# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 3 Problem Framework  
In this paper, we study a multi-agent system of \(N\) mobile agents. Each agent moves in \(\mathbb{R}^{m}\), with \(m\) equal either to 2 or 3, according to the following inertial equation \((i=1,...,N)\):  
\[\dot{\mathbf{r}}_{i}(t) =\mathbf{v}_{i}(t) \tag{5}\] \[\dot{\mathbf{v}}_{i}(t) =\frac{1}{M_{i}}\mathbf{u}_{i}(t),\]  
where \(\mathbf{r}_{i}(t)\in\mathbb{R}^{m}\) denotes the position of the \(i\)-th agent, \(\mathbf{v}_{i}(t)\in\mathbb{R}^{m}\) its speed, and \(M_{i}\) its inertial mass. Each agent can set a force input equal to \(\mathbf{u}_{i}(t)\in\mathbb{R}^{m}\) to steer its motion according to a control goal. Equation (5) are largely adopted models in vehicle platooning and in multi-vehicle cooperative control [3; 4; 7]. It is worth mentioning that, considering \(m=2\), and hence in the common special case of a team of robots moving in the plane, in [22], it is shown that the above Equation (5) are well suited to describe the motion of a team of mobile robots with non-holonomic constraints as follows:  
\[\left\{ \begin{array}{rl}\dot{x}_{i}(t)=v_{i}(t)\cos(\theta_{i}(t)),\\ \dot{y}_{i}(t)=v_{i}(t)\sin(\theta_{i}(t)),\\ \dot{\theta}_{i}(t)=\omega_{i}(t),\\ m_{i}\dot{\upsilon}_{i}(t)=f_{i}(t),\\ J_{i}\dot{\omega}_{i}(t)=\tau_{i}(t),\end{array} \right. \tag{6}\]  
upon the following transformation:  
\[ \begin{bmatrix}f_{i}\\ \tau_{i}\end{bmatrix} =\left[ \begin{array}{cc}\frac{1}{m_{i}}\cos(\theta_{i})& -\frac{d_{i}}{J_{i}}\sin(\theta_{i})\\ \frac{1}{m_{i}}\sin(\theta_{i})&\frac{d_{i}}{J_{i}}\cos(\theta_{i})\end{array}  \right] \begin{bmatrix}u_{xi}+v_{i}\omega_{i}\sin(\theta_{i})+d_{i}\omega_{i}^{2 }\cos(\theta_{i})\\ u_{yi}+v_{i}\omega_{i}\cos(\theta_{i})+d_{i}\omega_{i}^{2}\sin(\theta_{i}) \end{bmatrix} , \tag{7}\]  
where \(u_{xi}\) and \(u_{yi}\) are the control signals of system (6) and (7), which result in a set of equations as Equation (5); all the details can be found in the reference paper [22]. It is supposed that agents are able to sense the position of a subset of the teammates that are referred to as _neighbor agents_. For the sake of simplicity, we assume that the sensing is bidirectional, namely \(i\) are able to sense \(j\) if and only if \(j\) is able to sense the position of \(i\) too, though the proposed approach applies to directed networks as well. The neighbor agents, namely those that are able to sense among themselves, can be effectively described through the _adjacency matrix_\(\mathcal{A}\in\mathbb{R}^{N\times N}\) of a suitable communication graph \(\mathcal{G}\). In particular, matrix \(\mathcal{A}\in\mathbb{R}^{N\times N}\) is built upon the rule \([\mathcal{A}]_{ij}=1\) if agent \(j\) and \(i\) are able to exchange information between themselves, and \([\mathcal{A}]_{ij}=0\) otherwise. In the following, we assume that agents agree on the consensus protocol, as in [5], so that we assume that each agent sets its local input as:  
\[\mathbf{u}_{i}^{c}(t)=M_{i}\sum_{j=1}^{N}[\mathcal{A}]_{ij}k_{ij}[(\mathbf{r}_ {j}(t)-\mathbf{r}_{i}(t))+\gamma(\mathbf{v}_{j}-\mathbf{v}_{i})]. \tag{8}\]  
System (5) under protocol (8) is a very popular model for the analytical description of the collective motion of a team of mobile agents, thus representing a widely adopted model in vehicular networks [4] and coordination of mobile robots [23]. Indeed, it describes the motion of the whole team, especially when collective motion aggregates, converging to a common steady-state position in the plane.