# In-flight Wind Field Identification and Prediction of Parafoil Systems
## 2 Parafoil System Model  
A schematic of parafoil system is shown in Figure 1. Due to the materials of the suspension lines and the connecting belt are usually high in strength and small in deformation so that the elastic deformation of the ropes can be neglected, thereby limiting the relative rolling motion between the parafoil and the payload. Considering the relative pitch and relative yaw motions between two bodies, an eight-degree-of-freedom dynamic model of parafoil systems is established.  
Let \(V_{p}=\left[\begin{array}{cc}v_{x,p}&v_{y,p}&v_{z,p}\\ \end{array}\right]^{T}\) and \(W_{p}=\left[\begin{array}{cc}w_{x,p}&w_{y,p}&w_{z,p}\\ \end{array}\right]^{T}\) denote the speed and angular velocity of the parafoil, respectively. \(V_{l}=\left[\begin{array}{cc}v_{x,l}&v_{y,l}&v_{z,l}\\ \end{array}\right]^{T}\) and \(W_{l}=\left[\begin{array}{cc}w_{x,l}&w_{y,l}&w_{z,l}\\ \end{array}\right]^{T}\) denote the speed and angular velocity of the payload, respectively. Using the momentum and momentum momentum theorems to analyze the parafoil and the payload, we have  
\[\frac{\mathrm{d}\mathrm{T}_{p}}{\mathrm{d}t}+W_{p}\times T_{p}=F_{p}^{zero}+F _{p}^{f}+F_{p}^{G}+F_{p}^{t} \tag{1}\]  
\[\frac{\mathrm{d}\mathrm{H}_{p}}{\mathrm{d}t}+W_{p}\times H_{p}+V_{p}\times T_ {p}= \tag{2}\]  
\[\frac{\mathrm{d}\mathrm{T}_{l}}{\mathrm{d}t}+W_{l}\times T_{l}=F_{l}^{zero}+F _{l}^{t}+F_{l}^{G} \tag{3}\]  
\[\frac{\mathrm{d}\mathrm{H}_{l}}{\mathrm{d}t}+W_{l}\times H_{l}=M_{l}^{zero}+ M_{l}^{f}+M_{l}^{t} \tag{4}\]  
where \(T\) and \(H\) represent the momentum and the momentum moment, respectively. \(F\) and \(M\) represent force and moment, respectively. Subscripts \(p\) and \(l\) represent the analysis of the parafoil and the  
Figure 1: The schematic of a parafoil system.  
payload, respectively, and superscripts _aero_, \(f\), \(G\) and \(t\) represent the aerodynamic force, the friction force, the gravity and the tension of suspension lines, respectively.  
Taking the apparent mass of the parafoil system into consideration, the momentum \(P\) and the momentum moment \(H\) of the parafoil and the payload are expressed as  
\[\left[\begin{array}{c}T_{p}\\ H_{p}\end{array}\right]=\left[A_{a}+A_{r}\right]\left[\begin{array}{c}V_{p}\\ W_{p}\end{array}\right] \tag{5}\]  
\[\left[\begin{array}{c}T_{l}\\ H_{l}\end{array}\right]=\left[\begin{array}{cc}m_{l}&0\\ 0&J_{l}\end{array}\right]\left[\begin{array}{c}V_{l}\\ W_{l}\end{array}\right] \tag{6}\]  
where \(A_{a}\) and \(A_{r}\) are the real mass and the apparent mass matrix of the parafoil and payload, respectively. \(m_{l}\) and \(J_{l}\) are the mass and moment of inertia of the payload, respectively.  
The parafoil and the payload are connected by lines, and the constraint relationships between them are  
\[V_{l}+W_{l}\times L_{l-c}=V_{p}+W_{p}\times L_{p-c} \tag{7}\]  
\[W_{l}=W_{p}+\tau_{s}+\kappa_{p} \tag{8}\]  
where \(\tau_{s}\) and \(\kappa_{p}\) denote the relative yaw angle and relative pitch angle of the parafoil and the payload, respectively, which are written in vector forms as \(\tau_{s}=\left[\begin{array}{cc}0&0&\psi_{r}\end{array}\right]^{T}\) and \(\kappa_{p}=\left[\begin{array}{cc}0&\theta_{r}&0\end{array}\right]^{T}\). And \(L_{l-c}\) and \(L_{p-c}\) denote the distance between the center of payload \(l\) and the center of parafoil \(p\) to the center of the two suspension points \(C\), respectively.  
Through Equations (1)-(8), the eight-degree-of-freedom dynamic model of the parafoil system can be obtained. For the detailed modeling process, see [2].