## 2 System Dynamic Model  
A quadrotor is a four-rotor, under-actuated system as depicted in Figure 2. The quadrotor's linear and angular positions are represented by \( \begin{bmatrix}x&y&z\end{bmatrix} ^{T}\) and \( \begin{bmatrix}\phi&\theta&\psi\end{bmatrix} ^{T}\), respectively. The thrust forces produced by the rotors are represented by \(T_{i}(i=1,2,3,4)\). The adjacent rotors spin in opposite directions. Two rotors (1 & 3) rotate clockwise, whereas rotors (3 & 4) spin anticlockwise. A quadrotor with variable payload can be mathematically modeled using the [PERSON] and Euler equations. The quadrotor's translational dynamics with time-varying mass is given by [21; 22]  
\[\sum F=\frac{d}{dt}(P)=m\frac{d}{dt}(v)+v\frac{d}{dt}(m) \tag{1}\]where \(\sum F\) stands for the resultant force, \(P\) represents the linear momentum, m denotes the quadrotor's mass with variable payload, and \(v\) indicates the quadrotor's translational velocity. Equation (1) can be written as  
\[ \begin{cases}m\ddot{x}=-(sint\acute{e}cos\phi cos\psi+sint\acute{y}sint\acute{ \phi})T-\acute{n}t\dot{x}-W_{fx}\\ m\ddot{y}=-(cos\phi sint\acute{y}sint\acute{\theta}-cos\phi sint\acute{\phi})T- \acute{n}\dot{y}-W_{fy}\\ m\ddot{z}=-cos\phi cos\phi T+mg-m\dot{z}-W_{fz}\end{cases}  \tag{2}\]  
where  
\[T=\sum_{i=1}^{4}T_{i}=T_{1}+T_{2}+T_{3}+T_{4} \tag{3}\]  
\(T\) represents the thrust force exerted by the rotors, whereas \(W_{fx}\), \(W_{fy}\), and \(W_{fz}\) represent the wind forces in \(x\), \(y\), and \(z\) directions, respectively. The wind force can be expressed as [23]  
\[ \begin{bmatrix}W_{fx}\\ W_{fy}\\ W_{fz}\end{bmatrix} = \begin{bmatrix}k_{s}|\dot{x}-\dot{x}_{w}|(\dot{x}-\dot{x}_{ w})\\ k_{s}|\dot{y}-\dot{y}_{w}|(\dot{y}-\dot{y}_{w})\\ k_{u}|\dot{z}-z_{w}|(\dot{z}-z_{w})\end{bmatrix}  \tag{4}\]  
\( \begin{bmatrix}\dot{x}&\dot{y}&\dot{z}\end{bmatrix} ^{T}\) and \( \begin{bmatrix}\dot{x}_{w}&\dot{y}_{w}&\dot{z}_{w}\end{bmatrix} ^{T}\) show the translational and wind velocity, and the parameters \(k_{s}\) and \(k_{u}\) indicate the drag factors on the lower-upper faces and sides of the quadrotor, respectively. Furthermore, referring to [21; 22], the quadrotor's rotational dynamics with time-varying inertia is given by  
\[\sum M=J\frac{d}{dt}(\mathcal{W})+\mathcal{W}\frac{d}{dt}(J)+\mathcal{W}\times \mathcal{W} \tag{5}\]  
where \(\sum M\) indicates the net moment, \(J\) denotes the inertia tensor, and \(\mathcal{W}\) shows the angular speed. Equation (5) can be written as  
\[ \begin{cases}I_{x}\ddot{\phi}=(I_{y}-I_{z})\dot{\theta}\dot{\psi}-I_{x}\dot{ \phi}+J_{r}\Omega_{r}\dot{\theta}+\tau_{\phi}\\ I_{y}\dot{\theta}=(I_{z}-I_{x})\dot{\psi}\dot{\theta}-I_{y}\dot{\theta}-J_{r} \Omega_{r}\dot{\psi}+\tau_{\phi}\\ I_{z}\ddot{\psi}=(I_{x}-I_{y})\dot{\phi}\dot{\theta}-I_{z}\dot{\psi}+\tau_{ \psi}\end{cases}  \tag{6}\]  
Figure 2: The X-configuration of quadrotor. where \(\tau_{\phi}\), \(\tau_{\phi}\), and \(\tau_{\psi}\) represent the rolling, pitching, and yawning moments, respectively. \(J_{r}\) represents the rotor inertia, \(\Omega_{r}=\Omega_{1}+\Omega_{3}-\Omega_{2}-\Omega_{4}\), while \(\Omega_{i}(i=1,2,3,4)\) indicates the speed of rotors. Control signals are defined for moments and thrust force, which are as follows:  
\[ \begin{bmatrix}\tau_{\phi}\\ \tau_{\psi}\\ T\end{bmatrix} = \begin{bmatrix}U_{1}\\ U_{2}\\ U_{3}\\ U_{4}\end{bmatrix}  \tag{7}\]  
The relationship between control signals and motor commands is as follows [24; 25]:  
\[ \begin{bmatrix}\cap_{1}\\ \cap_{2}\\ \cap_{3}\\ \cap_{4}\end{bmatrix} = \begin{bmatrix}\frac{1}{2\sqrt{2}K_{f}}&\frac{1}{2\sqrt {2}K_{f}}&-\frac{1}{4K_{m}}&\frac{1}{4K_{f}}\\ -\frac{1}{2\sqrt{2}K_{f}}&\frac{1}{2\sqrt{2}K_{f}}&\frac{1}{4K_{m}}&\frac{1}{4K _{f}}\\ -\frac{1}{2\sqrt{2}K_{f}}&-\frac{1}{2\sqrt{2}K_{f}}&-\frac{1}{4K_{m}}&\frac{1}{4K _{f}}\\ \frac{1}{2\sqrt{2}K_{f}}&-\frac{1}{2\sqrt{2}K_{f}}&\frac{1}{4K_{m}}&\frac{1}{4K _{f}}\end{bmatrix}\begin{bmatrix}U_{1}\\ U_{2}\\ U_{3}\\ U_{4}\end{bmatrix}