## 3 Formation Tracking Control of Nonholonomic Mobile Robots
### Control Design
 \tag{17}\]  
We can rewrite the expressions of \(e_{ix}\) and \(e_{iy}\) as \(e_{ix}=\xi_{xi}-s_{xr}+\varrho_{ix}-\sum\limits_{i=1}^{m_{i}}E_{ix}-(1-\mu_{ i})\big{(}\rho_{rx}\hat{a}_{rx,i}-\hat{d}_{rx,i}-s_{xr}\big{)}\) and \(e_{iy}=\xi_{yi}-s_{yr}+\varrho_{iy}-\sum\limits_{i=1}^{m_{i}}E_{iy}-(1-\mu_{ i})\big{(}\rho_{ry}\hat{a}_{ry,i}-\hat{d}_{ry,i}-s_{yr}\big{)}\). Then substitute \(e_{ix}\) and \(e_{iy}\) into \(\dot{V}\) to obtain  
\[ \begin{split}\dot{V}=&-k_{1}\sum\limits_{i=1}^{N}P_ {i}\Bigg{(}\Big{(}\xi_{xi}-s_{xr}+\varrho_{ix}-\sum\limits_{i=1}^{m_{i}}E_{ix} \Big{)}z_{ix}+\Big{(}\xi_{yi}-s_{yr}+\varrho_{iy}-\sum\limits_{i=1}^{m_{i}}E_{ iy}\Big{)}z_{iy}\Bigg{)}-k_{2}\sum\limits_{i=1}^{N}\xi_{i\theta}^{2}\\ &+\sum\limits_{i=1}^{N}\left(1-\mu_{i}\right)\times\frac{P_{i}}{ \gamma_{ri}}\Big{(}-\tilde{a}_{rx,i}\hat{a}_{rx,i}-\tilde{d}_{rx,i}\hat{d}_{rx,i}+\gamma_{ri}\big{(}\rho_{rx}\hat{a}_{rx,i}-\hat{d}_{rx,i}-s_{xr}\big{)}z_{ix }\Big{)}\\ &+\sum\limits_{i=1}^{N}\left(1-\mu_{i}\right)\times\frac{P_{i}}{ \gamma_{ri}}\Big{(}-\tilde{a}_{ry,i}\hat{a}_{ry,i}-\tilde{d}_{ry,i}^{2}\hat{ d}_{ry,i}^{2}+\gamma_{ri}\big{(}\rho_{ry}\hat{a}_{ry,i}-\hat{d}_{ry,i}-s_{yr}\big{)}z_{ iy}\Big{)}\end{split}  \tag{18}\]  
By selecting the parameter estimators of Equation (15), we can obtain  
\[\begin{split}\dot{V}&=-k_{1}\sum\limits_{i=1}^{N}P_ {i}\Bigg{(}\Big{(}\xi_{xi}-s_{xr}+\varrho_{ix}-\sum\limits_{i=1}^{m_{i}}E_{ix} \Big{)}z_{ix}+\Big{(}\xi_{yi}-s_{yr}+\varrho_{iy}-\sum\limits_{i=1}^{m_{i}}E_ {iy}\Big{)}z_{iy}\Bigg{)}-k_{2}\sum\limits_{i=1}^{N}\xi_{i\theta}^{2}\\ &\text{Let }\xi_{ix}=s_{xi}-s_{xr}+\varrho_{ix}-\sum\limits_{i=1}^{m_{i}}E_ {i},\xi_{iy}=s_{yi}-s_{yr}+\varrho_{iy}-\sum\limits_{i=1}^{m_{i}}E_{i}\text{. Hence, }z_{ix}\) and \(z_{iy}\) can be rewritten as \(z_{ix}=\sum\limits_{j=1}^{N}a_{ij}\Big{(}\xi_{ix}-\xi_{jx}\Big{)}+\mu_{i}\xi_{ix}\) and \(z_{iy}=\sum\limits_{j=1}^{N}a_{ij}\Big{(}\xi_{iy}-\xi_{jy}\Big{)}+\mu_{i}\xi_{ iy}\).  
Combining Lemma 1, the derivative of \(V\) can be obtained as  
\[\dot{V}=-\frac{k_{1}}{2}\left(\xi_{x}^{T}Q\xi_{x}+\xi_{y}^{T}Q\xi_{y}\right)-k_{2 }\xi_{\theta}^{T}\xi_{\theta} \tag{19}\]  
where \(\xi_{x}=[\xi_{1x},\cdots,\xi_{Nx}]\), \(\xi_{y}=[\xi_{1y},\cdots,\xi_{Ny}]\) and \(\xi_{\theta}=[\xi_{1\theta},\cdots,\xi_{N\theta}]\). **Theorem 1**. : _For multiple nonholonomic mobile robots (2) satisfying Assumptions 1-4, apply controller (13) and parameter estimator (15) to it. Then, the formation errors of each robot can be guaranteed to meet the following requirements:_  
\[ \begin{split}&\lim\limits_{t\to\infty}s_{xi}-s_{xr}+\varrho_{ix}- \sum\limits_{i=1}^{m_{i}}E_{ix}\leq\sqrt{2}\varpi_{i1}\\ &\lim\limits_{t\to\infty}s_{yi}-s_{yr}+\varrho_{iy}-\sum\limits _{i=1}^{m_{i}}E_{iy}\leq\sqrt{2}\varpi_{i1}\\ \end{split}  \tag{20}\]  
\[\lim\limits_{t\to\infty}\bar{\theta}_{i}-\theta_{r}\leq\sqrt{2}\varpi_{i2}\]  
Proof. : By applying [PERSON]'s lemma, we can easily get that \(\xi_{ix}\), \(\xi_{iy}\) and \(\xi_{\theta}\) will converge to zero asymptotically. Hence, we can find that \(\lim_{t\to\infty}s_{xi}-s_{xr}-\sum\limits_{i=1}^{m_{i}}E_{i}=-\varrho_{ix}\), \(\lim_{t\to\infty}s_{yi}-s_{yr}-\sum\limits_{i=1}^{m_{i}}E_{i}=-\varrho_{iy}\) and \(\lim_{t\to\infty}\theta_{i}-\theta_{r}=0\). From (3)-(5), the following equations can be obtained  
\[||(s_{xi}-\bar{s}_{xi},s_{yi}-\bar{s}_{yi})||\leq\sqrt{2\varpi_{i1}^{2}}, \quad|\theta_{i}-\bar{\theta}_{i}|\leq\varpi_{i2} \tag{21}\]  
and then it follows  
\[ \begin{split}&|\bar{s}_{xi}-s_{xr}+\varrho_{ix}-\sum\limits _{i=1}^{m_{i}}E_{ix}|\leq|\bar{s}_{xi}-s_{xi}|+D_{ix}\\ &|\bar{s}_{yi}-s_{yr}+\varrho_{iy}-\sum\limits_{i=1}^{m_{i}}E_{ iy}|\leq|\bar{s}_{yi}-s_{yi}|+D_{iy}\\ &|\bar{\theta}_{i}-\theta_{r}|\leq|\bar{\theta}_{i}-\theta_{i}|+ |\theta_{i}-\theta_{r}|\end{split}  \tag{22}\]  
where \(D_{ix}=|s_{xi}-s_{xr}+\varrho_{ix}-\sum\limits_{i=1}^{m_{i}}E_{ix}|\), \(D_{iy}=|s_{yi}-s_{yr}+\varrho_{iy}-\sum\limits_{i=1}^{m_{i}}E_{iy}|\).