Bipedal Locomotion: Resolving the Failure Boundary

September 9, 2026

Part 3. With the gait-phase confound resolved, mapping the mathematical geometry of the failure boundary for the two leading candidates: intermediate checkpoint 2250 vs converged 5999.

Sep 9: Fine-Grained Boundary Sweep

A single success rate collapses two completely different physical strategies into an arbitrary scalar. To see the actual boundary shape, swept lateral push velocity Δvy\Delta v_y from 0.5500.550 to 0.7000.700 m/s in fine 0.0250.025 m/s increments.

Grid: 7 velocities ×\times 2 friction levels (μ∈{0.6,1.0}\mu \in \{0.6, 1.0\}) = 14 cells, N=100N=100 paired seeds per cell across the two policies = 2,800 rollouts total.

Wall time: 202 seconds, local. Runs 20260909T010313Z-5abe (2250) and 20260909T010448Z-a6f1 (5999).

Sep 9: The Discovery of Two Opposite Geometries

The data reveals that these two policies do not differ by a simple threshold. They have opposite topological structures:

  1. The Rigid Cliff (2250):
    • Near-perfect from 0.5500.550 to 0.6000.600 m/s: 2 falls in 600 rollouts.
    • Catastrophic collapse past 0.6250.625 m/s: 52%→96%→100%52\% \to 96\% \to 100\% failure at μ=1.0\mu=1.0.
  2. Compliant Graceful Degradation (5999):
    • Concedes 5%–16%5\%\text{--}16\% failure at low perturbations.
    • Degrades smoothly across the envelope: 16%→25%→33%→45%16\% \to 25\% \to 33\% \to 45\% at μ=1.0\mu=1.0.

The full diff reads 8 safer, 5 worse, 1 no significant change over the 14 tests. The five worse are the low-push cells where 2250 is nearly perfect and 5999 is not, and the no-change cell at 0.5750.575 m/s, μ=0.6\mu=0.6 is 0% against 5% in a cell that can only detect an 8 pp rise.

Model 2250 (Left) drops off a rigid cliff at 0.650 m/s; Model 5999 (Right) takes a compliant step and survives. One rollout each.

The Crossover

The crossover occurs strictly in the interval [0.600,0.625][0.600, 0.625] m/s.

  • If you benchmark at 0.5500.550 m/s, 2250 is declared unconditionally superior.
  • If you benchmark at 0.6500.650 m/s, 5999 is declared unconditionally superior (96%96\% vs 25%25\% failure at μ=1.0\mu=1.0).

Both statements are true scalars; both are false models of the system. The boundary geometry is the real finding.