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We explore controllability in flapping flight for micro aerial vehicles (MAVs), inch-size robots capable of autonomous flight. Differently from previous work, we focus on a MAV with very limited wing kinematics and simple input control schemes. In particular, in the first part we show how an MAV provided with a pair of wings, each with a single degree of freedom and passive rotation, can still ensure controllability. This is obtained by combining two ideas. The first idea is to parameterize wing trajectory based on biomimetic principles, i.e. principles that are directly inspired by observation of real insect flight. The second idea is to treat flapping flight within the framework of high frequency control and to apply averaging theory arguments in order to prove controllability. The results obtained set flapping flight as a compelling example of high frequency control present in nature, and shed light on some of the reasons of superior maneuverability observed in flapping flight. Then, in the second part we show that controllability is still guaranteed even when the wing-thorax dynamics is included and the electromechanical structure is driven by a pulse width modulation (PWM) scheme where only its amplitude, period and duty cycle are controllable on a wingbeat-by-wingbeat basis. However, in this case our modeling clearly shows some tradeoffs between controllability and lift generation efficiency, which seem consistent with observations in real insect flight.