# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 1 Introduction
First, as regards discontinuous protocols, the upper bound of the convergence time is usually so far that it is not informative in practical applications. In addition, such protocols require a useless high input effort. As regards the use of time-varying gains, they are usually designed to take values in a prescribed time span, and that no action is taken afterwards, so it makes this strategy prone to disturbances. Moreover, such controllers usually provide an unbounded gain in a neighborhood of the time instant, especially in the presence of noise or other uncertainties [11]. In general, these techniques require tailored and intricated computation by each node at each time instant, and they are not effective in practical applications when errors and disturbances must be accounted for. In this paper, we introduce and develop a brand new approach to achieving finite time formation attainment, which is based on the combination of adopting standard consensus protocols and an appropriate leader motion. Such leader motion is the consequence of an additional input injected by the leader node and it is computed by properly elaborating the past leader motion through a dedicated algorithm run by the leader node that we call in the following the _leader motion mechanism_. The main goal of the paper is to make the team reach a desired formation in finite time through the motion of the leader. The idea of exploiting the leader evolution to obtain a better team performance was introduced in [18; 19] for nodes with first-order dynamics and processor networks. Inspired by this, in this paper, we derive necessary and/or sufficient conditions under which a leader motion can induce finite time formation for a team of vehicles. However, the extension of this approach from the simple First-order agents hides several difficulties, as already pointed out in [20]. Basically, when dealing with second-order models, which are more suited to describing inertial relations, as in mechanical systems and networks of mobile devices, the control problem is much trickier and multi-objective. One main peculiarity of the proposed approach is that the collective goal is achieved through the leader motion, while the other nodes run some basic consensus protocols. This is in striking contrast with other techniques, which usually require tailored and intricated protocols that followers must follow [11]. The idea that one node may take the role of a leader and improve the overall performance through additional algorithms based on local data and not interfering with the underlying consensus protocols leading the group to the same agreements have been studied also in [21] to obtain robustness against the disruptive effects of faults and errors, even though the leader is not aware of the final consensus value. The paper is organized as follows. After a brief introduction of the notation adopted in this paper reported in the following, in Section 2, the mathematical setting of the problem is thoroughly introduced. In Section 3, the main results of the paper are derived; first, the architecture of the leader motion mechanism is introduced, then the steady-state behavior is thoroughly studied. After that, in Section 4, the dynamics of the overall system are studied to obtain the design strategies that allow us to accomplish the formation goal in finite time. Finally, Section 6, with a set of simulation results closes the paper, showing the effectiveness of formation control for a multi-agent system of six agents.