# Target Tracking Based on a Nonsingular Fast Terminal Sliding Mode Guidance Law by Fixed-Wing UAV
## 2 Problem Formulation  
### UAV Dynamic Model  
The vehicle dynamics are based on a simplified flight dynamics model called coordinated flight vehicle (CFV) widely used in most literature [25; 26]. The CFV model is described by ordinary differential equations of reasonable complexity, and captures the nonlinear dynamics of maneuvering flight at the same time. We assume that the UAV has a reliable flight control system, which effectively controls the aerodynamic surfaces to accurately track velocity and turn rate commands. Therefore, the UAV's motion can be simplified on the horizontal plane in the presence of environmental wind, which can be described by the following equations:\[\left\{\begin{aligned} \dot{x}_{a}&=v_{a}\cos\psi_{a}+W_{x},\\ \dot{y}_{a}&=v_{a}\sin\psi_{a}+W_{y},\\ \dot{v}_{a}&=u_{a},\\ \dot{y}_{a}&=\omega_{\psi},\end{aligned}\right. \tag{1}\]  
where \(x_{a},y_{a}\) are the UAV's horizontal position in inertial coordinate system; \(\dot{x}_{a},\dot{y}_{a}\) denotes the UAV's velocity component in the \(x\)- and \(y\)-axis, respectively; \(\psi_{a}\in(-\pi,\pi]\) is the UAV's heading angle which is the included angle between velocity direction and \(x\)-axis; \(u_{a}\) denotes the acceleration command and \(v_{a}\) is the airspeed, which is set to a constant value; and \(\omega_{\psi}\) represents the command of heading rate, which is the control object in this paper, with the constraint  
\[|\omega_{\psi}|\leq\dot{\psi}_{\max}. \tag{2}\]  
For the single-UAV-single-target tracking problem in this study, we suppose that the UAV speed is constant and track the target by holding the altitude constant through the zero climb rate. In addition, \((W_{x},W_{y})^{T}\) is the wind disturbance, and it is easy to get that \(W_{x}=v_{w}\cos\psi_{w}\) and \(W_{y}=v_{w}\sin\psi_{w}\), where \(v_{w}\) and \(\psi_{w}\) denote the speed and the direction of wind disturbance. In the actual application of UAV, the airspeed \(v_{a}\) and the heading angle \(\psi_{a}\) cannot be obtained easily due to UAV's scale constraints. For the micro-UAV, the groundspeed \(v_{g}\) and the course angle \(\chi_{a}\) can be achieved through the external sensors. Therefore, Equation (1) can be rewritten as  
\[\left\{\begin{aligned} \dot{x}_{a}&=v_{g}\cos\chi_{a}, \\ \dot{y}_{a}&=v_{g}\sin\chi_{a},\\ \dot{v}_{g}&=u_{g},\\ \dot{\chi}_{a}&=u_{\chi},\end{aligned}\right.\]  
the relation between the airspeed \(v_{a}\) and the groundspeed \(v_{g}\) can be constructed as  
\[v_{g}=(W_{x}\cos\chi_{a}+W_{y}\sin\chi_{a})+\sqrt{v_{a}^{2}-(W_{x}\sin\chi_{a }-W_{y}\cos\chi_{a})^{2}},\]  
and the course rate inputs with the restriction \(|u_{\chi}|\leq\hat{\chi}_{\max}\) can be given as follows:  
\[\hat{\chi}_{a}=u_{\chi}=\mathcal{L}(\psi_{a})\omega_{\psi}, \tag{3}\]  
where  
\[\mathcal{L}(\psi_{a})=\frac{v_{a}^{2}+v_{a}(W_{x}\cos\psi_{a}+W_{y}\sin\psi_{ a})}{v_{a}^{2}+W_{x}^{2}+W_{y}^{2}+2v_{a}(W_{x}\cos\psi_{a}+W_{y}\sin\psi_{ a})}.\]  
Note that, in the absence of wind, (i.e., \(W_{x}=0,W_{y}=0\)), \(\mathcal{L}(\psi_{a})=1\). From the inequality \(0\leq\sqrt{W_{x}^{2}+W_{y}^{2}}<v_{a}\), we can get that \(\mathcal{L}(\psi_{a})\in(1/2,1]\).