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Flapping-wing mechanisms inspired by biological insects have the potential to enable a new class of small, highly maneuverable aerial robots with hovering capabilities. In order for such devices to operate without an external power source, it is necessary to address a complex system design challenge: the integration of all of the required components on board the robot. This paper discusses the flight energetics of flapping-wing robotic insects with the goal of selecting design parameters that enable power autonomy and maximize flight time. The subsystems of the robot are analyzed both from a broad perspective and using a detailed set of models for a piezoelectrically driven two-wing design. The models are used to perform a system-level optimization for the maximum flight time permitted by current technology, compare the resulting robot configurations to biological insects across several key metrics, and discuss the effect of performance gains in various subsystems of the robot.