## 3 Controller Design
### Circular Motion Control  
We first propose a circular motion control law that guides each robot to converge to its own desired circle without considering phase angle and angular spacing on the circle. The design idea of this control scheme is that by rotating a fixed reference vector around the target point conterclockwise \(\mathbf{p}_{0}\), a rotation change circular trajectory can be obtained. Then, the circular trajectory can be used as a reference value, and the robot tracks the reference value for circular motion. The control objective can be described as follows :  
\[\mathbf{\bar{p}}_{i}(t)\to\ R(\varphi_{i}(t))\mathbf{\bar{p}}_{i}(t_{0})\rho_{i}^{d}, \hskip 14.226378 pti\to\infty \tag{9}\]  
where \(\ R(\varphi_{i}(t))\in\mathbb{R}^{2\times 2}\) is a rotation matrix of steering angle \(\phi_{i}(t)\in(0,2\pi]\) and \(\mathbf{\bar{p}}_{i}(t_{0})\) is the unit direction vector of \(\mathbf{\bar{p}}_{i}(t_{0})\) at the initial time \(t_{0}\), which are defined as  
\[\phi_{i}(t)=\varphi_{i}(t)-\varphi_{i}(t_{0}),\hskip 14.226378 pti\geq t_{0} \tag{10}\]The circular motion control law is designed as follows:  
\[\mathbf{u}_{i}(t)=\mathbf{v}_{0}(t)-k_{p}\Big{[}\mathbf{\bar{p}}_{i}(t)-\mathbf{R}(\phi_{i}(t)) \mathbf{\bar{p}}_{i}(t_{0})\rho_{i}^{d}\Big{]}+\dot{\mathbf{R}}(\phi_{i}(t))\mathbf{\bar{p}}_ {i}(t_{0})\rho_{i}^{d} \tag{13}\]  
where \(k_{p}\) is a position constant, which is used to adjust the convergence speed of circular trajectory tracking for all robots.  
**Theorem 1**.: _Assuming \(\phi_{i}(t)\) is a first-order continuous derivative function. Under the control law (13), robot i can converge to a circle with radius \(\rho_{i}^{d}\) centered on a static or moving target and maintain the set distance with the target, namely \(\rho_{i}(t)\to\rho_{i}^{d}\) as \(t\to\infty\)._  
Proof.: Denote the position error as  
\[\mathbf{e}_{p,i}(t)=\mathbf{\bar{p}}_{i}(t)-\mathbf{R}(\phi_{i}(t))\mathbf{\bar{p}}_{i}(t_{0}) \rho_{i}^{d} \tag{14}\]  
If (14) converges to zero, the error differential equation can be written as  
\[\dot{\mathbf{e}}_{p,i}(t)=-k_{p}\mathbf{e}_{p,i}(t) \tag{15}\]  
Substituting (6), (7) and (14) into (15) gives  
\[\dot{\mathbf{p}}_{i}^{\prime}(t)=\mathbf{v}_{0}(t)-k_{p}\Big{[}\mathbf{\bar{p}}_{i}(t)-\bm {R}(\phi_{i}(t))\mathbf{\bar{p}}_{i}(t_{0})\rho_{i}^{d}\Big{]}+\dot{\mathbf{R}}(\phi_{ i}(t))\mathbf{\bar{p}}_{i}(t_{0})\rho_{i}^{d} \tag{16}\]  
where  
\[\dot{\mathbf{R}}(\phi_{i}(t))=\dot{\phi}_{i}(t)\left[\begin{array}{cc}-\sin\phi _{i}(t)&-\cos\phi_{i}(t)\\ \cos\phi_{i}(t)&-\sin\phi_{i}(t)\end{array}\right] \tag{17}\]  
Figure 2: The structure of the designed controller scheme.  
According to the robot dynamics model (4), the controller (13) is obtained by (16). In addition, (15) can be solved as \(\mathbf{e}_{p,i}(t)=\exp(-k_{p}t)\mathbf{e}_{p,i}(t_{0})\). The position error \(\mathbf{e}_{p,i}(t)\) could converge to \(\mathbf{0}\) as \(t\rightarrow\infty\) because of \(k_{p}\) as a positive constant and \(\mathbf{e}_{p,i}(t_{0})\) as a positive constant vector. So, the control objective (9) has been proved. Especially, when \(\dot{\mathbf{\phi}}_{i}(t)\
eq 0\), all robots can rotate around the target uniformly.