Effects of abdomen undulation in energy consumption and stability for monarch butterfly.

The flight of monarch butterflies is characterized by a relatively large wing, flapping at a relatively low frequency coupled with abdomen undulation. This paper presents the dynamics of a flapping wing flyer that can be applied to the coupled motion of the wing, body, and abdomen at the monarch butterfly scale, which is formulated directly on the configuration manifold. The resulting thorax and abdomen motion as well as the resultant forces are consistent with the flight of a live monarch butterfly. Based on these, beneficial effects of the abdomen undulation in the flight of monarch butterflies are illustrated. For both hover and forward-climbing trajectories, the abdomen undulation results in a reduction of the energy and power consumption. Furthermore, the Floquet stability analysis shows that the periodic orbits associated with both flight modes are stable. In particular, the abdomen undulation improves the stability. Compared to the dynamics of hawkmoth, bumblebee, and fruitfly models, the monarch possesses superior stability properties.