# Cooperative Standoff Tracking of Moving Targets Using Modified Lyapunov Vector Field Guidance
## 4 Design of Cooperative Tracking Guidance Laws  
### Design of Vector Field for Fast Convergence Target Tracking  
When the UAV searches for the ground target, the target can be detected via an onboard camera when the UAV is far away from the target. Therefore, in this paper, we focus on the situation where the UAV is out of the standoff circle when it receives the 'tracking' command. In the analysis in the previous section, smaller guidance parameter \(c\) makes the vector field more radial toward the desired circle. Although faster arrival at the standoff circle, it may result in greater time wasting on stabilization by making the aircraft repeatedly traverse the circle, as shown in Figure 6.  
Two reasons may explain the event that trajectory intersects with the standoff circle. One reason is that the vector field will generate a non-flyable path for the aircraft with an inappropriate guidance parameter \(c\). The other is that the starting position for the 'tracking mission' is too close to the standoff circle with the course pointing to the standoff circle. As for the first reason, the extremum of the curvature near the standoff circle may exceed the curvature constraint of the UAV, despite the UAV turns at the maximum bank angle, and the trajectory intersects with the standoff circle, which leads to crossing over the circle several times. Therefore,  
Figure 5: Vector field curvature \(\kappa\) versus \(r\) in different \(c\) (\(rd=300\) m). Figure 6: Trajectories with different guidance parameter \(c\). in order to reduce the convergence time, and put an end to crossing over the circle, the guidance parameter must be small enough but satisfies the curvature constraint. The optimization problem can be expressed as:  
\[\left\{ \begin{array}{c}c=\underset{c}{\operatorname{argmax}}(\max(\left| \kappa(c,r)\right|)-\kappa_{\max})\\ \text{s.t.}\quad\max(\left|\kappa(c,r)\right|)<\kappa_{\max}\\ \text{where}\quad r\in(r_{d},+\infty)\end{array} \right.\]  
It is too complex to calculate derivative of Equation (11) with respect to \(r\), so an optimization algorithm based on the gradient descent method is developed to find \(\max(\left|\kappa(c,r)\right|)\). In this way, the optimal guidance parameter can be obtained based on the search method below (Algorithm 1). ```
Input: standoff radius \(r_{d}\); maximum turning rate \(\dot{\psi}_{\max}\); UAV cruising speed \(s\); grid point of parameter \(c_{1},c_{2},\cdots,c_{m}\); Output: optimal parameter \(c_{opt}\);  Calculate the maximum curvature \(\kappa_{\max}\) using Equation (11); \(d_{\min}=10\); \(c_{opt}=2\); for\(i=1\) to \(m\)do  Obtain curvature function \(\kappa_{c_{i}}\) using Equation (11)  Find the extremum \(|\kappa^{c}|_{\max}\) of the curvature function \(\kappa_{c_{i}}\) using the gradient descent method where \(r\in(r_{d},+\infty)\) if\(|\kappa^{c}|_{\max}>\kappa_{\max}\)then  continue; endif \(d=\kappa_{\max}-|\kappa^{c}|_{\max}\); if\(d<d_{\min}\)then \(c_{opt}=c_{i}\); \(d_{\min}=d\); endif endfor
```  
**Algorithm 1**Optimal Guidance Parameter Search  
To cope with another problem, the inverse Lyapunov vector field (ILVF) is introduced. The ILVF is used to guide the UAV to fly away from the standoff circle. The ILVF can be obtained by changing the negative sign of \(\dot{r}\) in Equation (4) to a positive one. \[\left[ \begin{array}{c}\dot{r}\\ r\dot{\theta}\end{array} \right]=\frac{s}{\sqrt{r^{4}+(c^{2}-2)r^{2}r^{2}_{d}+r^ {4}_{d}}}\left[ \begin{array}{c}r^{2}-{r_{d}}^{2}\\ {c}\cdot r\cdot{r_{d}}\end{array} \right] \tag{13}\]  
The control manner of ILVF can be rewritten into Equation (14) by transforming Equation (13) into a cartesian coordinate system  
\[\left[ \begin{array}{c}\dot{x}\\ \dot{y}\end{array} \right]=\frac{s}{r\sqrt{r^{4}+(c^{2}-2)r^{2}_{d}r^{2}+r^{4}_ {d}}}\left[ \begin{array}{c}(r^{2}-r^{2}_{d})x_{r}-c\cdot r\cdot r_{d}\cdot y _{r}\\ (r^{2}-r^{2}_{d})y_{r}+c\cdot r\cdot r_{d}\cdot x_{r}\end{array}