## 3 Trajectory Tracking Controller Design Based on LADRC
### Guidance Law Design for Horizontal Trajectory  
In Figure 2, \((x,y,z)\) represents the current position of the parafoil system, while \((x_{d}(\varpi),y_{d}(\varpi),z_{d}(\varpi))\) denotes the target position, where \(\varpi\) is the trajectory variable. By considering \((x_{d},y_{d},z_{d})\) as the origin, we establish a target coordinate system denoted as \(O_{\rm{p}}x_{\rm{p}}y_{\rm{p}}Z_{\rm{p}}\). Then, the horizontal trajectory tracking errors \(x_{e}\) and \(y_{e}\) in \(O_{\rm{p}}X_{\rm{p}}Y_{\rm{p}}Z_{\rm{p}}\) can be expressed as  
\[\begin{cases}x_{e}=(x-x_{d})\cos(\psi_{d})+(y-y_{d})\sin(\psi_{d})\\ y_{e}=-(x-x_{d})\sin(\psi_{d})+(y-y_{d})\cos(\psi_{d})\end{cases} \tag{6}\]  
where \(\psi_{d}(\varpi)=\arctan(\dot{y}_{d}(\varpi)/\dot{x}_{d}(\varpi))\).  
In this paper, the guidance law for the horizontal trajectory is designed as  
\[\psi_{\rm{g}}=\psi_{d}+\arctan(-\alpha y_{e})-\chi \tag{7}\]  
Figure 2: Schematic diagram of trajectory tracking for a powered parafoil system.  
where \(\alpha\) is a positive constant, \(\psi_{g}\) is the guided yaw angle, and \(\chi=arctan(v/u)\) is the sideslip angle. Then, the following theorem arises.  
Under the premise that the actual yaw angle \(\psi\) tracks the guided yaw angle \(\psi_{g}\) very well, the horizontal trajectory tracking error can gradually converge to 0 with the desired forward speed  
\[u_{d}=U\cos(\psi+\chi-\psi_{d})+kx_{e} \tag{8}\]  
where \(k>0\) and \(U=\sqrt{u^{2}+v^{2}}\).  
Proof of Theorem 1.: Construct a Lyapunov function as \(V=\frac{1}{2}x_{e}^{2}+\frac{1}{2}y_{e}^{2}\); then, its derivative is derived as  
\[\dot{V}=x_{e}\dot{x}_{e}+y_{e}\dot{y}_{e} \tag{9}\]  
According to Equation (6), the following expression can be derived,  
\[\begin{cases}\dot{x}_{e}=U\cos(\psi+\chi-\psi_{d})-u_{d}+\dot{\psi}_{d}y_{e}\\ \dot{y}_{e}=U\sin(\psi+\chi-\psi_{d})-\dot{\psi}_{d}x_{e}\end{cases} \tag{10}\]  
where the desired forward speed has the following expression:  
\[u_{d}=\dot{\omega}\sqrt{\dot{x}_{d}^{2}(\dot{\omega})+\dot{y}_{d}^{2}(\dot{ \omega})} \tag{11}\]  
With Equation (10), Equation (9) can be rearranged as  
\[\dot{V}=x_{e}U\cos(\psi+\chi-\psi_{d})-x_{e}u_{d}+y_{e}U\sin(\psi+\chi-\psi_{d}) \tag{12}\]  
Further, with \(\psi=\psi_{g}\) and substituting Equation (8) into Equation (12), the following inequality can be obtained  
\[\begin{split}\dot{V}&=-kx_{e}^{2}+y_{e}U\sin( \arctan(-\alpha y_{e}))\\ &=-kx_{e}^{2}-\alpha y_{e}^{2}\\ &\leq 0\end{split} \tag{13}\]  
The guidance law presented in Equation (7) effectively merges the lateral error \(y_{e}\) with the yaw angle \(\psi\) to achieve convergence of the lateral error. By altering the flight direction, this approach effectively addresses the issue of underactuation. Furthermore, by combining Equation (8) and Equation (11), we can derive the following rules for \(\dot{\omega}\):  
\[\dot{\omega}\!=\!\frac{U\cos(\psi+\chi-\psi_{d})+kx_{e}}{\sqrt{\dot{x}_{d}^{2} (\dot{\omega})+\dot{y}_{d}^{2}(\dot{\omega})}} \tag{14}\]  
To conclude, when the yaw angle \(\psi\) can accurately track the guided yaw angle \(\psi_{g}\) stated in Equation (7), and the parameter \(\omega\) satisfies Equation (14), it is possible to achieve horizontal trajectory tracking.