Approximation of Finite Rigid Body Motions

Abstract

It is well-known that there is no integrable relation between the twist of a rigid body and its finite motion, since the angular velocity components are non-holonomic velocity coordinates. 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. It is shown that a k-th order accurate reconstruction requires the first k – 1 time derivative of the velocity. As an application the reconstruction formulas are used for the rotation update in a momentum preserving time stepping scheme for time integration of the dynamic Euler equations.