Time integrator based on rescaled Rodrigues parameters

We develop an explicit, second-order, variational time integrator for full body dynamics that preserves the momenta of the continuous dynamics, such as linear and angular momenta, and exhibits near-conservation of total energy over exponentially long times. In order to achieve these properties, we parametrize the space of rotations using exponential local coordinates represented by a rescaled form of the Rodrigues rotation vector and we systematically derive the time integrator from a discrete Lagrangian function that yields discrete Euler-Lagrange equations amenable to explicit, closed-form solutions. By restricting attention to spherical bodies and Lagrangian functions with a quadratic kinetic energy and potential energies that solely depend on positions and attitudes, we show that the discrete Lagrangian map exhibits the same mathematical structure, up to terms of second order, of explicit Newmark or velocity Verlet algorithms, both known to be variational time integrators. These preserving properties, together with linear convergence of trajectories and quadratic convergence of total energy, are born out by two examples, namely the dynamics on $\mathrm{SO}(3)$ of a three-dimensional pendulum, and the nonlinear dynamics on $\mathrm{SE}(3)^n$ that results from the impact of a particle-binder torus against a rigid wall.