# Differentially-Driven Robots Moving in Formation--Leader-Follower Approach
## 5 Experimental Verification
### Experimental Results
#### 5.2.1 Experiment EX1  
Polynomial trajectory in EX1 was generated based on the equations  
\[ \begin{split} x_{0}(t)=T_{s}t\\ y_{0}(t)=a_{1}(T_{s}t)^{6}+a_{2}(T_{s}t)^{5}+a_{3}(T_{s}t)^{4}+a_{4 }(T_{s}t)^{3}+a_{5}(T_{s}t)^{2}+a_{6}(T_{s}t)+a_{7}\end{split}  \tag{19}\]  
with \(T_{s}=0.1\) (time scaling coefficient) and \(a_{1}=-0.0019\), \(a_{2}=0.0492\), \(a_{3}=-0.4806\), \(a_{4}=2.1653\), \(a_{5}=-4.2752\), \(a_{6}=2.6646\), \(a_{7}=-0.0005\). The coefficients have been chosen to ensure the feasibility of experiments involving trajectory tracking in available laboratory space. The obtained results for EX1 are presented in Figure 6.  
In the first graph, in Figure 6, the \(xy\)-plot of the robots motion path is shown. In the next ones, time plots of selected signals are presented. The time duration of the experiment was around \(27\) s and it was related with the speed of the robots and the available laboratory area. Figure 6,c show \(x\) and \(y\) position coordinates while Figure 6 orientation of the robots. In Figure 6, the position and orientation errors are presented, respectively. Taking into account these plots one can see that all robots properly follow a trajectory within an assigned formation structure and position errors convergence close to zero. In Figure 6,i linear and angular velocity of the calculated control signals is plotted whereas in Figure 6,k the angular velocities of the robot wheels are depicted. One can see that during the experiment velocities of the wheels reached fixed limit. Figure 6 shows 'freeze' signals which equals \(1\), indicate that the reference signals was frozen to avoid collision between robots. During the transient state of the motion collision avoidance mechanism was activated what is presented by the 'freeze' signals plot. Freezing the reference signals can be easily observed as resulting in some chattering of the linear velocity and especially in wheels velocity signals. Collision avoidance behavior prevented the robots from coming closer than \(0.5\) m to each other (see Figure 6). The threshold distance of 'freeze' signal activation was equal to the radius of the collision avoidance area \(\bar{R}_{j}=0.6\) m (dashed green line). As robots get closer to each other, the distance collision avoidance blocks cause repulsion between robots. As all robots maintained a distance greater than \(\bar{r}_{j}=0.2\) m (dashed red line) between each other, no collision occurred. Figure 6 shows signals:  
\[Q_{i}(v_{i})=\left\{ \begin{array}{ll}1&\text{for}\ \ |v_{i}|<v_{t}\\ 0&\text{otherwise}\end{array} \right. \tag{20}\]  
that represent activation of the procedure described in Assumption 4. The value of the threshold \(v_{t}\) was set to \(0.005\) m/s. This procedure was activated twice during the experiment, for a short period of time. It temporarily disrupts the robot's motion, but prevents it from getting stuck. The collision avoidance component of the control is shown in Figure 6. It is a norm of the sum of all collision avoidance vectors acting on the \(i\)-th robot:  
\[A_{i}=\sum_{j=1,j\
eq i}^{N}\left\|\left[\frac{\partial V_{aij}}{\partial e_{ ix}}\ \ \frac{\partial V_{aij}}{\partial e_{iy}}\right]\right\|. \tag{21}\]  
The signals are non-zero only if the robots are close enough to each other (at a distance less than \(\bar{R}_{j}\)). Figure 6 presents the situation when the robots reach their trajectories, the relative distances between them are close to desired value equal to \(||[d_{x}^{i-1,j}\ \ d_{y}^{i-1,j}]^{T}||=1\) m. The green dashed line denotes the distance \(R_{i}\) and red dashed line is the distance \(r_{i}\) at which the collision would occur. As can be seen, the distances between all the robots remain safe during the experiment, reaching values only slightly smaller than \(\bar{R}_{j}\). In addition, comparing Figure 6, Figure 6 and Figure 6, it is clear that collision avoidance and 'freeze' signals are activated only when the distance between robots is less then \(\bar{R}_{j}\). Figure 6: _Cont_.