SE(3)上的指数映射和Cayley映射综述：与广义泊松方程和柔体多体系统的李群积分相关
Review of the Exponential and Cayley Map on SE(3) as relevant for Lie Group Integration of the Generalized Poisson Equation and Flexible Multibody Systems
AI总结：
本文总结了SE(3)上指数和Cayley映射的闭式关系及证明，用于李群积分方案，并提出了一种计算效率更高的隐式generalized-alpha方案。
摘要：
SE(3)上的指数映射和Cayley映射是刚体和柔体系统李群积分方案中使用的流行坐标映射。此类几何积分器是Munthe-Kaas和generalized-alpha方案，它们涉及各自坐标映射的微分及其方向导数。过去二十年报道的相关闭式表达式散见于文献中，且部分未提供证明。本文提供了一份参考文献，总结了所有相关的闭式关系及证明，包括指数映射和Cayley映射的右平凡化微分及其方向导数（类似于Hessian）。后者促成了基于Cayley映射的刚体/柔体多体系统隐式generalized-alpha方案，具有更高的计算效率。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
