Integrating finite rotations

We study the integration of problems of evolution in the rotation group. Instead of attacking the problem in the nonlinear differential manifold SO(3) (pure rotational dynamics), as is usually done, we derive equations for the complete problem of motion (translational and rotational dynamics) on an extended manifold. We develop a generalization of Runge-Kutta methods that, by design, ensures that the solution will remain on the manifold for any choice of the tableau. This is obtained through configuration updates performed via the exponential map. We show how certain terms can be approximated, while retaining the order of accuracy of the scheme, and how the method conserves the total momentum of the system. Within this framework, we develop two nonlinearly unconditionally stable time integration schemes, that are associated with discrete laws of conservation/dissipation of the total energy. The dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well-known conventional methods. We present numerical results to support our analysis, and we develop a complete application of this methodology to the nonlinear dynamics of three-dimensional rods undergoing large displacements and finite rotations, under the assumption of small strains.