This is an executable numerical illustration of Exact Restoration and Historical Memory in Delayed Priority Networks with Three-Dimensional Motion, G. Bolduc, v0.6. A simulation does not constitute a mathematical proof.
This numerical laboratory belongs to the delayed-priority/3D study only. It illustrates selected mechanisms, not a proof of the full article or physical validation.
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 mode sets every applied offset rᵢ to zero. The ideal references converge to the common point μZ; positive-radius physical bodies retain avoidance and contact response and therefore need not have zero tracking error. 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