# Coupled-Error-Based Formation Control for Rapid Formation Completion by Omni-Directional Robots
## 3 Coupled-Error-Based Formation Control Method  
This paper focuses on rapidly forming formations, even if the overall system's trajectory tracking velocity is reduced. Nonetheless, as the formation is completed, the entire  
Figure 4: The multi-robot system with omni-directional robots. system should move along the trajectory. Therefore, as in [6], this paper employs a coupled error that integrates the information from its own trajectory (or the desired position generated by its own leader) and the followers as in the following. \[E_{i}^{x}=e_{i}^{x}-qe_{i+1}^{x},\ E_{i}^{y}=e_{i}^{y}-qe_{i+1}^{y}. \tag{9}\]  
where \(q>0\). The time derivatives of the coupled error in (9) are  
\[\dot{E}_{i}^{x} =\tfrac{d}{dt}\big{\{}e_{i}^{x}-qe_{i+1}^{x}\big{\}} \tag{10a}\] \[=(v_{i-1}^{x}-v_{i}^{y})-q\big{(}v_{i}^{x}-v_{i+1}^{x}\big{)}\,\] \[=-(q+1)v_{i}^{x}+v_{i-1}^{x}+qv_{i+i}^{x}\] \[\dot{E}_{i}^{y} =\tfrac{d}{dt}\left\{e_{i}^{y}-qe_{i+1}^{y}\right\}\] (10b) \[=\left(v_{i-1}^{y}-v_{i}^{y}\right)-q\Big{(}v_{i}^{y}-v_{i+1}^{y} \Big{)}\] \[=-(q+1)v_{i}^{y}+v_{i-1}^{y}+qv_{i+1}^{y}\]  
Here, since the coupled error in (9) and (10) use the information of the following robot, the last robot's coupled error is described as  
\[E_{n}^{x}=e_{n}^{x},\ E_{n}^{y}=e_{n}^{y}. \tag{11}\]  
where \(n\) is the index of the last follower. To ensure the relationship between the position error and coupled error, we refer to Lemma 1 in [6]. According to Lemma 1, it can be shown that if the coupled error defined in Equations (9)-(11) is bounded, then the position errors can also be bounded, and if the coupled error becomes zero, then the position errors will also be zero. **Theorem 1**. : _(Equivalence of the Boundedness and Convergences of the Coupled Error and Each Position Error Toward Zero): If the coupled error is bounded and becomes zero then, position error of each robot is bounded and becomes zero for all \(i=1\), \(2\), \(\ldots\), \(n\) at the same time._  
Proof. : To show that coupled error \(E_{i}^{x}\) and \(E_{i}^{y}\) are bounded and become zero if and only if \(e_{i}^{x}\) and \(e_{i}^{y}\) are bounded and become zero for all \(i=1\), \(2\), \(\ldots\), \(n\) at the same time, we can describe the coupled error \(E_{i}^{x}\) and \(E_{i}^{y}\), and \(e_{i}^{x}\) and \(e_{i}^{y}\) as \(\mathbf{E}=Q\mathbf{e}\), where  
\[\mathbf{e}_{x}= \begin{bmatrix}e_{1}^{x}&e_{2}^{x}&\cdots&e_{n}^{x }\end{bmatrix} ^{T},\ \mathbf{e}_{y}= \begin{bmatrix}e_{1}^{y}&e_{2}^{y}&\cdots&e_{n}^{y} \end{bmatrix} ^{T}\] \[\mathbf{E}_{x}= \begin{bmatrix}E_{1}^{x}&E_{2}^{x}&\cdots&E_{n}^{x }\end{bmatrix} ^{T},\ \mathbf{E}_{y}= \begin{bmatrix}E_{1}^{y}&E_{2}^{y}&\cdots&E_{n}^{y} \end{bmatrix} ^{T}\] \[Q= \begin{bmatrix}1&-q&0&\cdots&0\\ 0&1&-q&\cdots&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&\cdots&0&1&-q\\ 0&0&\cdots&0&1\end{bmatrix} .\]  
Since \(q>0\) is constant and \(Q\) is invertible, \(\|\mathbf{e}_{x}\|=\|Q^{-1}\|\|\|\mathbf{E}_{x}\|\). If \(\|\mathbf{E}_{x}\|\leq\overline{E}\), \(\|\mathbf{e}_{x}\|\leq\|Q^{-1}\|\overline{E}\). Also, if \(\overline{E}=0\), then it follows the equivalence of \(\mathbf{E}_{x}=0\) and \(\mathbf{e}_{x}=0\), and \(\mathbf{E}_{y}=0\) and \(\mathbf{e}_{y}=0\). \(\square\)  
As shown in Theorem 1 we should design the control law to make the CE converge to zero. The objective of the control law proposed in this paper is to make the coupled error zero. From Figure 4 and Equations (5)-(10), the following control algorithm is proposed:  
\[v_{i}^{x}=X_{i}+kE_{i}^{x},\ v_{i}^{y}=Y_{i}+kE_{i}^{y}, \tag{12}\]where \(X_{i}=\frac{1}{(1+q)}\big{(}v_{i-1}^{x}+qv_{i+1}^{x}\big{)}\), \(Y_{i}=\frac{1}{(1+q)}\Big{(}v_{i-1}^{y}+qv_{i+1}^{y}\Big{)}\), and \(k>0\) constant. In the case of \(i=n\), the formation control law can be expressed as  
\[v_{i}^{x}=v_{i-1}^{x}+ke_{i}^{x},\ v_{i}^{y}=v_{i-1}^{y}+ke_{i}^{y}. \tag{13}\]  
By using the proposed formation control law in (12) and (13), the coupled error can be bounded and converge to zero as time goes on, as in the following theorem. **Theorem 2**.