# 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)
### Tailless Flapping Wing Micro Air Vehicles
#### 3.2.2 System Identification via Frequency Approach of _Drosphilla melanogaster_  
Frequency-based system identification methodologies were studied in [116; 117; 142; 143]. In this section, the methodology in [143] for the estimation of stability and control derivatives is outlined. Since the flight dynamic equations, in general, are very complex, the original quasi-steady model proposes the lift and drag as functions of the instantaneous wing motion. Therefore, some basic assumptions are made to proceed to the controllability analysis. The first assumption portrays the quasi-steady model [88] as an experimental fit to the scaled dynamics of the flapping flier. However, the actual dynamics contain the lift and drag forces as functions of instantaneous wing motion, the aerodynamic forces are functions of the body motion, and the mechanism was the focus of the study in [144; 145; 146; 142], which gave birth to passive aerodynamic damping. The second assumption is that the instantaneous lift and drag forces on the rigid body are approximated by wing stroke average forces, and furnish the stability and control derivatives of the aircraft. In this scenario, the time dependency is removed via the averaging theory [147]. Consequently, a linear time-invariant (LTI) system of the following form is equipped. \[\dot{x}=Ax+Bu, \tag{27}\]where \(A\in R^{n\times n}\) represents the state matrix; \(B\in R^{n\times p}\) is the control derivative matrix; and \(u\in L_{2}^{p}[0,\,\infty)\) indicates the controlled input history and \(x\in L_{2}^{n}[0,\infty)\) points to the state history of the system. The aforementioned components smoothen the way to system identification. To obtain stability and control derivatives and to perform nonlinear simulations, the system was considered as a rigid body in combination with quasi-steady aerodynamics and the state perturbations caused by the state deviation from the equilibrium. Similar work can be found in [142] where the quasi-steady aerodynamics formulation was treated as state perturbation from a reference equilibrium condition via proper definition of the wing tip speed and the wing angle attack. The current subsection focused on the study in [143], with the aerodynamic model outlined in [142], which included the effects of the rotational and translational rigid body motions. In the underlying strategy, the trim kinematics [99,142] were treated as periodic, represented by \(\phi(t)\), in a stroke plane, which makes an angle \(\beta\) with the horizontal. Both the left and right wing were actuated simultaneously with the same control input for the said longitudinal motions. The control inputs were defined as stroke plane inclination \(\beta_{c}=\frac{1}{2}\left(\beta_{R}+\beta_{L}\right)\)[117,148], stroke plane offset \(\phi_{0 ff}=\frac{1}{2}\left(\phi_{0 ff_{R}}+\phi_{0 ff_{L}}\right)\)[143,148], and asymmetric wing angle \(\alpha_{ad}=\frac{1}{2}(\alpha_{u}-\alpha_{d})\)[117,143,148]. These inputs are shown in Figure 5.  
Note that the input \(\beta_{c}\), which physically represents the tilting of the stroke plane, is used to generate the pitch moment and forward force [142], \(\phi_{off}\) physically represents the force shift of the wing sweep for pitch moment generation and \(\alpha_{au}\) in a similar way represents the upstroke or downstroke asymmetry in wing angle relative to the stroke plane. This input is used to generate the forward force. The longitudinal dynamics of the insect were excited by applying the chirp signal at each input. The overall transfer function was found via the spectral analysis of the input transfer function \(G_{I}(\omega)\) and the output transfer function \(G_{O}(\omega)\). \[G_{s}(\omega)=\frac{G_{O}(\omega)}{G_{I}(\omega)}, \tag{28}\]  
For the sake of completion, the input/output response in terms of magnitude, phase, and coherence for \(\phi_{0 ff}\) to \(\theta\) is shown in Figure 6. The haltere-based feedback on the pitch axis was used to make the system matrix Hurwitz.