# Cooperative Standoff Tracking of Moving Targets Using Modified Lyapunov Vector Field Guidance
## 4 Design of Cooperative Tracking Guidance Laws
### Modification of the Guidance Law for Moving Targets Tracking  
When a team of UAVs are tracking a ground moving target, the motion state of the target must be considered. As a correction item, the state of the target is used to modify the desired course of the UAV. If the UAV moves in the designed course in the local coordinate system of the target, the relationship between the modified velocity and target velocity is:  
\[\left\{ \begin{array}{c}v_{g}=v_{t}+\lambda v_{d}\\ v_{\min}\leq\left|v_{g}\right|\leq v_{\max}\end{array} \right. \tag{15}\]  
where \(v_{d}\) is the desired velocity, \(v_{t}=\left[v_{x},v_{y}\right]^{T}\) is the velocity estimation of the target, \(v_{g}\) is the corrected desired speed in the global coordinate system, and \(\lambda\) is the correction factor. Every UAV in the team could position the ground moving target using onboard sensors, while the result is not accurate enough for subsequent operations, such as guiding shells to strike or UAVs to track it stably. In this section, the state of the target is estimated based on the data fusion technology. A classic interactive multi-model is integrated with the unscented Kalman filter (IMM-UKF) to improve the geolocation precision. UKF is an improvement of EKF, which is important in integrating information [22]. Furthermore, a classic federated filter is adopted to reduce estimation errors by integrating independent sensor information from UAVs. The framework of IMM-UKF with federated filter is shown in Figure 8. When tracking a moving target, every UAV localizes the target with a relatively low accuracy. The localization result will be processed using the IMM-UKF, which works well in estimating the state of the moving target. Then, the main filter collects the results of the local filter and distributes estimate results after time update and data fusion. The data fusion process with federated filter is introduced in Appendix A.  
Figure 7: Lyapunov vector field and inverse Lyapunov vector field with \(rd=50\) m and \(c=0.7\). (**a**) Lyapunov vector field; (**b**) Inverse Lyapunov vector field. In order to depict the state in a more accurate manner, three motion models are adopted in the IMM-UKF algorithm. Suppose state variables are  
\[X=[x,v_{x},y,v_{y},a_{x},a_{y}]^{T}\]  
where \(x\), \(y\) are the position variables, \(v_{x}\), \(v_{y}\) are the velocity variables, \(a_{x}\) and \(a_{y}\) are the acceleration variables. Motion models adopted in the paper will be introduced. The constant velocity model is used to model the target state in the constant velocity. All process noise mentioned below are zero-mean white Gaussian noise. \[X_{t}[k+1]=A_{cv}X[k]+\omega_{cv}[k] \tag{16}\]  
where  
\[A_{cv}=\left[ \begin{array}{cccccc}1&T&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&T&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&1\end{array} \right] \tag{17}\]  
\(A_{cv}\) is the state transition matrix, and \(\omega_{cv}[k]\) is the process noise. The acceleration model can be expressed as  
\[X_{t}[k+1]=A_{ca}X[k]+\omega_{ca}[k] \tag{18}\]  
Figure 8: The framework of the state estimator. where  
\[A_{\alpha}=\left[ \begin{array}{cccccc}1&T&0&0&T^{2}/2&0\\ 0&1&0&0&T&0\\ 0&0&1&T&0&T^{2}/2\\ 0&0&0&1&0&T\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array} \right] \tag{19}\]  
\(A_{\alpha}\) is the state transition matrix, and \(\omega_{\alpha}[k]\) is the process noise. Similarly, the turning model can be rewritten as  
\[X_{t}[k+1]=A_{s}X[k]+\omega_{s}[k] \tag{20}\]  
where  
\[A_{s}=\left[ \begin{array}{cccccc}1&\frac{\sin(\omega T)}{\omega}&0&0&\frac{( 1-\cos(\omega T))}{\omega^{2}}&0\\ 0&\cos(\omega T)&0&0&\frac{\sin(\omega T)}{\omega}&0\\ 0&0&1&\frac{\sin(\omega T)}{\omega}&0&\frac{(1-\cos(\omega T))}{\omega^{2}} \\ 0&0&0&\cos(\omega T)&0&\frac{\sin(\omega T)}{\omega}\\ 0&-\omega\sin(\omega T)&0&0&\cos(\omega T)&0\\ 0&0&0&-\omega\sin(\omega T)&0&\cos(\omega T)\end{array} \right] \tag{21}\]  
\(A_{s}\) is the state transition matrix, and \(\omega_{s}[k]\) is the process noise. The estimation process is introduced in Appendix B.