# In-flight Wind Field Identification and Prediction of Parafoil Systems
## 3 Wind Field Identification  
When there is no wind in the environment, the parafoil system is in a gliding flight state under no manipulation or synchronously pulling two steering ropes. When a single steering rope is pulled down at a constant state (also refers to unilateral deflection), the parafoil system is turning to the pulling side. Moreover, in windy conditions, the flight trajectory of the parafoil system drifts with winds, which relates to the speed and direction of the wind. Therefore, it is possible to identify the information of the ambient wind field through the dynamic change of flight trajectories of the parafoil system. In this section, an identification methods of wind fields is investigated [26]. The velocity vector diagram of the parafoil system is shown in Figure 2[12], which illustrates a vector triangle of wind velocity, constituted by \(V_{a}\), \(V_{w}\) and \(V\). \(V_{a}\) denotes the velocity of parafoil systems with respect to the air, named as air velocity. \(V_{w}\) denotes the wind velocity, and \(V\) denotes the velocity of parafoil systems with respect to the ground, named as ground velocity. \(\beta\) is the sideslip angle, and \(\psi\) denotes the yaw angle. \(\chi_{a}\), \(\chi_{w}\) and \(\chi\) are angles between \(V_{a}\), \(V_{w}\), \(V\) and north direction, respectively. From Figure 2, the ground velocity \(V\) of the parafoil system is equal to the sum vectors of wind velocity \(V_{w}\) and air velocity \(V_{a}\). Assuming that the parafoil system keeps unilateral deflection within a certain time period, and air velocity \(V_{a}\) remain unchanged [1], by the vector triangle, we obtain  
\[\left\{ \begin{array}{l}\frac{\mathrm{d}\chi_{i}}{\mathrm{d}t}=V_{w,x}+V_{a }\cos(\chi_{a,i})\\ \frac{\mathrm{d}\chi_{i}}{\mathrm{d}t}=V_{w,y}+V_{a}\sin(\chi_{a,i})\end{array} \right. \tag{9}\]  
where \(\mathrm{i}\in\{1,2,...,n\}\) denotes for the sampling time sequence. \(x\) and \(y\) denotes the directions of south-north and east-west, respectively. \(\mathrm{d}x_{i}/\mathrm{d}t\) and \(\mathrm{d}y_{i}/\mathrm{d}t\) denote the ground velocity of the parafoil system in \(x\) and \(y\) directions at the i-th sampling time, respectively. \(V_{w,x}\) and \(V_{w,y}\) denotes the components of the velocity of horizontal wind (i.e., cross wind) in \(x\) and \(y\) directions, respectively. Further, Equation (9) is rewritten as  
\[ \begin{array}{c}V_{a}^{2}=(\frac{\mathrm{d}x_{i}}{\mathrm{d}t}-V_{w,x})^{2}+( \frac{\mathrm{d}y_{i}}{\mathrm{d}t}-V_{w,y})^{2}\\ =V_{i}^{2}+V_{w}^{2}-2(\frac{\mathrm{d}x_{i}}{\mathrm{d}t}V_{w,x}+\frac{\mathrm{ d}y_{i}}{\mathrm{d}t}V_{w,y})\end{array}  \tag{10}\]  
where \(V_{i}^{2}=\left(\frac{dx_{i}}{dt}\right)^{2}+\left(\frac{dy_{i}}{dt}\right)^{2}\) denotes the ground velocity of the parafoil system at the \(i\)-th sampling time. By Equation (10), we get  
\[ \begin{array}{c}V_{a}^{2}-E(V_{a}^{2})=0\\ =V_{i}^{2}-E(V_{i}^{2})-\\ 2((\frac{\mathrm{d}x_{i}}{\mathrm{d}t}-E(\frac{\mathrm{d}x_{i}}{\mathrm{d}t})) V_{w,x}+(\frac{\mathrm{d}y_{i}}{\mathrm{d}t}-E(\frac{\mathrm{d}y_{i}}{\mathrm{d}t})) V_{w,y})\end{array}  \tag{11}\]  
where \(E\) denotes for taking expected value. For simplicity, let  
\[ \begin{array}{c}E(V_{i}^{2})=\mu_{V2}\\ E(\frac{\mathrm{d}x_{i}}{\mathrm{d}t})=\mu_{x_{i}}\\ E(\frac{\mathrm{d}y_{i}}{\mathrm{d}t})=\mu_{y_{i}}\end{array}  \tag{12}\]  
Now, the wind field identification of parafoil systems becomes solving linear regression by Equation (13)  
\[\left[ \begin{array}{cc}\frac{\mathrm{d}x_{1}}{\mathrm{d}t}-\mu_{\hat{x}}& \frac{\mathrm{d}y_{1}}{\mathrm{d}t}-\mu_{\hat{y}}\\ \vdots&\vdots\\ \frac{\mathrm{d}x_{n}}{\mathrm{d}t}-\mu_{\hat{x}}&\frac{\mathrm{d}y_{n}}{ \mathrm{d}t}-\mu_{\hat{y}}\end{array} \right]\left[ \begin{array}{c}V_{w,x}\\ V_{w,y}\end{array} \right]=\frac{1}{2}\left[ \begin{array}{c}V_{1}^{2}-\mu_{V2 }\\ \vdots\\ V_{n}^{2}-\mu_{V2}\end{array} \right] \tag{13}\]  
It is worth mentioning that Equation (13) is not a proper matrix multiplication, and expresses a linear equation set of iterative calculation.