# Leader-Follower Formation Control of Wheeled Mobile Robots without Attitude Measurements
## 5 Stability Analysis
The trajectories of the robots obtained during the two experiments with the desired Cartesian trajectories are shown in Figures 3 and 4, respectively. The figures also shown the robots' positions at the time instants \(t=0\) [s], \(t=13\) [s] for the first experiment and \(t=0\) [s] and \(t=20\) [s] for the second experiment. In both cases, after the transient response the robots successfully achieve the convoy formation. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Robot** & \(\mathbf{k_{ai}}\) & \(  
In order to assess the performance of the attitude observer and control laws we compute the orientation errors \(\widehat{\theta}_{i}=\theta_{i}-\theta_{di}\) and \(\widehat{\theta}_{i}=\theta_{i}-\widehat{\theta}_{i}\) where \(\theta_{i}\) is the angle measured by the cameras and the estimated angle \(\widehat{\theta}_{i}\) is extracted from \(\widehat{\mathbf{\Theta}}_{i}\) as follows  
\[\widehat{\theta}_{i}=\text{atan2}(\widehat{\Theta}_{2i},\widehat{\Theta}_{1i})\]  
where \(\text{atan2}(\cdot,\cdot)\) is the two argument arctangent function and \(\widehat{\Theta}_{i}=\text{col}(\widehat{\Theta}_{1i},\widehat{\Theta}_{2i})\). The time evolution of the position and attitude tracking errors obtained in each experiment are shown in Figures 5 and 6. It is observed in the Figures that despite the unmodeled dynamics and discretization of the control laws, a good tracking was achieved. The time evolution of the time delays are shown in Figures 7 and 8. Notice that for the circular path the delays converge to a constant value while for the second trajectory the time delays change slowly while their magnitude increases at the points of the curve with greater curvature (see Figure 4b). This behavior was expected since at this points the robots come closer to each other. Figure 4: Trajectory of the robot group following the lemniscate curve: (**a**) Robots at \(t=0\) s, and (**b**) Robots at \(t=30\) s.  
Figure 3: Trajectory of the robot group following the circular path: Robots at the time instants \(t=0\) and \(t=15\) s.  
In order to assess the performance of the proposed algorithm, the RMS error is computed for the distance and orientation of each robot to its desired trajectory, additionally to the RMS error of the estimation of the orientation observer is presented. Table 2 collects the results for the circular desired trajectory while Table 3 shows the results corresponding to the lemniscate curve desired trajectory experiment. On both cases it is observed that the RMS distance error is bellow 0.04 m, while RMS orientation error is under 0.155 radians. Figure 8: Time evolution of the delay for the second trajectory. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Robot** & **Distance [m]** & **Orientation [rad]** & **Observation [rad]** \\ \hline Leader & 0.0330 & 0.1533 & 0.2115 \\ Follower 1 & 0.0336 & 0.0806 & 0.1217 \\ Follower 2 & 0.0367 & 0.1416 & 0.1949 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: RMS errors of distance, orientation and observed orientation of the robots in the circular trajectory experiment. Figure 7: Time evolution of the delay for the first trajectory.