  Inspired by flight characteristics captured from live Monarch butterflies, an
optimal control problem is presented while accounting the effects of
low-frequency flapping and abdomen undulation. A flapping-wing aerial vehicle
is modeled as an articulated rigid body, and its dynamics are developed
according to Lagrangian mechanics on an abstract Lie group. This provides an
elegant, global formulation of the dynamics for flapping-wing aerial vehicles,
avoiding complexities and singularities associated with local coordinates. This
is utilized to identify an optimal periodic motion that minimizes energy
variations, and an optimal control is formulated to stabilize the periodic
motion. Furthermore, the outcome of this paper can be applied to optimal
control for any Lagrangian system on a Lie group with a configuration-dependent
inertia.
