## 3 Main Results
### Nonlinear Controller Design and Stability Analysis  
As can be seen from (6), a high-order controller is required considering the maneuvers during formation acquisition and keeping. In this section, we search for a nonlinear controller without input saturation. Using adaptive parameters, model (3) can be rewritten as \(\mathbf{u}_{*}=\mathbf{\ell}_{f}\mathbf{(K,\mathcal{L})}\mathbf{F}-(\mathbf{P}\circ(\mathbf{A}\otimes \mathbf{1}_{n\times 1}))1_{n\times 1}\), where \(\mathbf{F}_{i}(\mathbf{\Theta}_{i})=\left[\mathbf{X}_{t}^{i},\mathbf{q}_{t}^{i},\mathbf{u}_{t}^{i }\right]^{\mathsf{T}}\), and \(\mathbf{\ell}_{f}\mathbf{(K,\mathcal{L})}\) indicates the adaptive adjusting function of \(\mathbf{K}\) and \(\mathbf{\mathcal{L}}\). \[\mathbf{\ell}_{f}\mathbf{(K,\mathcal{L})}=-\left[\mathbf{l}_{ij}\mathbf{k}_{ij}\right]\in\mathbb{ R}^{Nn\times Nm} \tag{8}\]  
Now, we propose the controller with Proposition 1. **Proposition 1**. : _For multi-agent systems with strongly connected undirected graph \(\mathcal{G}\), the principle of \(\mathbf{\ell}_{f}\mathbf{(K,\mathcal{L})}\) and \(\mathbf{P}\) in (8), the controller defined in Equation (3) can guarantee the finite-time stability of the system regardless of communication interference._  
Figure 1: Coordinate transformation. \[\left\{ \begin{array}{l}\mathbf{k}_{ii}=k_{ij}^{1}[\mathbf{C_{K}},\mathbf{0}_{2\times 4}]\\ \\ +\frac{k_{ij}^{2}}{l_{\overline{u}}v_{t}^{i}}\left[\begin{array}{c}\sum v_{i}^{ i}a_{ij}k_{s},-\sum v_{i}^{i}a_{ij}k_{c},1,\mathbf{0}_{1\times 3}\\ \sum a_{ij}k_{c},\sum a_{ij}k_{s},0,\mathbf{0}_{1\times 3}\end{array}\right]\\ \\ \mathbf{k}_{ij}=\left[k_{ij}^{1}\mathbf{C_{K}},\mathbf{0}_{2},\ \ -\sin\left(\psi_{r}^{i} \right),k_{ij}^{1}\right]\\ \\ +\frac{k_{ij}^{2}}{v_{t}^{i}}\left[\begin{array}{c}-v_{t}^{i}k_{s},v_{t}^{ i}k_{c},v_{t}^{i}cos\left(\psi_{r}^{i}\right),\mathbf{0}_{1\times 3}\\ -k_{c},-k_{s}-\sin\left(\psi_{r}^{i}\right),\mathbf{0}_{1\times 3}\end{array} \right]\\ \\ \mathbf{C_{K}}=\frac{1}{v_{t}^{i}}\left[\begin{array}{c}v_{t}^{i}cos\left(\psi_ {t}^{i}\right),v_{t}^{i}sin\left(\psi_{t}^{i}\right)\\ -sin\left(\psi_{t}^{i}\right),cos\left(\psi_{t}^{i}\right)\end{array}\right] \\ \\ \mathbf{w}_{ij}=k_{ij}^{1}\mathbf{C_{K}}+\frac{k_{ij}^{2}}{v_{t}^{2}}\left[\begin{array} []{c}-v_{t}^{j}k_{s},v_{t}^{i}k_{c}\\ -k_{c},-k_{s}\end{array}\right]\\ \\ \end{array} \right. \tag{9}\]  
_where \(k_{s}=sin\left(\psi_{t}^{i}\right)\dot{\psi}_{t}^{j},k_{c}=cos\left(\psi_{t}^ {i}\right)\dot{\psi}_{t}^{j}\), and \(k_{ij}^{1}\) and \(k_{ij}^{2}\)are positive real numbers._  
Before proceeding to the proof of Proposition 1, we introduce Lemma 3. Lemma 3 (25; 26): _If quadratic scalar function \(V(t)\) concerning the state of the system satisfies the inequality \(V(t_{k+1})\leq V(t_{k})\leq V_{0}<\infty\)\(\forall t_{0}<t_{k}<t_{k+1}\), the system is progressively stable, where \(V_{0}\) is the value at the initial time \(t_{0}\). The inequality is equivalent to \(V(t)\leq 0\), and the equality holds when \(t_{k}\geq\tau\), where \(\tau>t_{0}\) is a positive real number._  
Lemma 4 (27; 28): _If a power scalar function \(V(\mathbf{x})\) is continuously differentiable along \(\mathbf{x}\) with \(\mathbf{x}\in\mathbb{R}^{n}\), and the first-order derivative satisfies \(V(\mathbf{x})\leq-\gamma_{1}V^{\alpha}(\mathbf{x})-\gamma_{1}V(\mathbf{x})-\gamma_{1}V^{ \beta}(\mathbf{x})+b_{V}\), where \(\sum\limits_{k=1,2,3}\gamma_{k}\
eq 0\),\(0<\alpha<1<\beta\), \(b_{V}\) is a bounded positive number. Then \(\mathbf{x}\) will approach the equilibrium state in a finite time._  
Next, we give the proof of Proposition 1.