This laboratory retains the delayed-priority model and distinguishes three nominal motion laws from the measured rigid-drone realization. New runs begin with 300 active logical drones and 2,500 powered physical chassis on the hovering reserve spiral. Before disturbance activation, the ideal reference law and active physical flight targets use the same unshifted coordinates. The separate 8 m altitude belongs only to powered standby bodies.
Nominal contracts and switching
The first disturbance activation initializes the separate nominal recovery state from the current physical positions and velocities in that same unshifted coordinate frame. Both latches start at zero, as in the supplied reset rule. It does not reset a physical drone. The sampled-latch law is the default wind response: ẋᵢ = vᵢ and v̇ᵢ = K(rᵢ + qᵢ + ŝ − xᵢ) − Cvᵢ + dᵢ, with K = ab and C = a + b. Its common guard runs at 120 Hz in the order plant and endpoint impulses, guard with q⁻, then priority update. Inactive outward motion triggers recomputation. An active latch recomputes until every unsettled body is returning, then holds its translation for at least 0.24 seconds of enabled returning callbacks.
One live admission at a time switches to the coherent common-mode shield. Incumbent offsets remain fixed throughout capture. Before a complete new-cohort snapshot commits, the old zero-sum incumbent correction is held and the single newcomer receives the declared zero waiting correction. At commitment the shield applies hᵢ − mean incumbent h, computed from the same immutable source-time table. Release requires bounded relative position and velocity throughout an entire eligible history window, including still-pending snapshots. Standard release changes no position or velocity and returns to ordinary full-cohort coherent recovery. A second admission, a departure, or a formation/gain edit is rejected until capture releases, because the supplied addendum proves one newcomer with a fixed incumbent cohort and fixed offsets.
These addendums are separate nominal contracts. The interface does not claim a theorem for arbitrary switching between them. Outside shield capture, a formation-offset edit, objective-mode edit, gain edit, or departure ends the current nominal theorem interval and explicitly initializes a new one while retaining physical position and velocity. The sampled latch therefore never silently crosses a cohort change with an old latch state. The offline round-trip restoration experiment does not activate the live-admission shield. The coherent controller uses complete same-time origin snapshots and checks L < h and κ = 2aH/e < 1. The latch theorem requires a fixed finite cohort after the final intervention, convergent goals and vanishing disturbance integrals; the coherent theorem requires its own differential-forcing hypotheses. Sustained heterogeneous wind can prevent settling.
The proofs concern ideal translational states. Rigid-body actuation, saturation, contact impulses, the Rendezvous-only physical acceleration projection and virtual safety-velocity interventions, finite fluid grids and binary64 roundoff remain measured realizations. The uploaded addendums cite a parent source hash different from the installed v05 file; both identities are retained in the implementation note. No byte-identity or formal-verification claim is made. This implementation and its finite tests were prepared with AI assistance; no independent review is claimed.
1. Delayed priority states
wᵢ(k + 1) = (1 − c dᵢ) wᵢ(k) + c Σⱼ wⱼ(k − τᵢⱼ)
Equation (5) is evaluated synchronously from complete initialized histories. The graph is simple, connected and undirected; communication delays are directional nonnegative integers. The gain satisfies 0 < c < 1/dmax. The endpoint is computed from the history-weighted invariant in equation (7), never substituted for the evolving state. The delay-free and singleton cases retain their stated meanings.
The laboratory uses an explicit numerical stopping rule rather than claiming finite-time equality: consensus is declared when the maximum absolute deviation from the predicted endpoint across every active delay layer remains at or below ε for the selected hold time. The default is ε = 10−8 for 2 simulated seconds. The priority rule is unchanged. Automatic pause also waits for exact-reference convergence, drone tracking and speed thresholds, so a formation change is not stopped while bodies are still moving.
2. Exact reference motion
cᵢ = rᵢ + wᵢ Z uᵢ = a b(cᵢ − xᵢ) − (a + b)vᵢ
Equations (36) and (38) generate a separate ideal double-integrator reference. Each held-goal interval uses the analytic exponential map, including a stable equal-gain limit. The cell hypothesis requires both xᵢ and yᵢ = xᵢ + vᵢ/b inside rᵢ + BR. Formation mode uses fixed offsets rᵢ and a translated endpoint xᵢ − rᵢ → μZ. Multiple planar and three-dimensional formation layouts are selectable. The article's finite-body separation bound additionally requires separated reference cells.
