Swarm Robotics and Consensus

A swarm is many simple robots with no leader and no central plan, each following local rules based only on its neighbors. Surprisingly, global order can emerge. In the Vicsek model, each robot just turns toward the average heading of robots nearby, plus a little noise, and the whole group can come to move as one.

Number of robots Sensing range Heading noise

The Vicsek model

Each robot $i$ moves at constant speed and updates its heading to the average of its neighbors within range $r$, plus random noise $\eta$:

$$\theta_i(t+1) = \operatorname{atan2}\!\Big(\sum_{j \in \mathcal{N}_i} \sin\theta_j,\ \sum_{j \in \mathcal{N}_i} \cos\theta_j\Big) + \eta\,\xi_i$$

The order parameter $\phi = \frac{1}{N}\big\lVert \sum_i (\cos\theta_i, \sin\theta_i)\big\rVert$ is 1 when everyone agrees and near 0 when headings are random.

What to notice

With a large sensing range, information spreads quickly and the swarm soon moves as one. With a short range or few robots, separate flocks form, each agreeing internally but going its own way. Too much noise and order never appears. No robot knows the global state, yet consensus emerges. The same averaging idea is used for multi-robot formation control and for distributed estimation.