Bottom-up butterfly model with thorax-pitch control and wing-pitch flexibility.

The diversity in butterfly morphology has attracted many people around the world since ancient times. Despite morphological diversity, the wing and body kinematics of butterflies have several common features. In the present study, we constructed a bottom-up butterfly model, whose morphology and kinematics are simplified while preserving the important features of butterflies. The present bottom-up butterfly model is composed of two trapezoidal wings and a rod-shaped body with a thorax and abdomen. Its wings are flapped downward in the downstroke and backward in the upstroke by changing the geometric angle of attack (AOA). The geometric AOA is determined by the thorax-pitch and wing-pitch angles. The thorax-pitch angle is actively controlled by abdominal undulation, and the wing-pitch angle is passively determined because of a rotary spring representing the basalar and subalar muscles connecting the wings and thorax. We investigated the effectiveness of abdominal undulation for thorax-pitch control and how wing-pitch flexibility affects aerodynamic-force generation and thorax-pitch control, through numerical simulations using the immersed boundary-lattice Boltzmann method. As a result, the thorax-pitch angle perfectly follows the desired angle through abdominal undulation. In addition, there is an optimal wing-pitch flexibility that maximizes the flying speed in both the forward and upward directions, but the effect of wing-pitch flexibility on thorax-pitch control is not significant. Finally, we compared the flight behavior of the present bottom-up butterfly model with that of an actual butterfly. It was found that the present model does not reproduce reasonable body kinematics but can provide reasonable aerodynamics in butterfly flights.