Energy-preserving, Adaptive Time-Step Lie Group Variational Integrators for Rigid Body Motion in SE(3)

Conservation laws and geometry of the configuration space play an important role in the study of rigid body systems evolving on a non-Euclidean manifold. Lie group variational integrators are structure-preserving numerical methods that respect both symplectic structure and geometry of the configuration space. In this paper, we present an adaptive time step Lie group variational integrator for the full (translation and rotation) rigid body motion in Lagrangian and Hamiltonian forms. The numerical integrator is obtained from a discrete variational principle using extended Lagrangian mechanics where time variations are considered in addition to the configuration variable variations. The derived adaptive algorithm is symplectic, momentum-preserving, and conserves the system energy via time adaptation. In addition, due to the Lie group approach, the discrete trajectory preserves the Lie group geometry of the configuration space. We apply this method to a conservative underwater vehicle motion to illustrate the advantages of using adaptive time-stepping in Lie group variational integrators.