# Albatross-Like Utilization of Wind Gradient for Unpowered Flight of Fixed-Wing Aircraft
## 2 Dynamic Model  
A wind shear is generated by the wind boundary layer phenomenon, and its mathematical model can take different forms. The following example is presented in the Military Specification MIL-F-8785C [40]:  
\[W=W_{6}\frac{\ln\left(\frac{z}{z_{0}}\right)}{\ln\left(\frac{6}{z_{0}}\right)},0.9\;\mathrm{m}<h<300\;\mathrm{m} \tag{1}\]  
where \(W\) denotes the mean wind velocity, \(W_{6}\) denotes the wind velocity at an altitude of 6 m, \(z\) is the altitude, and \(z_{0}\) is generally set to 0.5 m.  
A wind shear can also be modeled by an exponential wind velocity profile [36]:  
\[W=W_{R}\left(\frac{z}{Z_{R}}\right)^{p} \tag{2}\]  
where \(W_{R}\) denotes the reference wind velocity at altitude \(Z_{R}\), and \(p\) denotes the slope of the boundary layer. Let us assume a horizontal wind, the magnitude of which varies with the altitude [36]. The wind gradient can be expressed as follows:  
\[G_{w}(z)=\frac{dW}{dz} \tag{3}\]  
There are two common approaches to the modelling of dynamic soaring, namely, those proposed by [PERSON] [20] and [PERSON] [21]. The latter utilizes Cartesian coordinates, while the former uses spherical coordinates. A non-inertial coordinate frame that moves with the wind at the aircraft was adopted in the present study (see Figure 2). The \(z\)-axis of the frame points skyward. If the wind velocity increases with the altitude, the \(x\)-axis points in the opposite direction of the wind; otherwise, it points in the direction of the wind. The \(y\)-axis is determined by the right-hand principle. In Figure 2, \(W\) is the wind velocity vector at the aircraft position, and \(O_{0}\)-\(x_{0}y_{0}z_{0}\) represents the origin at the initial time. The problem formulation and modeling procedures adopted in this study are similar to those of some relevant previous works [16; 21; 23; 36]. The aircraft is under the action of its weight, aerodynamics, and inertial force. However, the model equations differ from ordinary equations in that the atmosphere-fixed frame is non-inertial. Further, the model transforms the wind gradient into an inertial force. Figure 1 illustrates an intuitive model that facilitates an understanding of the generation of the inertial force. Because the wind speed below the wave is nearly zero, and that above the wave has a certain magnitude, the wind can be considered to comprise two layers separated by a wind shear, around which a large gradient occurs. Let us assume that the airspeed relative to the albatross shown in Figure 1 is 17 m/s in the wind layer below the wave, and the wind speed above the wave is 7 m/s higher than that below the wave. If the albatross faces the wind and crosses the wind shear from the lower layer to the upper layer, the airspeed relative to it would suddenly increase to 24 m/s because there is no time for a gradual change in the ground speed. The wind gradient thus causes the albatross to gain extra airspeed, which is transformed into an inertial force, referred to as a dynamic soaring thrust. The dynamic model can be directly taken from [PERSON]'s [27] or [PERSON]'s [36] work as follows:  
\[m\dot{v}=-\frac{1}{2}\rho sC_{D}v^{2}-mg\sin\theta-m\frac{dW}{dt}\cos\theta \sin(\psi-\pi/2)\]  
\[mv\cos\theta\dot{\psi}=-\frac{1}{2}\rho sC_{L}v^{2}\sin\varphi-m\frac{dW}{dt} \cos(\psi-\pi/2)\]  
\[mv\dot{\theta}=\frac{1}{2}\rho sC_{L}v^{2}\cos\varphi-mg\cos\theta+m\frac{dW}{ dt}\sin\theta\sin(\psi-\pi/2)\text{ and }\]  
\[\dot{z}=v\sin\theta\]  
where \(v\) denotes the airspeed, \(\theta\) denotes the pitch angle of the flight path, \(\varphi\) denotes the bank angle and \(\psi\) denotes the yaw angle measured anticlockwise from \(x\)-axis. For the parameters, \(g\) is the gravitational acceleration, \(\rho\) is the density of air, \(s\) the wing area, \(m\) is the aircraft mass, and \(C_{L}\) and \(C_{D}\) are respectively the lift and drag coefficients.