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Energy-preserving, Adaptive Time-Step Lie Group Variational Integrators for the Attitude Dynamics of a Rigid Body

In this paper, we present an adaptive time step Lie group variational integrator for the attitude dynamics of a rigid body. Lie group variational integrators are geometric numerical integrators that preserve the Hamiltonian system structures and group structures concurrently. Here, the extended Lagrangian mechanics framework is used where time is treated as a dynamic variable and the numerical integrator is obtained from the discretized variational principle. The resulting adaptive algorithm conserves the total energy exactly, as well as the structures of the configuration manifold, symmetry, and symplecticity. Numerical examples of an uncontrolled 3D pendulum are presented to show the superior numerical performance of the adaptive algorithm compared to fixed time step Lie group variational integrator.