocities during the integration procedure, a constraint stabilization projection method based on constrained least-square minimization algorithm is introduced. Two numerical examples, heavy top dynamics and satellite with mounted 5-DOF manipulator, are presented. The proposed Lie-group DAE-index-1 integration scheme is easy-to-use for an MBS with kinematical constraints of general type, and it is especially suitable for dynamics of mechanical systems with large 3D rotations where standard (vector space) formulations might be inefficient due to kinematical singularities (three-parameter-based rotational coordinates) or additional kinematical constraints (redundant quaternion formulations).