## 3 Formation Tracking Control of Nonholonomic Mobile Robots
### Formation Control Objective  
Here, we express the desired trajectory of the system in terms of the components in the \(X\) and \(Y\) directions shown below.  
\[s_{xr}(t)=a_{rx}\rho_{rx}(t)+d_{rx}\qquad s_{yr}(t)=a_{ry}\rho_{ry}(t)+d_{ry} \tag{9}\]  
where \(\rho_{rx}(t)\) and \(\rho_{ry}(t)\) are the basis functions and are assumed to be known by all robots, while the constant parameters \(a_{rx}\), \(a_{ry}\), \(d_{rx}\) and \(d_{ry}\) can not be obtained by all robots. Additionally, \(\theta_{r}(t)=\arctan\left(\frac{i\varphi}{s_{xr}}\right)\) represents the reference trajectory for each robot orientation, because \(\rho_{rx}(t)\) and \(\rho_{ry}(t)\) are known to all robots. Then, by calculation, \(\dot{s}_{xr}\) and \(\dot{s}_{yr}\) are both constants. Hence, \(\theta_{r}(t)\) is known by all the followers.  
The control goal of this paper is to design distributed adaptive formation controllers so that all robots, while considering obstacle avoidance, follow a desired trajectory in the \(X\)-\(Y\) plane by keeping specified distances from the desired trajectory, i.e.,  
\[\lim\limits_{t\to\infty}\left[\dot{s}_{xi}(t)-s_{xr}(t)-\sum \limits_{i=1}^{m_{i}}E_{ix}\right]=-\varrho_{ix} \tag{10}\] \[\lim\limits_{t\to\infty}\left[\dot{s}_{yi}(t)-s_{yr}(t)-\sum \limits_{i=1}^{m_{i}}E_{iy}\right]=-\varrho_{iy}\] (11) \[\lim\limits_{t\to\infty}\left[\ddot{\theta}_{i}(t)-\theta_{r}(t) \right]=0 \tag{12}\]  
**Remark 1**: _In this paper, when the robot detects an obstacle, the robot avoids the obstacle first, and then forms a given formation after avoiding the obstacle. From the previous analysis, we can know that when the robot does not detect the obstacle, the terms \(\sum\limits_{i=1}^{m_{i}}E_{ix}\) and \(\sum\limits_{i=1}^{m_{i}}E_{iy}\) are equal to zero, so the expressions (10) and (11) become the regular formation expressions as in [34]. When there is an obstacle, the formation formed by multiple robots will change, that is, the distance between the robots will change, so the obstacle avoidance term is introduced into Equations (10) and (11)._  
**Remark 2**: _We consider the leader-follower formation strategy, where the desired trajectory is expected to act as the leader to guide the overall movement of the robot crowd, and each robot in the swarm will track its adjacent robot as a reference model. We refer to the reference model as the leader and the robots following it as followers. Notice about that in the leader-follower formation structure, any follower can be assigned as the intermediate leader of another robot, so there will be many pairs of leaders and followers in such a structure. This method does not need global knowledge and computation, and can be extended in the case of multiple agents. Suppose that for each robot \(R_{i}\)  
Figure 5: Potential field region.  
\(i\in N\), there is a fixed and unique leader \(R_{iL}\), \(i_{L}\
eq i\), \(i_{L}\in\bar{N}\) in the leader-follower structure. Thus, in the leader-follower formation with \(N\) mobile robots, they will be decomposed into \(N-1\) pairs of leaders and followers._  
From the above, we can already know that the graph \(\mathcal{G}\) stands for the communication status between \(N\) robots, and Assumption 2 is made. In order to realize the formation control goal, it is necessary to make the following assumption.  
**Assumption 3**.: \(\rho_{rx}\)_, \(\rho_{ry}\), \(\dot{\rho}_{rx}\), \(\dot{\rho}_{ry}\) and \(\dot{\rho}_{rx}\), \(\dot{\rho}_{ry}\) are bounded and piece-wise continuous bounded. Additionally, all robots know them._  
**Remark 3**.: _It is worth mentioning that the design trajectory shown in Equation (9) is similar to the designed trajectory that has been shown in many studies, including but not limited to [2, 35, 36]. Therefore, by looking up similar literature, we can know Assumption 3 is reasonable._  
**Assumption 4**.: _The neighbors of robot \(i\) can obtain the required distances \(\varrho_{ix}\) and \(\varrho_{iy}\)._