## 3 Controller Design
### Custom Phasing and Spacing Control  
Next, in order to ensure that the relative angular spacing between inter-robots is maintained within the desired values, the angular spacing of the desired formation can be carried out by changing the steering angles. According to (10), we can obtain \(\dot{\mathbf{\phi}}_{i}(t)=\dot{\mathbf{\phi}}_{i}(t)\). The control law (13) can be rewritten as  
\[\mathbf{u}_{i}(t)=\mathbf{v}_{0}(t)-k_{p}\Big{[}\mathbf{\tilde{p}}_{i}(t)-\mathbf{R}(\phi_{i}( t))\mathbf{\tilde{p}}_{i}(t_{0})\rho_{i}^{d}\Big{]}+\dot{\mathbf{\phi}}_{i}(t)\mathbf{R}^{*}( \phi_{i}(t))\mathbf{\tilde{p}}_{i}(t_{0})\rho_{i}^{d} \tag{18}\]  
where,  
\[\mathbf{R}^{*}(\phi_{i}(t))=\left[\begin{array}{cc}-\sin\phi_{i}(t)&-\cos\phi_ {i}(t)\\ \cos\phi_{i}(t)&-\sin\phi_{i}(t)\end{array}\right] \tag{19}\]  
Considering the practical applications, it is sometimes necessary to determine the position of the robots on their own circles. For example, when a multi-robot system monitors an area of interest, the accurate position of each robot needs to be pre-planned to monitor a specific region. Therefore, it is necessary to design a phase positioning and spacing assignment control law, so that the robots can reach the specified position on their circles. We introduce a nonlinear function into the controller (18), which is described as follows:  
\[\dot{\mathbf{\phi}}_{i}(t)=k_{\varphi}\sin\left[\left(\varphi_{1}^{d}+\sum_{j=1}^ {i-1}\Delta\varphi_{j}^{d}\right)-\left(\varphi_{1}(t)+\sum_{j=1}^{i-1}\Delta \varphi_{j}(t)\right)\right] \tag{20}\]  
where \(k_{\varphi}\) is a position constant, which is used to adjust the convergence speed of phase angle and angular spacing of the robots.  
**Theorem 2**.: _Under the circular motion control law (18), the phase positioning and spacing-assignment control law of all robots are satisfied (20). The phase angles of all robots on the circles can converge to the desired values calculated by desired initial phase \(\varphi_{1}^{d}\) of robot 1 and the desired relative angular spacing \(\mathbf{\varphi}^{d}\), that is, \(\lim\limits_{t\rightarrow\infty}\varphi_{i}(t)=\varphi_{1}^{d}+\sum\limits_{j =1}^{i-1}\Delta\varphi_{j}^{d}\) for \(i=1,2,\cdots,n\), which contains two parts: the phase positioning convergence of the robot 1, that is, \(\lim\limits_{t\rightarrow\infty}\varphi_{1}(t)=\varphi_{1}^{d}\), and the relative angular spacing convergence of all robots, that is, \(\lim\limits_{t\rightarrow\infty}\Delta\mathbf{\varphi}(t)=\Delta\mathbf{\varphi}^{d}\)._  
Proof.: Let \(\varphi_{i}(t)=\varphi_{1}(t)+\sum\limits_{j=1}^{i-1}\Delta\varphi_{j}(t)\). We choose a positive definite Lyapunov function candidate  
\[V=k_{\varphi}\left[1-\cos\left(\varphi_{1}^{d}+\sum_{j=1}^{i-1}\Delta\varphi_ {j}^{d}-\varphi_{i}(t)\right)\right] \tag{21}\]  
Differentiating Equation (20) with respect to time as  
\[\dot{V}=-k_{\varphi}\sin\left(\varphi_{1}^{d}+\sum_{j=1}^{i-1}\Delta\varphi_ {j}^{d}-\varphi_{i}(t)\right)\dot{\varphi}_{i}(t)=-k_{\varphi}{}^{2}\text{sin}^ {2}\Bigg{(}\varphi_{1}^{d}+\sum_{j=1}^{i-1}\Delta\varphi_{j}^{d}-\varphi_{i}(t )\Bigg{)}\leq 0 \tag{22}\]With (22), \(\dot{V}\leq 0\) is guaranteed to be negative definite. Let \(\dot{V}=0\); there are the relative equilibrium points \(\varphi_{i}(t)=\varphi_{1}^{d}+\sum\limits_{j=1}^{i-1}\Delta\varphi_{j}^{d}\) for \(i=1,2,...,n\). This shows that the phase convergence of each robot is guaranteed by control laws (20).  
To sum up, it is shown that the circular formation controller is demonstrated as stable by the control law (18) and (20).