Rendezvous sets every mathematical offset rᵢ to zero and retains the exact individual ideal reference targets, which converge to μZ under the declared hypotheses. A separate physical safety adapter uses Formation spacing to choose each drone's desired safety-bubble radius and the adjustable Rendezvous tolerance to brake inward acceleration before contact, without moving the mathematical endpoint. Positive-radius bodies cannot occupy the same point; collision-free physical convergence is not implied by the nominal theorem, and finite actuation, pre-existing overlaps or disabled avoidance can still cause contact. Stored formation coordinates remain available for returning to formation and for separated initialization. Mode changes preserve the priority history, invariant, reference position and velocity. Live geometry changes are interventions; the cell hypotheses are displayed again. Generated formations reflow after ordinary population edits without moving surviving states instantaneously. The Restart button starts a new experiment and scatters only the rigid bodies with a recorded fresh random seed; exact references use their initialization rules and retain explicit cell-premise checks. These scattered rigid bodies do not inherit the paper’s cell-invariance guarantee.
3. Rigid drone extension
m v̇ = −m g e₃ + T R e₃ − D(v − vwind) + Fcontact
J ω̇ = τ − ω × Jω − Dωω
Every visible drone has physical position, velocity, a unit quaternion, angular velocity, mass and rotational inertia. A controller follows the reference with finite body-axis thrust, torque, motor response, tilt and command limits. Predictive avoidance and rigid spherical contact envelopes affect the physical drones only. Commanded speed and acceleration limits are actuator requests; measured motion can exceed them under infeasible settings. No convergence flag snaps bodies, zeros their velocity or modifies priority rows.
The flight extension is informed by Lee, Leok & McClamroch (2010) and the PX4 controller architecture. It does not inherit the article's ideal-motion safety guarantee. Tracking error, avoidance interventions, saturation and contact are measured explicitly. These rigid bodies use a declared spherical collision envelope, rather than blade-resolved aerodynamics.
4. Runtime edits and restoration
Numeric sliders apply while dragging. Their ×10 and ÷10 controls change the displayed interval while preserving the parameter. Probability domains, the gain bound and explicit resource budgets remain enforced. Graph, delay, gain and history changes begin a new fixed-operator epoch using the retained numerical state. Rescale c with graph preserves c dmax during generated graph and ordinary membership changes; the resulting c remains visible. With rescaling disabled, an inadmissible graph is rejected without mutation. Communication links and formation geometry are separate: an already-agreed state can remain stationary under a different graph.
Delays that require unavailable history are rejected. Admission and departure apply the article's offline atomic history maps to the delay registers. Retired physical bodies remain in the scene and receive a stationary hold reference. This is not the article's causal packet protocol or its staged finite-parking construction. Arbitrary perpetual membership changes have no convergence claim here. The restoration laboratory computes its round trip from the input matrices, finite stay, insertion masses and newcomer preference. A computed q* outside the strictly positive simplex is reported as infeasible. The predicted endpoint is used for comparison only.
5. Original simulation and source lineage
The supplied swarm(5).zip contains the original 2D Swarm Lab v07 and its motion.js. That model uses moving discs, bounded preferred speed and acceleration, reciprocal velocity constraints, and a swept-circle collision guard. The guard can apply an instantaneous emergency stop. This 3D implementation carries forward variable-speed motion, predictive relative-velocity avoidance and swept collision detection, while expressing drone motion through forces, torque and rigid contact impulses. The 2D planner's epoch-mean destination is replaced by each agent's actual delayed local priority goal, as required by the uploaded v0.6 article. The archive has been inspected; byte identity with the separate Windows path is not established.
Reproducibility
The numerical engine and its readable source are contained in this standalone HTML page. Save an experiment to retain settings and the current state; export the measured trace and intervention log for analysis. The original scientific verification/data package is not supplied by this hosted page. Numerical thresholds affect labels and plots only.
Download standalone laboratory (index.html) · Companion article source identities