On mathematical modelling of insect flight dynamics in the context of micro air vehicles

This paper discusses several aspects of mathematical modelling relevant to the flight dynamics of insect flight in the context of insect-like flapping wing micro air vehicles (MAVs). MAVs are defined as flying vehicles ca six inch in size (hand-held) and are developed to reconnoitre in confined spaces (inside buildings, tunnels etc). This requires power-efficient, highly-manoeuvrable, low-speed flight with stable hover. All of these attributes are present in insect flight and hence the focus of reproducing the functionality of insect flight by engineering means. This can only be achieved if qualitative insight is accompanied by appropriate quantitative analysis, especially in the context of flight dynamics, as flight dynamics underpin the desirable manoeuvrability. We consider two aspects of mathematical modelling for insect flight dynamics. The first one is theoretical (computational), as opposed to empirical, generation of the aerodynamic data required for the six-degrees-of-freedom equations of motion. For these purposes we first explain insect wing kinematics and the salient features of the corresponding flow. In this context, we show that aerodynamic modelling is a feasible option for certain flight regimes, focussing on a successful example of modelling hover. Such modelling progresses from first principles of fluid mechanics, but relies on simplifications justified by the known flow phenomenology and/or geometric and kinematic symmetries. In particular, this is relevant to six types of fundamental manoeuvres, which we define as those steady flight conditions for which only one component of both the translational and rotational body velocities is non-zero (and constant). The second aspect of mathematical modelling for insect flight dynamics addressed here deals with the periodic character of the aerodynamic force and moment production. This leads to consideration of the types of solutions of nonlinear equations forced by nonlinear oscillations. In particular, the existence of non-periodic solutions of equations of motion is of practical interest, since this allows steady recitilinear flight. Progress in both aspects of mathematical modelling for insect flight will require further advances in aerodynamics of insect-like flapping. Improved aerodynamic modelling and computational fluid dynamics (CFD) calculations are required. These theoretical advances must be accompanied by further flow visualisation and measurement to validate both the aerodynamic modelling and CFD predictions.