On the relation between rotation increments in different tangent spaces

In computational mechanics, finite rotations are often represented by rotation vectors. Rotation vector increments corresponding to different tangent spaces are generally related by a linear operator, known as the tangential transformation T. In this note, we derive the higher order terms that are usually left out in linear relation. The exact nonlinear relation is also presented. Errors via the linearized T are numerically estimated. While the concept of T arises out of the nonlinear characteristics of the rotation manifold, it has been derived via tensor analysis in the context of computational mechanics ( Cardona and Géradin, 1988). We investigate the operator T from a Lie group perspective, which provides a better insight and a 1–1 correspondence between approaches based on tensor analysis and the standard matrix Lie group theory.