# Cooperative Standoff Tracking of Moving Targets Using Modified Lyapunov Vector Field Guidance
## 4 Design of Cooperative Tracking Guidance Laws
### Phase Keeping for Cooperative Tracking  
Besides standoff tracking of a ground target, it is of vital importance to control the intervehicle phase. On the one hand, the unpredicted maneuver of a vehicle can be observed when UAVs are distributed evenly on the circle. On the other hand, the precision of target localization can be enhanced by fusing independent sensor information.  
We develop a new controller based on the speed controller from Frew E W [7], which can achieve faster convergence. Different from the original method, we set an ideal seat for each UAV. These ideal seats are evenly spaced around the circle. Then, the speed controller is utilized to guide each UAV to its ideal seat by controlling the speed. As a result, the team of UAVs will be evenly distributed on the circle. Without loss of generality, we adopt three UAVs to perform the tracking mission, and UAV1, UAV2, and UAV3 are arranged counterclockwise, as shown in Figure 9.  
Figure 9 shows a team of three-UAVs, with corresponding phase angles \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) defined relative to the instantaneous tracking circle. Consider a phasing Lyapunov function  
\[V_{p}=\sum_{i=1}^{3}\left(\Delta\theta_{i}\right)^{2} \tag{22}\]  
where \(\Delta\theta_{i}=\theta_{di}-\theta_{i}\quad i=1,2,3\) denotes the unwrapped difference between the phase angle and the desired angle. The method for calculating the desired angle will be introduced later. The time derivation of this function is  
\[\frac{dV_{p}}{dt}=2\sum_{i=1}^{3}\Delta\theta_{i}\cdot(\theta_{di}-\dot{\theta }_{i}) \tag{23}\]Then, we choose the angular speed commands  
\[\dot{\theta}_{i}=k(\frac{r_{d}}{r_{i}})^{2}(\theta_{di}-\theta_{i})+\frac{v_{uv} }{r_{d}}\quad i=1,2,3 \tag{24}\]  
where \(k\) is a positive proportional gain. Since the desired seat circles around the UAV cruising at \(v_{uv}\), the desired angle speed is  
\[\dot{\theta}_{di}=\frac{v_{uv}}{r_{d}}\quad i=1,2,3 \tag{25}\]  
Then, Equation (24) results in  
\[\frac{dV_{p}}{dt}=-2k(\frac{r_{d}}{r})^{2}\sum_{i=1}^{3}{(\theta_{di}-\theta_{ i})^{2}}=-2k(\frac{r_{d}}{r})^{2}V_{p} \tag{26}\]  
Note that \(\dot{V}_{p}\) is negative semi-definite, which ensures that \(V_{p}\) converges exponentially to zero, hence the relative angle \(\Delta\theta_{i}=\theta_{di}-\theta_{i}\quad i=1,2,3\) converges exponentially to zero, which means the UAV phase angle converges to the desired angle.  
The corresponding speed control commands are then  
\[v_{di}=k(\frac{r_{d}}{r_{i}})^{2}(\theta_{di}-\theta_{i})r_{d}+v_{uv}\quad i =1,2,3 \tag{27}\]  
The term \((r_{d}/r_{i})^{2}\) is introduced to restrain the phase keeping operation when the UAVs are far away from the target [7].  
In the above algorithm, the choice of the desired seat is of vital importance. The optimal choice of the desired seats allow UAVs to reduce their speed, as well as the burden on the flight controller. As shown in Figure 9, the total angle difference between the desired seat and the corresponding UAV is  
\[D_{p}=|\theta_{d1}-\theta_{1}|+|\theta_{d2}-\theta_{2}|+|\theta_{d3}-\theta_{ 3}| \tag{28}\]  
where \(\theta_{d2}=\theta_{d1}+2\pi/3\),\(\theta_{d3}=\theta_{d1}+4\pi/3\). When \(\theta_{d1}=(\theta_{1}+\theta_{2}+\theta_{3}-2\pi)/3\), \(D_{p}\) reaches the minimize value. Thus, the desired phase for each UAV in any instantaneous frame can be calculated.