Optimal transition of flapping wing micro-air vehicles from hovering to forward flight

Abstract

In this work, we formulate a minimum-time optimal control problem to steer a FWMAV dynamical system from a hovering condition to forward flight with a prescribed forward speed using time-periodic and averaged dynamics formulations. For the averaged dynamics representation, we optimize the back and forth flapping angle and the up and down-stroke angles of attack of the wing. We represent the flapping angle via a generic periodic function with some parameters that determine the waveform of the flapping angle over the cycle. We formulate the optimal control problem such that the cost functional is the final time, and the slowly time-varying parameters of the flapping angle wave form and the up and down-stroke angles of attack are considered inputs to the averaged dynamics. On the other hand, the instantaneous the flapping speed and wing pitching angle are considered direct inputs to the time periodic system. The problem is then to steer the averaged dynamics from the hovering fixed point (origin) to a prescribed average forward speed, and the time periodic dynamics from the hovering periodic orbit to the orbit of the forward flight condition. We show that the averaging is not suitable for the steering between hovering and forward flight and that time-periodic dynamics are required for the controller to achieve proper transition. Also, we investigated the effect of using the time-averaged stability derivatives obtained using a computational fluid dynamics simulation versus using the time-varying hovering derivatives.