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

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