Murmur: A Delay-Tolerant Swarm Robotics Testbed

1. Problem statement

Five Arboria papers map where decentralized coordination breaks under communication delay. The main results are:

• the collapse is primitive-independent;
• its onset scales as $d_c\,\alpha \approx 2.8$;
• it is governed by staleness, not throughput;
• cheap peer-state prediction pushes it outward.

Every paper ends with the same caveat: a simulation-based algorithmic result, not physical-device validation.

Murmur is designed to remove that caveat. It is a small physical swarm in which delay and loss are the controlled variables, ground truth comes from outside the robots, and the coordination code is the same code path the simulator scores.

The question it answers is: does the delay cliff measured in Leviathan appear at the same normalized delay on real robots, with real radios, real actuation lag and real sensing noise?

2. Systems approach

3. Concept of operations

A run is fully described by a manifest: task (rendezvous, consensus, formation-hold), primitive (gossip-consensus, flocking, CRDT-intent), gain $\alpha$, delay $d$, loss probability $p$, predictor, and a seed. Maneuver.Map already produces exactly this manifest and seed tree for simulated runs, so a physical run and its simulated twin share one identifier.

4. Requirements

5. Architecture

The key interface decision is where delay and loss are injected. Murmur injects them on the receive side: each receiver flips its own loss coin and holds packets in its own FIFO. That makes loss independent per link, which SR-4 requires. It also mirrors Leviathan's delivered-edge model exactly: the simulator decides per link whether a message arrives, then gates coordination on the links that did.

6. Theory and models

6.1 Robot kinematics

Each robot is a differential-drive unicycle with wheel radius $r$ and track width $L$:

Coordination primitives emit a velocity $u_i \in \mathbb{R}^2$ for a point-mass agent, as they do in simulation. A unicycle cannot follow that directly, so Murmur controls a point a distance $\ell$ ahead of the axle, $p = (x + \ell\cos\theta,\; y + \ell\sin\theta)$. That point's dynamics are fully actuated:

This feedback linearization makes the robot behave like the simulator's point mass, up to actuator limits. It is what lets the same primitive run unchanged on both platforms.

6.2 Coordination under delay

For gossip-consensus rendezvous, robot $i$ at control step $k$ acts on the newest delivered state of each neighbour:

The prediction term is optional and is used only in the predictor experiments (constant-velocity shown here; Kalman is the alternative). The robot's own state is always fresh. Only its view of its peers is stale.

Analytic anchor. In the fully delayed scalar form $x[k+1] = x[k] - q\,x[k-d]$, which applies per Laplacian eigenvalue with $q = \alpha\lambda$, the system is asymptotically stable if and only if

This is the Levin–May condition. It predicts the inverse scaling $d_c \propto 1/\alpha$ that the simulation measured as $d_c\,\alpha \approx 2.8$. The constant differs because $Q$ measures task quality rather than stability, and because the robot's own term is undelayed. The scaling law is what Murmur tests. The constant is what it calibrates.

6.3 Mapping simulation steps to physical time

Delays are specified in steps, so the comparison is dimensionless. The arena only has to be large enough that a robot cannot cross it within the longest delay. At 0.2 m/s × 4 s = 0.8 m, the 2.4 m arena satisfies that.

7. Detailed design

Bill of materials (per robot)

Arena infrastructure adds roughly $250: an OV9281 global-shutter camera, a Raspberry Pi 5 ground-truth server, a ceiling frame and a matte floor.

8. Budgets

8.1 Native latency budget (SR-5)

Channel load is $8 \times 10\,\text{Hz} \times 0.5\,\text{ms} \approx 4\%$ of airtime, far below the point where contention would add tail latency.

8.2 Power budget (SR-1)

A 2S 500 mAh pack holds about 3.7 Wh, which gives ≈ 2.4 h of runtime: five times the 30-minute requirement.

8.3 Ground-truth accuracy (SR-2)

An OV9281 (1280 × 800) mounted 2.5 m up with a 2.6 m field of view gives ≈ 2 mm per pixel. AprilTag corner refinement is sub-pixel, so ≤ 5 mm position error is realistic. It still has to be shown against a surveyed grid before any run counts (verification test VT-2).

9. Trade studies

Radio. Scored 1–5, weighted:

ESP-NOW wins on latency and cost. Its weak determinism is acceptable because SR-5 bounds native latency below one step, and every packet's real age is logged anyway.

Ground truth:

10. Failure modes and effects

Severity (S), occurrence (O) and detection (D) are scored 1–10. RPN is the risk priority number, S × O × D.

11. Verification and validation

Validation experiments, each with a falsifiable hypothesis:

• E1, calibration. At zero delay, rendezvous reaches $Q \ge 0.95$. This mirrors the papers' Vicsek sanity check: the substrate is trusted before it is used.
• E2, H1: the cliff transfers. The physical $d_c$ lies within ±25% of the simulated $d_c$ at the same α. The run budget is 11 delays × 3 primitives × 5 seeds × 3 min ≈ 8 h of robot time.
• E3, H2: the scaling law transfers. $d_c\,\alpha$ stays constant within ±20% across a 4× range of α.
• E4, H3: staleness, not throughput. At fixed $d$, $Q$ varies by less than 0.05 across $p \in [0, 0.9]$.
• E5, prediction transfers. The constant-velocity predictor recovers at least half the lost $Q$ at $d = 2 d_c$.

12. Risks and roadmap

The main technical risk is actuation lag, since a real robot does not respond instantly as the simulated point mass does. It adds an effective delay the simulator does not model. Mitigation: measure the step response in phase 1 and add it to Leviathan's agent model, rather than hiding it.