## 4 Main Results
### Impact-Angle Control Part  
In the impact control part, the guidance laws for the leader and followers are designed accordingly. First, the nonsingular terminal sliding-mode guidance law [33] for the leader is presented. Consider the sliding-mode surface:  
\[s_{1}=x_{31}+k_{1}sig^{\lambda}(x_{41}) \tag{30}\]  
where \(k_{1}>0\), \(2>\lambda>1\). Based on (30), a finite-time convergence guidance law can be designed as:  
\[u_{q1}=x_{11}\bigg{(}-2\frac{x_{21}x_{41}}{x_{11}}+\frac{1}{k_{1}\lambda}sig^{ 2-\lambda}(S_{1})+k_{2}sig^{\eta}(S_{1})+hsign(S_{1})\bigg{)} \tag{31}\]  
where \(k_{2}>0\), and \(h>0\).  
Then, in light of the study in [34], a novel predefined-time consensus guidance law is proposed so that the followers simultaneously attack the target at the desired impact angle. A time-varying sliding-mode surface is designed as follows:  
\[S_{i}=t_{go}e^{\sum\limits_{i=1}^{N}a_{ij}(x_{4i}-x_{3j})}\sum\limits_{j=1}^{ N}a_{ij}(x_{4i}-x_{4j})+n(e^{\sum\limits_{i=1}^{N}a_{ij}(x_{3i}-x_{3j})}-1) \tag{32}\]  
where \(\mu>0\). Moreover, a predefined-time guidance law is designed as:  
\[u_{qi}=x_{1i}\Bigg{(}-2\frac{x_{2i}x_{4i}}{x_{1i}}+\frac{n}{t_{f}-t}\sum \limits_{j=1}^{N}a_{ij}(x_{4i}-x_{4j})+\big{(}\sum\limits_{j=1}^{N}a_{ij}(x_{ 4i}-x_{4j})\big{)}^{2}+k_{3}S_{i}+k_{4}sign(S_{i})\Bigg{)} \tag{33}\]  
**Theorem 4**.: _In the second-order nonlinear system (23), if the guidance law is designed as in (33), then the state variables \(x_{3}\)and \(x_{4}\)will simultaneously converge to zero at \(t=t_{f}\)._  
**Proof of Theorem 4.** The proof is provided in Appendix C. \(\square\)  
**Note 2.** In (33), the discontinuous control term (\(k_{4}sign(S_{i})\)) is designed to compensate for the target maneuver in normal LOS directions. However, the discontinuous control term may cause control-input chattering. In the numerical simulations, we used an approximate function to obtain continuous guidance commands, as follows:  
\[f(u)=\frac{1-e^{vu}}{1+e^{vu}} \tag{34}\]  
where \(v>0\) is a constant.  
**Remark 4**.: _As is shown in (33), the convergence time of the rate of the LOS in the proposed PTCG is designed as the \(t_{go}\), which means that the attack angle will converge to the expected value exactly when the interception impact appears. It is worth noting that the convergence time can be adjusted by replacing \(t_{f}-t\) with the convergence time (\(t_{c}\)). The convergence time (\(t_{c}\)) can be predetermined arbitrarily and independently of the system parameters or constants. This is the main superiority of the predefined-time guidance law._  
**Remark 5**.: _The existing study [35] on the predefined-time cooperative-guidance law focuses on its effectiveness at avoiding collisions between interceptors by the rapid convergence of the LOS. However, previous guidance laws, such as the finite-time cooperative-guidance law [23] and fixed-time cooperative-guidance law [20], can guarantee the rapid convergence of the rate of the LOS. Compared with such methods [20; 23], there is no significant advantage with the predefined-time cooperative-guidance law. In this paper, the superiority of the predefined-time cooperative-guidance law is applied to reduce the saturation of the control input. A controllable slower convergence rate is realized without the parameter design. Moreover, PTCG reduces the step of designing the reasonable convergence time. These factors facilitate the engineering realization._