## 2 Dynamical Model
### Control Problem Description  
In the above, \(P_{d}^{E}=\begin{bmatrix}p_{dx}^{e}&p_{dy}^{e}&p_{dz}^{e}\end{bmatrix}^{T}\) and \(V_{d}^{E}=\begin{bmatrix}v_{dx}^{e}&v_{dy}^{e}&v_{dz}^{e}\end{bmatrix}^{T}\) represent the desired position and velocity of the quadrotor in the inertial frame, respectively. \(\Theta_{d}^{E}=\begin{bmatrix}\phi_{d}&\theta_{d}&\psi_{d}\end{bmatrix}^{T}\) represents the desired Euler angles of the quadrotor, and \(\Omega_{d}^{B}=\begin{bmatrix}\rho_{d}&\mathbf{q}_{d}&\mathbf{r}_{d}\end{bmatrix} ^{T}\) represents the desired angular velocity of the quadrotor in the body frame. Among these variables, \(P_{d}^{E}\) and \(\psi_{d}\) serve as inputs to the entire control system. The desired roll angle \(\phi_{d}\) and pitch angle \(\theta_{d}\) are obtained by the outer-loop bioinspired backstepping position control system to achieve trajectory tracking. Specifically, the desired position \(P_{d}^{E}\) and yaw angle \(\psi_{d}\) of the quadrotor are determined based on the tracking trajectory at time \(t_{0}\). Figure 2 depicts the relationship between tracking errors in the body and inertial coordinate systems during the tracking process. Here, the position error is represented by \(e_{P}=P_{d}^{E}-P^{E}\), the velocity error by \(e_{V}=V_{d}^{E}-V^{E}\), the Euler angle error by \(e_{\Theta}=\Theta_{d}^{E}-\Theta^{E}\), and the body angle error by \(e_{\Omega}=\Omega_{d}^{B}-\Omega^{B}\). The outputs of the relevant controllers are denoted as \(u_{P}=\begin{bmatrix}u_{pi}\end{bmatrix}\in\mathbb{R}^{3\times 1}\), \(u_{V}=\begin{bmatrix}u_{pi}\end{bmatrix}\in\mathbb{R}^{3\times 1}\), \(u_{\Theta}=\begin{bmatrix}u_{\Theta i}\end{bmatrix}\in\mathbb{R}^{3\times 1}\), and \(u_{\Omega}=\begin{bmatrix}u_{\Omega i}\end{bmatrix}\in\mathbb{R}^{3\times 1}\). Specifically, \(u_{P}\), \(u_{\Theta}\), and \(u_{vi}(i=1,2)\) correspond to the inputs of the velocity controller, angle controller, and angular velocity controller, respectively. They satisfy \(V_{d}^{E}=u_{P}\), \(\phi_{cd}=u_{v1}\), \(\theta_{cd}=u_{v2}\), and \(\Omega_{d}^{B}=u_{\Theta}\). \(u_{v3}\) and \(u_{\Omega}\) represent the control signal inputs for the quadrotor, which drive the four motors to generate total thrust and torque for following the desired trajectory. In this paper, the actual quadrotor dynamical system is treated as a nominal system with the addition of an equivalent disturbance. Therefore, the quadrotor dynamical model can be rewritten as follows:  
\[\dot{P}^{E}=u_{P}+\Delta_{p}\]  
\[\dot{V}^{E}=G_{V}u_{V}-G+\Delta_{v}\]  
\[\dot{\Theta}^{E}=u_{\Theta}+\Delta_{\Theta}\]  
\[\dot{\Omega}^{B}=G_{\Omega}u_{\Omega}+I^{-1}\Big{[}G_{a}-\Omega^{B}\times \Big{(}I\cdot\Omega^{B}\Big{)}\Big{]}+\Delta_{\Omega} \tag{5}\]  
where \(G=\begin{bmatrix}0&0&g\end{bmatrix}^{T}\); \(G_{V}=diag(g,-g,k_{1}/m)\); \(G_{\Omega}=I^{-1}diag(k_{cp},k_{cq},k_{\sigma r})\);\(k_{t}\), \(k_{cp}\), \(k_{cq}\), and \(k_{\sigma r}\) are gain constants of the controller satisfying \(T_{K}=k_{t}u_{\varpi}\), \(\tau_{bp}=k_{cp}u_{\Omega p}\), \(\tau_{bq}=k_{cq}u_{\Omega q}\), and \(\tau_{br}=k_{\sigma r}u_{\Omega r}\); and \(\Delta_{i}(i=p,v,\Theta,\Omega)\) represents external disturbances and aerodynamics, satisfying:  
\[\Delta_{p}=V^{E}-u_{P}\]  
\[\Delta_{v}=\frac{1}{m}(F_{d}+dv)\]  
\[\Delta_{\Theta}=R_{\Theta}\Omega^{B}-u_{\Theta}\]  
\[\Delta_{\Omega}=I^{-1}M_{d} \tag{6}\]