# Event-Triggered Fault-Tolerant ADRC for Variable-Load Quadrotor with Prescribed Performance
## 2 Modeling of Variable-Load Quadrotor
\begin{cases}\ddot{x}=&[U_{1}(\cos\psi\sin\theta\cos\phi+\sin\psi\sin\phi)-k _{a}\dot{x}]/(M+m)\\ \ddot{y}=&[U_{1}(\sin\psi\sin\theta\cos\phi-\cos\psi\sin\phi)-k_{a}\dot{y}]/(M +m)\\ \dot{z}=&[U_{1}(\cos\theta\cos\phi)-k_{a}\dot{z}]/(M+m)-g\\ \ddot{\phi}=&[U_{2}+ml_{y}l_{z}(\dot{\theta}^{2}-\dot{\psi}^{2})+(I_{y}-J_{z} +ml_{z}^{2}-ml_{y}^{2})\dot{\phi}\dot{\psi}+ml_{x}l_{z}(\dot{\phi}\dot{\theta}+ \ddot{\psi})+ml_{x}l_{y}(\ddot{\theta}-\dot{\phi}\dot{\psi})]/[J_{x}+m(l_{y}^{ 2}+l_{z}^{2})]\\ \ddot{\theta}=&[U_{3}+ml_{x}l_{z}(\dot{\psi}^{2}-\dot{\psi}^{2})+(J_{z}-J_{x} +ml_{x}^{2}-ml_{z}^{2})\dot{\phi}\dot{\psi}+ml_{x}l_{y}(\dot{\theta}\dot{\psi}+ \ddot{\phi})+ml_{y}l_{z}(\ddot{\psi}-\dot{\phi}\dot{\theta})]/[J_{y}+m(l_{x}^{ 2}+l_{z}^{2})]\\ \ddot{\psi}=&[U_{4}+ml_{x}l_{y}(\dot{\phi}^{2}-\dot{\theta}^{2})+(J_{x}-J_{y} +ml_{y}^{2}-ml_{x}^{2})\dot{\phi}\dot{\theta}+ml_{y}l_{z}(\dot{\phi}\dot{\psi}+ \ddot{\theta})+ml_{x}l_{z}(\dot{\phi}-\dot{\phi}\dot{\theta})]/[J_{z}+m(l_{x}^{ 2}+l_{y}^{2})]\end{cases} . \tag{11}\]  
The actuator system, comprising four electronic speed controllers (ESCs) with corresponding brushless motor-propeller assemblies, serves as the critical actuation core in quadrotor flight control. During operational missions, various actuator failures may occur due to mechanical wear, electronic component malfunctions, or external disturbances, which directly alter the dynamic characteristics of the quadrotor. In quadrotor systems, actuator faults typically manifest as the following:  
* Loss of Effectiveness: Manifested as a decrease in motor thrust output efficiency; potential causes include insufficient power supply voltage, reduced ESC drive capability, or propeller damage. * Bias Fault: Characterized by a constant offset in motor output; potential causes include sensor zero drift or ESC calibration errors. When the actuator has the above faults, the actual lift it provides can be written as \[F_{i}^{\rm F}=a_{i}^{\rm F}F_{i}+b_{i}^{\rm F},\] (12) where \(F_{i}^{\rm F}\) denotes the actual lift force generated by the actuator. The efficiency factor \(a_{i}^{\rm F}\in[0,1]\) represents the effectiveness of the actuator, while \(b_{i}^{\rm F}\) indicates the bias factor. The system operates under the following conditions: when \(a_{i}^{\rm F}=1\) and \(b_{i}^{\rm F}=0\), the quadrotor is in normal operation; when \(a_{i}^{\rm F}\in[0,1)\) and \(b_{i}^{\rm F}=0\), the quadrotor has loss of effectiveness in the actuator; and when \(a_{i}^{\rm F}=1\) with \(b_{i}^{\rm F}\
eq 0\), a bias fault is present in the system. According to Equations (10) and (12), the actual control inputs of the variable-load quadrotor in the case of actuator failure can be obtained as  
\[ \begin{bmatrix}F_{I}^{\rm F}\\ \tau_{x}^{\rm F}\\ \tau_{y}^{\rm F}\\ \tau_{z}^{\rm F}\end{bmatrix} = \begin{bmatrix}1&1&1&1\\ -\frac{\sqrt{2}}{2}l&\frac{\sqrt{2}}{2}l&\frac{\sqrt{2}}{2}l&-\frac{\sqrt{2}}{ 2}l\\ \frac{\sqrt{2}}{2}l&\frac{\sqrt{2}}{2}l&-\frac{\sqrt{2}}{2}l&-\frac{\sqrt{2}}{ 2}l\\ k_{c}&-k_{c}&k_{c}&-k_{c}\end{bmatrix}\begin{bmatrix}F_{I}^{\rm F}\\ F_{I}^{\rm F}\\ F_{J}^{\rm F}\end{bmatrix} . \tag{13}\]  
Combining Equations (6)-(13), the dynamic model of the variable-load quadrotor with actuator faults can be obtained as  
\[