## 2 Mathematical Model of Quadrotor  
In this section, the dynamic model of the quadrotor will be presented. The quadrotor is a type of rotorcraft UAV driven by four rotors located at the end of the body frame with equal distance from the center mass of the quadrotor's body. It is typically low-cost and small in size with a unique capability of vertical, stationary, and low-speed flight [5]. Unlike most helicopters, the quadrotor uses two sets of identical fixed pitch propellers that spin in the opposite direction; one pair rotating in a clockwise direction and one pair rotating in a counterclockwise direction [9]. With this configuration, the torque produced by the rotor pair can be eliminated. The quadrotor is an underactuated system because it has six degrees of freedom (DOF) (three translational and three rotational) that need to be controlled by four control inputs, the thrust force, and the aerodynamic torques [45]. Therefore, modeling such a system is not an easy task. Before that, the following assumptions need to be considered [70]:  
1. The structure of the quadrotor is rigid and symmetric. 2. The propellers are rigid. 3. Thrust and drag forces are proportional to the square of the rotor's speed. 4. The center of mass and the origin of the coordinate system coincide. The reference frame that describes the absolute position of the quadrotor in the three-dimensional space \(E\left(e_{x},\,\,e_{y},\,\,e_{z}\right)\) is denoted by \(B\left(b_{x},\,\,b_{y},\,\,b_{z}\right)\) that is attached to the center of the quadrotor and the inertial/earth frame is denoted by \(E\big{(}e_{x},\ e_{y},\ e_{z}\big{)}\) that is fixed on a specific point in space. By using Newton-Euler formulism, the complete dynamic model of the quadrotor can be expressed as follows [70]: \(q=(x,\ y,\ z,\ \phi,\ \theta,\ \psi)\ \in\ (x,\ y,\ z)(\phi,\ \theta,\ \psi)\)  
\[ \begin{array}{l}\ddot{x}=(cos\phi sin\epsilon cos\psi+sin\phi sin\psi)\frac{1} {m}U_{1}\\ \ddot{y}=(cos\phi sin\phi sin\psi-sin\phi cos\psi)\frac{1}{m}U_{1}\\ \qquad\qquad\ddot{z}=-g+(cos\phi cos\theta)\frac{1}{m}U_{1}\\ \ddot{\phi}=\dot{\phi}\dot{\psi}\Big{(}\frac{l_{y}-l_{z}}{l_{z}}\Big{)}-\frac {l_{x}}{l_{x}}\dot{\theta}\Omega+\frac{1}{l_{x}}U_{2}\\ \ddot{\theta}=\dot{\phi}\dot{\psi}\Big{(}\frac{l_{x}-l_{y}}{l_{y}}\Big{)}+ \frac{l_{x}}{l_{y}}\dot{\phi}\Omega+\frac{1}{l_{y}}U_{3}\\ \qquad\qquad\ddot{\psi}=\dot{\phi}\dot{\phi}\Big{(}\frac{l_{x}-l_{y}}{l_{z}} \Big{)}+\frac{1}{l_{z}}U_{4}\end{array}  \tag{1}\]  
where \(m\) is the total mass of the quadrotor, \(I_{xyz}\) is the quadrotor's inertia in the \(x,\ y,\) and \(z\)-axis, respectively, \(g\) is the constant gravitational acting on the quadrotor's body, \(J_{r}\) is the inertia of the rotor, \(\Omega\) is the disturbance and can be described as \(\Omega=-\omega_{1}+\omega_{2}-\omega_{3}+\omega_{4}\), and \(U_{i}(i=1,\ 2,\ 3,\ 4)\) is the control input to the quadrotor and can be presented as in Equation (2) below:  
\[ \begin{array}{l}U_{1}=u_{T}=b\big{(}\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3 }^{2}+\omega_{4}^{2}\big{)}\\ U_{2}=\tau_{\phi}=lb\big{(}-\omega_{2}^{2}+\omega_{4}^{2}\big{)}\\ U_{3}=\tau_{\theta}=lb\big{(}-\omega_{1}^{2}+\omega_{3}^{2}\big{)}\\ U_{4}=\tau_{\psi}=d\big{(}\omega_{1}^{2}-\omega_{2}^{2}+\omega_{3}^{2}-\omega_ {4}^{2}\big{)}\end{array}  \tag{2}\]  
where \(b\) is the thrust coefficient, \(d\) is the drag coefficient, and \(w\) is the \(i\)th rotor's angular speed. In this study, the quadrotor parameters are taken from 20 different journals available in the literature. The parameters and the respective references are presented in Table 1 below. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Reference**} & \multirow{2}{*}{ \begin{tabular}{c} **Mass, m** \\ **(kg)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Arm Length,** \\ **L (meter)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Inertia** \\ **x-Axis, Ix** \\ **(kgm\({}^{\star}\))** \\ \end{tabular}  } & \multirow{2}{*}{