# 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 Normal Direction of LOS  
The last two equations in the missile-target movement model (3) are taken as the missile-target sight angle system and written in the form of the LOS overload constrained as:  
\[\left\{ \begin{array}{l}\dot{x}_{3i}=x_{4i}\\ \dot{x}_{4i}=-\frac{2x_{3i}}{x_{1i}}x_{4i}-\frac{sat(u_{qi})}{x_{1i}}+\frac{w _{qi}}{x_{1i}}\end{array} \right. \tag{10}\]  
where the saturation function \(sat(u_{qi})\) is defined as:  
\[sat(u_{qi})= \begin{cases}u_{qi},&\left|u_{qi}\right|\leq u_{qimax}\\ \text{sgn}(u_{qi})\cdot u_{qimax},&\left|u_{qi}\right|>u_{qimax}\end{cases}  \tag{11}\]  
where \(u_{qimax}\) is the ultimate overload that missile \(i\) can provide in the direction of the LOS. The sliding surface with the LOS constraint is usually designed as \(s=k\dot{q}+q\), where \(k\) is the coefficient of LOS change rate, and the proportion of LOS and LOS rate in the sliding surface is constant. Under the constraint of zero LOS, the trajectory will bend sharply due to the rapid convergence of the LOS. To obtain a smoother trajectory, the following adaptive sliding surface is designed:  
\[s_{i}=K_{i}\dot{q}_{i}+q_{i} \tag{12}\]  
where \(K_{i}\) is the adaptive coefficient, which is obtained from the following equation:  
\[K_{i}=10^{\sim}\left(\frac{r_{i}}{r_{i0}}\right)^{2}-0.8 \tag{13}\]  
In Equation (13), \(r_{i0}\) and \(r_{i}\) are the initial missile-target distance and the real-time missile-target distance, respectively. This design sets the LOS rate \(\dot{q}_{i}\) as the main guide term when the missile-target distance is far to avoid an overly curved trajectory. When the missile-target distance is close, the LOS error \(q_{i}\) is the main guide term to make a rapid response to the target maneuver. Considering the simplicity of the controller expression, \(K_{i}\) is regarded as a constant in the derivation of the controller. **Remark 3**. : _When \(i\) takes different values, the expression of the sliding mode variable \(s_{i}\) is the same. For readability, \(s_{i}\) in the following text does not have a subscript._  
For the sliding mode control with the LOS constraint, it is hoped that the approach speed of the sliding mode surface can be controlled, and the chattering of the control output is small to ensure the flight reliability of the missile. Based on this consideration, this paper selects a fast power reaching law as follows [38]:  
\[\dot{s}=-k_{1}s-k_{2}|s|^{\beta_{i}}\text{sgn}(s) \tag{14}\]  
where \(k_{1}>0\), \(k_{2}>0\), and \(0<\beta_{i}<1\). This reaching law is a combination of the exponential reaching law and power reaching law. It can be seen from Equation (14) that \(-k_{1}s\) plays a major role in the initial stage of guidance because the sliding mode variable \(s\) is large; as the missile approaches the target and the guidance error decreases, the approach speed primarily depends on \(-k_{2}|s|^{\beta_{i}}\text{sgn}(s)\). Choosing smaller \(k_{1}\) and \(\beta_{i}\) values and a larger \(k_{2}\) value, we can achieve a relatively smooth trajectory and can make a quick responses to the target maneuver when approaching the target. Combining Equations (12) and (14), and LOS system (10), the LOS normal direction controller is obtained as:  
\[u_{qi}=x_{1i}\left(-\frac{2x_{2i}}{x_{1i}}x_{4i}+\frac{x_{4i}+k_{1}s+k_{2}|s|^{ \beta_{i}}\text{sgn}(s)}{K_{i}}+\frac{w_{qi}}{x_{1i}}\right) \tag{15}\]  
Note that Equation (15) contains the uncertainty variables \(w_{qi}/x_{1i}\), which can be regarded as the disturbance in the system, where \(w_{qi}\) is the normal acceleration of the target in the LOS, which is difficult to observe directly. An effective method is to use the observer to estimate the disturbance in the system. **Lemma 3** ([39]). : _For first-order SISO nonlinear systems,_  
\[\dot{\sigma}=u+d \tag{16}\]  
_where \(\sigma\) is the state variable, \(u\) is the control input, and \(d\) is the system interference. An inhomogeneous high-order sliding mode disturbance observer of the form_  
\[\left\{