## 2 Dynamics of Powered Parafoil and Problem Formulation  
Figure 1 shows the structure of a powered parafoil and its three coordinate frames, that is, \(O_{d}X_{d}Y_{d}Z_{d}\) represents the earth reference frame, \(O_{s}X_{s}Y_{s}Z_{s}\) represents the parafoil reference frame, and \(O_{w}X_{w}Y_{w}Z_{w}\) represents the payload reference frame. The transformation between the earth reference frame and the parafoil reference frame is achieved through three Euler angles \(\left[\psi\quad\theta\quad\phi\right]^{\mathsf{T}}\); that is, \(\psi\) denotes the yaw angle, \(\theta\) denotes the pitch angle, and \(\phi\) denotes the roll angle. The study in this paper focuses on the altitude control of a powered parafoil that involves the relative motion between a parafoil and a payload. The dynamics model is established according to the Kirchhoff motion equation. The specific modeling process and model parameters can be found in [20]. The inclination angle \(\sigma\) of the powered parafoil is defined as the angle between the velocity and the horizontal plane, shown as  
\[\sigma=\arctan\left(\frac{-\dot{z}}{\sqrt{\dot{x}^{2}+\dot{y}^{2}}}\right), \tag{1}\]  
where \(\left[x\quad y\quad z\right]^{\mathsf{T}}\) is the position vector in the earth reference frame. **Lemma 1** ([13]). : _The vertical error \(e_{h}\) between the powered parafoil and the desired point \(\textbf{p}_{p}\) is global uniformly asymptotically stable and local exponential stable, if the inclination angle of the powered parafoil changes with the following guidance law_  
\[\sigma_{d}=\arctan\left(\frac{e_{h}}{k_{h}}\right), \tag{2}\]  
_where \(e_{h}=H_{d}-H\) and \(k_{h}\) is an adjustable parameter. \(H_{d}\) and \(H\) are the desired altitude and actual altitude of the powered parafoil, respectively._  
According to Lemma 1, the altitude control of the powered parafoil can be converted to the control of the inclination angle. The velocity transformation from the parafoil frame to the earth frame can be described as  
\[ \begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{z}\end{bmatrix} =\textbf{R}_{p-e} \begin{bmatrix}u_{s}\\ v_{s}\\ w_{s}\end{bmatrix} , \tag{3}\]  
where \( \begin{bmatrix}u_{s}&v_{s}&w_{s}\end{bmatrix} ^{\text{T}}\) is the velocity vector in the parafoil frame and the transformation matrix  
\[\textbf{R}_{p-e}= \begin{bmatrix}\cos\theta\cos\psi&\sin\phi\sin\theta\cos\psi -\cos\phi\sin\psi&\cos\phi\sin\theta\cos\psi+\sin\phi\sin\psi\\ \cos\theta\sin\psi&\sin\phi\sin\theta\sin\psi+\cos\phi\cos\psi&\cos\phi\sin \theta\sin\psi-\sin\phi\cos\psi\\ -\sin\theta&\sin\phi\cos\theta&\cos\phi\cos\theta\end{bmatrix} . \tag{4}\]  
If the motion of the longitudinal plane is considered, then it is easy to obtain  
\[v_{s}=0,y=0,\psi=0,\phi=0. \tag{5}\]  
Figure 1: The structure of the powered parafoil and coordinate frames. Substituting (4) and (5) into (3) yields  
\[\dot{x}=u_{s}\cos\theta+w_{s}\sin\theta, \tag{6}\]  
\[\dot{z}=-u_{s}\sin\theta+w_{s}\cos\theta. \tag{7}\]  
Choose the inclination angle \(\sigma\) of the powered parafoil as the state variable of the system. According to (1), (6), and (7), the first derivative of \(\sigma\) is  
\[\dot{\sigma}=\dot{\theta}+\frac{\dot{u}_{s}w_{s}-u_{s}\dot{w}_{s}}{u_{s}^{2}+w_ {s}^{2}}. \tag{8}\]  
Let \(f_{s}\) be the derivative of \(\frac{\dot{u}_{s}w_{s}-u_{s}\dot{w}_{s}}{u_{s}^{2}+w_{s}^{2}}\). The second derivative of \(\sigma\) is obtained as  
\[\ddot{\sigma}=\ddot{\theta}+f_{s}. \tag{9}\]  
For the altitude control of the powered parafoil, the only control variable that affects the flight velocity is the thrust \(u\) provided by the power propulsion. According to the nonlinear dynamics model of the powered parafoil [20], the thrust control variable is coupled in \(\dot{f_{s}}\), which increases the difficulty in the design of the controller. To facilitate the design of the control system, the control variable should be separated out. Rewrite (9) as  
\[\dot{\sigma} =\ddot{\theta}+f_{s}-bu+bu \tag{10}\] \[=f+bu,\]  
where \(f=\ddot{\theta}+f_{s}-bu\) is viewed as a total disturbance.