Multi-Agent Systems (MAS)

Structure

Large-scale system
Graph indicates coupling
Nodes are described by „simple“ dynamical subsystems

Goal

Achieve global behavior from local interaction

Problem

Scalability and limited information $\Rightarrow$ Distributed algorithms, local interaction among subsystems

Rendezvous Problem

Given a team of $N$ robots, how should they move to meet at the same location, using only local information?

Model

$\begin{align*}&\text{Positions:} & x_i \in \mathbb{R}^2, i = 1, \dots, N\\&\text{Agent} i: & \dot{x}_i = u_i, i=1, \dots, N\\&\text{Only information: distance to other robots} & x_i - x_j, \hspace{3mm} j=1, \dots, N\\&\text{Control law:} & u_i = g_i(x_i - x_1, \cdots, x_i-x_N)\end{align*}$

How to design $g_i$ ?

Assume: we have two robots $N=2$ with positions $x_1, x_2$
Robot dynamics:

$\begin{align*}\dot{x}_1 &= - \gamma_1(x_1 - x_2)\\\dot{x}_2 &= - \gamma_2(x_2 - x_1), \hspace{3mm} \gamma_1, \gamma_2 >0\end{align*}$

Rendezvous between $x_1(0)$ and $x_2(0)$

$\gamma_1 > \gamma_2 \Rightarrow$ closer to $x_2(0)$
$\gamma_1 < \gamma_2 \Rightarrow$ closer to $x_2(0)$
$\gamma_1 = \gamma_2 \Rightarrow$ rendezvous in the middle $\frac{x_1(0) + x_2(0)}{2}$