## 3 Formation Controller Design
### Tracking Controller
Then, substituting (22) and (23) into (27), the derivative of \(V_{i}\) can be calculated as follows:  
\[\dot{V_{i}}= \dot{\epsilon}_{xi}\epsilon_{xi}+\dot{\epsilon}_{yi}\epsilon_{yi} +\frac{1}{2}\dot{\epsilon}_{\theta i}\sin\frac{\epsilon_{\theta i}}{2}+\dot{ \Gamma_{i}} \tag{29}\] \[\leq \dot{\epsilon}_{xi}\epsilon_{xi}+\dot{\epsilon}_{yi}\epsilon_{yi }+\frac{1}{2}\dot{\epsilon}_{\theta i}\sin\frac{\epsilon_{\theta i}}{2}\] \[= \left[\epsilon_{xi}\left(\
u_{di}-\
u_{ti}\right)+2\sin\frac{ \epsilon_{\theta i}}{2}\left(\
u_{di}\epsilon_{yi}\cos\frac{\epsilon_{ \theta i}}{2}\right.\right.\] \[\left.\left.-\
u_{di}\epsilon_{xi}\sin\frac{\epsilon_{\theta i}}{ 2}+\frac{1}{4}\omega_{di}-\frac{1}{4}\omega_{ti}\right)\right]\] \[= -c_{1i}\epsilon_{xi}^{2}-2c_{2i}\sin\frac{\epsilon_{\theta i}}{ 2}\]  
As \(V_{i}\geq 0,\dot{V_{i}}\leq 0\), then \(V\) is bounded, which demonstrates that \(\epsilon_{xi}\) and \(\epsilon_{yi}\) are bounded in \(t\). Moreover, \(\dot{V_{i}}\) is uniformly continuous and bounded, so \(\dot{V_{i}}\to 0\), with regard to [PERSON]'s Lemma; then, \(\epsilon_{xi}\to 0\), \(\sin\frac{\epsilon_{\theta i}}{2}\to 0\) as \(t\rightarrow\infty\). To show \(\epsilon_{yi}\to 0\) when \(\epsilon_{xi}\to 0\), \(\sin\frac{\epsilon_{\theta i}}{2}\to 0\), define  
\[f(t)=\epsilon_{yi}\sin\epsilon_{\theta i} \tag{30}\]  
Since \(\epsilon_{yi}\) is bounded and \(\sin\frac{\epsilon_{\theta i}}{2}\to 0\), then \(f(t)\to 0\). Taking the time derivative of \(f(t)\),  
\[\dot{f}(t)= \dot{\epsilon}_{yi}\sin\epsilon_{\theta i}+\epsilon_{yi}(\omega_ {ri}-\omega_{ti})\cos\epsilon_{\theta i} \tag{31}\] \[= (-\omega_{i}\epsilon_{xi}+\
u_{ri}\sin\epsilon_{\theta i})\sin \epsilon_{\theta i}+\epsilon_{yi}(\omega_{ri}-\omega_{i})\cos\epsilon_{\theta i}\] \[= -4\
u_{ri}\epsilon_{yi}^{2}\cos\frac{\epsilon_{\theta i}}{2}+ \sin\frac{\epsilon_{\theta i}}{2}\left[\epsilon_{yi}\cos\epsilon_{\theta i}(4 \
u_{ri}\epsilon_{xi}-4c_{2i})\right.\] \[\left.+2\cos\frac{\epsilon_{\theta i}}{2}(-\omega_{i}\epsilon_{xi} +\
u_{ri}\sin\epsilon_{\theta i})\right]\] \[\leq \dot{\epsilon}_{xi}\epsilon_{xi}+\dot{\epsilon}_{yi}\epsilon_{yi }+\frac{1}{2}\dot{\epsilon}_{\theta i}\sin\frac{\epsilon_{\theta i}}{2}\]0  
where \(f_{1}(t),f_{2}(t)\) are defined as follows:  
\[\leq \dot{\epsilon}_{xi}\epsilon_{xi}+\dot{\epsilon}_{yi}\epsilon_{yi }+\frac{1}{2}\dot{\epsilon}_{\theta i}\sin\frac{\epsilon_{\theta i}}{2}\]1 \[\leq \dot{\epsilon}_{xi}\epsilon_{xi}+\dot{\epsilon}_{yi}\epsilon_{yi }+\frac{1}{2}\dot{\epsilon}_{\theta i}\sin\frac{\epsilon_{\theta i}}{2}\]2 \[\leq \dot{\epsilon}_{xi}\epsilon_{xi}+\dot{\epsilon}_{yi}\epsilon_{yi }+\frac{1}{2}\dot{\epsilon}_{\theta i}\sin\frac{\epsilon_{\theta i}}{2}\]3  
where \(\omega_{ri}\) can be substituted by (23). Since \(\sin\frac{e_{gi}}{2}\to 0\), \(f_{2}(t)\to 0\). \(f_{1}(t)\) is uniformly continuous in t, the following can be derived:  
\[\leq \dot{\epsilon}_{xi}\epsilon_{xi}+\dot{\epsilon}_{yi}\epsilon_{yi }+\frac{1}{2}\dot{\epsilon}_{\theta i}\sin\frac{\epsilon_{\theta i}}{2}\]4  
We note that \(v_{ri}\
eq 0\). Thus,  
\[\leq \dot{\epsilon}_{xi}\epsilon_{xi}+\dot{\epsilon}_{yi}\epsilon_{yi }+\frac{1}{2}\dot{\epsilon}_{\theta i}\sin\frac{\epsilon_{\theta i}}{2}\]5  
Thus, we can see that \(\epsilon_{yi}\to 0\) when \(\epsilon_{xi}\to 0\), \(\sin\frac{e_{gi}}{2}\to 0\), the stability of the closed-loop system by applying controller \(\left[ \begin{array}{cc}v_{i}&\omega_{i}\end{array} \right]^{T}\) is guaranteed and the controller can achieve good tracking performance. When the \(i\)th robot is outside the detection region and no collision occurs, the reference trajectories \(\left[ \begin{array}{cc}x_{ri}&y_{ri}\end{array} \right]^{T}\) are equal to the desired \(\left[ \begin{array}{cc}x_{di}&y_{di}\end{array} \right]^{T}\). Thus, with Theorem 1, \(N\) mobile robots can move along the reference trajectories, where the tracking errors can converge to \(0\) in the situation of no collision.