Minimum-time transition of FWMAVs from hovering to forward flight

Unlike conventional airplanes, flapping-wing micro-air-vehicles (FWMAVs) move their wings continuously with respect to the body. These new degrees of freedom for the wings (wing kinematics) provide more room for optimal design of these miniature vehicles that are prone to stringent weight and power constraints. However, very few attempts have aimed to provide maneuverability-optimum wing kinematics. In general, the shapes of the kinematic functions are assumed from the outset and their level of control authority is assessed at a later stage. In this work, we formulate a minimum- time optimal control problem to steer the FWMAV dynamical system from hovering configuration to forward flight with a prescribed forward speed. Assuming horizontal stroke plane and a piece-wise constant variation for the wing pitching angle, only the waveform of one degree-of-freedom for the wing is optimized (back and forth flapping). Since the flapping angle is periodic, we represent it via a truncated Fourier series. The number of Fourier terms is discussed. The optimal control problem is formulated such that the cost functional is the final time, the slowly time-varying Fourier coefficients of the flapping angle are the inputs to be optimized along with the angles of attack (i.e., design variables), and the goal is to steer the averaged dynamics from the hovering configuration (origin) to a prescribed forward speed.