# 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
#### 3.2.1 System Identification of In-Flight Dynamics of Robotic Bee  
In this section, the top-down approach to system identification of Robotic Bee [5] is presented. A physics-based model was developed and describes the effects of aerodynamic drag to the translational and rotational dynamics of vertically taking-off FWMAVs. The system identification strategies outlined in [55; 126; 127; 128; 136] aimed to quantify the aerodynamic effects and their response to the actuators' commands in stable air vehicles. However, in unstable systems, closed-loop treatment is carried out to keep away the undesirable coupling between input commands and dynamic responses in [137; 138]. In the context of close loop identification, the Grey box model identification is one of the suitable strategies [138]. In order to carry out system identification, the original nonlinear model is presented in such a way that the vector of unknown parameters can be evaluated via linear regression. This strategy provides certain benefits such as global extremum and analytic solutions. This helps in eliminating the tuning of the optimizing parameters and eliminating the need of a local search that may increase the relative computational complexity [139]. At this point, the in-flight aerodynamic model could be obtained by approximating the instantaneous drag, \(D\), on the two wings, with high Reynolds number law [140; 141]. \[D=2\times\frac{1}{2}\rho SC_{d}v^{2}, \tag{14}\]  
where \(\rho\) is the density of the fluid; \(S\) is the surface area of the wing; \(C_{d}\) is the coefficient of drag; and \(v\) is the speed of the wing relative to air. In the case of deviation of the robot from hovering and with additional damping force, the drag force contributes to the rotational damping, thus, the alternate form of (14) is [5]  
\[D\approx mbu+m\alpha\omega \tag{15}\]where \(m\) is the mass of the robot; \(b\) is the normalized drag force; \(u\) is the speed in forward direction; \(\alpha\) is the angle of attack; and \(\omega\) is the rotational speed of the robot. The resulting drag moment can be expressed as follows:  
\[M\approx Jc\omega+J\beta u, \tag{16}\]  
where \(c\) and \(\beta\) are the rotational damping coefficients normalized by inertia. Due to (15) and (16), the viscous forces and torque from the wings, which dominate the drag forces and are shown critical to the open loop dynamics and stability of FWMAVs [34; 141; 142], can also be treated as linear functions. The translational dynamic equation of the robot can be modeled as follows:  
\[m\ddot{X}=m\Gamma\dot{Z}+W+F_{a}, \tag{17}\]  
where \(F_{a}\) represents the lumped aerodynamic forces; \(W=mg\) represents the gravitational force; \(\Gamma\) is the normalized thrust with dimension of acceleration; and \(\dot{Z}\) represents the axis along the body of the robots. It includes the additional aerodynamic force due to drag forces on the wings and viscous forces on the body that can be further modeled as linear functions of translational and angular velocities. Equation (17), with explicit expression of \(F_{a}\), can be re-expressed as follows:  
\[\frac{d}{dt}\dot{X}+\omega\times\dot{X}=\left[ \begin{array}{c}0\\ 0\\ \Gamma\end{array} \right]-g\left[ \begin{array}{c}R_{31}\\ R_{32}\\ R_{32}\end{array} \right]+\left[ \begin{array}{c}-b_{x}\dot{x}-\alpha_{x} \omega_{y}\\ -b_{y}\dot{y}+\alpha_{y}\omega_{x}\\ -b_{z}z\end{array} \right], \tag{18}\]  
where \(\mathrm{b_{i}}\) and \(\alpha_{i}\) are contributed from the body velocity and angular velocity with \(i\in\{x,y,\)\(z\}\). When the FWMAV is not in perfect hovering, then it is affected by additional torques, therefore, the resultant torque on the robot can be expressed by the following rotational dynamic equations:  
\[\sum\tau=J\frac{d\omega}{dt}+\omega\times J\omega=\tau_{c}+\tau_{a}+J_{d}, \tag{19}\]  
where \(J_{d}\) is the unknown normalized disturbance. Note that in this presentation, the roll and pitch were focused and the aerodynamic contributions were modeled as linear in its variables.