# Target Tracking Based on a Nonsingular Fast Terminal Sliding Mode Guidance Law by Fixed-Wing UAV
## 2 Problem Formulation
### Relative Motion Model  
We suppose that the model of the target is similar to the UAV model as shown in Equation (1) with adding the speed command \(\psi_{t}=a_{t}\). Namely, the acceleration \(a_{t}\) and turning rate \(\omega_{t}\) of target are defined as the command for target maneuver strategy. The motion state of target can be summarized as \((x_{t},y_{t},v_{t},\psi_{t})\). Due to the fact that the detailed control information cannot be obtained by UAV, the extended Kalman filter (EKF) [27] is utilized to estimate the motion of target in this paper as follows:  
\[\left\{ \begin{aligned} \mathbf{X}_{k+1}&=\mathbf{F}_{t}( \mathbf{X}_{k})+\boldsymbol{\zeta}_{k},\\ \mathbf{Z}_{k+1}&=\mathbf{H}_{t}(\mathbf{X}_{k})+ \boldsymbol{\varrho}_{k},\end{aligned} \right. \tag{4}\]where motion state in EKF can be redefined as \(\mathbf{X}_{k}=(x_{t},\dot{x}_{t},y_{t},\dot{y}_{t})^{T}\) with \(\dot{x}_{t}=v_{t}\cos\psi_{t}\) and \(\dot{y}_{t}=v_{t}\sin\psi_{t}\). Therefore, the state equation can be expressed as [28]  
\[\mathbf{F}_{t}= \begin{bmatrix}1&T_{s}&0&0\\ 0&1&0&0\\ 0&0&1&T_{s}\\ 0&0&0&1\end{bmatrix} ,\]  
where \(T_{s}\) is sampling time. In addition, \(\xi_{k}\) with \(P(\zeta)\sim N(0,\mathbf{Q})\) is the zero-mean white Gaussian noise, and \(\mathbf{Q}=diag(V(\zeta_{x}),V(\zeta_{x}),V(\zeta_{y}),V(\zeta_{y}))\) is the corresponding covariance matrixes. \(V(\kappa)\) is the variance of \(\kappa\). Assuming that the sensor is fixed at the inertial frame origin, so that the observed value can be defined as \(\mathbf{Z}_{k}=(\hat{r}_{t},\hat{x}_{t})^{T}\). \(\hat{r}_{t}\) is the observed distance between the target and the sensor, while \(\hat{x}_{t}\) is the azimuth angle. The observation equation \(\mathbf{H}_{t}\) in Equation (4) can be given as  
\[\mathbf{H}_{t}(\mathbf{X}_{k})=\left\{ \begin{aligned} \hat{r}_{t}&=\sqrt{x_{t}^{2}+y_{t}^{2}}+ \varrho_{r},\\ \hat{x}_{t}&=\arctan(y_{t}/x_{t})+\varrho_{a},\end{aligned} \right.\]  
where \(\varrho_{k}\) with \(P(\varrho)\sim N(0,\mathbf{P})\) is the zero-mean white Gaussian observation noise, and \(\mathbf{P}=diag(V(\varrho_{r}),V(\varrho_{a}))\) is the corresponding covariance matrixes. Defining the target reference frame by setting the origin of frame at target inertial position, such that the target is relatively static. Figure 1 depicts the relative position between the target and UAV in inertial reference frame and target reference coordinate. The relative motion model between the two vehicles can be described by [29]  
\[ \begin{aligned} \dot{r}&=v_{a}\cdot\sin(\psi_{a}- \lambda)-v_{t}\cdot\sin(\psi_{t}-\lambda)+v_{w}\cdot\sin(\psi_{w}-\lambda), \\ (r+R_{d})\dot{\lambda}&=-v_{a}\cdot\cos(\psi_{a}- \lambda)+v_{t}\cdot\cos(\psi_{a}-\lambda)-v_{w}\cdot\cos(\psi_{w}-\lambda), \end{aligned}  \tag{5}\]  
where \(R_{d}\) is the desired separation distance between target and UAV; \(r=L-R_{d}\) denotes the distance error, where \(L=\sqrt{x_{\hat{r}}^{2}+y_{\hat{r}}^{2}}=\sqrt{(x-x_{t})^{2}+(y-y_{t})^{2}}\) is defined as the distance of two vehicles. In addition, \(\lambda=\theta-\pi/2\in(0,2\pi]\) is the auxiliary parameter, and \(\theta=\arctan\left((y-y_{t})\big{/}(x-x_{t})\right)\) represents the azimuth angle of UAV in the target-frame. Figure 1: Tracking geometry between the target and unmanned aerial vehicle (UAV). It is noteworthy that the target may flee away from the monitoring range of UAV when the speed of target is greater than the maximum speed of the UAV. Hence, it should be limited in Remark 1 in order to avoid task failure. The UAV and wind speeds are assumed to satisfy \(0\leq v_{t}+v_{w}<v_{a}\). (Target Tracking Problem) The objective of the target tracking problem is to design the guidance input \(\omega_{\psi}\) such that the following conditions are satisfied for the UAV in finite time \(T_{f}\):  
\[ \begin{split}\parallel r(t)\parallel&\mapsto 0,\\ \parallel\dot{r}(t)\parallel&\mapsto 0,\qquad t \mapsto T_{f},\\ r(t)\cdot\dot{r}(t)\leq& 0.\end{split}