# 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)
### Tailed Flapping Wing Micro-Air Vehicles
#### 3.1.1 System Identification of DelFly II Based on Least Mean Square Estimation
In this work, the nonlinear model was approximated with a linear model by following [55,130,131] for sufficient small excitations in the neighborhood of the stationary trimmed regime. It was assumed that the dynamics of the flyer could be modeled as [PERSON]'s Euler equations of motion while further assuming that the aircraft body is rigid with constant mass and no changing inertia, subject to no flapping in the absence of wind. Note that the flapping was modeled as a thrust force. The aerodynamic model structure was chosen as follows:  
\[F_{i}=C_{F_{0}}+\sum_{S=1}^{n}C_{F_{s}}\text{-}S;\ \ S=\left\{\phi_{r},\ \theta_{p},\psi_{y},\ u,\ v,\ w,\ p,\ q,r,\ \alpha,\beta,\gamma,\delta_{f},\ \delta_{e},\ \delta_{r}\right\} \tag{3}\]  
where \(C_{F_{0}}\) is the affine coefficient; \(C_{F_{s}}\) the states' coefficient vector; and \(S\) is the state vector. These coefficients as well as the states were estimated via the discrete time differentiation. Figure 4: The DelFly II schematic diagram [93]. Equation (3) represents the full model structure so that the aerodynamic forces and moments are expressed as linear functions of states. The reduced model is expressed in a form, which only needs the measurable available states. The core objective of the reduced model was to use it in the flight controller design. The reduced model structure appears as follows:  
\[ \begin{split} X&=X_{0}+X_{q}\theta+X_{\theta}\theta+X_{ \delta_{e}}\delta_{e}+X_{\delta_{f}}\delta_{f},\;Y=Y_{0}+Y_{P}P+Y_{\phi}\phi+Y_{ \delta_{e}}\delta_{r}+Y_{\delta_{f}}\delta_{f},\\ &\quad Z=Z_{0}+Z_{q}\theta+Z_{\theta}\theta+Z_{\delta_{e}}\delta_{ e}+Z_{\delta_{f}}\delta_{f},\;L=L_{0}+L_{\theta}\theta+L_{\delta_{e}}\delta_{r}+ l_{\delta_{f}}\delta_{f},\\ &\quad M=M_{0}+M_{\theta}\theta+M_{\delta_{e}}\delta_{e}+M_{ \delta_{f}}\delta_{f},\;N=N_{0}+N_{\theta}\theta+N_{\delta_{r}}\delta_{r}+N_{ \delta_{f}}\delta_{f}.\end{split}  \tag{4}\]  
The parameter estimation was carried out via the conventional least square estimator. The parameter estimation of the aforesaid forces and moments can be recovered, for example, as follows:  
\[ \begin{split}\left[\begin{array}{c}\hat{\chi}_{0}\\ \hat{\chi}_{q}\\ \hat{\chi}_{\theta}\\ \hat{\chi}_{\delta_{e}}\\ \hat{\chi}_{\delta_{f}}\\ \end{array}\right]&=\left(R^{T}R\right)^{-1}R^{T}X,\quad\quad R=\left[ \begin{array}{cccc}1&q(1)&\theta(1)&\delta_{e}(1)&\delta_{f}(1)\\ 1&q(2)&\theta(2)&\delta_{e}(2)&\delta_{f}(2)\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 1&q(N)&\theta(N)&\delta_{e}(N)&\delta_{f}(N)\\ \end{array}\right]\end{split}  \tag{5}\]  
where \(R\) is the regression matrix with \(N\) observations. The input profile on the elevator to conduct five longitudinal flight maneuvers in the near hover and slow forward flight scenarios included step, doublet, and triplet commands. The parameters were estimated for different flight scenarios. The proximity in the estimated parameter was helpful in choosing a single model for the control design purpose. The estimated parameters are displayed in Table 1[93]. The model validation was also conducted via two different strategies. In the first method, the validation of the maneuvers was done. In this case, the estimated parameters in the identification cycle were utilized to forecast the aerodynamic forces and moments of other maneuvers. On the other hand, in the second scenario, the validation was done via simulations. In this case, the estimated parameters were used in the nonlinear dynamic simulator, then the states and flight path were reconstructed and compared with the actual flight test data. The system identification and model validation were carried out via the Grey box method and the parameters were estimated for two different dynamic models, which were named as full order and reduced order for different flight regimes. It was concluded that the developed models were successfully obtained. However, further study is still required to achieve the prediction of the aerodynamic forces and moments with considerable accuracy for a longer period of time and full autonomous flight missions.