## 2 Model of the Parafoil Delivery System  
The parafoil delivery system is a very precise system, which has a unique structure. As shown in Figure 1, the parafoil consists of two parts: a flexible parafoil canopy and a payload. These two structures are connected by the lifting ropes and two controllable ropes. The controllable ropes are attached to the rear of the canopy. The shape of the canopy can be changed by manipulating the length of the controllable ropes. This phenomenon is so-called flap deflection. This method is used as the control input in the horizontal direction during the flight of the parafoil delivery system. With the flap deflection, the yaw angle of the parafoil delivery system can be controlled. Due to its good maneuverability, the parafoil delivery system has greatly improved the landing accuracy than the traditional parachute.  
The four main processes of flight tests are shown in Figure 2. The UAV system first carries the parafoil to the target altitude, and when it reaches the target altitude, the UAV releases it in the air. The whole flight process of the parafoil is controlled by the remote  
Figure 1: Parafoil delivery system.  
control device. Data during flight, such as control quantities and yaw angles, are used to analyze the dynamics of parafoil. The physical parameters of the parafoil are shown in Table 1.  
Figure 3 records the data of a real flight trajectory of the parafoil. Figure 2(a),b visually show the horizontal trajectory and 3D trajectory of the parafoil, respectively. The data in Figure 2(c) records the horizontal and vertical flight velocities of the parafoil at different times. The glide-ratio data are recorded in Figure 2(d), and glide-ratio is defined as the ratio between the forward distance and the altitude drop during flight. The results show that, during the whole parafoil flight, the average vertical velocity and horizontal velocity are 3.7 m/s and 7.5 m/s, respectively. The average glide-ratio during flight is 2.1. The minimum turning radius of parafoil delivery system is 20 m. By analyzing the above real parafoil flight data, we built a 4-DOF model of parafoil:  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value/Unit** \\ \hline Span & 2/m \\ Chord & 0.8/m \\ Area of canopy & 3/m\({}^{2}\) \\ Length of suspending ropes & 1.4/m \\ Mass of payload & 12.5/kg \\ Mass of canopy & 0.9/kg \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical parameters of parafoil.  
Figure 2: Process of flight test.  
\[\left\{\begin{array}{c}\dot{x}=v_{xy}\cos\varphi+v_{vx}\\ \dot{y}=v_{xy}\sin\varphi+v_{wy}\\ \dot{y}=v_{z}\\ \dot{\varphi}=\omega+u\end{array}\right. \tag{1}\]  
where \([x,y,z]\) denotes the position of the parafoil system, \(\left[v_{ux},v_{uy},v_{wz}\right]\) represents the component of wind velocity on the \(x\)-axis, \(y\)-axis, and \(z\)-axis, \(v_{xy}\) denotes the horizontal velocity, and \(v_{z}\) represents the vertical velocity of the parafoil system. In this work, it is assumed that the value of the component \(v_{wz}\) of wind speed on the \(z\)-axis is equal to 0, so \(z=v_{z}\). \(\varphi\) represents the yaw angle of the parafoil system, \(\dot{\varphi}\) is the angular velocity, \(\omega\) represents the accumulated angular velocity of the parafoil, and \(u\) represents the control input of parafoil system during flight, which is the angular acceleration.