The weight and power constraints imposed on flapping wing micro-air-vehicles necessitate optimal design of the flapping kinematics. To date, the approach adopted for kinematics optimization has been to assume specific functions for the Euler angles describing the wing motion with respect to the body. Then, optimization is performed on the parameters of these functions. In another approach, a number of instants over the flapping cycle are selected and optimization is performed on the magnitude of the Euler angles at these instants. This latter approach provides more freedom for the variations of the Euler angles rather than confining them to certain patterns. Yet, in both approaches, finite-dimensional optimization is adopted and, as such, additional constraints are imposed. Considering that the problem is an infinite-dimensional optimization problem, we use, in this work, the calculus of variations to obtain true optimality. The combination of the quasi-steady aerodynamics and the calculus of variations approach yields an upper bound for the flapping wing aerodynamic performance. This bound can be used as a basis for evaluating the performance of any realistic design by assessing the degree of closeness between that design and the true optimal performance.
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