# 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
#### 3.1.3 System Identification of Kinkade Slow Hawk Ornithopter  
Grauer, in [135], presents the flight dynamics modeling of an avian-based onirthopter that was performed based on nonlinear multi body kinematics and dynamics, which contain the tail and wing aerodynamics and gravitational effects. In this review, we were interested in the system identification approaches, which were performed to obtain the tail and wing aerodynamics through wind tunnel data and free flight data, respectively. The system identification performed was based on a linear least-squares estimation to model the longitudinal aerodynamic forces and torques of the ornithopter. Step-wise regression and equation error in the time domain were utilized to determine the model structure. Once the model structure was determined, parameter estimation was performed using the maximum likelihood cost function,  
\[J(\phi)=\frac{1}{2}\sum_{i=1}^{n}\mathbf{e}^{T}\mathbf{W}^{-1}\mathbf{e},\]  
where \(\phi\) is the vector that contains all the unknown parameters; \(\mathbf{e}\) is the residual vector between the outputs and measurements; and \(\mathbf{W}\) is the inverse noise covariance matrix. This estimator was simplified by the use of equation-error and output-error methods. The equation-error requires process noise assumption, and uses the state derivatives as the measurements, which resulted in a linear estimation problem. Meanwhile, output-error utilized measurement noise and the states as the measurements. This method resulted in a nonlinear and iterative solution that was relatively more accurate than that of the equation-error. In obtaining the tail aerodynamics, wings were removed and the ornithopter was mounted on a fixed platform. The tail aerodynamics were assumed independent of those of the wings. The system identification was performed using wind tunnel data from four different tests with different free stream velocities and varied angle of attack and sideslip angle. The outcomes of the identification were aerodynamics parameters with a nonlinear model structure. The parameters were obtained through parameter estimation using time-domain equation error [135]:  
\[ \begin{array}{l}C_{x}=C_{x0}+C_{x_{2}}a^{2},\\ C_{y}=C_{y\mu\beta}a\beta,\\ C_{z}=C_{z0}+C_{z\alpha}a+C_{z\beta}\beta,\\ C_{l}=C_{l_{\beta}}\beta+C_{l_{\mu\beta}}\beta,\\ C_{m}=C_{m0}+C_{m\alpha}a+C_{m\beta}\beta+C_{m\beta}\beta^{2}+C_{m_{\alpha}V} \alpha V,\\ C_{n}=C_{n\beta}\beta.\end{array} \]  
The wing aerodynamics were modeled through system identification using flight data, which were obtained using a visual tracking system. The ornithopter, fitted with markers on its body, was flown straight and at a level flight with minimized atmospheric disturbances. System identification was conducted to obtain the mathematical models for the heave force, longitudinal force, and pitching moments that are generated by wings. The generalized aerodynamic forces due to each wing were given as [135],  
\[\mathbf{a}_{wing}(\mathbf{p},\mathbf{v})=\tau-\left[\mathbf{M}(\mathbf{p}) \mathbf{v}+\mathbf{C}(\mathbf{p},\mathbf{v})\mathbf{v}+\mathbf{g}(\mathbf{p}) +\mathbf{a}_{tail}(\mathbf{p},\mathbf{v})\right],\]  
where \(\mathbf{p}\) and \(\mathbf{v}\) are the generalized position and velocity states, respectively; \(\mathbf{M(p)}\) is a generalized mass matrix; \(\mathbf{C(p},\mathbf{v})\) contains nonlinear coupling forces arising from centripetal and Coriolis accelerations; \(\mathbf{g(p)}\) describes gravitational effects; and \(\mathbf{a(p},\mathbf{v})\) describes aerodynamic effects. The forcing vector \(\mathbf{\tau}\) corresponds to forces on the fuselage center of mass,torques on the fuselage center of mass, and torques on the wings. Time-domain equation error and step-wise regression were also utilized in determining the model structure, which was nonlinear in terms of the state variables [135].