# Cooperative Guidance Law for the Mother-Cabin of the Anti-UAV Cluster Mother-Son Missile
## 3 Multi-Missile Cooperative Guidance Law Design
### Guidance Law Design in the LOS Direction
#### 3.1.2 Impact Time and Speed Consistency Control of Multi-Missiles  
According to Lemma 2, the consistency protocol is used to design a controller \(u_{i}\) to control the convergence consistency of the multi-agent state \(x_{i}\). However, the consistency protocol can only control the convergence of a single state. Scholars often use this consistency protocol to realize the time coordination of multiple missiles. However, the operational pattern of this paper requires two constraints: multiple mother-cabins must reach the \"release point\" at the same time, and the speed must converge to a consistent value. One way to achieve this goal is to design controllers for missile speed and time-to-go to control the two states and to achieve consistency, respectively. However, if the two controllers are superimposed in the LOS direction at the same time, coupling will inevitably occur. As a result, the \"consistent convergence in finite time\" of the consensus protocol no longer holds, and the finite-time convergence of the controllers cannot be proven. In view of this, we will not directly apply the consistency protocol (5), but instead try to imitate the form of Equation (5) to design the LOS direction controller hoping to achieve the convergence of both time-to-go and speed. Due to the zero LOS constraint, the missile will fly to the same altitude as the target in a short time. In addition, there is speed coordination among multiple missiles. Theoretically, the time coordination can be achieved only by taking the relative distance \(r_{i}\) between the missile and the target as the coordination variable. Therefore, under the condition that multiple missiles are expected to impact the target at the same speed and time, the missile speed \(v_{i}\) and the missile-target distance \(r_{i}\) are taken as the coordination variables, and the controller is obtained by imitating the form of consistency protocol (5):  
\[\overline{u}_{ri}=\text{sgn}\left(\sum_{j=1}^{n}\alpha_{ij}\big{[}(r_{i}-r_{j} )+(v_{i}-v_{j})\big{]}\right)\times\left|\sum_{j=1}^{n}\alpha_{ij}\big{[}(r_{ i}-r_{j})+(v_{i}-v_{j})\big{]}\right|^{\alpha_{i}} \tag{6}\]  
where \(\alpha_{ij}\) represents the communication capability between missile \(i\) and \(j\), \(\alpha_{i}\) is a constant, and \(0<\alpha_{i}<1\). During the test, it was found that directly taking the speed as the coordination variable in the controller (6) resulted in a large overshoot and frequent switching, which in turn led to a longer convergence time. In this regard, it was experimentally determined that adding the time-to-go coordination variable \(t_{go,i}\) could effectively suppress the overshoot, and replacing the coordination variable \(v_{i}\) with \(\dot{r}_{i}\) could make the output of the controller smoother. The resulting controller has a simple structure, the controller output is small in the middle and rear sections of the guidance, has a smooth gradient, and can ensure that the missile speed and time-to-go converge to the same value upon impact. The designed controller is as follows:  
\[\overline{u}_{ri}=\text{sgn} \left(\sum_{j=1}^{n}\alpha_{ij}\big{[}(r_{i}-r_{j})+(\dot{r}_{i}- \dot{r}_{j})+V_{\text{m}}(t_{go,i}-t_{go,j})\big{]}\right)\times \tag{7}\] \[\left|\sum_{j=1}^{n}\alpha_{ij}\big{[}(r_{i}-r_{j})+(\dot{r}_{i}- \dot{r}_{j})+V_{\text{m}}(t_{go,i}-t_{go,j})\big{]}\right|^{\alpha_{i}}\]  
where \(V_{\text{m}}\) is the average velocity of the missile, which is regarded as a constant in the controller and can be set after the performance parameters of the missile are determined. During the actual missile target approach, the derivative of the distance between the missile and the target changes slightly, and \(t_{go,i}\) can be calculated by the following equation [16]:  
\[t_{go,i}=-\frac{r_{i}}{\dot{r}_{i}} \tag{8}\]  
This time-to-go calculation method is simple and effective and is suitable for small maneuvering targets such as UAV clusters. The overload constraint is added to the controller (7) to obtain the LOS direction controller as follows:  
\[u_{ri}=