A Lie Algebra Approach to Lie Group Time Integration of Constrained Systems

Abstract

Lie group integrators preserve by construction the Lie group structure of a nonlinear configuration space. In multibody dynamics, they support a representation of (large) rotations in a Lie group setting that is free of singularities. The resulting equations of motion are differential equations on a manifold with tangent spaces being parametrized by the corresponding Lie algebra. In the present paper, we discuss the time discretization of these equations of motion by a generalized-\(\alpha \) Lie group integrator for constrained systems and show how to exploit in this context the linear structure of the Lie algebra. This linear structure allows a very natural definition of the generalized-\(\alpha \) Lie group integrator, an efficient practical implementation and a very detailed error analysis. Furthermore, the Lie algebra approach may be combined with analytical transformations that help to avoid an undesired order reduction phenomenon in generalized-\(\alpha \) time integration. After a tutorial-like step-by-step introduction to the generalized-\(\alpha \) Lie group integrator, we investigate its convergence behaviour and develop a novel initialization scheme to achieve second-order accuracy in the application to constrained systems. The theoretical results are illustrated by a comprehensive set of numerical tests for two Lie group formulations of a rotating heavy top.