## 4 Simulation  
In this section, we present simulations tested on five AUVs that demonstrate the efficacy of the theoretical results. The multi-AUV's interaction topology is shown in Figure 5, and had 0-1 weights. Define the following:  
\[\left\{ \begin{array}{l}\theta_{i}(t)=[x_{iX}(t),v_{iX}(t),x_{iY}(t),v_{iY}(t )]^{T}\\ m_{i}(t)=[m_{ixX}(t),m_{iyX}(t),m_{ixY}(t),m_{iyY}(t)]^{T}\end{array} \right., \tag{54}\]  
So, the dynamics of each AUV in the case through Kronecker product with \(n=2\) are shown:  
\[\dot{\theta}(t)=\Big{(}I_{N}\otimes\Big{(}B_{2}K_{1}+B_{1}B_{2}^{T}\Big{)} \Big{)}\theta_{i}(t)+(I_{N}\otimes B_{1})u_{i}(t), \tag{55}\]  
Simulation 1:  
Define \(m_{i}(t)(i=2,3,\ldots,N)\) as follows:  
\[\left\{ \begin{array}{l}g_{i}(t)\!=\!\text{sign}(\sin(\omega t/2\!+\!\pi(i-1) \,/5))\\ m_{ixX}(t)=r\cos((\omega t+2\pi(i-1)/5)-1)g_{i}(t)\\ m_{ixX}(t)=-\omega r\sin((\omega t+2\pi(i-1)/5)-1)g_{i}(t)\\ m_{ixY}(t)=r\sin(\omega t+2\pi(i-1)/5)\\ m_{ixY}(t)=\omega r\cos(\omega t+2\pi(i-1)/5)\end{array} \right., \tag{56}\]  
where \(r=7m\) and \(\omega=0.214\text{rad}/s\). When the multi-AUV system successfully had obtained the required time-varying formation configuration, the entire formation exhibited a highly coordinated, dynamic motion pattern. In this formation, a total of five AUVs were engaged in action, and they sailed strictly according to the preset figure-eight pattern. This mode not only required each AUV to have a high degree of autonomous navigation, but also required precise communication and coordination between them to ensure the stable operation of the entire fleet. Note that to ensure that there was no collision between the AUVs within the formation while maintaining sufficient spacing to facilitate the transmission of information and the execution of tasks, these AUVs maintained a phase separation of 0.4 T radians on the trajectory. This design not only considered the overall stability of the formation, but also the independence and flexibility of the AUV so that the entire formation could effectively achieve various tasks in the intricate underwater environment. The motion modes of the formation center were chosen at \(-0.58+1.26j\) and \(-0.58-1.26j\) with \(j^{2}=-1\), \(K_{1}=I_{2}\otimes[-2,-1.2]\), meaning that when the multi-AUVs established the time-varying formation, the formation center was stationary. Figure 5: Directed interaction topology \(G\) about the five AUVs. According to the Theorem 2, we designed \(K_{2}=I_{2}\otimes[0.3416,0.7330]\). Moreover, the initial states of the five AUVs were as follows:  
\[\left\{ \begin{array}{l}\theta_{1}(0)=[0.16,0.03,-0.07,0.03]^{T}\\ \theta_{2}(0)=[-4.92,-0.08,\ 6.38,-0.04]^{T}\\ \theta_{3}(0)=[-12.37,-0.26,\ 4.08,-0.03]^{T}\\ \theta_{4}(0)=[-12.73,\ 0.03,-4.56,-0.04]^{T}\\ \theta_{5}(0)=[4.63,0.05,6.9,\ 0.02]^{T}\\ \end{array} \right., \tag{57}\]  
Figure 6 shows the state trajectories of the multi-AUV system in the simulation and the desired \(c(t)\). In this Figure, the AUVs' and \(c(0)\)'s initial states are represented by circles, and the five AUVs' and \(c(t)\)'s final states are shown as the following different patterns: AUV1 is a square, AUV2 is a diamond, AUV3 is an upside-down triangle, AUV4 is a standard triangle, AUV5 is a left-pointing triangle, and \(c(t)\) is pentagram. Figure 6: The trajectories of five AUVs and \(c(t)\) in the simulation. (**a**) The position trajectories of five AUVs in the simulation; (**b**) the velocity trajectories of five AUVs in the simulation. Figure 7 describes the snapshots of the \(c(t)\) with the 16 to 21 second positions and velocities of the five AUVs. Choosing applicable initial states, we realized the desired time-varying formation transformation in the simulation, and the AUVs did not collide in this process. Simulation 2:  
The motion modes of the formation center function in this simulation were assigned to move periodically. Define \(m(t)\) as follows:  
\[\left\{