## 3 Mathematical Model of Formation Operation Control
### Leader-Follower System Model  
The simple diagram of the leader-follower system model is shown in Figure 7. According to Equations (20)-(22), the state error between leader formation multi-robot systems in this article is Equation (23).  
Then  
\[\left(\begin{array}{c}x_{ie}\\ y_{ie}\\ \theta_{ie}\end{array}\right)=\left(\begin{array}{ccc}\cos\theta_{i}&\sin \theta_{i}&0\\ -\sin\theta_{i}&\cos\theta_{i}&0\\ 0&0&1\end{array}\right)\left(\begin{array}{c}x_{i}^{d}-x_{i}\\ y_{i}^{d}-y_{i}\\ \theta_{i}^{d}-\theta_{i}\end{array}\right), \tag{23}\]  
where \(x_{ie}\), \(y_{ie}\) and \(\theta_{ie}\) are the state errors of the system. The ideal state quantities of the follower robot are defined as \(x_{i}^{d}\), \(y_{i}^{d}\) and \(\theta_{i}^{d}\). \(x_{i}\), \(y_{i}\) and \(\theta_{i}\) are the state quantities of the \(i\)-\(th\) robot.  
Figure 7: Simple diagram of leader-follower system model.  
The virtual robot proposed in the previous section is used as error analysis. Equation (24) can be obtained from the coordinate relative position relationship between the virtual leader robot and the following robot. Then  
\[\left(\begin{array}{c}x_{ie}\\ y_{ie}\\ \theta_{ie}\end{array}\right)=\left(\begin{array}{ccc}\cos\theta_{f}&\sin \theta_{f}&0\\ -\sin\theta_{f}&\cos\theta_{f}&0\\ 0&0&1\end{array}\right)\left(\begin{array}{c}x_{v}-x_{f}\\ y_{v}-y_{f}\\ \theta_{v}-\theta_{f}\end{array}\right), \tag{24}\]  
Sorting out the state error, deriving Equation (24), and substituting Equations (21) and (20), the following Equation (25) can be obtained. Then  
\[\left\{\begin{array}{ll}\dot{x}_{ie}&=\omega_{f}y_{ie}+\cos\theta_{f}(v_{v} \cos\theta_{v}+d\omega_{v}\sin\theta_{v})-\cos\theta_{f}\Big{(}v_{f}\cos \theta_{f}+d\omega_{f}\sin\theta_{f}\Big{)}\\ &\quad+\sin\theta_{f}(v_{v}\sin\theta_{v}-d\omega_{v}\cos\theta_{v})-\sin \theta_{f}\Big{(}v_{f}\sin\theta_{f}-d\omega_{f}\cos\theta_{f}\Big{)}\\ &=\omega_{f}y_{ie}+v_{v}\cos\theta_{ie}+\omega_{v}d\sin\theta_{ie}-v_{f}\\ \dot{y}_{ie}&=-\omega_{f}x_{ie}-\sin\theta_{f}(v_{v}\cos\theta_{v}+d\omega_{v} \sin\theta_{v})-\sin\theta_{f}\Big{(}v_{f}\cos\theta_{f}+d\omega_{f}\sin \theta_{f}\Big{)}\\ &\quad+\cos\theta_{f}(v_{v}\sin\theta_{v}-d\omega_{v}\cos\theta_{v})-\cos \theta_{f}\Big{(}v_{f}\sin\theta_{f}-d\omega_{f}\cos\theta_{f}\Big{)}\\ &=-\omega_{f}x_{ie}+v_{v}\sin\theta_{ie}+\omega_{f}d-\omega_{v}d\cos\theta_{ ie}\\ \dot{\theta}&=\omega_{v}-\omega_{f}\end{array}\right., \tag{25}\]  
The state error after coordinate transformation can be described as Equation (26). Then  
\[\left(\begin{array}{c}\dot{x}_{ie}\\ \dot{y}_{ie}\\ \dot{\theta}_{ie}\end{array}\right)=\left(\begin{array}{ccc}\cos\theta_{ie }&-1&d\sin\theta_{ie}&y_{ie}\\ \sin\theta_{ie}&0&-d\cos\theta_{ie}&d-x_{ie}\\ 0&0&1&-1\end{array}\right)\left(\begin{array}{c}v_{v}\\ v_{f}\\ \omega_{v}\\ \omega_{f}\end{array}\right), \tag{26}\]  
The input of the following robot \(\mathbf{R}_{f}\) are \(\upsilon\) and \(\omega\). \(\upsilon\) and \(\omega\) can be adjusted to satisfy the following Equation (27). Then  
\[\left\{\begin{array}{ll}\lim_{t\to t_{0}}|x_{ie}|=0\\ \lim_{t\to t_{0}}|y_{ie}|=0\\ \lim_{t\to t_{0}}|\theta_{ie}|=0\end{array}\right., \tag{27}\]  
In order to make the tracking error converge to zero, the multi-robot formation control law based on the leader-follower method can be designed as Equation (28). Then  
\[\left\{\begin{array}{ll}v_{f}=v_{v}\cos\theta_{ie}+k_{x}(x_{ie}-d(1-\cos \theta_{ie}))-k_{\theta}\theta_{ie}\omega_{f}\\ \omega_{f}=\omega_{v}+v_{v}\big{(}k_{y}k_{a}(y_{ie}+d\sin\theta_{ie}+k_{\theta} \theta_{ie})\big{)}+\frac{k_{b}}{k_{a}}\sin\theta_{ie}\end{array}\right., \tag{28}\]  
where, \(k_{x},k_{y},k_{\theta},k_{a}\) and \(k_{b}\) are the positive real number, and \(k_{a}+k_{b}=1\).