# Review on System Identification and Mathematical Modeling of Flapping Wing Micro-Aerial Vehicles
## 3 System Identification and Mathematical Modeling of Flapping Wing Micro Air Vehicles (FWMAVs)  
The ultimate objective, in the development of FWMAVs, is securing an autonomous flight in order to utilize them for the different tasks mentioned earlier. Therefore, a first step toward the establishment of an autonomous flight is the modeling of the flapping kinematics, aerodynamics, and flight dynamics of FWMAVs. This task is particularly challenging due to inherent instability, nonlinear, time-varying, and periodic nature of the vehicles' dynamics, especially during dynamic aerial maneuvers such as sharp and abrupt turns, obstacle avoidance, and mid-air prey capturing. The aerodynamics and flight dynamics of such maneuvers, even for natural fliers, have not been fully uncovered yet. Additionally, on-board sensors, flapping mechanism, and actuation systems may add complexities to the overall dynamics of the MAVs. Furthermore, for insect-mimicking FWMAVs, due to their small size, the selection of hardware for their avionics and flight control systems are limited. However, in this review, we focused on system identification approaches in order to model the FWMAVs without actually going through the details on hardware selection. In general, two complementary strategies are applied to model FWMAVs. These strategies are named as bottom-up and top-down [94]. The bottom-up approach mainly focuses on the fundamental approaches to explain and model the behavior and interaction of air motion with the flapping wings. This is essentially modeling with the first principlesbased on the well-studied aerodynamics and wing kinematics of natural flapping fliers (see [50] for a detailed summary). The practical system development of bio-inspired FWMAVs remains the later part in such developments. On the other hand, in the top-down approach, the construction and developments of the real MAVs take place prior to the theoretical developments. In order to obtain the mathematical model(s) of the FWMAVs, the first step is the system identification, which means a strategy for acquiring a known mathematical model (for instance, aerodynamic modeling, system stability derivatives, and system's control derivatives) of the system while utilizing the available measured data of the system's input and output signals produced by the MAVs. These available signals may be measured either in time domain or in frequency domain. The main advantages of performing system identification in modeling are that the process uses real-time data and can take into account design changes. The latter is particularly useful for control system design. Through the process, the value of specific design parameters and their effects on the dynamics of MAVs can be determined, and the structure of the mathematical models obtained is not unique, which gives the freedom to the engineers/researchers to select the models that suit the objectives of the modeling. For examples, these models may be suitable for (hardware-in-the-loop) simulations, analysis, and/or control. With regard to this stage, the FWMAVs and their flight have been enlightened in a general way, and now the objective is to present the system identification in detail. It is generally conducted either for equivalent linear systems or core nonlinear systems. This may further be classified into time-domain and frequency-domain based identification for tailed and tailless FWMAVs. It is obvious that the dynamic modeling of FWMAVs always considers the mechanism kinematics of linkages, which connect the motors and actuators to FWs [113], aerodynamic forces [114], and nonlinear phenomena like saturation and hysteresis [115]. Consequently, it results in different equations and realizations of the dynamics. The linear modeling and identification of fixed wing aircraft were extensively studied in [55; 114]. The identification strategies for modeling FWMAVs or natural fliers vary from system to system because of their varying kinematics and flight dynamics.