Honest Limits: Bearing Margins, Mirror Flips, and the Z2 Gauge Symmetry
Publishing an engineering limit unprompted is what makes every other number credible. Here is why dual-tag UWB ranging carries an exact mathematical reflection symmetry, what resolves it, and what cannot be resolved by ranging alone.
1. The Mathematical Origin: Reflection as an Exact Isometry
In our leader-follower architecture, the leader drone carries two UWB antennas mounted along its body axis with a baseline separation of 0.45 m.
When a follower ranges against this pair, two-way time-of-flight pins its radial distance to centimeters, but its angular bearing only to:
Consider a follower positioned at lateral offset (x, y). Its reflection across the leader's tag axis sits at (x, -y). Because reflection is an exact Euclidean isometry, its distance to any tag sitting on the axis at (a, 0) is mathematically identical:
This is not an approximate tie. In our simulator, the residual gap between true position and reflected position scores0.000E+000 sigma. For an anchor-only system, every one of the2^N follower configurations fits the measurements with identical cost.
Measured Swarm Departures: Anchor-Only vs. One Peer Link
Scored across 40 independent seeds per cell, stationary leader, 120 s simulated flight (departure defined as session RMS > 10 m):
| Fleet Size (Bearing 90°) | 3 Units | 5 Units | 8 Units |
|---|---|---|---|
| Anchor-Only (Dual-Tag) | 2 / 40 (5%) | 40 / 40 (100% fail) | 40 / 40 (100% fail) |
| One Peer Link per Unit | 0 / 40 (0%) | 0 / 40 (0%) | 0 / 40 (0%) |
2. What Resolves the Relative Ambiguity: One Peer Link
Reflecting two followers together preserves their mutual distance. However, reflecting follower iwhile keeping follower j in place changes their mutual distance by roughly2 × |y|—metres.
By adding just one peer ranging link per follower around an index ring, couplings land approximately 2,700 standard deviations above radio noise. As measured in our 40-seed test sweep:
- Swarm departures for 5-unit formations plummet from 40/40 down to 0/40.
- Swarm departures for 8-unit formations plummet from 40/40 down to 0/40.
- Mean session RMS improves from 58–92 m in failing runs to 0.9–4.1 m.
3. The Critical Honest Limit: Global Z2 Gauge Symmetry
Here is the limit that many academic publications omit: ranging alone can NEVER resolve the global flip.
If you reflect the entire fleet simultaneously across the leader's tag axis, every follower-to-anchor distance is unchanged (due to the collinear tag bar), and every follower-to-follower peer distance is unchanged (due to Euclidean isometry). The cost difference between the true swarm and the globally reflected swarm remains exactly zero:
An optimizer presented with an exact tie produces random output. Therefore, an external bearing source is strictly required. This is why the Luftschar swarm module integrates a wide-angle optical infrared camera tracking an active IR beacon on the leader. The camera supplies an external bearing field h_i, completely breaking the global Z2 gauge symmetry.
4. The 55°–60° Bearing Margin and Knife-Edge Instability
Early theoretical models assumed a 45° bearing margin would suffice to keep followers on their intended branch. Empirical sweeps across multi-angle angular stability boundaries disproved this:
- True Safe Margin: Formations must maintain a 55°–60° angular clearance off the leader's axis to prevent process noise from triggering stochastic branch hopping.
- Knife-Edge Flips at 29° / 31°: Sweeping bearing across 30° reveals non-monotonic instability. Small lateral excursions at 29° and 31° invert lateral velocity demands, accelerating followers away from their assigned slots.
- Yaw Rate Limit: While individual quadcopters can yaw at 90 deg/s, commanded leader turning rates above 2.0 deg/s cause follower lag to cross the reflection threshold. Trajectory generators must cap turning rates at 2 deg/s.