# Quadrotor Cascade Control System Design Based on Linear Active Disturbance Rejection Control
## 2 Mathematical Models  
The quadrotor UAV is a system with six-degrees-of-freedom driven by four motors, and its translational and rotational motions are achieved through motor speed regulation. \(x\), \(y\) and \(z\) are used to control the position channel of the quadrotor; \(\theta\), \(\phi\) and \(\psi\) are used to control the attitude channel of the quadrotor. In order to obtain an accurate quadrotor dynamics model [26], the internal friction of the airframe is neglected, the quadrotor is considered as a rigid structure, the earth coordinate system \(O_{Y}:\{O_{Y}X_{Y}Y_{Y}Z_{Y}\}\) and the airframe coordinate system \(O_{R}:\{O_{R}X_{R}Y_{R}Z_{R}\}\) are established, and the structure diagram and reference coordinate system of the quadrotor are shown in Figure 1.  
The transformation relationship of the quadrotor from the Earth coordinate system \(O_{Y}:\{O_{Y}X_{Y}Y_{Y}Z_{Y}\}\) to the airframe coordinate system \(O_{R}:\{O_{R}X_{R}Y_{R}Z_{R}\}\) is expressed in terms of the attitude matrix \(O_{Q}\)[27] as:  
\[ \begin{array}{ll}O_{Q}&=\begin{bmatrix}Q_{1}&Q_{2}&Q_{3}\end{bmatrix}\\ &=\begin{bmatrix}\cos\theta\cos\psi&\sin\theta\sin\phi\cos\psi-\sin\phi\sin \phi&\sin\theta\cos\phi\cos\psi+\sin\psi\sin\phi\\ \cos\theta\sin\psi&\sin\theta\sin\psi+\cos\psi\cos\phi&\sin\theta\sin\psi\cos \phi-\cos\phi\cos\psi\\ -\sin\theta&\sin\varphi\cos\theta&\cos\phi\cos\theta\end{bmatrix}\end{array}  \tag{1}\]  
We introduce \(T_{1}\), \(T_{2}\), \(T_{3}\) and \(T_{4}\) to denote the lift generated by the four motors; therefore, the flight dynamics of the quadrotor can be expressed as:  
\[T_{r}= \begin{bmatrix}T_{Xr}\\ T_{Yr}\\ T_{Zr}\end{bmatrix} = \begin{bmatrix}0\\ 0\\ T_{1}+T_{2}+T_{3}+T_{4}\end{bmatrix}  \tag{2}\]  
Figure 1: Construction and frame of tilt quadrotor. In the fuselage coordinate system \(O_{R}:\left\{O_{R}X_{R}Y_{R}Z_{R}\right\}\), bringing in the rotation matrix and associating Equations (1) and (3), the lift of the quadrotor can be expressed as:  
\[T_{q}= \begin{bmatrix}T_{Xq}\\ T_{Yq}\\ T_{Zq}\end{bmatrix} =O_{Q}\cdot T_{r} \tag{3}\]  
In the Earth coordinate system \(O_{Y}:\left\{O_{Y}X_{Y}Y_{Y}Z_{Y}\right\}\), the mathematical model of the position channel is expressed as:  
\[\left\{ \begin{array}{l}m\ddot{x}=T_{Xq}-J_{s}\dot{x}\\ m\dot{y}=T_{Yq}-J_{s}y\\ m\ddot{z}=T_{Zq}-mg-J_{s}\dot{z}\end{array} \right. \tag{4}\]  
where \(J_{s}\) indicates the torque coefficient; \(m\) represents the mass of the quadrotor; \(g\) represents the acceleration of gravity. Further, the channels for attitude control can be expressed as:  
\[\left\{ \begin{array}{l}\ddot{\theta}=c\left(-T_{1}+T_{2}+T_{3}-T_{4}-J_{s} \dot{\theta}\right)/h_{1}+(h_{3}-h_{1})\dot{\varphi}\dot{\psi}/h_{2}\\ \ddot{\varphi}=c\left(-T_{1}+T_{2}+T_{3}-T_{4}-J_{s}\dot{\varphi}\right)/h_{ 2}+(h_{2}-h_{3})\dot{\theta}\dot{\psi}/h_{1}\\ \ddot{\psi}=\left[J_{f}(-T_{1}+T_{2}-T_{3}+T_{4})-J_{s}\dot{\psi}\right]/h_{ 3}+(h_{1}-h_{2})\dot{\theta}\dot{\varphi}/h_{3}\end{array} \right. \tag{5}\]  
where \(h_{1}\), \(h_{2}\), and \(h_{3}\) express the rotational inertia of the \(x\), \(y\) and \(z\)-axis, respectively; \(J_{f}\) is the coefficient of air resistance; \(c\) is the distance between aircraft center and rotor center. For the convenience of the study, we introduce the dummy variable \(\Gamma_{i}\) to replace \(T_{i}\) and the dummy variable \(\Gamma_{x}\), \(\Gamma_{y}\), \(\Gamma_{z}\) to simplify the mathematical model of the quadrotor position motion. where \(i=1,2,3,4\), the intermediate variables of the simplified model are expressed as:  
\[ \begin{bmatrix}\Gamma_{1}\\ \Gamma_{2}\\ \Gamma_{3}\\ \Gamma_{4}\end{bmatrix} = \begin{bmatrix}1/m&1/m&1/m&1/m\\ -c/h_{2}&c/h_{2}&c/h_{2}&-c/h_{2}\\ -c/h_{1}&c/h_{1}&c/h_{1}&-c/h_{1}\\ -l_{o}/p_{3}&l_{o}/h_{3}&-l_{o}/h_{3}&l_{o}/h_{3}\end{bmatrix}\begin{bmatrix} T_{1}\\ T_{2}\\ T_{3}\\ T_{4}\end{bmatrix}  \tag{6}\]  
\[ \begin{bmatrix}\Gamma_{x}\\ \Gamma_{y}\\ \Gamma_{z}\end{bmatrix} =\frac{i=1}{m}\times