## 4 Numerical Simulation  
This section validates the theory. TVF tracking control is achieved under controller (5), according to Equations (6) and (7). The jointly connected topology is illustrated in Figure 1, which shows all possible topologies \(\{\tilde{G}_{1},\tilde{G}_{2},\tilde{G}_{3},\tilde{G}_{4},\tilde{G}_{5},\tilde {G}_{6}\}\). All possible communication topology diagrams are switched periodically in order of \(\tilde{G}_{1}\to\tilde{G}_{2}\to\tilde{G}_{3}\to\tilde{G}_{4}\to\tilde{G}_{5} \to\tilde{G}_{6}\to\tilde{G}_{1}\cdots\). As shown in Figure 2, a switching period of 2 s is divided into six switching times, and a communication structure diagram is used during each switching time. The simulation considers the non-holonomic mobile robot model as shown in Figure 3. All intelligent bodies have the same structure and motion model, and are described by the following kinematic equations: \(\dot{\tilde{x}}_{xi}=\bar{\sigma}_{i}\cos\theta_{i^{\prime}}\dot{\tilde{x}}_{yi }=\bar{\sigma}_{i}\sin\theta_{i^{\prime}}\dot{\theta}_{i}=\bar{r}_{i}\). Among them, \((\tilde{x}_{xi},\tilde{x}_{yi}),\bar{v}_{i},\theta_{i},\bar{r}_{i}\), respectively, represent the center position, linear velocity, heading angle and rotational velocity of the \(i\)-th robot. For the analysis of non-complete mobile robots, by analogy with Reference (36), a fixed point deviating from the center of the wheel is taken as \((\bar{x}_{xi},\bar{x}_{yi})\), which is taken as the inertial position of the \(i\)-th non-holonomic mobile robot, where \( \begin{bmatrix}\bar{x}_{xi}\\ x_{yi}\end{bmatrix} = \begin{bmatrix}\bar{x}_{xi}\\ x_{yi}\end{bmatrix} +d \begin{bmatrix}\cos\theta_{i}\\ \sin\theta_{i}\end{bmatrix} \). Secondly, we can obtain \( \begin{bmatrix}\dot{\bar{x}}_{xi}\\ \bar{x}_{yi}\end{bmatrix} = \begin{bmatrix}\cos\theta_{i}&-d\sin\theta_{i}\\ \sin\theta_{i}&d\cos\theta_{i}\end{bmatrix}\begin{bmatrix}\bar{v}_{i}\\ r_{i}\end{bmatrix} \), where \( \begin{bmatrix}\bar{v}_{i}\\ \bar{r}_{i}\end{bmatrix} = \begin{bmatrix}\cos\theta_{i}&\sin\theta_{i}\\ -\sin\theta_{i}/d&\cos\theta_{i}/d\end{bmatrix}\begin{bmatrix}\bar{v}_{xi}\\ \bar{v}_{yi}\end{bmatrix} \), \(d\
eq 0\). Finally, we define \(\bar{v}_{xi}=u_{xi},\bar{v}_{yi}=u_{yi}\), which, respectively, represent the linear velocity components along the X and Y directions. Figure 1: Diagram of possible communication topologies for the agent system. Figure 2: Switching signal. Using incomplete feedback linearization, the kinematic model is converted into a dynamic equation. Consider a multi-agent system consisting of five agents and one leader. The state of each agent is defined as \(x_{i}=\left[\bar{x}_{xi}^{T},\bar{v}_{xi}^{T},\bar{x}_{yi}^{T},\bar{v}_{yi}^{T} \right]^{T}\), and the control input is \(u_{i}=\left[u_{xi}^{T},u_{yi}^{T}\right]^{T}\). Assume that the coefficient matrices (1) and (2) of the system are as follows:  
\[A= \begin{bmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix} \quad B= \begin{bmatrix}0&0\\ 1&0\\ 0&0\\ 0&1\end{bmatrix} \]  
The nonlinear parameter \(\vartheta(\varepsilon_{i}(t))\) of the follower is  
\[ \begin{bmatrix}0.1\cos((x_{i1}(t)-h_{i1}(t))\\ 0.1\cos(x_{i2}(t)-h_{i2}(t))\\ 0.1\cos(x_{i3}(t)-h_{i3}(t))\\ 0.1\cos(x_{i4}(t)-h_{i4}(t))\end{bmatrix} \]  
The leader's nonlinear parameter \(\vartheta(x_{0}(t))\) is  
\[[0.1\cos(x_{01}(t)),0.1\cos(x_{02}(t)),0.1\cos(x_{03}(t)),0.1\cos(x_{04}(t))] ^{\mathrm{T}}\]  
The follower agent needs to complete the regular pentagonal TVF tracking, and the desired TVF configuration direction is as follows:  
\[h_{i}(t)= \begin{bmatrix}\sin\Bigl{(}t+\frac{2(i-1)}{5}\pi\Bigr{)}\\ \cos\Bigl{(}t+\frac{2(i-1)}{5}\pi\Bigr{)}\\ \cos\Bigl{(}t+\frac{2(i-1)}{5}\pi\Bigr{)}\\ -\sin\Bigl{(}t+\frac{2(i-1)}{5}\pi\Bigr{)}\end{bmatrix} i=1,2,3,4\]  
Figure 3: Structure diagram of the nonholonomic mobile robot. The formation compensation vector \(\xi_{i}(t)\) calculated according to Formula (6) is as follows:  
\[\xi_{i}(t)= \begin{bmatrix}-2\sin\!\left(t+\frac{2(i-1)}{5}\pi\right)\\ -2\cos\!\left(t+\frac{2(i-1)}{5}\pi\right)\end{bmatrix}