# Target Tracking Based on a Nonsingular Fast Terminal Sliding Mode Guidance Law by Fixed-Wing UAV
## 4 Simulation Results  
In this section, the numerical simulation results are presented to demonstrate the effectiveness of the proposed guidance law. Three sets of simulations are performed: (1) when the target is stationary; (2) when the target is moving with constant velocity; and (3) when the target is moving with changing acceleration and heading angle rate. These simulations demonstrate successful target tracking capability in all cases. The necessary parameters of target tracking are listed in Table 1.  
In order to make full verification for the method in this paper, the proposed NFTSM guidance law is compared with a traditional SMC in [18]. In the traditional SMC method, the sliding manifold can be designed as  
\[s=\tilde{y}_{a}+c_{1}\arctan c_{2}r,\qquad c_{1},c_{2}>0. \tag{34}\]  
In addition, the guidance law is proposed as  
\[\omega_{\psi}=\lambda+\tilde{y}_{a}^{d}-\frac{c_{1}c_{2}\dot{r}}{1+c_{2}^{2}r^ {2}}-c_{3}\text{sat}\left(\frac{s}{\epsilon}\right),\qquad c_{3}>0, \tag{35}\]  
where \(\text{sat}(y)\) is a high-slope saturation function, which is constructed to eliminate chattering by SMC method, defining as follows:  
\[\text{sat}(t)=\left\{\begin{array}{ll}y,&\quad\text{if}|y|\leq 1,\\ sign(y),&\quad\text{if}|y|>1,\end{array}\right.\]  
The coefficients of the NFTSM guidance law proposed in this paper and the traditional SMC method in Equation (35) are listed in Table 2, and another set of coefficients (\(\alpha_{2}\) and \(\beta_{2}\)), not performing good enough, is given as a contrast (NFTSM\({}_{1}\)).  
In this simulation, we get a desired trajectory of the moving target by Equation (1) whose subscript \(a\) is replaced as \(t\). In order to simulate the actual situation, we let a virtual sensor observe the information of the target at the inertial frame origin. The motion of the target is estimated by using the EKF. The parameters in Equation (4) can be set as follows: let the sampling time \(T_{s}=0.5\) s; set the corresponding covariance matrixes of the process noise and measurement noise are \(\mathbf{Q}=diag(0.2,0.01,0.2,0.01)\) and \(\mathbf{P}=diag(0.5,0.015)\), respectively. Thus, we can get the estimation of state: \(x_{t},y_{t},v_{t}=\sqrt{\dot{x}_{t}^{2}+\dot{y}_{t}^{2}},\dot{y}_{t}=\arctan \left(\dot{y}_{t}/\dot{x}_{t}\right)\).  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Initial Condition** & **Value** \\ \hline Sampling time, \(\Delta T\)(s) & 0.5 \\ UAV initial position, \([x,y]\)(m) & [\(-\)200,200] \\ Constant airspeed, \(v_{a}\)(m/s) & 13 \\ Maximum turn rate, \(\dot{\psi}_{\text{max}}(^{\circ}/s)\) & 5 \\ Desired distance, \(R_{d}\)(m) & 200 \\ Target initial position, \([x_{t},y_{t}]\)(m) & [10,10] \\ Target initial velocity, \(v_{t}\)(m/s) & 2 \\ Wind speed, \([W_{x},W_{y}]\)(m/s) & [0.1,0.2] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of target tracking, UAV: unmanned aerial vehicle.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Algorithm** & & & & & \\ \hline NFTSM & \(\begin{array}{ll}\alpha_{2}=0.5,&\alpha_{3}=0.2,&\eta=0.001,\\ k_{1}=0.05,&k_{2}=12,&k_{3}=6,\end{array}\) & \(\begin{array}{ll}\beta_{2}=0.2,&\beta_{3}=2/3\\ \gamma_{1}=9/10\end{array}\) \\ \hline TSM & \(c_{1}=0.2,&c_{2}=0.05,&c_{3}=0.005,&\epsilon=0.2\) \\ \hline NFTSM\({}_{1}\) & \(\begin{array}{ll}\alpha_{2}=2,&\alpha_{3}=0.2,&\eta=0.001,\\ k_{1}=0.05,&k_{2}=12,&k_{3}=6,\end{array}\) & \(\begin{array}{ll}\eta_{2}=3,&\beta_{3}=2/3\\ \gamma_{1}=9/10\end{array}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Coefficients of the different guidance laws, TSM: terminal sliding mode, NFTSM: nonsingular fast TSM, NFTSM\({}_{1}\): NFTSM with another set of coefficients.