## 2 Problem Formulation  
Considering a multi-robot system composed of \(n\) (\(n\geq 2\)) nonholonomic mobile robots, the kinematics model of robot \(i\) (\(i=1,2,\ldots,n\)) with pure rolling and non-slipping can be described as follows:  
\[\dot{x}_{i}=v_{i}\cos\theta_{i},\quad\dot{y}_{i}=v_{i}\sin\theta_{i},\quad\dot {\theta}_{i}=\omega_{i} \tag{1}\]  
where \(\mathbf{q}_{i}=[\mathbf{x}_{i},y_{i},\theta_{i}]^{T}\in\mathbb{R}^{3}\) denotes the pose of robot \(i\) in the global frame \(X\)-\(O\)-\(Y\), and where \(\mathbf{p}_{i}=[x_{i},y_{i}]^{T}\in\mathbb{R}^{2}\) and \(\theta_{i}\in(-\pi,\pi]\) denote its position and heading angle, respectively. \(v_{i}\in\mathbb{R}\) and \(\omega_{i}\in\mathbb{R}\) are its linear speed and angular speed, respectively. Considering the real linear and angular speed constraints, the speeds of robot \(i\) meet \(v_{i}\in[-v_{\max},\omega_{\max}]\) and \(\omega_{i}\in[-\omega_{\max},\omega_{\max}]\), where \(v_{\max}\in\mathbb{R}^{+}\) and \(\omega_{\max}\in\mathbb{R}^{+}\) are its maximum speed bounds. Here, we assume the dynamics of a target as a single-integral model  
\[\dot{\mathbf{p}}_{0}=\mathbf{v}_{0} \tag{2}\]where \(\mathbf{p}_{0}=[x_{0},y_{0}]^{T}\in\mathbb{R}^{2}\) and \(\mathbf{v}_{0}=[v_{i}^{x},v_{i}^{y}]^{T}\in\mathbb{R}^{2}\) denote the position and speed of the target in the global frame, and \(v_{i}^{x}\) and \(v_{i}^{y}\) are the speeds of the target in the \(x\) and \(y\) directions, respectively. Inspired by the related work [28], in this paper, we choose a bias point \(\mathbf{p}_{i}^{\prime}=[x_{i}^{\prime},y_{i}^{\prime}]^{T}\) from the center point \(\mathbf{p}_{i}\) of two wheels axis along the robot's orientation axis to distance \(d\ (d\
eq 0)\), as shown in Figure 1. For this reason, we assume that the bias point can be placed in some distance or angle measurement sensors, such as Lidar, camera, and others, so that the measured date can be used directly without further conversion. In that case, the controller based on distance or angle information can be simply designed. Therefore, we consider to move the bias point \(\mathbf{p}_{i}^{\prime}\) into a circular formation instead of moving point \(\mathbf{p}_{i}\), which has been studied in [29; 30; 31; 32], and the bias point is defined as  
\[\left[ \begin{array}{c}x_{i}^{\prime}\\ y_{i}^{\prime}\end{array} \right]=\left[ \begin{array}{c}x_{i}\\ y_{i}\end{array} \right]+d\left[ \begin{array}{c}\cos\theta_{i}\\ \sin\theta_{i}\end{array} \right] \tag{3}\]  
The robot \(i\) is also described as a single-integral dynamics model  
\[\dot{\mathbf{p}}_{i}^{\prime}=\mathbf{u}_{i} \tag{4}\]  
where \(\mathbf{u}_{i}=[u_{i}^{x},u_{i}^{y}]^{T}\in\mathbb{R}^{2}\) denotes the control input of robot \(i\), and \(u_{i}^{x}\) and \(u_{i}^{y}\) are its control inputs in the \(x\) and \(y\) directions of the global frame, respectively. Since the robots' motion is controlled directly by linear speed and angular speed or speeds of two driving wheels, it is necessary to transform the control input into the linear and angular speeds [20], which is given by  
\[\left[ \begin{array}{c}v_{i}\\ \omega_{i}\end{array} \right]=\left[ \begin{array}{cc}\cos\theta_{i}&\sin \theta_{i}\\ -\frac{1}{d}\sin\theta_{i}&\frac{1}{d}\cos\theta_{i}\end{array} \right]\left[ \begin{array}{c}u_{i}^{x}\\ u_{i}^{y}\end{array} \right] \tag{5}\]  
In order to easily establish the mathematical model of the controller, we introduce the following variables. Let \(\bar{\mathbf{p}}_{i}\) as the relative position between robot \(i\) and the target; it is expressed as follows  
\[\bar{\mathbf{p}}_{i}=\mathbf{p}_{i}^{\prime}-\mathbf{p}_{0} \tag{6}\]  
The relative velocity \(\bar{\mathbf{v}}_{i}\) between robot \(i\) and the target can be described as follows:  
\[\bar{\mathbf{v}}_{i}=\mathbf{v}_{i}^{\prime}-\mathbf{v}_{0} \tag{7}\]  
Figure 1: Illustration of the cooperative target enclosing.