# Fractional-Order Sliding Mode with Active Disturbance Rejection Control for UAVs
## 2 Quadrotor Dynamics Model  
The given UAV dynamics model is shown in Figure 1, which consists of four independent motors. The propellers are screwed above the motors, and the motor mounts are connected to the flight controller section through the arms to form a rigid cross-frame as a whole. The state vector \(E=[e^{x},e^{y},e^{z}]\) denotes the fixed ground inertial coordinate system, and \(B=[x,y,z]\) denotes the quadrotor body inertial coordinate system. The fixed ground inertial coordinate system and the airframe coordinate system are both right-handed coordinate systems, with x pointing to the east, y pointing to the north, and z pointing to the vertical, upward direction relative to the ground in the fixed ground inertial coordinate system, and O is the origin of the coordinate system.  
Because the UAV has six degrees of freedom (DOF; three positional degrees of freedom and three attitude angular degrees of freedom) in three dimensions but only four rotor rotational speeds on the input side to adjust the lift, the quadrotor is a typical nonlinear underactuated system. Numerous factors need to be considered when modeling it: thrust, moment, gravity, moment of inertia, counter torque, etc. [16]. Referring to the model in [17; 18], the dynamics model of a quadrotor UAV can be described as follows:  
\[\begin{cases}\ddot{x}=\dfrac{1}{m}(\cos\phi\sin\theta\cos\psi+\sin\phi\sin\psi) u_{1}-\dfrac{K_{1}\dot{x}}{m}\\ \ddot{y}=\dfrac{1}{m}(\cos\phi\sin\theta\sin\psi-\sin\phi\cos\psi)u_{1}-\dfrac{ K_{2}\dot{y}}{m}\\ \ddot{z}=\dfrac{1}{m}(\cos\phi\cos\theta)u_{1}-g-\dfrac{K_{3}\dot{z}}{m}\\ \ddot{\phi}=\dot{\phi}\psi\dfrac{I_{y}-I_{z}}{I_{x}}+\dfrac{J_{r}}{I_{x}}\dot {\sigma}\Omega_{r}+\dfrac{I}{I_{x}}u_{2}-\dfrac{K_{4}l}{I_{x}}\dot{\phi}\\ \ddot{\theta}=\dot{\psi}\dot{\phi}\dfrac{I_{z}-I_{x}}{I_{y}}-\dfrac{J_{r}}{I_ {y}}\dot{\phi}\Omega_{r}+\dfrac{I}{I_{y}}u_{3}-\dfrac{K_{5}l}{I_{y}}\dot{\theta }\\ \ddot{\psi}=\dot{\phi}\dot{\theta}\dfrac{I_{x}-I_{y}}{I_{z}}+\dfrac{1}{I_{z}}u _{4}-\dfrac{K_{6}}{I_{z}}\dot{\psi}\end{cases} \tag{1}\]  
where \([\phi,\theta,\psi]\) represent the roll, pitch, and yaw Euler angles, respectively, and \(\phi\in(-\pi/2,\pi/2)\), \(\theta\in(-\pi/2,\pi/2)\), \(\psi\in(-\pi/2,\pi/2)\). The drag coefficient \(K_{i}(i=1,\cdot\cdot\cdot,6)\) is a positive constant; \(I_{x},I_{y},I_{Z}\) represent the inertia of the UAV; \(J_{r}\) is the rotational inertia of the propeller; \(\Omega_{r}=\Omega_{1}-\Omega_{2}+\Omega_{3}-\Omega_{4},\Omega_{i}\)\((i=1,2,3,4)\) denotes the rotational speed of the propeller; \(u_{1}\) denotes the total thrust in the z-axis; \(u_{2}\) and \(u_{3}\) denote the inputs for the roll angle and the pitch angle, respectively; \(u_{4}\) denotes the yaw moment. \(u_{1}=(F_{1}+F_{2}+F_{3}+F_{4}),u_{2}=(-F_{2}+F_{4}),u_{3}=(-F_{1}+F_{3}),\)\(u_{4}=\)\(d(-F_{1}+F_{2}-F_{3}+F_{4})/b\), where \(F_{i}=b\Omega_{i}^{2}(i=1,2,3,4)\) denotes the thrust generated by the four rotors as a real control input to the powertrain model, \(b\) denotes the lift coefficient, and \(d\) denotes the force to moment proportionality factor, the exact values of which are shown in Table 1.  
Figure 1: Quadrotor-UAV.