## 3 Equation of Motion  
The motion of the aircraft system is expressed as 6 degrees of freedom. The state derivative equation of the aircraft can be derived using [PERSON]'s law of motion and the following equation is attained:  
\[\left[\begin{array}{c}\dot{v}_{ARS}\\ \dot{\omega}\end{array}\right]=I_{A}\left[\begin{array}{c}F_{B}\\ M_{B}\end{array}\right]-J_{B}\left[\begin{array}{c}v_{ARS}\\ \omega\end{array}\right] \tag{3}\]  
where  
\[\left\{\begin{array}{l}J_{A}=\left[\begin{array}{cc}J_{A11}&J_{A12}\\ J_{A21}&J_{A22}\end{array}\right]=\left[\begin{array}{cc}mI_{3\times 3}&-M\\ \Omega_{D}&I\end{array}\right]^{-1},\\ J_{B}=\left[\begin{array}{cc}J_{B11}&J_{B12}\\ J_{B21}&J_{B22}\end{array}\right]=J_{A}\left[\begin{array}{cc}m\Omega_{D}&- \Omega_{D}M\\ \Omega_{D}M&\Omega_{D}I-V_{D}M\end{array}\right],\\ V_{D}=\left[\begin{array}{cc}0&-w&v\\ w&0&-u\\ -v&u&0\end{array}\right],\;I_{NERT}=\left[\begin{array}{cc}I_{XX}&-I_{XY}&-I_ {XZ}\\ -I_{XY}&I_{YY}&-I_{YZ}\\ -I_{XZ}&-I_{YZ}&I_{ZZ}\end{array}\right],\\ M=\left[\begin{array}{cc}0&-m\Delta z&m\Delta y\\ m\Delta z&0&-m\Delta x\\ -m\Delta y&m\Delta x&0\end{array}\right],\;\Omega_{D}=\left[\begin{array}{cc}0 &-r&q\\ r&0&-p\\ -q&p&0\end{array}\right].\end{array}\right. \tag{4}\]  
In the above equation, \(\Delta x\), \(\Delta y\), and \(\Delta z\) are the distances of the center of gravity (CG) location from the aircraft reference system (ARS) frame; \(\omega=\left[p\;\;q\;\;r\right]^{T}\) is the body rate vector along the ARS frame; \(v_{ARS}=\left[u\;\;v\;\;w\right]^{T}\) is the body velocity vector of ARS frame; \(I_{NERT}\) is the moment of inertia matrix of the aircraft; and \(I_{N\times N}\) is the identity matrix of order \(N\). The ARS frame is the reference body frame to which sensors are attached. Normally, it is chosen as close to the CG as possible. Figure 1 shows the inertial frame \(O_{I}\), ARS frame \(O_{ARS}\), and CG point \(p_{CG}\). The detailed derivation of the above equations can be found in [23; 24; 25; 26], and is described in Appendix A.  
Figure 1: Inertial frame, aircraft reference frame, CG point, and corresponding \(x\), \(y\), and \(z\) basis vectors.  
The \(F_{B}=\left[F_{x},\;F_{y},\;F_{z}\right]^{T}\) and \(M_{B}=\left[M_{x},\;M_{y},\;M_{z}\right]^{T}\) are the external forces and moments, respectively, acting on the CG and they are  
\[\left\{\begin{array}{l}F_{x}=q_{\infty}\overline{S}C_{x}(\alpha,q,\delta_{e}) \\ F_{y}=q_{\infty}\overline{S}C_{y}(\alpha,q,\delta_{e})\\ F_{z}=q_{\infty}\overline{S}C_{z}(\alpha,\beta,p,r,\delta_{x},\delta_{r})\\ M_{x}=q_{\infty}\overline{S}b_{w}\left(C_{l_{trim}}+C_{l_{da}}\delta_{a}+C_{l _{dr}}\delta_{r}\right)\\ M_{y}=q_{\infty}\overline{S}c_{w}\left(C_{m_{trim}}+C_{m_{sk}}\delta_{e} \right)\\ M_{z}=q_{\infty}\overline{S}b_{w}\left(C_{n_{trim}}+C_{n_{sa}}\delta_{a}+C_{n _{fx}}\delta_{r}\right)\end{array}\right., \tag{5}\]  
where \(q_{\infty}=1/2\rho V_{T}^{2}\) is a dynamic pressure with density \(\rho\) and total speed \(V_{T}\); \(\overline{S}\) is the wing area; \(b_{w}\) and \(c_{w}\) are the wing span and chord length; \(C_{x},\;C_{y}\), and \(C_{z}\) are the translational aerodynamic coefficients; \(C_{l_{trim}},\;C_{m_{trim}}\), and \(C_{n_{trim}}\) are the trim rotational aerodynamic coefficients; \(C_{l_{ka}},\;C_{l_{gs}},\;C_{m_{kg}},\;C_{m_{kg}},\;C_{n_{sa}}\) and \(C_{n_{fx}}\) are the actuator rotational aerodynamic coefficients; \(\alpha\) is the angle of attack; \(p,\;q,\;r\) are the body rates along the \(x,\;y,\;z\) ARS axes; and \(\delta_{a},\;\delta_{e},\;\delta_{r}\) are the aileron, elevator, and rudder actuator deflection. The detailed derivation of the above equations can be found in [24].