# Cooperative Standoff Tracking of Moving Targets Using Modified Lyapunov Vector Field Guidance
## 3 Lyapunov Vector Field Guidance Framework  
Lyapunov vector field guidance proposed by [PERSON] [6] and its modified version are utilized to lead the fixed-wing UAV to approach the desired standoff distance and circle around a target in the local coordinate system. A nondimensional parameter is introduced to improve the convergence performance of LVFG [10]. The basic framework is  
\[\left[ \begin{array}{c}\dot{r}\\ r\dot{\theta}\end{array} \right]=\frac{s}{\sqrt{r^{4}+(c^{2}-2)r^{2}r_{d}^{2}+r_ {d}^{4}}}\left[ \begin{array}{c}-(r^{2}-r_{d}^{2})\\ c\cdot r\cdot r_{d}\end{array} \right] \tag{4}\]  
where \(c\) is a parameter, which can be used to adjust the speed of the generated field converging to the standoff circle. If \(c=2\), the vector field is the original version of LVFG. In the above equation, \(R=\left|\dot{r}\right|\) represents the speed of radial convergence to desired circle, and can be called a contraction component, while \(T=\left|r\dot{\theta}\right|\) denotes circulation component, which denotes the tangent speed, as shown in Figure 3.  
To discuss the influence of \(c\) on the contraction and circulation, \(k\) can be represented as  
\[k=\frac{r}{r_{d}} \tag{5}\]  
Substituting Equation (5) into Equation (4), we get  
\[R=\left|\frac{s(k^{2}-1)}{\sqrt{k^{4}+(c^{2}-2)k^{2}+1}}\right| \tag{6}\]  
\[T=\left|\frac{sck}{\sqrt{k^{4}+(c^{2}-2)k^{2}+1}}\right| \tag{7}\]  
Figure 2: Scenario of phase keeping in standoff target tracking. Curves of contraction and circulation components with \(c\) as a constant are shown in Figure 4. Without loss of generality, we set \(s\) as a constant at 1. As shown in curves in Figure 4a, as \(k\) goes from \(k=0\) or \(k=\infty\) to \(k=1\), meaning that the UAV approaches the standoff circle, the contraction component keeps decreasing, and the larger \(c\) is, the faster \(R\) goes down from 1, which means it spends more time converging to the desired circle. The opposite situation is shown in Figure 4b. The circulation component goes up more steeply as \(c\) is less when k is near \(k=1\), which denotes that as \(c\) becomes smaller, the convergence speed in standoff tracking is improved significantly. In the analysis of Shun Sun [12], when c becomes larger, the vector field converges slowly but can keep a distance from the prescribed radius stably; on the contrary, if c becomes smaller, the convergence time reduces sharply. However, in this situation, it is time-consuming on stabilization, where the UAV travels in and out of the loiter circle. Shun Sun copes with this problem by designing a series of guidance functions and searching for the optimal one before the UAV performs the mission. Obviously, offline search cannot adapt to the application scenario where the target is uncooperative with UAVs  
Figure 4: Curves of contraction and circulation components versus \(k\) in different \(c\). (**a**) Curves of contraction component versus \(k\); (**b**) Curves of circulation component versus \(k\). Figure 3: Construction component and circulation component. owing to the computing burden. In light of the above work, we will design a guidance function simultaneously taking into account both fast convergence and stable tracking performance in an online manner, which means we can adaptively balance the converging speed and stable tracking in real time. Due to the maximum turning rate of the fixed-wing UAV, the curvature of the route must be analyzed. Transforming Equation (4) into a Cartesian coordinate system, the control manner of Lyapunov vector field guidance can be rewritten into Equation (8). \[\left[ \begin{array}{c}\dot{x}\\ \dot{y}\end{array} \right]=-\frac{s}{r\sqrt{r^{4}+(c^{2}-2)r_{d}^{2}r^{2}+r_{d} ^{4}}}\left[ \begin{array}{c}(r^{2}-r_{d}^{2})x_{r}+c\cdot r\cdot r_{d}\cdot y _{r}\\ (r^{2}-r_{d}^{2})y_{r}-c\cdot r\cdot r_{d}\cdot x_{r}\end{array}