# Event-Triggered Fault-Tolerant ADRC for Variable-Load Quadrotor with Prescribed Performance
## 2 Modeling of Variable-Load Quadrotor
By treating the quadrotor UAV and its payload as a single rigid body, according to [PERSON]'s law, the displacement dynamics of the variable-load quadrotor in the inertial coordinate system \(E\) can be derived as  
\[ \begin{cases}\ddot{x}=(F_{x}-k_{a}\dot{x})/(M+m)\\ \ddot{y}=(F_{y}-k_{a}\dot{y})/(M+m)\\ \ddot{z}=(F_{z}-k_{a}\dot{z})/(M+m)-g\end{cases} . \tag{6}\]  
For the rotational dynamics of the quadrotor, the moment expression for the cross-configuration can be formulated as  
\[\tau= \begin{bmatrix}\tau_{x}&\tau_{y}&\tau_{z}\end{bmatrix} ^{T}= \begin{bmatrix} l(-F_{1}+F_{2}+F_{3}-F_{4})/\sqrt{2}\\ l(F_{1}+F_{2}-F_{3}-F_{4})/\sqrt{2}\\ k_{c}(F_{1}-F_{2}+F_{3}-F_{4})\end{bmatrix} , \tag{7}\]  
where \(l\) is the length of the rotor arm and \(k_{c}\) is the moment coefficient. We set \( \begin{bmatrix}p&q&r\end{bmatrix} ^{T}\) as the rotational velocity in the body coordinate system. The shifting center of mass during flight transforms the quadrotor into an asymmetric structure, resulting in time-varying moment of inertia matrices. For a quadrotor with variable payload, its inertia matrix can be expressed as  
\[J_{v}= \begin{bmatrix}J_{xx}&-J_{xy}&-J_{xz}\\ -J_{yx}&J_{yy}&-J_{yz}\\ -J_{zx}&-J_{zy}&J_{zz}\end{bmatrix} , \tag{8}\]where  
\[ \begin{cases}J_{xx}=J_{x}+m(l_{y}^{2}+l_{z}^{2})\\ J_{yy}=J_{y}+m(l_{x}^{2}+l_{z}^{2})\\ J_{zz}=J_{z}+m(l_{x}^{2}+l_{y}^{2})\\ J_{xy}=J_{yx}=ml_{x}l_{y}\\ J_{xz}=J_{zx}=ml_{x}l_{z}\\ J_{yz}=J_{zy}=ml_{y}l_{z}\end{cases} ,\]  
\(l_{x}\), \(l_{y}\), and \(l_{z}\) denote the offset distance between the center of gravity and the center of the body. When the quadrotor maintains stable flight with minimal attitude variations, the Euler angle rates in the inertial frame can be approximated by the body-frame angular velocities: \(\left[\dot{\phi}\quad\dot{\theta}\quad\dot{\psi}\right]^{T}\approx\left[p \quad q\quad r\right]^{T}\). This linear approximation permits derivation of the quadrotor rotational dynamics through the Euler equations. The rotational dynamics of the quadrotor can be derived from Euler equations of motion as  
\[ \begin{cases}\ddot{\phi}=&\tau_{x}/J_{xx}+[J_{yz}(\dot{\theta}^{2}-\dot{\psi}^{ 2})+(J_{yy}-J_{zz})\dot{\theta}\dot{\psi}+J_{xz}(\dot{\phi}\dot{\phi}+\ddot{ \psi})+J_{xy}(\dot{\theta}-\dot{\phi}\dot{\psi})]/J_{xx}\\ \ddot{\theta}=&\tau_{y}/J_{yy}+[J_{xx}(\dot{\psi}^{2}-\dot{\phi}^{2})+(J_{zz}- J_{xx})\dot{\phi}\dot{\psi}+J_{xy}(\dot{\psi}\dot{\phi}+\ddot{\psi})+J_{yz}(\ddot{ \psi}-\dot{\phi}\dot{\theta})]/J_{yy}\\ \ddot{\psi}=&\tau_{z}/J_{zz}+[J_{xy}(\dot{\psi}^{2}-\dot{\theta}^{2})+(J_{xx}- J_{yy})\dot{\phi}\dot{\theta}+J_{yz}(\dot{\phi}\dot{\psi}+\ddot{\theta})+J_{xz}( \dot{\phi}-\dot{\phi}\dot{\psi})]/J_{zz}\end{cases} . \tag{9}\]  
The displacement Equation (6) and rotational Equation (9) collectively constitute the dynamic model of the variable-load quadrotor. By introducing virtual control variables \(U_{1}\),\(U_{2}\),\(U_{3}\),\(U_{4}\) into the control system, where \(\left[U_{1}\quad U_{2}\quad U_{3}\quad U_{4}\right]^{T}=\left[F_{l}\quad\tau _{x}\quad\tau_{y}\quad\tau_{z}\right]^{T}\), and then combining Equations (3) and (7), the control inputs to the quadrotor can be obtained as  
\[ \begin{bmatrix}U_{1}\\ U_{2}\\ U_{3}\\ U_{4}\end{bmatrix} = \begin{bmatrix}F_{l}\\ \tau_{x}\\ \tau_{y}\\ \tau_{z}\end{bmatrix} = \begin{bmatrix}1&1&1&1&1\\ -\frac{l}{\sqrt{2}}&\frac{l}{\sqrt{2}}&\frac{l}{\sqrt{2}}&-\frac{l}{\sqrt{2}} \\ \frac{l}{\sqrt{2}}&\frac{l}{\sqrt{2}}&\frac{l}{\sqrt{2}}&\frac{l}{\sqrt{2}} \\ k_{c}&-k_{c}&k_{c}&-k_{c}\end{bmatrix}\begin{bmatrix}F_{1}\\ F_{2}\\ F_{3}\\ F_{4}\end{bmatrix} . \tag{10}\]  
The control model of the variable-load quadrotor can be given as  
\[