# 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  
The Delfly II, shown in Figure 4, is a bioinspired ornithopter, which is configured with four flapping wings and an inverted 'T' tail. Its weight is 17 g and is capable of hovering flight as well as forward flight at the speed of 8 \(ms^{-1}\)[93]. The experimental flight was conducted in an experimental chamber, which was equipped with a tracking system capable of recording the position of a small reflective marker at 200 Hz frequency. The position and altitude were tracked via the use of eight markers placed on its structure. A set of longitudinal elevator input with step, doublets, and triplets with a reference time of 0.3 s was programmed into its autopilot to acquire a full dynamic response over a set of frequencies. The framework for the system identification was defined as follows. The first step toward the system identification was observation/estimation of the state variables of the system from the provided position data. This process was named as flight path reconstruction in [129]. Then, these estimated and available data were used to reconstruct the indirectly measurable states such as angle of attack and refinement of the estimated states. The whole reconstruction was carried out in two reference frames. The tracking system frame was named as the inertial reference frame and was indicated with a subscript \(I\) at the respective coordinate axes so that the positive \(\overset{\rightarrow}{Z}_{I}\) was pointing upward. The frame fixed with body was identified by a subscript b at the respective frame. The positive was described along the forward path, \(\overset{\rightarrow}{\
i}_{b}=normalized\)\((\overset{\rightarrow}{i}_{b}\times\overset{\rightarrow}{W}_{LR})\), where \(\overset{\rightarrow}{i}_{b}\) is the unit vector along \(\overset{\rightarrow}{x}_{b}\) and \(\overset{\rightarrow}{W}_{LR}\) is a unit vector from the left to right wing marker vector. Furthermore, \(\overset{\rightarrow}{y}_{b}\) was defined along the \((\overset{\rightarrow}{x}_{b}\times\overset{\rightarrow}{z}_{b})\). Consequently, two reference frames were defined for the analysis and reconstruction of the states. The Euler angles were considered according to the standard literature approach of yaw (\(\psi\)), pitch (\(\theta\)), and roll (\(\phi\)). Any vector \(\overset{\rightarrow}{v}_{l}\) in the inertial reference frame can be defined in the body frame coordinated via the following relation. \[\overset{\rightarrow}{v}_{b}=R_{bl}\overset{\rightarrow}{v}_{l}, \tag{1}\]  
where \(R_{bl}\) is the standard rotation matrix [6,7]. It is witnessed that in the near hover configuration of the DelFly, the yaw angle (\(\psi\)) and roll angle (\(\phi\)) observed an abrupt variation of (180\({}^{\circ}\)). The Euler angles' yaw (\(\psi\)), pitch (\(\theta\)), and roll (\(\phi\)) were converted into new coordinates (\(\phi_{r}\)), (\(\theta_{p}\)), and (\(\psi_{y}\)), which can be above (90\({}^{\circ}\)) and to keep them valid for the aircraft equation of motion [6,7]. In addition, the dynamic equations were allowed, via the above coordinated conversion, to remain valid for the pitch angles in the interval (\(-\pi/2\), \(3\pi/2\)). From the available data, state reconstruction including the angular rates, angular accelerations, linear velocities, and linear acceleration was accomplished via the discrete time approach, and the effects of high oscillatory modes were reduced by using a 3 rd order Butterworth low pass filter with the cut-off frequency of 30 Hz. The angular velocities were recovered via the following equation:  
\[\left[ \begin{array}{c}p\\ q\\ r\end{array} \right]_{t}=\left[ \begin{array}{c}\phi_{rt}-\phi_{rt-1}\\ \theta_{pt}-\theta_{pt-1}\\ \psi_{yt}-\psi_{yt-1}\end{array} \right]\times\tau=\left[ \begin{array}{c} \Delta\phi_{r}\\ \Delta\theta_{p}\\ \Delta\psi_{y}\end{array} \right]\times\tau. \tag{2}\]  
To obtain the full nonlinear dynamic equations for the entire flight envelope of the aircraft is quite challenging.