## 5 MATLAB Simulation Results  
In this section, we demonstrate the effectiveness of our chasing controller utilizing simulations. \((x(k),y(k))-\mathbf{r}_{k}^{e}\) represents sensor measurement noise and is a vector with two elements. Each element in this vector has a Gaussian distribution with mean 0 and covariance 0.01. To predict the evader's location two steps forward in time, we utilize recent \(K=5\) measurements. Initially, the evader is at \((0,5000)\), and the pursuer is at the origin. \(T=2\) s. In addition, the initial speed of the pursuer is 150 m/s. In the terminal phase, \(v_{k}^{p}\) decreases with respect to \(k\) until \(v_{k}^{p}\) converges to 110 m/s. The pursuer's maximum acceleration is \(a_{m}=10\) m/s\({}^{2}\). Initially, the pursuer's heading is \(\mathbf{u}_{0}=(1,0)\). In the simulations, we set the maximum turn rate of the pursuer as \(q=\pi/6\) radians per second. MATLAB simulation runs for finite time, at the end of which the distance between the pursuer and the evader is less than 100 m. Recall that the evader's motion model is presented in (2). The authors of [8] introduced Frenet-Serret frames [33] to model the movement of an evader. \[\mathbf{r}_{k+1}^{e} = \mathbf{r}_{k}^{e}+T*\
u_{k}^{e}*\mathbf{x}_{k}^{e}\] \[\mathbf{x}_{k+1}^{e} = \mathbf{x}_{k}^{e}+T*\
u_{k}^{e}*u_{k}^{e}*\mathbf{y}_{k}^{e}\] \[\mathbf{y}_{k+1}^{e} = \mathbf{y}_{k}^{e}-T*\
u_{k}^{e}*u_{k}^{e}*\mathbf{x}_{k}^{e}, \tag{38}\]  
where \(u_{k}^{e}\) is the steering (i.e., curvature) control of the evader at time index \(k\), and \(v_{k}^{e}\) is the speed of the evader at time index \(k\). Recall that \(\mathbf{r}_{k}^{e}\) is the location of the evader at time index \(k\). Moreover, \(\mathbf{x}_{k}^{e}\) is the unit tangent vector to the trajectory of the evader at time index \(k\), and \(\mathbf{y}_{k}^{e}\) is the corresponding unit normal vector at time index \(k\). We utilize (38) to simulate the motion of the evader. As the first scenario, we set the evader's speed as \(v_{k}^{e}=100\) in m/s. In addition, the evader does not maneuver, i.e., \(u_{k}^{e}=0\). Figure 4 shows pursuer and evader trajectories. In this figure, the pursuer is depicted with red points, and the evader is depicted with blue points. Initially, the pursuer turns towards the evader with the maximum turn rate \(q\). Thereafter, the pursuer converges to the motion camouflage state. Figure 4: The system behavior for a constant velocity evader with \(u_{k}^{e}=0\) and \(v_{k}^{e}=100\). Figure 3: An illustration of \(\mu_{k+1}\). In Figure 5, \(1=\frac{\textbf{r}^{p}-\textbf{r}^{p}}{\|\textbf{r}^{r}-\textbf{r}^{p}\|}\) is plotted. \(UnitX-proj\) and \(UnitY-proj\) indicate the projection of **1** on the x-axis and y-axis respectively. The change of these values with respect to time indicates the rotation rate of LOS at every time index. For a pursuit-evasion system in the motion camouflage state, **1** converges. Hence, \(UnitX-proj\) and \(UnitY-proj\) also converge. Figure 5 depicts that a pursuit-evasion system converges to the motion camouflage state. The chasing controller is designed to converge to the motion camouflage state at time index 0. But, we set \(q=\pi/6\) radians per second in the simulation. Due to the constraint on the maximum turn rate, the pursuer cannot converge to the motion camouflage state at time index 0. The pursuer turns with the maximum turn rate until reaching the motion camouflage state. As the pursuer reaches the motion camouflage state, it stays in the state. The top figure in Figure 6 depicts the pursuer's speed with respect to time. See that in the terminal phase, the pursuer adjusts its speed so that it can capture the evader with low speed. Since the relative speed of the evader with respect to the pursuer is low, the probability to the evader is also low. The bottom figure in Figure 6 depicts the distance between the evader and the pursuer with respect to time. Even though the pursuer slows down in the terminal phase, the distance monotonically decreases as time passes.