# Cooperative Standoff Tracking of Moving Targets Using Modified Lyapunov Vector Field Guidance
## 5 Experiments and Results
### Numerical Simulation
#### 5.1.1 Standoff Tracking of Stationary Targets  
The numerical simulation of standoff tracking of the stationary target is carried out to verify the superiority of the proposed algorithm over other guidance laws. Stable tracking of stationary targets is the base of tracking moving targets and cooperative tracking.  
In this paper, methods proposed by [PERSON] [12] and [PERSON] [7] are compared with our guidance laws in the same simulation configuration. Two simulations are carried out with different initial UAV position. In simulation A, the initial position of the UAV is far away from the standoff circle, while in simulation B, it is near the standoff circle. Supposing UAV state vectors are \([x,y,\psi,v]^{T}\), where \(x\), \(y\) are the coordinates of the UAV position, \(\psi\) is the current heading angle, and \(v\) is the UAV speed.  
In Simulation A, the initial state vector of the UAV is \([600,600,1.25\pi,18]^{T}\). As shown in Figure 10a, obviously, the proposed guidance law guides the UAV faster to the standoff circle. The UAV guided by the proposed guidance law settles to the standoff circle at \(t=53\) s, while [PERSON]'s method at \(t=112\) s and [PERSON]'s method at \(t=210\) s in Figure 10b. Reasons can be found by analyzing Figure 10c: The proposed guidance law guides the UAV to fly at a bigger course rate when approaching the standoff circle, resulting in faster settling to the desired distance. In contrast, the other two methods guide the UAV in a relatively gentle manner. UAVs fly at a smaller course rate, squandering time on the path to the standoff circle.  
In simulation B, the initial UAV state vector is \([300,300,1.5\pi,18]^{T}\), which is close to the standoff circle. The trajectories are shown in Figure 11a. When the initial position of the UAV is near the standoff circle, the trajectory will intersect with the circle if the UAV obeys the guidance generated by the normal Lyapunov vector field. The distance error curvatures of this situation are denoted as the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(R\) & standoff distance & 300 m \\ \(s\) & cruising speed of UAVs & 20 m/s \\ \(s_{\text{min}}\) & minimum speed of UAVs & 13 m/s \\ \(s_{\text{max}}\) & maximum speed of UAVs & 25 m/s \\ \(a_{\text{min}}\) & minimum acceleration of UAVs & \(-2\) m/s\({}^{2}\) \\ \(a_{\text{max}}\) & maximum acceleration of UAVs & 2 m/s\({}^{2}\) \\ \(\dot{\psi}_{\text{max}}\) & maximum course rate & 0.2 rad/s \\ \hline \hline \end{tabular}
\end{table}
Table 1: Key parameters in the numerical simulation.  
Figure 10: Experiments results with respect to an initial UAV position far away from the standoff circle. (**a**) Trajectories of UAVs; (**b**) Distance error versus time; (**c**) Course rate versus time.  
cyan line and blue line in Figure 11b, where the values of the interval (10,110) are negative. In this situation, the proposed guidance law guides the UAV to fly away from the standoff circle temporarily using the inverse Lyapunov vector field and go back when the UAV is able to converge to the standoff circle without intersecting with the standoff circle, whose distance error curvature is denoted as the red line in Figure 11b. The red line corresponding to the proposed method has no negative distance error, which means that the UAV trajectory does not intersect with the standoff circle. As a side effect, the UAV flies at the maximum turning rate before settling to the standoff circle, as shown in Figure 11c.  
In this experiment, the guidance law generated by our algorithm is adopted to guide the UAV when it is at different distances from the desired circle. When the UAV is far away from the standoff circle, the trajectory converges to the circle faster than the compared algorithm and circles around the target stably without intersecting with the desired circle. When the UAV is close to the circle, our algorithm guides the UAV to fly away from the circle temporarily. In this way, the trajectory will not intersect with the circle.