# Signal Source Localization of Multiple Robots Using an Event-Triggered Communication Scheme
## 2 Preliminaries  
### Dynamics of Mobile Robots  
For mobile robots, such as Qbot in Figure 2, the dynamics can be described by:  
\[\left(\begin{array}{c}\dot{r}_{xi}\\ r_{yi}\\ \dot{\theta}_{i}\\ \dot{v}_{i}\\ \dot{\omega}_{i}\end{array}\right) = \left(\begin{array}{c}\
u_{i}\text{cos}\theta_{i}\\ \
u_{i}\text{sin}\theta_{i}\\ \omega_{i}\\ 0\\ 0\end{array}\right)+\left(\begin{array}{cc}0&0\\ 0&0\\ 0&0\\ \frac{1}{m_{i}}&0\\ 0&\frac{1}{l_{i}}\end{array}\right)\left(\begin{array}{c}F_{i}\\ \tau_{i}\end{array}\right) \tag{1}\]  
where \(r_{i}=(r_{xi},r_{yi})^{T}\) is the position of the \(i-\)th robot; \(\theta_{i}\) denotes the orientation; \(\
u_{i}\) is the linear velocity; \(\omega_{i}\) is the angular velocity; \(\tau_{i}\) is the torque; \(F_{i}\) is the force; \(m_{i}\) is the mass; and \(J_{i}\) is the moment of inertia. Let \(y_{i}=(r_{i},\theta_{i},\
u_{i},\omega_{i})^{T}\) be the state of the \(i-\)th robot and \(I_{i}=(F_{i},\tau_{i})^{T}\) be the control input.  
Figure 2: The Qbot robot.  
Because the nonholonomic systems cannot be stabilized with continuous static state feedback, we use the \"hand position\" instead of \"center position\" of the robot [37]. It should be pointed out that \"hand position\" is a position and lies a fixed offset \(L_{i}\) from the \"center position\". The line between between \"hand position\" and \"center position\" is perpendicular to the wheel axis (see [37]). Let (2) be the dynamics of the \"hand position\" of the robot.  
\[\left\{\begin{array}{l}\dot{x}_{i}=v_{i}\\ \dot{v}_{i}=u_{i}\quad i\in\{1,2,\ldots,n\}\end{array}\right. \tag{2}\]  
where \(x_{i}\) and \(v_{i}\), respectively, denote the position and the velocity for the robot \(i\) at the \"hand position\" and \(n\) is the number of robots. The relationship between the \"hand position\" and the \"center position\" can be described by:  
\[x_{i} = r_{i}+L_{i}\left(\begin{array}{l}\cos\theta_{i}\\ \text{sin}\theta_{i}\end{array}\right) \tag{3}\] \[v_{i} = \left(\begin{array}{l}\cos\theta_{i}&-L_{i}\text{sin}\theta\\ \text{sin}\theta_{i}&L_{i}\text{cos}\theta_{i}\end{array}\right)\left( \begin{array}{l}v_{i}\\ \omega_{i}\end{array}\right) \tag{4}\]  
According to (3) and (4), we can obtain the position and the velocity of the \"hand position\" of the robot and then calculate the control law \(u_{i}\) for the double-integrator system (2). Finally, we can obtain the control input (5) for the system (1) [37]:  
\[I_{i} = \left(\begin{array}{cc}\frac{1}{m_{i}}\text{cos}\theta_{i}&- \frac{L_{i}}{l_{i}}\text{sin}\theta_{i}\\ \frac{1}{m_{i}}\text{sin}\theta_{i}&\frac{L_{i}}{l_{i}}\text{cos}\theta_{i} \end{array}\right)^{-1}\left[u_{i}-\left(\begin{array}{l}-v_{i}\omega_{i} \text{sin}\theta_{i}-L_{i}\omega_{i}^{2}\text{cos}\theta_{i}\\ v_{i}\omega_{i}\text{cos}\theta_{i}-L_{i}\omega_{i}^{2}\text{sin}\theta_{i} \end{array}\right)\right] \tag{5}\]  
Usually, the applied torques for the left wheel and the right wheel can be calculated by:  
\[\tau_{l} = \frac{J_{wheel}}{b}\left(\frac{F_{i}}{m_{i}}-\frac{\tau_{l}l}{2I _{i}}\right) \tag{6}\] \[\tau_{r} = \frac{J_{wheel}}{b}\left(\frac{F_{i}}{m_{i}}+\frac{\tau_{l}l}{2I _{i}}\right) \tag{7}\]  
where \(b\) is the radius of the wheel; \(l\) denotes the axis length between two wheels; \(J_{wheel}\) is the moment of inertia of the wheel; \(\tau_{l}\) and \(\tau_{r}\) refer to the applied torques for the left wheel and the right wheel, respectively.  
Furthermore, the virtual leader is designed, and its dynamics is given as:  
\[\dot{x}_{0}(t)=v_{0}(t) \tag{8}\]  
where \(v_{0}(t)=v_{0}\) is a constant.  
**Remark 1**.: _It should be pointed out that the virtual leader is introduced to help the robot reach velocity consensus, and one can also control the final convergence velocity by setting \(v_{0}\)._