## 3 Dynamic Model of UAV System and Tracking Differentiator  
### UAV Dynamics and Problem Formulation  
A rotorcraft UAV has a mechanism to generate the required forces and torques. The nonlinear dynamics in the body-fixed frame, as shown in Figure 1, are expressed as  
\[\dot{P}=v, \tag{11}\] \[m\dot{v}=R_{t}F-mg\varepsilon_{z}+F_{d},\]  
\[\dot{\Theta}=R_{r}\omega \tag{12}\] \[J\dot{\omega}=-\omega\times J\omega+\tau+\tau_{d},\]  
where \(P=[x,\,y,\,z]^{T}\) is the position vector, \(\Theta=[\phi,\ \theta,\ \psi]^{T}\) is the orientation Euler angle vector, \(v=[u,v,w]^{T}\) and \(\omega=[p,\ q,\ r]^{T}\) are the velocity vectors in the body frame, respectively. \(m\) denotes the body mass including payload, \(J=diag(J_{\phi},\,J_{\theta},\,J_{\psi})\) denotes the moment of inertia vector, and \(\varepsilon_{z}=[0\ 0\ 1]^{T}\). \(F_{d}\) and \(\tau_{d}\) denote the uncertainties along the translation and rotational axes directions, respectively, including the gyroscopic effect, parameter change, payload variation, rotor fluctuation, and aerodynamic drag. A rotary transformation matrix \(R_{t}\) from the body frame to the navigation frame and \(\omega\) are given as follows:  
\[R_{t}=R_{\varphi}R_{R}R_{\phi}\] \[=\begin{bmatrix}\phi\sigma\psi&s\phi s\theta\varepsilon\psi+c \phi\varsigma\psi&c\phi s\theta\varepsilon\psi+s\phi s\psi\\ c\delta s\psi&s\phi s\psi+c\phi c\varphi&c\phi s\theta\varepsilon\psi-s\phi c \psi\\ -s\vartheta&s\varphi c\vartheta&c\phi c\theta\end{bmatrix}, \tag{13}\] \[\omega=R_{t}\dot{\varphi}=\begin{bmatrix}1&0&-s\vartheta\\ 0&c\phi&c\delta s\phi\\ 0&-s\phi&c\delta c\phi\end{bmatrix}\begin{bmatrix}\dot{\phi}\\ \dot{\psi}\end{bmatrix}, \tag{14}\]  
where \(\varphi=\left[\phi\ \theta\ \psi\right]^{T}\) and let \(\sin(\cdot)\) and \(\cos(\cdot)\) to be denoted as \(s(\cdot)\) and \(c(\cdot)\), respectively.  
A quadrotor is an underactuated system because it has six degree of freedom and four control inputs, which are the total thrust \(u_{z}\), produced by the four rotors, and the torques \((u_{\phi},\,u_{\theta},u_{\psi})\) are obtained by varying the rotor speeds. \(F=[0,\,0,\,u_{z}]^{T}\) is the resulting lift force vector and \(\tau=[u_{\phi},\,u_{\theta},\,u_{\psi}]^{T}\) is the torque vectors in the navigation frame acting on the airframe. The control input \((u_{z},\,u_{\phi},\,u_{\theta},\,u_{\psi})\) and the rotor speeds \((\omega_{1},\,\omega_{2},\,\omega_{3},\,\omega_{4})\) have the relationship expressed by  
\[\begin{bmatrix}u_{z}\\ u_{\phi}\\ u_{\theta}\\ u_{\psi}\end{bmatrix}=\begin{bmatrix}b&b&b&b\\ 0&-lb&0&lb\\ -lb&0&lb&0\\ d&-d&d&-d\end{bmatrix}\begin{bmatrix}\omega_{2}^{2}\\ \omega_{2}^{2}\\ \omega_{3}^{2}\\ \omega_{4}^{2}\end{bmatrix}, \tag{15}\]  
where \(b>0\) and \(d>0\) are constants and \(l\) is the distance from the rotors to the center of mass. Therefore, the following translational and rotational dynamics are obtained:  
\[\ddot{x}=\frac{1}{m_{0}}(c\phi s\theta c\psi+s\phi s\psi)u_{z}+f_{dx}, \tag{16}\]  
\[\ddot{y}=\frac{1}{m_{0}}(c\phi s\theta s\psi-s\phi c\psi)u_{z}+f_{dy}, \tag{17}\]  
\[\ddot{z}=\frac{1}{m_{0}}c\phi c\phi u_{z}-g+f_{dz}, \tag{18}\]  
\[\ddot{\phi}=\frac{I_{y}-J_{z}}{J_{x}}\dot{\phi}\dot{y}+\frac{J_{r}}{J_{x}}\dot {\phi}\Omega_{r}+\frac{l}{J_{x}}u_{\phi}+\tau_{d\phi}, \tag{19}\]  
\[\ddot{\theta}=\frac{I_{z}-J_{x}}{J_{y}}\dot{\psi}\dot{\phi}-\frac{I_{r}}{J_{y} }\dot{\phi}\Omega_{r}+\frac{l}{J_{y}}u_{\theta}+\tau_{d\theta}, \tag{20}\]  
\[\ddot{\psi}=\frac{J_{x}-J_{y}}{J_{z}}\dot{\phi}\dot{\phi}+\frac{l}{J_{z}}u_{ \psi}+\tau_{d\psi}, \tag{21}\]  
where \(m_{0}\) denotes the nominal body mass and \(f_{di}\), \(i=x,\,y,\,z\), and \(\tau_{dj}\) denote the uncertainties at each axis.