Virtual Structure Formation

Problem Statement

Virtual-structure control treats the swarm as one rigid body with assigned slots for each agent. It enables precise geometric formations during coordinated maneuvers.

Model and Formulation

Each agent tracks:

$$p_i^{ref}(t)=p_{vs}(t)+R_{vs}(t)r_i$$

where p_{vs}, R_{vs} define the structure pose and r_i is agent slot offset.

Practical Notes

• Feed the slot velocity forward, not just its position. Damping against the world frame instead of against the moving slot leaves a standing formation error of k_d·v_body / k_p — proportional to how fast the structure travels, and easy to mistake for a gain that needs raising. With the feed-forward wired up, formation error on the atlas demo drops from 2.11 m to 0.16 m.

• For a rotating structure the slot velocity includes the tangential ω × r term, and the slot acceleration the centripetal ω × (ω × r). A rigid body that yaws with its path needs both.

• Do not stack a multiplicative velocity decay on top of the PD's own k_d. The decay behaves as unmodelled drag: it demands a steady force to hold a steady velocity, which the PD can only produce from standing error. The agent model here is a plain double integrator.

• Works best with reliable relative localization.

• Slot assignment should minimize crossing paths during reconfiguration.

• Tracking gains should account for heterogeneous agent dynamics.

Implementation and Execution

python -m uav_sim.simulations.swarm.virtual_structure

Evidence

References

• Lewis and Tan, High Precision Formation Control Using Virtual Structures (1997)
• Beard et al., Coordination Variables and Consensus Building in Multiple Vehicle Systems

Related Algorithms

• Consensus Formation
• Leader-Follower