SE(3)上の指数写像とCayley写像のレビュー：一般化Poisson方程式と柔軟マルチボディシステムのリー群積分に関連して
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
剛体・柔軟マルチボディシステムのリー群積分で使われるSE(3)上の指数写像とCayley写像の閉形式表現と証明をまとめ、Cayley写像に基づく効率的な陰的generalized-alphaスキームを提案する。
著者: Andreas Mueller
分類: math.DG, cs.RO, math-ph, math.MP, physics.comp-ph
原文アブストラクト
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.
