control//multi-agent control//formation control

Formation control is the coordination of a group of vehicles so that they hold a prescribed geometric shape (a line of inspection drones, a ring around a target, a wedge of ground robots) while the group as a whole moves or decides where to be. The standard method is the consensus protocol run on offsets: if drone \(i\) should sit at position \(h_i\) relative to the formation's centre, the agents agree on \(x_i-h_i\) instead of on \(x_i\),

Formation control is the coordination of a group of vehicles so that they hold a prescribed geometric shape (a line of inspection drones, a ring around a target, a wedge of ground robots) while the group as a whole moves or decides where to be. The standard method is the consensus protocol run on offsets: if drone iii should sit at position hih_ihi​ relative to the formation's centre, the agents agree on xi−hix_i-h_ixi​−hi​ instead of on xix_ixi​,

x˙i=∑j∈Niaij [(xj−hj)−(xi−hi)].\dot x_i=\sum_{j\in\mathcal N_i}a_{ij}\,\bigl[(x_j-h_j)-(x_i-h_i)\bigr].x˙i​=j∈Ni​∑​aij​[(xj​−hj​)−(xi​−hi​)].

All the xi−hix_i-h_ixi​−hi​ converge to the same point, so each xi−xjx_i-x_jxi​−xj​ converges to hi−hjh_i-h_jhi​−hj​: the offsets fix the shape and the agreement decides its position, which lands on the average of where the drones started relative to their slots. Moving the formation means adding a common velocity reference or letting one agent lead (leader-follower).

The offsets only mean something in a shared frame.

The hih_ihi​ are written in a common reference, typically GNSS coordinates and a common heading, so a drone with a bad compass or a biased GNSS receiver puts its slot in the wrong place and the shape inherits the error.

Without a common frame a formation can be defined by inter-agent distances alone, which leads to a different theory, graph rigidity: which sets of distance constraints fix a shape up to rotation and translation (a triangle is rigid; a square held by its four sides can shear into a rhombus).

It inherits the speed and fragility of consensus: convergence at the rate of λ2\lambda_2λ2​ of the communication graph, a delay bound set by its largest eigenvalue, and a shape that comes apart if the graph splits.

The formation rule produces references, never thrust. It runs at 1 to 10 Hz and each drone tracks its reference with its own cascade control or LQR at about 50 Hz, with collision avoidance as a separate layer with priority (time-scale separation).

A drone light show reaches the same geometry by a very different route, with precomputed trajectories and no negotiation in flight (swarm robotics).