# Electro-Aero-Mechanical Model of Piezoelectric Direct-Driven Flapping-Wing Actuator
## 1 Introduction  
Micro aerial vehicles (MAVs) have recently attracted significant attention owing to the increasing demand for automated transport, precision agriculture, and unmanned exploration/rescue missions. Among various types of MAVs, bioinspired flapping-wing robots have attracted particular interest owing to their potential to mimic the high agility and robustness of natural insects or birds. In particular, insect or hummingbird life-scale MAVs are considered useful because their extremely low weight and small size can enable safe flight and widespread applications. There are many candidate actuation principles for flapping-wing MAVs, including piezoelectric, electromagnetic, shape memory alloy, and electrostatic. Piezoelectric actuation appears most suitable for small-scale vehicles owing to its high power-to-weight ratio, and some studies have already used piezoelectric actuators. For example, Harvard University [1; 2; 3] developed \"RoboBee\" and demonstrated stable tethered flight of a piezoelectric-actuated vehicle with a 30-mm wing span. This robot is equipped with piezoelectric Pb(Zr,Ti)O\({}_{3}\) (PZT) bimorph actuators and lever mechanisms to amplify and transmit the output displacement of the actuator to wings. To realize fully untethered flight, an energy-efficient design is important. Because a flapping-wing actuator includes an electrical system, aerodynamic effects, and mechanical parts, a model integrating these three physics is required for detailed characterization. Several studies have reported mechanical-aerodynamic coupling models with nonlinear aerodynamic force [4; 5] using semi-empirical quasi-steady aerodynamic theory [6; 7; 8; 9]. The reported results showed that the models were in good agreement with experiments. In contrast to aerodynamics, the integration of electrical models has received less attention. For instance, a research group at UC Berkeley first presented an integrated model of a flapping-wing actuator that integrated the electric, aerodynamic, and mechanical parts to estimate the lift force and power consumption [10; 11; 12]. However, in their model, the aerodynamic force was approximated as a linear damping although it is basically nonlinear. The Harvard University research group considered the scaling of their robot on lift force, weight, and power consumption [13]. Because they focused on refining the design based on a specific prototype, their approach was heuristic, and they have not presented a deterministic model. Another group reported system modelling based on the measured response of the actuator [14]. This approach was very effective for designing a control system with an existing actuator; however, it was difficult to apply to the prediction of a newly designed actuator. A research group at the University of Southern California developed an MAV called \"Robo Raven\", a larger, bird-scale vehicle, and reported the dynamic modelling of motor-driven flapping-wing MAVs and validation with flight data [15; 16]. Another group at the California Institute of Technology developed \"Robobat\", a bat-inspired MAV, and reported aerodynamics-based model optimization of the lift force [17]. However, these studies did not confirm the scalability of their models down to insect-scale vehicles. We believe that constructing an integrated model for an insect-scale actuation system can significantly contribute to the development of flapping-wing MAVs. [PERSON] et al. analyzed the conceptual design of insect-scale flapping-wing MAVs with a generalized (unspecified) actuator and aerodynamic force models [18]. They theoretically reported on how flight endurance was limited by power consumption without any experiments. In light of the abovementioned previous studies, in the present study, we focused on the experimental validation of a flapping-wing actuator based on a piezoelectric actuation system. We present a model that integrates a flapping-wing actuator with a piezoelectric unimorph actuator and a several-tens-of-millimeter-long wings, as reported previously [19; 20].