## 2 Preliminaries and Problem Formulation
### Problem Formulation  
Consider \(N\),\(N\geq 2\), mobile robots, labeled as \(p_{1},p_{2},\ldots,p_{N}\) and a free moving target labeled \(c\) to be around on an obstacle-free plane, as shown in Figure 1. All robots are initially located on the plane randomly and no two robots are collinear with the moving center point \(c\). From the standpoint of the system analysis, the position of each robot \(p_{i},i\in 1,2,\ldots,N\), denoted by \([x_{i},y_{i}],i\in 1,2,\ldots,N\) in a prior Cartesian coordinates system. Without loss of generality, all the robots is tagged counterclockwise by \(p_{1},p_{2},\ldots,p_{N}\) around target \(c\), as shown in Figure 1.  
Let \(G=(\mathcal{V},\mathcal{E})\) be the neighboring relationships among robots in the system, where \(\mathcal{V}=\{1,2,\ldots,N\}\) is a set of robots, \(\mathcal{E}=\{(1,2),(2,3),\ldots,(N-1,N),(N,1)\}\) stands for a set edges. That is, each robot has only two adjacent neighboring robots, which are in front or behind it. The set of two neighbors of the robot \(p_{i}\) can be denoted by \(\mathcal{N}_{i}=\{i^{-},i^{+}\}\), where  
\[i^{+}=\begin{cases}i+1&\text{when}\quad i=1,2,\ldots,N-1,\\ 1&\text{when}\quad i=N,\end{cases} \tag{1}\]  
and  
\[i^{-}=\begin{cases}N&\text{when}\quad i=1,\\ i-1&\text{when}\quad i=2,3,\ldots,N.\end{cases} \tag{2}\]  
Define \(\mathbf{r}_{i}(t)=[x_{i}(t),y_{i}(t)]^{T}\in\mathbb{R}^{2}\) as the position of the robot \(p_{i}\), and \(r_{c}(t)=[x_{c}(t),y_{c}(t)]^{T}\in\mathbb{R}^{2}\) denotes the positions of target \(c\) on the plane, at time \(t\). Then, each robot \(p_{i}\)'s model, similar to [39], can be described as  
\[\dot{\mathbf{r}}_{i}(t)=u_{i}(t)+g_{i}(x_{i}(t),t),\quad i=1,2,\ldots,N, \tag{3}\]  
Figure 1: Formation of \(N\) robots on a plane. (**a**) Robots initially locate on a plane. (**b**) Robots form a predetermined formation around the target.  
We can obtain that the order preservation condition is sufficient but not necessary for the collision avoidance condition.  
Given two \(N\)-dimensional vectors  
\[\alpha^{*}=[\alpha_{1}^{*},\alpha_{2}^{*},\ldots,\alpha_{N}^{*}],(\forall i\in \mathcal{V},\alpha_{i}^{*}>0,\sum_{i\in\mathcal{V}}\alpha_{i}^{*}=2\pi) \tag{9}\]  
and  
\[R^{*}=[R_{1},R_{2},\ldots,R_{N}],(\forall i\in\mathcal{V},R_{i}>0) \tag{10}\]  
Suppose that each robot \(p_{i}\) can only get the original information of related variables with respect to \(\bar{\mathbf{r}}_{i}(t)\), \(\bar{\mathbf{v}}_{i}(t)\), \(\bar{\mathbf{r}}_{i^{+}}(t)\), \(\bar{\mathbf{v}}_{i^{+}}(t)\), \(\bar{\mathbf{r}}_{i^{-}}(t)\), \(\bar{\mathbf{v}}_{i^{-}}(t)\), and then a parameter vector which can be presented in a synthesis form  
\[\bar{X}_{i}(t)=[\bar{\mathbf{r}}_{i}(t),\bar{\mathbf{v}}_{i}(t),\bar{\mathbf{ r}}_{i^{+}}(t),\bar{\mathbf{v}}_{i^{+}}(t),\bar{\mathbf{r}}_{i^{-}}(t),\bar{ \mathbf{v}}_{i^{-}}(t)].\]  
Then, a flexible distributed protocol is proposed as  
\[u_{i}(t)=F(\bar{X}_{i}),\quad i\in\{1,2,\ldots,N\}.\]  
Moreover, a concerned formation problem on the plane is formulated as follows.  
**Definition 3** (Formation problem on a plane).: _For any admissible formations that can be characterized by \(\alpha^{*}\) and \(R^{*}\), the distributed control protocol \(u_{i}=F(\bar{X}_{i})\), \(i=1,2,\ldots,N\) is designed, so that under any initial conditions, when \(N\) robots are arranged counterclockwise on a plane, the system's solution converges to the equilibrium point \(r^{*}\) satisfies (11). Namely, all mobile robots globally approach their desired locations asymptotically to form a desired formation._  
\[\left\{\begin{array}{l}a)\lim_{t\rightarrow+\infty}\bar{r}(t)=R^{*}\\ b)\lim_{t\rightarrow+\infty}\bar{\alpha}(t)=\alpha^{*}\end{array}\right. \tag{11}\]