The exponential and Cayley map on SE(3) are the prevailing coordinate maps used in Lie group integration schemes for rigid body and flexible body systems. Such geometric integrators are the Munthe-Kaas and generalized-alpha schemes, which involve the differential and its directional derivative of the respective coordinate map. Relevant closed form expressions, which were reported over the last two decades, are scattered in the literature, and some are reported without proof. This paper provides a reference summarizing all relevant closed form relations along with the relevant proofs. including the right-trivialized differential of the exponential and Cayley map and their directional derivatives (resembling the Hessian). The latter gives rise to an implicit generalized-alpha scheme for rigid/flexible multibody systems in terms of the Cayley map with improved computational efficiency.

翻译：SE(3)上的指数映射与Cayley映射是刚体及柔性体系统李群积分格式中常用的坐标映射。此类几何积分器包含Munthe-Kaas格式与广义alpha格式，涉及相应坐标映射的微分及其方向导数。过去二十年间报道的相关闭式表达式散见于文献，部分论证缺失。本文汇集所有相关闭式关系式及其证明，包括指数映射与Cayley映射的右平凡化微分及其方向导数（类海森矩阵）。后者基于Cayley映射构建了适用于刚体/柔性多体系统的隐式广义alpha格式，显著提升了计算效率。