σ-modified Lie group generalized-α methods for constrained multibody systems

Efficient and accurate time integration methods are crucial for real-time simulation, optimization and control of constrained multibody systems. This paper presents new Lie group generalized-$\alpha$ methods that improve accuracy for multibody systems with large rotations. The proposed methods extend the widely used geom1 scheme by Brüls and Cardona by introducing a $\sigma$-modification that allows to systematically eliminate a Lie group-specific part of the leading error term without compromising second-order accuracy or zero stability. While optimal accuracy is achieved for a specific choice of $\sigma$, the special case $\sigma =1$ offers notable algorithmic simplicity and minimal computational overhead. The original geom1 scheme is recovered by setting $\sigma =0$. Several numerical benchmarks demonstrate the potential of the proposed Lie group integrators compared to both the original geom1 method and conventional formulations based on Euler parameters or Cardan/Tait-Bryan angles.