# 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
#### 3.2.3 Computational Fluid Dynamic Approach Based System Identification  
The aerodynamics and system modeling of the insects and insect mimicking FWMAVs have attracted a wide number of researchers [150; 151; 152; 34; 46; 85; 153; 154; 34]. In recent developments, the dynamic flight stability of insects has gained much attention [86]. They focused on a formal quantitative study regarding the dynamic flight stability of desert locust _Schistocera Gregaria_. In their work, they estimated the stability derivatives via experimental study \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Standard Deviation** \\ \hline \(X_{u}\) & \(-12.3\) & \(4.0\%\) \\ \(Z_{w}\) & \(-4.7\) & \(9.1\%\) \\ \(M_{u}\) & \(547\) & \(3.6\%\) \\ \(M_{q}\) & \(-33.3\) & \(4.6\%\) \\ \(X_{\Phi_{lf}}\) & \(12.7\) & \(12.2\%\) \\ \(X_{\beta}\) & \(20.7\) & \(7.6\%\) \\ \(Z_{\Phi}\) & \(-17.4\) & \(4.8\%\) \\ \(M_{\Phi_{lf}}\) & \(6028\) & \(3.6\%\) \\ \(M_{\delta}\) & \(-2826\) & \(3.8\%\) \\ \(X_{\kappa ud}\) & \(-12360\) & \(5.0\%\) \\ \(M_{a\,ud}\) & \(-87.74\) & \(10.7\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: This indicates the estimated stability and control derivatives [143]. Figure 6: The system identification block diagram for the full nonlinear model of _Drosophila Melanogaster_ flight dynamics [117]. that were further studied for eigenvalue analysis to recognize different stability modes like the unstable oscillatory mode, stable fast subsidence, and slow subsidence mode. Note that the experiment carried out was subjected to some limitations such as that the insect had to be tethered and the reference flight might not have been in equilibrium. In practice, the presence of such limitations may make the estimation of some morphological parameters impossible. In order to pave the way to estimate stability derivatives for all conditions and different flight maneuvers, [112] presented a CFD approach to measure the stability derivatives of a hovering bumblebee while capturing the flight data used in the study of [84; 85], and the stability derivatives of beetle-mimicking FWMAV in [155]. In this study, they first determined the equilibrium condition, while taking account of the rigid body assumption, and utilized the flow equations and solution methodology presented in [48] to compute the stability derivatives for the longitudinal flight dynamics. In the stability analysis, once again, they identified the same mode as were concluded in [86]. In [156], the work of [112] was extended to carry out a formal analysis of the stability, controllability, and the stabilizing control of the hovering insect. The stability analysis resulted in the same modes as identified in [86]. Therefore, a control action in the feedback loop was needed to stabilize the unstable oscillatory mode, and stability augmentation was pursued for the slow subsidence mode. The stabilization was done by feeding back a controlled input (i.e., the changes in the mean stroke angle as a function of pitch attitude, pitch rate, and horizontal velocity), whereas the second case was handled via a feedback control input (i.e., the changes in the stroke amplitude as a function of vertical velocity). This study was fascinating for further study in different flight envelopes. In the context of longitudinal stability analysis, in [111], the dynamic stability of the beetle-mimicking FWMAVs [43] was investigated. The authors followed the trim condition determination strategy of [86], and CFD based stability derivatives were computed. The stability analysis was concluded as very similar to [86; 111]. In order to determine the flight stability, one needs to have the identified dynamics for both longitudinal and lateral motions. Currently, a quantitative analysis for the lateral dynamic stability of a hovering insect is addressed in [157] via the CFD strategy [111; 156]. In this case, they focused on the drone fly instead of the bumblebee.