## 2 Dynamic Modeling Introduction of Powered Parafoil  
To provide a more precise depiction of the parafoil's motion characteristics, this paper adopts an eight-DOF dynamic model [5]. The model encompasses the parafoil canopy's slew, sway, yaw, heave, pitch, and roll movements, along with the relative pitch and yaw motions between the parafoil canopy and the payload. To facilitate the modeling of the parafoil system, three coordinate systems are introduced: the ground coordinate system, denoted as \(O_{d}x_{d}y_{d}z_{d}\); the parafoil coordinate system, denoted as \(O_{s}x_{s}y_{s}z_{s}\); and the payload coordinate system, denoted as \(O_{w}x_{w}y_{w}z_{w}\). These coordinate systems are illustrated in Figure 1.  
First of all, based on Figure 1, we can observe that a powered parafoil system comprises a parafoil canopy, a payload, parafoil ropes, and a thrust device. During flight, the paracord can adjust its direction by pulling it either to the left or right. The thrust device, powered by the propeller, generates the necessary upward lift for the parafoil. Hence, compared to the conventional unpowered parafoil, the powered parafoil inves  
Figure 1: Schematic diagram of parafoil system coordinates. exhibits enhanced flexibility. Subsequently, we will briefly introduce the dynamic equation of the powered parafoil. By analyzing the forces acting on the canopy and the payload, the following momentum equation can be obtained,  
\[ \begin{cases}\dfrac{\partial P_{s}}{\partial t}+W_{s}\times P_{s}=F_{s}^{aero}+F _{s}^{G}+F_{s}^{t}\\ \dfrac{\partial P_{w}}{\partial t}+W_{w}\times P_{w}=F_{w}^{aero}+F_{w}^{G}+F_{ w}^{t}+F_{w}^{th}\end{cases}  \tag{1}\]  
where the subscripts \(s\) and \(w\) refer to the parafoil canopy and the payload, respectively, indicating the variables associated with each component. \(P_{s}\) and \(P_{w}\) represent the momentums, which are calculated as the product of the mass and velocity of the canopy or payload, respectively. In addition, the right side of the equation represents the force exerted on both the canopy and the payload. The canopy experiences the combined effects of aerodynamic force \(F_{s}^{aero}\), gravity \(F_{s}^{G}\), and the tension in the connecting ropes \(F_{s}^{t}\). Similarly, the payload undergoes these three forces, but it is further influenced by the propeller's thrust \(F_{w}^{th}\). The velocity vectors are denoted as \(V_{s}=[u_{s},v_{s},w_{s}]^{T}\) and \(V_{w}=[u_{w},v_{w},w_{w}]^{T}\), respectively. Similarly, the angular velocity vectors are represented by \(W_{s}=[p_{s},q_{s},r_{s}]^{T}\) and \(W_{w}=[p_{w},q_{w},r_{w}]^{T}\). For the specific parameters of the model in this paper, please refer to [5; 35]. Equation (2) is the momentum moment equation corresponding to Equation (1),  
\[ \begin{cases}\dfrac{\partial H_{s}}{\partial t}+W_{s}\times H_{s}+V_{s}\times P _{s}=M_{s}^{aro}+M_{s}^{G}+M_{s}^{f}+M_{s}^{t}\\ \dfrac{\partial H_{w}}{\partial t}+W_{w}\times H_{w}=M_{w}^{aero}+M_{w}^{f}+M_ {w}^{t}\end{cases}  \tag{2}\]  
where the moment of momentum \(H\) is obtained by multiplying the moment of inertia matrix with the angular velocity. Indeed, by utilizing Equations (1) and (2), we can derive the dynamic equation of the powered parafoil system. This equation encompasses the state variables \(V_{s}\), \(V_{w}\), \(W_{s}\) and \(W_{w}\), along with the relative yaw angle \(\psi_{r}\) and relative pitch angle \(\theta_{r}\) existing between the canopy and the payload. Furthermore, the system comprises two control variables: flap deflection, expressed as \(u_{1}(-10\leq u_{1}\leq 10\) cm), and thrust, expressed as \(u_{2}(0\leq u_{2}\leq 120\) N). \(u_{1}\) and \(u_{2}\) represent the variables for flap deflection and thrust, respectively. The acceptable range for these variables is determined by the capabilities of the steering gear and thrust motor. The saturation value is intentionally set to safeguard the equipment. Flap deflection alters the flight direction, while thrust generates lift for the parafoil, enabling climbing.