Approximation of finite rigid body motions from velocity fields

It is well-known that there is no integrable relation between the twist of a rigid body and its finite motion. Moreover, the reconstruction of the body's motion requires to solve a set of differential equations on the rigid body motion group. This is usually avoided by introducing local parameters (e.g. Euler angles) so that the problem becomes an ordinary differential equation on a vector space (e.g. kinematic Euler equations). In this paper the original problem on the motion group is treated. A family of approximation formulas is presented that allow reconstructing large rigid body motions from a given velocity field up to a desired order, where a k-th order reconstruction requires the first k - 1 time derivatives of the velocity. Such reconstruction formulas could be used whenever the velocity field is accessible. As an example the formulas are applied to the rotation update in a momentum preserving time stepping scheme for the dynamic Euler equations.