An Overview of Lie Group Variational Integrators and Their Applications to Optimal Control

We introduce a general framework for the construction of variational integrators of arbitrarily high-order that incorporate Lie group techniques to automatically remain on a Lie group, while retaining the geometric structure-preserving properties characteristic of variational integrators, including symplecticity, momentum-preservation, and good long-time energy behavior. This is achieved by constructing G-invariant discrete Lagrangians in the context of Lie group methods through the use of natural charts and interpolation at the level of the Lie algebra. In the presence of symmetry, the reduction of these G-invariant Lagrangians yield a higher-order analogue of discrete Euler–Poincaré reduction. As an illustrative example, we consider the full body problem from orbital mechanics, which is concerned with the dynamics of rigid bodies in space interacting under their mutual gravitational potential. The importance of simultaneously preserving the symplectic and Lie group properties of the full body dynamics is demonstrated in numerical simulations comparing Lie group variational integrators with integrators that are not symplectic or do not preserve the Lie group structure. Lastly, we demonstrate the application of Lie group variational integrators to the construction of optimal control algorithms on Lie groups, and describe a modified scheme that improves the numerical efficiency of the computation, while maintaining the accuracy of the computed solutions.