# Event-Triggered Fault-Tolerant ADRC for Variable-Load Quadrotor with Prescribed Performance
## 2 Modeling of Variable-Load Quadrotor  
To accurately describe the motion state of the quadrotor UAV, two coordinate systems were established as shown in Figure 2: the body coordinate frame \(B(o_{b},x_{b},y_{b},z_{b})\) and the inertial coordinate frame \(E(o_{e},x_{e},y_{e},z_{e})\). The system possesses six degrees of freedom containing both position coordinates \((x,y,z)\) and attitude angles \((\phi,\theta,\psi)\). These Euler angles represent roll about \(x_{e}\), pitch about  
Figure 2: Body and inertial coordinate frame for quadrotor. \(y_{e}\), and yaw about \(z_{e}\) axes, with counterclockwise rotation defined as positive. The rotation matrix \(R_{B-E}\) can be expressed as  
\[R_{B-E}=(R_{\phi}R_{\theta}R_{\psi})^{T}, \tag{1}\]  
where  
\[R_{\phi}= \begin{bmatrix}1&0&0\\ 0&\cos\phi&-\sin\phi\\ 0&\sin\phi&\cos\phi\end{bmatrix} ,R_{\theta}= \begin{bmatrix}\cos\theta&0&\sin \theta\\ 0&1&0\\ -\sin\theta&0&\cos\theta\end{bmatrix} ,R_{\psi}= \begin{bmatrix}\cos\psi&-\sin \psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{bmatrix} .\]  
According to the cross structure of the frame, the quadrotor UAV could be categorized into the \"+\" type and \"x\" type. The model in this study was built based on the \"x\" type structure, which is shown in Figure 3.  
The following assumptions were made for modeling the dynamics:  
**Assumption 1**. : _The airframe of quadrotor aircraft is typically constructed from lightweight yet rigid materials such as carbon fiber or aluminum alloy. Under normal operational loads, the elastic deformation of these structures can be considered negligible. The quadrotor is rigidly attached and exhibits no elastic deformation._  
**Assumption 2**. : _Most quadrotors are symmetrically designed. The load-free quadrotor has a uniformly distributed mass, with its center of gravity aligned with the of the body frame._  
**Assumption 3**. : _The variable payload is rigidly connected to the quadrotor._  
**Remark 1**. : _The structural rigidity assumption may lose validity under extreme operational conditions, particularly when subjected to high-frequency vibrations, impact loads, or substantial deformations. Assumption 3 considered a rigid connection between the payload and the quadrotor airframe, with non-rigid coupling scenarios explicitly excluded. This modeling approach presents certain limitations._  
Based on the physical structure of the quadrotor, it can be observed that the aircraft is primarily influenced by two forces: the lift forces generated by the four rotors and the gravity. The expression for gravitational force in the inertial frame is  
\[M_{e}= \begin{bmatrix}0&0&(M+m)g\end{bmatrix} ^{T}, \tag{2}\]  
Figure 3: Structure diagram of the quadrotor. where \(M\) is the mass of the quadrotor body, \(m\) is the mass of the load, and \(g\) is the gravitational acceleration. In the body coordinate system, the lift of the rotor can be expressed as  
\[F_{b}= \begin{bmatrix}0&0&F_{l}\end{bmatrix} ^{T}= \begin{bmatrix}0&0&\sum_{i=1}^{4 }F_{i}\end{bmatrix} ^{T}, \tag{3}\]  
where \(F_{l}\) denotes the total lift of the rotor and \(F_{i}(i=1,\cdots,4)\) denotes the lift provided by a single rotor. Combining Equations (1) and (3), the expression for lift in the inertial coordinate system can be obtained:  
\[F_{e}= \begin{bmatrix}F_{x}&F_{y}&F_{z}\end{bmatrix} ^{T}=R_{B-E}F_{b}=F_{l} \begin{bmatrix}\cos\psi\sin\theta\cos\phi+\sin\psi\sin\phi\\ \sin\psi\sin\theta\cos\phi-\cos\psi\sin\phi\\ \cos\theta\cos\phi\end{bmatrix} . \tag{4}\]  
During flight, the air resistance experienced by a quadrotor is closely related to its flight velocity, which can be expressed as  
\[F_{a}= \begin{bmatrix}k_{a}\dot{x}&k_{a}\dot{y}&k_{a}\dot{z}\end{bmatrix} ^{T}, \tag{5}\]  
where \(k_{a}\) is the air resistance coefficient. To simplify the analysis, the air resistance coefficient was set to be the same in all directions.