Connectivity Annulus

Problem Statement

Two barriers that pull opposite ways. SafeDistanceBarrier pushes every pair apart; ConnectivityBarrier pulls every pair together. Run them at once and the fleet is squeezed into a shell — close enough to talk, far enough not to touch — without anything having planned that shape. It falls out of the intersection of two safe sets.

Model and Formulation

$h_{\text{sep}}=\Vert \mathrm{\Delta }p{\Vert }^2-d^2$ and $h_{\text{conn}}=R^2-\Vert \mathrm{\Delta }p{\Vert }^2$ — the same quantity with opposite sign. Both have relative degree two, both become rows of one QP, and a solution satisfies both simultaneously.

Tuning and Failure Modes

• Start inside the safe set. A CBF guarantees forward invariance: it holds a set you are already in and cannot undo a violated initial condition. Seeding at random put three pairs inside the safe distance at t = 0, and the run then reported that as its minimum — which reads as a barrier failure when it is nothing of the kind.
• Size the initial ring against both bounds. On a circle of radius $R$ with $N$ agents the closest pair is $2R\sin (\pi /N)$ and the furthest is $2R$. Sizing off the separation alone put diametric pairs at 28 m against a 26 m radio: satisfying one barrier by breaking the other.
• Infeasibility is a real outcome. Ask for more separation than the range allows and no input satisfies both.
• A static scene proves nothing. Pushing outward against a fixed range reaches equilibrium in about thirteen seconds and then nothing moves. Flying the formation along a figure-8 instead put it in motion but pulled it together: the guide attraction beat the outward push, the furthest pair collapsed to half the range, and the connectivity barrier stopped being active at all — the shell, the entire point, quietly disappeared. Varying the range rather than the position keeps both barriers working.

Evidence

quantity	value	limit
closest pair	6.00 m	safe distance 6.0 m
furthest pair	31.94 m	comm range 32.0 m (peak)

The radio range breathes between 12 m and 32 m and the shell breathes with it: the fleet expands to fill whatever range it is given and contracts as that range collapses, never once closing inside the safe distance. Nothing schedules that — the barrier is simply time-varying and the QP re-solves against it each step.

The impossible case reports infeasible on 2200/2200 steps rather than returning something unsafe.

References

• Ames et al., CBF-based Quadratic Programs (2017)
• Borrmann et al., Control Barrier Certificates for Safe Swarm Behavior (2015)

Related Algorithms

• Position Exchange
• Connectivity Maintenance